toPreorder 📖 | CompOp | 17805 math, 2 bridgemath: countable_image_lt_image_Ioi_within, le_iff_eq_or_lt, Finset.val_strictMono, TopCat.Presheaf.generateEquivalenceOpensLe_functor'_obj_obj, ENNReal.instCanonicallyOrderedAdd, Metric.nhdsWithin_basis_eball, Prod.Lex.toLex_lt_toLex', TopologicalSpace.OpenNhds.coe_id, SuccOrder.hasBasis_nhds_Ioc, AddSubmonoid.pointwise_smul_le_pointwise_smul_iff₀, PredOrder.nhdsGT_eq_nhdsNE, SSet.OneTruncation₂.nerveEquiv_apply, Set.preimage_div_const_Ico, Filter.le_map_of_right_inverse, Finset.toLeft_monotone, HahnSeries.order_lt_iff_exists, ENNReal.exists_nat_mul_gt, Filter.IsApproximateUnit.iff_le_nhds_one, Affine.Simplex.sSameSide_affineSpan_faceOpposite_point_left_iff, Submodule.comap_lt_comap_iff_of_surjective, MeasureTheory.exists_pos_preimage_ball, lt_or_gt_iff_ne', CategoryTheory.Triangulated.SpectralObject.Hom.comm, Cardinal.lt_ord_succ_card, Function.Embedding.encard_le, IsLocalization.AtPrime.coe_primeSpectrumOrderIso_symm_apply_asIdeal, AlgebraicGeometry.Scheme.toSpecΓ_apply, partialSups_apply, LinearOrder.isSuccArchimedean_iff_isPredArchimedean, HomotopyCategory.spectralObjectMappingCone_δ'_app, EMetric.diam_pos_iff', Order.coheight_add_one_le, le_iff_forall_lt_one_mul_le, Set.finite_diff_iUnion_Ioo, CategoryTheory.Classifier.pullback_χ_obj_mk_truth, EReal.nhds_top_basis, tsub_lt_self_iff, upperBounds_supClosure, SpectrumRestricts.le_nnreal_iff, MeasureTheory.exists_lt_lowerSemicontinuous_integral_gt_nnreal, HurwitzZeta.hasSum_int_hurwitzZetaOdd, AlgebraicGeometry.Scheme.Opens.ι_image_basicOpen, Finset.bot_eq_empty, nonZeroDivisors_le_comap_nonZeroDivisors_of_injective, CategoryTheory.Limits.kernelSubobjectMap_arrow_assoc, FractionalIdeal.le_spanSingleton_mul_iff, Multiset.Ico_sub_Ico_right, isArtinian_iff, AddUnits.orderEmbeddingVal_apply, AlgebraicGeometry.Γ_map_morphismRestrict, Ideal.quotientEquivAlgOfEq_coe_eq_factorₐ, MvPolynomial.vanishingIdeal_anti_mono, one_le_div_iff, 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CategoryTheory.ComposableArrows.Exact.cokerIsoKer'_hom_inv_id_assoc, AlgebraicGeometry.Scheme.Hom.germ_stalkMap_assoc, comap_gauge_nhds_zero_le, Dense.exists_ge, Fin.image_castLE_Icc, Order.krullDim_le_of_strictComono_and_surj, ValuationSubring.valuation_lt_one_iff, LieIdeal.map_bracket_le, Valuation.leSubmodule_v_le_of_mem, TopCat.presheafToType_map, AddSubgroup.addSubgroupOf_eq_top, Algebra.Extension.Cotangent.span_eq_top_of_span_eq_ker, TopCat.Sheaf.interUnionPullbackCone_snd, OrdinalApprox.lfpApprox_mono_mid, Filter.lift'_mono', Filter.map_prodMap_coprod_le, IntermediateField.fieldRange_le, iSupIndep.injOn_iInf, YoungDiagram.transposeOrderIso_symm_apply, OrderIso.pnatIsoNat_symm_apply, Heyting.isRegular_top, TopologicalSpace.Opens.map_coe, GroupSeminorm.comp_mul_le, IntermediateField.LinearDisjoint.iff_inf_eq_bot, AlgebraicGeometry.Scheme.Hom.image_preimage_eq_opensRange_inter, AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToSection_apply, MeasureTheory.iInf_mul_le_lintegral, 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AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_counit_app, ZFSet.subset_vonNeumann, Fin.preimage_val_Ici_val, NNReal.HolderTriple.one_div_nonneg', MeasurableSpace.comap_const, AffineMap.restrict.bijective, SetLike.lt_iff_le_and_exists, le_nhds_iff_adhp_of_cauchy, Valuation.supp_quot_supp, cfcₙ_norm_nonneg, Submonoid.le_centralizer_centralizer, CategoryTheory.Subobject.underlyingIso_arrow_assoc, Valuation.HasExtension.val_map_le_iff, AlgebraicGeometry.Scheme.IsQuasiAffine.toIsImmersion, isUpperSet_iff_forall_lt, Subgroup.gc_map_comap, AlgebraicGeometry.StructureSheaf.globalSectionsIso_inv, Set.compl_ordConnectedSection_ordSeparatingSet_mem_nhdsGE, NonemptyInterval.mem_coe_interval, CategoryTheory.ShortComplex.mapToComposableArrows_app_0, ENat.lt_add_one_iff', MeasureTheory.Measure.le_comap_apply, EisensteinSeries.hasSum_e2Summand_symmetricIcc, biSup_le_eq_sup, CategoryTheory.Subobject.hasColimitsOfSize, MeasureTheory.eLpNorm_smul_le_mul_eLpNorm, CategoryTheory.ComposableArrows.IsComplex.zero'_assoc, OrderEmbedding.birkhoffFinset_inf, TopologicalSpace.Opens.mem_top, CategoryTheory.Subfunctor.Subpresheaf.equalizer_le, AlgebraicGeometry.Scheme.ofRestrict_appIso, AlgebraicGeometry.iSup_affineOpens_eq_top, AlgebraicGeometry.isAffineOpen_top, Valuation.isEquiv_iff_val_sub_one_lt_one, Submodule.set_smul_le_iff, Finset.sum_mono_set, eq_zero_or_nnnorm_pos, iUnion_Iic_eq_Iio_iSup, Filter.le_map_iff, HahnSeries.mem_cardSuppLTSubfield, SimpleGraph.deleteFar_iff, nonneg_of_mem_doublyStochastic, Valuation.one_apply_lt_one_iff, nhds_bot_basis, CStarAlgebra.mem_Icc_algebraMap_iff_norm_le, dense_iff_exists_between, exists_floor, Ordinal.lift_type_lt, ENNReal.rpow_lt_top_iff_of_pos, ringKrullDim_succ_le_ringKrullDim_polynomial, countable_image_gt_image_Ioi_within, FirstOrder.Language.Substructure.closure_eq_of_isRelational, AlgebraicGeometry.RingedSpace.basicOpen_res, not_monotone_not_antitone_iff_exists_le_le, sign_mul_self, Finset.exists_one_lt_of_prod_one_of_exists_ne_one', Ordinal.sInf_compl_lt_lift_ord_succ, LowerSet.compl_map, quasiconcaveOn_iff_min_le, MeasureTheory.hittingBtwn_apply_mono_right, CategoryTheory.Classifier.SubobjectRepresentableBy.pullback_homEquiv_symm_obj_Ω₀, AlgebraicGeometry.LocallyRingedSpace.restrict_presheaf_obj, Num.cast_lt, LowerSet.erase_le, affineSpan_le_toAffineSubspace_span, Ordinal.typein_le_typein, CFC.monotoneOn_one_sub_one_add_inv, EReal.coe_ennreal_le_coe_ennreal_iff, Set.abs_sub_right_of_mem_uIcc, Set.neg_Icc, not_isCofinal_iff, NonemptyInterval.coe_inv_interval, ModularFormClass.qExpansionFormalMultilinearSeries_radius, Ordinal.pred_succ, ENNReal.exists_inv_two_pow_lt, CategoryTheory.Pairwise.diagram_obj, YoungDiagram.cells_subset_iff, Submodule.le_annihilator_iff, Set.toENat_cardinalMk_subtype, MvPolynomial.degrees_monomial, WithLp.edist_fst_le, Subalgebra.toSubmodule_injective, convexHull_insert, Set.Ioi_subset_Ioo_union_Ici, Antitone.Icc, PartENat.ofENat_le, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv, PrimeSpectrum.asIdeal_bot, le_iInf₂_add, Ultrafilter.clusterPt_iff, Set.Infinite.exists_gt, Ideal.span_singleton_mul_le_span_singleton_mul, ENat.self_le_mul_left, lt_or_gt_of_ne, AlgebraicGeometry.Scheme.Hom.quasiFiniteLocus_comp, PrincipalSeg.map_pred, CategoryTheory.MorphismProperty.isomorphisms_le_of_containsIdentities, CategoryTheory.MorphismProperty.isStableUnderLimitsOfShape_iff_limitsOfShape_le, AlgebraicGeometry.Scheme.app_eq, PNat.map_subtype_embedding_Ico, MeasureTheory.aecover_Ioi_of_Ici, EReal.sub_nonneg, Set.preimage_const_add_Ioo, nnnorm_le_insert', Filter.Tendsto.le_comap, Cardinal.toENat_eq_natCast_iff, Ordinal.mem_range_typein_iff, Metric.isSeparated_insert_of_notMem, cocompact_le_atBot, Finset.card_strictMono, Fin.isAddFreimanIso_Iio, subset_closedConvexHull, Cardinal.ofENat_lt_lift, MeasureTheory.GridLines.T_lmarginal_antitone, nnnorm_le_nnnorm_add_nnnorm_div', Filter.tendsto_inv_atTop_iff, MonotoneOn.of_map_inf, MeasureTheory.unifTight_iff_real, MeasureTheory.hittingBtwn_mono_left, MeasureTheory.extend_iUnion_le_tsum_nat, Ordinal.IsAcc.isSuccLimit, UpperSet.Ici_zero, AddSubmonoid.comap_le_comap_iff_of_surjective, inv_atBot₀, AlgebraicTopology.NormalizedMooreComplex.obj_d, Polynomial.isEquivalent_atBot_div, DiscreteQuotient.ofLE_refl, inv_le_mabs, MeasureTheory.Measure.sub_top, RootPairing.Base.not_nonneg_iff_neg_of_sum_mem_range_root, Geometry.SimplicialComplex.vertex_mem_convexHull_iff, Ordinal.gamma_le_gamma, ENat.card_le_card_of_injective, CategoryTheory.Sieve.generate_mono, Matroid.eRk_lt_encard_iff_dep, AlgebraicGeometry.Scheme.support_nilradical, IsAbsoluteValue.abv_nonneg', EReal.neg_nonneg, tendsto_ceil_right', Submodule.span_sInf_le, MeasureTheory.lintegral_pow_le_pow_lintegral_fderiv, Set.Iic_add_bij, covBy_iff_le_iff_lt_right, AlgebraicGeometry.opensCone_pt, Filter.tendsto_comp_inv_atBot_iff, partialSups_const_add, WithTop.Ico_coe_coe, groupHomology.boundaries₂_le_cycles₂, LieIdeal.comap_incl_eq_top, List.support_formPerm_le, closedConvexHull_min, CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinsetLimitCone_isLimit_lift, Fin.Ico_add_one_eq_Ioo, ENNReal.le_liminf_mul, Valuation.mem_maximalIdeal_iff, Filter.bliminf_sup_le_and_aux_right, Disjoint.le_bot, Topology.IsOpenEmbedding.functor_isContinuous, LinOrd.ofHom_apply, convex_iff_ordConnected, SSet.horn₃₁.desc.multicofork_π_two, RootPairing.RootPositiveForm.zero_lt_posForm_apply_root, Finsupp.card_Ioc, Ordinal.add_succ, PowerSeries.le_order_subst_left', le_sdiff_right, AlgebraicGeometry.Scheme.zeroLocus_empty_eq_univ, MeasureTheory.integrable_iff_integrableAtFilter_atBot, GroupNorm.le_def, Order.krullDim_lt_coe_iff, nhdsWithin_le_comap, Multiset.erase_lt, Finset.instAddLeftMono, AlgebraicGeometry.morphismRestrict_base_coe, Fin.map_castLEEmb_Ioc, convexHull_multiset_sum, FiniteArchimedeanClass.mem_addSubgroup_iff, ConvexOn.monotoneOn_slope_lt, Nucleus.monotone, abs_sub_round, eq_sInf_coatoms, partialSups_succ', WellOrderExtension.wellFoundedLT, Cardinal.isStrongLimit_preBeth, NonUnitalAlgebra.adjoin_le, Fin.finsetImage_castLE_Ioo, MeasureTheory.Egorov.tendstoUniformlyOn_diff_iUnionNotConvergentSeq, Algebra.TensorProduct.includeLeft_map_center_le, Valuation.mem_leSubmodule_iff, isPiSystem_image_Iic, abs_add_eq_add_abs_iff, Finsupp.toMultiset_strictMono, partialSups_mono, Multiset.Ico_sub_Ico_left, Submodule.comm_trans_lTensorOne, AlgebraicGeometry.StructureSheaf.comap_id, Left.pow_lt_one_iff', Set.preimage_div_const_Icc, Set.pairwise_disjoint_Ico_zsmul, Geometry.SimplicialComplex.convexHull_subset_space, Multiset.le_singleton, PositiveLinearMap.apply_le_of_isSelfAdjoint, IsPredArchimedean.isAtom_findAtom, Ordinal.card_le_ofNat, IntermediateField.coe_algebraMap_over_bot, Function.Injective.map_atTop_finset_prod_eq, IdealFilter.ringFilterBasis_sets, Subsemiring.closure_mono, LinearGrowth.linearGrowthInf_add_le', WithBot.orderIsoPUnitSumLex_symm_inl, Set.disjoint_or_nonempty_inter, Finset.prod_one_sub_ordered, lt_iff_not_ge, LinearMap.IsPositive.ne_zero_iff, PredOrder.isOpen_iff, Cardinal.ofENat_lt_ofNat, Subgroup.toSubmonoid_le, RingCon.toCon_eq_bot, CategoryTheory.MorphismProperty.hasOfPostcompProperty_iff_le_diagonal, ProbabilityTheory.gaussianReal_const_mul, Mathlib.Meta.Positivity.pos_of_isNNRat, ContinuousLinearMapWOT.le_nhds_iff_forall_inner_apply_le_nhds, Finset.le_fold_max, Fin.preimage_castAdd_Ico_castAdd, deriv_abs, Finset.Ioo_map_sectR, NonUnitalStarAlgebra.mem_bot, minimal_ge_iff, Finset.Ioi_disjUnion_Iio, Nucleus.himp_apply, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverse_obj_π_app, IsDedekindDomain.HeightOneSpectrum.iInf_localization_eq_bot, MeasureTheory.SimpleFunc.measure_support_lt_top_of_integrable, compl_le, CFC.inv_nonneg, CategoryTheory.Retract.injectiveDimension_le, Flag.isMax_coe, isOpen_lt_prod, IsUnit.isStrictlyPositive_star_right_conjugate_iff, grade_bot, WithLp.prod_nndist_eq_of_L2, IsNoetherian.iff_rank_lt_aleph0, Ordinal.Iio_one_default_eq, Ordinal.lsub_pos, CategoryTheory.Precoverage.toGrothendieck_le_iff_le_toPrecoverage, NNRat.le_def, Ideal.height_le_iff_exists_minimalPrimes, comp_le_uniformity3, AlgebraicGeometry.Scheme.Hom.instIsContinuousOpensOpensFunctorGrothendieckTopologyCarrierCarrierCommRingCat, Real.nnnorm_deriv_mulExpNegMulSq_le_one, IntermediateField.lift_le, IsCompact.exists_isLeast, Subgroup.MapSubtype.orderIso_apply_coe, OrderIso.withTopCongr_apply, CategoryTheory.ComposableArrows.isIso_iff₂, Seminorm.bot_eq_zero, InfTopHom.inf_apply, Submonoid.coe_invOrderIso_apply, CStarAlgebra.nonneg_iff_exists_isSelfAdjoint_and_eq_mul_self, ProperCone.dual_sUnion, Fin.map_revPerm_Ioc, CategoryTheory.Dial.tensorObjImpl_rel, NonUnitalStarSubalgebra.centralizer_le, AlgebraicGeometry.IsAffineOpen.map_fromSpec, NonUnitalStarSubalgebra.topologicalClosure_adjoin_le_centralizer_centralizer, inf_right_le_sup_left, Ordinal.nmul_le_iff₃, GroupCone.oneLE_toSubmonoid, SimpleGraph.le_chromaticNumber_iff_colorable, MeasureTheory.aecover_Ioi_of_Ioi, OmegaCompletePartialOrder.ContinuousHom.ωSup_apply, Fin.image_val_Iic, ENNReal.exists_nnreal_pos_mul_lt, enorm_sub_le, lt_compl_self, FirstOrder.Language.PartialEquiv.monotone_symm, NNRat.not_cast_lt_zero, Finset.Icc_add_one_left_eq_Ioc, IsLocalization.AtPrime.coe_orderIsoOfPrime_apply_coe, WithTop.continuousOn_untopD, ConvexCone.mem_positive, Localization.mk_le_mk, Algebra.adjoin_singleton_intCast, MulArchimedeanClass.subgroup_strictAntiOn, Fin.map_castAddEmb_Ico, IsFractionRing.coeSubmodule_strictMono, Submodule.map_dualCoannihilator_le, AlgebraicGeometry.Scheme.germ_stalkClosedPointTo_Spec_fromSpecStalk_assoc, Fin.predAbove_left_monotone, LinearGrowth.le_linearGrowthSup_add, perfectClosure.map_le_of_algHom, nsmul_neg_iff, Module.End.mem_invtSubmodule, TopCat.Presheaf.Pushforward.id_inv_app, concaveOn_iff_slope_anti_adjacent, strictConvex_Iic, NonemptyInterval.snd_neg, Filter.pi_le_pi, sub_le_neg_add_iff, min_bot_left, Ideal.mul_le_right, le_sInf_iff, SimpleGraph.DeleteFar.le_card_sub_card, IsCompl.inf_left_eq_bot_iff, nhdsSet_Ioo, AddSubmonoid.prod_le_iff, instTrichotomousLt, AffineSubspace.setOf_sSameSide_eq_image2, LowerAdjoint.mem_iff, EReal.mul_nonpos_iff, IsSolvable.commutator_lt_of_ne_bot, Set.Intersecting.isUpperSet, Nat.Ioc_filter_modEq_cast, Ordinal.sup_eq_lsub_iff_succ, ClosureOperator.ofCompletePred_apply, AntilipschitzWith.hausdorffMeasure_preimage_le, Cardinal.aleph_succ, le_nhdsAdjoint_iff, CategoryTheory.ComposableArrows.fourδ₁Toδ₀_app_zero, ENNReal.toReal_pos_iff, List.sortedLE_mergeSort, AlgebraicGeometry.IsLocallyArtinian.isArtinianRing_presheaf_obj, Set.iUnionLift_unary, monovary_inv, RingEquiv.idealComapOrderIso_apply, Hyperreal.epsilon_lt_of_pos, AlgebraicGeometry.Scheme.Opens.toSpecΓ_naturality, PseudoMetric.IsUltra.sup, TopCat.Presheaf.germ_stalkPullbackHom, Subalgebra.pi_top, Finset.uIcc_toDual, Finset.Ico_subset_Ico_union_Ico, AlgebraicGeometry.Scheme.Opens.toSpecΓ_SpecMap_presheaf_map_top, Lattice.BicartSq.commSq, lowerClosure_infs, NatOrdinal.instAddLeftReflectLE, Finset.le_inf_div, BooleanSubalgebra.map_le_iff_le_comap, mem_subalgebraEquivIntermediateField, skyscraperPresheaf_obj, CategoryTheory.Triangulated.SpectralObject.ω₂_obj_distinguished, Fin.finsetImage_addNat_Ioi, PrincipalSeg.map_succ, Fin.attachFin_Iic, OrderIso.withTopCongr_refl, MeasureTheory.integral_Ici_eq_integral_Ioi, AlgebraicGeometry.LocallyRingedSpace.notMem_prime_iff_unit_in_stalk, MeasureTheory.MemLp.exists_simpleFunc_eLpNorm_sub_lt, AddSubmonoid.comap_strictMono_of_surjective, sSup_univ, OrderEmbedding.mulRight_apply, SSet.modelCategoryQuillen.I_le_monomorphisms, Nat.succPNat_mono, strictAntiOn_of_lt_sub_one, AlgebraicGeometry.Scheme.restrictFunctorΓ_inv_app, List.getD_max?_eq_unbotD_maximum, MeasureTheory.MemLp.toLp_val, isCompact_setOf_finiteMeasure_le_of_isCompact, IsNonarchimedean.multiset_powerset_image_add, ClosureOperator.closure_iSup₂_closure, Set.Ioc_union_Ioc_left, Set.Iio_succ_eq_Iic, supClosure_eq_self, MonotoneOn.Icc, Subfield.relfinrank_eq_one_iff, ENNReal.rpow_lt_rpow_iff, EReal.toENNReal_eq_zero_iff, imageToKernel_unop, ContinuousAt.finset_sup, Set.einfsep_pair_le_left, Subfield.closure_le, CategoryTheory.nerve.δ₂_zero, Ordinal.isInitialIso_apply, Filter.blimsup_and_le_inf, Padic.withValUniformEquiv_norm_le_one_iff, AlgebraicGeometry.Scheme.Modules.restrictAdjunction_counit_app_app, AlgebraicGeometry.ProjectiveSpectrum.Proj.toStalk_specStalkEquiv, Commute.cfcₙ_nnreal, AlgebraicGeometry.Scheme.mem_grothendieckTopology_iff, Int.cast_one_le_of_pos, Valued.toNormedField.norm_le_one_iff, ArchimedeanOrder.le_def, SimpleGraph.le_chromaticNumber_iff_coloring, AlgebraicGeometry.instGeometricallyIrreducibleMorphismRestrict, tendsto_cobounded_iff_meromorphicOrderAt_neg, EMetric.le_infEdist, NumberField.RingOfIntegers.HeightOneSpectrum.one_lt_absNorm_nnreal, AddMonoidAlgebra.supDegree_single, AlgebraicGeometry.instIsDomainCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTopOfIsIntegral, MeasurableSpace.comap_bot, Geometry.SimplicialComplex.mem_space_iff, NormedCommGroup.cauchySeq_iff, AntilipschitzWith.mul_le_nndist, PMF.toOuterMeasure_mono, AddSubsemigroup.closure_le_centralizer_centralizer, nullMeasurableSet_Ioc, Finset.image_orderEmbOfFin_univ, Set.Finite.lt_csInf_iff, OrderIso.isCoatom_iff, ProbabilityTheory.Kernel.partialTraj_succ_of_le, three_pos, Int.preimage_Ioc, Finset.Icc_map_sectL, FirstOrder.Language.Substructure.gc_map_comap, Filter.neg_atTop, Ordinal.not_bddAbove_fp_family, MeasureTheory.Lp.cauchySeq_Lp_iff_cauchySeq_eLpNorm, MeasureTheory.OuterMeasure.mkMetric_top, Set.Icc_self, codisjoint_bot, MeasureTheory.MeasuredSets.sub_le_edist, TopCat.Presheaf.generateEquivalenceOpensLe_unitIso, PNat.find_le, nhds_le_nhdsSet, AlgebraicGeometry.StructureSheaf.comap_comp, Finset.Iio_filter_lt, hasDerivWithinAt_Ioi_iff_Ici, AlgebraicGeometry.Scheme.Opens.mem_ι_image_iff, AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_hom_eq_germ, NumberField.hermiteTheorem.minkowskiBound_lt_boundOfDiscBdd, Cardinal.two_le_iff_one_lt, NatOrdinal.lt_mul_iff, ENNReal.log_le_log_iff, Topology.IsGeneratedBy.le_generatedBy, one_le_pow_mul_abs_eval_div, isProperMap_iff_ultrafilter, GradedAlgebra.exists_finset_adjoin_eq_top_and_homogeneous_ne_zero, Finset.tendsto_Ico_neg_atTop_atTop, Set.tsub_encard_le_encard_diff, IsCauSeq.bounded', ContractingWith.edist_le_of_fixedPoint, mulRightMono_of_mulRightStrictMono, AlgebraicGeometry.HasAffineProperty.affineAnd_le_affineAnd, Matrix.eRank_le_card_width, SeminormedGroup.disjoint_nhds, IntermediateField.map_le_iff_le_comap, SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj, MeasureTheory.Iio_ae_eq_Iic', AffineBasis.tot, Ordinal.liftOnWellOrder_type, finSumEquivOfFinset_inr, tendsto_inv_nhdsLE, Finset.add_sum_eq_sum_insertNone, LowerSet.Iic_inj, InfPrime.finset_inf_le, Module.Baer.extensionOfMax_le, AddMonoidHom.liftOfRightInverse_comp_apply, Ordinal.blsub_congr, AlgebraicGeometry.Scheme.Hom.exists_isIso_morphismRestrict_toNormalization, MeasureTheory.ae_le_const_iff_forall_gt_measure_zero, Fin.preimage_natAdd_Ioi_natAdd, Seminorm.finset_sup_apply, IsWellFounded.wellOrderExtension.isWellFounded_lt, Algebra.mem_top, Subgroup.pointwise_smul_subset_iff, IsPGroup.sylow_mem_fixedPoints_iff, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_symm_apply, MeasureTheory.Content.sup_le, MeasureTheory.OuterMeasure.le_boundedBy', Polynomial.div_tendsto_atTop_leadingCoeff_div_of_degree_eq, AlgebraicGeometry.morphismRestrict_ι_assoc, Set.pairwise_disjoint_Ioo_mul_zpow, SubMulAction.closure_le, max_def_lt, CategoryTheory.instMonoMap'KernelCokernelCompSequenceOfNatNat, NonemptyInterval.coe_neg_interval, IsOrderedMonoid.toMulRightMono, AddCircle.card_torsion_le_of_isSMulRegular, isLUB_iff_sSup_eq, Cardinal.aleph_pos, Filter.liminf_const_top, WithIdealFilter.mem_nhds_zero_iff, min_lt_min_right_iff, MeasureTheory.NullMeasurableSet.exists_isOpen_symmDiff_lt, Fin.preimage_rev_Ico, Finset.dens_map_le, Order.succ_lt_succ_iff, Ordinal.liftInitialSeg_coe, AlgebraicGeometry.functionField_isFractionRing_of_isAffineOpen, pi_nnnorm_le_iff, AlgebraicGeometry.Scheme.IdealSheafData.ideal_sSup, LieSubmodule.toSubmodule_le_toSubmodule, MeasureTheory.setLIntegral_le_iSup_mul, SuccOrder.limitRecOn_succ, Finsupp.card_Ico, List.maximum_cons, CategoryTheory.injectiveDimension_le_iff, Ideal.le_span_singleton_mul_iff, measure_le_lintegral_thickenedIndicatorAux, Nonneg.isStrictOrderedRing, AlgebraicGeometry.Scheme.fromSpecStalk_toSpecΓ_assoc, Ordinal.iSup_typein_succ, AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_fromSpec, ProperCone.innerDual_univ, Finsupp.toColex_monotone, Affine.Simplex.sign_excenterWeights_singleton_pos, DiscreteQuotient.proj_bot_inj, CauSeq.le_lim, CategoryTheory.Limits.equalizerSubobject_arrow'_assoc, WithBot.subtypeOrderIso_symm_apply, SetTheory.Game.addLeftMono, CStarAlgebra.inr_mem_Icc_iff_norm_le, ENat.card_lt_top, AlgebraicGeometry.Scheme.Modules.pushforwardCongr_hom_app_app, integralClosure_le_completeIntegralClosure, InformationTheory.mul_log_le_klDiv, lebesgue_number_lemma_of_emetric_nhds, Ordinal.existsAddOfLE, LinearGrowth.linearGrowthSup_add_le, Finset.uIcc_map_sectR, setOf_riemmanianEDist_lt_subset_nhds, MeasureTheory.Measure.outerMeasure_le_iff, Set.range_IciExtend, LinearOrderedCommGroup.tendsto_nhds, LinearMap.le_ker_iff_comp_subtype_eq_zero, Finset.le_sup'_iff, AddMonoidAlgebra.le_infDegree_add, PrincipalSeg.isLowerSet_range, TopologicalSpace.Closeds.complOrderIso_symm_apply, antitone_continuousOn, AlgebraicGeometry.AffineSpace.SpecIso_hom_appTop, Fin.finsetImage_addNat_Ici, ArchimedeanClass.closedBallAddSubgroup_top, ValuationSubring.valuation_le_iff, List.foldr_max_of_ne_nil, ArchimedeanClass.addSubgroup_antitone, MeasureTheory.hittingAfter_bot_le_iff, Nat.succPNat_le_succPNat, Finset.min_eq_inf_withTop, Ordinal.mul_le_nmul, OrderHom.le_map_sup_fixedPoints, IdealFilter.isPFilter_gabrielComposition, AlgebraicGeometry.LocallyRingedSpace.basicOpen_eq_bot_iff_forall_evaluation_eq_zero, NumberField.mixedEmbedding.one_le_convexBodyLTFactor, Nat.ceil_pos, Multiset.le_iff_count, MeasureTheory.sum_addContent_le_of_subset, UniformFun.edist_eval_le, isSublattice_latticeClosure, Set.iUnion_Ioc_eq_Ioi_self_iff, Filter.map_mono, PrimeSpectrum.equivSubtype_apply_coe, Eq.cmp_eq_eq', AlgebraicGeometry.exists_isAffineOpen_preimage_eq, edist_le_pi_edist, mabs_le, add_right_strictMono_of_ne_top, le_dimH_of_hausdorffMeasure_ne_zero, AlgebraicGeometry.Scheme.Modules.pushforward_obj_obj, strictMono_nhdsSet, Filter.bliminf_sup_le_and_aux_left, mapClusterPt_leftLim, Ordinal.blsub_le_iff, DividedPowers.IsDPMorphism.ideal_comp, MeasureTheory.Measure.pi_Ioc_ae_eq_pi_Icc, ENNReal.lintegral_mul_prod_norm_pow_le, Set.Icc.coe_zero, le_mabs', CFC.exists_pos_algebraMap_le_iff, SSet.ofSimplex_le_skeleton, IncidenceAlgebra.eulerChar_prod, Metric.mem_thickening_iff_infEDist_lt, MeasureTheory.addEquivAddHaarChar_pos, Finset.addConst_le_inv_dens, Cardinal.isSuccLimit_ord, AlgebraicGeometry.Scheme.IdealSheafData.one_eq_top, PrincipalSeg.isMin_apply_iff, sdiff_le_iff', IsOpen.exists_subset_affineIndependent_span_eq_top, HahnSeries.coe_cardSuppLTAddSubmonoid, SSet.Subcomplex.toSSetFunctor_map, Order.instSubsingletonSuccOrder, Filter.le_sub_iff, Finset.card_Ico_mul_right, AlgebraicGeometry.Spec.locallyRingedSpaceObj_presheaf_map, Pi.wcovBy_iff, Submonoid.fg_iff_add_fg, nhdsWithin_uIoo_right_le_nhdsNE, TopologicalSpace.Opens.coe_inclusion', Finset.fold_min_lt, nndist_pi_le_iff, MvPowerSeries.le_order_mul, Sublattice.map_equiv_eq_comap_symm, OrderRingHom.eq_id, AddSubgroup.map_inf_le, SSet.iSup_subcomplexOfSimplex_prod_eq_top, AlgebraicGeometry.Scheme.Hom.preimage_bot, monotoneOn_iff_forall_lt, Subgroup.subset_pointwise_smul_iff, OrderEmbedding.le_map_sup, Set.InjOn.filter_map_Iic, AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_inv_eq_germ_assoc, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.app_invApp, Multiset.union_le_iff, LowerSet.coe_subset_coe, Ordinal.omega_lt_omega, mem_nhdsLT_iff_exists_Ico_subset, TopologicalSpace.OpenNhds.inclusionMapIso_hom, IntermediateField.coe_bot, Ordinal.omega0_lt_omega_one, isSimpleOrder_iff_isAtom_top, UCompactlyGeneratedSpace.le_compactlyGenerated, LieModule.posFittingComp_le_iInf_lowerCentralSeries, Part.Fix.exists_fix_le_approx, lowerPolar_anti, Pi.card_Ici, Subgroup.le_normalClosure, stdSimplexHomeomorphUnitInterval_symm_apply_coe, FractionalIdeal.le_dual_iff, addLeftReflectLT_of_addLeftMono, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.inv_naturality, Cardinal.ofENat_le_ofNat, Submodule.mulMap_one_right_eq, LowerAdjoint.closed_eq_range_close, OrdinalApprox.gfp_mem_range_gfpApprox, CompleteLattice.isCompactElement_finsetSup, biSup_inf_le_biSup_inf, minimalPrimes_eq_minimals, AlgebraicGeometry.Scheme.mem_basicOpen', Real.toNNReal_le_toNNReal_iff', Ordinal.le_invVeblen₂_iff, MeasureTheory.Measure.mutuallySingular_tfae, Ring.isField_iff_isSimpleOrder_ideal, sup_compl_le_himp, Cardinal.toENat_ofENat, PNat.Icc_eq_finset_subtype, ConditionallyCompleteLinearOrder.toCompactIccSpace, SMulPosReflectLE_iff_smulPosReflectLT, Submonoid.closure_singleton_le_iff_mem, EReal.image_coe_Ioc, CategoryTheory.Subobject.inf_le_left, Set.Ici.coe_inf, AddSubsemigroup.coe_toSubsemigroup_symm_apply, Fin.finsetImage_castSucc_Ioo, Valuation.IsRankOneDiscrete.exists_generator_lt_one', LieSubalgebra.le_normalizer, Monotone.Ici, ENat.le_ceil_self, signHom_apply, MeasureTheory.Ico_ae_eq_Icc', Set.sdiff_singleton_wcovBy, Polynomial.div_tendsto_atBot_leadingCoeff_div_of_degree_eq, ContinuousMap.le_def, isPreconnected_iff_ordConnected, EMetric.nhdsWithin_basis_closed_eball, CategoryTheory.Subgroupoid.disconnect_le, Pi.toColex_monotone, AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_morphismRestrict_assoc, Ordinal.toNatOrdinal_eq_one, not_supPrime, sInf_lt_iff, WellFoundedGT.monotone_chain_condition, Set.ncard_le_encard, himp_triangle, idealFactorsEquivOfQuotEquiv_mem_normalizedFactors_of_mem_normalizedFactors, AlgebraicGeometry.Scheme.zeroLocus_eq_univ_iff_subset_nilradical, upperClosure_image, List.max_prod_le, CategoryTheory.Limits.MultispanIndex.toLinearOrderMultispanIso_inv_app, OrderAddMonoidHom.inl_apply, AlgebraicGeometry.Scheme.Opens.topIso_inv, AddSubgroup.mem_toSubgroup', Convex.convex_remove_iff_notMem_convexHull_remove, Opens.pretopology_toGrothendieck, ProjectiveSpectrum.gc_homogeneousIdeal, eventually_enorm_mfderivWithin_symm_extChartAt_lt, Bornology.IsBounded.subset_Icc_sInf_sSup, continuousOn_floor, iSupIndep.disjoint_biSup, Prod.Lex.lt_iff', Finset.Icc_min_max, upperSetIsoLowerSet_apply, EReal.exp_strictMono, MeasureTheory.setLIntegral_pos_iff, Finset.Icc_eq_image_powerset, Subgroup.op_le_op_iff, LowerAdjoint.closure_union_closure, AlgebraicGeometry.ProjectiveSpectrum.Proj.toSpec_preimage_basicOpen, TwoSidedIdeal.ringCon_le_iff, Finset.ordConnected_range_coe, Field.Emb.Cardinal.directed_filtration, AlgebraicGeometry.Proj.pow_apply, Set.Ioc_union_Ioc_right, EMetric.isOpen_iff, SimpleGraph.chromaticNumber_pos, BoxIntegral.IntegrationParams.henstock_le_mcShane, Fin.image_succ_Iic, RCLike.pos_iff, exists_enorm_lt', Ordinal.type_lt_iff, HahnSeries.isPWO_support, Matroid.eRk_singleton_le, CategoryTheory.Sieve.mem_iff_pullback_eq_top, Subgroup.normalCore_eq_iSup, Submodule.restrictScalars_monotone, Ordinal.IsNormal.apply_le_nfp, bot_sdiff, Ordinal.succ_one, AntitoneOn.Ici, Set.Intersecting.isUpperSet', IntermediateField.lift_insepDegree_bot', MeasureTheory.Measure.measure_subtype_coe_le_comap, ciSup_partialSups_eq', Complex.not_le_iff, Finset.grade_eq, Ideal.exists_minimal_isPrimary_decomposition_of_isPrimary_decomposition, NonUnitalSubring.toNonUnitalSubsemiring_strictMono, MeasureTheory.smul_le_stoppedValue_hittingBtwn, Filter.disjoint_comap_iff, Order.krullDim_nonneg_iff, MeasureTheory.hittingBtwn_univ, Set.image_neg_Ico, instIsCountablyGenerated_atBot, Matroid.eRk_mono, NonUnitalAlgHom.range_comp_le_range, Filter.tendsto_neg_atBot_atTop, TopCat.Presheaf.SheafConditionEqualizerProducts.piOpens.hom_ext_iff, OrderMonoidIso.withZero_symm_apply_symm_apply, Ordinal.sub_eq_zero_iff_le, RootPairing.rootSpan_dualAnnihilator_le_ker_rootForm, Polynomial.eventually_atBot_not_isRoot, Complementeds.coe_le_coe, MeasureTheory.measure_le_setLAverage_pos, inf_mem_infClosure, sSup_Iio_eq_self_iff_isSuccPrelimit, convexHull_eq_iInter, WithLp.enorm_snd_le, Finset.inf_eq_top_iff, ofAddUnits_addUnits_gc, MeasureTheory.exists_notMem_null_le_lintegral, CategoryTheory.Subobject.map_comp, Submonoid.mem_pos, Decidable.eq_iff_le_not_lt, HahnEmbedding.Partial.orderTop_eq_archimedeanClassMk, Fin.range_natAdd_eq_Ici, ENNReal.log_lt_zero_iff, Ring.krullDimLE_iff, Set.OrdConnected.IicExtend, NonUnitalAlgebra.coe_bot, Finset.Icc_erase_left, LSeries_injOn, CongruenceSubgroup.exists_Gamma_le_conj', Filter.le_smul_iff, Int.strictAntiOn_natAbs, OrderMonoidIso.withZero_symm_apply_apply, WithTop.Ioc_coe_coe, Finset.isAtom_iff, Finset.uIcc_eq_union, Continuous.upperSemicontinuous, ringKrullDim_eq_one_iff_of_isLocalRing_isDomain, bddAbove_iff_exists_ge, Matrix.specialUnitaryGroup_le_unitaryGroup, AddSubgroup.characteristic_iff_comap_le, OrderEmbedding.infIrredUpperSet_surjective, MeasureTheory.measure_diff_le_iff_le_add, ENNReal.rpow_add_le_add_rpow, Cardinal.ord.orderEmbedding_coe, AddSubmonoid.unop_le_unop_iff, Metric.tendsto_atTop, AlgebraicGeometry.Spec_presheaf, MeasureTheory.SimpleFunc.measure_support_lt_top_of_memLp, isCoatomic_dual_iff_isAtomic, MeasureTheory.Integrable.measure_norm_ge_lt_top, smulPosMono_iff_smulPosStrictMono, SubAddAction.closure_mono, Ordinal.omega0_lt_preOmega_iff, tendsto_smul_comp_nat_floor_of_tendsto_mul, Fin.preimage_castSucc_Iic_castSucc, List.minimum_of_length_pos_le_iff, Ideal.instCanLiftTwoSidedIdealCoeOrderHomAsIdealIsTwoSided, Equiv.Perm.disjoint_of_disjoint_support, strictMono_of_lt_add_one, iInf_of_empty, Disjoint.dual, Ordinal.mulRightMono, Field.Emb.Cardinal.instInverseSystemWithTopToTypeOrdRankAlgHomSubtypeMemIntermediateFieldCoeOrderEmbeddingFiltrationAlgebraicClosureEmbFunctor, ClosedSubmodule.orthogonal_gc, CategoryTheory.Sieve.generate_functorPullback_le, AddValuation.top_iff, Fin.map_castLEEmb_Iic, latticeClosure_univ, scottContinuous_inf_right, FirstOrder.Language.Substructure.fg_closure_singleton, Seminorm.bddAbove_range_iff, sSup_eq_bot, inseparable_top, Real.tendsto_toNNReal_atTop_iff, monotone_hausdorffEntourage, TwoSidedIdeal.span_mono, SimpleGraph.Subgraph.coeSubgraph_le, imageToKernel'_kernelSubobjectIso, exists_seq_strictMono_tendsto_nhdsWithin, Hyperreal.neg_lt_of_tendsto_zero_of_pos, Fin.card_Ici, nhdsWithin_Ioi_isMeasurablyGenerated, Ordinal.instAddRightMono, Metric.mem_closedEBall, AddGroupCone.nonneg_toAddSubmonoid, Prop.codisjoint_iff, nnnorm_cfcₙ_lt_iff, CategoryTheory.Subobject.isPullback_aux, cauchySeq_finset_iff_nat_tsum_vanishing, ENat.gc_toENNReal_floor, NonemptyInterval.mem_pure, AddSubgroup.le_prod_iff, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.invApp_app_assoc, Set.preimage_div_const_Iic, AlgebraicGeometry.Scheme.IdealSheafData.le_map_iff_comap_le, convex_Iic, Fin.prod_Ico_succ, AlgebraicGeometry.Scheme.Hom.resLE_eq_morphismRestrict, WithTop.Ico_coe_top, Profinite.NobelingProof.coe_πs, Module.length_of_free, Metric.hausdorffEDist_iUnion_le, Submonoid.le_pi_iff, ContinuousLinearMap.IsPositive.inner_nonneg_right, AbsoluteValue.IsEuclidean.sub_mod_lt, MvPolynomial.le_vanishingIdeal_zeroLocus, TensorAlgebra.adjoin_range_ι, isLocallyClosed_Ioi, Order.not_isPredPrelimit_iff, ofUnits_le_iff_le_units, LinearMap.range_domRestrict_le_range, Filter.NeBot.zero_div_nonneg, NonUnitalStarAlgebra.coe_bot, NonUnitalStarAlgebra.mem_top, PNat.Ioo_eq_finset_subtype, lowerClosure_eq_top_iff, MeasureTheory.MemLp.evariance_lt_top, WithZeroTopology.nhds_zero, ENNReal.exists_nat_pos_mul_gt, LowerSet.prod_self_lt_prod_self, CategoryTheory.preservesFiniteColimits_liftToFinset, FormalMultilinearSeries.le_changeOriginSeries_radius, sup_eq_sup_iff_left, sbtw_iff_mem_image_Ioo_and_ne, PrimeSpectrum.equivSubtype_symm_apply_asIdeal, ProbabilityTheory.sum_prob_mem_Ioc_le, Fin.orderHom_injective_iff, IsCompactlyGenerated.BooleanGenerators.isAtom, Polynomial.degreeLT_mono, Set.Icc_succ_left_eq_Ioc, CategoryTheory.ComposableArrows.twoδ₁Toδ₀_app_zero, Field.Emb.Cardinal.filtration_succ, Set.image_div_const_Iio, AlgHom.equalizer_eq_top, PosSMulStrictMono.toPosSMulMono, ArchimedeanClass.FiniteResidueField.ofArchimedean_apply, Finset.inf_lt_iff, Fin.finsetImage_natAdd_Icc, IsCompl.bihimp_eq_bot, NFA.pumping_lemma, AddCircle.equivIccQuot_comp_mk_eq_toIcoMod, List.nnnorm_prod_le', Set.IciExtend_self, AlgebraicGeometry.StructureSheaf.toOpen_comp_comap_apply, MeasureTheory.Measure.isTopologicalBasis_isOpen_lt_top, nnnorm_nsmul_le, Ordinal.lift_lt, IntermediateField.isCyclotomicExtension_le_of_dvd, Set.le_einfsep_of_forall_dist_le, ProperCone.coe_positive, AddSubgroup.le_centralizer_iff, Valuation.mem_ltIdeal_iff, CauSeq.abv_pos_of_not_limZero, Finset.sup_add_le, SimplexCategory.const_comp, iSup_false, NNReal.sqrt_le_iff_le_sq, SetTheory.Game.birthday_sub_le, ENNReal.tsum_iUnion_le_tsum, Valuation.val_le_one_or_val_inv_lt_one, Mathlib.Tactic.Order.ToInt.toInt_le_toInt, Matrix.eRank_le_card_height, ZFSet.lt_rank_iff, PosSMulReflectLE_iff_posSMulReflectLT, Con.le_iff, algebraicClosure.algebraicClosure_eq_bot, Nonneg.nontrivial, Ordinal.apply_le_nfp, PrimeSpectrum.discreteTopology_iff_finite_isMaximal_and_sInf_le_nilradical, Cardinal.range_ofENat, Wbtw.right_mem_image_Ici_of_left_ne, Module.End.independent_genEigenspace, MeasureTheory.measure_biUnion_le, Finset.box_succ_disjUnion, le_addRothNumber_product, MvPolynomial.weightedHomogeneousSubmodule_mul, negPart_eq_zero', Ordinal.mod_le, AffineSubspace.comap_bot, Opens.mayerVietorisSquare_X₃, Nat.cast_pos, ENat.card_pos, PUnit.bot_eq, Ideal.height_lt_top, Set.image_mulSingle_Ioo, TwoSidedIdeal.orderIsoRingCon_apply, BoundedLatticeHom.coe_comp_sup_hom', MeasureTheory.monotone_spanningSets, Subgroup.isCoatom_map, isPreconnected_Ici, smulPosStrictMono_iff_SMulPosReflectLE, Fin.map_addNatEmb_Ici, exists_enorm_lt, Cardinal.lift_eq_aleph_one, AlgebraicGeometry.Scheme.inv_app, segment_eq_image₂, preordToPartOrdCompToDualIsoToDualCompPreordToPartOrd_inv_app_hom_coe, Filter.Tendsto.partialSups_apply, Function.Surjective.le_map_cofinite, MeasureTheory.lintegral_indicator_one_le, iInf_false, ENNReal.exists_pos_sum_of_countable', Valuation.leSubmodule_monotone, MeasureTheory.Measure.measure_univ_pos, CategoryTheory.ComposableArrows.scMap_τ₁, Int.fract_add_fract_le, AlgebraicGeometry.Scheme.Hom.id_appTop, sum_div_card_sq_le_sum_sq_div_card, Subgroup.map_le_range, CategoryTheory.Presheaf.imageSieve_app, Finsupp.bot_eq_zero, OrderIso.invENNReal_symm_apply, FirstOrder.Language.Substructure.comap_strictMono_of_surjective, Set.Icc_min_max, PredOrder.isPredLimit_of_mem_frontier, IsCoatomic.exists_coatom, AlgebraicGeometry.Proj.basicOpenIsoAway_hom, two_mul_le_add_sq, IsUnit.star_right_conjugate_nonneg_iff, smoothSheafCommRing.ι_evalHom_apply, CategoryTheory.instIsSimpleOrderSubobjectOfSimple, IsDedekindDomain.HeightOneSpectrum.mem_adicCompletionIntegers, SetTheory.PGame.grundyValue_eq_iff_equiv_nim, strictAnti_vecEmpty, IsOrderedRing.toPosMulMono, CategoryTheory.Functor.mapComposableArrowsObjMk₁Iso_inv_app, TopCat.Presheaf.germ_exist_of_isBasis, MeasureTheory.eLpNorm_const_lt_top_iff_enorm, isLeast_csInf, Fin.finsetImage_val_Ioi, SimplexCategory.δ_comp_σ_of_gt'_assoc, interior_Ici, cauchySeq_iff_tendsto_dist_atTop_0, ENNReal.monotone_rpow_of_nonneg, Set.einfsep_lt_top_iff, Submodule.mul_one_div_le_one, ENat.toENNReal_strictMono, Int.cast_le_neg_one_of_neg, Topology.IsScott.isOpen_iff_scottContinuous_mem, Set.Icc_subset_Ico_union_Icc, Ordinal.not_lt_zero, uniformity_basis_edist, LieModule.derivedSeries_le_lowerCentralSeries, Module.End.invtSubmodule.disjoint_mk_iff, fact_one_le_two_ennreal, IsProperMap.ultrafilter_le_nhds_of_tendsto, NNReal.sqrt_mul_lt_half_add_of_ne, AddSubgroup.zmultiples_le, continuousWithinAt_inter_Ioi_iff_Ici, LinOrd.hom_id, Set.Ioo_inter_Iio, bot_lt_iSup, AlgebraicGeometry.image_morphismRestrict_preimage, Set.image_update_Ioo, CategoryTheory.Limits.MultispanIndex.toLinearOrderMultispanIso_hom_app, iInf_eq_top, LieModule.exists_genWeightSpace_zero_le_ker_of_isNoetherian, WithBot.le_of_forall_lt_iff_le, Set.Ioi_diff_Ici, LinearMap.polar_antitone, enorm_add₃_le, AlgebraicGeometry.Scheme.IdealSheafData.gc, Finset.val_lt_iff, ProbabilityTheory.indep_limsup_atTop_self, Matrix.det_vandermonde, Ordinal.card_le_aleph, covariant_lt_of_covariant_le_of_contravariant_eq, Order.length_le_coheight, Ordinal.isAcc_iff, posPart_eq_self, SetLike.isCoatom_iff, Int.padicValuation_lt_one_iff, Dynamics.netMaxcard_le_coverMincard, PrimeSpectrum.denseRange_comap_iff_ker_le_nilRadical, Order.height_add_one_le, bot_lt_isotypicComponent, CategoryTheory.Limits.kernelSubobject_arrow'_apply, Finset.set_encard_biUnion_le, Filter.eventuallyConst_atTop, IsGaloisGroup.fixingSubgroup_top, RingPreordering.toSubsemiring_strictMono, AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_hom_apply_asIdeal, HasCardinalLT.Set.functor_obj, ValuationRing.le_total_ideal, monotoneOn_of_sub_one_le, Multiset.sup_zero, MeromorphicNFAt.meromorphicOrderAt_nonneg_iff_analyticAt, CategoryTheory.Subfunctor.range_le_equalizer_iff, AddSubgroup.prod_mono_left, Interval.coe_sInf, RingHom.liftOfSurjective_comp, AlgebraicGeometry.Scheme.zeroLocus_mul, ENat.one_lt_card_iff_nontrivial, isCompl_ofDual_iff, CategoryTheory.Sieve.functorPushforward_ofObjects_le, AlgebraicGeometry.Scheme.Hom.le_ker_comp, PrimeSpectrum.gc, Set.projIic_surjective, CategoryTheory.Limits.CompleteLattice.hasFiniteLimits_of_semilatticeInf_orderTop, strictMonoOn_of_lt_add_one, CategoryTheory.Presieve.isSheaf_bot, invOf_pos, MeasureTheory.hasFiniteIntegral_iff_ofNNReal, Nonneg.isOrderedRing, addRightStrictMono_of_addRightReflectLE, Ordinal.omega_zero, MeasureTheory.le_eLpNorm_of_bddBelow, le_abs_self, IsGreatest.nnnorm_cfcₙ, MeasureTheory.Measure.le_ae_join, Complex.convexHull_reProdIm, Fin.orderPred_succ, Fin.prod_Ioo_castLE, AlgebraicGeometry.Proj.mul_apply, AlgebraicGeometry.StructureSheaf.comap_id', adjoin_le_integralClosure, TopologicalSpace.Opens.map_functor_eq', min_lt_min_left_iff, Ordinal.toNimber_symm_eq, AddSubmonoid.le_pointwise_smul_iff, Finset.Icc_succ_succ, Set.disjoint_union_right, Ordinal.bsup_id_succ, LatticeHom.coe_withTopWithBot, PrimeSpectrum.vanishingIdeal_strict_anti_mono_iff, EReal.neg_strictAnti, CategoryTheory.Subobject.factorThru_eq_zero, nhds_basis_Icc_pos, EMetric.diam_mono, Subsemigroup.op_le_op_iff, TopologicalSpace.OpenNhds.comp_apply, Finset.prod_Icc_eq_prod_Ico_mul, Filter.covariant_div, Submonoid.closure_pow_le, nullMeasurableSet_Ioo, tsum_geometric_lt_top, Valuation.Integers.isPrincipal_iff_exists_isGreatest, Ordinal.liftPrincipalSeg_top', le_iff_atom_le_imp, AlgebraicGeometry.Scheme.Modules.germ_restrictStalkNatIso_inv_app, lowerCentralSeries_antitone, completeSpace_iff_ultrafilter, InitialSeg.toOrderEmbedding_apply, Cardinal.one_lt_aleph0, nnnorm_sum_le, Ordinal.sup_succ_le_lsub, Set.encard_preimage_val_le_encard_left, LieIdeal.gc_map_comap, Dynamics.coverMincard_finite_of_isCompact_invariant, Real.dimH_lt_top, Subsemiring.gc_map_comap, MeasurableSpace.generateFrom_empty, Set.accumulate_bot, tendsto_inv_nhdsGE_inv, nhds_basis_abs_sub_lt, eq_iff_ge_not_gt, RootPairing.range_polarization_domRestrict_le_span_coroot, SetSemiring.mulRightMono, Finset.one_le_mulConst_self, Finsupp.le_weight_of_ne_zero', ENNReal.mulRightOrderIso_symm_apply, instWellQuasiOrderedLEOfWellFoundedLT, ENNReal.ofReal_le_ofReal_iff, ContinuousLinearMap.ker_le_ker_iff_range_le_range, Dynamics.coverEntropy_iUnion_le, egauge_ball_le_of_one_lt_norm, Ideal.subset_pointwise_smul_iff, AddSubgroup.map_eq_range_iff, mem_doublyStochastic_iff_sum, Ideal.IsPrime.prod_le, lt_iff_le_and_ne', FirstOrder.Language.PartialEquiv.dom_le_dom, Fin.attachFin_Ioo_eq_Ioi, Set.sub_mem_Ioc_iff_left, IsGaloisGroup.fixedPoints_top, SimpleGraph.Walk.edist_le, Set.strictMonoOn_projIic, LowerSet.sdiff_le_left, inv_atTop₀, PartENat.eq_natCast_sub_of_add_eq_natCast, Matroid.eRank_le_encard_add_eRk_compl, LinearOrderedCommGroupWithZero.inl_mul_inr_eq_coe_toLex, LaurentSeries.valuation_le_iff_coeff_lt_eq_zero, ENNReal.natCast_lt_top, Pi.covBy_iff_exists_right_eq, RingCon.comap_bot, Ordinal.iSup_le_lsub, BoxIntegral.TaggedPrepartition.isHenstock_single, Finpartition.IsEquipartition.card_biUnion_offDiag_le', AlgebraicGeometry.SpecMap_preimage_basicOpen, CategoryTheory.Subobject.ofMkLEMk_comp_ofMkLEMk_assoc, Matrix.le_one_of_mem_colStochastic, LieSubmodule.lcs_add_le_iff, MeasureTheory.condLExp_bot', MeasureTheory.AddContent.measure_eq, UpperSet.coe_add, BoundedLatticeHom.dual_id, Ordinal.card_iSup_Iio_le_sum_card, OrderIso.isNormal, exists_seq_strictAnti_tendsto, CategoryTheory.Functor.WellOrderInductionData.Extension.map_zero, Subsemigroup.map_le_iff_le_comap, Ordinal.isOpen_singleton_iff, Equiv.Perm.support_cycleOf_le, Lattice.BicartSq.le₁₂, Localization.localRingHom_mk', Int.card_Ioo, orderBornology_isBounded, AddCommMonoid.primaryComponent.disjoint, FirstOrder.Language.Hom.range_le_iff_comap, exposed_point_def, ENNReal.lintegral_mul_le_one_of_lintegral_rpow_eq_one, RingCon.matrix_strictMono_of_nonempty, CategoryTheory.Classifier.SubobjectRepresentableBy.iso_inv_hom_left_comp, isOpenMap_iff_nhds_le, lowerSemicontinuous_iff_isClosed_preimage, Metric.ediam_closedEBall_le, Set.ordConnected_iff_uIcc_subset_left, Ideal.bot_lt_of_maximal, isGLB_infClosure, lt_tsub_iff_right, isWellOrder_lt, Nat.ofNat_le, Finset.subtype_mono, upperCentralSeries_mono, IsClosed.upperClosure, LipschitzWith.ediam_image_le, antisymmetrization_fibration, NatOrdinal.toOrdinal_min, Multiset.card_Ico, AlgebraicGeometry.Scheme.ideal_ker_le_ker_ΓSpecIso_inv_comp, Ordinal.enum_symm_apply_coe, Real.toNNReal_le_ofNat, Finset.Ioc_erase_right, UnitAddTorus.mFourierSubalgebra_coe, Ne.not_ge_or_not_le, Metric.mem_cthickening_iff, Module.mem_support_iff_of_finite, FiniteArchimedeanClass.mk_lt_mk, OrderMonoidHom.mul_apply, infPrime_iff_not_isMax, Algebra.toSubsemiring_eq_top, IntermediateField.botEquiv_symm, Continuous.finset_sup, Subalgebra.op_bot, Submodule.set_smul_mono_left, CStarAlgebra.mem_Icc_iff_norm_le_one, YoungDiagram.coe_bot, Monotone.Ioo, smoothSheafCommRing.eval_germ, frattini_le_comap_frattini_of_surjective, partialSups_eq_sup'_range, LowerAdjoint.le_iff_subset, Filter.map_mul_right_nhdsGT, Fin.image_natAdd_Icc, Fin.find_mono_of_le, EReal.addLECancellable_coe, SetRel.core_mono, Order.krullDim_pos_iff, DividedPowers.SubDPIdeal.le_iff, Ordinal.toGame_lt_iff, max_add_add_le_max_add_max, Finset.sum_Ico_le_sum, MeasureTheory.OuterMeasure.trim_top, AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_surjective, tendsto_fract_left', Interval.coe_iInf₂, MulEquiv.comapSubgroup_symm_apply, Set.inv_Ico, AbsoluteValue.abs_abv_sub_le_abv_sub, ExpGrowth.expGrowthInf_le_iff, zpow_left_injOn₀, ContinuousAt.finset_inf, EReal.image_coe_Ico, Fin.preimage_rev_Ici, Order.mem_range_succ_or_isSuccPrelimit, Filter.totallyBounded_iff_filter, LieSubalgebra.submodule_span_le_lieSpan, monovary_iff_mul_rearrangement, CategoryTheory.Sieve.pullback_monotone, LieModule.disjoint_genWeightSpaceOf, DFinsupp.toLex_monotone, Ordinal.IsNormal.lt_iff, RingPreordering.toSubsemiring_lt_toSubsemiring, SetTheory.PGame.grundyValue_nim_add_nim, RootPairing.invtRootSubmodule.eq_top_iff, NNReal.inner_le_Lp_mul_Lq_hasSum, deltaGenerated_le, HasFPowerSeriesWithinAt.eventually, IsLocalRing.closedPoint_mem_iff, Finset.map_truncatedInf, Nimber.succ_def, Int.cast_le, Metric.isSeparated_iff_setRelIsSeparated, Order.krullDim_le_one_iff, MeasureTheory.Measure.toSignedMeasure_le_toSignedMeasure_iff, IsUnifLocDoublingMeasure.eventually_measure_mul_le_scalingConstantOf_mul, Filter.bind_smallSets_gc, Ordinal.deriv_strictMono, nilradical_le_prime, LSeriesSummable.abscissaOfAbsConv_le, Matroid.isRkFinite_iff_cRk_lt_aleph0, AlgebraicGeometry.Scheme.isoSpec_inv_toSpecΓ, NNReal.rpow_add_rpow_le_add, Filter.bliminf_sup_le_and, QuotientAddGroup.leftRel_eq_top, SimpleGraph.ediam_eq_top, IsDedekindDomain.HeightOneSpectrum.intValuation_le_one, OrderHom.lfp_le_gfp, Set.preimage_const_sub_Iic, NNReal.coe_le_one, Part.toUnitMono_coe, le_iff_forall_rat_lt_imp_le, Affine.Simplex.wSameSide_affineSpan_faceOpposite_iff, EReal.neg_lt_comm, ENNReal.lt_iff_exists_add_pos_lt, smul_nonneg_iff_neg_imp_nonpos, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.instIsIsoInvApp, Fin.finsetImage_val_Ioo, Affine.Simplex.wOppSide_affineSpan_faceOpposite_point_right_iff, Filter.disjoint_cocompact_left, WeierstrassCurve.valuation_Δ_aux_eq_of_isIntegral, ValuationRing.TFAE, ContinuousLinearMap.opNNNorm_le_iff, TopologicalSpace.Opens.isCoatom_iff, Nat.preimage_Iio, MvPowerSeries.LinearTopology.basis_le_iff, Finset.Nonempty.sup_eq_top_iff, Ordinal.enum_zero_le, MonomialOrder.degree_sub_le, Module.Basis.flag_le_ker_dual, LieModule.lowerCentralSeries_one_inf_center_le_ker_traceForm, IsArtinian.eventuallyConst_of_isArtinian, Submonoid.coe_invOrderIso_symm_apply, Codisjoint.dual, Eq.cmp_eq_eq, negPart_eq_neg', AlgebraicGeometry.SheafedSpace.IsOpenImmersion.invApp_app_apply, le_iInf_const, IsNonarchimedean.apply_sum_le_sup_of_isNonarchimedean, Surreal.Multiplication.mulOption_lt_mul_iff_P3, disjoint_nhds_cocompact, le_of_toZ_le, Algebra.adjoin_singleton_algebraMap, Finset.inf'_lt_iff, lowerClosure_image, MeasureTheory.Integrable.exists_hasCompactSupport_lintegral_sub_le, RingCon.matrix_top, PSet.le_succ_rank_sUnion, EReal.toENNReal_pos_iff, CategoryTheory.TransfiniteCompositionOfShape.fac_assoc, monovary_inv_left, ENNReal.instOrderedSub, SSet.Subcomplex.range_eq_top, Language.instMulRightMono, Polynomial.exists_min_root, Submodule.le_linearEquiv_of_sSup_eq_top, MeasureTheory.SignedMeasure.exists_compl_positive_negative, Multiset.le_sup, Hyperreal.infinitesimal_def, isPiSystem_Ioi, Submodule.FG.lTensor.directedSystem, Ideal.Filtration.smul_le, ENNReal.ofReal_toReal_le, TopologicalSpace.Opens.map_obj, Finset.sum_Ico_int_sub, Set.inv_Ioc, MeasureTheory.Measure.LebesgueDecomposition.iSup_monotone, AlgHom.fieldRange_eq_map, IntermediateField.relfinrank_bot_right, SimpleGraph.vertexCoverNum_le_iff, Ne.lt_sup_or_lt_sup, Field.exists_primitive_element, ContFract.convs_eq_convs', isGalois_iff_isGalois_top, polynomialFunctions.le_equalizer, OrderIso.map_pred, TopologicalSpace.Opens.comp_apply, IntermediateField.relrank_eq_one_iff, PartENat.lt_find_iff, posMulMono_iff_covariant_pos, min_mul_min_le_min_mul_mul', ArchimedeanClass.FiniteResidueField.mk_eq_zero, le_zero_iff, Multiset.mem_powerset, Seminorm.coe_le_coe, TwoSidedIdeal.instIsTwoSidedCoeOrderHomIdealAsIdeal, Fin.preimage_succ_Ioi_succ, AlgebraicGeometry.isIso_ΓSpec_adjunction_unit_app_basicOpen, MeasureTheory.lintegral_prod_lintegral_pow_le, Finset.lubell_yamamoto_meshalkin_inequality_sum_card_div_choose, List.foldr_inf_eq_inf_toFinset, Multiset.Icc_eq, convexJoin_segments, PosSMulReflectLT.of_pos, Affine.Simplex.sign_signedInfDist_lineMap_incenter_touchpoint, AlgebraicGeometry.isNoetherian_iff_of_finite_affine_openCover, Submodule.one_le_traceDual_one, NonemptyInterval.length_nonneg, ENNReal.zero_lt_top, Submodule.comap_le_comap_smul, Filter.le_principal_iff, Nat.le_of_ceil_le, Coheyting.boundary_top, AlgebraicGeometry.Scheme.ι_toIso_inv, PrincipalSeg.isSuccLimit_apply_iff, Subgroup.pi_le_iff, nnnorm_le_add_nnnorm_add, MeasureTheory.VectorMeasure.trim_eq_self, UpperSet.Ici_lt_top, LTSeries.exists_relSeries_covBy_and_head_eq_bot_and_last_eq_bot, Finset.Colex.toColex_lt_toColex_iff_max'_mem, derivedSeries_le_map_derivedSeries, Pi.card_Iic, nnnorm_le_mul_nnnorm_add, AlgebraicGeometry.Scheme.IdealSheafData.radical_bot, FractionalIdeal.coe_le_coe, StieltjesFunction.measure_Ioo, Finset.Ioi_insert, Dynamics.coverMincard_image_le, EReal.sign_eq_and_abs_eq_iff_eq, Subgroup.subgroupOf_eq_top, FirstOrder.Language.Substructure.small_closure, Ordinal.lt_mul_div_add, starConvex_iff_ordConnected, MeasureTheory.measure_support_eapprox_lt_top, le_egauge_pi, Fin.image_natAdd_Ico, Order.pred_injective, Set.Ico_add_Icc_subset, AddGroupSeminormClass.toNonnegHomClass, Set.chainHeight_le_encard, Valuation.Compatible.srel_iff_lt, Ordinal.omega_le_omega, one_lt_mabs, MeasureTheory.exists_setLAverage_le, Matrix.posDef_iff_dotProduct_mulVec, exists_le_isAssociatedPrime_of_isNoetherianRing, BoxIntegral.Box.disjoint_withBotCoe, Metric.infEDist_le_edist_of_mem, egauge_add_add_le, Set.Ioo.lt_one, CategoryTheory.Subfunctor.range_toRange, TopologicalSpace.OpenNhds.val_apply, UnitAddTorus.mFourierSubalgebra_closure_eq_top, Convex.Ioo_subset_of_mem_closure, Function.sSup_div_semiconj, CategoryTheory.Limits.CompleteLattice.finite_coproduct_eq_finset_sup, HVertexOperator.coeff_isPWOsupport, InfiniteGalois.mem_bot_iff_fixed, FractionalIdeal.one_le, eq_top_iff, exists_pos_right_iff_sameRay, Finset.lt_sup'_iff, NNRat.cast_le, Frm.Iso.mk_hom, CategoryTheory.MorphismProperty.pushouts_le, Valuation.IsEquiv.valuedCompletion_le_one_iff, antivaryOn_comm, Finset.monotone_iff_forall_le_cons, enorm_indicator_le_of_subset, LocalSubring.isMax_iff, Submodule.map_span_le, AlgebraicGeometry.SmoothOfRelativeDimension.exists_isStandardSmoothOfRelativeDimension, IsSemisimpleModule.jacobson_le_annihilator, Finset.card_insertNone, Ordinal.IsNormal.nfp_le_apply, MvPowerSeries.min_lexOrder_le, hnot_le_iff_codisjoint_left, Ordinal.enum_lt_enum, setOf_liouvilleWith_subset_aux, Algebra.toSubring_eq_top, Ordinal.isSuccLimit_omega0, AlgebraicGeometry.Scheme.Hom.preimageIso_hom_ι_assoc, CategoryTheory.Limits.CompleteLattice.colimit_eq_iSup, AddCon.toSetoid_top, Pi.lex_le_iff_of_unique, enorm_prod_le, NNReal.orderIsoRpow_apply, Set.Iic_iInf, WellFoundedGT.iSup_eq_monotonicSequenceLimit, ProbabilityTheory.condIndep_bot_left, Order.Ioo_eq_empty_iff_le_succ, CStarAlgebra.span_nonneg_inter_unitBall, mul_self_le_one_iff, Pi.Colex.le_sSup_apply, LinearOrderedAddCommGroupWithTop.add_eq_top, CategoryTheory.isArtinianObject_iff_not_strictAnti, WithBot.orderIsoPUnitSumLex_symm_inr, UniformSpace.comp, Set.Iic_diff_Iio, Fin.map_natAddEmb_Ici, Order.Ideal.IsProper.notMem_or_compl_notMem, Coheyting.boundary_le_hnot, OmegaCompletePartialOrder.ContinuousHom.ωSup_apply_ωSup, OrderEmbedding.preimage_uIoc, TopCat.GlueData.ofOpenSubsets_toGlueData_U, CategoryTheory.Functor.WellOrderInductionData.surjective, OmegaCompletePartialOrder.OrderHom.ωSup_coe, LieModule.genWeightSpaceChain_def', TopologicalSpace.Opens.instIsContinuousCompGrothendieckTopology, Manifold.riemannianEDist_le_pathELength, Module.End.genEigenspace_directed, Valuation.isEquiv_iff_val_lt_val, NNReal.agmSequences_zero, Finset.subConst_le_addConst_sq, NumberField.adjoin_eq_top_of_infinitePlace_lt, lowerClosure_one, NonUnitalSubring.closure_preimage_le, Set.Nonempty.eq_Icc_iff_int, LinearGrowth.le_linearGrowthSup_add', LieAlgebra.IsSemisimple.booleanGenerators, CategoryTheory.Localization.LeftBousfield.galoisConnection, LieSubmodule.lie_le_iff, Ordinal.isInitialIso_symm_apply_coe, LowerSet.Iio_strictMono, FirstOrder.Language.Substructure.cg_def, MonoidHom.coe_toMultiplicative_range, le_iff_forall_pos_le_add, lt_emultiplicity_of_lt_multiplicity, Cardinal.mk_monotone, ENat.monotone_map_iff, ENNReal.limsup_mul_le, Subgroup.le_centralizer_iff, Decidable.eq_iff_ge_not_gt, OrderAddMonoidHom.fst_comp_inl, Finset.le_inf'_iff, leOnePart_le_one, SSet.horn₃₂.desc.multicofork_π_one, NonUnitalSubalgebra.topologicalClosure_map_le, CategoryTheory.ObjectProperty.isomorphisms_le_isoModSerre, mul_nonpos_iff_pos_imp_nonpos, NonemptyInterval.add_eq_zero_iff, Finset.sup'_Iic, ContinuousLinearMap.ebound, IsSublattice.latticeClosure_eq, MeasureTheory.lowerCrossingTime_le, InnerProductSpace.gramSchmidt_bot, Set.Iio_add_Iic_subset, TopologicalSpace.OpenNhds.inclusionMapIso_inv, ENNReal.tendsto_pow_atTop_nhds_zero_iff, Order.not_isPredPrelimit_iff', Polynomial.IsSplittingField.splits_iff, Cardinal.succ_natCast, sup_biInf_le_biInf_sup, MeasureTheory.SimpleFunc.nearestPtInd_succ, Multiset.card_Ioc_eq_card_Icc_sub_one, himp_inf_le, Set.Ioc_subset_uIoc', Filter.NeBot.zero_smul_nonneg, sub_inv_antitoneOn_Ioi, Filter.addRightMono, Subring.toAddSubgroup_mono, AlgebraicGeometry.Scheme.IdealSheafData.equivOfIsAffine_apply, Set.disjoint_singleton_right, Partition.bot_notMem, CategoryTheory.GrothendieckTopology.monotone_close, Submodule.spanRank_toENat_eq_iInf_encard, List.sortedGT_iff_nodup_and_sortedGE, MeasureTheory.Measure.haarScalarFactor_pos_of_isHaarMeasure, Subsemigroup.opEquiv_symm_apply, isLeast_sdiff, AlgebraicGeometry.structureSheafInType.mul_apply, two_inv_lt_one, TopCat.Presheaf.germ_stalkSpecializes_apply, SSet.Subcomplex.topIso_inv_app_coe, one_notMem_iff, SSet.prodStdSimplex.orderHomOfSimplex_coe, Subgroup.ofUnits_strictMono, Matroid.le_eRk_iff, MeasureTheory.exists_upperSemicontinuous_le_lintegral_le, NNReal.agm_lt_max_of_ne, SimpleGraph.hasse_preconnected_of_pred, max_lt_max_left_iff, variationOnFromTo.antitoneOn, ENNReal.orderIsoRpow_symm_apply, OrderIso.essSup_apply, Finset.sortedGT_sort, PartENat.coe_lt_coe, Finset.mem_uIcc', nnnorm_le_nnnorm_add_nnnorm_sub, Ordinal.isInitial_preOmega, sign_nonneg_iff, Finset.tendsto_Ioc_neg_atTop_atTop, WithTop.pow_right_strictMono, min_abs_abs_le_abs_min, Multiset.lt_iff_cons_le, Submonoid.le_comap_map, MeasureTheory.Measure.le_sum_apply, LieAlgebra.derivedSeriesOfIdeal_succ_le, Subspace.dualAnnihilator_dualAnnihilator_eq, SimpleGraph.regularityReduced_adj, OrderHom.antisymmetrization_apply, Set.sub_mem_Ico_iff_left, Submodule.le_dualAnnihilator_dualCoannihilator, Hyperreal.coe_lt_omega, UpperSet.mem_map, himp_bot, Set.Icc.nonneg, Fin.strictMono_succAbove, AlgebraicGeometry.StructureSheaf.exists_const, Finset.card_Ico_add_right, LinOrd.dual_map, IsOpenMap.functorNhds_obj_coe, Besicovitch.exists_closedBall_covering_tsum_measure_le, Real.le_toNNReal_iff_coe_le, SetSemiring.up_lt_up, t1Space_iff_disjoint_nhds_pure, nhds_basis_Ioo_one_lt, skyscraperPresheafCocone_pt, OrderIso.sumLexIioIci_symm_apply_Ici, MeasureTheory.eLpNorm_add_lt_top, Finsupp.Lex.addLeftMono, CategoryTheory.Subobject.top_arrow_isIso, AddOpposite.op_le_one, Ioo_subset_closure_interior, NNReal.pi_pos, Set.bounded_gt_inter_ge, Cardinal.ofNat_lt_lift_iff, bot_lt_affineSpan, ConditionallyCompleteLinearOrderedField.toZeroLEOneClass, Finset.ofDual_inf, Filter.tendsto_Ioo_pure_bot, AlgebraicGeometry.Scheme.Modules.Hom.zero_app, biUnion_range_succ_disjointed, OrderIso.dualAntisymmetrization_symm_apply, pow_le_pow_iff_left, PrimeSpectrum.denseRange_comap_iff_minimalPrimes, Finsupp.Lex.single_strictAnti, MeasureTheory.hittingAfter_mono, LatticeHom.withTopWithBot_id, Submodule.iSup_map_single_le, Subring.op_le_op_iff, Valuation.isNontrivial_iff_exists_lt_one, Module.jacobson_lt_top, IntermediateField.extendScalars_injective, Order.le_succ_and_le_iff, TopCat.Sheaf.pushforward_map, FractionalIdeal.coeIdeal_le_one, polynomialFunctions.starClosure_topologicalClosure, Coheyting.hnot_boundary, Subsemiring.op_le_op_iff, nullMeasurableSet_Ioi, Ideal.map_under_le_under_map, EReal.preimage_coe_Ioi, OrderHom.map_lfp, Subgroup.IsSubnormal.iff_eq_top_or_exists, Valuation.Integers.valuation_irreducible_lt_one, RingHom.liftOfSurjective_comp_apply, Finset.le_truncatedSup, UpperSet.map_map, AlgebraicGeometry.Scheme.IdealSheafData.equivOfIsAffine_symm_apply, WithZeroTopology.nhds_zero_of_units, CategoryTheory.IsGrothendieckAbelian.generatingMonomorphisms.exists_ordinal, Subring.le_op_iff, AlgebraicGeometry.Scheme.congr_app, CategoryTheory.GrothendieckTopology.trivial_covering, Nimber.mul_le_of_forall_ne, TopologicalSpace.Opens.infLELeft_apply_mk, AlgebraicGeometry.Scheme.IdealSheafData.ideal_mul, Set.Ico_subset_Icc_union_Ico, Ordinal.add_log_le_log_mul, CategoryTheory.Subgroupoid.le_iff, hnot_symmDiff_self, ProperCone.mem_map, AddValuation.map_add', AffineEquiv.span_eq_top_iff, mem_nhdsGT_iff_exists_Ioc_subset, AlgebraicGeometry.IsZariskiLocalAtTarget.restrict, Sublattice.map_le_iff_le_comap, borel_eq_generateFrom_Ici, GroupSeminorm.lt_def, Topology.RelCWComplex.coe_skeletonLT, Num.cast_le, invOf_nonneg, Submodule.one_mem_div, CompleteLattice.wellFoundedGT_iff_isSupFiniteCompact, Order.Ideal.coe_sup, AugmentedSimplexCategory.eqToHom_toOrderHom, AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.structurePresheafInCommRing_obj_carrier, AddEquiv.coe_comapAddSubgroup, AlgebraicGeometry.Scheme.SpecΓIdentity_hom_app, t2Space_antitone, LinearEquiv.dilatransvections_pow_mono, Filter.tendsto_mabs_atTop_atTop, AffineSubspace.direction_eq_top_iff_of_nonempty, Finset.ordConnected_range_val, MeasureTheory.eLpNorm_mono_real, ENat.le_floor, Dynamics.netEntropyEntourage_le_coverEntropyEntourage, NNReal.not_summable_iff_tendsto_nat_atTop, Filter.disjoint_principal_left, Fin.sum_sum_eq_sum_triangle_add, LieSubmodule.map_inf_le, MeasureTheory.mul_meas_ge_le_pow_eLpNorm', noTopOrder_iff_noMaxOrder, Filter.HasBasis.cauchySeq_iff, NonUnitalAlgebra.adjoin_le_centralizer_centralizer, strictConvex_Icc, antivary_iff_smul_rearrangement, Nonneg.mk_nnratCast, isSeparatedMap_iff_disjoint_nhds, Set.Ioo_inter_Ioi, LowerSet.bot_lt_Iic, MeasureTheory.Measure.haar.addHaarContent_outerMeasure_self_pos, TopologicalSpace.Opens.isBasis_iff_nbhd, AlgebraicGeometry.locallyQuasiFinite_iff, StarSubalgebra.bot_toSubalgebra, FirstOrder.Language.Substructure.mem_closure, Subsemigroup.comap_le_comap_iff_of_surjective, ENNReal.div_lt_div_iff_left, Finset.card_Ioo_finset, CategoryTheory.MorphismProperty.colimitsOfShape_le_coproducts, ContDiffOn.dimH_image_le, zpow_two_nonneg, WithTop.Ioo_coe_top, AlgebraicGeometry.Scheme.preimage_comp, GenContFract.succ_nth_fib_le_of_nth_den, Ultrafilter.inf_neBot_iff, Module.End.maxGenEigenspace_eq, one_le_mul_self_iff, IsLocalization.coe_primeSpectrumOrderIso_symm_apply_asIdeal, smoothSheaf.ι_evalHom_apply, List.coe_maximum_of_length_pos, IsCompl.symmDiff_eq_top, negPart_anti, Filter.coprod_inf_prod_le, CategoryTheory.Limits.CoproductsFromFiniteFiltered.finiteSubcoproductsCocone_pt, IntermediateField.Lifts.lt_iff_le_carrier_ne, RCLike.nonpos_iff, SemilatInfCat_dual_comp_forget_to_partOrd, iSupIndep.pairwiseDisjoint, AlgebraicGeometry.Scheme.Cover.RelativeGluingData.toBase_preimage_eq_opensRange_ι, convexHull_union_neg_eq_absConvexHull, Subalgebra.mulMap_toLinearMap, IsUltrametricDist.nnnorm_pow_le, RCLike.nonpos_iff_exists_ofReal, upperClosure_sups, CategoryTheory.Triangulated.SpectralObject.ω₂_obj_mor₃, Module.End.eigenspace_restrict_le_eigenspace, NNReal.sqrt_sq, lt_inv_self_iff, ArchimedeanClass.mk_le_mk, NNRat.lt_def, Submonoid.pointwise_smul_le_pointwise_smul_iff₀, SetTheory.Game.not_le, exists_lub_Iio, Polynomial.div_tendsto_zero_iff_degree_lt, Set.Iio_top, tendsto_floor_right_pure, TopCat.Presheaf.SheafConditionEqualizerProducts.fork_π_app_walkingParallelPair_one, FractionalIdeal.inv_le_inv_iff, IntermediateField.adjoin.mono, BoxIntegral.Box.Ioo_subset_coe, Finset.tendsto_Ioo_atBot_prod_atTop, EReal.instPosMulMono, IsCompl.le_sup_right_iff_inf_left_le, Algebra.elemental.le_iff_mem, Metric.ediam_thickening_le, MeasureTheory.aecover_Iio_of_Iic, Nat.preimage_Ici, AddAction.stabilizer_inf_stabilizer_le_stabilizer_apply₂, ProbabilityTheory.evariance_lt_top_iff_memLp, Submodule.dualAnnihilator_map_dualMap_le, upperSemicontinuousAt_iff_frequently, Subfield.splits_bot, AlgebraicGeometry.IsFinite.instMorphismRestrict, Seminorm.lt_def, NonemptyInterval.toLex_strictMono, cfcHom_nonneg_iff, PartitionOfUnity.nonneg', Finsupp.Colex.isStrictOrder, ConditionallyCompleteLinearOrder.compare_eq_compareOfLessAndEq, BoxIntegral.Box.splitUpper_le, add_top, LieModule.map_genWeightSpace_le, smul_nonpos_iff, smul_inf_le, LieModule.trivial_iff_le_maximal_trivial, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.app_inv_app', four_pos, le_comap_range_lTensor, AlgebraicGeometry.StructureSheaf.toOpen_comp_comap, Multiset.isAtom_singleton, isClosedMap_iff_comap_nhds_le, CategoryTheory.Subgroupoid.inclusion_refl, induced_const, AddMonoid.minOrder_le_addOrderOf, isClosed_nonneg, mabs_div_le, Finset.Colex.cons_lt_cons, Set.image_mulSingle_Icc_right, EuclideanSpace.nndist_eq, AddSubmonoid.pi_le_iff, CategoryTheory.Regular.frobeniusStrongEpiMonoFactorisation_I, Algebra.toSubring_bot, IsCompact.isLeast_sInf, ringKrullDim_quotient, CategoryTheory.ComposableArrows.threeδ₂Toδ₁_app_one, AlgHom.le_equalizer, Localization.localRingHom_injective_of_primesOver_eq_singleton, NNReal.sqrt_pos, TopCat.Presheaf.pushforwardToOfIso_app, MulOpposite.one_le_op, Nat.count_le_setENCard, Nat.one_div_le_one_div, MvPowerSeries.le_weightedOrder_mul, AlgebraicGeometry.affineLocally_iff_forall_isAffineOpen, ContractingWith.one_sub_K_pos', Subgroup.toSubmonoid_mono, CategoryTheory.MorphismProperty.IsInvertedBy.iff_map_le_isomorphisms, IdemCommSemiring.bot_le, Filter.principal_mono, MeasureTheory.Measure.mkMetric_apply, IsOrderedRing.toZeroLEOneClass, EReal.expOrderIso_symm, Cardinal.isNormal_preAleph, Finset.antitone_iff_forall_insert_le, affineSpan_singleton_union_vadd_eq_top_of_span_eq_top, MeasureTheory.exists_notMem_null_le_laverage, AddMonoidAlgebra.supDegree_add_le, Cardinal.bddAbove_ord_image_iff, MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable, nhdsSet_le, FractionalIdeal.instMulPosStrictMonoNonZeroDivisors, InitialSeg.monotone, Set.unbounded_gt_inter_gt, RootPairing.RootPositiveForm.zero_lt_posForm_iff, IsSimpleRing.tfae, partialSups_add_one', integralClosure_eq_top_iff, ENNReal.pow_lt_top_iff, sup_le_inf, Subsemiring.homogeneousCore_mono, SimpleGraph.two_le_chromaticNumber_of_adj, PrincipalSeg.map_bot, RootPairing.setOf_root_add_zsmul_mem_eq_Icc, Filter.inv_atTop, interior_convexHull_nonempty_iff_affineSpan_eq_top, LinearOrderedField.lt_inducedMap_iff, Finset.max_sum_le, Subsemiring.le_op_iff, Order.height_pos, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.ofRestrict_invApp, MvPolynomial.schwartz_zippel_totalDegree, EReal.image_coe_Iio, Matrix.det_projVandermonde, OrderAddMonoidHom.add_apply, TopologicalSpace.Opens.complOrderIso_apply, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivCounitIso_hom_app_hom, AddSubmonoid.multiples_le, Convex.mapsTo_lineMap, SetTheory.PGame.birthday_moveLeft_lt, isGLB_Ioi, measurableSet_le, Finset.max'_le_iff, Set.image_div_const_Ici, OrderHom.gfp_const_inf_le, Right.pow_lt_one_iff, Interval.length_sub_le, Filter.bliminf_antitone, IsStrictOrderedRing.toMulPosStrictMono, isOpen_lt, isProperMap_iff_ultrafilter_of_t2, Set.preimage_div_const_Iio, continuousOn_fract, Order.isSuccPrelimit_iff_succ_lt, ENNReal.mul_le_mul_right, Cardinal.toENat_eq_zero, CompleteSublattice.top_mem, PointedCone.coe_map, Affine.Simplex.sign_touchpointWeights_empty, ContinuousLinearMap.exists_nonlinearRightInverse_of_surjective, ProbabilityTheory.HasSubgaussianMGF.const_mul, LowerAdjoint.idempotent, Homeomorph.opensCongr_apply, AlgebraicGeometry.Scheme.isoSpec_inv_preimage_zeroLocus, Subalgebra.LinearDisjoint.inf_eq_bot_of_commute, MvPolynomial.le_weightedTotalDegree, SimpleGraph.card_le_chromaticNumber_iff_forall_surjective, MeasureTheory.measure_integral_le_pos, Ordinal.epsilon_zero_lt_gamma, Finite.ciInf_le, AddSubgroup.IsSubnormal.lt_normal, thickenedIndicatorAux_lt_top, Coheyting.hnot_eq_top_iff_exists_boundary, Real.nnnorm_exp_I_mul_ofReal_sub_one_le, Fin.finsetImage_castAdd_Iio, MeasureTheory.measure_pos_iff_nonempty_of_isAddLeftInvariant, NormedField.exists_nnnorm_lt_one, AddSubgroup.closure_le_centralizer_centralizer, StieltjesFunction.mono', EMetric.infEdist_pos_iff_notMem_closure, EMetric.diam_le_iff, Nonneg.canonicallyOrderedAdd, NonemptyInterval.coe_dual, ENNReal.lintegral_mul_rpow_le_lintegral_rpow_mul_lintegral_rpow, hnot_hnot_le, thickenedIndicatorAux_mono, Fin.preimage_castSucc_Ico_castSucc, MeasureTheory.Content.le_outerMeasure_compacts, CategoryTheory.Limits.kernelSubobjectIsoComp_hom_arrow, NNReal.le_sqrt_iff_sq_le, MvPowerSeries.le_order_prod, CategoryTheory.Functor.WellOrderInductionData.sectionsMk_val_op_bot, TopologicalSpace.eq_top_iff_forall_inseparable, BoundedLENhdsClass.of_closedIciTopology, SemilatticeInf.inf_le_right, isLUB_iSup, MulArchimedeanClass.top_eq_mk_iff, NNReal.sqrt_eq_rpow, sSup_empty, Submonoid.adjoin_eq_span_of_eq_span, InitialSeg.map_pred, CStarAlgebra.mem_Icc_iff_nnnorm_le_one, LieSubmodule.disjoint_toSubmodule, Ioc_subset_biUnion_Ioc, OrderRingHom.apply_eq_self, Topology.CWComplex.iUnion_openCell_eq_skeletonLT, MeasureTheory.measure_spanningSets_lt_top, AlgebraicGeometry.ProjectiveSpectrum.Proj.toOpen_toSpec_val_c_app_assoc, TopCat.Presheaf.germ_res', CategoryTheory.SmallObject.SuccStruct.Iteration.mkOfLimit.inductiveSystem_obj, Hyperreal.InfiniteNeg.lt_zero, AlgebraicGeometry.PresheafedSpace.Γ_obj_op, Sym2.sortEquiv_apply_coe, Finset.orderIsoOfFin_symm_apply, Submonoid.comap_strictMono_of_surjective, Metric.exists_continuous_nnreal_forall_closedBall_subset, tsub_pos_iff_lt, emultiplicity_lt_iff_not_dvd, Ordinal.div_le, CategoryTheory.MorphismProperty.monotone_map, le_sup_left, Order.le_krullDim_iff, TwoSidedIdeal.bot_asIdeal, Set.Ioc.coe_pow, AddSubmonoid.coe_toSubmonoid_apply, CStarAlgebra.nonneg_iff_eq_sqrt_mul_sqrt, OrderDual.instIsStrictOrderedModule, CategoryTheory.ComposableArrows.threeδ₁Toδ₀_app_zero, Cardinal.toENat_eq_ofNat, TopologicalSpace.Opens.map_comp_obj_unop, IntermediateField.adjoin_zero, mem_absConvexHull_iff, CategoryTheory.Limits.imageSubobject_arrow_comp_eq_zero, CauSeq.const_lt, Ordinal.one_toPGame_leftMoves_default_eq, MvPolynomial.idealOfVars_eq_restrictSupportIdeal, AlgebraicGeometry.Spec.locallyRingedSpaceObj_presheaf', IsLocalRing.maximalIdeal_le_jacobson, Set.preimage_const_sub_Icc, TopologicalSpace.Clopens.exists_prod_subset, FractionalIdeal.le_div_iff_mul_le, CanonicallyOrderedAdd.toOrderedSub, BoxIntegral.Box.monotone_face, MeasureTheory.ae_measure_preimage_mul_right_lt_top, Nat.image_cast_int_Icc, Multiset.sub_le_iff_le_add', Ultrafilter.coe_le_coe, ConcaveOn.convex_ge, ENNReal.nhds_top', Ordinal.toGame_one, MeasureTheory.enorm_ae_le_eLpNormEssSup, exists_nonneg_left_iff_sameRay, EReal.le_coe_toReal, Profinite.NobelingProof.spanFunctorIsoIndexFunctor_hom_app_hom_hom_apply_coe, sInf_sup_le_iInf_sup, SimpleGraph.three_le_egirth, instNoMaxOrderOfNontrivial, Fin.preimage_val_Ioc_val, Multiset.mem_powersetCardAux, Valuation.Integers.map_le_one, Dynamics.IsDynNetIn.card_le_netMaxcard, TopCat.Sheaf.interUnionPullbackCone_fst, nnnorm_le_insert, emultiplicity_le_emultiplicity_of_dvd_right, FormalMultilinearSeries.radius_le_smul, CategoryTheory.Subobject.map_top, AlgebraicGeometry.functionField_isScalarTower, OpenSubgroup.toSubgroup_le, Nat.image_cast_int_Iic, Set.Ioo.one_minus_pos, leOnePart_eq_one', InnerProductSpace.gramSchmidt_def, Ring.jacobson_smul_top_le, Orientation.map_eq_neg_iff_det_neg, PUnit.le, AddSubsemigroup.le_op_iff, uniformity_le_symm, Cardinal.lift_lt_ofNat_iff, IntermediateField.adjoin_univ, PiLp.nnnorm_eq_of_L2, Filter.disjoint_comap_iff_map', CategoryTheory.MorphismProperty.transfiniteCompositionsOfShape_le_transfiniteCompositions, descending_central_series_ge_lower, Cardinal.preAleph_nat, Disjoint.le_symmDiff_sup_symmDiff_right, neg_abs_le_neg, IsAtomic.eq_bot_or_exists_atom_le, imageToKernel_op, Finset.inf_id_eq_sInf, NonUnitalSubsemiring.toAddSubmonoid_strictMono, Filter.tendsto_Ioc_pure_bot, AlgebraicGeometry.Scheme.Modules.toOpen_fromTildeΓ_app, TopCat.Presheaf.stalkPushforward_germ_apply, IsDedekindDomain.HeightOneSpectrum.intValuation_le_pow_iff_mem, Finset.mulConst_le_card, InfiniteGalois.toAlgEquivAux_eq_proj_of_mem, Submonoid.pointwise_smul_le_pointwise_smul_iff, MeasureTheory.meas_eLpNormEssSup_lt, OrderHom.coe_antisymmetrization, quasilinearOn_iff_monotoneOn_or_antitoneOn, MeasurableSet.exists_isCompact_isClosed_diff_lt, Filter.comap_cofinite_le, Part.Fix.approx_mem_approxChain, TopCat.GlueData.MkCore.V_id, Finset.Icc_subset_uIcc', ENNReal.toNNReal_le_toNNReal, AffineSubspace.linear_topEquiv, NNReal.half_le_self, Int.preimage_Ioi, ProbabilityTheory.Kernel.bound_le_one, Subalgebra.algebra_isAlgebraic_bot_right, UniformSpace.toTopologicalSpace_top, MeasureTheory.MemLp.meas_ge_lt_top, ProjectiveSpectrum.sup_vanishingIdeal_le, factorThruImageSubobject_comp_imageToKernel, MonomialOrder.degree_reduce_lt, IsNonarchimedean.finset_image_add, right_eq_sup, ENNReal.limsup_liminf_le_liminf_limsup, IsStrictOrderedModule.toIsOrderedModule, PSet.rank_mono, Fin.finsetImage_castSucc_Ioc, Archimedean.embedReal_apply, MeasureTheory.Ioo_ae_eq_Icc, Order.Ici_pred, StalkSkyscraperPresheafAdjunctionAuxs.germ_fromStalk, tendsto_inv_nhdsLE_inv, ENNReal.div_pos_iff, SSet.prodStdSimplex.objEquiv_apply_fst, SimpleGraph.singletonSubgraph_le_iff, SSet.stdSimplex.map_id, idealFactorsFunOfQuotHom_comp, NNReal.agm_le_agmSequences_snd, Set.covBy_iff_exists_sdiff_singleton, le_himp_comm, inv_le_one_iff₀, Polynomial.abs_div_tendsto_atTop_of_degree_gt, MeasureTheory.isFiniteMeasure_iff, Set.image_div_const_Ico, EReal.iSup_add_le_add_iSup, InitialSeg.map_succ, IsExposed.refl, add_tsub_cancel_iff_le, instWellFoundedLTSubtypeSetFinite, CategoryTheory.Equivalence.toOrderIso_apply, LowerAdjoint.closure_sup_closure_right, cfc_nonneg_iff, wbtw_lineMap_iff, iSup_partialSups_eq, Ring.DimensionLEOne.not_lt_lt, MulAction.le_stabilizer_iff_smul_le, isSimpleModule_iff_isAtom, Module.End.maxGenEigenspace_eq_genEigenspace_finrank, Continuous.cfc_nnreal', Ordinal.toNimber_min, MeasureTheory.Measure.hausdorffMeasure_le_liminf_sum, Nimber.add_le_of_forall_ne, emultiplicity_pos_of_dvd, mulRothNumber_map_mul_left, NonemptyInterval.snd_inv, Prod.Lex.toLexOrderHom_coe, CategoryTheory.TransfiniteCompositionOfShape.iic_isoBot, ENNReal.tendsto_nhds_top_iff_nnreal, AlgebraicGeometry.Scheme.exists_germ_injective, MeasureTheory.Measure.haar.addHaarContent_outerMeasure_closure_pos, HahnEmbedding.Partial.archimedeanClassMk_eq_iff, Filter.le_cofinite_iff_compl_singleton_mem, AddAction.map_stabilizer_le, exists_rat_gt, zero_lt_three, CategoryTheory.ComposableArrows.threeδ₃Toδ₂_app_one, Ideal.primeHeight_lt_top, LinearOrderedCommGroup.Subgroup.genLTOne_lt_one, Prod.Lex.toLex_le_toLex', T25Space.t2_5, Module.End.mapsTo_genEigenspace_of_comm, ValuativeRel.ValueGroupWithZero.mk_pos, MonoidWithZeroHom.fst_mono, Submodule.pow_smul_top_le, Submodule.closure_le_toAddSubmonoid_span, Subgroup.commutator_le_right, Set.Ico_mul_Ioc_subset', OpenAddSubgroup.toAddSubgroup_le, NNReal.HolderTriple.nonneg', Nucleus.map_himp_le, StarOrderedRing.lt_iff, SemilatSupCat.Iso.mk_inv_toFun, le_mabs, spectrum.spectralRadius_le_liminf_pow_nnnorm_pow_one_div, CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinsetColimIso_aux_assoc, OrderAddMonoidHom.addCommute_inlₗ_inrₗ, CategoryTheory.Subfunctor.Subpresheaf.preimage_eq_top_iff, CategoryTheory.ComposableArrows.whiskerLeft_obj, Odd.pow_nonneg_iff, Filter.le_pure_iff, Topology.IsUpper.isTopologicalBasis_insert_univ_subbasis, AddSubsemigroup.coe_toSubsemigroup_apply, Fin.sum_Icc_castAdd, IsNonarchimedean.apply_natCast_le_one_of_isNonarchimedean, NonUnitalSubalgebra.starClosure_le_iff, Subalgebra.comm_trans_rTensorBot, NNRat.cast_mono, disjoint_compl_right_iff, EMetric.exists_continuous_nnreal_forall_closedBall_subset, AddLocalization.mkOrderEmbedding_apply, UpperSet.coe_ssubset_coe, IsLeftCancelMul.mulLeftStrictMono_of_mulLeftMono, Mathlib.Meta.NormNum.isNat_ordinalLT_true, AffineSubspace.not_wSameSide_bot, add_upperClosure, inf_left_le_sup_left, PowerSeries.order_mul_ge, SimpleGraph.farFromTriangleFree_iff, SimpleGraph.chromaticNumber_le_sum_right, AlgebraicGeometry.Scheme.bot_mem_precoverage, Nat.lt_succ_floor, List.SortedGT.strictAnti, bddOrd_dual_comp_forget_to_bipointed, Finset.sup_erase_bot, not_lt_top_iff, AddSubsemigroup.center_le_centralizer, symmDiff_top', MonomialOrder.zero_le, PointedCone.subset_dual_dual, Subgroup.le_normalizer_comap, CompositionSeries.lt_last_of_mem_eraseLast, Finset.Colex.lt_iff_exists_forall_lt, sbtw_iff_left_ne_and_right_mem_image_Ioi, Geometry.SimplicialComplex.convexHull_inter_convexHull, CategoryTheory.Sieve.top_apply, Fin.sum_Ico_castSucc, Complementeds.mk_bot, AlgebraicGeometry.StructureSheaf.instIsLocalizedModuleObjOppositeOpensCarrierTopValStructureSheafInTypeOpBasicOpenPowersToOpenₗ, NNReal.sqrt_zero, MultilinearMap.map_piecewise_sub_map_piecewise, Ordinal.le_add_left, Subalgebra.algebra_isAlgebraic_bot_left_iff, Set.notMem_Iic, EReal.coe_nonpos, Fin.image_addNat_Ici, Interval.mem_pure, AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace_app_eq, HasCardinalLT.Set.cocone_pt, CategoryTheory.TransfiniteCompositionOfShape.ofArrowIso_isoBot, PrimeSpectrum.vanishingIdeal_anti_mono_iff, WithLp.nndist_fst_le, CategoryTheory.Limits.HasIterationOfShape.hasColimitsOfShape, Nat.ceil_eq_zero, PartENat.lt_coe_iff, TopCat.Presheaf.germ_res_apply, Even.pow_nonneg, MeasureTheory.Measure.univ_pi_Ioo_ae_eq_Icc, Algebra.EssFiniteType.adjoin_mem_finset, one_lt_inv_iff₀, zpow_neg_two_nonneg, Multiset.le_inf, ProbabilityTheory.cond_pos_of_inter_ne_zero, Order.succ_eq_iInf, Int.floor_eq_zero_iff, integrableOn_Ioc_iff_integrableOn_Ioo', MeasureTheory.meas_le_ae_eq_meas_lt, Affine.Simplex.affineSpan_face_le, Fin.preimage_natAdd_Icc_natAdd, inv_strictAntiOn, Order.instSubsingletonPredOrder, NonUnitalStarAlgHom.nnnorm_apply_le, TwoSidedIdeal.orderIsoMatrix_apply_ringCon_r, Ordinal.pred_le_self, NNReal.young_inequality, tendsto_inv_nhdsGT_inv, Set.uIoo_eq_union, IsCompl.left_le_iff, Ordinal.cof_eq_one_iff_is_succ, ENNReal.zpow_lt_top, IsLocalization.AtPrime.coe_orderIsoOfPrime_symm_apply_coe, AffineBasis.convexHull_eq_nonneg_coord, CFC.rpow_def, lt_inf_iff, NNReal.map_coe_atTop, AlgebraicGeometry.Scheme.Hom.preimage_top, partialSups_const_mul, tendsto_inv_nhdsWithin_Iic_inv, LinearMap.IsPositive.isPositive_smul_iff, Filter.sSup_lowerBounds, UniformFun.mono, LieModule.iSup_ucs_le_genWeightSpace_zero, ZFSet.vonNeumann_subset_vonNeumann_iff, IsIsotypic.submodule_linearEquiv_fun, Polynomial.IsSplittingField.adjoin_rootSet, TopologicalSpace.Closeds.closure_le, Ordinal.div_lt, CategoryTheory.ComposableArrows.Exact.cokerIsoKer'_hom, MulAction.stabilizer_inf_stabilizer_le_stabilizer_sdiff, inv_lt_zero', upperClosure_interior_subset', codisjoint_iff_compl_le_left, MeasureTheory.measure_univ_le_add_compl, ContinuousAlternatingMap.le_opNNNorm, AddCon.toSetoid_bot, ProbabilityTheory.Kernel.isIrreducible_iff, AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_snd_assoc, Subring.exists_le_valuationSubring_of_isIntegrallyClosedIn, MeasureTheory.SimpleFunc.measure_preimage_lt_top_of_integrable, Set.Ico_mul_Icc_subset', Ordinal.range_omega, AlgebraicGeometry.Scheme.Hom.toNormalization_app_preimage, IsModularLattice.complementedLattice_Ici, LinearOrderedField.inducedOrderRingIso_self, Multiset.card_Ico_eq_card_Icc_sub_one, HahnSeries.le_orderTop_of_leadingCoeff_eq, Ideal.subset_union_prime', AddSubsemigroup.op_le_iff, Matrix.dotProduct_star_self_pos_iff, nndist_add_add_le, Ideal.jacobson_bot_polynomial_le_sInf_map_maximal, AddSubmonoid.closure_le_centralizer_centralizer, AlgebraicGeometry.Proj.one_apply, AlgebraicGeometry.Scheme.Hom.toImage_app_injective, Order.not_isPredPrelimit, Ideal.quotientEquivAlgOfEq_coe_eq_factor, HeytAlg.Iso.mk_inv, AlgebraicGeometry.Scheme.isoSpec_hom, Representation.Coinvariants.le_comap_ker, Cardinal.lift_succ, PrincipalSeg.le_iff_le, AlgebraicGeometry.Scheme.ι_toIso_inv_assoc, InitialSeg.isNormal, pos_add_self_iff, mfderivWithin_projIcc_one, ENat.forall_natCast_le_iff_le, WithBot.lt_add_one_iff, UniformConvergenceCLM.topologicalSpace_mono, Equiv.Perm.disjoint_closure_of_disjoint_support, AlgebraicGeometry.Scheme.Hom.id_preimage, AddSemigroupIdeal.closure_le, AlgebraicGeometry.Flat.instMorphismRestrict, Finset.min'_le, AlgebraicGeometry.StructureSheaf.to_basicOpen_epi, intervalIntegral.integral_pos_iff_support_of_nonneg_ae', ContinuousLinearMapWOT.le_nhds_iff_forall_dual_apply_le_nhds, Mathlib.Tactic.Linarith.zero_lt_one, OrderDual.instIsCoatomistic, Fin.preimage_succ_Icc_succ, BddOrd.coe_id, Submonoid.ofUnits_units_le, Finset.mul_prod_Ioi_eq_prod_Ici, cauchySeq_finset_of_summable_norm, Seminorm.bddAbove_of_absorbent, AlgebraicGeometry.IsAffineOpen.opensRange_fromSpec, Set.Infinite.exists_lt, Cardinal.ofENat_mono, PartENat.pos_iff_one_le, tendsto_ceil_left_pure_ceil, FirstOrder.Language.Substructure.fg_closure, indicator_biUnion_finset_eventuallyEq, AlgebraicGeometry.Scheme.germ_stalkClosedPointTo_Spec_fromSpecStalk, Int.comap_cast_atTop, Set.einfsep_zero, Order.le_iff_eq_or_succ_le, ENNReal.lintegral_prod_norm_pow_le, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.app_inv_app', tendstoUniformly_tsum_of_cofinite_eventually, AlgebraicGeometry.Scheme.IdealSheafData.ideal_le_ker_glueDataObjι, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.isoRestrict_hom_c_app, le_himp_himp, Set.preimage_const_sub_Ioo, measurable_lt, TopCat.Presheaf.germ_stalkPullbackInv_assoc, Ordinal.not_bddAbove_isInitial, Valuation.ltAddSubgroup_monotone, Subgroup.instCovariantClassHSMulLe, Unitization.inr_le_iff, thickenedIndicatorAux_subset, Submodule.le_div_iff_mul_le, MeasureTheory.exists_le_lowerSemicontinuous_lintegral_ge, OrderIso.infIrredUpperSet_symm_apply, OrderHom.iInf_apply, Real.toNNReal_lt_toNNReal_iff_of_nonneg, AlgebraicGeometry.instQuasiCompactMorphismRestrict, le_dimH_of_hausdorffMeasure_eq_top, hasBasis_nhdsSet_Ici_Ioi, Polynomial.tendsto_atTop_iff_leadingCoeff_nonneg, Order.not_isPredLimit, AntitoneOn.Icc, AlgebraicGeometry.Scheme.eqToHom_c_app, BoundedLatticeHom.symm_dual_comp, Polynomial.IsDistinguishedAt.degree_eq_coe_lift_order_map, ENNReal.lt_add_right, Valuation.ltIdeal_v_le_of_mem, SimpleGraph.Subgraph.le_induce_top_verts, OrderIso.map_inf, Filter.sets_subset_sets, SchwartzMap.eLpNorm_lt_top, BddOrd.hom_id, Set.encard_preimage_val_le_encard_right, trdeg_lt_aleph0, IsRightCancelAdd.addRightReflectLE_of_addRightReflectLT, TopCat.GlueData.ofOpenSubsets_toGlueData_t, CategoryTheory.Presieve.map_functorPullback, ENNReal.tsum_biUnion_le, segment_subset_convexHull, Ordinal.lt_mul_succ_div, Set.preimage_const_mul_Icc, Set.preimage_add_const_Ioc, Set.Iio_inj, PseudoMetric.IsUltra.isTransitiveRel_closedBall, NNReal.HolderTriple.left_pos, DirichletCharacter.zetaMul_prime_pow_nonneg, MeasureTheory.aecover_Ioo_of_Ioc, Fin.preimage_succ_Iic_succ, HahnEmbedding.Seed.baseEmbedding_strictMono, TopologicalSpace.Closeds.coe_eq_empty, TopologicalSpace.Opens.complOrderIso_symm_apply, AlgebraicGeometry.IsAffineOpen.self_le_iSup_basicOpen_iff, Part.Fix.approx_le_fix, EReal.add_lt_top, AlgebraicGeometry.Scheme.zeroLocus_biInf, Subfield.mk_le_mk, PNat.le_of_dvd, Mathlib.Meta.NormNum.isRat_lt_true, Booleanisation.comp_lt_comp, ENNReal.hasBasis_nhds_of_ne_top, AlgebraicGeometry.PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π, Mathlib.Tactic.Order.ToInt.toInt_nle_toInt, AlgebraicGeometry.PresheafedSpace.componentwiseDiagram_map, ProbabilityTheory.Kernel.indep_limsup_atBot_self, AlgebraicGeometry.PresheafedSpace.stalkMap_germ_assoc, CategoryTheory.TransfiniteCompositionOfShape.ofOrderIso_incl, HasStrictFDerivAt.abs, Metric.coveringNumber_two_mul_le_externalCoveringNumber, PowerSeries.le_order_smul, Matrix.linfty_opNNNorm_mulVec, IsCompact.sSup_lt_iff_of_continuous, AlgebraicGeometry.Proj.basicOpenToSpec_app_top, SimpleGraph.componentComplFunctor_obj, Positive.val_one, EMetric.mem_ball, sSupIndep.pairwiseDisjoint, Equiv.Perm.alternatingGroup_le_of_index_le_two, AlgebraicGeometry.Scheme.IdealSheafData.comap_mono, convexHull_toCone_eq_sInf, NonUnitalSubalgebra.prod_top, Nat.cast_nonneg, cauchySeq_finset_iff_nat_tprod_vanishing, monovaryOn_inv, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.ofRestrict_invApp, FormalMultilinearSeries.le_radius_of_summable_nnnorm, ediam_smul_le, Fin.image_natAdd_Ici, le_iff_forall_one_lt_lt_mul', Ideal.toTwoSided_asIdeal, CHSH_inequality_of_comm, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.inv_naturality, Metric.infEDist_le_infEDist_add_edist, max_bot_left, AlgebraicGeometry.Scheme.Hom.image_preimage_le, lowerSemicontinuous_iff_le_liminf, TopologicalSpace.NoetherianSpace.exists_finset_irreducible, RingHom.strictMono_comap_of_surjective, ENNReal.inv_lt_iff_inv_lt, isAtomic_iff_forall_isAtomic_Iic, isWellOrder_gt, Set.preimage_const_add_Ioi, AlgebraicGeometry.Scheme.AffineZariskiSite.generate_presieveOfSections, CategoryTheory.MorphismProperty.colimitsOfShape_le, EMetric.mem_closedBall, Valued.isClopen_ball, Finset.image_add_left_Ioo, Metric.disjoint_cobounded_nhdsSet, FinPartOrd.dualEquiv_unitIso, Balanced.convexHull, NonUnitalStarAlgHom.range_comp_le_range, Finset.Colex.toColex_strictMono, AlgebraicGeometry.Scheme.Hom.image_top_eq_opensRange, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.invApp_app, Finsupp.iInf_ker_lapply_le_bot, frontier_Ici, Set.sdiff_singleton_covBy, Set.Icc.disjoint_iff, Finset.pow_right_monotone, tendsto_ceil_atTop, isAtomistic_iff, CategoryTheory.ComposableArrows.fourδ₄Toδ₃_app_two, Heyting.IsRegular.disjoint_compl_right_iff, CategoryTheory.Limits.hasColimitsOfShape_of_initialSeg, Ordinal.div_opow_log_pos, ConditionallyCompleteLinearOrderedField.toPosMulStrictMono, gt_trichotomy, Multiset.le_ndinsert_self, mulPosMono_iff_covariant_pos, MonotoneOn.of_map_sup, Finset.max_mono, LowerSet.Iic_iInf, NonemptyFinLinOrd.epi_iff_surjective, Quantale.rightMulResiduation_le_iff_mul_le, ValuativeRel.ValueGroupWithZero.embed_strictMono, Valuation.IsEquiv.pos_iff, exists_lt_of_directed_le, disjoint_bot_right, Subgroup.Normal.quotient_commutative_iff_commutator_le, PNat.card_Icc, IsAbsoluteValue.abv_nonneg, MeasureTheory.eLpNormEssSup_mono_measure, Types.monoOverEquivalenceSet_inverse_map, Fin.castSuccOrderEmb_toEmbedding, subset_absConvexHull, Filter.inv_nhdsGT, RootPairing.Base.pos_or_neg_of_sum_smul_root_mem, EReal.neg_le_neg_iff, ApproximatesLinearOn.norm_fderiv_sub_le, ENNReal.one_le_inv, Ordinal.one_add_ofNat, RCLike.toIsStrictOrderedModule, Fin.sum_Iic_cast, Set.monotone_projIic, PartENat.add_one_le_iff_lt, TopCat.Presheaf.generateEquivalenceOpensLe_inverse'_obj_obj_right_as, List.mem_argmin_iff, CategoryTheory.Subobject.inf_comp_left, exists_lt_mul_self, Prod.mk_wcovBy_mk_iff_right, Ne.lt_top', Metric.cthickening_eq_preimage_infEdist, MonomialOrder.toSyn_strictMono, Multiset.le_dedup, NNReal.rpow_zero_pos, inf_sup_le, sSup_atoms_le_eq, isUnit_le_nonZeroDivisors, Set.image_const_div_Ici, PowerSeries.order_finite_iff_ne_zero, Finset.Ico_diff_Ico_left, Fin.orderPred_zero, FiniteArchimedeanClass.val_mk, IsOpenMap.functor_faithful, MeasurableSpace.map_const, bot_covBy_top, Equiv.Perm.sign_eq_prod_prod_Ioi, Set.Ici.isAtom_iff, ProperCone.coe_bot, LinearMap.iInf_ker_proj_le_iSup_range_single, Set.unbounded_le_inter_not_le, Ordinal.nadd_le_nadd_iff_right, Module.one_lt_rank_of_one_lt_finrank, UpperSet.Ici_iSup, Int.self_le_zpow_clog, Fin.map_addNatEmb_Ioi, instTrichotomousLe, Polynomial.isEquivalent_atTop_lead, AlgebraicGeometry.Scheme.Hom.fromNormalization_preimage, Order.IsNormal.monotone, Multiset.mem_powersetCard, Equiv.Perm.support_pow_le, DistribLattice.le_sup_inf, Ordinal.isOpen_iff, CategoryTheory.IsFiltered.exists_directed, nhdsSet_mono, Monotone.of_map_inf_le_left, Polynomial.le_trailingDegree_C, MulEquiv.mapSubgroup_apply, NonUnitalStarAlgebra.coe_top, Geometry.SimplicialComplex.isRelLowerSet_faces, AlgebraicGeometry.Scheme.toSpecΓ_naturality_assoc, NNReal.agmSequences_fst_le_agm, MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge, Set.image_inv_Ici, Set.bounded_ge_inter_not_ge, PseudoMetric.IsUltra.isTrans_closedBall, StarSubalgebra.topologicalClosure_map_le, Order.Ideal.principal_bot, Finset.card_div_choose_le_card_shadow_div_choose, balancedHull_subset_convexHull_union_neg, Equiv.Perm.prod_Ioi_comp_eq_sign_mul_prod, MonomialOrder.div_single, ENNReal.coe_mem_upperBounds, TopologicalSpace.Closeds.gc, Finset.Ioo_eq_filter_ssubsets, AlgebraicGeometry.Scheme.Modules.pushforwardComp_inv_app_app, LinearOrder.supConvergenceClass, Ideal.le_radical, SimpleGraph.chromaticNumber_mono_of_embedding, Ideal.Filtration.Stable.bounded_difference, hasSumUniformly_iff_tendstoUniformly, SameRay.exists_eq_smul_add, LinOrd.hom_inv_apply, le_one_of_mem_doublyStochastic, LieSubalgebra.toSubmodule_le_toSubmodule, PreValuationRing.iff_ideal_total, Polynomial.isLittleO_atTop_of_degree_lt, IsCompact.exists_open_superset_measure_lt_top', Set.unbounded_le_inter_le, le_minSmoothness, LinearMap.map_le_map_iff', ringKrullDim_succ_le_ringKrullDim_powerseries, LowerSet.coe_map, sSupHom.coe_bot, Nimber.toOrdinal_max, Sublocale.top_mem, Nonneg.segment_eq_uIcc, Subring.toSubmonoid_mono, AlgebraicGeometry.SurjectiveOnStalks.iff_of_isAffine, CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinsetObj_obj, Monotone.of_map_inf, CategoryTheory.ShortComplex.mapToComposableArrows_app_1, Prop.isCoatom_iff, Module.rank_pos_iff_of_free, monotoneOn_descPochhammer_eval, IntermediateField.adjoin_le_iff, Ideal.comap_le_map_of_inverse, LieSubmodule.toSubmodule_orderEmbedding_apply, MeasureTheory.OuterMeasure.le_pi, Algebra.span_le_adjoin, Cardinal.nsmul_lt_aleph0_iff_of_ne_zero, AlgebraicGeometry.SpecMap_ΓSpecIso_hom, ENNReal.half_lt_self, CategoryTheory.IsGrothendieckAbelian.isomorphisms_le_pushouts_generatingMonomorphisms, Submonoid.le_topologicalClosure, Nat.image_cast_int_Ioc, CategoryTheory.Limits.kernelSubobject_arrow_assoc, disjoint_rootsOfUnity_of_coprime, MeasureTheory.measure_lt_top, RCLike.re_nonneg_of_nonneg, Subring.toSubsemiring_mono, Set.image_mulSingle_Ioo_right, ArchimedeanClass.mk_add_lt_mk_left_iff, Subalgebra.center_eq_top, ValuationSubring.isMax_toLocalSubring, pow_two_nonneg, EReal.preimage_coe_Ioo, Tuple.monotone_proj, AlgebraicGeometry.IsAffineOpen.isLocalization_basicOpen, Convex.convexHull_union, MeasureTheory.setIntegral_mono_on_ae₀, CategoryTheory.MorphismProperty.transfiniteCompositionsOfShape_pushouts_coproducts_le_llp_rlp, isBot_iff_eq_bot, Finset.monotone_iff_forall_le_insert, FiniteArchimedeanClass.mem_ballAddSubgroup_iff, Metric.infEdist_le_infEdist_thickening_add, closedConvexHull_eq_closure_convexHull, MeasureTheory.Egorov.measure_inter_notConvergentSeq_eq_zero, Fin.finsetImage_castAdd_Iic, Nonneg.toNonneg_le, instIsOrderedVAddOfIsOrderedAddMonoid, NonUnitalSubalgebra.inclusion_self, MonotoneOn.Iio, Finset.mem_convexHull', AddGroupNorm.le_def, MulArchimedeanClass.mk_eq_mk, QuotientGroup.con_le_iff, ENNReal.lintegral_mul_norm_pow_le, Fin.finsetImage_succ_Iio, IsIntegrallyClosedIn.integralClosure_eq_bot, CategoryTheory.Sieve.overEquiv_le_overEquiv_iff, MvPowerSeries.le_weightedOrder_prod, FractionalIdeal.le_div_iff_of_ne_zero, CategoryTheory.ComposableArrows.IsComplex.opcyclesToCycles_fac_assoc, Ordinal.le_iff_derivFamily, Multiset.inf_le, subset_convexHull, MeasureTheory.aecover_Ioc_of_Ioo, AlgebraicGeometry.Scheme.zeroLocus_iUnion, SSet.iSup_skeleton, Set.indicator_le_indicator_nonneg, SSet.OneTruncation₂.ofNerve₂.natIso_inv_app_obj_obj, Dynamics.one_le_coverMincard_iff, FractionalIdeal.instMulPosReflectLENonZeroDivisors, Dynamics.coverEntropy_monotone, Finset.orderEmbOfFin_eq_orderEmbOfFin_iff, CategoryTheory.Subobject.bot_eq_initial_to, iSup_iInf_le_iInf_iSup, integralClosure_idem, AlgebraicGeometry.IsReduced.component_reduced, UpperSet.coe_div, LinearMap.range_le_bot_iff, div_neg_iff, NormedField.exists_one_lt_enorm, Ideal.factors_decreasing, PrimeSpectrum.le_basicOpen_pow, Set.bounded_lt_inter_not_lt, Polynomial.abs_div_tendsto_atBot_atTop_of_degree_gt, span_le_of_C_coeff_mem, Finset.lt_eq_subset, Multiset.instCanonicallyOrderedAdd, ENNReal.ofReal_lt_ofReal_iff_of_nonneg, measurableSet_Ico, Valuation.Integers.le_of_dvd, geom_sum_pos_iff, ENat.ceil_lt_add_one, PrimeSpectrum.iSup_basicOpen_eq_top_iff, NNReal.image_coe_Ico, EMetric.exists_ball_subset_ball, MeasureTheory.IsFundamentalDomain.measure_le_of_pairwise_disjoint, ClosedSubmodule.mem_bot, OrderRingIso.eq_refl, div_le_one_iff, AlgebraicGeometry.SheafedSpace.comp_hom_c_app, Convex.convexHull_subset_iff, IsNormalClosure.adjoin_rootSet, RCLike.ofReal_nonneg, GenContFract.abs_sub_convergents_le', isClosed_Ioc_iff, Finpartition.mem_avoid, Cardinal.preAleph_le_aleph, abs_sub_le, NNRat.num_pos, Set.preimage_mul_const_Ico, AffineBasis.interior_convexHull, CategoryTheory.MorphismProperty.coproducts_monotone, Metric.thickening_eq_preimage_infEDist, IsLocalMinOn.not_nhds_le_map, PartENat.ofNat_lt_top, DividedPowers.SubDPIdeal.isSubideal, biSup_mono, partialSups_add_one, instIsAbelianGaloisSubtypeMemIntermediateFieldBot, Ordinal.CNF.sorted, AffineBasis.tot', Module.mapEvalEquiv_apply, Module.Basis.flag_mono, antitone_Ioi, Finset.insertNone_eraseNone, eVariationOn.nonempty_monotone_mem, Dynamics.coverMincard_monotone_time, Set.sub_mem_Ioo_iff_right, CFC.abs_nonneg, LieSubmodule.orderIsoMapComap_apply, Mathlib.Tactic.Qify.nnratCast_le, IsLocalRing.exists_maximalIdeal_pow_le_of_finite_quotient, LatticeHom.withTopWithBot_comp, Filter.mem_nhds_iff, AlgebraicGeometry.IsIntegralHom.isIntegral_app, Subgroup.pointwise_smul_le_pointwise_smul_iff₀, ENNReal.iInter_Ici_coe_nat, AlgebraicGeometry.Scheme.ker_ideal_of_isPullback_of_isOpenImmersion, isOpen_Ioo, integrableOn_Icc_iff_integrableOn_Ico, Rat.intCast_le_cast, Set.add_mem_Ico_iff_left, mabs_le', Ordinal.type_le_iff, FractionalIdeal.le_div_iff_of_nonzero, one_div_strictAntiOn, CFC.continuousOn_sqrt, Set.subsingleton_isTop, AddSubgroup.op_le_iff, MeasureTheory.OuterMeasure.isCaratheodory_iff_le', MulArchimedeanClass.mk_left_le_mk_mul_iff, Finset.sum_pos_iff, CategoryTheory.Subgroupoid.le_comap_map, Cardinal.lt_power_cof, Submodule.torsionBySet_le_torsionBySet_of_subset, Cardinal.liftInitialSeg_toFun, CategoryTheory.Sieve.functorPullback_bot, himp_top, Set.Finite.infClosure, CanonicallyOrderedAdd.multiset_prod_pos, Field.Emb.Cardinal.two_le_deg, nhdsNE_le_cofinite, TopCat.Sheaf.id_app, SimplexCategory.mkOfSucc_homToOrderHom_one, Int.cast_nonneg_iff, RootPairing.exists_ge_zero_eq_rootForm, ofDual_preimage_latticeClosure, lt_add_one, NNReal.instPosSMulStrictMono, Measurable.exists_continuous, Fin.map_castAddEmb_Ici, AbsoluteValue.isEquiv_iff_lt_one_iff, Int.cast_lt_zero, TopologicalSpace.Opens.isOpenEmbedding_obj_top, Multiset.disjSum_strictMono_left, Finset.inf_prodMap, Submodule.span_smul_le, Int.le_ceil, MulArchimedeanClass.mem_subgroup_iff, Fin.map_finCongr_Ioo, AddSubmonoid.closure_image_isAddIndecomposable_baseOf, Ordinal.iSup_succ, AlgebraicGeometry.LocallyRingedSpace.Γ_obj_op, Ideal.pointwise_smul_subset_iff, Algebra.mem_bot, SemilatInfCat.dual_obj_isOrderBot, Ordinal.opow_mul_add_pos, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_appTop_coord, Fintype.prod_mono', Fin.top_eq_last, CFC.one_le_iff, Finset.pow_right_strictMonoOn, Subring.integralClosure_subring_le_iff, VitaliFamily.FineSubfamilyOn.measure_le_tsum, Nat.Prime.emultiplicity_le_emultiplicity_choose_add, CategoryTheory.preservesFiniteLimits_liftToFinset, Ordinal.log_le_self, tendsto_zpow_atTop_zero, StarOrderedRing.pos_iff, Valuation.val_lt_one_iff, Set.pairwise_disjoint_Ioc_zsmul, Quaternion.normSq_le_zero, MulArchimedeanClass.subsemigroup_strictAnti, Finset.local_lubell_yamamoto_meshalkin_inequality_div, abs_le, NatOrdinal.add_le_iff, SetTheory.PGame.lt_iff_game_lt, Odd.zpow_nonpos_iff, isCoatom_compl, instAntisymmGe, Int.fract_pos, Finset.sup_div_le, AddAction.stabilizer_le_aestabilizer, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.isIso_of_subset, Topology.upperSet_le_scott, Finsupp.span_le_supported_biUnion_support, MeasureTheory.restrict_Ico_eq_restrict_Ioc, CategoryTheory.Sieve.pullback_pushforward_le, MeasureTheory.Measure.prod_prod_le, SimpleGraph.infinite_iff_in_eventualRange, CategoryTheory.Limits.kernelSubobjectMap_comp, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.ofRestrict_invApp_apply, Order.coheight_le_krullDim, Finpartition.parts_map, LinearPMap.le_graph_iff, GroupSeminormClass.toNonnegHomClass, LinearMap.exists_map_addHaar_eq_smul_addHaar, Submodule.le_orthogonal_orthogonal, Projectivization.Subspace.span_le_span, nhds_basis_zero_abs_lt, uniformity_basis_edist_le, OrdinalApprox.gfpApprox_le, Ordinal.coe_toZFSet, nhds_eq_iInf_abs_sub, MeasureTheory.nndist_integral_add_measure_le_lintegral, AlgebraicGeometry.Scheme.Hom.image_preimage_eq_opensRange_inf, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_over_assoc, Valuation.val_le_one_or_val_inv_le_one, AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToΓ_ΓToStalk, biSup_symmDiff_biSup_le, ValuationSubring.pointwise_smul_le_pointwise_smul_iff, Set.Icc_subset_uIcc, Finset.Colex.erase_lt_erase, Sublattice.mk_le_mk, CategoryTheory.Sieve.generate_bot, Fin.natAddOrderEmb_toEmbedding, measurableSet_Iio, WithTop.tendsto_untop, Set.sub_mem_Icc_iff_right, Ordinal.typein_top, mabs_inf_div_inf_le_mabs, CategoryTheory.Limits.kernelSubobject_comp_mono_isIso, star_pos_iff, Metric.tendsto_atTop', PrimeSpectrum.zeroLocus_subset_zeroLocus_iff, Finset.Colex.erase_le_erase, Ideal.exists_ltSeries_length_eq_height, EMetric.isUniformEmbedding_iff, LieSubmodule.lie_inf, Filter.mem_cardinaleGenerate_iff, TopHom.coe_inf, RootPairing.RootPositiveForm.posForm_apply_root_root_le_zero_iff, ArchimedeanClass.mk_left_le_mk_sub_iff, AlgebraicGeometry.StructureSheaf.comap_const, Ideal.instCovariantClassHSMulLe, SSet.Subcomplex.eq_top_iff_of_hasDimensionLT, ProperCone.mem_dual, AlgebraicGeometry.Scheme.Hom.image_iSup₂, BddOrd.ext_iff, MeasureTheory.martingale_iff, AlgHom.range_comp_le_range, MeasureTheory.Content.innerContent_mono', ProperCone.innerDual_sUnion, ENat.map_natCast_strictMono, AlgebraicGeometry.Scheme.isoSpec_hom_naturality, Set.image_add_const_Iic, Nimber.toOrdinal_min, mul_sub_mul_div_mul_nonpos_iff, Matroid.eRk_lt_encard_iff_dep_of_finite, algebraMap_strictMono, AlgebraicGeometry.Scheme.zeroLocus_singleton, AddSubsemigroup.toSubsemigroup'_closure, Ideal.prod_le_inf, pure_le_nhds, ENNReal.add_le_add_iff_right, OrderIso.setIsotypicComponents_apply, SummationFilter.instLeAtTopSymmetricIco, IsContDiffImplicitAt.one_le, Polynomial.abs_div_tendsto_atTop_atTop_of_degree_gt, FirstOrder.Language.Substructure.inclusion_self, Subgroup.map_inf_le, leOnePart_lt, AlgebraicGeometry.PresheafedSpace.comp_c_app, Order.krullDim_nonpos_iff_forall_isMax, CompleteSemilatticeSup.le_sSup, Filter.HasBasis.disjoint_iff_right, Field.Emb.Cardinal.isLeast_leastExt, sub_half_lt_round, ValuativeRel.nonempty_orderIso_withZeroMul_int_iff, indicator_enorm_le_enorm_self, Finset.divConst_le_card, Finset.coe_wcovBy_coe, Finset.one_le_mulConst, iInf₂_eq_top, IsLocalization.AtPrime.coe_primeSpectrumOrderIso_apply_coe_asIdeal, induced_topology_pure, SimpleGraph.lt_sup_edge, ScottContinuous.inf₂, Ordinal.invVeblen₂_le_iff, CategoryTheory.Subobject.mk_lt_mk_of_comm, AlgebraicGeometry.Scheme.toSpecΓ_isoSpec_inv_assoc, HomotopicalAlgebra.trivialCofibrations_sub_cofibrations, AlgebraicGeometry.LocallyRingedSpace.component_nontrivial, FirstOrder.Language.Substructure.small_bot, AddSubsemigroup.map_le_map_iff_of_injective, MeasureTheory.integrableOn_const_iff, Filter.map₂_distrib_le_left, Localization.mkOrderEmbedding_apply, Set.Ioc_disjoint_Ioc, PiCountable.min_edist_le_edist_pi, AlgebraicGeometry.Scheme.IdealSheafData.ideal_map, FractionalIdeal.le_zero_iff, CompletelyDistribLattice.sdiff_le_iff, disjoint_top, eq_one_or_nnnorm_pos, cmp_add_left, ENNReal.log_strictMono, MeasureTheory.hasFiniteIntegral_def, Algebra.QuasiFiniteAt.exists_fg_and_exists_notMem_and_awayMap_bijective, InitialSeg.map_bot, le_hnot_inf_hnot, AlgebraicGeometry.IsOpenImmersion.range_pullbackSnd, Ordinal.enum_le_enum', MeasureTheory.measure_inter_lt_top_of_right_ne_top, Submodule.IsPrincipal.contentIdeal_le_span_iff_dvd, alternatingGroup.isCoatom_stabilizer_singleton, Finset.le_inf_image₂, Set.Ioo_subset_Ioc_union_Ico, OrderHom.sSup_apply, Valuation.map_add, WellFoundedGT.finite_ne_bot_of_iSupIndep, OrderDual.instIsAtomic, AlgebraicGeometry.Proj.sub_apply, GaloisInsertion.isCoatom_iff, Set.einfsep_le_edist_of_mem, AlgebraicGeometry.isLocallyArtinian_iff_of_isOpenCover, Subalgebra.mem_toSubmodule, SimplexCategory.Hom.mk_toOrderHom_apply, FormalMultilinearSeries.nnnorm_changeOriginSeries_le_tsum, PrimeSpectrum.coe_preimageOrderIsoFiber_symm_apply_coe_asIdeal, MeasureTheory.Content.le_innerContent, AlgebraicGeometry.IsImmersion.instMorphismRestrict, rank_submodule_le_one_iff, sdiff_le_hnot, Set.disjoint_left, ArchimedeanClass.mk_le_mk_smul, AddSubgroup.characteristic_iff_le_map, MeasureTheory.exists_isSigmaFiniteSet_measure_ge, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverseObj_π_app, ite_le_sup, ExpGrowth.expGrowthInf_mul_le, MulArchimedeanClass.mk_lt_mk, MeasureTheory.SignedMeasure.exists_isCompl_positive_negative, pred_min, essInf_measure_zero, nnnorm_mul_le, div_le_inv_mul_iff, Set.disjoint_singleton_left, CategoryTheory.Subobject.inf_arrow_factors_right, PNat.factorMultiset_le_iff, NonemptyInterval.coe_nsmul_interval, EReal.nhds_bot_basis, Geometry.SimplicialComplex.inter_subset_convexHull, convexJoin_segment_singleton, ENNReal.zpow_pos, SameRay.exists_eq_smul, SimpleGraph.chromaticNumber_le_of_forall_imp, min_eq_right_iff, Associates.prod_le_prod_iff_le, CategoryTheory.MorphismProperty.colimitsOfShape_discrete_le_llp_rlp, HahnEmbedding.Partial.toOrderAddMonoidHom_apply, AlgebraicGeometry.Scheme.Hom.comp_app, Polynomial.SplittingFieldAux.adjoin_rootSet, isComplete_iff_ultrafilter, Set.disjoint_right, MeasureTheory.memLp_const_iff, monovaryOn_comm, Valuation.HasExtension.val_map_lt_iff, Matroid.isClosed_iff_isFlat, ofAddUnits_le_iff_le_addUnits, Ordinal.apply_lt_nfpFamily_iff, ProfiniteGrp.toLimit_surjective, PNat.tendsto_comp_val_iff, antivaryOn_inv_right, CategoryTheory.Limits.equalizerSubobject_arrow_comp, CategoryTheory.Limits.LimitPresentation.self_π, Order.IsNormal.ext, openSegment_eq_Ioo', LSeries_deriv_eqOn, Subalgebra.range_isScalarTower_toAlgHom, Ordinal.deriv_succ, RingCon.subsingleton_quotient, cocompact_le_atTop, MvPowerSeries.lexOrder_def_of_ne_zero, MonomialOrder.degree_add_le, Filter.le_vsub_iff, Filter.map₂_distrib_le_right, Submonoid.iSup_map_mulSingle_le, AlgebraicGeometry.instIsScalarTowerObjOppositeOpensCarrierTopValStructureSheafInType, CategoryTheory.Subobject.prod_eq_inf, ENNReal.image_coe_Iic, Valued.is_topological_valuation, Fin.finsetImage_castLE_Ico, indicator_le_thickenedIndicatorAux, ClosureOperator.le_closure_iff, ENNReal.mul_le_mul_iff_left, PseudoMetric.IsUltra.bot, Ordinal.bfamilyOfFamily'_typein, InfHomClass.toOrderHomClass, ENNReal.exists_pos_mul_lt, CategoryTheory.TransfiniteCompositionOfShape.ici_F, LSeries.abscissaOfAbsConv_le_of_le_const, Ordinal.invVeblen₁_le, Pi.toLex_update_le_self_iff, HahnEmbedding.Seed.strictMono_coeff, Set.image_sub_const_Iic, iSup_inf_le_inf_iSup, MeasureTheory.Measure.mkMetric_le_liminf_tsum, CompactlySupportedContinuousMap.instAddRightMono, TopCat.Presheaf.generateEquivalenceOpensLe_inverse'_obj_obj_hom, Filter.set_eventuallyLE_iff_inf_principal_le, interior_Iio, BddLat.forget_semilatSup_partOrd_eq_forget_bddOrd_partOrd, CategoryTheory.Sieve.generate_top, Finset.le_inf_add, min_le_max, SemilatticeInf.instIsCodirectedOrder, Bornology.isVonNBounded_iff_absorbing_le, Cardinal.aleph_one_le_continuum, EMetric.nhdsWithin_basis_eball, TopologicalSpace.Opens.mapId_hom_app, one_le_mul_mabs, Set.OrdConnected.vadd, MulAction.IsBlock.stabilizer_le, LinearMap.nonneg_iff_isPositive, AlgebraicGeometry.IsOpenImmersion.range_pullback_fst_of_right, upperClosure_interior_subset, StarSubalgebra.mem_bot, MeasureTheory.Measure.hausdorffMeasure_mono, Set.Ioi_diff_Ioi, negPart_pos_iff, Cardinal.lift_monotone, PowerSeries.le_order_subst_right', WellFoundedGT.rank_strictAnti, Metric.measure_closedBall_pos_iff, isPiSystem_image_Ioi, one_le_nhds_iff, Nimber.pos_iff_ne_zero, Order.Iio_succ_eq_insert, IsValuativeTopology.isOpen_ball, cfcHom_le_iff, Matrix.frobenius_nnnorm_one, NatOrdinal.succ_def, Finset.erase_wcovBy, gc_upperPolar_lowerPolar, TopologicalSpace.Opens.mapComp_hom_app, SSet.Subcomplex.topIso_inv_ι, LocallyBoundedMap.comap_cobounded_le', AlgebraicGeometry.Scheme.toSpecΓ_naturality, AlgebraicGeometry.PresheafedSpace.GlueData.snd_invApp_t_app_assoc, EReal.one_le_exp_iff, SSet.prodStdSimplex.strictMono_orderHomOfSimplex_iff, MeasureTheory.OuterMeasure.mono, continuous_iff_lower_upperSemicontinuous, OmegaCompletePartialOrder.ωSup_le_iff, Topology.RelCWComplex.iUnion_openCell_eq_skeleton, ConcaveOn.antitoneOn_slope_gt, MeasureTheory.measure_sigmaFiniteSetGE_le, PartOrd.coe_id, AlgebraicGeometry.Scheme.Hom.quasiFiniteLocus_eq_top, Bornology.le_cofinite, convexJoin_singleton_segment, ContinuousMap.HomotopyRel.trans_apply, convexHull_diam, CompositionSeries.lt_succ, PointedCone.dual_union, MeasureTheory.Measure.mem_cofinite, MvPowerSeries.le_weightedOrder_map, cauchySeq_finset_of_norm_bounded_eventually, AddSubmonoid.neg_le, thickenedIndicator_le_one, Filter.tendsto_inv_atTop_atBot, antivary_inv, Hyperreal.not_infinite_iff_exist_lt_gt, le_emultiplicity_map, IntermediateField.fixingSubgroup_top, LSeries_analyticOnNhd, disjoint_toDual_iff, Antitone.Ico, MulPosMono.toMulPosStrictMono, MulHom.mclosure_preimage_le, add_eq_left_iff_le, ENNReal.instSMulPosMonoNNReal, Dense.Iio_eq_biUnion, smoothSheaf.ι_evalHom_assoc, ENNReal.bot_eq_zero, BoxIntegral.Prepartition.distortion_le_of_mem, MeasureTheory.projectiveFamilyContent_diff, BddOrd.Iso.mk_hom, Stream'.Seq.length'_le_iff, Sublattice.mk_lt_mk, Sublattice.gc_map_comap, Int.floor_lt, monotone_vecEmpty, Prod.mk_covBy_mk_iff_right, mul_nonneg_iff, ringKrullDim_nonneg_of_nontrivial, AffineSubspace.coe_eq_univ_iff, le_iSup₂, toAntisymmetrization_lt_toAntisymmetrization_iff, Pi.toLex_update_lt_self_iff, CategoryTheory.Sieve.equalizer_self, WithBot.eq_top_iff_forall_ge, WithBot.Ioc_bot_coe, Order.lt_succ_iff_eq_or_gt, lowerClosure_interior_subset', Submodule.prod_le_iff, Field.Emb.Cardinal.succEquiv_coherence, Profinite.isIso_indexCone_lift, IntermediateField.bot_toSubalgebra, Profinite.NobelingProof.GoodProducts.spanFin, TopologicalSpace.Opens.mk_univ, MeasureTheory.VectorMeasure.restrict_le_zero_of_not_measurable, CategoryTheory.Localization.essSurj_mapComposableArrows_of_hasRightCalculusOfFractions, Codisjoint.hnot_le_left, SSet.stdSimplex.face_eq_ofSimplex, Ordinal.veblenWith_right_strictMono, Ordinal.nadd_le_nadd_iff_left, Fin.preimage_addNat_Iic_addNat, Subgroup.ofUnits_le_ofUnits_iff, top_bihimp, ENNReal.essSup_add_le, one_div_neg, AlgebraicGeometry.Scheme.Hom.ι_toNormalization, hasDerivWithinAt_abs, RCLike.toPosMulReflectLT, CategoryTheory.MorphismProperty.isStableUnderRetracts_iff_retracts_le, ENNReal.mul_le_iff_le_inv, Algebra.toSubmodule_bot, Metric.ediam_pos_iff', OrderIso.map_sInf_eq_sInf_symm_preimage, Prod.ωSup_zip, NNRat.coe_pos, Ordinal.succ_zero, StdSimplex.ext_iff, Subgroup.map_le_map_iff_of_injective, CategoryTheory.Pretopology.mem_toGrothendieck, LowerSet.Iic_one, Ordinal.card_le_preAleph, Fin.prod_Ioo_castAdd, openSegment_eq_image', TopologicalSpace.IrreducibleCloseds.map_strictMono_of_isInducing, MeasurableSet.exists_isCompact_lt_add, Cardinal.noMaxOrder, Finset.sup'_mono, abs_sub, Cardinal.ofENat_lt_aleph0, MeasureTheory.sfiniteSeq_le, AlgebraicGeometry.LocallyRingedSpace.Γ_Spec_left_triangle, AlgebraicGeometry.StructureSheaf.comap_apply, subset_closedAbsConvexHull, Cardinal.aleph_mul_aleph0, Submodule.map_le_map_iff_of_injective, Algebra.map_bot, ConcaveOn.convex_strict_hypograph, Hyperreal.tendsto_iff_forall, mapClusterPt_atTop_nsmul_tfae, gc_upperBounds_lowerBounds, ENNReal.not_top_le_coe, polynomialFunctions.starClosure_le_equalizer, MonoidHom.closure_preimage_le, MvPowerSeries.nat_le_order, CategoryTheory.instWellFoundedGTSubobjectOfIsNoetherianObject, CategoryTheory.MorphismProperty.multiplicativeClosure_monotone, inf_top_eq, ENNReal.exists_nat_gt, Dynamics.netEntropyEntourage_antitone, nnnorm_mul₃_le, nilradical_le_jacobson, Valuation.one_lt_val_iff, le_nhds_iff, IsTopologicalGroup.isOpenMap_iff_nhds_one, RootPairing.exist_set_root_not_disjoint_and_le_ker_coroot'_of_invtSubmodule, EReal.exists_nat_ge_mul, Orientation.map_eq_iff_det_pos, SSet.horn₃₁.desc.multicofork_pt, smul_nonpos_iff_pos_imp_nonpos, Set.image_update_Icc, Subfield.mem_bot_iff_pow_eq_self, MulAction.le_stabilizer_smul_left, LatticeHom.withTopWithBot'_toFun, archimedean_iff_nat_le, Filter.subsingleton_iff_exists_le_pure, monovaryOn_neg, HomogeneousIdeal.toAddSubmonoid_irrelevant_le, SetRel.image_core_gc, AlgebraicGeometry.Scheme.IdealSheafData.ideal_pow, Codisjoint.top_le, Set.image_const_add_Ioc, IsOpenMap.functorFullOfMono, MeasureTheory.eLpNorm_mono_ae, εNFA.pumping_lemma, Filter.map_mapsTo_Iic_iff_mapsTo, Nat.mono_cast, AlgebraicGeometry.RingedSpace.mem_top_basicOpen, SuccOrder.nhdsLE_eq_nhds, AffineSubspace.le_def', CategoryTheory.Functor.relativelyRepresentable.isomorphisms_le, Subfield.isTotallyReal_bot, Pi.colex_le_iff_of_unique, ENNReal.div_lt_div_iff_right, le_iff_forall_neg_add_le, HahnEmbedding.ArchimedeanStrata.ball_sup_stratum_eq, OmegaCompletePartialOrder.instOrderHomClassContinuousHom, Finset.Ico_disjoint_Ico_consecutive, AlgebraicGeometry.Scheme.Opens.germ_stalkIso_inv, compl_sup_compl_le, StarAlgebra.elemental.le_of_mem, PrimeSpectrum.isMax_iff, generateFrom_Icc_mem_le_borel, ContractingWith.apriori_edist_iterate_efixedPoint_le', Fin.finsetImage_succ_Ico, ContinuousOn.finset_inf, Fin.image_natAdd_Ioo, Quaternion.normSq_nonneg, Nat.le_floor_iff', countable_setOf_isolated_right, PMF.toOuterMeasure_caratheodory, OrderRingIso.subsingleton_right, IsSolvable.commutator_lt_top_of_nontrivial, sup_strictMonoOn_Icc_inf, Metric.disjoint_nhds_cobounded, Filter.lift'_inf_le, NNReal.strictConcaveOn_sqrt, Set.sub_mem_Ico_zero_iff_right, Filter.Eventually.lt_top_iff_ne_top, EReal.nhds_top', not_infIrred, compare_gt_iff_gt, Set.bounded_le_inter_not_le, LinearMap.span_preimage_le, mabs_mul_three, ENat.one_le_card_iff_nonempty, Finset.Ici_mul_Ioi_subset', NormedField.exists_lt_enorm, List.le_sum_of_mem, Fin.rev_top, Nat.image_cast_int_Ici, NNReal.arith_mean_le_rpow_mean, Submodule.length_lt, MeasureTheory.AEEqFun.inf_le_left, LinearMap.IsPositive.inner_nonneg_left, Set.range_add_eq_image_Ici, SSet.skeleton_le_skeletonOfMono, ValuationSubring.valuation_lt_one_or_eq_one, IntermediateField.relfinrank_top_right, Concept.intent_subset_intent_iff, upperCentralSeries.map, SSet.prodStdSimplex.objEquiv_δ_apply, SetTheory.PGame.toRightMovesNim_one_symm, FractionalIdeal.spanSingleton_le_iff_mem, Ultrafilter.of_le, Finset.card_eraseNone_le, Set.Icc_subset_Icc_union_Ioc, Filter.HasBasis.limsup_eq_sInf_iUnion_iInter, Filter.HasBasis.isLittleOTVS_iff, compl_anti, Nonneg.noMaxOrder, conjneg_pos, Profinite.NobelingProof.GoodProducts.smaller_factorization, Set.unbounded_lt_inter_le, Cardinal.isInaccessible_def, Filter.disjoint_atTop_principal_Iio, AlgebraicGeometry.Scheme.IdealSheafData.zero_eq_bot, MeasureTheory.mulEquivHaarChar_pos, Set.pairwise_disjoint_Ioo_add_intCast, Order.succ_top, Affine.Simplex.affineCombination_mem_interior_face_iff_mem_Ioo, Order.coheight_eq_coe_add_one_iff, latticeClosure_mono, Set.Finite.isCompact_convexHull, LowerAdjoint.mem_closed_iff_closure_le, Set.Ico_succ_succ_eq_Ioc, LieSubalgebra.lieSpan_mono, instIsStrictTotalOrderLt, IndiscreteTopology.eq_top, OrderIso.map_iSup, Submonoid.toAddSubmonoid_closure, image_latticeClosure', Multiset.add_le_add_iff_right, EMetric.uniformContinuous_iff, rank_lt_rank_dual', Fintype.sup_disjointed, WithZeroMulInt.toNNReal_pos, Order.krullDim_nonpos_iff_forall_isMin, LocallyFiniteOrder.orderAddMonoidHom_bijective, iInf₂_le, isConnected_Iio, SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj, NNReal.sqrt_le_sqrt, CategoryTheory.ComposableArrows.precomp_obj, MeasureTheory.eLpNorm_indicator_sub_le_of_dist_bdd, AlgebraicGeometry.IsAffineOpen.ideal_le_iff, Algebra.adjoin_singleton_zero, List.minimum_of_length_pos_le_of_mem, Nat.ceil_mono, Nimber.add_trichotomy, TopCat.Presheaf.IsSheaf.isSheafUniqueGluing_types, Tuple.sort_eq_refl_iff_monotone, Set.image_add_const_Iio, le_iff_posPart_negPart, liminf_eq_top, antitone_le, convexHull_add, BoundedContinuousFunction.lintegral_nnnorm_le, ENNReal.natCast_le_ofNNReal, Set.Ioo_add_one_right_eq_Ioc, CategoryTheory.MorphismProperty.retracts_le, Filter.zero_smul_filter_nonpos, Ordinal.lt_nfpFamily_iff, EReal.bot_lt_coe, Finset.divConst_le_mulConst_sq, FiniteMulArchimedeanClass.withTopOrderIso_apply_coe, CategoryTheory.ComposableArrows.Precomp.map_zero_succ_succ, WithTop.instBorelSpace, Module.length_pos_iff, Finsupp.Lex.isStrictOrder, EisensteinSeries.tsum_symmetricIco_linear_sub_linear_add_one_eq_zero, Finset.inf_sup, AlgebraicGeometry.Scheme.Hom.congr_app, Order.le_and_le_succ_iff, EReal.abs_mul_sign, instIsOrderedModule, SimpleGraph.vertexCoverNum_le_vertexCoverNum_of_injective, Set.Iio_subset_Iic_iff, AlgebraicGeometry.Scheme.Hom.preimage_iSup, WithTop.Icc_coe_coe, UpperSet.prod_le_prod_iff, Subring.closure_mono, Finsupp.Colex.addRightMono, AlgebraicGeometry.Scheme.Hom.toPartialMap_domain, Set.Icc.codisjoint_iff, OrderIso.withTopCongr_trans, OrderEmbedding.coe_birkhoffFinset, NNReal.isGreatest_Lp, UpperSet.Ici_sup, PNat.lt_add_right, Algebra.coe_bot, MeasureTheory.hittingBtwn_mono, Finset.eraseNone_union, Fin.sum_Ioo_succ, Real.one_lt_toNNReal, PartENat.instWellFoundedLT, RootPairing.zero_lt_pairingIn_iff', NonUnitalSubsemiring.gc_map_comap, AddValuation.self_le_supp_comap, CompleteLinearOrder.top_sdiff, Setoid.injective_iff_ker_bot, Nat.preimage_floor_of_ne_zero, Matroid.Matroidᵣ.le_iff, affineSpan_pair_le_of_mem_of_mem, Set.Ioc_subset_Ioc_union_Icc, HahnEmbedding.Partial.isWF_support_evalCoeff, Language.instMulLeftMono, PartOrd.dual_map, Submodule.topologicalClosure_map, Set.projIci_eq_self, Order.isPredPrelimit_of_pred_ne, strictMono_vecEmpty, Submodule.toConvexCone_top, OrderDual.instSMulPosStrictMono, PartOrdEmb.comp_apply, CategoryTheory.ComposableArrows.threeδ₂Toδ₁_app_two, CategoryTheory.OrthogonalReflection.iteration_map_succ, Polygon.edgeSet_eq_image_edgePath, Set.covBy_insert, MeasureTheory.condLExp_bot, Ordinal.omega0_pos, TopologicalSpace.Opens.map_id_obj', HahnEmbedding.Partial.orderTop_eq_iff, Mathlib.Meta.Positivity.nonneg_of_isNNRat, Fin.sum_Iic_castAdd, AlgebraicGeometry.Scheme.Opens.toSpecΓ_naturality_assoc, LieSubmodule.coe_map_toEnd_le, FractionalIdeal.mul_right_mono, Polynomial.tendsto_atBot_iff_leadingCoeff_nonpos, List.coe_le_maximum_iff, MeasureTheory.lintegral_const_mul_le, Pi.Lex.sInf_apply_le, tendsto_neg_nhdsGT_neg, PowerSeries.map_constantCoeff_le_self_of_X_mem, Mathlib.Tactic.ENatToNat.not_lt_top, Filter.isBoundedUnder_le_sup, Finset.lubell_yamamoto_meshalkin_inequality_sum_inv_choose, le_himp_iff_left, Ideal.IsPrime.inf_le, PNat.find_mono, sSupHom.bot_apply, Order.pred_le_iff_eq_or_le, ShrinkingLemma.PartialRefinement.exists_gt, Fin.preimage_castAdd_Ici_castAdd, AddSubmonoid.toNatSubmodule_toAddSubmonoid, Order.lt_succ_iff_eq_or_lt, CategoryTheory.Subfunctor.Subpresheaf.image_le_iff, LinearMap.exists_antilipschitzWith, Continuous.exists_forall_le_of_hasCompactSupport, HahnSeries.SummableFamily.isPWO_iUnion_support, Ordinal.not_bddAbove_principal, mulRothNumber_spec, AlgebraicGeometry.IsAffineOpen.isoSpec_inv, le_iff_lt_or_eq', derivedSeries_antitone, Ordinal.eq_zero_or_opow_omega0_le_of_mul_eq_right, Set.image_const_sub_Ico, ArchimedeanClass.top_eq_mk_iff, MeasureTheory.Measure.LebesgueDecomposition.iSup_le_le, NNReal.isCoboundedUnder_le_toReal, Multiset.prod_X_sub_C_dvd_iff_le_roots, AlgebraicGeometry.Scheme.Hom.id_app, TopCat.Presheaf.comp_app, Monoid.minOrder_le_natCard, Multiset.sup_coe, AlgebraicGeometry.Scheme.Γ_obj, SemigroupIdeal.instWellFoundedGT, Codisjoint.sup_lt_right_of_left_ne_bot, Mathlib.Meta.Positivity.nonneg_of_isNat, CategoryTheory.Presieve.isSheafFor_top_sieve, MeasureTheory.Measure.smul_measure_isAddInvariant_le_of_isCompact_closure, Ideal.Quotient.factorₐ_refl, LowerSet.total_le, WithTop.untop₀_nonneg, Ideal.map_le_comap_of_inv_on, SSet.Subcomplex.toSSetFunctor_obj, LinearMap.bot_lt_ker_of_det_eq_zero, Finset.eraseNone_none, SimplexCategory.δ_comp_σ_of_gt', le_iff_le_iff_lt_iff_lt, LocalSubring.toSubring_mono, Part.bind_le, OrderMonoidWithZeroHom.mul_comp, Subsemiring.unop_le_unop_iff, OrderIso.supIrredLowerSet_symm_apply, AlgebraicGeometry.isLocallyNoetherian_iff_of_affine_openCover, AlgebraicGeometry.Scheme.kerAdjunction_unit_app, ProbabilityTheory.condIndep_limsup_atTop_self, Filter.OrderBot.atBot_eq, compl_lt_self, AddMonoid.le_minOrder, nndist_smul_le, OrderIso.ofRelIsoLT_apply, IntermediateField.Lifts.exists_lift_of_splits', tendsto_ceil_right_pure_add_one, CategoryTheory.IsCardinalFiltered.exists_cardinal_directed, AlgebraicGeometry.Scheme.IdealSheafData.instIsEmptyCarrierCarrierCommRingCatSubschemeTop, Fin.preimage_cast_Ici, ProjectiveSpectrum.subset_zeroLocus_iff_le_vanishingIdeal, ENat.card_le_one_iff_subsingleton, Finset.inf_apply, CategoryTheory.Classifier.SubobjectRepresentableBy.homEquiv_eq, PosMulReflectLT.toPosMulReflectLE, NatOrdinal.lt_add_iff, SSet.hasDimensionLT_subcomplex_top_iff, Ideal.IsPrime.inf_le', LieSubmodule.isNilpotent_iff_exists_self_le_ucs, RCLike.re_monotone, AddSubmonoid.toNatSubmodule_closure, Set.image_update_Ico_right, Hyperreal.isSt_iff_abs_sub_lt_delta, PrincipalSeg.isSuccPrelimit_apply_iff, meromorphicOrderAt_add, Algebra.sInf_toSubmodule, Subgroup.IsSubnormal.exists_normal_and_le_and_lt_top_of_ne, IntermediateField.restrictScalars_top, SSet.Subcomplex.instSubsingletonHomToSSetBot, nhds_top_basis_Ici, MeasureTheory.Content.lt_top, AlgebraicGeometry.Scheme.IdealSheafData.mem_support_iff_of_mem, edist_le_ofReal, BoundedContinuousFunction.lintegral_lt_top_of_nnreal, Fintype.sum_mono, mapClusterPt_iff_ultrafilter, MeasureTheory.OuterMeasure.map_comap_le, Cardinal.toENat_lt_natCast_iff, TopCat.Presheaf.germ_stalkSpecializes_assoc, CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinsetColimitCocone_isColimit_desc, abs_max_sub_max_le_abs, finite_integral_rpow_sub_one_pow_aux, CStarAlgebra.span_nonneg, Finset.isLUB_sup_id, groupCohomology.coboundaries₂_le_cocycles₂, Finsupp.DegLex.single_lt_iff, ESeminormedAddMonoid.enorm_add_le, ArithmeticFunction.vonMangoldt.abscissaOfAbsConv_residueClass_le_one, EuclideanGeometry.Sphere.orthRadius_center, NumberField.Units.span_basisOfIsMaxRank, PartOrdEmb.hom_inv_apply, Set.projIic_surjOn, Cardinal.card_le_iff, Filter.NeBot.smul_zero_nonneg, Representation.invtSubmodule.coe_top, EReal.preimage_coe_Ioo_top, LaurentSeries.val_le_one_iff_eq_coe, convexOn_iff_forall_pos, Module.supportDim_le_of_injective, ProjectiveSpectrum.basicOpen_mul_le_left, Surreal.Multiplication.mulOption_lt_mul_of_equiv, MulPosReflectLE.toMulPosStrictMono, convex_Ico, upperHemicontinuousWithinAt_iff_preimage_Iic, Finset.top_eq_univ, Nucleus.idempotent', ProbabilityTheory.Kernel.HasSubgaussianMGF.const_mul, imageToKernel_comp_right, Fin.image_castLE_Iio, LightProfinite.lightToProfinite_map_proj_eq, PrimeSpectrum.isIdempotentElemEquivClopens_symm_bot, EMetric.infEdist_le_hausdorffEdist_of_mem, DistLat.Iso.mk_inv, TopologicalSpace.isOpen_top_iff, OrderAddMonoidHom.addCommute_inl_inr, Polynomial.le_trailingDegree_monomial, Finset.disjSum_strictMono_left, CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinsetObj_obj, Finset.Colex.singleton_le_toColex, CategoryTheory.Coverage.toGrothendieck_eq_sInf, EisensteinSeries.G2_eq_tsum_symmetricIco, sdiff_le_iff_left, Cardinal.small_Iio, ChainCompletePartialOrder.IsAdmissible.base_isLeast, AlgebraicGeometry.IsLocallyArtinian.isArtinianRing_of_isAffine, ENNReal.iUnion_Iic_coe_nat, AlgebraicGeometry.StructureSheaf.toPushforwardStalkAlgHom_apply, AlgebraicGeometry.Scheme.Hom.germ_stalkMap_apply, SetTheory.PGame.short_birthday, Cardinal.nat_lt_aleph0, AlgebraicGeometry.Scheme.preimage_basicOpen_top, Cardinal.natCast_add_one_le_iff, coe_nnHolderNorm_le_eHolderNorm, SimpleGraph.edgeSet_ssubset_edgeSet, MonomialOrder.le_degree, ENNReal.tendsto_tsum_compl_atTop_zero, Set.Iio_diff_Iio, isLUB_pair, Fin.top_eq_zero, nhds_basis_Icc_one_lt, Module.Basis.orientation_eq_iff_det_pos, WithTop.Ioo_coe_coe, sq_le_sq, MvPolynomial.adjoin_range_X, Nimber.add_def, Submodule.traceDual_top', List.sum_min_le, sup_top_eq, Fin.attachFin_Icc, affineSpan_convexHull, Order.succ_eq_zero, LinOrd.inv_hom_apply, PartOrdEmb.hom_comp, max_lt_max_right_iff, AlgebraicGeometry.isSchemeTheoreticallyDominant_iff, mul_nonneg_iff_pos_imp_nonneg, List.nnnorm_prod_le, AlgebraicGeometry.Scheme.ΓSpecIso_inv, ENNReal.div_self_le_one, Ordinal.liftPrincipalSeg_coe, MeasureTheory.Content.innerContent_iSup_nat, AlgebraicGeometry.Scheme.Hom.app_appIso_inv_assoc, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquiv_counitIso, Polynomial.degree_comp, Subgroup.closure_le, Finset.pluennecke_ruzsa_inequality_nsmul_add, MeasureTheory.measure_le_inter_add_diff, ADEInequality.lt_six, Fin.map_finCongr_Iic, Finsupp.Colex.wellFoundedLT, MeasureTheory.IsSetSemiring.isSetRing_supClosure, MeasureTheory.AEEqFun.inf_le_right, Cardinal.mk_strictMonoOn, Multiset.instExistsAddOfLE, Finset.prod_le_prod_of_subset', MeasureTheory.condExpIndSMul_empty, Ideal.sup_pow_add_le_pow_sup_pow, MeasureTheory.prob_le_one, Finset.abs_sum_le_sum_abs, AddSubgroup.toZModSubmodule_toAddSubgroup, Finpartition.ofErase_parts, Order.krullDim_int, Finsupp.Colex.addRightStrictMono, Function.locallyFinsuppWithin.posPart_add, frontier_Ioi, isOpen_setOf_disjoint_nhds_nhds, IsGalois.intermediateFieldEquivSubgroup_apply, NonUnitalAlgebra.range_id, Set.Finite.encard_lt_top, AlgebraicGeometry.IsAffineOpen.basicOpen_fromSpec_app, ENNReal.lintegral_rpow_add_le_add_eLpNorm_mul_lintegral_rpow_add, SimpleGraph.eccent_le_one_iff, invOf_lt_zero, PrincipalSeg.apply_covBy_apply_iff, FloorRing.gc_ceil_coe, LieModule.disjoint_lowerCentralSeries_maxTrivSubmodule_iff, LieSubmodule.comap_incl_eq_top, MeasureTheory.measure_union_lt_top_iff, nnnorm_le_mul_nnnorm_add', TopologicalSpace.OpenNhds.map_obj, iUnion_disjointed, EReal.bot_lt_add_iff, Filter.mem_cocardinal, LipschitzOnWith.comp_eVariationOn_le, TopCat.Presheaf.stalkPushforward_germ_assoc, Partition.pairwiseDisjoint, MeasureTheory.lintegral_nnnorm_condExpL2_indicator_le, SSet.prodStdSimplex.objEquiv_naturality, imageToKernel_comp_hom_inv_comp, PartENat.toWithTop_lt, IntermediateField.mem_top, OrderIso.isCompl, ENNReal.ofReal_lt_iff_lt_toReal, Ideal.relNorm_le_comap, Filter.bliminf_sup_le_or_aux_left, ENat.card_lt_top_of_finite, ENat.lift_one, leOnePart_le_one', CategoryTheory.ComposableArrows.opEquivalence_unitIso_inv_app, Cardinal.mul_natCast_strictMono, TopCat.Presheaf.pushforward_map_app, MeasureTheory.upperCrossingTime_mono, AlgebraicGeometry.Scheme.Hom.coequifibered_normalizationDiagramMap, PredOrder.nhdsGE_eq_nhds, bot_symmDiff, Filter.map_neg_atTop, NonemptyInterval.coe_pow_interval, continuousWithinAt_iff_lower_upperSemicontinuousWithinAt, Submodule.inf_genEigenspace, TopologicalSpace.OpenNhds.op_map_id_obj, NatOrdinal.bot_eq_zero, Cardinal.cast_toNat_eq_iff_lt_aleph0, MeasureTheory.OuterMeasure.map_le_restrict_range, Besicovitch.TauPackage.monotone_iUnionUpTo, AbsoluteValue.apply_nat_le_self, Int.ceil_eq_iff, Matroid.eRk_le_eRk_add_eRk_diff, BoxIntegral.IntegrationParams.henstock_le_riemann, BoxIntegral.Box.nndist_le_distortion_mul, ArchimedeanClass.mk_lt_mk, Fin.preimage_cast_Iic, AlgebraicGeometry.Scheme.isoSpec_hom_naturality_assoc, posPart_nonneg, Submodule.smul_le_self_of_tower, Cardinal.nat_lt_continuum, Subalgebra.le_centralizer_iff, Set.OrdConnected.smul, AlgebraicGeometry.Scheme.isoSpec_inv_image_zeroLocus, AlgebraicGeometry.Scheme.IdealSheafData.range_glueDataObjι_ι, Set.disjoint_singleton, ZMod.valMinAbs_mem_Ioc, MeasureTheory.SimpleFunc.nnnorm_approxOn_le, Ordinal.bsup_succ_eq_blsub, AlgebraicGeometry.Scheme.germToFunctionField_injective, Ordinal.opow_log_le_self, cauchy_iff_exists_le_nhds, CategoryTheory.Limits.factorThruKernelSubobject_comp_arrow, HomotopicalAlgebra.trivialCofibrations_sub_weakEquivalences, CategoryTheory.ShortComplex.Exact.isIso_imageToKernel, ENat.floor_le_ceil, Finset.truncatedInf_le, strictMono_of_sub_one_lt, Cardinal.aleph_add_aleph, FractionalIdeal.one_le_dual_one, add_left_strictMono_of_ne_top, Dynamics.coverMincard_closure_le, Submodule.span_le_restrictScalars, EReal.instMulPosMono, ProjectiveSpectrum.not_irrelevant_le, isPreconnected_Iic, neg_abs_le, Fin.map_revPerm_Ico, CategoryTheory.Localization.LeftBousfield.le_W_iff, ProfiniteGrp.denseRange_toLimit, instPosMulStrictMonoIdeal, Ordinal.le_nadd_self, oneLePart_eq_ite_lt, Subfield.closure_mono, Mathlib.Tactic.Bound.Nat.one_le_cast_of_le, Affine.Simplex.affineCombination_mem_closedInterior_face_iff_nonneg, CategoryTheory.Subfunctor.IsGeneratedBy.ofSection_le, MvPowerSeries.nat_le_weightedOrder, MeasureTheory.measure_lintegral_le_pos, Finset.minimal_iff_forall_diff_singleton, Rat.toNNRat_lt_toNNRat_iff', upperBounds_Iio, Field.primitive_element_iff_algHom_eq_of_eval', Set.encard_union_le, MeasureTheory.aecover_Icc_of_Ioc, LocalSubring.exists_le_valuationSubring, PrimeMultiset.prod_dvd_iff', ValuativeExtension.mapValueGroupWithZero_strictMono, CStarAlgebra.nonneg_iff_isSelfAdjoint_and_negPart_eq_zero, Nat.cast_add_one_pos, Ordinal.instZeroLEOneClass, ProfiniteGrp.closedSubgroup_eq_sInf_open, Ordinal.lsub_const, Ideal.pow_succ_lt_pow, borel_eq_generateFrom_Icc, Finset.bddAbove, sdiff_self, Module.End.genEigenspace_top_eq_maxUnifEigenspaceIndex, Antitone.Ioc, PNat.not_lt_one, CFC.posPart_eq_zero_iff, MeasureTheory.hittingBtwn_apply_anti, AlgebraicGeometry.Scheme.Modules.germ_restrictStalkNatIso_hom_app, codisjoint_iff, le_of_inf_eq', CategoryTheory.MorphismProperty.le_ind, BoxIntegral.Box.measure_coe_lt_top, Finset.le_sup_dite_pos, Set.encard_strictMono, CategoryTheory.GrothendieckTopology.top_mem, Ordinal.lt_one_iff_zero, Polynomial.trailingDegree_le_trailingDegree, Fin.image_castSucc_Ioc, ENNReal.iSup_coe_lt_top, Submodule.exists_le_ker_of_notMem, IsDedekindDomain.integer_empty, Ordinal.nmul_succ, instIsOrderedCancelVAddOfIsOrderedAddCancelMonoid, IsDedekindDomain.primesOverEquivPrimesOver_inertiagDeg_eq, Set.biUnion_le, Ordinal.toNatOrdinal_min, CategoryTheory.Subobject.sInf_le, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_counit_app', Submodule.height_strictMono, ThreeAPFree.addRothNumber_eq, Ordinal.isSuccLimit_add_iff, CategoryTheory.Triangulated.TStructure.le_monotone, HahnSeries.mem_cardSuppLTSubring, exists_wellOrder, EMetric.cauchySeq_iff, Fin.finsetImage_castAdd_Ico, AffineSubspace.not_sSameSide_bot, Set.image_single_Icc_left, unitInterval.symm_projIcc, cmp_self_eq_eq, CategoryTheory.not_strictMono_of_isNoetherianObject, Rat.preimage_cast_Ioi, Finset.covBy_iff_exists_cons, WithBot.Ico_bot_coe, Subgroup.FG.finset_sup, Order.isMin_or_mem_range_succ_or_isSuccLimit, sdiff_lt_left, cocompact_eq_atTop, ArchimedeanClass.FiniteElement.not_isUnit_iff_mk_pos, exists_lt_add_iff_lt_left, Hyperreal.archimedeanClassMk_neg_of_tendsto_atBot, LieIdeal.le_killingCompl_top_of_isLieAbelian, CategoryTheory.MorphismProperty.transfiniteCompositionsOfShape_le, Polynomial.abs_tendsto_atTop_iff, Order.Ideal.coe_sup_eq, Cardinal.preAleph_omega0, Real.toNNReal_le_one, CategoryTheory.Subobject.factorThru_mk_self, adjoin_eq_top_of_conductor_eq_top, Finset.min_le, IsCompact.isLUB_sSup, MeasureTheory.Content.innerContent_exists_compact, sInf_le_sInf, mul_star_self_nonneg, MeasureTheory.Measure.MeasureDense.fin_meas_approx, NNReal.closedBall_zero_eq_Icc, ENNReal.ofNat_le_ofReal, eHolderNorm_lt_top, SimpleGraph.Walk.toSubgraph_bypass_le_toSubgraph, TopologicalSpace.le_def, Topology.IsGeneratedBy.iff_le_generatedBy, Ring.DimensionLeOne.prime_le_prime_iff_eq, imageToKernel_arrow_assoc, CategoryTheory.Abelian.subobjectIsoSubobjectOp_apply, Cardinal.toENat_lift, AlgebraicGeometry.isIso_morphismRestrict_iff_isIso_app, Ordinal.lt_gamma0, PNat.Ico_eq_finset_subtype, cmp_eq_cmp_symm, Set.inv_Ioi, Set.image_update_Icc_left, extDeriv_apply_vectorField, Set.Icc.one_sub_le_one, CategoryTheory.SmallObject.SuccStruct.restrictionLTOfCoconeIso_inv_app, Submodule.strictMono_comap_prod_map, Ordinal.foldr_le_nfpFamily, CategoryTheory.Subfunctor.image_le_iff, Int.instIsSuccArchimedean, CStarAlgebra.isStrictlyPositive_iff_exists_isUnit_and_isSelfAdjoint_and_eq_mul_self, HomotopicalAlgebra.trivialFibrations_sub_fibrations, PartENat.lt_add_one_iff_lt, Module.End.map_genEigenrange_le, Submodule.mem_annihilator', hasDerivAt_iff_tendsto_slope_left_right, IntermediateField.normal_iff_normalClosure_le, AlgebraicGeometry.morphismRestrict_base, Filter.IsIncreasingApproximateUnit.eventually_nonneg, Dynamics.coverEntropyEntourage_le_netEntropyEntourage, Submodule.le_one_toAddSubmonoid, sign_zero, CategoryTheory.ShortComplex.exact_iff_isIso_imageToKernel, MeasureTheory.lintegral_le_iSup_mul, Matroid.eRk_compl_insert_union_add_eRk_compl_insert_inter_le, Function.Lex.wellFoundedLT, HomologicalComplex.HomologySequence.instEpiMap'ComposableArrows₃OfNatNat, Metric.infEDist_le_hausdorffEDist_of_mem, RootPairing.setOf_root_sub_zsmul_mem_eq_Icc, Asymptotics.isBigOTVS_iff_smallSets, min_le_iff, disjoint_partialSups_right, LocalSubring.exists_le_valuationSubring_of_isIntegrallyClosedIn, tendsto_floor_right_pure_floor, biInf_le_eq_inf, Finset.Ioc_sub_one_left_eq_Icc, CStarAlgebra.nonneg_iff_exists_nonneg_and_eq_mul_self, NNReal.lt_iff_exists_rat_btwn, Fin.attachFin_Ico_eq_Ici, OrderIso.finsetSetFinite_symm_apply, Ordinal.bot_eq_zero, IsOpen.exists_Ioo_subset, Int.cofinite_eq, Subalgebra.eq_bot_of_finrank_one, Ne.le_iff_lt, WithZero.lt_mul_exp_iff_le, CauSeq.bounded', StarSubsemiring.toSubsemiring_le_iff, UpperSet.map_refl, posMulReflectLT_iff_contravariant_pos, AlgebraicGeometry.Scheme.Opens.ι_image_basicOpen_topIso_inv, IsSelfAdjoint.neg_algebraMap_norm_le_self, MeasureTheory.lintegral_iUnion_le, convexHull_basis_eq_stdSimplex, Set.Nonempty.ordConnected_iff_of_bdd', fixingAddSubmonoid_fixedPoints_gc, IntermediateField.relrank_top_right, Hyperreal.isSt_sSup, AlgebraicGeometry.Scheme.zeroLocus_setMul, Filter.tendsto_const_mul_atBot_iff, AlgebraicGeometry.SpecMap_ΓSpecIso_inv_toSpecΓ, CategoryTheory.IsGrothendieckAbelian.generatingMonomorphisms_le_monomorphisms, Field.Emb.Cardinal.noMaxOrder_rank_toType, ENNReal.ofReal_le_iff_le_toReal, Order.Ioo_pred_left, Set.biUnion_Ico_eq_Iio_self_iff, Interval.one_mem_one, PosSMulMono.of_pos, PSet.rank_lt_of_mem, Ideal.IsPrime.multiset_prod_map_le, MeasureTheory.exists_pos_measure_of_cover, ENNReal.div_le_iff_le_mul, Set.OrdConnected.image_ennreal_ofReal, CategoryTheory.ComposableArrows.exact_iff_δ₀, CategoryTheory.Subobject.mk_le_mk_of_comm, IsPredArchimedean.pred_findAtom, disjoint_nhds_nhds_iff_not_specializes, MeasureTheory.AddContent.supClosure_apply_finpartition, Convex.radon_partition, Finset.Ico_add_one_left_eq_Ioo, Submodule.le_traceDual_traceDual, Subgroup.normalClosure_subset_iff, Units.orderEmbeddingVal_apply, AffineBasis.affineSpan_eq_top_of_toMatrix_left_inv, EMetric.infEdist_closure_pos_iff_notMem_closure, Continuous.exists_forall_ge_of_hasCompactSupport, ProbabilityTheory.Kernel.ae_lt_top_of_comp_ne_top, EMetric.tendsto_nhds, Alexandrov.principals_map, le_nonZeroDivisors_iff_isRegular, NNRat.one_le_cast, iSupIndep_pair, Finset.max'_subset, MvPowerSeries.le_lexOrder_iff, Fin.card_Ico, ENNReal.rpow_arith_mean_le_arith_mean_rpow, mem_codiscrete, List.minimum_append, Set.image_sub_const_Iio, AlgebraicGeometry.Scheme.toSpecΓ_preimage_zeroLocus, Subalgebra.finrank_toSubmodule, Pi.le_toLex_update_self_iff, Fin.finsetImage_cast_Ioi, Ordinal.pred_le_iff_le_succ, PosSMulMono.nnrat_of_rat, Cardinal.beth_succ, PrimeSpectrum.sup_vanishingIdeal_le, BoxIntegral.Box.measure_Icc_lt_top, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctorObj_π_app, edist_lt_ofReal, Finset.truncatedSup_empty, List.le_maximum_of_length_pos_of_mem, MeasureTheory.aecover_Ioc_of_Icc, Set.pairwise_disjoint_Ioc_add_zsmul, RCLike.ofReal_le_ofReal, WithSeminorms.uniformEquicontinuous_iff_exists_continuous_seminorm, OrderMonoidIso.withZeroUnits_symm_apply, SimplexCategory.toTop_map, Finset.isGLB_inf, continuous_bot, TwoSidedIdeal.orderIsoIsTwoSided_symm_apply, AlgebraicGeometry.Scheme.Hom.preimage_image_eq, ENat.add_one_le_iff, map_restrict_ae_le_map_indicator_ae, Set.Iio_mul_bij, LieIdeal.derivedSeries_map_le, Set.Ioi_inj, Filter.le_generate_iff, Int.card_Ioc, ArchimedeanClass.orderHom_zero, Fin.image_castAdd_Ioo, Set.image_single_Ioc, Int.preimage_Iio, Ordinal.mem_range_omega_iff, Submodule.restrictScalars_lt, MeasureTheory.setLIntegral_nnnorm_condExpL2_indicator_le, Algebra.coe_top, MeasureTheory.Measure.rnDeriv_lt_top, Matroid.lt_eq_isStrictMinor, IntermediateField.LinearDisjoint.bot_left, IsOpen.exists_iUnion_isClosed, Polynomial.exists_max_root, Language.le_iff, intervalIntegral.FTCFilter.le_nhds, Int.floor_le_iff, NNReal.instCanonicallyOrderedAdd, CauchyFilter.cauchyFilter_eq, Rat.preimage_cast_Icc, Cardinal.continuum_lt_lift, TopologicalSpace.Opens.inclusion'_top_functor, LowerSet.Iic_injective, LinOrd.Iso.mk_hom, Set.image_sub_const_Icc, TopCat.Presheaf.stalk_open_algebraMap, Set.Ico_subset_Ico_union_Ico, Ordinal.nat_le_card, FirstOrder.Language.Substructure.cg_bot, ENat.iInf_coe_lt_top, IntermediateField.isTotallyReal_bot, Finset.prod_sub_ordered, MonoidWithZeroHom.inl_strictMono, Metric.hausdorffEDist_le_ediam, mapClusterPt_atTop_zsmul_iff_nsmul, Cardinal.toNat_monotoneOn, AlgebraicGeometry.isBasis_preimage_isAffineOpen, ENat.instOrderTopology, MeasureTheory.Measure.mkMetric_top, SMulPosStrictMono.toSMulPosMono, EReal.mem_nhds_bot_iff, UpperSet.Ici_sSup, ProbabilityTheory.avgRisk_le_iSup_risk, AlgebraicGeometry.Scheme.Hom.app_surjective, Ordinal.mem_toZFSet_iff, generateFrom_Ico_mem_le_borel, units_smul_eq_self_iff, NormedSpace.normalize_smul, SetTheory.PGame.lt_birthday_iff, ENat.ceil_lt_top, MvPowerSeries.one_le_order_iff_constCoeff_eq_zero, Nonneg.coe_nnratCast, Real.toNNReal_monotone, sign_mul_abs, AlgebraicGeometry.coe_opensRestrict_symm_apply, HahnSeries.addVal_le_of_coeff_ne_zero, SeparatedNhds.disjoint_nhdsSet, AddAction.IsBlock.stabilizer_le, Set.Icc_pred_right_eq_Ico, SimpleGraph.farFromTriangleFree.le_card_sub_card, FractionalIdeal.le_one_iff_exists_coeIdeal, TopCat.GlueData.instIsIsoT, IsNoetherian.wf, ENNReal.inv_mul_le_iff, min_eq_iff, Finsupp.supported_comap_lmapDomain, TopCat.Presheaf.presheafEquivOfIso_inverse_obj_map, Set.Ici_top, TopCat.Presheaf.pushforwardEq_hom_app, Set.Finite.csSup_lt_iff, ContinuousAlternatingMap.alternatizeUncurryFin_alternatizeUncurryFinCLM_comp_apply, AbsoluteValue.IsEquiv.le_one_iff, SSet.Subcomplex.eq_top_iff_contains_nonDegenerate, Set.Ici_subset_Icc_union_Ici, lowerHemicontinuous_iff_isOpen_compl_preimage_Iic_compl, WithBot.ge_of_forall_gt_iff_ge, max_top_right, Interval.coe_dual, AlgebraicGeometry.affineLocally_iff_affineOpens_le, Matroid.cRk_mono, Affine.Simplex.ExcenterExists.sign_signedInfDist_touchpoint, DiscreteQuotient.proj_bot_bijective, cauchySeq_finset_iff_vanishing_norm, Finset.card_Iio_finset, AlgebraicGeometry.Scheme.fromSpecStalk_appTop, nhdsGT_basis, NonUnitalSubsemiring.closure_le_centralizer_centralizer, PrimeSpectrum.isIdempotentElemEquivClopens_one_sub, MeasureTheory.IsSetSemiring.mem_supClosure_iff, MeasureTheory.Measure.restrict_biUnion_le, AlgebraicGeometry.RingedSpace.basicOpen_pow, Fin.Ioo_add_one_eq_Ioc, Finsupp.rangeIcc_toFun, ZFSet.rank_mono, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.inv_naturality_assoc, le_iInf_add, Cardinal.instNoMaxOrder, Finset.sup_mul_le, FirstOrder.Language.Substructure.fg_def, Order.height_le_krullDim, Subalgebra.isIdempotentElem_toSubmodule, Filter.principal_le_lift', ArchimedeanClass.mem_ball_iff, NucleusClass.idempotent, Finset.max_eq_sup_coe, PNat.sub_coe, SimpleGraph.isUniform_singleton, nndist_le_pi_nndist, SupClosed.infClosure, PrimeSpectrum.basicOpen_le_basicOpen_iff_algebraMap_isUnit, AlgebraicGeometry.pullbackRestrictIsoRestrict_inv_fst, Acc.rank_lt_of_rel, Finset.Ico_inter_Ico_consecutive, IsNonarchimedean.multiset_image_add_of_nonempty, TopCat.presheafToTop_obj, Subalgebra.unop_top, Int.floor_le_sub_one_iff, Topology.IsUpper.isTopologicalSpace_basis, AddSubsemigroup.centralizer_le, Algebra.TensorProduct.includeRight_map_center_le, OrderMonoidIso.withZeroUnits_apply, nnnorm_zpow_le_mul_norm, ENNReal.rpow_inv_le_iff, Filter.Tendsto.finset_sup_nhds, compl_inf_eq_bot, CategoryTheory.TransfiniteCompositionOfShape.ofComposableArrows_F, NatOrdinal.lt_one_iff_zero, Cardinal.toENat_le_ofNat, tendsto_intCast_atBot_sup_atTop_cobounded, Asymptotics.isEquivalent_nat_ceil, disjoint_nhdsSet_principal, InverseSystem.PEquivOn.nat, units_smul_eq_neg_iff, SemilatticeInf.inf_le_left, Matroid.eRk_insert_le_add_one, Subsemigroup.gc_map_comap, AlgebraicGeometry.Scheme.Opens.toScheme_presheaf_obj, isPiSystem_Ioc, Filter.tendsto_neg_atTop_atBot, MeasureTheory.MemLp.meas_ge_lt_top'_enorm, nonempty_orderEmbedding_of_finite_infinite, ProperCone.innerDual_zero, Cardinal.lt_omega_iff_card_lt, Submonoid.map_le_iff_le_comap, BddOrd.dual_map, Profinite.NobelingProof.Products.eval_eq, TwoSidedIdeal.opOrderIso_symm_apply, le_iff_exists_inf, Disjoint.filter_principal, AlgebraicGeometry.Scheme.IdealSheafData.vanishingIdeal_ideal, WithTop.orderIsoSumLexPUnit_toLex, succ_min, ArchimedeanClass.mk_monotoneOn, AffineSubspace.le_def, EReal.range_coe_ennreal, SupBotHom.bot_apply, Ideal.span_singleton_le_span_singleton, StarAlgebra.elemental.le_iff_mem, Set.chainHeight_mono, AlgebraicGeometry.PresheafedSpace.Γ_map_op, Subalgebra.lTensorBot_one_tmul, MeasureTheory.Content.outerMeasure_interior_compacts, StarAlgebra.adjoin_nonUnitalStarSubalgebra_eq_span, WithBot.succ_eq_bot, Fin.sum_Ioi_cast, Submonoid.centralizer_le, Subalgebra.isSimpleOrder_of_finrank, Int.floor_nonneg, Subgroup.fg_iff_add_fg, Pairwise.disjoint_extend_bot, Ideal.height_le_height_add_of_liesOver, ofAdd_bot, Valuation.IsEquiv.one_lt_iff_one_lt, ContinuousWithinAt.partialSups_apply, addRothNumber_union_le, Flag.coe_covBy_coe, Finset.filter_fst_eq_antidiagonal, ENNReal.one_lt_top, CompactlySupportedContinuousMap.nnrealPart_add_le_add_nnrealPart, ArchimedeanClass.mk_sub_lt_mk_right_iff, Polynomial.div_tendsto_atBot_zero_of_degree_lt, Path.Homotopy.hcomp_apply, Subgroup.le_topologicalClosure, NNReal.comap_coe_atTop, Projectivization.Subspace.span_empty, MeasureTheory.ProbabilityMeasure.apply_mono, Fin.map_castLEEmb_Ico, NNReal.rpow_sum_le_const_mul_sum_rpow, Order.pred_top_le_iff, CategoryTheory.Dial.comp_le_lemma, Filter.comap_cocompact_le, IsDedekindDomain.HeightOneSpectrum.intValuation_le_pow_iff_dvd, SimpleGraph.Subgraph.le_induce_union_left, ONote.lt_def, hasProdLocallyUniformlyOn_iff_tendstoLocallyUniformlyOn, PiLp.nndist_apply_le, MeasureTheory.Measure.prod_apply_le, Iio_mem_nhdsSet_Iic_iff, ProbabilityTheory.Kernel.le_compProd_apply, PNat.addLeftReflectLT, AddSubmonoid.smul_le, Metric.isSeparated_insert, abs_sub_le_iff, Nat.one_le_cast, le_nhds_of_cauchy_adhp, sInf_eq_argmin_on, map_derivedSeries_le_derivedSeries, SupClosed.directedOn, BoundedContinuousFunction.nndist_set_exists, OrderedSemiring.toPosSMulMonoNat, IsDedekindDomain.selmerGroup.monotone, span_coeff_minpolyDiv, AddMonoidHom.mclosure_preimage_le, Filter.neg_le_self, Fin.sum_Ico_succ, Order.isPredPrelimit_iff, LieModule.exists_genWeightSpace_le_ker_of_isNoetherian, Finset.nsmul_right_monotone, MeasureTheory.Measure.countable_meas_pos_of_disjoint_iUnion₀, Set.add_mem_Ico_iff_right, limsup_eq_bot, nhdsWithinLE_sup_nhdsWithinGT, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.invApp_app_apply, AlgebraicGeometry.Scheme.id_appTop, midpoint_le_right, Int.cast_strictMono, Set.projIic_coe, OrderMonoidHom.mul_comp, strongRankCondition_iff_forall_rank_lt_aleph0, GenContFract.IntFractPair.nth_stream_fr_lt_one, SMulPosMono.toSMulPosReflectLT, Set.unbounded_inter_ge, Set.image_inv_Ioo, Ordinal.veblen_zero_lt_veblen_zero, MvPowerSeries.le_order_subst, IntermediateField.Lifts.eq_iff_le_carrier_eq, Cardinal.lift_lt_ofENat, covBy_iff_lt_iff_le_left, LipschitzWith.projIcc, min_lt_iff, BoundedContinuousFunction.Lp_nnnorm_le, Module.Basis.addSubgroupOfClosure_repr_apply, le_of_inf_eq, CategoryTheory.MorphismProperty.transfiniteCompositionsOfShape_monotone, Subsemiring.prod_mono_right, EMetric.diam_union', MvPowerSeries.le_weightedOrder_pow, Interval.subset_coe_map, bihimp_le_iff_left, Fin.preimage_val_Ioi_val, Set.image_inv_Icc, IntermediateField.exists_finset_maximalFor_isTranscendenceBasis_separableClosure, AddOpposite.one_le_op, sSup_compact_eq_top, Cardinal.le_preAleph_ord, Finset.Colex.toColex_sdiff_lt_toColex_sdiff, MeasureTheory.laverage_lt_top, iSupIndep.supIndep', Filter.Tendsto.map_mapsTo_Iic, MonotoneOn.Ici, Subalgebra.mulMap_bot_left_eq, Submodule.toAddSubmonoid_strictMono, Cardinal.aleph1_le_lift, Cardinal.lt_univ, Set.preimage_const_add_Ici, dist_le_coe, Set.Icc_eq_singleton_iff, CategoryTheory.simple_iff_subobject_isSimpleOrder, ciInf_le', RootPairing.zero_lt_pairingIn_iff, PrincipalSeg.monotone, le_emultiplicity_of_le_multiplicity, Subgroup.le_centralizer, ENNReal.inv_le_iff_le_mul, Order.Ioi_pred, LinearOrderedAddCommGroupWithTop.sub_top, Polynomial.abs_tendsto_atTop, Finset.tendsto_Ioc_atBot_prod_atTop, CategoryTheory.MorphismProperty.isStableUnderCobaseChange_iff_pushouts_le, Ordinal.typein_lt_type, Subgroup.toSubmonoid_strictMono, ENNReal.ofReal_add_le, AlgebraicGeometry.Scheme.Spec_map_presheaf_map_eqToHom, Filter.bliminf_or_le_inf, CategoryTheory.injectiveDimension_ge_iff, AlgebraicGeometry.Scheme.Opens.toSpecΓ_appTop_assoc, Metric.cobounded_le_cocompact, Multiset.map_add_left_Ico, CStarAlgebra.instOrderClosedTopology, CategoryTheory.Limits.MultispanIndex.toLinearOrder_fst, ENNReal.cancel_of_ne, MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ, MeasureTheory.Measure.pi_prod_map_IocProdIoc, Set.Ioi_subset_Ioc_union_Ioi, Fin.isAddFreimanIso_Iic, Ordinal.apply_lt_nfp, Order.Ideal.isMaximal_iff_isCoatom, CauSeq.lt_lim, AddMonoidAlgebra.le_inf_support_add, PSet.rank_insert, iSupIndep.supIndep, closedUnderRestriction_iff_id_le, MeasureTheory.measure_singleton_lt_top, tendsto_fract_right', coLindelof_le_cofinite, Ordinal.toZFSet_strictMono, TopologicalSpace.Opens.map_functor_eq, Matroid.closure_eq_subtypeClosure, Set.vadd_Icc, ValuationClass.map_add_le_max, MeasureTheory.measure_pos_iff_nonempty_of_vaddInvariant, biInf_lt_iff, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivCounitIso_inv_app_hom, Nat.castOrderEmbedding_apply, Matrix.IsIrreducible.nonneg, LinearMap.comap_leq_ker_subToSupQuotient, MeasureTheory.lintegral_abs_det_fderiv_le_addHaar_image, Fin.preimage_succ_Ioc_succ, ENNReal.ofReal_lt_top, Fin.image_succ_Ico, sign_pos, isPiSystem_Ico_mem, Set.Ioo_min_max, IsCompact.exists_isLUB, UniformSpace.sInf_le, ValuationSubring.valuation_le_one, MeasureTheory.hahn_decomposition, CategoryTheory.Limits.MultispanIndex.toLinearOrder_snd, PNat.natPred_monotone, Ideal.height_le_height_add_one_of_mem, SuccOrder.nhds_eq_pure, PowerSeries.le_order_map, Ideal.le_of_dvd, Fin.finsetImage_cast_Iio, Affine.Simplex.wSameSide_affineSpan_faceOpposite_point_left_iff, Matroid.spanning_iff_eRk_le', AddCon.le_iff, CategoryTheory.Limits.instHasIterationOfShapeElemIic, Multiset.map_add_left_Ioc, Subgroup.zpowers_le, Filter.lift_iInf_le, Ordinal.card_le_one, ENNReal.rpow_zero_pos, WellFoundedLT.toIsPredArchimedean, Ideal.exists_le_prime_disjoint, OrderIsoClass.toSupBotHomClass, antilipschitzWith_iff_le_mul_nndist, BoundedLatticeHom.coe_comp_lattice_hom', AffineSubspace.bot_coe, FormalMultilinearSeries.compAlongComposition_nnnorm, Convex.interior_nonempty_iff_affineSpan_eq_top, UniqueFactorizationMonoid.normalizedFactors_pos, MeasureTheory.limsup_measure_closed_le_of_forall_tendsto_measure, Ordinal.lt_sub, Set.pairwise_disjoint_Ioc_mul_zpow, inv_le_self_iff, monovaryOn_iff_forall_smul_nonneg, Finset.dens_le_one, IntermediateField.instFiniteSubtypeMemBot, Ordinal.lift_succ, LowerSet.coe_iicsInfHom, OrderMonoidHom.commute_inl_inr, GroupCone.coe_oneLE, Ordinal.veblen_of_ne_zero, Int.ceil_eq_zero_iff, Ordinal.iSup_eq_lsub_or_succ_iSup_eq_lsub, CategoryTheory.GrothendieckTopology.closureOperator_isClosed, Set.image_const_add_Ioi, Finset.le_prod_max_one, NormedAddCommGroup.cauchySeq_iff, EReal.range_coe_eq_Ioo, Polynomial.adjoin_rootSet_eq_range, Ideal.image_subset_nonunits_valuationSubring, Fin.finsetImage_val_Iio, Set.pairwise_disjoint_Ico_mul_zpow, MeasureTheory.eLpNorm_one_le_of_le', AffineSubspace.parallel_bot_iff_eq_bot, FractionalIdeal.spanSingleton_mul_le_iff, IsGreatest.nnnorm_cfc, egauge_le_one, FractionalIdeal.absNorm_bot, covariant_le_iff_contravariant_lt, Filter.HasBasis.nhds, CategoryTheory.Presheaf.equalizerSieve_self_eq_top, Set.biInter_ge, MeasureTheory.Measure.sub_le_iff_le_add, ProbabilityTheory.IsFiniteKernel.bound_lt_top, CategoryTheory.TransfiniteCompositionOfShape.iic_isColimit, HahnEmbedding.IsPartial.strictMono, Metric.packingNumber_two_mul_le_externalCoveringNumber, Multiset.inf_zero, infClosed_infClosure, Filter.disjoint_iff, MonomialOrder.lex_lt_iff_of_unique, Cardinal.add_lt_aleph0_iff, emultiplicity_le_emultiplicity_iff, Finset.le_max_of_eq, monovaryOn_inv_right, TopCat.Sheaf.extend_hom_app, OrderIso.withBotCongr_symm_apply, MeasureTheory.lintegral_edist_triangle, nhdsWithin_compl_singleton_le, SummationFilter.hasSum_symmetricIoc_int_iff, bot_sup_eq, CompleteLattice.Iic_coatomic_of_compact_element, isLocallyClosed_Ico, Subsemiring.opEquiv_apply, EReal.image_coe_Iic, ClosedSubmodule.bot_orthogonal_eq_top, inf_le_ite, SetSemiring.mulLeftMono, frontier_lt_subset_eq, Subalgebra.separatesPoints_monotone, MeasurableSet.exists_isCompact_diff_lt, AlgebraicGeometry.Scheme.IdealSheafData.le_radical, EMetric.infEdist_anti, Ultrafilter.le_sup_iff, RingPreordering.toSubsemiring_mono, essSup_const_bot, MeasureTheory.lintegral_mul_prod_lintegral_pow_le, SimplexCategory.II.map'_map', Ideal.disjoint_map_primeCompl_iff_comap_le, PartOrdEmb.Limits.CoconePt.fac_apply, two_pos, not_bddBelow_iff, AlgebraicGeometry.IsAffineOpen.primeIdealOf_isMaximal_of_isClosed, SimpleGraph.hasse_prod, SeminormedAddGroup.disjoint_nhds_zero, Lattice.BicartSq.le₂₄, AlgebraicGeometry.Scheme.fromSpecStalk_toSpecΓ, Set.preimage_mul_const_Ioi, Order.mem_range_pred_or_isPredPrelimit, PrimeSpectrum.isMin_iff, le_partialSups_of_le, Filter.not_nonneg_sub_iff, bddAbove_slope_gt_of_mem_interior, upperSetIsoLowerSet_symm_apply, Cardinal.mk_embedding_eq_zero_iff_lift_lt, LieSubmodule.map_incl_lt_iff_lt_top, AlgebraicGeometry.Scheme.ker_toSpecΓ, Multiset.card_Ioc, CFC.negPart_eq_neg, iUnion_Icc_intCast, TopCat.Presheaf.map_germ_eq_Γgerm, strictMono_iff_map_pos, Algebra.top_toSubmodule, comp_partialSups, Set.coe_projIci, AlgebraicGeometry.Scheme.IdealSheafData.ideal_bot, Ordinal.sup_succ_eq_lsub, MeasureTheory.Content.mono, Subring.closure_le_centralizer_centralizer, MvPowerSeries.min_weightedOrder_le_add, MulEquiv.strictMono_subsemigroupCongr, SignType.neg_one_lt_one, Finset.Ico_eq_filter_ssubsets, Sbtw.right_mem_image_Ioi, ENNReal.orderIsoIicOneBirational_apply_coe, Set.eq_empty_or_encard_eq_top_or_encard_diff_singleton_lt, SemilinearMapClass.nnbound_of_continuous, DirectSum.coe_decompose_mul_of_left_mem, MeasureTheory.measure_symmDiff_le, Submodule.one_le_one_div, Nat.one_le_floor_iff, LinearMap.isPosSemidef_iff_posSemidef_toMatrix, PNat.ofNat_lt_ofNat, Asymptotics.IsLittleOTVS.exists_eventuallyLE_mul, RootPairing.zero_le_rootForm, Module.coe_length, ContinuousLinearMap.toExposed.isExposed, Int.card_fintype_Ioo_of_lt, Dense.exists_le', MonomialOrder.degLex_le_iff, ConvexIndependent.mem_convexHull_iff, enorm_indicator_le_enorm_self, SSet.Subcomplex.preimage_range, Filter.tendsto_zpow_atTop_atTop, Filter.iSup_ultrafilter_le_eq, Set.preimage_const_add_Ico, CategoryTheory.Subfunctor.equivalenceMonoOver_inverse_map, AlgebraicGeometry.Scheme.ker_of_isAffine, CategoryTheory.MorphismProperty.universally_le, Fin.Ico_add_one_eq_Icc, Polynomial.map_under_lt_comap_of_weaklyQuasiFiniteAt, Set.Ico_inter_Ico, NonemptyInterval.coe_zero, Finset.card_eraseNone_of_not_mem, nndist_triangle_right, FractionalIdeal.mul_left_strictMono, Filter.le_add_iff, NNReal.one_le_coe, Subalgebra.fg_of_submodule_fg, tendsto_PNat_val_atTop_atTop, IsPredArchimedean.findAtom_eq_bot, not_disjoint_segment_convexHull_triple, IntermediateField.extendScalars.orderIso_apply, CategoryTheory.GrothendieckTopology.PreservesSheafification.le, Tuple.graphEquiv₂_apply, Subgroup.characteristic_iff_comap_le, AddSubgroup.map_subtype_le, le_oneLePart, WithTop.orderIsoSumLexPUnit_symm_inl, AlgebraicGeometry.IsAffineOpen.algebraMap_Spec_obj, Filter.isBoundedUnder_ge_inv, Ordinal.type_le_iff', AlgebraicGeometry.isBasis_basicOpen, denselyOrdered_units_iff, PrincipalSeg.apply_wCovBy_apply_iff, TopologicalSpace.Opens.map_id_obj, MeasureTheory.Measure.withDensity_rnDeriv_le, UniformOnFun.edist_eval_le, OrderIso.withZeroUnits_symm_apply, Filter.Iic_pure, MeasureTheory.lintegral_nnnorm_condExpL2_le, CategoryTheory.IsGrothendieckAbelian.generatingMonomorphisms.exists_larger_subobject, Order.bot_lt_krullDim, OrderDual.instIsAtomistic, Filter.tendsto_atTop_pure, Order.isSuccLimit_iff, NonUnitalStarAlgebra.adjoin_le, ENat.eq_top_iff_forall_gt, CategoryTheory.Subobject.bot_factors_iff_zero, Int.card_box, Finset.coe_orderIsoOfFin_apply, NNRat.coe_lt_coe, QuadraticMap.toBilin_apply, CategoryTheory.Limits.CompleteLattice.colimitCocone_cocone_pt, AffineSubspace.lt_def, ProbabilityTheory.gaussianPDF_pos, ContinuousMap.lt_def, SimplexCategory.toTop₀_map, Real.toNNReal_le_toNNReal_iff, AlgebraicGeometry.Scheme.Hom.mem_preimage, Finset.nnnorm_prod_le', Fin.preimage_succ_Iio_succ, AddSubsemigroup.le_comap_map, Order.Ideal.IsMaximal.isPrime, CategoryTheory.Triangulated.SpectralObject.ω₂_map_hom₁, mem_nerve_nonDegenerate_iff_injective, MulArchimedeanClass.min_le_mk_mul, Order.isSuccPrelimit_iff, MeasureTheory.Measure.lintegral_rnDeriv_lt_top, isConnected_Ici, disjoint_iff, Dynamics.coverEntropyInfEntourage_le_coverEntropyInf, inf_le_iff, atBot_atTop_le_cocompact, Set.einfsep_anti, NNReal.sqrt_eq_iff_eq_sq, tendsto_floor_atTop, MonomialOrder.bot_eq_zero, SSet.prodStdSimplex.objEquiv_apply_snd, map_eq_top_iff, ProbabilityTheory.bayesRisk_compProd_le_bayesRisk, SupPrime.not_isMin, CategoryTheory.ComposableArrows.twoδ₂Toδ₁_app_zero, Finset.uIcc_product_uIcc, Finsupp.le_iff, CompleteSublattice.bot_mem, RealRMK.rieszMeasure_le_of_eq_one, Set.ordConnected_iff_disjoint_Ioo_empty, AnalyticAt.meromorphicOrderAt_nonneg, SpectrumRestricts.nnreal_lt_iff, isClosed_and_discrete_iff, LieModule.genWeightSpace_le_genWeightSpaceOf, convexHull_vadd, NNReal.HolderTriple.inv_nonneg, Pi.card_Ioc, Filter.monotone_mem, Dynamics.netEntropyInfEntourage_nonneg, topologicalKrullDim_subspace_le, ValuativeRel.IsDiscrete.has_maximal_element, Int.floor_mono, Ideal.homogeneousHull.gc, Filter.covariant_swap_div, AlgebraicGeometry.Scheme.Γevaluation_naturality_apply, Matroid.closure_mono, Subgroup.normal_subgroupOf_iff_le_normalizer_inf, cfc_nonpos_iff, ArchimedeanClass.mk_add_lt_mk_right_iff, CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinset_obj_map, MeasureTheory.enorm_indicatorConstLp_le, LinearMap.finrank_genEigenspace_le, Nonneg.val_unitsEquivPos_symm_apply_coe, FunctionField.InftyValuation.map_add_le_max', mabs_mul_le, Prod.codisjoint_iff, Set.offDiag_mono, AlgebraicGeometry.Scheme.basicOpen_le, AlgebraicGeometry.locallyOfFiniteType_iff, ProbabilityTheory.tsum_prob_mem_Ioi_lt_top, TopologicalSpace.Closeds.complOrderIso_apply, FirstOrder.Language.Substructure.lift_card_closure_le_card_term, r1Space_iff_inseparable_or_disjoint_nhds, Fin.finsetImage_cast_Icc, Int.ball_eq_Ioo, AlgebraicGeometry.SheafedSpace.comp_c_app, Subgroup.characteristic_iff_map_le, MulArchimedeanClass.mem_closedBallSubgroup_iff, FiniteMulArchimedeanClass.mem_ballSubgroup_iff, partialSups_eq_sup_range, exists_spanning_measurableSet_le, Cardinal.le_aleph_ord, Multiset.le_dedup_self, ONote.NFBelow.lt, CategoryTheory.ComposableArrows.whiskerLeftFunctor_obj_map, CliffordAlgebra.adjoin_range_ι, ValuationSubring.principalUnitGroup_le_principalUnitGroup, EReal.instCanLiftENNRealToERealLeOfNat, essSup_eq_sInf, Filter.map_mapsTo_Iic_iff_tendsto, WithTop.isOpenEmbedding_coe, Ordinal.IsInitial.card_lt_card, AlgebraicGeometry.Flat.flat_and_surjective_iff_faithfullyFlat_of_isAffine, bihimp_le_iff_right, Finset.covBy_iff_exists_insert, inv_mabs_le, MonomialOrder.toSyn_lt_iff_ne_zero, MeasureTheory.upperCrossingTime_le_lowerCrossingTime, EMetric.hausdorffEdist_le_ediam, ENNReal.lt_iff_exists_coe, Subgroup.comap_lt_comap_of_surjective, WeierstrassCurve.IsMinimal.val_Δ_maximal, TopCat.Presheaf.SheafConditionEqualizerProducts.w, le_egauge_smul_left, SimplexCategory.congr_toOrderHom_apply, CompleteLattice.isCompactElement_iff_exists_le_iSup_of_le_iSup, Cardinal.lt_one_iff_zero, tendsto_inv_nhdsGT_zero, MeasureTheory.dense_of_generateFrom_isSetSemiring, abs_sub_lt_iff, Algebra.elemental.le_centralizer_centralizer, Module.Basis.isMaxChain_range_flag, IsGaloisGroup.le_fixedPoints_iff_le_fixingSubgroup, NNReal.addLeftMono, Dynamics.coverMincard_finite_of_isCompact_uniformContinuous, HasFDerivAt.abs, Set.Ici_diff_Ioi_same, Multiset.le_union_right, supPrime_iff_not_isMin, MeasureTheory.enorm_integral_le_lintegral_enorm, ArchimedeanClass.stdPart_eq_sInf, ProbabilityTheory.Kernel.ae_kernel_lt_top, RCLike.instMulPosReflectLE, MeasureTheory.lintegral_edist_lt_top, Filter.tendsto_nhds_atBot, Filter.addLeftMono, AffineSubspace.setOf_wOppSide_eq_image2, LieModule.weight_vector_multiplication, AddSubgroup.le_op_iff, MeasureTheory.exists_simpleFunc_forall_lintegral_sub_lt_of_pos, AlgebraicGeometry.Scheme.Opens.toSpecΓ_SpecMap_presheaf_map_top_assoc, iUnion_Ioc_add_intCast, Ideal.height_le_card_of_mem_minimalPrimes_span, Finset.geomSum_ofColex_strictMono, integralClosure_le_algebraicClosure, SSet.Subcomplex.image_le_iff, LinearOrderedField.cutMap_coe, CategoryTheory.Subobject.pullback_id, Finset.sup'_lt_iff, AntitoneOn.of_map_sup, PiTensorProduct.injectiveSeminorm_le_projectiveSeminorm, ENNReal.le_coe_iff, SimpleGraph.chromaticNumber_le_card, AlgebraicGeometry.liftCoborder_app, AlgebraicGeometry.topologically_iso_le, AlgebraicGeometry.LocallyRingedSpace.coe_toΓSpecSheafedSpace_hom_base_hom_apply_asIdeal, Nat.lt_one_of_floor_lt_one, Set.Ioc_mul_Ico_subset', HasFDerivWithinAt.abs, tendsto_mulIndicator_biUnion_finset, CategoryTheory.Triangulated.SpectralObject.triangle_obj₂, NatOrdinal.toOrdinal_max, topologicalKrullDim_zero_of_discreteTopology, Odd.pow_neg_iff, MeasureTheory.measure_compl_le_add_iff, OrderEmbedding.birkhoffSet_inf, Nonneg.instNontrivialSubtypeLeOfNatOfAddLeftMono, IsOpen.measure_pos, Subsemigroup.closure_le_centralizer_centralizer, Ordinal.typein_lt_typein, ringKrullDim_quotient_le, Finset.codisjoint_left, le_pow_sup, FirstOrder.Language.DirectLimit.partialEquivLimit_comp_inclusion, Module.End.iSup_genEigenspace_eq, projIcc_eq_one, Set.isCoatom_iff, Filter.map_inf_le, MeasureTheory.hitting_eq_end_iff, HasSSubset.SSubset.lt, AlgebraicGeometry.Scheme.Opens.toSpecΓ_preimage_basicOpen, TopCat.Presheaf.isSheaf_iff_isSheaf_comp, Hyperreal.infinitePos_iff_infinite_and_pos, Monotone.Iio, Filter.countableGenerate_isGreatest, Matroid.eRk_inter_add_eRk_union_le, Set.exists_isOpen_lt_add, Cardinal.IsInaccessible.nat_lt, CategoryTheory.Limits.factorThruImageSubobject_comp_self_assoc, Codisjoint.eq_top_of_self, CategoryTheory.Limits.CompleteLattice.finiteColimitCocone_cocone_ι_app, Finset.lt_fold_min, Submonoid.le_pointwise_smul_iff₀, EisensteinSeries.tendsto_e2Summand_atTop_nhds_zero, Polynomial.Splits.adjoin_rootSet_eq_range, Pi.card_Icc, strictConvex_Ioo, IsAtomistic.isLUB_atoms, Ordinal.eq_nat_or_omega0_le, IntermediateField.finrank_eq_one_iff, ArchimedeanClass.ballAddSubgroup_top, Setoid.top_def, wcovBy_iff_eq_or_covBy, CategoryTheory.Subobject.pullback_obj, LieSubmodule.lie_le_inf, codisjoint_self, Ordinal.iterate_veblen_lt_gamma0, Set.add_mem_Icc_iff_right, exists_mem_interior_convexHull_affineBasis, AlgebraicGeometry.isIntegral_appTop_of_universallyClosed, IntermediateField.lift_top, LaurentSeries.exists_ratFunc_val_lt, PiLp.enorm_apply_le, sameRay_smul_left_iff, exists_nat_ge, zpow_left_strictMonoOn₀, IsSimpleModule.isAtom, ENNReal.coe_lt_coe, CategoryTheory.Presieve.ofArrows_le_iff, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.ofRestrict_invApp_apply, AddSubmonoid.le_prod_iff, Fin.finsetImage_castAdd_Ici, MeasureTheory.Measure.nonpos_iff_eq_zero', AlgebraicGeometry.SheafedSpace.id_hom_c_app, nnnorm_multiset_sum_le, cardinalInterFilter_aleph_one_iff, AddSubmonoid.disjoint_def', MulPosReflectLE.toMulPosReflectLT, Ordinal.nat_lt_card, Finpartition.parts_eq_empty_iff, Filter.isAtom_pure, Ideal.comp_quotientMap_eq_of_comp_eq, Set.Ico.coe_inf, Int.Ioo_eq_finset_map, AlgebraicGeometry.Scheme.zeroLocus_def, min_top_left, antitone_iff_map_nonpos, IdemSemiring.toMulLeftMono, Fin.image_castSucc_Icc, LocalSubring.eq_iInf_of_isIntegrallyClosedIn, Fin.finsetImage_cast_Ici, not_lt_iff_eq_or_lt, List.exists_map_eq_of_sorted_nonempty_iff_sbtw, covariant_lt_iff_contravariant_le, Set.preimage_const_mul_Iic, LowerSet.sdiff_lt_left, Int.lt_floor_add_one, Multiset.Ico_filter_lt, LieAlgebra.derivedSeriesOfIdeal_le_self, Finset.card_eraseNone_of_mem, Flag.isMin_coe, SetLike.covBy_iff', AddMonoidAlgebra.supDegree_zero, LinOrd.id_apply, IncidenceAlgebra.mu_ofDual, AlgebraicGeometry.StructureSheaf.isLocalizedModule_toPushforwardStalkAlgHom_aux, IsWellFounded.rank_lt_of_rel, DFinsupp.Lex.le_iff_of_unique, Module.Basis.flag_le_ker_coord, TopologicalSpace.generateFrom_anti, iSup_eq_top, CStarAlgebra.span_nonneg_inter_unitClosedBall, le_iInf_mul, inf_le_right, Submodule.iSup_dualAnnihilator_le_iInf, CategoryTheory.ComposableArrows.IsComplex.mono_cokerToKer', MeasureTheory.measure_laverage_le_pos, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_inv_assoc, Module.End.genEigenspace_inf_le_add, CliffordAlgebra.range_ι_le_evenOdd_one, Subgroup.commutator_le, mulPosStrictMono_iff_mulPosReflectLE, interior_Icc, UpperSet.symm_map, Finset.addConst_le_subConst_sq, Field.Emb.Cardinal.instIsSeparableSubtypeMemIntermediateFieldAdjoinImageToTypeOrdRankCompCoeBasisWellOrderedBasisLeastExtIioSingletonSet, MeasureTheory.hasFiniteIntegral_iff_edist, Finset.prod_Ico_int_div, AddSubgroup.ker_le_comap, HahnSeries.embDomainOrderEmbedding_apply, LowerSet.symm_map, TopHom.sup_apply, OrderEmbedding.infIrredUpperSet_apply, partialSups_monotone, WithBot.pred_coe, Submonoid.LocalizationMap.nonZeroDivisors_le_comap, isGLB_biInf, Multiset.map_fst_le_of_subset_toEnumFinset, ENNReal.tendsto_atTop, ArchimedeanClass.mk_map_nonneg_of_archimedean, Finset.offDiag_mono, NonUnitalSubsemiring.closure_mono, AlgebraicGeometry.ΓSpec.adjunction_counit_app, iInf_const_mono, essInf_antitone_measure, ZFSet.rank_powerset, IntermediateField.eq_bot_of_isAlgClosed_of_isAlgebraic, disjoint_bot_left, le_iInf_iff, ContractingWith.edist_efixedPoint_lt_top', SummationFilter.hasSum_symmetricIcc_iff, ENat.coe_toNat_le_self, CompleteLattice.MulticoequalizerDiagram.multispanIndex_right, edist_mul_mul_le, Set.IicExtend_apply, AlgebraicGeometry.StructureSheaf.algebraMap_self_map, Fin.sum_Ioo_castLE, isCoatomic_iff, Filter.sets_ssubset_sets, Module.End.eigenspace_le_genEigenspace, Matrix.l2_opNNNorm_mulVec, AddSubgroup.toIntSubmodule_symm, FinPartOrd.hom_hom_comp, EMetric.diam_ball, ConcaveOn.slope_anti, UV.le_of_mem_compression_of_notMem, LocallyFiniteOrder.orderAddMonoidHom_strictMono, AlgebraicGeometry.Scheme.Hom.preimage_smoothLocus_eq, FirstOrder.Language.LHom.coe_substructureReduct, AlgebraicGeometry.tilde.isoTop_hom, CategoryTheory.Subfunctor.range_eq_top, Order.isGLB_Ioi_iff_isPredPrelimit, AlgebraicGeometry.Scheme.restrict_presheaf_obj, Ordinal.lt_nmul_iff₃, LieSubmodule.monotone_normalizer, Mathlib.Meta.NormNum.isNat_le_false, borel_eq_top_of_discrete, PiLp.nnnorm_apply_le, IntermediateField.insepDegree_bot', supClosure_empty, FiniteMulArchimedeanClass.mem_closedBallSubgroup_iff, AlgebraicGeometry.ProjectiveSpectrum.Proj.isLocalization_atPrime, MeasureTheory.limsup_measure_closed_le_iff_liminf_measure_open_ge, IntermediateField.fixedField_antitone, Valued.hasBasis_nhds_zero, Lat.Iso.mk_inv, AddSubmonoid.le_comap_single_pi, Filter.map_pi_map_coprodᵢ_le, ClosedSubmodule.top_orthogonal_eq_bot, LieSubmodule.orderIsoMapComap_symm_apply, NNReal.map_coe_nhdsGT, FirstOrder.Language.Substructure.mem_closure_iff_exists_term, MonomialOrder.degree_le_iff, Fin.preimage_natAdd_Ioo_natAdd, Ordinal.lt_epsilon_zero, ArchimedeanClass.FiniteResidueField.ofArchimedean_inj, differentialIdeal_le_iff, Affine.Simplex.wOppSide_affineSpan_faceOpposite_iff, Fin.image_val_Ici, Set.Ico.coe_bot, isLUB_hasProd', isCoatom_bot, Set.indicator_nonpos_le_indicator, PrimitiveSpectrum.hull_finsetInf, ModuleCat.Tilde.toOpen_res, Dynamics.netEntropyInfEntourage_le_netEntropyEntourage, CFC.le_one_iff, ringKrullDim_succ_le_of_surjective, MeasureTheory.integral_Iic_eq_integral_Iio', Mathlib.Meta.Positivity.pos_of_isNat', LowerSet.coe_ssubset_coe, MulAction.isCoatom_stabilizer_iff_preprimitive, Submodule.spanRank_finite_iff_fg, antivaryOn_neg_left, StarRingEquivClass.instOrderIsoClass, BoxIntegral.Box.splitLower_le, Filter.HasBasis.cauchySeq_iff', Order.pred_succ, Hyperreal.epsilon_lt_pos, OrderIso.map_succ, abs_lt_iff_mul_self_lt, Set.image_const_add_Iic, MeasureTheory.Measure.le_iff', NonemptyFinLinOrd.hom_comp, Complex.antilipschitz_equivRealProd, Hyperreal.coe_nonneg, biSup_ge_eq_sup, AddSubsemigroup.le_prod_iff, MeasurableSet.exists_isClosed_lt_add, Finset.isGreatest_max', Besicovitch.exist_finset_disjoint_balls_large_measure, Set.Ico.coe_zero, CategoryTheory.Sieve.le_generate, GenContFract.zero_le_of_den, ZNum.cast_lt, MeasureTheory.Measure.mem_support_iff, MeasureTheory.measure_le_measure_union_left, CoheytingHomClass.map_top, AlgebraicGeometry.germ_comp_stalkToFiberRingHom, Monoid.CoprodI.lift_mrange_le, CategoryTheory.Limits.CompleteLattice.colimitCocone_cocone_ι_app, Heyting.Regular.coe_le_coe, OrderIso.sumLexIioIci_symm_apply_Iio, UniqueFactorizationMonoid.factors_pos, Fin.sum_Ioc_castAdd, Finset.tendsto_Ioo_neg, AddSubsemigroup.opEquiv_symm_apply, RealRMK.measure_le_of_isCompact_of_integral, MeasureTheory.OuterMeasure.mkMetric'.tendsto_pre, CompleteLattice.MulticoequalizerDiagram.multispanIndex_fst, AlgebraicGeometry.Scheme.IdealSheafData.mem_supportSet_iff, Fin.prod_Ici_cast, Set.neg_mem_Ioo_iff, CategoryTheory.Subfunctor.equivalenceMonoOver_functor_map, SameRay.exists_pos_right, NonUnitalStarAlgebra.adjoin_mono, Int.card_Ico_of_le, HahnEmbedding.ArchimedeanStrata.disjoint_ball_stratum, AntilipschitzWith.le_mul_nnnorm', CFC.negPart_nonneg, Submonoid.FG.finset_sup, BoxIntegral.Box.Ioo_ae_eq_Icc, ENNReal.ofReal_le_ofReal, AlgebraicGeometry.Scheme.Opens.toSpecΓ_appTop, lowerSemicontinuous_iff_frequently, FinBoolAlg.forgetToFinPartOrdFaithful, Filter.eventuallyLE_antisymm_iff, PointedCone.mem_dual, tendstoUniformlyOn_tsum, Fin.addRothNumber_eq_rothNumberNat, ENNReal.mulLeftOrderIso_toEquiv, QuotientAddGroup.strictMono_comap_prod_image, infIccOrderIsoIccSup_apply_coe, CategoryTheory.ComposableArrows.map'_comp, LowerSet.coe_sub, tendsto_neg_nhdsLT_neg, mem_subalgebraEquivIntermediateField_symm, AddSubgroup.toSubgroup_closure, CategoryTheory.ComposableArrows.IsComplex.zero_assoc, HahnEmbedding.ArchimedeanStrata.archimedeanClassMk_of_mem_stratum, AlgebraicGeometry.Scheme.IdealSheafData.mem_support_iff, Finset.uIcc_subset_uIcc_iff_le', SimpleGraph.vertexCoverNum_mono, MeasureTheory.integral_Icc_eq_integral_Ioo, NNReal.bddAbove_range_agmSequences_fst, ProjectiveSpectrum.as_ideal_le_as_ideal, Filter.disjoint_atBot_atTop, Set.Icc_diff_left, dotProduct_star_self_nonneg, enorm_mul₃_le', Subgroup.sylow_mem_fixedPoints_iff, Mathlib.Meta.Positivity.ofReal_nonneg, Submodule.span_prod_le, CategoryTheory.Subfunctor.Subpresheaf.image_top, Fin.preimage_castLE_Ioo_castLE, NNReal.image_coe_Iio, WithTop.continuous_coe, AlgebraicGeometry.PresheafedSpace.GlueData.ι_image_preimage_eq, enorm'_le_iff_norm_le, Nat.one_div_cast_pos, ENNReal.lt_rpow_inv_iff, compl_image_latticeClosure, HolderWith.ediam_image_le, Ideal.primeHeight_le_ringKrullDim, Metric.ediam_union_le, AlgebraicGeometry.Scheme.IdealSheafData.le_support_iff_le_vanishingIdeal, NNReal.isBoundedUnder_ge_toReal, EMetric.controlled_of_isUniformEmbedding, LSeries.abscissaOfAbsConv_le_one_of_isBigO_one, Ordinal.le_zero, upperClosure_mul, continuousOn_cfc_nnreal, Filter.le_cofinite_iff_eventually_ne, Set.Ioc_subset_Ioc_union_Ioc, closedAbsConvexHull_min, Equiv.finsetCongr_toEmbedding, Finset.Ico_inter_Ico, PartENat.lt_def, ENNReal.le_div_iff_mul_le, MeasureTheory.exists_le_setLAverage, Filter.Eventually.volume_pos_of_nhds_real, InfiniteGalois.mulEquivToLimit_symm_continuous, right_lt_sup, Finset.uIcc_subset_uIcc_iff_le, Ordinal.iterate_omega0_opow_lt_epsilon0, SSet.horn₃₁.desc.multicofork_π_two_assoc, MeasureTheory.Measure.lintegral_rnDeriv_le, instPosSMulMonoNatOfIsOrderedAddMonoid, Order.pred_le_pred_iff, Dynamics.IsDynCoverOf.coverMincard_le_card, lift_rank_lt_rank_dual', Interval.coe_zero, Dynamics.coverEntropyInfEntourage_le_coverEntropyEntourage, MeasureTheory.Lp.coeFn_le, CategoryTheory.MorphismProperty.le_pushouts, IsMulFreimanHom.mulRothNumber_mono, AddSubgroup.disjoint_def, ValuationSubring.primeSpectrumOrderEquiv_apply_coe_carrier, FormalMultilinearSeries.nnnorm_compContinuousLinearMap_le, Ordinal.lt_nmul_iff, PrimeSpectrum.basicOpen_zero, zpow_two_pos_of_ne_zero, Filter.tendsto_inv_atBot_atTop, IsStrictOrderedRing.toIsStrictOrderedModule, NonUnitalAlgebra.adjoin_le_iff, SimpleGraph.FarFromTriangleFree.nonpos, GroupCone.mem_oneLE, Nonneg.val_unitsHomeomorphPos_symm_apply_coe, Metric.externalCoveringNumber_mono_set, IsDedekindDomain.HeightOneSpectrum.valuation_div_le_one_iff, AlgebraicGeometry.ProjectiveSpectrum.Proj.toStalk_stalkMap_toSpec_assoc, LowerSet.supIrred_iff_of_finite, TopologicalSpace.Opens.gc, AlgebraicGeometry.opensDiagram_map, PointedCone.dual_empty, MeasureTheory.restrict_Iio_eq_restrict_Iic, CategoryTheory.SmallObject.SuccStruct.Iteration.mkOfLimit.inductiveSystem_map, Set.image_neg_Ioi, Cardinal.one_le_ofENat, GradedAlgebra.exists_finset_adjoin_eq_top_and_homogeneous, Finset.sup_Iic, Int.sub_one_lt_floor, smoothSheafCommRing.ι_evalHom, LieModule.lowerCentralSeriesLast_le_of_not_isTrivial, AlgebraicGeometry.Γ_restrict_isLocalization, Set.projIci_surjective, Ideal.height_le_spanRank_toENat_of_mem_minimal_primes, AddEquiv.symm_mapAddSubgroup, ArchimedeanClass.stdPart_map_real, SetSemiring.instCanonicallyOrderedAdd, IntermediateField.restrictScalars_eq_top_iff, Ordinal.instPosMulStrictMono, Finset.prod_one_add_ordered, MeasureTheory.OuterMeasure.map_iInf_le, RCLike.le_iff_re_im, Fin.preimage_castLE_Ioi_castLE, Tropical.add_eq_left_iff, Nat.ofNat_lt, instModuleFinite, Asymptotics.IsBigOTVS.exists_eventuallyLE, Hyperreal.infinitePos_iff_infinitesimal_inv_pos, AlgebraicGeometry.Scheme.Hom.appIso_inv_app_presheafMap, Metric.IsCover.externalCoveringNumber_le_encard, MeasureTheory.lintegral_eapprox_le_lintegral, NonUnitalSubsemiring.toSubsemigroup_strictMono, IsValuativeTopology.mem_nhds_iff', Subsemiring.toSubmonoid_mono, CategoryTheory.Limits.ProductsFromFiniteCofiltered.finiteSubproductsCocone_π_app_eq_sum, not_infPrime_top, HahnEmbedding.IsPartial.truncLT_mem_range, AddSubgroup.op_le_op_iff, nhdsSet_iInter_le, DividedPowers.isDPMorphism_iff, CategoryTheory.Subobject.map_pullback, Submodule.restrictScalars_le, Subalgebra.instCovariantClassHSMulLe, PrimitiveSpectrum.gc_closureOperator, MeasureTheory.Measure.restrict_mono_set, Equiv.Perm.sign_eq_prod_prod_Iio, CategoryTheory.GrothendieckTopology.bot_covers, Order.coheight_pos_of_lt_top, le_nhds_add, IsDedekindDomain.HeightOneSpectrum.valuation_lt_one_iff_dvd, CategoryTheory.projectiveDimension_lt_iff, Complementeds.coe_top, AbsoluteValue.le_sub, NNRat.cast_nonneg, RestrictedProduct.inclusion_eq_id, Algebra.adjoin_singleton_le, PMF.toOuterMeasure_apply_le_toMeasure_apply, PrimeSpectrum.isIdempotentElemEquivClopens_symm_inf, ConvexOn.convex_epigraph, WithTop.iInf_coe_eq_top, AffineSubspace.smul_bot, MeasureTheory.Ioi_ae_eq_Ici, SignType.not_one_lt, Submodule.toAddSubgroup_mono, CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinsetColimitCocone_cocone_ι_app, isDenseEmbedding_pure, AlgebraicGeometry.Scheme.Hom.app_eq, MeasureTheory.le_inducedOuterMeasure, TopologicalSpace.Opens.coe_eq_empty, Filter.le_one_iff, AlgebraicGeometry.PresheafedSpace.restrictStalkIso_hom_eq_germ, FormalMultilinearSeries.le_radius_compContinuousLinearMap, AlgebraicGeometry.morphismRestrict_ι, LinOrd.ofHom_id, AlgebraicGeometry.Scheme.Modules.restrictFunctorComp_hom_app_app, HasFPowerSeriesOnBall.r_le, Module.length_eq_top_iff_infiniteDimensionalOrder, upperHemicontinuousOn_iff_preimage_Iic, PrincipalSeg.cocone_ι_app, Finset.Ici_add_Ioi_subset, WithZero.withZeroUnitsEquiv_symm_strictMono, AlgebraicGeometry.Scheme.Hom.preimageIso_inv_ι, totallyBounded_absConvexHull, PNat.coe_le_coe, iUnion_Ico_add_intCast, Filter.nonpos_iff, mapClusterPt_rightLim, CategoryTheory.top_apply, Submonoid.pointwise_smul_le_iff₀, CategoryTheory.Pairwise.cocone_pt, CauchyFilter.inseparable_iff, TopCat.Presheaf.germ_res_assoc, AlgebraicGeometry.ProjectiveSpectrum.Proj.toStalk_stalkMap_toSpec, OrderIso.map_sSup_eq_sSup_symm_preimage, MeasureTheory.Submartingale.eLpNorm_stoppedAbove_le, AddGroupSeminorm.coe_lt_coe, Coheyting.boundary_sup_le, Submodule.toSubalgebra_toSubmodule, Order.not_isSuccPrelimit_iff', Affine.Simplex.wSameSide_affineSpan_faceOpposite_point_right_iff, mulRightReflectLT_of_mulRightMono, AddSubgroup.map_le_map_iff, MeasureTheory.Integrable.exists_boundedContinuous_lintegral_sub_le, AlgebraicGeometry.Scheme.Hom.apply_mem_image_iff, Set.Nonempty.ciSup_lt_iff, Ordinal.derivFamily_succ, AffineMap.lineMap_mono, Subsemiring.closure_le_centralizer_centralizer, NonemptyInterval.fst_inv, Finset.prod_Ico_add', ClosureOperator.ext_iff, iSup_inf_le_sSup_inf, Ordinal.le_self_nadd, Ordinal.IsNormal.id_le, MeasureTheory.SimpleFunc.le_sup_lintegral, NonUnitalAlgebra.toSubmodule_bot, instWellFoundedGTAntisymmetrizationLe, isLUB_Iio, ENNReal.tsum_union_le, Filter.disjoint_cofinite_right, MeasureTheory.measure_le_laverage_pos, convex_halfSpace_ge, MeasureTheory.VectorMeasure.neg_le_neg_iff, Finset.coe_sup', Alexandrov.principalOpen_le_iff, infClosure_singleton, Order.coheight_eq_coe_iff_maximal_le_coheight, SSet.horn₃₁.desc.multicofork_π_zero_assoc, ExpGrowth.expGrowthInf_monotone, Finset.card_Icc_finset, ContDiffMapSupportedIn.iteratedFDerivWithOrderLM_apply, ENat.ceil_le, nnnorm_apply_le_nnnorm_cfcₙ, Concept.swap_le_swap_iff, Subgroup.le_pi_iff, Subalgebra.le_centralizer_centralizer, MeasureTheory.OuterMeasure.trim_le_trim_iff, csInf_le', Ordinal.isNormal_iff_lt_succ_and_bsup_eq, Fin.map_revPerm_Ici, MeasureTheory.Measure.lt_iff', Set.add_mem_Ioc_iff_right, SSet.Subcomplex.image_monotone, FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm_apply_le, SSet.horn_obj, MulPosReflectLE.of_posMulReflectLT_of_mulPosMono, List.le_prod_of_mem, Set.Icc_succ_pred_eq_Ioo, AbsoluteValue.IsEquiv.lt_iff_lt, TopCat.Presheaf.germToPullbackStalk_stalkPullbackHom, IsDedekindDomain.HeightOneSpectrum.intValuation.map_add_le_max', BoundedLatticeHom.coe_comp_sup_hom, Nat.comap_cast_atTop, Dynamics.coverEntropyEntourage_image_le, Compactum.le_nhds_of_str_eq, Opens.pretopology_ofGrothendieck, Seminorm.bound_of_continuous, Set.right_mem_uIoc, Cardinal.aleph_le_aleph, ENat.epow_right_mono, SimpleGraph.chromaticNumber_le_iff_colorable, MeasureTheory.eLpNorm'_le_eLpNormEssSup, AddSubgroup.closure_mono, Ordinal.IsNormal.le_apply, MeasureTheory.lpMeasToLpTrimLie_symm_indicator, ScottContinuousOn.sup₂, absConvexHull_nonempty, Finsupp.Colex.wellFoundedLT_of_finite, Rat.cast_lt_natCast, CategoryTheory.ShortComplex.SnakeInput.composableArrowsFunctor_map, AlgebraicGeometry.Scheme.Modules.restrictFunctorComp_inv_app_app, IsDedekindDomain.primesOverEquivPrimesOver_apply, AlgebraicGeometry.ΓSpecIso_obj_hom, LinearOrderedAddCommGroupWithTop.neg_pos, ENNReal.le_inv_iff_le_inv, algebraMap_monotone, Wbtw.left_mem_image_Ici_of_right_ne, Fin.sum_Ioo_cast, LSeries.abscissaOfAbsConv_le_of_forall_lt_LSeriesSummable, not_supIrred_bot, RootPairing.setOf_root_add_zsmul_eq_Icc_of_linearIndependent, Finset.card_Ioo_eq_card_Icc_sub_two, CategoryTheory.ComposableArrows.opEquivalence_functor_obj_map, Finset.card_Iio_eq_card_Iic_sub_one, LowerSet.iicsInfHom_apply, le_separableClosure_iff, AddSubmonoid.iSup_map_single_le, Fin.prod_Ioc_castLE, IsLindelof.disjoint_nhdsSet_left, nnnorm_pow_le, Valuation.isEquiv_iff_val_le_one, BoxIntegral.TaggedPrepartition.distortion_le_of_mem, Finset.inf'_Ici, Partition.le_of_mem, NumberField.Units.dirichletUnitTheorem.map_logEmbedding_sup_torsion, CategoryTheory.GrothendieckTopology.Cover.preOneHypercover_sieve₁, Filter.disjoint_map, one_le_cfc_iff, Topology.isLawson_le_isScott, CategoryTheory.SmallObject.SuccStruct.Iteration.obj_bot, TopologicalSpace.Opens.map_comp_map, MeasureTheory.lintegral_liminf_le, Filter.HasBasis.le_basis_iff, Set.Iic.coe_inf, Fin.image_val_Ioc, Ordinal.lt_add_iff, sup_hnot_self, Set.Icc.coe_mul, le_of_sup_eq, Set.exists_min_image, Set.image_neg_Ici, CategoryTheory.Subobject.inf_eq_map_pullback', disjointed_add_one, SetTheory.PGame.grundyValue_le_of_forall_moveLeft, Nucleus.comp_eq_right_iff_le, Set.preimage_const_mul_Ioc, ENNReal.toNNReal_pos, Polynomial.abs_tendsto_atBot_iff, OrderMonoidIso.withZero_apply_symm_apply, FractionalIdeal.adjoinIntegral_coe, Order.height_pos_of_bot_lt, min_def', Metric.ediam_cthickening_le, SSet.skeleton_obj_eq_top, Fin.finsetImage_castSucc_Ico, Filter.IsIncreasingApproximateUnit.eventually_nnnorm, Set.Ioc.coe_top, DivisibleHull.instIsStrictOrderedModuleNNRat, le_nhdsAdjoint_iff', AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.inv_naturality_assoc, MeasureTheory.measure_le_lintegral_pos, Filter.tendsto_const_mul_pow_atTop_iff, ConvexOn.le_on_segment, CliffordAlgebra.one_le_evenOdd_zero, AlgebraicGeometry.Spec.germ_stalkMapIso_hom_assoc, CauchySeq.comp_injective, Fin.finsetImage_addNat_Ioc, Order.pred_lt_iff_eq_or_lt, AlgebraicGeometry.Scheme.IdealSheafData.le_ofIdeals_iff, absConvexHull_empty, PowerSeries.le_order_mul, PrimeSpectrum.iInf_localization_eq_bot, AlgebraicGeometry.Scheme.Hom.preimage_basicOpen_top, InfiniteGalois.isOpen_mulEquivToLimit_image_fixingSubgroup, Subalgebra.mul_toSubmodule, partialSups_mul_const, ENNReal.tendsto_pow_atTop_nhds_top_iff, Finset.gc_map_inr_toRight, iSup_nhds_le_uniformity, BddLat.forget_semilatInf_partOrd_eq_forget_bddOrd_partOrd, Real.toNNReal_le_natCast, Cardinal.small_Icc, IntermediateField.bot_eq_top_of_finrank_adjoin_eq_one, Complex.nhdsWithin_lt_le_nhdsWithin_stolzSet, wellQuasiOrderedLE_iff_wellFoundedLT, MeasureTheory.exists_notMem_null_lintegral_le, Order.isSuccPrelimit_of_succ_ne, MeasureTheory.eLpNorm'_smul_le_mul_eLpNorm', Finset.sup_empty, AlgebraicGeometry.StructureSheaf.IsLocalization.to_basicOpen, Subsemiring.op_le_iff, LinearPMap.HasCore.le_domain, MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq_spec, posSMulStrictMono_iff_PosSMulReflectLE, Algebra.adjoin_nonUnitalSubalgebra_eq_span, Set.pairwise_disjoint_Ico_add_intCast, upperSemicontinuousWithinAt_iff_limsup_le, AlgebraicGeometry.Scheme.Opens.germ_stalkIso_inv_assoc, CategoryTheory.MorphismProperty.strictMap_multiplicativeClosure_le, lowerSemicontinuous_iff_isClosed_epigraph, MeasureTheory.exists_measurable_le_forall_setLIntegral_eq, Set.isAtom_singleton, IsSimpleGroup.instIsSimpleOrderSubgroup, Finset.isCoatom_compl_singleton, Setoid.mk_eq_top, NatOrdinal.not_lt_zero, MeasureTheory.OuterMeasure.comap_top, le_iSup, hasBasis_nhdsSet_Iic_Iic, PNat.find_le_iff, CategoryTheory.instIsCardinalFilteredToTypeOrd, Cardinal.IsStrongLimit.isSuccLimit, Set.preimage_div_const_Ioi, MulEquiv.coe_comapSubgroup, ArchimedeanClass.ball_le_closedBall, Valuation.Integers.not_denselyOrdered_of_isPrincipalIdealRing, NonemptyInterval.coe_ssubset_coe, AlgebraicGeometry.Scheme.Modules.restrict_map, CategoryTheory.Subobject.le_sSup, isCompact_Ioo_iff, TopCat.Sheaf.objSupIsoProdEqLocus_hom_fst, CFC.nnrpow_def, imageToKernel_epi_comp, SetTheory.Game.birthday_ordinalToGame, CategoryTheory.Sieve.overEquiv_top, AlgebraicGeometry.Scheme.PartialMap.restrict_id, Fin.map_castLEEmb_Ioo, Multiset.le_add_sub, ContractingWith.edist_efixedPoint_le', atom_iff_nonzero_span, Submonoid.exists_minimal_closure_eq_top, Submodule.instMulLeftMono, ProbabilityTheory.iIndep_comap_mem_iff, HahnSeries.mem_cardSuppLTAddSubgroup, tendsto_ceil_right_pure_floor_add_one, Ideal.le_pow_ramificationIdx, Profinite.NobelingProof.contained_eq_proj, Rat.preimage_cast_Ico, Set.Ioc_subset_uIoc, IntermediateField.botContinuousSMul, quasiconvexOn_iff_le_max, AlgebraicGeometry.AffineSpace.spec_le_iff, Submodule.finrank_strictMono, ofAntisymmetrization_le_ofAntisymmetrization_iff, PartOrdEmb.ofHom_apply, ENat.ceil_lt, MeasureTheory.measure_le_setAverage_pos, separableClosure_le, absConvexHull_eq_self, Field.exists_primitive_element_of_finite_bot, Set.encard_lt_top_iff, AlgebraicGeometry.Scheme.IsLocallyDirected.homOfLE_tAux, CategoryTheory.GrothendieckTopology.close_eq_top_iff_mem, Subgroup.mem_toAddSubgroup', Valuation.HasExtension.val_map_le_one_iff, maximal_mem_iff, AddCircle.equivIccQuot_comp_mk_eq_toIocMod, MeasureTheory.setLIntegral_mono_ae, TopologicalSpace.Opens.functor_obj_map_obj, smoothSheafCommRing.evalHom_germ, le_iff_forall_pos_lt_add', Subalgebra.unop_bot, isComplemented_bot, IntermediateField.restrictScalars_le_iff, EReal.preimage_coe_Icc, ProjectiveSpectrum.as_ideal_lt_as_ideal, Set.kernImage_mono, Valuation.map_sub, FixedDetMatrices.reps_entries_le_m', Hyperreal.coe_pos, InitialSeg.image_Iio, CFC.negPart_eq_zero_iff, MeasureTheory.Measure.singularPart_le, Positive.val_mul, HahnSeries.support_truncLT, Rat.valuation_le_one_iff_den, AlgebraicGeometry.HasRingHomProperty.iff_appLE, ContinuousLinearMap.nonneg_iff_isPositive, Fin.sum_Icc_cast, Finset.single_le_sum_of_canonicallyOrdered, AffineSubspace.parallel_iff_direction_eq_and_eq_bot_iff_eq_bot, BoundedLatticeHom.coe_toSupBotHom, CategoryTheory.Sieve.arrows_eq_top_iff, infIooOrderIsoIooSup_symm_apply_coe, supIrred_iff_not_isMin, rank_pos_iff_nontrivial, SimpleGraph.edist_pos_of_ne, le_map_add_map_div', Subring.centralizer_le, LowerAdjoint.closure_sup_closure_left, MeasureTheory.Ico_ae_eq_Ioc, FormalMultilinearSeries.changeOrigin_radius, PosSMulReflectLT.toPosSMulMono, FormalMultilinearSeries.le_radius_of_bound, EReal.sign_mul, Hyperreal.archimedeanClassMk_neg_of_tendsto_atTop, ENNReal.ofReal_le_ofNat, MulArchimedeanClass.mk_mul_lt_mk_left_iff, Set.image_const_mul_Iio, monotone_iff_forall_lt', Valuation.mem_valuationSubring_iff, not_isMin_iff_ne_bot, Filter.le_mul_iff, MeasureTheory.ae_measure_preimage_mul_right_lt_top_of_ne_zero, LinearPMap.apply_comp_inclusion, Set.image_mulSingle_Ioo_left, ProperCone.innerDual_singleton, Filter.disjoint_cofinite_left, MeasureTheory.SimpleFunc.memLp_iff, HahnSeries.embDomain_single, Cardinal.lift_lt_univ', SetTheory.Game.le_birthday, Complementeds.codisjoint_coe, AlgebraicGeometry.StructureSheaf.toOpen_comp_comap_assoc, Fin.preimage_rev_Icc, mem_Icc_of_mem_stdSimplex, LinearMap.ker_prod_ker_le_ker_coprod, MeasureTheory.UniformIntegrable.spec', LinearOrderedAddCommGroupWithTop.sub_left_strictMono_of_ne_top, ProperCone.dual_univ, MulPosStrictMono.toMulPosMono, Ordinal.iterate_le_nfp, List.argmax_cons, disjoint_ofDual_iff, Set.IicExtend_coe, Set.image_div_const_Icc, tendsto_ceil_left, IsWellFounded.rank_eq_typein, Fin.image_succ_Ici, SSet.RelativeMorphism.le_preimage, Int.cast_lt, PSet.rank_sUnion_le, CategoryTheory.Subobject.mapIsoToOrderIso_apply, inv_le_leOnePart, Set.monotone_accumulate, IsAtomic.Set.Iic.isAtomic, continuousWithinAt_iff_continuous_left'_right', ArchimedeanClass.mk_natCast_nonneg, OrderHom.isLeast_lfp, Submodule.mk_mem_projectivization_iff, ENNReal.coe_le_coe, tendsto_nhds_iff_meromorphicOrderAt_nonneg, AlgebraicGeometry.IsAffineOpen.self_le_basicOpen_union_iff, Ordinal.isSuccPrelimit_zero, isGreatest_himp, Finset.orderEmbOfFin_zero, AlgebraicGeometry.exists_appTop_π_eq_of_isAffine_of_isLimit, ENNReal.zero_lt_log_iff, AddSubmonoid.toSubmonoid_closure, Dynamics.coverEntropyInf_image_le_of_uniformContinuous, strictMono_int_of_lt_succ, separableClosure.map_le_of_algHom, CStarAlgebra.isStrictlyPositive_iff_isStrictlyPositive_sqrt_and_eq_sqrt_mul_sqrt, Ordinal.CNF_sorted, Module.jacobson_pi_le, monotone_iff_map_nonneg, Set.Icc_inter_Icc, WithZero.le_max_iff, Set.encard_le_card, Set.inv_Ici, MeasureTheory.eLpNorm'_le_eLpNorm'_mul_rpow_measure_univ, RingEquiv.mapTwoSidedIdeal_symm, OrderIso.withTopCongr_symm, SimpleGraph.hasse_preconnected_of_succ, segment_eq_Icc', Convex.mem_extremePoints_iff_mem_diff_convexHull_diff, AddSubmonoid.addUnits_mono, ConvexCone.coe_strictlyPositive, Ordinal.IsNormal.apply_lt_nfp, CategoryTheory.Sieve.arrows_top, latticeClosure_prod, TopologicalSpace.Opens.map_comp_obj', PointedCone.dual_insert, isPreconnected_Icc, CategoryTheory.SmallObject.succStruct_prop_le_propArrow, ringKrullDim_le_iff_height_le, EMetric.nhds_basis_eball, Submodule.jacobson_smul_lt_top, Order.PartialIso.exists_across, affineSpan_le, ProbabilityTheory.Kernel.partialTraj_le_def, bddOrd_dual_comp_forget_to_partOrd, LinearPMap.sSup_apply, Submodule.map_strictMono_of_injective, HasFPowerSeriesAt.radius_pos, Nat.cast_lt_one, NonarchAddGroupSeminorm.lt_def, ContinuousOn.image_uIcc_eq_Icc, ProfiniteGrp.cone_pt, CategoryTheory.Classifier.χ_pullback_obj_mk_truth_arrow, Ideal.inertia_le_stabilizer, Frm.Iso.mk_inv, isNoetherian_iff_fg_wellFounded, NNRat.nonpos_iff_eq_zero, Set.image_mulSingle_Icc, TopologicalSpace.Opens.mapMapIso_functor, leOnePart_eq_ite_lt, Filter.tendsto_id', Ioc_subset_closure_interior, Cardinal.univ_pos, CategoryTheory.Limits.ColimitPresentation.self_diag, LipschitzOnWith.ediam_image2_le, AddSubmonoid.centralizer_le, Valuation.Integers.isIntegral_iff_v_le_one, CategoryTheory.Subobject.factorThru_arrow_assoc, QuotientAddGroup.rightRel_eq_top, disjointed_le_id, Set.sep_infs_le, Ordinal.le_nfp, LieAlgebra.IsSemisimple.sSup_atoms_eq_top, Submonoid.pi_le_iff, ENNReal.finite_const_le_of_tsum_ne_top, partialSups_succ, AlgebraicGeometry.Scheme.OpenCover.iSup_opensRange, IntermediateField.normal_iff_forall_fieldRange_le, Set.monotoneOn_or_antitoneOn_iff_uIcc, sbtw_iff_right_ne_and_left_mem_image_Ioi, Subring.pointwise_smul_subset_iff, ArchimedeanClass.mk_right_le_mk_sub_iff, one_half_lt_one, isCompl_iff, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.image_preimage_is_empty, Matroid.ofMatroid_le_iff, le_hasProd', Ordinal.omega0_le, Ordinal.bsup_eq_blsub_iff_succ, CategoryTheory.Limits.kernelSubobjectIso_comp_kernel_map, Set.preimage_const_add_Iic, CategoryTheory.ComposableArrows.Exact.opcyclesIsoCycles_hom_fac_assoc, Algebra.Generators.sq_ker_comp_le_ker_compLocalizationAwayAlgHom, Finsupp.mapDomain_mono, MeasureTheory.Measure.pi_Ico_ae_eq_pi_Icc, Set.uIcc_subset_uIcc_iff_le', Interval.add_eq_zero_iff, Set.sep_sups_le, Monotone.Ico, MeasureTheory.SimpleFunc.edist_approxOn_y0_le, AlgebraicGeometry.HasAffineProperty.restrict, AddSubsemigroup.op_le_op_iff, HomogeneousIdeal.irrelevant_le, MeasureTheory.memLp_const_iff_enorm, Asymptotics.IsLittleOTVS.exists_eventuallyLE_mul_ennreal, one_div_nonneg, upperSemicontinuous_iff_frequently, AffineSubspace.mem_top, tendsto_pow_div_pow_atTop_zero, Multiset.max_sum_le, IsCauSeq.bounded, NonemptyInterval.coe_subset_coe, upperSemicontinuousOn_iff_limsup_le, Set.not_disjoint_iff_nonempty_inter, Booleanisation.lift_le_lift, Subalgebra.gc_map_comap, AddSubmonoid.coe_negOrderIso_symm_apply, Valuation.Integers.isPrincipalIdealRing_iff_not_denselyOrdered, isOpen_Iio, InfClosed.infClosure_eq, Rat.toNNRat_lt_toNNRat_iff_of_nonneg, CategoryTheory.ComposableArrows.δ₀Functor_obj_map, Algebra.bijective_algebraMap_iff, nhds_basis_Ioo_pos, nullMeasurableSet_le, Submodule.isCoatom_comap_iff, NNRat.preimage_cast_Iic, Finset.add_sum_Iio_eq_sum_Iic, Finsupp.le_degree, SetTheory.PGame.moveRight_nim, BoundedOrder.instSubsingleton, Subalgebra.topEquiv_apply, BooleanAlgebra.inf_compl_le_bot, FormalMultilinearSeries.compPartialSumTarget_tendsto_atTop, ArchimedeanClass.FiniteResidueField.ofArchimedean_injective, EReal.exp_lt_exp_iff, Submodule.FG.lTensor.directLimit_apply', iInf_le_iff, MeasureTheory.integral_pos_iff_support_of_nonneg_ae, QuasispectrumRestricts.le_nnreal_iff, AddSubmonoid.gc_map_comap, ENNReal.hasBasis_nhds_of_ne_top', ProbabilityTheory.Kernel.HasSubgaussianMGF.add, Finset.exists_max_image, Digraph.toSimpleGraphStrict_mono, Set.Iic.complementedLattice_iff, lt_iInf_iff, hnot_top, lt_tsub_comm, Finsupp.card_Icc, isCoatomic_iff_forall_isCoatomic_Ici, MeasureTheory.exists_mem_forall_mem_nhdsWithin_pos_measure, WithLp.prod_nnnorm_eq_of_L2, inf_le_sup, inf_left_le_sup_right, Set.Finite.encard_lt_encard, EReal.sign_coe, MeasureTheory.levyProkhorovEDist_lt_top, LipschitzWith.edist_iterate_succ_le_geometric, AddCircle.continuous_equivIoc_symm, UpperSet.coe_iciSupHom, Field.exists_primitive_element_of_finite_top, CategoryTheory.Limits.pullback_factors_iff, max_def', ENNReal.orderIsoIicCoe_symm_apply_coe, Ordinal.toNatOrdinal_one, OrderDual.instSMulPosReflectLE, natCast_le_analyticOrderAt, CategoryTheory.Subobject.arrow_mono, CategoryTheory.Subobject.isIso_top_arrow, Cardinal.IsInaccessible.aleph0_lt, SSet.stdSimplex.face_obj, MeasureTheory.Measure.countable_meas_level_set_pos, Set.Icc.coe_nonneg, Affine.Simplex.sign_excenterWeights_singleton_neg, AnalyticAt.eventually_constant_or_nhds_le_map_nhds, MeasureTheory.OuterMeasure.trim_anti_measurableSpace, Ordinal.sup_le_lsub, AddSubmonoid.exists_minimal_closure_eq_top, Subsemigroup.toAddSubsemigroup_closure, NonemptyFinLinOrd.Iso.mk_inv, BoundedLatticeHomClass.toBoundedOrderHomClass, SSet.stdSimplex.nonDegenerateEquiv_apply_apply, Rat.cast_lt_intCast, bot_lt_iff_ne_bot, Dynamics.coverMincard_antitone, Valuation.RankOne.strictMono, OrderMonoidHom.inrₗ_apply, Fin.sum_Ici_cast, Subgroup.subgroupOf_eq_bot, Fin.map_revPerm_Icc, Finset.tendsto_Icc_neg_atTop_atTop, ConcaveOn.quasiconcaveOn, OrderIso.map_iInf, FirstOrder.Language.Substructure.map_comap_le, tendsto_inv_atBot_nhdsLT_zero, ENNReal.liminf_mul_le, OmegaCompletePartialOrder.ωScottContinuous.map_ωSup, Nat.floor_lt', EReal.le_liminf_add, IsOpenMap.nhds_le, FinPartOrd.dualEquiv_counitIso, Valuation.Integers.le_iff_dvd, IntermediateField.adjoin_simple_eq_top_iff_of_isAlgebraic, continuous_top, SimpleGraph.IsUniform.pos, MulAction.IsPreprimitive.isCoatom_stabilizer_of_isPreprimitive, nhdsSet_sInter_le, himp_le, segment_eq_image, Finset.none_mem_insertNone, ENat.ceil_pos, AlgebraicGeometry.Scheme.Γevaluation_naturality_assoc, CategoryTheory.MorphismProperty.TransfiniteCompositionOfShape.instIsIsoAppIncl, Filter.neg_nhdsGT, AlgebraicGeometry.Scheme.kerAdjunction_counit_app, UpperSet.infIrred_Ici, AlgebraicGeometry.opensDiagramι_app, instPosSMulStrictMono, Fin.sum_Iic_castLE, lebesgue_number_lemma_of_emetric_sUnion, Nat.cast_nonpos, IsCompact.disjoint_nhdsSet_left, essInf_const_top, inv_pos, NonUnitalStarAlgebra.gc, StrictConcaveOn.convex_gt, EisensteinSeries.hasSum_e2Summand_symmetricIco, UniformConvergenceCLM.uniformSpace_mono, CategoryTheory.ComposableArrows.fourδ₄Toδ₃_app_one, Finset.image_some_eraseNone, posPart_nonpos, MeasureTheory.tsum_measure_le_measure_univ, four_mul_le_sq_add, TopologicalSpace.Opens.mapIso_inv_app, BooleanSubalgebra.closure_mono, le_sup_sdiff, Set.Icc_diff_pi_univ_Ioc_subset, MonoidHom.coe_toAdditive_map, MeasureTheory.hitting_le, Multiset.add_le_add_iff_left, Seminorm.bddAbove_iff, Dynamics.netMaxcard_antitone, LaurentSeries.exists_Polynomial_intValuation_lt, Cardinal.beth_le_beth, HahnSeries.orderTop_of_ne, AlgebraicGeometry.Scheme.IdealSheafData.coe_support_ofIdealTop, uniformity_basis_edist_inv_nat, Ordinal.IsNormal.le_iff, CategoryTheory.Presieve.ofArrows_comp_le, HahnSeries.coeff_truncLT, LinearOrderedAddCommGroupWithTop.top_pos, AddSubgroup.map_subtype_lt_map_subtype, Finsupp.rangeIcc_apply, Hyperreal.infiniteNeg_iff_infinite_and_neg, Set.Ici.coe_sup, MonomialOrder.degree_sum_le, SSet.instHasDimensionLTToSSetBotSubcomplex, sup_eq_left, SimpleGraph.vertexCoverNum_le_encard_edgeSet, tendsto_ceil_atBot, zsmul_mono_right, CategoryTheory.Limits.CompleteLattice.hasLimits_of_completeLattice, Fin.image_succ_Ioc, AlgebraicGeometry.Scheme.IdealSheafData.support_antitone, nnnorm_mul_le', Sublattice.le_comap_iSup, Filter.tendsto_finset_prod_atTop, Order.krullDim_nonpos_of_subsingleton, Matroid.IsRkFinite.eRk_lt_top, MeasureTheory.SimpleFunc.edist_approxOn_mono, himp_eq_top_iff, isGLB_pair, Ioc_diff_botSet, IsUnifLocDoublingMeasure.measure_mul_le_scalingConstantOf_mul, CategoryTheory.MorphismProperty.bot_mem_precoverage, round_le, LinearOrderedCommGroup.wellFoundedOn_setOf_ge_gt_iff_nonempty_discrete, Matroid.Indep.encard_le_eRk_of_subset, Fin.preimage_castAdd_Ioo_castAdd, AddSubgroup.pi_le_iff, LinearOrderedField.inducedOrderRingHom_toFun, IsClosedMap.comap_nhds_le, Ordinal.mod_lt, IsCoatomistic.isGLB_coatoms, ValuationSubring.valuation_le_one_iff, PrincipalSeg.ordinal_type_lt, OrderIso.isAtom_iff, LieAlgebra.IsSimple.isAtom_top, AlgebraicGeometry.Scheme.component_nontrivial, Field.Emb.Cardinal.strictMono_leastExt, IntermediateField.rank_adjoin_eq_one_iff, CategoryTheory.Sieve.le_pullback_bind, Fin.prod_Ioc_castSucc, TopCat.Presheaf.presheafEquivOfIso_unitIso_hom_app_app, NNRat.natCast_le_cast, ArchimedeanClass.mk_zero, le_induced_generateFrom, CategoryTheory.MorphismProperty.isoClosure_le_iff, Finset.Ioc_inter_Ioc, ENNReal.logOrderIso_symm, iSupIndep_def'', ClosureOperator.le_closure, CategoryTheory.TransfiniteCompositionOfShape.map_isoBot, Antitone.Iio, Int.injOn_natAbs_Iic, Set.Ioc_diff_Iic, MeasureTheory.restrict_Ioo_eq_restrict_Icc, OrderedSMul.toPosSMulStrictMono, nnnorm_pos', List.sortedLT_iff_nodup_and_sortedLE, Finset.mul_prod_eq_prod_insertNone, Submodule.FG.rTensor.directLimit_apply, Matroid.eRank_le_encard_ground, MeasureTheory.Measure.restrict_apply_le, PrimeSpectrum.subset_zeroLocus_iff_le_vanishingIdeal, AlgebraicGeometry.Scheme.zeroLocus_iInf_of_nonempty, Odd.pos, ThreeGPFree.le_mulRothNumber, IsLocalExtrOn.not_nhds_le_map, Set.exists_max_image, NonUnitalSubring.closure_mono, LatticeHom.withTop'_toFun, CategoryTheory.Subobject.hasLimitsOfShape, Set.disjoint_insert_left, DFinsupp.Colex.le_iff_of_unique, TopologicalSpace.Opens.coe_overEquivalence_functor_obj, Ordinal.not_bddAbove_fp, SSet.Truncated.spine_map_vertex, partialSups_disjointed, LinearMap.BilinForm.inf_orthogonal_self_le_ker_restrict, MeasureTheory.JordanDecomposition.exists_compl_positive_negative, Filter.inv_nhdsLT, AlgebraicGeometry.Scheme.AffineZariskiSite.generate_presieveOfSections_mem_grothendieckTopology, Ideal.pi_le_pi_iff, Padic.AddValuation.map_add, Nat.preimage_ceil_of_ne_zero, PosMulReflectLT.toPosMulMono, Filter.map_add_atTop_eq, AlgebraicGeometry.ΓSpec.toOpen_comp_locallyRingedSpaceAdjunction_homEquiv_app, Ordinal.card_iSup_Iio_le_card_mul_iSup, Ordinal.zero_or_succ_or_isSuccLimit, AddSubgroup.le_normalizer, AlgebraicGeometry.PresheafedSpace.GlueData.opensImagePreimageMap_app, compl_lt_compl_iff_lt, rank_submodule_le_one_iff', AddGroupTopology.toTopologicalSpace_top, EReal.exp_monotone, top_inf_eq, sSup_diff_singleton_bot, PNat.map_subtype_embedding_Icc, MeasureTheory.integrable_iff_integrableAtFilter_atBot_atTop, HahnSeries.orderTop_le_of_coeff_ne_zero, Finset.le_inf'_image₂, Finset.Ioi_sub_one_eq_Ici, Set.unbounded_gt_inter_not_gt, PrimeSpectrum.vanishingIdeal_anti_mono, Ordinal.nfp_le_iff, SimplexCategory.id_toOrderHom, Order.exists_orderEmbedding_insert, Finset.tendsto_Ico_atBot_prod_atTop, AlgebraicGeometry.Scheme.basicOpen_pow, sdiff_sdiff_sdiff_le_sdiff, MeasureTheory.Measure.bind_apply_le, Submodule.ker_liftQ_eq_bot', NonUnitalSubsemiring.le_topologicalClosure, MeasureTheory.extend_mono, Hyperreal.lt_of_tendsto_atTop, PartOrdEmb.Limits.cocone_ι_app, NNRat.preimage_cast_Ioi, ENNReal.add_lt_add_iff_left, Rat.natCast_le_cast, IsUltrametricDist.nnnorm_mul_le_max, isCompact_Ioc_iff, Finset.map_add_left_Ico, Module.rank_pos_of_free, Submodule.orderIsoMapComap_apply', HahnSeries.embDomain_coeff, forall_ge_imp_ne_iff_lt, Finset.Iic_bot, AlgebraicGeometry.Scheme.Opens.toSpecΓ_top, MeasureTheory.LpAddConst_lt_top, fact_one_le_top_ennreal, FractionalIdeal.inv_le_dual, LipschitzOnWith.dimH_image_le, Finset.nonneg_addConst, upperHemicontinuousAt_iff_preimage_Iic, ConcaveOn.antitoneOn_slope_lt, Set.exists_lower_bound_image, SSet.Subcomplex.PairingCore.RankFunction.lt, lowerSemicontinuousOn_iff_isClosed_epigraph, Finsupp.instSMulPosReflectLT, ENat.ceil_add_le, IsLindelof.disjoint_nhdsSet_right, AlgebraicGeometry.Scheme.Hom.ker_apply, TopologicalSpace.Closeds.coe_bot, NNReal.agmSequences_fst_lt_snd_of_ne, Order.PFilter.antitone_principal, OrderType.type_eq_type, FractionalIdeal.isFractional_adjoin_integral, forall_gt_imp_ge_iff_le_of_dense, monotoneOn_of_le_add_one, SimpleGraph.vertexCoverNum_lt_card, Cardinal.ord_aleph, Set.einfsep_lt_iff, MeasureTheory.le_liminf_measure_open_of_forall_tendsto_measure, IntermediateField.adjoin_one, CircleDeg1Lift.iterate_monotone, IntermediateField.rank_top, ENat.lt_ceil, CategoryTheory.Subfunctor.le_sheafify, CategoryTheory.Subobject.factors_comp_arrow, StrictConvexOn.lt_on_openSegment, CompleteDistribLattice.top_sdiff, PrincipalSeg.strictMono, directedOn_of_sup_mem, MeasureTheory.Measure.eventually_cofinite, CategoryTheory.sheafBotEquivalence_inverse_map_val, Ideal.exists_radical_pow_le_of_fg, EReal.coe_le_coe, AlgebraicGeometry.Scheme.IdealSheafData.ideal_comap_of_isOpenImmersion, SimpleGraph.top_le_span_range_lapMatrix_ker_basis_aux, List.SortedGT.strictAnti_get, Ordinal.small_Iic, concaveOn_iff_div, lt_add_iff_lt_left_or_exists_lt, FormalMultilinearSeries.nnnorm_changeOriginSeries_apply_le_tsum, Surreal.Multiplication.mulOption_lt, Filter.map_add_left_nhdsLT, MeasureTheory.measureUnivNNReal_pos, Dense.exists_ge', CochainComplex.mappingConeCompTriangle_mor₃_naturality, grade_le_grade_iff, Tropical.add_eq_iff, OrdinalApprox.lfpApprox_ord_eq_lfp, enorm_le_iff_norm_le, NonarchAddGroupSeminorm.coe_lt_coe, ConvexCone.strictlyPositive_le_positive, AlgebraicGeometry.ProjIsoSpecTopComponent.fromSpec_toSpec, HasOuterApproxClosed.measure_le_lintegral, List.sortedGT_iff_strictAnti_get, Fin.prod_Ioo_cast, CategoryTheory.Sieve.generate_of_singleton_isSplitEpi, classifyingSpaceUniversalCover_map, AlgebraicGeometry.AffineSpace.SpecIso_inv_appTop_coord, CategoryTheory.Sieve.functorPullback_top, DFinsupp.toMultiset_le_toMultiset, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.app_inv_app'_assoc, Order.Ideal.isPrime_iff_mem_or_compl_mem, lowerClosure_interior_subset, Finsupp.card_Ioo, QuotientAddGroup.ker_le_range_iff, Set.image_const_add_Ioo, max_eq_top, Set.Iic_diff_Ioc, ProjectiveSpectrum.gc_set, Matroid.eRank_lt_top_iff, Pi.covBy_iff, Finset.truncatedSup_singleton, ClosedSubmodule.coe_bot, instIsUpperProp, MeasureTheory.measure_lt_top_of_isCompact_of_isMulLeftInvariant, Valuation.map_add_le_max', AlgebraicGeometry.germ_stalkClosedPointIso_hom_assoc, Subgroup.map_lt_map_iff_of_injective, Metric.mem_closedEBall', IsLeftCancelAdd.addLeftStrictMono_of_addLeftMono, CategoryTheory.Limits.imageSubobjectIso_comp_image_map, minimal_iff_isLeast, add_abs_nonneg, inv_nhdsGT_zero, Sublattice.prodEquiv_toEquiv, Rat.cast_lt, Set.inv_Icc, indicator_le_thickenedIndicator, Finset.Ioc_sub_one_right_eq_Ioo, Finset.Icc_succ_pred_eq_Ioo, ENNReal.mul_pos, Hyperreal.archimedeanClassMk_pos_of_tendsto, MvPolynomial.degrees_pow_le, EReal.mul_ne_top, MeasureTheory.VectorMeasure.le_iff', Finsupp.DegLex.single_antitone, AlgebraicGeometry.Scheme.IdealSheafData.support_eq_top_iff, LipschitzWith.edist_le_mul, WithTop.pred_eq_top, Filter.nhds_eq, Finsupp.Lex.single_lt_iff, Pi.card_Iio, convexHull_add_subset, Nimber.add_nat, RootPairing.posRootForm_posForm_pos_of_ne_zero, Filter.curry_le_prod, Fin.finsetImage_rev_Iic, TopologicalSpace.Opens.coe_finset_sup, tendsto_abs_nhdsNE_zero, NonUnitalAlgebra.top_toNonUnitalSubsemiring, Finset.powersetCard_map, PresentedGroup.closure_rels_subset_ker, MeasureTheory.eLpNorm_le_add_measure_left, monovary_iff_forall_smul_nonneg, AlgebraicGeometry.Scheme.AffineZariskiSite.restrictIsoSpec_hom_app, TopologicalSpace.OpenNhds.id_apply, Set.preimage_mul_const_Iic, PartENat.toWithTop_le, Subgroup.normal_le_normalCore, exists_isExtrOn_uIoo_of_tendsto, Finite.le_ciSup, TopologicalSpace.Opens.instIsSimpleOrderOfNonemptyOfIndiscreteTopology, Affine.Simplex.ExcenterExists.sign_signedInfDist_lineMap_excenter_touchpoint, Int.Ioc_filter_modEq_card, AlgebraicGeometry.PresheafedSpace.ColimitCoconeIsColimit.desc_c_naturality, instFiniteObjOpenNormalSubgroupStabilizerHomSurjectiveAuxFunctor, Topology.scottHausdorff_le_isLawson, mem_integralClosure_iff_mem_fg, IsInducing.topologicalKrullDim_le, MeasureTheory.IsSetSemiring.diff_mem_supClosure, Subrepresentation.submoduleSubrepresentationOrderIso_apply, SimpleGraph.nonuniformWitnesses_spec, MeasureTheory.Ioo_ae_eq_Ico, CategoryTheory.Subobject.factors_self, AlgebraicGeometry.instIsIsoSchemeMorphismRestrict, exists_int_lt, LowerSet.Iio_eq_bot, Polynomial.isBigO_atBot_of_degree_le, Filter.gc_comap_kernMap, Profinite.instEpiAppDiscreteQuotientCarrierToTopTotallyDisconnectedSpaceπAsLimitCone, Quotient.subsingleton_iff, SpectrumRestricts.nnreal_le_iff, Ordinal.nfp_le_apply, CategoryTheory.TransfiniteCompositionOfShape.fac, Ordinal.wellFoundedLT, addRothNumber_spec, Finset.prod_add_ordered, Ordinal.veblenWith_lt_veblenWith_iff, CategoryTheory.Limits.imageSubobject_zero_arrow, lt_abs, FirstOrder.Language.DefinableSet.le_iff, ENNReal.exists_nat_pos_inv_mul_lt, Set.not_monotoneOn_not_antitoneOn_iff_exists_le_le, Finset.dens_strictMono, DirichletCharacter.zetaMul_nonneg, Interval.length_sum_le, Setoid.bot_def, eventually_nhdsWithin_sign_eq_of_deriv_neg, tendsto_indicator_biUnion_finset, MeasureTheory.levyProkhorovEDist_triangle, ProbabilityTheory.IsAEKolmogorovProcess.kolmogorovCondition, CategoryTheory.Pairwise.diagram_map, nonzero_span_atom, OrderRingIso.subsingleton_left, LieIdeal.isLieAbelian_iff, smul_top_inf_eq_smul_of_isSMulRegular_on_quot, exists_abs_lt, CategoryTheory.Subfunctor.Subpresheaf.bot_obj, nhdsSetWithin_mono_right, LinearOrderedAddCommGroupWithTop.strictMono_add_left_of_ne_top, Multiset.monotone_filter_right, Subfield.bot_eq_of_zMod_algebra, ENNReal.log_monotone, MeasureTheory.Measure.count_apply_lt_top, t1Space_antitone, Sylow.exists_subgroup_card_pow_prime_le, borel_eq_generateFrom_Ioc, ConvexCone.disjoint_hull_left_of_convex, EReal.liminf_add_le, NonUnitalSubalgebra.starClosure_mono, Set.Ico_add_bij, List.SortedLE.monotone_get, Valued.isClosed_ball, NNReal.bot_eq_zero, addLeftMono_of_addLeftStrictMono, Matroid.eRk_insert_inter_add_eRk_insert_union_le, Set.sub_mem_Ioc_zero_iff_right, PowerBasis.adjoin_gen_eq_top, hasSumLocallyUniformlyOn_iff_tendstoLocallyUniformlyOn, Complex.arg_eq_zero_iff_zero_le, MeasureTheory.IntegrableOn.setLIntegral_lt_top, Filter.not_disjoint_self_iff, OmegaCompletePartialOrder.ωScottContinuous_iff_map_ωSup_of_orderHom, AlgebraicGeometry.Scheme.evaluation_eq_zero_iff_notMem_basicOpen, CategoryTheory.Subfunctor.equivalenceMonoOver_inverse_obj, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctor_obj_pt, ENNReal.mulRightOrderIso_apply, covBy_iff_le_iff_lt_left, WithZero.exists_ne_zero_and_le_and_le, EReal.abs_coe_lt_top, Tuple.self_comp_sort, BoundedLatticeHom.map_top', FormalMultilinearSeries.inv_le_ofScalars_radius_of_tendsto, isSimpleRing_iff, MeasureTheory.hittingAfter_eq_top_iff, OrderMonoidHom.inlₗ_mul_inrₗ_eq_toLex, SemigroupIdeal.closure_le, Nat.cast_nonneg', NNReal.sqrt_mul, MeasureTheory.Content.outerMeasure_le, CategoryTheory.Subgroupoid.IsNormal.generatedNormal_le, StieltjesFunction.measurableSet_Ioi, mulRightStrictMono_of_mulRightReflectLE, Submodule.toIntSubmodule_toAddSubgroup, Fin.sum_Ioc_castLE, Submodule.spanFinrank_span_le_encard, AlgebraicGeometry.Scheme.stalkMap_germ_assoc, Multiset.replicate_le_replicate, AddSubgroup.pointwise_smul_le_iff₀, Filter.comap_mono, antitoneOn_of_le_sub_one, Fin.sum_Iio_castLE, Submonoid.map_comap_le, Set.Ico.nonneg, AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_inv_eq_germ_apply, Fin.preimage_val_Iic_val, SemilatSupCat.dual_map, Ordinal.card_lt_nat, MeasureTheory.eLpNorm_le_mul_eLpNorm_of_ae_le_mul, Subgroup.map_subtype_le_map_subtype, Set.bdd_below_bdd_above_iff_subset_uIcc, TopCat.Presheaf.SheafConditionEqualizerProducts.piInters.hom_ext_iff, Filter.disjoint_of_disjoint_of_mem, IsAbsoluteValue.abv_add', StarSubalgebra.eq_top_iff, disjoint_map_orderIso_iff, Finset.Ioo_succ_right_eq_Ioc, AddQuantale.leftAddResiduation_le_iff_add_le, List.sortedLE_insertionSort, Set.Icc.coe_pow, ringKrullDim_le_ringKrullDim_quotient_add_card, SSet.mem_skeleton_obj_iff_of_nonDegenerate, Submodule.toAddSubmonoid_mono, Fin.preimage_castAdd_Ioi_castAdd, IsCompact.bddAbove, Set.bounded_ge_inter_ge, NonUnitalSubsemiring.toAddSubmonoid_mono, Even.pow_pos, TwoSidedIdeal.asIdeal_jacobson, le_abs, CategoryTheory.Limits.imageSubobject_zero, MeasureTheory.integral_Icc_eq_integral_Ioo', sameRay_smul_left_iff_of_ne, Finset.filter_infs_le, MeasureTheory.Measure.toSignedMeasure_toMeasureOfZeroLE, MeasureTheory.Ioc_ae_eq_Icc, DeltaGeneratedSpace.le_deltaGenerated, infClosure_eq_self, Submodule.le_topologicalClosure, Nonneg.instNonUnitalIsometricContinuousFunctionalCalculus, Subgroup.mk_le_mk, UpperSet.Ioi_eq_top, PartENat.one_lt_top, IicProdIoc_def, CategoryTheory.MorphismProperty.coproducts_le_iff, le_sInf_inter, closedConvexHull_closure_eq_closedConvexHull, IntermediateField.adjoin_root_eq_top, PNat.ofNat_le_ofNat, Finset.orderEmbOfFin_last, AlgebraicGeometry.SheafedSpace.ofRestrict_hom_c_app, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.inv_invApp, strictConvex_Iio, AddEquiv.mapAddSubgroup_symm_apply, LinearGrowth.linearGrowthSup_le_iff, star_lt_one_iff, IsArtinian.monotone_stabilizes, ENNReal.sum_lt_top, codisjoint_iff_le_sup, contravariant_le_of_contravariant_eq_and_lt, SimpleGraph.ConnectedComponent.maximal_connected_toSubgraph, Set.lt_eq_ssubset, finGaloisGroupMap.map_id, List.exists_map_eq_of_sorted_nonempty_iff_wbtw, FiniteMulArchimedeanClass.withTopOrderIso_symm_apply, Order.pred_bot, Finset.sum_Icc_eq_sum_Ico_add, Subgroup.coe_toAddSubgroup_symm_apply, FiniteArchimedeanClass.mem_closedBallAddSubgroup_iff, AffineSubspace.smul_top, ConvexCone.toPointedCone_top, TopCat.Presheaf.whiskerIsoMapGenerateCocone_inv_hom, OmegaCompletePartialOrder.ContinuousHom.coe_toOrderHom, IncidenceAlgebra.sum_Icc_mu_left, Submodule.orthogonal_gc, Nonneg.val_inv_unitsHomeomorphPos_symm_apply_coe, Module.lt_rank_of_lt_finrank, csInf_Ioi, Real.ofNat_lt_toNNReal, MeasureTheory.addHaar_image_le_lintegral_abs_det_fderiv_aux2, not_bddAbove_iff_isCofinal, Order.succ_eq_iff_isMax, Filter.map_atTop_finset_sum_le_of_sum_eq, Fin.finsetImage_val_Ioc, wcovBy_iff_covBy_or_eq, Int.card_Ico, ContDiffMapSupportedIn.structureMapLM_apply_withOrder, Set.Icc_diff_pi_univ_Ioo_subset, Subring.prod_mono_left, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.inv_naturality_assoc, PseudoMetric.IsUltra.isTransitiveRel_ball, Set.nsmul_right_monotone, Finset.Ico_erase_left, Ideal.mul_le, Submonoid.map_le_map_iff_of_injective, Filter.tendsto_finset_powerset_atTop_atTop, OrderIsoClass.toTopHomClass, Matroid.Indep.encard_le_eRank, Filter.comap_coLindelof_le, IsCompact.lt_sInf_iff_of_continuous, MeasurableSpace.measurableSet_bot_iff, AlgebraicGeometry.Scheme.Hom.appIso_inv_app, Submodule.mem_colon_iff_le, Valuation.srel_iff_lt, abs_mul_sign, RCLike.neg_iff, AlgebraicGeometry.Scheme.Hom.liftCoborder_preimage, InfiniteGalois.proj_adjoin_singleton_val, Subfield.map_bot, MeasureTheory.Measure.pi_prod_map_IicProdIoc, FractionalIdeal.le_inv_comm, Order.pred_lt_top, CategoryTheory.PreZeroHypercover.presieve₀_restrictIndex_le, TopCat.Presheaf.stalk_mono_of_mono, ZFSet.mem_vonNeumann', CategoryTheory.ComposableArrows.Exact.cokerIsoKer'_inv_hom_id, Int.card_Ioc_of_le, Set.iUnion_Ico_eq_Iio_self_iff, MeasureTheory.Filtration.rightCont_apply, LinearOrderedAddCommGroup.discrete_iff_not_denselyOrdered, isAtomistic_dual_iff_isCoatomistic, Filter.NeBot.zero_mul_nonneg, NNReal.rpow_arith_mean_le_arith_mean_rpow, Ordinal.derivFamily_strictMono, Continuous.cfc_nnreal_of_mem_nhdsSet, iInf_ne_top_subtype, CategoryTheory.subobject_simple_iff_isAtom, CategoryTheory.GrothendieckTopology.discrete_eq_top, TwoSidedIdeal.mem_fromIdeal, Finset.Ioc_succ_succ, meromorphicNFAt_iff_analyticAt_or, IsLocalization.ringEquivOfRingEquiv_eq_map, CategoryTheory.Limits.kernelSubobjectIsoComp_inv_arrow, Subgroup.prod_mono, TopCat.Presheaf.map_germ_eq_Γgerm_assoc, AlgebraicGeometry.Scheme.Opens.germ_stalkIso_hom, Complex.sq_nonneg_iff, Fin.map_revPerm_Ioi, ENat.natCast_lt_succ, upperSemicontinuousOn_iff_preimage_Iio, ContinuousMap.HomotopyWith.trans_apply, TopologicalSpace.Opens.op_map_id_obj, compl_image_latticeClosure_eq_of_compl_image_eq_self, NNReal.le_add_nndist, le_sup_right, covariantClass_le_of_lt, PartOrd.coe_comp, Module.length_submodule, FiniteArchimedeanClass.mem_ball_iff, Submodule.comm_trans_rTensorOne, Multiset.ndinter_le_left, AlgebraicGeometry.Scheme.Hom.opensRange_pullbackFst, NonUnitalStarAlgebra.top_toNonUnitalSubalgebra, partialSups_eq_sUnion_image, CategoryTheory.SimplicialObject.δ_comp_σ_of_gt'_assoc, NonemptyInterval.coe_one, not_isCofinal_iff_bddAbove, MeasureTheory.lintegral_indicator_const_le, Int.image_Ico_emod, Fin.Iio_last_eq_map, Polynomial.rootSet_derivative_subset_convexHull_rootSet, CoheytingHom.map_top', Ordinal.blsub_le_of_brange_subset, PNat.equivNonZeroDivisorsNat_symm_apply_coe, ProbabilityTheory.IsKolmogorovProcess.kolmogorovCondition, MeasureTheory.OuterMeasure.zero_caratheodory, RingPreordering.toSubsemiring_le_toSubsemiring, AddSubgroup.closure_add_le, ProbabilityTheory.inter_pos_of_cond_ne_zero, OrderAddMonoidHom.fstₗ_comp_inlₗ, Fin.map_addNatEmb_Ico, CategoryTheory.Subobject.map_bot, AlgebraicGeometry.Scheme.zeroLocus_radical, grade_eq_grade_iff, WithZero.exists_ne_zero_and_lt_and_lt, Filter.map_generate_le_generate_preimage_preimage, enorm_sub_le_lintegral_derivWithin_Icc_of_contDiffOn_Icc, CategoryTheory.TransfiniteCompositionOfShape.map_isColimit, Finset.fold_sup_univ, TopCat.Sheaf.interUnionPullbackConeLift_right, ZFSet.vonNeumann_mem_vonNeumann_iff, ComplementedLattice.isStronglyCoatomic', PowerSeries.one_le_order_iff_constCoeff_eq_zero, MeasureTheory.iSup_lintegral_measurable_le_eq_lintegral, ENNReal.HolderTriple.inv_le_inv, ConvexCone.mem_top, left_or_right_lt_sup, Fin.preimage_val_Ico_val, AlgebraicGeometry.Scheme.Modules.restrictFunctorCongr_inv_app_app, continuousAt_iff_continuous_left'_right', compl_ge, AddSubmonoid.le_comap_map, Fin.preimage_cast_Ioi, Subfield.gc_map_comap, Filter.le_lift, AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_ι_assoc, hasProdLocallyUniformly_iff_tendstoLocallyUniformly, OrderIso.subsingleton_of_wellFoundedGT', MulArchimedeanClass.mk_le_mk_iff_lt, Cauchy.le_nhds_lim, IsLprojection.coe_top, HahnSeries.coe_cardSuppLTAddSubgroup, FirstOrder.Language.Substructure.skolem₁_reduct_isElementary, CategoryTheory.Limits.CompleteLattice.limitCone_isLimit_lift, MeasureTheory.SimpleFunc.FinMeasSupp.iff_lintegral_lt_top, Metric.IsCover.coveringNumber_le_encard, QuotientAddGroup.strictMono_comap_prod_map, latticeClosure_singleton, List.SortedLE.monotone, Submonoid.le_comap_mulSingle_pi, CategoryTheory.ComposableArrows.whiskerLeftFunctor_obj_obj, Associates.factors_le, himp_inf_himp_inf_le, Filter.tendsto_mabs_atBot_atTop, AlgebraicGeometry.affineOpensRestrict_apply_coe_coe, OrderIso.sumLexIicIoi_symm_apply_Iic, Nat.succPNat_lt_succPNat, Matrix.IsHermitian.eigenvalues₀_antitone, le_mabs_self, AlgebraicGeometry.Scheme.mem_overGrothendieckTopology, CategoryTheory.ComposableArrows.opEquivalence_functor_map_app, Ordinal.one_lt_card, Finset.Ioc_map_sectL, WithSeminorms.uniformEquicontinuous_iff_bddAbove_and_continuous_iSup, List.sorted_gt_ofFn_iff, Fin.map_castLEEmb_Icc, JordanHolderLattice.lt_of_isMaximal, Nimber.small_Ioo, LinearOrderedAddCommGroupWithTop.add_lt_top, AlternatingMap.alternatizeUncurryFin_alternatizeUncurryFinLM_comp_apply, Cauchy.ultrafilter_of, Affine.Simplex.face_points, Ideal.height_le_ringKrullDim_quotient_add_one, CategoryTheory.Classifier.SubobjectRepresentableBy.isPullback, lt_of_forall_le_imp_ne, MeasureTheory.OuterMeasure.boundedBy_le, MeasureTheory.MemLp.integral_indicator_norm_ge_nonneg_le_of_meas, Filter.tendsto_abs_atBot_atTop, CategoryTheory.Limits.CompleteLattice.finite_colimit_eq_finset_univ_sup, ConvexCone.hull_min, AlgebraicGeometry.Scheme.AffineZariskiSite.presieveOfSections_surjective, CategoryTheory.sheafBotEquivalence_unitIso, antivary_neg, closure_convexHull_extremePoints, Finset.range_orderEmbOfFin, EReal.ge_of_forall_gt_iff_ge, MeasureTheory.lintegral_ofReal_le_lintegral_enorm, tendsto_nat_floor_div_atTop, Order.PFilter.inf_mem_iff, CategoryTheory.Limits.imageSubobject_le_mk, Units.mem_posSubgroup, IsTopologicalAddGroup.isOpenMap_iff_nhds_zero, TopologicalSpace.Opens.mapMapIso_unitIso, CategoryTheory.ObjectProperty.galoisConnection_isColocal, Nat.cast_lt_ofNat, AlgebraicGeometry.RingedSpace.basicOpen_le, openSegment_eq_image, TopologicalSpace.OpenNhds.map_id_obj_unop, Fin.prod_Iio_castSucc, Set.preimage_sub_const_Ico, cmp_eq_compare, Continuous.cfcₙ_nnreal, tendsto_inv_atTop_nhdsGT_zero, ENNReal.ofReal_le_ofReal_iff', Filter.comap_abs_atTop, ContinuousLinearMap.nnnorm_holder_apply_apply_le, AlgebraicGeometry.StructureSheaf.const_one, compl_sup_le_himp, LieSubalgebra.map_le_iff_le_comap, CategoryTheory.MorphismProperty.map_inverseImage_le, RCLike.ofReal_mul_pos_iff, MulArchimedeanOrder.lt_def, Ordinal.lt_blsub_iff, ValuativeRel.ValueGroupWithZero.mk_lt_mk, ENat.toENNRealOrderEmbedding_apply, Subsemigroup.closure_le, Finset.prod_sdiff_le_prod_sdiff, NNRat.natCast_lt_cast, CategoryTheory.Functor.mapComposableArrowsObjMk₂Iso_hom_app, Order.coheight_int, isCoatomistic_iff, Fin.val_strictMono, Fin.preimage_addNat_Ioi_addNat, CategoryTheory.Subobject.factorThru_add_sub_factorThru_right, AlgebraicGeometry.Scheme.Opens.ι_appIso, Submonoid.leftInv_leftInv_le, monovaryOn_iff_exists_monotoneOn, Filter.tendsto_iff_comap, algebraMap_lt_algebraMap, Set.neg_Ici, MeasureTheory.Content.innerContent_bot, Complex.neg_iff, SimpleGraph.Finsubgraph.coe_bot, integrableOn_Iic_iff_integrableOn_Iio, Set.Ioc_pred_right_eq_Ioo, MulPosReflectLT.toMulPosStrictMono, MeasureTheory.Ioo_ae_eq_Ico', IntermediateField.extendScalars_self, ContinuousWithinAt.finset_inf, Finset.covBy_cons, DivisibleHull.mk_le_mk, OmegaCompletePartialOrder.ContinuousHom.forall_forall_merge, Prod.swap_wcovBy_swap, MeasureTheory.FiniteMeasure.limsup_measure_closed_le_of_tendsto, ENNReal.le_rpow_inv_iff, compare_ge_iff_ge, MeasureTheory.le_measure_symmDiff, Filter.lift'_interior_le, wbtw_zero_one_iff, max_le_iff, Nat.lt_of_floor_lt, Subalgebra.adjoin_eq_span_basis, AlgebraicGeometry.IsAffineOpen.isoSpec_inv_toSpecΓ, one_lt_div_iff, PartOrdEmb.dual_map, MeasureTheory.OuterMeasure.top_caratheodory, CStarAlgebra.isStrictlyPositive_iff_eq_mul_star_self, Fin.preimage_addNat_Iio_addNat, NNReal.instIsOrderBornology, Multiset.inf_coe, Metric.cthickening_eq_preimage_infEDist, TopologicalSpace.Closeds.coe_finset_inf, Continuous.bddBelow_range_of_hasCompactSupport, Set.image_mulSingle_Icc_left, HurwitzZeta.hasSum_int_completedHurwitzZetaOdd, AlgebraicGeometry.StructureSheaf.globalSectionsIso_hom, Cardinal.ord_le, CategoryTheory.MorphismProperty.le_isLocal_isLocal, Fin.image_castAdd_Iic, CategoryTheory.SimplicialThickening.functor_map, ENNReal.addLeftReflectLT, MeasureTheory.integral_Ioc_eq_integral_Ioo', tendsto_real_toNNReal_atTop, CompleteLattice.MulticoequalizerDiagram.multispanIndex_left, Polynomial.degree_sum_le, BoundedLatticeHom.dual_symm_apply_toFun, setOf_isPreconnected_eq_of_ordered, LowerSet.Iic_strictMono, MonomialOrder.degree_mul_lt_iff_left_lt_of_ne_zero, Sublattice.prodEquiv_apply, ClosureOperator.closure_sup_closure_right, MeasureTheory.Measure.pi_Ioi_ae_eq_pi_Ici, AlgebraicGeometry.instLocallyQuasiFiniteMorphismRestrict, Ordinal.isNormal_preOmega, Field.Emb.Cardinal.equivSucc_coherence, Set.einfsep_insert_le, IntermediateField.finrank_top, CategoryTheory.ComposableArrows.sc'MapIso_hom, Fin.preimage_natAdd_Ioc_natAdd, NNRat.cast_le_one, antitone_iff_map_nonneg, MeasureTheory.FiniteMeasure.apply_iUnion_le, Finpartition.default_eq_empty, Finset.Iio_add_one_eq_Iic, MeasureTheory.eLpNorm_le_add_measure_right, MeasureTheory.Lp.eLpNorm_lim_le_liminf_eLpNorm, NNReal.add_rpow_le_rpow_add, mem_emetric_ball_one_iff, Finset.sup_eq_top_iff, hnot_bot, le_iInf_comp, LinearOrderedCommGroupWithZero.inl_eq_coe_inlₗ, Set.Ioi_insert, WithTop.orderIsoSumLexPUnit_top, TopologicalSpace.Opens.overEquivalence_unitIso_hom_app_left, WithTop.subtypeOrderIso_symm_apply, SSet.stdSimplex.coe_edge_down_toOrderHom, Dynamics.le_coverEntropyEntourage_image, CategoryTheory.Subobject.mk_eq_bot_iff_zero, CompleteSublattice.codisjoint_iff, FirstOrder.Language.DirectLimit.liftInclusion_of, Set.Ico.coe_eq_zero, LieIdeal.map_mono, Subgroup.center_le_normalizer, StarSubalgebra.gc_map_comap, AlgebraicGeometry.PresheafedSpace.GlueData.opensImagePreimageMap_app', AddMonoidHom.liftOfRightInverse_comp, MeasureTheory.Submartingale.eLpNorm_stoppedAbove_le', AlgebraicGeometry.IsAffineOpen.toSpecΓ_isoSpec_inv, Nat.emultiplicity_two_factorial_lt, Metric.le_infEDist, Fin.map_valEmbedding_Ici, AlgebraicGeometry.Scheme.IdealSheafData.bot_mul, CategoryTheory.ComposableArrows.opEquivalence_inverse_obj, KleeneAlgebra.one_le_kstar, EReal.eq_bot_iff_forall_lt, AlgebraicGeometry.Scheme.Hom.appIso_inv_naturality, Order.not_isMax_zero, Submodule.comap_strictMono_of_surjective, MonomialOrder.degree_smul_le, CategoryTheory.MorphismProperty.gc_llp_rlp, DFinsupp.card_Ioo, Rat.comap_cast_atBot, Set.preimage_sub_const_Iio, NNReal.sqrt_eq_zero, MulEquiv.coe_mapSubgroup, MeasureTheory.hittingAfter_univ, Filter.map_inv_atBot, ENNReal.lt_iff_exists_real_btwn, Module.mem_support_iff_exists_annihilator, SemilinearMapClass.ebound_of_continuous, max_eq_left_iff, Submonoid.closure_mono, Real.comap_toNNReal_atTop, CategoryTheory.Limits.CompleteLattice.finiteLimitCone_isLimit_lift, Finset.Colex.lt_iff_exists_filter_lt, Dynamics.coverEntropyEntourage_monotone, MeasureTheory.FiniteMeasure.apply_mono, LieModuleHom.le_ker_iff_map, Polynomial.cyclotomic_pos_and_nonneg, Nucleus.toSublocale_le_toSublocale, isAtomic_iff, Set.Finite.isClosed_convexHull, MeasureTheory.measure_lt_top_of_subset, Ordinal.preOmega_ofNat, le_compl_self, MeasureTheory.eLpNorm'_mono_ae, Set.image_const_add_Ici, MeasureTheory.ProbabilityMeasure.le_liminf_measure_open_of_tendsto, max_lt_iff, Set.Ioc_sub_one_right_eq_Ioo, monovary_neg, Set.Iic.disjoint_iff, CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinset_obj_map, SetTheory.PGame.default_nim_one_leftMoves_eq, AlgebraicGeometry.Scheme.basicOpen_add_le, AlgebraicGeometry.Scheme.Hom.comp_app_assoc, le_emultiplicity_of_pow_dvd, MeasureTheory.L1.nnnorm_integral_le, iSupIndep_def', AlgebraicGeometry.Scheme.comp_app, forall_lt_imp_ne_iff_le, le_ciInf_iff', RCLike.toZeroLEOneClass, PointedCone.convex, Finset.orderEmbOfFin_singleton, NonUnitalStarAlgebra.range_eq_top, SimplexCategory.eqToHom_toOrderHom, Finset.sup_const_le, OrderIso.compl_symm_apply, Tuple.antitone_pair_of_not_sorted, ENNReal.pow_lt_pow_left_iff, tendsto_inv_atBot_zero, le_iff_uniformContinuous_id, csSup_empty, ContinuousOn.upperSemicontinuousOn, TopologicalSpace.Opens.isOpenEmbedding, AddSubmonoid.le_pi_iff, nhdsLT_sup_nhdsGE, BooleanSubalgebra.comap_mono, exists_int_ge, separableClosure.eq_top_iff, IsNonarchimedean.nmul_le, NNReal.coe_lt_one, BoxIntegral.IntegrationParams.gp_le, Finset.inf_eq_bot_iff, Ordinal.preOmega_le_omega, AlgebraicGeometry.Scheme.Hom.finite_app, convexHull_empty, ProbabilityTheory.iIndepSet.iIndep_comap_mem, TopCat.Presheaf.stalk_hom_ext_iff, le_sdiff, ENNReal.HolderConjugate.one_le, TwoSidedIdeal.opOrderIso_apply, Polynomial.map_roots_le, Int.card_fintype_Icc_of_le, WithTop.iSup_coe_lt_top, AlgebraicGeometry.Scheme.AffineZariskiSite.coequifibered_iff_forall_isLocalizationAway, ENNReal.rpow_add_le_mul_rpow_add_rpow, AlgebraicGeometry.instGeometricallyReducedMorphismRestrict, Equiv.Perm.support_zpowers_of_mem_cycleFactorsFinset_le, HomotopyCategory.composableArrowsFunctor_obj, Finset.Icc_sub_one_right_eq_Ico, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_zeroLocus, Algebra.top_toSubring, Order.isSuccLimitRecOn_succ, RelHom.toOrderHom_coe, antivaryOn_iff_exists_monotoneOn_antitoneOn, HahnSeries.leadingCoeff_pos_iff, Fin.prod_Icc_castLE, AlgebraicGeometry.exists_preimage_eq, addRothNumber_le_ruzsaSzemerediNumber, AlgebraicGeometry.PresheafedSpace.restrictStalkIso_inv_eq_germ_apply, abs_max_sub_max_le_max, HomotopicalAlgebra.trivialFibrations_sub_weakEquivalences, MeasurableSet.exists_isOpen_diff_lt, AlgebraicGeometry.Scheme.evaluation_naturality, Set.Ico_inter_Ici, IsUnit.isStrictlyPositive_star_left_conjugate_iff, EReal.mul_ne_bot, countable_image_lt_image_Iio_within, le_algebraicClosure, Set.image_preimage, Module.Basis.exists_opNNNorm_le, Ordinal.cof_succ, absConvexHull_add_subset, CategoryTheory.Limits.MultispanShape.ofLinearOrder_L, MvPowerSeries.le_weightedOrder_smul, inv_neg'', Ideal.height_le_iff, AddSubmonoid.pointwise_smul_le_iff, ProperCone.dual_iUnion, Sylow.normalizer_le_centralizer_or_le_commutator, AlgebraicGeometry.Scheme.ker_eq_top_of_isEmpty, Alexandrov.principals_obj, MeasureTheory.trim_eq_self, SimpleGraph.chromaticNumber_le_two_iff_isBipartite, ConvexCone.IsGenerating.top_le_span, Real.volume_pi_le_prod_diam, MeasureTheory.ProbabilityMeasure.apply_le_one, NumberField.isTotallyReal_iff_le_maximalRealSubfield, IntermediateField.Lifts.lt_iff, NNReal.mul_self_sqrt, EReal.mul_nonneg_iff, AlgebraicGeometry.ΓSpec.left_triangle, ofBoolRing_le_ofBoolRing_iff, orderEmbeddingOfInjective_apply, LinearMap.annihilator_le_of_injective, contravariant_lt_of_covariant_le, Finset.eraseNone_map_some, CategoryTheory.Subfunctor.eq_top_iff_isIso, FormalMultilinearSeries.radius_le_of_le, CategoryTheory.Subfunctor.Subpresheaf.le_def, ZFSet.vonNeumann_strictMono, Polynomial.isBigO_of_degree_le, lt_subset_interior_le, AlgebraicGeometry.Scheme.Modules.toOpen_fromTildeΓ_app_assoc, Set.image_neg_Iio, Finset.pluennecke_ruzsa_inequality_pow_div_pow_mul, CategoryTheory.sheafBotEquivalence_inverse_obj_val, StrictConcaveOn.lt_on_openSegment, AlgebraicGeometry.Scheme.Hom.ideal_ker_le, AlgebraicGeometry.AffineSpace.comp_homOfVector, Set.neg_mem_Ioc_iff, ofBoolAlg_mul_ofBoolAlg_eq_left_iff, IntermediateField.top_toSubalgebra, ENNReal.lt_inv_iff_lt_inv, Subfield.roots_X_pow_char_sub_X_bot, Ordinal.type_subrel, Fin.preimage_castAdd_Iio_castAdd, AlgebraicGeometry.smooth_iff, GroupSeminormClass.toMulLEAddHomClass, HasCountableSeparatingOn.range_Ioi, Finset.Ico_eq_image_ssubsets, MeasureTheory.meas_ge_le_lintegral_div, HahnSeries.orderTop_lt_top, isCompact_setOf_finiteMeasure_le_of_compactSpace, HahnSeries.isPWO_support', Ring.KrullDimLE.mem_minimalPrimes_iff_le_of_isPrime, Fin.map_succEmb_Ioi, HasCountableSeparatingOn.range_Iio, ENNReal.ofNNReal_le_natCast, Multiset.map_single_le_powerset, nonneg_iff_isSelfAdjoint_and_spectrumRestricts, ENNReal.instDenselyOrdered, untrop_sum, AddSubgroup.map_eq_bot_iff, MeasureTheory.Measure.mutuallySingular_iff_disjoint, IsCompact.ultrafilter_le_nhds', Real.toNNReal_pos, Set.Ioo_subset_uIoo, EMetric.tendsto_nhds_nhds, ENat.floor_lt_top, isLocallyClosed_Ioc, continuousWithinAt_inter_Iio_iff_Iic, CategoryTheory.ComposableArrows.scMapIso_inv, CompleteBooleanAlgebra.le_sup_inf, Finset.Ioc_sub_one_sub_one_eq_Ico, not_linearIndependent_iffₒₛ, AlgebraicGeometry.Scheme.Hom.appIso_hom', TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquiv_inverse, IntermediateField.relrank_top_left, Filter.le_map₂_iff, AddValuation.map_zero, AlgebraicGeometry.Scheme.zeroLocus_inf, Multiset.exists_max_image, left_lt_sup, NNReal.strictMono_rpow_of_pos, strictMonoOn_mul_self, MeasureTheory.lintegral_indicator_le, ArchimedeanClass.min_le_mk_add, Positive.addLeftMono, HahnSeries.order_le_of_coeff_ne_zero, Submodule.lTensorOne'_tmul, one_le_kstar, Fin.image_val_Iio, Polynomial.le_trailingDegree_mul, Fin.finsetImage_rev_Ioc, Finset.addConst_le_card, mulLeftReflectLT_of_mulLeftMono, AddSubmonoid.closure_nsmul_le, Cardinal.isRegular_aleph_succ, AlgebraicGeometry.Scheme.IdealSheafData.ofIdeals_mono, NNReal.lt_rpow_inv_iff, AlgebraicGeometry.Scheme.instIsOpenImmersionToSpecΓOfIsQuasiAffine, div_nonpos_iff, NNReal.agmSequences_fst_monotone, Finset.coe_sup_of_nonempty, convexHull_pair, Cardinal.continuum_pos, Dynamics.coverEntropy_antitone, Disjoint.eq_bot_of_self, Finset.Colex.toColex_le_toColex, Units.inv_pos, Fin.map_finCongr_Ici, Order.Frame.le_himp_iff, Submodule.rTensorOne'_tmul_one, Set.Ioi_sub_one_eq_Ici, Fin.preimage_succ_Ico_succ, Cardinal.ord_le_ord, ArchimedeanClass.mk_le_mk_iff_lt, Cardinal.beth_mono, MonotoneOn.Iic, EReal.sub_lt_iff, ContinuousLinearMap.isLeast_opNNNorm, tendsto_floor_right, SemilatInfCat.coe_forget_to_partOrd, CategoryTheory.IsCardinalFiltered.exists_cardinal_directed.functorMap_id, EMetric.ordConnected_setOf_closedBall_subset, WithVal.lt_def, UpperSet.le_Ici, CategoryTheory.Subfunctor.equalizer_le, nhdsSet_le_iff, MeasureTheory.eLpNorm_le_eLpNorm_fderiv_one, Ordinal.gamma_pos, LTSeries.strictMono, Order.isNormal_iff_strictMono_and_continuous, WithTop.ge_of_forall_gt_iff_ge, IsGaloisGroup.fixedPoints_bot, InitialSeg.lt_apply_iff, Finset.coe_coeEmb, SimpleGraph.two_le_chromaticNumber_iff_ne_bot, isAtomic_of_complementedLattice, PNat.factorMultiset_le_iff', Submodule.isOrtho_iff_le, antivary_iff_exists_monotone_antitone, Set.bounded_gt_inter_gt, AlgebraicGeometry.germ_injective_of_isIntegral, HahnSeries.order_ofForallLtEqZero, iInf_le_iInf₂, List.Sorted.get_mono, Subgroup.commutator_pi_pi_le, AlgebraicGeometry.Scheme.isoSpec_inv_naturality_assoc, Metric.edist_le_ediam_of_mem, TopCat.Presheaf.pushforward_obj_obj, le_integralClosure_iff_isIntegral, Seminorm.const_isBounded, MeasureTheory.hittingBtwn_apply_mono_left, SetTheory.Game.bddBelow_of_small, Subalgebra.topologicalClosure_map_le, ContinuousOn.cfc_nnreal_of_mem_nhdsSet, FractionalIdeal.num_le_mul_inv, isPreconnected_Ioi, AlgebraicGeometry.IsAffineOpen.fromSpec_app_self, OrderIso.supIrredLowerSet_apply, Cardinal.ofNat_lt_aleph0, lebesgue_number_lemma_of_emetric_nhdsWithin', Set.preimage_const_div_Ioi, nhds_top, Con.toSetoid_eq_bot, Finset.single_le_prod_of_canonicallyOrdered, Subgroup.le_normalizer, ENNReal.inv_strictAnti, Subgroup.disjoint_def', Fin.card_Icc, Finset.le_inf_mul, Concept.swapEquiv_apply, preCantorSet_antitone, IsUltrametricDist.nnnorm_add_one_le_max_nnnorm_one, MeasureTheory.condExp_bot, LieModule.antitone_lowerCentralSeries, Ideal.pow_le_self, ProbabilityTheory.meas_ge_le_variance_div_sq, Set.powersetCard.ofFinEmbEquiv_symm_apply, Submodule.rTensorOne'_tmul, Set.le_einfsep_iff, Multiset.instAddLeftMono, SSet.OneTruncation₂.nerveEquiv_symm_apply_map, separatedNhds_iff_disjoint, IsValuativeTopology.mem_nhds_zero_iff, AlgebraicGeometry.IsOpenImmersion.opensEquiv_apply_coe, AlgebraicGeometry.isClosedImmersion_iff_isAffineHom, Ordinal.lt_lift_iff, atTop_hasCountableBasis_of_archimedean, TopologicalSpace.Opens.mem_bot, MeasureTheory.hittingBtwn_anti, Cardinal.powerlt_mono_left, MvPowerSeries.le_weightedOrder_subst, TwoSidedIdeal.mem_comap, CategoryTheory.MorphismProperty.retracts_transfiniteComposition_pushouts_coproducts_le_llp_rlp, AlgebraicGeometry.PresheafedSpace.restrictTopIso_hom, Finset.one_le_divConst, AddOpposite.unop_nonpos, HahnEmbedding.Partial.eval_smul, MeasureTheory.eLpNormEssSup_le_of_ae_bound, CategoryTheory.ComposableArrows.mk₁_obj, Subgroup.orderIsoCon_apply, Set.image_single_Ioo_left, multipliable_iff_cauchySeq_finset, Finset.sup_prodMap, himp_le_left, nsmul_le_nsmul_iff_right, IsCoatomic.eq_top_or_exists_le_coatom, OrdinalApprox.lfpApprox_monotone, antitone_iff_forall_lt', Fin.image_castAdd_Ici, Complex.nonpos_iff, Submonoid.powers_le_zpowers, Order.Ico_succ_right, Int.le_ceil_iff, Dynamics.coverMincard_union_le, ProperCone.dual_singleton, Ordinal.principal_mul_omega, disjoint_nhdsSet_nhdsSet, PartOrdEmb.hom_id, codisjoint_ofDual_iff, exists_root_adjoin_eq_top_of_isCyclic, ENNReal.nhdsGT_zero_neBot, grade_top, CStarAlgebra.isStrictlyPositive_iff_exists_isStrictlyPositive_and_eq_mul_self, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.app_inv_app'_assoc, Finset.Iic_pred_eq_Iio, ENat.gc_ceil_toENNReal, LieAlgebra.center_le_radical, MulArchimedeanClass.mk_antitoneOn, Finset.disjSum_mono_right, AlgebraicGeometry.AffineSpace.hom_ext_iff, MulArchimedeanClass.mk_monotoneOn, PartOrdEmb.ofHom_id, Topology.scottHausdorff_le_lawson, MonomialOrder.degree_mul_le, HasCompactSupport.enorm_le_lintegral_Ici_deriv, CategoryTheory.Subobject.isoOfEq_hom, Filter.bliminf_sup_le_inf_aux_left, iSup₂_add_le, iSupIndep.injOn, Cardinal.aleph0_lt_mk_iff, SSet.stdSimplex.isoNerve_hom_app_apply, Subgroup.map_eq_bot_iff, Seminorm.sSup_empty, MvPolynomial.restrictSupport_mono, Valuation.apply_posSubmonoid_pos, LowerAdjoint.closure_iSup₂_closure, Finset.truncatedInf_empty, AlgebraicGeometry.ΓSpec.unop_locallyRingedSpaceAdjunction_counit_app', Cardinal.aleph1_lt_lift, hasStrictDerivAt_abs, one_le_sq_iff_one_le_abs, TwoSidedIdeal.mem_asIdealOpposite, nndist_midpoint_midpoint_le', Equiv.Perm.support_zpow_le, Subsemiring.pointwise_smul_subset_iff, AlgebraicGeometry.LocallyRingedSpace.stalkMap_germ_assoc, UniformFun.edist_le, Submodule.map_subtype_embedding_eq, Real.toNNReal_lt_iff_lt_coe, Finset.Ico_add_Icc_subset, tsub_eq_zero_iff_le, Module.jacobson_le_of_eq_bot, Multiset.toEnumFinset_subset_iff, MeasureTheory.Measure.univ_pi_Ico_ae_eq_Icc, Set.image_const_add_Icc, Valuation.one_srel_iff, Pi.Colex.isStrictOrder, midpoint_le_left, Multiset.le_add_left, Fin.predAbove_right_monotone, LieModule.iInf_lcs_le_of_isNilpotent_quot, MeasureTheory.StronglyMeasurable.exists_spanning_measurableSet_norm_le, CategoryTheory.SmallObject.ιFunctorObj_eq, ProbabilityTheory.monotone_preCDF, Ideal.exists_le_prime_notMem_of_isIdempotentElem, Disjoint.le_compl_right, AlgebraicGeometry.Scheme.Hom.preimage_basicOpen, Set.Iic_diff_Iio_same, Heyting.Regular.coe_bot, NonUnitalRingHom.sclosure_preimage_le, nhdsSet_diagonal_le_uniformity, monovary_iff_smul_rearrangement, ModP.v_p_lt_preVal, EReal.preimage_coe_Ici, Filter.tendsto_toLeft_atTop, leOnePart_eq_one, Cardinal.toNat_strictMonoOn, List.minimum_cons, Subalgebra.mul_toSubmodule_le, OrderTop.upperBounds_univ, Finset.exists_sup_ge, MeasureTheory.stoppedProcess_eq, sum_Ioo_inv_sq_le, Real.toNNReal_lt_toNNReal_iff, instLawfulBCmpCompare_mathlib, le_total, le_himp_iff', FirstOrder.Language.Substructure.mem_closed_of_isRelational, EReal.image_coe_Ioo, Nat.ofNat_le_cast, negPart_eq_ite, Int.cast_nonneg, Subalgebra.unop_le_unop_iff, CategoryTheory.ComposableArrows.whiskerLeft_map, CFC.log_monotoneOn, MeasurableSpace.generateFrom_singleton_univ, Quantale.mul_bot, Nat.succPNat_strictMono, ValuationSubring.nonunits_le_nonunits, Set.compl_Ioi, ENNReal.sub_eq_sInf, Ordinal.le_enum_succ, HasSumUniformly.tendstoUniformly, IsGaloisGroup.ofDual_intermediateFieldEquivSubgroup_apply, monovary_comm, Filter.nonneg_sub_iff, FiniteArchimedeanClass.subsemigroup_eq_addSubgroup, inf_lt_left_or_right, MeasureTheory.setIntegral_mono_on_ae, AlgebraicGeometry.Scheme.IdealSheafData.subschemeι_app, NonUnitalSubalgebra.map_le, IncidenceAlgebra.sum_Icc_mu_right, CategoryTheory.Subobject.lower_comm, Stream'.Seq.at_least_as_long_as_coind, RCLike.nonneg_iff, Nat.ofNat_pos, Finite.exists_max, NNRat.preimage_cast_Icc, Subsemigroup.strictMono_topEquiv, bihimp_triangle, Set.Iic_inj, imageToKernel_arrow, Fin.finsetImage_rev_Ioi, LinearMap.iterateKer_coe, continuousOn_iff_lower_upperSemicontinuousOn, EReal.natCast_lt_iff, Cardinal.isRegular_succ, toZ_iterate_pred_ge, MvPolynomial.degrees_map_le, Submodule.le_comap_mkQ, Affine.Simplex.affineSpan_faceOpposite_le, BooleanSubalgebra.closure_le, Filter.Subsingleton.exists_le_pure, addRothNumber_empty, MeasureTheory.partialTraj_const, Module.Flat.instSubalgebraToSubmodule, monovary_iff_exists_antitone, Filter.antitone_seq_of_seq, IsLprojection.coe_bot, measure_le_lintegral_thickenedIndicator, infClosure_univ, Set.left_mem_uIoc, Subgroup.characteristic_iff_le_comap, CategoryTheory.antitone_chain_condition_of_isArtinianObject, Dynamics.coverEntropyInf_monotone, FormalMultilinearSeries.le_radius_of_summable, Module.End.disjoint_genEigenspace, Multiset.le_prod_of_mem, Module.length_le_of_surjective, OrdinalApprox.gfpApprox_antitone, Set.Iic_iInf₂, Set.mem_Icc_iff_abs_le, disjoint_nhds_atBot_iff, AlgebraicGeometry.Scheme.isoSpec_Spec_hom, Subgroup.iSup_comap_le, AddMonoid.Coprod.codisjoint_mrange_inl_mrange_inr, Opens.mayerVietorisSquare_X₂, ENat.lift_zero, NNReal.geom_mean_le_arith_mean4_weighted, BooleanSubalgebra.map_inf_le, SSet.skeleton_succ, MeasureTheory.Filtration.rightCont_def, Int.gc_coe_floor, ProbabilityTheory.gaussianReal_mul_const, AlgebraicGeometry.Scheme.basicOpen_res_eq, sSup_atoms_eq_top, PosSMulStrictMono.toPosSMulReflectLE, instWellFoundedLTAntisymmetrizationLe, ENat.one_le_iff_ne_zero_withTop, NNReal.tendsto_rpow_atTop, Real.disjoint_residual_ae, FirstOrder.Language.Substructure.fg_iff_exists_fin_generating_family, Submonoid.op_le_iff, IntermediateField.finrank_eq_one_iff_eq_top, partOrdEmb_dual_comp_forget_to_pardOrd, SSet.Subcomplex.iSup_ofSimplex_nonDegenerate_eq_top, Multiset.map_add_left_Ioo, add_le_add_iff_left_of_ne_top, MeasureTheory.SimpleFunc.FinMeasSupp.meas_preimage_singleton_ne_zero, Filter.HasBasis.ge_iff, IsCompl.isCoatom_iff_isAtom, ENNReal.mulLeftOrderIso_symm_apply, WithZeroTopology.isClosed_iff, List.minimum_of_length_pos_le_getElem, ConvexOn.quasiconvexOn, MeasureTheory.measure_inter_lt_top_of_left_ne_top, strictConvex_Ici, AlgebraicGeometry.StructureSheaf.isLocalizedModule_toPushforwardStalkAlgHom, MeasureTheory.Measure.pi_Ioo_ae_eq_pi_Icc, NNReal.tsum_comp_le_tsum_of_inj, le_compl_iff_le_compl, fixingSubmonoid_antitone, SupBotHom.coe_sup, Ordinal.mem_toPSet_iff, TopologicalSpace.Opens.coe_finset_inf, AlgebraicGeometry.Scheme.Hom.isAffineOpen_iff_of_isOpenImmersion, edist_triangle4, TopHom.inf_apply, MeasureTheory.exists_ne_forall_mem_nhds_pos_measure_preimage, CategoryTheory.Limits.kernelOrderHom_coe, Fin.prod_Iic_castSucc, MeasureTheory.VectorMeasure.le_iff, Sublattice.le_pi, subset_latticeClosure, Hyperreal.archimedeanClassMk_nonneg_of_tendsto, gc_sdiff_sup, Filter.bliminf_sup_le_inf_aux_right, rothNumberNat_def, Nonneg.instArchimedean, Cardinal.isNormal_aleph, Metric.ediam_eball_le, IsDedekindDomain.primesOverEquivPrimesOver_ramificationIdx_eq, TopCat.GlueData.MkCore.t_inv, IsOrderedCancelAddMonoid.toAddRightReflectLT, Set.image_single_Ioo_right, Finset.sup_sub_le, Order.height_eq_coe_add_one_iff, mul_pos_iff, Ordinal.mul_succ, one_le_thickenedIndicator_apply', Order.not_isMin_succ, Continuous.if_le, AlgebraicGeometry.Scheme.image_basicOpen, Subfield.subring_closure_le, HahnSeries.unit_aux, CategoryTheory.Subobject.inf_isPullback, SummationFilter.symmetricIcc_le_Conditional, Fin.preimage_castLE_Ioc_castLE, Function.isMinimalFor_argminOn, monovaryOn_neg_right, AlgebraicGeometry.tilde.isIso_toOpen_top, IsLocalization.algEquivOfAlgEquiv_eq_map, mem_range_nerve_σ_iff, AlgebraicGeometry.Scheme.IdealSheafData.map_mono, Nat.choose_lt_pow_div, convexJoin_subset_convexHull, SupHom.withTop'_toFun, ArchimedeanClass.mem_addSubgroup_iff, integrableOn_Ico_iff_integrableOn_Ioo', Function.isMinimalFor_argmin, Fin.finsetImage_succ_Iic, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.invApp_app, Ordinal.cmp_veblen, WithZeroTopology.hasBasis_nhds_zero, Fin.strictMono_addNat, Set.finite_isTop, Algebra.adjoin_toSubmodule_le, Part.mem_ωSup, HahnSeries.order_lt_order_of_eq_add_single, CompleteLinearOrder.himp_bot, strictAnti_iff_map_neg, Order.covBy_iff_add_one_eq, TopCat.Presheaf.SubmonoidPresheaf.map, CategoryTheory.isCofilteredOrEmpty_of_semilatticeInf, ArchimedeanClass.addSubgroup_eq_bot, addRothNumber_lt_of_forall_not_threeAPFree, Geometry.SimplicialComplex.disjoint_or_exists_inter_eq_convexHull, Hyperreal.epsilon_lt_of_neg, PrimeSpectrum.isEmbedding_tensorProductTo_of_surjectiveOnStalks_aux, Fin.preimage_rev_Ioi, CFC.monotone_rpow, subset_infClosure, antitone_vecCons, Metric.exists_eball_subset_eball, UpperSet.map_Ici, ENNReal.monotone_truncateToReal, Sylow.le_center_or_le_commutator, RootPairing.GeckConstruction.span_range_h_le_range_diagonal, Submodule.comap_subtype_eq_top, ProbabilityTheory.Kernel.rnDeriv_pos, ENNReal.le_iInf_mul, Multiset.disjSum_mono_right, SimpleGraph.Subgraph.verts_monotone, Valuation.mem_leAddSubgroup_iff, IsGalois.intermediateFieldEquivSubgroup_symm_apply_toDual, Set.Iic_injective, Subgroup.closure_mono, codisjoint_top_left, convexHull_eq_empty, Submodule.inf_iSup_genEigenspace, mulPosMono_iff_mulPosReflectLT, MeasureTheory.integral_Icc_eq_integral_Ico', ONote.omega0_le_oadd, wellFounded_antisymmetrization_iff, Cardinal.omega0_le_ord, ENNReal.coe_le_one_iff, tendsto_nat_ceil_atTop, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.app_invApp_assoc, BddOrd.forget_map, IsClosedEmbedding.topologicalKrullDim_le, AddEquiv.comapAddSubgroup_apply, Delone.DeloneSet.packingRadius_pos, sup_eq_bot_iff, CategoryTheory.Limits.LimitPresentation.self_diag, QuadraticMap.PosDef.nonneg, Polynomial.trailingDegree_le_of_ne_zero, BooleanSubalgebra.mk_le_mk, MeasureTheory.integrable_iff_integrableAtFilter_atTop, FirstOrder.Language.isExtensionPair_iff_exists_embedding_closure_singleton_sup, exists_nonneg_right_iff_sameRay, DiscreteQuotient.leComap_id_iff, MonomialOrder.le_add_right, FiniteMultiplicity.emultiplicity_le_of_multiplicity_le, Multiset.bot_eq_zero, iSupIndep_ne_bot, Finset.toRight_monotone, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.app_invApp, lowerClosure_zero, image_le_kernel, AddCircle.coe_image_Icc_eq, CharP.intCast_injOn_Ico, lt_top_iff_ne_top, AddSubgroup.relIndex_eq_one, isClosedMap_iff_comap_nhdsSet_le, NNReal.HolderTriple.lt, Ordinal.enum_typein, Finset.max_le_iff, Set.image_inv_Ioi, Ordinal.instAddLeftReflectLT, AlgebraicGeometry.PresheafedSpace.restrictStalkIso_hom_eq_germ_apply, MeasureTheory.OuterMeasure.trim_sum_ge, NonUnitalSubring.map_le_iff_le_comap, Polynomial.isEquivalent_atTop_div, Ideal.IsPrime.multiset_prod_le, Set.Ici.disjoint_iff, Subfield.extendScalars_top, isLocallyClosed_Iio, AlgebraicGeometry.HasRingHomProperty.appTop, Set.image_neg_Ioc, Matroid.eRk_le_iff, edist_triangle_right, GroupNorm.coe_le_coe, ENNReal.nnreal_smul_lt_top_iff, AlgebraicGeometry.LocallyRingedSpace.basicOpen_zero, FractionalIdeal.bot_eq_zero, CategoryTheory.Sieve.le_pushforward_pullback, CategoryTheory.Subobject.mk_arrow, RootPairing.invtRootSubmodule.eq_bot_iff, Submodule.toAddSubgroup_toIntSubmodule, AbsoluteValue.exists_one_lt_lt_one_pi_of_not_isEquiv, Set.Ioi_bot, Archimedean.embedReal_injective, IsPreconnected.ordConnected, SimpleGraph.le_chromaticNumber_iff_forall_surjective, min_abs_abs_le_abs_max, Multiset.le_ndinter, eq_bot_or_bot_lt, OrderMonoidIso.withZero_apply_apply, MeasureTheory.measure_isOpen_pos_of_vaddInvariant_of_compact_ne_zero, Int.cast_nonpos, abs_sub_abs_le_abs_sub, CFC.rpow_neg_one_eq_cfc_inv, exteriorPower.ιMultiDual_apply_ιMulti, AddSubgroup.le_centralizer_iff_isAddCommutative, nonemptyFinLinOrd_dual_comp_forget_to_linOrd, AffineSubspace.mk'_top, NormedSpace.equicontinuous_TFAE, Ideal.setOf_isPrincipal_wellFoundedOn_gt, Cardinal.card_lt_card_of_left_finite, Mathlib.Tactic.Bound.Nat.cast_pos_of_pos, CFC.monotoneOn_one_sub_one_add_inv_real, Profinite.NobelingProof.GoodProducts.span, MeasureTheory.measure_isOpen_pos_of_vaddInvariant_of_ne_zero, Topology.IsLower.tendsto_nhds_iff_lt, iInf_sup_iInf_le, dimH_image_le_of_locally_lipschitzOn, tsub_add_cancel_iff_le, Order.krullDim_enat, List.getD_min?_eq_untopD_minimum, MvPolynomial.le_coeffsIn_pow, Algebra.zariskisMainProperty_iff_exists_saturation_eq_top, isTotallyDisconnected_iff_lt, Filter.NeBot.le_one_iff, Ordinal.limitRecOn_succ, LinearOrderedField.coe_mem_cutMap_iff, Valuation.vle_iff_le, PartOrd.comp_apply, Finsupp.instPosSMulStrictMono, Module.End.genEigenspace_eq_genEigenspace_maxUnifEigenspaceIndex_of_le, NumberField.is_primitive_element_of_infinitePlace_lt, MvPolynomial.supported_strictMono, Order.height_le_coe_iff, IsUnit.star_left_conjugate_nonneg_iff, Order.Ioi_pred_eq_insert, Submodule.le_comap_map, ENNReal.not_lt_zero, SimpleGraph.not_isUniform_iff, clusterPt_iff_not_disjoint, Filter.exists_ultrafilter_iff, DFinsupp.Icc_eq, Matrix.l2_opNNNorm_mul, Finset.Colex.toColex_lt_singleton, nndist_vsub_vsub_le, NonemptyInterval.one_mem_one, Finset.card_Ico_finset, StarSubalgebra.coe_bot, OrderIso.sumLexIicIoi_symm_apply_Ioi, Interval.neg_bot, MonomialOrder.degree_pow_le, Finset.max_eq_sup_withBot, AlgebraicGeometry.IsAffineOpen.image_of_isOpenImmersion, instNoMinOrderOfNontrivial, Pi.sum_nnnorm_apply_le_nnnorm, Sublattice.prod_left_mono, cmp_mul_right', mapClusterPt_atTop_zpow_iff_pow, LieHom.ker_le_comap, AddSubgroup.map_le_map_iff_of_injective, UniformSpace.toTopologicalSpace_bot, forall_lt_imp_le_iff_le_of_dense, Ordinal.sub_le, NatOrdinal.mul_le_iff, CompHausLike.LocallyConstant.locallyConstantIsoContinuousMap_hom, Ordinal.instAddLeftStrictMono, MeasureTheory.SimpleFunc.integrable_iff, AddSubgroup.IsSubnormal.exists_normal_and_le_and_lt_top_of_ne, mul_self_pos, Codisjoint.hnot_le_right, AddSubsemigroup.strictMono_topEquiv, CompleteLattice.isSupClosedCompact_iff_wellFoundedGT, MeasureTheory.ae_le_eLpNormEssSup, Set.Ici.codisjoint_iff, Order.height_eq_coe_iff, CategoryTheory.Presheaf.isLocallySurjective_iff_range_sheafify_eq_top, List.exists_map_eq_of_sorted_iff_sbtw, MeasureTheory.integral_Ioc_eq_integral_Ioo, MonoidAlgebra.exists_finset_adjoin_eq_top, measurableSet_Ioi, AlgebraicGeometry.Scheme.isNilpotent_iff_basicOpen_eq_bot, LipschitzWith.comap_cobounded_le, NonUnitalSubalgebra.le_topologicalClosure, MeasureTheory.Measure.univ_pi_Ioi_ae_eq_Ici, PNat.add_one_le_iff, convexHull_singleton, IntermediateField.finrank_adjoin_eq_one_iff, bddBelow_iff_exists_le, Submodule.instCovariantClassHSMulLe, Int.preimage_Iic, fixedPoints_addSubgroup_antitone, Topology.IsUpperSet.isSheaf_of_isRightKanExtension, Set.image_sub_const_Ioi, Equiv.Perm.support_mul_le, Valued.toNormedField.norm_le_iff, Valuation.map_add', CategoryTheory.Limits.imageSubobject_arrow_comp_apply, NonUnitalSubring.prod_mono_left, CategoryTheory.Sieve.functorPushforward_monotone, NNReal.geom_mean_le_arith_mean2_weighted, one_le_inv_iff₀, OmegaCompletePartialOrder.ωScottContinuous_iff_monotone_map_ωSup, AlgebraicGeometry.IsIntegral.component_integral, algebraicClosure.map_le_of_algHom, Module.End.isFinitelySemisimple_iff, ClosureOperator.ofCompletePred_isClosed, InfClosed.codirectedOn, AddSubgroup.toAddSubmonoid_strictMono, AlgebraicGeometry.morphismRestrict_comp, MeasureTheory.hittingAfter_apply_anti, CategoryTheory.Subobject.isoOfEqMk_inv, NNReal.image_coe_Ioc, ENNReal.essSup_liminf_le, pi_nnnorm_le_iff', map_pos_of_ne_zero, Submodule.closure_le, Orientation.oangle_sign_smul_left, IntermediateField.normal_iff_forall_map_le', MvPolynomial.degrees_indicator, CategoryTheory.ComposableArrows.opEquivalence_unitIso_hom_app, Matrix.nnnorm_entry_le_entrywise_sup_nnnorm, CategoryTheory.ComposableArrows.mapFunctorArrows_app, MeasureTheory.OuterMeasure.isCaratheodory_iff_le, AddSubgroup.orderIsoAddCon_symm_apply_coe, exists_isLocalExtr_uIoo_of_tendsto, nnnorm_cfc_le_iff, AbsoluteValue.pos, iUnion_Icc_add_intCast, SimplexCategory.II.castSucc_mem_finset_iff, Set.image_neg_Ioo, PointedCone.mem_map, Cardinal.ofENat_le_ofENat, Subgroup.commutator_le_self, le_iSup_inf_iSup, IntermediateField.bot_eq_top_of_rank_adjoin_eq_one, Int.Icc_eq_finset_map, BddDistLat.dual_map, TopologicalSpace.Opens.coe_id, CategoryTheory.ShortComplex.mapToComposableArrows_comp, ProbabilityTheory.Kernel.partialTraj_succ_self, Subring.instCovariantClassHSMulLe, Metric.ediam_pos_iff, CategoryTheory.Subfunctor.equivalenceMonoOver_functor_obj, Pi.ωScottContinuous_uncurry, ENat.toENNReal_lt, Nucleus.coe_lt_coe, AlgebraicGeometry.IsProper.instMorphismRestrict, TopCat.presheafToTypes_map, set_has_minimal_iff_artinian, RingEquiv.mapTwoSidedIdeal_apply, MulArchimedeanClass.ballSubgroup_antitone, MeasureTheory.smul_le_stoppedValue_hitting, Metric.ordConnected_setOf_closedEBall_subset, Finset.Colex.singleton_lt_singleton, Subalgebra.LinearDisjoint.mulRightMap_ker_eq_bot_iff_linearIndependent, Set.isSimpleOrder_Iic_iff_isAtom, AbsoluteValue.IsEquiv.lt_one_iff, Finset.uIcc_prod_def, CategoryTheory.Functor.mapComposableArrowsObjMk₁Iso_hom_app, EReal.mul_neg_iff, sSup_eq_bot', EMetric.infEdist_le_infEdist_add_hausdorffEdist, Cardinal.card_typein_lt, ENNReal.inv_mul_le_one, ENNReal.nhds_top_basis, SeparationQuotient.inseparableSetoid_eq_top_iff, TopCat.Presheaf.pushforwardPullbackAdjunction_unit_app_app_germToPullbackStalk_assoc, AlgebraicGeometry.Scheme.isPullback_toSpecΓ_toSpecΓ, CompleteLinearOrder.le_himp_iff, IsDedekindDomain.HeightOneSpectrum.valuation_lt_one_iff_mem, CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinsetObj_map, AntitoneOn.Ioi, AddSubsemigroup.closure_singleton_le_iff_mem, enorm_add₄_le, AlgebraicGeometry.Scheme.IdealSheafData.mul_top, countable_image_gt_image_Iio_within, Mathlib.Meta.Positivity.nonneg_of_isRat, OrderHom.lfp_lfp, Cardinal.add_one_le_succ, bot_eq_one', ne_iff_lt_or_gt, AlgebraicGeometry.Scheme.instSubsingletonCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensBot, Polynomial.le_trailingDegree_C_mul_X_pow, SimpleGraph.edist_le_eccent, Profinite.NobelingProof.GoodProducts.linearIndependent_iff_smaller, PiTensorProduct.dualSeminorms_bounded, AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_hom_eq_germ_apply, Fin.prod_Ioi_cast, NonUnitalSubring.mk_le_mk, Fin.preimage_succ_Ioo_succ, UpperSet.coe_mul, partialSups_eq_biSup, Subgroup.commutator_eq_bot_iff_le_centralizer, CategoryTheory.Functor.mapComposableArrows_obj_map, CategoryTheory.GrothendieckTopology.OneHypercoverFamily.exists_oneHypercover, AlgebraicGeometry.Scheme.ΓSpecIso_naturality, Fin.finsetImage_castAdd_Ioi, vectorSpan_mono, not_infPrime, SuccOrder.forall_ne_bot_iff, Valuation.ltSubmodule_monotone, Pi.lt_toColex_update_self_iff, Cardinal.ofNat_lt_ofENat, Filter.OrderTop.atTop_eq, MeasureTheory.Egorov.notConvergentSeqLTIndex_spec, isPreconnected_Ico, convex_Ioo, CategoryTheory.MorphismProperty.map_le_iff, Cardinal.card_typein_toType_lt, MeasureTheory.integral_Ico_eq_integral_Ioo, Fin.prod_Iio_castAdd, Submodule.le_isotypicComponent, Metric.exists_continuous_nnreal_forall_closedEBall_subset, SimpleGraph.Colorable.chromaticNumber_le, Finsupp.wellFoundedLT', MeasureTheory.measure_pos_of_superset, AlgebraicGeometry.Scheme.restrictFunctorΓ_hom_app, add_le_iff, TopologicalSpace.Opens.set_range_inclusion', Partition.bot_notMem', SimpleGraph.edist_triangle, SimplexCategory.mono_iff_injective, Finset.dens_biUnion_le, HahnSeries.finiteArchimedeanClassOrderIsoLex_apply_snd, IsNoetherian.wellFoundedGT, CategoryTheory.ObjectProperty.IsStrongGenerator.subobject_eq_top, AntilipschitzWith.le_hausdorffMeasure_image, Associates.factors_mono, Multiset.le_inter_iff, AlgebraicGeometry.Scheme.Modules.restrictFunctorCongr_hom_app_app, LipschitzWith.mul_edist_le, ContinuousMap.nnnorm_lt_iff_of_nonempty, Ordinal.blsub_type, Metric.edist_le_infEDist_add_ediam, Affine.Simplex.affineCombination_mem_interior_iff, differentialIdeal_le_fractionalIdeal_iff, Fin.finsetImage_natAdd_Ico, Int.card_fintype_Ioc_of_le, AddSubgroup.codisjoint_addSubgroupOf_sup, AddLocalization.mk_le_mk, Set.Icc.instIsCancelMulZero, Set.Icc.mul_le_left, Cardinal.lift_eq_aleph1, Module.Basis.orientation_comp_linearEquiv_eq_iff_det_pos, PointedCone.coe_closure, Nucleus.range_subset_range, MeasureTheory.addContent_mono, IntermediateField.finrank_bot, Subgroup.disjoint_def, Subgroup.map_centralizer_le_centralizer_image, Finset.Ico_mul_Icc_subset', CategoryTheory.Limits.kernelSubobject_arrow_comp_apply, PNat.card_Ico, SMulPosReflectLT.toSMulPosReflectLE, EMetric.mem_ball', sameRay_neg_smul_left_iff_of_ne, sub_one_lt, symmDiff_self, Fin.sum_Ico_cast, le_symmDiff_iff_right, Multiset.ndunion_le, CFC.continuousOn_nnrpow, Set.empty_covBy_singleton, continuousAt_iff_continuous_left_right, symmDiff_triangle, Set.relatively_discrete_of_finite, Interval.coe_iInf, CategoryTheory.ComposableArrows.Exact.cokerIsoKer'_hom_inv_id, CategoryTheory.SmallObject.instIsIsoRightAppArrowMapToTypeOrdFunctorIterationFunctor, Finset.inf_dite_neg_le, nhds_le_uniformity, CategoryTheory.Pairwise.cocone_ι_app, AbsoluteValue.nonneg, Filter.NeBot.le_pure_iff, ConvexOn.convex_lt, TopCat.Sheaf.comp_app, conjneg_nonpos, sdiff_sdiff_self, Fin.castLEOrderEmb_toEmbedding, Orientation.oangle_sign_smul_add_smul_smul_add_smul, MulEquiv.mapSubgroup_symm_apply, ValuativeRel.RankLeOneStruct.strictMono, EReal.preimage_coe_Ioi_bot, LowerAdjoint.closure_sup_closure, integrableOn_Ico_iff_integrableOn_Ioo, AlgebraicGeometry.Scheme.Modules.restrictFunctorId_inv_app_app, smoothSheafCommRing.ι_forgetStalk_inv_apply, le_radius_cauchyPowerSeries, EMetric.infEdist_le_edist_add_infEdist, Birkhoff_inequalities, RootPairing.Base.not_nonpos_iff_pos_of_sum_mem_range_root, Dilation.ratio_pos, Ordinal.zero_opow_le, nnnorm_le_pi_nnnorm, Finset.Ici_succ_eq_Ioi, Finset.Ioc_mul_Ico_subset', BoxIntegral.Box.not_disjoint_coe_iff_nonempty_inter, Cardinal.small_Ioo, IntermediateField.fixingSubgroup_antitone, MonomialOrder.toSyn_monotone, MvPolynomial.degrees_sub_le, PowerBasis.adjoin_eq_top_of_gen_mem_adjoin, WittVector.ker_map_le_ker_mk_comp_ghostComponent, Summable.le_tsum', eventually_nhdsWithin_sign_eq_of_deriv_pos, Set.inv_mem_Ioc_iff, isCoatom_iff_eq_bot, Valuation.Integers.dvdNotUnit_iff_lt, WithTop.iInf_coe_lt_top, AbsoluteValue.pos_iff, Ideal.le_comap_of_ramificationIdx_ne_zero, LowerAdjoint.closure_union_closure_subset, AddSubgroup.map_le_iff_le_comap, MeasureTheory.eLpNorm_const_smul_le', PartOrdEmb.coe_comp, ENNReal.natCast_le_ofReal, NonUnitalAlgebra.eq_top_iff, Real.ofReal_le_enorm, QuasispectrumRestricts.lt_nnreal_iff, ENNReal.lintegral_Lp_add_le, MeasureTheory.piPremeasure_pi_mono, ProperCone.innerDual_union, AlgebraicGeometry.SheafedSpace.Γ_map, SimpleGraph.Subgraph.deleteEdges_le_of_le, Filter.map_mul_left_nhdsLT, LinearMap.le_def, sup_eq_maxDefault, nhds_eq_iInf_mabs_div, Finsupp.Colex.le_iff_of_unique, Fin.orderPred_apply, exists_le_mul_self, Algebra.algebraMapSubmonoid_le_comap, nndist_mul_mul_le, IsIntegrallyClosed.integralClosure_eq_bot, Padic.valuation_p_lt_one, Path.Homotopy.trans_apply, Finsupp.Lex.le_iff_of_unique, Finset.dens_le_dens_sdiff_add_dens, Order.pred_eq_sSup, LaurentSeries.intValuation_le_iff_coeff_lt_eq_zero, ENNReal.inv_le_inv, MeasureTheory.eLpNorm_indicator_const_le, Tropical.addLeftMono, NormedField.exists_one_lt_nnnorm, Prod.covBy_iff, nndist_triangle, Ideal.isSimpleOrder, PUnit.not_lt, BooleanSubalgebra.latticeClosure_subset_closure, continuous_iff_le_induced, Opens.coe_mayerVietorisSquare_X₁, CategoryTheory.Subobject.isoOfMkEq_inv, Fin.image_castSucc_Ici, AddSubgroup.toZModSubmodule_symm, IsIntegrallyClosed.integralClosure_eq_bot_iff, EReal.neg_lt_zero, AlgebraicGeometry.Scheme.Modules.pushforwardComp_hom_app_app, finite_integral_one_add_norm, sdiff_le_inf_hnot, Antitone.Ioi, SimpleGraph.DeleteFar.le_card_edgeFinset, wellFoundedLT_antisymmetrization_iff, Fin.finsetImage_val_Ico, Metric.infEDist_closure_pos_iff_notMem_closure, TopCat.Presheaf.IsSheaf.isSheafPreservesLimitPairwiseIntersections, Real.toNNReal_add_le, PointedCone.tmul_subset_maxTensorProduct, Ideal.span_singleton_le_iff_mem, algebraicClosure.le_restrictScalars, ENNReal.ofReal_le_one, BooleanAlgebra.le_iff_atom_le_imp, Subsemiring.toSubmonoid_strictMono, NNReal.agm_eq_agm_gm_am, Ordinal.instOrderTopology, CategoryTheory.Subobject.underlyingIso_arrow, Multiset.le_cons_erase, convex_convexHull, Disjoint.right_lt_sup_of_left_ne_bot, Subgroup.comap_sup_comap_le, ENNReal.half_le_self, MeasureTheory.FiniteMeasure.mass_comap_le, Finset.uIcc_map_sectL, TopologicalSpace.Opens.IsBasis.isOpenCover_mem_and_le, Fin.attachFin_uIcc, IsPGroup.bot_lt_center, CategoryTheory.ObjectProperty.le_isLocal_iff, AntilipschitzWith.le_mul_nndist, ENNReal.nhds_of_ne_top, MvPowerSeries.weightedOrder_mul_ge, Polynomial.IsSplittingField.adjoin_rootSet', Valuation.IsEquiv.lt_iff_lt, isGLB_iff_sInf_eq, Set.equitableOn_singleton, Interval.coe_coe, Finset.covBy_iff_card_sdiff_eq_one, IntermediateField.sepDegree_bot, Finset.monotone_filter_left, AlgebraicGeometry.Scheme.Hom.ker_eq_bot_of_isIso, ValuationSubring.unitGroup_strictMono, MvPowerSeries.le_order, AlgebraicGeometry.germ_stalkClosedPointIso_hom, MeasureTheory.exists_pos_ball, bddLat_dual_comp_forget_to_semilatSupCat, TopCat.Presheaf.isLocallySurjective_iff, Multiset.le_union_left, AbsoluteValue.IsEquiv.le_iff_le, StructureGroupoid.le_iff, sSup_lowerBounds_eq_sInf, Finset.dens_lt_dens, IsCompl.right_le_iff, Set.Iic_mul_bij, Function.argminOn_le, Subalgebra.starClosure_le_iff, ProbabilityTheory.Kernel.iIndep_comap_mem_iff, AddQuantale.add_bot, MvPolynomial.supported_mono, RootPairing.zero_le_posForm, StieltjesFunction.mono, monotone_int_of_le_succ, le_iff_reflTransGen_covBy, BumpCovering.nonneg', Finset.Iio_succ_eq_Iic, strictAnti_of_lt_sub_one, Set.unbounded_le_Ioi, ProbabilityTheory.bayesRisk_le_avgRisk, SimpleGraph.Subgraph.connected_iff_forall_exists_walk_subgraph, AlgebraicGeometry.Scheme.toSpecΓ_isoSpec_inv, Set.one_le_encard_insert, Submodule.rTensorOne_tmul, LieIdeal.comap_bracket_le, CategoryTheory.Subfunctor.orderIsoSubobject_apply, Ordinal.bddAbove_range, Int.Ico_filter_dvd_card, cauchySeq_finset_iff_prod_vanishing, AlgebraicGeometry.IsFinite.instHasAffinePropertyAndIsAffineFiniteCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTopHomAppTop, PNat.map_subtype_embedding_Ioo, Set.Nonempty.encard_pos, Set.compl_ordConnectedSection_ordSeparatingSet_mem_nhdsLE, convexHull_sum, PowerSeries.le_order_pow_of_constantCoeff_eq_zero, ProperCone.dual_zero, LowerSet.Iic_iInf₂, Fin.prod_Ico_cast, NNReal.rpow_lt_rpow_iff, Set.monotone_or_antitone_iff_uIcc, IsSimpleAddGroup.instIsSimpleOrderAddSubgroup, star_mul_self_nonneg, MonomialOrder.degLex_single_le_iff, AlgHom.IsArithFrobAt.le_comap, MeasureTheory.eLpNorm_sub_le', Cardinal.natCast_lt_aleph0, CategoryTheory.ComposableArrows.mkOfObjOfMapSucc_obj, Filter.neg_nhdsLT, strictConcaveOn_iff_slope_strict_anti_adjacent, coinduced_le_iff_le_induced, Order.length_le_height, tendsto_ratCast_atBot_iff, isLeast_pair, instTrichotomousGt, IntermediateField.adjoin_simple_le_iff, monotone_of_le_add_one, MeasureTheory.aecover_Icc_of_Icc, Ring.jacobson_le_of_isMaximal, LieIdeal.map_lowerCentralSeries_le, Set.pow_right_monotone, CategoryTheory.Triangulated.SpectralObject.mapTriangulatedFunctor_ω₁, LinearOrderedCommGroup.wellFoundedOn_setOf_le_lt_iff_nonempty_discrete, Ideal.le_spanNorm_spanNorm, posPart_pos_iff, Order.Iio_succ, Nimber.instWellFoundedLT, EReal.bot_lt_zero, AlgebraicGeometry.IsAffineOpen.instIsAffineToSchemeBasicOpen, AddSubgroup.comap_le_comap_of_surjective, AddCircle.coe_image_Ioc_eq, AlgebraicGeometry.Scheme.Modules.mapPresheaf_app, IntermediateField.finInsepDegree_bot', Module.Basis.flag_covBy, Finset.Ico_succ_succ, Nonneg.unitsHomeomorphPos_apply_coe, Cardinal.IsInaccessible.pos, TwoSidedIdeal.le_iff, Filter.tendsto_mul_const_atTop_iff, Finset.SupIndep.le_sup_iff, MeasureTheory.Measure.volume_subtype_coe_le_volume, eq_Icc_of_connected_compact, interior_Ioc, Filter.pure_le_principal, NormedField.exists_enorm_lt_one, ArchimedeanClass.map_mk_eq, MeasureTheory.Content.outerMeasure_pos_of_is_mul_left_invariant, AddSubmonoid.op_le_op_iff, Ideal.idealProdEquiv_symm_apply, Set.Ioi_pred_eq_Ici, CategoryTheory.MorphismProperty.le_coproducts, OrdinalApprox.gfpApprox_ord_eq_gfp, Filter.HasBasis.disjoint_iff, FinPartOrd.dual_map, UpperSet.coe_icisSupHom, MeasureTheory.all_ae_norm_ofReal_F_le_bound, AlgebraicGeometry.opensDiagram_obj, NNReal.rpow_inv_le_iff, Ideal.homogeneousHull_mono, MvPowerSeries.order_le, ENNReal.one_le_ofReal, ENNReal.nhdsGT_coe_neBot, spectrum.spectralRadius_pow_le, AlgebraicGeometry.PresheafedSpace.GlueData.f_invApp_f_app_assoc, le_iInf₂_iff, ENNReal.ofReal_lt_natCast, Multiset.Ico_inter_Ico, FirstOrder.Language.Substructure.closure_insert, Finset.Colex.toColex_lt_toColex_iff_exists_forall_lt, isBounded_convexHull, AlgebraicGeometry.Scheme.Hom.preservesLocalization_normalizationDiagramMap, Rat.preimage_cast_Iic, Cardinal.lift_lt, Finset.max'_eq_sorted_last, Finset.min'_subset, LinearLocallyFiniteOrder.instIsPredArchimedeanOfLocallyFiniteOrder, isExposed_empty, EReal.preimage_coe_Ioc, forall_lt_iff_le, SimpleGraph.IsVertexCover.vertexCoverNum_le, AlgebraicGeometry.Proj.basicOpen_mono, Submodule.toConvexCone_le_toConvexCone, Submodule.orderIsoMapComap_symm_apply', FinBoolAlg.hasForgetToFinPartOrd_forget₂_obj_carrier, nhdsGT_le_nhdsNE, Fin.map_succEmb_Ici, PredOrder.nhds_eq_pure, Submodule.orderIsoMapComapOfBijective_symm_apply, PrimeSpectrum.le_iff_mem_closure, add_lt_add_iff_left_of_ne_top, NNReal.isBoundedUnder_le_toReal, NNReal.continuous_sqrt, hasBasis_nhdsSet_Iic_Iio, LieSubmodule.map_comap_le, NatOrdinal.toOrdinal_natCast, SimplexCategory.II.map'_eq_last_iff, IntermediateField.restrictScalars_bot_eq_self, Set.Ioc_pred_left_eq_Icc, Module.End.mem_genEigenspace_top, ContinuousMultilinearMap.isLeast_opNNNorm, Multiset.instWellFoundedLT, Set.image_sub_const_Ioo, NNReal.HolderTriple.one_div_nonneg, Valued.closure_coe_completion_v_lt, List.minimum_eq_coe_iff, InnerProductSpace.nullSubmodule_le_ker_toDualMap_left, Dynamics.netEntropyInfEntourage_monotone, sup_bot_eq, Affine.Triangle.affineSpan_orthocenter_point_le_altitude, MulArchimedeanClass.mk_div_lt_mk_left_iff, Disjoint.le_symmDiff_sup_symmDiff_left, AddAction.fixingAddSubgroup_le_stabilizer, RingEquiv.idealComapOrderIso_symm_apply, Pi.Lex.wellFoundedLT, sign_mul, grade_lt_grade_iff, Language.instOrderedSub, BoundedLatticeHom.symm_dual_id, Cardinal.lt_wf, countable_setOf_covBy_left, posPart_eq_ite, Polynomial.isEquivalent_atBot_lead, Module.End.injOn_genEigenspace, Finset.mem_convexHull, Fin.find_congr, TopologicalSpace.Opens.mapMapIso_inverse, NNReal.image_coe_Ici, Fin.preimage_rev_Ioc, OrderEmbedding.preimage_uIcc, Codisjoint.map_orderIso, ZNum.cast_le, NonarchAddGroupNorm.le_def, ENNReal.mul_lt_top_iff, AddSubgroup.le_normalizer_map, Asymptotics.isBigO_atTop_iff_eventually_exists, pred_notMem_iff_isMin, EReal.image_coe_Ici, le_egauge_iff, OrderHom.isFixedPt_lfp, IsDiscreteValuationRing.addVal_add, StarAlgHom.adjoin_le_equalizer, AlgebraicGeometry.Spec.sheafedSpaceMap_hom_c_app, Finset.filter_snd_eq_antidiagonal, essSup_mono_measure, MeasureTheory.measure_le_measure_closure_of_levyProkhorovEDist_eq_zero, ConcaveOn.convex_hypograph, Subgroup.normalClosure_mono, SupHomClass.toOrderHomClass, SetTheory.PGame.equiv_nim_grundyValue, Ideal.map_le_comap_of_inverse, TopCat.Presheaf.Pushforward.comp_inv_app, AlgebraicGeometry.Scheme.toIso_inv_ι_assoc, Mathlib.Meta.NormNum.isRat_lt_false, Nonneg.coe_nnqsmul, AlgebraicGeometry.Scheme.instFullOppositeIdealSheafDataOverSubschemeFunctor, MvPolynomial.le_zeroLocus_iff_le_vanishingIdeal, TopCat.Presheaf.covering_presieve_eq_self, Mathlib.Meta.NormNum.isNat_ordinalLT_false, BooleanSubalgebra.closure_latticeClosure, AlgebraicGeometry.morphismRestrict_app', AlgebraicGeometry.SheafedSpace.restrictTopIso_inv, monotone_iff_map_nonpos, Module.Basis.orientation_comp_linearEquiv_eq_neg_iff_det_neg, mul_neg_iff, PrimeSpectrum.isIdempotentElemEquivClopens_symm_sup, Finset.nonneg_addConst_self, convexHull_neg, Nat.gc_ceil_coe, EReal.coe_neg', ContinuousMap.Homotopy.trans_apply, ENNReal.lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_eq_top, Ordinal.epsilon_succ_eq_nfp, monotoneOn_iff_forall_lt', MeasureTheory.tsum_meas_le_meas_iUnion_of_disjoint, Cardinal.aleph0_lt_mk, Cardinal.preBeth_lt_preBeth, Finset.Icc_pred_right_eq_Ico, AlgebraicGeometry.Scheme.evaluation_naturality_apply, Order.height_le_height_apply_of_strictMono, Module.End.genEigenspace_one, Profinite.NobelingProof.Products.lt_iff_lex_lt, upperHemicontinuous_iff_isOpen_preimage_Iic, Submodule.smul_le_right, Dynamics.coverEntropyInf_nonneg, Fin.map_natAddEmb_Icc, AlgebraicGeometry.opensCone_π_app, RingSubgroupsBasis.inter, FirstOrder.Language.Substructure.reduct_withConstants, convexIndependent_set_iff_notMem_convexHull_diff, Ideal.Filtration.antitone, AlgebraicGeometry.Scheme.toSpecΓ_appTop, Order.Ideal.top_toLowerSet, CategoryTheory.Dial.Hom.le, ofUnits_units_gc, mabs_div, OrderHom.gfp_gfp, minimal_mem_iff, MeasureTheory.le_iInf_lintegral, Valued.closure_coe_completion_v_mul_v_lt, le_egauge_inter, Fin.prod_Ioi_succ, Set.Ici_mul_Ioi_subset', ENNReal.tsum_le_tsum_comp_of_surjective, upperHemicontinuous_iff_preimage_Iic, MeasureTheory.Measure.le_liftLinear_apply, Asymptotics.isBigO_atTop_iff_eventually_exists_pos, MeasureTheory.setIntegral_neg_eq_setIntegral_nonpos, NNReal.min_le_agm, compl_le_self, Set.encard_iUnion_le_of_finite, not_lt_iff_eq_or_lt', Ordinal.lt_div, mul_self_lt_one_iff, Complex.inv_natCast_cpow_ofReal_pos, AlgebraicGeometry.Scheme.Hom.appIso_inv_naturality_assoc, Sublattice.comap_mono, Order.Coframe.top_sdiff, ONote.le_def, IntermediateField.fg_bot, RootPairing.RootPositiveForm.rootLength_pos, measurable_IicProdIoc, Finset.image_subset_Iic_sup, Algebra.inf_toSubmodule, le_nonZeroDivisorsLeft_iff_isLeftRegular, Dynamics.IsDynCoverOf.coverEntropyEntourage_le_log_card_div, addLeftStrictMono_of_addLeftReflectLE, AlgebraicGeometry.IsOpenImmersion.opensEquiv_symm_apply, Ideal.IsRadical.radical_le_iff, Antitone.Ici, Int.Ioc_eq_finset_map, uniformity_basis_edist_nnreal_le, Filter.limsInf_top, AlgebraicGeometry.Scheme.nilradical_eq_bot, PointedCone.dual_zero, Finset.sum_le_sum_of_subset, PrincipalSeg.cocone_pt, List.monotone_sum_take, nndist_triangle_left, CategoryTheory.Triangulated.SpectralObject.triangle_mor₂, zero_lt_one', SupPrime.le_sup, LinearEquiv.transvections_pow_mono, Hyperreal.infinitesimal_neg_iff_infiniteNeg_inv, Finset.sup'_le_iff, instIsStrictOrderedModuleOfIsCancelAdd, ContinuousWithinAt.finset_sup, MeasureTheory.Ico_ae_eq_Ioc', CategoryTheory.MorphismProperty.le_pullbacks, Set.uIcc_eq_union, imageToKernel_epi_of_zero_of_mono, convexHull_sphere_eq_closedBall, AlgebraicGeometry.Scheme.IdealSheafData.support_eq_bot_iff, Filter.mulLeftMono, add_eq_right_iff_le, Dynamics.coverEntropyEntourage_finite_of_isCompact_invariant, PMF.apply_lt_top, Subsemiring.opEquiv_symm_apply, CategoryTheory.Presheaf.isLocallySurjective_iff_range_sheafify_eq_top', ENat.LEInfty.out, Set.pairwise_disjoint_Ioo_zpow, smoothSheaf.obj_eq, MeasureTheory.restrict_Ioi_eq_restrict_Ici, Fin.finsetImage_cast_Ioo, NatOrdinal.instAddLeftStrictMono, CategoryTheory.ObjectProperty.epimorphisms_le_epiModSerre, Int.instIsPredArchimedean, HasFPowerSeriesAt.eventually, Cardinal.instWellFoundedLT, TopologicalSpace.Closeds.isAtom_iff, LowerSet.Iic_le, Subring.unop_le_unop_iff, inf_lt_sup, TopCat.subsheafToTypes_val, MeasureTheory.Measure.finiteAt_principal, GroupTopology.toTopologicalSpace_bot, AddSubmonoid.toNatSubmodule_symm, Hyperreal.infiniteNeg_def, CategoryTheory.MorphismProperty.antitone_rlp, MulArchimedeanClass.mk_right_le_mk_mul_iff, AddEquiv.strictMono_subsemigroupCongr, Ordinal.veblenWith_lt_veblenWith_iff_right, Fin.image_natAdd_Ioc, IsMulIndecomposable.baseOf_subset_one_lt, Order.krullDim_le_one_iff_of_boundedOrder, isAtomic_of_isCoatomic_of_complementedLattice_of_isModular, OrderAddMonoidHom.coe_add, TopologicalSpace.Opens.mapId_inv_app, gc_Ici_sInf, Fin.card_Iio, MvPolynomial.degrees_sum_le, Finset.codisjoint_right, Cardinal.ord_lt_omega0, AlgHom.range_eq_top, smul_nonpos_iff_neg_imp_nonneg, Equiv.Perm.centralizer_le_alternating_iff, AddSemigroupIdeal.instWellFoundedGT, LieAlgebra.derivedSeriesOfIdeal_antitone, LinearMap.polar_gc, Algebra.adjoin_singleton_one, CategoryTheory.ShortComplex.mapToComposableArrows_app_2, Set.Ico_diff_Iio, FloorSemiring.gc_ceil, AlgebraicGeometry.PresheafedSpace.comp_c_app_assoc, Order.Ideal.IsMaximal.isCoatom', NNReal.isCoboundedUnder_ge_toReal, CategoryTheory.Subfunctor.Subpresheaf.epi_iff_range_eq_top, Set.unbounded_ge_iff_unbounded_inter_ge, AntitoneOn.Iic, isPiSystem_Ico, Set.finite_iff_bddBelow, NonUnitalAlgebra.mem_top, Valued.cauchy_iff, sInf_iUnion_Ici, Filter.not_le, MeasureTheory.exists_measure_pos_of_not_measure_iUnion_null, IsCoatomic.Set.Ici.isCoatomic, LinearMap.hasEigenvalue_zero_tfae, LowerAdjoint.subset_closure, MeasureTheory.measure_mono_ae, CategoryTheory.MorphismProperty.IsStableUnderTransfiniteCompositionOfShape.le, addRothNumber_singleton, Ideal.comap_le_comap_iff_of_surjective, MvPolynomial.schwartz_zippel_sup_sum, CategoryTheory.HasLiftingProperty.transfiniteComposition.SqStruct.map_f', Filter.filter_injOn_Iic_iff_injOn, Algebra.adjoin_eq_span_of_subset, Set.Icc.le_one, MeasureTheory.le_extend, Matroid.eRk_union_le_eRk_add_eRk, Asymptotics.IsEquivalent.tendsto_atBot_iff, MeasureTheory.VectorMeasure.le_restrict_empty, MeasureTheory.eLpNorm'_le_eLpNorm'_of_exponent_le, lift_rank_lt_rank_dual, NonUnitalStarAlgebra.toNonUnitalSubalgebra_eq_top, Nonneg.unitsEquivPos_apply_coe, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.isoRestrict_hom_hom_c_app, AlgebraicGeometry.IsAffineOpen.iSup_basicOpen_eq_self_iff, SSet.Subcomplex.image_top, Matroid.eRk_le_eRk_inter_add_eRk_diff, UpperSet.lt_sdiff_left, Valuation.one_apply_le_one, CategoryTheory.ComposableArrows.naturality', Cardinal.toENatAux_gc, Real.toNNReal_lt_one, Finset.sum_insertNone, AlgebraicGeometry.Scheme.Hom.comp_appIso, LSeries.abscissaOfAbsConv_binop_le, AddCircle.card_torsion_le_of_isSMulRegular_int, enorm_pos, isLUB_atoms_top, EReal.sign_bot, LinearOrderedAddCommGroup.Subgroup.negGen_neg, circleMap_preimage_codiscrete, ENNReal.iUnion_Icc_coe_nat, mul_star_self_pos, OrderIso.withBotCongr_refl, AlgebraicGeometry.Scheme.basicOpen_one, SummationFilter.support_eq_univ_iff, HahnSeries.min_order_le_order_add, isCompact_iff_ultrafilter_le_nhds, PointedCone.minTensorProduct_le_maxTensorProduct, Subgroup.le_prod_iff, transfiniteIterate_bot, Ordinal.not_principal_iff, Set.Finite.supClosure, le_perfectClosure, ENNReal.mulRightOrderIso_toEquiv, CategoryTheory.TransfiniteCompositionOfShape.ici_isoBot, covBy_iff_lt_and_eq_or_eq, ContinuousLinearMap.nonlinearRightInverseOfSurjective_def, Set.compl_Iic, Set.Iic.coe_iInf, Fin.finsetImage_val_Ici, Filter.comap_mabs_atTop, Prod.wcovBy_iff, nhdsWithin_Iio_isMeasurablyGenerated, comap_coe_nhdsLT_eq_atTop_iff, Set.image_add_const_Ici, Cardinal.toNat_toENat, CompleteSublattice.coe_bot, PNat.Ioc_eq_finset_subtype, EReal.bot_lt_inv, IsRingFiltration.toIsFiltration, NNReal.exists_pos_sum_of_countable, Order.not_isSuccLimit, EMetric.isUniformEmbedding_iff', Mathlib.Meta.NormNum.isInt_le_false, Multiset.filter_le, Fin.preimage_natAdd_Ico_natAdd, LinearOrderedAddCommGroupWithTop.neg_top, LieSubmodule.ucs_eq_top_iff, Finset.Iic_eq_cons_Iio, LieSubmodule.top_lie_le_iff_le_normalizer, Pi.sum_nnnorm_apply_le_nnnorm', AffineSubspace.affineSpan_eq_top_iff_vectorSpan_eq_top_of_nonempty, Ideal.map_radical_le, MeasureTheory.OuterMeasure.iUnion_nat, ArchimedeanClass.mk_left_le_mk_add_iff, Ordinal.IsNormal.monotone, IsPGroup.exists_le_sylow, IntermediateField.adjoin_empty, Cardinal.aleph_max, Ideal.quotEquivOfEq_eq_factor, Finset.sort_sorted_lt, NonUnitalAlgebra.elemental.le_of_mem, Finset.le_card_falling_div_choose, MulArchimedeanClass.mk_one, NonemptyInterval.snd_sup, Cardinal.IsStrongLimit.isSuccPrelimit, IsAdicComplete.le_jacobson_bot, Convex.exists_subset_interior_convexHull_finset_of_isCompact, Interval.coe_top, PrincipalSeg.orderIsoIio_apply_coe, zero_lt_three', lt_sup_iff, Finset.Ioc_pred_right_eq_Ioo, DividedPowers.IsSubDPIdeal.isSubideal, SimplexCategory.rev_map_apply, Finset.Iic_eq_powerset, NonUnitalSubalgebra.centralizer_le, rieszContentAux_sup_le, exists_pos_left_iff_sameRay, Subgroup.map_subtype_lt_map_subtype, Metric.infEDist_anti, LSeries.abscissaOfAbsConv_add_le, Subgroup.le_closure_toSubmonoid, IntermediateField.normalClosureOperator_apply, PartENat.ne_zero_iff, Finset.Icc_erase_right, ConvexOn.slope_mono, AffineSubspace.topEquiv_symm_apply_coe, Subgroup.map_le_map_iff, Pi.Colex.sInf_apply_le, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.app_invApp, Filter.map_surjOn_Iic_iff_surjOn, Module.Basis.flag_lt_flag, Ideal.le_toIdeal_homogeneousHull, Finset.prod_min_le, Set.Ico_succ_left_eq_Ioo, Int.floor_eq_iff, FrameHom.coe_toInfTopHom, ciSup_of_empty, ArchimedeanClass.FiniteResidueField.mk_eq_mk, Set.Ici_iSup₂, Cardinal.isSuccLimit_aleph0, ArchimedeanClass.stdPart_monotoneOn, AnalyticAt.eventually_constant_or_nhds_le_map_nhds_aux, PrincipalSeg.lt_apply_iff, abs_nonpos_iff, Ordinal.toNimber_one, CauchySeq.tendsto_limUnder, MeasureTheory.hasFiniteIntegral_iff_ofReal, CompactlySupportedContinuousMap.le_def, Concept.intent_ssubset_intent_iff, AlgebraicGeometry.Scheme.Modules.Hom.id_app, monovaryOn_iff_smul_rearrangement, Odd.strictMono_pow, cfcₙHom_le_iff, cauchySeq_finset_iff_tprod_vanishing, Cardinal.ord_preAleph, SSet.orderEmbeddingN_apply, Fin.bot_eq_zero, convexHull_mono, lowerClosure_add, AddAction.stabilizer_inf_stabilizer_le_stabilizer_inter, map_div_le_add, Subsemigroup.mk_le_mk, NNRat.instCanonicallyOrderedAdd, NNReal.isOpen_Ico_zero, AlgebraicGeometry.targetAffineLocally_affineAnd_iff, Subsemigroup.monotone_comap, CStarAlgebra.mem_Icc_algebraMap_iff_nnnorm_le, AlgebraicGeometry.StructureSheaf.algebraMap_obj_top_bijective, MeasureTheory.OuterMeasure.comap_mono, Submodule.le_traceDual, Module.End.invtSubmodule.mk_eq_top_iff, Fin.strictMono_succ, Concept.extent_subset_extent_iff, ContractingWith.exists_fixedPoint, TopologicalSpace.Opens.inclusion'_map_eq_top, Subalgebra.linearDisjoint_iff, InfHom.withTop'_toFun, ProbabilityTheory.bayesRisk_le_iInf, Finset.prod_prod_Ioi_mul_eq_prod_prod_off_diag, AlgebraicGeometry.Scheme.Hom.appIso_inv_app_assoc, CategoryTheory.Subfunctor.le_def, InnerProductSpace.gramSchmidt_def', CategoryTheory.Limits.kernelSubobjectIso_comp_kernel_map_assoc, Finset.toDual_inf, closure_Iio, Fin.image_castSucc_Ioi, Interval.coe_one, Multiset.sub_le_iff_le_add, Fin.cast_strictMono, CategoryTheory.Subobject.instIsEquivalenceMonoOverRepresentative, Set.not_disjoint_iff, TopCat.Sheaf.pushforward_obj_val, IntermediateField.rank_bot, convex_absConvexHull, Ordinal.omega0_lt_omega1, Int.card_Icc_of_le, PMF.coe_le_one, RootPairing.pairingIn_le_zero_iff, NonUnitalSubalgebra.gc_map_comap, CategoryTheory.SmallObject.SuccStruct.extendToSucc.obj_eq, Subalgebra.centralizer_eq_top_iff_subset, Ordinal.bfamilyOfFamily_typein, LowerSet.iicInfHom_apply, Ordinal.pred_lt_iff_not_isSuccPrelimit, CauchyFilter.monotone_gen, Order.le_iff_eq_or_le_pred', PrimeSpectrum.basicOpen_mul_le_left, WithTop.tendsto_untopA, CommMonoid.primaryComponent.disjoint, Set.monotone_mem, Valuation.leIdeal_v_le_of_mem, NonUnitalSubsemiring.map_le_iff_le_comap, Nat.Ico_filter_modEq_cast, SimpleGraph.Walk.toSubgraph_le_iff, Fin.strictMono_iff_lt_succ, CategoryTheory.TransfiniteCompositionOfShape.iic_F, MeasureTheory.measure_lt_top_of_isCompact_of_isAddLeftInvariant, Set.image_inv_Iio, CategoryTheory.HasLiftingProperty.transfiniteComposition.SqStruct.w, Set.isCoatom_singleton_compl, Metric.ediam_mono, Set.disjoint_iff_inter_eq_empty, le_inf_iff, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_self, Finset.Colex.insert_le_insert, Order.PFilter.sInf_gc, iSup_eq_dif, OrderEmbedding.supIrredLowerSet_surjective, PartOrdEmb.inv_hom_apply, Finset.expect_mul_sq_le_sq_mul_sq, CategoryTheory.Limits.ProductsFromFiniteCofiltered.finiteSubproductsCone_pt, NNRat.coe_le_coe, ENNReal.div_le_iff', CategoryTheory.SmallObject.SuccStruct.restrictionLTOfCoconeIso_hom_app, Mathlib.Meta.Positivity.ofReal_pos, MeasureTheory.L1.nnnorm_Integral_le_one, tendsto_ceil_left_pure, self_mul_sign, CategoryTheory.Limits.CompleteLattice.pushout_eq_sup, ContinuousWithinAt.lowerSemicontinuousWithinAt, IsFractionRing.coeSubmodule_le_coeSubmodule, Ordinal.mk_toPGame, Polynomial.roots_expand_pow_map_iterateFrobenius_le, Seminorm.coe_bot, separableClosure_inf_perfectClosure, Cardinal.mk_strictMono, Metric.coveringNumber_pos_iff, Set.uIcc_subset_uIcc_iff_le, LieSubalgebra.instCanonicallyOrderedAdd, AlgebraicGeometry.Scheme.Hom.ker_eq_top_iff_isEmpty, convexHullAddMonoidHom_apply, Ordinal.isNormal_iff_lt_succ_and_blsub_eq, Prod.mk_covBy_mk_iff_left, CategoryTheory.instEpiMap'KernelCokernelCompSequenceOfNatNat, Submodule.goursatFst_prod_goursatSnd_le, cocompact_eq_atBot, Finset.card_Ioc_eq_card_Icc_sub_one, MeasureTheory.IsHahnDecomposition.ge_on_compl, AlgebraicGeometry.ProjectiveSpectrum.Proj.toSpec_base_apply_eq, eq_zero_or_pos, IsSemisimpleModule.sSup_simples_le, LieModule.lowerCentralSeriesLast_le_max_triv, Fin.card_Ioo, RingSeminormClass.toNonnegHomClass, ENat.toENNReal_mono, ExpGrowth.expGrowthSup_mul_le, MeasureTheory.levyProkhorovEDist_le_max_measure_univ, OrderTopology.to_orderClosedTopology, Submodule.comapMkQOrderEmbedding_eq, Ordinal.cof_eq_cof_toType, SummationFilter.instLeAtTopSymmetricIoc, RootPairing.pairingIn_lt_zero_iff, ne_iff_gt_iff_ge, Convex.mem_toCone, MeasureTheory.ae_restrict_le, Filter.comap_mul_comap_le, Right.inv_nonneg, AddSubgroup.opEquiv_symm_apply, LinearGrowth.linearGrowthInf_natCast_nonneg, Fin.map_valEmbedding_Icc, le_restrictScalars_separableClosure, NonemptyInterval.length_add, CategoryTheory.ComposableArrows.sc'Map_τ₁, Fin.map_castSuccEmb_Ici, LinearLocallyFiniteOrder.succFn_spec, AlgebraicGeometry.Scheme.zeroLocus_eq_univ_iff_subset_nilradical_of_isCompact, List.argmax_eq_some_iff, uniformity_basis_edist_inv_two_pow, absConvexHull_mono, inf_compl_self, ENNReal.lt_top_of_mul_ne_top_left, AddOpposite.op_nonpos, Cardinal.isStrongLimit_beth, ENat.floor_mono, MeasureTheory.Measure.add_top, MeasureTheory.Measure.LebesgueDecomposition.iSup_monotone', Heyting.Regular.coe_top, MonoidWithZeroHom.inr_mono, Coheyting.boundary_inf_le, biUnion_Iic_disjointed, Set.preimage_add_const_Ioo, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.instIsIsoCommRingCatInvApp, IsValuativeTopology.hasBasis_nhds_zero', CompleteLattice.IsSupClosedCompact.wellFoundedGT, LieAlgebra.derivedSeriesOfIdeal_add_le_add, inf_le_bihimp, grade_mono, AlgebraicGeometry.LocallyRingedSpace.Γevaluation_naturality_apply, iSup_const_mono, Set.Icc.isCompl_iff, RingHom.liftOfRightInverse_comp_apply, IsStrictOrderedRing.toNoMaxOrder, TopCat.Presheaf.stalkFunctor_map_germ_apply, AddSubgroup.ofAddUnits_mono, IsMulFreimanIso.mulRothNumber_congr, Finset.convexHull_eq, MeasureTheory.Measure.mem_support_iff_forall, Algebra.top_toSubsemiring, TopCat.Presheaf.germ_res_apply', Units.nnnorm_pos, Mathlib.Meta.NormNum.isRat_le_false, t2Space_iff_disjoint_nhds, Subalgebra.rTensorBot_tmul_one, CategoryTheory.ShortComplex.ab_exact_iff_ker_le_range, Ordinal.preOmega_zero, EMetric.tendsto_atTop, Set.neg_mem_Icc_iff, Fin.image_castAdd_Icc, Subgroup.closure_le_centralizer_centralizer, Set.Infinite.exists_lt_map_eq_of_mapsTo, FormalMultilinearSeries.le_radius_of_isBigO, CategoryTheory.Precoverage.mem_toGrothendieck_iff_of_isStableUnderComposition, Ideal.height_le_ringKrullDim_quotient_add_encard, OrderIso.dualAntisymmetrization_apply, Set.Iic_sub_one_eq_Iio, Multiset.nodup_iff_le, Fintype.coe_finsetOrderIsoSet_symm, PrimeSpectrum.exists_primeSpectrum_prod_le, MeasureTheory.eLpNorm_indicator_le_of_bound, TopCat.Opens.coverDense_inducedFunctor, Multiset.cons_le_of_notMem, Specializes.nhds_le_nhds, Subalgebra.map_le, SimplexCategory.mkOfSucc_homToOrderHom_zero, le_egauge_prod, AlgebraicGeometry.Scheme.IdealSheafData.map_top, IsLocalRing.PrimeSpectrum.asIdeal_top, sdiff_eq_bot_iff, CategoryTheory.monotone_chain_condition_of_isNoetherianObject, Dynamics.coverEntropy_image_le_of_uniformContinuousOn_invariant, ConvexCone.disjoint_coe, totallyBounded_convexHull, Cardinal.small_Ioc, Polynomial.div_tendsto_leadingCoeff_div_of_degree_eq, Topology.IsScott.scottHausdorff_le, ENNReal.lt_ofReal_iff_toReal_lt, Sylow.exists_subgroup_le_card_pow_prime_of_le_card, MeasureTheory.VectorMeasure.zero_le_restrict_not_measurable, CompletelyNormalSpace.completely_normal, AddSubsemigroup.comap_strictMono_of_surjective, Set.preimage_mul_const_Ici, CategoryTheory.MorphismProperty.le_llp_rlp, PseudoEpimorphismClass.toTopHomClass, PNat.find_lt_iff, inf_eq_inf_iff_right, NNReal.HolderTriple.pos', Equiv.Perm.toList_formPerm_nontrivial, AlgebraicGeometry.Scheme.Modules.Hom.isIso_iff_isIso_app, Filter.not_tendsto_pow_atTop_atBot, Fin.attachFin_Ico, Set.Ioc_union_Ioc_symm, le_neg_self_iff, upperSemicontinuousOn_iff_frequently, MeasureTheory.Measure.FiniteSpanningSetsIn.finite, ContinuousLinearEquiv.subsingleton_or_nnnorm_symm_pos, sdiff_le, NNReal.le_rpow_inv_iff, edist_lt_coe, Set.one_le_chainHeight_iff, Tactic.NormNum.isNNRat_nnrealSqrt_of_isNNRat, Set.Nonempty.absConvexHull, Ordinal.aleph0_le_card, imageToKernel_comp_left, enorm_lt_coe, Real.toNNReal_mono, MeasureTheory.restrict_Ico_eq_restrict_Icc, FractionalIdeal.mul_le, Additive.canonicallyOrderedAdd, NonUnitalSubsemiring.prod_mono_left, Finset.orderEmbOfCardLe_mem, le_egauge_smul_right, tangentBundleCore.isContMDiff, Module.FaithfullyFlat.range_le_ker_of_exact_rTensor, exists_isLocalExtr_uIoo, pow_dvd_iff_le_emultiplicity, SetLike.coe_ssubset_coe, ENat.epow_left_mono, Finset.Iic_erase, Submonoid.LocalizationMap.map_nonZeroDivisors_le, Finset.sum_Ioi_add_eq_sum_Ici, LieSubmodule.map_incl_le, AlgebraicGeometry.Spec_zeroLocus_eq_zeroLocus, SSet.face_le_horn, Interval.length_nonneg, Dense.borel_eq_generateFrom_Ico_mem, codisjoint_toDual_iff, Mathlib.Meta.NormNum.isNNRat_le_true, SSet.Subcomplex.image_preimage_le, SimpleGraph.cliqueNum_le_chromaticNumber, OrderHom.coe_iInf, IsSelfAdjoint.exp_nonneg, sdiff_eq_sInf, MeasureTheory.Measure.mutuallySingular_iff_disjoint_ae, MeasureTheory.lintegral_nnnorm_condExpL2_indicator_le_real, Ideal.IsPrime.mul_le, tendsto_intCast_atTop_iff, PointedCone.tmul_subset_minTensorProduct, Nonneg.coe_inv, TopologicalSpace.noetherianSpace_TFAE, Nucleus.coe_bot, MeasureTheory.lintegral_union_le, Polynomial.eventually_atTop_not_isRoot, AlgebraicGeometry.Scheme.Opens.ι_appTop, Order.pred_le_le_iff, Fin.prod_Iic_cast, Finset.fold_max_le, Filter.disjoint_principal_principal, SimplexCategory.II.map'_succAboveOrderEmb, Submodule.map_le_isotypicComponent, AlgebraicGeometry.Scheme.Modules.Hom.comp_app, AlgebraicGeometry.Scheme.Hom.map_mem_image_iff, Order.lt_succ_iff, LinearOrderedField.inducedMap_mono, Polynomial.natTrailingDegree_le_trailingDegree, CategoryTheory.Limits.CompleteLattice.finite_limit_eq_finset_univ_inf, Set.OrdConnected.isSuccArchimedean, Multiset.map_add_right_Ioo, Finset.sum_eraseNone, MeasureTheory.integral_pos_iff_support_of_nonneg, Fin.image_castAdd_Ioc, Condensed.instFinalOppositeDiscreteQuotientCarrierToTopTotallyDisconnectedSpaceCostructuredArrowFintypeCatProfiniteOpToProfiniteOpPtAsLimitConeFunctorOp, setOf_isPreconnected_subset_of_ordered, SimplexCategory.instReflectsIsomorphismsForgetOrderHomFinHAddNatLenOfNat, MeasureTheory.integral_Icc_eq_integral_Ico, ContinuousWithinAt.partialSups, Valuation.Integers.maximalIdeal_eq_setOf_le_v_algebraMap, RingSeminormClass.toSubmultiplicativeHomClass, tsirelson_inequality, Set.InvOn.one_sub_one_add_inv, AddMonoid.Coprod.codisjoint_range_inl_range_inr, TopologicalSpace.Opens.overEquivalence_counitIso_inv_app, nhdsSet_Iic, Ordinal.isInitial_succ, Ordinal.natCast_succ, AddOpposite.nonneg_op, RingHom.strictMono_specComap_of_surjective, MulAction.stabilizer_le_aestabilizer, Ultrafilter.le_nhds_lim, WithZero.instMulLeftReflectLT, Filter.tendsto_finset_preimage_atTop_atTop, Finset.sup_inf, exteriorPower.pairingDual_apply_apply_eq_one, OrderIso.sumLexIicIoi_apply_inr, CategoryTheory.isCofiltered_of_semilatticeInf_nonempty, Order.isLUB_Iio_iff_isSuccPrelimit, NNReal.rpow_add_rpow_le, cauchySeq_const, ContinuousLinearMap.le_def, mulPosReflectLE_iff_mulPosReflectLT, EMetric.edist_le_infEdist_add_ediam, Set.image_const_mul_Iic, Prod.swap_covBy_swap, StarAlgebra.elemental.le_centralizer_centralizer, LatticeHom.withTopWithBot_toFun, Submodule.le_span_singleton_iff, Ordinal.coe_preOmega, Set.Icc.mem_iff_one_sub_mem, Subring.toSubsemiring_strictMono, CategoryTheory.IsCardinalFiltered.exists_cardinal_directed.functor_obj, BoundedLatticeHom.map_bot', BoxIntegral.Box.monotone_upper, LieModule.isNilpotent_quotient_iff, ENat.le_ceil, Subfield.bot_eq_of_charZero, LinearOrderedAddCommGroup.wellFoundedOn_setOf_ge_gt_iff_nonempty_discrete, Ordering.compares_iff_of_compares_impl, AddGroupSeminorm.coe_le_coe, pred_max, Set.exists_upper_bound_image, tendsto_floor_left', ENNReal.lt_iff_exists_nnreal_btwn, Subring.le_pointwise_smul_iff₀, Polynomial.map_roots_le_of_injective, Ordinal.veblen_zero_le_veblen_zero, Finset.Icc_eq_singleton_iff, spectrum.limsup_pow_nnnorm_pow_one_div_le_spectralRadius, stdSimplexHomeomorphUnitInterval_apply_coe, edist_add_add_le, Multiplicative.mem_toSubgroup, Filter.comap_le_iff_le_kernMap, Cardinal.preAleph_max, Order.krullDim_eq_iSup_coheight, Set.Iic.coe_biInf, RootPairing.coxeterWeightIn_le_four, List.sortedLT_iff_strictMono_get, Finset.pluennecke_ruzsa_inequality_pow_div, Set.Iic.coe_sSup, Set.antitone_bforall, RegularSpace.regular, Order.succ_injective, LSeries.abscissaOfAbsConv_le_of_forall_lt_LSeriesSummable', Cardinal.beth_pos, Finset.sum_Ico_add_right_sub_eq, span_le_span_iff, IsIntegrallyClosedIn.integralClosure_eq_bot_iff, CategoryTheory.MorphismProperty.le_llp_iff_le_rlp, CategoryTheory.Subobject.ofMkLEMk_comp_ofMkLEMk, MulArchimedeanClass.map_mk_eq, Fin.finsetImage_addNat_Ico, OrderIso.map_ciSup', AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_snd, TopCat.Presheaf.presieveOfCovering.indexOfHom_spec, EuclideanGeometry.Sphere.direction_orthRadius_le_iff, CompletelyDistribLattice.le_himp_iff, AffineMap.map_top_of_surjective, Ordinal.typein_injective, PrimeSpectrum.finite_setOf_isMin, EReal.exp_le_one_iff, IsPGroup.disjoint_of_ne, ProbabilityTheory.gaussianReal_map_mul_const, ENNReal.iSup_coe_eq_top, wellFoundedGT_antisymmetrization_iff, EReal.sub_nonpos, MeasureTheory.integral_Iic_eq_integral_Iio, MeasureTheory.stoppedProcess_eq'', Set.preimage_const_div_Ici, dimH_le_of_hausdorffMeasure_ne_top, Set.Iic_union_Ioi, AlgebraicGeometry.Scheme.map_basicOpen, BoundedContinuousFunction.nndist_coe_le_nndist, IsLocalization.nonZeroDivisors_le_comap, extDerivWithin_apply_vectorField, Finset.truncatedInf_singleton, CategoryTheory.MorphismProperty.coproducts_le, MeasureTheory.integrableOn_Ioi_iff_integrableAtFilter_atTop_nhdsWithin, PowerSeries.min_order_le_order_add, CategoryTheory.Limits.FormalCoproduct.cechFunctor_map_app, Set.Ioi_True, Finset.coe_eraseNone, Set.image_const_sub_Ioi, MeasureTheory.exists_pos_setLIntegral_lt_of_measure_lt, AlgebraicGeometry.IsOpenImmersion.affineOpensEquiv_apply_coe_coe, TangentBundle.contMDiffVectorBundle, bddLat_dual_comp_forget_to_semilatInfCat, MeasureTheory.Measure.one_le_hausdorffMeasure_zero_of_nonempty, CategoryTheory.Subfunctor.Subpresheaf.top_obj, MeasureTheory.IsSetRing.partialSups_mem, sup_mem_supClosure, Topology.scottHausdorff_le_scott, IsStarProjection.le_one, TopologicalSpace.Opens.adjunction_counit_map_functor, MeasureTheory.Lp.instAddLeftMono, Set.Ici_inj, segment_eq_image', MeasureTheory.MemLp.exists_hasCompactSupport_eLpNorm_sub_le, CategoryTheory.OrthogonalReflection.iteration_map_succ_assoc, cmp_sub_zero, ContinuousOn.cfcₙ_nnreal_of_mem_nhdsSet, Set.image_const_div_Iio, sSupIndep.disjoint_sSup, MeasureTheory.Measure.rnDeriv_pos, Subring.op_le_iff, half_lt_self_iff, CategoryTheory.Triangulated.SpectralObject.Hom.comm_assoc, TopCat.Presheaf.generateEquivalenceOpensLe_functor, Ordinal.ord_cof_le, SimpleGraph.ediam_le_two_mul_radius, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivUnitIso_inv_app_hom, Cardinal.type_cardinal, AlgebraicGeometry.Scheme.IdealSheafData.opensRange_subschemeCover_map, IsCompl.inf_eq_bot, Cardinal.preBeth_le_preBeth, Set.image_mul_const_Iic, Nat.not_ofNat_lt_one, MeasureTheory.Measure.smul_measure_isMulInvariant_le_of_isCompact_closure, CategoryTheory.Sieve.functorPullback_monotone, FormalMultilinearSeries.le_comp_radius_of_summable, AlgebraicGeometry.Proj.zero_apply, Rat.preimage_cast_Ici, nnnorm_inner_le_nnnorm, NNReal.sqrt_eq_one, PartENat.natCast_lt_top, Set.disjoint_insert_right, NonUnitalIsometricContinuousFunctionalCalculus.isGreatest_nnnorm_quasispectrum, Set.pairwiseDisjoint_nhds, Fin.map_natAddEmb_Ico, ClosureOperator.closure_sup_closure_left, lowerBounds_infClosure, isPreconnected_Ioc, Filter.prod_le_prod, LowerSet.mem_map, Set.pairwise_disjoint_Ioc_intCast, ArchimedeanClass.ballAddSubgroup_antitone, FractionalIdeal.mul_one_div_le_one, lowerCentralSeries_pi_le, sbtw_mul_sub_add_iff, IsometricContinuousFunctionalCalculus.nnnorm_spectrum_le, gc_nhdsKer_interior, MeasureTheory.setLIntegral_const_lt_top, AddCircle.openPartialHomeomorphCoe_source, NonUnitalAlgebra.adjoin_univ, Positive.isOrderedMonoid, Subalgebra.map_toSubmodule, le_mulRothNumber_product, sign_apply, Finset.nsmul_right_strictMono, Polynomial.Splits.dvd_iff_roots_le_roots, HahnSeries.embDomainOrderAddMonoidHom_injective, Finset.card_Ioo_eq_card_Ioc_sub_one, PosNum.cast_le, sq_lt_one_iff_abs_lt_one, AlgebraicGeometry.isRetrocompact_basicOpen, one_lt_star_iff, UpperSet.Ici_one, AlgebraicGeometry.Scheme.Hom.preimageIso_hom_ι, MeasureTheory.OuterMeasure.le_comap_map, CategoryTheory.ComposableArrows.mk₀_obj, lowerCentralSeries.map, MulOpposite.one_le_unop, Finite.exists_min, Filter.isBoundedUnder_le_inv, Subgroup.ofUnits_mono, Homeomorph.opensCongr_symm, Profinite.NobelingProof.Products.lt_nil_empty, Continuous.cfc_nnreal, nnnorm_pow_le_mul_norm, StalkSkyscraperPresheafAdjunctionAuxs.germ_fromStalk_assoc, AlgebraicGeometry.affinePreimage, sq_sum_le_card_mul_sum_sq, tendsto_nat_floor_atTop, Prop.disjoint_iff, Order.pred_top_lt_iff, Ordinal.toZFSetIso_apply, Nimber.toOrdinal_eq_one, CategoryTheory.Triangulated.SpectralObject.ω₂_obj_mor₂, MeasureTheory.Integrable.measure_norm_gt_lt_top, Submodule.instCanonicallyOrderedAdd, mem_eball_one_iff, LowerAdjoint.closure_union_closure_right, Tuple.comp_sort_eq_comp_iff_monotone, Matrix.mem_rowStochastic_iff_sum, absConvex_convexClosedHull, upperSemicontinuousAt_iff_limsup_le, ENNReal.coe_lt_natCast, Finset.sum_Ico_add', isωSup_ωSup, PointedCone.subset_span, EReal.le_neg, AlgebraicGeometry.Scheme.isoSpec_inv_naturality, DividedPowers.DPMorphism.ideal_comp, EReal.bot_lt_coe_ennreal, Filter.disjoint_comap_iff_map, le_sdiff_iff, Ideal.pow_le_pow_right, star_lt_star_iff, Set.Ici_diff_Ioi, Finset.Ico_add_one_right_eq_Icc, Complex.zero_lt_real, Ordinal.bsup_le_of_brange_subset, IntermediateField.bot_eq_top_iff_finrank_eq_one, AlgebraicGeometry.IsAffineOpen.toSpecΓ_fromSpec_assoc, Sym2.sortEquiv_symm_apply, Ordinal.nadd_le_iff, BoxIntegral.TaggedPrepartition.distortion_filter_le, ENNReal.mul_div_le, Ico_mem_nhds_iff, FrameHom.map_sSup', Types.monoOverEquivalenceSet_functor_map, Submodule.orderIsoMapComap_symm_apply, QuadraticMap.PosDef.le_zero_iff, ContinuousOn.partialSups, ClosureOperator.closure_sup_closure, Ideal.radical_eq_sInf, ContinuousLinearMap.IsPositive.inner_nonneg_left, Finset.Colex.toColex_mono, CategoryTheory.Subobject.eq_of_comp_arrow_eq_iff, Ordinal.mem_range_veblen_iff_le_invVeblen₁, toENat_rank_span_set, DirectSum.coe_decompose_mul_of_right_mem, Ordinal.add_eq_right_iff_mul_omega0_le, Real.toNNReal_le_toNNReal, TopCat.Sheaf.objSupIsoProdEqLocus_inv_snd, le_sup_inf, CategoryTheory.Limits.imageSubobjectCompIso_inv_arrow_assoc, MeasureTheory.L2.eLpNorm_inner_lt_top, Ordinal.omega0_le_omega, PiLp.edist_apply_le, TopologicalSpace.Opens.overEquivalence_unitIso_inv_app_left, ENNReal.coe_pos, ProjectiveSpectrum.vanishingIdeal_anti_mono, MeasureTheory.Measure.countable_meas_pos_of_disjoint_of_meas_iUnion_ne_top, Finset.min'_lt_max', Set.image_single_Ioo, oneLePart_le_one, Seminorm.le_def, Set.Subsingleton.equitableOn, Equiv.Perm.support_swap_mul_ge_support_diff, ENNReal.le_tsum, ENNReal.rpow_add_rpow_le, Int.comap_cast_atBot, CategoryTheory.Regular.instIsRegularEpiFrobeniusMorphism, ENNReal.div_le_iff, Finset.wellFoundedLT, monovaryOn_neg_left, DividedPowers.isDPMorphism_def, CategoryTheory.ComposableArrows.twoδ₁Toδ₀_app_one, GenContFract.of_den_mono, SuccOrder.isOpen_iff, AlgebraicGeometry.Scheme.Hom.id_appIso, Fin.image_addNat_Icc, IntermediateField.adjoin_eq_bot_iff, IsSemisimpleModule.jacobson_le_ker, Pi.codisjoint_iff, Subring.le_topologicalClosure, iSup_inf_le_inf_sSup, Subsemigroup.monotone_map, Dynamics.coverEntropyInf_antitone, DivisibleHull.qsmul_def, Filter.tendsto_neg_atBot_iff, convexHull_eq_singleton, AbsoluteValue.nonnegHomClass, MulOpposite.op_nonpos, CategoryTheory.Subfunctor.Subpresheaf.range_le_equalizer_iff, MeasureTheory.SimpleFunc.simpleFunc_bot', convexHull_eq_zero, ClosureOperator.closure_iSup_closure, Finset.map_add_left_Ioo, smoothSheafCommRing.ι_forgetStalk_inv_assoc, Ordinal.mem_closure_tfae, Submodule.rTensorOne_tmul_one, sup_iInf_le_iInf_sup, skyscraperPresheafCocone_ι_app, Finset.mulConst_le_inv_dens, AlgebraicGeometry.Scheme.Hom.preimageIso_inv_ι_assoc, AlgebraicGeometry.Scheme.Hom.quasiFiniteLocus_eq_top_iff, CategoryTheory.Triangulated.SpectralObject.id_hom, LowerSet.compl_le_compl, Finset.Ioc_pred_pred_eq_Ico, LinearMap.IsIdempotentElem.comp_eq_left_iff, hasProdUniformly_iff_tendstoUniformly, ArchimedeanClass.out_top, sign_one, Ordinal.bsup_eq_blsub_or_succ_bsup_eq_blsub, ValuativeRel.uniformizer_inv_le_iff, zpow_left_mono, AlgebraicGeometry.instQuasiSeparatedToSpecΓOfQuasiSeparatedSpaceCarrierCarrierCommRingCat, PartOrd.inv_hom_apply, Valuation.one_vlt_iff, RingHom.sclosure_preimage_le, UpperSet.coe_zero, PartOrdEmb.id_apply, AlgebraicGeometry.IsAffineOpen.basicOpenSectionsToAffine_isIso, AlgebraicGeometry.isAffineHom_iff, neg_lt_self_iff, Sublattice.inclusion_rfl, Prop.isAtom_iff, IsValuativeTopology.isClopen_ball, ONote.split_add_lt, PrimeSpectrum.comapEquiv_apply, IsValuativeTopology.hasBasis_nhds, Set.Iic_pred_eq_Iio, Finset.le_inf_iff, CategoryTheory.Sieve.pullback_eq_top_of_mem, tendsto_neg_nhdsLE_neg, Matrix.le_one_of_mem_rowStochastic, ENNReal.exists_pos_sum_of_countable, ProbabilityTheory.Kernel.bound_lt_top, Set.encard_mono, Fin.preimage_castLE_Iic_castLE, exists_ge_of_linear, OrdinalApprox.lfpApprox_mono_left, abs_cases, openSegment_eq_image_lineMap, Ideal.eq_jacobson_iff_notMem, min_le_left, CategoryTheory.ObjectProperty.le_isColocal_iff, Polynomial.tendsto_nhds_iff, AlgebraicGeometry.Scheme.IdealSheafData.subschemeFunctor_obj, Affine.Simplex.sSameSide_affineSpan_faceOpposite_iff, Fin.preimage_addNat_Icc_addNat, dvd_iff_emultiplicity_pos, CategoryTheory.Limits.imageSubobjectMap_arrow, Nonneg.instContinuousFunctionalCalculus, AlgebraicGeometry.PresheafedSpace.map_id_c_app, toPreorder_injective_iff, compatiblePreserving_opens_map, antitone_int_of_succ_le, Finsupp.mem_rangeIcc_apply_iff, AddCircle.measurePreserving_equivIoc, PrimitiveSpectrum.preimage_upperClosure_compl_finset, AntilipschitzWith.edist_lt_top, CategoryTheory.GrothendieckTopology.trivial_eq_bot, Rat.castOrderEmbedding_apply, isPreconnected_Ioo, ENat.add_lt_add_iff_left, TopCat.Presheaf.germ_exist, Ordinal.not_isSuccLimit_zero, Cardinal.nat_lt_lift_iff, BoolAlg.dual_map, MeasureTheory.Submartingale.stoppedValue_leastGE_eLpNorm_le, Filter.lift_lift'_same_le_lift', StarSubalgebra.rangeS_le, AlgebraicGeometry.exists_smooth_of_formallySmooth_stalk, Dynamics.coverEntropyEntourage_le_log_coverMincard_div, biInf_ge_eq_inf, AlgebraicGeometry.Scheme.comp_appTop_assoc, Subgroup.ker_le_comap, Ordinal.toGame_zero, Submodule.le_traceDual_comm, Order.isPredPrelimit_iff_pred_ne, AddAction.le_stabilizer_vadd_right, CategoryTheory.Limits.imageSubobjectMap_arrow_assoc, nndist_pi_const_le, zpow_left_strictMono, ArchimedeanClass.subsemigroup_strictAnti, PartENat.eq_top_iff_forall_lt, antivary_inv_left, StarAlgebra.adjoin_le_centralizer_centralizer, FirstOrder.Language.PartialEquiv.monotone_cod, Set.encard_singleton_inter, MeasureTheory.one_le_prob_iff, Set.Ioo.mem_iff_one_sub_mem, Matrix.IsIrreducible.exists_pos, Subgroup.toAddSubgroup_closure, MeasureTheory.unifTight_iff_ennreal, FirstOrder.Language.Substructure.closure_empty, QuotientAddGroup.btw_coe_iff, PartENat.coe_succ_le_iff, addRightMono_of_addRightStrictMono, Finset.sum_card_slice_div_choose_le_one, TopCat.Presheaf.SheafConditionEqualizerProducts.res_π, Filter.le_seq, IsDedekindDomain.HeightOneSpectrum.intValuation_lt_one_iff_dvd, LocalSubring.le_ofPrime, Ordinal.isSuccLimit_iff, Subring.toAddSubgroup_strictMono, AlgebraicGeometry.Scheme.local_affine, Dynamics.coverMincard_le_pow, Prop.instWellFoundedLT, PartENat.coe_lt_iff, AlgebraicGeometry.Proj.basicOpen_one, TopCat.Presheaf.germ_res, Filter.Eventually.exists_Ioo_subset, span_lt_top_of_card_lt_finrank, wbtw_iff_right_eq_or_left_mem_image_Ici, EReal.add_iInf_le_iInf_add, Mathlib.Meta.NormNum.isRat_le_true, CompleteLattice.isCoatomistic_iff, SummationFilter.hasProd_symmetricIco_int_iff, Finset.Iic_mul_Iio_subset', Ordinal.veblen_eq_veblen_iff, MeasureTheory.hittingBtwn_mono_right, KleeneAlgebra.mul_kstar_le_kstar, QuotientGroup.strictMono_comap_prod_image, Ideal.height_le_iff_covBy, CauSeq.const_le, Subgroup.map_le_map_iff', Filter.limsup_bot, monotone_Iio, FiniteMulArchimedeanClass.mk_le_mk, AddEquiv.coe_mapAddSubgroup, Ordinal.add_le_nadd, AlgebraicGeometry.PresheafedSpace.GlueData.f_invApp_f_app, LieSubmodule.Quotient.map_mk'_eq_bot_le, ringKrullDim_quotient_succ_le_of_nonZeroDivisor, instIsCodirectedOrder, Set.Iic.coe_sup, MulEquiv.comapSubgroup_apply, isCompact_closedAbsConvexHull_of_totallyBounded, AddSubgroup.MapSubtype.orderIso_apply_coe, Set.preimage_mul_const_Iio, AddSubgroup.characteristic_iff_le_comap, Ordinal.pred_eq_iff_isSuccPrelimit, SummationFilter.hasSum_symmetricIco_int_iff, archimedean_iff_nat_lt, algebraMap_le_iff_le_spectrum, AlgebraicGeometry.Scheme.IdealSheafData.ideal_top, Filter.disjoint_pure_pure, Set.MapsTo.filter_map_Iic, AddSubgroup.closure_nsmul_le, MeasureTheory.OuterMeasure.le_trim, SimpleGraph.eccent_le_ediam, IntermediateField.lift_sepDegree_bot', Dynamics.coverMincard_mul_le_pow, Finset.instMulRightMono, IsStarProjection.mem_Icc, Filter.nhds_principal, Finset.inf_mono, StarRingHomClass.instOrderHomClass, le_analyticOrderAt_sub, Set.preimage_sub_const_Ioc, disjoint_compl_left_iff, EReal.neg_le_zero, Submonoid.le_prod_iff, Subring.closure_preimage_le, Fin.image_addNat_Ioi, Multiplicative.canonicallyOrderedMul, Rat.toNNRat_pos, tendsto_pow_neg_atTop, TopologicalSpace.Opens.adjunction_counit_app_self, TopCat.Presheaf.instIsLocalizationCarrierObjOppositeOpensCarrierCommRingCatObjLocalizationPresheaf, egauge_ball_one_le_of_one_lt_norm, Asymptotics.IsEquivalent.tendsto_atTop_iff, MulOpposite.unop_nonpos, WithZero.denselyOrdered_set_iff_subsingleton, supClosure_isClosed, le_analyticOrderAt_add, Set.Iic_subset_Iic_union_Icc, Submodule.span_range_natDegree_eq_adjoin, OrderIso.withBotCongr_apply, TopologicalSpace.Opens.mem_map, Ordinal.mod_opow_log_lt_self, EuclideanGeometry.Sphere.IsTangentAt.le_orthRadius, antivaryOn_inv, ValuationSubring.pointwise_smul_subset_iff, Ideal.exists_ltSeries_of_hasGoingDown, OrderMonoidHom.inlₗ_apply, pi_nnnorm_const_le', Bornology.le_cofinite', IsConnected.Icc_subset, Real.toNNReal_lt_ofNat, absConvexHull_subset_closedAbsConvexHull, abs_pos, Order.pred_eq_iSup, AbsoluteValue.IsEuclidean.map_lt_map_iff', EReal.coe_nonneg, Set.subsingleton_Icc_iff, Ordinal.mk_Iio_ordinal, OmegaCompletePartialOrder.ωScottContinuous.monotone_map_ωSup, Matroid.setOf_indep_eq, ExpGrowth.le_expGrowthInf_add, InitialSeg.lt_iff_lt, AddSubgroup.coe_toZModSubmodule, EReal.toReal_image_Ioo_bot_zero, CompletelyDistribLattice.top_sdiff, Set.bounded_lt_inter_lt, aemeasurable_uIoc_iff, SSet.stdSimplex.monotone_apply, Finset.strictMono_iff_forall_lt_insert, Dynamics.netMaxcard_monotone_time, Submonoid.pointwise_smul_subset_iff, idealFactorsEquivOfQuotEquiv_symm, infIrred_iff_not_isMax, Set.pairwise_disjoint_Ioo_zsmul, RestrictedProduct.isEmbedding_inclusion_top, LinearLocallyFiniteOrder.instIsSuccArchimedeanOfLocallyFiniteOrder, OrderIso.subsingleton_of_wellFoundedLT', Valuation.one_vle_iff, Ordinal.bddAbove_of_small, ENat.one_le_epow, Profinite.NobelingProof.πs_apply_apply, isIso_iff_eq, zero_lt_one, lowerSemicontinuousOn_iff_frequently, instTrichotomousGe, Set.Ioc.coe_pos, AffineSubspace.direction_iInf, Affine.Simplex.affineCombination_mem_interior_face_iff_pos, List.SortedGE.antitone, Set.image_add_const_Ico, Filter.tendsto_const_mul_pow_atBot_iff, HahnSeries.orderTop_of_ne_zero, wbtw_iff_left_eq_or_right_mem_image_Ici, CategoryTheory.ComposableArrows.mk₁_map, Fin.preimage_castLE_Icc_castLE, Ioo_subset_openSegment, toWeakSpace_closedConvexHull_eq, Filter.map_inv_atTop, CategoryTheory.GrothendieckTopology.OneHypercoverFamily.IsGenerating.le, Set.unbounded_lt_iff, Ideal.ker_le_comap, Set.preimage_add_const_Ioi, GeneralizedHeytingAlgebra.le_himp_iff, AddGroupTopology.toTopologicalSpace_le, AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalHom, abs_sub_map_le_div, Finset.Ioo_map_sectL, tsub_le_self, IsOpenMap.coverPreserving, groupoid_of_pregroupoid_le, MeasureTheory.Measure.pi_Ioo_ae_eq_pi_Ioc, CategoryTheory.Presieve.le_of_factorsThru_sieve, convexHull_pi, symm_le_uniformity, Submodule.IsOrtho.le, Subalgebra.op_top, Ordinal.card_le_nat, AlgebraicGeometry.Scheme.isoSpec_Spec, CategoryTheory.Limits.kernelSubobject_arrow, ThreeAPFree.le_addRothNumber, CanonicallyOrderedAddCommMonoid.single_le_sum, CategoryTheory.ComposableArrows.threeδ₁Toδ₀_app_two, OrderIsoClass.toBotHomClass, tendsto_neg_nhdsGE, ArchimedeanClass.ball_top, AlgebraicGeometry.ProjectiveSpectrum.Proj.toSpec_base_isIso, CategoryTheory.isNoetherianObject_iff_monotone_chain_condition, Cardinal.toENat_strictMonoOn, Ordinal.typein_apply, LSeries.abscissaOfAbsConv_convolution_le, Ordinal.lsub_le_of_range_subset, ENNReal.mul_le_mul_left, FirstOrder.Language.monotone_distinctConstantsTheory, AlgebraicClosure.le_maxIdeal, AlgebraicGeometry.instLocallyOfFiniteTypeMorphismRestrict, CategoryTheory.Subfunctor.Subpresheaf.range_id, SchwartzMap.eLpNorm_le_seminorm, ENNReal.iUnion_Ico_coe_nat, LieSubmodule.lieSpan_mono, InnerProductSpace.span_gramSchmidt_Iic, MeasureTheory.Content.outerMeasure_exists_open, Fin.castAddOrderEmb_toEmbedding, CategoryTheory.Limits.Types.isPushout_of_bicartSq, AffineSubspace.coe_eq_bot_iff, Filter.tendsto_comp_neg_atBot_iff, Set.Nontrivial.exists_lt, SimpleGraph.FarFromTriangleFree.lt_half, Cardinal.not_isSuccLimit_natCast, TopCat.Presheaf.Pushforward.id_hom_app, IsPRadical.ker_le, Set.isAtom_iff, Finset.Nonempty.inf_eq_bot_iff, round_eq_zero_iff, iSup₂_eq_bot, riemannZeta_pos_of_one_lt, pure_le_nhdsWithin, AddSubgroup.centralizer_le, essSup_measure_zero, OrderTop.instSubsingleton, NNRat.preimage_cast_Ici, Subring.gc_map_comap, Monotone.Ioc, AlgebraicGeometry.IsIntegralHom.instMorphismRestrict, Algebra.adjoin_singleton_natCast, dimH_mono, MeasureTheory.Lp.simpleFunc.coeFn_le, measure_Icc_lt_top, abs_sup_sub_sup_le_abs, TopCat.Presheaf.pullback_obj_obj_ext_iff, EReal.zero_lt_exp_iff, BddOrd.id_apply, SSet.Subcomplex.ofSimplex_le_iff, FormalMultilinearSeries.radius_le_radius_continuousLinearMap_comp, InitialSeg.instOrderHomClassLt, Set.preimage_sub_const_Ioi, Nimber.not_bddAbove_compl_of_small, Set.Ici_sSup, tendsto_natCast_atTop_iff, Order.krullDim_eq_iSup_height, Finset.min_eq_bot, MeasureTheory.SimpleFunc.monotone_eapprox, eVariationOn.edist_le, IntermediateField.LinearDisjoint.inf_eq_bot, isGLB_sInf, Filter.atBot_le_cofinite, Metric.measure_closedEBall_pos, TopCat.Opens.coverDense_iff_isBasis, Finset.coe_covBy_coe, Ideal.map_le_iff_le_comap, Ordinal.one_nadd, MeasureTheory.eLpNorm_const_smul_le, gc_upperClosure_coe, Order.isPredLimitRecOn_pred, FirstOrder.Language.Substructure.iSup_eq_closure, CategoryTheory.Subobject.ofLE_refl, Set.preimage_sub_const_Ici, AlgebraicGeometry.Scheme.ofRestrict_toLRSHom_c_app, Submonoid.closure_le_centralizer_centralizer, SimplexCategory.II.map'_eq_castSucc_iff, forall_gt_iff_le, Submodule.IsMinimalPrimaryDecomposition.minimal, ProjectiveSpectrum.homogeneousIdeal_le_vanishingIdeal_zeroLocus, CategoryTheory.Regular.frobeniusStrongEpiMonoFactorisation_e, FiniteArchimedeanClass.mk_le_mk, min_def, NonUnitalStarAlgebra.elemental.le_centralizer_centralizer, Subalgebra.centralizer_le, Coheyting.boundary_hnot_le, rank_submodule_eq_one_iff, CategoryTheory.SmallObject.iterationFunctorMapSuccAppArrowIso_hom_right_right_comp_assoc, Metric.mem_thickening_iff_exists_edist_lt, Ordinal.ToType.mk_symm_apply_coe, Filter.le_lift', AlgebraicGeometry.Scheme.Hom.preimage_sup, Order.Ideal.principal_le_iff, MonomialOrder.wf, PosMulStrictMono.toPosMulMono, cmp_eq_eq_iff, RootPairing.GeckConstruction.cartanSubalgebra_le_lieAlgebra, CategoryTheory.Subobject.ofMkLEMk_refl, NonUnitalStarSubalgebra.toNonUnitalSubalgebra_le_iff, Field.exists_primitive_element_iff_finite_intermediateField, Finset.fold_min_le, Set.Icc.coe_eq_zero, AddMonoid.le_minOrder_iff_forall_addSubgroup, ENat.floor_le_self, nonneg_nhds_iff, frontier_Ici_subset, Set.preimage_div_const_Ioc, mem_nerve_nonDegenerate_iff_strictMono, Set.projIic_self, TwoSidedIdeal.comap_comap, Valued.toNormedField.one_lt_norm_iff, LinOrd.forget_map, CategoryTheory.ComposableArrows.isComplex₂_iff, Subgroup.coe_toAddSubgroup_apply, MeasureTheory.condLExp_smul_le, MeasureTheory.restrict_Ioo_eq_restrict_Ioc, Ideal.homogeneousCore'_le, AffineMap.restrict.surjective, Finset.inf_dite_pos_le, CategoryTheory.ExtremalEpi.subobject_eq_top, AlgebraicGeometry.IsClosedImmersion.hasAffineProperty, Field.Emb.Cardinal.filtration_apply, Set.image_update_Ioc_right, LinearMap.range_prod_le, PrimeMultiset.prod_dvd_iff, AddOpposite.one_le_unop, Set.Ioc_pred_pred_eq_Ico, OmegaCompletePartialOrder.ContinuousHom.toOrderHom_eq_coe, MeasureTheory.Measure.QuasiMeasurePreserving.ae_map_le, Coheyting.boundary_le_boundary_sup_sup_boundary_inf_left, SupBotHom.sup_apply, Matrix.nonneg_of_mem_colStochastic, FirstOrder.Language.Substructure.map_bot, MeasureTheory.setLIntegral_mono_ae', Submodule.orderIsoMapComap_apply, enorm_sub_le_lintegral_deriv_of_contDiffOn_Icc, CategoryTheory.HasLiftingProperty.transfiniteComposition.SqStruct.sq, Finset.Ioo_pred_left_eq_Ioc, HahnSeries.order_of_ne, OrderIso.toRelIsoLT_ofRelIsoLT, Affine.Simplex.ExcenterExists.sign_touchpointWeights, EMetric.edist_le_diam_of_mem, PointedCone.ext_iff, AddSubsemigroup.closure_le, AddGroupSeminorm.comp_add_le, ProbabilityTheory.Kernel.compProd_apply_univ_le, SimpleGraph.Subgraph.deleteVerts_le, AlgebraicGeometry.Scheme.preimage_opensRange_toSpecΓ, Rat.cast_mono, LowerSet.erase_lt, SimpleGraph.eccent_pos_iff, Set.OrdConnected.preimage_coe_nnreal_ennreal, intervalIntegrable_const_iff, le_monotonicSequenceLimit, Cardinal.range_aleph, AlgebraicGeometry.instIsOpenImmersionMorphismRestrict, PiCountable.edist_le_two, Nat.image_cast_int_Ioi, IsDedekindDomain.HeightOneSpectrum.intValuation_zero_lt, sq_lt_sq, Nat.cast_le_ofNat, Int.lt_zpow_succ_log_self, IsSimpleRing.simple, Topology.IsScott.instUnivSetOfIsUpper, egauge_le_of_smul_mem, Set.image_neg_Icc, MeasureTheory.VectorMeasure.instAddLeftMono, BoxIntegral.Box.disjoint_splitLower_splitUpper, IntermediateField.gc_map_comap, Cardinal.cantor, add_sq_le, convex_Icc, Finset.le_dens_sdiff, NoBotOrder.to_noMinOrder, SetTheory.Game.addRightStrictMono, indiscreteTopology_iff, Subsemiring.pointwise_smul_le_pointwise_smul_iff, minimal_iff, strictConvex_Ioc, Finset.min'_lt_max'_of_card, toZ_iterate_succ, CategoryTheory.ShortComplex.moduleCatLeftHomologyData_descH_hom, IsValuativeTopology.hasBasis_nhds_zero, affineSpan_pair_le_of_left_mem, AffineMap.lineMap_anti, TopCat.Presheaf.generateEquivalenceOpensLe_counitIso, Ordinal.bddAbove_iff_small, AlgebraicGeometry.Scheme.Spec.algebraMap_residueFieldIso_inv, ENNReal.inv_lt_one, Subalgebra.bot_eq_top_iff_finrank_eq_one, Set.Ioc_diff_right, ValuationSubring.unitGroup_le_unitGroup, SSet.prodStdSimplex.objEquiv_map_apply, MeasureTheory.Measure.hausdorffMeasure_le_liminf_tsum, Filter.tendsto_finset_range, HasFPowerSeriesOnBall.radius_pos, LieSubmodule.wellFoundedLT_of_isArtinian, nhdsWithinLT_sup_nhdsWithinGT, Ideal.mem_normalizedFactors_iff, SSet.Subcomplex.le_iff_of_hasDimensionLT, Fin.predAboveOrderHom_coe, Polynomial.span_of_finite_le_degreeLT, posSMulMono_iff_posSMulReflectLT, AddOpposite.nonneg_unop, IsIntegralClosure.range_le_span_dualBasis, Subgroup.commutator_le_inf, AddSubmonoid.mem_closure_image_pos_iff, Ideal.matrix_jacobson_le, Subalgebra.mulMap_bot_right_eq, AlgebraicGeometry.ProjectiveSpectrum.Proj.mk_mem_toSpec_base_apply, Valuation.IsNontrivial_iff_exists_one_lt, MeasureTheory.Measure.le_map_apply, isLowerSet_setOf, AlgebraicGeometry.IsAffineOpen.exists_basicOpen_le, ringKrullDim_add_natCard_le_ringKrullDim_mvPolynomial, smul_iInf_le, WithLp.nnnorm_fst_le, instZeroLEOneClass, intervalIntegral.integral_pos_iff_support_of_nonneg_ae, iSup_disjointed, AddSubmonoid.mul_le, Matroid.ExchangeProperty.encard_diff_le_aux, LinOrd.ext_iff, isOpen_Ioi, Affine.Simplex.sign_touchpointWeights_singleton_pos, EReal.preimage_coe_Ioo_bot, FormalMultilinearSeries.lt_radius_of_isBigO, NonemptyFinLinOrd.hom_hom_id, List.sorted_le_ofFn_iff, Submodule.toAddSubgroup_strictMono, ENNReal.log_lt_top_iff, nhdsSet_Ici, ValuationSubring.principal_units_le_units, Cardinal.lift_lt_univ, CategoryTheory.MorphismProperty.le_retracts, forall_gt_imp_ne_iff_le, StarSubalgebra.topologicalClosure_mono, isCompact_Ico_iff, ContinuousAt.partialSups, AddCircle.eq_coe_Ico, EMetric.cauchySeq_iff_le_tendsto_0, CStarAlgebra.mul_star_le_algebraMap_norm_sq, AlgebraicGeometry.Scheme.eq_zeroLocus_of_isClosed_of_isAffine, Finset.isLUB_sup', Finset.sum_sdiff_le_sum_sdiff, Topology.scottHausdorff_le_lower, SetRel.image_mono, supClosure_singleton, Set.Iio_subset_Iio_iff, ExpGrowth.le_expGrowthInf_iff, MeasureTheory.measure_pos_iff_nonempty_of_smulInvariant, Finset.antidiagonal.snd_le, AddOpposite.unop_le_one, Preorder.instHasBinaryCoproducts, AddMonoidAlgebra.exists_finset_adjoin_eq_top, CategoryTheory.Limits.imageSubobject_comp_le, Orientation.oangle_sign_smul_right, Nat.ofNat_nonneg, TopCat.Sheaf.interUnionPullbackConeLift_left, disjointed_le, Cardinal.aleph0_le_preAleph, Submonoid.unop_le_unop_iff, AlgebraicTopology.NormalizedMooreComplex.d_squared, AlgebraicGeometry.Scheme.Opens.ι_preimage_self, NonUnitalSubalgebra.topologicalClosure_adjoin_le_centralizer_centralizer, Odd.zpow_nonneg_iff, ContinuousLinearMap.opENorm_comp_le, Finset.Ioi_pred_eq_Ici, Fin.image_succ_Ioi, Function.Injective.image_strictMono, CategoryTheory.Subfunctor.instIsIsoFunctorTypeιTop, Asymptotics.isEquivalent_nat_floor, StrictOrderedSemiring.toSMulPosStrictMonoNat, MeasureTheory.Lp.coeFn_nonneg, Cardinal.preAleph_succ, Module.End.genEigenspace_eq_iSup_genEigenspace_nat, TopCat.subpresheafToTypes_map_coe, BotHom.sup_apply, sup_le_iff, Multiset.map_mono, Ideal.Filtration.pow_smul_le_pow_smul, trop_iInf, AddSubgroup.FG.finset_sup, CategoryTheory.Subfunctor.bot_obj, Finset.min'_lt_of_mem_erase_min', Filter.lift_map_le, Set.neg_Ioc, MeasureTheory.Measure.ae_le_iff_absolutelyContinuous, AlgebraicGeometry.Scheme.toSpecΓ_image_zeroLocus, Ne.not_le_or_not_ge, strictAnti_iff_map_pos, Set.dissipate_bot, Valuation.mem_ltAddSubgroup_iff, ENNReal.orderIsoUnitIntervalBirational_apply_coe, le_iff_forall_pos_lt_add, mabs_mabs_div_mabs_le_mabs_div, Order.coheight_le_coheight_apply_of_strictMono, upperSemicontinuous_iff_limsup_le, AlgebraicGeometry.PresheafedSpace.ofRestrict_top_c, Cardinal.ofNat_le_ofENat, AlgebraicGeometry.instQuasiSeparatedMorphismRestrict, BooleanSubalgebra.mk_lt_mk, R1Space.specializes_or_disjoint_nhds, LowerAdjoint.le_closure_iff, Ordinal.strictMono_gamma, MeasureTheory.eventually_nhds_zero_measure_vadd_diff_lt, CategoryTheory.TransfiniteCompositionOfShape.ici_isColimit, le_or_gt, Set.Iic_inter_Iic, MeasureTheory.nnnorm_indicatorConstLp_le, AbsoluteValue.nonpos_iff, IsCompact.bddAbove_image, HasFPowerSeriesWithinOnBall.r_le, Codisjoint.bihimp_inf_bihimp_le_right, AlgebraicGeometry.Scheme.Hom.image_iSup, CategoryTheory.ComposableArrows.δlastFunctor_obj_obj, AlgebraicGeometry.LocallyRingedSpace.evaluation_naturality, image_latticeClosure, PNat.succ_eq_add_one, Alexandrov.lowerCone_π_app, Lattice.isStronglyAtomic, Ordinal.enum_succ_eq_top, ENNReal.ofNat_lt_ofReal, MvPolynomial.mem_degreesLE, Order.krullDim_eq_zero_iff_of_orderTop, enorm_le_coe, InverseSystem.piSplitLE_eq, AlgebraicGeometry.PresheafedSpace.stalkMap_germ_apply, partialSups_zero, Nimber.toOrdinal_one, iSupIndep.sup_indep_univ, FractionalIdeal.mul_left_mono, doublyStochastic_eq_convexHull_permMatrix, ENNReal.Lp_add_le, IsUltrametricDist.nnnorm_nsmul_le, Module.Basis.SmithNormalForm.le_ker_coord_of_notMem_range, Finset.sup_eq_iSup, NNReal.agmSequences_monotone_and_antitone, CFC.rpow_nonneg, AbsoluteValue.le_add, Submodule.top_le_span_range_iff_forall_exists_fun, wellFoundedGT_iff_monotone_chain_condition, MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part, mul_smul_one_lt_iff, Finset.some_mem_insertNone, Subgroup.comap_le_comap_of_surjective, min_def_lt', PNat.factorMultiset_mono, Cardinal.lt_aleph0_iff_subtype_finite, latticeClosure_eq_self, Subalgebra.op_le_iff, Equiv.Perm.alternatingGroup_le_of_isPreprimitive_of_isThreeCycle_mem, IsGalois.intermediateFieldEquivSubgroup_symm_apply, MulAction.map_stabilizer_le, cocompact_eq_atBot_atTop, Submodule.map₂_le, Cardinal.lt_lift_iff, Algebra.iInf_toSubmodule, ProbabilityTheory.bayesRisk_le_bayesRisk_comp, Metric.nhdsWithin_basis_closedEBall, PrimeSpectrum.nhdsOrderEmbedding_apply, TopologicalSpace.Opens.map_comp_eq, UpperSet.icisSupHom_apply, floorDiv_le_ceilDiv, isSimpleModule_iff, Set.disjoint_univ, fourierSubalgebra_coe, Module.length_pos, CategoryTheory.ObjectProperty.monomorphisms_le_monoModSerre, ENat.addLECancellable_coe, imageSubobjectIso_imageToKernel', Finsupp.coe_orderIsoMultiset_symm, le_himp_iff, NNReal.rpow_add_le_mul_rpow_add_rpow, CategoryTheory.Subobject.mk_lt_mk_iff_of_comm, Interval.disjoint_coe, PartOrd.ofHom_id, Fintype.sum_strictMono, LieAlgebra.LieIdeal.isNilpotent_iff_le_maxNilpotentIdeal, CategoryTheory.ComposableArrows.Exact.cokerIsoKer'_inv_hom_id_assoc, Set.image_single_Icc_right, SetLike.coe_strictMono, Types.monoOverEquivalenceSet_functor_obj, PosSMulReflectLT.toPosSMulReflectLE, AlgebraicGeometry.toSpecΓ_SpecMap_ΓSpecIso_inv_assoc, absConvexHull_eq_iInter, star_le_iff, mulRothNumber_empty, CategoryTheory.Limits.imageSubobject_arrow, Set.image_add_const_Ioo, nhdsLT_basis, ENNReal.ofReal_pos, AlgebraicGeometry.SheafedSpace.Γ_obj_op, AlgebraicGeometry.IsClosedImmersion.isIso_lift, Pi.card_Ico, scottContinuous_inf_left, CategoryTheory.Limits.imageSubobject_comp_le_epi_of_epi, Submodule.spanRank_toENat_eq_iInf_finset_card, zero_notMem_iff, FractionalIdeal.instMulRightMono, Ordinal.iterate_veblen_lt_gamma_zero, IsAbsoluteValue.abv_sum, OrderMonoidHom.coe_mul, Algebra.isAlgebraic_iff, Order.succ_le_succ_iff, ENNReal.exists_pos_tsum_mul_lt_of_countable, enorm_lt_top, antivaryOn_inv_left, MeasureTheory.integral_le_measure, Ordinal.typein_surjOn, trdeg_pos, TopologicalSpace.Closeds.coe_top, CompositionSeries.strictMono, NNReal.sum_mul_le_sqrt_mul_sqrt, AlgebraicGeometry.Scheme.Hom.comp_preimage, isSaddlePointOn_iff', MeasureTheory.withDensity_inv_same_le, AlgebraicGeometry.Scheme.kerFunctor_map, LucasLehmer.two_lt_q, CategoryTheory.Limits.FormalCoproduct.cech_map, HahnSeries.orderTop_nsmul_le_orderTop_pow, MeasureTheory.MemLp.integral_indicator_norm_ge_nonneg_le, latticeClosure_empty, CategoryTheory.Sieve.ofObjects_mono, ConvexOn.convex_le, Multiset.inter_le_ndinter, TopologicalGroup.isOpenMap_iff_nhds_one, variationOnFromTo.sub_self_monotoneOn, NatOrdinal.instAddLeftMono, isForwardInvariant_iff_isInvariant, Multiset.ndunion_le_add, Subgroup.strictPeriods_le_periods, InitialSeg.apply_wCovBy_apply_iff, WeierstrassCurve.isMinimal_iff, Finset.card_Ico_eq_card_Icc_sub_one, Finset.pluennecke_ruzsa_inequality_nsmul_sub_nsmul_sub, MeasureTheory.exists_lt_lowerSemicontinuous_integral_lt, AlgebraicGeometry.Scheme.Modules.instIsIsoAbApp, Submonoid.op_le_op_iff, Metric.exists_continuous_ennreal_forall_closedEBall_subset, CategoryTheory.MorphismProperty.TransfiniteCompositionOfShape.instIsIsoMapF, Set.Iio_ssubset_Iio_iff, Subalgebra.lTensorBot_symm_apply, monotone_stabilizes_iff_noetherian, Surreal.Multiplication.mulOption_lt_iff_P1, AlgebraicGeometry.Scheme.Hom.naturality_assoc, add_lowerClosure, trichotomy_of_mul_eq_mul, CategoryTheory.Limits.CompleteLattice.pullback_eq_inf, BoxIntegral.Box.Ioo_subset_Icc, Dense.borel_eq_generateFrom_Ico_mem_aux, SSet.horn.edge₃_coe_down, BoundedContinuousFunction.lintegral_of_real_lt_top, NNReal.lt_inv_iff_mul_lt, AlgebraicGeometry.Scheme.IdealSheafData.opensRange_glueDataObjMap, SimpleGraph.map_monotone, AlgebraicGeometry.IsAffineOpen.fromSpec_app_self_apply, Fin.orderSucc_last, NonUnitalStarAlgebra.comap_top, EReal.sub_self_le_zero, NonUnitalSubsemiring.toSubsemigroup_mono, Ordinal.lt_omega0_opow, SimpleGraph.edist_anti, IsSelfAdjoint.mul_self_nonneg, Filter.isBoundedUnder_le_neg, MeasureTheory.Measure.IicSnd_le_fst, Polynomial.not_ker_le_map_C_of_surjective_of_weaklyQuasiFiniteAt, MonoidHom.mclosure_preimage_le, DFinsupp.Lex.wellFoundedLT_of_finite, TopCat.Presheaf.pushforward_map_app', bot_covBy_iff, CategoryTheory.Dial.tensorUnit_rel, Function.locallyFinsuppWithin.negPart_add, OrderDual.instSMulPosMono, Decidable.le_iff_eq_or_lt', Real.one_le_toNNReal, AlgebraicGeometry.Scheme.Hom.iInf_ker_openCover_map_comp_apply, Finset.coe_inf_of_nonempty, ContractingWith.edist_inequality, InitialSeg.apply_covBy_apply_iff, Filter.mono_bliminf', bot_codisjoint, Nat.one_sub_one_div_cast_nonneg, map_pos_of_ne_one, OrderedSMul.toPosSMulReflectLT, EReal.coe_strictMono, EReal.coe_ennreal_lt_coe_ennreal_iff, CategoryTheory.Subobject.imageFactorisation_F_I, Cardinal.aleph_one_le_lift, Fin.map_castAddEmb_Ioc, LinearOrder.max_def, Set.Icc_add_one_left_eq_Ioc, Set.OrdConnected.IciExtend, Matrix.PosDef.trace_pos, FreeAlgebra.adjoin_range_ι, MeasureTheory.eLpNormEssSup_add_le, UniformSpace.comap_mono, NonemptyFinLinOrd.dual_map, le_csInf_iff', ENNReal.nhds_zero_basis, ContinuousMap.subalgebra_topologicalClosure_eq_top_of_separatesPoints, Set.finite_isBot, Subring.pointwise_smul_le_pointwise_smul_iff, tendstoUniformlyOn_tsum_of_cofinite_eventually, IsCompact.isGLB_sInf, ClosedSubmodule.gc_map_comap, strictConvex_Ico, BoxIntegral.Box.exists_seq_mono_tendsto, Submodule.dualAnnihilator_gc, MeasureTheory.integrableOn_Iio_iff_integrableAtFilter_atBot_nhdsWithin, PNat.addLeftReflectLE, Set.Finite.relatively_discrete, Finset.isDirected_le, disjoint_nhds_nhds_iff_not_inseparable, ENat.pow_lt_top_iff, CategoryTheory.SimplicialThickening.Path.left_le, max_top_left, ConvexCone.coe_positive, MeasureTheory.measure_iUnion_le, CategoryTheory.SmallObject.SuccStruct.ofCocone_obj_eq_pt, Finsupp.DegLex.single_le_iff, Finset.centroid_mem_convexHull, ComplementedLattice.isStronglyCoatomic, BoxIntegral.Box.withBotCoe_subset_iff, compl_le_himp, List.sortedLT_finRange, Pi.isAtom_iff_eq_single, Int.existsUnique_mem_box, Filter.hasBasis_cocardinal, antivary_neg_left, Profinite.NobelingProof.spanFinBasis.span, Con.toSetoid_bot, Set.Iio_add_one_eq_Iic, inf_lt_left, min_top_right, CategoryTheory.Subobject.inf_map, SetTheory.Game.addRightMono, HahnSeries.archimedeanClassOrderIsoWithTop_apply, Finset.untrop_sum, SSet.stdSimplex.const_down_toOrderHom, max_eq_bot, ClosedSubmodule.orthogonal_eq_top_iff, TopCat.Presheaf.pushforward_obj_map, supClosure_mono, Fin.finsetImage_succ_Ioo, CategoryTheory.ComposableArrows.scMapIso_hom, Subalgebra.inclusion_self, Set.OrdConnected.restrict, MeasureTheory.sigmaFinite_bot_iff, eventually_enorm_mfderiv_extChartAt_lt, FiniteMulArchimedeanClass.mem_subgroup_iff, CategoryTheory.TransfiniteCompositionOfShape.ofOrderIso_isoBot, SSet.Subcomplex.topIso_inv_ι_assoc, Set.Ico_add_one_left_eq_Ioo, PartENat.add_lt_add_iff_right, Multiset.dedup_le, AlgebraicGeometry.Scheme.inv_appTop, Valuation.Integers.wfDvdMonoid_iff_wellFounded_gt_on_v, CategoryTheory.ComposableArrows.opEquivalence_functor_obj_obj, Subalgebra.finiteDimensional_toSubmodule, Set.encard_le_coe_iff_finite_ncard_le, Ordinal.IsNormal.eq_iff_zero_and_succ, CategoryTheory.MorphismProperty.coproducts_le_llp_rlp, Submodule.gc_map_comap, bddBelow_slope_lt_of_mem_interior, Rat.preimage_cast_Iio, IntermediateField.finrank_top', ENNReal.ofReal_lt_coe_iff, Order.Coframe.sdiff_le_iff, Set.Iic.instIsCompactlyGenerated, AlgebraicGeometry.isLocalization_away_of_isAffine, Fin.attachFin_Ioo, Finset.Ico_filter_le, List.Sorted.get_strictMono, Dynamics.coverEntropy_image_le_of_uniformContinuous, ENNReal.range_coe', Set.unbounded_gt_iff, LowerAdjoint.closureOperator_apply, CategoryTheory.SemilatticeInf.tensorUnit, Rat.cast_nonneg, SemilatticeSup.instIsDirectedOrder, Positive.val_pow, MvPowerSeries.le_order_smul, NNRat.toNNRat_mono, wbtw_one_zero_iff, Polynomial.one_le_cauchyBound, Finset.map_truncatedSup, OrdinalApprox.exists_gfpApprox_eq_gfpApprox, Heyting.Regular.coe_toRegular, Cardinal.aleph0_lt_univ, AlgebraicGeometry.IsAffineOpen.ι_basicOpen_preimage, iSup_neg, MeasureTheory.setLIntegral_strict_mono, TopologicalSpace.Opens.map_iSup, TopCat.Presheaf.presheafEquivOfIso_unitIso_inv_app_app, WithTop.isEmbedding_coe, PSet.rank_powerset, CategoryTheory.Limits.CompleteLattice.hasFiniteColimits_of_semilatticeSup_orderBot, Finset.tendsto_Icc_neg, oneTangentSpaceIcc_def, OrderAddMonoidHom.inlₗ_apply, Mathlib.Meta.NormNum.isInt_lt_true, Fin.prod_Ioo_castSucc, Fin.map_revPerm_Ioo, Ordinal.isInitial_omega, disjoint_nhds_nhdsSet, AffineSubspace.direction_top, Submodule.mem_projectivization_iff_submodule_le, MeasureTheory.OuterMeasure.pi_pi_le, CategoryTheory.Limits.CompleteLattice.hasColimits_of_completeLattice, OrderMonoidHom.fstₗ_apply, ProfiniteGrp.cone_π_app, MeasureTheory.Measure.lintegral_bind_le, Set.Ioc_inter_Ioc, Submodule.instCovariantClassHSMulLe_1, Bornology.IsBounded.measure_lt_top, IntermediateField.inclusion_self, Ordinal.epsilon0_lt_gamma, star_lt_iff, FractionalIdeal.dual_le_dual, Affine.Simplex.span_eq_top, Finset.map_add_right_Ico, MeasureTheory.tendstoInMeasure_iff_measureReal_enorm, CategoryTheory.ComposableArrows.fourδ₂Toδ₁_app_three, MeasureTheory.Filtration.piFinset_eq_comap_restrict, himp_le_himp_himp_himp, Real.volume_le_diam, MeasureTheory.le_toMeasure_apply, LieSubmodule.lcs_le_iff, CategoryTheory.Triangulated.SpectralObject.distinguished', IntermediateField.lift_bot, Multiset.card_mono, PartENat.ofENat_lt, IsZGroup.commutator_lt, Finset.min'_eq_sorted_zero, Fintype.coe_finsetOrderIsoSet, MeasureTheory.Integrable.lintegral_lt_top, Ordinal.omega0_lt_epsilon, Cardinal.preBeth_succ, PowerSeries.le_order_pow, Polynomial.isBoundedUnder_abs_atBot_iff, PartOrdEmb.forget_map, Fin.preimage_castAdd_Icc_castAdd, MeasureTheory.ProbabilityMeasure.apply_iUnion_le, OrderEmbedding.addRight_apply, nnnorm_smul_le, Set.einfsep_lt_top, CategoryTheory.Sieve.functorPushforward_pullback_le, Cardinal.nat_succ, Mathlib.Meta.NormNum.isNat_ordinalLE_true, commutator_centralizer_commutator_le_center, negPart_lt, IsNonarchimedean.finset_image_add_of_nonempty, DivisibleHull.coeOrderAddMonoidHom_apply, PosSMulMono.toSMulPosMono, Order.one_le_krullDim_iff, MeasureTheory.OuterMeasure.mkMetric'.mono_pre_nat, LieSubmodule.instIsAtomicOfIsArtinian, le_iff_lt_or_eq, Multiset.card_Ioo_eq_card_Icc_sub_two, Part.fix_eq_ωSup, PosSMulStrictMono.nnrat_of_rat, UniformSpace.le_def, emultiplicity_prime_le_emultiplicity_image_by_factor_orderIso, BoundedLatticeHom.dual_apply_toFun, IsGalois.fixedField_top, HahnSeries.archimedeanClassMk_le_archimedeanClassMk_iff_of_orderTop_ofLex, InfHom.withBot'_toFun, WithBot.sInf_empty, AlgebraicGeometry.structureSheafInType.add_apply, Submodule.sup_dualAnnihilator_le_inf, Metric.hausdorffEDist_prod_le, Subsemigroup.closure_singleton_le_iff_mem, AddAction.stabilizer_inf_stabilizer_le_stabilizer_union, continuous_id_iff_le, mul_nonpos_iff, MeasureTheory.preVariation.sum_le, Subalgebra.toSubmodule_toSubalgebra, ProbabilityTheory.indep_limsup_atBot_self, Finset.min'_le_max', instMonoImageToKernel, Int.one_le_ceil_iff, Digraph.toSimpleGraphInclusive_mono, AbsConvex.absConvexHull_subset_iff, Submodule.inf_comap_le_comap_add, nhdsWithin_le_nhds, MeasureTheory.monotone_lintegral, le_nhds_of_cauchy_adhp_aux, UpperSet.iciSupHom_apply, Hyperreal.lt_neg_of_pos_of_infinitesimal, Set.Iio_union_Ioi, ENNReal.log_le_zero_iff, ZeroLEOneClass.factZeroLtOne, stdSimplex.zero_le, TopologicalSpace.Opens.eq_bot_or_top, Decidable.le_iff_lt_or_eq', MeasureTheory.sum_measure_le_measure_univ, IsWellFounded.rank_eq, NNReal.ball_zero_eq_Ico', map_partialSups, AlgebraicGeometry.morphismRestrict_appTop, nhdsSet_Iio, Submonoid.comap_le_comap_iff_of_surjective, lowerHemicontinuous_iff_isClosed_preimage_Iic, ProperCone.innerDual_insert, thickenedIndicator_subset, IsLocalRing.closed_point_mem_iff, Set.Iic_add_Iio_subset, AlgebraicGeometry.Scheme.Hom.exists_mem_and_isIso_morphismRestrict_toNormalization, Subgroup.map_subtype_le, CompleteLinearOrder.compare_eq_compareOfLessAndEq, IsNonarchimedean.nsmul_le, Set.pairwise_disjoint_Ioc_add_intCast, AbsoluteValue.hasBasis_uniformity, IdemSemiring.toCanonicallyOrderedAdd, AlgebraicGeometry.Scheme.IdealSheafData.ofIdealTop_ideal, squarefree_iff_emultiplicity_le_one, ArchimedeanClass.mk_le_mk_iff_ratCast, Subgroup.index_toAddSubgroup, Algebra.discr_powerBasis_eq_prod, InnerProductSpace.gramSchmidtOrthonormalBasis_inv_blockTriangular, AddMonoidHom.coe_toIntLinearMap_ker, Filter.generate_image_preimage_le_comap, NNRat.one_lt_cast, geom_sum_neg_iff, OrderIso.infIrredUpperSet_apply, PrimeSpectrum.le_iff_specializes, ContinuousLinearMap.le_opNNNorm, Nimber.not_lt_zero, NonUnitalStarSubalgebra.le_topologicalClosure, Set.Ico_disjoint_Ico, MeasureTheory.lintegral_withDensity_le_lintegral_mul, FirstOrder.Language.Substructure.monotone_map, MeasureTheory.SimpleFunc.FinMeasSupp.lintegral_lt_top, Submonoid.closure_mul_le, EReal.preimage_coe_Ico_top, sdiff_sdiff_le, ae_restrict_le_codiscreteWithin, LSeries.abscissaOfAbsConv_le_of_le_const_mul_rpow, ContinuousLinearMap.nonlinearRightInverseOfSurjective_nnnorm_pos, MeasureTheory.eLpNorm_lt_top_of_finite, ValuativeRel.IsNontrivial.exists_lt_one, MeasureTheory.Measure.restrict_mono_ae, RingHom.pNilradical_le_ker_of_perfectRing, Finset.prod_Iio_mul_eq_prod_Iic, Algebra.IsAlgebraic.isNormalClosure_iff, Filter.not_one_le_div_iff, Set.unbounded_le_iff, bddLat_dual_comp_forget_to_bddOrd, CategoryTheory.Triangulated.SpectralObject.triangle_mor₁, TopCat.Presheaf.stalkPushforward_germ, MeasureTheory.eLpNorm_one_condExp_le_eLpNorm, FractionalIdeal.mul_left_le_iff, NNReal.HolderTriple.right_pos, iInf_top, Finset.isLUB_sup, Metric.coveringNumber_subset_le, enorm_mul₄_le', AddSubgroup.gc_map_comap, OrderAddMonoidHom.snd_comp_inr, MeasurableSpace.gc_comap_map, not_isTop_iff_ne_top, Order.lt_add_one_iff, Finset.card_Ioi_eq_card_Ici_sub_one, MeasureTheory.hittingAfter_eq_sInf, BddLat.forget_lat_partOrd_eq_forget_bddOrd_partOrd, Cardinal.not_injective_limitation_set, measurable_le, AlgebraicGeometry.SheafedSpace.comp_hom_c_app', MeasureTheory.condLExp_bot_ae_eq, ENNReal.image_coe_Iio, Rat.toNNRat_add_le, ENNReal.div_lt_top, StarSubalgebra.toSubalgebra_le_iff, Tuple.eq_sort_iff, LowerSet.Iic_sInf, Fin.finsetImage_castSucc_Ioi, NNReal.one_le_sqrt, MeasureTheory.stoppedProcess_eq_of_mem_finset, MvPolynomial.degrees_prod_le, Order.coheight_le_iff', Filter.le_div_iff, edist_le_zero, sign_neg, OrderAddMonoidHom.comp_add, AbsoluteValue.exists_one_lt_lt_one_of_not_isEquiv, Order.krullDim_le_of_strictMono, AlgebraicGeometry.Scheme.Hom.preimage_opensRange, Ordinal.veblen_le_veblen_iff, Unitization.nnreal_cfcₙ_eq_cfc_inr, StarSubalgebra.toSubalgebra_eq_top, Algebra.adjoin_union_coe_submodule, List.sortedGT_ofFn_iff, AntilipschitzWith.dimH_preimage_le, Filter.map_le_map_iff, ProperCone.dual_flip_dual, CategoryTheory.Triangulated.TStructure.ge_antitone, Cardinal.continuum_toENat, interior_eq_nhds, Set.preimage_const_sub_Ioi, SupClosed.supClosure_eq, essInf_eq_sSup, exists_int_gt, inv_nhdsLT_zero, inv_nonneg, Ideal.exists_maximal_not_isPrincipal, AlgebraicGeometry.ΓSpecIso_hom_stalkClosedPointIso_inv, induced_topology_le_preorder, PosSMulMono.toPosSMulStrictMono, Ordinal.bsup_congr, WithTop.isGLB_sInf, Nat.range_cast_int, germ_skyscraperPresheafStalkOfSpecializes_hom_assoc, List.SortedLT.strictMono_get, CategoryTheory.Functor.instIsWellOrderContinuousCompFunctor, OrderIso.sumLexIicIoi_apply_inl, Subsemiring.toSubsemiring_homogeneousCore_le, FinBoolAlg.hasForgetToFinPartOrd_forget₂_obj_isFintype, StarSubalgebra.mem_top, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_apply, iInf_lt_iff, OrderDual.instSMulPosReflectLT, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctor_obj_π_app, CompositionSeries.head_le_of_mem, CategoryTheory.ComposableArrows.whiskerLeftFunctor_map_app, monovary_id_iff, Sbtw.left_mem_image_Ioi, SuccOrder.nhdsGT, ValuativeRel.le_uniformizer_iff, IsPGroup.le_or_disjoint_of_coprime, Ordinal.eq_zero_or_pos, CategoryTheory.GrothendieckTopology.bot_covering, MeasureTheory.preVariation.sum_le_preVariationFun_iUnion', lt_or_eq_or_gt_or_incompRel, AddSubmonoid.surjOn_iff_le_map, LinearOrderedCommGroupWithZero.inr_eq_coe_inrₗ, ProbabilityTheory.Kernel.withDensity_rnDeriv_le, SupHom.withBot'_toFun, Submodule.mulMap_one_left_eq, Ioc_mem_nhds_iff, lebesgue_number_lemma_of_emetric, Finset.Icc_succ_left_eq_Ioc, Field.Emb.Cardinal.deg_lt_aleph0, CategoryTheory.Functor.IsWellOrderContinuous.restriction_setIci, isAntichain_iff_forall_not_lt, MeasureTheory.Egorov.notConvergentSeq_antitone, OrderHom.map_gfp, EReal.sub_neg, Fin.map_valEmbedding_Ioi, BooleanSubalgebra.map_mono, LinearMap.range_le_iff_comap, tendsto_const_mul_pow_nhds_iff, Ordinal.small_Icc, NNReal.Lp_add_le_hasSum, isPiSystem_image_Iio, UpperSet.prod_self_le_prod_self, MeasureTheory.le_trim, Pi.wcovBy_iff_exists_left_eq, DFinsupp.Lex.isStrictOrder, Fin.finsetImage_natAdd_Ioi, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_basicOpen, signedDist_smul, AffineSubspace.isEmpty_bot, max_sub_max_le_max, iUnion_Ico_intCast, Set.finite_iff_bddAbove, Submodule.toAddSubmonoid_le, MeasureTheory.SimpleFunc.measure_support_lt_top_of_lintegral_ne_top, lowerCentralSeries_map_subtype_le, ProbabilityTheory.Kernel.rnDeriv_lt_top, Dynamics.netEntropyInfEntourage_le_coverEntropyInfEntourage, TopCat.Presheaf.germ_stalkPullbackHom_assoc, MeasureTheory.eLpNorm_mono_ae_real, ENNReal.ofReal_le_coe, uniformity_edist, GenContFract.le_of_succ_succ_get?_contsAux_b, TopCat.Presheaf.whiskerIsoMapGenerateCocone_hom_hom, AlgebraicGeometry.LocallyRingedSpace.Γ_map, nhdsWithin_le_of_mem, ringKrullDim_le_iff_isMaximal_height_le, Finset.max_eq_top, ProbabilityTheory.gaussianPDFReal_inv_mul, nhdsGE_basis_Icc, FirstOrder.Language.DirectLimit.Equiv_isup_symm_inclusion, Cardinal.natCast_mul_strictMono, Valuation.integer.v_irreducible_pos, Set.Ioo_subset_Ioc_union_Ioo, MeasureTheory.Measure.compl_mem_cofinite, Ordinal.veblen_le_veblen_iff_right, Ordinal.ofNat_lt_card, StieltjesFunction.measure_Icc, IntermediateField.adjoin_simple_eq_bot_iff, MeasureTheory.all_ae_ofReal_f_le_bound, IntermediateField.relfinrank_bot_left, Projectivization.Subspace.monotone_span, Finsupp.Lex.addRightStrictMono, AlgebraicGeometry.isCompactOpen_iff_eq_basicOpen_union, Matrix.PosSemidef.det_nonneg, Set.Ioi_subset_Ioi_iff, CategoryTheory.ComposableArrows.scMap_τ₃, TopCat.Presheaf.presheafEquivOfIso_functor_obj_obj, ENNReal.lintegral_mul_le_Lp_mul_Lq, Pi.isCoatomic, AlgebraicGeometry.PresheafedSpace.componentwiseDiagram_obj, LieModule.genWeightSpaceChain_def, nnnorm_le_nnnorm_add_nnnorm_div, PointedCone.dual_iUnion, FirstOrder.Language.Substructure.subset_closure, PrimeSpectrum.coe_primesOverOrderIsoFiber_apply_asIdeal, PrimitiveSpectrum.closedsGC_closureOperator, covBy_top_iff, Ideal.span_singleton_mul_left_mono, le_imp_le_of_lt_imp_lt, exists_subset_affineIndependent_affineSpan_eq_top, Asymptotics.IsBigOTVS.eventually_smallSets, Complex.isOpen_im_gt_EReal, ENNReal.coe_inv_le, affineSpan_le_of_subset_coe, Order.pred_eq_csSup, AlgebraicGeometry.Scheme.Cover.toPresieveOver_le_arrows_iff, Module.AEval.mem_mapSubmodule_symm_apply, Valuation.ltSubmodule_le_leSubmodule, ringKrullDim_le_ringKrullDim_add_card, OrderIso.invENNReal_apply, OrderMonoidHom.commute_inlₗ_inrₗ, dotProduct_self_star_nonneg, Subsemigroup.opEquiv_apply, isGreatest_compl, Sublattice.comap_equiv_eq_map_symm, HasFPowerSeriesWithinOnBall.radius_pos, t1Space_iff_disjoint_pure_nhds, AddSubmonoid.leftNeg_le_isAddUnit, nnnorm_apply_le_nnnorm_cfc, lt_iff_exists_add, CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinsetLimitCone_cone_pt, Cardinal.lift_le_aleph1, AlgebraicGeometry.Spec.algebraMap_stalkIso_inv, CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinset_map_app, Valuation.leIdeal_mono, FirstOrder.Language.Substructure.closure_iUnion, ConditionallyCompleteLinearOrder.isCompact_Icc, absConvex_absConvexHull, Cardinal.gc_ord_card, Set.Ioc_diff_Ioi, coLindelof_le_coclosedLindelof, PNat.card_fintype_Ioo, Subgroup.le_comap_map, Nimber.small_Ioc, nnnorm_pow_le', infIooOrderIsoIooSup_apply_coe, monotone_Iic, Set.Ici_injective, CategoryTheory.ComposableArrows.fourδ₂Toδ₁_app_zero, StrictConvexOn.convex_lt, Metric.infEdist_le_infEdist_cthickening_add, Set.neg_Iio, HahnSeries.order_smul_not_lt, FiniteMulArchimedeanClass.congrOrderIso_symm, Int.ceil_le, CanonicallyOrderedCommMonoid.single_le_prod, AddSubgroup.map_subtype_le_map_subtype, CategoryTheory.ComposableArrows.δlastFunctor_map_app, CategoryTheory.Classifier.SubobjectRepresentableBy.iso_inv_left_π_assoc, Finset.Icc_subset_uIcc, SimpleGraph.Subgraph.map_monotone, Set.strictMonoOn_projIci, Booleanisation.lift_lt_lift, CategoryTheory.Subgroupoid.generated_le_generatedNormal, MeasureTheory.MemLp.integral_indicator_norm_ge_le, Antisymmetrization.prodEquiv_symm_apply_mk, Tuple.antitone_pair_of_not_sorted', Set.not_ordConnected_inter_Icc_iff, MeasureTheory.iInf_le_lintegral, Polynomial.trailingDegree_lt_wf, AntitoneOn.Ioo, LSeries_differentiableOn, Filter.map_add_right_nhdsLT, le_iff_forall_one_lt_lt_mul, SimplexCategoryGenRel.simplicialEvalσ_of_isAdmissible, Set.finite_diff_iUnion_Ioo', ZFSet.rank_range, le_inf_sup, List.monotone_prod_take, AlgebraicGeometry.Scheme.Hom.ker_eq_bot, Set.sub_mem_Ico_iff_right, AlgebraicGeometry.LocallyRingedSpace.stalkMap_germ_apply, MeasureTheory.Measure.AbsolutelyContinuous.ae_le, MeasureTheory.Measure.comap_apply_le, Fin.finsetImage_val_Icc, antivaryOn_iff_smul_rearrangement, Finset.inf_of_mem, RingCon.matrix_bot, PointedCone.mem_span_set, Module.Basis.flag_wcovBy, Subsemiring.instCovariantClassHSMulLe, Submodule.le_orthogonal_iff_le_orthogonal, CompleteLinearOrder.le_total, AddSubgroup.coe_toIntSubmodule, CauSeq.lim_lt, mabs_div_pos, OrderEmbedding.birkhoffSet_sup, Fin.image_val_Ioi, powers_le_nonZeroDivisors_of_noZeroDivisors, iSup₂_iInf₂_le_iInf₂_iSup₂, AddSubgroup.zmultiples_le_of_mem, SetRel.preimage_mono, Multiset.Icc_self, SSet.Subcomplex.N.le_iff, CategoryTheory.projectiveDimension_le_iff, SemilatInfCat.Iso.mk_inv_toFun, BoundedContinuousFunction.nnnorm_coe_le_nnnorm, Multiset.le_ndunion_left, MeasureTheory.Content.outerMeasure_caratheodory, TopCat.GlueData.MkCore.t_id, Set.Iic_subset_Iic_union_Ioc, Finset.subConst_le_card, Mathlib.Meta.Positivity.pos_of_isNat, bot_lt_top, Algebra.adjoin_adjoin_coe_preimage, Set.image_const_div_Ioo, AlgebraicGeometry.HasAffineProperty.affineAnd_iff, dist_lt_coe, MeasureTheory.OuterMeasure.le_trim_iff, Set.image_const_div_Icc, mem_closure_iff_ultrafilter, PredOrder.nhdsLT, CompletePartialOrder.scottContinuous, CategoryTheory.Limits.CompleteLattice.finiteLimitCone_cone_pt, Prod.instSMulPosStrictMono, AlgebraicGeometry.IsOpenImmersion.app_eq_appIso_inv_app_of_comp_eq, Subring.integralClosure_le_iff, SimpleGraph.chromaticNumber_le_one_of_subsingleton, EMetric.infEdist_le_edist_of_mem, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.c_iso, Set.ordConnected_of_uIcc_subset_left, PNat.Prime.one_lt, AlgebraicGeometry.Scheme.IdealSheafData.ker_subschemeι_app, ProjectiveSpectrum.gc_ideal, IsOrderedCancelAddMonoid.toAddLeftReflectLE, RieszExtension.step, Subalgebra.le_topologicalClosure, AlgebraicGeometry.IsOpenImmersion.ΓIso_inv, TopologicalSpace.Closeds.coe_finset_sup, ENNReal.natCast_lt_ofReal, AlgebraicGeometry.Scheme.Modules.Hom.sub_app, instIsConcreteLE, PrimeSpectrum.isIdempotentElemEquivClopens_apply_toOpens, Topology.RelCWComplex.iUnion_openCell_eq_skeletonLT, PrimeSpectrum.basicOpen_isIdempotentElemEquivClopens_symm, SignType.neg_lt_neg_iff, Subring.pointwise_smul_le_iff₀, nndist_nnnorm_nnnorm_le', sq_nonpos_iff, HeytingHomClass.map_bot, Filter.closure_singleton, Part.mem_chain_of_mem_ωSup, Ordinal.toGame_nmul, Cardinal.toNat_injOn, Set.Ioo_pred_left_eq_Ioc, fg_adjoin_of_finite, AlgebraicGeometry.eq_top_of_sigmaSpec_subset_of_isCompact, lowerSemicontinuousWithinAt_iff_le_liminf, Int.fract_add_le, Cardinal.succ_def, AddLocalization.mk_lt_mk, MeasureTheory.measure_biUnion_finset_le, Metric.coveringNumber_le_packingNumber, AlgebraicGeometry.IsOpenImmersion.range_pullback_snd_of_left, CompleteSemilatticeInf.sInf_le, AlgebraicGeometry.LocallyRingedSpace.restrict_presheaf_map, Order.pred_lt_iff_ne_bot, HahnSeries.orderTop_add_le_mul, Set.range_projIic, Valuation.ltAddSubgroup_le_leAddSubgroup, AffineSubspace.map_comap_le, MeasureTheory.Content.outerMeasure_exists_compact, Fin.map_castLEEmb_Iio, Ring.jacobson_le_of_eq_bot, LipschitzWith.dimH_range_le, LinOrd.Iso.mk_inv, Filter.covariant_vadd, Fin.preimage_cast_Iio, CategoryTheory.TransfiniteCompositionOfShape.ofArrowIso_incl, Algebra.adjoin_top, AlgebraicGeometry.Scheme.instIsIsoSubschemeιBotIdealSheafData, CategoryTheory.IsGrothendieckAbelian.generatingMonomorphisms.lt_largerSubobject, TopologicalSpace.Opens.mk_empty, Module.End.genEigenspace_le_genEigenspace_maxUnifEigenspaceIndex, IsOpenMap.functorNhds_map, Ordinal.bsup_eq_blsub_iff_lt_bsup, CategoryTheory.Limits.equalizerSubobject_arrow, FrameHomClass.toInfTopHomClass, HurwitzZeta.hasSum_nat_hurwitzZetaOdd, NNReal.half_lt_self, IntermediateField.extendScalars.orderIso_symm_apply_coe, MeasureTheory.ae_restrict_uIoc_iff, MeasureTheory.Measure.cofinite_le_ae, Filter.NeBot.mul_zero_nonneg, LinearOrderedCommGroupWithZero.fst_comp_inl, IsCyclotomicExtension.le_of_dvd, upperClosure_eq_bot_iff, AddEquiv.mapAddSubgroup_apply, IncidenceAlgebra.mu_eq_neg_sum_Ioc_of_ne, ENNReal.mul_inv_le_iff, not_lt, Subgroup.pointwise_smul_le_iff₀, bihimp_self, ENat.le_coe_iff, IsCompact.exists_isMinOn, Finset.upShadow_monotone, MeasureTheory.Measure.zero_le, Module.FinitePresentation.exists_lift_equiv_of_isLocalizedModule, iSup_ne_bot_subtype, MeasureTheory.mul_meas_ge_le_lintegral₀, Set.add_mem_Icc_iff_left, abs_pow_sub_pow_le, ContDiffMapSupportedIn.structureMapCLM_apply_withOrder, Int.ceil_lt_add_one, PartENat.withTopEquiv_le, EMetric.diam_pos_iff, Monoid.le_minOrder_iff_forall_subgroup, BddOrd.Iso.mk_inv, HahnEmbedding.ArchimedeanStrata.isInternal_stratum', Set.inv_Ioo, Submonoid.mk_le_mk, AlgebraicGeometry.Proj.add_apply, Finset.sum_Ico_add, AlgebraicGeometry.PresheafedSpace.Γ_obj, IsLocalRing.Ring.KrullDimLE.eq_bot_or_eq_top, AffineSubspace.perpBisector_eq_top, Algebra.adjoin_le_centralizer_centralizer, Multiset.lt_cons_self, ENNReal.inner_le_weight_mul_Lp_of_nonneg, NNReal.inner_le_weight_mul_Lp, Fin.image_castAdd_Ioi, Ordinal.le_preOmega_self, SSet.Subcomplex.Pairing.WeakRankFunction.lt, Dense.borel_eq_generateFrom_Ioc_mem_aux, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctor_map_hom, le_symmDiff_sup_left, MeasureTheory.condExpIndL1Fin_disjoint_union, Int.image_fract, Multiset.exists_min_image, OrderRingIso.apply_eq_self, EisensteinSeries.tsum_tsum_symmetricIco_sub_eq, Surreal.nat_toGame, Finset.isAtom_singleton, IntermediateField.adjoin_intCast, Ordinal.isSuccPrelimit_lift, RCLike.nonneg_iff_exists_ofReal, Filter.map_mul_left_nhdsGT, ClosedSubmodule.toSubmodule_le_toSubmodule, CategoryTheory.Limits.imageSubobject_arrow_comp_assoc, mem_codiscreteWithin, Module.supportDim_le_ringKrullDim, Topology.IsOpenEmbedding.functor_obj_injective, Fin.orderSucc_apply, UpperSet.le_erase, Nimber.instZeroLEOneClass, NNReal.le_toNNReal_of_coe_le, Set.Iic_union_Ici, Fin.succAboveOrderEmb_toEmbedding, Algebra.adjoin_root_eq_top_of_isSplittingField, Finset.Iio_mul_Iic_subset', Ideal.subset_union_prime_finite, SSet.skeletonOfMono_zero, Matrix.PosSemidef.dotProduct_mulVec_nonneg, not_isMin_top, WellFoundedLT.rank_strictMono, Finset.filter_sups_le, AlgebraicGeometry.Scheme.Hom.ι_toNormalization_assoc, HomotopyCategory.composableArrowsFunctor_map, inf_sSup_eq_iSup_inf_sup_finset, PrincipalSeg.isNormal, CategoryTheory.Limits.isIso_kernelSubobject_zero_arrow, AffineSubspace.map_inf_le, inv_mabs_le_inv, Pi.isAtom_iff, CategoryTheory.Functor.instIsWellOrderContinuousCompFunctorEquivalence, Positive.coe_inv, ContinuousOn.finset_sup, Rat.cast_nonpos, Finset.sortedLT_sort, Interval.coe_subset_coe, BoxIntegral.Box.disjoint_coe, isLUB_hasSum', Set.preimage_const_sub_Ico, Cardinal.mul_lt_aleph0_iff, SimplexCategory.II.monotone_map', integrableOn_Icc_iff_integrableOn_Ioc', Hyperreal.lt_of_pos_of_infinitesimal, AddSubgroup.comap_lt_comap_of_surjective, OmegaCompletePartialOrder.ωScottContinuous.isLUB, archimedean_iff_rat_le, LinearMap.image_convexHull, CategoryTheory.Triangulated.SpectralObject.mapTriangulatedFunctor_δ, Finset.Icc_mul_Ico_subset', Fin.prod_Ico_castSucc, one_lt_sq_iff_one_lt_abs, BoundedContinuousFunction.nnnorm_le, ZFSet.rank_insert, tendsto_neg_nhdsGE_neg, AlgebraicGeometry.PresheafedSpace.toRestrictTop_c, Dynamics.coverEntropy_nonneg, Ideal.IsMaximal.out, ArchimedeanClass.FiniteResidueField.mk_le_mk, AlgebraicGeometry.IsLocalAtTarget.restrict, Ordinal.toGame_lf_iff, ENNReal.nhds_zero_basis_Iic, Topology.IsInducing.map_functorObj, OrderIso.subsingleton_of_wellFoundedLT, Filter.tendsto_pow_atTop_iff, UpperSet.prod_self_lt_prod_self, TopCat.Presheaf.germ_stalkPullbackInv, Set.encard_le_encard_iff_encard_diff_le_encard_diff, mem_nhdsGE_iff_exists_Ico_subset, PartOrdEmb.Limits.instPreservesColimitsOfShapeForgetOrderEmbeddingCarrier, Fin.strictMono_castSucc, RingHom.liftOfRightInverse_comp, Affine.Simplex.sign_touchpointWeights_singleton_neg, Algebra.surjective_algebraMap_iff, Set.Ioc_add_bij, Finset.add_sum_Ioi_eq_sum_Ici, SetSemiring.up_le_up, tendsto_natCast_atTop_atTop, Module.length_le_of_injective, CategoryTheory.ComposableArrows.sc'Map_τ₃, MvPolynomial.degrees_add_le, Set.image_update_Ioc_left, Multiset.map_le_map_iff, CategoryTheory.nerve.δ_obj, ENNReal.tsum_comp_le_tsum_of_injective, DFinsupp.lex_le_iff_of_unique, Ordinal.enum_le_enum, IntermediateField.sepDegree_bot', MeasureTheory.integral_Ico_eq_integral_Ioo', ProbabilityTheory.HasSubgaussianMGF.add, ProperCone.toPointedCone_bot, Set.Ico_add_one_add_one_eq_Ioc, NNReal.iInf_real_pos_eq_iInf_nnreal_pos, Cardinal.preAleph_zero, Ordinal.lsub_le_iff, MvPowerSeries.map_iterateFrobenius_expand, IsAtomic.exists_atom, UpperSet.coe_sub, Ultrafilter.le_of_inf_neBot, SSet.horn₃₂.desc.multicofork_π_zero_assoc, addRothNumber_map_add_left, Polynomial.supDegree_eq_degree, Sublattice.prod_right_mono, MeasureTheory.SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge, Order.succ_eq_succ_iff, ENNReal.le_ofReal_iff_toReal_le, CategoryTheory.Presheaf.isSheaf_bot, Ideal.mul_le_left, Ordinal.lt_wf, AlgebraicGeometry.Scheme.Hom.fromNormalization_app_assoc, ProbabilityTheory.Fernique.measure_le_mul_measure_gt_normThreshold_le_of_map_rotation_eq_self, MeasureTheory.exists_Lp_half, ENNReal.sum_le_tsum, le_of_forall_lt_imp_ne, Finset.sum_sum_Ioi_add_eq_sum_sum_off_diag, CStarAlgebra.conjugate_le_norm_smul', PartENat.withTopEquiv_lt, Valuation.IsNontrivial.exists_one_lt, AlgebraicGeometry.Scheme.isoSpec_inv_toSpecΓ_assoc, lowerClosure_le, Algebra.adjoin_eq_span, Set.einfsep_pos_of_finite, RCLike.ofReal_nonpos, Set.indicator_le_self, SimpleGraph.Walk.toSubgraph_le_induce_support, Filter.comap_neg_atBot, AddMonoidAlgebra.le_infDegree_mul, Ideal.IsPrime.pow_le_iff, CategoryTheory.Limits.WalkingMultispan.inclusionOfLinearOrder_obj, le_antisymm_iff, nhdsSet_prod_le, OrderIso.pnatIsoNat_apply, Set.Icc_diff_right, le_iff_nhds, LocallyFiniteOrder.orderAddMonoidHom_apply, FirstOrder.Language.Substructure.closure_union, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverse_obj_pt, Finset.Ioc_map_sectR, TopHom.coe_sup, AlgebraicGeometry.ProjectiveSpectrum.Proj.isIso_toSpec, Fin.map_succEmb_Ioo, one_half_pos, Set.image_mulSingle_Ico_left, MvPolynomial.degrees_mul_le, Filter.map_atTop_finset_prod_le_of_prod_eq, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.invApp_app_apply, ContinuousMap.ideal_gc, AlgebraicGeometry.Scheme.IdealSheafData.ideal_iInf, not_lt_zero', Ordinal.card_succ, IntermediateField.LinearDisjoint.bot_right, nhdsLT_le_nhdsNE, AlgebraicGeometry.stalkMap_toStalk_apply, NumberField.mixedEmbedding.minkowskiBound_lt_top, Fin.preimage_castSucc_Iio_castSucc, AlgebraicGeometry.IsAffineOpen.primeIdealOf_eq_map_closedPoint, edist_le_coe, AlgebraicGeometry.Scheme.Hom.isoImage_inv_ι, Module.supportDim_le_supportDim_quotSMulTop_succ, Filter.limsSup_bot, MonoidHom.coe_toAdditive_range, SimpleGraph.le_card_nonuniformWitness, CategoryTheory.SmallObject.SuccStruct.transfiniteCompositionOfShapeιIteration_incl, MeasureTheory.ae_measure_preimage_add_right_lt_top_of_ne_zero, Set.SurjOn.filter_map_Iic, Coheyting.boundary_le, Cardinal.lt_aleph0_iff_finite, AffineSubspace.gc_map_comap, tendsto_neg_nhdsGT, Ideal.radical_le_radical_iff, TopologicalSpace.Opens.leSupr_apply_mk, le_iff_eq_or_lt', Finset.Colex.le_def, Ordinal.sub_le_self, ValuationRing.iff_ideal_total, CliffordAlgebra.even_toSubmodule, Finset.sup_eq_bot_of_isEmpty, InnerProductSpace.gramSchmidt_def'', NNReal.geom_mean_le_arith_mean_weighted, Set.unbounded_ge_inter_gt, Set.OrdConnected.image_real_toNNReal, CategoryTheory.projectiveDimension_ge_iff, StieltjesFunction.length_eq, AffineSubspace.pointwise_vadd_bot, Field.primitive_element_iff_minpoly_degree_eq, NNReal.le_inv_iff_mul_le, mem_nhdsLT_iff_exists_Ioo_subset, SummationFilter.instLeAtTopSymmetricIcc, FiniteMulArchimedeanClass.min_le_mk_mul, AntilipschitzWith.le_mul_nnnorm, MonomialOrder.degree_prod_le, NonemptyInterval.length_sum, EMetric.mem_nhds_iff, LinearOrderedSemifield.toPosSMulStrictMono_rat, nndist_midpoint_midpoint_le, EMetric.hausdorffEdist_union_le, Module.End.genEigenspace_zero_nat, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivUnitIso_hom_app_hom, Algebra.toSubmodule_eq_top, Representation.le_comap_invariants, NonemptyInterval.fst_neg, AlgebraicGeometry.Scheme.stalkMap_germ_apply, antivary_iff_mul_rearrangement, AlgebraicGeometry.Scheme.IdealSheafData.range_glueDataObjι, AddSubmonoid.closure_add_le, Dynamics.coverEntropyEntourage_antitone, ENNReal.pow_le_pow_left_iff, CategoryTheory.Subobject.bot_arrow, MeasureTheory.Filtration.rightCont_eq, Set.infinite_iff_exists_gt, NNReal.image_coe_Iic, DFA.pumping_lemma, principal_inter_le_nhdsSetWithin, AlgebraicGeometry.Scheme.Hom.isoImage_inv_ι_assoc, LieModule.le_max_triv_iff_bracket_eq_bot, AlgebraicTopology.NormalizedMooreComplex.map_f, Fin.map_valEmbedding_Ioo, AlgebraicGeometry.PresheafedSpace.GlueData.opensImagePreimageMap_app_assoc, MeasureTheory.eLpNorm'_le_eLpNormEssSup_mul_rpow_measure_univ, HomogeneousIdeal.irrelevant_eq_span, AlgebraicGeometry.Scheme.Hom.instIsIsoCommRingCatAppObjOpensOpensFunctor, AlgHom.fieldRange_eq_top, Filter.eventually_cocardinal, Ordinal.aleph0_le_cof, Filter.covariant_sub, FiniteMultiplicity.emultiplicity_lt_of_multiplicity_lt, HahnSeries.leadingCoeff_nonneg_iff, IntermediateField.normal_iff_forall_map_le, Ordinal.le_iSup, succ_notMem_iff_isMax, Subring.mk_le_mk, Subgroup.map_eq_range_iff, WithZeroTopology.isOpen_iff, Order.pred_lt_pred_iff, Fin.sum_Iio_castAdd, Ne.lt_top, TwoSidedIdeal.top_asIdeal, iSup_emptyset, MeasureTheory.isTightMeasureSet_iff_exists_isCompact_measure_compl_le, le_csInf_iff'', toZ_iterate_pred, AlgebraicGeometry.Scheme.ker_morphismRestrict_ideal, CategoryTheory.Subobject.lowerEquivalence_counitIso, Matroid.eRk_le_eRank, ENat.lt_add_one_iff, PrimeSpectrum.gc_set, ENat.addLECancellable_of_ne_top, IsLocalization.orderIsoOfPrime_symm_apply_coe, EMetric.diam_image_le_iff, MulPosReflectLT.toMulPosMono, StarAlgebra.gc, Continuous.cfcₙ_nnreal_of_mem_nhdsSet, Ordinal.card_omega, CompleteLattice.MulticoequalizerDiagram.multispanIndex_snd, RootedTree.isPredArchimedean, Profinite.NobelingProof.isClosed_proj, MeasureTheory.ProbabilityMeasure.apply_union_le, Nimber.small_Icc, Fin.preimage_rev_Iio, tendsto_intCast_atBot_iff, AlgebraicGeometry.Scheme.Modules.pushforwardId_hom_app_app, Filter.map_mul_right_nhdsLT, List.prod_min_le, BoundedContinuousFunction.nnnorm_const_le, Nimber.instNoMaxOrder, AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_ι, Filter.isBoundedUnder_ge_inf, Finset.max'_eq_iff, TwoSidedIdeal.mem_asIdeal, ContinuousMapZero.le_def, tendsto_nat_ceil_div_atTop, AlgebraicGeometry.Scheme.Hom.germ_stalkMap, Subalgebra.center_le_centralizer, Finsupp.DegLex.isStrictOrder, Cardinal.ofENat_lt_ofENat, convexHull_univ, lebesgue_number_lemma_of_emetric_nhdsWithin, Finset.dens_mono, LinearOrderedField.inducedOrderRingIso_symm, ArchimedeanClass.mk_eq_mk, Submonoid.inv_le_inv, NNRat.instOrderedSub, IntermediateField.normalClosureOperator_isClosed, CategoryTheory.CosimplicialObject.δ_comp_σ_of_gt'_assoc, Fin.orderPred_eq, Subfield.extendScalars.orderIso_apply, Filter.nhds_mono, convexHull_ediam, WithSeminorms.tendsto_nhds_atTop, ProbabilityTheory.IsZeroOrMarkovKernel.bound_le_one, Ordinal.card_typein, SubmodulesRingBasis.leftMul, discreteTopology_bot, EReal.sign_top, Matrix.IsPrimitive.exists_pos_pow, sup_eq_right, AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_fst, Set.image_const_sub_Iic, Real.enorm_exp_I_mul_ofReal_sub_one_le, Subgroup.prod_le_iff, le_imp_eq_iff_le_imp_ge, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivUnitIsoApp_hom_hom, Subfield.map_le_iff_le_comap, Module.End.mem_invtSubmodule_symm_iff_le_map, IsCompact.measure_lt_top, Valuation.srel_one_iff, Finset.inf_empty, Ordinal.lift_typein_top, max_mul_mul_le_max_mul_max', DualNumber.range_inlAlgHom_sup_adjoin_eps, MeasureTheory.hitting_mono, edist_lt_top, AlgebraicGeometry.Scheme.toOpen_eq, LinearOrder.isSuccArchimedean_of_isPredArchimedean, OrderAddMonoidHom.inlₗ_add_inrₗ_eq_toLex, Continuous.strictMono_of_inj, Algebra.Norm.Transitivity.mul_auxMat_blockTriangular, CategoryTheory.ShortComplex.moduleCat_exact_iff_ker_sub_range, SimpleGraph.le_chromaticNumber_of_pairwise_adj, Cardinal.IsRegular.ord_pos, AlgebraicGeometry.Scheme.Hom.ker_toNormalization, AddSubgroup.le_normalizer_of_normal, isCoatomistic_dual_iff_isAtomistic, MonomialOrder.div_set, List.mem_argmax_iff, nnnorm_tsum_le, Ordinal.typein_ordinal, MulArchimedeanClass.mk_mul_lt_mk_right_iff, NNReal.orderIsoIccZeroCoe_symm_apply_coe, tendsto_tsum_compl_atTop_zero, isPurelyInseparable_iff_perfectClosure_eq_top, lowerClosure_vadd, Fin.map_castSuccEmb_Icc, Fin.cast_top, MonoidWithZeroHom.snd_mono, neg_le_negPart, monotone_le, Finset.eraseNone_eq_biUnion, Set.Iic_False, Ordinal.apply_le_nfpFamily, ENNReal.mul_lt_mul_iff_left, posMulReflectLE_iff_posMulReflectLT, AlgebraicGeometry.Scheme.Modules.restrict_obj, CategoryTheory.MorphismProperty.limitsOfShape_le, FirstOrder.Language.Structure.cg_iff, SemilatSupCat.Iso.mk_hom_toFun, Cardinal.isSuccLimit_iff, NonUnitalSubalgebra.map_topologicalClosure_le, Set.Iic.coe_iSup, nullMeasurableSet_lt', iSupIndep_def, compl_bihimp_self, MeasureTheory.Measure.restrict_le_self, RingCon.coe_bot, Module.End.genEigenspace_top, disjoint_pure_nhds, top_disjoint, Monotone.Ioi, infClosure_min, absConvexHull_min, Cardinal.lift_aleph, Set.Iio_True, Ordinal.toNatOrdinal_zero, fact_one_le_one_ennreal, BoundedLatticeHomClass.map_top, IntermediateField.rank_eq_one_iff, gc_inf_himp, Ordinal.div_pos, symmDiff_le_sup, Int.Icc_eq_pair, CategoryTheory.SmallObject.iterationFunctorMapSuccAppArrowIso_hom_left, OmegaCompletePartialOrder.ContinuousHom.monotone, eVariationOn.mono, HahnEmbedding.Partial.orderTop_eq_finiteArchimedeanClassMk, Valuation.IsUniformizer.val_pos, Set.Ioi_subset_Ici_iff, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctorObj_pt, PrimeSpectrum.comapEquiv_symm, Subgroup.prod_mono_left, Finset.orderEmbOfFin_apply, AbsoluteValue.sum_le, PowerSeries.le_order_subst, bot_himp, mkOfAddGroupCone_le_iff, PSet.rank_singleton, Matroid.rankFinite_iff_cRank_lt_aleph0, AlgebraicGeometry.IsOpenImmersion.app_ΓIso_hom_assoc, Ordinal.exists_lt_add_of_not_principal_add, one_le_star_iff, Set.unbounded_ge_inter_not_ge, CStarAlgebra.star_mul_le_algebraMap_norm_sq, MeasureTheory.aecover_Ioo_of_Ioo, Order.pred_eq_iff_isMin, ringKrullDim_lt_top, CStarAlgebra.span_nonneg_inter_closedBall, List.exists_map_eq_of_sorted_iff_wbtw, AntilipschitzWith.le_dimH_image, CStarAlgebra.isClosed_nonneg, Odd.pow_pos_iff, AlgebraicGeometry.PresheafedSpace.GlueData.snd_invApp_t_app, Ordinal.ofNat_le_card, SSet.spine_map_vertex, fderiv_norm_smul, cardinal_lt_aleph0_of_finiteDimensional, Fin.sum_Ioo_castAdd, Set.disjoint_iff_forall_ne, Part.ωScottContinuous_toUnitMono, abs_add', LinOrd.ofHom_comp, ContinuousAlternatingMap.isLeast_opNNNorm, derived_le_lower_central, MeasureTheory.Ioc_ae_eq_Icc', Ideal.asIdeal_toTwoSided, CategoryTheory.injectiveDimension_lt_iff, ENNReal.mem_Iio_self_add, Fin.image_castSucc_Ico, MeasureTheory.Ioo_ae_eq_Ioc, Finsupp.le_iff', HasFDerivAt.hasFDerivAt_norm_smul, NNRat.cast_pos, EMetric.measure_ball_pos, borel_eq_generateFrom_Ico, lowerSemicontinuousAt_iff_le_liminf, Set.coe_projIic, RingHom.IsIntegral.quotient, NatOrdinal.toOrdinal_eq_zero, tendsto_floor_atBot, CStarAlgebra.isStrictlyPositive_iff_isUnit_sqrt_and_eq_sqrt_mul_sqrt, Finset.bddBelow, MeasureTheory.eLpNorm_indicator_le, Subalgebra.fg_bot, Ordinal.omega0_le_preOmega_iff, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.inv_naturality, Ordinal.IsFundamentalSequence.ord_cof, Ideal.decomposition_erase_inf, Ordinal.invVeblen₂_le, inf_eq_top_iff, QuotientAddGroup.btw_coe_iff', Algebra.adjoin_univ, Fin.sum_Iio_cast, Filter.ker_mono, antivary_id_iff, NonUnitalStarSubalgebra.gc_map_comap, Multiset.Ico_filter_le, Fin.preimage_addNat_Ici_addNat, AlgHom.equalizer_same, CompleteLat.Iso.mk_hom, CategoryTheory.IsGrothendieckAbelian.generatingMonomorphisms.functorToMonoOver_map, List.le_maximum_of_mem', StarSubalgebra.topologicalClosure_adjoin_le_centralizer_centralizer, negPart_nonpos', AbsoluteValue.nonneg', OrderIso.essInf_apply, Set.preimage_const_add_Iio, SetSemiring.down_ssubset_down, setOf_riemannianEDist_lt_subset_nhds, MeasureTheory.Content.innerContent_iUnion_nat, TopologicalSpace.Opens.functor_map_eq_inf, LieIdeal.comap_mono, MeasureTheory.aecover_Ioc_of_Ioc, Ideal.span_singleton_mul_right_mono, TwoSidedIdeal.orderIsoMatrix_symm_apply_ringCon_r, eventually_riemannianEDist_le_edist_extChartAt, Cardinal.small_iff_lift_mk_lt_univ, AlgebraicGeometry.Scheme.Modules.restrictAdjunction_unit_app_app, OrdinalApprox.gfpApprox_mono_mid, Subgroup.map_le_iff_le_comap, CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinset_map_app, PosMulMono.toPosMulReflectLT, IsCompactlyGenerated.BooleanGenerators.sSup_le_sSup_iff_of_atoms, SimpleGraph.chromaticNumber_mono, Finset.Icc_diff_both, Finset.Ico_add_Ioc_subset, PNat.instWellFoundedLT, ENNReal.HolderTriple.le, Fin.Iio_castSucc, Ordinal.mulLeftMono, Filter.mem_interior, IntermediateField.eq_bot_of_isPurelyInseparable_of_isSeparable, MeasureTheory.lintegral_max, Finset.Icc_self, MeasureTheory.eLpNorm'_add_le_of_le_one, Projectivization.Subspace.span_le_subspace_iff, Order.isNormal_iff', CategoryTheory.Subobject.mk_eq_top_of_isIso, Cardinal.ofENat_strictMono, ClosedSubmodule.inf_orthogonal_eq_bot, Pi.lt_toLex_update_self_iff, PosNum.cast_pos, CategoryTheory.MorphismProperty.retracts_le_iff, Int.preimage_Ioo, AddSubsemigroup.map_comap_le, CategoryTheory.Subfunctor.image_top, OrderType.type_le_type_iff, CongruenceSubgroup.Gamma1_in_Gamma0, NNReal.coe_mono, one_lt_oneLePart_iff, LieSubmodule.wellFoundedGT_of_noetherian, PUnit.top_eq, Submodule.toSubMulAction_strictMono, LinearMap.map_le_range, CategoryTheory.SmallObject.πFunctorObj_eq, isPiSystem_Iio, TopologicalSpace.Closeds.coe_eq_univ, isLeast_hnot, Equiv.Perm.support_prod_le, Set.abs_sub_le_of_uIcc_subset_uIcc, Mathlib.Tactic.Qify.nnratCast_lt, WithLp.nnnorm_snd_le, ProbabilityTheory.iSup_bayesRisk_le_minimaxRisk, NonemptyFinLinOrd.id_apply, AlgebraicGeometry.Scheme.Hom.isIntegral_app, Finset.prod_mono_set', Interval.length_zero, LTSeries.head_add_length_le_int, Subsemigroup.le_comap_map, HasCardinalLT.Set.cocone_ι_app, RCLike.lt_iff_re_im, IsGaloisGroup.intermediateFieldEquivSubgroup_symm_apply, tendsto_ceil_left', hahnEmbedding_isOrderedModule_rat, AlgebraicGeometry.Scheme.Hom.app_eq_appLE, ProbabilityTheory.bayesRisk_le_minimaxRisk, Subalgebra.pi_toSubsemiring, AddValuation.map_sub, Set.Icc.coe_inf, Affine.Simplex.sign_signedInfDist_touchpoint_empty, Filter.disjoint_atTop_principal_Iic, Algebra.adjoin_mono, Finsupp.Lex.addLeftStrictMono, FiniteArchimedeanClass.congrOrderIso_symm, Fin.map_castAddEmb_Iio, EReal.preimage_coe_Iio, Valuation.RankOne.strictMono', mabs_mul_eq_mul_mabs_iff, ENat.ceil_mono, AffineIndependent.affineSpan_eq_top_iff_card_eq_finrank_add_one, ProjectiveSpectrum.basicOpen_zero, Filter.comap_inv_atBot, Filter.mul_add_subset, mabs_div_mabs_le_mabs_mul, CategoryTheory.MorphismProperty.transfiniteCompositionsOfShape_le_llp_rlp, PNat.lt_add_left, IntermediateField.adjoin_root_eq_top_of_isSplittingField, WithBot.orderIsoPUnitSumLex_toLex, AddSubgroup.opEquiv_apply, Subgroup.opEquiv_symm_apply, iSup_succ, SimpContFract.of_isContFract, CompleteDistribLattice.sdiff_le_iff, StieltjesFunction.outer_le_length, AlgebraicGeometry.ΓSpecIso_inv_ΓSpec_adjunction_homEquiv, Finset.Ici_eq_cons_Ioi, tendsto_const_mul_zpow_atTop_zero, AffineIndependent.convexHull_inter', Polynomial.ringKrullDim_le, Set.image_const_div_Ioi, Set.antitone_mem, ComplementedLattice.isStronglyAtomic', Order.isSuccPrelimit_iff_succ_ne, Int.addSubmonoid_closure_range_pow, Set.Ioi_inter_Ioo, Fin.image_castSucc_Ioo, Ordinal.instNoMaxOrder, Algebra.adjoin_le_iff, SSet.Subcomplex.le_iff_contains_nonDegenerate, GroupNorm.lt_def, max_eq_right_iff, MeasureTheory.Measure.mem_support_restrict, CategoryTheory.Limits.pullback_factors, AlgebraicGeometry.RingedSpace.zeroLocus_empty_eq_univ, Interval.coe_sSubset_coe, Set.Ici_subset_Icc_union_Ioi, map_eq_bot_iff, algebraicClosure.eq_top_iff, Alexandrov.principalOpen_le, Ordinal.bsup_succ_le_blsub, Filter.le_countableGenerate_iff_of_countableInterFilter, SameRay.exists_nonneg_left, Fin.Ioc_sub_one_eq_Icc, AffineSubspace.not_sOppSide_bot, AlgebraicGeometry.StructureSheaf.algebraMap_germ_assoc, RingFilterBasis.SubmodulesBasis.inter, TopCat.Presheaf.SubmonoidPresheaf.toLocalizationPresheaf_app, Finset.le_max', PredOrder.hasBasis_nhds_Ioc, AlgebraicGeometry.StructureSheaf.res_apply, AddSubsemigroup.mk_le_mk, Filter.tendsto_Ioc_uIcc_uIcc, Ordinal.lift_preOmega, Finset.map_add_left_Ioc, nhdsWithin_uIoo_left_le_nhdsNE, NonUnitalSubring.toSubsemigroup_mono, Ring.KrullDimLE.mem_minimalPrimes_iff, MeasureTheory.withDensity_pdf_le_map, AntitoneOn.Iio, Ordinal.succ_iSup_eq_lsub_iff, upperClosure_add_distrib, Fin.Iio_add_one_eq_Iic, isSimpleOrder_iff_isCoatom_bot, Module.fgSystem.instIsDirectedOrderSubtypeSubmoduleFG, MeasureTheory.MeasurePreserving.measure_preimage_le, MeasureTheory.ae_norm_ofReal_f_le_bound, ENNReal.sub_lt_self, LipschitzWith.nnorm_le_mul', TopologicalSpace.instWellFoundedLTClosedsOfNoetherianSpace, GenContFract.of_one_le_get?_partDen, Valuation.vle_one_iff, Set.inv_mem_Ioo_iff, Complex.not_le_zero_iff, CategoryTheory.Functor.mapPresheaf_map_c, Ordinal.omega0_lt_gamma, Nat.ofNat_pos', sign_nonpos_iff, AlgebraicGeometry.Scheme.Modules.Hom.add_app, Cardinal.aleph0_le_aleph, HahnSeries.mem_cardSuppLTAddSubmonoid, Set.Finite.exists_lt_map_eq_of_forall_mem, disjoint_partialSups_left, smul_nonneg_iff, LatticeHom.withBot'_toFun, Cardinal.toENat_le_iff_of_le_aleph0, Valued.integer.exists_nnnorm_lt_one, MeasureTheory.SimpleFunc.measure_preimage_lt_top_of_memLp, PNat.sub_le, Finset.intervalGapsWithin_snd_of_lt, Ordinal.zero_lt_card, algebraicClosure.adjoin_le, one_le_leOnePart, isConnected_Iic, FormalMultilinearSeries.div_le_radius_compContinuousLinearMap, Valuation.IsEquiv.val_sub_one_lt_one_iff, AddSubgroup.le_pi_iff, Filter.Realizer.le_iff, LowerSet.prod_self_le_prod_self, Nat.one_div_cast_nonneg, Fin.sum_Iio_castSucc, Subgroup.relIndex_eq_one, Set.Icc_sub_one_right_eq_Ico, ENat.coe_lt_coe, sign_pow, CompleteDistribLattice.MinimalAxioms.iInf_sup_le_sup_sInf, Ordinal.lift_type_le, Finset.exists_inf_le, SetTheory.Game.bddAbove_range_of_small, NonemptyInterval.coe_coeHom, IsOpen.exists_between_affineIndependent_span_eq_top, Set.powersetCard.mem_ofFinEmbEquiv_iff_mem_range, AddSubgroup.MapSubtype.orderIso_symm_apply, NonemptyInterval.coe_def, Subring.toSubmonoid_strictMono, noBotOrder_iff_noMinOrder, BoxIntegral.Prepartition.distortion_le_iff, Setoid.eq_top_iff, Set.preimage_const_sub_Ici, PowerSeries.order_le, iSup_inf_le_iSup_inf, Metric.mem_thickening_iff_infEdist_lt, Ordinal.IsInitial.card_le_card, CategoryTheory.ComposableArrows.Exact.cokerIsoKer_hom_fac_assoc, measurableSet_Ioo, Valuation.Integers.coe_span_singleton_eq_setOf_le_v_algebraMap, NNReal.instOrderTopology, LinearOrderedAddCommGroup.discrete_or_denselyOrdered, Cardinal.preAleph_le_preAleph, NNReal.agmSequences_fst_le_snd, MonomialOrder.degLex_lt_iff, InfiniteGalois.toAlgEquivAux_eq_liftNormal, TopologicalSpace.OpenNhds.isOpenEmbedding, EReal.preimage_coe_Ioo_bot_top, AffineSubspace.top_coe, Set.Iic_diff_right, MeasureTheory.eLpNorm_one_le_of_le, Cardinal.one_lt_two, IsValuativeTopology.hasBasis_nhds', disjoint_nhds_atTop, sInf_le, ENNReal.addLECancellable_iff_ne, LinearOrder.infConvergenceClass, ContinuousOn.surjOn_Icc, emultiplicity_lt_top, MeasureTheory.ae_lt_top', ProbabilityTheory.evariance_lt_top, Finset.Ioo_add_one_right_eq_Ioc, mkOfGroupCone_le_iff, Ideal.Filtration.pow_smul_le, Finset.tendsto_Ioo_neg_atTop_atTop, latticeClosure_idem, Multiset.max_prod_le, Set.image_single_Ico_left, Set.Ioc.coe_eq_one, Multiset.grade_eq, Set.einfsep_pair_le_right, AlgebraicGeometry.toSpecΓ_SpecMap_ΓSpecIso_inv, ClosureOperator.closure_top, Filter.lift'_map_le, Complex.pos_iff, SSet.mem_skeleton, uniformContinuous_iff, MeasureTheory.Measure.le_iff, NonUnitalSubalgebra.star_mono, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_inv, Field.primitive_element_iff_minpoly_natDegree_eq, OrderMonoidHom.snd_comp_inl, convexHull_zero, isAtomic_iff_isCoatomic, antivaryOn_neg, LowerAdjoint.closure_iSup_closure, HahnModule.orderTop_vAdd_le_orderTop_smul, Polynomial.sup_ker_aeval_le_ker_aeval_mul, NonemptyInterval.length_sub, ENNReal.instOrderTopology, ContinuousLinearMap.nndist_le_opNNNorm, FormalMultilinearSeries.le_radius_of_summable_norm, CStarAlgebra.star_left_conjugate_le_norm_smul, strictConvex_iff_ordConnected, le_map_add_map_sub', Ordinal.small_Ioc, Finsupp.supported_mono, CategoryTheory.Subobject.factorThru_add_sub_factorThru_left, Set.image_const_add_Iio, sInfHom.coe_top, Fin.card_Iic, AlgebraicGeometry.instIsIsoModulesSpecOfCarrierFromTildeΓUnitOpensCarrierCarrierCommRingCatRingCatSheaf, CategoryTheory.Regular.exists_inf_pullback_eq_exists_inf, Fin.prod_Icc_succ, CategoryTheory.Subobject.inf_def, Subalgebra.fg_top, Int.card_Icc, IsClosed.nhdsSet_le_sup, Ordinal.omega_strictMono, Finset.image_add_right_Ico, NNRat.ofNat_le_cast, Cardinal.toENat_eq_top, sameRay_smul_right_iff_of_ne, MvPolynomial.supported_le_supported_iff, CategoryTheory.Subobject.bot_eq_zero, Metric.pos_of_mem_eball, Set.Ioc.coe_le_one, StarOrderedRing.le_iff, Fin.image_castLE_Ico, comp_le_uniformity, NonUnitalAlgebra.toSubmodule_eq_top, Valuation.Integers.valuation_pos_iff_ne_zero, isDenseInducing_pure, CategoryTheory.Subobject.isoOfMkEq_hom, Multiset.inf_mono, FreeAddGroup.range_lift_le, AlgebraicGeometry.RingedSpace.basicOpen_mul, hasDerivAt_abs, CategoryTheory.Subobject.top_eq_id, Finsupp.coe_rangeIcc, EReal.lt_sub_iff_add_lt, Finset.map_add_right_Ioo, Real.toNNReal_le_toNNReal_iff_of_pos, Valued.isOpen_closedBall, SimplexCategory.II.map'_predAbove, symmDiff_eq_bot, Finset.toDual_sup, ClosureOperator.isClosed_iff_closure_le, AlgebraicGeometry.Scheme.Hom.inv_appTop, Submonoid.instCovariantClassHSMulLe, Finset.lt_inf'_iff, Submonoid.opEquiv_apply, ContractingWith.exists_fixedPoint', OrderHom.coe_inf, RealRMK.exists_open_approx, Int.floor_le_neg_one_iff, invOf_nonpos, SubmodulesRingBasis.inter, exists_square_le, SSet.iSup_skeletonOfMono, Matrix.linfty_opNNNorm_mul, AddSubgroup.map_lt_map_iff_of_injective, Ordinal.toPGame_lt_iff, Nimber.small_Iic, negPart_nonneg, Affine.Simplex.sign_excenterWeights_empty, IsPreconnected.mapsTo_Ioi_or_Iio, Matroid.eRk_lt_encard_of_dep_of_finite, MeasureTheory.hittingAfter_mem_set_of_ne_top, ENNReal.tendsto_nhds, Finset.intervalGapsWithin_succ_fst_of_lt, PrimeSpectrum.coe_preimageOrderIsoFiber_apply_asIdeal, Field.Emb.Cardinal.strictMono_filtration, convexHull_eq_union_convexHull_finite_subsets, ProbabilityTheory.bayesRisk_le_iInf', CategoryTheory.Sieve.le_functorPushforward_pullback, tendsto_fract_left, Interval.coe_bot, Ring.map_jacobson_le, IsClosed.nhdsSet_le_sup', compl_antitone, CategoryTheory.Limits.kernel_map_comp_kernelSubobjectIso_inv, Finset.sort_sorted_gt, Order.not_isSuccLimit_natCast, PNat.card_Ioo, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.app_invApp_assoc, Fin.preimage_natAdd_Iic_natAdd, CategoryTheory.Subobject.skeletal, AddSubsemigroup.unop_le_unop_iff, Ordinal.lt_pred_iff_succ_lt, Asymptotics.isBigOTVS_iff, CategoryTheory.Subobject.pullback_map_self, affineSpan_coe_preimage_eq_top, GenContFract.fib_le_of_contsAux_b, Finset.map_add_right_Ioc, CategoryTheory.MorphismProperty.retracts_transfiniteCompositionsOfShape_pushouts_coproducts_le_llp_rlp, SetLike.covBy_iff, CategoryTheory.Limits.imageSubobject_arrow'_assoc, LieSubmodule.ucs_le_of_normalizer_eq_self, ENNReal.mul_right_strictMono, compl_strictAnti, CompleteLattice.IsSupFiniteCompact.wellFoundedGT, AffineSubspace.pointwise_vadd_top, SimplexCategory.II.mem_finset_iff, monotone_iff_forall_covBy, RootPairing.Base.exists_root_eq_sum_int, CategoryTheory.Subobject.factorThru_arrow, Complementeds.mk_top, Fin.image_castLE_Iic, leOnePart_eq_inv, Finset.le_min_iff, MeasureTheory.measure_isOpen_pos_of_smulInvariant_of_compact_ne_zero, ENat.coe_le_coe, IntermediateField.isScalarTower_over_bot, Int.cast_mono, ProbabilityTheory.condVar_bot', ProbabilityTheory.Kernel.indep_limsup_atTop_self, MeasureTheory.setLIntegral_max, gc_sSup_Iic, MeasureTheory.AEEqFun.le_sup_right, TwoSidedIdeal.top_ringCon, upperSemicontinuous_iff_isClosed_preimage, OrderMonoidHom.snd_apply, AlgebraicGeometry.isIntegralHom_iff, StarSubalgebra.centralizer_le, antitone_iff_forall_covBy, Hyperreal.lt_of_tendsto_atBot, max_bot_right, Nat.one_sub_one_div_cast_le_one, TopCat.GlueData.ofOpenSubsets_toGlueData_V, MvPowerSeries.weightedOrder_le, Concept.strictAnti_intent, Finset.one_lt_prod_iff, LinearMap.range_comp_le_range, AffineSubspace.direction_affineSpan_pair_le_iff_exists_smul, ContractingWith.edist_efixedPoint_lt_top, Set.Icc.coe_one, ONote.fundamentalSequenceProp_inl_some, Mathlib.Meta.Positivity.nonneg_of_isNat', CompleteSpace.complete, Finset.Ioc_pred_left_eq_Icc, Filter.HasBasis.liminf_eq_sSup_iUnion_iInter, Valuation.IsEquiv.one_le_iff_one_le, Set.Ioo_succ_right_eq_Ioc, strictAnti_int_of_succ_lt, Filter.principal_le_iff, SupIrred.not_isMin, Ideal.homogeneousCore'_eq_sSup, Convex.toCone_isLeast, IsStarProjection.le_of_mul_eq_left, AffineSubspace.perpBisector_self, OmegaCompletePartialOrder.ContinuousHom.ωSup_bind, NNReal.image_coe_Ioi, posMulStrictMono_iff_posMulReflectLE, Cardinal.small_Iic, Algebra.adjoin_prod_le, Module.End.invtSubmodule.codisjoint_iff, MeasureTheory.setLIntegral_pdf_le_map, CategoryTheory.ComposableArrows.instIsIsoOfNatNatTwoδ₁Toδ₀, Pi.toColex_strictMono, CategoryTheory.Subobject.inf_arrow_factors_left, Set.Icc_diff_both, SignType.not_lt_neg_one, codisjoint_top_right, LieSubmodule.inf_lie, Finset.sumEquiv_apply_snd, Valuation.ltSubmodule_v_le_of_mem, OrderHom.isGreatest_gfp_le, AddSubsemigroup.monotone_comap, Metric.measure_eball_pos, Ordinal.sup_eq_lsub, HahnEmbedding.Seed.truncLT_mem_range_baseEmbedding, Order.isPredLimit_iff, Multiset.inter_le_right, AlgHom.adjoin_le_equalizer, Cardinal.mk_surjective_eq_zero_iff, Nimber.one_le_iff_ne_zero, egauge_le_of_mem_smul, NNReal.nhds_zero, MeasureTheory.measure_preimage_smul_le, Set.pairwise_disjoint_Ico_zpow, MeasureTheory.eLpNorm_mono, AlgebraicGeometry.RingedSpace.basicOpen_res_eq, Nat.cast_div_le, Set.Iio_diff_Iic, ENNReal.half_pos, Lat_dual_comp_forget_to_partOrd, AntitoneOn.of_map_inf, AlgebraicGeometry.Spec_zeroLocus, OrderMonoidHom.inl_apply, CategoryTheory.MorphismProperty.IsInvertedBy.iff_le_inverseImage_isomorphisms, Positive.isOrderedCancelMonoid, OrderedSemiring.toSMulPosMonoNat, Finsupp.addLeftReflectLE, AddMonoidHom.coe_toIntLinearMap_map, ProperCone.innerDual_empty, Fin.finsetImage_castSucc_Iic, Ordinal.veblenWith_le_veblenWith_iff, NonarchAddGroupNorm.coe_le_coe, OrderAddMonoidHom.add_comp, Set.Iic_True, AlgebraicGeometry.Scheme.basicOpen_mul, Filter.exists_ultrafilter_le, PiCountable.edist_lt_top, Int.floor_pos, SetTheory.PGame.le_iff_game_le, Fin.attachFin_Iio, CategoryTheory.Classifier.SubobjectRepresentableBy.iso_inv_hom_left_comp_assoc, Valued.mem_nhds_zero, ENNReal.essSup_mul_le, IntermediateField.bot_toSubfield, iSup_iInf_le, AlgebraicGeometry.Scheme.ΓSpecIso_inv_naturality, Metric.ordConnected_setOf_eball_subset, Odd.zpow_pos_iff, AddSubgroup.prod_le_iff, CategoryTheory.Subobject.isoOfEq_inv, ENNReal.add_le_add_iff_left, AlgebraicGeometry.Scheme.IdealSheafData.coe_support_eq_eq_iInter_zeroLocus, Metric.disjoint_cobounded_nhds, Fin.range_castAdd, abs_eq_neg_self, Metric.measure_closedBall_pos, sign_eq_zero_iff, RootPairing.root_add_zsmul_mem_range_iff, Flag.grade_coe, Nat.hyperfilter_le_atTop, Subgroup.closure_mul_le, CategoryTheory.ShortComplex.toComposableArrows_map, sSupIndep_pair, sameRay_neg_smul_right_iff_of_ne, MeasureTheory.lintegral_sub_le', Fin.sort_univ, AddSubgroup.pointwise_smul_le_pointwise_smul_iff₀, ENNReal.zero_le_log_iff, HahnSeries.orderTop_le_orderTop_smul, Continuous.le_induced, LocallyBoundedMapClass.comap_cobounded_le, compl_inf_self, Cardinal.one_lt_iff_nontrivial, Ideal.ne_top_iff_exists_maximal, Path.trans_apply, PNat.equivNonZeroDivisorsNat_apply_coe, Int.preimage_Ici, Nonneg.instIsOrderedModule, PseudoMetric.IsUltra.isTrans_ball, Complex.re_eq_neg_norm, Finset.Colex.lt_iff_max'_mem, Valued.isOpen_ball, IntermediateField.topEquiv_symm_apply_coe, AffineSubspace.span_empty, SimpleGraph.edist_le_ediam, ENat.add_one_pos, OmegaCompletePartialOrder.ωScottContinuous.monotone, sup_lt_iff, HahnSeries.orderTop_lt_iff_exists, AddCircle.liftIoc_eq_lift_Icc, LieSubmodule.lie_le_right, Nimber.lt_wf, abs_sub_nonpos, OrderDual.instIsOrderedModule, Ne.gt_or_lt, instIsLinearOrder_mathlib, AddSubmonoid.multiples_le_zmultiples, ArchimedeanClass.mk_map_of_archimedean, Real.natCastle_toNNReal', Subfield.closure_preimage_le, Set.image_inv_Ico, SSet.skeletonOfMono_obj_eq_top, Submodule.torsion_gc, top_sup_eq, measurableSet_le', biInf_sup_le_biInf_sup, Dense.exists_le, CauSeq.lim_le, enorm_div_le, InitialSeg.strictMono, MeasureTheory.measure_preimage_vadd_le, Fin.finsetImage_castSucc_Icc, EReal.instMulPosReflectLT, SuccOrder.isOpen_singleton_iff, Set.image_const_sub_Ici, Fin.sum_Icc_succ, CategoryTheory.Subobject.mapIsoToOrderIso_symm_apply, ValuationSubring.mem_principalUnitGroup_iff, iSup_Iio_eq_self_iff_isSuccPrelimit, MeasureTheory.mul_meas_ge_le_lintegral, AlgebraicGeometry.Scheme.AffineZariskiSite.instIsDenseSubsiteOpensCarrierCarrierCommRingCatGrothendieckTopologyGrothendieckTopologyToOpensFunctor, le_sSup_iff, Ordinal.iSup_eq_lsub_iff_lt_iSup, LocallyBoundedVariationOn.exists_monotoneOn_sub_monotoneOn, NonUnitalStarAlgebra.map_bot, monovary_iff_forall_mul_nonneg, AffineSubspace.map_bot, Metric.nhds_basis_eball, pow_lt_one_iff, Ordinal.cmp_veblenWith, Fin.map_natAddEmb_Ioi, Metric.uniformity_edist, BddOrd.hom_comp, le_iff_exists_sup, Fin.map_valEmbedding_Ico, Ideal.FG.isNilpotent_iff_le_nilradical, Filter.map_surjOn_Iic_iff_le_map, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.invApp_app, CategoryTheory.MorphismProperty.isomorphisms_le_pushouts, zero_lt_two', MeasureTheory.Measure.measure_pos_of_nonempty_interior, Valued.mem_nhds, Subalgebra.lTensorBot_tmul, Mathlib.Meta.Positivity.abs_pos_of_ne_zero, sSup_compact_le_eq, isPreconnected_Iio, OrderIso.map_top, CategoryTheory.isArtinianObject_iff_antitone_chain_condition, iSup₂_le_iff, PartENat.lt_add_iff_pos_right, Ordinal.lsub_typein, NNReal.monotone_nnrpow_const, LieSubmodule.gc_map_comap, Fin.map_castSuccEmb_Iio, ONote.NF.of_dvd_omega0_opow, OrderMonoidWithZeroHom.coe_mul, Ordinal.lt_gamma_zero, Cardinal.aleph0_lt_aleph_one, TopCat.GlueData.ofOpenSubsets_toGlueData_f, abs_sub_map_le_sub, LipschitzWith.nndist_le, Metric.hausdorffEDist_triangle, CategoryTheory.Subobject.factorThru_add, EMetric.mem_closedBall', Set.Ico_inter_Iio, CategoryTheory.ComposableArrows.threeδ₃Toδ₂_app_two, EReal.neg_pos, le_imp_eq_iff_le_imp_ge', LieModule.weightSpace_le_genWeightSpace, punctured_nhds_eq_nhdsWithin_sup_nhdsWithin, Order.coheight_le_coe_iff, fixingSubmonoid_fixedPoints_gc, Nat.one_div_pos_of_nat, FiniteArchimedeanClass.withTopOrderIso_symm_apply, NonUnitalAlgebra.adjoin_empty, ENNReal.cancel_coe, CategoryTheory.Limits.equalizerSubobject_of_self, FinPartOrd.comp_apply, Ordinal.IsNormal.strictMono, Archimedean.exists_orderAddMonoidHom_real_injective, Filter.HasBasis.disjoint_iff_left, AlgebraicGeometry.Scheme.zeroLocus_biInf_of_nonempty, SSet.iSup_range_eq_top_of_isColimit, EReal.coe_pos, AddMonoidAlgebra.supDegree_sub_lt_of_leadingCoeff_eq, Nonneg.instNonUnitalContinuousFunctionalCalculus, Finset.abs_expect_le, Set.Icc_mul_Ico_subset', AddSubmonoid.leftNeg_leftNeg_le, Multiset.monotone_filter_left, ENNReal.iUnion_Ioo_coe_nat, MeasureTheory.eventually_nhds_one_measure_smul_diff_lt, SSet.stdSimplex.objEquiv_toOrderHom_apply, Subgroup.isComplement'_iff_card_mul_and_disjoint, StarAlgebra.adjoin_le_iff, IntermediateField.rank_top', Ordinal.lt_bsup_of_ne_bsup, RelEmbedding.ordinal_type_le, AffineSubspace.direction_sInf, ProfiniteGrp.toLimit_injective, isUniformInducing_iff', ContinuousLinearMap.opNNNorm_comp_le, NNReal.rpow_add_le_add_rpow, SMulPosReflectLE.toSMulPosReflectLT, Subalgebra.topEquiv_symm_apply_coe, cfcHom_isStrictlyPositive_iff, Order.embedding_from_countable_to_dense, AlgebraicGeometry.Scheme.Opens.germ_stalkIso_hom_assoc, Submodule.FG.directedSystem, HahnEmbedding.Partial.mem_domain, Ordinal.opow_lt_veblen_opow_iff, Matrix.mem_colStochastic, MeasureTheory.addHaar_image_le_lintegral_abs_det_fderiv, Set.preimage_const_mul_Ico, IsLocalRing.le_maximalIdeal, TopCat.Presheaf.isSheaf_on_punit_iff_isTerminal, IsClosed.lowerClosure, Polynomial.roots.le_of_dvd, upperBounds_range_partialSups, AlgebraicGeometry.Scheme.SpecΓIdentity_inv_app, span_lt_of_subset_of_card_lt_finrank, partialSups_eq_ciSup_Iic, Fin.map_revPerm_Iio, AddSubgroup.orderIsoAddCon_apply, Prod.Lex.toLex_mono, zero_mem_absConvexHull, AddSubmonoid.le_topologicalClosure, PreTilt.valAux_add, Fin.addRothNumber_le_rothNumberNat, Ordinal.toNatOrdinal_eq_zero, Topology.IsScott.scott_eq_upper_of_completeLinearOrder, MemHolder.nnHolderNorm_add_le, SetTheory.Game.small_setOf_birthday_lt, Cardinal.one_lt_ofENat, MeasureTheory.OuterMeasure.boundedBy_top, kstar_mono, AlgebraicGeometry.Scheme.Pullback.diagonalCoverDiagonalRange_eq_top_of_injective, CategoryTheory.Limits.imageSubobjectCompIso_hom_arrow_assoc, Even.add_pow_le, intermediate_value_univ, ENNReal.inv_le_iff_inv_le, idealFactorsEquivOfQuotEquiv_is_dvd_iso, ProperCone.dual_dual_flip, WithBot.coe_biInf, Metric.isBounded_iff_nndist, Finset.notMem_Iio_self, Filter.tendsto_Icc_uIcc_uIcc, iInf_eq_dif, Finset.Iio_add_Iic_subset, CategoryTheory.Regular.frobeniusMorphism_isPullback, Set.MapsTo.egauge_le, Subalgebra.map_topologicalClosure_le, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverse_map_hom, convex_Ici, EisensteinSeries.tendsto_double_sum_S_act, Topology.CWComplex.iUnion_openCell_eq_skeleton, CategoryTheory.Subobject.lowerEquivalence_unitIso, Finset.Ioo_filter_lt, SimplexCategory.const_apply, AugmentedSimplexCategory.inl'_eval, SummationFilter.symmetricIcc_eq_symmetricIoo_int, sq_pos_iff, CategoryTheory.ComposableArrows.isoMk₀_inv_app, AddSubmonoid.op_le_iff, MeasureTheory.eLpNorm_le_eLpNorm_mul_eLpNorm'_of_norm, Hyperreal.infinite_iff_abs_lt_abs, NNReal.orderIsoRpow_symm_eq, StieltjesFunction.length_mono, Finset.isLowerSet_shatterer, CategoryTheory.ComposableArrows.fourδ₁Toδ₀_app_two, TopCat.Presheaf.isSheaf_iff_isSheaf_comp', ZMod.erdos_ginzburg_ziv_multiset, Fin.Icc_sub_one_eq_Ico, MeasureTheory.Measure.countable_meas_pos_of_disjoint_iUnion, BoxIntegral.Prepartition.monotone_restrict, LinearOrderedCommMonoidWithZero.zero_le, AlgebraicGeometry.StructureSheaf.algebraMap_germ, Finset.grade_multiset_eq, Set.Iic.codisjoint_iff, EReal.le_iff_sign, imageToKernel_arrow_apply, BoundedLatticeHom.coe_toBoundedOrderHom, Submodule.map_lt_map_iff_of_injective, AlgebraicGeometry.IsOpenImmersion.app_ΓIso_hom, Ordinal.bsup_le_iff, OrderHom.antisymmetrization_apply_mk, Fin.map_succEmb_Ico, ofDual_toAdd_eq_top_iff, FloorSemiring.eventually_mul_pow_lt_factorial_sub, WithTop.tendsto_untopD, ThreeGPFree.mulRothNumber_eq, Finset.nonneg_subConst_self, IsRightCancelMul.mulRightStrictMono_of_mulRightMono, Pi.Colex.wellFoundedLT, Multiset.mapEmbedding_apply, Localization.coe_algEquiv, max_cases, Fin.prod_Iic_castLE, CompleteSublattice.coe_top, Finset.box_succ_eq_sdiff, specializes_iff_nhds, TopologicalSpace.OpenNhds.inclusion_obj, SSet.horn₃₁.desc.multicofork_π_three_assoc, MeasureTheory.Ioo_ae_eq_Icc', Polynomial.isBoundedUnder_abs_atTop_iff, MeasureTheory.Measure.toOuterMeasure_le, SimpleGraph.subgraphOfAdj_le_of_adj, DFinsupp.rangeIcc_apply, Ordinal.veblenWith_succ, ENNReal.toNNReal_lt_toNNReal, instContinuousSMulSubtypeLeOfNat, Matrix.mem_colStochastic_iff_sum, Finset.min'_eq_iff, IsAssociatedPrime.annihilator_le, LinOrd.coe_id, CategoryTheory.Sieve.functorPushforward_bot, eVariationOn.lowerSemicontinuous_uniformOn, ConditionallyCompleteLinearOrder.le_total, Filter.covariant_smul, Finset.monotone_sym2, Multiset.Nodup.le_nsmul_iff_le, IntermediateField.topEquiv_apply, Ideal.map_height_le_one_of_mem_minimalPrimes, Filter.HasBasis.le_iff, Finset.covBy_insert, MeasureTheory.MemLp.meas_ge_lt_top', addRightReflectLT_of_addRightMono, NormedSpace.Dual.dual_norm_topology_le_weak_dual_topology, StarOrderedRing.toExistsAddOfLE, disjoint_principal_nhdsSet, Function.argmin_le, TopologicalSpace.generatedBy_le, HahnSeries.embDomainOrderAddMonoidHom_apply, partialSups_iff_forall, Set.Ici_diff_Ici, MulArchimedeanClass.out_top, NNRat.coe_mono, acc_antisymmetrization_iff, ProbabilityTheory.condIndep_bot_right, MeasureTheory.restrict_Ioo_eq_restrict_Ico, sq_nonneg, CategoryTheory.Limits.CompleteLattice.prod_eq_inf, regularSpace_generateFrom, MeasureTheory.projectiveFamilyContent_iUnion_le, Ordinal.lt_veblen_iff_invVeblen₁_le, CategoryTheory.ComposableArrows.fourδ₂Toδ₁_app_one, Order.Frame.himp_bot, convexOn_pow, infClosure_prod, Submodule.coe_mapIic_apply, AffineSubspace.instIsSimpleOrderOfSubsingleton, Set.abs_sub_left_of_mem_uIcc, PartOrdEmb.Hom.injective, Fin.preimage_val_Icc_val, AlgebraicGeometry.StructureSheaf.instIsLocalizedModuleCarrierStalkAbPresheafOpensCarrierTopModuleStructurePresheafPrimeComplAsIdealToStalkₗ, Submodule.submoduleOf_eq_top, exists_int_le, HasCardinalLT.Set.functor_map_coe, HahnEmbedding.Seed.mem_domain_baseEmbedding, MeasureTheory.hittingBtwn_le, one_le_pow_iff, CategoryTheory.Triangulated.SpectralObject.comp_hom, ENNReal.limsup_mul_le', Fin.finsetImage_succ_Icc, MeasureTheory.le_iInf₂_lintegral, ENNReal.nhdsLT_neBot, Finset.card_Ioc_finset, nhdsWithin_le_iff, MulArchimedeanClass.mk_left_le_mk_div_iff, SetTheory.PGame.toLeftMovesNim_symm_lt, Complex.zero_le_real, mem_bot_iff_intCast, Submonoid.gc_map_comap, Finset.isGLB_iff_isLeast, Filter.one_le_div_iff, ENNReal.mul_le_mul_iff_right, ProjectiveSpectrum.ideal_le_vanishingIdeal_zeroLocus, CategoryTheory.Subobject.top_factors, le_compl_iff_disjoint_left, EReal.mul_eq_bot, inf_lt_right, Set.image_inv_Ioc, Subgroup.center_le_centralizer, contravariant_le_iff_contravariant_lt_and_eq, Subgroup.goursatFst_prod_goursatSnd_le, le_symmDiff_iff_left, AffineSubspace.not_le_iff_exists, MeasureTheory.tendstoInMeasure_iff_tendsto_toNNReal, AlgebraicGeometry.basicOpen_eq_of_affine', LowerAdjoint.closure_top, Finset.le_eq_subset, Subsemigroup.coe_toAddSubsemigroup_symm_apply, AffineSpace.asymptoticNhds_le_cobounded, Finsupp.instSMulPosStrictMono, CategoryTheory.Subfunctor.equivalenceMonoOver_unitIso, CategoryTheory.MorphismProperty.le_leftBousfieldW_isLocal, Fin.preimage_rev_Iic, LinearIndependent.lt_aleph0_of_finiteDimensional, IntermediateField.adjoin_natCast, cfc_le_iff, Finset.sum_Icc_of_even_eq_range, convex_Iio, BoundedLatticeHomClass.toSupBotHomClass, convex_halfSpace_gt, Heyting.Regular.coe_lt_coe, Finsupp.Lex.single_antitone, MeasureTheory.meas_ge_le_mul_pow_eLpNorm_enorm, CategoryTheory.IsCardinalFiltered.exists_cardinal_directed.final_functor, AlgebraicGeometry.Scheme.Hom.coe_image, AlgebraicGeometry.AffineSpace.isoOfIsAffine_hom_appTop, AlgebraicGeometry.Scheme.Hom.coe_preimage, Topology.RelCWComplex.skeleton_monotone, Fin.preimage_rev_Ioo, Ordinal.toGame_inj, Specializes.pure_le_nhds, sameRay_neg_smul_right_iff, PartENat.lt_wf, RootPairing.RootPositiveForm.exists_pos_eq, AddMonoidHom.coe_toIntLinearMap_range, symmDiff_le_iff, CategoryTheory.ShortComplex.exact_iff_epi_imageToKernel, NonUnitalAlgebra.elemental.le_centralizer_centralizer, Metric.ediam_image_le_iff, Set.finite_iff_bddBelow_bddAbove, AlgebraicGeometry.Scheme.Opens.fromSpecStalkOfMem_toSpecΓ, Valued.integer.locallyFiniteOrder_units_mrange_of_isCompact_integer, TopologicalSpace.Opens.coe_bot, Cardinal.preAleph_lt_preAleph, Ordinal.univ_id, NNRat.preimage_cast_Ico, inf_eq_left, RCLike.ofReal_pos, Filter.comap_inv_atTop, SetSemiring.down_subset_down, PartENat.add_lt_add_iff_left, Ordinal.gamma_lt_gamma, MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos, SMulPosReflectLT.toSMulPosMono, Subalgebra.coe_pi, AlgebraicGeometry.Scheme.Opens.toScheme_presheaf_map, FiniteDimensional.exists_relation_sum_zero_pos_coefficient_of_finrank_succ_lt_card, mem_convexHull_iff_exists_fintype, ENNReal.not_lt_top, NNReal.Lp_add_le_tsum', ClosedSubmodule.map_le_iff_le_comap, Dynamics.netMaxcard_finite_iff, PartOrdEmb.ofHom_comp, Finsupp.lex_le_iff_of_unique, SSet.horn₃₁.desc.multicofork_π_zero, CategoryTheory.ComposableArrows.scMap_τ₂, Matrix.posSemidef_iff_dotProduct_mulVec, WithBot.subtypeOrderIso_apply_coe, Int.closedBall_eq_Icc, AffineSubspace.topEquiv_apply, Ordinal.lt_iSup_iff, Hyperreal.ofSeq_le_ofSeq, Finset.empty_covBy_singleton, Fin.prod_Ioc_castAdd, BoxIntegral.Prepartition.eventually_not_disjoint_imp_le_of_mem_splitMany, Ico_subset_closure_interior, MeasureTheory.Measure.univ_pi_Iio_ae_eq_Iic, MeasureTheory.Measure.measure_Ioo_eq_zero, IsSuccArchimedean.of_orderIso, Cardinal.aleph_mul_aleph, Set.mem_uIoc, MeasureTheory.Measure.exists_positive_of_not_mutuallySingular, AlgebraicGeometry.instIsDominantToSpecΓOfCompactSpaceCarrierCarrierCommRingCat, PseudoMetric.coe_sup, EMetric.tendsto_nhdsWithin_nhds, CStarAlgebra.nonneg_iff_eq_star_mul_self, Complementeds.coe_bot, CategoryTheory.MonoOver.subobjectMk_le_mk_of_hom, Fin.finsetImage_rev_Iio, Metric.infEDist_le_infEDist_thickening_add, Set.Ioc_union_Ioc_union_Ioc_cycle, Filter.monotone_smallSets, CategoryTheory.MorphismProperty.colimitsOfShape_le_of_final, neg_le_abs, Set.Iic_bot, Nat.lt_of_ceil_lt, SuccOrder.isSuccLimit_of_mem_frontier, Set.univ_disjoint, MeasureTheory.setLIntegral_nnnorm_condExpIndSMul_le, CategoryTheory.GrothendieckTopology.le_close, AlgebraicGeometry.Scheme.preimage_eq_top_of_closedPoint_mem, CategoryTheory.ComposableArrows.isIso_iff₀, tendsto_inv_nhdsGE, Real.volume_pi_le_diam_pow, rank_span_of_finset, Set.antitone_dissipate, Lattice.inf_le_left, IsLocalization.ker_map, absConvexHull_univ, Set.image_mulSingle_Ioc_right, le_iff_forall_lt_rat_imp_le, toZ_iterate_succ_le, smoothSheafCommRing.ι_forgetStalk_hom_apply, CompletelyPositiveMapClass.instOrderHomClass, Sublocale.toNucleus_le_toNucleus, iInf_extend_top, PNat.card_fintype_Ioc, ScottContinuous.sup₂, AlgebraicGeometry.Scheme.IdealSheafData.comap_bot, Submodule.span_mono, TopologicalSpace.Closeds.isClopen_singleton_bot, EReal.le_of_forall_lt_iff_le, isUniformEmbedding_iff', TopologicalSpace.OpenNhds.map_id_obj', Finset.card_Ioo_eq_card_Ico_sub_one, Ordinal.CNF.lt_snd, IsCyclotomicExtension.singleton_one, MeasureTheory.hittingBtwn_eq_sInf, InfTopHom.coe_top, TopCat.Sheaf.objSupIsoProdEqLocus_hom_snd, round_le_add_half, Module.End.eigenspace_le_maxGenEigenspace, DFinsupp.card_Ico, OrderDual.instIsCoatomic, SSet.finite_subcomplex_top_iff, AlgebraicGeometry.stalkToFiberRingHom_germ, Cardinal.ofENat_lt_nat, Finset.image_add_right_Icc, LinearOrderedCommGroup.to_noMinOrder, LTSeries.exists_relSeries_covBy, ENNReal.HolderConjugate.pos, Odd.zpow_neg_iff, Finset.Iic_add_Iio_subset, Ideal.le_comap_map, MeasureTheory.stoppedProcess_eq', inf_lt_iff, CategoryTheory.Subobject.eq_top_of_isIso_arrow, nullMeasurableSet_Iio, isMin_iff_eq_bot, AlgebraicGeometry.Scheme.isoSpec_Spec_inv, Set.encard_image_le, sdiff_le_comm, MeasureTheory.Measure.le_sum, AddGroupSeminormClass.toSubadditiveHomClass, convexHull_toCone_isLeast, Finset.Ioi_add_Ici_subset, AlgebraicGeometry.Scheme.IdealSheafData.supportSet_subset_zeroLocus, Pi.isAtomistic, CategoryTheory.Sieve.galoisConnection, le_imp_le_iff_lt_imp_lt, MeasureTheory.OuterMeasure.isGreatest_ofFunction, AddMonoidHom.coe_toMultiplicative_map, CategoryTheory.Limits.equalizerSubobject_arrow', AbsoluteValue.IsEquiv.one_lt_iff, Cardinal.beth_strictMono, Algebra.adjoin_restrictScalars, Filter.monotone_nhds, NNRat.cast_nonpos, Set.Ioc_sub_one_sub_one_eq_Ico, SimpleGraph.Subgraph.preconnected_iff_forall_exists_walk_subgraph, spectrum.spectralRadius_pow_le', OrderIso.map_sup, EReal.preimage_coe_Iio_top, SimpleGraph.ConnectedComponent.maximal_subgraph_connected_iff, IsAbsoluteValue.sub_abv_le_abv_sub, Int.le_floor, CliffordAlgebra.evenOdd_mul_le, measure_Ioo_lt_top, Hyperreal.omega_pos, Filter.cocompact_le_cofinite, AlgebraicGeometry.Proj.awayι_preimage_basicOpen, Ultrafilter.lim_eq_iff_le_nhds, Valuation.rel_one_iff, NNReal.bddAbove_natCast_image_iff, Topology.IsInducing.functorNhds_obj_coe, isAtomic_dual_iff_isCoatomic, Finset.prod_eraseNone, Set.partialSups_eq_accumulate, MeasureTheory.le_lintegral_add, Dynamics.coverEntropy_biUnion_le, Set.countable_isTop, alternatingGroup.isCoatom_stabilizer, Rat.cast_lt_zero, TwoSidedIdeal.span_le, CategoryTheory.SimplicialObject.δ_comp_σ_of_gt', Order.krullDim_le_one_iff_forall_isMax, NatOrdinal.zero_le, NNReal.addLeftReflectLT, AlgebraicGeometry.IsFinite.finite_app, Submonoid.monotone_map, posTangentConeAt_mono, EisensteinSeries.summable_e2Summand_symmetricIcc, Int.fract_eq_self, IncidenceAlgebra.mu_mul_zeta, List.minimum_anti, Set.Ico_add_one_right_eq_Icc, lowerSemicontinuousAt_iff_frequently, CategoryTheory.TransfiniteCompositionOfShape.ofComposableArrows_isoBot, Set.projIci_self, FractionalIdeal.one_le_extendedHomₐ_iff, top_add, CategoryTheory.Subobject.underlying_arrow_assoc, StarOrderedRing.nonneg_iff, MeasureTheory.Filtration.le_rightCont, Submonoid.subset_pointwise_smul_iff, NonUnitalAlgebra.coe_top, Subgroup.centralizer_le, Rat.cast_strictMono, IsPredArchimedean.findAtom_le, Int.exists_strictMono, exists_pred_iterate_or, Flag.chain_lt, Cardinal.one_lt_lift_iff, LinearOrderedCommGroupWithZero.wellFoundedOn_setOf_ge_gt_iff_nonempty_discrete_of_ne_zero, CategoryTheory.IsGrothendieckAbelian.generatingMonomorphisms.top_mem_range, NonemptyInterval.zero_mem_zero, Order.pred_eq_iff_covBy, TopologicalSpace.Opens.coe_overEquivalence_inverse_obj_left, Finset.eraseNone_image_some, fixedPoints_subgroup_antitone, SSet.Subcomplex.image_le_range, Ideal.Filtration.mono, QuotientGroup.rightRel_eq_top, le_isUnit_iff_zero_notMem, CategoryTheory.LocalizerMorphism.map, EMetric.nhds_basis_closed_eball, Matroid.eRk_union_le_encard_add_eRk, NonUnitalAlgebra.map_top, Module.End.generalized_eigenvec_disjoint_range_ker, Metric.infEDist_le_infEDist_add_hausdorffEDist, Nucleus.coe_le_coe, IntermediateField.isPurelyInseparable_bot, SetTheory.Game.neg_birthday_le, NonemptyFinLinOrd.comp_apply, Topology.IsInducing.functor_obj, AlgebraicGeometry.instSubsingletonCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensOfIsEmpty, ComplementedLattice.isStronglyAtomic, ENat.floor_lt, AlgebraicGeometry.exists_eq_pow_mul_of_isAffineOpen, AddSubmonoid.coe_toSubmonoid_symm_apply, LowerSet.prod_le_prod_iff, Cardinal.isRegular_aleph_one, AddQuantale.bot_add, Submodule.le_div_iff, PartOrdEmb.Hom.le_iff_le, affineSpan_pair_le_of_right_mem, Cardinal.ofENat_le_one, IsAddFreimanIso.addRothNumber_congr, MeasureTheory.OuterMeasure.ofFunction_le, SimpleGraph.right_nonuniformWitnesses_card, Hyperreal.infiniteNeg_iff_infinitesimal_inv_neg, EReal.nhds_bot', ENNReal.Ico_eq_Iio, Filter.bliminf_or_le_inf_aux_left, smoothSheafCommRing.ι_forgetStalk_hom_assoc, ZFSet.mem_vonNeumann_succ, ArchimedeanClass.mk_intCast_nonneg, FinPartOrd.Iso.mk_inv, CategoryTheory.Presieve.bind_ofArrows_le_bindOfArrows, countable_image_lt_image_Ioi, tendsto_inv_nhdsLT_zero, DFinsupp.Colex.wellFoundedLT_of_finite, AlgebraicGeometry.Scheme.germ_stalkClosedPointTo_assoc, Set.sub_mem_Ioo_zero_iff_right, Set.iUnionLift_binary, pow_div_pow_eventuallyEq_atTop, AlgebraicGeometry.Scheme.topIso_hom, HahnSeries.archimedeanClassMk_le_archimedeanClassMk_iff, EMetric.cauchy_iff, AlgebraicGeometry.Proj.awayMap_awayToSection, ContinuousLinearMap.isPositive_iff, DivisibleHull.archimedeanClassOrderIso_apply, Int.fract_lt_one, NNReal.HolderTriple.inv_pos', Filter.le_prod, UpperSet.map_Ioi, Order.le_pred_iff_eq_bot, Set.ordConnected_singleton, integrableOn_Ici_iff_integrableOn_Ioi, SemilatInfCat.dual_map, CategoryTheory.Triangulated.SpectralObject.mapTriangulatedFunctor_δ', addRothNumber_le, Hyperreal.archimedeanClassMk_omega_neg, frontier_le_subset_eq, himp_le_iff, Module.Basis.flag_le_iff, Equiv.Perm.ofSubtype_eq_iff, ENNReal.iInf_coe_lt_top, EMetric.uniformContinuousOn_iff, Filter.disjoint_atTop_atBot, posPart_lt, MeasurableSpace.generateFrom_singleton, LieModule.map_lowerCentralSeries_le, Order.krullDim_pos_iff_of_orderTop, MeasureTheory.Content.mono', sup_sInf_le_iInf_sup, Valuation.Integers.isPrincipal_iff_exists_eq_setOf_valuation_le, AlgebraicGeometry.finite_appTop_of_universallyClosed, AlgebraicGeometry.Scheme.AffineZariskiSite.toOpens_mono, Ordinal.instAddLeftReflectLE, AffineSubspace.span_univ, TopCat.Presheaf.presieveOfCovering.mem_grothendieckTopology, moduleTopology_le, IsCompact.exists_open_superset_measure_lt_top, IntermediateField.top_toSubfield, RCLike.inv_pos, zsmul_lt_zsmul_iff_right, PNat.card_fintype_Ico, CategoryTheory.Sieve.functor_galoisConnection, TopologicalSpace.IsOpenCover.iSup_eq_top, CategoryTheory.Subobject.pullback_obj_mk, AlgebraicGeometry.IsLocallyNoetherian.component_noetherian, Hyperreal.archimedeanClassMk_coe_nonneg, min_le_right, AddSubmonoid.closure_le, Matrix.mem_glpos, Ideal.radical_le_jacobson, Submodule.lt_sup_iff_notMem, CategoryTheory.TransfiniteCompositionOfShape.iic_incl_app, Mathlib.Meta.NormNum.isNat_lt_false, IsValuativeTopology.isClosed_ball, min_add_min_le_min_add_add, Sublocale.mk_le_mk, Nimber.mul_def, inf_bot_eq, PartENat.zero_lt_top, MonotoneOn.exists_tendsto_deriv_liminf_lintegral_enorm_le, AddCircle.continuous_equivIco_symm, AddSubgroup.prod_mono, MeasureTheory.lintegral_mono_set', Algebra.adjoin_le, AddSubgroup.ofAddUnits_strictMono, nhdsWithinLE_sup_nhdsWithinGE, Subgroup.toAddSubgroup_comap, Set.Ico.coe_mul, cfcₙ_norm_sq_nonneg, Cardinal.ofENat_pos, Int.Ico_filter_modEq_card, ZFSet.rank_le_iff, Module.length_eq_coheight, Multiset.cons_lt_cons_iff, SimpleGraph.radius_le_ediam, PMF.apply_pos_iff, Valuation.Integer.not_isUnit_iff_valuation_lt_one, Int.pred_eq_pred, MeasureTheory.measure_mono, one_div_pos, Ordinal.le_lift_iff, CategoryTheory.TransfiniteCompositionOfShape.ofOrderIso_F, edist_le_range_sum_edist, MeasureTheory.trim_trim, Matrix.IsPrimitive.nonneg, Matroid.eRk_union_le_eRk_add_encard, Finset.Ioo_sub_one_left_eq_Ioc, AlgebraicGeometry.IsIntegralHom.integral_app, OrderIso.isSimpleOrder_iff, IsLocalization.submonoid_map_le_is_unit, MeasureTheory.Lp.mem_Lp_iff_eLpNorm_lt_top, SeminormedGroup.disjoint_nhds_one, ENNReal.toReal_lt_toReal, Disjoint.le_compl_left, Fintype.exists_disjointed_le, Ordinal.iSup_Iio_eq_bsup, LinearOrderedAddCommMonoidWithTop.top_add', Subgroup.le_adjoinNegOne, MeasureTheory.Content.innerContent_pos_of_is_mul_left_invariant, AffineSubspace.map_eq_bot_iff, WithTop.iSup_coe_eq_top, instNonemptyObjOpenNormalSubgroupStabilizerHomSurjectiveAuxFunctor, Fin.finsetImage_addNat_Icc, MeasureTheory.Egorov.mem_notConvergentSeq_iff, Set.Ioi_add_bij, List.sorted_lt_ofFn_iff, Fin.map_castSuccEmb_Iic, Right.inv_pos, Fin.map_finCongr_Ioi, CategoryTheory.Limits.equalizerSubobject_arrow_comp_assoc, MeasureTheory.integral_Icc_eq_integral_Ioc', Subsemiring.toAddSubmonoid_mono, MeasureTheory.lintegral_pow_le_pow_lintegral_fderiv_aux, Interval.inv_bot, NonUnitalAlgebra.to_subring_eq_top, PointwiseConvergenceCLM.tendsto_nhds_atTop, Finset.one_le_divConst_self, Nat.image_cast_int_Ioo, Int.Ioc_filter_dvd_eq, Filter.disjoint_prod, regularSpace_iff, OrderIso.asBoolAlgAsBoolRing_apply, TopCat.Presheaf.SheafConditionEqualizerProducts.fork_pt, Int.preimage_Icc, CategoryTheory.Limits.kernelSubobject_arrow'_assoc, MeasureTheory.ae_const_le_iff_forall_lt_measure_zero, Submodule.FG.lTensor.directLimit_apply, enorm_tsum_le_tsum_enorm, Filter.disjoint_atBot_principal_Ioi, inv_lt_zero, Affine.Simplex.sSameSide_affineSpan_faceOpposite_point_right_iff, isPiSystem_Ioc_mem, RingOfIntegers.exponent_eq_one_iff, div_pos_iff, fixingAddSubmonoid_antitone, Ordinal.IsFundamentalSequence.succ, measurableSet_lt, Multiset.union_le_add, Fin.sum_Icc_castLE, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.invApp_app_assoc, FractionalIdeal.mul_right_le_iff, AddCircle.continuousAt_equivIoc, CStarModule.inner_mul_inner_swap_le, imageToKernel_zero_right, ENat.add_le_add_iff_right, MeasureTheory.Measure.FiniteAtFilter.eventually, Set.exists_isOpen_le_add, one_lt_of_ne_one, Pi.toColex_update_le_self_iff, le_nonZeroDivisors_of_noZeroDivisors, Fin.map_addNatEmb_Ioc, Profinite.NobelingProof.contained_proj, Complex.isOpen_im_lt_EReal, OrderEmbedding.addLeft_apply, isUpperSet_setOf, Iic_mem_nhdsSet_Iic_iff, Module.fgSystem.equiv_comp_of, Rat.cast_pos_of_pos, Nat.Prime.emultiplicity_factorial_le_div_pred, sSup_eq_top, UniqueFactorizationMonoid.le_emultiplicity_iff_replicate_le_normalizedFactors, eq_iff_le_not_lt, PointedCone.dual_le_dual, LinearOrderedCommGroupWithZero.fst_apply, TopCat.Presheaf.SheafConditionEqualizerProducts.w_apply, variationOnFromTo.monotoneOn, Cardinal.lift_le_aleph_one, Cardinal.toENat_le_natCast_iff, CategoryTheory.instWellFoundedLTSubobjectOfIsArtinianObject, pos_of_ne_zero, Nat.strictMono_cast, cauchy_davenport_minOrder_add, AlgebraicGeometry.specTargetImageFactorization_app_injective, Finset.antidiagonal.fst_le, WithTop.isLUB_sSup, SetTheory.PGame.moveLeft_toLeftMovesNim, RCLike.map_nonneg_iff, Set.toENat_cardinalMk, ContinuousMultilinearMap.opNNNorm_le_iff, inf_right_le_sup_right, ProbabilityTheory.Kernel.measure_le_bound, AlgebraicGeometry.Scheme.IsGermInjectiveAt.cond, Fin.map_succEmb_Ioc, Nat.one_lt_cast, MulArchimedeanClass.ballSubgroup_top, Subrepresentation.subrepresentationSubmoduleOrderIso_symm_apply, TopologicalSpace.IrreducibleCloseds.map_mono, Set.le_einfsep_image_iff, grade_injective, Finset.Ico_diff_Ico_right, Subalgebra.rTensorBot_symm_apply, cmp_eq_compareOfLessAndEq, Cardinal.nsmul_lt_aleph0_iff, Algebra.comap_top, CategoryTheory.ShortComplex.SnakeInput.composableArrowsFunctor_obj, IncidenceAlgebra.zeta_mul_mu, ArchimedeanClass.mem_closedBall_iff, Concept.swap_lt_swap_iff, OrderAddMonoidHom.inr_apply, Polynomial.div_tendsto_atBot_zero_iff_degree_lt, Finset.inf_erase_top, TopCat.Presheaf.presheafEquivOfIso_counitIso_inv_app_app, AlgebraicGeometry.Scheme.IdealSheafData.zeroLocus_inter_subset_supportSet, Dynamics.coverEntropyEntourage_nonneg, Set.bounded_le_inter_lt, Ordinal.nfpFamily_le_iff, MeasureTheory.Measure.le_map_apply_image, CategoryTheory.Subobject.underlyingIso_inv_top_arrow_assoc, AlgebraicGeometry.IsAffineOpen.preimage_of_isIso, IsModuleFiltration.toIsFiltration, countable_of_isolated_left', openSegment_eq_image₂, CategoryTheory.Subobject.isPullback, NNReal.image_coe_Ioo, RootPairing.coe_bot, Finset.sup_mono, Dense.exists_seq_strictAnti_tendsto, Multiset.inter_le_left, isOpen_implies_isOpen_iff, AlgebraicGeometry.PresheafedSpace.Γ_map, Set.range_projIci, Set.Ici_subset_Ico_union_Ici, Valuation.vlt_one_iff, AlgebraicGeometry.instIsClosedImmersionMorphismRestrict, MeasureTheory.tendstoUniformlyOn_of_ae_tendsto_of_measurable_edist, Submodule.comap_le_comap_iff_of_surjective, MeasurableSpace.generateMeasurableRec_mono, NonUnitalAlgebra.toNonUnitalSubring_eq_top, ENNReal.ofNat_lt_top, codisjoint_map_orderIso_iff, AlgebraicGeometry.Scheme.Hom.preimage_inf, AlgebraicTopology.NormalizedMooreComplex.obj_X, DFinsupp.mem_rangeIcc_apply_iff, edist_vadd_vadd_le, SSet.mem_skeletonOfMono_obj_iff_of_nonDegenerate, FractionalIdeal.instPosMulReflectLEIdealOfIsDedekindDomain, NonUnitalStarAlgebra.adjoin_le_iff, MeasureTheory.hittingBtwn_eq_end_iff, ProbabilityTheory.indep_bot_right, Finset.card_Iic_finset, Finset.le_min'_iff, Dynamics.coverMincard_le_netMaxcard, ONote.repr_le_repr, Set.Finite.latticeClosure, top_le_iff, abs_abs_sub_abs_le_abs_sub, extremePoints_convexHull_subset, Ordinal.one_le_iff_ne_zero, Order.coheight_coe_enat, AddSubgroup.prod_mono_right, TopologicalSpace.Opens.op_map_comp_obj, Ideal.comap_le_map_of_inv_on, AlgebraicGeometry.SheafedSpace.id_c_app, IsSimpleOrder.instIsCoatomistic, AlgebraicGeometry.isPullback_morphismRestrict, ENat.mul_left_strictMono, Ideal.ringJacobson_le_jacobson, PointedCone.mem_closure, Finset.intervalGapsWithin_fst_of_lt_lt, Module.End.mem_genEigenspace_one, mabs_mul', Finset.prod_sdiff_lt_prod_sdiff, Dynamics.netEntropyInfEntourage_le_coverEntropyInf, LinearGrowth.le_linearGrowthInf_iff, PartOrd.Iso.mk_hom, CategoryTheory.Limits.equalizerSubobject_arrow_assoc, mem_segment_iff_div, SimpleGraph.end_hom_mk_of_mk, Cardinal.enat_gc, EReal.tendsto_nhds_top_iff_real, le_iff_oneLePart_leOnePart, IsModularLattice.isModularLattice_Ici, EReal.nhds_bot, Fin.preimage_castSucc_Ici_castSucc, Ordinal.nadd_succ, MeasureTheory.aecover_Ico_of_Ioo, AlgebraicGeometry.Scheme.IdealSheafData.map_ideal', ArchimedeanClass.mk_ratCast_nonneg, LieSubalgebra.lieSpan_le, PrimeSpectrum.isIdempotentElemEquivClopens_symm_top, Hyperreal.tendsto_atBot_iff, Topology.IsScott.isOpen_iff_Iic_compl_or_univ, FiniteArchimedeanClass.addSubgroup_strictAnti, le_perfectClosure_iff, AlgebraicGeometry.StructureSheaf.toOpenₗ_top_bijective, MeasureTheory.Measure.ae_measure_lt_top, ProbabilityTheory.condVar_bot_ae_eq, Ordinal.veblen_lt_veblen_iff, Submodule.smul_comap_le_comap_smul, MvPolynomial.zeroLocus_vanishingIdeal_galoisConnection, Nat.one_le_ceil_iff, NE.ne.pos, AlgebraicGeometry.StructureSheaf.comap_id_eq_map, YoungDiagram.cells_ssubset_iff, AffineSubspace.direction_bot, Ordinal.instIsEmptyIioZero, AlgebraicGeometry.Scheme.Hom.appIso_hom, FractionalIdeal.bot_lt_mul_inv, PowerSeries.le_order, Subsemigroup.map_strictMono_of_injective, AddSubmonoid.map_le_map_iff_of_injective, Ideal.pointwise_smul_le_pointwise_smul_iff, Metric.ediam_le_of_forall_dist_le, preordToPartOrdCompToDualIsoToDualCompPreordToPartOrd_inv_app_hom_coe', EMetric.hausdorffEdist_triangle, CategoryTheory.Subobject.isIso_arrow_iff_eq_top, leOnePart_eq_ite, Submonoid.inv_le, Subspace.map_le_dualAnnihilator_dualAnnihilator, Ordinal.lt_nfp_iff, nucleusIsoSublocale.eq_toSublocale, Codisjoint.eq_iff, Condensed.isColimitLocallyConstantPresheafDiagram_desc_apply, CStarAlgebra.nonneg_iff_eq_mul_star_self, Even.zpow_pos_iff, Ideal.radical_iInf_le, FractionalIdeal.not_inv_le_one_of_ne_bot, Set.unbounded_lt_inter_not_lt, ClosedSubmodule.mem_top, CategoryTheory.Subfunctor.orderIsoSubobject_symm_apply, CFC.monotone_sqrt, Surreal.Multiplication.mulOption_lt_of_lt, LieIdeal.comap_map_le, FirstOrder.Language.PartialEquiv.cod_le_cod, SimpleGraph.nonuniformWitness_spec, FinPartOrd.hom_id, Right.nsmul_neg_iff, CategoryTheory.MorphismProperty.transfiniteCompositions_le, MeasureTheory.IsTightMeasureSet_iff_exists_isCompact_measure_compl_le, ENat.map_natCast_nonneg, TopologicalSpace.gc_generateFrom, ONote.fundamentalSequenceProp_inr, Sublattice.le_prod_iff, Set.isPWO_iff_isWF, Ideal.le_comap_pow, nnnorm_multiset_prod_le, TopologicalSpace.Closeds.isAtom_coe, le_symmDiff_sup_right, CategoryTheory.Sieve.functorInclusion_top_isIso, instIsAtomicFilter, two_mul_le_add_pow_two, AddSubgroup.disjoint_iff_add_eq_zero, ExpGrowth.le_expGrowthSup_mul', CategoryTheory.Subobject.ofLE_mk_le_mk_of_comm, Submodule.AddMonoidHom.coe_toIntLinearMap_comap, Dynamics.coverEntropyInfEntourage_image_le, Filter.add_mul_subset, upperClosure_add, NNReal.HolderTriple.pos, SimpleGraph.eccent_le_iff, mem_doublyStochastic, MeasureTheory.upperCrossingTime_zero', EuclideanGeometry.Sphere.orthRadius_le_orthRadius_iff, LieAlgebra.IsKilling.rootSpace_zsmul_add_ne_bot_iff_mem, LinearIndependent.lt_aleph0_of_finite, Geometry.SimplicialComplex.face_subset_face_iff, MeasureTheory.Measure.le_count_apply, NNReal.rpow_inv_lt_iff, Finsupp.card_Iic, PNat.find_min', OrdinalApprox.gfpApprox_mono_left, Int.ceil_pos, NNReal.inv_le, Ordinal.invVeblen₁_lt_iff, Fin.finsetImage_rev_Icc, Finset.map_add_left_Icc, MaximalSpectrum.iInf_localization_eq_bot, AlgHom.fieldRange_le_normalClosure, Finset.fold_max_lt, MeasureTheory.iSup₂_lintegral_le, CompleteBooleanAlgebra.top_le_sup_compl, Finset.prod_insertNone, WithTop.tendsto_coe_atTop, ENNReal.orderIsoIicCoe_apply_coe, AlgebraicGeometry.IsOpenImmersion.app_ΓIso_hom_apply, HomologicalComplex.HomologySequence.instMonoMap'ComposableArrows₃OfNatNat, Set.disjoint_empty, Fin.image_natAdd_Ioi, Tuple.eq_sort_iff', AffineSubspace.map_le_iff_le_comap, inf_eq_inf_iff_left, LSeries_analyticOn, abs_lt, InfiniteGalois.finGaloisGroupFunctor_map_proj_eq_proj, Set.one_lt_encard_iff, SSet.prodStdSimplex.nonDegenerate_iff_injective_objEquiv, nnnorm_commutator_sub_one_le, CStarAlgebra.conjugate_le_norm_smul, Set.Icc_add_one_sub_one_eq_Ioo, ONote.repr_lt_repr, AffineBasis.isUnit_toMatrix_iff, EMetric.mem_closure_iff, Continuous.finset_inf, Algebra.IsCentral.center_eq_bot, Ideal.map_inf_le, Filter.map_le_iff_le_comap, negPart_eq_zero, MvPowerSeries.order_mul_ge, FiniteMulArchimedeanClass.ballSubgroup_strictAnti, IsGaloisGroup.fixingSubgroup_bot, Convex.convexHull_eq, AlgebraicGeometry.Scheme.IdealSheafData.ideal_sup, AffineSubspace.sup_direction_le, Pi.covBy_iff_exists_left_eq, Set.monotone_preimage, Set.exists_seq_iSup_eq_top_iff_countable, compare_lt_iff_lt, Seminorm.isBounded_sup, BddLat.coe_forget_to_semilatSup, Finpartition.le, CategoryTheory.Limits.CoproductsFromFiniteFiltered.finiteSubcoproductsCocone_ι_app, Ordinal.toPGame_moveLeft, Polynomial.degree_sum_eq_of_disjoint, CFC.sqrt_eq_cfc, EReal.lt_neg_comm, ProbabilityTheory.gaussianPDF_lt_top, Finsupp.single_le_iff, ProperCone.mem_bot, Filter.le_lift'_closure, Subgroup.Normal.commutator_le_of_self_sup_commutative_eq_top, Fin.Iic_sub_one_eq_Iio, HahnSeries.lt_iff, AlgebraicGeometry.Scheme.evaluation_naturality_assoc, Fin.rev_zero_eq_top, Subgroup.map_comap_le, instIsCountablyGenerated_atTop, Finset.shatterer_eq, AlgebraicGeometry.AffineSpace.isoOfIsAffine_hom, Finset.instMulLeftMono, EReal.exp_lt_one_iff, EMetric.infEdist_lt_iff, IsAddFreimanHom.addRothNumber_mono, ArchimedeanClass.mk_le_mk_add_of_archimedean, Ordinal.preOmega_strictMono, convexHull_convexHull_union_right, MeasureTheory.OuterMeasure.trim_mono, FiberBundle.exists_trivialization_Icc_subset, AlgebraicGeometry.tilde.toOpen_res, DivisibleHull.mk_lt_mk, Alexandrov.projSup_obj, AlgebraicGeometry.tilde.toOpen_res_assoc, sup_inf_le, Fin.Ici_add_one_eq_Ioi, LinearMap.isCoatom_ker_of_surjective, sInfHomClass.toInfTopHomClass, Set.Icc_add_Ico_subset, LowerAdjoint.closure_inf_le, AddMonoidAlgebra.supDegree_mul_le, Submonoid.closure_image_isMulIndecomposable_baseOf, ArchimedeanOrder.lt_def, AffineSubspace.instNonemptySubtypeMemTop, Rat.toNNRat_le_toNNRat_iff, IsCompact.exists_isGLB, Rat.le_toNNRat_iff_coe_le, SetTheory.PGame.birthday_moveRight_lt, convex_Ioc, ZFSet.mem_vonNeumann, Cardinal.toENatAux_le_nat, Ordinal.principal_opow_omega, Finsupp.Colex.addLeftMono, Hyperreal.lt_of_tendsto_zero_of_pos, BoundedContinuousFunction.add_norm_nonneg, Multiset.le_zero, AlgebraicGeometry.Scheme.Hom.ι_fromNormalization_assoc, Filter.tendsto_atBot_pure, CStarAlgebra.isStrictlyPositive_iff_eq_star_mul_self, Finset.Colex.toColex_lt_toColex_of_ssubset, Finset.sup_eq_sSup_image, EMetric.hausdorffEdist_prod_le, integrableOn_Icc_iff_integrableOn_Ioo', SetRel.gc_leftDual_rightDual, SSet.stdSimplex.objMk_apply, NNReal.coe_lt_coe, isAtomistic_of_complementedLattice, Finset.Icc_add_one_sub_one_eq_Ioo, idealFactorsFunOfQuotHom_id, Cardinal.toENat_le_nat, ProbabilityTheory.indep_bot_left, SSet.Subcomplex.Pairing.RankFunction.lt, Rat.toNNRat_le_iff_le_coe, Prod.mk_wcovBy_mk_iff, CategoryTheory.ComposableArrows.twoδ₂Toδ₁_app_one, Set.preimage_const_mul_Ici, MeasureTheory.UnifTight.exists_measurableSet_indicator, eHolderNorm_add_le, PSet.lt_rank_iff, LinearMap.BilinForm.le_orthogonal_orthogonal, Rat.cast_le_intCast, Complex.UnitDisc.tendsto_pow_atTop_nhds_zero, MeasureTheory.lintegral_prod_le, EReal.lowerSemicontinuous_add, dense_or_discrete', Ideal.mul_le_inf, iSup_le_iSup_of_subset, Set.encard_pos, mem_nhdsLE_iff_exists_Icc_subset, AlgebraicGeometry.Scheme.emptyTo_c_app, TopCat.presheafToType_obj, MeasureTheory.Lp.eLpNorm_lt_top, Submodule.smul_le, NNRat.instOrderTopology, Order.krullDim_pos_iff_of_orderBot, Int.preimage_floor_singleton, SSet.horn₃₂.desc.multicofork_π_three, Partition.disjoint, ContinuousMultilinearMap.nnnorm_ofSubsingleton_id_le, MeasureTheory.Measure.lintegral_join_le, instMulPosStrictMonoIdeal, fixedPoints_antitone_addSubmonoid, Ordinal.card_preOmega, CategoryTheory.exists_simple_subobject, MeasureTheory.Content.outerMeasure_pos_of_is_add_left_invariant, Finset.Ioo_succ_succ, nnnorm_cfcₙ_le_iff, IsUltrametricDist.nnnorm_intCast_le_one, MeasureTheory.Measure.modularCharacterFun_pos, dimH_orthogonalProjection_le, Fin.prod_Ioi_zero, LinearMap.ker_sup_ker_le_ker_comp_of_commute, Set.image_single_Ico, Finset.map_orderEmbOfFin_univ, EReal.toReal_image_Ioo_zero_top, AlgebraicGeometry.Scheme.Opens.mem_basicOpen_toScheme, closure_Ico, Module.exists_ltSeries_support_isMaximal_last_of_ltSeries_support, wcovBy_eq_reflGen_covBy, Complex.sq_nonpos_iff, AlgebraicGeometry.affineAnd_apply, Fin.sum_Ioc_succ, NonarchAddGroupNorm.lt_def, SimpleGraph.egirth_le_length, AlgebraicGeometry.tilde.isUnit_algebraMap_end_basicOpen, abs_max_le_max_abs_abs, edist_pi_le_iff, instIsSimpleOrderIdeal, HahnSeries.SummableFamily.powers_toFun, isSolvable_iff_commutator_lt, Tropical.add_eq_right_iff, Ordinal.lsub_le_succ_iSup, CategoryTheory.ComposableArrows.fourδ₃Toδ₂_app_one, Subring.pointwise_smul_le_pointwise_smul_iff₀, nhdsLT_eq_bot_iff, left_le_midpoint, cauchy_iff_le, NNReal.rpow_arith_mean_le_arith_mean2_rpow, GroupSeminorm.coe_le_coe, Module.Basis.flag_le_flag, Fin.image_castLE_Ioc, Concept.intent_top, HomotopyCategory.spectralObjectMappingCone_ω₁, EReal.inv_lt_top, Module.End.genEigenspace_div, Multiset.erase_le_iff_le_cons, WithTop.top_mem_insertTop, AlgebraicGeometry.HasRingHomProperty.iff_exists_appLE_locally, Ideal.span_mono, tendsto_neg_nhdsLE, Fin.sum_Ico_castAdd, one_lt_two, LieModule.isNilpotent_iff_le_maxNilpotentSubmodule, Finset.exists_min_image, IntermediateField.adjoin_le_subfield, Set.unbounded_le_inter_lt, monovaryOn_id_iff, List.Perm.maximum_eq, norm_cfcₙ_one_sub_one_add_inv_lt_one, Int.mem_box, BoundedContinuousFunction.norm_sub_nonneg, OrderMonoidHom.fst_apply, IntermediateField.fg_top, isTop_iff_eq_top, Ordinal.typein_lt_self, MeasureTheory.integrableOn_singleton_iff, Sublattice.map_mono, MeasureTheory.Measure.ae_smul_measure_le, CategoryTheory.MorphismProperty.retracts_le_llp_rlp, AddSubgroup.coe_toSubgroup_apply, Seminorm.closedBall_bot, instDiscreteMeasurableSpace, Filter.HasBasis.isBigOTVS_iff, Valuation.IsEquiv.le_one_iff_le_one, AlgebraicGeometry.Scheme.Hom.image_le_opensRange, Submodule.lTensorOne_symm_apply, NNReal.monotone_rpow_of_nonneg, Set.encard_le_coe_iff, Surreal.zero_toGame, Finset.shadow_monotone, Nimber.small_Ico, countable_setOf_isolated_left_within, continuousOn_cfcₙ_nnreal, Finsupp.card_Iio, OrdinalApprox.exists_lfpApprox_eq_lfpApprox, Set.Iic_sInf, Subsemiring.prod_mono_left, Nat.one_lt_ofNat, Set.Iic_diff_Iic, Submodule.mul_le, AlgebraicGeometry.targetAffineLocally_affineAnd_iff', isPiSystem_Icc, Order.isPredPrelimitRecOn_pred, CategoryTheory.Limits.ProductsFromFiniteCofiltered.finiteSubproductsCone_π_app, MeasureTheory.OuterMeasure.top_apply', Filter.limsup_le_iSup, EMetric.nhds_eq, List.maximum_mono, Multiset.toDFinsupp_le_toDFinsupp, Units.inv_neg, AlgebraicGeometry.AffineSpace.reindex_appTop_coord, AlgebraicGeometry.Spec.algebraMap_stalkIso_inv_assoc, exists_gt_t2space, Prod.card_box_succ, CategoryTheory.ComposableArrows.opEquivalence_inverse_map, IsLocalRing.exists_maximalIdeal_pow_le_of_isArtinianRing_quotient, IntermediateField.finInsepDegree_bot, LinearMap.iterateRange_coe, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivUnitIsoApp_inv_hom, Ioi_mem_nhdsSet_Ici_iff, AlgebraicGeometry.smoothOfRelativeDimension_iff, MvPolynomial.supported_empty, MeasureTheory.Measure.measure_Ioo_pos, Submodule.le_orthogonalBilin_orthogonalBilin, cauchySeq_finset_of_norm_bounded, OrderMonoidHom.snd_comp_inr, Nonneg.coe_toNonneg, IsGaloisGroup.intermediateFieldEquivSubgroup_apply, Dynamics.coverEntropyInfEntourage_closure, Profinite.NobelingProof.GoodProducts.span_iff_products, hasBasis_nhdsSet_Ici_Ici, Set.preimage_add_const_Ico, AlgebraicGeometry.quasiCompact_iff_forall_isAffineOpen, HasCardinalLT.Set.instIsCardinalFiltered, Order.succ_eq_csInf, not_bddAbove_iff, pos_invOf_of_invertible_cast, MeasureTheory.MemLp.exists_boundedContinuous_eLpNorm_sub_le, Polynomial.abs_tendsto_atBot, Fin.prod_Icc_castSucc, List.foldr_min_of_ne_nil, Ordinal.lt_veblen, Set.image_const_sub_Iio, CategoryTheory.ComposableArrows.Exact.cokerIsoKer_hom_fac, ClosedSubmodule.toSubmodule_bot, AlgebraicGeometry.HasRingHomProperty.iff_exists_appLE, CompactlySupportedContinuousMap.lt_def, IsCyclotomicExtension.iff_adjoin_eq_top, FormalMultilinearSeries.min_radius_le_radius_add, not_supIrred, Set.infinite_iff_exists_lt, Valuation.integer.coe_span_singleton_eq_setOf_le_v_coe, MonotoneOn.Ico, Subsemiring.subset_pointwise_smul_iff, EMetric.infEdist_le_infEdist_add_edist, SummationFilter.conditional_filter_eq_map_Ici, ExpGrowth.expGrowthInf_mul_le', CategoryTheory.Subobject.underlyingIso_inv_top_arrow, summable_iff_cauchySeq_finset, Nonneg.isOrderedAddMonoid, MeasureTheory.measure_ball_lt_top, Ordinal.toZFSet_subset_toZFSet_iff, Nat.choose_le_pow_div, DFinsupp.Colex.total_le, Valuation.Integers.wellFounded_gt_on_v_iff_discrete_mrange, SemilatInfCat.Iso.mk_hom_toFun, Order.Ideal.coe_subset_coe, SimplexCategory.toType_apply, le_isotypicComponent_iff, AlgebraicGeometry.instSmoothMorphismRestrict, IsDedekindDomain.primesOverEquivPrimesOver_symm_apply, oneLePart_lt, Ordinal.nat_lt_omega0, PUnit.canonicallyOrderedAdd, Order.pred_eq_pred_iff, StieltjesFunction.measure_Ioc, AlgEquiv.fieldRange_eq_top, Fin.sum_Ioc_cast, LinearOrder.bot_topologicalSpace_eq_generateFrom, NonUnitalSubsemiring.closure_le, LinearMap.IsPositive.inner_nonneg_right, ScottContinuous.of_map_sSup, Submodule.mk_le_mk, AlgebraicGeometry.Scheme.IdealSheafData.ideal_mono, ENNReal.nhdsGT_ofNat_neBot, AlgebraicGeometry.RingedSpace.isUnit_res_basicOpen, Multiset.map_strictMono, AlgebraicGeometry.Scheme.empty_presheaf, AlgebraicGeometry.IsAffineOpen.isoSpec_hom_apply, PartOrdEmb.Iso.mk_hom, fixingAddSubgroup_fixedPoints_gc, TopCat.Presheaf.germ_res'_assoc, AddMonoidHom.closure_preimage_le, Equiv.Perm.isCoatom_stabilizer, AlgebraicGeometry.Scheme.Opens.ι_app_self, Finset.insertNone_nonempty, FractionalIdeal.instPosMulStrictMonoNonZeroDivisors, PNat.natPred_strictMono, specializes_iff_not_disjoint, LinearMap.ker_le_iff, Metric.hausdorffEDist_union_le, Finset.prod_Ioi_mul_eq_prod_Ici, MeasureTheory.SimpleFunc.simpleFunc_bot, pow_div_pow_eventuallyEq_atBot, ContinuousOn.lowerSemicontinuousOn, SimpleGraph.Subgraph.deleteEdges_le, AddSubgroup.le_zmultiples_iff, CategoryTheory.Limits.MultispanShape.ofLinearOrder_fst, InnerProductSpace.span_gramSchmidt_Iio, Subgroup.normalClosure_le_normal, Odd.pow_lt_pow, Field.Emb.Cardinal.iSup_adjoin_eq_top, ConvexCone.disjoint_hull_right_of_convex, TopologicalSpace.Opens.mapComp_inv_app, Filter.le_vadd_iff, MeasureTheory.integrableOn_Ici_iff_integrableAtFilter_atTop, AlgebraicGeometry.ΓSpec.toSpecΓ_of, toWellOrderExtension_strictMono, InitialSeg.isSuccLimit_apply_iff, Set.notMem_Iio, FiniteArchimedeanClass.coe_congrOrderIso_apply, ValuationSubring.nonunits_le, List.SortedLT.strictMono, Cardinal.powerlt_succ, Fin.preimage_addNat_Ioo_addNat, affineSpan_eq_bot, MeasureTheory.integrableOn_Iic_iff_integrableAtFilter_atBot, AlgebraicGeometry.Scheme.zeroLocus_span, CategoryTheory.Functor.mapComposableArrows_obj_obj, AddMonoidAlgebra.sup_support_add_le, Nat.nth_eq_orderEmbOfFin, Nonneg.noZeroDivisors, FirstOrder.Language.Substructure.map_le_iff_le_comap, one_lt_mul_iff, Set.Ioo.pos, CategoryTheory.Classifier.SubobjectRepresentableBy.iso_inv_left_comp, CategoryTheory.Sieve.pushforward_le_bind_of_mem, compl_le_compl_iff_le, Fin.image_castAdd_Ico, specializes_iff_pure, AlgebraicGeometry.StructureSheaf.toOpen_germ, TopCat.Presheaf.IsSheaf.isSheafPairwiseIntersections, TotallyBounded.nhds_vietoris_le_nhds_hausdorff, Ordinal.invVeblen₂_lt_iff, AddSubmonoid.FG.exists_minimal_closure_eq, Ultrafilter.exists_le, UpperSet.Ioi_strictMono, ProbabilityTheory.condVar_bot, LinearGrowth.linearGrowthInf_le_iff, AlgebraicGeometry.Scheme.IdealSheafData.ideal_iSup, IsPredArchimedean.of_orderIso, AddGroupSeminorm.le_def, AlgebraicGeometry.Scheme.Hom.naturality, antitoneOn_of_add_one_le, DiscreteQuotient.instSubsingletonQuotientTop, cauchySeq_finset_iff_sum_vanishing, zero_le', SameRay.exists_pos, DFinsupp.toColex_monotone, Filter.tendsto_Icc_pure_pure, FirstOrder.Language.Substructure.le_comap_map, MvPolynomial.le_degrees_add_right, MvPowerSeries.le_weightedOrder, Module.End.genEigenspace_eq_genEigenspace_finrank_of_le, NonemptyFinLinOrd.hom_hom_comp, nhdsLE_basis_Icc, OrderIso.compl_apply, cmp_eq_gt_iff, CategoryTheory.MorphismProperty.antitone_llp, Codisjoint.bihimp_inf_bihimp_le_left, tendsto_floor_left_pure_ceil_sub_one, MeasureTheory.MemLp.eLpNorm_indicator_norm_ge_pos_le, Filter.limsup_const_bot, Set.compl_Iio, Filter.tendsto_comp_neg_atTop_iff, SimpleGraph.FarFromTriangleFree.lt_one, symmDiff_hnot_self, Ordinal.toNatOrdinal_natCast, EReal.le_limsup_add, le_compl_comm, AlgebraicGeometry.Scheme.Hom.image_injective, HahnSeries.orderTop_self_sub_one_pos_iff, min_cases, SimpleGraph.left_nonuniformWitnesses_card, CategoryTheory.Subobject.inf_le_right, sInf_diff_singleton_top, Finset.lt_min'_iff, SimpleGraph.chromaticNumber_le_sum_left, OrderTop.tendsto_atTop_nhds, JacobsonNoether.exists_separable_and_not_isCentral', spectrum.spectralRadius_le_nnnorm, BooleanAlgebra.top_le_sup_compl, ENNReal.exists_inv_nat_lt, Filter.neg_le_iff_le_neg, Module.supportDim_le_supportDim_quotSMulTop_succ_of_mem_jacobson, MeasureTheory.measure_le_integral_pos, FirstOrder.Language.Substructure.coe_closure_eq_range_term_realize, MeasureTheory.Measure.MutuallySingular.disjoint_ae, PartENat.lt_add_one, DirectSum.coe_mul_of_apply, mabs_eq_inv_self, MeasureTheory.addContent_sUnion_le_sum, AlgebraicGeometry.exists_appTop_π_eq_of_isLimit, IsSumSq.nonneg, IsUltrametricDist.nnnorm_tprod_le, IsPredArchimedean.findAtom_bot, disjoint_nhds_pure, pow_two_pos_of_ne_zero, MeasureTheory.Measure.hausdorffMeasure_apply, LowerSet.map_refl, Set.Ioc_subset_Ioo_union_Icc, AddCircle.continuousAt_equivIco, isCofinal_setOf_imp_lt, PartENat.withTopEquiv_symm_le, Fin.prod_Ici_succ, AlgebraicGeometry.Proj.stalkIso'_germ, IsAbsoluteValue.abv_add, LinearMap.le_comap_isotypicComponent, TopCat.Presheaf.isGluing_iff_pairwise, Alexandrov.lowerCone_pt, SimplexCategory.toCat_obj, Finset.lt_sup_iff, Set.projIci_surjOn, Finpartition.mk_mem_sparsePairs, CategoryTheory.Subobject.underlyingIso_top_hom, segment_eq_image_lineMap, rank_pos_iff_exists_ne_zero, AlgebraicGeometry.targetAffineLocally_affineAnd_le, tendsto_atTop_nhds, cmp_eq_lt_iff, AlgebraicGeometry.exists_isFinite_morphismRestrict_of_finite_preimage_singleton, Dynamics.coverEntropyEntourage_closure, concaveOn_iff_forall_pos, LowerSet.coe_add, MeasureTheory.lintegral_mono_set, AlgebraicGeometry.IsAffine.affine, LowerSet.coe_div, AlgebraicGeometry.IsAffineOpen.instAwayCarrierObjOppositeOpensCarrierCarrierCommRingCatSpecPresheafOpOpensBasicOpen, tendsto_inv_nhdsLT_inv, le_kstar, ContinuousLinearMap.opNNNorm_le_of_lipschitz, Set.image_update_Ico_left, StarSubalgebra.coe_top, CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinsetLimitCone_cone_π_app, AlgebraicGeometry.AffineTargetMorphismProperty.IsLocal.to_basicOpen, Set.Icc_subset_uIcc', zero_lt_four, OrderAddMonoidHom.fst_comp_inr, Int.floor_lt_self_iff, Int.abs_le_floor_nnreal_iff, SimpleGraph.egirth_anti, le_sup_iff, AlgebraicGeometry.SheafedSpace.Γ_obj, AddMonoidAlgebra.supDegree_sub_le, CategoryTheory.Arrow.cechNerve_map, NatOrdinal.toOrdinal_symm_eq, MeasureTheory.lintegral_map_le, CategoryTheory.SmallObject.SuccStruct.iterationCocone_pt, ProperCone.dual_empty, Polynomial.div_tendsto_atTop_zero_of_degree_lt, LocallyFiniteOrder.denselyOrdered_iff_subsingleton, Fin.rev_last_eq_bot, Cardinal.aleph1_eq_lift, Int.Ico_eq_finset_map, Submodule.lTensorOne'_one_tmul, instSMulPosStrictMonoIntOfIsOrderedAddMonoid, iSupIndep_map_orderIso_iff, compl_le_hnot, Fin.strictMono_castAdd, ENNReal.add_rpow_le_rpow_add, AlgebraicGeometry.Scheme.IdealSheafData.vanishingIdeal_top, isLowerSet_iff_forall_lt, ZFSet.rank_sUnion_le, NonemptyInterval.length_pure, Metric.boundedSpace_iff_nndist, MeasureTheory.maximal_ineq, CategoryTheory.ShortComplex.toComposableArrows_obj, OrderRingHom.subsingleton, ENNReal.ofReal_lt_ofReal_iff, AlgebraicGeometry.Scheme.restrict_presheaf_map, AbsoluteValue.sub_le, FirstOrder.Language.Substructure.closed, MulArchimedeanClass.mem_ballSubgroup_iff, AlgebraicGeometry.Proj.basicOpen_zero, AlgebraicGeometry.Scheme.IdealSheafData.inclusion_id_assoc, FormalMultilinearSeries.radius_compContinuousLinearMap_le, MeasureTheory.eLpNormEssSup_lt_top_iff_isBoundedUnder, Cardinal.isSuccPrelimit_aleph0, PredOrder.nhdsLE, Finset.min_le_of_eq, Finset.le_sup_dite_neg, CompositionSeries.toList_sorted, Prod.ωSup_snd, Order.succ_pred, MeasureTheory.lintegral_coe_le_coe_iff_integral_le, Set.encard_lt_encard_iff_encard_diff_lt_encard_diff, isLocallyClosed_Ioo, LieIdeal.map_le_iff_le_comap, strictConcaveOn_iff_div, Int.gc_ceil_coe, LocallyFiniteOrder.orderAddMonoidHom_toAddMonoidHom, trichotomy_of_add_eq_add, SimplexCategory.concreteCategoryHom_id, ArchimedeanClass.mk_eq_top_iff, NatOrdinal.toOrdinal_one, Ideal.coe_piOrderIso_symm_apply, ArchimedeanClass.mk_le_add_mk_of_archimedean, Subgroup.closure_le_normalClosure, GroupSeminorm.le_def, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.ofRestrict_invApp, RootPairing.isIrreducible_iff_invtRootSubmodule, SimpleGraph.IsClique.card_le_chromaticNumber, Heyting.Regular.toRegular_coe, AlgebraicGeometry.formallyUnramified_iff, CategoryTheory.Subfunctor.range_subobjectMk_ι, CategoryTheory.ComposableArrows.IsComplex.epi_cokerToKer', BddOrd.hom_inv_apply, Order.isNormal_iff, TopologicalSpace.Opens.map_comp_obj, IntermediateField.finrank_bot', AlgebraicGeometry.Spec.locallyRingedSpaceObj_presheaf_map', Filter.iInf_le_liminf, CStarAlgebra.directedOn_nonneg_ball, PairReduction.antitone_logSizeBallSeq_add_one_subset, nhds_le_nhds_iff, Finset.inf'_le_iff, AlgebraicGeometry.Scheme.mem_basicOpen_top, sdiff_top, PowerSeries.IsWeierstrassFactorization.degree_eq_coe_lift_order_map, MeasureTheory.eLpNorm'_const_smul_le, isGreatest_pair, supClosure_infClosure, OrderIso.subsingleton_of_wellFoundedGT, ConvexCone.isGenerating_top, OrderAddMonoidHom.fstₗ_apply, ENNReal.mem_Ioo_self_sub_add, essSup_comp_le_essSup_map_measure, Equiv.Perm.cycleType_le_of_mem_cycleFactorsFinset, measure_Ico_lt_top, LieSubmodule.codisjoint_toSubmodule, ArchimedeanClass.mk_right_le_mk_add_iff, Rat.cast_le_natCast, mabs_max_le_max_mabs_mabs, Set.unbounded_lt_Ici, archimedean_iff_int_lt, Set.Ico_succ_right_eq_Icc, ValuationSubring.le_top, Module.length_ne_top_iff_finiteDimensionalOrder, Set.image_single_Icc, ArchimedeanClass.FiniteResidueField.mk_lt_mk, Quantale.bot_mul, SSet.N.le_iff, Ordinal.lt_lsub, Fin.prod_Iio_cast, NonemptyInterval.subset_coe_map, isCompl_iff', AlgebraicGeometry.IsAffineOpen.range_fromSpec, Finset.diag_mono, Subgroup.orderIsoCon_symm_apply_coe, ClosureOperator.closure_inf_le, MeasureTheory.hausdorffMeasure_orthogonalProjection_le, AlgebraicGeometry.IsOpenImmersion.image_preimage_eq_preimage_image_of_isPullback, MeasureTheory.Content.innerContent_pos_of_is_add_left_invariant, Set.neg_mem_Ico_iff, NNRat.Nonneg.coe_ofScientific, upperClosure_vadd, tendsto_const_mul_zpow_atTop_nhds_iff, HahnSeries.min_orderTop_le_orderTop_sub, is_simple_module_of_finrank_eq_one, CategoryTheory.Pretopology.toGrothendieck_bot, Complex.compl_Iic_zero, Ordinal.toNimber_zero, CategoryTheory.SmallObject.iterationFunctorObjObjRightIso_ιIteration_app_right, AlgebraicTopology.NormalizedMooreComplex.objX_zero, Module.Basis.flag_le_ker_coord_iff, Dynamics.coverEntropyInf_iUnion_le, Polynomial.div_tendsto_zero_of_degree_lt, cocompact_le_atBot_atTop, nhdsGT_eq_bot_iff, Matrix.posSemidef_diagonal_iff, ContinuousLinearMap.isPositive_iff', IsPreconnected.Icc_subset, PrimeSpectrum.basicOpen_mul_le_right, isSaddlePointOn_iff, absConvexHull_isClosed, nucleusIsoSublocale.symm_eq_toNucleus, summable_iff_cauchySeq_finset_and_tsum_mem, EMetric.cauchySeq_iff_NNReal, Metric.ediam_union_le_add_edist, Filter.atTop_finset_eq_iInf, cmp_mul_left', Partition.bot_lt_of_mem, AlgebraicGeometry.Scheme.Opens.ι_image_basicOpen', Cardinal.canLiftCardinalNat, Mathlib.Meta.Positivity.pos_of_isRat, min_lt_max, CategoryTheory.Regular.frobeniusStrongEpiMonoFactorisation_m, Ordinal.bsup_eq_blsub, mulLeftMono_of_mulLeftStrictMono, Cardinal.aleph0_lt_lift, EReal.preimage_coe_Iic, AlgebraicGeometry.LocallyRingedSpace.Γevaluation_naturality_assoc, NatOrdinal.toOrdinal_toNatOrdinal, Fin.preimage_castSucc_Icc_castSucc, ConcaveOn.min_le_of_mem_segment, infClosure_mono, Cardinal.lift_lt_aleph0, FiniteMulArchimedeanClass.subgroup_eq_bot, eventuallyConst_of_isNoetherian, ProbabilityTheory.meas_ge_le_evariance_div_sq, Prod.Lex.toLex_strictMono, Algebra.eq_top_iff, Filter.liminf_bot, instSMulPosStrictMonoNatOfIsOrderedCancelAddMonoid, interior_Ico, SMulPosStrictMono.toSMulPosReflectLE, RelEmbedding.orderEmbeddingOfLTEmbedding_apply, Prod.mk_covBy_mk_iff, Complex.not_lt_zero_iff, MeasureTheory.exists_setLIntegral_compl_lt, RootPairing.rootForm_pos_of_ne_zero, Subring.opEquiv_apply, Set.Ioi_False, univ_option, ESeminormedMonoid.enorm_mul_le, Set.encard_le_chainHeight_of_isChain, Finset.nnnorm_sum_le_sup_nnnorm, Fin.prod_Ico_castLE, TwoSidedIdeal.gc, AffineSubspace.le_comap_map, MeasureTheory.Lp.instOrderClosedTopologySubtypeAEEqFunMemAddSubgroupOfClosedIciTopology, Set.ordConnected_iff_uIcc_subset_right, enorm_sum_le, cmp_compares, AddSubmonoid.coe_toNatSubmodule, Cardinal.lt_aleph0_of_finite, EReal.toENNReal_ne_zero_iff, Ideal.pow_right_strictAnti, OrderIso.withZeroUnits_apply, Setoid.mk_eq_bot, LinearOrderedField.coe_inducedOrderRingIso, Fin.sum_Iic_castSucc, MeasureTheory.lowerCrossingTime_mono, TwoSidedIdeal.matrix_monotone, LinOrd.hom_comp, AddSubmonoid.mk_le_mk, AffineSpace.asymptoticNhds_eq_smul, exteriorPower.pairingDual_apply_apply_eq_one_zero, exists_seq_strictMono_tendsto, monotone_iff_forall_lt, NumberField.IsTotallyReal.le_maximalRealSubfield, IsRealClosed.nonneg_iff_isSquare, le_upperClosure, AffineSubspace.comap_top, CategoryTheory.IsCardinalFiltered.exists_cardinal_directed.isCardinalFiltered, LieSubmodule.le_normalizer, le_iInf_iSup, OrderIso.map_iInf₂, antivaryOn_id_iff, GaloisCoinsertion.isAtom_iff, UpperSet.infIrred_iff_of_finite, NormedAddGroupHom.ker_le_ker_completion, Profinite.NobelingProof.spanFunctorIsoIndexFunctor_inv_app, MeasureTheory.Measure.le_join_apply, CStarAlgebra.isStrictlyPositive_TFAE, MulArchimedeanClass.subgroup_antitone, WithZeroTopology.Iio_mem_nhds_zero, CategoryTheory.Classifier.SubobjectRepresentableBy.iso_inv_left_π, AddCircle.openPartialHomeomorphCoe_symm_apply, ENNReal.tendsto_atTop_zero, AlgebraicGeometry.LocallyRingedSpace.preimage_basicOpen, Subgroup.le_normalizer_of_normal, fourierSubalgebra_closure_eq_top, AlgebraicGeometry.ProjectiveSpectrum.Proj.stalkMap_toSpec, instIsStronglyAtomicOfWellFoundedLT, ascending_central_series_le_upper, AlgebraicGeometry.IsAffineOpen.isoSpec_inv_toSpecΓ_assoc, convexHull_convexHull_union_left, mulRothNumber_le, rootsOfUnity_le_of_dvd, MeasureTheory.UniformIntegrable.spec, Function.HasMaxCutPropertyAt.rows_lt_aux, le_compl_compl, Fin.prod_Icc_castAdd, MeasureTheory.measure_pos_iff_nonempty_of_isMulLeftInvariant, OrderIso.map_sSup, Part.Fix.approx_mono', top_notMem_iff, CStarAlgebra.isStrictlyPositive_iff_isSelfAdjoint_and_spectrum_pos, lt_mabs, MeasureTheory.SimpleFunc.measure_lt_top_of_memLp_indicator, MonotoneOn.Ioc, MeasureTheory.setIntegral_pos_iff_support_of_nonneg_ae, Types.monoOverEquivalenceSet_inverse_obj, GenContFract.IntFractPair.succ_nth_stream_b_le_nth_stream_fr_inv, SignType.pos_iff, SuccOrder.prelimitRecOn_succ, himp_self, nhdsSetWithin_mono_left, AddSubgroup.pointwise_smul_le_pointwise_smul_iff, IdemSemiring.bot_le, Filter.limsInf_bot, Inseparable.nhds_le_uniformity, Continuous.partialSups_apply, List.sorted_ge_ofFn_iff, Filter.isTopologicalBasis_Iic_principal, HasOuterApproxClosed.indicator_le_apprSeq, frontier_Iic, Convex.ordConnected, NNReal.sqrt_one, Finset.subConst_le_inv_dens, AlgebraicGeometry.AffineSpace.homOverEquiv_apply, Finset.mul_prod_Iio_eq_prod_Iic, NonUnitalStarAlgebra.range_id, MeasureTheory.pow_mul_meas_ge_le_eLpNorm, upperSemicontinuous_iff_IsClosed_hypograph, NatOrdinal.instWellFoundedLT, Set.Ici_add_Ioi_subset, AlgebraicGeometry.Scheme.Modules.instFullPresheafOfModulesToPresheafOfModules, SimpleGraph.componentComplFunctor_finite, MeasureTheory.Measure.exists_measure_inter_spanningSets_pos, CategoryTheory.isFiltered_of_semilatticeSup_nonempty, IntermediateField.Lifts.exists_lift_of_splits, VitaliFamily.FineSubfamilyOn.measure_le_tsum_of_absolutelyContinuous, RCLike.ofReal_mul_neg_iff, Ideal.finiteHeight_iff_lt, Nat.preimage_floor_zero, Module.End.genEigenspace_nat, NNReal.inner_le_Lp_mul_Lq_tsum', Subalgebra.rank_eq_one_iff, add_pos_iff, Finset.eraseNone_inter, AlgebraicGeometry.pullbackRestrictIsoRestrict_inv_fst_assoc, Order.height_int, CategoryTheory.ComposableArrows.δ₀Functor_obj_obj, Ideal.spanNorm_le_comap, UpperSet.compl_map, Submodule.exists_minimal_isPrimary_decomposition_of_isPrimary_decomposition, MeasureTheory.hittingAfter_le_iff, Fin.castSuccFunctor_obj, Affine.Simplex.ExcenterExists.signedInfDist_excenter, edist_pi_const_le, sdiff_bot, ENNReal.coe_le_iff, Multiset.sup_le, Icc_mem_nhds_iff, AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.relevant, CategoryTheory.Subfunctor.ofSection_le_iff, FormalMultilinearSeries.radius_le_radius_derivSeries, DiscreteTopology.eq_bot, AddSubsemigroup.map_le_iff_le_comap, LSeries.abscissaOfAbsConv_sub_le, antivaryOn_iff_forall_mul_nonpos, integrableOn_Icc_iff_integrableOn_Ioc, MeasureTheory.Measure.mkMetric_le_liminf_sum, Subalgebra.prod_top, IsSquare.nonneg, Ordinal.sInf_compl_lt_ord_succ, Finset.max_abv_sum_one_le_of_isNonarchimedean, pow_le_kstar, Set.Icc.one_sub_nonneg, AlgebraicGeometry.Scheme.basicOpen_res, Set.image_sub_const_Ici, PositiveLinearMap.instOrderHomClass, GroupTopology.toTopologicalSpace_le, convexHull_min, isFwInvariant_iff_isInvariant, NNReal.Lp_add_le, CategoryTheory.Sieve.overEquiv_symm_top, Fin.preimage_castSucc_Ioo_castSucc, MulArchimedeanClass.closedBallSubgroup_top, MeasureTheory.lintegral_enorm_le_liminf_of_tendsto, MeasureTheory.eLpNormEssSup_lt_top_of_ae_bound, AlgebraicGeometry.Scheme.Opens.fromSpecStalkOfMem_toSpecΓ_assoc, AlgebraicGeometry.StructureSheaf.toPushforwardStalk_comp, Finset.Iio_insert, CategoryTheory.Subobject.arrow_congr, BooleanSubalgebra.inclusion_rfl, NeZero.pos, Dense.exists_lt, Fin.image_succ_Icc, Set.preimage_const_div_Ioo, SuccOrder.nhdsGE, TopologicalSpace.Opens.mapIso_hom_app, lt_sSup_iff, TopCat.Sheaf.objSupIsoProdEqLocus_inv_fst, HasFPowerSeriesWithinOnBall.r_pos, Cardinal.ord_strictMono, smoothSheaf.ι_evalHom, nnnorm_commutator_units_sub_one_le, NonUnitalAlgebra.comap_top, ConvexCone.hull_le_iff, Ideal.span_singleton_lt_span_singleton, MeasureTheory.Measure.toOuterMeasure_top, Mathlib.Tactic.Order.ToInt.toInt_nlt_toInt, hnot_sup_self, NonUnitalStarAlgebra.map_top, ConvexCone.mem_strictlyPositive, InitialSeg.le_apply_iff, AlgebraicGeometry.Smooth.exists_isStandardSmooth, LinearMap.map_span_le, WithBot.Ico_coe_coe, Set.Ici_False, Order.PFilter.principal_le_iff, Interval.zero_mem_zero, Ideal.homogeneousCore'_mono, Decidable.le_iff_lt_or_eq, Int.abs_sub_lt_one_of_floor_eq_floor, Finset.sup_le_iff, Ordinal.add_mul_succ, frontier_Iic_subset, AddSubgroup.fg_iff_mul_fg, Submonoid.coe_toAddSubmonoid_symm_apply, fixingSubgroup_fixedPoints_gc, Ideal.matrix_strictMono_of_nonempty, ENNReal.iSup_lt_eq_self, Affine.Simplex.affineCombination_mem_closedInterior_face_iff_mem_Icc, AlgebraicGeometry.Scheme.comp_appTop, Multiset.card_Iic, convex_halfSpace_le, smoothSheafCommRing.ι_evalHom_assoc, Fin.Ioo_sub_one_eq_Ico, archimedean_iff_rat_lt, mabs_min_le_max_mabs_mabs, Polynomial.map_under_lt_comap_of_quasiFiniteAt, Valuation.HasExtension.mk_smul_mk, Filter.lift_lift_same_le_lift, Submonoid.coe_toAddSubmonoid_apply, ConvexOn.convex_strict_epigraph, Nat.floor_eq_zero, LieAlgebra.derivedSeries_lt_top_of_solvable, CFC.sqrt_algebraMap, Finset.Colex.singleton_le_singleton, Fin.preimage_castLE_Ici_castLE, Finset.inf_eq_sInf_image, NNReal.coe_pos, Subalgebra.coe_toSubmodule, BoundedLatticeHom.dual_comp, convexHull_isClosed, AlgebraicGeometry.Scheme.IdealSheafData.radical_ideal, MeasureTheory.Measure.hausdorffMeasure_le_one_of_subsingleton, AddSubgroup.mk_le_mk, LinearMap.le_eqLocus, fixingAddSubgroup_antitone, CompleteLattice.wellFoundedGT_characterisations, Order.coheight_eq_coe_iff, WithTop.denselyOrdered_set_iff_subsingleton, TopCat.Presheaf.Pushforward.comp_hom_app, Ideal.exists_le_maximal, CategoryTheory.Arrow.cechConerve_map, CategoryTheory.HasLiftingProperty.transfiniteComposition.SqStruct.w_assoc, IntermediateField.botEquiv_def, ValuationSubring.mem_nonunits_iff, ENNReal.one_lt_coe_iff, Subsemigroup.map_comap_le, LipschitzOnWith.hausdorffMeasure_image_le, Submodule.FG.jacobson_smul_lt, isLowerSet_iff_Iio_subset, Cardinal.toENat_comp_ofENat, iInf_sup_le_iInf_sup, one_lt_leOnePart_iff, AlgebraicGeometry.IsClosedImmersion.isAffine_surjective_of_isAffine, le_algebraicClosure_iff, Real.Angle.sign_toReal, IsAbsoluteValue.abs_abv_sub_le_abv_sub, Valued.toNormedField.norm_lt_iff, MeasureTheory.aecover_Ioc_of_Ico, CategoryTheory.sieve₁'_toPreOneHypercover_eq_top, PseudoMetric.coe_finsetSup, CategoryTheory.Subobject.inf_comp_left_assoc, NNRat.preimage_cast_Ioo, EReal.coe_lt_coe, Int.floor_le, Finset.Colex.toColex_lt_toColex, LinearMap.range_dualMap_le_dualAnnihilator_ker, ProperCone.dual_le_dual, subalgebra_top_finrank_eq_submodule_top_finrank, Set.preimage_const_div_Ico, CategoryTheory.ComposableArrows.map'_self, NonemptyInterval.toLex_mono, NNReal.HolderTriple.inv_pos, PowerSeries.IsWeierstrassFactorizationAt.degree_eq_coe_lift_order_map_of_ne_top, NatOrdinal.small_Ioc, AddEquiv.symm_comapAddSubgroup, CategoryTheory.ComposableArrows.Exact.isIso_cokerToKer', alternatingGroup.commutator_perm_le, IntermediateField.adjoin_eq_top_iff, AlgebraicGeometry.PresheafedSpace.restrictStalkIso_inv_eq_germ, isLUB_sSup, toAntisymmetrization_le_toAntisymmetrization_iff, Valuation.rel_iff_le, Int.exists_strictAnti, biInf_mono, sq_le_one_iff_abs_le_one, ENNReal.nhdsGT_nat_neBot, StarSubalgebra.map_le_iff_le_comap, FractionalIdeal.zero_le, gc_nhds, Order.le_iff_eq_or_succ_le', Polynomial.abs_isBoundedUnder_iff, PrimeSpectrum.basicOpen_le_basicOpen_iff, Mathlib.Tactic.Linarith.natCast_nonneg, div_le_egauge_ball, Finset.inf_eq_iInf, bot_lt_isotypicComponents, SSet.OneTruncation₂.nerveEquiv_symm_apply_obj, AlgebraicGeometry.StructureSheaf.toPushforwardStalk_comp_assoc, IsUltrametricDist.nnnorm_natCast_le_one, setOf_minimal_antichain, Cardinal.preAleph_le_preBeth, MulArchimedeanClass.mk_le_mk, MeasurableSpace.generateMeasurableRec_omega_one, CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinsetColimitCocone_cocone_pt, AlgebraicGeometry.StructureSheaf.toOpenₗ_eq_const, StieltjesFunction.iInf_Ioi_eq, Nat.one_le_ofNat, ProperCone.mem_positive, MeasureTheory.Measure.HasTemperateGrowth.exists_eLpNorm_lt_top, Set.Ici_succ_eq_Ioi, Valuation.Integers.integralClosure, MeasureTheory.AddContent.supClosure_apply_of_mem, Fin.range_natAdd_eq_Ioi, quasilinearOn_iff_mem_uIcc, enorm_pos', IntermediateField.toSubalgebra_lt_toSubalgebra, MeasureTheory.Egorov.measure_notConvergentSeq_tendsto_zero, monotone_nhdsSet, Ideal.primeCompl_le_nonZeroDivisors, ENNReal.tsum_geometric_two_encode_le_two, preordToPartOrdCompToDualIsoToDualCompPreordToPartOrd_hom_app_hom_coe, IntermediateField.finSepDegree_bot', Fintype.finsetOrderIsoSet_toEquiv, Set.preimage_const_sub_Ioc, Set.Iio_mul_Iic_subset', InfClosed.supClosure, IsCyclotomicExtension.Rat.adjoin_singleton_eq_top, lt_add_iff_lt_right_or_exists_lt, AlgebraicGeometry.Scheme.fromSpecStalk_app, AddMonoidHom.range_le_ker_iff, MeasureTheory.FiniteMeasure.apply_le_mass, sign_eq_sign_or_eq_neg, Finset.prod_Icc_succ_eq_mul_endpoints, star_le_one_iff, Subalgebra.opEquiv_symm_apply, Hyperreal.lt_of_st_lt, MulArchimedeanClass.mk_eq_top_iff, OrderEmbedding.birkhoffSet_apply, SMulPosMono.toSMulPosStrictMono, LocallyFiniteOrder.orderMonoidHom_strictMono, WithTop.subtypeOrderIso_apply_coe, AlgebraicGeometry.IsAffineOpen.isLocalization_stalk, Set.OrdConnected.preimage_coe_nnreal_real, Rat.toNNRat_lt_iff_lt_coe, Set.ordConnected_inter_Icc_iff, ContinuousLinearMap.ker_prod_ker_le_ker_coprod, Set.Ico_subset_Icc_union_Ioo, SSet.modelCategoryQuillen.J_le_monomorphisms, HeytingHom.map_bot', countable_image_gt_image_Ioi, ENat.strictMono_map_iff, IsOrderedCancelAddMonoid.toAddLeftReflectLT, Finset.Colex.toColex_sdiff_lt_toColex_sdiff', Set.le_einfsep_pair, Multiset.le_count_iff_replicate_le, MonomialOrder.lex_lt_iff, TopCat.GlueData.MkCore.t_inter, MeasureTheory.le_ae_restrict, Nat.lt_ceil, FloorRing.gc_coe_floor, antitone_Ici, SetTheory.Game.bddBelow_range_of_small, Fin.antitone_iff_succ_le, coe_lt_enorm, NonarchAddGroupSeminorm.le_def, ContinuousOn.partialSups_apply, PosMulStrictMono.toPosMulReflectLE, CategoryTheory.TransfiniteCompositionOfShape.ofArrowIso_isColimit, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_comp, MonomialOrder.eq_zero_iff, ENNReal.ofReal_lt_ofNat, Cardinal.ord_lt_ord, ENNReal.coe_mono, NonUnitalStarAlgebra.elemental.le_of_mem, ENNReal.add_lt_add_iff_right, UniqueFactorizationMonoid.dvd_iff_normalizedFactors_le_normalizedFactors, Algebra.IsUnramifiedAt.exists_notMem_forall_ne_mem_and_adjoin_eq_top, Ordinal.blsub_const, Subfield.relrank_eq_one_iff, CanonicallyOrderedAdd.prod_pos, CategoryTheory.Subobject.pullback_top, EReal.exp_lt_top_iff, CategoryTheory.MorphismProperty.isStableUnderTransfiniteCompositionOfShape_iff, stdSimplexEquivIcc_apply_coe, Finset.disjSum_mono_left, CompleteLattice.MulticoequalizerDiagram.multicofork_pt, Mathlib.Meta.Positivity.nnabs_pos_of_pos, AlgebraicGeometry.isField_of_universallyClosed, Fin.preimage_castAdd_Ioc_castAdd, AlgebraicGeometry.IsAffineOpen.isoSpec_hom_base_apply, polynomialFunctions.topologicalClosure, ConvexCone.mem_hull_of_convex, SimpleGraph.radius_le_eccent, Unitization.lift_range_le, NNReal.sqrt_le_one, CompositionSeries.total, HolderWith.nnholderNorm_le, PNat.natPred_lt_natPred, Filter.tendsto_div_const_atBot_iff, AlgebraicGeometry.Scheme.exists_cover_of_mem_grothendieckTopology, CircleDeg1Lift.pow_monotone, AddSubgroup.toIntSubmodule_toAddSubgroup, ENat.one_lt_top, Valuation.Integers.valuation_irreducible_pos, sbtw_lineMap_iff, Module.End.genEigenspace_le_maximal, EReal.expOrderIso_apply, AddSubmonoid.map_strictMono_of_injective, ENNReal.image_coe_Ici, Finset.image_add_right_Ioc, Fin.preimage_natAdd_Ici_natAdd, Multiset.ndinsert_le, nhdsLE_sup_nhdsGT, Fin.finsetImage_val_Iic, Nonneg.toNonneg_coe, Finset.mapEmbedding_apply, Set.pairwise_disjoint_Ico_intCast, AlgebraicGeometry.Scheme.IdealSheafData.ideal_inf, MeasureTheory.measure_iUnion_fintype_le, sInf_empty, ZFSet.rank_lt_of_mem, RingCon.range_mkₐ, iUnion_Ici_eq_Ioi_iInf, clusterPt_iff_lift'_closure, AlgebraicGeometry.LocallyRingedSpace.Γevaluation_eq_zero_iff_notMem_basicOpen, pow_left_strictMonoOn₀, HahnSeries.finiteArchimedeanClassOrderIsoLex_apply_fst, AlgebraicGeometry.ProjectiveSpectrum.Proj.toSpec_base_apply_eq_comap, disjoint_lift'_closure_nhds, Valuation.vlt_iff_lt, left_eq_sup, stronglyMeasurable_bot_iff, le_compl_iff_disjoint_right, ENNReal.bot_lt_log_iff, MeasureTheory.upperCrossingTime_le, lowerSemicontinuousOn_iff_preimage_Iic, Ordinal.small_Ioo, MonoidWithZeroHom.inr_strictMono, nullMeasurableSet_Ico, Finset.sup'_image₂_le, IsCompact.measure_lt_top_of_nhdsWithin, AddSubgroup.le_closure_toAddSubmonoid, FiniteArchimedeanClass.ballAddSubgroup_strictAnti, Cardinal.preAleph_pos, LowerSet.coe_iicInfHom, AlgebraicGeometry.Scheme.Spec.algebraMap_residueFieldIso_inv_assoc, Complex.nhdsWithin_stolzCone_le_nhdsWithin_stolzSet, LieModule.posFittingCompOf_le_lowerCentralSeries, one_lt_pow_iff, Fin.finsetImage_succ_Ici, Cardinal.beth_lt_beth, compl_top, OmegaCompletePartialOrder.ContinuousHom.forall_forall_merge', IntermediateField.eq_bot_of_isSepClosed_of_isSeparable, Ideal.height_le_height_add_encard_of_subset, antitone_of_le_sub_one, IsSimpleOrder.lt_top_iff_eq_bot, CompactlySupportedContinuousMap.toReal_nonneg, ENat.lt_floor, CategoryTheory.HasLiftingProperty.transfiniteComposition.sqFunctor_obj, setOf_riemmanianEDist_lt_subset_nhds', enorm_smul_le, Set.Finite.lt_aleph0, monotone_stabilizes_iff_artinian, Finset.Ico_map_sectR, closedAbsConvexHull_eq_closure_absConvexHull, FirstOrder.Language.DirectLimit.instDirectedSystemSubtypeMemSubstructureDomCoeOrderHomPartialEquivEmbeddingInclusion, smoothSheafCommRing.ι_forgetStalk_hom, WellFoundedLT.finite_ne_bot_of_iSupIndep, NNReal.ball_zero_eq_Ico, ne_iff_lt_iff_le, Nat.sub_one_lt_floor, Order.iso_of_countable_dense, MeasurableSpace.map_top, ZFSet.le_def, CategoryTheory.Subobject.epi_iff_mk_eq_top, disjoint_disjointed, Set.Icc.coe_eq_one, iSup_eq_if, Rat.preimage_cast_Ioo, DivisibleHull.archimedeanClassOrderIso_symm_apply, MeasureTheory.exists_notMem_null_laverage_le, NatOrdinal.small_Iio, Fin.finsetImage_castLE_Iic, zero_lt_four', Ordinal.ToType.mk_apply, CategoryTheory.IsCardinalFiltered.exists_cardinal_directed.aux, CStarModule.inner_self_nonneg, ENNReal.coe_strictMono, Multiset.le_cons_self, integrableOn_Iic_iff_integrableOn_Iio', CFilter.inf_le_left, Nonneg.coe_div, FirstOrder.Language.Hom.range_comp_le_range, nonneg_iff_isSelfAdjoint_and_quasispectrumRestricts, InfiniteGalois.mk_toAlgEquivAux, AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToSection_germ, le_one_iff_eq_one, borel_eq_generateFrom_Ioc_le, LinearMap.ker_le_ker_comp, iInter_Iic_eq_empty_iff, MeasureTheory.lintegral_pos_iff_support, NonUnitalSubsemiring.center_le_centralizer, AlgebraicGeometry.instGeometricallyIntegralMorphismRestrict, Valued.hasBasis_uniformity, LinearOrderedField.coe_lt_inducedMap_iff, AlgebraicGeometry.Scheme.IdealSheafData.subschemeι_app_surjective, Nonneg.nat_ceil_coe, Set.exists_lt_mem_inter_of_not_pairwiseDisjoint, AlgebraicGeometry.Scheme.preimage_zeroLocus, Cardinal.small_Ico, Set.preimage_const_div_Icc, SSet.OneTruncation₂.ofNerve₂.natIso_inv_app_obj_map, NonUnitalStarAlgebra.elemental.le_iff_mem, Set.Iio_add_bij, csSup_Iio, Sym2.inf_le_sup, Finsupp.Icc_eq, Set.Iic.isCompl_iff, RootPairing.coe_top, ArchimedeanClass.addSubgroup_strictAntiOn, ConditionallyCompleteLinearOrderBot.csSup_empty, Nat.floor_mono, CategoryTheory.StructuredArrow.projectSubobject_factors, Opens.coe_mayerVietorisSquare_X₄, MeasureTheory.exists_le_laverage, ArchimedeanClass.FiniteResidueField.ordConnected_preimage_mk, Cardinal.aleph0_pos, Nat.not_ofNat_le_one, Matroid.Dep.eRk_lt_encard, Valuation.mem_ltSubmodule_iff, AlgebraicGeometry.PresheafedSpace.congr_app, nhds_basis_Ioo, cfcₙ_nonneg_iff, FirstOrder.Language.Substructure.elementarySkolem₁Reduct.instSmall, one_lt_mul_self_iff, Real.toNNReal_le_iff_le_coe, SimpleGraph.Subgraph.le_induce_union, PartOrdEmb.Limits.instPreservesFilteredColimitsOfSizeForgetOrderEmbeddingCarrier, Set.Ioc_add_Ico_subset, MeasureTheory.Measure.ae_mono', StieltjesFunction.outer_Ioc, isConnected_Ioi, Module.End.mem_genEigenspace_nat, LieSubmodule.mapOrderEmbedding_apply, Algebra.map_top, CategoryTheory.Subfunctor.Subpresheaf.ofSection_le_iff, Submodule.le_comap_single_pi, OrderAddMonoidHom.inl_add_inr_eq_mk, Concept.extent_top, IsPreconnected.intermediate_value, AlgebraicGeometry.Scheme.AffineZariskiSite.cocone_ι_app, Set.image_div_const_Ioo, CategoryTheory.Subfunctor.top_obj, Set.Iio_inter_Iio, atBot_isCountablyGenerated_of_archimedean, TopologicalSpace.Opens.coe_disjoint, TopCat.Presheaf.presheafEquivOfIso_counitIso_hom_app_app, bot_bihimp, Metric.externalCoveringNumber_le_encard_self, TwoSidedIdeal.coeOrderEmbedding_apply, Order.Ideal.mem_sup, List.minimum_le_of_mem', Nat.le_ceil, SSet.δ_comp_σ_of_gt'_apply, ENNReal.div_pos, ModP.v_p_lt_val, CategoryTheory.MorphismProperty.le_isColocal_isColocal, Finset.inf'_mono, CategoryTheory.Subobject.underlyingIso_hom_comp_eq_mk, Multiset.instOrderedSub, Finset.sup_apply, Ordinal.blsub_one, PartOrdEmb.ext_iff, Subalgebra.eq_bot_of_rank_le_one, PNat.instNoMaxOrder, AddSubgroup.le_comap_map, sup_eq_sup_iff_right, CategoryTheory.Subobject.pullback_comp, isPiSystem_Ici, AlgebraicGeometry.Scheme.Hom.smoothLocus_eq_top, FiniteArchimedeanClass.submodule_strictAnti, MeasureTheory.aecover_Ioo_of_Ico, PNat.lt_add_one_iff, Ordinal.small_Iio, CategoryTheory.Sieve.functorPushforward_top, Prod.ωSupImpl_fst, ENNReal.lt_iff_exists_rat_btwn, Set.compl_Ici, LieHom.map_le_idealRange, LocalSubring.le_def, AntilipschitzWith.comap_nhds_le, Matroid.IsStrictMinor.lt, IsSimpleModule.toIsSimpleOrder, MeasureTheory.hittingAfter_apply_mono, HomogeneousIdeal.irrelevant_eq_iSup, Set.Iic.coe_biSup, HolderOnWith.edist_le, strictMono_vecCons, TwoSidedIdeal.lt_iff, NNRat.cast_lt_one, Field.Emb.Cardinal.adjoin_image_leastExt, ENNReal.image_coe_Ioi, div_nonneg_iff, CategoryTheory.Limits.CompleteLattice.instHasBinaryProductsOfOrderTop, AlgebraicGeometry.Scheme.IdealSheafData.coe_support_inter, Cardinal.isSuccPrelimit_zero, mul_nonneg_iff_neg_imp_nonpos, borel_eq_generateFrom_Ioi, CategoryTheory.Limits.imageSubobjectCompIso_inv_arrow, Ordinal.toPGame_moveLeft_hEq, Finset.tendsto_Ico_neg, Filter.blimsup_mono, ConvexSpace.assoc, OrderIso.sumLexIioIci_apply_inl, scottContinuous_iff_map_sSup, Subsemiring.le_topologicalClosure, zero_lt_iff, IsMultiplyPretransitive.alternatingGroup_le, Finset.val_wcovBy_val, Matroid.isFlat_iff_isClosed, Subspace.dualAnnihilator_le_dualAnnihilator_iff, GroupNorm.coe_lt_coe, BotHom.coe_sup, NonUnitalStarSubalgebra.center_eq_top, Set.projIci_coe, Mathlib.Tactic.Order.ToInt.exists_translation, HahnSeries.le_order_iff_forall, Localization.mk_lt_mk, Valuation.integer.v_irreducible_lt_one, Cardinal.ofENat_toENat_le, MeasureTheory.submartingale_iff_condExp_sub_nonneg, AlgebraicGeometry.IsAffineOpen.fromSpec_top, Set.image_inv_Iic, IntermediateField.bot_eq_top_of_finrank_adjoin_le_one, AddSubmonoid.disjoint_def, IsLocalMaxOn.not_nhds_le_map, Set.Ici_True, CategoryTheory.Presieve.isSeparatedFor_top, compare_of_injective_eq_compareOfLessAndEq, Submodule.lTensorOne_tmul, sum_Ioc_inv_sq_le_sub, SimplexCategory.comp_toOrderHom, EReal.sign_mul_inv_abs, BoxIntegral.Box.measurableSet_Ioo, WithTop.orderIsoSumLexPUnit_symm_inr, AlgebraicGeometry.IsSchemeTheoreticallyDominant.ker_eq_bot, AddCon.toSetoid_eq_bot, Cardinal.omega0_lt_ord, Int.natCast_strictMono, min_def_lt, AbsoluteValue.add_le, Left.pow_lt_one_iff, ne_iff_gt_or_lt, IntermediateField.relrank_bot_right, BddOrd.comp_apply, StarAlgebra.adjoin_le, IsClosed.mk_lt_continuum, LinearMap.isPositive_iff, MeasureTheory.SimpleFunc.exists_upperSemicontinuous_le_lintegral_le, MeasureTheory.lintegral_sub_le, le_of_sup_eq', Ordinal.typein_one_toType, IntermediateField.le_iff_le, Module.Relations.Solution.span_relation_le_ker_π, MeasureTheory.aecover_Iio_of_Iio, Finset.Ioc_eq_filter_powerset, Set.preimage_div_const_Ioo, Cardinal.succ_zero, Metric.infEDist_le_infEDist_cthickening_add, MeasureTheory.IsFiniteMeasureOnCompacts.lt_top_of_isCompact, Real.nnreal_trichotomy, AddSubgroup.toSubgroup'_closure, AddSubgroup.pointwise_smul_le_iff, AlgebraicGeometry.Scheme.IdealSheafData.support_top, MulAction.le_stabilizer_smul_right, Decidable.le_iff_eq_or_lt, Dynamics.netEntropyEntourage_nonneg, IsUltrametricDist.nnnorm_zpow_le, CategoryTheory.Sieve.functorPullback_pushforward_le, ContDiff.dimH_range_le, SetTheory.Game.bddAbove_of_small, Subalgebra.comm_trans_lTensorBot, PNat.card_Ioc, ConvexOn.monotoneOn_slope_gt, ArchimedeanClass.mem_ballAddSubgroup_iff, EMetric.ordConnected_setOf_ball_subset, Filter.map_sub_atTop_eq, CategoryTheory.Subobject.isIso_iff_mk_eq_top, HahnSeries.forallLTEqZero_supp_BddBelow, directedOn_of_inf_mem, ConvexCone.gc_hull_coe, Fin.preimage_castAdd_Iic_castAdd, FirstOrder.Language.PartialEquiv.monotone_dom, AddSemigroupIdeal.closure_mono, OrderedFinpartition.parts_strictMono, Algebra.discr_powerBasis_eq_prod'', VitaliFamily.ae_eventually_measure_pos, covariant_lt_of_contravariant_le, convexHull_prod, InfiniteGalois.fixedField_bot, Subgroup.codisjoint_subgroupOf_sup, Multiset.range_le, instWellFoundedGTUnit, Module.Baer.ExtensionOf.le, CompletelyDistribLattice.himp_bot, Set.Ioi.coe_sup, UpperSet.coe_map, IsCompl.inf_right_eq_bot_iff, Ordinal.succ_pos, ENNReal.tendsto_nhds_zero, MeasureTheory.Measure.sub_def, NonUnitalAlgebra.map_bot, CategoryTheory.Limits.kernelSubobject_arrow_comp_assoc, LieSubmodule.lieSpan_le, Set.bounded_le_inter_le, Filter.isBoundedUnder_le_abs, IsModularLattice.isModularLattice_Iic, EReal.coe_ennreal_strictMono, Set.image_add_const_Ioi, AlgebraicGeometry.Scheme.map_basicOpen_map, NormedSpace.expSeries_radius_pos, Asymptotics.isLittleOTVS_iff, ClosedSubmodule.coe_top, SimplexCategory.II.map'_id, AddSubgroup.map_comap_le, PNat.natPred_le_natPred, CompleteLattice.ωScottContinuous.top, CategoryTheory.Sieve.functorPushforward_le_iff_le_functorPullback, Fin.image_val_Ico, LinearOrderedCommMonoidWithZero.toMulPosStrictMono, Cycle.chain_mono, CompleteLattice.sInf_le, inf_strictMonoOn_Icc_sup, AlgebraicGeometry.affineOpensRestrict_symm_apply_coe, EMetric.tendsto_nhdsWithin_nhdsWithin, Rat.cast_pos, CategoryTheory.Subobject.pullback_self, Fin.preimage_castLE_Ico_castLE, nhdsSet_inter_le, OrderAddMonoidHom.snd_apply, ringKrullDim_add_enatCard_le_ringKrullDim_mvPolynomial, Set.encard_insert_le, Nat.pow_le_choose, Interval.mul_eq_one_iff, EisensteinSeries.tendsto_tsum_one_div_linear_sub_succ_eq, Multiset.le_replicate_iff, integrableOn_Icc_iff_integrableOn_Ioo, BoundedLatticeHom.coe_toInfTopHom, MvPolynomial.degrees_X', Nat.map_cast_int_atTop, AlgebraicGeometry.PresheafedSpace.id_c_app, affineSpan_mono, Order.le_iff_eq_or_le_pred, StarAlgHom.range_eq_map_top, OrderTopology.of_discreteTopology, Ideal.quotientMap_injective, ENNReal.toReal_le_toReal, Finset.isGLB_inf_id, Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes_of_isLocalRing, wcovBy_iff_le_and_eq_or_eq, OrderMonoidHom.fstₗ_comp_inlₗ, IsSepClosed.separableClosure_eq_bot_iff, Set.image_single_Ioc_left, EReal.lt_iff_exists_rat_btwn, MeasureTheory.OuterMeasure.isCaratheodory_partialSups, PosNum.one_le_cast, wellFoundedGT, Subfield.closure_empty, AlgebraicGeometry.Scheme.Hom.comp_appTop, NNRat.preimage_cast_Ioc, ProperCone.dual_union, EReal.preimage_coe_Ioc_bot, HeytingAlgebra.himp_bot, NonUnitalStarAlgebra.eq_top_iff, CategoryTheory.IsCardinalFiltered.exists_cardinal_directed.functor_map, MeasureTheory.le_hittingAfter, Valuation.IsNontrivial.exists_lt_one, isCompact_iff_ultrafilter_le_nhds', Sublattice.map_inf_le, Set.Ici.isCompl_iff, OmegaCompletePartialOrder.ContinuousHom.continuous, Disjoint.map_orderIso, antitone_lt, MeasureTheory.eLpNorm'_add_le, ENNReal.lintegral_Lp_add_le_of_le_one, IsSMulRegular.isSMulRegular_on_quot_iff_smul_top_inf_eq_smul, AlgebraicGeometry.IsOpenImmersion.opensRange_pullback_snd_of_left, LowerSet.instIsConcreteLE, Even.zpow_nonneg, mem_nhds_iff_exists_Ioo_subset, Subalgebra.bot_eq_top_of_finrank_eq_one, AlgebraicGeometry.IsIntegralHom.hasAffineProperty, LinearGrowth.linearGrowthInf_add_le, ediam_mul_le, Set.encard_le_one_iff, Order.height_le_iff, Ordinal.toNimber_toOrdinal, Cardinal.toENat_le_one, Finset.isCoatom_iff, AddValuation.mem_supp_iff, Set.OrdConnected.isPredArchimedean, Antitone.Iic, interior_Ioo, Finset.Iio_eq_ssubsets, map_sub_le_add, Submodule.mem_iSup_of_chain, PUnit.instDenselyOrdered, Cardinal.lt_aleph0, FirstOrder.Language.Hom.map_le_range, Fin.preimage_cast_Ioo, NonUnitalSubring.toAddSubgroup_strictMono, MeasureTheory.MemLp.eLpNorm_lt_top, DiscreteQuotient.proj_bot_eq, ENNReal.mul_inv_le_one, Field.Emb.Cardinal.instFiniteDimensionalSubtypeMemIntermediateFieldAdjoinImageToTypeOrdRankCompCoeBasisWellOrderedBasisLeastExtIioSingletonSet, InitialSeg.isSuccPrelimit_apply_iff, Subalgebra.restrictScalars_toSubmodule, neg_one_lt_zero, MeasureTheory.lintegral_const_lt_top, totallyBounded_iff_filter, nnnorm_le_nnnorm_add_nnnorm_sub', Affine.Simplex.sign_signedInfDist_incenter, maximal_iff, Affine.Simplex.ExcenterExists.sign_signedInfDist_excenter, LinearOrderedAddCommGroup.to_noMaxOrder, ENNReal.iUnion_Ioc_coe_nat, RingCon.toCon_eq_top, Order.Ideal.top_of_top_mem, FirstOrder.Language.Substructure.closure_withConstants_eq, OrderIso.apply_blimsup, Finsupp.Colex.single_le_iff, FinPartOrd_dual_comp_forget_to_partOrd, dimH_range_le_of_locally_lipschitzOn, MeasureTheory.aecover_Icc_of_Ioo, RCLike.neg_iff_exists_ofReal, AlgebraicGeometry.Scheme.germ_stalkClosedPointTo_Spec, MeasureTheory.OuterMeasure.map_top_of_surjective, PowerSeries.le_order_subst_right, Submodule.orderIsoMapComapOfBijective_apply, cfcₙ_nonpos_iff, Ordinal.cof_type_lt, isAtom_iff_eq_top, Submodule.le_dualAnnihilator_iff_le_dualCoannihilator, PartOrd.Iso.mk_inv, Ideal.spanNorm_mul_spanNorm_le, LinearOrderedAddCommGroup.isAddCyclic_iff_not_denselyOrdered, round_eq_iff, Ideal.bot_lt_annihilator_of_disjoint_nonZeroDivisors, SemidirectProduct.mulEquivSubgroup_apply, conjneg_nonneg, Metric.IsSeparated.encard_le_packingNumber, PNat.lt_find_iff, FiniteMulArchimedeanClass.subsemigroup_eq_subgroup, Set.Iio_subset_Iio_union_Ico, isCompl_bot_top, Multiset.le_sub_add, CategoryTheory.ComposableArrows.threeδ₁Toδ₀_app_one, AddSubsemigroup.map_strictMono_of_injective, Finset.pluennecke_ruzsa_inequality_pow_div_pow_div, SSet.horn₃₂.desc.multicofork_pt, Filter.IsAntitoneBasis.antitone, Cardinal.range_natCast, RCLike.to_complex_nonneg_iff, Ordinal.toNimber_max, Filter.disjoint_cocompact_right, OrderTopology.of_linearLocallyFinite, AlgebraicGeometry.Scheme.basicOpen_zero, Dynamics.coverEntropyInf_biUnion_le, StarAlgebra.adjoin_eq_span, MeasureTheory.preVariation.sum_le_preVariationFun_iUnion, Finset.Ioc_add_Ico_subset, CategoryTheory.nerve.ext_of_isThin_iff, Subring.opEquiv_symm_apply, LieIdeal.map_le, PointedCone.mem_comap, strictAnti_of_add_one_lt, Polynomial.span_le_degreeLE_of_finite, Submonoid.leftInv_le_isUnit, Filter.tendsto_div_const_atTop_iff, isClosed_Ico_iff, Finset.instAddRightMono, MeasureTheory.countable_meas_le_ne_meas_lt, AddSubgroup.disjoint_def', MeasureTheory.AddQuotientMeasureEqMeasurePreimage.covolume_ne_top, Submonoid.toAddSubmonoid'_closure, CategoryTheory.Limits.CompleteLattice.instHasBinaryCoproductsOfOrderBot, Real.toNNReal_lt_natCast, AlgebraicGeometry.Scheme.IdealSheafData.ideal_ofIdeals_le, Set.encard_le_encard_of_injOn, Dynamics.coverMincard_finite_iff, NNReal.geom_mean_le_arith_mean3_weighted, CategoryTheory.ComposableArrows.threeδ₃Toδ₂_app_zero, MeasureTheory.eLpNormEssSup_indicator_const_le, Submodule.IsOrtho.ge, Module.length_quotient, Interval.length_neg, MeasureTheory.Measure.measure_pos_of_mem_nhds, Module.length_eq_height, AlgebraicGeometry.Scheme.IdealSheafData.top_mul, Fin.Ioc_sub_one_eq_Ioo, Complementeds.coe_lt_coe, min_eq_left_iff, sdiff_le_iff, List.sortedGE_iff_antitone_get, DirichletCharacter.factorsThrough_iff_ker_unitsMap, NumberField.mixedEmbedding.volume_fundamentalDomain_latticeBasis, SimpleGraph.Copy.ofLE_refl, Set.Iio_subset_Iic_union_Ico, AlgebraicGeometry.Scheme.IdealSheafData.map_bot, StarOrderedRing.nonneg_iff_spectrum_nonneg, frontier_Iio, AlgebraicGeometry.Scheme.IsLocallyDirected.exists_of_pullback_V_V, Ordinal.canonicallyOrderedAdd, Ordinal.bsup_le_blsub, le_sup_sdiff_sup_sdiff, AlgebraicGeometry.Scheme.SpecMap_presheaf_map_eqToHom, FirstOrder.Language.DirectLimit.Equiv_isup_of_apply, ENNReal.rpow_le_rpow_iff, LowerSet.map_Iic, MeasureTheory.exists_measurable_le_lintegral_eq, Finset.image_add_right_Ioo, AddSubmonoid.pointwise_smul_le_pointwise_smul_iff, AlgebraicGeometry.IsAffineOpen.fromSpec_image_basicOpen, Set.infsep_pos, mem_emetric_ball_zero_iff, Hyperreal.IsSt.lt, Metric.nhds_basis_closedEBall, Set.Finite.einfsep_pos, AlgebraicGeometry.Scheme.mem_smallGrothendieckTopology, Ordinal.toZFSet_monotone, denselyOrdered_set_iff_subsingleton, IsOrderedMonoid.to_hasUpperLowerClosure, WithBot.Ioo_coe_coe, iInf_eq_bot, MonotoneOn.Ioo, Set.image_update_Ico, NNRat.cast_lt, complementedLattice_iff_isAtomistic, AlgebraicGeometry.instHasAffinePropertyIsomorphismsSchemeAndIsAffineIsIsoCommRingCatAppTop, AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_morphismRestrict, inf_le_left, Filter.tendsto_finset_Ici_atBot_atTop, NNReal.HolderTriple.inv_nonneg', ExpGrowth.expGrowthSup_le_iff, Filter.lt_pure_iff, PartENat.lt_find, infClosure_supClosure, Set.encard_tsub_one_le_encard_diff_singleton, Multipliable.le_tprod', Filter.cocompact_le_coclosedCompact, MulAction.stabilizer_inf_stabilizer_le_stabilizer_inter, pow_pos_iff, ENat.mul_le_mul_right_iff, NNReal.agmSequences_snd_antitone, CompleteLatticeHom.toOrderIsoRangeOfInjective_apply, Ordinal.mem_range_preOmega_iff, CategoryTheory.ComposableArrows.fourδ₂Toδ₁_app_two, HomotopicalAlgebra.RelativeCellComplex.Cells.hj, convexHull_eq_union, MeasureTheory.tendstoInMeasure_iff_enorm, CanonicallyOrderedAdd.mul_pos, EReal.sign_mul_abs, HahnSeries.ofPowerSeries_apply, LieAlgebra.toLieSubmodule_le_rootSpace_zero, Order.Ioo_eq_empty_iff_pred_le, eq_bot_of_singletons_open, Finset.map_add_right_Icc, DistLat.Iso.mk_hom, MulAction.stabilizer_inf_stabilizer_le_stabilizer_apply₂, DividedPowers.SubDPIdeal.sup_carrier_def, EReal.sub_pos, CategoryTheory.Limits.image_map_comp_imageSubobjectIso_inv, skyscraperPresheafCoconeOfSpecializes_ι_app, CauchyFilter.inseparable_lim_iff, lowerClosure_mul, le_egauge_ball_one, MeasureTheory.Lp.meas_ge_le_mul_pow_enorm, Cardinal.toENat_injOn, Mathlib.Meta.NormNum.isNat_lt_true, Set.Iio_False, ArchimedeanClass.min_le_mk_sub, Dynamics.coverEntropyEntourage_le_coverEntropy, BddLat.coe_forget_to_bddOrd, PredOrder.isOpen_singleton_iff, untrop_sum_eq_sInf_image, LinearIndepOn.encard_le_toENat_rank, TopologicalSpace.Opens.apply_mk, Finset.strictMono_iff_forall_lt_cons, Subsemiring.map_le_iff_le_comap, Interval.length_add_le, spectrum.le_nnnorm_of_mem, Filter.bliminf_or_le_inf_aux_right, SimpleGraph.singletonFinsubgraph_le_adj_left, Int.preimage_ceil_singleton, TopologicalSpace.Opens.infLELeft_apply, NonUnitalAlgebra.elemental.le_iff_mem, Matrix.PosSemidef.trace_nonneg, AbsoluteValue.subadditiveHomClass, antivary_iff_forall_mul_nonpos, pow_le_one_iff, AlgebraicGeometry.IsAffineOpen.basicOpen_union_eq_self_iff, Finset.tendsto_Ioc_neg, DivisibleHull.instIsStrictOrderedModuleRat, Fin.finsetImage_addNat_Ioo, le_inv_self_iff, LaurentSeries.valuation_le_iff_coeff_lt_log_eq_zero, Commute.cfc_nnreal, MeasureTheory.measure_average_le_pos, Order.le_and_pred_le_iff, List.SortedGE.antitone_get, MvPolynomial.supported_univ, AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_hom_eq_germ_assoc, Matroid.eRk_compl_union_add_eRk_compl_inter_le, Submodule.iSup_eq_toSubmodule_range, AlgebraicGeometry.isFinite_iff, antivaryOn_neg_right, Sylow.exists_subgroup_card_pow_succ, monotone_lt, TopCat.Presheaf.presheafEquivOfIso_inverse_obj_obj, Interval.coe_inf, conductor_eq_top_iff_adjoin_eq_top, EReal.sub_le_iff_le_add, Fin.castSuccFunctor_map, convexHull_eq, Fin.card_Ioc, inv_antitoneOn_Ioi, ENNReal.ofReal_lt_ofReal_iff', SSet.prodStdSimplex.nonDegenerate_iff_strictMono_objEquiv, FractionalIdeal.dual_inv_le, Types.monoOverEquivalenceSet_unitIso, GenContFract.le_of_succ_get?_den, RootPairing.RootPositiveForm.zero_lt_apply_root_root_iff, CategoryTheory.Triangulated.SpectralObject.ω₂_map_hom₂, Cardinal.toENat_eq_nat, exists_seq_strictAnti_tendsto_nhdsWithin, AlgebraicGeometry.etale_iff, Ideal.le_pow_of_le_ramificationIdx, AlgebraicGeometry.PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_inv_ι_app, isSimpleModule_iff_isCoatom, BddLat.coe_forget_to_semilatInf, AlgebraicGeometry.Scheme.IdealSheafData.inclusion_id, LinearMap.iSup_range_single_le_iInf_ker_proj, AbsoluteValue.IsEquiv.one_le_iff, Ordinal.omega_max, isOpen_iff_nhds, Topology.IsUpperSet.upperSet_le_upper, ENNReal.isOpen_Ico_zero, Subalgebra.isAlgebraic_bot_iff, isPiSystem_Ioo, Orientation.oangle_sign_smul_add_smul_right, le_sSup, MeasureTheory.aecover_Ico_of_Ioc, borel_eq_generateFrom_Iio, Rat.lt_toNNRat_iff_coe_lt, div_le_div_flip, NNReal.summable_iff_not_tendsto_nat_atTop, NormedField.denselyOrdered_range_nnnorm, CategoryTheory.HasLiftingProperty.transfiniteComposition.sqFunctor_map, Filter.disjoint_pure_atBot, CategoryTheory.Subpresheaf.le_sheafify, Order.KrullDimLE.krullDim_le, isAtom_top, ClosureOperator.monotone, gt_or_lt_of_ne, HasSolidNorm.orderClosedTopology, toPreorder_injective, LipschitzWith.hausdorffMeasure_image_le, Subring.subset_pointwise_smul_iff, Set.one_lt_encard_iff_nontrivial, Mathlib.Meta.Positivity.zpow_zero_pos, EReal.natCast_div_le, CanonicallyOrderedAdd.list_prod_pos, CategoryTheory.MorphismProperty.IsCardinalForSmallObjectArgument.preservesColimit, TwoSidedIdeal.bot_ringCon, RCLike.instOrderClosedTopology, EReal.preimage_coe_Ico, Finset.disjoint_box_succ_prod, AlgebraicGeometry.Scheme.germ_residue_assoc, EisensteinSeries.eisensteinSeries_tendstoLocallyUniformlyOn, instIsContinuousOpensCarrierMapGrothendieckTopology, GenContFract.zero_le_of_contsAux_b, Fin.preimage_castSucc_Ioc_castSucc, MvPolynomial.le_degrees_add_left, OrderHom.isGreatest_gfp, Finsupp.DegLex.wellFoundedLT, LinearOrderedAddCommGroup.Subgroup.exists_neg_generator, AddSubgroup.isLeast_of_closure_iff_eq_abs, Monotone.Iic, Filter.NeBot.nonneg_sub, Ordinal.CNF.fst_le_log, negPart_eq_neg, AddSubsemigroup.toSubsemigroup_closure, Ideal.pow_lt_self, MeasureTheory.measure_setAverage_le_pos, TopCat.Presheaf.generateEquivalenceOpensLe_inverse'_obj_obj_left, MeasureTheory.Integrable.measure_le_integral, sInf_upperBounds_eq_csSup, r1Space_iff_specializes_or_disjoint_nhds, ArchimedeanClass.eq_zero_or_top_of_archimedean, tendsto_pow_div_pow_atTop_atTop, BddOrd.ofHom_apply, Localization.le_comap_primeCompl_iff, Finset.Colex.le_iff_sdiff_subset_lowerClosure, lt_or_ge, LieSubmodule.map_le_map_iff, NatOrdinal.small_Ico, mabs_lt, Module.Basis.mem_toFlag, LinearPMap.le_graph_of_le, supClosure_idem, AdjoinRoot.adjoinRoot_eq_top, AlgebraicGeometry.isAffineOpen_bot, NonemptyInterval.fst_sup, algebraMap_mono, AlgebraicGeometry.Scheme.IsLocallyDirected.V_self, Multiset.le_add_right, List.foldr_sup_eq_sup_toFinset, Nucleus.coe_top, MeasurableSet.exists_isCompact_isClosed_lt_add, borel_eq_top_of_countable, ProbabilityTheory.measure_le_mul_measure_gt_le_of_map_rotation_eq_self, AlgebraicGeometry.Scheme.kerFunctor_obj, LinearLocallyFiniteOrder.le_succFn, ENNReal.iSup_mul_le, Subalgebra.isSimpleOrder_of_finrank_prime, ENNReal.natCast_lt_coe, Set.image_mulSingle_Ico_right, HeytAlg.Iso.mk_hom, MeasureTheory.measure_lt_top_of_isCompact_of_isAddLeftInvariant', CategoryTheory.IsGrothendieckAbelian.generatingMonomorphisms.functorToMonoOver_obj, AlgebraicGeometry.Scheme.Opens.ι_image_le, HasStrictFDerivAt.hasStrictFDerivAt_norm_smul, CategoryTheory.GrothendieckTopology.top_mem', min_le_emultiplicity_add, Polynomial.signVariations_eq_eraseLead_add_ite, TopCat.Presheaf.Γgerm_res_apply, Set.inv_mem_Icc_iff, IntermediateField.extendScalars_top, iInf₂_eq_bot, NonUnitalSubring.gc_map_comap, List.sortedGE_mergeSort, ContinuousLinearMap.coe_le_coe_iff, posSMulMono_iff_posSMulStrictMono, Ordinal.toNimber_eq_one, RootPairing.corootSpan_dualAnnihilator_le_ker_rootForm, Fin.rev_strictAnti, Finsupp.orderedSub, RCLike.instPosMulReflectLE, Valuation.mem_leIdeal_iff, AddSubsemigroup.comap_le_comap_iff_of_surjective, ciSup_false, Fin.finsetImage_castSucc_Ici, Set.Icc.coe_le_one, SupBotHom.coe_bot, perfectClosure.eq_bot_of_isSeparable, Submonoid.prod_le_iff, AbsoluteValue.IsEuclidean.map_lt_map_iff, Subsemiring.nonneg_toAddSubmonoid, rank_pos, FinBoolAlg.hasForgetToFinPartOrd_forget₂_map, ProbabilityTheory.IsFiniteKernel.exists_univ_le, Cardinal.aleph_eq_preAleph, WithZeroTopology.tendsto_zero, Fin.prod_Iio_castLE, sInf_upperBounds_eq_sSup, LinearOrderedAddCommGroupWithTop.sub_le_sub_iff_left_of_ne_top, MeasureTheory.lintegral_nnnorm_condExpIndSMul_le, Finset.Icc_add_Ico_subset, Subsemiring.toAddSubmonoid_strictMono, Int.card_fintype_Ico_of_le, subsingleton_iff_bot_eq_top, AlgebraicGeometry.StructureSheaf.comap_basicOpen, BoxIntegral.TaggedPrepartition.distortion_le_iff, lt_biSup_iff, convexHull_smul, Cardinal.mul_natCast_lt_mul_natCast, EReal.nhds_top, AlgebraicGeometry.IsOpenImmersion.opensRange_pullback_fst_of_right, AddSubsemigroup.opEquiv_apply, Real.nnabs_pos, IsNonarchimedean.multiset_image_add, Ideal.exists_comap_eq_of_mem_minimalPrimes, BooleanSubalgebra.mem_map_equiv, Ordinal.omega_pos, iSup_le_iff, MeasureTheory.measure_le_average_pos, HolderWith.edist_le, Padic.norm_rat_le_one_iff_padicValuation_le_one, Ideal.subset_union_prime, emultiplicity_pos_iff, NonemptyInterval.neg_pure, NonUnitalSubsemiring.prod_mono_right, Sublattice.map_symm_eq_iff_eq_map, ringKrullDim_le_ringKrullDim_add_spanFinrank, AlgebraicGeometry.Scheme.mem_basicOpen'', MeasureTheory.Measure.addModularCharacterFun_pos, MeasureTheory.eLpNorm'_const_smul_le', ConvexCone.coe_top, Ordinal.invVeblen₂_lt, Ordinal.to_leftMoves_one_toPGame_symm, AddSubgroup.surjOn_iff_le_map, Multiset.card_Icc, NoTopOrder.to_noMaxOrder, enorm_add_le, CategoryTheory.Triangulated.SpectralObject.triangle_obj₃, Finset.exists_pos_of_sum_zero_of_exists_nonzero, CStarAlgebra.inr_mem_Icc_iff_nnnorm_le, antitone_iff_forall_lt, Filter.covariant_vadd_filter, Nonneg.Icc_subset_segment, Ordinal.lt_omega0, Set.nontrivial_iff_exists_lt, Set.bounded_ge_inter_gt, idealFactorsFunOfQuotHom_coe_coe, posPart_mono, Set.notMem_Ici, Valued.isClosed_closedBall, MeasurableSpace.measurableSet_top, upperClosure_mul_distrib, Fin.finsetImage_rev_Ico, Equiv.Perm.OnCycleFactors.kerParam_range_le_centralizer, Submodule.orthogonal_le_orthogonal_iff, SimpleGraph.componentComplFunctor_nonempty_of_infinite, Multiset.powersetCard_le_powerset, CategoryTheory.Presheaf.equalizerSieve_eq_top_iff, BddOrd.ofHom_comp, LinearOrder.compare_eq_compareOfLessAndEq, Finset.Colex.cons_le_cons, SignType.lt_one_iff, Complex.real_le_real, ENat.succ_def, Submodule.orthogonal_le_iff_orthogonal_le, Set.Nonempty.convexHull, Submodule.span_singleton_le_iff_mem, ProbabilityTheory.Kernel.indep_bot_left, Set.countable_isBot, Finpartition.bot_notMem, AbsoluteValue.IsNontrivial.exists_abv_lt_one, LieAlgebra.center_le_maxNilpotentIdeal, Set.preimage_add_const_Iic, mem_nhdsGT_iff_exists_Ioo_subset, ComplementedLattice.instOrderDual, Submodule.map_comap_le, csSup_Iic, Cardinal.nat_le_ofENat, Finset.range_mono, continuousWithinAt_Iio_iff_Iic, instOrderedRingOfIsStrictOrderedRing, Set.smul_Icc, Lat.Iso.mk_hom, Finset.Ico_succ_left_eq_Ioo, Ordinal.le_add_right, Set.integer_valuation_le_one, AlgebraicGeometry.Scheme.AffineZariskiSite.cocone_pt, ProfiniteGrp.toLimitFun_continuous, CFC.posPart_nonneg, le_separableClosure', CategoryTheory.Limits.CompleteLattice.limit_eq_iInf, grade_covBy_grade_iff, Fin.preimage_val_Ioo_val, IntermediateField.fixedField_bot, lowerClosure_smul, Set.preimage_const_div_Iic, EisensteinSeries.summable_e2Summand_symmetricIco, Set.biUnion_ge, UpperSet.Ici_le_Ioi, Order.Frame.MinimalAxioms.inf_sSup_le_iSup_inf, Ordinal.nadd_lt_nadd_iff_right, interior_Iic, Valuation.coe_ltAddSubgroup, Matroid.ofMatroid_lt_iff, Cardinal.ofENat_toENat_eq_self, MeasureTheory.LevyProkhorov.le_convergenceInDistribution, isGLB_iInf, MeasureTheory.Measure.IicSnd_mono, PartOrd.hom_inv_apply, AddCircle.coe_image_Ico_eq, absConvexHull_eq_convexHull_balancedHull, inv_nonpos, Finset.sum_sym2_filter_not_isDiag, enorm_multisetSum_le, Ordinal.lt_invVeblen₂_iff, Filter.nhds_nhds, Finset.Iic_sub_one_eq_Iio, FirstOrder.Language.Substructure.map_closure, instSMulPosMonoNatOfIsOrderedAddMonoid, OrderIso.withBotCongr_trans, Ordinal.add_le_iff, Order.le_succ_bot_iff, List.max_sum_le, Pi.ωScottContinuous_curry, le_hasSum', MeasureTheory.eLpNorm_const_lt_top_iff, RootPairing.root_sub_zsmul_mem_range_iff, Finsupp.DegLex.le_iff, AlgebraicGeometry.tilde.toOpen_map_app_assoc, ENNReal.orderIsoRpow_apply, IntermediateField.toSubalgebra_strictMono, FirstOrder.Language.Substructure.monotone_comap, interior_Ioi, refl_le_uniformity, NonUnitalAlgebra.range_eq_top, Filter.tendsto_comp_inv_atTop_iff, CategoryTheory.ComposableArrows.mk₀_map, IsIntegral.fg_adjoin_singleton, WithTop.le_of_forall_lt_iff_le, StarSubalgebra.map_topologicalClosure_le, div_le_egauge_closedBall, Real.smoothTransition.projIcc, SSet.horn.primitiveEdge_coe_down, measure_Ioc_lt_top, MeasureTheory.Measure.le_dirac_apply, Set.Iio_insert, Set.image_const_add_Ico, Cardinal.preBeth_strictMono, CompactlySupportedContinuousMap.instAddLeftMono, LinearMap.mem_span_iff_bound, Set.Finite.convexHull_eq_image, Ordinal.veblen_right_strictMono, Algebra.discr_powerBasis_eq_prod', MonoidWithZeroHom.inl_mono, Dynamics.netEntropyEntourage_le_coverEntropy, Finset.image_add_left_Icc, StarOrderedRing.nonneg_iff_quasispectrum_nonneg, ENNReal.rpow_inv_lt_iff, Filter.HasAntitoneBasis.antitone, BooleanAlgebra.le_top, Metric.externalCoveringNumber_pos_iff, nhdsWithinLT_sup_nhdsWithinGE, SemidirectProduct.mulEquivSubgroup_symm_apply, clusterPt_iff_ultrafilter, MulAction.stabilizer_inf_stabilizer_le_stabilizer_union, MeasureTheory.Measure.countable_meas_level_set_pos₀, OrderIso.map_sInf, Affine.Simplex.sOppSide_affineSpan_faceOpposite_point_right_iff, OmegaCompletePartialOrder.ContinuousHom.coe_apply, SameRay.exists_pos_left, wellFounded_gt_exact_sequence, AddGroupCone.mem_nonneg, CategoryTheory.Subobject.inf_comp_right, MeasureTheory.Egorov.exists_notConvergentSeq_lt, AlgebraicGeometry.LocallyRingedSpace.Γevaluation_ne_zero_iff_mem_basicOpen, antivaryOn_iff_exists_antitoneOn_monotoneOn, ENNReal.one_half_lt_one, MulOpposite.op_nonneg, Equiv.Perm.isCoatom_stabilizer_of_ncard_lt_ncard_compl, CategoryTheory.ComposableArrows.fourδ₃Toδ₂_app_two, AlgebraicGeometry.basicOpen_eq_of_affine, Multiset.lt_singleton, AlgebraicGeometry.LocallyRingedSpace.Γ_map_op, Nat.cast_le, biInf_le_biSup, OrdinalApprox.lfp_mem_range_lfpApprox, Matrix.isIrreducible_iff, MvPolynomial.schwartz_zippel_sum_degreeOf, FiniteMulArchimedeanClass.val_mk, CategoryTheory.Subfunctor.Subpresheaf.range_comp_le, LieModule.posFittingCompOf_le_posFittingComp, Set.range_injOn_strictMono, FiniteArchimedeanClass.ball_strictAnti, CategoryTheory.Triangulated.SpectralObject.comp_hom_assoc, Finsupp.Lex.wellFoundedLT, MeasureTheory.AEEqFun.le_sup_left, Fin.map_castSuccEmb_Ioi, right_le_midpoint, Finset.sum_Iio_add_eq_sum_Iic, continuous_iff_coinduced_le, iSup_add_le, FrameHom.toFun_eq_coe, TopologicalSpace.OpenNhds.map_id_obj, compl_le_iff_compl_le, WithTop.Ioc_coe_top, AlgebraicGeometry.HasAffineProperty.affineAnd_le_isAffineHom, Filter.blimsup_sup_le_or, ProfiniteGrp.diagram_map, Fin.finsetImage_castSucc_Iio, NNReal.HolderTriple.one_div_pos, NatOrdinal.small_Ioo, Filter.codiscreteWithin.mono, Subgroup.le_centralizer_iff_isMulCommutative, Order.coheight_anti, Int.ceil_mono, NNReal.sqrt_div, Subsemiring.pointwise_smul_le_pointwise_smul_iff₀, MeasureTheory.ExistsSeqTendstoAe.exists_nat_measure_lt_two_inv, CategoryTheory.TransfiniteCompositionOfShape.ofComposableArrows_incl_app, not_denselyOrdered_withZero_int, Ideal.multiset_prod_le_inf, HahnSeries.zero_lt_orderTop_iff, WithBot.Ioo_bot_coe, infClosure_idem, Subgroup.prod_mono_right, Continuous.exists_forall_le_of_hasCompactMulSupport, PrimeSpectrum.coe_primesOverOrderIsoFiber_symm_apply_coe, one_div_nonpos, Filter.compl_diagonal_mem_prod, ArchimedeanClass.orderHom_top, ENat.exists_nat_gt, AlgebraicGeometry.Scheme.ΓSpecIso_naturality_assoc, Topology.IsLower.isTopologicalBasis_insert_univ_subbasis, top_himp, Fin.map_succEmb_Icc, mem_openSegment_iff_div, HomologicalComplex.HomologySequence.composableArrows₃Functor_map, FirstOrder.Language.instDirectedSystemSubtypeMemSubstructureCoeOrderHomEmbeddingInclusion, Matrix.nnnorm_le_iff, Submodule.torsionBy_le_torsionBy_of_dvd, HolderOnWith.ediam_image_inter_le, AlgebraicGeometry.IsAffineOpen.fromSpec_toSpecΓ_assoc, FirstOrder.Language.Substructure.comap_le_comap_iff_of_surjective, Finset.offDiag_filter_lt_eq_filter_le, Complex.adjoin_I, partialSups_bot, PointedCone.mem_toConvexCone, HahnEmbedding.Partial.eval_zero, MeasureTheory.lintegral_abs_det_fderiv_le_addHaar_image_aux2, AlgebraicGeometry.Scheme.Hom.comp_image, SSet.Subcomplex.range_eq_top_iff, TwoSidedIdeal.orderIsoIsTwoSided_apply_coe, TwoSidedIdeal.asIdeal_matrix, MeasureTheory.eLpNormEssSup_indicator_le, Submodule.starProjection_le_starProjection_iff, AlgebraicGeometry.IsOpenImmersion.range_pullbackFst, nhdsGE_basis, OrderedFinpartition.emb_strictMono, AlgebraicGeometry.IsAffineOpen.fromSpec_image_zeroLocus, Antisymmetrization.prodEquiv_apply_mk, measurable_IocProdIoc, Con.toSetoid_top, StarAlgebra.adjoin_mono, Fin.sum_Icc_castSucc, CochainComplex.mappingConeCompTriangle_mor₃_naturality_assoc, Pi.le_toColex_update_self_iff, NNReal.tendsto_coe_atTop, ENNReal.image_coe_Ioo, CFC.posPart_eq_self, Partition.coe_removeBot, enorm_mul_le', AffineSubspace.inclusion_rfl, Finset.prod_le_prod_of_ne_one', AlgebraicGeometry.Scheme.IdealSheafData.mem_supportSet_iff_of_mem, IsSemisimpleModule.eq_bot_or_exists_simple_le, Subgroup.toAddSubgroup'_closure, Cardinal.ofENat_toENat, Ordinal.typein_le_typein', Ordinal.succ_pred_eq_iff_not_isSuccPrelimit, AlgebraicGeometry.PresheafedSpace.toRestrictTop_base, StieltjesFunction.measure_Ico, AlgebraicGeometry.LocallyRingedSpace.isUnit_res_toΓSpecMapBasicOpen, AlgebraicGeometry.Scheme.affineBasisCover_map_range, Ideal.le_comap_sup, Prod.disjoint_iff, Ordinal.toGame_injective, Left.nsmul_neg_iff, IsNonarchimedean.apply_intCast_le_one_of_isNonarchimedean, Set.ordConnected_iff_uIcc_subset, Dynamics.coverEntropyInf_image_le_of_uniformContinuousOn_invariant, Filter.gc_map_comap, Filter.tendsto_nhds_atTop, Finset.strictAnti_iff_forall_cons_lt, CategoryTheory.Triangulated.SpectralObject.ω₂_obj_obj₂, Fin.Ioi_sub_one_eq_Ici, CountableInterFilter.toCardinalInterFilter, Finset.exists_inf_eq_iInf, InfiniteGalois.algEquivToLimit_continuous, TopCat.Presheaf.germ_stalkSpecializes, QuadraticMap.nonneg_pi_iff, le_himp, ENNReal.lt_top_of_tsum_ne_top, set_has_maximal_iff_noetherian, Module.End.ker_pow_le_ker_pow_finrank, CategoryTheory.Subobject.representative_coe, LowerAdjoint.closureOperator_isClosed, not_bot_lt_iff, NNRat.cast_le_natCast, Submodule.span_preimage_le, Heyting.isRegular_bot, antivaryOn_iff_forall_smul_nonpos, List.minimum_concat, AlgebraicGeometry.ΓSpec_adjunction_homEquiv_eq, Finset.inf_le, isClosed_le_of_isClosed_nonneg, Cardinal.aleph_one_eq_lift, Pi.wcovBy_iff_exists_right_eq, OmegaCompletePartialOrder.ContinuousHom.toMono_coe, ENNReal.coe_div_le, tendsto_floor_left, CategoryTheory.Limits.MultispanIndex.toLinearOrder_left, OrderAddMonoidHom.snd_comp_inl, ContinuousMap.elemental_id_eq_top, instIsPartialOrderLe, AlgebraicGeometry.Scheme.Opens.ι_app, Ordinal.enum_inj, Multiset.sum_min_le, CompleteLattice.isCompactElement_iff_exists_le_sSup_of_le_sSup, Fin.finsetImage_cast_Iic, abs_add_le, ediam_add_le, OrderEmbedding.isEmbedding_of_ordConnected, CompleteSublattice.disjoint_iff, ContinuousLinearEquiv.nnnorm_symm_pos, EReal.mul_pos_iff, MeasureTheory.Measure.toSphereBallBound_mul_measure_unitBall_le_toSphere_ball, LinearOrderedCommGroup.to_noMaxOrder, Cardinal.mk_surjective_eq_zero_iff_lift, convexHull_rangle_single_eq_stdSimplex, Set.Icc.mul_le_right, Opens.toPretopology_grothendieckTopology, FirstOrder.Language.Substructure.closure_le, Hyperreal.InfinitePos.pos, Function.Injective.encard_range, Set.biUnion_Ioc_eq_Ioi_self_iff, Unitization.starLift_range_le, Int.erdos_ginzburg_ziv_multiset, FormalMultilinearSeries.enorm_compContinuousLinearMap_le, groupCohomology.coboundaries₁_le_cocycles₁, Multiset.erase_le, AlgebraicGeometry.Scheme.Modules.pushforwardCongr_inv_app_app, HomogeneousIdeal.irrelevant_eq_closure, SimpleGraph.Copy.ofLE_comp, Order.krullDim_eq_one_iff_of_boundedOrder, Set.encard_le_one_iff_eq, sameRay_smul_right_iff, MeasureTheory.Measure.count_apply_lt_top', Matroid.eRk_le_one_iff, CategoryTheory.MorphismProperty.monotone_isoClosure, ProbabilityTheory.bayesRisk_le_bayesRisk_map, Fin.map_finCongr_Iio, Ne.bot_lt', AffineBasis.centroid_mem_interior_convexHull, HahnSeries.finiteArchimedeanClassOrderIso_apply, LinearOrderedAddCommGroupWithTop.strictMono_add_right_of_ne_top, MeasureTheory.OuterMeasure.map_top, BoundedLatticeHomClass.map_bot, Order.height_mono, CategoryTheory.Limits.kernel_map_comp_kernelSubobjectIso_inv_assoc, MeasureTheory.levyProkhorov_le_convergenceInDistribution, BddOrd.ofHom_id, AffineMap.image_convexHull, WithTop.tendsto_nhds_top_iff, Polynomial.contentIdeal_le_span_content, Fin.orderSucc_castSucc, Polynomial.isLittleO_atBot_of_degree_lt, m_Birkhoff_inequalities, Ordinal.sup_eq_lsub_or_sup_succ_eq_lsub, AlgebraicGeometry.Scheme.IdealSheafData.supportSet_inter, min_mabs_mabs_le_mabs_max, WithIdealFilter.mem_nhds_iff, AlgebraicGeometry.locallyOfFinitePresentation_iff, Fin.strictMono_castLE, hasProdUniformlyOn_iff_tendstoUniformlyOn, Finset.nnnorm_prod_le_sup_nnnorm, EMetric.exists_continuous_eNNReal_forall_closedBall_subset, AffineSpace.asymptoticNhds_eq_smul_vadd, MonomialOrder.div, CategoryTheory.ComposableArrows.fourδ₃Toδ₂_app_three, Int.strictMonoOn_natAbs, CFC.sqrt_nonneg, egauge_anti, Finset.pluennecke_ruzsa_inequality_pow_mul, Nimber.toOrdinal_symm_eq, Profinite.NobelingProof.injective_πs, pi_nnnorm_const_le, convexHull_union, Finset.nnnorm_prod_le, OrderIso.map_iSup₂, AlgebraicGeometry.Scheme.Hom.isoImage_hom_ι, AlgebraicGeometry.IsAffineOpen.isOpenImmersion_fromSpec, Algebra.FiniteType.out, subsingleton_iff_top_eq_bot, sq_pos_of_ne_zero, convexHull_sub, Hyperreal.infinitePos_def, ENNReal.HolderConjugate.lt_top_iff_one_lt, Topology.RelCWComplex.skeletonLT_I, FirstOrder.Language.DirectLimit.instDirectedSystemSubtypeMemSubstructureCodCoeOrderHomPartialEquivEmbeddingInclusion, Cardinal.aleph_toENat, star_nonpos_iff, MeasureTheory.eLpNorm_add_le', instOrderIsoClassContinuousLinearMapIdOfNonUnitalAlgEquivClassOfStarHomClassOfContinuousMapClass, Sublattice.le_comap_sup, Continuous.partialSups, Fin.finsetImage_castAdd_Icc, separableClosure_le_iff, AddSubgroup.IsSubnormal.iff_eq_top_or_exists, FirstOrder.Language.Substructure.mem_closed_iff, NNReal.instIsStrictOrderedModule, Order.height_le_iff', maximal_iff_isGreatest, cmp_div_one', Multiset.toDFinsupp_lt_toDFinsupp, polynomialFunctions_closure_eq_top', WithZeroTopology.orderClosedTopology, Submodule.le_traceDual_mul_iff, Int.cast_le_neg_one_or_one_le_cast_of_ne_zero, Filter.tendsto_mul_self_atTop, Irreducible.maximalIdeal_pow_eq_setOf_le_v_coe_pow, ENNReal.one_lt_ofReal, ENat.floor_eq_zero, Finset.lt_fold_max, tendsto_ceil_right, CategoryTheory.Limits.kernelSubobject_arrow_apply, Finset.inf_univ_eq_iInf, Order.krullDim_nonneg, MeasureTheory.Measure.measure_Ioi_pos, EuclideanSpace.nnnorm_eq, NNReal.nhds_zero_basis, AlgebraicGeometry.Scheme.Hom.isCompact_preimage, CStarAlgebra.star_right_conjugate_le_norm_smul, NonUnitalStarSubalgebra.map_le, AnalyticOnNhd.map_codiscreteWithin, le_partialSups, smul_nonneg_iff_pos_imp_nonneg, MulAction.fixingSubgroup_le_stabilizer, AntilipschitzWith.mul_le_edist, FractionalIdeal.instMulLeftMono, Ideal.homogeneousCore_eq_sSup, MonomialOrder.lex_le_iff, upperHemicontinuous_iff_isClosed_compl_preimage_Iic_compl, oneLePart_eq_self, Subsemigroup.le_op_iff, Cardinal.preBeth_mono, exists_pair_lt, Complex.nonneg_iff, AffineSubspace.affineSpan_eq_top_iff_vectorSpan_eq_top_of_nontrivial, exists_pos_left_iff_sameRay_and_ne_zero, Filter.map_neg_atBot, Finsupp.lsingle_range_le_ker_lapply, Monotone.Icc, cfc_le_one_iff, ValuationSubring.instCovariantClassHSMulLe, Finset.sumEquiv_apply_fst, ENNReal.tendsto_nhds_top_iff_nat, Cardinal.lift_lt_continuum, abs_add_three, IsUnifLocDoublingMeasure.one_le_scalingConstantOf, Cardinal.countable_iff_lt_aleph_one, sameRay_neg_smul_left_iff, SetTheory.Game.birthday_quot_le_pGameBirthday, OrderHom.iSup_apply, AlgebraicGeometry.instUniversallyClosedMorphismRestrict, Pi.uncurry_curry_ωScottContinuous, tendsto_neg_nhdsLT, closure_subset_closedConvexHull, Projectivization.Subspace.span_univ, Finset.image_mono, Manifold.riemannianEDist_triangle, Subsemigroup.center_le_centralizer, Set.Nontrivial.einfsep_lt_top, Ordinal.veblen_succ, AlgebraicGeometry.Scheme.toIso_inv_ι, CategoryTheory.SmallObject.prop_iterationFunctor_map_succ, AlgebraicGeometry.SheafedSpace.congr_hom_app, AlgebraicGeometry.Scheme.Hom.instIsIsoCommRingCatApp, Rat.cast_le, exists_isClopen_upper_or_lower_of_ne, Pi.isCompl_iff, isClosed_restrictScalars_separableClosure, OnePoint.ultrafilter_le_nhds_infty, Subring.map_le_iff_le_comap, Order.not_isMax_pred, stdSimplexEquivIcc_symm_apply_coe, ProbabilityTheory.condIndep_limsup_atBot_self, Set.functorToTypes_map, zsmul_le_zsmul_iff_right, AddSubgroup.iSup_comap_le, MvPolynomial.range_mapAlgHom, Nonneg.coe_natCast, PrimeSpectrum.asIdeal_lt_asIdeal, AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_preimage_basicOpen, Submodule.span_singleton_smul_le, Fin.map_castSuccEmb_Ico, FirstOrder.Language.Substructure.instIsEmptySubtypeMemBotOfConstants, Ordinal.instAddLeftMono, antivary_iff_forall_smul_nonpos, HolderOnWith.nndist_le, Finset.Ico_filter_lt, hasCardinalLT_iff_cardinal_mk_lt, Nonneg.nat_floor_coe, IsArtinian.set_has_minimal, PosMulReflectLE.toPosMulStrictMono, Ordinal.opow_succ, Filter.isBoundedUnder_ge_neg, Filter.rtendsto_iff_le_rcomap, le_posPart, IntermediateField.insepDegree_bot, continuousOn_cfc_nnreal_setProd, tendstoUniformly_tsum, FiniteArchimedeanClass.min_le_mk_add, MonoidHom.liftOfRightInverse_comp, LieSubalgebra.wellFoundedGT_of_noetherian, Fin.map_addNatEmb_Icc, Prod.Lex.prodUnique_apply, Finset.min_mono, Order.succ_eq_iff_covBy, Set.lt_iff_ssubset, mulIndicator_biUnion_finset_eventuallyEq, Ideal.comap_fiberIsoOfBijectiveResidueField_symm, Set.Ici_inter_Ici, Set.Countable.substructure_closure, bot_eq_zero'', lowerClosure_mul_distrib, MeasureTheory.exists_upperSemicontinuous_le_integral_le, AlgebraicGeometry.Scheme.Hom.inv_image, Fin.map_castSuccEmb_Ioo, IsAlgClosed.algebraicClosure_eq_bot_iff, Continuous.bddBelow_range_of_hasCompactMulSupport, AlgebraicGeometry.PresheafedSpace.restrict_presheaf, Multiset.map_add_right_Ioc, Finsupp.support_monotone, antivaryOn_iff_mul_rearrangement, Set.Ioo_sub_one_left_eq_Ioc, FirstOrder.Language.Substructure.subset_closure_withConstants, ClosedSubmodule.toSubmodule_top, Fin.preimage_castSucc_Ioi_castSucc, Nat.inv_pos_of_nat, IntermediateField.rank_bot', OpenSubgroup.toOpens_top, disjoint_iff_inf_le, CategoryTheory.Subobject.inf_eq_map_pullback, HahnSeries.min_le_min_add, Part.Fix.mem_iff, iSup_const_le, ContinuousLinearMap.nnbound, Finset.sup_bot, Valuation.self_le_supp_comap, CategoryTheory.Equivalence.toOrderIso_symm_apply, CategoryTheory.Limits.imageSubobject_arrow_comp, convexHull_list_sum, Set.not_monotoneOn_not_antitoneOn_iff_exists_lt_lt, Module.Basis.toFlag_carrier, iSupIndep_iff_supIndep, AlgebraicGeometry.Scheme.germ_residue, ENNReal.nhds_top, Order.coheight_pos, Ordinal.add_one_eq_succ, Filter.le_map, CategoryTheory.MorphismProperty.transfiniteCompositions_pushouts_coproducts_le_llp_rlp, transGen_wcovBy_eq_reflTransGen_covBy, MeasureTheory.addContent_biUnion_le, atBot_hasCountableBasis_of_archimedean, Set.preimage_div_const_Ici, BotHom.inf_apply, iInf_eq_if, Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes, Disjoint.eq_iff, CompleteLattice.isAtomistic_iff, Filter.EventuallyLE.measure_le, Set.Finite.ncard_strictMonoOn, AddHom.mclosure_preimage_le, AlgebraicGeometry.IsOpenImmersion.le_monomorphisms, LieSubmodule.map_le_iff_le_comap, Order.Ideal.mem_sInf, IntermediateField.le_sup_toSubalgebra, exists_Icc_mem_subset_of_mem_nhds, SummationFilter.symmetricIcc_eq_map_Icc_nat, measurableSet_Ioc, TopologicalSpace.Opens.map_top, Complex.real_lt_real, IsOrderedCancelMonoid.toMulLeftReflectLT, HasCompactSupport.exist_eLpNorm_sub_le_of_continuous, Ordinal.card_lt_aleph0, MonomialOrder.degree_sPolynomial_le, IntermediateField.gc, AlgebraicGeometry.IsPreimmersion.instMorphismRestrict, AddSubgroup.le_pointwise_smul_iff, OrderEmbedding.range_inj, Nat.lt_of_lt_floor, Topology.lawson_le_lower, NNReal.agmSequences_succ', EReal.sign_neg, CategoryTheory.MorphismProperty.le_def, CategoryTheory.Functor.imageSieve_map, LinearMap.annihilator_le_of_surjective, Set.Ioc.coe_one, Finset.covBy_iff_exists_erase, Ordinal.nfpFamily_le_apply, LinearOrderedField.cutMap_self, add_self_nonpos_iff, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.app_inv_app', WithTop.succ_coe, CFC.nnrpow_nonneg, AlgebraicGeometry.PresheafedSpace.GlueData.ιInvApp_π, Ideal.exists_spanRank_eq_and_height_eq, SMulPosReflectLT.of_pos, sSup_iUnion_Iic, uniformity_basis_edist_nnreal, TopCat.Presheaf.presheafEquivOfIso_functor_map_app, SSet.stdSimplex.face_le_face_iff, WithVal.le_def, Submodule.span_monotone, ValuationSubring.coe_primeSpectrumOrderEquiv_symm_apply_asIdeal, Dynamics.le_coverEntropyInfEntourage_image, HasCountableSeparatingOn.range_Ici, Finset.pluennecke_ruzsa_inequality_nsmul_sub, DiscreteQuotient.ofLE_refl_apply, borel_anti, NE.ne.one_lt, CategoryTheory.Subfunctor.Subpresheaf.subobjectMk_range_arrow, NNReal.one_lt_coe, partialSups_eq_biUnion_range, addRothNumber_Ico, edist_triangle_left, AlgebraicGeometry.Scheme.Hom.opensRange_comp, Ordinal.IsAcc.pos, SimpleGraph.Subgraph.comap_monotone, Set.Ioc_inter_Ioi, Nat.floor_eq_iff', Filter.exists_antitone_seq, sSup_le_iff, CStarAlgebra.nonneg_TFAE, AddSubgroup.goursatFst_prod_goursatSnd_le, List.argmin_eq_some_iff, nndist_nnnorm_nnnorm_le, AddSubgroup.index_toSubgroup, IsDiscreteValuationRing.addVal_le_iff_dvd, Mathlib.Meta.NormNum.isNNRat_le_false, isPiSystem_Icc_mem, IsCompl.dual, AddSubsemigroup.closure_mono, kstar_eq_self, Ordinal.IsNormal.le_iff_deriv, MeasureTheory.Measure.toSphereBallBound_pos, WithBot.coe_biSup, Topology.IsInducing.le_functorObj_iff, AlgebraicGeometry.Scheme.Hom.ι_fromNormalization, cauchySeq_iff_tendsto, Fin.finsetImage_castLE_Ioc, Set.Iio.coe_inf, AlgebraicGeometry.LocallyRingedSpace.toΓSpec_preimage_zeroLocus_eq, Cardinal.IsRegular.nat_lt, MeasureTheory.FinStronglyMeasurable.fin_support_approx, AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_inv_eq_germ, Valuation.HasExtension.val_map_lt_one_iff, AlgebraicGeometry.exists_finite_imageι_comp_morphismRestrict_of_finite_image_preimage, CategoryTheory.MorphismProperty.pushouts_le_llp_rlp, LinearOrderedAddCommGroupWithTop.sub_self_nonneg, FiniteGaloisIntermediateField.le_iff, Ideal.homogeneousCore_mono, AlgebraicGeometry.genericPoint_eq_bot_of_affine, CategoryTheory.ComposableArrows.Precomp.map_comp, DividedPowers.SubDPIdeal.lt_iff, Metric.packingNumber_pos_iff, LinearOrderedField.cutMap_bddAbove, AlgebraicGeometry.LocallyRingedSpace.SpecΓIdentity_hom_app, CategoryTheory.TransfiniteCompositionOfShape.map_incl, OrderMonoidHom.fst_comp_inr, ArchimedeanClass.orderHom_mk, UpperSet.le_sdiff_left, covBy_iff_lt_iff_le_right, Set.image_div_const_Ioc, CategoryTheory.Limits.CompleteLattice.limitCone_cone_π_app, ultrafilter_converges_iff, WithBot.denselyOrdered_set_iff_subsingleton, MeasureTheory.Measure.sigmaFinite_iff_measure_singleton_lt_top, ContinuousAlternatingMap.nnnorm_ofSubsingleton_id_le, Submodule.map_le_iff_le_comap, Set.Ici.subtype_functor_final, QuadraticMap.posDef_iff_nonneg, AlgebraicGeometry.affineLocally_le, Decidable.ne_iff_lt_iff_le, Metric.isBounded_of_abs_le, TopologicalSpace.Opens.overEquivalence_inverse_obj_right_as, Set.Iic_subset_Iio_union_Icc, MeasureTheory.Measure.sub_le, CategoryTheory.Sieve.pullback_top, one_le_mabs, AlgebraicGeometry.instIsQuasicoherentOpensCarrierCarrierCommRingCatSpecTilde, AddGroupNorm.lt_def, Set.Icc_add_bij, IicProdIoc_comp_restrict₂, imageToKernel_zero_left, ContinuousLinearMap.le_opENorm, CategoryTheory.SimplicialThickening.Path.le_right, IntermediateField.insepDegree_top, StieltjesFunction.length_Ioc, Order.lt_succ_bot_iff, Set.image_const_sub_Ioo, Finset.Ico_map_sectL, Continuous.lowerSemicontinuous, Order.pred_lt_iff, NNReal.closedBall_zero_eq_Icc', MvPolynomial.IsSymmetric.antitone_supDegree, LinearOrder.le_total, Set.encard_le_encard, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_hom, ENat.toENNReal_le, Valuation.one_le_val_iff, Profinite.NobelingProof.GoodProducts.range_equiv_smaller_toFun_bijective, Cardinal.toENat_eq_iff_of_le_aleph0, ZFSet.rank_pair, separableClosure_le_separableClosure_iff, FormalMultilinearSeries.radius_pi_le, Subring.eq_iInf_of_isIntegrallyClosedIn, QuotientGroup.ker_le_range_iff, KleeneAlgebra.kstar_mul_le_kstar, FiniteMulArchimedeanClass.mk_lt_mk, Module.map_jacobson_le, Submonoid.closure_image_one_lt_and_isMulIndecomposable, BooleanSubalgebra.le_comap_sup, ProjectiveSpectrum.basicOpen_one, Real.coe_sqrt, ENNReal.one_le_coe_iff, TopologicalSpace.Opens.apply_def, Concept.strictMono_extent, Filter.blimsup_false, Order.Ideal.coe_inf, Fin.image_succ_Iio, Set.Ioo_disjoint_Ioo, MeasureTheory.eLpNorm_condExpL2_le, Filter.comap_le_comap_iff, Metric.externalCoveringNumber_le_coveringNumber, Fin.prod_Ioc_succ, MulArchimedeanClass.mk_right_le_mk_div_iff, EMetric.diam_union, Order.index_le_height, PointedCone.isClosed_dual, CategoryTheory.Triangulated.SpectralObject.triangle_obj₁, cfc_isStrictlyPositive_iff, MonoidHom.range_le_ker_iff, IsCompact.bddBelow_image, Pi.disjoint_iff, MeasureTheory.SimpleFunc.eapprox_lt_top, Mathlib.Meta.Positivity.pow_zero_pos, convexHull_nonempty_iff, ConvexCone.coe_hull_of_convex, Ordinal.add_le_add_iff_right, BoundedLatticeHom.coe_comp_inf_hom', Prod.instSMulPosReflectLT, Finset.prod_Icc_of_even_eq_range, eq_bot_iff, RestrictedProduct.topologicalSpace_eq_iSup, Fin.finsetImage_natAdd_Ioo, four_mul_le_pow_two_add, MulOpposite.unop_le_one, AlgebraicGeometry.Scheme.Γevaluation_naturality, ENNReal.tsum_mono_subtype, leOnePart_eq_inv', ENNReal.add_lt_top, MeasureTheory.eLpNorm_sub_le_of_dist_bdd, Valuation.Compatible.vle_iff_le, Set.RightInvOn.filter_map_Iic, AddSubgroup.le_topologicalClosure, Subsemigroup.unop_le_unop_iff, MeasureTheory.condExp_bot', iSup_symmDiff_iSup_le, Mathlib.Tactic.PNatToNat.coe_le_coe, Ultrafilter.le_cofinite_or_eq_pure, SetTheory.PGame.nim_grundyValue, Iotop_subset_Ioc, AlgebraicGeometry.IsSeparated.instMorphismRestrict, Order.Ideal.sup_mem_iff, exists_nat_gt, Multiset.covBy_cons, PNat.le_one_iff, IsGalois.tfae, StarConvex.ordConnected, FormalMultilinearSeries.le_radius_of_eventually_le, Subfield.card_bot, atTop_le_cocompact, WithZeroTopology.isOpen_Iio, csInf_Ici, AlgebraicGeometry.basicOpen_eq_bot_iff, CategoryTheory.ComposableArrows.δ₀Functor_map_app, Submodule.one_le, iSupIndep_iff_supIndep_univ, RingCon.coe_top, MeasurableSpace.generateMeasurableRec_omega1, Finsupp.Lex.addRightMono, AddQuantale.instAddLeftMono, HasProdUniformlyOn.tendstoUniformlyOn, zero_lt_two, OpenAddSubgroup.toOpens_top, SimpleGraph.le_egirth, CategoryTheory.Limits.CompleteLattice.finiteColimitCocone_cocone_pt, infClosure_isClosed, AlgebraicGeometry.Scheme.IdealSheafData.ker_glueDataObjι_appTop, ENNReal.biInf_le_nhds, ZFSet.IsOrdinal.rank_lt_iff_mem, SummationFilter.LeAtTop.le_atTop, Int.fract_nonneg, MonomialOrder.degree_sub_leadingTerm_lt_degree, CategoryTheory.Sieve.uliftFunctorInclusion_top_isIso, Prod.ωSup_fst, AddSubmonoid.map_comap_le, ContinuousAlternatingMap.opNNNorm_le_iff, AlgebraicGeometry.Spec.germ_stalkMapIso_hom, Int.Ico_filter_modEq_eq, Filter.Iic_principal, NonemptyInterval.coe_sup_interval, FiniteDimensional.subalgebra_toSubmodule, hasSumUniformlyOn_iff_tendstoUniformlyOn, TopologicalSpace.Opens.mapMapIso_counitIso, HasProdUniformly.tendstoUniformly, CategoryTheory.Limits.CoproductsFromFiniteFiltered.finiteSubcoproductsCocone_ι_app_eq_sum, projIcc_eq_zero, EReal.zero_lt_top, tendsto_intCast_atTop_cobounded, AlgebraicGeometry.Scheme.Hom.appIso_inv_app_apply, Real.tendsto_toNNReal_atTop, PartOrd.forget_map, mulRothNumber_lt_of_forall_not_threeGPFree, nhds_bot_basis_Iic, Filter.nhds_atTop, FractionalIdeal.inv_le_comm, NNReal.instDenselyOrdered, LieAlgebra.maxNilpotentIdeal_le_radical, NNReal.inner_le_Lp_mul_Lq_tsum, Ordinal.toZFSet_succ, Order.krullDim_le_one_iff_forall_isMin, Multiset.zero_le, mabs_div_lt_iff, measurableSet_lt', AddSubgroup.toIntSubmodule_closure, Matrix.mem_rowStochastic, Ici_mem_nhdsSet_Ici_iff, IntermediateField.isSimpleOrder_of_finrank_prime, NNReal.instSMulPosStrictMono, Positive.coe_zpow, Int.injOn_natAbs_Ici, Finset.sup_univ_eq_iSup, SummationFilter.hasProd_symmetricIcc_iff, WithTop.continuousOn_untopA, Finsupp.degree_mono, nnnorm_prod_le, IntermediateField.LinearDisjoint.eq_bot_of_self, ENNReal.iInter_Ioi_coe_nat, Topology.IsLower.isTopologicalSpace_basis, NNReal.tendsto_tsum_compl_atTop_zero, AddSubsemigroup.monotone_map, OrderEmbedding.map_inf_le, nhds_top_basis, LinearMap.IsSymmetricProjection.le_iff_range_le_range, Hyperreal.abs_lt_real_iff_infinitesimal, Int.preimage_fract, zpow_le_zpow_iff_left, Set.image_add_const_Icc, inv_lt_self_iff, instLawfulOrderSup_mathlib, Filter.le_iff_forall_inf_principal_compl, Set.inv_Iic, RootPairing.Base.exists_eq_sum_and_forall_sum_mem_of_isPos, Set.monotone_image, MeasureTheory.Lp.simpleFunc.denseRange_coeSimpleFuncNonnegToLpNonneg, MeasureTheory.OuterMeasureClass.measure_iUnion_nat_le, SubRootedTree.bot_mem_iff, NonUnitalStarSubalgebra.prod_top, Finset.tendsto_Icc_atBot_prod_atTop, Int.Ioc_filter_modEq_eq, OrderIso.setIsotypicComponents_symm_apply, Finset.inf_top, AddSubgroup.mem_toZModSubmodule, AlgebraicGeometry.Scheme.restrictFunctor_obj_left, Int.addSubmonoid_closure_range_mul_self, ProbabilityTheory.Kernel.indep_bot_right, IsLocalRing.le_maximalIdeal_of_isPrime, MeasureTheory.exists_eLpNorm_indicator_le, HomogeneousIdeal.toIdeal_irrelevant_le, List.sortedLE_ofFn_iff, Complex.le_def, Set.image_update_Ioc, isLUB_biSup, Finset.sup_image₂_le, mul_upperClosure, Ordinal.preOmega_le_preOmega, Subfield.extendScalars.orderIso_symm_apply_coe, lt_imp_lt_of_le_imp_le, NNReal.sum_sqrt_mul_sqrt_le, OrderMonoidHom.inr_apply, HahnSeries.zero_le_orderTop_iff, OrderHom.sInf_apply, TopologicalSpace.Opens.toTopCat_map, IncidenceAlgebra.mu_prod_mu, separableClosure.separableClosure_eq_bot, ProfiniteGrp.isIso_toLimit, CStarAlgebra.span_nonneg_inter_ball, LinearOrderedAddCommGroupWithTop.top_add', Fin.preimage_cast_Ico, Metric.encard_le_of_isSeparated, Hyperreal.infinitesimal_pos_iff_infinitePos_inv, Cardinal.mk_lt_aleph0, SemilatticeSup.le_sup_right, Multiset.abs_sum_le_sum_abs, antitone_of_add_one_le, Multiset.ndinter_le_right, NNReal.mulLeftMono, ModP.preVal_add, IsCompl.codisjoint, CategoryTheory.MorphismProperty.isStableUnderColimitsOfShape_iff_colimitsOfShape_le, oneLePart_eq_ite, le_separableClosure, AlgebraicGeometry.Spec.fromSpecStalk_eq, AlgebraicGeometry.quasiCompact_iff_forall_affine, IsCompl.isAtom_iff_isCoatom, NonUnitalAlgebra.top_toSubring, subset_supClosure, skyscraperPresheafCoconeOfSpecializes_pt, CStarAlgebra.isBasis_nonneg_sections, LieSubmodule.gc_lcs_ucs, ContinuousMapZero.elemental_eq_top, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_over, Module.End.hasGenEigenvector_iff, Matroid.eRk_le_encard, Dynamics.netEntropyInfEntourage_antitone, Filter.comap_add_comap_le, Ordinal.veblenWith_eq_veblenWith_iff, Set.image_single_Ico_right, MulArchimedeanClass.orderHom_mk, SemilatSupCat_dual_comp_forget_to_partOrd, Ordinal.epsilon_pos, Fin.rev_anti, PEquiv.le_def, NonarchAddGroupSeminorm.coe_le_coe, MeasureTheory.iSup_lintegral_le, IsCompact.exists_isGreatest, Ideal.map_comap_le, edist_le_Ico_sum_edist, IntermediateField.fg_top_iff, NNReal.HolderConjugate.sub_one_pos, Order.pred_le_and_le_iff, LinearOrder.isPredArchimedean_of_isSuccArchimedean, Set.Iic_mul_Iio_subset', eq_iff_atom_le_iff, Cardinal.mk_lt_aleph0_iff, Ico_subset_biUnion_Ico, CategoryTheory.Subobject.inf_pullback, AlgebraicGeometry.Spec.locallyRingedSpaceObj_presheaf, Pi.Lex.le_sSup_apply, Subgroup.zpowers_le_of_mem, AddSubmonoid.coe_negOrderIso_apply, Set.wcovBy_insert, NonemptyInterval.coe_pure, HomologicalComplex.HomologySequence.composableArrows₃Functor_obj, hnot_anti, HolderWith.eHolderNorm_le, abs_le_one_iff_mul_self_le_one, MeasureTheory.Filtration.IsRightContinuous.RC, CategoryTheory.Precoverage.le_of_zeroHypercover, CompactlySupportedContinuousMap.monotone_of_nnreal, nullMeasurableSet_lt, AlgebraicGeometry.instLocallyOfFinitePresentationMorphismRestrict, Set.range_IicExtend, AlgebraicGeometry.PresheafedSpace.GlueData.π_ιInvApp_π, MeasureTheory.integral_Ici_eq_integral_Ioi', Finset.subset_Iic_sup_id, CategoryTheory.TransfiniteCompositionOfShape.map_F, stdSimplex.le_one, le_iInf₂_mul, SameRay.exists_nonneg_right, StarOrderedRing.isStrictlyPositive_iff_spectrum_pos, antivary_comm, Ordinal.CNF_fst_le_log, EReal.neg_le, HahnSeries.leadingCoeff_neg_iff, TopCat.Presheaf.stalkFunctor_map_germ, CategoryTheory.SimplicialThickening.Path.le, ENat.tendsto_nhds_top_iff_natCast_lt, CategoryTheory.ComposableArrows.sc'Map_τ₂, CategoryTheory.ComposableArrows.mkOfObjOfMapSucc_exists, ValuationSubring.le_ofPrime, monovary_neg_right, upperPolar_anti, Subgroup.closure_pow_le, MeasureTheory.le_addContent_diff, CategoryTheory.ComposableArrows.isoMk₀_hom_app, AlgebraicGeometry.Scheme.IdealSheafData.ideal_biInf, FirstOrder.Language.LHom.mem_substructureReduct, Ideal.matrix_monotone, Finset.le_iff_subset, CharP.quotient_iff_le_ker_natCast, CategoryTheory.Subgroupoid.map_le_iff_le_comap, covBy_iff_wcovBy_and_ne, Subsemigroup.closure_mono, CategoryTheory.GrothendieckTopology.top_covers, Filter.limsup_eq, Ordinal.lift_omega, CategoryTheory.Presieve.factorsThru_top, Submonoid.disjoint_def', NonUnitalAlgebra.mem_bot, MeasureTheory.measure_setLAverage_le_pos, Order.IsNormal.strictMono, iUnion_Ioc_intCast, MeasureTheory.hasFiniteIntegral_iff_norm, Fin.preimage_addNat_Ico_addNat, ContinuousMap.compactOpen_le_induced, inv_lt_one_iff₀, le_of_forall_gt_imp_ne, IntermediateField.toSubalgebra_le_toSubalgebra, AlgebraicGeometry.LocallyRingedSpace.Γevaluation_naturality, convexOn_iff_div, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_basicOpen', not_isMax_bot, pairwise_disjoint_nhds, Multiset.singleton_le, AlgebraicGeometry.Scheme.basicOpen_restrict, CompleteLattice.le_sSup, Ordinal.preOmega_omega0, CategoryTheory.ComposableArrows.functorArrows_map, left_eq_inf, ge_antisymm_iff, Nonneg.coe_zpow, CategoryTheory.Subobject.inf_comp_right_assoc, compl_lt, induced_top, AlgebraicGeometry.structurePresheafInCommRingCat_obj_carrier, CategoryTheory.nerveMap_app, TopCat.Presheaf.germToPullbackStalk_stalkPullbackHom_assoc, AlgebraicGeometry.Scheme.IdealSheafData.support_bot, Ordinal.card_lt_ofNat, MonomialOrder.degree_monomial_le, closedAbsConvexHull_isClosed, CategoryTheory.MorphismProperty.le_isoClosure, AntitoneOn.Ioc, HasCardinalLT.Set.instIsFilteredOfFactIsRegular, ENNReal.le_limsup_mul, SupPrime.le_finset_sup, CategoryTheory.SimplicialThickening.functor_id, AlgebraicGeometry.Scheme.zariskiPrecoverage_le_qcPrecoverage, AlgebraicGeometry.Scheme.Hom.opensRange_of_isIso, SimpleGraph.edist_le, AlgebraicGeometry.IsAffineOpen.app_basicOpen_eq_away_map, Subsemiring.le_centralizer_centralizer, AlgebraicGeometry.LocallyRingedSpace.toΓSpecMapBasicOpen_eq, mul_lowerClosure, ProjectiveSpectrum.basicOpen_mul_le_right, TensorProduct.nndist_tmul_le, IsFractionRing.self_iff_nonZeroDivisors_le_isUnit, Submodule.lTensorOne_one_tmul, le_iSup_iff, CategoryTheory.Subobject.lowerEquivalence_inverse, Units.zero_lt, MeasureTheory.measure_isOpen_pos_of_smulInvariant_of_ne_zero, Quantale.instMulLeftMono, Set.range_injOn_strictAnti, OrderMonoidWithZeroHom.comp_mul, Polynomial.support_scaleRoots_le, PointedCone.mem_positive, Ordinal.iSup_le_iff, Set.image_const_div_Iic, Cardinal.aleph0_mul_aleph, Topology.IsLower.scottHausdorff_le, InfiniteGalois.limitToAlgEquiv_symm_apply, AffineSubspace.setOf_sOppSide_eq_image2, AddValuation.map_add, lipschitzExtensionConstant_pos, Module.End.mem_genEigenspace_zero, IsClosed.mk_lt_two_pow_mk_dense, Finset.Ioc_disjoint_Ioc, Set.image_update_Ioo_left, monovary_inv_right, Ordinal.preOmega_lt_preOmega, Valuation.ltIdeal_mono, Ordinal.left_le_veblen, sSup_inter_le, CategoryTheory.Subobject.lowerEquivalence_functor, Multiset.Nodup.le_dedup_iff_le, Cardinal.lift_lt_aleph1, Valuation.IsEquiv.lt_one_iff_lt_one, Submodule.restrictScalarsEmbedding_apply, IntermediateField.relfinrank_top_left, Set.encard_iUnion_le_of_fintype, MvPowerSeries.lexOrder_le_of_coeff_ne_zero, HahnSeries.support_embDomain_subset, ProbabilityTheory.gaussianPDFReal_mul, Subsemiring.pointwise_smul_le_iff₀, Subgroup.commutator_comm_le, CompactlySupportedContinuousMap.instMulRightMono, Fin.image_addNat_Ioc, Cardinal.ord_card_le, Order.Ideal.finsetSup_mem_iff, Polynomial.contentIdeal_mul_le_mul_contentIdeal, isOrderRightAdjoint_sSup, AlgebraicGeometry.StructureSheaf.instAwayObjOppositeOpensCarrierTopValStructureSheafInTypeOpBasicOpen, le_biSup, Polynomial.degreeLE_mono, Monoid.le_minOrder, maximal_le_iff, AlgebraicGeometry.Scheme.preimage_basicOpen, CategoryTheory.ComposableArrows.IsComplex.cokerToKer'_fac_assoc, Finset.disjSum_strictMono_right, CompleteLinearOrder.sdiff_le_iff, finsetInf'_mem_infClosure, iSupIndep.map_orderIso, CategoryTheory.ObjectProperty.galoisConnection_isLocal, ultrafilter_comap_pure_nhds, MeasureTheory.exists_measurable_le_setLIntegral_eq_of_integrable, Ordinal.iSup_eq_lsub, OrderEmbedding.ofAntisymmetrization_apply, csInf_le_csInf', Ordinal.not_bddAbove_compl_of_small, PNat.coe_lt_coe, abs_sub_pos, Finset.le_inf_const_le, iSup_of_empty, Set.one_le_encard_iff_nonempty, AlgebraicGeometry.Scheme.Opens.instIsIsoCommRingCatAppLEιTopToScheme, Set.Ioc.coe_mul, mem_convexHull_iff, Int.sign_eq_sign, disjoint_nhds_atTop_iff, Dynamics.dynEntourage_antitone, Ideal.height_le_spanRank_toENat, Archimedean.embedReal_one, Measurable.le', BoundedLatticeHom.coe_comp_inf_hom, NNReal.sqrt_mul_le_half_add, CategoryTheory.Subfunctor.Subpresheaf.range_subobjectMk_ι, Multiset.le_cons_of_notMem, Submodule.le_prod_iff, CategoryTheory.ComposableArrows.fourδ₁Toδ₀_app_one, borel_eq_generateFrom_Iic, Set.image_const_div_Ico, Cardinal.mk_biUnion_le_of_le, sInf_eq_top, Real.map_toNNReal_atTop, AlgebraicGeometry.IsAffineHom.isAffine_preimage, addRothNumber_map_add_right, comap_nonZeroDivisors_le_of_injective, Ordinal.veblen_lt_veblen_iff_right, ENNReal.limsup_add_le, MonomialOrder.degLex_single_lt_iff, ProperCone.innerDual_iUnion, List.sortedLT_ofFn_iff, iInf_neg, WithBot.Ioc_coe_coe, instIsPartialOrder_mathlib, Set.image_single_Ioc_right, Order.rev_index_le_coheight, IsOrderedAddMonoid.to_hasUpperLowerClosure, TopCat.Presheaf.toPushforwardOfIso_app, CategoryTheory.OrthogonalReflection.iteration_map_succ_surjectivity, sbtw_zero_one_iff, AlgebraicGeometry.IsAffineOpen.ΓSpecIso_hom_fromSpec_app, star_le_star_iff, Set.unbounded_ge_iff, List.zipWith_swap_prod_support, CategoryTheory.sheafBotEquivalence_counitIso, MeasureTheory.Measure.MutuallySingular.disjoint, CategoryTheory.Triangulated.SpectralObject.ω₂_obj_mor₁, strictConvexOn_iff_div, AntitoneOn.Ico, Complex.monotone_ofReal, GroupTopology.toTopologicalSpace_top, IncidenceAlgebra.eulerChar_orderDual, nhdsLE_sup_nhdsGE, mabs_mabs_div_mabs_le, NNRat.bddAbove_coe, Finsupp.toLex_monotone, CategoryTheory.Limits.WalkingMultispan.inclusionOfLinearOrder_map, AlgebraicGeometry.Scheme.exists_le_and_germ_injective, Set.sub_mem_Icc_zero_iff_right, Set.Ico.coe_lt_one, Fin.prod_Icc_cast, PartOrd.ofHom_apply, Fin.val_top, isPRadical_iff, EMetric.cauchySeq_iff', Matroid.IsMinor.le, PointedCone.mem_maxTensorProduct, Ordinal.nmul_le_iff, Fin.map_castAddEmb_Iic, Multiset.map_add_left_Icc, MeasureTheory.lintegral_liminf_le', Interval.bot_pow, MeasureTheory.aecover_Ico_of_Icc, iInf_lt_top, Con.toSetoid_eq_top, IsCompact.disjoint_nhdsSet_right, ENNReal.strictMono_rpow_of_pos, MvPolynomial.homogeneousSubmodule_mul, Fin.finsetImage_cast_Ioc, ENNReal.add_rpow_le_two_rpow_mul_rpow_add_rpow, Finset.isGLB_inf', AddSubmonoid.monotone_comap, Finset.card_eraseNone_eq_card_erase, List.getElem_le_maximum_of_length_pos, Cardinal.toNat_pos, tendsto_inv_nhdsLT, OrderIso.toCompleteLatticeHom_toFun, Fintype.prod_strictMono', concaveOn_iff_convex_hypograph, CFC.monotoneOn_cfcₙ_rpowIntegrand₀₁, MeasureTheory.OuterMeasure.dirac_caratheodory, rank_lt_rank_dual, PosSMulReflectLE.toPosSMulReflectLT, LowerAdjoint.closure_sup_closure_le, Int.padicValuation_le_one, EReal.coe_ennreal_nonneg, Module.fgSystem.instDirectedSystemSubtypeSubmoduleFGMemValCoeLinearMapId, Cardinal.lift_strictMono, inf_eq_right, Fin.map_valEmbedding_Iio, CategoryTheory.HasLiftingProperty.transfiniteComposition.SqStruct.w₁, MeasureTheory.Measure.ae_pi_le_pi, CategoryTheory.MorphismProperty.pullbacks_monotone, instIsStronglyCoatomicOfWellFoundedGT, AlgebraicGeometry.Scheme.restrictFunctor_map_left, Set.preimage_const_mul_Ioo, AddSubgroup.addSubgroupOf_eq_bot, Set.Ici_diff_left, IntermediateField.finSepDegree_top, EReal.coe_nnreal_lt_top, disjoint_self, CategoryTheory.IsGrothendieckAbelian.generatingMonomorphisms.largerSubobject_top, SSet.N.iSup_subcomplex_eq_top, Real.lt_toNNReal_iff_coe_lt, MeasureTheory.aecover_Ico_of_Ico, LieAlgebra.InvariantForm.atomistic, AbsoluteValue.listSum_le, Nucleus.top_apply, Valuation.isEquiv_iff_val_lt_one, AlgebraicGeometry.Scheme.affineOpenCover_X, IsOpen.affineSpan_eq_top, AlgebraicGeometry.Scheme.comp_app_assoc, Ordinal.lt_bsup, TopCat.Presheaf.generateEquivalenceOpensLe_functor'_map, Cardinal.natCast_eq_toENat_iff, le_max_iff, parallelepiped_eq_convexHull, LieSubmodule.map_le_range, CategoryTheory.Sieve.id_mem_iff_eq_top, Metric.uniformity_edist_aux, isIntegral_quotientMap_iff, Sylow.exists_subgroup_le_card_le, IsLocalFrameOn.generating, Fin.strictAnti_iff_succ_lt, PosMulReflectLT.toPosMulStrictMono, cauchy_davenport_minOrder_mul, nsmul_pos_iff, Set.not_equitableOn, Ordinal.iSup_eq_lsub_iff, Preorder.instHasBinaryProducts, BooleanAlgebra.bot_le, WellFoundedGT.toIsSuccArchimedean, Fin.sum_Ico_castLE, Nonneg.val_inv_unitsEquivPos_symm_apply_coe, LieModule.map_posFittingComp_le, AffineSubspace.not_wOppSide_bot, SequentiallyComplete.le_nhds_of_seq_tendsto_nhds, Valuation.IsUniformizer.val_lt_one, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.app_invApp_assoc, AlgebraicGeometry.instQuasiCompactToSpecΓOfCompactSpaceCarrierCarrierCommRingCat, Filter.HasBasis.disjoint_cobounded_iff, CompositionSeries.le_last_of_mem, ENat.eq_top_iff_forall_ge, Representation.invtSubmodule.coe_bot, MultilinearMap.map_sub_map_piecewise, SkyscraperPresheafFunctor.map'_app, Cardinal.ofENat_le_nat, Filter.mono_blimsup', IsAdjoinRoot.primitive_element_root, CategoryTheory.Subobject.functor_map, AlgebraicGeometry.LocallyRingedSpace.evaluation_naturality_assoc, OrdinalApprox.le_lfpApprox, Finsupp.Colex.single_lt_iff, AddMonoidAlgebra.supDegree_withBot_some_comp, Fin.prod_Iic_castAdd, Filter.map_comap_le, ContinuousAt.lowerSemicontinuousAt, AlgebraicGeometry.morphismRestrict_app, nhdsKer_mono, exists_le_of_linear, Monoid.Coprod.codisjoint_range_inl_range_inr, CategoryTheory.SmallObject.iterationFunctorMapSuccAppArrowIso_hom_right_right_comp, Nonneg.instMulArchimedean, strictAnti_vecCons, IsNonarchimedean.finset_powerset_image_add, Fin.Ioi_zero_eq_map, Fin.map_finCongr_Ioc, CategoryTheory.SimplicialObject.augmentedCechNerve_obj_left_map, ENNReal.coe_toNNReal_le_self, AlgebraicGeometry.AffineSpace.comp_homOfVector_assoc, ringHom_monotone, SMulPosReflectLE.toSMulPosStrictMono, Mathlib.Tactic.PNatToNat.coe_lt_coe, integrableOn_Ioc_iff_integrableOn_Ioo, monovaryOn_iff_exists_antitoneOn, MeasureTheory.FiniteMeasure.testAgainstNN_lipschitz_estimate, HolderOnWith.ediam_image_le, MonomialOrder.degree_sub_LTerm_lt, CategoryTheory.Retract.projectiveDimension_le, latticeClosure_isClosed, UniformOnFun.edist_le, Set.instPreservesColimitsOfShapeFunctorToTypesOfIsFilteredOrEmpty, IsLocalization.orderIsoOfPrime_apply_coe, NNRat.cast_lt_natCast, linOrd_dual_comp_forget_to_Lat, instIsStrictTotalOrderGt, PresheafOfModules.germ_ringCat_smul, SetTheory.PGame.grundyValue_le_of_forall_moveRight, Ideal.IsHomogeneous.radical_eq, mulPosReflectLT_iff_contravariant_pos, IntermediateField.normalClosure_le_iff_of_normal, Finset.lt_iff_ssubset, PointedCone.dual_flip_dual_dual_flip, abs_min_sub_min_le_max, countable_setOf_isolated_left, Finset.pluennecke_ruzsa_inequality_nsmul_sub_nsmul_add, AlgebraicGeometry.Scheme.PartialMap.le_domain_toRationalMap, Hyperreal.coe_le_coe, MulArchimedeanClass.min_le_mk_div, CategoryTheory.TransfiniteCompositionOfShape.ofComposableArrows_isColimit, Multiset.le_smul_dedup, AugmentedSimplexCategory.inr'_eval, Fin.map_natAddEmb_Ioo, Ordinal.toPGame_le_iff, LocallyFiniteOrder.orderMonoidWithZeroHom_strictMono, sSupHom.apply_blimsup_le, Metric.measure_ball_pos, AddSubgroup.toAddSubmonoid_le, OrderAddMonoidHom.fst_apply, star_mul_self_pos, Nimber.toOrdinal_zero, Affine.Simplex.interior_eq_image_Ioo, monovaryOn_iff_mul_rearrangement, ENat.mul_le_mul_left_iff, AffineSubspace.setOf_wSameSide_eq_image2, Order.le_le_succ_iff, ultrafilter_extend_eq_iff, ZFSet.lt_def, Equiv.Perm.mem_cycleFactorsFinset_support_le, strictMono_inf_prod_sup, BooleanSubalgebra.gc_map_comap, NatOrdinal.toOrdinal_zero, Polynomial.contentIdeal_le_contentIdeal_of_dvd, Filter.disjoint_atBot_principal_Ici, AlgebraicGeometry.Scheme.Modules.instIsRightAdjointPresheafOfModulesToPresheafOfModules, AlgebraicGeometry.Scheme.Opens.ι_image_top, Filter.coclosedCompact_le_cofinite, edist_vsub_vsub_le, AlgebraicGeometry.Scheme.Hom.toImage_app, Fin.image_succ_Ioo, BoundedLatticeHomClass.toInfTopHomClass, IntermediateField.Lifts.eq_iff, AddGroupCone.coe_nonneg, WithTop.some_mem_insertTop, HasSumUniformlyOn.tendstoUniformlyOn, mul_nonneg_of_three, AlgebraicGeometry.Scheme.Modules.restrictFunctorId_hom_app_app, TopCat.Sheaf.eq_of_locally_eq_iff, CategoryTheory.MorphismProperty.universally_mono, CompleteLattice.isCompactElement_iff_le_of_directed_sSup_le, CategoryTheory.MorphismProperty.presheaf_monomorphisms_le_monomorphisms, SemigroupIdeal.closure_mono, CategoryTheory.MorphismProperty.pushouts_le_iff, PartENat.lt_coe_succ_iff_le, ProperCone.mem_toPointedCone, Module.End.genEigenspace_le_genEigenspace_finrank, MonomialOrder.sPolynomial_lt_of_degree_ne_zero_of_degree_eq, AddAction.le_stabilizer_iff_vadd_le, Fin.sum_Ioi_zero, mabs_div_mabs_le_mabs_div, hnot_le_comm, min_mabs_mabs_le_mabs_min, MeasureTheory.Measure.addHaarScalarFactor_pos_of_isAddHaarMeasure, infIccOrderIsoIccSup_symm_apply_coe, imageToKernel_epi_of_epi_of_zero, mul_sub_mul_div_mul_neg_iff, Ring.le_comap_jacobson, AddSubgroup.comap_sup_comap_le, AntilipschitzWith.le_mul_ediam_image, Quantale.instMulRightMono, Cardinal.one_le_iff_pos, Ideal.span_prod_le, CategoryTheory.Limits.MultispanShape.ofLinearOrder_snd, Order.Iic_succ, isCompl_top_bot, Metric.disjoint_nhdsSet_cobounded, AlgebraicGeometry.Scheme.AffineZariskiSite.instIsLocallyFullOpensToOpensFunctorGrothendieckTopologyCarrierCarrierCommRingCat, ENNReal.rpow_lt_top_of_nonneg, PartOrd.hom_comp, intervalIntegral.FTCFilter.pure_le, MeasureTheory.Measure.trim_le, Subfield.extendScalars_self, PseudoMetric.finsetSup_apply, TopologicalGroup.isOpenMap_iff_nhds_zero, add_self_neg_iff, Subgroup.le_op_iff, Dense.exists_seq_strictMono_tendsto, Fin.Icc_add_one_eq_Ioc, AlgebraicGeometry.IsAffineOpen.isoSpec_hom, Equiv.Perm.monotone_iff, EReal.coe_ennreal_pos_iff_ne_zero, Module.End.mem_genEigenspace, Field.Emb.Cardinal.adjoin_basis_eq_top, ENNReal.image_coe_Ico, ENNReal.iInf_gt_eq_self, Fin.image_castSucc_Iic, Nat.ofNat_nonneg', ENNReal.ofReal_lt_one, iSup₂_le_iSup, AddSubmonoid.neg_le_neg, Heyting.IsRegular.disjoint_compl_left_iff, NonUnitalAlgebra.top_toSubmodule, Valuation.mem_integer_iff, Set.covBy_iff_exists_insert, le_nhds_of_unique_clusterPt, ZFSet.rank_singleton, NumberField.mixedEmbedding.one_le_convexBodyLT'Factor, PointedCone.dual_sUnion, ClosureOperator.eq_ofPred_closed, ENNReal.inv_lt_top, Ordinal.add_le_right_iff_mul_omega0_le, AlgebraicGeometry.Scheme.Modules.Hom.app_smul, not_isBot_iff_ne_bot, Fin.map_addNatEmb_Ioo, CategoryTheory.IsGrothendieckAbelian.generatingMonomorphisms.pushouts_ofLE_le_largerSubobject, iSup_eq_bot, Filter.le_iff_ultrafilter, Ideal.dvdNotUnit_iff_lt, convex_halfSpace_lt, FirstOrder.Language.DirectLimit.Equiv_isup_symm_inclusion_apply, ProperCone.innerDual_toSubmodule, Ordinal.iterate_omega0_opow_lt_epsilon_zero, Algebra.TensorProduct.adjoin_tmul_eq_top, Seminorm.coe_lt_coe, ENNReal.eventually_le_limsup, FinBddDistLat.dual_map, Subgroup.normalCore_le, Finset.image_add_left_Ico, DFinsupp.toMultiset_lt_toMultiset, AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace_app_spec, lowerClosure_add_distrib, posMulMono_iff_posMulStrictMono, MeasurableSet.exists_isOpen_symmDiff_lt, Set.Ioo_add_bij, Finset.maximal_iff_forall_insert, AlgebraicGeometry.Scheme.Γ_obj_op, Subalgebra.pointwise_smul_toSubmodule, Metric.cauchySeq_iff', LieSubalgebra.le_def, Set.preimage_sub_const_Iic, instZeroLEOneClassEReal, Subgroup.unop_le_unop_iff, CategoryTheory.ComposableArrows.naturality'_assoc, Set.powersetCard.mem_range_ofFinEmbEquiv_symm_iff_mem, nsmul_nonpos_iff, Fin.finsetImage_cast_Ico, ENNReal.inv_le_one, MeasureTheory.Filtration.rightCont_eq_of_neBot_nhdsGT, lt_or_lt_iff_ne, Finsupp.DegLex.monotone_degree, BddOrd.coe_comp, Cardinal.isSuccLimit_omega, Finset.card_uIcc_prod, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.inv_invApp, HahnSeries.orderTop_smul_not_lt, le_emultiplicity_iff_replicate_subperm_primeFactorsList, MeasureTheory.OuterMeasure.le_boundedBy, Matroid.one_le_eRank, Fin.preimage_addNat_Ioc_addNat, ArchimedeanClass.mk_antitoneOn, ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPoints, Finset.Ico_mul_Ioc_subset', mem_eball_zero_iff, BiheytingAlgebra.top_sdiff, ENNReal.mul_lt_mul_iff_right, SimpleGraph.componentComplFunctor_map, nontrivial_iff_lt, Nat.cast_lt, partialSups_add_one_eq_sup_disjointed, isOrderBornology_iff_eq_orderBornology, TopCat.Presheaf.IsSheaf.isSheafOpensLeCover, Int.succ_eq_succ, MeasureTheory.SimpleFunc.edist_nearestPt_le, Mathlib.Meta.NormNum.isNat_ordinalLE_false, Filter.hyperfilter_le_cofinite, Submodule.instMulRightMono, Set.preimage_add_const_Iio, MulEquiv.symm_comapSubgroup, CategoryTheory.MorphismProperty.isStableUnderBaseChange_iff_pullbacks_le, Cardinal.ord_le_omega0, Ordinal.self_le_succ_pred, OnePoint.le_nhds_infty, Set.inv_Iio, Subring.closure_le, groupHomology.boundaries₁_le_cycles₁, LipschitzWith.edist_lt_top, Subsemigroup.centralizer_le, UpperSet.coe_subset_coe, not_monotone_not_antitone_iff_exists_lt_lt, Rat.toNNRat_lt_toNNRat_iff, Set.preimage_sub_const_Icc, Cardinal.aleph_one_lt_lift, LieSubmodule.lcs_le_self, le_max_left, coverPreserving_opens_map, isPiSystem_Iic, csInf_univ, TopCat.Presheaf.IsSheaf.isSheafUniqueGluing, QuotientAddGroup.con_le_iff, ENNReal.image_coe_Icc, MeasureTheory.Measure.pi_Iio_ae_eq_pi_Iic, Int.instIsOrderBornology, MeasureTheory.VectorMeasure.le_restrict_univ_iff_le, CategoryTheory.Triangulated.SpectralObject.ω₂_map_hom₃, Order.not_isSuccPrelimit, disjoint_nhdsSet_nhds, Affine.Simplex.face_points', Ideal.height_le_card_of_mem_minimalPrimes_span_finset, IntermediateField.relrank_bot_left, Set.image_mulSingle_Ioc, nhdsGT_sup_nhdsWithin_singleton, CategoryTheory.Subobject.finset_inf_arrow_factors, ENNReal.young_inequality, SSet.Subcomplex.topIso_hom, Ordinal.typein_inj, MeasureTheory.MeasurePreserving.measure_symmDiff_preimage_iterate_le, absConvexHull_eq_empty, BoxIntegral.Prepartition.eventually_splitMany_inf_eq_filter, Set.image_neg_Iic, CategoryTheory.ComposableArrows.instIsIsoOfNatNatTwoδ₂Toδ₁, Fin.finsetImage_castAdd_Ioo, WithTop.Icc_coe_top, CompleteBooleanAlgebra.inf_compl_le_bot, Subalgebra.LinearDisjoint.eq_bot_of_commute_of_self, AlgebraicGeometry.Scheme.image_zeroLocus, SimpleGraph.Subgraph.map_le_iff_le_comap, tendsto_fract_right, HahnSeries.isWF_support, BumpCovering.le_one', iUnion_Iic_eq_Iic_iSup, LieHom.le_ker_iff, Cardinal.toENat_nat, tendsto_floor_left_pure_sub_one, Ideal.Quotient.factor_eq, MvPowerSeries.le_weightedOrder_subst_of_forall_ne_zero, Hyperreal.coe_lt_coe, LinearOrderedCommGroup.Subgroup.exists_generator_lt_one, Cardinal.IsRegular.pos, CategoryTheory.Sieve.pullback_ofObjects_eq_top, VitaliFamily.eventually_measure_lt_top, OrderMonoidHom.fst_comp_inl, HahnEmbedding.Partial.exists_domain_eq_top, Metric.coveringNumber_le_encard_self, CategoryTheory.isFilteredOrEmpty_of_semilatticeSup, Finset.le_sup', CategoryTheory.Sieve.pushforward_monotone, iSup₂_mul_le, AlgebraicGeometry.Scheme.basicOpen_eq_bot_iff_forall_evaluation_eq_zero, NatOrdinal.small_Icc, LinearMap.ker_le_range_iff, AlgebraicGeometry.HasRingHomProperty.iff_of_source_openCover, cmp_add_right, IsOpenMap.coe_functor_obj, Topology.RelCWComplex.skeletonLT_monotone, div_lt_one_iff, AlgebraicGeometry.Scheme.Hom.isoImage_hom_ι_assoc, FiniteMulArchimedeanClass.subgroup_strictAnti, Set.add_mem_Ioc_iff_left, AlgebraicGeometry.LocallyRingedSpace.toΓSpecCBasicOpens_app, OrderIso.withTopCongr_symm_apply, MeasureTheory.SimpleFunc.approx_apply, Ordinal.veblenWith_of_ne_zero, AddSubmonoid.FG.finset_sup, PowerSeries.nat_le_order, AlgebraicGeometry.LocallyRingedSpace.evaluation_eq_zero_iff_notMem_basicOpen, StrictOrderedSemiring.toPosSMulStrictMonoNat, LinearOrderedCommGroupWithZero.discrete_iff_not_denselyOrdered, Fin.image_addNat_Ico, isSMulRegular_on_quot_iff_lsmul_comap_le, FinBoolAlg.hasForgetToFinPartOrd_forget₂_obj_str, abs_abs_sub_abs_le, LieAlgebra.IsSemisimple.sSupIndep_isAtom, MeasureTheory.Measure.haar.haarContent_outerMeasure_closure_pos, SetTheory.PGame.moveRight_nim_heq, LinearMap.range_le_ker_iff, Filter.inv_le_self, RealRMK.le_rieszMeasure_tsupport_subset, monotone_of_sub_one_le, FinPartOrd.hom_comp, Nimber.le_zero, Subalgebra.finrank_bot, AlgebraicGeometry.Proj.res_apply, AnalyticAt.map_nhdsNE, zpow_left_monoOn₀, Multiset.le_iff_exists_add, CategoryTheory.CostructuredArrow.projectQuotient_factors, AddGroupNorm.coe_lt_coe, CompactlySupportedContinuousMap.instMulLeftMono, IdealFilter.isTorsionQuot_def, Order.pred_lt_iff_eq_or_gt, AlgebraicGeometry.Scheme.IdealSheafData.radical_top, MulEquiv.symm_mapSubgroup, LinearOrderedCommGroup.discrete_or_denselyOrdered, Set.preimage_mul_const_Ioo, Set.Ioc.le_one, TopCat.Presheaf.app_injective_iff_stalkFunctor_map_injective, AlgebraicGeometry.StructureSheaf.comapₗ_eq_localRingHom, AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.preimage_basicOpen, strictMono_iff_map_neg, SubRootedTree.mem_iff, Ordinal.sub_ne_zero_iff_lt, Finset.dens_union_le, MonotoneOn.Ioi, IsCompact.isGreatest_sSup, FirstOrder.Language.Substructure.cg_iff_empty_or_exists_nat_generating_family, ENNReal.le_inv_iff_mul_le, Polynomial.roots_countP_pos_le_signVariations, UpperSet.Ici_inj, AlgebraicGeometry.Scheme.Opens.topIso_hom, Valued.isClopen_closedBall, ValuationSubring.subset_pointwise_smul_iff, MeasureTheory.SimpleFunc.finMeasSupp_iff, Dynamics.coverEntropyInfEntourage_nonneg, FractionalIdeal.instCanonicallyOrderedAdd, DividedPowers.dpow_span_isSubideal, Set.pairwise_disjoint_Ico_add_zsmul, MeasureTheory.le_measure_diff, PNat.map_subtype_embedding_Ioc, Fin.finsetImage_natAdd_Ici, Set.OrdConnected.image_coe_nnreal_ennreal, MeasureTheory.SimpleFunc.finset_sup_apply, IsPRadical.ker_le', CategoryTheory.Limits.kernelSubobject_arrow_comp, SimplexCategory.toCat_map, Dense.Ioi_eq_biUnion, Finset.strictMono_sym2, fixedPoints_antitone, Part.fix_eq_ωSup_of_ωScottContinuous, upperSemicontinuousOn_iff_isClosed_hypograph, Nimber.toOrdinal_toNimber, Ideal.height_le_ringKrullDim_of_ne_top, ProbabilityTheory.Kernel.comp_apply_univ_le, MeasureTheory.OuterMeasure.top_apply, Nat.one_le_cast_iff_ne_zero, Order.Ideal.coe_top, AddQuantale.rightAddResiduation_le_iff_add_le, TopologicalSpace.Opens.overEquivalence_counitIso_hom_app, Concept.swapEquiv_symm_apply, CategoryTheory.ComposableArrows.opEquivalence_counitIso_hom_app_app, Subalgebra.prod_toSubmodule, CategoryTheory.Limits.CompleteLattice.colimitCocone_isColimit_desc, Submonoid.disjoint_def, Submodule.smul_top_le_comap_smul_top, AlgebraicGeometry.StructureSheaf.toOpen_res, Ordinal.le_one_iff, Int.cast_pos, Filter.sInter_nhds, AlgebraicGeometry.PresheafedSpace.restrictStalkIso_hom_eq_germ_assoc, Subgroup.disjoint_iff_mul_eq_one, IsStrictOrderedRing.toZeroLEOneClass, posPart_eq_zero, Nimber.small_Iio, PosMulMono.toPosMulStrictMono, Multiset.sup_mono, mem_exposedPoints_iff_exposed_singleton, bot_eq_zero, CategoryTheory.presheafIsGeneratedBy_of_isFinite, Module.End.invtSubmodule.mk_eq_bot_iff, Subalgebra.le_op_iff, abs_eq_self, Filter.IsApproximateUnit.iff_neBot_and_le_nhds_one, ZMod.fieldRange_castHom_eq_bot, CategoryTheory.Limits.CompleteLattice.finiteColimitCocone_isColimit_desc, tendsto_intCast_atTop_atTop, AlgebraicGeometry.Scheme.ofRestrict_app, RingHom.field_closure_preimage_le, AlgebraicGeometry.structurePresheafInModuleCat_obj_carrier, NonemptyFinLinOrd.mono_iff_injective, Subgroup.IsSubnormal.lt_normal, lt_neg_self_iff, AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace_app_spec_assoc, AlgebraicGeometry.Scheme.IsLocallyDirected.homOfLE_tAux_assoc, nhdsLT_sup_nhdsGT, AlgebraicGeometry.SheafedSpace.congr_app, OrderBot.lowerBounds_univ, IsSimpleOrder.bot_lt_iff_eq_top, AlgebraicGeometry.Scheme.OpenCover.restrict_X, exists_lt_of_directed_ge, Rat.preimage_cast_Ioc, NonUnitalStarSubalgebra.map_topologicalClosure_le, Subalgebra.restrictScalars_top, iUnion_Iio_eq_univ_iff, PresheafOfModules.germ_smul, isLUB_supClosure, Ordinal.natCast_lt_epsilon, Submodule.smul_le_span, WithTop.untop₀_le_untop₀_iff, AlgebraicGeometry.Scheme.IdealSheafData.mul_bot, QuotientGroup.leftRel_eq_top, LinearOrderedCommGroupWithZero.inl_apply, Set.bounded_lt_inter_le, ExpGrowth.le_expGrowthSup_mul, AlgebraicGeometry.Scheme.zeroLocus_univ, nhdsLT_neBot, IntermediateField.le_normalClosure, ZLattice.sum_piFinset_Icc_rpow_le, ArchimedeanClass.mk_sub_lt_mk_left_iff, WithTop.continuous_untop, AlgebraicGeometry.LocallyRingedSpace.toΓSpecCApp_spec, PrimeSpectrum.basicOpen_one, CStarAlgebra.hasBasis_approximateUnit, AddSubmonoid.opEquiv_symm_apply, CategoryTheory.SmallObject.preservesColimit, Decidable.ne_iff_gt_iff_ge, Filter.tendsto_abs_atTop_atTop, iUnion_Ici_eq_Ici_iInf, Multiset.replicate_mono, NonarchAddGroupNorm.coe_lt_coe, Fin.finsetImage_rev_Ioo, Valuation.leAddSubgroup_monotone, InitialSeg.ordinal_type_le, MeasureTheory.Ioi_ae_eq_Ici', TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquiv_functor, Submonoid.surjOn_iff_le_map, Cardinal.isInaccesible_def, Matrix.nonneg_of_mem_rowStochastic, Convex.mem_toCone', ValuationSubring.le_total_ideal, Cardinal.toNat_ne_zero, sbtw_one_zero_iff, AddSubgroup.le_pointwise_smul_iff₀, LinearOrderedCommGroupWithZero.inr_apply, AlgebraicGeometry.Scheme.Hom.image_le_image_iff, NonUnitalStarAlgebra.adjoin_le_starAlgebra_adjoin, Fin.sum_Ioc_castSucc, Finsupp.Lex.single_le_iff, Set.subsingleton_isBot, isPiSystem_Ioo_mem, Concept.extent_ssubset_extent_iff, Orientation.oangle_sign_smul_add_smul_left, MeasureTheory.measure_le_measure_union_right, Ideal.toIdeal_homogeneousCore_le, Finset.le_inf_sub, ProbabilityTheory.Kernel.HasSubgaussianMGF.measure_univ_le_one, Subsemigroup.comap_strictMono_of_surjective, bihimp_hnot_self, SimpleGraph.IsContained.vertexCoverNum_le_vertexCoverNum, IncidenceAlgebra.mu_toDual, Multiset.replicate_le_coe, Filter.isOpen_iff, CategoryTheory.Sieve.generate_le_iff, Dense.exists_gt, CategoryTheory.ComposableArrows.functorArrows_obj, Fin.orderSucc_eq, Ideal.IsPrime.radical_le_iff, Ordinal.instAddRightReflectLT, ENNReal.inner_le_Lp_mul_Lq, AlgebraicGeometry.Scheme.Hom.comp_appTop_assoc, convexOn_zpow, SpectrumRestricts.lt_nnreal_iff, MeasureTheory.exists_upperSemicontinuous_lt_integral_gt, Finset.fold_inf_univ, IsRightCancelMul.mulRightReflectLE_of_mulRightReflectLT, gc_lowerClosure_coe, AffineSubspace.eq_bot_or_nonempty, Finpartition.parts_extendOfLE_of_eq, Ideal.le_comap_mul, MeasureTheory.Measure.top_add, AlgebraicGeometry.AffineSpace.map_appTop_coord, LowerSet.Iic_inf, Module.End.genEigenspace_zero, top_symmDiff', not_linearIndepOn_finset_iffₒₛ, Finset.pow_right_strictMono, Cardinal.mk_iUnion_Ordinal_lift_le_of_le, Filter.disjoint_pure_atTop, Set.sub_mem_Ioc_iff_right, Polynomial.le_trailingDegree_X_pow, continuousWithinAt_iff_continuous_left_right, Matroid.toENat_cRank_eq, Filter.neg_le_neg_iff, mabs_sup_div_sup_le_mabs, MeasureTheory.Measure.countable_meas_pos_of_disjoint_of_meas_iUnion_ne_top₀, CategoryTheory.Subobject.map_obj_injective, integrableOn_Icc_iff_integrableOn_Ico', List.minimum_le_coe_iff, AlgebraicGeometry.Scheme.toSpecΓ_preimage_basicOpen, isCofinal_iff_top_mem, PrimeSpectrum.iSup_basicOpen_eq_top_iff', CategoryTheory.isNoetherianObject_iff_not_strictMono, measurableSet_bddBelow_range, NNReal.rpow_le_rpow_iff, Fin.succFunctor_obj, Filter.bliminf_false, PosSMulStrictMono.toSMulPosStrictMono, AddSubgroup.unop_le_unop_iff, MeasureTheory.Measure.FiniteAtFilter.exists_mem_basis, Set.preimage_const_mul_Iio, Hyperreal.st_eq_sSup, isFullyInvariant_iff_le_imp_isotypicComponent_le, Nonneg.toNonneg_lt, MeasureTheory.eLpNorm_lt_top_iff_lintegral_rpow_enorm_lt_top, PosSMulMono.toPosSMulReflectLT, nnnorm_sub_le, abs_min_le_max_abs_abs, Finset.orderEmbOfFin_mem, Ordinal.lt_veblen_invVeblen₁, Affine.Simplex.map_subtype_restrict, NonUnitalSubring.prod_mono_right, Fin.finsetImage_succ_Ioc, ConditionallyCompleteLinearOrderedField.toMulPosStrictMono, CategoryTheory.Limits.FormalCoproduct.cech_obj, Ordinal.toNatOrdinal_toOrdinal, MeasureTheory.Content.innerContent_mono, Filter.tendsto_finset_Iic_atTop_atTop, FirstOrder.Language.Substructure.map_le_map_iff_of_injective, pos_iff_ne_zero, Filter.tendsto_neg_atTop_iff, Set.Finite.encard_biUnion_le, Finset.max_prod_le, IsPredArchimedean.isAtom_findAtom_iff, InfHom.mk_le_mk, ContinuousWithinAt.upperSemicontinuousWithinAt, MvPowerSeries.le_order_pow, AddSubmonoid.le_pointwise_smul_iff₀, MeasureTheory.aecover_Icc_of_Ico, lebesgue_number_lemma_of_emetric_nhds', MeasureTheory.hitting_eq_sInf, NNReal.HolderTriple.nonneg, MeasurableSet.exists_isClosed_diff_lt, Filter.NeBot.one_le_div, mulRothNumber_map_mul_right, ContinuousMultilinearMap.le_opNNNorm, TopCat.Presheaf.isIso_iff_stalkFunctor_map_iso, AlgebraicGeometry.Scheme.Hom.app_appIso_inv, LinearOrder.min_def, MeasureTheory.Measure.setLIntegral_rnDeriv_le, AlgebraicGeometry.isIso_fromTildeΓ_iff, top_sdiff', Finset.sup_eq_bot_iff, OrderHom.map_inf_fixedPoints_le, ConvexCone.mem_toPointedCone, Set.projIic_eq_self, AbsoluteValue.sub_le_add, CategoryTheory.IsGrothendieckAbelian.generatingMonomorphisms.le_largerSubobject, Submodule.map_add_le, continuousAt_sign_of_ne_zero, isClosed_closedAbsConvexHull, IntermediateField.mem_bot, Affine.Simplex.restrict_map_subtype, tendsto_floor_right', MeasureTheory.Content.outerMeasure_lt_top_of_isCompact, Associates.prod_le, UniformFun.gc, iSup_comp_le, Fin.preimage_castLE_Iio_castLE, InfTopHom.coe_inf, MeasureTheory.preVariation.sum_le_preVariationFun_of_subset, TopCat.Sheaf.interUnionPullbackCone_pt, LieSubmodule.submodule_span_le_lieSpan, LieSubmodule.le_comap_map, IsValuativeTopology.mem_nhds_iff, Finset.sum_sdiff_lt_sum_sdiff, OrderIso.map_bot, abs_nonneg, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquiv_unitIso, Fin.map_castAddEmb_Icc, Order.isPredPrelimit_iff_lt_pred, ENNReal.coe_lt_one_iff, Filter.comap_neg_atTop, AddSubgroup.center_le_normalizer, stdSimplexEquivIcc_one, Finset.prod_Ico_add_right_sub_eq, NumberField.Units.dirichletUnitTheorem.logEmbedding_ker, Set.Ico.coe_nonneg, one_pos, CategoryTheory.Limits.factorThruKernelSubobject_comp_kernelSubobjectIso, AddSubmonoid.ofAddUnits_addUnits_le, Algebra.elemental.le_of_mem, List.sortedLE_iff_monotone_get, OmegaCompletePartialOrder.OrderHom.omegaCompletePartialOrder_ωSup_coe, Complex.re_eq_norm, Submonoid.map_strictMono_of_injective, Int.lt_floor_iff, lowerSemicontinuousWithinAt_iff_frequently, SetTheory.PGame.moveLeft_nim_heq, IsCompl.disjoint, Ordinal.preOmega_natCast, MeasureTheory.SimpleFunc.edist_approxOn_le, MulArchimedeanClass.subgroup_eq_bot, nhds_basis_mabs_div_lt, AddAction.stabilizer_inf_stabilizer_le_stabilizer_sdiff, Valuation.pos_iff, LinearOrderedCommGroup.discrete_iff_not_denselyOrdered, CategoryTheory.ComposableArrows.fourδ₄Toδ₃_app_zero, AlgebraicGeometry.PresheafedSpace.restrictTopIso_inv, add_div_two_lt_right, sSup_le_sSup, Valued.norm_pos_iff_valuation_pos, forall_le_imp_ne_iff_lt, isLUB_atoms_le, Submodule.covBy_span_singleton_sup, Nat.ceil_eq_iff, Set.bounded_gt_inter_not_gt, SSet.horn₃₂.desc.multicofork_π_zero, Algebra.IsCentral.out, exists_zero_lt, Fin.preimage_cast_Icc, List.maximum_eq_coe_iff, polynomialFunctions_closure_eq_top, Ordinal.blsub_le_bsup_succ, Order.height_enat, AlgebraicGeometry.Scheme.Hom.normalizationObjIso_hom_val, AlgebraicGeometry.Scheme.openCoverBasicOpenTop_f, partialSups_le_iff, MeasureTheory.exists_measurable_le_withDensity_eq, Cardinal.preBeth_pos, Filter.isOpen_Iic_principal, MeasureTheory.OuterMeasure.le_ofFunction, Ordinal.IsNormal.le_iff_eq, Rat.natCast_lt_cast, LowerAdjoint.closure_is_closed, IsDedekindDomain.integer_univ, SeminormedAddGroup.disjoint_nhds, SSet.stdSimplex.objEquiv_symm_apply, MeasureTheory.distribHaarChar_pos, Set.unbounded_le_Ici, EReal.image_coe_Ioi, CategoryTheory.ComposableArrows.Mk₁.map_comp, InfTopHom.top_apply, PseudoEMetricSpace.uniformity_edist, bihimp_top, CategoryTheory.Limits.CompleteLattice.finiteLimitCone_cone_π_app, CategoryTheory.ComposableArrows.IsComplex.cokerToKer'_fac, Submodule.map_le_localized₀, Set.isSimpleOrder_Ici_iff_isCoatom, Equiv.Perm.support_le_prod_of_mem, Real.ofNat_le_toNNReal, star_nonneg_iff, PartOrdEmb.Limits.instPreservesColimitForgetOrderEmbeddingCarrier, CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinset_obj_obj, Submonoid.coe_pos, OrderAddMonoidHom.inrₗ_apply, PredOrder.prelimitRecOn_pred, PseudoMetric.sup_apply, GaloisCoinsertion.u_bot, IsCompl.le_left_iff, NNReal.image_coe_Icc, closureOperator_gi_self, FinPartOrd.Iso.mk_hom, AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace_hom_c_app, smulPosMono_iff_smulPosReflectLT, Ordinal.toLeftMovesToPGame_symm_lt, SSet.horn₃₂.desc.multicofork_π_three_assoc, AlgebraicGeometry.Scheme.IdealSheafData.comap_top, IsCompact.disjoint_nhdsSet_nhds, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverseObj_pt, AlgebraicGeometry.Scheme.Hom.appLE_eq_app, MeasureTheory.IsSetSemiring.Ioc, List.sortedGE_insertionSort, Multiset.cons_le_cons_iff, ProjectiveSpectrum.le_iff_mem_closure, supClosure_prod, Ideal.homogeneousHull_eq_sInf, IsAdjoinRoot.adjoin_root_eq_top, Flag.coe_wcovBy_coe, Fin.map_castAddEmb_Ioo, Types.monoOverEquivalenceSet_counitIso, Ordinal.toZFSetIso_symm_apply, LocallyFiniteOrder.orderAddMonoidEquiv_apply, Submodule.toAddSubgroup_le, NonUnitalAlgebra.toNonUnitalSubring_top, SemilatSupCat.coe_forget_to_partOrd, iInf_le_iSup, atTop_hasAntitoneBasis_of_archimedean, BoxIntegral.IntegrationParams.MemBaseSet.exists_common_compl, Cardinal.ord_le_type, MeasureTheory.mul_meas_ge_le_pow_eLpNorm, sInf_le_sSup, Right.sign_neg, iInf_emptyset, Ideal.span_singleton_absNorm_le, bddAbove_range_partialSups, List.le_maximum_of_length_pos_iff, Set.compl_Ioc, LinearMap.injective_iff_antilipschitz, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.invApp_app_assoc, CategoryTheory.CostructuredArrow.unop_left_comp_underlyingIso_hom_unop, Equiv.Perm.prod_Iio_comp_eq_sign_mul_prod, Multiset.mem_powersetAux, Cardinal.lt_univ', Fin.preimage_succ_Ici_succ, Set.LeftInvOn.filter_map_Iic, Set.Icc.coe_sup, AffineIndependent.convexHull_inter, IsOrderedMonoid.toMulLeftMono, MeasureTheory.Measure.haar.haarContent_outerMeasure_self_pos, Order.Ideal.mem_inf, Set.sUnion_powerset_gc, IsModularLattice.complementedLattice_Iic, PNat.one_le, CategoryTheory.Subobject.instFaithfulPullback, AddSubgroup.center_le_centralizer, Ordinal.isSuccLimit_of_mem_frontier, AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec.image_basicOpen_eq_basicOpen, Ordinal.lsub_unique, ENNReal.instPosSMulStrictMonoNNReal, CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinset_obj_obj, List.coe_minimum_of_length_pos, Fin.finsetImage_castAdd_Ioc, Order.lt_succ_iff_ne_top, exists_le_add_iff_le_right, LinOrd.comp_apply, NatOrdinal.toOrdinal_eq_one, IsStarProjection.le_of_mul_eq_right, Finset.Ici_add_one_eq_Ioi, SSet.horn₃₂.desc.multicofork_π_one_assoc, UpperSet.lt_erase, isPiSystem_image_Ici, IsSelfAdjoint.le_algebraMap_norm_self, AlgebraicGeometry.IsOpenImmersion.app_eq_invApp_app_of_comp_eq_aux, AlgebraicGeometry.isField_of_isIntegral_of_subsingleton, finsetSup'_mem_supClosure, IsWellFounded.wellOrderExtension.isWellOrder_lt, CategoryTheory.Subobject.underlyingIso_arrow_apply, MeasureTheory.upcrossings_lt_top_iff, EReal.le_sub_iff_add_le, Odd.geom_sum_pos, Interval.bot_nsmul, Multiset.disjSum_strictMono_right, Set.einfsep_pos, abs_le', LinOrd.coe_comp, Subgroup.surjOn_iff_le_map, LinearMap.le_ker_iff_map, MeasureTheory.Measure.exists_isOpen_measure_lt_top, IsDedekindDomain.HeightOneSpectrum.valuation_le_one, IsCompact.exists_isOpen_lt_add, SetTheory.PGame.moveLeft_nim, AddMonoidAlgebra.infDegree_withTop_some_comp, BotHom.coe_inf, Submodule.wcovBy_span_singleton_sup, ENat.one_lt_card, ENat.self_le_mul_right, Cardinal.mul_lt_aleph0_iff_of_ne_zero, LaurentPolynomial.degree_T_le, AlgebraicGeometry.exists_of_res_eq_of_qcqs, CategoryTheory.MorphismProperty.pullbacks_le, ENat.add_le_add_iff_left, Order.length_le_height_last, AlgebraicGeometry.PresheafedSpace.map_comp_c_app, PosSMulMono.toPosSMulReflectLE, Order.Ideal.IsPrime.isMaximal, TopCat.Presheaf.coveringOfPresieve_apply, Filter.Tendsto.finset_inf_nhds, ENNReal.tendsto_coe_nhds_top, tendsto_pow_atTop_nhds_zero_iff, OrderIso.ofRelIsoLT_symm, TopologicalSpace.Opens.not_nonempty_iff_eq_bot, IsPreconnected.mem_intervals, IsStrictOrderedRing.toPosMulStrictMono, Multiset.le_sum_of_mem, Set.Ioi_inter_Ioi, Dense.borel_eq_generateFrom_Ioc_mem, NNReal.HolderTriple.inv_lt_inv, Subsemigroup.le_prod_iff, ENat.mul_right_strictMono, Subalgebra.star_mono, AlgebraicGeometry.ProjectiveSpectrum.Proj.toOpen_toSpec_val_c_app, CategoryTheory.SmallObject.iterationFunctorObjObjRightIso_ιIteration_app_right_assoc, AbsoluteValue.add_le', MeasureTheory.exists_lintegral_le, iInf_le, CompositionSeries.head_le, tendsto_intCast_atBot_cobounded, Finset.max'_lt_iff, comap_coe_nhdsGT_eq_atBot_iff, Finset.sup_id_eq_sSup, QuotientGroup.strictMono_comap_prod_map, Finset.listMap_orderEmbOfFin_finRange, FinBoolAlg.dual_map, strictConvex_Ioi, partOrd_dual_comp_forget_to_preord, PartOrd.ofHom_comp, infClosure_empty, TopCat.subpresheafToTypes_obj, MeasureTheory.setLAverage_lt_top, TopCat.Presheaf.submonoidPresheafOfStalk_obj, Multiset.card_Ioo, Subgroup.commutator_le_left, nndist_vadd_vadd_le, SubmodulesBasis.inter, Polynomial.div_tendsto_atTop_zero_iff_degree_lt, Valued.toNormedField.norm_lt_one_iff, Module.mem_support_iff_of_span_eq_top, le_min_iff, Set.add_mem_Ioo_iff_right, Algebra.Norm.Transitivity.auxMat_blockTriangular, ENNReal.div_lt_iff, ringKrullDim_le_of_surjective, MeasureTheory.projectiveFamilyContent_mono, instDenselyOrderedEReal, Quantale.leftMulResiduation_le_iff_mul_le, IsSimpleOrder.instIsAtomic, nnnorm_add_le, MvPowerSeries.le_lexOrder_mul, Subgroup.inf_normalizer_le_normalizer_inf, Finset.inf_Ici, TopCat.Presheaf.stalkFunctor_map_germ_apply', OrderIso.asBoolAlgAsBoolRing_symm_apply, Nonneg.coe_smul, Multiset.le_ndunion_right, AlgebraicGeometry.StructureSheaf.algebraMap_pushforward_stalk, IsLocalRing.finite_quotient_iff, PrimeSpectrum.isIdempotentElemEquivClopens_mul, convex_closedConvexHull, SimpleGraph.Walk.three_le_chromaticNumber_of_odd_loop, NNReal.coe_le_coe, Ideal.leadingCoeffNth_mono, CategoryTheory.PreOneHypercover.sieve₀_trivial, UpperSet.total_le, closure_subset_closedAbsConvexHull, continuousOn_cfcₙ_nnreal_setProd, instLawfulOrderInf_mathlib, DivisorChain.exists_chain_of_prime_pow, Nat.add_one_le_ceil_iff, HahnSeries.mem_orderTopSubOnePos_iff, LieAlgebra.LieIdeal.solvable_iff_le_radical, Continuous.exists_forall_ge_of_hasCompactMulSupport, Submodule.decomposition_erase_inf, spectrum.spectralRadius_le_pow_nnnorm_pow_one_div, Metric.packingNumber_le_encard_self, NNReal.holderConjugate_iff, SummationFilter.unconditional_filter, ENNReal.mulLeftOrderIso_apply, Subalgebra.LinearDisjoint.mulLeftMap_ker_eq_bot_iff_linearIndependent_op, denselyOrdered_iff_denselyOrdered_units_and_nontrivial_units, Filter.tendsto_toRight_atTop, Set.Ioc.codisjoint_iff, EMetric.mem_nhdsWithin_iff, Finset.monotone_preimage, PartOrdEmb.coe_id, Module.End.mem_invtSubmodule_iff_map_le, Filter.map_iInf_le, Module.rank_lt_aleph0, CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinsetColimIso_aux, MeasureTheory.Ico_ae_eq_Icc, CategoryTheory.Subfunctor.preimage_eq_top_iff, CategoryTheory.ShortComplex.mapToComposableArrows_id, SuccOrder.nhdsLT_eq_nhdsNE, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.inv_invApp, MeasureTheory.Measure.rnDeriv_pos', Module.End.invtSubmodule.codisjoint_mk_iff, Set.image_div_const_Ioi, ENNReal.tsum_iUnion_le, AlgebraicGeometry.isLocalization_basicOpen_of_qcqs, DFinsupp.card_Icc, IntermediateField.fixingSubgroup_bot, AlgebraicGeometry.Scheme.IdealSheafData.strictMono_ideal, MvPolynomial.coeffsIn_le, Subalgebra.rangeS_le, CategoryTheory.ComposableArrows.precomp_map, CategoryTheory.TransfiniteCompositionOfShape.ici_incl, Subgroup.le_pointwise_smul_iff₀, CategoryTheory.Subfunctor.Subpresheaf.eq_top_iff_isIso, TopCat.GlueData.MkCore.cocycle, iSup₂_eq_top, finGaloisGroupMap.map_comp, Filter.covariant_smul_filter, Ordinal.lt_nadd_iff, CategoryTheory.ComposableArrows.fourδ₁Toδ₀_app_three, SubMulAction.closure_mono, ENNReal.toNNReal_pos_iff, Fin.sum_Ioo_castSucc, NNReal.sqrt_mul_self, Matrix.PosDef.det_pos, CategoryTheory.Dial.tensorUnitImpl_rel, Order.sub_one_lt_iff, ZSpan.mem_fundamentalDomain, exists_succ_iterate_or, Set.image_mulSingle_Ico, Submonoid.FG.exists_minimal_closure_eq, Subgroup.op_le_iff, Cardinal.mem_range_aleph_iff, Module.Basis.isChain_range_flag, CategoryTheory.Abelian.subobjectIsoSubobjectOp_symm_apply, MeasureTheory.aecover_Ioo_of_Icc, ofAdd_toDual_eq_zero_iff, Set.Iio_subset_Iic_union_Ioo, minSmoothness_monotone, Set.Ici_eq_singleton_iff_isTop, MeasureTheory.Lp.pow_mul_meas_ge_le_enorm, Order.krullDim_of_isSimpleOrder, symmDiff_bot, Ordinal.lsub_le_sup_succ, iSup_mul_le, Submodule.height_lt_top, Real.log_of_ne_zero, Submodule.FG.rTensor.directLimit_apply', MeasureTheory.condLExp_add_le, MeasureTheory.AddContent.supClosure_apply, TopologicalSpace.OpenNhds.inclusionMapIso_inv_app, Set.disjoint_range_iff, AlgebraicGeometry.Scheme.PartialMap.restrict_id_hom, MulOpposite.op_le_one, Cardinal.mk_embedding_eq_zero_iff_lt, ArchimedeanClass.mk_map_of_archimedean', CategoryTheory.ComposableArrows.IsComplex.zero, Cardinal.ord_mono, Module.End.genEigenspace_le_smul, CategoryTheory.MorphismProperty.TransfiniteCompositionOfShape.mem_map, max_def, Set.image_sub_const_Ioc, FirstOrder.Language.Hom.eqOn_closure, CFC.continuousOn_rpow, Order.lt_one_iff_nonpos, Finset.exists_sup_eq_iSup, OrderIso.isSimpleOrder, inf_himp_le, Subgroup.isLeast_of_closure_iff_eq_mabs, setOf_maximal_antichain, AlgebraicGeometry.LocallyRingedSpace.evaluation_naturality_apply, MvPowerSeries.min_order_le_add, CategoryTheory.Presieve.isSheafFor_top, Order.succ_le_iff_eq_top, IsLocalRing.local_hom_TFAE, Nonneg.orderedSub, CategoryTheory.ComposableArrows.IsComplex.cokerToKer_fac, Set.image_div_const_Iic, Cardinal.aleph_lt_aleph, AddCircle.liftIco_eq_lift_Icc, MeasureTheory.Measure.lintegral_rnDeriv_lt_top_of_measure_ne_top, AlgebraicGeometry.IsAffineOpen.fromSpec_toSpecΓ, PNat.card_fintype_Icc, isNoetherian_iff, right_eq_inf, MeasureTheory.ae_restrict_uIoc_eq, Finset.Nonempty.dens_pos, Subalgebra.bot_eq_top_iff_rank_eq_one, one_le_thickenedIndicator_apply, HenselianRing.jac, AlgebraicGeometry.map_injective_of_isIntegral, ringKrullDim_le_ringKrullDim_quotSMulTop_succ, Nat.cast_pos', eVariationOn.lowerSemicontinuous, Prod.ωSupImpl_snd, MeasureTheory.Measure.restrict_union_le, Module.supportDim_quotSMulTop_succ_le_of_notMem_minimalPrimes, succ_max, CategoryTheory.Subobject.underlying_arrow, IntermediateField.coe_top, FiniteArchimedeanClass.addSubgroup_eq_bot, WithZero.min_le_iff, AbsoluteValue.IsNontrivial.exists_abv_gt_one, AlgebraicGeometry.exists_eq_pow_mul_of_isCompact_of_isQuasiSeparated, AlgebraicGeometry.AffineSpace.homOfVector_appTop_coord, edist_smul_le, CategoryTheory.ConcreteCategory.injective_le_monomorphisms, SSet.stdSimplex.nonDegenerateEquiv_symm_apply_coe, Set.Ici_iSup, Subrepresentation.subrepresentationSubmoduleOrderIso_apply, mabs_div_le_iff, WithZeroTopology.nhds_eq_update, ValuativeRel.uniformizer_lt_one, AlgHom.equalizer_toSubmodule, Metric.ediam_le_iff, AlgebraicGeometry.sourceAffineLocally_morphismRestrict, nhdsGT_neBot, Set.Iic.isCoatom_iff, HasCardinalLT.Set.isFiltered_of_aleph0_le, Monotone.of_map_sup, BiheytingAlgebra.sdiff_le_iff, Manifold.pathELength_mono, edist_pos, archimedean_iff_int_le, antitone_vecEmpty, OrderIso.finsetSetFinite_apply_coe, Fin.map_natAddEmb_Ioc, enorm_multisetProd_le, Submonoid.center_le_centralizer, min_eq_top, countable_setOf_covBy_right, Order.Ideal.coe_ssubset_coe, Ideal.coe_piOrderIso_apply, Polynomial.mul_contentIdeal_le_radical_contentIdeal_mul, QuotientGroup.le_comap_mk', isGalois_bot, max_def_lt', Prod.mk_wcovBy_mk_iff_left, AlgebraicGeometry.Scheme.IsQuasiAffine.isBasis_basicOpen, PowerSeries.le_order_prod, ValuativeRel.uniformizer_pos, Subalgebra.rTensorBot_tmul, AlgebraicGeometry.Proj.stalkIso'_symm_mk, EMetric.isUniformInducing_iff, nnnorm_le_add_nnnorm_add', monotone_vecCons, Cardinal.zero_lt_lift_iff, HahnEmbedding.Partial.exists_isMax, sup_sdiff_injOn, mulRothNumber_union_le, Ordinal.CNF_lt_snd, Archimedean.mkRat_mem_ratLt, CauSeq.bounded, AddGroupSeminorm.lt_def, Submodule.set_smul_le, Finset.box_succ_union_prod, Module.AEval.mem_mapSubmodule_apply, Prop.instWellFoundedGT, MeasureTheory.eLpNorm_restrict_le, FormalMultilinearSeries.le_radius_of_tendsto, CategoryTheory.Subfunctor.Subpresheaf.IsGeneratedBy.ofSection_le, FreeGroup.range_lift_le, AlgebraicGeometry.structureSheafInType.smul_apply, OrderHom.isFixedPt_gfp, CategoryTheory.Triangulated.SpectralObject.ω₂_obj_obj₁, Cardinal.nat_lt_univ, AddAction.le_stabilizer_vadd_left, NNRat.cast_le_ofNat, ENNReal.rpow_arith_mean_le_arith_mean2_rpow, EMetric.measure_closedBall_pos, Set.preimage_const_add_Ioc, exists_uIoo_isExtrOn_uIcc, TopologicalSpace.OpenNhds.apply_mk, Fin.zero_eq_top, eq_top_or_lt_top, AlgebraicGeometry.Etale.instMorphismRestrict, EReal.mul_eq_top, NNReal.Lp_add_le_tsum, Ordinal.type_toType, NNReal.inv_lt_one_iff, MeasureTheory.Egorov.measure_iUnionNotConvergentSeq, le_add_iff_lt_right_or_exists_le, ExpGrowth.le_expGrowthInf_mul, CategoryTheory.Subgroupoid.galoisConnection_map_comap, LowerSet.supIrred_Iic, instContinuousConstSMulSubtypeLeOfNat, inv_nonpos', Cardinal.mk_biUnion_le_of_le_lift, TestFunction.topologicalSpace_le_iff, HasOuterApproxClosed.apprSeq_apply_le_one, AlgebraicGeometry.Scheme.isoSpec_image_zeroLocus, Module.Basis.flag_strictMono, Subsemigroup.toAddSubsemigroup'_closure, LinearOrderedAddCommGroup.isAddCyclic_iff_nonempty_equiv_int, IsOrderedRing.toMulPosMono, tendsto_tprod_compl_atTop_one, disjointed_partialSups, Pell.IsFundamental.y_strictMono, ClosureOperator.closure_sup_closure_le, HahnSeries.orderTop_embDomain, Set.image_add_const_Ioc, ENNReal.Icc_mem_nhds, ZFSet.vonNeumann_succ, Set.neg_Iic, Valuation.IsRankOneDiscrete.exists_generator_lt_one, SignType.neg_iff, iSup_extend_bot, Module.le_comap_jacobson, Ordinal.nmul_eq_mul, CategoryTheory.Limits.cokernelOrderHom_coe, Set.Finite.convexHull_eq, BoundedGENhdsClass.of_closedIicTopology, Archimedean.embedRealFun_strictMono, TopCat.Presheaf.generateEquivalenceOpensLe_inverse'_map, ContractingWith.apriori_edist_iterate_efixedPoint_le, AddSubgroup.le_normalizer_comap, Subalgebra.pi_toSubmodule, CategoryTheory.MorphismProperty.pushouts_monotone, Ordinal.succ_iSup_le_lsub_iff, hnot_le_iff_codisjoint_right, FormalMultilinearSeries.ofScalars_radius_ge_inv_of_tendsto, Fin.map_castSuccEmb_Ioc, AlgebraicGeometry.PresheafedSpace.GlueData.π_ιInvApp_eq_id, MeasureTheory.OuterMeasure.map_ofFunction_le, Finset.erase_covBy, supClosed_supClosure, OrderHom.isLeast_lfp_le, Algebra.range_id, NNReal.sqrt_inv, AddValuation.supp_quot_supp, sSupHomClass.toSupBotHomClass, Sbtw.mem_image_Ioo, LinearGrowth.le_linearGrowthSup_iff, MeasureTheory.Measure.measure_Iio_pos, Metric.uniformCauchySeqOn_iff, ExpGrowth.le_expGrowthSup_iff, Ordinal.toZFSet_mem_toZFSet_iff, BddOrd.inv_hom_apply, Module.supportDim_le_of_surjective, le_nonZeroDivisorsRight_iff_isRightRegular, MeasureTheory.hittingAfter_anti, NumberField.mixedEmbedding.minkowskiBound_pos, Set.IsPWO.of_linearOrder, Profinite.NobelingProof.Products.instWellFoundedLT, AlgebraicGeometry.Scheme.IdealSheafData.ideal_map_of_isAffineHom, mabs_div_le_one, Sublattice.prodEquiv_symm_apply, Polynomial.SplittingField.adjoin_rootSet, MulArchimedeanClass.mk_div_lt_mk_right_iff, Order.length_le_coheight_head, Multiset.le_filter, AddSubmonoid.opEquiv_apply, egauge_lt_iff, Matrix.eRank_submatrix_le, CategoryTheory.MorphismProperty.le_colimitsOfShape_punit, AlgebraicGeometry.isCompact_and_isOpen_iff_finite_and_eq_biUnion_basicOpen, isSplittingField_iff_intermediateField, Finset.mem_eraseNone, hasSumLocallyUniformly_iff_tendstoLocallyUniformly, Ne.inf_lt_or_inf_lt, half_le_self_iff, ExpGrowth.expGrowthInf_le_expGrowthSup, RingHom.toKerIsLocalization_isLocalizedModule, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.app_inv_app'_assoc, PrimeSpectrum.exists_primeSpectrum_prod_le_and_ne_bot_of_domain, Nimber.lt_one_iff_zero, IsAbsoluteValue.abv_sub_le, OrderHom.coe_iSup, Finset.dens_le_dens, IntermediateField.FG.exists_finset_maximalFor_isTranscendenceBasis_separableClosure, BooleanAlgebra.eq_iff_atom_le_iff, AddSubgroup.toSubgroup_comap, DFinsupp.Lex.total_le, Ordinal.le_div, NonUnitalSubring.toSubsemigroup_strictMono, NatOrdinal.small_Iic, inf_eq_minDefault, Complex.sum_div_factorial_le, OrderHom.iterate_sup_le_sup_iff, compl_gt, Set.ncard_strictMono, Submodule.le_pow_toAddSubmonoid, iSup_bot, AlgebraicGeometry.IsAffineOpen.toSpecΓ_isoSpec_inv_assoc, Set.empty_disjoint, AlgebraicGeometry.PresheafedSpace.restrict_top_presheaf, Finset.val_le_iff_val_subset, zsmul_strictMono_right, Hyperreal.ofSeq_lt_ofSeq, Module.End.genEigenspace_restrict, ContinuousAt.upperSemicontinuousAt, orderTopology_of_ordConnected, MeasureTheory.Measure.lt_iff, finSumEquivOfFinset_inl, ENNReal.inv_pos, SimpleGraph.chromaticNumber_mono_of_hom, Fin.prod_prod_eq_prod_triangle_mul, isGalois_iff_isGalois_bot, exists_one_lt', le_comap_range_rTensor, Multiset.isAtom_iff, Set.Ioi_union_left, one_le_nnnorm_one, Fin.rev_bot, CategoryTheory.ComposableArrows.homMk₀_app, AlgebraicGeometry.Scheme.Modules.pushforward_map_app, CategoryTheory.TransfiniteCompositionOfShape.ofOrderIso_isColimit, AddSubgroup.coe_toSubgroup_symm_apply, Set.image_const_div_Ioc, AddSubsemigroup.le_topologicalClosure, cauchySeq_of_controlled, BddLat.dual_map, ClosureOperator.isClosed_top, exists_rat_lt, Prod.isCompl_iff, Interval.length_pure, CategoryTheory.SimplicialThickening.functor_obj_as, instIsDirectedOrder, Set.Ioo_subset_uIoo', Multiset.map_add_right_Ico, Set.Ioc_sub_one_left_eq_Icc, Cardinal.toENat_ne_top, Disjoint.eq_bot, ENNReal.ofReal_le_of_le_toReal, Submodule.le_traceDual_iff_map_le_one, NonemptyInterval.coe_top, PartOrd.ext_iff, ENat.ceil_le_floor_add_one, CategoryTheory.HasLiftingProperty.transfiniteComposition.SqStruct.w₂, MeasureTheory.Lp.simpleFunc.coeFn_nonneg, NonUnitalRingHom.closure_preimage_le, IsCompact.exists_isMaxOn, TrivSqZeroExt.range_inlAlgHom_sup_adjoin_range_inr, AlgebraicGeometry.coe_opensRestrict_apply_coe, AddMonoidAlgebra.supDegree_prod_le, YoungDiagram.transposeOrderIso_apply, ENNReal.one_lt_inv, Mathlib.Meta.NormNum.isNat_le_true, PrimeSpectrum.coe_isIdempotentElemEquivClopens_apply, Monotone.of_left_le_map_sup, IsCyclic.normalizer_le_centralizer, Cardinal.aleph0_lt_continuum, MeasureTheory.Filtration.mono', AlgebraicGeometry.Scheme.AffineZariskiSite.presieveOfSections_eq_ofArrows, CategoryTheory.MorphismProperty.retracts_monotone, Order.Ioc_pred_left, SimplexCategory.epi_iff_surjective, isClosed_Ioo_iff, MulArchimedeanOrder.le_def, Fintype.not_linearIndependent_iffₒₛ, Polynomial.adjoin_X, HasFPowerSeriesOnBall.r_pos, convex_Ioi, Bool.wcovBy_iff, MeasureTheory.SimpleFunc.monotone_approx, Finset.gc_map_inl_toLeft, eq_sSup_atoms, LE.total', Set.neg_Ico, WithLp.enorm_fst_le, nonneg_add_self_iff, zpow_lt_zpow_iff_left, LieAlgebra.IsSimple.isAtom_iff_eq_top, Metric.mem_eball', isClosed_closedConvexHull, AffineSubspace.bot_parallel_iff_eq_bot, AlgebraicGeometry.Scheme.Hom.inv_app, antitoneOn_iff_forall_lt, sub_le_sub_flip, ValuationSubring.primeSpectrumEquiv_apply, MeasureTheory.Content.innerContent_le, Odd.pow_nonpos_iff, Polynomial.cyclotomic_pos, Matroid.eRk_submod, le_bihimp_iff, AlgebraicGeometry.PresheafedSpace.GlueData.snd_invApp_t_app', Fin.map_revPerm_Iic, bihimp_eq_top, Dense.borel_eq_generateFrom_Icc_mem, Finset.Iic_diff_Ioc, cfcₙHom_nonneg_iff, OrderHom.map_lfp_comp, AlgebraicGeometry.Scheme.restrictFunctor_obj_hom, PrincipalSeg.le_apply_iff, Set.disjoint_union_left, LieSubalgebra.sInf_glb, Fin.sum_Ioi_succ, FirstOrder.Language.Substructure.fg_bot, pNilradical_le_nilradical, AlgebraicGeometry.Scheme.Hom.isQuasiSeparated_preimage, AlgebraicGeometry.RingedSpace.mem_basicOpen, ArchimedeanClass.ball_antitone, Multiset.lt_replicate_succ, PartENat.withTopEquiv_symm_lt, Set.Ioo_inter_Ioo, CategoryTheory.ComposableArrows.fourδ₄Toδ₃_app_three, SummationFilter.hasProd_symmetricIoc_int_iff, ContractingWith.edist_efixedPoint_le, Order.covBy_iff_sub_one_eq, AlgebraicGeometry.Scheme.Hom.app_injective, Finset.Ico_succ_succ_eq_Ioc, MeasureTheory.hasFiniteIntegral_iff_enorm, CategoryTheory.Functor.mapPresheaf_obj_presheaf, Int.Ico_filter_dvd_eq, Monoid.minOrder_le_orderOf, MvPolynomial.weightedTotalDegree_zero, EReal.exp_le_exp_iff, MeasureTheory.FiniteMeasure.apply_union_le, CFC.le_posPart, Ring.jacobson_lt_top, Order.krullDim_eq_zero_iff_of_orderBot, nhdsSet_Ioi, CompleteDistribLattice.MinimalAxioms.inf_sSup_le_iSup_inf, CategoryTheory.ConcreteCategory.surjective_le_epimorphisms, AlgebraicGeometry.Scheme.IdealSheafData.subschemeFunctor_map, Finset.image_add_left_Ioc, EReal.natCast_le_iff, Cardinal.succ_pos, CategoryTheory.Subobject.representative_arrow, pow_inf_le, Ordinal.small_Ico, Lattice.isStronglyCoatomic, NonarchimedeanAddGroup.cauchySeq_sum_of_tendsto_cofinite_zero, Submodule.mem_invtSubmodule_reflection_iff, TopCat.Presheaf.mono_iff_stalk_mono, CategoryTheory.Limits.kernelSubobjectMap_arrow, Submonoid.opEquiv_symm_apply, Set.Ioi_mul_Ici_subset', Left.sign_neg, AlgebraicGeometry.Scheme.Modules.inv_app, Affine.Simplex.sOppSide_affineSpan_faceOpposite_iff, measurableSet_bddAbove_range, FirstOrder.Language.DirectLimit.cod_partialEquivLimit, LowerSet.Iic_zero, Finset.le_max, LieModule.range_traceForm_le_span_weight, Function.Injective.map_atTop_finset_sum_eq, LieAlgebra.le_zeroRootSubalgebra, FinPartOrd.id_apply, Filter.Tendsto.partialSups, Specializes.not_disjoint, MeasureTheory.Measure.le_restrict_apply, sub_inv_antitoneOn_Iio, HahnSeries.min_orderTop_le_orderTop_add, instWellFoundedGTWithZeroMultiplicativeIntLeOne, Set.Ico.disjoint_iff, HolderOnWith.ediam_image_le_of_subset, CategoryTheory.HasLiftingProperty.transfiniteComposition.SqStruct.w₁_assoc, ENNReal.ofReal_le_natCast, Finset.ofDual_sup, SummationFilter.conditional_filter_eq_map_Iic, Order.Ideal.IsMaximal.isCoatom, PNat.one_le_find, lieModule_lcs_map_le, bihimp_bot, Order.PFilter.principal_le_principal_iff, Finset.sum_Icc_succ_eq_add_endpoints, Valuation.val_le_one_iff, Dynamics.one_le_netMaxcard_iff, Finset.sum_le_sum_Ioc, iInf_le_iInf_of_subset, Mathlib.Meta.NormNum.isInt_lt_false, TopologicalSpace.Opens.coe_top, TopCat.Presheaf.ext_iff, Subfield.isGLB_sInf, AlgebraicGeometry.Scheme.Opens.toSpecΓ_preimage_zeroLocus, Set.image_sub_const_Ico, isCoatomic_of_isAtomic_of_complementedLattice_of_isModular, partialSups_add_const, Filter.tendsto_mul_const_atBot_iff, Subsemiring.mem_nonneg, NNReal.dist_gm_am_le, HahnSeries.leadingCoeff_nonpos_iff, EReal.eq_top_iff_forall_lt, Filter.tendsto_finset_image_atTop_atTop, StieltjesFunction.right_continuous, Odd.pow_le_pow, Set.instOrdConnectedOrdConnectedComponent, Topology.IsUpperSet.instProp, IsOrderedCancelMonoid.toMulLeftReflectLE, FormalMultilinearSeries.compPartialSumTarget_tendsto_prod_atTop, SSet.horn₃₁.desc.multicofork_π_three, SetLike.isAtom_iff, ENNReal.tsum_biUnion_le_tsum, Ordinal.IsInitial.mem_range_preOmega, MeasureTheory.hittingAfter_lt_iff, Set.add_mem_Ioo_iff_left, coe_le_enorm, PrimitiveSpectrum.gc, MeasureTheory.measure_closedBall_lt_top, Antitone.Ioo, atTop_isCountablyGenerated_of_archimedean, Finset.eraseNone_empty, Complex.isOpen_re_gt_EReal, Finset.eraseNone_insertNone, Finpartition.exists_le_of_le, Cardinal.card_lt_card_of_right_finite, Filter.map₂_inf_subset_right, Complex.neg_re_eq_norm, Cardinal.natCast_le_toENat_iff, Polynomial.trailingDegree_one_le, Cardinal.lt_ord, CategoryTheory.Subfunctor.epi_iff_range_eq_top, Module.annihilator_le_of_mem_support, List.sortedGE_ofFn_iff, Set.Ioi_subset_Ioc_union_Ici, Nonneg.mk_natCast, min_bot_right, InitialSeg.le_iff_le, Submodule.span_le, OrderBot.instSubsingleton, PrimeSpectrum.isIdempotentElemEquivClopens_symm_compl, Submodule.instCovariantClassSetHSMulLe, Cardinal.lt_aleph0_iff_set_finite, IntermediateField.finrank_adjoin_simple_eq_one_iff, AlgebraicGeometry.Scheme.exists_isOpenCover_and_isAffine, RingCon.matrix_monotone, Int.card_Ioo_of_lt, LinearMap.BilinForm.le_flip_dualSubmodule, PMF.toMeasure_mono, Valued.toNormedField.one_le_norm_iff, Finset.sumEquiv_symm_apply, map_rootsOfUnity, Fin.image_val_Icc, MeasureTheory.setLIntegral_le_lintegral, Ordinal.nhds_eq_pure, Set.preimage_mul_const_Icc, Pi.Lex.isStrictOrder, Sublattice.mem_map_equiv, CompleteLattice.ωScottContinuous.bot, EReal.sign_mul_inv_abs', countable_image_gt_image_Iio, WithBot.Icc_coe_coe, Set.Ioc.coe_sup, Stream'.Seq.lt_length'_iff, ENNReal.logOrderIso_apply, Topology.IsOpenEmbedding.compatiblePreserving, Set.Finite.encard_strictMonoOn, le_iff_forall_one_lt_le_mul, CategoryTheory.PreOneHypercover.sieve₁_trivial, StarSubalgebra.top_toSubalgebra, Nat.cast_le_one, Order.Coframe.MinimalAxioms.iInf_sup_le_sup_sInf, Lattice.BicartSq.le₃₄, Ordinal.succ_nadd, AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_bijective, ENat.add_lt_add_iff_right, IsUltrametricDist.nnnorm_add_le_max, uniformity_pseudoedist, ModuleCat.toKernelSubobject_arrow, wellFoundedLT_toType, FractionalIdeal.num_le, LinearPMap.domain_mono, Subgroup.le_normalizer_map, HasOuterApproxClosed.exAppr, InitialSeg.coe_toOrderEmbedding, one_lt_ltOnePart_iff, Monoid.Coprod.codisjoint_mrange_inl_mrange_inr, AlgebraicGeometry.LocallyRingedSpace.comp_c_app, IsModularLattice.complementedLattice_Icc, Submonoid.units_mono, AffineSubspace.sup_direction_lt_of_nonempty_of_inter_empty, FinPartOrd.hom_hom_id, Nat.cast_inv_le_one, CategoryTheory.Limits.imageSubobject_arrow', lowerSemicontinuousOn_iff_le_liminf, MemHolder.eHolderNorm_lt_top, Finset.divConst_le_inv_dens, StrictConvex.ordConnected, instIsOrderedSMulOfIsOrderedMonoid, multiplicity_le_emultiplicity, ExpGrowth.expGrowthSup_monotone, Order.not_isSuccPrelimit_iff, exists_eq_sum_perm_of_mem_doublyStochastic, le_egauge_closedBall_one, ZMod.valMinAbs_spec, CategoryTheory.Subobject.factorThru_zero, Bornology.comap_cobounded_le_iff, Hyperreal.gt_of_neg_of_infinitesimal, NNReal.agm_lt_agmSequences_snd_of_ne, conjneg_neg', TopCat.Presheaf.presheafEquivOfIso_functor_obj_map, IsCompact.le_nhds_of_unique_clusterPt, mul_kstar_le_kstar, AlgebraicGeometry.Scheme.isoOfEq_hom, SSet.Subcomplex.PairingCore.WeakRankFunction.lt, LinearOrderedCommGroupWithZero.wellFoundedOn_setOf_le_lt_iff_nonempty_discrete_of_ne_zero, CategoryTheory.MorphismProperty.le_multiplicativeClosure, Metric.infEDist_lt_iff, NonUnitalStarSubalgebra.inclusion_self, cauchySeq_finset_iff_tsum_vanishing, EMetric.exists_pos_forall_lt_edist, Filter.HasBasis.mem_measureSupport, CategoryTheory.Subfunctor.Subpresheaf.range_eq_top, Fin.map_finCongr_Icc, AlgebraicGeometry.LocallyRingedSpace.toΓSpec_preimage_basicOpen_eq, instIndiscreteTopology, Multiset.toFinsupp_strictMono, Int.add_one_le_ceil_iff, DirectSum.coe_of_mul_apply, AlgebraicGeometry.StructureSheaf.algebraMap_germ_apply, SupHom.mk_le_mk, Nat.lt_floor_add_one, Complex.re_nonneg_iff_nonneg, MulPosStrictMono.toMulPosReflectLE, Finsupp.Lex.wellFoundedLT_of_finite, NNReal.bddAbove_coe, BooleanSubalgebra.le_comap_iSup, Subalgebra.LinearDisjoint.eq_bot_of_self, Subring.center_le_centralizer, OrderIso.sumLexIioIci_apply_inr, AlgebraicGeometry.Scheme.stalkMap_germ, Rat.comap_cast_atTop, OrderIsoClass.toInfTopHomClass, MeasureTheory.ProbabilityMeasure.limsup_measure_closed_le_of_tendsto, LowerSet.map_map, Mathlib.Meta.NormNum.isInt_le_true, MonomialOrder.degree_sPolynomial_lt_sup_degree, ENNReal.ae_le_essSup, ENNReal.lintegral_Lp_mul_le_Lq_mul_Lr, IsLocalization.map_id_mk', nnnorm_pos, add_lt_add_iff_right_of_ne_top, Dynamics.coverEntropyInfEntourage_monotone, NormedField.exists_lt_nnnorm, Seminorm.finset_sup_le_sum, Order.Ideal.coe_sInf, NonUnitalAlgebra.toNonUnitalSubsemiring_eq_top, Module.Basis.addSubgroupOfClosure_apply, NonUnitalSubring.toNonUnitalSubsemiring_mono, Subgroup.IsSubnormal.isSubnormal_iff, TopologicalSpace.Opens.coe_eq_univ, Set.encard_le_one_iff_subsingleton, Module.End.invtSubmodule.disjoint_iff, sInf_eq_bot, Submonoid.closure_le, HahnEmbedding.Seed.domain_baseEmbedding, ENNReal.ofReal_mono, AlgebraicGeometry.Scheme.AffineZariskiSite.mem_grothendieckTopology_iff_sectionsOfPresieve, CompleteLat.Iso.mk_inv, NNRat.cast_lt_ofNat, Nat.one_div_lt_one_div, MeasureTheory.integral_Icc_eq_integral_Ioc, Set.preimage_const_mul_Ioi, balanced_absConvexHull, Cardinal.aleph_toNat, AlgebraicGeometry.IsAffineOpen.isoSpec_inv_ι, Field.primitive_element_iff_algHom_eq_of_eval, Fin.succFunctor_map, CFC.neg_negPart_le, StarSubalgebra.le_topologicalClosure, Real.toNNReal_lt_toNNReal_iff', Order.bot_lt_krullDim_iff, Booleanisation.comp_le_comp, convexHull_subset_affineSpan, ClosureOperator.idempotent, reflTransGen_wcovBy_eq_reflTransGen_covBy, MeasureTheory.L2.eLpNorm_rpow_two_norm_lt_top, AddMonoid.minOrder_le_natCard, wellFounded_lt_exact_sequence, inf_compl_eq_bot, UniformOnFun.hasBasis_uniformity_of_basis_aux₂, MonoidHom.liftOfRightInverse_comp_apply, Set.monotone_powerset, le_nhds_mul, antivary_neg_right, convexHull_eq_self, abs_le_iff_mul_self_le, IsDedekindDomain.HeightOneSpectrum.ideal_ne_top_iff_exists, imageToKernel_comp_mono, Finset.Icc_eq_filter_powerset, PowerSeries.le_weightedOrder_subst, AlgebraicGeometry.IsAffineOpen.isoSpec_hom_appTop, Finset.antitone_iff_forall_cons_le, Subalgebra.op_le_op_iff, AlgebraicGeometry.PresheafedSpace.restrictStalkIso_inv_eq_germ_assoc, AddSubmonoid.center_le_centralizer, CategoryTheory.Subobject.isoOfEqMk_hom, IntermediateField.Lifts.le_iff, Algebra.IsAlgebraic.normalClosure_le_iSup_adjoin, ENNReal.lt_div_iff_mul_lt, PNat.mod_le, SubAddAction.closure_le, Ordinal.range_preOmega, separableClosure.eq_bot_iff, Continuous.bddAbove_range_of_hasCompactSupport, Surreal.one_toGame, PartOrd.hom_id, cauchySeq_finset_of_geometric_bound, AlgebraicGeometry.Scheme.IdealSheafData.map_gc, Set.image_mulSingle_Ioc_left, continuousWithinAt_Ioi_iff_Ici, Subalgebra.LinearDisjoint.bot_left, ArchimedeanClass.mem_closedBallAddSubgroup_iff, AlgebraicGeometry.Scheme.IdealSheafData.map_ideal_basicOpen, AlgebraicGeometry.Scheme.AffineZariskiSite.toOpensFunctor_obj, lt_of_forall_ge_imp_ne, Set.Iio_union_right, convexOn_iff_convex_epigraph, Ordinal.one_lt_omega0, PointedCone.closure_eq, Finset.disjiUnion_Iic_disjointed, Subalgebra.topologicalClosure_adjoin_le_centralizer_centralizer, sign_eq_neg_one_iff, FiniteArchimedeanClass.withTopOrderIso_apply_coe, Filter.le_comap_top, Order.height_eq_coe_iff_minimal_le_height, EisensteinSeries.tsum_symmetricIco_tsum_eq_S_act, Finset.sum_le_sum_of_ne_zero, isUpperSet_iff_Ioi_subset, Subalgebra.rank_bot, TopCat.Presheaf.presheafEquivOfIso_inverse_map_app, CategoryTheory.SimplicialThickening.functor_comp, Multiset.instAddLeftReflectLE, LinearOrderedField.mem_cutMap_iff, AlgebraicGeometry.instIsIsoModulesSpecOfCarrierFromTildeΓFreeOpensCarrierCarrierCommRingCat, LieSubmodule.lie_le_left, instAntisymmLe, AlgebraicGeometry.RingedSpace.zeroLocus_singleton, TopCat.Presheaf.stalkFunctor_map_germ_assoc, dense_or_discrete, MeasureTheory.restrict_Ioc_eq_restrict_Icc, PosNum.cast_lt, HolderWith.nndist_le, ClosureOperator.isClosed_closure, Module.End.pos_finrank_genEigenspace_of_hasEigenvalue, CategoryTheory.Subfunctor.range_id, Set.Iio_injective, antitoneOn_iff_forall_lt', Finset.orderEmbOfFin_compl_singleton_apply, AddSubmonoid.fg_iff_mul_fg, Subfield.le_topologicalClosure, Finset.val_le_iff, ENat.lt_coe_add_one_iff, Filter.le_pi_principal, Dense.borel_eq_generateFrom_Icc_mem_aux, AddSubgroup.le_centralizer, Dense.topology_eq_generateFrom, TopologicalSpace.OpenNhds.inclusionMapIso_hom_app, Asymptotics.isLittleO_pow_pow_atTop_of_lt, MeasureTheory.MemLp.meas_ge_lt_top_enorm, EReal.toENNReal_add_le, DFinsupp.Colex.isStrictOrder, IsOrderedRing.toIsOrderedModule, NNReal.agmSequences_succ, CircleDeg1Lift.translationNumber_mono, Metric.isBounded_of_abs_lt, Ordinal.preOmega_max, IsLocalization.coe_primeSpectrumOrderIso_apply_coe_asIdeal, monovary_neg_left, Cardinal.finset_card_lt_aleph0, FirstOrder.Language.HomClass.strictMono, Set.image_mul_const_Iio, binomialSeries_radius_ge_one, Finset.mulConst_le_divConst_sq, partialSups_disjoint_of_disjoint, CategoryTheory.Limits.kernelSubobject_arrow', le_sdiff_sup, Fin.map_succEmb_Iic, CategoryTheory.Subgroupoid.full_mono, nnnorm_zsmul_le, Complex.mem_slitPlane_iff_not_le_zero, nhdsSetWithin_prod_le, ENNReal.lt_top_of_mul_ne_top_right, ENat.card_le_one, trop_sInf_image, negPart_eq_ite_lt, Pi.card_Ioo, Polynomial.le_trailingDegree_X, Module.length_eq_rank, Coheyting.boundary_bot, Fin.finsetImage_rev_Ici, supClosure_univ, Ordinal.one_le_card, Set.Ioo.one_minus_lt_one, WithTop.nhds_coe, TopologicalSpace.Opens.mapIso_refl, Ordinal.nadd_one, NNReal.young_inequality_real, Subalgebra.LinearDisjoint.bot_right, CategoryTheory.Limits.CompleteLattice.finite_product_eq_finset_inf, CFilter.inf_le_right, nhdsWithin_mono, strictAntiOn_of_add_one_lt, ENNReal.orderIsoIicOneBirational_symm_apply, pure_le_nhds_iff, IntermediateField.rank_adjoin_simple_eq_one_iff, Pi.card_Ioi, FiniteMulArchimedeanClass.coe_congrOrderIso_apply, ProbabilityTheory.Kernel.setLIntegral_rnDeriv_le, Tactic.NormNum.isNat_nnrealSqrt, Scott.IsOpen.isUpperSet, Finset.wcovBy_insert, Ideal.le_jacobson, Nat.image_cast_int_Ico, CategoryTheory.ComposableArrows.fourδ₃Toδ₂_app_zero, Concept.extent_bot, ClosureOperator.id_apply, SetTheory.PGame.nim_add_nim_equiv, Submonoid.powers_le, CategoryTheory.HasLiftingProperty.transfiniteComposition.SqStruct.w₂_assoc, Ordinal.veblen_zero_strictMono, InitialSeg.isLowerSet_range, RootPairing.range_polarizationIn_le_span_coroot, wbtw_mul_sub_add_iff, Order.one_le_iff_pos, EMetric.pos_of_mem_ball, NNRat.preimage_cast_Iio, compare_iff, CauSeq.const_pos, Dynamics.le_coverMincard_image, Matroid.le_eq_isMinor, ConcaveOn.convex_gt, Filter.tendsto_pow_atTop, Fin.preimage_cast_Ioc, AlgebraicGeometry.Scheme.Hom.fromNormalization_app, PseudoMetric.nonneg, CategoryTheory.Limits.CompleteLattice.coprod_eq_sup, Ioo_subset_Iotop, separableClosure.adjoin_eq_of_isAlgebraic_of_isSeparable, Polynomial.roots_expand_map_frobenius_le, ContinuousMap.idealOfSet_isMaximal_iff, Filter.map_add_left_nhdsGT, Filter.pure_le_iff, AddCon.toSetoid_eq_top, BoundedContinuousFunction.apply_le_nndist_zero, MeasureTheory.addContent_le_sum_of_subset_sUnion, Set.inv_mem_Ico_iff, IsCompl.disjoint_right_iff, Ideal.gc_map_comap, MonomialOrder.degree_sub_leadingTerm_lt_iff, Ordinal.le_add_sub, Submodule.map_inf_le, Ordinal.toGame_nadd, NNRat.ofNat_lt_cast, SetTheory.PGame.moveRight_toRightMovesNim, lowerClosure_mono, disjoint_nhds_atBot, TwoSidedIdeal.matrix_strictMono_of_nonempty, OrderHom.coe_sup, IsLeftCancelAdd.addLeftReflectLE_of_addLeftReflectLT, Set.biInter_le, Summable.tsum_ofReal_lt_top, le_iff_compact_le_imp, ENNReal.mul_self_lt_top_iff, balancedHull_convexHull_subseteq_absConvexHull, MonoidHom.noncommCoprod_injective, Fin.prod_Ioc_cast, not_isMax_iff_ne_top, PrimeSpectrum.le_vanishingIdeal_zeroLocus, Ne.lt_or_gt, Matrix.posDef_diagonal_iff, abs_inf_sub_inf_le_abs, AlgebraicGeometry.Scheme.Hom.eqToHom_app, Submodule.map_subtype_le, Fin.preimage_val_Iio_val, AddSubgroup.inf_normalizer_le_normalizer_inf, Set.pairwise_disjoint_Ioo_intCast, notMem_iff_exists_ne_and_isConjRoot, sign_eq_one_iff, Matroid.spanning_iff_eRk_le, exists_le_add_iff_le_left, Set.preimage_const_sub_Iio, Finsupp.coe_orderIsoMultiset, MeasureTheory.measure_union_le, Set.Ioi_injective, TopCat.presheafToTypes_obj, Set.unbounded_lt_inter_lt, Finset.strictAnti_iff_forall_lt_insert, PartOrd.id_apply, MeasureTheory.lintegral_mul_const_le, BoundedContinuousFunction.nndist_le_two_nnnorm, Set.image_update_Ioo_right, mem_uniformity_edist, BoundedContinuousFunction.NNReal.upper_bound, Ordinal.right_le_veblen, Pi.isAtomic, WithBot.some_mem_insertBot, instRepresentablyFlatOpensCarrierMap, ContinuousAt.partialSups_apply, MeasureTheory.Ioo_ae_eq_Ioc', AddSubgroup.closure_le, codiscrete_le_cofinite, NonemptyInterval.inv_pure, MeasureTheory.ae_measure_preimage_add_right_lt_top, MeasureTheory.VectorMeasure.restrict_le_restrict_iff, Submodule.le_dualCoannihilator_dualAnnihilator, Ne.bot_lt, Cardinal.toENat_lt_top, FractionalIdeal.mul_right_strictMono, AlgebraicGeometry.instIsAffineHomιBasicOpen, MonomialOrder.degree_sub_leadingTerm_le, Coheyting.boundary_le_boundary_sup_sup_boundary_inf_right, HasFiniteFPowerSeriesAt.eventually, Algebra.gc, Topology.IsLowerSet.lowerSet_le_lower, left_lt_add_div_two, Profinite.isIso_asLimitCone_lift, MeasurableSpace.monotone_map, Subalgebra.rank_toSubmodule, Metric.exists_real_pos_lt_infEDist_of_notMem_closure, CStarAlgebra.le_nnnorm_of_mem_quasispectrum, Order.bot_lt_succ, Finset.Colex.insert_lt_insert, LieIdeal.comap_incl_eq_bot, Cardinal.succ_aleph0, AlgebraicGeometry.Scheme.id_app, LowerSet.coe_mul, Valuation.exists_div_eq_of_surjective, le_abs', Valuation.Integers.maximalIdeal_pow_eq_setOf_le_v_algebraMap_pow, upperClosure_smul, AlgebraicGeometry.Scheme.IdealSheafData.comap_map_le, LinearGrowth.le_linearGrowthInf_add, ENNReal.lt_top_of_sum_ne_top, CStarAlgebra.exists_sum_four_nonneg, Codisjoint.eq_top, Submonoid.mem_closure_image_one_lt_iff, Cardinal.lt_aleph0_iff_fintype, algebraMap_le_algebraMap, compl_bot, WithBot.one_le_iff_pos, AlgebraicGeometry.IsClosedImmersion.isIso_iff_ker_eq_bot, RingCone.coe_nonneg, cfcₙ_le_iff, LinearOrderedCommMonoidWithZero.toPosMulStrictMono, CategoryTheory.Limits.instEpiFactorThruImageSubobjectOfHasEqualizers, setOf_riemannianEDist_lt_subset_nhds', Nucleus.le_apply', LieIdeal.map_eq_bot_iff, Module.FinitePresentation.exists_basis_localizedModule_powers, AddSubmonoid.monotone_map, Algebra.adjoin_empty, Nat.ceil_le, NNReal.map_coe_nhdsGE, gc_coinduced_induced, OrderMonoidHom.comp_mul, Submodule.lt_top_of_finrank_lt_finrank, CategoryTheory.Subfunctor.range_comp_le, Subalgebra.fg_bot_toSubmodule, nhdsLE_basis, AddSubmonoid.toSubmonoid'_closure, Alexandrov.projSup_map, isAtom_compl, RelEmbedding.toOrderHom_injective, Algebra.botEquiv_symm_apply, Subsemiring.coe_nonneg, Int.preimage_Ico, Finset.card_mono, MeasureTheory.IsHahnDecomposition.le_on, DFinsupp.card_Ioc, le_add_iff_lt_left_or_exists_le, NNReal.sqrt_lt_sqrt, Filter.le_pi, one_le_formalMultilinearSeries_geometric_radius, AlgebraicGeometry.flat_iff, ContinuousLinearMap.le_opNorm_enorm, HahnSeries.le_orderTop_iff_forall, CategoryTheory.Subobject.hasLimitsOfSize, specializes_TFAE, MeasurableSpace.generateFrom_singleton_empty, SetLike.coe_mono, Seminorm.comp_add_le, MvPowerSeries.le_order_pow_of_constantCoeff_eq_zero, NonUnitalStarAlgebra.toNonUnitalSubalgebra_bot, Topology.IsInducing.functorNhds_map, MonoidHom.coe_toAdditive_ker, PiLp.nndist_eq_of_L2, Set.Ico_diff_left, AffineSubspace.notMem_bot, TopCat.Presheaf.SheafConditionEqualizerProducts.res_π_apply, CategoryTheory.Subfunctor.equivalenceMonoOver_counitIso, CategoryTheory.MorphismProperty.transfiniteCompositions_monotone, IsUltrametricDist.nnnorm_zsmul_le, MeasureTheory.IsFiniteMeasure.measure_univ_lt_top, Abelianization.commutator_subset_ker, MeasureTheory.eLpNorm_le_of_ae_bound, Filter.mulRightMono, AlgebraicGeometry.Scheme.zeroLocus_map_of_eq, IsCompl.disjoint_left_iff, Projectivization.Subspace.mem_submodule_iff, NNReal.inner_le_Lp_mul_Lq, AlgebraicGeometry.IsAffineOpen.toSpecΓ_fromSpec, AddSubmonoid.closure_singleton_le_iff_mem, Valuation.Integers.isPrincipalIdealRing_iff_not_denselyOrdered_mrange, GroupSeminorm.coe_lt_coe, lt_trichotomy, NucleusClass.le_apply, IsAlgClosed.dvd_iff_roots_le_roots, bot_inf_eq, ENNReal.nhdsGT_one_neBot, Fin.map_finCongr_Ico, WithTop.isLUB_sSup', Multiset.map_add_right_Icc, TwoSidedIdeal.coe_asIdeal, SetTheory.Game.addLeftStrictMono, latticeClosure_min, Set.unbounded_lt_Ioi, Cardinal.lift_lt_nat_iff, Complementeds.disjoint_coe, Finsupp.DegLex.single_strictAnti, WithBot.bot_mem_insertBot, List.trop_minimum, Finset.Ici_top, TopCat.Presheaf.pushforwardPullbackAdjunction_unit_app_app_germToPullbackStalk, ENNReal.rpow_add_rpow_le_add, not_supPrime_bot, Pi.Icc_eq, Fin.finsetImage_succ_Ioi, Set.Icc_subset_Icc_union_Icc, Finset.lt_max'_of_mem_erase_max', ENNReal.coe_lt_top, Ordinal.toNatOrdinal_max, WithBot.Icc_bot_coe, OrderHom.partBind_coe, Multiset.monotone_sym2, AlgebraicGeometry.IsAffineOpen.fromSpec_primeIdealOf, upperSemicontinuousWithinAt_iff_frequently, Filter.specializes_iff_le, Set.IciExtend_apply, Subfield.extendScalars_injective, Set.uIoc_eq_union, not_infIrred_top, Finset.map_some_eraseNone, Set.IicExtend_self, ZFSet.le_succ_rank_sUnion, convexHull_range_eq_exists_affineCombination, Matrix.posSemidef_iff_isHermitian_and_spectrum_nonneg, MeasureTheory.SimpleFunc.finMeasSupp_iff_support, mabs_le_one, convexIndependent_iff_notMem_convexHull_diff, Matrix.isPrimitive_iff, Real.arcsin_projIcc, NonemptyFinLinOrd.Iso.mk_hom, nonpos_iff_eq_zero, CategoryTheory.Localization.essSurj_mapComposableArrows, AddAction.isCoatom_stabilizer_iff_preprimitive, AddSubmonoid.map_le_iff_le_comap, exists_glb_Ioi, Order.one_lt_height_iff, stdSimplex_eq_inter, LaurentPolynomial.degree_C_le, Subring.prod_mono_right, NonUnitalIsometricContinuousFunctionalCalculus.nnnorm_quasispectrum_le, ClosureOperator.closure_isGLB, OrderHom.map_gfp_comp, LinearMap.exists_map_addHaar_eq_smul_addHaar', Real.natCast_le_toNNReal, Set.ordConnected_uIoc, Order.le_succ_iff_eq_or_le, Nimber.toOrdinal_eq_zero, lowerBounds_Ioi, MeasureTheory.eLpNorm_smul_le_eLpNorm_top_mul_eLpNorm, CategoryTheory.ComposableArrows.isIso_iff₁, CategoryTheory.CosimplicialObject.δ_comp_σ_of_gt', Ordinal.le_iff_deriv, RingHom.closure_preimage_le, countable_setOf_isolated_right_within, tendsto_mabs_nhdsNE_one, MeasureTheory.iInf_mul_le_setLIntegral, lt_max_iff, LieModule.coe_lowerCentralSeries_ideal_le, Set.preimage_const_div_Ioc, HahnSeries.le_order_smul, ofAntisymmetrization_lt_ofAntisymmetrization_iff, PointedCone.dual_univ, CategoryTheory.Limits.factorThruImageSubobject_comp_self, OmegaCompletePartialOrder.ContinuousHom.map_ωSup', egauge_le_of_smul_mem_of_ne, mulPosMono_iff_mulPosStrictMono, Finset.sum_mul_sq_le_sq_mul_sq, Ordinal.natCast_lt_gamma, EReal.coe_toReal_le, MeasureTheory.measure_sigmaFiniteSetGE_ge, EMetric.exists_real_pos_lt_infEdist_of_notMem_closure, FractionalIdeal.coe_ideal_le_self_mul_inv, TopologicalSpace.Opens.val_apply, Pi.toLex_strictMono, IsDedekindDomain.HeightOneSpectrum.intValuation_lt_one_iff_mem, LinearMap.range_add_le, Prod.instPosSMulStrictMono, DistribLattice.mem_ideal_sup_principal, MvPolynomial.restrictTotalDegree_le_restrictDegree, Set.monotone_projIci, iSup_lt_iff, Ordinal.toGame_natCast, IntermediateField.relfinrank_eq_one_iff, ENNReal.inv_lt_inv, EReal.coe_lt_coe_iff, Order.coheight_le_iff, FirstOrder.Language.Substructure.lift_card_closure_le, natCast_le_analyticOrderAt_iff_iteratedDeriv_eq_zero, GeneralizedCoheytingAlgebra.sdiff_le_iff, Nat.ofNat_lt_cast, ENat.iSup_coe_lt_top, FractionalIdeal.instPosMulReflectLENonZeroDivisors, AlgebraicGeometry.Scheme.homOfLE_rfl, Valuation.Integers.dvd_iff_le, Ideal.comap_jacobson, Subgroup.characteristic_iff_le_map, List.maximum_concat, MeasureTheory.OuterMeasure.map_mono, AddAction.IsPreprimitive.isCoatom_stabilizer_of_isPreprimitive, Cardinal.toENat_congr, CategoryTheory.Limits.instIsWellOrderContinuousCompOfPreservesWellOrderContinuousOfShape, inv_antitoneOn_Iio, AlgebraicGeometry.IsOpenImmersion.lift_app, Polynomial.emultiplicity_le_one_of_separable, MeasureTheory.condExp_bot_ae_eq, OrderMonoidWithZeroHom.mul_apply, AddSubgroup.map_le_map_iff', EReal.mem_nhds_top_iff, ENat.range_natCast, Order.Ideal.isPrime_iff_mem_or_mem, Finset.sum_min_le, CategoryTheory.nerve.δ₂_two, Ordinal.toPGameEmbedding_apply, WithLp.edist_snd_le, Polynomial.eventually_no_roots, AlgebraicGeometry.Scheme.AffineZariskiSite.instIsCoverDenseOpensToOpensFunctorGrothendieckTopologyCarrierCarrierCommRingCat, AddSubmonoid.closure_mono, Fin.sum_Ici_succ, MeasureTheory.tsum_meas_le_meas_iUnion_of_disjoint₀, lt_min_iff, Nucleus.bot_apply, FiniteArchimedeanClass.mem_closedBall_iff, CategoryTheory.Limits.kernelSubobject_comp_le, FirstOrder.Language.Substructure.closure_eq, Interval.coe_pure, PSet.rank_pair, TopologicalSpace.le_generateFrom_iff_subset_isOpen, Fin.map_valEmbedding_Iic, Dynamics.coverEntropyInf_le_coverEntropy, Ordinal.le_omega_self, NonUnitalSubsemiring.centralizer_le, Multiset.le_bind, DiscreteQuotient.proj_bot_injective, MeasureTheory.OuterMeasure.restrict_le_self, FirstOrder.Language.DirectLimit.le_partialEquivLimit, antivary_iff_exists_antitone_monotone, LieSubalgebra.gc_map_comap, Even.pow_pos_iff, CategoryTheory.Sieve.generate_of_contains_isSplitEpi, le_generateFrom, AlgebraicGeometry.IsAffineOpen.isoSpec_inv_ι_assoc, EReal.limsup_add_le, germ_skyscraperPresheafStalkOfSpecializes_hom, compare_le_iff_le, Subalgebra.rank_top, Set.sub_mem_Ioo_iff_left, antivary_inv_right, Seminorm.ball_bot, Cardinal.aleph_zero, Set.Ici_add_bij, LieIdeal.map_comap_le, AddSubgroup.characteristic_iff_map_le, Int.Ioc_filter_dvd_card, Ordinal.lift_le, discreteTopology_iff_orderTopology_of_pred_succ, AlgebraicGeometry.HasRingHomProperty.iff_of_isAffine, IsGaloisGroup.intermediateFieldEquivSubgroup_symm_apply_toDual, Additive.mem_toAddSubgroup, strictMonoOn_of_sub_one_lt, LinearMap.IsSymmetric.eigenvalues_antitone, NNReal.lintegral_mul_le_Lp_mul_Lq, Subalgebra.bot_eq_top_of_rank_eq_one, RingCon.toCon_bot, CategoryTheory.ComposableArrows.IsComplex.zero', Ordinal.isNormal_iff_strictMono_and_continuous, Metric.mem_eball, AddSubsemigroup.gc_map_comap, toZ_le_iff, AntilipschitzWith.ediam_preimage_le, Finset.Icc_map_sectR, Valuation.one_rel_iff, MulArchimedeanClass.orderHom_top, UniformSpace.Core.comp, EReal.coe_lt_top, Valuation.le_one_of_subsingleton, NonnegSpectrumClass.iff_spectrum_nonneg, Ordinal.isSuccLimit_iff_omega0_dvd, SetTheory.Game.not_lf, Fin.image_castAdd_Iio, Set.Intersecting.bot_notMem, tendsto_inv_nhdsGT, MeasureTheory.OuterMeasureClass.measure_mono, Set.image_update_Icc_right, Unitization.inr_nonneg_iff, IsClosedMap.comap_nhdsSet_le, Metric.boundedSpace_iff_edist, mabs_cases, Nat.preimage_Ico, GenContFract.abs_sub_convs_le, Metric.exists_pos_forall_lt_edist, CategoryTheory.Functor.WellOrderInductionData.Extension.zero_val, instIsLinearOrderGe, OrderEmbedding.birkhoffFinset_sup, PowerSeries.le_order_subst_left, Part.Fix.approx_mono, Ordinal.gamma_succ_eq_nfp, NNRat.castOrderEmbedding_apply, PointedCone.dual_eq_iInter_dual_singleton, continuousAt_iff_lower_upperSemicontinuousAt, sInfHom.top_apply, CategoryTheory.MorphismProperty.transfiniteCompositions_le_llp_rlp, Filter.map₂_inf_subset_left, FractionalIdeal.extendedHomₐ_le_one_iff, Subalgebra.opEquiv_apply, UniqueFactorizationMonoid.dvdNotUnit_iff_normalizedFactors_lt_normalizedFactors, separableClosure.adjoin_le, PartOrdEmb.Iso.mk_inv, ArchimedeanClass.closedBall_top, Set.preimage_add_const_Ici, Complex.orderClosedTopology, MeasureTheory.Submartingale.stoppedValue_leastGE_eLpNorm_le', PosSMulStrictMono.toPosSMulReflectLT, CategoryTheory.Subfunctor.Subpresheaf.range_toRange, ENat.one_le_iff_ne_zero, AlgebraicGeometry.Scheme.IdealSheafData.le_def, SetSemiring.addLeftMono, Topology.IsInducing.functor_map, MeasureTheory.exists_laverage_le, AlgebraicGeometry.SheafedSpace.restrictTopIso_hom, IsGalois.mem_bot_iff_fixed, WithLp.nndist_snd_le, Finsupp.Colex.single_strictMono, differentiableWithinAt_Ioi_iff_Ici, MeasureTheory.setLIntegral_lt_top_of_isCompact, NonarchimedeanHomClass.map_add_le_max, Pi.partialSups_apply, Matrix.dotProduct_self_star_pos_iff, IdemSemiring.toMulRightMono, FirstOrder.Language.Substructure.mem_closure_iff_of_isRelational, PNat.addLeftStrictMono, lt_one_add, SimpleGraph.Subgraph.induce_mono_right, MulOpposite.unop_nonneg, InnerProductSpace.nullSubmodule_le_ker_toDualMap_right, FiniteGaloisIntermediateField.adjoin_simple_le_iff, Ultrafilter.le_of_inf_neBot', Subalgebra.finrank_eq_one_iff, MeasureTheory.ae_lt_top, countable_image_lt_image_Iio, bot_notMem_iff, Ordinal.lt_lsub_iff, IsCompact.exists_sSup_image_eq_and_ge, IsOrderedAddMonoid.toAddLeftMono, ENNReal.coe_lt_ofReal, IsOrderedAddMonoid.toAddRightMono, TopCat.Presheaf.isSheaf_of_isOpenEmbedding, InfPrime.inf_le, Ordinal.veblenWith_left_monotone, separableClosure.le_restrictScalars, AlgebraicGeometry.Scheme.AffineZariskiSite.restrictIsoSpec_inv_app, Finset.sup_of_mem, SetTheory.PGame.toRightMovesNim_symm_lt, Submodule.neg_le_neg, Ideal.comap_le_iff_le_map, AddSubgroup.normal_addSubgroupOf_iff_le_normalizer_inf, AlgebraicGeometry.ΓSpec.toSpecΓ_unop, PointedCone.dual_dual_flip_dual, FirstOrder.Language.Substructure.closure_mono, Subgroup.le_zpowers_iff, Polynomial.IsRoot.norm_lt_cauchyBound, MeasureTheory.measure_lt_top_of_isCompact_of_isMulLeftInvariant', ONote.NFBelow.repr_lt, AbsoluteValue.exists_lt_one_one_le_of_not_isEquiv, monovary_iff_exists_monotone, MeasureTheory.OuterMeasure.toMeasure_top, nnnorm_le_pi_nnnorm', Pi.toColex_update_lt_self_iff, Ordinal.le_of_dvd, Order.Ideal.IsPrime.mem_or_compl_mem, AlgebraicGeometry.isClosedImmersion_diagonal_restrict_diagonalCoverDiagonalRange, PointedCone.coe_comap, WithZero.withZeroUnitsEquiv_strictMono, LinearOrderedAddCommGroupWithTop.sub_lt_sub_iff_left_of_ne_top, MeasureTheory.MemLp.eLpNorm_indicator_norm_ge_le, Set.sub_mem_Icc_iff_left, NNReal.bddAbove_range_natCast_iff, AddEquiv.comapAddSubgroup_symm_apply, CategoryTheory.Sieve.ofArrows_le_ofObjects, Ordinal.le_nfpFamily, hahnEmbedding_isOrderedAddMonoid, TopologicalSpace.Opens.IsBasis.le_iff, IntermediateField.algebra_adjoin_le_adjoin, List.maximum_append, TestFunction.originalTop_le, ENat.lt_one_iff_eq_zero, Matroid.toENat_cRk_eq, Finset.Ioi_mul_Ici_subset', nnnorm_div_le, HasCountableSeparatingOn.range_Iic, CategoryTheory.Subfunctor.subobjectMk_range_arrow, tsub_pos_iff_not_le, exists_lt_mem_inter_of_not_pairwise_disjoint, Complex.not_lt_iff, star_neg_iff, BoundedContinuousFunction.lintegral_le_edist_mul, LinearOrderedCommGroupWithZero.discrete_or_denselyOrdered, Nonneg.existsAddOfLE, principal_le_nhdsSet, instPosSMulStrictMonoNatOfIsOrderedAddMonoid, IsSimpleOrder.instIsCoatomic, GenContFract.IntFractPair.nth_stream_fr_nonneg, ConcaveOn.ge_on_segment, FirstOrder.Language.DirectLimit.rangeLiftInclusion, Ordinal.principal_mul_iff_le_two_or_omega0_opow_opow, IntermediateField.map_bot, Finset.isLeast_min', IdealFilter.isUniform_iff_exists_ringFilterBasis, ENat.floor_le, Submodule.toSubMulAction_mono, EMetric.diam_closedBall, subalgebra_top_rank_eq_submodule_top_rank, ConvexOn.le_max_of_mem_segment, Subsemigroup.op_le_iff, isComplemented_top, Multiset.card_strictMono, List.argmin_cons, nhds_le_nhdsSet_iff, Subrepresentation.submoduleSubrepresentationOrderIso_symm_apply, MvPowerSeries.lexOrder_mul_ge, Fin.image_castSucc_Iio, Tuple.monotone_sort, le_algebraicClosure', PrimeSpectrum.basicOpen_eq_bot_iff, NNReal.sq_sqrt, OrderIsoClass.toBoundedOrderHomClass, MeasureTheory.MemLp.eLpNorm_mk_lt_top, NNReal.tendsto_pow_atTop_nhds_zero_iff, Set.Ioi_ssubset_Ioi_iff, Set.encard_le_encard_diff_add_encard, ProbabilityTheory.Kernel.partialTraj_eq_prod, Ordinal.lt_epsilon0, IsUltrametricDist.nnnorm_tsum_le, ringKrullDim_le_ringKrullDim_quotient_add_encard, CategoryTheory.Dial.tensorObj_rel, Fin.map_valEmbedding_Ioc, AlgebraicGeometry.Scheme.IdealSheafData.vanishingIdeal_bot, Finpartition.parts_inf, Complex.arg_eq_pi_iff_lt_zero, AlgebraicGeometry.SheafedSpace.Γ_map_op, MeasureTheory.eLpNormEssSup_const_smul_le', Ordinal.toGame_le_iff, MeasureTheory.GridLines.T_insert_le_T_lmarginal_singleton, Ordinal.veblen_pos, Ideal.map_eq_bot_iff_le_ker, SSet.S.le_def, Icc_subset_segment, AlgebraicGeometry.tilde.instIsLocalizedModuleCarrierCarrierObjOppositeOpensCarrierCarrierCommRingCatSpecModuleCatPresheafModulesSheafModulesSpecToSheafOpBasicOpenPowersHomToOpen, EisensteinSeries.eisensteinSeries_tendstoLocallyUniformly, AlgebraicGeometry.Scheme.isNilpotent_iff_basicOpen_eq_bot_of_isCompact, MeasurableSpace.monotone_comap, Continuous.coinduced_le, MeasureTheory.exists_pos_lintegral_lt_of_sigmaFinite, Finset.mem_uIcc, ENNReal.sub_lt_self_iff, LinearOrderedAddCommGroup.to_noMinOrder, Finset.Ici_erase, Order.krullDimLE_iff, Cardinal.lift_preAleph, Filter.tendsto_const_mul_atTop_iff, MeasureTheory.Measure.zero_le_toSignedMeasure, EReal.instOrderTopology, CategoryTheory.PreZeroHypercover.Hom.sieve₀_le_sieve₀, bot_eq_one, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.instIsIsoInvApp, Ordinal.omega_eq_preOmega, Ideal.le_comap_pow_ramificationIdx, PNat.le_find_iff, Set.image_const_sub_Icc, CategoryTheory.Subobject.hasFiniteLimits, Cardinal.lift_lt_aleph_one, contMDiffOn_projIcc, SSet.skeletonOfMono_succ, NNReal.agm_le_max, Mathlib.Meta.NormNum.isNNRat_lt_true, Cardinal.toENat_eq_one, Order.isSuccPrelimitRecOn_succ, Dynamics.dynEntourage_monotone, Finset.inf'_le, YoungDiagram.notMem_bot, Ordinal.toNatOrdinal_symm_eq, NonemptyFinLinOrd.hom_id, MeasureTheory.Lp.eLpNorm'_lim_le_liminf_eLpNorm', Filter.le_def, NonarchimedeanGroup.cauchySeq_prod_of_tendsto_cofinite_one, kstar_mul_le_kstar, OrderIso.isCompl_iff, MeasureTheory.MemLp.exists_eLpNorm_indicator_compl_lt, CategoryTheory.Subobject.map_id, posPart_eq_ite_lt, le_algebraMap_iff_spectrum_le, skyscraperPresheaf_map, Finset.max_abv_sum_one_le, Matrix.PosDef.dotProduct_mulVec_pos, Set.Ioo.coe_mul, EReal.neg_lt_neg_iff, monotone_closure, AlgebraicGeometry.Scheme.IdealSheafData.le_map_comap, mem_nhdsLE_iff_exists_Ioc_subset, EReal.coe_ennreal_pos, TopCat.Presheaf.generateEquivalenceOpensLe_inverse, Order.succ_eq_sInf, AlgebraicGeometry.Scheme.IdealSheafData.supportSet_eq_iInter_zeroLocus, ENat.not_lt_zero, MulPosMono.toMulPosReflectLT, MonomialOrder.degree_sub_LTerm_le, NonemptyInterval.length_neg, Metric.infEDist_le_edist_add_infEDist, Finset.le_sup, MeasureTheory.Lp.mul_meas_ge_le_pow_enorm, Fin.monotone_iff_le_succ, FirstOrder.Language.Substructure.closure_univ, MulPosReflectLT.toMulPosReflectLE, Rat.padicValuation_le_one_iff, Finset.Ico_succ_right_eq_Icc, MeasureTheory.OuterMeasure.mkMetric'.le_pre, MeasureTheory.Measure.rnDeriv_le_one_iff_le, Continuous.cfcₙ_nnreal', RingCon.toCon_top, StalkSkyscraperPresheafAdjunctionAuxs.toSkyscraperPresheaf_app, Ordinal.monotone_gamma, exists_lt_add_iff_lt_right, Ring.not_isField_iff_exists_ideal_bot_lt_and_lt_top, NNReal.HolderTriple.one_div_pos', MonomialOrder.lex_le_iff_of_unique, LinearOrderedCommGroup.isCyclic_iff_nonempty_equiv_int, Subgroup.relIndex_toAddSubgroup, CategoryTheory.Limits.kernelSubobjectMap_arrow_apply, Filter.map_le_map_iff_of_injOn, Measurable.lt, Subgroup.IsComplement'.disjoint, Fin.prod_Ioo_succ, Fin.attachFin_Ioc, AffineSubspace.lt_iff_le_and_exists, posMulMono_iff_posMulReflectLT, tendsto_ratCast_atTop_iff, AlgebraicGeometry.Scheme.Modules.map_smul, Set.disjoint_iff, Subsemigroup.coe_toAddSubsemigroup_apply, PredOrder.colimitRecOn_pred, Ordinal.mul_div_le, LSeries.abscissaOfAbsConv_le_of_isBigO_rpow, ArchimedeanClass.stdPart_eq_sSup, AlgebraicGeometry.RingedSpace.mem_basicOpen', Submodule.coe_iSup_of_chain, IsOpen.measure_pos_iff, OrderEmbedding.mulLeft_apply, Module.mapEvalEquiv_symm_apply, Submodule.surjOn_iff_le_map, max_eq_iff, Nonneg.instIsometricContinuousFunctionalCalculus, PosMulReflectLE.toPosMulReflectLT, IsDedekindDomain.HeightOneSpectrum.notMem_adicCompletionIntegers, AlgebraicGeometry.Scheme.IdealSheafData.glueDataObjHom_id, MeasureTheory.Content.sup_le', bot_eq_zero', tendsto_smul_comp_nat_floor_of_tendsto_nsmul, Hyperreal.gt_of_tendsto_zero_of_neg, Cardinal.mk_Iio_ord_toType, FirstOrder.Language.DirectLimit.dom_partialEquivLimit, AddSubgroup.IsSubnormal.isSubnormal_iff, Metric.thickening_eq_preimage_infEdist, Hyperreal.epsilon_pos, Submodule.map_le_smul_top, EReal.tendsto_nhds_bot_iff_real, PosSMulReflectLE.toPosSMulStrictMono, ENNReal.mul_left_strictMono, Ultrafilter.isAtom, Filter.monotone_principal, ENNReal.rpow_sum_le_const_mul_sum_rpow, Ideal.homogeneousCore.gc, LaurentPolynomial.degree_C_mul_T_le, Multiset.prod_min_le, IsSimpleOrder.instIsAtomistic, Set.IciExtend_coe, PrincipalSeg.image_Iio, LinearOrderedCommGroup.isCyclic_iff_not_denselyOrdered, instDenselyOrderedUnits, AddSubmonoid.pointwise_smul_le_iff₀, LipschitzWith.nnorm_le_mul, EReal.instPosMulReflectLT, LE.total, upperSemicontinuousOn_iff_preimage_Ici, AlgebraicGeometry.Scheme.mem_basicOpen, NNReal.Icc_subset_segment, QuotientAddGroup.le_comap_mk', Finset.prod_Ico_add, Hyperreal.archimedeanClassMk_epsilon_pos, LTSeries.monotone, AlgebraicGeometry.IsOpenImmersion.affineOpensEquiv_symm_apply_coe, Ordinal.top_typein, Finset.dens_pos, Real.natCast_lt_toNNReal, NNReal.holderTriple_iff, Set.mem_uIcc, LowerAdjoint.closure_union_closure_left, Multiset.disjSum_mono_left, Subobject.presheaf_map, Subgroup.IsSubnormal.exists_chain, ProbabilityTheory.Kernel.instAddLeftMono, CategoryTheory.SemilatticeInf.tensorObj, Ne.ge_iff_gt, NonemptyInterval.mul_eq_one_iff, AlgebraicGeometry.sourceLocalClosure.le, IsAbsoluteValue.abv_pos, disjoint_nhds_nhds, SetTheory.PGame.toLeftMovesNim_one_symm, AlgebraicGeometry.Scheme.affineBasicOpen_le, WithTop.eq_bot_iff_forall_le, PointedCone.dual_singleton, Finpartition.avoid_parts_val, isCyclic_tfae, FirstOrder.Language.Substructure.map_strictMono_of_injective, TensorProduct.edist_tmul_le, AlgebraicGeometry.SheafedSpace.comp_c_app', Finset.mem_insertNone, Finset.nonneg_subConst, FirstOrder.Language.Substructure.closure_image, MeasureTheory.eLpNorm_smul_le_eLpNorm_mul_eLpNorm_top, NonUnitalAlgebra.adjoin_mono, separableClosure.eq_bot_of_isPurelyInseparable, Nonneg.isOrderedCancelAddMonoid, Subalgebra.eq_bot_of_isPurelyInseparable_of_isSeparable, biInf_le, NNRat.cast_lt_zero, AbsConvex.absConvexHull_eq, SimpleGraph.comap_monotone, Fin.map_castAddEmb_Ioi, Subgroup.opEquiv_apply, mulLeftStrictMono_of_mulLeftReflectLE, MeasureTheory.Measure.restrict_iUnion_le, CategoryTheory.MorphismProperty.TransfiniteCompositionOfShape.mem_incl_app, LinearOrderedSemiField.toDenselyOrdered, MonoidHom.coe_toMultiplicative_ker, Filter.mem_cardinalGenerate_iff, IsCompl.le_right_iff, MeasureTheory.QuotientMeasureEqMeasurePreimage.covolume_ne_top, Set.Nonempty.not_disjoint, WithBot.orderIsoPUnitSumLex_bot, mulRothNumber_singleton, PointedCone.salient_iff_inter_neg_eq_singleton, AddSubmonoid.le_op_iff, Ordinal.sup_typein_succ, OmegaCompletePartialOrder.isLUB_range_ωSup, SSet.stdSimplex.isoNerve_inv_app_apply, BoxIntegral.Box.antitone_lower, Subalgebra.finite_bot, Filter.atTop_le_cofinite, IntermediateField.finSepDegree_bot, affineSpan_eq_top_iff_nonempty_of_subsingleton, LieSubmodule.gc_top_lie_normalizer, exists_pos_right_iff_sameRay_and_ne_zero, Hyperreal.tendsto_atTop_iff, SimpleGraph.edist_le_one_iff_adj_or_eq, closedConvexHull_isClosed, IsAddIndecomposable.baseOf_subset_pos, Set.powersetCard.ofFinEmbEquiv_apply, Ordinal.liftPrincipalSeg_top, AlgebraicGeometry.SpecMap_ΓSpecIso_inv_toSpecΓ_assoc, LinearOrderedCommMonoidWithZero.toZeroLeOneClass, OrderIso.apply_bliminf, MeasureTheory.lintegral_le_iSup, CategoryTheory.not_strictAnti_of_isArtinianObject, Ordinal.enum_type, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_inv_apply, ENNReal.mul_pos_iff, GaloisInsertion.l_top, AlgebraicGeometry.PresheafedSpace.ofRestrict_c_app, IsGalois.ofDual_intermediateFieldEquivSubgroup_apply, MeasureTheory.pdf.ae_lt_top, Set.notMem_uIoc, Ideal.span_singleton_mul_le_iff, Topology.IsUpper.tendsto_nhds_iff_lt, BoxIntegral.TaggedPrepartition.isPartition_single, kstar_eq_one, integrableOn_Ici_iff_integrableOn_Ioi', LinearMap.ker_le_comap, Mathlib.Tactic.Order.ToInt.toInt_lt_toInt, Complex.lt_def, Finset.nsmul_right_strictMonoOn, supClosure_min, EReal.lt_iff_exists_real_btwn, Nonneg.instIsStrictOrderedModule, MeasureTheory.addContent_union_le, LinearOrderedField.toPosSMulStrictMono_rat, UniformSpace.Core.refl, InitialSeg.isMin_apply_iff, AlgebraicGeometry.LocallyRingedSpace.toΓSpecCApp_iff, Filter.disjoint_principal_right, ENat.coe_lt_top, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.ofRestrict_invApp_apply, Subsemiring.le_pointwise_smul_iff₀, Ordinal.nadd_lt_nadd_iff_left, AlgebraicGeometry.IsAffineOpen.preimage, Filter.NeBot.div_zero_nonneg, Set.PartiallyWellOrderedOn.ProdLex_iff, CategoryTheory.ComposableArrows.IsComplex.cokerToKer_fac_assoc, MonomialOrder.degree_sPolynomial, Polynomial.isBigO_atTop_of_degree_le, CategoryTheory.GrothendieckTopology.top_covering, CategoryTheory.Triangulated.SpectralObject.ω₂_obj_obj₃, one_le_oneLePart, Cardinal.not_isSuccLimit_zero, thickenedIndicatorAux_le_one, Ordinal.right_le_veblenWith, Finset.val_covBy_val, AddSubgroup.map_le_range, Nat.preimage_ceil_zero, Cardinal.natCast_mul_lt_natCast_mul, NonUnitalSubring.le_topologicalClosure, nhdsGE_basis_Ico, Lattice.inf_le_right, Ordinal.nadd_eq_add, CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinsetObj_map, HahnSeries.BddBelow_zero, PrincipalSeg.lt_iff_lt, Continuous.bddAbove_range_of_hasCompactMulSupport, IsCompact.exists_sInf_image_eq_and_le, Cardinal.nat_lt_ofENat, polynomialFunctions.comap_compRightAlgHom_iccHomeoI, alternatingGroup.isCoatom_stabilizer_of_ncard_lt_ncard_compl, Matroid.setOf_dep_eq, Ideal.le_colon, closedAbsConvexHull_closure_eq_closedAbsConvexHull, Fin.strictMono_natAdd, Prod.Lex.le_iff', Filter.limsSup_top, hahnEmbedding_isOrderedModule, Ordinal.one_toType_eq, CategoryTheory.MorphismProperty.multiplicativeClosure_le_iff, AddSubgroup.toAddSubmonoid_mono, Localization.coe_algEquiv_symm, AddMonoidAlgebra.supDegree_sum_le, SimpleGraph.singletonFinsubgraph_le_adj_right, Ordinal.isNormal_omega, Metric.infEDist_pos_iff_notMem_closure, Delone.DeloneSet.coveringRadius_pos, List.Perm.minimum_eq, Fin.succ_top, AlgebraicGeometry.LocallyRingedSpace.Γ_obj, LinearOrderedAddCommGroup.tendsto_nhds, AlgebraicGeometry.Spec.sheafedSpaceObj_presheaf, YoungDiagram.transpose_le_iff, Set.preimage_mul_const_Ioc, ENNReal.image_coe_Ioc, CategoryTheory.MorphismProperty.le_transfiniteCompositions, Filter.covariant_swap_sub, CategoryTheory.ComposableArrows.δlastFunctor_obj_map, AlgebraicGeometry.Spec.map_app, sInf_le_iff, closure_Ioc, Subgroup.MapSubtype.orderIso_symm_apply, upperClosure_anti, NonUnitalStarAlgebra.adjoin_le_centralizer_centralizer, Set.Ico_add_Ioc_subset, LinearMap.range_lt_top_of_det_eq_zero, Finset.Ico_add_one_add_one_eq_Ioc, LinearMap.comap_le_comap_iff, Set.Ioc_inter_Iic, FractionalIdeal.coeIdeal_le_coeIdeal, Fin.finsetImage_castLE_Icc, Set.Iio_inter_Ioo, CoheytingAlgebra.top_sdiff, restrict₂_comp_IicProdIoc, thickenedIndicator_mono, NNReal.instOrderedSub, AddGroupTopology.toTopologicalSpace_bot, Finset.isLUB_iff_isGreatest, AlgebraicGeometry.Scheme.Spec_fromSpecStalk, IntermediateField.sepDegree_top, ProbabilityTheory.Kernel.traj_eq_prod, MeasurableSpace.generateMeasurable_eq_rec, RCLike.ofReal_lt_ofReal, CategoryTheory.Limits.pullback_equalizer, IsGaloisGroup.fixedPoints_eq_bot, ProfiniteGrp.diagram_obj, MonomialOrder.degree_X_le_single, NatOrdinal.lt_wf, IsStarProjection.nonneg, PNat.lt_succ_self, algebraMap_le_cfc_iff, PointedCone.dual_span, AddSubgroup.relIndex_toSubgroup, Ordinal.toPGame_moveLeft', Profinite.NobelingProof.succ_mono, Submodule.comap_subtype_le_iff, stdSimplexEquivIcc_zero, Complex.rectangle_eq_convexHull, Ordinal.apply_lt_veblenWith_apply_iff, LowerAdjoint.closure_iUnion_closure, MeasureTheory.Iio_ae_eq_Iic, instIsLinearOrderLe, dirSupInacc_iff_forall_sSup, Filter.le_comap_map, Int.cobounded_eq, AlgebraicGeometry.IsAffineOpen.ideal_ext_iff, ZModModule.exists_submodule_subset_card_le, Fin.image_val_Ioo, NonUnitalSubring.closure_le, WithBot.add_one_le_iff, tendsto_zero_geometric_tsum_pnat, monovaryOn_inv_left, Subsemigroup.map_le_map_iff_of_injective, FirstOrder.Language.Substructure.cg_closure_singleton, Matrix.PosSemidef.diag_nonneg, LieModule.disjoint_genWeightSpace, one_lt_iff_ne_one, Subgroup.isCoatom_comap, AlgebraicGeometry.morphismRestrict_id, IsNonarchimedean.add_pow_le, PrimeSpectrum.asIdeal_le_asIdeal, Set.Icc.instZeroLEOneClass, Fin.prod_Ico_castAdd, Int.fract_eq_iff, Matroid.Matroidᵣ.lt_iff, Nucleus.le_apply, PseudoEMetricSpace.edist_triangle, closure_Ioi, convexOn_iff_slope_mono_adjacent, AddQuantale.instAddRightMono, Polynomial.isIntegral_isLocalization_polynomial_quotient, UpperSet.compl_le_compl, SimpleGraph.edgeFinset_ssubset_edgeFinset, IsLocalization.map_nonZeroDivisors_le, lt_tsub_iff_left, Lattice.BicartSq.le₁₃, Nat.image_cast_int_Iio, Set.mulIndicator_le_self, UpperSet.Ici_iSup₂, ENNReal.lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_ne_top, Filter.inv_le_iff_le_inv, CategoryTheory.Limits.CompleteLattice.limitCone_cone_pt, lt_iff_exists_mul, Set.ordConnected_uIcc, mem_nhdsGE_iff_exists_Icc_subset, MeasureTheory.exists_continuous_eLpNorm_sub_le_of_closed, AlgebraicGeometry.tilde.toOpen_map_app, MeasureTheory.tendstoUniformlyOn_of_ae_tendsto, Affine.Simplex.sOppSide_affineSpan_faceOpposite_point_left_iff, mabs_eq_self, MeasureTheory.AEStronglyMeasurable.aestronglyMeasurable_uIoc_iff, AddSubgroup.map_centralizer_le_centralizer_image, IsCoercive.antilipschitz, OrderIso.withBotCongr_symm, IsLprojection.le_def, OrderIso.ofRelIsoLT_toRelIsoLT, CategoryTheory.ComposableArrows.Precomp.map_one_one, add_le_add_iff_right_of_ne_top, MeasureTheory.Content.contentRegular_exists_compact, convexHull_subset_closedConvexHull, closure_Ioo, MvPowerSeries.le_order_map, CategoryTheory.Limits.ColimitPresentation.self_ι, isComplete_iff_ultrafilter', MeasureTheory.Measure.InnerRegularWRT.exists_subset_lt_add, atBot_le_cocompact, IsStarProjection.one_sub_nonneg, EisensteinSeries.tsum_symmetricIco_tsum_sub_eq, CategoryTheory.MorphismProperty.transfiniteCompositions_le_iff, Submodule.neg_le, Filter.le_pure_iff', ClosureOperator.id_isClosed, Hyperreal.coeRingHom_toFun, SSet.skeleton_zero, Order.Ideal.principal_top, Even.zpow_pos, TopologicalSpace.Opens.map_id_eq, AlgebraicGeometry.IsAffineOpen.primeIdealOf_genericPoint, FirstOrder.Language.Structure.fg_iff, FirstOrder.Language.Substructure.cg_closure, Ideal.isMaximal_def, MeasureTheory.preVariation.mono, Subsemiring.center_le_centralizer, RCLike.ofReal_lt_zero, Algebra.ZariskisMainProperty.exists_fg_and_exists_notMem_and_awayMap_bijective, AlgebraicGeometry.LocallyRingedSpace.stalkMap_germ, Pi.toLex_monotone, CategoryTheory.Limits.kernelSubobjectMap_id, Dynamics.coverMincard_monotone_subset, Nat.floor_pos, Ultrafilter.disjoint_iff_not_le, himp_eq_sSup, CategoryTheory.Functor.mapComposableArrows_map_app, AlgebraicGeometry.Scheme.Hom.id_image, StieltjesFunction.right_continuous', sdiff_triangle, AlgebraicGeometry.IsAffineOpen.mem_ideal_iff, CategoryTheory.Limits.kernelSubobject_zero, MeasureTheory.IsAddFundamentalDomain.measure_le_of_pairwise_disjoint, Subsemigroup.le_topologicalClosure, AlgebraicGeometry.Scheme.toSpecΓ_base, lt_iSup_iff, AlgebraicGeometry.Smooth.iff_forall_exists_isStandardSmooth, leOnePart_anti, Multiset.card_Ioo_eq_card_Ico_sub_one, Ordinal.veblen_left_monotone, AntilipschitzWith.comap_uniformity_le, instIsPartialOrderGe, UpperSet.Ici_strictMono, Interval.mem_pure_self, HahnEmbedding.Partial.toOrderAddMonoidHom_injective, monovaryOn_iff_forall_mul_nonneg
bridge: BoxIntegral.IntegrationParams.MemBaseSet.exists_compl, BoxIntegral.IntegrationParams.MemBaseSet.distortion_le
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