Documentation Verification Report

Defs

📁 Source: Mathlib/Order/ConditionallyCompleteLattice/Defs.lean

Statistics

MetricCount
DefinitionsConditionallyCompleteLattice, toInfSet, toLattice, toSupSet, ConditionallyCompleteLinearOrder, toConditionallyCompleteLattice, toDecidableEq, toDecidableLE, toDecidableLT, toOrd, ConditionallyCompleteLinearOrderBot, toConditionallyCompleteLinearOrder, toOrderBot, conditionallyCompleteLinearOrderBot, conditionallyCompleteLatticeOfLatticeOfsInf, conditionallyCompleteLatticeOfLatticeOfsSup, conditionallyCompleteLatticeOfsInf, conditionallyCompleteLatticeOfsSup
18
TheoremscsInf_le, csSup_le, le_csInf, le_csSup, compare_eq_compareOfLessAndEq, csInf_of_not_bddBelow, csSup_of_not_bddAbove, le_total, csSup_empty
9
Total27

ConditionallyCompleteLattice

Definitions

NameCategoryTheorems
toInfSet 📖CompOp
16 mathmath: Int.csInf_mem, Int.csInf_eq_least_of_bdd, ConditionallyCompleteLinearOrder.csInf_of_not_bddBelow, le_csInf, csInf_le_of_le, Int.csInf_eq_leastOfBdd, Int.csInf_of_not_bddBelow, IsGLB.csInf_eq, csInf_le_iff, isGLB_csInf, le_csInf, csInf_le, csInf_le_csInf, Int.csInf_of_not_bdd_below, csInf_le, Int.csInf_empty
toLattice 📖CompOp
1104 mathmath: NonUnitalSubalgebra.coe_starClosure, MeasureTheory.Measure.MeasureDense.approx, Submodule.finrank_sup_add_finrank_inf_eq, Subalgebra.LinearDisjoint.inf_eq_bot, Nat.iInf_le_succ', csSup_pair, LinearMap.quotientInfEquivSupQuotient_symm_apply_eq_zero_iff, AlgebraicGeometry.descendsAlong_isOpenImmersion_surjective_inf_flat_inf_quasicompact', AffineSubspace.direction_inf, Set.Iic.coe_sInf, AffineSubspace.direction_sup_eq_sup_direction, OrderEmbedding.supIrredLowerSet_apply, continuous_sup_rng_right, Nat.iSup_lt_succ', Submodule.supIndep_torsionBySet_ideal, Algebra.lift_cardinalMk_adjoin_le, SymmetricAlgebra.rank_eq, Subgroup.sup_eq_closure_mul, continuousMul_inf, TopCat.Sheaf.interUnionPullbackCone_snd, Set.Finite.einfsep, IntermediateField.LinearDisjoint.iff_inf_eq_bot, AlgebraicGeometry.Scheme.Hom.image_preimage_eq_opensRange_inter, IntermediateField.normal_inf, Cardinal.powerlt_max, AlgebraicGeometry.descendsAlong_universallyClosed_surjective_inf_flat_inf_quasicompact, LinearMap.map_le_map_iff, StarSubalgebra.map_inf, RootPairing.instNontrivialSubtypeSubmoduleMemSublatticeInvtRootSubmodule, AlgebraicGeometry.RingedSpace.basicOpen_res, Finpartition.isPartition_parts, Set.einfsep_of_fintype, AddGroupTopology.toTopologicalSpace_inf, Ordinal.max_zero_left, IntermediateField.extendScalars_sup, Cardinal.mk_list_eq_max_mk_aleph0, Submodule.toConvexCone_inf, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverse_obj_π_app, Cardinal.mk_list_le_max, Module.End.mem_invtSubmodule, CategoryTheory.ObjectProperty.HasCardinalLT.sup, Subalgebra.LinearDisjoint.sup_free_of_free, NonUnitalStarAlgebra.mem_sup_right, AffineSubspace.direction_inf_of_mem, continuous_sup_dom, IntermediateField.relrank_comap_comap_eq_relrank_inf, Filter.bliminf_inf_not, Finset.inf'_univ_eq_ciInf, Filter.blimsup_and_le_inf, IsCompactlyGenerated.BooleanGenerators.complementedLattice_of_sSup_eq_top, StarSubalgebra.mem_sup_left, MeasureTheory.measureReal_symmDiff_le, CategoryTheory.IsCardinalFiltered.exists_cardinal_directed.D₁_W, Cardinal.mk_multiset_of_nonempty, RootPairing.Base.forall_mem_support_invtSubmodule_iff, LinearMap.IsProj.mem_invtSubmodule_iff, Submodule.mem_sup', MeasureTheory.NullMeasurableSet.exists_isOpen_symmDiff_lt, MeasureTheory.exists_measure_symmDiff_lt_of_generateFrom_isSetRing, MeasureTheory.hittingAfter_bot_le_iff, TopologicalSpace.Opens.mem_sup, LinearMap.IsIdempotentElem.commute_iff, LinearPMap.domain_supSpanSingleton, CompleteLattice.isCompactElement_finsetSup, Ideal.sum_eq_sup, Subalgebra.comap_map_eq, Subalgebra.finrank_sup_eq_finrank_right_mul_finrank_of_free, Opens.pretopology_toGrothendieck, CompleteSublattice.isComplemented_iff, ciSup_partialSups_eq', MeasureTheory.hittingBtwn_univ, AlgebraicGeometry.IsAffineOpen.isCompact_pullback_inf, Submodule.isOrtho_sup_right, OrderEmbedding.infIrredUpperSet_surjective, induced_inf, ClosedSubmodule.toSubmodule_sup, GroupTopology.toTopologicalSpace_inf, Submodule.fst_sup_snd, IsCompactlyGenerated.BooleanGenerators.sSup_inter, NonUnitalAlgebra.mem_sup_left, FirstOrder.Language.Substructure.map_sup, DualNumber.range_lift, NonUnitalAlgebra.coe_inf, Cardinal.mul_le_max_of_aleph0_le_right, LocallyConvexSpace.inf, LinearEquiv.map_mem_invtSubmodule_conj_iff, Filter.cardinalInterFilter_sup, Cardinal.sum_pow_le_max_aleph0, FiniteGaloisIntermediateField.instIsGaloisSubtypeMemIntermediateFieldMax, StarSubalgebra.inf_toSubalgebra, MonoidAlgebra.cardinalMk_of_infinite, ConvexCone.coe_inf, csInf_insert, NatOrdinal.toOrdinal_min, Submodule.mem_invtSubmodule_reflection_of_mem, RootPairing.invtRootSubmodule.bot_mem, Measurable.sup_of_right, RootPairing.invtRootSubmodule.eq_top_iff, Filter.bliminf_sup_le_and, AddSubgroup.normal_add, MeasureTheory.measure_symmDiff_eq_top, Cardinal.mul_eq_max_of_aleph0_le_right, cbiSup_eq_of_not_forall, Topology.IsLower.toContinuousInf, NNReal.segment_eq_uIcc, FiniteGaloisIntermediateField.instFiniteDimensionalSubtypeMemIntermediateFieldMin_1, continuous_inf_dom_left₂, AlgebraicGeometry.isPullback_opens_inf, LieSubalgebra.span_union, ENNReal.iInf_ennreal, Subalgebra.rank_sup_eq_rank_left_mul_rank_of_free, Finset.isWF_sup, Cardinal.mk_finsupp_lift_of_infinite, Submodule.coe_sup, ExpGrowth.expGrowthSup_add, ConditionallyCompleteLinearOrderedField.toZeroLEOneClass, TensorProduct.quotientTensorQuotientEquiv_symm_apply_mk_tmul, Subfield.map_sup, biUnion_range_succ_disjointed, AddSubgroup.add_normal, AlgebraicGeometry.isImmersion_eq_inf, Subgroup.mul_normal, Ideal.comap_finsetInf, Submodule.colon_finsetInf, AlgebraicGeometry.Scheme.IdealSheafData.support_mul, Representation.invtSubmodule.nontrivial_iff, Nat.iInf_lt_succ', ConditionallyCompleteLinearOrder.compare_eq_compareOfLessAndEq, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivCounitIso_hom_app_hom, Subfield.cardinalMk_closure_le_max, Subgroup.coe_mul_of_left_le_normalizer_right, AlgebraicGeometry.descendsAlong_isomorphisms_surjective_inf_flat_inf_quasicompact, Cardinal.mul_eq_left_iff, NumberField.isTotallyReal_sup, continuousAdd_inf, Algebra.map_sup, CategoryTheory.GrothendieckTopology.arrow_intersect, AddSubmonoid.sup_mul, CategoryTheory.GrothendieckTopology.intersection_covering_iff, continuous_inf_dom_right₂, TopCat.Sheaf.interUnionPullbackCone_fst, Module.End.mem_invtSubmodule_iff_mapsTo, NonUnitalSubalgebra.prod_inf_prod, AlgebraicGeometry.IsSeparated.eq_valuativeCriterion, generateFrom_union, iSup_partialSups_eq, isFiniteLength_iff_exists_compositionSeries, Ordinal.toNimber_min, isOpen_sup, Sylow.normalizer_sup_eq_top', NonUnitalAlgebra.mem_sup_right, Subalgebra.unop_sup, AlgebraicGeometry.IsClosedImmersion.eq_proper_inf_monomorphisms, LinearMap.range_domRestrict_eq_range_iff, TopologicalSpace.Closeds.instContinuousSup, Subalgebra.prod_inf_prod, Submodule.comap_map_sup_of_comap_le, TopCat.pullback_topology, LinearMap.span_singleton_sup_orthogonal_eq_top, IntermediateField.inf_toSubalgebra, OrderIso.infIrredUpperSet_symm_apply, Projectivization.Subspace.span_sup, MeasureTheory.measureReal_symmDiff_eq, IsSemisimpleModule.toComplementedLattice, Algebraic.cardinalMk_le_max, FirstOrder.Language.Substructure.map_sup_comap_of_surjective, TopologicalSpace.NoetherianSpace.exists_finset_irreducible, UniformContinuous.sup_closeds, ConditionallyCompleteLinearOrderedField.toPosMulStrictMono, CategoryTheory.MorphismProperty.IsLocalAtSource.inf, NonUnitalAlgebra.mem_inf, AddSubmonoid.smul_sup, SSet.Subcomplex.degenerate_eq_top_iff, Ideal.smul_sup, Nimber.toOrdinal_max, Algebra.IsAlgebraic.cardinalMk_le_max, hasCardinalLT_subtype_max, Ideal.isPrimary_finsetInf, FirstOrder.Language.Substructure.comap_sup_map_of_injective, Submodule.neg_sup, Submodule.sup_eq_range, topologicalGroup_inf, Submodule.sub_mem_sup, Set.Finset.coe_einfsep, Filter.inf_limsup, Algebra.adjoin_union, topologicalAddGroup_inf, AlgebraicGeometry.Scheme.Hom.image_preimage_eq_opensRange_inf, Ideal.mem_sup_left, SSet.Subcomplex.preimage_min, Nimber.toOrdinal_min, Ideal.prod_le_inf, Submodule.mem_finset_inf, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverseObj_π_app, AddSubgroup.relIndex_map_map, ContinuousLinearMap.IsIdempotentElem.ker_mem_invtSubmodule, AffineSubspace.direction_affineSpan_insert, Finset.sup_univ_eq_ciSup, AffineSubspace.vectorSpan_union_of_mem_of_mem, IntermediateField.inf_relrank_right, beattySeq'_symmDiff_beattySeq_pos, IsAddFoelner.tendsto_meas_vadd_symmDiff_vadd, RootPairing.mem_invtRootSubmodule_iff, MeasureTheory.abs_measureReal_sub_le_measureReal_symmDiff', PrimeSpectrum.basicOpen_mul, Ordinal.card_omega0_opow, Submodule.smul_sup', HahnEmbedding.ArchimedeanStrata.ball_sup_stratum_eq, continuousNeg_inf, IntermediateField.sup_toSubalgebra_of_isAlgebraic_left, TopCat.prod_topology, Subrepresentation.toSubmodule_sup, IntermediateField.restrictScalars_sup, Submodule.finiteDimensional_finset_sup, FiniteGaloisIntermediateField.instFiniteDimensionalSubtypeMemIntermediateFieldMin, NNReal.image_coe_uIcc, Submodule.submoduleOf_sup_of_le, IsFoelner.tendsto_meas_smul_symmDiff, Dynamics.coverEntropyEntourage_union, MeasurableSpace.generateFrom_sup_generateFrom, OrderIso.supIrredLowerSet_symm_apply, Submonoid.sup_eq_closure_mul, Ideal.IsPrime.inf_le', Representation.invtSubmodule.coe_top, MeasureTheory.OuterMeasure.trim_sup, TopologicalSpace.Opens.mem_inf, Set.definable_finset_sup, AddSubgroup.relIndex_sup_left, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquiv_counitIso, MeasureTheory.IsSetSemiring.isSetRing_supClosure, Nucleus.mem_toSublocale, AlgebraicGeometry.instDescendsAlongSchemeMinMorphismPropertySurjectiveFlatLocallyOfFinitePresentationOfQuasiCompactOfIsZariskiLocalAtTarget, TopologicalSpace.IsTopologicalBasis.inf_induced, Submodule.finiteDimensional_sup, FreeAlgebra.cardinalMk_le_max, LinearMap.map_coprod_prod, Cardinal.aleph_add_aleph, AlgebraicGeometry.HasAffineProperty.affineAnd_eq_of_propertyIsLocal, AlgebraicGeometry.Etale.instHasOfPostcompPropertySchemeMinMorphismPropertyLocallyOfFiniteTypeFormallyUnramified, Ordinal.toNatOrdinal_min, LinearMap.IsIdempotentElem.ker_mem_invtSubmodule_iff, Subgroup.FG.finset_sup, MeasureTheory.Measure.MeasureDense.fin_meas_approx, MeasureTheory.edist_indicatorConstLp_eq_enorm, MonoidAlgebra.cardinalMk_lift_of_infinite, Subfield.extendScalars_inf, AlgebraicGeometry.IsFinite.eq_proper_inf_locallyQuasiFinite, Subalgebra.LinearDisjoint.rank_sup_of_free, Ordinal.iSup_sum, Ideal.mul_left_self_sup, StarSubalgebra.mul_mem_sup, MeasureTheory.hittingBtwn_bot_le_iff, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctorObj_π_app, WType.cardinalMk_le_max_aleph0_of_finite', MeasureTheory.IsSetSemiring.mem_supClosure_iff, CategoryTheory.ObjectProperty.IsLocal.inf, ConditionallyCompleteLinearOrderedField.toIsOrderedCancelAddMonoid, DirectedOn.sSup_inf_eq, Algebra.TensorProduct.map_ker, Submodule.map_sup, MeasurableSpace.map_inf, Dynamics.coverEntropy_union, continuousInv_inf, NonUnitalAlgebra.mul_mem_sup, Setoid.IsPartition.finpartition_parts, iSupIndep.supIndep', Filter.bliminf_or_le_inf, Ideal.isPrimary_finset_inf, FirstOrder.Language.card_functions_sum_skolem₁_le, PSet.rank_insert, iSupIndep.supIndep, Subalgebra.op_inf, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivCounitIso_inv_app_hom, LinearMap.comap_leq_ker_subToSupQuotient, ProbabilityTheory.condExpKernel_ae_eq_condExp', Submodule.map_mkQ_eq_top, IntermediateField.isCyclotomicExtension_lcm_sup, Submodule.eq_top_of_disjoint, IntermediateField.sup_toSubalgebra_of_isAlgebraic, SimpleGraph.chromaticNumber_sum, Module.End.invtSubmodule.isCompl_iff, AddSubmonoid.neg_sup, MeasureTheory.measure_symmDiff_le, Filter.bliminf_or_eq_inf, AffineSubspace.inf_coe, Filter.limsup_sup_filter, continuous_inf_dom_right, AffineSubspace.map_inf_eq, AddMonoidAlgebra.cardinalMk_of_infinite', Submonoid.smul_sup, CompleteLattice.isCompactElement_iff_exists_le_iSup_of_le_iSup, MeasureTheory.dense_of_generateFrom_isSetSemiring, AffineSubspace.span_union, Cardinal.mul_eq_max, LinearMap.quotientInfEquivSupQuotient_apply_mk, Sublocale.coe_inf, NatOrdinal.toOrdinal_max, AffineSubspace.coe_inf, Cardinal.mk_freeMonoid, Cardinal.mk_bounded_set_le, rel_sup_mul, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_inv_assoc, OrderEmbedding.infIrredUpperSet_apply, MeasureTheory.measure_neg_vadd_symmDiff, Algebra.map_inf, IsRetrocompact.finsetInf', IntermediateField.fg_sup, Submonoid.FG.finset_sup, MeasurableSpace.cardinal_generateMeasurableRec_le, complementedLattice_of_sSup_atoms_eq_top, IntermediateField.inf_relfinrank_right, Algebra.sup_toSubsemiring, LowerSet.supIrred_iff_of_finite, AddSubgroup.coe_add_of_left_le_normalizer_right, PrimitiveSpectrum.gc_closureOperator, beattySeq_symmDiff_beattySeq'_pos, Irrational.beattySeq_symmDiff_beattySeq_pos, Affine.Simplex.mongePlane_def, Subalgebra.mem_starClosure, Submodule.IsMinimalPrimaryDecomposition.inf_eq, ConvexCone.mem_inf, Opens.pretopology_ofGrothendieck, Subalgebra.finrank_right_dvd_finrank_sup_of_free, NumberField.Units.dirichletUnitTheorem.map_logEmbedding_sup_torsion, WType.cardinalMk_le_max_aleph0_of_finite, AlgebraicGeometry.Proj.basicOpen_mul, Subalgebra.mul_toSubmodule, IntermediateField.extendScalars_inf, Filter.liminf_sup_filter, Ideal.mul_sup, TopCat.Sheaf.objSupIsoProdEqLocus_hom_fst, Polynomial.sup_aeval_range_eq_top_of_isCoprime, TopologicalSpace.Opens.functor_obj_map_obj, Finset.sup'_eq_csSup_image, AddSubgroup.index_map, Cardinal.mk_list_eq_max, IsSemisimpleModule.sup, exists_compositionSeries_of_isNoetherian_isArtinian, Finsupp.supported_union, EReal.min_neg_neg, Subgroup.relIndex_map_map, Polynomial.sup_ker_aeval_eq_ker_aeval_mul_of_coprime, TopologicalSpace.Closeds.lipschitz_sup, Cardinal.mk_freeAddGroup, UpperSet.infIrred_Ici, pullbackTopology_def, Submodule.mul_sup, RootPairing.invtRootSubmodule.top_mem, AlgebraicGeometry.Scheme.pretopology_le_inf, LinearMap.sup_range_inl_inr, cbiInf_eq_of_not_forall, Cardinal.mk_finsupp_nat, LinearMap.span_singleton_sup_ker_eq_top, Nucleus.coe_toSublocale, Nat.iInf_le_succ, Seminorm.closedBall_finset_sup, Topology.IsGeneratedBy.sup, Filter.blimsup_sup_not, TopologicalSpace.Opens.coe_finset_sup, MeasureTheory.IsSetSemiring.diff_mem_supClosure, MeasureTheory.OuterMeasure.le_add_caratheodory, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctor_obj_pt, Topology.IsLocallyConstructible.finsetInf, TopCat.Presheaf.SheafConditionEqualizerProducts.piInters.hom_ext_iff, AddMonoidAlgebra.cardinalMk_lift_of_infinite, Ideal.sup_mul_right_self, Submodule.comap_finsetInf, Submodule.comap_sup_map_of_injective, IntermediateField.lift_inf, Subalgebra.mulMap'_surjective, partialSups_eq_sUnion_image, AddSubgroup.closure_add_le, TopCat.Sheaf.interUnionPullbackConeLift_right, Submodule.coe_finsetInf, Subgroup.relIndex_sup_right, MeasurableSpace.comap_prodMk, MeasureTheory.le_measure_symmDiff, FiniteGaloisIntermediateField.instIsSeparableSubtypeMemIntermediateFieldMin_1, TopologicalSpace.Closeds.coe_finset_inf, Module.End.invtSubmodule.one, IntermediateField.coe_inf, MeasurableSet.bihimp, IntermediateField.restrictScalars_adjoin_eq_sup, Nucleus.toSublocale_le_toSublocale, AlgebraicGeometry.Scheme.basicOpen_add_le, csSup_empty, IsPGroup.to_sup_of_normal_left', MeasureTheory.measure_symmDiff_inv_smul, Algebra.mem_sup_left, LinearMap.quotientInfEquivSupQuotient_injective, isSemisimpleModule_iff, IntermediateField.mem_inf, IntermediateField.cardinalMk_adjoin_le, Submodule.dualAnnihilator_sup_eq, Submonoid.inv_sup, instIsModularLatticeSubgroup, MeasureTheory.OuterMeasure.sup_apply, Sublattice.setLike_mem_inf, Subgroup.toSubmonoid_zpowers, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquiv_inverse, Subalgebra.finrank_left_dvd_finrank_sup_of_free, Cardinal.mk_freeAddMonoid, NumberField.Units.regOfFamily_div_regulator, AddSubmonoid.mul_sup, CategoryTheory.MorphismProperty.HasCardinalLT.sup, Submodule.dualCoannihilator_sup_eq, QuotientAddGroup.comap_map_mk', OrderIso.supIrredLowerSet_apply, Cardinal.mul_le_max_of_aleph0_le_left, Cardinal.powerlt_min, Cardinal.power_nat_le_max, iSupIndep_iff_supIndep_of_injOn, AffineIndependent.inf_affineSpan_eq_affineSpan_inter, Subalgebra.mul_toSubmodule_le, AffineSubspace.direction_sup, AlgebraicGeometry.Scheme.Hom.appLE_appIso_inv, AlgebraicGeometry.Scheme.PartialMap.equiv_iff_of_isSeparated, Algebra.IsAlgebraic.lift_cardinalMk_le_max, SSet.skeleton_succ, Sublattice.ext_mem_iff, TopologicalSpace.IsTopologicalBasis.inf, TopologicalSpace.Opens.coe_finset_inf, AddSubgroup.coe_add_of_right_le_normalizer_left, complementedLattice_of_isAtomistic, RootPairing.corootSpan_mem_invtSubmodule_coreflection, FirstOrder.Language.ClosedUnder.inf, FirstOrder.Language.isExtensionPair_iff_exists_embedding_closure_singleton_sup, R1Space.inf, NumberField.IsCMField.closure_realFundSystem_sup_torsion, RootPairing.invtRootSubmodule.eq_bot_iff, Module.Basis.flag_succ, IntermediateField.LinearDisjoint.finrank_sup, MonoidAlgebra.cardinalMk_of_infinite', LinearGrowth.linearGrowthSup_sup, Sublattice.mem_subtype, AlgebraicGeometry.Scheme.grothendieckTopology_eq_inf, MeasureTheory.Measure.inf_apply, Module.End.isFinitelySemisimple_iff, Subgroup.FG.sup, ENNReal.iSup_ennreal, IntermediateField.restrictScalars_inf, partialSups_eq_biSup, dist_mulIndicator, Submodule.fg_finset_sup, Algebra.TensorProduct.productMap_range, AlgebraicGeometry.Scheme.IdealSheafData.vanishingIdeal_sup, LinearGrowth.linearGrowthInf_inf, Relation.cutExpand_le_invImage_lex, HahnSeries.cardSupp_div_le, NonUnitalAlgebra.map_sup, AddSubgroup.ofAddUnits_inf, InfiniteGalois.restrict_fixedField, Ideal.Filtration.sup_N, OpenAddSubgroup.toOpens_inf, isFoelner_iff, TopCat.Presheaf.IsSheaf.isSheafPreservesLimitPairwiseIntersections, EReal.coe_toENNReal_eq_max, Submodule.sup_smul, Ordinal.max_eq_zero, Subgroup.ofUnits_inf, NumberField.Units.closure_fundSystem_sup_torsion_eq_top, Filter.sup_limsup, MeasureTheory.StronglyMeasurable.integral_condExpKernel', IsFoelner.tendsto_meas_smul_symmDiff_smul, FirstOrder.Language.Substructure.closure_insert, Cardinal.mk_freeRing, Ideal.mem_span_singleton_sup, LieSubmodule.sup_toSubmodule, Ideal.finset_inf_span_singleton, KaehlerDifferential.kerTotal_map, ProbabilityTheory.condExp_ae_eq_integral_condExpKernel', Cardinal.add_eq_max', IsRetrocompact.finsetSup, Ordinal.card_opow_le_of_omega0_le_right, Algebra.inf_toSubmodule, IntermediateField.sup_def, ciSup_or', CategoryTheory.ObjectProperty.instIsTriangulatedClosed₂MinOfIsClosedUnderIsomorphisms, IsPGroup.to_sup_of_normal_right, MeasurableSpace.CountablyGenerated.sup, VonNeumannAlgebra.IsIdempotentElem.mem_iff, TensorProduct.quotientTensorQuotientEquiv_apply_tmul_mk_tmul_mk, Cardinal.mul_mk_eq_max, MeasureTheory.IsStoppingTime.measurableSpace_min, LSeries.abscissaOfAbsConv_binop_le, MeasureTheory.tendsto_measure_symmDiff_preimage_nhds_zero, IntermediateField.finrank_sup_le, Set.Iic.coe_iInf, nhds_inf, Ideal.map_sup_comap_of_surjective, Cardinal.aleph_max, LSeries.abscissaOfAbsConv_add_le, ciSup_of_empty, OreLocalization.cardinalMk_le_max, OrderEmbedding.supIrredLowerSet_surjective, separableClosure_inf_perfectClosure, Subgroup.normal_mul, Subalgebra.unop_inf, biUnion_Iic_disjointed, ExpGrowth.expGrowthInf_inf, Ordinal.card_opow_omega0, Cardinal.add_eq_max, AlgebraicGeometry.HasRingHomProperty.inf, TopologicalSpace.Compacts.coe_finset_sup, NonUnitalStarAlgebra.mul_mem_sup, Ideal.map_sup, MeasureTheory.hitting_bot_le_iff, Subalgebra.LinearDisjoint.rank_inf_eq_one_of_flat_of_inj, FirstOrder.Language.Substructure.CG.sup, AffineSubspace.nonempty_sup_left, LinearMap.range_coprod, Subalgebra.finrank_sup_eq_finrank_left_mul_finrank_of_free, Cardinal.preAleph_max, Set.Iic.coe_biInf, Subalgebra.finrank_sup_le_of_free, Ideal.comap_map_quotientMk, MeasureTheory.IsSetRing.partialSups_mem, Subalgebra.starClosure_toSubalgebra, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivUnitIso_inv_app_hom, Submodule.comap_map_mkQ, OpenAddSubgroup.toAddSubgroup_sup, Directed.iSup_inf_eq, isPreconnected_uIcc, TopCat.Sheaf.objSupIsoProdEqLocus_inv_snd, ProbabilityTheory.condExpKernel_def, nndist_mulIndicator, MeasureTheory.eLpNorm_indicator_sub_indicator, NonUnitalStarAlgebra.mem_sup_left, ciSup_partialSups_eq, le_analyticOrderAt_sub, le_analyticOrderAt_add, ProjectiveSpectrum.basicOpen_mul, AlexandrovDiscrete.sup, ExpGrowth.le_expGrowthInf_add, Submodule.inf_orthogonal, IntermediateField.relfinrank_inf_mul_relfinrank, LSeries.abscissaOfAbsConv_convolution_le, VonNeumannAlgebra.IsStarProjection.mem_iff, Polynomial.cardinalMk_eq_max, TopologicalSpace.Closeds.coe_inf, Language.IsRegular.inf, IntermediateField.LinearDisjoint.inf_eq_bot, AlgebraicGeometry.isOpenImmersion_eq_inf, ClosedSubmodule.mem_sup, Submodule.IsMinimalPrimaryDecomposition.minimal, AlgebraicGeometry.Scheme.Hom.preimage_sup, MeasurableSet.symmDiff, Nat.iInf_lt_succ, Subalgebra.finite_sup, IsAlgClosed.cardinal_le_max_transcendence_basis', Ideal.comap_map_of_surjective, AddSubmonoid.coe_sup, TopCat.Sheaf.interUnionPullbackConeLift_left, Ideal.quotientInfEquivQuotientProd_snd, AddSubgroup.sup_normal, AlgebraicGeometry.IsClosedImmersion.eq_inf, AddSubgroup.FG.finset_sup, continuousSMul_inf, AlgebraicGeometry.Scheme.Hom.appLE_appIso_inv_assoc, IntermediateField.map_comap_eq, Submodule.restrictScalars_sup, iSupIndep.sup_indep_univ, IsDedekindDomain.inf_prime_pow_eq_prod, NonUnitalStarSubalgebra.prod_inf_prod, MeasureTheory.IsStoppingTime.measurableSpace_min_const, TopologicalSpace.Closeds.coe_sup, csSup_insert, CategoryTheory.MorphismProperty.pretopology_inf, Subalgebra.rank_sup_eq_rank_right_mul_rank_of_free, AlgebraicGeometry.Scheme.IdealSheafData.mul_inf, AlgebraicGeometry.targetAffineLocally_affineAnd_eq_affineLocally, SetTheory.PGame.birthday_def, Finset.isPWO_sup, Finset.partiallyWellOrderedOn_sup, exists_sSupIndep_disjoint_sSup_atoms, FirstOrder.Language.Term.card_sigma, CategoryTheory.Precoverage.sup_mem_coverings, LinearMap.eqOn_sup, Submodule.add_mem_sup, Submodule.sup_orthogonal_inf_of_hasOrthogonalProjection, Algebra.TensorProduct.map_range, Cardinal.mk_bounded_subset_le, AlgebraicGeometry.Scheme.basicOpen_appLE, MeasurableSpace.inf_le_invariants_comp, DeltaGeneratedSpace.sup, Submodule.sup_dualAnnihilator_le_inf, RegularSpace.inf, OrderIso.infIrredUpperSet_apply, CategoryTheory.Precoverage.IsStableUnderSup.sup_mem_coverings, Submonoid.closure_mul_le, IntermediateField.sup_toSubalgebra_of_left, Submodule.supIndep_torsionBy, SSet.Subcomplex.BicartSq.isPushout, Units.topology_eq_inf, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctor_obj_π_app, Projectivization.Subspace.sup_span, LinearMap.IsIdempotentElem.range_mem_invtSubmodule, Submodule.smul_sup, ClosedSubmodule.coe_inf, Cardinal.add_mk_eq_max, Module.AEval.mem_mapSubmodule_symm_apply, AlgebraicGeometry.isPreimmersion_eq_inf, EReal.max_neg_neg, AddMonoidAlgebra.cardinalMk_lift_of_infinite', LinearPMap.domain_sup, DFA.acceptsFrom_inter, IntermediateField.LinearDisjoint.rank_sup, Subalgebra.LinearDisjoint.rank_inf_eq_one_of_flat_left_of_inj, AlgebraicGeometry.IsOpenImmersion.ΓIso_inv, TopologicalSpace.Closeds.coe_finset_sup, MeasureTheory.measure_inv_smul_symmDiff, HomogeneousIdeal.toIdeal_sup, IntermediateField.rank_sup_le_of_isAlgebraic, AlgebraicGeometry.isProper_eq, Sublocale.inf_mem, Subalgebra.FG.sup, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctor_map_hom, instIsModularLatticeAddSubgroup, Submodule.isQuotientEquivQuotientPrime_iff, inf_sSup_eq_iSup_inf_sup_finset, AffineSubspace.map_inf_le, Subalgebra.LinearDisjoint.finrank_sup_of_free, ZFSet.rank_insert, Subgroup.smul_sup, FirstOrder.Language.Substructure.closure_union, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverse_obj_pt, Submodule.span_union, IsRetrocompact.finsetSup', Submonoid.coe_sup, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivUnitIso_hom_app_hom, AddSubmonoid.closure_add_le, AffineSubspace.finiteDimensional_sup, StarSubalgebra.map_sup, CategoryTheory.GrothendieckTopology.intersection_covering, Submodule.rank_add_le_rank_add_rank, Submodule.sup_mul, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivUnitIsoApp_hom_hom, Module.End.mem_invtSubmodule_symm_iff_le_map, DualNumber.range_inlAlgHom_sup_adjoin_eps, Submodule.spanRank_sup_le_sum_spanRank, Cardinal.mk_freeGroup, FirstOrder.Language.Substructure.FG.sup, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctorObj_pt, AlgebraicGeometry.IsOpenImmersion.app_ΓIso_hom_assoc, IntermediateField.relfinrank_mul_relfinrank_eq_inf_relfinrank, OpenSubgroup.toOpens_inf, TopologicalSpace.Opens.functor_map_eq_inf, Ordinal.card_opow_eq_of_omega0_le_right, AlgebraicGeometry.universallyClosed_eq_universallySpecializing, MeasurableSpace.cardinal_measurableSet_le, RootPairing.root_mem_submodule_iff_of_add_mem_invtSubmodule, Subgroup.relIndex_sup_left, MeasurableSpace.cardinal_generateMeasurable_le, IsAlgClosed.cardinal_le_max_transcendence_basis, CategoryTheory.Pretopology.mem_inf, completelyRegularSpace_inf, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_inv, Polynomial.sup_ker_aeval_le_ker_aeval_mul, AlgebraicGeometry.RingedSpace.basicOpen_mul, Module.End.span_orbit_mem_invtSubmodule, AlgebraicGeometry.Flat.surjective_descendsAlong_surjective_inf_flat_inf_quasicompact, MeasureTheory.Filtration.coeFn_sup, AffineSubspace.nonempty_sup_right, IntermediateField.relfinrank_inf_mul_relfinrank_of_le, Ideal.ofList_append, AlgebraicGeometry.IsProper.eq_valuativeCriterion, Module.End.invtSubmodule.codisjoint_iff, Filter.inf_liminf, Topology.IsUpper.toContinuousInf, AlgebraicGeometry.IsClosedImmersion.eq_isFinite_inf_mono, AlgebraicGeometry.Scheme.basicOpen_mul, Algebra.EssFiniteType.aux, Subgroup.closure_mul_le, AddSubmonoid.sup_eq_closure_add, MeasurableSpace.comap_sup, Submodule.sup_orthogonal_of_hasOrthogonalProjection, ContinuousLinearMap.IsIdempotentElem.range_mem_invtSubmodule_iff, IntermediateField.relrank_inf_mul_relrank_of_le, Measurable.sup_of_left, partialSups_eq_ciSup_Iic, MeasureTheory.MeasuredSets.dist_def, Subalgebra.rank_sup_le_of_free, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverse_map_hom, AlgebraicGeometry.UniversallyClosed.eq_valuativeCriterion, AlgebraicGeometry.IsOpenImmersion.app_ΓIso_hom, Representation.invtSubmodule.top_mem, ConditionallyCompleteLinearOrder.le_total, limsup_max, Algebra.isIntegral_sup, QuotientGroup.comap_map_mk', disjointed_eq_inf_compl, Cardinal.aleph_mul_aleph, dist_indicator, LinearMap.comap_eq_sup_ker_of_disjoint, liminf_finset_inf', Sublocale.toNucleus_le_toNucleus, TopCat.Sheaf.objSupIsoProdEqLocus_hom_snd, Subalgebra.op_sup, ContinuousLinearMap.IsIdempotentElem.commute_iff, SSet.Subcomplex.preimage_max, Submodule.finite_finset_sup, Set.partialSups_eq_accumulate, IsPGroup.to_sup_of_normal_right', Submodule.finrank_add_le_finrank_add_finrank, Ordinal.card_opow_le_of_omega0_le_left, Ideal.mem_quotient_iff_mem_sup, Subalgebra.LinearDisjoint.val_mulMap_tmul, AlgebraicGeometry.isomorphisms_eq_isOpenImmersion_inf_surjective, Subgroup.sup_normal, Subalgebra.mulMap_range, Submodule.lt_sup_iff_notMem, PrimitiveSpectrum.kernel_hull, Set.Mapsto.mem_invtSubmodule, ContinuousLinearMap.IsIdempotentElem.range_mem_invtSubmodule, Module.End.mem_invtSubmodule_iff_forall_mem_of_mem, Subfield.extendScalars_sup, FirstOrder.Language.Term.card_le, cauchy_davenport_minOrder_add, Ideal.IsMaximal.coprime_of_ne, Ideal.add_eq_sup, RootPairing.coe_bot, Module.End.invtSubmodule.top_mem, AlgebraicGeometry.Scheme.Hom.preimage_inf, Ordinal.card_opow_eq_of_omega0_le_left, Subalgebra.LinearDisjoint.rank_inf_eq_one_of_flat_right_of_inj, NonUnitalAlgebra.inf_toSubmodule, nucleusIsoSublocale.eq_toSublocale, IntermediateField.isSeparable_sup, StarSubalgebra.coe_inf, Submonoid.FG.sup, MeasureTheory.upperCrossingTime_zero', Submodule.IsLattice.sup, Submodule.complementedLattice, AlgebraicGeometry.IsOpenImmersion.app_ΓIso_hom_apply, AlgebraicGeometry.Scheme.IdealSheafData.ideal_sup, AffineSubspace.sup_direction_le, CompleteSublattice.mk'_carrier, continuousVAdd_inf, AlgebraicGeometry.geometrically_inf, Submodule.map₂_sup_left, Submodule.sup_eq_top_iff, LinearMap.ker_sup_ker_le_ker_comp_of_commute, coinduced_sup, LinearMap.range_ofIsCompl, Subalgebra.centralizer_coe_sup, csInf_union, MeasureTheory.abs_measureReal_sub_le_measureReal_symmDiff, Cardinal.add_mk_eq_max', DirectedOn.inf_sSup_eq, CategoryTheory.PreZeroHypercover.presieve₀_sum, MvPolynomial.cardinalMk_le_max_lift, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivUnitIsoApp_inv_hom, Subalgebra.coe_starClosure, AlgebraicGeometry.Scheme.Hom.appLE_appIso_inv_apply, MeasureTheory.measure_symmDiff_eq, Ideal.ofList_cons_smul, Sylow.normalizer_sup_eq_top, Finset.sup'_id_eq_csSup, AlgebraicGeometry.IsFinite.eq_isProper_inf_isAffineHom, Subring.smul_sup, max_aleph0_card_le_rank_fun_nat, FreeAlgebra.cardinalMk_le_max_lift, NonUnitalStarAlgebra.coe_inf, IntermediateField.lift_cardinalMk_adjoin_le, MeasureTheory.SignedMeasure.of_symmDiff_compl_positive_negative, LinearMap.coe_quotientInfToSupQuotient, Subfield.closure_union, RootPairing.isIrreducible_iff_invtRootSubmodule, NonUnitalStarAlgebra.map_sup, Directed.inf_iSup_eq, nucleusIsoSublocale.symm_eq_toNucleus, csSup_inter_le, AffineSubspace.map_sup, Submodule.map_sup_comap_of_surjective, UpperSet.infIrred_iff_of_finite, ProbabilityTheory.condExpKernel_eq, le_csInf_inter, Set.mabs_mulIndicator_symmDiff, LSeries.abscissaOfAbsConv_sub_le, AlgebraicGeometry.Scheme.basicOpen_res, TopCat.Sheaf.objSupIsoProdEqLocus_inv_fst, Nucleus.restrict_toSublocale, AddSubgroup.ofAddUnits_sup_addUnits, RootPairing.rootSpan_mem_invtSubmodule_reflection, Subalgebra.map_comap_eq, Submodule.span_sup, Cardinal.powerlt_aleph0_le, Ideal.quotientInfEquivQuotientProd_fst, EMetric.Closeds.lipschitz_sup, MonoidAlgebra.cardinalMk_lift_of_infinite', MeasureTheory.AddContent.supClosure_apply_of_mem, Algebra.mem_sup_right, TopologicalSpace.Opens.coe_sup, Projectivization.Subspace.span_union, IntermediateField.rank_sup_le, AddSubgroup.toAddSubmonoid_zmultiples, Filter.bliminf_not_inf, MeasureTheory.ae_eq_set_symmDiff, MeasureTheory.measure_symmDiff_neg_vadd, generateFrom_union_isOpen, LinearMap.coprod_map_prod, CompleteSublattice.comap_carrier, Cardinal.add_eq_left_iff, RootPairing.coe_top, ConditionallyCompleteLinearOrderBot.csSup_empty, Seminorm.closedBall_finset_sup', LinearMap.rank_comp_le, Finset.inf'_id_eq_csInf, TopologicalSpace.Closeds.uniformContinuous_sup, Cardinal.mk_finsupp_of_infinite, isArtinian_sup, Subsemiring.smul_sup, Algebra.coe_inf, nndist_indicator, limsup_finset_sup, Module.End.isSemisimple_iff', Subgroup.coe_mul_of_right_le_normalizer_left, Filter.limsup_piecewise, MeasureTheory.OuterMeasure.isCaratheodory_partialSups, SSet.stdSimplex.face_inter_face, FiniteGaloisIntermediateField.instIsGaloisSubtypeMemIntermediateFieldMin, Cardinal.mul_eq_max_of_aleph0_le_left, Ordinal.toNimber_max, IntermediateField.lift_relrank_comap_comap_eq_lift_relrank_inf, IsAddFoelner.tendsto_meas_vadd_symmDiff, Finset.wellFoundedOn_sup, UniformSpace.toTopologicalSpace_inf, complementedLattice_iff_isAtomistic, TensorProduct.map_ker, CompleteLatticeHom.toOrderIsoRangeOfInjective_apply, Ideal.eq_inf_of_isPrime_inf, TopologicalSpace.Opens.infLELeft_apply, Sublocale.infClosed, IntermediateField.adjoin_union, IntermediateField.inf_toSubfield, Ordinal.omega_max, Polynomial.cardinalMk_le_max, Submodule.finset_inf_coe, Submodule.sup_toAddSubmonoid, StarSubalgebra.mem_sup_right, Submodule.mem_finsetInf, ciSup_false, AffineSubspace.mem_inf_iff, ConditionallyCompleteLinearOrderedField.toIsStrictOrderedRing, BooleanSubalgebra.map_sup, Finset.inf'_eq_csInf_image, LinearMap.map_eq_top_iff, liminf_finset_inf, PrimitiveSpectrum.hull_sSup, isNoetherian_sup, Submodule.mem_sup, NonUnitalStarAlgebra.map_inf, IntermediateField.lift_sup, CategoryTheory.MorphismProperty.precoverage_inf, Algebra.cardinalMk_adjoin_le, Filter.blimsup_sup_le_or, Ideal.multiset_prod_le_inf, Algebra.mul_mem_sup, AlgebraicGeometry.Scheme.IdealSheafData.inf_mul, Ideal.le_comap_sup, Nat.iSup_lt_succ, MeasureTheory.exists_measure_symmDiff_lt_of_generateFrom_isSetSemiring, CompleteLattice.isCompactElement_iff_exists_le_sSup_of_le_sSup, NumberField.Units.finiteIndex_iff_sup_torsion_finiteIndex, Opens.toPretopology_grothendieckTopology, Filter.blimsup_not_sup, Submodule.instIsModularLattice, StarSubalgebra.mem_inf, Ideal.ofList_cons, LinearMap.IsIdempotentElem.range_mem_invtSubmodule_iff, CategoryTheory.ObjectProperty.instIsTriangulatedMinOfIsClosedUnderIsomorphisms, Sublattice.le_comap_sup, FiniteGaloisIntermediateField.instIsSeparableSubtypeMemIntermediateFieldMin, AddSubgroup.sup_eq_closure_add, Submodule.exists_mem_sup, Subalgebra.val_mulMap'_tmul, MeasureTheory.SimpleFunc.iSup_approx_apply, IntermediateField.sup_toSubalgebra_of_right, Submodule.span_insert, Ideal.comap_map_of_surjective', Cardinal.sum_pow_eq_max_aleph0, AlgHom.eqOn_sup, AlgebraicGeometry.isomorphisms_eq_stalkwise, DFA.accepts_inter, Submodule.FG.sup, iSupIndep_iff_supIndep, Set.definable_finset_inf, IntermediateField.le_sup_toSubalgebra, Sublocale.range_toNucleus, partialSups_eq_biUnion_range, AlgebraicGeometry.IsFinite.eq_inf, Filter.cardinalInterFilter_inf, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_hom, ZFSet.rank_pair, BooleanSubalgebra.le_comap_sup, IsPGroup.to_sup_of_normal_left, AlgebraicGeometry.GeometricallyIntegral.eq_geometricallyReduced_inf_geometricallyIrreducible, ExpGrowth.expGrowthSup_sup, Filter.blimsup_or_eq_sup, iSupIndep_iff_supIndep_univ, PrimitiveSpectrum.hull_iSup, IntermediateField.map_inf, TopologicalSpace.Opens.coe_inf, Module.End.invtSubmodule.zero, Subgroup.index_map, CategoryTheory.IsCardinalFiltered.exists_cardinal_directed.D₃_W, Topology.IsLocallyConstructible.finsetInf', Cardinal.mul_eq_max', Ideal.fst_comp_quotientInfEquivQuotientProd, IntermediateField.sup_toSubfield, Module.End.invtSubmodule.id, IntermediateField.relfinrank_comap_comap_eq_relfinrank_inf, Ideal.span_union, AddSubmonoid.addSubmonoid_smul_sup, LinearEquiv.map_mem_invtSubmodule_iff, IntermediateField.sup_toSubalgebra_of_isAlgebraic_right, continuous_sup_rng_left, CompleteSublattice.map_carrier, IntermediateField.inf_relfinrank_left, Ideal.IsHomogeneous.sup, continuous_inf_rng, Ordinal.max_zero_right, IntermediateField.map_sup, AlgebraicGeometry.IsAffineOpen.sup_of_disjoint, IsTranscendenceBasis.lift_cardinalMk_eq_max_lift, IsTranscendenceBasis.lift_rank_eq_max_lift, Submodule.rank_sup_add_rank_inf_eq, MvRatFunc.rank_eq_max_lift, Seminorm.ball_finset_sup, cauchy_davenport_minOrder_mul, IntermediateField.inf_relrank_left, Submodule.finite_sup, Representation.invtSubmodule.coe_bot, MvPolynomial.cardinalMk_le_max, Representation.invtSubmodule.instNontrivialSubtypeSubmoduleMemSublattice, Submodule.mem_sup_left, Cardinal.mk_freeCommRing, Cardinal.mk_freeAbelianGroup, limsup_finset_sup', AlgebraicGeometry.isPullback_opens_inf_le, MeasurableSet.exists_isOpen_symmDiff_lt, IntermediateField.relrank_inf_mul_relrank, Ideal.radical_finset_inf, Algebra.mem_inf, MeasureTheory.Filtration.coeFn_inf, MeasureTheory.MeasurePreserving.measure_symmDiff_preimage_iterate_le, Submodule.mem_sup_right, Submodule.sup_set_smul, AddSubmonoid.FG.finset_sup, LinearMap.IsIdempotentElem.ker_mem_invtSubmodule, MeasureTheory.IsSetRing.finsetSup_mem, Submodule.sup_span, NonUnitalSubalgebra.starClosure_toNonUnitalSubalgebra, MeasureTheory.IsSetSemiring.exists_finpartition_diff, LinearPMap.supSpanSingleton_apply_mk, HahnSeries.cardSupp_hsum_powers_le, AlgebraicGeometry.Scheme.IdealSheafData.support_sup, Ideal.mem_sup_right, Sublocale.toNucleus_apply, Ideal.sup_mul, Set.abs_indicator_symmDiff, EuclideanGeometry.orthogonalProjection_sup_of_orthogonalProjection_eq, Cardinal.add_le_max, Algebra.sup_def, Ideal.snd_comp_quotientInfEquivQuotientProd, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquiv_functor, Nat.iSup_le_succ', ConditionallyCompleteLinearOrderedField.toMulPosStrictMono, LieSubalgebra.add_eq_sup, Submodule.map_add_le, CategoryTheory.IsCardinalFiltered.exists_cardinal_directed.D₄_W, TopCat.Sheaf.interUnionPullbackCone_pt, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquiv_unitIso, Filter.liminf_piecewise, Submodule.covBy_span_singleton_sup, Ordinal.card_opow_le, MeasureTheory.MeasuredSets.edist_def, ClosedSubmodule.coe_sup, Sublattice.setLike_mem_coe, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverseObj_pt, Ideal.sup_mul_left_self, MeasureTheory.measure_symmDiff_eq_zero_iff, Subgroup.IsComplement'.sup_eq_top, Cardinal.mk_finsupp_lift_of_infinite', Submodule.wcovBy_span_singleton_sup, AlgebraicGeometry.IsIntegralHom.eq_universallyClosed_inf_isAffineHom, Finset.sup'_univ_eq_ciSup, Module.End.invtSubmodule.bot_mem, AddSubgroup.FG.sup, IntermediateField.relrank_mul_relrank_eq_inf_relrank, IsRetrocompact.finsetInf, Module.End.mem_invtSubmodule_iff_map_le, NonUnitalAlgebra.adjoin_union, ContinuousLinearMap.range_coprod, generateFrom_inter, Ideal.mul_right_self_sup, MeasureTheory.AddContent.supClosure_apply, Submodule.prod_sup_prod, AlgebraicGeometry.Scheme.pretopology_eq_inf, MeasurableSpace.measurableSet_inf, AddSubmonoid.FG.sup, Module.AEval.mem_mapSubmodule_apply, Seminorm.ball_finset_sup', Cardinal.mul_le_max, LowerSet.supIrred_Iic, MeasurableSpace.measurableSet_sup, Sublattice.setLike_mem_sup, AddOreLocalization.cardinalMk_le_max, HahnSeries.cardSupp_inv_le, Filter.sup_liminf, Submodule.add_eq_sup, PolynormableSpace.inf, CategoryTheory.IsCardinalFiltered.exists_cardinal_directed.D₂_W, Subspace.dualAnnihilator_inf_eq, TrivSqZeroExt.range_inlAlgHom_sup_adjoin_range_inr, Representation.mem_invtSubmodule, KaehlerDifferential.kerTotal_map', IntermediateField.normal_sup, Algebra.inf_toSubsemiring, ENNReal.image_coe_uIcc, edist_indicator, AddMonoidAlgebra.cardinalMk_of_infinite, AddSubgroup.relIndex_sup_right, Submodule.sup_toAddSubgroup, CategoryTheory.PreZeroHypercover.presieve₀_add, NonUnitalAlgebra.inf_toNonUnitalSubsemiring, Ordinal.lsub_sum, Submodule.mem_invtSubmodule_reflection_iff, CategoryTheory.MorphismProperty.IsLocalAtTarget.inf, MeasureTheory.OuterMeasure.map_sup, MeasureTheory.NullMeasurableSet.symmDiff, liminf_min, isAddFoelner_iff, PrimitiveSpectrum.gc, LinearMap.quotientInfEquivSupQuotient_surjective, csInf_pair, IsCyclotomicExtension.lcm_sup, SequentialSpace.sup, AffineSubspace.sup_direction_lt_of_nonempty_of_inter_empty, Ideal.span_insert, FreeAlgebra.cardinalMk_eq_max, Submodule.map₂_sup_right, ZFSet.rank_union, CompleteSublattice.subtype_apply, Module.End.invtSubmodule.disjoint_iff, Module.End.isSemisimple_iff, AddUnits.topology_eq_inf, csSup_union, IntermediateField.finiteDimensional_sup, Submodule.isOrtho_sup_left, ContinuousLinearMap.IsIdempotentElem.ker_mem_invtSubmodule_iff, Submodule.isPrimary_finsetInf, Ordinal.preOmega_max, NonUnitalSubalgebra.mem_starClosure, IntermediateField.fixingSubgroup_sup, AlgebraicGeometry.ValuativeCriterion.eq, MeasureTheory.dist_indicatorConstLp_eq_norm, Cardinal.add_eq_right_iff, OpenSubgroup.toSubgroup_sup, Subgroup.ofUnits_sup_units, LieSubmodule.instIsModularLattice, Cardinal.mk_finsupp_of_infinite', MvPolynomial.cardinalMk_eq_max_lift, continuous_inf_dom_left, Ordinal.toNatOrdinal_max, NonUnitalAlgebra.map_inf, Submodule.comap_sup_of_injective, LinearMap.range_add_le, FirstOrder.Language.Substructure.lift_card_closure_le, Sublattice.map_sup, IntermediateField.isPurelyInseparable_sup, NonUnitalStarAlgebra.mem_inf, Submodule.small_sup, PSet.rank_pair, analyticOrderAt_add_of_ne, CategoryTheory.IsCardinalFiltered.exists_cardinal_directed.Diagram.sup_W, FreeAlgebra.cardinalMk_eq_max_lift, AlgebraicGeometry.descendsAlong_universallyOpen_surjective_inf_flat_inf_quasicompact, LinearMap.prod_eq_sup_map, CategoryTheory.IsCardinalFiltered.exists_cardinal_directed.Diagram.sup_P, Nat.iSup_le_succ, Affine.Simplex.altitude_def, TrivSqZeroExt.range_liftAux, SSet.skeletonOfMono_succ, LinearMap.lift_rank_comp_le, LinearMap.surjective_domRestrict_iff, AffineSubspace.comap_inf, MvPolynomial.cardinalMk_eq_max, Representation.invtSubmodule.bot_mem, edist_mulIndicator, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_inv_apply, Algebra.Subalgebra.restrictScalars_adjoin, AlgebraicGeometry.descendsAlong_universallyInjective_surjective_inf_flat_inf_quasicompact, Algebraic.cardinalMk_lift_le_max, FirstOrder.Language.BoundedFormula.card_le, NonUnitalStarAlgebra.inf_toNonUnitalSubalgebra, ProbabilityTheory.condExpKernel_apply_eq_condDistrib, Algebra.rank_adjoin_le, LinearMap.ker_comp_eq_of_commute_of_disjoint_ker, TopologicalSpace.Opens.mk_inf_mk, LinearMap.BilinForm.span_singleton_sup_orthogonal_eq_top, rel_sup_add, setOf_isOpen_sup, Submodule.comap_map_eq
toSupSet 📖CompOp
17 mathmath: ConditionallyCompleteLinearOrder.csSup_of_not_bddAbove, Int.csSup_eq_greatestOfBdd, le_csSup_iff, csSup_le, le_csSup_of_le, Int.csSup_of_not_bddAbove, Int.csSup_eq_greatest_of_bdd, IsLUB.csSup_eq, Int.csSup_mem, ConditionallyCompleteLinearOrderBot.csSup_empty, le_csSup, le_csSup, isLUB_csSup, Int.csSup_of_not_bdd_above, csSup_le_csSup, csSup_le, Int.csSup_empty

Theorems

NameKindAssumesProvesValidatesDepends On
csInf_le 📖mathematicalBddBelow
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Lattice.toSemilatticeSup
toLattice
Set
Set.instMembership
InfSet.sInf
toInfSet
——
csSup_le 📖mathematicalSet.Nonempty
Set
Set.instMembership
upperBounds
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Lattice.toSemilatticeSup
toLattice
SupSet.sSup
toSupSet
——
le_csInf 📖mathematicalSet.Nonempty
Set
Set.instMembership
lowerBounds
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Lattice.toSemilatticeSup
toLattice
InfSet.sInf
toInfSet
——
le_csSup 📖mathematicalBddAbove
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Lattice.toSemilatticeSup
toLattice
Set
Set.instMembership
SupSet.sSup
toSupSet
——

ConditionallyCompleteLinearOrder

Definitions

NameCategoryTheorems
toConditionallyCompleteLattice 📖CompOp
1256 mathmath: Ordinal.iSup_eq_iSup, Order.coheight_add_one_le, LinearGrowth.linearGrowthSup_zero, IsClosed.sSup_mem, LinearGrowth.linearGrowthSup_top, Ordinal.iSup_eq_bsup, tendsto_zero_iff_meromorphicOrderAt_pos, ProbabilityTheory.bayesRisk_of_subsingleton, Measurable.limsup, Filter.Tendsto.limsup_eq, ENNReal.iInf_div_of_ne, Ordinal.lift_card_iSup_le_sum_card, SimpleGraph.ediam_le_iff, SimpleGraph.ediam_anti, Acc.rank_eq, Cardinal.mk_sUnion_le, Ordinal.iSup_pow_natCast, Order.LTSeries.length_le_krullDim, ENat.floor_pos, Module.rank_def, Filter.HasBasis.liminf_eq_ite, MeasureTheory.lintegral_def, SetTheory.PGame.grundyValue_eq_sInf_moveLeft, Continuous.strictMono_of_inj_boundedOrder', Algebra.lift_cardinalMk_adjoin_le, Ordinal.sup_eq_lsub_iff_lt_sup, SymmetricAlgebra.rank_eq, isSaddlePointOn_value, Order.krullDim_le_of_strictComono_and_surj, Set.Finite.einfsep, MeasureTheory.iInf_mul_le_lintegral, Cardinal.powerlt_max, Measurable.biSup, EMetric.hausdorffEdist_iUnion_le, ENNReal.add_sInf, ENat.sSup_eq_top_of_infinite, EisensteinSeries.hasSum_e2Summand_symmetricIcc, dimH_sUnion, Ordinal.sInf_compl_lt_lift_ord_succ, MeasureTheory.hittingBtwn_apply_mono_right, Set.einfsep_of_fintype, PartENat.ofENat_le, MeasureTheory.hittingBtwn_mono_left, RootPairing.Base.not_nonneg_iff_neg_of_sum_mem_range_root, lowerSemicontinuousOn_biSup, MeasureTheory.lintegral_iSup', ENNReal.le_liminf_mul, Cardinal.lift_iSup_le_lift_iSup, Order.krullDim_lt_coe_iff, Ordinal.max_zero_left, Cardinal.mk_list_eq_max_mk_aleph0, LinearGrowth.linearGrowthInf_add_le', ContinuousLinearMap.sSup_unitClosedBall_eq_nnnorm, Cardinal.mk_list_le_max, LinearGrowth.le_linearGrowthSup_add, CategoryTheory.ObjectProperty.HasCardinalLT.sup, NNReal.iSup_of_not_bddAbove, Ordinal.sup_eq_lsub_iff_succ, FormalMultilinearSeries.radius_pi_eq_iInf, ENNReal.ofNNReal_limsup, NatOrdinal.instAddLeftReflectLE, Padic.withValUniformEquiv_norm_le_one_iff, tendsto_cobounded_iff_meromorphicOrderAt_neg, lowerSemicontinuousWithinAt_iSup, Set.Finite.lt_csInf_iff, NNReal.mul_iSup_le, NatOrdinal.lt_mul_iff, ENat.iInf_mul_of_ne, Cardinal.mk_multiset_of_nonempty, ENat.smul_sSup, MeasureTheory.setLIntegral_le_iSup_mul, Ordinal.iSup_typein_succ, LinearGrowth.linearGrowthSup_add_le, MeasureTheory.hittingAfter_bot_le_iff, upperSemicontinuousAt_biInf, Metric.hausdorffEDist_def, csInf_mem, Ordinal.log_def, toCompactIccSpace, isPreconnected_iff_ordConnected, ciSup_partialSups_eq', Cardinal.iSup_of_empty, Order.krullDim_nonneg_iff, MeasureTheory.hittingBtwn_univ, ENNReal.limsup_const_mul_of_ne_top, sSup_Iio_eq_self_iff_isSuccPrelimit, IsOpen.measure_eq_iSup_isClosed, Cardinal.iSup_lt_of_isRegular, ENat.gc_toENNReal_floor, ContinuousLinearMap.nnnorm_def, Metric.hausdorffEDist_iUnion_le, ENNReal.iSup_pow_of_ne_zero, ENat.iSup_zero, MeasureTheory.OuterMeasure.iUnion_nat_of_monotone_of_tsum_ne_top, isPreconnected_Ici, Real.limsup_of_not_isBoundedUnder, Cardinal.mul_le_max_of_aleph0_le_right, NNReal.coe_sSup, MeasureTheory.lintegral_iSup_ae, isLeast_csInf, ENat.ceil_le_ceil, Order.length_le_coheight, Int.padicValuation_lt_one_iff, Order.height_add_one_le, MeromorphicNFAt.meromorphicOrderAt_nonneg_iff_analyticAt, Filter.cardinalInterFilter_sup, Cardinal.sum_pow_le_max_aleph0, Ordinal.sup_succ_le_lsub, MonoidAlgebra.cardinalMk_of_infinite, SimpleGraph.Walk.edist_le, LowerSemicontinuousOn.le_liminf, Ordinal.iSup_le_lsub, Ordinal.card_iSup_Iio_le_sum_card, MeasureTheory.Lp.eLpNorm_exponent_top_lim_le_liminf_eLpNorm_exponent_top, ENat.iInf_toNat, IsClosed.upperClosure, NatOrdinal.toOrdinal_min, Order.krullDim_pos_iff, NNReal.limsInf_of_not_isCobounded, Nimber.succ_def, Order.krullDim_le_one_iff, SimpleGraph.ediam_eq_top, Cardinal.mul_eq_max_of_aleph0_le_right, ENNReal.ofReal_iInf, MeasureTheory.OuterMeasure.coe_iSup, Cardinal.sum_le_mk_mul_iSup, Int.csInf_mem, MeasureTheory.Measure.LebesgueDecomposition.iSup_monotone, cbiSup_eq_of_not_forall, NNReal.segment_eq_uIcc, MeasureTheory.lintegral_iSup_directed, ProbabilityTheory.bayesRisk_of_subsingleton', Ideal.sup_primeHeight_eq_ringKrullDim, RootPairing.coe_chainBotCoeff_eq_sSup, ENNReal.iSup_add_iSup_le, setOf_liouvilleWith_subset_aux, Pi.Colex.le_sSup_apply, Set.Nonempty.eq_Icc_iff_int, LinearGrowth.le_linearGrowthSup_add', ENNReal.limsup_mul_le, Directed.measure_iUnion, Cardinal.mk_finsupp_lift_of_infinite, MeasureTheory.lowerCrossingTime_le, Submodule.spanRank_toENat_eq_iInf_encard, ExpGrowth.expGrowthSup_add, NatOrdinal.instIsOrderedMonoid, ConditionallyCompleteLinearOrderedField.toZeroLEOneClass, MeasureTheory.hittingAfter_mono, NatOrdinal.instIsOrderedRing, Nimber.mul_le_of_forall_ne, IsCompact.sSup_mem, LinearGrowth.linearGrowthInf_bot, ENat.le_floor, NNReal.coe_iSup, AEMeasurable.iInf, compare_eq_compareOfLessAndEq, IsCompact.isLeast_sInf, MeasureTheory.Measure.mkMetric_apply, ENNReal.mul_iInf, ENNReal.limsup_const_mul, RootPairing.setOf_root_add_zsmul_mem_eq_Icc, LinearOrderedField.lt_inducedMap_iff, MeasureTheory.Measure.iInf_IicSnd_gt, Order.height_pos, Subfield.cardinalMk_closure_le_max, MeasureTheory.biSup_measure_Iic, Cardinal.mul_eq_left_iff, ENNReal.toNNReal_sSup, Order.le_krullDim_iff, MeasureTheory.lintegral_iInf_ae, ENNReal.inv_sSup, MeasureTheory.lintegral_iInf', Int.csInf_eq_least_of_bdd, Cardinal.lift_iInf, ENNReal.limsup_liminf_le_liminf_limsup, limsInf_eq_of_le_nhds, Ordinal.toNimber_min, MeasureTheory.Measure.hausdorffMeasure_le_liminf_sum, Nimber.add_le_of_forall_ne, spectrum.spectralRadius_le_liminf_pow_nnnorm_pow_one_div, LinearGrowth.linearGrowthSup_bot, ContinuousLinearMap.sSup_sphere_eq_nnnorm, MeasureTheory.SimpleFunc.iSup_coe_eapprox, LinearOrderedField.inducedOrderRingIso_self, ENNReal.liminf_toReal_eq, MeasurableSet.measure_eq_iSup_isCompact_of_ne_top, Algebraic.cardinalMk_le_max, IsCompact.sSup_lt_iff_of_continuous, lowerSemicontinuousOn_iSup, lowerSemicontinuous_iff_le_liminf, LinearGrowth.linearGrowthInf_biInf, upperSemicontinuous_biInf, ConditionallyCompleteLinearOrderedField.toPosMulStrictMono, ENat.mul_iSup, ENat.sSup_eq_zero, RootPairing.Base.pos_or_neg_of_sum_smul_root_mem, IsClosed.sInf_mem, MeasureTheory.Measure.LebesgueDecomposition.iSup_succ_eq_sup, NNReal.iSup_eq_zero, Matroid.cRank_eq_iSup_cardinalMk_indep, Ordinal.card_sInf_range_compl_le_lift, Nimber.toOrdinal_max, MeasureTheory.Measure.LebesgueDecomposition.iSup_mem_measurableLE, Ordinal.iSup_le, Pi.Colex.sSup_apply, ENNReal.le_iInf₂_add_iInf₂, NNReal.natCast_iInf, Algebra.IsAlgebraic.cardinalMk_le_max, hasCardinalLT_subtype_max, ExpGrowth.expGrowthSup_def, MeasureTheory.limsup_lintegral_le, MeasureTheory.OuterMeasure.iInf_apply', Ordinal.iSup_succ, Ordinal.Principal.sSup, Set.Finset.coe_einfsep, NatOrdinal.add_le_iff, egauge_univ_pi, Eventually.le_linearGrowthInf, Order.coheight_le_krullDim, ProbabilityTheory.bayesRisk_eq_iInf_measure_of_subsingleton, Nimber.toOrdinal_min, Order.krullDim_nonpos_iff_forall_isMax, ValuativeRel.nonempty_orderIso_withZeroMul_int_iff, ENNReal.iInf_div, ExpGrowth.expGrowthSup_biSup, MeasureTheory.eLpNormEssSup_eq_iSup, MeasureTheory.exists_isSigmaFiniteSet_measure_ge, Filter.HasBasis.limsup_eq_ite, ENat.biSup_add', Finset.sup_univ_eq_ciSup, MeasureTheory.Measure.mkMetric_le_liminf_tsum, Nimber.pos_iff_ne_zero, NatOrdinal.succ_def, ENNReal.biSup_add, MeasureTheory.measure_sigmaFiniteSetGE_le, ENat.iInf_mul', ENNReal.sSup_div, ENNReal.essSup_add_le, Real.liminf_of_not_isBoundedUnder, Measurable.iSup, Pi.Lex.sInf_apply, Ordinal.card_omega0_opow, Ordinal.iSup_pow, EMetric.diam_insert, Cardinal.lift_iSup, Order.coheight_eq_coe_add_one_iff, Ordinal.deriv_limit, Order.krullDim_nonpos_iff_forall_isMin, isConnected_Iio, ENNReal.iInf_sum, Nimber.add_trichotomy, ENNReal.coe_sSup, liminf_eq_top, NNReal.image_coe_uIcc, EisensteinSeries.tsum_symmetricIco_linear_sub_linear_add_one_eq_zero, MeasureTheory.hittingBtwn_mono, Pi.Lex.sInf_apply_le, Dynamics.coverEntropyEntourage_union, Ordinal.card_sInf_range_compl_le, LinearGrowth.linearGrowthInf_top, MeasureTheory.Measure.LebesgueDecomposition.iSup_le_le, ENNReal.biSup_add_biSup_le, NatOrdinal.lt_add_iff, Real.limsSup_of_not_isCobounded, meromorphicOrderAt_add, EisensteinSeries.G2_eq_tsum_symmetricIco, Ordinal.sSup_eq_bsup, EReal.limsup_neg, Nimber.add_def, Cardinal.lift_sSup, SimpleGraph.eccent_le_one_iff, PartENat.toWithTop_lt, MeasureTheory.upperCrossingTime_mono, FreeAlgebra.cardinalMk_le_max, Int.csSup_eq_greatestOfBdd, ENat.floor_le_ceil, Cardinal.aleph_add_aleph, isPreconnected_Iic, Ordinal.iSup_add_nat, ENNReal.essSup_const_mul, MeasureTheory.hittingBtwn_apply_anti, Cardinal.mk_biUnion_le_lift, ENNReal.iSup_coe_lt_top, Ordinal.toNatOrdinal_min, IsCompact.isLUB_sSup, LinearGrowth.linearGrowthInf_const_mul_self, MeasureTheory.lintegral_le_iSup_mul, MonoidAlgebra.cardinalMk_lift_of_infinite, RootPairing.setOf_root_sub_zsmul_mem_eq_Icc, Cardinal.lift_iSup_le, Int.cofinite_eq, Set.Nonempty.ordConnected_iff_of_bdd', LowerSemicontinuousWithinAt.le_liminf, Ordinal.iSup_sum, ENat.iInf_coe_lt_top, ProbabilityTheory.avgRisk_le_iSup_risk, ENat.ceil_lt_top, WType.cardinalMk_le_max_aleph0_of_finite', Set.Finite.csSup_lt_iff, ENNReal.iSup_natCast, CategoryTheory.ObjectProperty.IsLocal.inf, exists_eq_ciInf_of_finite, ConditionallyCompleteLinearOrderedField.toIsOrderedCancelAddMonoid, Order.height_le_krullDim, ENat.iInf_mul, Topology.IsUpper.isTopologicalSpace_basis, ENNReal.ofReal_limsup, NatOrdinal.lt_one_iff_zero, tendsto_intCast_atBot_sup_atTop_cobounded, sInf_eq_argmin_on, NatOrdinal.instIsOrderedCancelAddMonoid, PseudoMetricSpace.dist_ofPreNNDist, Dynamics.coverEntropy_union, limsup_eq_bot, MvPowerSeries.le_order_subst, MeasureTheory.essSup_trim, ENat.add_iSup, ciInf_le', NNReal.iSup_mul_iSup_le, FirstOrder.Language.card_functions_sum_skolem₁_le, MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ, ENNReal.iInf_add_iInf, AEMeasurable.biInf, PSet.rank_insert, MeasureTheory.OuterMeasure.trim_eq_iInf', MeasureTheory.limsup_measure_closed_le_of_forall_tendsto_measure, SimpleGraph.chromaticNumber_eq_iInf, Ordinal.iSup_eq_lsub_or_succ_iSup_eq_lsub, ENat.mul_sSup, LinearGrowth.linearGrowthInf_zero, SimpleGraph.chromaticNumber_sum, SummationFilter.hasSum_symmetricIoc_int_iff, upperSemicontinuousOn_iInf, MeasurableSet.measure_eq_iSup_isCompact, Ordinal.sup_succ_eq_lsub, IsCompact.sInf_mem, LowerSemicontinuousAt.le_liminf, Ordinal.succ_log_def, Order.bot_lt_krullDim, isConnected_Ici, AnalyticAt.meromorphicOrderAt_nonneg, Ordinal.sSup_ord, topologicalKrullDim_subspace_le, Int.ball_eq_Ioo, AddMonoidAlgebra.cardinalMk_of_infinite', essSup_eq_sInf, Eventually.linearGrowthSup_le, MeasureTheory.upperCrossingTime_le_lowerCrossingTime, MeasureTheory.Content.outerMeasure_eq_iInf, Cardinal.mul_eq_max, NatOrdinal.toOrdinal_max, topologicalKrullDim_zero_of_discreteTopology, ENNReal.liminf_sub_const, MeasureTheory.hitting_eq_end_iff, EisensteinSeries.tendsto_e2Summand_atTop_nhds_zero, Cardinal.mk_freeMonoid, Cardinal.mk_bounded_set_le, Ordinal.enumOrd_zero, ciInf_mem, ENNReal.smul_iSup, Order.krullDim_eq_iSup_coheight_of_nonempty, ENNReal.iSup_add, LinearGrowth.linearGrowthSup_le_of_eventually_le, ENat.coe_sSup, ENNReal.iInf_mul_of_ne, SummationFilter.hasSum_symmetricIcc_iff, MeasureTheory.measure_biUnion_eq_iSup, RootPairing.coe_chainTopCoeff_eq_sSup, MeasureTheory.limsup_measure_closed_le_iff_liminf_measure_open_ge, Ordinal.iSup_mul_nat, ENNReal.tsum_iSup_eq, ENat.sInf_eq_zero, MeasurableSpace.cardinal_generateMeasurableRec_le, Order.krullDim_eq_iSup_length, Int.csInf_eq_leastOfBdd, Ideal.sup_primeHeight_of_maximal_eq_ringKrullDim, Finset.Nonempty.ciSup_mem_image, Order.coheight_pos_of_lt_top, egauge_pi, WithTop.iInf_coe_eq_top, essSup_comp_quotientAddGroup_mk, ENNReal.add_iInf, Set.Nonempty.ciSup_lt_iff, Order.coheight_eq_coe_iff_maximal_le_coheight, ENat.ceil_le, csInf_le', NNReal.agm_eq_ciSup, RootPairing.setOf_root_add_zsmul_eq_Icc_of_linearIndependent, SetTheory.PGame.grundyValue_eq_sInf_moveRight, ENNReal.toNNReal_iSup, Antitone.measure_iUnion, MeasureTheory.lintegral_liminf_le, SetTheory.PGame.grundyValue_le_of_forall_moveLeft, WType.cardinalMk_le_max_aleph0_of_finite, Order.height_pos_of_bot_lt, upperSemicontinuousWithinAt_iff_limsup_le, ENNReal.tsum_eq_iSup_nat', Metric.ediam_eq_sSup, NatOrdinal.not_lt_zero, ENNReal.inv_iSup, Int.csInf_of_not_bddBelow, ENat.ceil_lt, MeasureTheory.OuterMeasure.iInf_apply, UpperSemicontinuousWithinAt.limsup_le, SimpleGraph.edist_pos_of_ne, tendsto_nhds_iff_meromorphicOrderAt_nonneg, Cardinal.mk_list_eq_max, isPreconnected_Icc, EReal.min_neg_neg, upperSemicontinuousOn_iff_limsup_le, LinearGrowth.linearGrowthSup_biSup, Cardinal.mk_freeAddGroup, Ordinal.sup_le_lsub, ENNReal.liminf_mul_le, EReal.le_liminf_add, Ordinal.iSup_add_natCast, FormalMultilinearSeries.radius_inv_eq_limsup, ENat.ceil_pos, EisensteinSeries.hasSum_e2Summand_symmetricIco, Set.einfsep_insert, MeasureTheory.hitting_le, ENat.add_sSup, ENNReal.tsum_eq_iSup_sum, Order.krullDim_nonpos_of_subsingleton, LinearOrderedField.inducedOrderRingHom_toFun, Ordinal.IsNormal.map_iSup, NNReal.sSup_of_not_bddAbove, Padic.AddValuation.map_add, Order.coheight_eq_iSup_head_eq, Ordinal.card_iSup_Iio_le_card_mul_iSup, cbiInf_eq_of_not_forall, ENat.ceil_add_le, Cardinal.mk_finsupp_nat, MeasureTheory.le_liminf_measure_open_of_forall_tendsto_measure, ENat.lt_ceil, LinearGrowth.linearGrowthSup_inv, LinearGrowth.linearGrowthInf_const, LinearOrderedField.le_inducedMap_mul_self_of_mem_cutMap, MeasureTheory.OuterMeasure.biInf_apply, ENNReal.essSup_piecewise, ENNReal.iInf_add, ProbabilityTheory.bayesRisk_const_of_nonempty, ENat.coe_iSup, ENat.smul_iSup, PartENat.toWithTop_le, Finset.Nonempty.csSup_mem, IsInducing.topologicalKrullDim_le, ENNReal.mul_iInf', EReal.liminf_add_le, MeasureTheory.hittingAfter_eq_top_iff, AddMonoidAlgebra.cardinalMk_lift_of_infinite, MeasureTheory.measure_iInter_eq_iInf_measure_iInter_le, LinearGrowth.linearGrowthSup_le_iff, Cardinal.mul_ciSup, Measurable.limsup', IsCompact.lt_sInf_iff_of_continuous, meromorphicNFAt_iff_analyticAt_or, NNReal.iSup_pow_of_ne_zero, Order.krullDim_eq_iSup_height_of_nonempty, ExpGrowth.le_expGrowthInf_comp, ENNReal.liminf_const_sub, EMetric.infEdist_biUnion, MeasureTheory.iSup_lintegral_measurable_le_eq_lintegral, ENNReal.iInf_add_iInf_of_monotone, Nimber.small_Ioo, Ordinal.sup_eq_bsup, LinearGrowth.linearGrowthSup_const_mul_self, Cardinal.ciSup_mul, MeasureTheory.FiniteMeasure.limsup_measure_closed_le_of_tendsto, setOf_isPreconnected_eq_of_ordered, Set.einfsep_insert_le, MeasureTheory.IsAddFundamentalDomain.essSup_measure_restrict, Ordinal.Principal.iSup, ENat.sub_iSup, ENat.iInf_eq_top_of_isEmpty, MeasureTheory.Lp.eLpNorm_lim_le_liminf_eLpNorm, Set.Nonempty.csSup_mem, Cardinal.ord_eq_Inf, ENNReal.biSup_add', ENat.floor_lt_ceil, MeasureTheory.ProbabilityMeasure.le_liminf_measure_open_of_tendsto, le_ciInf_iff', csSup_empty, ENat.iInf_coe_eq_top, WithTop.iSup_coe_lt_top, ENNReal.iSup_add_iSup, lowerSemicontinuous_iSup, IntermediateField.cardinalMk_adjoin_le, untrop_sum, ENNReal.coe_essSup, ENat.floor_lt_top, Cardinal.mk_freeAddMonoid, ENNReal.iSup_sub, ENNReal.tsum_eq_iSup_sum', limsSup_eq_of_le_nhds, MeasureTheory.hittingBtwn_apply_mono_left, isPreconnected_Ioi, MeasureTheory.hittingBtwn_anti, MvPowerSeries.le_weightedOrder_subst, Cardinal.mul_le_max_of_aleph0_le_left, Cardinal.powerlt_min, ENat.gc_ceil_toENNReal, Cardinal.power_nat_le_max, MeasureTheory.measure_sigmaFiniteSetWRT', le_limsup_mul, ENat.exists_eq_iInf, IsCompact.measure_eq_iInf_isOpen, ENNReal.sub_eq_sInf, Ordinal.opow_limit, Finset.Nonempty.csInf_eq_min', MeasureTheory.OuterMeasure.ofFunction_eq_iInf_mem, Algebra.IsAlgebraic.lift_cardinalMk_le_max, Cardinal.preAleph_limit, Real.limsup_of_not_isCoboundedUnder, Order.height_eq_iSup_lt_height, Order.height_eq_coe_add_one_iff, dimH_eq_iInf, ENNReal.le_iInf_mul, IsClosedEmbedding.topologicalKrullDim_le, Real.liminf_of_not_isCoboundedUnder, NormedSpace.equicontinuous_TFAE, Order.krullDim_enat, MonoidAlgebra.cardinalMk_of_infinite', isTotallyDisconnected_iff_lt, Order.height_le_coe_iff, LinearGrowth.linearGrowthSup_sup, IsOpen.measure_eq_iSup_isCompact, PiLp.nnnorm_eq_ciSup, NatOrdinal.mul_le_iff, Cardinal.lift_iSup_le_sum, MeasureTheory.ae_bdd_liminf_atTop_rpow_of_eLpNorm_bdd, MeasureTheory.limsup_trim, Monotone.measure_iInter, Order.height_eq_coe_iff, MeasureTheory.Measure.inf_apply, MeasureTheory.hittingAfter_apply_anti, ENNReal.essSup_liminf_le, SimpleGraph.edist_le_eccent, isPreconnected_Ico, UpperSemicontinuousOn.limsup_le, SimpleGraph.edist_triangle, LinearGrowth.linearGrowthInf_inf, Relation.cutExpand_le_invImage_lex, HahnSeries.cardSupp_div_le, upperSemicontinuous_iInf, RootPairing.Base.not_nonpos_iff_pos_of_sum_mem_range_root, WithTop.iInf_coe_lt_top, ENat.coe_iInf, dimH_bUnion, EReal.coe_toENNReal_eq_max, Cardinal.sum_le_lift_mk_mul_iSup, ConditionallyCompleteLinearOrderedField.to_archimedean, ENNReal.inv_iInf, Ordinal.max_eq_zero, Ordinal.bsup_eq_sup, Order.length_le_height, ENat.mul_iInf_of_ne, eq_Icc_of_connected_compact, EMetric.infEdist_iUnion, Cardinal.mk_freeRing, Multiset.iSup_mem_map_of_ne_zero, sInf_mem_closure, ENNReal.toNNReal_iInf, Order.height_le_height_apply_of_strictMono, Cardinal.add_eq_max', MeasureTheory.eLpNormEssSup_eq_essSup_enorm, MeasureTheory.inducedOuterMeasure_eq_iInf, MeasureTheory.lintegral_iInf, MeasureTheory.le_iInf_lintegral, Ordinal.card_opow_le_of_omega0_le_right, NNReal.le_iInf_mul_iInf, ENNReal.le_iInf_mul_iInf, LinearGrowth.linearGrowthSup_eventually_monotone, ciSup_or', trdeg_eq_iSup_cardinalMk_isTranscendenceBasis, CategoryTheory.ObjectProperty.instIsTriangulatedClosed₂MinOfIsClosedUnderIsomorphisms, NatOrdinal.instAddLeftStrictMono, Cardinal.mul_mk_eq_max, Order.krullDim_le_one_iff_of_boundedOrder, Ordinal.iSup_mul_natCast, LowerSemicontinuous.le_liminf, Ordinal.mem_closure_iff_iSup, Cardinal.sInf_eq_zero_iff, sInf_iUnion_Ici, LinearGrowth.linearGrowthSup_const, NNReal.iSup_div, LSeries.abscissaOfAbsConv_binop_le, MeasureTheory.measure_iUnion_of_tendsto_zero, ENat.iSup_natCast, MeasureTheory.Lp.eLpNorm'_lim_eq_lintegral_liminf, Ordinal.iSup_lt_ord_lift, Cardinal.aleph_max, LSeries.abscissaOfAbsConv_add_le, Pi.Colex.sInf_apply_le, ciSup_of_empty, CategoryTheory.ObjectProperty.HasCardinalLT.iSup, ProbabilityTheory.bayesRisk_le_iInf, PiLp.edist_eq_iSup, OreLocalization.cardinalMk_le_max, Manifold.riemannianEDist_def, Finset.Nonempty.csSup_eq_max', LinearGrowth.linearGrowthInf_natCast_nonneg, ENNReal.tsum_eq_limsup_sum_nat, ENat.floor_mono, MeasureTheory.Measure.LebesgueDecomposition.iSup_monotone', IsOpen.measure_eq_biSup_integral_continuous, ExpGrowth.expGrowthInf_inf, Ordinal.card_opow_omega0, Ordinal.cof_eq_sInf_lsub, Cardinal.add_eq_max, MeasureTheory.Measure.restrict_iUnion_apply_eq_iSup, UpperSemicontinuousAt.limsup_le, EReal.le_limsup_mul, LinearOrderedField.inducedMap_mono, setOf_isPreconnected_subset_of_ordered, ENNReal.toReal_limsup, EReal.limsup_mul_le, ENat.le_ceil, LinearGrowth.linearGrowthSup_comp_nonneg, spectrum.limsup_pow_nnnorm_pow_one_div_le_spectralRadius, Cardinal.preAleph_max, Order.krullDim_eq_iSup_coheight, ENNReal.add_iSup, OrderIso.map_ciSup', ENNReal.iSup_coe_eq_top, SimpleGraph.ediam_le_two_mul_radius, isPreconnected_Ioc, SimpleGraph.edist_eq_sInf, upperSemicontinuousAt_iff_limsup_le, isPreconnected_uIcc, ENNReal.ofNNReal_liminf, EReal.le_liminf_mul, ENat.iSup_mul, Ordinal.mem_closure_tfae, Cardinal.iSup_lt_ord_lift_of_isRegular, isPreconnected_Ioo, SummationFilter.hasProd_symmetricIco_int_iff, MeasureTheory.hittingBtwn_mono_right, BoundedContinuousFunction.nndist_eq_iSup, SummationFilter.hasSum_symmetricIco_int_iff, SimpleGraph.eccent_le_ediam, le_analyticOrderAt_sub, le_analyticOrderAt_add, MeasureTheory.OuterMeasure.sSup_apply, NNReal.natCast_iSup, ExpGrowth.le_expGrowthInf_add, Measurable.liminf', SimpleGraph.ediam_eq_iSup_iSup_edist, Cardinal.iSup_lt_lift_of_isRegular, Ordinal.iSup'_eq_bsup, LSeries.abscissaOfAbsConv_convolution_le, Measurable.sInf, Polynomial.cardinalMk_eq_max, Nimber.not_bddAbove_compl_of_small, Order.krullDim_eq_iSup_height, ContinuousLinearMap.sSup_unit_ball_eq_nnnorm, UniformOnFun.edist_def', Monotone.linearGrowthInf_nonneg, MeasureTheory.tendsto_measure_sigmaFiniteSetGE, SimpleGraph.eccent_pos_iff, Topology.IsScott.instUnivSetOfIsUpper, IsAlgClosed.cardinal_le_max_transcendence_basis', MeasureTheory.Measure.hausdorffMeasure_le_liminf_tsum, hasCardinalLT_subtype_iSup, ENat.sSup_add, Cardinal.beth_limit, Filter.Tendsto.liminf_eq, ExpGrowth.expGrowthSup_comp_le, trop_iInf, Order.coheight_le_coheight_apply_of_strictMono, upperSemicontinuous_iff_limsup_le, ENNReal.biSup_add_biSup_le', Cardinal.mk_biUnion_le, MeasureTheory.lintegral_iSup_directed_of_measurable, MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part, Submodule.spanRank_toENat_eq_iInf_finset_card, isSaddlePointOn_iff', NatOrdinal.instAddLeftMono, Ordinal.iSup_sequence_lt_omega_one, SetTheory.PGame.birthday_def, SimpleGraph.edist_anti, ENNReal.mul_sSup, ENNReal.iSup_add_iSup_of_monotone, FirstOrder.Language.Term.card_sigma, ENNReal.mul_iInf_of_ne, le_csInf_iff', IsCompact.isGLB_sInf, MeasureTheory.Measure.LebesgueDecomposition.iSup_mem_measurableLE', Finset.untrop_sum, ENNReal.ofReal_essSup, ENat.coe_sInf, Cardinal.mk_bounded_subset_le, ENNReal.sSup_mul, ENNReal.iSup_zero, PartENat.ofENat_lt, Order.one_le_krullDim_iff, Cardinal.sum_le_iSup_lift, Set.einfsep_eq_iInf, IsWellFounded.rank_eq, Ordinal.iSup_typein_limit, ENat.iSup_eq_zero, essSup_comp_quotientGroup_mk, EReal.liminf_neg, Nimber.not_lt_zero, Ordinal.iSup_lt_lift, Real.limsInf_of_not_isBounded, LinearGrowth.linearGrowthInf_eventually_monotone, MeasureTheory.lintegral_eq_nnreal, UniformFun.edist_def, Cardinal.mk_iUnion_le_lift, Order.coheight_le_iff', Order.krullDim_le_of_strictMono, essInf_eq_sSup, WithTop.isGLB_sInf, ENNReal.iInf_mul_iInf, MeasureTheory.SimpleFunc.lintegral_restrict_iUnion_of_directed, NNReal.le_mul_iInf, ENNReal.limsup_eq_zero_iff, MeasureTheory.IsFundamentalDomain.essSup_measure_restrict, Cardinal.add_mk_eq_max, ExpGrowth.expGrowthInf_iInf, isCompact_Icc, EReal.liminf_mul_le, Ordinal.derivFamily_limit, BoundedContinuousFunction.nnnorm_eq_iSup_nnnorm, Nimber.small_Ioc, EReal.max_neg_neg, AddMonoidAlgebra.cardinalMk_lift_of_infinite', MeasureTheory.OuterMeasure.sInf_apply, MeasureTheory.iInf_le_lintegral, Int.csSup_of_not_bddAbove, ZFSet.rank_range, iSup₂_iInf₂_le_iInf₂_iSup₂, ENNReal.add_biSup', lowerSemicontinuousWithinAt_iff_le_liminf, Cardinal.succ_def, Monotone.expGrowthInf_comp_le, PartENat.withTopEquiv_le, Int.csSup_eq_greatest_of_bdd, EisensteinSeries.tsum_tsum_symmetricIco_sub_eq, Ordinal.iSup_ord, ENNReal.hasSum, ZFSet.rank_insert, ENat.iSup_add, Ordinal.sup_eq_sup, PartENat.withTopEquiv_lt, StieltjesFunction.length_eq, ENNReal.add_sSup, Ordinal.le_iSup, MeasureTheory.measure_eq_iInf', le_csInf_iff'', ENNReal.iInf_mul', Nimber.small_Icc, MeasureTheory.lintegral_iInf_directed_of_measurable, Ordinal.iSup_eq_zero_iff, LinearOrderedField.inducedOrderRingIso_symm, MeasureTheory.hitting_mono, Cardinal.sInf_empty, Measurable.liminf, Continuous.strictMono_of_inj, Metric.infEDist_iUnion, Cardinal.mk_freeGroup, ENNReal.iSup_div, Set.Nonempty.ciSup_mem_image, LinearGrowth.linearGrowthInf_iInf, liminf_mul_le, lowerSemicontinuousAt_iff_le_liminf, UniformOnFun.edist_def, MeasureTheory.Measure.iInf_rat_gt_prod_Iic, MeasureTheory.OuterMeasure.boundedBy_apply, Ordinal.card_opow_eq_of_omega0_le_right, ENNReal.limsup_add_of_right_tendsto_zero, ProbabilityTheory.iSup_bayesRisk_le_minimaxRisk, LinearGrowth.linearGrowthInf_monotone, ENat.ceil_mono, MeasurableSpace.cardinal_measurableSet_le, iSup_succ, Ordinal.sup_eq_bsup', Ordinal.succ_iSup_eq_lsub_iff, MeasurableSpace.cardinal_generateMeasurable_le, ENat.mul_iInf, isConnected_Iic, SimpleGraph.diam_def, ENat.sSup_mul, Cardinal.sum_le_lift_mk_mul_iSup_lift, IsAlgClosed.cardinal_le_max_transcendence_basis, ENat.iSup_coe_eq_top, Nimber.small_Iic, MeasureTheory.hittingAfter_mem_set_of_ne_top, ProbabilityTheory.bayesRisk_le_iInf', ENNReal.toReal_essSup, BoundedContinuousFunction.edist_eq_iSup, RootPairing.Base.exists_root_eq_sum_int, Ordinal.sup_eq_lsub, Nimber.one_le_iff_ne_zero, ExpGrowth.expGrowthInf_biInf, LinearGrowth.linearGrowthSup_iSup, LinearGrowth.linearGrowthInf_comp_nonneg, ENNReal.essSup_mul_le, Ordinal.lift_card_sInf_compl_le, RootPairing.root_add_zsmul_mem_range_iff, SimpleGraph.edist_le_ediam, iSup_Iio_eq_self_iff_isSuccPrelimit, Ordinal.iSup_eq_lsub_iff_lt_iSup, ENNReal.toReal_sSup, isPreconnected_Iio, ENNReal.sub_iSup, Order.coheight_le_coe_iff, ContinuousMap.nnnorm_eq_iSup_nnnorm, Antitone.measure_iInter, MeasureTheory.ae_bdd_liminf_atTop_of_eLpNorm_bdd, IsClosed.lowerClosure, ENNReal.liminf_add_of_right_tendsto_zero, Topology.IsScott.scott_eq_upper_of_completeLinearOrder, MeasureTheory.Measure.InnerRegularWRT.measure_eq_iSup, EisensteinSeries.tendsto_double_sum_S_act, Ordinal.IsNormal.map_iSup_of_bddAbove, SummationFilter.symmetricIcc_eq_symmetricIoo_int, Cardinal.preBeth_limit, IsCompact.continuous_sInf, ProbabilityTheory.bayesRisk_const_of_neZero, le_total, MeasureTheory.measure_eq_iInf, MeasureTheory.hittingBtwn_le, ENNReal.limsup_mul_le', MeasureTheory.le_iInf₂_lintegral, upperSemicontinuousOn_biInf, MeasureTheory.Measure.iSup_restrict_spanningSets, ContinuousMap.nndist_eq_iSup, NNReal.limsSup_of_not_isBounded, ENNReal.le_iInf_add_iInf, MeasureTheory.lintegral_iSup, Int.closedBall_eq_Icc, Ordinal.lt_iSup_iff, Cardinal.aleph_mul_aleph, ENNReal.iInf_mul, liminf_finset_inf', Ordinal.IsNormal.map_sSup_of_bddAbove, ENNReal.toReal_iInf, Order.krullDim_le_one_iff_forall_isMax, NatOrdinal.zero_le, EisensteinSeries.summable_e2Summand_symmetricIcc, ENNReal.iSup_eq_zero, ENNReal.sInf_add, Ordinal.card_opow_le_of_omega0_le_left, Ordinal.bsup'_eq_iSup, ENat.floor_lt, Ordinal.IsNormal.apply_omega0, MeasurableSet.measure_eq_iSup_isClosed_of_ne_top, ENNReal.iInf_coe_lt_top, Order.krullDim_pos_iff_of_orderTop, Nimber.mul_def, MonotoneOn.exists_tendsto_deriv_liminf_lintegral_enorm_le, MeasureTheory.eLpNormEssSup_count, ProbabilityTheory.bayesRisk_const, SimpleGraph.radius_le_ediam, Ordinal.iSup_Iio_eq_bsup, WithTop.iSup_coe_eq_top, NNReal.coe_iInf, MeasureTheory.OuterMeasure.iUnion_of_tendsto_zero, FirstOrder.Language.Term.card_le, AEMeasurable.iSup, cauchy_davenport_minOrder_add, WithTop.isLUB_sSup, ENNReal.smul_sSup, SimpleGraph.chromaticNumber_eq_biInf, Ordinal.iSup_eq_of_range_eq, MeasureTheory.hittingBtwn_eq_end_iff, Int.csSup_mem, Ordinal.card_opow_eq_of_omega0_le_left, Order.coheight_coe_enat, ZFSet.rank_iUnion, ENNReal.sSup_add, Ordinal.IsNormal.map_sSup, LinearGrowth.le_linearGrowthInf_iff, Topology.IsScott.isOpen_iff_Iic_compl_or_univ, SimpleGraph.eccent_le_iff, MeasureTheory.upperCrossingTime_zero', MeasureTheory.iSup₂_lintegral_le, Order.krullDim_eq_iSup_height_add_coheight_of_nonempty, Ordinal.iSup_sequence_lt_omega1, FiberBundle.exists_trivialization_Icc_subset, ENNReal.tsum_eq_iSup_nat, AlgebraicGeometry.geometrically_inf, ENNReal.iInf₂_add, Ordinal.bsup_eq_iSup, LinearOrderedField.inducedMap_nonneg, upperSemicontinuousWithinAt_biInf, Order.krullDim_pos_iff_of_orderBot, SimpleGraph.ediam_def, Ordinal.lsub_le_succ_iSup, Cardinal.add_mk_eq_max', NNReal.limsup_of_not_isBoundedUnder, ENNReal.essSup_indicator_eq_essSup_restrict, Cardinal.ciSup_mul_ciSup, Nimber.small_Ico, Cardinal.lift_sInf, AEMeasurable.biSup, MeasureTheory.OuterMeasure.top_apply', MvPolynomial.cardinalMk_le_max_lift, ExpGrowth.expGrowthInf_def, Real.limsInf_of_not_isCobounded, Measurable.iInf, LinearGrowth.linearGrowthInf_le_of_eventually_le, ENNReal.tsum_eq_liminf_sum_nat, LinearGrowth.linearGrowthInf_le_iff, NNReal.iSup_empty, EReal.le_limsup_add, IsUltrametricDist.nnnorm_tprod_le, MeasureTheory.Measure.hausdorffMeasure_apply, PartENat.withTopEquiv_symm_le, LinearGrowth.linearGrowthSup_comp_le, Monotone.linearGrowthInf_comp_le, max_aleph0_card_le_rank_fun_nat, FreeAlgebra.cardinalMk_le_max_lift, IntermediateField.lift_cardinalMk_adjoin_le, ProbabilityTheory.bayesRisk_discard, ENNReal.iSup_pow, Ideal.sup_height_eq_ringKrullDim, Monotone.linearGrowthInf_comp_mul, isSaddlePointOn_iff, Ordinal.mem_iff_iSup_of_isClosed, NNReal.mul_iSup, EMetric.hausdorffEdist_def, ENNReal.sub_iInf, ENNReal.essSup_restrict_eq_of_support_subset, LinearOrderedField.coe_inducedOrderRingIso, MeasureTheory.lowerCrossingTime_mono, Cardinal.add_ciSup, MeasureTheory.hittingAfter_le_iff, Set.measure_eq_iInf_isOpen, LSeries.abscissaOfAbsConv_sub_le, MeasureTheory.Measure.mkMetric_le_liminf_sum, Ordinal.sInf_compl_lt_ord_succ, MeasureTheory.lintegral_enorm_le_liminf_of_tendsto, ENNReal.iSup_lt_eq_self, MeasureTheory.OuterMeasure.sInf_apply', Order.coheight_eq_coe_iff, Cardinal.powerlt_aleph0_le, BoundedContinuousFunction.nndist_eq, ENNReal.finsetSum_iSup_of_monotone, NatOrdinal.small_Ioc, IsCompact.measure_eq_biInf_integral_hasCompactSupport, MonoidAlgebra.cardinalMk_lift_of_infinite', NNReal.liminf_of_not_isCoboundedUnder, MeasureTheory.Lp.eLpNorm_exponent_top_lim_eq_essSup_liminf, exists_eq_ciSup_of_finite, MeasureTheory.SimpleFunc.iSup_eapprox_apply, SimpleGraph.radius_le_eccent, MeasureTheory.ae_eq_set_symmDiff, Dense.ciInf', ENat.toENNReal_iSup, Ordinal.isClosed_iff_iSup, MeasureTheory.upperCrossingTime_le, Ordinal.bsup_eq_sup', ENat.lt_floor, NatOrdinal.small_Iio, Cardinal.add_eq_left_iff, LinearOrderedField.coe_lt_inducedMap_iff, ConditionallyCompleteLinearOrderBot.csSup_empty, LinearMap.rank_comp_le, isConnected_Ioi, Cardinal.mk_finsupp_of_infinite, MeasureTheory.hittingAfter_apply_mono, NNReal.sInf_empty, IsCompact.continuous_sSup, limsup_finset_sup, dimH_iUnion, NNReal.mul_iInf, NNReal.toReal_limsup, EisensteinSeries.tendsto_tsum_one_div_linear_sub_succ_eq, Monotone.measure_iUnion, MeasureTheory.le_hittingAfter, LinearGrowth.linearGrowthInf_add_le, Order.height_le_iff, Cardinal.mul_eq_max_of_aleph0_le_left, limsInf_nhds, NNReal.iInf_mul, Frequently.linearGrowthInf_le, Ordinal.toNimber_max, Cardinal.lift_iSup_le_lift_iSup', untrop_sum_eq_sInf_image, Ordinal.sup_typein_limit, Ordinal.omega_max, Order.KrullDimLE.krullDim_le, NatOrdinal.small_Ico, le_liminf_mul, Polynomial.cardinalMk_le_max, List.iSup_mem_map_of_ne_nil, ENNReal.iSup_mul_le, Monotone.linearGrowthInf_comp, ENat.add_biSup, NNReal.iInf_empty, ciSup_false, ConditionallyCompleteLinearOrderedField.toIsStrictOrderedRing, ENNReal.liminf_add_of_left_tendsto_zero, Padic.norm_rat_le_one_iff_padicValuation_le_one, liminf_finset_inf, MeasureTheory.OuterMeasure.sInfGen_def, EisensteinSeries.summable_e2Summand_symmetricIco, Finset.Nonempty.csInf_mem, Directed.measure_iInter, RootPairing.root_sub_zsmul_mem_range_iff, ENat.mul_iInf', Algebra.cardinalMk_adjoin_le, NatOrdinal.small_Ioo, Order.coheight_anti, Cardinal.sum_le_iSup, EMetric.diam_eq_sSup, NNReal.le_iInf_add_iInf, Measurable.sSup, Order.height_mono, Ordinal.sup_eq_lsub_or_sup_succ_eq_lsub, Cardinal.iInf_eq_zero_iff, CategoryTheory.ObjectProperty.instIsTriangulatedMinOfIsClosedUnderIsomorphisms, Order.height_le_iff', Order.krullDim_nonneg, MeasureTheory.measure_iUnion_eq_iSup_accumulate, lowerSemicontinuousAt_iSup, Cardinal.lift_iSup_le_iff, Cardinal.sum_pow_eq_max_aleph0, Monotone.linearGrowthSup_nonneg, ENNReal.hasProd_iInf_prod, Pi.Colex.sInf_apply, Order.coheight_pos, SummationFilter.symmetricIcc_eq_map_Icc_nat, Measurable.biInf, sSup_iUnion_Iic, ENat.iInf_eq_zero, Metric.ediam_iUnion_mem_option, Monotone.linearGrowthSup_comp_mul, Filter.cardinalInterFilter_inf, ZFSet.rank_pair, Monotone.le_linearGrowthSup_comp, Order.index_le_height, ExpGrowth.expGrowthSup_sup, Order.coheight_eq, tendsto_intCast_atTop_cobounded, Cardinal.aleph_limit, Order.krullDim_le_one_iff_forall_isMin, SummationFilter.hasProd_symmetricIcc_iff, Topology.IsLower.isTopologicalSpace_basis, bsupr_limsup_dimH, ENNReal.limsup_const_sub, ENNReal.add_iInf₂, Cardinal.mul_eq_max', ENat.floor_le_floor, MeasureTheory.iSup_lintegral_le, Pi.Lex.le_sSup_apply, upperSemicontinuousAt_iInf, upperSemicontinuousWithinAt_iInf, MeasureTheory.OuterMeasure.biInf_apply', Finset.Nonempty.ciSup_eq_max'_image, IsCompact.exists_sInf_image_eq, ENNReal.le_limsup_mul, SimpleGraph.edist_le, Ordinal.iSup_le_iff, Cardinal.iSup_lt_ord_of_isRegular, Int.csSup_of_not_bdd_above, Ordinal.iSup_eq_lsub, csInf_le_csInf', Ordinal.max_zero_right, ENNReal.limsup_add_le, Order.rev_index_le_coheight, IsTranscendenceBasis.lift_cardinalMk_eq_max_lift, IsTranscendenceBasis.lift_rank_eq_max_lift, MvRatFunc.rank_eq_max_lift, MeasureTheory.lintegral_liminf_le', Monotone.linearGrowthSup_comp, Int.padicValuation_le_one, MeasureTheory.OuterMeasure.iSup_apply, ENNReal.iSup_mul, cauchy_davenport_minOrder_mul, Ordinal.iSup_eq_lsub_iff, IsCompact.exists_sSup_image_eq, lowerSemicontinuousWithinAt_biSup, ENat.sSup_eq_zero', MvPolynomial.cardinalMk_le_max, Cardinal.ciSup_add_ciSup, SetTheory.PGame.grundyValue_le_of_forall_moveRight, ENNReal.limsup_sub_const, ENNReal.inv_liminf, NNReal.agm_eq_ciInf, Cardinal.mk_freeCommRing, ENNReal.finsetSum_iSup, Pi.Lex.sSup_apply, Ordinal.iSup_natCast, lowerSemicontinuousAt_biSup, ENNReal.iInf_gt_eq_self, Cardinal.mk_freeAbelianGroup, limsup_finset_sup', ENNReal.eventually_le_limsup, ENNReal.inv_limsup, Ordinal.iSup_iterate_eq_nfp, Int.instIsOrderBornology, NNReal.toReal_liminf, NatOrdinal.small_Icc, Real.limsSup_of_not_isBounded, Nimber.le_zero, Set.einfsep_iUnion_mem_option, Set.chainHeight_eq_iSup, ENNReal.toNNReal_sInf, IsCompact.isGreatest_sSup, HahnSeries.cardSupp_hsum_powers_le, Nimber.small_Iio, Cardinal.add_le_max, MeasureTheory.OuterMeasure.trim_eq_iInf, ZLattice.sum_piFinset_Icc_rpow_le, ContinuousMap.enorm_eq_iSup_enorm, ENNReal.inv_sInf, ENNReal.coe_iInf, Ordinal.apply_omega0_of_isNormal, ConditionallyCompleteLinearOrderedField.toMulPosStrictMono, Order.height_eq_iSup_last_eq, ENat.toENNReal_iInf, BoundedContinuousFunction.enorm_eq_iSup_enorm, ExpGrowth.expGrowthSup_iSup, Order.height_enat, Ordinal.card_opow_le, ENNReal.limsup_toReal_eq, dimH_def, NNReal.le_iInf_mul, Cardinal.mk_finsupp_lift_of_infinite', Order.length_le_height_last, Order.coheight_eq_iSup_gt_coheight, IsPreconnected.mem_intervals, LinearGrowth.linearGrowthSup_monotone, tendsto_intCast_atBot_cobounded, NNReal.iSup_mul, LinearGrowth.linearGrowthInf_le_linearGrowthSup_of_frequently_le, LinearGrowth.le_linearGrowthInf_comp, ENNReal.tprod_eq_iInf_prod, Ordinal.lsub_le_sup_succ, MeasureTheory.lpNorm_exponent_top_eq_essSup, CategoryTheory.IsCardinalFiltered.exists_cardinal_directed.Diagram.iSup_P, MeasureTheory.AECover.iSup_lintegral_eq_of_countably_generated, MeasureTheory.OuterMeasure.ofFunction_apply, Cardinal.mul_le_max, ProbabilityTheory.bayesRisk_const', Pell.IsFundamental.y_strictMono, Dense.ciSup', NNReal.iSup_pow, Ordinal.succ_iSup_le_lsub_iff, AddOreLocalization.cardinalMk_le_max, HahnSeries.cardSupp_inv_le, LinearGrowth.le_linearGrowthSup_iff, MeasureTheory.hittingAfter_anti, Order.length_le_coheight_head, Nimber.lt_one_iff_zero, NatOrdinal.small_Iic, MeasureTheory.setLIntegral_iUnion_of_directed, ENat.ceil_le_floor_add_one, MeasureTheory.lintegral_eq_iSup_eapprox_lintegral, ENNReal.coe_iSup, ENNReal.image_coe_uIcc, AddMonoidAlgebra.cardinalMk_of_infinite, FormalMultilinearSeries.radius_eq_liminf, PartENat.withTopEquiv_symm_lt, SummationFilter.hasProd_symmetricIoc_int_iff, Ordinal.IsNormal.apply_of_isSuccLimit, ENNReal.iInf_div', ENNReal.limsup_add_of_left_tendsto_zero, ProbabilityTheory.IdentDistrib.essSup_eq, Ordinal.lsub_sum, Ordinal.iSup_lt_ord, Int.csInf_of_not_bdd_below, MeasureTheory.hittingAfter_lt_iff, ENat.add_biSup', ENNReal.essSup_eq_zero_iff, lowerSemicontinuousOn_iff_le_liminf, limsup_mul_le, MeasureTheory.ProbabilityMeasure.limsup_measure_closed_le_of_tendsto, FreeAlgebra.cardinalMk_eq_max, ENNReal.ae_le_essSup, ZFSet.rank_union, Mathlib.Meta.Positivity.natCeil_pos, ENat.biSup_add, Order.bot_lt_krullDim_iff, ENNReal.iInf_coe_eq_top, ENNReal.iSup₂_pow_of_ne_zero, Cardinal.mk_iUnion_le, Order.height_eq_coe_iff_minimal_le_height, EisensteinSeries.tsum_symmetricIco_tsum_eq_S_act, Ordinal.preOmega_max, Int.csSup_empty, trop_sInf_image, Cardinal.iSup_le_sum, Cardinal.add_eq_right_iff, SimpleGraph.eccent_def, Cardinal.iSup_mk_le_mk_iUnion, iSup_limsup_dimH, LinearGrowth.le_linearGrowthInf_add, ZFSet.iSup_card_le_card_iUnion, Ordinal.card_iSup_le_sum_card, Cardinal.mk_finsupp_of_infinite', MvPolynomial.cardinalMk_eq_max_lift, UpperSemicontinuous.limsup_le, Cardinal.ciSup_add, Ordinal.iSup_lt, ENNReal.coe_sInf, Ordinal.toNatOrdinal_max, Order.one_lt_height_iff, MeasureTheory.iInf_mul_le_setLIntegral, MeasureTheory.measure_sigmaFiniteSetGE_ge, ContinuousMap.edist_eq_iSup, Order.coheight_le_iff, FirstOrder.Language.Substructure.lift_card_closure_le, ENat.iSup_coe_lt_top, Metric.infEDist_biUnion, PSet.rank_pair, EReal.limsup_add_le, analyticOrderAt_add_of_ne, NNReal.iSup_mul_le, egauge_pi', MeasureTheory.OuterMeasure.iSup_sInfGen_nonempty, FreeAlgebra.cardinalMk_eq_max_lift, ENNReal.toReal_iSup, IsCompact.exists_sSup_image_eq_and_ge, NNReal.iInf_const_zero, lowerSemicontinuous_biSup, ClassGroup.exists_mem_finset_approx', ENat.floor_le, sSup_mem_closure, CategoryTheory.IsCardinalFiltered.exists_cardinal_directed.Diagram.sup_P, IsUltrametricDist.nnnorm_tsum_le, Order.krullDimLE_iff, ENNReal.add_biSup, Metric.ediam_insert, MeasureTheory.Lp.eLpNorm'_lim_le_liminf_eLpNorm', LinearMap.lift_rank_comp_le, ClassGroup.exists_mem_finsetApprox, Rat.padicValuation_le_one_iff, PiLp.nndist_eq_iSup, Set.Nonempty.csInf_mem, MvPolynomial.cardinalMk_eq_max, Ordinal.sInf_empty, Frequently.le_linearGrowthSup, Ordinal.sup_typein_succ, MeasureTheory.Measure.iSup_restrict_spanningSets_of_measurableSet, SimpleGraph.edist_le_one_iff_adj_or_eq, MeasureTheory.lintegral_le_iSup, Algebraic.cardinalMk_lift_le_max, FirstOrder.Language.BoundedFormula.card_le, IsCompact.exists_sInf_image_eq_and_le, SimpleGraph.radius_eq_iInf_iSup_edist, ENNReal.mul_iSup, Algebra.rank_adjoin_le, EMetric.diam_iUnion_mem_option, Int.cobounded_eq, LinearGrowth.linearGrowthInf_neg, NNReal.coe_sInf, limsSup_nhds, Monotone.le_expGrowthSup_comp, ENNReal.toReal_sInf, Int.csInf_empty, EisensteinSeries.tsum_symmetricIco_tsum_sub_eq
toDecidableEq 📖CompOp
1 mathmath: compare_eq_compareOfLessAndEq
toDecidableLE 📖CompOp—
toDecidableLT 📖CompOp
1 mathmath: compare_eq_compareOfLessAndEq
toOrd 📖CompOp
1 mathmath: compare_eq_compareOfLessAndEq

Theorems

NameKindAssumesProvesValidatesDepends On
compare_eq_compareOfLessAndEq 📖mathematical—toOrd
Preorder.toLT
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Lattice.toSemilatticeSup
ConditionallyCompleteLattice.toLattice
toConditionallyCompleteLattice
toDecidableLT
toDecidableEq
——
csInf_of_not_bddBelow 📖mathematicalBddBelow
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Lattice.toSemilatticeSup
ConditionallyCompleteLattice.toLattice
toConditionallyCompleteLattice
InfSet.sInf
ConditionallyCompleteLattice.toInfSet
Set
Set.instEmptyCollection
——
csSup_of_not_bddAbove 📖mathematicalBddAbove
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Lattice.toSemilatticeSup
ConditionallyCompleteLattice.toLattice
toConditionallyCompleteLattice
SupSet.sSup
ConditionallyCompleteLattice.toSupSet
Set
Set.instEmptyCollection
——
le_total 📖mathematical—Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Lattice.toSemilatticeSup
ConditionallyCompleteLattice.toLattice
toConditionallyCompleteLattice
——

ConditionallyCompleteLinearOrderBot

Definitions

NameCategoryTheorems
toConditionallyCompleteLinearOrder 📖CompOp
1035 mathmath: Ordinal.iSup_eq_iSup, Order.coheight_add_one_le, LinearGrowth.linearGrowthSup_zero, IsClosed.sSup_mem, LinearGrowth.linearGrowthSup_top, Ordinal.iSup_eq_bsup, ProbabilityTheory.bayesRisk_of_subsingleton, ENNReal.iInf_div_of_ne, Ordinal.lift_card_iSup_le_sum_card, SimpleGraph.ediam_le_iff, SimpleGraph.ediam_anti, Acc.rank_eq, Cardinal.mk_sUnion_le, Ordinal.iSup_pow_natCast, Order.LTSeries.length_le_krullDim, ENat.floor_pos, Module.rank_def, MeasureTheory.lintegral_def, SetTheory.PGame.grundyValue_eq_sInf_moveLeft, Algebra.lift_cardinalMk_adjoin_le, Ordinal.sup_eq_lsub_iff_lt_sup, SymmetricAlgebra.rank_eq, isSaddlePointOn_value, Order.krullDim_le_of_strictComono_and_surj, Set.Finite.einfsep, MeasureTheory.iInf_mul_le_lintegral, Cardinal.powerlt_max, EMetric.hausdorffEdist_iUnion_le, ENNReal.add_sInf, ENat.sSup_eq_top_of_infinite, dimH_sUnion, Ordinal.sInf_compl_lt_lift_ord_succ, Set.einfsep_of_fintype, PartENat.ofENat_le, lowerSemicontinuousOn_biSup, MeasureTheory.lintegral_iSup', ENNReal.le_liminf_mul, Cardinal.lift_iSup_le_lift_iSup, Order.krullDim_lt_coe_iff, Ordinal.max_zero_left, Cardinal.mk_list_eq_max_mk_aleph0, LinearGrowth.linearGrowthInf_add_le', ContinuousLinearMap.sSup_unitClosedBall_eq_nnnorm, Cardinal.mk_list_le_max, LinearGrowth.le_linearGrowthSup_add, CategoryTheory.ObjectProperty.HasCardinalLT.sup, NNReal.iSup_of_not_bddAbove, Ordinal.sup_eq_lsub_iff_succ, FormalMultilinearSeries.radius_pi_eq_iInf, ENNReal.ofNNReal_limsup, NatOrdinal.instAddLeftReflectLE, lowerSemicontinuousWithinAt_iSup, NNReal.mul_iSup_le, NatOrdinal.lt_mul_iff, ENat.iInf_mul_of_ne, Cardinal.mk_multiset_of_nonempty, ENat.smul_sSup, MeasureTheory.setLIntegral_le_iSup_mul, Ordinal.iSup_typein_succ, LinearGrowth.linearGrowthSup_add_le, MeasureTheory.hittingAfter_bot_le_iff, upperSemicontinuousAt_biInf, Metric.hausdorffEDist_def, Ordinal.log_def, Cardinal.iSup_of_empty, Order.krullDim_nonneg_iff, ENNReal.limsup_const_mul_of_ne_top, sSup_Iio_eq_self_iff_isSuccPrelimit, IsOpen.measure_eq_iSup_isClosed, Cardinal.iSup_lt_of_isRegular, ENat.gc_toENNReal_floor, ContinuousLinearMap.nnnorm_def, Metric.hausdorffEDist_iUnion_le, ENNReal.iSup_pow_of_ne_zero, ENat.iSup_zero, MeasureTheory.OuterMeasure.iUnion_nat_of_monotone_of_tsum_ne_top, Cardinal.mul_le_max_of_aleph0_le_right, NNReal.coe_sSup, MeasureTheory.lintegral_iSup_ae, ENat.ceil_le_ceil, Order.length_le_coheight, Order.height_add_one_le, Filter.cardinalInterFilter_sup, Cardinal.sum_pow_le_max_aleph0, Ordinal.sup_succ_le_lsub, MonoidAlgebra.cardinalMk_of_infinite, SimpleGraph.Walk.edist_le, LowerSemicontinuousOn.le_liminf, Ordinal.iSup_le_lsub, Ordinal.card_iSup_Iio_le_sum_card, MeasureTheory.Lp.eLpNorm_exponent_top_lim_le_liminf_eLpNorm_exponent_top, ENat.iInf_toNat, NatOrdinal.toOrdinal_min, Order.krullDim_pos_iff, NNReal.limsInf_of_not_isCobounded, Nimber.succ_def, Order.krullDim_le_one_iff, SimpleGraph.ediam_eq_top, Cardinal.mul_eq_max_of_aleph0_le_right, ENNReal.ofReal_iInf, MeasureTheory.OuterMeasure.coe_iSup, Cardinal.sum_le_mk_mul_iSup, MeasureTheory.Measure.LebesgueDecomposition.iSup_monotone, NNReal.segment_eq_uIcc, MeasureTheory.lintegral_iSup_directed, ProbabilityTheory.bayesRisk_of_subsingleton', Ideal.sup_primeHeight_eq_ringKrullDim, ENNReal.iSup_add_iSup_le, Pi.Colex.le_sSup_apply, LinearGrowth.le_linearGrowthSup_add', ENNReal.limsup_mul_le, Directed.measure_iUnion, Cardinal.mk_finsupp_lift_of_infinite, MeasureTheory.lowerCrossingTime_le, Submodule.spanRank_toENat_eq_iInf_encard, ExpGrowth.expGrowthSup_add, NatOrdinal.instIsOrderedMonoid, NatOrdinal.instIsOrderedRing, Nimber.mul_le_of_forall_ne, LinearGrowth.linearGrowthInf_bot, ENat.le_floor, NNReal.coe_iSup, MeasureTheory.Measure.mkMetric_apply, ENNReal.mul_iInf, ENNReal.limsup_const_mul, MeasureTheory.Measure.iInf_IicSnd_gt, Order.height_pos, Subfield.cardinalMk_closure_le_max, MeasureTheory.biSup_measure_Iic, Cardinal.mul_eq_left_iff, ENNReal.toNNReal_sSup, Order.le_krullDim_iff, MeasureTheory.lintegral_iInf_ae, ENNReal.inv_sSup, MeasureTheory.lintegral_iInf', Cardinal.lift_iInf, ENNReal.limsup_liminf_le_liminf_limsup, Ordinal.toNimber_min, MeasureTheory.Measure.hausdorffMeasure_le_liminf_sum, Nimber.add_le_of_forall_ne, spectrum.spectralRadius_le_liminf_pow_nnnorm_pow_one_div, LinearGrowth.linearGrowthSup_bot, ContinuousLinearMap.sSup_sphere_eq_nnnorm, MeasureTheory.SimpleFunc.iSup_coe_eapprox, ENNReal.liminf_toReal_eq, MeasurableSet.measure_eq_iSup_isCompact_of_ne_top, Algebraic.cardinalMk_le_max, lowerSemicontinuousOn_iSup, lowerSemicontinuous_iff_le_liminf, LinearGrowth.linearGrowthInf_biInf, upperSemicontinuous_biInf, ENat.mul_iSup, ENat.sSup_eq_zero, IsClosed.sInf_mem, MeasureTheory.Measure.LebesgueDecomposition.iSup_succ_eq_sup, NNReal.iSup_eq_zero, Matroid.cRank_eq_iSup_cardinalMk_indep, Ordinal.card_sInf_range_compl_le_lift, Nimber.toOrdinal_max, MeasureTheory.Measure.LebesgueDecomposition.iSup_mem_measurableLE, Ordinal.iSup_le, Pi.Colex.sSup_apply, ENNReal.le_iInf₂_add_iInf₂, NNReal.natCast_iInf, Algebra.IsAlgebraic.cardinalMk_le_max, hasCardinalLT_subtype_max, ExpGrowth.expGrowthSup_def, MeasureTheory.limsup_lintegral_le, MeasureTheory.OuterMeasure.iInf_apply', Ordinal.iSup_succ, Ordinal.Principal.sSup, Set.Finset.coe_einfsep, NatOrdinal.add_le_iff, egauge_univ_pi, Eventually.le_linearGrowthInf, Order.coheight_le_krullDim, ProbabilityTheory.bayesRisk_eq_iInf_measure_of_subsingleton, Nimber.toOrdinal_min, Order.krullDim_nonpos_iff_forall_isMax, ENNReal.iInf_div, ExpGrowth.expGrowthSup_biSup, MeasureTheory.eLpNormEssSup_eq_iSup, MeasureTheory.exists_isSigmaFiniteSet_measure_ge, ENat.biSup_add', Finset.sup_univ_eq_ciSup, MeasureTheory.Measure.mkMetric_le_liminf_tsum, Nimber.pos_iff_ne_zero, NatOrdinal.succ_def, ENNReal.biSup_add, MeasureTheory.measure_sigmaFiniteSetGE_le, ENat.iInf_mul', ENNReal.sSup_div, ENNReal.essSup_add_le, Pi.Lex.sInf_apply, Ordinal.card_omega0_opow, Ordinal.iSup_pow, EMetric.diam_insert, Cardinal.lift_iSup, Order.coheight_eq_coe_add_one_iff, Ordinal.deriv_limit, Order.krullDim_nonpos_iff_forall_isMin, ENNReal.iInf_sum, Nimber.add_trichotomy, ENNReal.coe_sSup, liminf_eq_top, NNReal.image_coe_uIcc, Pi.Lex.sInf_apply_le, Dynamics.coverEntropyEntourage_union, Ordinal.card_sInf_range_compl_le, LinearGrowth.linearGrowthInf_top, MeasureTheory.Measure.LebesgueDecomposition.iSup_le_le, ENNReal.biSup_add_biSup_le, NatOrdinal.lt_add_iff, Ordinal.sSup_eq_bsup, EReal.limsup_neg, Nimber.add_def, Cardinal.lift_sSup, SimpleGraph.eccent_le_one_iff, PartENat.toWithTop_lt, MeasureTheory.upperCrossingTime_mono, FreeAlgebra.cardinalMk_le_max, ENat.floor_le_ceil, Cardinal.aleph_add_aleph, Ordinal.iSup_add_nat, ENNReal.essSup_const_mul, Cardinal.mk_biUnion_le_lift, ENNReal.iSup_coe_lt_top, Ordinal.toNatOrdinal_min, LinearGrowth.linearGrowthInf_const_mul_self, MeasureTheory.lintegral_le_iSup_mul, MonoidAlgebra.cardinalMk_lift_of_infinite, Cardinal.lift_iSup_le, LowerSemicontinuousWithinAt.le_liminf, Ordinal.iSup_sum, ENat.iInf_coe_lt_top, ProbabilityTheory.avgRisk_le_iSup_risk, ENat.ceil_lt_top, WType.cardinalMk_le_max_aleph0_of_finite', ENNReal.iSup_natCast, CategoryTheory.ObjectProperty.IsLocal.inf, Order.height_le_krullDim, ENat.iInf_mul, Topology.IsUpper.isTopologicalSpace_basis, ENNReal.ofReal_limsup, NatOrdinal.lt_one_iff_zero, NatOrdinal.instIsOrderedCancelAddMonoid, PseudoMetricSpace.dist_ofPreNNDist, Dynamics.coverEntropy_union, limsup_eq_bot, MvPowerSeries.le_order_subst, MeasureTheory.essSup_trim, ENat.add_iSup, ciInf_le', NNReal.iSup_mul_iSup_le, FirstOrder.Language.card_functions_sum_skolem₁_le, MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ, ENNReal.iInf_add_iInf, PSet.rank_insert, MeasureTheory.OuterMeasure.trim_eq_iInf', MeasureTheory.limsup_measure_closed_le_of_forall_tendsto_measure, SimpleGraph.chromaticNumber_eq_iInf, Ordinal.iSup_eq_lsub_or_succ_iSup_eq_lsub, ENat.mul_sSup, LinearGrowth.linearGrowthInf_zero, SimpleGraph.chromaticNumber_sum, upperSemicontinuousOn_iInf, MeasurableSet.measure_eq_iSup_isCompact, Ordinal.sup_succ_eq_lsub, LowerSemicontinuousAt.le_liminf, Ordinal.succ_log_def, Order.bot_lt_krullDim, Ordinal.sSup_ord, topologicalKrullDim_subspace_le, AddMonoidAlgebra.cardinalMk_of_infinite', Eventually.linearGrowthSup_le, MeasureTheory.upperCrossingTime_le_lowerCrossingTime, MeasureTheory.Content.outerMeasure_eq_iInf, Cardinal.mul_eq_max, NatOrdinal.toOrdinal_max, topologicalKrullDim_zero_of_discreteTopology, ENNReal.liminf_sub_const, Cardinal.mk_freeMonoid, Cardinal.mk_bounded_set_le, Ordinal.enumOrd_zero, ENNReal.smul_iSup, Order.krullDim_eq_iSup_coheight_of_nonempty, ENNReal.iSup_add, LinearGrowth.linearGrowthSup_le_of_eventually_le, ENat.coe_sSup, ENNReal.iInf_mul_of_ne, MeasureTheory.measure_biUnion_eq_iSup, MeasureTheory.limsup_measure_closed_le_iff_liminf_measure_open_ge, Ordinal.iSup_mul_nat, ENNReal.tsum_iSup_eq, ENat.sInf_eq_zero, MeasurableSpace.cardinal_generateMeasurableRec_le, Order.krullDim_eq_iSup_length, Ideal.sup_primeHeight_of_maximal_eq_ringKrullDim, Finset.Nonempty.ciSup_mem_image, Order.coheight_pos_of_lt_top, egauge_pi, WithTop.iInf_coe_eq_top, essSup_comp_quotientAddGroup_mk, ENNReal.add_iInf, Set.Nonempty.ciSup_lt_iff, Order.coheight_eq_coe_iff_maximal_le_coheight, ENat.ceil_le, csInf_le', NNReal.agm_eq_ciSup, SetTheory.PGame.grundyValue_eq_sInf_moveRight, ENNReal.toNNReal_iSup, Antitone.measure_iUnion, MeasureTheory.lintegral_liminf_le, SetTheory.PGame.grundyValue_le_of_forall_moveLeft, WType.cardinalMk_le_max_aleph0_of_finite, Order.height_pos_of_bot_lt, upperSemicontinuousWithinAt_iff_limsup_le, ENNReal.tsum_eq_iSup_nat', Metric.ediam_eq_sSup, NatOrdinal.not_lt_zero, ENNReal.inv_iSup, ENat.ceil_lt, MeasureTheory.OuterMeasure.iInf_apply, UpperSemicontinuousWithinAt.limsup_le, SimpleGraph.edist_pos_of_ne, Cardinal.mk_list_eq_max, EReal.min_neg_neg, upperSemicontinuousOn_iff_limsup_le, LinearGrowth.linearGrowthSup_biSup, Cardinal.mk_freeAddGroup, Ordinal.sup_le_lsub, ENNReal.liminf_mul_le, EReal.le_liminf_add, Ordinal.iSup_add_natCast, FormalMultilinearSeries.radius_inv_eq_limsup, ENat.ceil_pos, Set.einfsep_insert, ENat.add_sSup, ENNReal.tsum_eq_iSup_sum, Order.krullDim_nonpos_of_subsingleton, Ordinal.IsNormal.map_iSup, NNReal.sSup_of_not_bddAbove, Order.coheight_eq_iSup_head_eq, Ordinal.card_iSup_Iio_le_card_mul_iSup, ENat.ceil_add_le, Cardinal.mk_finsupp_nat, MeasureTheory.le_liminf_measure_open_of_forall_tendsto_measure, ENat.lt_ceil, LinearGrowth.linearGrowthSup_inv, LinearGrowth.linearGrowthInf_const, MeasureTheory.OuterMeasure.biInf_apply, ENNReal.essSup_piecewise, ENNReal.iInf_add, ProbabilityTheory.bayesRisk_const_of_nonempty, ENat.coe_iSup, ENat.smul_iSup, PartENat.toWithTop_le, IsInducing.topologicalKrullDim_le, ENNReal.mul_iInf', EReal.liminf_add_le, AddMonoidAlgebra.cardinalMk_lift_of_infinite, MeasureTheory.measure_iInter_eq_iInf_measure_iInter_le, LinearGrowth.linearGrowthSup_le_iff, Cardinal.mul_ciSup, NNReal.iSup_pow_of_ne_zero, Order.krullDim_eq_iSup_height_of_nonempty, ExpGrowth.le_expGrowthInf_comp, ENNReal.liminf_const_sub, EMetric.infEdist_biUnion, MeasureTheory.iSup_lintegral_measurable_le_eq_lintegral, ENNReal.iInf_add_iInf_of_monotone, Nimber.small_Ioo, Ordinal.sup_eq_bsup, LinearGrowth.linearGrowthSup_const_mul_self, Cardinal.ciSup_mul, MeasureTheory.FiniteMeasure.limsup_measure_closed_le_of_tendsto, Set.einfsep_insert_le, MeasureTheory.IsAddFundamentalDomain.essSup_measure_restrict, Ordinal.Principal.iSup, ENat.sub_iSup, ENat.iInf_eq_top_of_isEmpty, MeasureTheory.Lp.eLpNorm_lim_le_liminf_eLpNorm, Cardinal.ord_eq_Inf, ENNReal.biSup_add', ENat.floor_lt_ceil, MeasureTheory.ProbabilityMeasure.le_liminf_measure_open_of_tendsto, le_ciInf_iff', csSup_empty, ENat.iInf_coe_eq_top, WithTop.iSup_coe_lt_top, ENNReal.iSup_add_iSup, lowerSemicontinuous_iSup, IntermediateField.cardinalMk_adjoin_le, ENNReal.coe_essSup, ENat.floor_lt_top, Cardinal.mk_freeAddMonoid, ENNReal.iSup_sub, ENNReal.tsum_eq_iSup_sum', MvPowerSeries.le_weightedOrder_subst, Cardinal.mul_le_max_of_aleph0_le_left, Cardinal.powerlt_min, ENat.gc_ceil_toENNReal, Cardinal.power_nat_le_max, MeasureTheory.measure_sigmaFiniteSetWRT', ENat.exists_eq_iInf, IsCompact.measure_eq_iInf_isOpen, ENNReal.sub_eq_sInf, Ordinal.opow_limit, MeasureTheory.OuterMeasure.ofFunction_eq_iInf_mem, Algebra.IsAlgebraic.lift_cardinalMk_le_max, Cardinal.preAleph_limit, Order.height_eq_iSup_lt_height, Order.height_eq_coe_add_one_iff, dimH_eq_iInf, ENNReal.le_iInf_mul, IsClosedEmbedding.topologicalKrullDim_le, NormedSpace.equicontinuous_TFAE, Order.krullDim_enat, MonoidAlgebra.cardinalMk_of_infinite', Order.height_le_coe_iff, LinearGrowth.linearGrowthSup_sup, IsOpen.measure_eq_iSup_isCompact, PiLp.nnnorm_eq_ciSup, NatOrdinal.mul_le_iff, Cardinal.lift_iSup_le_sum, MeasureTheory.ae_bdd_liminf_atTop_rpow_of_eLpNorm_bdd, MeasureTheory.limsup_trim, Monotone.measure_iInter, Order.height_eq_coe_iff, MeasureTheory.Measure.inf_apply, ENNReal.essSup_liminf_le, SimpleGraph.edist_le_eccent, UpperSemicontinuousOn.limsup_le, SimpleGraph.edist_triangle, LinearGrowth.linearGrowthInf_inf, Relation.cutExpand_le_invImage_lex, HahnSeries.cardSupp_div_le, upperSemicontinuous_iInf, WithTop.iInf_coe_lt_top, ENat.coe_iInf, dimH_bUnion, EReal.coe_toENNReal_eq_max, Cardinal.sum_le_lift_mk_mul_iSup, ENNReal.inv_iInf, Ordinal.max_eq_zero, Ordinal.bsup_eq_sup, Order.length_le_height, ENat.mul_iInf_of_ne, EMetric.infEdist_iUnion, Cardinal.mk_freeRing, Multiset.iSup_mem_map_of_ne_zero, sInf_mem_closure, ENNReal.toNNReal_iInf, Order.height_le_height_apply_of_strictMono, Cardinal.add_eq_max', MeasureTheory.eLpNormEssSup_eq_essSup_enorm, MeasureTheory.inducedOuterMeasure_eq_iInf, MeasureTheory.lintegral_iInf, MeasureTheory.le_iInf_lintegral, Ordinal.card_opow_le_of_omega0_le_right, NNReal.le_iInf_mul_iInf, ENNReal.le_iInf_mul_iInf, LinearGrowth.linearGrowthSup_eventually_monotone, ciSup_or', trdeg_eq_iSup_cardinalMk_isTranscendenceBasis, CategoryTheory.ObjectProperty.instIsTriangulatedClosed₂MinOfIsClosedUnderIsomorphisms, NatOrdinal.instAddLeftStrictMono, Cardinal.mul_mk_eq_max, Order.krullDim_le_one_iff_of_boundedOrder, Ordinal.iSup_mul_natCast, LowerSemicontinuous.le_liminf, Ordinal.mem_closure_iff_iSup, Cardinal.sInf_eq_zero_iff, LinearGrowth.linearGrowthSup_const, NNReal.iSup_div, LSeries.abscissaOfAbsConv_binop_le, MeasureTheory.measure_iUnion_of_tendsto_zero, ENat.iSup_natCast, MeasureTheory.Lp.eLpNorm'_lim_eq_lintegral_liminf, Ordinal.iSup_lt_ord_lift, Cardinal.aleph_max, LSeries.abscissaOfAbsConv_add_le, Pi.Colex.sInf_apply_le, ciSup_of_empty, CategoryTheory.ObjectProperty.HasCardinalLT.iSup, ProbabilityTheory.bayesRisk_le_iInf, PiLp.edist_eq_iSup, OreLocalization.cardinalMk_le_max, Manifold.riemannianEDist_def, LinearGrowth.linearGrowthInf_natCast_nonneg, ENNReal.tsum_eq_limsup_sum_nat, ENat.floor_mono, MeasureTheory.Measure.LebesgueDecomposition.iSup_monotone', IsOpen.measure_eq_biSup_integral_continuous, ExpGrowth.expGrowthInf_inf, Ordinal.card_opow_omega0, Ordinal.cof_eq_sInf_lsub, Cardinal.add_eq_max, MeasureTheory.Measure.restrict_iUnion_apply_eq_iSup, UpperSemicontinuousAt.limsup_le, EReal.le_limsup_mul, ENNReal.toReal_limsup, EReal.limsup_mul_le, ENat.le_ceil, LinearGrowth.linearGrowthSup_comp_nonneg, spectrum.limsup_pow_nnnorm_pow_one_div_le_spectralRadius, Cardinal.preAleph_max, Order.krullDim_eq_iSup_coheight, ENNReal.add_iSup, OrderIso.map_ciSup', ENNReal.iSup_coe_eq_top, SimpleGraph.ediam_le_two_mul_radius, SimpleGraph.edist_eq_sInf, upperSemicontinuousAt_iff_limsup_le, ENNReal.ofNNReal_liminf, EReal.le_liminf_mul, ENat.iSup_mul, Ordinal.mem_closure_tfae, Cardinal.iSup_lt_ord_lift_of_isRegular, BoundedContinuousFunction.nndist_eq_iSup, SimpleGraph.eccent_le_ediam, le_analyticOrderAt_sub, le_analyticOrderAt_add, MeasureTheory.OuterMeasure.sSup_apply, NNReal.natCast_iSup, ExpGrowth.le_expGrowthInf_add, SimpleGraph.ediam_eq_iSup_iSup_edist, Cardinal.iSup_lt_lift_of_isRegular, Ordinal.iSup'_eq_bsup, LSeries.abscissaOfAbsConv_convolution_le, Polynomial.cardinalMk_eq_max, Nimber.not_bddAbove_compl_of_small, Order.krullDim_eq_iSup_height, ContinuousLinearMap.sSup_unit_ball_eq_nnnorm, UniformOnFun.edist_def', Monotone.linearGrowthInf_nonneg, MeasureTheory.tendsto_measure_sigmaFiniteSetGE, SimpleGraph.eccent_pos_iff, Topology.IsScott.instUnivSetOfIsUpper, IsAlgClosed.cardinal_le_max_transcendence_basis', MeasureTheory.Measure.hausdorffMeasure_le_liminf_tsum, hasCardinalLT_subtype_iSup, ENat.sSup_add, Cardinal.beth_limit, ExpGrowth.expGrowthSup_comp_le, Order.coheight_le_coheight_apply_of_strictMono, upperSemicontinuous_iff_limsup_le, ENNReal.biSup_add_biSup_le', Cardinal.mk_biUnion_le, MeasureTheory.lintegral_iSup_directed_of_measurable, MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part, Submodule.spanRank_toENat_eq_iInf_finset_card, isSaddlePointOn_iff', NatOrdinal.instAddLeftMono, Ordinal.iSup_sequence_lt_omega_one, SetTheory.PGame.birthday_def, SimpleGraph.edist_anti, ENNReal.mul_sSup, ENNReal.iSup_add_iSup_of_monotone, FirstOrder.Language.Term.card_sigma, ENNReal.mul_iInf_of_ne, MeasureTheory.Measure.LebesgueDecomposition.iSup_mem_measurableLE', ENNReal.ofReal_essSup, ENat.coe_sInf, Cardinal.mk_bounded_subset_le, ENNReal.sSup_mul, ENNReal.iSup_zero, PartENat.ofENat_lt, Order.one_le_krullDim_iff, Cardinal.sum_le_iSup_lift, Set.einfsep_eq_iInf, IsWellFounded.rank_eq, Ordinal.iSup_typein_limit, ENat.iSup_eq_zero, essSup_comp_quotientGroup_mk, EReal.liminf_neg, Nimber.not_lt_zero, Ordinal.iSup_lt_lift, LinearGrowth.linearGrowthInf_eventually_monotone, MeasureTheory.lintegral_eq_nnreal, UniformFun.edist_def, Cardinal.mk_iUnion_le_lift, Order.coheight_le_iff', Order.krullDim_le_of_strictMono, WithTop.isGLB_sInf, ENNReal.iInf_mul_iInf, MeasureTheory.SimpleFunc.lintegral_restrict_iUnion_of_directed, NNReal.le_mul_iInf, ENNReal.limsup_eq_zero_iff, MeasureTheory.IsFundamentalDomain.essSup_measure_restrict, Cardinal.add_mk_eq_max, ExpGrowth.expGrowthInf_iInf, EReal.liminf_mul_le, Ordinal.derivFamily_limit, BoundedContinuousFunction.nnnorm_eq_iSup_nnnorm, Nimber.small_Ioc, EReal.max_neg_neg, AddMonoidAlgebra.cardinalMk_lift_of_infinite', MeasureTheory.OuterMeasure.sInf_apply, MeasureTheory.iInf_le_lintegral, ZFSet.rank_range, iSup₂_iInf₂_le_iInf₂_iSup₂, ENNReal.add_biSup', lowerSemicontinuousWithinAt_iff_le_liminf, Cardinal.succ_def, Monotone.expGrowthInf_comp_le, PartENat.withTopEquiv_le, Ordinal.iSup_ord, ENNReal.hasSum, ZFSet.rank_insert, ENat.iSup_add, Ordinal.sup_eq_sup, PartENat.withTopEquiv_lt, StieltjesFunction.length_eq, ENNReal.add_sSup, Ordinal.le_iSup, MeasureTheory.measure_eq_iInf', le_csInf_iff'', ENNReal.iInf_mul', Nimber.small_Icc, MeasureTheory.lintegral_iInf_directed_of_measurable, Ordinal.iSup_eq_zero_iff, Cardinal.sInf_empty, Metric.infEDist_iUnion, Cardinal.mk_freeGroup, ENNReal.iSup_div, Set.Nonempty.ciSup_mem_image, LinearGrowth.linearGrowthInf_iInf, lowerSemicontinuousAt_iff_le_liminf, UniformOnFun.edist_def, MeasureTheory.Measure.iInf_rat_gt_prod_Iic, MeasureTheory.OuterMeasure.boundedBy_apply, Ordinal.card_opow_eq_of_omega0_le_right, ENNReal.limsup_add_of_right_tendsto_zero, ProbabilityTheory.iSup_bayesRisk_le_minimaxRisk, LinearGrowth.linearGrowthInf_monotone, ENat.ceil_mono, MeasurableSpace.cardinal_measurableSet_le, iSup_succ, Ordinal.sup_eq_bsup', Ordinal.succ_iSup_eq_lsub_iff, MeasurableSpace.cardinal_generateMeasurable_le, ENat.mul_iInf, SimpleGraph.diam_def, ENat.sSup_mul, Cardinal.sum_le_lift_mk_mul_iSup_lift, IsAlgClosed.cardinal_le_max_transcendence_basis, ENat.iSup_coe_eq_top, Nimber.small_Iic, ProbabilityTheory.bayesRisk_le_iInf', ENNReal.toReal_essSup, BoundedContinuousFunction.edist_eq_iSup, Ordinal.sup_eq_lsub, Nimber.one_le_iff_ne_zero, ExpGrowth.expGrowthInf_biInf, LinearGrowth.linearGrowthSup_iSup, LinearGrowth.linearGrowthInf_comp_nonneg, ENNReal.essSup_mul_le, Ordinal.lift_card_sInf_compl_le, SimpleGraph.edist_le_ediam, iSup_Iio_eq_self_iff_isSuccPrelimit, Ordinal.iSup_eq_lsub_iff_lt_iSup, ENNReal.toReal_sSup, ENNReal.sub_iSup, Order.coheight_le_coe_iff, ContinuousMap.nnnorm_eq_iSup_nnnorm, Antitone.measure_iInter, MeasureTheory.ae_bdd_liminf_atTop_of_eLpNorm_bdd, ENNReal.liminf_add_of_right_tendsto_zero, Topology.IsScott.scott_eq_upper_of_completeLinearOrder, MeasureTheory.Measure.InnerRegularWRT.measure_eq_iSup, Ordinal.IsNormal.map_iSup_of_bddAbove, Cardinal.preBeth_limit, ProbabilityTheory.bayesRisk_const_of_neZero, MeasureTheory.measure_eq_iInf, ENNReal.limsup_mul_le', MeasureTheory.le_iInf₂_lintegral, upperSemicontinuousOn_biInf, MeasureTheory.Measure.iSup_restrict_spanningSets, ContinuousMap.nndist_eq_iSup, NNReal.limsSup_of_not_isBounded, ENNReal.le_iInf_add_iInf, MeasureTheory.lintegral_iSup, Ordinal.lt_iSup_iff, Cardinal.aleph_mul_aleph, ENNReal.iInf_mul, Ordinal.IsNormal.map_sSup_of_bddAbove, ENNReal.toReal_iInf, Order.krullDim_le_one_iff_forall_isMax, NatOrdinal.zero_le, ENNReal.iSup_eq_zero, ENNReal.sInf_add, Ordinal.card_opow_le_of_omega0_le_left, Ordinal.bsup'_eq_iSup, ENat.floor_lt, Ordinal.IsNormal.apply_omega0, MeasurableSet.measure_eq_iSup_isClosed_of_ne_top, ENNReal.iInf_coe_lt_top, Order.krullDim_pos_iff_of_orderTop, Nimber.mul_def, MonotoneOn.exists_tendsto_deriv_liminf_lintegral_enorm_le, MeasureTheory.eLpNormEssSup_count, ProbabilityTheory.bayesRisk_const, SimpleGraph.radius_le_ediam, Ordinal.iSup_Iio_eq_bsup, WithTop.iSup_coe_eq_top, NNReal.coe_iInf, MeasureTheory.OuterMeasure.iUnion_of_tendsto_zero, FirstOrder.Language.Term.card_le, cauchy_davenport_minOrder_add, WithTop.isLUB_sSup, ENNReal.smul_sSup, SimpleGraph.chromaticNumber_eq_biInf, Ordinal.iSup_eq_of_range_eq, Ordinal.card_opow_eq_of_omega0_le_left, Order.coheight_coe_enat, ZFSet.rank_iUnion, ENNReal.sSup_add, Ordinal.IsNormal.map_sSup, LinearGrowth.le_linearGrowthInf_iff, Topology.IsScott.isOpen_iff_Iic_compl_or_univ, SimpleGraph.eccent_le_iff, MeasureTheory.upperCrossingTime_zero', MeasureTheory.iSup₂_lintegral_le, Order.krullDim_eq_iSup_height_add_coheight_of_nonempty, Ordinal.iSup_sequence_lt_omega1, ENNReal.tsum_eq_iSup_nat, AlgebraicGeometry.geometrically_inf, ENNReal.iInf₂_add, Ordinal.bsup_eq_iSup, upperSemicontinuousWithinAt_biInf, Order.krullDim_pos_iff_of_orderBot, SimpleGraph.ediam_def, Ordinal.lsub_le_succ_iSup, Cardinal.add_mk_eq_max', NNReal.limsup_of_not_isBoundedUnder, ENNReal.essSup_indicator_eq_essSup_restrict, Cardinal.ciSup_mul_ciSup, Nimber.small_Ico, Cardinal.lift_sInf, MeasureTheory.OuterMeasure.top_apply', MvPolynomial.cardinalMk_le_max_lift, ExpGrowth.expGrowthInf_def, LinearGrowth.linearGrowthInf_le_of_eventually_le, ENNReal.tsum_eq_liminf_sum_nat, LinearGrowth.linearGrowthInf_le_iff, NNReal.iSup_empty, EReal.le_limsup_add, IsUltrametricDist.nnnorm_tprod_le, MeasureTheory.Measure.hausdorffMeasure_apply, PartENat.withTopEquiv_symm_le, LinearGrowth.linearGrowthSup_comp_le, Monotone.linearGrowthInf_comp_le, max_aleph0_card_le_rank_fun_nat, FreeAlgebra.cardinalMk_le_max_lift, IntermediateField.lift_cardinalMk_adjoin_le, ProbabilityTheory.bayesRisk_discard, ENNReal.iSup_pow, Ideal.sup_height_eq_ringKrullDim, Monotone.linearGrowthInf_comp_mul, isSaddlePointOn_iff, Ordinal.mem_iff_iSup_of_isClosed, NNReal.mul_iSup, EMetric.hausdorffEdist_def, ENNReal.sub_iInf, ENNReal.essSup_restrict_eq_of_support_subset, MeasureTheory.lowerCrossingTime_mono, Cardinal.add_ciSup, Set.measure_eq_iInf_isOpen, LSeries.abscissaOfAbsConv_sub_le, MeasureTheory.Measure.mkMetric_le_liminf_sum, Ordinal.sInf_compl_lt_ord_succ, MeasureTheory.lintegral_enorm_le_liminf_of_tendsto, ENNReal.iSup_lt_eq_self, MeasureTheory.OuterMeasure.sInf_apply', Order.coheight_eq_coe_iff, Cardinal.powerlt_aleph0_le, BoundedContinuousFunction.nndist_eq, ENNReal.finsetSum_iSup_of_monotone, NatOrdinal.small_Ioc, IsCompact.measure_eq_biInf_integral_hasCompactSupport, MonoidAlgebra.cardinalMk_lift_of_infinite', NNReal.liminf_of_not_isCoboundedUnder, MeasureTheory.Lp.eLpNorm_exponent_top_lim_eq_essSup_liminf, MeasureTheory.SimpleFunc.iSup_eapprox_apply, SimpleGraph.radius_le_eccent, MeasureTheory.ae_eq_set_symmDiff, ENat.toENNReal_iSup, Ordinal.isClosed_iff_iSup, MeasureTheory.upperCrossingTime_le, Ordinal.bsup_eq_sup', ENat.lt_floor, NatOrdinal.small_Iio, Cardinal.add_eq_left_iff, csSup_empty, LinearMap.rank_comp_le, Cardinal.mk_finsupp_of_infinite, NNReal.sInf_empty, dimH_iUnion, NNReal.mul_iInf, NNReal.toReal_limsup, Monotone.measure_iUnion, LinearGrowth.linearGrowthInf_add_le, Order.height_le_iff, Cardinal.mul_eq_max_of_aleph0_le_left, NNReal.iInf_mul, Frequently.linearGrowthInf_le, Ordinal.toNimber_max, Cardinal.lift_iSup_le_lift_iSup', Ordinal.sup_typein_limit, Ordinal.omega_max, Order.KrullDimLE.krullDim_le, NatOrdinal.small_Ico, Polynomial.cardinalMk_le_max, List.iSup_mem_map_of_ne_nil, ENNReal.iSup_mul_le, Monotone.linearGrowthInf_comp, ENat.add_biSup, NNReal.iInf_empty, ciSup_false, ENNReal.liminf_add_of_left_tendsto_zero, MeasureTheory.OuterMeasure.sInfGen_def, Directed.measure_iInter, ENat.mul_iInf', Algebra.cardinalMk_adjoin_le, NatOrdinal.small_Ioo, Order.coheight_anti, Cardinal.sum_le_iSup, EMetric.diam_eq_sSup, NNReal.le_iInf_add_iInf, Order.height_mono, Ordinal.sup_eq_lsub_or_sup_succ_eq_lsub, Cardinal.iInf_eq_zero_iff, CategoryTheory.ObjectProperty.instIsTriangulatedMinOfIsClosedUnderIsomorphisms, Order.height_le_iff', Order.krullDim_nonneg, MeasureTheory.measure_iUnion_eq_iSup_accumulate, lowerSemicontinuousAt_iSup, Cardinal.lift_iSup_le_iff, Cardinal.sum_pow_eq_max_aleph0, Monotone.linearGrowthSup_nonneg, ENNReal.hasProd_iInf_prod, Pi.Colex.sInf_apply, Order.coheight_pos, ENat.iInf_eq_zero, Metric.ediam_iUnion_mem_option, Monotone.linearGrowthSup_comp_mul, Filter.cardinalInterFilter_inf, ZFSet.rank_pair, Monotone.le_linearGrowthSup_comp, Order.index_le_height, ExpGrowth.expGrowthSup_sup, Order.coheight_eq, Cardinal.aleph_limit, Order.krullDim_le_one_iff_forall_isMin, Topology.IsLower.isTopologicalSpace_basis, bsupr_limsup_dimH, ENNReal.limsup_const_sub, ENNReal.add_iInf₂, Cardinal.mul_eq_max', ENat.floor_le_floor, MeasureTheory.iSup_lintegral_le, Pi.Lex.le_sSup_apply, upperSemicontinuousAt_iInf, upperSemicontinuousWithinAt_iInf, MeasureTheory.OuterMeasure.biInf_apply', ENNReal.le_limsup_mul, SimpleGraph.edist_le, Ordinal.iSup_le_iff, Cardinal.iSup_lt_ord_of_isRegular, Ordinal.iSup_eq_lsub, csInf_le_csInf', Ordinal.max_zero_right, ENNReal.limsup_add_le, Order.rev_index_le_coheight, IsTranscendenceBasis.lift_cardinalMk_eq_max_lift, IsTranscendenceBasis.lift_rank_eq_max_lift, MvRatFunc.rank_eq_max_lift, MeasureTheory.lintegral_liminf_le', Monotone.linearGrowthSup_comp, MeasureTheory.OuterMeasure.iSup_apply, ENNReal.iSup_mul, cauchy_davenport_minOrder_mul, Ordinal.iSup_eq_lsub_iff, lowerSemicontinuousWithinAt_biSup, ENat.sSup_eq_zero', MvPolynomial.cardinalMk_le_max, Cardinal.ciSup_add_ciSup, SetTheory.PGame.grundyValue_le_of_forall_moveRight, ENNReal.limsup_sub_const, ENNReal.inv_liminf, NNReal.agm_eq_ciInf, Cardinal.mk_freeCommRing, ENNReal.finsetSum_iSup, Pi.Lex.sSup_apply, Ordinal.iSup_natCast, lowerSemicontinuousAt_biSup, ENNReal.iInf_gt_eq_self, Cardinal.mk_freeAbelianGroup, ENNReal.eventually_le_limsup, ENNReal.inv_limsup, Ordinal.iSup_iterate_eq_nfp, NNReal.toReal_liminf, NatOrdinal.small_Icc, Nimber.le_zero, Set.einfsep_iUnion_mem_option, Set.chainHeight_eq_iSup, ENNReal.toNNReal_sInf, HahnSeries.cardSupp_hsum_powers_le, Nimber.small_Iio, Cardinal.add_le_max, MeasureTheory.OuterMeasure.trim_eq_iInf, ContinuousMap.enorm_eq_iSup_enorm, ENNReal.inv_sInf, ENNReal.coe_iInf, Ordinal.apply_omega0_of_isNormal, Order.height_eq_iSup_last_eq, ENat.toENNReal_iInf, BoundedContinuousFunction.enorm_eq_iSup_enorm, ExpGrowth.expGrowthSup_iSup, Order.height_enat, Ordinal.card_opow_le, ENNReal.limsup_toReal_eq, dimH_def, NNReal.le_iInf_mul, Cardinal.mk_finsupp_lift_of_infinite', Order.length_le_height_last, Order.coheight_eq_iSup_gt_coheight, LinearGrowth.linearGrowthSup_monotone, NNReal.iSup_mul, LinearGrowth.linearGrowthInf_le_linearGrowthSup_of_frequently_le, LinearGrowth.le_linearGrowthInf_comp, ENNReal.tprod_eq_iInf_prod, Ordinal.lsub_le_sup_succ, CategoryTheory.IsCardinalFiltered.exists_cardinal_directed.Diagram.iSup_P, MeasureTheory.AECover.iSup_lintegral_eq_of_countably_generated, MeasureTheory.OuterMeasure.ofFunction_apply, Cardinal.mul_le_max, ProbabilityTheory.bayesRisk_const', NNReal.iSup_pow, Ordinal.succ_iSup_le_lsub_iff, AddOreLocalization.cardinalMk_le_max, HahnSeries.cardSupp_inv_le, LinearGrowth.le_linearGrowthSup_iff, Order.length_le_coheight_head, Nimber.lt_one_iff_zero, NatOrdinal.small_Iic, MeasureTheory.setLIntegral_iUnion_of_directed, ENat.ceil_le_floor_add_one, MeasureTheory.lintegral_eq_iSup_eapprox_lintegral, ENNReal.coe_iSup, ENNReal.image_coe_uIcc, AddMonoidAlgebra.cardinalMk_of_infinite, FormalMultilinearSeries.radius_eq_liminf, PartENat.withTopEquiv_symm_lt, Ordinal.IsNormal.apply_of_isSuccLimit, ENNReal.iInf_div', ENNReal.limsup_add_of_left_tendsto_zero, Ordinal.lsub_sum, Ordinal.iSup_lt_ord, ENat.add_biSup', ENNReal.essSup_eq_zero_iff, lowerSemicontinuousOn_iff_le_liminf, MeasureTheory.ProbabilityMeasure.limsup_measure_closed_le_of_tendsto, FreeAlgebra.cardinalMk_eq_max, ENNReal.ae_le_essSup, ZFSet.rank_union, Mathlib.Meta.Positivity.natCeil_pos, ENat.biSup_add, Order.bot_lt_krullDim_iff, ENNReal.iInf_coe_eq_top, ENNReal.iSup₂_pow_of_ne_zero, Cardinal.mk_iUnion_le, Order.height_eq_coe_iff_minimal_le_height, Ordinal.preOmega_max, Cardinal.iSup_le_sum, Cardinal.add_eq_right_iff, SimpleGraph.eccent_def, Cardinal.iSup_mk_le_mk_iUnion, iSup_limsup_dimH, LinearGrowth.le_linearGrowthInf_add, ZFSet.iSup_card_le_card_iUnion, Ordinal.card_iSup_le_sum_card, Cardinal.mk_finsupp_of_infinite', MvPolynomial.cardinalMk_eq_max_lift, UpperSemicontinuous.limsup_le, Cardinal.ciSup_add, Ordinal.iSup_lt, ENNReal.coe_sInf, Ordinal.toNatOrdinal_max, Order.one_lt_height_iff, MeasureTheory.iInf_mul_le_setLIntegral, MeasureTheory.measure_sigmaFiniteSetGE_ge, ContinuousMap.edist_eq_iSup, Order.coheight_le_iff, FirstOrder.Language.Substructure.lift_card_closure_le, ENat.iSup_coe_lt_top, Metric.infEDist_biUnion, PSet.rank_pair, EReal.limsup_add_le, analyticOrderAt_add_of_ne, NNReal.iSup_mul_le, egauge_pi', MeasureTheory.OuterMeasure.iSup_sInfGen_nonempty, FreeAlgebra.cardinalMk_eq_max_lift, ENNReal.toReal_iSup, NNReal.iInf_const_zero, lowerSemicontinuous_biSup, ENat.floor_le, sSup_mem_closure, CategoryTheory.IsCardinalFiltered.exists_cardinal_directed.Diagram.sup_P, IsUltrametricDist.nnnorm_tsum_le, Order.krullDimLE_iff, ENNReal.add_biSup, Metric.ediam_insert, MeasureTheory.Lp.eLpNorm'_lim_le_liminf_eLpNorm', LinearMap.lift_rank_comp_le, PiLp.nndist_eq_iSup, MvPolynomial.cardinalMk_eq_max, Ordinal.sInf_empty, Frequently.le_linearGrowthSup, Ordinal.sup_typein_succ, MeasureTheory.Measure.iSup_restrict_spanningSets_of_measurableSet, SimpleGraph.edist_le_one_iff_adj_or_eq, MeasureTheory.lintegral_le_iSup, Algebraic.cardinalMk_lift_le_max, FirstOrder.Language.BoundedFormula.card_le, SimpleGraph.radius_eq_iInf_iSup_edist, ENNReal.mul_iSup, Algebra.rank_adjoin_le, EMetric.diam_iUnion_mem_option, LinearGrowth.linearGrowthInf_neg, NNReal.coe_sInf, Monotone.le_expGrowthSup_comp, ENNReal.toReal_sInf
toOrderBot 📖CompOp
16 mathmath: MeasureTheory.hittingAfter_bot_le_iff, MeasureTheory.lowerCrossingTime_le, Finset.sup_univ_eq_ciSup, MeasureTheory.upperCrossingTime_mono, MeasureTheory.hittingBtwn_bot_le_iff, MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ, MeasureTheory.upperCrossingTime_le_lowerCrossingTime, csSup_empty, ciSup_of_empty, MeasureTheory.hitting_bot_le_iff, MeasureTheory.upperCrossingTime_zero', MeasureTheory.lowerCrossingTime_mono, MeasureTheory.upperCrossingTime_le, csSup_empty, MeasureTheory.StronglyAdapted.isStoppingTime_leastGE, ciSup_false

Theorems

NameKindAssumesProvesValidatesDepends On
csSup_empty 📖mathematical—SupSet.sSup
ConditionallyCompleteLattice.toSupSet
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
toConditionallyCompleteLinearOrder
Set
Set.instEmptyCollection
Bot.bot
OrderBot.toBot
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Lattice.toSemilatticeSup
ConditionallyCompleteLattice.toLattice
toOrderBot
——

WellFoundedLT

Definitions

NameCategoryTheorems
conditionallyCompleteLinearOrderBot 📖CompOp—

(root)

Definitions

NameCategoryTheorems
ConditionallyCompleteLattice 📖CompData—
ConditionallyCompleteLinearOrder 📖CompData—
ConditionallyCompleteLinearOrderBot 📖CompData—
conditionallyCompleteLatticeOfLatticeOfsInf 📖CompOp—
conditionallyCompleteLatticeOfLatticeOfsSup 📖CompOp—
conditionallyCompleteLatticeOfsInf 📖CompOp—
conditionallyCompleteLatticeOfsSup 📖CompOp—

---

← Back to Index