Documentation Verification Report

OmegaCompletePartialOrder

📁 Source: Mathlib/Order/OmegaCompletePartialOrder.lean

Statistics

MetricCount
DefinitionsinstOmegaCompletePartialOrder, instFunLikeNat, instInhabited, instLE, instMembership, map, pair, zip, ContinuousHom, apply, apply, bind, comp, const, copy, flip, id, inst, instInhabited, map, ofFun, seq, toMono, toOrderHom, ωSup, omegaCompletePartialOrder, ωSup, iterateChain, instFunLikeContinuousHom, instPartialOrderContinuousHom, subtype, toPartialOrder, «term_→𝒄_», ωScottContinuous, ωSup, omegaCompletePartialOrder, ωSup, instOmegaCompletePartialOrder, ωSupImpl, instOmegaCompletePartialOrderForall
40
Theoremsbot, iSup, inf, sSup, sup, top, directed, exists_of_mem_map, ext, ext_iff, instOrderHomClassNat, isChain_range, map_coe, map_comp, map_id, map_le_map, mem_map, mem_map_iff, pair_succ, pair_zero, pair_zip_pair, range_pair, zip_coe, apply_apply, apply_mono, bind_apply, coe_apply, coe_inj, coe_mk, coe_toOrderHom, comp_apply, comp_assoc, comp_id, congr_arg, congr_fun, const_apply, continuous, copy_apply, ext, ext_iff, flip_apply, forall_forall_merge, forall_forall_merge', id_apply, id_comp, map_apply, map_ωSup', monotone, ofFun_apply, seq_apply, toMono_coe, toOrderHom_eq_coe, ωScottContinuous, bind, map, seq, ωScottContinuous_apply, ωSup_apply, ωSup_apply_ωSup, ωSup_bind, ωSup_def, omegaCompletePartialOrder_ωSup_coe, ωSup_coe, ωSup_iterate_le_fixedPoint, ωSup_iterate_le_prefixedPoint, ωSup_iterate_mem_fixedPoint, instOrderHomClassContinuousHom, isLUB_range_ωSup, le_ωSup, le_ωSup_of_le, apply, apply₂, comp, const, fun_comp, fun_const, fun_id, id, isLUB, map_ωSup, map_ωSup_of_orderHom, monotone, monotone_map_ωSup, of_apply₂, of_map_ωSup_of_orderHom, of_monotone_map_ωSup, ωScottContinuous_iff_apply₂, ωScottContinuous_iff_map_ωSup_of_orderHom, ωScottContinuous_iff_monotone_map_ωSup, ωSup_eq_of_isLUB, ωSup_le, ωSup_le_iff, ωSup_le_ωSup_of_le, ωSup_total, eq_of_chain, mem_chain_of_mem_ωSup, mem_ωSup, ωSup_eq_none, ωSup_eq_some, prodMk, ωScottContinuous_fst, ωScottContinuous_snd, ωSupImpl_fst, ωSupImpl_snd, ωSup_fst, ωSup_snd, ωSup_zip, ωScottContinuous
108
Total148

CompleteLattice

Definitions

NameCategoryTheorems
instOmegaCompletePartialOrder 📖CompOp
721 mathmath: IsLocalization.AtPrime.coe_primeSpectrumOrderIso_symm_apply_asIdeal, Dynamics.netEntropyEntourage_monotone, Matroid.eRk_lt_top_iff, Dynamics.netMaxcard_monotone_subset, Set.ncard_mono, Matroid.exists_isBasis_disjoint_isBasis_of_subset, NonUnitalAlgebra.gc, Metric.disjoint_ball_infDist, convexIndependent_set_iff_inter_convexHull_subset, SimpleGraph.vertexCoverNum_le_card_sub_one, Dynamics.coverEntropyInfEntourage_antitone, Ideal.height_le_spanFinrank, OrdinalApprox.lfpApprox_mono_mid, IsLinearMap.image_convexHull, LocalizedModule.subsingleton_iff_disjoint, ringKrullDim_succ_le_ringKrullDim_polynomial, FirstOrder.Language.Substructure.closure_eq_of_isRelational, AlgebraicGeometry.Scheme.isEmpty_pullback_iff, convexHull_insert, subset_closedConvexHull, Geometry.SimplicialComplex.vertex_mem_convexHull_iff, Matroid.eRk_lt_encard_iff_dep, SimpleGraph.isBipartiteWith_neighborSet_disjoint, closedConvexHull_min, PowerSeries.le_order_subst_left', PrimeSpectrum.localization_specComap_range, convexHull_multiset_sum, Geometry.SimplicialComplex.convexHull_subset_space, CategoryTheory.Retract.injectiveDimension_le, Ideal.height_le_iff_exists_minimalPrimes, BoxIntegral.Box.disjoint_splitCenterBox, SimpleGraph.le_chromaticNumber_iff_colorable, Matroid.isBase_compl_iff_maximal_disjoint_isBase, AlgebraicGeometry.Scheme.mem_grothendieckTopology_iff, SimpleGraph.le_chromaticNumber_iff_coloring, Geometry.SimplicialComplex.mem_space_iff, AlgebraicGeometry.HasAffineProperty.affineAnd_le_affineAnd, MeasureTheory.exists_decomposition_of_monotoneOn_hasDerivWithinAt, CategoryTheory.injectiveDimension_le_iff, OrderHom.le_map_sup_fixedPoints, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.app_invApp, OrdinalApprox.gfp_mem_range_gfpApprox, Convex.convex_remove_iff_notMem_convexHull_remove, ProjectiveSpectrum.gc_homogeneousIdeal, SimpleGraph.chromaticNumber_pos, Matroid.eRk_singleton_le, Matroid.eRk_mono, convexHull_eq_iInter, Ring.krullDimLE_iff, FirstOrder.Language.Substructure.fg_closure_singleton, MeasureTheory.pairwise_disjoint_fundamentalInterior, Ideal.height_lt_top, MeasureTheory.monotone_spanningSets, LinearMap.polar_antitone, Dynamics.netMaxcard_le_coverMincard, HasCardinalLT.Set.functor_obj, PrimeSpectrum.gc, Complex.convexHull_reProdIm, Dynamics.coverMincard_finite_of_isCompact_invariant, MeasureTheory.SeparableSpace.exists_measurable_partition_diam_le, Matroid.eRank_le_encard_add_eRk_compl, Matroid.IsBasis.contract_dep_iff, OrderHom.lfp_le_gfp, AddAction.orbit.pairwiseDisjoint, SimpleGraph.vertexCoverNum_le_iff, Topology.RelCWComplex.pairwiseDisjoint', convexJoin_segments, Matroid.contract_isCocircuit_iff, Dynamics.coverMincard_image_le, FirstOrder.Language.Substructure.small_closure, Set.chainHeight_le_encard, Topology.RelCWComplex.disjoint_openCell_of_ne, FirstOrder.Language.Substructure.cg_def, Cardinal.mk_monotone, AddAction.IsBlock.disjoint_vadd_left, Matroid.le_eRk_iff, OrderIso.essSup_apply, OrderedFinpartition.disjoint, OrderHom.map_lfp, exists_partition_approximatesLinearOn_of_hasFDerivWithinAt, exists_dist_slope_lt_pairwiseDisjoint_hasSum, Ideal.primeHeight_mono, LinearEquiv.dilatransvections_pow_mono, FirstOrder.Language.Substructure.mem_closure, Besicovitch.exist_disjoint_covering_families, IsLocalization.coe_primeSpectrumOrderIso_symm_apply_asIdeal, MeasureTheory.exists_null_pairwise_disjoint_diff, convexHull_union_neg_eq_absConvexHull, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.app_inv_app', AddMonoid.minOrder_le_addOrderOf, Localization.localRingHom_injective_of_primesOver_eq_singleton, SimpleGraph.two_le_chromaticNumber_of_adj, interior_convexHull_nonempty_iff_affineSpan_eq_top, OrderHom.gfp_const_inf_le, SimpleGraph.card_le_chromaticNumber_iff_forall_surjective, Matroid.IsBasis.contract_indep_iff, mem_absConvexHull_iff, SimpleGraph.three_le_egirth, Dynamics.IsDynNetIn.card_le_netMaxcard, Perfect.small_diam_splitting, Ideal.primeHeight_lt_top, SimpleGraph.chromaticNumber_le_sum_right, Geometry.SimplicialComplex.convexHull_inter_convexHull, HasCardinalLT.Set.cocone_pt, IsLocalization.AtPrime.coe_orderIsoOfPrime_symm_apply_coe, AffineBasis.convexHull_eq_nonneg_coord, FirstOrder.Language.Substructure.fg_closure, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.app_inv_app', segment_subset_convexHull, Metric.coveringNumber_two_mul_le_externalCoveringNumber, convexHull_toCone_eq_sInf, lowerSemicontinuous_iff_le_liminf, MulAction.IsBlock.disjoint_smul_of_ne, Balanced.convexHull, subset_absConvexHull, Ideal.height_strict_mono_of_is_prime, Matroid.IsMinor.exists_eq_contract_delete_disjoint, balancedHull_subset_convexHull_union_neg, SimpleGraph.chromaticNumber_mono_of_embedding, le_minSmoothness, ringKrullDim_succ_le_ringKrullDim_powerseries, Matroid.delete_eq_self_iff, CategoryTheory.IsGrothendieckAbelian.isomorphisms_le_pushouts_generatingMonomorphisms, Convex.convexHull_union, closedConvexHull_eq_closure_convexHull, Finset.mem_convexHull', subset_convexHull, Dynamics.one_le_coverMincard_iff, Dynamics.coverEntropy_monotone, MulAction.isBlock_iff_smul_eq_or_disjoint, Convex.convexHull_subset_iff, AffineBasis.interior_convexHull, Matroid.contract_eq_self_iff, Dynamics.coverMincard_monotone_time, AlgebraicIndependent.adjoin_iff_disjoint, OrdinalApprox.gfpApprox_le, Matroid.eRk_lt_encard_iff_dep_of_finite, IsContDiffImplicitAt.one_le, MeasureTheory.SignedMeasure.exists_isCompl_positive_negative, Geometry.SimplicialComplex.inter_subset_convexHull, convexJoin_segment_singleton, SimpleGraph.chromaticNumber_le_of_forall_imp, Matroid.isClosed_iff_isFlat, PowerSeries.le_order_subst_right', convexJoin_singleton_segment, convexHull_diam, Matroid.Coindep.delete_spanning_iff, ringKrullDim_nonneg_of_nontrivial, subset_closedAbsConvexHull, Dynamics.netEntropyEntourage_antitone, Fin.Embedding.exists_embedding_disjoint_range_of_add_le_Nat_card, Finset.centerMass_id_mem_convexHull, Submodule.length_lt, Set.Finite.isCompact_convexHull, Finset.centerMass_id_mem_convexHull_of_nonpos, convexHull_add, Module.length_pos_iff, SimpleGraph.vertexCoverNum_le_vertexCoverNum_of_injective, Monoid.minOrder_le_natCard, AddMonoid.le_minOrder, Module.supportDim_le_of_injective, Metric.externalCoveringNumber_le_one_of_ediam_le, affineSpan_convexHull, AlgebraicGeometry.Scheme.Hom.app_appIso_inv_assoc, Cardinal.mk_strictMonoOn, MeasureTheory.disjoint_addFundamentalInterior_addFundamentalFrontier, Besicovitch.TauPackage.monotone_iUnionUpTo, Matroid.eRk_le_eRk_add_eRk_diff, Dynamics.coverMincard_closure_le, MeasureTheory.hittingBtwn_apply_anti, CategoryTheory.MorphismProperty.le_ind, Set.encard_strictMono, Submodule.height_strictMono, Subgroup.leftCoset_cover_filter_FiniteIndex_aux, Matroid.eRk_compl_insert_union_add_eRk_compl_insert_inter_le, convexHull_basis_eq_stdSimplex, Ideal.disjoint_powers_iff_notMem, CategoryTheory.IsGrothendieckAbelian.generatingMonomorphisms_le_monomorphisms, Matroid.Indep.contract_dep_iff, Convex.radon_partition, IsOpen.exists_iUnion_isClosed, Metric.externalCoveringNumber_anti, Matroid.cRk_mono, ConvexOn.bddAbove_convexHull, FirstOrder.Language.Substructure.fg_def, Matroid.eRk_insert_le_add_one, Set.chainHeight_mono, Ideal.height_le_height_add_of_liesOver, Subgroup.pairwiseDisjoint_leftCoset_cover_const_of_index_eq, MvPowerSeries.le_order_subst, CategoryTheory.injectiveDimension_ge_iff, AddAction.IsBlock.disjoint_vadd_vadd_set, Matroid.closure_eq_subtypeClosure, Ideal.height_le_height_add_one_of_mem, Matroid.spanning_iff_eRk_le', Metric.packingNumber_two_mul_le_externalCoveringNumber, Ideal.disjoint_map_primeCompl_iff_comap_le, AddAction.IsBlock.vadd_eq_vadd_or_disjoint, ConvexIndependent.mem_convexHull_iff, not_disjoint_segment_convexHull_triple, convexHull_vadd, Matroid.closure_mono, FirstOrder.Language.Substructure.lift_card_closure_le_card_term, Dynamics.coverMincard_finite_of_isCompact_uniformContinuous, Ideal.height_le_card_of_mem_minimalPrimes_span, SimpleGraph.chromaticNumber_le_card, AlgebraicGeometry.topologically_iso_le, ringKrullDim_quotient_le, Matroid.eRk_inter_add_eRk_union_le, exists_mem_interior_convexHull_affineBasis, Ideal.disjoint_nonZeroDivisors_of_mem_minimalPrimes, essInf_antitone_measure, FirstOrder.Language.Substructure.mem_closure_iff_exists_term, ringKrullDim_succ_le_of_surjective, Besicovitch.exist_finset_disjoint_balls_large_measure, SimpleGraph.vertexCoverNum_mono, Ideal.primeHeight_le_ringKrullDim, closedAbsConvexHull_min, Dynamics.IsDynCoverOf.coverMincard_le_card, Metric.externalCoveringNumber_mono_set, Ideal.height_le_spanRank_toENat_of_mem_minimal_primes, Metric.IsCover.externalCoveringNumber_le_encard, CategoryTheory.projectiveDimension_lt_iff, totallyBounded_absConvexHull, Matroid.dual_indep_iff_exists', ContDiffMapSupportedIn.iteratedFDerivWithOrderLM_apply, Besicovitch.exists_disjoint_closedBall_covering_ae_of_finiteMeasure_aux, SimpleGraph.chromaticNumber_le_iff_colorable, absConvexHull_nonempty, MeasureTheory.disjoint_fundamentalInterior_fundamentalFrontier, Ideal.primeHeight_strict_mono, absConvexHull_empty, AddAction.IsBlock.pairwiseDisjoint_range_vadd, upperSemicontinuousWithinAt_iff_limsup_le, absConvexHull_eq_self, OrderHom.isLeast_lfp, MulAction.orbit.eq_or_disjoint, Convex.mem_extremePoints_iff_mem_diff_convexHull_diff, CategoryTheory.SmallObject.succStruct_prop_le_propArrow, ringKrullDim_le_iff_height_le, IsLocalization.isPrime_iff_isPrime_disjoint, SubMulAction.disjoint_val_image, upperSemicontinuousOn_iff_limsup_le, natCast_le_analyticOrderAt, Dynamics.coverMincard_antitone, NonUnitalStarAlgebra.gc, Dynamics.netMaxcard_antitone, SimpleGraph.vertexCoverNum_le_encard_edgeSet, AddAction.IsBlock.disjoint_vadd_set_vadd, Matroid.IsRkFinite.eRk_lt_top, Matroid.Indep.encard_le_eRk_of_subset, AlgebraicGeometry.isCompl_range_inl_inr, Metric.disjoint_closedEBall_of_lt_infEDist, Matroid.eRank_le_encard_ground, SimpleGraph.vertexCoverNum_lt_card, OrdinalApprox.lfpApprox_ord_eq_lfp, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.app_inv_app'_assoc, ProjectiveSpectrum.gc_set, Matroid.eRank_lt_top_iff, convexHull_add_subset, ConvexCone.disjoint_hull_left_of_convex, Matroid.eRk_insert_inter_add_eRk_insert_union_le, Ideal.height_mono, EisensteinSeries.pairwise_disjoint_gammaSet, Submodule.spanFinrank_span_le_encard, ringKrullDim_le_ringKrullDim_quotient_add_card, closedConvexHull_closure_eq_closedConvexHull, ContDiffMapSupportedIn.structureMapLM_apply_withOrder, Matroid.Indep.encard_le_eRank, MulAction.IsBlock.disjoint_smul_right, Polynomial.rootSet_derivative_subset_convexHull_rootSet, BoxIntegral.unitPartition.disjoint, Metric.IsCover.coveringNumber_le_encard, AlgebraicGeometry.Scheme.mem_overGrothendieckTopology, Ideal.height_le_ringKrullDim_quotient_add_one, closure_convexHull_extremePoints, MulAction.IsBlock.disjoint_smul_smul_set, Dynamics.coverEntropyEntourage_monotone, Set.Finite.isClosed_convexHull, convexHull_empty, absConvexHull_add_subset, Ideal.height_le_iff, Finset.centerMass_mem_convexHull, SimpleGraph.chromaticNumber_le_two_iff_isBipartite, convexHull_pair, SimpleGraph.two_le_chromaticNumber_iff_ne_bot, preCantorSet_antitone, MeasureTheory.hittingBtwn_anti, MvPowerSeries.le_weightedOrder_subst, OrdinalApprox.lfpApprox_monotone, Dynamics.coverMincard_union_le, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.app_inv_app'_assoc, Matroid.contract_spanning_iff, MeasureTheory.stoppedProcess_eq, FirstOrder.Language.Substructure.mem_closed_of_isRelational, Topology.CWComplex.pairwiseDisjoint', Dynamics.coverEntropyInf_monotone, Module.length_le_of_surjective, OrdinalApprox.gfpApprox_antitone, FirstOrder.Language.Substructure.fg_iff_exists_fin_generating_family, MulAction.IsBlock.disjoint_smul_set_smul, convexJoin_subset_convexHull, Geometry.SimplicialComplex.disjoint_or_exists_inter_eq_convexHull, convexHull_eq_empty, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.app_invApp_assoc, ringKrullDim_le_ringKrullDim_quotient_add_spanFinrank, FirstOrder.Language.isExtensionPair_iff_exists_embedding_closure_singleton_sup, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.app_invApp, Matroid.eRk_le_iff, SimpleGraph.le_chromaticNumber_iff_forall_surjective, Vitali.exists_disjoint_subfamily_covering_enlargement_closedBall, MvPolynomial.supported_strictMono, AddAction.IsBlock.disjoint_vadd_right, convexHull_singleton, MeasureTheory.hittingAfter_apply_anti, OrderHom.lfp_lfp, SimpleGraph.Colorable.chromaticNumber_le, convex_convexHull, convexHull_sum, OrdinalApprox.gfpApprox_ord_eq_gfp, FirstOrder.Language.Substructure.closure_insert, isBounded_convexHull, SimpleGraph.IsVertexCover.vertexCoverNum_le, AddAction.isBlock_iff_vadd_eq_or_disjoint, Dynamics.netEntropyInfEntourage_monotone, Set.exists_union_disjoint_cardinal_eq_of_even, Finset.mem_convexHull, OrderHom.isFixedPt_lfp, essSup_mono_measure, convexHull_neg, convexIndependent_set_iff_notMem_convexHull_diff, OrderHom.gfp_gfp, LinearEquiv.transvections_pow_mono, convexHull_sphere_eq_closedBall, ENat.LEInfty.out, LinearMap.polar_gc, Matroid.eRk_union_le_eRk_add_eRk, Matroid.IsCircuit.disjoint_coloops, Matroid.eRk_le_eRk_inter_add_eRk_diff, Subgroup.IsComplement.pairwiseDisjoint_smul, Convex.exists_subset_interior_convexHull_finset_of_isCompact, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.app_invApp, convexHull_mono, VitaliFamily.FineSubfamilyOn.covering_disjoint_subtype, convex_absConvexHull, Cardinal.mk_strictMono, Metric.coveringNumber_pos_iff, convexHullAddMonoidHom_apply, absConvexHull_mono, Finset.convexHull_eq, Ideal.height_le_ringKrullDim_quotient_add_encard, MulAction.IsBlock.smul_eq_smul_or_disjoint, Ideal.primeHeight_add_one_le_of_lt, ConvexCone.disjoint_coe, totallyBounded_convexHull, Set.one_le_chainHeight_iff, Set.Nonempty.absConvexHull, tangentBundleCore.isContMDiff, SimpleGraph.cliqueNum_le_chromaticNumber, AddAction.IsBlock.disjoint_vadd_of_ne, TangentBundle.contMDiffVectorBundle, Topology.RelCWComplex.disjoint_interior_base_closedCell, absConvex_convexClosedHull, upperSemicontinuousAt_iff_limsup_le, convexHull_eq_singleton, convexHull_eq_zero, Set.encard_mono, OrdinalApprox.lfpApprox_mono_left, SimpleGraph.IsBipartiteWith.disjoint, FirstOrder.Language.Substructure.closure_empty, Dynamics.coverMincard_le_pow, Ideal.height_le_iff_covBy, ringKrullDim_quotient_succ_le_of_nonZeroDivisor, isCompact_closedAbsConvexHull_of_totallyBounded, Dynamics.coverMincard_mul_le_pow, le_analyticOrderAt_sub, le_analyticOrderAt_add, VitaliFamily.FineSubfamilyOn.exists_disjoint_covering_ae, absConvexHull_subset_closedAbsConvexHull, Matroid.setOf_indep_eq, Dynamics.netMaxcard_monotone_time, toWeakSpace_closedConvexHull_eq, convexHull_pi, FirstOrder.Language.monotone_distinctConstantsTheory, FirstOrder.Language.Substructure.iSup_eq_closure, AddMonoid.le_minOrder_iff_forall_addSubgroup, disjoint_ball_closedBall_iff, ringKrullDim_add_natCard_le_ringKrullDim_mvPolynomial, upperSemicontinuous_iff_limsup_le, SimpleGraph.isBipartiteWith_neighborSet_disjoint', doublyStochastic_eq_convexHull_permMatrix, Module.length_pos, absConvexHull_eq_iInter, isSaddlePointOn_iff', VitaliFamily.covering, exists_open_convex_of_notMem, Finset.centroid_mem_convexHull, AddAction.isBlock_iff_disjoint_vadd_of_ne, Vitali.exists_disjoint_subfamily_covering_enlargement, Matroid.coindep_contract_iff, AbsConvex.absConvexHull_subset_iff, AddSubgroup.pairwiseDisjoint_leftCoset_cover_const_of_index_eq, Metric.coveringNumber_subset_le, Topology.RelCWComplex.disjointBase, MeasureTheory.hittingAfter_eq_sInf, MeasureTheory.Egorov.notConvergentSeq_antitone, OrderHom.map_gfp, ringKrullDim_le_iff_isMaximal_height_le, Projectivization.Subspace.monotone_span, FirstOrder.Language.Substructure.subset_closure, AlgebraicGeometry.Scheme.Cover.toPresieveOver_le_arrows_iff, ringKrullDim_le_ringKrullDim_add_card, FirstOrder.Language.Substructure.closure_iUnion, Metric.disjoint_closedBall_of_lt_infDist, absConvex_absConvexHull, Metric.frontier_thickening_disjoint, iSup₂_iInf₂_le_iInf₂_iSup₂, CategoryTheory.projectiveDimension_le_iff, MulAction.isBlock_iff_smul_eq_smul_or_disjoint, SimpleGraph.chromaticNumber_le_one_of_subsingleton, AddAction.isBlock_iff_pairwiseDisjoint_range_vadd, ProjectiveSpectrum.gc_ideal, lowerSemicontinuousWithinAt_iff_le_liminf, Metric.coveringNumber_le_packingNumber, Vitali.exists_disjoint_covering_ae, ContDiffMapSupportedIn.structureMapCLM_apply_withOrder, Monoid.le_minOrder_iff_forall_subgroup, Module.supportDim_le_ringKrullDim, MulAction.IsBlock.disjoint_smul_left, Set.Infinite.exists_union_disjoint_cardinal_eq_of_infinite, LinearMap.image_convexHull, Module.length_le_of_injective, FirstOrder.Language.Substructure.closure_union, MulAction.isBlock_iff_disjoint_smul_of_ne, ContinuousMap.ideal_gc, Module.supportDim_le_supportDim_quotSMulTop_succ, CategoryTheory.projectiveDimension_ge_iff, Dynamics.coverEntropyEntourage_antitone, Besicovitch.exists_disjoint_closedBall_covering_ae, AddSubgroup.IsComplement.pairwiseDisjoint_vadd, Matroid.eRk_le_eRank, PrimeSpectrum.gc_set, IsLocalization.orderIsoOfPrime_symm_apply_coe, StarAlgebra.gc, Matroid.dual_indep_iff_exists, convexHull_univ, convexHull_ediam, disjoint_closedBall_closedBall_iff, SimpleGraph.le_chromaticNumber_of_pairwise_adj, FirstOrder.Language.Structure.cg_iff, absConvexHull_min, PowerSeries.le_order_subst, ringKrullDim_lt_top, CategoryTheory.injectiveDimension_lt_iff, lowerSemicontinuousAt_iff_le_liminf, AddAction.isBlock_iff_vadd_eq_vadd_or_disjoint, OrderIso.essInf_apply, OrdinalApprox.gfpApprox_mono_mid, SimpleGraph.chromaticNumber_mono, HasCardinalLT.Set.cocone_ι_app, IsLocalization.disjoint_comap_iff, AffineIndependent.convexHull_inter', Polynomial.ringKrullDim_le, AddAction.orbit.eq_or_disjoint, convexHull_zero, Matroid.eRk_lt_encard_of_dep_of_finite, convexHull_eq_union_convexHull_finite_subsets, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.app_invApp_assoc, Set.ncard_union_eq_iff, OrderHom.isGreatest_gfp_le, Topology.IsScott.ωScottContinuous_iff_continuous, Matroid.delete_dep_iff, MeasureTheory.exists_subordinate_pairwise_disjoint, zero_mem_absConvexHull, Ideal.map_height_le_one_of_mem_minimalPrimes, Matroid.contract_spanning_iff', Set.exists_union_disjoint_cardinal_eq_iff, HasCardinalLT.Set.functor_map_coe, affineCombination_mem_convexHull, Matroid.Indep.contract_indep_iff, mem_convexHull_iff_exists_fintype, Metric.frontier_cthickening_disjoint, Dynamics.netMaxcard_finite_iff, MulAction.orbit.pairwiseDisjoint, absConvexHull_univ, MeasureTheory.hittingBtwn_eq_sInf, MeasureTheory.stoppedProcess_eq', Topology.RelCWComplex.disjoint_skeleton_openCell, convexHull_toCone_isLeast, posTangentConeAt_mono, Matroid.eRk_union_le_encard_add_eRk, Matroid.eRk_union_le_eRk_add_encard, MeasureTheory.AEDisjoint.exists_disjoint_diff, cauchy_davenport_minOrder_add, MeasurableSpace.generateMeasurableRec_mono, disjoint_ball_ball_iff, Dynamics.coverMincard_le_netMaxcard, extremePoints_convexHull_subset, MvPolynomial.zeroLocus_vanishingIdeal_galoisConnection, Topology.RelCWComplex.disjointBase', Geometry.SimplicialComplex.face_subset_face_iff, OrdinalApprox.gfpApprox_mono_left, Convex.convexHull_eq, convexHull_convexHull_union_right, Subgroup.pairwiseDisjoint_leftCoset_cover_of_sum_inv_index_eq_one, SimpleGraph.egirth_le_length, PiNat.disjoint_cylinder_of_longestPrefix_lt, HasCardinalLT.Set.instIsCardinalFiltered, SimpleGraph.disjoint_image_val_universalVerts, ConvexCone.disjoint_hull_right_of_convex, MeasureTheory.Measure.exists_eq_disjoint_finiteSpanningSetsIn, EMetric.disjoint_closedBall_of_lt_infEdist, SimpleGraph.chromaticNumber_le_sum_left, Module.supportDim_le_supportDim_quotSMulTop_succ_of_mem_jacobson, FirstOrder.Language.Substructure.coe_closure_eq_range_term_realize, AlgebraicGeometry.targetAffineLocally_affineAnd_le, SimpleGraph.egirth_anti, Ideal.disjoint_primeCompl_of_liesOver, FirstOrder.Language.Substructure.closed, SimpleGraph.IsClique.card_le_chromaticNumber, essSup_comp_le_essSup_map_measure, MulAction.IsBlock.pairwiseDisjoint_range_smul, MulAction.isBlock_iff_pairwiseDisjoint_range_smul, isSaddlePointOn_iff, absConvexHull_isClosed, Set.encard_le_chainHeight_of_isChain, Set.exists_union_disjoint_ncard_eq_of_even, convexHull_convexHull_union_left, Ideal.finiteHeight_iff_lt, convexHull_min, Topology.RelCWComplex.Subcomplex.disjoint_openCell_subcomplex_of_not_mem, height_le_ringKrullDim_quotient_add_spanFinrank, convexHull_isClosed, Matroid.delete_isBasis_iff, AlgebraicGeometry.Scheme.exists_cover_of_mem_grothendieckTopology, Ideal.height_le_height_add_encard_of_subset, closedAbsConvexHull_eq_closure_absConvexHull, Matroid.Dep.eRk_lt_encard, Metric.externalCoveringNumber_le_encard_self, Matroid.isFlat_iff_isClosed, Matroid.IsCircuit.isCocircuit_disjoint_or_nontrivial_inter, PMF.toOuterMeasure_apply_eq_zero_iff, ConvexCone.gc_hull_coe, convexHull_prod, ωScottContinuous.top, ringKrullDim_add_enatCard_le_ringKrullDim_mvPolynomial, Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes_of_isLocalRing, FirstOrder.Language.Substructure.closure_withConstants_eq, PowerSeries.le_order_subst_right, Metric.IsSeparated.encard_le_packingNumber, Dynamics.coverMincard_finite_iff, AddSubgroup.leftCoset_cover_filter_FiniteIndex_aux, AlgebraicGeometry.Scheme.mem_smallGrothendieckTopology, convexHull_eq_union, Topology.RelCWComplex.disjoint_skeletonLT_openCell, Matroid.eRk_compl_union_add_eRk_compl_inter_le, convexHull_eq, OrderHom.isGreatest_gfp, convexHull_smul, ModelWithCorners.disjoint_interior_boundary, ringKrullDim_le_ringKrullDim_add_spanFinrank, Set.Nonempty.convexHull, absConvexHull_eq_convexHull_balancedHull, FirstOrder.Language.Substructure.map_closure, Set.Finite.convexHull_eq_image, Metric.externalCoveringNumber_pos_iff, OrdinalApprox.lfp_mem_range_lfpApprox, AlgebraicGeometry.HasAffineProperty.affineAnd_le_isAffineHom, mem_convexHull_of_exists_fintype, convexHull_rangle_single_eq_stdSimplex, FirstOrder.Language.Substructure.closure_le, Matroid.eRk_le_one_iff, AffineBasis.centroid_mem_interior_convexHull, AffineMap.image_convexHull, convexHull_union, convexHull_sub, FirstOrder.Language.Substructure.mem_closed_iff, SubAddAction.disjoint_val_image, closure_subset_closedConvexHull, MeasureTheory.SimpleFunc.iSup_approx_apply, ConcaveOn.bddBelow_convexHull, Set.Countable.substructure_closure, MeasureTheory.exists_pair_mem_lattice_not_disjoint_vadd, FirstOrder.Language.Substructure.subset_closure_withConstants, convexHull_list_sum, Matroid.Indep.contract_isBase_iff, Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes, Set.Finite.ncard_strictMonoOn, AlgebraicGeometry.IsOpenImmersion.le_monomorphisms, IntermediateField.gc, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.app_inv_app', Ideal.height_le_height_add_spanFinrank_of_le, Metric.packingNumber_pos_iff, AlgebraicGeometry.affineLocally_le, MulAction.IsBlock.smul_eq_or_disjoint, Metric.externalCoveringNumber_le_coveringNumber, convexHull_nonempty_iff, SimpleGraph.le_egirth, RealRMK.range_cut_partition, Besicovitch.exists_disjoint_closedBall_covering_ae_aux, Metric.encard_le_of_isSeparated, Ideal.exists_disjoint_powers_of_span_eq_top, Matroid.eRk_le_encard, Dynamics.netEntropyInfEntourage_antitone, MeasureTheory.pairwise_disjoint_addFundamentalInterior, Topology.RelCWComplex.pairwiseDisjoint, Matroid.Coindep.delete_isBase_iff, closedAbsConvexHull_isClosed, HasCardinalLT.Set.instIsFilteredOfFactIsRegular, Monoid.le_minOrder, mem_convexHull_iff, Ideal.height_le_spanRank_toENat, Matroid.dualIndepMatroid_Indep, parallelepiped_eq_convexHull, cauchy_davenport_minOrder_mul, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.app_invApp_assoc, OrdinalApprox.le_lfpApprox, PrimeSpectrum.localization_comap_range, CategoryTheory.Retract.projectiveDimension_le, IsLocalization.orderIsoOfPrime_apply_coe, VitaliFamily.FineSubfamilyOn.covering_disjoint, Matroid.one_le_eRank, Ideal.height_le_card_of_mem_minimalPrimes_span_finset, absConvexHull_eq_empty, Matroid.IsBasis'.contract_dep_iff, MvPowerSeries.le_weightedOrder_subst_of_forall_ne_zero, Metric.coveringNumber_le_encard_self, Matroid.delete_indep_iff, FirstOrder.Language.Substructure.cg_iff_empty_or_exists_nat_generating_family, Ideal.height_le_ringKrullDim_of_ne_top, Matroid.Indep.disjoint_loops, SimpleGraph.IsContained.vertexCoverNum_le_vertexCoverNum, MeasureTheory.hitting_eq_sInf, AlgebraicGeometry.Scheme.Hom.app_appIso_inv, OrderHom.map_inf_fixedPoints_le, isClosed_closedAbsConvexHull, essSup_mono_measure', AffineIndependent.convexHull_inter, Urysohns.CU.disjoint_C_support_lim, ringKrullDim_le_of_surjective, convex_closedConvexHull, SimpleGraph.Walk.three_le_chromaticNumber_of_odd_loop, closure_subset_closedAbsConvexHull, Metric.packingNumber_le_encard_self, IsLocalization.map_algebraMap_ne_top_iff_disjoint, AlgebraicGeometry.Scheme.codisjoint_zeroLocus, minSmoothness_monotone, Submodule.height_lt_top, FirstOrder.Language.Hom.eqOn_closure, ringKrullDim_le_ringKrullDim_quotSMulTop_succ, Module.supportDim_quotSMulTop_succ_le_of_notMem_minimalPrimes, HasCardinalLT.Set.isFiltered_of_aleph0_le, Submodule.coe_scott_continuous, OrderHom.isFixedPt_gfp, Matroid.delete_isCircuit_iff, Set.Finite.convexHull_eq, OrderHom.isLeast_lfp_le, Module.supportDim_le_of_surjective, MeasureTheory.hittingAfter_anti, SimpleGraph.ConnectedComponent.Represents.disjoint_supp_of_notMem, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.app_inv_app'_assoc, Set.ncard_strictMono, SimpleGraph.chromaticNumber_mono_of_hom, isClosed_closedConvexHull, Matroid.eRk_submod, OrderHom.map_lfp_comp, Monoid.minOrder_le_orderOf, Dynamics.one_le_netMaxcard_iff, NumberField.InfinitePlace.disjoint_isReal_isComplex, Metric.coveringNumber_anti, Vitali.exists_disjoint_covering_ae', ωScottContinuous.bot, Set.Finite.encard_strictMonoOn, AddSubgroup.pairwiseDisjoint_leftCoset_cover_of_sum_neg_index_eq_zero, Matroid.IsBasis'.contract_indep_iff, lowerSemicontinuousOn_iff_le_liminf, Dynamics.coverEntropyInfEntourage_monotone, balanced_absConvexHull, convexHull_subset_affineSpan, AddMonoid.minOrder_le_natCard, convexHull_eq_self, PowerSeries.le_weightedOrder_subst, disjoint_closedBall_ball_iff, IsLocalization.coe_primeSpectrumOrderIso_apply_coe_asIdeal, Dynamics.le_coverMincard_image, balancedHull_convexHull_subseteq_absConvexHull, Matroid.spanning_iff_eRk_le, PiNat.exists_disjoint_cylinder, Algebra.gc, NumberField.mixedEmbedding.disjoint_negAt_plusPart, PMF.toMeasure_apply_eq_zero_iff, convexHull_range_eq_exists_affineCombination, convexIndependent_iff_notMem_convexHull_diff, OrderHom.map_gfp_comp, Topology.RelCWComplex.disjoint_base_iUnion_openCell, AddAction.IsBlock.vadd_eq_or_disjoint, FirstOrder.Language.Substructure.lift_card_closure_le, natCast_le_analyticOrderAt_iff_iteratedDeriv_eq_zero, Polynomial.emultiplicity_le_one_of_separable, FirstOrder.Language.Substructure.closure_eq, PowerSeries.le_order_subst_left, FirstOrder.Language.Substructure.mem_closure_iff_of_isRelational, FirstOrder.Language.Substructure.closure_mono, ringKrullDim_le_ringKrullDim_quotient_add_encard, FirstOrder.Language.Substructure.closure_univ, AlgebraicGeometry.sourceLocalClosure.le, FirstOrder.Language.Substructure.closure_image, AbsConvex.absConvexHull_eq, closedConvexHull_isClosed, Topology.RelCWComplex.disjoint_interior_base_iUnion_closedCell, Matroid.setOf_dep_eq, closedAbsConvexHull_closure_eq_closedAbsConvexHull, AlgebraicGeometry.Spec.map_app, Complex.rectangle_eq_convexHull, FirstOrder.Language.Substructure.cg_closure_singleton, Metric.coveringNumber_le_one_of_ediam_le, convexHull_subset_closedConvexHull, FirstOrder.Language.Structure.fg_iff, FirstOrder.Language.Substructure.cg_closure, Dynamics.coverMincard_monotone_subset, Finset.centerMass_mem_convexHull_of_nonpos

CompleteLattice.ωScottContinuous

Theorems

NameKindAssumesProvesValidatesDepends On
bot 📖mathematicalOmegaCompletePartialOrder.ωScottContinuous
CompleteLattice.instOmegaCompletePartialOrder
Bot.bot
Pi.instBotForall
OrderBot.toBot
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Lattice.toSemilatticeSup
CompleteLattice.toLattice
BoundedOrder.toOrderBot
CompleteLattice.toBoundedOrder
sSup_empty
sSup
instIsEmptyFalse
iSup 📖mathematicalOmegaCompletePartialOrder.ωScottContinuous
CompleteLattice.instOmegaCompletePartialOrder
iSup
Pi.supSet
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
OmegaCompletePartialOrder.ωScottContinuous.of_monotone_map_ωSup
Monotone.iSup
OmegaCompletePartialOrder.ωScottContinuous.monotone
eq_of_forall_ge_iff
iSup_apply
OmegaCompletePartialOrder.ωScottContinuous.map_ωSup
forall_swap
inf 📖mathematicalOmegaCompletePartialOrder.ωScottContinuous
CompleteLattice.instOmegaCompletePartialOrder
CompleteLinearOrder.toCompleteLattice
Pi.instMinForall_mathlib
SemilatticeInf.toMin
Lattice.toSemilatticeInf
CompleteLattice.toLattice
OmegaCompletePartialOrder.ωScottContinuous.of_monotone_map_ωSup
Monotone.inf
OmegaCompletePartialOrder.ωScottContinuous.monotone
eq_of_forall_ge_iff
OmegaCompletePartialOrder.ωScottContinuous.map_ωSup
le_trans
OrderHom.mono
le_max_left
le_max_right
sSup 📖mathematicalOmegaCompletePartialOrder.ωScottContinuous
CompleteLattice.instOmegaCompletePartialOrder
SupSet.sSup
Pi.supSet
CompleteSemilatticeSup.toSupSet
CompleteLattice.toCompleteSemilatticeSup
sSup_eq_iSup
iSup
sup 📖mathematicalOmegaCompletePartialOrder.ωScottContinuous
CompleteLattice.instOmegaCompletePartialOrder
Pi.instMaxForall_mathlib
SemilatticeSup.toMax
Lattice.toSemilatticeSup
CompleteLattice.toLattice
sSup_pair
sSup
top 📖mathematicalOmegaCompletePartialOrder.ωScottContinuous
CompleteLattice.instOmegaCompletePartialOrder
Top.top
Pi.instTopForall
OrderTop.toTop
Preorder.toLE
PartialOrder.toPreorder
SemilatticeSup.toPartialOrder
Lattice.toSemilatticeSup
CompleteLattice.toLattice
BoundedOrder.toOrderTop
CompleteLattice.toBoundedOrder
OmegaCompletePartialOrder.ωScottContinuous.of_monotone_map_ωSup
monotone_const
eq_of_forall_ge_iff

OmegaCompletePartialOrder

Definitions

NameCategoryTheorems
ContinuousHom 📖CompData
26 mathmath: ContinuousHom.const_apply, ContinuousHom.ωSup_apply, ContinuousHom.ωSup_apply_ωSup, instOrderHomClassContinuousHom, ContinuousHom.ωScottContinuous, ContinuousHom.bind_apply, ContinuousHom.seq_apply, ContinuousHom.coe_toOrderHom, ContinuousHom.coe_mk, ContinuousHom.forall_forall_merge, ContinuousHom.ofFun_apply, ContinuousHom.toOrderHom_eq_coe, ContinuousHom.id_apply, ContinuousHom.congr_fun, ContinuousHom.Prod.apply_apply, ContinuousHom.flip_apply, ContinuousHom.monotone, ContinuousHom.comp_apply, ContinuousHom.ext_iff, ContinuousHom.ωSup_def, ContinuousHom.congr_arg, ContinuousHom.forall_forall_merge', ContinuousHom.continuous, ContinuousHom.coe_apply, ContinuousHom.toMono_coe, ContinuousHom.map_apply
instFunLikeContinuousHom 📖CompOp
27 mathmath: ContinuousHom.const_apply, ContinuousHom.ωSup_apply, ContinuousHom.ωSup_apply_ωSup, instOrderHomClassContinuousHom, ContinuousHom.ωScottContinuous, ContinuousHom.apply_mono, ContinuousHom.bind_apply, ContinuousHom.seq_apply, ContinuousHom.ωScottContinuous_apply, ContinuousHom.coe_toOrderHom, ContinuousHom.coe_mk, ContinuousHom.forall_forall_merge, ContinuousHom.ofFun_apply, ContinuousHom.toOrderHom_eq_coe, ContinuousHom.id_apply, ContinuousHom.congr_fun, ContinuousHom.Prod.apply_apply, ContinuousHom.flip_apply, ContinuousHom.monotone, ContinuousHom.comp_apply, ContinuousHom.ext_iff, ContinuousHom.congr_arg, ContinuousHom.forall_forall_merge', ContinuousHom.continuous, ContinuousHom.coe_apply, ContinuousHom.toMono_coe, ContinuousHom.map_apply
instPartialOrderContinuousHom 📖CompOp
5 mathmath: ContinuousHom.ωSup_apply, ContinuousHom.ωSup_apply_ωSup, ContinuousHom.forall_forall_merge, ContinuousHom.forall_forall_merge', ContinuousHom.toMono_coe
subtype 📖CompOp
toPartialOrder 📖CompOp
751 mathmath: IsLocalization.AtPrime.coe_primeSpectrumOrderIso_symm_apply_asIdeal, Dynamics.netEntropyEntourage_monotone, Matroid.eRk_lt_top_iff, Dynamics.netMaxcard_monotone_subset, Set.ncard_mono, Matroid.exists_isBasis_disjoint_isBasis_of_subset, NonUnitalAlgebra.gc, Metric.disjoint_ball_infDist, convexIndependent_set_iff_inter_convexHull_subset, le_ωSup, SimpleGraph.vertexCoverNum_le_card_sub_one, Dynamics.coverEntropyInfEntourage_antitone, Ideal.height_le_spanFinrank, OrdinalApprox.lfpApprox_mono_mid, IsLinearMap.image_convexHull, LocalizedModule.subsingleton_iff_disjoint, ringKrullDim_succ_le_ringKrullDim_polynomial, FirstOrder.Language.Substructure.closure_eq_of_isRelational, AlgebraicGeometry.Scheme.isEmpty_pullback_iff, convexHull_insert, subset_closedConvexHull, Geometry.SimplicialComplex.vertex_mem_convexHull_iff, Matroid.eRk_lt_encard_iff_dep, SimpleGraph.isBipartiteWith_neighborSet_disjoint, closedConvexHull_min, PowerSeries.le_order_subst_left', PrimeSpectrum.localization_specComap_range, convexHull_multiset_sum, Geometry.SimplicialComplex.convexHull_subset_space, CategoryTheory.Retract.injectiveDimension_le, Ideal.height_le_iff_exists_minimalPrimes, BoxIntegral.Box.disjoint_splitCenterBox, SimpleGraph.le_chromaticNumber_iff_colorable, ContinuousHom.ωSup_apply, Matroid.isBase_compl_iff_maximal_disjoint_isBase, AlgebraicGeometry.Scheme.mem_grothendieckTopology_iff, SimpleGraph.le_chromaticNumber_iff_coloring, Geometry.SimplicialComplex.mem_space_iff, AlgebraicGeometry.HasAffineProperty.affineAnd_le_affineAnd, MeasureTheory.exists_decomposition_of_monotoneOn_hasDerivWithinAt, CategoryTheory.injectiveDimension_le_iff, OrderHom.le_map_sup_fixedPoints, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.app_invApp, OrdinalApprox.gfp_mem_range_gfpApprox, Convex.convex_remove_iff_notMem_convexHull_remove, ProjectiveSpectrum.gc_homogeneousIdeal, SimpleGraph.chromaticNumber_pos, Matroid.eRk_singleton_le, Matroid.eRk_mono, convexHull_eq_iInter, Ring.krullDimLE_iff, FirstOrder.Language.Substructure.fg_closure_singleton, MeasureTheory.pairwise_disjoint_fundamentalInterior, Ideal.height_lt_top, MeasureTheory.monotone_spanningSets, LinearMap.polar_antitone, Dynamics.netMaxcard_le_coverMincard, HasCardinalLT.Set.functor_obj, PrimeSpectrum.gc, Complex.convexHull_reProdIm, Dynamics.coverMincard_finite_of_isCompact_invariant, MeasureTheory.SeparableSpace.exists_measurable_partition_diam_le, Matroid.eRank_le_encard_add_eRk_compl, Matroid.IsBasis.contract_dep_iff, OrderHom.lfp_le_gfp, AddAction.orbit.pairwiseDisjoint, SimpleGraph.vertexCoverNum_le_iff, Topology.RelCWComplex.pairwiseDisjoint', convexJoin_segments, Matroid.contract_isCocircuit_iff, Dynamics.coverMincard_image_le, FirstOrder.Language.Substructure.small_closure, Set.chainHeight_le_encard, Topology.RelCWComplex.disjoint_openCell_of_ne, ContinuousHom.ωSup_apply_ωSup, OrderHom.ωSup_coe, FirstOrder.Language.Substructure.cg_def, Cardinal.mk_monotone, AddAction.IsBlock.disjoint_vadd_left, Matroid.le_eRk_iff, OrderIso.essSup_apply, OrderedFinpartition.disjoint, OrderHom.map_lfp, exists_partition_approximatesLinearOn_of_hasFDerivWithinAt, exists_dist_slope_lt_pairwiseDisjoint_hasSum, Ideal.primeHeight_mono, LinearEquiv.dilatransvections_pow_mono, FirstOrder.Language.Substructure.mem_closure, Besicovitch.exist_disjoint_covering_families, IsLocalization.coe_primeSpectrumOrderIso_symm_apply_asIdeal, MeasureTheory.exists_null_pairwise_disjoint_diff, convexHull_union_neg_eq_absConvexHull, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.app_inv_app', AddMonoid.minOrder_le_addOrderOf, Localization.localRingHom_injective_of_primesOver_eq_singleton, SimpleGraph.two_le_chromaticNumber_of_adj, interior_convexHull_nonempty_iff_affineSpan_eq_top, OrderHom.gfp_const_inf_le, SimpleGraph.card_le_chromaticNumber_iff_forall_surjective, Matroid.IsBasis.contract_indep_iff, mem_absConvexHull_iff, SimpleGraph.three_le_egirth, Dynamics.IsDynNetIn.card_le_netMaxcard, Perfect.small_diam_splitting, Ideal.primeHeight_lt_top, SimpleGraph.chromaticNumber_le_sum_right, Geometry.SimplicialComplex.convexHull_inter_convexHull, HasCardinalLT.Set.cocone_pt, IsLocalization.AtPrime.coe_orderIsoOfPrime_symm_apply_coe, AffineBasis.convexHull_eq_nonneg_coord, FirstOrder.Language.Substructure.fg_closure, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.app_inv_app', segment_subset_convexHull, Metric.coveringNumber_two_mul_le_externalCoveringNumber, convexHull_toCone_eq_sInf, lowerSemicontinuous_iff_le_liminf, MulAction.IsBlock.disjoint_smul_of_ne, Balanced.convexHull, subset_absConvexHull, Ideal.height_strict_mono_of_is_prime, Matroid.IsMinor.exists_eq_contract_delete_disjoint, balancedHull_subset_convexHull_union_neg, SimpleGraph.chromaticNumber_mono_of_embedding, le_minSmoothness, ringKrullDim_succ_le_ringKrullDim_powerseries, Matroid.delete_eq_self_iff, CategoryTheory.IsGrothendieckAbelian.isomorphisms_le_pushouts_generatingMonomorphisms, Convex.convexHull_union, closedConvexHull_eq_closure_convexHull, Finset.mem_convexHull', subset_convexHull, Dynamics.one_le_coverMincard_iff, Dynamics.coverEntropy_monotone, MulAction.isBlock_iff_smul_eq_or_disjoint, Convex.convexHull_subset_iff, AffineBasis.interior_convexHull, Matroid.contract_eq_self_iff, Dynamics.coverMincard_monotone_time, AlgebraicIndependent.adjoin_iff_disjoint, OrdinalApprox.gfpApprox_le, Matroid.eRk_lt_encard_iff_dep_of_finite, IsContDiffImplicitAt.one_le, MeasureTheory.SignedMeasure.exists_isCompl_positive_negative, Geometry.SimplicialComplex.inter_subset_convexHull, convexJoin_segment_singleton, SimpleGraph.chromaticNumber_le_of_forall_imp, Matroid.isClosed_iff_isFlat, PowerSeries.le_order_subst_right', ωSup_le_iff, convexJoin_singleton_segment, convexHull_diam, Matroid.Coindep.delete_spanning_iff, ringKrullDim_nonneg_of_nontrivial, Prod.ωSup_zip, subset_closedAbsConvexHull, Dynamics.netEntropyEntourage_antitone, Fin.Embedding.exists_embedding_disjoint_range_of_add_le_Nat_card, instOrderHomClassContinuousHom, Finset.centerMass_id_mem_convexHull, Submodule.length_lt, Set.Finite.isCompact_convexHull, Finset.centerMass_id_mem_convexHull_of_nonpos, convexHull_add, Module.length_pos_iff, SimpleGraph.vertexCoverNum_le_vertexCoverNum_of_injective, Monoid.minOrder_le_natCard, AddMonoid.le_minOrder, Module.supportDim_le_of_injective, Metric.externalCoveringNumber_le_one_of_ediam_le, affineSpan_convexHull, AlgebraicGeometry.Scheme.Hom.app_appIso_inv_assoc, Cardinal.mk_strictMonoOn, MeasureTheory.disjoint_addFundamentalInterior_addFundamentalFrontier, Besicovitch.TauPackage.monotone_iUnionUpTo, Matroid.eRk_le_eRk_add_eRk_diff, Dynamics.coverMincard_closure_le, MeasureTheory.hittingBtwn_apply_anti, CategoryTheory.MorphismProperty.le_ind, Set.encard_strictMono, Submodule.height_strictMono, Subgroup.leftCoset_cover_filter_FiniteIndex_aux, Matroid.eRk_compl_insert_union_add_eRk_compl_insert_inter_le, convexHull_basis_eq_stdSimplex, Ideal.disjoint_powers_iff_notMem, CategoryTheory.IsGrothendieckAbelian.generatingMonomorphisms_le_monomorphisms, Matroid.Indep.contract_dep_iff, Convex.radon_partition, IsOpen.exists_iUnion_isClosed, Metric.externalCoveringNumber_anti, Matroid.cRk_mono, ConvexOn.bddAbove_convexHull, FirstOrder.Language.Substructure.fg_def, Matroid.eRk_insert_le_add_one, Set.chainHeight_mono, Ideal.height_le_height_add_of_liesOver, Subgroup.pairwiseDisjoint_leftCoset_cover_const_of_index_eq, MvPowerSeries.le_order_subst, CategoryTheory.injectiveDimension_ge_iff, AddAction.IsBlock.disjoint_vadd_vadd_set, Matroid.closure_eq_subtypeClosure, Ideal.height_le_height_add_one_of_mem, Matroid.spanning_iff_eRk_le', Metric.packingNumber_two_mul_le_externalCoveringNumber, Ideal.disjoint_map_primeCompl_iff_comap_le, AddAction.IsBlock.vadd_eq_vadd_or_disjoint, ConvexIndependent.mem_convexHull_iff, not_disjoint_segment_convexHull_triple, convexHull_vadd, Matroid.closure_mono, FirstOrder.Language.Substructure.lift_card_closure_le_card_term, Dynamics.coverMincard_finite_of_isCompact_uniformContinuous, Ideal.height_le_card_of_mem_minimalPrimes_span, SimpleGraph.chromaticNumber_le_card, AlgebraicGeometry.topologically_iso_le, ringKrullDim_quotient_le, Matroid.eRk_inter_add_eRk_union_le, exists_mem_interior_convexHull_affineBasis, Ideal.disjoint_nonZeroDivisors_of_mem_minimalPrimes, essInf_antitone_measure, FirstOrder.Language.Substructure.mem_closure_iff_exists_term, ringKrullDim_succ_le_of_surjective, Besicovitch.exist_finset_disjoint_balls_large_measure, SimpleGraph.vertexCoverNum_mono, Ideal.primeHeight_le_ringKrullDim, closedAbsConvexHull_min, Dynamics.IsDynCoverOf.coverMincard_le_card, Metric.externalCoveringNumber_mono_set, Ideal.height_le_spanRank_toENat_of_mem_minimal_primes, Metric.IsCover.externalCoveringNumber_le_encard, CategoryTheory.projectiveDimension_lt_iff, totallyBounded_absConvexHull, Matroid.dual_indep_iff_exists', ContDiffMapSupportedIn.iteratedFDerivWithOrderLM_apply, Besicovitch.exists_disjoint_closedBall_covering_ae_of_finiteMeasure_aux, SimpleGraph.chromaticNumber_le_iff_colorable, absConvexHull_nonempty, MeasureTheory.disjoint_fundamentalInterior_fundamentalFrontier, Ideal.primeHeight_strict_mono, absConvexHull_empty, AddAction.IsBlock.pairwiseDisjoint_range_vadd, upperSemicontinuousWithinAt_iff_limsup_le, absConvexHull_eq_self, OrderHom.isLeast_lfp, MulAction.orbit.eq_or_disjoint, Convex.mem_extremePoints_iff_mem_diff_convexHull_diff, CategoryTheory.SmallObject.succStruct_prop_le_propArrow, ringKrullDim_le_iff_height_le, IsLocalization.isPrime_iff_isPrime_disjoint, SubMulAction.disjoint_val_image, upperSemicontinuousOn_iff_limsup_le, natCast_le_analyticOrderAt, Dynamics.coverMincard_antitone, ωScottContinuous.map_ωSup, NonUnitalStarAlgebra.gc, Dynamics.netMaxcard_antitone, SimpleGraph.vertexCoverNum_le_encard_edgeSet, AddAction.IsBlock.disjoint_vadd_set_vadd, Matroid.IsRkFinite.eRk_lt_top, Matroid.Indep.encard_le_eRk_of_subset, AlgebraicGeometry.isCompl_range_inl_inr, Metric.disjoint_closedEBall_of_lt_infEDist, Matroid.eRank_le_encard_ground, SimpleGraph.vertexCoverNum_lt_card, OrdinalApprox.lfpApprox_ord_eq_lfp, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.app_inv_app'_assoc, ProjectiveSpectrum.gc_set, Matroid.eRank_lt_top_iff, convexHull_add_subset, ConvexCone.disjoint_hull_left_of_convex, Matroid.eRk_insert_inter_add_eRk_insert_union_le, ωScottContinuous_iff_map_ωSup_of_orderHom, Ideal.height_mono, EisensteinSeries.pairwise_disjoint_gammaSet, Submodule.spanFinrank_span_le_encard, ringKrullDim_le_ringKrullDim_quotient_add_card, closedConvexHull_closure_eq_closedConvexHull, ContinuousHom.coe_toOrderHom, ContDiffMapSupportedIn.structureMapLM_apply_withOrder, Matroid.Indep.encard_le_eRank, MulAction.IsBlock.disjoint_smul_right, Polynomial.rootSet_derivative_subset_convexHull_rootSet, BoxIntegral.unitPartition.disjoint, Metric.IsCover.coveringNumber_le_encard, AlgebraicGeometry.Scheme.mem_overGrothendieckTopology, Ideal.height_le_ringKrullDim_quotient_add_one, closure_convexHull_extremePoints, MulAction.IsBlock.disjoint_smul_smul_set, ContinuousHom.forall_forall_merge, Dynamics.coverEntropyEntourage_monotone, Set.Finite.isClosed_convexHull, convexHull_empty, absConvexHull_add_subset, Ideal.height_le_iff, Finset.centerMass_mem_convexHull, SimpleGraph.chromaticNumber_le_two_iff_isBipartite, convexHull_pair, SimpleGraph.two_le_chromaticNumber_iff_ne_bot, preCantorSet_antitone, MeasureTheory.hittingBtwn_anti, MvPowerSeries.le_weightedOrder_subst, OrdinalApprox.lfpApprox_monotone, Dynamics.coverMincard_union_le, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.app_inv_app'_assoc, Matroid.contract_spanning_iff, MeasureTheory.stoppedProcess_eq, FirstOrder.Language.Substructure.mem_closed_of_isRelational, Topology.CWComplex.pairwiseDisjoint', Dynamics.coverEntropyInf_monotone, Module.length_le_of_surjective, OrdinalApprox.gfpApprox_antitone, FirstOrder.Language.Substructure.fg_iff_exists_fin_generating_family, MulAction.IsBlock.disjoint_smul_set_smul, convexJoin_subset_convexHull, Geometry.SimplicialComplex.disjoint_or_exists_inter_eq_convexHull, convexHull_eq_empty, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.app_invApp_assoc, ringKrullDim_le_ringKrullDim_quotient_add_spanFinrank, FirstOrder.Language.isExtensionPair_iff_exists_embedding_closure_singleton_sup, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.app_invApp, Matroid.eRk_le_iff, SimpleGraph.le_chromaticNumber_iff_forall_surjective, Vitali.exists_disjoint_subfamily_covering_enlargement_closedBall, MvPolynomial.supported_strictMono, AddAction.IsBlock.disjoint_vadd_right, convexHull_singleton, ωScottContinuous_iff_monotone_map_ωSup, MeasureTheory.hittingAfter_apply_anti, Pi.ωScottContinuous_uncurry, OrderHom.lfp_lfp, SimpleGraph.Colorable.chromaticNumber_le, convex_convexHull, convexHull_sum, OrdinalApprox.gfpApprox_ord_eq_gfp, FirstOrder.Language.Substructure.closure_insert, isBounded_convexHull, SimpleGraph.IsVertexCover.vertexCoverNum_le, AddAction.isBlock_iff_vadd_eq_or_disjoint, Dynamics.netEntropyInfEntourage_monotone, Set.exists_union_disjoint_cardinal_eq_of_even, Finset.mem_convexHull, OrderHom.isFixedPt_lfp, essSup_mono_measure, convexHull_neg, convexIndependent_set_iff_notMem_convexHull_diff, OrderHom.gfp_gfp, LinearEquiv.transvections_pow_mono, convexHull_sphere_eq_closedBall, ENat.LEInfty.out, LinearMap.polar_gc, Matroid.eRk_union_le_eRk_add_eRk, Matroid.IsCircuit.disjoint_coloops, Matroid.eRk_le_eRk_inter_add_eRk_diff, Subgroup.IsComplement.pairwiseDisjoint_smul, Convex.exists_subset_interior_convexHull_finset_of_isCompact, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.app_invApp, convexHull_mono, VitaliFamily.FineSubfamilyOn.covering_disjoint_subtype, convex_absConvexHull, Cardinal.mk_strictMono, Metric.coveringNumber_pos_iff, convexHullAddMonoidHom_apply, absConvexHull_mono, Finset.convexHull_eq, Ideal.height_le_ringKrullDim_quotient_add_encard, MulAction.IsBlock.smul_eq_smul_or_disjoint, Ideal.primeHeight_add_one_le_of_lt, ConvexCone.disjoint_coe, totallyBounded_convexHull, Set.one_le_chainHeight_iff, Set.Nonempty.absConvexHull, tangentBundleCore.isContMDiff, SimpleGraph.cliqueNum_le_chromaticNumber, AddAction.IsBlock.disjoint_vadd_of_ne, TangentBundle.contMDiffVectorBundle, Topology.RelCWComplex.disjoint_interior_base_closedCell, absConvex_convexClosedHull, upperSemicontinuousAt_iff_limsup_le, isωSup_ωSup, convexHull_eq_singleton, convexHull_eq_zero, Set.encard_mono, OrdinalApprox.lfpApprox_mono_left, SimpleGraph.IsBipartiteWith.disjoint, FirstOrder.Language.Substructure.closure_empty, Dynamics.coverMincard_le_pow, Ideal.height_le_iff_covBy, ringKrullDim_quotient_succ_le_of_nonZeroDivisor, isCompact_closedAbsConvexHull_of_totallyBounded, Dynamics.coverMincard_mul_le_pow, le_analyticOrderAt_sub, le_analyticOrderAt_add, VitaliFamily.FineSubfamilyOn.exists_disjoint_covering_ae, absConvexHull_subset_closedAbsConvexHull, ωScottContinuous.monotone_map_ωSup, Matroid.setOf_indep_eq, Dynamics.netMaxcard_monotone_time, toWeakSpace_closedConvexHull_eq, convexHull_pi, FirstOrder.Language.monotone_distinctConstantsTheory, FirstOrder.Language.Substructure.iSup_eq_closure, AddMonoid.le_minOrder_iff_forall_addSubgroup, disjoint_ball_closedBall_iff, ContinuousHom.toOrderHom_eq_coe, ringKrullDim_add_natCard_le_ringKrullDim_mvPolynomial, upperSemicontinuous_iff_limsup_le, SimpleGraph.isBipartiteWith_neighborSet_disjoint', doublyStochastic_eq_convexHull_permMatrix, Module.length_pos, absConvexHull_eq_iInter, isSaddlePointOn_iff', VitaliFamily.covering, exists_open_convex_of_notMem, Finset.centroid_mem_convexHull, AddAction.isBlock_iff_disjoint_vadd_of_ne, Vitali.exists_disjoint_subfamily_covering_enlargement, Matroid.coindep_contract_iff, AbsConvex.absConvexHull_subset_iff, AddSubgroup.pairwiseDisjoint_leftCoset_cover_const_of_index_eq, Metric.coveringNumber_subset_le, Topology.RelCWComplex.disjointBase, MeasureTheory.hittingAfter_eq_sInf, MeasureTheory.Egorov.notConvergentSeq_antitone, OrderHom.map_gfp, ringKrullDim_le_iff_isMaximal_height_le, Projectivization.Subspace.monotone_span, FirstOrder.Language.Substructure.subset_closure, AlgebraicGeometry.Scheme.Cover.toPresieveOver_le_arrows_iff, ringKrullDim_le_ringKrullDim_add_card, FirstOrder.Language.Substructure.closure_iUnion, Metric.disjoint_closedBall_of_lt_infDist, absConvex_absConvexHull, Metric.frontier_thickening_disjoint, iSup₂_iInf₂_le_iInf₂_iSup₂, CategoryTheory.projectiveDimension_le_iff, MulAction.isBlock_iff_smul_eq_smul_or_disjoint, SimpleGraph.chromaticNumber_le_one_of_subsingleton, AddAction.isBlock_iff_pairwiseDisjoint_range_vadd, ProjectiveSpectrum.gc_ideal, lowerSemicontinuousWithinAt_iff_le_liminf, Metric.coveringNumber_le_packingNumber, Vitali.exists_disjoint_covering_ae, ContDiffMapSupportedIn.structureMapCLM_apply_withOrder, Monoid.le_minOrder_iff_forall_subgroup, Module.supportDim_le_ringKrullDim, MulAction.IsBlock.disjoint_smul_left, Set.Infinite.exists_union_disjoint_cardinal_eq_of_infinite, ωScottContinuous.isLUB, LinearMap.image_convexHull, Module.length_le_of_injective, FirstOrder.Language.Substructure.closure_union, MulAction.isBlock_iff_disjoint_smul_of_ne, ContinuousMap.ideal_gc, Module.supportDim_le_supportDim_quotSMulTop_succ, CategoryTheory.projectiveDimension_ge_iff, Dynamics.coverEntropyEntourage_antitone, Besicovitch.exists_disjoint_closedBall_covering_ae, AddSubgroup.IsComplement.pairwiseDisjoint_vadd, Matroid.eRk_le_eRank, PrimeSpectrum.gc_set, IsLocalization.orderIsoOfPrime_symm_apply_coe, StarAlgebra.gc, Matroid.dual_indep_iff_exists, convexHull_univ, convexHull_ediam, disjoint_closedBall_closedBall_iff, SimpleGraph.le_chromaticNumber_of_pairwise_adj, FirstOrder.Language.Structure.cg_iff, absConvexHull_min, ContinuousHom.monotone, PowerSeries.le_order_subst, ringKrullDim_lt_top, CategoryTheory.injectiveDimension_lt_iff, lowerSemicontinuousAt_iff_le_liminf, AddAction.isBlock_iff_vadd_eq_vadd_or_disjoint, OrderIso.essInf_apply, OrdinalApprox.gfpApprox_mono_mid, SimpleGraph.chromaticNumber_mono, HasCardinalLT.Set.cocone_ι_app, IsLocalization.disjoint_comap_iff, AffineIndependent.convexHull_inter', Polynomial.ringKrullDim_le, AddAction.orbit.eq_or_disjoint, convexHull_zero, Matroid.eRk_lt_encard_of_dep_of_finite, convexHull_eq_union_convexHull_finite_subsets, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.app_invApp_assoc, Set.ncard_union_eq_iff, ContinuousHom.ωSup_bind, OrderHom.isGreatest_gfp_le, ωScottContinuous.monotone, Matroid.delete_dep_iff, MeasureTheory.exists_subordinate_pairwise_disjoint, zero_mem_absConvexHull, Ideal.map_height_le_one_of_mem_minimalPrimes, Matroid.contract_spanning_iff', Set.exists_union_disjoint_cardinal_eq_iff, HasCardinalLT.Set.functor_map_coe, affineCombination_mem_convexHull, Matroid.Indep.contract_indep_iff, mem_convexHull_iff_exists_fintype, Metric.frontier_cthickening_disjoint, Dynamics.netMaxcard_finite_iff, MulAction.orbit.pairwiseDisjoint, absConvexHull_univ, MeasureTheory.hittingBtwn_eq_sInf, MeasureTheory.stoppedProcess_eq', Topology.RelCWComplex.disjoint_skeleton_openCell, convexHull_toCone_isLeast, posTangentConeAt_mono, Matroid.eRk_union_le_encard_add_eRk, Matroid.eRk_union_le_eRk_add_encard, MeasureTheory.AEDisjoint.exists_disjoint_diff, cauchy_davenport_minOrder_add, MeasurableSpace.generateMeasurableRec_mono, disjoint_ball_ball_iff, Dynamics.coverMincard_le_netMaxcard, extremePoints_convexHull_subset, MvPolynomial.zeroLocus_vanishingIdeal_galoisConnection, Topology.RelCWComplex.disjointBase', Geometry.SimplicialComplex.face_subset_face_iff, OrdinalApprox.gfpApprox_mono_left, Convex.convexHull_eq, convexHull_convexHull_union_right, Subgroup.pairwiseDisjoint_leftCoset_cover_of_sum_inv_index_eq_one, SimpleGraph.egirth_le_length, PiNat.disjoint_cylinder_of_longestPrefix_lt, HasCardinalLT.Set.instIsCardinalFiltered, SimpleGraph.disjoint_image_val_universalVerts, ConvexCone.disjoint_hull_right_of_convex, MeasureTheory.Measure.exists_eq_disjoint_finiteSpanningSetsIn, EMetric.disjoint_closedBall_of_lt_infEdist, SimpleGraph.chromaticNumber_le_sum_left, Module.supportDim_le_supportDim_quotSMulTop_succ_of_mem_jacobson, FirstOrder.Language.Substructure.coe_closure_eq_range_term_realize, AlgebraicGeometry.targetAffineLocally_affineAnd_le, SimpleGraph.egirth_anti, Ideal.disjoint_primeCompl_of_liesOver, FirstOrder.Language.Substructure.closed, Prod.ωSup_snd, SimpleGraph.IsClique.card_le_chromaticNumber, essSup_comp_le_essSup_map_measure, MulAction.IsBlock.pairwiseDisjoint_range_smul, MulAction.isBlock_iff_pairwiseDisjoint_range_smul, isSaddlePointOn_iff, absConvexHull_isClosed, Set.encard_le_chainHeight_of_isChain, Set.exists_union_disjoint_ncard_eq_of_even, convexHull_convexHull_union_left, Ideal.finiteHeight_iff_lt, convexHull_min, Topology.RelCWComplex.Subcomplex.disjoint_openCell_subcomplex_of_not_mem, height_le_ringKrullDim_quotient_add_spanFinrank, convexHull_isClosed, Matroid.delete_isBasis_iff, AlgebraicGeometry.Scheme.exists_cover_of_mem_grothendieckTopology, ContinuousHom.forall_forall_merge', Ideal.height_le_height_add_encard_of_subset, closedAbsConvexHull_eq_closure_absConvexHull, Matroid.Dep.eRk_lt_encard, Metric.externalCoveringNumber_le_encard_self, Prod.ωSupImpl_fst, Matroid.isFlat_iff_isClosed, Matroid.IsCircuit.isCocircuit_disjoint_or_nontrivial_inter, PMF.toOuterMeasure_apply_eq_zero_iff, ConvexCone.gc_hull_coe, convexHull_prod, ringKrullDim_add_enatCard_le_ringKrullDim_mvPolynomial, Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes_of_isLocalRing, ContinuousHom.continuous, FirstOrder.Language.Substructure.closure_withConstants_eq, PowerSeries.le_order_subst_right, Metric.IsSeparated.encard_le_packingNumber, Dynamics.coverMincard_finite_iff, AddSubgroup.leftCoset_cover_filter_FiniteIndex_aux, AlgebraicGeometry.Scheme.mem_smallGrothendieckTopology, convexHull_eq_union, Topology.RelCWComplex.disjoint_skeletonLT_openCell, Matroid.eRk_compl_union_add_eRk_compl_inter_le, convexHull_eq, OrderHom.isGreatest_gfp, convexHull_smul, ModelWithCorners.disjoint_interior_boundary, ringKrullDim_le_ringKrullDim_add_spanFinrank, Set.Nonempty.convexHull, absConvexHull_eq_convexHull_balancedHull, FirstOrder.Language.Substructure.map_closure, Pi.ωScottContinuous_curry, Set.Finite.convexHull_eq_image, Metric.externalCoveringNumber_pos_iff, ContinuousHom.coe_apply, OrdinalApprox.lfp_mem_range_lfpApprox, AlgebraicGeometry.HasAffineProperty.affineAnd_le_isAffineHom, mem_convexHull_of_exists_fintype, ContinuousHom.toMono_coe, convexHull_rangle_single_eq_stdSimplex, FirstOrder.Language.Substructure.closure_le, Matroid.eRk_le_one_iff, AffineBasis.centroid_mem_interior_convexHull, AffineMap.image_convexHull, convexHull_union, convexHull_sub, FirstOrder.Language.Substructure.mem_closed_iff, SubAddAction.disjoint_val_image, Pi.uncurry_curry_ωScottContinuous, closure_subset_closedConvexHull, MeasureTheory.SimpleFunc.iSup_approx_apply, ConcaveOn.bddBelow_convexHull, Set.Countable.substructure_closure, MeasureTheory.exists_pair_mem_lattice_not_disjoint_vadd, FirstOrder.Language.Substructure.subset_closure_withConstants, convexHull_list_sum, Matroid.Indep.contract_isBase_iff, Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes, Set.Finite.ncard_strictMonoOn, AlgebraicGeometry.IsOpenImmersion.le_monomorphisms, IntermediateField.gc, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.app_inv_app', Ideal.height_le_height_add_spanFinrank_of_le, Metric.packingNumber_pos_iff, AlgebraicGeometry.affineLocally_le, MulAction.IsBlock.smul_eq_or_disjoint, Metric.externalCoveringNumber_le_coveringNumber, convexHull_nonempty_iff, SimpleGraph.le_egirth, Prod.ωSup_fst, RealRMK.range_cut_partition, Besicovitch.exists_disjoint_closedBall_covering_ae_aux, Metric.encard_le_of_isSeparated, Ideal.exists_disjoint_powers_of_span_eq_top, Matroid.eRk_le_encard, Dynamics.netEntropyInfEntourage_antitone, MeasureTheory.pairwise_disjoint_addFundamentalInterior, Topology.RelCWComplex.pairwiseDisjoint, Matroid.Coindep.delete_isBase_iff, closedAbsConvexHull_isClosed, HasCardinalLT.Set.instIsFilteredOfFactIsRegular, Monoid.le_minOrder, mem_convexHull_iff, Ideal.height_le_spanRank_toENat, Matroid.dualIndepMatroid_Indep, parallelepiped_eq_convexHull, cauchy_davenport_minOrder_mul, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.app_invApp_assoc, OrdinalApprox.le_lfpApprox, PrimeSpectrum.localization_comap_range, CategoryTheory.Retract.projectiveDimension_le, IsLocalization.orderIsoOfPrime_apply_coe, VitaliFamily.FineSubfamilyOn.covering_disjoint, Matroid.one_le_eRank, Ideal.height_le_card_of_mem_minimalPrimes_span_finset, absConvexHull_eq_empty, Matroid.IsBasis'.contract_dep_iff, MvPowerSeries.le_weightedOrder_subst_of_forall_ne_zero, Metric.coveringNumber_le_encard_self, Matroid.delete_indep_iff, FirstOrder.Language.Substructure.cg_iff_empty_or_exists_nat_generating_family, Ideal.height_le_ringKrullDim_of_ne_top, Matroid.Indep.disjoint_loops, SimpleGraph.IsContained.vertexCoverNum_le_vertexCoverNum, MeasureTheory.hitting_eq_sInf, AlgebraicGeometry.Scheme.Hom.app_appIso_inv, OrderHom.map_inf_fixedPoints_le, isClosed_closedAbsConvexHull, OrderHom.omegaCompletePartialOrder_ωSup_coe, essSup_mono_measure', AffineIndependent.convexHull_inter, Urysohns.CU.disjoint_C_support_lim, ringKrullDim_le_of_surjective, convex_closedConvexHull, SimpleGraph.Walk.three_le_chromaticNumber_of_odd_loop, closure_subset_closedAbsConvexHull, Metric.packingNumber_le_encard_self, IsLocalization.map_algebraMap_ne_top_iff_disjoint, AlgebraicGeometry.Scheme.codisjoint_zeroLocus, minSmoothness_monotone, Submodule.height_lt_top, FirstOrder.Language.Hom.eqOn_closure, ringKrullDim_le_ringKrullDim_quotSMulTop_succ, Prod.ωSupImpl_snd, Module.supportDim_quotSMulTop_succ_le_of_notMem_minimalPrimes, HasCardinalLT.Set.isFiltered_of_aleph0_le, OrderHom.isFixedPt_gfp, Matroid.delete_isCircuit_iff, Set.Finite.convexHull_eq, OrderHom.isLeast_lfp_le, Module.supportDim_le_of_surjective, MeasureTheory.hittingAfter_anti, SimpleGraph.ConnectedComponent.Represents.disjoint_supp_of_notMem, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.app_inv_app'_assoc, Set.ncard_strictMono, SimpleGraph.chromaticNumber_mono_of_hom, isClosed_closedConvexHull, Matroid.eRk_submod, OrderHom.map_lfp_comp, Monoid.minOrder_le_orderOf, Dynamics.one_le_netMaxcard_iff, NumberField.InfinitePlace.disjoint_isReal_isComplex, Metric.coveringNumber_anti, Vitali.exists_disjoint_covering_ae', Set.Finite.encard_strictMonoOn, AddSubgroup.pairwiseDisjoint_leftCoset_cover_of_sum_neg_index_eq_zero, Matroid.IsBasis'.contract_indep_iff, lowerSemicontinuousOn_iff_le_liminf, Dynamics.coverEntropyInfEntourage_monotone, balanced_absConvexHull, convexHull_subset_affineSpan, AddMonoid.minOrder_le_natCard, convexHull_eq_self, PowerSeries.le_weightedOrder_subst, disjoint_closedBall_ball_iff, IsLocalization.coe_primeSpectrumOrderIso_apply_coe_asIdeal, Scott.IsOpen.isUpperSet, Dynamics.le_coverMincard_image, balancedHull_convexHull_subseteq_absConvexHull, Matroid.spanning_iff_eRk_le, PiNat.exists_disjoint_cylinder, Algebra.gc, NumberField.mixedEmbedding.disjoint_negAt_plusPart, PMF.toMeasure_apply_eq_zero_iff, convexHull_range_eq_exists_affineCombination, convexIndependent_iff_notMem_convexHull_diff, OrderHom.map_gfp_comp, Topology.RelCWComplex.disjoint_base_iUnion_openCell, ContinuousHom.map_ωSup', AddAction.IsBlock.vadd_eq_or_disjoint, FirstOrder.Language.Substructure.lift_card_closure_le, natCast_le_analyticOrderAt_iff_iteratedDeriv_eq_zero, Polynomial.emultiplicity_le_one_of_separable, FirstOrder.Language.Substructure.closure_eq, PowerSeries.le_order_subst_left, FirstOrder.Language.Substructure.mem_closure_iff_of_isRelational, FirstOrder.Language.Substructure.closure_mono, ringKrullDim_le_ringKrullDim_quotient_add_encard, FirstOrder.Language.Substructure.closure_univ, AlgebraicGeometry.sourceLocalClosure.le, FirstOrder.Language.Substructure.closure_image, AbsConvex.absConvexHull_eq, isLUB_range_ωSup, closedConvexHull_isClosed, Topology.RelCWComplex.disjoint_interior_base_iUnion_closedCell, Matroid.setOf_dep_eq, closedAbsConvexHull_closure_eq_closedAbsConvexHull, AlgebraicGeometry.Spec.map_app, Complex.rectangle_eq_convexHull, FirstOrder.Language.Substructure.cg_closure_singleton, Metric.coveringNumber_le_one_of_ediam_le, convexHull_subset_closedConvexHull, FirstOrder.Language.Structure.fg_iff, FirstOrder.Language.Substructure.cg_closure, Dynamics.coverMincard_monotone_subset, Finset.centerMass_mem_convexHull_of_nonpos
«term_→𝒄_» 📖CompOp
ωScottContinuous 📖MathDef
20 mathmath: ωScottContinuous_iff_apply₂, ScottContinuous.ωScottContinuous, ContinuousHom.ωScottContinuous, Prod.ωScottContinuous_fst, Prod.ωScottContinuous_snd, ωScottContinuous_iff_map_ωSup_of_orderHom, ωScottContinuous_iff_monotone_map_ωSup, Pi.ωScottContinuous_uncurry, ωScottContinuous.const, Topology.IsScott.ωScottContinuous_iff_continuous, ωScottContinuous.id, ωScottContinuous.of_map_ωSup_of_orderHom, ωScottContinuous.fun_const, ωScottContinuous.of_monotone_map_ωSup, CompleteLattice.ωScottContinuous.top, Pi.ωScottContinuous_curry, ωScottContinuous.apply, Submodule.coe_scott_continuous, CompleteLattice.ωScottContinuous.bot, ωScottContinuous.fun_id
ωSup 📖CompOp
34 mathmath: le_ωSup, ContinuousHom.ωSup_apply, ωSup_total, fixedPoints.ωSup_iterate_le_prefixedPoint, ContinuousHom.ωSup_apply_ωSup, OrderHom.ωSup_coe, ωScottContinuous.map_ωSup_of_orderHom, ωSup_le_iff, Prod.ωSup_zip, ωScottContinuous.map_ωSup, ωScottContinuous_iff_map_ωSup_of_orderHom, Part.mem_ωSup, ωSup_le, ωScottContinuous_iff_monotone_map_ωSup, isωSup_ωSup, ωScottContinuous.monotone_map_ωSup, Part.fix_eq_ωSup, ωSup_le_ωSup_of_le, ωScottContinuous.isLUB, ContinuousHom.ωSup_bind, Prod.ωSup_snd, ContinuousHom.ωSup_def, Prod.ωSupImpl_fst, ContinuousHom.continuous, Prod.ωSup_fst, fixedPoints.ωSup_iterate_le_fixedPoint, ωSup_eq_of_isLUB, Part.fix_eq_ωSup_of_ωScottContinuous, OrderHom.omegaCompletePartialOrder_ωSup_coe, Prod.ωSupImpl_snd, le_ωSup_of_le, fixedPoints.ωSup_iterate_mem_fixedPoint, ContinuousHom.map_ωSup', isLUB_range_ωSup

Theorems

NameKindAssumesProvesValidatesDepends On
instOrderHomClassContinuousHom 📖mathematicalOrderHomClass
ContinuousHom
Preorder.toLE
PartialOrder.toPreorder
toPartialOrder
instFunLikeContinuousHom
OrderHom.mono
isLUB_range_ωSup 📖mathematicalIsLUB
Preorder.toLE
PartialOrder.toPreorder
toPartialOrder
Set.range
DFunLike.coe
Chain
Chain.instFunLikeNat
ωSup
le_ωSup
ωSup_le
le_ωSup 📖mathematicalPreorder.toLE
PartialOrder.toPreorder
toPartialOrder
DFunLike.coe
Chain
Chain.instFunLikeNat
ωSup
le_ωSup_of_le 📖mathematicalPreorder.toLE
PartialOrder.toPreorder
toPartialOrder
DFunLike.coe
Chain
Chain.instFunLikeNat
ωSuple_trans
le_ωSup
ωScottContinuous_iff_apply₂ 📖mathematicalωScottContinuous
instOmegaCompletePartialOrderForall
ωScottContinuous.apply₂
ωScottContinuous.of_apply₂
ωScottContinuous_iff_map_ωSup_of_orderHom 📖mathematicalωScottContinuous
DFunLike.coe
OrderHom
PartialOrder.toPreorder
toPartialOrder
OrderHom.instFunLike
ωSup
Chain.map
ωScottContinuous_iff_monotone_map_ωSup
OrderHom.monotone'
ωScottContinuous_iff_monotone_map_ωSup 📖mathematicalωScottContinuous
ωSup
Chain.map
PartialOrder.toPreorder
toPartialOrder
ωScottContinuous.monotone
ωScottContinuous.map_ωSup
Chain.map_coe
Set.range_comp
ωSup_eq_of_isLUB
isLUB_range_ωSup
ωSup_eq_of_isLUB 📖mathematicalIsLUB
Preorder.toLE
PartialOrder.toPreorder
toPartialOrder
Set.range
DFunLike.coe
Chain
Chain.instFunLikeNat
ωSuple_antisymm_iff
le_ωSup
ωSup_le_iff
ωSup_le 📖mathematicalPreorder.toLE
PartialOrder.toPreorder
toPartialOrder
DFunLike.coe
Chain
Chain.instFunLikeNat
ωSup
ωSup_le_iff 📖mathematicalPreorder.toLE
PartialOrder.toPreorder
toPartialOrder
ωSup
DFunLike.coe
Chain
Chain.instFunLikeNat
le_ωSup
ωSup_le
ωSup_le_ωSup_of_le 📖mathematicalChain
PartialOrder.toPreorder
toPartialOrder
Chain.instLE
Preorder.toLE
ωSup
ωSup_le
le_trans
le_ωSup
ωSup_total 📖mathematicalPreorder.toLE
PartialOrder.toPreorder
toPartialOrder
DFunLike.coe
Chain
Chain.instFunLikeNat
ωSupby_cases
ωSup_le
le_ωSup_of_le

OmegaCompletePartialOrder.Chain

Definitions

NameCategoryTheorems
instFunLikeNat 📖CompOp
16 mathmath: OmegaCompletePartialOrder.le_ωSup, zip_coe, OmegaCompletePartialOrder.ωSup_le_iff, pair_zero, range_pair, OmegaCompletePartialOrder.ContinuousHom.forall_forall_merge, directed, ext_iff, instOrderHomClassNat, OmegaCompletePartialOrder.ωScottContinuous.isLUB, Scott.isωSup_iff_isLUB, map_coe, OmegaCompletePartialOrder.ContinuousHom.forall_forall_merge', isChain_range, pair_succ, OmegaCompletePartialOrder.isLUB_range_ωSup
instInhabited 📖CompOp
instLE 📖CompOp
1 mathmath: map_le_map
instMembership 📖CompOp
4 mathmath: Part.Fix.approx_mem_approxChain, Part.mem_ωSup, Part.mem_chain_of_mem_ωSup, mem_map_iff
map 📖CompOp
22 mathmath: map_id, OmegaCompletePartialOrder.ContinuousHom.ωSup_apply, OmegaCompletePartialOrder.OrderHom.ωSup_coe, OmegaCompletePartialOrder.ωScottContinuous.map_ωSup_of_orderHom, mem_map, map_le_map, OmegaCompletePartialOrder.ωScottContinuous.map_ωSup, OmegaCompletePartialOrder.ωScottContinuous_iff_map_ωSup_of_orderHom, OmegaCompletePartialOrder.ωScottContinuous_iff_monotone_map_ωSup, OmegaCompletePartialOrder.ωScottContinuous.monotone_map_ωSup, OmegaCompletePartialOrder.ωScottContinuous.isLUB, OmegaCompletePartialOrder.ContinuousHom.ωSup_bind, map_coe, Prod.ωSup_snd, Prod.ωSupImpl_fst, OmegaCompletePartialOrder.ContinuousHom.continuous, Prod.ωSup_fst, mem_map_iff, OmegaCompletePartialOrder.OrderHom.omegaCompletePartialOrder_ωSup_coe, Prod.ωSupImpl_snd, OmegaCompletePartialOrder.ContinuousHom.map_ωSup', map_comp
pair 📖CompOp
4 mathmath: pair_zip_pair, pair_zero, range_pair, pair_succ
zip 📖CompOp
4 mathmath: pair_zip_pair, OmegaCompletePartialOrder.ContinuousHom.ωSup_apply_ωSup, zip_coe, Prod.ωSup_zip

Theorems

NameKindAssumesProvesValidatesDepends On
directed 📖mathematicalDirected
Preorder.toLE
DFunLike.coe
OmegaCompletePartialOrder.Chain
instFunLikeNat
directedOn_range
IsChain.directedOn
instReflLe
isChain_range
exists_of_mem_map 📖mathematicalOmegaCompletePartialOrder.Chain
instMembership
map
DFunLike.coe
OrderHom
OrderHom.instFunLike
ext 📖DFunLike.coe
OmegaCompletePartialOrder.Chain
instFunLikeNat
DFunLike.ext'
ext_iff 📖mathematicalDFunLike.coe
OmegaCompletePartialOrder.Chain
instFunLikeNat
ext
instOrderHomClassNat 📖mathematicalOrderHomClass
OmegaCompletePartialOrder.Chain
Preorder.toLE
instFunLikeNat
OrderHom.instOrderHomClass
isChain_range 📖mathematicalIsChain
Preorder.toLE
Set.range
DFunLike.coe
OmegaCompletePartialOrder.Chain
instFunLikeNat
Monotone.isChain_range
OrderHomClass.mono
instOrderHomClassNat
map_coe 📖mathematicalDFunLike.coe
OmegaCompletePartialOrder.Chain
instFunLikeNat
map
OrderHom
OrderHom.instFunLike
map_comp 📖mathematicalmap
OrderHom.comp
map_id 📖mathematicalmap
OrderHom.id
OrderHom.comp_id
map_le_map 📖mathematicalOrderHom
Preorder.toLE
OrderHom.instPreorder
OmegaCompletePartialOrder.Chain
instLE
map
mem_map 📖mathematicalOmegaCompletePartialOrder.Chain
instMembership
map
DFunLike.coe
OrderHom
OrderHom.instFunLike
mem_map_iff 📖mathematicalOmegaCompletePartialOrder.Chain
instMembership
map
DFunLike.coe
OrderHom
OrderHom.instFunLike
exists_of_mem_map
mem_map
pair_succ 📖mathematicalPreorder.toLEDFunLike.coe
OmegaCompletePartialOrder.Chain
instFunLikeNat
pair
pair_zero 📖mathematicalPreorder.toLEDFunLike.coe
OmegaCompletePartialOrder.Chain
instFunLikeNat
pair
pair_zip_pair 📖mathematicalPreorder.toLEzip
pair
Prod.instPreorder
Prod.instLE_mathlib
Prod.le_def
Prod.le_def
OrderHom.ext
range_pair 📖mathematicalPreorder.toLESet.range
DFunLike.coe
OmegaCompletePartialOrder.Chain
instFunLikeNat
pair
Set
Set.instInsert
Set.instSingletonSet
Set.ext
zip_coe 📖mathematicalDFunLike.coe
OmegaCompletePartialOrder.Chain
Prod.instPreorder
instFunLikeNat
zip

OmegaCompletePartialOrder.ContinuousHom

Definitions

NameCategoryTheorems
bind 📖CompOp
1 mathmath: bind_apply
comp 📖CompOp
4 mathmath: comp_id, id_comp, comp_assoc, comp_apply
const 📖CompOp
1 mathmath: const_apply
copy 📖CompOp
1 mathmath: copy_apply
flip 📖CompOp
1 mathmath: flip_apply
id 📖CompOp
3 mathmath: comp_id, id_comp, id_apply
inst 📖CompOp
3 mathmath: ωSup_apply_ωSup, Prod.apply_apply, ωSup_def
instInhabited 📖CompOp
map 📖CompOp
1 mathmath: map_apply
ofFun 📖CompOp
1 mathmath: ofFun_apply
seq 📖CompOp
1 mathmath: seq_apply
toMono 📖CompOp
2 mathmath: ωSup_apply, toMono_coe
toOrderHom 📖CompOp
3 mathmath: coe_toOrderHom, toOrderHom_eq_coe, map_ωSup'
ωSup 📖CompOp
2 mathmath: ωSup_apply, ωSup_def

Theorems

NameKindAssumesProvesValidatesDepends On
apply_mono 📖mathematicalOmegaCompletePartialOrder.ContinuousHom
Preorder.toLE
PartialOrder.toPreorder
OmegaCompletePartialOrder.instPartialOrderContinuousHom
OmegaCompletePartialOrder.toPartialOrder
DFunLike.coe
OmegaCompletePartialOrder.instFunLikeContinuousHom
OrderHom.apply_mono
OmegaCompletePartialOrder.instOrderHomClassContinuousHom
bind_apply 📖mathematicalDFunLike.coe
OmegaCompletePartialOrder.ContinuousHom
Part
Part.omegaCompletePartialOrder
OmegaCompletePartialOrder.instFunLikeContinuousHom
bind
Part.bind
instOmegaCompletePartialOrderForall
coe_apply 📖mathematicalDFunLike.coe
OrderHom
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
OrderHom.instFunLike
OrderHomClass.toOrderHom
OmegaCompletePartialOrder.ContinuousHom
OmegaCompletePartialOrder.instFunLikeContinuousHom
OmegaCompletePartialOrder.instOrderHomClassContinuousHom
OmegaCompletePartialOrder.instOrderHomClassContinuousHom
coe_inj 📖DFunLike.coe
OmegaCompletePartialOrder.ContinuousHom
OmegaCompletePartialOrder.instFunLikeContinuousHom
DFunLike.ext'
coe_mk 📖mathematicalOrderHom.toFun
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
OmegaCompletePartialOrder.ωSup
OmegaCompletePartialOrder.Chain.map
DFunLike.coe
OmegaCompletePartialOrder.ContinuousHom
OmegaCompletePartialOrder.instFunLikeContinuousHom
OrderHom
OrderHom.instFunLike
coe_toOrderHom 📖mathematicalDFunLike.coe
OrderHom
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
OrderHom.instFunLike
toOrderHom
OmegaCompletePartialOrder.ContinuousHom
OmegaCompletePartialOrder.instFunLikeContinuousHom
comp_apply 📖mathematicalDFunLike.coe
OmegaCompletePartialOrder.ContinuousHom
OmegaCompletePartialOrder.instFunLikeContinuousHom
comp
comp_assoc 📖mathematicalcomp
comp_id 📖mathematicalcomp
id
congr_arg 📖mathematicalDFunLike.coe
OmegaCompletePartialOrder.ContinuousHom
OmegaCompletePartialOrder.instFunLikeContinuousHom
congr_fun 📖mathematicalDFunLike.coe
OmegaCompletePartialOrder.ContinuousHom
OmegaCompletePartialOrder.instFunLikeContinuousHom
DFunLike.congr_fun
const_apply 📖mathematicalDFunLike.coe
OmegaCompletePartialOrder.ContinuousHom
OmegaCompletePartialOrder.instFunLikeContinuousHom
const
continuous 📖mathematicalDFunLike.coe
OmegaCompletePartialOrder.ContinuousHom
OmegaCompletePartialOrder.instFunLikeContinuousHom
OmegaCompletePartialOrder.ωSup
OmegaCompletePartialOrder.Chain.map
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
OrderHomClass.toOrderHom
OmegaCompletePartialOrder.instOrderHomClassContinuousHom
OmegaCompletePartialOrder.ωScottContinuous.map_ωSup
ωScottContinuous
copy_apply 📖mathematicalDFunLike.coe
OmegaCompletePartialOrder.ContinuousHom
OmegaCompletePartialOrder.instFunLikeContinuousHom
copy
ext 📖DFunLike.coe
OmegaCompletePartialOrder.ContinuousHom
OmegaCompletePartialOrder.instFunLikeContinuousHom
DFunLike.ext
ext_iff 📖mathematicalDFunLike.coe
OmegaCompletePartialOrder.ContinuousHom
OmegaCompletePartialOrder.instFunLikeContinuousHom
ext
flip_apply 📖mathematicalDFunLike.coe
OmegaCompletePartialOrder.ContinuousHom
instOmegaCompletePartialOrderForall
OmegaCompletePartialOrder.instFunLikeContinuousHom
flip
forall_forall_merge 📖mathematicalPreorder.toLE
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
DFunLike.coe
OmegaCompletePartialOrder.ContinuousHom
OmegaCompletePartialOrder.instFunLikeContinuousHom
OmegaCompletePartialOrder.Chain
OmegaCompletePartialOrder.instPartialOrderContinuousHom
OmegaCompletePartialOrder.Chain.instFunLikeNat
le_trans
monotone
OrderHom.monotone
le_max_right
le_max_left
forall_forall_merge' 📖mathematicalPreorder.toLE
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
DFunLike.coe
OmegaCompletePartialOrder.ContinuousHom
OmegaCompletePartialOrder.instFunLikeContinuousHom
OmegaCompletePartialOrder.Chain
OmegaCompletePartialOrder.instPartialOrderContinuousHom
OmegaCompletePartialOrder.Chain.instFunLikeNat
forall_swap
forall_forall_merge
id_apply 📖mathematicalDFunLike.coe
OmegaCompletePartialOrder.ContinuousHom
OmegaCompletePartialOrder.instFunLikeContinuousHom
id
id_comp 📖mathematicalcomp
id
map_apply 📖mathematicalDFunLike.coe
OmegaCompletePartialOrder.ContinuousHom
Part
Part.omegaCompletePartialOrder
OmegaCompletePartialOrder.instFunLikeContinuousHom
map
Part.map
map_ωSup' 📖mathematicalOrderHom.toFun
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
toOrderHom
OmegaCompletePartialOrder.ωSup
OmegaCompletePartialOrder.Chain.map
monotone 📖mathematicalMonotone
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
DFunLike.coe
OmegaCompletePartialOrder.ContinuousHom
OmegaCompletePartialOrder.instFunLikeContinuousHom
OrderHom.monotone'
ofFun_apply 📖mathematicalOmegaCompletePartialOrder.ωScottContinuousDFunLike.coe
OmegaCompletePartialOrder.ContinuousHom
OmegaCompletePartialOrder.instFunLikeContinuousHom
ofFun
seq_apply 📖mathematicalDFunLike.coe
OmegaCompletePartialOrder.ContinuousHom
Part
Part.omegaCompletePartialOrder
OmegaCompletePartialOrder.instFunLikeContinuousHom
seq
Part.instMonad
toMono_coe 📖mathematicalDFunLike.coe
OrderHom
OmegaCompletePartialOrder.ContinuousHom
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
OmegaCompletePartialOrder.instPartialOrderContinuousHom
OrderHom.instPreorder
OrderHom.instFunLike
toMono
OrderHomClass.toOrderHom
OmegaCompletePartialOrder.instFunLikeContinuousHom
OmegaCompletePartialOrder.instOrderHomClassContinuousHom
toOrderHom_eq_coe 📖mathematicaltoOrderHom
OrderHomClass.toOrderHom
OmegaCompletePartialOrder.ContinuousHom
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
OmegaCompletePartialOrder.instFunLikeContinuousHom
OmegaCompletePartialOrder.instOrderHomClassContinuousHom
ωScottContinuous 📖mathematicalOmegaCompletePartialOrder.ωScottContinuous
DFunLike.coe
OmegaCompletePartialOrder.ContinuousHom
OmegaCompletePartialOrder.instFunLikeContinuousHom
OmegaCompletePartialOrder.ωScottContinuous.of_map_ωSup_of_orderHom
map_ωSup'
ωScottContinuous_apply 📖mathematicalOmegaCompletePartialOrder.ωScottContinuous
OmegaCompletePartialOrder.ContinuousHom
inst
DFunLike.coe
OmegaCompletePartialOrder.instFunLikeContinuousHom
OmegaCompletePartialOrder.ωScottContinuous.of_monotone_map_ωSup
OrderHom.apply_mono
OmegaCompletePartialOrder.ωScottContinuous.monotone
OmegaCompletePartialOrder.ωScottContinuous.map_ωSup
le_antisymm
OmegaCompletePartialOrder.ωSup_le
OmegaCompletePartialOrder.instOrderHomClassContinuousHom
continuous
OmegaCompletePartialOrder.le_ωSup_of_le
apply_mono
OrderHom.monotone
le_sup_left
le_sup_right
monotone
le_refl
ωSup_apply 📖mathematicalDFunLike.coe
OmegaCompletePartialOrder.ContinuousHom
OmegaCompletePartialOrder.instFunLikeContinuousHom
ωSup
OmegaCompletePartialOrder.ωSup
OmegaCompletePartialOrder.Chain.map
OrderHom
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
OrderHom.instPreorder
OmegaCompletePartialOrder.instPartialOrderContinuousHom
OmegaCompletePartialOrder.OrderHom.omegaCompletePartialOrder
toMono
OrderHom.apply
ωSup_apply_ωSup 📖mathematicalDFunLike.coe
OmegaCompletePartialOrder.ContinuousHom
OmegaCompletePartialOrder.instFunLikeContinuousHom
OmegaCompletePartialOrder.ωSup
inst
Prod.instOmegaCompletePartialOrder
Prod.apply
OmegaCompletePartialOrder.Chain.zip
PartialOrder.toPreorder
OmegaCompletePartialOrder.instPartialOrderContinuousHom
OmegaCompletePartialOrder.toPartialOrder
ωSup_bind 📖mathematicalOmegaCompletePartialOrder.ωSup
Part
Part.omegaCompletePartialOrder
OmegaCompletePartialOrder.Chain.map
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
OrderHom.partBind
Part.instMonad
instOmegaCompletePartialOrderForall
eq_of_forall_ge_iff
OmegaCompletePartialOrder.ωSup_le
OrderHom.mono
le_max_right
le_max_left
OmegaCompletePartialOrder.le_ωSup
ωSup_def 📖mathematicalOmegaCompletePartialOrder.ωSup
OmegaCompletePartialOrder.ContinuousHom
inst
ωSup

OmegaCompletePartialOrder.ContinuousHom.Prod

Definitions

NameCategoryTheorems
apply 📖CompOp
2 mathmath: OmegaCompletePartialOrder.ContinuousHom.ωSup_apply_ωSup, apply_apply

Theorems

NameKindAssumesProvesValidatesDepends On
apply_apply 📖mathematicalDFunLike.coe
OmegaCompletePartialOrder.ContinuousHom
Prod.instOmegaCompletePartialOrder
OmegaCompletePartialOrder.ContinuousHom.inst
OmegaCompletePartialOrder.instFunLikeContinuousHom
apply

OmegaCompletePartialOrder.ContinuousHom.Simps

Definitions

NameCategoryTheorems
apply 📖CompOp

OmegaCompletePartialOrder.ContinuousHom.ωScottContinuous

Theorems

NameKindAssumesProvesValidatesDepends On
bind 📖mathematicalOmegaCompletePartialOrder.ωScottContinuous
Part
Part.omegaCompletePartialOrder
instOmegaCompletePartialOrderForall
Part.instMonadOmegaCompletePartialOrder.ωScottContinuous.of_monotone_map_ωSup
Monotone.partBind
OmegaCompletePartialOrder.ωScottContinuous.monotone
OmegaCompletePartialOrder.ωScottContinuous.map_ωSup
OmegaCompletePartialOrder.ContinuousHom.ωSup_bind
map 📖mathematicalOmegaCompletePartialOrder.ωScottContinuous
Part
Part.omegaCompletePartialOrder
Part.instMonadmap_eq_bind_pure_comp
Part.instLawfulMonad
bind
OmegaCompletePartialOrder.ωScottContinuous.const
seq 📖mathematicalOmegaCompletePartialOrder.ωScottContinuous
Part
Part.omegaCompletePartialOrder
Part.instMonadPart.instLawfulMonad
bind
OmegaCompletePartialOrder.ωScottContinuous.of_apply₂
map

OmegaCompletePartialOrder.OrderHom

Definitions

NameCategoryTheorems
omegaCompletePartialOrder 📖CompOp
2 mathmath: OmegaCompletePartialOrder.ContinuousHom.ωSup_apply, omegaCompletePartialOrder_ωSup_coe
ωSup 📖CompOp
1 mathmath: ωSup_coe

Theorems

NameKindAssumesProvesValidatesDepends On
omegaCompletePartialOrder_ωSup_coe 📖mathematicalDFunLike.coe
OrderHom
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
OrderHom.instFunLike
OmegaCompletePartialOrder.ωSup
omegaCompletePartialOrder
OmegaCompletePartialOrder.Chain.map
OrderHom.instPreorder
OrderHom.apply
ωSup_coe 📖mathematicalDFunLike.coe
OrderHom
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
OrderHom.instFunLike
ωSup
OmegaCompletePartialOrder.ωSup
OmegaCompletePartialOrder.Chain.map
OrderHom.instPreorder
OrderHom.apply

OmegaCompletePartialOrder.fixedPoints

Definitions

NameCategoryTheorems
iterateChain 📖CompOp
3 mathmath: ωSup_iterate_le_prefixedPoint, ωSup_iterate_le_fixedPoint, ωSup_iterate_mem_fixedPoint

Theorems

NameKindAssumesProvesValidatesDepends On
ωSup_iterate_le_fixedPoint 📖mathematicalPreorder.toLE
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
DFunLike.coe
OmegaCompletePartialOrder.ContinuousHom
OmegaCompletePartialOrder.instFunLikeContinuousHom
Set
Set.instMembership
Function.fixedPoints
OmegaCompletePartialOrder.ωSup
iterateChain
OrderHomClass.toOrderHom
OmegaCompletePartialOrder.instOrderHomClassContinuousHom
ωSup_iterate_le_prefixedPoint
Eq.le
Function.mem_fixedPoints
ωSup_iterate_le_prefixedPoint 📖mathematicalPreorder.toLE
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
DFunLike.coe
OmegaCompletePartialOrder.ContinuousHom
OmegaCompletePartialOrder.instFunLikeContinuousHom
OmegaCompletePartialOrder.ωSup
iterateChain
OrderHomClass.toOrderHom
OmegaCompletePartialOrder.instOrderHomClassContinuousHom
OmegaCompletePartialOrder.ωSup_le
OmegaCompletePartialOrder.instOrderHomClassContinuousHom
Function.iterate_succ_apply'
le_trans
OmegaCompletePartialOrder.ContinuousHom.monotone
ωSup_iterate_mem_fixedPoint 📖mathematicalPreorder.toLE
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
DFunLike.coe
OmegaCompletePartialOrder.ContinuousHom
OmegaCompletePartialOrder.instFunLikeContinuousHom
Set
Set.instMembership
Function.fixedPoints
OmegaCompletePartialOrder.ωSup
iterateChain
OrderHomClass.toOrderHom
OmegaCompletePartialOrder.instOrderHomClassContinuousHom
OmegaCompletePartialOrder.instOrderHomClassContinuousHom
Function.mem_fixedPoints
Function.IsFixedPt.eq_1
OmegaCompletePartialOrder.ContinuousHom.continuous
le_antisymm
OmegaCompletePartialOrder.ωSup_le
Function.iterate_succ_apply'
OmegaCompletePartialOrder.le_ωSup
le_trans

OmegaCompletePartialOrder.ωScottContinuous

Theorems

NameKindAssumesProvesValidatesDepends On
apply 📖mathematicalOmegaCompletePartialOrder.ωScottContinuous
instOmegaCompletePartialOrderForall
apply₂
id
apply₂ 📖OmegaCompletePartialOrder.ωScottContinuous
instOmegaCompletePartialOrderForall
of_monotone_map_ωSup
monotone
map_ωSup
comp 📖OmegaCompletePartialOrder.ωScottContinuousof_monotone_map_ωSup
Monotone.comp
monotone
map_ωSup
const 📖mathematicalOmegaCompletePartialOrder.ωScottContinuousScottContinuousOn.const
fun_comp 📖OmegaCompletePartialOrder.ωScottContinuouscomp
fun_const 📖mathematicalOmegaCompletePartialOrder.ωScottContinuousconst
fun_id 📖mathematicalOmegaCompletePartialOrder.ωScottContinuousid
id 📖mathematicalOmegaCompletePartialOrder.ωScottContinuousScottContinuousOn.id
isLUB 📖mathematicalOmegaCompletePartialOrder.ωScottContinuousIsLUB
Preorder.toLE
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
Set.range
DFunLike.coe
OmegaCompletePartialOrder.Chain
OmegaCompletePartialOrder.Chain.instFunLikeNat
OmegaCompletePartialOrder.Chain.map
monotone
OmegaCompletePartialOrder.ωSup
monotone
Set.range_comp
Set.range_nonempty
IsChain.directedOn
instReflLe
OmegaCompletePartialOrder.Chain.isChain_range
OmegaCompletePartialOrder.isLUB_range_ωSup
map_ωSup 📖mathematicalOmegaCompletePartialOrder.ωScottContinuousOmegaCompletePartialOrder.ωSup
OmegaCompletePartialOrder.Chain.map
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
monotone
OmegaCompletePartialOrder.ωSup_eq_of_isLUB
monotone
isLUB
map_ωSup_of_orderHom 📖mathematicalOmegaCompletePartialOrder.ωScottContinuous
DFunLike.coe
OrderHom
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
OrderHom.instFunLike
OmegaCompletePartialOrder.ωSup
OmegaCompletePartialOrder.Chain.map
OmegaCompletePartialOrder.ωScottContinuous_iff_map_ωSup_of_orderHom
monotone 📖mathematicalOmegaCompletePartialOrder.ωScottContinuousMonotone
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
ScottContinuousOn.monotone
OmegaCompletePartialOrder.Chain.range_pair
monotone_map_ωSup 📖mathematicalOmegaCompletePartialOrder.ωScottContinuousOmegaCompletePartialOrder.ωSup
OmegaCompletePartialOrder.Chain.map
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
OmegaCompletePartialOrder.ωScottContinuous_iff_monotone_map_ωSup
of_apply₂ 📖mathematicalOmegaCompletePartialOrder.ωScottContinuousinstOmegaCompletePartialOrderForallof_monotone_map_ωSup
monotone
map_ωSup
of_map_ωSup_of_orderHom 📖mathematicalDFunLike.coe
OrderHom
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
OrderHom.instFunLike
OmegaCompletePartialOrder.ωSup
OmegaCompletePartialOrder.Chain.map
OmegaCompletePartialOrder.ωScottContinuousOmegaCompletePartialOrder.ωScottContinuous_iff_map_ωSup_of_orderHom
of_monotone_map_ωSup 📖mathematicalOmegaCompletePartialOrder.ωSup
OmegaCompletePartialOrder.Chain.map
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
OmegaCompletePartialOrder.ωScottContinuousOmegaCompletePartialOrder.ωScottContinuous_iff_monotone_map_ωSup

Part

Definitions

NameCategoryTheorems
omegaCompletePartialOrder 📖CompOp
6 mathmath: OmegaCompletePartialOrder.ContinuousHom.bind_apply, OmegaCompletePartialOrder.ContinuousHom.seq_apply, mem_ωSup, fix_eq_ωSup, OmegaCompletePartialOrder.ContinuousHom.ωSup_bind, OmegaCompletePartialOrder.ContinuousHom.map_apply
ωSup 📖CompOp
2 mathmath: ωSup_eq_none, ωSup_eq_some

Theorems

NameKindAssumesProvesValidatesDepends On
eq_of_chain 📖Part
OmegaCompletePartialOrder.Chain
PartialOrder.toPreorder
instPartialOrder
OmegaCompletePartialOrder.Chain.instMembership
some
le_total
OrderHom.monotone
eq_some_iff
mem_unique
mem_chain_of_mem_ωSup 📖mathematicalPart
instMembership
ωSup
OmegaCompletePartialOrder.Chain
PartialOrder.toPreorder
instPartialOrder
OmegaCompletePartialOrder.Chain.instMembership
some
eq_some_iff
mem_ωSup 📖mathematicalPart
instMembership
OmegaCompletePartialOrder.ωSup
omegaCompletePartialOrder
OmegaCompletePartialOrder.Chain
PartialOrder.toPreorder
instPartialOrder
OmegaCompletePartialOrder.Chain.instMembership
some
mem_chain_of_mem_ωSup
eq_of_chain
ωSup_eq_none 📖mathematicalPart
OmegaCompletePartialOrder.Chain
PartialOrder.toPreorder
instPartialOrder
OmegaCompletePartialOrder.Chain.instMembership
some
ωSup
none
ωSup_eq_some 📖mathematicalPart
OmegaCompletePartialOrder.Chain
PartialOrder.toPreorder
instPartialOrder
OmegaCompletePartialOrder.Chain.instMembership
some
ωSupeq_of_chain

Prod

Definitions

NameCategoryTheorems
instOmegaCompletePartialOrder 📖CompOp
8 mathmath: OmegaCompletePartialOrder.ContinuousHom.ωSup_apply_ωSup, ωSup_zip, ωScottContinuous_fst, ωScottContinuous_snd, OmegaCompletePartialOrder.ContinuousHom.Prod.apply_apply, ωSup_snd, ωSup_fst, ωScottContinuous.prodMk
ωSupImpl 📖CompOp
2 mathmath: ωSupImpl_fst, ωSupImpl_snd

Theorems

NameKindAssumesProvesValidatesDepends On
ωScottContinuous_fst 📖mathematicalOmegaCompletePartialOrder.ωScottContinuous
instOmegaCompletePartialOrder
ScottContinuousOn.fst
ωScottContinuous_snd 📖mathematicalOmegaCompletePartialOrder.ωScottContinuous
instOmegaCompletePartialOrder
ScottContinuousOn.snd
ωSupImpl_fst 📖mathematicalωSupImpl
OmegaCompletePartialOrder.ωSup
OmegaCompletePartialOrder.Chain.map
instPreorder
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
OrderHom.fst
ωSupImpl_snd 📖mathematicalωSupImpl
OmegaCompletePartialOrder.ωSup
OmegaCompletePartialOrder.Chain.map
instPreorder
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
OrderHom.snd
ωSup_fst 📖mathematicalOmegaCompletePartialOrder.ωSup
instOmegaCompletePartialOrder
OmegaCompletePartialOrder.Chain.map
instPreorder
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
OrderHom.fst
ωSup_snd 📖mathematicalOmegaCompletePartialOrder.ωSup
instOmegaCompletePartialOrder
OmegaCompletePartialOrder.Chain.map
instPreorder
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
OrderHom.snd
ωSup_zip 📖mathematicalOmegaCompletePartialOrder.ωSup
instOmegaCompletePartialOrder
OmegaCompletePartialOrder.Chain.zip
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder

Prod.ωScottContinuous

Theorems

NameKindAssumesProvesValidatesDepends On
prodMk 📖mathematicalOmegaCompletePartialOrder.ωScottContinuousProd.instOmegaCompletePartialOrderScottContinuousOn.prodMk
OmegaCompletePartialOrder.Chain.range_pair

ScottContinuous

Theorems

NameKindAssumesProvesValidatesDepends On
ωScottContinuous 📖mathematicalScottContinuous
PartialOrder.toPreorder
OmegaCompletePartialOrder.toPartialOrder
OmegaCompletePartialOrder.ωScottContinuousscottContinuousOn

(root)

Definitions

NameCategoryTheorems
instOmegaCompletePartialOrderForall 📖CompOp
10 mathmath: OmegaCompletePartialOrder.ωScottContinuous_iff_apply₂, OmegaCompletePartialOrder.ContinuousHom.bind_apply, OmegaCompletePartialOrder.ωScottContinuous.of_apply₂, Pi.ωScottContinuous_uncurry, Part.fix_eq_ωSup, OmegaCompletePartialOrder.ContinuousHom.flip_apply, Part.ωScottContinuous_toUnitMono, OmegaCompletePartialOrder.ContinuousHom.ωSup_bind, Pi.ωScottContinuous_curry, OmegaCompletePartialOrder.ωScottContinuous.apply

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