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Try doing some basic maths questions in the Lean Theorem Prover. Functions, real numbers, equivalence relations and groups. Click on README.md and then on "Open in CoCalc with one click".
Project: Xena
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import .all1run_cmd tactic.skip2apply_nolint3AddCommGroup4AddCommGroup.of5AddCommMon6AddCommMon.of7AddGroup8AddGroup.of9AddMon10AddMon.of11Cauchy.extend12Cauchy.gen13Class14CommRing.colimits.cocone_fun15CommRing.colimits.cocone_morphism16CommRing.colimits.colimit17CommRing.colimits.colimit_cocone18CommRing.colimits.colimit_is_colimit19CommRing.colimits.colimit_type20CommRing.colimits.desc_fun21CommRing.colimits.desc_fun_lift22CommRing.colimits.desc_morphism23CommRing.colimits.prequotient24CommRing.colimits.relation25CpltSepUniformSpace.to_UniformSpace26Gromov_Hausdorff.candidates_b_dist27Meas28Module.of29Mon.colimits.cocone_fun30Mon.colimits.cocone_morphism31Mon.colimits.colimit32Mon.colimits.colimit_cocone33Mon.colimits.colimit_is_colimit34Mon.colimits.colimit_type35Mon.colimits.desc_fun36Mon.colimits.desc_fun_lift37Mon.colimits.desc_morphism38Mon.colimits.prequotient39Mon.colimits.relation40Set.map_definable_aux41Set.mem42Set.mk43Set.subset44Top.colimit45Top.colimit_is_colimit46Top.limit47Top.limit_is_limit48Top.presheaf49Top.presheaf.pushforward50Top.presheaf.pushforward.comp51Top.presheaf.pushforward.id52Top.presheaf.pushforward_eq53Top.presheaf.stalk_pushforward54Well_order55abelianization56abelianization.lift57abelianization.lift.unique58abelianization.of59abv_sum_le_sum_abv60add_comm_group.is_Z_bilin61add_con.to_setoid62add_equiv.mk'63add_equiv.refl64add_equiv.symm65add_equiv.to_add_monoid_hom66add_equiv.to_equiv67add_equiv.trans68add_group.closure69add_group.in_closure70add_monoid.smul71add_monoid_hom.add72add_monoid_hom.comp73add_monoid_hom.id74add_monoid_hom.mk'75add_monoid_hom.neg76add_monoid_hom.zero77additive78adjoin_root79adjoin_root.lift80adjoin_root.mk81adjoin_root.of82adjoin_root.root83alg_hom.comp84alg_hom.id85alg_hom.range86alg_hom.to_ring_hom87algebra.adjoin88algebra.adjoin_eq_range89algebra.adjoin_singleton_eq_range90algebra.comap.comm_ring91algebra.comap.has_scalar92algebra.comap.of_comap93algebra.comap.ring94algebra.comap.to_comap95algebra.gi96algebra.lmul_left97algebra.lmul_right98algebra.of_id99algebra.to_comap100algebra.to_top101algebra_map102algebraic_geometry.PresheafedSpace.comp103algebraic_geometry.PresheafedSpace.id104algebraic_geometry.PresheafedSpace.stalk105algebraic_geometry.PresheafedSpace.stalk_map106alist.disjoint107alist.foldl108applicative_transformation109apply_fun_name110archimedean111archimedean.floor_ring112associated113associated.setoid114associates115associates.factor_set116associates.factor_set.coe_add117associates.factor_set.prod118associates.factors119associates.factors'120associates.mk121associates.out122associates.prime123associates.unique'124associates_int_equiv_nat125auto.add_conjuncts126auto.add_simps127auto.auto_config128auto.case_hyp129auto.case_option130auto.case_some_hyp131auto.case_some_hyp_aux132auto.classical_normalize_lemma_names133auto.common_normalize_lemma_names134auto.do_subst135auto.do_substs136auto.eelim137auto.eelims138auto.mk_hinst_lemmas139auto.normalize_hyp140auto.normalize_hyps141auto.normalize_negations142auto.preprocess_goal143auto.preprocess_hyps144auto.split_hyp145auto.split_hyps146auto.split_hyps_aux147auto.whnf_reducible148auto_cases_at149bicompl.bitraverse150bicompr.bitraverse151bifunctor152bifunctor.fst153bifunctor.snd154bilin_form.bilin_linear_map_equiv155bilin_form.to_linear_map156binder_eq_elim157binder_eq_elim.check158binder_eq_elim.check_eq159binder_eq_elim.old_conv160binder_eq_elim.pull161binder_eq_elim.push162bisequence163bitraversable164bounded_continuous_function.arzela_ascoli165bounded_continuous_function.arzela_ascoli₁166bounded_continuous_function.arzela_ascoli₂167bounded_continuous_function.const168bounded_continuous_function.dist_eq169bounded_continuous_function.dist_set_exists170bounded_continuous_function.equicontinuous_of_continuity_modulus171buffer.list_equiv_buffer172can_lift_attr173canonically_ordered_comm_semiring174card_subgroup_dvd_card175card_trivial176cardinal.aleph'.order_iso177cardinal.aleph_idx.initial_seg178cardinal.aleph_idx.order_iso179cardinal.cantor_function180cardinal.cantor_function_aux181cardinal.ord.order_embedding182cast_num183cast_pos_num184cast_znum185category_theory.Aut186category_theory.Kleisli187category_theory.Kleisli.comp_def188category_theory.Kleisli.id_def189category_theory.Kleisli.mk190category_theory.adjunction.adjunction_of_equiv_left191category_theory.adjunction.adjunction_of_equiv_right192category_theory.adjunction.cocones_iso193category_theory.adjunction.comp194category_theory.adjunction.cones_iso195category_theory.adjunction.core_hom_equiv196category_theory.adjunction.core_unit_counit197category_theory.adjunction.functoriality_counit198category_theory.adjunction.functoriality_counit'199category_theory.adjunction.functoriality_is_left_adjoint200category_theory.adjunction.functoriality_is_right_adjoint201category_theory.adjunction.functoriality_left_adjoint202category_theory.adjunction.functoriality_right_adjoint203category_theory.adjunction.functoriality_unit204category_theory.adjunction.functoriality_unit'205category_theory.adjunction.has_colimit_of_comp_equivalence206category_theory.adjunction.has_limit_of_comp_equivalence207category_theory.adjunction.id208category_theory.adjunction.left_adjoint_of_equiv209category_theory.adjunction.mk_of_hom_equiv210category_theory.adjunction.mk_of_unit_counit211category_theory.adjunction.right_adjoint_of_equiv212category_theory.as_iso213category_theory.category_struct214category_theory.comma215category_theory.comma.fst216category_theory.comma.map_left217category_theory.comma.map_left_comp218category_theory.comma.map_left_id219category_theory.comma.map_right220category_theory.comma.map_right_comp221category_theory.comma.map_right_id222category_theory.comma.nat_trans223category_theory.comma.snd224category_theory.comma_morphism225category_theory.core226category_theory.core.forget_functor_to_core227category_theory.core.inclusion228category_theory.coyoneda229category_theory.coyoneda.is_iso230category_theory.curry231category_theory.curry_obj232category_theory.currying233category_theory.discrete234category_theory.discrete.lift235category_theory.discrete.opposite236category_theory.epi237category_theory.eq_to_hom238category_theory.eq_to_iso239category_theory.equivalence.adjointify_η240category_theory.equivalence.counit241category_theory.equivalence.counit_inv242category_theory.equivalence.equivalence_of_fully_faithfully_ess_surj243category_theory.equivalence.ess_surj_of_equivalence244category_theory.equivalence.fun_inv_id_assoc245category_theory.equivalence.inv_fun_id_assoc246category_theory.equivalence.mk247category_theory.equivalence.refl248category_theory.equivalence.symm249category_theory.equivalence.to_adjunction250category_theory.equivalence.trans251category_theory.equivalence.unit252category_theory.equivalence.unit_inv253category_theory.ess_surj254category_theory.ess_surj.iso255category_theory.evaluation256category_theory.evaluation_uncurried257category_theory.full_subcategory_inclusion258category_theory.functor.adjunction259category_theory.functor.as_equivalence260category_theory.functor.associator261category_theory.functor.const262category_theory.functor.const.op_obj_op263category_theory.functor.const.op_obj_unop264category_theory.functor.flip265category_theory.functor.fun_inv_id266category_theory.functor.fun_obj_preimage_iso267category_theory.functor.inv268category_theory.functor.inv_fun_id269category_theory.functor.left_op270category_theory.functor.left_unitor271category_theory.functor.map_cocone_morphism272category_theory.functor.map_cone_inv273category_theory.functor.map_cone_morphism274category_theory.functor.map_iso275category_theory.functor.obj_preimage276category_theory.functor.of_function277category_theory.functor.op278category_theory.functor.op_hom279category_theory.functor.op_inv280category_theory.functor.right_op281category_theory.functor.right_unitor282category_theory.functor.sections283category_theory.functor.star284category_theory.functor.to_cocone285category_theory.functor.to_cone286category_theory.functor.ulift_down287category_theory.functor.ulift_down_up288category_theory.functor.ulift_up289category_theory.functor.ulift_up_down290category_theory.functor.unop291category_theory.has_hom292category_theory.has_hom.hom.op293category_theory.has_hom.hom.unop294category_theory.has_limits_of_reflective295category_theory.hom_of_element296category_theory.induced_category297category_theory.induced_functor298category_theory.is_equivalence.mk299category_theory.is_iso_of_fully_faithful300category_theory.is_iso_of_op301category_theory.is_left_adjoint302category_theory.is_right_adjoint303category_theory.iso304category_theory.iso.hom_congr305category_theory.iso.op306category_theory.iso.refl307category_theory.iso.symm308category_theory.iso.to_equiv309category_theory.iso.trans310category_theory.iso_whisker_left311category_theory.iso_whisker_right312category_theory.large_groupoid313category_theory.left_adjoint314category_theory.limits.binary_cofan315category_theory.limits.binary_cofan.mk316category_theory.limits.binary_fan317category_theory.limits.binary_fan.mk318category_theory.limits.cocone.equiv319category_theory.limits.cocone.extensions320category_theory.limits.cocone.of_cofork321category_theory.limits.cocone.of_pushout_cocone322category_theory.limits.cocone.whisker323category_theory.limits.cocone_left_op_of_cone324category_theory.limits.cocone_morphism325category_theory.limits.cocone_of_cone_left_op326category_theory.limits.cocones.forget327category_theory.limits.cocones.functoriality328category_theory.limits.cocones.precompose329category_theory.limits.cocones.precompose_comp330category_theory.limits.cocones.precompose_equivalence331category_theory.limits.cocones.precompose_id332category_theory.limits.coequalizer333category_theory.limits.coequalizer.desc334category_theory.limits.coequalizer.π335category_theory.limits.cofan336category_theory.limits.cofan.mk337category_theory.limits.cofork338category_theory.limits.cofork.of_cocone339category_theory.limits.cofor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putation.terminates_rec_on628con.to_setoid629const.bitraverse630constr_smul631cont632cont_t633cont_t.map634cont_t.monad_lift635cont_t.run636cont_t.with_cont_t637continuous.comap638continuous_map639continuous_map.coev640continuous_map.compact_open.gen641continuous_map.ev642continuous_map.induced643continuous_of_locally_uniform_limit_of_continuous644continuous_of_uniform_limit_of_continuous645conv.discharge_eq_lhs646conv.interactive.erw647conv.interactive.norm_cast648conv.interactive.norm_num649conv.interactive.norm_num1650conv.interactive.ring651conv.interactive.ring2652conv.repeat_count653conv.repeat_with_results654conv.replace_lhs655conv.slice656conv.slice_lhs657conv.slice_rhs658ctop.realizer.id659ctop.realizer.nhds660ctop.realizer.nhds_F661ctop.realizer.nhds_σ662ctop.realizer.of_equiv663ctop.to_realizer664decidable.lt_by_cases665decidable_linear_order.lift666decidable_linear_order_of_is_well_order667decidable_of_bool668decidable_of_iff669decidable_of_iff'670decidable_zero_symm671declaration.update_with_fun672dense_or_discrete673denumerable.equiv₂674denumerable.eqv675denumerable.lower676denumerable.lower'677denumerable.mk'678denumerable.of_encodable_of_infinite679denumerable.of_equiv680denumerable.of_nat681denumerable.pair682denumerable.raise683denumerable.raise'684denumerable.raise'_finset685dfinsupp686dfinsupp.decidable_eq687dfinsupp.erase688dfinsupp.lmk689dfinsupp.lsingle690dfinsupp.map_range_def691dfinsupp.map_range_single692dfinsupp.mk693dfinsupp.pre694dfinsupp.single695dfinsupp.subtype_domain_sum696dfinsupp.sum_apply697dfinsupp.support698dfinsupp.to_has_scalar699dfinsupp.to_module700dfinsupp.zip_with701dfinsupp.zip_with_def702dioph.pell_dioph703dioph.xn_dioph704direct_sum705direct_sum.component706direct_sum.id707direct_sum.lid708direct_sum.lmk709direct_sum.lof710direct_sum.lset_to_set711direct_sum.mk712direct_sum.of713direct_sum.set_to_set714direct_sum.to_group715direct_sum.to_module716directed_order717dlist.join718dlist.list_equiv_dlist719eckmann_hilton.comm_group720eckmann_hilton.comm_monoid721eckmann_hilton.is_unital722emetric.cauchy_iff723emetric.cauchy_seq_iff724emetric.cauchy_seq_iff'725emetric.exists_ball_subset_ball726emetric.is_open_iff727emetric.mem_closure_iff'728emetric.mem_nhds_iff729emetric.nhds_eq730emetric.tendsto_at_top731emetric.tendsto_nhds732emetric.totally_bounded_iff733emetric.totally_bounded_iff'734emetric.uniform_continuous_iff735emetric.uniform_embedding_iff736emetric.uniform_embedding_iff'737empty.elim738enat739encodable.choose740encodable.choose_x741encodable.decidable_eq_of_encodable742encodable.decidable_range_encode743encodable.decode2744encodable.decode_list745encodable.decode_multiset746encodable.decode_sigma747encodable.decode_subtype748encodable.decode_sum749encodable.encodable_of_list750encodable.encode_list751encodable.encode_multiset752encodable.encode_sigma753encodable.encode_subtype754encodable.encode_sum755encodable.equiv_range_encode756encodable.fintype_arrow757encodable.fintype_pi758encodable.of_inj759encodable.of_left_injection760encodable.of_left_inverse761encodable.trunc_encodable_of_fintype762ennreal.Icc_mem_nhds763ennreal.ennreal_equiv_nnreal764ennreal.ennreal_equiv_sum765ennreal.nhds_of_ne_top766ennreal.tendsto_at_top767ennreal.tendsto_nhds768eq_of_forall_dist_le769eq_of_forall_edist_le770eq_of_le_of_forall_ge_of_dense771eq_of_le_of_forall_le_of_dense772equiv.Pi_congr_right773equiv.Pi_curry774equiv.Prop_equiv_bool775equiv.add_comm_group776equiv.add_comm_monoid777equiv.add_comm_semigroup778equiv.add_group779equiv.add_left780equiv.add_monoid781equiv.add_right782equiv.add_semigroup783equiv.array_equiv_fin784equiv.arrow_arrow_equiv_prod_arrow785equiv.arrow_congr786equiv.arrow_prod_equiv_prod_arrow787equiv.arrow_punit_equiv_punit788equiv.bool_equiv_punit_sum_punit789equiv.bool_prod_equiv_sum790equiv.bool_prod_nat_equiv_nat791equiv.cast792equiv.comm_group793equiv.comm_monoid794equiv.comm_ring795equiv.comm_semigroup796equiv.comm_semiring797equiv.conj798equiv.d_array_equiv_fin799equiv.decidable_eq800equiv.decidable_eq_of_equiv801equiv.discrete_field802equiv.division_ring803equiv.domain804equiv.empty_arrow_equiv_punit805equiv.empty_equiv_pempty806equiv.empty_of_not_nonempty807equiv.empty_prod808equiv.empty_sum809equiv.equiv_congr810equiv.equiv_empty811equiv.equiv_pempty812equiv.false_arrow_equiv_punit813equiv.false_equiv_empty814equiv.false_equiv_pempty815equiv.field816equiv.fin_equiv_subtype817equiv.functor818equiv.group819equiv.has_add820equiv.has_inv821equiv.has_mul822equiv.has_neg823equiv.has_one824equiv.has_zero825equiv.inhabited_of_equiv826equiv.int_equiv_nat827equiv.int_equiv_nat_sum_nat828equiv.integral_domain829equiv.inv830equiv.is_lawful_traversable831equiv.is_lawful_traversable'832equiv.list_equiv_of_equiv833equiv.list_equiv_self_of_equiv_nat834equiv.list_nat_equiv_nat835equiv.map836equiv.monoid837equiv.mul_left838equiv.mul_right839equiv.nat_equiv_nat_sum_punit840equiv.nat_prod_nat_equiv_nat841equiv.nat_sum_nat_equiv_nat842equiv.nat_sum_punit_equiv_nat843equiv.neg844equiv.nonzero_comm_ring845equiv.of_bijective846equiv.option_equiv_sum_punit847equiv.pempty_arrow_equiv_punit848equiv.pempty_equiv_pempty849equiv.pempty_of_not_nonempty850equiv.pempty_prod851equiv.pempty_sum852equiv.perm.card_support_swap853equiv.perm.cycle_factors854equiv.perm.cycle_factors_aux855equiv.perm.cycle_of856equiv.perm.disjoint857equiv.perm.eq_swap_of_is_cycle_of_apply_apply_eq_self858equiv.perm.is_cycle859equiv.perm.is_cycle_swap860equiv.perm.is_cycle_swap_mul_aux₁861equiv.perm.is_swap862equiv.perm.of_subtype863equiv.perm.same_cycle864equiv.perm.sign_aux865equiv.perm.sign_aux2866equiv.perm.sign_aux3867equiv.perm.sign_bij_aux868equiv.perm.sign_cycle869equiv.perm.subtype_perm870equiv.perm.support871equiv.perm.support_swap872equiv.perm.swap_factors_aux873equiv.perm.trunc_swap_factors874equiv.perm_congr875equiv.pi_equiv_subtype_sigma876equiv.plift877equiv.pnat_equiv_nat878equiv.prod_assoc879equiv.prod_comm880equiv.prod_congr881equiv.prod_empty882equiv.prod_equiv_of_equiv_nat883equiv.prod_pempty884equiv.prod_punit885equiv.prod_sum_distrib886equiv.prop_equiv_punit887equiv.psigma_equiv_sigma888equiv.psum_equiv_sum889equiv.punit_arrow_equiv890equiv.punit_equiv_punit891equiv.punit_prod892equiv.refl893equiv.ring894equiv.semigroup895equiv.semiring896equiv.set.congr897equiv.set.empty898equiv.set.image899equiv.set.image_of_inj_on900equiv.set.insert901equiv.set.of_eq902equiv.set.pempty903equiv.set.prod904equiv.set.range905equiv.set.sep906equiv.set.singleton907equiv.set.sum_compl908equiv.set.union909equiv.set.union'910equiv.set.union_sum_inter911equiv.set.univ912equiv.set_congr913equiv.sigma_congr_left914equiv.sigma_congr_right915equiv.sigma_equiv_prod916equiv.sigma_equiv_prod_of_equiv917equiv.sigma_preimage_equiv918equiv.sigma_prod_distrib919equiv.sigma_subtype_preimage_equiv920equiv.sigma_subtype_preimage_equiv_subtype921equiv.subtype_congr922equiv.subtype_congr_prop923equiv.subtype_congr_right924equiv.subtype_equiv_of_subtype'925equiv.subtype_pi_equiv_pi926equiv.subtype_quotient_equiv_quotient_subtype927equiv.subtype_subtype_equiv_subtype_exists928equiv.subtype_subtype_equiv_subtype_inter929equiv.sum_arrow_equiv_prod_arrow930equiv.sum_assoc931equiv.sum_comm932equiv.sum_congr933equiv.sum_empty934equiv.sum_equiv_sigma_bool935equiv.sum_pempty936equiv.sum_prod_distrib937equiv.swap_core938equiv.symm939equiv.to_embedding940equiv.to_iso941equiv.to_pequiv942equiv.trans943equiv.traversable944equiv.traverse945equiv.true_equiv_punit946equiv.ulift947equiv.unique_congr948equiv.unique_of_equiv949equiv.units_equiv_ne_zero950equiv.vector_equiv_array951equiv.vector_equiv_fin952equiv.zero_ne_one_class953equiv_of_dim_eq_dim954equiv_of_unique_of_unique955equiv_punit_of_unique956equiv_type_constr957erased.bind958erased.choice959erased.equiv960erased.join961erased.mk962erased.out963erased.out_type964euclidean_domain965euclidean_domain.gcd966euclidean_domain.lcm967euclidean_domain.xgcd_aux968except_t.call_cc969except_t.mk_label970except_t.pass_aux971exists.classical_rec_on972exists_eq_elim973exists_forall_ge_and974exists_gpow_eq_one975exists_mem_ne_zero_of_dim_pos976exists_pow_eq_one977expr.flip_eq978expr.flip_iff979expr.get_pis980ext_param981ext_param_type982field.closure983field.direct_limit.discrete_field984field.direct_limit.field985field.direct_limit.inv986filter987filter.Liminf988filter.Liminf_le_Liminf989filter.Liminf_le_Liminf_of_le990filter.Liminf_le_Limsup991filter.Liminf_le_of_le992filter.Limsup993filter.filter_product.abs_def994filter.filter_product.of_seq995filter.filter_product.of_seq_one996filter.filter_product.of_seq_zero997filter.gi_generate998filter.hyperfilter999filter.infi_ne_bot_iff_of_directed1000filter.infi_ne_bot_iff_of_directed'1001filter.infi_ne_bot_of_directed1002filter.infi_ne_bot_of_directed'1003filter.infi_sets_eq1004filter.is_bounded_default1005filter.is_bounded_ge_of_bot1006filter.is_bounded_under1007filter.is_bounded_under_inf1008filter.is_cobounded_ge_of_top1009filter.is_cobounded_under1010filter.is_trans_ge1011filter.le_Liminf_of_le1012filter.lift_infi'1013filter.liminf1014filter.liminf_eq_supr_infi_of_nat1015filter.liminf_le_liminf1016filter.limsup1017filter.limsup_eq_infi_supr_of_nat1018filter.map_at_top_eq_of_gc1019filter.map_binfi_eq1020filter.map_div_at_top_eq_nat1021filter.map_infi_eq1022filter.mem_at_top_sets1023filter.mem_infi1024filter.mk_of_closure1025filter.monad1026filter.pcomap'1027filter.pmap1028filter.pointwise_add1029filter.pointwise_add_add_monoid1030filter.pointwise_mul1031filter.pointwise_mul_monoid1032filter.pointwise_one1033filter.pointwise_zero1034filter.ptendsto1035filter.ptendsto'1036filter.rcomap1037filter.rcomap'1038filter.realizer.of_eq1039filter.rmap1040filter.rtendsto1041filter.rtendsto'1042filter.tendsto_at_top'1043filter.tendsto_at_top_at_bot1044filter.tendsto_at_top_principal1045filter.ultrafilter.bind1046filter.ultrafilter.map1047filter.ultrafilter.pure1048fin.add_nat_val1049fin.cases1050fin.clamp1051fin.succ_rec1052fin.succ_rec_on1053fin2.cases'10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omm1282homeomorph.prod_congr1283homeomorph.refl1284homeomorph.sigma_prod_distrib1285homeomorph.symm1286homeomorph.trans1287hyperreal.epsilon_lt_pos1288hyperreal.gt_of_neg_of_infinitesimal1289hyperreal.infinite_neg_mul_of_not_infinitesimal_pos_infinite_neg1290hyperreal.infinite_pos_def1291hyperreal.infinite_pos_iff_infinite_and_pos1292hyperreal.infinite_pos_iff_infinite_of_nonneg1293hyperreal.infinite_pos_iff_infinite_of_pos1294hyperreal.infinite_pos_iff_infinitesimal_inv_pos1295hyperreal.infinite_pos_mul_of_infinite_pos_not_infinitesimal_pos1296hyperreal.infinitesimal_def1297hyperreal.infinitesimal_pos_iff_infinite_pos_inv1298hyperreal.is_st_iff_abs_sub_lt_delta1299hyperreal.lt_neg_of_pos_of_infinitesimal1300hyperreal.lt_of_pos_of_infinitesimal1301hyperreal.lt_of_tendsto_zero_of_pos1302hyperreal.neg_lt_of_tendsto_zero_of_pos1303hyperreal.pos_of_infinite_pos1304id.mk1305ideal1306ideal.closure1307ideal.is_coprime1308ideal.is_maximal1309ideal.is_prime1310ideal.is_principal1311ideal.is_principal.generator1312ideal.le_order_embedding_of_surjective1313ideal.lt_order_embedding_of_surjective1314ideal.map1315ideal.quotient1316ideal.quotient.lift1317ideal.quotient.map_mk1318ideal.quotient.mk1319ideal.quotient.nonzero_comm_ring1320ideal.quotient_inf_to_pi_quotient1321ideal.radical_pow1322ideal.span1323inducing1324infi_eq_elim1325infi_real_pos_eq_infi_nnreal_pos1326infinite1327infinite.nat_embedding1328initial_seg.le_add1329initial_seg.le_lt1330initial_seg.lt_or_eq1331int.bit_cases_on1332int.div_eq_div_of_mul_eq_mul1333int.eq_mul_div_of_mul_eq_mul_of_dvd_left1334int.even1335int.induction_on'1336int.modeq1337int.of_snum1338int.range1339int.shift21340int.to_nat'1341integral_closure1342irrational1343is_absolute_value.mem_uniformity1344is_add_group_hom.ker1345is_add_group_hom.to_linear_map1346is_add_subgroup.add_center1347is_add_subgroup.add_normalizer1348is_add_subgroup.group_equiv_quotient_times_subgroup1349is_add_subgroup.left_coset_equiv_subgroup1350is_add_subgroup.mem_trivial1351is_add_subgroup.normalizer_is_add_subgroup1352is_add_subgroup.trivial1353is_basis.coord_fun1354is_basis.repr1355is_basis.to_dual_flip1356is_cau_of_decreasing_bounded1357is_cau_of_mono_bounded1358is_cau_seq.cauchy₂1359is_cau_seq.cauchy₃1360is_cau_series_of_abv_le_cau1361is_closed_map1362is_comm_applicative1363is_conj1364is_cyclic1365is_cyclic.comm_group1366is_group_hom.ker1367is_integral1368is_lawful_bifunctor1369is_lawful_bitraversable1370is_lawful_monad_cont1371is_lawful_traversable1372is_linear_map1373is_linear_map.mk'1374is_local_ring1375is_local_ring_hom1376is_noetherian1377is_noetherian_iff_well_founded1378is_noetherian_ring1379is_null_measurable1380is_ring_anti_hom1381is_ring_hom.ker1382is_ring_hom.to_module1383is_seq_closed1384is_subfield1385is_subgroup.center1386is_subgroup.group_equiv_quotient_times_subgroup1387is_subgroup.left_coset_equiv_subgroup1388is_subgroup.mem_trivial1389is_subgroup.normalizer1390is_subgroup.normalizer_is_subgroup1391lattice.Inf_eq_bot1392lattice.complete_lattice.copy1393lattice.conditionally_complete_linear_order1394lattice.conditionally_complete_linear_order_bot1395lattice.fixed_points1396lattice.fixed_points.next1397lattice.fixed_points.next_fixed1398lattice.fixed_points.prev1399lattice.fixed_points.prev_fixed1400lattice.infi_eq_bot1401lazy_list.list_equiv_lazy_list1402lazy_list.traverse1403le_of_forall_ge_of_dense1404le_of_forall_le_of_dense1405lean.parser.get_includes1406lean.parser.get_namespace1407lean.parser.get_variables1408lebesgue_number_lemma_of_metric1409lebesgue_number_lemma_of_metric_sUnion1410left_add_coset1411left_add_coset_equiv1412left_coset1413left_coset_equiv1414lex1415linarith.add_exprs1416linarith.add_exprs_aux1417linarith.add_neg_eq_pfs1418linarith.cast_expr1419linarith.comp.add1420linarith.comp.coeff_of1421linarith.comp.is_contr1422linarith.comp.lt1423linarith.comp.scale1424linarith.comp_source1425linarith.comp_source.flatten1426linarith.comp_source.to_string1427linarith.elab_arg_list1428linarith.elim_all_vars1429linarith.elim_with_set1430linarith.expr_contains1431linarith.find_cancel_factor1432linarith.find_contr1433linarith.get_comps1434linarith.get_contr_lemma_name1435linarith.get_nat_comps1436linarith.get_rel_sides1437linarith.get_var_list1438linarith.get_vars1439linarith.guard_is_nat_prop1440linarith.guard_is_strict_int_prop1441linarith.ineq1442linarith.ineq.is_lt1443linarith.ineq.max1444linarith.ineq.to_string1445linarith.ineq_const_mul_nm1446linarith.ineq_const_nm1447linarith.ineq_pf_tp1448linarith.is_nat_int_coe1449linarith.is_numeric1450linarith.linarith_config1451linarith.linarith_monad1452linarith.linarith_monad.run1453linarith.list.mfind1454linarith.map_lt1455linarith.map_of_expr_mul_aux1456linarith.mk_cast_eq_and_nonneg_prfs1457linarith.mk_coe_nat_nonneg_prf1458linarith.mk_coe_nat_nonneg_prfs1459linarith.mk_int_pfs_of_nat_pf1460linarith.mk_lt_zero_pf_aux1461linarith.mk_neg_one_lt_zero_pf1462linarith.mk_non_strict_int_pf_of_strict_int_pf1463linarith.mk_prod_prf1464linarith.mk_single_comp_zero_pf1465linarith.monad.elim_var1466linarith.mul_eq1467linarith.mul_expr1468linarith.mul_neg1469linarith.mul_nonpos1470linarith.norm_hyp1471linarith.norm_hyp_aux1472linarith.parse_into_comp_and_expr1473linarith.partition_by_type1474linarith.partition_by_type_aux1475linarith.pcomp1476linarith.pcomp.add1477linarith.pcomp.is_contr1478linarith.pcomp.scale1479linarith.pelim_var1480linarith.preferred_type_of_goal1481linarith.rb_map.find_defeq1482linarith.rearr_comp1483linarith.rem_neg1484linarith.replace_nat_pfs1485linarith.replace_strict_int_pfs1486linarith.term_of_ineq_prf1487linarith.to_comp1488linarith.to_comp_fold1489linarith.update1490linarith.validate1491linear_equiv.to_equiv1492linear_equiv.to_linear_map1493linear_equiv_matrix1494linear_independent_monoid_hom1495linear_map1496linear_map.comp1497linear_map.compl₂1498linear_map.compr₂1499linear_map.flip1500linear_map.id1501linear_map.lcomp1502linear_map.lflip1503linear_map.llcomp1504linear_map.lsmul1505linear_map.map_finsupp_total1506linear_map.mk₂1507linear_map.to_bilin1508linear_order.lift1509list.choose1510list.choose_x1511list.comp_traverse1512list.erase_dupkeys1513list.erasep1514list.extractp1515list.find_indexes_aux1516list.forall₂1517list.func.add1518list.func.equiv1519list.func.get1520list.func.get_neg1521list.func.length_neg1522list.func.neg1523list.func.pointwise1524list.func.set1525list.func.sub1526list.head'1527list.insert_nth1528list.insert_nth_remove_nth_of_ge1529list.kextract1530list.kreplace1531list.lex1532list.map_head1533list.map_last1534list.map_with_index1535list.map_with_index_core1536list.mem_ext1537list.mmap_accuml1538list.mmap_accumr1539list.mpartition1540list.nodupkeys1541list.of_fn1542list.of_fn_aux1543list.of_fn_nth_val1544list.pairwise_gt_iota1545list.partition_map1546list.permutations_aux1547list.permutations_aux.rec1548list.permutations_aux21549list.reduce_option1550list.reverse_rec_on1551list.revzip1552list.scanr_aux1553list.split_on_p_aux1554list.sublists'_aux1555list.sublists_aux1556list.sublists_aux₁1557list.sublists_len1558list.sublists_len_aux1559list.take'1560list.tfae1561list.to_alist1562list.to_finmap1563list.transpose_aux1564list.traverse1565lists1566lists.atom1567lists.induction_mut1568lists.is_list1569lists.mem1570lists.of'1571lists.of_list1572lists.to_list1573lists'1574lists'.cons1575lists'.of_list1576lists'.to_list1577loc.to_string1578loc.to_string_aux1579local_of_is_local_ring1580local_of_nonunits_ideal1581local_of_unique_max_ideal1582local_of_unit_or_unit_one_sub1583local_ring1584local_ring.nonunits_ideal1585local_ring.residue_field1586local_ring.residue_field.map1587localization.at_prime1588localization.away1589localization.away.inv_self1590localization.away.lift1591localization.away_to_away_right1592localization.equiv_of_equiv1593localization.fraction_ring.eq_zero_of_ne_zero_of_mul_eq_zero1594localization.fraction_ring.equiv_of_equiv1595localization.fraction_ring.inv_aux1596localization.fraction_ring.map1597localization.le_order_embedding1598localization.lift1599localization.lift'1600localization.map1601localization.mk1602localization.non_zero_divisors1603localization.r1604localized_attr1605localized_cmd1606locally_finite.realizer1607magma.free_semigroup1608magma.free_semigroup.lift1609magma.free_semigroup.map1610magma.free_semigroup.of1611magma.free_semigroup.r1612manifold_core.local_homeomorph1613manifold_core.to_manifold1614matrix1615matrix.col1616matrix.diagonal1617matrix.minor1618matrix.mul1619matrix.mul_vec1620matrix.row1621matrix.sub_down1622matrix.sub_down_left1623matrix.sub_down_right1624matrix.sub_left1625matrix.sub_right1626matrix.sub_up1627matrix.sub_up_left1628matrix.sub_up_right1629matrix.transpose1630matrix.vec_mul1631matrix.vec_mul_vec1632max_le_add_of_nonneg1633measurable_equiv.cast1634measurable_equiv.prod_comm1635measurable_equiv.prod_congr1636measurable_equiv.prod_sum_distrib1637measurable_equiv.refl1638measurable_equiv.set.image1639measurable_equiv.set.prod1640measurable_equiv.set.range1641measurable_equiv.set.range_inl1642measurable_equiv.set.range_inr1643measurable_equiv.set.singleton1644measurable_equiv.set.univ1645measurable_equiv.sum_congr1646measurable_equiv.sum_prod_distrib1647measurable_equiv.sum_prod_sum1648measurable_equiv.symm1649measurable_equiv.trans1650measurable_space1651measurable_space.dynkin_system.generate1652measurable_space.dynkin_system.of_measurable_space1653measurable_space.dynkin_system.restrict_on1654measurable_space.dynkin_system.to_measurable_space1655measurable_space.gi_generate_from1656measurable_space.mk_of_closure1657measure_theory.ae_eq_fun.add1658measure_theory.ae_eq_fun.neg1659measure_theory.ae_eq_fun.smul1660measure_theory.lintegral_eq_nnreal1661measure_theory.measure1662measure_theory.measure.integral1663measure_theory.measure.is_complete1664measure_theory.measure.map1665measure_theory.measure.of_measurable1666measure_theory.measure.with_density1667measure_theory.measure'1668measure_theory.outer_measure1669measure_theory.outer_measure.Inf_gen1670measure_theory.outer_measure.map1671measure_theory.outer_measure.sum1672measure_theory.outer_measure.to_measure1673measure_theory.outer_measure.trim1674measure_theory.simple_func.eapprox1675measure_theory.simple_func.fin_vol_supp1676measure_theory.simple_func.ite1677measure_theory.volume1678mem_uniformity_edist1679metric.Hausdorff_dist_le_of_inf_dist1680metric.cauchy_iff1681metric.cauchy_seq_iff1682metric.cauchy_seq_iff'1683metric.completion.mem_uniformity_dist1684metric.completion.uniformity_dist1685metric.completion.uniformity_dist'1686metric.continuous_iff1687metric.continuous_iff'1688metric.diam_ball1689metric.diam_closed_ball1690metric.diam_le_of_forall_dist_le1691metric.exists_ball_subset_ball1692metric.exists_delta_of_continuous1693metric.glue_premetric1694metric.glue_space1695metric.is_open_iff1696metric.mem_ball_self1697metric.mem_closed_ball_self1698metric.mem_closure_iff'1699metric.mem_closure_range_iff1700metric.mem_nhds_iff1701metric.mem_of_closed'1702metric.mem_uniformity_dist1703metric.mem_uniformity_edist1704metric.nhds_eq1705metric.pos_of_mem_ball1706metric.second_countable_of_almost_dense_set1707metric.second_countable_of_countable_discretization1708metric.sum.dist1709metric.tendsto_at_top1710metric.tendsto_nhds1711metric.to_glue_l1712metric.to_glue_r1713metric.totally_bounded_iff1714metric.totally_bounded_of_finite_discretization1715metric.uniform_continuous_iff1716metric.uniform_embedding_iff1717metric.uniform_embedding_iff'1718metric.uniformity_dist1719metric.uniformity_dist'1720metric.uniformity_edist'1721metric_space.induced1722metric_space.replace_uniformity1723min_le_add_of_nonneg_left1724min_le_add_of_nonneg_right1725module.End1726module.core1727module.direct_limit1728module.direct_limit.totalize1729module.of_core1730monad_cont1731monad_cont.goto1732monad_cont.label1733monoid.foldl.get1734monoid.foldl.mk1735monoid.foldl.of_free_monoid1736monoid.foldr1737monoid.foldr.get1738monoid.foldr.mk1739monoid.foldr.of_free_monoid1740monoid.mfoldl1741monoid.mfoldl.get1742monoid.mfoldl.mk1743monoid.mfoldl.of_free_monoid1744monoid.mfoldr1745monoid.mfoldr.get1746monoid.mfoldr.mk1747monoid.mfoldr.of_free_monoid1748monoid_hom.one1749monotonicity1750mtry1751mul_action.comp_hom1752mul_action.fixed_points1753mul_action.mul_left_cosets1754mul_action.orbit1755mul_action.orbit_equiv_quotient_stabilizer1756mul_action.orbit_rel1757mul_action.stabilizer1758mul_action.to_perm1759mul_equiv.to_equiv1760mul_mono_nonpos1761multiplicative1762multiplicity.finite1763multiplicity.finite_mul_aux1764multiset.choose1765multiset.choose_x1766multiset.decidable_exists_multiset1767multiset.decidable_forall_multiset1768multiset.le_inf1769multiset.length_ndinsert_of_mem1770multiset.length_ndinsert_of_not_mem1771multiset.pi.cons1772multiset.pi.empty1773multiset.powerset1774multiset.powerset_aux1775multiset.powerset_aux'1776multiset.powerset_len1777multiset.powerset_len_aux1778multiset.rec_on1779multiset.sections1780multiset.strong_induction_on1781multiset.subsingleton_equiv1782multiset.sum1783multiset.sup_le1784multiset.to_finsupp1785multiset.traverse1786mv_polynomial.R1787mv_polynomial.evalᵢ1788mv_polynomial.evalₗ1789mv_polynomial.indicator1790mv_polynomial.restrict_degree1791mv_polynomial.restrict_total_degree1792mv_power_series.inv1793mv_power_series.trunc_fun1794mzip_with1795mzip_with'1796name.last_string1797nat.cases1798nat.dist_eq_sub_of_ge1799nat.elim1800nat.even1801nat.galois_connection_mul_div1802nat.min_fac_aux1803nat.modeq.chinese_remainder1804nat.partrec1805nat.partrec.code1806nat.partrec.code.const1807nat.partrec.code.curry1808nat.partrec.code.encode_code1809nat.partrec.code.eval1810nat.partrec.code.evaln1811nat.partrec.code.id1812nat.partrec.code.of_nat_code1813nat.partrec'.vec1814nat.primrec'.vec1815nat.rfind1816nat.rfind_opt1817nat.rfind_x1818nat.sqrt_aux1819nat.subtype.denumerable1820nat.subtype.of_nat1821nat.subtype.succ1822nat.to_pexpr1823nat.totient1824nat.unpaired1825nat.xgcd_aux1826nnreal.le_of_forall_epsilon_le1827nnreal.le_of_real_iff_coe_le1828nnreal.of_real1829nnreal.of_real_lt_iff_lt_coe1830nonempty_interior_of_Union_of_closed1831nonneg_comm_group.to_decidable_linear_ordered_comm_group1832nonneg_ring.to_linear_nonneg_ring1833nonunits1834norm_cast.derive1835norm_num.derive1836norm_num.derive11837norm_num.eval_div_ext1838norm_num.eval_ineq1839norm_num.eval_pow1840norm_num.eval_prime1841norm_num.min_fac_helper1842norm_num.prove_lt1843norm_num.prove_min_fac1844norm_num.prove_non_prime1845norm_num.prove_pos1846normal_add_subgroup1847normal_of_eq_add_cosets1848normal_of_eq_cosets1849normal_subgroup1850normalize1851normed_group.tendsto_nhds_zero1852not.elim1853null_measurable1854num.add1855num.bit1856num.bit01857num.bit11858num.cmp1859num.cmp_to_nat1860num.div1861num.div21862num.gcd1863num.gcd_aux1864num.land1865num.ldiff1866num.lor1867num.lxor1868num.mod1869num.mul1870num.nat_size1871num.of_nat'1872num.of_znum1873num.of_znum'1874num.one_bits1875num.ppred1876num.pred1877num.psub1878num.shiftl1879num.shiftr1880num.size1881num.sub1882num.sub'1883num.succ1884num.succ'1885num.test_bit1886num.to_znum1887num.to_znum_neg1888num.transfer1889num.transfer_rw1890nzsnum.bit01891nzsnum.bit11892nzsnum.drec'1893nzsnum.head1894nzsnum.not1895nzsnum.sign1896nzsnum.tail1897obviously.attr1898old_conv1899old_conv.apply1900old_conv.apply'1901old_conv.apply_lemmas1902old_conv.apply_lemmas_core1903old_conv.apply_propext_lemmas1904old_conv.apply_propext_lemmas_core1905old_conv.apply_propext_simp_set1906old_conv.apply_simp_set1907old_conv.applyc1908old_conv.bind1909old_conv.bottom_up1910old_conv.change1911old_conv.congr1912old_conv.congr_arg1913old_conv.congr_binder1914old_conv.congr_core1915old_conv.congr_fun1916old_conv.congr_rule1917old_conv.conversion1918old_conv.current_relation1919old_conv.dsimp1920old_conv.execute1921old_conv.fail1922old_conv.failed1923old_conv.find1924old_conv.find_pattern1925old_conv.findp1926old_conv.first1927old_conv.funext1928old_conv.funext'1929old_conv.head_beta1930old_conv.interactive.change1931old_conv.interactive.dsimp1932old_conv.interactive.find1933old_conv.interactive.itactic1934old_conv.interactive.trace_state1935old_conv.interactive.whnf1936old_conv.istep1937old_conv.lhs1938old_conv.lift_tactic1939old_conv.map1940old_conv.match_expr1941old_conv.match_pattern1942old_conv.mk_match_expr1943old_conv.orelse1944old_conv.propext'1945old_conv.pure1946old_conv.repeat1947old_conv.save_info1948old_conv.seq1949old_conv.skip1950old_conv.step1951old_conv.to_tactic1952old_conv.top_down1953old_conv.trace1954old_conv.trace_lhs1955old_conv.whnf1956old_conv_result1957omega.abort1958omega.add_ee1959omega.cancel1960omega.clause1961omega.clause.append1962omega.clause.holds1963omega.clause.sat1964omega.clause.unsat1965omega.clauses.sat1966omega.clauses.unsat1967omega.clear_unused_hyp1968omega.clear_unused_hyps1969omega.coeffs.val1970omega.coeffs.val_between1971omega.coeffs.val_except1972omega.coeffs_reduce1973omega.ee1974omega.ee.repr1975omega.ee_commit1976omega.ee_state1977omega.elim_eq1978omega.elim_eqs1979omega.elim_var1980omega.elim_var_aux1981omega.eq_elim1982omega.eqelim1983omega.factor1984omega.find_ees1985omega.find_min_coeff1986omega.find_min_coeff_core1987omega.find_neg_const1988omega.find_scalars1989omega.find_scalars_core1990omega.form_domain1991omega.form_wf1992omega.gcd1993omega.get_ees1994omega.get_eqs1995omega.get_gcd1996omega.get_les1997omega.goal_domain1998omega.goal_domain_aux1999omega.head_eq2000omega.int.canonize2001omega.int.desugar2002omega.int.dnf2003omega.int.dnf_core2004omega.int.form2005omega.int.form.equiv2006omega.int.form.fresh_index2007omega.int.form.holds2008omega.int.form.implies2009omega.int.form.induce2010omega.int.form.repr2011omega.int.form.sat2012omega.int.form.unsat2013omega.int.form.valid2014omega.int.is_nnf2015omega.int.neg_elim2016omega.int.neg_free2017omega.int.nnf2018omega.int.preterm2019omega.int.preterm.add_one2020omega.int.preterm.fresh_index2021omega.int.preterm.repr2022omega.int.preterm.val2023omega.int.prove_lia2024omega.int.prove_univ_close2025omega.int.push_neg2026omega.int.to_form2027omega.int.to_form_core2028omega.int.to_preterm2029omega.int.univ_close2030omega.is_lia_form2031omega.is_lia_term2032omega.is_lna_form2033omega.is_lna_term2034omega.lin_comb2035omega.nat.bools.or2036omega.nat.canonize2037omega.nat.desugar2038omega.nat.dnf2039omega.nat.dnf_core2040omega.nat.form2041omega.nat.form.equiv2042omega.nat.form.fresh_index2043omega.nat.form.holds2044omega.nat.form.implies2045omega.nat.form.induce2046omega.nat.form.neg_free2047omega.nat.form.repr2048omega.nat.form.sat2049omega.nat.form.sub_free2050omega.nat.form.sub_subst2051omega.nat.form.sub_terms2052omega.nat.form.unsat2053omega.nat.form.valid2054omega.nat.is_diff2055omega.nat.is_nnf2056omega.nat.neg_elim2057omega.nat.neg_elim_core2058omega.nat.nnf2059omega.nat.nonneg_consts2060omega.nat.nonneg_consts_core2061omega.nat.nonnegate2062omega.nat.preterm2063omega.nat.preterm.add_one2064omega.nat.preterm.fresh_index2065omega.nat.preterm.induce2066omega.nat.preterm.prove_sub_free2067omega.nat.preterm.repr2068omega.nat.preterm.sub_free2069omega.nat.preterm.sub_subst2070omega.nat.preterm.sub_terms2071omega.nat.preterm.val2072omega.nat.prove_lna2073omega.nat.prove_neg_free2074omega.nat.prove_sub_free2075omega.nat.prove_univ_close2076omega.nat.prove_unsat_neg_free2077omega.nat.prove_unsat_sub_free2078omega.nat.push_neg2079omega.nat.sub_elim2080omega.nat.sub_elim_core2081omega.nat.sub_fresh_index2082omega.nat.term.vars2083omega.nat.term.vars_core2084omega.nat.terms.vars2085omega.nat.to_form2086omega.nat.to_form_core2087omega.nat.to_preterm2088omega.nat.univ_close2089omega.preprocess2090omega.prove_forall_mem_eq_zero2091omega.prove_neg2092omega.prove_unsat2093omega.prove_unsat_ef2094omega.prove_unsat_lin_comb2095omega.prove_unsats2096omega.rev_lia2097omega.rev_lna2098omega.revert_cond2099omega.revert_cond_all2100omega.rhs2101omega.run2102omega.select_domain2103omega.set_ees2104omega.set_eqs2105omega.set_les2106omega.sgm2107omega.subst2108omega.sym_sym2109omega.symdiv2110omega.symmod2111omega.term2112omega.term.add2113omega.term.div2114omega.term.fresh_index2115omega.term.mul2116omega.term.neg2117omega.term.sub2118omega.term.to_string2119omega.term.val2120omega.term_domain2121omega.terms.fresh_index2122omega.trisect2123omega.type_domain2124omega.unsat_lin_comb2125omega.update2126omega.update_zero2127omega_int2128omega_nat2129onote.oadd_lt_oadd_12130onote.oadd_lt_oadd_22131onote.oadd_lt_oadd_32132onote.power_aux2133onote.to_string_aux12134open_add_subgroup2135open_add_subgroup.sum2136open_locale_cmd2137open_subgroup.prod2138opposite.op2139opposite.op_induction2140opposite.unop2141opt_minus2142option.cases_on'2143option.comp_traverse2144option.lift_or_get2145option.rel2146option.to_list2147option.traverse2148option_t.call_cc2149option_t.mk_label2150option_t.pass_aux2151order_embedding.collapse_F2152order_embedding.lt_embedding_of_le_embedding2153order_embedding.nat_gt2154order_embedding.nat_lt2155order_embedding.refl2156order_embedding.trans2157order_embedding.well_founded_iff_no_descending_seq2158order_iso.of_surjective2159order_iso.prod_lex_congr2160order_iso.refl2161order_iso.sum_lex_congr2162order_iso.symm2163order_iso.to_order_embedding2164order_iso.trans2165order_of_pos2166ordering.compares.eq_gt2167ordinal.CNF_rec2168ordinal.CNF_sorted2169ordinal.add_absorp_iff2170ordinal.initial_seg_out2171ordinal.lift.initial_seg2172ordinal.lift.principal_seg2173ordinal.limit_rec_on2174ordinal.order_iso_out2175ordinal.principal_seg_out2176ordinal.typein.principal_seg2177ordinal.typein_iso2178ordinal.unbounded_range_of_sup_ge2179pSet.definable.eq_mk2180pSet.definable.resp2181pSet.resp.equiv2182pSet.resp.eval_aux2183pSet.resp.f2184pSet.subset2185padic.complete'2186padic.exi_rat_seq_conv2187padic.lim_seq2188padic.padic_norm_e_lim_le2189padic.rat_dense2190padic.rat_dense'2191padic_norm.neg2192padic_norm_e.defn2193padic_norm_e.nonneg2194padic_norm_e.rat_norm2195padic_norm_z2196padic_seq2197padic_seq.norm_nonneg2198padic_seq.stationary_point_spec2199partial_order.lift2200partrec2201partrec₂2202pell.asq_pos2203pell.az2204pell.d_pos2205pell.dnsq2206pell.dvd_of_ysq_dvd2207pell.dz_val2208pell.eq_of_xn_modeq2209pell.eq_of_xn_modeq'2210pell.eq_of_xn_modeq_le2211pell.eq_of_xn_modeq_lem12212pell.eq_of_xn_modeq_lem22213pell.eq_of_xn_modeq_lem32214pell.eq_pell2215pell.eq_pell_lem2216pell.eq_pell_zd2217pell.eq_pow_of_pell2218pell.eq_pow_of_pell_lem2219pell.is_pell2220pell.is_pell_conj2221pell.is_pell_mul2222pell.is_pell_nat2223pell.is_pell_norm2224pell.is_pell_one2225pell.is_pell_pell_zd2226pell.matiyasevic2227pell.modeq_of_xn_modeq2228pell.n_lt_a_pow2229pell.n_lt_xn2230pell.pell_eq2231pell.pell_eqz2232pell.pell_val2233pell.pell_zd2234pell.pell_zd_add2235pell.pell_zd_im2236pell.pell_zd_re2237pell.pell_zd_sub2238pell.pell_zd_succ2239pell.pell_zd_succ_succ2240pell.x_increasing2241pell.x_pos2242pell.x_sub_y_dvd_pow2243pell.x_sub_y_dvd_pow_lem2244pell.xn2245pell.xn_add2246pell.xn_ge_a_pow2247pell.xn_modeq_x2n_add2248pell.xn_modeq_x2n_add_lem2249pell.xn_modeq_x2n_sub2250pell.xn_modeq_x2n_sub_lem2251pell.xn_modeq_x4n_add2252pell.xn_modeq_x4n_sub2253pell.xn_one2254pell.xn_succ2255pell.xn_succ_succ2256pell.xn_zero2257pell.xy_coprime2258pell.xy_modeq_of_modeq2259pell.xy_modeq_yn2260pell.xy_succ_succ2261pell.xz2262pell.xz_sub2263pell.xz_succ2264pell.xz_succ_succ2265pell.y_dvd_iff2266pell.y_increasing2267pell.y_mul_dvd2268pell.yn2269pell.yn_add2270pell.yn_ge_n2271pell.yn_modeq_a_sub_one2272pell.yn_modeq_two2273pell.yn_one2274pell.yn_succ2275pell.yn_succ_succ2276pell.yn_zero2277pell.ysq_dvd_yy2278pell.yz2279pell.yz_sub2280pell.yz_succ2281pell.yz_succ_succ2282pempty.elim2283pequiv.matrix_mul_apply2284pequiv.mul_matrix_apply2285pequiv.of_set2286pequiv.refl2287pequiv.single2288pequiv.single_mul_single_right2289pequiv.symm2290pequiv.to_matrix2291pequiv.trans2292pequiv.trans_single_of_eq_none2293perfect_closure.UMP2294perfect_closure.eq_iff'2295perfect_closure.frobenius_equiv2296perfect_closure.has_inv2297perfect_closure.of2298perfect_closure.r2299perms_of_finset2300perms_of_list2301pexpr.get_uninst_pis2302pfun.core2303pfun.equiv_subtype2304pfun.fix2305pfun.fix_induction2306pfun.graph'2307pfun.image2308pfun.preimage2309pfun.res2310pgame.add_lt_add2311pgame.numeric.lt_move_right2312pgame.numeric.move_left_lt2313pi.lex2314pi.linear_independent_std_basis2315pilex2316pmf.bernoulli2317pmf.bind2318pmf.map2319pmf.of_fintype2320pmf.of_multiset2321pmf.pure2322pmf.seq2323pmf.support2324pnat.div2325pnat.div_exact2326pnat.gcd2327pnat.gcd_a'2328pnat.gcd_b'2329pnat.gcd_d2330pnat.gcd_w2331pnat.gcd_x2332pnat.gcd_y2333pnat.gcd_z2334pnat.lcm2335pnat.mod2336pnat.mod_div2337pnat.prime2338pnat.xgcd2339pnat.xgcd_type.a2340pnat.xgcd_type.b2341pnat.xgcd_type.finish2342pnat.xgcd_type.flip2343pnat.xgcd_type.is_reduced'2344pnat.xgcd_type.is_special'2345pnat.xgcd_type.mk'2346pnat.xgcd_type.q2347pnat.xgcd_type.qp2348pnat.xgcd_type.r2349pnat.xgcd_type.succ₂2350pnat.xgcd_type.v2351pnat.xgcd_type.w2352pnat.xgcd_type.z2353polynomial.binom_expansion2354polynomial.coeff_coe_to_fun2355polynomial.comp2356polynomial.decidable_dvd_monic2357polynomial.degree_eq_iff_nat_degree_eq_of_pos2358polynomial.div2359polynomial.div_mod_by_monic_aux2360polynomial.eval_sub_factor2361polynomial.eval₂_zero2362polynomial.lcoeff2363polynomial.map_injective2364polynomial.mod2365polynomial.monomial2366polynomial.nonzero_comm_ring.of_polynomial_ne2367polynomial.nonzero_comm_semiring.of_polynomial_ne2368polynomial.pow_add_expansion2369polynomial.pow_sub_pow_factor2370polynomial.rec_on_horner2371polynomial.root_multiplicity2372polynomial.splits2373polynomial.tendsto_infinity2374pos_num.add2375pos_num.bit2376pos_num.cmp2377pos_num.cmp_to_nat2378pos_num.div'2379pos_num.divmod2380pos_num.divmod_aux2381pos_num.is_one2382pos_num.land2383pos_num.ldiff2384pos_num.lor2385pos_num.lxor2386pos_num.mod'2387pos_num.mul2388pos_num.nat_size2389pos_num.of_nat2390pos_num.of_nat_succ2391pos_num.of_znum2392pos_num.of_znum'2393pos_num.one_bits2394pos_num.pred2395pos_num.pred'2396pos_num.pred_to_nat2397pos_num.shiftl2398pos_num.shiftr2399pos_num.size2400pos_num.sqrt_aux2401pos_num.sqrt_aux12402pos_num.sub2403pos_num.sub'2404pos_num.succ2405pos_num.test_bit2406pos_num.transfer2407pos_num.transfer_rw2408power_series.inv2409power_series.inv.aux2410power_series.mk2411power_series.order_add_ge2412power_series.order_ge2413power_series.order_ge_nat2414power_series.order_mul_ge2415premetric_space2416preorder.lift2417primcodable.of_equiv2418primcodable.subtype2419prime_multiset.of_nat_multiset2420prime_multiset.of_pnat_list2421prime_multiset.of_pnat_multiset2422prime_multiset.prod2423prime_multiset.to_pnat_multiset2424principal_ideal_domain2425principal_ideal_domain.factors2426principal_seg2427principal_seg.equiv_lt2428principal_seg.lt_equiv2429principal_seg.lt_le2430principal_seg.top_lt_top2431principal_seg.trans2432principal_seg.trans_top2433prod.bitraverse2434prod.lex.decidable2435push_neg.normalize_negations2436push_neg.push_neg_at_goal2437push_neg.push_neg_at_hyp2438push_neg.whnf_reducible2439quot.hrec_on₂2440quotient.choice2441quotient.fin_choice2442quotient.fin_choice_aux2443quotient.hrec_on₂2444quotient.lift_on'2445quotient.lift_on₂'2446quotient.mk'2447quotient.out'2448quotient_add_group.eq_class_eq_left_coset2449quotient_add_group.inhabited2450quotient_add_group.ker_lift2451quotient_add_group.left_rel2452quotient_add_group.lift2453quotient_add_group.map2454quotient_add_group.mk2455quotient_add_group.quotient2456quotient_add_group.quotient_ker_equiv_of_surjective2457quotient_add_group.quotient_ker_equiv_range2458quotient_group.eq_class_eq_left_coset2459quotient_group.fintype2460quotient_group.inhabited2461quotient_group.left_rel2462quotient_group.lift2463quotient_group.map2464quotient_group.mk2465quotient_group.preimage_mk_equiv_subgroup_times_set2466quotient_group.quotient_ker_equiv_of_surjective2467quotient_group.quotient_ker_equiv_range2468rank2469rat.add2470rat.inv2471rat.le2472rat.mul2473rat.neg2474rat.nonneg2475rat.num_denom_cases_on2476rat.num_denom_cases_on'2477rat.repr2478rat.sqrt2479rat_add_continuous_lemma2480rat_inv_continuous_lemma2481rat_mul_continuous_lemma2482re_pred2483reader_t.call_cc2484reader_t.mk_label2485real2486real.Inf2487real.Sup2488real.comm_ring_aux2489real.cos2490real.cosh2491real.exp2492real.le2493real.le_mk_of_forall_le2494real.le_of_forall_epsilon_le2495real.mk2496real.mk_le_of_forall_le2497real.mk_near_of_forall_near2498real.of_near2499real.of_rat2500real.pi_gt_3142501real.pi_gt_sqrt_two_add_series2502real.pi_gt_three2503real.sin2504real.sinh2505real.sqrt2506real.sqrt_aux2507real.sqrt_two_add_series_nonneg2508real.sqrt_two_add_series_zero_nonneg2509real.tan2510real.tanh2511relation.comp2512relation.join2513relation.map2514relator.bi_total2515relator.bi_unique2516relator.left_total2517relator.left_unique2518relator.lift_fun2519relator.right_total2520relator.right_unique2521restate_axiom_cmd2522right_add_coset2523right_coset2524ring.closure2525ring.direct_limit2526ring_anti_equiv.refl2527ring_equiv2528ring_equiv.to_add_equiv2529ring_equiv.to_equiv2530ring_equiv.to_mul_equiv2531ring_hom.to_add_monoid_hom2532ring_hom.to_monoid_hom2533ring_invo.id2534ring_invo.to_ring_anti_equiv2535roption.equiv_option2536roption.get_or_else2537roption.restrict2538saturate_fun2539semiquot.bind2540semiquot.get2541semiquot.is_pure2542semiquot.map2543separated2544seq.bisim_o2545seq.cases_on2546seq.corec.F2547seq.is_bisimulation2548seq.mem2549sequence2550set.Union_eq_sigma_of_disjoint2551set.add_comm_monoid2552set.bUnion_eq_sigma_of_disjoint2553set.centralizer.add_submonoid2554set.comm_monoid2555set.countable.to_encodable2556set.enumerate2557set.fintype_bUnion2558set.fintype_bind2559set.fintype_insert'2560set.fintype_of_fintype_image2561set.fintype_subset2562set.kern_image2563set.pairwise_disjoint2564set.pi2565set.pointwise_add2566set.pointwise_add_add_monoid2567set.pointwise_add_add_semigroup2568set.pointwise_add_fintype2569set.pointwise_inv2570set.pointwise_mul2571set.pointwise_mul_action2572set.pointwise_mul_comm_semiring2573set.pointwise_mul_fintype2574set.pointwise_mul_monoid2575set.pointwise_mul_semigroup2576set.pointwise_mul_semiring2577set.pointwise_neg2578set.pointwise_one2579set.pointwise_zero2580set.seq2581set.sigma_to_Union2582set_fintype2583setoid.is_partition2584side2585side.other2586side.to_string2587simple_add_group2588simple_group2589snum.add2590snum.bit2591snum.bit02592snum.bit12593snum.bits2594snum.cadd2595snum.czadd2596snum.drec'2597snum.head2598snum.mul2599snum.neg2600snum.not2601snum.pred2602snum.rec'2603snum.sign2604snum.sub2605snum.succ2606snum.tail2607snum.test_bit2608state_t.call_cc2609state_t.mk_label2610strict_order.cof2611string.ltb2612string.map_tokens2613string.over_list2614string.split_on2615subalgebra2616subalgebra.comap2617subalgebra.fg2618subalgebra.to_submodule2619subalgebra.under2620subalgebra.val2621subgroup_units_cyclic2622submodule.annihilator2623submodule.colon2624submodule.fg2625submodule.subtype2626subrel.order_embedding2627subtype.restrict2628succeeds2629sum.bind2630sum.bitraverse2631sum.comp_traverse2632sum.elim2633sum.map2634sum.traverse2635sum.traverse_map2636sum_fin_sum_equiv2637summable_iff_vanishing_norm2638supr_eq_elim2639surreal.add2640swap_right2641sylow.fixed_points_mul_left_cosets_equiv_quotient2642sylow.mk_vector_prod_eq_one2643sylow.rotate_vectors_prod_eq_one2644sylow.vectors_prod_eq_one2645tactic.abel.add_g2646tactic.abel.cache2647tactic.abel.cache.app2648tactic.abel.cache.iapp2649tactic.abel.cache.int_to_expr2650tactic.abel.cache.mk_app2651tactic.abel.cache.mk_term2652tactic.abel.eval2653tactic.abel.eval'2654tactic.abel.eval_add2655tactic.abel.eval_atom2656tactic.abel.eval_neg2657tactic.abel.eval_smul2658tactic.abel.mk_cache2659tactic.abel.normal_expr2660tactic.abel.normal_expr.e2661tactic.abel.normal_expr.pp2662tactic.abel.normal_expr.refl_conv2663tactic.abel.normal_expr.term'2664tactic.abel.normal_expr.to_list2665tactic.abel.normal_expr.to_string2666tactic.abel.normal_expr.zero'2667tactic.abel.normalize2668tactic.abel.normalize_mode2669tactic.abel.smul2670tactic.abel.smulg2671tactic.abel.term2672tactic.abel.termg2673tactic.abstract_if_success2674tactic.add_coinductive_predicate2675tactic.add_coinductive_predicate.coind_pred2676tactic.add_coinductive_predicate.coind_pred.add_theorem2677tactic.add_coinductive_predicate.coind_pred.construct2678tactic.add_coinductive_predicate.coind_pred.corec_functional2679tactic.add_coinductive_predicate.coind_pred.destruct2680tactic.add_coinductive_predicate.coind_pred.func2681tactic.add_coinductive_predicate.coind_pred.func_g2682tactic.add_coinductive_predicate.coind_pred.f₁_l2683tactic.add_coinductive_predicate.coind_pred.f₂_l2684tactic.add_coinductive_predicate.coind_pred.impl_locals2685tactic.add_coinductive_predicate.coind_pred.impl_params2686tactic.add_coinductive_predicate.coind_pred.le2687tactic.add_coinductive_predicate.coind_pred.mono2688tactic.add_coinductive_predicate.coind_pred.pred2689tactic.add_coinductive_predicate.coind_pred.pred_g2690tactic.add_coinductive_predicate.coind_pred.rec'2691tactic.add_coinductive_predicate.coind_pred.u_params2692tactic.add_coinductive_predicate.coind_rule2693tactic.add_edge2694tactic.add_refl2695tactic.add_symm_proof2696tactic.alias.alias_attr2697tactic.alias.alias_cmd2698tactic.alias.alias_direct2699tactic.alias.alias_iff2700tactic.alias.get_alias_target2701tactic.alias.get_lambda_body2702tactic.alias.make_left_right2703tactic.alias.mk_iff_mp_app2704tactic.all_rewrites_using2705tactic.ancestor_attr2706tactic.apply_mod_cast2707tactic.assert_fresh2708tactic.assertv_fresh2709tactic.assoc_refl2710tactic.assoc_refl'2711tactic.assoc_rewrite2712tactic.assoc_rewrite_hyp2713tactic.assoc_rewrite_intl2714tactic.assoc_rewrite_target2715tactic.assoc_root2716tactic.assumption_mod_cast2717tactic.assumption_symm2718tactic.assumption_with2719tactic.calculated_Prop2720tactic.chain2721tactic.chain_core2722tactic.chain_eq_trans2723tactic.coinductive_predicate2724tactic.constr_to_prop2725tactic.contradiction_symm2726tactic.contradiction_with2727tactic.def_replacer_cmd2728tactic.derive_field2729tactic.derive_field_subtype2730tactic.derive_reassoc_proof2731tactic.elide.replace2732tactic.elide.unelide2733tactic.enum_assoc_subexpr2734tactic.enum_assoc_subexpr'2735tactic.exact_mod_cast2736tactic.explode2737tactic.explode.append_dep2738tactic.explode.args2739tactic.explode.core2740tactic.explode.entries2741tactic.explode.entries.add2742tactic.explode.entries.find2743tactic.explode.entries.head2744tactic.explode.entries.size2745tactic.explode.entry2746tactic.explode.format_aux2747tactic.explode.head'2748tactic.explode.may_be_proof2749tactic.explode.pad_right2750tactic.explode.status2751tactic.explode_cmd2752tactic.explode_expr2753tactic.expr_list_to_list_expr2754tactic.ext2755tactic.ext12756tactic.ext_parse2757tactic.ext_patt2758tactic.fill_args2759tactic.fin_cases_at2760tactic.fin_cases_at_aux2761tactic.find_ancestors2762tactic.f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