{"names": ["ADEInequality.admissible_D'", "AEMeasurable.add_measure", "AEMeasurable.coe_ereal_ennreal", "AEMeasurable.comp_aemeasurable", "AEMeasurable.comp_measurable", "AEMeasurable.const_inf", "AEMeasurable.const_smul", "AEMeasurable.div", "AEMeasurable.div_const", "AEMeasurable.ennreal_toReal", "AEMeasurable.exists_measurable_nonneg", "AEMeasurable.map_map_of_aemeasurable", "AEMeasurable.mono_measure", "AEMeasurable.mul", "AEMeasurable.pow", "AEMeasurable.re", "AList.insert_insert_of_ne", "AList.lookup_insert_eq_none", "AList.union_entries", "Abelianization.map_id", "AbsConvexOpenSets.coe_balanced", "AbsoluteValue.IsAdmissible.exists_approx", "AbsoluteValue.map_one_of_isLeftRegular", "Absorbs.eventually", "Absorbs.zero", "AbstractCompletion.continuous_coe", "AccPt.nhds_inter", "Action.hom_ext", "Action.rightUnitor_hom_hom", "AddAction.automorphize_smul_left", "AddAction.orbitZMultiplesEquiv_symm_apply'", "AddAction.stabilizer_vadd_eq_stabilizer_map_conj", "AddChar.coe_mk", "AddChar.ext", "AddChar.map_zero_eq_one", "AddChar.mulShift_zero", "AddChar.val_mem_rootsOfUnity", "AddChar.zmodChar_apply'", "AddCircle.addOrderOf_eq_pos_iff", "AddCircle.equivAddCircle_symm_apply_mk", "AddCircle.equivIccQuot_comp_mk_eq_toIcoMod", "AddCircle.exists_norm_eq_of_isOfFinAddOrder", "AddCircle.gcd_mul_addOrderOf_div_eq", "AddCircle.homeomorphCircle_apply", "AddCircle.liftIco_zero_coe_apply", "AddCircle.liftIco_zero_continuous", "AddCircle.norm_eq'", "AddCircle.volume_of_add_preimage_eq", "AddCommGroup.DirectLimit.of.zero_exact", "AddCommGroup.ModEq.add_left", "AddCommGroup.modEq_iff_eq_mod_zmultiples", "AddCommGroup.modEq_neg", "AddCommGroup.nsmul_modEq_nsmul", "AddCommGrp.biproductIsoPi_inv_comp_\u03c0", "AddCommGrp.kernelIsoKer_hom_comp_subtype", "AddConstMap.coe_mk", "AddConstMapClass.map_const_add", "AddConstMapClass.strictAnti_iff_Icc", "AddEquiv.toIntLinearEquiv_symm", "AddLECancellable.add_tsub_cancel_left", "AddLECancellable.add_tsub_tsub_cancel", "AddLECancellable.eq_tsub_of_add_eq", "AddLECancellable.le_tsub_iff_le_tsub", "AddLECancellable.mul_tsub", "AddMonoid.End.smul_apply", "AddMonoidAlgebra.divOf_add", "AddMonoidAlgebra.gradeBy.isInternal", "AddMonoidAlgebra.lift_apply'", "AddMonoidAlgebra.lift_unique", "AddMonoidAlgebra.mem_grade_iff", "AddMonoidAlgebra.mem_grade_iff'", "AddMonoidAlgebra.ne_zero_of_not_supDegree_le", "AddMonoidAlgebra.of'_divOf", "AddMonoidAlgebra.of'_mem_span", "AddMonoidAlgebra.sum_single_index", "AddMonoidAlgebra.supDegree_prod_le", "AddMonoidAlgebra.toDirectSum_zero", "AddMonoidHom.mulLeft_eq_mulRight_iff_forall_commute", "AddMonoidHom.toNatLinearMap_injective", "AddSubgroup.coe_toZModSubmodule", "AddSubgroup.quotient_norm_eq", "AddSubgroup.toZModSubmodule_toAddSubgroup", "AddSubgroup.zsmul_mem_zmultiples_iff_exists_sub_div", "AddSubmonoid.le_pointwise_smul_iff\u2080", "AddSubmonoid.mem_pointwise_smul_iff_inv_smul_mem\u2080", "AddSubmonoid.mul_le", "AddSubmonoid.mul_le_mul_left", "AddSubmonoid.mul_mem_mul", "AddSubmonoid.pow_eq_closure_pow_set", "AddSubsemigroup.toSubsemigroup_closure", "AddValuation.IsEquiv.map", "AddValuation.IsEquiv.symm", "AddValuation.ext_iff", "Additive.forall", "Additive.isAddHom", "AdicCompletion.eval\u2090_mk", "AdicCompletion.isAdicCauchy_iff", "AdicCompletion.mk_surjective", "AdicCompletion.of_apply", "AdicCompletion.transitionMap_comp_apply", "AdicCompletion.val_smul", "AdicCompletion.val_sum", "AdjoinRoot.Polynomial.quotQuotEquivComm_symm_mk_mk", "AdjoinRoot.coe_algEquivOfEq", "AdjoinRoot.coe_injective", "AdjoinRoot.isAdjoinRootMonic_map_eq_mk", "AdjoinRoot.isIntegral_root'", "AdjoinRoot.lift_root", "AdjoinRoot.minpoly_powerBasis_gen_of_monic", "AdjoinRoot.minpoly_root", "AdjoinRoot.smul_of", "Aesop.BuiltinRules.heq_iff_eq", "Affine.Simplex.centroid_eq_iff", "Affine.Simplex.circumradius_reindex", "Affine.Simplex.direction_mongePlane", "Affine.Triangle.orthocenter_replace_orthocenter_eq_point", "AffineBasis.card_eq_finrank_add_one", "AffineBasis.coord_apply_centroid", "AffineBasis.coord_apply_eq", "AffineBasis.ext", "AffineBasis.interior_convexHull", "AffineBasis.toMatrix_inv_vecMul_toMatrix", "AffineEquiv.apply_lineMap", "AffineEquiv.coe_mul", "AffineEquiv.range_eq", "AffineEquiv.trans_refl", "AffineEquiv.wSameSide_map_iff", "AffineIndependent.affineSpan_image_finset_eq_of_le_of_card_eq_finrank_add_one", "AffineIndependent.not_mem_affineSpan_diff", "AffineIsometry.angle_map", "AffineIsometry.coe_toAffineMap", "AffineIsometry.comp_assoc", "AffineIsometry.image_intrinsicInterior", "AffineIsometry.lipschitz", "AffineIsometryEquiv.coe_vaddConst_symm", "AffineIsometryEquiv.pointReflection_midpoint_left", "AffineIsometryEquiv.vadd_vsub", "AffineMap.add_linear", "AffineMap.coe_neg", "AffineMap.coe_smul", "AffineMap.coe_sub", "AffineMap.comp_assoc", "AffineMap.continuous_of_finiteDimensional", "AffineMap.differentiableOn", "AffineMap.differentiableWithinAt", "AffineMap.ext_linear_iff", "AffineMap.fst_lineMap", "AffineMap.hasDerivAt_lineMap", "AffineMap.homothety_eq_lineMap", "AffineMap.homothety_one", "AffineMap.isOpenMap_linear_iff", "AffineMap.lineMap_continuous", "AffineMap.lineMap_vadd_lineMap", "AffineMap.linearMap_vsub", "AffineMap.linear_surjective_iff", "AffineMap.pi_apply", "AffineMap.snd_linear", "AffineMap.vadd_apply", "AffineSubspace.SOppSide.wOppSide", "AffineSubspace.SSameSide.trans_wOppSide", "AffineSubspace.affineSpan_parallel_iff_vectorSpan_eq_and_eq_empty_iff_eq_empty", "AffineSubspace.bot_coe", "AffineSubspace.bot_ne_top", "AffineSubspace.coe_map", "AffineSubspace.comap_span", "AffineSubspace.direction_sup", "AffineSubspace.eq_bot_or_nonempty", "AffineSubspace.eq_univ_of_subsingleton_span_eq_top", "AffineSubspace.gc_map_comap", "AffineSubspace.isConnected_setOf_sOppSide", "AffineSubspace.map_sup", "AffineSubspace.mem_perpBisector_iff_dist_eq", "AffineSubspace.not_mem_bot", "AffineSubspace.parallel_comm", "AffineSubspace.sOppSide_iff_exists_right", "AffineSubspace.sOppSide_vadd_left_iff", "AffineSubspace.sSameSide_vadd_right_iff", "AffineSubspace.span_univ", "AffineSubspace.subtype\u2090\u1d62_toAffineMap", "AffineSubspace.sup_direction_lt_of_nonempty_of_inter_empty", "AffineSubspace.wOppSide_lineMap_left", "AffineSubspace.wOppSide_self_iff", "AffineSubspace.wSameSide_and_wOppSide_iff", "AkraBazziRecurrence.GrowsPolynomially.neg", "AkraBazziRecurrence.GrowsPolynomially.norm", "AkraBazziRecurrence.eventually_b_le_r", "AkraBazziRecurrence.eventually_one_add_smoothingFn_nonneg", "AkraBazziRecurrence.eventually_one_add_smoothingFn_r_pos", "AkraBazziRecurrence.growsPolynomially_deriv_rpow_p_mul_one_sub_smoothingFn", "AlgEquiv.bijective", "AlgEquiv.coe_coe_symm_apply_coe_apply", "AlgEquiv.coe_ringEquiv", "AlgEquiv.isPurelyInseparable", "AlgEquiv.map_neg", "AlgEquiv.restrictScalars_apply", "AlgEquiv.symm_apply_apply", "AlgEquiv.symm_bijective", "AlgEquiv.toEquiv_eq_coe", "AlgHom.Finite.comp", "AlgHom.FinitePresentation.id", "AlgHom.algebraicIndependent_iff", "AlgHom.coe_coe", "AlgHom.coe_comp", "AlgHom.id_algebraCat_comp", "Algebra.Etale.of_isLocalization_Away", "Algebra.FinitePresentation.ker_fg_of_mvPolynomial", "Algebra.FiniteType.baseChangeAux_surj", "Algebra.FormallySmooth.of_equiv", "Algebra.FormallyUnramified.lift_unique_of_ringHom", "Algebra.FormallyUnramified.localization_map", "Algebra.Generators.Cotangent.val_of", "Algebra.Generators.algebraMap_surjective", "Algebra.IsAlgebraic.algHomEquivAlgHomOfSplits_apply_apply", "Algebra.IsAlgebraic.normalClosure_le_iSup_adjoin", "Algebra.IsAlgebraic.perfectField", "Algebra.PowerBasis.norm_gen_eq_coeff_zero_minpoly", "Algebra.Presentation.span_range_relation_eq_ker_baseChange", "Algebra.TensorProduct.congr_refl", "Algebra.TensorProduct.map_comp", "Algebra.TensorProduct.map_id", "Algebra.TensorProduct.opAlgEquiv_symm_apply", "Algebra.TensorProduct.rid_symm_apply", "Algebra.TensorProduct.rid_toLinearEquiv", "Algebra.adjoin_eq", "Algebra.adjoin_eq_span", "Algebra.adjoin_induction''", "Algebra.adjoin_le", "Algebra.adjoin_le_iff", "Algebra.algebraMap_eq_smul_one", "Algebra.coe_algebraMap_ofSubsemiring", "Algebra.commutes", "Algebra.ext_id", "Algebra.fg_trans", "Algebra.intNorm_zero", "Algebra.isIntegral_norm", "Algebra.isIntegral_sup", "Algebra.leftMulMatrix_mulVec_repr", "Algebra.lmul_algebraMap", "Algebra.mem_sup_left", "Algebra.mem_top", "Algebra.norm_eq_matrix_det", "Algebra.norm_eq_zero_iff", "Algebra.prod_embeddings_eq_finrank_pow", "Algebra.trace_algebraMap_of_basis", "Algebra.trace_comp_trace", "Algebra.trace_eq_matrix_trace", "AlgebraCat.id_apply", "Algebraic.cardinal_mk_le_mul", "Algebraic.cardinal_mk_of_countable_of_charZero", "AlgebraicClosure.spanEval_ne_top", "AlgebraicClosureAux.ofStep_succ", "AlgebraicGeometry.AffineTargetMorphismProperty.isLocalOfOpenCoverImply", "AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_basicOpen", "AlgebraicGeometry.IsAffineOpen.isLocalization_stalk", "AlgebraicGeometry.IsAffineOpen.preimage_of_isIso", "AlgebraicGeometry.IsAffineOpen.\u03b9Opens_basicOpen_preimage", "AlgebraicGeometry.IsOpenImmersion.app_eq_invApp_app_of_comp_eq", "AlgebraicGeometry.IsOpenImmersion.app_eq_invApp_app_of_comp_eq_aux", "AlgebraicGeometry.IsOpenImmersion.isoOfRangeEq_inv_fac", "AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.inv_naturality", "AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.lift_range", "AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.of_stalk_iso", "AlgebraicGeometry.LocallyRingedSpace.to\u0393SpecSheafedSpace_app_eq", "AlgebraicGeometry.MorphismProperty.topologically_iso_le", "AlgebraicGeometry.PresheafedSpace.GlueData.\u03b9_image_preimage_eq", "AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.lift_uniq", "AlgebraicGeometry.PresheafedSpace.colimit_presheaf", "AlgebraicGeometry.PresheafedSpace.map_id_c_app", "AlgebraicGeometry.PresheafedSpace.ofRestrict_top_c", "AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff_of_mem_mul", "AlgebraicGeometry.ProjectiveSpectrum.Proj.toSpec_base_isIso", "AlgebraicGeometry.QuasiCompact.affineProperty_isLocal", "AlgebraicGeometry.RingedSpace.basicOpen_mul", "AlgebraicGeometry.RingedSpace.basicOpen_of_isUnit", "AlgebraicGeometry.Scheme.Pullback.pullbackFst\u03b9ToV_fst", "AlgebraicGeometry.Scheme.Pullback.pullbackFst\u03b9ToV_snd", "AlgebraicGeometry.Scheme.Spec_map_presheaf_map_eqToHom", "AlgebraicGeometry.Scheme.comp_coeBase", "AlgebraicGeometry.Scheme.comp_val", "AlgebraicGeometry.Scheme.preimage_basicOpen", "AlgebraicGeometry.Scheme.subcanonical_zariskiTopology", "AlgebraicGeometry.SheafedSpace.GlueData.\u03b9_isoPresheafedSpace_inv", "AlgebraicGeometry.SheafedSpace.IsOpenImmersion.image_preimage_is_empty", "AlgebraicGeometry.SheafedSpace.IsOpenImmersion.to_iso", "AlgebraicGeometry.Spec.map_base", "AlgebraicGeometry.Spec.preimage_map", "AlgebraicGeometry.Spec.toPresheafedSpace_obj", "AlgebraicGeometry.Spec_map_localization_isIso", "AlgebraicGeometry.StructureSheaf.IsFraction.eq_mk'", "AlgebraicGeometry.StructureSheaf.algebraMap_pushforward_stalk", "AlgebraicGeometry.StructureSheaf.comap_id", "AlgebraicGeometry.StructureSheaf.const_apply'", "AlgebraicGeometry.StructureSheaf.const_self", "AlgebraicGeometry.StructureSheaf.const_zero", "AlgebraicGeometry.StructureSheaf.germ_toOpen", "AlgebraicGeometry.StructureSheaf.isLocalizedModule_toPushforwardStalkAlgHom_aux", "AlgebraicGeometry.StructureSheaf.localizationToStalk_stalkSpecializes", "AlgebraicGeometry.StructureSheaf.stalkToFiberRingHom_germ", "AlgebraicGeometry.StructureSheaf.toBasicOpen_injective", "AlgebraicGeometry.StructureSheaf.toOpen_eq_const", "AlgebraicGeometry.exists_pow_mul_eq_zero_of_res_basicOpen_eq_zero_of_isCompact", "AlgebraicGeometry.isOpenImmersion_iff_stalk", "AlgebraicGeometry.isOpenImmersion_isLocalAtTarget", "AlgebraicGeometry.isOpenImmersion_stableUnderBaseChange", "AlgebraicGeometry.isReduced_of_isReduced_stalk", "AlgebraicGeometry.morphismRestrict_\u03b9", "AlgebraicGeometry.quasiCompact_stableUnderBaseChange", "AlgebraicGeometry.quasiSeparated_respectsIso", "AlgebraicGeometry.sourceAffineLocally_of_source_open_cover_aux", "AlgebraicGeometry.targetAffineLocally_of_openCover", "AlgebraicGeometry.universallyClosed_stableUnderBaseChange", "AlgebraicGeometry.\u0393Spec.SpecMap_\u0393SpecIso_hom", "AlgebraicGeometry.\u0393Spec.adjunction_unit_map_basicOpen", "AlgebraicGeometry.\u0393Spec.right_triangle", "AlgebraicGeometry.\u0393Spec.toOpen_unit_app_val_c_app'", "AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_C", "AlgebraicIndependent.ne_zero", "AlgebraicIndependent.repr_ker", "AlgebraicTopology.AlternatingFaceMapComplex.obj_d_eq", "AlgebraicTopology.DoldKan.Compatibility.equivalence\u2081UnitIso_eq", "AlgebraicTopology.DoldKan.HigherFacesVanish.comp_P_eq_self", "AlgebraicTopology.DoldKan.HigherFacesVanish.comp_\u03c3", "AlgebraicTopology.DoldKan.HigherFacesVanish.inclusionOfMooreComplexMap", "AlgebraicTopology.DoldKan.HigherFacesVanish.of_P", "AlgebraicTopology.DoldKan.MorphComponents.postComp_\u03c6", "AlgebraicTopology.DoldKan.N\u2082\u0393\u2082_inv_app_f_f", "AlgebraicTopology.DoldKan.PInfty_f_naturality", "AlgebraicTopology.DoldKan.PInfty_on_\u0393\u2080_splitting_summand_eq_self", "AlgebraicTopology.DoldKan.P_add_Q", "AlgebraicTopology.DoldKan.QInfty_f_0", "AlgebraicTopology.DoldKan.c_mk", "AlgebraicTopology.DoldKan.comp_P_eq_self_iff", "AlgebraicTopology.DoldKan.compatibility_N\u2082_N\u2081_karoubi", "AlgebraicTopology.DoldKan.identity_N\u2082_objectwise", "AlgebraicTopology.DoldKan.karoubi_PInfty_f", "AlgebraicTopology.DoldKan.toKaroubiCompN\u2082IsoN\u2081_inv_app", "AlgebraicTopology.NormalizedMooreComplex.objX_zero", "AlternatingMap.coe_multilinearMap_injective", "AlternatingMap.map_vecCons_add", "AnalyticAt.prod", "AnalyticOn.comp\u2082", "AnalyticOn.congr", "AnalyticOn.congr'", "AnalyticOn.eval_polynomial", "AnalyticOn.iterated_deriv", "And.exists", "AntilipschitzWith.le_mul_ediam_image", "AntilipschitzWith.subsingleton", "AntilipschitzWith.subtype_coe", "AntilipschitzWith.to_rightInverse", "AntisymmRel.trans", "Antitone.antitoneOn", "Antitone.ball", "Antitone.continuousWithinAt_Ioi_iff_rightLim_eq", "Antitone.eq_of_le_of_le", "Antitone.image_uIcc_subset", "Antitone.image_upperBounds_subset_lowerBounds_image", "Antitone.imp", "Antitone.isBoundedUnder_ge_comp", "Antitone.isBoundedUnder_ge_comp_iff", "Antitone.map_iInf_of_continuousAt'", "Antitone.map_iSup_of_continuousAt", "Antitone.map_iSup\u2082_le", "Antitone.map_sSup_of_continuousAt", "Antitone.tendsto_le_alternating_series", "Antitone.union", "AntitoneOn.image_lowerBounds_subset_upperBounds_image", "AntitoneOn.mem_lowerBounds_image_self", "AntitoneOn.min", "AntitoneOn.set_prod", "Antivary.comp_antitone_left", "Antivary.pow_right\u2080", "AntivaryOn.dual", "AntivaryOn.mul_left\u2080", "AntivaryOn.sum_mul_lt_sum_comp_perm_mul_iff", "ApplicativeTransformation.congr_arg", "ApplicativeTransformation.congr_fun", "ApproximatesLinearOn.antilipschitz", "ApproximatesLinearOn.mono_num", "ApproximatesLinearOn.toPartialHomeomorph_coe", "ArithmeticFunction.IsMultiplicative.eulerProduct_tprod", "ArithmeticFunction.IsMultiplicative.prodPrimeFactors", "ArithmeticFunction.IsMultiplicative.prodPrimeFactors_add_of_squarefree", "ArithmeticFunction.LSeriesSummable_mul", "ArithmeticFunction.cardFactors_apply_prime", "ArithmeticFunction.cardFactors_zero", "ArithmeticFunction.coe_inj", "ArithmeticFunction.coe_zeta_mul_moebius", "ArithmeticFunction.ext", "ArithmeticFunction.isMultiplicative_moebius", "ArithmeticFunction.moebius_apply_isPrimePow_not_prime", "ArithmeticFunction.moebius_ne_zero_iff_squarefree", "ArithmeticFunction.one_eq_delta", "ArithmeticFunction.prod_eq_iff_prod_pow_moebius_eq_of_nonzero", "ArithmeticFunction.sigma_zero_apply_prime_pow", "ArithmeticFunction.vonMangoldt_le_log", "ArithmeticFunction.zeta_mul_vonMangoldt", "Array.extract_extract", "Array.push_data", "Array.size_extract", "Array.size_push", "ArzelaAscoli.isCompact_closure_of_closedEmbedding", "Associated.trans", "Associates.count_ne_zero_iff_dvd", "Associates.count_self", "Associates.factors'_cong", "Associates.factors_one", "Associates.le_of_count_ne_zero", "Associates.mk_eq_mk_iff_associated", "Associates.normalize_out", "Associates.one_or_eq_of_le_of_prime", "Associates.prod_le_prod", "Associates.quot_out_zero", "Asymptotics.IsBigO.exists_mem_basis", "Asymptotics.IsBigO.integrableAtFilter", "Asymptotics.IsBigO.not_isLittleO", "Asymptotics.IsBigO.trans_tendsto_norm_atTop", "Asymptotics.IsBigOWith.mul", "Asymptotics.IsBigOWith.prod_left_snd", "Asymptotics.IsBigOWith.rpow", "Asymptotics.IsBigOWith.trans_le", "Asymptotics.IsEquivalent.symm", "Asymptotics.IsEquivalent.tendsto_nhds_iff", "Asymptotics.IsLittleO.add_isBigOWith", "Asymptotics.IsLittleO.insert", "Asymptotics.IsLittleO.prod_rightr", "Asymptotics.IsLittleO.right_isTheta_add'", "Asymptotics.IsLittleO.smul_isBigO", "Asymptotics.IsLittleO.sup", "Asymptotics.IsTheta.isBigO_congr_right", "Asymptotics.IsTheta.tendsto_zero_iff", "Asymptotics.SuperpolynomialDecay.mul_const", "Asymptotics.SuperpolynomialDecay.param_pow_mul", "Asymptotics.SuperpolynomialDecay.trans_abs_le", "Asymptotics.div_isBoundedUnder_of_isBigO", "Asymptotics.isBigOWith_fst_prod", "Asymptotics.isBigOWith_map", "Asymptotics.isBigOWith_neg_right", "Asymptotics.isBigOWith_norm_right", "Asymptotics.isBigO_abs_left", "Asymptotics.isBigO_atTop_iff_eventually_exists", "Asymptotics.isBigO_const_const", "Asymptotics.isBigO_const_smul_left", "Asymptotics.isBigO_fst_prod", "Asymptotics.isBigO_neg_right", "Asymptotics.isBigO_snd_prod", "Asymptotics.isLittleO_const_const_iff", "Asymptotics.isLittleO_const_left_of_ne", "Asymptotics.isLittleO_neg_left", "Asymptotics.isLittleO_norm_pow_id", "Asymptotics.superpolynomialDecay_iff_zpow_tendsto_zero", "Asymptotics.superpolynomialDecay_mul_param_pow_iff", "Balanced.smul_mem_mono", "Balanced.subset_smul", "Basis.basis_singleton_iff", "Basis.card_le_card_of_linearIndependent", "Basis.coe_mk", "Basis.coe_ofRepr", "Basis.coe_repr_symm", "Basis.exists_opNNNorm_le", "Basis.exists_opNorm_le", "Basis.ext_elem_iff", "Basis.ext_multilinear", "Basis.groupSMul_span_eq_top", "Basis.isScalarTower_of_nonempty", "Basis.le_span", "Basis.mem_center_iff", "Basis.mem_span_repr_support", "Basis.mk_eq_rank''", "Basis.ofEquivFun_repr_apply", "Basis.ofIsLocalizedModule_apply", "Basis.ofZlatticeBasis_apply", "Basis.orientation_isEmpty", "Basis.reindexFinsetRange_repr_self", "Basis.reindexRange_repr", "Basis.singleton_repr", "Basis.tensorProduct_apply", "Basis.tensorProduct_repr_tmul_apply", "Basis.toDual_apply", "Basis.toDual_range", "Basis.toMatrix_map_vecMul", "Batteries.BinomialHeap.Imp.Heap.realSize_tail?", "Batteries.DList.toList_append", "Batteries.DList.toList_empty", "Batteries.DList.toList_ofList", "Batteries.HashMap.Imp.WF.mapVal", "Batteries.LTCmp.cmp_iff_gt", "Batteries.LTCmp.eq_compareOfLessAndEq", "Batteries.LawfulCmp.compareOfLessAndEq", "Batteries.PairingHeapImp.Heap.deleteMin_fst", "Batteries.PairingHeapImp.Heap.noSibling_deleteMin", "Batteries.PairingHeapImp.Heap.noSibling_tail?", "Batteries.RBNode.All.balRight", "Batteries.RBNode.All_def", "Batteries.RBNode.Ordered.find?_some", "Batteries.RBNode.Ordered.memP_iff_lowerBound?", "Batteries.RBNode.Ordered.upperBound?_exists", "Batteries.RBNode.Path.Ordered.erase", "Batteries.RBNode.Path.Ordered.insertNew", "Batteries.RBNode.Path.insertNew_eq_insert", "Batteries.RBNode.RedRed.balance2", "Batteries.RBNode.Stream.foldl_eq_foldl_toList", "Batteries.RBNode.Stream.foldr_cons", "Batteries.RBNode.WF.modify", "Batteries.RBNode.WF.out", "Batteries.RBNode.balance2_All", "Batteries.RBNode.cmpEq_iff", "Batteries.RBNode.find?_reverse", "Batteries.RBNode.find?_some_eq_eq", "Batteries.RBNode.foldr_eq_foldr_toList", "Batteries.RBNode.insert_toList_zoom_node", "Batteries.RBNode.lowerBound?_of_some", "Batteries.RBNode.reverse_balRight", "Batteries.RBNode.reverse_eq_iff", "Batteries.RBNode.reverse_size", "Batteries.RBNode.upperBound?_mem", "Batteries.RBSet.find?_some", "Batteries.RBSet.foldr_eq_foldr_toList", "Batteries.RBSet.mem_congr", "Batteries.RBSet.mem_iff_find?", "Batteries.RBSet.mem_toList", "Batteries.RBSet.upperBoundP?_mem", "Batteries.TransCmp.ge_trans", "Batteries.TransOrd.instLexOrd", "Batteries.TransOrd.instOpposite", "Batteries.UnionFind.find_parent_1", "Batteries.UnionFind.parentD_linkAux", "Batteries.UnionFind.parent_lt", "Batteries.UnionFind.rankD_findAux", "Batteries.UnionFind.rankD_set", "Batteries.UnionFind.rootD_ext", "Batteries.UnionFind.rootD_lt", "Batteries.UnionFind.root_link", "BddAbove.convolutionExistsAt", "BddDistLat.coe_toBddLat", "Behrend.bound_aux", "Behrend.map_zero", "Bertrand.real_main_inequality", "Besicovitch.exists_disjoint_closedBall_covering_ae_of_finiteMeasure_aux", "BialgEquiv.coe_toEquiv_trans", "BialgHom.coe_coe", "BialgHom.id_toAlgHom", "BialgHom.id_toCoalgHom", "BialgHom.toAlgHom_toLinearMap", "BialgebraCat.forget\u2082_algebra_obj", "Bifunctor.mapEquiv_apply", "Bimod.LeftUnitorBimod.hom_left_act_hom'", "Bimod.TensorBimod.left_assoc'", "Bimod.TensorBimod.middle_assoc'", "Bimod.whiskerRight_id_bimod", "Bitraversable.tfst_tsnd", "Bool.bne_eq_xor", "Bool.eq_false_of_not_eq_true'", "Bool.eq_or_eq_not", "Bool.injective_iff", "Bool.not_true", "BooleanRing.sup_comm", "Booleanisation.comp_bot", "Booleanisation.comp_lt_comp", "Booleanisation.comp_sdiff_comp", "Booleanisation.comp_sup_comp", "Booleanisation.lift_sdiff_comp", "Bornology.IsBounded.bddBelow", "Bornology.IsBounded.reProdIm", "Bornology.IsVonNBounded.add", "Bornology.ext_iff", "Bornology.isBounded_biUnion", "Bornology.isBounded_iUnion", "Bornology.isBounded_image_fst_and_snd", "Bornology.isBounded_prod_self", "BotHom.cancel_right", "BoundedContinuousFunction.arzela_ascoli\u2081", "BoundedContinuousFunction.exists_norm_eq_restrict_eq_of_closed", "BoundedContinuousFunction.forall_coe_one_iff_one", "BoundedContinuousFunction.integral_const_sub", "BoundedContinuousFunction.lipschitz_comp", "BoundedContinuousFunction.mem_Lp", "BoundedContinuousFunction.mkOfCompact_star", "BoundedContinuousFunction.natCast_apply", "BoundedContinuousFunction.norm_normComp", "BoundedContinuousFunction.tietze_extension_step", "BoundedContinuousFunction.uniformContinuous_comp", "BoundedLatticeHom.coe_comp", "BoundedLatticeHom.coe_comp_inf_hom", "BoundedLatticeHom.coe_toBoundedOrderHom", "BoundedLatticeHom.symm_dual_comp", "BoundedOrderHom.coe_copy", "BoundedVariationOn.sub_le", "BoxIntegral.Box.Icc_def", "BoxIntegral.Box.coe_mk'", "BoxIntegral.Box.continuousOn_face_Icc", "BoxIntegral.Box.exists_mem_splitCenterBox", "BoxIntegral.Box.exists_seq_mono_tendsto", "BoxIntegral.Box.le_def", "BoxIntegral.Box.measurableSet_coe", "BoxIntegral.Box.nonempty_coe", "BoxIntegral.Box.splitLower_eq_self", "BoxIntegral.Box.volume_apply'", "BoxIntegral.HasIntegral.mono", "BoxIntegral.Integrable.dist_integralSum_integral_le_of_memBaseSet", "BoxIntegral.Integrable.to_subbox_aux", "BoxIntegral.Prepartition.distortion_biUnion", "BoxIntegral.Prepartition.eq_of_boxes_subset_iUnion_superset", "BoxIntegral.Prepartition.iUnion_toSubordinate", "BoxIntegral.Prepartition.iUnion_top", "BoxIntegral.Prepartition.mem_biUnion", "BoxIntegral.Prepartition.mem_splitCenter", "BoxIntegral.Prepartition.mem_split_iff'", "BoxIntegral.Prepartition.splitMany_insert", "BoxIntegral.Prepartition.upper_sub_lower_of_mem_splitCenter", "BoxIntegral.TaggedPrepartition.disjUnion_tag_of_mem_right", "BoxIntegral.TaggedPrepartition.iUnion_def", "BoxIntegral.TaggedPrepartition.iUnion_disjUnion", "BoxIntegral.TaggedPrepartition.isPartition_single_iff", "BoxIntegral.hasIntegral_GP_divergence_of_forall_hasDerivWithinAt", "BoxIntegral.hasIntegral_GP_pderiv", "BoxIntegral.integrable_neg", "Btw.btw.antisymm", "BumpCovering.IsSubordinate.mono", "BumpCovering.IsSubordinate.toPartitionOfUnity", "ByteArray.extract_data", "ByteArray.get_set_eq", "CPolynomialOn_const", "CancelDenoms.inv_subst", "CanonicallyOrderedCommSemiring.pow_pos", "CantorScheme.VanishingDiam.dist_lt", "Cardinal.add_one_of_aleph0_le", "Cardinal.aleph'_zero", "Cardinal.aleph0_le_mul_iff", "Cardinal.aleph0_le_mul_iff'", "Cardinal.aleph0_mul_aleph", "Cardinal.alephIdx.relIso_coe", "Cardinal.aleph_isNormal", "Cardinal.aleph_lt", "Cardinal.aleph_toNat", "Cardinal.bddAbove_iff_small", "Cardinal.cantor'", "Cardinal.cantorFunctionAux_nonneg", "Cardinal.cantorFunction_le", "Cardinal.card_lt_of_card_iUnion_lt", "Cardinal.exists_eq_of_iSup_eq_of_not_isSuccLimit", "Cardinal.infinite_pigeonhole_card_lt", "Cardinal.le_def", "Cardinal.lift_lift", "Cardinal.lift_lt", "Cardinal.lift_mk_shrink", "Cardinal.lift_ofENat", "Cardinal.lift_one", "Cardinal.lift_prod", "Cardinal.lift_succ", "Cardinal.lift_uzero", "Cardinal.map_mk", "Cardinal.mk_bounded_subset_le", "Cardinal.mk_compl_eq_mk_compl_infinite", "Cardinal.mk_eq_aleph0", "Cardinal.mk_eq_one", "Cardinal.mk_finset_of_infinite", "Cardinal.mk_finsupp_lift_of_infinite", "Cardinal.mk_finsupp_nat", "Cardinal.mk_le_aleph0", "Cardinal.mk_ne_zero_iff", "Cardinal.mk_perm_eq_self_power", "Cardinal.mk_preimage_of_injective_lift", "Cardinal.mul_eq_left_iff", "Cardinal.mul_lt_aleph0_iff", "Cardinal.mul_power", "Cardinal.natCast_le", "Cardinal.natCast_pow", "Cardinal.noMaxOrder", "Cardinal.ofENat_le_lift", "Cardinal.ofENat_mul", "Cardinal.ofNat_eq_lift_iff", "Cardinal.ofNat_le_ofENat", "Cardinal.ord_aleph0", "Cardinal.ord_le", "Cardinal.ord_nat", "Cardinal.ord_one", "Cardinal.ord_strictMono", "Cardinal.partENatOfENat_toENat", "Cardinal.power_add", "Cardinal.power_lt_aleph0", "Cardinal.power_one", "Cardinal.powerlt_le_powerlt_left", "Cardinal.prime_of_aleph0_le", "Cardinal.prod_const", "Cardinal.succ_zero", "Cardinal.sum_le_iSup", "Cardinal.sum_nat_eq_add_sum_succ", "Cardinal.toENat_eq_nat", "Cardinal.toENat_injOn", "Cardinal.toNat_add", "Cardinal.toNat_apply_of_aleph0_le", "Cardinal.toNat_eq_one", "Cardinal.toNat_le_iff_le_of_lt_aleph0", "Cardinal.toNat_mul", "Cardinal.toPartENat_lt_natCast_iff", "Cardinal.toPartENat_strictMonoOn", "Cardinal.zero_power", "Cardinal.zero_toNat", "CategoryTheory.Abelian.Exact.op", "CategoryTheory.Abelian.Exact.unop", "CategoryTheory.Abelian.Pseudoelement.apply_eq_zero_of_comp_eq_zero", "CategoryTheory.Abelian.Pseudoelement.pseudo_pullback", "CategoryTheory.Abelian.epi_snd_of_isLimit", "CategoryTheory.Abelian.exact_iff_exact_coimage_\u03c0", "CategoryTheory.Abelian.pseudoEqual_symm", "CategoryTheory.ActionCategory.\u03c0_map", "CategoryTheory.AdditiveFunctor.forget_obj_of", "CategoryTheory.Adjunction.CoreHomEquiv.homEquiv_naturality_right_symm_aux", "CategoryTheory.Adjunction.he'", "CategoryTheory.Adjunction.isLeftAdjoint", "CategoryTheory.Adjunction.map_injective", "CategoryTheory.Adjunction.rightAdjointUniq_refl", "CategoryTheory.Arrow.hom_inv_id_left", "CategoryTheory.Aut.Aut_mul_def", "CategoryTheory.BicartesianSq.of_has_biproduct\u2082", "CategoryTheory.Bicategory.Equivalence.left_triangle_hom", "CategoryTheory.Bicategory.Equivalence.right_triangle", "CategoryTheory.Bicategory.LeftLift.IsKan.fac", "CategoryTheory.Bicategory.LeftLift.IsKan.hom_ext", "CategoryTheory.Bicategory.leftZigzagIso_symm", "CategoryTheory.Bicategory.triangle_assoc_comp_left", "CategoryTheory.CartesianClosed.curry_id_eq_coev", "CategoryTheory.Center.leftUnitor_inv_f", "CategoryTheory.Center.tensor_fst", "CategoryTheory.Center.whiskerLeft_comm", "CategoryTheory.ChosenFiniteProducts.tensorHom_fst", "CategoryTheory.CommSq.HasLift.iff_unop", "CategoryTheory.CommSq.HasLift.mk'", "CategoryTheory.Comma.comp_right", "CategoryTheory.ComposableArrows.Exact.\u03b4\u2080", "CategoryTheory.ComposableArrows.Precomp.map_zero_zero", "CategoryTheory.ComposableArrows.Precomp.obj_zero", "CategoryTheory.ComposableArrows.ext\u2083", "CategoryTheory.ComposableArrows.ext\u2084", "CategoryTheory.ComposableArrows.homMk\u2082_app_one", "CategoryTheory.ComposableArrows.homMk\u2082_app_two", "CategoryTheory.ComposableArrows.homMk\u2082_app_zero", "CategoryTheory.ComposableArrows.homMk\u2085_app_two", "CategoryTheory.ComposableArrows.map'_eq_hom\u2081", "CategoryTheory.ComposableArrows.mk\u2084_surjective", "CategoryTheory.ComposableArrows.naturality'", "CategoryTheory.CosimplicialObject.\u03b4_comp_\u03c3_of_gt", "CategoryTheory.CosimplicialObject.\u03b4_comp_\u03c3_of_gt'", "CategoryTheory.CostructuredArrow.mk_right", "CategoryTheory.Coyoneda.fullyFaithful_preimage", "CategoryTheory.Discrete.mk_as", "CategoryTheory.Discrete.monoidal_tensorUnit_as", "CategoryTheory.Endofunctor.Algebra.Initial.right_inv", "CategoryTheory.Equivalence.Equivalence_mk'_counit", "CategoryTheory.Equivalence.funInvIdAssoc_inv_app", "CategoryTheory.Equivalence.hasSheafCompose", "CategoryTheory.Equivalence.map_injective_iff", "CategoryTheory.Equivalence.map_projective_iff", "CategoryTheory.Equivalence.preregular", "CategoryTheory.ExactPairing.coevaluation_evaluation''", "CategoryTheory.ExactPairing.evaluation_coevaluation''", "CategoryTheory.FreeMonoidalCategory.discrete_functor_obj_eq_as", "CategoryTheory.FreeMonoidalCategory.normalizeIsoApp_eq", "CategoryTheory.Functor.CommShift.ofIso_compatibility", "CategoryTheory.Functor.Elements.ext", "CategoryTheory.Functor.FullyFaithful.hasShift.map_add_hom_app", "CategoryTheory.Functor.IsCartesian.eq_of_fac", "CategoryTheory.Functor.IsCartesian.map_self", "CategoryTheory.Functor.IsCoverDense.compatiblePreserving", "CategoryTheory.Functor.IsCoverDense.isContinuous", "CategoryTheory.Functor.IsCoverDense.sheaf_eq_amalgamation", "CategoryTheory.Functor.IsLocalization.for_id", "CategoryTheory.Functor.Monoidal.associator_hom_app", "CategoryTheory.Functor.cofinal_of_isTerminal_colimit_comp_yoneda", "CategoryTheory.Functor.effectiveEpi_of_map", "CategoryTheory.Functor.eq_of_iso", "CategoryTheory.Functor.essImage.ofNatIso", "CategoryTheory.Functor.ext_of_iso", "CategoryTheory.Functor.final_iff_comp_final_full_faithful", "CategoryTheory.Functor.final_of_comp_full_faithful'", "CategoryTheory.Functor.final_of_exists_of_isFiltered_of_fullyFaithful", "CategoryTheory.Functor.flip_injective", "CategoryTheory.Functor.hasRightDerivedFunctor_iff", "CategoryTheory.Functor.hasRightDerivedFunctor_iff_of_iso", "CategoryTheory.Functor.homologySequence_epi_shift_map_mor\u2082_iff", "CategoryTheory.Functor.initial_iff_comp_equivalence", "CategoryTheory.Functor.initial_iff_equivalence_comp", "CategoryTheory.Functor.inv_fun_map", "CategoryTheory.Functor.lanAdjunction_counit_app", "CategoryTheory.Functor.lanAdjunction_unit", "CategoryTheory.Functor.mapIso_injective", "CategoryTheory.Functor.map_commSq", "CategoryTheory.Functor.map_hom_inv", "CategoryTheory.Functor.map_shiftFunctorCompIsoId_hom_app", "CategoryTheory.Functor.map_zsmul", "CategoryTheory.Functor.mem_eventualRange_iff", "CategoryTheory.Functor.preservesZeroMorphisms_of_map_zero_object", "CategoryTheory.Functor.reflectsMonomorphisms_of_preserves_of_reflects", "CategoryTheory.Functor.rightDerivedNatTrans_fac", "CategoryTheory.Functor.rightDerived_fac_app", "CategoryTheory.Functor.rotate_essImageDistTriang", "CategoryTheory.Functor.sum_map_inr", "CategoryTheory.Functor.uncurry_obj_curry_obj_flip_flip'", "CategoryTheory.Functor.\u03b9_biproductComparison'", "CategoryTheory.FunctorToTypes.binaryProductIso_inv_comp_snd", "CategoryTheory.FunctorToTypes.prod_ext", "CategoryTheory.GlueData.diagramIso_app_left", "CategoryTheory.GradedObject.eqToHom_proj", "CategoryTheory.GradedObject.mapBifunctorMapObj_ext", "CategoryTheory.GradedObject.mapBifunctor_triangle", "CategoryTheory.GradedObject.\u03b9MapObjOrZero_eq_zero", "CategoryTheory.GradedObject.\u03b9_descMapObj", "CategoryTheory.GradedObject.\u03b9_mapMap", "CategoryTheory.GrothendieckTopology.Cover.ext", "CategoryTheory.GrothendieckTopology.Plus.res_mk_eq_mk_pullback", "CategoryTheory.GrothendieckTopology.W_eq_inverseImage_isomorphisms_of_adjunction", "CategoryTheory.GrothendieckTopology.W_iff", "CategoryTheory.GrothendieckTopology.W_inverseImage_whiskeringLeft", "CategoryTheory.GrothendieckTopology.W_of_isLocallyBijective", "CategoryTheory.GrothendieckTopology.W_sheafToPreheaf_map_iff_isIso", "CategoryTheory.GrothendieckTopology.arrow_max", "CategoryTheory.GrothendieckTopology.bot_covers", "CategoryTheory.GrothendieckTopology.close_close", "CategoryTheory.GrothendieckTopology.isClosed_pullback", "CategoryTheory.GrothendieckTopology.isoSheafify_hom", "CategoryTheory.GrothendieckTopology.isoToPlus_inv", "CategoryTheory.GrothendieckTopology.over_forget_coverPreserving", "CategoryTheory.GrothendieckTopology.preservesSheafification_iff_of_adjunctions", "CategoryTheory.GrothendieckTopology.sheafToPresheaf_map_sheafComposeNatTrans_eq_sheafifyCompIso_inv", "CategoryTheory.GrothendieckTopology.sheafificationWhiskerRightIso_inv_app", "CategoryTheory.GrothendieckTopology.sheafifyMap_comp", "CategoryTheory.GrothendieckTopology.sheafify_hom_ext", "CategoryTheory.Groupoid.inv_eq_inv", "CategoryTheory.HasLiftingProperty.of_arrow_iso_right", "CategoryTheory.HasShift.shift_obj_obj", "CategoryTheory.HomOrthogonal.matrixDecomposition_comp", "CategoryTheory.Idempotents.DoldKan.equivalence_inverse", "CategoryTheory.Idempotents.Karoubi.HomologicalComplex.p_idem", "CategoryTheory.Idempotents.Karoubi.decomp_p", "CategoryTheory.InjectiveResolution.ofCocomplex_d_0_1", "CategoryTheory.IsCofiltered.eq_condition", "CategoryTheory.IsDetecting.mono", "CategoryTheory.IsDetector.def", "CategoryTheory.IsFiltered.bowtie", "CategoryTheory.IsFiltered.coeq\u2083_condition\u2082", "CategoryTheory.IsFiltered.of_hasFiniteColimits", "CategoryTheory.IsFiltered.toSup_commutes", "CategoryTheory.IsFilteredOrEmpty.of_equivalence", "CategoryTheory.IsFilteredOrEmpty.of_final", "CategoryTheory.IsFilteredOrEmpty.of_isRightAdjoint", "CategoryTheory.IsIso.inv_comp_eq", "CategoryTheory.IsPullback.inr_fst", "CategoryTheory.IsPullback.of_isLimit'", "CategoryTheory.IsPullback.of_is_bilimit'", "CategoryTheory.IsPullback.op", "CategoryTheory.IsPushout.of_horiz_isIso", "CategoryTheory.IsPushout.op", "CategoryTheory.IsPushout.unop", "CategoryTheory.IsTriangulated.mk'", "CategoryTheory.IsUniversalColimit.map_reflective", "CategoryTheory.Iso.conj_id", "CategoryTheory.Iso.eq_comp_inv", "CategoryTheory.Iso.homCongr_apply", "CategoryTheory.Iso.inv_eq_inv", "CategoryTheory.Iso.inv_ext'", "CategoryTheory.Iso.inv_hom_id_triangle_hom\u2083", "CategoryTheory.Iso.map_hom_inv_id", "CategoryTheory.Iso.map_inv_hom_id_app", "CategoryTheory.Iso.map_inv_hom_id_eval_app", "CategoryTheory.Iso.symm_inv", "CategoryTheory.Iso.symm_self_id_assoc", "CategoryTheory.Iso.to_eq", "CategoryTheory.Iso.trans_refl", "CategoryTheory.LeftExactFunctor.forget_map", "CategoryTheory.LeftExactFunctor.of_fst", "CategoryTheory.Limits.Bicone.toCone_pt", "CategoryTheory.Limits.BinaryBiproduct.bicone_inr", "CategoryTheory.Limits.BinaryCofan.IsColimit.hom_ext", "CategoryTheory.Limits.BinaryCofan.mk_inr", "CategoryTheory.Limits.BinaryFan.assocInv_fst", "CategoryTheory.Limits.BinaryFan.assoc_snd", "CategoryTheory.Limits.Cofork.IsColimit.homIso_natural", "CategoryTheory.Limits.CompleteLattice.finite_coproduct_eq_finset_sup", "CategoryTheory.Limits.Concrete.cokernel_funext", "CategoryTheory.Limits.Concrete.from_union_surjective_of_isColimit", "CategoryTheory.Limits.Concrete.isColimit_exists_rep", "CategoryTheory.Limits.Concrete.multiequalizer_ext", "CategoryTheory.Limits.Concrete.widePushout_exists_rep'", "CategoryTheory.Limits.Cotrident.coequalizer_ext", "CategoryTheory.Limits.Fork.IsLimit.existsUnique", "CategoryTheory.Limits.Fork.IsLimit.homIso_natural", "CategoryTheory.Limits.Fork.hom_comp_\u03b9", "CategoryTheory.Limits.HasBinaryProduct.swap", "CategoryTheory.Limits.HasBiproduct.of_hasCoproduct", "CategoryTheory.Limits.HasLimit.isoOfEquivalence_inv_\u03c0", "CategoryTheory.Limits.HasZeroObject.zeroIsoInitial_hom", "CategoryTheory.Limits.HasZeroObject.zeroIsoIsTerminal_inv", "CategoryTheory.Limits.IsColimit.comp_coconePointsIsoOfNatIso_inv", "CategoryTheory.Limits.IsLimit.hom_isIso", "CategoryTheory.Limits.IsLimit.uniq_cone_morphism", "CategoryTheory.Limits.IsZero.of_mono_zero", "CategoryTheory.Limits.KernelFork.mapOfIsLimit_\u03b9", "CategoryTheory.Limits.KernelFork.map_condition", "CategoryTheory.Limits.MonoCoprod.mono_binaryCofanSum_inl'", "CategoryTheory.Limits.MonoFactorisation.kernel_\u03b9_comp", "CategoryTheory.Limits.Multicofork.fst_app_right", "CategoryTheory.Limits.Multiequalizer.multifork_\u03b9", "CategoryTheory.Limits.Multifork.app_right_eq_\u03b9_comp_fst", "CategoryTheory.Limits.MultispanIndex.multispan_obj_right", "CategoryTheory.Limits.Pi.map'_eq", "CategoryTheory.Limits.Pi.reindex_inv_\u03c0", "CategoryTheory.Limits.PreservesPushout.inr_iso_inv", "CategoryTheory.Limits.PullbackCone.IsLimit.lift_fst", "CategoryTheory.Limits.PullbackCone.equalizer_ext", "CategoryTheory.Limits.PullbackCone.mono_fst_of_is_pullback_of_mono", "CategoryTheory.Limits.PullbackCone.snd_colimit_cocone", "CategoryTheory.Limits.PullbackCone.\u03c0_app_left", "CategoryTheory.Limits.PushoutCocone.IsColimit.hom_ext", "CategoryTheory.Limits.PushoutCocone.IsColimit.inl_desc", "CategoryTheory.Limits.Sigma.map_comp_map'", "CategoryTheory.Limits.Sigma.map_id", "CategoryTheory.Limits.SingleObj.Types.Quot.Rel.iff_orbitRel", "CategoryTheory.Limits.Types.Colimit.\u03b9_desc_apply'", "CategoryTheory.Limits.Types.Limit.lift_\u03c0_apply", "CategoryTheory.Limits.Types.binaryCofan_isColimit_iff", "CategoryTheory.Limits.Types.binaryProductIso_inv_comp_fst", "CategoryTheory.Limits.Types.colimit_sound", "CategoryTheory.Limits.Types.colimit_sound'", "CategoryTheory.Limits.Types.descFun_apply_spec", "CategoryTheory.Limits.Types.descSet_univ", "CategoryTheory.Limits.Types.jointly_surjective'", "CategoryTheory.Limits.Types.limit_ext_iff'", "CategoryTheory.Limits.WidePushout.eq_desc_of_comp_eq", "CategoryTheory.Limits.as_factorThruImage", "CategoryTheory.Limits.biprod.add_eq_lift_id_desc", "CategoryTheory.Limits.biprod.inlCokernelCofork_\u03c0", "CategoryTheory.Limits.biprod.inr_desc", "CategoryTheory.Limits.biprod.lift_snd", "CategoryTheory.Limits.biprod.mapBiprod_inv_map_desc", "CategoryTheory.Limits.biproduct.isoCoproduct_hom", "CategoryTheory.Limits.biproduct.lift_desc", "CategoryTheory.Limits.biproduct.map_desc", "CategoryTheory.Limits.biproduct.matrix_map", "CategoryTheory.Limits.biproduct.\u03b9_\u03c0_self", "CategoryTheory.Limits.coequalizer.cofork_\u03b9_app_one", "CategoryTheory.Limits.coequalizer.condition", "CategoryTheory.Limits.cokernel.map_desc", "CategoryTheory.Limits.cokernelIsoOfEq_trans", "CategoryTheory.Limits.cokernel_map_comp_cokernelComparison", "CategoryTheory.Limits.colimit.post_post", "CategoryTheory.Limits.colimit.pre_post", "CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation_inv_colimit_map", "CategoryTheory.Limits.colimit_\u03b9_zero_cokernel_desc", "CategoryTheory.Limits.coprod.symmetry", "CategoryTheory.Limits.cospanExt_app_right", "CategoryTheory.Limits.equalizer.lift_\u03b9", "CategoryTheory.Limits.hasCoproducts_of_opposite", "CategoryTheory.Limits.hasFiniteColimits_of_hasColimitsOfSize", "CategoryTheory.Limits.hasLimit_of_hasColimit_leftOp", "CategoryTheory.Limits.hasPushout_of_right_iso", "CategoryTheory.Limits.hasTerminal_of_hasInitial_op", "CategoryTheory.Limits.hasWidePullbacks_shrink", "CategoryTheory.Limits.has_colimits_of_hasCoequalizers_and_coproducts", "CategoryTheory.Limits.has_limits_of_hasEqualizers_and_products", "CategoryTheory.Limits.image.as_\u03b9", "CategoryTheory.Limits.image.compIso_inv_comp_image_\u03b9", "CategoryTheory.Limits.image.isImage_lift", "CategoryTheory.Limits.image.preComp_\u03b9", "CategoryTheory.Limits.imageSubobjectIso_comp_image_map", "CategoryTheory.Limits.imageSubobject_arrow_comp", "CategoryTheory.Limits.imageSubobject_le", "CategoryTheory.Limits.imageSubobject_le_mk", "CategoryTheory.Limits.inl_comp_pushoutSymmetry_hom", "CategoryTheory.Limits.inr_comp_pushoutComparison", "CategoryTheory.Limits.inr_of_isLimit", "CategoryTheory.Limits.limMap_\u03c0", "CategoryTheory.Limits.limitUncurryIsoLimitCompLim_hom_\u03c0_\u03c0", "CategoryTheory.Limits.map_id_left_eq_curry_map", "CategoryTheory.Limits.opCoproductIsoProduct'_inv_comp_inj", "CategoryTheory.Limits.parallelPair_map_left", "CategoryTheory.Limits.prod.lift_fst_snd", "CategoryTheory.Limits.prod.pentagon", "CategoryTheory.Limits.pullback.lift_fst", "CategoryTheory.Limits.pullbackConeOfLeftIso_\u03c0_app_left", "CategoryTheory.Limits.pullbackDiagonalMapIso_inv_fst", "CategoryTheory.Limits.pullback_diagonal_map_snd_fst_fst", "CategoryTheory.Limits.pullback_map_eq_pullbackFstFstIso_inv", "CategoryTheory.Limits.pushout.condition", "CategoryTheory.Limits.strongEpi_factorThruImage_of_strongEpiMonoFactorisation", "CategoryTheory.Limits.unop_zero", "CategoryTheory.Limits.wideCoequalizer.condition", "CategoryTheory.Limits.zero_of_comp_mono", "CategoryTheory.Linear.units_smul_comp", "CategoryTheory.Localization.Preadditive.add'_comp", "CategoryTheory.Localization.essSurj_mapArrow_of_hasRightCalculusOfFractions", "CategoryTheory.Localization.full_whiskeringLeft", "CategoryTheory.Localization.homEquiv_isoOfHom_inv", "CategoryTheory.LocalizerMorphism.IsRightDerivabilityStructure.mk'", "CategoryTheory.LocalizerMorphism.inverts", "CategoryTheory.LocalizerMorphism.isIso_iff_of_hasLeftResolutions", "CategoryTheory.LocalizerMorphism.isRightDerivabilityStructure_iff", "CategoryTheory.Mat.id_apply", "CategoryTheory.Mat_.add_apply", "CategoryTheory.Mat_.additiveObjIsoBiproduct_naturality'", "CategoryTheory.Mat_.equivalenceSelfOfHasFiniteBiproducts_functor", "CategoryTheory.Monad.Algebra.comp_eq_comp", "CategoryTheory.Monad.assoc", "CategoryTheory.Monad.beckCoequalizer_desc", "CategoryTheory.Monad.forget_creates_colimits_of_monad_preserves", "CategoryTheory.MonadHom.comp_toNatTrans", "CategoryTheory.Monoidal.whiskerRight_app", "CategoryTheory.MonoidalCategory.associator_inv_naturality_middle", "CategoryTheory.MonoidalCategory.hom_inv_id_tensor'", "CategoryTheory.MonoidalCategory.inv_hom_whiskerRight", "CategoryTheory.MonoidalCategory.prodMonoidal_leftUnitor_inv_fst", "CategoryTheory.MonoidalCategory.whiskerLeft_dite", "CategoryTheory.MonoidalClosed.id_tensor_pre_app_comp_ev", "CategoryTheory.MonoidalClosed.ofEquiv_uncurry_def", "CategoryTheory.MonoidalFunctor.mapAction_\u03b5_inv_hom", "CategoryTheory.MonoidalFunctor.mapAction_\u03bc_inv_hom", "CategoryTheory.MonoidalOpposite.unmop_inv_braiding", "CategoryTheory.MorphismProperty.FunctorialFactorizationData.fac_app", "CategoryTheory.MorphismProperty.IsInvertedBy.isoClosure_iff", "CategoryTheory.MorphismProperty.IsInvertedBy.of_le", "CategoryTheory.MorphismProperty.IsStableUnderProductsOfShape.mk", "CategoryTheory.MorphismProperty.LeftFraction\u2082Rel.fst", "CategoryTheory.MorphismProperty.Q_inverts", "CategoryTheory.MorphismProperty.RespectsIso.unop", "CategoryTheory.MorphismProperty.RightFraction.op_map", "CategoryTheory.MorphismProperty.map_eq_iff_postcomp", "CategoryTheory.MorphismProperty.map_inverseImage_le", "CategoryTheory.MorphismProperty.map_mem_map", "CategoryTheory.NatIso.cancel_natIso_hom_right_assoc", "CategoryTheory.NatTrans.app_sub", "CategoryTheory.NatTrans.epi_iff_epi_app", "CategoryTheory.NatTrans.removeLeftOp_id", "CategoryTheory.OplaxFunctor.map\u2082_eqToHom", "CategoryTheory.OverPresheafAux.OverArrows.map\u2081_map\u2082", "CategoryTheory.OverPresheafAux.OverArrows.map\u2081_val", "CategoryTheory.OverPresheafAux.counitBackward_counitForward", "CategoryTheory.OverPresheafAux.counitForward_naturality\u2081", "CategoryTheory.Paths.lift_cons", "CategoryTheory.PreGaloisCategory.evaluation_injective_of_isConnected", "CategoryTheory.PreGaloisCategory.exists_galois_representative", "CategoryTheory.PreGaloisCategory.has_decomp_connected_components_aux_conn", "CategoryTheory.PreOneHypercover.sieve\u2081_eq_pullback_sieve\u2081'", "CategoryTheory.Preadditive.exact_iff_homology'_zero", "CategoryTheory.Preadditive.exact_of_iso_of_exact", "CategoryTheory.Preadditive.isSeparating_iff", "CategoryTheory.Preadditive.mono_iff_injective'", "CategoryTheory.Preadditive.neg_comp_neg", "CategoryTheory.Preadditive.neg_iso_inv", "CategoryTheory.Presheaf.IsSheaf.isSeparated", "CategoryTheory.Presheaf.comp_isLocallySurjective_iff", "CategoryTheory.Presheaf.equalizerSieve_eq_top_iff", "CategoryTheory.Presheaf.isLocallyInjective_of_isLocallyInjective", "CategoryTheory.Presheaf.isLocallyInjective_of_isLocallyInjective_of_isLocallySurjective_fac", "CategoryTheory.Presheaf.isLocallySurjective_comp_iff", "CategoryTheory.Presheaf.isSheaf_coherent_of_hasPullbacks_of_comp", "CategoryTheory.Presheaf.isSheaf_coherent_of_projective_comp", "CategoryTheory.Presheaf.isSheaf_iff_isSheaf_comp", "CategoryTheory.PresheafOfGroups.Cochain\u2080.mul_apply", "CategoryTheory.PresheafOfGroups.OneCochain.mul_ev", "CategoryTheory.PresheafOfGroups.OneCocycle.one_toOneCochain", "CategoryTheory.Presieve.factorsThru_top", "CategoryTheory.Presieve.image_mem_functorPushforward", "CategoryTheory.Presieve.isSheaf_pretopology", "CategoryTheory.Presieve.piComparison_fac", "CategoryTheory.Pretriangulated.Opposite.distinguished_cocone_triangle", "CategoryTheory.Pretriangulated.Triangle.isZero\u2083_of_isZero\u2081\u2082", "CategoryTheory.Pretriangulated.Triangle.mor\u2081_eq_zero_of_mono\u2082", "CategoryTheory.Pretriangulated.Triangle.shift_distinguished", "CategoryTheory.Pretriangulated.contractible_distinguished\u2082", "CategoryTheory.Pretriangulated.exists_iso_of_arrow_iso", "CategoryTheory.Pretriangulated.mem_distTriang_op_iff", "CategoryTheory.Pretriangulated.mem_distTriang_op_iff'", "CategoryTheory.Pretriangulated.shift_unop_opShiftFunctorEquivalence_counitIso_hom_app", "CategoryTheory.ProjectiveResolution.complex_d_succ_comp", "CategoryTheory.Pseudofunctor.map\u2082_inv", "CategoryTheory.Quotient.LiftCommShift.iso_hom_app", "CategoryTheory.Quotient.compClosure_eq_self", "CategoryTheory.RanIsSheafOfIsCocontinuous.gluedSection_is_unique", "CategoryTheory.RelCat.rel_iso_iff", "CategoryTheory.RightExactFunctor.ofExact_obj", "CategoryTheory.Sheaf.Hom.mono_iff_presheaf_mono", "CategoryTheory.Sheaf.isLocallyBijective_iff_isIso'", "CategoryTheory.SheafOfTypes.Hom.ext'", "CategoryTheory.ShortComplex.Exact.map_of_preservesRightHomologyOf", "CategoryTheory.ShortComplex.Exact.op", "CategoryTheory.ShortComplex.HasLeftHomology.hasCokernel", "CategoryTheory.ShortComplex.LeftHomologyData.cyclesIso_inv_comp_iCycles", "CategoryTheory.ShortComplex.LeftHomologyData.op_g'", "CategoryTheory.ShortComplex.LeftHomologyData.\u03c0_descH", "CategoryTheory.ShortComplex.LeftHomologyMapData.congr_\u03c6K", "CategoryTheory.ShortComplex.RightHomologyData.exact_iff_mono_g'", "CategoryTheory.ShortComplex.RightHomologyData.liftH_\u03b9", "CategoryTheory.ShortComplex.RightHomologyData.opcyclesIso_inv_comp_descOpcycles", "CategoryTheory.ShortComplex.RightHomologyData.p_comp_opcyclesIso_inv", "CategoryTheory.ShortComplex.ShortExact.ab_injective_f", "CategoryTheory.ShortComplex.ShortExact.\u03b4_apply", "CategoryTheory.ShortComplex.SnakeInput.L\u2080_exact", "CategoryTheory.ShortComplex.SnakeInput.comp_f\u2080", "CategoryTheory.ShortComplex.ab_exact_iff_range_eq_ker", "CategoryTheory.ShortComplex.asIsoHomology\u03b9_inv_comp_homology\u03b9", "CategoryTheory.ShortComplex.cyclesIsoX\u2082_hom_inv_id", "CategoryTheory.ShortComplex.cyclesMap'_add", "CategoryTheory.ShortComplex.epi_\u03c4\u2082_of_exact_of_epi", "CategoryTheory.ShortComplex.exact_iff_epi_kernel_lift", "CategoryTheory.ShortComplex.hasHomology_of_epi_of_isIso_of_mono'", "CategoryTheory.ShortComplex.hasLeftHomology_of_epi_of_isIso_of_mono", "CategoryTheory.ShortComplex.i_cyclesMk", "CategoryTheory.ShortComplex.leftHomologyMap'_neg", "CategoryTheory.ShortComplex.leftRightHomologyComparison'_eq_descH", "CategoryTheory.ShortComplex.mapCyclesIso_hom_naturality", "CategoryTheory.ShortComplex.neg_\u03c4\u2082", "CategoryTheory.ShortComplex.opcyclesMap_add", "CategoryTheory.ShortComplex.opcycles_ext", "CategoryTheory.ShortComplex.quasiIso_iff", "CategoryTheory.ShortComplex.shortExact_iff_unop", "CategoryTheory.ShortComplex.toCycles_comp_homology\u03c0", "CategoryTheory.ShortComplex.zero_\u03c4\u2082", "CategoryTheory.Sieve.mem_ofObjects_iff", "CategoryTheory.Sieve.overEquiv_pullback", "CategoryTheory.Sieve.pullback_monotone", "CategoryTheory.Sieve.pushforward_comp", "CategoryTheory.Sigma.descUniq_hom_app", "CategoryTheory.SimplicialObject.Augmented.hom_ext", "CategoryTheory.SimplicialObject.eqToIso_refl", "CategoryTheory.SimplicialObject.\u03b4_comp_\u03b4_self", "CategoryTheory.SimplicialObject.\u03b4_comp_\u03c3_of_le", "CategoryTheory.SingleFunctors.shiftIso_add'_inv_app", "CategoryTheory.SingleFunctors.shiftIso_add_hom_app", "CategoryTheory.SplitEpi.epi", "CategoryTheory.StrongEpi.iff_of_arrow_iso", "CategoryTheory.StrongEpi.mk'", "CategoryTheory.StructuredArrow.IsUniversal.hom_desc", "CategoryTheory.StructuredArrow.isCoseparating_proj_preimage", "CategoryTheory.StructuredArrow.map_mk", "CategoryTheory.Subgroupoid.IsNormal.conj'", "CategoryTheory.Subgroupoid.IsNormal.generatedNormal_le", "CategoryTheory.Subgroupoid.generatedNormal_isNormal", "CategoryTheory.Subgroupoid.id_mem_of_nonempty_isotropy", "CategoryTheory.Subgroupoid.inclusion_faithful", "CategoryTheory.Subgroupoid.isThin_iff", "CategoryTheory.Subobject.bot_eq_initial_to", "CategoryTheory.Subobject.factorThru_right", "CategoryTheory.Subobject.factorThru_self", "CategoryTheory.Subobject.factors_add", "CategoryTheory.Subobject.map_comp", "CategoryTheory.Subobject.map_top", "CategoryTheory.Subobject.ofLEMk_comp_ofMkLE", "CategoryTheory.Subobject.ofLE_comp_ofLEMk", "CategoryTheory.Subobject.ofMkLE_comp_ofLE", "CategoryTheory.Subobject.sInf_le", "CategoryTheory.Subobject.sup_factors_of_factors_left", "CategoryTheory.Sum.swap_map_inr", "CategoryTheory.ThinSkeleton.map_iso_eq", "CategoryTheory.Triangulated.TStructure.GE_antitone", "CategoryTheory.Triangulated.TStructure.predicateShift_LE", "CategoryTheory.TwoSquare.guitartExact_iff_isConnected_downwards", "CategoryTheory.Under.map_obj_hom", "CategoryTheory.Under.w", "CategoryTheory.Yoneda.fullyFaithful_preimage", "CategoryTheory.areEqualizedByLocalization_iff", "CategoryTheory.asIso_inv", "CategoryTheory.associator_hom_apply", "CategoryTheory.balanced_opposite", "CategoryTheory.braiding_leftUnitor", "CategoryTheory.braiding_leftUnitor_aux\u2081", "CategoryTheory.cancel_mono", "CategoryTheory.coconeOfRepresentable_pt", "CategoryTheory.coconeOfRepresentable_\u03b9_app", "CategoryTheory.coherentTopology.mem_sieves_iff_hasEffectiveEpiFamily", "CategoryTheory.coherentTopology.mem_sieves_of_hasEffectiveEpiFamily", "CategoryTheory.coherentTopology.presheafIsLocallySurjective_iff", "CategoryTheory.comp_eq_zero_of_image_eq_kernel", "CategoryTheory.comp_ite", "CategoryTheory.composePath_comp'", "CategoryTheory.coyonedaEquiv_apply", "CategoryTheory.coyonedaEquiv_symm_map", "CategoryTheory.dcongr_arg", "CategoryTheory.epi_of_epi_fac", "CategoryTheory.epi_of_nonzero_to_simple", "CategoryTheory.eqToHom_refl", "CategoryTheory.eqToIso_map_trans", "CategoryTheory.exact_comp_iso", "CategoryTheory.exact_kernelSubobject_arrow", "CategoryTheory.extensiveTopology.isLocallySurjective_iff", "CategoryTheory.finitaryExtensive_of_preserves_and_reflects", "CategoryTheory.groupoid_of_isCodetecting_empty", "CategoryTheory.homOfElement_eq_iff", "CategoryTheory.hom_inv_id_apply", "CategoryTheory.initiallySmall_of_initial_of_essentiallySmall", "CategoryTheory.isCodetecting_unop_iff", "CategoryTheory.isFilteredOrEmpty_of_isCofilteredOrEmpty_op", "CategoryTheory.isIso_of_mono_of_epi", "CategoryTheory.isIso_of_mono_of_strongEpi", "CategoryTheory.isIso_prod_iff", "CategoryTheory.isPullback_of_cofan_isVanKampen", "CategoryTheory.isSeparating_iff_epi", "CategoryTheory.isSeparating_unop_iff", "CategoryTheory.isSeparator_unop_iff", "CategoryTheory.isSheaf_coherent", "CategoryTheory.leftDistributor_hom_comp_biproduct_\u03c0", "CategoryTheory.leftUnitor_inv_apply", "CategoryTheory.locallySmall_of_faithful", "CategoryTheory.monoidalOfHasFiniteCoproducts.associator_hom", "CategoryTheory.monoidalOfHasFiniteProducts.whiskerRight", "CategoryTheory.mop_hom_rightUnitor", "CategoryTheory.mop_tensorHom", "CategoryTheory.op_inv_leftUnitor", "CategoryTheory.predicateShift_predicateShift", "CategoryTheory.predicateShift_zero", "CategoryTheory.reflectsIsomorphisms_forget\u2082", "CategoryTheory.regularTopology.equalizerCondition_w'", "CategoryTheory.regularTopology.mem_sieves_of_hasEffectiveEpi", "CategoryTheory.regularTopology.subcanonical", "CategoryTheory.representable_of_natIso", "CategoryTheory.rightAdjointMate_comp", "CategoryTheory.shiftFunctorAdd'_add_zero", "CategoryTheory.shiftFunctorAdd'_zero_add_hom_app", "CategoryTheory.shiftFunctorAdd_add_zero_hom_app", "CategoryTheory.shiftFunctorAdd_assoc", "CategoryTheory.shiftFunctorAdd_assoc_inv_app", "CategoryTheory.shiftFunctorAdd_zero_add_inv_app", "CategoryTheory.shiftFunctorComm_eq", "CategoryTheory.tensorLeftHomEquiv_whiskerLeft_comp_evaluation", "CategoryTheory.toSheafify_sheafifyLift", "CategoryTheory.transferNatTransSelf_id", "CategoryTheory.types_comp_apply", "CategoryTheory.uncurry_expComparison", "CategoryTheory.unmop_leftUnitor", "CategoryTheory.unop_inv_associator", "CategoryTheory.whiskerRight_comp", "CategoryTheory.zag_symmetric", "CategoryTheory.zero_map", "CategoryTheory.\u03b5_hom_inv_app", "CategoryTheory.\u03b5_inv_hom_app", "CategoryTheory.\u03bc_inv_naturality\u1d63", "CategoryTheory.\u03bc_inv_naturality\u2097", "CauSeq.Completion.inv_mul_cancel", "CauSeq.cauchySeq", "CauSeq.coe_pow", "CauSeq.const_one", "CauSeq.const_pow", "CauSeq.const_sub", "CauSeq.le_antisymm", "CauSeq.le_of_exists", "CauSeq.lt_of_eq_of_lt", "CauSeq.mul_equiv_zero'", "CauSeq.rat_sup_continuous_lemma", "CauSeq.smul_equiv_smul", "CauSeq.sup_inf_distrib_left", "CauSeq.sup_lt", "Cauchy.mono", "CauchyFilter.inseparable_lim_iff", "CentroidHom.toEnd_smul", "ChainComplex.exactAt_succ_single_obj", "ChainComplex.isoHomology\u03b9\u2080_inv_naturality", "ChainComplex.single\u2080ObjXSelf", "CharP.char_ne_zero_of_finite", "CharP.quotient_iff_le_ker_natCast", "CharTwo.bit1_eq_one", "CharTwo.list_sum_mul_self", "CharTwo.neg_eq", "CharZero.charZero_iff_forall_prime_ne_zero", "CharacterModule.ext", "CircleDeg1Lift.commute_nat_add", "CircleDeg1Lift.floor_sub_le_translationNumber", "CircleDeg1Lift.iterate_pos_lt_iff", "CircleDeg1Lift.le_iterate_pos_iff", "CircleDeg1Lift.map_lt_of_translationNumber_lt_int", "CircleDeg1Lift.map_lt_of_translationNumber_lt_nat", "CircleDeg1Lift.mul_floor_map_zero_le_floor_iterate_zero", "CircleDeg1Lift.semiconj_of_isUnit_of_translationNumber_eq", "Class.coe_inter", "Class.coe_subset", "Class.eq_univ_of_forall", "Class.sUnion_empty", "ClassGroup.equiv_mk", "ClassGroup.exists_mk0_eq_mk0", "ClassGroup.mem_finsetApprox", "ClassGroup.mk0_eq_mk0_inv_iff", "Classical.not_imp", "Classical.not_not", "CliffordAlgebra.EquivEven.e0_mul_v_mul_e0", "CliffordAlgebra.EquivEven.involute_v", "CliffordAlgebra.changeForm_self_apply", "CliffordAlgebra.even.lift_\u03b9", "CliffordAlgebra.involute_prod_map_\u03b9", "CliffordAlgebra.lift_unique", "CliffordAlgebra.lift_\u03b9_apply", "CliffordAlgebra.ofBaseChange_tmul_\u03b9", "CliffordAlgebra.one_le_evenOdd_zero", "CliffordAlgebra.reverse_involute", "CliffordAlgebra.right_induction", "CliffordAlgebra.submodule_map_reverse_eq_comap", "CliffordAlgebra.toBaseChange_comp_ofBaseChange", "CliffordAlgebra.unop_reverseOp", "CliffordAlgebra.\u03b9_mul_\u03b9_add_swap_of_isOrtho", "ClopenUpperSet.ext", "ClosedEmbedding.LindelofSpace", "ClosureOperator.closure_sup_closure_le", "ClosureOperator.le_closure_iff", "CoalgEquiv.symm_toLinearEquiv", "CoalgEquiv.toEquiv_injective", "CoalgEquiv.toLinearEquiv_eq_coe", "CoalgHom.coe_comp", "CoalgHom.coe_mks", "CoalgHom.mk_coe", "CoalgHom.toLinearMap_eq_coe", "CoalgHomClass.counit_comp_apply", "CochainComplex.HomComplex.Cochain.add_v", "CochainComplex.HomComplex.Cochain.comp_assoc_of_first_is_zero_cochain", "CochainComplex.HomComplex.Cochain.comp_assoc_of_second_degree_eq_neg_third_degree", "CochainComplex.HomComplex.Cochain.comp_sub", "CochainComplex.HomComplex.Cochain.leftUnshift_leftShift", "CochainComplex.HomComplex.Cochain.ofHom_v", "CochainComplex.HomComplex.Cochain.rightShift_add", "CochainComplex.HomComplex.Cochain.rightShift_neg", "CochainComplex.HomComplex.Cochain.rightUnshift_zero", "CochainComplex.HomComplex.Cochain.units_smul_comp", "CochainComplex.HomComplex.Cochain.zero_comp", "CochainComplex.HomComplex.Cochain.\u03b4_rightUnshift", "CochainComplex.HomComplex.Cocycle.mem_iff", "CochainComplex.HomComplex.\u03b4_neg_one_cochain", "CochainComplex.ShiftSequence.shiftIso_hom_app", "CochainComplex.liftCycles_shift_homology\u03c0", "CochainComplex.mapBifunctorShift\u2081Iso_trans_mapBifunctorShift\u2082Iso", "CochainComplex.mappingCone.d_snd_v", "CochainComplex.mappingCone.liftCochain_descCochain", "CochainComplex.mappingCone.liftCochain_snd", "CochainComplex.mappingCone.map_id", "CochainComplex.mappingCone.ofHom_desc", "CochainComplex.mappingCone.quasiIso_descShortComplex", "CochainComplex.mappingCone.triangleMapOfHomotopy_comm\u2083", "CochainComplex.mappingConeCompHomotopyEquiv_comm\u2081", "CochainComplex.mappingConeHomOfDegreewiseSplitIso_inv_comp_triangle_mor\u2083", "CochainComplex.mk_X_2", "CochainComplex.prev", "CochainComplex.shiftFunctorAdd'_inv_app_f'", "CocompactMap.coe_id", "CocompactMap.isCompact_preimage_of_isClosed", "Codisjoint.map", "Codisjoint.ne_bot_of_ne_top'", "Codisjoint_comm", "CofiniteTopology.isOpen_iff", "CofiniteTopology.isOpen_iff'", "CoheytingHom.cancel_left", "CoheytingHom.cancel_right", "CommGroupWithZero.normalize_eq_one", "CommMon_.comp'", "CommRingCat.Colimits.quot_neg", "CommSemiRingCat.coe_ringHom_id", "CommSemiring.antipode_eq_id", "Commensurable.comm", "Commute.add_right", "Commute.function_commute_mul_left", "Commute.geom_sum\u2082_comm", "Commute.geom_sum\u2082_mul_add", "Commute.inv", "Commute.inv_add_inv", "Commute.inv_mul_cancel_assoc", "Commute.isNilpotent_add", "Commute.mul_self_eq_mul_self_iff", "Commute.mul_self_sub_mul_self_eq", "Commute.multiset_sum_right", "Commute.natCast_mul_natCast_mul", "Commute.neg_left", "Commute.pow_dvd_add_pow_of_pow_eq_zero_left", "Commute.pow_self", "Commute.smul_right", "Commute.unop", "Commute.zero_left", "Comon_.tensorObj_comul", "Comon_.tensorObj_counit", "Comp.seq_mk", "CompHaus.mono_iff_injective", "CompHausLike.epi_of_surjective", "CompTriple.comp", "CompactExhaustion.hasBasis_compactConvergenceUniformity", "CompactlySupportedContinuousMap.add_apply", "CompactlySupportedContinuousMap.coe_smulc", "CompactlySupportedContinuousMap.comp_id", "Compactum.cl_cl", "Complementeds.coe_injective", "Complementeds.coe_top", "Complementeds.disjoint_coe", "CompleteLattice.Independent.dfinsupp_sumAddHom_injective", "CompleteLattice.Independent.setIndependent_range", "CompleteLattice.Independent.subtype_ne_bot_le_rank", "CompleteLattice.Independent.supIndep'", "CompleteLattice.IsCompactElement.exists_finset_of_le_iSup", "CompleteLattice.WellFounded.finite_of_setIndependent", "CompleteLattice.independent_iff_dfinsupp_sumAddHom_injective", "CompleteLattice.independent_iff_pairwiseDisjoint", "CompleteLattice.independent_ne_bot_iff_independent", "CompleteLattice.independent_pempty", "CompleteLattice.sSup_continuous'", "CompleteLattice.wellFounded_characterisations", "CompleteLattice.wellFounded_iff_isSupFiniteCompact", "CompleteLatticeHom.cancel_right", "CompleteLatticeHom.coe_toBoundedLatticeHom", "Complex.Gamma_nat_eq_factorial", "Complex.Gamma_ne_zero", "Complex.Gamma_zero", "Complex.Gamma\u2102_one", "Complex.Gamma\u211d_ne_zero_of_re_pos", "Complex.HadamardThreeLines.eventuallyle", "Complex.HadamardThreeLines.norm_le_interpStrip_of_mem_verticalClosedStrip", "Complex.HadamardThreeLines.norm_le_interp_of_mem_verticalClosedStrip'", "Complex.HadamardThreeLines.norm_le_sSupNormIm", "Complex.HadamardThreeLines.sSupNormIm_eps_pos", "Complex.HadamardThreeLines.sSupNormIm_nonneg", "Complex.HasDerivAt.complex_of_real", "Complex.I_re", "Complex.IsExpCmpFilter.isLittleO_log_re_re", "Complex.UnitDisc.abs_ne_one", "Complex.UnitDisc.conj_zero", "Complex.abs_eq_sqrt_sq_add_sq", "Complex.abs_exp", "Complex.abs_im_eq_abs", "Complex.angle_mul_right", "Complex.arg_coe_angle_toReal_eq_arg", "Complex.arg_eq_nhds_of_re_neg_of_im_pos", "Complex.arg_inv_coe_angle", "Complex.arg_lt_pi_iff", "Complex.arg_neg_eq_arg_add_pi_of_im_neg", "Complex.arg_ofReal_of_nonneg", "Complex.betaIntegral_eval_one_right", "Complex.circleIntegral_div_sub_of_differentiable_on_off_countable", "Complex.circleIntegral_eq_zero_of_differentiable_on_off_countable", "Complex.circleIntegral_sub_inv_smul_of_differentiable_on_off_countable", "Complex.closure_setOf_lt_re", "Complex.compl_Iic_zero", "Complex.contDiff_sin", "Complex.continuous_re", "Complex.convexHull_reProdIm", "Complex.cos_eq_cos_iff", "Complex.cos_eq_neg_one_iff", "Complex.cos_eq_tsum'", "Complex.cos_nat_mul_two_pi_add_pi", "Complex.cos_ne_zero_iff", "Complex.cos_sub", "Complex.cos_zero", "Complex.cosh_sub_sinh", "Complex.cpow_mul_nat", "Complex.cpow_nat_mul'", "Complex.cpow_neg_one", "Complex.cpow_ofNat_mul", "Complex.differentiableOn_update_limUnder_of_isLittleO", "Complex.dist_conj_self", "Complex.dist_of_im_eq", "Complex.div_ratCast_re", "Complex.div_re", "Complex.edist_of_im_eq", "Complex.eventually_eq_or_eq_zero_of_isLocalMin_norm", "Complex.exp_add_pi_mul_I", "Complex.ext_abs_arg", "Complex.frontier_setOf_re_lt", "Complex.hasDerivAt_Gamma_one_half", "Complex.hasDerivAt_sinh", "Complex.hasStrictDerivAt_const_cpow", "Complex.im_ofReal_mul", "Complex.im_tsum", "Complex.inner", "Complex.integral_boundary_rect_eq_zero_of_differentiable_on_off_countable", "Complex.isCauSeq_abs_exp", "Complex.isCauSeq_exp", "Complex.isOpen_setOf_mem_nhds_and_isMaxOn_norm", "Complex.isometryOfOrthonormal_apply", "Complex.linearEquiv_det_conjAe", "Complex.log_exp", "Complex.log_re", "Complex.lt_def", "Complex.measurable_ofReal", "Complex.measurable_re", "Complex.mem_reProdIm", "Complex.mem_slitPlane_iff", "Complex.mul_angle_le_norm_sub", "Complex.mul_conj", "Complex.neg_im", "Complex.neg_pi_lt_arg", "Complex.neg_re_eq_abs", "Complex.nhdsWithin_stolzCone_le_nhdsWithin_stolzSet", "Complex.nndist_of_re_eq", "Complex.norm_cderiv_le", "Complex.norm_eqOn_closedBall_of_isMaxOn", "Complex.norm_log_natCast_le_rpow_div", "Complex.norm_log_sub_logTaylor_le", "Complex.norm_max_aux\u2083", "Complex.norm_natCast_cpow_of_pos", "Complex.ofRealCLM_apply", "Complex.ofReal_arctan", "Complex.ofReal_exp_ofReal_re", "Complex.ofReal_sin", "Complex.ofReal_zpow", "Complex.one_mem_slitPlane", "Complex.quotientMap_im", "Complex.range_exp_mul_I", "Complex.re_neg_ne_zero_of_one_lt_re", "Complex.restrictScalars_one_smulRight'", "Complex.sin_sub_pi", "Complex.sin_sub_two_pi", "Complex.sin_surjective", "Complex.sinh_zero", "Complex.slitPlane_ne_zero", "Complex.sub_im", "Complex.sum_div_factorial_le", "Complex.tan_conj", "Complex.tendstoUniformlyOn_deriv_of_cthickening_subset", "Complex.tendsto_limUnder_of_differentiable_on_punctured_nhds_of_isLittleO", "Complex.uniformContinuous_ringHom_eq_id_or_conj", "Complex.volume_sum_rpow_lt_one", "ComplexShape.add_rel", "Composition.invEmbedding_comp", "Composition.length_gather", "Composition.lt_sizeUpTo_index_succ", "Composition.ne_single_iff", "Composition.ones_length", "CompositionSeries.mem_eraseLast", "CompositionSeries.strictMono", "CompositionSeries.toList_sorted", "Computable.list_ofFn", "Computable.of_eq", "Computable.pred", "Computation.Equiv.trans", "Computation.LiftRelAux.ret_left", "Computation.Results.mem", "Computation.eq_thinkN'", "Computation.exists_of_liftRel_left", "Computation.get_promises", "Computation.liftRelAux_inl_inr", "Computation.liftRel_think_left", "Computation.mem_unique", "Computation.results_thinkN_pure", "Computation.results_think_iff", "Computation.terminates_parallel", "Computation.thinkN_equiv", "Con.conGen_eq", "Con.induction_on", "Con.induction_on\u2082", "Con.rel_mk", "Con.sInf_toSetoid", "ConcaveOn.add", "ConcaveOn.comp_affineMap", "ConcaveOn.convex_ge", "ConcaveOn.exists_le_of_centerMass", "ConcaveOn.slope_anti", "ConcaveOn.smul'", "Concept.bot_fst", "Concept.sSup_snd", "Concept.swap_swap", "Condensed.equalizerCondition_profinite", "Configuration.ProjectivePlane.two_lt_lineCount", "Conformal.comp", "ConjAct.smul_eq_mulAut_conj", "ConjAct.toConjAct_inv", "ContDiff.dimH_range_le", "ContDiff.exp", "ContDiff.one_of_succ", "ContDiffAt.add", "ContDiffAt.arctan", "ContDiffAt.cexp", "ContDiffAt.csin", "ContDiffAt.differentiableAt", "ContDiffAt.mem_toPartialHomeomorph_source", "ContDiffBump.convolution_eq_right", "ContDiffBump.nonneg'", "ContDiffBump.normed_le_div_measure_closedBall_rIn", "ContDiffBump.normed_sub", "ContDiffOn.arsinh", "ContDiffOn.congr", "ContDiffOn.dist", "ContDiffOn.log", "ContDiffOn.of_le", "ContDiffOn.sub", "ContDiffWithinAt.congr", "ContDiffWithinAt.div", "ContDiffWithinAt.sub", "ContMDiff.div", "ContMDiff.prod_map", "ContMDiffAt.clm_prodMap", "ContMDiffAt.mdifferentiableAt", "ContMDiffMap.mdifferentiable'", "ContMDiffOn.contMDiffOn_tangentMapWithin_aux", "ContMDiffOn.of_le", "ContMDiffOn.of_succ", "ContMDiffWithinAt.div_const", "ContMDiffWithinAt.insert", "ContMDiffWithinAt.of_succ", "ContMDiffWithinAt.prod_mk_space", "ContinuedFraction.convergents_eq_convergents'", "Continuous.IccExtend", "Continuous.ae_eq_iff_eq", "Continuous.bddAbove_range_of_hasCompactMulSupport", "Continuous.comp_lowerSemicontinuousOn", "Continuous.const_clm_comp", "Continuous.fourier_inversion_inv", "Continuous.image_connectedComponentIn_subset", "Continuous.inner", "Continuous.integrableAt_nhds", "Continuous.locallyIntegrable", "Continuous.lowerSemicontinuous", "Continuous.matrix_elem", "Continuous.stronglyMeasurable", "Continuous.subtype_val", "ContinuousAffineEquiv.constVAdd_coe", "ContinuousAffineEquiv.image_symm", "ContinuousAffineEquiv.symm_toAffineEquiv", "ContinuousAffineEquiv.trans_apply", "ContinuousAffineMap.coe_contLinear", "ContinuousAffineMap.coe_linear_eq_coe_contLinear", "ContinuousAffineMap.mk_coe", "ContinuousAffineMap.norm_image_zero_le", "ContinuousAffineMap.smul_contLinear", "ContinuousAlternatingMap.coe_toContinuousMultilinearMap", "ContinuousAlternatingMap.compContinuousLinearMap_apply", "ContinuousAlternatingMap.toAlternatingMap_injective", "ContinuousAt.cpow", "ContinuousAt.finset_sup'_apply", "ContinuousAt.mul", "ContinuousAt.prod", "ContinuousAt.rpow", "ContinuousLinearEquiv.apply_symm_apply", "ContinuousLinearEquiv.arrowCongr_apply", "ContinuousLinearEquiv.coe_funUnique", "ContinuousLinearEquiv.coe_toDiffeomorph", "ContinuousLinearEquiv.comp_hasFDerivWithinAt_iff'", "ContinuousLinearEquiv.coord_norm", "ContinuousLinearEquiv.differentiableAt", "ContinuousLinearEquiv.eq_symm_apply", "ContinuousLinearEquiv.fderivWithin", "ContinuousLinearEquiv.one_le_norm_mul_norm_symm", "ContinuousLinearMap.apply_norm_eq_sqrt_inner_adjoint_left", "ContinuousLinearMap.bound_of_antilipschitz", "ContinuousLinearMap.closed_complemented_range_of_isCompl_of_ker_eq_bot", "ContinuousLinearMap.coe_pow", "ContinuousLinearMap.coe_prodMap", "ContinuousLinearMap.coe_restrictScalarsL", "ContinuousLinearMap.coe_smul", "ContinuousLinearMap.compLeftContinuousCompact_apply", "ContinuousLinearMap.comp_hasFiniteFPowerSeriesOnBall", "ContinuousLinearMap.contMDiffAt", "ContinuousLinearMap.differentiableAt", "ContinuousLinearMap.equivProdOfSurjectiveOfIsCompl_apply", "ContinuousLinearMap.ext_iff", "ContinuousLinearMap.ext_ring", "ContinuousLinearMap.flip_smul", "ContinuousLinearMap.hasTemperateGrowth", "ContinuousLinearMap.image_rayleigh_eq_image_rayleigh_sphere", "ContinuousLinearMap.isAdjointPair_inner", "ContinuousLinearMap.isClosed_ker", "ContinuousLinearMap.isSelfAdjoint_iff_isSymmetric", "ContinuousLinearMap.is_weak_closed_closedBall", "ContinuousLinearMap.isometry_mul", "ContinuousLinearMap.le_opNNNorm", "ContinuousLinearMap.lipschitz", "ContinuousLinearMap.map_zero\u2082", "ContinuousLinearMap.mdifferentiable", "ContinuousLinearMap.nndist_le_opNNNorm", "ContinuousLinearMap.norm_adjoint_comp_self", "ContinuousLinearMap.norm_precompR_le", "ContinuousLinearMap.norm_smulRightL_le", "ContinuousLinearMap.opNNNorm_le_of_lipschitz", "ContinuousLinearMap.opNNNorm_mul_apply", "ContinuousLinearMap.opNorm_le_of_ball", "ContinuousLinearMap.opNorm_lsmul_apply_le", "ContinuousLinearMap.opNorm_subsingleton", "ContinuousLinearMap.opNorm_zero_iff", "ContinuousLinearMap.projKerOfRightInverse_apply_idem", "ContinuousLinearMap.quotientMap", "ContinuousLinearMap.reApplyInnerSelf_apply", "ContinuousLinearMap.sSup_unit_ball_eq_nnnorm", "ContinuousLinearMap.smulRight_apply", "ContinuousLinearMap.smul_comp", "ContinuousLinearMap.smul_compLpL", "ContinuousLinearMap.smul_def", "ContinuousLinearMap.tendsto_birkhoffAverage_orthogonalProjection", "ContinuousLinearMap.toSesqForm_apply_norm_le", "ContinuousLinearMap.toSpanSingleton_smul'", "ContinuousLinearMap.uniformity_eq_seminorm", "ContinuousLinearMap.zero_comp", "ContinuousMap.AlgHom.closure_ker_inter", "ContinuousMap.HomotopicRel.symm", "ContinuousMap.Homotopy.apply_zero", "ContinuousMap.Homotopy.eq_path_of_eq_image", "ContinuousMap.Homotopy.start_path", "ContinuousMap.HomotopyEquiv.coe_invFun", "ContinuousMap.HomotopyWith.coeFn_injective", "ContinuousMap.coe_zpow", "ContinuousMap.compactOpen_eq", "ContinuousMap.embedding_sigmaMk_comp", "ContinuousMap.exists_restrict_eq", "ContinuousMap.idealOfSet_closed", "ContinuousMap.ideal_isMaximal_iff", "ContinuousMap.inf'_apply", "ContinuousMap.inf_mem_subalgebra_closure", "ContinuousMap.liftCover_restrict", "ContinuousMap.mabs_apply", "ContinuousMap.not_mem_idealOfSet", "ContinuousMap.not_mem_setOfIdeal", "ContinuousMap.periodic_tsum_comp_add_zsmul", "ContinuousMap.smul_apply", "ContinuousMap.subalgebra_topologicalClosure_eq_top_of_separatesPoints", "ContinuousMap.summable_of_locally_summable_norm", "ContinuousMap.toLp_injective", "ContinuousMap.tsum_apply", "ContinuousMapZero.embedding_toContinuousMap", "ContinuousMultilinearMap.cPolyomialOn", "ContinuousMultilinearMap.coe_restrictScalars", "ContinuousMultilinearMap.le_of_opNorm_le", "ContinuousMultilinearMap.linearDeriv_apply", "ContinuousMultilinearMap.mkPiRing_zero", "ContinuousMultilinearMap.nnnorm_ofSubsingleton", "ContinuousMultilinearMap.norm_mkPiAlgebra", "ContinuousMultilinearMap.opNorm_add_le", "ContinuousMultilinearMap.opNorm_zero_iff", "ContinuousMultilinearMap.sub_apply", "ContinuousMultilinearMap.uncurryRight_norm", "ContinuousMultilinearMap.unit_le_opNorm", "ContinuousOn.continuousAt", "ContinuousOn.div_const", "ContinuousOn.exists_forall_ge'", "ContinuousOn.exists_isMinOn'", "ContinuousOn.if'", "ContinuousOn.image_uIcc_eq_Icc", "ContinuousOn.inv\u2080", "ContinuousOn.isSeparable_image", "ContinuousOn.snd", "ContinuousOn.strictAntiOn_of_injOn_Icc", "ContinuousOn.vsub", "ContinuousOpenMap.cancel_left", "ContinuousOpenMap.toFun_eq_coe", "ContinuousOrderHom.cancel_left", "ContinuousOrderHom.coe_id", "ContinuousSMul.of_basis_zero", "ContinuousWithinAt.compCM", "ContinuousWithinAt.inner", "ContinuousWithinAt.nnnorm'", "ContractingWith.aposteriori_dist_iterate_fixedPoint_le", "ContractingWith.dist_le_mul", "Convex.add", "Convex.affine_preimage", "Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt", "Convex.finsum_mem", "Convex.lipschitzOnWith_of_nnnorm_hasDerivWithin_le", "Convex.mapsTo_lineMap", "Convex.min_le_combo", "Convex.openSegment_interior_self_subset_interior", "Convex.smul_preimage", "Convex.subset_toCone", "ConvexBody.coe_add", "ConvexCone.coe_positive", "ConvexCone.coe_top", "ConvexCone.mem_closure", "ConvexCone.mem_mk", "ConvexCone.mem_strictlyPositive", "ConvexCone.pointed_of_nonempty_of_isClosed", "ConvexCone.salient_iff_not_flat", "ConvexIndependent.comp_embedding", "ConvexOn.le_max_of_mem_segment", "ConvexOn.lt_right_of_left_lt", "ConvexOn.openSegment_subset_strict_epigraph", "Coplanar.finiteDimensional_direction_affineSpan", "CoxeterMatrix.toCoxeterSystem_simple", "CoxeterSystem.IsReflection.pow_two", "CoxeterSystem.alternatingWord_succ'", "CoxeterSystem.getD_rightInvSeq", "CoxeterSystem.isLeftInversion_simple_iff_isLeftDescent", "CoxeterSystem.isReduced_take", "CoxeterSystem.isReflection_of_mem_leftInvSeq", "CoxeterSystem.length_alternatingWord", "CoxeterSystem.length_mul_mod_two", "CoxeterSystem.length_rightInvSeq", "CoxeterSystem.prod_alternatingWord_eq_mul_pow", "CoxeterSystem.prod_alternatingWord_eq_prod_alternatingWord_sub", "CoxeterSystem.simple_determines_coxeterSystem", "CoxeterSystem.simple_mul_simple_self", "CoxeterSystem.wordProd_append", "CstarRing.norm_mul_coe_unitary", "Cubic.a_of_eq", "Cubic.degree_of_d_ne_zero", "Cubic.disc_eq_prod_three_roots", "Cubic.natDegree_of_b_ne_zero", "Cubic.natDegree_of_zero", "Cycle.chain_of_pairwise", "Cycle.formPerm_subsingleton", "Cycle.lists_nil", "Cycle.map_eq_nil", "Cycle.next_reverse_eq_prev'", "Cycle.nodup_nil", "Cycle.reverse_coe", "Cycle.toFinset_eq_nil", "Cycle.toFinset_toMultiset", "Cycle.toMultiset_eq_nil", "DFA.eval_singleton", "DFinsupp.Lex.acc_of_single", "DFinsupp.add_eq_zero_iff", "DFinsupp.coe_lt_coe", "DFinsupp.coe_neg", "DFinsupp.comapDomain'_single", "DFinsupp.equivFunOnFintype_single", "DFinsupp.equivFunOnFintype_symm_single", "DFinsupp.erase_single", "DFinsupp.filter_single", "DFinsupp.induction", "DFinsupp.lt_def", "DFinsupp.mapRange_add", "DFinsupp.mapRange_smul", "DFinsupp.mem_rangeIcc_apply_iff", "DFinsupp.mk_of_not_mem", "DFinsupp.mk_zero", "DFinsupp.neLocus_sub_right", "DFinsupp.not_mem_neLocus", "DFinsupp.sigmaCurry_smul", "DFinsupp.single_eq_of_ne", "DFinsupp.single_left_injective", "DFinsupp.single_sub", "DFinsupp.smul_sum", "DFinsupp.subtypeDomain_sub", "DFinsupp.sup_apply", "DFinsupp.support_inf", "DFinsupp.support_mapRange", "DFinsupp.support_smul", "DFinsupp.support_subset_iff", "DFinsupp.support_sup", "DFinsupp.toMultiset_injective", "DFinsupp.update_eq_sub_add_single", "DFinsupp.wellFoundedLT", "DFinsupp.zsmul_apply", "DFunLike.ext'_iff", "DFunLike.subsingleton_cod", "DMatrix.map_add", "Decidable.le_iff_eq_or_lt", "Decidable.le_iff_lt_or_eq", "Decidable.not_forall\u2082", "DedekindDomain.ProdAdicCompletions.IsFiniteAdele.zero", "Dense.diff_finset", "Dense.smul", "DenseInducing.continuous_extend", "DenseInducing.mk'", "DenseRange.nonempty", "DenseRange.prod_map", "DenseRange.topologicalClosure_map_submodule", "Denumerable.list_ofNat_zero", "Derivation.coe_mk'", "Derivation.map_add", "Derivation.zero_apply", "DerivedCategory.HomologySequence.exact\u2081", "DiffContOnCl.mono", "Diffeomorph.contDiff", "Diffeomorph.contMDiffAt", "Diffeomorph.isLocalDiffeomorph", "Differentiable.comp_differentiableOn", "Differentiable.cos", "Differentiable.dist", "Differentiable.inverse", "Differentiable.log", "Differentiable.sin", "DifferentiableAt.div_const", "DifferentiableAt.hasFDerivAt", "DifferentiableAt.sinh", "DifferentiableOn.diffContOnCl_ball", "DifferentiableOn.div_const", "DifferentiableOn.log", "DifferentiableWithinAt.arctan", "DifferentiableWithinAt.ccos", "DifferentiableWithinAt.continuousWithinAt", "DifferentiableWithinAt.cosh", "DifferentiableWithinAt.zpow", "DihedralGroup.card_conjClasses_odd", "DihedralGroup.commProb_cons", "DihedralGroup.orderOf_r", "Dilation.congr_fun", "Dilation.id_comp", "Dilation.mapsTo_sphere", "Dilation.ratio_comp", "Dilation.ratio_unique", "DilationEquiv.symm_symm", "DilationEquiv.trans_refl", "Dioph.DiophList.forall", "Dioph.reindex_diophFn", "Dioph.sub_dioph", "DirectSum.Gmodule.one_smul'", "DirectSum.IsInternal.collectedBasis_mem", "DirectSum.IsInternal.submodule_independent", "DirectSum.coeAddMonoidHom_of", "DirectSum.coeLinearMap_of", "DirectSum.coe_decompose_mul_of_right_mem_of_not_le", "DirectSum.coe_of_mul_apply_of_not_le", "DirectSum.coe_toModule_eq_coe_toAddMonoid", "DirectSum.decomposeAlgEquiv_symm_apply", "DirectSum.decompose_of_mem_ne", "DirectSum.decompose_one", "DirectSum.fromAddMonoid_of", "DirectSum.lie_of", "DirectSum.list_prod_ofFn_of_eq_dProd", "DirectSum.sigmaCurry_apply", "DirectSum.sigmaLuncurry_apply", "DirectSum.toAddMonoid.unique", "Directed.exists_mem_subset_of_finset_subset_biUnion", "Directed.iSup_le", "DirectedOn.sSup_le", "DirichletCharacter.LSeriesSummable_mul", "DirichletCharacter.LSeriesSummable_twist_vonMangoldt", "DirichletCharacter.mul_convolution_distrib", "DiscreteQuotient.LEComap.mono", "DiscreteQuotient.ofLE_proj", "DiscreteQuotient.proj_surjective", "DiscreteValuationRing.addVal_eq_top_iff", "DiscreteValuationRing.associated_pow_irreducible", "DiscreteValuationRing.irreducible_of_span_eq_maximalIdeal", "DiscreteValuationRing.not_isField", "Disjoint.exists_thickenings", "Disjoint.inf_left", "Disjoint.inter_eq", "Disjoint.inter_right'", "Disjoint.mono", "Disjoint.symmDiff_right", "DistribMulActionHom.one_apply", "DomMulAct.coe_smul_addMonoidHom", "DomMulAct.mk_mul", "DomMulAct.mk_smul_mulDistribActionHom_apply", "DomMulAct.smul_linearMap_apply", "Doset.disjoint_out'", "Doset.rel_bot_eq_right_group_rel", "Doset.right_bot_eq_right_quot", "DoubleCentralizer.ext", "DoubleCentralizer.nnnorm_fst", "DoubleCentralizer.norm_fst", "DoubleCentralizer.range_toProd", "DoubleQuot.quotQuotEquivComm_symm", "DoubleQuot.quotQuotToQuotSup\u2090_toRingHom", "Down.mem_compression", "Down.mem_compression_of_insert_mem_compression", "DualNumber.algHom_ext", "DualNumber.eps_mul_eps", "DualNumber.lift_apply_eps", "Dvd.elim_left", "Dvd.intro", "EMetric.diam_pos_iff", "EMetric.diam_union", "EMetric.exists_continuous_nnreal_forall_closedBall_subset", "EMetric.exists_forall_closedBall_subset_aux\u2081", "EMetric.hausdorffEdist_image", "EMetric.mem_ball", "EMetric.mem_closedBall_self", "EMetric.mem_nhdsWithin_iff", "EMetric.nhds_basis_eball", "EMetric.tendsto_nhds", "EMetric.uniformInducing_iff", "ENNReal.add_halves", "ENNReal.add_lt_add_left", "ENNReal.aemeasurable_of_exist_almost_disjoint_supersets", "ENNReal.biSup_add'", "ENNReal.coe_finset_sup", "ENNReal.coe_inv_le", "ENNReal.coe_lt_one_iff", "ENNReal.coe_mem_upperBounds", "ENNReal.coe_natCast", "ENNReal.coe_ofNat", "ENNReal.coe_pow", "ENNReal.continuous_coe_iff", "ENNReal.continuous_const_mul", "ENNReal.dichotomy", "ENNReal.div_lt_of_lt_mul'", "ENNReal.div_mul_cancel", "ENNReal.div_zero", "ENNReal.exists_frequently_lt_of_liminf_ne_top'", "ENNReal.exists_inv_two_pow_lt", "ENNReal.exists_nat_gt", "ENNReal.hasSum_iff_tendsto_nat", "ENNReal.iSup_add_iSup_of_monotone", "ENNReal.iSup_coe_lt_top", "ENNReal.iSup_ne_top", "ENNReal.inv_map_iInf", "ENNReal.inv_pow", "ENNReal.le_ofReal_iff_toReal_le", "ENNReal.le_of_forall_pos_le_add", "ENNReal.lintegral_rpow_add_le_add_snorm_mul_lintegral_rpow_add", "ENNReal.log_bot_lt_iff", "ENNReal.log_homeomorph_apply", "ENNReal.log_one", "ENNReal.log_pow", "ENNReal.lt_rpow_one_div_iff", "ENNReal.max_mul", "ENNReal.mem_Iio_self_add", "ENNReal.monotone_truncateToReal", "ENNReal.mul_div_le", "ENNReal.mul_div_mul_left", "ENNReal.mul_lt_mul", "ENNReal.natCast_le_ofReal", "ENNReal.nhdsWithin_Ioi_ofNat_nebot", "ENNReal.ofReal_le_ofReal", "ENNReal.ofReal_le_ofReal_iff", "ENNReal.ofReal_pos", "ENNReal.one_le_rpow", "ENNReal.one_lt_inv", "ENNReal.one_rpow", "ENNReal.rpow_one", "ENNReal.rpow_sum_le_const_mul_sum_rpow", "ENNReal.sSup_add", "ENNReal.some_eq_coe'", "ENNReal.sub_half", "ENNReal.sub_lt_of_sub_lt", "ENNReal.sub_mul", "ENNReal.sum_lt_sum_of_nonempty", "ENNReal.summable", "ENNReal.tendsto_ofReal_atTop", "ENNReal.toNNReal_lt_toNNReal", "ENNReal.toReal_iInf", "ENNReal.toReal_le_coe_of_le_coe", "ENNReal.toReal_le_of_le_ofReal", "ENNReal.toReal_lt_toReal", "ENNReal.toReal_max", "ENNReal.toReal_nat", "ENNReal.toReal_nonneg", "ENNReal.toReal_ofReal", "ENNReal.toReal_ofReal'", "ENNReal.toReal_pos_iff", "ENNReal.top_ne_coe", "ENNReal.top_rpow_of_pos", "ENNReal.tsum_coe_eq", "ENNReal.tsum_coe_ne_top_iff_summable", "ENNReal.tsum_const_eq_top_of_ne_zero", "ENNReal.tsum_const_smul", "ENNReal.tsum_eq_add_tsum_ite", "ENNReal.tsum_eq_iSup_sum", "ENNReal.tsum_toNNReal_eq", "ENNReal.young_inequality", "ENat.coe_toNatHom", "ENat.recTopCoe_top", "ENat.toENNReal_max", "ENat.zero_lt_top", "EReal.add_eq_bot_iff", "EReal.bot_div_of_pos_ne_top", "EReal.coe_ennreal_eq_top_iff", "EReal.coe_ennreal_ofReal", "EReal.coe_ennreal_pow", "EReal.coe_neg", "EReal.coe_one", "EReal.induction\u2082_symm", "EReal.monotone_div_right_of_nonneg", "EReal.mul_div", "EReal.neg_strictAnti", "EReal.neg_top", "EReal.nhds_top'", "EReal.preimage_coe_Ioo_bot_top", "EReal.preimage_coe_Ioo_top", "EReal.sub_top", "EReal.toReal_top", "EReal.top_add_iff_ne_bot", "EReal.top_add_top", "EReal.top_mul_of_pos", "EReal.top_sub_bot", "EisensteinSeries.norm_eq_max_natAbs", "EllipticCurve.coe_inv_variableChange_\u0394'", "EllipticCurve.ofJ_0_of_three_eq_zero", "EllipticCurve.ofJ_j", "EllipticCurve.variableChange_j", "Embedding.closure_eq_preimage_closure_image", "Embedding.continuous_iff", "Embedding.measurableEmbedding", "Encodable.decode_list_zero", "Encodable.decode_unit_succ", "Encodable.encode_inj", "Encodable.mem_decode\u2082'", "Eq.trans", "EqualCharZero.of_algebraRat", "EqualCharZero.of_not_mixedCharZero", "EqualCharZero.pnatCast_eq_natCast", "Equicontinuous.tendsto_uniformFun_iff_pi", "EquicontinuousOn.comp", "EquicontinuousOn.mono", "Equiv.Perm.Disjoint.mem_imp", "Equiv.Perm.IsCycleOn.countable", "Equiv.Perm.IsSwap.of_subtype_isSwap", "Equiv.Perm.SameCycle.conj", "Equiv.Perm.SameCycle.cycleOf_apply", "Equiv.Perm.card_support_extend_domain", "Equiv.Perm.cycleOf_self_apply_zpow", "Equiv.Perm.cycleType_eq_zero", "Equiv.Perm.cycleType_le_of_mem_cycleFactorsFinset", "Equiv.Perm.cycleType_of_card_le_mem_cycleType_add_two", "Equiv.Perm.disjoint_iff_eq_or_eq", "Equiv.Perm.extendDomain_apply_not_subtype", "Equiv.Perm.extendDomain_mul", "Equiv.Perm.isConj_iff_cycleType_eq", "Equiv.Perm.isCycleOn_inv", "Equiv.Perm.isThreeCycle_sq_of_three_mem_cycleType_five", "Equiv.Perm.lcm_cycleType", "Equiv.Perm.mclosure_isSwap", "Equiv.Perm.mem_iff_ofSubtype_apply_mem", "Equiv.Perm.mem_support_cycleOf_iff'", "Equiv.Perm.mul_symm", "Equiv.Perm.ofSubtype_apply_of_mem", "Equiv.Perm.prod_comp", "Equiv.Perm.prod_comp'", "Equiv.Perm.signBijAux_surj", "Equiv.Perm.sign_ofSubtype", "Equiv.Perm.sign_one", "Equiv.Perm.sign_prodExtendRight", "Equiv.Perm.subsingleton_eq_refl", "Equiv.Perm.support_conj", "Equiv.Perm.support_prod_le", "Equiv.Perm.swap_induction_on", "Equiv.Perm.toCycle_eq_toList", "Equiv.Perm.viaFintypeEmbedding_apply_mem_range", "Equiv.Set.image_symm_apply", "Equiv.Set.rangeInr_apply_inr", "Equiv.ext_iff", "Equiv.finite_iff", "Equiv.forall_congr_right", "Equiv.hasProd_iff", "Equiv.inv_addLeft", "Equiv.mulEquiv_symm_apply", "Equiv.multipliable_iff_of_mulSupport", "Equiv.ofLeftInverse'_eq_ofInjective", "Equiv.optionEquivSumPUnit_none", "Equiv.optionEquivSumPUnit_some", "Equiv.option_symm_apply_none_iff", "Equiv.polishSpace_induced", "Equiv.pow_def", "Equiv.prodSumDistrib_apply_left", "Equiv.refl_apply", "Equiv.removeNone_aux_inv", "Equiv.subtypeQuotientEquivQuotientSubtype_mk", "Equiv.sumAssoc_symm_apply_inr_inr", "Equiv.sumProdDistrib_symm_apply_left", "Equiv.surjective_comp", "Equiv.symm_trans_swap_trans", "Equiv.toEmbedding_apply", "Equiv.toFun_inducedStructureEquiv_Symm", "Equiv.trans_assoc", "Equiv.trans_toPartialEquiv", "Equiv.uncountable_iff", "Equiv.uniformEmbedding", "Equiv.zpow_addLeft", "Euclidean.ball_subset_closedBall", "EuclideanDomain.dvd_lcm_left", "EuclideanDomain.gcd_eq_gcd_ab", "EuclideanDomain.gcd_val", "EuclideanDomain.lcm_dvd_iff", "EuclideanDomain.mod_add_div'", "EuclideanDomain.val_dvd_le", "EuclideanDomain.xgcd_val", "EuclideanGeometry.OrthocentricSystem.exists_circumradius_eq", "EuclideanGeometry.Sphere.center_ne_iff_ne_of_mem", "EuclideanGeometry.Sphere.dist_div_sin_oangle_eq_two_mul_radius", "EuclideanGeometry.Sphere.tan_div_two_smul_rotation_pi_div_two_vadd_midpoint_eq_center", "EuclideanGeometry.abs_oangle_left_toReal_lt_pi_div_two_of_dist_eq", "EuclideanGeometry.affineSpan_of_orthocentricSystem", "EuclideanGeometry.angle_eq_arcsin_of_angle_eq_pi_div_two", "EuclideanGeometry.center_eq_inversion'", "EuclideanGeometry.cos_eq_one_iff_angle_eq_zero", "EuclideanGeometry.cos_oangle_right_of_oangle_eq_pi_div_two", "EuclideanGeometry.dist_affineCombination", "EuclideanGeometry.dist_center_eq_dist_center_of_mem_sphere'", "EuclideanGeometry.dist_eq_iff_dist_orthogonalProjection_eq", "EuclideanGeometry.dist_smul_vadd_eq_dist", "EuclideanGeometry.exists_circumradius_eq_of_cospherical_subset", "EuclideanGeometry.exists_circumsphere_eq_of_cospherical", "EuclideanGeometry.exists_dist_eq_circumradius_of_subset_insert_orthocenter", "EuclideanGeometry.inner_pos_or_eq_of_dist_le_radius", "EuclideanGeometry.inner_vsub_vsub_of_mem_sphere_of_mem_sphere", "EuclideanGeometry.inter_eq_singleton_orthogonalProjection", "EuclideanGeometry.inversion_of_mem_sphere", "EuclideanGeometry.oangle_ne_zero_and_ne_pi_iff_affineIndependent", "EuclideanGeometry.reflection_eq_iff_orthogonalProjection_eq", "EuclideanGeometry.reflection_eq_self_iff", "EuclideanGeometry.right_ne_of_oangle_sign_eq_neg_one", "EuclideanGeometry.right_ne_of_oangle_sign_ne_zero", "EuclideanGeometry.sin_oangle_left_mul_dist_of_oangle_eq_pi_div_two", "EuclideanGeometry.sin_oangle_right_mul_dist_of_oangle_eq_pi_div_two", "EuclideanGeometry.tan_oangle_right_mul_dist_of_oangle_eq_pi_div_two", "EuclideanGeometry.vsub_orthogonalProjection_mem_direction", "EuclideanGeometry.wbtw_of_collinear_of_dist_center_le_radius", "EuclideanSpace.closedBall_zero_eq", "EulerProduct.prod_primesBelow_tsum_eq_tsum_smoothNumbers", "EulerSine.integral_cos_mul_cos_pow_aux", "EventuallyEq.countable_iInter", "EventuallyLE.countable_bUnion", "EventuallyMeasurable.congr", "ExceptT.goto_mkLabel", "ExistsContDiffBumpBase.u_le_one", "ExistsContDiffBumpBase.u_support", "ExistsContDiffBumpBase.y_le_one", "ExteriorAlgebra.induction", "ExteriorAlgebra.lhom_ext", "ExteriorAlgebra.map_apply_\u03b9Multi", "ExteriorAlgebra.\u03b9Inv_comp_map", "ExteriorAlgebra.\u03b9_add_mul_swap", "ExteriorAlgebra.\u03b9_mul_prod_list", "FdRep.char_mul_comm", "FermatLastTheoremForThreeGen.Solution.two_le_multiplicity", "FermatLastTheoremForThreeGen.ex_cube_add_cube_eq_and_isCoprime_and_not_dvd_and_dvd", "FermatLastTheoremForThreeGen.lambda_sq_dvd_c", "FiberBundle.chartedSpace_chartAt_symm_fst", "FiberBundle.surjective_proj", "Field.DirectLimit.inv_mul_cancel", "Field.DirectLimit.mul_inv_cancel", "Field.closure.isSubfield", "Field.closure.isSubmonoid", "Field.closure_subset", "Field.exists_primitive_element_of_finite_intermediateField", "Field.finSepDegree_mul_finSepDegree_of_isAlgebraic", "Field.sepDegree_mul_sepDegree_of_isAlgebraic", "Filter.EventuallyEq.differentiableWithinAt_iff", "Filter.EventuallyEq.fderivWithin_eq", "Filter.EventuallyEq.fun_comp", "Filter.EventuallyEq.inv", "Filter.EventuallyEq.iteratedFDerivWithin_eq", "Filter.EventuallyEq.self_of_nhdsSet", "Filter.EventuallyLE.inter", "Filter.Germ.const_div", "Filter.Germ.inductionOn\u2083", "Filter.Germ.map_id", "Filter.HasAntitoneBasis.comap", "Filter.HasBasis.clusterPt_iff", "Filter.HasBasis.comp_surjective", "Filter.HasBasis.equicontinuousWithinAt_iff", "Filter.HasBasis.forall_mem_mem", "Filter.HasBasis.inf", "Filter.HasBasis.inf_basis_neBot_iff", "Filter.HasBasis.le_basis_iff", "Filter.HasBasis.limsInf_eq_iSup_sInf", "Filter.HasBasis.nhds_of_one", "Filter.HasBasis.sup_principal", "Filter.HasBasis.tendsto_right_iff", "Filter.HasBasis.uniformContinuous_iff", "Filter.HasBasis.uniformEquicontinuousOn_iff_right", "Filter.HasBasis.uniformSpace_eq_bot", "Filter.IsBasis.filter_eq_generate", "Filter.IsBoundedUnder.bddAbove_range_of_cofinite", "Filter.IsBoundedUnder.mono_le", "Filter.NeBot.div", "Filter.NeBot.inv", "Filter.OrderIso.apply_blimsup", "Filter.Subsingleton.atBot_eq", "Filter.Tendsto.Ico", "Filter.Tendsto.atTop_div_const", "Filter.Tendsto.atTop_nsmul_neg_const", "Filter.Tendsto.congr_dist", "Filter.Tendsto.dist", "Filter.Tendsto.div_atTop", "Filter.Tendsto.eventually_forall_ge_atTop", "Filter.Tendsto.finset_sup_nhds_apply", "Filter.Tendsto.frequently_map", "Filter.Tendsto.fst", "Filter.Tendsto.inner", "Filter.Tendsto.inv_inv", "Filter.Tendsto.inv_tendsto_atTop", "Filter.Tendsto.nnnorm'", "Filter.Tendsto.subseq_mem_entourage", "Filter.Tendsto.update", "Filter.Tendsto.zero_mul_isBoundedUnder_le", "Filter.bddBelow_range_of_tendsto_atTop_atTop", "Filter.bliminf_eq_iSup_biInf", "Filter.bliminf_eq_liminf", "Filter.bliminf_false", "Filter.bliminf_or_le_inf", "Filter.bot_div", "Filter.cardinal_iInter_mem", "Filter.closure_singleton", "Filter.comap_abs_atTop", "Filter.comap_comm", "Filter.comap_embedding_atTop", "Filter.comap_neBot_iff_frequently", "Filter.comap_sSup", "Filter.coprod\u1d62_neBot_iff'", "Filter.disjoint_cofinite_right", "Filter.disjoint_prod", "Filter.empty_not_mem", "Filter.eventuallyConst_pred", "Filter.eventuallyEq_inf_principal_iff", "Filter.eventually_atBot_prod_self", "Filter.eventually_bot", "Filter.eventually_congr", "Filter.eventually_ge_atTop", "Filter.eventually_iSup", "Filter.eventually_iff", "Filter.eventually_one", "Filter.eventually_principal", "Filter.eventually_smallSets", "Filter.eventually_sub_nonneg", "Filter.frequently_cocardinal_mem", "Filter.frequently_curry_iff", "Filter.frequently_or_distrib", "Filter.frequently_prod_and", "Filter.generate_union", "Filter.hasBasis_biInf_principal", "Filter.hasBasis_cocardinal", "Filter.hasBasis_generate", "Filter.hasBasis_iInf_principal_finite", "Filter.iInf_neBot_iff_of_directed", "Filter.inter_eventuallyEq_left", "Filter.inter_mem", "Filter.inter_mem_inf", "Filter.inv.instNeBot", "Filter.inv_mem_inv", "Filter.isBounded_iff", "Filter.isBounded_le_map_of_bounded_range", "Filter.ker_pure", "Filter.le_div_iff", "Filter.le_prod", "Filter.lift'_inf_principal_eq", "Filter.lift'_map_le", "Filter.lift_iInf_of_map_univ", "Filter.lift_lift'_assoc", "Filter.lift_lift'_same_le_lift'", "Filter.lift_principal", "Filter.liminf_eq_iSup_iInf", "Filter.liminf_eq_iSup_iInf_of_nat", "Filter.limsSup_top", "Filter.limsup_const", "Filter.limsup_eq", "Filter.limsup_le_iSup", "Filter.limsup_piecewise", "Filter.map_eq_comap_of_inverse", "Filter.map_eq_map_iff_of_injOn", "Filter.map_id", "Filter.map_map\u2082_distrib_left", "Filter.map_neg_atBot", "Filter.map_prod_map_const_id_principal_coprod_principal", "Filter.map_pure", "Filter.map_sub_atTop_eq_nat", "Filter.map_swap4_eq_comap", "Filter.map\u2082_distrib_le_right", "Filter.map\u2082_left_comm", "Filter.map\u2082_mono_left", "Filter.map\u2082_vsub", "Filter.mem_coclosedCompact", "Filter.mem_cofinite", "Filter.mem_generate_of_mem", "Filter.mem_iInf'", "Filter.mem_iff_ultrafilter", "Filter.mem_lift_sets", "Filter.mem_map_seq_iff", "Filter.mem_top", "Filter.mono_blimsup'", "Filter.monotone_lift", "Filter.monotone_nhds", "Filter.mp_mem", "Filter.mul_add_subset", "Filter.mul_eq_one_iff", "Filter.mul_pure", "Filter.nhds_eq", "Filter.not_neBot", "Filter.ofSets_filter_eq_generate", "Filter.pi_inf_principal_pi_eq_bot", "Filter.pi_mono", "Filter.principal_coprod_principal", "Filter.principal_eq_iff_eq", "Filter.prod_comap_comap_eq", "Filter.prod_comm'", "Filter.prod_map_left", "Filter.prod_same_eq", "Filter.pure_seq_eq_map", "Filter.sInter_nhds", "Filter.set_eventuallyLE_iff_mem_inf_principal", "Filter.smallSets_top", "Filter.smul_filter_eq_bot_iff", "Filter.subseq_tendsto_of_neBot", "Filter.tendsto_Ioi_atTop", "Filter.tendsto_atBot_add_nonpos_left", "Filter.tendsto_atBot_atBot_of_monotone", "Filter.tendsto_atTop_add_const_left", "Filter.tendsto_atTop_atBot_of_antitone", "Filter.tendsto_atTop_diagonal", "Filter.tendsto_comap", "Filter.tendsto_comap_iff", "Filter.tendsto_comp_val_Iic_atBot", "Filter.tendsto_congr", "Filter.tendsto_const_mul_atBot_iff_neg", "Filter.tendsto_const_mul_atBot_iff_pos", "Filter.tendsto_const_mul_atTop_iff_pos", "Filter.tendsto_finset_preimage_atTop_atTop", "Filter.tendsto_id'", "Filter.tendsto_mul_const_atTop_of_neg", "Filter.tendsto_mul_const_atTop_of_pos", "Filter.tendsto_neg_atTop_iff", "Filter.tendsto_nhds_min_right", "Filter.tendsto_of_subseq_tendsto", "Filter.vsub_neBot_iff", "FilterBasis.generate", "Fin.append_cast_right", "Fin.append_rev", "Fin.bot_eq_zero", "Fin.card_filter_univ_succ", "Fin.card_fintype_uIcc", "Fin.castLE_inj", "Fin.castLE_rfl", "Fin.castPred_succAbove_castPred", "Fin.cast_eq_self", "Fin.cast_zero", "Fin.cons_zero", "Fin.dist_insertNth_insertNth", "Fin.eq_zero", "Fin.equiv_iff_eq", "Fin.exists_eq_add_of_le", "Fin.foldl_loop_eq", "Fin.foldr_succ", "Fin.getElem_list", "Fin.init_snoc", "Fin.le_iff_val_le_val", "Fin.mem_piFinset_succ", "Fin.modNat_rev", "Fin.mul_one'", "Fin.nonempty_embedding_iff", "Fin.ofNat''_eq_cast", "Fin.pi_lex_lt_cons_cons", "Fin.pos_iff_nonempty", "Fin.predAbove_castPred_self", "Fin.predAbove_last_castSucc", "Fin.predAbove_last_of_ne_last", "Fin.pred_last", "Fin.pred_succ", "Fin.pred_succAbove_pred", "Fin.prod_univ_two", "Fin.rev_involutive", "Fin.rev_le_iff", "Fin.rev_surjective", "Fin.snocCases_snoc", "Fin.snoc_mem_piFinset_snoc", "Fin.strictMono_addNat", "Fin.succAbove_of_lt_succ", "Fin.succAbove_succ_of_lt", "Fin.succAbove_succ_self", "Fin.succ_apply", "Fin.succ_one_eq_two'", "Fin.succ_succAbove_zero", "Fin.succ_zero_eq_one", "Fin.succ_zero_eq_one'", "Fin.sum_const", "Fin.sum_pow_mul_eq_add_pow", "Fin.tail_init_eq_init_tail", "Fin.zero_mul'", "FinEnum.mem_toList", "FinVec.map_eq", "Finite.card_eq_zero_iff", "Finite.card_eq_zero_of_injective", "Finite.card_eq_zero_of_surjective", "Finite.card_le_of_injective", "Finite.card_option", "Finite.exists_max", "Finite.induction_empty_option", "Finite.injective_iff_surjective", "FiniteDimensional.finrank_finsupp_self", "FiniteDimensional.finrank_pos_iff", "FiniteDimensional.of_locallyCompactSpace", "FiniteDimensional.rank_linearMap_self", "FiniteField.X_pow_card_sub_X_ne_zero", "FiniteField.even_card_iff_char_two", "FiniteField.odd_card_of_char_ne_two", "FiniteField.sum_subgroup_units", "Finmap.insert_singleton_eq", "Finmap.keys_empty", "Finmap.keys_erase_toFinset", "Finmap.keys_ext", "Finmap.lookup_isSome", "Finmap.mem_lookup_union", "Finmap.mem_union", "Finmap.toFinmap_cons", "Finpartition.IsEquipartition.card_biUnion_offDiag_le", "Finpartition.exists_le_of_le", "Finpartition.mem_part_ofSetoid_iff_rel", "Finpartition.parts_eq_empty_iff", "Finpartition.parts_nonempty_iff", "Finpartition.sum_card_parts", "Finset.Colex.toColex_sdiff_le_toColex_sdiff", "Finset.Icc_subset_Iic_self", "Finset.Ico_eq_filter_ssubsets", "Finset.Ico_filter_le_left", "Finset.Ico_filter_lt", "Finset.Ico_subset_Ici_self", "Finset.Iic_eq_Icc", "Finset.Ioc_subset_Ioo_right", "Finset.Ioo_subset_Ioo_right", "Finset.Nonempty.one_mem_div", "Finset.Nonempty.zero_mul", "Finset.Shatters.subset_iff", "Finset.SupIndep.product", "Finset.Superset.trans", "Finset.antidiagonal_congr'", "Finset.antitone_iff_forall_insert_le", "Finset.attach_map_val", "Finset.biUnion_mono", "Finset.biUnion_singleton_eq_self", "Finset.card_biUnion_le_card_mul", "Finset.card_eq_of_bijective", "Finset.card_eq_sum_ones", "Finset.card_image\u2082_iff", "Finset.card_le_mul_card_image_of_maps_to", "Finset.card_le_one_iff", "Finset.card_le_one_iff_subset_singleton", "Finset.card_map", "Finset.card_mul_eq_card_mul", "Finset.card_mul_singleton", "Finset.card_range", "Finset.card_sdiff", "Finset.card_subtype", "Finset.card_union_le", "Finset.centerMass_le_sup", "Finset.centerMass_subset", "Finset.centroid_pair_fin", "Finset.centroid_univ", "Finset.coe_disjUnion", "Finset.coe_pow", "Finset.coe_range", "Finset.coe_singleton", "Finset.compl_insert", "Finset.compl_singleton", "Finset.compls_empty", "Finset.disjSum_empty", "Finset.disjSups_assoc", "Finset.disjiUnion_val", "Finset.disjoint_erase_comm", "Finset.disjoint_of_erase_left", "Finset.disjoint_sdiff_inter", "Finset.disjoint_singleton_left", "Finset.disjoint_sup_right", "Finset.dvd_lcm", "Finset.empty_ssubset_singleton", "Finset.eq_of_mem_slice", "Finset.eq_singleton_iff_nonempty_unique_mem", "Finset.erase_biUnion", "Finset.erase_eq_empty_iff", "Finset.eventually_cofinite_nmem", "Finset.exists_card_eq", "Finset.exists_eq_pow_of_mul_eq_pow_of_coprime", "Finset.exists_subset_card_eq", "Finset.exists_sum_fiber_lt_of_maps_to_of_sum_lt_nsmul", "Finset.exists_superset_card_eq", "Finset.filter_filter", "Finset.filter_mem_eq_inter", "Finset.filter_not", "Finset.filter_subset", "Finset.finsuppAntidiag_empty_of_ne_zero", "Finset.fold_const", "Finset.fold_hom", "Finset.fold_max_add", "Finset.forall_mem_not_eq", "Finset.gcd_eq_gcd_image", "Finset.gcd_eq_zero_iff", "Finset.gcd_insert", "Finset.geomSum_ofColex_strictMono", "Finset.iInf_coe", "Finset.iInf_option_toFinset", "Finset.iSup_finset_image", "Finset.iSup_insert", "Finset.iSup_insert_update", "Finset.image_add_left_Ico", "Finset.image_diag_union_image_offDiag", "Finset.image_injective", "Finset.image_subset_image", "Finset.image\u2082_eq_empty_iff", "Finset.image\u2082_insert_left", "Finset.inf'_comp_eq_map", "Finset.inf'_image", "Finset.inf_mul_right", "Finset.inf_prodMap", "Finset.infs_subset_iff", "Finset.inr_mem_sumLexLift", "Finset.insertNone_eraseNone", "Finset.insert_inter_of_not_mem", "Finset.insert_sdiff_of_not_mem", "Finset.inter_eq_left", "Finset.inter_eq_right", "Finset.inter_union_distrib_right", "Finset.inv_op_smul_finset_distrib\u2080", "Finset.inv_univ", "Finset.isG\u03b4_compl", "Finset.lcm_mono_fun", "Finset.lcm_union", "Finset.le_inf'_image\u2082", "Finset.left_mem_uIcc", "Finset.lt_min'_iff", "Finset.map_cast_heq", "Finset.map_cons", "Finset.map_refl", "Finset.map_toDual_min", "Finset.max'_le_iff", "Finset.max_mem_image_coe", "Finset.mem_Ioo", "Finset.mem_attachFin", "Finset.mem_def", "Finset.mem_filter", "Finset.mem_image_of_mem", "Finset.mem_mul", "Finset.mem_powerset", "Finset.mem_range_iff_mem_finset_range_of_mod_eq", "Finset.mem_singleton_self", "Finset.mem_sort", "Finset.mem_sym_iff", "Finset.mem_uIcc", "Finset.memberSubfamily_memberSubfamily", "Finset.min'_lt_max'", "Finset.min_mono", "Finset.mulEnergy_empty_right", "Finset.mulEnergy_eq_card_filter", "Finset.nonMemberSubfamily_inter", "Finset.noncommProd_mul_distrib_aux", "Finset.noncommProd_singleton", "Finset.nonempty_coe_sort", "Finset.not_mem_of_lt_min", "Finset.not_mem_sdiff_of_mem_right", "Finset.ofDual_max'", "Finset.offDiag_card", "Finset.offDiag_singleton", "Finset.one_lt_card_iff", "Finset.one_mem_one", "Finset.op_smul_finset_subset_mul", "Finset.ordConnected_range_val", "Finset.orderEmbOfFin_apply", "Finset.pair_comm", "Finset.piecewise_cases", "Finset.piecewise_eq_of_mem", "Finset.piecewise_le_of_le_of_le", "Finset.piecewise_singleton", "Finset.pimage_congr", "Finset.pimage_inter", "Finset.pimage_subset", "Finset.pow_card_mul_prod", "Finset.pow_eq_prod_const", "Finset.powersetCard_eq_empty", "Finset.powerset_empty", "Finset.powerset_eq_univ", "Finset.preimage_mul_right_one", "Finset.prod_Icc_succ_top", "Finset.prod_Ico_add'", "Finset.prod_Ico_id_eq_factorial", "Finset.prod_antidiagonal_pow_choose_succ", "Finset.prod_bij'", "Finset.prod_div_prod_mem_mulSpan", "Finset.prod_eq_prod_Ico_succ_bot", "Finset.prod_erase", "Finset.prod_extend_by_one", "Finset.prod_finset_product_right", "Finset.prod_fn", "Finset.prod_inv_distrib", "Finset.prod_lt_prod_of_nonempty", "Finset.prod_mono_set'", "Finset.prod_mulIndicator_eq_prod_inter", "Finset.prod_nbij'", "Finset.prod_powersetCard", "Finset.prod_product_right'", "Finset.prod_range_one", "Finset.prod_range_zero", "Finset.prod_sdiff_eq_div", "Finset.prod_subtype", "Finset.prod_univ_sum", "Finset.prod_zpow", "Finset.product_biUnion", "Finset.product_image_fst", "Finset.product_image_snd", "Finset.product_sdiff_diag", "Finset.product_self_eq_disjiUnion_perm", "Finset.sdiff_singleton_eq_erase", "Finset.sdiff_union_erase_cancel", "Finset.seqRight_def", "Finset.set_biInter_singleton", "Finset.set_biUnion_finset_image", "Finset.set_biUnion_preimage_singleton", "Finset.shadow_empty", "Finset.shatterer_mono", "Finset.shatters_univ", "Finset.singleton_div", "Finset.singleton_inter_of_mem", "Finset.smul_finset_inter\u2080", "Finset.smul_finset_sdiff", "Finset.smul_mem_smul_finset_iff\u2080", "Finset.sort_sorted_gt", "Finset.sort_sorted_lt", "Finset.sorted_last_eq_max'", "Finset.sorted_last_eq_max'_aux", "Finset.ssubset_iff", "Finset.ssubset_iff_exists_cons_subset", "Finset.strictMono_iff_forall_lt_cons", "Finset.stronglyMeasurable_prod", "Finset.subset_biUnion_of_mem", "Finset.subset_div", "Finset.subset_erase", "Finset.subset_insert_iff", "Finset.subset_map_iff", "Finset.subset_product_image_snd", "Finset.subset_union_elim", "Finset.subset_vsub", "Finset.subtype_eq_empty", "Finset.sum_antidiagonal_choose_succ_mul", "Finset.sum_centroidWeightsIndicator", "Finset.sum_indicator_mod", "Finset.sum_pow", "Finset.sum_pow_mul_eq_add_pow", "Finset.sum_powerset_apply_card", "Finset.sum_schlomilch_le'", "Finset.sum_single_ite", "Finset.sum_smul_vsub_eq_weightedVSubOfPoint_sub", "Finset.sup'_inf_distrib_right", "Finset.sup'_mono", "Finset.sup'_product_left", "Finset.supIndep_attach", "Finset.sup_div_right", "Finset.sup_eq_biUnion", "Finset.sup_eq_iSup", "Finset.sup_inv", "Finset.sup_le_of_le_directed", "Finset.sup_mul_le_mul_sup_of_nonneg", "Finset.sup_sdiff_right", "Finset.sup_set_eq_biUnion", "Finset.support_prod", "Finset.sups_aux", "Finset.sups_union_left", "Finset.sym2_eq_image", "Finset.sym2_univ", "Finset.sym_nonempty", "Finset.union_div_inter_subset_union", "Finset.union_erase_of_mem", "Finset.union_mul_inter_subset_union", "Finset.union_product", "Finset.union_right_comm", "Finset.union_sdiff_of_subset", "Finset.union_subset", "Finset.union_subset_union_left", "Finset.union_val", "Finset.univ_map_subtype", "Finset.univ_mul_univ", "Finset.univ_pi_univ", "Finset.univ_sigma_univ", "Finset.update_piecewise_of_mem", "Finset.val_inj", "Finset.val_injective", "Finset.val_strictMono", "Finset.weightedVSubOfPoint_sdiff", "Finset.weightedVSubOfPoint_sdiff_sub", "Finset.zero_mem_box", "Finsupp.addHom_ext", "Finsupp.add_eq_zero_iff", "Finsupp.apply_single", "Finsupp.card_support_le_one'", "Finsupp.coe_lsum", "Finsupp.coe_mono", "Finsupp.coe_sumElim", "Finsupp.cons_zero_zero", "Finsupp.distribMulActionHom_ext'", "Finsupp.domLCongr_single", "Finsupp.embDomain_add", "Finsupp.erase_eq_sub_single", "Finsupp.le_iff'", "Finsupp.lift_symm_apply", "Finsupp.mapDomain_comp", "Finsupp.mapDomain_injOn", "Finsupp.mapDomain_injective", "Finsupp.mapRange.addEquiv_symm", "Finsupp.mapRange.addMonoidHom_toZeroHom", "Finsupp.mapRange.linearEquiv_trans", "Finsupp.mapRange_sub", "Finsupp.mem_splitSupport_iff_nonzero", "Finsupp.mk_mem_graph", "Finsupp.nonempty_neLocus_iff", "Finsupp.sigmaFinsuppAddEquivPiFinsupp_apply", "Finsupp.single_apply", "Finsupp.single_eq_single_iff", "Finsupp.smul_sum", "Finsupp.some_apply", "Finsupp.sumElim_inr", "Finsupp.sum_apply", "Finsupp.sum_curry_index", "Finsupp.sum_ite_self_eq", "Finsupp.sum_mul", "Finsupp.support_eq_empty", "Finsupp.support_inf_union_support_sup", "Finsupp.support_mapRange", "Finsupp.supported_union", "Finsupp.toFreeAbelianGroup_comp_toFinsupp", "Finsupp.totalOn_range", "Finsupp.total_fin_zero", "Finsupp.total_total", "Finsupp.update_eq_single_add_erase", "Finsupp.zero_apply", "Fintype.card_bool", "Fintype.card_congr'", "Fintype.card_eq_zero_iff", "Fintype.card_filter_piFinset_eq_of_mem", "Fintype.card_finset", "Fintype.card_le_card_of_rightTotal_unique", "Fintype.card_ofSubsingleton", "Fintype.card_shrink", "Fintype.card_subtype", "Fintype.card_subtype_or_disjoint", "Fintype.card_unit", "Fintype.card_units_int", "Fintype.exists_lt_card_fiber_of_nsmul_lt_card", "Fintype.finsetEquivSet_symm_apply", "Fintype.mem_piFinset", "Fintype.one_le_prod", "Fintype.piFinset_update_singleton_eq_filter_piFinset_eq", "Fintype.prod_ite_eq_ite_exists", "Fintype.prod_sum_elim", "Fintype.subtype_card", "Fintype.univ_pempty", "FirstOrder.Field.realize_eqZero", "FirstOrder.Language.BoundedFormula.listDecode_encode_list", "FirstOrder.Language.BoundedFormula.realize_constantsVarsEquiv", "FirstOrder.Language.BoundedFormula.realize_iff", "FirstOrder.Language.BoundedFormula.realize_liftAt", "FirstOrder.Language.BoundedFormula.realize_liftAt_one_self", "FirstOrder.Language.BoundedFormula.relabelAux_sum_inl", "FirstOrder.Language.BoundedFormula.relabel_all", "FirstOrder.Language.DefinableSet.coe_top", "FirstOrder.Language.DirectLimit.comp_unify", "FirstOrder.Language.ElementarilyEquivalent.symm", "FirstOrder.Language.Embedding.subtype_comp_codRestrict", "FirstOrder.Language.Equiv.map_fun", "FirstOrder.Language.Equiv.refl_apply", "FirstOrder.Language.Equiv.symm_apply_apply", "FirstOrder.Language.Formula.realize_relabel", "FirstOrder.Language.Formula.sup_semanticallyEquivalent_not_inf_not", "FirstOrder.Language.Hom.comp_id", "FirstOrder.Language.HomClass.strongHomClassOfIsAlgebraic", "FirstOrder.Language.LHom.comp_sumElim", "FirstOrder.Language.LHom.id_onBoundedFormula", "FirstOrder.Language.LHom.id_onTerm", "FirstOrder.Language.Structure.CG.range", "FirstOrder.Language.Structure.relMap_apply\u2082", "FirstOrder.Language.Substructure.closure_empty", "FirstOrder.Language.Substructure.closure_withConstants_eq", "FirstOrder.Language.Substructure.coe_closure_eq_range_term_realize", "FirstOrder.Language.Substructure.coe_copy", "FirstOrder.Language.Substructure.coe_inclusion", "FirstOrder.Language.Substructure.coe_inf", "FirstOrder.Language.Substructure.mem_map", "FirstOrder.Language.Substructure.subset_closure", "FirstOrder.Language.Term.realize_le", "FirstOrder.Language.Theory.CompleteType.setOf_mem_eq_univ_iff", "FirstOrder.Language.Theory.CompleteType.setOf_subset_eq_empty_iff", "FirstOrder.Language.Theory.completeTheory.subset", "FirstOrder.Language.Theory.isSatisfiable_of_isSatisfiable_onTheory", "FirstOrder.Language.Theory.model_singleton_iff", "FirstOrder.Language.constantsOnMap_isExpansionOn", "FirstOrder.Language.directed_distinctConstantsTheory", "FirstOrder.Language.exists_elementarilyEquivalent_card_eq", "FirstOrder.Language.orderLHom_leSymb", "FirstOrder.Language.realize_noBotOrder", "FirstOrder.Language.withConstants_funMap_sum_inr", "FirstOrder.Ring.realize_one", "FirstOrder.Sequence\u2082.sum_card", "FixedPoints.minpoly.eval\u2082'", "FixedPoints.minpoly.monic", "Flag.le_or_le", "Flag.mem_rangeFin", "Flow.ext", "Flow.isInvariant_omegaLimit", "Flow.map_zero_apply", "FormalMultilinearSeries.apply_eq_zero_of_lt_order", "FormalMultilinearSeries.changeOriginSeries_summable_aux\u2083", "FormalMultilinearSeries.changeOrigin_finite_of_finite", "FormalMultilinearSeries.comp_id", "FormalMultilinearSeries.constFormalMultilinearSeries_radius", "FormalMultilinearSeries.continuousOn_of_finite", "FormalMultilinearSeries.le_radius_of_bound", "FormalMultilinearSeries.leftInv_comp", "FormalMultilinearSeries.order_eq_find'", "FormalMultilinearSeries.order_eq_zero_iff", "FormalMultilinearSeries.radius_neg", "FormalMultilinearSeries.rightInv_coeff", "FractionalIdeal.absNorm_eq", "FractionalIdeal.adjoinIntegral_eq_one_of_isUnit", "FractionalIdeal.coeIdeal_eq_one", "FractionalIdeal.coeIdeal_le_coeIdeal'", "FractionalIdeal.coe_copy", "FractionalIdeal.coe_mul", "FractionalIdeal.constant_factor_ne_zero", "FractionalIdeal.exists_not_mem_one_of_ne_bot", "FractionalIdeal.finprod_heightOneSpectrum_factorization", "FractionalIdeal.finprod_heightOneSpectrum_factorization_principal", "FractionalIdeal.inv_anti_mono", "FractionalIdeal.isFractional", "FractionalIdeal.isFractional_of_fg", "FractionalIdeal.isFractional_span_iff", "FractionalIdeal.isNoetherian", "FractionalIdeal.le_zero_iff", "FractionalIdeal.mem_dual", "FractionalIdeal.mul_self_le_self", "FractionalIdeal.one_le", "FractionalIdeal.spanSingleton_div_self", "FractionalIdeal.zero_mem", "FreeAbelianGroup.map_neg", "FreeAbelianGroup.map_of", "FreeAbelianGroup.of_mul_of", "FreeAbelianGroup.seq_sub", "FreeAlgebra.mk_mul", "FreeGroup.Red.Step.cons_not", "FreeGroup.Red.Step.length", "FreeGroup.Red.append_append", "FreeGroup.Red.cons_cons", "FreeGroup.Red.to_append_iff", "FreeGroup.injective_lift_of_ping_pong", "FreeGroup.invRev_length", "FreeGroup.lift.of_eq", "FreeGroup.map.of", "FreeGroup.map_one", "FreeGroup.reduce.exact", "FreeGroup.reduce.min", "FreeGroup.reduce.red", "FreeLieAlgebra.liftAux_spec", "FreeMagma.length_toFreeSemigroup", "FreeMagma.toFreeSemigroup_comp_of", "FreeMagma.traverse_eq", "FreeMonoid.lift_ofList", "FreeMonoid.ofList_smul", "FreeNonUnitalNonAssocAlgebra.lift_symm_apply", "FreeRing.coe_add", "FreeSemigroup.lift_of", "FreeSemigroup.pure_bind", "Function.Antiperiodic.add_const", "Function.Antiperiodic.const_smul\u2080", "Function.Antiperiodic.int_odd_mul_antiperiodic", "Function.Bijective.prodMap", "Function.Commute.comp_iterate", "Function.Commute.comp_left", "Function.Commute.id_right", "Function.Commute.iterate_pos_eq_iff_map_eq", "Function.Commute.iterate_pos_le_iff_map_le'", "Function.Commute.iterate_pos_lt_of_map_lt'", "Function.Commute.minimalPeriod_of_comp_eq_mul_of_coprime", "Function.Commute.self_iterate", "Function.Embedding.cardinal_le", "Function.Embedding.coe_prodMap", "Function.Embedding.coe_quotientOut", "Function.Embedding.nonempty_iff_card_le", "Function.Embedding.toEquivRange_symm_apply_self", "Function.Embedding.uncountable", "Function.Exact.split_tfae'", "Function.FactorsThrough.comp_right", "Function.FromTypes.const_succ_apply", "Function.HasTemperateGrowth.const", "Function.HasTemperateGrowth.zero", "Function.Injective.encard_image", "Function.Injective.lieAlgebra_isNilpotent", "Function.Injective.prodMap", "Function.Injective.rightDistribClass", "Function.Injective.sbtw_map_iff", "Function.Injective2.eq_iff", "Function.IsFixedPt.apply", "Function.IsFixedPt.birkhoffAverage_eq", "Function.IsFixedPt.eq", "Function.IsFixedPt.perm_inv", "Function.IsPeriodicPt.iterate", "Function.LeftInverse.eq_rightInverse", "Function.LeftInverse.image_image", "Function.LeftInverse.list_map", "Function.Periodic.add_antiperiod_eq", "Function.Periodic.comp_addHom", "Function.Periodic.const_inv_smul\u2080", "Function.Periodic.const_sub", "Function.Periodic.exists_mem_Ico\u2080", "Function.Periodic.funext", "Function.Periodic.int_mul", "Function.RightInverse.filter_map", "Function.Semiconj.option_map", "Function.Semiconj.trans", "Function.Surjective.comp_exact_iff_exact", "Function.Surjective.image_preimage", "Function.Surjective.isBezout", "Function.Surjective.isEmpty", "Function.Surjective.isEngelian", "Function.Surjective.nonempty_preimage", "Function.Surjective.nontrivial", "Function.Surjective.preimage_injective", "Function.Surjective.uncountable", "Function.apply_invFun_apply", "Function.argminOn_le", "Function.comp_const", "Function.extend_inv", "Function.graph_def", "Function.graph_injective", "Function.invFunOn_injOn_image", "Function.iterate_mul", "Function.iterate_zero", "Function.leftInverse_invFun", "Function.mem_mulSupport", "Function.minimalPeriod_iterate_eq_div_gcd", "Function.monotoneOn_of_rightInvOn_of_mapsTo", "Function.mt", "Function.mulSupport_along_fiber_finite_of_finite", "Function.mulSupport_const", "Function.mulSupport_eq_preimage", "Function.mulSupport_prod_mk'", "Function.mulSupport_sup", "Function.mulSupport_update_one", "Function.ne_update_self_iff", "Function.nodup_periodicOrbit", "Function.not_surjective", "Function.not_surjective_Type", "Function.periodicOrbit_chain'", "Function.rightInverse_of_injective_of_leftInverse", "Function.support_inv'", "Function.symmetric_apply_eq_iff", "Function.uncurry_def", "Function.updateFinset_empty", "Function.updateFinset_updateFinset", "Function.update_comm", "Function.update_strictMono", "FunctionField.InftyValuation.map_add_le_max'", "Functor.map_map", "GaloisCoinsertion.isCoatom_iff'", "GaloisConnection.l_comm_iff_u_comm", "GaloisConnection.le_iff_le", "GaloisConnection.u_unique", "GaloisConnection.upperBounds_l_image", "GaloisInsertion.strictMono_u", "GaloisInsertion.u_injective", "Gamma0_det", "Gamma_mem", "Gamma_normal", "GaussianInt.natCast_natAbs_norm", "GaussianInt.toComplex_def'", "GaussianInt.toComplex_div_re", "GaussianInt.toComplex_neg", "GenLoop.Homotopic.refl", "GenLoop.fromLoop_symm_toLoop", "GeneralizedContinuedFraction.coe_toGeneralizedContinuedFraction", "GeneralizedContinuedFraction.compExactValue_correctness_of_stream_eq_some", "GeneralizedContinuedFraction.convergents_eq_convergents'", "GeneralizedContinuedFraction.denominators_recurrence", "GeneralizedContinuedFraction.exists_gcf_pair_rat_eq_nth_conts", "GeneralizedContinuedFraction.fib_le_of_continuantsAux_b", "GeneralizedContinuedFraction.of_convergents_eq_convergents'", "GeneralizedContinuedFraction.of_correctness_of_nth_stream_eq_none", "GeneralizedContinuedFraction.of_terminates_iff_of_rat_terminates", "GeneralizedContinuedFraction.succ_nth_convergent'_eq_squashGCF_nth_convergent'", "GeneralizedContinuedFraction.terminatedAt_iff_s_none", "GeneralizedContinuedFraction.terminated_stable", "GeneralizedContinuedFraction.zero_le_of_denom", "Geometry.SimplicialComplex.mem_space_iff", "Geometry.SimplicialComplex.mem_vertices", "Geometry.SimplicialComplex.vertex_mem_convexHull_iff", "GradedMonoid.list_prod_map_eq_dProd", "GradedMonoid.snd_smul", "GradedTensorProduct.tmul_one_mul_one_tmul", "GromovHausdorff.HD_bound_aux2", "GromovHausdorff.HD_lipschitz_aux2", "GromovHausdorff.candidates_lipschitz", "GromovHausdorff.toGHSpace_continuous", "GromovHausdorff.totallyBounded", "Group.conjugates_subset_normal", "Group.normalClosure.isSubgroup", "Group.normalClosure.is_normal", "GroupFilterBasis.one", "GroupSeminorm.toFun_eq_coe", "GroupTopology.toTopologicalSpace_le", "Grp.SurjectiveOfEpiAuxs.h_apply_fromCoset_nin_range", "Grp.epi_iff_range_eq_top", "Grp.mono_iff_injective", "GrpWithZero.coe_id", "HNNExtension.NormalWord.prod_group_smul", "HNNExtension.of_mul_inv_t", "HNNExtension.t_mul_of", "HahnSeries.SummableFamily.hsum_smul", "HahnSeries.coeff_order_ne_zero", "HahnSeries.embDomain_injective", "HahnSeries.min_order_le_order_add", "HahnSeries.ofPowerSeries_apply_coeff", "HahnSeries.ofPowerSeries_injective", "HahnSeries.orderTop_eq_top_iff", "HahnSeries.single_pow", "HahnSeries.support_single_subset", "HahnSeries.untop_orderTop_of_ne_zero", "Hamming.dist_eq_hammingDist", "Hamming.ofHamming_smul", "Hamming.ofHamming_toHamming", "HasCompactSupport.hasDerivAt_convolution_left", "HasDerivAt.comp_const_add", "HasDerivAt.comp_hasDerivWithinAt_of_eq", "HasDerivAt.hasDerivAtFilter", "HasDerivAt.hasDerivWithinAt", "HasDerivAt.hasGradientAt'", "HasDerivAt.lhopital_zero_left_on_Ioc", "HasDerivAt.pow", "HasDerivAtFilter.sub", "HasDerivWithinAt.differentiableWithinAt", "HasDerivWithinAt.liminf_right_norm_slope_le", "HasDerivWithinAt.liminf_right_slope_norm_le", "HasFDerivAt.clm_comp", "HasFDerivAt.congr_fderiv", "HasFDerivAt.const_mul", "HasFDerivAt.differentiableAt", "HasFDerivAt.of_local_left_inverse", "HasFDerivAt.sum", "HasFDerivAtFilter.add_const", "HasFDerivAtFilter.tendsto_nhds", "HasFDerivWithinAt.ccosh", "HasFDerivWithinAt.clm_comp", "HasFDerivWithinAt.comp_of_mem", "HasFDerivWithinAt.finset_prod", "HasFDerivWithinAt.isBigO_sub", "HasFDerivWithinAt.mono", "HasFDerivWithinAt.neg", "HasFDerivWithinAt.sin", "HasFPowerSeriesAt.eventually_eq_zero", "HasFPowerSeriesAt.locally_ne_zero", "HasFPowerSeriesAt.radius_pos", "HasFPowerSeriesOnBall.analyticAt", "HasFPowerSeriesOnBall.eventually_eq_zero", "HasFPowerSeriesOnBall.prod", "HasFPowerSeriesOnBall.uniform_geometric_approx'", "HasFTaylorSeriesUpTo.eq_iteratedFDeriv", "HasFTaylorSeriesUpTo.hasFTaylorSeriesUpToOn", "HasFTaylorSeriesUpTo.ofLe", "HasFTaylorSeriesUpToOn.differentiableOn", "HasFTaylorSeriesUpToOn.hasFDerivWithinAt", "HasFTaylorSeriesUpToOn.of_le", "HasFiniteFPowerSeriesOnBall.cPolynomialOn", "HasFiniteFPowerSeriesOnBall.eq_zero_of_bound_zero", "HasFiniteFPowerSeriesOnBall.fderiv'", "HasGradientWithinAt.congr_of_eventuallyEq_of_mem", "HasLineDerivAt.smul", "HasLineDerivAt.unique", "HasLineDerivWithinAt.congr", "HasLineDerivWithinAt.congr_mono", "HasMFDerivAt.comp", "HasMFDerivWithinAt.hasMFDerivAt", "HasOuterApproxClosed.apprSeq_apply_le_one", "HasProd.congr_fun", "HasProd.nat_mul_neg", "HasStrictDerivAt.comp", "HasStrictDerivAt.iterate", "HasStrictDerivAt.rpow", "HasStrictDerivAt.scomp", "HasStrictFDerivAt.comp_hasStrictDerivAt", "HasStrictFDerivAt.hasStrictDerivAt", "HasStrictFDerivAt.implicitFunction_apply_image", "HasStrictFDerivAt.implicitToPartialHomeomorphOfComplemented_self", "HasStrictFDerivAt.implicitToPartialHomeomorph_fst", "HasStrictFDerivAt.isBigO_sub_rev", "HasStrictFDerivAt.iterate", "HasStrictFDerivAt.mem_implicitToPartialHomeomorphOfComplemented_source", "HasStrictFDerivAt.smul_const", "HasStrictFDerivAt.sub", "HasSum.div_const", "HasSum.exists_hasSum_smul_of_apply_eq_zero", "Heyting.IsRegular.inf", "Heyting.Regular.coe_bot", "Heyting.Regular.coe_le_coe", "Heyting.Regular.coe_lt_coe", "HeytingHom.coe_id", "HeytingHom.toFun_eq_coe_aux", "Hindman.FP.finset_prod", "HolderOnWith.ediam_image_le", "HolderOnWith.hausdorffMeasure_image_le", "HolderWith.nndist_le", "Holor.cprankMax_1", "Holor.slice_unitVec_mul", "Holor.slice_zero", "HolorIndex.drop_take", "Homeomorph.comp_continuousAt_iff", "Homeomorph.compactSpace", "Homeomorph.contDiff_symm", "Homeomorph.continuous", "Homeomorph.image_closure", "Homeomorph.isLocalHomeomorph", "Homeomorph.preimage_interior", "Homeomorph.self_comp_symm", "Homeomorph.smulOfNeZero_symm_apply", "Homeomorph.t1Space", "Homeomorph.trans_toHomotopyEquiv", "HomogeneousIdeal.mem_iff", "HomogeneousLocalization.NumDenSameDeg.den_neg", "HomogeneousLocalization.isUnit_iff_isUnit_val", "HomogeneousLocalization.mk_pow", "HomogeneousLocalization.val_intCast", "HomogeneousLocalization.val_mul", "HomogeneousLocalization.val_natCast", "HomogeneousLocalization.val_sub", "HomologicalComplex.Hom.from_single\u2080_exact_f_d_at_zero", "HomologicalComplex.Hom.to_single\u2080_exact_at_succ", "HomologicalComplex.HomologySequence.composableArrows\u2083_exact", "HomologicalComplex.biprodXIso_hom_snd", "HomologicalComplex.biprod_inr_snd_f", "HomologicalComplex.cyclesMap_comp", "HomologicalComplex.cylinder.\u03b9\u2080_desc", "HomologicalComplex.cylinder.\u03c0Comp\u03b9\u2080Homotopy.biprod_lift_id_sub_id", "HomologicalComplex.cylinder.\u03c0Comp\u03b9\u2080Homotopy.nullHomotopicMap_eq", "HomologicalComplex.dFrom_comp_xNextIsoSelf", "HomologicalComplex.dTo_comp_dFrom", "HomologicalComplex.d_comp_d", "HomologicalComplex.d_pOpcycles", "HomologicalComplex.extend.d_none_eq_zero", "HomologicalComplex.fromOpcycles_d", "HomologicalComplex.homologyMap_zero", "HomologicalComplex.homotopyCofiber.desc_f", "HomologicalComplex.homotopyCofiber.ext_from_X", "HomologicalComplex.iCyclesIso_inv_hom_id", "HomologicalComplex.image_to_eq_image", "HomologicalComplex.isZero_single_comp_eval", "HomologicalComplex.kernel_eq_kernel", "HomologicalComplex.liftCycles_homology\u03c0_eq_zero_of_boundary", "HomologicalComplex.singleObjOpcyclesSelfIso_hom", "HomologicalComplex.singleObjOpcyclesSelfIso_hom_naturality", "HomologicalComplex\u2082.d\u2081_eq_zero", "HomologicalComplex\u2082.totalAux.d\u2082_eq'", "Homotopy.dNext_succ_chainComplex", "Homotopy.dNext_zero_chainComplex", "Homotopy.map_eq_of_inverts_homotopyEquivalences", "Homotopy.prevD_chainComplex", "HomotopyGroup.isUnital_auxGroup", "HurwitzKernelBounds.f_int_ofNat", "HurwitzKernelBounds.isBigO_atTop_F_nat_zero_sub", "HurwitzKernelBounds.isBigO_one_aux", "HurwitzZeta.completedCosZeta_one_sub", "HurwitzZeta.completedHurwitzZetaEven_eq", "HurwitzZeta.continuousOn_oddKernel", "HurwitzZeta.cosZeta_neg_two_mul_nat_add_one", "HurwitzZeta.cosZeta_zero", "HurwitzZeta.differentiableAt_cosZeta", "HurwitzZeta.evenKernel_def", "HurwitzZeta.evenKernel_eq_cosKernel_of_zero", "HurwitzZeta.expZeta_one_sub", "HurwitzZeta.hasSum_int_evenKernel\u2080", "HurwitzZeta.hurwitzZetaEven_neg_two_mul_nat_add_one", "HurwitzZeta.hurwitzZetaEven_one_sub_two_mul_nat", "HurwitzZeta.hurwitzZeta_neg_nat", "HurwitzZeta.jacobiTheta\u2082''_neg_left", "HurwitzZeta.tendsto_hurwitzZetaEven_sub_one_div_nhds_one", "Hyperreal.InfinitePos.neg", "Hyperreal.IsSt.map", "Hyperreal.IsSt.mul", "Hyperreal.epsilon_pos", "Hyperreal.infiniteNeg_mul_infinitePos", "Hyperreal.infiniteNeg_mul_of_not_infinitesimal_neg_infinitePos", "Hyperreal.infiniteNeg_neg", "Hyperreal.infinitePos_iff_infinite_and_pos", "Hyperreal.infinitePos_neg", "Hyperreal.infinite_of_infinitesimal_inv", "Hyperreal.infinitesimal_iff_infinite_inv", "Hyperreal.inv_epsilon", "Hyperreal.isSt_iff_abs_sub_lt_delta", "Hyperreal.isSt_of_tendsto", "Hyperreal.isSt_sSup", "Hyperreal.lt_of_tendsto_zero_of_pos", "Hyperreal.not_infinite_mul", "Hyperreal.not_real_of_infinitesimal_ne_zero", "Icc_subset_segment", "Ici_mem_nhdsSet_Ico", "Ici_mem_nhdsSet_Ioc", "Ico_ofDual", "Ideal.Filtration.Stable.of_le", "Ideal.Filtration.inf_submodule", "Ideal.IsHomogeneous.toIdeal_homogeneousCore_eq_self", "Ideal.IsHomogeneous.toIdeal_homogeneousHull_eq_self", "Ideal.IsHomogeneous.top", "Ideal.IsMaximal.exists_inv", "Ideal.IsPrime.homogeneousCore", "Ideal.IsPrime.inf_le", "Ideal.IsPrime.inf_le'", "Ideal.IsPrime.mul_mem_pow", "Ideal.IsPrime.multiset_prod_map_le", "Ideal.IsPrime.multiset_prod_mem_iff_exists_mem", "Ideal.MvPolynomial.isJacobson_MvPolynomial_fin", "Ideal.Quotient.exists_inv", "Ideal.Quotient.mk_comp_algebraMap", "Ideal.Quotient.mk_eq_mk", "Ideal.adic_module_basis", "Ideal.associatesEquivIsPrincipal_map_one", "Ideal.comap_isMaximal_of_surjective", "Ideal.comap_lt_comap_of_integral_mem_sdiff", "Ideal.comap_of_equiv", "Ideal.comap_sInf", "Ideal.eq_bot_or_top", "Ideal.eq_top_of_isUnit_mem", "Ideal.finite_setOf_absNorm_eq", "Ideal.finsuppTotal_apply_eq_of_fintype", "Ideal.iSup_iInf_eq_top_iff_pairwise", "Ideal.isCoprime_iff_codisjoint", "Ideal.isMaximal_comap_of_isIntegral_of_isMaximal", "Ideal.isMaximal_iff", "Ideal.isMaximal_of_isMaximal_disjoint", "Ideal.isOpen_of_isOpen_subideal", "Ideal.isPrime_map_C_iff_isPrime", "Ideal.kerLiftAlg_injective", "Ideal.le_of_localization_maximal", "Ideal.le_of_pow_le_prime", "Ideal.map.isMaximal", "Ideal.map_eq_iff_sup_ker_eq_of_surjective", "Ideal.map_eq_submodule_map", "Ideal.map_eq_top_iff_of_ker_le", "Ideal.map_le_comap_of_inverse", "Ideal.map_span", "Ideal.mem_isPrincipalSubmonoid_iff", "Ideal.mem_jacobson_bot", "Ideal.mem_prime_of_mul_mem_pow", "Ideal.mem_span_singleton_mul", "Ideal.mul_comm", "Ideal.mul_homogeneous_element_mem_of_mem", "Ideal.mul_le", "Ideal.mul_le_left", "Ideal.mul_mem_mul", "Ideal.mul_top", "Ideal.multiset_prod_eq_bot", "Ideal.multiset_prod_span_singleton", "Ideal.pi_quotient_surjective", "Ideal.prime_of_isPrime", "Ideal.quotientEquivAlgOfEq_symm", "Ideal.quotientMap_mk", "Ideal.quotientToQuotientRangePowQuotSucc_injective", "Ideal.quotient_map_mk\u2090", "Ideal.radical_eq_jacobson_iff_radical_quotient_eq_jacobson_bot", "Ideal.radical_pow", "Ideal.radical_sup", "Ideal.ramificationIdx_lt", "Ideal.range_finsuppTotal", "Ideal.rank_pow_quot", "Ideal.relNorm_singleton", "Ideal.spanNorm_localization", "Ideal.spanNorm_mul", "Ideal.span_eq_bot", "Ideal.span_eq_top_iff_finite", "Ideal.span_le", "Ideal.span_singleton_mul_eq_span_singleton_mul", "Ideal.sub_mem", "Ideal.subset_span", "Ideal.sum_ramification_inertia", "Ideal.sup_mul_eq_of_coprime_right", "Ideal.toCotangent_range", "Ideal.torsionOf_zero", "ImplicitFunctionData.left_map_implicitFunction", "Inducing.comp", "Inducing.dense_iff", "Inducing.hasProd_iff", "Inducing.isSeparable_preimage", "Inducing.mapClusterPt_iff", "Inducing.topologicalGroup", "InfHom.coe_comp", "InfHom.coe_const", "InfHom.comp_id", "InfHom.symm_dual_comp", "InfPrime.inf_le", "InfTopHom.coe_comp", "InfTopHom.coe_mk", "Infinite.exists_superset_card_eq", "InitialSeg.acc", "InnerProductGeometry.angle_neg_right", "InnerProductGeometry.norm_eq_of_angle_sub_eq_angle_sub_rev_of_angle_ne_pi", "InnerProductGeometry.norm_sub_sq_eq_norm_sq_add_norm_sq_iff_angle_eq_pi_div_two", "InnerProductGeometry.sin_angle_sub_of_inner_eq_zero", "InnerProductGeometry.tan_angle_add_of_inner_eq_zero", "InnerProductSpace.Core.inner_add_add_self", "InnerProductSpace.continuousLinearMapOfBilin_apply", "InnerProductSpaceable.add_left_aux3", "Inseparable.mem_measurableSet_iff", "Int.ModEq.cancel_right_div_gcd", "Int.ModEq.mul_left", "Int.ModEq.sub_left", "Int.ModEq.trans", "Int.abs_ediv_le_abs", "Int.add_sub_assoc", "Int.bitwise_diff", "Int.bodd_add", "Int.bodd_bit0", "Int.card_Ioc_of_le", "Int.card_fintype_Ico", "Int.cast_injOn_of_ringChar_ne_two", "Int.cast_ite", "Int.cast_lt_zero", "Int.cast_mono", "Int.cast_neg_natCast", "Int.cast_pos", "Int.cast_pow", "Int.ceil_intCast", "Int.ceil_natCast", "Int.ceil_ofNat", "Int.cobounded_eq", "Int.coe_nat_lcm", "Int.covBy_add_one", "Int.csInf_eq_least_of_bdd", "Int.dist_eq'", "Int.div2_val", "Int.div_eq_ediv_of_dvd", "Int.dvd_mul_of_div_dvd", "Int.dvd_sub_of_emod_eq", "Int.ediv_dvd_ediv", "Int.emod_lt_of_pos", "Int.exists_strictAnti", "Int.floor_fract", "Int.floor_sub_ofNat", "Int.fract_eq_iff", "Int.fract_int_add", "Int.gcd_dvd_gcd_mul_right_right", "Int.gcd_zero_left", "Int.injOn_natAbs_Iic", "Int.isSquare_natCast_iff", "Int.isUnit_iff_abs_eq", "Int.lcm_mul_left", "Int.lcm_one_right", "Int.ldiff_bit", "Int.le_add_one_iff", "Int.le_natCast_sub", "Int.log_of_left_le_one", "Int.log_of_right_le_zero", "Int.lt_ceil", "Int.lxor_bit", "Int.map_round", "Int.modEq_sub_fac", "Int.mod_nat_eq_sub_mul_floor_rat_div", "Int.mul_comm", "Int.natAbs_coe_sub_coe_le_of_le", "Int.natAbs_dvd_natAbs", "Int.natAbs_neg", "Int.natAbs_pow", "Int.natCast_eq_zero", "Int.natCast_succ_pos", "Int.ne_of_odd_add", "Int.negOnePow_eq_neg_one_iff", "Int.negOnePow_neg", "Int.ofAdd_mul", "Int.ofNat_ediv", "Int.ofNat_eq_coe", "Int.ofNat_isUnit", "Int.ofNat_zero", "Int.one_ne_zero", "Int.pow_eq", "Int.preimage_Iic", "Int.prime_two", "Int.rel_of_forall_rel_succ_of_lt", "Int.sq_of_coprime", "Int.sqrt_nonneg", "Int.strictMonoOn_natAbs", "Int.toNat_lt'", "Int.zero_shiftRight'", "IntermediateField.AdjoinSimple.isIntegral_gen", "IntermediateField.adjoin.mono", "IntermediateField.adjoin_eq_adjoin_pow_expChar_pow_of_isSeparable'", "IntermediateField.adjoin_finset_isCompactElement", "IntermediateField.adjoin_le_iff", "IntermediateField.algebraMap_mem", "IntermediateField.card_algHom_adjoin_integral", "IntermediateField.coe_inclusion", "IntermediateField.finSepDegree_adjoin_simple_eq_finrank_iff", "IntermediateField.finrank_bot'", "IntermediateField.gc_map_comap", "IntermediateField.isAlgebraic_adjoin_simple", "IntermediateField.isSeparable_adjoin_pair_of_separable", "IntermediateField.mem_mk", "IntermediateField.mem_toSubalgebra", "IntermediateField.pow_mem", "IntermediateField.restrictScalars_toSubalgebra", "IntermediateField.sInf_toSubalgebra", "IntermediateField.sepDegree_adjoin_eq_of_isAlgebraic_of_isPurelyInseparable", "IntermediateField.sepDegree_top", "IntermediateField.splits_of_splits", "IntermediateField.subset_adjoin", "IntermediateField.subsingleton_of_finrank_adjoin_le_one", "IntermediateField.sup_toSubalgebra_of_isAlgebraic", "Interval.bot_ne_one", "Interval.coe_coe", "Interval.coe_sInf", "Interval.div_bot", "Interval.dual_bot", "Interval.pure_injective", "IntervalIntegrable.div_const", "IntervalIntegrable.log", "IntervalIntegrable.norm", "InvImage.equivalence", "Ioi_ofDual", "Ioo_mem_nhdsWithin_Iio", "Irrational.div_int", "Irrational.ne_int", "Irrational.of_rat_mul", "Irrational.of_sub_nat", "Irreducible.dvd_iff", "IsAbsoluteValue.abv_pow", "IsAdjoinRoot.lift_self", "IsAdjoinRootMonic.coeff_apply", "IsAdjoinRootMonic.coeff_apply_le", "IsAdjoinRootMonic.isIntegral_root", "IsAlgClosed.algebraMap_surjective_of_isAlgebraic", "IsAntichain.eq'", "IsAntichain.image_iso_iff", "IsAntichain.interior_eq_empty", "IsAntichain.isWeakAntichain", "IsAntichain.not_lt", "IsAntichain.preimage_relIso", "IsAntichain.swap", "IsAntichain.to_dual_iff", "IsAntichain.volume_eq_zero", "IsAssociatedPrime.annihilator_le", "IsBaseChange.lift_eq", "IsBaseChange.of_lift_unique", "IsBezout.gcd_dvd_right", "IsBoundedBilinearMap.differentiable", "IsBoundedLinearMap.differentiableOn", "IsBoundedLinearMap.hasFDerivAt", "IsCauSeq.const", "IsCauSeq.of_abv_le", "IsChain.mono_rel", "IsClosed.Icc_subset_of_forall_exists_gt", "IsClosed.inter", "IsClosed.isCompact", "IsClosed.measurableSet", "IsCoatomic.exists_coatom", "IsCompact.closure", "IsCompact.compl_mem_sets_of_nhdsWithin", "IsCompact.disjoint_nhdsSet_left", "IsCompact.exists_forall_le'", "IsCompact.exists_infDist_eq_dist", "IsCompact.exists_isOpen_closure_subset", "IsCompact.exists_sSup_image_eq_and_ge", "IsCompact.image", "IsCompact.inter_left", "IsCompact.measure_eq_iInf_isOpen", "IsCompact.nonempty_sInter_of_directed_nonempty_isCompact_isClosed", "IsCompactOperator.continuous_comp", "IsCompactOperator.image_closedBall_subset_compact", "IsCompactOperator.image_subset_compact_of_bounded", "IsCompl.inf_left_eq_bot_iff", "IsCompl.isCoatom_iff_isAtom", "IsCompl.symm", "IsCompl.symmDiff_eq_top", "IsConnected.union", "IsCoprime.add_mul_left_left_iff", "IsCoprime.add_mul_right_right", "IsCoprime.mul_dvd", "IsCoprime.ne_zero", "IsCyclotomicExtension.Rat.five_pid", "IsCyclotomicExtension.discr_odd_prime", "IsCyclotomicExtension.neZero'", "IsCyclotomicExtension.norm_zeta_sub_one_of_isPrimePow", "IsCyclotomicExtension.norm_zeta_sub_one_of_prime_ne_two", "IsCyclotomicExtension.numberField", "IsDedekindDomain.HeightOneSpectrum.coe_smul_adicCompletionIntegers", "IsDedekindDomain.HeightOneSpectrum.valuationOfNeZeroToFun_eq", "IsDedekindDomain.HeightOneSpectrum.valuation_def", "IsDedekindDomain.HeightOneSpectrum.valuation_uniformizer_ne_zero", "IsDedekindDomain.inf_prime_pow_eq_prod", "IsDedekindDomain.isPrincipalIdealRing_localization_over_prime", "IsEquivalent.summable_iff_nat", "IsExposed.mono", "IsExtrOn.inter", "IsExtrOn.localize", "IsGLB.exists_seq_strictAnti_tendsto_of_not_mem", "IsGLB.mem_lowerBounds_of_tendsto", "IsGLB.mono", "IsGalois.card_aut_eq_finrank", "IsGalois.is_separable_splitting_field", "IsGalois.separable", "IsGreatest.union", "IsGreatest.unique", "IsG\u03b4.biInter", "IsG\u03b4.biUnion", "IsG\u03b4.dense_biUnion_interior_of_closed", "IsG\u03b4.iInter_of_isOpen", "IsHereditarilyLindelof.isLindelof_subset", "IsIdempotentElem.eq", "IsIntegral.of_finite", "IsIntegrallyClosedIn.integralClosure_eq_bot", "IsInvariantSubring.coe_subtypeHom'", "IsLUB.exists_between", "IsLeast.csInf_eq", "IsLeast.insert", "IsLeast.nonempty", "IsLeftRegular.mul_left_eq_zero_iff", "IsLeftRegular.pow_iff", "IsLeftRegular.subsingleton", "IsLindelof.elim_countable_subcover", "IsLindelof.elim_nhds_subcover", "IsLindelof.indexed_countable_subcover", "IsLocalExtr.neg", "IsLocalExtrOn.comp_antitone", "IsLocalHomeomorph.mk", "IsLocalMax.comp_mono", "IsLocalMax.min", "IsLocalMaxOn.isLocalMax", "IsLocalMin.neg", "IsLocalization.Away.sec_spec", "IsLocalization.coeSubmodule_span", "IsLocalization.coeSubmodule_strictMono", "IsLocalization.coeffIntegerNormalization_mem_support", "IsLocalization.eq_of_eq", "IsLocalization.isUnit_comp", "IsLocalization.lift_eq", "IsLocalization.lift_mem_adjoin_finsetIntegerMultiple", "IsLocalization.localization_isScalarTower_of_submonoid_le", "IsLocalization.map_injective_of_injective", "IsLocalization.mk'_self'", "IsLocalization.span_invSubmonoid", "IsLocalizedModule.iso_symm_apply'", "IsLocalizedModule.iso_symm_comp", "IsLocalizedModule.lift_apply", "IsLocalizedModule.map_LocalizedModules", "IsLocalizedModule.mk'_cancel'", "IsLocalizedModule.mk'_sub_mk'", "IsLocallyConstant.prod_mk", "IsLocallyConstant.range_finite", "IsLocallyInjective.isOpen_eqLocus", "IsLowerSet.isUpperSet_preimage_coe", "IsLowerSet.nonMemberSubfamily", "IsLowerSet.null_frontier", "IsLprojection.distrib_lattice_lemma", "IsLprojection.mul_compl_self", "IsMaxChain.isChain", "IsMaxChain.range_fin_of_covBy", "IsMaxChain.top_mem", "IsMaxFilter.comp_antitone", "IsMaxOn.isExtr", "IsMaxOn.norm_add_sameRay", "IsMaxOn.on_preimage", "IsMinFilter.comp_tendsto", "IsMonoidHom.inv", "IsMulCentral.left_comm", "IsMulFreimanHom.div", "IsMulFreimanHom.mul_eq_mul", "IsMulFreimanIso.threeGPFree_congr", "IsNilpotent.isUnit_sub_one", "IsNilpotent.of_pow", "IsNilpotent.smul", "IsNoetherian.range_finsetBasis", "IsNowhereDense.subset_of_closed_isNowhereDense", "IsOfFinOrder.orderOf_pos", "IsOpen.closure_inter", "IsOpen.div_right", "IsOpen.exists_smooth_support_eq", "IsOpen.interior_eq", "IsOpen.reProdIm", "IsOpen.uniqueMDiffOn", "IsOpen.uniqueMDiffWithinAt", "IsOpen.upperClosure", "IsOpenMap.isOpen_range", "IsOpenMap.nhds_le", "IsPGroup.of_bot", "IsPGroup.powEquiv_apply", "IsPRadical.comap_pNilradical", "IsPRadical.isPurelyInseparable", "IsPathConnected.exists_path_through_family'", "IsPerfectClosure.equiv_self_apply", "IsPicardLindelof.exists_forall_hasDerivWithinAt_Icc_eq", "IsPreconnected.intermediate_value_Iii", "IsPreconnected.subset_left_of_subset_union", "IsPreirreducible.isPreconnected", "IsPreirreducible.open_subset", "IsPrimePow.dvd", "IsPrimitiveRoot.eq_pow_of_pow_eq_one", "IsPrimitiveRoot.exists_neg_pow_of_isOfFinOrder", "IsPrimitiveRoot.mk_of_lt", "IsPrimitiveRoot.norm'_eq_one", "IsPrimitiveRoot.norm_eq_neg_one_pow", "IsPrimitiveRoot.nthRoots_nodup", "IsPrimitiveRoot.nthRoots_one_nodup", "IsPrimitiveRoot.orderOf", "IsPrimitiveRoot.powerBasis_gen_mem_adjoin_zeta_sub_one", "IsPrimitiveRoot.prime_norm_toInteger_sub_one_of_prime_pow_ne_two", "IsPrimitiveRoot.subOneIntegralPowerBasis_gen", "IsPrimitiveRoot.toInteger_sub_one_dvd_prime", "IsPrimitiveRoot.zeta_sub_one_prime_of_two_pow", "IsPrimitiveRoot.zmodEquivZPowers_symm_apply_pow", "IsProperMap.isClosedMap", "IsPurelyInseparable.tower_bot", "IsRegular.all", "IsRegular.ne_zero", "IsRelPrime.add_mul_right_left_iff", "IsRelPrime.mul_add_left_right_iff", "IsRelPrime.mul_left_iff", "IsRelPrime.prod_left_iff", "IsRightRegular.ne_zero", "IsRightRegular.pow_iff", "IsRingHom.map_neg", "IsSMulRegular.finsupp", "IsSMulRegular.mul_iff", "IsSMulRegular.not_zero_iff", "IsSMulRegular.smul", "IsSelfAdjoint.spectralRadius_eq_nnnorm", "IsSelfAdjoint.star_eq", "IsSemisimpleModule.annihilator_isRadical", "IsSeparable.insepDegree_eq", "IsSeparable.of_equiv_equiv", "IsSeqCompact.totallyBounded", "IsSl2Triple.symm", "IsStarNormal.spectralRadius_eq_nnnorm", "IsSubgroup.eq_trivial_iff", "IsSubgroup.mem_norm_comm_iff", "IsSublattice.prod", "IsSubmonoid.finset_prod_mem", "IsSubring.inter", "IsSymmOp.flip_eq", "IsTop.Iic_eq", "IsTop.fst", "IsTop.mono", "IsTopologicalBasis.iInf_induced", "IsUnifLocDoublingMeasure.one_le_scalingConstantOf", "IsUnit.dvd", "IsUnit.isOpenMap_smul", "IsUnit.isRegular", "IsUnit.map'", "IsUnit.unit_of_val_units", "IsUpperSet.Ioi_subset", "IsUpperSet.inter", "IsUpperSet.interior", "IsUpperSet.null_frontier", "Isometry.ediam_range", "Isometry.hausdorffMeasure_preimage", "Isometry.uniformEmbedding", "Isometry.uniformInducing", "IsometryEquiv.coe_toHomeomorph", "IsometryEquiv.comp_continuous_iff", "IsometryEquiv.completeSpace_iff", "IsometryEquiv.hausdorffMeasure_image", "IsometryEquiv.map_hausdorffMeasure", "IsometryEquiv.toDilationEquiv_apply", "IsometryEquiv.toEquiv_injective", "Joined.symm", "KaehlerDifferential.D_tensorProductTo", "KaehlerDifferential.range_kerCotangentToTensor", "LDL.lowerInv_eq_gramSchmidtBasis", "LDL.lowerInv_triangular", "LDL.lower_conj_diag", "LSeries.term_convolution", "LSeries.term_delta", "LSeriesHasSum.LSeries_eq", "LSeriesSummable.add", "LSeriesSummable.isBigO_rpow", "LSeriesSummable_of_abscissaOfAbsConv_lt_re", "LSeriesSummable_of_bounded_of_one_lt_real", "LSeries_zero", "LT.lt.ne'", "Lagrange.basis_ne_zero", "Lagrange.degree_basisDivisor_of_ne", "Lagrange.nodal_subgroup_eq_X_pow_card_sub_one", "Language.iSup_mul", "Language.kstar_def_nonempty", "LatticeHom.birkhoffFinset_injective", "LatticeHom.coe_toInfHom", "LatticeHom.withTopWithBot_apply", "LaurentPolynomial.T_mul", "LaurentPolynomial.algebraMap_apply", "LaurentPolynomial.degree_T", "LaurentPolynomial.induction_on'", "LaurentPolynomial.invert_symm", "LaurentPolynomial.isLocalization", "LaurentPolynomial.support_C_mul_T", "Left.inv_lt_self", "Left.min_le_max_of_mul_le_mul", "Left.mul_le_one_of_le_of_le", "Left.one_lt_inv_iff", "Left.one_lt_mul_of_le_of_lt_of_pos", "Left.one_lt_mul_of_lt_of_le", "LeftCancelMonoid.toMonoid_injective", "LeftInvariantDerivation.comp_L", "LeftInvariantDerivation.evalAt_apply", "LeftInvariantDerivation.map_smul", "LeftInvariantDerivation.map_sub", "LeftOrdContinuous.map_sSup", "Levenshtein.impl_cons", "Levenshtein.impl_cons_fst_zero", "Lex.exists", "Lex.forall", "LieAlgebra.ExtendScalars.bracket_leibniz_lie", "LieAlgebra.HasTrivialRadical.center_eq_bot", "LieAlgebra.InvariantForm.orthogonal_disjoint", "LieAlgebra.IsKilling.chainLength_zero_right", "LieAlgebra.IsKilling.chainTopCoeff_add_chainBotCoeff", "LieAlgebra.IsKilling.coe_corootSpace_eq_span_singleton", "LieAlgebra.IsKilling.ideal_eq_bot_of_isLieAbelian", "LieAlgebra.IsKilling.rootSpace_two_smul", "LieAlgebra.IsKilling.span_weight_eq_top", "LieAlgebra.IsSimple.eq_top_of_isAtom", "LieAlgebra.Orthogonal.mem_so", "LieAlgebra.Orthogonal.pb_inv", "LieAlgebra.abelian_radical_iff_solvable_is_abelian", "LieAlgebra.ad_apply", "LieAlgebra.derivedLength_zero", "LieAlgebra.derivedSeriesOfIdeal_succ", "LieAlgebra.hasTrivialRadical_iff_no_abelian_ideals", "LieAlgebra.isNilpotent_iff_forall", "LieAlgebra.nilpotent_of_nilpotent_quotient", "LieAlgebra.toEnd_pow_apply_mem", "LieAlgebra.zero_rootSpace_eq_top_of_nilpotent", "LieDerivation.lie_ad", "LieDerivation.smul_apply", "LieDerivation.zero_apply", "LieEquiv.coe_linearEquiv_injective", "LieHom.mem_range_self", "LieIdeal.comap_map_le", "LieIdeal.idealizer_eq_normalizer", "LieIdeal.incl_idealRange", "LieIdeal.mem_comap", "LieModule.Weight.weightSpace_ne_bot", "LieModule.chainBot_zero", "LieModule.chainTopCoeff_add_one", "LieModule.chainTop_isNonZero'", "LieModule.exists_forall_mem_corootSpace_smul_add_eq_zero", "LieModule.exists_isRegular_of_finrank_le_card", "LieModule.iInf_lcs_le_of_isNilpotent_quot", "LieModule.iSup_weightSpace_eq_top", "LieModule.isCompl_weightSpace_zero_posFittingComp", "LieModule.isNilpotent_range_toEnd_iff", "LieModule.isNilpotent_toEnd_of_isNilpotent\u2082", "LieModule.isTrivial_of_nilpotencyLength_le_one", "LieModule.lie_mem_maxGenEigenspace_toEnd", "LieModule.lie_mem_weightSpaceChain_of_weightSpace_eq_bot_left", "LieModule.lie_traceForm_eq_zero", "LieModule.map_posFittingComp_eq", "LieModule.map_posFittingComp_le", "LieModule.nilpotencyLength_eq_one_iff", "LieModule.posFittingCompOf_le_posFittingComp", "LieModule.shiftedWeightSpace.aux", "LieModule.shiftedWeightSpace.toEnd_eq", "LieModule.toModuleHom_apply", "LieModule.weightSpace_chainTopCoeff_add_one_zsmul_add", "LieModuleEquiv.toEquiv_mk", "LieModuleEquiv.trans_apply", "LieModuleHom.coe_restrictLie", "LieModuleHom.coe_toLinearMap", "LieModuleHom.map_top", "LieSubalgebra.coe_ad_pow", "LieSubalgebra.exists_nested_lieIdeal_coe_eq_iff", "LieSubalgebra.incl_range", "LieSubalgebra.lie_mem_sup_of_mem_normalizer", "LieSubalgebra.mem_normalizer_iff'", "LieSubalgebra.top_coe_submodule", "LieSubmodule.Quotient.mk_eq_zero", "LieSubmodule.coe_sub", "LieSubmodule.iInf_coe", "LieSubmodule.isCompactElement_lieSpan_singleton", "LieSubmodule.isNilpotent_iff_exists_self_le_ucs", "LieSubmodule.ker_incl", "LieSubmodule.lie_le_left", "LieSubmodule.lie_sup", "LieSubmodule.mono_lie", "LieSubmodule.normalizer_mono", "LieSubmodule.sInf_coe_toSubmodule'", "LieSubmodule.toEnd_restrict_eq_toEnd", "LieSubmodule.ucs_eq_self_of_normalizer_eq_self", "LightProfinite.finiteCoproduct.\u03b9_jointly_surjective", "LightProfinite.mono_iff_injective", "LineDifferentiableAt.smul", "LinearEquiv.arrowCongr_apply", "LinearEquiv.coe_lTensor_symm", "LinearEquiv.coe_rTensor", "LinearEquiv.coe_toContinuousLinearEquiv'", "LinearEquiv.coe_zero", "LinearEquiv.det_coe_symm", "LinearEquiv.det_mul_det_symm", "LinearEquiv.det_refl", "LinearEquiv.ext_iff", "LinearEquiv.lTensor_symm_tmul", "LinearEquiv.lTensor_trans_rTensor", "LinearEquiv.nonempty_equiv_iff_lift_rank_eq", "LinearEquiv.ofBijective_symm_apply_apply", "LinearEquiv.ofSubmodule'_symm_apply", "LinearEquiv.ofSubmodules_apply", "LinearEquiv.pow_apply", "LinearEquiv.self_trans_symm", "LinearEquiv.symm_apply_eq", "LinearEquiv.symm_flip", "LinearEquiv.symm_neg", "LinearEquiv.toAddMonoidHom_commutes", "LinearEquiv.trans_apply", "LinearIndependent.cardinal_le_rank'", "LinearIndependent.eq_of_smul_apply_eq_smul_apply", "LinearIndependent.fin_cons'", "LinearIndependent.inl_union_inr", "LinearIndependent.to_subtype_range'", "LinearIsometry.coe_id", "LinearIsometry.coe_mul", "LinearIsometry.dist_map", "LinearIsometry.id_apply", "LinearIsometry.norm_iteratedFDeriv_comp_left", "LinearIsometry.toContinuousLinearMap_inj", "LinearIsometryEquiv.coe_ofEq_apply", "LinearIsometryEquiv.comp_differentiableAt_iff", "LinearIsometryEquiv.comp_differentiable_iff", "LinearIsometryEquiv.lipschitz", "LinearIsometryEquiv.map_eq_zero_iff", "LinearIsometryEquiv.reflections_generate_dim", "LinearIsometryEquiv.symm_symm", "LinearLocallyFiniteOrder.le_succFn", "LinearMap.BilinForm.Nondegenerate.flip", "LinearMap.BilinForm.coe_zero", "LinearMap.BilinForm.compRight_compLeft", "LinearMap.BilinForm.comp_id_right", "LinearMap.BilinForm.dualSubmodule_flip_dualSubmodule_of_basis", "LinearMap.BilinForm.iIsOrtho_def", "LinearMap.BilinForm.isAdjointPair_id", "LinearMap.BilinForm.isAdjointPair_zero", "LinearMap.BilinForm.isSkewAdjoint_bracket", "LinearMap.BilinForm.isSkewAdjoint_iff_neg_self_adjoint", "LinearMap.BilinForm.mem_orthogonal_iff", "LinearMap.BilinForm.nondegenerate_toMatrix'_iff", "LinearMap.BilinForm.not_nondegenerate_zero", "LinearMap.BilinForm.smul_right_of_tower", "LinearMap.BilinForm.span_singleton_sup_orthogonal_eq_top", "LinearMap.BilinForm.sub_left", "LinearMap.BilinForm.sum_right", "LinearMap.BilinForm.toLin'Flip_apply", "LinearMap.IsAdjointPair.smul", "LinearMap.IsAlt.isRefl", "LinearMap.IsProj.eq_conj_prodMap", "LinearMap.IsRefl.ortho_comm", "LinearMap.IsSymmetric.diagonalization_apply_self_apply", "LinearMap.IsSymmetric.diagonalization_symm_apply", "LinearMap.IsSymmetric.directSum_decompose_apply", "LinearMap.aeval_eq_aeval_mod_charpoly", "LinearMap.bound_of_ball_bound", "LinearMap.charpoly_def", "LinearMap.charpoly_toMatrix", "LinearMap.coe_det", "LinearMap.commute_pow_left_of_commute", "LinearMap.compAlternatingMap_apply", "LinearMap.comp_apply", "LinearMap.continuous_of_isClosed_ker", "LinearMap.continuous_of_seq_closed_graph", "LinearMap.coprod_inl_inr", "LinearMap.detAux_def''", "LinearMap.det_toMatrix", "LinearMap.dualMap_comp_dualMap", "LinearMap.dualMap_def", "LinearMap.dualMap_injective_iff", "LinearMap.eventually_disjoint_ker_pow_range_pow", "LinearMap.ext_on", "LinearMap.ext_ring_op", "LinearMap.extendTo\ud835\udd5c'_apply_re", "LinearMap.finrank_range_of_inj", "LinearMap.fst_comp_inr", "LinearMap.fst_eq_coprod", "LinearMap.hasEigenvalue_zero_tfae", "LinearMap.im_inner_adjoint_mul_self_eq_zero", "LinearMap.inr_map_mul", "LinearMap.isAdjointPair_zero", "LinearMap.isNilpotent_mulRight_iff", "LinearMap.isometryOfInner_toLinearMap", "LinearMap.iterate_surjective", "LinearMap.ker_id_sub_eq_of_proj", "LinearMap.ker_le_comap", "LinearMap.ker_le_of_iterateMapComap_eq_succ", "LinearMap.ker_prodMap", "LinearMap.ker_stdBasis", "LinearMap.ker_zero", "LinearMap.lTensor_comp", "LinearMap.lTensor_neg", "LinearMap.lift_rank_comp_le", "LinearMap.lift_rank_le_of_surjective", "LinearMap.map_coprod_prod", "LinearMap.map_eq_top_iff", "LinearMap.map_smul\u2082", "LinearMap.mem_isPairSelfAdjointSubmodule", "LinearMap.minpoly_toMatrix'", "LinearMap.mul_eq_comp", "LinearMap.mul_eq_one_of_mul_eq_one", "LinearMap.nilRankAux_le", "LinearMap.nilRank_le_card", "LinearMap.ofIsCompl_eq", "LinearMap.ortho_smul_left", "LinearMap.ortho_smul_right", "LinearMap.orthogonal_span_singleton_eq_to_lin_ker", "LinearMap.polyCharpolyAux_map_aeval", "LinearMap.polyCharpoly_monic", "LinearMap.pow_apply_mem_of_forall_mem", "LinearMap.range_comp_of_range_eq_top", "LinearMap.range_dualMap_eq_dualAnnihilator_ker_of_surjective", "LinearMap.range_le_iff_comap", "LinearMap.separatingLeft_toLinearMap\u2082'_iff_det_ne_zero", "LinearMap.separatingLeft_toLinearMap\u2082'_of_det_ne_zero'", "LinearMap.separatingRight_iff_flip_ker_eq_bot", "LinearMap.splittingOfFinsuppSurjective_splits", "LinearMap.sub_apply", "LinearMap.sup_range_inl_inr", "LinearMap.tailing_le_tunnel", "LinearMap.toMatrix'_one", "LinearMap.toMatrixAlgEquiv_transpose_apply'", "LinearMap.toMatrix_apply'", "LinearMap.toMatrix\u2082'_compl\u2082", "LinearMap.toMvPolynomial_totalDegree_le", "LinearMap.trace_mul_cycle", "LinearMap.update_apply", "LinearOrder.concaveOn_of_lt", "LinearPMap.IsClosable.graph_closure_eq_closure_graph", "LinearPMap.eq_of_le_of_domain_eq", "LinearPMap.graph_map_snd_eq_range", "LinearPMap.inverse_domain", "LinearPMap.le_closure", "LinearPMap.map_zero", "LinearPMap.mem_domain_iff_of_eq_graph", "LinearPMap.mem_graph_iff", "LinearPMap.sSup_le", "Liouville.liouvilleWith", "Liouville.transcendental", "LiouvilleNumber.partialSum_succ", "LiouvilleWith.mono", "LipschitzOnWith.coordinate", "LipschitzOnWith.hausdorffMeasure_image_le", "LipschitzOnWith.le_add_mul", "LipschitzOnWith.mk_one", "LipschitzOnWith.of_le_add_mul'", "LipschitzWith.comp_boundedVariationOn", "LipschitzWith.completion_map", "LipschitzWith.dist_iterate_succ_le_geometric", "LipschitzWith.edist_lt_mul_of_lt", "LipschitzWith.integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul'", "LipschitzWith.list_prod", "LipschitzWith.min", "LipschitzWith.mk_one", "LipschitzWith.nnorm_le_mul'", "LipschitzWith.norm_compLp_sub_le", "LipschitzWith.uniformContinuous", "List.Chain'.count_false_le_count_true_add_one", "List.Chain'.count_not_eq_count", "List.Chain'.imp_head", "List.Duplicate.mono_sublist", "List.Duplicate.ne_singleton", "List.Ico.filter_le_of_bot", "List.Ico.filter_le_of_top_le", "List.Ico.filter_lt_of_ge", "List.IsPrefix.length_le", "List.IsRotated.map", "List.IsRotated.nodup_iff", "List.IsSuffix.eq_of_length", "List.IsSuffix.filter", "List.Nat.antidiagonalTuple_zero_right", "List.Nat.antidiagonal_succ'", "List.Nodup.diff_eq_filter", "List.Nodup.pairwise_coe", "List.Nodup.rotate_eq_self_iff", "List.Palindrome.of_reverse_eq", "List.Perm.filter", "List.Perm.filterMap", "List.Perm.kerase", "List.Perm.map", "List.Sublist.countP_le", "List.Sublist.erase_diff_erase_sublist", "List.Sublist.length_le", "List.Sublist.map", "List.Sublist.tail", "List.Subperm.count_le", "List.aestronglyMeasurable_prod'", "List.alternatingProd_cons_cons'", "List.argAux_self", "List.argmax_cons", "List.attach_map_coe'", "List.chain'_cons", "List.chain'_join", "List.choose_mem", "List.comp_map", "List.comp_traverse", "List.concat_eq_reverse_cons", "List.continuousAt_length", "List.countP_bind", "List.countP_cons_of_neg", "List.countP_map", "List.countP_nil", "List.count_not_add_count", "List.count_singleton'", "List.count_tail", "List.cyclicPermutations_of_ne_nil", "List.dProdIndex_nil", "List.dProd_cons", "List.dedup_cons_of_not_mem'", "List.destutter'_singleton", "List.diff_cons", "List.diff_erase", "List.disjoint_append_left", "List.disjoint_pmap", "List.disjoint_right", "List.dropWhile_idempotent", "List.drop_sum_join", "List.duplicate_iff_two_le_count", "List.eq_of_mem_map_const", "List.eq_of_prod_take_eq", "List.eraseP_map", "List.erase_append_right", "List.erase_orderedInsert", "List.exists_pw_disjoint_with_card", "List.exists_tfae", "List.ext_get?'", "List.ext_get_iff", "List.findIdx_eq_length", "List.findIdx_lt_length_of_exists", "List.findIdx_of_get?_eq_some", "List.foldl1_eq_foldr1", "List.foldl_append", "List.foldl_nil", "List.foldrRecOn_cons", "List.foldr_ext", "List.foldr_monotone", "List.foldr_reverse", "List.forM_append", "List.forall_mem_of_forall_mem_cons", "List.forall_zipWith", "List.forall\u2082_and_left", "List.forall\u2082_comp_perm_eq_perm_comp_forall\u2082", "List.forall\u2082_cons_left_iff", "List.forall\u2082_map_right_iff", "List.forall\u2082_refl", "List.formPerm_cons_concat_apply_last", "List.formPerm_nil", "List.get?_eq_getElem?", "List.get?_length", "List.get?_modifyNth", "List.getElem?_range'", "List.getElem?_rotate", "List.getElem_drop'", "List.getElem_insertNth_add_succ", "List.getElem_ofFn_go", "List.getElem_rotate", "List.getLastI_eq_getLast?", "List.get_append", "List.get_cyclicPermutations", "List.get_pmap", "List.get_tails", "List.head?_append_of_ne_nil", "List.headI_cons", "List.indexOf_append_of_not_mem", "List.indexOf_cons", "List.index_of_argmax", "List.infix_bind_of_mem", "List.infix_concat", "List.infix_cons", "List.inits_eq_initsTR", "List.inits_reverse", "List.insert_of_not_mem", "List.inter_eq_nil_iff_disjoint", "List.isRotated_comm", "List.isRotated_cyclicPermutations_iff", "List.isSubperm_iff", "List.join_eq_nil", "List.keys_kreplace", "List.kreplace_nodupKeys", "List.last_ofFn_succ", "List.length_attach", "List.length_cons", "List.length_eq_three", "List.length_inits", "List.length_iterate", "List.length_le_length_insertNth", "List.length_modifyNth", "List.length_permutationsAux2", "List.length_pos_iff_ne_nil", "List.length_range", "List.length_replicate", "List.length_scanl", "List.length_sigma", "List.length_sigma'", "List.length_split_lt", "List.length_take", "List.length_toAssocList", "List.lookupAll_nodup", "List.lt_findIdx_of_not", "List.lt_iff_lex_lt", "List.lt_length_left_of_zipWith", "List.mapIdxMAux'_eq_mapIdxMGo", "List.mapIdx_cons", "List.map_diff", "List.map_erase", "List.map_leftInverse_iff", "List.map_permutationsAux", "List.mem_cons_eq", "List.mem_inter_of_mem_of_mem", "List.mem_keys", "List.mem_keys_kerase_of_ne", "List.mem_map_swap", "List.mem_merge", "List.mem_of_mem_inter_right", "List.mem_orderedInsert", "List.mem_pair", "List.mem_permutationsAux2'", "List.mem_rotate", "List.mem_sublists", "List.mem_union_iff", "List.mem_union_left", "List.minimum_cons", "List.mk_mem_sym2", "List.modifyNth_eq_set_get", "List.modify_get?_length", "List.new_def_eq_old_def", "List.nextOr_singleton", "List.nil_prefix", "List.nil_subset", "List.nil_suffix", "List.nodupKeys_nil", "List.nodup_append", "List.nodup_cons", "List.nodup_iff_sublist", "List.nodup_permutations", "List.nodup_range", "List.nodup_sublists'", "List.norm_prod_le'", "List.not_exists_mem_nil", "List.not_mem_keys", "List.nthLe_insertNth_add_succ", "List.nthLe_insertNth_of_lt", "List.nthLe_of_mem", "List.ofFn_inj'", "List.orderedInsert_nil", "List.pairwise_append_comm", "List.pairwise_filter", "List.pairwise_sublists", "List.perm_append_left_iff", "List.perm_replicate_append_replicate", "List.perm_split", "List.permutationsAux2_fst", "List.prefix_append", "List.prefix_cons_inj", "List.prev_cons_cons_of_ne'", "List.prod_eq_pow_single", "List.prod_join", "List.prod_nil", "List.prod_reverse_noncomm", "List.prod_take_succ", "List.range'_concat", "List.range'_subset_right", "List.range_add", "List.range_subset", "List.ranges_disjoint", "List.ranges_join'", "List.ranges_length", "List.reduceOption_length_lt_iff", "List.rel_filterMap", "List.rel_mem", "List.rel_of_chain_cons", "List.reverse_foldl", "List.reverse_nil", "List.rotate_cons_succ", "List.rotate_nil", "List.rotate_one_eq_self_iff_eq_replicate", "List.rotate_singleton", "List.rtakeWhile_suffix", "List.scanl_cons", "List.singleton_eq", "List.snd_mem_of_mem_enumFrom", "List.splitOnP_cons", "List.splitOnP_nil", "List.sublist_eq_map_get", "List.sublist_join", "List.sublist_orderedInsert", "List.sublists'Aux_eq_array_foldl", "List.sublists'Aux_eq_map", "List.sublists'_map", "List.sublists_eq_sublists'", "List.subperm_of_subset", "List.subset_cons", "List.subset_insert", "List.sum_take_map_length_splitWrtComposition", "List.support_formPerm_le'", "List.surjective_head!", "List.sym_one_eq", "List.tail_append_of_ne_nil", "List.tails_eq_tailsTR", "List.takeD_eq_takeDTR", "List.takeD_nil", "List.takeWhile_prefix", "List.take_append_of_le_length", "List.tendsto_cons", "List.tendsto_insertNth'", "List.tendsto_nhds", "List.toFinsupp_singleton", "List.unzip_swap", "List.zipWith_flip", "List.zipWith_zipWith_left", "Lists'.ofList_subset", "Lists.Equiv.symm", "Lists.of_toList", "LocalRing.CotangentSpace.map_eq_top_iff", "LocalRing.closedPoint_mem_iff", "LocalRing.eq_maximalIdeal", "LocalRing.maximal_ideal_unique", "Localization.AtPrime.map_eq_maximalIdeal", "Localization.AtPrime.nilpotent_iff_mem_maximal_of_minimal", "Localization.liftOn_zero", "Localization.mk_eq_mk'", "Localization.mk_zero", "LocalizedModule.algebraMap_mk", "LocalizedModule.mk_cancel_common_right", "LocalizedModule.smul'_mul", "LocallyBoundedVariationOn.ae_differentiableWithinAt_of_mem", "LocallyBoundedVariationOn.ae_differentiableWithinAt_of_mem_real", "LocallyConstant.apply_eq_of_preconnectedSpace", "LocallyConstant.ker_comap\u2097", "LocallyConstant.one_apply", "LocallyConstant.piecewise_apply_right", "LocallyFinite.Realizer.to_locallyFinite", "LocallyFinite.continuous", "LocallyFinite.finite_nonempty_inter_compact", "LocallyFinite.finite_nonempty_of_compact", "LocallyFinite.option_elim'", "LocallyFinite.subset", "LocallyLipschitz.comp", "LocallyLipschitz.continuous", "LocallyLipschitz.pow_end", "LowerAdjoint.closure_sup_closure", "LowerAdjoint.closure_sup_closure_left", "LowerAdjoint.mem_iff", "LowerSemicontinuous.lowerSemicontinuousAt", "LowerSemicontinuous.measurable", "LowerSet.Iic_top", "LowerSet.coe_iInf", "LowerSet.coe_sup", "LowerSet.compl_iInf", "LowerSet.compl_le_compl", "LowerSet.mem_prod", "LowerSet.mem_top", "LowerSet.prod_eq_bot", "LowerSet.symm_map", "LucasLehmer.X.add_snd", "LucasLehmer.X.coe_mul", "LucasLehmer.X.ofNat_snd", "MDifferentiableAt.const_smul", "MDifferentiableOn.prod_mk_space", "MDifferentiableWithinAt.mdifferentiableAt", "Magma.AssocQuotient.induction_on", "Magma.AssocQuotient.quot_mk_assoc_left", "Mathlib.Meta.NormNum.Finset.sum_empty", "Mathlib.Meta.NormNum.IsInt.neg_to_eq", "Mathlib.Meta.NormNum.eq_of_true", "Mathlib.Meta.NormNum.isFibAux_one", "Mathlib.Meta.NormNum.isInt_ediv", "Mathlib.Meta.NormNum.isInt_emod_neg", "Mathlib.Meta.NormNum.isInt_emod_zero", "Mathlib.Meta.NormNum.isInt_le_false", "Mathlib.Meta.NormNum.isInt_zpow_pos", "Mathlib.Meta.NormNum.isNat_fib", "Mathlib.Meta.NormNum.isNat_intOfNat", "Mathlib.Meta.NormNum.isNat_minFac_1", "Mathlib.Meta.NormNum.isNat_prime_1", "Mathlib.Meta.NormNum.isRat_inv_neg", "Mathlib.Meta.NormNum.isRat_inv_neg_one", "Mathlib.Meta.NormNum.isRat_mkRat", "Mathlib.Meta.NormNum.jacobiSymNat.one_right", "Mathlib.Meta.NormNum.jacobiSymNat.qr\u2083", "Mathlib.Meta.NormNum.minFacHelper_3", "Mathlib.Meta.Positivity.max_ne", "Mathlib.Tactic.Abel.term_atomg", "Mathlib.Tactic.ComputeDegree.degree_eq_of_le_of_coeff_ne_zero'", "Mathlib.Tactic.ComputeDegree.natDegree_smul_le_of_le", "Mathlib.Tactic.PushNeg.not_forall_eq", "Mathlib.Tactic.PushNeg.not_lt_eq", "Mathlib.Tactic.Ring.cast_rat", "Mathlib.Tactic.Ring.mul_add", "Mathlib.Tactic.Ring.natCast_add", "Mathlib.Tactic.Ring.one_pow", "Mathlib.Tactic.Ring.sub_pf", "Mathlib.Tactic.RingNF.nat_rawCast_0", "Matrix.BlockTriangular.add", "Matrix.GeneralLinearGroup.coe_mul", "Matrix.IsDiag.smul", "Matrix.IsHermitian.add", "Matrix.IsHermitian.adjugate", "Matrix.IsHermitian.map", "Matrix.IsHermitian.spectral_theorem", "Matrix.IsHermitian.spectrum_toEuclideanLin", "Matrix.IsHermitian.star_eigenvectorUnitary_mulVec", "Matrix.IsSymm.ext", "Matrix.IsSymm.ext_iff", "Matrix.J_inv", "Matrix.Pivot.exists_isTwoBlockDiagonal_list_transvec_mul_mul_list_transvec", "Matrix.Pivot.listTransvecCol_mul_last_row", "Matrix.Pivot.listTransvecCol_mul_mul_listTransvecRow_last_col", "Matrix.PosDef.posSemidef_add", "Matrix.Represents.algebraMap", "Matrix.SpecialLinearGroup.det_coe", "Matrix.SpecialLinearGroup.isCoprime_row", "Matrix.TransvectionStruct.sumInl_toMatrix_prod_mul", "Matrix.UnitaryGroup.ext_iff", "Matrix.UnitaryGroup.mul_val", "Matrix.add_hadamard", "Matrix.add_kroneckerTMul", "Matrix.algebraMap_eq_diagonal", "Matrix.blockDiag'_transpose", "Matrix.blockDiag_smul", "Matrix.blockDiagonal'_map", "Matrix.blockDiagonal'_zero", "Matrix.blockDiagonal_map", "Matrix.circulant_neg", "Matrix.coe_detMonoidHom", "Matrix.coe_vecMulLinear", "Matrix.conjTransposeAddEquiv_symm", "Matrix.conjTranspose_inj", "Matrix.cons_val_one", "Matrix.cons_val_three", "Matrix.cons_val_zero", "Matrix.cons_vec_bit1_eq_alt1", "Matrix.det_conjTranspose", "Matrix.det_eq_of_forall_col_eq_smul_add_pred", "Matrix.det_fin_two_of", "Matrix.det_fromBlocks_zero\u2082\u2081", "Matrix.det_mul", "Matrix.det_mul_row", "Matrix.det_nonsing_inv", "Matrix.det_one", "Matrix.det_units_conj'", "Matrix.det_updateRow_add_self", "Matrix.det_vandermonde", "Matrix.det_vandermonde_ne_zero_iff", "Matrix.det_zero_of_row_eq", "Matrix.diagonal_comp_stdBasis", "Matrix.diagonal_map", "Matrix.diagonal_tsum", "Matrix.dotProduct_comm", "Matrix.dotProduct_diagonal'", "Matrix.dotProduct_zero", "Matrix.eta_fin_three", "Matrix.exists_right_inverse_iff_isUnit", "Matrix.exp_conj", "Matrix.exp_diagonal", "Matrix.fromBlocks_add", "Matrix.fromBlocks_submatrix_sum_swap_sum_swap", "Matrix.fromBlocks_transpose", "Matrix.fromColumns_toColumns", "Matrix.fromRows_mul", "Matrix.induction_on'", "Matrix.inv_fromBlocks_zero\u2081\u2082_of_isUnit_iff", "Matrix.inv_inv_of_invertible", "Matrix.isHermitian_one", "Matrix.isSymm_one", "Matrix.isSymm_transpose_mul_self", "Matrix.isUnit_det_of_left_inverse", "Matrix.isUnit_det_transpose", "Matrix.isUnit_diagonal", "Matrix.kroneckerMapBilinear_mul_mul", "Matrix.kroneckerTMul_smul", "Matrix.l2_opNorm_conjTranspose", "Matrix.linearIndependent_cols_iff_isUnit", "Matrix.linfty_opNNNorm_mul", "Matrix.linfty_opNNNorm_toMatrix", "Matrix.map_id'", "Matrix.map_natCast", "Matrix.map_op_smul'", "Matrix.mem_unitaryGroup_iff", "Matrix.mulVecLin_mul", "Matrix.mulVec_add", "Matrix.mulVec_cramer", "Matrix.mulVec_injective_iff", "Matrix.mulVec_stdBasisMatrix", "Matrix.mulVec_surjective_iff_exists_right_inverse", "Matrix.mul_adjugate", "Matrix.mul_nonsing_inv", "Matrix.mul_right_injective_of_inv", "Matrix.mul_smul", "Matrix.neg_dotProduct", "Matrix.neg_empty", "Matrix.norm_col", "Matrix.norm_row", "Matrix.one_apply", "Matrix.one_kronecker_one", "Matrix.pow_inv_comm'", "Matrix.pow_sub'", "Matrix.range_diagonal", "Matrix.rank_self_mul_transpose", "Matrix.reindex_refl_refl", "Matrix.represents_iff'", "Matrix.reverse_charpoly", "Matrix.right_inv_eq_right_inv", "Matrix.row_apply", "Matrix.row_cons", "Matrix.single_vecMul", "Matrix.star_dotProduct_star", "Matrix.stdBasis_eq_stdBasisMatrix", "Matrix.submatrix_add", "Matrix.submatrix_cons_row", "Matrix.toBilin'Aux_eq", "Matrix.toBilin'_apply", "Matrix.toBilin'_apply'", "Matrix.toBilin'_comp", "Matrix.toBlock_mul_eq_mul", "Matrix.toBlocks_fromBlocks\u2082\u2082", "Matrix.toEuclideanCLM_piLp_equiv_symm", "Matrix.toLinearMap\u2082'_apply'", "Matrix.toLinearMap\u2082_basisFun", "Matrix.toLinearMap\u209b\u2097\u2082'_stdBasis", "Matrix.toMvPolynomial_zero", "Matrix.toRows\u2082_fromRows", "Matrix.toSquareBlockProp_def", "Matrix.trace_add", "Matrix.trace_eq_sum_roots_charpoly_of_splits", "Matrix.trace_fin_one", "Matrix.trace_multiset_sum", "Matrix.updateColumn_ne", "Matrix.updateColumn_submatrix_equiv", "Matrix.vecMul_surjective_iff_exists_left_inverse", "Matrix.vecMul_surjective_of_invertible", "Matroid.Base.dep_of_ssubset", "Matroid.Base.exchange_base_of_indep'", "Matroid.Base.exists_insert_of_ssubset", "Matroid.Base.map", "Matroid.Base.ncard_diff_comm", "Matroid.Base.not_base_of_ssubset", "Matroid.Basis.basis'", "Matroid.Basis.basis_inter_ground", "Matroid.Basis.basis_of_basis_of_subset_of_subset", "Matroid.Basis.basis_restriction", "Matroid.Basis.eq_exchange_of_diff_eq_singleton", "Matroid.Basis.of_restriction", "Matroid.Basis.transfer", "Matroid.Indep.basis_of_maximal_subset", "Matroid.Indep.exists_base_subset_union_base", "Matroid.Indep.exists_basis_subset_union_basis", "Matroid.Indep.exists_bijOn_of_map", "Matroid.Indep.mapEmbedding", "Matroid.Restriction.antisymm", "Matroid.comapOn_base_iff_of_surjOn", "Matroid.comapOn_indep_iff", "Matroid.comapOn_preimage_eq", "Matroid.comap_basis_iff", "Matroid.dual_inj", "Matroid.emptyOn_ground", "Matroid.eq_of_restrictSubtype_eq", "Matroid.finite_or_infiniteRk", "Matroid.finite_setOf_matroid", "Matroid.finite_setOf_matroid'", "Matroid.ground_nonempty", "Matroid.indep_iff_forall_finite_subset_indep", "Matroid.loopyOn_empty", "Matroid.restrictSubtype_ground", "Matroid.restrict_strictRestriction", "Matroid.setOf_base_eq_maximals_setOf_indep", "Matroid.setOf_dual_base_eq", "Matroid.singleton_subset_ground", "Matroid.uniqueBaseOn_indep_iff", "MaximalSpectrum.iInf_localization_eq_bot", "Measurable.add_simpleFunc", "Measurable.ceil", "Measurable.dist", "Measurable.ite", "Measurable.lintegral_prod_left", "Measurable.lintegral_prod_right'", "Measurable.lmarginal", "Measurable.sinh", "Measurable.smul", "Measurable.stronglyMeasurable_add", "MeasurableEmbedding.aemeasurable_comp_iff", "MeasurableEmbedding.comap_eq", "MeasurableEmbedding.measurable_comp_iff", "MeasurableEmbedding.withDensity_ofReal_comap_apply_eq_integral_abs_deriv_mul", "MeasurableEmbedding.withDensity_ofReal_comap_apply_eq_integral_abs_deriv_mul'", "MeasurableEquiv.coe_mulLeft", "MeasurableEquiv.coe_sumPiEquivProdPi", "MeasurableEquiv.map_measurableEquiv_injective", "MeasurableEquiv.map_symm", "MeasurableEquiv.measurableEmbedding", "MeasurableEquiv.preimage_symm", "MeasurableEquiv.withDensity_ofReal_map_symm_apply_eq_integral_abs_deriv_mul'", "MeasurableSet.coe_sdiff", "MeasurableSet.image_of_measurable_injOn", "MeasurableSet.induction_on_open", "MeasurableSet.insert", "MeasurableSet.ite'", "MeasurableSet.map_coe_volume", "MeasurableSet.nullMeasurableSet", "MeasurableSpace.DynkinSystem.has_univ", "MeasurableSpace.comap_eq_generateFrom", "MeasurableSpace.comap_map_le", "MeasurableSpace.generateFrom_countableGeneratingSet", "MeasurableSpace.generateFrom_natGeneratingSequence", "MeasurableSpace.invariants_id", "MeasurableSpace.measurableSet_generateFrom_memPartition_iff", "MeasurableSpace.measurableSpace_iSup_eq", "MeasurableSpace.measurable_invariants_of_semiconj", "MeasureTheory.AECover.comp_tendsto", "MeasureTheory.AECover.iSup_lintegral_eq_of_countably_generated", "MeasureTheory.AECover.integrable_of_lintegral_nnnorm_bounded'", "MeasureTheory.AEDisjoint.comm", "MeasureTheory.AEDisjoint.diff_ae_eq_right", "MeasureTheory.AEEqFun.compMeasurePreserving_eq_mk", "MeasureTheory.AEEqFun.compQuasiMeasurePreserving_toGerm", "MeasureTheory.AEEqFun.mk_pow", "MeasureTheory.AEStronglyMeasurable'.neg", "MeasureTheory.AEStronglyMeasurable'.sub", "MeasureTheory.AEStronglyMeasurable.const_smul'", "MeasureTheory.AEStronglyMeasurable.convolution_integrand", "MeasureTheory.AEStronglyMeasurable.convolution_integrand'", "MeasureTheory.AEStronglyMeasurable.div", "MeasureTheory.AEStronglyMeasurable.im", "MeasureTheory.AEStronglyMeasurable.sum_measure", "MeasureTheory.Adapted.isStoppingTime_crossing", "MeasureTheory.BoundedContinuousFunction.integral_eq_integral_meas_le", "MeasureTheory.Content.contentRegular_exists_compact", "MeasureTheory.Content.innerContent_iUnion_nat", "MeasureTheory.Content.innerContent_mono'", "MeasureTheory.Content.lt_top", "MeasureTheory.Content.mono", "MeasureTheory.Content.outerMeasure_pos_of_is_mul_left_invariant", "MeasureTheory.Filtration.const_apply", "MeasureTheory.Filtration.stronglyMeasurable_limitProcess", "MeasureTheory.FiniteMeasure.coeFn_smul", "MeasureTheory.FiniteMeasure.restrict_apply", "MeasureTheory.FiniteMeasure.restrict_mass", "MeasureTheory.FiniteMeasure.smul_apply", "MeasureTheory.FiniteMeasure.tendsto_iff_forall_toWeakDualBCNN_tendsto", "MeasureTheory.FiniteMeasure.tendsto_iff_weak_star_tendsto", "MeasureTheory.FiniteMeasure.tendsto_map_of_tendsto_of_continuous", "MeasureTheory.FiniteMeasure.tendsto_testAgainstNN_of_tendsto_normalize_testAgainstNN_of_tendsto_mass", "MeasureTheory.FiniteMeasure.tendsto_zero_testAgainstNN_of_tendsto_zero_mass", "MeasureTheory.FiniteMeasure.testAgainstNN_lipschitz", "MeasureTheory.FiniteMeasure.testAgainstNN_one", "MeasureTheory.HasFiniteIntegral.const_mul", "MeasureTheory.Integrable.add_measure", "MeasureTheory.Integrable.aemeasurable", "MeasureTheory.Integrable.compProd_mk_left_ae", "MeasureTheory.Integrable.comp_div", "MeasureTheory.Integrable.comp_inv", "MeasureTheory.Integrable.comp_mul_left", "MeasureTheory.Integrable.condexpKernel_ae", "MeasureTheory.Integrable.congr", "MeasureTheory.Integrable.induction", "MeasureTheory.Integrable.intervalIntegrable", "MeasureTheory.Integrable.left_of_add_measure", "MeasureTheory.Integrable.pos_part", "MeasureTheory.Integrable.prod_mul", "MeasureTheory.Integrable.prod_right_ae", "MeasureTheory.Integrable.real_toNNReal", "MeasureTheory.Integrable.tendsto_ae_condexp", "MeasureTheory.Integrable.toL1_sub", "MeasureTheory.IntegrableAtFilter.eq_zero_of_tendsto", "MeasureTheory.IntegrableAtFilter.inf_of_left", "MeasureTheory.IntegrableOn.integrable_of_ae_not_mem_eq_zero", "MeasureTheory.IsFundamentalDomain.lintegral_eq_tsum", "MeasureTheory.IsFundamentalDomain.setIntegral_eq", "MeasureTheory.IsProbabilityMeasure.ae_neBot", "MeasureTheory.IsProbabilityMeasure.ne_zero", "MeasureTheory.IsStoppingTime.add_const_nat", "MeasureTheory.IsStoppingTime.measurableSet_inter_le_const_iff", "MeasureTheory.IsStoppingTime.measurableSet_inter_le_iff", "MeasureTheory.IsStoppingTime.measurableSet_min_iff", "MeasureTheory.JordanDecomposition.real_smul_posPart_nonneg", "MeasureTheory.JordanDecomposition.smul_negPart", "MeasureTheory.JordanDecomposition.zero_negPart", "MeasureTheory.L1.SimpleFunc.setToL1SCLM_zero_left", "MeasureTheory.L1.SimpleFunc.setToL1S_mono_left", "MeasureTheory.L1.integral_neg", "MeasureTheory.L1.integral_of_fun_eq_integral", "MeasureTheory.L1.ofReal_norm_eq_lintegral", "MeasureTheory.L1.setToL1_mono_left", "MeasureTheory.L2.inner_indicatorConstLp_eq_setIntegral_inner", "MeasureTheory.L2.smul_left'", "MeasureTheory.LocallyIntegrableOn.continuousOn_smul", "MeasureTheory.Lp.boundedContinuousFunction_dense", "MeasureTheory.Lp.coeFn_neg", "MeasureTheory.Lp.coeFn_negPart", "MeasureTheory.Lp.nnnorm_def", "MeasureTheory.Lp.nnnorm_eq_zero_iff", "MeasureTheory.Lp.nnnorm_zero", "MeasureTheory.Lp.simpleFunc.aestronglyMeasurable", "MeasureTheory.Lp.simpleFunc.induction", "MeasureTheory.Lp.simpleFunc.mem\u2112p", "MeasureTheory.Lp.simpleFunc.toLp_neg", "MeasureTheory.Lp.simpleFunc.uniformEmbedding", "MeasureTheory.LpAddConst_zero", "MeasureTheory.Measure.InnerRegular.exists_compact_not_null", "MeasureTheory.Measure.InnerRegular.map_iff", "MeasureTheory.Measure.InnerRegularWRT.measurableSet_of_isOpen", "MeasureTheory.Measure.InnerRegularWRT.rfl", "MeasureTheory.Measure.IsEverywherePos.IsGdelta_of_isMulLeftInvariant", "MeasureTheory.Measure.IsMulLeftInvariant.quotientMeasureEqMeasurePreimage_of_set", "MeasureTheory.Measure.MutuallySingular.measure_compl_nullSet", "MeasureTheory.Measure.MutuallySingular.sum_right", "MeasureTheory.Measure.MutuallySingular.withDensity", "MeasureTheory.Measure.QuasiMeasurePreserving.comp", "MeasureTheory.Measure.WeaklyRegular.restrict_of_measure_ne_top", "MeasureTheory.Measure.ae_ae_eq_curry_of_prod", "MeasureTheory.Measure.comap_apply\u2080", "MeasureTheory.Measure.compProd_add_left", "MeasureTheory.Measure.count_empty", "MeasureTheory.Measure.countable_meas_pos_of_disjoint_iUnion", "MeasureTheory.Measure.countable_meas_pos_of_disjoint_of_meas_iUnion_ne_top\u2080", "MeasureTheory.Measure.empty_of_count_eq_zero", "MeasureTheory.Measure.eventually_nonempty_inter_smul_of_density_one", "MeasureTheory.Measure.exists_positive_of_not_mutuallySingular", "MeasureTheory.Measure.ext_iff'", "MeasureTheory.Measure.ext_of_Ico_finite", "MeasureTheory.Measure.fst_prod", "MeasureTheory.Measure.haar.index_union_eq", "MeasureTheory.Measure.haar.prehaar_mem_haarProduct", "MeasureTheory.Measure.haar.prehaar_sup_le", "MeasureTheory.Measure.isEverywherePos_everywherePosSubset_of_measure_ne_top", "MeasureTheory.Measure.isHaarMeasure_map_of_isFiniteMeasure", "MeasureTheory.Measure.le_mkMetric", "MeasureTheory.Measure.map_congr", "MeasureTheory.Measure.map_fst_prod", "MeasureTheory.Measure.map_mono_of_aemeasurable", "MeasureTheory.Measure.measurePreserving_homeomorphUnitSphereProd", "MeasureTheory.Measure.measure_preimage_isMulLeftInvariant_eq_smul_of_hasCompactSupport", "MeasureTheory.Measure.measure_toMeasurable_add_inter_left", "MeasureTheory.Measure.mkMetric_top", "MeasureTheory.Measure.one_le_hausdorffMeasure_zero_of_nonempty", "MeasureTheory.Measure.pi_eq_generateFrom", "MeasureTheory.Measure.quasiMeasurePreserving_fst", "MeasureTheory.Measure.restrict_iUnion_ae", "MeasureTheory.Measure.restrict_le_self", "MeasureTheory.Measure.restrict_union_add_inter'", "MeasureTheory.Measure.rnDeriv_add", "MeasureTheory.Measure.rnDeriv_add_of_mutuallySingular", "MeasureTheory.Measure.rnDeriv_add_right_of_mutuallySingular'", "MeasureTheory.Measure.rnDeriv_withDensity_left", "MeasureTheory.Measure.setIntegral_comp_smul", "MeasureTheory.Measure.setIntegral_condKernel_univ_left", "MeasureTheory.Measure.setIntegral_toReal_rnDeriv'", "MeasureTheory.Measure.setIntegral_toReal_rnDeriv_eq_withDensity", "MeasureTheory.Measure.setLIntegral_compProd", "MeasureTheory.Measure.singularPart_zero", "MeasureTheory.Measure.smul_finite", "MeasureTheory.Measure.snd_univ", "MeasureTheory.Measure.sub_le", "MeasureTheory.Measure.sum_apply_eq_zero", "MeasureTheory.Measure.sum_apply_eq_zero'", "MeasureTheory.Measure.sum_comp_equiv", "MeasureTheory.Measure.sum_zero", "MeasureTheory.Measure.toSignedMeasure_zero", "MeasureTheory.Measure.trimmed", "MeasureTheory.Measure.withDensity_rnDeriv_le", "MeasureTheory.MeasurePreserving.comp_right_iff", "MeasureTheory.MeasurePreserving.mul_right", "MeasureTheory.MeasurePreserving.prod", "MeasureTheory.MeasurePreserving.skew_product", "MeasureTheory.MeasurePreserving.trans", "MeasureTheory.Mem\u2112p.exists_simpleFunc_snorm_sub_lt", "MeasureTheory.Mem\u2112p.im", "MeasureTheory.Mem\u2112p.induction", "MeasureTheory.Mem\u2112p.integral_indicator_norm_ge_le", "MeasureTheory.Mem\u2112p.of_measure_le_smul", "MeasureTheory.Mem\u2112p.of_nnnorm_le_mul", "MeasureTheory.Mem\u2112p.piecewise", "MeasureTheory.Mem\u2112p.right_of_add_measure", "MeasureTheory.Mem\u2112p.snormEssSup_indicator_norm_ge_eq_zero", "MeasureTheory.Mem\u2112p.snorm_indicator_le", "MeasureTheory.Mem\u2112p.snorm_lt_top", "MeasureTheory.Mem\u2112p.variance_eq_of_integral_eq_zero", "MeasureTheory.NullMeasurable.aemeasurable", "MeasureTheory.NullMeasurable.measure_preimage_eq_measure_restrict_preimage_of_ae_compl_eq_const", "MeasureTheory.NullMeasurableSet.compl_iff", "MeasureTheory.NullMeasurableSet.fundamentalFrontier", "MeasureTheory.NullMeasurableSet.inter", "MeasureTheory.NullMeasurableSet.left_of_prod", "MeasureTheory.NullMeasurableSet.right_of_prod", "MeasureTheory.NullMeasurableSet.union_null", "MeasureTheory.OuterMeasure.biInf_apply", "MeasureTheory.OuterMeasure.boundedBy_eq_ofFunction", "MeasureTheory.OuterMeasure.boundedBy_eq_self", "MeasureTheory.OuterMeasure.comap_mono", "MeasureTheory.OuterMeasure.exists_measurable_superset_eq_trim", "MeasureTheory.OuterMeasure.iUnion", "MeasureTheory.OuterMeasure.iUnion_of_tendsto_zero", "MeasureTheory.OuterMeasure.isometry_comap_mkMetric", "MeasureTheory.OuterMeasure.map_apply", "MeasureTheory.OuterMeasure.measure_inter_union", "MeasureTheory.OuterMeasure.mkMetric_nnreal_smul", "MeasureTheory.OuterMeasure.sInfGen_def", "MeasureTheory.OuterMeasure.sSup_apply", "MeasureTheory.OuterMeasure.smul_dirac_apply", "MeasureTheory.ProbabilityMeasure.coe_mk", "MeasureTheory.SignedMeasure.eq_rnDeriv", "MeasureTheory.SignedMeasure.im_toComplexMeasure", "MeasureTheory.SignedMeasure.null_of_totalVariation_zero", "MeasureTheory.SignedMeasure.toComplexMeasure_apply", "MeasureTheory.SimpleFunc.FinMeasSupp.map\u2082", "MeasureTheory.SimpleFunc.FinMeasSupp.meas_preimage_singleton_ne_zero", "MeasureTheory.SimpleFunc.FinMeasSupp.mul", "MeasureTheory.SimpleFunc.coe_restrict", "MeasureTheory.SimpleFunc.iSup_approx_apply", "MeasureTheory.SimpleFunc.integrable_approxOn", "MeasureTheory.SimpleFunc.integral_eq_lintegral'", "MeasureTheory.SimpleFunc.map_coe_nnreal_restrict", "MeasureTheory.SimpleFunc.measure_lt_top_of_mem\u2112p_indicator", "MeasureTheory.SimpleFunc.mem_range_of_measure_ne_zero", "MeasureTheory.SimpleFunc.mem\u2112p_approxOn_range", "MeasureTheory.SimpleFunc.mem\u2112p_iff", "MeasureTheory.SimpleFunc.mem\u2112p_iff_integrable", "MeasureTheory.SimpleFunc.norm_setToSimpleFunc_le_sum_mul_norm", "MeasureTheory.SimpleFunc.piecewise_univ", "MeasureTheory.SimpleFunc.pow_apply", "MeasureTheory.SimpleFunc.restrict_const_lintegral", "MeasureTheory.SimpleFunc.setToSimpleFunc_add_left'", "MeasureTheory.SimpleFunc.simpleFunc_bot", "MeasureTheory.SimpleFunc.support_eq", "MeasureTheory.SimpleFunc.tendsto_approxOn_range_Lp_snorm", "MeasureTheory.StronglyMeasurable.comp_measurable", "MeasureTheory.StronglyMeasurable.finStronglyMeasurable_of_set_sigmaFinite", "MeasureTheory.StronglyMeasurable.integral_kernel_prod_left'", "MeasureTheory.StronglyMeasurable.inv", "MeasureTheory.StronglyMeasurable.measurableSet_le", "MeasureTheory.StronglyMeasurable.mul", "MeasureTheory.StronglyMeasurable.of_uncurry_right", "MeasureTheory.Submartingale.condexp_sub_nonneg", "MeasureTheory.Submartingale.expected_stoppedValue_mono", "MeasureTheory.Supermartingale.condexp_ae_le", "MeasureTheory.Supermartingale.smul_nonneg", "MeasureTheory.TendstoInMeasure.exists_seq_tendstoInMeasure_atTop", "MeasureTheory.UnifIntegrable.add", "MeasureTheory.VectorMeasure.AbsolutelyContinuous.neg_left", "MeasureTheory.VectorMeasure.ext", "MeasureTheory.VectorMeasure.ext_iff", "MeasureTheory.adapted_const", "MeasureTheory.addHaar_image_eq_zero_of_det_fderivWithin_eq_zero", "MeasureTheory.ae_all_iff", "MeasureTheory.ae_eq_restrict_biUnion_iff", "MeasureTheory.ae_mono", "MeasureTheory.ae_of_ae_map", "MeasureTheory.ae_of_all", "MeasureTheory.ae_restrict_iff'", "MeasureTheory.ae_restrict_mem\u2080", "MeasureTheory.ae_smul_measure", "MeasureTheory.ae_zero", "MeasureTheory.aecover_Icc_of_Ico", "MeasureTheory.aecover_Ioo_of_Ioc", "MeasureTheory.aedisjoint_compl_right", "MeasureTheory.aemeasurable_withDensity_ennreal_iff'", "MeasureTheory.all_ae_tendsto_ofReal_norm", "MeasureTheory.compl_mem_measurableCylinders", "MeasureTheory.condexpInd_nonneg", "MeasureTheory.condexpL1CLM_lpMeas", "MeasureTheory.condexpL1_add", "MeasureTheory.condexpL1_congr_ae", "MeasureTheory.condexp_ae_eq_condexpL1", "MeasureTheory.condexp_ae_eq_restrict_of_measurableSpace_eq_on", "MeasureTheory.condexp_condexp_of_le", "MeasureTheory.condexp_of_not_le", "MeasureTheory.condexp_stopping_time_ae_eq_restrict_eq", "MeasureTheory.condexp_stronglyMeasurable_mul\u2080", "MeasureTheory.continuousAt_of_dominated", "MeasureTheory.convolution_smul", "MeasureTheory.countable_meas_le_ne_meas_lt", "MeasureTheory.dist_convolution_le", "MeasureTheory.eventually_mul_left_iff", "MeasureTheory.exists_nat_measurableEquiv_range_coe_fin_of_finite", "MeasureTheory.exists_null_frontiers_thickening", "MeasureTheory.finStronglyMeasurable_iff_measurable", "MeasureTheory.finStronglyMeasurable_zero", "MeasureTheory.frequently_ae_iff", "MeasureTheory.fundamentalInterior_smul", "MeasureTheory.hahn_decomposition", "MeasureTheory.hasFiniteIntegral_iff_norm", "MeasureTheory.hasFiniteIntegral_zero_measure", "MeasureTheory.hausdorffMeasure_homothety_image", "MeasureTheory.hausdorffMeasure_pi_real", "MeasureTheory.hausdorffMeasure_prod_real", "MeasureTheory.hitting_eq_hitting_of_exists", "MeasureTheory.hitting_le", "MeasureTheory.hitting_le_iff_of_lt", "MeasureTheory.hitting_mem_set_of_hitting_lt", "MeasureTheory.inducedOuterMeasure_eq_iInf", "MeasureTheory.inducedOuterMeasure_preimage", "MeasureTheory.injective_dirac", "MeasureTheory.insert_ae_eq_self", "MeasureTheory.integrableOn_Ioi_of_intervalIntegral_norm_bounded", "MeasureTheory.integrableOn_finite_iUnion", "MeasureTheory.integrableOn_iUnion_of_summable_integral_norm", "MeasureTheory.integrableOn_image_iff_integrableOn_abs_det_fderiv_smul", "MeasureTheory.integrable_average", "MeasureTheory.integrable_condexp", "MeasureTheory.integrable_condexpL2_indicator", "MeasureTheory.integrable_count_iff", "MeasureTheory.integrable_smul_const", "MeasureTheory.integrable_withDensity_iff", "MeasureTheory.integrable_withDensity_iff_integrable_smul'", "MeasureTheory.integral_Icc_eq_integral_Ico", "MeasureTheory.integral_Icc_eq_integral_Ioo'", "MeasureTheory.integral_Ico_eq_integral_Ioo'", "MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonpos", "MeasureTheory.integral_average", "MeasureTheory.integral_average_sub", "MeasureTheory.integral_deriv_mul_eq_sub", "MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable", "MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable_aux\u2082", "MeasureTheory.integral_fintype_prod_eq_prod", "MeasureTheory.integral_llr_tilted_left", "MeasureTheory.integral_mono", "MeasureTheory.integral_piecewise", "MeasureTheory.integral_singleton", "MeasureTheory.intervalIntegral_tendsto_integral", "MeasureTheory.isStoppingTime_const", "MeasureTheory.ite_ae_eq_of_measure_compl_zero", "MeasureTheory.laverage_mem_openSegment_compl_self", "MeasureTheory.le_hitting", "MeasureTheory.levyProkhorovDist_le_of_forall_le", "MeasureTheory.lintegral_abs_det_fderiv_eq_addHaar_image", "MeasureTheory.lintegral_biUnion_finset\u2080", "MeasureTheory.lintegral_count'", "MeasureTheory.lintegral_iInf'", "MeasureTheory.lintegral_iInf_ae", "MeasureTheory.lintegral_iSup", "MeasureTheory.lintegral_indicator_const_comp", "MeasureTheory.lintegral_le_meas", "MeasureTheory.lintegral_mono'", "MeasureTheory.lintegral_mul_const'", "MeasureTheory.lintegral_mul_const''", "MeasureTheory.lintegral_nnnorm_condexpL2_indicator_le_real", "MeasureTheory.lintegral_nnnorm_neg", "MeasureTheory.lintegral_strict_mono", "MeasureTheory.lintegral_strict_mono_of_ae_le_of_frequently_ae_lt", "MeasureTheory.lintegral_withDensity_eq_lintegral_mul", "MeasureTheory.lintegral_withDensity_eq_lintegral_mul_non_measurable\u2080", "MeasureTheory.lintegral_zero_fun", "MeasureTheory.llr_smul_right", "MeasureTheory.lmarginal_eq_of_subset", "MeasureTheory.lpMeas.ae_eq_zero_of_forall_setIntegral_eq_zero", "MeasureTheory.lpMeasSubgroupToLpTrim_ae_eq", "MeasureTheory.lpMeasToLpTrim_smul", "MeasureTheory.lpTrimToLpMeas_ae_eq", "MeasureTheory.meas_le_lintegral\u2080", "MeasureTheory.measurableCylinders.measurableSet", "MeasureTheory.measurePreserving_mul_prod_inv_right", "MeasureTheory.measurePreserving_piFinSuccAbove", "MeasureTheory.measurePreserving_prod_mul_swap_right", "MeasureTheory.measureUnivNNReal_pos", "MeasureTheory.measure_eq_measure_smaller_of_between_null_diff", "MeasureTheory.measure_eq_zero_iff_eq_empty_of_smulInvariant", "MeasureTheory.measure_integral_le_pos", "MeasureTheory.measure_le_setAverage_pos", "MeasureTheory.measure_lt_one_eq_integral_div_gamma", "MeasureTheory.measure_map_restrict_apply", "MeasureTheory.measure_mul_laverage", "MeasureTheory.measure_mul_setLaverage", "MeasureTheory.measure_null_iff_singleton", "MeasureTheory.measure_null_of_locally_null", "MeasureTheory.measure_smul", "MeasureTheory.measure_union\u2080'", "MeasureTheory.measure_unitBall_eq_integral_div_gamma", "MeasureTheory.mem_disjointed_spanningSetsIndex", "MeasureTheory.mem_lpMeasSubgroup_iff_aeStronglyMeasurable'", "MeasureTheory.mem_lpMeas_iff_aeStronglyMeasurable'", "MeasureTheory.mem\u2112p_indicator_const", "MeasureTheory.norm_setIntegral_le_of_norm_le_const_ae''", "MeasureTheory.ofReal_average", "MeasureTheory.ofReal_toReal_ae_eq", "MeasureTheory.one_le_div_ae", "MeasureTheory.pdf.IsUniform.measure_preimage", "MeasureTheory.pdf.IsUniform.pdf_toReal_ae_eq", "MeasureTheory.pdf.IsUniform.toMeasurable", "MeasureTheory.pdf.congr", "MeasureTheory.piPremeasure_pi'", "MeasureTheory.prob_compl_eq_one_sub\u2080", "MeasureTheory.prob_le_one", "MeasureTheory.progMeasurable_const", "MeasureTheory.progMeasurable_min_stopping_time", "MeasureTheory.quasiMeasurePreserving_div_left_of_right_invariant", "MeasureTheory.restrict_compl_sigmaFiniteSet", "MeasureTheory.restrict_compl_sigmaFiniteSet_eq_zero_or_top", "MeasureTheory.rnDeriv_ae_eq_condexp", "MeasureTheory.rnDeriv_tilted_right", "MeasureTheory.sFinite_withDensity_of_sigmaFinite_of_measurable", "MeasureTheory.setIntegral_congr_ae", "MeasureTheory.setIntegral_congr_set_ae", "MeasureTheory.setIntegral_le_nonneg", "MeasureTheory.setIntegral_mono_set", "MeasureTheory.setIntegral_nonneg", "MeasureTheory.setIntegral_nonpos_ae", "MeasureTheory.setIntegral_tilted'", "MeasureTheory.setLIntegral_const_lt_top", "MeasureTheory.setLaverage_congr_fun", "MeasureTheory.setLaverage_const", "MeasureTheory.setToFun_congr_left", "MeasureTheory.setToFun_zero_left'", "MeasureTheory.snorm'_add_le_of_le_one", "MeasureTheory.snorm'_congr_norm_ae", "MeasureTheory.snorm'_mono_nnnorm_ae", "MeasureTheory.snorm'_neg", "MeasureTheory.snormEssSup_const_smul", "MeasureTheory.snormEssSup_map_measure", "MeasureTheory.snormEssSup_measure_zero", "MeasureTheory.snormEssSup_mono_measure", "MeasureTheory.snorm_congr_ae", "MeasureTheory.snorm_const_lt_top_iff", "MeasureTheory.snorm_indicator_const", "MeasureTheory.snorm_indicator_le_of_bound", "MeasureTheory.snorm_le_snorm_fderiv_of_le", "MeasureTheory.snorm_map_measure", "MeasureTheory.snorm_one_eq_lintegral_nnnorm", "MeasureTheory.snorm_top_piecewise", "MeasureTheory.stoppedProcess_eq''", "MeasureTheory.stoppedValue_const", "MeasureTheory.stoppedValue_lowerCrossingTime", "MeasureTheory.stoppedValue_stoppedValue_leastGE", "MeasureTheory.submartingale_of_expected_stoppedValue_mono", "MeasureTheory.submartingale_of_setIntegral_le", "MeasureTheory.supermartingale_of_setIntegral_succ_le", "MeasureTheory.tendsto_integral_norm_approxOn_sub", "MeasureTheory.tendsto_integral_smul_of_tendsto_average_norm_sub", "MeasureTheory.tendsto_measure_iInter'", "MeasureTheory.tendsto_setToFun_filter_of_dominated_convergence", "MeasureTheory.toMeasure_apply\u2080", "MeasureTheory.tsum_measure_le_measure_univ", "MeasureTheory.unifIntegrable_congr_ae", "MeasureTheory.unifIntegrable_of", "MeasureTheory.unifIntegrable_zero_meas", "MeasureTheory.uniformIntegrable_average", "MeasureTheory.uniformIntegrable_subsingleton", "MeasureTheory.union_ae_eq_univ_of_ae_eq_univ_left", "MeasureTheory.upcrossingStrat_le_one", "MeasureTheory.upcrossingStrat_nonneg", "MeasureTheory.upcrossingsBefore_bot", "MeasureTheory.upcrossingsBefore_mono", "MeasureTheory.upperCrossingTime_stabilize'", "MeasureTheory.upperCrossingTime_succ", "MeasureTheory.vadd_ae_eq_self_of_mem_zmultiples", "MeasureTheory.volume_sum_rpow_lt", "MeasureTheory.weightedSMul_empty", "MeasureTheory.withDensity_apply", "MeasureTheory.withDensity_const", "MeasureTheory.withDensity_inv_same\u2080", "MeasureTheory.withDensity\u1d65_add'", "MeasureTheory.withDensity\u1d65_smul'", "MeasureTheory.zero_convolution", "Mem\u2113p.add", "MeromorphicOn.congr", "MeromorphicOn.mul", "Metric.Sigma.isometry_mk", "Metric.Sum.dist_eq_glueDist", "Metric.ball_disjoint_ball", "Metric.ball_subset_interior_closedBall", "Metric.closedBall_zero", "Metric.closure_eq_iInter_cthickening", "Metric.comap_dist_right_atTop", "Metric.continuousWithinAt_iff", "Metric.controlled_of_uniformEmbedding", "Metric.cthickening_subset_thickening", "Metric.denseRange_iff", "Metric.dist_lt_add_of_nonempty_ball_inter_ball", "Metric.dist_lt_add_of_nonempty_closedBall_inter_ball", "Metric.eventually_isCompact_closedBall", "Metric.exists_continuous_real_forall_closedBall_subset", "Metric.exists_isLocalMin_mem_ball", "Metric.glueDist_glued_points", "Metric.hausdorffDist_le_of_mem_dist", "Metric.hausdorffDist_nonneg", "Metric.hausdorffDist_zero_iff_closure_eq_closure", "Metric.isBounded_Ico", "Metric.isBounded_range_of_tendsto_cofinite_uniformity", "Metric.isCobounded_iff_closedBall_compl_subset", "Metric.nonempty_iInter_of_nonempty_biInter", "Metric.preimage_smul_ball", "Metric.secondCountable_of_countable_discretization", "Metric.sphere_subset_closedBall", "Metric.tendstoLocallyUniformly_iff", "Metric.tendsto_nhdsWithin_nhdsWithin", "Metric.thickening_cthickening_subset", "Metric.thickening_eq_biUnion_ball", "Metric.thickening_iUnion", "Metric.uniformContinuousOn_iff", "Metric.uniformInducing_iff", "Metric.uniformity_basis_dist", "ModP.preVal_add", "ModP.preVal_mul", "ModP.preVal_zero", "ModelWithCorners.coe_extChartAt_transDiffeomorph", "ModelWithCorners.extChartAt_transDiffeomorph_target", "ModelWithCorners.hasMFDerivWithinAt_symm", "ModelWithCorners.leftInverse", "ModelWithCorners.symm_comp_self", "ModelWithCorners.symm_map_nhdsWithin_range", "ModularCyclotomicCharacter.toFun_spec'", "ModularCyclotomicCharacter.toFun_spec''", "ModularForm.coe_add", "ModularForm.slash_mul", "ModularForm.toSlashInvariantForm_intCast", "ModularGroup.g_eq_of_c_eq_one", "ModularGroup.re_T_zpow_smul", "Module.Baer.ExtensionOfMaxAdjoin.extendIdealTo_wd", "Module.Baer.injective", "Module.DirectLimit.lift_of", "Module.DirectLimit.of.zero_exact", "Module.DualBases.coe_dualBasis", "Module.End.IsSemisimple.minpoly_squarefree", "Module.End.IsSemisimple.sub_of_commute", "Module.End.disjoint_genEigenspace", "Module.End.exists_eigenvalue", "Module.End.genEigenspace_restrict", "Module.End.hasEigenvalue_of_hasEigenvector", "Module.End.hasGenEigenvalue_iff_hasEigenvalue", "Module.End.injOn_genEigenspace", "Module.End.isSemisimple_iff", "Module.End.lie_apply", "Module.End.maxGenEigenspace_eq", "Module.End.mem_maxGenEigenspace", "Module.End_algebraMap_isUnit_inv_apply_eq_iff", "Module.Finite.equiv", "Module.Finite.injective_of_surjective_endomorphism", "Module.Finite_of_isLocalization", "Module.Flat.iff_characterModule_baer", "Module.Flat.iff_characterModule_injective", "Module.Flat.iff_lTensor_preserves_injective_linearMap", "Module.Flat.of_forall_isTrivialRelation", "Module.Free.rank_eq_mk_of_infinite_lt", "Module.Quotient.mk_smul_mk", "Module.Ray.someVector_ray", "Module.equiv", "Module.exists_nontrivial_relation_sum_zero_of_finrank_succ_lt_card", "Module.finitePresentation_of_surjective", "Module.finite_def", "Module.finite_of_rank_eq_nat", "Module.finrank_le_one_iff_top_isPrincipal", "Module.forall_dual_apply_eq_zero_iff", "Module.free_of_finite_type_torsion_free", "Module.injOn_dualMap_subtype_span_range_range", "Module.injective_iff_injective_object", "Module.isTorsionBy_quotient_iff", "Module.preReflection_apply_self", "Module.reflection_symm", "ModuleCat.ExtendScalars.map'_comp", "ModuleCat.FilteredColimits.colimitModule.add_smul", "ModuleCat.Free.\u03bc_natural", "ModuleCat.MonoidalCategory.associator_naturality_aux", "ModuleCat.MonoidalCategory.hexagon_reverse", "ModuleCat.cokernel_\u03c0_cokernelIsoRangeQuotient_hom", "ModuleCat.epi_iff_range_eq_top", "ModuleCat.ihom_ev_app", "ModuleCat.ker_eq_bot_of_mono", "ModuleCat.mono_iff_ker_eq_bot", "ModuleCat.monoidalClosed_pre_app", "ModuleCat.restrictScalarsComp'App_inv_apply", "ModuleCat.restrictScalarsId'App_inv_apply", "ModuleCat.toKernelSubobject_arrow", "Mon_.associator_hom_hom", "Monoid.Coprod.fst_apply_inl", "Monoid.Coprod.lift_comp_inl", "Monoid.Coprod.lift_swap", "Monoid.Coprod.mk_of_inl", "Monoid.Coprod.prod_mk_fst_snd", "Monoid.CoprodI.NeWord.toList_getLast?", "Monoid.CoprodI.Word.mem_of_mem_equivPair_tail", "Monoid.CoprodI.lift_word_ping_pong", "Monoid.CoprodI.of_apply", "Monoid.PushoutI.NormalWord.base_smul_eq_smul", "Monoid.PushoutI.NormalWord.prod_smul_empty", "Monoid.PushoutI.NormalWord.rcons_injective", "Monoid.PushoutI.hom_ext", "Monoid.closure_mono", "Monoid.exponent_dvd_of_forall_orderOf_dvd", "Monoid.exponent_eq_max'_orderOf", "Monoid.exponent_eq_zero_of_order_zero", "Monoid.isTorsionFree_iff_torsion_eq_bot", "MonoidAlgebra.domCongr_apply", "MonoidAlgebra.exists_finset_adjoin_eq_top", "MonoidAlgebra.exists_leftInverse_of_injective", "MonoidAlgebra.mul_single_one_apply", "MonoidAlgebra.single_add", "MonoidAlgebra.single_mul_single", "MonoidAlgebra.sum_single_index", "MonoidAlgebra.support_gen_of_gen", "MonoidAlgebra.support_single_mul_subset", "MonoidHom.coe_coe", "MonoidHom.compl\u2082_apply", "MonoidHom.coprod_comp_inl", "MonoidHom.dfinsupp_prod_apply", "MonoidHom.isOpenMap_of_sigmaCompact", "MonoidHom.ker_one", "MonoidHom.ker_rangeRestrict", "MonoidHom.map_mul", "MonoidHom.map_neg", "MonoidHom.noncommCoprod_comp_inl", "MonoidHom.noncommCoprod_inl_inr", "MonoidHom.noncommPiCoprod_mrange", "MonoidHom.ofLeftInverse_symm_apply", "MonoidHom.one_comp", "MonoidHom.prod_unique", "MonoidHom.toMulHom_coe", "MonoidHomClass.isometry_iff_norm", "MonoidWithZeroHom.ext_nat", "Monotone.Ioc", "Monotone.ciSup_mem_iInter_Icc_of_antitone", "Monotone.covariant_of_const'", "Monotone.image_upperBounds_subset_upperBounds_image", "Monotone.map_finset_min'", "Monotone.map_limsInf_of_continuousAt", "Monotone.seq_le_seq", "Monotone.seq_lt_seq_of_lt_of_le", "Monotone.tendsto_alternating_series_of_tendsto_zero", "MonotoneOn.Ici", "MonotoneOn.Ioc", "MonotoneOn.exists_monotone_extension", "MonotoneOn.locallyBoundedVariationOn", "MonotoneOn.map_bddBelow", "MonotoneOn.map_min", "MonotoneOn.mapsTo_Icc", "MonotoneOn.set_prod", "MonotoneOn.union", "Monovary.div_left\u2080", "Monovary.div_right", "MonovaryOn.comp_antitoneOn_right", "MonovaryOn.dual_right", "MonovaryOn.sum_comp_perm_mul_le_sum_mul", "MonovaryOn.sum_smul_comp_perm_lt_sum_smul_iff", "MulAction.IsBlock.iff_top", "MulAction.IsBlock.mk_mem", "MulAction.Supports.mono", "MulAction.bijective\u2080", "MulAction.fixedBy_mul", "MulAction.le_stabilizer_smul_left", "MulAction.mem_stabilizer_iff", "MulAction.orbit.coe_smul", "MulAction.orbit.pairwiseDisjoint", "MulAction.orbitEquivQuotientStabilizer_symm_apply", "MulAction.orbitRel.Quotient.mem_subgroup_orbit_iff'", "MulAction.orbitRel_le_snd", "MulAction.orbit_eq_iff", "MulAction.set_mem_fixedBy_iff", "MulAction.smul_inv_mem_fixedBy_iff_mem_fixedBy", "MulAction.smul_mem_of_set_mem_fixedBy", "MulAction.smul_orbit_eq_orbit_smul", "MulActionHom.congr_fun", "MulActionHom.ofEq_apply", "MulActionWithZero.subsingleton", "MulChar.IsQuadratic.inv", "MulChar.one_apply", "MulChar.pow_card_eq_one", "MulDistribMulAction.toMonoidHom_apply", "MulEquiv.ext", "MulEquiv.isDomain", "MulEquiv.map_ne_one_iff", "MulHom.cancel_right", "MulHom.coe_comp", "MulHom.coe_copy_eq", "MulHom.prod_comp_prodMap", "MulHom.toFun_eq_coe", "MulLECancellable.inj", "MulLECancellable.le_mul_iff_one_le_left", "MulOpposite.coe_opLinearEquiv_symm_addEquiv", "MulOpposite.comap_unop_nhds", "MulOpposite.nndist_op", "MulOpposite.norm_op", "MulOpposite.unop_smul_eq_unop_smul_unop", "MulPosMono.toMulPosStrictMono", "MultilinearMap.alternatization_apply", "MultilinearMap.bound_of_shell_of_norm_map_coord_zero", "MultilinearMap.coe_injective", "MultilinearMap.compLinearMap_apply", "MultilinearMap.curryFinFinset_apply_const_aux", "MultilinearMap.curryFinFinset_symm_apply_piecewise_const", "MultilinearMap.map_add_eq_map_add_linearDeriv_add", "MultilinearMap.map_add_sub_map_add_sub_linearDeriv", "MultilinearMap.map_zero", "MultilinearMap.smulRight_apply", "MultilinearMap.sum_apply", "Multipliable.comp_injective", "Multipliable.map_tprod", "Multipliable.mul", "Multiset.Disjoint.symm", "Multiset.Ico_add_Ico_eq_Ico", "Multiset.Ico_filter_le_of_right_le", "Multiset.Ico_filter_lt", "Multiset.Ico_filter_lt_of_right_le", "Multiset.Ico_self", "Multiset.Ioc_eq_zero_iff", "Multiset.Ioo_self", "Multiset.Nodup.filter", "Multiset.Nodup.ndinsert", "Multiset.add_sigma", "Multiset.aemeasurable_prod", "Multiset.antidiagonal_eq_map_powerset", "Multiset.bind_assoc", "Multiset.card_Ico", "Multiset.card_Ico_eq_card_Icc_sub_one", "Multiset.card_disjSum", "Multiset.card_eq_three", "Multiset.card_sections", "Multiset.card_singleton", "Multiset.coe_attach", "Multiset.coe_bind", "Multiset.coe_dedup", "Multiset.coe_foldr", "Multiset.coe_mk", "Multiset.coe_sub", "Multiset.cons_inter_of_pos", "Multiset.count_add", "Multiset.count_eq_zero", "Multiset.count_inter", "Multiset.count_map_eq_count'", "Multiset.count_replicate", "Multiset.count_sum", "Multiset.count_sum'", "Multiset.count_univ", "Multiset.disjoint_cons_right", "Multiset.disjoint_finset_sum_right", "Multiset.eq_of_mem_map_const", "Multiset.eq_replicate_card", "Multiset.eq_zero_of_subset_zero", "Multiset.erase_cons_tail_of_mem", "Multiset.esymm_neg", "Multiset.exists_smul_of_dvd_count", "Multiset.filterMap_filterMap", "Multiset.filter_zero", "Multiset.finset_sum_eq_sup_iff_disjoint", "Multiset.fold_cons_right", "Multiset.fold_dedup_idem", "Multiset.grade_eq", "Multiset.inf_coe", "Multiset.lcm_mono", "Multiset.le_prod_of_submultiplicative_on_pred", "Multiset.length_toList", "Multiset.lists_coe", "Multiset.lt_replicate_succ", "Multiset.map_cons", "Multiset.map_hcongr", "Multiset.map_traverse", "Multiset.map_union", "Multiset.map_univ_coeEmbedding", "Multiset.max_le_of_forall_le", "Multiset.mem_coe", "Multiset.mem_cons", "Multiset.mem_cons_self", "Multiset.mem_disjSum", "Multiset.mem_filter", "Multiset.mem_filterMap", "Multiset.mem_filter_of_mem", "Multiset.mem_map_of_mem", "Multiset.mem_powersetCardAux", "Multiset.mem_product", "Multiset.mem_sigma", "Multiset.mem_toFinset", "Multiset.mk_mem_sym2_iff", "Multiset.ndinter_cons_of_not_mem", "Multiset.ndunion_le", "Multiset.nodup_bind", "Multiset.nodup_cons", "Multiset.nodup_iff_count_eq_one", "Multiset.noncommFold_eq_fold", "Multiset.noncommFoldr_cons", "Multiset.pow_card_le_prod", "Multiset.powersetAux'_nil", "Multiset.powersetCardAux_cons", "Multiset.powersetCard_coe", "Multiset.powersetCard_le_powerset", "Multiset.powersetCard_mono", "Multiset.prod_eq_foldr", "Multiset.prod_eq_one_iff", "Multiset.prod_eq_zero_iff", "Multiset.prod_hom_rel", "Multiset.prod_hom\u2082", "Multiset.prod_lt_prod_of_nonempty'", "Multiset.product_singleton", "Multiset.range_subset", "Multiset.rel_bind", "Multiset.rel_eq_refl", "Multiset.replicate_add", "Multiset.self_mem_range_succ", "Multiset.sup_powerset_len", "Multiset.support_sum_subset", "Multiset.sym2_eq_zero_iff", "Multiset.toDFinsupp_inj", "Multiset.toFinsupp_eq_iff", "Multiset.toList_eq_singleton_iff", "Multiset.union_le", "MvFunctor.LiftR_def", "MvPFunctor.M.bisim_lemma", "MvPFunctor.wMk_eq", "MvPFunctor.wRec_eq", "MvPolynomial.C_add", "MvPolynomial.C_eq_coe_nat", "MvPolynomial.C_injective", "MvPolynomial.C_mul_X_pow_eq_monomial", "MvPolynomial.IsHomogeneous.eq_zero_of_forall_eval_eq_zero_of_le_card", "MvPolynomial.IsHomogeneous.rename_isHomogeneous", "MvPolynomial.IsWeightedHomogeneous.add", "MvPolynomial.aevalTower_C", "MvPolynomial.aevalTower_ofId", "MvPolynomial.aeval_def", "MvPolynomial.aeval_map_algebraMap", "MvPolynomial.aeval_zero'", "MvPolynomial.algebraTensorAlgEquiv_symm_X", "MvPolynomial.bind\u2081_X_left", "MvPolynomial.coeffs_eq_empty_of_subsingleton", "MvPolynomial.constantCoeff_monomial", "MvPolynomial.counitNat_X", "MvPolynomial.counit_X", "MvPolynomial.degrees_C", "MvPolynomial.degrees_indicator", "MvPolynomial.divMonomial_zero", "MvPolynomial.eq_zero_iff", "MvPolynomial.eval_prod", "MvPolynomial.eval\u2082_assoc", "MvPolynomial.eval\u2082_cast_comp", "MvPolynomial.exists_finset_rename\u2082", "MvPolynomial.finsum_weightedHomogeneousComponent", "MvPolynomial.frobenius_zmod", "MvPolynomial.funext_fin", "MvPolynomial.homogeneousComponent_eq_zero", "MvPolynomial.induction_on_monomial", "MvPolynomial.isHomogeneous_X_pow", "MvPolynomial.isLocalization_C_mk'", "MvPolynomial.linearMap_ext", "MvPolynomial.map_map", "MvPolynomial.map_mapRange_eq_iff", "MvPolynomial.map_mvPolynomial_eq_eval\u2082", "MvPolynomial.map_restrict_dom_eval\u2097", "MvPolynomial.mem_supported", "MvPolynomial.noZeroDivisors_of_finite", "MvPolynomial.pderiv_C", "MvPolynomial.pderiv_X_self", "MvPolynomial.pderiv_eq_zero_of_not_mem_vars", "MvPolynomial.polynomial_eval_eval\u2082", "MvPolynomial.psum_zero", "MvPolynomial.quotient_map_C_eq_zero", "MvPolynomial.radical_le_vanishingIdeal_zeroLocus", "MvPolynomial.rename_comp_bind\u2081", "MvPolynomial.rename_expand", "MvPolynomial.sum_C", "MvPolynomial.sum_homogeneousComponent", "MvPolynomial.support_X_mul", "MvPolynomial.support_esymm'", "MvPolynomial.support_esymm''", "MvPolynomial.totalDegree_finset_prod", "MvPolynomial.totalDegree_finset_sum", "MvPolynomial.totalDegree_pow", "MvPolynomial.vanishingIdeal_zeroLocus_eq_radical", "MvPolynomial.vars_rename", "MvPolynomial.weightedHomogeneousComponent_eq_zero", "MvPolynomial.weightedHomogeneousComponent_weighted_homogeneous_polynomial", "MvPolynomial.weightedTotalDegree_zero", "MvPolynomial.zeroLocus_top", "MvPolynomial.zero_not_mem_coeffs", "MvPowerSeries.add_mul", "MvPowerSeries.algebraMap_apply'", "MvPowerSeries.coeff_X", "MvPowerSeries.coeff_invOfUnit", "MvPowerSeries.coeff_mul", "MvPowerSeries.coeff_prod", "MvPowerSeries.inv_mul_cancel", "MvPowerSeries.map_C", "MvPowerSeries.mul_invOfUnit", "MvQPF.Cofix.bisim_aux", "MvQPF.Cofix.bisim\u2082", "MvQPF.Fix.rec_eq", "MvQPF.wEquiv.symm", "NNRat.cast_divNat_of_ne_zero", "NNRat.cast_zpow_of_ne_zero", "NNRat.coe_le_coe", "NNRat.coe_natCast", "NNRat.natCast_eq_divNat", "NNRat.preimage_cast_Icc", "NNRat.toNNRat_coe_nat", "NNRat.toNNRat_sum_of_nonneg", "NNReal.IsConjExponent.inv_inv", "NNReal.bot_eq_zero", "NNReal.coe_le_one", "NNReal.coe_lt_coe", "NNReal.coe_ofScientific", "NNReal.coe_prod", "NNReal.div_le_div_left_of_le", "NNReal.div_le_iff'", "NNReal.div_le_of_le_mul'", "NNReal.div_lt_iff", "NNReal.forall", "NNReal.indicator_summable", "NNReal.inv_lt_inv", "NNReal.le_iInf_mul", "NNReal.nndist_zero_eq_val", "NNReal.nndist_zero_eq_val'", "NNReal.one_le_coe", "NNReal.orderIsoIccZeroCoe_symm_apply_coe", "NNReal.rpow_add_rpow_le", "NNReal.rpow_sum_le_const_mul_sum_rpow", "NNReal.sInf_empty", "NNReal.sSup_of_not_bddAbove", "NNReal.sq_sqrt", "NNReal.sqrt_eq_one", "NNReal.sqrt_one", "NNReal.sum_mul_le_sqrt_mul_sqrt", "NNReal.summable_Lp_add", "NNReal.summable_iff_not_tendsto_nat_atTop", "NNReal.tendsto_coe'", "NNReal.tendsto_sum_nat_add", "NNReal.zero_rpow", "Nat.Coprime.eq_one_of_dvd", "Nat.Coprime.pow_left", "Nat.Ico_filter_coprime_le", "Nat.Ico_zero_eq_range", "Nat.Ioc_filter_dvd_card_eq_div", "Nat.Ioc_filter_modEq_cast", "Nat.ModEq.mul_right_cancel'", "Nat.ModEq.refl", "Nat.ModEq.symm", "Nat.Partrec.Code.encode_lt_comp", "Nat.Partrec.Code.eval_curry", "Nat.Partrec.Code.evaln_bound", "Nat.Partrec.Code.exists_code", "Nat.Partrec.Code.rec_prim'", "Nat.Prime.eq_two_or_odd'", "Nat.Prime.multiplicity_choose'", "Nat.Prime.pow_eq_iff", "Nat.Prime.prod_properDivisors", "Nat.Prime.sub_one_mul_multiplicity_factorial", "Nat.Primrec'.add", "Nat.Primrec'.prim_iff\u2082", "Nat.Primrec'.sqrt", "Nat.Primrec.swap", "Nat.Subtype.toFunAux_eq", "Nat.WithBot.add_eq_zero_iff", "Nat.WithBot.coe_nonneg", "Nat.add_div_right", "Nat.add_eq_left", "Nat.add_factorial_le_factorial_add", "Nat.add_factorial_succ_le_factorial_add_succ", "Nat.add_factorial_succ_lt_factorial_add_succ", "Nat.add_left_comm", "Nat.add_mod_eq_add_mod_left", "Nat.add_pos_iff_pos_or_pos", "Nat.add_two_le_nth_prime", "Nat.and_forall_succ", "Nat.and_two_pow", "Nat.bit0_eq_zero", "Nat.bit0_inj", "Nat.bit0_lt_bit1_iff", "Nat.bit1_lt_bit0_iff", "Nat.bit1_mod_two", "Nat.bits_append_bit", "Nat.bodd_bit0", "Nat.card_Iic", "Nat.card_Iio", "Nat.card_Ioo", "Nat.card_eq_card_units_add_one", "Nat.card_eq_two_iff'", "Nat.card_fintypeIcc", "Nat.card_fintypeIoc", "Nat.card_image_of_injective", "Nat.card_ne_zero", "Nat.card_pos_iff", "Nat.card_preimage_of_injOn", "Nat.card_submonoidPowers", "Nat.cast_bit1", "Nat.cast_descFactorial", "Nat.cast_div", "Nat.cast_dvd_cast", "Nat.cast_le", "Nat.cast_multiset_sum", "Nat.cast_pos'", "Nat.cauchy_induction_two_mul", "Nat.ceil_congr", "Nat.ceil_intCast", "Nat.ceil_one", "Nat.choose_eq_factorial_div_factorial", "Nat.choose_le_centralBinom", "Nat.choose_le_middle", "Nat.choose_le_pow", "Nat.choose_succ_self_right", "Nat.choose_symm", "Nat.clog_anti_left", "Nat.clog_mono_right", "Nat.cond_decide_mod_two", "Nat.cons_self_properDivisors", "Nat.coprime_div_gcd_of_squarefree", "Nat.coprime_factorizationLCMLeft_factorizationLCMRight", "Nat.coprime_fintype_prod_left_iff", "Nat.coprime_mul_left_add_right", "Nat.coprime_one_left_iff", "Nat.count_add'", "Nat.count_modEq_card", "Nat.count_mono_left", "Nat.decreasingInduction_succ_left", "Nat.descFactorial_eq_factorial_mul_choose", "Nat.descFactorial_self", "Nat.descFactorial_zero", "Nat.digits_inj_iff", "Nat.digits_lt_base'", "Nat.dist_cast_rat", "Nat.dist_comm", "Nat.dist_zero_right", "Nat.div2_bit0", "Nat.div2_zero", "Nat.div_dvd_of_dvd", "Nat.div_eq_iff_eq_of_dvd_dvd", "Nat.div_mod_unique", "Nat.div_mul_div_le_div", "Nat.divisors_inj", "Nat.divisors_subset_properDivisors", "Nat.doubleFactorial_le_factorial", "Nat.dvd_ceilRoot_pow", "Nat.dvd_of_factors_subperm", "Nat.eq_of_testBit_eq", "Nat.eq_sq_add_sq_iff_eq_sq_mul", "Nat.eq_zero_of_gcd_eq_zero_right", "Nat.eq_zero_of_mul_eq_zero", "Nat.even_add", "Nat.even_or_odd'", "Nat.exists_coprime", "Nat.factorial_dvd_descFactorial", "Nat.factorial_ne_zero", "Nat.factorization_choose_of_lt_three_mul", "Nat.factorization_eq_factors_multiset", "Nat.factorization_eq_of_coprime_right", "Nat.factorization_eq_zero_iff_remainder", "Nat.factorization_mul_of_coprime", "Nat.factorization_one", "Nat.factorization_prime_le_iff_dvd", "Nat.factorization_zero", "Nat.factors_eq_nil", "Nat.factors_one", "Nat.fib_add", "Nat.fib_bit1", "Nat.fib_coprime_fib_succ", "Nat.fib_strictMonoOn", "Nat.fib_succ_eq_succ_sum", "Nat.filter_multiset_Ico_card_eq_of_periodic", "Nat.findGreatest_of_ne_zero", "Nat.floorRoot_eq_zero", "Nat.floorRoot_one_right", "Nat.floor_add_nat", "Nat.floor_le_floor", "Nat.forall_lt_succ", "Nat.gcdA_zero_left", "Nat.gcdB_zero_right", "Nat.gcd_add_mul_right_right", "Nat.gcd_mul_left_add_left", "Nat.gcd_mul_right_add_left", "Nat.greatestFib_eq_zero", "Nat.greatestFib_pos", "Nat.greatestFib_sub_fib_greatestFib_le_greatestFib", "Nat.half_le_of_sub_le_half", "Nat.iSup_lt_succ'", "Nat.image_cast_int_Iic", "Nat.infinite_setOf_prime", "Nat.insert_self_properDivisors", "Nat.isCoprime_iff_coprime", "Nat.isZeckendorfRep_zeckendorf", "Nat.lcm_pos", "Nat.leRecOn_self", "Nat.leRecOn_succ_left", "Nat.le_findGreatest", "Nat.le_mul_self", "Nat.le_of_lt_succ", "Nat.le_sub_of_add_le", "Nat.left_le_pair", "Nat.left_ne_zero_of_mem_divisorsAntidiagonal", "Nat.log_div_mul_self", "Nat.log_lt_of_lt_pow", "Nat.lt_add_of_zero_lt_left", "Nat.lt_div_mul_add", "Nat.lt_find_iff", "Nat.lt_of_mul_lt_mul_left", "Nat.lt_of_testBit", "Nat.lt_succ", "Nat.lt_succ_sqrt", "Nat.lt_two_pow", "Nat.map_div_right_divisors", "Nat.mem_divisors", "Nat.mem_factors_iff_dvd", "Nat.mem_factors_mul", "Nat.mem_rfind", "Nat.mem_smoothNumbers", "Nat.mem_smoothNumbersUpTo", "Nat.mod_add_mod", "Nat.mod_lcm", "Nat.mod_modEq", "Nat.mod_two_ne_zero", "Nat.monotone_primeCounting", "Nat.mul_add_mod'", "Nat.mul_def", "Nat.mul_dvd_mul_iff_right", "Nat.multinomial_spec", "Nat.noZeroSMulDivisors", "Nat.not_dvd_iff_between_consec_multiples", "Nat.not_le_of_gt", "Nat.nth_eq_orderIsoOfNat", "Nat.nth_false", "Nat.odd_mul", "Nat.odd_sub'", "Nat.ofAdd_mul", "Nat.ofDigits_div_eq_ofDigits_tail", "Nat.ofDigits_eq_sum_mapIdx", "Nat.ofDigits_zmod", "Nat.ofNat_le_cast", "Nat.one_le_cast", "Nat.ord_compl_dvd_ord_compl_iff_dvd", "Nat.ord_compl_mul", "Nat.pair_unpair", "Nat.periodic_gcd", "Nat.pos_of_mem_divisors", "Nat.pow_factorization_choose_le", "Nat.pow_le_pow_of_le_left", "Nat.preimage_Ioo", "Nat.prime_def_lt''", "Nat.prime_iff_card_units", "Nat.prime_mul_iff", "Nat.probablePrime_iff_modEq", "Nat.prod_factors", "Nat.prod_pow_factorization_choose", "Nat.prod_pow_factorization_eq_self", "Nat.prod_range_factorial_succ", "Nat.psp_from_prime_gt_p", "Nat.psub'_eq_psub", "Nat.rec_zero", "Nat.rfind_dom", "Nat.right_ne_zero_of_mem_divisorsAntidiagonal", "Nat.sSup_def", "Nat.shiftLeft'_false", "Nat.shiftLeft_sub", "Nat.shiftRight_add", "Nat.size_bit", "Nat.smoothNumbers_mono", "Nat.sqrt_pos", "Nat.squarefree_pow_iff", "Nat.sub_eq_iff_eq_add", "Nat.sub_le", "Nat.sub_le_iff_le_add'", "Nat.succ.injEq", "Nat.succPNat_mono", "Nat.sum_totient", "Nat.sum_totient'", "Nat.totient_prime_pow", "Nat.two_dvd_ne_zero", "Nat.two_not_dvd_two_mul_sub_one", "Nat.uniformEmbedding_coe_rat", "Nat.uniformEmbedding_coe_real", "Nat.xor_cancel_left", "Nat.xor_comm", "Nat.xor_self", "Nat.zero_descFactorial_succ", "Nat.zero_eq", "Nat.zero_lt_two", "NatOrdinal.lt_wf", "NatOrdinal.toOrdinal_eq_one", "Ne.bot_lt'", "Ne.ne_or_ne", "NeZero.nat_of_neZero", "NeZero.pos_of_neZero_natCast", "NonUnitalAlgHom.coe_mulHom_mk", "NonUnitalAlgHom.ext_iff", "NonUnitalAlgHom.map_mul", "NonUnitalAlgHom.map_zero", "NonUnitalAlgebra.adjoin_union", "NonUnitalAlgebra.range_top_iff_surjective", "NonUnitalAlgebra.to_subring_eq_top", "NonUnitalCommRing.ext_iff", "NonUnitalRingHom.coe_rangeRestrict", "NonUnitalRingHom.eqLocus_same", "NonUnitalRingHom.ext", "NonUnitalRingHom.map_srange", "NonUnitalRingHom.snd_comp_prod", "NonUnitalStarAlgHom.copy_eq", "NonUnitalStarAlgHom.inr_apply", "NonUnitalStarAlgHom.restrictScalars_apply", "NonUnitalStarAlgebra.coe_iInf", "NonUnitalStarSubalgebra.centralizer_univ", "NonUnitalStarSubalgebra.coe_eq_zero", "NonUnitalStarSubalgebra.coe_toNonUnitalSubalgebra", "NonUnitalStarSubalgebra.ext", "NonUnitalStarSubalgebra.mem_map", "NonUnitalStarSubalgebra.toNonUnitalSubalgebra_injective", "NonUnitalSubalgebra.coe_comap", "NonUnitalSubalgebra.coe_neg", "NonUnitalSubalgebra.coe_smul", "NonUnitalSubalgebra.iSupLift_mk", "NonUnitalSubalgebra.mem_carrier", "NonUnitalSubalgebra.mem_center_iff", "NonUnitalSubalgebra.prod_mono", "NonUnitalSubalgebra.toNonUnitalSubring_injective", "NonUnitalSubalgebra.toNonUnitalSubsemiring_injective", "NonUnitalSubring.closure_preimage_le", "NonUnitalSubring.coe_mk'", "NonUnitalSubring.eq_top_iff'", "NonUnitalSubring.mem_bot", "NonUnitalSubring.toNonUnitalSubsemiring_mono", "NonUnitalSubsemiring.comap_equiv_eq_map_symm", "NonUnitalSubsemiring.toAddSubmonoid_strictMono", "NonarchAddGroupSeminorm.le_def", "NonemptyInterval.coe_pure_interval", "NonemptyInterval.coe_sub_interval", "NonemptyInterval.mem_pure_self", "NonemptyInterval.one_mem_one", "NonemptyInterval.sub_mem_sub", "NonemptyInterval.toProd_one", "Nonneg.coe_zero", "Nonneg.coe_zpow", "NormOneClass.induced", "NormedAddCommGroup.cauchy_series_of_le_geometric''", "NormedAddGroupHom.IsQuotient.norm_lift", "NormedAddGroupHom.coe_toAddMonoidHom", "NormedAddGroupHom.completion_comp", "NormedAddGroupHom.le_of_opNorm_le", "NormedAddGroupHom.norm_comp_le", "NormedAddGroupHom.norm_completion", "NormedRing.inverse_add_nth_order", "NormedRing.inverse_one_sub_nth_order'", "NormedSpace.Dual.isClosed_image_polar_of_mem_nhds", "NormedSpace.cobounded_neBot", "NormedSpace.eq_iff_forall_dual_eq", "NormedSpace.exists_lt_norm", "NormedSpace.expSeries_apply_eq'", "NormedSpace.exp_hasFPowerSeriesAt_zero", "NormedSpace.isVonNBounded_iff'", "NormedSpace.sphere_nonempty_rclike", "Num.bit_to_nat", "Num.cast_to_int", "Num.cmp_to_nat", "Num.lor_eq_or", "Num.mem_ofZNum'", "Num.ofInt'_toZNum", "Num.ofZNum_toNat", "Num.toZNum_inj", "NumberField.InfinitePlace.ComplexEmbedding.exists_comp_symm_eq_of_comp_eq", "NumberField.InfinitePlace.card_complex_embeddings", "NumberField.InfinitePlace.embedding_mk_eq", "NumberField.InfinitePlace.even_card_aut_of_not_isUnramifiedIn", "NumberField.InfinitePlace.even_nat_card_aut_of_not_isUnramified", "NumberField.InfinitePlace.isUnramifiedIn_comap", "NumberField.InfinitePlace.isUnramified_iff_card_stabilizer_eq_one", "NumberField.InfinitePlace.isUnramified_iff_stabilizer_eq_bot", "NumberField.InfinitePlace.nrRealPlaces_eq_one_of_finrank_eq_one", "NumberField.InfinitePlace.prod_eq_abs_norm", "NumberField.RingOfIntegers.isIntegral", "NumberField.Units.coe_neg_one", "NumberField.Units.regulator_eq_det'", "NumberField.canonicalEmbedding.conj_apply", "NumberField.canonicalEmbedding.integerLattice.inter_ball_finite", "NumberField.canonicalEmbedding.integralBasis_repr_apply", "NumberField.canonicalEmbedding.latticeBasis_apply", "NumberField.finite_of_discr_bdd", "NumberField.isUnit_iff_norm", "NumberField.mixedEmbedding.convexBodyLT_mem", "NumberField.mixedEmbedding.convexBodySumFun_continuous", "NumberField.mixedEmbedding.convexBodySumFun_smul", "NumberField.mixedEmbedding.exists_ne_zero_mem_ideal_lt'", "NumberField.mixedEmbedding.indexEquiv_apply_ofIsComplex_snd", "NumberField.mixedEmbedding.latticeBasis_repr_apply", "NumberField.mixedEmbedding.mem_rat_span_latticeBasis", "NumberField.mixedEmbedding.normAtPlace_apply_isReal", "NumberField.mixedEmbedding.normAtPlace_neg", "NumberField.mixedEmbedding.normAtPlace_smul", "NumberField.mixedEmbedding.norm_real", "ONote.eq_of_cmp_eq", "ONote.fastGrowing_one", "ONote.fastGrowing\u03b5\u2080_two", "ONote.oadd_add", "ONote.oadd_lt_oadd_1", "ONote.oadd_mul_nfBelow", "ONote.ofNat_one", "ONote.repr_opow", "ONote.scale_eq_mul", "ONote.split_dvd", "ONote.zero_lt_one", "Odd.pos", "Odd.pow_nonpos_iff", "OfNat.ofNat_eq_ofNat", "OmegaCompletePartialOrder.Continuous.of_bundled", "OmegaCompletePartialOrder.ContinuousHom.congr_fun", "OmegaCompletePartialOrder.ContinuousHom.forall_forall_merge'", "OmegaCompletePartialOrder.ContinuousHom.\u03c9Sup_apply_\u03c9Sup", "OmegaCompletePartialOrder.id_continuous'", "OneHom.comp_apply", "OneHom.comp_assoc", "OneMemClass.coe_one", "OnePoint.continuous_iff_from_nat", "OnePoint.hasBasis_nhds_infty", "OnePoint.inseparable_coe", "OnePoint.not_inseparable_coe_infty", "OnePoint.tendsto_nhds_infty'", "OpenEmbedding.isLocalHomeomorph", "OpenEmbedding.open_iff_image_open", "OpenEmbedding.prod", "OpenSubgroup.coe_prod", "OpenSubgroup.mem_toOpens", "Option.bind_pmap", "Option.bnot_comp_isSome", "Option.exists_ne_none", "Option.getD_default_eq_iget", "Option.map_pbind", "Option.mem_map\u2082_iff", "Option.orElse_eq_some", "Option.pmap_eq_some_iff", "Or.elim3", "Or.intro_right", "Order.Cofinal.above_mem", "Order.Ico_pred_left", "Order.Ideal.PrimePair.compl_F_eq_I", "Order.Ideal.ext", "Order.Ideal.nonempty", "Order.Ideal.top_toLowerSet", "Order.Ioo_pred_left_of_not_isMin", "Order.Ioo_succ_right_of_not_isMax", "Order.IsPredLimit.lt_pred", "Order.PFilter.directed", "Order.isPredLimit_of_pred_lt", "Order.isSuccLimitRecOn_succ'", "Order.iso_of_countable_dense", "Order.le_succ_bot_iff", "Order.lt_succ_iff", "Order.lt_succ_iff_of_not_isMax", "Order.not_isSuccLimit_iff'", "Order.pred_eq_pred_iff", "Order.pred_lt_iff_eq_or_lt_of_not_isMin", "Order.pred_strictMono", "Order.sequenceOfCofinals.monotone", "Order.succ_le_succ", "Order.succ_mono", "OrderDual.instPartialOrder.dual_dual", "OrderDual.le_toDual", "OrderEmbedding.acc", "OrderEmbedding.birkhoffFinset_sup", "OrderEmbedding.image_Icc", "OrderEmbedding.le_iff_le", "OrderEmbedding.le_map_sup", "OrderEmbedding.map_inf_le", "OrderHom.apply_mono", "OrderHom.le_def", "OrderHom.le_prevFixed", "OrderHom.monotone", "OrderHom.nextFixed_le_iff", "OrderHom.orderHom_eq_id", "OrderIso.bijective", "OrderIso.concaveOn_symm", "OrderIso.convexOn_symm", "OrderIso.isAtomic_iff", "OrderIso.isBoundedUnder_ge_comp", "OrderIso.le_iff_le", "OrderIso.map_csInf'", "OrderIso.mulLeft_symm", "OrderIso.ofRelIsoLT_symm", "OrderIso.preimage_Ico", "OrderIso.rightOrdContinuous", "OrderIso.symm_apply_eq", "OrderIso.tendsto_atBot", "OrderIso.tendsto_atTop_iff", "OrderMonoidWithZeroHom.coe_monoidWithZeroHom", "OrderMonoidWithZeroHom.coe_mul", "OrderMonoidWithZeroHom.comp_apply", "OrderMonoidWithZeroHom.id_comp", "OrderRingIso.self_trans_symm", "OrderRingIso.trans_apply", "OrderTop.upperBounds_univ", "Ordering.compares_swap", "Ordinal.CNF_lt_snd", "Ordinal.IsFundamentalSequence.monotone", "Ordinal.IsNormal.cof_eq", "Ordinal.apply_lt_nfpBFamily", "Ordinal.bddAbove_of_small", "Ordinal.bfamilyOfFamily_typein", "Ordinal.blsub_le_bsup_succ", "Ordinal.bmex_lt_ord_succ_card", "Ordinal.bsup_eq_blsub_of_lt_succ_limit", "Ordinal.bsup_id_limit", "Ordinal.bsup_one", "Ordinal.card_le_card", "Ordinal.card_omega", "Ordinal.cof_sup_le", "Ordinal.enumOrd_def", "Ordinal.enumOrd_def'", "Ordinal.enumOrd_def'_nonempty", "Ordinal.enum_lt_enum", "Ordinal.enum_zero_eq_bot", "Ordinal.eq_zero_or_pos", "Ordinal.exists_blsub_cof", "Ordinal.exists_of_lt_bmex", "Ordinal.foldr_le_nfpFamily", "Ordinal.fp_iff_derivBFamily", "Ordinal.fp_iff_derivFamily", "Ordinal.has_succ_of_type_succ_lt", "Ordinal.le_add_sub", "Ordinal.le_mex_of_forall", "Ordinal.le_of_nadd_le_nadd_left", "Ordinal.le_sup", "Ordinal.lift_down", "Ordinal.lift_one", "Ordinal.lift_type_eq", "Ordinal.limitRecOn_limit", "Ordinal.lsub_lt_ord", "Ordinal.lt_mul_of_limit", "Ordinal.lt_nfpFamily", "Ordinal.lt_nmul_iff\u2083", "Ordinal.max_zero_left", "Ordinal.mem_closure_iff_bsup", "Ordinal.mul_lt_of_lt_div", "Ordinal.natCast_mod", "Ordinal.natCast_pos", "Ordinal.nat_lt_limit", "Ordinal.nfp_le", "Ordinal.nmul_assoc", "Ordinal.nmul_le_nmul_of_nonneg_right", "Ordinal.nmul_nadd", "Ordinal.nmul_nadd_lt\u2083'", "Ordinal.not_succ_isLimit", "Ordinal.omega_le", "Ordinal.omega_lt_omega1", "Ordinal.one_add_of_omega_le", "Ordinal.one_le_iff_ne_zero", "Ordinal.one_lt_card", "Ordinal.one_toPGame_leftMoves_default_eq", "Ordinal.opow_mul_add_lt_opow_mul_succ", "Ordinal.opow_omega", "Ordinal.ord_cof_eq", "Ordinal.principal_add_iff_add_left_eq_self", "Ordinal.principal_add_isLimit", "Ordinal.principal_mul_iff_mul_left_eq", "Ordinal.principal_opow_omega", "Ordinal.sup_eq_lsub_iff_lt_sup", "Ordinal.sup_iterate_eq_nfp", "Ordinal.sup_mul_nat", "Ordinal.toNatOrdinal_one", "Ordinal.toNatOrdinal_toOrdinal", "Ordinal.toPGame_add", "Ordinal.toPGame_eq_iff", "Ordinal.toPGame_equiv_iff", "Ordinal.toPGame_le", "Ordinal.toPGame_moveLeft'", "Ordinal.type_eq_zero_iff_isEmpty", "Ordinal.type_ne_zero_of_nonempty", "Ordinal.typein_one_out", "Ordinal.univ_id", "Ordinal.zero_lt_card", "Ordinal.zero_opow", "Ordinal.zero_or_succ_or_limit", "OrdinalApprox.gfpApprox_le", "OrdinalApprox.le_lfpApprox", "OrdinalApprox.lfp_mem_range_lfpApprox", "Ordnode.Bounded.dual", "Ordnode.Bounded.dual_iff", "Ordnode.Raised.dist_le'", "Ordnode.Sized.node'", "Ordnode.Valid'.balance'_aux", "Ordnode.Valid'.of_lt", "Ordnode.dual_balanceL", "Ordnode.insertWith.valid", "Ordnode.node4L_size", "OreLocalization.add_comm", "OreLocalization.ind", "OreLocalization.inv_zero", "OreLocalization.mul_smul", "OreLocalization.oreDiv_add_char'", "OreLocalization.right_distrib", "OreLocalization.smul_div_one", "OreLocalization.universalMulHom_commutes", "Orientation.eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero", "Orientation.inner_eq_norm_mul_norm_mul_cos_oangle", "Orientation.inner_mul_inner_add_areaForm_mul_areaForm'", "Orientation.inner_rightAngleRotation_right", "Orientation.inner_rotation_pi_div_two_left_smul", "Orientation.inner_smul_rotation_pi_div_two_right", "Orientation.kahler_apply_self", "Orientation.map_neg", "Orientation.ne_of_oangle_eq_pi_div_two", "Orientation.ne_of_oangle_ne_zero", "Orientation.norm_div_tan_oangle_add_left_of_oangle_eq_pi_div_two", "Orientation.oangle_eq_of_angle_eq_of_sign_eq", "Orientation.oangle_eq_pi_iff_oangle_rev_eq_pi", "Orientation.oangle_eq_zero_or_eq_pi_iff_right_eq_smul", "Orientation.oangle_sign_sub_right", "Orientation.oangle_sign_sub_smul_right", "Orientation.reindex_refl", "Orientation.rightAngleRotation_map_complex", "Orientation.two_zsmul_oangle_smul_right_self", "Orientation.volumeForm_zero_pos", "OrthogonalFamily.inner_right_fintype", "Orthonormal.exists_orthonormalBasis_extension_of_card_eq", "Orthonormal.isHilbertSum", "OrthonormalBasis.addHaar_eq_volume", "OrthonormalBasis.coe_toBasis_repr_apply", "OrthonormalBasis.det_to_matrix_orthonormalBasis_real", "OrthonormalBasis.reindex_toBasis", "OrthonormalBasis.sum_repr", "PEquiv.ext_iff", "PEquiv.symm_refl", "PEquiv.symm_trans_self", "PEquiv.toMatrix_apply", "PEquiv.trans_bot", "PEquiv.trans_refl", "PFun.comp_id", "PFun.dom_comp", "PFun.dom_iff_graph", "PFun.ext'", "PFun.fix_fwd_eq", "PFun.image_inter", "PFun.mem_image", "PFun.mem_prodLift", "PFun.preimage_mono", "PFun.preimage_subset_dom", "PFunctor.liftp_iff'", "PMF.apply_eq_zero_iff", "PMF.monad_seq_eq_seq", "PMF.ofFinset_apply_of_not_mem", "PMF.ofMultiset_apply", "PMF.ofMultiset_apply_of_not_mem", "PMF.pure_map", "PMF.support_pure", "PMF.toMeasure_ofFintype_apply", "PMF.toMeasure_uniformOfFinset_apply", "PMF.toOuterMeasure_apply_finset", "PMF.toOuterMeasure_bindOnSupport_apply", "PNat.Coprime.factor_eq_gcd_right", "PNat.Coprime.factor_eq_gcd_right_right", "PNat.Ico_eq_finset_subtype", "PNat.add_one", "PNat.coe_inj", "PNat.factorMultiset_gcd", "PNat.factorMultiset_le_iff", "PNat.find_min'", "PNat.find_spec", "PNat.gcdA'_coe", "PNat.gcd_eq_right_iff_dvd", "PNat.le_of_dvd", "PNat.lt_add_one_iff", "PNat.map_subtype_embedding_uIcc", "PNat.mk_ofNat", "PNat.mod_coe", "PNat.pos", "PSet.Resp.Equiv.euc", "PSigma.Lex.mono", "PSigma.lex_wf", "PUnit.le", "Padic.complete''", "Padic.exi_rat_seq_conv", "Padic.exi_rat_seq_conv_cauchy", "Padic.valuation_zero", "PadicInt.coe_add", "PadicInt.denseRange_intCast", "PadicInt.isCauSeq_nthHom", "PadicInt.lift_spec", "PadicInt.limNthHom_zero", "PadicInt.mkUnits_eq", "PadicInt.mk_zero", "PadicInt.norm_int_le_pow_iff_dvd", "PadicInt.valuation_p_pow_mul", "PadicInt.valuation_zero", "PadicSeq.norm_equiv", "Part.bind_map", "Part.bind_of_mem", "Part.div_mem_div", "Part.fix_eq", "Part.fix_le", "Part.map_map", "Part.map_none", "Part.mem_map_iff", "Part.mem_mk_iff", "Part.mul_mem_mul", "Part.of_toOption", "Part.one_def", "PartENat.card_congr", "PartENat.card_le_one_iff_subsingleton", "PartENat.dom_some", "PartENat.dom_zero", "PartENat.lt_coe_succ_iff_le", "PartENat.lt_find", "PartENat.lt_find_iff", "PartialEquiv.copy_eq", "PartialEquiv.image_symm_image_of_subset_target", "PartialEquiv.invOn", "PartialEquiv.leftInvOn", "PartialEquiv.left_inv", "PartialEquiv.pi_trans", "PartialEquiv.source_inter_preimage_inv_preimage", "PartialEquiv.symm_image_target_inter_eq'", "PartialEquiv.symm_source", "PartialEquiv.symm_target", "PartialEquiv.trans_assoc", "PartialEquiv.trans_refl_restr", "PartialEquiv.trans_symm_eq_symm_trans_symm", "PartialHomeomorph.IsImage.iff_symm_preimage_eq", "PartialHomeomorph.IsImage.leftInvOn_piecewise", "PartialHomeomorph.IsImage.mapsTo", "PartialHomeomorph.MDifferentiable.comp_symm_deriv", "PartialHomeomorph.MDifferentiable.mdifferentiableAt_symm", "PartialHomeomorph.coe_trans_symm", "PartialHomeomorph.continuousWithinAt_iff_continuousWithinAt_comp_left", "PartialHomeomorph.eqOnSource_refl", "PartialHomeomorph.eventually_left_inverse", "PartialHomeomorph.extend_coord_change_source_mem_nhdsWithin", "PartialHomeomorph.extend_image_source_inter", "PartialHomeomorph.extend_target_subset_range", "PartialHomeomorph.image_eq_target_inter_inv_preimage", "PartialHomeomorph.isLocalStructomorphWithinAt_source_iff", "PartialHomeomorph.isOpen_extend_source", "PartialHomeomorph.map_extend_symm_nhdsWithin_range", "PartialHomeomorph.map_source''", "PartialHomeomorph.mapsTo", "PartialHomeomorph.ofSet_trans", "PartialHomeomorph.prod_eq_prod_of_nonempty'", "PartialHomeomorph.prod_symm", "PartialHomeomorph.refl_symm", "PartialHomeomorph.self_trans_symm", "PartialHomeomorph.singleton_hasGroupoid", "PartialHomeomorph.source_inter_preimage_target_inter", "PartialHomeomorph.transHomeomorph_transHomeomorph", "PartialHomeomorph.trans_apply", "PartitionOfUnity.sum_nonneg", "Partrec.const'", "Partrec.nat_iff", "Partrec.to\u2082", "Partrec\u2082.comp", "Path.Homotopic.comp_pi_eq_pi_comp", "Path.Homotopy.reflTransSymmAux_mem_I", "Path.Homotopy.transAssocReparamAux_zero", "Path.delayReflLeft_one", "Path.map_id", "Path.refl_range", "Path.truncate_self", "Path.truncate_zero_one", "Pell.IsFundamental.d_pos", "Pell.Solution\u2081.coe_mk", "Pell.Solution\u2081.x_mk", "Pell.Solution\u2081.y_zpow_pos", "Pell.eq_pellZd", "Pell.eq_pell_lem", "Pell.isPell_mul", "Pell.pellZd_im", "Pell.pellZd_succ_succ", "Pell.x_sub_y_dvd_pow_lem", "Pell.xn_modEq_x2n_sub", "Pell.xy_modEq_of_modEq", "Pell.y_mul_dvd", "Pell.yn_add", "PerfectField.separable_iff_squarefree", "PerfectPairing.apply_toDualLeft_symm_apply", "PerfectRing.liftAux_apply", "PerfectRing.lift_comp_lift_apply_eq_self", "Perfection.frobenius_pthRoot", "PerfectionMap.hom_ext", "PerfectionMap.of", "Pi.Lex.noMaxOrder'", "Pi.ceilDiv_apply", "Pi.continuous_uncurry", "Pi.floorDiv_apply", "Pi.floorDiv_def", "Pi.hasProd", "Pi.induced_precomp", "Pi.isAtom_iff", "Pi.isPWO", "Pi.kstar_apply", "Pi.leOnePart_apply", "Pi.leOnePart_def", "Pi.mulSingle_eq_of_ne", "Pi.nonUnitalRingHom_injective", "Pi.sdiff_apply", "Pi.uniformContinuous_restrict", "Pi.uniformSpace_comap_restrict", "Pi.uniformity", "PiLp.basis_toMatrix_basisFun_mul", "PiLp.dist_eq_iSup", "PiLp.dist_eq_sum", "PiLp.edist_equiv_symm_single_same", "PiLp.norm_eq_ciSup", "PiLp.norm_eq_sum", "PiNat.apply_firstDiff_ne", "PiNat.cylinder_eq_res", "PiNat.dist_triangle_nonarch", "PiNat.exists_disjoint_cylinder", "PiNat.exists_retraction_of_isClosed", "PiNat.exists_retraction_subtype_of_isClosed", "PiNat.shortestPrefixDiff_pos", "PiTensorProduct.induction_on'", "PiTensorProduct.liftAux_tprod", "PiTensorProduct.liftIsometry_tprodL", "PiTensorProduct.lift_comp_reindex_symm", "PiTensorProduct.lift_reindex_symm", "PiTensorProduct.lifts_zero", "PiTensorProduct.mapL_apply", "PiTensorProduct.map_id", "PiTensorProduct.map_smul", "PiTensorProduct.mem_lifts_iff", "PiTensorProduct.norm_eval_le_projectiveSeminorm", "PiTensorProduct.projectiveSeminormAux_add_le", "PiTensorProduct.reindex_symm", "PiTensorProduct.reindex_tprod", "PicardLindelof.nonempty_Icc", "PointedCone.toConvexCone_positive", "Poly.coe_sub", "Poly.sub_apply", "Polynomial.C_add", "Polynomial.C_eq_algebraMap", "Polynomial.C_leadingCoeff_mul_prod_multiset_X_sub_C", "Polynomial.C_mul", "Polynomial.C_mul_monomial", "Polynomial.C_sub", "Polynomial.Chebyshev.U_add_two", "Polynomial.Chebyshev.U_one", "Polynomial.Chebyshev.aeval_T", "Polynomial.Chebyshev.induct", "Polynomial.Chebyshev.one_sub_X_sq_mul_U_eq_pol_in_T", "Polynomial.Gal.ext", "Polynomial.Gal.galActionHom_bijective_of_prime_degree'", "Polynomial.Gal.galActionHom_injective", "Polynomial.IsPrimitive.dvd_iff_fraction_map_dvd_fraction_map", "Polynomial.IsPrimitive.irreducible_of_irreducible_map_of_injective", "Polynomial.IsRoot.of_map", "Polynomial.IsSplittingField.adjoin_rootSet_eq_range", "Polynomial.IsUnitTrinomial.irreducible_of_coprime'", "Polynomial.IsWeaklyEisensteinAt.map", "Polynomial.Monic.isRegular", "Polynomial.Monic.leadingCoeff_not_mem", "Polynomial.Monic.mul_left_ne_zero", "Polynomial.Monic.natSepDegree_eq_one_iff", "Polynomial.Separable.map_irreducible_of_isPurelyInseparable", "Polynomial.Separable.unit_mul", "Polynomial.SplittingFieldAux.succ", "Polynomial.X_eq_normalize", "Polynomial.X_pow_sub_C_ne_zero", "Polynomial.X_pow_sub_one_separable_iff", "Polynomial.addHom_ext", "Polynomial.adjoin_X", "Polynomial.aevalTower_comp_C", "Polynomial.aeval_X_pow", "Polynomial.aeval_ofReal", "Polynomial.aroots_C", "Polynomial.aroots_X", "Polynomial.as_sum_range'", "Polynomial.ascPochhammer_smeval_neg_eq_descPochhammer", "Polynomial.associated_of_dvd_of_natDegree_le", "Polynomial.card_nthRoots", "Polynomial.card_roots_toFinset_le_derivative", "Polynomial.card_support_eq_two", "Polynomial.cardinal_mk_le_max", "Polynomial.coe_one", "Polynomial.coeff_C_mul_X", "Polynomial.coeff_C_mul_X_pow", "Polynomial.coeff_X_pow_mul", "Polynomial.coeff_comp_degree_mul_degree", "Polynomial.coeff_divByMonic_X_sub_C", "Polynomial.coeff_eq_zero_of_lt_natTrailingDegree", "Polynomial.coeff_eq_zero_of_natDegree_lt", "Polynomial.coeff_hermite_succ_succ", "Polynomial.coeff_isUnit_isNilpotent_of_isUnit", "Polynomial.coeff_mul_C", "Polynomial.coeff_mul_monomial", "Polynomial.coeff_restriction", "Polynomial.coeff_sum", "Polynomial.coeff_toSubring'", "Polynomial.coeff_update_apply", "Polynomial.coeff_update_same", "Polynomial.coeff_zero_multiset_prod", "Polynomial.comp_X_add_C_ne_zero_iff", "Polynomial.content_C", "Polynomial.content_eq_gcd_leadingCoeff_content_eraseLead", "Polynomial.continuous_eval\u2082", "Polynomial.cyclotomic'.monic", "Polynomial.cyclotomic_eq_prod_X_pow_sub_one_pow_moebius", "Polynomial.cyclotomic_irreducible_of_irreducible_pow", "Polynomial.degree_add_eq_of_leadingCoeff_add_ne_zero", "Polynomial.degree_add_eq_right_of_degree_lt", "Polynomial.degree_cyclotomic'", "Polynomial.degree_eq_bot", "Polynomial.degree_eq_card_roots'", "Polynomial.degree_mono", "Polynomial.degree_mul_le_of_le", "Polynomial.degree_pos_of_ne_zero_of_nonunit", "Polynomial.degree_pow_le_of_le", "Polynomial.degree_prod", "Polynomial.degree_quadratic_lt_degree_C_mul_X_cb", "Polynomial.degree_sub_lt", "Polynomial.degree_sum_fin_lt", "Polynomial.derivWithin", "Polynomial.derivative_X_add_C", "Polynomial.derivative_comp", "Polynomial.derivative_mul", "Polynomial.derivative_one", "Polynomial.differentiableAt", "Polynomial.divByMonic_eq_div", "Polynomial.divX_add", "Polynomial.div_modByMonic_unique", "Polynomial.div_tendsto_atTop_of_degree_gt'", "Polynomial.dvd_iff_content_dvd_content_and_primPart_dvd_primPart", "Polynomial.eq_of_dvd_of_natDegree_le_of_leadingCoeff", "Polynomial.eq_of_monic_of_associated", "Polynomial.eq_zero_of_degree_lt_of_eval_finset_eq_zero", "Polynomial.eq_zero_of_hasseDeriv_eq_zero", "Polynomial.erase_mem_lifts", "Polynomial.eval_intCast", "Polynomial.eval\u2082_C", "Polynomial.eval\u2082_X", "Polynomial.eval\u2082_at_natCast", "Polynomial.eval\u2082_at_ofNat", "Polynomial.eval\u2082_finset_prod", "Polynomial.eval\u2082_finset_sum", "Polynomial.eval\u2082_mul_eq_zero_of_left", "Polynomial.eval\u2082_pow", "Polynomial.eval\u2082_sum", "Polynomial.exists_approx_polynomial_aux", "Polynomial.expand_contract", "Polynomial.expand_eq_sum", "Polynomial.forall_eq_iff_forall_eq", "Polynomial.hasseDeriv_C", "Polynomial.hermite_one", "Polynomial.image_rootSet", "Polynomial.int_coeff_of_cyclotomic'", "Polynomial.int_cyclotomic_spec", "Polynomial.integralNormalization_eval\u2082_eq_zero", "Polynomial.irreducible_iff_roots_eq_zero_of_degree_le_three", "Polynomial.isCoprime_X_sub_C_of_isUnit_sub", "Polynomial.isPrimitive_iff_isUnit_of_C_dvd", "Polynomial.isRoot_cyclotomic_iff", "Polynomial.isRoot_gcd_iff_isRoot_left_right", "Polynomial.isRoot_of_aeval_algebraMap_eq_zero", "Polynomial.isUnitTrinomial_iff''", "Polynomial.isUnit_C", "Polynomial.isUnit_C_add_X_mul_iff", "Polynomial.isUnit_leadingCoeff_mul_left_eq_zero_iff", "Polynomial.isUnit_map", "Polynomial.isUnit_or_eq_zero_of_isUnit_integerNormalization_primPart", "Polynomial.iterate_comp_eval\u2082", "Polynomial.iterate_derivative_X_sub_pow", "Polynomial.le_degree_of_mem_supp", "Polynomial.le_natTrailingDegree", "Polynomial.leadingCoeff_add_of_degree_eq", "Polynomial.leadingCoeff_divByMonic_X_sub_C", "Polynomial.leadingCoeff_divByMonic_of_monic", "Polynomial.leadingCoeff_dvd_leadingCoeff", "Polynomial.leadingCoeff_map", "Polynomial.leadingCoeff_mul_monic", "Polynomial.leadingCoeff_multiset_prod'", "Polynomial.leadingCoeff_quadratic", "Polynomial.leadingCoeff_sub_of_degree_lt'", "Polynomial.lift_of_splits", "Polynomial.linearIndependent_powers_iff_aeval", "Polynomial.lt_natDegree_of_mem_eraseLead_support", "Polynomial.map_add", "Polynomial.map_contract", "Polynomial.map_injective", "Polynomial.map_list_prod", "Polynomial.map_mapRingHom_evalEval", "Polynomial.map_mod", "Polynomial.map_monic_eq_zero_iff", "Polynomial.map_multiset_prod", "Polynomial.map_restriction", "Polynomial.map_roots_le", "Polynomial.map_sub", "Polynomial.mem_degreeLT", "Polynomial.mem_iff_annIdealGenerator_dvd", "Polynomial.mem_roots", "Polynomial.mirror_natDegree", "Polynomial.mirror_natTrailingDegree", "Polynomial.modByMonic_X_sub_C_eq_C_eval", "Polynomial.modByMonic_add_div", "Polynomial.modByMonic_one", "Polynomial.monic_X_pow", "Polynomial.monomial_neg", "Polynomial.monomial_sub", "Polynomial.multiplicity_X_sub_C_finite", "Polynomial.multiplicity_le_one_of_separable", "Polynomial.multiset_prod_X_sub_C_nextCoeff", "Polynomial.natCast_comp", "Polynomial.natDegree_X_add_C", "Polynomial.natDegree_add_C", "Polynomial.natDegree_divX_eq_natDegree_tsub_one", "Polynomial.natDegree_expand", "Polynomial.natDegree_mul_le", "Polynomial.natDegree_multiset_sum_le", "Polynomial.natDegree_not_mem_eraseLead_support", "Polynomial.natDegree_quadratic", "Polynomial.natDegree_sub_eq_right_of_natDegree_lt", "Polynomial.natDegree_sub_le_iff_left", "Polynomial.natSepDegree_zero", "Polynomial.natTrailingDegree_natCast", "Polynomial.ne_zero_of_degree_ge_degree", "Polynomial.nextCoeff_X_sub_C", "Polynomial.nextCoeff_eq_zero", "Polynomial.nnqsmul_eq_C_mul", "Polynomial.nodup_roots", "Polynomial.normUnit_X", "Polynomial.not_isUnit_X_pow_sub_one", "Polynomial.not_isUnit_of_natDegree_pos", "Polynomial.not_monic_zero", "Polynomial.ofFinsupp_add", "Polynomial.ofFinsupp_sum", "Polynomial.ofFinsupp_zero", "Polynomial.opRingEquiv_symm_C_mul_X_pow", "Polynomial.pow_comp", "Polynomial.pow_mul_divByMonic_rootMultiplicity_eq", "Polynomial.prod_cyclotomic_eq_X_pow_sub_one", "Polynomial.prod_multiset_X_sub_C_dvd", "Polynomial.prod_roots_eq_coeff_zero_of_monic_of_split", "Polynomial.revAt_invol", "Polynomial.ringHom_ext'", "Polynomial.rootMultiplicity_eq_zero_iff", "Polynomial.rootMultiplicity_mul'", "Polynomial.rootOfSplits'_eq_rootOfSplits", "Polynomial.rootSet_X_pow", "Polynomial.root_mul_right_of_isRoot", "Polynomial.roots.le_of_dvd", "Polynomial.scaleRoots_eval\u2082_eq_zero_of_eval\u2082_div_eq_zero", "Polynomial.separable_gcd_right", "Polynomial.smeval_C", "Polynomial.smeval_X_pow", "Polynomial.smeval_add", "Polynomial.smeval_at_zero", "Polynomial.smul_X_eq_monomial", "Polynomial.smul_eq_C_mul", "Polynomial.span_le_degreeLE_of_finite", "Polynomial.splits_of_splits_mul'", "Polynomial.sub_one_lt_natAbs_cyclotomic_eval", "Polynomial.sub_one_pow_totient_lt_cyclotomic_eval", "Polynomial.sum_fin", "Polynomial.support_X_empty", "Polynomial.support_map_subset", "Polynomial.support_update_zero", "Polynomial.taylor_injective", "Polynomial.toFinsupp_mul", "Polynomial.toFinsupp_pow", "Polynomial.trailingDegree_X", "Polynomial.trinomial_leadingCoeff", "Polynomial.trinomial_middle_coeff", "Polynomial.trinomial_mirror", "Polynomial.zero_nmem_multiset_map_X_sub_C", "PolynomialModule.comp_eval", "PolynomialModule.lsingle_apply", "PosNum.add_one", "PosNum.natSize_to_nat", "PosNum.to_nat_pos", "PosSMulReflectLE_iff_posSMulReflectLT", "Positive.val_one", "PowerBasis.adjoin_gen_eq_top", "PowerBasis.coe_basis", "PowerBasis.equivOfMinpoly_symm", "PowerBasis.natDegree_minpolyGen", "PowerBasis.ofGenMemAdjoin'_dim", "PowerBasis.quotientEquivQuotientMinpolyMap_symm_apply_mk", "PowerBasis.trace_gen_eq_sum_roots", "PowerSeries.C_eq_algebraMap", "PowerSeries.X_pow_order_dvd", "PowerSeries.coeff_cos_bit1", "PowerSeries.coeff_mul_X_pow", "PowerSeries.coeff_mul_X_pow'", "PowerSeries.coeff_mul_one_sub_of_lt_order", "PowerSeries.coeff_zero_X", "PowerSeries.constantCoeff_smul", "PowerSeries.derivativeFun_coe", "PowerSeries.divided_by_X_pow_orderMul", "PowerSeries.eq_inv_iff_mul_eq_one", "PowerSeries.evalNegHom_X", "PowerSeries.ext_iff", "PowerSeries.invOneSubPow_eq_inv_one_sub_pow", "PowerSeries.map_X", "PowerSeries.natDegree_trunc_lt", "PowerSeries.sub_const_eq_shift_mul_X", "PowerSeries.subsingleton_iff", "PowerSeries.trunc_add", "PreErgodic.ae_eq_const_of_ae_eq_comp", "Prefunctor.congr_map", "Prefunctor.mapPath_comp", "Prefunctor.pathStar_injective", "Prefunctor.symmetrifyCostar", "PresentedGroup.to_group_eq_one_of_mem_closure", "PresheafOfModules.Derivation'.app_apply", "PresheafOfModules.isSheaf_of_isLimit", "Pretrivialization.coe_coe", "Pretrivialization.proj_surjOn_baseSet", "Primcodable.mem_range_encode", "Prime.dvd_prime_iff_associated", "Prime.dvd_prod_iff", "Prime.left_dvd_or_dvd_right_of_dvd_mul", "Prime.not_dvd_finset_prod", "Prime.not_dvd_mul", "PrimeMultiset.prod_inf", "PrimeMultiset.prod_ofPNatMultiset", "PrimeMultiset.prod_sup", "PrimeSpectrum.asIdeal_lt_asIdeal", "PrimeSpectrum.comap_asIdeal", "PrimeSpectrum.comap_inducing_of_surjective", "PrimeSpectrum.isBasis_basic_opens", "PrimeSpectrum.isOpen_basicOpen", "PrimeSpectrum.range_comap_of_surjective", "PrimeSpectrum.t1Space_iff_isField", "PrimeSpectrum.vanishingIdeal_iUnion", "PrimeSpectrum.zeroLocus_inf", "PrimeSpectrum.zeroLocus_subset_zeroLocus_singleton_iff", "Primrec.dom_bool", "Primrec.encode_iff", "Primrec.nat_add", "Primrec.nat_bodd", "Primrec.option_bind\u2081", "Primrec.option_some", "Primrec.option_some_iff", "Primrec.subtype_val", "Primrec.succ", "Primrec.vector_cons", "Primrec.vector_head", "Primrec.vector_tail", "PrimrecPred.comp", "PrimrecRel.of_eq", "Primrec\u2082.curry", "PrincipalSeg.coe_coe_fn'", "PrincipalSeg.down", "ProbabilityTheory.CondIndep.condIndepSet_of_measurableSet", "ProbabilityTheory.IdentDistrib.coe_nnreal_ennreal", "ProbabilityTheory.IdentDistrib.norm", "ProbabilityTheory.IdentDistrib.snorm_eq", "ProbabilityTheory.IndepFun.ae_eq", "ProbabilityTheory.IndepFun.cgf_add", "ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux_def'", "ProbabilityTheory.IsRatCondKernelCDF.iInf_rat_gt_eq", "ProbabilityTheory.IsRatCondKernelCDFAux.bddBelow_range", "ProbabilityTheory.aestronglyMeasurable'_integral_condexpKernel", "ProbabilityTheory.cgf_undef", "ProbabilityTheory.compProd_toKernel", "ProbabilityTheory.condCount_singleton", "ProbabilityTheory.condDistrib_ae_eq_of_measure_eq_compProd", "ProbabilityTheory.condDistrib_apply_of_ne_zero", "ProbabilityTheory.condIndepFun_iff_condexp_inter_preimage_eq_mul", "ProbabilityTheory.condIndepSet_iff", "ProbabilityTheory.condIndep_bot_left", "ProbabilityTheory.condIndep_iSup_of_disjoint", "ProbabilityTheory.condIndep_limsup_atBot_self", "ProbabilityTheory.condKernel_const", "ProbabilityTheory.cond_apply'", "ProbabilityTheory.condexpKernel_ae_eq_condexp", "ProbabilityTheory.condexpKernel_ae_eq_trim_condexp", "ProbabilityTheory.condexp_prod_ae_eq_integral_condDistrib", "ProbabilityTheory.defaultRatCDF_le_one", "ProbabilityTheory.defaultRatCDF_nonneg", "ProbabilityTheory.eq_condKernel_of_measure_eq_compProd", "ProbabilityTheory.evariance_def'", "ProbabilityTheory.evariance_mul", "ProbabilityTheory.gaussianPDFReal_mul", "ProbabilityTheory.gaussianPDFReal_sub", "ProbabilityTheory.gaussianPDF_pos", "ProbabilityTheory.gaussianPDF_zero_var", "ProbabilityTheory.geometricPMFReal_pos", "ProbabilityTheory.iCondIndep.iCondIndepSets", "ProbabilityTheory.iCondIndepSet.condIndep_generateFrom_le", "ProbabilityTheory.iCondIndepSet.condIndep_generateFrom_lt", "ProbabilityTheory.iCondIndep_iff", "ProbabilityTheory.iIndepFun.mgf_sum", "ProbabilityTheory.iIndepSet.indep_generateFrom_of_disjoint", "ProbabilityTheory.indepSet_empty_left", "ProbabilityTheory.indepSet_empty_right", "ProbabilityTheory.indepSets_singleton_iff", "ProbabilityTheory.indep_biSup_compl", "ProbabilityTheory.indep_iSup_limsup", "ProbabilityTheory.indep_iSup_of_directed_le", "ProbabilityTheory.integrable_toReal_condexpKernel", "ProbabilityTheory.integral_gaussianPDFReal_eq_one", "ProbabilityTheory.integral_stieltjesOfMeasurableRat", "ProbabilityTheory.isRatStieltjesPoint_unit_prod_iff", "ProbabilityTheory.kernel.IsFiniteKernel.integrable", "ProbabilityTheory.kernel.compProd_add_right", "ProbabilityTheory.kernel.const_zero", "ProbabilityTheory.kernel.density_fst_univ", "ProbabilityTheory.kernel.density_nonneg", "ProbabilityTheory.kernel.deterministic_apply", "ProbabilityTheory.kernel.finset_sum_apply", "ProbabilityTheory.kernel.finset_sum_apply'", "ProbabilityTheory.kernel.fst_prod", "ProbabilityTheory.kernel.iIndepFun.indepFun_prod_range_succ", "ProbabilityTheory.kernel.iIndepFun_iff_measure_inter_preimage_eq_mul", "ProbabilityTheory.kernel.iIndepSet_iff_meas_biInter", "ProbabilityTheory.kernel.iIndep_comap_mem_iff", "ProbabilityTheory.kernel.indep_limsup_atTop_self", "ProbabilityTheory.kernel.integral_fn_integral_add", "ProbabilityTheory.kernel.integral_withDensity", "ProbabilityTheory.kernel.isCondKernelCDF_condKernelCDF", "ProbabilityTheory.kernel.isFiniteKernel_withDensity_of_bounded", "ProbabilityTheory.kernel.lintegral_density", "ProbabilityTheory.kernel.lintegral_prodMkRight", "ProbabilityTheory.kernel.measurable_compProdFun", "ProbabilityTheory.kernel.measure_zero_or_one_of_measurableSet_limsup_atBot", "ProbabilityTheory.kernel.setLIntegral_compProd_univ_left", "ProbabilityTheory.kernel.singularPart_of_subset_mutuallySingularSetSlice", "ProbabilityTheory.kernel.sum_add", "ProbabilityTheory.kernel.tendsto_densityProcess_limitProcess", "ProbabilityTheory.kernel.tendsto_snorm_one_restrict_densityProcess_limitProcess", "ProbabilityTheory.kernel.withDensity_rnDeriv_eq_zero_iff_measure_eq_zero", "ProbabilityTheory.measure_cdf", "ProbabilityTheory.measure_ge_le_exp_cgf", "ProbabilityTheory.mgf_const", "ProbabilityTheory.mgf_nonneg", "ProbabilityTheory.mgf_zero_fun", "ProbabilityTheory.setIntegral_stieltjesOfMeasurableRat", "ProbabilityTheory.setIntegral_stieltjesOfMeasurableRat_rat", "ProbabilityTheory.stieltjesOfMeasurableRat_nonneg", "ProbabilityTheory.strong_law_ae", "ProbabilityTheory.sum_variance_truncation_le", "ProbabilityTheory.tendsto_integral_truncation", "ProbabilityTheory.tendsto_stieltjesOfMeasurableRat_atTop", "Prod.comul_comp_snd", "Prod.edist_eq", "Prod.fst_sSup", "Prod.gameAdd_le_lex", "Prod.inv_mk", "Prod.map_id", "Prod.mk_dvd_mk", "Prod.pow_fst", "Prod.smul_def", "Prod.smul_fst", "Prod.smul_mk", "Prod.smul_swap", "Prod.swap_iInf", "Prod.swap_swap_eq", "Prod.\u03c9Sup_zip", "Profinite.NobelingProof.Products.isGood_mono", "Profinite.NobelingProof.Products.isGood_nil", "Profinite.NobelingProof.Products.prop_of_isGood", "Profinite.NobelingProof.Products.rel_head!_of_mem", "Profinite.NobelingProof.isClosed_C1", "Profinite.NobelingProof.projRestricts_eq_id", "Profinite.NobelingProof.proj_prop_eq_self", "Profinite.NobelingProof.spanFinBasis.span", "Profinite.NobelingProof.succ_mono", "Profinite.NobelingProof.swapTrue_eq_true", "Profinite.coe_id", "Profinite.projective_of_extrDisc", "Profinite.pullback.hom_ext", "ProjectiveSpectrum.basicOpen_mul", "ProjectiveSpectrum.basicOpen_mul_le_left", "ProjectiveSpectrum.mem_compl_zeroLocus_iff_not_mem", "ProjectiveSpectrum.vanishingIdeal_singleton", "ProjectiveSpectrum.zeroLocus_iSup_homogeneousIdeal", "ProjectiveSpectrum.zeroLocus_mul_homogeneousIdeal", "ProjectiveSpectrum.zeroLocus_singleton_pow", "ProjectiveSpectrum.zeroLocus_sup_homogeneousIdeal", "Projectivization.Subspace.span_le_subspace_iff", "Projectivization.dependent_pair_iff_eq", "Projectivization.ind", "Projectivization.mk_eq_mk_iff'", "ProperCone.dual_dual", "Pullback.continuous_totalSpaceMk", "PythagoreanTriple.eq", "PythagoreanTriple.even_odd_of_coprime", "QPF.Cofix.bisim'", "QPF.comp_map", "QuadraticForm.IsometryEquiv.coe_toLinearEquiv", "QuadraticForm.anisotropic_of_prod", "QuadraticForm.associated_isOrtho", "QuadraticForm.associated_linMulLin", "QuadraticForm.complex_equivalent", "QuadraticForm.ext", "QuadraticForm.isOrtho_polarBilin", "QuadraticForm.linMulLinSelfPosDef", "QuadraticForm.linMulLin_apply", "QuadraticForm.map_neg", "QuadraticForm.polarBilin_comp", "QuadraticForm.polar_smul_left", "QuadraticForm.posDef_of_nonneg", "QuadraticForm.posDef_of_toMatrix'", "QuadraticForm.sum_apply", "QuadraticForm.tensorDistrib_tmul", "QuadraticForm.tmul_comp_tensorMap", "QuadraticForm.toFun_eq_coe", "QuadraticForm.toQuadraticForm_associated", "QuadraticModuleCat.forget\u2082_map_associator_hom", "QuasiErgodic.ae_eq_const_of_ae_eq_comp_ae", "QuasispectrumRestricts.comp", "QuasispectrumRestricts.nonUnitalStarAlgHom_id", "QuasispectrumRestricts.subset_preimage", "Quaternion.coe_imI", "Quaternion.ext", "Quaternion.fst_imJ_dualNumberEquiv_symm", "Quaternion.imK_nnratCast", "Quaternion.im_exp", "Quaternion.im_star", "Quaternion.inner_def", "Quaternion.mul_re", "Quaternion.natCast_imJ", "Quaternion.neg_re", "Quaternion.snd_re_dualNumberEquiv_symm", "Quaternion.star_imJ", "Quaternion.star_re", "Quaternion.sub_self_re", "QuaternionAlgebra.Basis.j_mul_k", "QuaternionAlgebra.Basis.lift_mul", "QuaternionAlgebra.coe_add", "QuaternionAlgebra.coe_commutes", "QuaternionAlgebra.coe_imI", "QuaternionAlgebra.coe_one", "QuaternionAlgebra.finrank_eq_four", "QuaternionAlgebra.intCast_imI", "QuaternionAlgebra.mul_imI", "QuaternionAlgebra.mul_star_eq_coe", "QuaternionAlgebra.smul_re", "QuaternionAlgebra.star_add_self'", "QuaternionAlgebra.sub_imK", "QuaternionGroup.a_mul_a", "Quiver.Hom.unop_mk", "Quiver.Path.cast_cons", "Quiver.Path.comp_inj", "Quiver.SingleObj.ext", "Quiver.SingleObj.toPrefunctor_comp", "QuotSMulTop.map_comp", "Quotient.inductionOn\u2083'", "Quotient.mk_eq_iff_out", "Quotient.out_eq'", "Quotient.surjective_Quotient_mk''", "QuotientAddGroup.integral_mul_eq_integral_automorphize_mul", "QuotientGroup.map_comp_map", "QuotientGroup.measurable_coe", "QuotientGroup.subgroup_eq_top_of_subsingleton", "RCLike.I_to_real", "RCLike.abs_re_le_norm", "RCLike.conj_eq_iff_real", "RCLike.conj_im", "RCLike.div_re_ofReal", "RCLike.ext_iff", "RCLike.imCLM_apply", "RCLike.intCast_im", "RCLike.inv_re", "RCLike.nonUnitalContinuousFunctionalCalculus", "RCLike.ofReal_eq_zero", "RCLike.ofReal_ofNat", "RCLike.ofReal_re", "RCLike.ofReal_sum", "RCLike.one_re", "RCLike.re_eq_self_of_le", "Rack.PreEnvelGroupRel.symm", "Rack.invAct_apply", "Rack.left_cancel_inv", "Rack.self_invAct_invAct_eq", "Rat.add_mul", "Rat.add_nonneg", "Rat.cast_def", "Rat.cast_mono", "Rat.cast_natCast", "Rat.cast_pos_of_pos", "Rat.cast_sub_of_ne_zero", "Rat.den_intCast", "Rat.divInt_div_divInt_cancel_right", "Rat.divInt_self", "Rat.fract_inv_num_lt_num_of_pos", "Rat.intCast_num", "Rat.inv_intCast_den_of_pos", "Rat.inv_natCast_den_of_pos", "Rat.isSquare_iff", "Rat.mkRat_eq_divInt", "Rat.mk_denom_ne_zero_of_ne_zero", "Rat.mk_num_ne_zero_of_ne_zero", "Rat.mul_comm", "Rat.mul_den", "Rat.mul_den_dvd", "Rat.nonneg_antisymm", "Rat.not_irrational", "Rat.num_den_mk", "Rat.num_nonpos", "Rat.num_pos", "Rat.numberField_discr", "Rat.ofInt_ofNat", "Rat.ofNat_den", "Rat.one_den", "Rat.preimage_cast_Ioi", "Rat.preimage_cast_uIcc", "Rat.zero_num", "RatFunc.coe_add", "RatFunc.coe_def", "RatFunc.coe_mapRingHom_eq_coe_map", "RatFunc.laurent_C", "RatFunc.mk_eq_div", "RatFunc.monic_denom", "RatFunc.num_div_dvd", "RatFunc.ofFractionRing_mul", "RatFunc.ofFractionRing_neg", "RatFunc.ofFractionRing_sub", "ReaderT.run_failure", "Real.Angle.abs_sin_eq_of_two_zsmul_eq", "Real.Angle.coe_expMapCircle", "Real.Angle.coe_nsmul", "Real.Angle.cos_antiperiodic", "Real.Angle.cos_eq_zero_iff", "Real.Angle.cos_sq_add_sin_sq", "Real.Angle.expMapCircle_coe", "Real.Angle.intCast_mul_eq_zsmul", "Real.Angle.sign_neg", "Real.Angle.sin_coe", "Real.Angle.tan_eq_of_two_zsmul_eq", "Real.Angle.toReal_neg_iff_sign_neg", "Real.Angle.toReal_pi_div_two", "Real.BohrMollerup.tendsto_logGammaSeq", "Real.Gamma_mul_Gamma_add_half", "Real.Gamma_one_half_eq", "Real.arccos_neg", "Real.arccos_nonneg", "Real.arcsin_eq_neg_pi_div_two", "Real.arcsin_lt_zero", "Real.arcsin_nonneg", "Real.arcsin_one", "Real.arctan_add", "Real.coe_toNNReal", "Real.concaveOn_negMulLog", "Real.contDiff_cos", "Real.continuousAt_const_rpow", "Real.continuousAt_rpow_const", "Real.continuousAt_rpow_of_ne", "Real.continuous_arccos", "Real.continuous_sin", "Real.continuous_sinh", "Real.convergent_succ", "Real.cos_add_int_mul_two_pi", "Real.cos_mem_Icc", "Real.cos_nonneg_of_neg_pi_div_two_le_of_le", "Real.cos_sq'", "Real.cos_sq_le_one", "Real.cosh_arsinh", "Real.cosh_sub_sinh", "Real.cosh_two_mul", "Real.deriv_cos'", "Real.differentiable_rpow_const", "Real.dist_le_of_mem_pi_Icc", "Real.dist_right_le_of_mem_uIcc", "Real.ediam_Icc", "Real.exists_rat_pow_btwn_rat", "Real.expNear_sub", "Real.exp_abs_le", "Real.exp_approx_start", "Real.exp_arsinh", "Real.exp_le_one_iff", "Real.exp_lt_exp", "Real.exp_neg_one_lt_d9", "Real.fderiv_fourierChar_neg_bilinear_left_apply", "Real.fderiv_fourierChar_neg_bilinear_right_apply", "Real.fourierIntegralInv_eq'", "Real.fourierIntegral_eq", "Real.fourierIntegral_real_eq_integral_exp_smul", "Real.geom_mean_le_arith_mean", "Real.hasStrictDerivAt_log", "Real.hasStrictDerivAt_rpow_const", "Real.hasStrictDerivAt_tan", "Real.iInf_of_not_bddBelow", "Real.iSup_mul_of_nonpos", "Real.image_tan_Ioo", "Real.injOn_cos", "Real.inv_rpow", "Real.inv_sqrt_one_add_tan_sq", "Real.isBigO_one_exp_comp", "Real.le_log_iff_exp_le", "Real.le_logb_iff_rpow_le_of_base_lt_one", "Real.le_sin_mul", "Real.le_tan", "Real.log_le_log", "Real.log_lt_sub_one_of_pos", "Real.log_nonpos", "Real.log_prod", "Real.log_zero", "Real.logb_div_base", "Real.logb_lt_logb", "Real.logb_lt_logb_iff_of_base_lt_one", "Real.logb_nonneg_iff_of_base_lt_one", "Real.logb_pos_iff_of_base_lt_one", "Real.lt_logb_iff_rpow_lt", "Real.lt_sInf_add_pos", "Real.lt_sin", "Real.map_exp_nhds", "Real.mk_one", "Real.monotone_arcsin", "Real.mul_le_sin", "Real.negMulLog_nonneg", "Real.norm_rpow_of_nonneg", "Real.not_summable_natCast_inv", "Real.ofCauchy_zero", "Real.one_lt_rpow_iff_of_pos", "Real.pow_sum_div_card_le_sum_pow", "Real.rank_rat_real", "Real.real_sqrt_lt_nat_sqrt_succ", "Real.rpow_add", "Real.rpow_def_of_nonpos", "Real.rpow_eq_zero_iff_of_nonneg", "Real.rpow_intCast_mul", "Real.rpow_inv_eq", "Real.rpow_inv_le_iff_of_pos", "Real.rpow_inv_lt_iff_of_neg", "Real.rpow_le_rpow", "Real.rpow_le_rpow_of_exponent_ge", "Real.rpow_lt_rpow_left_iff", "Real.rpow_neg_one", "Real.rpow_sub_one", "Real.rpow_sum_of_pos", "Real.sSup_smul_of_nonpos", "Real.self_le_sinh_iff", "Real.sign_apply_eq", "Real.sin_half_eq_sqrt", "Real.sin_le_one", "Real.sin_lt_sin_of_lt_of_le_pi_div_two", "Real.sin_pi_div_eight", "Real.sin_pi_div_two", "Real.sin_sub", "Real.sin_sub_sin", "Real.sinh_sub", "Real.sqrtTwoAddSeries_monotone_left", "Real.sqrt_div", "Real.sqrt_eq_cases", "Real.sqrt_mul'", "Real.sqrt_mul_self", "Real.sqrt_one", "Real.sqrt_pos", "Real.strictMonoOn_sin", "Real.sum_le_exp_of_nonneg", "Real.tan_lt_tan_of_lt_of_lt_pi_div_two", "Real.tendsto_comp_exp_atTop", "Real.tendsto_integral_gaussian_smul", "Real.toNNReal_add_le", "Real.toNNReal_div'", "Real.toNNReal_ofNat", "Real.two_mul_arctan_sub_pi", "Real.volume_pi_Ioo", "RegularExpression.one_def", "Rel.codom_inv", "Rel.core_union", "Rel.image_univ", "Rel.preimage_bot", "Rel.preimage_inv", "Rel.preimage_top", "RelEmbedding.coe_toRelHom", "RelEmbedding.injective", "RelEmbedding.isIrrefl", "RelIso.ext_iff", "RelIso.injective", "RelSeries.append_apply_left", "RelSeries.last_snoc'", "Relation.TransGen.head'", "Relation.TransGen.head'_iff", "Relation.acc_of_singleton", "Relation.symmetric_join", "Relation.transGen_eq_self", "Relation.transitive_transGen", "Relator.left_unique_of_rel_eq", "Rep.trivial_def", "Representation.asModuleEquiv_symm_map_smul", "Right.min_le_max_of_mul_le_mul", "Right.mul_lt_mul", "Right.mul_lt_one'", "RightDerivMeasurableAux.A_mem_nhdsWithin_Ioi", "RightDerivMeasurableAux.B_mem_nhdsWithin_Ioi", "RightOrdContinuous.le_iff", "Ring.DirectLimit.lift_unique", "Ring.choose_natCast", "Ring.closure_subset_iff", "Ring.factorial_nsmul_multichoose_eq_ascPochhammer", "Ring.inverse_add_inverse", "Ring.inverse_non_unit", "Ring.isArtinian_of_zero_eq_one", "Ring.mul_inverse_cancel", "RingCat.ext", "RingCon.coe_bot", "RingCon.multisetSum", "RingCon.op_iff", "RingCon.refl", "RingCon.ringConGen_mono", "RingEquiv.coe_prodComm", "RingEquiv.toRingHom_injective", "RingHom.FinitePresentation.of_comp_finiteType", "RingHom.IsIntegral.tower_bot", "RingHom.PropertyIsLocal.affineLocally_of_isOpenImmersion", "RingHom.PropertyIsLocal.ofLocalizationSpan", "RingHom.PropertyIsLocal.source_affine_openCover_iff", "RingHom.RespectsIso.is_localization_away_iff", "RingHom.charZero_iff", "RingHom.eq_of_eqOn_set_top", "RingHom.fieldRange_eq_map", "RingHom.id_apply", "RingHom.id_comp", "RingHom.kerLift_mk", "RingHom.ker_coe_equiv", "RingHom.ker_isRadical_iff_reduced_of_surjective", "RingHom.ker_ne_top", "RingHom.map_dotProduct", "RingHom.map_frobenius", "RingHom.map_one", "RingHom.map_one_ne_zero", "RingHom.prod_comp_prodMap", "RingHom.restrict_apply", "RingHom.sub_mem_ker_iff", "RingHomCompTriple.comp_apply", "RingHomInvPair.symm", "RingTheory.Sequence.IsWeaklyRegular.cons", "RootPairing.injOn_dualMap_subtype_span_root_coroot", "RootPairing.ne_zero", "SModEq.comap", "SModEq.top", "SMulPosStrictMono.lift", "SSet.Quasicategory.hornFilling", "SameRay.exists_eq_smul_add", "SameRay.exists_nonneg_right", "SameRay.norm_add", "SameRay.norm_sub", "SameRay.sameRay_comm", "SameRay.sameRay_map_iff", "SameRay.sameRay_nonneg_smul_right", "SatisfiesM.map", "SatisfiesM_ExceptT_eq", "SchwartzMap.integralCLM_apply", "SchwartzMap.norm_iteratedFDeriv_le_seminorm", "SemiNormedGrp.iso_isometry_of_normNoninc", "SemiNormedGrp\u2081.coe_comp", "SemiNormedGrp\u2081.coe_of", "SemiconjBy.inv_right_iff", "SemiconjBy.inv_symm_left_iff", "SemiconjBy.natCast_mul_left", "SemiconjBy.neg_one_left", "SemiconjBy.neg_one_right", "SemiconjBy.prod", "SemiconjBy.units_inv_right_iff", "SemidirectProduct.map_right", "SemidirectProduct.mk_eq_inl_mul_inr", "Semigroup.mem_center_iff", "SemilatInfCat.coe_forget_to_partOrd", "SemilatticeInf.ext", "SemilinearIsometryClass.nnnorm_map", "Seminorm.ball_sup", "Seminorm.ball_zero'", "Seminorm.closedBall_smul_ball", "Seminorm.cont_normedSpace_to_withSeminorms", "Seminorm.continuous_of_continuous_comp", "Seminorm.finset_sup_apply_le", "Seminorm.isBounded_const", "Seminorm.isBounded_sup", "Seminorm.norm_sub_map_le_sub", "Seminorm.smul_le_smul", "Seminorm.uniformContinuous'", "SeminormFamily.basisSets_add", "SeminormFamily.basisSets_iff", "SeminormFamily.basisSets_nonempty", "Semiquot.pure_le", "SemistandardYoungTableau.ext", "SeparatedNhds.symm", "SeparatingDual.exists_ne_zero", "SeparationQuotient.comap_mk_nhdsSet", "SeparationQuotient.completeSpace_iff", "SeparationQuotient.continuous_mk", "SeparationQuotient.map_mk", "Set.BijOn.exists_extend", "Set.BijOn.forall", "Set.Countable.exists_eq_range", "Set.Countable.isConnected_compl_of_one_lt_rank", "Set.Countable.isG\u03b4_compl", "Set.Countable.isLindelof_sUnion", "Set.EqOn.piecewise_ite'", "Set.Finite.coeSort_toFinset", "Set.Finite.diff", "Set.Finite.dinduction_on", "Set.Finite.exists_ge", "Set.Finite.exists_lt_map_eq_of_forall_mem", "Set.Finite.finite_subsets", "Set.Finite.iUnion", "Set.Finite.infClosure", "Set.Finite.interior_sInter", "Set.Finite.latticeClosure", "Set.Finite.mul", "Set.Finite.ofFinset", "Set.Finite.of_surjOn", "Set.Finite.prod", "Set.Finite.sInter", "Set.Finite.toFinset_eq_univ", "Set.Finite.toFinset_setOf", "Set.Finite.upperClosure", "Set.Icc.coe_pow", "Set.Icc.coe_zero", "Set.Icc.mk_zero", "Set.Icc_prod_eq", "Set.Icc_subset_uIcc", "Set.Icc_union_Icc'", "Set.Icc_union_Ioo_eq_Ico", "Set.IciExtend_of_mem", "Set.Ico_diff_Iio", "Set.Ico_infinite", "Set.Iic.coe_biSup", "Set.Iic.coe_bot", "Set.Iic_iInf", "Set.Iic_prod_Iic", "Set.Iic_union_Icc", "Set.Iic_union_Ioi", "Set.Iio_inter_Ici", "Set.Iio_inter_Ioi", "Set.Iio_union_Icc_eq_Iic", "Set.Iio_union_Ico", "Set.Iio_union_Ioo", "Set.Infinite.card_eq_zero", "Set.Infinite.mono", "Set.Infinite.nontrivial", "Set.Infinite.not_bddBelow", "Set.InjOn.image_eq_image_iff", "Set.InjOn.mono", "Set.Ioc.mk_one", "Set.Ioc_diff_Ioo_same", "Set.Ioc_union_Icc_eq_Ioc", "Set.Ioc_union_Ioc_left", "Set.Ioc_union_Ioi_eq_Ioi", "Set.Ioc_union_Ioo_eq_Ioo", "Set.Ioo.lt_one", "Set.Ioo.one_minus_lt_one", "Set.Ioo_def", "Set.Ioo_inter_Iio", "Set.Ioo_inter_Ioc_of_right_lt", "Set.Ioo_union_Ico_eq_Ioo", "Set.IsPWO.union", "Set.IsWF.min_union", "Set.LeftInvOn.comp", "Set.LeftInvOn.eq", "Set.LeftInvOn.mapsTo", "Set.MapsTo.image_subset", "Set.MapsTo.iterate_restrict", "Set.MapsTo.nonempty", "Set.MapsTo.piecewise_ite", "Set.MulAntidiagonal.eq_of_snd_eq_snd", "Set.Nat.encard_range", "Set.Nonempty.inr", "Set.Nonempty.intrinsicInterior", "Set.Nonempty.of_sUnion_eq_univ", "Set.Nonempty.subset_one_iff", "Set.Nonempty.to_subtype", "Set.Nonempty.vsub", "Set.Nontrivial.infsep_exists_of_finite", "Set.Nontrivial.infsep_of_fintype", "Set.Nontrivial.nonempty", "Set.OrdConnected.measurableSet", "Set.Pairwise.mono'", "Set.PairwiseDisjoint.biUnion", "Set.PairwiseDisjoint.image_finset_of_le", "Set.PairwiseDisjoint.insert", "Set.PartiallyWellOrderedOn.image_of_monotone_on", "Set.PartiallyWellOrderedOn.insert", "Set.Sized.empty_mem_iff", "Set.Subsingleton.equitableOn", "Set.Subsingleton.induction_on", "Set.SurjOn.rightInvOn_invFunOn", "Set.UniformEquicontinuousOn.closure", "Set.UniformEquicontinuousOn.uniformContinuousOn_of_mem", "Set.abs_sub_le_of_uIcc_subset_uIcc", "Set.add_mem_centralizer", "Set.biInter_le_succ", "Set.biInter_le_succ'", "Set.biInter_union", "Set.biUnion_finsetSigma'", "Set.biUnion_sigma'", "Set.biUnion_univ", "Set.bijective_iff_bijOn_univ", "Set.bounded_le_inter_lt", "Set.bounded_lt_Ioc", "Set.bounded_lt_inter_not_lt", "Set.card_le_card", "Set.card_range_of_injective", "Set.chainHeight_image", "Set.codRestrict_restrict", "Set.compl_Iic", "Set.compl_inter_self", "Set.diagonal_eq_univ_iff", "Set.diff_subset_comm", "Set.directedOn_sUnion", "Set.disjoint_compl_right_iff_subset", "Set.disjoint_diagonal_offDiag", "Set.disjoint_of_subset_right", "Set.disjoint_right", "Set.disjoint_sUnion_right", "Set.div_empty", "Set.div_iUnion", "Set.dual_Ioc", "Set.dual_ordConnectedSection", "Set.einfsep_empty", "Set.einfsep_insert_le", "Set.einfsep_pair_le_right", "Set.empty_card", "Set.empty_definable_iff", "Set.encard_eq_coe_toFinset_card", "Set.encard_le_one_iff", "Set.eqOn_refl", "Set.eq_of_mem_uIoc_of_mem_uIoc", "Set.eq_of_nonempty_of_subsingleton", "Set.eq_of_nonempty_of_subsingleton'", "Set.eq_singleton_or_nontrivial", "Set.equitableOn_iff_exists_eq_eq_add_one", "Set.equitableOn_iff_exists_le_le_add_one", "Set.exists_lower_bound_image", "Set.exists_ne_mem_inter_of_not_pairwiseDisjoint", "Set.exists_prod_set", "Set.exists_smaller_set", "Set.exists_subset_encard_eq", "Set.finite_of_ncard_pos", "Set.forall_image2_iff", "Set.iInter_and", "Set.iInter_eq_iInter_finset", "Set.iInter\u2082_mono", "Set.iInter\u2082_subset_of_subset", "Set.iUnionLift_binary", "Set.iUnion_Icc_left", "Set.iUnion_Ico_left", "Set.iUnion_Ioc_left", "Set.iUnion_Ioc_right", "Set.iUnion_diff", "Set.iUnion_eq_compl_iInter_compl", "Set.iUnion_eq_iUnion_finset", "Set.iUnion_eq_univ_iff", "Set.iUnion_iInter_of_antitone", "Set.iUnion_singleton_eq_range", "Set.iUnion_vsub", "Set.iUnion\u2082_eq_univ_iff", "Set.iUnion\u2082_prod_const", "Set.image2_iUnion_right", "Set.image2_iUnion\u2082_left", "Set.image2_inter_left", "Set.image2_inter_subset_left", "Set.image2_inter_union_subset", "Set.image2_singleton_right", "Set.image_add_const_Ioc", "Set.image_const_add_Icc", "Set.image_const_sub_Icc", "Set.image_const_sub_Ico", "Set.image_eta", "Set.image_inter_mulSupport_eq", "Set.image_mulSingle_Ioc_right", "Set.image_neg_Ico", "Set.image_preimage_eq_range_inter", "Set.image_singleton", "Set.image_subset_image_iff", "Set.image_subtype_val_Icc_Ici", "Set.image_subtype_val_Ico_Ioi", "Set.image_subtype_val_Ico_subset", "Set.image_subtype_val_Iio_Iio", "Set.image_subtype_val_Ioo", "Set.image_subtype_val_Ioo_subset", "Set.image_update_Ico_left", "Set.image_update_Ico_right", "Set.image_update_Ioc", "Set.indicator_mul_right", "Set.indicator_smul", "Set.infinite_coe_iff", "Set.infinite_image_iff", "Set.infs_comm", "Set.infs_singleton", "Set.infs_subset_right", "Set.infs_union_left", "Set.infs_union_right", "Set.infsep_pos_iff_nontrivial_of_finite", "Set.insert_nonempty", "Set.insert_subset_insert_iff", "Set.inter_comm", "Set.inter_eq_iInter", "Set.inter_insert_of_not_mem", "Set.inter_left_comm", "Set.inter_nonempty", "Set.inter_setOf_eq_sep", "Set.inter_smul_union_subset_union", "Set.inter_subset_inter_right", "Set.inter_univ", "Set.isUnit_iff", "Set.isWF_iff_no_descending_seq", "Set.ite_same", "Set.kernImage_compl", "Set.le_einfsep_iff", "Set.mapsTo_union", "Set.mapsTo_univ", "Set.maps_range_to", "Set.mem_Icc_of_Ioo", "Set.mem_diff_of_mem", "Set.mem_dite_univ_left", "Set.mem_image2_of_mem", "Set.mem_insert_iff", "Set.mem_of_mem_diff", "Set.mem_uIoc", "Set.mk_preimage_prod", "Set.mk_preimage_prod_left_eq_if", "Set.mk_preimage_prod_right_eq_if", "Set.monotone_powerset", "Set.monotone_projIic", "Set.mulIndicator_apply_eq_one", "Set.mulIndicator_const_preimage_eq_union", "Set.mulIndicator_le_self", "Set.mulIndicator_mul'", "Set.mulIndicator_mul_eq_left", "Set.mul_eq_one_iff", "Set.mul_iUnion\u2082", "Set.mul_singleton", "Set.ncard_diff_add_ncard_of_subset", "Set.ncard_eq_toFinset_card'", "Set.ncard_le_ncard_iff_ncard_diff_le_ncard_diff", "Set.ncard_lt_ncard", "Set.ncard_singleton", "Set.nonempty_iInter\u2082", "Set.nonempty_of_ssubset", "Set.not_mem_Ico_of_ge", "Set.not_mem_Ioi_self", "Set.not_mem_of_not_mem_sUnion", "Set.not_mem_uIoc", "Set.not_nonempty_empty", "Set.not_ordConnected_inter_Icc_iff", "Set.offDiag_empty", "Set.offDiag_union", "Set.one_le_encard_iff_nonempty", "Set.ordConnected_image", "Set.pair_diff_left", "Set.pairwise_disjoint_Ioo_intCast", "Set.pairwise_eq_iff_exists_eq", "Set.pi_nonempty_iff", "Set.pi_univ_Iic", "Set.pi_univ_Ioc_update_right", "Set.piecewise_insert", "Set.piecewise_mem_Icc", "Set.piecewise_univ", "Set.pow_subset_pow_of_one_mem", "Set.powerset_mono", "Set.preimage_add_const_Ico", "Set.preimage_add_const_uIcc", "Set.preimage_compl_eq_image_compl", "Set.preimage_const_add_Ici", "Set.preimage_const_mul_Icc_of_neg", "Set.preimage_const_mul_Ioc_of_neg", "Set.preimage_const_sub_Ico", "Set.preimage_iInter\u2082", "Set.preimage_inl_image_inr", "Set.preimage_mul_const_Iio", "Set.preimage_mul_const_uIcc", "Set.preimage_mul_left_one", "Set.preimage_neg_Ioo", "Set.preimage_preimage", "Set.preimage_range_inter", "Set.preimage_subtype_coe_eq_compl", "Set.preimage_val_sInter_eq_sInter", "Set.prod_eq_biUnion_left", "Set.prod_iInter", "Set.prod_iUnion", "Set.prod_sInter", "Set.prod_subset_prod_iff", "Set.prod_univ", "Set.projIcc_eq_left", "Set.projIcc_left", "Set.range_diag", "Set.range_enumerateCountable_of_mem", "Set.range_ite_subset'", "Set.range_list_getD", "Set.range_piecewise", "Set.range_smul_range", "Set.restrictPreimage_closedEmbedding", "Set.restrictPreimage_inducing", "Set.sInter_image", "Set.sInter_range", "Set.sInter_subset_of_mem", "Set.sUnion_subset", "Set.sep_eq_empty_iff_mem_false", "Set.sep_inter", "Set.sep_of_ncard_eq", "Set.setOf_mem_eq", "Set.sigmaToiUnion_surjective", "Set.sigma_mono", "Set.singleton_subset_iff", "Set.singleton_unop_op", "Set.sized_singleton", "Set.smul_mem_center", "Set.smul_mem_centralizer", "Set.smul_mem_smul_set_iff\u2080", "Set.smul_set_eq_empty", "Set.smul_set_subset_iff", "Set.smul_zero_subset", "Set.ssubset_iff_insert", "Set.sub_mem_Ioc_iff_left", "Set.subset_center_units", "Set.subset_compl_comm", "Set.subset_compl_iff_disjoint_left", "Set.subset_compl_iff_disjoint_right", "Set.subset_eq_empty", "Set.subset_iInter", "Set.subset_image_iff", "Set.subset_insert", "Set.subset_ite", "Set.subset_preimage_image", "Set.subset_range_enumerate", "Set.subset_sUnion_of_subset", "Set.sups_right_comm", "Set.sups_subset_right", "Set.toFinset_eq_univ", "Set.toFinset_smul_set", "Set.top_eq_univ", "Set.uIoc_comm", "Set.union_subset_iff", "Set.univ_mul", "Set.univ_nonempty", "Set.univ_pi_eq_iInter", "Set.univ_union", "Set.update_image", "Set.vsub_mem_vsub", "Set.zero_mem_center", "Set.zero_mem_centralizer", "SetLike.coe_set_eq", "SetLike.list_prod_map_mem_graded", "SetSemiring.up_le_up", "SetTheory.Game.PGame.equiv_iff_game_eq", "SetTheory.PGame.Impartial.equiv_iff_add_equiv_zero'", "SetTheory.PGame.Impartial.forall_rightMoves_fuzzy_iff_equiv_zero", "SetTheory.PGame.Impartial.lf_zero_iff", "SetTheory.PGame.Impartial.nonneg", "SetTheory.PGame.Impartial.nonpos", "SetTheory.PGame.Numeric.lt_moveRight", "SetTheory.PGame.add_le_add_right'", "SetTheory.PGame.add_lf_add_of_le_of_lf", "SetTheory.PGame.add_lf_add_right", "SetTheory.PGame.add_moveLeft_inr", "SetTheory.PGame.bddBelow_range_of_small", "SetTheory.PGame.birthday_one", "SetTheory.PGame.equiv_nim_grundyValue", "SetTheory.PGame.fuzzy_congr_imp", "SetTheory.PGame.fuzzy_of_equiv_of_fuzzy", "SetTheory.PGame.grundyValue_eq_iff_equiv_nim", "SetTheory.PGame.isOption_neg_neg", "SetTheory.PGame.le_or_gf", "SetTheory.PGame.leftMoves_mk", "SetTheory.PGame.lf_of_lf_of_equiv", "SetTheory.PGame.lt_congr_imp", "SetTheory.PGame.lt_def", "SetTheory.PGame.lt_iff_le_and_lf", "SetTheory.PGame.mk_lf_of_le", "SetTheory.PGame.moveRight_neg'", "SetTheory.PGame.moveRight_neg_symm'", "SetTheory.PGame.moveRight_nim_hEq", "SetTheory.PGame.mul_assoc_equiv", "SetTheory.PGame.mul_moveLeft_inr", "SetTheory.PGame.neg_equiv_iff", "SetTheory.PGame.neg_lf_iff", "SetTheory.PGame.quot_mul_assoc", "SetTheory.PGame.quot_sub", "SetTheory.PGame.relabel_moveLeft'", "SetTheory.PGame.short_birthday", "SetTheory.PGame.zero_le_add_left_neg", "SetTheory.PGame.zero_lf_le", "Setoid.eq_iff_rel_eq", "Setoid.eq_top_iff", "Setoid.eqvGen_eq", "Setoid.finite_classes_ker", "Setoid.mapOfSurjective_eq_map", "Setoid.symm", "SheafOfModules.add_val", "Shrink.ext", "ShrinkingLemma.PartialRefinement.find_mem", "ShrinkingLemma.PartialRefinement.mem_find_carrier_iff", "Sigma.Ico_mk_mk", "Sigma.card_Ioc", "Sigma.curry_mul", "Sigma.eq", "Sigma.fst_injective_iff", "Sigma.isConnected_iff", "Sigma.nhds_mk", "Sigma.smul_def", "SignType.coe_mul", "SignType.mul_assoc", "SignType.pos_eq_one", "SimpleGraph.CliqueFree.anti", "SimpleGraph.Coloring.card_colorClasses_le", "SimpleGraph.Coloring.not_adj_of_mem_colorClass", "SimpleGraph.ComponentCompl.hom_refl", "SimpleGraph.ComponentCompl.not_mem_of_mem", "SimpleGraph.ComponentCompl.supp_injective", "SimpleGraph.Connected.map", "SimpleGraph.ConnectedComponent.iso_inv_image_comp_eq_iff_eq_map", "SimpleGraph.ConnectedComponent.sound", "SimpleGraph.Dart.ext", "SimpleGraph.FarFromTriangleFree.cliqueFinset_nonempty", "SimpleGraph.IsClique.map", "SimpleGraph.IsNClique.mono", "SimpleGraph.IsNClique.not_cliqueFree", "SimpleGraph.IsRegularOfDegree.degree_eq", "SimpleGraph.IsSRGWith.top", "SimpleGraph.IsTuranMaximal.isEquipartition", "SimpleGraph.IsTuranMaximal.not_adj_iff_part_eq", "SimpleGraph.Iso.comap_apply", "SimpleGraph.Iso.preconnected_iff", "SimpleGraph.Preconnected.set_univ_walk_nonempty", "SimpleGraph.Reachable.exists_walk_of_dist", "SimpleGraph.Reachable.trans", "SimpleGraph.Subgraph.Adj.of_spanningCoe", "SimpleGraph.Subgraph.coe_hom", "SimpleGraph.Subgraph.connected_iff", "SimpleGraph.Subgraph.deleteEdges_adj", "SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq", "SimpleGraph.Subgraph.deleteVerts_mono", "SimpleGraph.Subgraph.deleteVerts_verts", "SimpleGraph.Subgraph.edgeSet_iSup", "SimpleGraph.Subgraph.edgeSet_top", "SimpleGraph.Subgraph.iInf_adj", "SimpleGraph.Subgraph.loopless", "SimpleGraph.Subgraph.map_monotone", "SimpleGraph.Subgraph.neighborSet_subset", "SimpleGraph.Subgraph.spanningCoe_top", "SimpleGraph.Subgraph.subgraphOfAdj_eq_induce", "SimpleGraph.Subgraph.top_induce_pair_connected_of_adj", "SimpleGraph.TripartiteFromTriangles.Graph.in\u2080\u2082_iff", "SimpleGraph.TripartiteFromTriangles.Graph.in\u2081\u2080_iff", "SimpleGraph.TripartiteFromTriangles.locallyLinear", "SimpleGraph.Walk.IsCycle.ne_bot", "SimpleGraph.Walk.IsPath.cons", "SimpleGraph.Walk.IsPath.of_append_left", "SimpleGraph.Walk.IsPath.tail", "SimpleGraph.Walk.IsTrail.isEulerian_of_forall_mem", "SimpleGraph.Walk.IsTrail.of_append_left", "SimpleGraph.Walk.IsTrail.of_cons", "SimpleGraph.Walk.adj_getVert_succ", "SimpleGraph.Walk.append_assoc", "SimpleGraph.Walk.append_concat", "SimpleGraph.Walk.append_nil", "SimpleGraph.Walk.cons_append", "SimpleGraph.Walk.darts_takeUntil_subset", "SimpleGraph.Walk.edges_append", "SimpleGraph.Walk.end_mem_verts_toSubgraph", "SimpleGraph.Walk.exists_boundary_dart", "SimpleGraph.Walk.isHamiltonianCycle_iff_isCycle_and_support_count_tail_eq_one", "SimpleGraph.Walk.length_bypass_le", "SimpleGraph.Walk.length_edges", "SimpleGraph.Walk.length_eq_zero_iff", "SimpleGraph.Walk.length_support", "SimpleGraph.Walk.mapLe_isPath", "SimpleGraph.Walk.map_fst_darts", "SimpleGraph.Walk.nil_iff_eq_nil", "SimpleGraph.Walk.reverse_transfer", "SimpleGraph.Walk.rotate_edges", "SimpleGraph.Walk.start_mem_support", "SimpleGraph.Walk.toSubgraph_connected", "SimpleGraph.Walk.toSubgraph_le_induce_support", "SimpleGraph.adj_injective", "SimpleGraph.adj_symm", "SimpleGraph.boxProd_neighborFinset", "SimpleGraph.card_edgeFinset_sup_edge", "SimpleGraph.coe_recolorOfEmbedding", "SimpleGraph.colorable_of_fintype", "SimpleGraph.commonNeighbors_top_eq", "SimpleGraph.degMatrix_mulVec_apply", "SimpleGraph.deleteEdges_le", "SimpleGraph.deleteEdges_sdiff_eq_of_le", "SimpleGraph.dotProduct_mulVec_adjMatrix", "SimpleGraph.edgeSet_nonempty", "SimpleGraph.edgeSet_top", "SimpleGraph.edge_adj", "SimpleGraph.exists_girth_eq_length", "SimpleGraph.farFromTriangleFree_of_disjoint_triangles", "SimpleGraph.fromRel_adj", "SimpleGraph.iSup_adj", "SimpleGraph.incMatrix_mul_transpose", "SimpleGraph.incidenceSet_inter_incidenceSet_of_adj", "SimpleGraph.induce_eq_coe_induce_top", "SimpleGraph.induce_sUnion_connected_of_pairwise_not_disjoint", "SimpleGraph.infinite_iff_in_eventualRange", "SimpleGraph.interedges_empty_left", "SimpleGraph.isNClique_bot_iff", "SimpleGraph.isNClique_empty", "SimpleGraph.isNClique_map_iff", "SimpleGraph.isTree_iff_existsUnique_path", "SimpleGraph.isTuranMaximal_of_iso", "SimpleGraph.isUniform_comm", "SimpleGraph.isUniform_singleton", "SimpleGraph.map_map", "SimpleGraph.mem_commonNeighbors", "SimpleGraph.nonuniformWitnesses_spec", "SimpleGraph.not_cliqueFree_card_of_top_embedding", "SimpleGraph.not_mem_commonNeighbors_left", "SimpleGraph.sdiff_compl_neighborFinset_inter_eq", "SimpleGraph.triangleRemovalBound_mul_cube_lt", "SimpleGraph.triangle_removal_aux", "SimpleGraph.turanGraph_cliqueFree", "SimplicialObject.Split.cofan_inj_naturality_symm", "SimplicialObject.Split.congr_f", "SimplicialObject.Splitting.cofan_inj_comp_app", "SlashInvariantForm.add_apply", "SlimCheck.Gen.chooseNatLt_aux", "SmoothAt.coordChangeL", "SmoothAt.div", "SmoothAt.snd", "SmoothBumpCovering.isSubordinate_toBumpCovering", "SmoothBumpFunction.eqOn_source", "SmoothBumpFunction.eventuallyEq_one_of_dist_lt", "SmoothBumpFunction.isOpen_support", "SmoothBumpFunction.mem_Icc", "SmoothMap.coe_smul", "SmoothOn.inv\u2080", "SmoothPartitionOfUnity.sum_finsupport_smul_eq_finsum", "SmoothWithinAt.comp'", "Specialization.map_id", "Specialization.ofEquiv_symm", "Specializes.const_smul", "Specializes.fst", "Specializes.joinedIn", "SpectralMap.copy_eq", "SpectralMap.ext", "SpectrumRestricts.algebraMap_image", "SpectrumRestricts.image", "SpectrumRestricts.spectralRadius_eq", "Squarefree.pow_dvd_of_squarefree_of_pow_succ_dvd_mul_left", "StalkSkyscraperPresheafAdjunctionAuxs.fromStalk_to_skyscraper", "StarAlgHom.comp_apply", "StarAlgHom.realContinuousMapOfNNReal_apply_comp_toReal", "StarAlgHom.snd_prod", "StarAlgHom.subtype_comp_codRestrict", "StarAlgebra.self_mem_adjoin_singleton", "StarConvex.add_left", "StarConvex.prod", "StarOrderedRing.mul_le_mul_of_nonneg_left", "StarOrderedRing.nonneg_iff", "StarSubalgebra.isClosed_topologicalClosure", "StarSubalgebra.mem_sup_right", "StarSubalgebra.smul_mem", "Std.Range.numElems_le_iff", "StieltjesFunction.id_leftLim", "StieltjesFunction.measure_Iic", "StieltjesFunction.measure_Ioo", "Stirling.factorial_isEquivalent_stirling", "Stream'.Seq.dropn_tail", "Stream'.Seq.eq_of_bisim", "Stream'.Seq.get?_cons_succ", "Stream'.Seq.join_cons_nil", "Stream'.Seq.map_tail", "Stream'.Seq.tail_nil", "Stream'.WSeq.Equiv.refl", "Stream'.WSeq.LiftRel.refl", "Stream'.WSeq.cons_congr", "Stream'.WSeq.destruct_map", "Stream'.WSeq.exists_of_liftRel_right", "Stream'.WSeq.get?_terminates_le", "Stream'.WSeq.head_some_of_head_tail_some", "Stream'.WSeq.join_cons", "Stream'.WSeq.liftRel_dropn_destruct", "Stream'.WSeq.map_id", "Stream'.WSeq.map_nil", "Stream'.WSeq.mem_of_mem_tail", "Stream'.WSeq.not_mem_nil", "Stream'.WSeq.productive_congr", "Stream'.WSeq.tail_think", "Stream'.WSeq.think_congr", "Stream'.WSeq.toList_cons", "Stream'.WSeq.toList_nil", "Stream'.get_const", "Stream'.get_map", "Stream'.get_succ_iterate'", "Stream'.get_zip", "Stream'.head_drop", "Stream'.iterate_eq", "Stream'.mem_const", "Stream'.tails_eq_iterate", "Stream'.take_succ", "StrictAnti.injective", "StrictAnti.ite", "StrictAntiOn.mapsTo_Iio", "StrictConcaveOn.add_const", "StrictConcaveOn.eq_of_map_sum_eq", "StrictConcaveOn.strictAntiOn_derivWithin", "StrictConvex.add", "StrictConvex.add_smul_mem", "StrictConvexOn.add_const", "StrictConvexOn.congr", "StrictConvexOn.secant_strict_mono", "StrictConvexOn.slope_lt_of_hasDerivWithinAt_Iio", "StrictMono.embedding_of_ordConnected", "StrictMono.image_Ico_subset", "StrictMono.image_Iio_subset", "StrictMono.induced_topology_eq_preorder", "StrictMono.sign_comp", "StrictMono.strictAnti_iterate_of_map_lt", "StrictMonoOn.comp_strictAntiOn", "StrictMonoOn.mono", "String.Iterator.Valid.next", "String.Iterator.Valid.prevn", "String.Iterator.ValidFor.next", "String.Iterator.ValidFor.out'", "String.Iterator.ValidFor.setCurr'", "String.Iterator.forward_eq_nextn", "String.Pos.ext_iff", "String.Pos.zero_ne_addChar", "String.data_drop", "String.extract.go\u2081_append_right", "String.extract_cons_addChar", "String.foldlAux_of_valid", "String.leftpad_length", "String.offsetOfPos_of_valid", "String.utf8Len_le_of_infix", "String.utf8PrevAux_of_valid", "String.valid_toSubstring", "StronglyMeasurableAtFilter.eventually", "StructureGroupoid.LocalInvariantProp.liftPropAt_congr_iff_of_eventuallyEq", "StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_source", "StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_target", "StructureGroupoid.compatible_of_mem_maximalAtlas", "SubMulAction.image_inclusion", "Subadditive.apply_mul_add_le", "Subadditive.tendsto_lim", "Subalgebra.add_mem", "Subalgebra.coe_pointwise_smul", "Subalgebra.fg_iff_finiteType", "Subalgebra.fg_of_fg_map", "Subalgebra.fg_top", "Subalgebra.finiteDimensional_sup", "Subalgebra.finiteDimensional_toSubmodule", "Subalgebra.iSupLift_mk", "Subalgebra.inclusion_self", "Subalgebra.mem_center_iff", "Subalgebra.toSubsemiring_subtype", "Subalgebra.topologicalClosure_star_comm", "Subalgebra.unop_op", "Subalgebra.unop_sSup", "Subbimodule.smul_mem'", "Subfield.closure_preimage_le", "Subfield.coe_div", "Subfield.coe_map", "Subfield.coe_toSubring", "Subfield.coe_top", "Subfield.comap_comap", "Subfield.sInf_toSubring", "Subfield.sub_mem", "SubfieldClass.ofScientific_mem", "Subgroup.IsComplement'.index_eq_card", "Subgroup.IsComplement'.symm", "Subgroup.IsComplement.equiv_fst_eq_mul_inv", "Subgroup.IsComplement.existsUnique", "Subgroup.IsComplement.rightCosetEquivalence_equiv_snd", "Subgroup.MemLeftTransversals.inv_mul_toFun_mem", "Subgroup.Normal.map", "Subgroup.SchurZassenhausInduction.step1", "Subgroup.SchurZassenhausInduction.step2", "Subgroup.bot_subgroupOf", "Subgroup.centralizer_le", "Subgroup.closure_eq_bot_iff", "Subgroup.closure_induction''", "Subgroup.coeSubtype", "Subgroup.coe_iInf", "Subgroup.coe_inclusion", "Subgroup.coe_norm", "Subgroup.coe_one", "Subgroup.comap_equiv_eq_map_symm", "Subgroup.commutator_commutator_eq_bot_of_rotate", "Subgroup.commute_of_normal_of_disjoint", "Subgroup.div_mem", "Subgroup.eq_one_of_smul_eq_one", "Subgroup.exists_right_transversal", "Subgroup.iSup_induction", "Subgroup.independent_of_coprime_order", "Subgroup.index_comap", "Subgroup.index_iInf_le", "Subgroup.inf_subgroupOf_inf_normal_of_right", "Subgroup.inv_subset_closure", "Subgroup.isCoatom_comap_of_surjective", "Subgroup.isOpen_of_openSubgroup", "Subgroup.list_prod_mem", "Subgroup.map_inf_eq", "Subgroup.map_injective", "Subgroup.mem_carrier", "Subgroup.mem_leftTransversals_iff_existsUnique_quotient_mk''_eq", "Subgroup.mem_sSup_of_mem", "Subgroup.mem_subgroupOf", "Subgroup.mem_units_iff_val_mem", "Subgroup.mul_injective_of_disjoint", "Subgroup.ne_bot_iff_exists_ne_one", "Subgroup.nontrivial_iff_exists_ne_one", "Subgroup.nontrivial_iff_ne_bot", "Subgroup.pi_le_iff", "Subgroup.prod_le_iff", "Subgroup.properlyDiscontinuousSMul_opposite_of_tendsto_cofinite", "Subgroup.relindex_bot_left_eq_card", "Subgroup.relindex_iInf_le", "Subgroup.smul_bot", "Subgroup.smul_mem_pointwise_smul_iff\u2080", "Subgroup.topologicalClosure_minimal", "Subgroup.unit_mem_of_mem_ofUnits", "Subgroup.unop_toSubmonoid", "Subgroup.val_multiset_prod", "Sublattice.coe_eq_empty", "Sublattice.coe_inf'", "Sublattice.coe_subtype", "Sublattice.coe_top", "Sublattice.comap_id", "Sublattice.comap_inf", "Sublattice.mem_sInf", "Sublattice.subsingleton_iff", "Submodule.FG.stabilizes_of_iSup_eq", "Submodule.IsOrtho.map", "Submodule.closedComplemented_bot", "Submodule.closure_induction", "Submodule.coe_copy", "Submodule.coe_neg", "Submodule.coe_subtypeL'", "Submodule.coe_toSubMulAction", "Submodule.colon_top", "Submodule.comap_equiv_eq_map_symm", "Submodule.comap_fst", "Submodule.comap_surjective_of_injective", "Submodule.disjoint_def'", "Submodule.disjoint_span_singleton'", "Submodule.dualCopairing_apply", "Submodule.dualRestrict_def", "Submodule.exists_le_ker_of_lt_top", "Submodule.fg_iff_finiteDimensional", "Submodule.fg_of_fg_map_of_fg_inf_ker", "Submodule.finite_span_isCompactElement", "Submodule.finrank_add_finrank_orthogonal", "Submodule.inclusion_apply", "Submodule.inf_iSup_genEigenspace", "Submodule.isClosed_or_dense_of_isCoatom", "Submodule.isOrtho_iff_inner_eq", "Submodule.isOrtho_sup_right", "Submodule.isTorsion'_iff_torsion'_eq_top", "Submodule.lTensorOne_one_tmul", "Submodule.le_comap_mkQ", "Submodule.le_one_toAddSubmonoid", "Submodule.map_div", "Submodule.map_mkQ_eq_top", "Submodule.map_op_mul", "Submodule.map_span_le", "Submodule.map_strictMono_of_injective", "Submodule.map_subtype_le", "Submodule.map_subtype_range_inclusion", "Submodule.map_subtype_top", "Submodule.map_zero", "Submodule.map\u2082_bot_right", "Submodule.map\u2082_sup_right", "Submodule.mem_bot", "Submodule.mem_dualCoannihilator", "Submodule.mem_one", "Submodule.mem_orthogonal'", "Submodule.mem_orthogonal_singleton_iff_inner_left", "Submodule.mem_sSup_iff_exists_finset", "Submodule.mem_sSup_of_directed", "Submodule.mk_quotientEquivOfIsCompl_apply", "Submodule.mulMap_one_left_eq", "Submodule.mul_bot", "Submodule.mul_induction_on", "Submodule.mul_mem_mul", "Submodule.mul_mem_mul_rev", "Submodule.nontrivial_span_singleton", "Submodule.one_eq_range", "Submodule.orthogonal_orthogonal_eq_closure", "Submodule.prodComm_trans_prodEquivOfIsCompl", "Submodule.prodEquivOfIsCompl_symm_apply_snd_eq_zero", "Submodule.quotientEquivOfIsCompl_apply_mk_coe", "Submodule.range_liftQ", "Submodule.range_map_nonempty", "Submodule.range_snd", "Submodule.restrictScalars_mem", "Submodule.restrictScalars_top", "Submodule.sInf_orthogonal", "Submodule.smul_bot", "Submodule.smul_mem_pointwise_smul", "Submodule.smul_mem_smul", "Submodule.smul_mono", "Submodule.smul_span", "Submodule.span_attach_biUnion", "Submodule.span_eq_span", "Submodule.span_insert_zero", "Submodule.span_mono", "Submodule.span_smul_le", "Submodule.span_span", "Submodule.submodule_eq_sSup_le_nonzero_spans", "Submodule.subtype_comp_inclusion", "Submodule.sup_eq_sup_smul_of_le_smul_of_le_jacobson", "Submodule.sup_orthogonal_of_completeSpace", "Submodule.toConvexCone_le_iff", "Submodule.toNonUnitalSubalgebra_mk", "Submodule.toSubMulAction_strictMono", "Submodule.toSubalgebra_mk", "Submodule.top_eq_ofList_cons_smul_iff", "Submodule.topologicalClosure_map", "Submodule.torsion_int", "Submodule.wcovBy_span_singleton_sup", "SubmodulesBasis.nonarchimedean", "Submonoid.LocalizationMap.lift_injective_iff", "Submonoid.LocalizationMap.lift_mk'", "Submonoid.LocalizationMap.lift_spec_mul", "Submonoid.LocalizationMap.map_map", "Submonoid.LocalizationMap.mk'_eq_iff_eq'", "Submonoid.LocalizationMap.mulEquivOfLocalizations_apply", "Submonoid.LocalizationMap.mulEquivOfMulEquiv_eq", "Submonoid.LocalizationMap.sec_spec", "Submonoid.LocalizationMap.surj\u2082", "Submonoid.LocalizationWithZeroMap.leftCancelMulZero_of_le_isLeftRegular", "Submonoid.closure_induction'", "Submonoid.coe_center", "Submonoid.coe_subtype", "Submonoid.coe_topologicalClosure", "Submonoid.comap_injective_of_surjective", "Submonoid.comap_strictMono_of_surjective", "Submonoid.dense_induction", "Submonoid.inv_mem_units", "Submonoid.le_op_iff", "Submonoid.mem_units_of_val_mem_inv_val_mem", "Submonoid.mrange_inr", "Submonoid.mul_leftInvEquiv", "Submonoid.multiset_prod_mem", "Submonoid.nontrivial_iff", "Submonoid.op_top", "Submonoid.sup_eq_range", "Submonoid.top_closure_mul_self_eq", "Submonoid.units_iInf\u2082", "SubmonoidClass.finsupp_prod_mem", "Subring.center_le_centralizer", "Subring.closure_eq", "Subring.coe_add", "Subring.coe_comap", "Subring.list_prod_mem", "Subring.mem_center_iff", "Subring.mem_inf", "Subring.pointwise_smul_toAddSubgroup", "Subring.pow_mem", "Subring.prod_mono", "Subring.range_subtype", "Subring.toAddSubgroup_strictMono", "Subring.unop_inf", "Subsemigroup.centralizer_univ", "Subsemigroup.coe_set_mk", "Subsemigroup.comap_id", "Subsemigroup.comap_inf", "Subsemigroup.iSup_eq_closure", "Subsemigroup.map_equiv_top", "Subsemigroup.map_inf_comap_of_surjective", "Subsemigroup.map_map", "Subsemigroup.mem_iSup", "Subsemigroup.mem_map", "Subsemigroup.mem_map_of_mem", "Subsemigroup.mk_le_mk", "Subsemiring.add_mem", "Subsemiring.closure_empty", "Subsemiring.coe_pointwise_smul", "Subsemiring.coe_toSubmonoid", "Subsemiring.gc_map_comap", "Subsemiring.le_topologicalClosure", "Subsemiring.op_sSup", "Subsemiring.op_unop", "Subsemiring.pointwise_smul_subset_iff", "Subsemiring.rangeS_subtype", "Subsingleton.elim", "Subsingleton.measurableSet", "Subsingleton.monotone", "Subsingleton.set_cases", "Subspace.dualAnnihilator_inj", "Subspace.quotAnnihilatorEquiv_apply", "Substring.Valid.extract", "Substring.ValidFor.nextn_stop", "Substring.ValidFor.prevn", "Subtype.coe_le_coe", "Subtype.coe_ne_coe", "Subtype.exists_set_subtype", "Subtype.ext_val", "Subtype.image_preimage_coe", "Subtype.mk_le_mk", "Subtype.val_injective", "Sum.Ioo_inr_inl", "Sum.Lex.Ioo_inr_inr", "Sum.Lex.inl_bot", "Sum.Lex.inl_strictMono", "Sum.Lex.toLex_lt_toLex", "Sum.elim_one_one", "Sum.eq_left_getLeft_of_isLeft", "Sum.eq_right_getRight_of_isRight", "Sum.inl_strictMono", "Sum.isLeft_inl", "Sum.map_bijective", "Summable.of_norm_bounded", "SupBotHom.coe_toBotHom", "SupBotHom.dual_id", "SupBotHom.id_comp", "SupHom.symm_dual_id", "Surreal.Multiplication.P3_comm", "Surreal.Multiplication.P4_neg_left", "Surreal.Multiplication.ih1", "Surreal.Multiplication.numeric_option_mul_option", "Surreal.dyadicMap_apply", "Surreal.nsmul_pow_two_powHalf'", "Sylow.conj_eq_normalizer_conj_of_mem_centralizer", "Sylow.pointwise_smul_def", "Sylow.pow_dvd_card_of_pow_dvd_card", "Sym.coe_attach", "Sym.cons_swap", "Sym.ext", "Sym.mem_map", "Sym2.IsDiag.mem_range_diag", "Sym2.card_image_offDiag", "Sym2.congr_left", "Sym2.forall", "Sym2.rel_iff", "SymAlg.unsym_neg", "SymOptionSuccEquiv.decode_inr", "SymmetricRel.inter", "SymplecticGroup.transpose_mem", "SzemerediRegularity.edgeDensity_chunk_not_uniform", "SzemerediRegularity.eps_le_card_star_div", "SzemerediRegularity.m_add_one_div_m_le_one_add", "SzemerediRegularity.m_le_card_of_mem_chunk_parts", "SzemerediRegularity.one_sub_eps_mul_card_nonuniformWitness_le_card_star", "T2Space.isSeparatedMap", "T_pos", "Tactic.NormNum.int_not_isCoprime_helper", "Tactic.NormNum.isInt_isCoprime", "Tactic.NormNum.nat_gcd_helper_2'", "Tactic.ReduceModChar.CharP.intCast_eq_mod", "TangentBundle.coe_chartAt_symm_fst", "TangentBundle.symmL_model_space", "TangentBundle.tangentMap_tangentBundle_pure", "TangentBundle.trivializationAt_symmL", "TendstoLocallyUniformly.tendstoLocallyUniformlyOn", "TendstoLocallyUniformlyOn.congr", "TendstoLocallyUniformlyOn.tendsto_at", "TendstoUniformly.mul", "TendstoUniformly.tendsto_comp", "TensorAlgebra.\u03b9_inj", "TensorPower.algebraMap\u2080_mul_algebraMap\u2080", "TensorPower.gMul_def", "TensorProduct.AlgebraTensorModule.congr_one", "TensorProduct.AlgebraTensorModule.dualDistrib_apply", "TensorProduct.LieModule.liftLie_apply", "TensorProduct.directSum_lof_tmul_lof", "TensorProduct.directSum_symm_lof_tmul", "TensorProduct.exists_finite_submodule_of_finite", "TensorProduct.ext'", "TensorProduct.ext_threefold", "TensorProduct.finsuppScalarLeft_apply_tmul_apply", "TensorProduct.forall_vanishesTrivially_iff_forall_FG_rTensor_injective", "TensorProduct.lift.tmul", "TensorProduct.lift_comp_map", "TensorProduct.map_ker", "TensorProduct.map_map_assoc_symm", "TensorProduct.map\u2082_eq_range_lift_comp_mapIncl", "TensorProduct.prodLeft_tmul", "TensorProduct.span_tmul_eq_top", "TensorProduct.sum_tmul_eq_zero_of_vanishesTrivially", "TensorProduct.tmul_of_gradedMul_of_tmul", "TensorProduct.toDirectLimit_tmul_of", "TensorProduct.vanishesTrivially_iff_sum_tmul_eq_zero", "ThreeGPFree.smul_set", "ThreeGPFree.smul_set\u2080", "Thunk.prod_get_snd", "TopCat.GlueData.range_fromOpenSubsetsGlue", "TopCat.GlueData.rel_equiv", "TopCat.PrelocalPredicate.sheafifyOf", "TopCat.Presheaf.covering_presieve_eq_self", "TopCat.Presheaf.ext", "TopCat.Presheaf.locally_surjective_iff_surjective_on_stalks", "TopCat.Presheaf.toType_isSheaf", "TopCat.epi_iff_surjective", "TopCat.induced_of_isLimit", "TopCat.isSheaf_of_isLimit", "TopCat.prodIsoProd_inv_fst", "TopCat.pullbackIsoProdSubtype_hom_apply", "TopCat.pullbackIsoProdSubtype_inv_fst_apply", "TopCat.pullbackIsoProdSubtype_inv_snd_apply", "TopCat.pullbackSnd_apply", "TopCat.pullback_map_openEmbedding_of_open_embeddings", "TopCommRingCat.coe_of", "TopHom.cancel_left", "TopHom.symm_dual_id", "TopologicalGroup.tendstoLocallyUniformlyOn_iff", "TopologicalSemiring.toTopologicalRing", "TopologicalSpace.Clopens.coe_bot", "TopologicalSpace.Closeds.coe_finset_inf", "TopologicalSpace.Closeds.compl_compl", "TopologicalSpace.Closeds.mem_singleton", "TopologicalSpace.IsTopologicalBasis.inseparable_iff", "TopologicalSpace.IsTopologicalBasis.isOpen_iff", "TopologicalSpace.IsTopologicalBasis.subset_of_forall_subset", "TopologicalSpace.NoetherianSpace.exists_open_ne_empty_le_irreducibleComponent", "TopologicalSpace.OpenNhds.map_id_obj_unop", "TopologicalSpace.Opens.CompleteCopy.dist_val_le_dist", "TopologicalSpace.Opens.coe_sup", "TopologicalSpace.Opens.gc", "TopologicalSpace.Opens.isBasis_iff_cover", "TopologicalSpace.Opens.map_top", "TopologicalSpace.Opens.not_nonempty_iff_eq_bot", "TopologicalSpace.Opens.openEmbedding_obj_top", "TopologicalSpace.PositiveCompacts.carrier_eq_coe", "TopologicalSpace.PositiveCompacts.coe_mk", "TopologicalSpace.PositiveCompacts.coe_prod", "TopologicalSpace.PositiveCompacts.map_comp", "TopologicalSpace.ext_iff_isClosed", "TopologicalSpace.isOpen_top_iff", "TopologicalSpace.le_def", "TopologicalSpace.nhds_generateFrom", "Topology.IsLower.topology_eq", "Topology.IsLowerSet.isClosed_iff_isUpper", "Topology.IsScott.closure_singleton", "Topology.WithLawson.ofLawson_toLawson", "Topology.WithLower.toLower_ofLower", "Topology.WithLowerSet.map_id", "Topology.WithLowerSet.ofLowerSet_toLowerSet", "Topology.WithLowerSet.toLowerSet_ofLowerSet", "Topology.WithUpperSet.toUpperSet_ofUpperSet", "Topology.WithUpperSet.toUpperSet_specializes_toUpperSet", "Topology.scottHausdorff_le_lawson", "Topology.scottHausdorff_le_scott", "Traversable.comp_sequence", "Traversable.foldlm_toList", "Traversable.foldr.ofFreeMonoid_comp_of", "Traversable.foldrm.ofFreeMonoid_comp_of", "Traversable.naturality'", "Traversable.traverse_map'", "Tree.height_le_numNodes", "TrivSqZeroExt.continuous_snd", "TrivSqZeroExt.exp_def_of_smul_comm", "TrivSqZeroExt.fst_inr", "TrivSqZeroExt.inl_pow", "TrivSqZeroExt.lift_comp_inrHom", "TrivSqZeroExt.mul_inl_eq_op_smul", "TrivSqZeroExt.mul_inv_cancel", "TrivSqZeroExt.mul_inv_rev", "TrivSqZeroExt.nhds_def", "TrivSqZeroExt.snd_mk", "TrivSqZeroExt.snd_pow_of_smul_comm", "Trivialization.continuousAt_of_comp_left", "Trivialization.continuousLinearEquivAt_prod", "Trivialization.linear", "Trivialization.linearMapAt_def_of_mem", "Trivialization.mem_target", "Trivialization.mk_symm", "Trivialization.preimageHomeomorph_apply", "Trivialization.prod_apply", "Trivialization.smoothOn_iff", "Tropical.add_eq_zero_iff", "Tropical.rightInverse_trop", "Tropical.tropOrderIso_coe_fn", "Trunc.finChoice_eq", "Trunc.ind", "TruncatedWittVector.card", "TruncatedWittVector.out_injective", "Tuple.antitone_pair_of_not_sorted", "Tuple.eq_sort_iff'", "Tuple.graph.card", "Tuple.graphEquiv\u2082_apply", "Tuple.lt_card_ge_iff_apply_ge_of_antitone", "Tuple.unique_monotone", "Turing.BlankRel.equivalence", "Turing.ListBlank.exists_cons", "Turing.PartrecToTM2.K'.elim_rev", "Turing.PartrecToTM2.K'.elim_update_stack", "Turing.PartrecToTM2.copy_ok", "Turing.PartrecToTM2.trNormal_respects", "Turing.PartrecToTM2.trStmts\u2081_trans", "Turing.PartrecToTM2.tr_pred", "Turing.PartrecToTM2.tr_ret_halt", "Turing.Reaches\u2080.single", "Turing.Reaches\u2080.tail'", "Turing.Reaches\u2081.to\u2080", "Turing.TM1to1.supportsStmt_write", "Turing.TM1to1.tr_supports", "Turing.TM2to1.addBottom_modifyNth", "Turing.TM2to1.addBottom_nth_snd", "Turing.Tape.map_mk\u2081", "Turing.Tape.mk'_nth_nat", "Turing.ToPartrec.Code.head_eval", "Turing.proj_map_nth", "Turing.reaches_total", "Turing.tr_eval_rev", "Turing.tr_reaches_rev", "TwoPointing.nonempty_two_pointing_iff", "TwoPointing.sum_snd", "TwoUniqueProds.of_injective_mulHom", "TwoUniqueProds.of_mulOpposite", "TypeVec.comp_id", "TypeVec.dropFun_from_append1_drop_last", "TypeVec.lastFun_from_append1_drop_last", "TypeVec.prod_snd_mk", "UInt32.toUInt16_toNat", "UInt32.val_ofNat", "UInt64.toNat_add", "UInt8.le_antisymm", "ULift.closedEmbedding_down", "ULift.edist_up_up", "ULift.nnnorm_up", "ULift.one_down", "ULift.up_sup", "ULift.up_surjective", "ULiftable.down_up", "USize.toNat_div", "UV.disjoint_of_mem_compression_of_not_mem", "Ultrafilter.comap_inf_principal_neBot_of_image_mem", "Ultrafilter.continuous_mul_left", "Ultrafilter.ext", "Ultrafilter.frequently_iff_eventually", "Ultrafilter.ofComapInfPrincipal_eq_of_map", "UniformCauchySeqOn.prod_map", "UniformConvergenceCLM.topologicalSpace_mono", "UniformEquiv.apply_symm_apply", "UniformEquiv.coe_prodComm", "UniformFun.gc", "UniformFun.iInf_eq", "UniformFun.isClosed_setOf_continuous", "UniformInducing.denseInducing", "UniformInducing.equicontinuousWithinAt_iff", "UniformInducing.uniformContinuous", "UniformOnFun.isCountablyGenerated_uniformity", "UniformOnFun.toFun_div", "UniformOnFun.toFun_smul", "UniformOnFun.uniformSpace_eq_iInf_precomp_of_cover", "UniformOnFun.uniformSpace_eq_inf_precomp_of_cover", "UniformSpace.Completion.Continuous.mul", "UniformSpace.Completion.coe_isometry", "UniformSpace.Completion.coe_one", "UniformSpace.Completion.comap_coe_eq_uniformity", "UniformSpace.Completion.extension_coe", "UniformSpace.comap_inf", "UniformSpace.compactSpace_iff_seqCompactSpace", "UniformSpace.hasBasis_nhds_prod", "UniformSpace.mem_closure_iff_ball", "UniformSpace.mem_closure_iff_symm_ball", "UniformSpace.metrizable_uniformity", "UniformSpace.replaceTopology_eq", "UniformSpace.toTopologicalSpace_iInf", "UnionFind.model'", "Unique.eq_default", "UniqueDiffWithinAt.univ_pi", "UniqueFactorizationMonoid.associated_iff_normalizedFactors_eq_normalizedFactors", "UniqueFactorizationMonoid.count_normalizedFactors_eq'", "UniqueFactorizationMonoid.factors_one", "UniqueFactorizationMonoid.normalizedFactors_of_irreducible_pow", "UniqueFactorizationMonoid.normalizedFactors_zero", "UniqueMDiffOn.tangentBundle_proj_preimage", "UniqueMul.mulHom_preimage", "UniqueProds.toTwoUniqueProds_of_group", "UnitalShelf.act_self_act_eq", "UnitalShelf.assoc", "Unitization.antilipschitzWith_addEquiv", "Unitization.fst_inl", "Unitization.inl_smul", "Unitization.inr_star", "Unitization.nnnorm_eq_sup", "Unitization.starLift_range_le", "Units.add_divp", "Units.coe_map'", "Units.continuousOn_zpow\u2080_spectrum", "Units.divp_add", "Units.inv_eq_of_mul_eq_one_left", "Units.inv_eq_of_mul_eq_one_right", "Units.mem_posSubgroup", "Units.mul_left_dvd", "Units.mul_right_inj", "Units.ne_zero", "Units.nnnorm_pos", "UpperHalfPlane.ModularGroup.SL_neg_smul", "UpperHalfPlane.dist_coe_le", "UpperHalfPlane.im_smul", "UpperHalfPlane.modular_T_zpow_smul", "UpperHalfPlane.verticalStrip_anti_right", "UpperHalfPlane.zero_form_isBoundedAtImInfty", "UpperSemicontinuous.add'", "UpperSemicontinuousOn.add'", "UpperSemicontinuousOn.mono", "UpperSet.coe_compl", "UpperSet.coe_iSup\u2082", "UpperSet.coe_map", "UpperSet.compl_iInf", "UpperSet.compl_iSup", "UpperSet.compl_sInf", "UpperSet.erase_inf_Ici", "UpperSet.lt_erase", "UpperSet.mem_iSup_iff", "UpperSet.prod_le_prod_iff", "UpperSet.prod_sup", "Urysohns.CU.continuous_lim", "Urysohns.CU.lim_eq_midpoint", "Valuation.Integers.le_of_dvd", "Valuation.IsEquiv.trans", "Valuation.isEquiv_iff_valuationSubring", "Valuation.isEquiv_of_val_le_one", "Valuation.isEquiv_valuation_valuationSubring", "Valuation.map_add'", "Valuation.map_neg", "Valuation.map_one", "Valuation.mem_valuationSubring_iff", "Valuation.onQuot_comap_eq", "ValuationRing.iff_isInteger_or_isInteger", "ValuationRing.tFAE", "ValuationRing.unique_irreducible", "ValuationSubring.mapOfLE_valuation_apply", "ValuationSubring.ofPrime_idealOfLE", "ValuationSubring.pointwise_smul_le_pointwise_smul_iff", "ValuationSubring.valuation_surjective", "Vector.get_cons_nil", "Vector.head_ofFn", "Vector.inductionOn_cons", "Vector.mapAccumr_eq_map_of_unused_state", "Vector.mapAccumr_snoc", "Vector.mapAccumr\u2082_comm", "Vector.mapAccumr\u2082_mapAccumr_left", "Vector.mapAccumr\u2082_mapAccumr\u2082_left_right", "Vector.map\u2082_map_right", "Vector.mem_map_succ_iff", "Vector.mem_succ_iff", "Vector.replicate_succ", "Vector.reverse_get_zero", "Vector.scanl_val", "Vector.snoc_cons", "Vector.toList_length", "Vector.toList_map", "Vector.toList_set", "Vector3.append_add", "VectorBundleCore.localTriv_symmL", "VectorFourier.fourierIntegral_continuous", "VectorFourier.fourierIntegral_continuousLinearMap_apply", "VitaliFamily.FineSubfamilyOn.index_countable", "VitaliFamily.FineSubfamilyOn.measure_diff_biUnion", "VitaliFamily.FineSubfamilyOn.measure_le_tsum", "VitaliFamily.ae_eventually_measure_zero_of_singular", "WCovBy.covBy_of_lt", "WType.cardinal_mk_le_of_le", "Wbtw.map", "Wbtw.oangle_sign_eq_of_ne_right", "Wbtw.oangle\u2083\u2081\u2082_eq_zero", "Wbtw.trans_sbtw_right", "Wbtw.wOppSide\u2081\u2083", "WeakDual.CharacterSpace.coe_coe", "WeakDual.eval_continuous", "WeakFEPair.hf_modif_FE", "WeakFEPair.hf_zero", "WeakSpace.coe_map", "WeierstrassCurve.Affine.CoordinateRing.coe_norm_smul_basis", "WeierstrassCurve.Affine.CoordinateRing.smul", "WeierstrassCurve.Affine.CoordinateRing.smul_basis_eq_zero", "WeierstrassCurve.Affine.Point.add_of_Y_eq", "WeierstrassCurve.Affine.Point.add_self_of_Y_eq", "WeierstrassCurve.Affine.Point.toClass_eq_zero", "WeierstrassCurve.Affine.Point.zero_def", "WeierstrassCurve.Affine.Y_eq_of_Y_ne", "WeierstrassCurve.Affine.map_addY", "WeierstrassCurve.Affine.negY_negY", "WeierstrassCurve.Affine.nonsingular_add", "WeierstrassCurve.Affine.polynomial_ne_zero", "WeierstrassCurve.Affine.slope_of_Y_ne_eq_eval", "WeierstrassCurve.Jacobian.Point.add_def", "WeierstrassCurve.Jacobian.Point.toAffineLift_add", "WeierstrassCurve.Jacobian.Point.toAffineLift_zero", "WeierstrassCurve.Jacobian.X_ne_zero_of_Z_eq_zero", "WeierstrassCurve.Jacobian.Y_eq_iff'", "WeierstrassCurve.Jacobian.Y_eq_negY_of_Y_eq", "WeierstrassCurve.Jacobian.addMap_of_Y_eq", "WeierstrassCurve.Jacobian.addY_neg", "WeierstrassCurve.Jacobian.addY_self", "WeierstrassCurve.Jacobian.add_smul_of_equiv", "WeierstrassCurve.Jacobian.dblXYZ_of_Z_eq_zero", "WeierstrassCurve.Jacobian.dblX_of_Z_eq_zero", "WeierstrassCurve.Jacobian.equiv_some_of_Z_ne_zero", "WeierstrassCurve.Jacobian.isUnit_dblU_of_Y_eq", "WeierstrassCurve.Jacobian.map_negDblY", "WeierstrassCurve.Jacobian.map_polynomial", "WeierstrassCurve.Jacobian.negDblY_smul", "WeierstrassCurve.Jacobian.nonsingularLift_addMap", "WeierstrassCurve.Jacobian.nonsingularLift_iff", "WeierstrassCurve.Jacobian.nonsingularLift_zero", "WeierstrassCurve.Jacobian.nonsingular_iff_of_Y_eq_negY", "WeierstrassCurve.Jacobian.nonsingular_smul", "WeierstrassCurve.Jacobian.nonsingular_some", "WeierstrassCurve.Projective.nonsingularLift_iff", "WeierstrassCurve.Projective.nonsingularLift_zero", "WeierstrassCurve.Projective.nonsingular_iff_affine_of_Z_ne_zero", "WeierstrassCurve.Projective.nonsingular_some", "WeierstrassCurve.baseChange_\u03a8Sq", "WeierstrassCurve.coeff_pre\u03a8'_ne_zero", "WeierstrassCurve.coeff_\u03a6", "WeierstrassCurve.coeff_\u03a6_ne_zero", "WeierstrassCurve.coeff_\u03a8Sq_ne_zero", "WeierstrassCurve.leadingCoeff_\u03a6", "WeierstrassCurve.map_b\u2088", "WeierstrassCurve.map_c\u2086", "WeierstrassCurve.map_injective", "WeierstrassCurve.natDegree_coeff_\u03a8Sq_ofNat", "WeierstrassCurve.natDegree_pre\u03a8\u2084_le", "WeierstrassCurve.natDegree_\u03a8\u2083", "WeierstrassCurve.pre\u03a8'_zero", "WeierstrassCurve.pre\u03a8_neg", "WeierstrassCurve.pre\u03a8_zero", "WeierstrassCurve.variableChange_c\u2084", "WeierstrassCurve.\u03a6_ne_zero", "WeierstrassCurve.\u03a8_even", "WeierstrassCurve.\u03a8_zero", "WeierstrassCurve.\u03c6_one", "WellFounded.exists_well_order_ge", "WellFounded.fixF_eq_fixFC", "WellFounded.lt_succ", "WellFoundedGT.fix_eq", "WithBot.coe_strictMono", "WithBot.mul_eq_bot_iff", "WithBot.ne_bot_iff_exists", "WithBot.ofDual_lt_iff", "WithBot.preimage_coe_Ioc", "WithBot.preimage_coe_Ioo_bot", "WithBot.preimage_coe_bot", "WithBot.unbot'_le_iff", "WithBot.unbot_le_iff", "WithLp.prod_lipschitzWith_equiv", "WithLp.prod_norm_eq_card", "WithLp.prod_norm_eq_of_nat", "WithOne.map_id", "WithSeminorms.T1_of_separating", "WithSeminorms.hasBasis", "WithSeminorms.hasBasis_zero_ball", "WithSeminorms.mem_nhds_iff", "WithTop.Ico_coe_coe", "WithTop.add_le_add_iff_right", "WithTop.coe_mono", "WithTop.iInf_empty", "WithTop.isLUB_sSup", "WithTop.map_le_iff", "WithTop.map_ofDual", "WithTop.mul_lt_top", "WithTop.ofDual_le_iff", "WithTop.one_eq_coe", "WithTop.one_le_coe", "WithTop.prod_lt_top", "WithTop.some_le_some", "WithTop.top_mul", "WithTop.top_mul'", "WithZero.coe_le_one", "WithZero.monoidWithZeroHom_ext", "WithZeroTopology.Iio_mem_nhds", "WithZeroTopology.isOpen_iff", "WittVector.RecursionBase.solution_pow", "WittVector.aeval_verschiebung_poly'", "WittVector.bind\u2081_verschiebungPoly_wittPolynomial", "WittVector.coeff_add_of_disjoint", "WittVector.coeff_mk", "WittVector.coeff_p_pow", "WittVector.coeff_truncate", "WittVector.constantCoeff_wittMul", "WittVector.constantCoeff_wittSub", "WittVector.ext_iff", "WittVector.frobeniusPoly_zmod", "WittVector.frobenius_verschiebung", "WittVector.ghostFun_natCast", "WittVector.ghostFun_sub", "WittVector.init_pow", "WittVector.iterate_frobenius_coeff", "WittVector.iterate_verschiebung_mul", "WittVector.iterate_verschiebung_mul_coeff", "WittVector.mulN_coeff", "WittVector.truncate_lift", "WittVector.verschiebungFun_coeff_succ", "WittVector.verschiebung_shift", "WittVector.wittPolyProdRemainder_vars", "YoungDiagram.cells_bot", "YoungDiagram.mk_mem_col_iff", "ZFSet.empty_isTransitive", "ZFSet.isTransitive_iff_subset_powerset", "ZFSet.map_fval", "ZFSet.regularity", "ZFSet.sInter_empty", "ZFSet.sUnion_empty", "ZFSet.toSet_pair", "ZMod.cast_cast_zmod_of_le", "ZMod.cast_mul", "ZMod.cast_zmod_eq_zero_iff_of_le", "ZMod.eisenstein_lemma_aux\u2081", "ZMod.eq_zero_of_gcd_ne_one", "ZMod.exponent", "ZMod.inv_zero", "ZMod.natCast_self'", "ZMod.ne_zero_of_gcd_eq_one", "ZMod.neg_one_ne_one", "ZMod.nonsquare_iff_jacobiSym_eq_neg_one", "ZMod.pow_card_sub_one", "ZMod.ringHom_eq_of_ker_eq", "ZMod.unitsMap_def", "ZMod.valMinAbs_def_zero", "ZMod.valMinAbs_spec", "ZMod.val_one", "ZMod.\u03c7\u2088'_int_eq_if_mod_eight", "ZNum.abs_to_nat", "ZNum.cast_lt", "ZeroAtInftyContinuousMap.coe_copy", "ZeroAtInftyContinuousMap.copy_eq", "Zlattice.module_finite", "Zlattice.rank", "Zspan.floor_eq_self_of_mem", "Zspan.repr_ceil_apply", "Zspan.span_top", "Zsqrtd.divides_sq_eq_zero", "Zsqrtd.divides_sq_eq_zero_z", "Zsqrtd.intCast_dvd", "Zsqrtd.nonneg_muld", "Zsqrtd.norm_nonneg", "Zsqrtd.ofNat_re", "Zsqrtd.one_re", "Zsqrtd.smuld_val", "abs_eq_self", "absorbs_biUnion_finset", "absorbs_neg_neg", "act_rel_of_rel_of_act_rel", "add_eq_left_iff_le", "add_halves'", "add_tsub_cancel_iff_le", "add_tsub_cancel_of_le", "aeSeq.aeSeq_eq_fun_ae", "ae_eq_const_or_norm_integral_lt_of_norm_le_const", "aemeasurable_deriv", "aemeasurable_indicator_iff\u2080", "aestronglyMeasurable_const_smul_iff\u2080", "aestronglyMeasurable_iUnion_iff", "affineIndependent_iff_not_collinear_set", "affineIndependent_iff_of_fintype", "affineIndependent_of_ne_of_mem_of_mem_of_not_mem", "affineSegment_image", "affineSegment_same", "affineSegment_vsub_const_image", "algebraMap_star_comm", "algebraicIndependent_comp_subtype", "algebraicIndependent_sUnion_of_directed", "alternatingGroup.isThreeCycle_isConj", "analyticAt_const", "analyticAt_mul", "and_congr_left", "and_imp", "antisymmRel_iff_eq", "antisymm_iff", "antisymm_of", "antitoneOn_iff_forall_lt", "antitoneOn_of_deriv_nonpos", "antitone_iff_map_nonpos", "antitone_of_deriv_nonpos", "antivaryOn_inv_left", "antivary_const_left", "antivary_inv_left", "antivary_inv_right", "antivary_inv\u2080", "apply_fdifferential", "approximatesLinearOn_empty", "associatedPrimes.eq_empty_of_subsingleton", "associated_isUnit_mul_left_iff", "associated_mul_isUnit_right_iff", "associated_mul_unit_right_iff", "autEquivZmod_symm_apply_intCast", "aux", "aux1", "ball_add_closedBall", "banach_steinhaus_iSup_nnnorm", "basis_toMatrix_basisFun_mul", "bddAbove_Icc", "bddAbove_prod", "bddBelow_Ici", "bddBelow_Ioo", "bddBelow_bddAbove_iff_subset_Icc", "bddLat_dual_comp_forget_to_bddOrd", "bernoulli'_one", "bernoulli'_spec", "bernoulliFourierCoeff_eq", "bernsteinPolynomial.sum", "biInf_finsetSigma", "biSup_sigma'", "bihimp_assoc", "bihimp_def", "bihimp_iff_iff", "bihimp_inf_sup", "birkhoffAverage_one", "birkhoffSum_one", "birkhoffSum_succ", "bit1_injective", "borel_eq_top_of_countable", "bot_ne_top", "boundariesToCycles'_naturality", "boundedSpace_induced_iff", "bounded_stdSimplex", "btw_rfl", "card_eq_pow_finrank", "card_le_of_surjective'", "cardinal_sInter_mem", "cast_cast", "catalan_three", "cauchySeq_finset_iff_nat_tprod_vanishing", "cauchySeq_of_le_geometric", "cauchySeq_of_le_tendsto_0", "cauchySeq_tendsto_of_complete", "cauchy_iff_exists_le_nhds", "cauchy_map_of_uniformCauchySeqOn_fderiv", "cauchy_product", "ceilDiv_one", "cfc_apply", "cfc_apply_pi", "cfc_comp_polynomial", "cfc_nonneg_iff", "cfc_nonpos_iff", "cfc_smul", "cfc_star_id", "cfc\u2099_apply", "cfc\u2099_comp_star", "charZero_of_inj_zero", "charpoly_leftMulMatrix", "chartAt_self_eq", "ciInf_le_of_le", "ciSup_of_empty", "ciSup_subsingleton", "circleIntegrable_iff", "circleIntegral.integral_sub", "circleMap_zero_radius", "circle_def", "closedBall_one_div_singleton", "closedBall_one_mul_singleton", "closedBall_prod_same", "closedBall_sub_ball", "closedEmbedding_inl", "closedEmbedding_subtype_val", "closedEmbedding_update", "closure_Ioc", "closure_closure", "closure_diff_intrinsicFrontier", "closure_empty_iff", "closure_eq_cluster_pts", "closure_iUnion", "closure_image_closure", "closure_induced", "closure_minimal", "closure_thickening", "cmp_eq_gt_iff", "cmp_ofDual", "cmp_swap", "codisjoint_assoc", "coe_algEquiv_lpBCF", "coe_comp_nnnorm'", "coe_div_coe", "coe_inv_circle", "coe_iterateFrobenius", "coe_nnreal_ennreal_nndist", "collinear_iff_rank_le_one", "collinear_insert_insert_insert_left_of_mem_affineSpan_pair", "commProb_eq_zero_of_infinite", "commutatorElement_def", "commute_lmul_sq_rmul", "comp_le_uniformity", "comp_le_uniformity3", "compactCovering_subset", "compare_le_iff_le", "compl_bot", "compl_compl", "compl_sSup", "compl_sup_compl_le", "compl_surjective", "concaveOn_of_slope_anti_adjacent", "conductor_subset_adjoin", "conformalFactorAt_pos", "congr", "conjugate_nonneg", "conjugate_pos", "connectedComponentIn_eq", "connectedComponentIn_eq_empty", "connectedComponentIn_subset", "connectedComponents_preimage_singleton", "constantCoeff_wittStructureRat_zero", "contDiffAt_const", "contDiffAt_succ_iff_hasFDerivAt", "contDiffGroupoid_prod", "contDiffOn_succ_iff_deriv_of_isOpen", "contDiffOn_top_iff_fderivWithin", "contDiffOn_top_iff_fderiv_of_isOpen", "contDiffWithinAt_fst", "contDiff_euclidean", "contDiff_inner", "contDiff_mul", "contDiff_top", "contDiff_zero_fun", "contMDiffAt_extChartAt'", "contMDiffAt_fst", "contMDiffAt_prod_iff", "contMDiffOn_chart", "contMDiffOn_congr", "contMDiffOn_finset_prod'", "contMDiffOn_snd", "contMDiff_iff", "contMDiff_prod_iff", "continuousAt_const", "continuousAt_cpow", "continuousAt_gauge_zero", "continuousAt_iff_continuous_left_right", "continuousAt_jacobiTheta\u2082", "continuousAt_matrix_inv", "continuousAt_of_locally_lipschitz", "continuousLinearMapToWeakSpace_bijective", "continuousOn_iff_isClosed", "continuousOn_integral_bilinear_of_locally_integrable_of_compact_support", "continuousOn_tangentCoordChange", "continuousWithinAt_const_smul_iff\u2080", "continuousWithinAt_diff_self", "continuous_def", "continuous_iInf_dom", "continuous_iSup_rng", "continuous_iff_ultrafilter", "continuous_inclusion", "continuous_induced_rng", "continuous_isLeft", "continuous_isRight", "continuous_le_rng", "continuous_matrix", "continuous_of_continuousAt_one\u2082", "continuous_of_mulTSupport", "continuous_of_uniform_approx_of_continuous", "continuous_vsub", "convexHull_sum", "convexJoin_right_comm", "convexOn_dist", "convexOn_iff_convex_epigraph", "convexOn_rpow", "convexOn_univ_dist", "convex_halfspace_im_lt", "convex_halfspace_re_lt", "convex_iff_div", "convex_openSegment", "convex_stdSimplex", "countable_sInter_mem", "coverPreserving_opens_map", "cpow_eq_nhds'", "cross_apply", "csInf_Ici", "csInf_Ioc", "csInf_Ioo", "csInf_le", "csSup_mem_of_not_isSuccLimit'", "dNext_eq_zero", "denseRange_pure", "dense_compl_singleton_iff_not_open", "dense_discrete", "dense_iff_inter_open", "dense_pi", "dense_sUnion_interior_of_closed", "derivWithin_ccos", "derivWithin_const", "derivWithin_pi", "derivWithin_pow'", "derivWithin_sum", "derivWithin_zero_of_isolated", "deriv_circleMap", "deriv_circleMap_ne_zero", "deriv_const_add", "deriv_id'", "deriv_mem_iff", "deriv_pow", "deriv_pow'", "deriv_sin", "deriv_smul_const", "deriv_sqrt_mul_log'", "derivedSeries_one", "descPochhammer_map", "descPochhammer_zero", "devosMulRel_of_le_of_le", "diffContOnCl_const", "diff_mem_nhdsWithin_compl", "differentiableAt_comp_neg_iff", "differentiableAt_const", "differentiableAt_fst", "differentiableAt_jacobiTheta\u2082_fst", "differentiableOn_id", "differentiableOn_sub_const_iff", "differentiableWithinAt_dslope_of_ne", "differentiableWithinAt_inverse", "differentiable_const_add_iff", "differentiable_euclidean", "differentialIdeal_le_iff", "directed_id", "discreteTopology_iff_locallyInjective", "discreteTopology_iff_nhds_ne", "discreteTopology_of_isOpen_singleton_one", "disjoint_comm", "disjoint_compl_right_iff", "disjoint_disjointed", "disjoint_lift'_closure_nhds", "disjoint_nhds_atTop_iff", "disjoint_sSup_iff", "disjoint_sup_left", "disjointed_subset", "dist_center_homothety", "dist_dist_dist_le_right", "dist_le_Ico_sum_of_dist_le", "dist_le_range_sum_dist", "dist_le_tsum_dist_of_tendsto\u2080", "dist_le_tsum_of_dist_le_of_tendsto\u2080", "dist_midpoint_midpoint_le", "dist_mono_right_pi", "dist_mul_left", "dist_nndist", "dist_ofMul", "dist_pi_const_le", "dist_vadd_left", "dite_dite_comm", "dite_eq_ite", "dite_prop_iff_and", "div_add_div_same", "div_eq_of_eq_mul'", "div_le_iff_of_neg'", "div_le_of_nonneg_of_le_mul", "div_le_one_iff", "div_left_inj", "div_lt_iff_lt_mul", "div_lt_one", "div_mul_cancel_left", "div_mul_cancel_of_imp", "div_mul_div_comm", "div_ne_one_of_ne", "div_neg_iff", "div_nonneg", "div_sq_cancel", "div_sub_div_same", "dslope_sub_smul", "dualTensorHom_apply", "dvd_gcd_iff", "dvd_iff_exists_eq_mul_left", "dvd_mul_of_dvd_left", "dvd_sub", "dvd_zsmul_of_dvd", "eVariationOn.add_point", "eVariationOn.eq_zero_iff", "edist_le_of_edist_le_geometric_two_of_tendsto", "edist_le_pi_edist", "edist_mulIndicator", "edist_naturalParameterization_eq_zero", "edist_nndist", "edist_ofAdd", "edist_ofDual", "edist_triangle_left", "edist_vsub_vsub_le", "egauge_zero_zero", "embedding_prod_mk", "eq_iff_atom_le_iff", "eq_iff_eq_of_div_eq_div", "eq_iff_eq_of_lt_iff_lt_of_gt_iff_gt", "eq_iff_le_not_lt", "eq_inv_mul_iff_mul_eq", "eq_ite_iff", "eq_max", "eq_of_forall_lt_rat_iff_lt", "eq_of_has_deriv_right_eq", "eq_of_heq", "eq_of_le_of_not_lt", "eq_of_mul_inv_eq_one", "eq_of_nndist_eq_zero", "eq_one_div_of_mul_eq_one_left", "eq_one_or_one_lt", "eq_or_lt_of_le", "eq_orthogonalProjection_of_eq_submodule", "eq_singleton_bot_of_sSup_eq_bot_of_nonempty", "eq_star_iff_eq_star", "eq_zero_or_locallyCompactSpace_of_support_subset_isCompact_of_group", "equicontinuousWithinAt_finite", "even_bit0", "eventually_homothety_mem_of_mem_interior", "eventually_le_of_tendsto_lt", "existsUnique_sub_zsmul_mem_Ico", "exists_Ioc_subset_of_mem_nhds", "exists_borelSpace_of_countablyGenerated_of_separatesPoints", "exists_continuous_zero_one_of_isClosed", "exists_embedding_euclidean_of_compact", "exists_eq_ciInf_of_not_isPredLimit", "exists_eq_left", "exists_finset_piecewise_mem_of_mem_nhds", "exists_forall_closed_ball_dist_add_le_two_mul_sub", "exists_linearIndependent_of_le_rank", "exists_linearIndependent_of_lt_rank", "exists_mem_nhds_isCompact_isClosed", "exists_monotone_Icc_subset_open_cover_Icc", "exists_nhds_split_inv", "exists_nontrivial_relation_sum_zero_of_not_affine_ind", "exists_of_schnirelmannDensity_eq_zero", "exists_partition_approximatesLinearOn_of_hasFDerivWithinAt", "exists_prop_congr", "exists_smooth_tsupport_subset", "exists_subset_iUnion_closure_subset", "exists_sum_eq_one_iff_pairwise_coprime'", "expChar_of_injective_algebraMap", "exp_neg_mul_rpow_isLittleO_exp_neg", "extChartAt_image_nhd_mem_nhds_of_boundaryless", "extChartAt_target", "extentClosure_iUnion", "extremePoints_convexHull_subset", "extremePoints_empty", "extremePoints_prod", "false_and_iff", "false_or", "fderiv.comp", "fderiv.comp_deriv", "fderivWithin.comp", "fderivWithin_const_apply", "fderivWithin_eq_fderiv", "fderivWithin_eventually_congr_set", "fderivWithin_fderivWithin", "fderivWithin_fst", "fderivWithin_inter", "fderivWithin_mul_const", "fderivWithin_of_mem", "fderiv_mem_iff", "fderiv_sinh", "fdifferential_comp", "fermatLastTheoremFor_zero", "filter_collapse_eq", "finAddFlip_apply_natAdd", "finCongr_symm_apply_coe", "finEquivPowers_apply", "finEquivZPowers_apply", "finSuccEquiv'_symm_none", "finite_ofLocalizationSpan", "finite_zpowers", "finprod_curry\u2083", "finprod_div_distrib", "finprod_emb_domain'", "finprod_eq_dif", "finprod_mem_def", "finprod_mem_insert", "finprod_mem_insert'", "finprod_mem_singleton", "finprod_mem_univ", "finprod_of_infinite_mulSupport", "finprod_true", "finrank_eq_one_iff'", "finrank_quotient_span_eq_natDegree_norm", "finrank_span_eq_card", "finsuppTensorFinsupp'_single_tmul_single", "finsuppTensorFinsuppLid_self", "finsuppTensorFinsuppLid_single_tmul_single", "finsuppTensorFinsupp_apply", "fixedPoints_subgroup_sup", "fixingSubmonoid_fixedPoints_gc", "floorDiv_le_ceilDiv", "forall_apply_eq_imp_iff'", "forall_imp_iff_exists_imp", "forall\u2082_congr", "four_functions_theorem_aux", "fourier_coe_apply", "frequently_nhdsWithin_iff", "frobenius_natCast", "frontier_Ico", "frontier_Ioi'", "frontier_ball", "frontier_inter_open_inter", "frontier_inter_subset", "frontier_union_subset", "gal_X_pow_sub_C_isSolvable_aux", "gauge_add_le", "gauge_def", "gauge_le_one_iff_mem_closure", "gauge_mono", "gc_floorDiv_mul", "gc_upperClosure_coe", "gcd_pow_left_dvd_pow_gcd", "generateFrom_iUnion", "generateFrom_union", "geom_lt", "geom_sum_Ico", "gold_pos", "grade_toDual", "gronwallBound_continuous_\u03b5", "groupCohomology.dZero_ker_eq_invariants", "groupCohomology.groupCohomology\u03c0_comp_isoH0_hom", "groupCohomology.mem_twoCocycles_iff", "groupCohomology.oneCocycles_map_inv", "groupCohomology.resolution.ofMulAction_free", "groupCohomology.toCocycles_comp_isoOneCocycles_hom", "groupCohomology.twoCocycles_\u03c1_map_inv_sub_map_inv", "hallMatchingsOn.nonempty", "hammingNorm_ne_zero_iff", "hasCompactMulSupport_iff_eventuallyEq", "hasConstantSpeedOnWith_of_subsingleton", "hasDerivAt_bernoulliFun", "hasDerivAt_exp_of_mem_ball", "hasDerivAt_iff_isLittleO", "hasDerivAt_tsum_of_isPreconnected", "hasDerivWithinAt_id", "hasDerivWithinAt_iff_tendsto_slope", "hasDerivWithinAt_inter", "hasFDerivAt_id", "hasFDerivAt_integral_of_dominated_of_fderiv_le", "hasFDerivAt_single", "hasFDerivAt_stereoInvFunAux", "hasFDerivWithinAt_inter'", "hasFDerivWithinAt_inv", "hasFDerivWithinAt_of_nmem_closure", "hasFTaylorSeriesUpToOn_zero_iff", "hasGradientAt_iff_isLittleO_nhds_zero", "hasGradientWithinAt_congr_set", "hasLineDerivAt_iff_tendsto_slope_zero", "hasLineDerivWithinAt_smul_iff", "hasLineDerivWithinAt_univ", "hasMFDerivAt_snd", "hasMFDerivWithinAt_inter", "hasProd_unique", "hasStrictDerivAt_pi", "hasStrictFDerivAt_iff_hasStrictDerivAt", "hasSum_fourier_series_of_summable", "hasSum_geometric_of_abs_lt_one", "hasSum_zeta_two", "hausdorffMeasure_of_dimH_lt", "hausdorffMeasure_of_lt_dimH", "himp_le_himp_himp_himp", "himp_ne_right", "hnot_bot", "hnot_hnot_le", "hnot_hnot_sdiff_distrib", "holley", "hom_coe_pow", "homology'.condition_\u03c0'", "homology'.map_eq_lift_desc'_right", "homology'.\u03c0'_map", "homology'.\u03c0_map_apply", "iInf_eq_iInf_finset'", "iInf_iSup_ge_nat_add", "iInf_inf_eq", "iInf_le_iInf_of_subset", "iInf_nat_gt_zero_eq", "iInf_option", "iInf_pair", "iInf_sigma'", "iInf_subtype'", "iInf_sum", "iInter_Iic_eq_empty_iff", "iSup_const_le", "iSup_extend_bot", "iSup_nhds_le_uniformity", "iSup_option", "iSup_sigma'", "iSup_singleton", "iSup_split", "iSup_subtype''", "iSup_union", "iUnion_Ici_eq_Ioi_iInf", "iUnion_Ioc_add_intCast", "id_compRel", "idealFactorsFunOfQuotHom_comp", "im_inner_eq_norm_sub_i_smul_mul_self_sub_norm_add_i_smul_mul_self_div_four", "image2_lowerBounds_lowerBounds_subset", "image2_upperBounds_lowerBounds_subset_lowerBounds_image2", "image_le_of_liminf_slope_right_le_deriv_boundary", "image_parallelepiped", "image_segment", "image_smul_set", "imaginaryPart_smul", "imp_not_comm", "indicator_ae_eq_zero_of_restrict_ae_eq_zero", "induced_orderTopology", "induced_topology_eq_preorder", "inducing_id", "inducing_induced", "infEdist_smul\u2080", "inf_congr_right", "inf_covBy_of_covBy_sup_right", "inf_eq_and_sup_eq_iff", "inf_le_left", "inf_le_of_right_le", "inf_le_sup", "inf_lt_left_or_right", "inf_top_eq", "inf_uniformity", "inhomogeneousCochains.d_eq", "injective_of_le_imp_le", "inner_self_eq_norm_sq_to_K", "inner_smul_right", "inner_zero_left", "inseparable_iff_ker_uniformity", "inseparable_of_nhdsWithin_eq", "insert_mem_nhdsWithin_insert", "integrableOn_Ioi_cpow_iff", "integrableOn_peak_smul_of_integrableOn_of_tendsto", "integral_cos_sq", "integral_exp_neg_rpow", "integral_gaussian", "integral_gaussian_complex", "integral_one_div_of_neg", "integral_sin_pow", "integral_sin_pow_three", "integral_sin_sq_mul_cos", "interior_Icc", "interior_convexHull_nonempty_iff_affineSpan_eq_top", "interior_inter", "interior_mono", "interior_prod_eq", "intermediate_value_univ\u2082", "intervalIntegrable_iff'", "intervalIntegral.continuousAt_of_dominated_interval", "intervalIntegral.continuousOn_primitive_Icc", "intervalIntegral.deriv_integral_left", "intervalIntegral.integral_comp_add_right", "intervalIntegral.integral_deriv_mul_eq_sub_of_hasDerivWithinAt", "intervalIntegral.integral_finset_sum", "intervalIntegral.integral_hasStrictDerivAt_of_tendsto_ae_right", "intervalIntegral.integral_mono_interval", "intervalIntegral.integral_ofReal", "intervalIntegral.norm_integral_le_of_norm_le_const_ae", "intervalIntegral.smul_integral_comp_mul_add", "intervalIntegral.sub_le_integral_of_hasDeriv_right_of_le_Ico", "intrinsicClosure_subset_affineSpan", "intrinsicInterior_union_intrinsicFrontier", "invOf_mul_self_assoc", "invOf_nonneg", "inv_eq_of_root_of_coeff_zero_ne_zero", "inv_gold", "inv_inf", "inv_le_div_iff_le_mul", "inv_lt'", "inv_lt_of_mabs_lt", "inv_mul_cancel_comm", "inv_mul_lt_iff_lt_mul", "inv_neg'", "inv_neg_one", "inv_pow_le_inv_pow_of_le", "inv_pow_sub_of_lt", "inv_smul_le_iff_of_neg", "inv_sub_inv", "irrational_int_mul_iff", "irrational_nrt_of_n_not_dvd_multiplicity", "irrational_sqrt_of_multiplicity_odd", "irrational_sqrt_rat_iff", "irreducibleComponent_mem_irreducibleComponents", "irreducible_iff_prime", "irreducible_iff_prime_of_exists_unique_irreducible_factors", "irreducible_or_factor", "irrefl_of", "isAdjointPair_toLinearMap\u2082", "isAtomic_iff_forall_isAtomic_Iic", "isAtomistic_dual_iff_isCoatomistic", "isBot_iff_eq_bot", "isCancelMulZero_iff_isDomain_or_subsingleton", "isClosedMap_const", "isClosedMap_fst_of_compactSpace", "isClosed_and_discrete_iff", "isClosed_iff_frequently", "isClosed_induced_iff", "isClosed_nonneg", "isClosed_sum_iff", "isCompact_closure_singleton", "isCompact_diagonal", "isCompact_pi_infinite", "isConnected_iff_sUnion_disjoint_open", "isCoprime_group_smul_right", "isCoprime_mul_unit_left_left", "isCoprime_self", "isCoprime_zero_right", "isCornerFree_iff", "isCycle_finRotate", "isCyclic_of_orderOf_eq_card", "isEmpty_pprod", "isEmpty_prod", "isEmpty_sigma", "isExtrFilter_const", "isExtrFilter_dual_iff", "isGLB_Ioi", "isGLB_Ioo", "isGalois_of_isSplittingField_X_pow_sub_C", "isGreatest_Icc", "isGreatest_union_iff", "isIntegralCurveAt_comp_add", "isIntegral_iff_isIntegral_closure_finite", "isIntegral_localization'", "isIntegrallyClosedIn_iff", "isIrreducible_singleton", "isLUB_pair", "isLeast_csInf", "isLindelof_iUnion", "isLittleO_exp_neg_mul_rpow_atTop", "isLittleO_log_rpow_rpow_atTop", "isLittleO_pow_const_const_pow_of_one_lt", "isLocalMinOn_const", "isLowerSet_preimage_toDual_iff", "isLowerSet_setOf", "isMaxFilter_dual_iff", "isMax_top", "isMinFilter_dual_iff", "isMulFreimanHom_const", "isNilpotent_of_pos_nilpotencyClass", "isNilpotent_pi_of_bounded_class", "isNilpotent_sum", "isNormalSubgroup_of_commGroup", "isOpenMap_inl", "isOpenMap_quotient_mk'_mul", "isOpen_analyticAt", "isOpen_extChartAt_source", "isOpen_iff_mem_nhds", "isOpen_iff_ultrafilter", "isOpen_pi_iff", "isOpen_pi_iff'", "isPiSystem_Iic", "isPiSystem_image_Iic", "isPreconnected_Iic", "isPreconnected_iff_ordConnected", "isPurelyInseparable_iff", "isPurelyInseparable_of_finSepDegree_eq_one", "isRadical_iff_pow_one_lt", "isRelPrime_of_no_nonunits_factors", "isRelPrime_zero_right", "isRoot_of_unity_iff", "isSMulRegular_of_isSMulRegular_on_submodule_on_quotient", "isSMulRegular_on_submodule_iff_mem_imp_smul_eq_zero_imp_eq_zero", "isSeparable_tower_bot_of_isSeparable", "isSigmaCompact_iUnion", "isSigmaCompact_univ", "isSplittingField_iff_intermediateField", "isSquare_ofAdd_iff", "isSublattice_empty", "isSublattice_singleton", "isSubmonoid_iUnion_of_directed", "isTop_ofDual_iff", "isTop_top", "isTopologicalBasis_pi", "isTotallyDisconnected_iff_lt", "isTotallyDisconnected_singleton", "isUnit_gcd_one_right", "isUnit_map_of_leftInverse", "isUnit_of_map_unit", "isUnit_of_mul_isUnit_left", "isUpperSet_iInter", "isUpperSet_preimage_toDual_iff", "ite_pow", "ite_zero_smul", "iterateFrobeniusEquiv_add", "iterateFrobenius_mul_apply", "iterateFrobenius_zero", "iterate_map_inv", "iteratedDerivWithin_neg", "iteratedDerivWithin_succ", "iteratedDeriv_succ", "iteratedFDerivWithin_eventually_congr_set", "iteratedFDerivWithin_one_apply", "iteratedFDerivWithin_succ_const", "iteratedFDerivWithin_univ", "jacobiSym.eq_one_or_neg_one", "jacobiSym.mod_right", "jacobiSym.one_left", "jacobiSym.quadratic_reciprocity_one_mod_four", "jacobiSym.zero_left", "jacobiTheta_eq_tsum_nat", "jacobiTheta\u2082_add_left'", "jacobiTheta\u2082_add_right", "joinM_map_map", "latticeClosure_mono", "leOnePart_div_oneLePart", "leOnePart_one", "le_antisymm_iff", "le_compl_self", "le_csInf_iff", "le_div_iff_of_neg", "le_gauge_of_subset_closedBall", "le_gronwallBound_of_liminf_deriv_right_le", "le_iff_exists_nonneg_add", "le_iff_forall_one_lt_le_mul", "le_iff_forall_one_lt_lt_mul", "le_kstar", "le_max_of_sq_le_mul", "le_max_right", "le_mul_iff_one_le_right", "le_mul_inv_iff_mul_le", "le_mul_left", "le_mul_of_le_left", "le_multiplicity_iff_replicate_subperm_factors", "le_nhds_iff", "le_of_forall_ge_of_dense", "le_of_max_le_left", "le_of_mul_le_mul_right'", "le_of_mul_le_of_one_le_right", "le_of_smul_le_smul_left", "le_of_smul_le_smul_right", "le_of_span_le_span", "le_one_of_mul_le_left", "le_padicValNat_iff_replicate_subperm_factors", "le_partialSups_of_le", "le_sInf_inter", "lebesgue_number_lemma_of_metric", "leftInverse_mul_right_inv_mul", "left_eq_sup", "left_lt_lineMap_iff_lt", "left_subset_compRel", "legendreSym.at_one", "legendreSym.at_zero", "legendreSym.eq_one_iff'", "legendreSym.eq_one_of_sq_sub_mul_sq_eq_zero", "legendreSym.mod", "legendreSym.quadratic_reciprocity'", "legendreSym.quadratic_reciprocity_one_mod_four", "liftOfDerivationToSquareZero_mk_apply", "limsup_const_sub", "lineDerivWithin_congr_set", "lineDeriv_smul", "lineDifferentiableAt_smul_iff", "lineMap_mono_left", "lineMap_strict_mono_left", "linearEquiv_det_rotation", "linearIndependent_empty_type", "linearIndependent_equiv", "linearIndependent_fin_cons", "linearIndependent_iUnion_of_directed", "linearIndependent_iff''", "linearIndependent_set_iff_affineIndependent_vadd_union_singleton", "linearIndependent_unique_iff", "lipschitzWith_iff_dist_le_mul", "list_foldl'", "locallyConvexSpace_iff_zero", "locallyLipschitz_inv_iff", "lowerBounds_union", "lowerClosure_singleton", "lowerSemicontinuousAt_biSup", "lowerSemicontinuousWithinAt_tsum", "lowerSemicontinuous_iff_isOpen_preimage", "lp.coeFn_smul", "lp.ext", "lp.inner_single_right", "lp.norm_eq_tsum_rpow", "ltByCases_gt", "lt_add_of_tsub_lt_left", "lt_imp_lt_of_le_imp_le", "lt_irrefl", "lt_mem_nhds", "lt_mul_iff_one_lt_right", "lt_mul_of_one_lt_right", "lt_of_lt_of_le'", "lt_of_mul_lt_mul_of_le\u2080", "lt_of_mul_lt_of_one_le_of_nonneg_left", "lt_of_mul_self_lt_mul_self", "lt_of_one_div_lt_one_div", "lt_one_of_mul_lt_right", "lt_or_eq_of_le", "lt_tsub_of_add_lt_left", "mabs_eq_mabs", "mabs_mul_eq_oneLePart_sq", "manyOneReducible_toNat", "mapClusterPt_iff", "mapClusterPt_iff_ultrafilter", "mapClusterPt_of_comp", "map_bihimp", "map_bind", "map_coe_atBot_of_Ioo_subset", "map_comap_subtype_coe", "map_commutatorElement", "map_comp_zpow'", "map_continuousAt", "map_derivedSeries_le_derivedSeries", "map_extChartAt_nhdsWithin'", "map_extChartAt_nhdsWithin_eq_image'", "map_extChartAt_symm_nhdsWithin'", "map_finsupp_prod", "map_iSup\u2082", "map_inv_le_iff", "map_le_lineMap_iff_slope_le_slope_right", "map_list_prod", "map_mul_eq_one", "map_mul_left_nhds\u2080", "map_ne_one_iff", "map_snd_nhds", "map_uniformity_set_coe", "matPolyEquiv_map_smul", "matrixEquivTensor_apply", "max_assoc", "max_div_div_right'", "max_lt", "max_mul_mul_le_max_mul_max'", "mdifferentiableAt_fst", "mdifferentiableWithinAt_fst", "mdifferentiableWithinAt_inter'", "mdifferentiable_iff_differentiable", "mdifferentiable_of_mem_atlas", "measurableSet_Ioc", "measurableSet_closedBall", "measurableSet_eball", "measurableSet_eq", "measurableSet_eq_fun'", "measurableSet_of_differentiableAt_with_param", "measurableSet_prod", "measurableSet_region_between_co", "measurableSet_region_between_oc", "measurable_ereal_toReal", "measurable_iSup", "measurable_id", "measurable_inv_iff", "measurable_limsup", "measurable_measure_prod_mk_left_finite", "measurable_of_isClosed", "measurable_of_restrict_of_restrict_compl", "measurable_sSup", "measurable_swap", "measure_Ioo_lt_top", "mellin_const_smul", "mem_adjoin_of_dvd_coeff_of_dvd_aeval", "mem_affineSpan", "mem_ball_one_iff", "mem_closedBall_iff_norm'''", "mem_closure_iff_nhds_one", "mem_closure_iff_seq_limit", "mem_closure_iff_ultrafilter", "mem_conjugatesOf_self", "mem_lowerBounds_image2", "mem_nhdsWithin", "mem_nhdsWithin_iff_eventually", "mem_nhdsWithin_of_mem_nhds", "mem_nhdsWithin_subtype", "mem_nhds_induced", "mem_nonunits_iff", "mem_of_mem_measurableAtom", "mem_omegaLimit_singleton_iff_map_cluster_point", "mem_openSegment_of_ne_left_right", "mem_perfectClosure_iff", "mem_permsOfList_iff", "mem_residual_iff", "mem_rightCoset_rightCoset", "mem_skewAdjointMatricesSubmodule", "mem_span_finset", "mem_tangentConeAt_of_pow_smul", "mem\u2113p_zero", "mfderiv_prod_left", "midpoint_eq_iff", "midpoint_pointReflection_right", "min_abs_abs_le_abs_min", "min_def", "min_eq_left_iff", "min_le_left", "min_lt_min", "min_lt_of_left_lt", "min_mul_min_le_min_mul_mul'", "minimal_nonempty_open_subsingleton", "minimals_antichain", "minimals_of_symm", "minimals_union", "minpoly.ToAdjoin.injective", "minpoly.eq_X_sub_C'", "minpoly.eq_of_irreducible", "minpoly.eq_of_irreducible_of_monic", "minpoly.map_eq_of_separable_of_isPurelyInseparable", "minpolyDiv_spec", "minpoly_algHom_toLinearMap", "mk_mem_nonZeroDivisors_associates", "modelWithCorners_prod_coe_symm", "monotoneOn_of_hasDerivWithinAt_nonneg", "monotone_app", "monotone_nat_of_le_succ", "monotone_sInf_of_monotone", "monotone_vecEmpty", "monovaryOn_iff_forall_mul_nonneg", "monovaryOn_iff_mul_rearrangement", "monovary_comm", "monovary_id_iff", "monovary_inv_right\u2080", "mt", "mulEquivOfOrderOfEq_apply_gen", "mulIndicator_eventuallyLE_mulIndicator", "mulPosStrictMono_iff_mulPosReflectLE", "mulRothNumber_le", "mulRothNumber_map_mul_left", "mulTSupport_eq_empty_iff", "mul_div", "mul_div_div_cancel", "mul_dvd_mul_right", "mul_eq_of_eq_div'", "mul_eq_one_iff_inv_eq", "mul_inv_le_one_iff", "mul_inv_rev", "mul_le_mul", "mul_le_mul_of_nonpos_right", "mul_le_mul_right\u2080", "mul_le_of_le_of_le_one_of_nonneg", "mul_le_of_mul_le_left", "mul_le_one\u2080", "mul_left_surjective", "mul_lt_mul", "mul_lt_mul_of_le_of_lt'", "mul_lt_of_le_of_lt_one_of_pos", "mul_lt_of_lt_one_of_le", "mul_lt_of_one_lt_right", "mul_mem_cancel_right", "mul_min_of_nonneg", "mul_nonpos_of_nonneg_of_nonpos", "mul_not_mem_of_exponent_two", "mul_not_mem_of_orderOf_eq_two", "mul_one", "mul_one_add_self", "mul_right_iterate", "mul_right_iterate_apply_one", "mul_right_surjective\u2080", "mul_smul_one", "mul_sub_mul_div_mul_nonpos_iff", "mul_zpow", "multiplesAddHom_apply", "multiplicative_of_symmetric_of_isTotal", "multiplicity.eq_of_associated_left", "multiplicity.eq_of_associated_right", "multiplicity.finite_int_iff", "multiplicity.min_le_multiplicity_add", "multiplicity.multiplicity_add_of_gt", "multiplicity.multiplicity_le_multiplicity_of_dvd_right", "multiplicity.multiplicity_map_eq", "multiplicity.pow_sub_pow_of_prime", "multiplicity_normalizedFactorsEquivSpanNormalizedFactors_eq_multiplicity", "nat_abs_sum_le", "nat_sub_dvd_pow_sub_pow", "ncs_tendsto_lim", "ne_iff_lt_or_gt", "ne_of_eq_of_ne", "ne_of_ne_of_eq'", "ne_zero_of_dvd_ne_zero", "ne_zero_of_eq_one", "ne_zero_or_ne_zero_of_nat_coprime", "neg_div'", "neg_of_mul_neg_left", "neg_of_mul_pos_left", "neg_one_pow_eq_one_iff_even", "neg_one_pow_eq_or", "neg_one_sq", "neg_pow", "neg_pow_bit1", "nhdsSet_Ioc", "nhdsSet_diagonal", "nhdsSet_diagonal_eq_uniformity", "nhdsSet_univ", "nhdsWithin_Icc_eq_nhdsWithin_Iic", "nhdsWithin_eq_map_subtype_coe", "nhdsWithin_insert", "nhdsWithin_neBot", "nhdsWithin_prod_eq", "nhdsWithin_restrict", "nhdsWithin_restrict'", "nhdsWithin_subtype_eq_bot_iff", "nhds_basis_Ioo_pos", "nhds_basis_Ioo_pos_of_pos", "nhds_eq_comap_uniformity", "nhds_eq_nhds_emetric_ball", "nhds_false", "nhds_injective", "nhds_of_nhdsWithin_of_nhds", "nhds_pi", "nhds_toAdd", "nhds_top", "nndist_edist", "nndist_homothety_center", "nndist_toDual", "noncompactSpace_of_neBot", "nonempty_linearEquiv_of_lift_rank_eq", "nonempty_measurable_superset", "nonempty_sections_of_finite_cofiltered_system.init", "nonpos_of_mul_nonpos_right", "norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero", "norm_fderiv_iteratedFDeriv", "norm_fst_le", "norm_image_sub_le_of_norm_deriv_le_segment", "norm_inner_div_norm_mul_norm_eq_one_iff", "norm_iteratedFDerivWithin_clm_apply_const", "norm_iteratedFDerivWithin_comp_le", "norm_iteratedFDerivWithin_eq_norm_iteratedDerivWithin", "norm_iteratedFDeriv_fderiv", "norm_le_gronwallBound_of_norm_deriv_right_le", "norm_mul", "norm_sum_le", "norm_sup_le_add", "normal_exists_closure_subset", "normalizeScaleRoots_coeff_mul_leadingCoeff_pow", "normalize_eq", "not_and_or", "not_covBy", "not_covBy_of_lt_of_lt", "not_false_eq_true", "not_forall_not", "not_iff_not", "not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter_aux", "not_isCyclic_iff_exponent_eq_prime", "not_isRightRegular_zero_iff", "not_isTop", "not_mulDissociated_iff_exists_disjoint", "not_small_ordinal", "npow_mul_comm", "odd_add_one_self", "ofAdd_smul", "ofDual_sInf", "ofLex_one", "ofNat_mem", "of_not_not", "omegaLimit_def", "omegaLimit_iInter", "omegaLimit_mono_right", "one_add_div", "one_add_one_eq_two", "one_div_le_of_neg", "one_div_lt_one_div_of_neg_of_lt", "one_div_ne_zero", "one_div_one", "one_div_pos", "one_div_zpow", "one_half_le_sum_primes_ge_one_div", "one_half_lt_one", "one_le_of_le_mul_right", "one_lt_dite", "one_lt_inv", "one_lt_natDegree_of_irrational_root", "openSegment_eq_image", "openSegment_eq_image'", "openSegment_same", "opensMeasurableSpace_iff_forall_measurableSet", "or_assoc", "or_congr", "or_iff_right", "or_self", "orderOf_dvd_natCard", "orderOf_generator_eq_natCard", "orderOf_one", "orthogonalProjectionFn_mem", "orthogonalProjection_eq_linear_proj", "orthogonalProjection_orthogonalProjection_of_le", "orthonormal_empty", "orthonormal_subtype_iff_ite", "pNilradical_prime", "padicNorm.add_eq_max_of_ne", "padicNorm.int_eq_one_iff", "padicNorm.of_int", "padicNorm.sum_lt", "padicNorm.zero_of_padicNorm_eq_zero", "padicValNat.pow_add_pow", "padicValNat_def'", "padicValNat_factorial_mul", "partialSups_succ", "pathConnectedSpace_iff_univ", "pcontinuous_iff'", "perfectField_iff_splits_of_natSepDegree_eq_one", "pi_Ico_mem_nhds'", "pi_Iio_mem_nhds'", "pi_Ioc_mem_nhds'", "pi_dvd_iff", "pi_generateFrom_eq", "piiUnionInter_singleton", "polynomialFunctions.eq_adjoin_X", "posSMulStrictMono_iff_PosSMulReflectLE", "posTangentConeAt_mono", "pow_bit0_pos_iff", "pow_inv_comm\u2080", "pow_lt_pow_left'", "pow_mod_card", "pow_mod_orderOf", "pow_mul_comm", "pow_pos", "pow_right_mono", "pow_right_strictAnti", "pow_sub_one_mul", "preStoneCechExtend_extends", "pred_max", "preimage_smul_set\u209b\u2097", "prevD_comp_right", "prime_dvd_char_iff_dvd_card", "prod_eq_generateFrom", "prod_mem", "prod_mul_tprod_subtype_compl", "projIcc_eq_one", "pure_le_nhds", "quasiIso'_of_comp_left", "quasiIso'_of_comp_right", "quasispectrum.mem_of_not_quasiregular", "rTensor.inverse_of_rightInverse_apply", "range_two_mul_add_one", "rank_eq_one_iff", "rank_prod", "rank_self", "rank_span_of_finset", "rank_submodule_le_one_iff", "rank_tensorProduct'", "realPart_add_I_smul_imaginaryPart", "real_inner_add_add_self", "rec_heq_of_heq", "rec_subsingleton", "reflTransGen_of_pred_of_ge", "reflection_bot", "rel_iSup_tprod", "rescale_to_shell_semi_normed", "riemannZeta_def_of_ne_zero", "riemannZeta_eulerProduct", "riemannZeta_one_ne_zero", "riesz_lemma_of_norm_lt", "right_eq_midpoint_iff", "ringExpChar.eq", "root_X_pow_sub_C_eq_zero_iff", "rootsOfUnity.integer_power_of_ringEquiv", "roots_mem_integralClosure", "rotation_apply", "round_add_one", "round_eq_zero_iff", "round_nat_add", "round_two_inv", "sInfHom.coe_comp", "sInfHom.comp_id", "sInfHom.id_comp", "sInf_image", "sInf_le", "sInf_mem_of_not_isPredLimit", "sSupHom.symm_dual_comp", "sSup_image2_eq_sSup_sSup", "sameRay_iff_of_norm_eq", "sameRay_neg_smul_left_iff_of_ne", "sbtw_of_btw_not_btw", "sbtw_pointReflection_of_ne", "schnirelmannDensity_le_of_not_mem", "schnirelmannDensity_setOf_modeq_one", "sdiff_eq_left", "sdiff_eq_right", "sdiff_inf_self_left", "sdiff_le_comm", "sdiff_right_comm", "sdiff_self", "segment_eq_image", "segment_eq_image_lineMap", "segment_subset_iff", "segment_symm", "selfAdjoint.val_pow", "selfAdjoint.val_qsmul", "selfAdjoint.val_ratCast", "selfZPow_of_neg", "semiconjBy_star_star_star", "separableClosure.adjoin_eq_of_isAlgebraic", "separableClosure.adjoin_eq_of_isAlgebraic_of_isSeparable", "seqClosure_eq_closure", "set_smul_mem_nhds_smul", "set_smul_mem_nhds_zero_iff", "sigmaFinsuppEquivDFinsupp_smul", "sign_eq_neg_one_iff", "sign_finRotate", "simple_iff_isSimpleModule'", "single_le_finprod", "singleton_mem_nhdsWithin_of_mem_discrete", "singleton_mul_closedBall", "singletons_open_iff_discrete", "skewAdjoint.smul_mem", "skewAdjointMatricesLieSubalgebraEquiv_apply", "small_congr", "small_of_surjective", "smoothAt_fst", "smoothAt_hom_bundle", "smoothWithinAt_pi_space", "smooth_mul", "smulAddHom_apply", "smul_add_smul_lt_smul_add_smul", "smul_eq_zero_of_left", "smul_lt_of_lt_one_left", "smul_lt_smul'", "smul_lt_smul_iff_of_pos_left", "smul_max_of_nonpos", "smul_mul_smul", "smul_ne_zero_iff_ne", "smul_neg", "smul_neg_iff_of_pos_left", "smul_nonpos_iff_neg_imp_nonneg", "smul_nonpos_of_nonpos_of_nonneg", "smul_pow", "smul_singleton_mem_nhds_of_sigmaCompact", "snd_himp", "span_eq_iUnion_nat", "span_eq_top_localization_localization", "span_le_span_iff", "spec_cfc\u2099Aux", "spectrum.gelfandTransform_eq", "spectrum.hasFPowerSeriesOnBall_inverse_one_sub_smul", "spectrum.inv_mem_resolventSet", "spectrum.limsup_pow_nnnorm_pow_one_div_le_spectralRadius", "spectrum.nonempty", "spectrum.pow_image_subset", "spectrum.units_conjugate'", "spectrum.zero_not_mem_iff", "sphere_ext_iff", "sphere_prod", "sq_eq_sq", "sq_le_one_iff_abs_le_one", "sqrt_one_add_norm_sq_le", "squeeze_one_norm", "starConvex_pi", "starConvex_sInter", "starRingEnd_self_apply", "star_intCast_smul", "star_le_star_iff", "stdSimplex_of_subsingleton", "stereographic'_source", "strictAntiOn_Iic_of_succ_lt", "strictAnti_nat_of_succ_lt", "strictAnti_of_deriv_neg", "strictAnti_vecEmpty", "strictConcaveOn_of_slope_strict_anti_adjacent", "strictConvexOn_of_slope_strict_mono_adjacent", "strictMono_of_odd_strictMonoOn_nonneg", "stronglyMeasurable_deriv_with_param", "sub_div'", "sub_lt_of_abs_sub_lt_left", "sub_pow_expChar_pow_of_commute", "sub_self_div_two", "sub_smul_slope_vadd", "subalgebra_top_finrank_eq_submodule_top_finrank", "subset_convexJoin_right", "subset_sInf_of_within", "suffixLevenshtein_length", "sum_div_card_sq_le_sum_sq_div_card", "summable_iff_cauchySeq_finset_and_tsum_mem", "summable_jacobiTheta\u2082_term_fderiv_iff", "summable_jacobiTheta\u2082_term_iff", "summable_matrix_blockDiagonal", "summable_of_summable_hasFDerivAt", "sup_comm", "sup_eq_left", "sup_eq_right", "sup_eq_sdiff_sup_sdiff_sup_inf", "sup_himp_self_left", "sup_le", "sup_le_sup_right", "sup_right_idem", "sup_sdiff_injOn", "surjective_algebraMap_of_linearMap", "symmDiff_comm", "symmDiff_le_iff", "symmDiff_left_inj", "symmDiff_right_injective", "symmDiff_right_surjective", "symmDiff_sdiff_eq_sup", "symmDiff_sdiff_inf", "symmDiff_self", "symmDiff_symmDiff_left", "t0Space_iff_uniformity'", "t1Space_TFAE", "t2Quotient.unique_lift", "tangentBundleModelSpaceHomeomorph_coe", "tangentConeAt.lim_zero", "tangentMapWithin_congr", "taylorWithinEval_self", "taylor_within_zero_eval", "tendsto_Ioo_atBot", "tendsto_add_one_pow_atTop_atTop_of_pos", "tendsto_atBot_isGLB", "tendsto_atTop_ciInf", "tendsto_ceil_left'", "tendsto_div_comap_self", "tendsto_floor_left'", "tendsto_floor_left_pure_sub_one", "tendsto_floor_right'", "tendsto_fract_left'", "tendsto_iff_forall_eval_tendsto_topDualPairing", "tendsto_indicator_const_apply_iff_eventually'", "tendsto_multiset_prod", "tendsto_natCast_atTop_iff", "tendsto_nat_ceil_div_atTop", "tendsto_nat_floor_mul_div_atTop", "tendsto_nhdsWithin_range", "tendsto_nhds_bot_mono", "tendsto_norm_cobounded_atTop'", "tendsto_of_monotone", "tendsto_of_tendsto_of_tendsto_of_le_of_le'", "tendsto_pow_atTop_nhdsWithin_zero_of_lt_one", "tendsto_pow_const_mul_const_pow_of_abs_lt_one", "tendsto_rpow_abs_mul_exp_neg_mul_sq_cocompact", "tendsto_zero_of_norm_tendsto_zero", "thickenedIndicatorAux_le_one", "thickenedIndicatorAux_lt_top", "threeGPFree_smul_set\u2080", "toIcoDiv_add_right", "toIcoDiv_apply_left", "toIcoDiv_apply_right", "toIcoDiv_neg", "toIcoMod_eq_toIcoMod", "toIcoMod_le_toIocMod", "toIcoMod_mem_Ico", "toIocDiv_add_zsmul", "toIocDiv_eq_neg_floor", "toIocDiv_sub_eq_toIocDiv_add'", "toIocMod_eq_sub", "toIocMod_sub_zsmul'", "toIocMod_toIocMod", "toLex_inv", "toMul_sub", "toReal_coe_nnnorm'", "toRingHom_injective", "toTopologicalSpace_prod", "toUnits_val_apply", "toZ_iterate_succ_le", "toZ_mono", "toZ_of_lt", "top_inf_eq", "top_mem_upperBounds", "top_sdiff'", "torusIntegral_dim0", "torusMap_eq_center_iff", "totallyBounded_subset", "totallySeparatedSpace_of_t1_of_basis_clopen", "tprod_image", "tprod_le_tprod", "tprod_ne_one_iff", "tprod_setProd_singleton_right", "tprod_singleton", "tprod_subtype_le", "tprod_subtype_mulSupport", "transGen_wcovBy_of_le", "trichotomous_of", "true_imp_iff", "tsub_add_tsub_comm", "tsub_eq_zero_iff_le", "tsub_le_iff_right", "tsub_le_tsub_right", "tsub_min", "tsub_tsub_cancel_of_le", "tsub_tsub_eq_min", "tsub_tsub_le", "tsum_const_smul'", "tsupport_mul_subset_right", "ultrafilter_isOpen_basic", "uniformCauchySeqOnFilter_of_fderiv", "uniformCauchySeqOn_iff_uniformCauchySeqOnFilter", "uniformContinuousOn_univ", "uniformContinuous_of_const", "uniformContinuous_of_tendsto_one", "uniformContinuous_subtype_val", "uniformEmbedding_subtype_val", "uniformEmbedding_translate_mul", "uniformSpace_comap_id", "uniformity_basis_edist_nnreal_le", "uniformity_hasBasis_closure", "uniformity_pseudoedist", "uniqueDiffOn_Ici", "uniqueDiffOn_empty", "uniqueDiffWithinAt_convex", "uniqueElim_const", "unique_unit_speed", "unitInterval.half_le_symm_iff", "unitInterval.one_mem", "unitary.coe_star_mul_self", "unitary.spectrum.unitary_conjugate", "unop_nonZeroSMulDivisors_mulOpposite_eq_nonZeroDivisors", "upperBounds_mono_mem", "upperCentralSeries_nilpotencyClass", "upperCentralSeries_one", "upperClosure_mul_distrib", "upperClosure_one", "upperSemicontinuousAt_ciInf", "upperSemicontinuousOn_biInf", "upperSemicontinuousWithinAt_univ_iff", "upperSemicontinuous_iInf", "upperSemicontinuous_iff_limsup_le", "uzpow_natCast", "uzpow_neg", "vadd_right_mem_affineSpan_pair", "wbtw_pointReflection", "wellFounded_gt_exact_sequence", "wellFounded_iff_wellFounded_subrel", "wellFounded_of_isEmpty", "with_gaugeSeminormFamily", "wittStructureInt_prop", "wittStructureInt_rename", "wittStructureRat_rec_aux", "xor_false", "xor_self", "zero_dvd_iff", "zero_eq_dist", "zero_le_one", "zero_lt_iff", "zero_mem_nonunits", "zero_mem\u2113p", "zero_pow_eq", "zero_pow_eq_zero", "zero_pow_le_one", "zero_smul", "zeta_nat_eq_tsum_of_gt_one", "zorn_nonempty_partialOrder\u2080", "zorn_preorder", "zpow_abs_eq_one", "zpow_eq_one_iff_modEq", "zpow_le_iff_le", "zpow_mem", "zpow_mod_card", "zpow_one", "zpow_one_add\u2080", "zpow_pos_of_pos", "zpowers_eq_top_of_prime_card", "\u03b5NFA.evalFrom_append_singleton"], "statements": ["theorem admissible_D' (n : \u2115+) : Admissible (D' n)", "@[measurability]\ntheorem add_measure {f : \u03b1 \u2192 \u03b2} (h\u03bc : AEMeasurable f \u03bc) (h\u03bd : AEMeasurable f \u03bd) :\n    AEMeasurable f (\u03bc + \u03bd)", "@[measurability, fun_prop]\ntheorem AEMeasurable.coe_ereal_ennreal {f : \u03b1 \u2192 \u211d\u22650\u221e} {\u03bc : Measure \u03b1} (hf : AEMeasurable f \u03bc) :\n    AEMeasurable (fun x => (f x : EReal)) \u03bc", "theorem comp_aemeasurable {f : \u03b1 \u2192 \u03b4} {g : \u03b4 \u2192 \u03b2} (hg : AEMeasurable g (\u03bc.map f))\n    (hf : AEMeasurable f \u03bc) : AEMeasurable (g \u2218 f) \u03bc", "theorem comp_measurable {f : \u03b1 \u2192 \u03b4} {g : \u03b4 \u2192 \u03b2} (hg : AEMeasurable g (\u03bc.map f))\n    (hf : Measurable f) : AEMeasurable (g \u2218 f) \u03bc", "@[measurability]\ntheorem AEMeasurable.const_inf (hf : AEMeasurable f \u03bc) (c : M) :\n    AEMeasurable (fun x => c \u2293 f x) \u03bc", "@[to_additive (attr := measurability)]\ntheorem AEMeasurable.const_smul (hf : AEMeasurable g \u03bc) (c : M) : AEMeasurable (c \u2022 g) \u03bc", "@[to_additive (attr := fun_prop, aesop safe 20 apply (rule_sets := [Measurable]))]\ntheorem AEMeasurable.div [MeasurableDiv\u2082 G] (hf : AEMeasurable f \u03bc) (hg : AEMeasurable g \u03bc) :\n    AEMeasurable (fun a => f a / g a) \u03bc", "@[to_additive (attr := measurability)]\ntheorem AEMeasurable.div_const [MeasurableDiv G] (hf : AEMeasurable f \u03bc) (c : G) :\n    AEMeasurable (fun x => f x / c) \u03bc", "@[measurability, fun_prop]\ntheorem AEMeasurable.ennreal_toReal {f : \u03b1 \u2192 \u211d\u22650\u221e} {\u03bc : Measure \u03b1} (hf : AEMeasurable f \u03bc) :\n    AEMeasurable (fun x => ENNReal.toReal (f x)) \u03bc", "theorem exists_measurable_nonneg {\u03b2} [Preorder \u03b2] [Zero \u03b2] {m\u03b2 : MeasurableSpace \u03b2} {f : \u03b1 \u2192 \u03b2}\n    (hf : AEMeasurable f \u03bc) (f_nn : \u2200\u1d50 t \u2202\u03bc, 0 \u2264 f t) : \u2203 g, Measurable g \u2227 0 \u2264 g \u2227 f =\u1d50[\u03bc] g", "theorem map_map_of_aemeasurable {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} (hg : AEMeasurable g (Measure.map f \u03bc))\n    (hf : AEMeasurable f \u03bc) : (\u03bc.map f).map g = \u03bc.map (g \u2218 f)", "theorem mono_measure (h : AEMeasurable f \u03bc) (h' : \u03bd \u2264 \u03bc) : AEMeasurable f \u03bd", "@[to_additive (attr := fun_prop, aesop safe 20 apply (rule_sets := [Measurable]))]\ntheorem AEMeasurable.mul [MeasurableMul\u2082 M] (hf : AEMeasurable f \u03bc) (hg : AEMeasurable g \u03bc) :\n    AEMeasurable (fun a => f a * g a) \u03bc", "@[aesop safe 20 apply (rule_sets := [Measurable])]\ntheorem AEMeasurable.pow (hf : AEMeasurable f \u03bc) (hg : AEMeasurable g \u03bc) :\n    AEMeasurable (fun x => f x ^ g x) \u03bc", "@[measurability]\ntheorem AEMeasurable.re (hf : AEMeasurable f \u03bc) : AEMeasurable (fun x => RCLike.re (f x)) \u03bc", "theorem insert_insert_of_ne {a a'} {b : \u03b2 a} {b' : \u03b2 a'} (s : AList \u03b2) (h : a \u2260 a') :\n    ((s.insert a b).insert a' b').entries ~ ((s.insert a' b').insert a b).entries", "@[simp] theorem lookup_insert_eq_none {l : AList \u03b2} {k k' : \u03b1} {v : \u03b2 k} :\n    (l.insert k v).lookup k' = none \u2194 (k' \u2260 k) \u2227 l.lookup k' = none", "@[simp]\ntheorem union_entries {s\u2081 s\u2082 : AList \u03b2} : (s\u2081 \u222a s\u2082).entries = kunion s\u2081.entries s\u2082.entries", "@[simp]\ntheorem map_id : map (MonoidHom.id G) = MonoidHom.id (Abelianization G)", "theorem coe_balanced (s : AbsConvexOpenSets \ud835\udd5c E) : Balanced \ud835\udd5c (s : Set E)", "theorem exists_approx {\u03b9 : Type*} [Fintype \u03b9] {\u03b5 : \u211d} (h\u03b5 : 0 < \u03b5) {b : R} (hb : b \u2260 0)\n    (h : abv.IsAdmissible) (A : Fin (h.card \u03b5 ^ Fintype.card \u03b9).succ \u2192 \u03b9 \u2192 R) :\n    \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 k, (abv (A i\u2081 k % b - A i\u2080 k % b) : \u211d) < abv b \u2022 \u03b5", "theorem map_one_of_isLeftRegular (h : IsLeftRegular (abv 1)) : abv 1 = 1", "protected lemma eventually (h : Absorbs M s t) : \u2200\u1da0 a in cobounded M, t \u2286 a \u2022 s := h", "protected lemma zero [Zero E] [SMulZeroClass M E] {s : Set E} (hs : 0 \u2208 s) : Absorbs M s 0", "theorem continuous_coe : Continuous \u03b9", "theorem AccPt.nhds_inter {x : \u03b1} {U : Set \u03b1} (h_acc : AccPt x (\ud835\udcdf C)) (hU : U \u2208 \ud835\udcdd x) :\n    AccPt x (\ud835\udcdf (U \u2229 C))", "@[ext]\nlemma hom_ext {M N : Action V G} (\u03c6\u2081 \u03c6\u2082 : M \u27f6 N) (h : \u03c6\u2081.hom = \u03c6\u2082.hom) : \u03c6\u2081 = \u03c6\u2082", "theorem rightUnitor_hom_hom {X : Action V G} : Hom.hom (\u03c1_ X).hom = (\u03c1_ X.V).hom", "lemma AddAction.automorphize_smul_left [AddGroup \u03b1] [AddAction \u03b1 \u03b2]  (f : \u03b2 \u2192 M)\n    (g : Quotient (AddAction.orbitRel \u03b1 \u03b2) \u2192 R) :\n    AddAction.automorphize ((g \u2218 (@Quotient.mk' _ (_))) \u2022 f)\n      = g \u2022 (AddAction.automorphize f : Quotient (AddAction.orbitRel \u03b1 \u03b2) \u2192 M)", "theorem _root_.AddAction.orbitZMultiplesEquiv_symm_apply' {\u03b1 \u03b2 : Type*} [AddGroup \u03b1] (a : \u03b1)\n    [AddAction \u03b1 \u03b2] (b : \u03b2) (k : \u2124) :\n    (AddAction.orbitZMultiplesEquiv a b).symm k =\n      k \u2022 (\u27e8a, mem_zmultiples a\u27e9 : zmultiples a) +\u1d65 \u27e8b, AddAction.mem_orbit_self b\u27e9", "theorem stabilizer_vadd_eq_stabilizer_map_conj (g : G) (a : \u03b1) :\n    stabilizer G (g +\u1d65 a) = (stabilizer G a).map (AddAut.conj g).toAddMonoidHom", "@[simp] lemma coe_mk (f : A \u2192 M)\n    (map_zero_eq_one' : f 0 = 1) (map_add_eq_mul' : \u2200 a b : A, f (a + b) = f a * f b) :\n    AddChar.mk f map_zero_eq_one' map_add_eq_mul' = f", "@[ext] lemma ext (f g : AddChar A M) (h : \u2200 x : A, f x = g x) : f = g", "@[simp] lemma map_zero_eq_one (\u03c8 : AddChar A M) : \u03c8 0 = 1 := \u03c8.map_zero_eq_one'", "@[simp]\ntheorem mulShift_zero (\u03c8 : AddChar R M) : mulShift \u03c8 0 = 1", "lemma val_mem_rootsOfUnity (\u03c6 : AddChar R R') (a : R) (h : 0 < ringChar R) :\n    (\u03c6.val_isUnit a).unit \u2208 rootsOfUnity (ringChar R).toPNat' R'", "theorem zmodChar_apply' {n : \u2115} [NeZero n] {\u03b6 : C} (h\u03b6 : \u03b6 ^ n = 1) (a : \u2115) :\n    zmodChar n h\u03b6 a = \u03b6 ^ a", "theorem addOrderOf_eq_pos_iff {u : AddCircle p} {n : \u2115} (h : 0 < n) :\n    addOrderOf u = n \u2194 \u2203 m < n, m.gcd n = 1 \u2227 \u2191(\u2191m / \u2191n * p) = u", "@[simp]\ntheorem equivAddCircle_symm_apply_mk (hp : p \u2260 0) (hq : q \u2260 0) (x : \ud835\udd5c) :\n    (equivAddCircle p q hp hq).symm (x : \ud835\udd5c) = (x * (q\u207b\u00b9 * p) : \ud835\udd5c)", "theorem equivIccQuot_comp_mk_eq_toIcoMod :\n    equivIccQuot p a \u2218 Quotient.mk'' = fun x =>\n      Quot.mk _ \u27e8toIcoMod hp.out a x, Ico_subset_Icc_self <| toIcoMod_mem_Ico _ _ x\u27e9", "theorem exists_norm_eq_of_isOfFinAddOrder {u : AddCircle p} (hu : IsOfFinAddOrder u) :\n    \u2203 k : \u2115, \u2016u\u2016 = p * (k / addOrderOf u)", "theorem gcd_mul_addOrderOf_div_eq {n : \u2115} (m : \u2115) (hn : 0 < n) :\n    m.gcd n * addOrderOf (\u2191(\u2191m / \u2191n * p) : AddCircle p) = n", "theorem homeomorphCircle_apply (hT : T \u2260 0) (x : AddCircle T) :\n    homeomorphCircle hT x = toCircle x", "theorem liftIco_zero_coe_apply {f : \ud835\udd5c \u2192 B} {x : \ud835\udd5c} (hx : x \u2208 Ico 0 p) : liftIco p 0 f \u2191x = f x", "theorem liftIco_zero_continuous [TopologicalSpace B] {f : \ud835\udd5c \u2192 B} (hf : f 0 = f p)\n    (hc : ContinuousOn f <| Icc 0 p) : Continuous (liftIco p 0 f)", "theorem norm_eq' (hp : 0 < p) {x : \u211d} : \u2016(x : AddCircle p)\u2016 = p * |p\u207b\u00b9 * x - round (p\u207b\u00b9 * x)|", "theorem volume_of_add_preimage_eq (s I : Set <| AddCircle T) (u x : AddCircle T)\n    (hu : IsOfFinAddOrder u) (hs : (u +\u1d65 s : Set <| AddCircle T) =\u1d50[volume] s)\n    (hI : I =\u1d50[volume] ball x (T / (2 * addOrderOf u))) :\n    volume s = addOrderOf u \u2022 volume (s \u2229 I)", "theorem of.zero_exact [IsDirected \u03b9 (\u00b7 \u2264 \u00b7)] [DirectedSystem G fun i j h => f i j h] (i x)\n    (h : of G f i x = 0) : \u2203 j hij, f i j hij x = 0", "protected theorem add_left (c : \u03b1) (h : a \u2261 b [PMOD p]) : c + a \u2261 c + b [PMOD p]", "theorem modEq_iff_eq_mod_zmultiples : a \u2261 b [PMOD p] \u2194 (b : \u03b1 \u29f8 AddSubgroup.zmultiples p) = a", "@[simp]\ntheorem modEq_neg : a \u2261 b [PMOD -p] \u2194 a \u2261 b [PMOD p]", "@[simp]\ntheorem nsmul_modEq_nsmul [NoZeroSMulDivisors \u2115 \u03b1] (hn : n \u2260 0) :\n    n \u2022 a \u2261 n \u2022 b [PMOD n \u2022 p] \u2194 a \u2261 b [PMOD p]", "@[simp, elementwise]\ntheorem biproductIsoPi_inv_comp_\u03c0 (f : J \u2192 AddCommGrp.{u}) (j : J) :\n    (biproductIsoPi f).inv \u226b biproduct.\u03c0 f j = Pi.evalAddMonoidHom (fun j => f j) j", "@[simp]\ntheorem kernelIsoKer_hom_comp_subtype {G H : AddCommGrp.{u}} (f : G \u27f6 H) :\n    (kernelIsoKer f).hom \u226b AddSubgroup.subtype f.ker = kernel.\u03b9 f", "@[scoped simp] theorem coe_mk (f : G \u2192 H) (hf) : \u21d1(mk f hf : G \u2192+c[a, b] H) = f", "@[scoped simp]\ntheorem map_const_add [AddCommSemigroup G] [Add H] [AddConstMapClass F G H a b]\n    (f : F) (x : G) : f (a + x) = f x + b", "theorem strictAnti_iff_Icc [LinearOrderedAddCommGroup G] [Archimedean G] [OrderedAddCommGroup H]\n    [AddConstMapClass F G H a b] {f : F} (ha : 0 < a) (l : G) :\n    StrictAnti f \u2194 StrictAntiOn f (Icc l (l + a))", "@[simp]\ntheorem toIntLinearEquiv_symm : e.toIntLinearEquiv.symm = e.symm.toIntLinearEquiv", "@[simp]\nprotected theorem add_tsub_cancel_left (ha : AddLECancellable a) : a + b - a = b", "@[simp]\nprotected theorem add_tsub_tsub_cancel (hac : AddLECancellable (a - c)) (h : c \u2264 a) :\n    a + b - (a - c) = b + c", "protected theorem eq_tsub_of_add_eq (hc : AddLECancellable c) (h : a + c = b) : a = b - c", "protected theorem le_tsub_iff_le_tsub (ha : AddLECancellable a) (hc : AddLECancellable c)\n    (h\u2081 : a \u2264 b) (h\u2082 : c \u2264 b) : a \u2264 b - c \u2194 c \u2264 b - a", "protected theorem mul_tsub (h : AddLECancellable (a * c)) : a * (b - c) = a * b - a * c", "theorem smul_apply (r : R) (f : AddMonoid.End A) (x : A) : (r \u2022 f) x = r \u2022 f x", "theorem divOf_add (x : k[G]) (a b : G) : x /\u1d52\u1da0 (a + b) = x /\u1d52\u1da0 a /\u1d52\u1da0 b", "theorem gradeBy.isInternal : DirectSum.IsInternal (gradeBy R f)", "theorem lift_apply' (F : Multiplicative G \u2192* A) (f : MonoidAlgebra k G) :\n    lift k G A F f = f.sum fun a b => algebraMap k A b * F (Multiplicative.ofAdd a)", "theorem lift_unique (F : k[G] \u2192\u2090[k] A) (f : MonoidAlgebra k G) :\n    F f = f.sum fun a b => b \u2022 F (single a 1)", "theorem mem_grade_iff (m : M) (a : R[M]) : a \u2208 grade R m \u2194 a.support \u2286 {m}", "theorem mem_grade_iff' (m : M) (a : R[M]) :\n    a \u2208 grade R m \u2194 a \u2208 (LinearMap.range (Finsupp.lsingle m : R \u2192\u2097[R] M \u2192\u2080 R) :\n      Submodule R R[M])", "theorem ne_zero_of_not_supDegree_le {b : B} (h : \u00ac p.supDegree D \u2264 b) : p \u2260 0", "theorem of'_divOf (a : G) : of' k G a /\u1d52\u1da0 a = 1", "theorem of'_mem_span [Nontrivial R] {m : M} {S : Set M} :\n    of' R M m \u2208 span R (of' R M '' S) \u2194 m \u2208 S", "@[simp]\ntheorem sum_single_index {N} [AddCommMonoid N] {a : G} {b : k} {h : G \u2192 k \u2192 N}\n    (h_zero : h a 0 = 0) :\n    (single a b).sum h = h a b", "theorem supDegree_prod_le {R A B : Type*} [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B]\n    [SemilatticeSup B] [OrderBot B]\n    [CovariantClass B B (\u00b7 + \u00b7) (\u00b7 \u2264 \u00b7)] [CovariantClass B B (Function.swap (\u00b7 + \u00b7)) (\u00b7 \u2264 \u00b7)]\n    {D : A \u2192 B} (hzero : D 0 = 0) (hadd : \u2200 a1 a2, D (a1 + a2) = D a1 + D a2)\n    {\u03b9} {s : Finset \u03b9} {f : \u03b9 \u2192 R[A]} :\n    (\u220f i \u2208 s, f i).supDegree D \u2264 \u2211 i \u2208 s, (f i).supDegree D", "@[simp]\ntheorem toDirectSum_zero [Semiring M] : (0 : AddMonoidAlgebra M \u03b9).toDirectSum = 0", "lemma AddMonoidHom.mulLeft_eq_mulRight_iff_forall_commute {a : R} :\n    mulLeft a = mulRight a \u2194 \u2200 b, Commute a b", "theorem AddMonoidHom.toNatLinearMap_injective [AddCommMonoid M] [AddCommMonoid M\u2082] :\n    Function.Injective (@AddMonoidHom.toNatLinearMap M M\u2082 _ _)", "@[simp]\ntheorem coe_toZModSubmodule (S : AddSubgroup M) : (toZModSubmodule n S : Set M) = S", "theorem AddSubgroup.quotient_norm_eq {S : AddSubgroup M} (x : M \u29f8 S) :\n    \u2016x\u2016 = sInf (norm '' { m : M | (m : M \u29f8 S) = x })", "@[simp]\ntheorem toZModSubmodule_toAddSubgroup (S : AddSubgroup M) :\n    (toZModSubmodule n S).toAddSubgroup = S", "theorem zsmul_mem_zmultiples_iff_exists_sub_div {r : R} {z : \u2124} (hz : z \u2260 0) :\n    z \u2022 r \u2208 AddSubgroup.zmultiples p \u2194\n      \u2203 k : Fin z.natAbs, r - (k : \u2115) \u2022 (p / z : R) \u2208 AddSubgroup.zmultiples p", "theorem le_pointwise_smul_iff\u2080 {a : \u03b1} (ha : a \u2260 0) {S T : AddSubmonoid A} :\n    S \u2264 a \u2022 T \u2194 a\u207b\u00b9 \u2022 S \u2264 T", "theorem mem_pointwise_smul_iff_inv_smul_mem\u2080 {a : \u03b1} (ha : a \u2260 0) (S : AddSubmonoid A) (x : A) :\n    x \u2208 a \u2022 S \u2194 a\u207b\u00b9 \u2022 x \u2208 S", "theorem mul_le {M N P : AddSubmonoid R} : M * N \u2264 P \u2194 \u2200 m \u2208 M, \u2200 n \u2208 N, m * n \u2208 P", "theorem mul_le_mul_left {M N P : AddSubmonoid R} (h : M \u2264 N) : M * P \u2264 N * P", "theorem mul_mem_mul {M N : AddSubmonoid R} {m n : R} (hm : m \u2208 M) (hn : n \u2208 N) : m * n \u2208 M * N", "theorem pow_eq_closure_pow_set (s : AddSubmonoid R) (n : \u2115) :\n    s ^ n = closure ((s : Set R) ^ n)", "theorem AddSubsemigroup.toSubsemigroup_closure (S : Set A) :\n    AddSubsemigroup.toSubsemigroup (AddSubsemigroup.closure S) =\n      Subsemigroup.closure (Multiplicative.toAdd \u207b\u00b9' S)", "theorem map {v' : AddValuation R \u0393\u2080} (f : \u0393\u2080 \u2192+ \u0393'\u2080) (ht : f \u22a4 = \u22a4) (hf : Monotone f)\n    (inf : Injective f) (h : v.IsEquiv v') : (v.map f ht hf).IsEquiv (v'.map f ht hf)", "@[symm]\ntheorem symm (h : v\u2081.IsEquiv v\u2082) : v\u2082.IsEquiv v\u2081", "theorem ext_iff {v\u2081 v\u2082 : AddValuation R \u0393\u2080} : v\u2081 = v\u2082 \u2194 \u2200 (r : R), v\u2081 r = v\u2082 r", "@[simp]\nprotected lemma \u00abforall\u00bb {p : Additive \u03b1 \u2192 Prop} : (\u2200 a, p a) \u2194 \u2200 a, p (ofMul a) := Iff.rfl", "theorem Additive.isAddHom [Mul \u03b1] [Mul \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : IsMulHom f) :\n    @IsAddHom (Additive \u03b1) (Additive \u03b2) _ _ f", "@[simp]\ntheorem eval\u2090_mk (n : \u2115) (x : AdicCauchySequence I R) :\n    eval\u2090 I n (mk I R x) = Ideal.Quotient.mk (I ^ n) (x.val n)", "theorem isAdicCauchy_iff (f : \u2115 \u2192 M) :\n    IsAdicCauchy I M f \u2194 \u2200 n, f n \u2261 f (n + 1) [SMOD (I ^ n \u2022 \u22a4 : Submodule R M)]", "theorem mk_surjective : Function.Surjective (mk I M)", "@[simp]\ntheorem of_apply (x : M) (n : \u2115) : (of I M x).1 n = mkQ (I ^ n \u2022 \u22a4 : Submodule R M) x", "@[simp]\ntheorem transitionMap_comp_apply {m n k : \u2115} (hmn : m \u2264 n) (hnk : n \u2264 k)\n    (x : M \u29f8 (I ^ k \u2022 \u22a4 : Submodule R M)) :\n    transitionMap I M hmn (transitionMap I M hnk x) = transitionMap I M (hmn.trans hnk) x", "theorem val_smul (n : \u2115) (r : R) (f : AdicCompletion I M) : (r \u2022 f).val n = r \u2022 f.val n", "@[simp]\ntheorem val_sum {\u03b1 : Type*} (s : Finset \u03b1) (f : \u03b1 \u2192 AdicCompletion I M) (n : \u2115) :\n    (Finset.sum s f).val n = Finset.sum s (fun a \u21a6 (f a).val n)", "@[simp]\ntheorem Polynomial.quotQuotEquivComm_symm_mk_mk (p : R[X]) :\n    (Polynomial.quotQuotEquivComm I f).symm (Ideal.Quotient.mk (span\n    ({(Ideal.Quotient.mk (I.map C)) f} : Set (R[X] \u29f8 (I.map C)))) (Ideal.Quotient.mk (I.map C) p)) =\n      Ideal.Quotient.mk (span {f.map (Ideal.Quotient.mk I)}) (p.map (Ideal.Quotient.mk I))", "theorem coe_algEquivOfEq {p q : K[X]} (hp : p \u2260 0) (h_eq : p = q) :\n    (algEquivOfEq hp h_eq).toFun = liftHom p (root q) (by rw [h_eq, aeval_eq, mk_self])", "theorem coe_injective (h : degree f \u2260 0) : Function.Injective ((\u2191) : K \u2192 AdjoinRoot f)", "@[simp]\ntheorem isAdjoinRootMonic_map_eq_mk (hf : f.Monic) :\n    (AdjoinRoot.isAdjoinRootMonic f hf).map = AdjoinRoot.mk f", "theorem isIntegral_root' (hg : g.Monic) : IsIntegral R (root g)", "@[simp]\ntheorem lift_root : lift i a h (root f) = a", "theorem minpoly_powerBasis_gen_of_monic (hf : f.Monic) (hf' : f \u2260 0 := hf.ne_zero) :\n    minpoly K (powerBasis hf').gen = f", "theorem minpoly_root (hf : f \u2260 0) : minpoly K (root f) = f * C f.leadingCoeff\u207b\u00b9", "theorem smul_of [DistribSMul S R] [IsScalarTower S R R] (a : S) (x : R) :\n    a \u2022 of f x = of f (a \u2022 x)", "@[aesop (rule_sets := [builtin]) norm simp]\ntheorem heq_iff_eq (x y : \u03b1) : HEq x y \u2194 x = y", "@[simp]\ntheorem centroid_eq_iff [CharZero k] {n : \u2115} (s : Simplex k P n) {fs\u2081 fs\u2082 : Finset (Fin (n + 1))}\n    {m\u2081 m\u2082 : \u2115} (h\u2081 : fs\u2081.card = m\u2081 + 1) (h\u2082 : fs\u2082.card = m\u2082 + 1) :\n    fs\u2081.centroid k s.points = fs\u2082.centroid k s.points \u2194 fs\u2081 = fs\u2082", "@[simp]\ntheorem circumradius_reindex {m n : \u2115} (s : Simplex \u211d P m) (e : Fin (m + 1) \u2243 Fin (n + 1)) :\n    (s.reindex e).circumradius = s.circumradius", "theorem direction_mongePlane {n : \u2115} (s : Simplex \u211d P (n + 2)) {i\u2081 i\u2082 : Fin (n + 3)} :\n    (s.mongePlane i\u2081 i\u2082).direction =\n      (\u211d \u2219 s.points i\u2081 -\u1d65 s.points i\u2082)\u15ee \u2293 vectorSpan \u211d (Set.range s.points)", "theorem orthocenter_replace_orthocenter_eq_point {t\u2081 t\u2082 : Triangle \u211d P} {i\u2081 i\u2082 i\u2083 j\u2081 j\u2082 j\u2083 : Fin 3}\n    (hi\u2081\u2082 : i\u2081 \u2260 i\u2082) (hi\u2081\u2083 : i\u2081 \u2260 i\u2083) (hi\u2082\u2083 : i\u2082 \u2260 i\u2083) (hj\u2081\u2082 : j\u2081 \u2260 j\u2082) (hj\u2081\u2083 : j\u2081 \u2260 j\u2083)\n    (hj\u2082\u2083 : j\u2082 \u2260 j\u2083) (h\u2081 : t\u2082.points j\u2081 = t\u2081.orthocenter) (h\u2082 : t\u2082.points j\u2082 = t\u2081.points i\u2082)\n    (h\u2083 : t\u2082.points j\u2083 = t\u2081.points i\u2083) : t\u2082.orthocenter = t\u2081.points i\u2081", "theorem card_eq_finrank_add_one [Fintype \u03b9] (b : AffineBasis \u03b9 k P) :\n    Fintype.card \u03b9 = FiniteDimensional.finrank k V + 1", "@[simp]\ntheorem coord_apply_centroid [CharZero k] (b : AffineBasis \u03b9 k P) {s : Finset \u03b9} {i : \u03b9}\n    (hi : i \u2208 s) : b.coord i (s.centroid k b) = (s.card : k)\u207b\u00b9", "@[simp]\ntheorem coord_apply_eq (i : \u03b9) : b.coord i (b i) = 1", "@[ext]\ntheorem ext {b\u2081 b\u2082 : AffineBasis \u03b9 k P} (h : (b\u2081 : \u03b9 \u2192 P) = b\u2082) : b\u2081 = b\u2082", "theorem AffineBasis.interior_convexHull {\u03b9 E : Type*} [Finite \u03b9] [NormedAddCommGroup E]\n    [NormedSpace \u211d E] (b : AffineBasis \u03b9 \u211d E) :\n    interior (convexHull \u211d (range b)) = {x | \u2200 i, 0 < b.coord i x}", "@[simp]\ntheorem toMatrix_inv_vecMul_toMatrix (x : P) :\n    b.coords x \u1d65* (b.toMatrix b\u2082)\u207b\u00b9 = b\u2082.coords x", "@[simp]\ntheorem apply_lineMap (e : P\u2081 \u2243\u1d43[k] P\u2082) (a b : P\u2081) (c : k) :\n    e (AffineMap.lineMap a b c) = AffineMap.lineMap (e a) (e b) c", "@[simp]\ntheorem coe_mul (e e' : P\u2081 \u2243\u1d43[k] P\u2081) : \u21d1(e * e') = e \u2218 e'", "@[simp]\ntheorem range_eq (e : P\u2081 \u2243\u1d43[k] P\u2082) : range e = univ", "@[simp]\ntheorem trans_refl (e : P\u2081 \u2243\u1d43[k] P\u2082) : e.trans (refl k P\u2082) = e", "@[simp]\ntheorem _root_.AffineEquiv.wSameSide_map_iff {s : AffineSubspace R P} {x y : P} (f : P \u2243\u1d43[R] P') :\n    (s.map \u2191f).WSameSide (f x) (f y) \u2194 s.WSameSide x y", "theorem AffineIndependent.affineSpan_image_finset_eq_of_le_of_card_eq_finrank_add_one\n    [DecidableEq P] {p : \u03b9 \u2192 P}\n    (hi : AffineIndependent k p) {s : Finset \u03b9} {sp : AffineSubspace k P}\n    [FiniteDimensional k sp.direction] (hle : affineSpan k (s.image p : Set P) \u2264 sp)\n    (hc : Finset.card s = finrank k sp.direction + 1) : affineSpan k (s.image p : Set P) = sp", "theorem AffineIndependent.not_mem_affineSpan_diff [Nontrivial k] {p : \u03b9 \u2192 P}\n    (ha : AffineIndependent k p) (i : \u03b9) (s : Set \u03b9) : p i \u2209 affineSpan k (p '' (s \\ {i}))", "@[simp]\ntheorem _root_.AffineIsometry.angle_map {V\u2082 P\u2082 : Type*} [NormedAddCommGroup V\u2082]\n    [InnerProductSpace \u211d V\u2082] [MetricSpace P\u2082] [NormedAddTorsor V\u2082 P\u2082]\n    (f : P \u2192\u1d43\u2071[\u211d] P\u2082) (p\u2081 p\u2082 p\u2083 : P) : \u2220 (f p\u2081) (f p\u2082) (f p\u2083) = \u2220 p\u2081 p\u2082 p\u2083", "@[simp]\ntheorem coe_toAffineMap : \u21d1f.toAffineMap = f", "theorem comp_assoc (f : P\u2083 \u2192\u1d43\u2071[\ud835\udd5c] P\u2084) (g : P\u2082 \u2192\u1d43\u2071[\ud835\udd5c] P\u2083) (h : P \u2192\u1d43\u2071[\ud835\udd5c] P\u2082) :\n    (f.comp g).comp h = f.comp (g.comp h)", "@[simp]\ntheorem image_intrinsicInterior (\u03c6 : P \u2192\u1d43\u2071[\ud835\udd5c] Q) (s : Set P) :\n    intrinsicInterior \ud835\udd5c (\u03c6 '' s) = \u03c6 '' intrinsicInterior \ud835\udd5c s", "protected theorem lipschitz : LipschitzWith 1 f", "@[simp]\ntheorem coe_vaddConst_symm (p : P) : \u21d1(vaddConst \ud835\udd5c p).symm = fun p' => p' -\u1d65 p", "@[simp]\ntheorem pointReflection_midpoint_left (x y : P) : pointReflection \u211d (midpoint \u211d x y) x = y", "theorem vadd_vsub {f : P \u2192 P\u2082} (hf : Isometry f) {p : P} {g : V \u2192 V\u2082}\n    (hg : \u2200 v, g v = f (v +\u1d65 p) -\u1d65 f p) : Isometry g", "@[simp]\ntheorem add_linear (f g : P1 \u2192\u1d43[k] V2) : (f + g).linear = f.linear + g.linear", "@[simp, norm_cast]\ntheorem coe_neg (f : P1 \u2192\u1d43[k] V2) : \u21d1(-f) = -f", "@[simp, norm_cast]\ntheorem coe_smul (c : R) (f : P1 \u2192\u1d43[k] V2) : \u21d1(c \u2022 f) = c \u2022 \u21d1f", "@[simp, norm_cast]\ntheorem coe_sub (f g : P1 \u2192\u1d43[k] V2) : \u21d1(f - g) = f - g", "theorem comp_assoc (f\u2083\u2084 : P3 \u2192\u1d43[k] P4) (f\u2082\u2083 : P2 \u2192\u1d43[k] P3) (f\u2081\u2082 : P1 \u2192\u1d43[k] P2) :\n    (f\u2083\u2084.comp f\u2082\u2083).comp f\u2081\u2082 = f\u2083\u2084.comp (f\u2082\u2083.comp f\u2081\u2082)", "theorem AffineMap.continuous_of_finiteDimensional (f : PE \u2192\u1d43[\ud835\udd5c] PF) : Continuous f", "protected theorem differentiableOn : DifferentiableOn \ud835\udd5c f s", "protected theorem differentiableWithinAt : DifferentiableWithinAt \ud835\udd5c f s x", "theorem ext_linear_iff {f g : P1 \u2192\u1d43[k] P2} : f = g \u2194 (f.linear = g.linear) \u2227 (\u2203 p, f p = g p)", "@[simp]\ntheorem fst_lineMap (p\u2080 p\u2081 : P1 \u00d7 P2) (c : k) : (lineMap p\u2080 p\u2081 c).1 = lineMap p\u2080.1 p\u2081.1 c", "theorem hasDerivAt_lineMap :  HasDerivAt (lineMap a b) (b - a) x", "theorem homothety_eq_lineMap (c : P1) (r : k) (p : P1) : homothety c r p = lineMap c p r", "@[simp]\ntheorem homothety_one (c : P1) : homothety c (1 : k) = id k P1", "theorem AffineMap.isOpenMap_linear_iff {f : P \u2192\u1d43[\ud835\udd5c] P\u2082} : IsOpenMap f.linear \u2194 IsOpenMap f", "@[continuity]\ntheorem lineMap_continuous [TopologicalSpace R] [ContinuousSMul R F] {p v : F} :\n    Continuous (lineMap p v : R \u2192\u1d43[R] F)", "theorem lineMap_vadd_lineMap (v\u2081 v\u2082 : V1) (p\u2081 p\u2082 : P1) (c : k) :\n    lineMap v\u2081 v\u2082 c +\u1d65 lineMap p\u2081 p\u2082 c = lineMap (v\u2081 +\u1d65 p\u2081) (v\u2082 +\u1d65 p\u2082) c", "@[simp]\ntheorem linearMap_vsub (f : P1 \u2192\u1d43[k] P2) (p1 p2 : P1) : f.linear (p1 -\u1d65 p2) = f p1 -\u1d65 f p2", "@[simp]\ntheorem linear_surjective_iff (f : P1 \u2192\u1d43[k] P2) :\n    Function.Surjective f.linear \u2194 Function.Surjective f", "@[simp]\ntheorem pi_apply (c : P1) (i : \u03b9) : pi fp c i = fp i c", "@[simp]\ntheorem snd_linear : (snd : P1 \u00d7 P2 \u2192\u1d43[k] P2).linear = LinearMap.snd k V1 V2", "@[simp]\ntheorem vadd_apply (f : P1 \u2192\u1d43[k] V2) (g : P1 \u2192\u1d43[k] P2) (p : P1) : (f +\u1d65 g) p = f p +\u1d65 g p", "theorem SOppSide.wOppSide {s : AffineSubspace R P} {x y : P} (h : s.SOppSide x y) :\n    s.WOppSide x y", "theorem SSameSide.trans_wOppSide {s : AffineSubspace R P} {x y z : P} (hxy : s.SSameSide x y)\n    (hyz : s.WOppSide y z) : s.WOppSide x z", "theorem affineSpan_parallel_iff_vectorSpan_eq_and_eq_empty_iff_eq_empty {s\u2081 s\u2082 : Set P} :\n    affineSpan k s\u2081 \u2225 affineSpan k s\u2082 \u2194 vectorSpan k s\u2081 = vectorSpan k s\u2082 \u2227 (s\u2081 = \u2205 \u2194 s\u2082 = \u2205)", "@[simp]\ntheorem bot_coe : ((\u22a5 : AffineSubspace k P) : Set P) = \u2205", "theorem bot_ne_top : (\u22a5 : AffineSubspace k P) \u2260 \u22a4", "@[simp]\ntheorem coe_map (s : AffineSubspace k P\u2081) : (s.map f : Set P\u2082) = f '' s", "theorem comap_span (f : P\u2081 \u2243\u1d43[k] P\u2082) (s : Set P\u2082) :\n    (affineSpan k s).comap (f : P\u2081 \u2192\u1d43[k] P\u2082) = affineSpan k (f \u207b\u00b9' s)", "theorem direction_sup {s1 s2 : AffineSubspace k P} {p1 p2 : P} (hp1 : p1 \u2208 s1) (hp2 : p2 \u2208 s2) :\n    (s1 \u2294 s2).direction = s1.direction \u2294 s2.direction \u2294 k \u2219 p2 -\u1d65 p1", "theorem eq_bot_or_nonempty (Q : AffineSubspace k P) : Q = \u22a5 \u2228 (Q : Set P).Nonempty", "theorem eq_univ_of_subsingleton_span_eq_top {s : Set P} (h\u2081 : s.Subsingleton)\n    (h\u2082 : affineSpan k s = \u22a4) : s = (univ : Set P)", "theorem gc_map_comap (f : P\u2081 \u2192\u1d43[k] P\u2082) : GaloisConnection (map f) (comap f)", "theorem isConnected_setOf_sOppSide {s : AffineSubspace \u211d P} {x : P} (hx : x \u2209 s)\n    (h : (s : Set P).Nonempty) : IsConnected { y | s.SOppSide x y }", "theorem map_sup (s t : AffineSubspace k P\u2081) (f : P\u2081 \u2192\u1d43[k] P\u2082) : (s \u2294 t).map f = s.map f \u2294 t.map f", "theorem mem_perpBisector_iff_dist_eq : c \u2208 perpBisector p\u2081 p\u2082 \u2194 dist c p\u2081 = dist c p\u2082", "theorem not_mem_bot (p : P) : p \u2209 (\u22a5 : AffineSubspace k P)", "theorem parallel_comm {s\u2081 s\u2082 : AffineSubspace k P} : s\u2081 \u2225 s\u2082 \u2194 s\u2082 \u2225 s\u2081", "theorem sOppSide_iff_exists_right {s : AffineSubspace R P} {x y p\u2082 : P} (h : p\u2082 \u2208 s) :\n    s.SOppSide x y \u2194 x \u2209 s \u2227 y \u2209 s \u2227 \u2203 p\u2081 \u2208 s, SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)", "theorem sOppSide_vadd_left_iff {s : AffineSubspace R P} {x y : P} {v : V} (hv : v \u2208 s.direction) :\n    s.SOppSide (v +\u1d65 x) y \u2194 s.SOppSide x y", "theorem sSameSide_vadd_right_iff {s : AffineSubspace R P} {x y : P} {v : V} (hv : v \u2208 s.direction) :\n    s.SSameSide x (v +\u1d65 y) \u2194 s.SSameSide x y", "@[simp]\ntheorem span_univ : affineSpan k (Set.univ : Set P) = \u22a4", "@[simp]\ntheorem subtype\u2090\u1d62_toAffineMap (s : AffineSubspace \ud835\udd5c P) [Nonempty s] :\n    s.subtype\u2090\u1d62.toAffineMap = s.subtype", "theorem sup_direction_lt_of_nonempty_of_inter_empty {s1 s2 : AffineSubspace k P}\n    (h1 : (s1 : Set P).Nonempty) (h2 : (s2 : Set P).Nonempty) (he : (s1 \u2229 s2 : Set P) = \u2205) :\n    s1.direction \u2294 s2.direction < (s1 \u2294 s2).direction", "theorem wOppSide_lineMap_left {s : AffineSubspace R P} {x : P} (y : P) (h : x \u2208 s) {t : R}\n    (ht : t \u2264 0) : s.WOppSide (lineMap x y t) y", "@[simp]\ntheorem wOppSide_self_iff {s : AffineSubspace R P} {x : P} : s.WOppSide x x \u2194 x \u2208 s", "theorem wSameSide_and_wOppSide_iff {s : AffineSubspace R P} {x y : P} :\n    s.WSameSide x y \u2227 s.WOppSide x y \u2194 x \u2208 s \u2228 y \u2208 s", "protected lemma neg {f : \u211d \u2192 \u211d} (hf : GrowsPolynomially f) : GrowsPolynomially (-f)", "protected lemma norm (hf : GrowsPolynomially f) : GrowsPolynomially (fun x => \u2016f x\u2016)", "lemma eventually_b_le_r : \u2200\u1da0 (n:\u2115) in atTop, \u2200 i, (b i : \u211d) * n - (n / log n ^ 2) \u2264 r i n", "lemma eventually_one_add_smoothingFn_nonneg : \u2200\u1da0 (n:\u2115) in atTop, 0 \u2264 1 + \u03b5 n", "lemma eventually_one_add_smoothingFn_r_pos : \u2200\u1da0 (n:\u2115) in atTop, \u2200 i, 0 < 1 + \u03b5 (r i n)", "lemma growsPolynomially_deriv_rpow_p_mul_one_sub_smoothingFn (p : \u211d) :\n    GrowsPolynomially fun x => \u2016deriv (fun z => z ^ p * (1 - \u03b5 z)) x\u2016", "protected theorem bijective : Function.Bijective e", "theorem coe_coe_symm_apply_coe_apply {F : Type*} [EquivLike F A\u2081 A\u2082] [AlgEquivClass F R A\u2081 A\u2082]\n    (f : F) (x : A\u2081) :\n    (f : A\u2081 \u2243\u2090[R] A\u2082).symm (f x) = x", "@[simp, norm_cast]\ntheorem coe_ringEquiv : ((e : A\u2081 \u2243+* A\u2082) : A\u2081 \u2192 A\u2082) = e", "theorem AlgEquiv.isPurelyInseparable (e : K \u2243\u2090[F] E) [IsPurelyInseparable F K] :\n    IsPurelyInseparable F E", "@[deprecated map_neg (since := \"2024-06-20\")]\nprotected theorem map_neg (x) : e (-x) = -e x", "theorem restrictScalars_apply (f : A \u2243\u2090[S] B) (x : A) : f.restrictScalars R x = f x", "@[simp]\ntheorem symm_apply_apply (e : A\u2081 \u2243\u2090[R] A\u2082) : \u2200 x, e.symm (e x) = x", "theorem symm_bijective : Function.Bijective (symm : (A\u2081 \u2243\u2090[R] A\u2082) \u2192 A\u2082 \u2243\u2090[R] A\u2081)", "@[simp]\ntheorem toEquiv_eq_coe : e.toEquiv = e", "theorem comp {g : B \u2192\u2090[R] C} {f : A \u2192\u2090[R] B} (hg : g.Finite) (hf : f.Finite) : (g.comp f).Finite", "theorem id : FinitePresentation (AlgHom.id R A)", "theorem AlgHom.algebraicIndependent_iff (f : A \u2192\u2090[R] A') (hf : Injective f) :\n    AlgebraicIndependent R (f \u2218 x) \u2194 AlgebraicIndependent R x", "@[simp]\nprotected theorem coe_coe {F : Type*} [FunLike F A B] [AlgHomClass F R A B] (f : F) :\n    \u21d1(f : A \u2192\u2090[R] B) = f", "@[simp]\ntheorem coe_comp (\u03c6\u2081 : B \u2192\u2090[R] C) (\u03c6\u2082 : A \u2192\u2090[R] B) : \u21d1(\u03c6\u2081.comp \u03c6\u2082) = \u03c6\u2081 \u2218 \u03c6\u2082", "@[simp] theorem AlgHom.id_algebraCat_comp\n    {R} [CommRing R] {G : Type u} [Ring G] [Algebra R G] {H : AlgebraCat.{u} R} (f : G \u2192\u2090[R] H) :\n    AlgHom.comp (\ud835\udfd9 H) f = f", "theorem of_isLocalization_Away (r : R) [IsLocalization.Away r A] : Etale R A where", "theorem ker_fg_of_mvPolynomial {n : \u2115} (f : MvPolynomial (Fin n) R \u2192\u2090[R] A)\n    (hf : Function.Surjective f) [FinitePresentation R A] : f.toRingHom.ker.FG", "theorem baseChangeAux_surj {\u03c3 : Type*} {f : MvPolynomial \u03c3 R \u2192\u2090[R] A} (hf : Function.Surjective f) :\n    Function.Surjective (Algebra.TensorProduct.map (AlgHom.id B B) f)", "theorem of_equiv [FormallySmooth R A] (e : A \u2243\u2090[R] B) : FormallySmooth R B", "theorem lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]\n    (f : B \u2192+* C) (hf : IsNilpotent <| RingHom.ker f) (g\u2081 g\u2082 : A \u2192\u2090[R] B)\n    (h : f.comp \u2191g\u2081 = f.comp (g\u2082 : A \u2192+* B)) : g\u2081 = g\u2082", "theorem localization_map [FormallyUnramified R S] :\n    FormallyUnramified R\u2098 S\u2098", "@[simp] lemma val_of : (of w).val = w := rfl", "lemma algebraMap_surjective : Function.Surjective (algebraMap P.Ring S) := (\u27e8_, P.aeval_val_\u03c3 \u00b7\u27e9)", "theorem Algebra.IsAlgebraic.algHomEquivAlgHomOfSplits_apply_apply (L : Type*) [Field L]\n    [Algebra F L] [Algebra L A] [IsScalarTower F L A]\n    (hL : \u2200 x : K, (minpoly F x).Splits (algebraMap F L)) (f : K \u2192\u2090[F] L) (x : K) :\n    Algebra.IsAlgebraic.algHomEquivAlgHomOfSplits A L hL f x = algebraMap L A (f x)", "lemma normalClosure_le_iSup_adjoin :\n    normalClosure F K L \u2264 \u2a06 x : K, IntermediateField.adjoin F ((minpoly F x).rootSet L)", "theorem Algebra.IsAlgebraic.perfectField {K L : Type*} [Field K] [Field L] [Algebra K L]\n    [Algebra.IsAlgebraic K L] [PerfectField K] : PerfectField L", "theorem PowerBasis.norm_gen_eq_coeff_zero_minpoly (pb : PowerBasis R S) :\n    norm R pb.gen = (-1) ^ pb.dim * coeff (minpoly R pb.gen) 0", "private lemma span_range_relation_eq_ker_baseChange :\n    Ideal.span (Set.range fun i \u21a6 (MvPolynomial.map (algebraMap R T)) (P.relation i)) =\n      RingHom.ker (aeval (R := T) (S\u2081 := T \u2297[R] S) P.baseChange.val)", "@[simp]\ntheorem congr_refl : congr (.refl : A \u2243\u2090[S] A) (.refl : C \u2243\u2090[R] C) = .refl", "theorem map_comp (f\u2082 : B \u2192\u2090[S] C) (f\u2081 : A \u2192\u2090[S] B) (g\u2082 : E \u2192\u2090[R] F) (g\u2081 : D \u2192\u2090[R] E) :\n    map (f\u2082.comp f\u2081) (g\u2082.comp g\u2081) = (map f\u2082 g\u2082).comp (map f\u2081 g\u2081)", "@[simp]\ntheorem map_id : map (.id S A) (.id R C) = .id S _", "theorem opAlgEquiv_symm_apply (x : (A \u2297[R] B)\u1d50\u1d52\u1d56) :\n    (opAlgEquiv R S A B).symm x =\n      _root_.TensorProduct.map (opLinearEquiv R).toLinearMap (opLinearEquiv R).toLinearMap x.unop", "@[simp]\ntheorem rid_symm_apply (a : A) : (TensorProduct.rid R S A).symm a = a \u2297\u209c 1", "@[simp] theorem rid_toLinearEquiv :\n    (TensorProduct.rid R S A).toLinearEquiv = AlgebraTensorModule.rid R S A", "theorem adjoin_eq (S : Subalgebra R A) : adjoin R \u2191S = S", "theorem adjoin_eq_span : Subalgebra.toSubmodule (adjoin R s) = span R (Submonoid.closure s)", "@[elab_as_elim]\ntheorem adjoin_induction'' {x : A} (hx : x \u2208 adjoin R s)\n    {p : (x : A) \u2192 x \u2208 adjoin R s \u2192 Prop} (mem : \u2200 x (h : x \u2208 s), p x (subset_adjoin h))\n    (algebraMap : \u2200 (r : R), p (algebraMap R A r) (algebraMap_mem _ r))\n    (add : \u2200 x hx y hy, p x hx \u2192 p y hy \u2192 p (x + y) (add_mem hx hy))\n    (mul : \u2200 x hx y hy, p x hx \u2192 p y hy \u2192 p (x * y) (mul_mem hx hy)) :\n    p x hx", "theorem adjoin_le {S : Subalgebra R A} (H : s \u2286 S) : adjoin R s \u2264 S", "theorem adjoin_le_iff {S : Subalgebra R A} : adjoin R s \u2264 S \u2194 s \u2286 S", "theorem algebraMap_eq_smul_one (r : R) : algebraMap R A r = r \u2022 (1 : A)", "theorem coe_algebraMap_ofSubsemiring (S : Subsemiring R) : (algebraMap S R : S \u2192 R) = Subtype.val", "theorem commutes (r : R) (x : A) : algebraMap R A r * x = x * algebraMap R A r", "@[ext high]\ntheorem ext_id (f g : R \u2192\u2090[R] A) : f = g", "theorem fg_trans (h1 : (adjoin R s).toSubmodule.FG) (h2 : (adjoin (adjoin R s) t).toSubmodule.FG) :\n    (adjoin R (s \u222a t)).toSubmodule.FG", "@[simp]\nlemma Algebra.intNorm_zero : Algebra.intNorm A B 0 = 0", "theorem isIntegral_norm [Algebra R L] [Algebra R K] [IsScalarTower R K L] [IsSeparable K L]\n    [FiniteDimensional K L] {x : L} (hx : IsIntegral R x) : IsIntegral R (norm K x)", "theorem Algebra.isIntegral_sup {S T : Subalgebra R A} :\n    Algebra.IsIntegral R (S \u2294 T : Subalgebra R A) \u2194\n      Algebra.IsIntegral R S \u2227 Algebra.IsIntegral R T", "theorem leftMulMatrix_mulVec_repr (x y : S) :\n    leftMulMatrix b x *\u1d65 b.repr y = b.repr (x * y)", "theorem lmul_algebraMap (x : R) : Algebra.lmul R A (algebraMap R A x) = Algebra.lsmul R R A x", "theorem mem_sup_left {S T : Subalgebra R A} : \u2200 {x : A}, x \u2208 S \u2192 x \u2208 S \u2294 T", "@[simp]\ntheorem mem_top {x : A} : x \u2208 (\u22a4 : Subalgebra R A)", "theorem norm_eq_matrix_det [Fintype \u03b9] [DecidableEq \u03b9] (b : Basis \u03b9 R S) (s : S) :\n    norm R s = Matrix.det (Algebra.leftMulMatrix b s)", "@[simp]\ntheorem norm_eq_zero_iff [IsDomain R] [IsDomain S] [Module.Free R S] [Module.Finite R S] {x : S} :\n    norm R x = 0 \u2194 x = 0", "theorem prod_embeddings_eq_finrank_pow [Algebra L F] [IsScalarTower K L F] [IsAlgClosed E]\n    [IsSeparable K F] [FiniteDimensional K F] (pb : PowerBasis K L) :\n    \u220f \u03c3 : F \u2192\u2090[K] E, \u03c3 (algebraMap L F pb.gen) =\n      ((@Finset.univ _ (PowerBasis.AlgHom.fintype pb)).prod\n        fun \u03c3 : L \u2192\u2090[K] E => \u03c3 pb.gen) ^ finrank L F", "theorem trace_algebraMap_of_basis (x : R) : trace R S (algebraMap R S x) = Fintype.card \u03b9 \u2022 x", "@[simp]\ntheorem trace_comp_trace [Algebra K T] [Algebra L T] [IsScalarTower K L T] [FiniteDimensional K L]\n    [FiniteDimensional L T] : (trace K L).comp ((trace L T).restrictScalars K) = trace K T", "theorem trace_eq_matrix_trace [DecidableEq \u03b9] (b : Basis \u03b9 R S) (s : S) :\n    trace R S s = Matrix.trace (Algebra.leftMulMatrix b s)", "@[simp]\ntheorem id_apply (m : M) : (\ud835\udfd9 M : M \u2192 M) m = m", "theorem cardinal_mk_le_mul : #{ x : A // IsAlgebraic R x } \u2264 #R[X] * \u2135\u2080", "@[simp]\ntheorem cardinal_mk_of_countable_of_charZero [CharZero A] [IsDomain R] :\n    #{ x : A // IsAlgebraic R x } = \u2135\u2080", "theorem spanEval_ne_top : spanEval k \u2260 \u22a4", "theorem ofStep_succ (n : \u2115) : (ofStep k (n + 1)).comp (toStepSucc k n) = ofStep k n", "theorem AffineTargetMorphismProperty.isLocalOfOpenCoverImply (P : AffineTargetMorphismProperty)\n    (hP : P.toProperty.RespectsIso)\n    (H : \u2200 {X Y : Scheme.{u}} (f : X \u27f6 Y),\n      (\u2203 (\ud835\udcb0 : Scheme.OpenCover.{u} Y) (_ : \u2200 i, IsAffine (\ud835\udcb0.obj i)),\n        \u2200 i : \ud835\udcb0.J, P (pullback.snd : (\ud835\udcb0.pullbackCover f).obj i \u27f6 \ud835\udcb0.obj i)) \u2192\n        \u2200 {U : Scheme} (g : U \u27f6 Y) [IsAffine U] [IsOpenImmersion g],\n          P (pullback.snd : pullback f g \u27f6 U)) :\n    P.IsLocal", "theorem fromSpec_preimage_basicOpen :\n    hU.fromSpec \u207b\u00b9\u1d41 X.basicOpen f = PrimeSpectrum.basicOpen f", "theorem isLocalization_stalk (x : U) :\n    IsLocalization.AtPrime (X.presheaf.stalk x) (hU.primeIdealOf x).asIdeal", "theorem IsAffineOpen.preimage_of_isIso {X Y : Scheme} {U : Opens Y.carrier} (hU : IsAffineOpen U)\n    (f : X \u27f6 Y) [IsIso f] : IsAffineOpen (f \u207b\u00b9\u1d41 U)", "theorem \u03b9Opens_basicOpen_preimage (r : \u0393(X, \u22a4)):\n    IsAffineOpen (Scheme.\u03b9Opens (X.basicOpen r) \u207b\u00b9\u1d41 U)", "theorem app_eq_invApp_app_of_comp_eq {X Y U : Scheme.{u}} (f : Y \u27f6 U) (g : U \u27f6 X) (fg : Y \u27f6 X)\n    (H : fg = f \u226b g) [h : IsOpenImmersion g] (V : Opens U) :\n    f.app V = g.invApp V \u226b fg.app (g ''\u1d41 V) \u226b Y.presheaf.map\n      (eqToHom <| IsOpenImmersion.app_eq_invApp_app_of_comp_eq_aux f g fg H V).op", "theorem app_eq_invApp_app_of_comp_eq_aux {X Y U : Scheme.{u}} (f : Y \u27f6 U) (g : U \u27f6 X) (fg : Y \u27f6 X)\n    (H : fg = f \u226b g) [h : IsOpenImmersion g] (V : Opens U) :\n    f \u207b\u00b9\u1d41 V = fg \u207b\u00b9\u1d41 (g ''\u1d41 V)", "@[simp, reassoc]\nlemma isoOfRangeEq_inv_fac {X Y Z : Scheme.{u}} (f : X \u27f6 Z) (g : Y \u27f6 Z)\n    [IsOpenImmersion f] [IsOpenImmersion g] (e : Set.range f.1.base = Set.range g.1.base) :\n    (isoOfRangeEq f g e).inv \u226b f = g", "@[reassoc (attr := simp)]\ntheorem inv_naturality {U V : (Opens X)\u1d52\u1d56} (i : U \u27f6 V) :\n    X.presheaf.map i \u226b H.invApp (unop V) =\n      H.invApp (unop U) \u226b Y.presheaf.map (opensFunctor f |>.op.map i)", "theorem lift_range (H' : Set.range g.1.base \u2286 Set.range f.1.base) :\n    Set.range (lift f g H').1.base = f.1.base \u207b\u00b9' Set.range g.1.base", "theorem of_stalk_iso {X Y : LocallyRingedSpace} (f : X \u27f6 Y) (hf : OpenEmbedding f.1.base)\n    [stalk_iso : \u2200 x : X.1, IsIso (LocallyRingedSpace.stalkMap f x)] :\n    LocallyRingedSpace.IsOpenImmersion f", "theorem to\u0393SpecSheafedSpace_app_eq :\n    X.to\u0393SpecSheafedSpace.c.app (op (basicOpen r)) = X.to\u0393SpecCApp r", "lemma MorphismProperty.topologically_iso_le\n    (hP : \u2200 {\u03b1 \u03b2 : Type u} [TopologicalSpace \u03b1] [TopologicalSpace \u03b2] (f : \u03b1 \u2243\u209c \u03b2), P f) :\n    MorphismProperty.isomorphisms Scheme \u2264 (MorphismProperty.topologically P)", "theorem \u03b9_image_preimage_eq (i j : D.J) (U : Opens (D.U i).carrier) :\n    (Opens.map (\ud835\udda3.\u03b9 j).base).obj ((D.\u03b9_openEmbedding i).isOpenMap.functor.obj U) =\n      (opensFunctor (D.f j i)).obj\n        ((Opens.map (\ud835\udda3.t j i).base).obj ((Opens.map (\ud835\udda3.f i j).base).obj U))", "theorem lift_uniq (H : Set.range g.base \u2286 Set.range f.base) (l : Y \u27f6 X) (hl : l \u226b f = g) :\n    l = lift f g H", "@[simp]\ntheorem colimit_presheaf (F : J \u2964 PresheafedSpace.{_, _, v} C) :\n    (colimit F).presheaf = limit (pushforwardDiagramToColimit F).leftOp", "@[simp]\ntheorem map_id_c_app (F : J \u2964 PresheafedSpace.{_, _, v} C) (j) (U) :\n    (F.map (\ud835\udfd9 j)).c.app (op U) =\n      (Pushforward.id (F.obj j).presheaf).inv.app (op U) \u226b\n        (pushforwardEq (by simp) (F.obj j).presheaf).hom.app\n          (op U)", "theorem ofRestrict_top_c (X : PresheafedSpace C) :\n    (X.ofRestrict (Opens.openEmbedding \u22a4)).c =\n      eqToHom\n        (by\n          rw [restrict_top_presheaf, \u2190 Presheaf.Pushforward.comp_eq]\n          erw [Iso.inv_hom_id]\n          rw [Presheaf.Pushforward.id_eq])", "theorem mem_carrier_iff_of_mem_mul (hm : 0 < m)\n    (q : Spec.T A\u2070_ f) (a : A) {n} (hn : a \u2208 \ud835\udc9c (n * m)) :\n    a \u2208 carrier f_deg q \u2194 (HomogeneousLocalization.mk \u27e8m * n, \u27e8a, mul_comm n m \u25b8 hn\u27e9,\n        \u27e8f ^ n, by rw [mul_comm]; mem_tac\u27e9, \u27e8_, rfl\u27e9\u27e9 : A\u2070_ f) \u2208 q.asIdeal", "lemma toSpec_base_isIso {f} {m} (f_deg : f \u2208 \ud835\udc9c m) (hm : 0 < m) :\n    IsIso (toSpec \ud835\udc9c f).1.base", "theorem QuasiCompact.affineProperty_isLocal : (QuasiCompact.affineProperty : _).IsLocal", "@[simp]\ntheorem basicOpen_mul {U : Opens X} (f g : X.presheaf.obj (op U)) :\n    X.basicOpen (f * g) = X.basicOpen f \u2293 X.basicOpen g", "theorem basicOpen_of_isUnit {U : Opens X} {f : X.presheaf.obj (op U)} (hf : IsUnit f) :\n    X.basicOpen f = U", "@[simp, reassoc]\ntheorem pullbackFst\u03b9ToV_fst (i j : \ud835\udcb0.J) :\n    pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b pullback.fst = pullback.snd", "@[simp, reassoc]\ntheorem pullbackFst\u03b9ToV_snd (i j : \ud835\udcb0.J) :\n    pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b pullback.snd = pullback.fst \u226b pullback.snd", "theorem Scheme.Spec_map_presheaf_map_eqToHom {X : Scheme} {U V : Opens X} (h : U = V) (W) :\n    (Spec.map (X.presheaf.map (eqToHom h).op)).app W = eqToHom (by cases h; dsimp; simp)", "@[simp, reassoc] theorem comp_coeBase {X Y Z : Scheme} (f : X \u27f6 Y) (g : Y \u27f6 Z) :\n    (f \u226b g).val.base = f.val.base \u226b g.val.base", "@[reassoc]\ntheorem comp_val {X Y Z : Scheme} (f : X \u27f6 Y) (g : Y \u27f6 Z) : (f \u226b g).val = f.val \u226b g.val", "@[simp]\ntheorem preimage_basicOpen {X Y : Scheme} (f : X \u27f6 Y) {U : Opens Y} (r : \u0393(Y, U)) :\n    f \u207b\u00b9\u1d41 (Y.basicOpen r) = X.basicOpen (f.app U r)", "lemma subcanonical_zariskiTopology : Sheaf.Subcanonical zariskiTopology", "theorem \u03b9_isoPresheafedSpace_inv (i : D.J) :\n    D.toPresheafedSpaceGlueData.toGlueData.\u03b9 i \u226b D.isoPresheafedSpace.inv = \ud835\udda3.\u03b9 i", "theorem image_preimage_is_empty (j : Discrete \u03b9) (h : i \u2260 j) (U : Opens (F.obj i)) :\n    (Opens.map (colimit.\u03b9 (F \u22d9 SheafedSpace.forgetToPresheafedSpace) j).base).obj\n        ((Opens.map (preservesColimitIso SheafedSpace.forgetToPresheafedSpace F).inv.base).obj\n          ((sigma_\u03b9_openEmbedding F i).isOpenMap.functor.obj U)) =\n      \u22a5", "theorem to_iso [h' : Epi f.base] : IsIso f", "lemma Spec.map_base : (Spec.map f).1.base = PrimeSpectrum.comap f := rfl", "@[simp] lemma Spec.preimage_map : Spec.preimage (Spec.map \u03c6) = \u03c6", "@[simp]\ntheorem Spec.toPresheafedSpace_obj (R : CommRingCat.{u}\u1d52\u1d56) :\n    Spec.toPresheafedSpace.obj R = (Spec.sheafedSpaceObj (unop R)).toPresheafedSpace", "theorem Spec_map_localization_isIso (R : CommRingCat.{u}) (M : Submonoid R)\n    (x : PrimeSpectrum (Localization M)) :\n    IsIso\n      (PresheafedSpace.stalkMap\n        (Spec.toPresheafedSpace.map (CommRingCat.ofHom (algebraMap R (Localization M))).op) x)", "theorem IsFraction.eq_mk' {U : Opens (PrimeSpectrum.Top R)} {f : \u2200 x : U, Localizations R x}\n    (hf : IsFraction f) :\n    \u2203 r s : R,\n      \u2200 x : U,\n        \u2203 hs : s \u2209 x.1.asIdeal,\n          f x =\n            IsLocalization.mk' (Localization.AtPrime _) r\n              (\u27e8s, hs\u27e9 : (x : PrimeSpectrum.Top R).asIdeal.primeCompl)", "theorem algebraMap_pushforward_stalk :\n    algebraMap R ((Spec.topMap f _* (structureSheaf S).1).stalk p) =\n      f \u226b StructureSheaf.toPushforwardStalk f p", "theorem comap_id {U V : Opens (PrimeSpectrum.Top R)} (hUV : U = V) :\n    (comap (RingHom.id R) U V fun p hpV => by rwa [hUV, PrimeSpectrum.comap_id]) =\n      eqToHom (show (structureSheaf R).1.obj (op U) = _ by rw [hUV])", "theorem const_apply' (f g : R) (U : Opens (PrimeSpectrum.Top R))\n    (hu : \u2200 x \u2208 U, g \u2208 (x : PrimeSpectrum.Top R).asIdeal.primeCompl) (x : U)\n    (hx : g \u2208 (x : PrimeSpectrum.Top R).asIdeal.primeCompl) :\n    (const R f g U hu).1 x = IsLocalization.mk' _ f \u27e8g, hx\u27e9", "theorem const_self (f : R) (U hu) : const R f f U hu = 1", "theorem const_zero (f : R) (U hu) : const R 0 f U hu = 0", "@[simp]\ntheorem germ_toOpen (U : Opens (PrimeSpectrum.Top R)) (x : U) (f : R) :\n    (structureSheaf R).presheaf.germ x (toOpen R U f) = toStalk R x f", "theorem isLocalizedModule_toPushforwardStalkAlgHom_aux (y) :\n    \u2203 x : S \u00d7 p.asIdeal.primeCompl, x.2 \u2022 y = toPushforwardStalkAlgHom R S p x.1", "@[simp, reassoc, elementwise]\ntheorem localizationToStalk_stalkSpecializes {R : Type*} [CommRing R] {x y : PrimeSpectrum R}\n    (h : x \u2933 y) :\n    StructureSheaf.localizationToStalk R y \u226b (structureSheaf R).presheaf.stalkSpecializes h =\n      CommRingCat.ofHom (PrimeSpectrum.localizationMapOfSpecializes h) \u226b\n        StructureSheaf.localizationToStalk R x", "@[simp]\ntheorem stalkToFiberRingHom_germ (U : Opens (PrimeSpectrum.Top R)) (x : U)\n    (s : (structureSheaf R).1.obj (op U)) :\n    stalkToFiberRingHom R x ((structureSheaf R).presheaf.germ x s) = s.1 x", "theorem toBasicOpen_injective (f : R) : Function.Injective (toBasicOpen R f)", "theorem toOpen_eq_const (U : Opens (PrimeSpectrum.Top R)) (f : R) :\n    toOpen R U f = const R f 1 U fun x _ => (Ideal.ne_top_iff_one _).1 x.2.1", "theorem exists_pow_mul_eq_zero_of_res_basicOpen_eq_zero_of_isCompact (X : Scheme.{u})\n    {U : Opens X} (hU : IsCompact U.1) (x f : \u0393(X, U))\n    (H : x |_ X.basicOpen f = 0) : \u2203 n : \u2115, f ^ n * x = 0", "theorem isOpenImmersion_iff_stalk {f : X \u27f6 Y} : IsOpenImmersion f \u2194\n    OpenEmbedding f.1.base \u2227 \u2200 x, IsIso (PresheafedSpace.stalkMap f.1 x)", "theorem isOpenImmersion_isLocalAtTarget : PropertyIsLocalAtTarget @IsOpenImmersion", "theorem isOpenImmersion_stableUnderBaseChange :\n    MorphismProperty.StableUnderBaseChange @IsOpenImmersion", "theorem isReduced_of_isReduced_stalk [\u2200 x : X, _root_.IsReduced (X.presheaf.stalk x)] :\n    IsReduced X", "@[simp, reassoc]\ntheorem morphismRestrict_\u03b9 {X Y : Scheme.{u}} (f : X \u27f6 Y) (U : Opens Y) :\n    (f \u2223_ U) \u226b Scheme.\u03b9Opens U = Scheme.\u03b9Opens (f \u207b\u00b9\u1d41 U) \u226b f", "theorem quasiCompact_stableUnderBaseChange : MorphismProperty.StableUnderBaseChange @QuasiCompact", "theorem quasiSeparated_respectsIso : MorphismProperty.RespectsIso @QuasiSeparated", "theorem sourceAffineLocally_of_source_open_cover_aux (h\u2081 : RingHom.RespectsIso @P)\n    (h\u2083 : RingHom.OfLocalizationSpanTarget @P) {X Y : Scheme.{u}} (f : X \u27f6 Y) (U : X.affineOpens)\n    (s : Set (X.presheaf.obj (op U.1))) (hs : Ideal.span s = \u22a4)\n    (hs' : \u2200 r : s, P (Scheme.\u0393.map (Scheme.\u03b9Opens (X.basicOpen r.1) \u226b f).op)) :\n    P (Scheme.\u0393.map (Scheme.\u03b9Opens U \u226b f).op)", "theorem targetAffineLocally_of_openCover {P : AffineTargetMorphismProperty} (hP : P.IsLocal)\n    {X Y : Scheme} (f : X \u27f6 Y) (\ud835\udcb0 : Y.OpenCover) [\u2200 i, IsAffine (\ud835\udcb0.obj i)]\n    (h\ud835\udcb0 : \u2200 i, P (pullback.snd : (\ud835\udcb0.pullbackCover f).obj i \u27f6 \ud835\udcb0.obj i)) :\n    targetAffineLocally P f", "theorem universallyClosed_stableUnderBaseChange : StableUnderBaseChange @UniversallyClosed", "@[simp]\ntheorem SpecMap_\u0393SpecIso_hom (R : CommRingCat.{u}) :\n    Spec.map ((Scheme.\u0393SpecIso R).hom) = adjunction.unit.app (Spec R)", "lemma adjunction_unit_map_basicOpen (X : Scheme.{u}) (r : \u0393(X, \u22a4)) :\n    (\u0393Spec.adjunction.unit.app X \u207b\u00b9\u1d41 (PrimeSpectrum.basicOpen r)) = X.basicOpen r", "theorem right_triangle (R : CommRingCat) :\n    identityTo\u0393Spec.app (Spec.toLocallyRingedSpace.obj <| op R) \u226b\n        Spec.toLocallyRingedSpace.map (Spec\u0393Identity.inv.app R).op =\n      \ud835\udfd9 _", "@[reassoc (attr := simp)]\ntheorem toOpen_unit_app_val_c_app' {X : Scheme.{u}} (U : Opens (PrimeSpectrum \u0393(X, \u22a4))) :\n    toOpen \u0393(X, \u22a4) U \u226b (adjunction.unit.app X).app U =\n      X.presheaf.map (homOfLE (by exact le_top)).op", "theorem AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_C\n    (hx : AlgebraicIndependent R x) (r) :\n    hx.mvPolynomialOptionEquivPolynomialAdjoin (C r) = Polynomial.C (algebraMap _ _ r)", "theorem ne_zero [Nontrivial R] (i : \u03b9) : x i \u2260 0", "theorem AlgebraicIndependent.repr_ker :\n    RingHom.ker (hx.repr : adjoin R (range x) \u2192+* MvPolynomial \u03b9 R) = \u22a5", "@[simp]\ntheorem obj_d_eq (X : SimplicialObject C) (n : \u2115) :\n    (AlternatingFaceMapComplex.obj X).d (n + 1) n\n      = \u2211 i : Fin (n + 2), (-1 : \u2124) ^ (i : \u2115) \u2022 X.\u03b4 i", "theorem equivalence\u2081UnitIso_eq : (equivalence\u2081 hF).unitIso = equivalence\u2081UnitIso hF", "@[reassoc]\ntheorem comp_P_eq_self {Y : C} {n q : \u2115} {\u03c6 : Y \u27f6 X _[n + 1]} (v : HigherFacesVanish q \u03c6) :\n    \u03c6 \u226b (P q).f (n + 1) = \u03c6", "theorem HigherFacesVanish.comp_\u03c3 {Y : C} {X : SimplicialObject C} {n b q : \u2115} {\u03c6 : Y \u27f6 X _[n + 1]}\n    (v : HigherFacesVanish q \u03c6) (hnbq : n + 1 = b + q) :\n    HigherFacesVanish q\n      (\u03c6 \u226b\n        X.\u03c3 \u27e8b, by\n          simp only [hnbq, Nat.lt_add_one_iff, le_add_iff_nonneg_right, zero_le]\u27e9)", "theorem HigherFacesVanish.inclusionOfMooreComplexMap (n : \u2115) :\n    HigherFacesVanish (n + 1) ((inclusionOfMooreComplexMap X).f (n + 1))", "theorem of_P : \u2200 q n : \u2115, HigherFacesVanish q ((P q).f (n + 1) : X _[n + 1] \u27f6 X _[n + 1])", "@[simp]\ntheorem postComp_\u03c6 : (f.postComp h).\u03c6 = f.\u03c6 \u226b h", "@[simp]\ntheorem N\u2082\u0393\u2082_inv_app_f_f (X : Karoubi (ChainComplex C \u2115)) (n : \u2115) :\n    (N\u2082\u0393\u2082.inv.app X).f.f n =\n      X.p.f n \u226b ((\u0393\u2080.splitting X.X).cofan _).inj (Splitting.IndexSet.id (op [n]))", "@[reassoc (attr := simp)]\ntheorem PInfty_f_naturality (n : \u2115) {X Y : SimplicialObject C} (f : X \u27f6 Y) :\n    f.app (op [n]) \u226b PInfty.f n = PInfty.f n \u226b f.app (op [n])", "@[reassoc (attr := simp)]\ntheorem PInfty_on_\u0393\u2080_splitting_summand_eq_self (K : ChainComplex C \u2115) {n : \u2115} :\n    ((\u0393\u2080.splitting K).cofan _).inj (Splitting.IndexSet.id (op [n])) \u226b\n      (PInfty : K[\u0393\u2080.obj K] \u27f6 _).f n =\n      ((\u0393\u2080.splitting K).cofan _).inj (Splitting.IndexSet.id (op [n]))", "theorem P_add_Q (q : \u2115) : P q + Q q = \ud835\udfd9 K[X]", "@[simp]\ntheorem QInfty_f_0 : (QInfty.f 0 : X _[0] \u27f6 X _[0]) = 0", "theorem c_mk (i j : \u2115) (h : j + 1 = i) : c.Rel i j", "theorem comp_P_eq_self_iff {Y : C} {n q : \u2115} {\u03c6 : Y \u27f6 X _[n + 1]} :\n    \u03c6 \u226b (P q).f (n + 1) = \u03c6 \u2194 HigherFacesVanish q \u03c6", "theorem compatibility_N\u2082_N\u2081_karoubi :\n    N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor =\n      karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n        N\u2081 \u22d9 (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n            Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse _", "theorem identity_N\u2082_objectwise (P : Karoubi (SimplicialObject C)) :\n    (N\u2082\u0393\u2082.inv.app (N\u2082.obj P) : N\u2082.obj P \u27f6 N\u2082.obj (\u0393\u2082.obj (N\u2082.obj P))) \u226b\n    N\u2082.map (\u0393\u2082N\u2082.natTrans.app P) = \ud835\udfd9 (N\u2082.obj P)", "theorem karoubi_PInfty_f {Y : Karoubi (SimplicialObject C)} (n : \u2115) :\n    ((PInfty : K[(karoubiFunctorCategoryEmbedding _ _).obj Y] \u27f6 _).f n).f =\n      Y.p.app (op [n]) \u226b (PInfty : K[Y.X] \u27f6 _).f n", "@[simp]\nlemma toKaroubiCompN\u2082IsoN\u2081_inv_app (X : SimplicialObject C) :\n    (toKaroubiCompN\u2082IsoN\u2081.inv.app X).f = PInfty := rfl", "theorem objX_zero : objX X 0 = \u22a4", "theorem coe_multilinearMap_injective :\n    Function.Injective ((\u2191) : M [\u22c0^\u03b9]\u2192\u2097[R] N \u2192 MultilinearMap R (fun _ : \u03b9 => M) N)", "theorem map_vecCons_add {n : \u2115} (f : M [\u22c0^Fin n.succ]\u2192\u2097[R] N) (m : Fin n \u2192 M) (x y : M) :\n    f (Matrix.vecCons (x + y) m) = f (Matrix.vecCons x m) + f (Matrix.vecCons y m)", "lemma AnalyticAt.prod {e : E} {f : E \u2192 F} {g : E \u2192 G}\n    (hf : AnalyticAt \ud835\udd5c f e) (hg : AnalyticAt \ud835\udd5c g e) :\n    AnalyticAt \ud835\udd5c (fun x \u21a6 (f x, g x)) e", "theorem AnalyticOn.comp\u2082 {h : F \u00d7 G \u2192 H} {f : E \u2192 F} {g : E \u2192 G} {s : Set (F \u00d7 G)} {t : Set E}\n    (ha : AnalyticOn \ud835\udd5c h s) (fa : AnalyticOn \ud835\udd5c f t) (ga : AnalyticOn \ud835\udd5c g t)\n    (m : \u2200 x, x \u2208 t \u2192 (f x, g x) \u2208 s) : AnalyticOn \ud835\udd5c (fun x \u21a6 h (f x, g x)) t", "theorem AnalyticOn.congr {s : Set E} (hs : IsOpen s) (hf : AnalyticOn \ud835\udd5c f s) (hg : s.EqOn f g) :\n    AnalyticOn \ud835\udd5c g s", "theorem AnalyticOn.congr' {s : Set E} (hf : AnalyticOn \ud835\udd5c f s) (hg : f =\u1da0[\ud835\udcdd\u02e2 s] g) :\n    AnalyticOn \ud835\udd5c g s", "theorem AnalyticOn.eval_polynomial {A} [NormedCommRing A] [NormedAlgebra \ud835\udd5c A] (p : A[X]) :\n    AnalyticOn \ud835\udd5c (eval \u00b7 p) Set.univ", "theorem AnalyticOn.iterated_deriv [CompleteSpace F] (h : AnalyticOn \ud835\udd5c f s) (n : \u2115) :\n    AnalyticOn \ud835\udd5c (_root_.deriv^[n] f) s", "theorem And.exists {p q : Prop} {f : p \u2227 q \u2192 Prop} : (\u2203 h, f h) \u2194 \u2203 hp hq, f \u27e8hp, hq\u27e9", "theorem le_mul_ediam_image (hf : AntilipschitzWith K f) (s : Set \u03b1) : diam s \u2264 K * diam (f '' s)", "protected theorem subsingleton {\u03b1 \u03b2} [EMetricSpace \u03b1] [PseudoEMetricSpace \u03b2] {f : \u03b1 \u2192 \u03b2}\n    (h : AntilipschitzWith 0 f) : Subsingleton \u03b1", "theorem subtype_coe (s : Set \u03b1) : AntilipschitzWith 1 ((\u2191) : s \u2192 \u03b1)", "theorem to_rightInverse (hf : AntilipschitzWith K f) {g : \u03b2 \u2192 \u03b1} (hg : Function.RightInverse g f) :\n    LipschitzWith K g", "@[trans]\ntheorem AntisymmRel.trans [IsTrans \u03b1 r] {a b c : \u03b1} (hab : AntisymmRel r a b)\n    (hbc : AntisymmRel r b c) : AntisymmRel r a c", "protected theorem Antitone.antitoneOn (hf : Antitone f) (s : Set \u03b1) : AntitoneOn f s", "theorem Antitone.ball {P : \u03b2 \u2192 \u03b1 \u2192 Prop} {s : Set \u03b2} (hP : \u2200 x \u2208 s, Antitone (P x)) :\n    Antitone fun y => \u2200 x \u2208 s, P x y", "theorem continuousWithinAt_Ioi_iff_rightLim_eq :\n    ContinuousWithinAt f (Ioi x) x \u2194 rightLim f x = f x", "theorem Antitone.eq_of_le_of_le {a\u2081 a\u2082 : \u03b1} (h_anti : Antitone f) (h_fa : f a\u2081 = f a\u2082) {i : \u03b1}\n    (h\u2081 : a\u2081 \u2264 i) (h\u2082 : i \u2264 a\u2082) : f i = f a\u2081", "lemma _root_.Antitone.image_uIcc_subset (hf : Antitone f) : f '' uIcc a b \u2286 uIcc (f a) (f b)", "theorem image_upperBounds_subset_lowerBounds_image : f '' upperBounds s \u2286 lowerBounds (f '' s)", "theorem Antitone.imp (hf : Antitone f) (h : a \u2264 b) : f b \u2264 f a", "theorem _root_.Antitone.isBoundedUnder_ge_comp [Preorder \u03b1] [Preorder \u03b2] {l : Filter \u03b3} {u : \u03b3 \u2192 \u03b1}\n    {v : \u03b1 \u2192 \u03b2} (hv : Antitone v) (hl : l.IsBoundedUnder (\u00b7 \u2264 \u00b7) u) :\n    l.IsBoundedUnder (\u00b7 \u2265 \u00b7) (v \u2218 u)", "theorem Antitone.isBoundedUnder_ge_comp_iff [Nonempty \u03b2] [LinearOrder \u03b2] [Preorder \u03b3] [NoMinOrder \u03b3]\n    {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} {l : Filter \u03b1} (hg : Antitone g) (hg' : Tendsto g atTop atBot) :\n    IsBoundedUnder (\u00b7 \u2265 \u00b7) l (g \u2218 f) \u2194 IsBoundedUnder (\u00b7 \u2264 \u00b7) l f", "theorem Antitone.map_iInf_of_continuousAt' {\u03b9 : Sort*} [Nonempty \u03b9] {f : \u03b1 \u2192 \u03b2} {g : \u03b9 \u2192 \u03b1}\n    (Cf : ContinuousAt f (iInf g)) (Af : Antitone f)\n    (bdd : BddBelow (range g)", "theorem Antitone.map_iSup_of_continuousAt {\u03b9 : Sort*} {f : \u03b1 \u2192 \u03b2} {g : \u03b9 \u2192 \u03b1}\n    (Cf : ContinuousAt f (iSup g)) (Af : Antitone f) (fbot : f \u22a5 = \u22a4) :\n    f (\u2a06 i, g i) = \u2a05 i, f (g i)", "theorem Antitone.map_iSup\u2082_le [CompleteLattice \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : Antitone f) (s : \u2200 i, \u03ba i \u2192 \u03b1) :\n    f (\u2a06 (i) (j), s i j) \u2264 \u2a05 (i) (j), f (s i j)", "theorem Antitone.map_sSup_of_continuousAt {f : \u03b1 \u2192 \u03b2} {s : Set \u03b1} (Cf : ContinuousAt f (sSup s))\n    (Af : Antitone f) (fbot : f \u22a5 = \u22a4) : f (sSup s) = sInf (f '' s)", "theorem Antitone.tendsto_le_alternating_series\n    (hfl : Tendsto (fun n \u21a6 \u2211 i \u2208 range n, (-1) ^ i * f i) atTop (\ud835\udcdd l))\n    (hfa : Antitone f) (k : \u2115) : l \u2264 \u2211 i \u2208 range (2 * k + 1), (-1) ^ i * f i", "theorem Antitone.union [Preorder \u03b2] {f g : \u03b2 \u2192 Set \u03b1} (hf : Antitone f) (hg : Antitone g) :\n    Antitone fun x => f x \u222a g x", "theorem image_lowerBounds_subset_upperBounds_image :\n    f '' (lowerBounds s \u2229 t) \u2286 upperBounds (f '' s)", "theorem mem_lowerBounds_image_self : a \u2208 upperBounds t \u2192 a \u2208 t \u2192 f a \u2208 lowerBounds (f '' t)", "protected theorem min [Preorder \u03b1] [LinearOrder \u03b2] {f g : \u03b1 \u2192 \u03b2} {s : Set \u03b1} (hf : AntitoneOn f s)\n    (hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s", "theorem _root_.AntitoneOn.set_prod (hf : AntitoneOn f s) (hg : AntitoneOn g s) :\n    AntitoneOn (fun x => f x \u00d7\u02e2 g x) s", "theorem Antivary.comp_antitone_left (h : Antivary f g) (hf : Antitone f') : Monovary (f' \u2218 f) g", "lemma Antivary.pow_right\u2080 (hg : 0 \u2264 g) (hfg : Antivary f g) (n : \u2115) : Antivary f (g ^ n)", "theorem AntivaryOn.dual : AntivaryOn f g s \u2192 AntivaryOn (toDual \u2218 f) (toDual \u2218 g) s", "lemma AntivaryOn.mul_left\u2080 (hf\u2081 : \u2200 i \u2208 s, 0 \u2264 f\u2081 i) (hf\u2082 : \u2200 i \u2208 s, 0 \u2264 f\u2082 i)\n    (h\u2081 : AntivaryOn f\u2081 g s) (h\u2082 : AntivaryOn f\u2082 g s) : AntivaryOn (f\u2081 * f\u2082) g s", "theorem AntivaryOn.sum_mul_lt_sum_comp_perm_mul_iff (hfg : AntivaryOn f g s)\n    (h\u03c3 : {x | \u03c3 x \u2260 x} \u2286 s) :\n    \u2211 i \u2208 s, f i * g i < \u2211 i \u2208 s, f (\u03c3 i) * g i \u2194 \u00acAntivaryOn (f \u2218 \u03c3) g s", "protected theorem congr_arg (\u03b7 : ApplicativeTransformation F G) {\u03b1 : Type u} {x y : F \u03b1}\n    (h : x = y) : \u03b7 x = \u03b7 y", "protected theorem congr_fun (\u03b7 \u03b7' : ApplicativeTransformation F G) (h : \u03b7 = \u03b7') {\u03b1 : Type u}\n    (x : F \u03b1) : \u03b7 x = \u03b7' x", "protected theorem antilipschitz (hf : ApproximatesLinearOn f (f' : E \u2192L[\ud835\udd5c] F) s c)\n    (hc : Subsingleton E \u2228 c < N\u207b\u00b9) : AntilipschitzWith (N\u207b\u00b9 - c)\u207b\u00b9 (s.restrict f)", "theorem mono_num (hc : c \u2264 c') (hf : ApproximatesLinearOn f f' s c) :\n    ApproximatesLinearOn f f' s c'", "@[simp]\ntheorem toPartialHomeomorph_coe (hf : ApproximatesLinearOn f (f' : E \u2192L[\ud835\udd5c] F) s c)\n    (hc : Subsingleton E \u2228 c < N\u207b\u00b9) (hs : IsOpen s) :\n    (hf.toPartialHomeomorph f s hc hs : E \u2192 F) = f", "nonrec theorem IsMultiplicative.eulerProduct_tprod {f : ArithmeticFunction R}\n    (hf : f.IsMultiplicative) (hsum : Summable (\u2016f \u00b7\u2016)) :\n    \u220f' p : Primes, \u2211' e, f (p ^ e) = \u2211' n, f n", "@[arith_mult]\ntheorem prodPrimeFactors [CommMonoidWithZero R] (f : \u2115 \u2192 R) :\n    IsMultiplicative (prodPrimeFactors f)", "theorem prodPrimeFactors_add_of_squarefree [CommSemiring R] {f g : ArithmeticFunction R}\n    (hf : IsMultiplicative f) (hg : IsMultiplicative g) {n : \u2115} (hn : Squarefree n) :\n    \u220f\u1d56 p \u2223 n, (f + g) p = (f * g) n", "lemma LSeriesSummable_mul {f g : ArithmeticFunction \u2102} {s : \u2102} (hf : LSeriesSummable \u2197f s)\n    (hg : LSeriesSummable \u2197g s) :\n    LSeriesSummable \u2197(f * g) s", "@[simp]\ntheorem cardFactors_apply_prime {p : \u2115} (hp : p.Prime) : \u03a9 p = 1", "lemma cardFactors_zero : \u03a9 0 = 0", "theorem coe_inj {f g : ArithmeticFunction R} : (f : \u2115 \u2192 R) = g \u2194 f = g", "@[simp]\ntheorem coe_zeta_mul_moebius : (\u03b6 * \u03bc : ArithmeticFunction \u2124) = 1", "@[ext]\ntheorem ext \u2983f g : ArithmeticFunction R\u2984 (h : \u2200 x, f x = g x) : f = g", "@[arith_mult]\ntheorem isMultiplicative_moebius : IsMultiplicative \u03bc", "theorem moebius_apply_isPrimePow_not_prime {n : \u2115} (hn : IsPrimePow n) (hn' : \u00acn.Prime) :\n    \u03bc n = 0", "theorem moebius_ne_zero_iff_squarefree {n : \u2115} : \u03bc n \u2260 0 \u2194 Squarefree n", "lemma ArithmeticFunction.one_eq_delta : \u2197(1 : ArithmeticFunction \u2102) = \u03b4", "theorem prod_eq_iff_prod_pow_moebius_eq_of_nonzero [CommGroupWithZero R] {f g : \u2115 \u2192 R}\n    (hf : \u2200 n : \u2115, 0 < n \u2192 f n \u2260 0) (hg : \u2200 n : \u2115, 0 < n \u2192 g n \u2260 0) :\n    (\u2200 n > 0, \u220f i \u2208 n.divisors, f i = g n) \u2194\n      \u2200 n > 0, \u220f x \u2208 n.divisorsAntidiagonal, g x.snd ^ \u03bc x.fst = f n", "theorem sigma_zero_apply_prime_pow {p i : \u2115} (hp : p.Prime) : \u03c3 0 (p ^ i) = i + 1", "theorem vonMangoldt_le_log : \u2200 {n : \u2115}, \u039b n \u2264 Real.log (n : \u211d)", "@[simp]\ntheorem zeta_mul_vonMangoldt : (\u03b6 : ArithmeticFunction \u211d) * \u039b = log", "theorem extract_extract {a : Array \u03b1} (h : s1 + e2 \u2264 e1) :\n    (a.extract s1 e1).extract s2 e2 = a.extract (s1 + s2) (s1 + e2)", "@[simp] theorem push_data (arr : Array \u03b1) (a : \u03b1) : (arr.push a).data = arr.data ++ [a]", "@[simp] theorem size_extract (as : Array \u03b1) (start stop : Nat) :\n    (as.extract start stop).size = min stop as.size - start", "@[simp] theorem size_push (a : Array \u03b1) (v : \u03b1) : (push a v).size = a.size + 1", "theorem ArzelaAscoli.isCompact_closure_of_closedEmbedding [TopologicalSpace \u03b9] [T2Space \u03b1]\n    {\ud835\udd16 : Set (Set X)} (\ud835\udd16_compact : \u2200 K \u2208 \ud835\udd16, IsCompact K)\n    (F_clemb : ClosedEmbedding (UniformOnFun.ofFun \ud835\udd16 \u2218 F))\n    {s : Set \u03b9} (s_eqcont : \u2200 K \u2208 \ud835\udd16, EquicontinuousOn (F \u2218 ((\u2191) : s \u2192 \u03b9)) K)\n    (s_pointwiseCompact : \u2200 K \u2208 \ud835\udd16, \u2200 x \u2208 K, \u2203 Q, IsCompact Q \u2227 \u2200 i \u2208 s, F i x \u2208 Q) :\n    IsCompact (closure s)", "@[trans]\nprotected theorem trans [Monoid \u03b1] : \u2200 {x y z : \u03b1}, x ~\u1d64 y \u2192 y ~\u1d64 z \u2192 x ~\u1d64 z", "theorem count_ne_zero_iff_dvd {a p : \u03b1} (ha0 : a \u2260 0) (hp : Irreducible p) :\n    (Associates.mk p).count (Associates.mk a).factors \u2260 0 \u2194 p \u2223 a", "theorem count_self [Nontrivial \u03b1] [DecidableEq (Associates \u03b1)] {p : Associates \u03b1}\n    (hp : Irreducible p) : p.count p.factors = 1", "theorem factors'_cong {a b : \u03b1} (h : a ~\u1d64 b) : factors' a = factors' b", "@[simp]\ntheorem factors_one [Nontrivial \u03b1] : factors (1 : Associates \u03b1) = 0", "theorem le_of_count_ne_zero {m p : Associates \u03b1} (h0 : m \u2260 0) (hp : Irreducible p) :\n    count p m.factors \u2260 0 \u2192 p \u2264 m", "theorem mk_eq_mk_iff_associated [Monoid \u03b1] {a b : \u03b1} : Associates.mk a = Associates.mk b \u2194 a ~\u1d64 b", "@[simp 1100]\ntheorem normalize_out (a : Associates \u03b1) : normalize a.out = a.out", "theorem one_or_eq_of_le_of_prime {p m : Associates \u03b1} (hp : Prime p) (hle : m \u2264 p) :\n    m = 1 \u2228 m = p", "theorem prod_le_prod {p q : Multiset (Associates \u03b1)} (h : p \u2264 q) : p.prod \u2264 q.prod", "@[simp]\ntheorem quot_out_zero : Quot.out (0 : Associates \u03b1) = 0", "theorem IsBigO.exists_mem_basis {\u03b9} {p : \u03b9 \u2192 Prop} {s : \u03b9 \u2192 Set \u03b1} (h : f =O[l] g')\n    (hb : l.HasBasis p s) :\n    \u2203 c > 0, \u2203 i : \u03b9, p i \u2227 \u2200 x \u2208 s i, \u2016f x\u2016 \u2264 c * \u2016g' x\u2016", "theorem _root_.Asymptotics.IsBigO.integrableAtFilter [IsMeasurablyGenerated l]\n    (hf : f =O[l] g) (hfm : StronglyMeasurableAtFilter f l \u03bc) (hg : IntegrableAtFilter g l \u03bc) :\n    IntegrableAtFilter f l \u03bc", "theorem IsBigO.not_isLittleO (h : f'' =O[l] g') (hf : \u2203\u1da0 x in l, f'' x \u2260 0) :\n    \u00acg' =o[l] f''", "theorem Asymptotics.IsBigO.trans_tendsto_norm_atTop {\u03b1 : Type*} {u v : \u03b1 \u2192 \ud835\udd5c} {l : Filter \u03b1}\n    (huv : u =O[l] v) (hu : Tendsto (fun x => \u2016u x\u2016) l atTop) :\n    Tendsto (fun x => \u2016v x\u2016) l atTop", "theorem IsBigOWith.mul {f\u2081 f\u2082 : \u03b1 \u2192 R} {g\u2081 g\u2082 : \u03b1 \u2192 \ud835\udd5c} {c\u2081 c\u2082 : \u211d} (h\u2081 : IsBigOWith c\u2081 l f\u2081 g\u2081)\n    (h\u2082 : IsBigOWith c\u2082 l f\u2082 g\u2082) :\n    IsBigOWith (c\u2081 * c\u2082) l (fun x => f\u2081 x * f\u2082 x) fun x => g\u2081 x * g\u2082 x", "theorem IsBigOWith.prod_left_snd (h : IsBigOWith c l (fun x => (f' x, g' x)) k') :\n    IsBigOWith c l g' k'", "theorem IsBigOWith.rpow (h : IsBigOWith c l f g) (hc : 0 \u2264 c) (hr : 0 \u2264 r) (hg : 0 \u2264\u1da0[l] g) :\n    IsBigOWith (c ^ r) l (fun x => f x ^ r) fun x => g x ^ r", "theorem IsBigOWith.trans_le (hfg : IsBigOWith c l f g) (hgk : \u2200 x, \u2016g x\u2016 \u2264 \u2016k x\u2016) (hc : 0 \u2264 c) :\n    IsBigOWith c l f k", "@[symm]\ntheorem IsEquivalent.symm (h : u ~[l] v) : v ~[l] u", "theorem IsEquivalent.tendsto_nhds_iff {c : \u03b2} (huv : u ~[l] v) :\n    Tendsto u l (\ud835\udcdd c) \u2194 Tendsto v l (\ud835\udcdd c)", "theorem IsLittleO.add_isBigOWith (h\u2081 : f\u2081 =o[l] g) (h\u2082 : IsBigOWith c\u2081 l f\u2082 g) (hc : c\u2081 < c\u2082) :\n    IsBigOWith c\u2082 l (fun x => f\u2081 x + f\u2082 x) g", "protected theorem IsLittleO.insert [TopologicalSpace \u03b1] {x : \u03b1} {s : Set \u03b1} {g : \u03b1 \u2192 E'}\n    {g' : \u03b1 \u2192 F'} (h1 : g =o[\ud835\udcdd[s] x] g') (h2 : g x = 0) : g =o[\ud835\udcdd[insert x s] x] g'", "theorem IsLittleO.prod_rightr (h : f =o[l] g') : f =o[l] fun x => (f' x, g' x)", "theorem IsLittleO.right_isTheta_add' {f\u2081 f\u2082 : \u03b1 \u2192 E'} (h : f\u2081 =o[l] f\u2082) :\n    f\u2082 =\u0398[l] (f\u2082 + f\u2081)", "theorem IsLittleO.smul_isBigO (h\u2081 : k\u2081 =o[l] k\u2082) (h\u2082 : f' =O[l] g') :\n    (fun x => k\u2081 x \u2022 f' x) =o[l] fun x => k\u2082 x \u2022 g' x", "theorem IsLittleO.sup (h : f =o[l] g) (h' : f =o[l'] g) : f =o[l \u2294 l'] g", "theorem IsTheta.isBigO_congr_right (h : g' =\u0398[l] k') : f =O[l] g' \u2194 f =O[l] k'", "theorem IsTheta.tendsto_zero_iff (h : f'' =\u0398[l] g'') :\n    Tendsto f'' l (\ud835\udcdd 0) \u2194 Tendsto g'' l (\ud835\udcdd 0)", "theorem SuperpolynomialDecay.mul_const [ContinuousMul \u03b2] (hf : SuperpolynomialDecay l k f) (c : \u03b2) :\n    SuperpolynomialDecay l k fun n => f n * c", "theorem SuperpolynomialDecay.param_pow_mul (hf : SuperpolynomialDecay l k f) (n : \u2115) :\n    SuperpolynomialDecay l k (k ^ n * f)", "theorem SuperpolynomialDecay.trans_abs_le (hf : SuperpolynomialDecay l k f)\n    (hfg : \u2200 x, |g x| \u2264 |f x|) : SuperpolynomialDecay l k g", "theorem div_isBoundedUnder_of_isBigO {\u03b1 : Type*} {l : Filter \u03b1} {f g : \u03b1 \u2192 \ud835\udd5c} (h : f =O[l] g) :\n    IsBoundedUnder (\u00b7 \u2264 \u00b7) l fun x => \u2016f x / g x\u2016", "theorem isBigOWith_fst_prod : IsBigOWith 1 l f' fun x => (f' x, g' x)", "@[simp]\ntheorem isBigOWith_map {k : \u03b2 \u2192 \u03b1} {l : Filter \u03b2} :\n    IsBigOWith c (map k l) f g \u2194 IsBigOWith c l (f \u2218 k) (g \u2218 k)", "@[simp]\ntheorem isBigOWith_neg_right : (IsBigOWith c l f fun x => -g' x) \u2194 IsBigOWith c l f g'", "@[simp]\ntheorem isBigOWith_norm_right : (IsBigOWith c l f fun x => \u2016g' x\u2016) \u2194 IsBigOWith c l f g'", "@[simp]\ntheorem isBigO_abs_left : (fun x => |u x|) =O[l] g \u2194 u =O[l] g", "theorem isBigO_atTop_iff_eventually_exists {\u03b1 : Type*} [SemilatticeSup \u03b1] [Nonempty \u03b1]\n    {f : \u03b1 \u2192 E} {g : \u03b1 \u2192 F} : f =O[atTop] g \u2194 \u2200\u1da0 n\u2080 in atTop, \u2203 c, \u2200 n \u2265 n\u2080, \u2016f n\u2016 \u2264 c * \u2016g n\u2016", "theorem isBigO_const_const (c : E) {c' : F''} (hc' : c' \u2260 0) (l : Filter \u03b1) :\n    (fun _x : \u03b1 => c) =O[l] fun _x => c'", "theorem isBigO_const_smul_left {c : \ud835\udd5c} (hc : c \u2260 0) : (fun x => c \u2022 f' x) =O[l] g \u2194 f' =O[l] g", "theorem isBigO_fst_prod : f' =O[l] fun x => (f' x, g' x)", "@[simp]\ntheorem isBigO_neg_right : (f =O[l] fun x => -g' x) \u2194 f =O[l] g'", "theorem isBigO_snd_prod : g' =O[l] fun x => (f' x, g' x)", "@[simp 1001] theorem isLittleO_const_const_iff [NeBot l] {d : E''} {c : F''} :\n    ((fun _x => d) =o[l] fun _x => c) \u2194 d = 0", "theorem isLittleO_const_left_of_ne {c : E''} (hc : c \u2260 0) :\n    (fun _x => c) =o[l] g \u2194 Tendsto (fun x => \u2016g x\u2016) l atTop", "@[simp]\ntheorem isLittleO_neg_left : (fun x => -f' x) =o[l] g \u2194 f' =o[l] g", "theorem isLittleO_norm_pow_id {n : \u2115} (h : 1 < n) :\n    (fun x : E' => \u2016x\u2016 ^ n) =o[\ud835\udcdd 0] fun x => x", "theorem superpolynomialDecay_iff_zpow_tendsto_zero (hk : Tendsto k l atTop) :\n    SuperpolynomialDecay l k f \u2194 \u2200 z : \u2124, Tendsto (fun a : \u03b1 => k a ^ z * f a) l (\ud835\udcdd 0)", "theorem superpolynomialDecay_mul_param_pow_iff (hk : Tendsto k l atTop) (n : \u2115) :\n    SuperpolynomialDecay l k (f * k ^ n) \u2194 SuperpolynomialDecay l k f", "theorem Balanced.smul_mem_mono [SMulCommClass \ud835\udd5d \ud835\udd5c E] (hs : Balanced \ud835\udd5d s) {a : \ud835\udd5c} {b : \ud835\udd5d}\n    (ha : a \u2022 x \u2208 s) (hba : \u2016b\u2016 \u2264 \u2016a\u2016) : b \u2022 x \u2208 s", "theorem Balanced.subset_smul (hA : Balanced \ud835\udd5c A) (ha : 1 \u2264 \u2016a\u2016) : A \u2286 a \u2022 A", "theorem basis_singleton_iff {R M : Type*} [Ring R] [Nontrivial R] [AddCommGroup M] [Module R M]\n    [NoZeroSMulDivisors R M] (\u03b9 : Type*) [Unique \u03b9] :\n    Nonempty (Basis \u03b9 R M) \u2194 \u2203 x \u2260 0, \u2200 y : M, \u2203 r : R, r \u2022 x = y", "theorem Basis.card_le_card_of_linearIndependent {\u03b9 : Type*} [Fintype \u03b9] (b : Basis \u03b9 R M)\n    {\u03b9' : Type*} [Fintype \u03b9'] {v : \u03b9' \u2192 M} (hv : LinearIndependent R v) :\n    Fintype.card \u03b9' \u2264 Fintype.card \u03b9", "@[simp]\ntheorem coe_mk : \u21d1(Basis.mk hli hsp) = v", "@[simp]\ntheorem coe_ofRepr (e : M \u2243\u2097[R] \u03b9 \u2192\u2080 R) : \u21d1(ofRepr e) = fun i => e.symm (Finsupp.single i 1)", "@[simp]\ntheorem coe_repr_symm : \u2191b.repr.symm = Finsupp.total \u03b9 M R b", "theorem Basis.exists_opNNNorm_le {\u03b9 : Type*} [Finite \u03b9] (v : Basis \u03b9 \ud835\udd5c E) :\n    \u2203 C > (0 : \u211d\u22650), \u2200 {u : E \u2192L[\ud835\udd5c] F} (M : \u211d\u22650), (\u2200 i, \u2016u (v i)\u2016\u208a \u2264 M) \u2192 \u2016u\u2016\u208a \u2264 C * M", "theorem Basis.exists_opNorm_le {\u03b9 : Type*} [Finite \u03b9] (v : Basis \u03b9 \ud835\udd5c E) :\n    \u2203 C > (0 : \u211d), \u2200 {u : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 i, \u2016u (v i)\u2016 \u2264 M) \u2192 \u2016u\u2016 \u2264 C * M", "theorem ext_elem_iff {x y : M} : x = y \u2194 \u2200 i, b.repr x i = b.repr y i", "theorem Basis.ext_multilinear [Finite \u03b9] {f g : MultilinearMap R (fun _ : \u03b9 => M\u2082) M\u2083} {\u03b9\u2081 : Type*}\n    (e : Basis \u03b9\u2081 R M\u2082) (h : \u2200 v : \u03b9 \u2192 \u03b9\u2081, (f fun i => e (v i)) = g fun i => e (v i)) : f = g", "theorem groupSMul_span_eq_top {G : Type*} [Group G] [DistribMulAction G R] [DistribMulAction G M]\n    [IsScalarTower G R M] {v : \u03b9 \u2192 M} (hv : Submodule.span R (Set.range v) = \u22a4) {w : \u03b9 \u2192 G} :\n    Submodule.span R (Set.range (w \u2022 v)) = \u22a4", "theorem Basis.isScalarTower_of_nonempty {\u03b9} [Nonempty \u03b9] (b : Basis \u03b9 S A) : IsScalarTower R S S", "theorem Basis.le_span {J : Set M} (v : Basis \u03b9 R M) (hJ : span R J = \u22a4) : #(range v) \u2264 #J", "lemma Basis.mem_center_iff {A}\n    [Semiring R] [NonUnitalNonAssocSemiring A]\n    [Module R A] [SMulCommClass R A A] [SMulCommClass R R A] [IsScalarTower R A A]\n    (b : Basis \u03b9 R A) {z : A} :\n    z \u2208 Set.center A \u2194\n      (\u2200 i, Commute (b i) z) \u2227 \u2200 i j,\n        z * (b i * b j) = (z * b i) * b j\n          \u2227 (b i * z) * b j = b i * (z * b j)\n          \u2227 (b i * b j) * z = b i * (b j * z)", "theorem mem_span_repr_support (m : M) : m \u2208 span R (b '' (b.repr m).support)", "theorem Basis.mk_eq_rank'' {\u03b9 : Type v} (v : Basis \u03b9 R M) : #\u03b9 = Module.rank R M", "@[simp]\ntheorem Basis.ofEquivFun_repr_apply [Finite \u03b9] (e : M \u2243\u2097[R] \u03b9 \u2192 R) (x : M) (i : \u03b9) :\n    (Basis.ofEquivFun e).repr x i = e x i", "@[simp]\ntheorem Basis.ofIsLocalizedModule_apply (i : \u03b9) : b.ofIsLocalizedModule R\u209b S f i = f (b i)", "@[simp]\ntheorem Basis.ofZlatticeBasis_apply (i : \u03b9) :\n    b.ofZlatticeBasis K L i =  b i", "@[simp]\ntheorem orientation_isEmpty [IsEmpty \u03b9] (b : Basis \u03b9 R M) :\n    b.orientation = positiveOrientation", "theorem reindexFinsetRange_repr_self (i : \u03b9) :\n    b.reindexFinsetRange.repr (b i) =\n      Finsupp.single \u27e8b i, Finset.mem_image_of_mem b (Finset.mem_univ i)\u27e9 1", "@[simp]\ntheorem reindexRange_repr (x : M) (i : \u03b9) (h := Set.mem_range_self i) :\n    b.reindexRange.repr x \u27e8b i, h\u27e9 = b.repr x i", "@[simp]\ntheorem singleton_repr (\u03b9 R : Type*) [Unique \u03b9] [Semiring R] (x i) :\n    (Basis.singleton \u03b9 R).repr x i = x", "@[simp]\ntheorem Basis.tensorProduct_apply (b : Basis \u03b9 R M) (c : Basis \u03ba R N) (i : \u03b9) (j : \u03ba) :\n    Basis.tensorProduct b c (i, j) = b i \u2297\u209c c j", "@[simp]\ntheorem Basis.tensorProduct_repr_tmul_apply (b : Basis \u03b9 R M) (c : Basis \u03ba R N) (m : M) (n : N)\n    (i : \u03b9) (j : \u03ba) :\n    (Basis.tensorProduct b c).repr (m \u2297\u209c n) (i, j) = b.repr m i * c.repr n j", "theorem toDual_apply (i j : \u03b9) : b.toDual (b i) (b j) = if i = j then 1 else 0", "theorem toDual_range [Finite \u03b9] : LinearMap.range b.toDual = \u22a4", "theorem toMatrix_map_vecMul {S : Type*} [Ring S] [Algebra R S] [Fintype \u03b9] (b : Basis \u03b9 R S)\n    (v : \u03b9' \u2192 S) : b \u1d65* ((b.toMatrix v).map <| algebraMap R S) = v", "theorem Heap.realSize_tail? {s : Heap \u03b1} : s.tail? le = some s' \u2192\n    s.realSize = s'.realSize + 1", "theorem toList_append (l\u2081 l\u2082 : DList \u03b1) : toList (l\u2081 ++ l\u2082) = toList l\u2081 ++ toList l\u2082", "theorem toList_empty : toList (@empty \u03b1) = []", "theorem toList_ofList (l : List \u03b1) : DList.toList (DList.ofList l) = l", "theorem WF.mapVal {\u03b1 \u03b2 \u03b3} {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3} [BEq \u03b1] [Hashable \u03b1]\n    {m : Imp \u03b1 \u03b2} (H : WF m) : WF (mapVal f m)", "theorem LTCmp.cmp_iff_gt [LT \u03b1] [LTCmp (\u03b1 := \u03b1) cmp] : cmp x y = .gt \u2194 y < x", "theorem LTCmp.eq_compareOfLessAndEq\n    [LT \u03b1] [DecidableEq \u03b1] [BEq \u03b1] [LawfulBEq \u03b1] [BEqCmp cmp] [LTCmp cmp]\n    (x y : \u03b1) [Decidable (x < y)] : cmp x y = compareOfLessAndEq x y", "protected theorem LawfulCmp.compareOfLessAndEq\n    [LT \u03b1] [DecidableRel (LT.lt (\u03b1 := \u03b1))] [DecidableEq \u03b1] [BEq \u03b1] [LawfulBEq \u03b1] [LE \u03b1]\n    (lt_irrefl : \u2200 x : \u03b1, \u00acx < x)\n    (lt_trans : \u2200 {x y z : \u03b1}, x < y \u2192 y < z \u2192 x < z)\n    (not_lt : \u2200 {x y : \u03b1}, \u00acx < y \u2194 y \u2264 x)\n    (le_antisymm : \u2200 {x y : \u03b1}, x \u2264 y \u2192 y \u2264 x \u2192 x = y) :\n    LawfulCmp (\u03b1 := \u03b1) (compareOfLessAndEq \u00b7 \u00b7)", "theorem Heap.deleteMin_fst : ((s : Heap \u03b1).deleteMin le).map (\u00b7.1) = s.head?", "theorem Heap.noSibling_deleteMin {s : Heap \u03b1} (eq : s.deleteMin le = some (a, s')) :\n    s'.NoSibling", "theorem Heap.noSibling_tail? {s : Heap \u03b1} : s.tail? le = some s' \u2192\n    s'.NoSibling", "protected theorem All.balRight\n    (hl : l.All p) (hv : p v) (hr : r.All p) : (balRight l v r).All p", "theorem All_def {t : RBNode \u03b1} : t.All p \u2194 \u2200 x \u2208 t, p x", "theorem Ordered.find?_some [@TransCmp \u03b1 cmp] [IsStrictCut cmp cut] (ht : Ordered cmp t) :\n    t.find? cut = some x \u2194 x \u2208 t \u2227 cut x = .eq", "theorem Ordered.memP_iff_lowerBound? [@TransCmp \u03b1 cmp] [IsCut cmp cut] (ht : Ordered cmp t) :\n    t.MemP cut \u2194 \u2203 x, t.lowerBound? cut = some x \u2227 cut x = .eq", "theorem Ordered.upperBound?_exists [@TransCmp \u03b1 cmp] [IsCut cmp cut] (h : Ordered cmp t) :\n    (\u2203 x, t.upperBound? cut = some x) \u2194 \u2203 x \u2208 t, cut x \u2260 .gt", "theorem Ordered.erase : \u2200 {path : Path \u03b1} {t : RBNode \u03b1},\n    path.Ordered cmp \u2192 t.Ordered cmp \u2192 t.All (path.RootOrdered cmp) \u2192 (path.erase t).Ordered cmp", "theorem Ordered.insertNew {path : Path \u03b1} (hp : path.Ordered cmp) (vp : path.RootOrdered cmp v) :\n    (path.insertNew v).Ordered cmp", "theorem insertNew_eq_insert (h : zoom (cmp v) t = (nil, path)) :\n    path.insertNew v = (t.insert cmp v).setBlack", "protected theorem RedRed.balance2 {l : RBNode \u03b1} {v : \u03b1} {r : RBNode \u03b1}\n    (hl : l.Balanced c n) (hr : r.RedRed p n) : \u2203 c, (balance2 l v r).Balanced c (n + 1)", "theorem foldl_eq_foldl_toList {t : RBNode.Stream \u03b1} : t.foldl f init = t.toList.foldl f init", "theorem foldr_cons (t : RBNode.Stream \u03b1) (l) : t.foldr (\u00b7::\u00b7) l = t.toList ++ l", "theorem WF.modify {t : RBNode \u03b1}\n    (H : (t.zoom cut).1.OnRoot fun x => cmpEq cmp (f x) x)\n    (h : WF cmp t) : WF cmp (t.modify cut f)", "theorem WF.out {t : RBNode \u03b1} (h : t.WF cmp) : t.Ordered cmp \u2227 \u2203 c n, t.Balanced c n", "@[simp] theorem balance2_All {l : RBNode \u03b1} {v : \u03b1} {r : RBNode \u03b1} :\n    (balance2 l v r).All p \u2194 p v \u2227 l.All p \u2227 r.All p", "theorem cmpEq_iff [TransCmp cmp] : cmpEq cmp x y \u2194 cmp x y = .eq", "@[simp] theorem find?_reverse (t : RBNode \u03b1) (cut : \u03b1 \u2192 Ordering) :\n    t.reverse.find? cut = t.find? (cut \u00b7 |>.swap)", "theorem find?_some_eq_eq {t : RBNode \u03b1} : x \u2208 t.find? cut \u2192 cut x = .eq", "theorem foldr_eq_foldr_toList {t : RBNode \u03b1} : t.foldr f init = t.toList.foldr f init", "theorem insert_toList_zoom_node {t : RBNode \u03b1} (ht : Balanced t c n)\n    (e : zoom (cmp v) t = (node c' l v' r, p)) :\n    (t.insert cmp v).toList = p.withList (node c l v r).toList", "theorem lowerBound?_of_some {t : RBNode \u03b1} : \u2203 x, t.lowerBound? cut (some y) = some x", "@[simp] theorem reverse_balRight (l : RBNode \u03b1) (v : \u03b1) (r : RBNode \u03b1) :\n    (balRight l v r).reverse = balLeft r.reverse v l.reverse", "theorem reverse_eq_iff {t t' : RBNode \u03b1} : t.reverse = t' \u2194 t = t'.reverse", "@[simp] theorem reverse_size (t : RBNode \u03b1) : t.reverse.size = t.size", "theorem upperBound?_mem {t : RBNode \u03b1} (h : t.upperBound? cut = some x) : x \u2208 t", "theorem find?_some [@TransCmp \u03b1 cmp] {t : RBSet \u03b1 cmp} :\n    t.find? x = some y \u2194 y \u2208 toList t \u2227 cmp x y = .eq", "theorem foldr_eq_foldr_toList {t : RBSet \u03b1 cmp} : t.foldr f init = t.toList.foldr f init", "theorem mem_congr [@TransCmp \u03b1 cmp] {t : RBSet \u03b1 cmp} (h : cmp x y = .eq) : x \u2208 t \u2194 y \u2208 t", "theorem mem_iff_find? [@TransCmp \u03b1 cmp] {t : RBSet \u03b1 cmp} :\n    x \u2208 t \u2194 \u2203 y, t.find? x = some y", "theorem mem_toList {t : RBSet \u03b1 cmp} : x \u2208 toList t \u2194 x \u2208 t.1", "theorem upperBoundP?_mem [@OrientedCmp \u03b1 cmp] {t : RBSet \u03b1 cmp}\n    (h : t.upperBoundP? cut = some x) : x \u2208 t", "theorem ge_trans (h\u2081 : cmp x y \u2260 .lt) (h\u2082 : cmp y z \u2260 .lt) : cmp x z \u2260 .lt", "theorem TransOrd.instLexOrd [Ord \u03b1] [Ord \u03b2]\n    [TransOrd \u03b1] [TransOrd \u03b2] : @TransOrd (\u03b1 \u00d7 \u03b2) lexOrd", "theorem TransOrd.instOpposite [ord : Ord \u03b1] [inst : TransOrd \u03b1] : @TransOrd _ ord.opposite", "@[simp] theorem find_parent_1 (self : UnionFind) (x : Fin self.size) :\n    (self.find x).1.parent x = self.rootD x", "theorem parentD_linkAux {self} {x y : Fin self.size} :\n    parentD (linkAux self x y) i =\n    if x.1 = y then\n      parentD self i\n    else\n      if (self.get y).rank < (self.get x).rank then\n        if y = i then x else parentD self i\n      else\n        if x = i then y else parentD self i", "theorem parent_lt (self : UnionFind) (i : Nat) : self.parent i < self.size \u2194 i < self.size", "theorem rankD_findAux {self : UnionFind} {x : Fin self.size} :\n    rankD (findAux self x).s i = self.rank i", "theorem rankD_set {arr : Array UFNode} {x v i} :\n    rankD (arr.set x v) i = if x.1 = i then v.rank else rankD arr i", "theorem rootD_ext {m1 m2 : UnionFind}\n    (H : \u2200 x, m1.parent x = m2.parent x) {x} : m1.rootD x = m2.rootD x", "theorem rootD_lt {self : UnionFind} {x : Nat} : self.rootD x < self.size \u2194 x < self.size", "theorem root_link {self : UnionFind} {x y : Fin self.size}\n    (xroot : self.parent x = x) (yroot : self.parent y = y) :\n    \u2203 r, (r = x \u2228 r = y) \u2227 \u2200 i,\n      (link self x y yroot).rootD i =\n      if self.rootD i = x \u2228 self.rootD i = y then r.1 else self.rootD i", "theorem _root_.BddAbove.convolutionExistsAt [MeasurableAdd\u2082 G] [SigmaFinite \u03bc] {x\u2080 : G} {s : Set G}\n    (hbg : BddAbove ((fun i => \u2016g i\u2016) '' ((fun t => x\u2080 - t) \u207b\u00b9' s))) (hs : MeasurableSet s)\n    (h2s : (support fun t => L (f t) (g (x\u2080 - t))) \u2286 s) (hf : IntegrableOn f s \u03bc)\n    (hmg : AEStronglyMeasurable g \u03bc) : ConvolutionExistsAt f g x\u2080 L \u03bc", "@[simp]\ntheorem coe_toBddLat (X : BddDistLat) : \u21a5X.toBddLat = \u21a5X", "theorem bound_aux (hd : d \u2260 0) (hn : 2 \u2264 n) :\n    (d ^ (n - 2 :) / n : \u211d) \u2264 rothNumberNat ((2 * d - 1) ^ n)", "theorem map_zero (d : \u2115) (a : Fin 0 \u2192 \u2115) : map d a = 0", "theorem real_main_inequality {x : \u211d} (x_large : (512 : \u211d) \u2264 x) :\n    x * (2 * x) ^ \u221a(2 * x) * 4 ^ (2 * x / 3) \u2264 4 ^ x", "theorem exists_disjoint_closedBall_covering_ae_of_finiteMeasure_aux (\u03bc : Measure \u03b1)\n    [IsFiniteMeasure \u03bc] (f : \u03b1 \u2192 Set \u211d) (s : Set \u03b1)\n    (hf : \u2200 x \u2208 s, \u2200 \u03b4 > 0, (f x \u2229 Ioo 0 \u03b4).Nonempty) :\n    \u2203 t : Set (\u03b1 \u00d7 \u211d), t.Countable \u2227 (\u2200 p \u2208 t, p.1 \u2208 s) \u2227 (\u2200 p \u2208 t, p.2 \u2208 f p.1) \u2227\n      \u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 t), closedBall p.1 p.2) = 0 \u2227\n        t.PairwiseDisjoint fun p => closedBall p.1 p.2", "@[simp]\ntheorem coe_toEquiv_trans : (e\u2081\u2082 : A \u2243 B).trans e\u2082\u2083 = (e\u2081\u2082.trans e\u2082\u2083 : A \u2243 C)", "@[simp]\nprotected theorem coe_coe {F : Type*} [FunLike F A B] [BialgHomClass F R A B] (f : F) :\n    \u21d1(f : A \u2192\u2090c[R] B) = f", "@[simp]\ntheorem id_toAlgHom : BialgHom.id R A = AlgHom.id R A", "@[simp]\ntheorem id_toCoalgHom : BialgHom.id R A = CoalgHom.id R A", "theorem toAlgHom_toLinearMap (f : A \u2192\u2090c[R] B) :\n    ((f : A \u2192\u2090[R] B) : A \u2192\u2097[R] B) = f", "@[simp]\ntheorem forget\u2082_algebra_obj (X : BialgebraCat R) :\n    (forget\u2082 (BialgebraCat R) (AlgebraCat R)).obj X = AlgebraCat.of R X", "@[simp]\ntheorem mapEquiv_apply (h : \u03b1 \u2243 \u03b2) (h' : \u03b1' \u2243 \u03b2') (x : F \u03b1 \u03b1') :\n    (mapEquiv F h h' : F \u03b1 \u03b1' \u2243 F \u03b2 \u03b2') x = bimap h h' x", "theorem hom_left_act_hom' :\n    ((regular R).tensorBimod P).actLeft \u226b hom P = (R.X \u25c1 hom P) \u226b P.actLeft", "theorem left_assoc' :\n    (R.mul \u25b7 _) \u226b actLeft P Q = (\u03b1_ R.X R.X _).hom \u226b (R.X \u25c1 actLeft P Q) \u226b actLeft P Q", "theorem middle_assoc' :\n    (actLeft P Q \u25b7 T.X) \u226b actRight P Q =\n      (\u03b1_ R.X _ T.X).hom \u226b (R.X \u25c1 actRight P Q) \u226b actLeft P Q", "theorem whiskerRight_id_bimod {X Y : Mon_ C} {M N : Bimod X Y} (f : M \u27f6 N) :\n    whiskerRight f (regular Y) = (rightUnitorBimod M).hom \u226b f \u226b (rightUnitorBimod N).inv", "@[higher_order tfst_comp_tsnd]\ntheorem tfst_tsnd {\u03b1\u2080 \u03b1\u2081 \u03b2\u2080 \u03b2\u2081} (f : \u03b1\u2080 \u2192 F \u03b1\u2081) (f' : \u03b2\u2080 \u2192 G \u03b2\u2081) (x : t \u03b1\u2080 \u03b2\u2080) :\n    Comp.mk (tfst f <$> tsnd f' x)\n      = bitraverse (Comp.mk \u2218 pure \u2218 f) (Comp.mk \u2218 map pure \u2218 f') x", "theorem bne_eq_xor : bne = xor", "theorem eq_false_of_not_eq_true' {a : Bool} : !a = true \u2192 a = false", "lemma eq_or_eq_not : \u2200 a b, a = b \u2228 a = !b", "@[simp]\ntheorem injective_iff {\u03b1 : Sort*} {f : Bool \u2192 \u03b1} : Function.Injective f \u2194 f false \u2260 f true", "@[simp] theorem Bool.not_true  : (!true) = false", "theorem sup_comm (a b : \u03b1) : a \u2294 b = b \u2294 a", "@[simp] lemma comp_bot : comp (\u22a5 : \u03b1) = \u22a4 := rfl", "@[simp] lemma comp_lt_comp : comp a < comp b \u2194 b < a := \u27e8by rintro \u27e8_\u27e9; assumption, LT.comp\u27e9", "@[simp] lemma comp_sdiff_comp (a b : \u03b1) : comp a \\ comp b = lift (b \\ a) := rfl", "@[simp] lemma comp_sup_comp (a b : \u03b1) : comp a \u2294 comp b = comp (a \u2293 b) := rfl", "@[simp] lemma lift_sdiff_comp (a b : \u03b1) : lift a \\ comp b = lift (a \u2293 b) := rfl", "protected lemma Bornology.IsBounded.bddBelow (hs : IsBounded s) : BddBelow s", "theorem Bornology.IsBounded.reProdIm (hs : IsBounded s) (ht : IsBounded t) : IsBounded (s \u00d7\u2102 t)", "protected theorem IsVonNBounded.add (hs : IsVonNBounded \ud835\udd5c s) (ht : IsVonNBounded \ud835\udd5c t) :\n    IsVonNBounded \ud835\udd5c (s + t)", "lemma Bornology.ext_iff (t t' : Bornology \u03b1) :\n    t = t' \u2194 @Bornology.cobounded \u03b1 t = @Bornology.cobounded \u03b1 t'", "theorem isBounded_biUnion {s : Set \u03b9} {f : \u03b9 \u2192 Set \u03b1} (hs : s.Finite) :\n    IsBounded (\u22c3 i \u2208 s, f i) \u2194 \u2200 i \u2208 s, IsBounded (f i)", "@[simp]\ntheorem isBounded_iUnion [Finite \u03b9] {s : \u03b9 \u2192 Set \u03b1} :\n    IsBounded (\u22c3 i, s i) \u2194 \u2200 i, IsBounded (s i)", "theorem isBounded_image_fst_and_snd {s : Set (\u03b1 \u00d7 \u03b2)} :\n    IsBounded (Prod.fst '' s) \u2227 IsBounded (Prod.snd '' s) \u2194 IsBounded s", "theorem isBounded_prod_self : IsBounded (s \u00d7\u02e2 s) \u2194 IsBounded s", "@[simp]\ntheorem cancel_right {g\u2081 g\u2082 : BotHom \u03b2 \u03b3} {f : BotHom \u03b1 \u03b2} (hf : Surjective f) :\n    g\u2081.comp f = g\u2082.comp f \u2194 g\u2081 = g\u2082", "theorem arzela_ascoli\u2081 [CompactSpace \u03b2] (A : Set (\u03b1 \u2192\u1d47 \u03b2)) (closed : IsClosed A)\n    (H : Equicontinuous ((\u2191) : A \u2192 \u03b1 \u2192 \u03b2)) : IsCompact A", "theorem exists_norm_eq_restrict_eq_of_closed {s : Set Y} (f : s \u2192\u1d47 \u211d) (hs : IsClosed s) :\n    \u2203 g : Y \u2192\u1d47 \u211d, \u2016g\u2016 = \u2016f\u2016 \u2227 g.restrict s = f", "@[to_additive]\ntheorem forall_coe_one_iff_one (f : \u03b1 \u2192\u1d47 \u03b2) : (\u2200 x, f x = 1) \u2194 f = 1", "lemma integral_const_sub (f : X \u2192\u1d47 \u211d) (c : \u211d) :\n    \u222b x, (const X c - f) x \u2202\u03bc = ENNReal.toReal (\u03bc (Set.univ)) \u2022 c - \u222b x, f x \u2202\u03bc", "theorem lipschitz_comp {G : \u03b2 \u2192 \u03b3} {C : \u211d\u22650} (H : LipschitzWith C G) :\n    LipschitzWith C (comp G H : (\u03b1 \u2192\u1d47 \u03b2) \u2192 \u03b1 \u2192\u1d47 \u03b3)", "theorem mem_Lp (f : \u03b1 \u2192\u1d47 E) : f.toContinuousMap.toAEEqFun \u03bc \u2208 Lp E p \u03bc", "theorem _root_.BoundedContinuousFunction.mkOfCompact_star [CompactSpace \u03b1] (f : C(\u03b1, \u03b2)) :\n    mkOfCompact (star f) = star (mkOfCompact f)", "@[simp]\ntheorem natCast_apply [NatCast \u03b2] (n : \u2115) (x : \u03b1) : (n : \u03b1 \u2192\u1d47 \u03b2) x = n", "@[simp]\ntheorem norm_normComp : \u2016f.normComp\u2016 = \u2016f\u2016", "theorem tietze_extension_step (f : X \u2192\u1d47 \u211d) (e : C(X, Y)) (he : ClosedEmbedding e) :\n    \u2203 g : Y \u2192\u1d47 \u211d, \u2016g\u2016 \u2264 \u2016f\u2016 / 3 \u2227 dist (g.compContinuous e) f \u2264 2 / 3 * \u2016f\u2016", "theorem uniformContinuous_comp {G : \u03b2 \u2192 \u03b3} {C : \u211d\u22650} (H : LipschitzWith C G) :\n    UniformContinuous (comp G H : (\u03b1 \u2192\u1d47 \u03b2) \u2192 \u03b1 \u2192\u1d47 \u03b3)", "@[simp]\ntheorem coe_comp (f : BoundedLatticeHom \u03b2 \u03b3) (g : BoundedLatticeHom \u03b1 \u03b2) :\n    (f.comp g : \u03b1 \u2192 \u03b3) = f \u2218 g", "theorem coe_comp_inf_hom (f : BoundedLatticeHom \u03b2 \u03b3) (g : BoundedLatticeHom \u03b1 \u03b2) :\n    (f.comp g : InfHom \u03b1 \u03b3) = (f : InfHom \u03b2 \u03b3).comp g", "@[simp] lemma coe_toBoundedOrderHom (f : BoundedLatticeHom \u03b1 \u03b2) : \u21d1f.toBoundedOrderHom = f := rfl", "@[simp]\ntheorem symm_dual_comp (g : BoundedLatticeHom \u03b2\u1d52\u1d48 \u03b3\u1d52\u1d48) (f : BoundedLatticeHom \u03b1\u1d52\u1d48 \u03b2\u1d52\u1d48) :\n    BoundedLatticeHom.dual.symm (g.comp f) =\n      (BoundedLatticeHom.dual.symm g).comp (BoundedLatticeHom.dual.symm f)", "@[simp]\ntheorem coe_copy (f : BoundedOrderHom \u03b1 \u03b2) (f' : \u03b1 \u2192 \u03b2) (h : f' = f) : \u21d1(f.copy f' h) = f'", "theorem _root_.BoundedVariationOn.sub_le {f : \u03b1 \u2192 \u211d} {s : Set \u03b1} (h : BoundedVariationOn f s)\n    {x y : \u03b1} (hx : x \u2208 s) (hy : y \u2208 s) : f x - f y \u2264 (eVariationOn f s).toReal", "theorem Icc_def : Box.Icc I = Icc I.lower I.upper", "@[simp]\ntheorem coe_mk' (l u : \u03b9 \u2192 \u211d) : (mk' l u : Set (\u03b9 \u2192 \u211d)) = pi univ fun i \u21a6 Ioc (l i) (u i)", "theorem continuousOn_face_Icc {X} [TopologicalSpace X] {n} {f : (Fin (n + 1) \u2192 \u211d) \u2192 X}\n    {I : Box (Fin (n + 1))} (h : ContinuousOn f (Box.Icc I)) {i : Fin (n + 1)} {x : \u211d}\n    (hx : x \u2208 Icc (I.lower i) (I.upper i)) :\n    ContinuousOn (f \u2218 i.insertNth x) (Box.Icc (I.face i))", "@[simp]\ntheorem exists_mem_splitCenterBox {I : Box \u03b9} {x : \u03b9 \u2192 \u211d} : (\u2203 s, x \u2208 I.splitCenterBox s) \u2194 x \u2208 I", "theorem exists_seq_mono_tendsto (I : Box \u03b9) :\n    \u2203 J : \u2115 \u2192o Box \u03b9,\n      (\u2200 n, Box.Icc (J n) \u2286 Box.Ioo I) \u2227\n        Tendsto (lower \u2218 J) atTop (\ud835\udcdd I.lower) \u2227 Tendsto (upper \u2218 J) atTop (\ud835\udcdd I.upper)", "theorem le_def : I \u2264 J \u2194 \u2200 x \u2208 I, x \u2208 J", "theorem measurableSet_coe : MeasurableSet (I : Set (\u03b9 \u2192 \u211d))", "theorem nonempty_coe : Set.Nonempty (I : Set (\u03b9 \u2192 \u211d))", "@[simp]\ntheorem splitLower_eq_self : I.splitLower i x = I \u2194 I.upper i \u2264 x", "@[simp]\ntheorem volume_apply' (I : Box \u03b9) :\n    ((volume : Measure (\u03b9 \u2192 \u211d)) I).toReal = \u220f i, (I.upper i - I.lower i)", "theorem HasIntegral.mono {l\u2081 l\u2082 : IntegrationParams} (h : HasIntegral I l\u2081 f vol y) (hl : l\u2082 \u2264 l\u2081) :\n    HasIntegral I l\u2082 f vol y", "theorem dist_integralSum_integral_le_of_memBaseSet (h : Integrable I l f vol) (h\u2080 : 0 < \u03b5)\n    (h\u03c0 : l.MemBaseSet I c (h.convergenceR \u03b5 c) \u03c0) (h\u03c0p : \u03c0.IsPartition) :\n    dist (integralSum f vol \u03c0) (integral I l f vol) \u2264 \u03b5", "theorem to_subbox_aux (h : Integrable I l f vol) (hJ : J \u2264 I) :\n    \u2203 y : F, HasIntegral J l f vol y \u2227\n      Tendsto (integralSum f vol) (l.toFilteriUnion I (Prepartition.single I J hJ)) (\ud835\udcdd y)", "theorem distortion_biUnion (\u03c0 : Prepartition I) (\u03c0i : \u2200 J, Prepartition J) :\n    (\u03c0.biUnion \u03c0i).distortion = \u03c0.boxes.sup fun J => (\u03c0i J).distortion", "theorem eq_of_boxes_subset_iUnion_superset (h\u2081 : \u03c0\u2081.boxes \u2286 \u03c0\u2082.boxes) (h\u2082 : \u03c0\u2082.iUnion \u2286 \u03c0\u2081.iUnion) :\n    \u03c0\u2081 = \u03c0\u2082", "@[simp]\ntheorem iUnion_toSubordinate (\u03c0 : Prepartition I) (r : (\u03b9 \u2192 \u211d) \u2192 Ioi (0 : \u211d)) :\n    (\u03c0.toSubordinate r).iUnion = \u03c0.iUnion", "@[simp]\ntheorem iUnion_top : (\u22a4 : Prepartition I).iUnion = I", "@[simp]\ntheorem mem_biUnion : J \u2208 \u03c0.biUnion \u03c0i \u2194 \u2203 J' \u2208 \u03c0, J \u2208 \u03c0i J'", "@[simp]\ntheorem mem_splitCenter : J \u2208 splitCenter I \u2194 \u2203 s, I.splitCenterBox s = J", "theorem mem_split_iff' : J \u2208 split I i x \u2194\n    (J : Set (\u03b9 \u2192 \u211d)) = \u2191I \u2229 { y | y i \u2264 x } \u2228 (J : Set (\u03b9 \u2192 \u211d)) = \u2191I \u2229 { y | x < y i }", "@[simp]\ntheorem splitMany_insert (I : Box \u03b9) (s : Finset (\u03b9 \u00d7 \u211d)) (p : \u03b9 \u00d7 \u211d) :\n    splitMany I (insert p s) = splitMany I s \u2293 split I p.1 p.2", "theorem upper_sub_lower_of_mem_splitCenter (h : J \u2208 splitCenter I) (i : \u03b9) :\n    J.upper i - J.lower i = (I.upper i - I.lower i) / 2", "theorem disjUnion_tag_of_mem_right (h : Disjoint \u03c0\u2081.iUnion \u03c0\u2082.iUnion) (hJ : J \u2208 \u03c0\u2082) :\n    (\u03c0\u2081.disjUnion \u03c0\u2082 h).tag J = \u03c0\u2082.tag J", "theorem iUnion_def : \u03c0.iUnion = \u22c3 J \u2208 \u03c0, \u2191J", "@[simp]\ntheorem iUnion_disjUnion (h : Disjoint \u03c0\u2081.iUnion \u03c0\u2082.iUnion) :\n    (\u03c0\u2081.disjUnion \u03c0\u2082 h).iUnion = \u03c0\u2081.iUnion \u222a \u03c0\u2082.iUnion", "theorem isPartition_single_iff (hJ : J \u2264 I) (h : x \u2208 Box.Icc I) :\n    (single I J hJ x h).IsPartition \u2194 J = I", "theorem hasIntegral_GP_divergence_of_forall_hasDerivWithinAt\n    (f : (Fin (n + 1) \u2192 \u211d) \u2192 Fin (n + 1) \u2192 E)\n    (f' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] (Fin (n + 1) \u2192 E))\n    (s : Set (Fin (n + 1) \u2192 \u211d)) (hs : s.Countable)\n    (Hs : \u2200 x \u2208 s, ContinuousWithinAt f (Box.Icc I) x)\n    (Hd : \u2200 x \u2208 (Box.Icc I) \\ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x) :\n    HasIntegral.{0, u, u} I GP (fun x => \u2211 i, f' x (Pi.single i 1) i) BoxAdditiveMap.volume\n      (\u2211 i,\n        (integral.{0, u, u} (I.face i) GP (fun x => f (i.insertNth (I.upper i) x) i)\n            BoxAdditiveMap.volume -\n          integral.{0, u, u} (I.face i) GP (fun x => f (i.insertNth (I.lower i) x) i)\n            BoxAdditiveMap.volume))", "theorem hasIntegral_GP_pderiv (f : (Fin (n + 1) \u2192 \u211d) \u2192 E)\n    (f' : (Fin (n + 1) \u2192 \u211d) \u2192 (Fin (n + 1) \u2192 \u211d) \u2192L[\u211d] E) (s : Set (Fin (n + 1) \u2192 \u211d))\n    (hs : s.Countable) (Hs : \u2200 x \u2208 s, ContinuousWithinAt f (Box.Icc I) x)\n    (Hd : \u2200 x \u2208 (Box.Icc I) \\ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x) (i : Fin (n + 1)) :\n    HasIntegral.{0, u, u} I GP (fun x => f' x (Pi.single i 1)) BoxAdditiveMap.volume\n      (integral.{0, u, u} (I.face i) GP (fun x => f (i.insertNth (I.upper i) x))\n          BoxAdditiveMap.volume -\n        integral.{0, u, u} (I.face i) GP (fun x => f (i.insertNth (I.lower i) x))\n          BoxAdditiveMap.volume)", "@[simp]\ntheorem integrable_neg : Integrable I l (-f) vol \u2194 Integrable I l f vol", "theorem Btw.btw.antisymm {a b c : \u03b1} (h : btw a b c) : btw c b a \u2192 a = b \u2228 b = c \u2228 c = a", "theorem IsSubordinate.mono {f : BumpCovering \u03b9 X s} {U V : \u03b9 \u2192 Set X} (hU : f.IsSubordinate U)\n    (hV : \u2200 i, U i \u2286 V i) : f.IsSubordinate V", "theorem IsSubordinate.toPartitionOfUnity {f : BumpCovering \u03b9 X s} {U : \u03b9 \u2192 Set X}\n    (h : f.IsSubordinate U) : f.toPartitionOfUnity.IsSubordinate U", "@[simp] theorem extract_data (a : ByteArray) (start stop) :\n    (a.extract start stop).data = a.data.extract start stop", "@[simp] theorem get_set_eq (a : ByteArray) (i : Fin a.size) (v : UInt8) : (a.set i v)[i.val] = v", "theorem CPolynomialOn_const {v : F} {s : Set E} : CPolynomialOn \ud835\udd5c (fun _ => v) s", "theorem inv_subst {\u03b1} [Field \u03b1] {n k e : \u03b1} (h2 : e \u2260 0) (h3 : n * e = k) :\n    k * (e \u207b\u00b9) = n", "lemma pow_pos (ha : 0 < a) (n : \u2115) : 0 < a ^ n := pos_iff_ne_zero.2 <| pow_ne_zero _ ha.ne'", "theorem VanishingDiam.dist_lt (hA : VanishingDiam A) (\u03b5 : \u211d) (\u03b5_pos : 0 < \u03b5) (x : \u2115 \u2192 \u03b2) :\n    \u2203 n : \u2115, \u2200 (y) (_ : y \u2208 A (res x n)) (z) (_ : z \u2208 A (res x n)), dist y z < \u03b5", "@[simp]\ntheorem add_one_of_aleph0_le {c} (h : \u2135\u2080 \u2264 c) : c + 1 = c", "@[simp]\ntheorem aleph'_zero : aleph' 0 = 0", "theorem aleph0_le_mul_iff {a b : Cardinal} : \u2135\u2080 \u2264 a * b \u2194 a \u2260 0 \u2227 b \u2260 0 \u2227 (\u2135\u2080 \u2264 a \u2228 \u2135\u2080 \u2264 b)", "theorem aleph0_le_mul_iff' {a b : Cardinal.{u}} : \u2135\u2080 \u2264 a * b \u2194 a \u2260 0 \u2227 \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 a \u2227 b \u2260 0", "@[simp]\ntheorem aleph0_mul_aleph (o : Ordinal) : \u2135\u2080 * aleph o = aleph o", "@[simp]\ntheorem alephIdx.relIso_coe : (alephIdx.relIso : Cardinal \u2192 Ordinal) = alephIdx", "theorem aleph_isNormal : IsNormal (ord \u2218 aleph)", "@[simp]\ntheorem aleph_lt {o\u2081 o\u2082 : Ordinal} : aleph o\u2081 < aleph o\u2082 \u2194 o\u2081 < o\u2082", "@[simp]\ntheorem aleph_toNat (o : Ordinal) : toNat (aleph o) = 0", "theorem bddAbove_iff_small {s : Set Cardinal.{u}} : BddAbove s \u2194 Small.{u} s", "theorem cantor' (a) {b : Cardinal} (hb : 1 < b) : a < b ^ a", "theorem cantorFunctionAux_nonneg (h : 0 \u2264 c) : 0 \u2264 cantorFunctionAux c f n", "theorem cantorFunction_le (h1 : 0 \u2264 c) (h2 : c < 1) (h3 : \u2200 n, f n \u2192 g n) :\n    cantorFunction c f \u2264 cantorFunction c g", "@[simp]\ntheorem card_lt_of_card_iUnion_lt {\u03b9 : Type u} {\u03b1 : Type u} {t : \u03b9 \u2192 Set \u03b1} {c : Cardinal}\n    (h : #(\u22c3 i, t i) < c) (i : \u03b9) : #(t i) < c", "lemma exists_eq_of_iSup_eq_of_not_isSuccLimit\n    {\u03b9 : Type u} (f : \u03b9 \u2192 Cardinal.{v}) (\u03c9 : Cardinal.{v})\n    (h\u03c9 : \u00ac Order.IsSuccLimit \u03c9)\n    (h : \u2a06 i : \u03b9, f i = \u03c9) : \u2203 i, f i = \u03c9", "theorem infinite_pigeonhole_card_lt {\u03b2 \u03b1 : Type u} (f : \u03b2 \u2192 \u03b1) (w : #\u03b1 < #\u03b2) (w' : \u2135\u2080 \u2264 #\u03b1) :\n    \u2203 a : \u03b1, #\u03b1 < #(f \u207b\u00b9' {a})", "theorem le_def (\u03b1 \u03b2 : Type u) : #\u03b1 \u2264 #\u03b2 \u2194 Nonempty (\u03b1 \u21aa \u03b2)", "@[simp]\ntheorem lift_lift.{u_1} (a : Cardinal.{u_1}) : lift.{w} (lift.{v} a) = lift.{max v w} a", "@[simp]\ntheorem lift_lt {a b : Cardinal.{u}} : lift.{v, u} a < lift.{v, u} b \u2194 a < b", "@[simp, nolint simpNF]\ntheorem lift_mk_shrink (\u03b1 : Type u) [Small.{v} \u03b1] :\n    Cardinal.lift.{max u w} #(Shrink.{v} \u03b1) = Cardinal.lift.{max v w} #\u03b1", "@[simp, norm_cast] lemma lift_ofENat : \u2200 m : \u2115\u221e, lift.{u, v} m = m\n  | (m : \u2115) => lift_natCast m\n  | \u22a4 => lift_aleph0", "@[simp]\ntheorem lift_one : lift 1 = 1", "@[simp]\ntheorem lift_prod {\u03b9 : Type u} (c : \u03b9 \u2192 Cardinal.{v}) :\n    lift.{w} (prod c) = prod fun i => lift.{w} (c i)", "@[simp]\ntheorem lift_succ (a) : lift.{v, u} (succ a) = succ (lift.{v, u} a)", "theorem lift_uzero (a : Cardinal.{u}) : lift.{0} a = a", "@[simp]\ntheorem map_mk (f : Type u \u2192 Type v) (hf : \u2200 \u03b1 \u03b2, \u03b1 \u2243 \u03b2 \u2192 f \u03b1 \u2243 f \u03b2) (\u03b1 : Type u) :\n    map f hf #\u03b1 = #(f \u03b1)", "theorem mk_bounded_subset_le {\u03b1 : Type u} (s : Set \u03b1) (c : Cardinal.{u}) :\n    #{ t : Set \u03b1 // t \u2286 s \u2227 #t \u2264 c } \u2264 max #s \u2135\u2080 ^ c", "theorem mk_compl_eq_mk_compl_infinite {\u03b1 : Type*} [Infinite \u03b1] {s t : Set \u03b1} (hs : #s < #\u03b1)\n    (ht : #t < #\u03b1) : #(s\u1d9c : Set \u03b1) = #(t\u1d9c : Set \u03b1)", "@[simp]\ntheorem mk_eq_aleph0 (\u03b1 : Type*) [Countable \u03b1] [Infinite \u03b1] : #\u03b1 = \u2135\u2080", "theorem mk_eq_one (\u03b1 : Type u) [Unique \u03b1] : #\u03b1 = 1", "@[simp]\ntheorem mk_finset_of_infinite (\u03b1 : Type u) [Infinite \u03b1] : #(Finset \u03b1) = #\u03b1", "@[simp]\ntheorem mk_finsupp_lift_of_infinite (\u03b1 : Type u) (\u03b2 : Type v) [Infinite \u03b1] [Zero \u03b2] [Nontrivial \u03b2] :\n    #(\u03b1 \u2192\u2080 \u03b2) = max (lift.{v} #\u03b1) (lift.{u} #\u03b2)", "theorem mk_finsupp_nat (\u03b1 : Type u) [Nonempty \u03b1] : #(\u03b1 \u2192\u2080 \u2115) = max #\u03b1 \u2135\u2080", "@[simp]\ntheorem mk_le_aleph0 [Countable \u03b1] : #\u03b1 \u2264 \u2135\u2080", "theorem mk_ne_zero_iff {\u03b1 : Type u} : #\u03b1 \u2260 0 \u2194 Nonempty \u03b1", "theorem mk_perm_eq_self_power : #(Equiv.Perm \u03b1) = #\u03b1 ^ #\u03b1", "theorem mk_preimage_of_injective_lift {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (s : Set \u03b2)\n    (h : Injective f) : lift.{v} #(f \u207b\u00b9' s) \u2264 lift.{u} #s", "theorem mul_eq_left_iff {a b : Cardinal} : a * b = a \u2194 max \u2135\u2080 b \u2264 a \u2227 b \u2260 0 \u2228 b = 1 \u2228 a = 0", "theorem mul_lt_aleph0_iff {a b : Cardinal} : a * b < \u2135\u2080 \u2194 a = 0 \u2228 b = 0 \u2228 a < \u2135\u2080 \u2227 b < \u2135\u2080", "theorem mul_power {a b c : Cardinal} : (a * b) ^ c = a ^ c * b ^ c", "@[norm_cast]\ntheorem natCast_le {m n : \u2115} : (m : Cardinal) \u2264 n \u2194 m \u2264 n", "@[norm_cast]\ntheorem natCast_pow {m n : \u2115} : (\u2191(m ^ n) : Cardinal) = (\u2191m : Cardinal) ^ (\u2191n : Cardinal)", "theorem noMaxOrder {c} (h : \u2135\u2080 \u2264 c) : NoMaxOrder c.ord.out.\u03b1", "@[simp] lemma ofENat_le_lift {x : Cardinal.{v}} {m : \u2115\u221e} : m \u2264 lift.{u} x \u2194 m \u2264 x", "@[simp] lemma ofENat_mul (m n : \u2115\u221e) : ofENat (m * n) = m * n", "@[simp]\ntheorem ofNat_eq_lift_iff {a : Cardinal.{u}} {n : \u2115} [n.AtLeastTwo] :\n    (no_index (OfNat.ofNat n : Cardinal)) = lift.{v} a \u2194 (OfNat.ofNat n : Cardinal) = a", "@[simp]\nlemma ofNat_le_ofENat {m : \u2115} [m.AtLeastTwo] {n : \u2115\u221e} :\n    no_index (OfNat.ofNat m : Cardinal) \u2264 n \u2194 OfNat.ofNat m \u2264 n := nat_le_ofENat", "@[simp]\ntheorem ord_aleph0 : ord.{u} \u2135\u2080 = \u03c9", "theorem ord_le {c o} : ord c \u2264 o \u2194 c \u2264 o.card", "@[simp]\ntheorem ord_nat (n : \u2115) : ord n = n", "@[simp]\ntheorem ord_one : ord 1 = 1", "@[mono]\ntheorem ord_strictMono : StrictMono ord", "@[simp]\ntheorem partENatOfENat_toENat (c : Cardinal) : (toENat c : PartENat) = toPartENat c", "theorem power_add {a b c : Cardinal} : a ^ (b + c) = a ^ b * a ^ c", "theorem power_lt_aleph0 {a b : Cardinal} (ha : a < \u2135\u2080) (hb : b < \u2135\u2080) : a ^ b < \u2135\u2080", "@[simp]\ntheorem power_one {a : Cardinal.{u}} : a ^ (1 : Cardinal) = a", "theorem powerlt_le_powerlt_left {a b c : Cardinal} (h : b \u2264 c) : a ^< b \u2264 a ^< c", "@[simp]\ntheorem prime_of_aleph0_le (ha : \u2135\u2080 \u2264 a) : Prime a", "@[simp]\ntheorem prod_const (\u03b9 : Type u) (a : Cardinal.{v}) :\n    (prod fun _ : \u03b9 => a) = lift.{u} a ^ lift.{v} #\u03b9", "@[simp]\ntheorem succ_zero : succ (0 : Cardinal) = 1", "theorem sum_le_iSup {\u03b9 : Type u} (f : \u03b9 \u2192 Cardinal.{u}) : sum f \u2264 #\u03b9 * iSup f", "theorem sum_nat_eq_add_sum_succ (f : \u2115 \u2192 Cardinal.{u}) :\n    Cardinal.sum f = f 0 + Cardinal.sum fun i => f (i + 1)", "@[simp] lemma toENat_eq_nat {a : Cardinal} {n : \u2115} : toENat a = n \u2194 a = n := toENatAux_eq_nat", "lemma toENat_injOn : InjOn toENat (Iic \u2135\u2080) := toENat_strictMonoOn.injOn", "@[simp]\ntheorem toNat_add (hc : c < \u2135\u2080) (hd : d < \u2135\u2080) : toNat (c + d) = toNat c + toNat d", "theorem toNat_apply_of_aleph0_le {c : Cardinal} (h : \u2135\u2080 \u2264 c) : toNat c = 0", "@[simp]\ntheorem toNat_eq_one : toNat c = 1 \u2194 c = 1", "theorem toNat_le_iff_le_of_lt_aleph0 (hc : c < \u2135\u2080) (hd : d < \u2135\u2080) :\n    toNat c \u2264 toNat d \u2194 c \u2264 d", "theorem toNat_mul (x y : Cardinal) : toNat (x * y) = toNat x * toNat y", "@[simp]\ntheorem _root_.Cardinal.toPartENat_lt_natCast_iff {n : \u2115} {c : Cardinal} :\n    toPartENat c < \u2191n \u2194 c < \u2191n", "theorem toPartENat_strictMonoOn : StrictMonoOn toPartENat (Set.Iic \u2135\u2080)", "@[simp]\ntheorem zero_power {a : Cardinal} : a \u2260 0 \u2192 (0 : Cardinal) ^ a = 0", "theorem zero_toNat : toNat 0 = 0", "theorem Exact.op (h : Exact f g) : Exact g.op f.op", "theorem Exact.unop {X Y Z : C\u1d52\u1d56} (g : X \u27f6 Y) (f : Y \u27f6 Z) (h : Exact g f) : Exact f.unop g.unop", "theorem apply_eq_zero_of_comp_eq_zero {P Q R : C} (f : Q \u27f6 R) (a : P \u27f6 Q) : a \u226b f = 0 \u2192 f a = 0", "theorem pseudo_pullback {P Q R : C} {f : P \u27f6 R} {g : Q \u27f6 R} {p : P} {q : Q} :\n    f p = g q \u2192\n      \u2203 s, (pullback.fst : pullback f g \u27f6 P) s = p \u2227 (pullback.snd : pullback f g \u27f6 Q) s = q", "theorem epi_snd_of_isLimit [Epi f] {s : PullbackCone f g} (hs : IsLimit s) : Epi s.snd", "theorem exact_iff_exact_coimage_\u03c0 : Exact f g \u2194 Exact f (coimage.\u03c0 g)", "theorem pseudoEqual_symm {P : C} : Symmetric (PseudoEqual P)", "@[simp]\ntheorem \u03c0_map (p q : ActionCategory M X) (f : p \u27f6 q) : (\u03c0 M X).map f = f.val", "theorem AdditiveFunctor.forget_obj_of (F : C \u2964 D) [F.Additive] :\n    (AdditiveFunctor.forget C D).obj (AdditiveFunctor.of F) = F", "@[simp]\ntheorem homEquiv_naturality_right_symm_aux (f : X \u27f6 G.obj Y) (g : Y \u27f6 Y') :\n    F.map f \u226b (adj.homEquiv (G.obj Y) Y').symm (G.map g) = (adj.homEquiv X Y).symm f \u226b g", "private theorem he' {X Y Y'} (f g) : (e X Y').symm (f \u226b G.map g) = (e X Y).symm f \u226b g", "lemma isLeftAdjoint : F.IsLeftAdjoint := \u27e8_, \u27e8adj\u27e9\u27e9", "theorem map_injective (adj : F \u22a3 G) [F.PreservesMonomorphisms] (I : D) (hI : Injective I) :\n    Injective (G.obj I)", "@[simp]\ntheorem rightAdjointUniq_refl {F : C \u2964 D} {G : D \u2964 C} (adj1 : F \u22a3 G) :\n    (rightAdjointUniq adj1 adj1).hom = \ud835\udfd9 _", "@[reassoc (attr := simp)]\nlemma hom_inv_id_left (e : f \u2245 g) : e.hom.left \u226b e.inv.left = \ud835\udfd9 _", "theorem Aut_mul_def (f g : Aut X) : f * g = g.trans f", "@[simp]\ntheorem of_has_biproduct\u2082 [HasBinaryBiproduct X Y] :\n    BicartesianSq (0 : 0 \u27f6 X) (0 : 0 \u27f6 Y) biprod.inl biprod.inr", "theorem left_triangle_hom (e : a \u224c b) :\n    leftZigzag e.unit.hom e.counit.hom = (\u03bb_ e.hom).hom \u226b (\u03c1_ e.hom).inv", "theorem right_triangle (e : a \u224c b) :\n    rightZigzagIso e.unit e.counit = \u03c1_ e.inv \u226a\u226b (\u03bb_ e.inv).symm", "@[reassoc (attr := simp)]\ntheorem fac (H : IsKan t) (s : LeftLift f g) :\n    t.unit \u226b H.desc s \u25b7 f = s.unit", "theorem hom_ext (H : IsKan t) {k : c \u27f6 b} {\u03c4 \u03c4' : t.lift \u27f6 k}\n    (w : t.unit \u226b \u03c4 \u25b7 f = t.unit \u226b \u03c4' \u25b7 f) : \u03c4 = \u03c4'", "@[simp]\ntheorem leftZigzagIso_symm : (leftZigzagIso \u03b7 \u03b5).symm = rightZigzagIso \u03b5.symm \u03b7.symm", "theorem triangle_assoc_comp_left (f : a \u27f6 b) (g : b \u27f6 c) :\n    (\u03b1_ f (\ud835\udfd9 b) g).hom \u226b f \u25c1 (\u03bb_ g).hom = (\u03c1_ f).hom \u25b7 g", "theorem curry_id_eq_coev (A X : C) [Exponentiable A] : curry (\ud835\udfd9 _) = (exp.coev A).app X", "@[simp]\ntheorem leftUnitor_inv_f (X : Center C) : Hom.f (\u03bb_ X).inv = (\u03bb_ X.1).inv", "@[simp]\ntheorem tensor_fst (X Y : Center C) : (X \u2297 Y).1 = X.1 \u2297 Y.1", "@[reassoc]\ntheorem whiskerLeft_comm (X : Center C) {Y\u2081 Y\u2082 : Center C} (f : Y\u2081 \u27f6 Y\u2082) (U : C) :\n    (X.1 \u25c1 f.f) \u25b7 U \u226b ((tensorObj X Y\u2082).2.\u03b2 U).hom =\n      ((tensorObj X Y\u2081).2.\u03b2 U).hom \u226b U \u25c1 X.1 \u25c1 f.f", "@[reassoc (attr := simp)]\nlemma tensorHom_fst {X\u2081 X\u2082 Y\u2081 Y\u2082 : C} (f : X\u2081 \u27f6 X\u2082) (g : Y\u2081 \u27f6 Y\u2082) :\n    (f \u2297 g) \u226b fst _ _ = fst _ _ \u226b f := lift_fst _ _", "theorem iff_unop {A B X Y : C\u1d52\u1d56} {f : A \u27f6 X} {i : A \u27f6 B} {p : X \u27f6 Y} {g : B \u27f6 Y}\n    (sq : CommSq f i p g) : HasLift sq \u2194 HasLift sq.unop", "theorem mk' (l : sq.LiftStruct) : HasLift sq", "@[simp]\ntheorem comp_right : (f \u226b g).right = f.right \u226b g.right", "lemma Exact.\u03b4\u2080 {S : ComposableArrows C (n + 2)} (hS : S.Exact) :\n    S.\u03b4\u2080.Exact", "@[simp]\nlemma map_zero_zero : map F f 0 0 (by simp) = \ud835\udfd9 X := rfl", "@[simp]\nlemma obj_zero : obj F X 0 = X := rfl", "lemma ext\u2083 {f g : ComposableArrows C 3}\n    (h\u2080 : f.obj' 0 = g.obj' 0) (h\u2081 : f.obj' 1 = g.obj' 1) (h\u2082 : f.obj' 2 = g.obj' 2)\n    (h\u2083 : f.obj' 3 = g.obj' 3)\n    (w\u2080 : f.map' 0 1 = eqToHom h\u2080 \u226b g.map' 0 1 \u226b eqToHom h\u2081.symm)\n    (w\u2081 : f.map' 1 2 = eqToHom h\u2081 \u226b g.map' 1 2 \u226b eqToHom h\u2082.symm)\n    (w\u2082 : f.map' 2 3 = eqToHom h\u2082 \u226b g.map' 2 3 \u226b eqToHom h\u2083.symm) : f = g", "lemma ext\u2084 {f g : ComposableArrows C 4}\n    (h\u2080 : f.obj' 0 = g.obj' 0) (h\u2081 : f.obj' 1 = g.obj' 1) (h\u2082 : f.obj' 2 = g.obj' 2)\n    (h\u2083 : f.obj' 3 = g.obj' 3) (h\u2084 : f.obj' 4 = g.obj' 4)\n    (w\u2080 : f.map' 0 1 = eqToHom h\u2080 \u226b g.map' 0 1 \u226b eqToHom h\u2081.symm)\n    (w\u2081 : f.map' 1 2 = eqToHom h\u2081 \u226b g.map' 1 2 \u226b eqToHom h\u2082.symm)\n    (w\u2082 : f.map' 2 3 = eqToHom h\u2082 \u226b g.map' 2 3 \u226b eqToHom h\u2083.symm)\n    (w\u2083 : f.map' 3 4 = eqToHom h\u2083 \u226b g.map' 3 4 \u226b eqToHom h\u2084.symm) :\n    f = g", "@[simp]\nlemma homMk\u2082_app_one : (homMk\u2082 app\u2080 app\u2081 app\u2082 w\u2080 w\u2081).app 1 = app\u2081 := rfl", "@[simp]\nlemma homMk\u2082_app_two : (homMk\u2082 app\u2080 app\u2081 app\u2082 w\u2080 w\u2081).app \u27e82, by valid\u27e9 = app\u2082 := rfl", "@[simp]\nlemma homMk\u2082_app_zero : (homMk\u2082 app\u2080 app\u2081 app\u2082 w\u2080 w\u2081).app 0 = app\u2080 := rfl", "@[simp]\nlemma homMk\u2085_app_two :\n    (homMk\u2085 app\u2080 app\u2081 app\u2082 app\u2083 app\u2084 app\u2085 w\u2080 w\u2081 w\u2082 w\u2083 w\u2084).app \u27e82, by valid\u27e9 = app\u2082 := rfl", "lemma map'_eq_hom\u2081 (F : ComposableArrows C 1) : F.map' 0 1 = F.hom := rfl", "lemma mk\u2084_surjective (X : ComposableArrows C 4) :\n    \u2203 (X\u2080 X\u2081 X\u2082 X\u2083 X\u2084 : C) (f\u2080 : X\u2080 \u27f6 X\u2081) (f\u2081 : X\u2081 \u27f6 X\u2082) (f\u2082 : X\u2082 \u27f6 X\u2083) (f\u2083 : X\u2083 \u27f6 X\u2084),\n      X = mk\u2084 f\u2080 f\u2081 f\u2082 f\u2083", "@[reassoc]\nlemma naturality' (\u03c6 : F \u27f6 G) (i j : \u2115) (hij : i \u2264 j", "@[reassoc]\ntheorem \u03b4_comp_\u03c3_of_gt {n} {i : Fin (n + 2)} {j : Fin (n + 1)} (H : Fin.castSucc j < i) :\n    X.\u03b4 i.succ \u226b X.\u03c3 (Fin.castSucc j) = X.\u03c3 j \u226b X.\u03b4 i", "@[reassoc]\ntheorem \u03b4_comp_\u03c3_of_gt' {n} {i : Fin (n + 3)} {j : Fin (n + 2)} (H : j.succ < i) :\n    X.\u03b4 i \u226b X.\u03c3 j =\n      X.\u03c3 (j.castLT ((add_lt_add_iff_right 1).mp (lt_of_lt_of_le H i.is_le))) \u226b\n        X.\u03b4 (i.pred <|\n          fun (hi : i = 0) => by simp only [Fin.not_lt_zero, hi] at H)", "@[simp]\ntheorem mk_right (f : S.obj Y \u27f6 T) : (mk f).right = \u27e8\u27e8\u27e9\u27e9", "lemma fullyFaithful_preimage {X Y : C\u1d52\u1d56} (f : coyoneda.obj X \u27f6 coyoneda.obj Y) :\n    fullyFaithful.preimage f = (f.app X.unop (\ud835\udfd9 X.unop)).op := rfl", "@[simp]\ntheorem Discrete.mk_as {\u03b1 : Type u\u2081} (X : Discrete \u03b1) : Discrete.mk X.as = X", "@[to_additive (attr := simp) Discrete.addMonoidal_tensorUnit_as]\nlemma Discrete.monoidal_tensorUnit_as : (\ud835\udfd9_ (Discrete M)).as = 1 := rfl", "theorem right_inv : A.str \u226b strInv h = \ud835\udfd9 _", "@[simp]\ntheorem Equivalence_mk'_counit (functor inverse unit_iso counit_iso f) :\n    (\u27e8functor, inverse, unit_iso, counit_iso, f\u27e9 : C \u224c D).counit = counit_iso.hom", "@[simp]\ntheorem funInvIdAssoc_inv_app (e : C \u224c D) (F : C \u2964 E) (X : C) :\n    (funInvIdAssoc e F).inv.app X = F.map (e.unit.app X)", "theorem hasSheafCompose : J.HasSheafCompose F where", "theorem map_injective_iff (P : C) : Injective (F.functor.obj P) \u2194 Injective P", "theorem map_projective_iff (P : C) : Projective (F.functor.obj P) \u2194 Projective P", "theorem preregular : Preregular D", "lemma coevaluation_evaluation'' :\n    Y \u25c1 \u03b7_ X Y \u2297\u226b \u03b5_ X Y \u25b7 Y = \u2297\ud835\udfd9", "lemma evaluation_coevaluation'' :\n    \u03b7_ X Y \u25b7 X \u2297\u226b X \u25c1 \u03b5_ X Y = \u2297\ud835\udfd9", "@[simp]\ntheorem discrete_functor_obj_eq_as : (Discrete.functor f).obj X = f X.as", "theorem normalizeIsoApp_eq :\n    \u2200 (X : F C) (n : N C), normalizeIsoApp C X n = normalizeIsoApp' C X n.as", "lemma ofIso_compatibility :\n    letI := ofIso e A\n    NatTrans.CommShift e.hom A", "lemma Functor.Elements.ext {F : C \u2964 Type w} (x y : F.Elements) (h\u2081 : x.fst = y.fst)\n    (h\u2082 : F.map (eqToHom h\u2081) x.snd = y.snd) : x = y", "@[simp]\nlemma map_add_hom_app (a b : A) (X : C) :\n    F.map ((add hF s i a b).hom.app X) =\n      (i (a + b)).hom.app X \u226b (shiftFunctorAdd D a b).hom.app (F.obj X) \u226b\n        ((i a).inv.app X)\u27e6b\u27e7' \u226b (i b).inv.app ((s a).obj X)", "lemma eq_of_fac {\u03c8 \u03c8' : a' \u27f6 a} [IsHomLift p (\ud835\udfd9 R) \u03c8]\n    [IsHomLift p (\ud835\udfd9 R) \u03c8'] (hcomp : \u03c8 \u226b \u03c6 = \u03c6') (hcomp' : \u03c8' \u226b \u03c6 = \u03c6') : \u03c8 = \u03c8'", "@[simp]\nlemma map_self : IsCartesian.map p f \u03c6 \u03c6 = \ud835\udfd9 a", "lemma compatiblePreserving [Faithful G] : CompatiblePreserving K G", "lemma isContinuous [Faithful G] (Hp : CoverPreserving J K G) : G.IsContinuous J K", "theorem sheaf_eq_amalgamation (\u2131 : Sheaf K A) {X : A} {U : D} {T : Sieve U} (hT)\n    (x : FamilyOfElements _ T) (hx) (t) (h : x.IsAmalgamation t) :\n    t = (\u2131.cond X T hT).amalgamate x hx", "theorem IsLocalization.for_id (hW : W \u2264 MorphismProperty.isomorphisms C) : (\ud835\udfed C).IsLocalization W", "@[simp]\nlemma associator_hom_app (F\u2081 F\u2082 F\u2083 : J \u2964 C) (j : J) :\n    (\u03b1_ F\u2081 F\u2082 F\u2083).hom.app j = (\u03b1_ _ _ _).hom", "theorem cofinal_of_isTerminal_colimit_comp_yoneda\n    (h : IsTerminal (colimit (F \u22d9 yoneda))) : Final F", "lemma effectiveEpi_of_map (F : C \u2964 D) [F.ReflectsEffectiveEpis] {X Y : C} (f : X \u27f6 Y)\n    (h : EffectiveEpi (F.map f)) : EffectiveEpi f", "theorem Functor.eq_of_iso {F\u2081 F\u2082 : D \u2964 C} [Quiver.IsThin C] (hC : Skeletal C) (hF : F\u2081 \u2245 F\u2082) :\n    F\u2081 = F\u2082", "theorem essImage.ofNatIso {F' : C \u2964 D} (h : F \u2245 F') {Y : D} (hY : Y \u2208 essImage F) :\n    Y \u2208 essImage F'", "lemma ext_of_iso {F G : C \u2964 D} (e : F \u2245 G) (hobj : \u2200 X, F.obj X = G.obj X)\n    (happ : \u2200 X, e.hom.app X = eqToHom (hobj X)) : F = G", "theorem final_iff_comp_final_full_faithful [Final G] [Full G] [Faithful G] :\n    Final F \u2194 Final (F \u22d9 G)", "theorem final_of_comp_full_faithful' [Full G] [Faithful G] [Final (F \u22d9 G)] : Final G", "theorem Functor.final_of_exists_of_isFiltered_of_fullyFaithful [IsFilteredOrEmpty D] [F.Full]\n    [F.Faithful] (h : \u2200 d, \u2203 c, Nonempty (d \u27f6 F.obj c)) : Final F", "lemma flip_injective {F\u2081 F\u2082 : B \u2964 C \u2964 D} (h : F\u2081.flip = F\u2082.flip) :\n    F\u2081 = F\u2082", "lemma hasRightDerivedFunctor_iff :\n    F.HasRightDerivedFunctor W \u2194 HasLeftKanExtension L F", "lemma hasRightDerivedFunctor_iff_of_iso :\n    HasRightDerivedFunctor F W \u2194 HasRightDerivedFunctor F' W", "lemma homologySequence_epi_shift_map_mor\u2082_iff :\n    Epi ((F.shift n\u2080).map T.mor\u2082) \u2194 F.homologySequence\u03b4 T n\u2080 n\u2081 h = 0", "theorem initial_iff_comp_equivalence [IsEquivalence G] : Initial F \u2194 Initial (F \u22d9 G)", "theorem initial_iff_equivalence_comp [IsEquivalence F] : Initial G \u2194 Initial (F \u22d9 G)", "@[simp]\ntheorem inv_fun_map (F : C \u2964 D) [IsEquivalence F] (X Y : C) (f : X \u27f6 Y) :\n    F.inv.map (F.map f) = F.asEquivalence.unitInv.app X \u226b f \u226b F.asEquivalence.unit.app Y", "lemma lanAdjunction_counit_app (G : D \u2964 H) :\n    (L.lanAdjunction H).counit.app G =\n      descOfIsLeftKanExtension (L.lan.obj (L \u22d9 G)) (L.lanUnit.app (L \u22d9 G)) G (\ud835\udfd9 (L \u22d9 G))", "@[simp]\nlemma lanAdjunction_unit : (L.lanAdjunction H).unit = L.lanUnit", "theorem mapIso_injective (F : C \u2964 D) [Faithful F] :\n    Function.Injective <| (F.mapIso : (X \u2245 Y) \u2192 (F.obj X \u2245 F.obj Y))", "theorem map_commSq (s : CommSq f g h i) : CommSq (F.map f) (F.map g) (F.map h) (F.map i)", "@[reassoc]\ntheorem map_hom_inv (F : C \u2964 D) {X Y : C} (f : X \u27f6 Y) [IsIso f] :\n    F.map f \u226b F.map (inv f) = \ud835\udfd9 (F.obj X)", "@[simp, reassoc]\nlemma map_shiftFunctorCompIsoId_hom_app [F.CommShift A] (X : C) (a b : A) (h : a + b = 0) :\n    F.map ((shiftFunctorCompIsoId C a b h).hom.app X) =\n      (F.commShiftIso b).hom.app (X\u27e6a\u27e7) \u226b ((F.commShiftIso a).hom.app X)\u27e6b\u27e7' \u226b\n        (shiftFunctorCompIsoId D a b h).hom.app (F.obj X)", "theorem map_zsmul {X Y : C} {f : X \u27f6 Y} {r : \u2124} : F.map (r \u2022 f) = r \u2022 F.map f", "theorem mem_eventualRange_iff {x : F.obj j} :\n    x \u2208 F.eventualRange j \u2194 \u2200 \u2983i\u2984 (f : i \u27f6 j), x \u2208 range (F.map f)", "theorem preservesZeroMorphisms_of_map_zero_object (i : F.obj 0 \u2245 0) : PreservesZeroMorphisms F where", "theorem reflectsMonomorphisms_of_preserves_of_reflects (F : C \u2964 D) (G : D \u2964 E)\n    [PreservesMonomorphisms G] [ReflectsMonomorphisms (F \u22d9 G)] : ReflectsMonomorphisms F", "@[reassoc (attr := simp)]\nlemma rightDerivedNatTrans_fac (\u03c4 : F \u27f6 F') :\n    \u03b1 \u226b whiskerLeft L (rightDerivedNatTrans RF RF' \u03b1 \u03b1' W \u03c4) = \u03c4 \u226b \u03b1'", "@[reassoc (attr := simp)]\nlemma rightDerived_fac_app (G : D \u2964 H) (\u03b2 : F \u27f6 L \u22d9 G) (X : C):\n    \u03b1.app X \u226b (RF.rightDerivedDesc \u03b1 W G \u03b2).app (L.obj X) = \u03b2.app X", "lemma rotate_essImageDistTriang [Preadditive D] [L.Additive]\n    [\u2200 (n : \u2124), (shiftFunctor D n).Additive] (T : Triangle D) :\n  T \u2208 L.essImageDistTriang \u2194 T.rotate \u2208 L.essImageDistTriang", "@[simp]\ntheorem sum_map_inr (F : A \u2964 B) (G : C \u2964 D) {c c' : C} (f : inr c \u27f6 inr c') :\n    (F.sum G).map f = G.map f", "lemma uncurry_obj_curry_obj_flip_flip' (F\u2081 : B \u2964 C) (F\u2082 : D \u2964 E) (G : C \u00d7 E \u2964 H) :\n    uncurry.obj (F\u2081 \u22d9 (F\u2082 \u22d9 (curry.obj G).flip).flip) = (F\u2081.prod F\u2082) \u22d9 G", "@[reassoc (attr := simp)]\ntheorem \u03b9_biproductComparison' (j : J) :\n    biproduct.\u03b9 _ j \u226b biproductComparison' F f = F.map (biproduct.\u03b9 f j)", "@[simp]\nlemma binaryProductIso_inv_comp_snd :\n    (binaryProductIso F G).inv \u226b Limits.prod.snd = prod.snd", "@[ext]\nlemma prod_ext {a : C} (z w : (prod F G).obj a) (h1 : z.1 = w.1) (h2 : z.2 = w.2) :\n    z = w := Prod.ext h1 h2", "@[simp]\ntheorem diagramIso_app_left (i : D.J \u00d7 D.J) :\n    (D.diagramIso F).app (WalkingMultispan.left i) = Iso.refl _", "@[simp]\ntheorem eqToHom_proj {I : Type*} {x x' : GradedObject I C} (h : x = x') (i : I) :\n    (eqToHom h : x \u27f6 x') i = eqToHom (Function.funext_iff.mp h i)", "@[ext]\nlemma mapBifunctorMapObj_ext {X : GradedObject I C\u2081} {Y : GradedObject J C\u2082} {A : C\u2083} {k : K}\n    [HasMap (((mapBifunctor F I J).obj X).obj Y) p]\n    {f g : mapBifunctorMapObj F p X Y k \u27f6 A}\n    (h : \u2200 (i : I) (j : J) (hij : p \u27e8i, j\u27e9 = k),\n      \u03b9MapBifunctorMapObj F p X Y i j k hij \u226b f = \u03b9MapBifunctorMapObj F p X Y i j k hij \u226b g) :\n    f = g", "lemma mapBifunctor_triangle :\n    (mapBifunctorAssociator associator \u03c4.\u03c1\u2081\u2082 \u03c4.\u03c1\u2082\u2083 X\u2081 ((single\u2080 I\u2082).obj X\u2082) X\u2083).hom \u226b\n    mapBifunctorMapMap G \u03c0 (\ud835\udfd9 X\u2081) (mapBifunctorLeftUnitor F\u2082 X\u2082 e\u2082 \u03c4.p\u2082\u2083 \u03c4.h\u2083 X\u2083).hom =\n      mapBifunctorMapMap G \u03c0 (mapBifunctorRightUnitor F\u2081 X\u2082 e\u2081 \u03c4.p\u2081\u2082 \u03c4.h\u2081 X\u2081).hom (\ud835\udfd9 X\u2083)", "lemma \u03b9MapObjOrZero_eq_zero (h : p i \u2260 j) : X.\u03b9MapObjOrZero p i j = 0 := dif_neg h", "@[reassoc (attr := simp)]\nlemma \u03b9_descMapObj {A : C} {j : J}\n    (\u03c6 : \u2200 (i : I) (_ : p i = j), X i \u27f6 A) (i : I) (hi : p i = j) :\n    X.\u03b9MapObj p i j hi \u226b X.descMapObj p \u03c6 = \u03c6 i hi", "@[reassoc (attr := simp)]\nlemma \u03b9_mapMap (i : I) (j : J) (hij : p i = j) :\n    X.\u03b9MapObj p i j hij \u226b mapMap \u03c6 p j = \u03c6 i \u226b Y.\u03b9MapObj p i j hij", "@[ext]\ntheorem ext (S T : J.Cover X) (h : \u2200 \u2983Y\u2984 (f : Y \u27f6 X), S f \u2194 T f) : S = T", "theorem res_mk_eq_mk_pullback {Y X : C} {P : C\u1d52\u1d56 \u2964 D} {S : J.Cover X} (x : Meq P S) (f : Y \u27f6 X) :\n    (J.plusObj P).map f.op (mk x) = mk (x.pullback f)", "lemma W_eq_inverseImage_isomorphisms_of_adjunction :\n    J.W = (MorphismProperty.isomorphisms _).inverseImage G", "lemma W_iff {P\u2081 P\u2082 : C\u1d52\u1d56 \u2964 A} (f : P\u2081 \u27f6 P\u2082) :\n    J.W f \u2194 IsIso ((presheafToSheaf J A).map f)", "lemma W_inverseImage_whiskeringLeft :\n    K.W.inverseImage ((whiskeringLeft D\u1d52\u1d56 C\u1d52\u1d56 A).obj G.op) = J.W", "lemma W_of_isLocallyBijective [Presheaf.IsLocallyInjective J f]\n    [Presheaf.IsLocallySurjective J f] : J.W f", "lemma W_sheafToPreheaf_map_iff_isIso {F\u2081 F\u2082 : Sheaf J A} (\u03c6 : F\u2081 \u27f6 F\u2082) :\n    J.W ((sheafToPresheaf J A).map \u03c6) \u2194 IsIso \u03c6", "theorem arrow_max (f : Y \u27f6 X) (S : Sieve X) (hf : S f) : J.Covers S f", "theorem bot_covers (S : Sieve X) (f : Y \u27f6 X) : (\u22a5 : GrothendieckTopology C).Covers S f \u2194 S f", "@[simp]\ntheorem close_close {X : C} (S : Sieve X) : J\u2081.close (J\u2081.close S) = J\u2081.close S", "theorem isClosed_pullback {X Y : C} (f : Y \u27f6 X) (S : Sieve X) :\n    J\u2081.IsClosed S \u2192 J\u2081.IsClosed (S.pullback f)", "@[simp]\ntheorem isoSheafify_hom {P : C\u1d52\u1d56 \u2964 D} (hP : Presheaf.IsSheaf J P) :\n    (J.isoSheafify hP).hom = J.toSheafify P", "@[simp]\ntheorem isoToPlus_inv (hP : Presheaf.IsSheaf J P) :\n    (J.isoToPlus P hP).inv = J.plusLift (\ud835\udfd9 _) hP", "lemma over_forget_coverPreserving (X : C) :\n    CoverPreserving (J.over X) J (Over.forget X) where\n  cover_preserve hS := hS", "lemma GrothendieckTopology.preservesSheafification_iff_of_adjunctions :\n    J.PreservesSheafification F \u2194 \u2200 (P : C\u1d52\u1d56 \u2964 A),\n      IsIso (G\u2082.map (whiskerRight (adj\u2081.unit.app P) F))", "lemma sheafToPresheaf_map_sheafComposeNatTrans_eq_sheafifyCompIso_inv (P : C\u1d52\u1d56 \u2964 D) :\n    (sheafToPresheaf J E).map\n      ((sheafComposeNatTrans J F (plusPlusAdjunction J D) (plusPlusAdjunction J E)).app P) =\n      (sheafifyCompIso J F P).inv", "@[simp]\ntheorem sheafificationWhiskerRightIso_inv_app :\n    (J.sheafificationWhiskerRightIso F).inv.app P = (J.sheafifyCompIso F P).inv", "@[simp]\ntheorem sheafifyMap_comp {P Q R : C\u1d52\u1d56 \u2964 D} (\u03b7 : P \u27f6 Q) (\u03b3 : Q \u27f6 R) :\n    J.sheafifyMap (\u03b7 \u226b \u03b3) = J.sheafifyMap \u03b7 \u226b J.sheafifyMap \u03b3", "theorem sheafify_hom_ext {P Q : C\u1d52\u1d56 \u2964 D} (\u03b7 \u03b3 : J.sheafify P \u27f6 Q) (hQ : Presheaf.IsSheaf J Q)\n    (h : J.toSheafify P \u226b \u03b7 = J.toSheafify P \u226b \u03b3) : \u03b7 = \u03b3", "@[simp]\ntheorem Groupoid.inv_eq_inv (f : X \u27f6 Y) : Groupoid.inv f = CategoryTheory.inv f", "theorem of_arrow_iso_right {A B X Y X' Y' : C} (i : A \u27f6 B) {p : X \u27f6 Y} {p' : X' \u27f6 Y'}\n    (e : Arrow.mk p \u2245 Arrow.mk p') [hip : HasLiftingProperty i p] : HasLiftingProperty i p'", "@[simp]\ntheorem HasShift.shift_obj_obj (n : A) (X : C) : (HasShift.shift.obj \u27e8n\u27e9).obj X = X\u27e6n\u27e7", "theorem matrixDecomposition_comp (o : HomOrthogonal s) {\u03b1 \u03b2 \u03b3 : Type} [Finite \u03b1] [Fintype \u03b2]\n    [Finite \u03b3] {f : \u03b1 \u2192 \u03b9} {g : \u03b2 \u2192 \u03b9} {h : \u03b3 \u2192 \u03b9} (z : (\u2a01 fun a => s (f a)) \u27f6 \u2a01 fun b => s (g b))\n    (w : (\u2a01 fun b => s (g b)) \u27f6 \u2a01 fun c => s (h c)) (i : \u03b9) :\n    o.matrixDecomposition (z \u226b w) i = o.matrixDecomposition w i * o.matrixDecomposition z i", "theorem equivalence_inverse : (equivalence : SimplicialObject C \u224c _).inverse = \u0393", "@[simp, reassoc]\ntheorem p_idem : P.p.f n \u226b P.p.f n = P.p.f n", "theorem decomp_p (P : Karoubi C) : (toKaroubi C).map P.p = decompId_p P \u226b decompId_i P", "lemma ofCocomplex_d_0_1 :\n    (ofCocomplex Z).d 0 1 = d (Injective.\u03b9 Z)", "@[reassoc]\ntheorem eq_condition {j j' : C} (f f' : j \u27f6 j') : eqHom f f' \u226b f = eqHom f f' \u226b f'", "theorem IsDetecting.mono {\ud835\udca2 : Set C} (h\ud835\udca2 : IsDetecting \ud835\udca2) {\u210b : Set C} (h\ud835\udca2\u210b : \ud835\udca2 \u2286 \u210b) :\n    IsDetecting \u210b", "theorem IsDetector.def {G : C} :\n    IsDetector G \u2192 \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), (\u2200 h : G \u27f6 Y, \u2203! h', h' \u226b f = h) \u2192 IsIso f", "theorem bowtie {j\u2081 j\u2082 k\u2081 k\u2082 : C} (f\u2081 : j\u2081 \u27f6 k\u2081) (g\u2081 : j\u2081 \u27f6 k\u2082) (f\u2082 : j\u2082 \u27f6 k\u2081) (g\u2082 : j\u2082 \u27f6 k\u2082) :\n    \u2203 (s : C) (\u03b1 : k\u2081 \u27f6 s) (\u03b2 : k\u2082 \u27f6 s), f\u2081 \u226b \u03b1 = g\u2081 \u226b \u03b2 \u2227 f\u2082 \u226b \u03b1 = g\u2082 \u226b \u03b2", "theorem coeq\u2083_condition\u2082 {j\u2081 j\u2082 : C} (f g h : j\u2081 \u27f6 j\u2082) :\n    g \u226b coeq\u2083Hom f g h = h \u226b coeq\u2083Hom f g h", "theorem of_hasFiniteColimits [HasFiniteColimits C] : IsFiltered C", "theorem toSup_commutes {X Y : C} (mX : X \u2208 O) (mY : Y \u2208 O) {f : X \u27f6 Y}\n    (mf : (\u27e8X, Y, mX, mY, f\u27e9 : \u03a3' (X Y : C) (_ : X \u2208 O) (_ : Y \u2208 O), X \u27f6 Y) \u2208 H) :\n    f \u226b toSup O H mY = toSup O H mX", "theorem of_equivalence (h : C \u224c D) : IsFilteredOrEmpty D", "theorem IsFilteredOrEmpty.of_final (F : C \u2964 D) [Final F] [IsFilteredOrEmpty C] :\n    IsFilteredOrEmpty D where", "theorem of_isRightAdjoint (R : C \u2964 D) [R.IsRightAdjoint] : IsFilteredOrEmpty D", "@[simp]\ntheorem inv_comp_eq (\u03b1 : X \u27f6 Y) [IsIso \u03b1] {f : X \u27f6 Z} {g : Y \u27f6 Z} : inv \u03b1 \u226b f = g \u2194 f = \u03b1 \u226b g", "@[simp]\ntheorem inr_fst (X Y : C) [HasBinaryBiproduct X Y] :\n    IsPullback biprod.inr (0 : Y \u27f6 0) biprod.fst (0 : 0 \u27f6 X)", "theorem of_isLimit' (w : CommSq fst snd f g) (h : Limits.IsLimit w.cone) :\n    IsPullback fst snd f g", "theorem of_is_bilimit' {b : BinaryBicone X Y} (h : b.IsBilimit) :\n    IsPullback (0 : 0 \u27f6 X) (0 : 0 \u27f6 Y) b.inl b.inr", "theorem op (h : IsPullback fst snd f g) : IsPushout g.op f.op snd.op fst.op", "theorem of_horiz_isIso [IsIso f] [IsIso inr] (sq : CommSq f g inl inr) : IsPushout f g inl inr", "theorem op (h : IsPushout f g inl inr) : IsPullback inr.op inl.op g.op f.op", "theorem unop {Z X Y P : C\u1d52\u1d56} {f : Z \u27f6 X} {g : Z \u27f6 Y} {inl : X \u27f6 P} {inr : Y \u27f6 P}\n    (h : IsPushout f g inl inr) : IsPullback inr.unop inl.unop g.unop f.unop", "lemma IsTriangulated.mk' (h : \u2200 \u2983X\u2081' X\u2082' X\u2083' : C\u2984 (u\u2081\u2082' : X\u2081' \u27f6 X\u2082') (u\u2082\u2083' : X\u2082' \u27f6 X\u2083'),\n    \u2203 (X\u2081 X\u2082 X\u2083 Z\u2081\u2082 Z\u2082\u2083 Z\u2081\u2083 : C) (u\u2081\u2082 : X\u2081 \u27f6 X\u2082) (u\u2082\u2083 : X\u2082 \u27f6 X\u2083) (e\u2081 : X\u2081' \u2245 X\u2081) (e\u2082 : X\u2082' \u2245 X\u2082)\n    (e\u2083 : X\u2083' \u2245 X\u2083) (_ : u\u2081\u2082' \u226b e\u2082.hom = e\u2081.hom \u226b u\u2081\u2082)\n    (_ : u\u2082\u2083' \u226b e\u2083.hom = e\u2082.hom \u226b u\u2082\u2083)\n    (v\u2081\u2082 : X\u2082 \u27f6 Z\u2081\u2082) (w\u2081\u2082 : Z\u2081\u2082 \u27f6 X\u2081\u27e61\u27e7) (h\u2081\u2082 : Triangle.mk u\u2081\u2082 v\u2081\u2082 w\u2081\u2082 \u2208 distTriang C)\n    (v\u2082\u2083 : X\u2083 \u27f6 Z\u2082\u2083) (w\u2082\u2083 : Z\u2082\u2083 \u27f6 X\u2082\u27e61\u27e7) (h\u2082\u2083 : Triangle.mk u\u2082\u2083 v\u2082\u2083 w\u2082\u2083 \u2208 distTriang C)\n    (v\u2081\u2083 : X\u2083 \u27f6 Z\u2081\u2083) (w\u2081\u2083 : Z\u2081\u2083 \u27f6 X\u2081\u27e61\u27e7)\n      (h\u2081\u2083 : Triangle.mk (u\u2081\u2082 \u226b u\u2082\u2083) v\u2081\u2083 w\u2081\u2083 \u2208 distTriang C),\n        Nonempty (Octahedron rfl h\u2081\u2082 h\u2082\u2083 h\u2081\u2083)) :\n    IsTriangulated C where\n  octahedron_axiom {X\u2081' X\u2082' X\u2083' Z\u2081\u2082' Z\u2082\u2083' Z\u2081\u2083' u\u2081\u2082' u\u2082\u2083' u\u2081\u2083'} comm'\n    {v\u2081\u2082' w\u2081\u2082'} h\u2081\u2082' {v\u2082\u2083' w\u2082\u2083'} h\u2082\u2083' {v\u2081\u2083' w\u2081\u2083'} h\u2081\u2083'", "theorem IsUniversalColimit.map_reflective\n    {Gl : C \u2964 D} {Gr : D \u2964 C} (adj : Gl \u22a3 Gr) [Gr.Full] [Gr.Faithful]\n    {F : J \u2964 D} {c : Cocone (F \u22d9 Gr)}\n    (H : IsUniversalColimit c)\n    [\u2200 X (f : X \u27f6 Gl.obj c.pt), HasPullback (Gr.map f) (adj.unit.app c.pt)]\n    [\u2200 X (f : X \u27f6 Gl.obj c.pt), PreservesLimit (cospan (Gr.map f) (adj.unit.app c.pt)) Gl] :\n    IsUniversalColimit (Gl.mapCocone c)", "@[simp]\ntheorem conj_id : \u03b1.conj (\ud835\udfd9 X) = \ud835\udfd9 Y", "theorem eq_comp_inv (\u03b1 : X \u2245 Y) {f : Z \u27f6 Y} {g : Z \u27f6 X} : g = f \u226b \u03b1.inv \u2194 g \u226b \u03b1.hom = f", "@[simp]\ntheorem homCongr_apply {X Y X\u2081 Y\u2081 : C} (\u03b1 : X \u2245 X\u2081) (\u03b2 : Y \u2245 Y\u2081) (f : X \u27f6 Y) :\n    \u03b1.homCongr \u03b2 f = \u03b1.inv \u226b f \u226b \u03b2.hom", "theorem inv_eq_inv (f g : X \u2245 Y) : f.inv = g.inv \u2194 f.hom = g.hom", "@[aesop apply safe (rule_sets := [CategoryTheory])]\ntheorem inv_ext' {f : X \u2245 Y} {g : Y \u27f6 X} (hom_inv_id : f.hom \u226b g = \ud835\udfd9 X) : g = f.inv", "@[reassoc (attr := simp)]\nlemma _root_.CategoryTheory.Iso.inv_hom_id_triangle_hom\u2083 {A B : Triangle C} (e : A \u2245 B) :\n    e.inv.hom\u2083 \u226b e.hom.hom\u2083 = \ud835\udfd9 _", "@[reassoc (attr := simp)]\nlemma map_hom_inv_id (F : C \u2964 D) :\n    F.map e.hom \u226b F.map e.inv = \ud835\udfd9 _", "@[reassoc (attr := simp)]\ntheorem map_inv_hom_id_app {X Y : C} (e : X \u2245 Y) (F : C \u2964 D \u2964 E) (Z : D) :\n    (F.map e.inv).app Z \u226b (F.map e.hom).app Z = \ud835\udfd9 _", "@[reassoc (attr := simp)]\nlemma map_inv_hom_id_eval_app (e : X \u2245 Y) (F : C \u2964 D \u2964 E) (j : J) (Y : D) :\n    (F.map (e.inv j)).app Y \u226b (F.map (e.hom j)).app Y = \ud835\udfd9 _", "@[simp]\ntheorem symm_inv (\u03b1 : X \u2245 Y) : \u03b1.symm.inv = \u03b1.hom", "@[simp]\ntheorem symm_self_id_assoc (\u03b1 : X \u2245 Y) (\u03b2 : Y \u2245 Z) : \u03b1.symm \u226a\u226b \u03b1 \u226a\u226b \u03b2 = \u03b2", "theorem Iso.to_eq {x y : X} (f : x \u2245 y) : x = y", "@[simp]\ntheorem trans_refl (\u03b1 : X \u2245 Y) : \u03b1 \u226a\u226b Iso.refl Y = \u03b1", "@[simp]\ntheorem LeftExactFunctor.forget_map {F G : C \u2964\u2097 D} (\u03b1 : F \u27f6 G) :\n    (LeftExactFunctor.forget C D).map \u03b1 = \u03b1", "@[simp]\ntheorem LeftExactFunctor.of_fst (F : C \u2964 D) [PreservesFiniteLimits F] :\n    (LeftExactFunctor.of F).obj = F", "@[simp]\ntheorem toCone_pt (B : Bicone F) : B.toCone.pt = B.pt", "@[simp]\ntheorem BinaryBiproduct.bicone_inr : (BinaryBiproduct.bicone X Y).inr = biprod.inr", "theorem BinaryCofan.IsColimit.hom_ext {W X Y : C} {s : BinaryCofan X Y} (h : IsColimit s)\n    {f g : s.pt \u27f6 W} (h\u2081 : s.inl \u226b f = s.inl \u226b g) (h\u2082 : s.inr \u226b f = s.inr \u226b g) : f = g", "@[simp]\ntheorem BinaryCofan.mk_inr {P : C} (\u03b9\u2081 : X \u27f6 P) (\u03b9\u2082 : Y \u27f6 P) : (BinaryCofan.mk \u03b9\u2081 \u03b9\u2082).inr = \u03b9\u2082", "@[simp]\ntheorem BinaryFan.assocInv_fst {X Y Z : C} {sXY : BinaryFan X Y} (P : IsLimit sXY)\n    {sYZ : BinaryFan Y Z} (s : BinaryFan X sYZ.pt) :\n    (BinaryFan.assocInv P s).fst = P.lift (BinaryFan.mk s.fst (s.snd \u226b sYZ.fst))", "@[simp]\ntheorem BinaryFan.assoc_snd {X Y Z : C} {sXY : BinaryFan X Y} {sYZ : BinaryFan Y Z}\n    (Q : IsLimit sYZ) (s : BinaryFan sXY.pt Z) :\n    (BinaryFan.assoc Q s).snd = Q.lift (BinaryFan.mk (s.fst \u226b sXY.snd) s.snd)", "theorem Cofork.IsColimit.homIso_natural {X Y : C} {f g : X \u27f6 Y} {t : Cofork f g} {Z Z' : C}\n    (q : Z \u27f6 Z') (ht : IsColimit t) (k : t.pt \u27f6 Z) :\n    (Cofork.IsColimit.homIso ht _ (k \u226b q) : Y \u27f6 Z') =\n      (Cofork.IsColimit.homIso ht _ k : Y \u27f6 Z) \u226b q", "theorem finite_coproduct_eq_finset_sup [SemilatticeSup \u03b1] [OrderBot \u03b1] {\u03b9 : Type u} [Fintype \u03b9]\n    (f : \u03b9 \u2192 \u03b1) : \u2210 f = Fintype.elems.sup f", "theorem cokernel_funext {C : Type*} [Category C] [HasZeroMorphisms C] [ConcreteCategory C]\n    {M N K : C} {f : M \u27f6 N} [HasCokernel f] {g h : cokernel f \u27f6 K}\n    (w : \u2200 n : N, g (cokernel.\u03c0 f n) = h (cokernel.\u03c0 f n)) : g = h", "theorem Concrete.from_union_surjective_of_isColimit {D : Cocone F} (hD : IsColimit D) :\n    let ff : (\u03a3j : J, F.obj j) \u2192 D.pt := fun a => D.\u03b9.app a.1 a.2\n    Function.Surjective ff", "theorem Concrete.isColimit_exists_rep {D : Cocone F} (hD : IsColimit D) (x : D.pt) :\n    \u2203 (j : J) (y : F.obj j), D.\u03b9.app j y = x", "theorem multiequalizer_ext {I : MulticospanIndex.{w} C} [HasMultiequalizer I]\n    [PreservesLimit I.multicospan (forget C)] (x y : \u2191(multiequalizer I))\n    (h : \u2200 t : I.L, Multiequalizer.\u03b9 I t x = Multiequalizer.\u03b9 I t y) : x = y", "theorem widePushout_exists_rep' {B : C} {\u03b1 : Type _} [Nonempty \u03b1] {X : \u03b1 \u2192 C}\n    (f : \u2200 j : \u03b1, B \u27f6 X j) [HasWidePushout.{v} B X f] [PreservesColimit (wideSpan B X f) (forget C)]\n    (x : \u2191(widePushout B X f)) : \u2203 (i : \u03b1) (y : X i), \u03b9 f i y = x", "theorem Cotrident.coequalizer_ext [Nonempty J] (s : Cotrident f) {W : C} {k l : s.pt \u27f6 W}\n    (h : s.\u03c0 \u226b k = s.\u03c0 \u226b l) : \u2200 j : WalkingParallelFamily J, s.\u03b9.app j \u226b k = s.\u03b9.app j \u226b l", "theorem Fork.IsLimit.existsUnique {s : Fork f g} (hs : IsLimit s) {W : C} (k : W \u27f6 X)\n    (h : k \u226b f = k \u226b g) : \u2203! l : W \u27f6 s.pt, l \u226b Fork.\u03b9 s = k", "theorem Fork.IsLimit.homIso_natural {X Y : C} {f g : X \u27f6 Y} {t : Fork f g} (ht : IsLimit t)\n    {Z Z' : C} (q : Z' \u27f6 Z) (k : Z \u27f6 t.pt) :\n    (Fork.IsLimit.homIso ht _ (q \u226b k) : Z' \u27f6 X) = q \u226b (Fork.IsLimit.homIso ht _ k : Z \u27f6 X)", "@[reassoc (attr := simp)]\ntheorem Fork.hom_comp_\u03b9 {s t : Fork f g} (f : s \u27f6 t) : f.hom \u226b t.\u03b9 = s.\u03b9", "theorem HasBinaryProduct.swap (P Q : C) [HasBinaryProduct P Q] : HasBinaryProduct Q P", "theorem HasBiproduct.of_hasCoproduct (f : J \u2192 C) [HasCoproduct f] : HasBiproduct f", "@[simp]\ntheorem HasLimit.isoOfEquivalence_inv_\u03c0 {F : J \u2964 C} [HasLimit F] {G : K \u2964 C} [HasLimit G]\n    (e : J \u224c K) (w : e.functor \u22d9 G \u2245 F) (j : J) :\n    (HasLimit.isoOfEquivalence e w).inv \u226b limit.\u03c0 F j =\n    limit.\u03c0 G (e.functor.obj j) \u226b w.hom.app j", "@[simp]\ntheorem zeroIsoInitial_hom [HasInitial C] : zeroIsoInitial.hom = (0 : 0 \u27f6 \u22a5_ C)", "@[simp]\ntheorem zeroIsoIsTerminal_inv {X : C} (t : IsTerminal X) : (zeroIsoIsTerminal t).inv = 0", "@[reassoc]\ntheorem comp_coconePointsIsoOfNatIso_inv {F G : J \u2964 C} {s : Cocone F} {t : Cocone G}\n    (P : IsColimit s) (Q : IsColimit t) (w : F \u2245 G) (j : J) :\n    t.\u03b9.app j \u226b (coconePointsIsoOfNatIso P Q w).inv = w.inv.app j \u226b s.\u03b9.app j", "theorem hom_isIso {s t : Cone F} (P : IsLimit s) (Q : IsLimit t) (f : s \u27f6 t) : IsIso f", "theorem uniq_cone_morphism {s t : Cone F} (h : IsLimit t) {f f' : s \u27f6 t} : f = f'", "theorem of_mono_zero (X Y : C) [Mono (0 : X \u27f6 Y)] : IsZero X", "@[reassoc (attr := simp)]\nlemma mapOfIsLimit_\u03b9 (kf : KernelFork f) {kf' : KernelFork f'} (hf' : IsLimit kf')\n    (\u03c6 : Arrow.mk f \u27f6 Arrow.mk f') :\n    kf.mapOfIsLimit hf' \u03c6 \u226b kf'.\u03b9 = kf.\u03b9 \u226b \u03c6.left", "@[reassoc (attr := simp)]\nlemma map_condition : G.map c.\u03b9 \u226b G.map f = 0", "lemma mono_binaryCofanSum_inl' [MonoCoprod C] (inl : c\u2081.pt \u27f6 c.pt)\n    (hinl : \u2200 (i\u2081 : I\u2081), c\u2081.inj i\u2081 \u226b inl = c.inj (Sum.inl i\u2081)) :\n    Mono inl", "@[simp]\ntheorem MonoFactorisation.kernel_\u03b9_comp [HasKernel f] (F : MonoFactorisation f) :\n    kernel.\u03b9 f \u226b F.e = 0", "@[simp]\ntheorem fst_app_right (a) : K.\u03b9.app (WalkingMultispan.left a) = I.fst a \u226b K.\u03c0 _", "@[simp]\ntheorem multifork_\u03b9 (a) : (Multiequalizer.multifork I).\u03b9 a = Multiequalizer.\u03b9 I a", "@[simp]\ntheorem app_right_eq_\u03b9_comp_fst (b) :\n    K.\u03c0.app (WalkingMulticospan.right b) = K.\u03b9 (I.fstTo b) \u226b I.fst b", "@[simp]\ntheorem multispan_obj_right (b) : I.multispan.obj (WalkingMultispan.right b) = I.right b", "lemma Pi.map'_eq {f : \u03b1 \u2192 C} {g : \u03b2 \u2192 C} [HasProduct f] [HasProduct g] {p p' : \u03b2 \u2192 \u03b1}\n    {q : \u2200 (b : \u03b2), f (p b) \u27f6 g b} {q' : \u2200 (b : \u03b2), f (p' b) \u27f6 g b} (hp : p = p')\n    (hq : \u2200 (b : \u03b2), eqToHom (hp \u25b8 rfl) \u226b q b = q' b) : Pi.map' p q = Pi.map' p' q'", "@[reassoc (attr := simp)]\ntheorem Pi.reindex_inv_\u03c0 (b : \u03b2) : (Pi.reindex \u03b5 f).inv \u226b Pi.\u03c0 (f \u2218 \u03b5) b = Pi.\u03c0 f (\u03b5 b)", "@[reassoc (attr := simp)]\ntheorem PreservesPushout.inr_iso_inv :\n    G.map pushout.inr \u226b (PreservesPushout.iso G f g).inv = pushout.inr", "@[reassoc (attr := simp)]\nlemma IsLimit.lift_fst {t : PullbackCone f g} (ht : IsLimit t) {W : C} (h : W \u27f6 X) (k : W \u27f6 Y)\n    (w : h \u226b f = k \u226b g) : IsLimit.lift ht h k w \u226b fst t = h := ht.fac _ _", "theorem equalizer_ext (t : PullbackCone f g) {W : C} {k l : W \u27f6 t.pt} (h\u2080 : k \u226b fst t = l \u226b fst t)\n    (h\u2081 : k \u226b snd t = l \u226b snd t) : \u2200 j : WalkingCospan, k \u226b t.\u03c0.app j = l \u226b t.\u03c0.app j", "theorem mono_fst_of_is_pullback_of_mono {t : PullbackCone f g} (ht : IsLimit t) [Mono g] :\n    Mono t.fst", "@[simp]\ntheorem PullbackCone.snd_colimit_cocone {X Y Z : C} (f : X \u27f6 Z) (g : Y \u27f6 Z)\n    [HasLimit (cospan f g)] : PullbackCone.snd (limit.cone (cospan f g)) = pullback.snd", "@[simp]\ntheorem \u03c0_app_left (c : PullbackCone f g) : c.\u03c0.app WalkingCospan.left = c.fst", "theorem IsColimit.hom_ext {t : PushoutCocone f g} (ht : IsColimit t) {W : C} {k l : t.pt \u27f6 W}\n    (h\u2080 : inl t \u226b k = inl t \u226b l) (h\u2081 : inr t \u226b k = inr t \u226b l) : k = l", "@[reassoc (attr := simp)]\nlemma IsColimit.inl_desc {t : PushoutCocone f g} (ht : IsColimit t) {W : C} (h : Y \u27f6 W) (k : Z \u27f6 W)\n    (w : f \u226b h = g \u226b k) : inl t \u226b IsColimit.desc ht h k w = h", "lemma Sigma.map_comp_map' {f g : \u03b1 \u2192 C} {h : \u03b2 \u2192 C} [HasCoproduct f] [HasCoproduct g]\n    [HasCoproduct h] (p : \u03b1 \u2192 \u03b2) (q : \u2200 (a : \u03b1), f a \u27f6 g a) (q' : \u2200 (a : \u03b1), g a \u27f6 h (p a)) :\n    Sigma.map q \u226b Sigma.map' p q' = Sigma.map' p (fun a => q a \u226b q' a)", "@[simp]\nlemma Sigma.map_id {f : \u03b1 \u2192 C} [HasCoproduct f] : Sigma.map (fun a => \ud835\udfd9 (f a)) = \ud835\udfd9 (\u2210 f)", "lemma Types.Quot.Rel.iff_orbitRel (x y : J.obj (SingleObj.star G)) :\n    Types.Quot.Rel J \u27e8SingleObj.star G, x\u27e9 \u27e8SingleObj.star G, y\u27e9\n    \u2194 Setoid.Rel (MulAction.orbitRel G (J.obj (SingleObj.star G))) x y", "@[simp]\ntheorem Colimit.\u03b9_desc_apply' (F : J \u2964 Type v) (s : Cocone F) (j : J) (x : F.obj j) :\n    colimit.desc F s (colimit.\u03b9 F j x) = s.\u03b9.app j x", "theorem Limit.lift_\u03c0_apply (s : Cone F) (j : J) (x : s.pt) :\n    limit.\u03c0 F j (limit.lift F s x) = s.\u03c0.app j x", "theorem binaryCofan_isColimit_iff {X Y : Type u} (c : BinaryCofan X Y) :\n    Nonempty (IsColimit c) \u2194\n      Injective c.inl \u2227 Injective c.inr \u2227 IsCompl (Set.range c.inl) (Set.range c.inr)", "@[elementwise (attr := simp)]\ntheorem binaryProductIso_inv_comp_fst (X Y : Type u) :\n    (binaryProductIso X Y).inv \u226b Limits.prod.fst = _root_.Prod.fst", "theorem colimit_sound {j j' : J} {x : F.obj j} {x' : F.obj j'} (f : j \u27f6 j')\n    (w : F.map f x = x') : colimit.\u03b9 F j x = colimit.\u03b9 F j' x'", "theorem colimit_sound' {j j' : J} {x : F.obj j} {x' : F.obj j'} {j'' : J}\n    (f : j \u27f6 j'') (f' : j' \u27f6 j'') (w : F.map f x = F.map f' x') :\n    colimit.\u03b9 F j x = colimit.\u03b9 F j' x'", "lemma descFun_apply_spec {x : c.pt} {y : lc.pt} : descFun hc lc x = y \u2194 x \u2208 descSet hc {y}", "lemma descSet_univ : descSet hc (@Set.univ lc.pt) = Set.univ", "theorem jointly_surjective' (x : colimit F) :\n    \u2203 j y, colimit.\u03b9 F j y = x", "theorem limit_ext_iff' (F : J \u2964 Type v) (x y : limit F) :\n    x = y \u2194 \u2200 j, limit.\u03c0 F j x = limit.\u03c0 F j y", "theorem eq_desc_of_comp_eq (g : widePushout _ _ arrows \u27f6 X) :\n    (\u2200 j : J, \u03b9 arrows j \u226b g = fs j) \u2192 head arrows \u226b g = f \u2192 g = desc f fs w", "@[simp]\ntheorem as_factorThruImage : (Image.monoFactorisation f).e = factorThruImage f", "theorem biprod.add_eq_lift_id_desc [HasBinaryBiproduct X X] :\n    f + g = biprod.lift (\ud835\udfd9 X) (\ud835\udfd9 X) \u226b biprod.desc f g", "@[simp]\ntheorem biprod.inlCokernelCofork_\u03c0 : Cofork.\u03c0 (biprod.inlCokernelCofork X Y) = biprod.snd", "@[reassoc (attr := simp)]\ntheorem biprod.inr_desc {W X Y : C} [HasBinaryBiproduct X Y] (f : X \u27f6 W) (g : Y \u27f6 W) :\n    biprod.inr \u226b biprod.desc f g = g", "@[reassoc (attr := simp)]\ntheorem biprod.lift_snd {W X Y : C} [HasBinaryBiproduct X Y] (f : W \u27f6 X) (g : W \u27f6 Y) :\n    biprod.lift f g \u226b biprod.snd = g", "theorem biprod.mapBiprod_inv_map_desc (f : X \u27f6 W) (g : Y \u27f6 W) :\n    (F.mapBiprod X Y).inv \u226b F.map (biprod.desc f g) = biprod.desc (F.map f) (F.map g)", "@[simp]\ntheorem biproduct.isoCoproduct_hom {f : J \u2192 C} [HasBiproduct f] :\n    (biproduct.isoCoproduct f).hom = biproduct.desc (Sigma.\u03b9 f)", "@[reassoc]\ntheorem biproduct.lift_desc {T U : C} {g : \u2200 j, T \u27f6 f j} {h : \u2200 j, f j \u27f6 U} :\n    biproduct.lift g \u226b biproduct.desc h = \u2211 j : J, g j \u226b h j", "@[reassoc (attr := simp)]\ntheorem biproduct.map_desc {f g : J \u2192 C} [HasBiproduct f] [HasBiproduct g] (p : \u2200 j, f j \u27f6 g j)\n    {P : C} (k : \u2200 j, g j \u27f6 P) :\n    biproduct.map p \u226b biproduct.desc k = biproduct.desc fun j => p j \u226b k j", "@[reassoc]\ntheorem biproduct.matrix_map {f : J \u2192 C} {g : K \u2192 C} {h : K \u2192 C} (m : \u2200 j k, f j \u27f6 g k)\n    (n : \u2200 k, g k \u27f6 h k) :\n    biproduct.matrix m \u226b biproduct.map n = biproduct.matrix fun j k => m j k \u226b n k", "@[reassoc] theorem biproduct.\u03b9_\u03c0_self (f : J \u2192 C) [HasBiproduct f] (j : J) :\n    biproduct.\u03b9 f j \u226b biproduct.\u03c0 f j = \ud835\udfd9 _", "theorem coequalizer.cofork_\u03b9_app_one : (coequalizer.cofork f g).\u03b9.app one = coequalizer.\u03c0 f g", "@[reassoc]\ntheorem coequalizer.condition : f \u226b coequalizer.\u03c0 f g = g \u226b coequalizer.\u03c0 f g", "theorem cokernel.map_desc {X Y Z X' Y' Z' : C} (f : X \u27f6 Y) [HasCokernel f] (g : Y \u27f6 Z)\n    (w : f \u226b g = 0) (f' : X' \u27f6 Y') [HasCokernel f'] (g' : Y' \u27f6 Z') (w' : f' \u226b g' = 0) (p : X \u27f6 X')\n    (q : Y \u27f6 Y') (r : Z \u27f6 Z') (h\u2081 : f \u226b q = p \u226b f') (h\u2082 : g \u226b r = q \u226b g') :\n    cokernel.map f f' p q h\u2081 \u226b cokernel.desc f' g' w' = cokernel.desc f g w \u226b r", "@[simp]\ntheorem cokernelIsoOfEq_trans {f g h : X \u27f6 Y} [HasCokernel f] [HasCokernel g] [HasCokernel h]\n    (w\u2081 : f = g) (w\u2082 : g = h) :\n    cokernelIsoOfEq w\u2081 \u226a\u226b cokernelIsoOfEq w\u2082 = cokernelIsoOfEq (w\u2081.trans w\u2082)", "@[reassoc]\ntheorem cokernel_map_comp_cokernelComparison {X' Y' : C} [HasCokernel f] [HasCokernel (G.map f)]\n    (g : X' \u27f6 Y') [HasCokernel g] [HasCokernel (G.map g)] (p : X \u27f6 X') (q : Y \u27f6 Y')\n    (hpq : f \u226b q = p \u226b g) :\n    cokernel.map (G.map f) (G.map g) (G.map p) (G.map q) (by rw [\u2190 G.map_comp, hpq, G.map_comp]) \u226b\n        cokernelComparison _ G =\n      cokernelComparison _ G \u226b G.map (cokernel.map f g p q hpq)", "@[simp]\ntheorem colimit.post_post {E : Type u''} [Category.{v''} E] (H : D \u2964 E)\n    [h : HasColimit ((F \u22d9 G) \u22d9 H)] : haveI : HasColimit (F \u22d9 G \u22d9 H) := h\n    colimit.post (F \u22d9 G) H \u226b H.map (colimit.post F G) = colimit.post F (G \u22d9 H)", "theorem colimit.pre_post {D : Type u'} [Category.{v'} D] (E : K \u2964 J) (F : J \u2964 C) (G : C \u2964 D)\n    [HasColimit F] [HasColimit (E \u22d9 F)] [HasColimit (F \u22d9 G)] [h : HasColimit ((E \u22d9 F) \u22d9 G)] :\n    haveI : HasColimit (E \u22d9 F \u22d9 G) := h\n    colimit.post (E \u22d9 F) G \u226b G.map (colimit.pre F E) =\n      colimit.pre (F \u22d9 G) E \u226b colimit.post F G", "@[reassoc (attr := simp)]\ntheorem colimitObjIsoColimitCompEvaluation_inv_colimit_map [HasColimitsOfShape J C] (F : J \u2964 K \u2964 C)\n    {i j : K} (f : i \u27f6 j) :\n    (colimitObjIsoColimitCompEvaluation _ _).inv \u226b (colimit F).map f =\n      colimMap (whiskerLeft _ ((evaluation _ _).map f)) \u226b\n        (colimitObjIsoColimitCompEvaluation _ _).inv", "@[reassoc (attr := simp)]\nlemma colimit_\u03b9_zero_cokernel_desc {C : Type*} [Category C]\n    [HasZeroMorphisms C] {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z) (h : f \u226b g = 0) [HasCokernel f] :\n    colimit.\u03b9 (parallelPair f 0) WalkingParallelPair.zero \u226b cokernel.desc f g h = 0", "theorem coprod.symmetry (P Q : C) : (coprod.braiding P Q).hom \u226b (coprod.braiding Q P).hom = \ud835\udfd9 _", "@[simp]\ntheorem cospanExt_app_right : (cospanExt iX iY iZ wf wg).app WalkingCospan.right = iY", "@[reassoc]\ntheorem equalizer.lift_\u03b9 {W : C} (k : W \u27f6 X) (h : k \u226b f = k \u226b g) :\n    equalizer.lift k h \u226b equalizer.\u03b9 f g = k", "theorem hasCoproducts_of_opposite [HasProducts.{v\u2082} C\u1d52\u1d56] : HasCoproducts.{v\u2082} C", "lemma hasFiniteColimits_of_hasColimitsOfSize [HasColimitsOfSize.{v', u'} C] :\n    HasFiniteColimits C where\n  out := fun J hJ hJ' =>\n    haveI := hasColimitsOfSizeShrink.{0, 0} C\n    let F := @FinCategory.equivAsType J (@FinCategory.fintypeObj J hJ hJ') hJ hJ'\n    @hasColimitsOfShape_of_equivalence (@FinCategory.AsType J (@FinCategory.fintypeObj J hJ hJ'))\n    (@FinCategory.categoryAsType J (@FinCategory.fintypeObj J hJ hJ') hJ hJ') _ _ J hJ F _", "theorem hasLimit_of_hasColimit_leftOp (F : J \u2964 C\u1d52\u1d56) [HasColimit F.leftOp] : HasLimit F", "theorem hasPushout_of_right_iso : HasPushout f g", "theorem hasTerminal_of_hasInitial_op [HasInitial C\u1d52\u1d56] : HasTerminal C", "theorem hasWidePullbacks_shrink [HasWidePullbacks.{max w w'} C] : HasWidePullbacks.{w} C", "theorem has_colimits_of_hasCoequalizers_and_coproducts [HasCoproducts.{w} C] [HasCoequalizers C] :\n    HasColimitsOfSize.{w, w} C where", "theorem has_limits_of_hasEqualizers_and_products [HasProducts.{w} C] [HasEqualizers C] :\n    HasLimitsOfSize.{w, w} C", "@[simp]\ntheorem image.as_\u03b9 : (Image.monoFactorisation f).m = image.\u03b9 f", "@[reassoc (attr := simp)]\ntheorem image.compIso_inv_comp_image_\u03b9 [HasImage f] [IsIso g] :\n    (image.compIso f g).inv \u226b image.\u03b9 f = image.\u03b9 (f \u226b g) \u226b inv g", "@[simp]\ntheorem image.isImage_lift (F : MonoFactorisation f) : (Image.isImage f).lift F = image.lift F", "@[reassoc (attr := simp)]\ntheorem image.preComp_\u03b9 [HasImage g] [HasImage (f \u226b g)] :\n    image.preComp f g \u226b image.\u03b9 g = image.\u03b9 (f \u226b g)", "theorem imageSubobjectIso_comp_image_map {W X Y Z : C} {f : W \u27f6 X} [HasImage f] {g : Y \u27f6 Z}\n    [HasImage g] (sq : Arrow.mk f \u27f6 Arrow.mk g) [HasImageMap sq] :\n    (imageSubobjectIso _).hom \u226b image.map sq =\n      imageSubobjectMap sq \u226b (imageSubobjectIso _).hom", "@[reassoc (attr := simp), elementwise (attr := simp)]\ntheorem imageSubobject_arrow_comp : factorThruImageSubobject f \u226b (imageSubobject f).arrow = f", "theorem imageSubobject_le {A B : C} {X : Subobject B} (f : A \u27f6 B) [HasImage f] (h : A \u27f6 X)\n    (w : h \u226b X.arrow = f) : imageSubobject f \u2264 X", "theorem imageSubobject_le_mk {A B : C} {X : C} (g : X \u27f6 B) [Mono g] (f : A \u27f6 B) [HasImage f]\n    (h : A \u27f6 X) (w : h \u226b g = f) : imageSubobject f \u2264 Subobject.mk g", "@[reassoc (attr := simp)]\ntheorem inl_comp_pushoutSymmetry_hom [HasPushout f g] :\n    pushout.inl \u226b (pushoutSymmetry f g).hom = pushout.inr", "@[reassoc (attr := simp)]\ntheorem inr_comp_pushoutComparison (f : X \u27f6 Y) (g : X \u27f6 Z) [HasPushout f g]\n    [HasPushout (G.map f) (G.map g)] : pushout.inr \u226b pushoutComparison G f g = G.map pushout.inr", "theorem inr_of_isLimit {X Y : C} {t : BinaryBicone X Y} (ht : IsLimit t.toCone) :\n    t.inr = ht.lift (BinaryFan.mk 0 (\ud835\udfd9 Y))", "@[reassoc (attr := simp)]\ntheorem limMap_\u03c0 {F G : J \u2964 C} [HasLimit F] [HasLimit G] (\u03b1 : F \u27f6 G) (j : J) :\n    limMap \u03b1 \u226b limit.\u03c0 G j = limit.\u03c0 F j \u226b \u03b1.app j", "@[simp, reassoc]\ntheorem limitUncurryIsoLimitCompLim_hom_\u03c0_\u03c0 {j} {k} :\n    (limitUncurryIsoLimitCompLim F).hom \u226b limit.\u03c0 _ j \u226b limit.\u03c0 _ k = limit.\u03c0 _ (j, k)", "theorem map_id_left_eq_curry_map {j : J} {k k' : K} {f : k \u27f6 k'} :\n    F.map ((\ud835\udfd9 j, f) : (j, k) \u27f6 (j, k')) = ((curry.obj F).obj j).map f", "theorem opCoproductIsoProduct'_inv_comp_inj {c : Cofan Z} {f : Fan (op <| Z \u00b7)}\n    (hc : IsColimit c) (hf : IsLimit f) (b : \u03b1) :\n    (opCoproductIsoProduct' hc hf).inv \u226b (c.inj b).op = f.proj b", "@[simp]\ntheorem parallelPair_map_left (f g : X \u27f6 Y) : (parallelPair f g).map left = f", "@[simp]\ntheorem prod.lift_fst_snd {X Y : C} [HasBinaryProduct X Y] :\n    prod.lift prod.fst prod.snd = \ud835\udfd9 (X \u2a2f Y)", "@[reassoc]\ntheorem prod.pentagon [HasBinaryProducts C] (W X Y Z : C) :\n    prod.map (prod.associator W X Y).hom (\ud835\udfd9 Z) \u226b\n        (prod.associator W (X \u2a2f Y) Z).hom \u226b prod.map (\ud835\udfd9 W) (prod.associator X Y Z).hom =\n      (prod.associator (W \u2a2f X) Y Z).hom \u226b (prod.associator W X (Y \u2a2f Z)).hom", "@[reassoc]\ntheorem pullback.lift_fst {W X Y Z : C} {f : X \u27f6 Z} {g : Y \u27f6 Z} [HasPullback f g] (h : W \u27f6 X)\n    (k : W \u27f6 Y) (w : h \u226b f = k \u226b g) : pullback.lift h k w \u226b pullback.fst = h", "@[simp]\ntheorem pullbackConeOfLeftIso_\u03c0_app_left : (pullbackConeOfLeftIso f g).\u03c0.app left = g \u226b inv f", "@[reassoc (attr := simp)]\ntheorem pullbackDiagonalMapIso_inv_fst :\n    (pullbackDiagonalMapIso f i i\u2081 i\u2082).inv \u226b pullback.fst = pullback.fst \u226b i\u2081 \u226b pullback.fst", "@[reassoc (attr := simp)]\ntheorem pullback_diagonal_map_snd_fst_fst :\n    (pullback.snd :\n          pullback (diagonal f)\n              (map (i\u2081 \u226b snd) (i\u2082 \u226b snd) f f (i\u2081 \u226b fst) (i\u2082 \u226b fst) i (by simp [condition])\n                (by simp [condition])) \u27f6\n            _) \u226b\n        fst \u226b i\u2081 \u226b fst =\n      pullback.fst", "theorem pullback_map_eq_pullbackFstFstIso_inv {X Y S X' Y' S' : C} (f : X \u27f6 S) (g : Y \u27f6 S)\n    (f' : X' \u27f6 S') (g' : Y' \u27f6 S') (i\u2081 : X \u27f6 X') (i\u2082 : Y \u27f6 Y') (i\u2083 : S \u27f6 S') (e\u2081 : f \u226b i\u2083 = i\u2081 \u226b f')\n    (e\u2082 : g \u226b i\u2083 = i\u2082 \u226b g') [Mono i\u2083] :\n    pullback.map f g f' g' i\u2081 i\u2082 i\u2083 e\u2081 e\u2082 =\n      (pullbackFstFstIso f g f' g' i\u2081 i\u2082 i\u2083 e\u2081 e\u2082).inv \u226b pullback.snd \u226b pullback.fst", "@[reassoc]\ntheorem pushout.condition {X Y Z : C} {f : X \u27f6 Y} {g : X \u27f6 Z} [HasPushout f g] :\n    f \u226b (pushout.inl : Y \u27f6 pushout f g) = g \u226b pushout.inr", "theorem strongEpi_factorThruImage_of_strongEpiMonoFactorisation {X Y : C} {f : X \u27f6 Y} [HasImage f]\n    (F : StrongEpiMonoFactorisation f) : StrongEpi (factorThruImage f)", "@[simp] lemma unop_zero (X Y : C\u1d52\u1d56) : (0 : X \u27f6 Y).unop = 0 := rfl", "@[reassoc]\ntheorem wideCoequalizer.condition (j\u2081 j\u2082 : J) :\n    f j\u2081 \u226b wideCoequalizer.\u03c0 f = f j\u2082 \u226b wideCoequalizer.\u03c0 f", "theorem zero_of_comp_mono {X Y Z : C} {f : X \u27f6 Y} (g : Y \u27f6 Z) [Mono g] (h : f \u226b g = 0) : f = 0", "@[simp]\nlemma units_smul_comp {X Y Z : C} (r : R\u02e3) (f : X \u27f6 Y) (g : Y \u27f6 Z) :\n    (r \u2022 f) \u226b g = r \u2022 f \u226b g", "@[reassoc (attr := simp)]\nlemma add'_comp (f\u2081 f\u2082 : L.obj X \u27f6 L.obj Y) (g : L.obj Y \u27f6 L.obj Z) :\n    add' W f\u2081 f\u2082 \u226b g = add' W (f\u2081 \u226b g) (f\u2082 \u226b g)", "lemma Localization.essSurj_mapArrow_of_hasRightCalculusOfFractions :\n    L.mapArrow.EssSurj where\n  mem_essImage f", "lemma full_whiskeringLeft : ((whiskeringLeft C D E).obj L).Full", "lemma homEquiv_isoOfHom_inv (f : Y \u27f6 X) (hf : W f) :\n    homEquiv W L\u2081 L\u2082 (isoOfHom L\u2081 W f hf).inv = (isoOfHom L\u2082 W f hf).inv", "lemma mk' : \u03a6.IsRightDerivabilityStructure", "lemma inverts : W\u2081.IsInvertedBy (\u03a6.functor \u22d9 L\u2082)", "lemma isIso_iff_of_hasLeftResolutions [\u03a6.HasLeftResolutions] {F G : D\u2082 \u2964 H} (\u03b1 : F \u27f6 G) :\n    IsIso \u03b1 \u2194 \u2200 (X\u2081 : C\u2081), IsIso (\u03b1.app (L\u2082.obj (\u03a6.functor.obj X\u2081)))", "lemma isRightDerivabilityStructure_iff [\u03a6.HasRightResolutions] (e : \u03a6.functor \u22d9 L\u2082 \u2245 L\u2081 \u22d9 F) :\n    \u03a6.IsRightDerivabilityStructure \u2194 TwoSquare.GuitartExact e.hom", "theorem id_apply (M : Mat R) (i j : M) : (\ud835\udfd9 M : Matrix M M R) i j = if i = j then 1 else 0", "@[simp]\ntheorem add_apply {M N : Mat_ C} (f g : M \u27f6 N) (i j) : (f + g) i j = f i j + g i j", "@[reassoc]\ntheorem additiveObjIsoBiproduct_naturality' (F : Mat_ C \u2964 D) [Functor.Additive F] {M N : Mat_ C}\n    (f : M \u27f6 N) :\n    (additiveObjIsoBiproduct F M).inv \u226b F.map f =\n      biproduct.matrix (fun i j => F.map ((embedding C).map (f i j)) : _) \u226b\n        (additiveObjIsoBiproduct F N).inv", "@[simp]\ntheorem equivalenceSelfOfHasFiniteBiproducts_functor {C : Type (u\u2081 + 1)} [LargeCategory C]\n    [Preadditive C] [HasFiniteBiproducts C] :\n    (equivalenceSelfOfHasFiniteBiproducts C).functor = lift (\ud835\udfed C)", "@[simp]\ntheorem comp_eq_comp {A A' A'' : Algebra T} (f : A \u27f6 A') (g : A' \u27f6 A'') :\n    Algebra.Hom.comp f g = f \u226b g", "@[reassoc]\ntheorem Monad.assoc (T : Monad C) (X : C) :\n    (T : C \u2964 C).map (T.\u03bc.app X) \u226b T.\u03bc.app _ = T.\u03bc.app _ \u226b T.\u03bc.app _", "@[simp]\ntheorem beckCoequalizer_desc (s : Cofork (T.toFunctor.map X.a) (T.\u03bc.app X.A)) :\n    (beckCoequalizer X).desc s = T.\u03b7.app _ \u226b s.\u03c0", "theorem forget_creates_colimits_of_monad_preserves [PreservesColimitsOfShape J (T : C \u2964 C)]\n    (D : J \u2964 Algebra T) [HasColimit (D \u22d9 forget T)] : HasColimit D", "@[simp]\ntheorem MonadHom.comp_toNatTrans {T\u2081 T\u2082 T\u2083 : Monad C} (f : T\u2081 \u27f6 T\u2082) (g : T\u2082 \u27f6 T\u2083) :\n    (f \u226b g).toNatTrans = ((f.toNatTrans : _ \u27f6 (T\u2082 : C \u2964 C)) \u226b g.toNatTrans : (T\u2081 : C \u2964 C) \u27f6 T\u2083)", "@[simp]\ntheorem whiskerRight_app {F G F' : C \u2964 D} {\u03b1 : F \u27f6 G} {X} :\n    (\u03b1 \u25b7 F').app X = \u03b1.app X \u25b7 F'.obj X", "@[reassoc]\ntheorem associator_inv_naturality_middle (X : C) {Y Y' : C} (f : Y \u27f6 Y') (Z : C) :\n    X \u25c1 f \u25b7 Z \u226b (\u03b1_ X Y' Z).inv = (\u03b1_ X Y Z).inv \u226b (X \u25c1 f) \u25b7 Z", "@[reassoc (attr := simp)]\ntheorem hom_inv_id_tensor' {V W X Y Z : C} (f : V \u27f6 W) [IsIso f] (g : X \u27f6 Y) (h : Y \u27f6 Z) :\n    (f \u2297 g) \u226b (inv f \u2297 h) = (\ud835\udfd9 V \u2297 g) \u226b (\ud835\udfd9 V \u2297 h)", "@[reassoc (attr := simp)]\ntheorem inv_hom_whiskerRight {X Y : C} (f : X \u2245 Y) (Z : C) :\n    f.inv \u25b7 Z \u226b f.hom \u25b7 Z = \ud835\udfd9 (Y \u2297 Z)", "@[simp]\ntheorem prodMonoidal_leftUnitor_inv_fst (X : C\u2081 \u00d7 C\u2082) :\n    ((\u03bb_ X).inv : X \u27f6 \ud835\udfd9_ _ \u2297 X).1 = (\u03bb_ X.1).inv", "theorem whiskerLeft_dite {P : Prop} [Decidable P]\n    (X : C) {Y Z : C} (f : P \u2192 (Y \u27f6 Z)) (f' : \u00acP \u2192 (Y \u27f6 Z)) :\n      X \u25c1 (if h : P then f h else f' h) = if h : P then X \u25c1 f h else X \u25c1 f' h", "@[reassoc (attr := simp)]\ntheorem id_tensor_pre_app_comp_ev (f : B \u27f6 A) (X : C) :\n    B \u25c1 (pre f).app X \u226b (ihom.ev B).app X = f \u25b7 (A \u27f6[C] X) \u226b (ihom.ev A).app X", "theorem ofEquiv_uncurry_def {X Y Z : C} :\n    letI := ofEquiv F adj\n    \u2200 (f : Y \u27f6 (ihom X).obj Z), MonoidalClosed.uncurry f =\n      ((Iso.compInverseIso (H := adj.toEquivalence)\n          (MonoidalFunctor.commTensorLeft F X)).inv.app Y) \u226b\n            (adj.toEquivalence.symm.toAdjunction.homEquiv _ _).symm\n              (MonoidalClosed.uncurry ((adj.homEquiv _ _).symm f))", "@[simp]\ntheorem mapAction_\u03b5_inv_hom : (inv (F.mapAction G).\u03b5).hom = inv F.\u03b5", "@[simp]\ntheorem mapAction_\u03bc_inv_hom (X Y : Action V G) :\n    (inv ((F.mapAction G).\u03bc X Y)).hom = inv (F.\u03bc X.V Y.V)", "@[simp]\nlemma unmop_inv_braiding (X Y : C\u1d39\u1d52\u1d56) : (\u03b2_ X Y).inv.unmop = (\u03b2_ (unmop Y) (unmop X)).inv := rfl", "@[reassoc (attr := simp)]\nlemma fac_app {f : Arrow C} : data.i.app f \u226b data.p.app f = f.hom", "@[simp]\nlemma IsInvertedBy.isoClosure_iff (W : MorphismProperty C) (F : C \u2964 D) :\n    W.isoClosure.IsInvertedBy F \u2194 W.IsInvertedBy F", "lemma of_le (P Q : MorphismProperty C) (F : C \u2964 D) (hQ : Q.IsInvertedBy F) (h : P \u2264 Q) :\n    P.IsInvertedBy F", "lemma IsStableUnderProductsOfShape.mk (J : Type*)\n    (hW\u2080 : W.RespectsIso) [HasProductsOfShape J C]\n    (hW : \u2200 (X\u2081 X\u2082 : J \u2192 C) (f : \u2200 j, X\u2081 j \u27f6 X\u2082 j) (_ : \u2200 (j : J), W (f j)),\n      W (Pi.map f)) : W.IsStableUnderProductsOfShape J", "lemma fst : LeftFractionRel z\u2081.fst z\u2082.fst", "theorem _root_.CategoryTheory.MorphismProperty.Q_inverts : W.IsInvertedBy W.Q", "theorem RespectsIso.unop {P : MorphismProperty C\u1d52\u1d56} (h : RespectsIso P) : RespectsIso P.unop", "lemma RightFraction.op_map\n    {X Y : C} (\u03c6 : W.RightFraction X Y) (L : C \u2964 D) (hL : W.IsInvertedBy L) :\n    (\u03c6.map L hL).op = \u03c6.op.map L.op hL.op", "lemma MorphismProperty.map_eq_iff_postcomp {X Y : C} (f\u2081 f\u2082 : X \u27f6 Y) :\n    L.map f\u2081 = L.map f\u2082 \u2194 \u2203 (Z : C) (s : Y \u27f6 Z) (_ : W s), f\u2081 \u226b s = f\u2082 \u226b s", "lemma map_inverseImage_le (P : MorphismProperty D) (F : C \u2964 D) :\n    (P.inverseImage F).map F \u2264 P.isoClosure", "lemma map_mem_map (P : MorphismProperty C) (F : C \u2964 D) {X Y : C} (f : X \u27f6 Y) (hf : P f) :\n    (P.map F) (F.map f) := \u27e8X, Y, f, hf, \u27e8Iso.refl _\u27e9\u27e9", "@[simp]\ntheorem cancel_natIso_hom_right_assoc {W X X' : D} {Y : C} (f : W \u27f6 X) (g : X \u27f6 F.obj Y)\n    (f' : W \u27f6 X') (g' : X' \u27f6 F.obj Y) :\n    f \u226b g \u226b \u03b1.hom.app Y = f' \u226b g' \u226b \u03b1.hom.app Y \u2194 f \u226b g = f' \u226b g'", "@[simp]\ntheorem app_sub (X : C) (\u03b1 \u03b2 : F \u27f6 G) : (\u03b1 - \u03b2).app X = \u03b1.app X - \u03b2.app X", "theorem NatTrans.epi_iff_epi_app {F G : C \u2964 D} (\u03b7 : F \u27f6 G) : Epi \u03b7 \u2194 \u2200 c, Epi (\u03b7.app c)", "@[simp]\ntheorem removeLeftOp_id : NatTrans.removeLeftOp (\ud835\udfd9 F.leftOp) = \ud835\udfd9 F", "@[simp]\nlemma OplaxFunctor.map\u2082_eqToHom (F : OplaxFunctor B C) {a b : B} {f g : a \u27f6 b} (h : f = g) :\n    F.map\u2082 (eqToHom h) = eqToHom (F.congr_map h)", "@[simp]\nlemma map\u2081_map\u2082 {F G : C\u1d52\u1d56 \u2964 Type v} {\u03b7 : F \u27f6 A} {\u03bc : G \u27f6 A} (\u03b5 : F \u27f6 G)\n    (h\u03b5 : \u03b5 \u226b \u03bc = \u03b7) {X Y : C} {s : yoneda.obj X \u27f6 A} {t : yoneda.obj Y \u27f6 A} (f : X \u27f6 Y)\n    (hf : yoneda.map f \u226b t = s) (u : OverArrows \u03b7 t) :\n    (u.map\u2081 \u03b5 h\u03b5).map\u2082 f hf = (u.map\u2082 f hf).map\u2081 \u03b5 h\u03b5", "@[simp]\nlemma map\u2081_val {F G : C\u1d52\u1d56 \u2964 Type v} {\u03b7 : F \u27f6 A} {\u03bc : G \u27f6 A} {X : C}\n    (s : yoneda.obj X \u27f6 A) (u : OverArrows \u03b7 s) (\u03b5 : F \u27f6 G) (h\u03b5 : \u03b5 \u226b \u03bc = \u03b7) :\n    (u.map\u2081 \u03b5 h\u03b5).val = \u03b5.app _ u.val", "lemma counitBackward_counitForward (F : (CostructuredArrow yoneda A)\u1d52\u1d56 \u2964 Type v)\n    (s : CostructuredArrow yoneda A) : counitBackward F s \u2218 counitForward F s = id", "@[simp]\nlemma counitForward_naturality\u2081 {G : (CostructuredArrow yoneda A)\u1d52\u1d56 \u2964 Type v} (\u03b7 : F \u27f6 G)\n    (s : (CostructuredArrow yoneda A)\u1d52\u1d56) (x : F.obj s) : counitForward G s.unop (\u03b7.app s x) =\n      OverArrows.map\u2081 (counitForward F s.unop x) (yonedaCollectionPresheafMap\u2081 \u03b7) (by aesop_cat)", "@[simp]\ntheorem lift_cons {C} [Category C] (\u03c6 : V \u2964q C) {X Y Z : V} (p : Quiver.Path X Y) (f : Y \u27f6 Z) :\n    (lift \u03c6).map (p.cons f) = (lift \u03c6).map p \u226b \u03c6.map f", "lemma evaluation_injective_of_isConnected (A X : C) [IsConnected A] (a : F.obj A) :\n    Function.Injective (fun (f : A \u27f6 X) \u21a6 F.map f a)", "lemma exists_galois_representative (X : C) : \u2203 (A : C) (a : F.obj A),\n    IsGalois A \u2227 Function.Bijective (fun (f : A \u27f6 X) \u21a6 F.map f a)", "private lemma has_decomp_connected_components_aux_conn (X : C) [IsConnected X] :\n    \u2203 (\u03b9 : Type) (f : \u03b9 \u2192 C) (g : (i : \u03b9) \u2192 (f i) \u27f6 X) (_ : IsColimit (Cofan.mk X g)),\n    (\u2200 i, IsConnected (f i)) \u2227 Finite \u03b9", "lemma sieve\u2081_eq_pullback_sieve\u2081' {W : C} (p\u2081 : W \u27f6 E.X i\u2081) (p\u2082 : W \u27f6 E.X i\u2082)\n    (w : p\u2081 \u226b E.f i\u2081 = p\u2082 \u226b E.f i\u2082) :\n    E.sieve\u2081 p\u2081 p\u2082 = (E.sieve\u2081' i\u2081 i\u2082).pullback (pullback.lift _ _ w)", "theorem Preadditive.exact_iff_homology'_zero {A B C : V} (f : A \u27f6 B) (g : B \u27f6 C) :\n    Exact f g \u2194 \u2203 w : f \u226b g = 0, Nonempty (homology' f g w \u2245 0)", "theorem Preadditive.exact_of_iso_of_exact {A\u2081 B\u2081 C\u2081 A\u2082 B\u2082 C\u2082 : V} (f\u2081 : A\u2081 \u27f6 B\u2081) (g\u2081 : B\u2081 \u27f6 C\u2081)\n    (f\u2082 : A\u2082 \u27f6 B\u2082) (g\u2082 : B\u2082 \u27f6 C\u2082) (\u03b1 : Arrow.mk f\u2081 \u2245 Arrow.mk f\u2082) (\u03b2 : Arrow.mk g\u2081 \u2245 Arrow.mk g\u2082)\n    (p : \u03b1.hom.right = \u03b2.hom.left) (h : Exact f\u2081 g\u2081) : Exact f\u2082 g\u2082", "theorem Preadditive.isSeparating_iff (\ud835\udca2 : Set C) :\n    IsSeparating \ud835\udca2 \u2194 \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), (\u2200 G \u2208 \ud835\udca2, \u2200 (h : G \u27f6 X), h \u226b f = 0) \u2192 f = 0", "lemma Preadditive.mono_iff_injective' {X Y : C} (f : X \u27f6 Y) :\n    Mono f \u2194 Function.Injective ((forget C).map f)", "@[reassoc]\ntheorem neg_comp_neg : (-f) \u226b (-g) = f \u226b g", "@[simp]\nlemma neg_iso_inv (e : X \u2245 Y) : (-e).inv = -e.inv := rfl", "lemma Presheaf.IsSheaf.isSeparated {F : C\u1d52\u1d56 \u2964 A} [ConcreteCategory A]\n    [J.HasSheafCompose (forget A)] (hF : Presheaf.IsSheaf J F) :\n    Presheaf.IsSeparated J F", "lemma comp_isLocallySurjective_iff\n    {F\u2081 F\u2082 F\u2083 : C\u1d52\u1d56 \u2964 A} (f\u2081 : F\u2081 \u27f6 F\u2082) (f\u2082 : F\u2082 \u27f6 F\u2083)\n    [IsLocallySurjective J f\u2081] :\n    IsLocallySurjective J (f\u2081 \u226b f\u2082) \u2194 IsLocallySurjective J f\u2082", "@[simp]\nlemma equalizerSieve_eq_top_iff {F : C\u1d52\u1d56 \u2964 D} {X : C\u1d52\u1d56} (x y : F.obj X) :\n    equalizerSieve x y = \u22a4 \u2194 x = y", "lemma isLocallyInjective_of_isLocallyInjective [IsLocallyInjective J (\u03c6 \u226b \u03c8)] :\n    IsLocallyInjective J \u03c6 where\n  equalizerSieve_mem {X} x y h := equalizerSieve_mem J (\u03c6 \u226b \u03c8) x y (by simp [h])", "lemma isLocallyInjective_of_isLocallyInjective_of_isLocallySurjective_fac\n    {F\u2081 F\u2082 F\u2083 : C\u1d52\u1d56 \u2964 A} {f\u2081 : F\u2081 \u27f6 F\u2082} {f\u2082 : F\u2082 \u27f6 F\u2083} (f\u2083 : F\u2081 \u27f6 F\u2083) (fac : f\u2081 \u226b f\u2082 = f\u2083)\n    [IsLocallyInjective J f\u2083] [IsLocallySurjective J f\u2081] :\n    IsLocallyInjective J f\u2082", "lemma isLocallySurjective_comp_iff\n    {F\u2081 F\u2082 F\u2083 : C\u1d52\u1d56 \u2964 A} (f\u2081 : F\u2081 \u27f6 F\u2082) (f\u2082 : F\u2082 \u27f6 F\u2083)\n    [IsLocallyInjective J f\u2082] [IsLocallySurjective J f\u2082] :\n    IsLocallySurjective J (f\u2081 \u226b f\u2082) \u2194 IsLocallySurjective J f\u2081", "lemma isSheaf_coherent_of_hasPullbacks_of_comp [Preregular C] [FinitaryExtensive C]\n    [h : \u2200 {Y X : C} (f : Y \u27f6 X) [EffectiveEpi f], HasPullback f f]\n    [ReflectsFiniteLimits s]\n    (hF : IsSheaf (coherentTopology C) (F \u22d9 s)) : IsSheaf (coherentTopology C) F", "lemma isSheaf_coherent_of_projective_comp [Preregular C] [FinitaryExtensive C]\n    [\u2200 (X : C), Projective X] [PreservesFiniteProducts s]\n    (hF : IsSheaf (coherentTopology C) F) : IsSheaf (coherentTopology C) (F \u22d9 s)", "theorem isSheaf_iff_isSheaf_comp (s : A \u2964 B) [HasLimitsOfSize.{v\u2081, max v\u2081 u\u2081} A]\n    [PreservesLimitsOfSize.{v\u2081, max v\u2081 u\u2081} s] [s.ReflectsIsomorphisms] :\n    IsSheaf J P \u2194 IsSheaf J (P \u22d9 s)", "@[simp]\nlemma mul_apply (\u03b3\u2081 \u03b3\u2082 : ZeroCochain G U) (i : I) : (\u03b3\u2081 * \u03b3\u2082) i = \u03b3\u2081 i * \u03b3\u2082 i := rfl", "@[simp]\nlemma mul_ev (\u03b3\u2081 \u03b3\u2082 : OneCochain G U) (i j : I) {T : C} (a : T \u27f6 U i) (b : T \u27f6 U j) :\n    (\u03b3\u2081 * \u03b3\u2082).ev i j a b = \u03b3\u2081.ev i j a b * \u03b3\u2082.ev i j a b := rfl", "@[simp]\nlemma one_toOneCochain : (1 : OneCocycle G U).toOneCochain = 1 := rfl", "lemma factorsThru_top {X : C} (S : Presieve X) : S.FactorsThru \u22a4", "theorem image_mem_functorPushforward (R : Presieve X) {f : Y \u27f6 X} (h : R f) :\n    R.functorPushforward F (F.map f)", "theorem isSheaf_pretopology [HasPullbacks C] (K : Pretopology C) :\n    IsSheaf (K.toGrothendieck C) P \u2194 \u2200 {X : C} (R : Presieve X), R \u2208 K X \u2192 IsSheafFor P R", "theorem piComparison_fac :\n    have : HasCoproduct X := \u27e8\u27e8c, hc\u27e9\u27e9\n    piComparison F (fun x \u21a6 op (X x)) = F.map (opCoproductIsoProduct' hc (productIsProduct _)).inv \u226b\n    Equalizer.Presieve.Arrows.forkMap F X c.inj", "lemma distinguished_cocone_triangle {X Y : C\u1d52\u1d56} (f : X \u27f6 Y) :\n    \u2203 (Z : C\u1d52\u1d56) (g : Y \u27f6 Z) (h : Z \u27f6 X\u27e6(1 : \u2124)\u27e7),\n      Triangle.mk f g h \u2208 distinguishedTriangles C", "lemma isZero\u2083_of_isZero\u2081\u2082 (h\u2081 : IsZero T.obj\u2081) (h\u2082 : IsZero T.obj\u2082) : IsZero T.obj\u2083", "lemma mor\u2081_eq_zero_of_mono\u2082 (h : Mono T.mor\u2082) : T.mor\u2081 = 0 := (T.mor\u2081_eq_zero_iff_mono\u2082 hT).2 h", "lemma shift_distinguished (n : \u2124) :\n    (CategoryTheory.shiftFunctor (Triangle C) n).obj T \u2208 distTriang C", "lemma contractible_distinguished\u2082 (X : C) :\n    Triangle.mk (0 : X \u27f6 0) 0 (\ud835\udfd9 (X\u27e61\u27e7)) \u2208 distTriang C", "lemma exists_iso_of_arrow_iso (T\u2081 T\u2082 : Triangle C) (hT\u2081 : T\u2081 \u2208 distTriang C)\n    (hT\u2082 : T\u2082 \u2208 distTriang C) (e : Arrow.mk T\u2081.mor\u2081 \u2245 Arrow.mk T\u2082.mor\u2081) :\n    \u2203 (e' : T\u2081 \u2245 T\u2082), e'.hom.hom\u2081 = e.hom.left \u2227 e'.hom.hom\u2082 = e.hom.right", "lemma mem_distTriang_op_iff (T : Triangle C\u1d52\u1d56) :\n    (T \u2208 distTriang C\u1d52\u1d56) \u2194 ((triangleOpEquivalence C).inverse.obj T).unop \u2208 distTriang C", "lemma mem_distTriang_op_iff' (T : Triangle C\u1d52\u1d56) :\n    (T \u2208 distTriang C\u1d52\u1d56) \u2194 \u2203 (T' : Triangle C) (_ : T' \u2208 distTriang C),\n      Nonempty (T \u2245 (triangleOpEquivalence C).functor.obj (Opposite.op T'))", "lemma shift_unop_opShiftFunctorEquivalence_counitIso_hom_app (X : C\u1d52\u1d56) (n : \u2124) :\n    ((opShiftFunctorEquivalence C n).counitIso.hom.app X).unop\u27e6n\u27e7' =\n      ((opShiftFunctorEquivalence C n).unitIso.inv.app ((Opposite.op (X.unop\u27e6n\u27e7)))).unop", "theorem complex_d_succ_comp (n : \u2115) :\n    P.complex.d n (n + 1) \u226b P.complex.d (n + 1) (n + 2) = 0", "@[simp]\nlemma map\u2082_inv {f g : a \u27f6 b} (\u03b7 : f \u27f6 g) [IsIso \u03b7] : F.map\u2082 (inv \u03b7) = inv (F.map\u2082 \u03b7)", "@[simp]\nlemma iso_hom_app (a : A) (X : C) :\n    (iso F r hF a).hom.app ((functor r).obj X) =\n      (lift r F hF).map (((functor r).commShiftIso a).inv.app X) \u226b\n      (F.commShiftIso a).hom.app X", "@[simp]\ntheorem compClosure_eq_self [h : Congruence r] :\n    CompClosure r = r", "theorem gluedSection_is_unique (y) (hy : x.IsAmalgamation y) : y = gluedSection \u2131 hS hx", "theorem rel_iso_iff {X Y : RelCat} (r : X \u27f6 Y) :\n    IsIso (C := RelCat) r \u2194 \u2203 f : (Iso (C := Type) X Y), graphFunctor.map f.hom = r", "@[simp]\ntheorem RightExactFunctor.ofExact_obj (F : C \u2964\u2091 D) :\n    (RightExactFunctor.ofExact C D).obj F = \u27e8F.1, F.2.2\u27e9", "theorem Sheaf.Hom.mono_iff_presheaf_mono {F G : Sheaf J D} (f : F \u27f6 G) : Mono f \u2194 Mono f.1", "private lemma isLocallyBijective_iff_isIso' :\n    IsLocallyInjective f \u2227 IsLocallySurjective f \u2194 IsIso f", "@[ext]\ntheorem Hom.ext' {X Y : SheafOfTypes J} (f g : X \u27f6 Y) (w : f.val = g.val) : f = g", "lemma Exact.map_of_preservesRightHomologyOf (h : S.Exact) (F : C \u2964 D)\n    [F.PreservesZeroMorphisms] [F.PreservesRightHomologyOf S]\n    [(S.map F).HasHomology] : (S.map F).Exact", "lemma Exact.op (h : S.Exact) : S.op.Exact", "lemma hasCokernel [S.HasLeftHomology] [HasKernel S.g] :\n    HasCokernel (kernel.lift S.g S.f S.zero)", "@[reassoc (attr := simp)]\nlemma cyclesIso_inv_comp_iCycles : h.cyclesIso.inv \u226b S.iCycles = h.i", "@[simp] lemma op_g' (h : S.LeftHomologyData) :\n    h.op.g' = h.f'.op := rfl", "@[reassoc (attr := simp)]\nlemma \u03c0_descH (k : h.K \u27f6 A) (hk : h.f' \u226b k = 0) : h.\u03c0 \u226b h.descH k hk = k", "lemma congr_\u03c6K {\u03b3\u2081 \u03b3\u2082 : LeftHomologyMapData \u03c6 h\u2081 h\u2082} (eq : \u03b3\u2081 = \u03b3\u2082) : \u03b3\u2081.\u03c6K = \u03b3\u2082.\u03c6K", "lemma RightHomologyData.exact_iff_mono_g' [S.HasHomology] (h : RightHomologyData S) :\n    S.Exact \u2194 Mono h.g'", "@[reassoc (attr := simp)]\nlemma liftH_\u03b9 (k : A \u27f6 h.Q) (hk : k \u226b h.g' = 0) : h.liftH k hk \u226b h.\u03b9 = k", "@[reassoc (attr := simp)]\nlemma RightHomologyData.opcyclesIso_inv_comp_descOpcycles :\n    h.opcyclesIso.inv \u226b S.descOpcycles k hk = h.descQ k hk", "@[reassoc (attr := simp)]\nlemma p_comp_opcyclesIso_inv : h.p \u226b h.opcyclesIso.inv = S.pOpcycles", "lemma ShortExact.ab_injective_f (hS : S.ShortExact) :\n    Function.Injective S.f", "lemma \u03b4_apply (x\u2083 : (forget\u2082 C Ab).obj (S.X\u2083.X i))\n    (hx\u2083 : (forget\u2082 C Ab).map (S.X\u2083.d i j) x\u2083 = 0)\n    (x\u2082 : (forget\u2082 C Ab).obj (S.X\u2082.X i)) (hx\u2082 : (forget\u2082 C Ab).map (S.g.f i) x\u2082 = x\u2083)\n    (x\u2081 : (forget\u2082 C Ab).obj (S.X\u2081.X j))\n    (hx\u2081 : (forget\u2082 C Ab).map (S.f.f j) x\u2081 = (forget\u2082 C Ab).map (S.X\u2082.d i j) x\u2082)\n    (k : \u03b9) (hk : c.next j = k) :\n    (forget\u2082 C Ab).map (hS.\u03b4 i j hij)\n      ((forget\u2082 C Ab).map (S.X\u2083.homology\u03c0 i) (S.X\u2083.cyclesMk x\u2083 j (c.next_eq' hij) hx\u2083)) =\n        (forget\u2082 C Ab).map (S.X\u2081.homology\u03c0 j) (S.X\u2081.cyclesMk x\u2081 k hk (by\n          have := hS.mono_f\n          apply (Preadditive.mono_iff_injective (S.f.f k)).1 inferInstance\n          erw [\u2190 forget\u2082_comp_apply, \u2190 HomologicalComplex.Hom.comm, forget\u2082_comp_apply, hx\u2081,\n            \u2190 forget\u2082_comp_apply, HomologicalComplex.d_comp_d, Functor.map_zero, map_zero,\n            AddMonoidHom.zero_apply]))", "lemma L\u2080_exact : S.L\u2080.Exact", "@[simp, reassoc] lemma comp_f\u2080 : (f \u226b g).f\u2080 = f.f\u2080 \u226b g.f\u2080 := rfl", "lemma ab_exact_iff_range_eq_ker : S.Exact \u2194 S.f.range = S.g.ker", "@[reassoc (attr := simp)]\nlemma asIsoHomology\u03b9_inv_comp_homology\u03b9 (hg : S.g = 0) [S.HasHomology] :\n    (S.asIsoHomology\u03b9 hg).inv \u226b S.homology\u03b9 = \ud835\udfd9 _ := Iso.inv_hom_id _", "@[reassoc (attr := simp)]\nlemma cyclesIsoX\u2082_hom_inv_id (hg : S.g = 0) :\n    S.iCycles \u226b (S.cyclesIsoX\u2082 hg).inv = \ud835\udfd9 _ := (S.cyclesIsoX\u2082 hg).hom_inv_id", "@[simp]\nlemma cyclesMap'_add :\n    cyclesMap' (\u03c6 + \u03c6') h\u2081 h\u2082 = cyclesMap' \u03c6 h\u2081 h\u2082 +\n      cyclesMap' \u03c6' h\u2081 h\u2082", "lemma epi_\u03c4\u2082_of_exact_of_epi {S\u2081 S\u2082 : ShortComplex C} (\u03c6 : S\u2081 \u27f6 S\u2082)\n    (h\u2082 : S\u2082.Exact) [Epi S\u2081.g] [Epi S\u2082.g] [Epi \u03c6.\u03c4\u2081] [Epi \u03c6.\u03c4\u2083] : Epi \u03c6.\u03c4\u2082", "lemma exact_iff_epi_kernel_lift [S.HasHomology] [HasKernel S.g] :\n    S.Exact \u2194 Epi (kernel.lift S.g S.f S.zero)", "lemma hasHomology_of_epi_of_isIso_of_mono' (\u03c6 : S\u2081 \u27f6 S\u2082) [HasHomology S\u2082]\n    [Epi \u03c6.\u03c4\u2081] [IsIso \u03c6.\u03c4\u2082] [Mono \u03c6.\u03c4\u2083] : HasHomology S\u2081", "lemma hasLeftHomology_of_epi_of_isIso_of_mono (\u03c6 : S\u2081 \u27f6 S\u2082) [HasLeftHomology S\u2081]\n    [Epi \u03c6.\u03c4\u2081] [IsIso \u03c6.\u03c4\u2082] [Mono \u03c6.\u03c4\u2083] : HasLeftHomology S\u2082", "@[simp]\nlemma i_cyclesMk [S.HasHomology] (x\u2082 : (forget\u2082 C Ab).obj S.X\u2082)\n    (hx\u2082 : ((forget\u2082 C Ab).map S.g) x\u2082 = 0) :\n    (forget\u2082 C Ab).map S.iCycles (S.cyclesMk x\u2082 hx\u2082) = x\u2082", "@[simp]\nlemma leftHomologyMap'_neg :\n    leftHomologyMap' (-\u03c6) h\u2081 h\u2082 = -leftHomologyMap' \u03c6 h\u2081 h\u2082", "lemma leftRightHomologyComparison'_eq_descH :\n    leftRightHomologyComparison' h\u2081 h\u2082 =\n      h\u2081.descH (h\u2082.liftH (h\u2081.i \u226b h\u2082.p) (by simp))\n        (by rw [\u2190 cancel_mono h\u2082.\u03b9, assoc, RightHomologyData.liftH_\u03b9,\n          LeftHomologyData.f'_i_assoc, RightHomologyData.wp, zero_comp])", "@[reassoc]\nlemma mapCyclesIso_hom_naturality [S\u2081.HasLeftHomology] [S\u2082.HasLeftHomology]\n    [F.PreservesLeftHomologyOf S\u2081] [F.PreservesLeftHomologyOf S\u2082] :\n    cyclesMap (F.mapShortComplex.map \u03c6) \u226b (S\u2082.mapCyclesIso F).hom =\n      (S\u2081.mapCyclesIso F).hom \u226b F.map (cyclesMap \u03c6)", "@[simp] lemma neg_\u03c4\u2082 (\u03c6 : S\u2081 \u27f6 S\u2082) : (-\u03c6).\u03c4\u2082 = -\u03c6.\u03c4\u2082 := rfl", "@[simp]\nlemma opcyclesMap_add : opcyclesMap (\u03c6 + \u03c6') = opcyclesMap \u03c6 + opcyclesMap \u03c6'", "@[ext]\nlemma opcycles_ext {A : C} (f\u2081 f\u2082 : S.opcycles \u27f6 A)\n    (h : S.pOpcycles \u226b f\u2081 = S.pOpcycles \u226b f\u2082) : f\u2081 = f\u2082", "lemma quasiIso_iff (\u03c6 : S\u2081 \u27f6 S\u2082) :\n    QuasiIso \u03c6 \u2194 IsIso (homologyMap \u03c6)", "lemma shortExact_iff_unop (S : ShortComplex C\u1d52\u1d56) : S.ShortExact \u2194 S.unop.ShortExact", "@[reassoc (attr := simp)]\nlemma toCycles_comp_homology\u03c0 :\n    S.toCycles \u226b S.homology\u03c0 = 0", "@[simp] lemma zero_\u03c4\u2082 : Hom.\u03c4\u2082 (0 : S\u2081 \u27f6 S\u2082) = 0 := rfl", "lemma mem_ofObjects_iff {I : Type*} (Y : I \u2192 C) {Z X : C} (g : Z \u27f6 X) :\n    ofObjects Y X g \u2194 \u2203 (i : I), Nonempty (Z \u27f6 Y i)", "lemma overEquiv_pullback {X : C} {Y\u2081 Y\u2082 : Over X} (f : Y\u2081 \u27f6 Y\u2082) (S : Sieve Y\u2082) :\n    overEquiv _ (S.pullback f) = (overEquiv _ S).pullback f.left", "theorem pullback_monotone (f : Y \u27f6 X) : Monotone (Sieve.pullback f)", "theorem pushforward_comp {f : Y \u27f6 X} {g : Z \u27f6 Y} (R : Sieve Z) :\n    R.pushforward (g \u226b f) = (R.pushforward g).pushforward f", "@[simp]\nlemma descUniq_hom_app (q : (\u03a3i, C i) \u2964 D) (h : \u2200 i, incl i \u22d9 q \u2245 F i) (i : I) (X : C i) :\n    (descUniq F q h).hom.app \u27e8i, X\u27e9 = (h i).hom.app X", "@[ext]\nlemma hom_ext {X Y : Augmented C} (f g : X \u27f6 Y) (h\u2081 : f.left = g.left) (h\u2082 : f.right = g.right) :\n    f = g", "@[simp]\ntheorem eqToIso_refl {n : \u2115} (h : n = n) : X.eqToIso h = Iso.refl _", "@[reassoc]\ntheorem \u03b4_comp_\u03b4_self {n} {i : Fin (n + 2)} :\n    X.\u03b4 (Fin.castSucc i) \u226b X.\u03b4 i = X.\u03b4 i.succ \u226b X.\u03b4 i", "@[reassoc]\ntheorem \u03b4_comp_\u03c3_of_le {n} {i : Fin (n + 2)} {j : Fin (n + 1)} (H : i \u2264 Fin.castSucc j) :\n    X.\u03c3 j.succ \u226b X.\u03b4 (Fin.castSucc i) = X.\u03b4 i \u226b X.\u03c3 j", "lemma shiftIso_add'_inv_app (n m mn : A) (hnm : m + n = mn) (a a' a'' : A)\n    (ha' : n + a = a') (ha'' : m + a' = a'') (X : C) :\n    (F.shiftIso mn a a'' (by rw [\u2190 hnm, \u2190 ha'', \u2190 ha', add_assoc])).inv.app X =\n        (F.shiftIso n a a' ha').inv.app X \u226b\n        ((F.shiftIso m a' a'' ha'').inv.app X)\u27e6n\u27e7' \u226b\n      (shiftFunctorAdd' D m n mn hnm).inv.app ((F.functor a'').obj X)", "lemma shiftIso_add_hom_app (n m a a' a'' : A) (ha' : n + a = a') (ha'' : m + a' = a'') (X : C) :\n    (F.shiftIso (m + n) a a'' (by rw [add_assoc, ha', ha''])).hom.app X =\n      (shiftFunctorAdd D m n).hom.app ((F.functor a'').obj X) \u226b\n        ((F.shiftIso m a' a'' ha'').hom.app X)\u27e6n\u27e7' \u226b\n        (F.shiftIso n a a' ha').hom.app X", "theorem SplitEpi.epi {X Y : C} {f : X \u27f6 Y} (se : SplitEpi f) : Epi f", "theorem StrongEpi.iff_of_arrow_iso {A B A' B' : C} {f : A \u27f6 B} {g : A' \u27f6 B'}\n    (e : Arrow.mk f \u2245 Arrow.mk g) : StrongEpi f \u2194 StrongEpi g", "theorem StrongEpi.mk' {f : P \u27f6 Q} [Epi f]\n    (hf : \u2200 (X Y : C) (z : X \u27f6 Y)\n      (_ : Mono z) (u : P \u27f6 X) (v : Q \u27f6 Y) (sq : CommSq u f z v), sq.HasLift) :\n    StrongEpi f", "theorem hom_desc (h : IsUniversal f) {c : C} (\u03b7 : f.right \u27f6 c) :\n    \u03b7 = h.desc (mk <| f.hom \u226b T.map \u03b7)", "theorem isCoseparating_proj_preimage {\ud835\udca2 : Set C} (h\ud835\udca2 : IsCoseparating \ud835\udca2) :\n    IsCoseparating ((proj S T).obj \u207b\u00b9' \ud835\udca2)", "@[simp]\ntheorem map_mk {f : S' \u27f6 T.obj Y} (g : S \u27f6 S') : (map g).obj (mk f) = mk (g \u226b f)", "theorem IsNormal.conj' {S : Subgroupoid C} (Sn : IsNormal S) :\n    \u2200 {c d} (p : d \u27f6 c) {\u03b3 : c \u27f6 c}, \u03b3 \u2208 S.arrows c c \u2192 p \u226b \u03b3 \u226b Groupoid.inv p \u2208 S.arrows d d", "theorem IsNormal.generatedNormal_le {S : Subgroupoid C} (Sn : S.IsNormal) :\n    generatedNormal X \u2264 S \u2194 \u2200 c d, X c d \u2286 S.arrows c d", "theorem generatedNormal_isNormal : (generatedNormal X).IsNormal", "theorem id_mem_of_nonempty_isotropy (c : C) : c \u2208 objs S \u2192 \ud835\udfd9 c \u2208 S.arrows c c", "theorem inclusion_faithful {S T : Subgroupoid C} (h : S \u2264 T) (s t : S.objs) :\n    Function.Injective fun f : s \u27f6 t => (inclusion h).map f", "nonrec theorem isThin_iff : S.IsThin \u2194 \u2200 c : S.objs, Subsingleton (S.arrows c c)", "theorem bot_eq_initial_to {B : C} : (\u22a5 : Subobject B) = Subobject.mk (initial.to B)", "theorem factorThru_right {X Y Z : C} {P : Subobject Z} (f : X \u27f6 Y) (g : Y \u27f6 Z) (h : P.Factors g) :\n    f \u226b P.factorThru g h = P.factorThru (f \u226b g) (factors_of_factors_right f h)", "@[simp]\ntheorem factorThru_self {X : C} (P : Subobject X) (h) : P.factorThru P.arrow h = \ud835\udfd9 (P : C)", "theorem factors_add {X Y : C} {P : Subobject Y} (f g : X \u27f6 Y) (wf : P.Factors f)\n    (wg : P.Factors g) : P.Factors (f + g)", "theorem map_comp (f : X \u27f6 Y) (g : Y \u27f6 Z) [Mono f] [Mono g] (x : Subobject X) :\n    (map (f \u226b g)).obj x = (map g).obj ((map f).obj x)", "@[simp]\ntheorem map_top (f : X \u27f6 Y) [Mono f] : (map f).obj \u22a4 = Subobject.mk f", "@[reassoc (attr := simp)]\ntheorem ofLEMk_comp_ofMkLE {B A : C} (X : Subobject B) (f : A \u27f6 B) [Mono f] (Y : Subobject B)\n    (h\u2081 : X \u2264 mk f) (h\u2082 : mk f \u2264 Y) : ofLEMk X f h\u2081 \u226b ofMkLE f Y h\u2082 = ofLE X Y (h\u2081.trans h\u2082)", "@[reassoc (attr := simp)]\ntheorem ofLE_comp_ofLEMk {B A : C} (X Y : Subobject B) (f : A \u27f6 B) [Mono f] (h\u2081 : X \u2264 Y)\n    (h\u2082 : Y \u2264 mk f) : ofLE X Y h\u2081 \u226b ofLEMk Y f h\u2082 = ofLEMk X f (h\u2081.trans h\u2082)", "@[reassoc (attr := simp)]\ntheorem ofMkLE_comp_ofLE {B A\u2081 : C} (f : A\u2081 \u27f6 B) [Mono f] (X Y : Subobject B) (h\u2081 : mk f \u2264 X)\n    (h\u2082 : X \u2264 Y) : ofMkLE f X h\u2081 \u226b ofLE X Y h\u2082 = ofMkLE f Y (h\u2081.trans h\u2082)", "theorem sInf_le {A : C} (s : Set (Subobject A)) (f) (hf : f \u2208 s) : sInf s \u2264 f", "theorem sup_factors_of_factors_left {A B : C} {X Y : Subobject B} {f : A \u27f6 B} (P : X.Factors f) :\n    (X \u2294 Y).Factors f", "@[simp]\ntheorem swap_map_inr {X Y : D} {f : inr X \u27f6 inr Y} : (swap C D).map f = f", "theorem map_iso_eq {F\u2081 F\u2082 : D \u2964 C} (h : F\u2081 \u2245 F\u2082) : map F\u2081 = map F\u2082", "lemma GE_antitone : Antitone t.GE", "lemma predicateShift_LE (a n n' : \u2124) (hn' : a + n = n') :\n    (PredicateShift (t.LE n) a) = t.LE n'", "lemma guitartExact_iff_isConnected_downwards :\n    w.GuitartExact \u2194 \u2200 {X\u2082 : C\u2082} {X\u2083 : C\u2083} (g : R.obj X\u2082 \u27f6 B.obj X\u2083),\n      IsConnected (w.CostructuredArrowDownwards g)", "@[simp]\ntheorem map_obj_hom : ((map f).obj U).hom = f \u226b U.hom", "@[reassoc (attr := simp)]\ntheorem w {A B : Under X} (f : A \u27f6 B) : A.hom \u226b f.right = B.hom", "lemma fullyFaithful_preimage {X Y : C} (f : yoneda.obj X \u27f6 yoneda.obj Y) :\n    fullyFaithful.preimage f = f.app (op X) (\ud835\udfd9 X) := rfl", "lemma areEqualizedByLocalization_iff [L.IsLocalization W]:\n    AreEqualizedByLocalization W f g \u2194 L.map f = L.map g", "@[simp]\ntheorem asIso_inv (f : X \u27f6 Y) {_ : IsIso f} : (asIso f).inv = inv f", "@[simp]\ntheorem associator_hom_apply {X Y Z : Type u} {x : X} {y : Y} {z : Z} :\n    ((\u03b1_ X Y Z).hom : (X \u2297 Y) \u2297 Z \u2192 X \u2297 Y \u2297 Z) ((x, y), z) = (x, (y, z))", "theorem balanced_opposite [Balanced C] : Balanced C\u1d52\u1d56", "@[reassoc]\ntheorem braiding_leftUnitor (X : C) : (\u03b2_ X (\ud835\udfd9_ C)).hom \u226b (\u03bb_ X).hom = (\u03c1_ X).hom", "theorem braiding_leftUnitor_aux\u2081 (X : C) :\n    (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) X).hom \u226b\n        (\ud835\udfd9_ C \u25c1 (\u03b2_ X (\ud835\udfd9_ C)).inv) \u226b (\u03b1_ _ X _).inv \u226b ((\u03bb_ X).hom \u25b7 _) =\n      ((\u03bb_ _).hom \u25b7 X) \u226b (\u03b2_ X (\ud835\udfd9_ C)).inv", "theorem cancel_mono (f : X \u27f6 Y) [Mono f] {g h : Z \u27f6 X} : g \u226b f = h \u226b f \u2194 g = h", "@[simp]\ntheorem coconeOfRepresentable_pt (P : C\u1d52\u1d56 \u2964 Type u\u2081) : (coconeOfRepresentable P).pt = P", "theorem coconeOfRepresentable_\u03b9_app (P : C\u1d52\u1d56 \u2964 Type u\u2081) (j : P.Elements\u1d52\u1d56) :\n    (coconeOfRepresentable P).\u03b9.app j = yonedaEquiv.symm j.unop.2", "theorem coherentTopology.mem_sieves_iff_hasEffectiveEpiFamily (S : Sieve X) :\n    (S \u2208 GrothendieckTopology.sieves (coherentTopology C) X) \u2194\n    (\u2203 (\u03b1 : Type) (_ : Finite \u03b1) (Y : \u03b1 \u2192 C) (\u03c0 : (a : \u03b1) \u2192 (Y a \u27f6 X)),\n        EffectiveEpiFamily Y \u03c0 \u2227 (\u2200 a : \u03b1, (S.arrows) (\u03c0 a)) )", "theorem coherentTopology.mem_sieves_of_hasEffectiveEpiFamily (S : Sieve X) :\n    (\u2203 (\u03b1 : Type) (_ : Finite \u03b1) (Y : \u03b1 \u2192 C) (\u03c0 : (a : \u03b1) \u2192 (Y a \u27f6 X)),\n      EffectiveEpiFamily Y \u03c0 \u2227 (\u2200 a : \u03b1, (S.arrows) (\u03c0 a)) ) \u2192\n        (S \u2208 GrothendieckTopology.sieves (coherentTopology C) X)", "lemma coherentTopology.presheafIsLocallySurjective_iff {F G : C\u1d52\u1d56 \u2964 D} (f : F \u27f6 G)\n    [Preregular C] [FinitaryPreExtensive C] [PreservesFiniteProducts F] [PreservesFiniteProducts G]\n      [PreservesFiniteProducts (forget D)] :\n        Presheaf.IsLocallySurjective (coherentTopology C) f \u2194\n          Presheaf.IsLocallySurjective (regularTopology C) f", "theorem comp_eq_zero_of_image_eq_kernel {A B C : V} (f : A \u27f6 B) (g : B \u27f6 C)\n    (p : imageSubobject f = kernelSubobject g) : f \u226b g = 0", "theorem comp_ite {P : Prop} [Decidable P] {X Y Z : C} (f : X \u27f6 Y) (g g' : Y \u27f6 Z) :\n    (f \u226b if P then g else g') = if P then f \u226b g else f \u226b g'", "@[simp]\ntheorem composePath_comp' {X Y Z : Paths C} (f : X \u27f6 Y) (g : Y \u27f6 Z) :\n    composePath (f \u226b g) = composePath f \u226b composePath g", "theorem coyonedaEquiv_apply {X : C} {F : C \u2964 Type v\u2081} (f : coyoneda.obj (op X) \u27f6 F) :\n    coyonedaEquiv f = f.app X (\ud835\udfd9 X)", "lemma coyonedaEquiv_symm_map {X Y : C} (f : X \u27f6 Y) {F : C \u2964 Type v\u2081} (t : F.obj X) :\n    coyonedaEquiv.symm (F.map f t) = coyoneda.map f.op \u226b coyonedaEquiv.symm t", "theorem dcongr_arg {\u03b9 : Type*} {F G : \u03b9 \u2192 C} (\u03b1 : \u2200 i, F i \u27f6 G i) {i j : \u03b9} (h : i = j) :\n    \u03b1 i = eqToHom (congr_arg F h) \u226b \u03b1 j \u226b eqToHom (congr_arg G h.symm)", "theorem epi_of_epi_fac {X Y Z : C} {f : X \u27f6 Y} {g : Y \u27f6 Z} {h : X \u27f6 Z} [Epi h]\n    (w : f \u226b g = h) : Epi g", "theorem epi_of_nonzero_to_simple [HasEqualizers C] {X Y : C} [Simple Y] {f : X \u27f6 Y} [HasImage f]\n    (w : f \u2260 0) : Epi f", "@[simp]\ntheorem eqToHom_refl (X : C) (p : X = X) : eqToHom p = \ud835\udfd9 X", "@[simp]\ntheorem eqToIso_map_trans (F : C \u2964 D) {X Y Z : C} (p : X = Y) (q : Y = Z) :\n    F.mapIso (eqToIso p) \u226a\u226b F.mapIso (eqToIso q) = F.mapIso (eqToIso <| p.trans q)", "@[simp]\ntheorem exact_comp_iso [IsIso h] : Exact f (g \u226b h) \u2194 Exact f g", "theorem exact_kernelSubobject_arrow : Exact (kernelSubobject f).arrow f", "lemma extensiveTopology.isLocallySurjective_iff [FinitaryExtensive C]\n    {F G : Sheaf (extensiveTopology C) D} (f : F \u27f6 G)\n      [PreservesFiniteProducts (forget D)] : IsLocallySurjective f \u2194\n        \u2200 (X : C), Function.Surjective (f.val.app (op X))", "theorem finitaryExtensive_of_preserves_and_reflects (F : C \u2964 D) [FinitaryExtensive D]\n    [HasFiniteCoproducts C] [HasPullbacksOfInclusions C]\n    [PreservesPullbacksOfInclusions F]\n    [ReflectsLimitsOfShape WalkingCospan F] [PreservesColimitsOfShape (Discrete WalkingPair) F]\n    [ReflectsColimitsOfShape (Discrete WalkingPair) F] : FinitaryExtensive C", "theorem groupoid_of_isCodetecting_empty (h : IsCodetecting (\u2205 : Set C)) {X Y : C} (f : X \u27f6 Y) :\n    IsIso f", "theorem homOfElement_eq_iff {X : Type u} (x y : X) : homOfElement x = homOfElement y \u2194 x = y", "@[simp]\ntheorem hom_inv_id_apply {X Y : Type u} (f : X \u2245 Y) (x : X) : f.inv (f.hom x) = x", "theorem initiallySmall_of_initial_of_essentiallySmall [EssentiallySmall.{w} K] :\n    InitiallySmall.{w} J", "theorem isCodetecting_unop_iff {\ud835\udca2 : Set C\u1d52\u1d56} : IsCodetecting \ud835\udca2.unop \u2194 IsDetecting \ud835\udca2", "lemma isFilteredOrEmpty_of_isCofilteredOrEmpty_op [IsCofilteredOrEmpty C\u1d52\u1d56] : IsFilteredOrEmpty C", "theorem isIso_of_mono_of_epi [Balanced C] {X Y : C} (f : X \u27f6 Y) [Mono f] [Epi f] : IsIso f", "theorem isIso_of_mono_of_strongEpi (f : P \u27f6 Q) [Mono f] [StrongEpi f] : IsIso f", "theorem isIso_prod_iff {P Q : C} {S T : D} {f : (P, S) \u27f6 (Q, T)} :\n    IsIso f \u2194 IsIso f.1 \u2227 IsIso f.2", "theorem isPullback_of_cofan_isVanKampen [HasInitial C] {\u03b9 : Type*} {X : \u03b9 \u2192 C}\n    {c : Cofan X} (hc : IsVanKampenColimit c) (i j : \u03b9) [DecidableEq \u03b9] :\n    IsPullback (P := (if j = i then X i else \u22a5_ C))\n      (if h : j = i then eqToHom (if_pos h) else eqToHom (if_neg h) \u226b initial.to (X i))\n      (if h : j = i then eqToHom ((if_pos h).trans (congr_arg X h.symm))\n        else eqToHom (if_neg h) \u226b initial.to (X j))\n      (Cofan.inj c i) (Cofan.inj c j)", "theorem isSeparating_iff_epi (\ud835\udca2 : Set C)\n    [\u2200 A : C, HasCoproduct fun f : \u03a3G : \ud835\udca2, (G : C) \u27f6 A => (f.1 : C)] :\n    IsSeparating \ud835\udca2 \u2194 \u2200 A : C, Epi (Sigma.desc (@Sigma.snd \ud835\udca2 fun G => (G : C) \u27f6 A))", "theorem isSeparating_unop_iff (\ud835\udca2 : Set C\u1d52\u1d56) : IsSeparating \ud835\udca2.unop \u2194 IsCoseparating \ud835\udca2", "theorem isSeparator_unop_iff (G : C\u1d52\u1d56) : IsSeparator (unop G) \u2194 IsCoseparator G", "lemma isSheaf_coherent (P : C\u1d52\u1d56 \u2964 Type w) :\n    Presieve.IsSheaf (coherentTopology C) P \u2194\n    (\u2200 (B : C) (\u03b1 : Type) [Finite \u03b1] (X : \u03b1 \u2192 C) (\u03c0 : (a : \u03b1) \u2192 (X a \u27f6 B)),\n      EffectiveEpiFamily X \u03c0 \u2192 (Presieve.ofArrows X \u03c0).IsSheafFor P)", "@[reassoc (attr := simp)]\ntheorem leftDistributor_hom_comp_biproduct_\u03c0 {J : Type} [Fintype J] (X : C) (f : J \u2192 C) (j : J) :\n    (leftDistributor X f).hom \u226b biproduct.\u03c0 _ j = X \u25c1 biproduct.\u03c0 _ j", "@[simp]\ntheorem leftUnitor_inv_apply {X : Type u} {x : X} :\n    ((\u03bb_ X).inv : X \u27f6 \ud835\udfd9_ (Type u) \u2297 X) x = (PUnit.unit, x)", "theorem locallySmall_of_faithful {C : Type u} [Category.{v} C] {D : Type u'} [Category.{v'} D]\n    (F : C \u2964 D) [F.Faithful] [LocallySmall.{w} D] : LocallySmall.{w} C where", "theorem associator_hom (X Y Z : C) :\n    (\u03b1_ X Y Z).hom =\n      coprod.desc (coprod.desc coprod.inl (coprod.inl \u226b coprod.inr)) (coprod.inr \u226b coprod.inr)", "@[simp]\ntheorem whiskerRight {X Y : C} (f : X \u27f6 Y) (Z : C) : f \u25b7 Z = Limits.prod.map f (\ud835\udfd9 Z)", "@[simp] lemma mop_hom_rightUnitor (X : C) : (\u03c1_ X).hom.mop = (\u03bb_ (mop X)).hom := rfl", "@[simp] lemma mop_tensorHom {X\u2081 Y\u2081 X\u2082 Y\u2082 : C} (f : X\u2081 \u27f6 Y\u2081) (g : X\u2082 \u27f6 Y\u2082) :\n    (f \u2297 g).mop = g.mop \u2297 f.mop := rfl", "@[simp] lemma op_inv_leftUnitor (X : C) : (\u03bb_ X).inv.op = (\u03bb_ (op X)).hom := rfl", "lemma predicateShift_predicateShift (a b c : A) (h : a + b = c) [ClosedUnderIsomorphisms P] :\n    PredicateShift (PredicateShift P b) a = PredicateShift P c", "@[simp]\nlemma predicateShift_zero [ClosedUnderIsomorphisms P] : PredicateShift P (0 : A) = P", "theorem reflectsIsomorphisms_forget\u2082 [HasForget\u2082 C D] [(forget C).ReflectsIsomorphisms] :\n    (forget\u2082 C D).ReflectsIsomorphisms", "lemma equalizerCondition_w' (P : C\u1d52\u1d56 \u2964 Type*) {X B : C} (\u03c0 : X \u27f6 B)\n    [HasPullback \u03c0 \u03c0] : P.map \u03c0.op \u226b P.map (pullback.fst (f := \u03c0) (g := \u03c0)).op =\n    P.map \u03c0.op \u226b P.map (pullback.snd).op", "theorem mem_sieves_of_hasEffectiveEpi (S : Sieve X) :\n    (\u2203 (Y : C) (\u03c0 : Y \u27f6 X), EffectiveEpi \u03c0 \u2227 S.arrows \u03c0) \u2192 (S \u2208 (regularTopology C).sieves X)", "theorem subcanonical [Preregular C] : Sheaf.Subcanonical (regularTopology C)", "theorem representable_of_natIso (F : C\u1d52\u1d56 \u2964 Type v\u2081) {G} (i : F \u2245 G) [F.Representable] :\n    G.Representable", "theorem rightAdjointMate_comp {X Y Z : C} [HasRightDual X] [HasRightDual Y] {f : X \u27f6 Y}\n    {g : X\u1601 \u27f6 Z} :\n    f\u1601 \u226b g =\n      (\u03c1_ (Y\u1601)).inv \u226b\n        _ \u25c1 \u03b7_ X (X\u1601) \u226b _ \u25c1 (f \u2297 g) \u226b (\u03b1_ (Y\u1601) Y Z).inv \u226b \u03b5_ Y (Y\u1601) \u25b7 _ \u226b (\u03bb_ Z).hom", "lemma shiftFunctorAdd'_add_zero (a : A) :\n    shiftFunctorAdd' C a 0 a (add_zero a) = (Functor.rightUnitor _).symm \u226a\u226b\n    isoWhiskerLeft (shiftFunctor C a) (shiftFunctorZero C A).symm", "lemma shiftFunctorAdd'_zero_add_hom_app (a : A) (X : C) :\n    (shiftFunctorAdd' C 0 a a (zero_add a)).hom.app X =\n    ((shiftFunctorZero C A).inv.app X)\u27e6a\u27e7'", "lemma shiftFunctorAdd_add_zero_hom_app (a : A) (X : C) : (shiftFunctorAdd C a 0).hom.app X =\n    eqToHom (by dsimp; rw [add_zero]) \u226b (shiftFunctorZero C A).inv.app (X\u27e6a\u27e7)", "lemma shiftFunctorAdd_assoc (a\u2081 a\u2082 a\u2083 : A) :\n    shiftFunctorAdd C (a\u2081 + a\u2082) a\u2083 \u226a\u226b\n      isoWhiskerRight (shiftFunctorAdd C a\u2081 a\u2082) _ \u226a\u226b Functor.associator _ _ _ =\n    shiftFunctorAdd' C a\u2081 (a\u2082 + a\u2083) _ (add_assoc a\u2081 a\u2082 a\u2083).symm \u226a\u226b\n      isoWhiskerLeft _ (shiftFunctorAdd C a\u2082 a\u2083)", "@[reassoc]\nlemma shiftFunctorAdd_assoc_inv_app (a\u2081 a\u2082 a\u2083 : A) (X : C) :\n    ((shiftFunctorAdd C a\u2081 a\u2082).inv.app X)\u27e6a\u2083\u27e7' \u226b\n      (shiftFunctorAdd C (a\u2081 + a\u2082) a\u2083).inv.app X =\n    (shiftFunctorAdd C a\u2082 a\u2083).inv.app (X\u27e6a\u2081\u27e7) \u226b\n      (shiftFunctorAdd' C a\u2081 (a\u2082 + a\u2083) (a\u2081 + a\u2082 + a\u2083) (add_assoc _ _ _).symm).inv.app X", "lemma shiftFunctorAdd_zero_add_inv_app (a : A) (X : C) : (shiftFunctorAdd C 0 a).inv.app X =\n    ((shiftFunctorZero C A).hom.app X)\u27e6a\u27e7' \u226b eqToHom (by dsimp; rw [zero_add])", "lemma shiftFunctorComm_eq (i j k : A) (h : i + j = k) :\n    shiftFunctorComm C i j = (shiftFunctorAdd' C i j k h).symm \u226a\u226b\n      shiftFunctorAdd' C j i k (by rw [add_comm j i, h])", "@[simp]\ntheorem tensorLeftHomEquiv_whiskerLeft_comp_evaluation {Y Z : C} [HasLeftDual Z] (f : Y \u27f6 \u1601Z) :\n    (tensorLeftHomEquiv _ _ _ _) (Z \u25c1 f \u226b \u03b5_ _ _) = f \u226b (\u03c1_ _).inv", "@[reassoc (attr := simp)]\ntheorem toSheafify_sheafifyLift {P Q : C\u1d52\u1d56 \u2964 D} (\u03b7 : P \u27f6 Q) (hQ : Presheaf.IsSheaf J Q) :\n    toSheafify J P \u226b sheafifyLift J \u03b7 hQ = \u03b7", "@[simp]\ntheorem transferNatTransSelf_id : transferNatTransSelf adj\u2081 adj\u2081 (\ud835\udfd9 _) = \ud835\udfd9 _", "@[simp]\ntheorem types_comp_apply {X Y Z : Type u} (f : X \u27f6 Y) (g : Y \u27f6 Z) (x : X) : (f \u226b g) x = g (f x)", "theorem uncurry_expComparison (A B : C) :\n    CartesianClosed.uncurry ((expComparison F A).app B) =\n      inv (prodComparison F _ _) \u226b F.map ((exp.ev _).app _)", "@[simp] lemma unmop_leftUnitor (X : C\u1d39\u1d52\u1d56) : (\u03bb_ X).unmop = \u03c1_ (unmop X) := rfl", "@[simp] lemma unop_inv_associator (X Y Z : C\u1d52\u1d56) :\n    (\u03b1_ X Y Z).inv.unop = (\u03b1_ (unop X) (unop Y) (unop Z)).hom := rfl", "@[simp, reassoc]\ntheorem whiskerRight_comp {G H K : C \u2964 D} (\u03b1 : G \u27f6 H) (\u03b2 : H \u27f6 K) (F : D \u2964 E) :\n    whiskerRight (\u03b1 \u226b \u03b2) F = whiskerRight \u03b1 F \u226b whiskerRight \u03b2 F", "theorem zag_symmetric : Symmetric (@Zag J _)", "@[simp]\ntheorem _root_.CategoryTheory.zero_map [HasZeroObject D] [HasZeroMorphisms D] {X Y : C}\n    (f : X \u27f6 Y) : (0 : C \u2964 D).map f = 0", "@[reassoc (attr := simp)]\ntheorem \u03b5_hom_inv_app (X : C) : F.\u03b5.app X \u226b F.\u03b5Iso.inv.app X = \ud835\udfd9 _", "@[reassoc (attr := simp)]\ntheorem \u03b5_inv_hom_app (X : C) : F.\u03b5Iso.inv.app X \u226b F.\u03b5.app X = \ud835\udfd9 _", "@[reassoc (attr := simp)]\ntheorem \u03bc_inv_naturality\u1d63 {m n n' : M} (g : n \u27f6 n') (X : C) :\n    (F.\u03bcIso m n).inv.app X \u226b (F.map g).app ((F.obj m).obj X) =\n      (F.map (m \u25c1 g)).app X \u226b (F.\u03bcIso m n').inv.app X", "@[reassoc (attr := simp)]\ntheorem \u03bc_inv_naturality\u2097 {m n m' : M} (f : m \u27f6 m') (X : C) :\n    (F.\u03bcIso m n).inv.app X \u226b (F.obj n).map ((F.map f).app X) =\n      (F.map (f \u25b7 n)).app X \u226b (F.\u03bcIso m' n).inv.app X", "protected theorem inv_mul_cancel {x : (Cauchy abv)} : x \u2260 0 \u2192 x\u207b\u00b9 * x = 1", "theorem CauSeq.cauchySeq (f : CauSeq \u03b2 norm) : CauchySeq f", "@[simp, norm_cast]\ntheorem coe_pow (f : CauSeq \u03b2 abv) (n : \u2115) : \u21d1(f ^ n) = (f : \u2115 \u2192 \u03b2) ^ n", "@[simp]\ntheorem const_one : const 1 = 1", "theorem const_pow (x : \u03b2) (n : \u2115) : const (x ^ n) = const x ^ n", "theorem const_sub (x y : \u03b2) : const (x - y) = const x - const y", "theorem le_antisymm {f g : CauSeq \u03b1 abs} (fg : f \u2264 g) (gf : g \u2264 f) : f \u2248 g", "theorem le_of_exists {f g : CauSeq \u03b1 abs} (h : \u2203 i, \u2200 j \u2265 i, f j \u2264 g j) : f \u2264 g", "theorem lt_of_eq_of_lt {f g h : CauSeq \u03b1 abs} (fg : f \u2248 g) (gh : g < h) : f < h", "theorem mul_equiv_zero' (g : CauSeq _ abv) {f : CauSeq _ abv} (hf : f \u2248 0) : f * g \u2248 0", "theorem rat_sup_continuous_lemma {\u03b5 : \u03b1} {a\u2081 a\u2082 b\u2081 b\u2082 : \u03b1} :\n    abs (a\u2081 - b\u2081) < \u03b5 \u2192 abs (a\u2082 - b\u2082) < \u03b5 \u2192 abs (a\u2081 \u2294 a\u2082 - b\u2081 \u2294 b\u2082) < \u03b5", "theorem smul_equiv_smul {G : Type*} [SMul G \u03b2] [IsScalarTower G \u03b2 \u03b2] {f1 f2 : CauSeq \u03b2 abv} (c : G)\n    (hf : f1 \u2248 f2) : c \u2022 f1 \u2248 c \u2022 f2", "protected theorem sup_inf_distrib_left (a b c : CauSeq \u03b1 abs) : a \u2294 b \u2293 c = (a \u2294 b) \u2293 (a \u2294 c)", "protected theorem sup_lt {a b c : CauSeq \u03b1 abs} (ha : a < c) (hb : b < c) : a \u2294 b < c", "theorem Cauchy.mono {f g : Filter \u03b1} [hg : NeBot g] (h_c : Cauchy f) (h_le : g \u2264 f) : Cauchy g", "theorem inseparable_lim_iff [CompleteSpace \u03b1] {f g : CauchyFilter \u03b1} :\n    haveI := f.2.1.nonempty; Inseparable (lim f.1) (lim g.1) \u2194 Inseparable f g", "theorem toEnd_smul (m : M) (x : CentroidHom \u03b1) : (m \u2022 x).toEnd = m \u2022 x.toEnd", "lemma exactAt_succ_single_obj (A : C) (n : \u2115) :\n    ExactAt ((single\u2080 C).obj A) (n + 1)", "@[reassoc (attr := simp)]\nlemma isoHomology\u03b9\u2080_inv_naturality [L.HasHomology 0] :\n    K.isoHomology\u03b9\u2080.inv \u226b HomologicalComplex.homologyMap \u03c6 0 =\n      HomologicalComplex.opcyclesMap \u03c6 0 \u226b L.isoHomology\u03b9\u2080.inv", "@[simp]\nlemma single\u2080ObjXSelf (X : V) :\n    HomologicalComplex.singleObjXSelf (ComplexShape.down \u2115) 0 X = Iso.refl _ := rfl", "theorem char_ne_zero_of_finite (p : \u2115) [CharP R p] [Finite R] : p \u2260 0", "theorem quotient_iff_le_ker_natCast {R : Type*} [CommRing R] (n : \u2115) [CharP R n] (I : Ideal R) :\n    CharP (R \u29f8 I) n \u2194 I.comap (Nat.castRingHom R) \u2264 RingHom.ker (Nat.castRingHom R)", "@[simp]\ntheorem bit1_eq_one : (bit1 : R \u2192 R) = 1", "theorem list_sum_mul_self (l : List R) : l.sum * l.sum = (List.map (fun x => x * x) l).sum", "@[simp]\ntheorem neg_eq (x : R) : -x = x", "lemma charZero_iff_forall_prime_ne_zero [NonAssocRing R] [NoZeroDivisors R] [Nontrivial R] :\n    CharZero R \u2194 \u2200 p : \u2115, p.Prime \u2192 (p : R) \u2260 0", "@[ext] theorem ext {c c' : CharacterModule A} (h : \u2200 x, c x = c' x) : c = c'", "theorem commute_nat_add (n : \u2115) : Function.Commute f (n + \u00b7)", "theorem floor_sub_le_translationNumber (x : \u211d) : \u2191\u230af x - x\u230b \u2264 \u03c4 f", "theorem iterate_pos_lt_iff {x : \u211d} {m : \u2124} {n : \u2115} (hn : 0 < n) :\n    f^[n] x < x + n * m \u2194 f x < x + m", "theorem le_iterate_pos_iff {x : \u211d} {m : \u2124} {n : \u2115} (hn : 0 < n) :\n    x + n * m \u2264 f^[n] x \u2194 x + m \u2264 f x", "theorem map_lt_of_translationNumber_lt_int {n : \u2124} (h : \u03c4 f < n) (x : \u211d) : f x < x + n", "theorem map_lt_of_translationNumber_lt_nat {n : \u2115} (h : \u03c4 f < n) (x : \u211d) : f x < x + n", "theorem mul_floor_map_zero_le_floor_iterate_zero (n : \u2115) : \u2191n * \u230af 0\u230b \u2264 \u230af^[n] 0\u230b", "theorem semiconj_of_isUnit_of_translationNumber_eq {f\u2081 f\u2082 : CircleDeg1Lift} (h\u2081 : IsUnit f\u2081)\n    (h\u2082 : IsUnit f\u2082) (h : \u03c4 f\u2081 = \u03c4 f\u2082) : \u2203 F : CircleDeg1Lift, Semiconj F f\u2081 f\u2082", "@[simp, norm_cast]\ntheorem coe_inter (x y : ZFSet.{u}) : \u2191(x \u2229 y) = (x : Class.{u}) \u2229 y", "@[simp, norm_cast]\ntheorem coe_subset (x y : ZFSet.{u}) : (x : Class.{u}) \u2286 y \u2194 x \u2286 y", "theorem eq_univ_of_forall {A : Class.{u}} : (\u2200 x : ZFSet, A x) \u2192 A = univ", "@[simp]\ntheorem sUnion_empty : \u22c3\u2080 (\u2205 : Class.{u}) = (\u2205 : Class.{u})", "@[simp]\ntheorem ClassGroup.equiv_mk (K' : Type*) [Field K'] [Algebra R K'] [IsFractionRing R K']\n    (I : (FractionalIdeal R\u2070 K)\u02e3) :\n    ClassGroup.equiv K' (ClassGroup.mk I) =\n      QuotientGroup.mk' _ (Units.mapEquiv (\u2191(FractionalIdeal.canonicalEquiv R\u2070 K K')) I)", "theorem exists_mk0_eq_mk0 [IsDedekindDomain S] [Algebra.IsAlgebraic R L] (I : (Ideal S)\u2070) :\n    \u2203 J : (Ideal S)\u2070,\n      ClassGroup.mk0 I = ClassGroup.mk0 J \u2227\n        algebraMap _ _ (\u220f m \u2208 finsetApprox bS adm, m) \u2208 (J : Ideal S)", "@[simp]\ntheorem mem_finsetApprox {x : R} :\n    x \u2208 finsetApprox bS adm \u2194 \u2203 i j, i \u2260 j \u2227 distinctElems bS adm i - distinctElems bS adm j =\n    x", "theorem ClassGroup.mk0_eq_mk0_inv_iff [IsDedekindDomain R] {I J : (Ideal R)\u2070} :\n    ClassGroup.mk0 I = (ClassGroup.mk0 J)\u207b\u00b9 \u2194\n      \u2203 x \u2260 (0 : R), I * J = Ideal.span {x}", "@[simp] theorem not_imp : \u00ac(a \u2192 b) \u2194 a \u2227 \u00acb", "@[simp] theorem not_not : \u00ac\u00aca \u2194 a", "@[simp]\ntheorem e0_mul_v_mul_e0 (m : M) : e0 Q * v Q m * e0 Q = v Q m", "@[simp]\ntheorem involute_v (m : M) : involute (v Q m) = -v Q m", "theorem changeForm_self_apply (x : CliffordAlgebra Q) : changeForm (Q' := Q)\n    changeForm.zero_proof x = x", "theorem even.lift_\u03b9 (f : EvenHom Q A) (m\u2081 m\u2082 : M) :\n    even.lift Q f ((even.\u03b9 Q).bilin m\u2081 m\u2082) = f.bilin m\u2081 m\u2082", "theorem involute_prod_map_\u03b9 :\n    \u2200 l : List M, involute (l.map <| \u03b9 Q).prod = (-1 : R) ^ l.length \u2022 (l.map <| \u03b9 Q).prod", "@[simp]\ntheorem lift_unique (f : M \u2192\u2097[R] A) (cond : \u2200 m : M, f m * f m = algebraMap _ _ (Q m))\n    (g : CliffordAlgebra Q \u2192\u2090[R] A) : g.toLinearMap.comp (\u03b9 Q) = f \u2194 g = lift Q \u27e8f, cond\u27e9", "@[simp]\ntheorem lift_\u03b9_apply (f : M \u2192\u2097[R] A) (cond : \u2200 m, f m * f m = algebraMap _ _ (Q m)) (x) :\n    lift Q \u27e8f, cond\u27e9 (\u03b9 Q x) = f x", "@[simp] theorem ofBaseChange_tmul_\u03b9 (Q : QuadraticForm R V) (z : A) (v : V) :\n    ofBaseChange A Q (z \u2297\u209c \u03b9 Q v) = \u03b9 (Q.baseChange A) (z \u2297\u209c v)", "theorem one_le_evenOdd_zero : 1 \u2264 evenOdd Q 0", "theorem reverse_involute :\n    \u2200 a : CliffordAlgebra Q, reverse (involute a) = involute (reverse a)", "@[elab_as_elim]\ntheorem right_induction {P : CliffordAlgebra Q \u2192 Prop} (algebraMap : \u2200 r : R, P (algebraMap _ _ r))\n    (add : \u2200 x y, P x \u2192 P y \u2192 P (x + y)) (mul_\u03b9 : \u2200 m x, P x \u2192 P (x * \u03b9 Q m)) : \u2200 x, P x", "theorem submodule_map_reverse_eq_comap (p : Submodule R (CliffordAlgebra Q)) :\n    p.map (reverse : CliffordAlgebra Q \u2192\u2097[R] CliffordAlgebra Q) =\n      p.comap (reverse : CliffordAlgebra Q \u2192\u2097[R] CliffordAlgebra Q)", "theorem toBaseChange_comp_ofBaseChange (Q : QuadraticForm R V) :\n    (toBaseChange A Q).comp (ofBaseChange A Q) = AlgHom.id _ _", "@[simp] theorem unop_reverseOp (x : CliffordAlgebra Q) : (reverseOp x).unop = reverse x", "@[simp] theorem \u03b9_mul_\u03b9_add_swap_of_isOrtho {a b : M} (h : Q.IsOrtho a b) :\n    \u03b9 Q a * \u03b9 Q b + \u03b9 Q b * \u03b9 Q a = 0", "@[ext]\nprotected theorem ext {s t : ClopenUpperSet \u03b1} (h : (s : Set \u03b1) = t) : s = t", "protected theorem ClosedEmbedding.LindelofSpace [h : LindelofSpace Y] {f : X \u2192 Y}\n    (hf : ClosedEmbedding f) : LindelofSpace X", "theorem closure_sup_closure_le (x y : \u03b1) : c x \u2294 c y \u2264 c (x \u2294 y)", "theorem le_closure_iff : x \u2264 c y \u2194 c x \u2264 c y", "@[simp]\ntheorem symm_toLinearEquiv (e : A \u2243\u2097c[R] B) :\n    e.symm = (e : A \u2243\u2097[R] B).symm", "theorem toEquiv_injective : Function.Injective (toEquiv : (A \u2243\u2097c[R] B) \u2192 A \u2243 B)", "@[simp]\ntheorem toLinearEquiv_eq_coe (f : A \u2243\u2097c[R] B) : f.toLinearEquiv = f", "@[simp]\ntheorem coe_comp (\u03c6\u2081 : B \u2192\u2097c[R] C) (\u03c6\u2082 : A \u2192\u2097c[R] B) : \u21d1(\u03c6\u2081.comp \u03c6\u2082) = \u03c6\u2081 \u2218 \u03c6\u2082", "@[norm_cast]\ntheorem coe_mks {f : A \u2192 B} (h\u2081 h\u2082 h\u2083 h\u2084) : \u21d1(\u27e8\u27e8\u27e8f, h\u2081\u27e9, h\u2082\u27e9, h\u2083, h\u2084\u27e9 : A \u2192\u2097c[R] B) = f", "@[simp]\ntheorem mk_coe {f : A \u2192\u2097c[R] B} (h\u2081 h\u2082 h\u2083 h\u2084) : (\u27e8\u27e8\u27e8f, h\u2081\u27e9, h\u2082\u27e9, h\u2083, h\u2084\u27e9 : A \u2192\u2097c[R] B) = f", "@[simp]\ntheorem toLinearMap_eq_coe (f : A \u2192\u2097c[R] B) : f.toLinearMap = f", "@[simp]\ntheorem counit_comp_apply (f : F) (x : A) : counit (f x) = counit (R := R) x", "@[simp]\nlemma add_v {n : \u2124} (z\u2081 z\u2082 : Cochain F G n) (p q : \u2124) (hpq : p + n = q) :\n    (z\u2081 + z\u2082).v p q hpq = z\u2081.v p q hpq + z\u2082.v p q hpq := rfl", "@[simp]\nlemma comp_assoc_of_first_is_zero_cochain {n\u2082 n\u2083 n\u2082\u2083 : \u2124}\n    (z\u2081 : Cochain F G 0) (z\u2082 : Cochain G K n\u2082) (z\u2083 : Cochain K L n\u2083)\n    (h\u2082\u2083 : n\u2082 + n\u2083 = n\u2082\u2083) :\n    (z\u2081.comp z\u2082 (zero_add n\u2082)).comp z\u2083 h\u2082\u2083 = z\u2081.comp (z\u2082.comp z\u2083 h\u2082\u2083) (zero_add n\u2082\u2083)", "@[simp]\nlemma comp_assoc_of_second_degree_eq_neg_third_degree {n\u2081 n\u2082 n\u2081\u2082 : \u2124}\n    (z\u2081 : Cochain F G n\u2081) (z\u2082 : Cochain G K (-n\u2082)) (z\u2083 : Cochain K L n\u2082) (h\u2081\u2082 : n\u2081 + (-n\u2082) = n\u2081\u2082) :\n    (z\u2081.comp z\u2082 h\u2081\u2082).comp z\u2083\n      (show n\u2081\u2082 + n\u2082 = n\u2081 by rw [\u2190 h\u2081\u2082, add_assoc, neg_add_self, add_zero]) =\n      z\u2081.comp (z\u2082.comp z\u2083 (neg_add_self n\u2082)) (add_zero n\u2081)", "@[simp]\nprotected lemma comp_sub {n\u2081 n\u2082 n\u2081\u2082 : \u2124} (z\u2081 : Cochain F G n\u2081) (z\u2082 z\u2082' : Cochain G K n\u2082)\n    (h : n\u2081 + n\u2082 = n\u2081\u2082) : z\u2081.comp (z\u2082 - z\u2082') h = z\u2081.comp z\u2082 h - z\u2081.comp z\u2082' h", "@[simp]\nlemma leftUnshift_leftShift (a n' : \u2124) (hn' : n + a = n') :\n    (\u03b3.leftShift a n' hn').leftUnshift n hn' = \u03b3", "@[simp]\nlemma ofHom_v (\u03c6 : F \u27f6 G) (p : \u2124) : (ofHom \u03c6).v p p (add_zero p) = \u03c6.f p", "@[simp]\nlemma rightShift_add (a n' : \u2124) (hn' : n' + a = n) :\n    (\u03b3\u2081 + \u03b3\u2082).rightShift a n' hn' = \u03b3\u2081.rightShift a n' hn' + \u03b3\u2082.rightShift a n' hn'", "@[simp]\nlemma rightShift_neg (a n' : \u2124) (hn' : n' + a = n) :\n    (-\u03b3).rightShift a n' hn' = -\u03b3.rightShift a n' hn'", "@[simp]\nlemma rightUnshift_zero (a n' : \u2124) (hn' : n' + a = n) :\n    (0 : Cochain K (L\u27e6a\u27e7) n').rightUnshift n hn' = 0", "@[simp]\nlemma units_smul_comp {n\u2081 n\u2082 n\u2081\u2082 : \u2124} (k : R\u02e3) (z\u2081 : Cochain F G n\u2081) (z\u2082 : Cochain G K n\u2082)\n    (h : n\u2081 + n\u2082 = n\u2081\u2082) : (k \u2022 z\u2081).comp z\u2082 h = k \u2022 (z\u2081.comp z\u2082 h)", "@[simp]\nprotected lemma zero_comp {n\u2081 n\u2082 n\u2081\u2082 : \u2124} (z\u2082 : Cochain G K n\u2082)\n    (h : n\u2081 + n\u2082 = n\u2081\u2082) : (0 : Cochain F G n\u2081).comp z\u2082 h = 0", "lemma \u03b4_rightUnshift {a n' : \u2124} (\u03b3 : Cochain K (L\u27e6a\u27e7) n') (n : \u2124) (hn : n' + a = n)\n    (m m' : \u2124) (hm' : m' + a = m) :\n    \u03b4 n m (\u03b3.rightUnshift n hn) = a.negOnePow \u2022 (\u03b4 n' m' \u03b3).rightUnshift m hm'", "lemma mem_iff (hnm : n + 1 = m) (z : Cochain F G n) :\n    z \u2208 cocycle F G n \u2194 \u03b4 n m z = 0", "lemma \u03b4_neg_one_cochain (z : Cochain F G (-1)) :\n    \u03b4 (-1) 0 z = Cochain.ofHom (Homotopy.nullHomotopicMap'\n      (fun i j hij => z.v i j (by dsimp at hij; rw [\u2190 hij, add_neg_cancel_right])))", "lemma shiftIso_hom_app (n a a' : \u2124) (ha' : n + a = a') (K : CochainComplex C \u2124):\n    (shiftIso C n a a' ha').hom.app K =\n      ShortComplex.homologyMap ((shiftShortComplexFunctorIso C n a a' ha').hom.app K)", "@[reassoc]\nlemma liftCycles_shift_homology\u03c0\n    (K : CochainComplex C \u2124) {A : C} {n i : \u2124} (f : A \u27f6 (K\u27e6n\u27e7).X i) (j : \u2124)\n    (hj : (up \u2124).next i = j) (hf : f \u226b (K\u27e6n\u27e7).d i j = 0) (i' : \u2124) (hi' : n + i = i') (j' : \u2124)\n    (hj' : (up \u2124).next i' = j') :\n    (K\u27e6n\u27e7).liftCycles f j hj hf \u226b (K\u27e6n\u27e7).homology\u03c0 i =\n      K.liftCycles (f \u226b (K.shiftFunctorObjXIso n i i' (by omega)).hom) j' hj' (by\n        simp only [next] at hj hj'\n        obtain rfl : i' = i + n", "lemma mapBifunctorShift\u2081Iso_trans_mapBifunctorShift\u2082Iso :\n    mapBifunctorShift\u2081Iso K\u2081 (K\u2082\u27e6y\u27e7) F x \u226a\u226b\n      (CategoryTheory.shiftFunctor _ x).mapIso (mapBifunctorShift\u2082Iso K\u2081 K\u2082 F y) =\n      (x * y).negOnePow \u2022 (mapBifunctorShift\u2082Iso (K\u2081\u27e6x\u27e7) K\u2082 F y \u226a\u226b\n        (CategoryTheory.shiftFunctor _ y).mapIso (mapBifunctorShift\u2081Iso K\u2081 K\u2082 F x) \u226a\u226b\n          (shiftFunctorComm (CochainComplex D \u2124) x y).app _)", "@[reassoc]\nlemma d_snd_v (i j : \u2124) (hij : i + 1 = j) :\n    (mappingCone \u03c6).d i j \u226b (snd \u03c6).v j j (add_zero _) =\n      (fst \u03c6).1.v i j hij \u226b \u03c6.f j + (snd \u03c6).v i i (add_zero i) \u226b G.d i j", "@[simp]\nlemma liftCochain_descCochain :\n    (liftCochain \u03c6 \u03b1 \u03b2 h).comp (descCochain \u03c6 \u03b1' \u03b2' h') hp =\n      \u03b1.comp \u03b1' (by omega) + \u03b2.comp \u03b2' (by omega)", "@[simp]\nlemma liftCochain_snd :\n    (liftCochain \u03c6 \u03b1 \u03b2 h).comp (snd \u03c6) (add_zero n) = \u03b2", "lemma map_id : map \u03c6 \u03c6 (\ud835\udfd9 _) (\ud835\udfd9 _) (by rw [id_comp, comp_id]) = \ud835\udfd9 _", "@[simp]\nlemma ofHom_desc :\n    Cochain.ofHom (desc \u03c6 \u03b1 \u03b2 eq) = descCochain \u03c6 \u03b1 (Cochain.ofHom \u03b2) (neg_add_self 1)", "lemma quasiIso_descShortComplex : QuasiIso (descShortComplex S) where\n  quasiIsoAt n", "@[reassoc]\nlemma triangleMapOfHomotopy_comm\u2083 :\n    mapOfHomotopy H \u226b (triangle \u03c6\u2082).mor\u2083 = (triangle \u03c6\u2081).mor\u2083 \u226b a\u27e61\u27e7'", "@[reassoc]\nlemma mappingConeCompHomotopyEquiv_comm\u2081 :\n    inr (map f (f \u226b g) (\ud835\udfd9 X\u2081) g (by rw [id_comp])) \u226b\n      (mappingConeCompHomotopyEquiv f g).inv = (mappingConeCompTriangle f g).mor\u2082", "@[reassoc (attr := simp)]\nlemma mappingConeHomOfDegreewiseSplitIso_inv_comp_triangle_mor\u2083 :\n    (mappingConeHomOfDegreewiseSplitIso S \u03c3).inv \u226b\n      (mappingCone.triangle (homOfDegreewiseSplit S \u03c3)).mor\u2083 = -S.g\u27e6(1 : \u2124)\u27e7'", "@[simp]\ntheorem mk_X_2 : (mk X\u2080 X\u2081 X\u2082 d\u2080 d\u2081 s succ).X 2 = X\u2082", "@[simp]\ntheorem prev (\u03b1 : Type*) [AddGroup \u03b1] [One \u03b1] (i : \u03b1) : (ComplexShape.up \u03b1).prev i = i - 1", "lemma shiftFunctorAdd'_inv_app_f' (K : CochainComplex C \u2124) (a b ab : \u2124) (h : a + b = ab) (n : \u2124) :\n    ((CategoryTheory.shiftFunctorAdd' (CochainComplex C \u2124) a b ab h).inv.app K).f n =\n      (K.XIsoOfEq (by dsimp; rw [\u2190 h, add_assoc, add_comm a])).hom", "@[simp]\ntheorem coe_id : \u21d1(CocompactMap.id \u03b1) = id", "theorem isCompact_preimage_of_isClosed (f : CocompactMap \u03b1 \u03b2)\n    \u2983s : Set \u03b2\u2984 (hs : IsCompact s) (h's : IsClosed s) :\n    IsCompact (f \u207b\u00b9' s)", "theorem Codisjoint.map (h : Codisjoint a b) : Codisjoint (f a) (f b)", "lemma Codisjoint.ne_bot_of_ne_top' (h : Codisjoint a b) (hb : b \u2260 \u22a4) : a \u2260 \u22a5", "theorem Codisjoint_comm : Codisjoint a b \u2194 Codisjoint b a", "theorem isOpen_iff {s : Set (CofiniteTopology X)} : IsOpen s \u2194 s.Nonempty \u2192 s\u1d9c.Finite", "theorem isOpen_iff' {s : Set (CofiniteTopology X)} : IsOpen s \u2194 s = \u2205 \u2228 s\u1d9c.Finite", "@[simp]\ntheorem cancel_left (hg : Injective g) : g.comp f\u2081 = g.comp f\u2082 \u2194 f\u2081 = f\u2082", "@[simp]\ntheorem cancel_right (hf : Surjective f) : g\u2081.comp f = g\u2082.comp f \u2194 g\u2081 = g\u2082", "theorem normalize_eq_one {a : G\u2080} (h0 : a \u2260 0) : normalize a = 1", "@[simp]\nlemma comp' {A\u2081 A\u2082 A\u2083 : CommMon_ C} (f : A\u2081 \u27f6 A\u2082) (g : A\u2082 \u27f6 A\u2083) :\n    ((f \u226b g : A\u2081 \u27f6 A\u2083) : A\u2081.toMon_ \u27f6 A\u2083.toMon_) = @CategoryStruct.comp (Mon_ C) _ _ _ _ f g := rfl", "@[simp]\ntheorem quot_neg (x : Prequotient F) :\n    (by exact Quot.mk Setoid.r (neg x) : ColimitType F) = -(by exact Quot.mk Setoid.r x)", "@[simp] theorem coe_ringHom_id {X : CommSemiRingCat} :\n    @DFunLike.coe (X \u2192+* X) X (fun _ \u21a6 X) _ (\ud835\udfd9 X) = id", "@[simp]\ntheorem antipode_eq_id : antipode (R := R) (A := R) = .id", "theorem comm {H K : Subgroup G} : Commensurable H K \u2194 Commensurable K H", "@[simp]\ntheorem add_right [Distrib R] {a b c : R} : Commute a b \u2192 Commute a c \u2192 Commute a (b + c)", "@[to_additive]\ntheorem Commute.function_commute_mul_left (h : Commute a b) :\n    Function.Commute (a * \u00b7) (b * \u00b7)", "protected theorem Commute.geom_sum\u2082_comm {\u03b1 : Type u} [Semiring \u03b1] {x y : \u03b1} (n : \u2115)\n    (h : Commute x y) :\n    \u2211 i \u2208 range n, x ^ i * y ^ (n - 1 - i) = \u2211 i \u2208 range n, y ^ i * x ^ (n - 1 - i)", "protected theorem Commute.geom_sum\u2082_mul_add {x y : \u03b1} (h : Commute x y) (n : \u2115) :\n    (\u2211 i \u2208 range n, (x + y) ^ i * y ^ (n - 1 - i)) * x + y ^ n = (x + y) ^ n", "@[to_additive]\nprotected theorem inv (hab : Commute a b) : (a * b)\u207b\u00b9 = a\u207b\u00b9 * b\u207b\u00b9", "protected theorem Commute.inv_add_inv (hab : Commute a b) (ha : a \u2260 0) (hb : b \u2260 0) :\n    a\u207b\u00b9 + b\u207b\u00b9 = (a + b) / (a * b)", "@[to_additive]\nlemma inv_mul_cancel_assoc (h : Commute a b) : a\u207b\u00b9 * (b * a) = b", "theorem isNilpotent_add (hx : IsNilpotent x) (hy : IsNilpotent y) : IsNilpotent (x + y)", "theorem mul_self_eq_mul_self_iff [NonUnitalNonAssocRing R] [NoZeroDivisors R] {a b : R}\n    (h : Commute a b) : a * a = b * b \u2194 a = b \u2228 a = -b", "theorem mul_self_sub_mul_self_eq [NonUnitalNonAssocRing R] {a b : R} (h : Commute a b) :\n    a * a - b * b = (a + b) * (a - b)", "theorem multiset_sum_right (a : \u03b1) (h : \u2200 b \u2208 s, Commute a b) : Commute a s.sum", "@[simp] lemma natCast_mul_natCast_mul (h : Commute a b) (m n : \u2115) : Commute (m * a) (n * b)", "theorem neg_left : Commute a b \u2192 Commute (-a) b", "lemma pow_dvd_add_pow_of_pow_eq_zero_left (hx : x ^ n = 0) :\n    y ^ m \u2223 (x + y) ^ p", "@[to_additive]\ntheorem pow_self (a : M) (n : \u2115) : Commute (a ^ n) a", "@[to_additive]\nlemma Commute.smul_right [Mul \u03b1] [SMulCommClass M \u03b1 \u03b1] [IsScalarTower M \u03b1 \u03b1] {a b : \u03b1}\n    (h : Commute a b) (r : M) : Commute a (r \u2022 b)", "@[to_additive]\nnonrec theorem _root_.Commute.unop [Mul \u03b1] {x y : \u03b1\u1d50\u1d52\u1d56} (h : Commute x y) :\n    Commute (unop x) (unop y)", "@[simp]\ntheorem zero_left [MulZeroClass G\u2080] (a : G\u2080) : Commute 0 a", "theorem tensorObj_comul (A B : Comon_ C) :\n    (A \u2297 B).comul = (A.comul \u2297 B.comul) \u226b tensor_\u03bc C (A.X, A.X) (B.X, B.X)", "theorem tensorObj_counit (A B : Comon_ C) : (A \u2297 B).counit = (A.counit \u2297 B.counit) \u226b (\u03bb_ _).hom", "@[functor_norm]\ntheorem Comp.seq_mk {\u03b1 \u03b2 : Type w} {f : Type u \u2192 Type v} {g : Type w \u2192 Type u} [Applicative f]\n    [Applicative g] (h : f (g (\u03b1 \u2192 \u03b2))) (x : f (g \u03b1)) :\n    Comp.mk h <*> Comp.mk x = Comp.mk ((\u00b7 <*> \u00b7) <$> h <*> x)", "theorem mono_iff_injective {X Y : CompHaus.{u}} (f : X \u27f6 Y) : Mono f \u2194 Function.Injective f", "theorem epi_of_surjective {X Y : CompHausLike.{u} P} (f : X \u27f6 Y) (hf : Function.Surjective f) :\n    Epi f", "theorem comp {M N P : Type*}\n    {\u03c6 : M \u2192 N} {\u03c8 : N \u2192 P} :\n    CompTriple \u03c6 \u03c8  (\u03c8.comp \u03c6) where", "theorem _root_.CompactExhaustion.hasBasis_compactConvergenceUniformity {\u03b9 : Type*}\n    {p : \u03b9 \u2192 Prop} {V : \u03b9 \u2192 Set (\u03b2 \u00d7 \u03b2)} (K : CompactExhaustion \u03b1) (hb : (\ud835\udce4 \u03b2).HasBasis p V) :\n    HasBasis (\ud835\udce4 C(\u03b1, \u03b2)) (fun i : \u2115 \u00d7 \u03b9 \u21a6 p i.2) fun i \u21a6\n      {fg | \u2200 x \u2208 K i.1, (fg.1 x, fg.2 x) \u2208 V i.2}", "theorem add_apply [AddZeroClass \u03b2] [ContinuousAdd \u03b2] (f g : C_c(\u03b1, \u03b2)) : (f + g) x = f x + g x", "@[simp]\ntheorem coe_smulc [Zero \u03b2] [TopologicalSpace \u03b3] [SMulZeroClass \u03b3 \u03b2] [ContinuousSMul \u03b3 \u03b2]\n    {F : Type*} [FunLike F \u03b1 \u03b3] [ContinuousMapClass F \u03b1 \u03b3] (f : F) (g : C_c(\u03b1, \u03b2)) :\n    \u21d1(f \u2022 g) = fun x => f x \u2022 g x", "@[simp]\ntheorem comp_id (f : C_c(\u03b3, \u03b4)) : f.comp (CocompactMap.id \u03b3) = f", "private theorem cl_cl {X : Compactum} (A : Set X) : cl (cl A) \u2286 cl A", "theorem coe_injective : Injective ((\u2191) : Complementeds \u03b1 \u2192 \u03b1)", "@[simp, norm_cast]\ntheorem coe_top : ((\u22a4 : Complementeds \u03b1) : \u03b1) = \u22a4", "@[simp, norm_cast]\ntheorem disjoint_coe : Disjoint (a : \u03b1) b \u2194 Disjoint a b", "theorem Independent.dfinsupp_sumAddHom_injective {p : \u03b9 \u2192 AddSubgroup N} (h : Independent p) :\n    Function.Injective (sumAddHom fun i => (p i).subtype)", "theorem Independent.setIndependent_range (ht : Independent t) : SetIndependent <| range t", "theorem CompleteLattice.Independent.subtype_ne_bot_le_rank [Nontrivial R]\n    {V : \u03b9 \u2192 Submodule R M} (hV : CompleteLattice.Independent V) :\n    Cardinal.lift.{v} #{ i : \u03b9 // V i \u2260 \u22a5 } \u2264 Cardinal.lift.{w} (Module.rank R M)", "theorem CompleteLattice.Independent.supIndep' [CompleteLattice \u03b1] {f : \u03b9 \u2192 \u03b1} (s : Finset \u03b9)\n    (h : CompleteLattice.Independent f) : s.SupIndep f", "theorem IsCompactElement.exists_finset_of_le_iSup {k : \u03b1} (hk : IsCompactElement k) {\u03b9 : Type*}\n    (f : \u03b9 \u2192 \u03b1) (h : k \u2264 \u2a06 i, f i) : \u2203 s : Finset \u03b9, k \u2264 \u2a06 i \u2208 s, f i", "theorem WellFounded.finite_of_setIndependent (h : WellFounded ((\u00b7 > \u00b7) : \u03b1 \u2192 \u03b1 \u2192 Prop)) {s : Set \u03b1}\n    (hs : SetIndependent s) : s.Finite", "theorem independent_iff_dfinsupp_sumAddHom_injective (p : \u03b9 \u2192 AddSubgroup N) :\n    Independent p \u2194 Function.Injective (sumAddHom fun i => (p i).subtype)", "theorem independent_iff_pairwiseDisjoint {f : \u03b9 \u2192 \u03b1} : Independent f \u2194 Pairwise (Disjoint on f)", "@[simp]\ntheorem independent_ne_bot_iff_independent :\n    Independent (fun i : {i // t i \u2260 \u22a5} \u21a6 t i) \u2194 Independent t", "@[simp]\ntheorem independent_pempty (t : PEmpty \u2192 \u03b1) : Independent t", "theorem sSup_continuous' (s : Set (\u03b1 \u2192 \u03b2)) (hc : \u2200 f \u2208 s, Continuous' f) :\n    Continuous' (sSup s)", "theorem wellFounded_characterisations : List.TFAE\n    [WellFounded ((\u00b7 > \u00b7) : \u03b1 \u2192 \u03b1 \u2192 Prop),\n      IsSupFiniteCompact \u03b1, IsSupClosedCompact \u03b1, \u2200 k : \u03b1, IsCompactElement k]", "theorem wellFounded_iff_isSupFiniteCompact :\n    WellFounded ((\u00b7 > \u00b7) : \u03b1 \u2192 \u03b1 \u2192 Prop) \u2194 IsSupFiniteCompact \u03b1", "@[simp]\ntheorem cancel_right {g\u2081 g\u2082 : CompleteLatticeHom \u03b2 \u03b3} {f : CompleteLatticeHom \u03b1 \u03b2}\n    (hf : Surjective f) : g\u2081.comp f = g\u2082.comp f \u2194 g\u2081 = g\u2082", "@[simp] lemma coe_toBoundedLatticeHom (f : CompleteLatticeHom \u03b1 \u03b2) : \u21d1f.toBoundedLatticeHom = f", "theorem Gamma_nat_eq_factorial (n : \u2115) : Gamma (n + 1) = n !", "theorem Gamma_ne_zero {s : \u2102} (hs : \u2200 m : \u2115, s \u2260 -m) : Gamma s \u2260 0", "@[simp]\ntheorem Gamma_zero : Gamma 0 = 0", "@[simp]\nlemma Gamma\u2102_one : Gamma\u2102 1 = 1 / \u03c0", "lemma Gamma\u211d_ne_zero_of_re_pos {s : \u2102} (hs : 0 < re s) : Gamma\u211d s \u2260 0", "lemma eventuallyle (z : \u2102) (hB : BddAbove ((norm \u2218 f) '' (verticalClosedStrip 0 1)))\n    (hd : DiffContOnCl \u2102 f (verticalStrip 0 1)) (hz : z \u2208 verticalStrip 0 1) :\n    (fun _ : \u211d \u21a6 \u2016f z\u2016) \u2264\u1da0[\ud835\udcdd[>] 0]\n    (fun \u03b5 \u21a6 \u2016((\u03b5 + sSupNormIm f 0) ^ (1 - z) * (\u03b5 + sSupNormIm f 1) ^ z : \u2102)\u2016)", "lemma norm_le_interpStrip_of_mem_verticalClosedStrip (f : \u2102 \u2192 E) {z : \u2102}\n    (hz : z \u2208 verticalClosedStrip 0 1) (hd : DiffContOnCl \u2102 f (verticalStrip 0 1))\n    (hB : BddAbove ((norm \u2218 f) '' (verticalClosedStrip 0 1))) :\n    \u2016f z\u2016 \u2264 \u2016interpStrip f z\u2016", "lemma norm_le_interp_of_mem_verticalClosedStrip' (f : \u2102 \u2192 E) {z : \u2102} {a b : \u211d}\n    (hz : z \u2208 verticalClosedStrip 0 1) (hd : DiffContOnCl \u2102 f (verticalStrip 0 1))\n    (hB : BddAbove ((norm \u2218 f) '' (verticalClosedStrip 0 1)))\n    (ha : \u2200 z \u2208 re \u207b\u00b9' {0}, \u2016f z\u2016 \u2264 a) (hb : \u2200 z \u2208 re \u207b\u00b9' {1}, \u2016f z\u2016 \u2264 b) :\n    \u2016f z\u2016 \u2264 a ^ (1 - z.re) * b ^ z.re", "lemma norm_le_sSupNormIm (f : \u2102 \u2192 E) (z : \u2102) (hD : z \u2208 verticalClosedStrip 0 1)\n    (hB : BddAbove ((norm \u2218 f) '' (verticalClosedStrip 0 1))) :\n    \u2016f z\u2016 \u2264 sSupNormIm f (z.re)", "lemma sSupNormIm_eps_pos {\u03b5 : \u211d} (h\u03b5 : \u03b5 > 0) (x : \u211d) : 0 < \u03b5 + sSupNormIm f x", "lemma sSupNormIm_nonneg (x : \u211d) : 0 \u2264 sSupNormIm f x", "private lemma HasDerivAt.complex_of_real {f : \u2102 \u2192 \u2102} {g : \u211d \u2192 \u211d} {g' s : \u211d}\n    (hf : DifferentiableAt \u2102 f s) (hg : HasDerivAt g g' s) (hfg : \u2200 s : \u211d, f \u2191s = \u2191(g s)) :\n    HasDerivAt f \u2191g' s", "@[simp]\ntheorem I_re : I.re = 0", "theorem isLittleO_log_re_re (hl : IsExpCmpFilter l) : (fun z => Real.log z.re) =o[l] re", "theorem abs_ne_one (z : \ud835\udd3b) : abs (z : \u2102) \u2260 1", "@[simp]\ntheorem conj_zero : conj 0 = 0", "lemma abs_eq_sqrt_sq_add_sq (z : \u2102) : abs z = (z.re ^ 2 + z.im ^ 2).sqrt", "theorem abs_exp (z : \u2102) : abs (exp z) = Real.exp z.re", "@[simp]\nlemma abs_im_eq_abs {z : \u2102} : |z.im| = abs z \u2194 z.re = 0", "@[simp] lemma angle_mul_right (ha : a \u2260 0) (x y : \u2102) : angle (x * a) (y * a) = angle x y", "@[simp]\ntheorem arg_coe_angle_toReal_eq_arg (z : \u2102) : (arg z : Real.Angle).toReal = arg z", "theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) :\n    arg =\u1da0[\ud835\udcdd x] fun x => Real.arcsin ((-x).im / abs x) + \u03c0", "@[simp]\ntheorem arg_inv_coe_angle (x : \u2102) : (arg x\u207b\u00b9 : Real.Angle) = -arg x", "theorem arg_lt_pi_iff {z : \u2102} : arg z < \u03c0 \u2194 0 \u2264 z.re \u2228 z.im \u2260 0", "theorem arg_neg_eq_arg_add_pi_of_im_neg {x : \u2102} (hi : x.im < 0) : arg (-x) = arg x + \u03c0", "theorem arg_ofReal_of_nonneg {x : \u211d} (hx : 0 \u2264 x) : arg x = 0", "theorem betaIntegral_eval_one_right {u : \u2102} (hu : 0 < re u) : betaIntegral u 1 = 1 / u", "theorem circleIntegral_div_sub_of_differentiable_on_off_countable {R : \u211d} {c w : \u2102} {s : Set \u2102}\n    (hs : s.Countable) (hw : w \u2208 ball c R) {f : \u2102 \u2192 \u2102} (hc : ContinuousOn f (closedBall c R))\n    (hd : \u2200 z \u2208 ball c R \\ s, DifferentiableAt \u2102 f z) :\n    (\u222e z in C(c, R), f z / (z - w)) = 2 * \u03c0 * I * f w", "theorem circleIntegral_eq_zero_of_differentiable_on_off_countable {R : \u211d} (h0 : 0 \u2264 R) {f : \u2102 \u2192 E}\n    {c : \u2102} {s : Set \u2102} (hs : s.Countable) (hc : ContinuousOn f (closedBall c R))\n    (hd : \u2200 z \u2208 ball c R \\ s, DifferentiableAt \u2102 f z) : (\u222e z in C(c, R), f z) = 0", "theorem circleIntegral_sub_inv_smul_of_differentiable_on_off_countable {R : \u211d} {c w : \u2102} {f : \u2102 \u2192 E}\n    {s : Set \u2102} (hs : s.Countable) (hw : w \u2208 ball c R) (hc : ContinuousOn f (closedBall c R))\n    (hd : \u2200 x \u2208 ball c R \\ s, DifferentiableAt \u2102 f x) :\n    (\u222e z in C(c, R), (z - w)\u207b\u00b9 \u2022 f z) = (2 * \u03c0 * I : \u2102) \u2022 f w", "@[simp]\ntheorem closure_setOf_lt_re (a : \u211d) : closure { z : \u2102 | a < z.re } = { z | a \u2264 z.re }", "protected lemma compl_Iic_zero : (Set.Iic 0)\u1d9c = slitPlane := Set.ext fun _ \u21a6\n  mem_slitPlane_iff_not_le_zero.symm", "theorem contDiff_sin {n} : ContDiff \u2102 n sin", "@[continuity, fun_prop]\ntheorem continuous_re : Continuous re", "lemma convexHull_reProdIm (s t : Set \u211d) :\n    convexHull \u211d (s \u00d7\u2102 t) = convexHull \u211d s \u00d7\u2102 convexHull \u211d t", "theorem cos_eq_cos_iff {x y : \u2102} : cos x = cos y \u2194 \u2203 k : \u2124, y = 2 * k * \u03c0 + x \u2228 y = 2 * k * \u03c0 - x", "theorem cos_eq_neg_one_iff {x : \u2102} : cos x = -1 \u2194 \u2203 k : \u2124, \u03c0 + k * (2 * \u03c0) = x", "theorem Complex.cos_eq_tsum' (z : \u2102) :\n    Complex.cos z = \u2211' n : \u2115, (z * Complex.I) ^ (2 * n) / \u2191(2 * n)!", "theorem cos_nat_mul_two_pi_add_pi (n : \u2115) : cos (n * (2 * \u03c0) + \u03c0) = -1", "theorem cos_ne_zero_iff {\u03b8 : \u2102} : cos \u03b8 \u2260 0 \u2194 \u2200 k : \u2124, \u03b8 \u2260 (2 * k + 1) * \u03c0 / 2", "theorem cos_sub : cos (x - y) = cos x * cos y + sin x * sin y", "@[simp]\ntheorem cos_zero : cos 0 = 1", "@[simp]\ntheorem cosh_sub_sinh : cosh x - sinh x = exp (-x)", "lemma cpow_mul_nat (x y : \u2102) (n : \u2115) : x ^ (y * n) = (x ^ y) ^ n", "lemma cpow_nat_mul' {x : \u2102} {n : \u2115} (hlt : -\u03c0 < n * x.arg) (hle : n * x.arg \u2264 \u03c0) (y : \u2102) :\n    x ^ (n * y) = (x ^ n) ^ y", "theorem cpow_neg_one (x : \u2102) : x ^ (-1 : \u2102) = x\u207b\u00b9", "lemma cpow_ofNat_mul (x : \u2102) (n : \u2115) [n.AtLeastTwo] (y : \u2102) :\n    x ^ (no_index (OfNat.ofNat n) * y) = (x ^ y) ^ (OfNat.ofNat n : \u2115)", "theorem differentiableOn_update_limUnder_of_isLittleO {f : \u2102 \u2192 E} {s : Set \u2102} {c : \u2102} (hc : s \u2208 \ud835\udcdd c)\n    (hd : DifferentiableOn \u2102 f (s \\ {c}))\n    (ho : (fun z => f z - f c) =o[\ud835\udcdd[\u2260] c] fun z => (z - c)\u207b\u00b9) :\n    DifferentiableOn \u2102 (update f c (limUnder (\ud835\udcdd[\u2260] c) f)) s", "theorem dist_conj_self (z : \u2102) : dist (conj z) z = 2 * |z.im|", "theorem dist_of_im_eq {z w : \u2102} (h : z.im = w.im) : dist z w = dist z.re w.re", "@[simp] lemma div_ratCast_re (z : \u2102) (x : \u211a) : (z / x).re = z.re / x", "lemma div_re (z w : \u2102) : (z / w).re = z.re * w.re / normSq w + z.im * w.im / normSq w", "theorem edist_of_im_eq {z w : \u2102} (h : z.im = w.im) : edist z w = edist z.re w.re", "theorem eventually_eq_or_eq_zero_of_isLocalMin_norm {f : E \u2192 \u2102} {c : E}\n    (hf : \u2200\u1da0 z in \ud835\udcdd c, DifferentiableAt \u2102 f z) (hc : IsLocalMin (norm \u2218 f) c) :\n    (\u2200\u1da0 z in \ud835\udcdd c, f z = f c) \u2228 f c = 0", "@[simp]\ntheorem exp_add_pi_mul_I (z : \u2102) : exp (z + \u03c0 * I) = -exp z", "theorem ext_abs_arg {x y : \u2102} (h\u2081 : abs x = abs y) (h\u2082 : x.arg = y.arg) : x = y", "@[simp]\ntheorem frontier_setOf_re_lt (a : \u211d) : frontier { z : \u2102 | z.re < a } = { z | z.re = a }", "lemma hasDerivAt_Gamma_one_half : HasDerivAt Gamma (-\u221a\u03c0 * (\u03b3 + 2 * log 2)) (1 / 2)", "theorem hasDerivAt_sinh (x : \u2102) : HasDerivAt sinh (cosh x) x", "theorem hasStrictDerivAt_const_cpow {x y : \u2102} (h : x \u2260 0 \u2228 y \u2260 0) :\n    HasStrictDerivAt (fun y => x ^ y) (x ^ y * log x) y", "theorem im_ofReal_mul (r : \u211d) (z : \u2102) : (r * z).im = r * z.im", "theorem im_tsum {f : \u03b1 \u2192 \u2102} (h : Summable f) : (\u2211' a, f a).im = \u2211' a, (f a).im", "@[simp]\nprotected theorem Complex.inner (w z : \u2102) : \u27eaw, z\u27eb_\u211d = (conj w * z).re", "theorem integral_boundary_rect_eq_zero_of_differentiable_on_off_countable (f : \u2102 \u2192 E) (z w : \u2102)\n    (s : Set \u2102) (hs : s.Countable) (Hc : ContinuousOn f ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]))\n    (Hd : \u2200 x \u2208 Ioo (min z.re w.re) (max z.re w.re) \u00d7\u2102 Ioo (min z.im w.im) (max z.im w.im) \\ s,\n      DifferentiableAt \u2102 f x) :\n    (\u222b x : \u211d in z.re..w.re, f (x + z.im * I)) - (\u222b x : \u211d in z.re..w.re, f (x + w.im * I)) +\n      I \u2022 (\u222b y : \u211d in z.im..w.im, f (re w + y * I)) -\n      I \u2022 (\u222b y : \u211d in z.im..w.im, f (re z + y * I)) = 0", "theorem isCauSeq_abs_exp (z : \u2102) :\n    IsCauSeq _root_.abs fun n => \u2211 m \u2208 range n, abs (z ^ m / m.factorial)", "theorem isCauSeq_exp (z : \u2102) : IsCauSeq abs fun n => \u2211 m \u2208 range n, z ^ m / m.factorial", "theorem isOpen_setOf_mem_nhds_and_isMaxOn_norm {f : E \u2192 F} {s : Set E}\n    (hd : DifferentiableOn \u2102 f s) : IsOpen {z | s \u2208 \ud835\udcdd z \u2227 IsMaxOn (norm \u2218 f) s z}", "theorem Complex.isometryOfOrthonormal_apply (v : OrthonormalBasis (Fin 2) \u211d F) (z : \u2102) :\n    Complex.isometryOfOrthonormal v z = z.re \u2022 v 0 + z.im \u2022 v 1", "@[simp]\ntheorem linearEquiv_det_conjAe : LinearEquiv.det conjAe.toLinearEquiv = -1", "theorem log_exp {x : \u2102} (hx\u2081 : -\u03c0 < x.im) (hx\u2082 : x.im \u2264 \u03c0) : log (exp x) = x", "theorem log_re (x : \u2102) : x.log.re = x.abs.log", "theorem lt_def {z w : \u2102} : z < w \u2194 z.re < w.re \u2227 z.im = w.im", "@[measurability]\ntheorem measurable_ofReal : Measurable ((\u2191) : \u211d \u2192 \u2102)", "@[measurability]\ntheorem measurable_re : Measurable re", "theorem mem_reProdIm {z : \u2102} {s t : Set \u211d} : z \u2208 s \u00d7\u2102 t \u2194 z.re \u2208 s \u2227 z.im \u2208 t", "lemma mem_slitPlane_iff {z : \u2102} : z \u2208 slitPlane \u2194 0 < z.re \u2228 z.im \u2260 0 := Set.mem_setOf", "lemma mul_angle_le_norm_sub (hx : \u2016x\u2016 = 1) (hy : \u2016y\u2016 = 1) : 2 / \u03c0 * angle x y \u2264 \u2016x - y\u2016", "theorem mul_conj (z : \u2102) : z * conj z = normSq z", "@[simp]\ntheorem neg_im (z : \u2102) : (-z).im = -z.im", "theorem neg_pi_lt_arg (x : \u2102) : -\u03c0 < arg x", "@[simp]\nlemma neg_re_eq_abs {z : \u2102} : -z.re = abs z \u2194 z \u2264 0", "lemma nhdsWithin_stolzCone_le_nhdsWithin_stolzSet {s : \u211d} (hs : 0 < s) :\n    \u2203 M, \ud835\udcdd[stolzCone s] 1 \u2264 \ud835\udcdd[stolzSet M] 1", "theorem nndist_of_re_eq {z w : \u2102} (h : z.re = w.re) : nndist z w = nndist z.im w.im", "theorem norm_cderiv_le (hr : 0 < r) (hf : \u2200 w \u2208 sphere z r, \u2016f w\u2016 \u2264 M) :\n    \u2016cderiv r f z\u2016 \u2264 M / r", "theorem norm_eqOn_closedBall_of_isMaxOn {f : E \u2192 F} {z : E} {r : \u211d}\n    (hd : DiffContOnCl \u2102 f (ball z r)) (hz : IsMaxOn (norm \u2218 f) (ball z r) z) :\n    EqOn (norm \u2218 f) (const E \u2016f z\u2016) (closedBall z r)", "lemma norm_log_natCast_le_rpow_div (n : \u2115) {\u03b5 : \u211d} (h\u03b5 : 0 < \u03b5) : \u2016log n\u2016 \u2264 n ^ \u03b5 / \u03b5", "lemma norm_log_sub_logTaylor_le (n : \u2115) {z : \u2102} (hz : \u2016z\u2016 < 1) :\n    \u2016log (1 + z) - logTaylor (n + 1) z\u2016 \u2264 \u2016z\u2016 ^ (n + 1) * (1 - \u2016z\u2016)\u207b\u00b9 / (n + 1)", "theorem norm_max_aux\u2083 {f : \u2102 \u2192 F} {z w : \u2102} {r : \u211d} (hr : dist w z = r)\n    (hd : DiffContOnCl \u2102 f (ball z r)) (hz : IsMaxOn (norm \u2218 f) (ball z r) z) : \u2016f w\u2016 = \u2016f z\u2016", "lemma norm_natCast_cpow_of_pos {n : \u2115} (hn : 0 < n) (s : \u2102) :\n    \u2016(n : \u2102) ^ s\u2016 = (n : \u211d) ^ (s.re)", "@[simp]\ntheorem ofRealCLM_apply (x : \u211d) : ofRealCLM x = x", "@[simp, norm_cast]\ntheorem ofReal_arctan (x : \u211d) : (Real.arctan x : \u2102) = arctan x", "@[simp]\ntheorem ofReal_exp_ofReal_re (x : \u211d) : ((exp x).re : \u2102) = exp x", "@[simp, norm_cast]\ntheorem ofReal_sin (x : \u211d) : (Real.sin x : \u2102) = sin x", "@[simp, norm_cast]\ntheorem ofReal_zpow (r : \u211d) (n : \u2124) : ((r ^ n : \u211d) : \u2102) = (r : \u2102) ^ n", "@[simp] lemma one_mem_slitPlane : 1 \u2208 slitPlane := ofReal_mem_slitPlane.2 one_pos", "theorem quotientMap_im : QuotientMap im", "@[simp]\ntheorem range_exp_mul_I : (Set.range fun x : \u211d => exp (x * I)) = Metric.sphere 0 1", "lemma re_neg_ne_zero_of_one_lt_re {s : \u2102} (hs : 1 < s.re) : (-s).re \u2260 0", "theorem restrictScalars_one_smulRight' (x : E) :\n    ContinuousLinearMap.restrictScalars \u211d ((1 : \u2102 \u2192L[\u2102] \u2102).smulRight x : \u2102 \u2192L[\u2102] E) =\n      reCLM.smulRight x + I \u2022 imCLM.smulRight x", "theorem sin_sub_pi (x : \u2102) : sin (x - \u03c0) = -sin x", "theorem sin_sub_two_pi (x : \u2102) : sin (x - 2 * \u03c0) = sin x", "theorem sin_surjective : Function.Surjective sin", "@[simp]\ntheorem sinh_zero : sinh 0 = 0", "lemma slitPlane_ne_zero {z : \u2102} (hz : z \u2208 slitPlane) : z \u2260 0", "@[simp]\ntheorem sub_im (z w : \u2102) : (z - w).im = z.im - w.im", "theorem sum_div_factorial_le {\u03b1 : Type*} [LinearOrderedField \u03b1] (n j : \u2115) (hn : 0 < n) :\n    (\u2211 m \u2208 filter (fun k => n \u2264 k) (range j),\n      (1 / m.factorial : \u03b1)) \u2264 n.succ / (n.factorial * n)", "theorem tan_conj : tan (conj x) = conj (tan x)", "theorem tendstoUniformlyOn_deriv_of_cthickening_subset (hf : TendstoLocallyUniformlyOn F f \u03c6 U)\n    (hF : \u2200\u1da0 n in \u03c6, DifferentiableOn \u2102 (F n) U) {\u03b4 : \u211d} (h\u03b4 : 0 < \u03b4) (hK : IsCompact K)\n    (hU : IsOpen U) (hKU : cthickening \u03b4 K \u2286 U) :\n    TendstoUniformlyOn (deriv \u2218 F) (cderiv \u03b4 f) \u03c6 K", "theorem tendsto_limUnder_of_differentiable_on_punctured_nhds_of_isLittleO {f : \u2102 \u2192 E} {c : \u2102}\n    (hd : \u2200\u1da0 z in \ud835\udcdd[\u2260] c, DifferentiableAt \u2102 f z)\n    (ho : (fun z => f z - f c) =o[\ud835\udcdd[\u2260] c] fun z => (z - c)\u207b\u00b9) :\n    Tendsto f (\ud835\udcdd[\u2260] c) (\ud835\udcdd <| limUnder (\ud835\udcdd[\u2260] c) f)", "theorem Complex.uniformContinuous_ringHom_eq_id_or_conj (K : Subfield \u2102) {\u03c8 : K \u2192+* \u2102}\n    (hc : UniformContinuous \u03c8) : \u03c8.toFun = K.subtype \u2228 \u03c8.toFun = conj \u2218 K.subtype", "theorem Complex.volume_sum_rpow_lt_one {p : \u211d} (hp : 1 \u2264 p) :\n    volume {x : \u03b9 \u2192 \u2102 | \u2211 i, \u2016x i\u2016 ^ p < 1} =\n      .ofReal ((\u03c0 * Real.Gamma (2 / p + 1)) ^ card \u03b9 / Real.Gamma (2 * card \u03b9 / p + 1))", "lemma add_rel (r : I) {p q : I} (hpq : c.Rel p q) : c.Rel (r + p) (r + q)", "theorem invEmbedding_comp (i : Fin c.length) (j : Fin (c.blocksFun i)) :\n    (c.invEmbedding (c.embedding i j) : \u2115) = j", "theorem length_gather (a : Composition n) (b : Composition a.length) :\n    length (a.gather b) = b.length", "theorem lt_sizeUpTo_index_succ (j : Fin n) : (j : \u2115) < c.sizeUpTo (c.index j).succ", "theorem ne_single_iff {n : \u2115} (hn : 0 < n) {c : Composition n} :\n    c \u2260 single n hn \u2194 \u2200 i, c.blocksFun i < n", "@[simp]\ntheorem ones_length (n : \u2115) : (ones n).length = n", "theorem mem_eraseLast {s : CompositionSeries X} {x : X} (h : 0 < s.length) :\n    x \u2208 s.eraseLast \u2194 x \u2260 s.last \u2227 x \u2208 s", "protected theorem strictMono (s : CompositionSeries X) : StrictMono s", "theorem toList_sorted (s : CompositionSeries X) : s.toList.Sorted (\u00b7 < \u00b7)", "theorem list_ofFn :\n    \u2200 {n} {f : Fin n \u2192 \u03b1 \u2192 \u03c3},\n      (\u2200 i, Computable (f i)) \u2192 Computable fun a => List.ofFn fun i => f i a", "theorem of_eq {f g : \u03b1 \u2192 \u03c3} (hf : Computable f) (H : \u2200 n, f n = g n) : Computable g", "theorem pred : Computable Nat.pred", "@[trans]\ntheorem Equiv.trans {s t u : Computation \u03b1} : s ~ t \u2192 t ~ u \u2192 s ~ u", "@[simp]\ntheorem LiftRelAux.ret_left (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (C : Computation \u03b1 \u2192 Computation \u03b2 \u2192 Prop) (a cb) :\n    LiftRelAux R C (Sum.inl a) (destruct cb) \u2194 \u2203 b, b \u2208 cb \u2227 R a b", "theorem Results.mem {s : Computation \u03b1} {a n} : Results s a n \u2192 a \u2208 s", "theorem eq_thinkN' (s : Computation \u03b1) [_h : Terminates s] :\n    s = thinkN (pure (get s)) (length s)", "theorem exists_of_liftRel_left {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {ca cb} (H : LiftRel R ca cb) {a} (h : a \u2208 ca) :\n    \u2203 b, b \u2208 cb \u2227 R a b", "theorem get_promises : s ~> get s", "@[simp] lemma liftRelAux_inl_inr {a : \u03b1} {cb} :\n    LiftRelAux R C (Sum.inl a) (Sum.inr cb) = \u2203 b, b \u2208 cb \u2227 R a b", "@[simp]\ntheorem liftRel_think_left (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (ca : Computation \u03b1) (cb : Computation \u03b2) :\n    LiftRel R (think ca) cb \u2194 LiftRel R ca cb", "theorem mem_unique {s : Computation \u03b1} {a b : \u03b1} : a \u2208 s \u2192 b \u2208 s \u2192 a = b", "theorem results_thinkN_pure (a : \u03b1) (n) : Results (thinkN (pure a) n) a n", "@[simp]\ntheorem results_think_iff {s : Computation \u03b1} {a n} : Results (think s) a (n + 1) \u2194 Results s a n", "theorem terminates_parallel {S : WSeq (Computation \u03b1)} {c} (h : c \u2208 S) [T : Terminates c] :\n    Terminates (parallel S)", "theorem thinkN_equiv (s : Computation \u03b1) (n) : thinkN s n ~ s", "@[to_additive addConGen_eq \"The inductively defined smallest additive congruence relation\ncontaining a binary relation `r` equals the infimum of the set of additive congruence relations\ncontaining `r`.\"]\ntheorem conGen_eq (r : M \u2192 M \u2192 Prop) : conGen r = sInf { s : Con M | \u2200 x y, r x y \u2192 s x y }", "@[to_additive (attr := elab_as_elim) \"The inductive principle used to prove propositions about\nthe elements of a quotient by an additive congruence relation.\"]\nprotected theorem induction_on {C : c.Quotient \u2192 Prop} (q : c.Quotient) (H : \u2200 x : M, C x) : C q", "@[to_additive (attr := elab_as_elim) \"A version of `AddCon.induction_on` for predicates which take\ntwo arguments.\"]\nprotected theorem induction_on\u2082 {d : Con N} {C : c.Quotient \u2192 d.Quotient \u2192 Prop} (p : c.Quotient)\n    (q : d.Quotient) (H : \u2200 (x : M) (y : N), C x y) : C p q", "@[to_additive (attr := simp)]\ntheorem rel_mk {s : Setoid M} {h a b} : Con.mk s h a b \u2194 r a b", "@[to_additive \"The infimum of a set of additive congruence relations is the same as the infimum of\nthe set's image under the map to the underlying equivalence relation.\"]\ntheorem sInf_toSetoid (S : Set (Con M)) : (sInf S).toSetoid = sInf (toSetoid '' S)", "theorem ConcaveOn.add (hf : ConcaveOn \ud835\udd5c s f) (hg : ConcaveOn \ud835\udd5c s g) : ConcaveOn \ud835\udd5c s (f + g)", "theorem ConcaveOn.comp_affineMap {f : F \u2192 \u03b2} (g : E \u2192\u1d43[\ud835\udd5c] F) {s : Set F} (hf : ConcaveOn \ud835\udd5c s f) :\n    ConcaveOn \ud835\udd5c (g \u207b\u00b9' s) (f \u2218 g)", "theorem ConcaveOn.convex_ge (hf : ConcaveOn \ud835\udd5c s f) (r : \u03b2) : Convex \ud835\udd5c ({ x \u2208 s | r \u2264 f x })", "theorem ConcaveOn.exists_le_of_centerMass (h : ConcaveOn \ud835\udd5c s f) (hw\u2080 : \u2200 i \u2208 t, 0 \u2264 w i)\n    (hw\u2081 : 0 < \u2211 i \u2208 t, w i) (hp : \u2200 i \u2208 t, p i \u2208 s) : \u2203 i \u2208 t, f (p i) \u2264 f (t.centerMass w p)", "lemma ConcaveOn.slope_anti (hfc : ConcaveOn \ud835\udd5c s f) (hx : x \u2208 s) :\n    AntitoneOn (slope f x) (s \\ {x})", "lemma ConcaveOn.smul' (hf : ConcaveOn \ud835\udd5c s f) (hg : ConcaveOn \ud835\udd5c s g) (hf\u2080 : \u2200 \u2983x\u2984, x \u2208 s \u2192 0 \u2264 f x)\n    (hg\u2080 : \u2200 \u2983x\u2984, x \u2208 s \u2192 0 \u2264 g x) (hfg : AntivaryOn f g s) : ConcaveOn \ud835\udd5c s (f \u2022 g)", "@[simp]\ntheorem bot_fst : (\u22a5 : Concept \u03b1 \u03b2 r).fst = extentClosure r univ", "@[simp]\ntheorem sSup_snd (S : Set (Concept \u03b1 \u03b2 r)) : (sSup S).snd = \u22c2 c \u2208 S, (c : Concept _ _ _).snd", "@[simp]\ntheorem swap_swap (c : Concept \u03b1 \u03b2 r) : c.swap.swap = c", "theorem equalizerCondition_profinite (X : Sheaf (coherentTopology Profinite.{u}) A) :\n    EqualizerCondition X.val", "theorem two_lt_lineCount [Finite P] [Finite L] (p : P) : 2 < lineCount L p", "theorem comp {f : X \u2192 Y} {g : Y \u2192 Z} (hf : Conformal f) (hg : Conformal g) : Conformal (g \u2218 f)", "theorem smul_eq_mulAut_conj (g : ConjAct G) (h : G) : g \u2022 h = MulAut.conj (ofConjAct g) h", "@[simp]\ntheorem toConjAct_inv (x : G) : toConjAct x\u207b\u00b9 = (toConjAct x)\u207b\u00b9", "theorem ContDiff.dimH_range_le {f : E \u2192 F} (h : ContDiff \u211d 1 f) : dimH (range f) \u2264 finrank \u211d E", "theorem ContDiff.exp {n} (hf : ContDiff \u211d n f) : ContDiff \u211d n fun x => Real.exp (f x)", "theorem ContDiff.one_of_succ {n : \u2115} (h : ContDiff \ud835\udd5c (n + 1) f) : ContDiff \ud835\udd5c 1 f", "theorem ContDiffAt.add {f g : E \u2192 F} (hf : ContDiffAt \ud835\udd5c n f x) (hg : ContDiffAt \ud835\udd5c n g x) :\n    ContDiffAt \ud835\udd5c n (fun x => f x + g x) x", "theorem ContDiffAt.arctan (h : ContDiffAt \u211d n f x) : ContDiffAt \u211d n (fun x => arctan (f x)) x", "theorem ContDiffAt.cexp {n} (hf : ContDiffAt \ud835\udd5c n f x) :\n    ContDiffAt \ud835\udd5c n (fun x => Complex.exp (f x)) x", "theorem ContDiffAt.csin {n} (hf : ContDiffAt \u2102 n f x) :\n    ContDiffAt \u2102 n (fun x => Complex.sin (f x)) x", "theorem ContDiffAt.differentiableAt (h : ContDiffAt \ud835\udd5c n f x) (hn : 1 \u2264 n) :\n    DifferentiableAt \ud835\udd5c f x", "theorem mem_toPartialHomeomorph_source {n : \u2115\u221e} (hf : ContDiffAt \ud835\udd42 n f a)\n    (hf' : HasFDerivAt f (f' : E \u2192L[\ud835\udd42] F) a) (hn : 1 \u2264 n) :\n    a \u2208 (hf.toPartialHomeomorph f hf' hn).source", "theorem convolution_eq_right {x\u2080 : G} (hg : \u2200 x \u2208 ball x\u2080 \u03c6.rOut, g x = g x\u2080) :\n    (\u03c6 \u22c6[lsmul \u211d \u211d, \u03bc] g : G \u2192 E') x\u2080 = integral \u03bc \u03c6 \u2022 g x\u2080", "theorem nonneg' (x : E) : 0 \u2264 f x", "theorem normed_le_div_measure_closedBall_rIn (x : E) :\n    f.normed \u03bc x \u2264 1 / (\u03bc (closedBall c f.rIn)).toReal", "theorem normed_sub (x : E) : f.normed \u03bc (c - x) = f.normed \u03bc (c + x)", "theorem ContDiffOn.arsinh (h : ContDiffOn \u211d n f s) : ContDiffOn \u211d n (fun x => arsinh (f x)) s", "theorem ContDiffOn.congr (h : ContDiffOn \ud835\udd5c n f s) (h\u2081 : \u2200 x \u2208 s, f\u2081 x = f x) :\n    ContDiffOn \ud835\udd5c n f\u2081 s", "theorem ContDiffOn.dist (hf : ContDiffOn \u211d n f s) (hg : ContDiffOn \u211d n g s)\n    (hne : \u2200 x \u2208 s, f x \u2260 g x) : ContDiffOn \u211d n (fun y => dist (f y) (g y)) s", "theorem ContDiffOn.log {n} (hf : ContDiffOn \u211d n f s) (hs : \u2200 x \u2208 s, f x \u2260 0) :\n    ContDiffOn \u211d n (fun x => log (f x)) s", "theorem ContDiffOn.of_le (h : ContDiffOn \ud835\udd5c n f s) (hmn : m \u2264 n) : ContDiffOn \ud835\udd5c m f s", "theorem ContDiffOn.sub {s : Set E} {f g : E \u2192 F} (hf : ContDiffOn \ud835\udd5c n f s)\n    (hg : ContDiffOn \ud835\udd5c n g s) : ContDiffOn \ud835\udd5c n (fun x => f x - g x) s", "theorem ContDiffWithinAt.congr (h : ContDiffWithinAt \ud835\udd5c n f s x) (h\u2081 : \u2200 y \u2208 s, f\u2081 y = f y)\n    (hx : f\u2081 x = f x) : ContDiffWithinAt \ud835\udd5c n f\u2081 s x", "theorem ContDiffWithinAt.div [CompleteSpace \ud835\udd5c] {f g : E \u2192 \ud835\udd5c} {n} (hf : ContDiffWithinAt \ud835\udd5c n f s x)\n    (hg : ContDiffWithinAt \ud835\udd5c n g s x) (hx : g x \u2260 0) :\n    ContDiffWithinAt \ud835\udd5c n (fun x => f x / g x) s x", "theorem ContDiffWithinAt.sub {s : Set E} {f g : E \u2192 F} (hf : ContDiffWithinAt \ud835\udd5c n f s x)\n    (hg : ContDiffWithinAt \ud835\udd5c n g s x) : ContDiffWithinAt \ud835\udd5c n (fun x => f x - g x) s x", "@[to_additive]\ntheorem ContMDiff.div {f g : M \u2192 G} (hf : ContMDiff I' I n f) (hg : ContMDiff I' I n g) :\n    ContMDiff I' I n fun x => f x / g x", "theorem ContMDiff.prod_map (hf : ContMDiff I I' n f) (hg : ContMDiff J J' n g) :\n    ContMDiff (I.prod J) (I'.prod J') n (Prod.map f g)", "nonrec theorem ContMDiffAt.clm_prodMap {g : M \u2192 F\u2081 \u2192L[\ud835\udd5c] F\u2083} {f : M \u2192 F\u2082 \u2192L[\ud835\udd5c] F\u2084} {x : M}\n    (hg : ContMDiffAt I \ud835\udcd8(\ud835\udd5c, F\u2081 \u2192L[\ud835\udd5c] F\u2083) n g x) (hf : ContMDiffAt I \ud835\udcd8(\ud835\udd5c, F\u2082 \u2192L[\ud835\udd5c] F\u2084) n f x) :\n    ContMDiffAt I \ud835\udcd8(\ud835\udd5c, F\u2081 \u00d7 F\u2082 \u2192L[\ud835\udd5c] F\u2083 \u00d7 F\u2084) n (fun x => (g x).prodMap (f x)) x", "theorem ContMDiffAt.mdifferentiableAt (hf : ContMDiffAt I I' n f x) (hn : 1 \u2264 n) :\n    MDifferentiableAt I I' f x", "protected theorem mdifferentiable' (f : C^n\u27eeI, M; I', M'\u27ef) (hn : 1 \u2264 n) : MDifferentiable I I' f", "theorem ContMDiffOn.contMDiffOn_tangentMapWithin_aux {f : H \u2192 H'} {s : Set H}\n    (hf : ContMDiffOn I I' n f s) (hmn : m + 1 \u2264 n) (hs : UniqueMDiffOn I s) :\n    ContMDiffOn I.tangent I'.tangent m (tangentMapWithin I I' f s)\n      (\u03c0 E (TangentSpace I) \u207b\u00b9' s)", "theorem ContMDiffOn.of_le (hf : ContMDiffOn I I' n f s) (le : m \u2264 n) : ContMDiffOn I I' m f s", "theorem ContMDiffOn.of_succ {n : \u2115} (h : ContMDiffOn I I' n.succ f s) : ContMDiffOn I I' n f s", "@[to_additive]\ntheorem ContMDiffWithinAt.div_const (hf : ContMDiffWithinAt I' I n f s x) :\n    ContMDiffWithinAt I' I n (fun x \u21a6 f x / c) s x", "theorem ContMDiffWithinAt.insert (h : ContMDiffWithinAt I I' n f s x) :\n    ContMDiffWithinAt I I' n f (insert x s) x", "theorem ContMDiffWithinAt.of_succ {n : \u2115} (h : ContMDiffWithinAt I I' n.succ f s x) :\n    ContMDiffWithinAt I I' n f s x", "theorem ContMDiffWithinAt.prod_mk_space {f : M \u2192 E'} {g : M \u2192 F'}\n    (hf : ContMDiffWithinAt I \ud835\udcd8(\ud835\udd5c, E') n f s x) (hg : ContMDiffWithinAt I \ud835\udcd8(\ud835\udd5c, F') n g s x) :\n    ContMDiffWithinAt I \ud835\udcd8(\ud835\udd5c, E' \u00d7 F') n (fun x => (f x, g x)) s x", "nonrec theorem convergents_eq_convergents' [LinearOrderedField K] {c : ContinuedFraction K} :\n    (\u2191c : GeneralizedContinuedFraction K).convergents =\n    (\u2191c : GeneralizedContinuedFraction K).convergents'", "protected theorem Continuous.IccExtend {f : \u03b3 \u2192 Icc a b \u2192 \u03b2} {g : \u03b3 \u2192 \u03b1} (hf : Continuous \u21bff)\n    (hg : Continuous g) : Continuous fun a => IccExtend h (f a) (g a)", "theorem _root_.Continuous.ae_eq_iff_eq {f g : X \u2192 Y} (hf : Continuous f) (hg : Continuous g) :\n    f =\u1d50[\u03bc] g \u2194 f = g", "@[to_additive \" A continuous function with compact support is bounded above. \"]\ntheorem Continuous.bddAbove_range_of_hasCompactMulSupport [ClosedIciTopology \u03b1] [One \u03b1]\n    {f : \u03b2 \u2192 \u03b1} (hf : Continuous f) (h : HasCompactMulSupport f) : BddAbove (range f)", "theorem Continuous.comp_lowerSemicontinuousOn {g : \u03b3 \u2192 \u03b4} {f : \u03b1 \u2192 \u03b3} (hg : Continuous g)\n    (hf : LowerSemicontinuousOn f s) (gmon : Monotone g) : LowerSemicontinuousOn (g \u2218 f) s", "theorem _root_.Continuous.const_clm_comp {X} [TopologicalSpace X] {f : X \u2192 E \u2192SL[\u03c3\u2081\u2082] F}\n    (hf : Continuous f) (g : F \u2192SL[\u03c3\u2082\u2083] G) :\n    Continuous (fun x => g.comp (f x) : X \u2192 E \u2192SL[\u03c3\u2081\u2083] G)", "theorem Continuous.fourier_inversion_inv (h : Continuous f)\n    (hf : Integrable f) (h'f : Integrable (\ud835\udcd5 f)) :\n    \ud835\udcd5 (\ud835\udcd5\u207b f) = f", "theorem Continuous.image_connectedComponentIn_subset [TopologicalSpace \u03b2] {f : \u03b1 \u2192 \u03b2} {s : Set \u03b1}\n    {a : \u03b1} (hf : Continuous f) (hx : a \u2208 s) :\n    f '' connectedComponentIn s a \u2286 connectedComponentIn (f '' s) (f a)", "@[continuity]\ntheorem Continuous.inner (hf : Continuous f) (hg : Continuous g) : Continuous fun t => \u27eaf t, g t\u27eb", "theorem Continuous.integrableAt_nhds [TopologicalSpace \u03b1] [SecondCountableTopologyEither \u03b1 E]\n    [OpensMeasurableSpace \u03b1] {\u03bc : Measure \u03b1} [IsLocallyFiniteMeasure \u03bc] {f : \u03b1 \u2192 E}\n    (hf : Continuous f) (a : \u03b1) : IntegrableAtFilter f (\ud835\udcdd a) \u03bc", "theorem Continuous.locallyIntegrable [IsLocallyFiniteMeasure \u03bc] [SecondCountableTopologyEither X E]\n    (hf : Continuous f) : LocallyIntegrable f \u03bc", "theorem Continuous.lowerSemicontinuous {f : \u03b1 \u2192 \u03b3} (h : Continuous f) : LowerSemicontinuous f", "theorem Continuous.matrix_elem {A : X \u2192 Matrix m n R} (hA : Continuous A) (i : m) (j : n) :\n    Continuous fun x => A x i j", "theorem _root_.Continuous.stronglyMeasurable [MeasurableSpace \u03b1] [TopologicalSpace \u03b1]\n    [OpensMeasurableSpace \u03b1] [TopologicalSpace \u03b2] [PseudoMetrizableSpace \u03b2]\n    [h : SecondCountableTopologyEither \u03b1 \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : Continuous f) :\n    StronglyMeasurable f", "theorem Continuous.subtype_val {f : Y \u2192 Subtype p} (hf : Continuous f) :\n    Continuous fun x => (f x : X)", "lemma constVAdd_coe [ContinuousConstVAdd V\u2081 P\u2081] (v : V\u2081) :\n    (constVAdd k P\u2081 v).toAffineEquiv = .constVAdd k P\u2081 v := rfl", "@[simp]\ntheorem image_symm (f : P\u2081 \u2243\u1d43L[k] P\u2082) (s : Set P\u2082) : f.symm '' s = f \u207b\u00b9' s", "@[simp]\ntheorem symm_toAffineEquiv (e : P\u2081 \u2243\u1d43L[k] P\u2082) : e.toAffineEquiv.symm = e.symm.toAffineEquiv", "@[simp]\ntheorem trans_apply (e : P\u2081 \u2243\u1d43L[k] P\u2082) (e' : P\u2082 \u2243\u1d43L[k] P\u2083) (p : P\u2081) : e.trans e' p = e' (e p)", "@[simp]\ntheorem coe_contLinear (f : P \u2192\u1d2c[R] Q) : (f.contLinear : V \u2192 W) = f.linear", "theorem coe_linear_eq_coe_contLinear (f : P \u2192\u1d2c[R] Q) :\n    ((f : P \u2192\u1d43[R] Q).linear : V \u2192 W) = (\u21d1f.contLinear : V \u2192 W)", "@[simp]\ntheorem mk_coe (f : P \u2192\u1d2c[R] Q) (h) : (\u27e8(f : P \u2192\u1d43[R] Q), h\u27e9 : P \u2192\u1d2c[R] Q) = f", "theorem norm_image_zero_le : \u2016f 0\u2016 \u2264 \u2016f\u2016", "@[simp]\ntheorem smul_contLinear (t : R) (f : P \u2192\u1d2c[R] W) : (t \u2022 f).contLinear = t \u2022 f.contLinear", "@[simp]\ntheorem coe_toContinuousMultilinearMap : \u21d1f.toContinuousMultilinearMap = f", "@[simp]\ntheorem compContinuousLinearMap_apply (g : M [\u22c0^\u03b9]\u2192L[R] N) (f : M' \u2192L[R] M) (m : \u03b9 \u2192 M') :\n    g.compContinuousLinearMap f m = g (f \u2218 m)", "theorem toAlternatingMap_injective :\n    Injective (toAlternatingMap : (M [\u22c0^\u03b9]\u2192L[R] N) \u2192 (M [\u22c0^\u03b9]\u2192\u2097[R] N))", "nonrec theorem ContinuousAt.cpow (hf : ContinuousAt f a) (hg : ContinuousAt g a)\n    (h0 : f a \u2208 slitPlane) : ContinuousAt (fun x => f x ^ g x) a", "lemma ContinuousAt.finset_sup'_apply (hne : s.Nonempty) (hs : \u2200 i \u2208 s, ContinuousAt (f i) x) :\n    ContinuousAt (fun a \u21a6 s.sup' hne (f \u00b7 a)) x", "@[to_additive (attr := fun_prop)]\ntheorem ContinuousAt.mul {f g : X \u2192 M} {x : X} (hf : ContinuousAt f x) (hg : ContinuousAt g x) :\n    ContinuousAt (fun x => f x * g x) x", "@[fun_prop]\ntheorem ContinuousAt.prod {f : X \u2192 Y} {g : X \u2192 Z} {x : X} (hf : ContinuousAt f x)\n    (hg : ContinuousAt g x) : ContinuousAt (fun x => (f x, g x)) x", "nonrec theorem ContinuousAt.rpow (hf : ContinuousAt f x) (hg : ContinuousAt g x)\n    (h : f x \u2260 0 \u2228 0 < g x) : ContinuousAt (fun t => f t ^ g t) x", "@[simp]\ntheorem apply_symm_apply (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (c : M\u2082) : e (e.symm c) = c", "@[simp] lemma arrowCongr_apply (e\u2081 : E \u2243L[\ud835\udd5c] F) (e\u2082 : H \u2243L[\ud835\udd5c] G) (f : E \u2192L[\ud835\udd5c] H) (x : F) :\n    e\u2081.arrowCongr e\u2082 f x = e\u2082 (f (e\u2081.symm x)) := rfl", "@[simp]\ntheorem coe_funUnique : \u21d1(funUnique \u03b9 R M) = Function.eval default", "@[simp]\ntheorem coe_toDiffeomorph : \u21d1e.toDiffeomorph = e", "theorem comp_hasFDerivWithinAt_iff' {f : G \u2192 E} {s : Set G} {x : G} {f' : G \u2192L[\ud835\udd5c] F} :\n    HasFDerivWithinAt (iso \u2218 f) f' s x \u2194\n      HasFDerivWithinAt f ((iso.symm : F \u2192L[\ud835\udd5c] E).comp f') s x", "@[simp]\ntheorem coord_norm (x : E) (h : x \u2260 0) : \u2016coord \ud835\udd5c x h\u2016 = \u2016x\u2016\u207b\u00b9", "@[fun_prop]\nprotected theorem differentiableAt : DifferentiableAt \ud835\udd5c iso x", "theorem eq_symm_apply (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) {x y} : y = e.symm x \u2194 e y = x", "protected theorem fderivWithin (hxs : UniqueDiffWithinAt \ud835\udd5c s x) : fderivWithin \ud835\udd5c iso s x = iso", "theorem one_le_norm_mul_norm_symm [RingHomIsometric \u03c3\u2081\u2082] [Nontrivial E] (e : E \u2243SL[\u03c3\u2081\u2082] F) :\n    1 \u2264 \u2016(e : E \u2192SL[\u03c3\u2081\u2082] F)\u2016 * \u2016(e.symm : F \u2192SL[\u03c3\u2082\u2081] E)\u2016", "theorem apply_norm_eq_sqrt_inner_adjoint_left (A : E \u2192L[\ud835\udd5c] F) (x : E) :\n    \u2016A x\u2016 = \u221a(re \u27ea(A\u2020 \u2218L A) x, x\u27eb)", "theorem bound_of_antilipschitz (f : E \u2192SL[\u03c3] F) {K : \u211d\u22650} (h : AntilipschitzWith K f) (x) :\n    \u2016x\u2016 \u2264 K * \u2016f x\u2016", "theorem closed_complemented_range_of_isCompl_of_ker_eq_bot {F : Type*} [NormedAddCommGroup F]\n    [NormedSpace \ud835\udd5c F] [CompleteSpace F] (f : E \u2192L[\ud835\udd5c] F) (G : Submodule \ud835\udd5c F)\n    (h : IsCompl (LinearMap.range f) G) (hG : IsClosed (G : Set F)) (hker : ker f = \u22a5) :\n    IsClosed (LinearMap.range f : Set F)", "theorem coe_pow (f : M\u2081 \u2192L[R\u2081] M\u2081) (n : \u2115) : \u21d1(f ^ n) = f^[n]", "@[simp, norm_cast]\ntheorem coe_prodMap [Module R\u2081 M\u2082] [Module R\u2081 M\u2083] [Module R\u2081 M\u2084] (f\u2081 : M\u2081 \u2192L[R\u2081] M\u2082)\n    (f\u2082 : M\u2083 \u2192L[R\u2081] M\u2084) : \u2191(f\u2081.prodMap f\u2082) = (f\u2081 : M\u2081 \u2192\u2097[R\u2081] M\u2082).prodMap (f\u2082 : M\u2083 \u2192\u2097[R\u2081] M\u2084)", "@[simp]\ntheorem coe_restrictScalarsL : (restrictScalarsL \ud835\udd5c E F\u2097 \ud835\udd5c' \ud835\udd5c'' : (E \u2192L[\ud835\udd5c] F\u2097) \u2192\u2097[\ud835\udd5c''] E \u2192L[\ud835\udd5c'] F\u2097) =\n    restrictScalars\u2097 \ud835\udd5c E F\u2097 \ud835\udd5c' \ud835\udd5c''", "@[simp, norm_cast]\ntheorem coe_smul (c : S\u2082) (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) :\n    \u2191(c \u2022 f) = c \u2022 (f : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082)", "@[simp]\ntheorem ContinuousLinearMap.compLeftContinuousCompact_apply (g : \u03b2 \u2192L[\ud835\udd5c] \u03b3) (f : C(X, \u03b2)) (x : X) :\n    g.compLeftContinuousCompact X f x = g (f x)", "theorem ContinuousLinearMap.comp_hasFiniteFPowerSeriesOnBall (g : F \u2192L[\ud835\udd5c] G)\n    (h : HasFiniteFPowerSeriesOnBall f p x n r) :\n    HasFiniteFPowerSeriesOnBall (g \u2218 f) (g.compFormalMultilinearSeries p) x n r", "theorem ContinuousLinearMap.contMDiffAt (L : E \u2192L[\ud835\udd5c] F) {x} : ContMDiffAt \ud835\udcd8(\ud835\udd5c, E) \ud835\udcd8(\ud835\udd5c, F) n L x", "@[simp, fun_prop]\nprotected theorem ContinuousLinearMap.differentiableAt : DifferentiableAt \ud835\udd5c e x", "@[simp]\ntheorem equivProdOfSurjectiveOfIsCompl_apply {f : E \u2192L[\ud835\udd5c] F} {g : E \u2192L[\ud835\udd5c] G} (hf : range f = \u22a4)\n    (hg : range g = \u22a4) (hfg : IsCompl (ker f) (ker g)) (x : E) :\n    equivProdOfSurjectiveOfIsCompl f g hf hg hfg x = (f x, g x)", "theorem ext_iff {f g : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082} : f = g \u2194 \u2200 x, f x = g x", "@[ext]\ntheorem ext_ring [TopologicalSpace R\u2081] {f g : R\u2081 \u2192L[R\u2081] M\u2081} (h : f 1 = g 1) : f = g", "@[simp]\ntheorem flip_smul (c : \ud835\udd5c\u2083) (f : E \u2192SL[\u03c3\u2081\u2083] F \u2192SL[\u03c3\u2082\u2083] G) : (c \u2022 f).flip = c \u2022 f.flip", "lemma _root_.ContinuousLinearMap.hasTemperateGrowth (f : E \u2192L[\u211d] F) :\n    Function.HasTemperateGrowth f", "theorem image_rayleigh_eq_image_rayleigh_sphere {r : \u211d} (hr : 0 < r) :\n    rayleighQuotient T '' {0}\u1d9c = rayleighQuotient T '' sphere 0 r", "theorem isAdjointPair_inner (A : E \u2192L[\ud835\udd5c] F) :\n    LinearMap.IsAdjointPair (sesqFormOfInner : E \u2192\u2097[\ud835\udd5c] E \u2192\u2097\u22c6[\ud835\udd5c] \ud835\udd5c)\n      (sesqFormOfInner : F \u2192\u2097[\ud835\udd5c] F \u2192\u2097\u22c6[\ud835\udd5c] \ud835\udd5c) A (A\u2020)", "theorem isClosed_ker [T1Space M\u2082] [FunLike F M\u2081 M\u2082] [ContinuousSemilinearMapClass F \u03c3\u2081\u2082 M\u2081 M\u2082]\n    (f : F) :\n    IsClosed (ker f : Set M\u2081)", "theorem _root_.ContinuousLinearMap.isSelfAdjoint_iff_isSymmetric {A : E \u2192L[\ud835\udd5c] E} :\n    IsSelfAdjoint A \u2194 (A : E \u2192\u2097[\ud835\udd5c] E).IsSymmetric", "theorem is_weak_closed_closedBall (f\u2080 : E' \u2192SL[\u03c3\u2081\u2082] F) (r : \u211d) \u2983f : E' \u2192SL[\u03c3\u2081\u2082] F\u2984\n    (hf : \u21d1f \u2208 closure (((\u2191) : (E' \u2192SL[\u03c3\u2081\u2082] F) \u2192 E' \u2192 F) '' closedBall f\u2080 r)) :\n    f \u2208 closedBall f\u2080 r", "lemma isometry_mul : Isometry (mul \ud835\udd5c \ud835\udd5c')", "theorem le_opNNNorm : \u2016f x\u2016\u208a \u2264 \u2016f\u2016\u208a * \u2016x\u2016\u208a", "theorem lipschitz : LipschitzWith \u2016f\u2016\u208a f", "theorem map_zero\u2082 (f : M \u2192SL[\u03c1\u2081\u2082] F \u2192SL[\u03c3\u2081\u2082] G') (y : F) : f 0 y = 0", "protected theorem mdifferentiable : MDifferentiable \ud835\udcd8(\ud835\udd5c, E) \ud835\udcd8(\ud835\udd5c, E') f", "theorem nndist_le_opNNNorm (x y : E) : nndist (f x) (f y) \u2264 \u2016f\u2016\u208a * nndist x y", "theorem norm_adjoint_comp_self (A : E \u2192L[\ud835\udd5c] F) :\n    \u2016ContinuousLinearMap.adjoint A \u2218L A\u2016 = \u2016A\u2016 * \u2016A\u2016", "theorem norm_precompR_le (L : E \u2192L[\ud835\udd5c] F\u2097 \u2192L[\ud835\udd5c] G\u2097) : \u2016precompR E\u2097 L\u2016 \u2264 \u2016L\u2016", "lemma norm_smulRightL_le : \u2016smulRightL \ud835\udd5c E F\u2097\u2016 \u2264 1", "theorem opNNNorm_le_of_lipschitz {f : E \u2192SL[\u03c3\u2081\u2082] F} {K : \u211d\u22650} (hf : LipschitzWith K f) :\n    \u2016f\u2016\u208a \u2264 K", "@[simp]\nlemma opNNNorm_mul_apply (x : \ud835\udd5c') : \u2016mul \ud835\udd5c \ud835\udd5c' x\u2016\u208a = \u2016x\u2016\u208a", "theorem opNorm_le_of_ball {f : E \u2192SL[\u03c3\u2081\u2082] F} {\u03b5 : \u211d} {C : \u211d} (\u03b5_pos : 0 < \u03b5) (hC : 0 \u2264 C)\n    (hf : \u2200 x \u2208 ball (0 : E) \u03b5, \u2016f x\u2016 \u2264 C * \u2016x\u2016) : \u2016f\u2016 \u2264 C", "theorem opNorm_lsmul_apply_le (x : \ud835\udd5c') : \u2016(lsmul \ud835\udd5c \ud835\udd5c' x : E \u2192L[\ud835\udd5c] E)\u2016 \u2264 \u2016x\u2016", "@[simp, nontriviality]\ntheorem opNorm_subsingleton [Subsingleton E] : \u2016f\u2016 = 0", "theorem opNorm_zero_iff [RingHomIsometric \u03c3\u2081\u2082] : \u2016f\u2016 = 0 \u2194 f = 0", "@[simp]\ntheorem projKerOfRightInverse_apply_idem [TopologicalAddGroup M] (f\u2081 : M \u2192SL[\u03c3\u2081\u2082] M\u2082)\n    (f\u2082 : M\u2082 \u2192SL[\u03c3\u2082\u2081] M) (h : Function.RightInverse f\u2082 f\u2081) (x : LinearMap.ker f\u2081) :\n    f\u2081.projKerOfRightInverse f\u2082 h x = x", "protected theorem quotientMap (surj : Surjective f) : QuotientMap f", "theorem ContinuousLinearMap.reApplyInnerSelf_apply (T : E \u2192L[\ud835\udd5c] E) (x : E) :\n    T.reApplyInnerSelf x = re \u27eaT x, x\u27eb", "theorem sSup_unit_ball_eq_nnnorm {\ud835\udd5c \ud835\udd5c\u2082 E F : Type*} [NormedAddCommGroup E]\n    [SeminormedAddCommGroup F] [DenselyNormedField \ud835\udd5c] [NontriviallyNormedField \ud835\udd5c\u2082] {\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082}\n    [NormedSpace \ud835\udd5c E] [NormedSpace \ud835\udd5c\u2082 F] [RingHomIsometric \u03c3\u2081\u2082] (f : E \u2192SL[\u03c3\u2081\u2082] F) :\n    sSup ((fun x => \u2016f x\u2016\u208a) '' ball 0 1) = \u2016f\u2016\u208a", "@[simp]\ntheorem smulRight_apply {c : M\u2081 \u2192L[R] S} {f : M\u2082} {x : M\u2081} :\n    (smulRight c f : M\u2081 \u2192 M\u2082) x = c x \u2022 f", "@[simp]\ntheorem smul_comp (c : S\u2083) (h : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083) (f : M \u2192SL[\u03c3\u2081\u2082] M\u2082) :\n    (c \u2022 h).comp f = c \u2022 h.comp f", "theorem smul_compLpL [Fact (1 \u2264 p)] {\ud835\udd5c'} [NormedRing \ud835\udd5c'] [Module \ud835\udd5c' F] [BoundedSMul \ud835\udd5c' F]\n    [SMulCommClass \ud835\udd5c \ud835\udd5c' F] (c : \ud835\udd5c') (L : E \u2192L[\ud835\udd5c] F) : (c \u2022 L).compLpL p \u03bc = c \u2022 L.compLpL p \u03bc", "@[simp]\nprotected theorem smul_def (f : M\u2081 \u2192L[R\u2081] M\u2081) (a : M\u2081) : f \u2022 a = f a", "theorem ContinuousLinearMap.tendsto_birkhoffAverage_orthogonalProjection (f : E \u2192L[\ud835\udd5c] E)\n    (hf : \u2016f\u2016 \u2264 1) (x : E) :\n    Tendsto (birkhoffAverage \ud835\udd5c f _root_.id \u00b7 x) atTop\n      (\ud835\udcdd <| orthogonalProjection (LinearMap.eqLocus f 1) x)", "theorem toSesqForm_apply_norm_le {f : E \u2192L[\ud835\udd5c] E'} {v : E'} : \u2016toSesqForm f v\u2016 \u2264 \u2016f\u2016 * \u2016v\u2016", "theorem toSpanSingleton_smul' {\u03b1} [Monoid \u03b1] [DistribMulAction \u03b1 M\u2081] [ContinuousConstSMul \u03b1 M\u2081]\n    [SMulCommClass R\u2081 \u03b1 M\u2081] (c : \u03b1) (x : M\u2081) :\n    toSpanSingleton R\u2081 (c \u2022 x) = c \u2022 toSpanSingleton R\u2081 x", "private lemma uniformity_eq_seminorm :\n    \ud835\udce4 (E \u2192SL[\u03c3\u2081\u2082] F) = \u2a05 r > 0, \ud835\udcdf {f | \u2016f.1 - f.2\u2016 < r}", "@[simp]\ntheorem zero_comp (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) : (0 : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083).comp f = 0", "theorem AlgHom.closure_ker_inter {F S K A : Type*} [CommRing K] [Ring A] [Algebra K A]\n    [TopologicalSpace K] [T1Space K] [TopologicalSpace A] [ContinuousSub A] [ContinuousSMul K A]\n    [FunLike F A K] [AlgHomClass F K A K] [SetLike S A] [OneMemClass S A] [AddSubgroupClass S A]\n    [SMulMemClass S K A] (\u03c6 : F) (h\u03c6 : Continuous \u03c6) (s : S) :\n    closure (s \u2229 RingHom.ker \u03c6) = closure s \u2229 (ker \u03c6 : Set A)", "@[symm]\ntheorem symm \u2983f g : C(X, Y)\u2984 (h : HomotopicRel f g S) : HomotopicRel g f S", "@[simp]\ntheorem apply_zero (F : Homotopy f\u2080 f\u2081) (x : X) : F (0, x) = f\u2080 x", "theorem eq_path_of_eq_image :\n    (\u03c0\u2098 f).map \u27e6p\u27e7 = hcast (start_path hfg) \u226b (\u03c0\u2098 g).map \u27e6q\u27e7 \u226b hcast (end_path hfg).symm", "private theorem start_path : f x\u2080 = g x\u2082", "@[simp]\ntheorem coe_invFun (h : HomotopyEquiv X Y) : (\u21d1h.invFun : Y \u2192 X) = \u21d1h.symm", "theorem coeFn_injective : @Function.Injective (HomotopyWith f\u2080 f\u2081 P) (I \u00d7 X \u2192 Y) (\u21d1)", "@[to_additive (attr := norm_cast) (reorder := 7 8)]\ntheorem coe_zpow [Group \u03b2] [TopologicalGroup \u03b2] (f : C(\u03b1, \u03b2)) (z : \u2124) : \u21d1(f ^ z) = (\u21d1f) ^ z", "theorem compactOpen_eq : @compactOpen X Y _ _ =\n    .generateFrom (image2 (fun K U \u21a6 {f | MapsTo f K U}) {K | IsCompact K} {t | IsOpen t})", "theorem embedding_sigmaMk_comp [Nonempty X] :\n    Embedding (fun g : \u03a3 i, C(X, Y i) \u21a6 (sigmaMk g.1).comp g.2) where", "theorem ContinuousMap.exists_restrict_eq (f : C(s, Y)) : \u2203 (g : C(X, Y)), g.restrict s = f", "theorem idealOfSet_closed [T2Space R] (s : Set X) :\n    IsClosed (idealOfSet R s : Set C(X, R))", "theorem ideal_isMaximal_iff (I : Ideal C(X, \ud835\udd5c)) [hI : IsClosed (I : Set C(X, \ud835\udd5c))] :\n    I.IsMaximal \u2194 \u2203 x : X, idealOfSet \ud835\udd5c {x}\u1d9c = I", "lemma inf'_apply {\u03b9 : Type*} {s : Finset \u03b9} (H : s.Nonempty) (f : \u03b9 \u2192 C(\u03b1, \u03b2)) (a : \u03b1) :\n    s.inf' H f a = s.inf' H fun i \u21a6 f i a", "theorem inf_mem_subalgebra_closure (A : Subalgebra \u211d C(X, \u211d)) (f g : A) :\n    (f : C(X, \u211d)) \u2293 (g : C(X, \u211d)) \u2208 A.topologicalClosure", "theorem liftCover_restrict {i : \u03b9} : (liftCover S \u03c6 h\u03c6 hS).restrict (S i) = \u03c6 i", "@[to_additive (attr := simp)]\nlemma mabs_apply (f : C(\u03b1, \u03b2)) (x : \u03b1) : |f|\u2098 x = |f x|\u2098 := rfl", "theorem not_mem_idealOfSet {s : Set X} {f : C(X, R)} : f \u2209 idealOfSet R s \u2194 \u2203 x \u2208 s\u1d9c, f x \u2260 0", "theorem not_mem_setOfIdeal {I : Ideal C(X, R)} {x : X} :\n    x \u2209 setOfIdeal I \u2194 \u2200 \u2983f : C(X, R)\u2984, f \u2208 I \u2192 f x = 0", "theorem periodic_tsum_comp_add_zsmul [AddCommGroup X] [TopologicalAddGroup X] [AddCommMonoid Y]\n    [ContinuousAdd Y] [T2Space Y] (f : C(X, Y)) (p : X) :\n    Function.Periodic (\u21d1(\u2211' n : \u2124, f.comp (ContinuousMap.addRight (n \u2022 p)))) p", "@[to_additive]\ntheorem smul_apply [SMul R M] [ContinuousConstSMul R M] (c : R) (f : C(\u03b1, M)) (a : \u03b1) :\n    (c \u2022 f) a = c \u2022 f a", "theorem subalgebra_topologicalClosure_eq_top_of_separatesPoints (A : Subalgebra \u211d C(X, \u211d))\n    (w : A.SeparatesPoints) : A.topologicalClosure = \u22a4", "theorem summable_of_locally_summable_norm {\u03b9 : Type*} {F : \u03b9 \u2192 C(X, E)}\n    (hF : \u2200 K : Compacts X, Summable fun i => \u2016(F i).restrict K\u2016) : Summable F", "theorem toLp_injective [\u03bc.IsOpenPosMeasure] [NormedField \ud835\udd5c] [NormedSpace \ud835\udd5c E] :\n    Function.Injective (\u21d1(toLp p \u03bc \ud835\udd5c : C(\u03b1, E) \u2192L[\ud835\udd5c] Lp E p \u03bc))", "theorem tsum_apply [T2Space \u03b2] [AddCommMonoid \u03b2] [ContinuousAdd \u03b2] {\u03b3 : Type*} {f : \u03b3 \u2192 C(\u03b1, \u03b2)}\n    (hf : Summable f) (x : \u03b1) :\n    \u2211' i : \u03b3, f i x = (\u2211' i : \u03b3, f i) x", "lemma embedding_toContinuousMap : Embedding ((\u2191) : C(X, R)\u2080 \u2192 C(X, R)) where\n  induced := rfl\n  inj _ _ h := ext fun x \u21a6 congr($(h) x)", "lemma cPolyomialOn : CPolynomialOn \ud835\udd5c f \u22a4 := fun x _ \u21a6 f.cPolynomialAt x", "@[simp]\ntheorem coe_restrictScalars (f : ContinuousMultilinearMap A M\u2081 M\u2082) : \u21d1(f.restrictScalars R) = f", "theorem le_of_opNorm_le {C : \u211d} (h : \u2016f\u2016 \u2264 C) : \u2016f m\u2016 \u2264 C * \u220f i, \u2016m i\u2016", "@[simp]\nlemma linearDeriv_apply : f.linearDeriv x y = \u2211 i, f (Function.update x i (y i))", "theorem mkPiRing_zero : ContinuousMultilinearMap.mkPiRing R \u03b9 (0 : M) = 0", "@[simp]\ntheorem nnnorm_ofSubsingleton [Subsingleton \u03b9] (i : \u03b9) (f : G \u2192L[\ud835\udd5c] G') :\n    \u2016ofSubsingleton \ud835\udd5c G G' i f\u2016\u208a = \u2016f\u2016\u208a", "@[simp]\ntheorem norm_mkPiAlgebra [NormOneClass A] : \u2016ContinuousMultilinearMap.mkPiAlgebra \ud835\udd5c \u03b9 A\u2016 = 1", "theorem opNorm_add_le : \u2016f + g\u2016 \u2264 \u2016f\u2016 + \u2016g\u2016", "theorem opNorm_zero_iff : \u2016f\u2016 = 0 \u2194 f = 0", "@[simp]\ntheorem sub_apply (m : \u2200 i, M\u2081 i) : (f - f') m = f m - f' m", "@[simp]\ntheorem ContinuousMultilinearMap.uncurryRight_norm\n    (f : ContinuousMultilinearMap \ud835\udd5c (fun i : Fin n => Ei <| castSucc i) (Ei (last n) \u2192L[\ud835\udd5c] G)) :\n    \u2016f.uncurryRight\u2016 = \u2016f\u2016", "theorem unit_le_opNorm (h : \u2016m\u2016 \u2264 1) : \u2016f m\u2016 \u2264 \u2016f\u2016", "theorem ContinuousOn.continuousAt {f : \u03b1 \u2192 \u03b2} {s : Set \u03b1} {x : \u03b1} (h : ContinuousOn f s)\n    (hx : s \u2208 \ud835\udcdd x) : ContinuousAt f x", "theorem ContinuousOn.div_const (hf : ContinuousOn f s) (y : G\u2080) :\n    ContinuousOn (fun x => f x / y) s", "@[deprecated ContinuousOn.exists_isMaxOn' (since := \"2023-02-06\")]\ntheorem ContinuousOn.exists_forall_ge' [ClosedIciTopology \u03b1] {s : Set \u03b2} {f : \u03b2 \u2192 \u03b1}\n    (hf : ContinuousOn f s) (hsc : IsClosed s) {x\u2080 : \u03b2} (h\u2080 : x\u2080 \u2208 s)\n    (hc : \u2200\u1da0 x in cocompact \u03b2 \u2293 \ud835\udcdf s, f x \u2264 f x\u2080) : \u2203 x \u2208 s, \u2200 y \u2208 s, f y \u2264 f x", "theorem ContinuousOn.exists_isMinOn' [ClosedIicTopology \u03b1] {s : Set \u03b2} {f : \u03b2 \u2192 \u03b1}\n    (hf : ContinuousOn f s) (hsc : IsClosed s) {x\u2080 : \u03b2} (h\u2080 : x\u2080 \u2208 s)\n    (hc : \u2200\u1da0 x in cocompact \u03b2 \u2293 \ud835\udcdf s, f x\u2080 \u2264 f x) : \u2203 x \u2208 s, IsMinOn f s x", "theorem ContinuousOn.if' {s : Set \u03b1} {p : \u03b1 \u2192 Prop} {f g : \u03b1 \u2192 \u03b2} [\u2200 a, Decidable (p a)]\n    (hpf : \u2200 a \u2208 s \u2229 frontier { a | p a },\n      Tendsto f (\ud835\udcdd[s \u2229 { a | p a }] a) (\ud835\udcdd <| if p a then f a else g a))\n    (hpg :\n      \u2200 a \u2208 s \u2229 frontier { a | p a },\n        Tendsto g (\ud835\udcdd[s \u2229 { a | \u00acp a }] a) (\ud835\udcdd <| if p a then f a else g a))\n    (hf : ContinuousOn f <| s \u2229 { a | p a }) (hg : ContinuousOn g <| s \u2229 { a | \u00acp a }) :\n    ContinuousOn (fun a => if p a then f a else g a) s", "theorem image_uIcc_eq_Icc (h : ContinuousOn f [[a, b]]) :\n    f '' [[a, b]] = Icc (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))", "@[fun_prop]\ntheorem ContinuousOn.inv\u2080 (hf : ContinuousOn f s) (h0 : \u2200 x \u2208 s, f x \u2260 0) :\n    ContinuousOn (fun x => (f x)\u207b\u00b9) s", "theorem _root_.ContinuousOn.isSeparable_image [TopologicalSpace \u03b2] {f : \u03b1 \u2192 \u03b2} {s : Set \u03b1}\n    (hf : ContinuousOn f s) (hs : IsSeparable s) : IsSeparable (f '' s)", "@[fun_prop]\ntheorem ContinuousOn.snd {f : \u03b1 \u2192 \u03b2 \u00d7 \u03b3} {s : Set \u03b1} (hf : ContinuousOn f s) :\n    ContinuousOn (fun x => (f x).2) s", "theorem ContinuousOn.strictAntiOn_of_injOn_Icc {a b : \u03b1} {f : \u03b1 \u2192 \u03b4}\n    (hab : a \u2264 b) (hfab : f b \u2264 f a)\n    (hf_c : ContinuousOn f (Icc a b)) (hf_i : InjOn f (Icc a b)) :\n    StrictAntiOn f (Icc a b)", "theorem ContinuousOn.vsub {f g : \u03b1 \u2192 P} {s : Set \u03b1} (hf : ContinuousOn f s)\n    (hg : ContinuousOn g s) : ContinuousOn (f -\u1d65 g) s", "@[simp]\ntheorem cancel_left {g : \u03b2 \u2192CO \u03b3} {f\u2081 f\u2082 : \u03b1 \u2192CO \u03b2} (hg : Injective g) :\n    g.comp f\u2081 = g.comp f\u2082 \u2194 f\u2081 = f\u2082", "theorem toFun_eq_coe {f : \u03b1 \u2192CO \u03b2} : f.toFun = (f : \u03b1 \u2192 \u03b2)", "@[simp]\ntheorem cancel_left {g : \u03b2 \u2192Co \u03b3} {f\u2081 f\u2082 : \u03b1 \u2192Co \u03b2} (hg : Injective g) :\n    g.comp f\u2081 = g.comp f\u2082 \u2194 f\u2081 = f\u2082", "@[simp]\ntheorem coe_id : \u21d1(ContinuousOrderHom.id \u03b1) = id", "theorem _root_.ContinuousSMul.of_basis_zero {\u03b9 : Type*} [TopologicalRing R] [TopologicalSpace M]\n    [TopologicalAddGroup M] {p : \u03b9 \u2192 Prop} {b : \u03b9 \u2192 Set M} (h : HasBasis (\ud835\udcdd 0) p b)\n    (hsmul : \u2200 {i}, p i \u2192 \u2203 V \u2208 \ud835\udcdd (0 : R), \u2203 j, p j \u2227 V \u2022 b j \u2286 b i)\n    (hsmul_left : \u2200 (x\u2080 : R) {i}, p i \u2192 \u2203 j, p j \u2227 MapsTo (x\u2080 \u2022 \u00b7) (b j) (b i))\n    (hsmul_right : \u2200 (m\u2080 : M) {i}, p i \u2192 \u2200\u1da0 x in \ud835\udcdd (0 : R), x \u2022 m\u2080 \u2208 b i) : ContinuousSMul R M", "nonrec lemma _root_.ContinuousWithinAt.compCM (hg : ContinuousWithinAt g s a)\n    (hf : ContinuousWithinAt f s a) : ContinuousWithinAt (fun x \u21a6 (g x).comp (f x)) s a", "theorem ContinuousWithinAt.inner (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :\n    ContinuousWithinAt (fun t => \u27eaf t, g t\u27eb) s x", "@[to_additive ContinuousWithinAt.nnnorm]\ntheorem ContinuousWithinAt.nnnorm' {s : Set \u03b1} {a : \u03b1} (h : ContinuousWithinAt f s a) :\n    ContinuousWithinAt (fun x => \u2016f x\u2016\u208a) s a", "theorem aposteriori_dist_iterate_fixedPoint_le (x n) :\n    dist (f^[n] x) (fixedPoint f hf) \u2264 dist (f^[n] x) (f^[n + 1] x) / (1 - K)", "theorem dist_le_mul (x y : \u03b1) : dist (f x) (f y) \u2264 K * dist x y", "theorem Convex.add {t : Set E} (hs : Convex \ud835\udd5c s) (ht : Convex \ud835\udd5c t) : Convex \ud835\udd5c (s + t)", "theorem Convex.affine_preimage (f : E \u2192\u1d43[\ud835\udd5c] F) {s : Set F} (hs : Convex \ud835\udd5c s) : Convex \ud835\udd5c (f \u207b\u00b9' s)", "theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex \u211d s) {f : E \u2192 G}\n    (hder : \u2200\u1da0 y in \ud835\udcdd[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) :\n    \u2203 K, \u2203 t \u2208 \ud835\udcdd[s] x, LipschitzOnWith K f t", "theorem Convex.finsum_mem {\u03b9 : Sort*} {w : \u03b9 \u2192 R} {z : \u03b9 \u2192 E} {s : Set E} (hs : Convex R s)\n    (h\u2080 : \u2200 i, 0 \u2264 w i) (h\u2081 : \u2211\u1da0 i, w i = 1) (hz : \u2200 i, w i \u2260 0 \u2192 z i \u2208 s) :\n    (\u2211\u1da0 i, w i \u2022 z i) \u2208 s", "theorem lipschitzOnWith_of_nnnorm_hasDerivWithin_le {C : \u211d\u22650} (hs : Convex \u211d s)\n    (hf : \u2200 x \u2208 s, HasDerivWithinAt f (f' x) s x) (bound : \u2200 x \u2208 s, \u2016f' x\u2016\u208a \u2264 C) :\n    LipschitzOnWith C f s", "theorem Convex.mapsTo_lineMap (h : Convex \ud835\udd5c s) {x y : E} (hx : x \u2208 s) (hy : y \u2208 s) :\n    MapsTo (AffineMap.lineMap x y) (Icc (0 : \ud835\udd5c) 1) s", "theorem Convex.min_le_combo (x y : E) (ha : 0 \u2264 a) (hb : 0 \u2264 b) (hab : a + b = 1) :\n    min x y \u2264 a \u2022 x + b \u2022 y", "theorem Convex.openSegment_interior_self_subset_interior {s : Set E} (hs : Convex \ud835\udd5c s) {x y : E}\n    (hx : x \u2208 interior s) (hy : y \u2208 s) : openSegment \ud835\udd5c x y \u2286 interior s", "theorem Convex.smul_preimage (hs : Convex \ud835\udd5c s) (c : \ud835\udd5c) : Convex \ud835\udd5c ((fun z => c \u2022 z) \u207b\u00b9' s)", "theorem subset_toCone : s \u2286 hs.toCone s", "@[simp] theorem coe_add (K L : ConvexBody V) : (\u2191(K + L) : Set V) = (K : Set V) + L", "@[simp]\ntheorem coe_positive : \u2191(positive \ud835\udd5c E) = Set.Ici (0 : E)", "@[simp]\ntheorem coe_top : \u2191(\u22a4 : ConvexCone \ud835\udd5c E) = (univ : Set E)", "@[simp]\nprotected theorem mem_closure {K : ConvexCone \ud835\udd5c E} {a : E} :\n    a \u2208 K.closure \u2194 a \u2208 closure (K : Set E)", "@[simp]\ntheorem mem_mk {s : Set E} {h\u2081 h\u2082 x} : x \u2208 @mk \ud835\udd5c _ _ _ _ s h\u2081 h\u2082 \u2194 x \u2208 s", "@[simp]\ntheorem mem_strictlyPositive {x : E} : x \u2208 strictlyPositive \ud835\udd5c E \u2194 0 < x", "theorem ConvexCone.pointed_of_nonempty_of_isClosed (K : ConvexCone \u211d H) (ne : (K : Set H).Nonempty)\n    (hc : IsClosed (K : Set H)) : K.Pointed", "theorem salient_iff_not_flat (S : ConvexCone \ud835\udd5c E) : S.Salient \u2194 \u00acS.Flat", "theorem ConvexIndependent.comp_embedding {\u03b9' : Type*} (f : \u03b9' \u21aa \u03b9) {p : \u03b9 \u2192 E}\n    (hc : ConvexIndependent \ud835\udd5c p) : ConvexIndependent \ud835\udd5c (p \u2218 f)", "lemma ConvexOn.le_max_of_mem_segment (hf : ConvexOn \ud835\udd5c [x -[\ud835\udd5c] y] f) (hz : z \u2208 [x -[\ud835\udd5c] y]) :\n    f z \u2264 max (f x) (f y)", "theorem ConvexOn.lt_right_of_left_lt (hf : ConvexOn \ud835\udd5c s f) {x y z : E} (hx : x \u2208 s) (hy : y \u2208 s)\n    (hz : z \u2208 openSegment \ud835\udd5c x y) (hxz : f x < f z) : f z < f y", "theorem ConvexOn.openSegment_subset_strict_epigraph (hf : ConvexOn \ud835\udd5c s f) (p q : E \u00d7 \u03b2)\n    (hp : p.1 \u2208 s \u2227 f p.1 < p.2) (hq : q.1 \u2208 s \u2227 f q.1 \u2264 q.2) :\n    openSegment \ud835\udd5c p q \u2286 { p : E \u00d7 \u03b2 | p.1 \u2208 s \u2227 f p.1 < p.2 }", "theorem Coplanar.finiteDimensional_direction_affineSpan {s : Set P} (h : Coplanar k s) :\n    FiniteDimensional k (affineSpan k s).direction", "@[simp]\ntheorem _root_.CoxeterMatrix.toCoxeterSystem_simple (M : CoxeterMatrix B) :\n    M.toCoxeterSystem.simple = M.simple", "theorem pow_two : t ^ 2 = 1", "theorem alternatingWord_succ' (i i' : B) (m : \u2115) :\n    alternatingWord i i' (m + 1) = (if Even m then i' else i) :: alternatingWord i i' m", "theorem getD_rightInvSeq (\u03c9 : List B) (j : \u2115) :\n    (ris \u03c9).getD j 1 =\n      (\u03c0 (\u03c9.drop (j + 1)))\u207b\u00b9\n        * (Option.map (cs.simple) (\u03c9.get? j)).getD 1\n        * \u03c0 (\u03c9.drop (j + 1))", "@[simp]\ntheorem isLeftInversion_simple_iff_isLeftDescent (w : W) (i : B) :\n    cs.IsLeftInversion w (s i) \u2194 cs.IsLeftDescent w i", "theorem isReduced_take {\u03c9 : List B} (h\u03c9 : cs.IsReduced \u03c9) (j : \u2115) : cs.IsReduced (\u03c9.take j)", "theorem isReflection_of_mem_leftInvSeq (\u03c9 : List B) {t : W} (ht : t \u2208 lis \u03c9) :\n    cs.IsReflection t", "@[simp]\ntheorem length_alternatingWord (i i' : B) (m : \u2115) :\n    List.length (alternatingWord i i' m) = m", "theorem length_mul_mod_two (w\u2081 w\u2082 : W) : \u2113 (w\u2081 * w\u2082) % 2 = (\u2113 w\u2081 + \u2113 w\u2082) % 2", "@[simp] theorem length_rightInvSeq (\u03c9 : List B) : (ris \u03c9).length = \u03c9.length", "theorem prod_alternatingWord_eq_mul_pow (i i' : B) (m : \u2115) :\n    \u03c0 (alternatingWord i i' m) = (if Even m then 1 else s i') * (s i * s i') ^ (m / 2)", "theorem prod_alternatingWord_eq_prod_alternatingWord_sub (i i' : B) (m : \u2115) (hm : m \u2264 M i i' * 2) :\n    \u03c0 (alternatingWord i i' m) = \u03c0 (alternatingWord i' i (M i i' * 2 - m))", "theorem simple_determines_coxeterSystem :\n    Injective (simple : CoxeterSystem M W \u2192 B \u2192 W)", "@[simp]\ntheorem simple_mul_simple_self (i : B) : s i * s i = 1", "theorem wordProd_append (\u03c9 \u03c9' : List B) : \u03c0 (\u03c9 ++ \u03c9') = \u03c0 \u03c9 * \u03c0 \u03c9'", "@[simp]\ntheorem norm_mul_coe_unitary (A : E) (U : unitary E) : \u2016A * U\u2016 = \u2016A\u2016", "theorem a_of_eq (h : P.toPoly = Q.toPoly) : P.a = Q.a", "@[simp]\ntheorem degree_of_d_ne_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c = 0) (hd : P.d \u2260 0) :\n    P.toPoly.degree = 0", "theorem disc_eq_prod_three_roots (ha : P.a \u2260 0) (h3 : (map \u03c6 P).roots = {x, y, z}) :\n    \u03c6 P.disc = (\u03c6 P.a * \u03c6 P.a * (x - y) * (x - z) * (y - z)) ^ 2", "@[simp]\ntheorem natDegree_of_b_ne_zero (ha : P.a = 0) (hb : P.b \u2260 0) : P.toPoly.natDegree = 2", "@[simp]\ntheorem natDegree_of_zero : (0 : Cubic R).toPoly.natDegree = 0", "theorem chain_of_pairwise : (\u2200 a \u2208 s, \u2200 b \u2208 s, r a b) \u2192 Chain r s", "theorem formPerm_subsingleton (s : Cycle \u03b1) (h : Subsingleton s) : formPerm s h.nodup = 1", "@[simp]\ntheorem lists_nil : lists (@nil \u03b1) = [([] : List \u03b1)]", "@[simp]\ntheorem map_eq_nil {\u03b2 : Type*} (f : \u03b1 \u2192 \u03b2) (s : Cycle \u03b1) : map f s = nil \u2194 s = nil", "@[simp]\ntheorem next_reverse_eq_prev' (s : Cycle \u03b1) (hs : Nodup s.reverse) (x : \u03b1) (hx : x \u2208 s.reverse) :\n    s.reverse.next hs x hx = s.prev (nodup_reverse_iff.mp hs) x (mem_reverse_iff.mp hx)", "@[simp]\nnonrec theorem nodup_nil : Nodup (@nil \u03b1)", "@[simp]\ntheorem reverse_coe (l : List \u03b1) : (l : Cycle \u03b1).reverse = l.reverse", "@[simp]\ntheorem toFinset_eq_nil {s : Cycle \u03b1} : s.toFinset = \u2205 \u2194 s = Cycle.nil", "@[simp]\ntheorem toFinset_toMultiset (s : Cycle \u03b1) : s.toMultiset.toFinset = s.toFinset", "@[simp]\ntheorem toMultiset_eq_nil {s : Cycle \u03b1} : s.toMultiset = 0 \u2194 s = Cycle.nil", "@[simp]\ntheorem eval_singleton (a : \u03b1) : M.eval [a] = M.step M.start a", "theorem Lex.acc_of_single [DecidableEq \u03b9] [\u2200 (i) (x : \u03b1 i), Decidable (x \u2260 0)] (x : \u03a0\u2080 i, \u03b1 i) :\n    (\u2200 i \u2208 x.support, Acc (DFinsupp.Lex r s) <| single i (x i)) \u2192 Acc (DFinsupp.Lex r s) x", "@[simp]\ntheorem add_eq_zero_iff (f g : \u03a0\u2080 i, \u03b1 i) : f + g = 0 \u2194 f = 0 \u2227 g = 0", "@[simp, norm_cast] lemma coe_lt_coe : \u21d1f < g \u2194 f < g := Iff.rfl", "@[simp, norm_cast] lemma coe_neg [\u2200 i, AddGroup (\u03b2 i)] (g : \u03a0\u2080 i, \u03b2 i) : \u21d1(-g) = -g := rfl", "@[simp]\ntheorem comapDomain'_single [DecidableEq \u03b9] [DecidableEq \u03ba] [\u2200 i, Zero (\u03b2 i)] (h : \u03ba \u2192 \u03b9)\n    {h' : \u03b9 \u2192 \u03ba} (hh' : Function.LeftInverse h' h) (k : \u03ba) (x : \u03b2 (h k)) :\n    comapDomain' h hh' (single (h k) x) = single k x", "@[simp]\ntheorem equivFunOnFintype_single [Fintype \u03b9] (i : \u03b9) (m : \u03b2 i) :\n    (@DFinsupp.equivFunOnFintype \u03b9 \u03b2 _ _) (DFinsupp.single i m) = Pi.single i m", "@[simp]\ntheorem equivFunOnFintype_symm_single [Fintype \u03b9] (i : \u03b9) (m : \u03b2 i) :\n    (@DFinsupp.equivFunOnFintype \u03b9 \u03b2 _ _).symm (Pi.single i m) = DFinsupp.single i m", "theorem erase_single (j : \u03b9) (i : \u03b9) (x : \u03b2 i) :\n    (single i x).erase j = if i = j then 0 else single i x", "theorem filter_single (p : \u03b9 \u2192 Prop) [DecidablePred p] (i : \u03b9) (x : \u03b2 i) :\n    (single i x).filter p = if p i then single i x else 0", "protected theorem induction {p : (\u03a0\u2080 i, \u03b2 i) \u2192 Prop} (f : \u03a0\u2080 i, \u03b2 i) (h0 : p 0)\n    (ha : \u2200 (i b) (f : \u03a0\u2080 i, \u03b2 i), f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)) : p f", "lemma lt_def : f < g \u2194 f \u2264 g \u2227 \u2203 i, f i < g i := Pi.lt_def", "theorem mapRange_add (f : \u2200 i, \u03b2\u2081 i \u2192 \u03b2\u2082 i) (hf : \u2200 i, f i 0 = 0)\n    (hf' : \u2200 i x y, f i (x + y) = f i x + f i y) (g\u2081 g\u2082 : \u03a0\u2080 i, \u03b2\u2081 i) :\n    mapRange f hf (g\u2081 + g\u2082) = mapRange f hf g\u2081 + mapRange f hf g\u2082", "theorem mapRange_smul (f : \u2200 i, \u03b2\u2081 i \u2192 \u03b2\u2082 i) (hf : \u2200 i, f i 0 = 0) (r : R)\n    (hf' : \u2200 i x, f i (r \u2022 x) = r \u2022 f i x) (g : \u03a0\u2080 i, \u03b2\u2081 i) :\n    mapRange f hf (r \u2022 g) = r \u2022 mapRange f hf g", "theorem mem_rangeIcc_apply_iff : a \u2208 f.rangeIcc g i \u2194 f i \u2264 a \u2227 a \u2264 g i", "theorem mk_of_not_mem (hi : i \u2209 s) : (mk s x : \u2200 i, \u03b2 i) i = 0", "@[simp]\ntheorem mk_zero [\u2200 i, Zero (\u03b2 i)] {s : Finset \u03b9} : mk s (0 : \u2200 i : (\u2191s : Set \u03b9), \u03b2 i.1) = 0", "@[simp]\ntheorem neLocus_sub_right : neLocus (f\u2081 - g) (f\u2082 - g) = neLocus f\u2081 f\u2082", "theorem not_mem_neLocus {f g : \u03a0\u2080 a, N a} {a : \u03b1} : a \u2209 f.neLocus g \u2194 f a = g a", "@[simp]\ntheorem sigmaCurry_smul [Monoid \u03b3] [\u2200 i j, AddMonoid (\u03b4 i j)] [\u2200 i j, DistribMulAction \u03b3 (\u03b4 i j)]\n    (r : \u03b3) (f : \u03a0\u2080 (i : \u03a3 _, _), \u03b4 i.1 i.2) :\n    sigmaCurry (r \u2022 f) = r \u2022 sigmaCurry f", "theorem single_eq_of_ne {i i' b} (h : i \u2260 i') : (single i b : \u03a0\u2080 i, \u03b2 i) i' = 0", "theorem single_left_injective {b : \u2200 i : \u03b9, \u03b2 i} (h : \u2200 i, b i \u2260 0) :\n    Function.Injective (fun i => single i (b i) : \u03b9 \u2192 \u03a0\u2080 i, \u03b2 i)", "@[simp]\ntheorem single_sub {\u03b2 : \u03b9 \u2192 Type v} [\u2200 i, AddGroup (\u03b2 i)] (i : \u03b9) (x y : \u03b2 i) :\n    single i (x - y) = single i x - single i y", "theorem smul_sum {\u03b1 : Type*} [Monoid \u03b1] [\u2200 i, Zero (\u03b2 i)] [\u2200 (i) (x : \u03b2 i), Decidable (x \u2260 0)]\n    [AddCommMonoid \u03b3] [DistribMulAction \u03b1 \u03b3] {f : \u03a0\u2080 i, \u03b2 i} {h : \u2200 i, \u03b2 i \u2192 \u03b3} {c : \u03b1} :\n    c \u2022 f.sum h = f.sum fun a b => c \u2022 h a b", "@[simp]\ntheorem subtypeDomain_sub [\u2200 i, AddGroup (\u03b2 i)] {p : \u03b9 \u2192 Prop} [DecidablePred p]\n    {v v' : \u03a0\u2080 i, \u03b2 i} : (v - v').subtypeDomain p = v.subtypeDomain p - v'.subtypeDomain p", "theorem sup_apply [\u2200 i, SemilatticeSup (\u03b1 i)] (f g : \u03a0\u2080 i, \u03b1 i) (i : \u03b9) : (f \u2294 g) i = f i \u2294 g i", "@[simp]\ntheorem support_inf : (f \u2293 g).support = f.support \u2229 g.support", "theorem support_mapRange {f : \u2200 i, \u03b2\u2081 i \u2192 \u03b2\u2082 i} {hf : \u2200 i, f i 0 = 0} {g : \u03a0\u2080 i, \u03b2\u2081 i} :\n    (mapRange f hf g).support \u2286 g.support", "theorem support_smul {\u03b3 : Type w} [Semiring \u03b3] [\u2200 i, AddCommMonoid (\u03b2 i)] [\u2200 i, Module \u03b3 (\u03b2 i)]\n    [\u2200 (i : \u03b9) (x : \u03b2 i), Decidable (x \u2260 0)] (b : \u03b3) (v : \u03a0\u2080 i, \u03b2 i) :\n    (b \u2022 v).support \u2286 v.support", "theorem support_subset_iff {s : Set \u03b9} {f : \u03a0\u2080 i, \u03b2 i} : \u2191f.support \u2286 s \u2194 \u2200 i \u2209 s, f i = 0", "@[simp]\ntheorem support_sup : (f \u2294 g).support = f.support \u222a g.support", "theorem toMultiset_injective : Injective (toMultiset : (\u03a0\u2080 _a, \u2115) \u2192 Multiset \u03b1)", "theorem update_eq_sub_add_single {\u03b2 : \u03b9 \u2192 Type*} [\u2200 i, AddGroup (\u03b2 i)] (f : \u03a0\u2080 i, \u03b2 i) (i : \u03b9)\n    (b : \u03b2 i) : f.update i b = f - single i (f i) + single i b", "protected theorem DFinsupp.wellFoundedLT [\u2200 i, Zero (\u03b1 i)] [\u2200 i, Preorder (\u03b1 i)]\n    [\u2200 i, WellFoundedLT (\u03b1 i)] (hbot : \u2200 \u2983i\u2984 \u2983a : \u03b1 i\u2984, \u00aca < 0) : WellFoundedLT (\u03a0\u2080 i, \u03b1 i)", "theorem zsmul_apply [\u2200 i, AddGroup (\u03b2 i)] (b : \u2124) (v : \u03a0\u2080 i, \u03b2 i) (i : \u03b9) : (b \u2022 v) i = b \u2022 v i", "theorem ext'_iff {f g : F} : f = g \u2194 (f : \u2200 a : \u03b1, \u03b2 a) = (g : \u2200 a : \u03b1, \u03b2 a)", "lemma subsingleton_cod [\u2200 a, Subsingleton (\u03b2 a)] : Subsingleton F", "theorem map_add [\u2200 i j, AddMonoid (\u03b1 i j)] {\u03b2 : m \u2192 n \u2192 Type w} [\u2200 i j, AddMonoid (\u03b2 i j)]\n    (f : \u2200 \u2983i j\u2984, \u03b1 i j \u2192+ \u03b2 i j) (M N : DMatrix m n \u03b1) :\n    ((M + N).map fun i j => @f i j) = (M.map fun i j => @f i j) + N.map fun i j => @f i j", "protected theorem Decidable.le_iff_eq_or_lt [PartialOrder \u03b1] [@DecidableRel \u03b1 (\u00b7 \u2264 \u00b7)] {a b : \u03b1} :\n    a \u2264 b \u2194 a = b \u2228 a < b", "theorem le_iff_lt_or_eq {a b : \u03b1} : a \u2264 b \u2194 a < b \u2228 a = b", "protected theorem Decidable.not_forall\u2082 [Decidable (\u2203 x h, \u00acP x h)] [\u2200 x h, Decidable (P x h)] :\n    (\u00ac\u2200 x h, P x h) \u2194 \u2203 x h, \u00acP x h", "theorem zero : (0 : K_hat R K).IsFiniteAdele", "theorem Dense.diff_finset [T1Space X] [\u2200 x : X, NeBot (\ud835\udcdd[\u2260] x)] {s : Set X} (hs : Dense s)\n    (t : Finset X) : Dense (s \\ t)", "@[to_additive]\ntheorem Dense.smul (c : G) {s : Set \u03b1} (hs : Dense s) : Dense (c \u2022 s)", "theorem continuous_extend [T3Space \u03b3] {f : \u03b1 \u2192 \u03b3} (di : DenseInducing i)\n    (hf : \u2200 b, \u2203 c, Tendsto f (comap i (\ud835\udcdd b)) (\ud835\udcdd c)) : Continuous (di.extend f)", "theorem mk' (i : \u03b1 \u2192 \u03b2) (c : Continuous i) (dense : \u2200 x, x \u2208 closure (range i))\n    (H : \u2200 (a : \u03b1), \u2200 s \u2208 \ud835\udcdd a, \u2203 t \u2208 \ud835\udcdd (i a), \u2200 b, i b \u2208 t \u2192 b \u2208 s) : DenseInducing i", "theorem DenseRange.nonempty [h : Nonempty X] (hf : DenseRange f) : Nonempty \u03b1", "theorem DenseRange.prod_map {\u03b9 : Type*} {\u03ba : Type*} {f : \u03b9 \u2192 Y} {g : \u03ba \u2192 Z} (hf : DenseRange f)\n    (hg : DenseRange g) : DenseRange (Prod.map f g)", "theorem _root_.DenseRange.topologicalClosure_map_submodule [RingHomSurjective \u03c3\u2081\u2082]\n    [TopologicalSpace R\u2081] [TopologicalSpace R\u2082] [ContinuousSMul R\u2081 M\u2081] [ContinuousAdd M\u2081]\n    [ContinuousSMul R\u2082 M\u2082] [ContinuousAdd M\u2082] {f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082} (hf' : DenseRange f)\n    {s : Submodule R\u2081 M\u2081} (hs : s.topologicalClosure = \u22a4) :\n    (s.map (f : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082)).topologicalClosure = \u22a4", "@[simp]\ntheorem list_ofNat_zero : ofNat (List \u03b1) 0 = []", "@[simp]\ntheorem coe_mk' (D : A \u2192\u2097[R] M) (h) : \u21d1(mk' D h) = D", "protected theorem map_add : D (a + b) = D a + D b", "theorem zero_apply (a : A) : (0 : Derivation R A M) a = 0", "lemma exact\u2081 : (ShortComplex.mk _ _ (\u03b4_comp T hT n\u2080 n\u2081 h)).Exact", "protected theorem mono (h : DiffContOnCl \ud835\udd5c f s) (ht : t \u2286 s) : DiffContOnCl \ud835\udd5c f t", "protected theorem contDiff (h : E \u2243\u2098^n\u27ee\ud835\udcd8(\ud835\udd5c, E), \ud835\udcd8(\ud835\udd5c, E')\u27ef E') : ContDiff \ud835\udd5c n h", "protected theorem contMDiffAt (h : M \u2243\u2098^n\u27eeI, I'\u27ef M') {x} : ContMDiffAt I I' n h x", "lemma Diffeomorph.isLocalDiffeomorph (\u03a6 : M \u2243\u2098^n\u27eeI, J\u27ef N) : IsLocalDiffeomorph I J n \u03a6", "@[fun_prop]\ntheorem Differentiable.comp_differentiableOn {g : F \u2192 G} (hg : Differentiable \ud835\udd5c g)\n    (hf : DifferentiableOn \ud835\udd5c f s) : DifferentiableOn \ud835\udd5c (g \u2218 f) s", "@[simp]\ntheorem Differentiable.cos (hc : Differentiable \u211d f) : Differentiable \u211d fun x => Real.cos (f x)", "theorem Differentiable.dist (hf : Differentiable \u211d f) (hg : Differentiable \u211d g)\n    (hne : \u2200 x, f x \u2260 g x) : Differentiable \u211d fun y => dist (f y) (g y)", "@[simp, fun_prop]\ntheorem Differentiable.inverse (hf : Differentiable \ud835\udd5c h) (hz : \u2200 x, IsUnit (h x)) :\n    Differentiable \ud835\udd5c fun x => Ring.inverse (h x)", "@[simp]\ntheorem Differentiable.log (hf : Differentiable \u211d f) (hx : \u2200 x, f x \u2260 0) :\n    Differentiable \u211d fun x => log (f x)", "@[simp]\ntheorem Differentiable.sin (hc : Differentiable \u211d f) : Differentiable \u211d fun x => Real.sin (f x)", "@[simp]\ntheorem DifferentiableAt.div_const (hc : DifferentiableAt \ud835\udd5c c x) (d : \ud835\udd5c') :\n    DifferentiableAt \ud835\udd5c (fun x => c x / d) x", "theorem DifferentiableAt.hasFDerivAt (h : DifferentiableAt \ud835\udd5c f x) :\n    HasFDerivAt f (fderiv \ud835\udd5c f x) x", "@[simp]\ntheorem DifferentiableAt.sinh (hc : DifferentiableAt \u211d f x) :\n    DifferentiableAt \u211d (fun x => Real.sinh (f x)) x", "theorem DifferentiableOn.diffContOnCl_ball {U : Set E} {c : E} {R : \u211d} (hf : DifferentiableOn \ud835\udd5c f U)\n    (hc : closedBall c R \u2286 U) : DiffContOnCl \ud835\udd5c f (ball c R)", "theorem DifferentiableOn.div_const (hc : DifferentiableOn \ud835\udd5c c s) (d : \ud835\udd5c') :\n    DifferentiableOn \ud835\udd5c (fun x => c x / d) s", "theorem DifferentiableOn.log (hf : DifferentiableOn \u211d f s) (hx : \u2200 x \u2208 s, f x \u2260 0) :\n    DifferentiableOn \u211d (fun x => log (f x)) s", "theorem DifferentiableWithinAt.arctan (hf : DifferentiableWithinAt \u211d f s x) :\n    DifferentiableWithinAt \u211d (fun x => Real.arctan (f x)) s x", "theorem DifferentiableWithinAt.ccos (hf : DifferentiableWithinAt \u2102 f s x) :\n    DifferentiableWithinAt \u2102 (fun x => Complex.cos (f x)) s x", "@[fun_prop]\ntheorem DifferentiableWithinAt.continuousWithinAt (h : DifferentiableWithinAt \ud835\udd5c f s x) :\n    ContinuousWithinAt f s x", "theorem DifferentiableWithinAt.cosh (hf : DifferentiableWithinAt \u211d f s x) :\n    DifferentiableWithinAt \u211d (fun x => Real.cosh (f x)) s x", "theorem DifferentiableWithinAt.zpow (hf : DifferentiableWithinAt \ud835\udd5c f t a) (h : f a \u2260 0 \u2228 0 \u2264 m) :\n    DifferentiableWithinAt \ud835\udd5c (fun x => f x ^ m) t a", "lemma card_conjClasses_odd (hn : Odd n) :\n    Nat.card (ConjClasses (DihedralGroup n)) = (n + 3) / 2", "lemma commProb_cons (n : \u2115) (l : List \u2115) :\n    commProb (Product (n :: l)) = commProb (DihedralGroup n) * commProb (Product l)", "theorem orderOf_r [NeZero n] (i : ZMod n) : orderOf (r i) = n / Nat.gcd n i.val", "theorem congr_fun {f g : \u03b1 \u2192\u1d48 \u03b2} (h : f = g) (x : \u03b1) : f x = g x", "@[simp]\ntheorem id_comp (f : \u03b1 \u2192\u1d48 \u03b2) : (Dilation.id \u03b2).comp f = f", "theorem mapsTo_sphere (x : \u03b1) (r' : \u211d) :\n    MapsTo (f : \u03b1 \u2192 \u03b2) (Metric.sphere x r') (Metric.sphere (f x) (ratio f * r'))", "@[simp]\ntheorem ratio_comp [MetricSpace \u03b1] [Nontrivial \u03b1] [PseudoEMetricSpace \u03b2]\n    [PseudoEMetricSpace \u03b3] {g : \u03b2 \u2192\u1d48 \u03b3} {f : \u03b1 \u2192\u1d48 \u03b2} : ratio (g.comp f) = ratio g * ratio f", "theorem ratio_unique [DilationClass F \u03b1 \u03b2] {f : F} {x y : \u03b1} {r : \u211d\u22650} (h\u2080 : edist x y \u2260 0)\n    (htop : edist x y \u2260 \u22a4) (hr : edist (f x) (f y) = r * edist x y) : r = ratio f", "@[simp] theorem symm_symm (e : X \u2243\u1d48 Y) : e.symm.symm = e", "@[simp] theorem trans_refl (e : X \u2243\u1d48 Y) : e.trans (refl Y) = e", "theorem DiophList.forall (l : List (Set <| \u03b1 \u2192 \u2115)) (d : l.Forall Dioph) :\n    Dioph {v | l.Forall fun S : Set (\u03b1 \u2192 \u2115) => v \u2208 S}", "theorem reindex_diophFn {f : (\u03b1 \u2192 \u2115) \u2192 \u2115} (g : \u03b1 \u2192 \u03b2) (d : DiophFn f) :\n    DiophFn fun v => f (v \u2218 g)", "theorem sub_dioph : DiophFn fun v => f v - g v", "private theorem one_smul' [DecidableEq \u03b9A] [DecidableEq \u03b9B] [GMonoid A] [Gmodule A M]\n    (x : \u2a01 i, M i) :\n    (1 : \u2a01 i, A i) \u2022 x = x", "theorem IsInternal.collectedBasis_mem (h : IsInternal A) {\u03b1 : \u03b9 \u2192 Type*}\n    (v : \u2200 i, Basis (\u03b1 i) R (A i)) (a : \u03a3i, \u03b1 i) : h.collectedBasis v a \u2208 A a.1", "theorem IsInternal.submodule_independent (h : IsInternal A) : CompleteLattice.Independent A", "@[simp]\ntheorem coeAddMonoidHom_of {M S : Type*} [DecidableEq \u03b9] [AddCommMonoid M] [SetLike S M]\n    [AddSubmonoidClass S M] (A : \u03b9 \u2192 S) (i : \u03b9) (x : A i) :\n    DirectSum.coeAddMonoidHom A (of (fun i => A i) i x) = x", "@[simp]\ntheorem coeLinearMap_of (i : \u03b9) (x : A i) : DirectSum.coeLinearMap A (of (fun i \u21a6 A i) i x) = x", "theorem coe_decompose_mul_of_right_mem_of_not_le (b_mem : b \u2208 \ud835\udc9c i) (h : \u00aci \u2264 n) :\n    (decompose \ud835\udc9c (a * b) n : A) = 0", "theorem coe_of_mul_apply_of_not_le {i : \u03b9} (r : A i) (r' : \u2a01 i, A i) (n : \u03b9) (h : \u00aci \u2264 n) :\n    ((of (fun i => A i) i r * r') n : R) = 0", "theorem coe_toModule_eq_coe_toAddMonoid :\n    (toModule R \u03b9 N \u03c6 : (\u2a01 i, M i) \u2192 N) = toAddMonoid fun i \u21a6 (\u03c6 i).toAddMonoidHom", "@[simp]\nlemma decomposeAlgEquiv_symm_apply (a : \u2a01 i, \ud835\udc9c i) :\n    (decomposeAlgEquiv \ud835\udc9c).symm a = (decompose \ud835\udc9c).symm a := rfl", "theorem decompose_of_mem_ne {x : M} {i j : \u03b9} (hx : x \u2208 \u2133 i) (hij : i \u2260 j) :\n    (decompose \u2133 x j : M) = 0", "@[simp]\ntheorem decompose_one : decompose \ud835\udc9c (1 : A) = 1", "@[simp]\ntheorem fromAddMonoid_of (i : \u03b9) (f : \u03b3 \u2192+ \u03b2 i) : fromAddMonoid (of _ i f) = (of _ i).comp f", "@[simp]\ntheorem lie_of [DecidableEq \u03b9] {i j : \u03b9} (x : L i) (y : L j) :\n    \u2045of L i x, of L j y\u2046 = if hij : i = j then of L i \u2045x, hij.symm.recOn y\u2046 else 0", "theorem list_prod_ofFn_of_eq_dProd (n : \u2115) (f\u03b9 : Fin n \u2192 \u03b9) (fA : \u2200 a, A (f\u03b9 a)) :\n    (List.ofFn fun a => of A (f\u03b9 a) (fA a)).prod = of A _ ((List.finRange n).dProd f\u03b9 fA)", "@[simp]\ntheorem sigmaCurry_apply (f : \u2a01 i : \u03a3 _i, _, \u03b4 i.1 i.2) (i : \u03b9) (j : \u03b1 i) :\n    sigmaCurry f i j = f \u27e8i, j\u27e9", "@[simp]\ntheorem sigmaLuncurry_apply [\u2200 i, DecidableEq (\u03b1 i)] [\u2200 i j, DecidableEq (\u03b4 i j)]\n    (f : \u2a01 (i) (j), \u03b4 i j) (i : \u03b9) (j : \u03b1 i) : sigmaLuncurry R f \u27e8i, j\u27e9 = f i j", "theorem toAddMonoid.unique (f : \u2a01 i, \u03b2 i) : \u03c8 f = toAddMonoid (fun i => \u03c8.comp (of \u03b2 i)) f", "lemma Directed.exists_mem_subset_of_finset_subset_biUnion {\u03b1 \u03b9 : Type*} [Nonempty \u03b9]\n    {f : \u03b9 \u2192 Set \u03b1} (h : Directed (\u00b7 \u2286 \u00b7) f) {s : Finset \u03b1} (hs : (s : Set \u03b1) \u2286 \u22c3 i, f i) :\n    \u2203 i, (s : Set \u03b1) \u2286 f i", "protected lemma Directed.iSup_le (hf : Directed (\u00b7 \u2264 \u00b7) f) (ha : \u2200 i, f i \u2264 a) :  \u2a06 i, f i \u2264 a", "protected lemma DirectedOn.sSup_le (hd : DirectedOn (\u00b7 \u2264 \u00b7) d) (ha : \u2200 b \u2208 d, b \u2264 a) : sSup d \u2264 a", "lemma LSeriesSummable_mul {N : \u2115} (\u03c7 : DirichletCharacter \u2102 N) {f : \u2115 \u2192 \u2102} {s : \u2102}\n    (h : LSeriesSummable f s) :\n    LSeriesSummable (\u2197\u03c7 * f) s", "lemma LSeriesSummable_twist_vonMangoldt {N : \u2115} (\u03c7 : DirichletCharacter \u2102 N) {s : \u2102}\n    (hs : 1 < s.re) :\n    LSeriesSummable (\u2197\u03c7 * \u2197\u039b) s", "lemma mul_convolution_distrib {R : Type*} [CommSemiring R] {n : \u2115} (\u03c7 : DirichletCharacter R n)\n    (f g : \u2115 \u2192 R) :\n    (((\u03c7 \u00b7) : \u2115 \u2192 R) * f) \u235f (((\u03c7 \u00b7) : \u2115 \u2192 R) * g) = ((\u03c7 \u00b7) : \u2115 \u2192 R) * (f \u235f g)", "@[mono]\ntheorem LEComap.mono (h : LEComap f A B) (hA : A' \u2264 A) (hB : B \u2264 B') : LEComap f A' B'", "@[simp]\ntheorem ofLE_proj (h : A \u2264 B) (x : X) : ofLE h (A.proj x) = B.proj x", "theorem proj_surjective : Function.Surjective S.proj", "theorem addVal_eq_top_iff {a : R} : addVal R a = \u22a4 \u2194 a = 0", "theorem associated_pow_irreducible {x : R} (hx : x \u2260 0) {\u03d6 : R} (hirr : Irreducible \u03d6) :\n    \u2203 n : \u2115, Associated x (\u03d6 ^ n)", "theorem irreducible_of_span_eq_maximalIdeal {R : Type*} [CommRing R] [LocalRing R] [IsDomain R]\n    (\u03d6 : R) (h\u03d6 : \u03d6 \u2260 0) (h : maximalIdeal R = Ideal.span {\u03d6}) : Irreducible \u03d6", "theorem not_isField : \u00acIsField R", "theorem _root_.Disjoint.exists_thickenings (hst : Disjoint s t) (hs : IsCompact s)\n    (ht : IsClosed t) :\n    \u2203 \u03b4, 0 < \u03b4 \u2227 Disjoint (thickening \u03b4 s) (thickening \u03b4 t)", "theorem Disjoint.inf_left (h : Disjoint a b) : Disjoint (a \u2293 c) b", "theorem _root_.Disjoint.inter_eq : Disjoint s t \u2192 s \u2229 t = \u2205", "theorem inter_right' (u : Set \u03b1) (h : Disjoint s t) : Disjoint s (u \u2229 t)", "theorem Disjoint.mono (h\u2081 : a \u2264 b) (h\u2082 : c \u2264 d) : Disjoint b d \u2192 Disjoint a c", "protected theorem Disjoint.symmDiff_right (ha : Disjoint a b) (hb : Disjoint a c) :\n    Disjoint a (b \u2206 c)", "theorem one_apply (a : A) : (1 : A \u2192+[M] A) a = a", "theorem coe_smul_addMonoidHom (c : M\u1d48\u1d50\u1d43) (f : A \u2192+ B) : \u21d1(c \u2022 f) = c \u2022 \u21d1f", "@[to_additive (attr := simp)]\nlemma mk_mul [Mul M] (a b : M) : mk (a * b) = mk b * mk a := rfl", "@[simp]\ntheorem mk_smul_mulDistribActionHom_apply (c : M) (f : A \u2192+[N] B) (a : A) :\n    (mk c \u2022 f) a = f (c \u2022 a)", "theorem _root_.DomMulAct.smul_linearMap_apply (a : S'\u1d48\u1d50\u1d43) (f : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) (x : M) :\n    (a \u2022 f) x = f (DomMulAct.mk.symm a \u2022 x)", "theorem disjoint_out' {H K : Subgroup G} {a b : Quotient H.1 K} :\n    a \u2260 b \u2192 Disjoint (doset a.out' H K) (doset b.out' (H : Set G) K)", "theorem rel_bot_eq_right_group_rel (H : Subgroup G) :\n    (setoid \u2191H \u2191(\u22a5 : Subgroup G)).Rel = (QuotientGroup.rightRel H).Rel", "theorem right_bot_eq_right_quot (H : Subgroup G) :\n    Quotient (H.1 : Set G) (\u22a5 : Subgroup G) = _root_.Quotient (QuotientGroup.rightRel H)", "@[ext]\nlemma DoubleCentralizer.ext (\ud835\udd5c : Type u) (A : Type v) [NontriviallyNormedField \ud835\udd5c]\n    [NonUnitalNormedRing A] [NormedSpace \ud835\udd5c A] [SMulCommClass \ud835\udd5c A A] [IsScalarTower \ud835\udd5c A A]\n    (a b : \ud835\udcdc(\ud835\udd5c, A)) (h : a.toProd = b.toProd) : a = b", "@[simp]\ntheorem nnnorm_fst (a : \ud835\udcdc(\ud835\udd5c, A)) : \u2016a.fst\u2016\u208a = \u2016a\u2016\u208a", "@[simp]\ntheorem norm_fst (a : \ud835\udcdc(\ud835\udd5c, A)) : \u2016a.fst\u2016 = \u2016a\u2016", "theorem range_toProd :\n    Set.range toProd = { lr : (A \u2192L[\ud835\udd5c] A) \u00d7 (A \u2192L[\ud835\udd5c] A) | \u2200 x y, lr.2 x * y = x * lr.1 y }", "@[simp]\ntheorem quotQuotEquivComm_symm : (quotQuotEquivComm I J).symm = quotQuotEquivComm J I", "@[simp]\ntheorem quotQuotToQuotSup\u2090_toRingHom :\n    ((quotQuotToQuotSup\u2090 R I J) : _ \u29f8 map (Ideal.Quotient.mk\u2090 R I) J \u2192+* _) =\n      quotQuotToQuotSup I J", "theorem mem_compression : s \u2208 \ud835\udcd3 a \ud835\udc9c \u2194 s \u2208 \ud835\udc9c \u2227 s.erase a \u2208 \ud835\udc9c \u2228 s \u2209 \ud835\udc9c \u2227 insert a s \u2208 \ud835\udc9c", "theorem mem_compression_of_insert_mem_compression (h : insert a s \u2208 \ud835\udcd3 a \ud835\udc9c) : s \u2208 \ud835\udcd3 a \ud835\udc9c", "@[ext 1200]\nnonrec theorem algHom_ext \u2983f g : R[\u03b5] \u2192\u2090[R] A\u2984 (h\u03b5 : f \u03b5 = g \u03b5) : f = g", "@[simp]\ntheorem eps_mul_eps [Semiring R] : (\u03b5 * \u03b5 : R[\u03b5]) = 0", "@[simp] theorem lift_apply_eps\n    (fe : {fe : (A \u2192\u2090[R] B) \u00d7 B // fe.2 * fe.2 = 0 \u2227 \u2200 a, Commute fe.2 (fe.1 a)}) :\n    lift fe (\u03b5 : A[\u03b5]) = fe.val.2", "theorem Dvd.elim_left {P : Prop} (h\u2081 : a \u2223 b) (h\u2082 : \u2200 c, b = c * a \u2192 P) : P", "theorem Dvd.intro (c : \u03b1) (h : a * c = b) : a \u2223 b", "theorem diam_pos_iff : 0 < diam s \u2194 s.Nontrivial", "theorem diam_union {t : Set \u03b1} (xs : x \u2208 s) (yt : y \u2208 t) :\n    diam (s \u222a t) \u2264 diam s + edist x y + diam t", "theorem exists_continuous_nnreal_forall_closedBall_subset (hK : \u2200 i, IsClosed (K i))\n    (hU : \u2200 i, IsOpen (U i)) (hKU : \u2200 i, K i \u2286 U i) (hfin : LocallyFinite K) :\n    \u2203 \u03b4 : C(X, \u211d\u22650), (\u2200 x, 0 < \u03b4 x) \u2227 \u2200 (i), \u2200 x \u2208 K i, closedBall x (\u03b4 x) \u2286 U i", "theorem exists_forall_closedBall_subset_aux\u2081 (hK : \u2200 i, IsClosed (K i)) (hU : \u2200 i, IsOpen (U i))\n    (hKU : \u2200 i, K i \u2286 U i) (hfin : LocallyFinite K) (x : X) :\n    \u2203 r : \u211d, \u2200\u1da0 y in \ud835\udcdd x,\n      r \u2208 Ioi (0 : \u211d) \u2229 ENNReal.ofReal \u207b\u00b9' \u22c2 (i) (_ : y \u2208 K i), { r | closedBall y r \u2286 U i }", "theorem hausdorffEdist_image (h : Isometry \u03a6) :\n    hausdorffEdist (\u03a6 '' s) (\u03a6 '' t) = hausdorffEdist s t", "@[simp] theorem mem_ball : y \u2208 ball x \u03b5 \u2194 edist y x < \u03b5", "theorem mem_closedBall_self : x \u2208 closedBall x \u03b5", "theorem mem_nhdsWithin_iff : s \u2208 \ud835\udcdd[t] x \u2194 \u2203 \u03b5 > 0, ball x \u03b5 \u2229 t \u2286 s", "theorem nhds_basis_eball : (\ud835\udcdd x).HasBasis (fun \u03b5 : \u211d\u22650\u221e => 0 < \u03b5) (ball x)", "theorem tendsto_nhds {f : Filter \u03b2} {u : \u03b2 \u2192 \u03b1} {a : \u03b1} :\n    Tendsto u f (\ud835\udcdd a) \u2194 \u2200 \u03b5 > 0, \u2200\u1da0 x in f, edist (u x) a < \u03b5", "theorem uniformInducing_iff [PseudoEMetricSpace \u03b2] {f : \u03b1 \u2192 \u03b2} :\n    UniformInducing f \u2194 UniformContinuous f \u2227\n      \u2200 \u03b4 > 0, \u2203 \u03b5 > 0, \u2200 {a b : \u03b1}, edist (f a) (f b) < \u03b5 \u2192 edist a b < \u03b4", "@[simp]\nprotected theorem add_halves (a : \u211d\u22650\u221e) : a / 2 + a / 2 = a", "@[gcongr] protected theorem add_lt_add_left : a \u2260 \u221e \u2192 b < c \u2192 a + b < a + c", "theorem ENNReal.aemeasurable_of_exist_almost_disjoint_supersets {\u03b1 : Type*} {m : MeasurableSpace \u03b1}\n    (\u03bc : Measure \u03b1) (f : \u03b1 \u2192 \u211d\u22650\u221e)\n    (h : \u2200 (p : \u211d\u22650) (q : \u211d\u22650), p < q \u2192\n      \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227\n        { x | f x < p } \u2286 u \u2227 { x | (q : \u211d\u22650\u221e) < f x } \u2286 v \u2227 \u03bc (u \u2229 v) = 0) :\n    AEMeasurable f \u03bc", "theorem biSup_add' {\u03b9 : Sort*} {p : \u03b9 \u2192 Prop} (h : \u2203 i, p i) {f : \u03b9 \u2192 \u211d\u22650\u221e} :\n    (\u2a06 (i) (_ : p i), f i) + a = \u2a06 (i) (_ : p i), f i + a", "@[simp, norm_cast]\ntheorem coe_finset_sup {s : Finset \u03b1} {f : \u03b1 \u2192 \u211d\u22650} : \u2191(s.sup f) = s.sup fun x => (f x : \u211d\u22650\u221e)", "theorem coe_inv_le : (\u2191r\u207b\u00b9 : \u211d\u22650\u221e) \u2264 (\u2191r)\u207b\u00b9", "@[simp, norm_cast]\ntheorem coe_lt_one_iff : (\u2191p : \u211d\u22650\u221e) < 1 \u2194 p < 1", "theorem coe_mem_upperBounds {s : Set \u211d\u22650} :\n    \u2191r \u2208 upperBounds (ofNNReal '' s) \u2194 r \u2208 upperBounds s", "@[simp, norm_cast]\ntheorem coe_natCast (n : \u2115) : ((n : \u211d\u22650) : \u211d\u22650\u221e) = n", "@[simp, norm_cast]\ntheorem coe_ofNat (n : \u2115) [n.AtLeastTwo] :\n    ((no_index (OfNat.ofNat n) : \u211d\u22650) : \u211d\u22650\u221e) = OfNat.ofNat n", "@[simp, norm_cast] lemma coe_pow (x : \u211d\u22650) (n : \u2115) : (\u2191(x ^ n) : \u211d\u22650\u221e) = x ^ n := rfl", "theorem continuous_coe_iff {\u03b1} [TopologicalSpace \u03b1] {f : \u03b1 \u2192 \u211d\u22650} :\n    (Continuous fun a => (f a : \u211d\u22650\u221e)) \u2194 Continuous f", "protected theorem continuous_const_mul {a : \u211d\u22650\u221e} (ha : a \u2260 \u221e) : Continuous (a * \u00b7)", "protected theorem dichotomy (p : \u211d\u22650\u221e) [Fact (1 \u2264 p)] : p = \u221e \u2228 1 \u2264 p.toReal", "theorem div_lt_of_lt_mul' (h : a < b * c) : a / b < c", "protected theorem div_mul_cancel (h0 : a \u2260 0) (hI : a \u2260 \u221e) : b / a * a = b", "theorem div_zero (h : a \u2260 0) : a / 0 = \u221e", "theorem exists_frequently_lt_of_liminf_ne_top' {\u03b9 : Type*} {l : Filter \u03b9} {x : \u03b9 \u2192 \u211d}\n    (hx : liminf (fun n => (Real.nnabs (x n) : \u211d\u22650\u221e)) l \u2260 \u221e) : \u2203 R, \u2203\u1da0 n in l, R < x n", "theorem exists_inv_two_pow_lt (ha : a \u2260 0) : \u2203 n : \u2115, 2\u207b\u00b9 ^ n < a", "protected theorem exists_nat_gt {r : \u211d\u22650\u221e} (h : r \u2260 \u221e) : \u2203 n : \u2115, r < n", "theorem hasSum_iff_tendsto_nat {f : \u2115 \u2192 \u211d\u22650\u221e} (r : \u211d\u22650\u221e) :\n    HasSum f r \u2194 Tendsto (fun n : \u2115 => \u2211 i \u2208 Finset.range n, f i) atTop (\ud835\udcdd r)", "theorem iSup_add_iSup_of_monotone {\u03b9 : Type*} [SemilatticeSup \u03b9] {f g : \u03b9 \u2192 \u211d\u22650\u221e} (hf : Monotone f)\n    (hg : Monotone g) : iSup f + iSup g = \u2a06 a, f a + g a", "lemma iSup_coe_lt_top : \u2a06 i, (f i : \u211d\u22650\u221e) < \u22a4 \u2194 BddAbove (range f) := WithTop.iSup_coe_lt_top", "theorem iSup_ne_top [CompleteLattice \u03b1] (f : \u211d\u22650\u221e \u2192 \u03b1) :\n    \u2a06 (x) (_ : x \u2260 \u221e), f x = \u2a06 x : \u211d\u22650, f x", "theorem inv_map_iInf {\u03b9 : Sort*} {x : \u03b9 \u2192 \u211d\u22650\u221e} : (iInf x)\u207b\u00b9 = \u2a06 i, (x i)\u207b\u00b9", "protected theorem inv_pow : \u2200 {a : \u211d\u22650\u221e} {n : \u2115}, (a ^ n)\u207b\u00b9 = a\u207b\u00b9 ^ n", "theorem le_ofReal_iff_toReal_le {a : \u211d\u22650\u221e} {b : \u211d} (ha : a \u2260 \u221e) (hb : 0 \u2264 b) :\n    a \u2264 ENNReal.ofReal b \u2194 ENNReal.toReal a \u2264 b", "theorem le_of_forall_pos_le_add (h : \u2200 \u03b5 : \u211d\u22650, 0 < \u03b5 \u2192 b < \u221e \u2192 a \u2264 b + \u03b5) : a \u2264 b", "theorem lintegral_rpow_add_le_add_snorm_mul_lintegral_rpow_add {p q : \u211d}\n    (hpq : p.IsConjExponent q) {f g : \u03b1 \u2192 \u211d\u22650\u221e} (hf : AEMeasurable f \u03bc)\n    (hf_top : (\u222b\u207b a, f a ^ p \u2202\u03bc) \u2260 \u22a4) (hg : AEMeasurable g \u03bc) (hg_top : (\u222b\u207b a, g a ^ p \u2202\u03bc) \u2260 \u22a4) :\n    (\u222b\u207b a, (f + g) a ^ p \u2202\u03bc) \u2264\n      ((\u222b\u207b a, f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b a, g a ^ p \u2202\u03bc) ^ (1 / p)) *\n        (\u222b\u207b a, (f a + g a) ^ p \u2202\u03bc) ^ (1 / q)", "@[simp]\ntheorem log_bot_lt_iff {x : \u211d\u22650\u221e} : \u22a5 < log x \u2194 0 < x", "@[simp] theorem log_homeomorph_apply (x : \u211d\u22650\u221e) : log_homeomorph x = log x", "@[simp]\ntheorem log_one : log 1 = 0", "theorem log_pow {x : \u211d\u22650\u221e} {n : \u2115} : log (x ^ n) = (n : \u211d\u22650\u221e) * log x", "theorem lt_rpow_one_div_iff {x y : \u211d\u22650\u221e} {z : \u211d} (hz : 0 < z) : x < y ^ (1 / z) \u2194 x ^ z < y", "theorem max_mul : max a b * c = max (a * c) (b * c)", "theorem mem_Iio_self_add : x \u2260 \u221e \u2192 \u03b5 \u2260 0 \u2192 x \u2208 Iio (x + \u03b5)", "lemma monotone_truncateToReal {t : \u211d\u22650\u221e} (t_ne_top : t \u2260 \u221e) : Monotone (truncateToReal t)", "theorem mul_div_le : a * (b / a) \u2264 b", "protected theorem mul_div_mul_left (a b : \u211d\u22650\u221e) (hc : c \u2260 0) (hc' : c \u2260 \u22a4) :\n    c * a / (c * b) = a / b", "@[mono, gcongr]\ntheorem mul_lt_mul (ac : a < c) (bd : b < d) : a * b < c * d", "@[simp]\nlemma natCast_le_ofReal {n : \u2115} {p : \u211d} (hn : n \u2260 0) : n \u2264 ENNReal.ofReal p \u2194 n \u2264 p", "@[instance]\ntheorem nhdsWithin_Ioi_ofNat_nebot (n : \u2115) [n.AtLeastTwo] :\n    (\ud835\udcdd[>] (OfNat.ofNat n : \u211d\u22650\u221e)).NeBot", "@[gcongr]\ntheorem ofReal_le_ofReal {p q : \u211d} (h : p \u2264 q) : ENNReal.ofReal p \u2264 ENNReal.ofReal q", "@[simp]\ntheorem ofReal_le_ofReal_iff {p q : \u211d} (h : 0 \u2264 q) :\n    ENNReal.ofReal p \u2264 ENNReal.ofReal q \u2194 p \u2264 q", "@[simp]\ntheorem ofReal_pos {p : \u211d} : 0 < ENNReal.ofReal p \u2194 0 < p", "theorem one_le_rpow {x : \u211d\u22650\u221e} {z : \u211d} (hx : 1 \u2264 x) (hz : 0 < z) : 1 \u2264 x ^ z", "@[simp]\nprotected theorem one_lt_inv : 1 < a\u207b\u00b9 \u2194 a < 1", "@[simp]\ntheorem one_rpow (x : \u211d) : (1 : \u211d\u22650\u221e) ^ x = 1", "@[simp]\ntheorem rpow_one (x : \u211d\u22650\u221e) : x ^ (1 : \u211d) = x", "theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 \u2264 p) :\n    (\u2211 i \u2208 s, f i) ^ p \u2264 (card s : \u211d\u22650\u221e) ^ (p - 1) * \u2211 i \u2208 s, f i ^ p", "theorem sSup_add {s : Set \u211d\u22650\u221e} (hs : s.Nonempty) : sSup s + a = \u2a06 b \u2208 s, b + a", "@[simp] theorem some_eq_coe' (a : \u211d\u22650) : (WithTop.some a : \u211d\u22650\u221e) = (\u2191a : \u211d\u22650\u221e)", "theorem sub_half (h : a \u2260 \u221e) : a - a / 2 = a / 2", "theorem sub_lt_of_sub_lt (h\u2082 : c \u2264 a) (h\u2083 : a \u2260 \u221e \u2228 b \u2260 \u221e) (h\u2081 : a - b < c) : a - c < b", "theorem sub_mul (h : 0 < b \u2192 b < a \u2192 c \u2260 \u221e) : (a - b) * c = a * c - b * c", "theorem sum_lt_sum_of_nonempty {s : Finset \u03b1} (hs : s.Nonempty) {f g : \u03b1 \u2192 \u211d\u22650\u221e}\n    (Hlt : \u2200 i \u2208 s, f i < g i) : \u2211 i \u2208 s, f i < \u2211 i \u2208 s, g i", "@[simp]\nprotected theorem summable : Summable f", "theorem tendsto_ofReal_atTop : Tendsto ENNReal.ofReal atTop (\ud835\udcdd \u221e)", "@[simp]\ntheorem toNNReal_lt_toNNReal (ha : a \u2260 \u221e) (hb : b \u2260 \u221e) : a.toNNReal < b.toNNReal \u2194 a < b", "theorem toReal_iInf (hf : \u2200 i, f i \u2260 \u221e) : (iInf f).toReal = \u2a05 i, (f i).toReal", "theorem toReal_le_coe_of_le_coe {a : \u211d\u22650\u221e} {b : \u211d\u22650} (h : a \u2264 b) : a.toReal \u2264 b", "theorem toReal_le_of_le_ofReal {a : \u211d\u22650\u221e} {b : \u211d} (hb : 0 \u2264 b) (h : a \u2264 ENNReal.ofReal b) :\n    ENNReal.toReal a \u2264 b", "@[simp]\ntheorem toReal_lt_toReal (ha : a \u2260 \u221e) (hb : b \u2260 \u221e) : a.toReal < b.toReal \u2194 a < b", "theorem toReal_max (hr : a \u2260 \u221e) (hp : b \u2260 \u221e) :\n    ENNReal.toReal (max a b) = max (ENNReal.toReal a) (ENNReal.toReal b)", "@[simp, norm_cast]\ntheorem toReal_nat (n : \u2115) : (n : \u211d\u22650\u221e).toReal = n", "@[simp] theorem toReal_nonneg {a : \u211d\u22650\u221e} : 0 \u2264 a.toReal", "@[simp]\ntheorem toReal_ofReal {r : \u211d} (h : 0 \u2264 r) : (ENNReal.ofReal r).toReal = r", "theorem toReal_ofReal' {r : \u211d} : (ENNReal.ofReal r).toReal = max r 0", "theorem toReal_pos_iff : 0 < a.toReal \u2194 0 < a \u2227 a < \u221e", "@[simp] theorem top_ne_coe : \u221e \u2260 (r : \u211d\u22650\u221e)", "@[simp]\ntheorem top_rpow_of_pos {y : \u211d} (h : 0 < y) : (\u22a4 : \u211d\u22650\u221e) ^ y = \u22a4", "protected theorem tsum_coe_eq {f : \u03b1 \u2192 \u211d\u22650} (h : HasSum f r) : (\u2211' a, (f a : \u211d\u22650\u221e)) = r", "theorem tsum_coe_ne_top_iff_summable {f : \u03b2 \u2192 \u211d\u22650} : (\u2211' b, (f b : \u211d\u22650\u221e)) \u2260 \u221e \u2194 Summable f", "theorem tsum_const_eq_top_of_ne_zero {\u03b1 : Type*} [Infinite \u03b1] {c : \u211d\u22650\u221e} (hc : c \u2260 0) :\n    \u2211' _ : \u03b1, c = \u221e", "protected theorem tsum_const_smul {R} [SMul R \u211d\u22650\u221e] [IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e] (a : R) :\n    \u2211' i, a \u2022 f i = a \u2022 \u2211' i, f i", "theorem tsum_eq_add_tsum_ite {f : \u03b2 \u2192 \u211d\u22650\u221e} (b : \u03b2) :\n    \u2211' x, f x = f b + \u2211' x, ite (x = b) 0 (f x)", "protected theorem tsum_eq_iSup_sum : \u2211' a, f a = \u2a06 s : Finset \u03b1, \u2211 a \u2208 s, f a", "theorem tsum_toNNReal_eq {f : \u03b1 \u2192 \u211d\u22650\u221e} (hf : \u2200 a, f a \u2260 \u221e) :\n    (\u2211' a, f a).toNNReal = \u2211' a, (f a).toNNReal", "theorem young_inequality (a b : \u211d\u22650\u221e) {p q : \u211d} (hpq : p.IsConjExponent q) :\n    a * b \u2264 a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q", "@[simp, norm_cast] lemma coe_toNatHom : toNatHom = toNat := rfl", "@[simp]\ntheorem recTopCoe_top {C : \u2115\u221e \u2192 Sort*} (d : C \u22a4) (f : \u2200 a : \u2115, C a) :\n    @recTopCoe C d f \u22a4 = d", "@[simp]\ntheorem toENNReal_max (m n : \u2115\u221e) : \u2191(max m n) = (max m n : \u211d\u22650\u221e)", "@[simp]\ntheorem zero_lt_top : (0 : \u2115\u221e) < \u22a4", "@[simp]\ntheorem add_eq_bot_iff {x y : EReal} : x + y = \u22a5 \u2194 x = \u22a5 \u2228 y = \u22a5", "lemma bot_div_of_pos_ne_top {a : EReal} (h : 0 < a) (h' : a \u2260 \u22a4) : \u22a5 / a = \u22a5", "@[simp]\ntheorem coe_ennreal_eq_top_iff {x : \u211d\u22650\u221e} : (x : EReal) = \u22a4 \u2194 x = \u22a4", "@[simp]\ntheorem coe_ennreal_ofReal {x : \u211d} : (ENNReal.ofReal x : EReal) = max x 0", "@[simp, norm_cast]\ntheorem coe_ennreal_pow (x : \u211d\u22650\u221e) (n : \u2115) : (\u2191(x ^ n) : EReal) = (x : EReal) ^ n", "@[simp, norm_cast] theorem coe_neg (x : \u211d) : (\u2191(-x) : EReal) = -\u2191x", "@[simp, norm_cast]\ntheorem coe_one : ((1 : \u211d) : EReal) = 1", "@[elab_as_elim]\ntheorem induction\u2082_symm {P : EReal \u2192 EReal \u2192 Prop} (symm : \u2200 {x y}, P x y \u2192 P y x)\n    (top_top : P \u22a4 \u22a4) (top_pos : \u2200 x : \u211d, 0 < x \u2192 P \u22a4 x) (top_zero : P \u22a4 0)\n    (top_neg : \u2200 x : \u211d, x < 0 \u2192 P \u22a4 x) (top_bot : P \u22a4 \u22a5) (pos_bot : \u2200 x : \u211d, 0 < x \u2192 P x \u22a5)\n    (coe_coe : \u2200 x y : \u211d, P x y) (zero_bot : P 0 \u22a5) (neg_bot : \u2200 x : \u211d, x < 0 \u2192 P x \u22a5)\n    (bot_bot : P \u22a5 \u22a5) : \u2200 x y, P x y", "lemma monotone_div_right_of_nonneg {b : EReal} (h : 0 \u2264 b) : Monotone fun a \u21a6 a / b", "lemma mul_div (a b c : EReal) : a * (b / c) = (a * b) / c", "theorem neg_strictAnti : StrictAnti (- \u00b7 : EReal \u2192 EReal)", "@[simp]\ntheorem neg_top : -(\u22a4 : EReal) = \u22a5", "theorem nhds_top' : \ud835\udcdd (\u22a4 : EReal) = \u2a05 a : \u211d, \ud835\udcdf (Ioi \u2191a)", "@[simp]\nlemma preimage_coe_Ioo_bot_top : Real.toEReal \u207b\u00b9' Ioo \u22a5 \u22a4 = univ", "@[simp]\nlemma preimage_coe_Ioo_top (x : \u211d) : Real.toEReal \u207b\u00b9' Ioo x \u22a4 = Ioi x", "@[simp]\ntheorem sub_top (x : EReal) : x - \u22a4 = \u22a5", "@[simp]\ntheorem toReal_top : toReal \u22a4 = 0", "theorem top_add_iff_ne_bot {x : EReal} : \u22a4 + x = \u22a4 \u2194 x \u2260 \u22a5", "@[simp]\ntheorem top_add_top : (\u22a4 : EReal) + \u22a4 = \u22a4", "lemma top_mul_of_pos {x : EReal} (h : 0 < x) : \u22a4 * x = \u22a4", "@[simp]\ntheorem top_sub_bot : (\u22a4 : EReal) - \u22a5 = \u22a4", "lemma norm_eq_max_natAbs (x : Fin 2 \u2192 \u2124) : \u2016x\u2016 = max (x 0).natAbs (x 1).natAbs", "lemma coe_inv_variableChange_\u0394' : (E.variableChange C).\u0394'\u207b\u00b9 = C.u ^ 12 * E.\u0394'\u207b\u00b9", "lemma ofJ_0_of_three_eq_zero (h3 : (3 : F) = 0) :\n    ofJ 0 = @ofJ1728 _ _ (invertibleOfNonzero <| two_or_three_ne_zero.neg_resolve_right h3)", "lemma ofJ_j : (ofJ j).j = j", "@[simp]\nlemma variableChange_j : (E.variableChange C).j = E.j", "theorem Embedding.closure_eq_preimage_closure_image (hf : Embedding f) (s : Set X) :\n    closure s = f \u207b\u00b9' closure (f '' s)", "theorem Embedding.continuous_iff (hg : Embedding g) :\n    Continuous f \u2194 Continuous (g \u2218 f)", "protected theorem Embedding.measurableEmbedding {f : \u03b1 \u2192 \u03b2} (h\u2081 : Embedding f)\n    (h\u2082 : MeasurableSet (range f)) : MeasurableEmbedding f", "@[simp]\ntheorem decode_list_zero : decode (\u03b1 := List \u03b1) 0 = some []", "@[simp]\ntheorem decode_unit_succ (n) : decode (succ n) = (none : Option PUnit)", "@[simp]\ntheorem encode_inj [Encodable \u03b1] {a b : \u03b1} : encode a = encode b \u2194 a = b", "theorem mem_decode\u2082' [Encodable \u03b1] {n : \u2115} {a : \u03b1} :\n    a \u2208 decode\u2082 \u03b1 n \u2194 a \u2208 decode n \u2227 encode a = n", "theorem Eq.trans {\u03b1 : Sort u} {a b c : \u03b1} (h\u2081 : Eq a b) (h\u2082 : Eq b c) : Eq a c", "theorem of_algebraRat [Algebra \u211a R] : \u2200 I : Ideal R, I \u2260 \u22a4 \u2192 CharZero (R \u29f8 I)", "theorem of_not_mixedCharZero [CharZero R] (h : \u2200 p > 0, \u00acMixedCharZero R p) :\n    \u2200 I : Ideal R, I \u2260 \u22a4 \u2192 CharZero (R \u29f8 I)", "@[simp]\ntheorem pnatCast_eq_natCast [Fact (\u2200 I : Ideal R, I \u2260 \u22a4 \u2192 CharZero (R \u29f8 I))] (n : \u2115+) :\n    ((n : R\u02e3) : R) = \u2191n", "theorem Equicontinuous.tendsto_uniformFun_iff_pi [CompactSpace X]\n    (F_eqcont : Equicontinuous F) (\u2131 : Filter \u03b9) (f : X \u2192 \u03b1) :\n    Tendsto (UniformFun.ofFun \u2218 F) \u2131 (\ud835\udcdd <| UniformFun.ofFun f) \u2194\n    Tendsto F \u2131 (\ud835\udcdd f)", "theorem EquicontinuousOn.comp {F : \u03b9 \u2192 X \u2192 \u03b1} {S : Set X} (h : EquicontinuousOn F S) (u : \u03ba \u2192 \u03b9) :\n    EquicontinuousOn (F \u2218 u) S", "lemma EquicontinuousOn.mono {F : \u03b9 \u2192 X \u2192 \u03b1} {S T : Set X}\n    (H : EquicontinuousOn F T) (hST : S \u2286 T) : EquicontinuousOn F S", "theorem Disjoint.mem_imp (h : Disjoint f g) {x : \u03b1} (hx : x \u2208 f.support) : x \u2209 g.support", "protected theorem IsCycleOn.countable (hs : f.IsCycleOn s) : s.Countable", "theorem IsSwap.of_subtype_isSwap {p : \u03b1 \u2192 Prop} [DecidablePred p] {f : Perm (Subtype p)}\n    (h : f.IsSwap) : (ofSubtype f).IsSwap", "theorem SameCycle.conj : SameCycle f x y \u2192 SameCycle (g * f * g\u207b\u00b9) (g x) (g y)", "theorem SameCycle.cycleOf_apply : SameCycle f x y \u2192 cycleOf f x y = f y", "theorem card_support_extend_domain (f : \u03b1 \u2243 Subtype p) {g : Perm \u03b1} :\n    (g.extendDomain f).support.card = g.support.card", "@[simp]\ntheorem cycleOf_self_apply_zpow (f : Perm \u03b1) [DecidableRel f.SameCycle] (n : \u2124) (x : \u03b1) :\n    cycleOf f ((f ^ n) x) = cycleOf f x", "@[simp] theorem cycleType_eq_zero {\u03c3 : Perm \u03b1} : \u03c3.cycleType = 0 \u2194 \u03c3 = 1", "theorem cycleType_le_of_mem_cycleFactorsFinset {f g : Perm \u03b1} (hf : f \u2208 g.cycleFactorsFinset) :\n    f.cycleType \u2264 g.cycleType", "theorem cycleType_of_card_le_mem_cycleType_add_two {n : \u2115} {g : Perm \u03b1}\n    (hn2 : Fintype.card \u03b1 < n + 2) (hng : n \u2208 g.cycleType) : g.cycleType = {n}", "theorem disjoint_iff_eq_or_eq : Disjoint f g \u2194 \u2200 x : \u03b1, f x = x \u2228 g x = x", "theorem Perm.extendDomain_apply_not_subtype {b : \u03b2'} (h : \u00acp b) : e.extendDomain f b = b", "@[simp]\ntheorem extendDomain_mul (e e' : Perm \u03b1) :\n    e.extendDomain f * e'.extendDomain f = (e * e').extendDomain f", "theorem isConj_iff_cycleType_eq {\u03c3 \u03c4 : Perm \u03b1} : IsConj \u03c3 \u03c4 \u2194 \u03c3.cycleType = \u03c4.cycleType", "@[simp]\ntheorem isCycleOn_inv : f\u207b\u00b9.IsCycleOn s \u2194 f.IsCycleOn s", "theorem isThreeCycle_sq_of_three_mem_cycleType_five {g : Perm (Fin 5)} (h : 3 \u2208 cycleType g) :\n    IsThreeCycle (g * g)", "@[simp] theorem lcm_cycleType (\u03c3 : Perm \u03b1) : \u03c3.cycleType.lcm = orderOf \u03c3", "theorem mclosure_isSwap [Finite \u03b1] : Submonoid.closure { \u03c3 : Perm \u03b1 | IsSwap \u03c3 } = \u22a4", "theorem mem_iff_ofSubtype_apply_mem (f : Perm (Subtype p)) (x : \u03b1) :\n    p x \u2194 p ((ofSubtype f : \u03b1 \u2192 \u03b1) x)", "theorem mem_support_cycleOf_iff' (hx : f x \u2260 x) [DecidableEq \u03b1] [Fintype \u03b1] :\n    y \u2208 support (f.cycleOf x) \u2194 SameCycle f x y", "@[simp]\ntheorem mul_symm (e : Perm \u03b1) : e * e.symm = 1", "theorem ofSubtype_apply_of_mem (f : Perm (Subtype p)) (ha : p a) : ofSubtype f a = f \u27e8a, ha\u27e9", "@[to_additive]\ntheorem _root_.Equiv.Perm.prod_comp (\u03c3 : Equiv.Perm \u03b1) (s : Finset \u03b1) (f : \u03b1 \u2192 \u03b2)\n    (hs : { a | \u03c3 a \u2260 a } \u2286 s) : (\u220f x \u2208 s, f (\u03c3 x)) = \u220f x \u2208 s, f x", "@[to_additive]\ntheorem _root_.Equiv.Perm.prod_comp' (\u03c3 : Equiv.Perm \u03b1) (s : Finset \u03b1) (f : \u03b1 \u2192 \u03b1 \u2192 \u03b2)\n    (hs : { a | \u03c3 a \u2260 a } \u2286 s) : (\u220f x \u2208 s, f (\u03c3 x) x) = \u220f x \u2208 s, f x (\u03c3.symm x)", "theorem signBijAux_surj {n : \u2115} {f : Perm (Fin n)} :\n    \u2200 a \u2208 finPairsLT n, \u2203 b \u2208 finPairsLT n, signBijAux f b = a", "@[simp]\ntheorem sign_ofSubtype {p : \u03b1 \u2192 Prop} [DecidablePred p] (f : Equiv.Perm (Subtype p)) :\n    sign (ofSubtype f) = sign f", "theorem sign_one : sign (1 : Perm \u03b1) = 1", "@[simp]\ntheorem sign_prodExtendRight (a : \u03b1) (\u03c3 : Perm \u03b2) : sign (prodExtendRight a \u03c3) = sign \u03c3", "theorem Perm.subsingleton_eq_refl [Subsingleton \u03b1] (e : Perm \u03b1) : e = Equiv.refl \u03b1", "@[simp]\ntheorem support_conj : (\u03c3 * \u03c4 * \u03c3\u207b\u00b9).support = \u03c4.support.map \u03c3.toEmbedding", "theorem support_prod_le (l : List (Perm \u03b1)) : l.prod.support \u2264 (l.map support).foldr (\u00b7 \u2294 \u00b7) \u22a5", "@[elab_as_elim]\ntheorem swap_induction_on [Finite \u03b1] {P : Perm \u03b1 \u2192 Prop} (f : Perm \u03b1) :\n    P 1 \u2192 (\u2200 f x y, x \u2260 y \u2192 P f \u2192 P (swap x y * f)) \u2192 P f", "theorem toCycle_eq_toList (f : Perm \u03b1) (hf : IsCycle f) (x : \u03b1) (hx : f x \u2260 x) :\n    toCycle f hf = toList f x", "theorem Equiv.Perm.viaFintypeEmbedding_apply_mem_range {b : \u03b2} (h : b \u2208 Set.range f) :\n    e.viaFintypeEmbedding f b = f (e (f.invOfMemRange \u27e8b, h\u27e9))", "@[simp]\nprotected theorem image_symm_apply {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2) (s : Set \u03b1) (H : Injective f) (x : \u03b1)\n    (h : f x \u2208 f '' s) : (Set.image f s H).symm \u27e8f x, h\u27e9 = \u27e8x, H.mem_set_image.1 h\u27e9", "@[simp] lemma rangeInr_apply_inr (\u03b1 : Type*) {\u03b2 : Type*} (x : \u03b2) :\n    (rangeInr \u03b1 \u03b2) \u27e8.inr x, mem_range_self _\u27e9 = x", "theorem ext_iff {f g : Equiv \u03b1 \u03b2} : f = g \u2194 \u2200 x, f x = g x", "theorem Equiv.finite_iff (f : \u03b1 \u2243 \u03b2) : Finite \u03b1 \u2194 Finite \u03b2", "protected lemma forall_congr_right : (\u2200 a, q (e a)) \u2194 \u2200 b, q b", "@[to_additive]\ntheorem Equiv.hasProd_iff (e : \u03b3 \u2243 \u03b2) : HasProd (f \u2218 e) a \u2194 HasProd f a", "@[simp] lemma inv_addLeft : (Equiv.addLeft a)\u207b\u00b9 = Equiv.addLeft (-a) := Equiv.coe_inj.1 rfl", "@[to_additive]\ntheorem mulEquiv_symm_apply (e : \u03b1 \u2243 \u03b2) [Mul \u03b2] (b : \u03b2) :\n    letI := Equiv.mul e\n    (mulEquiv e).symm b = e.symm b", "@[to_additive]\ntheorem Equiv.multipliable_iff_of_mulSupport {g : \u03b3 \u2192 \u03b1} (e : mulSupport f \u2243 mulSupport g)\n    (he : \u2200 x : mulSupport f, g (e x) = f x) : Multipliable f \u2194 Multipliable g", "theorem ofLeftInverse'_eq_ofInjective {\u03b1 \u03b2 : Type*} (f : \u03b1 \u2192 \u03b2) (f_inv : \u03b2 \u2192 \u03b1)\n    (hf : LeftInverse f_inv f) : ofLeftInverse' f f_inv hf = ofInjective f hf.injective", "@[simp]\ntheorem optionEquivSumPUnit_none : optionEquivSumPUnit \u03b1 none = Sum.inr PUnit.unit", "@[simp]\ntheorem optionEquivSumPUnit_some (a) : optionEquivSumPUnit \u03b1 (some a) = Sum.inl a", "@[simp]\ntheorem option_symm_apply_none_iff : e.symm none = none \u2194 e none = none", "theorem _root_.Equiv.polishSpace_induced [t : TopologicalSpace \u03b2] [PolishSpace \u03b2] (f : \u03b1 \u2243 \u03b2) :\n    @PolishSpace \u03b1 (t.induced f)", "theorem pow_def {N : Type*} [Pow \u03b2 N] (n : N) (x : \u03b1) :\n    letI := e.pow N\n    x ^ n = e.symm (e x ^ n)", "@[simp]\ntheorem prodSumDistrib_apply_left (a : \u03b1) (b : \u03b2) :\n    prodSumDistrib \u03b1 \u03b2 \u03b3 (a, Sum.inl b) = Sum.inl (a, b)", "@[simp] theorem refl_apply (x : \u03b1) : Equiv.refl \u03b1 x = x", "theorem removeNone_aux_inv (x : \u03b1) : removeNone_aux e.symm (removeNone_aux e x) = x", "@[simp]\ntheorem subtypeQuotientEquivQuotientSubtype_mk (p\u2081 : \u03b1 \u2192 Prop)\n    [s\u2081 : Setoid \u03b1] [s\u2082 : Setoid (Subtype p\u2081)] (p\u2082 : Quotient s\u2081 \u2192 Prop) (hp\u2082 : \u2200 a, p\u2081 a \u2194 p\u2082 \u27e6a\u27e7)\n    (h : \u2200 x y : Subtype p\u2081, @Setoid.r _ s\u2082 x y \u2194 (x : \u03b1) \u2248 y)\n    (x hx) : subtypeQuotientEquivQuotientSubtype p\u2081 p\u2082 hp\u2082 h \u27e8\u27e6x\u27e7, hx\u27e9 = \u27e6\u27e8x, (hp\u2082 _).2 hx\u27e9\u27e7", "@[simp]\ntheorem sumAssoc_symm_apply_inr_inr {\u03b1 \u03b2 \u03b3} (c) : (sumAssoc \u03b1 \u03b2 \u03b3).symm (inr (inr c)) = inr c", "@[simp]\ntheorem sumProdDistrib_symm_apply_left (a : \u03b1 \u00d7 \u03b3) :\n    (sumProdDistrib \u03b1 \u03b2 \u03b3).symm (inl a) = (inl a.1, a.2)", "theorem surjective_comp (e : \u03b1 \u2243 \u03b2) (f : \u03b2 \u2192 \u03b3) : Surjective (f \u2218 e) \u2194 Surjective f", "@[simp]\ntheorem symm_trans_swap_trans [DecidableEq \u03b2] (a b : \u03b1) (e : \u03b1 \u2243 \u03b2) :\n    (e.symm.trans (swap a b)).trans e = swap (e a) (e b)", "theorem Equiv.toEmbedding_apply (a : \u03b1) : f.toEmbedding a = f a", "@[simp]\ntheorem toFun_inducedStructureEquiv_Symm (e : M \u2243 N) :\n    (by\n    letI : L.Structure N := inducedStructure e\n    exact DFunLike.coe (@inducedStructureEquiv L M N _ e).symm) = (e.symm : N \u2192 M)", "theorem trans_assoc {\u03b4} (ab : \u03b1 \u2243 \u03b2) (bc : \u03b2 \u2243 \u03b3) (cd : \u03b3 \u2243 \u03b4) :\n    (ab.trans bc).trans cd = ab.trans (bc.trans cd)", "@[simp, mfld_simps]\ntheorem trans_toPartialEquiv :\n    (e.trans e').toPartialEquiv = e.toPartialEquiv.trans e'.toPartialEquiv", "theorem Equiv.uncountable_iff (e : \u03b1 \u2243 \u03b2) : Uncountable \u03b1 \u2194 Uncountable \u03b2", "theorem Equiv.uniformEmbedding {\u03b1 \u03b2 : Type*} [UniformSpace \u03b1] [UniformSpace \u03b2] (f : \u03b1 \u2243 \u03b2)\n    (h\u2081 : UniformContinuous f) (h\u2082 : UniformContinuous f.symm) : UniformEmbedding f", "@[simp] lemma zpow_addLeft (n : \u2124) : Equiv.addLeft a ^ n = Equiv.addLeft (n \u2022 a)", "theorem ball_subset_closedBall {x : E} {r : \u211d} : ball x r \u2286 closedBall x r", "theorem dvd_lcm_left (x y : R) : x \u2223 lcm x y", "theorem gcd_eq_gcd_ab (a b : R) : (gcd a b : R) = a * gcdA a b + b * gcdB a b", "theorem gcd_val (a b : R) : gcd a b = gcd (b % a) a", "@[simp]\ntheorem lcm_dvd_iff {x y z : R} : lcm x y \u2223 z \u2194 x \u2223 z \u2227 y \u2223 z", "theorem mod_add_div' (m k : R) : m % k + m / k * k = m", "theorem val_dvd_le : \u2200 a b : R, b \u2223 a \u2192 a \u2260 0 \u2192 \u00aca \u227a b", "theorem xgcd_val (x y : R) : xgcd x y = (gcdA x y, gcdB x y)", "theorem OrthocentricSystem.exists_circumradius_eq {s : Set P} (ho : OrthocentricSystem s) :\n    \u2203 r : \u211d, \u2200 t : Triangle \u211d P, Set.range t.points \u2286 s \u2192 t.circumradius = r", "theorem Sphere.center_ne_iff_ne_of_mem {s\u2081 s\u2082 : Sphere P} {p : P} (hs\u2081 : p \u2208 s\u2081) (hs\u2082 : p \u2208 s\u2082) :\n    s\u2081.center \u2260 s\u2082.center \u2194 s\u2081 \u2260 s\u2082", "theorem dist_div_sin_oangle_eq_two_mul_radius {s : Sphere P} {p\u2081 p\u2082 p\u2083 : P} (hp\u2081 : p\u2081 \u2208 s)\n    (hp\u2082 : p\u2082 \u2208 s) (hp\u2083 : p\u2083 \u2208 s) (hp\u2081p\u2082 : p\u2081 \u2260 p\u2082) (hp\u2081p\u2083 : p\u2081 \u2260 p\u2083) (hp\u2082p\u2083 : p\u2082 \u2260 p\u2083) :\n    dist p\u2081 p\u2083 / |Real.Angle.sin (\u2221 p\u2081 p\u2082 p\u2083)| = 2 * s.radius", "theorem tan_div_two_smul_rotation_pi_div_two_vadd_midpoint_eq_center {s : Sphere P} {p\u2081 p\u2082 : P}\n    (hp\u2081 : p\u2081 \u2208 s) (hp\u2082 : p\u2082 \u2208 s) (h : p\u2081 \u2260 p\u2082) :\n    (Real.Angle.tan (\u2221 p\u2082 p\u2081 s.center) / 2) \u2022 o.rotation (\u03c0 / 2 : \u211d) (p\u2082 -\u1d65 p\u2081) +\u1d65\n      midpoint \u211d p\u2081 p\u2082 = s.center", "theorem abs_oangle_left_toReal_lt_pi_div_two_of_dist_eq {p\u2081 p\u2082 p\u2083 : P}\n    (h : dist p\u2081 p\u2082 = dist p\u2081 p\u2083) : |(\u2221 p\u2082 p\u2083 p\u2081).toReal| < \u03c0 / 2", "theorem affineSpan_of_orthocentricSystem {s : Set P} (ho : OrthocentricSystem s) {p : Fin 3 \u2192 P}\n    (hps : Set.range p \u2286 s) (hpi : Function.Injective p) :\n    affineSpan \u211d (Set.range p) = affineSpan \u211d s", "theorem angle_eq_arcsin_of_angle_eq_pi_div_two {p\u2081 p\u2082 p\u2083 : P} (h : \u2220 p\u2081 p\u2082 p\u2083 = \u03c0 / 2)\n    (h0 : p\u2081 \u2260 p\u2082 \u2228 p\u2083 \u2260 p\u2082) : \u2220 p\u2082 p\u2083 p\u2081 = Real.arcsin (dist p\u2081 p\u2082 / dist p\u2081 p\u2083)", "@[simp]\ntheorem center_eq_inversion' : c = inversion c R x \u2194 x = c \u2228 R = 0", "nonrec theorem cos_eq_one_iff_angle_eq_zero {p\u2081 p\u2082 p\u2083 : P} :\n    Real.cos (\u2220 p\u2081 p\u2082 p\u2083) = 1 \u2194 \u2220 p\u2081 p\u2082 p\u2083 = 0", "theorem cos_oangle_right_of_oangle_eq_pi_div_two {p\u2081 p\u2082 p\u2083 : P} (h : \u2221 p\u2081 p\u2082 p\u2083 = \u2191(\u03c0 / 2)) :\n    Real.Angle.cos (\u2221 p\u2082 p\u2083 p\u2081) = dist p\u2083 p\u2082 / dist p\u2081 p\u2083", "theorem dist_affineCombination {\u03b9 : Type*} {s : Finset \u03b9} {w\u2081 w\u2082 : \u03b9 \u2192 \u211d} (p : \u03b9 \u2192 P)\n    (h\u2081 : \u2211 i \u2208 s, w\u2081 i = 1) (h\u2082 : \u2211 i \u2208 s, w\u2082 i = 1) : by\n      have a\u2081 := s.affineCombination \u211d p w\u2081\n      have a\u2082 := s.affineCombination \u211d p w\u2082\n      exact dist a\u2081 a\u2082 * dist a\u2081 a\u2082 = (-\u2211 i\u2081 \u2208 s, \u2211 i\u2082 \u2208 s,\n        (w\u2081 - w\u2082) i\u2081 * (w\u2081 - w\u2082) i\u2082 * (dist (p i\u2081) (p i\u2082) * dist (p i\u2081) (p i\u2082))) / 2", "theorem dist_center_eq_dist_center_of_mem_sphere' {p\u2081 p\u2082 : P} {s : Sphere P} (hp\u2081 : p\u2081 \u2208 s)\n    (hp\u2082 : p\u2082 \u2208 s) : dist s.center p\u2081 = dist s.center p\u2082", "theorem dist_eq_iff_dist_orthogonalProjection_eq {s : AffineSubspace \u211d P} [Nonempty s]\n    [HasOrthogonalProjection s.direction] {p1 p2 : P} (p3 : P) (hp1 : p1 \u2208 s) (hp2 : p2 \u2208 s) :\n    dist p1 p3 = dist p2 p3 \u2194\n      dist p1 (orthogonalProjection s p3) = dist p2 (orthogonalProjection s p3)", "theorem dist_smul_vadd_eq_dist {v : V} (p\u2081 p\u2082 : P) (hv : v \u2260 0) (r : \u211d) :\n    dist (r \u2022 v +\u1d65 p\u2081) p\u2082 = dist p\u2081 p\u2082 \u2194 r = 0 \u2228 r = -2 * \u27eav, p\u2081 -\u1d65 p\u2082\u27eb / \u27eav, v\u27eb", "theorem exists_circumradius_eq_of_cospherical_subset {s : AffineSubspace \u211d P} {ps : Set P}\n    (h : ps \u2286 s) [Nonempty s] {n : \u2115} [FiniteDimensional \u211d s.direction]\n    (hd : finrank \u211d s.direction = n) (hc : Cospherical ps) :\n    \u2203 r : \u211d, \u2200 sx : Simplex \u211d P n, Set.range sx.points \u2286 ps \u2192 sx.circumradius = r", "theorem exists_circumsphere_eq_of_cospherical {ps : Set P} {n : \u2115} [FiniteDimensional \u211d V]\n    (hd : finrank \u211d V = n) (hc : Cospherical ps) :\n    \u2203 c : Sphere P, \u2200 sx : Simplex \u211d P n, Set.range sx.points \u2286 ps \u2192 sx.circumsphere = c", "theorem exists_dist_eq_circumradius_of_subset_insert_orthocenter {t : Triangle \u211d P}\n    (ho : t.orthocenter \u2209 Set.range t.points) {p : Fin 3 \u2192 P}\n    (hps : Set.range p \u2286 insert t.orthocenter (Set.range t.points)) (hpi : Function.Injective p) :\n    \u2203 c \u2208 affineSpan \u211d (Set.range t.points), \u2200 p\u2081 \u2208 Set.range p, dist p\u2081 c = t.circumradius", "theorem inner_pos_or_eq_of_dist_le_radius {s : Sphere P} {p\u2081 p\u2082 : P} (hp\u2081 : p\u2081 \u2208 s)\n    (hp\u2082 : dist p\u2082 s.center \u2264 s.radius) : 0 < \u27eap\u2081 -\u1d65 p\u2082, p\u2081 -\u1d65 s.center\u27eb \u2228 p\u2081 = p\u2082", "theorem inner_vsub_vsub_of_mem_sphere_of_mem_sphere {p\u2081 p\u2082 : P} {s\u2081 s\u2082 : Sphere P} (hp\u2081s\u2081 : p\u2081 \u2208 s\u2081)\n    (hp\u2082s\u2081 : p\u2082 \u2208 s\u2081) (hp\u2081s\u2082 : p\u2081 \u2208 s\u2082) (hp\u2082s\u2082 : p\u2082 \u2208 s\u2082) :\n    \u27eas\u2082.center -\u1d65 s\u2081.center, p\u2082 -\u1d65 p\u2081\u27eb = 0", "theorem inter_eq_singleton_orthogonalProjection {s : AffineSubspace \u211d P} [Nonempty s]\n    [HasOrthogonalProjection s.direction] (p : P) :\n    (s : Set P) \u2229 mk' p s.direction\u15ee = {\u2191(orthogonalProjection s p)}", "theorem inversion_of_mem_sphere (h : x \u2208 Metric.sphere c R) : inversion c R x = x", "theorem oangle_ne_zero_and_ne_pi_iff_affineIndependent {p\u2081 p\u2082 p\u2083 : P} :\n    \u2221 p\u2081 p\u2082 p\u2083 \u2260 0 \u2227 \u2221 p\u2081 p\u2082 p\u2083 \u2260 \u03c0 \u2194 AffineIndependent \u211d ![p\u2081, p\u2082, p\u2083]", "theorem reflection_eq_iff_orthogonalProjection_eq (s\u2081 s\u2082 : AffineSubspace \u211d P) [Nonempty s\u2081]\n    [Nonempty s\u2082] [HasOrthogonalProjection s\u2081.direction] [HasOrthogonalProjection s\u2082.direction]\n    (p : P) :\n    reflection s\u2081 p = reflection s\u2082 p \u2194\n      (orthogonalProjection s\u2081 p : P) = orthogonalProjection s\u2082 p", "theorem reflection_eq_self_iff {s : AffineSubspace \u211d P} [Nonempty s]\n    [HasOrthogonalProjection s.direction] (p : P) : reflection s p = p \u2194 p \u2208 s", "theorem right_ne_of_oangle_sign_eq_neg_one {p\u2081 p\u2082 p\u2083 : P} (h : (\u2221 p\u2081 p\u2082 p\u2083).sign = -1) : p\u2083 \u2260 p\u2082", "theorem right_ne_of_oangle_sign_ne_zero {p\u2081 p\u2082 p\u2083 : P} (h : (\u2221 p\u2081 p\u2082 p\u2083).sign \u2260 0) : p\u2083 \u2260 p\u2082", "theorem sin_oangle_left_mul_dist_of_oangle_eq_pi_div_two {p\u2081 p\u2082 p\u2083 : P}\n    (h : \u2221 p\u2081 p\u2082 p\u2083 = \u2191(\u03c0 / 2)) : Real.Angle.sin (\u2221 p\u2083 p\u2081 p\u2082) * dist p\u2081 p\u2083 = dist p\u2083 p\u2082", "theorem sin_oangle_right_mul_dist_of_oangle_eq_pi_div_two {p\u2081 p\u2082 p\u2083 : P}\n    (h : \u2221 p\u2081 p\u2082 p\u2083 = \u2191(\u03c0 / 2)) : Real.Angle.sin (\u2221 p\u2082 p\u2083 p\u2081) * dist p\u2081 p\u2083 = dist p\u2081 p\u2082", "theorem tan_oangle_right_mul_dist_of_oangle_eq_pi_div_two {p\u2081 p\u2082 p\u2083 : P}\n    (h : \u2221 p\u2081 p\u2082 p\u2083 = \u2191(\u03c0 / 2)) : Real.Angle.tan (\u2221 p\u2082 p\u2083 p\u2081) * dist p\u2083 p\u2082 = dist p\u2081 p\u2082", "theorem vsub_orthogonalProjection_mem_direction {s : AffineSubspace \u211d P} [Nonempty s]\n    [HasOrthogonalProjection s.direction] {p1 : P} (p2 : P) (hp1 : p1 \u2208 s) :\n    \u2191((\u27e8p1, hp1\u27e9 : s) -\u1d65 orthogonalProjection s p2 : s.direction) \u2208 s.direction", "theorem wbtw_of_collinear_of_dist_center_le_radius {s : Sphere P} {p\u2081 p\u2082 p\u2083 : P}\n    (h : Collinear \u211d ({p\u2081, p\u2082, p\u2083} : Set P)) (hp\u2081 : p\u2081 \u2208 s) (hp\u2082 : dist p\u2082 s.center \u2264 s.radius)\n    (hp\u2083 : p\u2083 \u2208 s) (hp\u2081p\u2083 : p\u2081 \u2260 p\u2083) : Wbtw \u211d p\u2081 p\u2082 p\u2083", "theorem EuclideanSpace.closedBall_zero_eq {n : Type*} [Fintype n] (r : \u211d) (hr : 0 \u2264 r) :\n    Metric.closedBall (0 : EuclideanSpace \u211d n) r = {x | \u2211 i, x i ^ 2 \u2264 r ^ 2}", "lemma prod_primesBelow_tsum_eq_tsum_smoothNumbers (hsum : Summable (\u2016f \u00b7\u2016)) (N : \u2115) :\n    \u220f p \u2208 N.primesBelow, \u2211' n : \u2115, f (p ^ n) = \u2211' m : N.smoothNumbers, f m", "theorem integral_cos_mul_cos_pow_aux (hn : 2 \u2264 n) (hz : z \u2260 0) :\n    (\u222b x in (0 : \u211d)..\u03c0 / 2, Complex.cos (2 * z * x) * (cos x : \u2102) ^ n) =\n      n / (2 * z) *\n        \u222b x in (0 : \u211d)..\u03c0 / 2, Complex.sin (2 * z * x) * sin x * (cos x : \u2102) ^ (n - 1)", "theorem EventuallyEq.countable_iInter [Countable \u03b9] {s t : \u03b9 \u2192 Set \u03b1} (h : \u2200 i, s i =\u1da0[l] t i) :\n    \u22c2 i, s i =\u1da0[l] \u22c2 i, t i", "theorem EventuallyLE.countable_bUnion {\u03b9 : Type*} {S : Set \u03b9} (hS : S.Countable)\n    {s t : \u2200 i \u2208 S, Set \u03b1} (h : \u2200 i hi, s i hi \u2264\u1da0[l] t i hi) :\n    \u22c3 i \u2208 S, s i \u2039_\u203a \u2264\u1da0[l] \u22c3 i \u2208 S, t i \u2039_\u203a", "theorem EventuallyMeasurable.congr\n    (hf : EventuallyMeasurable m l f) (hgf : g =\u1da0[l] f) : EventuallyMeasurable m l g", "theorem ExceptT.goto_mkLabel {\u03b1 \u03b2 \u03b5 : Type _} (x : Label (Except.{u, u} \u03b5 \u03b1) m \u03b2) (i : \u03b1) :\n    goto (ExceptT.mkLabel x) i = ExceptT.mk (Except.ok <$> goto x (Except.ok i))", "theorem u_le_one (x : E) : u x \u2264 1", "theorem u_support : support (u : E \u2192 \u211d) = ball 0 1", "theorem y_le_one {D : \u211d} (x : E) (Dpos : 0 < D) : y D x \u2264 1", "@[elab_as_elim]\ntheorem induction {C : ExteriorAlgebra R M \u2192 Prop}\n    (algebraMap : \u2200 r, C (algebraMap R (ExteriorAlgebra R M) r)) (\u03b9 : \u2200 x, C (\u03b9 R x))\n    (mul : \u2200 a b, C a \u2192 C b \u2192 C (a * b)) (add : \u2200 a b, C a \u2192 C b \u2192 C (a + b))\n    (a : ExteriorAlgebra R M) : C a", "@[ext]\ntheorem lhom_ext \u2983f g : ExteriorAlgebra R M \u2192\u2097[R] N\u2984\n    (h : \u2200 i, f.compAlternatingMap (\u03b9Multi R i) = g.compAlternatingMap (\u03b9Multi R i)) : f = g", "@[simp]\ntheorem map_apply_\u03b9Multi {n : \u2115} (f : M \u2192\u2097[R] N) (m : Fin n \u2192 M) :\n    map f (\u03b9Multi R n m) = \u03b9Multi R n (f \u2218 m)", "theorem \u03b9Inv_comp_map (f : M \u2192\u2097[R] N) :\n    \u03b9Inv.comp (map f).toLinearMap = f.comp \u03b9Inv", "@[simp]\ntheorem \u03b9_add_mul_swap (x y : M) : \u03b9 R x * \u03b9 R y + \u03b9 R y * \u03b9 R x = 0", "theorem \u03b9_mul_prod_list {n : \u2115} (f : Fin n \u2192 M) (i : Fin n) :\n    (\u03b9 R <| f i) * (List.ofFn fun i => \u03b9 R <| f i).prod = 0", "theorem char_mul_comm (V : FdRep k G) (g : G) (h : G) :\n    V.character (h * g) = V.character (g * h)", "lemma Solution.two_le_multiplicity : 2 \u2264 S.multiplicity", "lemma ex_cube_add_cube_eq_and_isCoprime_and_not_dvd_and_dvd :\n    \u2203 (a' b' : \ud835\udcde K), a' ^ 3 + b' ^ 3 = S'.u * S'.c ^ 3 \u2227 IsCoprime a' b' \u2227 \u00ac \u03bb \u2223 a' \u2227\n      \u00ac \u03bb \u2223 b' \u2227 \u03bb ^ 2 \u2223 a' + b'", "lemma lambda_sq_dvd_c : \u03bb ^ 2 \u2223 S'.c", "theorem FiberBundle.chartedSpace_chartAt_symm_fst (x : TotalSpace F E) (y : ModelProd HB F)\n    (hy : y \u2208 (chartAt (ModelProd HB F) x).target) :\n    ((chartAt (ModelProd HB F) x).symm y).proj = (chartAt HB x.proj).symm y.1", "theorem surjective_proj [Nonempty F] : Function.Surjective (\u03c0 F E)", "protected theorem inv_mul_cancel {p : Ring.DirectLimit G f} (hp : p \u2260 0) : inv G f p * p = 1", "protected theorem mul_inv_cancel {p : Ring.DirectLimit G f} (hp : p \u2260 0) : p * inv G f p = 1", "theorem closure.isSubfield : IsSubfield (closure S)", "theorem closure.isSubmonoid : IsSubmonoid (closure S)", "theorem closure_subset {T : Set F} (hT : IsSubfield T) (H : S \u2286 T) : closure S \u2286 T", "theorem exists_primitive_element_of_finite_intermediateField\n    [Finite (IntermediateField F E)] (K : IntermediateField F E) : \u2203 \u03b1 : E, F\u27ee\u03b1\u27ef = K", "theorem finSepDegree_mul_finSepDegree_of_isAlgebraic\n    [Algebra E K] [IsScalarTower F E K] [Algebra.IsAlgebraic E K] :\n    finSepDegree F E * finSepDegree E K = finSepDegree F K", "theorem sepDegree_mul_sepDegree_of_isAlgebraic (K : Type v) [Field K] [Algebra F K]\n    [Algebra E K] [IsScalarTower F E K] [Algebra.IsAlgebraic F E] :\n    sepDegree F E * sepDegree E K = sepDegree F K", "theorem Filter.EventuallyEq.differentiableWithinAt_iff (h : f\u2080 =\u1da0[\ud835\udcdd[s] x] f\u2081) (hx : f\u2080 x = f\u2081 x) :\n    DifferentiableWithinAt \ud835\udd5c f\u2080 s x \u2194 DifferentiableWithinAt \ud835\udd5c f\u2081 s x", "theorem Filter.EventuallyEq.fderivWithin_eq (hs : f\u2081 =\u1da0[\ud835\udcdd[s] x] f) (hx : f\u2081 x = f x) :\n    fderivWithin \ud835\udd5c f\u2081 s x = fderivWithin \ud835\udd5c f s x", "theorem EventuallyEq.fun_comp {f g : \u03b1 \u2192 \u03b2} {l : Filter \u03b1} (H : f =\u1da0[l] g) (h : \u03b2 \u2192 \u03b3) :\n    h \u2218 f =\u1da0[l] h \u2218 g", "@[to_additive]\ntheorem EventuallyEq.inv [Inv \u03b2] {f g : \u03b1 \u2192 \u03b2} {l : Filter \u03b1} (h : f =\u1da0[l] g) :\n    (fun x => (f x)\u207b\u00b9) =\u1da0[l] fun x => (g x)\u207b\u00b9", "theorem Filter.EventuallyEq.iteratedFDerivWithin_eq (h : f\u2081 =\u1da0[\ud835\udcdd[s] x] f) (hx : f\u2081 x = f x)\n    (n : \u2115) : iteratedFDerivWithin \ud835\udd5c n f\u2081 s x = iteratedFDerivWithin \ud835\udd5c n f s x", "nonrec theorem Filter.EventuallyEq.self_of_nhdsSet {f g : X \u2192 Y} (h : f =\u1da0[\ud835\udcdd\u02e2 s] g) : EqOn f g s", "@[mono]\ntheorem EventuallyLE.inter {s t s' t' : Set \u03b1} {l : Filter \u03b1} (h : s \u2264\u1da0[l] t) (h' : s' \u2264\u1da0[l] t') :\n    (s \u2229 s' : Set \u03b1) \u2264\u1da0[l] (t \u2229 t' : Set \u03b1)", "@[to_additive (attr := simp, norm_cast)]\ntheorem const_div [Div M] (a b : M) : (\u2191(a / b) : Germ l M) = \u2191a / \u2191b", "@[elab_as_elim]\ntheorem inductionOn\u2083 (f : Germ l \u03b2) (g : Germ l \u03b3) (h : Germ l \u03b4)\n    {p : Germ l \u03b2 \u2192 Germ l \u03b3 \u2192 Germ l \u03b4 \u2192 Prop}\n    (H : \u2200 (f : \u03b1 \u2192 \u03b2) (g : \u03b1 \u2192 \u03b3) (h : \u03b1 \u2192 \u03b4), p f g h) : p f g h", "@[simp]\ntheorem map_id : map id = (id : Germ l \u03b2 \u2192 Germ l \u03b2)", "protected theorem HasAntitoneBasis.comap {l : Filter \u03b1} {s : \u03b9'' \u2192 Set \u03b1}\n    (hf : HasAntitoneBasis l s) (m : \u03b2 \u2192 \u03b1) : HasAntitoneBasis (comap m l) (m \u207b\u00b9' s \u00b7)", "theorem Filter.HasBasis.clusterPt_iff {\u03b9X \u03b9F} {pX : \u03b9X \u2192 Prop} {sX : \u03b9X \u2192 Set X} {pF : \u03b9F \u2192 Prop}\n    {sF : \u03b9F \u2192 Set X} {F : Filter X} (hX : (\ud835\udcdd x).HasBasis pX sX) (hF : F.HasBasis pF sF) :\n    ClusterPt x F \u2194 \u2200 \u2983i\u2984, pX i \u2192 \u2200 \u2983j\u2984, pF j \u2192 (sX i \u2229 sF j).Nonempty", "theorem HasBasis.comp_surjective (h : l.HasBasis p s) {g : \u03b9' \u2192 \u03b9} (hg : Function.Surjective g) :\n    l.HasBasis (p \u2218 g) (s \u2218 g)", "theorem Filter.HasBasis.equicontinuousWithinAt_iff {\u03ba\u2081 \u03ba\u2082 : Type*} {p\u2081 : \u03ba\u2081 \u2192 Prop}\n    {s\u2081 : \u03ba\u2081 \u2192 Set X} {p\u2082 : \u03ba\u2082 \u2192 Prop} {s\u2082 : \u03ba\u2082 \u2192 Set (\u03b1 \u00d7 \u03b1)} {F : \u03b9 \u2192 X \u2192 \u03b1} {S : Set X} {x\u2080 : X}\n    (hX : (\ud835\udcdd[S] x\u2080).HasBasis p\u2081 s\u2081) (h\u03b1 : (\ud835\udce4 \u03b1).HasBasis p\u2082 s\u2082) :\n    EquicontinuousWithinAt F S x\u2080 \u2194\n      \u2200 k\u2082, p\u2082 k\u2082 \u2192 \u2203 k\u2081, p\u2081 k\u2081 \u2227 \u2200 x \u2208 s\u2081 k\u2081, \u2200 i, (F i x\u2080, F i x) \u2208 s\u2082 k\u2082", "theorem HasBasis.forall_mem_mem (h : HasBasis l p s) {x : \u03b1} :\n    (\u2200 t \u2208 l, x \u2208 t) \u2194 \u2200 i, p i \u2192 x \u2208 s i", "theorem HasBasis.inf {\u03b9 \u03b9' : Type*} {p : \u03b9 \u2192 Prop} {s : \u03b9 \u2192 Set \u03b1} {p' : \u03b9' \u2192 Prop}\n    {s' : \u03b9' \u2192 Set \u03b1} (hl : l.HasBasis p s) (hl' : l'.HasBasis p' s') :\n    (l \u2293 l').HasBasis (fun i : \u03b9 \u00d7 \u03b9' => p i.1 \u2227 p' i.2) fun i => s i.1 \u2229 s' i.2", "theorem HasBasis.inf_basis_neBot_iff (hl : l.HasBasis p s) (hl' : l'.HasBasis p' s') :\n    NeBot (l \u2293 l') \u2194 \u2200 \u2983i\u2984, p i \u2192 \u2200 \u2983i'\u2984, p' i' \u2192 (s i \u2229 s' i').Nonempty", "theorem HasBasis.le_basis_iff (hl : l.HasBasis p s) (hl' : l'.HasBasis p' s') :\n    l \u2264 l' \u2194 \u2200 i', p' i' \u2192 \u2203 i, p i \u2227 s i \u2286 s' i'", "theorem HasBasis.limsInf_eq_iSup_sInf {p : \u03b9 \u2192 Prop} {s : \u03b9 \u2192 Set \u03b1} {f : Filter \u03b1}\n    (h : f.HasBasis p s) : limsInf f = \u2a06 (i) (_ : p i), sInf (s i)", "@[to_additive]\ntheorem Filter.HasBasis.nhds_of_one {\u03b9 : Sort*} {p : \u03b9 \u2192 Prop} {s : \u03b9 \u2192 Set G}\n    (hb : HasBasis (\ud835\udcdd 1 : Filter G) p s) (x : G) :\n    HasBasis (\ud835\udcdd x) p fun i => { y | y / x \u2208 s i }", "theorem HasBasis.sup_principal (hl : l.HasBasis p s) (t : Set \u03b1) :\n    (l \u2294 \ud835\udcdf t).HasBasis p fun i => s i \u222a t", "theorem HasBasis.tendsto_right_iff (hlb : lb.HasBasis pb sb) :\n    Tendsto f la lb \u2194 \u2200 i, pb i \u2192 \u2200\u1da0 x in la, f x \u2208 sb i", "theorem Filter.HasBasis.uniformContinuous_iff {\u03b9'} [UniformSpace \u03b2] {p : \u03b9 \u2192 Prop}\n    {s : \u03b9 \u2192 Set (\u03b1 \u00d7 \u03b1)} (ha : (\ud835\udce4 \u03b1).HasBasis p s) {q : \u03b9' \u2192 Prop} {t : \u03b9' \u2192 Set (\u03b2 \u00d7 \u03b2)}\n    (hb : (\ud835\udce4 \u03b2).HasBasis q t) {f : \u03b1 \u2192 \u03b2} :\n    UniformContinuous f \u2194 \u2200 i, q i \u2192 \u2203 j, p j \u2227 \u2200 x y, (x, y) \u2208 s j \u2192 (f x, f y) \u2208 t i", "theorem Filter.HasBasis.uniformEquicontinuousOn_iff_right {p : \u03ba \u2192 Prop}\n    {s : \u03ba \u2192 Set (\u03b1 \u00d7 \u03b1)} {F : \u03b9 \u2192 \u03b2 \u2192 \u03b1} {S : Set \u03b2} (h\u03b1 : (\ud835\udce4 \u03b1).HasBasis p s) :\n    UniformEquicontinuousOn F S \u2194\n      \u2200 k, p k \u2192 \u2200\u1da0 xy : \u03b2 \u00d7 \u03b2 in \ud835\udce4 \u03b2 \u2293 \ud835\udcdf (S \u00d7\u02e2 S), \u2200 i, (F i xy.1, F i xy.2) \u2208 s k", "protected lemma _root_.Filter.HasBasis.uniformSpace_eq_bot {\u03b9 p} {s : \u03b9 \u2192 Set (\u03b1 \u00d7 \u03b1)}\n    {u : UniformSpace \u03b1} (h : \ud835\udce4[u].HasBasis p s) :\n    u = \u22a5 \u2194 \u2203 i, p i \u2227 Pairwise fun x y : \u03b1 \u21a6 (x, y) \u2209 s i", "theorem filter_eq_generate (h : IsBasis p s) : h.filter = generate { U | \u2203 i, p i \u2227 s i = U }", "theorem IsBoundedUnder.bddAbove_range_of_cofinite [Preorder \u03b2] [IsDirected \u03b2 (\u00b7 \u2264 \u00b7)] {f : \u03b1 \u2192 \u03b2}\n    (hf : IsBoundedUnder (\u00b7 \u2264 \u00b7) cofinite f) : BddAbove (range f)", "theorem IsBoundedUnder.mono_le [Preorder \u03b2] {l : Filter \u03b1} {u v : \u03b1 \u2192 \u03b2}\n    (hu : IsBoundedUnder (\u00b7 \u2264 \u00b7) l u) (hv : v \u2264\u1da0[l] u) : IsBoundedUnder (\u00b7 \u2264 \u00b7) l v", "@[to_additive]\nprotected theorem NeBot.div : NeBot f \u2192 NeBot g \u2192 NeBot (f / g)", "@[to_additive]\nprotected theorem NeBot.inv : f.NeBot \u2192 f\u207b\u00b9.NeBot", "theorem OrderIso.apply_blimsup [CompleteLattice \u03b3] (e : \u03b1 \u2243o \u03b3) :\n    DFunLike.coe e (blimsup u f p) = blimsup ((DFunLike.coe e) \u2218 u) f p", "@[nontriviality]\ntheorem Subsingleton.atBot_eq (\u03b1) [Subsingleton \u03b1] [Preorder \u03b1] : (atBot : Filter \u03b1) = \u22a4", "protected theorem Tendsto.Ico {l\u2081 l\u2082 : Filter \u03b1} [TendstoIxxClass Ico l\u2081 l\u2082] {lb : Filter \u03b2}\n    {u\u2081 u\u2082 : \u03b2 \u2192 \u03b1} (h\u2081 : Tendsto u\u2081 lb l\u2081) (h\u2082 : Tendsto u\u2082 lb l\u2081) :\n    Tendsto (fun x => Ico (u\u2081 x) (u\u2082 x)) lb l\u2082.smallSets", "theorem Tendsto.atTop_div_const (hr : 0 < r) (hf : Tendsto f l atTop) :\n    Tendsto (fun x => f x / r) l atTop", "theorem Tendsto.atTop_nsmul_neg_const {f : \u03b1 \u2192 \u2115} (hr : r < 0) (hf : Tendsto f l atTop) :\n    Tendsto (fun x => f x \u2022 r) l atBot", "theorem Filter.Tendsto.congr_dist {f\u2081 f\u2082 : \u03b9 \u2192 \u03b1} {p : Filter \u03b9} {a : \u03b1}\n    (h\u2081 : Tendsto f\u2081 p (\ud835\udcdd a)) (h : Tendsto (fun x => dist (f\u2081 x) (f\u2082 x)) p (\ud835\udcdd 0)) :\n    Tendsto f\u2082 p (\ud835\udcdd a)", "protected lemma Filter.Tendsto.dist {f g : \u03b2 \u2192 \u03b1} {x : Filter \u03b2} {a b : \u03b1}\n    (hf : Tendsto f x (\ud835\udcdd a)) (hg : Tendsto g x (\ud835\udcdd b)) :\n    Tendsto (fun x => dist (f x) (g x)) x (\ud835\udcdd (dist a b))", "theorem Filter.Tendsto.div_atTop {a : \ud835\udd5c} (h : Tendsto f l (\ud835\udcdd a)) (hg : Tendsto g l atTop) :\n    Tendsto (fun x => f x / g x) l (\ud835\udcdd 0)", "theorem Tendsto.eventually_forall_ge_atTop [Preorder \u03b2] {l : Filter \u03b1}\n    {p : \u03b2 \u2192 Prop} {f : \u03b1 \u2192 \u03b2} (hf : Tendsto f l atTop) (h_evtl : \u2200\u1da0 x in atTop, p x) :\n    \u2200\u1da0 x in l, \u2200 y, f x \u2264 y \u2192 p y", "lemma finset_sup_nhds_apply [SemilatticeSup L] [OrderBot L] [ContinuousSup L]\n    (hs : \u2200 i \u2208 s, Tendsto (f i) l (\ud835\udcdd (g i))) :\n    Tendsto (fun a \u21a6 s.sup (f \u00b7 a)) l (\ud835\udcdd (s.sup g))", "theorem Tendsto.frequently_map {l\u2081 : Filter \u03b1} {l\u2082 : Filter \u03b2} {p : \u03b1 \u2192 Prop} {q : \u03b2 \u2192 Prop}\n    (f : \u03b1 \u2192 \u03b2) (c : Filter.Tendsto f l\u2081 l\u2082) (w : \u2200 x, p x \u2192 q (f x)) (h : \u2203\u1da0 x in l\u2081, p x) :\n    \u2203\u1da0 y in l\u2082, q y", "theorem Tendsto.fst {h : Filter \u03b3} {m : \u03b1 \u2192 \u03b2 \u00d7 \u03b3} (H : Tendsto m f (g \u00d7\u02e2 h)) :\n    Tendsto (fun a \u21a6 (m a).1) f g", "theorem Filter.Tendsto.inner {f g : \u03b1 \u2192 E} {l : Filter \u03b1} {x y : E} (hf : Tendsto f l (\ud835\udcdd x))\n    (hg : Tendsto g l (\ud835\udcdd y)) : Tendsto (fun t => \u27eaf t, g t\u27eb) l (\ud835\udcdd \u27eax, y\u27eb)", "@[to_additive]\nprotected theorem Tendsto.inv_inv : Tendsto m f\u2081 f\u2082 \u2192 Tendsto m f\u2081\u207b\u00b9 f\u2082\u207b\u00b9", "theorem Filter.Tendsto.inv_tendsto_atTop (h : Tendsto f l atTop) : Tendsto f\u207b\u00b9 l (\ud835\udcdd 0)", "@[to_additive Filter.Tendsto.nnnorm]\ntheorem Filter.Tendsto.nnnorm' (h : Tendsto f l (\ud835\udcdd a)) : Tendsto (fun x => \u2016f x\u2016\u208a) l (\ud835\udcdd \u2016a\u2016\u208a)", "theorem Filter.Tendsto.subseq_mem_entourage {V : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)} (hV : \u2200 n, V n \u2208 \ud835\udce4 \u03b1) {u : \u2115 \u2192 \u03b1}\n    {a : \u03b1} (hu : Tendsto u atTop (\ud835\udcdd a)) : \u2203 \u03c6 : \u2115 \u2192 \u2115, StrictMono \u03c6 \u2227 (u (\u03c6 0), a) \u2208 V 0 \u2227\n      \u2200 n, (u <| \u03c6 (n + 1), u <| \u03c6 n) \u2208 V (n + 1)", "theorem Filter.Tendsto.update [DecidableEq \u03b9] {l : Filter Y} {f : Y \u2192 \u2200 i, \u03c0 i} {x : \u2200 i, \u03c0 i}\n    (hf : Tendsto f l (\ud835\udcdd x)) (i : \u03b9) {g : Y \u2192 \u03c0 i} {xi : \u03c0 i} (hg : Tendsto g l (\ud835\udcdd xi)) :\n    Tendsto (fun a => update (f a) i (g a)) l (\ud835\udcdd <| update x i xi)", "theorem Filter.Tendsto.zero_mul_isBoundedUnder_le {f g : \u03b9 \u2192 \u03b1} {l : Filter \u03b9}\n    (hf : Tendsto f l (\ud835\udcdd 0)) (hg : IsBoundedUnder (\u00b7 \u2264 \u00b7) l ((\u2016\u00b7\u2016) \u2218 g)) :\n    Tendsto (fun x => f x * g x) l (\ud835\udcdd 0)", "theorem bddBelow_range_of_tendsto_atTop_atTop [IsDirected \u03b1 (\u00b7 \u2265 \u00b7)] {u : \u2115 \u2192 \u03b1}\n    (hx : Tendsto u atTop atTop) : BddBelow (Set.range u)", "theorem bliminf_eq_iSup_biInf {f : Filter \u03b2} {p : \u03b2 \u2192 Prop} {u : \u03b2 \u2192 \u03b1} :\n    bliminf u f p = \u2a06 s \u2208 f, \u2a05 (b) (_ : p b \u2227 b \u2208 s), u b", "lemma bliminf_eq_liminf {f : Filter \u03b2} {u : \u03b2 \u2192 \u03b1} {p : \u03b2 \u2192 Prop} :\n    bliminf u f p = liminf u (f \u2293 \ud835\udcdf {x | p x})", "@[simp]\ntheorem bliminf_false {f : Filter \u03b2} {u : \u03b2 \u2192 \u03b1} : (bliminf u f fun _ => False) = \u22a4", "theorem bliminf_or_le_inf : (bliminf u f fun x => p x \u2228 q x) \u2264 bliminf u f p \u2293 bliminf u f q", "@[to_additive (attr := simp)]\ntheorem bot_div : \u22a5 / g = \u22a5", "theorem cardinal_iInter_mem {s : \u03b9 \u2192 Set \u03b1} (hic : #\u03b9 < c) :\n    (\u22c2 i, s i) \u2208 l \u2194 \u2200 i, s i \u2208 l", "@[simp]\nprotected theorem closure_singleton (l : Filter \u03b1) : closure {l} = Ici l", "@[simp]\ntheorem comap_abs_atTop : comap (abs : \u03b1 \u2192 \u03b1) atTop = atBot \u2294 atTop", "theorem comap_comm (G : Filter \u03b4) : comap \u03c6 (comap \u03c8 G) = comap \u03b8 (comap \u03c1 G)", "theorem comap_embedding_atTop [Preorder \u03b2] [Preorder \u03b3] {e : \u03b2 \u2192 \u03b3}\n    (hm : \u2200 b\u2081 b\u2082, e b\u2081 \u2264 e b\u2082 \u2194 b\u2081 \u2264 b\u2082) (hu : \u2200 c, \u2203 b, c \u2264 e b) : comap e atTop = atTop", "theorem comap_neBot_iff_frequently {f : Filter \u03b2} {m : \u03b1 \u2192 \u03b2} :\n    NeBot (comap m f) \u2194 \u2203\u1da0 y in f, y \u2208 range m", "theorem comap_sSup {s : Set (Filter \u03b2)} {m : \u03b1 \u2192 \u03b2} : comap m (sSup s) = \u2a06 f \u2208 s, comap m f", "theorem coprod\u1d62_neBot_iff' :\n    NeBot (Filter.coprod\u1d62 f) \u2194 (\u2200 i, Nonempty (\u03b1 i)) \u2227 \u2203 d, NeBot (f d)", "theorem disjoint_cofinite_right : Disjoint l cofinite \u2194 \u2203 s \u2208 l, Set.Finite s", "@[simp]\nlemma disjoint_prod {f' : Filter \u03b1} {g' : Filter \u03b2} :\n    Disjoint (f \u00d7\u02e2 g) (f' \u00d7\u02e2 g') \u2194 Disjoint f f' \u2228 Disjoint g g'", "@[simp]\ntheorem empty_not_mem (f : Filter \u03b1) [NeBot f] : \u00ac\u2205 \u2208 f", "theorem eventuallyConst_pred {p : \u03b1 \u2192 Prop} :\n    EventuallyConst p l \u2194 (\u2200\u1da0 x in l, p x) \u2228 (\u2200\u1da0 x in l, \u00acp x)", "theorem eventuallyEq_inf_principal_iff {F : Filter \u03b1} {s : Set \u03b1} {f g : \u03b1 \u2192 \u03b2} :\n    f =\u1da0[F \u2293 \ud835\udcdf s] g \u2194 \u2200\u1da0 x in F, x \u2208 s \u2192 f x = g x", "theorem eventually_atBot_prod_self [SemilatticeInf \u03b1] [Nonempty \u03b1] {p : \u03b1 \u00d7 \u03b1 \u2192 Prop} :\n    (\u2200\u1da0 x in atBot, p x) \u2194 \u2203 a, \u2200 k l, k \u2264 a \u2192 l \u2264 a \u2192 p (k, l)", "@[simp]\ntheorem eventually_bot {p : \u03b1 \u2192 Prop} : \u2200\u1da0 x in \u22a5, p x", "theorem eventually_congr {f : Filter \u03b1} {p q : \u03b1 \u2192 Prop} (h : \u2200\u1da0 x in f, p x \u2194 q x) :\n    (\u2200\u1da0 x in f, p x) \u2194 \u2200\u1da0 x in f, q x", "theorem eventually_ge_atTop [Preorder \u03b1] (a : \u03b1) : \u2200\u1da0 x in atTop, a \u2264 x", "@[simp]\ntheorem eventually_iSup {p : \u03b1 \u2192 Prop} {fs : \u03b9 \u2192 Filter \u03b1} :\n    (\u2200\u1da0 x in \u2a06 b, fs b, p x) \u2194 \u2200 b, \u2200\u1da0 x in fs b, p x", "theorem eventually_iff {f : Filter \u03b1} {P : \u03b1 \u2192 Prop} : (\u2200\u1da0 x in f, P x) \u2194 { x | P x } \u2208 f", "@[to_additive (attr := simp)]\ntheorem eventually_one {p : \u03b1 \u2192 Prop} : (\u2200\u1da0 x in 1, p x) \u2194 p 1", "@[simp]\ntheorem eventually_principal {a : Set \u03b1} {p : \u03b1 \u2192 Prop} : (\u2200\u1da0 x in \ud835\udcdf a, p x) \u2194 \u2200 x \u2208 a, p x", "theorem eventually_smallSets {p : Set \u03b1 \u2192 Prop} :\n    (\u2200\u1da0 s in l.smallSets, p s) \u2194 \u2203 s \u2208 l, \u2200 t, t \u2286 s \u2192 p t", "theorem eventually_sub_nonneg [OrderedRing \u03b2] {l : Filter \u03b1} {f g : \u03b1 \u2192 \u03b2} :\n    0 \u2264\u1da0[l] g - f \u2194 f \u2264\u1da0[l] g", "lemma frequently_cocardinal_mem {s : Set \u03b1} :\n    (\u2203\u1da0 x in cocardinal \u03b1 hreg, x \u2208 s) \u2194 c \u2264 #s := frequently_cocardinal", "theorem frequently_curry_iff {\u03b1 \u03b2 : Type*} {l : Filter \u03b1} {m : Filter \u03b2}\n    (p : (\u03b1 \u00d7 \u03b2) \u2192 Prop) : (\u2203\u1da0 x in l.curry m, p x) \u2194 \u2203\u1da0 x in l, \u2203\u1da0 y in m, p (x, y)", "@[simp]\ntheorem frequently_or_distrib {f : Filter \u03b1} {p q : \u03b1 \u2192 Prop} :\n    (\u2203\u1da0 x in f, p x \u2228 q x) \u2194 (\u2203\u1da0 x in f, p x) \u2228 \u2203\u1da0 x in f, q x", "theorem frequently_prod_and {p : \u03b1 \u2192 Prop} {q : \u03b2 \u2192 Prop} :\n    (\u2203\u1da0 x in f \u00d7\u02e2 g, p x.1 \u2227 q x.2) \u2194 (\u2203\u1da0 a in f, p a) \u2227 \u2203\u1da0 b in g, q b", "theorem generate_union {s t : Set (Set \u03b1)} :\n    Filter.generate (s \u222a t) = Filter.generate s \u2293 Filter.generate t", "theorem hasBasis_biInf_principal {s : \u03b2 \u2192 Set \u03b1} {S : Set \u03b2} (h : DirectedOn (s \u207b\u00b9'o (\u00b7 \u2265 \u00b7)) S)\n    (ne : S.Nonempty) : (\u2a05 i \u2208 S, \ud835\udcdf (s i)).HasBasis (fun i => i \u2208 S) s", "theorem hasBasis_cocardinal : HasBasis (cocardinal \u03b1 hreg) {s : Set \u03b1 | #s < c} compl", "theorem hasBasis_generate (s : Set (Set \u03b1)) :\n    (generate s).HasBasis (fun t => Set.Finite t \u2227 t \u2286 s) fun t => \u22c2\u2080 t", "theorem hasBasis_iInf_principal_finite {\u03b9 : Type*} (s : \u03b9 \u2192 Set \u03b1) :\n    (\u2a05 i, \ud835\udcdf (s i)).HasBasis (fun t : Set \u03b9 => t.Finite) fun t => \u22c2 i \u2208 t, s i", "theorem iInf_neBot_iff_of_directed {f : \u03b9 \u2192 Filter \u03b1} [Nonempty \u03b1] (hd : Directed (\u00b7 \u2265 \u00b7) f) :\n    NeBot (iInf f) \u2194 \u2200 i, NeBot (f i)", "theorem inter_eventuallyEq_left {s t : Set \u03b1} {l : Filter \u03b1} :\n    (s \u2229 t : Set \u03b1) =\u1da0[l] s \u2194 \u2200\u1da0 x in l, x \u2208 s \u2192 x \u2208 t", "theorem inter_mem {s t : Set \u03b1} (hs : s \u2208 f) (ht : t \u2208 f) : s \u2229 t \u2208 f", "theorem inter_mem_inf {\u03b1 : Type u} {f g : Filter \u03b1} {s t : Set \u03b1} (hs : s \u2208 f) (ht : t \u2208 g) :\n    s \u2229 t \u2208 f \u2293 g", "@[to_additive neg.instNeBot]\nlemma inv.instNeBot [NeBot f] : NeBot f\u207b\u00b9 := .inv \u2039_\u203a", "@[to_additive]\ntheorem inv_mem_inv (hs : s \u2208 f) : s\u207b\u00b9 \u2208 f\u207b\u00b9", "theorem isBounded_iff : f.IsBounded r \u2194 \u2203 s \u2208 f.sets, \u2203 b, s \u2286 { x | r x b }", "@[deprecated isBoundedUnder_of (since := \"2024-06-07\")]\nlemma Filter.isBounded_le_map_of_bounded_range {\u03b9 : Type*} (F : Filter \u03b9) {f : \u03b9 \u2192 \u211d}\n    (h : Bornology.IsBounded (Set.range f)) :\n    (F.map f).IsBounded (\u00b7 \u2264 \u00b7)", "@[simp] lemma ker_pure (a : \u03b1) : ker (pure a) = {a}", "@[to_additive (attr := simp)]\nprotected theorem le_div_iff : h \u2264 f / g \u2194 \u2200 \u2983s\u2984, s \u2208 f \u2192 \u2200 \u2983t\u2984, t \u2208 g \u2192 s / t \u2208 h", "theorem le_prod {f : Filter (\u03b1 \u00d7 \u03b2)} {g : Filter \u03b1} {g' : Filter \u03b2} :\n    (f \u2264 g \u00d7\u02e2 g') \u2194 Tendsto Prod.fst f g \u2227 Tendsto Prod.snd f g'", "theorem lift'_inf_principal_eq {h : Set \u03b1 \u2192 Set \u03b2} {s : Set \u03b2} :\n    f.lift' h \u2293 \ud835\udcdf s = f.lift' fun t => h t \u2229 s", "theorem lift'_map_le {g : Set \u03b2 \u2192 Set \u03b3} {m : \u03b1 \u2192 \u03b2} : (map m f).lift' g \u2264 f.lift' (g \u2218 image m)", "theorem lift_iInf_of_map_univ {f : \u03b9 \u2192 Filter \u03b1} {g : Set \u03b1 \u2192 Filter \u03b2}\n    (hg : \u2200 s t, g (s \u2229 t) = g s \u2293 g t) (hg' : g univ = \u22a4) :\n    (iInf f).lift g = \u2a05 i, (f i).lift g", "theorem lift_lift'_assoc {g : Set \u03b1 \u2192 Set \u03b2} {h : Set \u03b2 \u2192 Filter \u03b3} (hg : Monotone g)\n    (hh : Monotone h) : (f.lift' g).lift h = f.lift fun s => h (g s)", "theorem lift_lift'_same_le_lift' {g : Set \u03b1 \u2192 Set \u03b1 \u2192 Set \u03b2} :\n    (f.lift fun s => f.lift' (g s)) \u2264 f.lift' fun s => g s s", "theorem lift_principal {s : Set \u03b1} (hg : Monotone g) : (\ud835\udcdf s).lift g = g s", "theorem liminf_eq_iSup_iInf {f : Filter \u03b2} {u : \u03b2 \u2192 \u03b1} : liminf u f = \u2a06 s \u2208 f, \u2a05 a \u2208 s, u a", "theorem liminf_eq_iSup_iInf_of_nat {u : \u2115 \u2192 \u03b1} : liminf u atTop = \u2a06 n : \u2115, \u2a05 i \u2265 n, u i", "@[simp]\ntheorem limsSup_top : limsSup (\u22a4 : Filter \u03b1) = \u22a4", "@[simp]\ntheorem limsup_const {\u03b1 : Type*} [ConditionallyCompleteLattice \u03b2] {f : Filter \u03b1} [NeBot f]\n    (b : \u03b2) : limsup (fun _ => b) f = b", "theorem limsup_eq : limsup u f = sInf { a | \u2200\u1da0 n in f, u n \u2264 a }", "theorem limsup_le_iSup {f : Filter \u03b2} {u : \u03b2 \u2192 \u03b1} : limsup u f \u2264 \u2a06 n, u n", "lemma limsup_piecewise {s : Set \u03b2} [DecidablePred (\u00b7 \u2208 s)] {v} :\n    limsup (s.piecewise u v) f = blimsup u f (\u00b7 \u2208 s) \u2294 blimsup v f (\u00b7 \u2209 s)", "theorem map_eq_comap_of_inverse {f : Filter \u03b1} {m : \u03b1 \u2192 \u03b2} {n : \u03b2 \u2192 \u03b1} (h\u2081 : m \u2218 n = id)\n    (h\u2082 : n \u2218 m = id) : map m f = comap n f", "theorem map_eq_map_iff_of_injOn {f g : Filter \u03b1} {m : \u03b1 \u2192 \u03b2} {s : Set \u03b1} (hsf : s \u2208 f) (hsg : s \u2208 g)\n    (hm : InjOn m s) : map m f = map m g \u2194 f = g", "@[simp]\ntheorem map_id : Filter.map id f = f", "theorem map_map\u2082_distrib_left {n : \u03b3 \u2192 \u03b4} {m' : \u03b1' \u2192 \u03b2 \u2192 \u03b4} {n' : \u03b1 \u2192 \u03b1'}\n    (h_distrib : \u2200 a b, n (m a b) = m' (n' a) b) : (map\u2082 m f g).map n = map\u2082 m' (f.map n') g", "theorem map_neg_atBot : map (Neg.neg : \u03b2 \u2192 \u03b2) atBot = atTop", "theorem map_prod_map_const_id_principal_coprod_principal {\u03b1 \u03b2 \u03b9 : Type*} (a : \u03b1) (b : \u03b2) (i : \u03b9) :\n    map (Prod.map (fun _ : \u03b1 => b) id) ((\ud835\udcdf {a}).coprod (\ud835\udcdf {i})) =\n      \ud835\udcdf (({b} : Set \u03b2) \u00d7\u02e2 (univ : Set \u03b9))", "@[simp]\ntheorem map_pure (f : \u03b1 \u2192 \u03b2) (a : \u03b1) : map f (pure a) = pure (f a)", "theorem map_sub_atTop_eq_nat (k : \u2115) : map (fun a => a - k) atTop = atTop", "theorem map_swap4_eq_comap {f : Filter ((\u03b1 \u00d7 \u03b2) \u00d7 \u03b3 \u00d7 \u03b4)} :\n    map (fun p : (\u03b1 \u00d7 \u03b2) \u00d7 \u03b3 \u00d7 \u03b4 => ((p.1.1, p.2.1), (p.1.2, p.2.2))) f =\n      comap (fun p : (\u03b1 \u00d7 \u03b3) \u00d7 \u03b2 \u00d7 \u03b4 => ((p.1.1, p.2.1), (p.1.2, p.2.2))) f", "theorem map\u2082_distrib_le_right {m : \u03b4 \u2192 \u03b3 \u2192 \u03b5} {n : \u03b1 \u2192 \u03b2 \u2192 \u03b4} {m\u2081 : \u03b1 \u2192 \u03b3 \u2192 \u03b1'} {m\u2082 : \u03b2 \u2192 \u03b3 \u2192 \u03b2'}\n    {n' : \u03b1' \u2192 \u03b2' \u2192 \u03b5} (h_distrib : \u2200 a b c, m (n a b) c = n' (m\u2081 a c) (m\u2082 b c)) :\n    map\u2082 m (map\u2082 n f g) h \u2264 map\u2082 n' (map\u2082 m\u2081 f h) (map\u2082 m\u2082 g h)", "theorem map\u2082_left_comm {m : \u03b1 \u2192 \u03b4 \u2192 \u03b5} {n : \u03b2 \u2192 \u03b3 \u2192 \u03b4} {m' : \u03b1 \u2192 \u03b3 \u2192 \u03b4'} {n' : \u03b2 \u2192 \u03b4' \u2192 \u03b5}\n    (h_left_comm : \u2200 a b c, m a (n b c) = n' b (m' a c)) :\n    map\u2082 m f (map\u2082 n g h) = map\u2082 n' g (map\u2082 m' f h)", "theorem map\u2082_mono_left (h : g\u2081 \u2264 g\u2082) : map\u2082 m f g\u2081 \u2264 map\u2082 m f g\u2082", "@[simp]\ntheorem map\u2082_vsub : map\u2082 (\u00b7 -\u1d65 \u00b7) f g = f -\u1d65 g", "@[deprecated mem_coclosedCompact_iff (since := \"2024-02-16\")]\ntheorem mem_coclosedCompact : s \u2208 coclosedCompact X \u2194 \u2203 t, IsClosed t \u2227 IsCompact t \u2227 t\u1d9c \u2286 s", "@[simp]\ntheorem mem_cofinite {s : Set \u03b1} : s \u2208 @cofinite \u03b1 \u2194 s\u1d9c.Finite", "lemma mem_generate_of_mem {s : Set <| Set \u03b1} {U : Set \u03b1} (h : U \u2208 s) :\n    U \u2208 generate s := GenerateSets.basic h", "theorem mem_iInf' {\u03b9} {s : \u03b9 \u2192 Filter \u03b1} {U : Set \u03b1} :\n    (U \u2208 \u2a05 i, s i) \u2194\n      \u2203 I : Set \u03b9, I.Finite \u2227 \u2203 V : \u03b9 \u2192 Set \u03b1, (\u2200 i, V i \u2208 s i) \u2227\n        (\u2200 i \u2209 I, V i = univ) \u2227 (U = \u22c2 i \u2208 I, V i) \u2227 U = \u22c2 i, V i", "theorem mem_iff_ultrafilter : s \u2208 f \u2194 \u2200 g : Ultrafilter \u03b1, \u2191g \u2264 f \u2192 s \u2208 g", "theorem mem_lift_sets (hg : Monotone g) {s : Set \u03b2} : s \u2208 f.lift g \u2194 \u2203 t \u2208 f, s \u2208 g t", "theorem mem_map_seq_iff {f : Filter \u03b1} {g : Filter \u03b2} {m : \u03b1 \u2192 \u03b2 \u2192 \u03b3} {s : Set \u03b3} :\n    s \u2208 (f.map m).seq g \u2194 \u2203 t u, t \u2208 g \u2227 u \u2208 f \u2227 \u2200 x \u2208 u, \u2200 y \u2208 t, m x y \u2208 s", "@[simp]\ntheorem mem_top {s : Set \u03b1} : s \u2208 (\u22a4 : Filter \u03b1) \u2194 s = univ", "theorem mono_blimsup' (h : \u2200\u1da0 x in f, p x \u2192 u x \u2264 v x) : blimsup u f p \u2264 blimsup v f p", "theorem monotone_lift [Preorder \u03b3] {f : \u03b3 \u2192 Filter \u03b1} {g : \u03b3 \u2192 Set \u03b1 \u2192 Filter \u03b2} (hf : Monotone f)\n    (hg : Monotone g) : Monotone fun c => (f c).lift (g c)", "theorem monotone_nhds : Monotone (\ud835\udcdd : Filter \u03b1 \u2192 Filter (Filter \u03b1))", "theorem mp_mem (hs : s \u2208 f) (h : { x | x \u2208 s \u2192 x \u2208 t } \u2208 f) : t \u2208 f", "theorem mul_add_subset : f * (g + h) \u2264 f * g + f * h", "@[to_additive]\nprotected theorem mul_eq_one_iff : f * g = 1 \u2194 \u2203 a b, f = pure a \u2227 g = pure b \u2227 a * b = 1", "@[to_additive (attr := simp)]\ntheorem mul_pure : f * pure b = f.map (\u00b7 * b)", "theorem nhds_eq (l : Filter \u03b1) : \ud835\udcdd l = l.lift' (Iic \u2218 \ud835\udcdf)", "@[simp] theorem not_neBot {f : Filter \u03b1} : \u00acf.NeBot \u2194 f = \u22a5", "theorem ofSets_filter_eq_generate (s : Set (Set \u03b1)) :\n    (FilterBasis.ofSets s).filter = generate s", "@[simp]\ntheorem pi_inf_principal_pi_eq_bot [\u2200 i, NeBot (f i)] {I : Set \u03b9} :\n    pi f \u2293 \ud835\udcdf (Set.pi I s) = \u22a5 \u2194 \u2203 i \u2208 I, f i \u2293 \ud835\udcdf (s i) = \u22a5", "@[mono]\ntheorem pi_mono (h : \u2200 i, f\u2081 i \u2264 f\u2082 i) : pi f\u2081 \u2264 pi f\u2082", "theorem principal_coprod_principal (s : Set \u03b1) (t : Set \u03b2) :\n    (\ud835\udcdf s).coprod (\ud835\udcdf t) = \ud835\udcdf (s\u1d9c \u00d7\u02e2 t\u1d9c)\u1d9c", "@[simp] theorem principal_eq_iff_eq {s t : Set \u03b1} : \ud835\udcdf s = \ud835\udcdf t \u2194 s = t", "theorem prod_comap_comap_eq.{u, v, w, x} {\u03b1\u2081 : Type u} {\u03b1\u2082 : Type v} {\u03b2\u2081 : Type w} {\u03b2\u2082 : Type x}\n    {f\u2081 : Filter \u03b1\u2081} {f\u2082 : Filter \u03b1\u2082} {m\u2081 : \u03b2\u2081 \u2192 \u03b1\u2081} {m\u2082 : \u03b2\u2082 \u2192 \u03b1\u2082} :\n    comap m\u2081 f\u2081 \u00d7\u02e2 comap m\u2082 f\u2082 = comap (fun p : \u03b2\u2081 \u00d7 \u03b2\u2082 => (m\u2081 p.1, m\u2082 p.2)) (f\u2081 \u00d7\u02e2 f\u2082)", "theorem prod_comm' : f \u00d7\u02e2 g = comap Prod.swap (g \u00d7\u02e2 f)", "theorem prod_map_left (f : \u03b1 \u2192 \u03b2) (F : Filter \u03b1) (G : Filter \u03b3) :\n    map f F \u00d7\u02e2 G = map (Prod.map f id) (F \u00d7\u02e2 G)", "theorem prod_same_eq : f \u00d7\u02e2 f = f.lift' fun t : Set \u03b1 => t \u00d7\u02e2 t", "@[simp]\ntheorem pure_seq_eq_map (g : \u03b1 \u2192 \u03b2) (f : Filter \u03b1) : seq (pure g) f = f.map g", "theorem sInter_nhds (l : Filter \u03b1) : \u22c2\u2080 { s | s \u2208 \ud835\udcdd l } = Iic l", "theorem set_eventuallyLE_iff_mem_inf_principal {s t : Set \u03b1} {l : Filter \u03b1} :\n    s \u2264\u1da0[l] t \u2194 t \u2208 l \u2293 \ud835\udcdf s", "@[simp]\ntheorem smallSets_top : (\u22a4 : Filter \u03b1).smallSets = \u22a4", "@[to_additive (attr := simp)]\ntheorem smul_filter_eq_bot_iff : a \u2022 f = \u22a5 \u2194 f = \u22a5", "theorem subseq_tendsto_of_neBot {f : Filter \u03b1} [IsCountablyGenerated f] {u : \u2115 \u2192 \u03b1}\n    (hx : NeBot (f \u2293 map u atTop)) : \u2203 \u03b8 : \u2115 \u2192 \u2115, StrictMono \u03b8 \u2227 Tendsto (u \u2218 \u03b8) atTop f", "theorem tendsto_Ioi_atTop [SemilatticeSup \u03b1] {a : \u03b1} {f : \u03b2 \u2192 Ioi a} {l : Filter \u03b2} :\n    Tendsto f l atTop \u2194 Tendsto (fun x => (f x : \u03b1)) l atTop", "theorem tendsto_atBot_add_nonpos_left (hf : \u2200 x, f x \u2264 0) (hg : Tendsto g l atBot) :\n    Tendsto (fun x => f x + g x) l atBot", "theorem tendsto_atBot_atBot_of_monotone [Preorder \u03b1] [Preorder \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : Monotone f)\n    (h : \u2200 b, \u2203 a, f a \u2264 b) : Tendsto f atBot atBot", "theorem tendsto_atTop_add_const_left (C : \u03b2) (hf : Tendsto f l atTop) :\n    Tendsto (fun x => C + f x) l atTop", "theorem tendsto_atTop_atBot_of_antitone [Preorder \u03b1] [Preorder \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : Antitone f)\n    (h : \u2200 b, \u2203 a, f a \u2264 b) : Tendsto f atTop atBot", "theorem tendsto_atTop_diagonal [SemilatticeSup \u03b1] : Tendsto (fun a : \u03b1 => (a, a)) atTop atTop", "theorem tendsto_comap {f : \u03b1 \u2192 \u03b2} {x : Filter \u03b2} : Tendsto f (comap f x) x", "@[simp]\ntheorem tendsto_comap_iff {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} {a : Filter \u03b1} {c : Filter \u03b3} :\n    Tendsto f a (c.comap g) \u2194 Tendsto (g \u2218 f) a c", "@[simp, nolint simpNF] theorem tendsto_comp_val_Iic_atBot [SemilatticeInf \u03b1] {a : \u03b1} {f : \u03b1 \u2192 \u03b2} {l : Filter \u03b2} :\n    Tendsto (fun x : Iic a => f x) atBot l \u2194 Tendsto f atBot l", "theorem tendsto_congr {f\u2081 f\u2082 : \u03b1 \u2192 \u03b2} {l\u2081 : Filter \u03b1} {l\u2082 : Filter \u03b2} (h : \u2200 x, f\u2081 x = f\u2082 x) :\n    Tendsto f\u2081 l\u2081 l\u2082 \u2194 Tendsto f\u2082 l\u2081 l\u2082", "theorem tendsto_const_mul_atBot_iff_neg [NeBot l] (h : Tendsto f l atTop) :\n    Tendsto (fun x => r * f x) l atBot \u2194 r < 0", "theorem tendsto_const_mul_atBot_iff_pos [NeBot l] (h : Tendsto f l atBot) :\n    Tendsto (fun x => r * f x) l atBot \u2194 0 < r", "theorem tendsto_const_mul_atTop_iff_pos [NeBot l] (h : Tendsto f l atTop) :\n    Tendsto (fun x => r * f x) l atTop \u2194 0 < r", "theorem tendsto_finset_preimage_atTop_atTop {f : \u03b1 \u2192 \u03b2} (hf : Function.Injective f) :\n    Tendsto (fun s : Finset \u03b2 => s.preimage f (hf.injOn)) atTop atTop", "theorem tendsto_id' {x y : Filter \u03b1} : Tendsto id x y \u2194 x \u2264 y", "theorem tendsto_mul_const_atTop_of_neg (hr : r < 0) :\n    Tendsto (fun x => f x * r) l atTop \u2194 Tendsto f l atBot", "theorem tendsto_mul_const_atTop_of_pos (hr : 0 < r) :\n    Tendsto (fun x => f x * r) l atTop \u2194 Tendsto f l atTop", "@[simp]\ntheorem tendsto_neg_atTop_iff : Tendsto (fun x => -f x) l atTop \u2194 Tendsto f l atBot", "theorem Filter.tendsto_nhds_min_right {l : Filter \u03b2} {a : \u03b1} (h : Tendsto f l (\ud835\udcdd[<] a)) :\n    Tendsto (fun i => min a (f i)) l (\ud835\udcdd[<] a)", "theorem tendsto_of_subseq_tendsto {\u03b9 : Type*} {x : \u03b9 \u2192 \u03b1} {f : Filter \u03b1} {l : Filter \u03b9}\n    [l.IsCountablyGenerated]\n    (hxy : \u2200 ns : \u2115 \u2192 \u03b9, Tendsto ns atTop l \u2192\n      \u2203 ms : \u2115 \u2192 \u2115, Tendsto (fun n => x (ns <| ms n)) atTop f) :\n    Tendsto x l f", "@[simp]\ntheorem vsub_neBot_iff : (f -\u1d65 g : Filter \u03b1).NeBot \u2194 f.NeBot \u2227 g.NeBot", "protected theorem generate (B : FilterBasis \u03b1) : generate B.sets = B.filter", "@[simp] lemma append_cast_right {n m} {\u03b1 : Type*} (xs : Fin n \u2192 \u03b1) (ys : Fin m \u2192 \u03b1) (m' : \u2115)\n    (h : m' = m) :\n    Fin.append xs (ys \u2218 Fin.cast h) = Fin.append xs ys \u2218 (Fin.cast <| by rw [h])", "lemma append_rev {m n} {\u03b1 : Type*} (xs : Fin m \u2192 \u03b1) (ys : Fin n \u2192 \u03b1) (i : Fin (m + n)) :\n    append xs ys (rev i) = append (ys \u2218 rev) (xs \u2218 rev) (cast (Nat.add_comm ..) i)", "lemma bot_eq_zero (n : \u2115) : \u22a5 = (0 : Fin (n + 1)) := rfl", "theorem card_filter_univ_succ (p : Fin (n + 1) \u2192 Prop) [DecidablePred p] :\n    (univ.filter p).card =\n    if p 0 then (univ.filter (p \u2218 Fin.succ)).card + 1 else (univ.filter (p \u2218 Fin.succ)).card", "theorem card_fintype_uIcc : Fintype.card (Set.uIcc a b) = (b - a : \u2124).natAbs + 1", "@[simp] lemma castLE_inj {hmn : m \u2264 n} {a b : Fin m} : castLE hmn a = castLE hmn b \u2194 a = b", "@[simp] lemma castLE_rfl (n : \u2115) : Fin.castLE (le_refl n) = id", "lemma castPred_succAbove_castPred {a : Fin (n + 2)} {b : Fin (n + 1)} (ha : a \u2260 last (n + 1))\n    (hb : b \u2260 last n) (hk := succAbove_ne_last ha hb) :\n    (a.castPred ha).succAbove (b.castPred hb) = (a.succAbove b).castPred hk", "@[simp] lemma cast_eq_self (a : Fin n) : cast rfl a = a := rfl", "@[simp]\ntheorem cast_zero {n' : \u2115} [NeZero n] {h : n = n'} : cast h (0 : Fin n) =\n    by { haveI : NeZero n'", "@[simp]\ntheorem cons_zero : cons x p 0 = x", "@[simp]\nlemma Fin.dist_insertNth_insertNth {n : \u2115} {\u03b1 : Fin (n + 1) \u2192 Type*}\n    [\u2200 i, PseudoMetricSpace (\u03b1 i)] (i : Fin (n + 1)) (x y : \u03b1 i) (f g : \u2200 j, \u03b1 (i.succAbove j)) :\n    dist (i.insertNth x f) (i.insertNth y g) = max (dist x y) (dist f g)", "theorem eq_zero (n : Fin 1) : n = 0", "lemma equiv_iff_eq : Nonempty (Fin m \u2243 Fin n) \u2194 m = n", "theorem exists_eq_add_of_le {n : \u2115} {a b : Fin n} (h : a \u2264 b) : \u2203 k \u2264 b, b = a + k", "theorem foldl_loop_eq (f : \u03b1 \u2192 Fin n \u2192 \u03b1) (x) : foldl.loop n f x n = x", "theorem foldr_succ (f : Fin (n+1) \u2192 \u03b1 \u2192 \u03b1) (x) :\n    foldr (n+1) f x = f 0 (foldr n (fun i => f i.succ) x)", "@[simp] theorem getElem_list (i : Nat) (h : i < (list n).length) :\n    (list n)[i] = cast (length_list n) \u27e8i, h\u27e9", "@[simp]\ntheorem init_snoc : init (snoc p x) = p", "theorem le_iff_val_le_val {a b : Fin n} : a \u2264 b \u2194 (a : \u2115) \u2264 b", "lemma mem_piFinset_succ {x : \u2200 i, \u03b1 i} {s : \u2200 i, Finset (\u03b1 i)} :\n    x \u2208 piFinset s \u2194 x 0 \u2208 s 0 \u2227 tail x \u2208 piFinset (tail s)", "theorem modNat_rev (i : Fin (m * n)) : i.rev.modNat = i.modNat.rev", "protected theorem mul_one' [NeZero n] (k : Fin n) : k * 1 = k", "lemma nonempty_embedding_iff : Nonempty (Fin n \u21aa Fin m) \u2194 n \u2264 m", "@[simp]\ntheorem ofNat''_eq_cast (n : \u2115) [NeZero n] (a : \u2115) : (Fin.ofNat'' a : Fin n) = a", "theorem pi_lex_lt_cons_cons {x\u2080 y\u2080 : \u03b1 0} {x y : \u2200 i : Fin n, \u03b1 i.succ}\n    (s : \u2200 {i : Fin n.succ}, \u03b1 i \u2192 \u03b1 i \u2192 Prop) :\n    Pi.Lex (\u00b7 < \u00b7) (@s) (Fin.cons x\u2080 x) (Fin.cons y\u2080 y) \u2194\n      s x\u2080 y\u2080 \u2228 x\u2080 = y\u2080 \u2227 Pi.Lex (\u00b7 < \u00b7) (@fun i : Fin n \u21a6 @s i.succ) x y", "theorem pos_iff_nonempty {n : Nat} : 0 < n \u2194 Nonempty (Fin n)", "lemma predAbove_castPred_self (p : Fin (n + 1)) (hp : p \u2260 last n) :\n    (castPred p hp).predAbove p = castPred p hp := predAbove_castPred_of_le _ _ Fin.le_rfl hp", "@[simp] lemma predAbove_last_castSucc {i : Fin (n + 1)} : predAbove (last n) (i.castSucc) = i", "@[simp] lemma predAbove_last_of_ne_last {i : Fin (n + 2)} (hi : i \u2260 last (n + 1)) :\n    predAbove (last n) i = castPred i hi", "theorem pred_last (h := ext_iff.not.2 last_pos'.ne') :\n    pred (last (n + 1)) h = last n", "@[simp]\ntheorem pred_succ (i : Fin n) {h : i.succ \u2260 0} : i.succ.pred h = i", "lemma pred_succAbove_pred {a : Fin (n + 2)} {b : Fin (n + 1)} (ha : a \u2260 0) (hb : b \u2260 0)\n    (hk := succAbove_ne_zero ha hb) :\n    (a.pred ha).succAbove (b.pred hb) = (a.succAbove b).pred hk", "@[to_additive (attr := simp)]\ntheorem prod_univ_two [CommMonoid \u03b2] (f : Fin 2 \u2192 \u03b2) : \u220f i, f i = f 0 * f 1", "theorem rev_involutive : Involutive (rev : Fin n \u2192 Fin n)", "theorem rev_le_iff {i j : Fin n} : rev i \u2264 j \u2194 rev j \u2264 i", "theorem rev_surjective : Surjective (@rev n)", "@[simp] lemma snocCases_snoc\n    {P : (\u2200 i : Fin (n+1), \u03b1 i) \u2192 Sort*} (h : \u2200 x x\u2080, P (Fin.snoc x x\u2080))\n    (x : \u2200 i : Fin n, (Fin.init \u03b1) i) (x\u2080 : \u03b1 (Fin.last _)) :\n    snocCases h (Fin.snoc x x\u2080) = h x x\u2080", "lemma snoc_mem_piFinset_snoc {x : \u2200 i : Fin n, \u03b1 i.castSucc} {x\u2099 : \u03b1 (.last n)}\n    {s : \u2200 i : Fin n, Finset (\u03b1 i.castSucc)} {s\u2099 : Finset (\u03b1 $ .last n)} :\n    snoc x x\u2099 \u2208 piFinset (snoc s s\u2099) \u2194 x \u2208 piFinset s \u2227 x\u2099 \u2208 s\u2099", "lemma strictMono_addNat (m) : StrictMono ((addNat \u00b7 m) : Fin n \u2192 Fin (n + m))", "lemma succAbove_of_lt_succ (p : Fin (n + 1)) (i : Fin n) (h : p < succ i) :\n    p.succAbove i = succ i := succAbove_of_le_castSucc _ _ (le_castSucc_iff.mpr h)", "lemma succAbove_succ_of_lt (p i : Fin n) (h : p < i) : succAbove p.succ i = i.succ", "@[simp] lemma succAbove_succ_self (j : Fin n) : j.succ.succAbove j = j.castSucc", "@[simp]\ntheorem succ_apply {n : \u2115} (a) : SuccOrder.succ a = if a < Fin.last n then a + 1 else a", "@[simp]\ntheorem succ_one_eq_two' [NeZero n] : Fin.succ (1 : Fin (n + 1)) = 2", "@[simp] lemma succ_succAbove_zero {n : \u2115} [NeZero n] (i : Fin n) : succAbove i.succ 0 = 0", "@[simp] theorem succ_zero_eq_one : Fin.succ (0 : Fin (n + 1)) = 1", "@[simp]\ntheorem succ_zero_eq_one' [NeZero n] : Fin.succ (0 : Fin n) = 1", "theorem sum_const [AddCommMonoid \u03b1] (n : \u2115) (x : \u03b1) : \u2211 _i : Fin n, x = n \u2022 x", "theorem sum_pow_mul_eq_add_pow {n : \u2115} {R : Type*} [CommSemiring R] (a b : R) :\n    (\u2211 s : Finset (Fin n), a ^ s.card * b ^ (n - s.card)) = (a + b) ^ n", "theorem tail_init_eq_init_tail {\u03b2 : Type*} (q : Fin (n + 2) \u2192 \u03b2) :\n    tail (init q) = init (tail q)", "protected theorem zero_mul' [NeZero n] (k : Fin n) : (0 : Fin n) * k = 0", "@[simp]\ntheorem mem_toList [FinEnum \u03b1] (x : \u03b1) : x \u2208 toList \u03b1", "@[simp]\ntheorem map_eq (f : \u03b1 \u2192 \u03b2) {m} (v : Fin m \u2192 \u03b1) : map f v = f \u2218 v", "theorem card_eq_zero_iff [Finite \u03b1] : Nat.card \u03b1 = 0 \u2194 IsEmpty \u03b1", "theorem card_eq_zero_of_injective [Nonempty \u03b1] {f : \u03b1 \u2192 \u03b2} (hf : Function.Injective f)\n    (h : Nat.card \u03b1 = 0) : Nat.card \u03b2 = 0", "theorem card_eq_zero_of_surjective {f : \u03b1 \u2192 \u03b2} (hf : Function.Surjective f) (h : Nat.card \u03b2 = 0) :\n    Nat.card \u03b1 = 0", "theorem card_le_of_injective [Finite \u03b2] (f : \u03b1 \u2192 \u03b2) (hf : Function.Injective f) :\n    Nat.card \u03b1 \u2264 Nat.card \u03b2", "@[simp]\ntheorem card_option [Finite \u03b1] : Nat.card (Option \u03b1) = Nat.card \u03b1 + 1", "theorem Finite.exists_max [Finite \u03b1] [Nonempty \u03b1] [LinearOrder \u03b2] (f : \u03b1 \u2192 \u03b2) :\n    \u2203 x\u2080 : \u03b1, \u2200 x, f x \u2264 f x\u2080", "theorem Finite.induction_empty_option {P : Type u \u2192 Prop} (of_equiv : \u2200 {\u03b1 \u03b2}, \u03b1 \u2243 \u03b2 \u2192 P \u03b1 \u2192 P \u03b2)\n    (h_empty : P PEmpty) (h_option : \u2200 {\u03b1} [Fintype \u03b1], P \u03b1 \u2192 P (Option \u03b1)) (\u03b1 : Type u)\n    [Finite \u03b1] : P \u03b1", "theorem injective_iff_surjective {f : \u03b1 \u2192 \u03b1} : Injective f \u2194 Surjective f", "@[simp]\ntheorem finrank_finsupp_self {\u03b9 : Type v} [Fintype \u03b9] : finrank R (\u03b9 \u2192\u2080 R) = card \u03b9", "theorem FiniteDimensional.finrank_pos_iff [NoZeroSMulDivisors R M] :\n    0 < finrank R M \u2194 Nontrivial M", "theorem FiniteDimensional.of_locallyCompactSpace [LocallyCompactSpace E] :\n    FiniteDimensional \ud835\udd5c E", "theorem FiniteDimensional.rank_linearMap_self :\n    Module.rank S (M \u2192\u2097[R] S) = lift.{u'} (Module.rank R M)", "theorem X_pow_card_sub_X_ne_zero (hp : 1 < p) : (X ^ p - X : K'[X]) \u2260 0", "theorem even_card_iff_char_two : ringChar F = 2 \u2194 Fintype.card F % 2 = 0", "theorem odd_card_of_char_ne_two (hF : ringChar F \u2260 2) : Fintype.card F % 2 = 1", "@[simp]\ntheorem sum_subgroup_units [Ring K] [NoZeroDivisors K]\n    {G : Subgroup K\u02e3} [Fintype G] [Decidable (G = \u22a5)] :\n    \u2211 x : G, (x.val : K) = if G = \u22a5 then 1 else 0", "@[simp]\ntheorem insert_singleton_eq {a : \u03b1} {b b' : \u03b2 a} : insert a b (singleton a b') = singleton a b", "@[simp]\ntheorem keys_empty : (\u2205 : Finmap \u03b2).keys = \u2205", "@[simp]\ntheorem keys_erase_toFinset (a : \u03b1) (s : AList \u03b2) : keys \u27e6s.erase a\u27e7 = (keys \u27e6s\u27e7).erase a", "@[simp]\ntheorem keys_ext {s\u2081 s\u2082 : AList \u03b2} : keys \u27e6s\u2081\u27e7 = keys \u27e6s\u2082\u27e7 \u2194 s\u2081.keys ~ s\u2082.keys", "theorem lookup_isSome {a : \u03b1} {s : Finmap \u03b2} : (s.lookup a).isSome \u2194 a \u2208 s", "theorem mem_lookup_union {a} {b : \u03b2 a} {s\u2081 s\u2082 : Finmap \u03b2} :\n    b \u2208 lookup a (s\u2081 \u222a s\u2082) \u2194 b \u2208 lookup a s\u2081 \u2228 a \u2209 s\u2081 \u2227 b \u2208 lookup a s\u2082", "@[simp]\ntheorem mem_union {a} {s\u2081 s\u2082 : Finmap \u03b2} : a \u2208 s\u2081 \u222a s\u2082 \u2194 a \u2208 s\u2081 \u2228 a \u2208 s\u2082", "theorem toFinmap_cons (a : \u03b1) (b : \u03b2 a) (xs : List (Sigma \u03b2)) :\n    List.toFinmap (\u27e8a, b\u27e9 :: xs) = insert a b xs.toFinmap", "lemma IsEquipartition.card_biUnion_offDiag_le (h\u03b5 : 0 < \u03b5) (hP : P.IsEquipartition)\n    (hP' : 4 / \u03b5 \u2264 P.parts.card) : (P.parts.biUnion offDiag).card \u2264 \u03b5 / 2 * A.card ^ 2", "theorem exists_le_of_le {a b : \u03b1} {P Q : Finpartition a} (h : P \u2264 Q) (hb : b \u2208 Q.parts) :\n    \u2203 c \u2208 P.parts, c \u2264 b", "theorem mem_part_ofSetoid_iff_rel {s : Setoid \u03b1} [DecidableRel s.r] {b : \u03b1} :\n    b \u2208 (ofSetoid s).part a \u2194 s.r a b", "theorem parts_eq_empty_iff : P.parts = \u2205 \u2194 a = \u22a5", "theorem parts_nonempty_iff : P.parts.Nonempty \u2194 a \u2260 \u22a5", "theorem sum_card_parts : \u2211 i \u2208 P.parts, i.card = s.card", "lemma toColex_sdiff_le_toColex_sdiff (hus : u \u2286 s) (hut : u \u2286 t) :\n    toColex (s \\ u) \u2264 toColex (t \\ u) \u2194 toColex s \u2264 toColex t", "theorem Icc_subset_Iic_self : Icc a b \u2286 Iic b", "theorem Ico_eq_filter_ssubsets : Ico s t = t.ssubsets.filter (s \u2286 \u00b7)", "theorem Ico_filter_le_left {a b : \u03b1} [DecidablePred (\u00b7 \u2264 a)] (hab : a < b) :\n    ((Ico a b).filter fun x => x \u2264 a) = {a}", "@[simp]\ntheorem Ico_filter_lt (a b c : \u03b1) : ((Ico a b).filter fun x => x < c) = Ico a (min b c)", "theorem Ico_subset_Ici_self : Ico a b \u2286 Ici a", "theorem Iic_eq_Icc : Iic = Icc (\u22a5 : \u03b1)", "theorem Ioc_subset_Ioo_right (h : b\u2081 < b\u2082) : Ioc a b\u2081 \u2286 Ioo a b\u2082", "theorem Ioo_subset_Ioo_right (h : b\u2081 \u2264 b\u2082) : Ioo a b\u2081 \u2286 Ioo a b\u2082", "@[to_additive]\ntheorem Nonempty.one_mem_div (h : s.Nonempty) : (1 : \u03b1) \u2208 s / s", "theorem Nonempty.zero_mul (hs : s.Nonempty) : 0 * s = 0", "protected lemma Shatters.subset_iff (h : \ud835\udc9c.Shatters s) : t \u2286 s \u2194 \u2203 u \u2208 \ud835\udc9c, s \u2229 u = t", "theorem SupIndep.product {s : Finset \u03b9} {t : Finset \u03b9'} {f : \u03b9 \u00d7 \u03b9' \u2192 \u03b1}\n    (hs : s.SupIndep fun i => t.sup fun i' => f (i, i'))\n    (ht : t.SupIndep fun i' => s.sup fun i => f (i, i')) : (s \u00d7\u02e2 t).SupIndep f", "theorem Superset.trans {s\u2081 s\u2082 s\u2083 : Finset \u03b1} : s\u2081 \u2287 s\u2082 \u2192 s\u2082 \u2287 s\u2083 \u2192 s\u2081 \u2287 s\u2083", "lemma antidiagonal_congr' (hp : p \u2208 antidiagonal n) (hq : q \u2208 antidiagonal n) :\n    p = q \u2194 p.2 = q.2", "lemma antitone_iff_forall_insert_le : Antitone f \u2194 \u2200 s \u2983a\u2984, a \u2209 s \u2192 f (insert a s) \u2264 f s", "theorem attach_map_val {s : Finset \u03b1} : s.attach.map (Embedding.subtype _) = s", "lemma biUnion_mono (h : \u2200 a \u2208 s, t\u2081 a \u2286 t\u2082 a) : s.biUnion t\u2081 \u2286 s.biUnion t\u2082", "@[simp]\nlemma biUnion_singleton_eq_self [DecidableEq \u03b1] : s.biUnion (singleton : \u03b1 \u2192 Finset \u03b1) = s", "theorem card_biUnion_le_card_mul [DecidableEq \u03b2] (s : Finset \u03b9) (f : \u03b9 \u2192 Finset \u03b2) (n : \u2115)\n    (h : \u2200 a \u2208 s, (f a).card \u2264 n) : (s.biUnion f).card \u2264 s.card * n", "theorem card_eq_of_bijective (f : \u2200 i, i < n \u2192 \u03b1) (hf : \u2200 a \u2208 s, \u2203 i, \u2203 h : i < n, f i h = a)\n    (hf' : \u2200 i (h : i < n), f i h \u2208 s)\n    (f_inj : \u2200 i j (hi : i < n) (hj : j < n), f i hi = f j hj \u2192 i = j) : s.card = n", "theorem card_eq_sum_ones (s : Finset \u03b1) : s.card = \u2211 x \u2208 s, 1", "theorem card_image\u2082_iff :\n    (image\u2082 f s t).card = s.card * t.card \u2194 (s \u00d7\u02e2 t : Set (\u03b1 \u00d7 \u03b2)).InjOn fun x => f x.1 x.2", "theorem card_le_mul_card_image_of_maps_to {f : \u03b1 \u2192 \u03b2} {s : Finset \u03b1} {t : Finset \u03b2}\n    (Hf : \u2200 a \u2208 s, f a \u2208 t) (n : \u2115) (hn : \u2200 a \u2208 t, (s.filter fun x \u21a6 f x = a).card \u2264 n) :\n    s.card \u2264 n * t.card", "theorem card_le_one_iff : s.card \u2264 1 \u2194 \u2200 {a b}, a \u2208 s \u2192 b \u2208 s \u2192 a = b", "theorem card_le_one_iff_subset_singleton [Nonempty \u03b1] : s.card \u2264 1 \u2194 \u2203 x : \u03b1, s \u2286 {x}", "@[simp]\ntheorem card_map (f : \u03b1 \u21aa \u03b2) : (s.map f).card = s.card", "theorem card_mul_eq_card_mul [\u2200 a b, Decidable (r a b)]\n    (hm : \u2200 a \u2208 s, (t.bipartiteAbove r a).card = m)\n    (hn : \u2200 b \u2208 t, (s.bipartiteBelow r b).card = n) : s.card * m = t.card * n", "@[to_additive (attr := simp)]\ntheorem card_mul_singleton : (s * {a}).card = s.card", "@[simp]\ntheorem card_range (n : \u2115) : (range n).card = n", "theorem card_sdiff (h : s \u2286 t) : card (t \\ s) = t.card - s.card", "@[simp]\ntheorem card_subtype (p : \u03b1 \u2192 Prop) [DecidablePred p] (s : Finset \u03b1) :\n    (s.subtype p).card = (s.filter p).card", "theorem card_union_le (s t : Finset \u03b1) : (s \u222a t).card \u2264 s.card + t.card", "theorem centerMass_le_sup {s : Finset \u03b9} {f : \u03b9 \u2192 \u03b1} {w : \u03b9 \u2192 R} (hw\u2080 : \u2200 i \u2208 s, 0 \u2264 w i)\n    (hw\u2081 : 0 < \u2211 i \u2208 s, w i) :\n    s.centerMass w f \u2264 s.sup' (nonempty_of_ne_empty <| by rintro rfl; simp at hw\u2081) f", "theorem Finset.centerMass_subset {t' : Finset \u03b9} (ht : t \u2286 t') (h : \u2200 i \u2208 t', i \u2209 t \u2192 w i = 0) :\n    t.centerMass w z = t'.centerMass w z", "theorem centroid_pair_fin [Invertible (2 : k)] (p : Fin 2 \u2192 P) :\n    univ.centroid k p = (2\u207b\u00b9 : k) \u2022 (p 1 -\u1d65 p 0) +\u1d65 p 0", "theorem centroid_univ (s : Finset P) : univ.centroid k ((\u2191) : s \u2192 P) = s.centroid k id", "@[simp, norm_cast]\ntheorem coe_disjUnion {s t : Finset \u03b1} (h : Disjoint s t) :\n    (disjUnion s t h : Set \u03b1) = (s : Set \u03b1) \u222a t", "@[to_additive (attr := simp, norm_cast)]\ntheorem coe_pow (s : Finset \u03b1) (n : \u2115) : \u2191(s ^ n) = (s : Set \u03b1) ^ n", "@[simp, norm_cast]\ntheorem coe_range (n : \u2115) : (range n : Set \u2115) = Set.Iio n", "@[simp, norm_cast]\ntheorem coe_singleton (a : \u03b1) : (({a} : Finset \u03b1) : Set \u03b1) = {a}", "@[simp]\ntheorem compl_insert : (insert a s)\u1d9c = s\u1d9c.erase a", "theorem compl_singleton (a : \u03b1) : ({a} : Finset \u03b1)\u1d9c = univ.erase a", "@[simp] lemma compls_empty : (\u2205 : Finset \u03b1)\u1d9c\u02e2 = \u2205 := map_empty _", "@[simp]\ntheorem disjSum_empty : s.disjSum (\u2205 : Finset \u03b2) = s.map Embedding.inl", "theorem disjSups_assoc : \u2200 s t u : Finset \u03b1, s \u25cb t \u25cb u = s \u25cb (t \u25cb u)", "@[simp]\nlemma disjiUnion_val (s : Finset \u03b1) (t : \u03b1 \u2192 Finset \u03b2) (h) :\n    (s.disjiUnion t h).1 = s.1.bind fun a \u21a6 (t a).1 := rfl", "theorem disjoint_erase_comm : Disjoint (s.erase a) t \u2194 Disjoint s (t.erase a)", "theorem disjoint_of_erase_left (ha : a \u2209 t) (hst : Disjoint (s.erase a) t) : Disjoint s t", "theorem disjoint_sdiff_inter (s t : Finset \u03b1) : Disjoint (s \\ t) (s \u2229 t)", "@[simp]\ntheorem disjoint_singleton_left : Disjoint (singleton a) s \u2194 a \u2209 s", "protected theorem disjoint_sup_right : Disjoint a (s.sup f) \u2194 \u2200 \u2983i\u2984, i \u2208 s \u2192 Disjoint a (f i)", "theorem dvd_lcm {b : \u03b2} (hb : b \u2208 s) : f b \u2223 s.lcm f", "theorem empty_ssubset_singleton : (\u2205 : Finset \u03b1) \u2282 {a}", "theorem eq_of_mem_slice (h\u2081 : A \u2208 \ud835\udc9c # r\u2081) (h\u2082 : A \u2208 \ud835\udc9c # r\u2082) : r\u2081 = r\u2082", "theorem eq_singleton_iff_nonempty_unique_mem {s : Finset \u03b1} {a : \u03b1} :\n    s = {a} \u2194 s.Nonempty \u2227 \u2200 x \u2208 s, x = a", "lemma erase_biUnion (f : \u03b1 \u2192 Finset \u03b2) (s : Finset \u03b1) (b : \u03b2) :\n    (s.biUnion f).erase b = s.biUnion fun x \u21a6 (f x).erase b", "theorem erase_eq_empty_iff (s : Finset \u03b1) (a : \u03b1) : s.erase a = \u2205 \u2194 s = \u2205 \u2228 s = {a}", "theorem _root_.Finset.eventually_cofinite_nmem (s : Finset \u03b1) : \u2200\u1da0 x in cofinite, x \u2209 s", "lemma exists_card_eq [Infinite \u03b1] : \u2200 n : \u2115, \u2203 s : Finset \u03b1, s.card = n\n  | 0 => \u27e8\u2205, card_empty\u27e9\n  | n + 1 => by\n    classical\n    obtain \u27e8s, rfl\u27e9 := exists_card_eq n\n    obtain \u27e8a, ha\u27e9 := s.exists_not_mem\n    exact \u27e8insert a s, card_insert_of_not_mem ha\u27e9", "nonrec\ntheorem Finset.exists_eq_pow_of_mul_eq_pow_of_coprime {\u03b9 R : Type*} [CommSemiring R] [IsDomain R]\n    [GCDMonoid R] [Unique R\u02e3] {n : \u2115} {c : R} {s : Finset \u03b9} {f : \u03b9 \u2192 R}\n    (h : \u2200 i \u2208 s, \u2200 j \u2208 s, i \u2260 j \u2192 IsCoprime (f i) (f j))\n    (hprod : \u220f i \u2208 s, f i = c ^ n) : \u2200 i \u2208 s, \u2203 d : R, f i = d ^ n", "lemma exists_subset_card_eq (hns : n \u2264 s.card) : \u2203 t \u2286 s, t.card = n", "theorem exists_sum_fiber_lt_of_maps_to_of_sum_lt_nsmul (hf : \u2200 a \u2208 s, f a \u2208 t)\n    (hb : \u2211 x \u2208 s, w x < t.card \u2022 b) : \u2203 y \u2208 t, \u2211 x \u2208 s.filter fun x => f x = y, w x < b", "lemma Finset.exists_superset_card_eq [Fintype \u03b1] {n : \u2115} {s : Finset \u03b1} (hsn : s.card \u2264 n)\n    (hn\u03b1 : n \u2264 Fintype.card \u03b1) :\n    \u2203 t, s \u2286 t \u2227 t.card = n", "theorem filter_filter (s : Finset \u03b1) : (s.filter p).filter q = s.filter fun a => p a \u2227 q a", "theorem filter_mem_eq_inter {s t : Finset \u03b1} [\u2200 i, Decidable (i \u2208 t)] :\n    (s.filter fun i => i \u2208 t) = s \u2229 t", "theorem filter_not (s : Finset \u03b1) : (s.filter fun a => \u00acp a) = s \\ s.filter p", "@[simp]\ntheorem filter_subset (s : Finset \u03b1) : s.filter p \u2286 s", "theorem finsuppAntidiag_empty_of_ne_zero {n : \u03bc} (hn : n \u2260 0) :\n    finsuppAntidiag (\u2205 : Finset \u03b9) n = \u2205", "theorem fold_const [hd : Decidable (s = \u2205)] (c : \u03b2) (h : op c (op b c) = op b c) :\n    Finset.fold op b (fun _ => c) s = if s = \u2205 then b else op b c", "theorem fold_hom {op' : \u03b3 \u2192 \u03b3 \u2192 \u03b3} [Std.Commutative op'] [Std.Associative op'] {m : \u03b2 \u2192 \u03b3}\n    (hm : \u2200 x y, m (op x y) = op' (m x) (m y)) :\n    (s.fold op' (m b) fun x => m (f x)) = m (s.fold op b f)", "theorem fold_max_add [Add \u03b2] [CovariantClass \u03b2 \u03b2 (Function.swap (\u00b7 + \u00b7)) (\u00b7 \u2264 \u00b7)] (n : WithBot \u03b2)\n    (s : Finset \u03b1) : (s.fold max \u22a5 fun x : \u03b1 => \u2191(f x) + n) = s.fold max \u22a5 ((\u2191) \u2218 f) + n", "@[simp] lemma forall_mem_not_eq {s : Finset \u03b1} {a : \u03b1} : (\u2200 b \u2208 s, \u00ac a = b) \u2194 a \u2209 s", "theorem gcd_eq_gcd_image [DecidableEq \u03b1] : s.gcd f = (s.image f).gcd id", "theorem gcd_eq_zero_iff : s.gcd f = 0 \u2194 \u2200 x : \u03b2, x \u2208 s \u2192 f x = 0", "@[simp]\ntheorem gcd_insert [DecidableEq \u03b2] {b : \u03b2} :\n    (insert b s : Finset \u03b2).gcd f = GCDMonoid.gcd (f b) (s.gcd f)", "lemma geomSum_ofColex_strictMono (hn : 2 \u2264 n) : StrictMono fun s \u21a6 \u2211 k \u2208 ofColex s, n ^ k", "theorem iInf_coe [InfSet \u03b2] (f : \u03b1 \u2192 \u03b2) (s : Finset \u03b1) : \u2a05 x \u2208 (\u2191s : Set \u03b1), f x = \u2a05 x \u2208 s, f x", "theorem iInf_option_toFinset (o : Option \u03b1) (f : \u03b1 \u2192 \u03b2) : \u2a05 x \u2208 o.toFinset, f x = \u2a05 x \u2208 o, f x", "theorem iSup_finset_image {f : \u03b3 \u2192 \u03b1} {g : \u03b1 \u2192 \u03b2} {s : Finset \u03b3} :\n    \u2a06 x \u2208 s.image f, g x = \u2a06 y \u2208 s, g (f y)", "theorem iSup_insert (a : \u03b1) (s : Finset \u03b1) (t : \u03b1 \u2192 \u03b2) :\n    \u2a06 x \u2208 insert a s, t x = t a \u2294 \u2a06 x \u2208 s, t x", "theorem iSup_insert_update {x : \u03b1} {t : Finset \u03b1} (f : \u03b1 \u2192 \u03b2) {s : \u03b2} (hx : x \u2209 t) :\n    \u2a06 i \u2208 insert x t, Function.update f x s i = s \u2294 \u2a06 i \u2208 t, f i", "@[simp] lemma image_add_left_Ico (a b c : \u03b1) : (Ico a b).image (c + \u00b7) = Ico (c + a) (c + b)", "theorem image_diag_union_image_offDiag :\n    s.diag.image Sym2.mk \u222a s.offDiag.image Sym2.mk = s.sym2", "lemma image_injective (hf : Injective f) : Injective (image f)", "theorem image_subset_image {s\u2081 s\u2082 : Finset \u03b1} (h : s\u2081 \u2286 s\u2082) : s\u2081.image f \u2286 s\u2082.image f", "@[simp]\ntheorem image\u2082_eq_empty_iff : image\u2082 f s t = \u2205 \u2194 s = \u2205 \u2228 t = \u2205", "@[simp]\ntheorem image\u2082_insert_left [DecidableEq \u03b1] :\n    image\u2082 f (insert a s) t = (t.image fun b => f a b) \u222a image\u2082 f s t", "lemma inf'_comp_eq_map {s : Finset \u03b3} {f : \u03b3 \u21aa \u03b2} (g : \u03b2 \u2192 \u03b1) (hs : s.Nonempty) :\n    s.inf' hs (g \u2218 f) = (s.map f).inf' (map_nonempty.2 hs) g", "@[simp]\ntheorem inf'_image [DecidableEq \u03b2] {s : Finset \u03b3} {f : \u03b3 \u2192 \u03b2} (hs : (s.image f).Nonempty)\n    (g : \u03b2 \u2192 \u03b1)  :\n    (s.image f).inf' hs g = s.inf' hs.of_image (g \u2218 f)", "@[to_additive]\nlemma inf_mul_right [SemilatticeInf \u03b2] [OrderTop \u03b2] (s t : Finset \u03b1) (f : \u03b1 \u2192 \u03b2) :\n    inf (s * t) f = inf t fun y \u21a6 inf s (f <| \u00b7 * y)", "@[simp] lemma inf_prodMap (hs : s.Nonempty) (ht : t.Nonempty) (f : \u03b9 \u2192 \u03b1) (g : \u03ba \u2192 \u03b2) :\n    inf (s \u00d7\u02e2 t) (Prod.map f g) = (inf s f, inf t g)", "@[simp]\ntheorem infs_subset_iff : s \u22bc t \u2286 u \u2194 \u2200 a \u2208 s, \u2200 b \u2208 t, a \u2293 b \u2208 u", "lemma inr_mem_sumLexLift {c\u2082 : \u03b3\u2082} :\n    inr c\u2082 \u2208 sumLexLift f\u2081 f\u2082 g\u2081 g\u2082 a b \u2194\n      (\u2203 a\u2081 b\u2082, a = inl a\u2081 \u2227 b = inr b\u2082 \u2227 c\u2082 \u2208 g\u2082 a\u2081 b\u2082) \u2228\n        \u2203 a\u2082 b\u2082, a = inr a\u2082 \u2227 b = inr b\u2082 \u2227 c\u2082 \u2208 f\u2082 a\u2082 b\u2082", "@[simp]\ntheorem insertNone_eraseNone [DecidableEq (Option \u03b1)] (s : Finset (Option \u03b1)) :\n    insertNone (eraseNone s) = insert none s", "@[simp]\ntheorem insert_inter_of_not_mem {s\u2081 s\u2082 : Finset \u03b1} {a : \u03b1} (h : a \u2209 s\u2082) :\n    insert a s\u2081 \u2229 s\u2082 = s\u2081 \u2229 s\u2082", "theorem insert_sdiff_of_not_mem (s : Finset \u03b1) {t : Finset \u03b1} {x : \u03b1} (h : x \u2209 t) :\n    insert x s \\ t = insert x (s \\ t)", "@[simp] lemma inter_eq_left : s \u2229 t = s \u2194 s \u2286 t := inf_eq_left", "@[simp] lemma inter_eq_right : t \u2229 s = s \u2194 s \u2286 t := inf_eq_right", "theorem inter_union_distrib_right (s t u : Finset \u03b1) : s \u2229 t \u222a u = (s \u222a u) \u2229 (t \u222a u)", "@[simp] lemma inv_op_smul_finset_distrib\u2080 (a : \u03b1) (s : Finset \u03b1) : (op a \u2022 s)\u207b\u00b9 = a\u207b\u00b9 \u2022 s\u207b\u00b9", "@[to_additive (attr := simp)]\nlemma inv_univ [Fintype \u03b1] : (univ : Finset \u03b1)\u207b\u00b9 = univ", "theorem Finset.isG\u03b4_compl [T1Space X] (s : Finset X) : IsG\u03b4 (s\u1d9c : Set X)", "theorem lcm_mono_fun {g : \u03b2 \u2192 \u03b1} (h : \u2200 b \u2208 s, f b \u2223 g b) : s.lcm f \u2223 s.lcm g", "theorem lcm_union [DecidableEq \u03b2] : (s\u2081 \u222a s\u2082).lcm f = GCDMonoid.lcm (s\u2081.lcm f) (s\u2082.lcm f)", "@[simp (default + 1)] lemma le_inf'_image\u2082 {g : \u03b3 \u2192 \u03b4} {a : \u03b4} (h : (image\u2082 f s t).Nonempty) :\n    a \u2264 inf' (image\u2082 f s t) h g \u2194 \u2200 x \u2208 s, \u2200 y \u2208 t, a \u2264 g (f x y)", "theorem left_mem_uIcc : a \u2208 [[a, b]]", "@[simp]\ntheorem lt_min'_iff : x < s.min' H \u2194 \u2200 y \u2208 s, x < y", "@[simp]\ntheorem map_cast_heq {\u03b1 \u03b2} (h : \u03b1 = \u03b2) (s : Finset \u03b1) :\n    HEq (s.map (Equiv.cast h).toEmbedding) s", "@[simp]\ntheorem map_cons (f : \u03b1 \u21aa \u03b2) (a : \u03b1) (s : Finset \u03b1) (ha : a \u2209 s) :\n    (cons a s ha).map f = cons (f a) (s.map f) (by simpa using ha)", "@[simp]\ntheorem map_refl : s.map (Embedding.refl _) = s", "theorem map_toDual_min (s : Finset \u03b1) : s.min.map toDual = (s.image toDual).max", "@[simp]\ntheorem max'_le_iff {x} : s.max' H \u2264 x \u2194 \u2200 y \u2208 s, y \u2264 x", "theorem max_mem_image_coe {s : Finset \u03b1} (hs : s.Nonempty) :\n    s.max \u2208 (s.image (\u2191) : Finset (WithBot \u03b1))", "@[simp]\ntheorem mem_Ioo : x \u2208 Ioo a b \u2194 a < x \u2227 x < b", "@[simp]\ntheorem mem_attachFin {n : \u2115} {s : Finset \u2115} (h : \u2200 m \u2208 s, m < n) {a : Fin n} :\n    a \u2208 s.attachFin h \u2194 (a : \u2115) \u2208 s", "theorem mem_def {a : \u03b1} {s : Finset \u03b1} : a \u2208 s \u2194 a \u2208 s.1", "@[simp]\ntheorem mem_filter {s : Finset \u03b1} {a : \u03b1} : a \u2208 s.filter p \u2194 a \u2208 s \u2227 p a", "theorem mem_image_of_mem (f : \u03b1 \u2192 \u03b2) {a} (h : a \u2208 s) : f a \u2208 s.image f", "@[to_additive]\ntheorem mem_mul {x : \u03b1} : x \u2208 s * t \u2194 \u2203 y \u2208 s, \u2203 z \u2208 t, y * z = x", "@[simp]\ntheorem mem_powerset {s t : Finset \u03b1} : s \u2208 powerset t \u2194 s \u2286 t", "theorem mem_range_iff_mem_finset_range_of_mod_eq [DecidableEq \u03b1] {f : \u2124 \u2192 \u03b1} {a : \u03b1} {n : \u2115}\n    (hn : 0 < n) (h : \u2200 i, f (i % n) = f i) :\n    a \u2208 Set.range f \u2194 a \u2208 (Finset.range n).image (fun (i : \u2115) => f i)", "theorem mem_singleton_self (a : \u03b1) : a \u2208 ({a} : Finset \u03b1)", "@[simp]\ntheorem mem_sort {s : Finset \u03b1} {a : \u03b1} : a \u2208 sort r s \u2194 a \u2208 s", "@[simp]\ntheorem mem_sym_iff {m : Sym \u03b1 n} : m \u2208 s.sym n \u2194 \u2200 a \u2208 m, a \u2208 s", "@[simp]\ntheorem mem_uIcc : x \u2208 uIcc a b \u2194 a \u2293 b \u2264 x \u2227 x \u2264 a \u2294 b", "@[simp]\ntheorem memberSubfamily_memberSubfamily : (\ud835\udc9c.memberSubfamily a).memberSubfamily a = \u2205", "theorem min'_lt_max' {i j} (H1 : i \u2208 s) (H2 : j \u2208 s) (H3 : i \u2260 j) :\n    s.min' \u27e8i, H1\u27e9 < s.max' \u27e8i, H1\u27e9", "@[gcongr]\ntheorem min_mono {s t : Finset \u03b1} (st : s \u2286 t) : t.min \u2264 s.min", "@[to_additive (attr := simp)] lemma mulEnergy_empty_right : E\u2098[s, \u2205] = 0", "@[to_additive] lemma mulEnergy_eq_card_filter (s t : Finset \u03b1) :\n    E\u2098[s, t] = (((s \u00d7\u02e2 t) \u00d7\u02e2 s \u00d7\u02e2 t).filter fun ((a, b), c, d) \u21a6 a * b = c * d).card", "theorem nonMemberSubfamily_inter (a : \u03b1) (\ud835\udc9c \u212c : Finset (Finset \u03b1)) :\n    (\ud835\udc9c \u2229 \u212c).nonMemberSubfamily a = \ud835\udc9c.nonMemberSubfamily a \u2229 \u212c.nonMemberSubfamily a", "@[to_additive]\ntheorem noncommProd_mul_distrib_aux {s : Finset \u03b1} {f : \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b2}\n    (comm_ff : (s : Set \u03b1).Pairwise fun x y => Commute (f x) (f y))\n    (comm_gg : (s : Set \u03b1).Pairwise fun x y => Commute (g x) (g y))\n    (comm_gf : (s : Set \u03b1).Pairwise fun x y => Commute (g x) (f y)) :\n    (s : Set \u03b1).Pairwise fun x y => Commute ((f * g) x) ((f * g) y)", "@[to_additive (attr := simp)]\ntheorem noncommProd_singleton (a : \u03b1) (f : \u03b1 \u2192 \u03b2) :\n    noncommProd ({a} : Finset \u03b1) f\n        (by\n          norm_cast\n          exact Set.pairwise_singleton _ _) =\n      f a", "theorem nonempty_coe_sort {s : Finset \u03b1} : Nonempty (s : Type _) \u2194 s.Nonempty", "theorem not_mem_of_lt_min {s : Finset \u03b1} {a b : \u03b1} (h\u2081 : a < b) (h\u2082 : s.min = \u2191b) : a \u2209 s", "theorem not_mem_sdiff_of_mem_right (h : a \u2208 t) : a \u2209 s \\ t", "theorem ofDual_max' {s : Finset \u03b1\u1d52\u1d48} (hs : s.Nonempty) :\n    ofDual (max' s hs) = min' (s.image ofDual) (hs.image _)", "@[simp]\ntheorem offDiag_card : (offDiag s).card = s.card * s.card - s.card", "@[simp]\ntheorem offDiag_singleton : ({a} : Finset \u03b1).offDiag = \u2205", "theorem one_lt_card_iff : 1 < s.card \u2194 \u2203 a b, a \u2208 s \u2227 b \u2208 s \u2227 a \u2260 b", "@[to_additive]\ntheorem one_mem_one : (1 : \u03b1) \u2208 (1 : Finset \u03b1)", "@[to_additive]\ntheorem op_smul_finset_subset_mul : a \u2208 t \u2192 op a \u2022 s \u2286 s * t", "lemma ordConnected_range_val : Set.OrdConnected (Set.range val : Set <| Multiset \u03b1)", "theorem orderEmbOfFin_apply (s : Finset \u03b1) {k : \u2115} (h : s.card = k) (i : Fin k) :\n    s.orderEmbOfFin h i =\n      (s.sort (\u00b7 \u2264 \u00b7)).get \u27e8i, by rw [length_sort, h]; exact i.2\u27e9", "theorem pair_comm (a b : \u03b1) : ({a, b} : Finset \u03b1) = {b, a}", "lemma piecewise_cases {i} (p : \u03c0 i \u2192 Prop) (hf : p (f i)) (hg : p (g i)) :\n    p (s.piecewise f g i)", "@[simp]\nlemma piecewise_eq_of_mem {i : \u03b9} (hi : i \u2208 s) : s.piecewise f g i = f i", "lemma piecewise_le_of_le_of_le (hf : f \u2264 h) (hg : g \u2264 h) : s.piecewise f g \u2264 h := fun x =>\n  piecewise_cases s f g (\u00b7 \u2264 h x) (hf x) (hg x)", "lemma piecewise_singleton [DecidableEq \u03b9] (i : \u03b9) : piecewise {i} f g = update g i (f i)", "theorem pimage_congr (h\u2081 : s = t) (h\u2082 : \u2200 x \u2208 t, f x = g x) : s.pimage f = t.pimage g", "theorem pimage_inter [DecidableEq \u03b1] : (s \u2229 t).pimage f \u2286 s.pimage f \u2229 t.pimage f", "theorem pimage_subset {t : Finset \u03b2} : s.pimage f \u2286 t \u2194 \u2200 x \u2208 s, \u2200 y \u2208 f x, y \u2208 t", "@[to_additive card_nsmul_add_sum]\ntheorem pow_card_mul_prod {b : \u03b2} : b ^ s.card * \u220f a \u2208 s, f a = \u220f a \u2208 s, b * f a", "@[to_additive]\ntheorem pow_eq_prod_const (b : \u03b2) : \u2200 n, b ^ n = \u220f _k \u2208 range n, b", "@[simp]\nlemma powersetCard_eq_empty : powersetCard n s = \u2205 \u2194 s.card < n", "@[simp]\ntheorem powerset_empty : (\u2205 : Finset \u03b1).powerset = {\u2205}", "@[simp] lemma powerset_eq_univ : s.powerset = univ \u2194 s = univ", "@[to_additive (attr := simp)]\ntheorem preimage_mul_right_one : preimage 1 (\u00b7 * b) (mul_left_injective _).injOn = {b\u207b\u00b9}", "@[to_additive]\ntheorem prod_Icc_succ_top {a b : \u2115} (hab : a \u2264 b + 1) (f : \u2115 \u2192 M) :\n    (\u220f k in Icc a (b + 1), f k) = (\u220f k in Icc a b, f k) * f (b + 1)", "@[to_additive]\ntheorem prod_Ico_add' [OrderedCancelAddCommMonoid \u03b1] [ExistsAddOfLE \u03b1] [LocallyFiniteOrder \u03b1]\n    (f : \u03b1 \u2192 M) (a b c : \u03b1) : (\u220f x \u2208 Ico a b, f (x + c)) = \u220f x \u2208 Ico (a + c) (b + c), f x", "@[simp]\ntheorem prod_Ico_id_eq_factorial : \u2200 n : \u2115, (\u220f x \u2208 Ico 1 (n + 1), x) = n !", "@[to_additive sum_antidiagonal_choose_succ_nsmul]\ntheorem prod_antidiagonal_pow_choose_succ {M : Type*} [CommMonoid M] (f : \u2115 \u2192 \u2115 \u2192 M) (n : \u2115) :\n    (\u220f ij \u2208 antidiagonal (n + 1), f ij.1 ij.2 ^ (n + 1).choose ij.1) =\n      (\u220f ij \u2208 antidiagonal n, f ij.1 (ij.2 + 1) ^ n.choose ij.1) *\n        \u220f ij \u2208 antidiagonal n, f (ij.1 + 1) ij.2 ^ n.choose ij.2", "@[to_additive \"Reorder a sum.\n\nThe difference with `Finset.sum_bij` is that the bijection is specified with an inverse, rather than\nas a surjective injection.\n\nThe difference with `Finset.sum_nbij'` is that the bijection and its inverse are allowed to use\nmembership of the domains of the sums, rather than being non-dependent functions.\"]\ntheorem prod_bij' (i : \u2200 a \u2208 s, \u03ba) (j : \u2200 a \u2208 t, \u03b9) (hi : \u2200 a ha, i a ha \u2208 t)\n    (hj : \u2200 a ha, j a ha \u2208 s) (left_inv : \u2200 a ha, j (i a ha) (hi a ha) = a)\n    (right_inv : \u2200 a ha, i (j a ha) (hj a ha) = a) (h : \u2200 a ha, f a = g (i a ha)) :\n    \u220f x \u2208 s, f x = \u220f x \u2208 t, g x", "@[to_additive]\nlemma prod_div_prod_mem_mulSpan (ht : t \u2286 s) (hu : u \u2286 s) :\n    (\u220f a in t, a) / \u220f a in u, a \u2208 mulSpan s", "@[to_additive]\ntheorem prod_eq_prod_Ico_succ_bot {a b : \u2115} (hab : a < b) (f : \u2115 \u2192 M) :\n    \u220f k \u2208 Ico a b, f k = f a * \u220f k \u2208 Ico (a + 1) b, f k", "@[to_additive \"If a function applied at a point is 0, a sum is unchanged by\nremoving that point, if present, from a `Finset`.\"]\ntheorem prod_erase [DecidableEq \u03b1] (s : Finset \u03b1) {f : \u03b1 \u2192 \u03b2} {a : \u03b1} (h : f a = 1) :\n    \u220f x \u2208 s.erase a, f x = \u220f x \u2208 s, f x", "@[to_additive]\ntheorem prod_extend_by_one [DecidableEq \u03b1] (s : Finset \u03b1) (f : \u03b1 \u2192 \u03b2) :\n    \u220f i \u2208 s, (if i \u2208 s then f i else 1) = \u220f i \u2208 s, f i", "@[to_additive]\ntheorem prod_finset_product_right (r : Finset (\u03b1 \u00d7 \u03b3)) (s : Finset \u03b3) (t : \u03b3 \u2192 Finset \u03b1)\n    (h : \u2200 p : \u03b1 \u00d7 \u03b3, p \u2208 r \u2194 p.2 \u2208 s \u2227 p.1 \u2208 t p.2) {f : \u03b1 \u00d7 \u03b3 \u2192 \u03b2} :\n    \u220f p \u2208 r, f p = \u220f c \u2208 s, \u220f a \u2208 t c, f (a, c)", "@[to_additive \"An 'unapplied' analogue of `Finset.sum_apply`.\"]\ntheorem Finset.prod_fn {\u03b1 : Type*} {\u03b2 : \u03b1 \u2192 Type*} {\u03b3} [\u2200 a, CommMonoid (\u03b2 a)] (s : Finset \u03b3)\n    (g : \u03b3 \u2192 \u2200 a, \u03b2 a) : \u220f c \u2208 s, g c = fun a \u21a6 \u220f c \u2208 s, g c a", "@[to_additive (attr := simp)]\ntheorem prod_inv_distrib : (\u220f x \u2208 s, (f x)\u207b\u00b9) = (\u220f x \u2208 s, f x)\u207b\u00b9", "lemma prod_lt_prod_of_nonempty (hf : \u2200 i \u2208 s, 0 < f i) (hfg : \u2200 i \u2208 s, f i < g i)\n    (h_ne : s.Nonempty) :\n    \u220f i \u2208 s, f i < \u220f i \u2208 s, g i", "@[to_additive sum_mono_set]\ntheorem prod_mono_set' (f : \u03b9 \u2192 M) : Monotone fun s \u21a6 \u220f x \u2208 s, f x", "@[to_additive]\nlemma prod_mulIndicator_eq_prod_inter [DecidableEq \u03b9] (s t : Finset \u03b9) (f : \u03b9 \u2192 \u03b2) :\n    \u220f i \u2208 s, (t : Set \u03b9).mulIndicator f i = \u220f i \u2208 s \u2229 t, f i", "@[to_additive \"Reorder a sum.\n\nThe difference with `Finset.sum_nbij` is that the bijection is specified with an inverse, rather\nthan as a surjective injection.\n\nThe difference with `Finset.sum_bij'` is that the bijection and its inverse are non-dependent\nfunctions, rather than being allowed to use membership of the domains of the sums.\n\nThe difference with `Finset.sum_equiv` is that bijectivity is only required to hold on the domains\nof the sums, rather than on the entire types.\"]\nlemma prod_nbij' (i : \u03b9 \u2192 \u03ba) (j : \u03ba \u2192 \u03b9) (hi : \u2200 a \u2208 s, i a \u2208 t) (hj : \u2200 a \u2208 t, j a \u2208 s)\n    (left_inv : \u2200 a \u2208 s, j (i a) = a) (right_inv : \u2200 a \u2208 t, i (j a) = a)\n    (h : \u2200 a \u2208 s, f a = g (i a)) : \u220f x \u2208 s, f x = \u220f x \u2208 t, g x", "@[to_additive\n\"A sum over `Finset.powersetCard` which only depends on the size of the sets is constant.\"]\nlemma prod_powersetCard (n : \u2115) (s : Finset \u03b1) (f : \u2115 \u2192 \u03b2) :\n    \u220f t \u2208 powersetCard n s, f t.card = f n ^ s.card.choose n", "@[to_additive \"An uncurried version of `Finset.sum_product_right`\"]\ntheorem prod_product_right' {s : Finset \u03b3} {t : Finset \u03b1} {f : \u03b3 \u2192 \u03b1 \u2192 \u03b2} :\n    \u220f x \u2208 s \u00d7\u02e2 t, f x.1 x.2 = \u220f y \u2208 t, \u220f x \u2208 s, f x y", "@[to_additive sum_range_one]\ntheorem prod_range_one (f : \u2115 \u2192 \u03b2) : \u220f k \u2208 range 1, f k = f 0", "@[to_additive]\ntheorem prod_range_zero (f : \u2115 \u2192 \u03b2) : \u220f k \u2208 range 0, f k = 1", "@[to_additive (attr := simp)]\ntheorem prod_sdiff_eq_div (h : s\u2081 \u2286 s\u2082) :\n    \u220f x \u2208 s\u2082 \\ s\u2081, f x = (\u220f x \u2208 s\u2082, f x) / \u220f x \u2208 s\u2081, f x", "@[to_additive]\ntheorem prod_subtype {p : \u03b1 \u2192 Prop} {F : Fintype (Subtype p)} (s : Finset \u03b1) (h : \u2200 x, x \u2208 s \u2194 p x)\n    (f : \u03b1 \u2192 \u03b2) : \u220f a \u2208 s, f a = \u220f a : Subtype p, f a", "lemma prod_univ_sum [Fintype \u03b9] (t : \u2200 i, Finset (\u03ba i)) (f : \u2200 i, \u03ba i \u2192 \u03b1) :\n    \u220f i, \u2211 j \u2208 t i, f i j = \u2211 x \u2208 piFinset t, \u220f i, f i (x i)", "@[to_additive]\ntheorem prod_zpow (f : \u03b1 \u2192 \u03b2) (s : Finset \u03b1) (n : \u2124) : \u220f a \u2208 s, f a ^ n = (\u220f a \u2208 s, f a) ^ n", "@[simp]\ntheorem product_biUnion [DecidableEq \u03b3] (s : Finset \u03b1) (t : Finset \u03b2) (f : \u03b1 \u00d7 \u03b2 \u2192 Finset \u03b3) :\n    (s \u00d7\u02e2 t).biUnion f = s.biUnion fun a => t.biUnion fun b => f (a, b)", "theorem product_image_fst [DecidableEq \u03b1] (ht : t.Nonempty) : (s \u00d7\u02e2 t).image Prod.fst = s", "theorem product_image_snd [DecidableEq \u03b2] (ht : s.Nonempty) : (s \u00d7\u02e2 t).image Prod.snd = t", "theorem product_sdiff_diag : s \u00d7\u02e2 s \\ s.diag = s.offDiag", "theorem product_self_eq_disjiUnion_perm (hf : f.IsCycleOn s) :\n    s \u00d7\u02e2 s =\n      (range s.card).disjiUnion\n        (fun k => s.map \u27e8fun i => (i, (f ^ k) i), fun i j => congr_arg Prod.fst\u27e9)\n        (product_self_eq_disjiUnion_perm_aux hf)", "theorem sdiff_singleton_eq_erase (a : \u03b1) (s : Finset \u03b1) : s \\ singleton a = erase s a", "theorem sdiff_union_erase_cancel (hts : t \u2286 s) (ha : a \u2208 t) : s \\ t \u222a t.erase a = s.erase a", "@[simp]\ntheorem seqRight_def (s : Finset \u03b1) (t : Finset \u03b2) : s *> t = if s = \u2205 then \u2205 else t", "theorem set_biInter_singleton (a : \u03b1) (s : \u03b1 \u2192 Set \u03b2) : \u22c2 x \u2208 ({a} : Finset \u03b1), s x = s a", "theorem set_biUnion_finset_image {f : \u03b3 \u2192 \u03b1} {g : \u03b1 \u2192 Set \u03b2} {s : Finset \u03b3} :\n    \u22c3 x \u2208 s.image f, g x = \u22c3 y \u2208 s, g (f y)", "@[simp]\ntheorem set_biUnion_preimage_singleton (f : \u03b1 \u2192 \u03b2) (s : Finset \u03b2) :\n    \u22c3 y \u2208 s, f \u207b\u00b9' {y} = f \u207b\u00b9' s", "@[simp]\ntheorem shadow_empty : \u2202 (\u2205 : Finset (Finset \u03b1)) = \u2205", "lemma shatterer_mono (h : \ud835\udc9c \u2286 \u212c) : \ud835\udc9c.shatterer \u2286 \u212c.shatterer", "@[simp] lemma shatters_univ [Fintype \u03b1] : \ud835\udc9c.Shatters univ \u2194 \ud835\udc9c = univ", "@[to_additive (attr := simp)]\ntheorem singleton_div (a : \u03b1) : {a} / s = s.image (a / \u00b7)", "@[simp]\ntheorem singleton_inter_of_mem {a : \u03b1} {s : Finset \u03b1} (H : a \u2208 s) : {a} \u2229 s = {a}", "theorem smul_finset_inter\u2080 (ha : a \u2260 0) : a \u2022 (s \u2229 t) = a \u2022 s \u2229 a \u2022 t", "@[to_additive]\ntheorem smul_finset_sdiff : a \u2022 (s \\ t) = a \u2022 s \\ a \u2022 t", "@[simp]\ntheorem smul_mem_smul_finset_iff\u2080 (ha : a \u2260 0) : a \u2022 b \u2208 a \u2022 s \u2194 b \u2208 s", "theorem sort_sorted_gt (s : Finset \u03b1) : List.Sorted (\u00b7 > \u00b7) (sort (\u00b7 \u2265 \u00b7) s)", "theorem sort_sorted_lt (s : Finset \u03b1) : List.Sorted (\u00b7 < \u00b7) (sort (\u00b7 \u2264 \u00b7) s)", "theorem sorted_last_eq_max' {s : Finset \u03b1}\n    {h : (s.sort (\u00b7 \u2264 \u00b7)).length - 1 < (s.sort (\u00b7 \u2264 \u00b7)).length} :\n    (s.sort (\u00b7 \u2264 \u00b7)).get \u27e8(s.sort (\u00b7 \u2264 \u00b7)).length - 1, h\u27e9 =\n      s.max' (by rw [length_sort] at h; exact card_pos.1 (lt_of_le_of_lt bot_le h))", "theorem sorted_last_eq_max'_aux (s : Finset \u03b1)\n    (h : (s.sort (\u00b7 \u2264 \u00b7)).length - 1 < (s.sort (\u00b7 \u2264 \u00b7)).length) (H : s.Nonempty) :\n    (s.sort (\u00b7 \u2264 \u00b7)).get \u27e8(s.sort (\u00b7 \u2264 \u00b7)).length - 1, h\u27e9 = s.max' H", "theorem ssubset_iff : s \u2282 t \u2194 \u2203 a \u2209 s, insert a s \u2286 t", "theorem ssubset_iff_exists_cons_subset : s \u2282 t \u2194 \u2203 (a : _) (h : a \u2209 s), s.cons a h \u2286 t", "lemma strictMono_iff_forall_lt_cons : StrictMono f \u2194 \u2200 s \u2983a\u2984 ha, f s < f (cons a s ha)", "@[to_additive (attr := measurability)]\ntheorem _root_.Finset.stronglyMeasurable_prod {\u03b9 : Type*} {f : \u03b9 \u2192 \u03b1 \u2192 M} (s : Finset \u03b9)\n    (hf : \u2200 i \u2208 s, StronglyMeasurable (f i)) : StronglyMeasurable fun a => \u220f i \u2208 s, f i a", "lemma subset_biUnion_of_mem (u : \u03b1 \u2192 Finset \u03b2) {x : \u03b1} (xs : x \u2208 s) : u x \u2286 s.biUnion u", "@[to_additive\n      \"If a finset `u` is contained in the sum of two sets `s - t`, we can find two finsets\n      `s'`, `t'` such that `s' \u2286 s`, `t' \u2286 t` and `u \u2286 s' - t'`.\"]\ntheorem subset_div {s t : Set \u03b1} :\n    \u2191u \u2286 s / t \u2192 \u2203 s' t' : Finset \u03b1, \u2191s' \u2286 s \u2227 \u2191t' \u2286 t \u2227 u \u2286 s' / t'", "theorem subset_erase {a : \u03b1} {s t : Finset \u03b1} : s \u2286 t.erase a \u2194 s \u2286 t \u2227 a \u2209 s", "theorem subset_insert_iff {a : \u03b1} {s t : Finset \u03b1} : s \u2286 insert a t \u2194 erase s a \u2286 t", "theorem subset_map_iff {f : \u03b1 \u21aa \u03b2} {s : Finset \u03b2} {t : Finset \u03b1} :\n    s \u2286 t.map f \u2194 \u2203 u \u2286 t, s = u.map f", "theorem subset_product_image_snd [DecidableEq \u03b2] : (s \u00d7\u02e2 t).image Prod.snd \u2286 t", "theorem subset_union_elim {s : Finset \u03b1} {t\u2081 t\u2082 : Set \u03b1} (h : \u2191s \u2286 t\u2081 \u222a t\u2082) :\n    \u2203 s\u2081 s\u2082 : Finset \u03b1, s\u2081 \u222a s\u2082 = s \u2227 \u2191s\u2081 \u2286 t\u2081 \u2227 \u2191s\u2082 \u2286 t\u2082 \\ t\u2081", "theorem subset_vsub {s t : Set \u03b2} :\n    \u2191u \u2286 s -\u1d65 t \u2192 \u2203 s' t' : Finset \u03b2, \u2191s' \u2286 s \u2227 \u2191t' \u2286 t \u2227 u \u2286 s' -\u1d65 t'", "theorem subtype_eq_empty {p : \u03b1 \u2192 Prop} [DecidablePred p] {s : Finset \u03b1} :\n    s.subtype p = \u2205 \u2194 \u2200 x, p x \u2192 x \u2209 s", "theorem sum_antidiagonal_choose_succ_mul (f : \u2115 \u2192 \u2115 \u2192 R) (n : \u2115) :\n    (\u2211 ij \u2208 antidiagonal (n + 1), ((n + 1).choose ij.1 : R) * f ij.1 ij.2) =\n      (\u2211 ij \u2208 antidiagonal n, (n.choose ij.1 : R) * f ij.1 (ij.2 + 1)) +\n        \u2211 ij \u2208 antidiagonal n, (n.choose ij.2 : R) * f (ij.1 + 1) ij.2", "theorem sum_centroidWeightsIndicator [Fintype \u03b9] :\n    \u2211 i, s.centroidWeightsIndicator k i = \u2211 i \u2208 s, s.centroidWeights k i", "lemma Finset.sum_indicator_mod {R : Type*} [AddCommMonoid R] (m : \u2115) [NeZero m] (f : \u2115 \u2192 R) :\n    f = \u2211 a : ZMod m, {n : \u2115 | (n : ZMod m) = a}.indicator f", "theorem sum_pow [CommSemiring R] (x : \u03b1 \u2192 R) (n : \u2115) :\n    s.sum x ^ n = \u2211 k \u2208 s.sym n, k.val.multinomial * (k.val.map x).prod", "theorem sum_pow_mul_eq_add_pow (a b : \u03b1) (s : Finset \u03b9) :\n    (\u2211 t \u2208 s.powerset, a ^ t.card * b ^ (s.card - t.card)) = (a + b) ^ s.card", "theorem sum_powerset_apply_card {\u03b1 \u03b2 : Type*} [AddCommMonoid \u03b1] (f : \u2115 \u2192 \u03b1) {x : Finset \u03b2} :\n    \u2211 m \u2208 x.powerset, f m.card = \u2211 m \u2208 range (x.card + 1), x.card.choose m \u2022 f m", "theorem sum_schlomilch_le' (hf : \u2200 \u2983m n\u2984, 1 < m \u2192 m \u2264 n \u2192 f n \u2264 f m) (h_pos : \u2200 n, 0 < u n)\n    (hu : Monotone u) (n : \u2115) :\n    (\u2211 k \u2208 range n, (u (k + 1) - u k) \u2022 f (u (k + 1))) \u2264 \u2211 k \u2208 Ico (u 0 + 1) (u n + 1), f k", "theorem _root_.Finset.sum_single_ite [Fintype n] (a : R) (i : n) :\n    (\u2211 x : n, Finsupp.single x (if i = x then a else 0)) = Finsupp.single i a", "theorem sum_smul_vsub_eq_weightedVSubOfPoint_sub (w : \u03b9 \u2192 k) (p\u2081 p\u2082 : \u03b9 \u2192 P) (b : P) :\n    (\u2211 i \u2208 s, w i \u2022 (p\u2081 i -\u1d65 p\u2082 i)) =\n      s.weightedVSubOfPoint p\u2081 b w - s.weightedVSubOfPoint p\u2082 b w", "theorem sup'_inf_distrib_right (f : \u03b9 \u2192 \u03b1) (a : \u03b1) :\n    s.sup' hs f \u2293 a = s.sup' hs fun i => f i \u2293 a", "theorem sup'_mono {s\u2081 s\u2082 : Finset \u03b2} (h : s\u2081 \u2286 s\u2082) (h\u2081 : s\u2081.Nonempty):\n    s\u2081.sup' h\u2081 f \u2264 s\u2082.sup' (h\u2081.mono h) f", "theorem sup'_product_left {t : Finset \u03b3} (h : (s \u00d7\u02e2 t).Nonempty) (f : \u03b2 \u00d7 \u03b3 \u2192 \u03b1) :\n    (s \u00d7\u02e2 t).sup' h f = s.sup' h.fst fun i => t.sup' h.snd fun i' => f \u27e8i, i'\u27e9", "@[simp, nolint simpNF]\ntheorem supIndep_attach : (s.attach.SupIndep fun a => f a) \u2194 s.SupIndep f", "@[to_additive]\nlemma sup_div_right [SemilatticeSup \u03b2] [OrderBot \u03b2] (s t : Finset \u03b1) (f : \u03b1 \u2192 \u03b2) :\n    sup (s / t) f = sup t fun y \u21a6 sup s (f <| \u00b7 / y)", "theorem sup_eq_biUnion {\u03b1 \u03b2} [DecidableEq \u03b2] (s : Finset \u03b1) (t : \u03b1 \u2192 Finset \u03b2) :\n    s.sup t = s.biUnion t", "theorem sup_eq_iSup [CompleteLattice \u03b2] (s : Finset \u03b1) (f : \u03b1 \u2192 \u03b2) : s.sup f = \u2a06 a \u2208 s, f a", "@[to_additive (attr := simp)]\nlemma sup_inv [SemilatticeSup \u03b2] [OrderBot \u03b2] (s : Finset \u03b1) (f : \u03b1 \u2192 \u03b2) :\n    sup s\u207b\u00b9 f = sup s (f \u00b7\u207b\u00b9)", "theorem sup_le_of_le_directed {\u03b1 : Type*} [SemilatticeSup \u03b1] [OrderBot \u03b1] (s : Set \u03b1)\n    (hs : s.Nonempty) (hdir : DirectedOn (\u00b7 \u2264 \u00b7) s) (t : Finset \u03b1) :\n    (\u2200 x \u2208 t, \u2203 y \u2208 s, x \u2264 y) \u2192 \u2203 x \u2208 s, t.sup id \u2264 x", "theorem sup_mul_le_mul_sup_of_nonneg [OrderBot R] (s : Finset \u03b9) (ha : \u2200 i \u2208 s, 0 \u2264 a i)\n    (hb : \u2200 i \u2208 s, 0 \u2264 b i) : s.sup (a * b) \u2264 s.sup a * s.sup b", "theorem sup_sdiff_right {\u03b1 \u03b2 : Type*} [GeneralizedBooleanAlgebra \u03b1] (s : Finset \u03b2) (f : \u03b2 \u2192 \u03b1)\n    (a : \u03b1) : (s.sup fun b => f b \\ a) = s.sup f \\ a", "@[simp]\ntheorem sup_set_eq_biUnion (s : Finset \u03b1) (f : \u03b1 \u2192 Set \u03b2) : s.sup f = \u22c3 x \u2208 s, f x", "lemma support_prod (s : Finset \u03b9) (f : \u03b9 \u2192 \u03ba \u2192 M\u2080) :\n    support (fun j \u21a6 \u220f i \u2208 s, f i j) = \u22c2 i \u2208 s, support (f i)", "private lemma sups_aux : a \u2208 upperClosure \u2191(s \u22bb t) \u2194 a \u2208 upperClosure s \u2227 a \u2208 upperClosure t", "theorem sups_union_left : (s\u2081 \u222a s\u2082) \u22bb t = s\u2081 \u22bb t \u222a s\u2082 \u22bb t", "theorem sym2_eq_image : s.sym2 = (s \u00d7\u02e2 s).image Sym2.mk", "@[simp]\ntheorem sym2_univ [Fintype \u03b1] (inst : Fintype (Sym2 \u03b1) := Sym2.instFintype) :\n    (univ : Finset \u03b1).sym2 = univ", "@[simp]\ntheorem sym_nonempty : (s.sym n).Nonempty \u2194 n = 0 \u2228 s.Nonempty", "@[to_additive]\ntheorem union_div_inter_subset_union : (s\u2081 \u222a s\u2082) / (t\u2081 \u2229 t\u2082) \u2286 s\u2081 / t\u2081 \u222a s\u2082 / t\u2082", "theorem union_erase_of_mem (ha : a \u2208 s) (t : Finset \u03b1) : s \u222a t.erase a = s \u222a t", "@[to_additive]\ntheorem union_mul_inter_subset_union : (s\u2081 \u222a s\u2082) * (t\u2081 \u2229 t\u2082) \u2286 s\u2081 * t\u2081 \u222a s\u2082 * t\u2082", "@[simp]\ntheorem union_product [DecidableEq \u03b1] [DecidableEq \u03b2] : (s \u222a s') \u00d7\u02e2 t = s \u00d7\u02e2 t \u222a s' \u00d7\u02e2 t", "theorem union_right_comm (s t u : Finset \u03b1) : s \u222a t \u222a u = s \u222a u \u222a t", "theorem union_sdiff_of_subset (h : s \u2286 t) : s \u222a t \\ s = t", "theorem union_subset (hs : s \u2286 u) : t \u2286 u \u2192 s \u222a t \u2286 u", "@[gcongr]\ntheorem union_subset_union_left (h : s\u2081 \u2286 s\u2082) : s\u2081 \u222a t \u2286 s\u2082 \u222a t", "@[simp]\ntheorem union_val (s t : Finset \u03b1) : (s \u222a t).1 = s.1 \u222a t.1", "lemma univ_map_subtype [Fintype \u03b1] (p : \u03b1 \u2192 Prop) [DecidablePred p] [Fintype {a // p a}] :\n    univ.map (Function.Embedding.subtype p) = univ.filter p", "@[to_additive (attr := simp)]\ntheorem univ_mul_univ [Fintype \u03b1] : (univ : Finset \u03b1) * univ = univ", "@[simp]\ntheorem Finset.univ_pi_univ {\u03b1 : Type*} {\u03b2 : \u03b1 \u2192 Type*} [DecidableEq \u03b1] [Fintype \u03b1]\n    [\u2200 a, Fintype (\u03b2 a)] :\n    (Finset.univ.pi fun a : \u03b1 => (Finset.univ : Finset (\u03b2 a))) = Finset.univ", "@[simp] lemma Finset.univ_sigma_univ : univ.sigma (fun _ \u21a6 univ) = (univ : Finset (\u03a3 i, \u03ba i)) := rfl", "lemma update_piecewise_of_mem [DecidableEq \u03b9] {i : \u03b9} (hi : i \u2208 s) (v : \u03c0 i) :\n    update (s.piecewise f g) i v = s.piecewise (update f i v) g", "@[simp]\ntheorem val_inj {s t : Finset \u03b1} : s.1 = t.1 \u2194 s = t", "theorem val_injective : Injective (val : Finset \u03b1 \u2192 Multiset \u03b1)", "lemma val_strictMono : StrictMono (val : Finset \u03b1 \u2192 Multiset \u03b1) := fun _ _ \u21a6 val_lt_iff.2", "theorem weightedVSubOfPoint_sdiff [DecidableEq \u03b9] {s\u2082 : Finset \u03b9} (h : s\u2082 \u2286 s) (w : \u03b9 \u2192 k)\n    (p : \u03b9 \u2192 P) (b : P) :\n    (s \\ s\u2082).weightedVSubOfPoint p b w + s\u2082.weightedVSubOfPoint p b w =\n      s.weightedVSubOfPoint p b w", "theorem weightedVSubOfPoint_sdiff_sub [DecidableEq \u03b9] {s\u2082 : Finset \u03b9} (h : s\u2082 \u2286 s) (w : \u03b9 \u2192 k)\n    (p : \u03b9 \u2192 P) (b : P) :\n    (s \\ s\u2082).weightedVSubOfPoint p b w - s\u2082.weightedVSubOfPoint p b (-w) =\n      s.weightedVSubOfPoint p b w", "@[simp] lemma zero_mem_box : (0 : \u03b1) \u2208 box n \u2194 n = 0", "theorem addHom_ext [AddZeroClass N] \u2983f g : (\u03b1 \u2192\u2080 M) \u2192+ N\u2984\n    (H : \u2200 x y, f (single x y) = g (single x y)) : f = g", "@[simp]\ntheorem add_eq_zero_iff (f g : \u03b9 \u2192\u2080 \u03b1) : f + g = 0 \u2194 f = 0 \u2227 g = 0", "lemma apply_single [AddCommMonoid N] [AddCommMonoid P]\n    {F : Type*} [FunLike F N P] [AddMonoidHomClass F N P] (e : F)\n    (a : \u03b1) (n : N) (b : \u03b1) :\n    e ((single a n) b) = single a (e n) b", "theorem card_support_le_one' [Nonempty \u03b1] {f : \u03b1 \u2192\u2080 M} :\n    card f.support \u2264 1 \u2194 \u2203 a b, f = single a b", "@[simp]\ntheorem coe_lsum (f : \u03b1 \u2192 M \u2192\u2097[R] N) : (lsum S f : (\u03b1 \u2192\u2080 M) \u2192 N) = fun d => d.sum fun i => f i", "lemma coe_mono : Monotone (Finsupp.toFun : (\u03b9 \u2192\u2080 \u03b1) \u2192 \u03b9 \u2192 \u03b1) := fun _ _ \u21a6 id", "@[simp, norm_cast]\ntheorem coe_sumElim {\u03b1 \u03b2 \u03b3 : Type*} [Zero \u03b3] (f : \u03b1 \u2192\u2080 \u03b3) (g : \u03b2 \u2192\u2080 \u03b3) :\n    \u21d1(sumElim f g) = Sum.elim f g", "@[simp]\ntheorem cons_zero_zero : cons 0 (0 : Fin n \u2192\u2080 M) = 0", "@[ext]\ntheorem distribMulActionHom_ext' {f g : (\u03b1 \u2192\u2080 M) \u2192+[R] N}\n    (h : \u2200 a : \u03b1, f.comp (DistribMulActionHom.single a) = g.comp (DistribMulActionHom.single a)) :\n    f = g", "theorem domLCongr_single {\u03b1\u2081 : Type*} {\u03b1\u2082 : Type*} (e : \u03b1\u2081 \u2243 \u03b1\u2082) (i : \u03b1\u2081) (m : M) :\n    (Finsupp.domLCongr e : _ \u2243\u2097[R] _) (Finsupp.single i m) = Finsupp.single (e i) m", "@[simp]\ntheorem embDomain_add (f : \u03b1 \u21aa \u03b2) (v w : \u03b1 \u2192\u2080 M) :\n    embDomain f (v + w) = embDomain f v + embDomain f w", "theorem erase_eq_sub_single [AddGroup G] (f : \u03b1 \u2192\u2080 G) (a : \u03b1) : f.erase a = f - single a (f a)", "theorem le_iff' (f g : \u03b9 \u2192\u2080 \u03b1) {s : Finset \u03b9} (hf : f.support \u2286 s) : f \u2264 g \u2194 \u2200 i \u2208 s, f i \u2264 g i", "@[simp]\ntheorem lift_symm_apply (f) (x) : ((lift M R X).symm f) x = f (single x 1)", "theorem mapDomain_comp {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} :\n    mapDomain (g \u2218 f) v = mapDomain g (mapDomain f v)", "theorem mapDomain_injOn (S : Set \u03b1) {f : \u03b1 \u2192 \u03b2} (hf : Set.InjOn f S) :\n    Set.InjOn (mapDomain f : (\u03b1 \u2192\u2080 M) \u2192 \u03b2 \u2192\u2080 M) { w | (w.support : Set \u03b1) \u2286 S }", "theorem mapDomain_injective {f : \u03b1 \u2192 \u03b2} (hf : Function.Injective f) :\n    Function.Injective (mapDomain f : (\u03b1 \u2192\u2080 M) \u2192 \u03b2 \u2192\u2080 M)", "@[simp]\ntheorem mapRange.addEquiv_symm (f : M \u2243+ N) :\n    ((mapRange.addEquiv f).symm : (\u03b1 \u2192\u2080 _) \u2243+ _) = mapRange.addEquiv f.symm", "@[simp]\ntheorem mapRange.addMonoidHom_toZeroHom (f : M \u2192+ N) :\n    (mapRange.addMonoidHom f).toZeroHom = (mapRange.zeroHom f.toZeroHom : ZeroHom (\u03b1 \u2192\u2080 _) _)", "theorem mapRange.linearEquiv_trans (f : M \u2243\u2097[R] N) (f\u2082 : N \u2243\u2097[R] P) :\n    (mapRange.linearEquiv (f.trans f\u2082) : (\u03b1 \u2192\u2080 _) \u2243\u2097[R] _) =\n      (mapRange.linearEquiv f).trans (mapRange.linearEquiv f\u2082)", "theorem mapRange_sub [SubNegZeroMonoid G] [SubNegZeroMonoid H] {f : G \u2192 H} {hf : f 0 = 0}\n    (hf' : \u2200 x y, f (x - y) = f x - f y) (v\u2081 v\u2082 : \u03b1 \u2192\u2080 G) :\n    mapRange f hf (v\u2081 - v\u2082) = mapRange f hf v\u2081 - mapRange f hf v\u2082", "theorem mem_splitSupport_iff_nonzero (i : \u03b9) : i \u2208 splitSupport l \u2194 split l i \u2260 0", "theorem mk_mem_graph (f : \u03b1 \u2192\u2080 M) {a : \u03b1} (ha : a \u2208 f.support) : (a, f a) \u2208 f.graph", "@[simp]\ntheorem nonempty_neLocus_iff {f g : \u03b1 \u2192\u2080 N} : (f.neLocus g).Nonempty \u2194 f \u2260 g", "@[simp]\ntheorem sigmaFinsuppAddEquivPiFinsupp_apply {\u03b1 : Type*} {\u03b9s : \u03b7 \u2192 Type*} [AddMonoid \u03b1]\n    (f : (\u03a3j, \u03b9s j) \u2192\u2080 \u03b1) (j i) : sigmaFinsuppAddEquivPiFinsupp f j i = f \u27e8j, i\u27e9", "theorem single_apply [Decidable (a = a')] : single a b a' = if a = a' then b else 0", "theorem single_eq_single_iff (a\u2081 a\u2082 : \u03b1) (b\u2081 b\u2082 : M) :\n    single a\u2081 b\u2081 = single a\u2082 b\u2082 \u2194 a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082 \u2228 b\u2081 = 0 \u2227 b\u2082 = 0", "theorem smul_sum [Zero \u03b2] [AddCommMonoid M] [DistribSMul R M] {v : \u03b1 \u2192\u2080 \u03b2} {c : R} {h : \u03b1 \u2192 \u03b2 \u2192 M} :\n    c \u2022 v.sum h = v.sum fun a b => c \u2022 h a b", "@[simp]\ntheorem some_apply [Zero M] (f : Option \u03b1 \u2192\u2080 M) (a : \u03b1) : f.some a = f (Option.some a)", "theorem sumElim_inr {\u03b1 \u03b2 \u03b3 : Type*} [Zero \u03b3] (f : \u03b1 \u2192\u2080 \u03b3) (g : \u03b2 \u2192\u2080 \u03b3) (x : \u03b2) :\n    sumElim f g (Sum.inr x) = g x", "@[simp]\ntheorem sum_apply [Zero M] [AddCommMonoid N] {f : \u03b1 \u2192\u2080 M} {g : \u03b1 \u2192 M \u2192 \u03b2 \u2192\u2080 N} {a\u2082 : \u03b2} :\n    (f.sum g) a\u2082 = f.sum fun a\u2081 b => g a\u2081 b a\u2082", "theorem sum_curry_index (f : \u03b1 \u00d7 \u03b2 \u2192\u2080 M) (g : \u03b1 \u2192 \u03b2 \u2192 M \u2192 N) (hg\u2080 : \u2200 a b, g a b 0 = 0)\n    (hg\u2081 : \u2200 a b c\u2080 c\u2081, g a b (c\u2080 + c\u2081) = g a b c\u2080 + g a b c\u2081) :\n    (f.curry.sum fun a f => f.sum (g a)) = f.sum fun p c => g p.1 p.2 c", "theorem sum_ite_self_eq [DecidableEq \u03b1] {N : Type*} [AddCommMonoid N] (f : \u03b1 \u2192\u2080 N) (a : \u03b1) :\n    (f.sum fun x v => ite (a = x) v 0) = f a", "theorem Finsupp.sum_mul (b : S) (s : \u03b1 \u2192\u2080 R) {f : \u03b1 \u2192 R \u2192 S} :\n    s.sum f * b = s.sum fun a c => f a c * b", "@[simp]\ntheorem support_eq_empty {f : \u03b1 \u2192\u2080 M} : f.support = \u2205 \u2194 f = 0", "theorem support_inf_union_support_sup : (f \u2293 g).support \u222a (f \u2294 g).support = f.support \u222a g.support", "theorem support_mapRange {f : M \u2192 N} {hf : f 0 = 0} {g : \u03b1 \u2192\u2080 M} :\n    (mapRange f hf g).support \u2286 g.support", "theorem supported_union (s t : Set \u03b1) :\n    supported M R (s \u222a t) = supported M R s \u2294 supported M R t", "@[simp]\ntheorem Finsupp.toFreeAbelianGroup_comp_toFinsupp :\n    toFreeAbelianGroup.comp toFinsupp = AddMonoidHom.id (FreeAbelianGroup X)", "theorem totalOn_range (s : Set \u03b1) : LinearMap.range (Finsupp.totalOn \u03b1 M R v s) = \u22a4", "@[simp]\ntheorem total_fin_zero (f : Fin 0 \u2192 M) : Finsupp.total (Fin 0) M R f = 0", "theorem total_total {\u03b1 \u03b2 : Type*} (A : \u03b1 \u2192 M) (B : \u03b2 \u2192 \u03b1 \u2192\u2080 R) (f : \u03b2 \u2192\u2080 R) :\n    Finsupp.total \u03b1 M R A (Finsupp.total \u03b2 (\u03b1 \u2192\u2080 R) R B f) =\n      Finsupp.total \u03b2 M R (fun b => Finsupp.total \u03b1 M R A (B b)) f", "theorem update_eq_single_add_erase (f : \u03b1 \u2192\u2080 M) (a : \u03b1) (b : M) :\n    f.update a b = single a b + f.erase a", "theorem zero_apply {a : \u03b1} : (0 : \u03b1 \u2192\u2080 M) a = 0", "@[simp]\ntheorem Fintype.card_bool : Fintype.card Bool = 2", "@[congr]\ntheorem card_congr' {\u03b1 \u03b2} [Fintype \u03b1] [Fintype \u03b2] (h : \u03b1 = \u03b2) : card \u03b1 = card \u03b2", "theorem card_eq_zero_iff : card \u03b1 = 0 \u2194 IsEmpty \u03b1", "lemma card_filter_piFinset_eq_of_mem (s : \u2200 i, Finset (\u03b1 i)) (i : \u03b9) {a : \u03b1 i} (ha : a \u2208 s i) :\n    ((piFinset s).filter fun f \u21a6 f i = a).card = \u220f j \u2208 univ.erase i, (s j).card", "@[simp]\ntheorem Fintype.card_finset [Fintype \u03b1] : Fintype.card (Finset \u03b1) = 2 ^ Fintype.card \u03b1", "theorem card_le_card_of_rightTotal_unique (h\u2081 : RightTotal r) (h\u2082 : RightUnique r) :\n    Fintype.card \u03b2 \u2264 Fintype.card \u03b1", "@[simp]\ntheorem card_ofSubsingleton (a : \u03b1) [Subsingleton \u03b1] : @Fintype.card _ (ofSubsingleton a) = 1", "@[simp] lemma Fintype.card_shrink [Fintype (Shrink.{v} \u03b1)] : card (Shrink.{v} \u03b1) = card \u03b1", "theorem Fintype.card_subtype [Fintype \u03b1] (p : \u03b1 \u2192 Prop) [DecidablePred p] :\n    Fintype.card { x // p x } = ((Finset.univ : Finset \u03b1).filter p).card", "theorem Fintype.card_subtype_or_disjoint (p q : \u03b1 \u2192 Prop) (h : Disjoint p q) [Fintype { x // p x }]\n    [Fintype { x // q x }] [Fintype { x // p x \u2228 q x }] :\n    Fintype.card { x // p x \u2228 q x } = Fintype.card { x // p x } + Fintype.card { x // q x }", "theorem Fintype.card_unit : Fintype.card Unit = 1", "@[simp]\ntheorem Fintype.card_units_int : Fintype.card \u2124\u02e3 = 2", "theorem exists_lt_card_fiber_of_nsmul_lt_card (hb : card \u03b2 \u2022 b < card \u03b1) :\n    \u2203 y : \u03b2, b < (univ.filter fun x => f x = y).card", "@[simp] lemma finsetEquivSet_symm_apply (s : Set \u03b1) [Fintype s] :\n    finsetEquivSet.symm s = s.toFinset", "@[simp]\ntheorem mem_piFinset {t : \u2200 a, Finset (\u03b4 a)} {f : \u2200 a, \u03b4 a} : f \u2208 piFinset t \u2194 \u2200 a, f a \u2208 t a", "@[to_additive sum_nonneg]\nlemma one_le_prod (hf : 1 \u2264 f) : 1 \u2264 \u220f i, f i := Finset.one_le_prod' fun _ _ \u21a6 hf _", "lemma piFinset_update_singleton_eq_filter_piFinset_eq (s : \u2200 i, Finset (\u03b4 i)) (i : \u03b1) {a : \u03b4 i}\n    (ha : a \u2208 s i) :\n    piFinset (Function.update s i {a}) = (piFinset s).filter (fun f \u21a6 f i = a)", "@[to_additive]\nlemma prod_ite_eq_ite_exists (p : \u03b9 \u2192 Prop) [DecidablePred p] (h : \u2200 i j, p i \u2192 p j \u2192 i = j)\n    (a : \u03b1) : \u220f i, ite (p i) a 1 = ite (\u2203 i, p i) a 1", "@[to_additive]\ntheorem Fintype.prod_sum_elim (f : \u03b1\u2081 \u2192 M) (g : \u03b1\u2082 \u2192 M) :\n    \u220f x, Sum.elim f g x = (\u220f a\u2081, f a\u2081) * \u220f a\u2082, g a\u2082", "theorem subtype_card {p : \u03b1 \u2192 Prop} (s : Finset \u03b1) (H : \u2200 x : \u03b1, x \u2208 s \u2194 p x) :\n    @card { x // p x } (Fintype.subtype s H) = s.card", "theorem Fintype.univ_pempty : @univ PEmpty _ = \u2205", "@[simp] theorem realize_eqZero [CommRing K] [CompatibleRing K] (n : \u2115)\n    (v : Empty \u2192 K) : (Formula.Realize (eqZero n) v) \u2194 ((n : K) = 0)", "@[simp]\ntheorem listDecode_encode_list (l : List (\u03a3n, L.BoundedFormula \u03b1 n)) :\n    (listDecode (l.bind fun \u03c6 => \u03c6.2.listEncode)).1 = l.headI", "theorem realize_constantsVarsEquiv [L[[\u03b1]].Structure M] [(lhomWithConstants L \u03b1).IsExpansionOn M]\n    {n} {\u03c6 : L[[\u03b1]].BoundedFormula \u03b2 n} {v : \u03b2 \u2192 M} {xs : Fin n \u2192 M} :\n    (constantsVarsEquiv \u03c6).Realize (Sum.elim (fun a => \u2191(L.con a)) v) xs \u2194 \u03c6.Realize v xs", "@[simp]\ntheorem realize_iff : (\u03c6.iff \u03c8).Realize v xs \u2194 (\u03c6.Realize v xs \u2194 \u03c8.Realize v xs)", "theorem realize_liftAt {n n' m : \u2115} {\u03c6 : L.BoundedFormula \u03b1 n} {v : \u03b1 \u2192 M} {xs : Fin (n + n') \u2192 M}\n    (hmn : m + n' \u2264 n + 1) :\n    (\u03c6.liftAt n' m).Realize v xs \u2194\n      \u03c6.Realize v (xs \u2218 fun i => if \u2191i < m then Fin.castAdd n' i else Fin.addNat i n')", "@[simp]\ntheorem realize_liftAt_one_self {n : \u2115} {\u03c6 : L.BoundedFormula \u03b1 n} {v : \u03b1 \u2192 M}\n    {xs : Fin (n + 1) \u2192 M} : (\u03c6.liftAt 1 n).Realize v xs \u2194 \u03c6.Realize v (xs \u2218 castSucc)", "@[simp]\ntheorem relabelAux_sum_inl (k : \u2115) :\n    relabelAux (Sum.inl : \u03b1 \u2192 Sum \u03b1 (Fin n)) k = Sum.map id (natAdd n)", "@[simp]\ntheorem relabel_all (g : \u03b1 \u2192 Sum \u03b2 (Fin n)) {k} (\u03c6 : L.BoundedFormula \u03b1 (k + 1)) :\n    \u03c6.all.relabel g = (\u03c6.relabel g).all", "@[simp, norm_cast]\ntheorem coe_top : ((\u22a4 : L.DefinableSet A \u03b1) : Set (\u03b1 \u2192 M)) = univ", "theorem comp_unify {\u03b1 : Type*} {x : \u03b1 \u2192 \u03a3\u02e3 f} {i j : \u03b9} (ij : i \u2264 j)\n    (h : i \u2208 upperBounds (range (Sigma.fst \u2218 x))) :\n    f i j ij \u2218 unify f x i h = unify f x j\n      fun k hk => _root_.trans (mem_upperBounds.1 h k hk) ij", "@[symm]\nnonrec theorem symm (h : M \u2245[L] N) : N \u2245[L] M", "@[simp]\ntheorem subtype_comp_codRestrict (f : M \u21aa[L] N) (p : L.Substructure N) (h : \u2200 b, f b \u2208 p) :\n    p.subtype.comp (codRestrict p f h) = f", "@[simp]\ntheorem map_fun (\u03c6 : M \u2243[L] N) {n : \u2115} (f : L.Functions n) (x : Fin n \u2192 M) :\n    \u03c6 (funMap f x) = funMap f (\u03c6 \u2218 x)", "@[simp]\ntheorem refl_apply (x : M) : refl L M x = x", "@[simp]\ntheorem symm_apply_apply (f : M \u2243[L] N) (a : M) : f.symm (f a) = a", "@[simp]\ntheorem realize_relabel {\u03c6 : L.Formula \u03b1} {g : \u03b1 \u2192 \u03b2} {v : \u03b2 \u2192 M} :\n    (\u03c6.relabel g).Realize v \u2194 \u03c6.Realize (v \u2218 g)", "theorem sup_semanticallyEquivalent_not_inf_not :\n    T.SemanticallyEquivalent (\u03c6 \u2294 \u03c8) (\u03c6.not \u2293 \u03c8.not).not", "@[simp]\ntheorem comp_id (f : M \u2192[L] N) : f.comp (id L M) = f", "theorem HomClass.strongHomClassOfIsAlgebraic [L.IsAlgebraic] {F M N} [L.Structure M] [L.Structure N]\n    [FunLike F M N] [HomClass L F M N] : StrongHomClass L F M N where", "theorem comp_sumElim {L3 : Language} (\u03b8 : L' \u2192\u1d38 L3) :\n    \u03b8 \u2218\u1d38 \u03d5.sumElim \u03c8 = (\u03b8 \u2218\u1d38 \u03d5).sumElim (\u03b8 \u2218\u1d38 \u03c8)", "@[simp]\ntheorem id_onBoundedFormula :\n    ((LHom.id L).onBoundedFormula : L.BoundedFormula \u03b1 n \u2192 L.BoundedFormula \u03b1 n) = id", "@[simp]\ntheorem id_onTerm : ((LHom.id L).onTerm : L.Term \u03b1 \u2192 L.Term \u03b1) = id", "theorem CG.range {N : Type*} [L.Structure N] (h : CG L M) (f : M \u2192[L] N) : f.range.CG", "@[simp]\ntheorem relMap_apply\u2082 (r : r\u2082) (x y : M) :\n    @Structure.RelMap _ M (Structure.mk\u2082 c' f\u2081' f\u2082' r\u2081' r\u2082') 2 r ![x, y] = r\u2082' r x y", "@[simp]\ntheorem closure_empty : closure L (\u2205 : Set M) = \u22a5", "theorem closure_withConstants_eq :\n    closure (L[[A]]) s =\n      (closure L (A \u222a s)).withConstants ((A.subset_union_left).trans subset_closure)", "theorem coe_closure_eq_range_term_realize :\n    (closure L s : Set M) = range (@Term.realize L _ _ _ ((\u2191) : s \u2192 M))", "@[simp]\ntheorem coe_copy {s : Set M} (hs : s = S) : (S.copy s hs : Set M) = s", "@[simp]\ntheorem coe_inclusion {S T : L.Substructure M} (h : S \u2264 T) :\n    (inclusion h : S \u2192 T) = Set.inclusion h", "@[simp]\ntheorem coe_inf (p p' : L.Substructure M) :\n    ((p \u2293 p' : L.Substructure M) : Set M) = (p : Set M) \u2229 (p' : Set M)", "@[simp]\ntheorem mem_map {f : M \u2192[L] N} {S : L.Substructure M} {y : N} :\n    y \u2208 S.map f \u2194 \u2203 x \u2208 S, f x = y", "@[simp]\ntheorem subset_closure : s \u2286 closure L s", "@[simp]\ntheorem Term.realize_le [LE M] [L.OrderedStructure M] {t\u2081 t\u2082 : L.Term (Sum \u03b1 (Fin n))} {v : \u03b1 \u2192 M}\n    {xs : Fin n \u2192 M} :\n    (t\u2081.le t\u2082).Realize v xs \u2194 t\u2081.realize (Sum.elim v xs) \u2264 t\u2082.realize (Sum.elim v xs)", "theorem setOf_mem_eq_univ_iff (\u03c6 : L[[\u03b1]].Sentence) :\n    { p : T.CompleteType \u03b1 | \u03c6 \u2208 p } = Set.univ \u2194 (L.lhomWithConstants \u03b1).onTheory T \u22a8\u1d47 \u03c6", "theorem setOf_subset_eq_empty_iff (S : L[[\u03b1]].Theory) :\n    { p : T.CompleteType \u03b1 | S \u2286 \u2191p } = \u2205 \u2194\n      \u00ac((L.lhomWithConstants \u03b1).onTheory T \u222a S).IsSatisfiable", "theorem completeTheory.subset [MT : M \u22a8 T] : T \u2286 L.completeTheory M", "theorem isSatisfiable_of_isSatisfiable_onTheory {L' : Language.{w, w'}} (\u03c6 : L \u2192\u1d38 L')\n    (h : (\u03c6.onTheory T).IsSatisfiable) : T.IsSatisfiable", "theorem model_singleton_iff {\u03c6 : L.Sentence} : M \u22a8 ({\u03c6} : L.Theory) \u2194 M \u22a8 \u03c6", "theorem constantsOnMap_isExpansionOn {f : \u03b1 \u2192 \u03b2} {f\u03b1 : \u03b1 \u2192 M} {f\u03b2 : \u03b2 \u2192 M} (h : f\u03b2 \u2218 f = f\u03b1) :\n    @LHom.IsExpansionOn _ _ (LHom.constantsOnMap f) M (constantsOn.structure f\u03b1)\n      (constantsOn.structure f\u03b2)", "theorem directed_distinctConstantsTheory :\n    Directed (\u00b7 \u2286 \u00b7) (L.distinctConstantsTheory : Set \u03b1 \u2192 L[[\u03b1]].Theory)", "theorem exists_elementarilyEquivalent_card_eq (M : Type w') [L.Structure M] [Infinite M]\n    (\u03ba : Cardinal.{w}) (h1 : \u2135\u2080 \u2264 \u03ba) (h2 : lift.{w} L.card \u2264 Cardinal.lift.{max u v} \u03ba) :\n    \u2203 N : CategoryTheory.Bundled L.Structure, (M \u2245[L] N) \u2227 #N = \u03ba", "@[simp]\ntheorem orderLHom_leSymb [L.IsOrdered] :\n    (orderLHom L).onRelation leSymb = (leSymb : L.Relations 2)", "@[simp]\ntheorem realize_noBotOrder [h : NoBotOrder M] : M \u22a8 Language.order.noBotOrderSentence", "@[simp]\ntheorem withConstants_funMap_sum_inr {a : \u03b1} {x : Fin 0 \u2192 M} :\n    @funMap (L[[\u03b1]]) M _ 0 (Sum.inr a : L[[\u03b1]].Functions 0) x = L.con a", "@[simp]\ntheorem realize_one (v : \u03b1 \u2192 R) : Term.realize v (1 : ring.Term \u03b1) = 1", "@[simp]\ntheorem sum_card : Cardinal.sum (fun i => #(Sequence\u2082 a\u2080 a\u2081 a\u2082 i)) = #a\u2080 + #a\u2081 + #a\u2082", "theorem eval\u2082' :\n    Polynomial.eval\u2082 (Subfield.subtype <| FixedPoints.subfield G F) x (minpoly G F x) = 0", "theorem monic : (minpoly G F x).Monic", "protected theorem le_or_le (s : Flag \u03b1) (ha : a \u2208 s) (hb : b \u2208 s) : a \u2264 b \u2228 b \u2264 a", "@[simp] theorem Flag.mem_rangeFin {x h0 hlast hcovBy} :\n    x \u2208 rangeFin f h0 hlast hcovBy \u2194 \u2203 k, f k = x", "@[ext]\ntheorem ext : \u2200 {\u03d5\u2081 \u03d5\u2082 : Flow \u03c4 \u03b1}, (\u2200 t x, \u03d5\u2081 t x = \u03d5\u2082 t x) \u2192 \u03d5\u2081 = \u03d5\u2082", "theorem isInvariant_omegaLimit (hf : \u2200 t, Tendsto (t + \u00b7) f f) : IsInvariant \u03d5 (\u03c9 f \u03d5 s)", "theorem map_zero_apply (x : \u03b1) : \u03d5 0 x = x", "theorem apply_eq_zero_of_lt_order (hp : n < p.order) : p n = 0", "theorem changeOriginSeries_summable_aux\u2083 {r : \u211d\u22650} (hr : \u2191r < p.radius) (k : \u2115) :\n    Summable fun l : \u2115 => \u2016p.changeOriginSeries k l\u2016\u208a * r ^ l", "lemma changeOrigin_finite_of_finite (p : FormalMultilinearSeries \ud835\udd5c E F) {n : \u2115}\n    (hn : \u2200 (m : \u2115), n \u2264 m \u2192 p m = 0) {k : \u2115} (hk : n \u2264 k) :\n    p.changeOrigin x k = 0", "@[simp]\ntheorem comp_id (p : FormalMultilinearSeries \ud835\udd5c E F) : p.comp (id \ud835\udd5c E) = p", "@[simp]\ntheorem constFormalMultilinearSeries_radius {v : F} :\n    (constFormalMultilinearSeries \ud835\udd5c E v).radius = \u22a4", "protected theorem FormalMultilinearSeries.continuousOn_of_finite\n    (p : FormalMultilinearSeries \ud835\udd5c E F) {n : \u2115} (hn : \u2200 m, n \u2264 m \u2192 p m = 0) :\n    Continuous p.sum", "theorem le_radius_of_bound (C : \u211d) {r : \u211d\u22650} (h : \u2200 n : \u2115, \u2016p n\u2016 * (r : \u211d) ^ n \u2264 C) :\n    (r : \u211d\u22650\u221e) \u2264 p.radius", "theorem leftInv_comp (p : FormalMultilinearSeries \ud835\udd5c E F) (i : E \u2243L[\ud835\udd5c] F)\n    (h : p 1 = (continuousMultilinearCurryFin1 \ud835\udd5c E F).symm i) : (leftInv p i).comp p = id \ud835\udd5c E", "theorem order_eq_find' [DecidablePred fun n => p n \u2260 0] (hp : p \u2260 0) :\n    p.order = Nat.find (FormalMultilinearSeries.ne_iff.mp hp)", "theorem order_eq_zero_iff (hp : p \u2260 0) : p.order = 0 \u2194 p 0 \u2260 0", "@[simp]\ntheorem radius_neg (p : FormalMultilinearSeries \ud835\udd5c E F) : (-p).radius = p.radius", "theorem rightInv_coeff (p : FormalMultilinearSeries \ud835\udd5c E F) (i : E \u2243L[\ud835\udd5c] F) (n : \u2115) (hn : 2 \u2264 n) :\n    p.rightInv i n =\n      -(i.symm : F \u2192L[\ud835\udd5c] E).compContinuousMultilinearMap\n          (\u2211 c \u2208 ({c | 1 < Composition.length c}.toFinset : Finset (Composition n)),\n            p.compAlongComposition (p.rightInv i) c)", "theorem absNorm_eq (I : FractionalIdeal R\u2070 K) :\n    absNorm I = (Ideal.absNorm I.num : \u211a) / |Algebra.norm \u2124 (I.den : R)|", "theorem FractionalIdeal.adjoinIntegral_eq_one_of_isUnit [Algebra A K] [IsFractionRing A K] (x : K)\n    (hx : IsIntegral A x) (hI : IsUnit (adjoinIntegral A\u2070 x hx)) : adjoinIntegral A\u2070 x hx = 1", "@[simp]\ntheorem coeIdeal_eq_one {I : Ideal R} : (I : FractionalIdeal R\u2070 K) = 1 \u2194 I = 1", "theorem coeIdeal_le_coeIdeal' [IsLocalization S P] (h : S \u2264 nonZeroDivisors R) {I J : Ideal R} :\n    (I : FractionalIdeal S P) \u2264 J \u2194 I \u2264 J", "@[simp]\ntheorem coe_copy (p : FractionalIdeal S P) (s : Set P) (hs : s = \u2191p) : \u2191(p.copy s hs) = s", "@[simp, norm_cast]\ntheorem coe_mul (I J : FractionalIdeal S P) : (\u2191(I * J) : Submodule R P) = I * J", "theorem constant_factor_ne_zero {R} [CommRing R] {K : Type*} [Field K] [Algebra R K]\n    [IsFractionRing R K] {I : FractionalIdeal R\u2070 K} (hI : I \u2260 0) {a : R} {J : Ideal R}\n    (haJ : I = spanSingleton R\u2070 ((algebraMap R K) a)\u207b\u00b9 * \u2191J) :\n    (Ideal.span {a} : Ideal R) \u2260 0", "theorem exists_not_mem_one_of_ne_bot [IsDedekindDomain A] {I : Ideal A} (hI0 : I \u2260 \u22a5)\n    (hI1 : I \u2260 \u22a4) : \u2203 x \u2208 (I\u207b\u00b9 : FractionalIdeal A\u2070 K), x \u2209 (1 : FractionalIdeal A\u2070 K)", "theorem finprod_heightOneSpectrum_factorization {I : FractionalIdeal R\u2070 K} (hI : I \u2260 0) {a : R}\n    {J : Ideal R} (haJ : I = spanSingleton R\u2070 ((algebraMap R K) a)\u207b\u00b9 * \u2191J) :\n    \u220f\u1da0 v : HeightOneSpectrum R, (v.asIdeal : FractionalIdeal R\u2070 K) ^\n      ((Associates.mk v.asIdeal).count (Associates.mk J).factors -\n        (Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors : \u2124) = I", "theorem finprod_heightOneSpectrum_factorization_principal {I : FractionalIdeal R\u2070 K} (hI : I \u2260 0)\n    (k : K) (hk : I = spanSingleton R\u2070 k) :\n    \u220f\u1da0 v : HeightOneSpectrum R, (v.asIdeal : FractionalIdeal R\u2070 K) ^\n      ((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {choose\n          (mk'_surjective R\u2070 k)} : Ideal R)).factors -\n        (Associates.mk v.asIdeal).count (Associates.mk ((Ideal.span {(\u2191(choose\n          (choose_spec (mk'_surjective R\u2070 k)) : \u21a5R\u2070) : R)}) : Ideal R)).factors : \u2124) = I", "theorem inv_anti_mono (hI : I \u2260 0) (hJ : J \u2260 0) (hIJ : I \u2264 J) : J\u207b\u00b9 \u2264 I\u207b\u00b9", "protected theorem isFractional (I : FractionalIdeal S P) : IsFractional S (I : Submodule R P)", "theorem isFractional_of_fg {I : Submodule R P} (hI : I.FG) : IsFractional S I", "theorem isFractional_span_iff {s : Set P} :\n    IsFractional S (span R s) \u2194 \u2203 a \u2208 S, \u2200 b : P, b \u2208 s \u2192 IsInteger R (a \u2022 b)", "theorem isNoetherian [IsNoetherianRing R\u2081] (I : FractionalIdeal R\u2081\u2070 K) : IsNoetherian R\u2081 I", "@[simp]\ntheorem le_zero_iff {I : FractionalIdeal S P} : I \u2264 0 \u2194 I = 0", "lemma mem_dual {x} :\n    x \u2208 dual A K I \u2194 \u2200 a \u2208 I, traceForm K L x a \u2208 (algebraMap A K).range", "theorem mul_self_le_self {I : FractionalIdeal S P} (hI : I \u2264 1) : I * I \u2264 I", "@[simp]\ntheorem one_le {I : FractionalIdeal S P} : 1 \u2264 I \u2194 (1 : P) \u2208 I", "theorem spanSingleton_div_self {x : K} (hx : x \u2260 0) :\n    spanSingleton R\u2081\u2070 x / spanSingleton R\u2081\u2070 x = 1", "lemma zero_mem (I : FractionalIdeal S P) : 0 \u2208 I := I.coeToSubmodule.zero_mem", "@[simp]\nprotected theorem map_neg (f : \u03b1 \u2192 \u03b2) (x : FreeAbelianGroup \u03b1) : f <$> (-x) = -f <$> x", "@[simp]\ntheorem map_of (f : \u03b1 \u2192 \u03b2) (y : \u03b1) : f <$> of y = of (f y)", "@[simp]\ntheorem of_mul_of (x y : \u03b1) : of x * of y = of (x * y)", "@[simp]\ntheorem seq_sub (f : FreeAbelianGroup (\u03b1 \u2192 \u03b2)) (x y : FreeAbelianGroup \u03b1) :\n    f <*> x - y = (f <*> x) - (f <*> y)", "private theorem mk_mul (x y : Pre R X) :\n    Quot.mk (Rel R X) (x * y) = (HMul.hMul (self := instHMul (\u03b1 := FreeAlgebra R X))\n    (Quot.mk (Rel R X) x) (Quot.mk (Rel R X) y))", "@[to_additive (attr := simp)]\ntheorem Step.cons_not {x b} : Red.Step ((x, b) :: (x, !b) :: L) L", "@[to_additive \"Predicate asserting that the word `w\u2081` can be reduced to `w\u2082` in one step, i.e. there\n  are words `w\u2083 w\u2084` and letter `x` such that `w\u2081 = w\u2083 + x + (-x) + w\u2084` and `w\u2082 = w\u2083w\u2084`\"]\ntheorem Step.length : \u2200 {L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)}, Step L\u2081 L\u2082 \u2192 L\u2082.length + 2 = L\u2081.length", "@[to_additive]\ntheorem append_append (h\u2081 : Red L\u2081 L\u2083) (h\u2082 : Red L\u2082 L\u2084) : Red (L\u2081 ++ L\u2082) (L\u2083 ++ L\u2084)", "@[to_additive]\ntheorem cons_cons {p} : Red L\u2081 L\u2082 \u2192 Red (p :: L\u2081) (p :: L\u2082)", "@[to_additive]\ntheorem to_append_iff : Red L (L\u2081 ++ L\u2082) \u2194 \u2203 L\u2083 L\u2084, L = L\u2083 ++ L\u2084 \u2227 Red L\u2083 L\u2081 \u2227 Red L\u2084 L\u2082", "theorem _root_.FreeGroup.injective_lift_of_ping_pong : Function.Injective (FreeGroup.lift a)", "@[to_additive (attr := simp)]\ntheorem invRev_length : (invRev L\u2081).length = L\u2081.length", "@[to_additive]\ntheorem lift.of_eq (x : FreeGroup \u03b1) : lift FreeGroup.of x = x", "@[to_additive (attr := simp)]\ntheorem map.of {x} : map f (of x) = of (f x)", "@[to_additive (attr := simp)]\ntheorem map_one (f : \u03b1 \u2192 \u03b2) : f <$> (1 : FreeGroup \u03b1) = 1", "@[to_additive \"If two words have a common maximal reduction, then they correspond to the same\n  element in the additive free group.\"]\ntheorem reduce.exact (H : reduce L\u2081 = reduce L\u2082) : mk L\u2081 = mk L\u2082", "@[to_additive \"The second theorem that characterises the function `reduce`: the maximal reduction of\n  a word only reduces to itself.\"]\ntheorem reduce.min (H : Red (reduce L\u2081) L\u2082) : reduce L\u2081 = L\u2082", "@[to_additive \"The first theorem that characterises the function `reduce`: a word reduces to its\n  maximal reduction.\"]\ntheorem reduce.red : Red L (reduce L)", "theorem liftAux_spec (f : X \u2192 L) (a b : lib R X) (h : FreeLieAlgebra.Rel R X a b) :\n    liftAux R f a = liftAux R f b", "@[to_additive (attr := simp)]\ntheorem length_toFreeSemigroup (x : FreeMagma \u03b1) : (toFreeSemigroup x).length = x.length", "@[to_additive (attr := simp)]\ntheorem toFreeSemigroup_comp_of : @toFreeSemigroup \u03b1 \u2218 of = FreeSemigroup.of", "@[to_additive (attr := simp)]\ntheorem traverse_eq (x) : FreeMagma.traverse F x = traverse F x", "@[to_additive (attr := simp)]\ntheorem lift_ofList (f : \u03b1 \u2192 M) (l : List \u03b1) : lift f (ofList l) = (l.map f).prod", "@[to_additive]\ntheorem ofList_smul (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (l : List \u03b1) (b : \u03b2) :\n    haveI := mkMulAction f\n    ofList l \u2022 b = l.foldr f b", "@[simp]\ntheorem lift_symm_apply (F : FreeNonUnitalNonAssocAlgebra R X \u2192\u2099\u2090[R] A) :\n    (lift R).symm F = F \u2218 of R", "@[simp, norm_cast]\nprotected theorem coe_add (x y : FreeRing \u03b1) : \u2191(x + y) = (x : FreeCommRing \u03b1) + y", "@[to_additive (attr := simp)]\ntheorem lift_of (x : \u03b1) : lift f (of x) = f x", "@[to_additive (attr := simp)]\ntheorem pure_bind (f : \u03b1 \u2192 FreeSemigroup \u03b2) (x) : pure x >>= f = f x", "theorem Antiperiodic.add_const [AddCommSemigroup \u03b1] [Neg \u03b2] (h : Antiperiodic f c) (a : \u03b1) :\n    Antiperiodic (fun x => f (x + a)) c", "theorem Antiperiodic.const_smul\u2080 [AddCommMonoid \u03b1] [Neg \u03b2] [DivisionSemiring \u03b3] [Module \u03b3 \u03b1]\n    (h : Antiperiodic f c) {a : \u03b3} (ha : a \u2260 0) : Antiperiodic (fun x => f (a \u2022 x)) (a\u207b\u00b9 \u2022 c)", "theorem Antiperiodic.int_odd_mul_antiperiodic [Ring \u03b1] [InvolutiveNeg \u03b2] (h : Antiperiodic f c)\n    (n : \u2124) : Antiperiodic f (n * (2 * c) + c)", "theorem Bijective.prodMap (hf : Bijective f) (hg : Bijective g) : Bijective (map f g)", "theorem comp_iterate (h : Commute f g) (n : \u2115) : (f \u2218 g)^[n] = f^[n] \u2218 g^[n]", "nonrec theorem comp_left (h : Commute f g) (h' : Commute f' g) : Commute (f \u2218 f') g", "theorem id_right : Commute f id", "theorem iterate_pos_eq_iff_map_eq (h : Commute f g) (hf : Monotone f) (hg : StrictMono g) {x n}\n    (hn : 0 < n) : f^[n] x = g^[n] x \u2194 f x = g x", "theorem iterate_pos_le_iff_map_le' (h : Commute f g) (hf : StrictMono f) (hg : Monotone g) {x n}\n    (hn : 0 < n) : f^[n] x \u2264 g^[n] x \u2194 f x \u2264 g x", "theorem iterate_pos_lt_of_map_lt' (h : Commute f g) (hf : StrictMono f) (hg : Monotone g) {x}\n    (hx : f x < g x) {n} (hn : 0 < n) : f^[n] x < g^[n] x", "theorem Commute.minimalPeriod_of_comp_eq_mul_of_coprime {g : \u03b1 \u2192 \u03b1} (h : Commute f g)\n    (hco : Coprime (minimalPeriod f x) (minimalPeriod g x)) :\n    minimalPeriod (f \u2218 g) x = minimalPeriod f x * minimalPeriod g x", "theorem self_iterate (n : \u2115) : Commute f f^[n]", "theorem _root_.Function.Embedding.cardinal_le {\u03b1 \u03b2 : Type u} (f : \u03b1 \u21aa \u03b2) : #\u03b1 \u2264 #\u03b2", "@[simp]\ntheorem coe_prodMap {\u03b1 \u03b2 \u03b3 \u03b4 : Type*} (e\u2081 : \u03b1 \u21aa \u03b2) (e\u2082 : \u03b3 \u21aa \u03b4) :\n    e\u2081.prodMap e\u2082 = Prod.map e\u2081 e\u2082", "@[simp]\ntheorem coe_quotientOut (\u03b1) [Setoid \u03b1] : \u2191(quotientOut \u03b1) = Quotient.out", "theorem nonempty_iff_card_le [Fintype \u03b1] [Fintype \u03b2] :\n    Nonempty (\u03b1 \u21aa \u03b2) \u2194 Fintype.card \u03b1 \u2264 Fintype.card \u03b2", "@[simp]\ntheorem Function.Embedding.toEquivRange_symm_apply_self (a : \u03b1) :\n    f.toEquivRange.symm \u27e8f a, Set.mem_range_self a\u27e9 = a", "protected lemma Function.Embedding.uncountable [Uncountable \u03b1] (f : \u03b1 \u21aa \u03b2) : Uncountable \u03b2", "theorem Exact.split_tfae' (h : Function.Exact f g) :\n    List.TFAE [\n      Function.Injective f \u2227 \u2203 l, g \u2218\u2097 l = LinearMap.id,\n      Function.Surjective g \u2227 \u2203 l, l \u2218\u2097 f = LinearMap.id,\n      \u2203 e : N \u2243\u2097[R] M \u00d7 P, f = e.symm \u2218\u2097 LinearMap.inl R M P \u2227 g = LinearMap.snd R M P \u2218\u2097 e]", "theorem comp_right {\u03b1 \u03b2 \u03b3 \u03b4 : Sort*} {f : \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b3} (h : FactorsThrough g f) (g' : \u03b4 \u2192 \u03b1) :\n    FactorsThrough (g \u2218 g') (f \u2218 g')", "theorem const_succ_apply {n} (p : Fin (n + 1) \u2192 Type u) {\u03c4 : Type u} (t : \u03c4)\n    (x : p 0) : const p t x = const (vecTail p) t", "lemma _root_.Function.HasTemperateGrowth.const (c : F) :\n    Function.HasTemperateGrowth (fun _ : E \u21a6 c)", "lemma _root_.Function.HasTemperateGrowth.zero :\n    Function.HasTemperateGrowth (fun _ : E \u21a6 (0 : F))", "theorem _root_.Function.Injective.encard_image (hf : f.Injective) (s : Set \u03b1) :\n    (f '' s).encard = s.encard", "theorem Function.Injective.lieAlgebra_isNilpotent [h\u2081 : IsNilpotent R L'] {f : L \u2192\u2097\u2045R\u2046 L'}\n    (h\u2082 : Function.Injective f) : IsNilpotent R L", "theorem Injective.prodMap (hf : Injective f) (hg : Injective g) : Injective (map f g)", "theorem rightDistribClass [Mul \u03b1] [Add \u03b1] [RightDistribClass \u03b1] (add : \u2200 x y, f (x + y) = f x + f y)\n    (mul : \u2200 x y, f (x * y) = f x * f y) : RightDistribClass \u03b2 where", "theorem Function.Injective.sbtw_map_iff {x y z : P} {f : P \u2192\u1d43[R] P'} (hf : Function.Injective f) :\n    Sbtw R (f x) (f y) (f z) \u2194 Sbtw R x y z", "theorem eq_iff (hf : Injective2 f) {a\u2081 a\u2082 b\u2081 b\u2082} : f a\u2081 b\u2081 = f a\u2082 b\u2082 \u2194 a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082", "protected theorem apply {x : \u03b1} (hx : IsFixedPt f x) : IsFixedPt f (f x)", "theorem Function.IsFixedPt.birkhoffAverage_eq [CharZero R] {f : \u03b1 \u2192 \u03b1} {x : \u03b1} (h : IsFixedPt f x)\n    (g : \u03b1 \u2192 M) {n : \u2115} (hn : n \u2260 0) : birkhoffAverage R f g n x = g x", "protected theorem eq (hf : IsFixedPt f x) : f x = x", "protected theorem perm_inv (h : IsFixedPt e x) : IsFixedPt (\u21d1e\u207b\u00b9) x", "protected theorem iterate (hf : IsPeriodicPt f n x) (m : \u2115) : IsPeriodicPt f^[m] n x", "theorem LeftInverse.eq_rightInverse {f : \u03b1 \u2192 \u03b2} {g\u2081 g\u2082 : \u03b2 \u2192 \u03b1} (h\u2081 : LeftInverse g\u2081 f)\n    (h\u2082 : RightInverse g\u2082 f) : g\u2081 = g\u2082", "theorem LeftInverse.image_image {g : \u03b2 \u2192 \u03b1} (h : LeftInverse g f) (s : Set \u03b1) :\n    g '' (f '' s) = s", "theorem _root_.Function.LeftInverse.list_map {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b1} (h : LeftInverse f g) :\n    LeftInverse (map f) (map g)", "theorem Periodic.add_antiperiod_eq [AddGroup \u03b1] [Neg \u03b2] (h1 : Periodic f c\u2081)\n    (h2 : Antiperiodic f c\u2082) : f (c\u2081 + c\u2082) = -f 0", "theorem Periodic.comp_addHom [Add \u03b1] [Add \u03b3] (h : Periodic f c) (g : AddHom \u03b3 \u03b1) (g_inv : \u03b1 \u2192 \u03b3)\n    (hg : RightInverse g_inv g) : Periodic (f \u2218 g) (g_inv c)", "theorem Periodic.const_inv_smul\u2080 [AddCommMonoid \u03b1] [DivisionSemiring \u03b3] [Module \u03b3 \u03b1]\n    (h : Periodic f c) (a : \u03b3) : Periodic (fun x => f (a\u207b\u00b9 \u2022 x)) (a \u2022 c)", "theorem Periodic.const_sub [AddCommGroup \u03b1] (h : Periodic f c) (a : \u03b1) :\n    Periodic (fun x => f (a - x)) c", "theorem Periodic.exists_mem_Ico\u2080 [LinearOrderedAddCommGroup \u03b1] [Archimedean \u03b1] (h : Periodic f c)\n    (hc : 0 < c) (x) : \u2203 y \u2208 Ico 0 c, f x = f y", "protected theorem Periodic.funext [Add \u03b1] (h : Periodic f c) : (fun x => f (x + c)) = f", "protected theorem Periodic.int_mul [Ring \u03b1] (h : Periodic f c) (n : \u2124) : Periodic f (n * c)", "nonrec theorem _root_.Function.RightInverse.filter_map {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b1}\n    (hfg : RightInverse g f) : RightInverse (map g) (map f)", "theorem option_map {f : \u03b1 \u2192 \u03b2} {ga : \u03b1 \u2192 \u03b1} {gb : \u03b2 \u2192 \u03b2} (h : Semiconj f ga gb) :\n    Semiconj (Option.map f) (Option.map ga) (Option.map gb)", "protected theorem trans (hab : Semiconj fab ga gb) (hbc : Semiconj fbc gb gc) :\n    Semiconj (fbc \u2218 fab) ga gc", "lemma Surjective.comp_exact_iff_exact {p : M' \u2192\u2097[R] M} (h : Surjective p) :\n    Exact (f \u2218\u2097 p) g \u2194 Exact f g", "theorem Surjective.image_preimage (hf : Surjective f) (s : Set \u03b2) : f '' (f \u207b\u00b9' s) = s", "theorem _root_.Function.Surjective.isBezout {S : Type v} [CommRing S] (f : R \u2192+* S)\n    (hf : Function.Surjective f) [IsBezout R] : IsBezout S", "theorem Function.Surjective.isEmpty [IsEmpty \u03b1] {f : \u03b1 \u2192 \u03b2} (hf : f.Surjective) : IsEmpty \u03b2", "theorem Function.Surjective.isEngelian {f : L \u2192\u2097\u2045R\u2046 L\u2082} (hf : Function.Surjective f)\n    (h : LieAlgebra.IsEngelian.{u\u2081, u\u2082, u\u2084} R L) : LieAlgebra.IsEngelian.{u\u2081, u\u2083, u\u2084} R L\u2082", "@[simp]\ntheorem Surjective.nonempty_preimage (hf : Surjective f) {s : Set \u03b2} :\n    (f \u207b\u00b9' s).Nonempty \u2194 s.Nonempty", "protected theorem Function.Surjective.nontrivial [Nontrivial \u03b2] {f : \u03b1 \u2192 \u03b2}\n    (hf : Function.Surjective f) : Nontrivial \u03b1", "theorem Surjective.preimage_injective (hf : Surjective f) : Injective (preimage f)", "protected theorem Function.Surjective.uncountable [Uncountable \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : Surjective f) :\n    Uncountable \u03b1", "theorem apply_invFun_apply {\u03b1 \u03b2 : Type*} {f : \u03b1 \u2192 \u03b2} {a : \u03b1} :\n    f (@invFun _ _ \u27e8a\u27e9 f (f a)) = f a", "theorem argminOn_le (s : Set \u03b1) {a : \u03b1} (ha : a \u2208 s) (hs : s.Nonempty := Set.nonempty_of_mem ha) :\n    f (argminOn f h s hs) \u2264 f a", "@[simp] theorem comp_const (f : \u03b2 \u2192 \u03c6) (b : \u03b2) : f \u2218 const \u03b1 b = const \u03b1 (f b)", "@[to_additive]\ntheorem extend_inv [Inv \u03b3] (f : \u03b1 \u2192 \u03b2) (g : \u03b1 \u2192 \u03b3) (e : \u03b2 \u2192 \u03b3) :\n    Function.extend f g\u207b\u00b9 e\u207b\u00b9 = (Function.extend f g e)\u207b\u00b9", "@[simp] lemma graph_def (f : \u03b1 \u2192 \u03b2) (x y) : f.graph x y \u2194 (f x = y) := Iff.rfl", "theorem graph_injective : Injective (graph : (\u03b1 \u2192 \u03b2) \u2192 Rel \u03b1 \u03b2)", "theorem _root_.Function.invFunOn_injOn_image [Nonempty \u03b1] (f : \u03b1 \u2192 \u03b2) (s : Set \u03b1) :\n    Set.InjOn (invFunOn f s) (f '' s)", "theorem iterate_mul (m : \u2115) : \u2200 n, f^[m * n] = f^[m]^[n]", "@[simp]\ntheorem iterate_zero : f^[0] = id", "theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f", "@[to_additive (attr := simp)]\ntheorem mem_mulSupport {f : \u03b1 \u2192 M} {x : \u03b1} : x \u2208 mulSupport f \u2194 f x \u2260 1", "theorem minimalPeriod_iterate_eq_div_gcd (h : n \u2260 0) :\n    minimalPeriod f^[n] x = minimalPeriod f x / Nat.gcd (minimalPeriod f x) n", "theorem monotoneOn_of_rightInvOn_of_mapsTo {\u03b1 \u03b2 : Type*} [PartialOrder \u03b1] [LinearOrder \u03b2]\n    {\u03c6 : \u03b2 \u2192 \u03b1} {\u03c8 : \u03b1 \u2192 \u03b2} {t : Set \u03b2} {s : Set \u03b1} (h\u03c6 : MonotoneOn \u03c6 t)\n    (\u03c6\u03c8s : Set.RightInvOn \u03c8 \u03c6 s) (\u03c8ts : Set.MapsTo \u03c8 s t) : MonotoneOn \u03c8 s", "protected theorem Function.mt {a b : Prop} : (a \u2192 b) \u2192 \u00acb \u2192 \u00aca", "@[to_additive (attr := simp)]\nlemma mulSupport_along_fiber_finite_of_finite (f : \u03b1 \u00d7 \u03b2 \u2192 \u03b3) (a : \u03b1) (h : (mulSupport f).Finite) :\n    (mulSupport fun b \u21a6 f (a, b)).Finite", "@[to_additive]\ntheorem mulSupport_const {c : M} (hc : c \u2260 1) : (mulSupport fun _ : \u03b1 => c) = Set.univ", "@[to_additive]\ntheorem mulSupport_eq_preimage (f : \u03b1 \u2192 M) : mulSupport f = f \u207b\u00b9' {1}\u1d9c", "@[to_additive support_prod_mk']\ntheorem mulSupport_prod_mk' (f : \u03b1 \u2192 M \u00d7 N) :\n    mulSupport f = (mulSupport fun x => (f x).1) \u222a mulSupport fun x => (f x).2", "@[to_additive]\nlemma mulSupport_sup [SemilatticeSup M] (f g : \u03b1 \u2192 M) :\n    mulSupport (fun x \u21a6 f x \u2294 g x) \u2286 mulSupport f \u222a mulSupport g", "@[to_additive]\ntheorem mulSupport_update_one [DecidableEq \u03b1] (f : \u03b1 \u2192 M) (x : \u03b1) :\n    mulSupport (update f x 1) = mulSupport f \\ {x}", "lemma ne_update_self_iff : f \u2260 update f a b \u2194 f a \u2260 b := eq_update_self_iff.not", "theorem nodup_periodicOrbit : (periodicOrbit f x).Nodup", "lemma not_surjective : Surjective Not := not_involutive.surjective", "theorem not_surjective_Type {\u03b1 : Type u} (f : \u03b1 \u2192 Type max u v) : \u00acSurjective f", "theorem periodicOrbit_chain' (r : \u03b1 \u2192 \u03b1 \u2192 Prop) {f : \u03b1 \u2192 \u03b1} {x : \u03b1} (hx : x \u2208 periodicPts f) :\n    (periodicOrbit f x).Chain r \u2194 \u2200 n, r (f^[n] x) (f^[n + 1] x)", "theorem rightInverse_of_injective_of_leftInverse {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b1} (injf : Injective f)\n    (lfg : LeftInverse f g) : RightInverse f g", "@[simp] lemma support_inv' (f : \u03b9 \u2192 G\u2080) : support f\u207b\u00b9 = support f := support_inv _", "@[simp]\nlemma symmetric_apply_eq_iff {\u03b1 : Sort*} {f : \u03b1 \u2192 \u03b1} : Symmetric (f \u00b7 = \u00b7) \u2194 Involutive f", "theorem uncurry_def {\u03b1 \u03b2 \u03b3} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) : uncurry f = fun p \u21a6 f p.1 p.2", "@[simp] theorem updateFinset_empty {y} : updateFinset x \u2205 y = x", "theorem updateFinset_updateFinset {s t : Finset \u03b9} (hst : Disjoint s t)\n    {y : \u2200 i : \u21a5s, \u03c0 i} {z : \u2200 i : \u21a5t, \u03c0 i} :\n    updateFinset (updateFinset x s y) t z =\n    updateFinset x (s \u222a t) (Equiv.piFinsetUnion \u03c0 hst \u27e8y, z\u27e9)", "theorem update_comm {\u03b1} [DecidableEq \u03b1] {\u03b2 : \u03b1 \u2192 Sort*} {a b : \u03b1} (h : a \u2260 b) (v : \u03b2 a) (w : \u03b2 b)\n    (f : \u2200 a, \u03b2 a) : update (update f a v) b w = update (update f b w) a v", "theorem update_strictMono : StrictMono (update f i)", "theorem InftyValuation.map_add_le_max' (x y : RatFunc Fq) :\n    inftyValuationDef Fq (x + y) \u2264 max (inftyValuationDef Fq x) (inftyValuationDef Fq y)", "@[functor_norm]\ntheorem Functor.map_map (m : \u03b1 \u2192 \u03b2) (g : \u03b2 \u2192 \u03b3) (x : f \u03b1) : g <$> m <$> x = (g \u2218 m) <$> x", "theorem isCoatom_iff' [OrderTop \u03b1] [OrderTop \u03b2] [IsCoatomic \u03b2] {l : \u03b1 \u2192 \u03b2} {u : \u03b2 \u2192 \u03b1}\n    (gi : GaloisCoinsertion l u) (htop : l \u22a4 = \u22a4) (h_coatom : \u2200 b, IsCoatom b \u2192 l (u b) = b)\n    (a : \u03b1) : IsCoatom (l a) \u2194 IsCoatom a", "theorem l_comm_iff_u_comm {X : Type*} [PartialOrder X] {Y : Type*} [Preorder Y] {Z : Type*}\n    [Preorder Z] {W : Type*} [PartialOrder W] {lYX : X \u2192 Y} {uXY : Y \u2192 X}\n    (hXY : GaloisConnection lYX uXY) {lWZ : Z \u2192 W} {uZW : W \u2192 Z} (hZW : GaloisConnection lWZ uZW)\n    {lWY : Y \u2192 W} {uYW : W \u2192 Y} (hWY : GaloisConnection lWY uYW) {lZX : X \u2192 Z} {uXZ : Z \u2192 X}\n    (hXZ : GaloisConnection lZX uXZ) :\n    (\u2200 w : W, uXZ (uZW w) = uXY (uYW w)) \u2194 \u2200 x : X, lWZ (lZX x) = lWY (lYX x)", "theorem le_iff_le {a : \u03b1} {b : \u03b2} : l a \u2264 b \u2194 a \u2264 u b", "theorem u_unique {l' : \u03b1 \u2192 \u03b2} {u' : \u03b2 \u2192 \u03b1} (gc' : GaloisConnection l' u') (hl : \u2200 a, l a = l' a)\n    {b : \u03b2} : u b = u' b", "theorem upperBounds_l_image (s : Set \u03b1) : upperBounds (l '' s) = u \u207b\u00b9' upperBounds s", "theorem strictMono_u [Preorder \u03b1] [Preorder \u03b2] (gi : GaloisInsertion l u) : StrictMono u", "theorem u_injective [Preorder \u03b1] [PartialOrder \u03b2] (gi : GaloisInsertion l u) : Injective u", "theorem Gamma0_det (N : \u2115) (A : Gamma0 N) : (A.1.1.det : ZMod N) = 1", "@[simp]\ntheorem Gamma_mem (N : \u2115) (\u03b3 : SL(2, \u2124)) : \u03b3 \u2208 Gamma N \u2194 ((\u2191\u2098\u03b3 0 0 : \u2124) : ZMod N) = 1 \u2227\n    ((\u2191\u2098\u03b3 0 1 : \u2124) : ZMod N) = 0 \u2227 ((\u2191\u2098\u03b3 1 0 : \u2124) : ZMod N) = 0 \u2227 ((\u2191\u2098\u03b3 1 1 : \u2124) : ZMod N) = 1", "theorem Gamma_normal (N : \u2115) : Subgroup.Normal (Gamma N)", "@[simp]\ntheorem natCast_natAbs_norm {\u03b1 : Type*} [Ring \u03b1] (x : \u2124[i]) : (x.norm.natAbs : \u03b1) = x.norm", "theorem toComplex_def' (x y : \u2124) : ((\u27e8x, y\u27e9 : \u2124[i]) : \u2102) = x + y * I", "theorem toComplex_div_re (x y : \u2124[i]) : ((x / y : \u2124[i]) : \u2102).re = round (x / y : \u2102).re", "theorem toComplex_neg (x : \u2124[i]) : ((-x : \u2124[i]) : \u2102) = -x", "@[refl]\ntheorem refl (f : \u03a9^ N X x) : Homotopic f f", "theorem fromLoop_symm_toLoop {i : N} {p : \u03a9^ N X x} : fromLoop i (toLoop i p).symm = symmAt i p", "@[simp, norm_cast]\ntheorem coe_toGeneralizedContinuedFraction {g : GeneralizedContinuedFraction \u03b1} :\n    (g : GeneralizedContinuedFraction \u03b2) =\n      \u27e8(g.h : \u03b2), (g.s.map (\u2191) : Stream'.Seq <| Pair \u03b2)\u27e9", "theorem compExactValue_correctness_of_stream_eq_some :\n    \u2200 {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n \u2192\n      v = compExactValue ((of v).continuantsAux n) ((of v).continuantsAux <| n + 1) ifp_n.fr", "theorem convergents_eq_convergents' [LinearOrderedField K]\n    (s_pos : \u2200 {gp : Pair K} {m : \u2115}, m < n \u2192 g.s.get? m = some gp \u2192 0 < gp.a \u2227 0 < gp.b) :\n    g.convergents n = g.convergents' n", "theorem denominators_recurrence {gp : Pair K} {ppredB predB : K}\n    (succ_nth_s_eq : g.s.get? (n + 1) = some gp) (nth_denom_eq : g.denominators n = ppredB)\n    (succ_nth_denom_eq : g.denominators (n + 1) = predB) :\n    g.denominators (n + 2) = gp.b * predB + gp.a * ppredB", "theorem exists_gcf_pair_rat_eq_nth_conts :\n    \u2203 conts : Pair \u211a, (of v).continuants n = (conts.map (\u2191) : Pair K)", "theorem fib_le_of_continuantsAux_b :\n    n \u2264 1 \u2228 \u00ac(of v).TerminatedAt (n - 2) \u2192 (fib n : K) \u2264 ((of v).continuantsAux n).b", "theorem of_convergents_eq_convergents' : (of v).convergents = (of v).convergents'", "theorem of_correctness_of_nth_stream_eq_none (nth_stream_eq_none : IntFractPair.stream v n = none) :\n    v = (of v).convergents (n - 1)", "theorem of_terminates_iff_of_rat_terminates {v : K} {q : \u211a} (v_eq_q : v = (q : K)) :\n    (of v).Terminates \u2194 (of q).Terminates", "theorem succ_nth_convergent'_eq_squashGCF_nth_convergent' :\n    g.convergents' (n + 1) = (squashGCF g n).convergents' n", "theorem terminatedAt_iff_s_none : g.TerminatedAt n \u2194 g.s.get? n = none", "theorem terminated_stable (n_le_m : n \u2264 m) (terminated_at_n : g.TerminatedAt n) :\n    g.TerminatedAt m", "theorem zero_le_of_denom : 0 \u2264 (of v).denominators n", "theorem mem_space_iff : x \u2208 K.space \u2194 \u2203 s \u2208 K.faces, x \u2208 convexHull \ud835\udd5c (s : Set E)", "theorem mem_vertices : x \u2208 K.vertices \u2194 {x} \u2208 K.faces", "theorem vertex_mem_convexHull_iff (hx : x \u2208 K.vertices) (hs : s \u2208 K.faces) :\n    x \u2208 convexHull \ud835\udd5c (s : Set E) \u2194 x \u2208 s", "theorem GradedMonoid.list_prod_map_eq_dProd (l : List \u03b1) (f : \u03b1 \u2192 GradedMonoid A) :\n    (l.map f).prod = GradedMonoid.mk _ (l.dProd (fun i => (f i).1) fun i => (f i).2)", "@[simp] theorem snd_smul [\u2200 i, SMul \u03b1 (A i)] (a : \u03b1) (x : GradedMonoid A) :\n    (a \u2022 x).snd = a \u2022 x.snd", "theorem tmul_one_mul_one_tmul (a\u2081 : A) (b\u2082 : B) :\n    (a\u2081 \u1d4d\u2297\u209c[R] (1 : B) * (1 : A) \u1d4d\u2297\u209c[R] b\u2082 : \ud835\udc9c \u1d4d\u2297[R] \u212c) = (a\u2081 : A) \u1d4d\u2297\u209c (b\u2082 : B)", "private theorem HD_bound_aux2 (f : Cb X Y) (C : \u211d) :\n    BddAbove (range fun y : Y => \u2a05 x, f (inl x, inr y) + C)", "private theorem HD_lipschitz_aux2 (f g : Cb X Y) :\n    (\u2a06 y, \u2a05 x, f (inl x, inr y)) \u2264 (\u2a06 y, \u2a05 x, g (inl x, inr y)) + dist f g", "private theorem candidates_lipschitz (fA : f \u2208 candidates X Y) :\n    LipschitzWith (2 * maxVar X Y) f", "theorem toGHSpace_continuous :\n    Continuous (NonemptyCompacts.toGHSpace : NonemptyCompacts X \u2192 GHSpace)", "theorem totallyBounded {t : Set GHSpace} {C : \u211d} {u : \u2115 \u2192 \u211d} {K : \u2115 \u2192 \u2115}\n    (ulim : Tendsto u atTop (\ud835\udcdd 0)) (hdiam : \u2200 p \u2208 t, diam (univ : Set (GHSpace.Rep p)) \u2264 C)\n    (hcov : \u2200 p \u2208 t, \u2200 n : \u2115, \u2203 s : Set (GHSpace.Rep p),\n      (#s) \u2264 K n \u2227 univ \u2286 \u22c3 x \u2208 s, ball x (u n)) :\n    TotallyBounded t", "theorem conjugates_subset_normal {N : Subgroup G} [tn : N.Normal] {a : G} (h : a \u2208 N) :\n    conjugatesOf a \u2286 N", "theorem normalClosure.isSubgroup (s : Set G) : IsSubgroup (normalClosure s)", "theorem normalClosure.is_normal : IsNormalSubgroup (normalClosure s)", "@[to_additive]\ntheorem one {U : Set G} : U \u2208 B \u2192 (1 : G) \u2208 U", "@[to_additive (attr := simp)]\ntheorem toFun_eq_coe : p.toFun = p", "@[to_additive (attr := simp)]\ntheorem toTopologicalSpace_le {x y : GroupTopology \u03b1} :\n    x.toTopologicalSpace \u2264 y.toTopologicalSpace \u2194 x \u2264 y", "theorem h_apply_fromCoset_nin_range (x : B) (hx : x \u2208 f.range) (b : B) (hb : b \u2209 f.range) :\n    h x (fromCoset \u27e8b \u2022 f.range, b, rfl\u27e9) = fromCoset \u27e8(x * b) \u2022 \u2191f.range, x * b, rfl\u27e9", "theorem epi_iff_range_eq_top : Epi f \u2194 f.range = \u22a4", "@[to_additive]\ntheorem mono_iff_injective : Mono f \u2194 Function.Injective f", "lemma coe_id {X : GrpWithZero} : (\ud835\udfd9 X : X \u2192 X) = id := rfl", "@[simp]\ntheorem prod_group_smul (g : G) (w : NormalWord d) :\n    (g \u2022 w).prod \u03c6 = of g * (w.prod \u03c6)", "theorem of_mul_inv_t (a : A) :\n    (of (a : G) : HNNExtension G A B \u03c6) * t\u207b\u00b9 = t\u207b\u00b9 * of (\u03c6 a : G)", "theorem t_mul_of (a : A) :\n    t * (of (a : G) : HNNExtension G A B \u03c6) = of (\u03c6 a : G) * t", "@[simp]\ntheorem hsum_smul {x : HahnSeries \u0393 R} {s : SummableFamily \u0393 R \u03b1} : (x \u2022 s).hsum = x * s.hsum", "theorem coeff_order_ne_zero {x : HahnSeries \u0393 R} (hx : x \u2260 0) : x.coeff x.order \u2260 0", "theorem embDomain_injective {f : \u0393 \u21aao \u0393'} :\n    Function.Injective (embDomain f : HahnSeries \u0393 R \u2192 HahnSeries \u0393' R)", "theorem min_order_le_order_add {\u0393} [Zero \u0393] [LinearOrder \u0393] {x y : HahnSeries \u0393 R}\n    (hxy : x + y \u2260 0) : min x.order y.order \u2264 (x + y).order", "theorem ofPowerSeries_apply_coeff (x : PowerSeries R) (n : \u2115) :\n    (ofPowerSeries \u0393 R x).coeff n = PowerSeries.coeff R n x", "theorem ofPowerSeries_injective : Function.Injective (ofPowerSeries \u0393 R)", "theorem orderTop_eq_top_iff {x : HahnSeries \u0393 R} : orderTop x = \u22a4 \u2194 x = 0", "@[simp]\ntheorem single_pow (a : \u0393) (n : \u2115) (r : R) : single a r ^ n = single (n \u2022 a) (r ^ n)", "theorem support_single_subset : support (single a r) \u2286 {a}", "theorem untop_orderTop_of_ne_zero {x : HahnSeries \u0393 R} (hx : x \u2260 0) :\n    WithTop.untop x.orderTop (ne_zero_iff_orderTop.mp hx) =\n      x.isWF_support.min (support_nonempty_iff.2 hx)", "@[simp, push_cast]\ntheorem dist_eq_hammingDist (x y : Hamming \u03b2) :\n    dist x y = hammingDist (ofHamming x) (ofHamming y)", "@[simp]\ntheorem ofHamming_smul [\u2200 i, SMul \u03b1 (\u03b2 i)] {r : \u03b1} {x : Hamming \u03b2} :\n    ofHamming (r \u2022 x) = r \u2022 ofHamming x", "@[simp]\ntheorem ofHamming_toHamming (x : \u2200 i, \u03b2 i) : ofHamming (toHamming x) = x", "theorem _root_.HasCompactSupport.hasDerivAt_convolution_left [IsNegInvariant \u03bc]\n    (hcf : HasCompactSupport f\u2080) (hf : ContDiff \ud835\udd5c 1 f\u2080) (hg : LocallyIntegrable g\u2080 \u03bc) (x\u2080 : \ud835\udd5c) :\n    HasDerivAt (f\u2080 \u22c6[L, \u03bc] g\u2080) ((deriv f\u2080 \u22c6[L, \u03bc] g\u2080) x\u2080) x\u2080", "lemma HasDerivAt.comp_const_add {\ud835\udd5c : Type*} [NontriviallyNormedField \ud835\udd5c] (a x : \ud835\udd5c) {\ud835\udd5c' : Type*}\n    [NormedAddCommGroup \ud835\udd5c'] [NormedSpace \ud835\udd5c \ud835\udd5c'] {h : \ud835\udd5c \u2192 \ud835\udd5c'} {h' : \ud835\udd5c'}\n    (hh : HasDerivAt h h' (a + x)) :\n    HasDerivAt (fun x \u21a6 h (a + x)) h' x", "theorem HasDerivAt.comp_hasDerivWithinAt_of_eq (hh\u2082 : HasDerivAt h\u2082 h\u2082' y)\n    (hh : HasDerivWithinAt h h' s x) (hy : y = h x) :\n    HasDerivWithinAt (h\u2082 \u2218 h) (h\u2082' * h') s x", "theorem HasDerivAt.hasDerivAtFilter (h : HasDerivAt f f' x) (hL : L \u2264 \ud835\udcdd x) :\n    HasDerivAtFilter f f' x L", "theorem HasDerivAt.hasDerivWithinAt (h : HasDerivAt f f' x) : HasDerivWithinAt f f' s x", "theorem HasDerivAt.hasGradientAt' (h : HasDerivAt g g' u) :\n    HasGradientAt g g' u", "theorem lhopital_zero_left_on_Ioc (hff' : \u2200 x \u2208 Ioo a b, HasDerivAt f (f' x) x)\n    (hgg' : \u2200 x \u2208 Ioo a b, HasDerivAt g (g' x) x) (hcf : ContinuousOn f (Ioc a b))\n    (hcg : ContinuousOn g (Ioc a b)) (hg' : \u2200 x \u2208 Ioo a b, g' x \u2260 0) (hfb : f b = 0) (hgb : g b = 0)\n    (hdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[<] b) l) :\n    Tendsto (fun x => f x / g x) (\ud835\udcdd[<] b) l", "theorem HasDerivAt.pow (hc : HasDerivAt c c' x) :\n    HasDerivAt (fun y => c y ^ n) ((n : \ud835\udd5c) * c x ^ (n - 1) * c') x", "theorem HasDerivAtFilter.sub (hf : HasDerivAtFilter f f' x L) (hg : HasDerivAtFilter g g' x L) :\n    HasDerivAtFilter (fun x => f x - g x) (f' - g') x L", "theorem HasDerivWithinAt.differentiableWithinAt (h : HasDerivWithinAt f f' s x) :\n    DifferentiableWithinAt \ud835\udd5c f s x", "theorem HasDerivWithinAt.liminf_right_norm_slope_le (hf : HasDerivWithinAt f f' (Ici x) x)\n    (hr : \u2016f'\u2016 < r) : \u2203\u1da0 z in \ud835\udcdd[>] x, \u2016z - x\u2016\u207b\u00b9 * \u2016f z - f x\u2016 < r", "theorem HasDerivWithinAt.liminf_right_slope_norm_le (hf : HasDerivWithinAt f f' (Ici x) x)\n    (hr : \u2016f'\u2016 < r) : \u2203\u1da0 z in \ud835\udcdd[>] x, (z - x)\u207b\u00b9 * (\u2016f z\u2016 - \u2016f x\u2016) < r", "@[fun_prop]\ntheorem HasFDerivAt.clm_comp (hc : HasFDerivAt c c' x) (hd : HasFDerivAt d d' x) :\n    HasFDerivAt (fun y => (c y).comp (d y))\n      ((compL \ud835\udd5c F G H (c x)).comp d' + ((compL \ud835\udd5c F G H).flip (d x)).comp c') x", "theorem HasFDerivAt.congr_fderiv (h : HasFDerivAt f f' x) (h' : f' = g') : HasFDerivAt f g' x", "@[fun_prop]\ntheorem HasFDerivAt.const_mul (ha : HasFDerivAt a a' x) (b : \ud835\udd38) :\n    HasFDerivAt (fun y => b * a y) (b \u2022 a') x", "@[fun_prop]\ntheorem HasFDerivAt.differentiableAt (h : HasFDerivAt f f' x) : DifferentiableAt \ud835\udd5c f x", "theorem HasFDerivAt.of_local_left_inverse {f : E \u2192 F} {f' : E \u2243L[\ud835\udd5c] F} {g : F \u2192 E} {a : F}\n    (hg : ContinuousAt g a) (hf : HasFDerivAt f (f' : E \u2192L[\ud835\udd5c] F) (g a))\n    (hfg : \u2200\u1da0 y in \ud835\udcdd a, f (g y) = y) : HasFDerivAt g (f'.symm : F \u2192L[\ud835\udd5c] E) a", "@[fun_prop]\ntheorem HasFDerivAt.sum (h : \u2200 i \u2208 u, HasFDerivAt (A i) (A' i) x) :\n    HasFDerivAt (fun y => \u2211 i \u2208 u, A i y) (\u2211 i \u2208 u, A' i) x", "theorem HasFDerivAtFilter.add_const (hf : HasFDerivAtFilter f f' x L) (c : F) :\n    HasFDerivAtFilter (fun y => f y + c) f' x L", "theorem HasFDerivAtFilter.tendsto_nhds (hL : L \u2264 \ud835\udcdd x) (h : HasFDerivAtFilter f f' x L) :\n    Tendsto f L (\ud835\udcdd (f x))", "theorem HasFDerivWithinAt.ccosh (hf : HasFDerivWithinAt f f' s x) :\n    HasFDerivWithinAt (fun x => Complex.cosh (f x)) (Complex.sinh (f x) \u2022 f') s x", "@[fun_prop]\ntheorem HasFDerivWithinAt.clm_comp (hc : HasFDerivWithinAt c c' s x)\n    (hd : HasFDerivWithinAt d d' s x) :\n    HasFDerivWithinAt (fun y => (c y).comp (d y))\n      ((compL \ud835\udd5c F G H (c x)).comp d' + ((compL \ud835\udd5c F G H).flip (d x)).comp c') s x", "@[fun_prop]\ntheorem HasFDerivWithinAt.comp_of_mem {g : F \u2192 G} {g' : F \u2192L[\ud835\udd5c] G} {t : Set F}\n    (hg : HasFDerivWithinAt g g' t (f x)) (hf : HasFDerivWithinAt f f' s x)\n    (hst : Tendsto f (\ud835\udcdd[s] x) (\ud835\udcdd[t] f x)) : HasFDerivWithinAt (g \u2218 f) (g'.comp f') s x", "theorem HasFDerivWithinAt.finset_prod [DecidableEq \u03b9] {x : E}\n    (hg : \u2200 i \u2208 u, HasFDerivWithinAt (g i) (g' i) s x) :\n    HasFDerivWithinAt (\u220f i \u2208 u, g i \u00b7) (\u2211 i \u2208 u, (\u220f j \u2208 u.erase i, g j x) \u2022 g' i) s x", "nonrec theorem HasFDerivWithinAt.isBigO_sub {f : E \u2192 F} {s : Set E} {x\u2080 : E} {f' : E \u2192L[\ud835\udd5c] F}\n    (h : HasFDerivWithinAt f f' s x\u2080) : (f \u00b7 - f x\u2080) =O[\ud835\udcdd[s] x\u2080] (\u00b7 - x\u2080)", "nonrec theorem HasFDerivWithinAt.mono (h : HasFDerivWithinAt f f' t x) (hst : s \u2286 t) :\n    HasFDerivWithinAt f f' s x", "@[fun_prop]\nnonrec theorem HasFDerivWithinAt.neg (h : HasFDerivWithinAt f f' s x) :\n    HasFDerivWithinAt (fun x => -f x) (-f') s x", "theorem HasFDerivWithinAt.sin (hf : HasFDerivWithinAt f f' s x) :\n    HasFDerivWithinAt (fun x => Real.sin (f x)) (Real.cos (f x) \u2022 f') s x", "theorem HasFPowerSeriesAt.eventually_eq_zero\n    (hf : HasFPowerSeriesAt f (0 : FormalMultilinearSeries \ud835\udd5c E F) x) : \u2200\u1da0 z in \ud835\udcdd x, f z = 0", "theorem locally_ne_zero (hp : HasFPowerSeriesAt f p z\u2080) (h : p \u2260 0) : \u2200\u1da0 z in \ud835\udcdd[\u2260] z\u2080, f z \u2260 0", "theorem HasFPowerSeriesAt.radius_pos (hf : HasFPowerSeriesAt f p x) : 0 < p.radius", "theorem HasFPowerSeriesOnBall.analyticAt (hf : HasFPowerSeriesOnBall f p x r) : AnalyticAt \ud835\udd5c f x", "theorem HasFPowerSeriesOnBall.eventually_eq_zero\n    (hf : HasFPowerSeriesOnBall f (0 : FormalMultilinearSeries \ud835\udd5c E F) x r) :\n    \u2200\u1da0 z in \ud835\udcdd x, f z = 0", "lemma HasFPowerSeriesOnBall.prod {e : E} {f : E \u2192 F} {g : E \u2192 G} {r s : \u211d\u22650\u221e}\n    {p : FormalMultilinearSeries \ud835\udd5c E F} {q : FormalMultilinearSeries \ud835\udd5c E G}\n    (hf : HasFPowerSeriesOnBall f p e r) (hg : HasFPowerSeriesOnBall g q e s) :\n    HasFPowerSeriesOnBall (fun x \u21a6 (f x, g x)) (p.prod q) e (min r s) where\n  r_le", "theorem HasFPowerSeriesOnBall.uniform_geometric_approx' {r' : \u211d\u22650}\n    (hf : HasFPowerSeriesOnBall f p x r) (h : (r' : \u211d\u22650\u221e) < r) :\n    \u2203 a \u2208 Ioo (0 : \u211d) 1, \u2203 C > 0, \u2200 y \u2208 Metric.ball (0 : E) r', \u2200 n,\n      \u2016f (x + y) - p.partialSum n y\u2016 \u2264 C * (a * (\u2016y\u2016 / r')) ^ n", "theorem HasFTaylorSeriesUpTo.eq_iteratedFDeriv\n    (h : HasFTaylorSeriesUpTo n f p) {m : \u2115} (hmn : (m : \u2115\u221e) \u2264 n) (x : E) :\n    p x m = iteratedFDeriv \ud835\udd5c m f x", "theorem HasFTaylorSeriesUpTo.hasFTaylorSeriesUpToOn (h : HasFTaylorSeriesUpTo n f p) (s : Set E) :\n    HasFTaylorSeriesUpToOn n f p s", "theorem HasFTaylorSeriesUpTo.ofLe (h : HasFTaylorSeriesUpTo n f p) (hmn : m \u2264 n) :\n    HasFTaylorSeriesUpTo m f p", "theorem HasFTaylorSeriesUpToOn.differentiableOn (h : HasFTaylorSeriesUpToOn n f p s) (hn : 1 \u2264 n) :\n    DifferentiableOn \ud835\udd5c f s", "theorem HasFTaylorSeriesUpToOn.hasFDerivWithinAt (h : HasFTaylorSeriesUpToOn n f p s) (hn : 1 \u2264 n)\n    (hx : x \u2208 s) : HasFDerivWithinAt f (continuousMultilinearCurryFin1 \ud835\udd5c E F (p x 1)) s x", "theorem HasFTaylorSeriesUpToOn.of_le (h : HasFTaylorSeriesUpToOn n f p s) (hmn : m \u2264 n) :\n    HasFTaylorSeriesUpToOn m f p s", "theorem HasFiniteFPowerSeriesOnBall.cPolynomialOn (hf : HasFiniteFPowerSeriesOnBall f p x n r) :\n    CPolynomialOn \ud835\udd5c f (EMetric.ball x r)", "theorem HasFiniteFPowerSeriesOnBall.eq_zero_of_bound_zero\n    (hf : HasFiniteFPowerSeriesOnBall f pf x 0 r) : \u2200 y \u2208 EMetric.ball x r, f y = 0", "theorem HasFiniteFPowerSeriesOnBall.fderiv' (h : HasFiniteFPowerSeriesOnBall f p x n r) :\n    HasFiniteFPowerSeriesOnBall (fderiv \ud835\udd5c f) p.derivSeries x (n - 1) r", "theorem HasGradientWithinAt.congr_of_eventuallyEq_of_mem (h : HasGradientWithinAt f f' s x)\n    (h\u2081 : f\u2081 =\u1da0[\ud835\udcdd[s] x] f) (hx : x \u2208 s) : HasGradientWithinAt f\u2081 f' s x", "theorem HasLineDerivAt.smul (h : HasLineDerivAt \ud835\udd5c f f' x v) (c : \ud835\udd5c) :\n    HasLineDerivAt \ud835\udd5c f (c \u2022 f') x (c \u2022 v)", "theorem HasLineDerivAt.unique (h\u2080 : HasLineDerivAt \ud835\udd5c f f\u2080' x v) (h\u2081 : HasLineDerivAt \ud835\udd5c f f\u2081' x v) :\n    f\u2080' = f\u2081'", "theorem HasLineDerivWithinAt.congr (h : HasLineDerivWithinAt \ud835\udd5c f f' s x v) (hs : EqOn f\u2081 f s)\n    (hx : f\u2081 x = f x) : HasLineDerivWithinAt \ud835\udd5c f\u2081 f' s x v", "theorem HasLineDerivWithinAt.congr_mono (h : HasLineDerivWithinAt \ud835\udd5c f f' s x v) (ht : EqOn f\u2081 f t)\n    (hx : f\u2081 x = f x) (h\u2081 : t \u2286 s) : HasLineDerivWithinAt \ud835\udd5c f\u2081 f' t x v", "theorem HasMFDerivAt.comp (hg : HasMFDerivAt I' I'' g (f x) g') (hf : HasMFDerivAt I I' f x f') :\n    HasMFDerivAt I I'' (g \u2218 f) x (g'.comp f')", "theorem HasMFDerivWithinAt.hasMFDerivAt (h : HasMFDerivWithinAt I I' f s x f') (hs : s \u2208 \ud835\udcdd x) :\n    HasMFDerivAt I I' f x f'", "lemma apprSeq_apply_le_one (n : \u2115) (x : X) :\n    hF.apprSeq n x \u2264 1", "@[to_additive]\nlemma HasProd.congr_fun (hf : HasProd f a) (h : \u2200 x : \u03b2, g x = f x) : HasProd g a", "@[to_additive]\ntheorem HasProd.nat_mul_neg {f : \u2124 \u2192 M} (hf : HasProd f m) :\n    HasProd (fun n : \u2115 \u21a6 f n * f (-n)) (m * f 0)", "theorem HasStrictDerivAt.comp (hh\u2082 : HasStrictDerivAt h\u2082 h\u2082' (h x)) (hh : HasStrictDerivAt h h' x) :\n    HasStrictDerivAt (h\u2082 \u2218 h) (h\u2082' * h') x", "protected nonrec theorem HasStrictDerivAt.iterate {f : \ud835\udd5c \u2192 \ud835\udd5c} {f' : \ud835\udd5c}\n    (hf : HasStrictDerivAt f f' x) (hx : f x = x) (n : \u2115) :\n    HasStrictDerivAt f^[n] (f' ^ n) x", "theorem _root_.HasStrictDerivAt.rpow {f g : \u211d \u2192 \u211d} {f' g' : \u211d} (hf : HasStrictDerivAt f f' x)\n    (hg : HasStrictDerivAt g g' x) (h : 0 < f x) : HasStrictDerivAt (fun x => f x ^ g x)\n      (f' * g x * f x ^ (g x - 1) + g' * f x ^ g x * Real.log (f x)) x", "theorem HasStrictDerivAt.scomp (hg : HasStrictDerivAt g\u2081 g\u2081' (h x)) (hh : HasStrictDerivAt h h' x) :\n    HasStrictDerivAt (g\u2081 \u2218 h) (h' \u2022 g\u2081') x", "theorem HasStrictFDerivAt.comp_hasStrictDerivAt (hl : HasStrictFDerivAt l l' (f x))\n    (hf : HasStrictDerivAt f f' x) : HasStrictDerivAt (l \u2218 f) (l' f') x", "protected theorem HasStrictFDerivAt.hasStrictDerivAt {f' : \ud835\udd5c \u2192L[\ud835\udd5c] F} :\n    HasStrictFDerivAt f f' x \u2192 HasStrictDerivAt f (f' 1) x", "@[simp]\ntheorem implicitFunction_apply_image (hf : HasStrictFDerivAt f f' a) (hf' : range f' = \u22a4) :\n    hf.implicitFunction f f' hf' (f a) 0 = a", "@[simp]\ntheorem implicitToPartialHomeomorphOfComplemented_self (hf : HasStrictFDerivAt f f' a)\n    (hf' : range f' = \u22a4) (hker : (ker f').ClosedComplemented) :\n    hf.implicitToPartialHomeomorphOfComplemented f f' hf' hker a = (f a, 0)", "@[simp]\ntheorem implicitToPartialHomeomorph_fst (hf : HasStrictFDerivAt f f' a) (hf' : range f' = \u22a4)\n    (x : E) : (hf.implicitToPartialHomeomorph f f' hf' x).fst = f x", "theorem HasStrictFDerivAt.isBigO_sub_rev {f' : E \u2243L[\ud835\udd5c] F}\n    (hf : HasStrictFDerivAt f (f' : E \u2192L[\ud835\udd5c] F) x) :\n    (fun p : E \u00d7 E => p.1 - p.2) =O[\ud835\udcdd (x, x)] fun p : E \u00d7 E => f p.1 - f p.2", "@[fun_prop]\nprotected theorem HasStrictFDerivAt.iterate {f : E \u2192 E} {f' : E \u2192L[\ud835\udd5c] E}\n    (hf : HasStrictFDerivAt f f' x) (hx : f x = x) (n : \u2115) :\n    HasStrictFDerivAt f^[n] (f' ^ n) x", "theorem mem_implicitToPartialHomeomorphOfComplemented_source (hf : HasStrictFDerivAt f f' a)\n    (hf' : range f' = \u22a4) (hker : (ker f').ClosedComplemented) :\n    a \u2208 (hf.implicitToPartialHomeomorphOfComplemented f f' hf' hker).source", "@[fun_prop]\ntheorem HasStrictFDerivAt.smul_const (hc : HasStrictFDerivAt c c' x) (f : F) :\n    HasStrictFDerivAt (fun y => c y \u2022 f) (c'.smulRight f) x", "@[fun_prop]\ntheorem HasStrictFDerivAt.sub (hf : HasStrictFDerivAt f f' x) (hg : HasStrictFDerivAt g g' x) :\n    HasStrictFDerivAt (fun x => f x - g x) (f' - g') x", "theorem HasSum.div_const (h : HasSum f a) (b : \u03b1) : HasSum (fun i \u21a6 f i / b) (a / b)", "theorem exists_hasSum_smul_of_apply_eq_zero (hs : HasSum (fun m => z ^ m \u2022 a m) s)\n    (ha : \u2200 k < n, a k = 0) : \u2203 t : E, z ^ n \u2022 t = s \u2227 HasSum (fun m => z ^ m \u2022 a (m + n)) t", "theorem IsRegular.inf (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a \u2293 b)", "@[simp, norm_cast]\ntheorem coe_bot : ((\u22a5 : Regular \u03b1) : \u03b1) = \u22a5", "@[simp, norm_cast]\ntheorem coe_le_coe {a b : Regular \u03b1} : (a : \u03b1) \u2264 b \u2194 a \u2264 b", "@[simp, norm_cast]\ntheorem coe_lt_coe {a b : Regular \u03b1} : (a : \u03b1) < b \u2194 a < b", "@[simp]\ntheorem coe_id : \u21d1(HeytingHom.id \u03b1) = id", "@[simp]\ntheorem toFun_eq_coe_aux {f : HeytingHom \u03b1 \u03b2} : (\u2191f.toLatticeHom) = \u21d1f", "@[to_additive]\ntheorem FP.finset_prod {M} [CommMonoid M] (a : Stream' M) (s : Finset \u2115) (hs : s.Nonempty) :\n    (s.prod fun i => a.get i) \u2208 FP a", "theorem ediam_image_le (hf : HolderOnWith C r f s) :\n    EMetric.diam (f '' s) \u2264 (C : \u211d\u22650\u221e) * EMetric.diam s ^ (r : \u211d)", "theorem hausdorffMeasure_image_le (h : HolderOnWith C r f s) (hr : 0 < r) {d : \u211d} (hd : 0 \u2264 d) :\n    \u03bcH[d] (f '' s) \u2264 (C : \u211d\u22650\u221e) ^ d * \u03bcH[r * d] s", "theorem nndist_le (hf : HolderWith C r f) (x y : X) :\n    nndist (f x) (f y) \u2264 C * nndist x y ^ (r : \u211d)", "theorem cprankMax_1 [Monoid \u03b1] [AddMonoid \u03b1] {x : Holor \u03b1 ds} (h : CPRankMax1 x) :\n    CPRankMax 1 x", "theorem slice_unitVec_mul [Ring \u03b1] {i : \u2115} {j : \u2115} (hid : i < d) (x : Holor \u03b1 ds) :\n    slice (unitVec d j \u2297 x) i hid = if i = j then x else 0", "theorem slice_zero [Zero \u03b1] (i : \u2115) (hid : i < d) : slice (0 : Holor \u03b1 (d :: ds)) i hid = 0", "theorem drop_take : \u2200 t : HolorIndex (ds\u2081 ++ ds\u2082 ++ ds\u2083), t.assocRight.drop.take = t.take.drop", "theorem comp_continuousAt_iff (h : X \u2243\u209c Y) (f : Z \u2192 X) (z : Z) :\n    ContinuousAt (h \u2218 f) z \u2194 ContinuousAt f z", "protected theorem compactSpace [CompactSpace X] (h : X \u2243\u209c Y) : CompactSpace Y where", "theorem Homeomorph.contDiff_symm [CompleteSpace E] (f : E \u2243\u209c F) {f\u2080' : E \u2192 E \u2243L[\ud835\udd5c] F}\n    (hf\u2080' : \u2200 a, HasFDerivAt f (f\u2080' a : E \u2192L[\ud835\udd5c] F) a) (hf : ContDiff \ud835\udd5c n (f : E \u2192 F)) :\n    ContDiff \ud835\udd5c n (f.symm : F \u2192 E)", "@[continuity, fun_prop]\nprotected theorem continuous (h : X \u2243\u209c Y) : Continuous h", "theorem image_closure (h : X \u2243\u209c Y) (s : Set X) : h '' closure s = closure (h '' s)", "theorem Homeomorph.isLocalHomeomorph (f : X \u2243\u209c Y) : IsLocalHomeomorph f", "theorem preimage_interior (h : X \u2243\u209c Y) (s : Set Y) : h \u207b\u00b9' interior s = interior (h \u207b\u00b9' s)", "@[simp]\ntheorem self_comp_symm (h : X \u2243\u209c Y) : h \u2218 h.symm = id", "@[simp]\ntheorem Homeomorph.smulOfNeZero_symm_apply {c : G\u2080} (hc : c \u2260 0) :\n    \u21d1(Homeomorph.smulOfNeZero c hc).symm = (c\u207b\u00b9 \u2022 \u00b7 : \u03b1 \u2192 \u03b1)", "protected theorem t1Space [T1Space X] (h : X \u2243\u209c Y) : T1Space Y", "@[simp]\ntheorem trans_toHomotopyEquiv (h\u2080 : X \u2243\u209c Y) (h\u2081 : Y \u2243\u209c Z) :\n    (h\u2080.trans h\u2081).toHomotopyEquiv = h\u2080.toHomotopyEquiv.trans h\u2081.toHomotopyEquiv", "@[simp]\ntheorem HomogeneousIdeal.mem_iff {I : HomogeneousIdeal \ud835\udc9c} {x : A} : x \u2208 I.toIdeal \u2194 x \u2208 I", "@[simp]\ntheorem den_neg (c : NumDenSameDeg \ud835\udc9c x) : ((-c).den : A) = c.den", "theorem isUnit_iff_isUnit_val (f : HomogeneousLocalization.AtPrime \ud835\udc9c \ud835\udd2d) :\n    IsUnit f.val \u2194 IsUnit f", "@[simp] lemma mk_pow (i : NumDenSameDeg \ud835\udc9c x) (n : \u2115) : mk (i ^ n) = mk i ^ n := rfl", "@[simp]\ntheorem val_intCast (n : \u2124) : (n : HomogeneousLocalization \ud835\udc9c x).val = n", "@[simp]\ntheorem val_mul : \u2200 y1 y2 : HomogeneousLocalization \ud835\udc9c x, (y1 * y2).val = y1.val * y2.val", "@[simp]\ntheorem val_natCast (n : \u2115) : (n : HomogeneousLocalization \ud835\udc9c x).val = n", "@[simp]\ntheorem val_sub (y1 y2 : HomogeneousLocalization \ud835\udc9c x) : (y1 - y2).val = y1.val - y2.val", "theorem from_single\u2080_exact_f_d_at_zero [hf : QuasiIso' f] : Exact (f.f 0) (X.d 0 1)", "theorem to_single\u2080_exact_at_succ [hf : QuasiIso' f] (n : \u2115) :\n    Exact (X.d (n + 2) (n + 1)) (X.d (n + 1) n)", "lemma composableArrows\u2083_exact [CategoryWithHomology C] :\n    (composableArrows\u2083 K i j).Exact", "@[reassoc (attr := simp)]\nlemma biprodXIso_hom_snd (i : \u03b9) :\n    (biprodXIso K L i).hom \u226b biprod.snd = (biprod.snd : K \u229e L \u27f6 L).f i", "@[reassoc (attr := simp)]\nlemma biprod_inr_snd_f (i : \u03b9) :\n    (biprod.inr : L \u27f6 K \u229e L).f i \u226b (biprod.snd : K \u229e L \u27f6 L).f i = \ud835\udfd9 _", "@[reassoc]\nlemma cyclesMap_comp : cyclesMap (\u03c6 \u226b \u03c8) i = cyclesMap \u03c6 i \u226b cyclesMap \u03c8 i", "@[reassoc (attr := simp)]\nlemma \u03b9\u2080_desc : \u03b9\u2080 K \u226b desc \u03c6\u2080 \u03c6\u2081 h = \u03c6\u2080", "lemma biprod_lift_id_sub_id : biprod.lift (\ud835\udfd9 K) (-\ud835\udfd9 K) = biprod.inl - biprod.inr", "lemma nullHomotopicMap_eq : nullHomotopicMap K = \u03c0 K \u226b \u03b9\u2080 K - \ud835\udfd9 _", "@[reassoc (attr := simp)]\ntheorem dFrom_comp_xNextIsoSelf {i : \u03b9} (h : \u00acc.Rel i (c.next i)) :\n    C.dFrom i \u226b (C.xNextIsoSelf h).hom = 0", "@[simp 1100]\ntheorem dTo_comp_dFrom (j : \u03b9) : C.dTo j \u226b C.dFrom j = 0", "@[reassoc (attr := simp)]\ntheorem d_comp_d (C : HomologicalComplex V c) (i j k : \u03b9) : C.d i j \u226b C.d j k = 0", "@[reassoc (attr := simp)]\nlemma d_pOpcycles [K.HasHomology j] : K.d i j \u226b K.pOpcycles j = 0", "lemma d_none_eq_zero (i j : Option \u03b9) (hi : i = none) :\n    d K i j = 0", "@[reassoc (attr := simp)]\nlemma fromOpcycles_d :\n    K.fromOpcycles i j \u226b K.d j k = 0", "@[simp]\nlemma homologyMap_zero : homologyMap (0 : K \u27f6 L) i = 0", "lemma desc_f (j k : \u03b9) (hjk : c.Rel j k) :\n    (desc \u03c6 \u03b1 h\u03b1).f j = fstX \u03c6 j _ hjk \u226b h\u03b1.hom _ j + sndX \u03c6 j \u226b \u03b1.f j", "lemma ext_from_X (i j : \u03b9) (hij : c.Rel j i) {A : C} {f g : X \u03c6 j \u27f6 A}\n    (h\u2081 : inlX \u03c6 i j hij \u226b f = inlX \u03c6 i j hij \u226b g) (h\u2082 : inrX \u03c6 j \u226b f = inrX \u03c6 j \u226b g) :\n    f = g", "@[reassoc (attr := simp)]\nlemma iCyclesIso_inv_hom_id :\n    (K.iCyclesIso i j hj h).inv \u226b K.iCycles i = \ud835\udfd9 _", "theorem image_to_eq_image [HasImages V] [HasEqualizers V] {i j : \u03b9} (r : c.Rel i j) :\n    imageSubobject (C.dTo j) = imageSubobject (C.d i j)", "lemma isZero_single_comp_eval (j i : \u03b9) (hi : i \u2260 j) : IsZero (single V c j \u22d9 eval V c i)", "theorem kernel_eq_kernel [HasKernels V] {i j j' : \u03b9} (r : c.Rel i j) (r' : c.Rel i j') :\n    kernelSubobject (C.d i j) = kernelSubobject (C.d i j')", "@[reassoc]\nlemma liftCycles_homology\u03c0_eq_zero_of_boundary {A : C} (k : A \u27f6 K.X i) (j : \u03b9)\n    (hj : c.next i = j) {i' : \u03b9} (x : A \u27f6 K.X i') (hx : k = x \u226b K.d i' i) :\n    K.liftCycles k j hj (by rw [hx, assoc, K.d_comp_d, comp_zero]) \u226b K.homology\u03c0 i = 0", "@[reassoc]\nlemma singleObjOpcyclesSelfIso_hom :\n    (singleObjOpcyclesSelfIso c j A).hom =\n      (singleObjXSelf c j A).inv \u226b ((single C c j).obj A).pOpcycles j := rfl", "@[reassoc (attr := simp)]\nlemma singleObjOpcyclesSelfIso_hom_naturality :\n    (singleObjOpcyclesSelfIso c j A).hom \u226b opcyclesMap ((single C c j).map f) j  =\n      f \u226b (singleObjOpcyclesSelfIso c j B).hom", "lemma d\u2081_eq_zero (h : \u00ac c\u2081.Rel i\u2081 (c\u2081.next i\u2081)) :\n    K.d\u2081 c\u2081\u2082 i\u2081 i\u2082 i\u2081\u2082 = 0", "lemma d\u2082_eq' (i\u2081 : I\u2081) {i\u2082 i\u2082' : I\u2082} (h : c\u2082.Rel i\u2082 i\u2082') (i\u2081\u2082 : I\u2081\u2082) :\n    K.d\u2082 c\u2081\u2082 i\u2081 i\u2082 i\u2081\u2082 = ComplexShape.\u03b5\u2082 c\u2081 c\u2082 c\u2081\u2082 \u27e8i\u2081, i\u2082\u27e9 \u2022 ((K.X i\u2081).d i\u2082 i\u2082' \u226b\n    K.toGradedObject.\u03b9MapObjOrZero (ComplexShape.\u03c0 c\u2081 c\u2082 c\u2081\u2082) \u27e8i\u2081, i\u2082'\u27e9 i\u2081\u2082)", "@[simp 1100]\ntheorem dNext_succ_chainComplex (f : \u2200 i j, P.X i \u27f6 Q.X j) (i : \u2115) :\n    dNext (i + 1) f = P.d _ _ \u226b f i (i + 1)", "@[simp 1100]\ntheorem dNext_zero_chainComplex (f : \u2200 i j, P.X i \u27f6 Q.X j) : dNext 0 f = 0", "lemma _root_.Homotopy.map_eq_of_inverts_homotopyEquivalences\n    {\u03c6\u2080 \u03c6\u2081 : F \u27f6 G} (h : Homotopy \u03c6\u2080 \u03c6\u2081) (hc : \u2200 j, \u2203 i, c.Rel i j)\n    [\u2200 i, HasBinaryBiproduct (F.X i) (F.X i)]\n    [HasHomotopyCofiber (biprod.lift (\ud835\udfd9 F) (-\ud835\udfd9 F))]\n    {D : Type*} [Category D] (H : HomologicalComplex C c \u2964 D)\n    (hH : (homotopyEquivalences C c).IsInvertedBy H) :\n    H.map \u03c6\u2080 = H.map \u03c6\u2081", "@[simp 1100]\ntheorem prevD_chainComplex (f : \u2200 i j, P.X i \u27f6 Q.X j) (j : \u2115) :\n    prevD j f = f j (j + 1) \u226b Q.d _ _", "theorem isUnital_auxGroup (i : N) :\n    EckmannHilton.IsUnital (auxGroup i).mul (\u27e6const\u27e7 : HomotopyGroup N X x) where", "lemma f_int_ofNat (k : \u2115) {a : \u211d} (ha : 0 \u2264 a) (t : \u211d) (n : \u2115) :\n    f_int k a t (Int.ofNat n) = f_nat k a t n", "lemma isBigO_atTop_F_nat_zero_sub {a : \u211d} (ha : 0 \u2264 a) : \u2203 p, 0 < p \u2227\n    (fun t \u21a6 F_nat 0 a t - (if a = 0 then 1 else 0)) =O[atTop] fun t \u21a6 exp (-p * t)", "private lemma isBigO_one_aux :\n    IsBigO atTop (fun t : \u211d \u21a6 (1 - rexp (-\u03c0 * t))\u207b\u00b9) (fun _ \u21a6 (1 : \u211d))", "lemma completedCosZeta_one_sub (a : UnitAddCircle) (s : \u2102) :\n    completedCosZeta a (1 - s) = completedHurwitzZetaEven a s", "lemma completedHurwitzZetaEven_eq (a : UnitAddCircle) (s : \u2102) :\n    completedHurwitzZetaEven a s =\n    completedHurwitzZetaEven\u2080 a s - (if a = 0 then 1 else 0) / s - 1 / (1 - s)", "lemma continuousOn_oddKernel (a : UnitAddCircle) : ContinuousOn (oddKernel a) (Ioi 0)", "theorem cosZeta_neg_two_mul_nat_add_one (a : UnitAddCircle) (n : \u2115) :\n    cosZeta a (-2 * (n + 1)) = 0", "lemma HurwitzZeta.cosZeta_zero : cosZeta 0 = riemannZeta", "lemma differentiableAt_cosZeta (a : UnitAddCircle) {s : \u2102} (hs' : s \u2260 1 \u2228 a \u2260 0) :\n    DifferentiableAt \u2102 (cosZeta a) s", "lemma evenKernel_def (a x : \u211d) :\n    \u2191(evenKernel \u2191a x) = cexp (-\u03c0 * a ^ 2 * x) * jacobiTheta\u2082 (a * I * x) (I * x)", "lemma evenKernel_eq_cosKernel_of_zero : evenKernel 0 = cosKernel 0", "lemma expZeta_one_sub (a : UnitAddCircle) {s : \u2102} (hs : \u2200 (n : \u2115), s \u2260 1 - n) :\n    expZeta a (1 - s) = (2 * \u03c0) ^ (-s) * Gamma s *\n    (exp (\u03c0 * I * s / 2) * hurwitzZeta a s + exp (-\u03c0 * I * s / 2) * hurwitzZeta (-a) s)", "lemma hasSum_int_evenKernel\u2080 (a : \u211d) {t : \u211d} (ht : 0 < t) :\n    HasSum (fun n : \u2124 \u21a6 if n + a = 0 then 0 else rexp (-\u03c0 * (n + a) ^ 2 * t))\n    (evenKernel a t - if (a : UnitAddCircle) = 0 then 1 else 0)", "theorem hurwitzZetaEven_neg_two_mul_nat_add_one (a : UnitAddCircle) (n : \u2115) :\n    hurwitzZetaEven a (-2 * (n + 1)) = 0", "theorem hurwitzZetaEven_one_sub_two_mul_nat (hk : k \u2260 0) (hx : x \u2208 Icc (0 : \u211d) 1) :\n    hurwitzZetaEven x (1 - 2 * k) =\n      -1 / (2 * k) * ((Polynomial.bernoulli (2 * k)).map (algebraMap \u211a \u2102)).eval (x : \u2102)", "theorem hurwitzZeta_neg_nat (hk : k \u2260 0) (hx : x \u2208 Icc (0 : \u211d) 1) :\n    hurwitzZeta x (-k) =\n    -1 / (k + 1) * ((Polynomial.bernoulli (k + 1)).map (algebraMap \u211a \u2102)).eval (x : \u2102)", "lemma jacobiTheta\u2082''_neg_left (z \u03c4 : \u2102) : jacobiTheta\u2082'' (-z) \u03c4 = -jacobiTheta\u2082'' z \u03c4", "lemma tendsto_hurwitzZetaEven_sub_one_div_nhds_one (a : UnitAddCircle) :\n    Tendsto (fun s \u21a6 hurwitzZetaEven a s - 1 / (s - 1) / Gamma\u211d s) (\ud835\udcdd 1)\n    (\ud835\udcdd (hurwitzZetaEven a 1))", "theorem InfinitePos.neg {x : \u211d*} : InfinitePos x \u2192 InfiniteNeg (-x)", "theorem IsSt.map {x : \u211d*} {r : \u211d} (hxr : IsSt x r) {f : \u211d \u2192 \u211d} (hf : ContinuousAt f r) :\n    IsSt (x.map f) (f r)", "theorem IsSt.mul {x y : \u211d*} {r s : \u211d} (hxr : IsSt x r) (hys : IsSt y s) : IsSt (x * y) (r * s)", "theorem epsilon_pos : 0 < \u03b5", "theorem infiniteNeg_mul_infinitePos {x y : \u211d*} :\n    InfiniteNeg x \u2192 InfinitePos y \u2192 InfiniteNeg (x * y)", "theorem infiniteNeg_mul_of_not_infinitesimal_neg_infinitePos {x y : \u211d*} :\n    \u00acInfinitesimal x \u2192 x < 0 \u2192 InfinitePos y \u2192 InfiniteNeg (x * y)", "@[simp] theorem infiniteNeg_neg {x : \u211d*} : InfiniteNeg (-x) \u2194 InfinitePos x", "theorem infinitePos_iff_infinite_and_pos {x : \u211d*} : InfinitePos x \u2194 Infinite x \u2227 0 < x", "@[simp] theorem infinitePos_neg {x : \u211d*} : InfinitePos (-x) \u2194 InfiniteNeg x", "theorem infinite_of_infinitesimal_inv {x : \u211d*} (h0 : x \u2260 0) (hi : Infinitesimal x\u207b\u00b9) :\n    Infinite x", "theorem infinitesimal_iff_infinite_inv {x : \u211d*} (h : x \u2260 0) : Infinitesimal x \u2194 Infinite x\u207b\u00b9", "@[simp]\ntheorem inv_epsilon : \u03b5\u207b\u00b9 = \u03c9", "theorem isSt_iff_abs_sub_lt_delta {x : \u211d*} {r : \u211d} : IsSt x r \u2194 \u2200 \u03b4 : \u211d, 0 < \u03b4 \u2192 |x - \u2191r| < \u03b4", "theorem isSt_of_tendsto {f : \u2115 \u2192 \u211d} {r : \u211d} (hf : Tendsto f atTop (\ud835\udcdd r)) : IsSt (ofSeq f) r", "theorem isSt_sSup {x : \u211d*} (hni : \u00acInfinite x) : IsSt x (sSup { y : \u211d | (y : \u211d*) < x })", "theorem lt_of_tendsto_zero_of_pos {f : \u2115 \u2192 \u211d} (hf : Tendsto f atTop (\ud835\udcdd 0)) :\n    \u2200 {r : \u211d}, 0 < r \u2192 ofSeq f < (r : \u211d*)", "theorem not_infinite_mul {x y : \u211d*} (hx : \u00acInfinite x) (hy : \u00acInfinite y) : \u00acInfinite (x * y)", "theorem not_real_of_infinitesimal_ne_zero (x : \u211d*) : Infinitesimal x \u2192 x \u2260 0 \u2192 \u2200 r : \u211d, x \u2260 r", "theorem Icc_subset_segment : Icc x y \u2286 [x -[\ud835\udd5c] y]", "theorem Ici_mem_nhdsSet_Ico (h : a < b) : Ici a \u2208 \ud835\udcdd\u02e2 (Ico b c)", "theorem Ici_mem_nhdsSet_Ioc (h : a \u2264 b) : Ici a \u2208 \ud835\udcdd\u02e2 (Ioc b c)", "theorem Ico_ofDual (a b : \u03b1\u1d52\u1d48) : Ico (ofDual a) (ofDual b) = (Ioc b a).map ofDual.toEmbedding", "theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)\n    {F' : I.Filtration M} (hf : F' \u2264 F) : F'.Stable", "theorem inf_submodule : (F \u2293 F').submodule = F.submodule \u2293 F'.submodule", "theorem Ideal.IsHomogeneous.toIdeal_homogeneousCore_eq_self (h : I.IsHomogeneous \ud835\udc9c) :\n    (I.homogeneousCore \ud835\udc9c).toIdeal = I", "theorem Ideal.IsHomogeneous.toIdeal_homogeneousHull_eq_self (h : I.IsHomogeneous \ud835\udc9c) :\n    (Ideal.homogeneousHull \ud835\udc9c I).toIdeal = I", "theorem top : Ideal.IsHomogeneous \ud835\udc9c \u22a4", "theorem IsMaximal.exists_inv {I : Ideal \u03b1} (hI : I.IsMaximal) {x} (hx : x \u2209 I) :\n    \u2203 y, \u2203 i \u2208 I, y * x + i = 1", "theorem Ideal.IsPrime.homogeneousCore {I : Ideal A} (h : I.IsPrime) :\n    (I.homogeneousCore \ud835\udc9c).toIdeal.IsPrime", "theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I \u2293 J \u2264 P \u2194 I \u2264 P \u2228 J \u2264 P", "theorem IsPrime.inf_le' {s : Finset \u03b9} {f : \u03b9 \u2192 Ideal R} {P : Ideal R} (hp : IsPrime P) :\n    s.inf f \u2264 P \u2194 \u2203 i \u2208 s, f i \u2264 P", "theorem Ideal.IsPrime.mul_mem_pow (I : Ideal R) [hI : I.IsPrime] {a b : R} {n : \u2115}\n    (h : a * b \u2208 I ^ n) : a \u2208 I \u2228 b \u2208 I ^ n", "theorem IsPrime.multiset_prod_map_le {s : Multiset \u03b9} (f : \u03b9 \u2192 Ideal R) {P : Ideal R}\n    (hp : IsPrime P) : (s.map f).prod \u2264 P \u2194 \u2203 i \u2208 s, f i \u2264 P", "theorem IsPrime.multiset_prod_mem_iff_exists_mem {I : Ideal R} (hI : I.IsPrime) (s : Multiset R) :\n    s.prod \u2208 I \u2194 \u2203 p \u2208 s, p \u2208 I", "theorem isJacobson_MvPolynomial_fin {R : Type u} [CommRing R] [H : IsJacobson R] :\n    \u2200 n : \u2115, IsJacobson (MvPolynomial (Fin n) R)", "theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :\n    \u2200 {a : R \u29f8 I}, a \u2260 0 \u2192 \u2203 b : R \u29f8 I, a * b = 1", "@[simp]\ntheorem Quotient.mk_comp_algebraMap (I : Ideal A) :\n    (Quotient.mk I).comp (algebraMap R\u2081 A) = algebraMap R\u2081 (A \u29f8 I)", "@[simp]\ntheorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R \u29f8 I) = mk I x", "theorem adic_module_basis :\n    I.ringFilterBasis.SubmodulesBasis fun n : \u2115 => I ^ n \u2022 (\u22a4 : Submodule R M)", "@[simp]\ntheorem associatesEquivIsPrincipal_map_one :\n    (associatesEquivIsPrincipal R 1 : Ideal R) = 1", "theorem comap_isMaximal_of_surjective {K : Ideal S} [H : IsMaximal K] : IsMaximal (comap f K)", "theorem comap_lt_comap_of_integral_mem_sdiff [Algebra R S] [hI : I.IsPrime] (hIJ : I \u2264 J) {x : S}\n    (mem : x \u2208 (J : Set S) \\ I) (integral : IsIntegral R x) :\n    I.comap (algebraMap R S) < J.comap (algebraMap R S)", "@[simp]\ntheorem comap_of_equiv (I : Ideal R) (f : R \u2243+* S) :\n    (I.comap (f.symm : S \u2192+* R)).comap (f : R \u2192+* S) = I", "theorem comap_sInf (s : Set (Ideal S)) : (sInf s).comap f = \u2a05 I \u2208 s, (I : Ideal S).comap f", "theorem eq_bot_or_top : I = \u22a5 \u2228 I = \u22a4", "theorem eq_top_of_isUnit_mem {x} (hx : x \u2208 I) (h : IsUnit x) : I = \u22a4", "theorem finite_setOf_absNorm_eq [CharZero S] {n : \u2115} (hn : 0 < n) :\n    {I : Ideal S | Ideal.absNorm I = n}.Finite", "theorem finsuppTotal_apply_eq_of_fintype [Fintype \u03b9] (f : \u03b9 \u2192\u2080 I) :\n    finsuppTotal \u03b9 M I v f = \u2211 i, (f i : R) \u2022 v i", "theorem iSup_iInf_eq_top_iff_pairwise {t : Finset \u03b9} (h : t.Nonempty) (I : \u03b9 \u2192 Ideal R) :\n    (\u2a06 i \u2208 t, \u2a05 (j) (_ : j \u2208 t) (_ : j \u2260 i), I j) = \u22a4 \u2194\n      (t : Set \u03b9).Pairwise fun i j => I i \u2294 I j = \u22a4", "theorem isCoprime_iff_codisjoint : IsCoprime I J \u2194 Codisjoint I J", "theorem isMaximal_comap_of_isIntegral_of_isMaximal [Algebra.IsIntegral R S] (I : Ideal S)\n    [hI : I.IsMaximal] : IsMaximal (I.comap (algebraMap R S))", "theorem isMaximal_iff {I : Ideal \u03b1} :\n    I.IsMaximal \u2194 (1 : \u03b1) \u2209 I \u2227 \u2200 (J : Ideal \u03b1) (x), I \u2264 J \u2192 x \u2209 I \u2192 x \u2208 J \u2192 (1 : \u03b1) \u2208 J", "theorem isMaximal_of_isMaximal_disjoint [IsJacobson R] (I : Ideal R) (hI : I.IsMaximal)\n    (hy : y \u2209 I) : (map (algebraMap R S) I).IsMaximal", "theorem isOpen_of_isOpen_subideal {U I : Ideal R} (h : U \u2264 I) (hU : IsOpen (U : Set R)) :\n    IsOpen (I : Set R)", "theorem isPrime_map_C_iff_isPrime (P : Ideal R) :\n    IsPrime (map (C : R \u2192+* R[X]) P : Ideal R[X]) \u2194 IsPrime P", "theorem kerLiftAlg_injective (f : A \u2192\u2090[R\u2081] B) : Function.Injective (kerLiftAlg f)", "theorem Ideal.le_of_localization_maximal {I J : Ideal R}\n    (h : \u2200 (P : Ideal R) (hP : P.IsMaximal),\n      Ideal.map (algebraMap R (Localization.AtPrime P)) I \u2264\n        Ideal.map (algebraMap R (Localization.AtPrime P)) J) :\n    I \u2264 J", "theorem Ideal.le_of_pow_le_prime {I P : Ideal R} [hP : P.IsPrime] {n : \u2115} (h : I ^ n \u2264 P) :\n    I \u2264 P", "theorem map.isMaximal {I : Ideal R} (H : IsMaximal I) : IsMaximal (map f I)", "theorem map_eq_iff_sup_ker_eq_of_surjective {I J : Ideal R} (f : R \u2192+* S)\n    (hf : Function.Surjective f) : map f I = map f J \u2194 I \u2294 RingHom.ker f = J \u2294 RingHom.ker f", "theorem map_eq_submodule_map (f : R \u2192+* S) [h : RingHomSurjective f] (I : Ideal R) :\n    I.map f = Submodule.map f.toSemilinearMap I", "lemma map_eq_top_iff_of_ker_le\n    (f : R \u2192+* S) {I : Ideal R} (hf\u2081 : RingHom.ker f \u2264 I) (hf\u2082 : f.IsIntegral) :\n    I.map f = \u22a4 \u2194 I = \u22a4", "theorem map_le_comap_of_inverse (g : G) (I : Ideal R) (h : Function.LeftInverse g f) :\n    I.map f \u2264 I.comap g", "theorem map_span (f : F) (s : Set R) : map f (span s) = span (f '' s)", "theorem mem_isPrincipalSubmonoid_iff {I : Ideal R} :\n    I \u2208 isPrincipalSubmonoid R \u2194 IsPrincipal I", "theorem mem_jacobson_bot {x : R} : x \u2208 jacobson (\u22a5 : Ideal R) \u2194 \u2200 y, IsUnit (x * y + 1)", "theorem Ideal.mem_prime_of_mul_mem_pow [IsDedekindDomain S] {P : Ideal S} [P_prime : P.IsPrime]\n    (hP : P \u2260 \u22a5) {i : \u2115} {a b : S} (a_not_mem : a \u2209 P ^ (i + 1)) (ab_mem : a * b \u2208 P ^ (i + 1)) :\n    b \u2208 P", "theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x \u2208 span {y} * I \u2194 \u2203 z \u2208 I, y * z = x", "protected theorem mul_comm : I * J = J * I", "theorem Ideal.mul_homogeneous_element_mem_of_mem {I : Ideal A} (r x : A) (hx\u2081 : Homogeneous \ud835\udc9c x)\n    (hx\u2082 : x \u2208 I) (j : \u03b9) : GradedRing.proj \ud835\udc9c j (r * x) \u2208 I", "theorem mul_le : I * J \u2264 K \u2194 \u2200 r \u2208 I, \u2200 s \u2208 J, r * s \u2208 K", "theorem mul_le_left : I * J \u2264 J", "theorem mul_mem_mul {r s} (hr : r \u2208 I) (hs : s \u2208 J) : r * s \u2208 I * J", "@[simp]\ntheorem mul_top : I * \u22a4 = I", "@[simp]\nlemma multiset_prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} :\n    s.prod = \u22a5 \u2194 \u22a5 \u2208 s", "@[simp]\ntheorem multiset_prod_span_singleton (m : Multiset R) :\n    (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R)", "lemma pi_quotient_surjective {R : Type*} [CommRing R] {\u03b9 : Type*} [Finite \u03b9] {I : \u03b9 \u2192 Ideal R}\n    (hf : Pairwise fun i j \u21a6 IsCoprime (I i) (I j)) (x : (i : \u03b9) \u2192 R \u29f8 I i) :\n    \u2203 r : R, \u2200 i, r = x i", "theorem Ideal.prime_of_isPrime {P : Ideal A} (hP : P \u2260 \u22a5) (h : IsPrime P) : Prime P", "@[simp]\ntheorem quotientEquivAlgOfEq_symm {I J : Ideal A} (h : I = J) :\n    (quotientEquivAlgOfEq R\u2081 h).symm = quotientEquivAlgOfEq R\u2081 h.symm", "@[simp]\ntheorem quotientMap_mk {J : Ideal R} {I : Ideal S} {f : R \u2192+* S} {H : J \u2264 I.comap f} {x : R} :\n    quotientMap I f H (Quotient.mk J x) = Quotient.mk I (f x)", "theorem quotientToQuotientRangePowQuotSucc_injective [IsDedekindDomain S] [P.IsPrime]\n    {i : \u2115} (hi : i < e) {a : S} (a_mem : a \u2208 P ^ i) (a_not_mem : a \u2209 P ^ (i + 1)) :\n    Function.Injective (quotientToQuotientRangePowQuotSucc f p P a_mem)", "@[simp]\ntheorem quotient_map_mk\u2090 {I : Ideal A} (J : Ideal B) (f : A \u2192\u2090[R\u2081] B) (H : I \u2264 J.comap f) {x : A} :\n    quotientMap\u2090 J f H (Quotient.mk I x) = Quotient.mk\u2090 R\u2081 J (f x)", "theorem radical_eq_jacobson_iff_radical_quotient_eq_jacobson_bot :\n    I.radical = I.jacobson \u2194 radical (\u22a5 : Ideal (R \u29f8 I)) = jacobson \u22a5", "lemma radical_pow : \u2200 {n}, n \u2260 0 \u2192 radical (I ^ n) = radical I\n  | 1, _ => by simp\n  | n + 2, _ => by rw [pow_succ, radical_mul, radical_pow n.succ_ne_zero, inf_idem]", "theorem radical_sup : radical (I \u2294 J) = radical (radical I \u2294 radical J)", "theorem ramificationIdx_lt {n : \u2115} (hgt : \u00acmap f p \u2264 P ^ n) : ramificationIdx f p P < n", "theorem range_finsuppTotal :\n    LinearMap.range (finsuppTotal \u03b9 M I v) = I \u2022 Submodule.span R (Set.range v)", "theorem rank_pow_quot [IsDedekindDomain S] [p.IsMaximal] [P.IsPrime] (hP0 : P \u2260 \u22a5)\n    (i : \u2115) (hi : i \u2264 e) :\n    Module.rank (R \u29f8 p) (Ideal.map (Ideal.Quotient.mk (P ^ e)) (P ^ i)) =\n      (e - i) \u2022 Module.rank (R \u29f8 p) (S \u29f8 P)", "@[simp]\ntheorem relNorm_singleton (r : S) : relNorm R (span ({r} : Set S)) = span {Algebra.norm R r}", "theorem spanNorm_localization (I : Ideal S) [Module.Finite R S] [Module.Free R S] (M : Submonoid R)\n    {R\u2098 : Type*} (S\u2098 : Type*) [CommRing R\u2098] [Algebra R R\u2098] [CommRing S\u2098] [Algebra S S\u2098]\n    [Algebra R\u2098 S\u2098] [Algebra R S\u2098] [IsScalarTower R R\u2098 S\u2098] [IsScalarTower R S S\u2098]\n    [IsLocalization M R\u2098] [IsLocalization (Algebra.algebraMapSubmonoid S M) S\u2098] :\n    spanNorm R\u2098 (I.map (algebraMap S S\u2098)) = (spanNorm R I).map (algebraMap R R\u2098)", "theorem spanNorm_mul (I J : Ideal S) : spanNorm R (I * J) = spanNorm R I * spanNorm R J", "theorem span_eq_bot {s : Set \u03b1} : span s = \u22a5 \u2194 \u2200 x \u2208 s, (x : \u03b1) = 0", "theorem span_eq_top_iff_finite (s : Set \u03b1) :\n    span s = \u22a4 \u2194 \u2203 s' : Finset \u03b1, \u2191s' \u2286 s \u2227 span (s' : Set \u03b1) = \u22a4", "theorem span_le {s : Set \u03b1} {I} : span s \u2264 I \u2194 s \u2286 I", "theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) :\n    span {x} * I = span {y} * J \u2194\n      (\u2200 zI \u2208 I, \u2203 zJ \u2208 J, x * zI = y * zJ) \u2227 \u2200 zJ \u2208 J, \u2203 zI \u2208 I, x * zI = y * zJ", "protected theorem sub_mem : a \u2208 I \u2192 b \u2208 I \u2192 a - b \u2208 I", "theorem subset_span {s : Set \u03b1} : s \u2286 span s", "theorem sum_ramification_inertia (K L : Type*) [Field K] [Field L] [IsDedekindDomain R]\n    [Algebra R K] [IsFractionRing R K] [Algebra S L] [IsFractionRing S L] [Algebra K L]\n    [Algebra R L] [IsScalarTower R S L] [IsScalarTower R K L] [IsNoetherian R S]\n    [IsIntegralClosure S R L] [p.IsMaximal] (hp0 : p \u2260 \u22a5) :\n    (\u2211 P \u2208 (factors (map (algebraMap R S) p)).toFinset,\n        ramificationIdx (algebraMap R S) p P * inertiaDeg (algebraMap R S) p P) =\n      finrank K L", "theorem sup_mul_eq_of_coprime_right (h : I \u2294 K = \u22a4) : I \u2294 J * K = I \u2294 J", "theorem toCotangent_range : LinearMap.range I.toCotangent = \u22a4", "@[simp]\ntheorem torsionOf_zero : torsionOf R M (0 : M) = \u22a4", "theorem left_map_implicitFunction :\n    \u2200\u1da0 p : F \u00d7 G in \ud835\udcdd (\u03c6.prodFun \u03c6.pt), \u03c6.leftFun (\u03c6.implicitFunction p.1 p.2) = p.1", "protected theorem Inducing.comp (hg : Inducing g) (hf : Inducing f) :\n    Inducing (g \u2218 f)", "theorem dense_iff (hf : Inducing f) {s : Set X} :\n    Dense s \u2194 \u2200 x, f x \u2208 closure (f '' s)", "@[to_additive]\nprotected theorem Inducing.hasProd_iff [CommMonoid \u03b3] [TopologicalSpace \u03b3] {G}\n    [FunLike G \u03b1 \u03b3] [MonoidHomClass G \u03b1 \u03b3] {g : G} (hg : Inducing g) (f : \u03b2 \u2192 \u03b1) (a : \u03b1) :\n    HasProd (g \u2218 f) (g a) \u2194 HasProd f a", "protected theorem _root_.Inducing.isSeparable_preimage {f : \u03b2 \u2192 \u03b1} [TopologicalSpace \u03b2]\n    (hf : Inducing f) {s : Set \u03b1} (hs : IsSeparable s) : IsSeparable (f \u207b\u00b9' s)", "theorem mapClusterPt_iff (hf : Inducing f) {x : X} {l : Filter X} :\n    MapClusterPt (f x) l f \u2194 ClusterPt x l", "@[to_additive]\nprotected theorem Inducing.topologicalGroup {F : Type*} [Group H] [TopologicalSpace H]\n    [FunLike F H G] [MonoidHomClass F H G] (f : F) (hf : Inducing f) : TopologicalGroup H", "@[simp]\ntheorem coe_comp (f : InfHom \u03b2 \u03b3) (g : InfHom \u03b1 \u03b2) : (f.comp g : \u03b1 \u2192 \u03b3) = f \u2218 g", "@[simp]\ntheorem coe_const (b : \u03b2) : \u21d1(const \u03b1 b) = Function.const \u03b1 b", "@[simp] theorem comp_id (f : InfHom \u03b1 \u03b2) : f.comp (InfHom.id \u03b1) = f", "@[simp]\ntheorem symm_dual_comp (g : SupHom \u03b2\u1d52\u1d48 \u03b3\u1d52\u1d48) (f : SupHom \u03b1\u1d52\u1d48 \u03b2\u1d52\u1d48) :\n    InfHom.dual.symm (g.comp f) =\n      (InfHom.dual.symm g).comp (InfHom.dual.symm f)", "theorem InfPrime.inf_le (ha : InfPrime a) : b \u2293 c \u2264 a \u2194 b \u2264 a \u2228 c \u2264 a", "@[simp]\ntheorem coe_comp (f : InfTopHom \u03b2 \u03b3) (g : InfTopHom \u03b1 \u03b2) : (f.comp g : \u03b1 \u2192 \u03b3) = f \u2218 g", "@[simp] lemma coe_mk (f : InfHom \u03b1 \u03b2) (hf) : \u21d1(mk f hf) = f := rfl", "theorem exists_superset_card_eq [Infinite \u03b1] (s : Finset \u03b1) (n : \u2115) (hn : s.card \u2264 n) :\n    \u2203 t : Finset \u03b1, s \u2286 t \u2227 t.card = n", "protected theorem acc (f : r \u227ci s) (a : \u03b1) : Acc r a \u2194 Acc s (f a)", "theorem angle_neg_right (x y : V) : angle x (-y) = \u03c0 - angle x y", "theorem norm_eq_of_angle_sub_eq_angle_sub_rev_of_angle_ne_pi {x y : V}\n    (h : angle x (x - y) = angle y (y - x)) (hpi : angle x y \u2260 \u03c0) : \u2016x\u2016 = \u2016y\u2016", "theorem norm_sub_sq_eq_norm_sq_add_norm_sq_iff_angle_eq_pi_div_two (x y : V) :\n    \u2016x - y\u2016 * \u2016x - y\u2016 = \u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016 \u2194 angle x y = \u03c0 / 2", "theorem sin_angle_sub_of_inner_eq_zero {x y : V} (h : \u27eax, y\u27eb = 0) (h0 : x \u2260 0 \u2228 y \u2260 0) :\n    Real.sin (angle x (x - y)) = \u2016y\u2016 / \u2016x - y\u2016", "theorem tan_angle_add_of_inner_eq_zero {x y : V} (h : \u27eax, y\u27eb = 0) :\n    Real.tan (angle x (x + y)) = \u2016y\u2016 / \u2016x\u2016", "theorem inner_add_add_self (x y : F) : \u27eax + y, x + y\u27eb = \u27eax, x\u27eb + \u27eax, y\u27eb + \u27eay, x\u27eb + \u27eay, y\u27eb", "@[simp]\ntheorem continuousLinearMapOfBilin_apply (v w : E) : \u27eaB\u266f v, w\u27eb = B v w", "private theorem add_left_aux3 (y z : E) :\n    \u20162 \u2022 z + y\u2016 * \u20162 \u2022 z + y\u2016 = 2 * (\u2016y + z\u2016 * \u2016y + z\u2016 + \u2016z\u2016 * \u2016z\u2016) - \u2016y\u2016 * \u2016y\u2016", "theorem Inseparable.mem_measurableSet_iff {x y : \u03b3} (h : Inseparable x y) {s : Set \u03b3}\n    (hs : MeasurableSet s) : x \u2208 s \u2194 y \u2208 s", "theorem cancel_right_div_gcd (hm : 0 < m) (h : a * c \u2261 b * c [ZMOD m]) :\n    a \u2261 b [ZMOD m / gcd m c]", "@[gcongr] protected theorem mul_left (c : \u2124) (h : a \u2261 b [ZMOD n]) : c * a \u2261 c * b [ZMOD n]", "@[gcongr] protected theorem sub_left (c : \u2124) (h : a \u2261 b [ZMOD n]) : c - a \u2261 c - b [ZMOD n]", "@[trans]\nprotected theorem trans : a \u2261 b [ZMOD n] \u2192 b \u2261 c [ZMOD n] \u2192 a \u2261 c [ZMOD n]", "theorem abs_ediv_le_abs : \u2200 a b : \u2124, |a / b| \u2264 |a|", "protected theorem add_sub_assoc (a b c : Int) : a + b - c = a + (b - c)", "theorem bitwise_diff : (bitwise fun a b => a && not b) = ldiff", "@[simp]\ntheorem bodd_add (m n : \u2124) : bodd (m + n) = xor (bodd m) (bodd n)", "@[simp, deprecated (since := \"2023-01-02\")]\ntheorem bodd_bit0 (n : \u2124) : bodd (bit0 n) = false", "theorem card_Ioc_of_le (h : a \u2264 b) : ((Ioc a b).card : \u2124) = b - a", "theorem card_fintype_Ico : Fintype.card (Set.Ico a b) = (b - a).toNat", "lemma Int.cast_injOn_of_ringChar_ne_two {R : Type*} [NonAssocRing R] [Nontrivial R]\n    (hR : ringChar R \u2260 2) : ({0, 1, -1} : Set \u2124).InjOn ((\u2191) : \u2124 \u2192 R)", "@[simp, norm_cast]\ntheorem cast_ite [AddGroupWithOne \u03b1] (P : Prop) [Decidable P] (m n : \u2124) :\n    ((ite P m n : \u2124) : \u03b1) = ite P (m : \u03b1) (n : \u03b1)", "@[simp] lemma cast_lt_zero : (n : R) < 0 \u2194 n < 0", "lemma cast_mono : Monotone (Int.cast : \u2124 \u2192 R)", "@[norm_cast]\ntheorem cast_neg_natCast {R} [DivisionRing R] (n : \u2115) : ((-n : \u2124) : R) = -n", "@[simp] lemma cast_pos : (0 : R) < n \u2194 0 < n", "@[simp, norm_cast] lemma cast_pow {R : Type*} [Ring R] (n : \u2124) (m : \u2115) :\n    \u2191(n ^ m) = (n ^ m : R)", "@[simp]\ntheorem ceil_intCast (z : \u2124) : \u2308(z : \u03b1)\u2309 = z", "@[simp]\ntheorem ceil_natCast (n : \u2115) : \u2308(n : \u03b1)\u2309 = n", "@[simp]\ntheorem ceil_ofNat (n : \u2115) [n.AtLeastTwo] : \u2308(no_index (OfNat.ofNat n : \u03b1))\u2309 = n", "@[simp]\ntheorem cobounded_eq : Bornology.cobounded \u2124 = atBot \u2294 atTop", "protected theorem coe_nat_lcm (m n : \u2115) : Int.lcm \u2191m \u2191n = Nat.lcm m n", "@[simp]\ntheorem covBy_add_one (z : \u2124) : z \u22d6 z + 1", "theorem csInf_eq_least_of_bdd {s : Set \u2124} [DecidablePred (\u00b7 \u2208 s)] (b : \u2124) (Hb : \u2200 z \u2208 s, b \u2264 z)\n    (Hinh : \u2203 z : \u2124, z \u2208 s) : sInf s = leastOfBdd b Hb Hinh", "theorem dist_eq' (m n : \u2124) : dist m n = |m - n|", "theorem div2_val : \u2200 n, div2 n = n / 2", "theorem div_eq_ediv_of_dvd {a b : Int} (h : b \u2223 a) : a.div b = a / b", "lemma dvd_mul_of_div_dvd (h : b \u2223 a) (hdiv : a / b \u2223 c) : a \u2223 b * c", "theorem dvd_sub_of_emod_eq {a b c : Int} (h : a % b = c) : b \u2223 a - c", "lemma ediv_dvd_ediv : \u2200 {a b c : \u2124}, a \u2223 b \u2192 b \u2223 c \u2192 b / a \u2223 c / a\n  | a, _, _, \u27e8b, rfl\u27e9, \u27e8c, rfl\u27e9 =>\n    if az : a = 0 then by simp [az]\n    else by\n      rw [Int.mul_ediv_cancel_left _ az, Int.mul_assoc, Int.mul_ediv_cancel_left _ az];\n        apply Int.dvd_mul_right", "theorem emod_lt_of_pos (a : Int) {b : Int} (H : 0 < b) : a % b < b", "theorem exists_strictAnti : \u2203 f : \u2124 \u2192 \u03b1, StrictAnti f", "@[simp]\ntheorem floor_fract (a : \u03b1) : \u230afract a\u230b = 0", "@[simp]\ntheorem floor_sub_ofNat (a : \u03b1) (n : \u2115) [n.AtLeastTwo] :\n    \u230aa - (no_index (OfNat.ofNat n))\u230b = \u230aa\u230b - OfNat.ofNat n", "theorem fract_eq_iff {a b : \u03b1} : fract a = b \u2194 0 \u2264 b \u2227 b < 1 \u2227 \u2203 z : \u2124, a - b = z", "@[simp]\ntheorem fract_int_add (m : \u2124) (a : \u03b1) : fract (\u2191m + a) = fract a", "theorem gcd_dvd_gcd_mul_right_right (i j k : \u2124) : gcd i j \u2223 gcd i (j * k)", "@[simp]\ntheorem gcd_zero_left (i : \u2124) : gcd 0 i = natAbs i", "theorem injOn_natAbs_Iic : InjOn natAbs (Iic 0)", "@[norm_cast, simp]\ntheorem isSquare_natCast_iff {n : \u2115} : IsSquare (n : \u2124) \u2194 IsSquare n", "theorem isUnit_iff_abs_eq {x : \u2124} : IsUnit x \u2194 abs x = 1", "theorem lcm_mul_left {m n k : \u2124} : (m * n).lcm (m * k) = natAbs m * n.lcm k", "@[simp]\ntheorem lcm_one_right (i : \u2124) : lcm i 1 = natAbs i", "@[simp]\ntheorem ldiff_bit (a m b n) : ldiff (bit a m) (bit b n) = bit (a && not b) (ldiff m n)", "lemma le_add_one_iff : m \u2264 n + 1 \u2194 m \u2264 n \u2228 m = n + 1", "theorem le_natCast_sub (m n : \u2115) : (m - n : \u2124) \u2264 \u2191(m - n : \u2115)", "theorem log_of_left_le_one {b : \u2115} (hb : b \u2264 1) (r : R) : log b r = 0", "theorem log_of_right_le_zero (b : \u2115) {r : R} (hr : r \u2264 0) : log b r = 0", "theorem lt_ceil : z < \u2308a\u2309 \u2194 (z : \u03b1) < a", "@[simp]\ntheorem lxor_bit (a m b n) : Int.xor (bit a m) (bit b n) = bit (xor a b) (Int.xor m n)", "theorem map_round (f : F) (hf : StrictMono f) (a : \u03b1) : round (f a) = round a", "theorem modEq_sub_fac {a b n : \u2124} (c : \u2124) (ha : a \u2261 b [ZMOD n]) : a - n * c \u2261 b [ZMOD n]", "theorem Int.mod_nat_eq_sub_mul_floor_rat_div {n : \u2124} {d : \u2115} : n % d = n - d * \u230a(n : \u211a) / d\u230b", "protected theorem mul_comm (a b : Int) : a * b = b * a", "theorem natAbs_coe_sub_coe_le_of_le {a b n : \u2115} (a_le_n : a \u2264 n) (b_le_n : b \u2264 n) :\n    natAbs (a - b : \u2124) \u2264 n", "@[simp] theorem natAbs_dvd_natAbs {a b : Int} : natAbs a \u2223 natAbs b \u2194 a \u2223 b", "@[simp] theorem natAbs_neg : \u2200 (a : Int), natAbs (-a) = natAbs a", "lemma natAbs_pow (n : \u2124) (k : \u2115) : Int.natAbs (n ^ k) = Int.natAbs n ^ k", "@[simp, nolint simpNF] lemma natCast_eq_zero {n : \u2115} : (n : \u2124) = 0 \u2194 n = 0", "lemma natCast_succ_pos (n : \u2115) : 0 < (n.succ : \u2124) := natCast_pos.2 n.succ_pos", "lemma ne_of_odd_add (h : Odd (m + n)) : m \u2260 n := fun hnot \u21a6 by simp [hnot, parity_simps] at h", "lemma negOnePow_eq_neg_one_iff (n : \u2124) : n.negOnePow = -1 \u2194 Odd n", "@[simp]\nlemma negOnePow_neg (n : \u2124) : (-n).negOnePow = n.negOnePow", "@[simp] lemma ofAdd_mul (a b : \u2124) : ofAdd (a * b) = ofAdd a ^ b := (toAdd_zpow ..).symm", "@[simp, norm_cast] theorem ofNat_ediv (m n : Nat) : (\u2191(m / n) : Int) = \u2191m / \u2191n", "@[simp] theorem ofNat_eq_coe : Int.ofNat n = Nat.cast n", "@[norm_cast]\nlemma ofNat_isUnit {n : \u2115} : IsUnit (n : \u2124) \u2194 IsUnit n", "@[simp] theorem ofNat_zero : ((0 : Nat) : Int) = 0", "protected lemma one_ne_zero : (1 : \u2124) \u2260 0", "@[simp] lemma pow_eq (m : \u2124) (n : \u2115) : m.pow n = m ^ n := rfl", "@[simp]\ntheorem preimage_Iic : ((\u2191) : \u2124 \u2192 \u03b1) \u207b\u00b9' Set.Iic a = Set.Iic \u230aa\u230b", "theorem prime_two : Prime (2 : \u2124)", "theorem Int.rel_of_forall_rel_succ_of_lt (r : \u03b2 \u2192 \u03b2 \u2192 Prop) [IsTrans \u03b2 r] {f : \u2124 \u2192 \u03b2}\n    (h : \u2200 n, r (f n) (f (n + 1))) \u2983a b : \u2124\u2984 (hab : a < b) : r (f a) (f b)", "theorem sq_of_coprime {a b c : \u2124} (h : IsCoprime a b) (heq : a * b = c ^ 2) :\n    \u2203 a0 : \u2124, a = a0 ^ 2 \u2228 a = -a0 ^ 2", "theorem sqrt_nonneg (n : \u2124) : 0 \u2264 sqrt n", "theorem strictMonoOn_natAbs : StrictMonoOn natAbs (Ici 0)", "lemma toNat_lt' {n : \u2115} (hn : n \u2260 0) : m.toNat < n \u2194 m < n", "@[simp]\ntheorem zero_shiftRight' (n : \u2124) : 0 >>> n = 0", "@[simp]\ntheorem AdjoinSimple.isIntegral_gen : IsIntegral F (AdjoinSimple.gen F \u03b1) \u2194 IsIntegral F \u03b1", "@[mono]\ntheorem adjoin.mono (T : Set E) (h : S \u2286 T) : adjoin F S \u2264 adjoin F T", "theorem adjoin_eq_adjoin_pow_expChar_pow_of_isSeparable' [IsSeparable F E] (S : Set E)\n    (q : \u2115) [ExpChar F q] (n : \u2115) : adjoin F S = adjoin F ((\u00b7 ^ q ^ n) '' S)", "theorem adjoin_finset_isCompactElement (S : Finset E) :\n    IsCompactElement (adjoin F S : IntermediateField F E)", "@[simp]\ntheorem adjoin_le_iff {S : Set E} {T : IntermediateField F E} : adjoin F S \u2264 T \u2194 S \u2264 T", "theorem algebraMap_mem (x : K) : algebraMap K L x \u2208 S", "theorem card_algHom_adjoin_integral (h : IsIntegral F \u03b1) (h_sep : (minpoly F \u03b1).Separable)\n    (h_splits : (minpoly F \u03b1).Splits (algebraMap F K)) :\n    @Fintype.card (F\u27ee\u03b1\u27ef \u2192\u2090[F] K) (fintypeOfAlgHomAdjoinIntegral F h) = (minpoly F \u03b1).natDegree", "@[simp]\ntheorem coe_inclusion {E F : IntermediateField K L} (hEF : E \u2264 F) (e : E) :\n    (inclusion hEF e : L) = e", "theorem finSepDegree_adjoin_simple_eq_finrank_iff (\u03b1 : E) (halg : IsAlgebraic F \u03b1) :\n    finSepDegree F F\u27ee\u03b1\u27ef = finrank F F\u27ee\u03b1\u27ef \u2194 (minpoly F \u03b1).Separable", "@[simp] theorem finrank_bot' : finrank (\u22a5 : IntermediateField F E) E = finrank F E", "theorem gc_map_comap (f :L \u2192\u2090[K] L') : GaloisConnection (map f) (comap f)", "theorem isAlgebraic_adjoin_simple {x : L} (hx : IsIntegral K x) : Algebra.IsAlgebraic K K\u27eex\u27ef", "theorem IntermediateField.isSeparable_adjoin_pair_of_separable {x y : E}\n    (hx : (minpoly F x).Separable) (hy : (minpoly F y).Separable) :\n    IsSeparable F F\u27eex, y\u27ef", "@[simp]\ntheorem mem_mk (s : Subsemiring L) (hK : \u2200 x, algebraMap K L x \u2208 s) (hi) (x : L) :\n    x \u2208 IntermediateField.mk (Subalgebra.mk s hK) hi \u2194 x \u2208 s", "@[simp]\ntheorem mem_toSubalgebra (s : IntermediateField K L) (x : L) : x \u2208 s.toSubalgebra \u2194 x \u2208 s", "protected theorem pow_mem {x : L} (hx : x \u2208 S) (n : \u2124) : x ^ n \u2208 S", "@[simp]\ntheorem restrictScalars_toSubalgebra {E : IntermediateField L' L} :\n    (E.restrictScalars K).toSubalgebra = E.toSubalgebra.restrictScalars K", "@[simp]\ntheorem sInf_toSubalgebra (S : Set (IntermediateField F E)) :\n    (sInf S).toSubalgebra = sInf (toSubalgebra '' S)", "theorem IntermediateField.sepDegree_adjoin_eq_of_isAlgebraic_of_isPurelyInseparable\n    (S : Set K) [Algebra.IsAlgebraic F (adjoin F S)] [IsPurelyInseparable F E] :\n    sepDegree E (adjoin E S) = sepDegree F (adjoin F S)", "@[simp]\ntheorem sepDegree_top : sepDegree F (\u22a4 : IntermediateField E K) = sepDegree F K", "theorem IntermediateField.splits_of_splits (h : p.Splits (algebraMap K L))\n    (hF : \u2200 x \u2208 p.rootSet L, x \u2208 F) : p.Splits (algebraMap K F)", "theorem subset_adjoin : S \u2286 adjoin F S", "theorem subsingleton_of_finrank_adjoin_le_one [FiniteDimensional F E]\n    (h : \u2200 x : E, finrank F F\u27eex\u27ef \u2264 1) : Subsingleton (IntermediateField F E)", "theorem sup_toSubalgebra_of_isAlgebraic\n    (halg : Algebra.IsAlgebraic K E1 \u2228 Algebra.IsAlgebraic K E2) :\n    (E1 \u2294 E2).toSubalgebra = E1.toSubalgebra \u2294 E2.toSubalgebra", "@[to_additive] lemma bot_ne_one : (\u22a5 : Interval \u03b1) \u2260 1 := bot_ne_pure", "@[simp, norm_cast]\ntheorem coe_coe (s : NonemptyInterval \u03b1) : ((s : Interval \u03b1) : Set \u03b1) = s", "@[simp, norm_cast]\ntheorem coe_sInf [@DecidableRel \u03b1 (\u00b7 \u2264 \u00b7)] (S : Set (Interval \u03b1)) :\n    \u2191(sInf S) = \u22c2 s \u2208 S, (s : Set \u03b1)", "@[to_additive existing (attr := simp)]\ntheorem div_bot : s / \u22a5 = \u22a5", "@[simp]\ntheorem dual_bot : dual (\u22a5 : Interval \u03b1) = \u22a5", "theorem pure_injective : Injective (pure : \u03b1 \u2192 Interval \u03b1)", "@[simp]\ntheorem div_const {\ud835\udd5c : Type*} {f : \u211d \u2192 \ud835\udd5c} [NormedField \ud835\udd5c] (h : IntervalIntegrable f \u03bc a b)\n    (c : \ud835\udd5c) : IntervalIntegrable (fun x => f x / c) \u03bc a b", "@[simp]\ntheorem _root_.IntervalIntegrable.log (hf : ContinuousOn f [[a, b]])\n    (h : \u2200 x : \u211d, x \u2208 [[a, b]] \u2192 f x \u2260 0) :\n    IntervalIntegrable (fun x => log (f x)) \u03bc a b", "theorem norm (h : IntervalIntegrable f \u03bc a b) : IntervalIntegrable (fun x => \u2016f x\u2016) \u03bc a b", "theorem InvImage.equivalence {\u03b1 : Sort u} {\u03b2 : Sort v} (r : \u03b2 \u2192 \u03b2 \u2192 Prop) (f : \u03b1 \u2192 \u03b2)\n    (h : Equivalence r) : Equivalence (InvImage r f)", "theorem Ioi_ofDual (a : \u03b1\u1d52\u1d48) : Ioi (ofDual a) = (Iio a).map ofDual.toEmbedding", "theorem Ioo_mem_nhdsWithin_Iio (H : b \u2208 Ioc a c) : Ioo a c \u2208 \ud835\udcdd[<] b", "theorem div_int (h : Irrational x) {m : \u2124} (hm : m \u2260 0) : Irrational (x / m)", "theorem ne_int (h : Irrational x) (m : \u2124) : x \u2260 m", "theorem of_rat_mul : Irrational (q * x) \u2192 Irrational x", "theorem of_sub_nat (m : \u2115) (h : Irrational (x - m)) : Irrational x", "theorem Irreducible.dvd_iff [Monoid \u03b1] {x y : \u03b1} (hx : Irreducible x) :\n    y \u2223 x \u2194 IsUnit y \u2228 Associated x y", "theorem abv_pow [Nontrivial R] (abv : R \u2192 S) [IsAbsoluteValue abv] (a : R) (n : \u2115) :\n    abv (a ^ n) = abv a ^ n", "theorem lift_self (h : IsAdjoinRoot S f) :\n    h.lift (algebraMap R S) h.root h.aeval_root = RingHom.id S", "theorem coeff_apply (h : IsAdjoinRootMonic S f) (z : S) (i : \u2115) :\n    h.coeff z i = if hi : i < natDegree f then h.basis.repr z \u27e8i, hi\u27e9 else 0", "theorem coeff_apply_le (h : IsAdjoinRootMonic S f) (z : S) (i : \u2115) (hi : natDegree f \u2264 i) :\n    h.coeff z i = 0", "theorem isIntegral_root (h : IsAdjoinRootMonic S f) : IsIntegral R h.root", "@[deprecated algebraMap_surjective_of_isIntegral (since := \"2024-05-08\")]\ntheorem algebraMap_surjective_of_isAlgebraic {k K : Type*} [Field k] [Ring K] [IsDomain K]\n    [IsAlgClosed k] [Algebra k K] [Algebra.IsAlgebraic k K] :\n    Function.Surjective (algebraMap k K)", "protected theorem eq' (hs : IsAntichain r s) {a b : \u03b1} (ha : a \u2208 s) (hb : b \u2208 s) (h : r b a) :\n    a = b", "theorem image_iso_iff [LE \u03b1] [LE \u03b2] {\u03c6 : \u03b1 \u2243o \u03b2} :\n    IsAntichain (\u00b7 \u2264 \u00b7) (\u03c6 '' s) \u2194 IsAntichain (\u00b7 \u2264 \u00b7) s", "lemma IsAntichain.interior_eq_empty [\u2200 x : \u03b1, (\ud835\udcdd[<] x).NeBot] {s : Set \u03b1}\n    (hs : IsAntichain (\u00b7 \u2264 \u00b7) s) : interior s = \u2205", "protected theorem IsAntichain.isWeakAntichain (hs : IsAntichain (\u00b7 \u2264 \u00b7) s) : IsWeakAntichain s", "theorem IsAntichain.not_lt (hs : IsAntichain (\u00b7 \u2264 \u00b7) s) (ha : a \u2208 s) (hb : b \u2208 s) : \u00aca < b", "theorem preimage_relIso {t : Set \u03b2} (hs : IsAntichain r' t) (\u03c6 : r \u2243r r') :\n    IsAntichain r (\u03c6 \u207b\u00b9' t)", "theorem swap (hs : IsAntichain r s) : IsAntichain (swap r) s", "theorem to_dual_iff [LE \u03b1] : IsAntichain (\u00b7 \u2264 \u00b7) s \u2194 @IsAntichain \u03b1\u1d52\u1d48 (\u00b7 \u2264 \u00b7) s", "theorem IsAntichain.volume_eq_zero [Nonempty \u03b9] (hs : IsAntichain (\u00b7 \u2264 \u00b7) s) : volume s = 0", "theorem IsAssociatedPrime.annihilator_le (h : IsAssociatedPrime I M) :\n    (\u22a4 : Submodule R M).annihilator \u2264 I", "nonrec theorem IsBaseChange.lift_eq (g : M \u2192\u2097[R] Q) (x : M) : h.lift g (f x) = g x", "theorem IsBaseChange.of_lift_unique\n    (h : \u2200 (Q : Type max v\u2081 v\u2082 v\u2083) [AddCommMonoid Q],\n      \u2200 [Module R Q] [Module S Q], \u2200 [IsScalarTower R S Q],\n        \u2200 g : M \u2192\u2097[R] Q, \u2203! g' : N \u2192\u2097[S] Q, (g'.restrictScalars R).comp f = g) :\n    IsBaseChange S f", "theorem gcd_dvd_right : gcd x y \u2223 y", "@[fun_prop]\ntheorem IsBoundedBilinearMap.differentiable (h : IsBoundedBilinearMap \ud835\udd5c b) : Differentiable \ud835\udd5c b", "@[fun_prop]\ntheorem IsBoundedLinearMap.differentiableOn (h : IsBoundedLinearMap \ud835\udd5c f) : DifferentiableOn \ud835\udd5c f s", "@[fun_prop]\ntheorem IsBoundedLinearMap.hasFDerivAt (h : IsBoundedLinearMap \ud835\udd5c f) :\n    HasFDerivAt f h.toContinuousLinearMap x", "lemma const (x : \u03b2) : IsCauSeq abv fun _ \u21a6 x", "lemma of_abv_le (n : \u2115) (hm : \u2200 m, n \u2264 m \u2192 abv (f m) \u2264 a m) :\n    IsCauSeq abs (fun n \u21a6 \u2211 i \u2208 range n, a i) \u2192 IsCauSeq abv fun n \u21a6 \u2211 i \u2208 range n, f i", "theorem IsChain.mono_rel {r' : \u03b1 \u2192 \u03b1 \u2192 Prop} (h : IsChain r s) (h_imp : \u2200 x y, r x y \u2192 r' x y) :\n    IsChain r' s", "theorem IsClosed.Icc_subset_of_forall_exists_gt {a b : \u03b1} {s : Set \u03b1} (hs : IsClosed (s \u2229 Icc a b))\n    (ha : a \u2208 s) (hgt : \u2200 x \u2208 s \u2229 Ico a b, \u2200 y \u2208 Ioi x, (s \u2229 Ioc x y).Nonempty) : Icc a b \u2286 s", "theorem IsClosed.inter (h\u2081 : IsClosed s\u2081) (h\u2082 : IsClosed s\u2082) : IsClosed (s\u2081 \u2229 s\u2082)", "theorem IsClosed.isCompact [CompactSpace X] (h : IsClosed s) : IsCompact s", "theorem IsClosed.measurableSet (h : IsClosed s) : MeasurableSet s", "lemma IsCoatomic.exists_coatom [OrderTop \u03b1] [Nontrivial \u03b1] [IsCoatomic \u03b1] : \u2203 a : \u03b1, IsCoatom a", "protected theorem IsCompact.closure {K : Set X} (hK : IsCompact K) : IsCompact (closure K)", "theorem IsCompact.compl_mem_sets_of_nhdsWithin (hs : IsCompact s) {f : Filter X}\n    (hf : \u2200 x \u2208 s, \u2203 t \u2208 \ud835\udcdd[s] x, t\u1d9c \u2208 f) : s\u1d9c \u2208 f", "theorem IsCompact.disjoint_nhdsSet_left {l : Filter X} (hs : IsCompact s) :\n    Disjoint (\ud835\udcdd\u02e2 s) l \u2194 \u2200 x \u2208 s, Disjoint (\ud835\udcdd x) l", "theorem IsCompact.exists_forall_le' [ClosedIicTopology \u03b1] [NoMaxOrder \u03b1] {f : \u03b2 \u2192 \u03b1}\n    {s : Set \u03b2} (hs : IsCompact s) (hf : ContinuousOn f s) {a : \u03b1} (hf' : \u2200 b \u2208 s, a < f b) :\n    \u2203 a', a < a' \u2227 \u2200 b \u2208 s, a' \u2264 f b", "theorem _root_.IsCompact.exists_infDist_eq_dist (h : IsCompact s) (hne : s.Nonempty) (x : \u03b1) :\n    \u2203 y \u2208 s, infDist x s = dist x y", "theorem IsCompact.exists_isOpen_closure_subset {K U : Set X} (hK : IsCompact K) (hU : U \u2208 \ud835\udcdd\u02e2 K) :\n    \u2203 V, IsOpen V \u2227 K \u2286 V \u2227 closure V \u2286 U", "theorem IsCompact.exists_sSup_image_eq_and_ge [ClosedIciTopology \u03b1] {s : Set \u03b2}\n    (hs : IsCompact s) (ne_s : s.Nonempty) {f : \u03b2 \u2192 \u03b1} (hf : ContinuousOn f s) :\n    \u2203 x \u2208 s, sSup (f '' s) = f x \u2227 \u2200 y \u2208 s, f y \u2264 f x", "theorem IsCompact.image {f : X \u2192 Y} (hs : IsCompact s) (hf : Continuous f) : IsCompact (f '' s)", "theorem IsCompact.inter_left (ht : IsCompact t) (hs : IsClosed s) : IsCompact (s \u2229 t)", "protected lemma _root_.IsCompact.measure_eq_iInf_isOpen [InnerRegularCompactLTTop \u03bc]\n    [IsLocallyFiniteMeasure \u03bc] [R1Space \u03b1] [BorelSpace \u03b1] {K : Set \u03b1} (hK : IsCompact K) :\n    \u03bc K = \u2a05 (U : Set \u03b1) (_ : K \u2286 U) (_ : IsOpen U), \u03bc U", "theorem IsCompact.nonempty_sInter_of_directed_nonempty_isCompact_isClosed\n    {S : Set (Set X)} [hS : Nonempty S] (hSd : DirectedOn (\u00b7 \u2287 \u00b7) S) (hSn : \u2200 U \u2208 S, U.Nonempty)\n    (hSc : \u2200 U \u2208 S, IsCompact U) (hScl : \u2200 U \u2208 S, IsClosed U) : (\u22c2\u2080 S).Nonempty", "theorem IsCompactOperator.continuous_comp {f : M\u2081 \u2192 M\u2082} (hf : IsCompactOperator f) {g : M\u2082 \u2192 M\u2083}\n    (hg : Continuous g) : IsCompactOperator (g \u2218 f)", "theorem IsCompactOperator.image_closedBall_subset_compact [ContinuousConstSMul \ud835\udd5c\u2082 M\u2082]\n    {f : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082} (hf : IsCompactOperator f) (r : \u211d) :\n    \u2203 K : Set M\u2082, IsCompact K \u2227 f '' Metric.closedBall 0 r \u2286 K", "theorem IsCompactOperator.image_subset_compact_of_bounded [ContinuousConstSMul \ud835\udd5c\u2082 M\u2082]\n    {f : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082} (hf : IsCompactOperator f) {S : Set M\u2081} (hS : Bornology.IsBounded S) :\n    \u2203 K : Set M\u2082, IsCompact K \u2227 f '' S \u2286 K", "theorem inf_left_eq_bot_iff (h : IsCompl y z) : x \u2293 y = \u22a5 \u2194 x \u2264 z", "theorem isCoatom_iff_isAtom : IsCoatom a \u2194 IsAtom b", "@[symm]\nprotected theorem symm (h : IsCompl x y) : IsCompl y x", "theorem IsCompl.symmDiff_eq_top {a b : \u03b1} (h : IsCompl a b) : a \u2206 b = \u22a4", "theorem IsConnected.union {s t : Set \u03b1} (H : (s \u2229 t).Nonempty) (Hs : IsConnected s)\n    (Ht : IsConnected t) : IsConnected (s \u222a t)", "theorem add_mul_left_left_iff {x y z : R} : IsCoprime (x + y * z) y \u2194 IsCoprime x y", "theorem add_mul_right_right {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime x (y + z * x)", "theorem IsCoprime.mul_dvd (H : IsCoprime x y) (H1 : x \u2223 z) (H2 : y \u2223 z) : x * y \u2223 z", "theorem IsCoprime.ne_zero [Nontrivial R] {p : Fin 2 \u2192 R} (h : IsCoprime (p 0) (p 1)) : p \u2260 0", "theorem five_pid [IsCyclotomicExtension {5} \u211a K] : IsPrincipalIdealRing (\ud835\udcde K)", "theorem discr_odd_prime [IsCyclotomicExtension {p} K L] [hp : Fact (p : \u2115).Prime]\n    (h\u03b6 : IsPrimitiveRoot \u03b6 p) (hirr : Irreducible (cyclotomic p K)) (hodd : p \u2260 2) :\n    discr K (h\u03b6.powerBasis K).basis = (-1) ^ (((p : \u2115) - 1) / 2) * p ^ ((p : \u2115) - 2)", "protected\ntheorem neZero' [IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : \u2115) : A)", "theorem norm_zeta_sub_one_of_isPrimePow (hn : IsPrimePow (n : \u2115)) [IsCyclotomicExtension {n} K L]\n    (hirr : Irreducible (cyclotomic (n : \u2115) K)) (h : n \u2260 2) :\n    norm K (zeta n K L - 1) = (n : \u2115).minFac", "theorem norm_zeta_sub_one_of_prime_ne_two [Fact (p : \u2115).Prime]\n    [IsCyclotomicExtension {p} K L] (hirr : Irreducible (cyclotomic p K)) (h : p \u2260 2) :\n    norm K (zeta p K L - 1) = p", "theorem numberField [h : NumberField K] [Finite S] [IsCyclotomicExtension S K L] : NumberField L", "@[simp]\ntheorem coe_smul_adicCompletionIntegers (r : R) (x : v.adicCompletionIntegers K) :\n    (\u2191(r \u2022 x) : v.adicCompletion K) = r \u2022 (x : v.adicCompletion K)", "@[simp]\ntheorem valuationOfNeZeroToFun_eq (x : K\u02e3) :\n    (v.valuationOfNeZeroToFun x : \u2124\u2098\u2080) = v.valuation (x : K)", "theorem valuation_def (x : K) :\n    v.valuation x =\n      v.intValuation.extendToLocalization\n        (fun r hr => Set.mem_compl (v.int_valuation_ne_zero' \u27e8r, hr\u27e9)) K x", "theorem valuation_uniformizer_ne_zero : Classical.choose (v.valuation_exists_uniformizer K) \u2260 0", "theorem IsDedekindDomain.inf_prime_pow_eq_prod {\u03b9 : Type*} (s : Finset \u03b9) (f : \u03b9 \u2192 Ideal R)\n    (e : \u03b9 \u2192 \u2115) (prime : \u2200 i \u2208 s, Prime (f i))\n    (coprime : \u2200\u1d49 (i \u2208 s) (j \u2208 s), i \u2260 j \u2192 f i \u2260 f j) :\n    (s.inf fun i => f i ^ e i) = \u220f i \u2208 s, f i ^ e i", "theorem IsDedekindDomain.isPrincipalIdealRing_localization_over_prime [IsDomain S] :\n    IsPrincipalIdealRing S\u209a", "theorem IsEquivalent.summable_iff_nat {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n    [FiniteDimensional \u211d E] {f : \u2115 \u2192 E} {g : \u2115 \u2192 E} (h : f ~[atTop] g) : Summable f \u2194 Summable g", "protected theorem mono (hC : IsExposed \ud835\udd5c A C) (hBA : B \u2286 A) (hCB : C \u2286 B) : IsExposed \ud835\udd5c B C", "theorem IsExtrOn.inter (hf : IsExtrOn f s a) (t) : IsExtrOn f (s \u2229 t) a", "theorem IsExtrOn.localize (hf : IsExtrOn f s a) : IsLocalExtrOn f s a", "theorem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem {t : Set \u03b1} {x : \u03b1}\n    [IsCountablyGenerated (\ud835\udcdd x)] (htx : IsGLB t x) (not_mem : x \u2209 t) (ht : t.Nonempty) :\n    \u2203 u : \u2115 \u2192 \u03b1, StrictAnti u \u2227 (\u2200 n, x < u n) \u2227 Tendsto u atTop (\ud835\udcdd x) \u2227 \u2200 n, u n \u2208 t", "theorem IsGLB.mem_lowerBounds_of_tendsto [Preorder \u03b3] [TopologicalSpace \u03b3] [OrderClosedTopology \u03b3]\n    {f : \u03b1 \u2192 \u03b3} {s : Set \u03b1} {a : \u03b1} {b : \u03b3} (hf : MonotoneOn f s) (ha : IsGLB s a)\n    (hb : Tendsto f (\ud835\udcdd[s] a) (\ud835\udcdd b)) : b \u2208 lowerBounds (f '' s)", "theorem IsGLB.mono (ha : IsGLB s a) (hb : IsGLB t b) (hst : s \u2286 t) : b \u2264 a", "theorem card_aut_eq_finrank [FiniteDimensional F E] [IsGalois F E] :\n    Fintype.card (E \u2243\u2090[F] E) = finrank F E", "theorem is_separable_splitting_field [FiniteDimensional F E] [IsGalois F E] :\n    \u2203 p : F[X], p.Separable \u2227 p.IsSplittingField F E", "theorem separable [IsGalois F E] (x : E) : (minpoly F x).Separable", "theorem IsGreatest.union [LinearOrder \u03b3] {a b : \u03b3} {s t : Set \u03b3} (ha : IsGreatest s a)\n    (hb : IsGreatest t b) : IsGreatest (s \u222a t) (max a b)", "theorem IsGreatest.unique (Ha : IsGreatest s a) (Hb : IsGreatest s b) : a = b", "theorem IsG\u03b4.biInter {s : Set \u03b9} (hs : s.Countable) {t : \u2200 i \u2208 s, Set X}\n    (ht : \u2200 (i) (hi : i \u2208 s), IsG\u03b4 (t i hi)) : IsG\u03b4 (\u22c2 i \u2208 s, t i \u2039_\u203a)", "theorem IsG\u03b4.biUnion {s : Set \u03b9} (hs : s.Finite) {f : \u03b9 \u2192 Set X} (h : \u2200 i \u2208 s, IsG\u03b4 (f i)) :\n    IsG\u03b4 (\u22c3 i \u2208 s, f i)", "theorem IsG\u03b4.dense_biUnion_interior_of_closed {t : Set \u03b1} {s : Set X} (hs : IsG\u03b4 s) (hd : Dense s)\n    (ht : t.Countable) {f : \u03b1 \u2192 Set X} (hc : \u2200 i \u2208 t, IsClosed (f i)) (hU : s \u2286 \u22c3 i \u2208 t, f i) :\n    Dense (\u22c3 i \u2208 t, interior (f i))", "theorem IsG\u03b4.iInter_of_isOpen [Countable \u03b9'] {f : \u03b9' \u2192 Set X} (hf : \u2200 i, IsOpen (f i)) :\n    IsG\u03b4 (\u22c2 i, f i)", "lemma IsHereditarilyLindelof.isLindelof_subset (hs : IsHereditarilyLindelof s) (ht : t \u2286 s) :\n    IsLindelof t := hs t ht", "theorem eq {p : M} (h : IsIdempotentElem p) : p * p = p", "theorem IsIntegral.of_finite [Module.Finite R B] (x : B) : IsIntegral R x", "@[simp]\ntheorem integralClosure_eq_bot [NoZeroSMulDivisors R A] [Nontrivial A] : integralClosure R A = \u22a5", "@[simp]\ntheorem IsInvariantSubring.coe_subtypeHom' :\n    ((IsInvariantSubring.subtypeHom M U).toRingHom : U \u2192+* R') = U.subtype", "theorem IsLUB.exists_between (h : IsLUB s a) (hb : b < a) : \u2203 c \u2208 s, b < c \u2227 c \u2264 a", "theorem IsLeast.csInf_eq (H : IsLeast s a) : sInf s = a", "protected theorem IsLeast.insert [LinearOrder \u03b3] (a) {b} {s : Set \u03b3} (hs : IsLeast s b) :\n    IsLeast (insert a s) (min a b)", "theorem IsLeast.nonempty (h : IsLeast s a) : s.Nonempty", "@[simp] lemma IsLeftRegular.mul_left_eq_zero_iff (hb : IsLeftRegular b) : b * a = 0 \u2194 a = 0", "theorem IsLeftRegular.pow_iff {n : \u2115} (n0 : 0 < n) : IsLeftRegular (a ^ n) \u2194 IsLeftRegular a", "theorem IsLeftRegular.subsingleton (h : IsLeftRegular (0 : R)) : Subsingleton R", "theorem IsLindelof.elim_countable_subcover {\u03b9 : Type v} (hs : IsLindelof s) (U : \u03b9 \u2192 Set X)\n    (hUo : \u2200 i, IsOpen (U i)) (hsU : s \u2286 \u22c3 i, U i) :\n    \u2203 r : Set \u03b9, r.Countable \u2227 (s \u2286 \u22c3 i \u2208 r, U i)", "theorem IsLindelof.elim_nhds_subcover (hs : IsLindelof s) (U : X \u2192 Set X)\n    (hU : \u2200 x \u2208 s, U x \u2208 \ud835\udcdd x) :\n    \u2203 t : Set X, t.Countable \u2227 (\u2200 x \u2208 t, x \u2208 s) \u2227 s \u2286 \u22c3 x \u2208 t, U x", "theorem IsLindelof.indexed_countable_subcover {\u03b9 : Type v} [Nonempty \u03b9]\n    (hs : IsLindelof s) (U : \u03b9 \u2192 Set X) (hUo : \u2200 i, IsOpen (U i)) (hsU : s \u2286 \u22c3 i, U i) :\n    \u2203 f : \u2115 \u2192 \u03b9, s \u2286 \u22c3 n, U (f n)", "nonrec theorem IsLocalExtr.neg (hf : IsLocalExtr f a) : IsLocalExtr (fun x => -f x) a", "nonrec theorem IsLocalExtrOn.comp_antitone (hf : IsLocalExtrOn f s a) {g : \u03b2 \u2192 \u03b3}\n    (hg : Antitone g) : IsLocalExtrOn (g \u2218 f) s a", "theorem mk (h : \u2200 x : X, \u2203 e : PartialHomeomorph X Y, x \u2208 e.source \u2227 Set.EqOn f e e.source) :\n    IsLocalHomeomorph f", "nonrec theorem IsLocalMax.comp_mono (hf : IsLocalMax f a) {g : \u03b2 \u2192 \u03b3} (hg : Monotone g) :\n    IsLocalMax (g \u2218 f) a", "nonrec theorem IsLocalMax.min (hf : IsLocalMax f a) (hg : IsLocalMax g a) :\n    IsLocalMax (fun x => min (f x) (g x)) a", "theorem IsLocalMaxOn.isLocalMax (hf : IsLocalMaxOn f s a) (hs : s \u2208 \ud835\udcdd a) : IsLocalMax f a", "nonrec theorem IsLocalMin.neg (hf : IsLocalMin f a) : IsLocalMax (fun x => -f x) a", "lemma sec_spec (s : S) : s * (algebraMap R S) (x ^ (IsLocalization.Away.sec x s).2) =\n    algebraMap R S (IsLocalization.Away.sec x s).1", "@[simp]\ntheorem coeSubmodule_span (s : Set R) :\n    coeSubmodule S (Ideal.span s) = Submodule.span R (algebraMap R S '' s)", "@[mono]\ntheorem coeSubmodule_strictMono (h : M \u2264 nonZeroDivisors R) :\n    StrictMono (coeSubmodule S : Ideal R \u2192 Submodule R S)", "theorem coeffIntegerNormalization_mem_support (p : S[X]) (i : \u2115)\n    (h : coeffIntegerNormalization M p i \u2260 0) : i \u2208 p.support", "theorem eq_of_eq {g : R \u2192+* P} (hg : \u2200 y : M, IsUnit (g y)) {x y}\n    (h : (algebraMap R S) x = (algebraMap R S) y) : g x = g y", "theorem isUnit_comp (j : S \u2192+* P) (y : M) : IsUnit (j.comp (algebraMap R S) y)", "@[simp]\ntheorem lift_eq (x : R) : lift hg ((algebraMap R S) x) = g x", "theorem IsLocalization.lift_mem_adjoin_finsetIntegerMultiple [Algebra R S] [Algebra R S']\n    [IsScalarTower R S S'] [IsLocalization (M.map (algebraMap R S)) S'] (x : S) (s : Finset S')\n    (hx : algebraMap S S' x \u2208 Algebra.adjoin R (s : Set S')) :\n    \u2203 m : M, m \u2022 x \u2208\n      Algebra.adjoin R\n        (IsLocalization.finsetIntegerMultiple (M.map (algebraMap R S)) s : Set S)", "theorem localization_isScalarTower_of_submonoid_le (M N : Submonoid R) (h : M \u2264 N)\n    [IsLocalization M S] [IsLocalization N T] :\n    @IsScalarTower R S T _ (localizationAlgebraOfSubmonoidLe S T M N h).toSMul _", "theorem map_injective_of_injective (h : Function.Injective g) [IsLocalization (M.map g) Q] :\n    Function.Injective (map Q g M.le_comap_map : S \u2192 Q)", "@[simp]\ntheorem mk'_self' {x : M} : mk' S (x : R) x = 1", "theorem span_invSubmonoid : Submodule.span R (invSubmonoid M S : Set S) = \u22a4", "theorem iso_symm_apply' (m : M') (a : M) (b : S) (eq1 : b \u2022 m = f a) :\n    (iso S f).symm m = LocalizedModule.mk a b", "theorem iso_symm_comp : (iso S f).symm.toLinearMap.comp f = LocalizedModule.mkLinearMap S M", "@[simp]\ntheorem lift_apply (g : M \u2192\u2097[R] M'') (h) (x) :\n    lift S f g h (f x) = g x", "lemma map_LocalizedModules (g : M\u2080 \u2192\u2097[R] M\u2081) (m : M\u2080) (s : S) :\n    ((map S (mkLinearMap S M\u2080) (mkLinearMap S M\u2081)) g)\n    (LocalizedModule.mk m s) = LocalizedModule.mk (g m) s", "@[simp]\ntheorem mk'_cancel' (m : M) (s : S) : s \u2022 mk' f m s = f m", "theorem mk'_sub_mk' {M M' : Type*} [AddCommGroup M] [AddCommGroup M'] [Module R M] [Module R M']\n    (f : M \u2192\u2097[R] M') [IsLocalizedModule S f] (m\u2081 m\u2082 : M) (s\u2081 s\u2082 : S) :\n    mk' f m\u2081 s\u2081 - mk' f m\u2082 s\u2082 = mk' f (s\u2082 \u2022 m\u2081 - s\u2081 \u2022 m\u2082) (s\u2081 * s\u2082)", "theorem prod_mk {Y'} {f : X \u2192 Y} {f' : X \u2192 Y'} (hf : IsLocallyConstant f)\n    (hf' : IsLocallyConstant f') : IsLocallyConstant fun x => (f x, f' x)", "theorem range_finite [CompactSpace X] {f : X \u2192 Y} (hf : IsLocallyConstant f) :\n    (Set.range f).Finite", "theorem IsLocallyInjective.isOpen_eqLocus (he : f \u2218 g\u2081 = f \u2218 g\u2082) : IsOpen {a | g\u2081 a = g\u2082 a}", "lemma IsLowerSet.isUpperSet_preimage_coe (hs : IsLowerSet s) :\n    IsUpperSet ((\u2191) \u207b\u00b9' t : Set s) \u2194 \u2200 b \u2208 s, \u2200 c \u2208 t, c \u2264 b \u2192 b \u2208 t", "theorem IsLowerSet.nonMemberSubfamily (h : IsLowerSet (\ud835\udc9c : Set (Finset \u03b1))) :\n    IsLowerSet (\ud835\udc9c.nonMemberSubfamily a : Set (Finset \u03b1))", "theorem IsLowerSet.null_frontier (hs : IsLowerSet s) : volume (frontier s) = 0", "theorem distrib_lattice_lemma [FaithfulSMul M X] {P Q R : { P : M // IsLprojection X P }} :\n    ((\u2191P : M) + \u2191P\u1d9c * R) * (\u2191P + \u2191Q * \u2191R * \u2191P\u1d9c) = \u2191P + \u2191Q * \u2191R * \u2191P\u1d9c", "theorem mul_compl_self {P : { P : M // IsLprojection X P }} : (\u2191P : M) * \u2191P\u1d9c = 0", "theorem IsMaxChain.isChain (h : IsMaxChain r s) : IsChain r s", "theorem IsMaxChain.range_fin_of_covBy (h0 : f 0 = \u22a5) (hlast : f (.last n) = \u22a4)\n    (hcovBy : \u2200 k : Fin n, f k.castSucc \u2a7f f k.succ) :\n    IsMaxChain (\u00b7 \u2264 \u00b7) (range f)", "theorem IsMaxChain.top_mem [LE \u03b1] [OrderTop \u03b1] (h : IsMaxChain (\u00b7 \u2264 \u00b7) s) : \u22a4 \u2208 s", "theorem IsMaxFilter.comp_antitone (hf : IsMaxFilter f l a) {g : \u03b2 \u2192 \u03b3} (hg : Antitone g) :\n    IsMinFilter (g \u2218 f) l a", "theorem IsMaxOn.isExtr (h : IsMaxOn f s a) : IsExtrOn f s a", "theorem IsMaxOn.norm_add_sameRay (h : IsMaxOn (norm \u2218 f) s c) (hy : SameRay \u211d (f c) y) :\n    IsMaxOn (fun x => \u2016f x + y\u2016) s c", "theorem IsMaxOn.on_preimage (g : \u03b4 \u2192 \u03b1) {b : \u03b4} (hf : IsMaxOn f s (g b)) :\n    IsMaxOn (f \u2218 g) (g \u207b\u00b9' s) b", "theorem IsMinFilter.comp_tendsto {g : \u03b4 \u2192 \u03b1} {l' : Filter \u03b4} {b : \u03b4} (hf : IsMinFilter f l (g b))\n    (hg : Tendsto g l' l) : IsMinFilter (f \u2218 g) l' b", "@[to_additive\n      \"The negation of a map which preserves addition, preserves addition\n      when the target is commutative.\"]\ntheorem inv {\u03b1 \u03b2} [MulOneClass \u03b1] [CommGroup \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : IsMonoidHom f) :\n    IsMonoidHom fun a => (f a)\u207b\u00b9", "@[to_additive]\nprotected theorem left_comm (h : IsMulCentral a) (b c) : a * (b * c) = b * (a * c)", "@[to_additive] lemma IsMulFreimanHom.div {\u03b2 : Type*} [DivisionCommMonoid \u03b2] {B\u2081 B\u2082 : Set \u03b2}\n    {f\u2081 f\u2082 : \u03b1 \u2192 \u03b2} (h\u2081 : IsMulFreimanHom n A B\u2081 f\u2081) (h\u2082 : IsMulFreimanHom n A B\u2082 f\u2082) :\n    IsMulFreimanHom n A (B\u2081 / B\u2082) (f\u2081 / f\u2082) where\n  mapsTo a ha := div_mem_div (h\u2081.mapsTo ha) (h\u2082.mapsTo ha)\n  map_prod_eq_map_prod s t hsA htA hs ht h", "@[to_additive]\nlemma IsMulFreimanHom.mul_eq_mul (hf : IsMulFreimanHom 2 A B f) {a b c d : \u03b1}\n    (ha : a \u2208 A) (hb : b \u2208 A) (hc : c \u2208 A) (hd : d \u2208 A) (h : a * b = c * d) :\n    f a * f b = f c * f d", "@[to_additive]\nlemma IsMulFreimanIso.threeGPFree_congr (hf : IsMulFreimanIso 2 s t f) :\n    ThreeGPFree s \u2194 ThreeGPFree t where\n  mpr := hf.isMulFreimanHom.threeGPFree hf.bijOn.injOn\n  mp hs a hfa b hfb c hfc habc", "theorem IsNilpotent.isUnit_sub_one [Ring R] {r : R} (hnil : IsNilpotent r) : IsUnit (r - 1)", "theorem  IsNilpotent.of_pow [MonoidWithZero R] {x : R} {m : \u2115}\n    (h : IsNilpotent (x ^ m)) : IsNilpotent x", "lemma IsNilpotent.smul [MonoidWithZero R] [MonoidWithZero S] [MulActionWithZero R S]\n    [SMulCommClass R S S] [IsScalarTower R S S] {a : S} (ha : IsNilpotent a) (t : R) :\n    IsNilpotent (t \u2022 a)", "@[simp]\ntheorem range_finsetBasis [IsNoetherian K V] :\n    Set.range (finsetBasis K V) = Basis.ofVectorSpaceIndex K V", "lemma IsNowhereDense.subset_of_closed_isNowhereDense {s : Set X} (hs : IsNowhereDense s) :\n    \u2203 t : Set X, s \u2286 t \u2227 IsNowhereDense t \u2227 IsClosed t", "@[to_additive]\nprotected lemma IsOfFinOrder.orderOf_pos (h : IsOfFinOrder x) : 0 < orderOf x", "theorem IsOpen.closure_inter (h : IsOpen t) : closure s \u2229 t \u2286 closure (s \u2229 t)", "@[to_additive]\ntheorem IsOpen.div_right (hs : IsOpen s) : IsOpen (s / t)", "theorem IsOpen.exists_smooth_support_eq {s : Set E} (hs : IsOpen s) :\n    \u2203 f : E \u2192 \u211d, f.support = s \u2227 ContDiff \u211d \u22a4 f \u2227 Set.range f \u2286 Set.Icc 0 1", "theorem IsOpen.interior_eq (h : IsOpen s) : interior s = s", "theorem IsOpen.reProdIm (hs : IsOpen s) (ht : IsOpen t) : IsOpen (s \u00d7\u2102 t)", "theorem IsOpen.uniqueMDiffOn (hs : IsOpen s) : UniqueMDiffOn I s", "theorem IsOpen.uniqueMDiffWithinAt (hs : IsOpen s) (xs : x \u2208 s) : UniqueMDiffWithinAt I s x", "protected theorem IsOpen.upperClosure : IsOpen s \u2192 IsOpen (upperClosure s : Set \u03b1)", "theorem isOpen_range (hf : IsOpenMap f) : IsOpen (range f)", "theorem nhds_le (hf : IsOpenMap f) (x : X) : \ud835\udcdd (f x) \u2264 (\ud835\udcdd x).map f", "theorem of_bot : IsPGroup p (\u22a5 : Subgroup G)", "@[simp]\ntheorem powEquiv_apply {n : \u2115} (hn : p.Coprime n) (g : G) : hG.powEquiv hn g = g ^ n", "theorem IsPRadical.comap_pNilradical [IsPRadical i p] :\n    (pNilradical L p).comap i = pNilradical K p", "theorem IsPRadical.isPurelyInseparable [IsPRadical (algebraMap K L) p] :\n    IsPurelyInseparable K L", "theorem IsPathConnected.exists_path_through_family' {n : \u2115}\n    {s : Set X} (h : IsPathConnected s) (p : Fin (n + 1) \u2192 X) (hp : \u2200 i, p i \u2208 s) :\n    \u2203 (\u03b3 : Path (p 0) (p n)) (t : Fin (n + 1) \u2192 I), (\u2200 t, \u03b3 t \u2208 s) \u2227 \u2200 i, \u03b3 (t i) = p i", "theorem equiv_self_apply (x : L) : equiv i i p x = x", "theorem IsPicardLindelof.exists_forall_hasDerivWithinAt_Icc_eq [CompleteSpace E] {v : \u211d \u2192 E \u2192 E}\n    {tMin t\u2080 tMax : \u211d} (x\u2080 : E) {C R : \u211d} {L : \u211d\u22650}\n    (hpl : IsPicardLindelof v tMin t\u2080 tMax x\u2080 L R C) :\n    \u2203 f : \u211d \u2192 E, f t\u2080 = x\u2080 \u2227\n      \u2200 t \u2208 Icc tMin tMax, HasDerivWithinAt f (v t (f t)) (Icc tMin tMax) t", "theorem IsPreconnected.intermediate_value_Iii {s : Set X} (hs : IsPreconnected s) {l\u2081 l\u2082 : Filter X}\n    [NeBot l\u2081] [NeBot l\u2082] (hl\u2081 : l\u2081 \u2264 \ud835\udcdf s) (hl\u2082 : l\u2082 \u2264 \ud835\udcdf s) {f : X \u2192 \u03b1} (hf : ContinuousOn f s)\n    (ht\u2081 : Tendsto f l\u2081 atBot) (ht\u2082 : Tendsto f l\u2082 atTop) : univ \u2286 f '' s", "theorem IsPreconnected.subset_left_of_subset_union (hu : IsOpen u) (hv : IsOpen v)\n    (huv : Disjoint u v) (hsuv : s \u2286 u \u222a v) (hsu : (s \u2229 u).Nonempty) (hs : IsPreconnected s) :\n    s \u2286 u", "theorem IsPreirreducible.isPreconnected {s : Set \u03b1} (H : IsPreirreducible s) : IsPreconnected s", "theorem IsPreirreducible.open_subset {U : Set X} (ht : IsPreirreducible t) (hU : IsOpen U)\n    (hU' : U \u2286 t) : IsPreirreducible U", "theorem IsPrimePow.dvd {n m : \u2115} (hn : IsPrimePow n) (hm : m \u2223 n) (hm\u2081 : m \u2260 1) : IsPrimePow m", "theorem eq_pow_of_pow_eq_one {k : \u2115} {\u03b6 \u03be : R} (h : IsPrimitiveRoot \u03b6 k) (h\u03be : \u03be ^ k = 1)\n    (h0 : 0 < k) : \u2203 i < k, \u03b6 ^ i = \u03be", "theorem exists_neg_pow_of_isOfFinOrder [NumberField K] [IsCyclotomicExtension {n} \u211a K]\n    (hno : Odd (n : \u2115)) {\u03b6 x : K} (h\u03b6 : IsPrimitiveRoot \u03b6 n) (hx : IsOfFinOrder x) :\n    \u2203 r : \u2115, x = (-\u03b6) ^ r", "theorem mk_of_lt (\u03b6 : M) (hk : 0 < k) (h1 : \u03b6 ^ k = 1) (h : \u2200 l : \u2115, 0 < l \u2192 l < k \u2192 \u03b6 ^ l \u2260 1) :\n    IsPrimitiveRoot \u03b6 k", "theorem IsPrimitiveRoot.norm'_eq_one {\u03b6 : \u2102} {n : \u2115} (h : IsPrimitiveRoot \u03b6 n) (hn : n \u2260 0) :\n    \u2016\u03b6\u2016 = 1", "theorem norm_eq_neg_one_pow (h\u03b6 : IsPrimitiveRoot \u03b6 2) [IsDomain L] :\n    norm K \u03b6 = (-1 : K) ^ finrank K L", "theorem nthRoots_nodup {\u03b6 : R} {n : \u2115} (h : IsPrimitiveRoot \u03b6 n) {a : R} (ha : a \u2260 0) :\n    (nthRoots n a).Nodup", "theorem nthRoots_one_nodup {\u03b6 : R} {n : \u2115} (h : IsPrimitiveRoot \u03b6 n) :\n    (nthRoots n (1 : R)).Nodup", "protected theorem orderOf (\u03b6 : M) : IsPrimitiveRoot \u03b6 (orderOf \u03b6)", "theorem powerBasis_gen_mem_adjoin_zeta_sub_one :\n    (h\u03b6.powerBasis K).gen \u2208 adjoin K ({\u03b6 - 1} : Set L)", "lemma prime_norm_toInteger_sub_one_of_prime_pow_ne_two [IsCyclotomicExtension {p ^ (k + 1)} \u211a K]\n    (h\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))) (htwo : p ^ (k + 1) \u2260 2) :\n    Prime (Algebra.norm \u2124 (h\u03b6.toInteger - 1))", "@[simp]\ntheorem subOneIntegralPowerBasis_gen [IsCyclotomicExtension {p ^ k} \u211a K]\n    (h\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ k)) :\n    h\u03b6.subOneIntegralPowerBasis.gen =\n      \u27e8\u03b6 - 1, Subalgebra.sub_mem _ (h\u03b6.isIntegral (p ^ k).pos) (Subalgebra.one_mem _)\u27e9", "lemma toInteger_sub_one_dvd_prime [hcycl : IsCyclotomicExtension {p ^ (k + 1)} \u211a K]\n    (h\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))) : ((h\u03b6.toInteger - 1)) \u2223 p", "theorem zeta_sub_one_prime_of_two_pow [IsCyclotomicExtension {(2 : \u2115+) ^ (k + 1)} \u211a K]\n    (h\u03b6 : IsPrimitiveRoot \u03b6 \u2191((2 : \u2115+) ^ (k + 1))) :\n    Prime (h\u03b6.toInteger - 1)", "@[simp]\ntheorem zmodEquivZPowers_symm_apply_pow (i : \u2115) :\n    h.zmodEquivZPowers.symm (Additive.ofMul (\u27e8\u03b6 ^ i, i, rfl\u27e9 : Subgroup.zpowers \u03b6)) = i", "lemma IsProperMap.isClosedMap (h : IsProperMap f) : IsClosedMap f", "theorem IsPurelyInseparable.tower_bot [Algebra E K] [IsScalarTower F E K]\n    [IsPurelyInseparable F K] : IsPurelyInseparable F E", "@[to_additive \"If all additions cancel then every element is add-regular.\"]\ntheorem IsRegular.all [Mul R] [IsCancelMul R] (g : R) : IsRegular g", "theorem IsRegular.ne_zero [Nontrivial R] (la : IsRegular a) : a \u2260 0", "theorem add_mul_right_left_iff : IsRelPrime (x + z * y) y \u2194 IsRelPrime x y", "theorem mul_add_left_right_iff {x y z : R} : IsRelPrime x (x * z + y) \u2194 IsRelPrime x y", "theorem IsRelPrime.mul_left_iff : IsRelPrime (x * y) z \u2194 IsRelPrime x z \u2227 IsRelPrime y z", "theorem IsRelPrime.prod_left_iff : IsRelPrime (\u220f i \u2208 t, s i) x \u2194 \u2200 i \u2208 t, IsRelPrime (s i) x", "theorem IsRightRegular.ne_zero [Nontrivial R] (ra : IsRightRegular a) : a \u2260 0", "theorem IsRightRegular.pow_iff {n : \u2115} (n0 : 0 < n) :\n    IsRightRegular (a ^ n) \u2194 IsRightRegular a", "theorem map_neg (hf : IsRingHom f) : f (-x) = -f x", "theorem _root_.IsSMulRegular.finsupp [Zero M] [SMulZeroClass R M] {k : R}\n    (hk : IsSMulRegular M k) : IsSMulRegular (\u03b1 \u2192\u2080 M) k", "theorem mul_iff : IsSMulRegular M (a * b) \u2194 IsSMulRegular M a \u2227 IsSMulRegular M b", "theorem not_zero_iff : \u00acIsSMulRegular M (0 : R) \u2194 Nontrivial M", "theorem smul (ra : IsSMulRegular M a) (rs : IsSMulRegular M s) : IsSMulRegular M (a \u2022 s)", "theorem IsSelfAdjoint.spectralRadius_eq_nnnorm {a : A} (ha : IsSelfAdjoint a) :\n    spectralRadius \u2102 a = \u2016a\u2016\u208a", "theorem star_eq [Star R] {x : R} (hx : IsSelfAdjoint x) : star x = x", "theorem annihilator_isRadical (R) [CommRing R] [Module R M] [IsSemisimpleModule R M] :\n    (Module.annihilator R M).IsRadical", "theorem IsSeparable.insepDegree_eq [IsSeparable F E] : insepDegree F E = 1", "lemma IsSeparable.of_equiv_equiv {A\u2081 B\u2081 A\u2082 B\u2082 : Type*} [Field A\u2081] [Field B\u2081]\n    [Field A\u2082] [Field B\u2082] [Algebra A\u2081 B\u2081] [Algebra A\u2082 B\u2082] (e\u2081 : A\u2081 \u2243+* A\u2082) (e\u2082 : B\u2081 \u2243+* B\u2082)\n    (he : RingHom.comp (algebraMap A\u2082 B\u2082) \u2191e\u2081 = RingHom.comp \u2191e\u2082 (algebraMap A\u2081 B\u2081))\n    [IsSeparable A\u2081 B\u2081] : IsSeparable A\u2082 B\u2082", "protected theorem IsSeqCompact.totallyBounded (h : IsSeqCompact s) : TotallyBounded s", "lemma symm (ht : IsSl2Triple h e f) : IsSl2Triple (-h) f e where\n  h_ne_zero", "theorem IsStarNormal.spectralRadius_eq_nnnorm (a : A) [IsStarNormal a] :\n    spectralRadius \u2102 a = \u2016a\u2016\u208a", "@[to_additive]\ntheorem eq_trivial_iff {s : Set G} (hs : IsSubgroup s) : s = trivial G \u2194 \u2200 x \u2208 s, x = (1 : G)", "@[to_additive]\ntheorem mem_norm_comm_iff {s : Set G} (hs : IsNormalSubgroup s) {a b : G} : a * b \u2208 s \u2194 b * a \u2208 s", "lemma IsSublattice.prod {t : Set \u03b2} (hs : IsSublattice s) (ht : IsSublattice t) :\n    IsSublattice (s \u00d7\u02e2 t) := \u27e8hs.1.prod ht.1, hs.2.prod ht.2\u27e9", "@[to_additive\n      \"The sum of elements of an `AddSubmonoid` of an `AddCommMonoid` indexed by\n      a `Finset` is an element of the `AddSubmonoid`.\"]\ntheorem finset_prod_mem {M A} [CommMonoid M] {s : Set M} (hs : IsSubmonoid s) (f : A \u2192 M) :\n    \u2200 t : Finset A, (\u2200 b \u2208 t, f b \u2208 s) \u2192 (\u220f b \u2208 t, f b) \u2208 s", "theorem IsSubring.inter {S\u2081 S\u2082 : Set R} (hS\u2081 : IsSubring S\u2081) (hS\u2082 : IsSubring S\u2082) :\n    IsSubring (S\u2081 \u2229 S\u2082)", "theorem IsSymmOp.flip_eq (op) [IsSymmOp \u03b1 \u03b2 op] : flip op = op", "theorem _root_.IsTop.Iic_eq (h : IsTop a) : Iic a = univ", "theorem IsTop.fst (hx : IsTop x) : IsTop x.1", "theorem IsTop.mono (ha : IsTop a) (h : a \u2264 b) : IsTop b", "theorem IsTopologicalBasis.iInf_induced {\u03b2 : Type*} {\u03b9 : Type*} {X : \u03b9 \u2192 Type*}\n    [t : \u03a0 i, TopologicalSpace (X i)] {T : \u03a0 i, Set (Set (X i))}\n    (cond : \u2200 i, IsTopologicalBasis (T i)) (f : \u03a0 i, \u03b2 \u2192 X i) :\n    IsTopologicalBasis (t := \u2a05 i, induced (f i) (t i))\n      { S | \u2203 (U : \u2200 i, Set (X i)) (F : Finset \u03b9),\n        (\u2200 i, i \u2208 F \u2192 U i \u2208 T i) \u2227 S = \u22c2 (i) (_ : i \u2208 F), f i \u207b\u00b9' U i }", "@[simp]\ntheorem one_le_scalingConstantOf (K : \u211d) : 1 \u2264 scalingConstantOf \u03bc K", "@[simp]\ntheorem dvd : u \u2223 a", "nonrec theorem isOpenMap_smul (hc : IsUnit c) : IsOpenMap fun x : \u03b1 => c \u2022 x", "@[to_additive \"An additive unit in an additive monoid is add-regular.\"]\ntheorem IsUnit.isRegular (ua : IsUnit a) : IsRegular a", "theorem map' {f : M \u2192 N} (hf : IsMonoidHom f) {x : M} (h : IsUnit x) : IsUnit (f x)", "@[to_additive (attr := simp)]\ntheorem unit_of_val_units {a : M\u02e3} (h : IsUnit (a : M)) : h.unit = a", "theorem IsUpperSet.Ioi_subset (h : IsUpperSet s) \u2983a\u2984 (ha : a \u2208 s) : Ioi a \u2286 s", "theorem IsUpperSet.inter (hs : IsUpperSet s) (ht : IsUpperSet t) : IsUpperSet (s \u2229 t)", "protected theorem IsUpperSet.interior (h : IsUpperSet s) : IsUpperSet (interior s)", "theorem IsUpperSet.null_frontier (hs : IsUpperSet s) : volume (frontier s) = 0", "theorem ediam_range (hf : Isometry f) : EMetric.diam (range f) = EMetric.diam (univ : Set \u03b1)", "theorem hausdorffMeasure_preimage (hf : Isometry f) (hd : 0 \u2264 d \u2228 Surjective f) (s : Set Y) :\n    \u03bcH[d] (f \u207b\u00b9' s) = \u03bcH[d] (s \u2229 range f)", "protected theorem uniformEmbedding (hf : Isometry f) : UniformEmbedding f", "protected theorem uniformInducing (hf : Isometry f) : UniformInducing f", "@[simp]\ntheorem coe_toHomeomorph (h : \u03b1 \u2243\u1d62 \u03b2) : \u21d1h.toHomeomorph = h", "@[simp]\ntheorem comp_continuous_iff {\u03b3} [TopologicalSpace \u03b3] (h : \u03b1 \u2243\u1d62 \u03b2) {f : \u03b3 \u2192 \u03b1} :\n    Continuous (h \u2218 f) \u2194 Continuous f", "theorem completeSpace_iff (e : \u03b1 \u2243\u1d62 \u03b2) : CompleteSpace \u03b1 \u2194 CompleteSpace \u03b2", "@[simp]\ntheorem hausdorffMeasure_image (e : X \u2243\u1d62 Y) (d : \u211d) (s : Set X) : \u03bcH[d] (e '' s) = \u03bcH[d] s", "@[simp]\ntheorem map_hausdorffMeasure (e : X \u2243\u1d62 Y) (d : \u211d) : Measure.map e \u03bcH[d] = \u03bcH[d]", "@[simp]\nlemma _root_.IsometryEquiv.toDilationEquiv_apply (e : X \u2243\u1d62 Y) (x : X) :\n    e.toDilationEquiv x = e x", "theorem toEquiv_injective : Injective (toEquiv : (\u03b1 \u2243\u1d62 \u03b2) \u2192 (\u03b1 \u2243 \u03b2))", "@[symm]\ntheorem Joined.symm {x y : X} (h : Joined x y) : Joined y x", "theorem KaehlerDifferential.D_tensorProductTo (x : KaehlerDifferential.ideal R S) :\n    (KaehlerDifferential.D R S).tensorProductTo x =\n      (KaehlerDifferential.ideal R S).toCotangent x", "theorem KaehlerDifferential.range_kerCotangentToTensor\n    (h : Function.Surjective (algebraMap A B)) :\n    LinearMap.range (kerCotangentToTensor R A B) =\n      (LinearMap.ker (KaehlerDifferential.mapBaseChange R A B)).restrictScalars A", "theorem LDL.lowerInv_eq_gramSchmidtBasis :\n    LDL.lowerInv hS =\n      ((Pi.basisFun \ud835\udd5c n).toMatrix\n          (@gramSchmidtBasis \ud835\udd5c (n \u2192 \ud835\udd5c) _ (_ : _) (InnerProductSpace.ofMatrix hS.transpose) n _ _ _\n            (Pi.basisFun \ud835\udd5c n)))\u1d40", "theorem LDL.lowerInv_triangular {i j : n} (hij : i < j) : LDL.lowerInv hS i j = 0", "theorem LDL.lower_conj_diag : LDL.lower hS * LDL.diag hS * (LDL.lower hS)\u1d34 = S", "lemma term_convolution (f g : \u2115 \u2192 \u2102) (s : \u2102) (n : \u2115) :\n    term (f \u235f g) s n = \u2211 p \u2208 n.divisorsAntidiagonal, term f s p.1 * term g s p.2", "lemma term_delta (s : \u2102) (n : \u2115) : term \u03b4 s n = if n = 1 then 1 else 0", "lemma LSeriesHasSum.LSeries_eq {f : \u2115 \u2192 \u2102} {s a : \u2102}\n    (h : LSeriesHasSum f s a) : LSeries f s = a", "lemma LSeriesSummable.add {f g : \u2115 \u2192 \u2102} {s : \u2102} (hf : LSeriesSummable f s)\n    (hg : LSeriesSummable g s) :\n    LSeriesSummable (f + g) s", "lemma LSeriesSummable.isBigO_rpow {f : \u2115 \u2192 \u2102} {s : \u2102} (h : LSeriesSummable f s) :\n    f =O[atTop] fun n \u21a6 (n : \u211d) ^ s.re", "lemma LSeriesSummable_of_abscissaOfAbsConv_lt_re {f : \u2115 \u2192 \u2102} {s : \u2102}\n    (hs : abscissaOfAbsConv f < s.re) : LSeriesSummable f s", "theorem LSeriesSummable_of_bounded_of_one_lt_real {f : \u2115 \u2192 \u2102} {m : \u211d}\n    (h : \u2200 n \u2260 0, Complex.abs (f n) \u2264 m) {s : \u211d} (hs : 1 < s) :\n    LSeriesSummable f s", "@[simp]\nlemma LSeries_zero : LSeries 0 = 0", "theorem ne' [Preorder \u03b1] {x y : \u03b1} (h : x < y) : y \u2260 x", "theorem basis_ne_zero (hvs : Set.InjOn v s) (hi : i \u2208 s) : Lagrange.basis s v i \u2260 0", "theorem degree_basisDivisor_of_ne (hxy : x \u2260 y) : (basisDivisor x y).degree = 1", "@[simp] theorem nodal_subgroup_eq_X_pow_card_sub_one [IsDomain R]\n  (G : Subgroup R\u02e3) [Fintype G] :\n  nodal (G : Set R\u02e3).toFinset ((\u2191) : R\u02e3 \u2192 R) = X ^ (Fintype.card G) - 1", "theorem iSup_mul {\u03b9 : Sort v} (l : \u03b9 \u2192 Language \u03b1) (m : Language \u03b1) :\n    (\u2a06 i, l i) * m = \u2a06 i, l i * m", "theorem kstar_def_nonempty (l : Language \u03b1) :\n    l\u2217 = { x | \u2203 S : List (List \u03b1), x = S.join \u2227 \u2200 y \u2208 S, y \u2208 l \u2227 y \u2260 [] }", "lemma birkhoffFinset_injective : Injective (birkhoffFinset (\u03b1 := \u03b1))", "@[simp] lemma coe_toInfHom (f : LatticeHom \u03b1 \u03b2) : \u21d1f.toInfHom = f := rfl", "lemma withTopWithBot_apply (f : LatticeHom \u03b1 \u03b2) (a : WithTop <| WithBot \u03b1) :\n    f.withTopWithBot a = a.map (Option.map f) := rfl", "@[simp]\ntheorem T_mul (n : \u2124) (f : R[T;T\u207b\u00b9]) : T n * f = f * T n", "theorem algebraMap_apply {R A : Type*} [CommSemiring R] [Semiring A] [Algebra R A] (r : R) :\n    algebraMap R (LaurentPolynomial A) r = C (algebraMap R A r)", "@[simp]\ntheorem degree_T [Nontrivial R] (n : \u2124) : (T n : R[T;T\u207b\u00b9]).degree = n", "@[elab_as_elim]\nprotected theorem induction_on' {M : R[T;T\u207b\u00b9] \u2192 Prop} (p : R[T;T\u207b\u00b9])\n    (h_add : \u2200 p q, M p \u2192 M q \u2192 M (p + q)) (h_C_mul_T : \u2200 (n : \u2124) (a : R), M (C a * T n)) :\n    M p", "@[simp] lemma invert_symm : (invert (R := R)).symm = invert := rfl", "theorem isLocalization : IsLocalization (Submonoid.closure ({X} : Set R[X])) R[T;T\u207b\u00b9]", "theorem support_C_mul_T (a : R) (n : \u2124) : Finsupp.support (C a * T n) \u2286 {n}", "@[to_additive]\ntheorem Left.inv_lt_self (h : 1 < a) : a\u207b\u00b9 < a", "@[to_additive] lemma Left.min_le_max_of_mul_le_mul\n    [CovariantClass \u03b1 \u03b1 (\u00b7 * \u00b7) (\u00b7 < \u00b7)] [CovariantClass \u03b1 \u03b1 (swap (\u00b7 * \u00b7)) (\u00b7 \u2264 \u00b7)]\n    (h : a * b \u2264 c * d) : min a b \u2264 max c d", "theorem Left.mul_le_one_of_le_of_le [PosMulMono \u03b1] (ha : a \u2264 1) (hb : b \u2264 1) (a0 : 0 \u2264 a) :\n    a * b \u2264 1", "@[to_additive (attr := simp) Left.neg_pos_iff \"Uses `left` co(ntra)variant.\"]\ntheorem Left.one_lt_inv_iff : 1 < a\u207b\u00b9 \u2194 a < 1", "theorem Left.one_lt_mul_of_le_of_lt_of_pos [PosMulStrictMono \u03b1] (ha : 1 \u2264 a) (hb : 1 < b)\n    (a0 : 0 < a) : 1 < a * b", "@[to_additive Left.add_pos_of_pos_of_nonneg\n      \"Assumes left covariance.\n      The lemma assuming right covariance is `Right.add_pos_of_pos_of_nonneg`.\"]\ntheorem Left.one_lt_mul_of_lt_of_le [CovariantClass \u03b1 \u03b1 (\u00b7 * \u00b7) (\u00b7 \u2264 \u00b7)] {a b : \u03b1} (ha : 1 < a)\n    (hb : 1 \u2264 b) :\n    1 < a * b", "@[to_additive]\ntheorem LeftCancelMonoid.toMonoid_injective {M : Type u} :\n    Function.Injective (@LeftCancelMonoid.toMonoid M)", "theorem comp_L : (X f).comp (\ud835\udc73 I g) = X (f.comp (\ud835\udc73 I g))", "theorem evalAt_apply : evalAt g X f = (X f) g", "protected theorem map_smul : X (r \u2022 f) = r \u2022 X f", "protected theorem map_sub : X (f - f') = X f - X f'", "theorem map_sSup (hf : LeftOrdContinuous f) (s : Set \u03b1) : f (sSup s) = \u2a06 x \u2208 s, f x", "theorem impl_cons (w' : 0 < List.length ds) :\n    impl C (x :: xs) y \u27e8d :: ds, w\u27e9 =\n      let \u27e8r, w\u27e9 := impl C xs y \u27e8ds, w'\u27e9\n      \u27e8min (C.delete x + r[0]) (min (C.insert y + d) (C.substitute x y + ds[0])) :: r, by simp\u27e9", "theorem impl_cons_fst_zero (h) (w' : 0 < List.length ds) :\n    (impl C (x :: xs) y \u27e8d :: ds, w\u27e9).1[0] =\n      let \u27e8r, w\u27e9 := impl C xs y \u27e8ds, w'\u27e9\n      min (C.delete x + r[0]) (min (C.insert y + d) (C.substitute x y + ds[0]))", "@[simp] lemma Lex.exists {p : Lex \u03b1 \u2192 Prop} : (\u2203 a, p a) \u2194 \u2203 a, p (toLex a) := Iff.rfl", "@[simp] lemma Lex.forall {p : Lex \u03b1 \u2192 Prop} : (\u2200 a, p a) \u2194 \u2200 a, p (toLex a) := Iff.rfl", "private theorem bracket_leibniz_lie (x y : A \u2297[R] L) (z : A \u2297[R] M) :\n    \u2045x, \u2045y, z\u2046\u2046 = \u2045\u2045x, y\u2046, z\u2046 + \u2045y, \u2045x, z\u2046\u2046", "@[simp]\ntheorem HasTrivialRadical.center_eq_bot [HasTrivialRadical R L] : center R L = \u22a5", "lemma orthogonal_disjoint\n    (\u03a6 : LinearMap.BilinForm R L) (h\u03a6_nondeg : \u03a6.Nondegenerate) (h\u03a6_inv : \u03a6.lieInvariant L)\n    (hL : \u2200 I : LieIdeal R L, IsAtom I \u2192 \u00acIsLieAbelian I)\n    (I : LieIdeal R L) (hI : IsAtom I) :\n    Disjoint I (orthogonal \u03a6 h\u03a6_inv I)", "lemma chainLength_zero_right [Nontrivial L] : chainLength \u03b1 0 = 2", "lemma chainTopCoeff_add_chainBotCoeff :\n    chainTopCoeff \u03b1 \u03b2 + chainBotCoeff \u03b1 \u03b2 = chainLength \u03b1 \u03b2", "lemma coe_corootSpace_eq_span_singleton (\u03b1 : Weight K H L) :\n    (corootSpace \u03b1).toSubmodule = K \u2219 coroot \u03b1", "lemma ideal_eq_bot_of_isLieAbelian [IsDomain R] [IsPrincipalIdealRing R]\n    (I : LieIdeal R L) [IsLieAbelian I] : I = \u22a5", "lemma rootSpace_two_smul : rootSpace H (2 \u2022 \u03b1) = \u22a5", "@[simp]\nlemma span_weight_eq_top :\n    span K (range (Weight.toLinear K H L)) = \u22a4", "lemma eq_top_of_isAtom (I : LieIdeal R L) (hI : IsAtom I) : I = \u22a4", "@[simp]\ntheorem mem_so [Fintype n] (A : Matrix n n R) : A \u2208 so n R \u2194 A\u1d40 = -A", "theorem pb_inv [Invertible (2 : R)] : PB l R * Matrix.fromBlocks 1 0 0 (\u215f (PD l R)) = 1", "theorem abelian_radical_iff_solvable_is_abelian [IsNoetherian R L] :\n    IsLieAbelian (radical R L) \u2194 \u2200 I : LieIdeal R L, IsSolvable R I \u2192 IsLieAbelian I", "@[simp]\ntheorem LieAlgebra.ad_apply (x y : L) : LieAlgebra.ad R L x y = \u2045x, y\u2046", "theorem derivedLength_zero (I : LieIdeal R L) [hI : IsSolvable R I] :\n    derivedLengthOfIdeal R L I = 0 \u2194 I = \u22a5", "@[simp]\ntheorem derivedSeriesOfIdeal_succ (k : \u2115) :\n    derivedSeriesOfIdeal R L (k + 1) I =\n      \u2045derivedSeriesOfIdeal R L k I, derivedSeriesOfIdeal R L k I\u2046", "theorem hasTrivialRadical_iff_no_abelian_ideals :\n    HasTrivialRadical R L \u2194 \u2200 I : LieIdeal R L, IsLieAbelian I \u2192 I = \u22a5", "theorem LieAlgebra.isNilpotent_iff_forall [IsNoetherian R L] :\n    LieAlgebra.IsNilpotent R L \u2194 \u2200 x, _root_.IsNilpotent <| LieAlgebra.ad R L x", "theorem LieAlgebra.nilpotent_of_nilpotent_quotient {I : LieIdeal R L} (h\u2081 : I \u2264 center R L)\n    (h\u2082 : IsNilpotent R (L \u29f8 I)) : IsNilpotent R L", "lemma toEnd_pow_apply_mem {\u03c7\u2081 \u03c7\u2082 : H \u2192 R} {x : L} {m : M}\n    (hx : x \u2208 rootSpace H \u03c7\u2081) (hm : m \u2208 weightSpace M \u03c7\u2082) (n) :\n    (toEnd R L M x ^ n : Module.End R M) m \u2208 weightSpace M (n \u2022 \u03c7\u2081 + \u03c7\u2082)", "theorem zero_rootSpace_eq_top_of_nilpotent [IsNilpotent R L] :\n    rootSpace (\u22a4 : LieSubalgebra R L) 0 = \u22a4", "lemma lie_ad (x : L) (D : LieDerivation R L L) : \u2045ad R L x, D\u2046 = \u2045x, D\u2046", "theorem smul_apply (r : S) (D : LieDerivation R L M) : (r \u2022 D) a = r \u2022 D a", "theorem zero_apply (a : L) : (0 : LieDerivation R L M) a = 0", "theorem coe_linearEquiv_injective : Injective ((\u2191) : (L\u2081 \u2243\u2097\u2045R\u2046 L\u2082) \u2192 L\u2081 \u2243\u2097[R] L\u2082)", "theorem mem_range_self (x : L) : f x \u2208 f.range", "theorem comap_map_le : I \u2264 comap f (map f I)", "@[simp]\nlemma _root_.LieIdeal.idealizer_eq_normalizer (I : LieIdeal R L) :\n    I.idealizer = I.normalizer", "@[simp]\ntheorem incl_idealRange : I.incl.idealRange = I", "@[simp]\ntheorem mem_comap {x : L} : x \u2208 comap f J \u2194 f x \u2208 J", "lemma weightSpace_ne_bot (\u03c7 : Weight R L M) : weightSpace M \u03c7 \u2260 \u22a5 := \u03c7.weightSpace_ne_bot'", "@[simp] lemma chainBot_zero : chainBot 0 \u03b2 = \u03b2", "lemma chainTopCoeff_add_one :\n    letI := Classical.propDecidable\n    chainTopCoeff \u03b1 \u03b2 + 1 =\n      Nat.find (eventually_weightSpace_smul_add_eq_bot M \u03b1 \u03b2 h\u03b1).exists", "lemma chainTop_isNonZero' (h\u03b1' : weightSpace M \u03b1 \u2260 \u22a5) :\n    (chainTop \u03b1 \u03b2).IsNonZero", "lemma exists_forall_mem_corootSpace_smul_add_eq_zero\n    [IsDomain R] [IsPrincipalIdealRing R] [CharZero R] [NoZeroSMulDivisors R M] [IsNoetherian R M]\n    (h\u03b1 : \u03b1 \u2260 0) (h\u03c7 : weightSpace M \u03c7 \u2260 \u22a5) :\n    \u2203 a b : \u2124, 0 < b \u2227 \u2200 x \u2208 corootSpace \u03b1, (a \u2022 \u03b1 + b \u2022 \u03c7) x = 0", "lemma exists_isRegular_of_finrank_le_card (h : finrank R M \u2264 #R) :\n    \u2203 x : L, IsRegular R M x", "theorem iInf_lcs_le_of_isNilpotent_quot (h : IsNilpotent R L (M \u29f8 N)) :\n    \u2a05 k, lowerCentralSeries R L M k \u2264 N", "lemma iSup_weightSpace_eq_top [IsTriangularizable K L M] :\n    \u2a06 \u03c7 : L \u2192 K, weightSpace M \u03c7 = \u22a4", "lemma isCompl_weightSpace_zero_posFittingComp :\n    IsCompl (weightSpace M 0) (posFittingComp R L M)", "@[simp]\ntheorem isNilpotent_range_toEnd_iff :\n    IsNilpotent R (toEnd R L M).range M \u2194 IsNilpotent R L M", "theorem isNilpotent_toEnd_of_isNilpotent\u2082 [IsNilpotent R L M] (x y : L) :\n    _root_.IsNilpotent (toEnd R L M x \u2218\u2097 toEnd R L M y)", "theorem isTrivial_of_nilpotencyLength_le_one [IsNilpotent R L M] (h : nilpotencyLength R L M \u2264 1) :\n    IsTrivial L M", "lemma lie_mem_maxGenEigenspace_toEnd\n    {\u03c7\u2081 \u03c7\u2082 : R} {x y : L} {m : M} (hy : y \u2208 \ud835\udd4e(L, \u03c7\u2081, x)) (hm : m \u2208 \ud835\udd4e(M, \u03c7\u2082, x)) :\n    \u2045y, m\u2046 \u2208 \ud835\udd4e(M, \u03c7\u2081 + \u03c7\u2082, x)", "lemma lie_mem_weightSpaceChain_of_weightSpace_eq_bot_left [LieAlgebra.IsNilpotent R H]\n    (hp : weightSpace M (p \u2022 \u03b1 + \u03c7) = \u22a5)\n    {x : L} (hx : x \u2208 rootSpace H (-\u03b1))\n    {y : M} (hy : y \u2208 weightSpaceChain M \u03b1 \u03c7 p q) :\n    \u2045x, y\u2046 \u2208 weightSpaceChain M \u03b1 \u03c7 p q", "@[simp] lemma lie_traceForm_eq_zero (x : L) : \u2045x, traceForm R L M\u2046 = 0", "lemma map_posFittingComp_eq (e : M \u2243\u2097\u2045R,L\u2046 M\u2082) :\n    (posFittingComp R L M).map e = posFittingComp R L M\u2082", "lemma map_posFittingComp_le :\n    (posFittingComp R L M).map f \u2264 posFittingComp R L M\u2082", "@[simp]\ntheorem nilpotencyLength_eq_one_iff [Nontrivial M] :\n    nilpotencyLength R L M = 1 \u2194 IsTrivial L M", "lemma posFittingCompOf_le_posFittingComp (x : L) :\n    posFittingCompOf R M x \u2264 posFittingComp R L M", "private lemma aux [h : Nontrivial (shiftedWeightSpace R L M \u03c7)] : weightSpace M \u03c7 \u2260 \u22a5", "lemma toEnd_eq (x : L) :\n    toEnd R L (shiftedWeightSpace R L M \u03c7) x =\n    (shift R L M \u03c7).conj (toEnd R L (weightSpace M \u03c7) x - \u03c7 x \u2022 LinearMap.id)", "@[simp]\ntheorem toModuleHom_apply (x : L) (m : M) : toModuleHom R L M (x \u2297\u209c m) = \u2045x, m\u2046", "lemma weightSpace_chainTopCoeff_add_one_zsmul_add :\n    weightSpace M ((chainTopCoeff \u03b1 \u03b2 + 1 : \u2124) \u2022 \u03b1 + \u03b2 : L \u2192 R) = \u22a5", "@[simp]\ntheorem toEquiv_mk (f : M \u2192\u2097\u2045R,L\u2046 N) (g : N \u2192 M) (h\u2081 h\u2082) :\n    toEquiv (mk f g h\u2081 h\u2082 : M \u2243\u2097\u2045R,L\u2046 N) = Equiv.mk f g h\u2081 h\u2082", "@[simp]\ntheorem trans_apply (e\u2081 : M \u2243\u2097\u2045R,L\u2046 N) (e\u2082 : N \u2243\u2097\u2045R,L\u2046 P) (m : M) : (e\u2081.trans e\u2082) m = e\u2082 (e\u2081 m)", "@[simp]\ntheorem _root_.LieModuleHom.coe_restrictLie (f : M \u2192\u2097\u2045R,L\u2046 N) : \u21d1(f.restrictLie L') = f", "@[simp, norm_cast]\ntheorem coe_toLinearMap (f : M \u2192\u2097\u2045R,L\u2046 N) : ((f : M \u2192\u2097[R] N) : M \u2192 N) = f", "@[simp]\ntheorem map_top : LieSubmodule.map f \u22a4 = f.range", "lemma LieSubalgebra.coe_ad_pow (H : LieSubalgebra R L) (x y : H) (n : \u2115) :\n    ((ad R H x ^ n) y : L) = (ad R L x ^ n) y", "theorem exists_nested_lieIdeal_coe_eq_iff {K' : LieSubalgebra R L} (h : K \u2264 K') :\n    (\u2203 I : LieIdeal R K', \u2191I = ofLe h) \u2194 \u2200 x y : L, x \u2208 K' \u2192 y \u2208 K \u2192 \u2045x, y\u2046 \u2208 K", "@[simp]\ntheorem incl_range : K.incl.range = K", "theorem lie_mem_sup_of_mem_normalizer {x y z : L} (hx : x \u2208 H.normalizer) (hy : y \u2208 (R \u2219 x) \u2294 \u2191H)\n    (hz : z \u2208 (R \u2219 x) \u2294 \u2191H) : \u2045y, z\u2046 \u2208 (R \u2219 x) \u2294 \u2191H", "theorem mem_normalizer_iff' (x : L) : x \u2208 H.normalizer \u2194 \u2200 y : L, y \u2208 H \u2192 \u2045y, x\u2046 \u2208 H", "@[simp]\ntheorem top_coe_submodule : ((\u22a4 : LieSubalgebra R L) : Submodule R L) = \u22a4", "theorem mk_eq_zero {m : M} : mk' N m = 0 \u2194 m \u2208 N", "@[simp, norm_cast]\ntheorem coe_sub (m m' : N) : (\u2191(m - m') : M) = (m : M) - (m' : M)", "@[simp]\ntheorem iInf_coe {\u03b9} (p : \u03b9 \u2192 LieSubmodule R L M) : (\u2191(\u2a05 i, p i) : Set M) = \u22c2 i, \u2191(p i)", "lemma isCompactElement_lieSpan_singleton (m : M) :\n    CompleteLattice.IsCompactElement (lieSpan R L {m})", "theorem isNilpotent_iff_exists_self_le_ucs :\n    LieModule.IsNilpotent R L N \u2194 \u2203 k, N \u2264 (\u22a5 : LieSubmodule R L M).ucs k", "@[simp]\ntheorem ker_incl : N.incl.ker = \u22a5", "theorem lie_le_left : \u2045I, J\u2046 \u2264 I", "@[simp]\ntheorem lie_sup : \u2045I, N \u2294 N'\u2046 = \u2045I, N\u2046 \u2294 \u2045I, N'\u2046", "theorem mono_lie (h\u2081 : I \u2264 J) (h\u2082 : N \u2264 N') : \u2045I, N\u2046 \u2264 \u2045J, N'\u2046", "@[gcongr, mono]\ntheorem normalizer_mono (h : N\u2081 \u2264 N\u2082) : normalizer N\u2081 \u2264 normalizer N\u2082", "theorem sInf_coe_toSubmodule' (S : Set (LieSubmodule R L M)) :\n    (\u2191(sInf S) : Submodule R M) = \u2a05 N \u2208 S, (N : Submodule R M)", "@[simp]\ntheorem toEnd_restrict_eq_toEnd (h := N.toEnd_comp_subtype_mem x) :\n    (toEnd R L M x).restrict h = toEnd R L N x", "theorem ucs_eq_self_of_normalizer_eq_self (h : N\u2081.normalizer = N\u2081) (k : \u2115) : N\u2081.ucs k = N\u2081", "lemma finiteCoproduct.\u03b9_jointly_surjective (R : finiteCoproduct X) :\n    \u2203 (a : \u03b1) (r : X a), R = finiteCoproduct.\u03b9 X a r := \u27e8R.fst, R.snd, rfl\u27e9", "theorem mono_iff_injective {X Y : LightProfinite.{u}} (f : X \u27f6 Y) :\n    Mono f \u2194 Function.Injective f", "theorem LineDifferentiableAt.smul (h : LineDifferentiableAt \ud835\udd5c f x v) (c : \ud835\udd5c) :\n    LineDifferentiableAt \ud835\udd5c f x (c \u2022 v)", "@[simp]\ntheorem arrowCongr_apply {R M\u2081 M\u2082 M\u2082\u2081 M\u2082\u2082 : Sort _} [CommSemiring R] [AddCommMonoid M\u2081]\n    [AddCommMonoid M\u2082] [AddCommMonoid M\u2082\u2081] [AddCommMonoid M\u2082\u2082] [Module R M\u2081] [Module R M\u2082]\n    [Module R M\u2082\u2081] [Module R M\u2082\u2082] (e\u2081 : M\u2081 \u2243\u2097[R] M\u2082) (e\u2082 : M\u2082\u2081 \u2243\u2097[R] M\u2082\u2082) (f : M\u2081 \u2192\u2097[R] M\u2082\u2081)\n    (x : M\u2082) : arrowCongr e\u2081 e\u2082 f x = e\u2082 (f (e\u2081.symm x))", "@[simp] theorem coe_lTensor_symm : (lTensor M f).symm = (f.symm : P \u2192\u2097[R] N).lTensor M", "@[simp] theorem coe_rTensor : rTensor M f = (f : N \u2192\u2097[R] P).rTensor M", "@[simp]\ntheorem coe_toContinuousLinearEquiv' (e : E \u2243\u2097[\ud835\udd5c] F) : (e.toContinuousLinearEquiv : E \u2192 F) = e", "@[simp]\ntheorem coe_zero : \u21d1(0 : M \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082) = 0", "theorem LinearEquiv.det_coe_symm {\ud835\udd5c : Type*} [Field \ud835\udd5c] [Module \ud835\udd5c M] (f : M \u2243\u2097[\ud835\udd5c] M) :\n    LinearMap.det (f.symm : M \u2192\u2097[\ud835\udd5c] M) = (LinearMap.det (f : M \u2192\u2097[\ud835\udd5c] M))\u207b\u00b9", "@[simp]\ntheorem LinearEquiv.det_mul_det_symm {A : Type*} [CommRing A] [Module A M] (f : M \u2243\u2097[A] M) :\n    LinearMap.det (f : M \u2192\u2097[A] M) * LinearMap.det (f.symm : M \u2192\u2097[A] M) = 1", "@[simp]\ntheorem det_refl : LinearEquiv.det (LinearEquiv.refl R M) = 1", "theorem ext_iff : e = e' \u2194 \u2200 x, e x = e' x", "@[simp] theorem lTensor_symm_tmul : (f.lTensor M).symm (m \u2297\u209c p) = m \u2297\u209c f.symm p", "@[simp] theorem lTensor_trans_rTensor (f : M \u2243\u2097[R] P) (g : N \u2243\u2097[R] Q) :\n    g.lTensor M \u226a\u226b\u2097 f.rTensor Q = TensorProduct.congr f g", "theorem LinearEquiv.nonempty_equiv_iff_lift_rank_eq : Nonempty (M \u2243\u2097[R] M') \u2194\n    Cardinal.lift.{v'} (Module.rank R M) = Cardinal.lift.{v} (Module.rank R M')", "@[simp]\ntheorem ofBijective_symm_apply_apply [RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081] [RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082] {h} (x : M) :\n    (ofBijective f h).symm (f x) = x", "@[simp]\ntheorem ofSubmodule'_symm_apply [Module R M] [Module R\u2082 M\u2082] (f : M \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082)\n    (U : Submodule R\u2082 M\u2082) (x : U) : ((f.ofSubmodule' U).symm x : M) = f.symm (x : M\u2082)", "@[simp]\ntheorem ofSubmodules_apply {p : Submodule R M} {q : Submodule R\u2082 M\u2082} (h : p.map \u2191e = q) (x : p) :\n    \u2191(e.ofSubmodules p q h x) = e x", "theorem pow_apply (e : M \u2243\u2097[R] M) (n : \u2115) (m : M) : (e ^ n) m = e^[n] m", "@[simp]\ntheorem self_trans_symm (f : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082) : f.trans f.symm = LinearEquiv.refl R\u2081 M\u2081", "theorem symm_apply_eq {x y} : e.symm x = y \u2194 x = e y", "lemma symm_flip : e.flip.symm = e.symm.dualMap.trans (evalEquiv R M).symm := rfl", "@[simp]\ntheorem symm_neg : (neg R : M \u2243\u2097[R] M).symm = neg R", "theorem toAddMonoidHom_commutes : e.toLinearMap.toAddMonoidHom = e.toAddEquiv.toAddMonoidHom", "@[simp]\ntheorem trans_apply (c : M\u2081) : (e\u2081\u2082.trans e\u2082\u2083 : M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2083] M\u2083) c = e\u2082\u2083 (e\u2081\u2082 c)", "theorem cardinal_le_rank' {s : Set M}\n    (hs : LinearIndependent R (fun x => x : s \u2192 M)) : #s \u2264 Module.rank R M", "theorem LinearIndependent.eq_of_smul_apply_eq_smul_apply {M : Type*} [AddCommGroup M] [Module R M]\n    {v : \u03b9 \u2192 M} (li : LinearIndependent R v) (c d : R) (i j : \u03b9) (hc : c \u2260 0)\n    (h : c \u2022 v i = d \u2022 v j) : i = j", "theorem LinearIndependent.fin_cons' {m : \u2115} (x : M) (v : Fin m \u2192 M) (hli : LinearIndependent R v)\n    (x_ortho : \u2200 (c : R) (y : Submodule.span R (Set.range v)), c \u2022 x + y = (0 : M) \u2192 c = 0) :\n    LinearIndependent R (Fin.cons x v : Fin m.succ \u2192 M)", "theorem LinearIndependent.inl_union_inr {s : Set M} {t : Set M'}\n    (hs : LinearIndependent R (fun x => x : s \u2192 M))\n    (ht : LinearIndependent R (fun x => x : t \u2192 M')) :\n    LinearIndependent R (fun x => x : \u21a5(inl R M M' '' s \u222a inr R M M' '' t) \u2192 M \u00d7 M')", "theorem LinearIndependent.to_subtype_range' {\u03b9} {f : \u03b9 \u2192 M} (hf : LinearIndependent R f) {t}\n    (ht : range f = t) : LinearIndependent R ((\u2191) : t \u2192 M)", "@[simp]\ntheorem coe_id : ((id : E \u2192\u2097\u1d62[R] E) : E \u2192 E) = _root_.id", "@[simp]\ntheorem coe_mul (f g : E \u2192\u2097\u1d62[R] E) : \u21d1(f * g) = f \u2218 g", "@[simp]\ntheorem dist_map (x y : E) : dist (f x) (f y) = dist x y", "@[simp]\ntheorem id_apply (x : E) : (id : E \u2192\u2097\u1d62[R] E) x = x", "theorem LinearIsometry.norm_iteratedFDeriv_comp_left {f : E \u2192 F} (g : F \u2192\u2097\u1d62[\ud835\udd5c] G)\n    (hf : ContDiff \ud835\udd5c n f) (x : E) {i : \u2115} (hi : (i : \u2115\u221e) \u2264 n) :\n    \u2016iteratedFDeriv \ud835\udd5c i (g \u2218 f) x\u2016 = \u2016iteratedFDeriv \ud835\udd5c i f x\u2016", "@[simp]\ntheorem toContinuousLinearMap_inj {f g : E \u2192\u209b\u2097\u1d62[\u03c3\u2081\u2082] E\u2082} :\n    f.toContinuousLinearMap = g.toContinuousLinearMap \u2194 f = g", "@[simp]\ntheorem coe_ofEq_apply (h : p = q) (x : p) : (ofEq p q h x : E) = x", "theorem comp_differentiableAt_iff {f : G \u2192 E} {x : G} :\n    DifferentiableAt \ud835\udd5c (iso \u2218 f) x \u2194 DifferentiableAt \ud835\udd5c f x", "theorem comp_differentiable_iff {f : G \u2192 E} : Differentiable \ud835\udd5c (iso \u2218 f) \u2194 Differentiable \ud835\udd5c f", "protected theorem lipschitz : LipschitzWith 1 e", "theorem map_eq_zero_iff {x : E} : e x = 0 \u2194 x = 0", "theorem LinearIsometryEquiv.reflections_generate_dim [FiniteDimensional \u211d F] (\u03c6 : F \u2243\u2097\u1d62[\u211d] F) :\n    \u2203 l : List F, l.length \u2264 finrank \u211d F \u2227 \u03c6 = (l.map fun v => reflection (\u211d \u2219 v)\u15ee).prod", "@[simp]\ntheorem symm_symm : e.symm.symm = e", "theorem le_succFn (i : \u03b9) : i \u2264 succFn i", "lemma Nondegenerate.flip {B : BilinForm K V} (hB : B.Nondegenerate) :\n    B.flip.Nondegenerate", "@[deprecated (since := \"2024-04-14\")]\ntheorem coe_zero : \u21d1(0 : BilinForm R M) = 0", "@[simp]\ntheorem compRight_compLeft (B : BilinForm R M) (l r : M \u2192\u2097[R] M) :\n    (B.compRight r).compLeft l = B.comp l r", "@[simp]\ntheorem comp_id_right (B : BilinForm R M) (l : M \u2192\u2097[R] M) :\n    B.comp l LinearMap.id = B.compLeft l", "lemma dualSubmodule_flip_dualSubmodule_of_basis {\u03b9 : Type*} [Finite \u03b9]\n    (hB : B.Nondegenerate) (b : Basis \u03b9 S M) :\n    B.flip.dualSubmodule (B.dualSubmodule (Submodule.span R (Set.range b))) =\n      Submodule.span R (Set.range b)", "theorem iIsOrtho_def {n : Type w} {B : BilinForm R M} {v : n \u2192 M} :\n    B.iIsOrtho v \u2194 \u2200 i j : n, i \u2260 j \u2192 B (v i) (v j) = 0", "theorem isAdjointPair_id : IsAdjointPair B B 1 1", "theorem isAdjointPair_zero : IsAdjointPair B B' 0 0", "theorem LinearMap.BilinForm.isSkewAdjoint_bracket {f g : Module.End R M}\n    (hf : f \u2208 B.skewAdjointSubmodule) (hg : g \u2208 B.skewAdjointSubmodule) :\n    \u2045f, g\u2046 \u2208 B.skewAdjointSubmodule", "theorem isSkewAdjoint_iff_neg_self_adjoint (f : Module.End R\u2081 M\u2081) :\n    B\u2081.IsSkewAdjoint f \u2194 IsAdjointPair (-B\u2081) B\u2081 f f", "@[simp]\ntheorem mem_orthogonal_iff {N : Submodule R M} {m : M} :\n    m \u2208 B.orthogonal N \u2194 \u2200 n \u2208 N, IsOrtho B n m", "@[simp]\ntheorem nondegenerate_toMatrix'_iff {B : BilinForm R\u2083 (\u03b9 \u2192 R\u2083)} :\n    B.toMatrix'.Nondegenerate (R := R\u2083) (m := \u03b9) \u2194 B.Nondegenerate", "theorem not_nondegenerate_zero [Nontrivial M] : \u00ac(0 : BilinForm R M).Nondegenerate", "lemma smul_right_of_tower (r : S) (x y : M) : B x (r \u2022 y) = r \u2022 B x y", "theorem span_singleton_sup_orthogonal_eq_top {B : BilinForm K V} {x : V} (hx : \u00acB.IsOrtho x x) :\n    (K \u2219 x) \u2294 B.orthogonal (K \u2219 x) = \u22a4", "theorem sub_left (x y z : M\u2081) : B\u2081 (x - y) z = B\u2081 x z - B\u2081 y z", "theorem sum_right {\u03b1} (t : Finset \u03b1) (w : M) (g : \u03b1 \u2192 M) :\n    B w (\u2211 i \u2208 t, g i) = \u2211 i \u2208 t, B w (g i)", "theorem toLin'Flip_apply (A : BilinForm R M) (x : M) : toLinHomFlip (M := M) A x = fun y => A y x", "theorem IsAdjointPair.smul (c : R) (h : IsAdjointPair B B' f g) :\n    IsAdjointPair B B' (c \u2022 f) (c \u2022 g)", "theorem isRefl : B.IsRefl", "theorem IsProj.eq_conj_prodMap {f : E \u2192\u2097[R] E} (h : IsProj p f) :\n    f = (p.prodEquivOfIsCompl (ker f) h.isCompl).conj (prodMap id 0)", "theorem ortho_comm {x y} : IsOrtho B x y \u2194 IsOrtho B y x", "theorem diagonalization_apply_self_apply (v : E) (\u03bc : Eigenvalues T) :\n    hT.diagonalization (T v) \u03bc = (\u03bc : \ud835\udd5c) \u2022 hT.diagonalization v \u03bc", "@[simp]\ntheorem diagonalization_symm_apply (w : PiLp 2 fun \u03bc : Eigenvalues T => eigenspace T \u03bc) :\n    hT.diagonalization.symm w = \u2211 \u03bc, w \u03bc", "theorem directSum_decompose_apply [_hT : Fact T.IsSymmetric] (x : E) (\u03bc : Eigenvalues T) :\n    DirectSum.decompose (fun \u03bc : Eigenvalues T => eigenspace T \u03bc) x \u03bc =\n      orthogonalProjection (eigenspace T \u03bc) x", "theorem aeval_eq_aeval_mod_charpoly (p : R[X]) : aeval f p = aeval f (p %\u2098 f.charpoly)", "theorem bound_of_ball_bound {r : \u211d} (r_pos : 0 < r) (c : \u211d) (f : E \u2192\u2097[\ud835\udd5c] F\u2097)\n    (h : \u2200 z \u2208 Metric.ball (0 : E) r, \u2016f z\u2016 \u2264 c) : \u2203 C, \u2200 z : E, \u2016f z\u2016 \u2264 C * \u2016z\u2016", "theorem charpoly_def : f.charpoly = (toMatrix (chooseBasis R M) (chooseBasis R M) f).charpoly", "@[simp]\ntheorem charpoly_toMatrix {\u03b9 : Type w} [DecidableEq \u03b9] [Fintype \u03b9] (b : Basis \u03b9 R M) :\n    (toMatrix b b f).charpoly = f.charpoly", "theorem coe_det [DecidableEq M] :\n    \u21d1(LinearMap.det : (M \u2192\u2097[A] M) \u2192* A) =\n      if H : \u2203 s : Finset M, Nonempty (Basis s A M) then\n        LinearMap.detAux (Trunc.mk H.choose_spec.some)\n      else 1", "theorem commute_pow_left_of_commute\n    [Semiring R\u2082] [AddCommMonoid M\u2082] [Module R\u2082 M\u2082] {\u03c3\u2081\u2082 : R \u2192+* R\u2082}\n    {f : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082} {g : Module.End R M} {g\u2082 : Module.End R\u2082 M\u2082}\n    (h : g\u2082.comp f = f.comp g) (k : \u2115) : (g\u2082 ^ k).comp f = f.comp (g ^ k)", "@[simp]\ntheorem compAlternatingMap_apply (g : N \u2192\u2097[R] N\u2082) (f : M [\u22c0^\u03b9]\u2192\u2097[R] N) (m : \u03b9 \u2192 M) :\n    g.compAlternatingMap f m = g (f m)", "theorem comp_apply (x : M\u2081) : f.comp g x = f (g x)", "theorem LinearMap.continuous_of_isClosed_ker (l : E \u2192\u2097[\ud835\udd5c] \ud835\udd5c)\n    (hl : IsClosed (LinearMap.ker l : Set E)) :\n    Continuous l", "theorem LinearMap.continuous_of_seq_closed_graph\n    (hg : \u2200 (u : \u2115 \u2192 E) (x y), Tendsto u atTop (\ud835\udcdd x) \u2192 Tendsto (g \u2218 u) atTop (\ud835\udcdd y) \u2192 y = g x) :\n    Continuous g", "@[simp]\ntheorem coprod_inl_inr : coprod (inl R M M\u2082) (inr R M M\u2082) = LinearMap.id", "theorem detAux_def'' {\u03b9' : Type*} [Fintype \u03b9'] [DecidableEq \u03b9'] (tb : Trunc <| Basis \u03b9 A M)\n    (b' : Basis \u03b9' A M) (f : M \u2192\u2097[A] M) :\n    LinearMap.detAux tb f = Matrix.det (LinearMap.toMatrix b' b' f)", "@[simp]\ntheorem det_toMatrix (b : Basis \u03b9 A M) (f : M \u2192\u2097[A] M) :\n    Matrix.det (toMatrix b b f) = LinearMap.det f", "theorem LinearMap.dualMap_comp_dualMap {M\u2083 : Type*} [AddCommGroup M\u2083] [Module R M\u2083]\n    (f : M\u2081 \u2192\u2097[R] M\u2082) (g : M\u2082 \u2192\u2097[R] M\u2083) : f.dualMap.comp g.dualMap = (g.comp f).dualMap", "theorem LinearMap.dualMap_def (f : M\u2081 \u2192\u2097[R] M\u2082) : f.dualMap = Module.Dual.transpose (R := R) f", "@[simp]\ntheorem dualMap_injective_iff {f : V\u2081 \u2192\u2097[K] V\u2082} :\n    Function.Injective f.dualMap \u2194 Function.Surjective f", "theorem LinearMap.eventually_disjoint_ker_pow_range_pow (f : M \u2192\u2097[R] M) :\n    \u2200\u1da0 n in atTop, Disjoint (LinearMap.ker (f ^ n)) (LinearMap.range (f ^ n))", "theorem ext_on {s : Set M} {f g : F} (hv : span R s = \u22a4) (h : Set.EqOn f g s) : f = g", "@[ext high]\ntheorem ext_ring_op {\u03c3 : R\u1d50\u1d52\u1d56 \u2192+* S} {f g : R \u2192\u209b\u2097[\u03c3] M\u2083} (h : f (1 : R) = g (1 : R)) :\n    f = g", "@[simp]\ntheorem extendTo\ud835\udd5c'_apply_re (fr : F \u2192\u2097[\u211d] \u211d) (x : F) : re (fr.extendTo\ud835\udd5c' x : \ud835\udd5c) = fr x", "theorem LinearMap.finrank_range_of_inj {f : M \u2192\u2097[R] N} (hf : Function.Injective f) :\n    finrank R (LinearMap.range f) = finrank R M", "@[simp] theorem fst_comp_inr : fst R M M\u2082 \u2218\u2097 inr R M M\u2082 = 0", "theorem fst_eq_coprod : fst R M M\u2082 = coprod LinearMap.id 0", "lemma hasEigenvalue_zero_tfae (\u03c6 : Module.End K M) :\n    List.TFAE [\n      Module.End.HasEigenvalue \u03c6 0,\n      IsRoot (minpoly K \u03c6) 0,\n      constantCoeff \u03c6.charpoly = 0,\n      LinearMap.det \u03c6 = 0,\n      \u22a5 < ker \u03c6,\n      \u2203 (m : M), m \u2260 0 \u2227 \u03c6 m = 0 ]", "@[simp]\ntheorem im_inner_adjoint_mul_self_eq_zero (T : E \u2192\u2097[\ud835\udd5c] E) (x : E) :\n    im \u27eax, LinearMap.adjoint T (T x)\u27eb = 0", "theorem inr_map_mul (b\u2081 b\u2082 : B) :\n    LinearMap.inr R A B (b\u2081 * b\u2082) = LinearMap.inr R A B b\u2081 * LinearMap.inr R A B b\u2082", "theorem isAdjointPair_zero : IsAdjointPair B B' 0 0", "@[simp]\ntheorem isNilpotent_mulRight_iff (a : A) : IsNilpotent (mulRight R a) \u2194 IsNilpotent a", "@[simp]\ntheorem LinearMap.isometryOfInner_toLinearMap (f : E \u2192\u2097[\ud835\udd5c] E') (h) :\n    (f.isometryOfInner h).toLinearMap = f", "theorem iterate_surjective (h : Surjective f') : \u2200 n : \u2115, Surjective (f' ^ n)", "theorem ker_id_sub_eq_of_proj {f : E \u2192\u2097[R] p} (hf : \u2200 x : p, f x = x) :\n    ker (id - p.subtype.comp f) = p", "@[simp]\ntheorem ker_le_comap {p : Submodule R\u2082 M\u2082} (f : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082) :\n    ker f \u2264 p.comap f", "theorem ker_le_of_iterateMapComap_eq_succ (K : Submodule R N)\n    (m : \u2115) (heq : f.iterateMapComap i m K = f.iterateMapComap i (m + 1) K)\n    (hf : Surjective f) (hi : Injective i) : LinearMap.ker f \u2264 K", "theorem ker_prodMap (f : M \u2192\u2097[R] M\u2082) (g : M\u2083 \u2192\u2097[R] M\u2084) :\n    ker (LinearMap.prodMap f g) = Submodule.prod (ker f) (ker g)", "theorem ker_stdBasis (i : \u03b9) : ker (stdBasis R \u03c6 i) = \u22a5", "@[simp]\ntheorem ker_zero : ker (0 : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082) = \u22a4", "theorem lTensor_comp : (g.comp f).lTensor M = (g.lTensor M).comp (f.lTensor M)", "@[simp]\ntheorem lTensor_neg (f : N \u2192\u2097[R] P) : (-f).lTensor M = -f.lTensor M", "theorem lift_rank_comp_le (g : V \u2192\u2097[K] V') (f : V' \u2192\u2097[K] V'') :\n    Cardinal.lift.{v'} (rank (f.comp g)) \u2264\n      min (Cardinal.lift.{v'} (rank f)) (Cardinal.lift.{v''} (rank g))", "theorem LinearMap.lift_rank_le_of_surjective (f : M \u2192\u2097[R] M') (h : Surjective f) :\n    lift.{v} (Module.rank R M') \u2264 lift.{v'} (Module.rank R M)", "theorem map_coprod_prod (f : M \u2192\u2097[R] M\u2083) (g : M\u2082 \u2192\u2097[R] M\u2083) (p : Submodule R M)\n    (q : Submodule R M\u2082) : map (coprod f g) (p.prod q) = map f p \u2294 map g q", "theorem map_eq_top_iff {f : F} (hf : range f = \u22a4) {p : Submodule R M} :\n    p.map f = \u22a4 \u2194 p \u2294 LinearMap.ker f = \u22a4", "theorem map_smul\u2082 (f : M\u2082 \u2192\u2097[R] N\u2082 \u2192\u209b\u2097[\u03c3\u2081\u2082] P\u2082) (r : R) (x y) : f (r \u2022 x) y = r \u2022 f x y", "@[simp]\ntheorem mem_isPairSelfAdjointSubmodule (f : Module.End R M) :\n    f \u2208 isPairSelfAdjointSubmodule B F \u2194 IsPairSelfAdjoint B F f", "@[simp]\ntheorem minpoly_toMatrix' (f : (n \u2192 R) \u2192\u2097[R] n \u2192 R) : minpoly R (toMatrix' f) = minpoly R f", "theorem mul_eq_comp (f g : Module.End R M) : f * g = f.comp g", "theorem mul_eq_one_of_mul_eq_one [FiniteDimensional K V] {f g : V \u2192\u2097[K] V} (hfg : f * g = 1) :\n    g * f = 1", "lemma nilRankAux_le [Nontrivial R] (b : Basis \u03b9 R L) (b' : Basis \u03b9' R L) :\n    nilRankAux \u03c6 b \u2264 nilRankAux \u03c6 b'", "lemma nilRank_le_card (b : Basis \u03b9 R M) : nilRank \u03c6 \u2264 Fintype.card \u03b9", "theorem ofIsCompl_eq (h : IsCompl p q) {\u03c6 : p \u2192\u2097[R] F} {\u03c8 : q \u2192\u2097[R] F} {\u03c7 : E \u2192\u2097[R] F}\n    (h\u03c6 : \u2200 u, \u03c6 u = \u03c7 u) (h\u03c8 : \u2200 u, \u03c8 u = \u03c7 u) : ofIsCompl h \u03c6 \u03c8 = \u03c7", "theorem ortho_smul_left {B : V\u2081 \u2192\u209b\u2097[I\u2081] V\u2082 \u2192\u209b\u2097[I\u2082] V} {x y} {a : K\u2081} (ha : a \u2260 0) :\n    IsOrtho B x y \u2194 IsOrtho B (a \u2022 x) y", "theorem ortho_smul_right {B : V\u2081 \u2192\u209b\u2097[I\u2081] V\u2082 \u2192\u209b\u2097[I\u2082] V} {x y} {a : K\u2082} {ha : a \u2260 0} :\n    IsOrtho B x y \u2194 IsOrtho B x (a \u2022 y)", "theorem orthogonal_span_singleton_eq_to_lin_ker {B : V \u2192\u2097[K] V \u2192\u209b\u2097[J] V\u2082} (x : V) :\n    Submodule.orthogonalBilin (K \u2219 x) B = LinearMap.ker (B x)", "lemma polyCharpolyAux_map_aeval\n    (A : Type*) [CommRing A] [Algebra R A] [Module.Finite A (A \u2297[R] M)] [Module.Free A (A \u2297[R] M)]\n    (x : \u03b9 \u2192 A) :\n    (polyCharpolyAux \u03c6 b b\u2098).map (MvPolynomial.aeval x).toRingHom =\n      LinearMap.charpoly ((tensorProduct R A M M).comp (baseChange A \u03c6)\n        ((basis A b).repr.symm (Finsupp.equivFunOnFinite.symm x)))", "lemma polyCharpoly_monic : (polyCharpoly \u03c6 b).Monic", "theorem pow_apply_mem_of_forall_mem {p : Submodule R M} (n : \u2115) (h : \u2200 x \u2208 p, f' x \u2208 p) (x : M)\n    (hx : x \u2208 p) : (f' ^ n) x \u2208 p", "theorem range_comp_of_range_eq_top [RingHomSurjective \u03c4\u2081\u2082] [RingHomSurjective \u03c4\u2082\u2083]\n    [RingHomSurjective \u03c4\u2081\u2083] {f : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082} (g : M\u2082 \u2192\u209b\u2097[\u03c4\u2082\u2083] M\u2083) (hf : range f = \u22a4) :\n    range (g.comp f : M \u2192\u209b\u2097[\u03c4\u2081\u2083] M\u2083) = range g", "theorem range_dualMap_eq_dualAnnihilator_ker_of_surjective (f : M \u2192\u2097[R] M')\n    (hf : Function.Surjective f) : LinearMap.range f.dualMap = f.ker.dualAnnihilator", "theorem range_le_iff_comap [RingHomSurjective \u03c4\u2081\u2082] {f : F} {p : Submodule R\u2082 M\u2082} :\n    range f \u2264 p \u2194 comap f p = \u22a4", "theorem separatingLeft_toLinearMap\u2082'_iff_det_ne_zero {M : Matrix \u03b9 \u03b9 R\u2081} :\n    M.toLinearMap\u2082'.SeparatingLeft \u2194 M.det \u2260 0", "theorem separatingLeft_toLinearMap\u2082'_of_det_ne_zero' (M : Matrix \u03b9 \u03b9 R\u2081) (h : M.det \u2260 0) :\n    M.toLinearMap\u2082'.SeparatingLeft", "theorem separatingRight_iff_flip_ker_eq_bot {B : M\u2081 \u2192\u209b\u2097[I\u2081] M\u2082 \u2192\u209b\u2097[I\u2082] M} :\n    B.SeparatingRight \u2194 LinearMap.ker B.flip = \u22a5", "theorem splittingOfFinsuppSurjective_splits (f : M \u2192\u2097[R] \u03b1 \u2192\u2080 R) (s : Surjective f) :\n    f.comp (splittingOfFinsuppSurjective f s) = LinearMap.id", "@[simp]\ntheorem sub_apply (f g : M \u2192\u209b\u2097[\u03c3\u2081\u2082] N\u2082) (x : M) : (f - g) x = f x - g x", "theorem sup_range_inl_inr : (range <| inl R M M\u2082) \u2294 (range <| inr R M M\u2082) = \u22a4", "@[deprecated (since := \"2024-06-05\")]\ntheorem tailing_le_tunnel (f : M \u00d7 N \u2192\u2097[R] M) (i : Injective f) (n : \u2115) :\n    tailing f i n \u2264 OrderDual.ofDual (\u03b1 := Submodule R M) (tunnel f i n)", "@[simp]\ntheorem LinearMap.toMatrix'_one : LinearMap.toMatrix' (1 : (n \u2192 R) \u2192\u2097[R] n \u2192 R) = 1", "theorem LinearMap.toMatrixAlgEquiv_transpose_apply' (f : M\u2081 \u2192\u2097[R] M\u2081) (j : n) :\n    (LinearMap.toMatrixAlgEquiv v\u2081 f)\u1d40 j = v\u2081.repr (f (v\u2081 j))", "theorem LinearMap.toMatrix_apply' (f : M\u2081 \u2192\u2097[R] M\u2082) (i : m) (j : n) :\n    LinearMap.toMatrix v\u2081 v\u2082 f i j = v\u2082.repr (f (v\u2081 j)) i", "theorem LinearMap.toMatrix\u2082'_compl\u2082 (B : (n \u2192 R) \u2192\u2097[R] (m \u2192 R) \u2192\u2097[R] R) (f : (m' \u2192 R) \u2192\u2097[R] m \u2192 R) :\n    toMatrix\u2082' (B.compl\u2082 f) = toMatrix\u2082' B * toMatrix' f", "lemma toMvPolynomial_totalDegree_le (f : M\u2081 \u2192\u2097[R] M\u2082) (i : \u03b9\u2082) :\n    (f.toMvPolynomial b\u2081 b\u2082 i).totalDegree \u2264 1", "lemma trace_mul_cycle (f g h : M \u2192\u2097[R] M) :\n    trace R M (f * g * h) = trace R M (h * f * g)", "theorem update_apply (f : (i : \u03b9) \u2192 M\u2082 \u2192\u2097[R] \u03c6 i) (c : M\u2082) (i j : \u03b9) (b : M\u2082 \u2192\u2097[R] \u03c6 i) :\n    (update f i b j) c = update (fun i => f i c) i (b c) j", "theorem LinearOrder.concaveOn_of_lt (hs : Convex \ud835\udd5c s)\n    (hf : \u2200 \u2983x\u2984, x \u2208 s \u2192 \u2200 \u2983y\u2984, y \u2208 s \u2192 x < y \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192\n      a \u2022 f x + b \u2022 f y \u2264 f (a \u2022 x + b \u2022 y)) :\n    ConcaveOn \ud835\udd5c s f", "theorem IsClosable.graph_closure_eq_closure_graph {f : E \u2192\u2097.[R] F} (hf : f.IsClosable) :\n    f.graph.topologicalClosure = f.closure.graph", "theorem eq_of_le_of_domain_eq {f g : E \u2192\u2097.[R] F} (hle : f \u2264 g) (heq : f.domain = g.domain) :\n    f = g", "theorem graph_map_snd_eq_range (f : E \u2192\u2097.[R] F) :\n    f.graph.map (LinearMap.snd R E F) = LinearMap.range f.toFun", "theorem inverse_domain : (inverse f).domain = LinearMap.range f.toFun", "theorem le_closure (f : E \u2192\u2097.[R] F) : f \u2264 f.closure", "@[simp]\ntheorem map_zero (f : E \u2192\u2097.[R] F) : f 0 = 0", "theorem mem_domain_iff_of_eq_graph {f g : E \u2192\u2097.[R] F} (h : f.graph = g.graph) {x : E} :\n    x \u2208 f.domain \u2194 x \u2208 g.domain", "@[simp]\ntheorem mem_graph_iff (f : E \u2192\u2097.[R] F) {x : E \u00d7 F} :\n    x \u2208 f.graph \u2194 \u2203 y : f.domain, (\u2191y : E) = x.1 \u2227 f y = x.2", "protected theorem sSup_le {c : Set (E \u2192\u2097.[R] F)} (hc : DirectedOn (\u00b7 \u2264 \u00b7) c) {g : E \u2192\u2097.[R] F}\n    (hg : \u2200 f \u2208 c, f \u2264 g) : LinearPMap.sSup c hc \u2264 g", "protected theorem liouvilleWith (hx : Liouville x) (p : \u211d) : LiouvilleWith p x", "protected theorem transcendental {x : \u211d} (lx : Liouville x) : Transcendental \u2124 x", "theorem partialSum_succ (m : \u211d) (n : \u2115) :\n    partialSum m (n + 1) = partialSum m n + 1 / m ^ (n + 1)!", "theorem mono (h : LiouvilleWith p x) (hle : q \u2264 p) : LiouvilleWith q x", "theorem LipschitzOnWith.coordinate [PseudoMetricSpace \u03b1] (f : \u03b1 \u2192 \u2113^\u221e(\u03b9)) (s : Set \u03b1) (K : \u211d\u22650) :\n    LipschitzOnWith K f s \u2194 \u2200 i : \u03b9, LipschitzOnWith K (fun a : \u03b1 \u21a6 f a i) s", "theorem hausdorffMeasure_image_le (h : LipschitzOnWith K f s) {d : \u211d} (hd : 0 \u2264 d) :\n    \u03bcH[d] (f '' s) \u2264 (K : \u211d\u22650\u221e) ^ d * \u03bcH[d] s", "protected theorem le_add_mul {f : \u03b1 \u2192 \u211d} {K : \u211d\u22650} (h : LipschitzOnWith K f s) {x : \u03b1} (hx : x \u2208 s)\n    {y : \u03b1} (hy : y \u2208 s) : f x \u2264 f y + K * dist x y", "protected theorem mk_one (h : \u2200 x \u2208 s, \u2200 y \u2208 s, dist (f x) (f y) \u2264 dist x y) :\n    LipschitzOnWith 1 f s", "protected theorem of_le_add_mul' {f : \u03b1 \u2192 \u211d} (K : \u211d)\n    (h : \u2200 x \u2208 s, \u2200 y \u2208 s, f x \u2264 f y + K * dist x y) : LipschitzOnWith (Real.toNNReal K) f s", "theorem LipschitzWith.comp_boundedVariationOn {f : E \u2192 F} {C : \u211d\u22650} (hf : LipschitzWith C f)\n    {g : \u03b1 \u2192 E} {s : Set \u03b1} (h : BoundedVariationOn g s) : BoundedVariationOn (f \u2218 g) s", "theorem LipschitzWith.completion_map [PseudoMetricSpace \u03b2] {f : \u03b1 \u2192 \u03b2} {K : \u211d\u22650}\n    (h : LipschitzWith K f) : LipschitzWith K (Completion.map f)", "theorem dist_iterate_succ_le_geometric {f : \u03b1 \u2192 \u03b1} (hf : LipschitzWith K f) (x n) :\n    dist (f^[n] x) (f^[n + 1] x) \u2264 dist x (f x) * (K : \u211d) ^ n", "theorem edist_lt_mul_of_lt (h : LipschitzWith K f) (hK : K \u2260 0) (hr : edist x y < r) :\n    edist (f x) (f y) < K * r", "theorem integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul'\n    (hf : LipschitzWith C f) (h'f : HasCompactSupport f) (hg : Continuous g) (v : E) :\n    Tendsto (fun (t : \u211d) \u21a6 \u222b x, (t\u207b\u00b9 \u2022 (f (x + t \u2022 v) - f x)) * g x \u2202\u03bc) (\ud835\udcdd[>] 0)\n      (\ud835\udcdd (\u222b x, lineDeriv \u211d f x v * g x \u2202\u03bc))", "protected theorem list_prod (f : \u03b9 \u2192 Function.End \u03b1) (K : \u03b9 \u2192 \u211d\u22650)\n    (h : \u2200 i, LipschitzWith (K i) (f i)) : \u2200 l : List \u03b9, LipschitzWith (l.map K).prod (l.map f).prod", "protected theorem min (hf : LipschitzWith Kf f) (hg : LipschitzWith Kg g) :\n    LipschitzWith (max Kf Kg) fun x => min (f x) (g x)", "protected theorem mk_one (h : \u2200 x y, dist (f x) (f y) \u2264 dist x y) : LipschitzWith 1 f", "@[to_additive LipschitzWith.nnorm_le_mul]\ntheorem LipschitzWith.nnorm_le_mul' {f : E \u2192 F} {K : \u211d\u22650} (h : LipschitzWith K f) (hf : f 1 = 1)\n    (x) : \u2016f x\u2016\u208a \u2264 K * \u2016x\u2016\u208a", "theorem norm_compLp_sub_le (hg : LipschitzWith c g) (g0 : g 0 = 0) (f f' : Lp E p \u03bc) :\n    \u2016hg.compLp g0 f - hg.compLp g0 f'\u2016 \u2264 c * \u2016f - f'\u2016", "protected theorem uniformContinuous (hf : LipschitzWith K f) : UniformContinuous f", "theorem count_false_le_count_true_add_one (hl : Chain' (\u00b7 \u2260 \u00b7) l) :\n    count false l \u2264 count true l + 1", "theorem count_not_eq_count (hl : Chain' (\u00b7 \u2260 \u00b7) l) (h2 : Even (length l)) (b : Bool) :\n    count (!b) l = count b l", "theorem Chain'.imp_head {x y} (h : \u2200 {z}, R x z \u2192 R y z) {l} (hl : Chain' R (x :: l)) :\n    Chain' R (y :: l)", "theorem Duplicate.mono_sublist {l' : List \u03b1} (hx : x \u2208+ l) (h : l <+ l') : x \u2208+ l'", "theorem Duplicate.ne_singleton (h : x \u2208+ l) (y : \u03b1) : l \u2260 [y]", "@[simp]\ntheorem filter_le_of_bot {n m : \u2115} (hnm : n < m) : ((Ico n m).filter fun x => x \u2264 n) = [n]", "theorem filter_le_of_top_le {n m l : \u2115} (hml : m \u2264 l) : ((Ico n m).filter fun x => l \u2264 x) = []", "theorem filter_lt_of_ge {n m l : \u2115} (hlm : l \u2264 m) :\n    ((Ico n m).filter fun x => x < l) = Ico n l", "theorem IsPrefix.length_le (h : l\u2081 <+: l\u2082) : l\u2081.length \u2264 l\u2082.length", "theorem IsRotated.map {\u03b2 : Type*} {l\u2081 l\u2082 : List \u03b1} (h : l\u2081 ~r l\u2082) (f : \u03b1 \u2192 \u03b2) :\n    map f l\u2081 ~r map f l\u2082", "theorem IsRotated.nodup_iff (h : l ~r l') : Nodup l \u2194 Nodup l'", "theorem IsSuffix.eq_of_length (h : l\u2081 <:+ l\u2082) : l\u2081.length = l\u2082.length \u2192 l\u2081 = l\u2082", "theorem IsSuffix.filter (p : \u03b1 \u2192 Bool) \u2983l\u2081 l\u2082 : List \u03b1\u2984 (h : l\u2081 <:+ l\u2082) :\n    l\u2081.filter p <:+ l\u2082.filter p", "theorem antidiagonalTuple_zero_right : \u2200 k, antidiagonalTuple k 0 = [0]", "theorem antidiagonal_succ' {n : \u2115} :\n    antidiagonal (n + 1) = (antidiagonal n).map (Prod.map id Nat.succ) ++ [(n + 1, 0)]", "theorem Nodup.diff_eq_filter [DecidableEq \u03b1] :\n    \u2200 {l\u2081 l\u2082 : List \u03b1} (_ : l\u2081.Nodup), l\u2081.diff l\u2082 = l\u2081.filter (\u00b7 \u2209 l\u2082)", "@[simp]\ntheorem Nodup.pairwise_coe [IsSymm \u03b1 r] (hl : l.Nodup) :\n    { a | a \u2208 l }.Pairwise r \u2194 l.Pairwise r", "theorem Nodup.rotate_eq_self_iff {l : List \u03b1} (hl : l.Nodup) {n : \u2115} :\n    l.rotate n = l \u2194 n % l.length = 0 \u2228 l = []", "theorem of_reverse_eq {l : List \u03b1} : reverse l = l \u2192 Palindrome l", "theorem Perm.filter (p : \u03b1 \u2192 Bool) {l\u2081 l\u2082 : List \u03b1} (s : l\u2081 ~ l\u2082) :\n    filter p l\u2081 ~ filter p l\u2082", "theorem Perm.filterMap (f : \u03b1 \u2192 Option \u03b2) {l\u2081 l\u2082 : List \u03b1} (p : l\u2081 ~ l\u2082) :\n    filterMap f l\u2081 ~ filterMap f l\u2082", "theorem Perm.kerase {a : \u03b1} {l\u2081 l\u2082 : List (Sigma \u03b2)} (nd : l\u2081.NodupKeys) :\n    l\u2081 ~ l\u2082 \u2192 kerase a l\u2081 ~ kerase a l\u2082", "theorem Perm.map (f : \u03b1 \u2192 \u03b2) {l\u2081 l\u2082 : List \u03b1} (p : l\u2081 ~ l\u2082) : map f l\u2081 ~ map f l\u2082", "theorem Sublist.countP_le (s : l\u2081 <+ l\u2082) : countP p l\u2081 \u2264 countP p l\u2082", "theorem Sublist.erase_diff_erase_sublist {a : \u03b1} :\n    \u2200 {l\u2081 l\u2082 : List \u03b1}, l\u2081 <+ l\u2082 \u2192 (l\u2082.erase a).diff (l\u2081.erase a) <+ l\u2082.diff l\u2081", "theorem Sublist.length_le : l\u2081 <+ l\u2082 \u2192 length l\u2081 \u2264 length l\u2082", "theorem Sublist.map (f : \u03b1 \u2192 \u03b2) {l\u2081 l\u2082 : List \u03b1} (s : l\u2081 <+ l\u2082) : map f l\u2081 <+ map f l\u2082", "@[gcongr] protected theorem Sublist.tail : \u2200 {l\u2081 l\u2082 : List \u03b1}, l\u2081 <+ l\u2082 \u2192 tail l\u2081 <+ tail l\u2082", "theorem Subperm.count_le [DecidableEq \u03b1] {l\u2081 l\u2082 : List \u03b1} (s : l\u2081 <+~ l\u2082) (a) :\n    count a l\u2081 \u2264 count a l\u2082", "@[to_additive (attr := measurability)]\ntheorem _root_.List.aestronglyMeasurable_prod' (l : List (\u03b1 \u2192 M))\n    (hl : \u2200 f \u2208 l, AEStronglyMeasurable f \u03bc) : AEStronglyMeasurable l.prod \u03bc", "@[to_additive]\ntheorem alternatingProd_cons_cons' (a b : \u03b1) (l : List \u03b1) :\n    alternatingProd (a :: b :: l) = a * b\u207b\u00b9 * alternatingProd l", "@[simp]\ntheorem argAux_self (hr\u2080 : Irreflexive r) (a : \u03b1) : argAux r (some a) a = a", "theorem argmax_cons (f : \u03b1 \u2192 \u03b2) (a : \u03b1) (l : List \u03b1) :\n    argmax f (a :: l) =\n      Option.casesOn (argmax f l) (some a) fun c => if f a < f c then some c else some a", "theorem attach_map_coe' (l : List \u03b1) (f : \u03b1 \u2192 \u03b2) :\n    (l.attach.map fun (i : {i // i \u2208 l}) => f i) = l.map f", "@[simp]\ntheorem chain'_cons {x y l} : Chain' R (x :: y :: l) \u2194 R x y \u2227 Chain' R (y :: l)", "lemma chain'_join : \u2200 {L : List (List \u03b1)}, [] \u2209 L \u2192\n    (Chain' R L.join \u2194 (\u2200 l \u2208 L, Chain' R l) \u2227\n    L.Chain' (fun l\u2081 l\u2082 => \u2200\u1d49 (x \u2208 l\u2081.getLast?) (y \u2208 l\u2082.head?), R x y))\n| [], _ => by simp\n| [l], _ => by simp [join]\n| (l\u2081 :: l\u2082 :: L), hL => by\n    rw [mem_cons, not_or, \u2190 Ne] at hL\n    rw [join, chain'_append, chain'_join hL.2, forall_mem_cons, chain'_cons]\n    rw [mem_cons, not_or, \u2190 Ne] at hL\n    simp only [forall_mem_cons, and_assoc, join, head?_append_of_ne_nil _ hL.2.1.symm]\n    exact Iff.rfl.and (Iff.rfl.and <| Iff.rfl.and and_comm)", "theorem choose_mem (hp : \u2203 a, a \u2208 l \u2227 p a) : choose p l hp \u2208 l", "theorem comp_map (h : \u03b2 \u2192 \u03b3) (g : \u03b1 \u2192 \u03b2) (l : List \u03b1) : map (h \u2218 g) l = map h (map g l)", "protected theorem comp_traverse {\u03b1 \u03b2 \u03b3} (f : \u03b2 \u2192 F \u03b3) (g : \u03b1 \u2192 G \u03b2) (x : List \u03b1) :\n    List.traverse (Comp.mk \u2218 (f <$> \u00b7) \u2218 g) x =\n    Comp.mk (List.traverse f <$> List.traverse g x)", "theorem concat_eq_reverse_cons (a : \u03b1) (l : List \u03b1) : concat l a = reverse (a :: reverse l)", "theorem continuousAt_length : \u2200 l : List \u03b1, ContinuousAt List.length l", "lemma countP_bind (p : \u03b2 \u2192 Bool) (l : List \u03b1) (f : \u03b1 \u2192 List \u03b2) :\n    countP p (l.bind f) = sum (map (countP p \u2218 f) l)", "@[simp] theorem countP_cons_of_neg (l) (pa : \u00acp a) : countP p (a :: l) = countP p l", "@[simp] theorem countP_map (p : \u03b2 \u2192 Bool) (f : \u03b1 \u2192 \u03b2) :\n    \u2200 l, countP p (map f l) = countP (p \u2218 f) l", "@[simp] theorem countP_nil : countP p [] = 0", "@[simp]\ntheorem count_not_add_count (l : List Bool) (b : Bool) : count (!b) l + count b l = length l", "theorem count_singleton' (a b : \u03b1) : count a [b] = if a = b then 1 else 0", "theorem count_tail : \u2200 (l : List \u03b1) (a : \u03b1) (h : l \u2260 []),\n      l.tail.count a = l.count a - if a = l.head h then 1 else 0", "theorem cyclicPermutations_of_ne_nil (l : List \u03b1) (h : l \u2260 []) :\n    cyclicPermutations l = dropLast (zipWith (\u00b7 ++ \u00b7) (tails l) (inits l))", "@[simp]\ntheorem List.dProdIndex_nil (f\u03b9 : \u03b1 \u2192 \u03b9) : ([] : List \u03b1).dProdIndex f\u03b9 = 0", "@[simp]\ntheorem List.dProd_cons (f\u03b9 : \u03b1 \u2192 \u03b9) (fA : \u2200 a, A (f\u03b9 a)) (a : \u03b1) (l : List \u03b1) :\n    (a :: l).dProd f\u03b9 fA = (GradedMonoid.GMul.mul (fA a) (l.dProd f\u03b9 fA) : _)", "theorem dedup_cons_of_not_mem' {a : \u03b1} {l : List \u03b1} (h : a \u2209 dedup l) :\n    dedup (a :: l) = a :: dedup l", "@[simp]\ntheorem destutter'_singleton : [b].destutter' R a = if R a b then [a, b] else [a]", "@[simp] theorem diff_cons (l\u2081 l\u2082 : List \u03b1) (a : \u03b1) : l\u2081.diff (a :: l\u2082) = (l\u2081.erase a).diff l\u2082", "theorem diff_erase (l\u2081 l\u2082 : List \u03b1) (a : \u03b1) : (l\u2081.diff l\u2082).erase a = (l\u2081.erase a).diff l\u2082", "@[simp] theorem disjoint_append_left : Disjoint (l\u2081 ++ l\u2082) l \u2194 Disjoint l\u2081 l \u2227 Disjoint l\u2082 l", "theorem disjoint_pmap {p : \u03b1 \u2192 Prop} {f : \u2200 a : \u03b1, p a \u2192 \u03b2} {s t : List \u03b1}\n    (hs : \u2200 a \u2208 s, p a) (ht : \u2200 a \u2208 t, p a)\n    (hf : \u2200 (a a' : \u03b1) (ha : p a) (ha' : p a'), f a ha = f a' ha' \u2192 a = a')\n    (h : Disjoint s t) :\n    Disjoint (s.pmap f hs) (t.pmap f ht)", "theorem disjoint_right : Disjoint l\u2081 l\u2082 \u2194 \u2200 \u2983a\u2984, a \u2208 l\u2082 \u2192 a \u2209 l\u2081", "theorem dropWhile_idempotent : dropWhile p (dropWhile p l) = dropWhile p l", "lemma drop_sum_join (L : List (List \u03b1)) (i : \u2115) :\n    L.join.drop ((L.map length).take i).sum = (L.drop i).join", "theorem duplicate_iff_two_le_count [DecidableEq \u03b1] : x \u2208+ l \u2194 2 \u2264 count x l", "theorem eq_of_mem_map_const {b\u2081 b\u2082 : \u03b2} {l : List \u03b1} (h : b\u2081 \u2208 map (const \u03b1 b\u2082) l) :\n    b\u2081 = b\u2082", "@[to_additive]\nlemma eq_of_prod_take_eq [LeftCancelMonoid M] {L L' : List M} (h : L.length = L'.length)\n    (h' : \u2200 i \u2264 L.length, (L.take i).prod = (L'.take i).prod) : L = L'", "theorem eraseP_map (f : \u03b2 \u2192 \u03b1) : \u2200 (l : List \u03b2), (map f l).eraseP p = map f (l.eraseP (p \u2218 f))", "theorem erase_append_right [LawfulBEq \u03b1] {a : \u03b1} {l\u2081 : List \u03b1} (l\u2082 : List \u03b1) (h : a \u2209 l\u2081) :\n    (l\u2081 ++ l\u2082).erase a = (l\u2081 ++ l\u2082.erase a)", "theorem erase_orderedInsert [DecidableEq \u03b1] [IsRefl \u03b1 r] (x : \u03b1) (xs : List \u03b1) :\n    (xs.orderedInsert r x).erase x = xs", "theorem List.exists_pw_disjoint_with_card {\u03b1 : Type*} [Fintype \u03b1]\n    {c : List \u2115} (hc : c.sum \u2264 Fintype.card \u03b1) :\n    \u2203 o : List (List \u03b1),\n      o.map length = c \u2227 (\u2200 s \u2208 o, s.Nodup) \u2227 Pairwise List.Disjoint o", "theorem exists_tfae {\u03b1 : Type*} (l : List (\u03b1 \u2192 Prop)) (H : \u2200 a : \u03b1, (l.map (fun p \u21a6 p a)).TFAE) :\n    (l.map (fun p \u21a6 \u2203 a, p a)).TFAE", "theorem ext_get?' {l\u2081 l\u2082 : List \u03b1} (h' : \u2200 n < max l\u2081.length l\u2082.length, l\u2081.get? n = l\u2082.get? n) :\n    l\u2081 = l\u2082", "theorem ext_get_iff {l\u2081 l\u2082 : List \u03b1} :\n    l\u2081 = l\u2082 \u2194 l\u2081.length = l\u2082.length \u2227 \u2200 n h\u2081 h\u2082, get l\u2081 \u27e8n, h\u2081\u27e9 = get l\u2082 \u27e8n, h\u2082\u27e9", "theorem findIdx_eq_length {p : \u03b1 \u2192 Bool} {xs : List \u03b1} :\n    xs.findIdx p = xs.length \u2194 \u2200 x \u2208 xs, \u00acp x", "theorem findIdx_lt_length_of_exists {xs : List \u03b1} (h : \u2203 x \u2208 xs, p x) :\n    xs.findIdx p < xs.length", "theorem findIdx_of_get?_eq_some {xs : List \u03b1} (w : xs.get? (xs.findIdx p) = some y) : p y", "theorem foldl1_eq_foldr1 : \u2200 a b l, foldl f a (l ++ [b]) = foldr f b (a :: l)", "@[simp] theorem foldl_append {\u03b2 : Type _} (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (b) (l l' : List \u03b1) :\n    (l ++ l').foldl f b = l'.foldl f (l.foldl f b)", "@[simp] theorem foldl_nil : [].foldl f b = b", "@[simp]\ntheorem foldrRecOn_cons {C : \u03b2 \u2192 Sort*} (x : \u03b1) (l : List \u03b1) (op : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (b) (hb : C b)\n    (hl : \u2200 b, C b \u2192 \u2200 a \u2208 x :: l, C (op a b)) :\n    foldrRecOn (x :: l) op b hb hl =\n      hl _ (foldrRecOn l op b hb fun b hb a ha => hl b hb a (mem_cons_of_mem _ ha)) x\n        (mem_cons_self _ _)", "theorem foldr_ext (f g : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (b : \u03b2) {l : List \u03b1} (H : \u2200 a \u2208 l, \u2200 b : \u03b2, f a b = g a b) :\n    foldr f b l = foldr g b l", "theorem foldr_monotone [Preorder \u03b2] {f : \u03b1 \u2192 \u03b2 \u2192 \u03b2} (H : \u2200 a, Monotone (f a)) (l : List \u03b1) :\n    Monotone fun b \u21a6 l.foldr f b", "@[simp] theorem foldr_reverse (l : List \u03b1) (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (b) :\n    l.reverse.foldr f b = l.foldl (fun x y => f y x) b", "@[simp] theorem forM_append [Monad m] [LawfulMonad m] (l\u2081 l\u2082 : List \u03b1) (f : \u03b1 \u2192 m PUnit) :\n    (l\u2081 ++ l\u2082).forM f = (do l\u2081.forM f; l\u2082.forM f)", "theorem forall_mem_of_forall_mem_cons {p : \u03b1 \u2192 Prop} {a : \u03b1} {l : List \u03b1} (h : \u2200 x \u2208 a :: l, p x) :\n    \u2200 x \u2208 l, p x", "theorem forall_zipWith {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3} {p : \u03b3 \u2192 Prop} :\n    \u2200 {l\u2081 : List \u03b1} {l\u2082 : List \u03b2}, length l\u2081 = length l\u2082 \u2192\n      (Forall p (zipWith f l\u2081 l\u2082) \u2194 Forall\u2082 (fun x y => p (f x y)) l\u2081 l\u2082)", "theorem forall\u2082_and_left {p : \u03b1 \u2192 Prop} :\n    \u2200 l u, Forall\u2082 (fun a b => p a \u2227 R a b) l u \u2194 (\u2200 a \u2208 l, p a) \u2227 Forall\u2082 R l u", "theorem forall\u2082_comp_perm_eq_perm_comp_forall\u2082 : Forall\u2082 r \u2218r Perm = Perm \u2218r Forall\u2082 r", "theorem forall\u2082_cons_left_iff {a l u} :\n    Forall\u2082 R (a :: l) u \u2194 \u2203 b u', R a b \u2227 Forall\u2082 R l u' \u2227 u = b :: u'", "@[simp]\ntheorem forall\u2082_map_right_iff {f : \u03b3 \u2192 \u03b2} :\n    \u2200 {l u}, Forall\u2082 R l (map f u) \u2194 Forall\u2082 (fun a c => R a (f c)) l u", "theorem forall\u2082_refl [IsRefl \u03b1 R\u2090] (l : List \u03b1) : Forall\u2082 R\u2090 l l", "@[simp]\ntheorem formPerm_cons_concat_apply_last (x y : \u03b1) (xs : List \u03b1) :\n    formPerm (x :: (xs ++ [y])) y = x", "@[simp]\ntheorem formPerm_nil : formPerm ([] : List \u03b1) = 1", "@[simp] theorem get?_eq_getElem? (l : List \u03b1) (i : Nat) : l.get? i = l[i]?", "@[deprecated getElem?_length (since := \"2024-06-12\")]\ntheorem get?_length (l : List \u03b1) : l.get? l.length = none", "@[deprecated getElem?_modifyNth (since := \"2024-06-12\")]\ntheorem get?_modifyNth (f : \u03b1 \u2192 \u03b1) (n) (l : List \u03b1) (m) :\n    (modifyNth f n l).get? m = (fun a => if n = m then f a else a) <$> l.get? m", "theorem getElem?_range' (s step) :\n    \u2200 {m n : Nat}, m < n \u2192 (range' s n step)[m]? = some (s + step * m)", "theorem getElem?_rotate {l : List \u03b1} {n m : \u2115} (hml : m < l.length) :\n    (l.rotate n)[m]? = l[(m + n) % l.length]?", "theorem getElem_drop' (L : List \u03b1) {i : Nat} {j : Nat} {h : j < (L.drop i).length} :\n    (L.drop i)[j] = L[i + j]'(by\n      rw [Nat.add_comm]\n      exact Nat.add_lt_of_lt_sub (length_drop i L \u25b8 h))", "theorem getElem_insertNth_add_succ (l : List \u03b1) (x : \u03b1) (n k : \u2115) (hk' : n + k < l.length)\n    (hk : n + k + 1 < (insertNth n x l).length := (by\n      rwa [length_insertNth _ _ (by omega), Nat.succ_lt_succ_iff])) :\n    (insertNth n x l)[n + k + 1] = l[n + k]", "theorem getElem_ofFn_go {n} (f : Fin n \u2192 \u03b1) (i j h) (k) (hk) :\n    (ofFn.go f i j h)[k] = f \u27e8j + k, by simp at hk; omega\u27e9", "theorem getElem_rotate (l : List \u03b1) (n : \u2115) (k : Nat) (h : k < (l.rotate n).length) :\n    (l.rotate n)[k] =\n      l[(k + n) % l.length]'(mod_lt _ (length_rotate l n \u25b8 k.zero_le.trans_lt h))", "theorem getLastI_eq_getLast? [Inhabited \u03b1] : \u2200 l : List \u03b1, l.getLastI = l.getLast?.iget", "@[deprecated getElem_append (since := \"2024-06-12\")]\ntheorem get_append {l\u2081 l\u2082 : List \u03b1} (n : Nat) (h : n < l\u2081.length) :\n    (l\u2081 ++ l\u2082).get \u27e8n, length_append .. \u25b8 Nat.lt_add_right _ h\u27e9 = l\u2081.get \u27e8n, h\u27e9", "theorem get_cyclicPermutations (l : List \u03b1) (n : Fin (length (cyclicPermutations l))) :\n    (cyclicPermutations l).get n = l.rotate n", "theorem get_pmap {p : \u03b1 \u2192 Prop} (f : \u2200 a, p a \u2192 \u03b2) {l : List \u03b1} (h : \u2200 a \u2208 l, p a) {n : \u2115}\n    (hn : n < (pmap f l h).length) :\n    get (pmap f l h) \u27e8n, hn\u27e9 =\n      f (get l \u27e8n, @length_pmap _ _ p f l h \u25b8 hn\u27e9)\n        (h _ (get_mem l n (@length_pmap _ _ p f l h \u25b8 hn)))", "theorem get_tails (l : List \u03b1) (n : Fin (length (tails l))) : (tails l).get n = l.drop n", "theorem head?_append_of_ne_nil :\n    \u2200 (l\u2081 : List \u03b1) {l\u2082 : List \u03b1} (_ : l\u2081 \u2260 []), head? (l\u2081 ++ l\u2082) = head? l\u2081", "@[simp] theorem headI_cons [Inhabited \u03b1] {h : \u03b1} {t : List \u03b1} : (h :: t).headI = h", "theorem indexOf_append_of_not_mem {a : \u03b1} (h : a \u2209 l\u2081) :\n    indexOf a (l\u2081 ++ l\u2082) = l\u2081.length + indexOf a l\u2082", "theorem indexOf_cons [BEq \u03b1] :\n    (x :: xs : List \u03b1).indexOf y = bif x == y then 0 else xs.indexOf y + 1", "theorem index_of_argmax :\n    \u2200 {l : List \u03b1} {m : \u03b1}, m \u2208 argmax f l \u2192 \u2200 {a}, a \u2208 l \u2192 f m \u2264 f a \u2192 l.indexOf m \u2264 l.indexOf a", "theorem infix_bind_of_mem {a : \u03b1} {as : List \u03b1} (h : a \u2208 as) (f : \u03b1 \u2192 List \u03b1) :\n    f a <:+: as.bind f", "theorem infix_concat : l\u2081 <:+: l\u2082 \u2192 l\u2081 <:+: concat l\u2082 a", "theorem infix_cons : l\u2081 <:+: l\u2082 \u2192 l\u2081 <:+: a :: l\u2082", "@[csimp] theorem inits_eq_initsTR : @inits = @initsTR", "theorem inits_reverse (l : List \u03b1) : inits (reverse l) = reverse (map reverse l.tails)", "@[simp] theorem insert_of_not_mem {l : List \u03b1} (h : a \u2209 l) : l.insert a = a :: l", "theorem inter_eq_nil_iff_disjoint : l\u2081 \u2229 l\u2082 = [] \u2194 Disjoint l\u2081 l\u2082", "theorem isRotated_comm : l ~r l' \u2194 l' ~r l", "@[simp]\ntheorem isRotated_cyclicPermutations_iff {l l' : List \u03b1} :\n    l.cyclicPermutations ~r l'.cyclicPermutations \u2194 l ~r l'", "theorem isSubperm_iff {l\u2081 l\u2082 : List \u03b1} : l\u2081.isSubperm l\u2082 \u2194 l\u2081 <+~ l\u2082", "@[simp]\ntheorem join_eq_nil : \u2200 {L : List (List \u03b1)}, join L = [] \u2194 \u2200 l \u2208 L, l = []", "theorem keys_kreplace (a : \u03b1) (b : \u03b2 a) : \u2200 l : List (Sigma \u03b2), (kreplace a b l).keys = l.keys", "theorem kreplace_nodupKeys (a : \u03b1) (b : \u03b2 a) {l : List (Sigma \u03b2)} :\n    (kreplace a b l).NodupKeys \u2194 l.NodupKeys", "theorem last_ofFn_succ {n : \u2115} (f : Fin n.succ \u2192 \u03b1)\n    (h : ofFn f \u2260 [] := mt ofFn_eq_nil_iff.mp (Nat.succ_ne_zero _)) :\n    getLast (ofFn f) h = f (Fin.last _)", "@[simp]\ntheorem length_attach (L : List \u03b1) : L.attach.length = L.length", "@[simp] theorem length_cons {\u03b1} (a : \u03b1) (as : List \u03b1) : (cons a as).length = as.length + 1", "theorem length_eq_three {l : List \u03b1} : l.length = 3 \u2194 \u2203 a b c, l = [a, b, c]", "@[simp]\ntheorem length_inits (l : List \u03b1) : length (inits l) = length l + 1", "@[simp]\ntheorem length_iterate (f : \u03b1 \u2192 \u03b1) (a : \u03b1) (n : \u2115) : length (iterate f a n) = n", "theorem length_le_length_insertNth (l : List \u03b1) (x : \u03b1) (n : \u2115) :\n    l.length \u2264 (insertNth n x l).length", "theorem length_modifyNth (f : \u03b1 \u2192 \u03b1) : \u2200 n l, length (modifyNth f n l) = length l", "theorem length_permutationsAux2 (t : \u03b1) (ts : List \u03b1) (ys : List \u03b1) (f : List \u03b1 \u2192 \u03b2) :\n    length (permutationsAux2 t ts [] ys f).2 = length ys", "theorem length_pos_iff_ne_nil {l : List \u03b1} : 0 < length l \u2194 l \u2260 []", "@[simp] theorem length_range (n : Nat) : length (range n) = n", "@[simp] theorem length_replicate (n : Nat) (a : \u03b1) : (replicate n a).length = n", "theorem length_scanl : \u2200 a l, length (scanl f a l) = l.length + 1", "lemma length_sigma {\u03c3 : \u03b1 \u2192 Type*} (l\u2081 : List \u03b1) (l\u2082 : \u2200 a, List (\u03c3 a)) :\n    length (l\u2081.sigma l\u2082) = (l\u2081.map fun a \u21a6 length (l\u2082 a)).sum", "theorem length_sigma' (l\u2081 : List \u03b1) (l\u2082 : \u2200 a, List (\u03c3 a)) :\n    length (l\u2081.sigma l\u2082) = Nat.sum (l\u2081.map fun a \u21a6 length (l\u2082 a))", "theorem length_split_lt {a b} {l l\u2081 l\u2082 : List \u03b1} (h : split (a :: b :: l) = (l\u2081, l\u2082)) :\n    length l\u2081 < length (a :: b :: l) \u2227 length l\u2082 < length (a :: b :: l)", "@[simp] theorem length_take : \u2200 (i : Nat) (l : List \u03b1), length (take i l) = min i (length l)", "@[simp] theorem _root_.List.length_toAssocList (l : List (\u03b1 \u00d7 \u03b2)) :\n    l.toAssocList.length = l.length", "theorem lookupAll_nodup (a : \u03b1) {l : List (Sigma \u03b2)} (h : l.NodupKeys) : (lookupAll a l).Nodup", "theorem lt_findIdx_of_not {p : \u03b1 \u2192 Bool} {xs : List \u03b1} {i : \u2115} (h : i < xs.length)\n    (h2 : \u2200 j (hji : j \u2264 i), \u00acp (xs.get \u27e8j, hji.trans_lt h\u27e9)) : i < xs.findIdx p", "theorem lt_iff_lex_lt [LinearOrder \u03b1] (l l' : List \u03b1) : lt l l' \u2194 Lex (\u00b7 < \u00b7) l l'", "theorem lt_length_left_of_zipWith {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3} {i : \u2115} {l : List \u03b1} {l' : List \u03b2}\n    (h : i < (zipWith f l l').length) : i < l.length", "theorem mapIdxMAux'_eq_mapIdxMGo {\u03b1} (f : \u2115 \u2192 \u03b1 \u2192 m PUnit) (as : List \u03b1) (arr : Array PUnit) :\n    mapIdxMAux' f arr.size as = mapIdxM.go f as arr *> pure PUnit.unit", "@[simp]\ntheorem mapIdx_cons (l : List \u03b1) (f : \u2115 \u2192 \u03b1 \u2192 \u03b2) (a : \u03b1) :\n    mapIdx f (a :: l) = f 0 a :: mapIdx (fun i \u21a6 f (i + 1)) l", "@[simp]\ntheorem map_diff [DecidableEq \u03b2] {f : \u03b1 \u2192 \u03b2} (finj : Injective f) {l\u2081 l\u2082 : List \u03b1} :\n    map f (l\u2081.diff l\u2082) = (map f l\u2081).diff (map f l\u2082)", "theorem map_erase [DecidableEq \u03b2] {f : \u03b1 \u2192 \u03b2} (finj : Injective f) {a : \u03b1} (l : List \u03b1) :\n    map f (l.erase a) = (map f l).erase (f a)", "@[simp]\ntheorem map_leftInverse_iff {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b1} :\n    LeftInverse (map f) (map g) \u2194 LeftInverse f g", "theorem map_permutationsAux (f : \u03b1 \u2192 \u03b2) :\n    \u2200 ts is :\n    List \u03b1, map (map f) (permutationsAux ts is) = permutationsAux (map f ts) (map f is)", "theorem mem_cons_eq (a y : \u03b1) (l : List \u03b1) : (a \u2208 y :: l) = (a = y \u2228 a \u2208 l)", "theorem mem_inter_of_mem_of_mem (h\u2081 : a \u2208 l\u2081) (h\u2082 : a \u2208 l\u2082) : a \u2208 l\u2081 \u2229 l\u2082", "theorem mem_keys {a} {l : List (Sigma \u03b2)} : a \u2208 l.keys \u2194 \u2203 b : \u03b2 a, Sigma.mk a b \u2208 l", "@[simp]\ntheorem mem_keys_kerase_of_ne {a\u2081 a\u2082} {l : List (Sigma \u03b2)} (h : a\u2081 \u2260 a\u2082) :\n    a\u2081 \u2208 (kerase a\u2082 l).keys \u2194 a\u2081 \u2208 l.keys", "@[simp 1100]\ntheorem mem_map_swap (x : \u03b1) (y : \u03b2) (xs : List (\u03b1 \u00d7 \u03b2)) :\n    (y, x) \u2208 map Prod.swap xs \u2194 (x, y) \u2208 xs", "@[simp]\ntheorem mem_merge {s : \u03b1 \u2192 \u03b1 \u2192 Bool} : x \u2208 merge s l r \u2194 x \u2208 l \u2228 x \u2208 r", "theorem mem_of_mem_inter_right (h : a \u2208 l\u2081 \u2229 l\u2082) : a \u2208 l\u2082", "@[simp]\ntheorem mem_orderedInsert {a b : \u03b1} {l : List \u03b1} :\n    a \u2208 orderedInsert r b l \u2194 a = b \u2228 a \u2208 l", "lemma mem_pair {a b c : \u03b1} : a \u2208 [b, c] \u2194 a = b \u2228 a = c", "theorem mem_permutationsAux2' {t : \u03b1} {ts : List \u03b1} {ys : List \u03b1} {l : List \u03b1} :\n    l \u2208 (permutationsAux2 t ts [] ys id).2 \u2194\n      \u2203 l\u2081 l\u2082, l\u2082 \u2260 [] \u2227 ys = l\u2081 ++ l\u2082 \u2227 l = l\u2081 ++ t :: l\u2082 ++ ts", "@[simp]\ntheorem mem_rotate : \u2200 {l : List \u03b1} {a : \u03b1} {n : \u2115}, a \u2208 l.rotate n \u2194 a \u2208 l", "@[simp]\ntheorem mem_sublists {s t : List \u03b1} : s \u2208 sublists t \u2194 s <+ t", "@[simp] theorem mem_union_iff [LawfulBEq \u03b1] {x : \u03b1} {l\u2081 l\u2082 : List \u03b1} :\n    x \u2208 l\u2081 \u222a l\u2082 \u2194 x \u2208 l\u2081 \u2228 x \u2208 l\u2082", "theorem mem_union_left (h : a \u2208 l\u2081) (l\u2082 : List \u03b1) : a \u2208 l\u2081 \u222a l\u2082", "theorem minimum_cons (a : \u03b1) (l : List \u03b1) : minimum (a :: l) = min \u2191a (minimum l)", "theorem mk_mem_sym2 {xs : List \u03b1} {a b : \u03b1} (ha : a \u2208 xs) (hb : b \u2208 xs) :\n    s(a, b) \u2208 xs.sym2", "theorem modifyNth_eq_set_get (f : \u03b1 \u2192 \u03b1) {n} {l : List \u03b1} (h) :\n    l.modifyNth f n = l.set n (f (l.get \u27e8n, h\u27e9))", "@[simp] theorem modify_get?_length (f : \u03b1 \u2192 \u03b1) : \u2200 n l, length (modifyNth f n l) = length l", "protected theorem new_def_eq_old_def :\n    \u2200 (f : \u2115 \u2192 \u03b1 \u2192 \u03b2) (l : List \u03b1), l.mapIdx f = List.oldMapIdx f l", "@[simp]\ntheorem nextOr_singleton (x y d : \u03b1) : nextOr [y] x d = d", "theorem nil_prefix (l : List \u03b1) : [] <+: l", "@[simp] theorem nil_subset (l : List \u03b1) : [] \u2286 l", "theorem nil_suffix (l : List \u03b1) : [] <:+ l", "@[simp]\ntheorem nodupKeys_nil : @NodupKeys \u03b1 \u03b2 []", "theorem nodup_append {l\u2081 l\u2082 : List \u03b1} :\n    Nodup (l\u2081 ++ l\u2082) \u2194 Nodup l\u2081 \u2227 Nodup l\u2082 \u2227 Disjoint l\u2081 l\u2082", "@[simp]\ntheorem nodup_cons {a : \u03b1} {l : List \u03b1} : Nodup (a :: l) \u2194 a \u2209 l \u2227 Nodup l", "theorem nodup_iff_sublist {l : List \u03b1} : Nodup l \u2194 \u2200 a, \u00ac[a, a] <+ l", "theorem nodup_permutations (s : List \u03b1) (hs : Nodup s) : Nodup s.permutations", "theorem nodup_range (n : \u2115) : Nodup (range n)", "@[simp]\ntheorem nodup_sublists' {l : List \u03b1} : Nodup (sublists' l) \u2194 Nodup l", "theorem List.norm_prod_le' : \u2200 {l : List \u03b1}, l \u2260 [] \u2192 \u2016l.prod\u2016 \u2264 (l.map norm).prod", "theorem not_exists_mem_nil (p : \u03b1 \u2192 Prop) : \u00ac\u2203 x \u2208 @nil \u03b1, p x", "theorem not_mem_keys {a} {l : List (Sigma \u03b2)} : a \u2209 l.keys \u2194 \u2200 b : \u03b2 a, Sigma.mk a b \u2209 l", "@[deprecated get_insertNth_add_succ (since := \"2023-01-05\")]\ntheorem nthLe_insertNth_add_succ : \u2200 (l : List \u03b1) (x : \u03b1) (n k : \u2115) (hk' : n + k < l.length)\n    (hk : n + k + 1 < (insertNth n x l).length := (by\n      rwa [length_insertNth _ _ (by omega), Nat.succ_lt_succ_iff])),\n    (insertNth n x l).nthLe (n + k + 1) hk = nthLe l (n + k) hk'", "@[deprecated get_insertNth_of_lt (since := \"2023-01-05\")]\ntheorem nthLe_insertNth_of_lt : \u2200 (l : List \u03b1) (x : \u03b1) (n k : \u2115), k < n \u2192 \u2200 (hk : k < l.length)\n    (hk' : k < (insertNth n x l).length := hk.trans_le (length_le_length_insertNth _ _ _)),\n    (insertNth n x l).nthLe k hk' = l.nthLe k hk", "@[deprecated get_of_mem (since := \"2023-01-05\")]\ntheorem nthLe_of_mem {a} {l : List \u03b1} (h : a \u2208 l) : \u2203 n h, nthLe l n h = a", "theorem ofFn_inj' {m n : \u2115} {f : Fin m \u2192 \u03b1} {g : Fin n \u2192 \u03b1} :\n    ofFn f = ofFn g \u2194 (\u27e8m, f\u27e9 : \u03a3n, Fin n \u2192 \u03b1) = \u27e8n, g\u27e9", "@[simp]\ntheorem orderedInsert_nil (a : \u03b1) : [].orderedInsert r a = [a]", "theorem pairwise_append_comm {R : \u03b1 \u2192 \u03b1 \u2192 Prop} (s : \u2200 {x y}, R x y \u2192 R y x) {l\u2081 l\u2082 : List \u03b1} :\n    Pairwise R (l\u2081 ++ l\u2082) \u2194 Pairwise R (l\u2082 ++ l\u2081)", "theorem pairwise_filter (p : \u03b1 \u2192 Prop) [DecidablePred p] {l : List \u03b1} :\n    Pairwise R (filter p l) \u2194 Pairwise (fun x y => p x \u2192 p y \u2192 R x y) l", "theorem pairwise_sublists {R} {l : List \u03b1} (H : Pairwise R l) :\n    Pairwise (fun l\u2081 l\u2082 => Lex R (reverse l\u2081) (reverse l\u2082)) (sublists l)", "theorem perm_append_left_iff {l\u2081 l\u2082 : List \u03b1} : \u2200 l, l ++ l\u2081 ~ l ++ l\u2082 \u2194 l\u2081 ~ l\u2082", "theorem perm_replicate_append_replicate {l : List \u03b1} {a b : \u03b1} {m n : \u2115} (h : a \u2260 b) :\n    l ~ replicate m a ++ replicate n b \u2194 count a l = m \u2227 count b l = n \u2227 l \u2286 [a, b]", "theorem perm_split : \u2200 {l l\u2081 l\u2082 : List \u03b1}, split l = (l\u2081, l\u2082) \u2192 l ~ l\u2081 ++ l\u2082", "theorem permutationsAux2_fst (t : \u03b1) (ts : List \u03b1) (r : List \u03b2) :\n    \u2200 (ys : List \u03b1) (f : List \u03b1 \u2192 \u03b2), (permutationsAux2 t ts r ys f).1 = ys ++ ts", "@[simp] theorem prefix_append (l\u2081 l\u2082 : List \u03b1) : l\u2081 <+: l\u2081 ++ l\u2082", "@[simp]\ntheorem prefix_cons_inj (a) : a :: l\u2081 <+: a :: l\u2082 \u2194 l\u2081 <+: l\u2082", "theorem prev_cons_cons_of_ne' (y z : \u03b1) (h : x \u2208 y :: z :: l) (hy : x \u2260 y) (hz : x = z) :\n    prev (y :: z :: l) x h = y", "@[to_additive]\nlemma prod_eq_pow_single [DecidableEq M] (a : M) (h : \u2200 a', a' \u2260 a \u2192 a' \u2208 l \u2192 a' = 1) :\n    l.prod = a ^ l.count a", "@[to_additive (attr := simp)]\ntheorem prod_join {l : List (List M)} : l.join.prod = (l.map List.prod).prod", "@[to_additive (attr := simp)]\ntheorem prod_nil : ([] : List M).prod = 1", "@[to_additive \"A non-commutative variant of `List.sum_reverse`\"]\ntheorem prod_reverse_noncomm : \u2200 L : List G, L.reverse.prod = (L.map fun x => x\u207b\u00b9).prod\u207b\u00b9", "@[to_additive (attr := simp)]\ntheorem prod_take_succ :\n    \u2200 (L : List M) (i : \u2115) (p : i < L.length), (L.take (i + 1)).prod = (L.take i).prod * L[i]", "theorem range'_concat (s n : Nat) : range' s (n + 1) step = range' s n step ++ [s + step * n]", "theorem range'_subset_right {s m n : Nat} (step0 : 0 < step) :\n    range' s m step \u2286 range' s n step \u2194 m \u2264 n", "theorem range_add (a b : Nat) : range (a + b) = range a ++ (range b).map (a + \u00b7)", "@[simp]\ntheorem range_subset {m n : Nat} : range m \u2286 range n \u2194 m \u2264 n", "theorem ranges_disjoint (l : List \u2115) :\n    Pairwise Disjoint (ranges l)", "lemma ranges_join' : \u2200 l : List \u2115, l.ranges.join = range (Nat.sum l)\n  | [] => rfl\n  | a :: l => by simp only [sum_cons, join, \u2190 map_join, ranges_join', range_add]", "theorem ranges_length (l : List \u2115) :\n    l.ranges.map length = l", "theorem reduceOption_length_lt_iff {l : List (Option \u03b1)} :\n    l.reduceOption.length < l.length \u2194 none \u2208 l", "theorem rel_filterMap : ((R \u21d2 Option.Rel P) \u21d2 Forall\u2082 R \u21d2 Forall\u2082 P) filterMap filterMap", "theorem rel_mem (hr : BiUnique R) : (R \u21d2 Forall\u2082 R \u21d2 Iff) (\u00b7 \u2208 \u00b7) (\u00b7 \u2208 \u00b7)", "theorem rel_of_chain_cons {a b : \u03b1} {l : List \u03b1} (p : Chain R a (b :: l)) : R a b", "@[simp]\ntheorem reverse_foldl {l : List \u03b1} : reverse (foldl (fun t h => h :: t) [] l) = l", "@[simp] theorem reverse_nil : reverse ([] : List \u03b1) = []", "theorem rotate_cons_succ (l : List \u03b1) (a : \u03b1) (n : \u2115) :\n    (a :: l : List \u03b1).rotate (n + 1) = (l ++ [a]).rotate n", "@[simp]\ntheorem rotate_nil (n : \u2115) : ([] : List \u03b1).rotate n = []", "theorem rotate_one_eq_self_iff_eq_replicate [Nonempty \u03b1] {l : List \u03b1} :\n    l.rotate 1 = l \u2194 \u2203 a : \u03b1, l = List.replicate l.length a", "@[simp]\ntheorem rotate_singleton (x : \u03b1) (n : \u2115) : [x].rotate n = [x]", "theorem rtakeWhile_suffix : l.rtakeWhile p <:+ l", "@[simp]\ntheorem scanl_cons : scanl f b (a :: l) = [b] ++ scanl f (f b a) l", "theorem singleton_eq (x : \u03b1) : ({x} : List \u03b1) = [x]", "theorem snd_mem_of_mem_enumFrom {x : \u2115 \u00d7 \u03b1} {n : \u2115} {l : List \u03b1} (h : x \u2208 enumFrom n l) : x.2 \u2208 l", "@[simp]\ntheorem splitOnP_cons (x : \u03b1) (xs : List \u03b1) :\n    (x :: xs).splitOnP p =\n      if p x then [] :: xs.splitOnP p else (xs.splitOnP p).modifyHead (cons x)", "@[simp]\ntheorem splitOnP_nil : [].splitOnP p = [[]]", "theorem sublist_eq_map_get (h : l' <+ l) : \u2203 is : List (Fin l.length),\n    l' = map (get l) is \u2227 is.Pairwise (\u00b7 < \u00b7)", "lemma sublist_join (L : List (List \u03b1)) {s : List \u03b1} (hs : s \u2208 L) :\n    s.Sublist L.join", "theorem sublist_orderedInsert (x : \u03b1) (xs : List \u03b1) : xs <+ xs.orderedInsert r x", "theorem sublists'Aux_eq_array_foldl (a : \u03b1) : \u2200 (r\u2081 r\u2082 : List (List \u03b1)),\n    sublists'Aux a r\u2081 r\u2082 = ((r\u2081.toArray).foldl (init := r\u2082.toArray)\n      (fun r l => r.push (a :: l))).toList", "theorem sublists'Aux_eq_map (a : \u03b1) (r\u2081 : List (List \u03b1)) : \u2200 (r\u2082 : List (List \u03b1)),\n    sublists'Aux a r\u2081 r\u2082 = r\u2082 ++ map (cons a) r\u2081", "theorem sublists'_map (f : \u03b1 \u2192 \u03b2) : \u2200 (l : List \u03b1),\n    sublists' (map f l) = map (map f) (sublists' l)", "theorem sublists_eq_sublists' (l : List \u03b1) : sublists l = map reverse (sublists' (reverse l))", "theorem subperm_of_subset (d : Nodup l\u2081) (H : l\u2081 \u2286 l\u2082) : l\u2081 <+~ l\u2082", "@[simp] theorem subset_cons (a : \u03b1) (l : List \u03b1) : l \u2286 a :: l", "theorem subset_insert (a : \u03b1) (l : List \u03b1) : l \u2286 l.insert a", "theorem sum_take_map_length_splitWrtComposition (l : List \u03b1) (c : Composition l.length) (i : \u2115) :\n    (((l.splitWrtComposition c).map length).take i).sum = c.sizeUpTo i", "theorem support_formPerm_le' : { x | formPerm l x \u2260 x } \u2264 l.toFinset", "theorem surjective_head! [Inhabited \u03b1] : Surjective (@head! \u03b1 _)", "theorem sym_one_eq : xs.sym 1 = xs.map (\u00b7 ::\u209b .nil)", "theorem tail_append_of_ne_nil (l l' : List \u03b1) (h : l \u2260 []) : (l ++ l').tail = l.tail ++ l'", "@[csimp] theorem tails_eq_tailsTR : @tails = @tailsTR", "@[csimp] theorem takeD_eq_takeDTR : @takeD = @takeDTR", "@[simp] theorem takeD_nil (n) (a : \u03b1) : takeD n [] a = replicate n a", "theorem takeWhile_prefix (p : \u03b1 \u2192 Bool) : l.takeWhile p <+: l", "theorem take_append_of_le_length {l\u2081 l\u2082 : List \u03b1} {n : Nat} (h : n \u2264 l\u2081.length) :\n    (l\u2081 ++ l\u2082).take n = l\u2081.take n", "theorem List.tendsto_cons {a : \u03b1} {l : List \u03b1} :\n    Tendsto (fun p : \u03b1 \u00d7 List \u03b1 => List.cons p.1 p.2) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd l) (\ud835\udcdd (a::l))", "theorem tendsto_insertNth' {a : \u03b1} :\n    \u2200 {n : \u2115} {l : List \u03b1},\n      Tendsto (fun p : \u03b1 \u00d7 List \u03b1 => insertNth n p.1 p.2) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd l) (\ud835\udcdd (insertNth n a l))", "theorem tendsto_nhds {\u03b2 : Type*} {f : List \u03b1 \u2192 \u03b2} {r : List \u03b1 \u2192 Filter \u03b2}\n    (h_nil : Tendsto f (pure []) (r []))\n    (h_cons :\n      \u2200 l a,\n        Tendsto f (\ud835\udcdd l) (r l) \u2192\n          Tendsto (fun p : \u03b1 \u00d7 List \u03b1 => f (p.1::p.2)) (\ud835\udcdd a \u00d7\u02e2 \ud835\udcdd l) (r (a::l))) :\n    \u2200 l, Tendsto f (\ud835\udcdd l) (r l)", "theorem toFinsupp_singleton (x : M) [DecidablePred (getD [x] \u00b7 0 \u2260 0)] :\n    toFinsupp [x] = Finsupp.single 0 x", "theorem unzip_swap (l : List (\u03b1 \u00d7 \u03b2)) : unzip (l.map Prod.swap) = (unzip l).swap", "@[simp]\ntheorem zipWith_flip (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) : \u2200 as bs, zipWith (flip f) bs as = zipWith f as bs", "theorem zipWith_zipWith_left (f : \u03b4 \u2192 \u03b3 \u2192 \u03b5) (g : \u03b1 \u2192 \u03b2 \u2192 \u03b4) :\n    \u2200 (la : List \u03b1) (lb : List \u03b2) (lc : List \u03b3),\n      zipWith f (zipWith g la lb) lc = zipWith3 (fun a b c => f (g a b) c) la lb lc", "theorem ofList_subset {l\u2081 l\u2082 : List (Lists \u03b1)} (h : l\u2081 \u2286 l\u2082) :\n    Lists'.ofList l\u2081 \u2286 Lists'.ofList l\u2082", "@[symm]\ntheorem Equiv.symm {l\u2081 l\u2082 : Lists \u03b1} (h : l\u2081 ~ l\u2082) : l\u2082 ~ l\u2081", "theorem of_toList : \u2200 {l : Lists \u03b1}, IsList l \u2192 ofList (toList l) = l", "lemma CotangentSpace.map_eq_top_iff [IsNoetherianRing R] {M : Submodule R (maximalIdeal R)} :\n    M.map (maximalIdeal R).toCotangent = \u22a4 \u2194 M = \u22a4", "theorem closedPoint_mem_iff (U : TopologicalSpace.Opens <| PrimeSpectrum R) :\n    closedPoint R \u2208 U \u2194 U = \u22a4", "theorem eq_maximalIdeal {I : Ideal R} (hI : I.IsMaximal) : I = maximalIdeal R", "theorem maximal_ideal_unique : \u2203! I : Ideal R, I.IsMaximal", "theorem AtPrime.map_eq_maximalIdeal :\n    Ideal.map (algebraMap R (Localization.AtPrime I)) I =\n      LocalRing.maximalIdeal (Localization I.primeCompl)", "theorem nilpotent_iff_mem_maximal_of_minimal {x : _} :\n    IsNilpotent x \u2194 x \u2208 LocalRing.maximalIdeal (Localization I.primeCompl)", "theorem liftOn_zero {p : Type*} (f : M \u2192 S \u2192 p) (H) : liftOn 0 f H = f 0 1", "theorem mk_eq_mk' : (mk : R \u2192 M \u2192 Localization M) = IsLocalization.mk' (Localization M)", "theorem mk_zero (x : S) : mk 0 (x : S) = 0", "theorem algebraMap_mk {A : Type*} [Semiring A] [Algebra R A] (a : R) (s : S) :\n    algebraMap _ _ (Localization.mk a s) = mk (algebraMap R A a) s", "@[simp]\ntheorem mk_cancel_common_right (s s' : S) (m : M) : mk (s' \u2022 m) (s * s') = mk m s", "theorem smul'_mul {A : Type*} [Semiring A] [Algebra R A] (x : T) (p\u2081 p\u2082 : LocalizedModule S A) :\n    x \u2022 p\u2081 * p\u2082 = x \u2022 (p\u2081 * p\u2082)", "theorem ae_differentiableWithinAt_of_mem {f : \u211d \u2192 V} {s : Set \u211d}\n    (h : LocallyBoundedVariationOn f s) : \u2200\u1d50 x, x \u2208 s \u2192 DifferentiableWithinAt \u211d f s x", "theorem ae_differentiableWithinAt_of_mem_real {f : \u211d \u2192 \u211d} {s : Set \u211d}\n    (h : LocallyBoundedVariationOn f s) : \u2200\u1d50 x, x \u2208 s \u2192 DifferentiableWithinAt \u211d f s x", "theorem apply_eq_of_preconnectedSpace [PreconnectedSpace X] (f : LocallyConstant X Y) (x y : X) :\n    f x = f y", "lemma ker_comap\u2097 [Semiring R] [AddCommMonoid Z] [Module R Z] (f : C(X, Y))\n    (hfs : Function.Surjective f) :\n    LinearMap.ker (comap\u2097 R f : LocallyConstant Y Z \u2192\u2097[R] LocallyConstant X Z) = \u22a5", "@[to_additive]\ntheorem one_apply [One Y] (x : X) : (1 : LocallyConstant X Y) x = 1", "@[simp]\nlemma piecewise_apply_right {C\u2081 C\u2082 : Set X} (h\u2081 : IsClosed C\u2081) (h\u2082 : IsClosed C\u2082)\n    (h : C\u2081 \u222a C\u2082 = Set.univ) (f : LocallyConstant C\u2081 Z) (g : LocallyConstant C\u2082 Z)\n    (hfg : \u2200 (x : X) (hx : x \u2208 C\u2081 \u2229 C\u2082), f \u27e8x, hx.1\u27e9 = g \u27e8x, hx.2\u27e9)\n    [DecidablePred (\u00b7 \u2208 C\u2081)] (x : X) (hx : x \u2208 C\u2082) :\n    piecewise h\u2081 h\u2082 h f g hfg x = g \u27e8x, hx\u27e9", "theorem LocallyFinite.Realizer.to_locallyFinite [TopologicalSpace \u03b1] {F : Ctop.Realizer \u03b1}\n    {f : \u03b2 \u2192 Set \u03b1} (R : LocallyFinite.Realizer F f) : LocallyFinite f", "protected theorem continuous {g : X \u2192 Y} (hf : LocallyFinite f) (h_cov : \u22c3 i, f i = univ)\n    (h_cl : \u2200 i, IsClosed (f i)) (h_cont : \u2200 i, ContinuousOn g (f i)) :\n    Continuous g", "theorem LocallyFinite.finite_nonempty_inter_compact {f : \u03b9 \u2192 Set X}\n    (hf : LocallyFinite f) (hs : IsCompact s) : { i | (f i \u2229 s).Nonempty }.Finite", "theorem LocallyFinite.finite_nonempty_of_compact [CompactSpace X] {f : \u03b9 \u2192 Set X}\n    (hf : LocallyFinite f) : { i | (f i).Nonempty }.Finite", "theorem LocallyFinite.option_elim' (hf : LocallyFinite f) (s : Set X) :\n    LocallyFinite (Option.elim' s f)", "protected theorem subset (hf : LocallyFinite f) (hg : \u2200 i, g i \u2286 f i) : LocallyFinite g", "protected lemma comp  {f : \u03b2 \u2192 \u03b3} {g : \u03b1 \u2192 \u03b2}\n    (hf : LocallyLipschitz f) (hg : LocallyLipschitz g) : LocallyLipschitz (f \u2218 g)", "protected theorem continuous {f : \u03b1 \u2192 \u03b2} (hf : LocallyLipschitz f) : Continuous f", "protected theorem pow_end {f : Function.End \u03b1} (h : LocallyLipschitz f) :\n    \u2200 n : \u2115, LocallyLipschitz (f ^ n : Function.End \u03b1)", "theorem closure_sup_closure (x y : \u03b1) : u (l (u (l x) \u2294 u (l y))) = u (l (x \u2294 y))", "theorem closure_sup_closure_left (x y : \u03b1) : u (l (u (l x) \u2294 y)) = u (l (x \u2294 y))", "theorem mem_iff (s : Set \u03b2) (x : \u03b2) : x \u2208 l s \u2194 \u2200 S : \u03b1, s \u2286 S \u2192 x \u2208 S", "theorem LowerSemicontinuous.lowerSemicontinuousAt (h : LowerSemicontinuous f) (x : \u03b1) :\n    LowerSemicontinuousAt f x", "theorem LowerSemicontinuous.measurable [TopologicalSpace \u03b4] [OpensMeasurableSpace \u03b4] {f : \u03b4 \u2192 \u03b1}\n    (hf : LowerSemicontinuous f) : Measurable f", "@[simp]\nnonrec theorem Iic_top [OrderTop \u03b1] : Iic (\u22a4 : \u03b1) = \u22a4", "@[simp, norm_cast]\ntheorem coe_iInf (f : \u03b9 \u2192 LowerSet \u03b1) : (\u2191(\u2a05 i, f i) : Set \u03b1) = \u22c2 i, f i", "@[simp, norm_cast]\ntheorem coe_sup (s t : LowerSet \u03b1) : (\u2191(s \u2294 t) : Set \u03b1) = (s : Set \u03b1) \u222a t", "protected theorem compl_iInf (f : \u03b9 \u2192 LowerSet \u03b1) : (\u2a05 i, f i).compl = \u2a05 i, (f i).compl", "@[simp]\ntheorem compl_le_compl : s.compl \u2264 t.compl \u2194 s \u2264 t", "@[simp]\ntheorem mem_prod {s : LowerSet \u03b1} {t : LowerSet \u03b2} : x \u2208 s \u00d7\u02e2 t \u2194 x.1 \u2208 s \u2227 x.2 \u2208 t", "@[simp]\ntheorem mem_top : a \u2208 (\u22a4 : LowerSet \u03b1)", "@[simp]\ntheorem prod_eq_bot : s \u00d7\u02e2 t = \u22a5 \u2194 s = \u22a5 \u2228 t = \u22a5", "@[simp]\ntheorem symm_map (f : \u03b1 \u2243o \u03b2) : (map f).symm = map f.symm", "@[simp]\ntheorem add_snd (x y : X q) : (x + y).2 = x.2 + y.2", "@[norm_cast]\ntheorem coe_mul (n m : \u2124) : ((n * m : \u2124) : X q) = (n : X q) * (m : X q)", "@[simp] theorem ofNat_snd (n : \u2115) [n.AtLeastTwo] :\n    (no_index (OfNat.ofNat n) : X q).snd = 0", "theorem MDifferentiableAt.const_smul (hf : MDifferentiableAt I \ud835\udcd8(\ud835\udd5c, E') f z) (s : \ud835\udd5c) :\n    MDifferentiableAt I \ud835\udcd8(\ud835\udd5c, E') (s \u2022 f) z", "theorem MDifferentiableOn.prod_mk_space {f : M \u2192 E'} {g : M \u2192 E''}\n    (hf : MDifferentiableOn I \ud835\udcd8(\ud835\udd5c, E') f s) (hg : MDifferentiableOn I \ud835\udcd8(\ud835\udd5c, E'') g s) :\n    MDifferentiableOn I \ud835\udcd8(\ud835\udd5c, E' \u00d7 E'') (fun x => (f x, g x)) s", "theorem MDifferentiableWithinAt.mdifferentiableAt (h : MDifferentiableWithinAt I I' f s x)\n    (hs : s \u2208 \ud835\udcdd x) : MDifferentiableAt I I' f x", "@[to_additive (attr := elab_as_elim, induction_eliminator)]\nprotected theorem induction_on {C : AssocQuotient \u03b1 \u2192 Prop} (x : AssocQuotient \u03b1)\n    (ih : \u2200 x, C (of x)) : C x", "@[to_additive]\ntheorem quot_mk_assoc_left (x y z w : \u03b1) :\n    Quot.mk (AssocRel \u03b1) (x * (y * z * w)) = Quot.mk _ (x * (y * (z * w)))", "protected lemma Finset.sum_empty {\u03b2 \u03b1 : Type*} [CommSemiring \u03b2] (f : \u03b1 \u2192 \u03b2) :\n    IsNat (Finset.sum \u2205 f) 0", "theorem IsInt.neg_to_eq {\u03b1} [Ring \u03b1] {n} :\n    {a a' : \u03b1} \u2192 IsInt a (.negOfNat n) \u2192 n = a' \u2192 a = -a'", "theorem eq_of_true {a b : Prop} (ha : a) (hb : b) : a = b", "theorem isFibAux_one : IsFibAux 1 1 1", "lemma isInt_ediv {a b q m a' : \u2124} {b' r : \u2115}\n    (ha : IsInt a a') (hb : IsNat b b')\n    (hm : q * b' = m) (h : r + m = a') (h\u2082 : Nat.blt r b' = true) :\n    IsInt (a / b) q := \u27e8by\n  obtain \u27e8\u27e8rfl\u27e9, \u27e8rfl\u27e9\u27e9 := ha, hb\n  simp only [Nat.blt_eq] at h\u2082; simp only [\u2190 h, \u2190 hm, Int.cast_id]\n  rw [Int.add_mul_ediv_right _ _ (Int.ofNat_ne_zero.2 ((Nat.zero_le ..).trans_lt h\u2082).ne')]\n  rw [Int.ediv_eq_zero_of_lt, zero_add] <;> [simp; simpa using h\u2082]\u27e9", "lemma isInt_emod_neg {a b : \u2124} {r : \u2115} (h : IsNat (a % -b) r) : IsNat (a % b) r", "lemma isInt_emod_zero : \u2200 {a b r : \u2124}, IsInt a r \u2192 IsNat b (nat_lit 0) \u2192 IsInt (a % b) r\n  | _, _, _, e, \u27e8rfl\u27e9 => by simp [e]", "theorem isInt_le_false [OrderedRing \u03b1] [Nontrivial \u03b1] {a b : \u03b1} {a' b' : \u2124}\n    (ha : IsInt a a') (hb : IsInt b b') (h : decide (b' < a')) : \u00aca \u2264 b", "theorem isInt_zpow_pos {\u03b1 : Type*} [DivisionRing \u03b1] {a : \u03b1} {b : \u2124} {nb ne : \u2115}\n    (pb : IsNat b nb) (pe' : IsInt (a ^ nb) (Int.negOfNat ne)) :\n    IsInt (a ^ b) (Int.negOfNat ne)", "theorem isNat_fib : {x nx z : \u2115} \u2192 IsNat x nx \u2192 Nat.fib nx = z \u2192 IsNat (Nat.fib x) z", "theorem isNat_intOfNat : {n n' : \u2115} \u2192 IsNat n n' \u2192 IsNat (Int.ofNat n) n'", "theorem isNat_minFac_1 : {a : \u2115} \u2192 IsNat a (nat_lit 1) \u2192 IsNat a.minFac 1", "theorem isNat_prime_1 : {n : \u2115} \u2192 IsNat n (nat_lit 1) \u2192 \u00ac n.Prime", "theorem isRat_inv_neg {\u03b1} [DivisionRing \u03b1] [CharZero \u03b1] {a : \u03b1} {n d : \u2115} :\n    IsRat a (.negOfNat (Nat.succ n)) d \u2192 IsRat a\u207b\u00b9 (.negOfNat d) (Nat.succ n)", "theorem isRat_inv_neg_one {\u03b1} [DivisionRing \u03b1] : {a : \u03b1} \u2192\n    IsInt a (.negOfNat (nat_lit 1)) \u2192 IsInt a\u207b\u00b9 (.negOfNat (nat_lit 1))", "theorem isRat_mkRat : {a na n : \u2124} \u2192 {b nb d : \u2115} \u2192 IsInt a na \u2192 IsNat b nb \u2192\n    IsRat (na / nb : \u211a) n d \u2192 IsRat (mkRat a b) n d", "theorem jacobiSymNat.one_right (a : \u2115) : jacobiSymNat a 1 = 1", "theorem jacobiSymNat.qr\u2083 (a b : \u2115) (r : \u2124) (ha : a % 4 = 3) (hb : b % 4 = 3)\n    (hr : jacobiSymNat b a = r) : jacobiSymNat a b = -r", "theorem minFacHelper_3 {n k k' : \u2115} (e : k + 2 = k') (nk : (n % k).beq 0 = false)\n    (h : MinFacHelper n k) : MinFacHelper n k'", "private lemma max_ne (ha : a \u2260 c) (hb : b \u2260 c) : max a b \u2260 c", "theorem term_atomg {\u03b1} [AddCommGroup \u03b1] (x : \u03b1) : x = termg 1 x 0", "theorem degree_eq_of_le_of_coeff_ne_zero' {deg m o : WithBot \u2115} {c : R} {p : R[X]}\n    (h_deg_le : degree p \u2264 m) (coeff_eq : coeff p (WithBot.unbot' 0 deg) = c)\n    (coeff_ne_zero : c \u2260 0) (deg_eq_deg : m = deg) (coeff_eq_deg : o = deg) :\n    degree p = deg", "theorem natDegree_smul_le_of_le {n : \u2115} {a : R} {f : R[X]} (hf : natDegree f \u2264 n) :\n    natDegree (a \u2022 f) \u2264 n", "theorem not_forall_eq : (\u00ac \u2200 x, s x) = (\u2203 x, \u00ac s x)", "theorem not_lt_eq (a b : \u03b2) : (\u00ac (a < b)) = (b \u2264 a)", "theorem cast_rat {R} [DivisionRing R] {a : R} : IsRat a n d \u2192 a = Rat.rawCast n d + 0", "theorem mul_add (_ : (a : R) * b\u2081 = c\u2081) (_ : a * b\u2082 = c\u2082) (_ : c\u2081 + 0 + c\u2082 = d) :\n    a * (b\u2081 + b\u2082) = d", "theorem natCast_add (_ : ((a\u2081 : \u2115) : R) = b\u2081) (_ : ((a\u2082 : \u2115) : R) = b\u2082) :\n    ((a\u2081 + a\u2082 : \u2115) : R) = b\u2081 + b\u2082", "theorem one_pow (b : \u2115) : ((nat_lit 1).rawCast : R) ^ b = (nat_lit 1).rawCast", "theorem sub_pf {R} [Ring R] {a b c d : R}\n    (_ : -b = c) (_ : a + c = d) : a - b = d", "theorem nat_rawCast_0 : (Nat.rawCast 0 : R) = 0", "theorem BlockTriangular.add (hM : BlockTriangular M b) (hN : BlockTriangular N b) :\n    BlockTriangular (M + N) b", "@[simp]\ntheorem coe_mul : \u2191(A * B) = (\u2191A : Matrix n n R) * (\u2191B : Matrix n n R)", "theorem IsDiag.smul [Monoid R] [AddMonoid \u03b1] [DistribMulAction R \u03b1] (k : R) {A : Matrix n n \u03b1}\n    (ha : A.IsDiag) : (k \u2022 A).IsDiag", "@[simp]\ntheorem IsHermitian.add {A B : Matrix n n \u03b1} (hA : A.IsHermitian) (hB : B.IsHermitian) :\n    (A + B).IsHermitian", "theorem IsHermitian.adjugate [Fintype m] [DecidableEq m] {A : Matrix m m \u03b1} (hA : A.IsHermitian) :\n    A.adjugate.IsHermitian", "@[simp]\ntheorem IsHermitian.map {A : Matrix n n \u03b1} (h : A.IsHermitian) (f : \u03b1 \u2192 \u03b2)\n    (hf : Function.Semiconj f star star) : (A.map f).IsHermitian", "theorem spectral_theorem :\n    A = (eigenvectorUnitary hA : Matrix n n \ud835\udd5c) * diagonal (RCLike.ofReal \u2218 hA.eigenvalues)\n      * (star (eigenvectorUnitary hA : Matrix n n \ud835\udd5c))", "theorem spectrum_toEuclideanLin : spectrum \ud835\udd5c (toEuclideanLin A) = spectrum \ud835\udd5c A", "theorem star_eigenvectorUnitary_mulVec (j : n) :\n    (star (eigenvectorUnitary hA : Matrix n n \ud835\udd5c)) *\u1d65 \u21d1(hA.eigenvectorBasis j) = Pi.single j 1", "theorem IsSymm.ext {A : Matrix n n \u03b1} : (\u2200 i j, A j i = A i j) \u2192 A.IsSymm", "theorem IsSymm.ext_iff {A : Matrix n n \u03b1} : A.IsSymm \u2194 \u2200 i j, A j i = A i j", "theorem J_inv : (J l R)\u207b\u00b9 = -J l R", "theorem exists_isTwoBlockDiagonal_list_transvec_mul_mul_list_transvec\n    (M : Matrix (Sum (Fin r) Unit) (Sum (Fin r) Unit) \ud835\udd5c) :\n    \u2203 L L' : List (TransvectionStruct (Sum (Fin r) Unit) \ud835\udd5c),\n      IsTwoBlockDiagonal ((L.map toMatrix).prod * M * (L'.map toMatrix).prod)", "theorem listTransvecCol_mul_last_row (i : Sum (Fin r) Unit) :\n    ((listTransvecCol M).prod * M) (inr unit) i = M (inr unit) i", "theorem listTransvecCol_mul_mul_listTransvecRow_last_col (hM : M (inr unit) (inr unit) \u2260 0)\n    (i : Fin r) :\n    ((listTransvecCol M).prod * M * (listTransvecRow M).prod) (inr unit) (inl i) = 0", "protected lemma posSemidef_add {A : Matrix m m R} {B : Matrix m m R}\n    (hA : A.PosSemidef) (hB : B.PosDef) : (A + B).PosDef", "theorem Matrix.Represents.algebraMap (r : R) :\n    (algebraMap _ (Matrix \u03b9 \u03b9 R) r).Represents b (algebraMap _ (Module.End R M) r)", "@[simp]\ntheorem det_coe : det \u2191\u2098A = 1", "lemma isCoprime_row (A : SL(2, R)) (i : Fin 2): IsCoprime (A i 0) (A i 1)", "@[simp]\ntheorem sumInl_toMatrix_prod_mul [Fintype n] [Fintype p] (M : Matrix n n R)\n    (L : List (TransvectionStruct n R)) (N : Matrix p p R) :\n    (L.map (toMatrix \u2218 sumInl p)).prod * fromBlocks M 0 0 N =\n      fromBlocks ((L.map toMatrix).prod * M) 0 0 N", "theorem ext_iff (A B : unitaryGroup n \u03b1) : A = B \u2194 \u2200 i j, A i j = B i j", "@[simp] theorem mul_val : \u2191(A * B) = A.1 * B.1", "theorem add_hadamard [Distrib \u03b1] : (B + C) \u2299 A = B \u2299 A + C \u2299 A", "theorem add_kroneckerTMul (A\u2081 A\u2082 : Matrix l m \u03b1) (B : Matrix n p \u03b1) :\n    (A\u2081 + A\u2082) \u2297\u2096\u209c[R] B = A\u2081 \u2297\u2096\u209c B + A\u2082 \u2297\u2096\u209c B", "theorem algebraMap_eq_diagonal (r : R) :\n    algebraMap R (Matrix n n \u03b1) r = diagonal (algebraMap R (n \u2192 \u03b1) r)", "@[simp]\ntheorem blockDiag'_transpose (M : Matrix (\u03a3i, m' i) (\u03a3i, n' i) \u03b1) (k : o) :\n    blockDiag' M\u1d40 k = (blockDiag' M k)\u1d40", "@[simp]\ntheorem blockDiag_smul {R : Type*} [Monoid R] [AddMonoid \u03b1] [DistribMulAction R \u03b1] (x : R)\n    (M : Matrix (m \u00d7 o) (n \u00d7 o) \u03b1) : blockDiag (x \u2022 M) = x \u2022 blockDiag M", "theorem blockDiagonal'_map (M : \u2200 i, Matrix (m' i) (n' i) \u03b1) (f : \u03b1 \u2192 \u03b2) (hf : f 0 = 0) :\n    (blockDiagonal' M).map f = blockDiagonal' fun k => (M k).map f", "@[simp]\ntheorem blockDiagonal'_zero : blockDiagonal' (0 : \u2200 i, Matrix (m' i) (n' i) \u03b1) = 0", "theorem blockDiagonal_map (M : o \u2192 Matrix m n \u03b1) (f : \u03b1 \u2192 \u03b2) (hf : f 0 = 0) :\n    (blockDiagonal M).map f = blockDiagonal fun k => (M k).map f", "theorem circulant_neg [Neg \u03b1] [Sub n] (v : n \u2192 \u03b1) : circulant (-v) = -circulant v", "@[simp]\ntheorem coe_detMonoidHom : (detMonoidHom : Matrix n n R \u2192 R) = det", "theorem Matrix.coe_vecMulLinear [Fintype m] (M : Matrix m n R) :\n    (M.vecMulLinear : _ \u2192 _) = M.vecMul", "@[simp]\ntheorem conjTransposeAddEquiv_symm [AddMonoid \u03b1] [StarAddMonoid \u03b1] :\n    (conjTransposeAddEquiv m n \u03b1).symm = conjTransposeAddEquiv n m \u03b1", "@[simp] theorem conjTranspose_inj [InvolutiveStar \u03b1] {A B : Matrix m n \u03b1} : A\u1d34 = B\u1d34 \u2194 A = B", "@[simp]\ntheorem cons_val_one (x : \u03b1) (u : Fin m.succ \u2192 \u03b1) : vecCons x u 1 = vecHead u", "@[simp]\nlemma cons_val_three (x : \u03b1) (u : Fin m.succ.succ.succ \u2192 \u03b1) :\n    vecCons x u 3 = vecHead (vecTail (vecTail u))", "@[simp]\ntheorem cons_val_zero (x : \u03b1) (u : Fin m \u2192 \u03b1) : vecCons x u 0 = x", "@[simp]\ntheorem cons_vec_bit1_eq_alt1 (x : \u03b1) (u : Fin n \u2192 \u03b1) (i : Fin (n + 1)) :\n    vecCons x u (bit1 i) = vecAlt1 rfl (vecAppend rfl (vecCons x u) (vecCons x u)) i", "@[simp]\ntheorem det_conjTranspose [StarRing R] (M : Matrix m m R) : det M\u1d34 = star (det M)", "theorem det_eq_of_forall_col_eq_smul_add_pred {n : \u2115} {A B : Matrix (Fin (n + 1)) (Fin (n + 1)) R}\n    (c : Fin n \u2192 R) (A_zero : \u2200 i, A i 0 = B i 0)\n    (A_succ : \u2200 (i) (j : Fin n), A i j.succ = B i j.succ + c j * A i (Fin.castSucc j)) :\n    det A = det B", "@[simp]\ntheorem det_fin_two_of (a b c d : R) : Matrix.det !![a, b; c, d] = a * d - b * c", "@[simp]\ntheorem det_fromBlocks_zero\u2082\u2081 (A : Matrix m m R) (B : Matrix m n R) (D : Matrix n n R) :\n    (Matrix.fromBlocks A B 0 D).det = A.det * D.det", "@[simp]\ntheorem det_mul (M N : Matrix n n R) : det (M * N) = det M * det N", "theorem det_mul_row (v : n \u2192 R) (A : Matrix n n R) :\n    det (of fun i j => v j * A i j) = (\u220f i, v i) * det A", "@[simp]\ntheorem det_nonsing_inv : A\u207b\u00b9.det = Ring.inverse A.det", "@[simp]\ntheorem det_one : det (1 : Matrix n n R) = 1", "theorem det_units_conj' (M : (Matrix m m R)\u02e3) (N : Matrix m m R) :\n    det ((\u2191M\u207b\u00b9 : Matrix _ _ _) * N * (\u2191M : Matrix _ _ _)) = det N", "theorem det_updateRow_add_self (A : Matrix n n R) {i j : n} (hij : i \u2260 j) :\n    det (updateRow A i (A i + A j)) = det A", "theorem det_vandermonde {n : \u2115} (v : Fin n \u2192 R) :\n    det (vandermonde v) = \u220f i : Fin n, \u220f j \u2208 Ioi i, (v j - v i)", "theorem det_vandermonde_ne_zero_iff [IsDomain R] {n : \u2115} {v : Fin n \u2192 R} :\n    det (vandermonde v) \u2260 0 \u2194 Function.Injective v", "theorem det_zero_of_row_eq (i_ne_j : i \u2260 j) (hij : M i = M j) : M.det = 0", "theorem diagonal_comp_stdBasis (w : n \u2192 R) (i : n) :\n    (diagonal w).toLin'.comp (LinearMap.stdBasis R (fun _ : n => R) i) =\n      w i \u2022 LinearMap.stdBasis R (fun _ : n => R) i", "@[simp]\ntheorem diagonal_map [Zero \u03b1] [Zero \u03b2] {f : \u03b1 \u2192 \u03b2} (h : f 0 = 0) {d : n \u2192 \u03b1} :\n    (diagonal d).map f = diagonal fun m => f (d m)", "theorem Matrix.diagonal_tsum [DecidableEq n] [T2Space R] {f : X \u2192 n \u2192 R} :\n    diagonal (\u2211' x, f x) = \u2211' x, diagonal (f x)", "theorem dotProduct_comm [AddCommMonoid \u03b1] [CommSemigroup \u03b1] (v w : m \u2192 \u03b1) : v \u2b1d\u1d65 w = w \u2b1d\u1d65 v", "@[simp]\ntheorem dotProduct_diagonal' (i : m) : (v \u2b1d\u1d65 fun j => diagonal w j i) = v i * w i", "@[simp]\ntheorem dotProduct_zero : v \u2b1d\u1d65 0 = 0", "theorem eta_fin_three (A : Matrix (Fin 3) (Fin 3) \u03b1) :\n    A = !![A 0 0, A 0 1, A 0 2;\n           A 1 0, A 1 1, A 1 2;\n           A 2 0, A 2 1, A 2 2]", "theorem exists_right_inverse_iff_isUnit : (\u2203 B, A * B = 1) \u2194 IsUnit A", "theorem exp_conj (U : Matrix m m \ud835\udd38) (A : Matrix m m \ud835\udd38) (hy : IsUnit U) :\n    exp \ud835\udd42 (U * A * U\u207b\u00b9) = U * exp \ud835\udd42 A * U\u207b\u00b9", "theorem exp_diagonal (v : m \u2192 \ud835\udd38) : exp \ud835\udd42 (diagonal v) = diagonal (exp \ud835\udd42 v)", "theorem fromBlocks_add [Add \u03b1] (A : Matrix n l \u03b1) (B : Matrix n m \u03b1) (C : Matrix o l \u03b1)\n    (D : Matrix o m \u03b1) (A' : Matrix n l \u03b1) (B' : Matrix n m \u03b1) (C' : Matrix o l \u03b1)\n    (D' : Matrix o m \u03b1) : fromBlocks A B C D + fromBlocks A' B' C' D' =\n      fromBlocks (A + A') (B + B') (C + C') (D + D')", "theorem fromBlocks_submatrix_sum_swap_sum_swap {l m n o \u03b1 : Type*} (A : Matrix n l \u03b1)\n    (B : Matrix n m \u03b1) (C : Matrix o l \u03b1) (D : Matrix o m \u03b1) :\n    (fromBlocks A B C D).submatrix Sum.swap Sum.swap = fromBlocks D C B A", "theorem fromBlocks_transpose (A : Matrix n l \u03b1) (B : Matrix n m \u03b1) (C : Matrix o l \u03b1)\n    (D : Matrix o m \u03b1) : (fromBlocks A B C D)\u1d40 = fromBlocks A\u1d40 C\u1d40 B\u1d40 D\u1d40", "@[simp]\nlemma fromColumns_toColumns (A : Matrix m (n\u2081 \u2295 n\u2082) R) :\n    fromColumns A.toColumns\u2081 A.toColumns\u2082 = A", "@[simp]\nlemma fromRows_mul (A\u2081 : Matrix m\u2081 n R) (A\u2082 : Matrix m\u2082 n R) (B : Matrix n m R) :\n    fromRows A\u2081 A\u2082 * B = fromRows (A\u2081 * B) (A\u2082 * B)", "@[elab_as_elim]\nprotected theorem induction_on' [Finite m] [Finite n] {P : Matrix m n \u03b1 \u2192 Prop} (M : Matrix m n \u03b1)\n    (h_zero : P 0) (h_add : \u2200 p q, P p \u2192 P q \u2192 P (p + q))\n    (h_std_basis : \u2200 (i : m) (j : n) (x : \u03b1), P (stdBasisMatrix i j x)) : P M", "theorem inv_fromBlocks_zero\u2081\u2082_of_isUnit_iff (A : Matrix m m \u03b1) (C : Matrix n m \u03b1) (D : Matrix n n \u03b1)\n    (hAD : IsUnit A \u2194 IsUnit D) :\n    (fromBlocks A 0 C D)\u207b\u00b9 = fromBlocks A\u207b\u00b9 0 (-(D\u207b\u00b9 * C * A\u207b\u00b9)) D\u207b\u00b9", "@[simp]\ntheorem inv_inv_of_invertible [Invertible A] : A\u207b\u00b9\u207b\u00b9 = A", "@[simp]\ntheorem isHermitian_one [DecidableEq n] : (1 : Matrix n n \u03b1).IsHermitian", "@[simp]\ntheorem isSymm_one [DecidableEq n] [Zero \u03b1] [One \u03b1] : (1 : Matrix n n \u03b1).IsSymm", "theorem isSymm_transpose_mul_self [Fintype n] [CommSemiring \u03b1] (A : Matrix n n \u03b1) :\n    (A\u1d40 * A).IsSymm", "theorem isUnit_det_of_left_inverse (h : B * A = 1) : IsUnit A.det", "theorem isUnit_det_transpose (h : IsUnit A.det) : IsUnit A\u1d40.det", "@[simp]\ntheorem isUnit_diagonal {v : n \u2192 \u03b1} : IsUnit (diagonal v) \u2194 IsUnit v", "theorem kroneckerMapBilinear_mul_mul [CommSemiring R] [Fintype m] [Fintype m']\n    [NonUnitalNonAssocSemiring \u03b1] [NonUnitalNonAssocSemiring \u03b2] [NonUnitalNonAssocSemiring \u03b3]\n    [Module R \u03b1] [Module R \u03b2] [Module R \u03b3] (f : \u03b1 \u2192\u2097[R] \u03b2 \u2192\u2097[R] \u03b3)\n    (h_comm : \u2200 a b a' b', f (a * b) (a' * b') = f a a' * f b b') (A : Matrix l m \u03b1)\n    (B : Matrix m n \u03b1) (A' : Matrix l' m' \u03b2) (B' : Matrix m' n' \u03b2) :\n    kroneckerMapBilinear f (A * B) (A' * B') =\n      kroneckerMapBilinear f A A' * kroneckerMapBilinear f B B'", "theorem kroneckerTMul_smul (r : R) (A : Matrix l m \u03b1) (B : Matrix n p \u03b1) :\n    A \u2297\u2096\u209c[R] (r \u2022 B) = r \u2022 A \u2297\u2096\u209c B", "lemma l2_opNorm_conjTranspose [DecidableEq m] (A : Matrix m n \ud835\udd5c) : \u2016A\u1d34\u2016 = \u2016A\u2016", "theorem linearIndependent_cols_iff_isUnit {A : Matrix m m K} :\n    LinearIndependent K (fun i \u21a6 A\u1d40 i) \u2194 IsUnit A", "theorem linfty_opNNNorm_mul (A : Matrix l m \u03b1) (B : Matrix m n \u03b1) : \u2016A * B\u2016\u208a \u2264 \u2016A\u2016\u208a * \u2016B\u2016\u208a", "@[simp] lemma linfty_opNNNorm_toMatrix (f : (n \u2192 \u03b1) \u2192L[\u03b1] (m \u2192 \u03b1)) :\n    \u2016LinearMap.toMatrix' (\u2191f : (n \u2192 \u03b1) \u2192\u2097[\u03b1] (m \u2192 \u03b1))\u2016\u208a = \u2016f\u2016\u208a", "@[simp]\ntheorem map_id' (M : Matrix m n \u03b1) : M.map (\u00b7) = M", "protected theorem map_natCast [DecidableEq n] [AddMonoidWithOne \u03b1] [AddMonoidWithOne \u03b2]\n    {f : \u03b1 \u2192 \u03b2} (h : f 0 = 0) (d : \u2115) :\n    (d : Matrix n n \u03b1).map f = diagonal (fun _ => f d)", "theorem map_op_smul' [Mul \u03b1] [Mul \u03b2] (f : \u03b1 \u2192 \u03b2) (r : \u03b1) (A : Matrix n n \u03b1)\n    (hf : \u2200 a\u2081 a\u2082, f (a\u2081 * a\u2082) = f a\u2081 * f a\u2082) :\n    (MulOpposite.op r \u2022 A).map f = MulOpposite.op (f r) \u2022 A.map f", "theorem mem_unitaryGroup_iff : A \u2208 Matrix.unitaryGroup n \u03b1 \u2194 A * star A = 1", "@[simp]\ntheorem Matrix.mulVecLin_mul [Fintype m] (M : Matrix l m R) (N : Matrix m n R) :\n    Matrix.mulVecLin (M * N) = (Matrix.mulVecLin M).comp (Matrix.mulVecLin N)", "theorem mulVec_add [Fintype n] (A : Matrix m n \u03b1) (x y : n \u2192 \u03b1) :\n    A *\u1d65 (x + y) = A *\u1d65 x + A *\u1d65 y", "@[simp]\ntheorem mulVec_cramer (A : Matrix n n \u03b1) (b : n \u2192 \u03b1) : A *\u1d65 cramer A b = A.det \u2022 b", "theorem Matrix.mulVec_injective_iff {R : Type*} [CommRing R] {M : Matrix m n R} :\n    Function.Injective M.mulVec \u2194 LinearIndependent R (fun i \u21a6 M\u1d40 i)", "theorem mulVec_stdBasisMatrix [Fintype m] (i : n) (j : m) (c : \u03b1) (x : m \u2192 \u03b1) :\n    mulVec (stdBasisMatrix i j c) x = Function.update (0 : n \u2192 \u03b1) i (c * x j)", "theorem mulVec_surjective_iff_exists_right_inverse [Finite m] [Fintype n] {A : Matrix m n R} :\n    Function.Surjective A.mulVec \u2194 \u2203 B : Matrix n m R, A * B = 1", "theorem mul_adjugate (A : Matrix n n \u03b1) : A * adjugate A = A.det \u2022 (1 : Matrix n n \u03b1)", "@[simp]\ntheorem mul_nonsing_inv (h : IsUnit A.det) : A * A\u207b\u00b9 = 1", "lemma mul_right_injective_of_inv (A : Matrix m n \u03b1) (B : Matrix n m \u03b1) (h : A * B = 1) :\n    Function.Injective (fun x : Matrix m l \u03b1 => B * x)", "@[simp]\ntheorem mul_smul [Fintype n] [Monoid R] [DistribMulAction R \u03b1] [SMulCommClass R \u03b1 \u03b1]\n    (M : Matrix m n \u03b1) (a : R) (N : Matrix n l \u03b1) : M * (a \u2022 N) = a \u2022 (M * N)", "@[simp]\ntheorem neg_dotProduct : -v \u2b1d\u1d65 w = -(v \u2b1d\u1d65 w)", "@[simp]\ntheorem neg_empty (v : Fin 0 \u2192 \u03b1) : -v = ![]", "@[simp]\ntheorem norm_col (v : m \u2192 \u03b1) : \u2016col \u03b9 v\u2016 = \u2016v\u2016", "@[simp]\ntheorem norm_row (v : n \u2192 \u03b1) : \u2016row \u03b9 v\u2016 = \u2016v\u2016", "theorem one_apply {i j} : (1 : Matrix n n \u03b1) i j = if i = j then 1 else 0", "theorem one_kronecker_one [MulZeroOneClass \u03b1] [DecidableEq m] [DecidableEq n] :\n    (1 : Matrix m m \u03b1) \u2297\u2096 (1 : Matrix n n \u03b1) = 1", "theorem pow_inv_comm' (A : M) (m n : \u2115) : A\u207b\u00b9 ^ m * A ^ n = A ^ n * A\u207b\u00b9 ^ m", "theorem pow_sub' (A : M) {m n : \u2115} (ha : IsUnit A.det) (h : n \u2264 m) :\n    A ^ (m - n) = A ^ m * (A ^ n)\u207b\u00b9", "theorem range_diagonal [DecidableEq m] (w : m \u2192 K) :\n    LinearMap.range (toLin' (diagonal w)) =\n      \u2a06 i \u2208 { i | w i \u2260 0 }, LinearMap.range (LinearMap.stdBasis K (fun _ => K) i)", "@[simp]\ntheorem rank_self_mul_transpose (A : Matrix m n R) : (A * A\u1d40).rank = A.rank", "theorem reindex_refl_refl (A : Matrix m n \u03b1) : reindex (Equiv.refl _) (Equiv.refl _) A = A", "theorem Matrix.represents_iff' {A : Matrix \u03b9 \u03b9 R} {f : Module.End R M} :\n    A.Represents b f \u2194 \u2200 j, \u2211 i : \u03b9, A i j \u2022 b i = f (b j)", "lemma reverse_charpoly (M : Matrix n n R) :\n    M.charpoly.reverse = M.charpolyRev", "theorem right_inv_eq_right_inv (h : A * B = 1) (g : A * C = 1) : B = C", "@[simp]\ntheorem row_apply (v : n \u2192 \u03b1) (i : \u03b9) (j) : row \u03b9 v i j = v j", "@[simp]\ntheorem row_cons (x : \u03b1) (u : Fin m \u2192 \u03b1) : row \u03b9 (vecCons x u) = of fun _ => vecCons x u", "@[simp]\ntheorem single_vecMul [Fintype m] [DecidableEq m] [NonUnitalNonAssocSemiring R] (M : Matrix m n R)\n    (i : m) (x : R) : Pi.single i x \u1d65* M = fun j => x * M i j", "theorem star_dotProduct_star : star v \u2b1d\u1d65 star w = star (w \u2b1d\u1d65 v)", "theorem stdBasis_eq_stdBasisMatrix (i : m) (j : n) [DecidableEq m] [DecidableEq n] :\n    stdBasis R m n (i, j) = stdBasisMatrix i j (1 : R)", "theorem submatrix_add [Add \u03b1] (A B : Matrix m n \u03b1) :\n    ((A + B).submatrix : (l \u2192 m) \u2192 (o \u2192 n) \u2192 Matrix l o \u03b1) = A.submatrix + B.submatrix", "@[simp]\ntheorem submatrix_cons_row (A : Matrix m' n' \u03b1) (i : m') (row : Fin m \u2192 m') (col : o' \u2192 n') :\n    submatrix A (vecCons i row) col = vecCons (fun j => A i (col j)) (submatrix A row col)", "@[simp]\ntheorem Matrix.toBilin'Aux_eq (M : Matrix n n R\u2082) : Matrix.toBilin'Aux M = Matrix.toBilin' M", "theorem Matrix.toBilin'_apply (M : Matrix n n R\u2082) (x y : n \u2192 R\u2082) :\n    Matrix.toBilin' M x y = \u2211 i, \u2211 j, x i * M i j * y j", "theorem Matrix.toBilin'_apply' (M : Matrix n n R\u2082) (v w : n \u2192 R\u2082) :\n    Matrix.toBilin' M v w = Matrix.dotProduct v (M *\u1d65 w)", "theorem Matrix.toBilin'_comp (M : Matrix n n R\u2082) (P Q : Matrix n o R\u2082) :\n    M.toBilin'.comp (Matrix.toLin' P) (Matrix.toLin' Q) = Matrix.toBilin' (P\u1d40 * M * Q)", "theorem toBlock_mul_eq_mul {m n k : Type*} [Fintype n] (p : m \u2192 Prop) (q : k \u2192 Prop)\n    (A : Matrix m n R) (B : Matrix n k R) :\n    (A * B).toBlock p q = A.toBlock p \u22a4 * B.toBlock \u22a4 q", "@[simp]\ntheorem toBlocks_fromBlocks\u2082\u2082 (A : Matrix n l \u03b1) (B : Matrix n m \u03b1) (C : Matrix o l \u03b1)\n    (D : Matrix o m \u03b1) : (fromBlocks A B C D).toBlocks\u2082\u2082 = D", "@[simp]\nlemma toEuclideanCLM_piLp_equiv_symm (A : Matrix n n \ud835\udd5c) (x : n \u2192 \ud835\udd5c) :\n    toEuclideanCLM (n := n) (\ud835\udd5c := \ud835\udd5c) A ((WithLp.equiv _ _).symm x) =\n      (WithLp.equiv _ _).symm (toLin' A x)", "theorem Matrix.toLinearMap\u2082'_apply' (M : Matrix n m R) (v : n \u2192 R) (w : m \u2192 R) :\n    Matrix.toLinearMap\u2082' M v w = Matrix.dotProduct v (M *\u1d65 w)", "theorem Matrix.toLinearMap\u2082_basisFun :\n    Matrix.toLinearMap\u2082 (Pi.basisFun R n) (Pi.basisFun R m) = Matrix.toLinearMap\u2082'", "@[simp]\ntheorem Matrix.toLinearMap\u209b\u2097\u2082'_stdBasis (M : Matrix n m R) (i : n) (j : m) :\n    Matrix.toLinearMap\u209b\u2097\u2082' \u03c3\u2081 \u03c3\u2082 M (LinearMap.stdBasis R\u2081 (fun _ => R\u2081) i 1)\n      (LinearMap.stdBasis R\u2082 (fun _ => R\u2082) j 1) = M i j", "@[simp]\nlemma toMvPolynomial_zero : (0 : Matrix m n R).toMvPolynomial = 0", "@[simp]\nlemma toRows\u2082_fromRows  (A\u2081 : Matrix m\u2081 n R) (A\u2082 : Matrix m\u2082 n R) :\n    toRows\u2082 (fromRows A\u2081 A\u2082) = A\u2082 := rfl", "theorem toSquareBlockProp_def (M : Matrix m m \u03b1) (p : m \u2192 Prop) :\n    toSquareBlockProp M p = of (fun i j : { a // p a } => M \u2191i \u2191j)", "@[simp]\ntheorem trace_add (A B : Matrix n n R) : trace (A + B) = trace A + trace B", "theorem trace_eq_sum_roots_charpoly_of_splits (hAps : A.charpoly.Splits (RingHom.id R)) :\n    A.trace = (Matrix.charpoly A).roots.sum", "theorem trace_fin_one (A : Matrix (Fin 1) (Fin 1) R) : trace A = A 0 0", "@[simp]\ntheorem trace_multiset_sum (s : Multiset (Matrix n n R)) : trace s.sum = (s.map trace).sum", "@[simp]\ntheorem updateColumn_ne [DecidableEq n] {j' : n} (j_ne : j' \u2260 j) :\n    updateColumn M j c i j' = M i j'", "theorem updateColumn_submatrix_equiv [DecidableEq o] [DecidableEq n] (A : Matrix m n \u03b1) (j : o)\n    (c : l \u2192 \u03b1) (e : l \u2243 m) (f : o \u2243 n) : updateColumn (A.submatrix e f) j c =\n    (A.updateColumn (f j) fun i => c (e.symm i)).submatrix e f", "theorem vecMul_surjective_iff_exists_left_inverse [Fintype m] [Finite n] {A : Matrix m n R} :\n    Function.Surjective A.vecMul \u2194 \u2203 B : Matrix n m R, B * A = 1", "theorem vecMul_surjective_of_invertible (A : Matrix m m R) [Invertible A] :\n    Function.Surjective A.vecMul", "theorem Base.dep_of_ssubset (hB : M.Base B) (h : B \u2282 X) (hX : X \u2286 M.E", "theorem Base.exchange_base_of_indep' (hB : M.Base B) (he : e \u2208 B) (hf : f \u2209 B)\n    (hI : M.Indep (insert f B \\ {e})) : M.Base (insert f B \\ {e})", "theorem Base.exists_insert_of_ssubset (hB : M.Base B) (hIB : I \u2282 B) (hB' : M.Base B') :\n    \u2203 e \u2208 B' \\ I, M.Indep (insert e I)", "lemma Base.map {B : Set \u03b1} (hB : M.Base B) {f : \u03b1 \u2192 \u03b2} (hf) : (M.map f hf).Base (f '' B)", "theorem Base.ncard_diff_comm (hB\u2081 : M.Base B\u2081) (hB\u2082 : M.Base B\u2082) :\n    (B\u2081 \\ B\u2082).ncard = (B\u2082 \\ B\u2081).ncard", "theorem Base.not_base_of_ssubset (hB : M.Base B) (hX : X \u2282 B) : \u00ac M.Base X", "theorem Basis.basis' (hI : M.Basis I X) : M.Basis' I X", "theorem Basis.basis_inter_ground (hI : M.Basis I X) : M.Basis I (X \u2229 M.E)", "theorem Basis.basis_of_basis_of_subset_of_subset (hI : M.Basis I X) (hJ : M.Basis J Y) (hJX : J \u2286 X)\n    (hIY : I \u2286 Y) : M.Basis I Y", "theorem Basis.basis_restriction (hI : M.Basis I X) (hNM : N \u2264r M) (hX : X \u2286 N.E) : N.Basis I X", "theorem Basis.eq_exchange_of_diff_eq_singleton (hI : M.Basis I X) (hJ : M.Basis J X)\n    (hIJ : I \\ J = {e}) : \u2203 f \u2208 J \\ I, J = insert f I \\ {e}", "theorem Basis.of_restriction (hI : N.Basis I X) (hNM : N \u2264r M) : M.Basis I X", "theorem Basis.transfer (hIX : M.Basis I X) (hJX : M.Basis J X) (hXY : X \u2286 Y) (hJY : M.Basis J Y) :\n    M.Basis I Y", "theorem Indep.basis_of_maximal_subset (hI : M.Indep I) (hIX : I \u2286 X)\n    (hmax : \u2200 \u2983J\u2984, M.Indep J \u2192 I \u2286 J \u2192 J \u2286 X \u2192 J \u2286 I) (hX : X \u2286 M.E", "theorem Indep.exists_base_subset_union_base (hI : M.Indep I) (hB : M.Base B) :\n    \u2203 B', M.Base B' \u2227 I \u2286 B' \u2227 B' \u2286 I \u222a B", "theorem Indep.exists_basis_subset_union_basis (hI : M.Indep I) (hIX : I \u2286 X) (hJ : M.Basis J X) :\n    \u2203 I', M.Basis I' X \u2227 I \u2286 I' \u2227 I' \u2286 I \u222a J", "lemma Indep.exists_bijOn_of_map {I : Set \u03b2} (hf) (hI : (M.map f hf).Indep I) :\n    \u2203 I\u2080, M.Indep I\u2080 \u2227 BijOn f I\u2080 I", "lemma Indep.mapEmbedding (hI : M.Indep I) (f : \u03b1 \u21aa \u03b2) : (M.mapEmbedding f).Indep (f '' I)", "theorem Restriction.antisymm {M' : Matroid \u03b1} (h : M \u2264r M') (h' : M' \u2264r M) : M = M'", "lemma comapOn_base_iff_of_surjOn (h : SurjOn f E N.E) :\n    (N.comapOn E f).Base B \u2194 (N.Base (f '' B) \u2227 InjOn f B \u2227 B \u2286 E)", "@[simp] lemma comapOn_indep_iff :\n    (N.comapOn E f).Indep I \u2194 (N.Indep (f '' I) \u2227 InjOn f I \u2227 I \u2286 E)", "lemma comapOn_preimage_eq (N : Matroid \u03b2) (f : \u03b1 \u2192 \u03b2) : N.comapOn (f \u207b\u00b9' N.E) f = N.comap f", "@[simp] lemma comap_basis_iff {I X : Set \u03b1} :\n    (N.comap f).Basis I X \u2194 N.Basis (f '' I) (f '' X) \u2227 I.InjOn f \u2227 I \u2286 X", "@[simp] theorem dual_inj {M\u2081 M\u2082 : Matroid \u03b1} : M\u2081\u2736 = M\u2082\u2736 \u2194 M\u2081 = M\u2082", "@[simp] theorem emptyOn_ground : (emptyOn \u03b1).E = \u2205", "lemma eq_of_restrictSubtype_eq {N : Matroid \u03b1} (hM : M.E = E) (hN : N.E = E)\n    (h : M.restrictSubtype E = N.restrictSubtype E) : M = N", "theorem finite_or_infiniteRk (M : Matroid \u03b1) : FiniteRk M \u2228 InfiniteRk M", "theorem finite_setOf_matroid {E : Set \u03b1} (hE : E.Finite) : {M : Matroid \u03b1 | M.E \u2286 E}.Finite", "theorem finite_setOf_matroid' {E : Set \u03b1} (hE : E.Finite) : {M : Matroid \u03b1 | M.E = E}.Finite", "theorem ground_nonempty (M : Matroid \u03b1) [M.Nonempty] : M.E.Nonempty", "theorem indep_iff_forall_finite_subset_indep {M : Matroid \u03b1} [Finitary M] :\n    M.Indep I \u2194 \u2200 J, J \u2286 I \u2192 J.Finite \u2192 M.Indep J", "@[simp] theorem loopyOn_empty (\u03b1 : Type*) : loopyOn (\u2205 : Set \u03b1) = emptyOn \u03b1", "@[simp] lemma restrictSubtype_ground : (M.restrictSubtype X).E = univ", "theorem restrict_strictRestriction {M : Matroid \u03b1} (hR : R \u2282 M.E) : M \u21be R <r M", "theorem setOf_base_eq_maximals_setOf_indep : {B | M.Base B} = maximals (\u00b7 \u2286 \u00b7) {I | M.Indep I}", "theorem setOf_dual_base_eq : {B | M\u2736.Base B} = (fun X \u21a6 M.E \\ X) '' {B | M.Base B}", "@[aesop unsafe 10% (rule_sets := [Matroid])]\nprivate theorem singleton_subset_ground (he : e \u2208 M.E) : {e} \u2286 M.E", "theorem uniqueBaseOn_indep_iff (hIE : I \u2286 E) : (uniqueBaseOn I E).Indep J \u2194 J \u2286 I", "theorem iInf_localization_eq_bot : (\u2a05 v : MaximalSpectrum R,\n    Localization.subalgebra.ofField K _ v.asIdeal.primeCompl_le_nonZeroDivisors) = \u22a5", "theorem _root_.Measurable.add_simpleFunc\n    {E : Type*} {_ : MeasurableSpace \u03b1} [MeasurableSpace E] [AddGroup E] [MeasurableAdd E]\n    {g : \u03b1 \u2192 E} (hg : Measurable g) (f : SimpleFunc \u03b1 E) :\n    Measurable (g + (f : \u03b1 \u2192 E))", "@[measurability]\ntheorem Measurable.ceil [OpensMeasurableSpace R] {f : \u03b1 \u2192 R} (hf : Measurable f) :\n    Measurable fun x => \u2308f x\u2309", "@[measurability, fun_prop]\ntheorem Measurable.dist {f g : \u03b2 \u2192 \u03b1} (hf : Measurable f) (hg : Measurable g) :\n    Measurable fun b => dist (f b) (g b)", "theorem Measurable.ite {p : \u03b1 \u2192 Prop} {_ : DecidablePred p} (hp : MeasurableSet { a : \u03b1 | p a })\n    (hf : Measurable f) (hg : Measurable g) : Measurable fun x => ite (p x) (f x) (g x)", "theorem Measurable.lintegral_prod_left [SFinite \u03bc] {f : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e}\n    (hf : Measurable (uncurry f)) : Measurable fun y => \u222b\u207b x, f x y \u2202\u03bc", "theorem Measurable.lintegral_prod_right' [SFinite \u03bd] :\n    \u2200 {f : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e}, Measurable f \u2192 Measurable fun x => \u222b\u207b y, f (x, y) \u2202\u03bd", "theorem _root_.Measurable.lmarginal (hf : Measurable f) : Measurable (\u222b\u22ef\u222b\u207b_s, f \u2202\u03bc)", "@[measurability]\ntheorem Measurable.sinh : Measurable fun x => Real.sinh (f x)", "@[to_additive (attr := fun_prop, aesop safe 20 apply (rule_sets := [Measurable]))]\ntheorem Measurable.smul [MeasurableSMul\u2082 M \u03b2] (hf : Measurable f) (hg : Measurable g) :\n    Measurable fun x => f x \u2022 g x", "theorem _root_.Measurable.stronglyMeasurable_add\n    {\u03b1 E : Type*} {_ : MeasurableSpace \u03b1} [AddGroup E] [TopologicalSpace E]\n    [MeasurableSpace E] [BorelSpace E] [ContinuousAdd E] [PseudoMetrizableSpace E]\n    {g f : \u03b1 \u2192 E} (hg : Measurable g) (hf : StronglyMeasurable f) :\n    Measurable (f + g)", "theorem MeasurableEmbedding.aemeasurable_comp_iff {g : \u03b2 \u2192 \u03b3} (hg : MeasurableEmbedding g)\n    {\u03bc : Measure \u03b1} : AEMeasurable (g \u2218 f) \u03bc \u2194 AEMeasurable f \u03bc", "@[simp] theorem comap_eq (hf : MeasurableEmbedding f) : MeasurableSpace.comap f \u2039_\u203a = \u2039_\u203a", "theorem measurable_comp_iff (hg : MeasurableEmbedding g) : Measurable (g \u2218 f) \u2194 Measurable f", "lemma _root_.MeasurableEmbedding.withDensity_ofReal_comap_apply_eq_integral_abs_deriv_mul\n    {f : \u211d \u2192 \u211d} (hf : MeasurableEmbedding f) {s : Set \u211d} (hs : MeasurableSet s)\n    {g : \u211d \u2192 \u211d} (hg : \u2200\u1d50 x, x \u2208 f '' s \u2192 0 \u2264 g x) (hg_int : IntegrableOn g (f '' s))\n    {f' : \u211d \u2192 \u211d} (hf' : \u2200 x \u2208 s, HasDerivWithinAt f (f' x) s x) :\n    (volume.withDensity (fun x \u21a6 ENNReal.ofReal (g x))).comap f s\n      = ENNReal.ofReal (\u222b x in s, |f' x| * g (f x))", "lemma _root_.MeasurableEmbedding.withDensity_ofReal_comap_apply_eq_integral_abs_deriv_mul'\n    {f : \u211d \u2192 \u211d} (hf : MeasurableEmbedding f) {s : Set \u211d} (hs : MeasurableSet s)\n    {f' : \u211d \u2192 \u211d} (hf' : \u2200 x, HasDerivAt f (f' x) x)\n    {g : \u211d \u2192 \u211d} (hg : 0 \u2264\u1d50[volume] g) (hg_int : Integrable g) :\n    (volume.withDensity (fun x \u21a6 ENNReal.ofReal (g x))).comap f s\n      = ENNReal.ofReal (\u222b x in s, |f' x| * g (f x))", "@[to_additive (attr := simp)]\ntheorem coe_mulLeft (g : G) : \u21d1(mulLeft g) = (g * \u00b7)", "theorem coe_sumPiEquivProdPi (\u03b1 : \u03b4 \u2295 \u03b4' \u2192 Type*) [\u2200 i, MeasurableSpace (\u03b1 i)] :\n    \u21d1(MeasurableEquiv.sumPiEquivProdPi \u03b1) = Equiv.sumPiEquivProdPi \u03b1", "theorem map_measurableEquiv_injective (e : \u03b1 \u2243\u1d50 \u03b2) : Injective (Measure.map e)", "lemma map_symm (e : \u03b2 \u2243\u1d50 \u03b1) : \u03bc.map e.symm = \u03bc.comap e", "protected theorem measurableEmbedding (e : \u03b1 \u2243\u1d50 \u03b2) : MeasurableEmbedding e where", "lemma preimage_symm (e : \u03b1 \u2243\u1d50 \u03b2) (s : Set \u03b1) : e.symm \u207b\u00b9' s = e '' s := (image_eq_preimage _ _).symm", "lemma _root_.MeasurableEquiv.withDensity_ofReal_map_symm_apply_eq_integral_abs_deriv_mul'\n    (f : \u211d \u2243\u1d50 \u211d) {s : Set \u211d} (hs : MeasurableSet s)\n    {f' : \u211d \u2192 \u211d} (hf' : \u2200 x, HasDerivAt f (f' x) x)\n    {g : \u211d \u2192 \u211d} (hg : 0 \u2264\u1d50[volume] g) (hg_int : Integrable g) :\n    (volume.withDensity (fun x \u21a6 ENNReal.ofReal (g x))).map f.symm s\n      = ENNReal.ofReal (\u222b x in s, |f' x| * g (f x))", "@[simp]\ntheorem coe_sdiff (s t : Subtype (MeasurableSet : Set \u03b1 \u2192 Prop)) : \u2191(s \\ t) = (s : Set \u03b1) \\ t", "theorem _root_.MeasurableSet.image_of_measurable_injOn {f : \u03b3 \u2192 \u03b1}\n    [MeasurableSpace.CountablySeparated \u03b1]\n    [MeasurableSpace \u03b3] [StandardBorelSpace \u03b3]\n    (hs : MeasurableSet s) (f_meas : Measurable f) (f_inj : InjOn f s) :\n    MeasurableSet (f '' s)", "theorem MeasurableSet.induction_on_open [TopologicalSpace \u03b1] [MeasurableSpace \u03b1] [BorelSpace \u03b1]\n    {C : Set \u03b1 \u2192 Prop} (h_open : \u2200 U, IsOpen U \u2192 C U)\n    (h_compl : \u2200 t, MeasurableSet t \u2192 C t \u2192 C t\u1d9c)\n    (h_union :\n      \u2200 f : \u2115 \u2192 Set \u03b1,\n        Pairwise (Disjoint on f) \u2192 (\u2200 i, MeasurableSet (f i)) \u2192 (\u2200 i, C (f i)) \u2192 C (\u22c3 i, f i)) :\n    \u2200 \u2983t\u2984, MeasurableSet t \u2192 C t", "@[measurability]\nprotected theorem MeasurableSet.insert {s : Set \u03b1} (hs : MeasurableSet s) (a : \u03b1) :\n    MeasurableSet (insert a s)", "theorem MeasurableSet.ite' {s t : Set \u03b1} {p : Prop} (hs : p \u2192 MeasurableSet s)\n    (ht : \u00acp \u2192 MeasurableSet t) : MeasurableSet (ite p s t)", "theorem MeasurableSet.map_coe_volume {s : Set \u03b1} (hs : MeasurableSet s) :\n    volume.map ((\u2191) : s \u2192 \u03b1) = restrict volume s", "@[simp]\ntheorem _root_.MeasurableSet.nullMeasurableSet (h : MeasurableSet s) : NullMeasurableSet s \u03bc", "theorem has_univ : d.Has univ", "theorem comap_eq_generateFrom (m : MeasurableSpace \u03b2) (f : \u03b1 \u2192 \u03b2) :\n    m.comap f = generateFrom { t | \u2203 s, MeasurableSet s \u2227 f \u207b\u00b9' s = t }", "theorem comap_map_le : (m.map f).comap f \u2264 m", "lemma generateFrom_countableGeneratingSet [m : MeasurableSpace \u03b1] [h : CountablyGenerated \u03b1] :\n    generateFrom (countableGeneratingSet \u03b1) = m", "lemma generateFrom_natGeneratingSequence (\u03b1 : Type*) [m : MeasurableSpace \u03b1]\n    [CountablyGenerated \u03b1] : generateFrom (range (natGeneratingSequence _)) = m", "@[simp]\ntheorem invariants_id : invariants (id : \u03b1 \u2192 \u03b1) = \u2039MeasurableSpace \u03b1\u203a", "lemma measurableSet_generateFrom_memPartition_iff (t : \u2115 \u2192 Set \u03b1) (n : \u2115) (s : Set \u03b1) :\n    MeasurableSet[generateFrom (memPartition t n)] s\n      \u2194 \u2203 S : Finset (Set \u03b1), \u2191S \u2286 memPartition t n \u2227 s = \u22c3\u2080 S", "theorem measurableSpace_iSup_eq (m : \u03b9 \u2192 MeasurableSpace \u03b1) :\n    \u2a06 n, m n = generateFrom { s | \u2203 n, MeasurableSet[m n] s }", "theorem measurable_invariants_of_semiconj {fa : \u03b1 \u2192 \u03b1} {fb : \u03b2 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b2} (hg : Measurable g)\n    (hfg : Semiconj g fa fb) : @Measurable _ _ (invariants fa) (invariants fb) g", "theorem AECover.comp_tendsto {\u03b1 \u03b9 \u03b9' : Type*} [MeasurableSpace \u03b1] {\u03bc : Measure \u03b1} {l : Filter \u03b9}\n    {l' : Filter \u03b9'} {\u03c6 : \u03b9 \u2192 Set \u03b1} (h\u03c6 : AECover \u03bc l \u03c6) {u : \u03b9' \u2192 \u03b9} (hu : Tendsto u l' l) :\n    AECover \u03bc l' (\u03c6 \u2218 u) where", "theorem AECover.iSup_lintegral_eq_of_countably_generated [Nonempty \u03b9] [l.NeBot]\n    [l.IsCountablyGenerated] {\u03c6 : \u03b9 \u2192 Set \u03b1} (h\u03c6 : AECover \u03bc l \u03c6) {f : \u03b1 \u2192 \u211d\u22650\u221e}\n    (hfm : AEMeasurable f \u03bc) : \u2a06 i : \u03b9, \u222b\u207b x in \u03c6 i, f x \u2202\u03bc = \u222b\u207b x, f x \u2202\u03bc", "theorem AECover.integrable_of_lintegral_nnnorm_bounded' [l.NeBot] [l.IsCountablyGenerated]\n    {\u03c6 : \u03b9 \u2192 Set \u03b1} (h\u03c6 : AECover \u03bc l \u03c6) {f : \u03b1 \u2192 E} (I : \u211d\u22650) (hfm : AEStronglyMeasurable f \u03bc)\n    (hbounded : \u2200\u1da0 i in l, (\u222b\u207b x in \u03c6 i, \u2016f x\u2016\u208a \u2202\u03bc) \u2264 I) : Integrable f \u03bc", "protected theorem comm : AEDisjoint \u03bc s t \u2194 AEDisjoint \u03bc t s", "theorem diff_ae_eq_right (h : AEDisjoint \u03bc s t) : (t \\ s : Set \u03b1) =\u1d50[\u03bc] t", "theorem compMeasurePreserving_eq_mk (g : \u03b2 \u2192\u2098[\u03bd] \u03b3) (hf : MeasurePreserving f \u03bc \u03bd) :\n    g.compMeasurePreserving f hf =\n      mk (g \u2218 f) (g.aestronglyMeasurable.comp_quasiMeasurePreserving hf.quasiMeasurePreserving)", "@[simp]\ntheorem compQuasiMeasurePreserving_toGerm [MeasurableSpace \u03b2] {f : \u03b1 \u2192 \u03b2} {\u03bd}\n    (g : \u03b2 \u2192\u2098[\u03bd] \u03b3) (hf : Measure.QuasiMeasurePreserving f \u03bc \u03bd) :\n    (g.compQuasiMeasurePreserving f hf).toGerm = g.toGerm.compTendsto f hf.tendsto_ae", "@[simp]\ntheorem mk_pow (f : \u03b1 \u2192 \u03b3) (hf) (n : \u2115) :\n    (mk f hf : \u03b1 \u2192\u2098[\u03bc] \u03b3) ^ n =\n      mk (f ^ n) ((_root_.continuous_pow n).comp_aestronglyMeasurable hf)", "theorem neg [AddGroup \u03b2] [TopologicalAddGroup \u03b2] {f : \u03b1 \u2192 \u03b2} (hfm : AEStronglyMeasurable' m f \u03bc) :\n    AEStronglyMeasurable' m (-f) \u03bc", "theorem sub [AddGroup \u03b2] [TopologicalAddGroup \u03b2] {f g : \u03b1 \u2192 \u03b2} (hfm : AEStronglyMeasurable' m f \u03bc)\n    (hgm : AEStronglyMeasurable' m g \u03bc) : AEStronglyMeasurable' m (f - g) \u03bc", "@[to_additive (attr := measurability)]\nprotected theorem const_smul' {\ud835\udd5c} [SMul \ud835\udd5c \u03b2] [ContinuousConstSMul \ud835\udd5c \u03b2]\n    (hf : AEStronglyMeasurable f \u03bc) (c : \ud835\udd5c) : AEStronglyMeasurable (fun x => c \u2022 f x) \u03bc", "theorem AEStronglyMeasurable.convolution_integrand (hf : AEStronglyMeasurable f \u03bd)\n    (hg : AEStronglyMeasurable g \u03bc) :\n    AEStronglyMeasurable (fun p : G \u00d7 G => L (f p.2) (g (p.1 - p.2))) (\u03bc.prod \u03bd)", "theorem AEStronglyMeasurable.convolution_integrand' [MeasurableAdd\u2082 G]\n    [MeasurableNeg G] [SigmaFinite \u03bd] (hf : AEStronglyMeasurable f \u03bd)\n    (hg : AEStronglyMeasurable g <| map (fun p : G \u00d7 G => p.1 - p.2) (\u03bc.prod \u03bd)) :\n    AEStronglyMeasurable (fun p : G \u00d7 G => L (f p.2) (g (p.1 - p.2))) (\u03bc.prod \u03bd)", "@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]\nprotected theorem div [Group \u03b2] [TopologicalGroup \u03b2] (hf : AEStronglyMeasurable f \u03bc)\n    (hg : AEStronglyMeasurable g \u03bc) : AEStronglyMeasurable (f / g) \u03bc", "protected theorem im {f : \u03b1 \u2192 \ud835\udd5c} (hf : AEStronglyMeasurable f \u03bc) :\n    AEStronglyMeasurable (fun x => RCLike.im (f x)) \u03bc", "theorem sum_measure [PseudoMetrizableSpace \u03b2] {m : MeasurableSpace \u03b1} {\u03bc : \u03b9 \u2192 Measure \u03b1}\n    (h : \u2200 i, AEStronglyMeasurable f (\u03bc i)) : AEStronglyMeasurable f (Measure.sum \u03bc)", "theorem Adapted.isStoppingTime_crossing (hf : Adapted \u2131 f) :\n    IsStoppingTime \u2131 (upperCrossingTime a b f N n) \u2227\n      IsStoppingTime \u2131 (lowerCrossingTime a b f N n)", "lemma BoundedContinuousFunction.integral_eq_integral_meas_le\n    {\u03b1 : Type*} [MeasurableSpace \u03b1] [TopologicalSpace \u03b1] [OpensMeasurableSpace \u03b1]\n    (f : \u03b1 \u2192\u1d47 \u211d) (\u03bc : Measure \u03b1) [IsFiniteMeasure \u03bc] (f_nn : 0 \u2264\u1d50[\u03bc] f) :\n    \u222b \u03c9, f \u03c9 \u2202\u03bc = \u222b t in Ioc 0 \u2016f\u2016, ENNReal.toReal (\u03bc {a : \u03b1 | t \u2264 f a})", "theorem contentRegular_exists_compact (H : ContentRegular \u03bc) (K : TopologicalSpace.Compacts G)\n    {\u03b5 : NNReal} (h\u03b5 : \u03b5 \u2260 0) :\n    \u2203 K' : TopologicalSpace.Compacts G, K.carrier \u2286 interior K'.carrier \u2227 \u03bc K' \u2264 \u03bc K + \u03b5", "theorem innerContent_iUnion_nat [R1Space G] \u2983U : \u2115 \u2192 Set G\u2984\n    (hU : \u2200 i : \u2115, IsOpen (U i)) :\n    \u03bc.innerContent \u27e8\u22c3 i : \u2115, U i, isOpen_iUnion hU\u27e9 \u2264 \u2211' i : \u2115, \u03bc.innerContent \u27e8U i, hU i\u27e9", "theorem innerContent_mono' \u2983U V : Set G\u2984 (hU : IsOpen U) (hV : IsOpen V) (h2 : U \u2286 V) :\n    \u03bc.innerContent \u27e8U, hU\u27e9 \u2264 \u03bc.innerContent \u27e8V, hV\u27e9", "theorem lt_top (K : Compacts G) : \u03bc K < \u221e", "theorem mono (K\u2081 K\u2082 : Compacts G) (h : (K\u2081 : Set G) \u2286 K\u2082) : \u03bc K\u2081 \u2264 \u03bc K\u2082", "@[to_additive]\ntheorem outerMeasure_pos_of_is_mul_left_invariant [Group G] [TopologicalGroup G]\n    (h3 : \u2200 (g : G) {K : Compacts G}, \u03bc (K.map _ <| continuous_mul_left g) = \u03bc K) (K : Compacts G)\n    (hK : \u03bc K \u2260 0) {U : Set G} (h1U : IsOpen U) (h2U : U.Nonempty) : 0 < \u03bc.outerMeasure U", "@[simp]\ntheorem const_apply {m' : MeasurableSpace \u03a9} {hm' : m' \u2264 m} (i : \u03b9) : const \u03b9 m' hm' i = m'", "theorem stronglyMeasurable_limitProcess : StronglyMeasurable[\u2a06 n, \u2131 n] (limitProcess f \u2131 \u03bc)", "@[simp, norm_cast]\ntheorem coeFn_smul [IsScalarTower R \u211d\u22650 \u211d\u22650] (c : R) (\u03bc : FiniteMeasure \u03a9) :\n    (\u21d1(c \u2022 \u03bc) : Set \u03a9 \u2192 \u211d\u22650) = c \u2022 (\u21d1\u03bc : Set \u03a9 \u2192 \u211d\u22650)", "theorem restrict_apply (\u03bc : FiniteMeasure \u03a9) (A : Set \u03a9) {s : Set \u03a9} (s_mble : MeasurableSet s) :\n    (\u03bc.restrict A) s = \u03bc (s \u2229 A)", "theorem restrict_mass (\u03bc : FiniteMeasure \u03a9) (A : Set \u03a9) : (\u03bc.restrict A).mass = \u03bc A", "@[simp]\ntheorem smul_apply [IsScalarTower R \u211d\u22650 \u211d\u22650] (c : R) (\u03bc : FiniteMeasure \u03a9) (s : Set \u03a9) :\n    (c \u2022 \u03bc) s = c \u2022 \u03bc s", "theorem tendsto_iff_forall_toWeakDualBCNN_tendsto {\u03b3 : Type*} {F : Filter \u03b3}\n    {\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9} {\u03bc : FiniteMeasure \u03a9} :\n    Tendsto \u03bcs F (\ud835\udcdd \u03bc) \u2194\n      \u2200 f : \u03a9 \u2192\u1d47 \u211d\u22650, Tendsto (fun i => (\u03bcs i).toWeakDualBCNN f) F (\ud835\udcdd (\u03bc.toWeakDualBCNN f))", "theorem tendsto_iff_weak_star_tendsto {\u03b3 : Type*} {F : Filter \u03b3} {\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9}\n    {\u03bc : FiniteMeasure \u03a9} :\n    Tendsto \u03bcs F (\ud835\udcdd \u03bc) \u2194 Tendsto (fun i => (\u03bcs i).toWeakDualBCNN) F (\ud835\udcdd \u03bc.toWeakDualBCNN)", "lemma tendsto_map_of_tendsto_of_continuous {\u03b9 : Type*} {L : Filter \u03b9}\n    (\u03bds : \u03b9 \u2192 FiniteMeasure \u03a9) (\u03bd : FiniteMeasure \u03a9) (lim : Tendsto \u03bds L (\ud835\udcdd \u03bd))\n    {f : \u03a9 \u2192 \u03a9'} (f_cont : Continuous f) :\n    Tendsto (fun i \u21a6 (\u03bds i).map f) L (\ud835\udcdd (\u03bd.map f))", "theorem tendsto_testAgainstNN_of_tendsto_normalize_testAgainstNN_of_tendsto_mass {\u03b3 : Type*}\n    {F : Filter \u03b3} {\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9}\n    (\u03bcs_lim : Tendsto (fun i => (\u03bcs i).normalize) F (\ud835\udcdd \u03bc.normalize))\n    (mass_lim : Tendsto (fun i => (\u03bcs i).mass) F (\ud835\udcdd \u03bc.mass)) (f : \u03a9 \u2192\u1d47 \u211d\u22650) :\n    Tendsto (fun i => (\u03bcs i).testAgainstNN f) F (\ud835\udcdd (\u03bc.testAgainstNN f))", "theorem tendsto_zero_testAgainstNN_of_tendsto_zero_mass {\u03b3 : Type*} {F : Filter \u03b3}\n    {\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9} (mass_lim : Tendsto (fun i => (\u03bcs i).mass) F (\ud835\udcdd 0)) (f : \u03a9 \u2192\u1d47 \u211d\u22650) :\n    Tendsto (fun i => (\u03bcs i).testAgainstNN f) F (\ud835\udcdd 0)", "theorem testAgainstNN_lipschitz (\u03bc : FiniteMeasure \u03a9) :\n    LipschitzWith \u03bc.mass fun f : \u03a9 \u2192\u1d47 \u211d\u22650 => \u03bc.testAgainstNN f", "@[simp]\ntheorem testAgainstNN_one (\u03bc : FiniteMeasure \u03a9) : \u03bc.testAgainstNN 1 = \u03bc.mass", "theorem HasFiniteIntegral.const_mul [NormedRing \ud835\udd5c] {f : \u03b1 \u2192 \ud835\udd5c} (h : HasFiniteIntegral f \u03bc) (c : \ud835\udd5c) :\n    HasFiniteIntegral (fun x => c * f x) \u03bc", "theorem Integrable.add_measure {f : \u03b1 \u2192 \u03b2} (h\u03bc : Integrable f \u03bc) (h\u03bd : Integrable f \u03bd) :\n    Integrable f (\u03bc + \u03bd)", "theorem Integrable.aemeasurable [MeasurableSpace \u03b2] [BorelSpace \u03b2] {f : \u03b1 \u2192 \u03b2}\n    (hf : Integrable f \u03bc) : AEMeasurable f \u03bc", "theorem _root_.MeasureTheory.Integrable.compProd_mk_left_ae \u2983f : \u03b2 \u00d7 \u03b3 \u2192 E\u2984\n    (hf : Integrable f ((\u03ba \u2297\u2096 \u03b7) a)) : \u2200\u1d50 x \u2202\u03ba a, Integrable (fun y => f (x, y)) (\u03b7 (a, x))", "theorem Integrable.comp_div {g : \u211d \u2192 F} (hg : Integrable g) {R : \u211d} (hR : R \u2260 0) :\n    Integrable fun x => g (x / R)", "@[to_additive]\ntheorem Integrable.comp_inv [IsInvInvariant \u03bc] {f : G \u2192 F} (hf : Integrable f \u03bc) :\n    Integrable (fun t => f t\u207b\u00b9) \u03bc", "@[to_additive]\ntheorem Integrable.comp_mul_left {f : G \u2192 F} [IsMulLeftInvariant \u03bc] (hf : Integrable f \u03bc) (g : G) :\n    Integrable (fun t => f (g * t)) \u03bc", "theorem _root_.MeasureTheory.Integrable.condexpKernel_ae (hf_int : Integrable f \u03bc) :\n    \u2200\u1d50 \u03c9 \u2202\u03bc, Integrable f (condexpKernel \u03bc m \u03c9)", "theorem Integrable.congr {f g : \u03b1 \u2192 \u03b2} (hf : Integrable f \u03bc) (h : f =\u1d50[\u03bc] g) : Integrable g \u03bc", "@[elab_as_elim]\ntheorem Integrable.induction (P : (\u03b1 \u2192 E) \u2192 Prop)\n    (h_ind : \u2200 (c : E) \u2983s\u2984, MeasurableSet s \u2192 \u03bc s < \u221e \u2192 P (s.indicator fun _ => c))\n    (h_add :\n      \u2200 \u2983f g : \u03b1 \u2192 E\u2984,\n        Disjoint (support f) (support g) \u2192 Integrable f \u03bc \u2192 Integrable g \u03bc \u2192 P f \u2192 P g \u2192 P (f + g))\n    (h_closed : IsClosed { f : \u03b1 \u2192\u2081[\u03bc] E | P f })\n    (h_ae : \u2200 \u2983f g\u2984, f =\u1d50[\u03bc] g \u2192 Integrable f \u03bc \u2192 P f \u2192 P g) :\n    \u2200 \u2983f : \u03b1 \u2192 E\u2984, Integrable f \u03bc \u2192 P f", "theorem MeasureTheory.Integrable.intervalIntegrable (hf : Integrable f \u03bc) :\n    IntervalIntegrable f \u03bc a b", "theorem Integrable.left_of_add_measure {f : \u03b1 \u2192 \u03b2} (h : Integrable f (\u03bc + \u03bd)) : Integrable f \u03bc", "theorem Integrable.pos_part {f : \u03b1 \u2192 \u211d} (hf : Integrable f \u03bc) :\n    Integrable (fun a => max (f a) 0) \u03bc", "theorem Integrable.prod_mul {L : Type*} [RCLike L] {f : \u03b1 \u2192 L} {g : \u03b2 \u2192 L} (hf : Integrable f \u03bc)\n    (hg : Integrable g \u03bd) : Integrable (fun z : \u03b1 \u00d7 \u03b2 => f z.1 * g z.2) (\u03bc.prod \u03bd)", "theorem Integrable.prod_right_ae [SigmaFinite \u03bc] \u2983f : \u03b1 \u00d7 \u03b2 \u2192 E\u2984 (hf : Integrable f (\u03bc.prod \u03bd)) :\n    \u2200\u1d50 x \u2202\u03bc, Integrable (fun y => f (x, y)) \u03bd", "theorem Integrable.real_toNNReal {f : \u03b1 \u2192 \u211d} (hf : Integrable f \u03bc) :\n    Integrable (fun x => ((f x).toNNReal : \u211d)) \u03bc", "theorem Integrable.tendsto_ae_condexp (hg : Integrable g \u03bc)\n    (hgmeas : StronglyMeasurable[\u2a06 n, \u2131 n] g) :\n    \u2200\u1d50 x \u2202\u03bc, Tendsto (fun n => (\u03bc[g|\u2131 n]) x) atTop (\ud835\udcdd (g x))", "theorem toL1_sub (f g : \u03b1 \u2192 \u03b2) (hf : Integrable f \u03bc) (hg : Integrable g \u03bc) :\n    toL1 (f - g) (hf.sub hg) = toL1 f hf - toL1 g hg", "lemma IntegrableAtFilter.eq_zero_of_tendsto\n    (h : IntegrableAtFilter f l \u03bc) (h' : \u2200 s \u2208 l, \u03bc s = \u221e) {a : E}\n    (hf : Tendsto f l (\ud835\udcdd a)) : a = 0", "theorem IntegrableAtFilter.inf_of_left (hl : IntegrableAtFilter f l \u03bc) :\n    IntegrableAtFilter f (l \u2293 l') \u03bc", "theorem IntegrableOn.integrable_of_ae_not_mem_eq_zero (hf : IntegrableOn f s \u03bc)\n    (h't : \u2200\u1d50 x \u2202\u03bc, x \u2209 s \u2192 f x = 0) : Integrable f \u03bc", "@[to_additive]\ntheorem lintegral_eq_tsum (h : IsFundamentalDomain G s \u03bc) (f : \u03b1 \u2192 \u211d\u22650\u221e) :\n    \u222b\u207b x, f x \u2202\u03bc = \u2211' g : G, \u222b\u207b x in g \u2022 s, f x \u2202\u03bc", "@[to_additive]\nprotected theorem setIntegral_eq (hs : IsFundamentalDomain G s \u03bc) (ht : IsFundamentalDomain G t \u03bc)\n    {f : \u03b1 \u2192 E} (hf : \u2200 (g : G) (x), f (g \u2022 x) = f x) : \u222b x in s, f x \u2202\u03bc = \u222b x in t, f x \u2202\u03bc", "theorem IsProbabilityMeasure.ae_neBot [IsProbabilityMeasure \u03bc] : NeBot (ae \u03bc)", "theorem IsProbabilityMeasure.ne_zero (\u03bc : Measure \u03b1) [IsProbabilityMeasure \u03bc] : \u03bc \u2260 0", "theorem add_const_nat {f : Filtration \u2115 m} {\u03c4 : \u03a9 \u2192 \u2115} (h\u03c4 : IsStoppingTime f \u03c4) {i : \u2115} :\n    IsStoppingTime f fun \u03c9 => \u03c4 \u03c9 + i", "theorem measurableSet_inter_le_const_iff (h\u03c4 : IsStoppingTime f \u03c4) (s : Set \u03a9) (i : \u03b9) :\n    MeasurableSet[h\u03c4.measurableSpace] (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i}) \u2194\n      MeasurableSet[(h\u03c4.min_const i).measurableSpace] (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 i})", "theorem measurableSet_inter_le_iff [TopologicalSpace \u03b9] [SecondCountableTopology \u03b9]\n    [OrderTopology \u03b9] [MeasurableSpace \u03b9] [BorelSpace \u03b9] (h\u03c4 : IsStoppingTime f \u03c4)\n    (h\u03c0 : IsStoppingTime f \u03c0) (s : Set \u03a9) :\n    MeasurableSet[h\u03c4.measurableSpace] (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9}) \u2194\n      MeasurableSet[(h\u03c4.min h\u03c0).measurableSpace] (s \u2229 {\u03c9 | \u03c4 \u03c9 \u2264 \u03c0 \u03c9})", "theorem measurableSet_min_iff (h\u03c4 : IsStoppingTime f \u03c4) (h\u03c0 : IsStoppingTime f \u03c0) (s : Set \u03a9) :\n    MeasurableSet[(h\u03c4.min h\u03c0).measurableSpace] s \u2194\n      MeasurableSet[h\u03c4.measurableSpace] s \u2227 MeasurableSet[h\u03c0.measurableSpace] s", "theorem real_smul_posPart_nonneg (r : \u211d) (hr : 0 \u2264 r) :\n    (r \u2022 j).posPart = r.toNNReal \u2022 j.posPart", "@[simp]\ntheorem smul_negPart (r : \u211d\u22650) : (r \u2022 j).negPart = r \u2022 j.negPart", "@[simp]\ntheorem zero_negPart : (0 : JordanDecomposition \u03b1).negPart = 0", "@[simp]\ntheorem setToL1SCLM_zero_left (hT : DominatedFinMeasAdditive \u03bc (0 : Set \u03b1 \u2192 E \u2192L[\u211d] F) C)\n    (f : \u03b1 \u2192\u2081\u209b[\u03bc] E) : setToL1SCLM \u03b1 E \u03bc hT f = 0", "theorem setToL1S_mono_left {T T' : Set \u03b1 \u2192 E \u2192L[\u211d] G''} (hTT' : \u2200 s x, T s x \u2264 T' s x)\n    (f : \u03b1 \u2192\u2081\u209b[\u03bc] E) : setToL1S T f \u2264 setToL1S T' f", "@[integral_simps]\ntheorem integral_neg (f : \u03b1 \u2192\u2081[\u03bc] E) : integral (-f) = -integral f", "@[simp, nolint simpNF]\ntheorem L1.integral_of_fun_eq_integral {f : \u03b1 \u2192 G} (hf : Integrable f \u03bc) :\n    \u222b a, (hf.toL1 f) a \u2202\u03bc = \u222b a, f a \u2202\u03bc", "theorem ofReal_norm_eq_lintegral (f : \u03b1 \u2192\u2081[\u03bc] \u03b2) :\n    ENNReal.ofReal \u2016f\u2016 = \u222b\u207b x, (\u2016f x\u2016\u208a : \u211d\u22650\u221e) \u2202\u03bc", "theorem setToL1_mono_left {T T' : Set \u03b1 \u2192 E \u2192L[\u211d] G''} {C C' : \u211d}\n    (hT : DominatedFinMeasAdditive \u03bc T C) (hT' : DominatedFinMeasAdditive \u03bc T' C')\n    (hTT' : \u2200 s x, T s x \u2264 T' s x) (f : \u03b1 \u2192\u2081[\u03bc] E) : setToL1 hT f \u2264 setToL1 hT' f", "theorem inner_indicatorConstLp_eq_setIntegral_inner (f : Lp E 2 \u03bc) (hs : MeasurableSet s) (c : E)\n    (h\u03bcs : \u03bc s \u2260 \u221e) : (\u27eaindicatorConstLp 2 hs h\u03bcs c, f\u27eb : \ud835\udd5c) = \u222b x in s, \u27eac, f x\u27eb \u2202\u03bc", "private theorem smul_left' (f g : \u03b1 \u2192\u2082[\u03bc] E) (r : \ud835\udd5c) : \u27ear \u2022 f, g\u27eb = conj r * inner f g", "theorem continuousOn_smul [LocallyCompactSpace X] [T2Space X] {\ud835\udd5c : Type*} [NormedField \ud835\udd5c]\n    [SecondCountableTopologyEither X \ud835\udd5c] [NormedSpace \ud835\udd5c E] {f : X \u2192 E} {g : X \u2192 \ud835\udd5c} {s : Set X}\n    (hs : IsOpen s) (hf : LocallyIntegrableOn f s \u03bc) (hg : ContinuousOn g s) :\n    LocallyIntegrableOn (fun x => g x \u2022 f x) s \u03bc", "theorem boundedContinuousFunction_dense [SecondCountableTopologyEither \u03b1 E] [Fact (1 \u2264 p)]\n    (hp : p \u2260 \u221e) [\u03bc.WeaklyRegular] :\n    Dense (boundedContinuousFunction E p \u03bc : Set (Lp E p \u03bc))", "theorem coeFn_neg (f : Lp E p \u03bc) : \u21d1(-f) =\u1d50[\u03bc] -f", "theorem coeFn_negPart (f : Lp \u211d p \u03bc) : \u2200\u1d50 a \u2202\u03bc, negPart f a = -min (f a) 0", "theorem nnnorm_def (f : Lp E p \u03bc) : \u2016f\u2016\u208a = ENNReal.toNNReal (snorm f p \u03bc)", "theorem nnnorm_eq_zero_iff {f : Lp E p \u03bc} (hp : 0 < p) : \u2016f\u2016\u208a = 0 \u2194 f = 0", "@[simp]\ntheorem nnnorm_zero : \u2016(0 : Lp E p \u03bc)\u2016\u208a = 0", "protected theorem aestronglyMeasurable (f : Lp.simpleFunc E p \u03bc) :\n    AEStronglyMeasurable (toSimpleFunc f) \u03bc", "@[elab_as_elim]\nprotected theorem induction (hp_pos : p \u2260 0) (hp_ne_top : p \u2260 \u221e) {P : Lp.simpleFunc E p \u03bc \u2192 Prop}\n    (h_ind :\n      \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u03bc s < \u221e),\n        P (Lp.simpleFunc.indicatorConst p hs h\u03bcs.ne c))\n    (h_add :\n      \u2200 \u2983f g : \u03b1 \u2192\u209b E\u2984,\n        \u2200 hf : Mem\u2112p f p \u03bc,\n          \u2200 hg : Mem\u2112p g p \u03bc,\n            Disjoint (support f) (support g) \u2192\n              P (Lp.simpleFunc.toLp f hf) \u2192\n                P (Lp.simpleFunc.toLp g hg) \u2192 P (Lp.simpleFunc.toLp f hf + Lp.simpleFunc.toLp g hg))\n    (f : Lp.simpleFunc E p \u03bc) : P f", "protected theorem mem\u2112p (f : Lp.simpleFunc E p \u03bc) : Mem\u2112p (toSimpleFunc f) p \u03bc", "theorem toLp_neg (f : \u03b1 \u2192\u209b E) (hf : Mem\u2112p f p \u03bc) : toLp (-f) hf.neg = -toLp f hf", "protected theorem uniformEmbedding : UniformEmbedding ((\u2191) : Lp.simpleFunc E p \u03bc \u2192 Lp E p \u03bc)", "theorem LpAddConst_zero : LpAddConst 0 = 1", "theorem exists_compact_not_null [InnerRegular \u03bc] : (\u2203 K, IsCompact K \u2227 \u03bc K \u2260 0) \u2194 \u03bc \u2260 0", "protected theorem map_iff [BorelSpace \u03b1] [MeasurableSpace \u03b2] [TopologicalSpace \u03b2]\n    [BorelSpace \u03b2] (f : \u03b1 \u2243\u209c \u03b2) :\n    InnerRegular (Measure.map f \u03bc) \u2194 InnerRegular \u03bc", "theorem measurableSet_of_isOpen [OuterRegular \u03bc] (H : InnerRegularWRT \u03bc p IsOpen)\n    (hd : \u2200 \u2983s U\u2984, p s \u2192 IsOpen U \u2192 p (s \\ U)) :\n    InnerRegularWRT \u03bc p fun s => MeasurableSet s \u2227 \u03bc s \u2260 \u221e", "theorem rfl {p : Set \u03b1 \u2192 Prop} : InnerRegularWRT \u03bc p p", "@[to_additive]\nlemma IsEverywherePos.IsGdelta_of_isMulLeftInvariant\n    {k : Set G} (h : \u03bc.IsEverywherePos k) (hk : IsCompact k) (h'k : IsClosed k) :\n    IsG\u03b4 k", "@[to_additive\n\"Assume that a measure `\u03bc` is `IsAddLeftInvariant`, that the action of `\u0393` on `G` has a\nmeasurable fundamental domain `s` with positive finite volume, and that there is a single measurable\nset `V \u2286 G \u29f8 \u0393` along which the pullback of `\u03bc` and `\u03bd` agree (so the scaling is right). Then\n`\u03bc` satisfies `AddQuotientMeasureEqMeasurePreimage`. The main tool of the proof is the uniqueness of\nleft invariant measures, if normalized by a single positive finite-measured set.\"]\ntheorem MeasureTheory.Measure.IsMulLeftInvariant.quotientMeasureEqMeasurePreimage_of_set {s : Set G}\n    (fund_dom_s : IsFundamentalDomain \u0393.op s \u03bd) {V : Set (G \u29f8 \u0393)}\n    (meas_V : MeasurableSet V) (neZeroV : \u03bc V \u2260 0) (hV : \u03bc V = \u03bd (\u03c0 \u207b\u00b9' V \u2229 s))\n    (neTopV : \u03bc V \u2260 \u22a4) : QuotientMeasureEqMeasurePreimage \u03bd \u03bc", "@[simp]\nlemma measure_compl_nullSet (h : \u03bc \u27c2\u2098 \u03bd) : \u03bd h.nullSet\u1d9c = 0 := h.choose_spec.2.2", "@[simp]\ntheorem sum_right {\u03b9 : Type*} [Countable \u03b9] {\u03bd : \u03b9 \u2192 Measure \u03b1} : \u03bc \u27c2\u2098 sum \u03bd \u2194 \u2200 i, \u03bc \u27c2\u2098 \u03bd i", "lemma Measure.MutuallySingular.withDensity {\u03bd : Measure \u03b1} {f : \u03b1 \u2192 \u211d\u22650\u221e} (h : \u03bc \u27c2\u2098 \u03bd) :\n    \u03bc.withDensity f \u27c2\u2098 \u03bd", "protected theorem comp {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} (hg : QuasiMeasurePreserving g \u03bcb \u03bcc)\n    (hf : QuasiMeasurePreserving f \u03bca \u03bcb) : QuasiMeasurePreserving (g \u2218 f) \u03bca \u03bcc", "theorem restrict_of_measure_ne_top [BorelSpace \u03b1] [WeaklyRegular \u03bc] {A : Set \u03b1}\n    (h'A : \u03bc A \u2260 \u221e) : WeaklyRegular (\u03bc.restrict A)", "theorem ae_ae_eq_curry_of_prod {f g : \u03b1 \u00d7 \u03b2 \u2192 \u03b3} (h : f =\u1d50[\u03bc.prod \u03bd] g) :\n    \u2200\u1d50 x \u2202\u03bc, curry f x =\u1d50[\u03bd] curry g x", "theorem comap_apply\u2080 [MeasurableSpace \u03b1] (f : \u03b1 \u2192 \u03b2) (\u03bc : Measure \u03b2) (hfi : Injective f)\n    (hf : \u2200 s, MeasurableSet s \u2192 NullMeasurableSet (f '' s) \u03bc)\n    (hs : NullMeasurableSet s (comap f \u03bc)) : comap f \u03bc s = \u03bc (f '' s)", "lemma compProd_add_left (\u03bc \u03bd : Measure \u03b1) [SFinite \u03bc] [SFinite \u03bd] (\u03ba : kernel \u03b1 \u03b2)\n    [IsSFiniteKernel \u03ba] :\n    (\u03bc + \u03bd) \u2297\u2098 \u03ba = \u03bc \u2297\u2098 \u03ba + \u03bd \u2297\u2098 \u03ba", "theorem count_empty : count (\u2205 : Set \u03b1) = 0", "theorem countable_meas_pos_of_disjoint_iUnion {\u03b9 : Type*} {_ : MeasurableSpace \u03b1} {\u03bc : Measure \u03b1}\n    [SFinite \u03bc] {As : \u03b9 \u2192 Set \u03b1} (As_mble : \u2200 i : \u03b9, MeasurableSet (As i))\n    (As_disj : Pairwise (Disjoint on As)) : Set.Countable { i : \u03b9 | 0 < \u03bc (As i) }", "theorem countable_meas_pos_of_disjoint_of_meas_iUnion_ne_top\u2080 {\u03b9 : Type*} {_ : MeasurableSpace \u03b1}\n    (\u03bc : Measure \u03b1) {As : \u03b9 \u2192 Set \u03b1} (As_mble : \u2200 i : \u03b9, NullMeasurableSet (As i) \u03bc)\n    (As_disj : Pairwise (AEDisjoint \u03bc on As)) (Union_As_finite : \u03bc (\u22c3 i, As i) \u2260 \u221e) :\n    Set.Countable { i : \u03b9 | 0 < \u03bc (As i) }", "theorem empty_of_count_eq_zero [MeasurableSingletonClass \u03b1] (hsc : count s = 0) : s = \u2205", "theorem eventually_nonempty_inter_smul_of_density_one (s : Set E) (x : E)\n    (h : Tendsto (fun r => \u03bc (s \u2229 closedBall x r) / \u03bc (closedBall x r)) (\ud835\udcdd[>] 0) (\ud835\udcdd 1)) (t : Set E)\n    (ht : MeasurableSet t) (h't : \u03bc t \u2260 0) :\n    \u2200\u1da0 r in \ud835\udcdd[>] (0 : \u211d), (s \u2229 ({x} + r \u2022 t)).Nonempty", "theorem exists_positive_of_not_mutuallySingular (\u03bc \u03bd : Measure \u03b1) [IsFiniteMeasure \u03bc]\n    [IsFiniteMeasure \u03bd] (h : \u00ac\u03bc \u27c2\u2098 \u03bd) :\n    \u2203 \u03b5 : \u211d\u22650, 0 < \u03b5 \u2227\n      \u2203 E : Set \u03b1,\n        MeasurableSet E \u2227 0 < \u03bd E \u2227 0 \u2264[E] \u03bc.toSignedMeasure - (\u03b5 \u2022 \u03bd).toSignedMeasure", "theorem ext_iff' : \u03bc\u2081 = \u03bc\u2082 \u2194 \u2200 s, \u03bc\u2081 s = \u03bc\u2082 s", "theorem ext_of_Ico_finite {\u03b1 : Type*} [TopologicalSpace \u03b1] {m : MeasurableSpace \u03b1}\n    [SecondCountableTopology \u03b1] [LinearOrder \u03b1] [OrderTopology \u03b1] [BorelSpace \u03b1] (\u03bc \u03bd : Measure \u03b1)\n    [IsFiniteMeasure \u03bc] (h\u03bc\u03bd : \u03bc univ = \u03bd univ) (h : \u2200 \u2983a b\u2984, a < b \u2192 \u03bc (Ico a b) = \u03bd (Ico a b)) :\n    \u03bc = \u03bd", "@[simp]\nlemma fst_prod [IsProbabilityMeasure \u03bd] : (\u03bc.prod \u03bd).fst = \u03bc", "@[to_additive addIndex_union_eq]\ntheorem index_union_eq (K\u2081 K\u2082 : Compacts G) {V : Set G} (hV : (interior V).Nonempty)\n    (h : Disjoint (K\u2081.1 * V\u207b\u00b9) (K\u2082.1 * V\u207b\u00b9)) :\n    index (K\u2081.1 \u222a K\u2082.1) V = index K\u2081.1 V + index K\u2082.1 V", "@[to_additive]\ntheorem prehaar_mem_haarProduct (K\u2080 : PositiveCompacts G) {U : Set G} (hU : (interior U).Nonempty) :\n    prehaar (K\u2080 : Set G) U \u2208 haarProduct (K\u2080 : Set G)", "@[to_additive]\ntheorem prehaar_sup_le {K\u2080 : PositiveCompacts G} {U : Set G} (K\u2081 K\u2082 : Compacts G)\n    (hU : (interior U).Nonempty) :\n    prehaar (K\u2080 : Set G) U (K\u2081 \u2294 K\u2082) \u2264 prehaar (K\u2080 : Set G) U K\u2081 + prehaar (K\u2080 : Set G) U K\u2082", "lemma isEverywherePos_everywherePosSubset_of_measure_ne_top\n    [OpensMeasurableSpace \u03b1] [InnerRegularCompactLTTop \u03bc] (hs : MeasurableSet s) (h's : \u03bc s \u2260 \u221e) :\n    \u03bc.IsEverywherePos (\u03bc.everywherePosSubset s)", "@[to_additive\n\"The image of a finite additive Haar measure under a continuous surjective additive group\nhomomorphism is again an additive Haar measure. See also `isAddHaarMeasure_map`.\"]\ntheorem isHaarMeasure_map_of_isFiniteMeasure\n    [BorelSpace G] [TopologicalGroup G] {H : Type*} [Group H]\n    [TopologicalSpace H] [MeasurableSpace H] [BorelSpace H] [TopologicalGroup H] [IsFiniteMeasure \u03bc]\n    (f : G \u2192* H) (hf : Continuous f) (h_surj : Surjective f) :\n    IsHaarMeasure (Measure.map f \u03bc)", "theorem le_mkMetric (m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e) (\u03bc : Measure X) (\u03b5 : \u211d\u22650\u221e) (h\u2080 : 0 < \u03b5)\n    (h : \u2200 s : Set X, diam s \u2264 \u03b5 \u2192 \u03bc s \u2264 m (diam s)) : \u03bc \u2264 mkMetric m", "theorem map_congr {f g : \u03b1 \u2192 \u03b2} (h : f =\u1d50[\u03bc] g) : Measure.map f \u03bc = Measure.map g \u03bc", "@[simp] lemma map_fst_prod : Measure.map Prod.fst (\u03bc.prod \u03bd) = (\u03bd univ) \u2022 \u03bc", "theorem MeasureTheory.Measure.map_mono_of_aemeasurable {f : \u03b1 \u2192 \u03b4} (h : \u03bc \u2264 \u03bd)\n    (hf : AEMeasurable f \u03bd) : \u03bc.map f \u2264 \u03bd.map f", "theorem measurePreserving_homeomorphUnitSphereProd :\n    MeasurePreserving (homeomorphUnitSphereProd E) (\u03bc.comap (\u2191))\n      (\u03bc.toSphere.prod (volumeIoiPow (dim E - 1)))", "@[to_additive measure_preimage_isAddLeftInvariant_eq_smul_of_hasCompactSupport\n\"Two left invariant measures give the same mass to level sets of continuous compactly supported\nfunctions, up to the scalar `addHaarScalarFactor \u03bc' \u03bc`.\n\nAuxiliary lemma in the proof of the more general\n`measure_isAddInvariant_eq_smul_of_isCompact_closure`, which works for any set with\ncompact closure.\"]\nlemma measure_preimage_isMulLeftInvariant_eq_smul_of_hasCompactSupport\n    (\u03bc' \u03bc : Measure G) [IsHaarMeasure \u03bc] [IsFiniteMeasureOnCompacts \u03bc'] [IsMulLeftInvariant \u03bc']\n    {f : G \u2192 \u211d} (hf : Continuous f) (h'f : HasCompactSupport f) :\n    \u03bc' (f \u207b\u00b9' {1}) = haarScalarFactor \u03bc' \u03bc \u2022 \u03bc (f \u207b\u00b9' {1})", "theorem measure_toMeasurable_add_inter_left {s t : Set \u03b1} (hs : MeasurableSet s)\n    (ht : (\u03bc + \u03bd) t \u2260 \u221e) : \u03bc (toMeasurable (\u03bc + \u03bd) t \u2229 s) = \u03bc (t \u2229 s)", "@[simp]\ntheorem mkMetric_top : (mkMetric (fun _ => \u221e : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e) : Measure X) = \u22a4", "theorem one_le_hausdorffMeasure_zero_of_nonempty {s : Set X} (h : s.Nonempty) : 1 \u2264 \u03bcH[0] s", "theorem pi_eq_generateFrom {C : \u2200 i, Set (Set (\u03b1 i))}\n    (hC : \u2200 i, generateFrom (C i) = by apply_assumption) (h2C : \u2200 i, IsPiSystem (C i))\n    (h3C : \u2200 i, (\u03bc i).FiniteSpanningSetsIn (C i)) {\u03bc\u03bd : Measure (\u2200 i, \u03b1 i)}\n    (h\u2081 : \u2200 s : \u2200 i, Set (\u03b1 i), (\u2200 i, s i \u2208 C i) \u2192 \u03bc\u03bd (pi univ s) = \u220f i, \u03bc i (s i)) :\n    Measure.pi \u03bc = \u03bc\u03bd", "theorem quasiMeasurePreserving_fst : QuasiMeasurePreserving Prod.fst (\u03bc.prod \u03bd) \u03bc", "theorem restrict_iUnion_ae [Countable \u03b9] {s : \u03b9 \u2192 Set \u03b1} (hd : Pairwise (AEDisjoint \u03bc on s))\n    (hm : \u2200 i, NullMeasurableSet (s i) \u03bc) : \u03bc.restrict (\u22c3 i, s i) = sum fun i => \u03bc.restrict (s i)", "theorem restrict_le_self : \u03bc.restrict s \u2264 \u03bc", "theorem restrict_union_add_inter' (hs : MeasurableSet s) (t : Set \u03b1) :\n    \u03bc.restrict (s \u222a t) + \u03bc.restrict (s \u2229 t) = \u03bc.restrict s + \u03bc.restrict t", "lemma rnDeriv_add (\u03bd\u2081 \u03bd\u2082 \u03bc : Measure \u03b1) [IsFiniteMeasure \u03bd\u2081] [IsFiniteMeasure \u03bd\u2082]\n    [\u03bd\u2081.HaveLebesgueDecomposition \u03bc] [\u03bd\u2082.HaveLebesgueDecomposition \u03bc]\n    [(\u03bd\u2081 + \u03bd\u2082).HaveLebesgueDecomposition \u03bc] :\n    (\u03bd\u2081 + \u03bd\u2082).rnDeriv \u03bc =\u1d50[\u03bc] \u03bd\u2081.rnDeriv \u03bc + \u03bd\u2082.rnDeriv \u03bc", "lemma rnDeriv_add_of_mutuallySingular (\u03bd\u2081 \u03bd\u2082 \u03bc : Measure \u03b1)\n    [SigmaFinite \u03bd\u2081] [SigmaFinite \u03bd\u2082] [SigmaFinite \u03bc] (h : \u03bd\u2082 \u27c2\u2098 \u03bc) :\n    (\u03bd\u2081 + \u03bd\u2082).rnDeriv \u03bc =\u1d50[\u03bc] \u03bd\u2081.rnDeriv \u03bc", "lemma rnDeriv_add_right_of_mutuallySingular' {\u03bd' : Measure \u03b1}\n    [SigmaFinite \u03bc] [SigmaFinite \u03bd] [SigmaFinite \u03bd']\n    (h\u03bc\u03bd' : \u03bc \u27c2\u2098 \u03bd') (h\u03bd\u03bd' : \u03bd \u27c2\u2098 \u03bd') :\n    \u03bc.rnDeriv (\u03bd + \u03bd') =\u1d50[\u03bd] \u03bc.rnDeriv \u03bd", "lemma rnDeriv_withDensity_left {\u03bc \u03bd : Measure \u03b1} [SigmaFinite \u03bc] [SigmaFinite \u03bd]\n    (hf\u03bc : AEMeasurable f \u03bc) (hf\u03bd : AEMeasurable f \u03bd)\n    (hf_ne_top : \u2200\u1d50 x \u2202\u03bc, f x \u2260 \u221e) :\n    (\u03bc.withDensity f).rnDeriv \u03bd =\u1d50[\u03bd] fun x \u21a6 f x * \u03bc.rnDeriv \u03bd x", "theorem setIntegral_comp_smul (f : E \u2192 F) {R : \u211d} (s : Set E) (hR : R \u2260 0) :\n    \u222b x in s, f (R \u2022 x) \u2202\u03bc = |(R ^ finrank \u211d E)\u207b\u00b9| \u2022 \u222b x in R \u2022 s, f x \u2202\u03bc", "lemma setIntegral_condKernel_univ_left {t : Set \u03a9} (ht : MeasurableSet t)\n    (hf : IntegrableOn f (Set.univ \u00d7\u02e2 t) \u03c1) :\n    \u222b b, \u222b \u03c9 in t, f (b, \u03c9) \u2202(\u03c1.condKernel b) \u2202\u03c1.fst = \u222b x in Set.univ \u00d7\u02e2 t, f x \u2202\u03c1", "lemma setIntegral_toReal_rnDeriv' [SigmaFinite \u03bc] [HaveLebesgueDecomposition \u03bc \u03bd]\n    (h\u03bc\u03bd : \u03bc \u226a \u03bd) {s : Set \u03b1} (hs : MeasurableSet s) :\n    \u222b x in s, (\u03bc.rnDeriv \u03bd x).toReal \u2202\u03bd = (\u03bc s).toReal", "lemma setIntegral_toReal_rnDeriv_eq_withDensity [SigmaFinite \u03bc] [SFinite \u03bd] (s : Set \u03b1) :\n    \u222b x in s, (\u03bc.rnDeriv \u03bd x).toReal \u2202\u03bd = (\u03bd.withDensity (\u03bc.rnDeriv \u03bd) s).toReal", "lemma setLIntegral_compProd [SFinite \u03bc] [IsSFiniteKernel \u03ba]\n    {f : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e} (hf : Measurable f)\n    {s : Set \u03b1} (hs : MeasurableSet s) {t : Set \u03b2} (ht : MeasurableSet t) :\n    \u222b\u207b x in s \u00d7\u02e2 t, f x \u2202(\u03bc \u2297\u2098 \u03ba) = \u222b\u207b a in s, \u222b\u207b b in t, f (a, b) \u2202(\u03ba a) \u2202\u03bc", "@[simp]\ntheorem singularPart_zero (\u03bd : Measure \u03b1) : (0 : Measure \u03b1).singularPart \u03bd = 0", "theorem Measure.smul_finite (\u03bc : Measure \u03b1) [IsFiniteMeasure \u03bc] {c : \u211d\u22650\u221e} (hc : c \u2260 \u221e) :\n    IsFiniteMeasure (c \u2022 \u03bc)", "theorem snd_univ : \u03c1.snd univ = \u03c1 univ", "theorem sub_le : \u03bc - \u03bd \u2264 \u03bc", "@[simp]\ntheorem sum_apply_eq_zero [Countable \u03b9] {\u03bc : \u03b9 \u2192 Measure \u03b1} {s : Set \u03b1} :\n    sum \u03bc s = 0 \u2194 \u2200 i, \u03bc i s = 0", "theorem sum_apply_eq_zero' {\u03bc : \u03b9 \u2192 Measure \u03b1} {s : Set \u03b1} (hs : MeasurableSet s) :\n    sum \u03bc s = 0 \u2194 \u2200 i, \u03bc i s = 0", "@[simp] lemma sum_comp_equiv {\u03b9 \u03b9' : Type*} (e : \u03b9' \u2243 \u03b9) (m : \u03b9 \u2192 Measure \u03b1) :\n    sum (m \u2218 e) = sum m", "@[simp]\nlemma sum_zero : Measure.sum (fun (_ : \u03b9) \u21a6 (0 : Measure \u03b1)) = 0", "@[simp]\ntheorem toSignedMeasure_zero : (0 : Measure \u03b1).toSignedMeasure = 0", "theorem trimmed (\u03bc : Measure \u03b1) : \u03bc.toOuterMeasure.trim = \u03bc.toOuterMeasure", "theorem withDensity_rnDeriv_le (\u03bc \u03bd : Measure \u03b1) : \u03bd.withDensity (\u03bc.rnDeriv \u03bd) \u2264 \u03bc", "protected theorem comp_right_iff {g : \u03b1 \u2192 \u03b2} {e : \u03b3 \u2243\u1d50 \u03b1} (h : MeasurePreserving e \u03bcc \u03bca) :\n    MeasurePreserving (g \u2218 e) \u03bcc \u03bcb \u2194 MeasurePreserving g \u03bca \u03bcb", "@[to_additive]\ntheorem MeasurePreserving.mul_right (\u03bc : Measure G) [IsMulRightInvariant \u03bc] (g : G) {X : Type*}\n    [MeasurableSpace X] {\u03bc' : Measure X} {f : X \u2192 G} (hf : MeasurePreserving f \u03bc' \u03bc) :\n    MeasurePreserving (fun x => f x * g) \u03bc' \u03bc", "protected theorem prod [SFinite \u03bca] [SFinite \u03bcc] {f : \u03b1 \u2192 \u03b2} {g : \u03b3 \u2192 \u03b4}\n    (hf : MeasurePreserving f \u03bca \u03bcb) (hg : MeasurePreserving g \u03bcc \u03bcd) :\n    MeasurePreserving (Prod.map f g) (\u03bca.prod \u03bcc) (\u03bcb.prod \u03bcd)", "theorem skew_product [SFinite \u03bca] [SFinite \u03bcc] {f : \u03b1 \u2192 \u03b2} (hf : MeasurePreserving f \u03bca \u03bcb)\n    {g : \u03b1 \u2192 \u03b3 \u2192 \u03b4} (hgm : Measurable (uncurry g)) (hg : \u2200\u1d50 x \u2202\u03bca, map (g x) \u03bcc = \u03bcd) :\n    MeasurePreserving (fun p : \u03b1 \u00d7 \u03b3 => (f p.1, g p.1 p.2)) (\u03bca.prod \u03bcc) (\u03bcb.prod \u03bcd)", "protected theorem trans {e : \u03b1 \u2243\u1d50 \u03b2} {e' : \u03b2 \u2243\u1d50 \u03b3}\n    {\u03bca : Measure \u03b1} {\u03bcb : Measure \u03b2} {\u03bcc : Measure \u03b3}\n    (h : MeasurePreserving e \u03bca \u03bcb) (h' : MeasurePreserving e' \u03bcb \u03bcc) :\n    MeasurePreserving (e.trans e') \u03bca \u03bcc", "theorem _root_.MeasureTheory.Mem\u2112p.exists_simpleFunc_snorm_sub_lt {E : Type*}\n    [NormedAddCommGroup E] {f : \u03b2 \u2192 E} {\u03bc : Measure \u03b2} (hf : Mem\u2112p f p \u03bc) (hp_ne_top : p \u2260 \u221e)\n    {\u03b5 : \u211d\u22650\u221e} (h\u03b5 : \u03b5 \u2260 0) : \u2203 g : \u03b2 \u2192\u209b E, snorm (f - \u21d1g) p \u03bc < \u03b5 \u2227 Mem\u2112p g p \u03bc", "theorem Mem\u2112p.im (hf : Mem\u2112p f p \u03bc) : Mem\u2112p (fun x => RCLike.im (f x)) p \u03bc", "@[elab_as_elim]\ntheorem Mem\u2112p.induction [_i : Fact (1 \u2264 p)] (hp_ne_top : p \u2260 \u221e) (P : (\u03b1 \u2192 E) \u2192 Prop)\n    (h_ind : \u2200 (c : E) \u2983s\u2984, MeasurableSet s \u2192 \u03bc s < \u221e \u2192 P (s.indicator fun _ => c))\n    (h_add : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, Disjoint (support f) (support g) \u2192 Mem\u2112p f p \u03bc \u2192 Mem\u2112p g p \u03bc \u2192\n      P f \u2192 P g \u2192 P (f + g))\n    (h_closed : IsClosed { f : Lp E p \u03bc | P f })\n    (h_ae : \u2200 \u2983f g\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u03bc \u2192 P f \u2192 P g) :\n    \u2200 \u2983f : \u03b1 \u2192 E\u2984, Mem\u2112p f p \u03bc \u2192 P f", "theorem Mem\u2112p.integral_indicator_norm_ge_le (hf : Mem\u2112p f 1 \u03bc) (hmeas : StronglyMeasurable f)\n    {\u03b5 : \u211d} (h\u03b5 : 0 < \u03b5) :\n    \u2203 M : \u211d, (\u222b\u207b x, \u2016{ x | M \u2264 \u2016f x\u2016\u208a }.indicator f x\u2016\u208a \u2202\u03bc) \u2264 ENNReal.ofReal \u03b5", "theorem Mem\u2112p.of_measure_le_smul {\u03bc' : Measure \u03b1} (c : \u211d\u22650\u221e) (hc : c \u2260 \u221e) (h\u03bc'_le : \u03bc' \u2264 c \u2022 \u03bc)\n    {f : \u03b1 \u2192 E} (hf : Mem\u2112p f p \u03bc) : Mem\u2112p f p \u03bc'", "theorem Mem\u2112p.of_nnnorm_le_mul {f : \u03b1 \u2192 E} {g : \u03b1 \u2192 F} {c : \u211d\u22650} (hg : Mem\u2112p g p \u03bc)\n    (hf : AEStronglyMeasurable f \u03bc) (hfg : \u2200\u1d50 x \u2202\u03bc, \u2016f x\u2016\u208a \u2264 c * \u2016g x\u2016\u208a) : Mem\u2112p f p \u03bc", "protected lemma Mem\u2112p.piecewise [DecidablePred (\u00b7 \u2208 s)] {g}\n    (hs : MeasurableSet s) (hf : Mem\u2112p f p (\u03bc.restrict s)) (hg : Mem\u2112p g p (\u03bc.restrict s\u1d9c)) :\n    Mem\u2112p (s.piecewise f g) p \u03bc", "theorem Mem\u2112p.right_of_add_measure {f : \u03b1 \u2192 E} (h : Mem\u2112p f p (\u03bc + \u03bd)) : Mem\u2112p f p \u03bd", "theorem Mem\u2112p.snormEssSup_indicator_norm_ge_eq_zero (hf : Mem\u2112p f \u221e \u03bc)\n    (hmeas : StronglyMeasurable f) :\n    \u2203 M : \u211d, snormEssSup ({ x | M \u2264 \u2016f x\u2016\u208a }.indicator f) \u03bc = 0", "theorem Mem\u2112p.snorm_indicator_le (hp_one : 1 \u2264 p) (hp_top : p \u2260 \u221e) (hf : Mem\u2112p f p \u03bc) {\u03b5 : \u211d}\n    (h\u03b5 : 0 < \u03b5) :\n    \u2203 (\u03b4 : \u211d) (h\u03b4 : 0 < \u03b4), \u2200 s, MeasurableSet s \u2192 \u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192\n      snorm (s.indicator f) p \u03bc \u2264 ENNReal.ofReal \u03b5", "theorem Mem\u2112p.snorm_lt_top {f : \u03b1 \u2192 E} (hfp : Mem\u2112p f p \u03bc) : snorm f p \u03bc < \u221e", "theorem _root_.MeasureTheory.Mem\u2112p.variance_eq_of_integral_eq_zero (hX : Mem\u2112p X 2 \u03bc)\n    (hXint : \u03bc[X] = 0) : variance X \u03bc = \u03bc[X ^ (2 : Nat)]", "lemma MeasureTheory.NullMeasurable.aemeasurable {f : \u03b1 \u2192 \u03b2}\n    [hc : MeasurableSpace.CountablyGenerated \u03b2] (h : NullMeasurable f \u03bc) : AEMeasurable f \u03bc", "lemma NullMeasurable.measure_preimage_eq_measure_restrict_preimage_of_ae_compl_eq_const\n    {\u03b2 : Type*} [MeasurableSpace \u03b2] {b : \u03b2} {f : \u03b1 \u2192 \u03b2} {s : Set \u03b1}\n    (f_mble : NullMeasurable f (\u03bc.restrict s)) (hs : f =\u1d50[Measure.restrict \u03bc s\u1d9c] (fun _ \u21a6 b))\n    {t : Set \u03b2} (t_mble : MeasurableSet t) (ht : b \u2209 t) :\n    \u03bc (f \u207b\u00b9' t) = \u03bc.restrict s (f \u207b\u00b9' t)", "@[simp]\ntheorem compl_iff : NullMeasurableSet s\u1d9c \u03bc \u2194 NullMeasurableSet s \u03bc", "@[to_additive MeasureTheory.NullMeasurableSet.addFundamentalFrontier]\nprotected theorem NullMeasurableSet.fundamentalFrontier (hs : NullMeasurableSet s \u03bc) :\n    NullMeasurableSet (fundamentalFrontier G s) \u03bc", "@[simp]\nprotected theorem inter (hs : NullMeasurableSet s \u03bc) (ht : NullMeasurableSet t \u03bc) :\n    NullMeasurableSet (s \u2229 t) \u03bc", "lemma _root_.MeasureTheory.NullMeasurableSet.left_of_prod {s : Set \u03b1} {t : Set \u03b2}\n    (h : NullMeasurableSet (s \u00d7\u02e2 t) (\u03bc.prod \u03bd)) (ht : \u03bd t \u2260 0) : NullMeasurableSet s \u03bc", "lemma _root_.MeasureTheory.NullMeasurableSet.right_of_prod {s : Set \u03b1} {t : Set \u03b2}\n    (h : NullMeasurableSet (s \u00d7\u02e2 t) (\u03bc.prod \u03bd)) (hs : \u03bc s \u2260 0) : NullMeasurableSet t \u03bd", "protected theorem union_null (hs : NullMeasurableSet s \u03bc) (ht : \u03bc t = 0) :\n    NullMeasurableSet (s \u222a t) \u03bc", "theorem biInf_apply {\u03b9} {I : Set \u03b9} (hI : I.Nonempty) (m : \u03b9 \u2192 OuterMeasure \u03b1) (s : Set \u03b1) :\n    (\u2a05 i \u2208 I, m i) s = \u2a05 (t : \u2115 \u2192 Set \u03b1) (_ : s \u2286 iUnion t), \u2211' n, \u2a05 i \u2208 I, m i (t n)", "theorem boundedBy_eq_ofFunction (m_empty : m \u2205 = 0) (s : Set \u03b1) :\n    boundedBy m s = OuterMeasure.ofFunction m m_empty s", "@[simp]\ntheorem boundedBy_eq_self (m : OuterMeasure \u03b1) : boundedBy m = m", "@[mono]\ntheorem comap_mono {\u03b2} (f : \u03b1 \u2192 \u03b2) : Monotone (comap f)", "theorem exists_measurable_superset_eq_trim (m : OuterMeasure \u03b1) (s : Set \u03b1) :\n    \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 m t = m.trim s", "@[deprecated measure_iUnion_le (since := \"2024-05-14\")]\nprotected theorem iUnion (m : OuterMeasure \u03b1) {\u03b2} [Countable \u03b2] (s : \u03b2 \u2192 Set \u03b1) :\n    m (\u22c3 i, s i) \u2264 \u2211' i, m (s i)", "theorem iUnion_of_tendsto_zero {\u03b9} (m : OuterMeasure \u03b1) {s : \u03b9 \u2192 Set \u03b1} (l : Filter \u03b9) [NeBot l]\n    (h0 : Tendsto (fun k => m ((\u22c3 n, s n) \\ s k)) l (\ud835\udcdd 0)) : m (\u22c3 n, s n) = \u2a06 n, m (s n)", "theorem isometry_comap_mkMetric (m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e) {f : X \u2192 Y} (hf : Isometry f)\n    (H : Monotone m \u2228 Surjective f) : comap f (mkMetric m) = mkMetric m", "@[simp]\ntheorem map_apply {\u03b2} (f : \u03b1 \u2192 \u03b2) (m : OuterMeasure \u03b1) (s : Set \u03b2) : map f m s = m (f \u207b\u00b9' s)", "theorem measure_inter_union (h : s\u2081 \u2229 s\u2082 \u2286 \u2205) (h\u2081 : IsCaratheodory m s\u2081) {t : Set \u03b1} :\n    m (t \u2229 (s\u2081 \u222a s\u2082)) = m (t \u2229 s\u2081) + m (t \u2229 s\u2082)", "theorem mkMetric_nnreal_smul (m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e) {c : \u211d\u22650} (hc : c \u2260 0) :\n    (mkMetric (c \u2022 m) : OuterMeasure X) = c \u2022 mkMetric m", "theorem sInfGen_def (m : Set (OuterMeasure \u03b1)) (t : Set \u03b1) :\n    sInfGen m t = \u2a05 (\u03bc : OuterMeasure \u03b1) (_ : \u03bc \u2208 m), \u03bc t", "@[simp]\ntheorem sSup_apply (ms : Set (OuterMeasure \u03b1)) (s : Set \u03b1) :\n    (sSup ms) s = \u2a06 m \u2208 ms, (m : OuterMeasure \u03b1) s", "theorem smul_dirac_apply (a : \u211d\u22650\u221e) (b : \u03b1) (s : Set \u03b1) :\n    (a \u2022 dirac b) s = indicator s (fun _ => a) b", "@[simp, norm_cast] lemma coe_mk (\u03bc : Measure \u03a9) (h\u03bc) : toMeasure \u27e8\u03bc, h\u03bc\u27e9 = \u03bc := rfl", "theorem eq_rnDeriv (t : SignedMeasure \u03b1) (f : \u03b1 \u2192 \u211d) (hfi : Integrable f \u03bc)\n    (ht\u03bc : t \u27c2\u1d65 \u03bc.toENNRealVectorMeasure) (hadd : s = t + \u03bc.withDensity\u1d65 f) :\n    f =\u1d50[\u03bc] s.rnDeriv \u03bc", "theorem _root_.MeasureTheory.SignedMeasure.im_toComplexMeasure (s t : SignedMeasure \u03b1) :\n    ComplexMeasure.im (SignedMeasure.toComplexMeasure s t) = t", "theorem null_of_totalVariation_zero (s : SignedMeasure \u03b1) {i : Set \u03b1}\n    (hs : s.totalVariation i = 0) : s i = 0", "theorem _root_.MeasureTheory.SignedMeasure.toComplexMeasure_apply\n    {s t : SignedMeasure \u03b1} {i : Set \u03b1} : s.toComplexMeasure t i = \u27e8s i, t i\u27e9", "protected theorem map\u2082 [Zero \u03b4] (hf : f.FinMeasSupp \u03bc) {g : \u03b1 \u2192\u209b \u03b3} (hg : g.FinMeasSupp \u03bc)\n    {op : \u03b2 \u2192 \u03b3 \u2192 \u03b4} (H : op 0 0 = 0) : ((pair f g).map (Function.uncurry op)).FinMeasSupp \u03bc", "theorem meas_preimage_singleton_ne_zero (h : f.FinMeasSupp \u03bc) {y : \u03b2} (hy : y \u2260 0) :\n    \u03bc (f \u207b\u00b9' {y}) < \u221e", "protected theorem mul {\u03b2} [MonoidWithZero \u03b2] {f g : \u03b1 \u2192\u209b \u03b2} (hf : f.FinMeasSupp \u03bc)\n    (hg : g.FinMeasSupp \u03bc) : (f * g).FinMeasSupp \u03bc", "@[simp]\ntheorem coe_restrict (f : \u03b1 \u2192\u209b \u03b2) {s : Set \u03b1} (hs : MeasurableSet s) :\n    \u21d1(restrict f s) = indicator s f", "theorem iSup_approx_apply [TopologicalSpace \u03b2] [CompleteLattice \u03b2] [OrderClosedTopology \u03b2] [Zero \u03b2]\n    [MeasurableSpace \u03b2] [OpensMeasurableSpace \u03b2] (i : \u2115 \u2192 \u03b2) (f : \u03b1 \u2192 \u03b2) (a : \u03b1) (hf : Measurable f)\n    (h_zero : (0 : \u03b2) = \u22a5) : \u2a06 n, (approx i f n : \u03b1 \u2192\u209b \u03b2) a = \u2a06 (k) (_ : i k \u2264 f a), i k", "theorem integrable_approxOn [BorelSpace E] {f : \u03b2 \u2192 E} {\u03bc : Measure \u03b2} (fmeas : Measurable f)\n    (hf : Integrable f \u03bc) {s : Set E} {y\u2080 : E} (h\u2080 : y\u2080 \u2208 s) [SeparableSpace s]\n    (hi\u2080 : Integrable (fun _ => y\u2080) \u03bc) (n : \u2115) : Integrable (approxOn f fmeas s y\u2080 h\u2080 n) \u03bc", "theorem integral_eq_lintegral' {f : \u03b1 \u2192\u209b E} {g : E \u2192 \u211d\u22650\u221e} (hf : Integrable f \u03bc) (hg0 : g 0 = 0)\n    (ht : \u2200 b, g b \u2260 \u221e) :\n    (f.map (ENNReal.toReal \u2218 g)).integral \u03bc = ENNReal.toReal (\u222b\u207b a, g (f a) \u2202\u03bc)", "theorem map_coe_nnreal_restrict (f : \u03b1 \u2192\u209b \u211d\u22650) (s : Set \u03b1) :\n    (f.restrict s).map ((\u2191) : \u211d\u22650 \u2192 \u211d) = (f.map (\u2191)).restrict s", "theorem measure_lt_top_of_mem\u2112p_indicator (hp_pos : p \u2260 0) (hp_ne_top : p \u2260 \u221e) {c : E} (hc : c \u2260 0)\n    {s : Set \u03b1} (hs : MeasurableSet s) (hcs : Mem\u2112p ((const \u03b1 c).piecewise s hs (const \u03b1 0)) p \u03bc) :\n    \u03bc s < \u22a4", "theorem mem_range_of_measure_ne_zero {f : \u03b1 \u2192\u209b \u03b2} {x : \u03b2} {\u03bc : Measure \u03b1} (H : \u03bc (f \u207b\u00b9' {x}) \u2260 0) :\n    x \u2208 f.range", "theorem mem\u2112p_approxOn_range [BorelSpace E] {f : \u03b2 \u2192 E} {\u03bc : Measure \u03b2} (fmeas : Measurable f)\n    [SeparableSpace (range f \u222a {0} : Set E)] (hf : Mem\u2112p f p \u03bc) (n : \u2115) :\n    Mem\u2112p (approxOn f fmeas (range f \u222a {0}) 0 (by simp) n) p \u03bc", "theorem mem\u2112p_iff {f : \u03b1 \u2192\u209b E} (hp_pos : p \u2260 0) (hp_ne_top : p \u2260 \u221e) :\n    Mem\u2112p f p \u03bc \u2194 \u2200 y, y \u2260 0 \u2192 \u03bc (f \u207b\u00b9' {y}) < \u221e", "theorem mem\u2112p_iff_integrable {f : \u03b1 \u2192\u209b E} (hp_pos : p \u2260 0) (hp_ne_top : p \u2260 \u221e) :\n    Mem\u2112p f p \u03bc \u2194 Integrable f \u03bc", "theorem norm_setToSimpleFunc_le_sum_mul_norm (T : Set \u03b1 \u2192 F \u2192L[\u211d] F') {C : \u211d}\n    (hT_norm : \u2200 s, MeasurableSet s \u2192 \u2016T s\u2016 \u2264 C * (\u03bc s).toReal) (f : \u03b1 \u2192\u209b F) :\n    \u2016f.setToSimpleFunc T\u2016 \u2264 C * \u2211 x \u2208 f.range, (\u03bc (f \u207b\u00b9' {x})).toReal * \u2016x\u2016", "@[simp]\ntheorem piecewise_univ (f g : \u03b1 \u2192\u209b \u03b2) : piecewise univ MeasurableSet.univ f g = f", "theorem pow_apply [Monoid \u03b2] (n : \u2115) (f : \u03b1 \u2192\u209b \u03b2) (a : \u03b1) : (f ^ n) a = f a ^ n", "theorem restrict_const_lintegral (c : \u211d\u22650\u221e) {s : Set \u03b1} (hs : MeasurableSet s) :\n    ((const \u03b1 c).restrict s).lintegral \u03bc = c * \u03bc s", "theorem setToSimpleFunc_add_left' (T T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F)\n    (h_add : \u2200 s, MeasurableSet s \u2192 \u03bc s < \u221e \u2192 T'' s = T s + T' s) {f : \u03b1 \u2192\u209b E}\n    (hf : Integrable f \u03bc) : setToSimpleFunc T'' f = setToSimpleFunc T f + setToSimpleFunc T' f", "theorem simpleFunc_bot {\u03b1} (f : @SimpleFunc \u03b1 \u22a5 \u03b2) [Nonempty \u03b2] : \u2203 c, \u2200 x, f x = c", "theorem support_eq [MeasurableSpace \u03b1] [Zero \u03b2] (f : \u03b1 \u2192\u209b \u03b2) :\n    support f = \u22c3 y \u2208 f.range.filter fun y => y \u2260 0, f \u207b\u00b9' {y}", "theorem tendsto_approxOn_range_Lp_snorm [BorelSpace E] {f : \u03b2 \u2192 E} (hp_ne_top : p \u2260 \u221e)\n    {\u03bc : Measure \u03b2} (fmeas : Measurable f) [SeparableSpace (range f \u222a {0} : Set E)]\n    (hf : snorm f p \u03bc < \u221e) :\n    Tendsto (fun n => snorm (\u21d1(approxOn f fmeas (range f \u222a {0}) 0 (by simp) n) - f) p \u03bc)\n      atTop (\ud835\udcdd 0)", "theorem comp_measurable [TopologicalSpace \u03b2] {_ : MeasurableSpace \u03b1} {_ : MeasurableSpace \u03b3}\n    {f : \u03b1 \u2192 \u03b2} {g : \u03b3 \u2192 \u03b1} (hf : StronglyMeasurable f) (hg : Measurable g) :\n    StronglyMeasurable (f \u2218 g)", "theorem finStronglyMeasurable_of_set_sigmaFinite [TopologicalSpace \u03b2] [Zero \u03b2]\n    {m : MeasurableSpace \u03b1} {\u03bc : Measure \u03b1} (hf_meas : StronglyMeasurable f) {t : Set \u03b1}\n    (ht : MeasurableSet t) (hft_zero : \u2200 x \u2208 t\u1d9c, f x = 0) (ht\u03bc : SigmaFinite (\u03bc.restrict t)) :\n    FinStronglyMeasurable f \u03bc", "theorem StronglyMeasurable.integral_kernel_prod_left' \u2983f : \u03b2 \u00d7 \u03b1 \u2192 E\u2984 (hf : StronglyMeasurable f) :\n    StronglyMeasurable fun y => \u222b x, f (x, y) \u2202\u03ba y", "@[to_additive (attr := measurability)]\nprotected theorem inv [Inv \u03b2] [ContinuousInv \u03b2] (hf : StronglyMeasurable f) :\n    StronglyMeasurable f\u207b\u00b9", "theorem measurableSet_le {m : MeasurableSpace \u03b1} [TopologicalSpace \u03b2] [Preorder \u03b2]\n    [OrderClosedTopology \u03b2] [PseudoMetrizableSpace \u03b2] {f g : \u03b1 \u2192 \u03b2} (hf : StronglyMeasurable f)\n    (hg : StronglyMeasurable g) : MeasurableSet { a | f a \u2264 g a }", "@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]\nprotected theorem mul [Mul \u03b2] [ContinuousMul \u03b2] (hf : StronglyMeasurable f)\n    (hg : StronglyMeasurable g) : StronglyMeasurable (f * g)", "theorem of_uncurry_right [TopologicalSpace \u03b2] {_ : MeasurableSpace \u03b1} {_ : MeasurableSpace \u03b3}\n    {f : \u03b1 \u2192 \u03b3 \u2192 \u03b2} (hf : StronglyMeasurable (uncurry f)) {y : \u03b3} :\n    StronglyMeasurable fun x => f x y", "theorem Submartingale.condexp_sub_nonneg {f : \u03b9 \u2192 \u03a9 \u2192 \u211d} (hf : Submartingale f \u2131 \u03bc) {i j : \u03b9}\n    (hij : i \u2264 j) : 0 \u2264\u1d50[\u03bc] \u03bc[f j - f i|\u2131 i]", "theorem Submartingale.expected_stoppedValue_mono [SigmaFiniteFiltration \u03bc \ud835\udca2]\n    (hf : Submartingale f \ud835\udca2 \u03bc) (h\u03c4 : IsStoppingTime \ud835\udca2 \u03c4) (h\u03c0 : IsStoppingTime \ud835\udca2 \u03c0) (hle : \u03c4 \u2264 \u03c0)\n    {N : \u2115} (hbdd : \u2200 \u03c9, \u03c0 \u03c9 \u2264 N) : \u03bc[stoppedValue f \u03c4] \u2264 \u03bc[stoppedValue f \u03c0]", "theorem condexp_ae_le [LE E] (hf : Supermartingale f \u2131 \u03bc) {i j : \u03b9} (hij : i \u2264 j) :\n    \u03bc[f j|\u2131 i] \u2264\u1d50[\u03bc] f i", "theorem smul_nonneg {f : \u03b9 \u2192 \u03a9 \u2192 F} {c : \u211d} (hc : 0 \u2264 c) (hf : Supermartingale f \u2131 \u03bc) :\n    Supermartingale (c \u2022 f) \u2131 \u03bc", "theorem TendstoInMeasure.exists_seq_tendstoInMeasure_atTop {u : Filter \u03b9} [NeBot u]\n    [IsCountablyGenerated u] {f : \u03b9 \u2192 \u03b1 \u2192 E} {g : \u03b1 \u2192 E} (hfg : TendstoInMeasure \u03bc f u g) :\n    \u2203 ns : \u2115 \u2192 \u03b9, TendstoInMeasure \u03bc (fun n => f (ns n)) atTop g", "protected theorem add (hf : UnifIntegrable f p \u03bc) (hg : UnifIntegrable g p \u03bc) (hp : 1 \u2264 p)\n    (hf_meas : \u2200 i, AEStronglyMeasurable (f i) \u03bc) (hg_meas : \u2200 i, AEStronglyMeasurable (g i) \u03bc) :\n    UnifIntegrable (f + g) p \u03bc", "theorem neg_left {M : Type*} [AddCommGroup M] [TopologicalSpace M] [TopologicalAddGroup M]\n    {v : VectorMeasure \u03b1 M} {w : VectorMeasure \u03b1 N} (h : v \u226a\u1d65 w) : -v \u226a\u1d65 w", "@[ext]\ntheorem ext {s t : VectorMeasure \u03b1 M} (h : \u2200 i : Set \u03b1, MeasurableSet i \u2192 s i = t i) : s = t", "theorem ext_iff (v w : VectorMeasure \u03b1 M) : v = w \u2194 \u2200 i : Set \u03b1, MeasurableSet i \u2192 v i = w i", "theorem adapted_const (f : Filtration \u03b9 m) (x : \u03b2) : Adapted f fun _ _ => x", "theorem addHaar_image_eq_zero_of_det_fderivWithin_eq_zero\n    (hf' : \u2200 x \u2208 s, HasFDerivWithinAt f (f' x) s x) (h'f' : \u2200 x \u2208 s, (f' x).det = 0) :\n    \u03bc (f '' s) = 0", "theorem ae_all_iff {\u03b9 : Sort*} [Countable \u03b9] {p : \u03b1 \u2192 \u03b9 \u2192 Prop} :\n    (\u2200\u1d50 a \u2202\u03bc, \u2200 i, p a i) \u2194 \u2200 i, \u2200\u1d50 a \u2202\u03bc, p a i", "theorem ae_eq_restrict_biUnion_iff (s : \u03b9 \u2192 Set \u03b1) {t : Set \u03b9} (ht : t.Countable) (f g : \u03b1 \u2192 \u03b4) :\n    f =\u1d50[\u03bc.restrict (\u22c3 i \u2208 t, s i)] g \u2194 \u2200 i \u2208 t, f =\u1d50[\u03bc.restrict (s i)] g", "@[mono]\ntheorem ae_mono (h : \u03bc \u2264 \u03bd) : ae \u03bc \u2264 ae \u03bd", "theorem ae_of_ae_map {f : \u03b1 \u2192 \u03b2} (hf : AEMeasurable f \u03bc) {p : \u03b2 \u2192 Prop} (h : \u2200\u1d50 y \u2202\u03bc.map f, p y) :\n    \u2200\u1d50 x \u2202\u03bc, p (f x)", "theorem ae_of_all {p : \u03b1 \u2192 Prop} (\u03bc : F) : (\u2200 a, p a) \u2192 \u2200\u1d50 a \u2202\u03bc, p a", "theorem ae_restrict_iff' {p : \u03b1 \u2192 Prop} (hs : MeasurableSet s) :\n    (\u2200\u1d50 x \u2202\u03bc.restrict s, p x) \u2194 \u2200\u1d50 x \u2202\u03bc, x \u2208 s \u2192 p x", "theorem ae_restrict_mem\u2080 (hs : NullMeasurableSet s \u03bc) : \u2200\u1d50 x \u2202\u03bc.restrict s, x \u2208 s", "theorem ae_smul_measure {p : \u03b1 \u2192 Prop} [Monoid R] [DistribMulAction R \u211d\u22650\u221e]\n    [IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e] (h : \u2200\u1d50 x \u2202\u03bc, p x) (c : R) : \u2200\u1d50 x \u2202c \u2022 \u03bc, p x", "@[simp]\ntheorem ae_zero {_m0 : MeasurableSpace \u03b1} : ae (0 : Measure \u03b1) = \u22a5", "theorem aecover_Icc_of_Ico (ha : Tendsto a l (\ud835\udcdd A)) (hb : Tendsto b l (\ud835\udcdd B)) :\n    AECover (\u03bc.restrict <| Icc A B) l fun i => Ico (a i) (b i)", "theorem aecover_Ioo_of_Ioc : AECover (\u03bc.restrict <| Ioo A B) l fun i => Ioc (a i) (b i)", "theorem aedisjoint_compl_right : AEDisjoint \u03bc s s\u1d9c", "theorem aemeasurable_withDensity_ennreal_iff' {f : \u03b1 \u2192 \u211d\u22650}\n    (hf : AEMeasurable f \u03bc) {g : \u03b1 \u2192 \u211d\u22650\u221e} :\n    AEMeasurable g (\u03bc.withDensity fun x => (f x : \u211d\u22650\u221e)) \u2194\n      AEMeasurable (fun x => (f x : \u211d\u22650\u221e) * g x) \u03bc", "theorem all_ae_tendsto_ofReal_norm (h : \u2200\u1d50 a \u2202\u03bc, Tendsto (fun n => F n a) atTop <| \ud835\udcdd <| f a) :\n    \u2200\u1d50 a \u2202\u03bc, Tendsto (fun n => ENNReal.ofReal \u2016F n a\u2016) atTop <| \ud835\udcdd <| ENNReal.ofReal \u2016f a\u2016", "theorem compl_mem_measurableCylinders (hs : s \u2208 measurableCylinders \u03b1) :\n    s\u1d9c \u2208 measurableCylinders \u03b1", "theorem condexpInd_nonneg {E} [NormedLatticeAddCommGroup E] [NormedSpace \u211d E] [OrderedSMul \u211d E]\n    (hs : MeasurableSet s) (h\u03bcs : \u03bc s \u2260 \u221e) (x : E) (hx : 0 \u2264 x) : 0 \u2264 condexpInd E hm \u03bc s x", "theorem condexpL1CLM_lpMeas (f : lpMeas F' \u211d m 1 \u03bc) :\n    condexpL1CLM F' hm \u03bc (f : \u03b1 \u2192\u2081[\u03bc] F') = \u2191f", "theorem condexpL1_add (hf : Integrable f \u03bc) (hg : Integrable g \u03bc) :\n    condexpL1 hm \u03bc (f + g) = condexpL1 hm \u03bc f + condexpL1 hm \u03bc g", "theorem condexpL1_congr_ae (hm : m \u2264 m0) [SigmaFinite (\u03bc.trim hm)] (h : f =\u1d50[\u03bc] g) :\n    condexpL1 hm \u03bc f = condexpL1 hm \u03bc g", "theorem condexp_ae_eq_condexpL1 (hm : m \u2264 m0) [h\u03bcm : SigmaFinite (\u03bc.trim hm)] (f : \u03b1 \u2192 F') :\n    \u03bc[f|m] =\u1d50[\u03bc] condexpL1 hm \u03bc f", "theorem condexp_ae_eq_restrict_of_measurableSpace_eq_on {m m\u2082 m0 : MeasurableSpace \u03b1}\n    {\u03bc : Measure \u03b1} (hm : m \u2264 m0) (hm\u2082 : m\u2082 \u2264 m0) [SigmaFinite (\u03bc.trim hm)]\n    [SigmaFinite (\u03bc.trim hm\u2082)] (hs_m : MeasurableSet[m] s)\n    (hs : \u2200 t, MeasurableSet[m] (s \u2229 t) \u2194 MeasurableSet[m\u2082] (s \u2229 t)) :\n    \u03bc[f|m] =\u1d50[\u03bc.restrict s] \u03bc[f|m\u2082]", "theorem condexp_condexp_of_le {m\u2081 m\u2082 m0 : MeasurableSpace \u03b1} {\u03bc : Measure \u03b1} (hm\u2081\u2082 : m\u2081 \u2264 m\u2082)\n    (hm\u2082 : m\u2082 \u2264 m0) [SigmaFinite (\u03bc.trim hm\u2082)] : \u03bc[\u03bc[f|m\u2082]|m\u2081] =\u1d50[\u03bc] \u03bc[f|m\u2081]", "theorem condexp_of_not_le (hm_not : \u00acm \u2264 m0) : \u03bc[f|m] = 0", "theorem condexp_stopping_time_ae_eq_restrict_eq [FirstCountableTopology \u03b9]\n    [SigmaFiniteFiltration \u03bc \u2131] (h\u03c4 : IsStoppingTime \u2131 \u03c4)\n    [SigmaFinite (\u03bc.trim h\u03c4.measurableSpace_le)] (i : \u03b9) :\n    \u03bc[f|h\u03c4.measurableSpace] =\u1d50[\u03bc.restrict {x | \u03c4 x = i}] \u03bc[f|\u2131 i]", "theorem condexp_stronglyMeasurable_mul\u2080 {f g : \u03b1 \u2192 \u211d} (hf : AEStronglyMeasurable' m f \u03bc)\n    (hfg : Integrable (f * g) \u03bc) (hg : Integrable g \u03bc) : \u03bc[f * g|m] =\u1d50[\u03bc] f * \u03bc[g|m]", "theorem continuousAt_of_dominated {F : X \u2192 \u03b1 \u2192 G} {x\u2080 : X} {bound : \u03b1 \u2192 \u211d}\n    (hF_meas : \u2200\u1da0 x in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc)\n    (h_bound : \u2200\u1da0 x in \ud835\udcdd x\u2080, \u2200\u1d50 a \u2202\u03bc, \u2016F x a\u2016 \u2264 bound a) (bound_integrable : Integrable bound \u03bc)\n    (h_cont : \u2200\u1d50 a \u2202\u03bc, ContinuousAt (fun x => F x a) x\u2080) :\n    ContinuousAt (fun x => \u222b a, F x a \u2202\u03bc) x\u2080", "theorem convolution_smul [SMulCommClass \u211d \ud835\udd5c F] {y : \ud835\udd5c} : f \u22c6[L, \u03bc] y \u2022 g = y \u2022 (f \u22c6[L, \u03bc] g)", "theorem countable_meas_le_ne_meas_lt (g : \u03b1 \u2192 R) :\n    {t : R | \u03bc {a : \u03b1 | t \u2264 g a} \u2260 \u03bc {a : \u03b1 | t < g a}}.Countable", "theorem dist_convolution_le {f : G \u2192 \u211d} {x\u2080 : G} {R \u03b5 : \u211d} {z\u2080 : E'} (h\u03b5 : 0 \u2264 \u03b5)\n    (hf : support f \u2286 ball (0 : G) R) (hnf : \u2200 x, 0 \u2264 f x) (hintf : \u222b x, f x \u2202\u03bc = 1)\n    (hmg : AEStronglyMeasurable g \u03bc) (hg : \u2200 x \u2208 ball x\u2080 R, dist (g x) z\u2080 \u2264 \u03b5) :\n    dist ((f \u22c6[lsmul \u211d \u211d, \u03bc] g : G \u2192 E') x\u2080) z\u2080 \u2264 \u03b5", "@[to_additive]\ntheorem eventually_mul_left_iff (\u03bc : Measure G) [IsMulLeftInvariant \u03bc] (t : G) {p : G \u2192 Prop} :\n    (\u2200\u1d50 x \u2202\u03bc, p (t * x)) \u2194 \u2200\u1d50 x \u2202\u03bc, p x", "theorem exists_nat_measurableEquiv_range_coe_fin_of_finite [Finite \u03b1] :\n    \u2203 n : \u2115, Nonempty (\u03b1 \u2243\u1d50 range ((\u2191) : Fin n \u2192 \u211d))", "theorem exists_null_frontiers_thickening (\u03bc : Measure \u03a9) [SigmaFinite \u03bc] (s : Set \u03a9) :\n    \u2203 rs : \u2115 \u2192 \u211d,\n      Tendsto rs atTop (\ud835\udcdd 0) \u2227 \u2200 n, 0 < rs n \u2227 \u03bc (frontier (Metric.thickening (rs n) s)) = 0", "theorem finStronglyMeasurable_iff_measurable {_m0 : MeasurableSpace \u03b1} (\u03bc : Measure \u03b1)\n    [SigmaFinite \u03bc] : FinStronglyMeasurable f \u03bc \u2194 Measurable f", "theorem finStronglyMeasurable_zero {\u03b1 \u03b2} {m : MeasurableSpace \u03b1} {\u03bc : Measure \u03b1} [Zero \u03b2]\n    [TopologicalSpace \u03b2] : FinStronglyMeasurable (0 : \u03b1 \u2192 \u03b2) \u03bc", "theorem frequently_ae_iff {p : \u03b1 \u2192 Prop} : (\u2203\u1d50 a \u2202\u03bc, p a) \u2194 \u03bc { a | p a } \u2260 0", "@[to_additive (attr := simp) MeasureTheory.addFundamentalInterior_vadd]\ntheorem fundamentalInterior_smul [Group H] [MulAction H \u03b1] [SMulCommClass H G \u03b1] (g : H) :\n    fundamentalInterior G (g \u2022 s) = g \u2022 fundamentalInterior G s", "theorem hahn_decomposition [IsFiniteMeasure \u03bc] [IsFiniteMeasure \u03bd] :\n    \u2203 s,\n      MeasurableSet s \u2227\n        (\u2200 t, MeasurableSet t \u2192 t \u2286 s \u2192 \u03bd t \u2264 \u03bc t) \u2227 \u2200 t, MeasurableSet t \u2192 t \u2286 s\u1d9c \u2192 \u03bc t \u2264 \u03bd t", "theorem hasFiniteIntegral_iff_norm (f : \u03b1 \u2192 \u03b2) :\n    HasFiniteIntegral f \u03bc \u2194 (\u222b\u207b a, ENNReal.ofReal \u2016f a\u2016 \u2202\u03bc) < \u221e", "@[simp]\ntheorem hasFiniteIntegral_zero_measure {m : MeasurableSpace \u03b1} (f : \u03b1 \u2192 \u03b2) :\n    HasFiniteIntegral f (0 : Measure \u03b1)", "theorem hausdorffMeasure_homothety_image {d : \u211d} (hd : 0 \u2264 d) (x : P) {c : \ud835\udd5c} (hc : c \u2260 0)\n    (s : Set P) : \u03bcH[d] (AffineMap.homothety x c '' s) = \u2016c\u2016\u208a ^ d \u2022 \u03bcH[d] s", "@[simp]\ntheorem hausdorffMeasure_pi_real {\u03b9 : Type*} [Fintype \u03b9] :\n    (\u03bcH[Fintype.card \u03b9] : Measure (\u03b9 \u2192 \u211d)) = volume", "@[simp]\ntheorem hausdorffMeasure_prod_real : (\u03bcH[2] : Measure (\u211d \u00d7 \u211d)) = volume", "theorem hitting_eq_hitting_of_exists {m\u2081 m\u2082 : \u03b9} (h : m\u2081 \u2264 m\u2082)\n    (h' : \u2203 j \u2208 Set.Icc n m\u2081, u j \u03c9 \u2208 s) : hitting u s n m\u2081 \u03c9 = hitting u s n m\u2082 \u03c9", "theorem hitting_le {m : \u03b9} (\u03c9 : \u03a9) : hitting u s n m \u03c9 \u2264 m", "theorem hitting_le_iff_of_lt [IsWellOrder \u03b9 (\u00b7 < \u00b7)] {m : \u03b9} (i : \u03b9) (hi : i < m) :\n    hitting u s n m \u03c9 \u2264 i \u2194 \u2203 j \u2208 Set.Icc n i, u j \u03c9 \u2208 s", "theorem hitting_mem_set_of_hitting_lt [IsWellOrder \u03b9 (\u00b7 < \u00b7)] {m : \u03b9} (hl : hitting u s n m \u03c9 < m) :\n    u (hitting u s n m \u03c9) \u03c9 \u2208 s", "theorem inducedOuterMeasure_eq_iInf (s : Set \u03b1) :\n    inducedOuterMeasure m P0 m0 s = \u2a05 (t : Set \u03b1) (ht : P t) (_ : s \u2286 t), m t ht", "theorem inducedOuterMeasure_preimage (f : \u03b1 \u2243 \u03b1) (Pm : \u2200 s : Set \u03b1, P (f \u207b\u00b9' s) \u2194 P s)\n    (mm : \u2200 (s : Set \u03b1) (hs : P s), m (f \u207b\u00b9' s) ((Pm _).mpr hs) = m s hs) {A : Set \u03b1} :\n    inducedOuterMeasure m P0 m0 (f \u207b\u00b9' A) = inducedOuterMeasure m P0 m0 A", "lemma injective_dirac [SeparatesPoints \u03b1] :\n    Function.Injective (fun (x : \u03b1) \u21a6 dirac x) := fun x y x_ne_y \u21a6 by rwa [\u2190 dirac_eq_dirac_iff]", "theorem insert_ae_eq_self (a : \u03b1) (s : Set \u03b1) : (insert a s : Set \u03b1) =\u1d50[\u03bc] s", "theorem integrableOn_Ioi_of_intervalIntegral_norm_bounded (I a : \u211d)\n    (hfi : \u2200 i, IntegrableOn f (Ioc a (b i)) \u03bc) (hb : Tendsto b l atTop)\n    (h : \u2200\u1da0 i in l, (\u222b x in a..b i, \u2016f x\u2016 \u2202\u03bc) \u2264 I) : IntegrableOn f (Ioi a) \u03bc", "@[simp]\ntheorem integrableOn_finite_iUnion [Finite \u03b2] {t : \u03b2 \u2192 Set \u03b1} :\n    IntegrableOn f (\u22c3 i, t i) \u03bc \u2194 \u2200 i, IntegrableOn f (t i) \u03bc", "theorem integrableOn_iUnion_of_summable_integral_norm {f : X \u2192 E} {s : \u03b9 \u2192 Set X}\n    (hs : \u2200 i : \u03b9, MeasurableSet (s i)) (hi : \u2200 i : \u03b9, IntegrableOn f (s i) \u03bc)\n    (h : Summable fun i : \u03b9 => \u222b x : X in s i, \u2016f x\u2016 \u2202\u03bc) : IntegrableOn f (iUnion s) \u03bc", "theorem integrableOn_image_iff_integrableOn_abs_det_fderiv_smul (hs : MeasurableSet s)\n    (hf' : \u2200 x \u2208 s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) (g : E \u2192 F) :\n    IntegrableOn g (f '' s) \u03bc \u2194 IntegrableOn (fun x => |(f' x).det| \u2022 g (f x)) s \u03bc", "theorem integrable_average [IsFiniteMeasure \u03bc] {f : \u03b1 \u2192 \u03b2} :\n    Integrable f ((\u03bc univ)\u207b\u00b9 \u2022 \u03bc) \u2194 Integrable f \u03bc", "theorem integrable_condexp : Integrable (\u03bc[f|m]) \u03bc", "theorem integrable_condexpL2_indicator (hm : m \u2264 m0) [SigmaFinite (\u03bc.trim hm)]\n    (hs : MeasurableSet s) (h\u03bcs : \u03bc s \u2260 \u221e) (x : E') :\n    Integrable (\u03b2 := E') (condexpL2 E' \ud835\udd5c hm (indicatorConstLp 2 hs h\u03bcs x)) \u03bc", "lemma integrable_count_iff :\n    Integrable f Measure.count \u2194 Summable (\u2016f \u00b7\u2016)", "theorem integrable_smul_const {f : \u03b1 \u2192 \ud835\udd5c} {c : E} (hc : c \u2260 0) :\n    Integrable (fun x => f x \u2022 c) \u03bc \u2194 Integrable f \u03bc", "theorem integrable_withDensity_iff {f : \u03b1 \u2192 \u211d\u22650\u221e} (hf : Measurable f) (hflt : \u2200\u1d50 x \u2202\u03bc, f x < \u221e)\n    {g : \u03b1 \u2192 \u211d} : Integrable g (\u03bc.withDensity f) \u2194 Integrable (fun x => g x * (f x).toReal) \u03bc", "theorem integrable_withDensity_iff_integrable_smul' {f : \u03b1 \u2192 \u211d\u22650\u221e} (hf : Measurable f)\n    (hflt : \u2200\u1d50 x \u2202\u03bc, f x < \u221e) {g : \u03b1 \u2192 E} :\n    Integrable g (\u03bc.withDensity f) \u2194 Integrable (fun x => (f x).toReal \u2022 g x) \u03bc", "theorem integral_Icc_eq_integral_Ico : \u222b t in Icc x y, f t \u2202\u03bc = \u222b t in Ico x y, f t \u2202\u03bc", "theorem integral_Icc_eq_integral_Ioo' (hx : \u03bc {x} = 0) (hy : \u03bc {y} = 0) :\n    \u222b t in Icc x y, f t \u2202\u03bc = \u222b t in Ioo x y, f t \u2202\u03bc", "theorem integral_Ico_eq_integral_Ioo' (hx : \u03bc {x} = 0) :\n    \u222b t in Ico x y, f t \u2202\u03bc = \u222b t in Ioo x y, f t \u2202\u03bc", "theorem integral_Ioi_of_hasDerivAt_of_nonpos (hcont : ContinuousWithinAt g (Ici a) a)\n    (hderiv : \u2200 x \u2208 Ioi a, HasDerivAt g (g' x) x) (g'neg : \u2200 x \u2208 Ioi a, g' x \u2264 0)\n    (hg : Tendsto g atTop (\ud835\udcdd l)) : \u222b x in Ioi a, g' x = l - g a", "theorem integral_average (\u03bc : Measure \u03b1) [IsFiniteMeasure \u03bc] (f : \u03b1 \u2192 E) :\n    \u222b _, \u2a0d a, f a \u2202\u03bc \u2202\u03bc = \u222b x, f x \u2202\u03bc", "theorem integral_average_sub [IsFiniteMeasure \u03bc] (hf : Integrable f \u03bc) :\n    \u222b x, \u2a0d a, f a \u2202\u03bc - f x \u2202\u03bc = 0", "theorem integral_deriv_mul_eq_sub [CompleteSpace A]\n    (hu : \u2200 x, HasDerivAt u (u' x) x) (hv : \u2200 x, HasDerivAt v (v' x) x)\n    (huv : Integrable (u' * v + u * v'))\n    (h_bot : Tendsto (u * v) atBot (\ud835\udcdd a')) (h_top : Tendsto (u * v) atTop (\ud835\udcdd b')) :\n    \u222b (x : \u211d), u' x * v x + u x * v' x = b' - a'", "theorem integral_divergence_of_hasFDerivWithinAt_off_countable (hle : a \u2264 b)\n    (f : \u211d\u207f\u207a\u00b9 \u2192 E\u207f\u207a\u00b9)\n    (f' : \u211d\u207f\u207a\u00b9 \u2192 \u211d\u207f\u207a\u00b9 \u2192L[\u211d] E\u207f\u207a\u00b9)\n    (s : Set \u211d\u207f\u207a\u00b9) (hs : s.Countable) (Hc : ContinuousOn f (Icc a b))\n    (Hd : \u2200 x \u2208 (Set.pi univ fun i => Ioo (a i) (b i)) \\ s, HasFDerivAt f (f' x) x)\n    (Hi : IntegrableOn (fun x => \u2211 i, f' x (e i) i) (Icc a b)) :\n    (\u222b x in Icc a b, \u2211 i, f' x (e i) i) =\n      \u2211 i : Fin (n + 1),\n        ((\u222b x in face i, f (frontFace i x) i) - \u222b x in face i, f (backFace i x) i)", "theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux\u2082 (I : Box (Fin (n + 1)))\n    (f : \u211d\u207f\u207a\u00b9 \u2192 E\u207f\u207a\u00b9)\n    (f' : \u211d\u207f\u207a\u00b9 \u2192 \u211d\u207f\u207a\u00b9 \u2192L[\u211d] E\u207f\u207a\u00b9)\n    (s : Set \u211d\u207f\u207a\u00b9) (hs : s.Countable) (Hc : ContinuousOn f (Box.Icc I))\n    (Hd : \u2200 x \u2208 Box.Ioo I \\ s, HasFDerivAt f (f' x) x)\n    (Hi : IntegrableOn (\u2211 i, f' \u00b7 (e i) i) (Box.Icc I)) :\n    (\u222b x in Box.Icc I, \u2211 i, f' x (e i) i) =\n      \u2211 i : Fin (n + 1),\n        ((\u222b x in Box.Icc (I.face i), f (i.insertNth (I.upper i) x) i) -\n          \u222b x in Box.Icc (I.face i), f (i.insertNth (I.lower i) x) i)", "theorem integral_fintype_prod_eq_prod (\u03b9 : Type*) [Fintype \u03b9] {E : \u03b9 \u2192 Type*}\n    (f : (i : \u03b9) \u2192 E i \u2192 \ud835\udd5c) [\u2200 i, MeasureSpace (E i)] [\u2200 i, SigmaFinite (volume : Measure (E i))] :\n    \u222b x : (i : \u03b9) \u2192 E i, \u220f i, f i (x i) = \u220f i, \u222b x, f i x", "lemma integral_llr_tilted_left [IsProbabilityMeasure \u03bc] [SigmaFinite \u03bd]\n    (h\u03bc\u03bd : \u03bc \u226a \u03bd) (hf : Integrable f \u03bc) (h_int : Integrable (llr \u03bc \u03bd) \u03bc)\n    (hf\u03bc : Integrable (fun x \u21a6 exp (f x)) \u03bc) (hf\u03bd : AEMeasurable f \u03bd) :\n    \u222b x, llr (\u03bc.tilted f) \u03bd x \u2202\u03bc = \u222b x, llr \u03bc \u03bd x \u2202\u03bc + \u222b x, f x \u2202\u03bc - log (\u222b x, exp (f x) \u2202\u03bc)", "@[mono]\ntheorem integral_mono {f g : \u03b1 \u2192 \u211d} (hf : Integrable f \u03bc) (hg : Integrable g \u03bc) (h : f \u2264 g) :\n    \u222b a, f a \u2202\u03bc \u2264 \u222b a, g a \u2202\u03bc", "theorem integral_piecewise [DecidablePred (\u00b7 \u2208 s)] (hs : MeasurableSet s) (hf : IntegrableOn f s \u03bc)\n    (hg : IntegrableOn g s\u1d9c \u03bc) :\n    \u222b x, s.piecewise f g x \u2202\u03bc = \u222b x in s, f x \u2202\u03bc + \u222b x in s\u1d9c, g x \u2202\u03bc", "theorem integral_singleton [MeasurableSingletonClass \u03b1] {\u03bc : Measure \u03b1} (f : \u03b1 \u2192 E) (a : \u03b1) :\n    \u222b a in {a}, f a \u2202\u03bc = (\u03bc {a}).toReal \u2022 f a", "theorem intervalIntegral_tendsto_integral (hfi : Integrable f \u03bc) (ha : Tendsto a l atBot)\n    (hb : Tendsto b l atTop) : Tendsto (fun i => \u222b x in a i..b i, f x \u2202\u03bc) l (\ud835\udcdd <| \u222b x, f x \u2202\u03bc)", "theorem isStoppingTime_const [Preorder \u03b9] (f : Filtration \u03b9 m) (i : \u03b9) :\n    IsStoppingTime f fun _ => i", "theorem ite_ae_eq_of_measure_compl_zero {\u03b3} (f : \u03b1 \u2192 \u03b3) (g : \u03b1 \u2192 \u03b3)\n    (s : Set \u03b1) [DecidablePred (\u00b7 \u2208 s)] (hs_zero : \u03bc s\u1d9c = 0) :\n    (fun x => ite (x \u2208 s) (f x) (g x)) =\u1d50[\u03bc] f", "theorem laverage_mem_openSegment_compl_self [IsFiniteMeasure \u03bc] (hs : NullMeasurableSet s \u03bc)\n    (hs\u2080 : \u03bc s \u2260 0) (hsc\u2080 : \u03bc s\u1d9c \u2260 0) :\n    \u2a0d\u207b x, f x \u2202\u03bc \u2208 openSegment \u211d\u22650\u221e (\u2a0d\u207b x in s, f x \u2202\u03bc) (\u2a0d\u207b x in s\u1d9c, f x \u2202\u03bc)", "theorem le_hitting {m : \u03b9} (hnm : n \u2264 m) (\u03c9 : \u03a9) : n \u2264 hitting u s n m \u03c9", "lemma levyProkhorovDist_le_of_forall_le\n    (\u03bc \u03bd : Measure \u03a9) [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bd] {\u03b4 : \u211d} (\u03b4_nn : 0 \u2264 \u03b4)\n    (h : \u2200 \u03b5 B, \u03b4 < \u03b5 \u2192 MeasurableSet B \u2192 \u03bc B \u2264 \u03bd (thickening \u03b5 B) + ENNReal.ofReal \u03b5) :\n    levyProkhorovDist \u03bc \u03bd \u2264 \u03b4", "theorem lintegral_abs_det_fderiv_eq_addHaar_image (hs : MeasurableSet s)\n    (hf' : \u2200 x \u2208 s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) :\n    (\u222b\u207b x in s, ENNReal.ofReal |(f' x).det| \u2202\u03bc) = \u03bc (f '' s)", "theorem lintegral_biUnion_finset\u2080 {s : Finset \u03b2} {t : \u03b2 \u2192 Set \u03b1}\n    (hd : Set.Pairwise (\u2191s) (AEDisjoint \u03bc on t)) (hm : \u2200 b \u2208 s, NullMeasurableSet (t b) \u03bc)\n    (f : \u03b1 \u2192 \u211d\u22650\u221e) : \u222b\u207b a in \u22c3 b \u2208 s, t b, f a \u2202\u03bc = \u2211 b \u2208 s, \u222b\u207b a in t b, f a \u2202\u03bc", "theorem lintegral_count' {f : \u03b1 \u2192 \u211d\u22650\u221e} (hf : Measurable f) : \u222b\u207b a, f a \u2202count = \u2211' a, f a", "theorem lintegral_iInf' {f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e} (h_meas : \u2200 n, AEMeasurable (f n) \u03bc)\n    (h_anti : \u2200\u1d50 a \u2202\u03bc, Antitone (fun i \u21a6 f i a)) (h_fin : \u222b\u207b a, f 0 a \u2202\u03bc \u2260 \u221e) :\n    \u222b\u207b a, \u2a05 n, f n a \u2202\u03bc = \u2a05 n, \u222b\u207b a, f n a \u2202\u03bc", "theorem lintegral_iInf_ae {f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e} (h_meas : \u2200 n, Measurable (f n))\n    (h_mono : \u2200 n : \u2115, f n.succ \u2264\u1d50[\u03bc] f n) (h_fin : \u222b\u207b a, f 0 a \u2202\u03bc \u2260 \u221e) :\n    \u222b\u207b a, \u2a05 n, f n a \u2202\u03bc = \u2a05 n, \u222b\u207b a, f n a \u2202\u03bc", "theorem lintegral_iSup {f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e} (hf : \u2200 n, Measurable (f n)) (h_mono : Monotone f) :\n    \u222b\u207b a, \u2a06 n, f n a \u2202\u03bc = \u2a06 n, \u222b\u207b a, f n a \u2202\u03bc", "theorem lintegral_indicator_const_comp {m\u03b2 : MeasurableSpace \u03b2} {f : \u03b1 \u2192 \u03b2} {s : Set \u03b2}\n    (hf : Measurable f) (hs : MeasurableSet s) (c : \u211d\u22650\u221e) :\n    \u222b\u207b a, s.indicator (fun _ => c) (f a) \u2202\u03bc = c * \u03bc (f \u207b\u00b9' s)", "lemma lintegral_le_meas {s : Set \u03b1} {f : \u03b1 \u2192 \u211d\u22650\u221e} (hf : \u2200 a, f a \u2264 1) (h'f : \u2200 a \u2208 s\u1d9c, f a = 0) :\n    \u222b\u207b a, f a \u2202\u03bc \u2264 \u03bc s", "@[mono]\ntheorem lintegral_mono' {m : MeasurableSpace \u03b1} \u2983\u03bc \u03bd : Measure \u03b1\u2984 (h\u03bc\u03bd : \u03bc \u2264 \u03bd) \u2983f g : \u03b1 \u2192 \u211d\u22650\u221e\u2984\n    (hfg : f \u2264 g) : \u222b\u207b a, f a \u2202\u03bc \u2264 \u222b\u207b a, g a \u2202\u03bd", "theorem lintegral_mul_const' (r : \u211d\u22650\u221e) (f : \u03b1 \u2192 \u211d\u22650\u221e) (hr : r \u2260 \u221e) :\n    \u222b\u207b a, f a * r \u2202\u03bc = (\u222b\u207b a, f a \u2202\u03bc) * r", "theorem lintegral_mul_const'' (r : \u211d\u22650\u221e) {f : \u03b1 \u2192 \u211d\u22650\u221e} (hf : AEMeasurable f \u03bc) :\n    \u222b\u207b a, f a * r \u2202\u03bc = (\u222b\u207b a, f a \u2202\u03bc) * r", "theorem lintegral_nnnorm_condexpL2_indicator_le_real (hs : MeasurableSet s) (h\u03bcs : \u03bc s \u2260 \u221e)\n    (ht : MeasurableSet[m] t) (h\u03bct : \u03bc t \u2260 \u221e) :\n    \u222b\u207b a in t, \u2016(condexpL2 \u211d \u211d hm (indicatorConstLp 2 hs h\u03bcs 1) : \u03b1 \u2192 \u211d) a\u2016\u208a \u2202\u03bc \u2264 \u03bc (s \u2229 t)", "theorem lintegral_nnnorm_neg {f : \u03b1 \u2192 \u03b2} : (\u222b\u207b a, \u2016(-f) a\u2016\u208a \u2202\u03bc) = \u222b\u207b a, \u2016f a\u2016\u208a \u2202\u03bc", "theorem lintegral_strict_mono {f g : \u03b1 \u2192 \u211d\u22650\u221e} (h\u03bc : \u03bc \u2260 0) (hg : AEMeasurable g \u03bc)\n    (hfi : \u222b\u207b x, f x \u2202\u03bc \u2260 \u221e) (h : \u2200\u1d50 x \u2202\u03bc, f x < g x) : \u222b\u207b x, f x \u2202\u03bc < \u222b\u207b x, g x \u2202\u03bc", "theorem lintegral_strict_mono_of_ae_le_of_frequently_ae_lt {f g : \u03b1 \u2192 \u211d\u22650\u221e} (hg : AEMeasurable g \u03bc)\n    (hfi : \u222b\u207b x, f x \u2202\u03bc \u2260 \u221e) (h_le : f \u2264\u1d50[\u03bc] g) (h : \u2203\u1d50 x \u2202\u03bc, f x \u2260 g x) :\n    \u222b\u207b x, f x \u2202\u03bc < \u222b\u207b x, g x \u2202\u03bc", "theorem lintegral_withDensity_eq_lintegral_mul (\u03bc : Measure \u03b1) {f : \u03b1 \u2192 \u211d\u22650\u221e}\n    (h_mf : Measurable f) :\n    \u2200 {g : \u03b1 \u2192 \u211d\u22650\u221e}, Measurable g \u2192 \u222b\u207b a, g a \u2202\u03bc.withDensity f = \u222b\u207b a, (f * g) a \u2202\u03bc", "theorem lintegral_withDensity_eq_lintegral_mul_non_measurable\u2080 (\u03bc : Measure \u03b1) {f : \u03b1 \u2192 \u211d\u22650\u221e}\n    (hf : AEMeasurable f \u03bc) (h'f : \u2200\u1d50 x \u2202\u03bc, f x < \u221e) (g : \u03b1 \u2192 \u211d\u22650\u221e) :\n    \u222b\u207b a, g a \u2202\u03bc.withDensity f = \u222b\u207b a, (f * g) a \u2202\u03bc", "theorem lintegral_zero_fun : lintegral \u03bc (0 : \u03b1 \u2192 \u211d\u22650\u221e) = 0", "lemma llr_smul_right [IsFiniteMeasure \u03bc] [Measure.HaveLebesgueDecomposition \u03bc \u03bd]\n    (h\u03bc\u03bd : \u03bc \u226a \u03bd) (c : \u211d\u22650\u221e) (hc : c \u2260 0) (hc_ne_top : c \u2260 \u221e) :\n    llr \u03bc (c \u2022 \u03bd) =\u1d50[\u03bc] fun x \u21a6 llr \u03bc \u03bd x - log c.toReal", "theorem lmarginal_eq_of_subset {f g : (\u2200 i, \u03c0 i) \u2192 \u211d\u22650\u221e} (hst : s \u2286 t)\n    (hf : Measurable f) (hg : Measurable g) (hfg : \u222b\u22ef\u222b\u207b_s, f \u2202\u03bc = \u222b\u22ef\u222b\u207b_s, g \u2202\u03bc) :\n    \u222b\u22ef\u222b\u207b_t, f \u2202\u03bc = \u222b\u22ef\u222b\u207b_t, g \u2202\u03bc", "theorem lpMeas.ae_eq_zero_of_forall_setIntegral_eq_zero (hm : m \u2264 m0) (f : lpMeas E' \ud835\udd5c m p \u03bc)\n    (hp_ne_zero : p \u2260 0) (hp_ne_top : p \u2260 \u221e)\n    (hf_int_finite : \u2200 s, MeasurableSet[m] s \u2192 \u03bc s < \u221e \u2192 IntegrableOn (f : Lp E' p \u03bc) s \u03bc)\n    (hf_zero : \u2200 s : Set \u03b1, MeasurableSet[m] s \u2192 \u03bc s < \u221e \u2192 \u222b x in s, (f : Lp E' p \u03bc) x \u2202\u03bc = 0) :\n    f =\u1d50[\u03bc] (0 : \u03b1 \u2192 E')", "theorem lpMeasSubgroupToLpTrim_ae_eq (hm : m \u2264 m0) (f : lpMeasSubgroup F m p \u03bc) :\n    lpMeasSubgroupToLpTrim F p \u03bc hm f =\u1d50[\u03bc] f", "theorem lpMeasToLpTrim_smul (hm : m \u2264 m0) (c : \ud835\udd5c) (f : lpMeas F \ud835\udd5c m p \u03bc) :\n    lpMeasToLpTrim F \ud835\udd5c p \u03bc hm (c \u2022 f) = c \u2022 lpMeasToLpTrim F \ud835\udd5c p \u03bc hm f", "theorem lpTrimToLpMeas_ae_eq (hm : m \u2264 m0) (f : Lp F p (\u03bc.trim hm)) :\n    lpTrimToLpMeas F \ud835\udd5c p \u03bc hm f =\u1d50[\u03bc] f", "lemma meas_le_lintegral\u2080 {f : \u03b1 \u2192 \u211d\u22650\u221e} (hf : AEMeasurable f \u03bc)\n    {s : Set \u03b1} (hs : \u2200 x \u2208 s, 1 \u2264 f x) : \u03bc s \u2264 \u222b\u207b a, f a \u2202\u03bc", "theorem measurableCylinders.measurableSet (ht : t \u2208 measurableCylinders \u03b1) :\n    MeasurableSet (measurableCylinders.set ht)", "@[to_additive measurePreserving_add_prod_neg_right\n\"The map `(x, y) \u21a6 (x + y, - x)` is measure-preserving.\"]\ntheorem measurePreserving_mul_prod_inv_right [IsMulRightInvariant \u03bc] [IsMulRightInvariant \u03bd] :\n    MeasurePreserving (fun z : G \u00d7 G => (z.1 * z.2, z.1\u207b\u00b9)) (\u03bc.prod \u03bd) (\u03bc.prod \u03bd)", "theorem measurePreserving_piFinSuccAbove {n : \u2115} {\u03b1 : Fin (n + 1) \u2192 Type u}\n    {m : \u2200 i, MeasurableSpace (\u03b1 i)} (\u03bc : \u2200 i, Measure (\u03b1 i)) [\u2200 i, SigmaFinite (\u03bc i)]\n    (i : Fin (n + 1)) :\n    MeasurePreserving (MeasurableEquiv.piFinSuccAbove \u03b1 i) (Measure.pi \u03bc)\n      ((\u03bc i).prod <| Measure.pi fun j => \u03bc (i.succAbove j))", "@[to_additive measurePreserving_prod_add_swap_right\n\" The map `(x, y) \u21a6 (y, x + y)` sends the measure `\u03bc \u00d7 \u03bd` to `\u03bd \u00d7 \u03bc`. \"]\ntheorem measurePreserving_prod_mul_swap_right [IsMulRightInvariant \u03bc] :\n    MeasurePreserving (fun z : G \u00d7 G => (z.2, z.1 * z.2)) (\u03bc.prod \u03bd) (\u03bd.prod \u03bc)", "theorem measureUnivNNReal_pos [IsFiniteMeasure \u03bc] (h\u03bc : \u03bc \u2260 0) : 0 < measureUnivNNReal \u03bc", "theorem measure_eq_measure_smaller_of_between_null_diff {s\u2081 s\u2082 s\u2083 : Set \u03b1} (h12 : s\u2081 \u2286 s\u2082)\n    (h23 : s\u2082 \u2286 s\u2083) (h_nulldiff : \u03bc (s\u2083 \\ s\u2081) = 0) : \u03bc s\u2081 = \u03bc s\u2082", "@[to_additive]\ntheorem measure_eq_zero_iff_eq_empty_of_smulInvariant (h\u03bc : \u03bc \u2260 0) (hU : IsOpen U) :\n    \u03bc U = 0 \u2194 U = \u2205", "theorem measure_integral_le_pos (hf : Integrable f \u03bc) : 0 < \u03bc {x | \u222b a, f a \u2202\u03bc \u2264 f x}", "theorem measure_le_setAverage_pos (h\u03bc : \u03bc s \u2260 0) (h\u03bc\u2081 : \u03bc s \u2260 \u221e) (hf : IntegrableOn f s \u03bc) :\n    0 < \u03bc ({x \u2208 s | f x \u2264 \u2a0d a in s, f a \u2202\u03bc})", "theorem MeasureTheory.measure_lt_one_eq_integral_div_gamma {p : \u211d} (hp : 0 < p) :\n    \u03bc {x : E | g x < 1} =\n      .ofReal ((\u222b (x : E), Real.exp (- (g x) ^ p) \u2202\u03bc) / Real.Gamma (finrank \u211d E / p + 1))", "@[to_additive addMeasure_map_restrict_apply]\nlemma measure_map_restrict_apply (s : Set \u03b1) {U : Set (Quotient \u03b1_mod_G)}\n    (meas_U : MeasurableSet U) :\n    (\u03bc.restrict s).map \u03c0 U = \u03bc ((\u03c0 \u207b\u00b9' U) \u2229 s)", "@[simp]\ntheorem measure_mul_laverage [IsFiniteMeasure \u03bc] (f : \u03b1 \u2192 \u211d\u22650\u221e) :\n    \u03bc univ * \u2a0d\u207b x, f x \u2202\u03bc = \u222b\u207b x, f x \u2202\u03bc", "theorem measure_mul_setLaverage (f : \u03b1 \u2192 \u211d\u22650\u221e) (h : \u03bc s \u2260 \u221e) :\n    \u03bc s * \u2a0d\u207b x in s, f x \u2202\u03bc = \u222b\u207b x in s, f x \u2202\u03bc", "lemma measure_null_iff_singleton (hs : s.Countable) : \u03bc s = 0 \u2194 \u2200 x \u2208 s, \u03bc {x} = 0", "theorem measure_null_of_locally_null [TopologicalSpace \u03b1] [SecondCountableTopology \u03b1]\n    (s : Set \u03b1) (hs : \u2200 x \u2208 s, \u2203 u \u2208 \ud835\udcdd[s] x, \u03bc u = 0) : \u03bc s = 0", "@[to_additive (attr := simp)]\ntheorem measure_smul (s : Set \u03b1) : \u03bc (c \u2022 s) = \u03bc s", "theorem measure_union\u2080' (hs : NullMeasurableSet s \u03bc) (hd : AEDisjoint \u03bc s t) :\n    \u03bc (s \u222a t) = \u03bc s + \u03bc t", "theorem MeasureTheory.measure_unitBall_eq_integral_div_gamma {E : Type*} {p : \u211d}\n    [NormedAddCommGroup E] [NormedSpace \u211d E] [FiniteDimensional \u211d E] [MeasurableSpace E]\n    [BorelSpace E] (\u03bc : Measure E) [IsAddHaarMeasure \u03bc] (hp : 0 < p) :\n    \u03bc (Metric.ball 0 1) =\n      .ofReal ((\u222b (x : E), Real.exp (- \u2016x\u2016 ^ p) \u2202\u03bc) / Real.Gamma (finrank \u211d E / p + 1))", "theorem mem_disjointed_spanningSetsIndex (\u03bc : Measure \u03b1) [SigmaFinite \u03bc] (x : \u03b1) :\n    x \u2208 disjointed (spanningSets \u03bc) (spanningSetsIndex \u03bc x)", "theorem mem_lpMeasSubgroup_iff_aeStronglyMeasurable' {m m0 : MeasurableSpace \u03b1} {\u03bc : Measure \u03b1}\n    {f : Lp F p \u03bc} : f \u2208 lpMeasSubgroup F m p \u03bc \u2194 AEStronglyMeasurable' m f \u03bc", "theorem mem_lpMeas_iff_aeStronglyMeasurable' {m m0 : MeasurableSpace \u03b1} {\u03bc : Measure \u03b1}\n    {f : Lp F p \u03bc} : f \u2208 lpMeas F \ud835\udd5c m p \u03bc \u2194 AEStronglyMeasurable' m f \u03bc", "theorem mem\u2112p_indicator_const (p : \u211d\u22650\u221e) (hs : MeasurableSet s) (c : E) (h\u03bcsc : c = 0 \u2228 \u03bc s \u2260 \u221e) :\n    Mem\u2112p (s.indicator fun _ => c) p \u03bc", "theorem norm_setIntegral_le_of_norm_le_const_ae'' {C : \u211d} (hs : \u03bc s < \u221e) (hsm : MeasurableSet s)\n    (hC : \u2200\u1d50 x \u2202\u03bc, x \u2208 s \u2192 \u2016f x\u2016 \u2264 C) : \u2016\u222b x in s, f x \u2202\u03bc\u2016 \u2264 C * (\u03bc s).toReal", "theorem ofReal_average {f : \u03b1 \u2192 \u211d} (hf : Integrable f \u03bc) (hf\u2080 : 0 \u2264\u1d50[\u03bc] f) :\n    ENNReal.ofReal (\u2a0d x, f x \u2202\u03bc) = (\u222b\u207b x, ENNReal.ofReal (f x) \u2202\u03bc) / \u03bc univ", "theorem ofReal_toReal_ae_eq {f : \u03b1 \u2192 \u211d\u22650\u221e} (hf : \u2200\u1d50 x \u2202\u03bc, f x < \u221e) :\n    (fun x => ENNReal.ofReal (f x).toReal) =\u1d50[\u03bc] f", "@[to_additive sub_nonneg_ae]\nlemma one_le_div_ae {\u03b2 : Type*} [Group \u03b2] [LE \u03b2]\n    [CovariantClass \u03b2 \u03b2 (Function.swap (\u00b7 * \u00b7)) (\u00b7 \u2264 \u00b7)] (f g : \u03b1 \u2192 \u03b2) :\n    1 \u2264\u1d50[\u03bc] g / f \u2194 f \u2264\u1d50[\u03bc] g", "theorem measure_preimage {X : \u03a9 \u2192 E} {s : Set E} (hns : \u03bc s \u2260 0) (hnt : \u03bc s \u2260 \u221e)\n    (hu : IsUniform X s \u2119 \u03bc) {A : Set E} (hA : MeasurableSet A) :\n    \u2119 (X \u207b\u00b9' A) = \u03bc (s \u2229 A) / \u03bc s", "theorem pdf_toReal_ae_eq {X : \u03a9 \u2192 E} {s : Set E} (hms : MeasurableSet s)\n    (hX : IsUniform X s \u2119 \u03bc) :\n    (fun x => (pdf X \u2119 \u03bc x).toReal) =\u1d50[\u03bc] fun x =>\n      (s.indicator ((\u03bc s)\u207b\u00b9 \u2022 (1 : E \u2192 \u211d\u22650\u221e)) x).toReal", "protected theorem toMeasurable {X : \u03a9 \u2192 E} {s : Set E} (hu : IsUniform X s \u2119 \u03bc) :\n    IsUniform X (toMeasurable \u03bc s) \u2119 \u03bc", "protected theorem congr {X Y : \u03a9 \u2192 E} (hXY : X =\u1d50[\u2119] Y) : pdf X \u2119 \u03bc = pdf Y \u2119 \u03bc", "theorem piPremeasure_pi' {s : \u2200 i, Set (\u03b1 i)} : piPremeasure m (pi univ s) = \u220f i, m i (s i)", "lemma prob_compl_eq_one_sub\u2080 (h : NullMeasurableSet s \u03bc) : \u03bc s\u1d9c = 1 - \u03bc s", "theorem prob_le_one [IsProbabilityMeasure \u03bc] : \u03bc s \u2264 1", "theorem progMeasurable_const [MeasurableSpace \u03b9] (f : Filtration \u03b9 m) (b : \u03b2) :\n    ProgMeasurable f (fun _ _ => b : \u03b9 \u2192 \u03a9 \u2192 \u03b2)", "theorem progMeasurable_min_stopping_time [MetrizableSpace \u03b9] (h\u03c4 : IsStoppingTime f \u03c4) :\n    ProgMeasurable f fun i \u03c9 => min i (\u03c4 \u03c9)", "@[to_additive]\ntheorem quasiMeasurePreserving_div_left_of_right_invariant [IsMulRightInvariant \u03bc] (g : G) :\n    QuasiMeasurePreserving (fun h : G => g / h) \u03bc \u03bc", "lemma restrict_compl_sigmaFiniteSet [SFinite \u03bc] :\n    \u03bc.restrict \u03bc.sigmaFiniteSet\u1d9c = \u221e \u2022 \u03bc.toFinite.restrict \u03bc.sigmaFiniteSet\u1d9c", "lemma restrict_compl_sigmaFiniteSet_eq_zero_or_top (\u03bc : Measure \u03b1) [SFinite \u03bc] (s : Set \u03b1) :\n    \u03bc.restrict \u03bc.sigmaFiniteSet\u1d9c s = 0 \u2228 \u03bc.restrict \u03bc.sigmaFiniteSet\u1d9c s = \u221e", "theorem rnDeriv_ae_eq_condexp {hm : m \u2264 m0} [h\u03bcm : SigmaFinite (\u03bc.trim hm)] {f : \u03b1 \u2192 \u211d}\n    (hf : Integrable f \u03bc) :\n    SignedMeasure.rnDeriv ((\u03bc.withDensity\u1d65 f).trim hm) (\u03bc.trim hm) =\u1d50[\u03bc] \u03bc[f|m]", "lemma rnDeriv_tilted_right (\u03bc \u03bd : Measure \u03b1) [SigmaFinite \u03bc] [SigmaFinite \u03bd]\n    (hf : Integrable (fun x \u21a6 exp (f x)) \u03bd) :\n    \u03bc.rnDeriv (\u03bd.tilted f)\n      =\u1d50[\u03bd] fun x \u21a6 ENNReal.ofReal (exp (- f x) * \u222b x, exp (f x) \u2202\u03bd) * \u03bc.rnDeriv \u03bd x", "lemma sFinite_withDensity_of_sigmaFinite_of_measurable (\u03bc : Measure \u03b1) [SigmaFinite \u03bc]\n    {f : \u03b1 \u2192 \u211d\u22650\u221e} (hf : Measurable f) :\n    SFinite (\u03bc.withDensity f)", "theorem setIntegral_congr_ae (hs : MeasurableSet s) (h : \u2200\u1d50 x \u2202\u03bc, x \u2208 s \u2192 f x = g x) :\n    \u222b x in s, f x \u2202\u03bc = \u222b x in s, g x \u2202\u03bc", "theorem setIntegral_congr_set_ae (hst : s =\u1d50[\u03bc] t) : \u222b x in s, f x \u2202\u03bc = \u222b x in t, f x \u2202\u03bc", "theorem setIntegral_le_nonneg {s : Set X} (hs : MeasurableSet s) (hf : StronglyMeasurable f)\n    (hfi : Integrable f \u03bc) : \u222b x in s, f x \u2202\u03bc \u2264 \u222b x in {y | 0 \u2264 f y}, f x \u2202\u03bc", "theorem setIntegral_mono_set (hfi : IntegrableOn f t \u03bc) (hf : 0 \u2264\u1d50[\u03bc.restrict t] f)\n    (hst : s \u2264\u1d50[\u03bc] t) : \u222b x in s, f x \u2202\u03bc \u2264 \u222b x in t, f x \u2202\u03bc", "theorem setIntegral_nonneg (hs : MeasurableSet s) (hf : \u2200 x, x \u2208 s \u2192 0 \u2264 f x) :\n    0 \u2264 \u222b x in s, f x \u2202\u03bc", "theorem setIntegral_nonpos_ae (hs : MeasurableSet s) (hf : \u2200\u1d50 x \u2202\u03bc, x \u2208 s \u2192 f x \u2264 0) :\n    \u222b x in s, f x \u2202\u03bc \u2264 0", "lemma setIntegral_tilted' (f : \u03b1 \u2192 \u211d) (g : \u03b1 \u2192 E) {s : Set \u03b1} (hs : MeasurableSet s) :\n    \u222b x in s, g x \u2202(\u03bc.tilted f) = \u222b x in s, (exp (f x) / \u222b x, exp (f x) \u2202\u03bc) \u2022 (g x) \u2202\u03bc", "theorem setLIntegral_const_lt_top [IsFiniteMeasure \u03bc] (s : Set \u03b1) {c : \u211d\u22650\u221e} (hc : c \u2260 \u221e) :\n    \u222b\u207b _ in s, c \u2202\u03bc < \u221e", "theorem setLaverage_congr_fun (hs : MeasurableSet s) (h : \u2200\u1d50 x \u2202\u03bc, x \u2208 s \u2192 f x = g x) :\n    \u2a0d\u207b x in s, f x \u2202\u03bc = \u2a0d\u207b x in s, g x \u2202\u03bc", "theorem setLaverage_const (hs\u2080 : \u03bc s \u2260 0) (hs : \u03bc s \u2260 \u221e) (c : \u211d\u22650\u221e) : \u2a0d\u207b _x in s, c \u2202\u03bc = c", "theorem setToFun_congr_left (hT : DominatedFinMeasAdditive \u03bc T C)\n    (hT' : DominatedFinMeasAdditive \u03bc T' C') (h : T = T') (f : \u03b1 \u2192 E) :\n    setToFun \u03bc T hT f = setToFun \u03bc T' hT' f", "theorem setToFun_zero_left' (hT : DominatedFinMeasAdditive \u03bc T C)\n    (h_zero : \u2200 s, MeasurableSet s \u2192 \u03bc s < \u221e \u2192 T s = 0) : setToFun \u03bc T hT f = 0", "theorem snorm'_add_le_of_le_one {f g : \u03b1 \u2192 E} (hf : AEStronglyMeasurable f \u03bc) (hq0 : 0 \u2264 q)\n    (hq1 : q \u2264 1) : snorm' (f + g) q \u03bc \u2264 (2 : \u211d\u22650\u221e) ^ (1 / q - 1) * (snorm' f q \u03bc + snorm' g q \u03bc)", "theorem snorm'_congr_norm_ae {f g : \u03b1 \u2192 F} (hfg : \u2200\u1d50 x \u2202\u03bc, \u2016f x\u2016 = \u2016g x\u2016) :\n    snorm' f q \u03bc = snorm' g q \u03bc", "theorem snorm'_mono_nnnorm_ae {f : \u03b1 \u2192 F} {g : \u03b1 \u2192 G} (hq : 0 \u2264 q) (h : \u2200\u1d50 x \u2202\u03bc, \u2016f x\u2016\u208a \u2264 \u2016g x\u2016\u208a) :\n    snorm' f q \u03bc \u2264 snorm' g q \u03bc", "@[simp]\ntheorem snorm'_neg {f : \u03b1 \u2192 F} : snorm' (-f) q \u03bc = snorm' f q \u03bc", "theorem snormEssSup_const_smul (c : \ud835\udd5c) (f : \u03b1 \u2192 F) :\n    snormEssSup (c \u2022 f) \u03bc = (\u2016c\u2016\u208a : \u211d\u22650\u221e) * snormEssSup f \u03bc", "theorem snormEssSup_map_measure (hg : AEStronglyMeasurable g (Measure.map f \u03bc))\n    (hf : AEMeasurable f \u03bc) : snormEssSup g (Measure.map f \u03bc) = snormEssSup (g \u2218 f) \u03bc", "@[simp]\ntheorem snormEssSup_measure_zero {f : \u03b1 \u2192 F} : snormEssSup f (0 : Measure \u03b1) = 0", "@[mono]\ntheorem snormEssSup_mono_measure (f : \u03b1 \u2192 F) (h\u03bc\u03bd : \u03bd \u226a \u03bc) : snormEssSup f \u03bd \u2264 snormEssSup f \u03bc", "theorem snorm_congr_ae {f g : \u03b1 \u2192 F} (hfg : f =\u1d50[\u03bc] g) : snorm f p \u03bc = snorm g p \u03bc", "theorem snorm_const_lt_top_iff {p : \u211d\u22650\u221e} {c : F} (hp_ne_zero : p \u2260 0) (hp_ne_top : p \u2260 \u221e) :\n    snorm (fun _ : \u03b1 => c) p \u03bc < \u221e \u2194 c = 0 \u2228 \u03bc Set.univ < \u221e", "theorem snorm_indicator_const {c : G} (hs : MeasurableSet s) (hp : p \u2260 0) (hp_top : p \u2260 \u221e) :\n    snorm (s.indicator fun _ => c) p \u03bc = \u2016c\u2016\u208a * \u03bc s ^ (1 / p.toReal)", "theorem snorm_indicator_le_of_bound {f : \u03b1 \u2192 \u03b2} (hp_top : p \u2260 \u221e) {\u03b5 : \u211d} (h\u03b5 : 0 < \u03b5) {M : \u211d}\n    (hf : \u2200 x, \u2016f x\u2016 < M) :\n    \u2203 (\u03b4 : \u211d) (h\u03b4 : 0 < \u03b4), \u2200 s,\n      MeasurableSet s \u2192 \u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (s.indicator f) p \u03bc \u2264 ENNReal.ofReal \u03b5", "theorem snorm_le_snorm_fderiv_of_le [FiniteDimensional \u211d F]\n    {u : E \u2192 F} {s : Set E} (hu : ContDiff \u211d 1 u) (h2u : u.support \u2286 s)\n    {p q : \u211d\u22650} (hp : 1 \u2264 p) (h2p : p < finrank \u211d E)\n    (hpq : p\u207b\u00b9 - (finrank \u211d E : \u211d)\u207b\u00b9 \u2264 (q : \u211d)\u207b\u00b9)\n    (hs : Bornology.IsBounded s) :\n    snorm u q \u03bc \u2264 snormLESNormFDerivOfLeConst F \u03bc s p q * snorm (fderiv \u211d u) p \u03bc", "theorem snorm_map_measure (hg : AEStronglyMeasurable g (Measure.map f \u03bc)) (hf : AEMeasurable f \u03bc) :\n    snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc", "theorem snorm_one_eq_lintegral_nnnorm {f : \u03b1 \u2192 F} : snorm f 1 \u03bc = \u222b\u207b x, \u2016f x\u2016\u208a \u2202\u03bc", "lemma snorm_top_piecewise {s : Set \u03b1} (f g : \u03b1 \u2192 E) [DecidablePred (\u00b7 \u2208 s)]\n    (hs : MeasurableSet s) :\n    snorm (Set.piecewise s f g) \u221e \u03bc\n      = max (snorm f \u221e (\u03bc.restrict s)) (snorm g \u221e (\u03bc.restrict s\u1d9c))", "theorem stoppedProcess_eq'' [LinearOrder \u03b9] [LocallyFiniteOrderBot \u03b9] [AddCommMonoid E] (n : \u03b9) :\n    stoppedProcess u \u03c4 n = Set.indicator {a | n \u2264 \u03c4 a} (u n) +\n      \u2211 i \u2208 Finset.Iio n, Set.indicator {\u03c9 | \u03c4 \u03c9 = i} (u i)", "theorem stoppedValue_const (u : \u03b9 \u2192 \u03a9 \u2192 \u03b2) (i : \u03b9) : (stoppedValue u fun _ => i) = u i", "theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n \u03c9 \u2260 N) :\n    stoppedValue f (lowerCrossingTime a b f N n) \u03c9 \u2264 a", "theorem stoppedValue_stoppedValue_leastGE (f : \u2115 \u2192 \u03a9 \u2192 \u211d) (\u03c0 : \u03a9 \u2192 \u2115) (r : \u211d) {n : \u2115}\n    (h\u03c0n : \u2200 \u03c9, \u03c0 \u03c9 \u2264 n) : stoppedValue (fun i => stoppedValue f (leastGE f r i)) \u03c0 =\n      stoppedValue (stoppedProcess f (leastGE f r n)) \u03c0", "theorem submartingale_of_expected_stoppedValue_mono [IsFiniteMeasure \u03bc] (hadp : Adapted \ud835\udca2 f)\n    (hint : \u2200 i, Integrable (f i) \u03bc) (hf : \u2200 \u03c4 \u03c0 : \u03a9 \u2192 \u2115, IsStoppingTime \ud835\udca2 \u03c4 \u2192 IsStoppingTime \ud835\udca2 \u03c0 \u2192\n      \u03c4 \u2264 \u03c0 \u2192 (\u2203 N, \u2200 \u03c9, \u03c0 \u03c9 \u2264 N) \u2192 \u03bc[stoppedValue f \u03c4] \u2264 \u03bc[stoppedValue f \u03c0]) :\n    Submartingale f \ud835\udca2 \u03bc", "theorem submartingale_of_setIntegral_le [IsFiniteMeasure \u03bc] {f : \u03b9 \u2192 \u03a9 \u2192 \u211d} (hadp : Adapted \u2131 f)\n    (hint : \u2200 i, Integrable (f i) \u03bc) (hf : \u2200 i j : \u03b9,\n      i \u2264 j \u2192 \u2200 s : Set \u03a9, MeasurableSet[\u2131 i] s \u2192 \u222b \u03c9 in s, f i \u03c9 \u2202\u03bc \u2264 \u222b \u03c9 in s, f j \u03c9 \u2202\u03bc) :\n    Submartingale f \u2131 \u03bc", "theorem supermartingale_of_setIntegral_succ_le [IsFiniteMeasure \u03bc] {f : \u2115 \u2192 \u03a9 \u2192 \u211d}\n    (hadp : Adapted \ud835\udca2 f) (hint : \u2200 i, Integrable (f i) \u03bc)\n    (hf : \u2200 i, \u2200 s : Set \u03a9, MeasurableSet[\ud835\udca2 i] s \u2192 \u222b \u03c9 in s, f (i + 1) \u03c9 \u2202\u03bc \u2264 \u222b \u03c9 in s, f i \u03c9 \u2202\u03bc) :\n    Supermartingale f \ud835\udca2 \u03bc", "theorem tendsto_integral_norm_approxOn_sub [MeasurableSpace E] [BorelSpace E] {f : \u03b1 \u2192 E}\n    (fmeas : Measurable f) (hf : Integrable f \u03bc) [SeparableSpace (range f \u222a {0} : Set E)] :\n    Tendsto (fun n \u21a6 \u222b x, \u2016SimpleFunc.approxOn f fmeas (range f \u222a {0}) 0 (by simp) n x - f x\u2016 \u2202\u03bc)\n      atTop (\ud835\udcdd 0)", "theorem tendsto_integral_smul_of_tendsto_average_norm_sub\n    {\u03b9 : Type*} {a : \u03b9 \u2192 Set \u03b1} {l : Filter \u03b9} {f : \u03b1 \u2192 E} {c : E} {g : \u03b9 \u2192 \u03b1 \u2192 \u211d} (K : \u211d)\n    (hf : Tendsto (fun i \u21a6 \u2a0d y in a i, \u2016f y - c\u2016 \u2202\u03bc) l (\ud835\udcdd 0))\n    (f_int : \u2200\u1da0 i in l, IntegrableOn f (a i) \u03bc)\n    (hg : Tendsto (fun i \u21a6 \u222b y, g i y \u2202\u03bc) l (\ud835\udcdd 1))\n    (g_supp : \u2200\u1da0 i in l, Function.support (g i) \u2286 a i)\n    (g_bound : \u2200\u1da0 i in l, \u2200 x, |g i x| \u2264 K / (\u03bc (a i)).toReal) :\n    Tendsto (fun i \u21a6 \u222b y, g i y \u2022 f y \u2202\u03bc) l (\ud835\udcdd c)", "theorem tendsto_measure_iInter' {\u03b1 \u03b9 : Type*} [MeasurableSpace \u03b1] {\u03bc : Measure \u03b1} [Countable \u03b9]\n    [Preorder \u03b9] [IsDirected \u03b9 (\u00b7 \u2264 \u00b7)] {f : \u03b9 \u2192 Set \u03b1} (hm : \u2200 i, MeasurableSet (f i))\n    (hf : \u2203 i, \u03bc (f i) \u2260 \u221e) :\n    Tendsto (fun i \u21a6 \u03bc (\u22c2 j \u2208 {j | j \u2264 i}, f j)) atTop (\ud835\udcdd (\u03bc (\u22c2 i, f i)))", "theorem tendsto_setToFun_filter_of_dominated_convergence (hT : DominatedFinMeasAdditive \u03bc T C) {\u03b9}\n    {l : Filter \u03b9} [l.IsCountablyGenerated] {fs : \u03b9 \u2192 \u03b1 \u2192 E} {f : \u03b1 \u2192 E} (bound : \u03b1 \u2192 \u211d)\n    (hfs_meas : \u2200\u1da0 n in l, AEStronglyMeasurable (fs n) \u03bc)\n    (h_bound : \u2200\u1da0 n in l, \u2200\u1d50 a \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a) (bound_integrable : Integrable bound \u03bc)\n    (h_lim : \u2200\u1d50 a \u2202\u03bc, Tendsto (fun n => fs n a) l (\ud835\udcdd (f a))) :\n    Tendsto (fun n => setToFun \u03bc T hT (fs n)) l (\ud835\udcdd <| setToFun \u03bc T hT f)", "theorem toMeasure_apply\u2080 (m : OuterMeasure \u03b1) (h : ms \u2264 m.caratheodory) {s : Set \u03b1}\n    (hs : NullMeasurableSet s (m.toMeasure h)) : m.toMeasure h s = m s", "theorem tsum_measure_le_measure_univ {s : \u03b9 \u2192 Set \u03b1} (hs : \u2200 i, MeasurableSet (s i))\n    (H : Pairwise (Disjoint on s)) : (\u2211' i, \u03bc (s i)) \u2264 \u03bc (univ : Set \u03b1)", "theorem unifIntegrable_congr_ae {p : \u211d\u22650\u221e} {f g : \u03b9 \u2192 \u03b1 \u2192 \u03b2} (hfg : \u2200 n, f n =\u1d50[\u03bc] g n) :\n    UnifIntegrable f p \u03bc \u2194 UnifIntegrable g p \u03bc", "theorem unifIntegrable_of (hp : 1 \u2264 p) (hp' : p \u2260 \u221e) {f : \u03b9 \u2192 \u03b1 \u2192 \u03b2}\n    (hf : \u2200 i, AEStronglyMeasurable (f i) \u03bc)\n    (h : \u2200 \u03b5 : \u211d, 0 < \u03b5 \u2192 \u2203 C : \u211d\u22650,\n      \u2200 i, snorm ({ x | C \u2264 \u2016f i x\u2016\u208a }.indicator (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5) :\n    UnifIntegrable f p \u03bc", "theorem unifIntegrable_zero_meas [MeasurableSpace \u03b1] {p : \u211d\u22650\u221e} {f : \u03b9 \u2192 \u03b1 \u2192 \u03b2} :\n    UnifIntegrable f p (0 : Measure \u03b1)", "theorem uniformIntegrable_average\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n    (hp : 1 \u2264 p) {f : \u2115 \u2192 \u03b1 \u2192 E} (hf : UniformIntegrable f p \u03bc) :\n    UniformIntegrable (fun (n : \u2115) => (n : \u211d)\u207b\u00b9 \u2022 (\u2211 i \u2208 Finset.range n, f i)) p \u03bc", "theorem uniformIntegrable_subsingleton [Subsingleton \u03b9] (hp_one : 1 \u2264 p) (hp_top : p \u2260 \u221e)\n    (hf : \u2200 i, Mem\u2112p (f i) p \u03bc) : UniformIntegrable f p \u03bc", "theorem union_ae_eq_univ_of_ae_eq_univ_left (h : s =\u1d50[\u03bc] univ) : (s \u222a t : Set \u03b1) =\u1d50[\u03bc] univ", "theorem upcrossingStrat_le_one : upcrossingStrat a b f N n \u03c9 \u2264 1", "theorem upcrossingStrat_nonneg : 0 \u2264 upcrossingStrat a b f N n \u03c9", "@[simp]\ntheorem upcrossingsBefore_bot [Preorder \u03b9] [OrderBot \u03b9] [InfSet \u03b9] {a b : \u211d} {f : \u03b9 \u2192 \u03a9 \u2192 \u211d}\n    {\u03c9 : \u03a9} : upcrossingsBefore a b f \u22a5 \u03c9 = \u22a5", "theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N \u03c9 => upcrossingsBefore a b f N \u03c9", "theorem upperCrossingTime_stabilize' (hnm : n \u2264 m) (hn : N \u2264 upperCrossingTime a b f N n \u03c9) :\n    upperCrossingTime a b f N m \u03c9 = N", "theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) \u03c9 =\n    hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n \u03c9) N \u03c9) N \u03c9", "theorem vadd_ae_eq_self_of_mem_zmultiples {G : Type u} {\u03b1 : Type w} {s : Set \u03b1}\n    {m : MeasurableSpace \u03b1} [AddGroup G] [AddAction G \u03b1] [MeasurableSpace G] [MeasurableVAdd G \u03b1]\n    {\u03bc : Measure \u03b1} [VAddInvariantMeasure G \u03b1 \u03bc] {x y : G}\n    (hs : (x +\u1d65 s : Set \u03b1) =\u1d50[\u03bc] s) (hy : y \u2208 AddSubgroup.zmultiples x) :\n    (y +\u1d65 s : Set \u03b1) =\u1d50[\u03bc] s", "theorem MeasureTheory.volume_sum_rpow_lt [Nonempty \u03b9] {p : \u211d} (hp : 1 \u2264 p) (r : \u211d) :\n    volume {x : \u03b9 \u2192 \u211d | (\u2211 i, |x i| ^ p) ^ (1 / p) < r} = (.ofReal r) ^ card \u03b9 *\n      .ofReal ((2 * Gamma (1 / p + 1)) ^ card \u03b9 / Gamma (card \u03b9 / p + 1))", "@[simp]\ntheorem weightedSMul_empty {m : MeasurableSpace \u03b1} (\u03bc : Measure \u03b1) :\n    weightedSMul \u03bc \u2205 = (0 : F \u2192L[\u211d] F)", "@[simp]\ntheorem withDensity_apply (f : \u03b1 \u2192 \u211d\u22650\u221e) {s : Set \u03b1} (hs : MeasurableSet s) :\n    \u03bc.withDensity f s = \u222b\u207b a in s, f a \u2202\u03bc", "@[simp]\ntheorem withDensity_const (c : \u211d\u22650\u221e) : \u03bc.withDensity (fun _ \u21a6 c) = c \u2022 \u03bc", "lemma withDensity_inv_same\u2080 {\u03bc : Measure \u03b1} {f : \u03b1 \u2192 \u211d\u22650\u221e}\n    (hf : AEMeasurable f \u03bc) (hf_ne_zero : \u2200\u1d50 x \u2202\u03bc, f x \u2260 0) (hf_ne_top : \u2200\u1d50 x \u2202\u03bc, f x \u2260 \u221e) :\n    (\u03bc.withDensity f).withDensity (fun x \u21a6 (f x)\u207b\u00b9) = \u03bc", "theorem withDensity\u1d65_add' (hf : Integrable f \u03bc) (hg : Integrable g \u03bc) :\n    (\u03bc.withDensity\u1d65 fun x => f x + g x) = \u03bc.withDensity\u1d65 f + \u03bc.withDensity\u1d65 g", "theorem withDensity\u1d65_smul' {\ud835\udd5c : Type*} [NontriviallyNormedField \ud835\udd5c] [NormedSpace \ud835\udd5c E]\n    [SMulCommClass \u211d \ud835\udd5c E] (f : \u03b1 \u2192 E) (r : \ud835\udd5c) :\n    (\u03bc.withDensity\u1d65 fun x => r \u2022 f x) = r \u2022 \u03bc.withDensity\u1d65 f", "@[simp]\ntheorem zero_convolution : 0 \u22c6[L, \u03bc] g = 0", "theorem add {f g : \u2200 i, E i} (hf : Mem\u2113p f p) (hg : Mem\u2113p g p) : Mem\u2113p (f + g) p", "lemma congr (h_eq : Set.EqOn f g U) (hu : IsOpen U) : MeromorphicOn g U", "lemma mul : MeromorphicOn (s * t) U := fun x hx \u21a6 (hs x hx).mul (ht x hx)", "theorem isometry_mk (i : \u03b9) : Isometry (Sigma.mk i : E i \u2192 \u03a3k, E k)", "theorem Sum.dist_eq_glueDist {p q : X \u2295 Y} (x : X) (y : Y) :\n    Sum.dist p q =\n      glueDist (fun _ : Unit => Nonempty.some \u27e8x\u27e9) (fun _ : Unit => Nonempty.some \u27e8y\u27e9) 1 p q", "theorem ball_disjoint_ball (h : \u03b4 + \u03b5 \u2264 dist x y) : Disjoint (ball x \u03b4) (ball y \u03b5)", "theorem ball_subset_interior_closedBall : ball x \u03b5 \u2286 interior (closedBall x \u03b5)", "@[simp] theorem closedBall_zero : closedBall x 0 = {x}", "theorem closure_eq_iInter_cthickening (E : Set \u03b1) :\n    closure E = \u22c2 (\u03b4 : \u211d) (_ : 0 < \u03b4), cthickening \u03b4 E", "@[simp]\ntheorem comap_dist_right_atTop (c : \u03b1) : comap (dist \u00b7 c) atTop = cobounded \u03b1", "theorem continuousWithinAt_iff [PseudoMetricSpace \u03b2] {f : \u03b1 \u2192 \u03b2} {a : \u03b1} {s : Set \u03b1} :\n    ContinuousWithinAt f s a \u2194\n      \u2200 \u03b5 > 0, \u2203 \u03b4 > 0, \u2200 {x : \u03b1}, x \u2208 s \u2192 dist x a < \u03b4 \u2192 dist (f x) (f a) < \u03b5", "theorem controlled_of_uniformEmbedding [PseudoMetricSpace \u03b2] {f : \u03b1 \u2192 \u03b2} (h : UniformEmbedding f) :\n    (\u2200 \u03b5 > 0, \u2203 \u03b4 > 0, \u2200 {a b : \u03b1}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5) \u2227\n      \u2200 \u03b4 > 0, \u2203 \u03b5 > 0, \u2200 {a b : \u03b1}, dist (f a) (f b) < \u03b5 \u2192 dist a b < \u03b4", "theorem cthickening_subset_thickening {\u03b4\u2081 : \u211d\u22650} {\u03b4\u2082 : \u211d} (hlt : (\u03b4\u2081 : \u211d) < \u03b4\u2082) (E : Set \u03b1) :\n    cthickening \u03b4\u2081 E \u2286 thickening \u03b4\u2082 E", "theorem denseRange_iff {f : \u03b2 \u2192 \u03b1} : DenseRange f \u2194 \u2200 x, \u2200 r > 0, \u2203 y, dist x (f y) < r", "theorem dist_lt_add_of_nonempty_ball_inter_ball (h : (ball x \u03b5\u2081 \u2229 ball y \u03b5\u2082).Nonempty) :\n    dist x y < \u03b5\u2081 + \u03b5\u2082", "theorem dist_lt_add_of_nonempty_closedBall_inter_ball (h : (closedBall x \u03b5\u2081 \u2229 ball y \u03b5\u2082).Nonempty) :\n    dist x y < \u03b5\u2081 + \u03b5\u2082", "lemma eventually_isCompact_closedBall [WeaklyLocallyCompactSpace \u03b1] (x : \u03b1) :\n    \u2200\u1da0 r in \ud835\udcdd (0 : \u211d), IsCompact (closedBall x r)", "theorem exists_continuous_real_forall_closedBall_subset (hK : \u2200 i, IsClosed (K i))\n    (hU : \u2200 i, IsOpen (U i)) (hKU : \u2200 i, K i \u2286 U i) (hfin : LocallyFinite K) :\n    \u2203 \u03b4 : C(X, \u211d), (\u2200 x, 0 < \u03b4 x) \u2227 \u2200 (i), \u2200 x \u2208 K i, closedBall x (\u03b4 x) \u2286 U i", "theorem Metric.exists_isLocalMin_mem_ball [PseudoMetricSpace \u03b1] [ProperSpace \u03b1] [TopologicalSpace \u03b2]\n    [ConditionallyCompleteLinearOrder \u03b2] [OrderTopology \u03b2] {f : \u03b1 \u2192 \u03b2} {a z : \u03b1} {r : \u211d}\n    (hf : ContinuousOn f (closedBall a r)) (hz : z \u2208 closedBall a r)\n    (hf1 : \u2200 z' \u2208 sphere a r, f z < f z') : \u2203 z \u2208 ball a r, IsLocalMin f z", "theorem glueDist_glued_points [Nonempty Z] (\u03a6 : Z \u2192 X) (\u03a8 : Z \u2192 Y) (\u03b5 : \u211d) (p : Z) :\n    glueDist \u03a6 \u03a8 \u03b5 (.inl (\u03a6 p)) (.inr (\u03a8 p)) = \u03b5", "theorem hausdorffDist_le_of_mem_dist {r : \u211d} (hr : 0 \u2264 r) (H1 : \u2200 x \u2208 s, \u2203 y \u2208 t, dist x y \u2264 r)\n    (H2 : \u2200 x \u2208 t, \u2203 y \u2208 s, dist x y \u2264 r) : hausdorffDist s t \u2264 r", "theorem hausdorffDist_nonneg : 0 \u2264 hausdorffDist s t", "theorem hausdorffDist_zero_iff_closure_eq_closure (fin : hausdorffEdist s t \u2260 \u22a4) :\n    hausdorffDist s t = 0 \u2194 closure s = closure t", "theorem isBounded_Ico (a b : \u03b1) : IsBounded (Ico a b)", "theorem isBounded_range_of_tendsto_cofinite_uniformity {f : \u03b2 \u2192 \u03b1}\n    (hf : Tendsto (Prod.map f f) (.cofinite \u00d7\u02e2 .cofinite) (\ud835\udce4 \u03b1)) : IsBounded (range f)", "theorem isCobounded_iff_closedBall_compl_subset {s : Set \u03b1} (c : \u03b1) :\n    IsCobounded s \u2194 \u2203 (r : \u211d), (Metric.closedBall c r)\u1d9c \u2286 s", "theorem nonempty_iInter_of_nonempty_biInter [CompleteSpace \u03b1] {s : \u2115 \u2192 Set \u03b1}\n    (hs : \u2200 n, IsClosed (s n)) (h's : \u2200 n, IsBounded (s n)) (h : \u2200 N, (\u22c2 n \u2264 N, s n).Nonempty)\n    (h' : Tendsto (fun n => diam (s n)) atTop (\ud835\udcdd 0)) : (\u22c2 n, s n).Nonempty", "@[to_additive (attr := simp)]\ntheorem preimage_smul_ball (c : G) (x : X) (r : \u211d) : (c \u2022 \u00b7) \u207b\u00b9' ball x r = ball (c\u207b\u00b9 \u2022 x) r", "theorem secondCountable_of_countable_discretization {\u03b1 : Type u} [MetricSpace \u03b1]\n    (H : \u2200 \u03b5 > (0 : \u211d), \u2203 (\u03b2 : Type*) (_ : Encodable \u03b2) (F : \u03b1 \u2192 \u03b2),\n      \u2200 x y, F x = F y \u2192 dist x y \u2264 \u03b5) :\n    SecondCountableTopology \u03b1", "theorem sphere_subset_closedBall : sphere x \u03b5 \u2286 closedBall x \u03b5", "theorem tendstoLocallyUniformly_iff [TopologicalSpace \u03b2] {F : \u03b9 \u2192 \u03b2 \u2192 \u03b1} {f : \u03b2 \u2192 \u03b1}\n    {p : Filter \u03b9} :\n    TendstoLocallyUniformly F f p \u2194\n      \u2200 \u03b5 > 0, \u2200 x : \u03b2, \u2203 t \u2208 \ud835\udcdd x, \u2200\u1da0 n in p, \u2200 y \u2208 t, dist (f y) (F n y) < \u03b5", "theorem tendsto_nhdsWithin_nhdsWithin [PseudoMetricSpace \u03b2] {t : Set \u03b2} {f : \u03b1 \u2192 \u03b2} {a b} :\n    Tendsto f (\ud835\udcdd[s] a) (\ud835\udcdd[t] b) \u2194\n      \u2200 \u03b5 > 0, \u2203 \u03b4 > 0, \u2200 {x : \u03b1}, x \u2208 s \u2192 dist x a < \u03b4 \u2192 f x \u2208 t \u2227 dist (f x) b < \u03b5", "@[simp]\ntheorem thickening_cthickening_subset (\u03b5 : \u211d) (h\u03b4 : 0 \u2264 \u03b4) (s : Set \u03b1) :\n    thickening \u03b5 (cthickening \u03b4 s) \u2286 thickening (\u03b5 + \u03b4) s", "theorem thickening_eq_biUnion_ball {\u03b4 : \u211d} {E : Set X} : thickening \u03b4 E = \u22c3 x \u2208 E, ball x \u03b4", "@[simp]\ntheorem thickening_iUnion (\u03b4 : \u211d) (f : \u03b9 \u2192 Set \u03b1) :\n    thickening \u03b4 (\u22c3 i, f i) = \u22c3 i, thickening \u03b4 (f i)", "theorem uniformContinuousOn_iff [PseudoMetricSpace \u03b2] {f : \u03b1 \u2192 \u03b2} {s : Set \u03b1} :\n    UniformContinuousOn f s \u2194\n      \u2200 \u03b5 > 0, \u2203 \u03b4 > 0, \u2200 x \u2208 s, \u2200 y \u2208 s, dist x y < \u03b4 \u2192 dist (f x) (f y) < \u03b5", "nonrec theorem uniformInducing_iff [PseudoMetricSpace \u03b2] {f : \u03b1 \u2192 \u03b2} :\n    UniformInducing f \u2194 UniformContinuous f \u2227\n      \u2200 \u03b4 > 0, \u2203 \u03b5 > 0, \u2200 {a b : \u03b1}, dist (f a) (f b) < \u03b5 \u2192 dist a b < \u03b4", "theorem uniformity_basis_dist :\n    (\ud835\udce4 \u03b1).HasBasis (fun \u03b5 : \u211d => 0 < \u03b5) fun \u03b5 => { p : \u03b1 \u00d7 \u03b1 | dist p.1 p.2 < \u03b5 }", "theorem preVal_add (x y : ModP K v O hv p) :\n    preVal K v O hv p (x + y) \u2264 max (preVal K v O hv p x) (preVal K v O hv p y)", "theorem preVal_mul {x y : ModP K v O hv p} (hxy0 : x * y \u2260 0) :\n    preVal K v O hv p (x * y) = preVal K v O hv p x * preVal K v O hv p y", "theorem preVal_zero : preVal K v O hv p 0 = 0", "theorem coe_extChartAt_transDiffeomorph (x : M) :\n    \u21d1(extChartAt (I.transDiffeomorph e) x) = e \u2218 extChartAt I x", "theorem extChartAt_transDiffeomorph_target (x : M) :\n    (extChartAt (I.transDiffeomorph e) x).target = e.symm \u207b\u00b9' (extChartAt I x).target", "theorem hasMFDerivWithinAt_symm {x} (hx : x \u2208 range I) :\n    HasMFDerivWithinAt \ud835\udcd8(\ud835\udd5c, E) I I.symm (range I) x (ContinuousLinearMap.id _ _)", "protected theorem leftInverse : LeftInverse I.symm I", "@[simp, mfld_simps]\ntheorem symm_comp_self : I.symm \u2218 I = id", "theorem symm_map_nhdsWithin_range (x : H) : map I.symm (\ud835\udcdd[range I] I x) = \ud835\udcdd x", "theorem toFun_spec' (g : L \u2243+* L) {n : \u2115+} {t : L\u02e3} (ht : t \u2208 rootsOfUnity n L) :\n    g t = t ^ (\u03c7\u2080 n g).val", "theorem toFun_spec'' (g : L \u2243+* L) {n : \u2115+} {t : L} (ht : IsPrimitiveRoot t n) :\n    g t = t ^ (\u03c7\u2080 n g).val", "@[simp]\ntheorem coe_add (f g : ModularForm \u0393 k) : \u21d1(f + g) = f + g", "private theorem slash_mul (k : \u2124) (A B : GL(2, \u211d)\u207a) (f : \u210d \u2192 \u2102) :\n    f \u2223[k](A * B) = (f \u2223[k]A) \u2223[k]B", "lemma toSlashInvariantForm_intCast (\u0393 : Subgroup SL(2, \u2124)) (z : \u2124) :\n    (z : ModularForm \u0393 0).toSlashInvariantForm = z := rfl", "theorem g_eq_of_c_eq_one (hc : (\u2191\u2098g) 1 0 = 1) : g = T ^ (\u2191\u2098g) 0 0 * S * T ^ (\u2191\u2098g) 1 1", "theorem re_T_zpow_smul (n : \u2124) : (T ^ n \u2022 z).re = z.re + n", "theorem ExtensionOfMaxAdjoin.extendIdealTo_wd (h : Module.Baer R Q) {y : N} (r r' : R)\n    (eq1 : r \u2022 y = r' \u2022 y) : ExtensionOfMaxAdjoin.extendIdealTo i f h y r =\n    ExtensionOfMaxAdjoin.extendIdealTo i f h y r'", "protected theorem injective (h : Module.Baer R Q) : Module.Injective R Q where", "theorem lift_of {i} (x) : lift R \u03b9 G f g Hg (of R \u03b9 G f i x) = g i x", "theorem of.zero_exact [IsDirected \u03b9 (\u00b7 \u2264 \u00b7)] {i x} (H : of R \u03b9 G f i x = 0) :\n    \u2203 j hij, f i j hij x = (0 : G j)", "theorem coe_dualBasis [_root_.Finite \u03b9] : \u21d1h.basis.dualBasis = \u03b5", "theorem IsSemisimple.minpoly_squarefree : Squarefree (minpoly K f)", "theorem IsSemisimple.sub_of_commute : (f - g).IsSemisimple", "lemma disjoint_genEigenspace [NoZeroSMulDivisors R M]\n    (f : End R M) {\u03bc\u2081 \u03bc\u2082 : R} (h\u03bc : \u03bc\u2081 \u2260 \u03bc\u2082) (k l : \u2115) :\n    Disjoint (f.genEigenspace \u03bc\u2081 k) (f.genEigenspace \u03bc\u2082 l)", "theorem exists_eigenvalue [IsAlgClosed K] [FiniteDimensional K V] [Nontrivial V] (f : End K V) :\n    \u2203 c : K, f.HasEigenvalue c", "theorem genEigenspace_restrict (f : End R M) (p : Submodule R M) (k : \u2115) (\u03bc : R)\n    (hfp : \u2200 x : M, x \u2208 p \u2192 f x \u2208 p) :\n    genEigenspace (LinearMap.restrict f hfp) \u03bc k =\n      Submodule.comap p.subtype (f.genEigenspace \u03bc k)", "theorem hasEigenvalue_of_hasEigenvector {f : End R M} {\u03bc : R} {x : M} (h : HasEigenvector f \u03bc x) :\n    HasEigenvalue f \u03bc", "@[simp]\ntheorem hasGenEigenvalue_iff_hasEigenvalue {f : End R M} {\u03bc : R} {k : \u2115} (hk : 0 < k) :\n    f.HasGenEigenvalue \u03bc k \u2194 f.HasEigenvalue \u03bc", "lemma injOn_genEigenspace [NoZeroSMulDivisors R M] (f : End R M) :\n    InjOn (\u2a06 k, f.genEigenspace \u00b7 k) {\u03bc | \u2a06 k, f.genEigenspace \u03bc k \u2260 \u22a5}", "lemma isSemisimple_iff :\n    f.IsSemisimple \u2194 \u2200 p : Submodule R M, p \u2264 p.comap f \u2192 \u2203 q, q \u2264 q.comap f \u2227 IsCompl p q", "@[simp] lemma Module.End.lie_apply (f : Module.End R M) (m : M) : \u2045f, m\u2046 = f m := rfl", "theorem maxGenEigenspace_eq [h : IsNoetherian R M] (f : End R M) (\u03bc : R) :\n    maxGenEigenspace f \u03bc =\n      f.genEigenspace \u03bc (maxGenEigenspaceIndex f \u03bc)", "@[simp]\ntheorem mem_maxGenEigenspace (f : End R M) (\u03bc : R) (m : M) :\n    m \u2208 f.maxGenEigenspace \u03bc \u2194 \u2203 k : \u2115, ((f - \u03bc \u2022 (1 : End R M)) ^ k) m = 0", "theorem End_algebraMap_isUnit_inv_apply_eq_iff {x : R}\n    (h : IsUnit (algebraMap R (Module.End S M) x)) (m m' : M) :\n    (\u2191(h.unit\u207b\u00b9) : Module.End S M) m = m' \u2194 m = x \u2022 m'", "theorem equiv [Finite R M] (e : M \u2243\u2097[R] N) : Finite R N", "@[deprecated OrzechProperty.injective_of_surjective_endomorphism (since := \"2024-05-30\")]\ntheorem Module.Finite.injective_of_surjective_endomorphism {R : Type*} [CommRing R] {M : Type*}\n    [AddCommGroup M] [Module R M] [Finite R M] (f : M \u2192\u2097[R] M)\n    (f_surj : Function.Surjective f) : Function.Injective f", "lemma Module.Finite_of_isLocalization (R S R\u209a S\u209a) [CommSemiring R] [CommRing S] [CommRing R\u209a]\n    [CommRing S\u209a] [Algebra R S] [Algebra R R\u209a] [Algebra R S\u209a] [Algebra S S\u209a] [Algebra R\u209a S\u209a]\n    [IsScalarTower R S S\u209a] [IsScalarTower R R\u209a S\u209a] (M : Submonoid R) [IsLocalization M R\u209a]\n    [IsLocalization (Algebra.algebraMapSubmonoid S M) S\u209a] [hRS : Module.Finite R S] :\n    Module.Finite R\u209a S\u209a", "theorem iff_characterModule_baer : Flat R M \u2194 Module.Baer R (CharacterModule M)", "theorem iff_characterModule_injective [Small.{v} R] :\n    Flat R M \u2194 Module.Injective R (CharacterModule M)", "lemma iff_lTensor_preserves_injective_linearMap [Small.{v} R] :\n    Flat R M \u2194\n    \u2200 \u2983N N' : Type v\u2984 [AddCommGroup N] [AddCommGroup N'] [Module R N] [Module R N']\n      (L : N \u2192\u2097[R] N'), Function.Injective L \u2192 Function.Injective (L.lTensor M)", "theorem of_forall_isTrivialRelation (hfx : \u2200 {\u03b9 : Type u} [Fintype \u03b9] {f : \u03b9 \u2192 R} {x : \u03b9 \u2192 M},\n    \u2211 i, f i \u2022 x i = 0 \u2192 IsTrivialRelation f x) : Flat R M", "lemma rank_eq_mk_of_infinite_lt [Infinite R] (h_lt : lift.{v} #R < lift.{u} #M) :\n    Module.rank R M = #M", "lemma Quotient.mk_smul_mk (r : R) (m : M) :\n    Ideal.Quotient.mk I r \u2022\n      Submodule.Quotient.mk (p := (I \u2022 \u22a4 : Submodule R M)) m =\n      Submodule.Quotient.mk (p := (I \u2022 \u22a4 : Submodule R M)) (r \u2022 m)", "@[simp]\ntheorem someVector_ray (x : Module.Ray R M) : rayOfNeZero R _ x.someVector_ne_zero = x", "lemma equiv (e : M \u2243\u2097[R] N) : IsReflexive R N where\n  bijective_dual_eval'", "theorem Module.exists_nontrivial_relation_sum_zero_of_finrank_succ_lt_card\n    {t : Finset M} (h : finrank R M + 1 < t.card) :\n    \u2203 f : M \u2192 R, \u2211 e \u2208 t, f e \u2022 e = 0 \u2227 \u2211 e \u2208 t, f e = 0 \u2227 \u2203 x \u2208 t, f x \u2260 0", "lemma Module.finitePresentation_of_surjective [h : Module.FinitePresentation R M] (l : M \u2192\u2097[R] N)\n    (hl : Function.Surjective l) (hl' : (LinearMap.ker l).FG) :\n    Module.FinitePresentation R N", "theorem finite_def {R M} [Semiring R] [AddCommMonoid M] [Module R M] :\n    Finite R M \u2194 (\u22a4 : Submodule R M).FG", "theorem Module.finite_of_rank_eq_nat [Module.Free R M] {n : \u2115} (h : Module.rank R M = n) :\n    Module.Finite R M", "theorem Module.finrank_le_one_iff_top_isPrincipal [Module.Free K V] [Module.Finite K V] :\n    finrank K V \u2264 1 \u2194 (\u22a4 : Submodule K V).IsPrincipal", "theorem forall_dual_apply_eq_zero_iff (v : V) : (\u2200 \u03c6 : Module.Dual K V, \u03c6 v = 0) \u2194 v = 0", "theorem Module.free_of_finite_type_torsion_free [_root_.Finite \u03b9] {s : \u03b9 \u2192 M}\n    (hs : span R (range s) = \u22a4) [NoZeroSMulDivisors R M] : Module.Free R M", "lemma injOn_dualMap_subtype_span_range_range {\u03b9 : Type*} [NoZeroSMulDivisors \u2124 M]\n    {r : \u03b9 \u21aa M} {c : \u03b9 \u2192 Dual R M} (hfin : (range r).Finite)\n    (h_two : \u2200 i, c i (r i) = 2)\n    (h_mapsTo : \u2200 i, MapsTo (preReflection (r i) (c i)) (range r) (range r)) :\n    InjOn (span R (range r)).subtype.dualMap (range c)", "theorem Module.injective_iff_injective_object :\n    Module.Injective R Q \u2194\n    CategoryTheory.Injective (ModuleCat.of R Q)", "theorem isTorsionBy_quotient_iff (N : Submodule R M) (r : R) :\n    IsTorsionBy R (M\u29f8N) r \u2194 \u2200 x, r \u2022 x \u2208 N", "lemma preReflection_apply_self (h : f x = 2) :\n    preReflection x f x = - x", "@[simp]\nlemma reflection_symm (h : f x = 2) :\n    (reflection h).symm = reflection h", "theorem map'_comp {M\u2081 M\u2082 M\u2083 : ModuleCat.{v} R} (l\u2081\u2082 : M\u2081 \u27f6 M\u2082) (l\u2082\u2083 : M\u2082 \u27f6 M\u2083) :\n    map' f (l\u2081\u2082 \u226b l\u2082\u2083) = map' f l\u2081\u2082 \u226b map' f l\u2082\u2083", "private theorem colimitModule.add_smul (r s : R) (x : (M F)) : (r + s) \u2022 x = r \u2022 x + s \u2022 x", "theorem \u03bc_natural {X Y X' Y' : Type u} (f : X \u27f6 Y) (g : X' \u27f6 Y') :\n    ((free R).map f \u2297 (free R).map g) \u226b (\u03bc R Y Y').hom = (\u03bc R X X').hom \u226b (free R).map (f \u2297 g)", "private theorem associator_naturality_aux {X\u2081 X\u2082 X\u2083 : Type*} [AddCommMonoid X\u2081] [AddCommMonoid X\u2082]\n    [AddCommMonoid X\u2083] [Module R X\u2081] [Module R X\u2082] [Module R X\u2083] {Y\u2081 Y\u2082 Y\u2083 : Type*}\n    [AddCommMonoid Y\u2081] [AddCommMonoid Y\u2082] [AddCommMonoid Y\u2083] [Module R Y\u2081] [Module R Y\u2082]\n    [Module R Y\u2083] (f\u2081 : X\u2081 \u2192\u2097[R] Y\u2081) (f\u2082 : X\u2082 \u2192\u2097[R] Y\u2082) (f\u2083 : X\u2083 \u2192\u2097[R] Y\u2083) :\n    \u2191(assoc R Y\u2081 Y\u2082 Y\u2083) \u2218\u2097 map (map f\u2081 f\u2082) f\u2083 = map f\u2081 (map f\u2082 f\u2083) \u2218\u2097 \u2191(assoc R X\u2081 X\u2082 X\u2083)", "@[simp]\ntheorem hexagon_reverse (X Y Z : ModuleCat.{u} R) :\n    (\u03b1_ X Y Z).inv \u226b (braiding _ Z).hom \u226b (\u03b1_ Z X Y).inv =\n      X \u25c1 (Y.braiding Z).hom \u226b (\u03b1_ X Z Y).inv \u226b (X.braiding Z).hom \u25b7 Y", "@[simp, elementwise]\ntheorem cokernel_\u03c0_cokernelIsoRangeQuotient_hom :\n    cokernel.\u03c0 f \u226b (cokernelIsoRangeQuotient f).hom = f.range.mkQ", "theorem epi_iff_range_eq_top : Epi f \u2194 LinearMap.range f = \u22a4", "theorem ihom_ev_app (M N : ModuleCat.{u} R) :\n    (ihom.ev M).app N = TensorProduct.uncurry _ _ _ _ LinearMap.id.flip", "theorem ker_eq_bot_of_mono [Mono f] : LinearMap.ker f = \u22a5", "theorem mono_iff_ker_eq_bot : Mono f \u2194 LinearMap.ker f = \u22a5", "theorem monoidalClosed_pre_app {M N : ModuleCat.{u} R} (P : ModuleCat.{u} R) (f : N \u27f6 M) :\n    (MonoidalClosed.pre f).app P = LinearMap.lcomp R _ f", "lemma restrictScalarsComp'App_inv_apply (M : ModuleCat R\u2083) (x : M) :\n    (restrictScalarsComp'App f g gf hgf M).inv x = x", "lemma restrictScalarsId'App_inv_apply (M : ModuleCat R) (x : M) :\n    (restrictScalarsId'App f hf M).inv x = x", "@[simp]\ntheorem toKernelSubobject_arrow {M N : ModuleCat R} {f : M \u27f6 N} (x : LinearMap.ker f) :\n    (kernelSubobject f).arrow (toKernelSubobject x) = x.1", "@[simp]\ntheorem associator_hom_hom (X Y Z : Mon_ C) : (\u03b1_ X Y Z).hom.hom = (\u03b1_ X.X Y.X Z.X).hom", "@[to_additive (attr := simp)] theorem fst_apply_inl (x : M) : fst (inl x : M \u2217 N) = x", "@[to_additive (attr := simp)]\ntheorem lift_comp_inl (f : M \u2192* P) (g : N \u2192* P) : (lift f g).comp inl = f", "@[to_additive (attr := simp)]\ntheorem lift_swap (f : M \u2192* P) (g : N \u2192* P) (x : N \u2217 M) : lift f g (swap N M x) = lift g f x", "@[to_additive (attr := simp)]\ntheorem mk_of_inl (x : M) : (mk (of (.inl x)) : M \u2217 N) = inl x", "@[to_additive (attr := simp)]\ntheorem prod_mk_fst_snd (x : M \u2217 N) : (fst x, snd x) = toProd x", "@[simp]\ntheorem toList_getLast? {i j} (w : NeWord M i j) : w.toList.getLast? = Option.some \u27e8j, w.last\u27e9", "theorem mem_of_mem_equivPair_tail {i j : \u03b9} {w : Word M} (m : M i) :\n    (\u27e8i, m\u27e9 \u2208 (equivPair j w).tail.toList) \u2192 \u27e8i, m\u27e9 \u2208 w.toList", "theorem lift_word_ping_pong {i j k} (w : NeWord H i j) (hk : j \u2260 k) :\n    lift f w.prod \u2022 X k \u2286 X i", "theorem of_apply {i} (m : M i) : of m = Con.mk' _ (FreeMonoid.of <| Sigma.mk i m)", "theorem base_smul_eq_smul (h : H) (w : NormalWord d) :\n    base \u03c6 h \u2022 w = h \u2022 w", "theorem prod_smul_empty (w : NormalWord d) : w.prod \u2022 empty = w", "theorem rcons_injective {i : \u03b9} : Function.Injective (rcons (d := d) i)", "@[ext 1199]\ntheorem hom_ext {f g : PushoutI \u03c6 \u2192* K}\n    (h : \u2200 i, f.comp (of i : G i \u2192* _) = g.comp (of i : G i \u2192* _))\n    (hbase : f.comp (base \u03c6) = g.comp (base \u03c6)) : f = g", "@[to_additive\n      \"Given subsets `t` and `s` of an `AddMonoid M`, if `s \u2286 t`, the `AddSubmonoid`\n      of `M` generated by `s` is contained in the `AddSubmonoid` generated by `t`.\"]\ntheorem closure_mono {s t : Set M} (h : s \u2286 t) : Closure s \u2286 Closure t", "@[to_additive (attr := deprecated (since := \"2024-01-27\"))]\ntheorem exponent_dvd_of_forall_orderOf_dvd (n : \u2115) (h : \u2200 g : G, orderOf g \u2223 n) : exponent G \u2223 n", "@[to_additive]\ntheorem exponent_eq_max'_orderOf [Fintype G] :\n    exponent G = ((@Finset.univ G _).image orderOf).max' \u27e81, by simp\u27e9", "@[to_additive exponent_eq_zero_addOrder_zero]\ntheorem exponent_eq_zero_of_order_zero {g : G} (hg : orderOf g = 0) : exponent G = 0", "@[to_additive]\nlemma isTorsionFree_iff_torsion_eq_bot {G} [CommGroup G] :\n    IsTorsionFree G \u2194 CommGroup.torsion G = \u22a5", "@[simp] theorem domCongr_apply (e : G \u2243* H) (f : MonoidAlgebra A G) (h : H) :\n    domCongr k A e f h = f (e.symm h)", "theorem exists_finset_adjoin_eq_top [h : FiniteType R (MonoidAlgebra R M)] :\n    \u2203 G : Finset M, Algebra.adjoin R (of R M '' G) = \u22a4", "theorem exists_leftInverse_of_injective (f : V \u2192\u2097[MonoidAlgebra k G] W)\n    (hf : LinearMap.ker f = \u22a5) :\n    \u2203 g : W \u2192\u2097[MonoidAlgebra k G] V, g.comp f = LinearMap.id", "theorem mul_single_one_apply [MulOneClass G] (f : MonoidAlgebra k G) (r : k) (x : G) :\n    (HMul.hMul (\u03b2 := MonoidAlgebra k G) f (single 1 r)) x = f x * r", "theorem single_add (a : G) (b\u2081 b\u2082 : k) : single a (b\u2081 + b\u2082) = single a b\u2081 + single a b\u2082", "@[simp]\ntheorem single_mul_single [Mul G] {a\u2081 a\u2082 : G} {b\u2081 b\u2082 : k} :\n    single a\u2081 b\u2081 * single a\u2082 b\u2082 = single (a\u2081 * a\u2082) (b\u2081 * b\u2082)", "@[simp]\ntheorem sum_single_index {N} [AddCommMonoid N] {a : G} {b : k} {h : G \u2192 k \u2192 N}\n    (h_zero : h a 0 = 0) :\n    (single a b).sum h = h a b", "theorem support_gen_of_gen {S : Set (MonoidAlgebra R M)} (hS : Algebra.adjoin R S = \u22a4) :\n    Algebra.adjoin R (\u22c3 f \u2208 S, of R M '' (f.support : Set M)) = \u22a4", "theorem support_single_mul_subset [DecidableEq G] [Mul G] (f : MonoidAlgebra k G) (r : k) (a : G) :\n    (single a r * f : MonoidAlgebra k G).support \u2286 Finset.image (a * \u00b7) f.support", "@[to_additive (attr := simp)]\ntheorem MonoidHom.coe_coe [MonoidHomClass F M N] (f : F) : ((f : M \u2192* N) : M \u2192 N) = f", "@[to_additive (attr := simp)]\ntheorem compl\u2082_apply [MulOneClass M] [MulOneClass N] [CommMonoid P] [MulOneClass Q]\n    (f : M \u2192* N \u2192* P) (g : Q \u2192* N) (m : M) (q : Q) : (compl\u2082 f g) m q = f m (g q)", "@[to_additive (attr := simp)]\ntheorem coprod_comp_inl : (f.coprod g).comp (inl M N) = f", "@[to_additive]\ntheorem dfinsupp_prod_apply [Monoid R] [CommMonoid S] (f : \u03a0\u2080 i, \u03b2 i) (g : \u2200 i, \u03b2 i \u2192 R \u2192* S)\n    (r : R) : (f.prod g) r = f.prod fun a b => (g a b) r", "@[to_additive]\ntheorem MonoidHom.isOpenMap_of_sigmaCompact\n    {H : Type*} [Group H] [TopologicalSpace H] [BaireSpace H] [T2Space H] [ContinuousMul H]\n    (f : G \u2192* H) (hf : Function.Surjective f) (h'f : Continuous f) :\n    IsOpenMap f", "@[to_additive (attr := simp)]\ntheorem ker_one : (1 : G \u2192* M).ker = \u22a4", "@[to_additive (attr := simp)]\ntheorem ker_rangeRestrict (f : G \u2192* N) : ker (rangeRestrict f) = ker f", "@[to_additive \"If `f` is an additive monoid homomorphism then `f (a + b) = f a + f b`.\"]\nprotected theorem MonoidHom.map_mul [MulOneClass M] [MulOneClass N] (f : M \u2192* N) (a b : M) :\n    f (a * b) = f a * f b", "@[simp]\ntheorem map_neg (x : R) : f (-x) = f x", "@[to_additive (attr := simp)]\ntheorem noncommCoprod_comp_inl : (f.noncommCoprod g comm).comp (inl M N) = f", "@[to_additive (attr := simp)]\ntheorem noncommCoprod_inl_inr {M N : Type*} [Monoid M] [Monoid N]:\n    (inl M N).noncommCoprod (inr M N) commute_inl_inr = id (M \u00d7 N)", "@[to_additive]\ntheorem noncommPiCoprod_mrange :\n    MonoidHom.mrange (noncommPiCoprod \u03d5 hcomm) = \u2a06 i : \u03b9, MonoidHom.mrange (\u03d5 i)", "@[to_additive (attr := simp)]\ntheorem ofLeftInverse_symm_apply {f : G \u2192* N} {g : N \u2192* G} (h : Function.LeftInverse g f)\n    (x : f.range) : (ofLeftInverse h).symm x = g x", "@[to_additive (attr := simp)]\ntheorem one_comp [MulOneClass M] [MulOneClass N] [MulOneClass P] (f : M \u2192* N) :\n    (1 : N \u2192* P).comp f = 1", "@[to_additive (attr := simp) prod_unique]\ntheorem prod_unique (f : M \u2192* N \u00d7 P) : ((fst N P).comp f).prod ((snd N P).comp f) = f", "@[to_additive (attr := simp)]\ntheorem MonoidHom.toMulHom_coe [MulOneClass M] [MulOneClass N] (f : M \u2192* N) :\n    f.toMulHom.toFun = f", "@[to_additive]\ntheorem MonoidHomClass.isometry_iff_norm [MonoidHomClass \ud835\udcd5 E F] (f : \ud835\udcd5) :\n    Isometry f \u2194 \u2200 x, \u2016f x\u2016 = \u2016x\u2016", "@[ext]\ntheorem MonoidWithZeroHom.ext_nat {f g : \u2115 \u2192*\u2080 A} : (\u2200 {n : \u2115}, 0 < n \u2192 f n = g n) \u2192 f = g", "protected theorem Monotone.Ioc (hf : Monotone f) (hg : Antitone g) :\n    Antitone fun x => Ioc (f x) (g x)", "theorem Monotone.ciSup_mem_iInter_Icc_of_antitone [SemilatticeSup \u03b2] {f g : \u03b2 \u2192 \u03b1} (hf : Monotone f)\n    (hg : Antitone g) (h : f \u2264 g) : (\u2a06 n, f n) \u2208 \u22c2 n, Icc (f n) (g n)", "theorem Monotone.covariant_of_const' {\u03bc : N \u2192 N \u2192 N} [CovariantClass N N (swap \u03bc) (\u00b7 \u2264 \u00b7)]\n    (hf : Monotone f) (m : N) : Monotone (f <| \u03bc \u00b7 m)", "theorem image_upperBounds_subset_upperBounds_image : f '' upperBounds s \u2286 upperBounds (f '' s)", "lemma _root_.Monotone.map_finset_min' [LinearOrder \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : Monotone f) {s : Finset \u03b1}\n    (h : s.Nonempty) : f (s.min' h) = (s.image f).min' (h.image f)", "theorem Monotone.map_limsInf_of_continuousAt {F : Filter R} [NeBot F] {f : R \u2192 S}\n    (f_incr : Monotone f) (f_cont : ContinuousAt f F.limsInf)\n    (bdd_above : F.IsBounded (\u00b7 \u2264 \u00b7)", "theorem seq_le_seq (hf : Monotone f) (n : \u2115) (h\u2080 : x 0 \u2264 y 0) (hx : \u2200 k < n, x (k + 1) \u2264 f (x k))\n    (hy : \u2200 k < n, f (y k) \u2264 y (k + 1)) : x n \u2264 y n", "theorem seq_lt_seq_of_lt_of_le (hf : Monotone f) (n : \u2115) (h\u2080 : x 0 < y 0)\n    (hx : \u2200 k < n, x (k + 1) < f (x k)) (hy : \u2200 k < n, f (y k) \u2264 y (k + 1)) : x n < y n", "theorem Monotone.tendsto_alternating_series_of_tendsto_zero (hfa : Monotone f)\n    (hf0 : Tendsto f atTop (\ud835\udcdd 0)) :\n    \u2203 l, Tendsto (fun n \u21a6 \u2211 i \u2208 range n, (-1) ^ i * f i) atTop (\ud835\udcdd l)", "protected theorem MonotoneOn.Ici (hf : MonotoneOn f s) : AntitoneOn (fun x => Ici (f x)) s", "protected theorem MonotoneOn.Ioc (hf : MonotoneOn f s) (hg : AntitoneOn g s) :\n    AntitoneOn (fun x => Ioc (f x) (g x)) s", "theorem MonotoneOn.exists_monotone_extension (h : MonotoneOn f s) (hl : BddBelow (f '' s))\n    (hu : BddAbove (f '' s)) : \u2203 g : \u03b1 \u2192 \u03b2, Monotone g \u2227 EqOn f g s", "theorem MonotoneOn.locallyBoundedVariationOn {f : \u03b1 \u2192 \u211d} {s : Set \u03b1} (hf : MonotoneOn f s) :\n    LocallyBoundedVariationOn f s", "theorem map_bddBelow : (lowerBounds s \u2229 t).Nonempty \u2192 BddBelow (f '' s)", "theorem MonotoneOn.map_min (hf : MonotoneOn f s) (ha : a \u2208 s) (hb : b \u2208 s) : f (min a b) =\n    min (f a) (f b)", "lemma MonotoneOn.mapsTo_Icc (h : MonotoneOn f (Icc a b)) : MapsTo f (Icc a b) (Icc (f a) (f b))", "theorem _root_.MonotoneOn.set_prod (hf : MonotoneOn f s) (hg : MonotoneOn g s) :\n    MonotoneOn (fun x => f x \u00d7\u02e2 g x) s", "theorem MonotoneOn.union [Preorder \u03b2] {f g : \u03b2 \u2192 Set \u03b1} {s : Set \u03b2} (hf : MonotoneOn f s)\n    (hg : MonotoneOn g s) : MonotoneOn (fun x => f x \u222a g x) s", "lemma Monovary.div_left\u2080 (hf\u2081 : 0 \u2264 f\u2081) (hf\u2082 : StrongLT 0 f\u2082) (h\u2081 : Monovary f\u2081 g)\n    (h\u2082 : Antivary f\u2082 g) : Monovary (f\u2081 / f\u2082) g", "@[to_additive] lemma Monovary.div_right (h\u2081 : Monovary f g\u2081) (h\u2082 : Antivary f g\u2082) :\n    Monovary f (g\u2081 / g\u2082)", "theorem MonovaryOn.comp_antitoneOn_right (h : MonovaryOn f g s) (hg : AntitoneOn g' (g '' s)) :\n    AntivaryOn f (g' \u2218 g) s", "theorem MonovaryOn.dual_right : MonovaryOn f g s \u2192 AntivaryOn f (toDual \u2218 g) s", "theorem MonovaryOn.sum_comp_perm_mul_le_sum_mul (hfg : MonovaryOn f g s)\n    (h\u03c3 : {x | \u03c3 x \u2260 x} \u2286 s) : \u2211 i \u2208 s, f (\u03c3 i) * g i \u2264 \u2211 i \u2208 s, f i * g i", "theorem MonovaryOn.sum_smul_comp_perm_lt_sum_smul_iff (hfg : MonovaryOn f g s)\n    (h\u03c3 : {x | \u03c3 x \u2260 x} \u2286 s) :\n    \u2211 i \u2208 s, f i \u2022 g (\u03c3 i) < \u2211 i \u2208 s, f i \u2022 g i \u2194 \u00acMonovaryOn f (g \u2218 \u03c3) s", "theorem IsBlock.iff_top (B : Set X) :\n    IsBlock G B \u2194 IsBlock (\u22a4 : Subgroup G) B", "theorem IsBlock.mk_mem {B : Set X} :\n    IsBlock G B \u2194 \u2200 (g : G) (a : X) (_ : a \u2208 B) (_ : g \u2022 a \u2208 B), g \u2022 B = B", "@[to_additive]\ntheorem Supports.mono (h : s \u2286 t) (hs : Supports G s b) : Supports G t b", "protected theorem MulAction.bijective\u2080 (ha : a \u2260 0) : Function.Bijective (a \u2022 \u00b7 : \u03b2 \u2192 \u03b2)", "@[to_additive]\ntheorem fixedBy_mul (m\u2081 m\u2082 : M) : fixedBy \u03b1 m\u2081 \u2229 fixedBy \u03b1 m\u2082 \u2286 fixedBy \u03b1 (m\u2081 * m\u2082)", "@[to_additive]\nlemma le_stabilizer_smul_left [SMul \u03b1 \u03b2] [IsScalarTower G \u03b1 \u03b2] (a : \u03b1) (b : \u03b2) :\n    stabilizer G a \u2264 stabilizer G (a \u2022 b)", "@[to_additive (attr := simp)]\ntheorem mem_stabilizer_iff {a : \u03b1} {g : G} : g \u2208 stabilizer G a \u2194 g \u2022 a = a", "@[to_additive (attr := simp)]\ntheorem orbit.coe_smul {a : \u03b1} {m : M} {a' : orbit M a} : \u2191(m \u2022 a') = m \u2022 (a' : \u03b1)", "theorem orbit.pairwiseDisjoint :\n    (Set.range fun x : X => orbit G x).PairwiseDisjoint id", "@[to_additive (attr := simp)]\ntheorem orbitEquivQuotientStabilizer_symm_apply (b : \u03b2) (a : \u03b1) :\n    ((orbitEquivQuotientStabilizer \u03b1 b).symm a : \u03b2) = a \u2022 b", "@[to_additive]\nlemma orbitRel.Quotient.mem_subgroup_orbit_iff' {H : Subgroup G} {x : orbitRel.Quotient G \u03b1}\n    {a b : x.orbit} {c : \u03b1} (h : (\u27e6a\u27e7 : orbitRel.Quotient H x.orbit) = \u27e6b\u27e7) :\n    (a : \u03b1) \u2208 MulAction.orbit H c \u2194 (b : \u03b1) \u2208 MulAction.orbit H c", "@[to_additive]\nlemma orbitRel_le_snd :\n    orbitRel G (\u03b1 \u00d7 \u03b2) \u2264 (orbitRel G \u03b2).comap Prod.snd", "@[to_additive]\ntheorem orbit_eq_iff {a b : \u03b1} : orbit G a = orbit G b \u2194 a \u2208 orbit G b", "@[to_additive \"If a set `s : Set \u03b1` is in `fixedBy (Set \u03b1) g`, then all points of `s` will stay in\n`s` after being moved by `g`.\"]\ntheorem set_mem_fixedBy_iff (s : Set \u03b1) (g : G) :\n    s \u2208 fixedBy (Set \u03b1) g \u2194 \u2200 x, g \u2022 x \u2208 s \u2194 x \u2208 s", "@[to_additive]\ntheorem smul_inv_mem_fixedBy_iff_mem_fixedBy {a : \u03b1} {g : G} :\n    g\u207b\u00b9 \u2022 a \u2208 fixedBy \u03b1 g \u2194 a \u2208 fixedBy \u03b1 g", "theorem smul_mem_of_set_mem_fixedBy {s : Set \u03b1} {g : G} (s_in_fixedBy : s \u2208 fixedBy (Set \u03b1) g)\n    {x : \u03b1} : g \u2022 x \u2208 s \u2194 x \u2208 s", "lemma smul_orbit_eq_orbit_smul (N : Subgroup G) [nN : N.Normal] (a : X) (g : G) :\n    g \u2022 orbit N a = orbit N (g \u2022 a)", "protected theorem congr_fun {f g : X \u2192\u2091[\u03c6] Y} (h : f = g) (x : X) :\n    f x = g x", "@[simp]\ntheorem ofEq_apply {\u03c6' : M \u2192 N} (h : \u03c6 = \u03c6') (f : X \u2192\u2091[\u03c6] Y) (a : X) :\n    (f.ofEq h) a = f a", "protected lemma MulActionWithZero.subsingleton\n    [MulActionWithZero R M] [Subsingleton R] : Subsingleton M", "theorem IsQuadratic.inv {\u03c7 : MulChar R R'} (h\u03c7 : \u03c7.IsQuadratic) : \u03c7\u207b\u00b9 = \u03c7", "lemma one_apply {x : R} (hx : IsUnit x) : (1 : MulChar R R') x = 1 := one_apply_coe hx.unit", "protected lemma pow_card_eq_one [Fintype M\u02e3] (\u03c7 : MulChar M R) : \u03c7 ^ (Fintype.card M\u02e3) = 1", "@[simp]\ntheorem MulDistribMulAction.toMonoidHom_apply (r : M) (x : A) :\n    MulDistribMulAction.toMonoidHom A r x = r \u2022 x", "@[to_additive (attr := ext)\n  \"Two additive isomorphisms agree if they are defined by the same underlying function.\"]\ntheorem ext {f g : MulEquiv M N} (h : \u2200 x, f x = g x) : f = g", "protected theorem isDomain {A : Type*} (B : Type*) [Semiring A] [Semiring B] [IsDomain B]\n    (e : A \u2243* B) : IsDomain A", "@[to_additive]\ntheorem map_ne_one_iff (h : M \u2243* N) {x : M} : h x \u2260 1 \u2194 x \u2260 1", "@[to_additive]\ntheorem MulHom.cancel_right [Mul M] [Mul N] [Mul P] {g\u2081 g\u2082 : N \u2192\u2099* P} {f : M \u2192\u2099* N}\n    (hf : Function.Surjective f) : g\u2081.comp f = g\u2082.comp f \u2194 g\u2081 = g\u2082", "@[to_additive (attr := simp)]\ntheorem MulHom.coe_comp [Mul M] [Mul N] [Mul P] (g : N \u2192\u2099* P) (f : M \u2192\u2099* N) :\n    \u2191(g.comp f) = g \u2218 f", "@[to_additive]\ntheorem MulHom.coe_copy_eq {_ : Mul M} {_ : Mul N} (f : M \u2192\u2099* N) (f' : M \u2192 N) (h : f' = f) :\n    f.copy f' h = f", "@[to_additive prod_comp_prodMap]\ntheorem prod_comp_prodMap (f : P \u2192\u2099* M) (g : P \u2192\u2099* N) (f' : M \u2192\u2099* M') (g' : N \u2192\u2099* N') :\n    (f'.prodMap g').comp (f.prod g) = (f'.comp f).prod (g'.comp g)", "@[to_additive (attr := simp)]\ntheorem MulHom.toFun_eq_coe [Mul M] [Mul N] (f : M \u2192\u2099* N) : f.toFun = f", "@[to_additive]\nprotected theorem inj [Mul \u03b1] [PartialOrder \u03b1] {a b c : \u03b1} (ha : MulLECancellable a) :\n    a * b = a * c \u2194 b = c", "@[to_additive]\nprotected theorem le_mul_iff_one_le_left [MulOneClass \u03b1] [i : IsSymmOp \u03b1 \u03b1 (\u00b7 * \u00b7)]\n    [CovariantClass \u03b1 \u03b1 (\u00b7 * \u00b7) (\u00b7 \u2264 \u00b7)] {a b : \u03b1} (ha : MulLECancellable a) :\n    a \u2264 b * a \u2194 1 \u2264 b", "@[simp]\ntheorem coe_opLinearEquiv_symm_addEquiv :\n    ((opLinearEquiv R : M \u2243\u2097[R] M\u1d50\u1d52\u1d56).symm : M\u1d50\u1d52\u1d56 \u2243+ M) = opAddEquiv.symm", "@[to_additive (attr := simp)]\ntheorem comap_unop_nhds (x : M) : comap (unop : M\u1d50\u1d52\u1d56 \u2192 M) (\ud835\udcdd x) = \ud835\udcdd (op x)", "@[to_additive (attr := simp)]\nlemma nndist_op (x y : \u03b1) : nndist (op x) (op y) = nndist x y := rfl", "lemma norm_op [SeminormedAddGroup E] (a : E) : \u2016MulOpposite.op a\u2016 = \u2016a\u2016 := rfl", "theorem unop_smul_eq_unop_smul_unop [SMul M \u03b1] [SMul M\u1d50\u1d52\u1d56 \u03b1] [IsCentralScalar M \u03b1] (r : M\u1d50\u1d52\u1d56)\n    (a : \u03b1\u1d50\u1d52\u1d56) : unop (r \u2022 a) = unop r \u2022 unop a", "theorem MulPosMono.toMulPosStrictMono [MulPosMono \u03b1] : MulPosStrictMono \u03b1", "theorem alternatization_apply (m : MultilinearMap R (fun _ : \u03b9 => M) N') (v : \u03b9 \u2192 M) :\n    alternatization m v = \u2211 \u03c3 : Perm \u03b9, Equiv.Perm.sign \u03c3 \u2022 m.domDomCongr \u03c3 v", "theorem bound_of_shell_of_norm_map_coord_zero (hf\u2080 : \u2200 {m i}, \u2016m i\u2016 = 0 \u2192 \u2016f m\u2016 = 0)\n    {\u03b5 : \u03b9 \u2192 \u211d} {C : \u211d} (h\u03b5 : \u2200 i, 0 < \u03b5 i) {c : \u03b9 \u2192 \ud835\udd5c} (hc : \u2200 i, 1 < \u2016c i\u2016)\n    (hf : \u2200 m : \u2200 i, E i, (\u2200 i, \u03b5 i / \u2016c i\u2016 \u2264 \u2016m i\u2016) \u2192 (\u2200 i, \u2016m i\u2016 < \u03b5 i) \u2192 \u2016f m\u2016 \u2264 C * \u220f i, \u2016m i\u2016)\n    (m : \u2200 i, E i) : \u2016f m\u2016 \u2264 C * \u220f i, \u2016m i\u2016", "theorem coe_injective : Injective ((\u2191) : MultilinearMap R M\u2081 M\u2082 \u2192 (\u2200 i, M\u2081 i) \u2192 M\u2082)", "@[simp]\ntheorem compLinearMap_apply (g : MultilinearMap R M\u2081' M\u2082) (f : \u2200 i, M\u2081 i \u2192\u2097[R] M\u2081' i)\n    (m : \u2200 i, M\u2081 i) : g.compLinearMap f m = g fun i => f i (m i)", "@[simp]\ntheorem curryFinFinset_apply_const_aux {k l n : \u2115} {s : Finset (Fin n)} (hk : s.card = k)\n    (hl : s\u1d9c.card = l) (f : MultilinearMap R (fun _ : Fin n => M') M\u2082) (x y : M') :\n    (f fun i => Sum.elim (fun _ => x) (fun _ => y) ((\u21d1 (Equiv.symm (finSumEquivOfFinset hk hl))) i))\n      = f (s.piecewise (fun _ => x) fun _ => y)", "theorem curryFinFinset_symm_apply_piecewise_const {k l n : \u2115} {s : Finset (Fin n)} (hk : s.card = k)\n    (hl : s\u1d9c.card = l)\n    (f : MultilinearMap R (fun _ : Fin k => M') (MultilinearMap R (fun _ : Fin l => M') M\u2082))\n    (x y : M') :\n    (curryFinFinset R M\u2082 M' hk hl).symm f (s.piecewise (fun _ => x) fun _ => y) =\n      f (fun _ => x) fun _ => y", "lemma map_add_eq_map_add_linearDeriv_add [DecidableEq \u03b9] [Fintype \u03b9] (x h : (i : \u03b9) \u2192 M\u2081 i) :\n    f (x + h) = f x + f.linearDeriv x h +\n      \u2211 s \u2208 univ.powerset.filter (2 \u2264 \u00b7.card), f (s.piecewise h x)", "lemma map_add_sub_map_add_sub_linearDeriv [DecidableEq \u03b9] [Fintype \u03b9] (x h h' : (i : \u03b9) \u2192 M\u2081 i) :\n    f (x + h) - f (x + h') - f.linearDeriv x (h - h') =\n    \u2211 s \u2208 univ.powerset.filter (2 \u2264 \u00b7.card), (f (s.piecewise h x) - f (s.piecewise h' x))", "@[simp]\ntheorem map_zero [Nonempty \u03b9] : f 0 = 0", "@[simp]\ntheorem smulRight_apply (f : MultilinearMap R M\u2081 R) (z : M\u2082) (m : \u2200 i, M\u2081 i) :\n    f.smulRight z m = f m \u2022 z", "theorem sum_apply {\u03b1 : Type*} (f : \u03b1 \u2192 MultilinearMap R M\u2081 M\u2082) (m : \u2200 i, M\u2081 i) {s : Finset \u03b1} :\n    (\u2211 a \u2208 s, f a) m = \u2211 a \u2208 s, f a m", "@[to_additive]\ntheorem Multipliable.comp_injective {i : \u03b3 \u2192 \u03b2} (hf : Multipliable f) (hi : Injective i) :\n    Multipliable (f \u2218 i)", "@[to_additive]\ntheorem Multipliable.map_tprod [CommMonoid \u03b3] [TopologicalSpace \u03b3] [T2Space \u03b3] (hf : Multipliable f)\n    {G} [FunLike G \u03b1 \u03b3] [MonoidHomClass G \u03b1 \u03b3] (g : G) (hg : Continuous g) :\n    g (\u220f' i, f i) = \u220f' i, g (f i)", "@[to_additive]\ntheorem Multipliable.mul (hf : Multipliable f) (hg : Multipliable g) :\n    Multipliable fun b \u21a6 f b * g b", "@[symm]\ntheorem Disjoint.symm {s t : Multiset \u03b1} (d : Disjoint s t) : Disjoint t s", "theorem Ico_add_Ico_eq_Ico {a b c : \u03b1} (hab : a \u2264 b) (hbc : b \u2264 c) :\n    Ico a b + Ico b c = Ico a c", "theorem Ico_filter_le_of_right_le [DecidablePred (b \u2264 \u00b7)] :\n    ((Ico a b).filter fun x => b \u2264 x) = \u2205", "@[simp]\ntheorem Ico_filter_lt (a b c : \u03b1) : ((Ico a b).filter fun x => x < c) = Ico a (min b c)", "theorem Ico_filter_lt_of_right_le [DecidablePred (\u00b7 < c)] (hbc : b \u2264 c) :\n    ((Ico a b).filter fun x => x < c) = Ico a b", "theorem Ico_self : Ico a a = 0", "@[simp]\ntheorem Ioc_eq_zero_iff : Ioc a b = 0 \u2194 \u00aca < b", "theorem Ioo_self : Ioo a a = 0", "theorem Nodup.filter (p : \u03b1 \u2192 Prop) [DecidablePred p] {s} : Nodup s \u2192 Nodup (filter p s)", "theorem Nodup.ndinsert (a : \u03b1) : Nodup s \u2192 Nodup (ndinsert a s)", "@[simp]\ntheorem add_sigma (s t : Multiset \u03b1) (u : \u2200 a, Multiset (\u03c3 a)) :\n    (s + t).sigma u = s.sigma u + t.sigma u", "@[to_additive (attr := measurability)]\ntheorem Multiset.aemeasurable_prod (s : Multiset (\u03b1 \u2192 M)) (hs : \u2200 f \u2208 s, AEMeasurable f \u03bc) :\n    AEMeasurable (fun x => (s.map fun f : \u03b1 \u2192 M => f x).prod) \u03bc", "theorem antidiagonal_eq_map_powerset [DecidableEq \u03b1] (s : Multiset \u03b1) :\n    s.antidiagonal = s.powerset.map fun t \u21a6 (s - t, t)", "theorem bind_assoc {s : Multiset \u03b1} {f : \u03b1 \u2192 Multiset \u03b2} {g : \u03b2 \u2192 Multiset \u03b3} :\n    (s.bind f).bind g = s.bind fun a => (f a).bind g", "theorem card_Ico :\n    (Finset.Ico s t).card = \u220f i \u2208 s.toFinset \u222a t.toFinset, (t.count i + 1 - s.count i) - 1", "theorem card_Ico_eq_card_Icc_sub_one (a b : \u03b1) : card (Ico a b) = card (Icc a b) - 1", "@[simp]\ntheorem card_disjSum : Multiset.card (s.disjSum t) = Multiset.card s + Multiset.card t", "theorem card_eq_three {s : Multiset \u03b1} : card s = 3 \u2194 \u2203 x y z, s = {x, y, z}", "theorem card_sections {s : Multiset (Multiset \u03b1)} : card (Sections s) = prod (s.map card)", "@[simp]\ntheorem card_singleton (a : \u03b1) : card ({a} : Multiset \u03b1) = 1", "@[simp]\ntheorem coe_attach (l : List \u03b1) : @Eq (Multiset { x // x \u2208 l }) (@attach \u03b1 l) l.attach", "@[simp]\ntheorem coe_bind (l : List \u03b1) (f : \u03b1 \u2192 List \u03b2) : (@bind \u03b1 \u03b2 l fun a => f a) = l.bind f", "@[simp]\ntheorem coe_dedup (l : List \u03b1) : @dedup \u03b1 _ l = l.dedup", "@[simp]\ntheorem coe_foldr (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H : LeftCommutative f) (b : \u03b2) (l : List \u03b1) :\n    foldr f H b l = l.foldr f b", "theorem Multiset.coe_mk {x : \u03b1} {i : Fin (m.count x)} : \u2191(m.mkToType x i) = x", "@[simp]\ntheorem coe_sub (s t : List \u03b1) : (s - t : Multiset \u03b1) = (s.diff t : List \u03b1)", "@[simp]\ntheorem cons_inter_of_pos {a} (s : Multiset \u03b1) {t} : a \u2208 t \u2192 (a ::\u2098 s) \u2229 t = a ::\u2098 s \u2229 t.erase a", "@[simp]\ntheorem count_add (a : \u03b1) : \u2200 s t, count a (s + t) = count a s + count a t", "@[simp] lemma count_eq_zero {a : \u03b1} : count a s = 0 \u2194 a \u2209 s := count_ne_zero.not_right", "@[simp]\ntheorem count_inter (a : \u03b1) (s t : Multiset \u03b1) : count a (s \u2229 t) = min (count a s) (count a t)", "theorem count_map_eq_count' [DecidableEq \u03b2] (f : \u03b1 \u2192 \u03b2) (s : Multiset \u03b1) (hf : Function.Injective f)\n    (x : \u03b1) : (s.map f).count (f x) = s.count x", "theorem count_replicate (a b : \u03b1) (n : \u2115) : count a (replicate n b) = if a = b then n else 0", "theorem count_sum [DecidableEq \u03b1] {m : Multiset \u03b2} {f : \u03b2 \u2192 Multiset \u03b1} {a : \u03b1} :\n    count a (map f m).sum = sum (m.map fun b => count a <| f b)", "theorem count_sum' {s : Finset \u03b2} {a : \u03b1} {f : \u03b2 \u2192 Multiset \u03b1} :\n    count a (\u2211 x \u2208 s, f x) = \u2211 x \u2208 s, count a (f x)", "@[simp]\ntheorem count_univ [DecidableEq \u03b1] (a : \u03b1) : count a Finset.univ.val = 1", "@[simp]\ntheorem disjoint_cons_right {a : \u03b1} {s t : Multiset \u03b1} :\n    Disjoint s (a ::\u2098 t) \u2194 a \u2209 s \u2227 Disjoint s t", "theorem disjoint_finset_sum_right {\u03b2 : Type*} {i : Finset \u03b2} {f : \u03b2 \u2192 Multiset \u03b1}\n    {a : Multiset \u03b1} : Multiset.Disjoint a (i.sum f) \u2194 \u2200 b \u2208 i, Multiset.Disjoint a (f b)", "theorem eq_of_mem_map_const {b\u2081 b\u2082 : \u03b2} {l : List \u03b1} (h : b\u2081 \u2208 map (Function.const \u03b1 b\u2082) l) :\n    b\u2081 = b\u2082", "theorem eq_replicate_card {a : \u03b1} {s : Multiset \u03b1} : s = replicate (card s) a \u2194 \u2200 b \u2208 s, b = a", "theorem eq_zero_of_subset_zero {s : Multiset \u03b1} (h : s \u2286 0) : s = 0", "theorem erase_cons_tail_of_mem (h : a \u2208 s) :\n    (b ::\u2098 s).erase a = b ::\u2098 s.erase a", "theorem esymm_neg (s : Multiset R) (k : \u2115) : (map Neg.neg s).esymm k = (-1) ^ k * esymm s k", "theorem exists_smul_of_dvd_count (s : Multiset \u03b1) {k : \u2115}\n    (h : \u2200 a : \u03b1, a \u2208 s \u2192 k \u2223 Multiset.count a s) : \u2203 u : Multiset \u03b1, s = k \u2022 u", "theorem filterMap_filterMap (f : \u03b1 \u2192 Option \u03b2) (g : \u03b2 \u2192 Option \u03b3) (s : Multiset \u03b1) :\n    filterMap g (filterMap f s) = filterMap (fun x => (f x).bind g) s", "@[simp]\ntheorem filter_zero : filter p 0 = 0", "theorem finset_sum_eq_sup_iff_disjoint [DecidableEq \u03b1] {\u03b2 : Type*} {i : Finset \u03b2}\n    {f : \u03b2 \u2192 Multiset \u03b1} :\n    i.sum f = i.sup f \u2194\n      \u2200\u1d49 (x \u2208 i) (y \u2208 i), x \u2260 y \u2192 Multiset.Disjoint (f x) (f y)", "theorem fold_cons_right (b a : \u03b1) (s : Multiset \u03b1) : (a ::\u2098 s).fold op b = s.fold op b * a", "@[simp]\ntheorem fold_dedup_idem [DecidableEq \u03b1] [hi : Std.IdempotentOp op] (s : Multiset \u03b1) (b : \u03b1) :\n    (dedup s).fold op b = s.fold op b", "@[simp] lemma grade_eq (m : Multiset \u03b1) : grade \u2115 m = card m := rfl", "@[simp]\ntheorem inf_coe (l : List \u03b1) : inf (l : Multiset \u03b1) = l.foldr (\u00b7 \u2293 \u00b7) \u22a4", "theorem lcm_mono {s\u2081 s\u2082 : Multiset \u03b1} (h : s\u2081 \u2286 s\u2082) : s\u2081.lcm \u2223 s\u2082.lcm", "@[to_additive le_sum_of_subadditive_on_pred]\nlemma le_prod_of_submultiplicative_on_pred (f : \u03b1 \u2192 \u03b2)\n    (p : \u03b1 \u2192 Prop) (h_one : f 1 = 1) (hp_one : p 1)\n    (h_mul : \u2200 a b, p a \u2192 p b \u2192 f (a * b) \u2264 f a * f b) (hp_mul : \u2200 a b, p a \u2192 p b \u2192 p (a * b))\n    (s : Multiset \u03b1) (hps : \u2200 a, a \u2208 s \u2192 p a) : f s.prod \u2264 (s.map f).prod", "@[simp]\ntheorem length_toList (s : Multiset \u03b1) : s.toList.length = card s", "@[simp]\ntheorem lists_coe (l : List \u03b1) : lists (l : Multiset \u03b1) = l.permutations.toFinset", "theorem lt_replicate_succ {m : Multiset \u03b1} {x : \u03b1} {n : \u2115} :\n    m < replicate (n + 1) x \u2194 m \u2264 replicate n x", "@[simp]\ntheorem map_cons (f : \u03b1 \u2192 \u03b2) (a s) : map f (a ::\u2098 s) = f a ::\u2098 map f s", "theorem map_hcongr {\u03b2' : Type v} {m : Multiset \u03b1} {f : \u03b1 \u2192 \u03b2} {f' : \u03b1 \u2192 \u03b2'} (h : \u03b2 = \u03b2')\n    (hf : \u2200 a \u2208 m, HEq (f a) (f' a)) : HEq (map f m) (map f' m)", "theorem map_traverse {G : Type* \u2192 Type _} [Applicative G] [CommApplicative G] {\u03b1 \u03b2 \u03b3 : Type _}\n    (g : \u03b1 \u2192 G \u03b2) (h : \u03b2 \u2192 \u03b3) (x : Multiset \u03b1) :\n    Functor.map (Functor.map h) (traverse g x) = traverse (Functor.map h \u2218 g) x", "@[simp]\ntheorem map_union [DecidableEq \u03b2] {f : \u03b1 \u2192 \u03b2} (finj : Function.Injective f) {s t : Multiset \u03b1} :\n    map f (s \u222a t) = map f s \u222a map f t", "theorem Multiset.map_univ_coeEmbedding (m : Multiset \u03b1) :\n    (Finset.univ : Finset m).map m.coeEmbedding = m.toEnumFinset", "lemma max_le_of_forall_le {\u03b1 : Type*} [LinearOrder \u03b1] [OrderBot \u03b1] (l : Multiset \u03b1)\n    (n : \u03b1) (h : \u2200 x \u2208 l, x \u2264 n) : l.fold max \u22a5 \u2264 n", "@[simp]\ntheorem mem_coe {a : \u03b1} {l : List \u03b1} : a \u2208 (l : Multiset \u03b1) \u2194 a \u2208 l", "@[simp]\ntheorem mem_cons {a b : \u03b1} {s : Multiset \u03b1} : a \u2208 b ::\u2098 s \u2194 a = b \u2228 a \u2208 s", "theorem mem_cons_self (a : \u03b1) (s : Multiset \u03b1) : a \u2208 a ::\u2098 s", "theorem mem_disjSum : x \u2208 s.disjSum t \u2194 (\u2203 a, a \u2208 s \u2227 inl a = x) \u2228 \u2203 b, b \u2208 t \u2227 inr b = x", "@[simp]\ntheorem mem_filter {a : \u03b1} {s} : a \u2208 filter p s \u2194 a \u2208 s \u2227 p a", "@[simp]\ntheorem mem_filterMap (f : \u03b1 \u2192 Option \u03b2) (s : Multiset \u03b1) {b : \u03b2} :\n    b \u2208 filterMap f s \u2194 \u2203 a, a \u2208 s \u2227 f a = some b", "theorem mem_filter_of_mem {a : \u03b1} {l} (m : a \u2208 l) (h : p a) : a \u2208 filter p l", "theorem mem_map_of_mem (f : \u03b1 \u2192 \u03b2) {a : \u03b1} {s : Multiset \u03b1} (h : a \u2208 s) : f a \u2208 map f s", "@[simp]\ntheorem mem_powersetCardAux {n} {l : List \u03b1} {s} : s \u2208 powersetCardAux n l \u2194 s \u2264 \u2191l \u2227 card s = n", "@[simp] lemma mem_product : \u2200 {p : \u03b1 \u00d7 \u03b2}, p \u2208 @product \u03b1 \u03b2 s t \u2194 p.1 \u2208 s \u2227 p.2 \u2208 t\n  | (a, b) => by simp [product, and_left_comm]", "@[simp] lemma mem_sigma : \u2200 {p : \u03a3a, \u03c3 a}, p \u2208 @Multiset.sigma \u03b1 \u03c3 s t \u2194 p.1 \u2208 s \u2227 p.2 \u2208 t p.1\n  | \u27e8a, b\u27e9 => by simp [Multiset.sigma, and_assoc, and_left_comm]", "@[simp]\ntheorem mem_toFinset {a : \u03b1} {s : Multiset \u03b1} : a \u2208 s.toFinset \u2194 a \u2208 s", "theorem mk_mem_sym2_iff {m : Multiset \u03b1} {a b : \u03b1} :\n    s(a, b) \u2208 m.sym2 \u2194 a \u2208 m \u2227 b \u2208 m", "@[simp]\ntheorem ndinter_cons_of_not_mem {a : \u03b1} (s : Multiset \u03b1) {t : Multiset \u03b1} (h : a \u2209 t) :\n    ndinter (a ::\u2098 s) t = ndinter s t", "theorem ndunion_le {s t u : Multiset \u03b1} : ndunion s t \u2264 u \u2194 s \u2286 u \u2227 t \u2264 u", "@[simp] lemma nodup_bind :\n    Nodup (bind s f) \u2194 (\u2200 a \u2208 s, Nodup (f a)) \u2227 s.Pairwise fun a b => Disjoint (f a) (f b)", "@[simp]\ntheorem nodup_cons {a : \u03b1} {s : Multiset \u03b1} : Nodup (a ::\u2098 s) \u2194 a \u2209 s \u2227 Nodup s", "theorem nodup_iff_count_eq_one [DecidableEq \u03b1] : Nodup s \u2194 \u2200 a \u2208 s, count a s = 1", "theorem noncommFold_eq_fold (s : Multiset \u03b1) [Std.Commutative op] (a : \u03b1) :\n    noncommFold op s (fun x _ y _ _ => Std.Commutative.comm x y) a = fold op a s", "theorem noncommFoldr_cons (s : Multiset \u03b1) (a : \u03b1) (h h') (b : \u03b2) :\n    noncommFoldr f (a ::\u2098 s) h b = f a (noncommFoldr f s h' b)", "@[to_additive card_nsmul_le_sum]\nlemma pow_card_le_prod (h : \u2200 x \u2208 s, a \u2264 x) : a ^ card s \u2264 s.prod", "@[simp]\ntheorem powersetAux'_nil : powersetAux' (@nil \u03b1) = [0]", "@[simp]\ntheorem powersetCardAux_cons (n : \u2115) (a : \u03b1) (l : List \u03b1) :\n    powersetCardAux (n + 1) (a :: l) =\n      powersetCardAux (n + 1) l ++ List.map (cons a) (powersetCardAux n l)", "theorem powersetCard_coe (n) (l : List \u03b1) :\n    @powersetCard \u03b1 n l = ((sublistsLen n l).map (\u2191) : List (Multiset \u03b1))", "theorem powersetCard_le_powerset (n : \u2115) (s : Multiset \u03b1) : powersetCard n s \u2264 powerset s", "theorem powersetCard_mono (n : \u2115) {s t : Multiset \u03b1} (h : s \u2264 t) :\n    powersetCard n s \u2264 powersetCard n t", "@[to_additive]\ntheorem prod_eq_foldr (s : Multiset \u03b1) :\n    prod s = foldr (\u00b7 * \u00b7) (fun x y z => by simp [mul_left_comm]) 1 s", "@[to_additive] lemma prod_eq_one_iff : m.prod = 1 \u2194 \u2200 x \u2208 m, x = (1 : \u03b1)", "@[simp] lemma prod_eq_zero_iff : s.prod = 0 \u2194 (0 : \u03b1) \u2208 s", "@[to_additive]\ntheorem prod_hom_rel [CommMonoid \u03b2] (s : Multiset \u03b9) {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {f : \u03b9 \u2192 \u03b1} {g : \u03b9 \u2192 \u03b2}\n    (h\u2081 : r 1 1) (h\u2082 : \u2200 \u2983a b c\u2984, r b c \u2192 r (f a * b) (g a * c)) :\n    r (s.map f).prod (s.map g).prod", "@[to_additive]\ntheorem prod_hom\u2082 [CommMonoid \u03b2] [CommMonoid \u03b3] (s : Multiset \u03b9) (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3)\n    (hf : \u2200 a b c d, f (a * b) (c * d) = f a c * f b d) (hf' : f 1 1 = 1) (f\u2081 : \u03b9 \u2192 \u03b1)\n    (f\u2082 : \u03b9 \u2192 \u03b2) : (s.map fun i => f (f\u2081 i) (f\u2082 i)).prod = f (s.map f\u2081).prod (s.map f\u2082).prod", "@[to_additive sum_lt_sum_of_nonempty]\nlemma prod_lt_prod_of_nonempty' (hs : s \u2260 \u2205) (hfg : \u2200 i \u2208 s, f i < g i) :\n    (s.map f).prod < (s.map g).prod", "@[simp]\ntheorem product_singleton : ({a} : Multiset \u03b1) \u00d7\u02e2 ({b} : Multiset \u03b2) = {(a, b)}", "theorem range_subset {m n : \u2115} : range m \u2286 range n \u2194 m \u2264 n", "theorem rel_bind {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {p : \u03b3 \u2192 \u03b4 \u2192 Prop} {s t} {f : \u03b1 \u2192 Multiset \u03b3}\n    {g : \u03b2 \u2192 Multiset \u03b4} (h : (r \u21d2 Rel p) f g) (hst : Rel r s t) :\n    Rel p (s.bind f) (t.bind g)", "theorem rel_eq_refl {s : Multiset \u03b1} : Rel (\u00b7 = \u00b7) s s", "theorem replicate_add (m n : \u2115) (a : \u03b1) : replicate (m + n) a = replicate m a + replicate n a", "theorem self_mem_range_succ (n : \u2115) : n \u2208 range (n + 1)", "theorem sup_powerset_len {\u03b1 : Type*} [DecidableEq \u03b1] (x : Multiset \u03b1) :\n    (Finset.sup (Finset.range (card x + 1)) fun k => x.powersetCard k) = x.powerset", "theorem Multiset.support_sum_subset [AddCommMonoid M] (s : Multiset (\u03b9 \u2192\u2080 M)) :\n    s.sum.support \u2286 (s.map Finsupp.support).sup", "@[simp]\ntheorem sym2_eq_zero_iff {m : Multiset \u03b1} : m.sym2 = 0 \u2194 m = 0", "@[simp]\ntheorem toDFinsupp_inj : toDFinsupp s = toDFinsupp t \u2194 s = t", "theorem toFinsupp_eq_iff {s : Multiset \u03b1} {f : \u03b1 \u2192\u2080 \u2115} :\n    toFinsupp s = f \u2194 s = Finsupp.toMultiset f", "@[simp]\ntheorem toList_eq_singleton_iff {a : \u03b1} {m : Multiset \u03b1} : m.toList = [a] \u2194 m = {a}", "theorem union_le (h\u2081 : s \u2264 u) (h\u2082 : t \u2264 u) : s \u222a t \u2264 u", "theorem LiftR_def (x y : F \u03b1) :\n    LiftR' R x y \u2194\n      \u2203 u : F (Subtype_ R),\n        (TypeVec.prod.fst \u229a subtypeVal R) <$$> u = x \u2227\n          (TypeVec.prod.snd \u229a subtypeVal R) <$$> u = y", "theorem M.bisim_lemma {\u03b1 : TypeVec n} {a\u2081 : (mp P).A} {f\u2081 : (mp P).B a\u2081 \u27f9 \u03b1} {a' : P.A}\n    {f' : (P.B a').drop \u27f9 \u03b1} {f\u2081' : (P.B a').last \u2192 M P \u03b1}\n    (e\u2081 : M.dest P \u27e8a\u2081, f\u2081\u27e9 = \u27e8a', splitFun f' f\u2081'\u27e9) :\n    \u2203 (g\u2081' : _)(e\u2081' : PFunctor.M.dest a\u2081 = \u27e8a', g\u2081'\u27e9),\n      f' = M.pathDestLeft P e\u2081' f\u2081 \u2227\n        f\u2081' = fun x : (last P).B a' => \u27e8g\u2081' x, M.pathDestRight P e\u2081' f\u2081 x\u27e9", "theorem wMk_eq {\u03b1 : TypeVec n} (a : P.A) (f : P.last.B a \u2192 P.last.W) (g' : P.drop.B a \u27f9 \u03b1)\n    (g : \u2200 j : P.last.B a, P.WPath (f j) \u27f9 \u03b1) :\n    (P.wMk a g' fun i => \u27e8f i, g i\u27e9) = \u27e8\u27e8a, f\u27e9, P.wPathCasesOn g' g\u27e9", "theorem wRec_eq {\u03b1 : TypeVec n} {C : Type*}\n    (g : \u2200 a : P.A, P.drop.B a \u27f9 \u03b1 \u2192 (P.last.B a \u2192 P.W \u03b1) \u2192 (P.last.B a \u2192 C) \u2192 C) (a : P.A)\n    (f' : P.drop.B a \u27f9 \u03b1) (f : P.last.B a \u2192 P.W \u03b1) :\n    P.wRec g (P.wMk a f' f) = g a f' f fun i => P.wRec g (f i)", "theorem C_add : (C (a + a') : MvPolynomial \u03c3 R) = C a + C a'", "theorem C_eq_coe_nat (n : \u2115) : (C \u2191n : MvPolynomial \u03c3 R) = n", "theorem C_injective (\u03c3 : Type*) (R : Type*) [CommSemiring R] :\n    Function.Injective (C : R \u2192 MvPolynomial \u03c3 R)", "theorem C_mul_X_pow_eq_monomial {s : \u03c3} {a : R} {n : \u2115} :\n    C a * X s ^ n = monomial (Finsupp.single s n) a", "lemma eq_zero_of_forall_eval_eq_zero_of_le_card\n    (hF : F.IsHomogeneous n) (h : \u2200 r : \u03c3 \u2192 R, eval r F = 0) (hnR : n \u2264 #R) :\n    F = 0", "theorem rename_isHomogeneous {f : \u03c3 \u2192 \u03c4} (h : \u03c6.IsHomogeneous n):\n    (rename f \u03c6).IsHomogeneous n", "theorem add {w : \u03c3 \u2192 M} (h\u03c6 : IsWeightedHomogeneous w \u03c6 n) (h\u03c8 : IsWeightedHomogeneous w \u03c8 n) :\n    IsWeightedHomogeneous w (\u03c6 + \u03c8) n", "@[simp]\ntheorem aevalTower_C (x : R) : aevalTower g y (C x) = g x", "@[simp]\ntheorem aevalTower_ofId :\n    aevalTower (Algebra.ofId S A) = (aeval : (\u03c3 \u2192 A) \u2192 MvPolynomial \u03c3 S \u2192\u2090[S] A)", "theorem aeval_def (p : MvPolynomial \u03c3 R) : aeval f p = eval\u2082 (algebraMap R S\u2081) f p", "theorem aeval_map_algebraMap (x : \u03c3 \u2192 B) (p : MvPolynomial \u03c3 R) :\n    aeval x (map (algebraMap R A) p) = aeval x p", "@[simp]\ntheorem aeval_zero' (p : MvPolynomial \u03c3 R) :\n    aeval (fun _ => 0 : \u03c3 \u2192 S\u2081) p = algebraMap _ _ (constantCoeff p)", "@[simp]\nlemma algebraTensorAlgEquiv_symm_X (s : \u03c3) :\n    (algebraTensorAlgEquiv R A).symm (X s) = 1 \u2297\u209c X s", "@[simp]\ntheorem bind\u2081_X_left : bind\u2081 (X : \u03c3 \u2192 MvPolynomial \u03c3 R) = AlgHom.id R _", "@[nontriviality]\nlemma coeffs_eq_empty_of_subsingleton [Subsingleton R] (p : MvPolynomial \u03c3 R) : p.coeffs = \u2205", "theorem constantCoeff_monomial [DecidableEq \u03c3] (d : \u03c3 \u2192\u2080 \u2115) (r : R) :\n    constantCoeff (monomial d r) = if d = 0 then r else 0", "@[simp]\ntheorem counitNat_X (a : A) : counitNat A (X a) = a", "@[simp]\ntheorem counit_X (r : R) : counit R (X r) = r", "theorem degrees_C (a : R) : degrees (C a : MvPolynomial \u03c3 R) = 0", "theorem degrees_indicator (c : \u03c3 \u2192 K) :\n    degrees (indicator c) \u2264 \u2211 s : \u03c3, (Fintype.card K - 1) \u2022 {s}", "theorem divMonomial_zero (x : MvPolynomial \u03c3 R) : x /\u1d50\u1d52\u207f\u1d52\u1d50\u2071\u1d43\u02e1 0 = x", "theorem eq_zero_iff {p : MvPolynomial \u03c3 R} : p = 0 \u2194 \u2200 d, coeff d p = 0", "@[to_additive existing]\ntheorem eval_prod {\u03b9 : Type*} (s : Finset \u03b9) (f : \u03b9 \u2192 MvPolynomial \u03c3 R) (g : \u03c3 \u2192 R) :\n    eval g (\u220f i \u2208 s, f i) = \u220f i \u2208 s, eval g (f i)", "theorem eval\u2082_assoc (q : S\u2082 \u2192 MvPolynomial \u03c3 R) (p : MvPolynomial S\u2082 R) :\n    eval\u2082 f (fun t => eval\u2082 f g (q t)) p = eval\u2082 f g (eval\u2082 C q p)", "theorem eval\u2082_cast_comp (f : \u03c3 \u2192 \u03c4) (c : \u2124 \u2192+* R) (g : \u03c4 \u2192 R) (p : MvPolynomial \u03c3 \u2124) :\n    eval\u2082 c (g \u2218 f) p = eval\u2082 c g (rename f p)", "theorem exists_finset_rename\u2082 (p\u2081 p\u2082 : MvPolynomial \u03c3 R) :\n    \u2203 (s : Finset \u03c3) (q\u2081 q\u2082 : MvPolynomial s R), p\u2081 = rename (\u2191) q\u2081 \u2227 p\u2082 = rename (\u2191) q\u2082", "theorem finsum_weightedHomogeneousComponent :\n    (finsum fun m => weightedHomogeneousComponent w m \u03c6) = \u03c6", "theorem frobenius_zmod (f : MvPolynomial \u03c3 (ZMod p)) : frobenius _ p f = expand p f", "private theorem funext_fin {n : \u2115} {p : MvPolynomial (Fin n) R}\n    (h : \u2200 x : Fin n \u2192 R, eval x p = 0) : p = 0", "theorem homogeneousComponent_eq_zero (h : \u03c6.totalDegree < n) : homogeneousComponent n \u03c6 = 0", "theorem induction_on_monomial {M : MvPolynomial \u03c3 R \u2192 Prop} (h_C : \u2200 a, M (C a))\n    (h_X : \u2200 p n, M p \u2192 M (p * X n)) : \u2200 s a, M (monomial s a)", "lemma _root_.MvPolynomial.isHomogeneous_X_pow (i : \u03c3) (n : \u2115) :\n    (X (R := R) i ^ n).IsHomogeneous n", "lemma isLocalization_C_mk' (a : R) (m : M) :\n    C (IsLocalization.mk' S a m) = IsLocalization.mk' (MvPolynomial \u03c3 S) (C (\u03c3 := \u03c3) a)\n      \u27e8C m, Submonoid.mem_map_of_mem C m.property\u27e9", "@[ext]\ntheorem linearMap_ext {M : Type*} [AddCommMonoid M] [Module R M] {f g : MvPolynomial \u03c3 R \u2192\u2097[R] M}\n    (h : \u2200 s, f \u2218\u2097 monomial s = g \u2218\u2097 monomial s) : f = g", "theorem map_map [CommSemiring S\u2082] (g : S\u2081 \u2192+* S\u2082) (p : MvPolynomial \u03c3 R) :\n    map g (map f p) = map (g.comp f) p", "theorem map_mapRange_eq_iff (f : R \u2192+* S\u2081) (g : S\u2081 \u2192 R) (hg : g 0 = 0) (\u03c6 : MvPolynomial \u03c3 S\u2081) :\n    map f (Finsupp.mapRange g hg \u03c6) = \u03c6 \u2194 \u2200 d, f (g (coeff d \u03c6)) = coeff d \u03c6", "theorem map_mvPolynomial_eq_eval\u2082 {S : Type*} [CommRing S] [Finite \u03c3] (\u03d5 : MvPolynomial \u03c3 R \u2192+* S)\n    (p : MvPolynomial \u03c3 R) :\n    \u03d5 p = MvPolynomial.eval\u2082 (\u03d5.comp MvPolynomial.C) (fun s => \u03d5 (MvPolynomial.X s)) p", "theorem map_restrict_dom_eval\u2097 : (restrictDegree \u03c3 K (Fintype.card K - 1)).map (eval\u2097 K \u03c3) = \u22a4", "theorem mem_supported : p \u2208 supported R s \u2194 \u2191p.vars \u2286 s", "theorem noZeroDivisors_of_finite (R : Type u) (\u03c3 : Type v) [CommSemiring R] [Finite \u03c3]\n    [NoZeroDivisors R] : NoZeroDivisors (MvPolynomial \u03c3 R)", "theorem pderiv_C {i : \u03c3} : pderiv i (C a) = 0", "@[simp]\ntheorem pderiv_X_self (i : \u03c3) : pderiv i (X i : MvPolynomial \u03c3 R) = 1", "theorem pderiv_eq_zero_of_not_mem_vars {i : \u03c3} {f : MvPolynomial \u03c3 R} (h : i \u2209 f.vars) :\n    pderiv i f = 0", "theorem polynomial_eval_eval\u2082 [CommSemiring R] [CommSemiring S]\n    {x : S} (f : R \u2192+* Polynomial S) (g : \u03c3 \u2192 Polynomial S) (p : MvPolynomial \u03c3 R) :\n    Polynomial.eval x (eval\u2082 f g p) =\n      eval\u2082 ((Polynomial.evalRingHom x).comp f) (fun s => Polynomial.eval x (g s)) p", "@[simp]\ntheorem psum_zero : psum \u03c3 R 0 = Fintype.card \u03c3", "theorem quotient_map_C_eq_zero {I : Ideal R} {i : R} (hi : i \u2208 I) :\n    (Ideal.Quotient.mk (Ideal.map (C : R \u2192+* MvPolynomial \u03c3 R) I :\n      Ideal (MvPolynomial \u03c3 R))).comp C i = 0", "theorem radical_le_vanishingIdeal_zeroLocus (I : Ideal (MvPolynomial \u03c3 k)) :\n    I.radical \u2264 vanishingIdeal (zeroLocus I)", "theorem rename_comp_bind\u2081 {\u03c5 : Type*} (f : \u03c3 \u2192 MvPolynomial \u03c4 R) (g : \u03c4 \u2192 \u03c5) :\n    (rename g).comp (bind\u2081 f) = bind\u2081 fun i => rename g <| f i", "@[simp]\ntheorem rename_expand (f : \u03c3 \u2192 \u03c4) (p : \u2115) (\u03c6 : MvPolynomial \u03c3 R) :\n    rename f (expand p \u03c6) = expand p (rename f \u03c6)", "@[simp]\ntheorem sum_C {A : Type*} [AddCommMonoid A] {b : (\u03c3 \u2192\u2080 \u2115) \u2192 R \u2192 A} (w : b 0 0 = 0) :\n    sum (C a) b = b 0 a", "theorem sum_homogeneousComponent :\n    (\u2211 i \u2208 range (\u03c6.totalDegree + 1), homogeneousComponent i \u03c6) = \u03c6", "@[simp]\ntheorem support_X_mul (s : \u03c3) (p : MvPolynomial \u03c3 R) :\n    (X s * p).support = p.support.map (addLeftEmbedding (Finsupp.single s 1))", "theorem support_esymm' (n : \u2115) [DecidableEq \u03c3] [Nontrivial R] :\n    (esymm \u03c3 R n).support =\n      (powersetCard n (univ : Finset \u03c3)).biUnion fun t => {\u2211 i \u2208 t, Finsupp.single i 1}", "theorem support_esymm'' (n : \u2115) [DecidableEq \u03c3] [Nontrivial R] :\n    (esymm \u03c3 R n).support =\n      (powersetCard n (univ : Finset \u03c3)).biUnion fun t =>\n        (Finsupp.single (\u2211 i \u2208 t, Finsupp.single i 1) (1 : R)).support", "theorem totalDegree_finset_prod {\u03b9 : Type*} (s : Finset \u03b9) (f : \u03b9 \u2192 MvPolynomial \u03c3 R) :\n    (s.prod f).totalDegree \u2264 \u2211 i \u2208 s, (f i).totalDegree", "theorem totalDegree_finset_sum {\u03b9 : Type*} (s : Finset \u03b9) (f : \u03b9 \u2192 MvPolynomial \u03c3 R) :\n    (s.sum f).totalDegree \u2264 Finset.sup s fun i => (f i).totalDegree", "theorem totalDegree_pow (a : MvPolynomial \u03c3 R) (n : \u2115) :\n    (a ^ n).totalDegree \u2264 n * a.totalDegree", "@[simp]\ntheorem vanishingIdeal_zeroLocus_eq_radical (I : Ideal (MvPolynomial \u03c3 k)) :\n    vanishingIdeal (zeroLocus I) = I.radical", "theorem vars_rename [DecidableEq \u03c4] (f : \u03c3 \u2192 \u03c4) (\u03c6 : MvPolynomial \u03c3 R) :\n    (rename f \u03c6).vars \u2286 \u03c6.vars.image f", "theorem weightedHomogeneousComponent_eq_zero [SemilatticeSup M] [OrderBot M]\n    (h : weightedTotalDegree w \u03c6 < n) : weightedHomogeneousComponent w n \u03c6 = 0", "theorem weightedHomogeneousComponent_weighted_homogeneous_polynomial [DecidableEq M] (m n : M)\n    (p : MvPolynomial \u03c3 R) (h : p \u2208 weightedHomogeneousSubmodule R w n) :\n    weightedHomogeneousComponent w m p = if m = n then p else 0", "theorem weightedTotalDegree_zero (w : \u03c3 \u2192 M) :\n    weightedTotalDegree w (0 : MvPolynomial \u03c3 R) = \u22a5", "@[simp]\ntheorem zeroLocus_top : zeroLocus (\u22a4 : Ideal (MvPolynomial \u03c3 k)) = \u22a5", "lemma zero_not_mem_coeffs (p : MvPolynomial \u03c3 R) : 0 \u2209 p.coeffs", "protected theorem add_mul (\u03c6\u2081 \u03c6\u2082 \u03c6\u2083 : MvPowerSeries \u03c3 R) : (\u03c6\u2081 + \u03c6\u2082) * \u03c6\u2083 = \u03c6\u2081 * \u03c6\u2083 + \u03c6\u2082 * \u03c6\u2083", "theorem algebraMap_apply' (p : MvPolynomial \u03c3 R) :\n    algebraMap (MvPolynomial \u03c3 R) (MvPowerSeries \u03c3 A) p = map \u03c3 (algebraMap R A) p", "theorem coeff_X [DecidableEq \u03c3] (n : \u03c3 \u2192\u2080 \u2115) (s : \u03c3) :\n    coeff R n (X s : MvPowerSeries \u03c3 R) = if n = single s 1 then 1 else 0", "theorem coeff_invOfUnit [DecidableEq \u03c3] (n : \u03c3 \u2192\u2080 \u2115) (\u03c6 : MvPowerSeries \u03c3 R) (u : R\u02e3) :\n    coeff R n (invOfUnit \u03c6 u) =\n      if n = 0 then \u2191u\u207b\u00b9\n      else\n        -\u2191u\u207b\u00b9 *\n          \u2211 x \u2208 antidiagonal n,\n            if x.2 < n then coeff R x.1 \u03c6 * coeff R x.2 (invOfUnit \u03c6 u) else 0", "theorem coeff_mul [DecidableEq \u03c3] :\n    coeff R n (\u03c6 * \u03c8) = \u2211 p \u2208 antidiagonal n, coeff R p.1 \u03c6 * coeff R p.2 \u03c8", "theorem coeff_prod [DecidableEq \u03c3]\n    (f : \u03b9 \u2192 MvPowerSeries \u03c3 R) (d : \u03c3 \u2192\u2080 \u2115) (s : Finset \u03b9) :\n    coeff R d (\u220f j \u2208 s, f j) =\n      \u2211 l \u2208 finsuppAntidiag s d,\n        \u220f i \u2208 s, coeff R (l i) (f i)", "@[simp]\nprotected theorem inv_mul_cancel (\u03c6 : MvPowerSeries \u03c3 k) (h : constantCoeff \u03c3 k \u03c6 \u2260 0) :\n    \u03c6\u207b\u00b9 * \u03c6 = 1", "@[simp]\ntheorem map_C (a : R) : map \u03c3 f (C \u03c3 R a) = C \u03c3 S (f a)", "theorem mul_invOfUnit (\u03c6 : MvPowerSeries \u03c3 R) (u : R\u02e3) (h : constantCoeff \u03c3 R \u03c6 = u) :\n    \u03c6 * invOfUnit \u03c6 u = 1", "private theorem Cofix.bisim_aux {\u03b1 : TypeVec n} (r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop) (h' : \u2200 x, r x x)\n    (h : \u2200 x y, r x y \u2192\n      appendFun id (Quot.mk r) <$$> Cofix.dest x = appendFun id (Quot.mk r) <$$> Cofix.dest y) :\n    \u2200 x y, r x y \u2192 x = y", "theorem Cofix.bisim\u2082 {\u03b1 : TypeVec n} (r : Cofix F \u03b1 \u2192 Cofix F \u03b1 \u2192 Prop)\n    (h : \u2200 x y, r x y \u2192 LiftR' (RelLast' \u03b1 r) (Cofix.dest x) (Cofix.dest y)) :\n    \u2200 x y, r x y \u2192 x = y", "theorem Fix.rec_eq {\u03b2 : Type u} (g : F (append1 \u03b1 \u03b2) \u2192 \u03b2) (x : F (append1 \u03b1 (Fix F \u03b1))) :\n    Fix.rec g (Fix.mk x) = g (appendFun id (Fix.rec g) <$$> x)", "theorem wEquiv.symm {\u03b1 : TypeVec n} (x y : q.P.W \u03b1) : WEquiv x y \u2192 WEquiv y x", "@[norm_cast] lemma cast_divNat_of_ne_zero (a : \u2115) {b : \u2115} (hb : (b : \u03b1) \u2260 0) :\n    divNat a b = (a / b : \u03b1)", "theorem cast_zpow_of_ne_zero {K} [DivisionSemiring K] (q : \u211a\u22650) (z : \u2124) (hq : (q.num : K) \u2260 0) :\n    NNRat.cast (q ^ z) = (NNRat.cast q : K) ^ z", "@[norm_cast] theorem coe_le_coe : (p : \u211a) \u2264 q \u2194 p \u2264 q", "@[simp, norm_cast] lemma coe_natCast (n : \u2115) : (\u2191(\u2191n : \u211a\u22650) : \u211a) = n := rfl", "lemma natCast_eq_divNat (n : \u2115) : (n : \u211a\u22650) = divNat n 1 := (num_divNat_den _).symm", "@[simp]\ntheorem preimage_cast_Icc (p q : \u211a\u22650) : (\u2191) \u207b\u00b9' Icc (p : K) q = Icc p q", "@[simp]\ntheorem toNNRat_coe_nat (n : \u2115) : toNNRat n = n", "theorem toNNRat_sum_of_nonneg {s : Finset \u03b1} {f : \u03b1 \u2192 \u211a} (hf : \u2200 a, a \u2208 s \u2192 0 \u2264 f a) :\n    (\u2211 a \u2208 s, f a).toNNRat = \u2211 a \u2208 s, (f a).toNNRat", "protected lemma inv_inv (ha : a \u2260 0) (hb : b \u2260 0) (hab : a + b = 1) :\n    a\u207b\u00b9.IsConjExponent b\u207b\u00b9", "theorem bot_eq_zero : (\u22a5 : \u211d\u22650) = 0", "@[simp, norm_cast] lemma coe_le_one : (r : \u211d) \u2264 1 \u2194 r \u2264 1", "@[simp, norm_cast] lemma coe_lt_coe : (r\u2081 : \u211d) < r\u2082 \u2194 r\u2081 < r\u2082 := Iff.rfl", "@[simp, norm_cast]\nprotected theorem coe_ofScientific (m : \u2115) (s : Bool) (e : \u2115) :\n    \u2191(OfScientific.ofScientific m s e : \u211d\u22650) = (OfScientific.ofScientific m s e : \u211d)", "@[norm_cast]\ntheorem coe_prod {\u03b1} {s : Finset \u03b1} {f : \u03b1 \u2192 \u211d\u22650} : \u2191(\u220f a \u2208 s, f a) = \u220f a \u2208 s, (f a : \u211d)", "theorem div_le_div_left_of_le {a b c : \u211d\u22650} (c0 : c \u2260 0) (cb : c \u2264 b) :\n    a / b \u2264 a / c", "nonrec theorem div_le_iff' {a b r : \u211d\u22650} (hr : r \u2260 0) : a / r \u2264 b \u2194 a \u2264 r * b", "theorem div_le_of_le_mul' {a b c : \u211d\u22650} (h : a \u2264 b * c) : a / b \u2264 c", "theorem div_lt_iff {a b r : \u211d\u22650} (hr : r \u2260 0) : a / r < b \u2194 a < b * r", "protected theorem \u00abforall\u00bb {p : \u211d\u22650 \u2192 Prop} :\n    (\u2200 x : \u211d\u22650, p x) \u2194 \u2200 (x : \u211d) (hx : 0 \u2264 x), p \u27e8x, hx\u27e9", "theorem indicator_summable {f : \u03b1 \u2192 \u211d\u22650} (hf : Summable f) (s : Set \u03b1) :\n    Summable (s.indicator f)", "theorem inv_lt_inv {x y : \u211d\u22650} (hx : x \u2260 0) (h : x < y) : y\u207b\u00b9 < x\u207b\u00b9", "theorem le_iInf_mul {a : \u211d\u22650} {g : \u03b9 \u2192 \u211d\u22650} {h : \u211d\u22650} (H : \u2200 i, a \u2264 g i * h) : a \u2264 iInf g * h", "@[simp]\nlemma NNReal.nndist_zero_eq_val (z : \u211d\u22650) : nndist 0 z = z", "@[simp]\nlemma NNReal.nndist_zero_eq_val' (z : \u211d\u22650) : nndist z 0 = z", "@[simp, norm_cast] lemma one_le_coe : 1 \u2264 (r : \u211d) \u2194 1 \u2264 r", "@[simp]\ntheorem orderIsoIccZeroCoe_symm_apply_coe (a : \u211d\u22650) (b : Set.Iic a) :\n    ((orderIsoIccZeroCoe a).symm b : \u211d) = b", "theorem rpow_add_rpow_le {p q : \u211d} (a b : \u211d\u22650) (hp_pos : 0 < p) (hpq : p \u2264 q) :\n    (a ^ q + b ^ q) ^ (1 / q) \u2264 (a ^ p + b ^ p) ^ (1 / p)", "theorem rpow_sum_le_const_mul_sum_rpow (f : \u03b9 \u2192 \u211d\u22650) {p : \u211d} (hp : 1 \u2264 p) :\n    (\u2211 i \u2208 s, f i) ^ p \u2264 (card s : \u211d\u22650) ^ (p - 1) * \u2211 i \u2208 s, f i ^ p", "@[simp]\ntheorem sInf_empty : sInf (\u2205 : Set \u211d\u22650) = 0", "nonrec theorem sSup_of_not_bddAbove {s : Set \u211d\u22650} (hs : \u00acBddAbove s) : SupSet.sSup s = 0", "@[simp] lemma sq_sqrt (x : \u211d\u22650) : sqrt x ^ 2 = x := sqrt.symm_apply_apply _", "@[simp] lemma sqrt_eq_one : sqrt x = 1 \u2194 x = 1", "@[simp] lemma sqrt_one : sqrt 1 = 1", "lemma sum_mul_le_sqrt_mul_sqrt (s : Finset \u03b9) (f g : \u03b9 \u2192 \u211d\u22650) :\n    \u2211 i \u2208 s, f i * g i \u2264 sqrt (\u2211 i \u2208 s, f i ^ 2) * sqrt (\u2211 i \u2208 s, g i ^ 2)", "theorem summable_Lp_add {f g : \u03b9 \u2192 \u211d\u22650} {p : \u211d} (hp : 1 \u2264 p) (hf : Summable fun i => f i ^ p)\n    (hg : Summable fun i => g i ^ p) : Summable fun i => (f i + g i) ^ p", "theorem summable_iff_not_tendsto_nat_atTop {f : \u2115 \u2192 \u211d\u22650} :\n    Summable f \u2194 \u00acTendsto (fun n : \u2115 => \u2211 i \u2208 Finset.range n, f i) atTop atTop", "theorem tendsto_coe' {f : Filter \u03b1} [NeBot f] {m : \u03b1 \u2192 \u211d\u22650} {x : \u211d} :\n    Tendsto (fun a => m a : \u03b1 \u2192 \u211d) f (\ud835\udcdd x) \u2194 \u2203 hx : 0 \u2264 x, Tendsto m f (\ud835\udcdd \u27e8x, hx\u27e9)", "theorem tendsto_sum_nat_add (f : \u2115 \u2192 \u211d\u22650) : Tendsto (fun i => \u2211' k, f (k + i)) atTop (\ud835\udcdd 0)", "@[simp]\ntheorem zero_rpow {x : \u211d} (h : x \u2260 0) : (0 : \u211d\u22650) ^ x = 0", "theorem Coprime.eq_one_of_dvd {k m : Nat} (H : Coprime k m) (d : k \u2223 m) : k = 1", "theorem Coprime.pow_left (n : Nat) (H1 : Coprime m k) : Coprime (m ^ n) k", "theorem Ico_filter_coprime_le {a : \u2115} (k n : \u2115) (a_pos : 0 < a) :\n    ((Ico k (k + n)).filter (Coprime a)).card \u2264 totient a * (n / a + 1)", "@[simp]\ntheorem Ico_zero_eq_range : Ico 0 = range", "theorem Ioc_filter_dvd_card_eq_div (n p : \u2115) : ((Ioc 0 n).filter fun x => p \u2223 x).card = n / p", "lemma Ioc_filter_modEq_cast {v : \u2115} : ((Ioc a b).filter (\u00b7 \u2261 v [MOD r])).map castEmbedding =\n    (Ioc (a : \u2124) (b : \u2124)).filter (\u00b7 \u2261 v [ZMOD r])", "protected theorem mul_right_cancel' {a b c m : \u2115} (hc : c \u2260 0) :\n    a * c \u2261 b * c [MOD m * c] \u2192 a \u2261 b [MOD m]", "@[refl]\nprotected theorem refl (a : \u2115) : a \u2261 a [MOD n]", "@[symm]\nprotected theorem symm : a \u2261 b [MOD n] \u2192 b \u2261 a [MOD n]", "theorem encode_lt_comp (cf cg) :\n    encode cf < encode (comp cf cg) \u2227 encode cg < encode (comp cf cg)", "@[simp]\ntheorem eval_curry (c n x) : eval (curry c n) x = eval c (Nat.pair n x)", "theorem evaln_bound : \u2200 {k c n x}, x \u2208 evaln k c n \u2192 n < k", "theorem exists_code {f : \u2115 \u2192. \u2115} : Nat.Partrec f \u2194 \u2203 c : Code, eval c = f", "theorem rec_prim' {\u03b1 \u03c3} [Primcodable \u03b1] [Primcodable \u03c3] {c : \u03b1 \u2192 Code} (hc : Primrec c) {z : \u03b1 \u2192 \u03c3}\n    (hz : Primrec z) {s : \u03b1 \u2192 \u03c3} (hs : Primrec s) {l : \u03b1 \u2192 \u03c3} (hl : Primrec l) {r : \u03b1 \u2192 \u03c3}\n    (hr : Primrec r) {pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3} (hpr : Primrec\u2082 pr)\n    {co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3} (hco : Primrec\u2082 co) {pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3}\n    (hpc : Primrec\u2082 pc) {rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3} (hrf : Primrec\u2082 rf) :\n    let PR (a) cf cg hf hg := pr a (cf, cg, hf, hg)\n    let CO (a) cf cg hf hg := co a (cf, cg, hf, hg)\n    let PC (a) cf cg hf hg := pc a (cf, cg, hf, hg)\n    let RF (a) cf hf := rf a (cf, hf)\n    let F (a : \u03b1) (c : Code) : \u03c3", "theorem Prime.eq_two_or_odd' {p : \u2115} (hp : Prime p) : p = 2 \u2228 Odd p", "theorem multiplicity_choose' {p n k b : \u2115} (hp : p.Prime) (hnb : log p (n + k) < b) :\n    multiplicity p (choose (n + k) k) =\n      ((Ico 1 b).filter fun i => p ^ i \u2264 k % p ^ i + n % p ^ i).card", "theorem Prime.pow_eq_iff {p a k : \u2115} (hp : p.Prime) : a ^ k = p \u2194 a = p \u2227 k = 1", "@[to_additive (attr := simp)]\ntheorem Prime.prod_properDivisors {\u03b1 : Type*} [CommMonoid \u03b1] {p : \u2115} {f : \u2115 \u2192 \u03b1} (h : p.Prime) :\n    \u220f x \u2208 p.properDivisors, f x = f 1", "theorem sub_one_mul_multiplicity_factorial {n p : \u2115} (hp : p.Prime) :\n     (p - 1) * (multiplicity p n !).get (finite_nat_iff.mpr \u27e8hp.ne_one, factorial_pos n\u27e9) =\n     n - (p.digits n).sum", "theorem add : @Primrec' 2 fun v => v.head + v.tail.head", "theorem prim_iff\u2082 {f : \u2115 \u2192 \u2115 \u2192 \u2115} : (@Primrec' 2 fun v => f v.head v.tail.head) \u2194 Primrec\u2082 f", "theorem sqrt : @Primrec' 1 fun v => v.head.sqrt", "protected theorem swap : Nat.Primrec (unpaired (swap Nat.pair))", "private theorem toFunAux_eq (x : s) : toFunAux x = ((Finset.range x).filter (\u00b7 \u2208 s)).card", "theorem add_eq_zero_iff {n m : WithBot \u2115} : n + m = 0 \u2194 n = 0 \u2227 m = 0", "theorem coe_nonneg {n : \u2115} : 0 \u2264 (n : WithBot \u2115)", "@[simp] theorem add_div_right (x : Nat) {z : Nat} (H : 0 < z) : (x + z) / z = (x / z) + 1", "@[simp, nolint simpNF] protected lemma add_eq_left : a + b = a \u2194 b = 0", "theorem add_factorial_le_factorial_add (i : \u2115) {n : \u2115} (n1 : 1 \u2264 n) : i + n ! \u2264 (i + n)!", "theorem add_factorial_succ_le_factorial_add_succ (i : \u2115) (n : \u2115) :\n    i + (n + 1)! \u2264 (i + (n + 1))!", "theorem add_factorial_succ_lt_factorial_add_succ {i : \u2115} (n : \u2115) (hi : 2 \u2264 i) :\n    i + (n + 1)! < (i + n + 1)!", "protected theorem add_left_comm (n m k : Nat) : n + (m + k) = m + (n + k)", "lemma add_mod_eq_add_mod_left (c : \u2115) (H : a % d = b % d) : (c + a) % d = (c + b) % d", "lemma add_pos_iff_pos_or_pos : 0 < m + n \u2194 0 < m \u2228 0 < n", "lemma add_two_le_nth_prime (n : \u2115) : n + 2 \u2264 nth Prime n", "lemma and_forall_succ : p 0 \u2227 (\u2200 n, p (n + 1)) \u2194 \u2200 n, p n", "lemma and_two_pow (n i : \u2115) : n &&& 2 ^ i = (n.testBit i).toNat * 2 ^ i", "protected theorem bit0_eq_zero {n : \u2115} : bit0 n = 0 \u2194 n = 0", "protected theorem bit0_inj : \u2200 {n m : \u2115}, bit0 n = bit0 m \u2192 n = m", "@[simp]\nlemma bit0_lt_bit1_iff : bit0 m < bit1 n \u2194 m \u2264 n", "@[simp]\nlemma bit1_lt_bit0_iff : bit1 m < bit0 n \u2194 m < n", "theorem bit1_mod_two : bit1 n % 2 = 1", "@[simp]\ntheorem bits_append_bit (n : \u2115) (b : Bool) (hn : n = 0 \u2192 b = true) :\n    (bit b n).bits = b :: n.bits", "@[simp]\ntheorem bodd_bit0 (n) : bodd (bit0 n) = false", "@[simp]\nlemma card_Iic : (Iic b).card = b + 1", "@[simp]\ntheorem card_Iio : (Iio b).card = b", "@[simp]\ntheorem card_Ioo : (Ioo a b).card = b - a - 1", "theorem Nat.card_eq_card_units_add_one [GroupWithZero \u03b1] [Finite \u03b1] :\n    Nat.card \u03b1 = Nat.card \u03b1\u02e3 + 1", "theorem card_eq_two_iff' (x : \u03b1) : Nat.card \u03b1 = 2 \u2194 \u2203! y, y \u2260 x", "theorem card_fintypeIcc : Fintype.card (Set.Icc a b) = b + 1 - a", "theorem card_fintypeIoc : Fintype.card (Set.Ioc a b) = b - a", "lemma card_image_of_injective {f : \u03b1 \u2192 \u03b2} (hf : Injective f) (s : Set \u03b1) :\n    Nat.card (f '' s) = Nat.card s := card_image_of_injOn hf.injOn", "lemma card_ne_zero : Nat.card \u03b1 \u2260 0 \u2194 Nonempty \u03b1 \u2227 Finite \u03b1", "lemma card_pos_iff : 0 < Nat.card \u03b1 \u2194 Nonempty \u03b1 \u2227 Finite \u03b1", "lemma card_preimage_of_injOn {f : \u03b1 \u2192 \u03b2} {s : Set \u03b2} (hf : (f \u207b\u00b9' s).InjOn f) (hsf : s \u2286 range f) :\n    Nat.card (f \u207b\u00b9' s) = Nat.card s", "@[to_additive \"See also `addOrder_eq_card_multiples`.\"]\nlemma Nat.card_submonoidPowers : Nat.card (powers a) = orderOf a", "@[norm_cast, deprecated (since := \"2022-11-19\")]\ntheorem cast_bit1 [AddMonoidWithOne R] (n : \u2115) : ((bit1 n : \u2115) : R) = bit1 (n : R)", "theorem cast_descFactorial :\n    (a.descFactorial b : S) = (ascPochhammer S b).eval (a - (b - 1) : S)", "@[simp]\ntheorem cast_div [DivisionSemiring \u03b1] {m n : \u2115} (n_dvd : n \u2223 m) (hn : (n : \u03b1) \u2260 0) :\n    ((m / n : \u2115) : \u03b1) = m / n", "lemma cast_dvd_cast (h : m \u2223 n) : (m : \u03b1) \u2223 (n : \u03b1) := map_dvd (Nat.castRingHom \u03b1) h", "@[simp, norm_cast]\ntheorem cast_le : (m : \u03b1) \u2264 n \u2194 m \u2264 n", "@[simp, norm_cast]\nlemma cast_multiset_sum [AddCommMonoidWithOne \u03b2] (s : Multiset \u2115) :\n    (\u2191s.sum : \u03b2) = (s.map (\u2191)).sum", "@[simp low]\ntheorem cast_pos' {n : \u2115} : (0 : \u03b1) < n \u2194 0 < n", "theorem Nat.cauchy_induction_two_mul (seed : \u2115) (hs : P seed.succ)\n    (hm : \u2200 x, seed < x \u2192 P x \u2192 P (2 * x)) (n : \u2115) : P n", "theorem ceil_congr (h : \u2200 n : \u2115, a \u2264 n \u2194 b \u2264 n) : \u2308a\u2309\u208a = \u2308b\u2309\u208a", "@[simp]\ntheorem ceil_intCast {\u03b1 : Type*} [LinearOrderedRing \u03b1] [FloorSemiring \u03b1] (z : \u2124) :\n    \u2308(z : \u03b1)\u2309\u208a = z.toNat", "@[simp]\ntheorem ceil_one : \u2308(1 : \u03b1)\u2309\u208a = 1", "theorem choose_eq_factorial_div_factorial {n k : \u2115} (hk : k \u2264 n) :\n    choose n k = n ! / (k ! * (n - k)!)", "theorem choose_le_centralBinom (r n : \u2115) : choose (2 * n) r \u2264 centralBinom n", "theorem choose_le_middle (r n : \u2115) : choose n r \u2264 choose n (n / 2)", "theorem choose_le_pow (r n : \u2115) : (n.choose r : \u03b1) \u2264 (n ^ r : \u03b1) / r !", "@[simp]\ntheorem choose_succ_self_right : \u2200 n : \u2115, (n + 1).choose n = n + 1", "@[simp]\ntheorem choose_symm {n k : \u2115} (hk : k \u2264 n) : choose n (n - k) = choose n k", "@[mono]\ntheorem clog_anti_left {b c n : \u2115} (hc : 1 < c) (hb : c \u2264 b) : clog b n \u2264 clog c n", "@[mono]\ntheorem clog_mono_right (b : \u2115) {n m : \u2115} (h : n \u2264 m) : clog b n \u2264 clog b m", "theorem cond_decide_mod_two (x : \u2115) [d : Decidable (x % 2 = 1)] :\n    cond (@decide (x % 2 = 1) d) 1 0 = x % 2", "theorem cons_self_properDivisors (h : n \u2260 0) :\n    cons n (properDivisors n) properDivisors.not_self_mem = divisors n", "lemma coprime_div_gcd_of_squarefree (hm : Squarefree m) (hn : n \u2260 0) : Coprime (m / gcd m n) n", "lemma coprime_factorizationLCMLeft_factorizationLCMRight :\n    (factorizationLCMLeft a b).Coprime (factorizationLCMRight a b)", "theorem coprime_fintype_prod_left_iff [Fintype \u03b9] {s : \u03b9 \u2192 \u2115} {x : \u2115} :\n    Coprime (\u220f i, s i) x \u2194 \u2200 i, Coprime (s i) x", "@[simp]\ntheorem coprime_mul_left_add_right (m n k : \u2115) : Coprime m (m * k + n) \u2194 Coprime m n", "theorem coprime_one_left_iff (n : \u2115) : Coprime 1 n \u2194 True", "theorem count_add' (a b : \u2115) : count p (a + b) = count (fun k \u21a6 p (k + b)) a + count p b", "theorem count_modEq_card (v : \u2115) :\n    b.count (\u00b7 \u2261 v [MOD r]) = b / r + if v % r < b % r then 1 else 0", "theorem count_mono_left {n : \u2115} (hpq : \u2200 k, p k \u2192 q k) : count p n \u2264 count q n", "lemma decreasingInduction_succ_left {P : \u2115 \u2192 Sort*} (h : \u2200 n, P (n + 1) \u2192 P n)\n    (smn : m + 1 \u2264 n) (mn : m \u2264 n) (hP : P n) :\n    decreasingInduction h mn hP = h m (decreasingInduction h smn hP)", "theorem descFactorial_eq_factorial_mul_choose (n k : \u2115) : n.descFactorial k = k ! * n.choose k", "theorem descFactorial_self : \u2200 n : \u2115, n.descFactorial n = n !", "@[simp]\ntheorem descFactorial_zero (n : \u2115) : n.descFactorial 0 = 1", "@[simp]\ntheorem digits_inj_iff {b n m : \u2115} : b.digits n = b.digits m \u2194 n = m", "theorem digits_lt_base' {b m : \u2115} : \u2200 {d}, d \u2208 digits (b + 2) m \u2192 d < b + 2", "@[norm_cast, simp]\ntheorem Nat.dist_cast_rat (x y : \u2115) : dist (x : \u211a) y = dist x y", "theorem dist_comm (n m : \u2115) : dist n m = dist m n", "theorem dist_zero_right (n : \u2115) : dist n 0 = n", "@[simp]\ntheorem div2_bit0 (n) : div2 (bit0 n) = n", "@[simp] lemma div2_zero : div2 0 = 0 := rfl", "lemma div_dvd_of_dvd (h : n \u2223 m) : m / n \u2223 m := \u27e8n, (Nat.div_mul_cancel h).symm\u27e9", "lemma div_eq_iff_eq_of_dvd_dvd (hn : n \u2260 0) (ha : a \u2223 n) (hb : b \u2223 n) : n / a = n / b \u2194 a = b", "protected lemma div_mod_unique (h : 0 < b) :\n    a / b = d \u2227 a % b = c \u2194 c + b * d = a \u2227 c < b", "lemma div_mul_div_le_div (a b c : \u2115) : a / c * b / a \u2264 b / c", "@[simp]\ntheorem divisors_inj {a b : \u2115} : a.divisors = b.divisors \u2194 a = b", "theorem divisors_subset_properDivisors {m : \u2115} (hzero : n \u2260 0) (h : m \u2223 n) (hdiff : m \u2260 n) :\n    divisors m \u2286 properDivisors n", "lemma doubleFactorial_le_factorial : \u2200 n, n\u203c \u2264 n !\n  | 0 => le_rfl\n  | n + 1 => by\n    rw [factorial_eq_mul_doubleFactorial]; exact Nat.le_mul_of_pos_right _ n.doubleFactorial_pos", "lemma dvd_ceilRoot_pow (hn : n \u2260 0) : a \u2223 ceilRoot n a ^ n", "theorem dvd_of_factors_subperm {a b : \u2115} (ha : a \u2260 0) (h : a.factors <+~ b.factors) : a \u2223 b", "theorem eq_of_testBit_eq {x y : Nat} (pred : \u2200i, testBit x i = testBit y i) : x = y", "theorem Nat.eq_sq_add_sq_iff_eq_sq_mul {n : \u2115} :\n    (\u2203 x y : \u2115, n = x ^ 2 + y ^ 2) \u2194 \u2203 a b : \u2115, n = a ^ 2 * b \u2227 IsSquare (-1 : ZMod b)", "theorem eq_zero_of_gcd_eq_zero_right {m n : Nat} (H : gcd m n = 0) : n = 0", "theorem eq_zero_of_mul_eq_zero : \u2200 {n m : \u2115}, n * m = 0 \u2192 n = 0 \u2228 m = 0", "@[parity_simps] lemma even_add : Even (m + n) \u2194 (Even m \u2194 Even n)", "lemma even_or_odd' (n : \u2115) : \u2203 k, n = 2 * k \u2228 n = 2 * k + 1", "theorem exists_coprime (m n : Nat) :\n    \u2203 m' n', Coprime m' n' \u2227 m = m' * gcd m n \u2227 n = n' * gcd m n", "theorem factorial_dvd_descFactorial (n k : \u2115) : k ! \u2223 n.descFactorial k", "theorem factorial_ne_zero (n : \u2115) : n ! \u2260 0", "theorem factorization_choose_of_lt_three_mul (hp' : p \u2260 2) (hk : p \u2264 k) (hk' : p \u2264 n - k)\n    (hn : n < 3 * p) : (choose n k).factorization p = 0", "theorem factorization_eq_factors_multiset (n : \u2115) :\n    n.factorization = Multiset.toFinsupp (n.factors : Multiset \u2115)", "theorem factorization_eq_of_coprime_right {p a b : \u2115} (hab : Coprime a b) (hpb : p \u2208 b.factors) :\n    (a * b).factorization p = b.factorization p", "theorem factorization_eq_zero_iff_remainder {p r : \u2115} (i : \u2115) (pp : p.Prime) (hr0 : r \u2260 0) :\n    \u00acp \u2223 r \u2194 (p * i + r).factorization p = 0", "theorem factorization_mul_of_coprime {a b : \u2115} (hab : Coprime a b) :\n    (a * b).factorization = a.factorization + b.factorization", "@[simp]\ntheorem factorization_one : factorization 1 = 0", "theorem factorization_prime_le_iff_dvd {d n : \u2115} (hd : d \u2260 0) (hn : n \u2260 0) :\n    (\u2200 p : \u2115, p.Prime \u2192 d.factorization p \u2264 n.factorization p) \u2194 d \u2223 n", "@[simp]\ntheorem factorization_zero : factorization 0 = 0", "@[simp]\ntheorem factors_eq_nil (n : \u2115) : n.factors = [] \u2194 n = 0 \u2228 n = 1", "@[simp]\ntheorem factors_one : factors 1 = []", "theorem fib_add (m n : \u2115) : fib (m + n + 1) = fib m * fib n + fib (m + 1) * fib (n + 1)", "theorem fib_bit1 (n : \u2115) : fib (bit1 n) = fib (n + 1) ^ 2 + fib n ^ 2", "theorem fib_coprime_fib_succ (n : \u2115) : Nat.Coprime (fib n) (fib (n + 1))", "lemma fib_strictMonoOn : StrictMonoOn fib (Set.Ici 2)\n  | _m + 2, _, _n + 2, _, hmn => fib_add_two_strictMono <| lt_of_add_lt_add_right hmn", "theorem fib_succ_eq_succ_sum (n : \u2115) : fib (n + 1) = (\u2211 k \u2208 Finset.range n, fib k) + 1", "theorem filter_multiset_Ico_card_eq_of_periodic (n a : \u2115) (p : \u2115 \u2192 Prop) [DecidablePred p]\n    (pp : Periodic p a) : card (filter p (Ico n (n + a))) = a.count p", "theorem findGreatest_of_ne_zero (h : Nat.findGreatest P n = m) (h0 : m \u2260 0) : P m", "@[simp] lemma floorRoot_eq_zero : floorRoot n a = 0 \u2194 n = 0 \u2228 a = 0", "@[simp] lemma floorRoot_one_right (hn : n \u2260 0) : floorRoot n 1 = 1", "theorem floor_add_nat (ha : 0 \u2264 a) (n : \u2115) : \u230aa + n\u230b\u208a = \u230aa\u230b\u208a + n", "@[gcongr]\ntheorem floor_le_floor : \u2200 x y : \u03b1, x \u2264 y \u2192 \u230ax\u230b\u208a \u2264 \u230ay\u230b\u208a", "lemma forall_lt_succ : (\u2200 m < n + 1, p m) \u2194 (\u2200 m < n, p m) \u2227 p n", "@[simp]\ntheorem gcdA_zero_left {s : \u2115} : gcdA 0 s = 0", "@[simp]\ntheorem gcdB_zero_right {s : \u2115} (h : s \u2260 0) : gcdB s 0 = 0", "@[simp]\ntheorem gcd_add_mul_right_right (m n k : \u2115) : gcd m (n + k * m) = gcd m n", "@[simp]\ntheorem gcd_mul_left_add_left (m n k : \u2115) : gcd (n * k + m) n = gcd m n", "@[simp]\ntheorem gcd_mul_right_add_left (m n k : \u2115) : gcd (k * n + m) n = gcd m n", "@[simp] lemma greatestFib_eq_zero : greatestFib n = 0 \u2194 n = 0", "@[simp] lemma greatestFib_pos : 0 < greatestFib n \u2194 0 < n", "lemma greatestFib_sub_fib_greatestFib_le_greatestFib (hn : n \u2260 0) :\n    greatestFib (n - fib (greatestFib n)) \u2264 greatestFib n - 2", "lemma half_le_of_sub_le_half (h : a - b \u2264 a / 2) : a / 2 \u2264 b", "theorem iSup_lt_succ' (u : \u2115 \u2192 \u03b1) (n : \u2115) : \u2a06 k < n + 1, u k = u 0 \u2294 \u2a06 k < n, u (k + 1)", "theorem image_cast_int_Iic (a : \u2115) : (\u2191) '' Iic a = Icc (0 : \u2124) a", "theorem infinite_setOf_prime : { p | Prime p }.Infinite", "theorem insert_self_properDivisors (h : n \u2260 0) : insert n (properDivisors n) = divisors n", "theorem Nat.isCoprime_iff_coprime {m n : \u2115} : IsCoprime (m : \u2124) n \u2194 Nat.Coprime m n", "lemma isZeckendorfRep_zeckendorf : \u2200 n, (zeckendorf n).IsZeckendorfRep\n  | 0 => by simp only [zeckendorf_zero, IsZeckendorfRep_nil]\n  | n + 1 => by\n    rw [zeckendorf_succ, IsZeckendorfRep, List.cons_append]\n    refine (isZeckendorfRep_zeckendorf _).cons' (fun a ha \u21a6 ?_)\n    obtain h | h := eq_zero_or_pos (n + 1 - fib (greatestFib (n + 1)))\n    \u00b7 simp only [h, zeckendorf_zero, nil_append, head?_cons, Option.mem_some_iff] at ha\n      subst ha\n      exact le_greatestFib.2 le_add_self\n    rw [zeckendorf_of_pos h, cons_append, head?_cons, Option.mem_some_iff] at ha\n    subst a\n    exact add_le_of_le_tsub_right_of_le (le_greatestFib.2 le_add_self)\n      (greatestFib_sub_fib_greatestFib_le_greatestFib n.succ_ne_zero)", "theorem lcm_pos {m n : \u2115} : 0 < m \u2192 0 < n \u2192 0 < m.lcm n", "lemma leRecOn_self {C : \u2115 \u2192 Sort*} {n} {next : \u2200 {k}, C k \u2192 C (k + 1)} (x : C n) :\n    (leRecOn n.le_refl next x : C n) = x", "lemma leRecOn_succ_left {C : \u2115 \u2192 Sort*} {n m} (h1 : n \u2264 m) (h2 : n + 1 \u2264 m)\n    {next : \u2200 {k}, C k \u2192 C (k + 1)} (x : C n) :\n    (leRecOn h2 next (next x) : C m) = (leRecOn h1 next x : C m)", "lemma le_findGreatest (hmb : m \u2264 n) (hm : P m) : m \u2264 Nat.findGreatest P n", "lemma le_mul_self : \u2200 n : \u2115, n \u2264 n * n\n  | 0 => Nat.le_refl _\n  | n + 1 => by simp [Nat.mul_add]", "theorem Nat.le_of_lt_succ {m n : Nat} : LT.lt m (succ n) \u2192 LE.le m n", "theorem le_sub_of_add_le {a b c : Nat} (h : a + b \u2264 c) : a \u2264 c - b", "theorem left_le_pair (a b : \u2115) : a \u2264 pair a b", "lemma left_ne_zero_of_mem_divisorsAntidiagonal {p : \u2115 \u00d7 \u2115} (hp : p \u2208 n.divisorsAntidiagonal) :\n    p.1 \u2260 0", "@[simp]\ntheorem log_div_mul_self (b n : \u2115) : log b (n / b * b) = log b n", "theorem log_lt_of_lt_pow {b x y : \u2115} (hy : y \u2260 0) : y < b ^ x \u2192 log b y < x", "theorem Nat.lt_add_of_zero_lt_left (a b : Nat) (h : 0 < b) : a < a + b", "lemma lt_div_mul_add (hb : 0 < b) : a < a / b * b + b", "@[simp] lemma lt_find_iff (h : \u2203 n : \u2115, p n) (n : \u2115) : n < Nat.find h \u2194 \u2200 m \u2264 n, \u00ac p m", "protected theorem lt_of_mul_lt_mul_left {a b c : Nat} (h : a * b < a * c) : b < c", "theorem lt_of_testBit {n m : \u2115} (i : \u2115) (hn : testBit n i = false) (hm : testBit m i = true)\n    (hnm : \u2200 j, i < j \u2192 testBit n j = testBit m j) : n < m", "theorem lt_succ : m < succ n \u2194 m \u2264 n", "lemma lt_succ_sqrt (n : \u2115) : n < succ (sqrt n) * succ (sqrt n) := (sqrt_isSqrt n).right", "lemma lt_two_pow (n : \u2115) : n < 2 ^ n := lt_pow_self (by decide) n", "theorem map_div_right_divisors :\n    n.divisors.map \u27e8fun d => (d, n / d), fun p\u2081 p\u2082 => congr_arg Prod.fst\u27e9 =\n      n.divisorsAntidiagonal", "@[simp]\ntheorem mem_divisors {m : \u2115} : n \u2208 divisors m \u2194 n \u2223 m \u2227 m \u2260 0", "theorem mem_factors_iff_dvd {n p : \u2115} (hn : n \u2260 0) (hp : Prime p) : p \u2208 factors n \u2194 p \u2223 n", "theorem mem_factors_mul {a b : \u2115} (ha : a \u2260 0) (hb : b \u2260 0) {p : \u2115} :\n    p \u2208 (a * b).factors \u2194 p \u2208 a.factors \u2228 p \u2208 b.factors", "@[simp]\ntheorem mem_rfind {p : \u2115 \u2192. Bool} {n : \u2115} :\n    n \u2208 rfind p \u2194 true \u2208 p n \u2227 \u2200 {m : \u2115}, m < n \u2192 false \u2208 p m", "lemma mem_smoothNumbers {n m : \u2115} : m \u2208 smoothNumbers n \u2194 m \u2260 0 \u2227 \u2200 p \u2208 factors m, p < n", "lemma mem_smoothNumbersUpTo {N k n : \u2115} :\n    n \u2208 smoothNumbersUpTo N k \u2194 n \u2264 N \u2227 n \u2208 smoothNumbers k", "@[simp] theorem mod_add_mod (m n k : Nat) : (m % n + k) % n = (m + k) % n", "theorem mod_lcm (hn : a \u2261 b [MOD n]) (hm : a \u2261 b [MOD m]) : a \u2261 b [MOD lcm n m]", "theorem mod_modEq (a n) : a % n \u2261 a [MOD n]", "@[simp] lemma mod_two_ne_zero : \u00acn % 2 = 0 \u2194 n % 2 = 1", "theorem monotone_primeCounting : Monotone primeCounting", "lemma mul_add_mod' (a b c : \u2115) : (a * b + c) % b = c % b", "@[simp] lemma mul_def : Nat.mul m n = m * n := rfl", "protected lemma mul_dvd_mul_iff_right (hc : 0 < c) : a * c \u2223 b * c \u2194 a \u2223 b", "theorem multinomial_spec : (\u220f i \u2208 s, (f i)!) * multinomial s f = (\u2211 i \u2208 s, f i)!", "theorem Nat.noZeroSMulDivisors : NoZeroSMulDivisors \u2115 M", "lemma not_dvd_iff_between_consec_multiples (n : \u2115) {a : \u2115} (ha : 0 < a) :\n    (\u2203 k : \u2115, a * k < n \u2227 n < a * (k + 1)) \u2194 \u00aca \u2223 n", "protected theorem not_le_of_gt {n m : Nat} (h : n > m) : \u00ac n \u2264 m", "theorem nth_eq_orderIsoOfNat (hf : (setOf p).Infinite) :\n    nth p = (\u2191) \u2218 @Nat.Subtype.orderIsoOfNat (setOf p) hf.to_subtype", "@[simp] theorem nth_false (n : \u2115) : nth (fun _ \u21a6 False) n = 0", "lemma odd_mul : Odd (m * n) \u2194 Odd m \u2227 Odd n", "lemma odd_sub' (h : n \u2264 m) : Odd (m - n) \u2194 (Odd n \u2194 Even m)", "@[simp] lemma ofAdd_mul (a b : \u2115) : ofAdd (a * b) = ofAdd a ^ b := (toAdd_pow _ _).symm", "lemma ofDigits_div_eq_ofDigits_tail {p : \u2115} (hpos : 0 < p) (digits : List \u2115)\n    (w\u2081 : \u2200 l \u2208 digits, l < p) : ofDigits p digits / p = ofDigits p digits.tail", "theorem ofDigits_eq_sum_mapIdx (b : \u2115) (L : List \u2115) :\n    ofDigits b L = (L.mapIdx fun i a => a * b ^ i).sum", "theorem ofDigits_zmod (b : \u2124) (k : \u2115) (L : List \u2115) : ofDigits b L % k = ofDigits (b % k) L % k", "@[simp]\ntheorem ofNat_le_cast : (no_index (OfNat.ofNat m : \u03b1)) \u2264 n \u2194 (OfNat.ofNat m : \u2115) \u2264 n", "@[simp, norm_cast]\ntheorem one_le_cast : 1 \u2264 (n : \u03b1) \u2194 1 \u2264 n", "theorem ord_compl_dvd_ord_compl_iff_dvd (a b : \u2115) :\n    (\u2200 p : \u2115, ord_compl[p] a \u2223 ord_compl[p] b) \u2194 a \u2223 b", "theorem ord_compl_mul (a b p : \u2115) : ord_compl[p] (a * b) = ord_compl[p] a * ord_compl[p] b", "@[simp]\ntheorem pair_unpair (n : \u2115) : pair (unpair n).1 (unpair n).2 = n", "theorem periodic_gcd (a : \u2115) : Periodic (gcd a) a", "theorem pos_of_mem_divisors {m : \u2115} (h : m \u2208 n.divisors) : 0 < m", "theorem pow_factorization_choose_le (hn : 0 < n) : p ^ (choose n k).factorization p \u2264 n", "theorem pow_le_pow_of_le_left {n m : Nat} (h : n \u2264 m) : \u2200 (i : Nat), n^i \u2264 m^i", "@[simp]\ntheorem preimage_Ioo {a b : \u03b1} (ha : 0 \u2264 a) :\n    (Nat.cast : \u2115 \u2192 \u03b1) \u207b\u00b9' Set.Ioo a b = Set.Ioo \u230aa\u230b\u208a \u2308b\u2309\u208a", "theorem prime_def_lt'' {p : \u2115} : Prime p \u2194 2 \u2264 p \u2227 \u2200 m, m \u2223 p \u2192 m = 1 \u2228 m = p", "theorem prime_iff_card_units (p : \u2115) [Fintype (ZMod p)\u02e3] :\n    p.Prime \u2194 Fintype.card (ZMod p)\u02e3 = p - 1", "theorem prime_mul_iff {a b : \u2115} : Nat.Prime (a * b) \u2194 a.Prime \u2227 b = 1 \u2228 b.Prime \u2227 a = 1", "theorem probablePrime_iff_modEq (n : \u2115) {b : \u2115} (h : 1 \u2264 b) :\n    ProbablePrime n b \u2194 b ^ (n - 1) \u2261 1 [MOD n]", "theorem prod_factors : \u2200 {n}, n \u2260 0 \u2192 List.prod (factors n) = n", "theorem prod_pow_factorization_choose (n k : \u2115) (hkn : k \u2264 n) :\n    (\u220f p \u2208 Finset.range (n + 1), p ^ (Nat.choose n k).factorization p) = choose n k", "theorem prod_pow_factorization_eq_self {f : \u2115 \u2192\u2080 \u2115} (hf : \u2200 p : \u2115, p \u2208 f.support \u2192 Prime p) :\n    (f.prod (\u00b7 ^ \u00b7)).factorization = f", "@[simp]\ntheorem prod_range_factorial_succ (n : \u2115) : \u220f x \u2208 range n, (x + 1)! = sf n", "private theorem psp_from_prime_gt_p {b : \u2115} (b_ge_two : 2 \u2264 b) {p : \u2115} (p_prime : p.Prime)\n    (p_gt_two : 2 < p) : p < psp_from_prime b p", "theorem psub'_eq_psub (m n) : psub' m n = psub m n", "@[simp]\nlemma rec_zero {C : \u2115 \u2192 Sort*} (h0 : C 0) (h : \u2200 n, C n \u2192 C (n + 1)) : Nat.rec h0 h 0 = h0 := rfl", "@[simp]\ntheorem rfind_dom {p : \u2115 \u2192. Bool} :\n    (rfind p).Dom \u2194 \u2203 n, true \u2208 p n \u2227 \u2200 {m : \u2115}, m < n \u2192 (p m).Dom", "lemma right_ne_zero_of_mem_divisorsAntidiagonal {p : \u2115 \u00d7 \u2115} (hp : p \u2208 n.divisorsAntidiagonal) :\n    p.2 \u2260 0", "theorem sSup_def {s : Set \u2115} (h : \u2203 n, \u2200 a \u2208 s, a \u2264 n) :\n    sSup s = @Nat.find (fun n \u21a6 \u2200 a \u2208 s, a \u2264 n) _ h", "@[simp]\nlemma shiftLeft'_false : \u2200 n, shiftLeft' false m n = m <<< n\n  | 0 => rfl\n  | n + 1 => by\n    have : 2 * (m * 2^n) = 2^(n+1)*m", "lemma shiftLeft_sub : \u2200 (m : Nat) {n k}, k \u2264 n \u2192 m <<< (n - k) = (m <<< n) >>> k", "theorem shiftRight_add (m n : Nat) : \u2200 k, m >>> (n + k) = (m >>> n) >>> k", "@[simp]\ntheorem size_bit {b n} (h : bit b n \u2260 0) : size (bit b n) = succ (size n)", "@[gcongr] lemma smoothNumbers_mono {N M : \u2115} (hNM : N \u2264 M) : N.smoothNumbers \u2286 M.smoothNumbers", "lemma sqrt_pos : 0 < sqrt n \u2194 0 < n", "theorem squarefree_pow_iff {n k : \u2115} (hn : n \u2260 1) (hk : k \u2260 0) :\n    Squarefree (n ^ k) \u2194 Squarefree n \u2227 k = 1", "protected theorem sub_eq_iff_eq_add {c : Nat} (h : b \u2264 a) : a - b = c \u2194 a = c + b", "@[simp] theorem sub_le (n m : Nat) : n - m \u2264 n", "protected theorem sub_le_iff_le_add' {a b c : Nat} : a - b \u2264 c \u2194 a \u2264 b + c", "theorem Nat.succ.injEq (u v : Nat) : (u.succ = v.succ) = (u = v)", "@[mono]\ntheorem succPNat_mono : Monotone succPNat", "theorem sum_totient (n : \u2115) : n.divisors.sum \u03c6 = n", "theorem sum_totient' (n : \u2115) : (\u2211 m \u2208 (range n.succ).filter (\u00b7 \u2223 n), \u03c6 m) = n", "theorem totient_prime_pow {p : \u2115} (hp : p.Prime) {n : \u2115} (hn : 0 < n) :\n    \u03c6 (p ^ n) = p ^ (n - 1) * (p - 1)", "@[simp] lemma two_dvd_ne_zero : \u00ac2 \u2223 n \u2194 n % 2 = 1", "lemma two_not_dvd_two_mul_sub_one : \u2200 {n}, 0 < n \u2192 \u00ac2 \u2223 2 * n - 1\n  | n + 1, _ => two_not_dvd_two_mul_add_one n", "theorem Nat.uniformEmbedding_coe_rat : UniformEmbedding ((\u2191) : \u2115 \u2192 \u211a)", "theorem uniformEmbedding_coe_real : UniformEmbedding ((\u2191) : \u2115 \u2192 \u211d)", "theorem xor_cancel_left (n m : \u2115) : n ^^^ (n ^^^ m) = m", "protected lemma xor_comm (n m : \u2115) : n ^^^ m = m ^^^ n", "@[simp]\ntheorem xor_self (n : \u2115) : n ^^^ n = 0", "theorem zero_descFactorial_succ (k : \u2115) : (0 : \u2115).descFactorial (k + 1) = 0", "@[simp] theorem zero_eq : Nat.zero = 0", "protected theorem zero_lt_two : 0 < 2", "theorem lt_wf : @WellFounded NatOrdinal (\u00b7 < \u00b7)", "@[simp]\ntheorem toOrdinal_eq_one (a) : toOrdinal a = 1 \u2194 a = 1", "theorem Ne.bot_lt' (h : \u22a5 \u2260 a) : \u22a5 < a", "theorem Ne.ne_or_ne {x y : \u03b1} (z : \u03b1) (h : x \u2260 y) : x \u2260 z \u2228 y \u2260 z", "theorem NeZero.nat_of_neZero {R S} [Semiring R] [Semiring S]\n    {F} [FunLike F R S] [RingHomClass F R S] (f : F)\n    {n : \u2115} [hn : NeZero (n : S)] : NeZero (n : R)", "lemma pos_of_neZero_natCast (R) [AddMonoidWithOne R] {n : \u2115} [NeZero (n : R)] : 0 < n", "@[norm_cast]\ntheorem coe_mulHom_mk (f : A \u2192\u209b\u2099\u2090[\u03c6] B) (h\u2081 h\u2082 h\u2083 h\u2084) :\n    ((\u27e8\u27e8\u27e8f, h\u2081\u27e9, h\u2082, h\u2083\u27e9, h\u2084\u27e9 : A \u2192\u209b\u2099\u2090[\u03c6] B) : A \u2192\u2099* B) = \u27e8f, h\u2084\u27e9", "theorem ext_iff {f g : A \u2192\u209b\u2099\u2090[\u03c6] B} : f = g \u2194 \u2200 x, f x = g x", "protected theorem map_mul (f : A \u2192\u209b\u2099\u2090[\u03c6] B) (x y : A) : f (x * y) = f x * f y", "protected theorem map_zero (f : A \u2192\u209b\u2099\u2090[\u03c6] B) : f 0 = 0", "theorem adjoin_union (s t : Set A) : adjoin R (s \u222a t) = adjoin R s \u2294 adjoin R t", "theorem range_top_iff_surjective (f : A \u2192\u2099\u2090[R] B) :\n    NonUnitalAlgHom.range f = (\u22a4 : NonUnitalSubalgebra R B) \u2194 Function.Surjective f", "@[simp]\ntheorem to_subring_eq_top {R A : Type*} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} : S.toNonUnitalSubring = \u22a4 \u2194 S = \u22a4", "theorem ext_iff {inst\u2081 inst\u2082 : NonUnitalCommRing R} :\n    inst\u2081 = inst\u2082 \u2194\n      (local_hAdd[R, inst\u2081] = local_hAdd[R, inst\u2082]) \u2227\n      (local_hMul[R, inst\u2081] = local_hMul[R, inst\u2082])", "@[simp]\ntheorem coe_rangeRestrict (f : R \u2192\u2099+* S) (x : R) : (f.rangeRestrict x : S) = f x", "@[simp]\ntheorem eqLocus_same (f : R \u2192\u2099+* S) : f.eqLocus f = \u22a4", "@[ext]\ntheorem ext \u2983f g : \u03b1 \u2192\u2099+* \u03b2\u2984 : (\u2200 x, f x = g x) \u2192 f = g", "theorem map_srange (g : S \u2192\u2099+* T) (f : R \u2192\u2099+* S) : map g (srange f) = srange (g.comp f)", "@[simp]\ntheorem snd_comp_prod : (snd S T).comp (f.prod g) = g", "theorem copy_eq (f : A \u2192\u22c6\u2099\u2090[R] B) (f' : A \u2192 B) (h : f' = f) : f.copy f' h = f", "theorem inr_apply (x : B) : inr R A B x = (0, x)", "@[simp]\nlemma restrictScalars_apply (f : A \u2192\u22c6\u2099\u2090[S] B) (x : A) : f.restrictScalars R x = f x := rfl", "@[simp, norm_cast]\ntheorem coe_iInf {\u03b9 : Sort*} {S : \u03b9 \u2192 NonUnitalStarSubalgebra R A} :\n    (\u2191(\u2a05 i, S i) : Set A) = \u22c2 i, S i", "@[simp]\ntheorem centralizer_univ : centralizer R Set.univ = center R A", "protected theorem coe_eq_zero {x : S} : (x : A) = 0 \u2194 x = 0", "@[simp]\ntheorem coe_toNonUnitalSubalgebra (S : NonUnitalStarSubalgebra R A) :\n    (\u2191S.toNonUnitalSubalgebra : Set A) = S", "@[ext]\ntheorem ext {S T : NonUnitalStarSubalgebra R A} (h : \u2200 x : A, x \u2208 S \u2194 x \u2208 T) : S = T", "@[simp]\ntheorem mem_map {S : NonUnitalStarSubalgebra R A} {f : F} {y : B} :\n    y \u2208 map f S \u2194 \u2203 x \u2208 S, f x = y", "theorem toNonUnitalSubalgebra_injective :\n    Function.Injective\n      (toNonUnitalSubalgebra : NonUnitalStarSubalgebra R A \u2192 NonUnitalSubalgebra R A)", "@[simp, norm_cast]\ntheorem coe_comap (S : NonUnitalSubalgebra R B) (f : F) : (comap f S : Set A) = f \u207b\u00b9' (S : Set B)", "protected theorem coe_neg {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A]\n    {S : NonUnitalSubalgebra R A} (x : S) : (\u2191(-x) : A) = -\u2191x", "@[simp, norm_cast]\ntheorem coe_smul [Semiring R'] [SMul R' R] [Module R' A] [IsScalarTower R' R A] (r : R') (x : S) :\n    \u2191(r \u2022 x) = r \u2022 (x : A)", "@[simp]\ntheorem iSupLift_mk {i : \u03b9} (x : K i) (hx : (x : A) \u2208 T) :\n    iSupLift K dir f hf T hT \u27e8x, hx\u27e9 = f i x", "theorem mem_carrier {s : NonUnitalSubalgebra R A} {x : A} : x \u2208 s.carrier \u2194 x \u2208 s", "theorem mem_center_iff {a : A} : a \u2208 center R A \u2194 \u2200 b : A, b * a = a * b", "theorem prod_mono {S T : NonUnitalSubalgebra R A} {S\u2081 T\u2081 : NonUnitalSubalgebra R B} :\n    S \u2264 T \u2192 S\u2081 \u2264 T\u2081 \u2192 prod S S\u2081 \u2264 prod T T\u2081", "theorem toNonUnitalSubring_injective :\n    Function.Injective (toNonUnitalSubring : NonUnitalSubalgebra R A \u2192 NonUnitalSubring A)", "theorem toNonUnitalSubsemiring_injective :\n    Function.Injective\n      (toNonUnitalSubsemiring : NonUnitalSubalgebra R A \u2192 NonUnitalSubsemiring A)", "theorem closure_preimage_le (f : F) (s : Set S) :\n    closure ((f : R \u2192 S) \u207b\u00b9' s) \u2264 (closure s).comap f", "@[simp]\ntheorem coe_mk' {s : Set R} {sm : Subsemigroup R} (hm : \u2191sm = s) {sa : AddSubgroup R}\n    (ha : \u2191sa = s) : (NonUnitalSubring.mk' s sm sa hm ha : Set R) = s", "theorem eq_top_iff' (A : NonUnitalSubring R) : A = \u22a4 \u2194 \u2200 x : R, x \u2208 A", "theorem mem_bot {x : R} : x \u2208 (\u22a5 : NonUnitalSubring R) \u2194 x = 0", "@[mono]\ntheorem toNonUnitalSubsemiring_mono :\n    Monotone (toNonUnitalSubsemiring : NonUnitalSubring R \u2192 NonUnitalSubsemiring R)", "theorem comap_equiv_eq_map_symm (f : R \u2243+* S) (K : NonUnitalSubsemiring S) :\n    K.comap (f : R \u2192\u2099+* S) = K.map f.symm", "@[mono]\ntheorem toAddSubmonoid_strictMono :\n    StrictMono (toAddSubmonoid : NonUnitalSubsemiring R \u2192 AddSubmonoid R)", "theorem le_def : p \u2264 q \u2194 (p : E \u2192 \u211d) \u2264 q", "@[simp, norm_cast]\ntheorem coe_pure_interval (a : \u03b1) : (pure a : Interval \u03b1) = Interval.pure a", "@[simp]\ntheorem coe_sub_interval : (\u2191(s - t) : Interval \u03b1) = s - t", "theorem mem_pure_self (a : \u03b1) : a \u2208 pure a", "@[to_additive]\ntheorem one_mem_one : (1 : \u03b1) \u2208 (1 : NonemptyInterval \u03b1)", "theorem sub_mem_sub (ha : a \u2208 s) (hb : b \u2208 t) : a - b \u2208 s - t", "@[to_additive (attr := simp) toProd_zero]\ntheorem toProd_one : (1 : NonemptyInterval \u03b1).toProd = 1", "@[simp, norm_cast]\nprotected theorem coe_zero [Zero \u03b1] [Preorder \u03b1] : ((0 : { x : \u03b1 // 0 \u2264 x }) : \u03b1) = 0", "@[simp, norm_cast]\nprotected theorem coe_zpow (a : { x : \u03b1 // 0 \u2264 x }) (n : \u2124) :\n    ((a ^ n : { x : \u03b1 // 0 \u2264 x }) : \u03b1) = (a : \u03b1) ^ n", "theorem NormOneClass.induced {F : Type*} (R S : Type*) [Ring R] [SeminormedRing S]\n    [NormOneClass S] [FunLike F R S] [RingHomClass F R S] (f : F) :\n    @NormOneClass R (SeminormedRing.induced R S f).toNorm _", "theorem NormedAddCommGroup.cauchy_series_of_le_geometric'' {C : \u211d} {u : \u2115 \u2192 \u03b1} {N : \u2115} {r : \u211d}\n    (hr\u2080 : 0 < r) (hr\u2081 : r < 1) (h : \u2200 n \u2265 N, \u2016u n\u2016 \u2264 C * r ^ n) :\n    CauchySeq fun n \u21a6 \u2211 k \u2208 range (n + 1), u k", "theorem IsQuotient.norm_lift {f : NormedAddGroupHom M N} (hquot : IsQuotient f) {\u03b5 : \u211d} (h\u03b5 : 0 < \u03b5)\n    (n : N) : \u2203 m : M, f m = n \u2227 \u2016m\u2016 < \u2016n\u2016 + \u03b5", "@[simp]\ntheorem coe_toAddMonoidHom : \u21d1f.toAddMonoidHom = f", "theorem NormedAddGroupHom.completion_comp (f : NormedAddGroupHom G H) (g : NormedAddGroupHom H K) :\n    g.completion.comp f.completion = (g.comp f).completion", "theorem le_of_opNorm_le {c : \u211d} (h : \u2016f\u2016 \u2264 c) (x : V\u2081) : \u2016f x\u2016 \u2264 c * \u2016x\u2016", "theorem norm_comp_le (g : NormedAddGroupHom V\u2082 V\u2083) (f : NormedAddGroupHom V\u2081 V\u2082) :\n    \u2016g.comp f\u2016 \u2264 \u2016g\u2016 * \u2016f\u2016", "@[simp]\ntheorem NormedAddGroupHom.norm_completion (f : NormedAddGroupHom G H) : \u2016f.completion\u2016 = \u2016f\u2016", "theorem inverse_add_nth_order (x : R\u02e3) (n : \u2115) :\n    \u2200\u1da0 t in \ud835\udcdd 0, inverse ((x : R) + t) =\n      (\u2211 i \u2208 range n, (-\u2191x\u207b\u00b9 * t) ^ i) * \u2191x\u207b\u00b9 + (-\u2191x\u207b\u00b9 * t) ^ n * inverse (x + t)", "theorem inverse_one_sub_nth_order' (n : \u2115) {t : R} (ht : \u2016t\u2016 < 1) :\n    inverse ((1 : R) - t) = (\u2211 i \u2208 range n, t ^ i) + t ^ n * inverse (1 - t)", "theorem _root_.NormedSpace.Dual.isClosed_image_polar_of_mem_nhds {s : Set E}\n    (s_nhd : s \u2208 \ud835\udcdd (0 : E)) :\n    IsClosed (((\u2191) : Dual \ud835\udd5c E \u2192 E \u2192 \ud835\udd5c) '' NormedSpace.polar \ud835\udd5c s)", "protected lemma NormedSpace.cobounded_neBot : NeBot (cobounded E)", "theorem eq_iff_forall_dual_eq {x y : E} : x = y \u2194 \u2200 g : Dual \ud835\udd5c E, g x = g y", "theorem NormedSpace.exists_lt_norm (c : \u211d) : \u2203 x : E, c < \u2016x\u2016", "theorem expSeries_apply_eq' (x : \ud835\udd38) :\n    (fun n => expSeries \ud835\udd42 \ud835\udd38 n fun _ => x) = fun n => (n !\u207b\u00b9 : \ud835\udd42) \u2022 x ^ n", "theorem exp_hasFPowerSeriesAt_zero : HasFPowerSeriesAt (exp \ud835\udd42) (expSeries \ud835\udd42 \ud835\udd38) 0", "theorem isVonNBounded_iff' {s : Set E} :\n    Bornology.IsVonNBounded \ud835\udd5c s \u2194 \u2203 r : \u211d, \u2200 x \u2208 s, \u2016x\u2016 \u2264 r", "theorem NormedSpace.sphere_nonempty_rclike [Nontrivial E] {r : \u211d} (hr : 0 \u2264 r) :\n    Nonempty (sphere (0 : E) r)", "theorem bit_to_nat (b n) : (bit b n : \u2115) = Nat.bit b n", "@[simp, norm_cast]\ntheorem cast_to_int {\u03b1} [AddGroupWithOne \u03b1] (n : Num) : ((n : \u2124) : \u03b1) = n", "theorem cmp_to_nat : \u2200 m n, (Ordering.casesOn (cmp m n) ((m : \u2115) < n) (m = n) ((n : \u2115) < m) : Prop)", "@[simp] lemma lor_eq_or (p q : Num) : p.lor q = p ||| q := rfl", "theorem mem_ofZNum' : \u2200 {m : Num} {n : ZNum}, m \u2208 ofZNum' n \u2194 n = toZNum m", "theorem ofInt'_toZNum : \u2200 n : \u2115, toZNum n = ZNum.ofInt' n", "@[simp]\ntheorem ofZNum_toNat : \u2200 n : ZNum, (ofZNum n : \u2115) = Int.toNat n", "theorem toZNum_inj {m n : Num} : m.toZNum = n.toZNum \u2194 m = n", "lemma ComplexEmbedding.exists_comp_symm_eq_of_comp_eq [Algebra k K] [IsGalois k K] (\u03c6 \u03c8 : K \u2192+* \u2102)\n    (h : \u03c6.comp (algebraMap k K) = \u03c8.comp (algebraMap k K)) :\n    \u2203 \u03c3 : K \u2243\u2090[k] K, \u03c6.comp \u03c3.symm = \u03c8", "theorem card_complex_embeddings :\n    card { \u03c6 : K \u2192+* \u2102 // \u00acComplexEmbedding.IsReal \u03c6 } = 2 * NrComplexPlaces K", "theorem embedding_mk_eq (\u03c6 : K \u2192+* \u2102) :\n    embedding (mk \u03c6) = \u03c6 \u2228 embedding (mk \u03c6) = ComplexEmbedding.conjugate \u03c6", "lemma even_card_aut_of_not_isUnramifiedIn [IsGalois k K] [FiniteDimensional k K]\n    {w : InfinitePlace k} (hw : \u00ac w.IsUnramifiedIn K) :\n    Even (Fintype.card <| K \u2243\u2090[k] K)", "lemma even_nat_card_aut_of_not_isUnramified [IsGalois k K] (hw : \u00ac IsUnramified k w) :\n    Even (Nat.card <| K \u2243\u2090[k] K)", "lemma isUnramifiedIn_comap [IsGalois k K] {w : InfinitePlace K} :\n    (w.comap (algebraMap k K)).IsUnramifiedIn K \u2194 w.IsUnramified k", "lemma isUnramified_iff_card_stabilizer_eq_one [IsGalois k K] :\n    IsUnramified k w \u2194 Nat.card (Stab w) = 1", "lemma isUnramified_iff_stabilizer_eq_bot [IsGalois k K] : IsUnramified k w \u2194 Stab w = \u22a5", "theorem nrRealPlaces_eq_one_of_finrank_eq_one (h : finrank \u211a K = 1) :\n    NrRealPlaces K = 1", "theorem prod_eq_abs_norm (x : K) :\n    \u220f w : InfinitePlace K, w x ^ mult w = abs (Algebra.norm \u211a x)", "theorem isIntegral (x : \ud835\udcde K) : IsIntegral \u2124 x", "theorem coe_neg_one : ((-1 : (\ud835\udcde K)\u02e3) : K) = (-1 : K)", "theorem regulator_eq_det' (e : {w : InfinitePlace K // w \u2260 w\u2080} \u2243 Fin (rank K)) :\n    regulator K = |(Matrix.of fun i \u21a6 (logEmbedding K) (fundSystem K (e i))).det|", "theorem conj_apply {x : ((K \u2192+* \u2102) \u2192 \u2102)} (\u03c6 : K \u2192+* \u2102)\n    (hx : x \u2208 Submodule.span \u211d (Set.range (canonicalEmbedding K))) :\n    conj (x \u03c6) = x (ComplexEmbedding.conjugate \u03c6)", "theorem integerLattice.inter_ball_finite [NumberField K] (r : \u211d) :\n    ((integerLattice K : Set ((K \u2192+* \u2102) \u2192 \u2102)) \u2229 Metric.closedBall 0 r).Finite", "theorem integralBasis_repr_apply [NumberField K] (x : K) (i : Free.ChooseBasisIndex \u2124 (\ud835\udcde K)) :\n    (latticeBasis K).repr (canonicalEmbedding K x) i = (integralBasis K).repr x i", "@[simp]\ntheorem latticeBasis_apply [NumberField K] (i : Free.ChooseBasisIndex \u2124 (\ud835\udcde K)) :\n    latticeBasis K i = (canonicalEmbedding K) (integralBasis K i)", "theorem _root_.NumberField.finite_of_discr_bdd :\n    {K : { F : IntermediateField \u211a A // FiniteDimensional \u211a F} |\n      haveI :  NumberField K := @NumberField.mk _ _ inferInstance K.prop\n      |discr K| \u2264 N }.Finite", "theorem NumberField.isUnit_iff_norm [NumberField K] {x : \ud835\udcde K} :\n    IsUnit x \u2194 |(RingOfIntegers.norm \u211a x : \u211a)| = 1", "theorem convexBodyLT_mem {x : K} :\n    mixedEmbedding K x \u2208 (convexBodyLT K f) \u2194 \u2200 w : InfinitePlace K, w x < f w", "theorem convexBodySumFun_continuous :\n    Continuous (convexBodySumFun : (E K) \u2192 \u211d)", "theorem convexBodySumFun_smul (c : \u211d) (x : E K) :\n    convexBodySumFun (c \u2022 x) = |c| * convexBodySumFun x", "theorem exists_ne_zero_mem_ideal_lt' (w\u2080 : {w : InfinitePlace K // IsComplex w})\n    (h : minkowskiBound K I < volume (convexBodyLT' K f w\u2080)) :\n    \u2203 a \u2208 (I : FractionalIdeal (\ud835\udcde K)\u2070 K), a \u2260 0 \u2227 (\u2200 w : InfinitePlace K, w \u2260 w\u2080 \u2192 w a < f w) \u2227\n      |(w\u2080.val.embedding a).re| < 1 \u2227 |(w\u2080.val.embedding a).im| < (f w\u2080 : \u211d) ^ 2", "@[simp]\ntheorem indexEquiv_apply_ofIsComplex_snd (w : {w : InfinitePlace K // IsComplex w}) :\n    (indexEquiv K) (Sum.inr \u27e8w, 1\u27e9) = ComplexEmbedding.conjugate w.val.embedding", "theorem latticeBasis_repr_apply (x : K) (i : ChooseBasisIndex \u2124 (\ud835\udcde K)) :\n    (latticeBasis K).repr (mixedEmbedding K x) i = (integralBasis K).repr x i", "theorem mem_rat_span_latticeBasis (x : K) :\n    mixedEmbedding K x \u2208 Submodule.span \u211a (Set.range (latticeBasis K))", "theorem normAtPlace_apply_isReal {w : InfinitePlace K} (hw : IsReal w) (x : E K):\n    normAtPlace w x = \u2016x.1 \u27e8w, hw\u27e9\u2016", "theorem normAtPlace_neg (w : InfinitePlace K) (x : E K)  :\n    normAtPlace w (- x) = normAtPlace w x", "theorem normAtPlace_smul (w : InfinitePlace K) (x : E K) (c : \u211d) :\n    normAtPlace w (c \u2022 x) = |c| * normAtPlace w x", "theorem norm_real (c : \u211d) :\n    mixedEmbedding.norm ((fun _ \u21a6 c, fun _ \u21a6 c) : (E K)) = |c| ^ finrank \u211a K", "theorem eq_of_cmp_eq : \u2200 {o\u2081 o\u2082}, cmp o\u2081 o\u2082 = Ordering.eq \u2192 o\u2081 = o\u2082", "@[simp]\ntheorem fastGrowing_one : fastGrowing 1 = fun n => 2 * n", "theorem fastGrowing\u03b5\u2080_two : fastGrowing\u03b5\u2080 2 = 2048", "theorem oadd_add (e n a o) : oadd e n a + o = addAux e n (a + o)", "theorem oadd_lt_oadd_1 {e\u2081 n\u2081 o\u2081 e\u2082 n\u2082 o\u2082} (h\u2081 : NF (oadd e\u2081 n\u2081 o\u2081)) (h : e\u2081 < e\u2082) :\n    oadd e\u2081 n\u2081 o\u2081 < oadd e\u2082 n\u2082 o\u2082", "theorem oadd_mul_nfBelow {e\u2081 n\u2081 a\u2081 b\u2081} (h\u2081 : NFBelow (oadd e\u2081 n\u2081 a\u2081) b\u2081) :\n    \u2200 {o\u2082 b\u2082}, NFBelow o\u2082 b\u2082 \u2192 NFBelow (oadd e\u2081 n\u2081 a\u2081 * o\u2082) (repr e\u2081 + b\u2082)", "@[simp 1200]\ntheorem ofNat_one : ofNat 1 = 1", "theorem repr_opow (o\u2081 o\u2082) [NF o\u2081] [NF o\u2082] : repr (o\u2081 ^ o\u2082) = repr o\u2081 ^ repr o\u2082", "theorem scale_eq_mul (x) [NF x] : \u2200 (o) [NF o], scale x o = oadd x 1 0 * o", "theorem split_dvd {o o' m} [NF o] (h : split o = (o', m)) : \u03c9 \u2223 repr o'", "protected theorem zero_lt_one : (0 : ONote) < 1", "lemma Odd.pos [Nontrivial \u03b1] : Odd a \u2192 0 < a", "lemma Odd.pow_nonpos_iff (hn : Odd n) : a ^ n \u2264 0 \u2194 a \u2264 0", "@[simp] lemma ofNat_eq_ofNat {m n : \u2115} [m.AtLeastTwo] [n.AtLeastTwo] :\n    (no_index (ofNat m) : R) = no_index (ofNat n) \u2194 (ofNat m : \u2115) = ofNat n", "theorem Continuous.of_bundled (f : \u03b1 \u2192 \u03b2) (hf : Monotone f) (hf' : Continuous \u27e8f, hf\u27e9) :\n    Continuous' f", "theorem congr_fun {f g : \u03b1 \u2192\ud835\udc84 \u03b2} (h : f = g) (x : \u03b1) : f x = g x", "@[simp]\ntheorem forall_forall_merge' (c\u2080 : Chain (\u03b1 \u2192\ud835\udc84 \u03b2)) (c\u2081 : Chain \u03b1) (z : \u03b2) :\n    (\u2200 j i : \u2115, (c\u2080 i) (c\u2081 j) \u2264 z) \u2194 \u2200 i : \u2115, (c\u2080 i) (c\u2081 i) \u2264 z", "theorem \u03c9Sup_apply_\u03c9Sup (c\u2080 : Chain (\u03b1 \u2192\ud835\udc84 \u03b2)) (c\u2081 : Chain \u03b1) :\n    \u03c9Sup c\u2080 (\u03c9Sup c\u2081) = Prod.apply (\u03c9Sup (c\u2080.zip c\u2081))", "theorem id_continuous' : Continuous' (@id \u03b1)", "@[to_additive]\ntheorem OneHom.comp_apply [One M] [One N] [One P] (g : OneHom N P) (f : OneHom M N) (x : M) :\n    g.comp f x = g (f x)", "@[to_additive \"Composition of additive monoid homomorphisms is associative.\"]\ntheorem OneHom.comp_assoc {Q : Type*} [One M] [One N] [One P] [One Q]\n    (f : OneHom M N) (g : OneHom N P) (h : OneHom P Q) :\n    (h.comp g).comp f = h.comp (g.comp f)", "@[to_additive (attr := simp, norm_cast)]\ntheorem coe_one : ((1 : S') : M\u2081) = 1", "lemma continuous_iff_from_nat {Y : Type*} [TopologicalSpace Y] (f : OnePoint \u2115 \u2192 Y) :\n    Continuous f \u2194 Tendsto (fun x : \u2115 \u21a6 f x) atTop (\ud835\udcdd (f \u221e))", "theorem hasBasis_nhds_infty :\n    (\ud835\udcdd (\u221e : OnePoint X)).HasBasis (fun s : Set X => IsClosed s \u2227 IsCompact s) fun s =>\n      (\u2191) '' s\u1d9c \u222a {\u221e}", "@[simp, norm_cast]\ntheorem inseparable_coe {x y : X} : Inseparable (x : OnePoint X) y \u2194 Inseparable x y", "theorem not_inseparable_coe_infty {x : X} : \u00acInseparable (x : OnePoint X) \u221e", "theorem tendsto_nhds_infty' {\u03b1 : Type*} {f : OnePoint X \u2192 \u03b1} {l : Filter \u03b1} :\n    Tendsto f (\ud835\udcdd \u221e) l \u2194 Tendsto f (pure \u221e) l \u2227 Tendsto (f \u2218 (\u2191)) (coclosedCompact X) l", "theorem OpenEmbedding.isLocalHomeomorph (hf : OpenEmbedding f) : IsLocalHomeomorph f", "theorem OpenEmbedding.open_iff_image_open (hf : OpenEmbedding f) {s : Set X} :\n    IsOpen s \u2194 IsOpen (f '' s)", "protected theorem OpenEmbedding.prod {f : X \u2192 Y} {g : Z \u2192 W} (hf : OpenEmbedding f)\n    (hg : OpenEmbedding g) : OpenEmbedding fun x : X \u00d7 Z => (f x.1, g x.2)", "@[to_additive (attr := simp, norm_cast)]\ntheorem coe_prod (U : OpenSubgroup G) (V : OpenSubgroup H) :\n    (U.prod V : Set (G \u00d7 H)) = (U : Set G) \u00d7\u02e2 (V : Set H)", "@[to_additive (attr := simp, norm_cast)]\ntheorem mem_toOpens : g \u2208 (U : Opens G) \u2194 g \u2208 U", "theorem bind_pmap {\u03b1 \u03b2 \u03b3} {p : \u03b1 \u2192 Prop} (f : \u2200 a, p a \u2192 \u03b2) (x : Option \u03b1) (g : \u03b2 \u2192 Option \u03b3) (H) :\n    pmap f x H >>= g = x.pbind fun a h \u21a6 g (f a (H _ h))", "@[simp]\nlemma bnot_comp_isSome : (! \u00b7) \u2218 @Option.isSome \u03b1 = Option.isNone", "theorem exists_ne_none {p : Option \u03b1 \u2192 Prop} : (\u2203 x \u2260 none, p x) \u2194 (\u2203 x : \u03b1, p x)", "theorem getD_default_eq_iget [Inhabited \u03b1] (o : Option \u03b1) :\n    o.getD default = o.iget", "theorem map_pbind (f : \u03b2 \u2192 \u03b3) (x : Option \u03b1) (g : \u2200 a, a \u2208 x \u2192 Option \u03b2) :\n    Option.map f (x.pbind g) = x.pbind fun a H \u21a6 Option.map f (g a H)", "theorem mem_map\u2082_iff {c : \u03b3} : c \u2208 map\u2082 f a b \u2194 \u2203 a' b', a' \u2208 a \u2227 b' \u2208 b \u2227 f a' b' = c", "theorem orElse_eq_some (o o' : Option \u03b1) (x : \u03b1) :\n    (o <|> o') = some x \u2194 o = some x \u2228 o = none \u2227 o' = some x", "theorem pmap_eq_some_iff {hf} {y : \u03b2} :\n    pmap f x hf = some y \u2194 \u2203 (a : \u03b1) (H : x = some a), f a (hf a H) = y", "theorem Or.elim3 {c d : Prop} (h : a \u2228 b \u2228 c) (ha : a \u2192 d) (hb : b \u2192 d) (hc : c \u2192 d) : d", "theorem Or.intro_right (a : Prop) (h : b) : Or a b", "theorem above_mem : D.above x \u2208 D", "theorem Ico_pred_left (h : pred a < b) : Ico (pred a) b = insert (pred a) (Ico a b)", "theorem compl_F_eq_I : (IF.F : Set P)\u1d9c = IF.I", "@[ext]\ntheorem ext {s t : Ideal P} : (s : Set P) = t \u2192 s = t", "protected theorem nonempty (s : Ideal P) : (s : Set P).Nonempty", "@[simp]\ntheorem top_toLowerSet : (\u22a4 : Ideal P).toLowerSet = \u22a4", "theorem Ioo_pred_left_of_not_isMin (ha : \u00acIsMin a) : Ioo (pred a) b = Ico a b", "theorem Ioo_succ_right_of_not_isMax (hb : \u00acIsMax b) : Ioo a (succ b) = Ioc a b", "theorem IsPredLimit.lt_pred (h : IsPredLimit a) : a < b \u2192 a < pred b", "theorem directed : DirectedOn (\u00b7 \u2265 \u00b7) (F : Set P)", "theorem isPredLimit_of_pred_lt (H : \u2200 a > b, pred a < b) : IsPredLimit b", "theorem isSuccLimitRecOn_succ' (hs : \u2200 a, \u00acIsMax a \u2192 C (succ a)) (hl : \u2200 a, IsSuccLimit a \u2192 C a)\n    {b : \u03b1} (hb : \u00acIsMax b) : @isSuccLimitRecOn \u03b1 _ _ C (succ b) hs hl = hs b hb", "theorem iso_of_countable_dense [Countable \u03b1] [DenselyOrdered \u03b1] [NoMinOrder \u03b1] [NoMaxOrder \u03b1]\n    [Nonempty \u03b1] [Countable \u03b2] [DenselyOrdered \u03b2] [NoMinOrder \u03b2] [NoMaxOrder \u03b2] [Nonempty \u03b2] :\n    Nonempty (\u03b1 \u2243o \u03b2)", "theorem le_succ_bot_iff : a \u2264 succ \u22a5 \u2194 a = \u22a5 \u2228 a = succ \u22a5", "@[simp]\ntheorem lt_succ_iff : a < succ b \u2194 a \u2264 b", "theorem lt_succ_iff_of_not_isMax (ha : \u00acIsMax a) : b < succ a \u2194 b \u2264 a", "theorem not_isSuccLimit_iff' : \u00acIsSuccLimit a \u2194 a \u2208 range (@succ \u03b1 _ _)", "@[simp]\ntheorem pred_eq_pred_iff : pred a = pred b \u2194 a = b", "theorem pred_lt_iff_eq_or_lt_of_not_isMin (ha : \u00acIsMin a) : pred a < b \u2194 a = b \u2228 a < b", "theorem pred_strictMono : StrictMono (pred : \u03b1 \u2192 \u03b1)", "theorem sequenceOfCofinals.monotone : Monotone (sequenceOfCofinals p \ud835\udc9f)", "@[simp, mono]\ntheorem succ_le_succ (h : a \u2264 b) : succ a \u2264 succ b", "theorem succ_mono : Monotone (succ : \u03b1 \u2192 \u03b1)", "theorem instPartialOrder.dual_dual (\u03b1 : Type*) [H : PartialOrder \u03b1] :\n    OrderDual.instPartialOrder \u03b1\u1d52\u1d48 = H", "theorem le_toDual [LE \u03b1] {a : \u03b1\u1d52\u1d48} {b : \u03b1} : a \u2264 toDual b \u2194 b \u2264 ofDual a", "protected theorem acc (a : \u03b1) : Acc (\u00b7 < \u00b7) (f a) \u2192 Acc (\u00b7 < \u00b7) a", "@[simp] lemma birkhoffFinset_sup (a b : \u03b1) :\n    birkhoffFinset (a \u2294 b) = birkhoffFinset a \u222a birkhoffFinset b", "theorem image_Icc (e : \u03b1 \u21aao \u03b2) (he : OrdConnected (range e)) (x y : \u03b1) :\n    e '' Icc x y = Icc (e x) (e y)", "@[simp]\ntheorem le_iff_le {a b} : f a \u2264 f b \u2194 a \u2264 b", "theorem OrderEmbedding.le_map_sup [SemilatticeSup \u03b1] [SemilatticeSup \u03b2] (f : \u03b1 \u21aao \u03b2) (x y : \u03b1) :\n    f x \u2294 f y \u2264 f (x \u2294 y)", "theorem OrderEmbedding.map_inf_le [SemilatticeInf \u03b1] [SemilatticeInf \u03b2] (f : \u03b1 \u21aao \u03b2) (x y : \u03b1) :\n    f (x \u2293 y) \u2264 f x \u2293 f y", "@[mono]\ntheorem apply_mono {f g : \u03b1 \u2192o \u03b2} {x y : \u03b1} (h\u2081 : f \u2264 g) (h\u2082 : x \u2264 y) : f x \u2264 g y", "theorem le_def {f g : \u03b1 \u2192o \u03b2} : f \u2264 g \u2194 \u2200 x, f x \u2264 g x", "theorem le_prevFixed {x : \u03b1} (hx : f x \u2264 x) {y : fixedPoints f} (h : \u2191y \u2264 x) :\n    y \u2264 f.prevFixed x hx", "protected theorem monotone (f : \u03b1 \u2192o \u03b2) : Monotone f", "@[simp]\ntheorem nextFixed_le_iff {x : \u03b1} (hx : x \u2264 f x) {y : fixedPoints f} :\n    f.nextFixed x hx \u2264 y \u2194 x \u2264 y", "theorem orderHom_eq_id [Subsingleton \u03b1] (g : \u03b1 \u2192o \u03b1) : g = OrderHom.id", "protected theorem bijective (e : \u03b1 \u2243o \u03b2) : Function.Bijective e", "theorem OrderIso.concaveOn_symm (f : \u03b1 \u2243o \u03b2) (hf : ConvexOn \ud835\udd5c univ f) :\n    ConcaveOn \ud835\udd5c univ f.symm", "theorem OrderIso.convexOn_symm (f : \u03b1 \u2243o \u03b2) (hf : ConcaveOn \ud835\udd5c univ f) :\n    ConvexOn \ud835\udd5c univ f.symm", "protected theorem isAtomic_iff [OrderBot \u03b1] [OrderBot \u03b2] (f : \u03b1 \u2243o \u03b2) :\n    IsAtomic \u03b1 \u2194 IsAtomic \u03b2", "@[simp]\ntheorem _root_.OrderIso.isBoundedUnder_ge_comp [Preorder \u03b1] [Preorder \u03b2] (e : \u03b1 \u2243o \u03b2) {l : Filter \u03b3}\n    {u : \u03b3 \u2192 \u03b1} : (IsBoundedUnder (\u00b7 \u2265 \u00b7) l fun x => e (u x)) \u2194 IsBoundedUnder (\u00b7 \u2265 \u00b7) l u", "theorem le_iff_le (e : \u03b1 \u2243o \u03b2) {x y : \u03b1} : e x \u2264 e y \u2194 x \u2264 y", "theorem map_csInf' (e : \u03b1 \u2243o \u03b2) {s : Set \u03b1} (hne : s.Nonempty) (hbdd : BddBelow s) :\n    e (sInf s) = sInf (e '' s)", "@[to_additive (attr := simp)]\ntheorem OrderIso.mulLeft_symm (a : \u03b1) : (OrderIso.mulLeft a).symm = OrderIso.mulLeft a\u207b\u00b9", "@[simp]\ntheorem ofRelIsoLT_symm {\u03b1 \u03b2} [PartialOrder \u03b1] [PartialOrder \u03b2]\n    (e : ((\u00b7 < \u00b7) : \u03b1 \u2192 \u03b1 \u2192 Prop) \u2243r ((\u00b7 < \u00b7) : \u03b2 \u2192 \u03b2 \u2192 Prop)) :\n    (ofRelIsoLT e).symm = ofRelIsoLT e.symm", "@[simp]\ntheorem preimage_Ico (e : \u03b1 \u2243o \u03b2) (a b : \u03b2) : e \u207b\u00b9' Ico a b = Ico (e.symm a) (e.symm b)", "protected theorem rightOrdContinuous : RightOrdContinuous e", "theorem symm_apply_eq (e : \u03b1 \u2243o \u03b2) {x : \u03b1} {y : \u03b2} : e.symm y = x \u2194 y = e x", "theorem tendsto_atBot (e : \u03b1 \u2243o \u03b2) : Tendsto e atBot atBot", "@[simp]\ntheorem tendsto_atTop_iff {l : Filter \u03b3} {f : \u03b3 \u2192 \u03b1} (e : \u03b1 \u2243o \u03b2) :\n    Tendsto (fun x => e (f x)) l atTop \u2194 Tendsto f l atTop", "@[simp]\ntheorem coe_monoidWithZeroHom (f : \u03b1 \u2192*\u2080o \u03b2) : \u21d1(f : \u03b1 \u2192*\u2080 \u03b2) = f", "@[simp]\ntheorem coe_mul (f g : \u03b1 \u2192*\u2080o \u03b2) : \u21d1(f * g) = f * g", "@[simp]\ntheorem comp_apply (f : \u03b2 \u2192*\u2080o \u03b3) (g : \u03b1 \u2192*\u2080o \u03b2) (a : \u03b1) : (f.comp g) a = f (g a)", "@[simp]\ntheorem id_comp (f : \u03b1 \u2192*\u2080o \u03b2) : (OrderMonoidWithZeroHom.id \u03b2).comp f = f", "@[simp]\ntheorem self_trans_symm (e : \u03b1 \u2243+*o \u03b2) : e.trans e.symm = OrderRingIso.refl \u03b1", "@[simp]\ntheorem trans_apply (f : \u03b1 \u2243+*o \u03b2) (g : \u03b2 \u2243+*o \u03b3) (a : \u03b1) : f.trans g a = g (f a)", "@[simp]\ntheorem OrderTop.upperBounds_univ [PartialOrder \u03b3] [OrderTop \u03b3] :\n    upperBounds (univ : Set \u03b3) = {\u22a4}", "theorem compares_swap [LT \u03b1] {a b : \u03b1} {o : Ordering} : o.swap.Compares a b \u2194 o.Compares b a", "theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal \u00d7 Ordinal} : x \u2208 CNF b o \u2192 0 < x.2", "protected theorem monotone (hf : IsFundamentalSequence a o f) {i j : Ordinal} (hi : i < o)\n    (hj : j < o) (hij : i \u2264 j) : f i hi \u2264 f j hj", "theorem IsNormal.cof_eq {f} (hf : IsNormal f) {a} (ha : IsLimit a) : cof (f a) = cof a", "theorem apply_lt_nfpBFamily (H : \u2200 i hi, IsNormal (f i hi)) {a b} (hb : b < nfpBFamily.{u, v} o f a)\n    (i hi) : f i hi b < nfpBFamily.{u, v} o f a", "theorem bddAbove_of_small (s : Set Ordinal.{u}) [h : Small.{u} s] : BddAbove s", "@[simp]\ntheorem bfamilyOfFamily_typein {\u03b9} (f : \u03b9 \u2192 \u03b1) (i) :\n    bfamilyOfFamily f (typein _ i) (typein_lt_type _ i) = f i", "theorem blsub_le_bsup_succ {o : Ordinal.{u}} (f : \u2200 a < o, Ordinal.{max u v}) :\n    blsub.{_, v} o f \u2264 succ (bsup.{_, v} o f)", "theorem bmex_lt_ord_succ_card {o : Ordinal.{u}} (f : \u2200 a < o, Ordinal.{u}) :\n    bmex.{_, u} o f < (succ o.card).ord", "theorem bsup_eq_blsub_of_lt_succ_limit {o : Ordinal.{u}} (ho : IsLimit o)\n    {f : \u2200 a < o, Ordinal.{max u v}} (hf : \u2200 a ha, f a ha < f (succ a) (ho.2 a ha)) :\n    bsup.{_, v} o f = blsub.{_, v} o f", "theorem bsup_id_limit {o : Ordinal} : (\u2200 a < o, succ a < o) \u2192 (bsup.{u, u} o fun x _ => x) = o", "@[simp]\ntheorem bsup_one (f : \u2200 a < (1 : Ordinal), Ordinal) : bsup 1 f = f 0 zero_lt_one", "theorem card_le_card {o\u2081 o\u2082 : Ordinal} : o\u2081 \u2264 o\u2082 \u2192 card o\u2081 \u2264 card o\u2082", "@[simp]\ntheorem card_omega : card \u03c9 = \u2135\u2080", "theorem cof_sup_le {\u03b9} {f : \u03b9 \u2192 Ordinal} (H : \u2200 i, f i < sup.{u, u} f) :\n    cof (sup.{u, u} f) \u2264 #\u03b9", "theorem enumOrd_def (o) : enumOrd S o = sInf (S \u2229 { b | \u2200 c, c < o \u2192 enumOrd S c < b })", "theorem enumOrd_def' (o) :\n    enumOrd S o = sInf (S \u2229 Set.Ici (blsub.{u, u} o fun a _ => enumOrd S a))", "theorem enumOrd_def'_nonempty (hS : Unbounded (\u00b7 < \u00b7) S) (a) : (S \u2229 Set.Ici a).Nonempty", "theorem enum_lt_enum {r : \u03b1 \u2192 \u03b1 \u2192 Prop} [IsWellOrder \u03b1 r] {o\u2081 o\u2082 : Ordinal} (h\u2081 : o\u2081 < type r)\n    (h\u2082 : o\u2082 < type r) : r (enum r o\u2081 h\u2081) (enum r o\u2082 h\u2082) \u2194 o\u2081 < o\u2082", "theorem enum_zero_eq_bot {o : Ordinal} (ho : 0 < o) :\n    enum (\u00b7 < \u00b7) 0 (by rwa [type_lt]) =\n      haveI H := outOrderBotOfPos ho\n      \u22a5", "theorem eq_zero_or_pos : \u2200 a : Ordinal, a = 0 \u2228 0 < a", "theorem exists_blsub_cof (o : Ordinal) :\n    \u2203 f : \u2200 a < (cof o).ord, Ordinal, blsub.{u, u} _ f = o", "theorem exists_of_lt_bmex {o : Ordinal} {f : \u2200 a < o, Ordinal} {a} (ha : a < bmex o f) :\n    \u2203 i hi, f i hi = a", "theorem foldr_le_nfpFamily (f : \u03b9 \u2192 Ordinal \u2192 Ordinal)\n    (a l) : List.foldr f a l \u2264 nfpFamily.{u, v} f a", "theorem fp_iff_derivBFamily (H : \u2200 i hi, IsNormal (f i hi)) {a} :\n    (\u2200 i hi, f i hi a = a) \u2194 \u2203 b, derivBFamily.{u, v} o f b = a", "theorem fp_iff_derivFamily (H : \u2200 i, IsNormal (f i)) {a} :\n    (\u2200 i, f i a = a) \u2194 \u2203 o, derivFamily.{u, v} f o = a", "theorem has_succ_of_type_succ_lt {\u03b1} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} [wo : IsWellOrder \u03b1 r]\n    (h : \u2200 a < type r, succ a < type r) (x : \u03b1) : \u2203 y, r x y", "theorem le_add_sub (a b : Ordinal) : a \u2264 b + (a - b)", "theorem le_mex_of_forall {\u03b9 : Type u} {f : \u03b9 \u2192 Ordinal.{max u v}} {a : Ordinal}\n    (H : \u2200 b < a, \u2203 i, f i = b) : a \u2264 mex.{_, v} f", "theorem le_of_nadd_le_nadd_left : \u2200 {a b c}, a \u266f b \u2264 a \u266f c \u2192 b \u2264 c", "theorem le_sup {\u03b9 : Type u} (f : \u03b9 \u2192 Ordinal.{max u v}) : \u2200 i, f i \u2264 sup.{_, v} f", "theorem lift_down {a : Ordinal.{u}} {b : Ordinal.{max u v}} (h : b \u2264 lift.{v,u} a) :\n    \u2203 a', lift.{v,u} a' = b", "@[simp]\ntheorem lift_one : lift 1 = 1", "theorem lift_type_eq {\u03b1 : Type u} {\u03b2 : Type v} {r s} [IsWellOrder \u03b1 r] [IsWellOrder \u03b2 s] :\n    lift.{max v w} (type r) = lift.{max u w} (type s) \u2194 Nonempty (r \u2243r s)", "@[simp]\ntheorem limitRecOn_limit {C} (o H\u2081 H\u2082 H\u2083 h) :\n    @limitRecOn C o H\u2081 H\u2082 H\u2083 = H\u2083 o h fun x _h => @limitRecOn C x H\u2081 H\u2082 H\u2083", "theorem lsub_lt_ord {\u03b9} {f : \u03b9 \u2192 Ordinal} {c : Ordinal} (h\u03b9 : #\u03b9 < c.cof) :\n    (\u2200 i, f i < c) \u2192 lsub.{u, u} f < c", "theorem lt_mul_of_limit {a b c : Ordinal} (h : IsLimit c) : a < b * c \u2194 \u2203 c' < c, a < b * c'", "theorem lt_nfpFamily {a b} : a < nfpFamily.{u, v} f b \u2194 \u2203 l, a < List.foldr f b l", "theorem lt_nmul_iff\u2083 :\n    d < a \u2a33 b \u2a33 c \u2194\n      \u2203 a' < a, \u2203 b' < b, \u2203 c' < c,\n        d \u266f a' \u2a33 b' \u2a33 c \u266f a' \u2a33 b \u2a33 c' \u266f a \u2a33 b' \u2a33 c' \u2264\n          a' \u2a33 b \u2a33 c \u266f a \u2a33 b' \u2a33 c \u266f a \u2a33 b \u2a33 c' \u266f a' \u2a33 b' \u2a33 c'", "theorem max_zero_left : \u2200 a : Ordinal, max 0 a = a", "theorem mem_closure_iff_bsup :\n    a \u2208 closure s \u2194\n      \u2203 (o : Ordinal) (_ho : o \u2260 0) (f : \u2200 a < o, Ordinal),\n        (\u2200 i hi, f i hi \u2208 s) \u2227 bsup.{u, u} o f = a", "theorem mul_lt_of_lt_div {a b c : Ordinal} : a < b / c \u2192 c * a < b", "@[simp, norm_cast]\ntheorem natCast_mod (m n : \u2115) : ((m % n : \u2115) : Ordinal) = m % n", "theorem natCast_pos {n : \u2115} : (0 : Ordinal) < n \u2194 0 < n", "theorem nat_lt_limit {o} (h : IsLimit o) : \u2200 n : \u2115, \u2191n < o", "theorem nfp_le {a b} : (\u2200 n, f^[n] a \u2264 b) \u2192 nfp f a \u2264 b", "theorem nmul_assoc : \u2200 a b c, a \u2a33 b \u2a33 c = a \u2a33 (b \u2a33 c)", "theorem nmul_le_nmul_of_nonneg_right (h\u2081 : a \u2264 b) (h\u2082 : 0 \u2264 c) : a \u2a33 c \u2264 b \u2a33 c", "theorem nmul_nadd : \u2200 a b c, a \u2a33 (b \u266f c) = a \u2a33 b \u266f a \u2a33 c", "theorem nmul_nadd_lt\u2083' {a' b' c' : Ordinal} (ha : a' < a) (hb : b' < b) (hc : c' < c) :\n    a' \u2a33 (b \u2a33 c) \u266f a \u2a33 (b' \u2a33 c) \u266f a \u2a33 (b \u2a33 c') \u266f a' \u2a33 (b' \u2a33 c') <\n      a \u2a33 (b \u2a33 c) \u266f a' \u2a33 (b' \u2a33 c) \u266f a' \u2a33 (b \u2a33 c') \u266f a \u2a33 (b' \u2a33 c')", "theorem not_succ_isLimit (o) : \u00acIsLimit (succ o)", "theorem omega_le {o : Ordinal} : \u03c9 \u2264 o \u2194 \u2200 n : \u2115, \u2191n \u2264 o", "lemma omega_lt_omega1 : \u03c9 < \u03c9\u2081 := ord_aleph0.symm.trans_lt (ord_lt_ord.mpr (aleph0_lt_aleph_one))", "@[simp]\ntheorem one_add_of_omega_le {o} (h : \u03c9 \u2264 o) : 1 + o = o", "theorem one_le_iff_ne_zero {o : Ordinal} : 1 \u2264 o \u2194 o \u2260 0", "@[simp]\ntheorem one_lt_card {o} : 1 < card o \u2194 1 < o", "@[simp]\ntheorem one_toPGame_leftMoves_default_eq :\n    (default : (toPGame 1).LeftMoves) = @toLeftMovesToPGame 1 \u27e80, Set.mem_Iio.mpr zero_lt_one\u27e9", "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n    b ^ u * v + w < b ^ u * succ v", "theorem opow_omega {a : Ordinal} (a1 : 1 < a) (h : a < omega) : (a^omega) = omega", "theorem ord_cof_eq (r : \u03b1 \u2192 \u03b1 \u2192 Prop) [IsWellOrder \u03b1 r] :\n    \u2203 S, Unbounded r S \u2227 type (Subrel r S) = (cof (type r)).ord", "theorem principal_add_iff_add_left_eq_self {o : Ordinal} :\n    Principal (\u00b7 + \u00b7) o \u2194 \u2200 a < o, a + o = o", "theorem principal_add_isLimit {o : Ordinal} (ho\u2081 : 1 < o) (ho : Principal (\u00b7 + \u00b7) o) :\n    o.IsLimit", "theorem principal_mul_iff_mul_left_eq {o : Ordinal} :\n    Principal (\u00b7 * \u00b7) o \u2194 \u2200 a, 0 < a \u2192 a < o \u2192 a * o = o", "theorem principal_opow_omega : Principal (\u00b7^\u00b7) omega", "theorem sup_eq_lsub_iff_lt_sup {\u03b9 : Type u} (f : \u03b9 \u2192 Ordinal.{max u v}) :\n    sup.{_, v} f = lsub.{_, v} f \u2194 \u2200 i, f i < sup.{_, v} f", "@[simp]\ntheorem sup_iterate_eq_nfp (f : Ordinal.{u} \u2192 Ordinal.{u}) :\n    (fun a => sup fun n : \u2115 => f^[n] a) = nfp f", "@[simp]\ntheorem sup_mul_nat (o : Ordinal) : (sup fun n : \u2115 => o * n) = o * \u03c9", "@[simp]\ntheorem toNatOrdinal_one : toNatOrdinal 1 = 1", "@[simp]\ntheorem toNatOrdinal_toOrdinal (a : Ordinal) :  NatOrdinal.toOrdinal (toNatOrdinal a) = a", "theorem toPGame_add : \u2200 a b : Ordinal.{u}, a.toPGame + b.toPGame \u2248 (a \u266f b).toPGame", "@[simp]\ntheorem toPGame_eq_iff {a b : Ordinal} : a.toPGame = b.toPGame \u2194 a = b", "@[simp]\ntheorem toPGame_equiv_iff {a b : Ordinal} : (a.toPGame \u2248 b.toPGame) \u2194 a = b", "theorem toPGame_le {a b : Ordinal} (h : a \u2264 b) : a.toPGame \u2264 b.toPGame", "@[simp]\ntheorem toPGame_moveLeft' {o : Ordinal} (i) :\n    o.toPGame.moveLeft i = (toLeftMovesToPGame.symm i).val.toPGame", "@[simp]\ntheorem type_eq_zero_iff_isEmpty [IsWellOrder \u03b1 r] : type r = 0 \u2194 IsEmpty \u03b1", "theorem type_ne_zero_of_nonempty (r) [IsWellOrder \u03b1 r] [h : Nonempty \u03b1] : type r \u2260 0", "@[simp]\ntheorem typein_one_out (x : (1 : Ordinal).out.\u03b1) :\n    @typein _ (\u00b7 < \u00b7) (isWellOrder_out_lt _) x = 0", "theorem univ_id : univ.{u, u + 1} = @type Ordinal (\u00b7 < \u00b7) _", "@[simp]\ntheorem zero_lt_card {o} : 0 < card o \u2194 0 < o", "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a \u2260 0) : (0 : Ordinal) ^ a = 0", "theorem zero_or_succ_or_limit (o : Ordinal) : o = 0 \u2228 (\u2203 a, o = succ a) \u2228 IsLimit o", "theorem gfpApprox_le {a : Ordinal}: gfpApprox f x a \u2264 x", "theorem le_lfpApprox {a : Ordinal} : x \u2264 lfpApprox f x a", "theorem lfp_mem_range_lfpApprox : lfp f \u2208 Set.range (lfpApprox f \u22a5)", "theorem Bounded.dual :\n    \u2200 {t : Ordnode \u03b1} {o\u2081 o\u2082}, Bounded t o\u2081 o\u2082 \u2192 @Bounded \u03b1\u1d52\u1d48 _ (dual t) o\u2082 o\u2081", "theorem Bounded.dual_iff {t : Ordnode \u03b1} {o\u2081 o\u2082} :\n    Bounded t o\u2081 o\u2082 \u2194 @Bounded \u03b1\u1d52\u1d48 _ (.dual t) o\u2082 o\u2081", "theorem Raised.dist_le' {n m} (H : Raised n m) : Nat.dist m n \u2264 1", "theorem Sized.node' {l x r} (hl : @Sized \u03b1 l) (hr : Sized r) : Sized (node' l x r)", "theorem Valid'.balance'_aux {l} {x : \u03b1} {r o\u2081 o\u2082} (hl : Valid' o\u2081 l x) (hr : Valid' x r o\u2082)\n    (H\u2081 : 2 * @size \u03b1 r \u2264 9 * size l + 5 \u2228 size r \u2264 3)\n    (H\u2082 : 2 * @size \u03b1 l \u2264 9 * size r + 5 \u2228 size l \u2264 3) : Valid' o\u2081 (@balance' \u03b1 l x r) o\u2082", "theorem Valid'.of_lt {t : Ordnode \u03b1} {x : \u03b1} {o\u2081 o\u2082} (H : Valid' o\u2081 t o\u2082) (h\u2081 : Bounded nil o\u2081 x)\n    (h\u2082 : All (\u00b7 < x) t) : Valid' o\u2081 t x", "theorem dual_balanceL (l : Ordnode \u03b1) (x : \u03b1) (r : Ordnode \u03b1) :\n    dual (balanceL l x r) = balanceR (dual r) x (dual l)", "theorem insertWith.valid [IsTotal \u03b1 (\u00b7 \u2264 \u00b7)] [@DecidableRel \u03b1 (\u00b7 \u2264 \u00b7)] (f : \u03b1 \u2192 \u03b1) (x : \u03b1)\n    (hf : \u2200 y, x \u2264 y \u2227 y \u2264 x \u2192 x \u2264 f y \u2227 f y \u2264 x) {t} (h : Valid t) : Valid (insertWith f x t)", "theorem node4L_size {l x m y r} (hm : Sized m) :\n    size (@node4L \u03b1 l x m y r) = size l + size m + size r + 2", "protected theorem add_comm (x y : X[S\u207b\u00b9]) : x + y = y + x", "@[to_additive (attr := elab_as_elim, cases_eliminator, induction_eliminator)]\nprotected theorem ind {\u03b2 : X[S\u207b\u00b9] \u2192 Prop}\n    (c : \u2200 (r : X) (s : S), \u03b2 (r /\u2092 s)) : \u2200 q, \u03b2 q", "protected theorem inv_zero : (0 : R[R\u2070\u207b\u00b9])\u207b\u00b9 = 0", "@[to_additive]\nprotected theorem mul_smul (x y : R[S\u207b\u00b9]) (z : X[S\u207b\u00b9]) : (x * y) \u2022 z = x \u2022 y \u2022 z", "theorem oreDiv_add_char' {r r' : X} (s s' : S) (rb : R) (sb : R)\n    (h : sb * s = rb * s') (h' : sb * s \u2208 S) :\n    r /\u2092 s + r' /\u2092 s' = (sb \u2022 r + rb \u2022 r') /\u2092 \u27e8sb * s, h'\u27e9", "theorem right_distrib (x y z : R[S\u207b\u00b9]) : (x + y) * z = x * z + y * z", "@[to_additive (attr := simp)]\ntheorem smul_div_one {p : R} {r : X} {s : S} : (p /\u2092 s) \u2022 (r /\u2092 1) = (p \u2022 r) /\u2092 s", "@[to_additive]\ntheorem universalMulHom_commutes {r : R} : universalMulHom f fS hf (numeratorHom r) = f r", "theorem eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero {x y : V}\n    (h : (o.oangle x y).sign = 0) :\n    x = 0 \u2228 y = 0 \u2228 InnerProductGeometry.angle x y = 0 \u2228 InnerProductGeometry.angle x y = \u03c0", "theorem inner_eq_norm_mul_norm_mul_cos_oangle (x y : V) :\n    \u27eax, y\u27eb = \u2016x\u2016 * \u2016y\u2016 * Real.Angle.cos (o.oangle x y)", "theorem inner_mul_inner_add_areaForm_mul_areaForm' (a x : E) :\n    \u27eaa, x\u27eb \u2022 inner\u209b\u2097 \u211d a + \u03c9 a x \u2022 \u03c9 a = \u2016a\u2016 ^ 2 \u2022 inner\u209b\u2097 \u211d x", "@[simp]\ntheorem inner_rightAngleRotation_right (x y : E) : \u27eax, J y\u27eb = -\u03c9 x y", "@[simp]\ntheorem inner_rotation_pi_div_two_left_smul (x : V) (r : \u211d) :\n    \u27eao.rotation (\u03c0 / 2 : \u211d) x, r \u2022 x\u27eb = 0", "@[simp]\ntheorem inner_smul_rotation_pi_div_two_right (x : V) (r : \u211d) :\n    \u27eax, r \u2022 o.rotation (\u03c0 / 2 : \u211d) x\u27eb = 0", "@[simp]\ntheorem kahler_apply_self (x : E) : o.kahler x x = \u2016x\u2016 ^ 2", "@[simp]\nprotected theorem Orientation.map_neg {\u03b9 : Type*} (f : M \u2243\u2097[R] N) (x : Orientation R M \u03b9) :\n    Orientation.map \u03b9 f (-x) = -Orientation.map \u03b9 f x", "theorem ne_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = (\u03c0 / 2 : \u211d)) : x \u2260 y", "theorem ne_of_oangle_ne_zero {x y : V} (h : o.oangle x y \u2260 0) : x \u2260 y", "theorem norm_div_tan_oangle_add_left_of_oangle_eq_pi_div_two {x y : V}\n    (h : o.oangle x y = \u2191(\u03c0 / 2)) : \u2016x\u2016 / Real.Angle.tan (o.oangle (x + y) y) = \u2016y\u2016", "theorem oangle_eq_of_angle_eq_of_sign_eq {w x y z : V}\n    (h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z)\n    (hs : (o.oangle w x).sign = (o.oangle y z).sign) : o.oangle w x = o.oangle y z", "theorem oangle_eq_pi_iff_oangle_rev_eq_pi {x y : V} : o.oangle x y = \u03c0 \u2194 o.oangle y x = \u03c0", "theorem oangle_eq_zero_or_eq_pi_iff_right_eq_smul {x y : V} :\n    o.oangle x y = 0 \u2228 o.oangle x y = \u03c0 \u2194 x = 0 \u2228 \u2203 r : \u211d, y = r \u2022 x", "@[simp]\ntheorem oangle_sign_sub_right (x y : V) : (o.oangle x (y - x)).sign = (o.oangle x y).sign", "@[simp]\ntheorem oangle_sign_sub_smul_right (x y : V) (r : \u211d) :\n    (o.oangle x (y - r \u2022 x)).sign = (o.oangle x y).sign", "@[simp]\ntheorem Orientation.reindex_refl : (Orientation.reindex R M <| Equiv.refl \u03b9) = Equiv.refl _", "theorem rightAngleRotation_map_complex (f : E \u2243\u2097\u1d62[\u211d] \u2102)\n    (hf : Orientation.map (Fin 2) f.toLinearEquiv o = Complex.orientation) (x : E) :\n    f (J x) = I * f x", "@[simp]\ntheorem two_zsmul_oangle_smul_right_self (x : V) {r : \u211d} : (2 : \u2124) \u2022 o.oangle x (r \u2022 x) = 0", "@[simp]\ntheorem volumeForm_zero_pos [_i : Fact (finrank \u211d E = 0)] :\n    Orientation.volumeForm (positiveOrientation : Orientation \u211d E (Fin 0)) =\n      AlternatingMap.constLinearEquivOfIsEmpty 1", "theorem OrthogonalFamily.inner_right_fintype [Fintype \u03b9] (l : \u2200 i, G i) (i : \u03b9) (v : G i) :\n    \u27eaV i v, \u2211 j : \u03b9, V j (l j)\u27eb = \u27eav, l i\u27eb", "theorem Orthonormal.exists_orthonormalBasis_extension_of_card_eq {\u03b9 : Type*} [Fintype \u03b9]\n    (card_\u03b9 : finrank \ud835\udd5c E = Fintype.card \u03b9) {v : \u03b9 \u2192 E} {s : Set \u03b9}\n    (hv : Orthonormal \ud835\udd5c (s.restrict v)) : \u2203 b : OrthonormalBasis \u03b9 \ud835\udd5c E, \u2200 i \u2208 s, b i = v i", "theorem Orthonormal.isHilbertSum {v : \u03b9 \u2192 E} (hv : Orthonormal \ud835\udd5c v)\n    (hsp : \u22a4 \u2264 (span \ud835\udd5c (Set.range v)).topologicalClosure) :\n    IsHilbertSum \ud835\udd5c (fun _ : \u03b9 => \ud835\udd5c) fun i => LinearIsometry.toSpanSingleton \ud835\udd5c E (hv.1 i)", "theorem OrthonormalBasis.addHaar_eq_volume {\u03b9 F : Type*} [Fintype \u03b9] [NormedAddCommGroup F]\n    [InnerProductSpace \u211d F] [FiniteDimensional \u211d F] [MeasurableSpace F] [BorelSpace F]\n    (b : OrthonormalBasis \u03b9 \u211d F) :\n    b.toBasis.addHaar = volume", "@[simp]\nprotected theorem coe_toBasis_repr_apply (b : OrthonormalBasis \u03b9 \ud835\udd5c E) (x : E) (i : \u03b9) :\n    b.toBasis.repr x i = b.repr x i", "theorem OrthonormalBasis.det_to_matrix_orthonormalBasis_real :\n    a.toBasis.det b = 1 \u2228 a.toBasis.det b = -1", "@[simp]\ntheorem reindex_toBasis (b : OrthonormalBasis \u03b9 \ud835\udd5c E) (e : \u03b9 \u2243 \u03b9') :\n    (b.reindex e).toBasis = b.toBasis.reindex e", "protected theorem sum_repr (b : OrthonormalBasis \u03b9 \ud835\udd5c E) (x : E) : \u2211 i, b.repr x i \u2022 b i = x", "theorem ext_iff {f g : \u03b1 \u2243. \u03b2} : f = g \u2194 \u2200 x, f x = g x", "@[simp]\ntheorem symm_refl : (PEquiv.refl \u03b1).symm = PEquiv.refl \u03b1", "theorem symm_trans_self (f : \u03b1 \u2243. \u03b2) : f.symm.trans f = ofSet { b | (f.symm b).isSome }", "@[simp]\ntheorem toMatrix_apply [DecidableEq n] [Zero \u03b1] [One \u03b1] (f : m \u2243. n) (i j) :\n    toMatrix f i j = if j \u2208 f i then (1 : \u03b1) else 0", "@[simp]\ntheorem trans_bot (f : \u03b1 \u2243. \u03b2) : f.trans (\u22a5 : \u03b2 \u2243. \u03b3) = \u22a5", "@[simp]\ntheorem trans_refl (f : \u03b1 \u2243. \u03b2) : f.trans (PEquiv.refl \u03b2) = f", "@[simp]\ntheorem comp_id (f : \u03b1 \u2192. \u03b2) : f.comp (PFun.id \u03b1) = f", "@[simp]\ntheorem dom_comp (f : \u03b2 \u2192. \u03b3) (g : \u03b1 \u2192. \u03b2) : (f.comp g).Dom = g.preimage f.Dom", "theorem dom_iff_graph (f : \u03b1 \u2192. \u03b2) (x : \u03b1) : x \u2208 f.Dom \u2194 \u2203 y, (x, y) \u2208 f.graph", "theorem ext' {f g : \u03b1 \u2192. \u03b2} (H1 : \u2200 a, a \u2208 Dom f \u2194 a \u2208 Dom g) (H2 : \u2200 a p q, f.fn a p = g.fn a q) :\n    f = g", "theorem fix_fwd_eq {f : \u03b1 \u2192. Sum \u03b2 \u03b1} {a a' : \u03b1} (ha' : Sum.inr a' \u2208 f a) : f.fix a = f.fix a'", "theorem image_inter (s t : Set \u03b1) : f.image (s \u2229 t) \u2286 f.image s \u2229 f.image t", "theorem mem_image (y : \u03b2) (s : Set \u03b1) : y \u2208 f.image s \u2194 \u2203 x \u2208 s, y \u2208 f x", "theorem mem_prodLift {f : \u03b1 \u2192. \u03b2} {g : \u03b1 \u2192. \u03b3} {x : \u03b1} {y : \u03b2 \u00d7 \u03b3} :\n    y \u2208 f.prodLift g x \u2194 y.1 \u2208 f x \u2227 y.2 \u2208 g x", "theorem preimage_mono {s t : Set \u03b2} (h : s \u2286 t) : f.preimage s \u2286 f.preimage t", "theorem preimage_subset_dom (s : Set \u03b2) : f.preimage s \u2286 f.Dom", "theorem liftp_iff' {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) (a : P.A) (f : P.B a \u2192 \u03b1) :\n    @Liftp.{u} P.Obj _ \u03b1 p \u27e8a, f\u27e9 \u2194 \u2200 i, p (f i)", "theorem apply_eq_zero_iff (p : PMF \u03b1) (a : \u03b1) : p a = 0 \u2194 a \u2209 p.support", "theorem monad_seq_eq_seq {\u03b1 \u03b2 : Type u} (q : PMF (\u03b1 \u2192 \u03b2)) (p : PMF \u03b1) : q <*> p = q.seq p", "theorem ofFinset_apply_of_not_mem {a : \u03b1} (ha : a \u2209 s) : ofFinset f s h h' a = 0", "@[simp]\ntheorem ofMultiset_apply (a : \u03b1) : ofMultiset s hs a = s.count a / (Multiset.card s)", "theorem ofMultiset_apply_of_not_mem {a : \u03b1} (ha : a \u2209 s) : ofMultiset s hs a = 0", "theorem pure_map (a : \u03b1) : (pure a).map f = pure (f a)", "@[simp]\ntheorem support_pure : (pure a).support = {a}", "@[simp]\ntheorem toMeasure_ofFintype_apply [MeasurableSpace \u03b1] (hs : MeasurableSet s) :\n    (ofFintype f h).toMeasure s = \u2211' x, s.indicator f x", "@[simp]\ntheorem toMeasure_uniformOfFinset_apply [MeasurableSpace \u03b1] (ht : MeasurableSet t) :\n    (uniformOfFinset s hs).toMeasure t = (s.filter (\u00b7 \u2208 t)).card / s.card", "@[simp]\ntheorem toOuterMeasure_apply_finset (s : Finset \u03b1) : p.toOuterMeasure s = \u2211 x \u2208 s, p x", "@[simp]\ntheorem toOuterMeasure_bindOnSupport_apply :\n    (p.bindOnSupport f).toOuterMeasure s =\n      \u2211' a, p a * if h : p a = 0 then 0 else (f a h).toOuterMeasure s", "theorem Coprime.factor_eq_gcd_right {a b m n : \u2115+} (cop : m.Coprime n) (am : a \u2223 m) (bn : b \u2223 n) :\n    a = (b * a).gcd m", "theorem Coprime.factor_eq_gcd_right_right {a b m n : \u2115+} (cop : m.Coprime n) (am : a \u2223 m)\n    (bn : b \u2223 n) : a = m.gcd (b * a)", "theorem Ico_eq_finset_subtype : Ico a b = (Ico (a : \u2115) b).subtype fun n : \u2115 => 0 < n", "theorem add_one (a : \u2115+) : a + 1 = succPNat a", "@[simp, norm_cast]\ntheorem coe_inj {m n : \u2115+} : (m : \u2115) = n \u2194 m = n", "theorem factorMultiset_gcd (m n : \u2115+) :\n    factorMultiset (gcd m n) = factorMultiset m \u2293 factorMultiset n", "theorem factorMultiset_le_iff {m n : \u2115+} : factorMultiset m \u2264 factorMultiset n \u2194 m \u2223 n", "protected theorem find_min' {m : \u2115+} (hm : p m) : PNat.find h \u2264 m", "protected theorem find_spec : p (PNat.find h)", "theorem gcdA'_coe : (gcdA' a b : \u2115) = gcdW a b + gcdX a b", "theorem gcd_eq_right_iff_dvd {m n : \u2115+} : m \u2223 n \u2194 n.gcd m = m", "theorem le_of_dvd {m n : \u2115+} : m \u2223 n \u2192 m \u2264 n", "theorem lt_add_one_iff : \u2200 {a b : \u2115+}, a < b + 1 \u2194 a \u2264 b", "theorem map_subtype_embedding_uIcc : (uIcc a b).map (Embedding.subtype _) = uIcc \u2191a \u2191b", "@[simp]\nlemma mk_ofNat (n : \u2115) (h : 0 < n) :\n    @Eq \u2115+ (\u27e8no_index (OfNat.ofNat n), h\u27e9 : \u2115+) (haveI : NeZero n := \u27e8h.ne'\u27e9; OfNat.ofNat n)", "theorem mod_coe (m k : \u2115+) :\n    (mod m k : \u2115) = ite ((m : \u2115) % (k : \u2115) = 0) (k : \u2115) ((m : \u2115) % (k : \u2115))", "@[simp]\ntheorem pos (n : \u2115+) : 0 < (n : \u2115)", "protected theorem Resp.Equiv.euc :\n    \u2200 {n} {a b c : Resp n}, Resp.Equiv a b \u2192 Resp.Equiv c b \u2192 Resp.Equiv a c", "theorem Lex.mono {r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop} {s\u2081 s\u2082 : \u2200 i, \u03b1 i \u2192 \u03b1 i \u2192 Prop}\n    (hr : \u2200 a b, r\u2081 a b \u2192 r\u2082 a b) (hs : \u2200 i a b, s\u2081 i a b \u2192 s\u2082 i a b) {a b : \u03a3' i, \u03b1 i}\n    (h : Lex r\u2081 s\u2081 a b) : Lex r\u2082 s\u2082 a b", "theorem lex_wf (ha : WellFounded r) (hb : \u2200 x, WellFounded (s x)) : WellFounded (Lex r s)", "protected theorem le : a \u2264 b", "theorem complete'' : \u2203 q : \u211a_[p], \u2200 \u03b5 > 0, \u2203 N, \u2200 i \u2265 N, padicNormE (f i - q : \u211a_[p]) < \u03b5", "theorem exi_rat_seq_conv {\u03b5 : \u211a} (h\u03b5 : 0 < \u03b5) :\n    \u2203 N, \u2200 i \u2265 N, padicNormE (f i - (limSeq f i : \u211a_[p]) : \u211a_[p]) < \u03b5", "theorem exi_rat_seq_conv_cauchy : IsCauSeq (padicNorm p) (limSeq f)", "@[simp]\ntheorem valuation_zero : valuation (0 : \u211a_[p]) = 0", "@[simp, norm_cast]\ntheorem coe_add (z1 z2 : \u2124_[p]) : ((z1 + z2 : \u2124_[p]) : \u211a_[p]) = z1 + z2", "theorem denseRange_intCast : DenseRange (Int.cast : \u2124 \u2192 \u2124_[p])", "theorem isCauSeq_nthHom (r : R) : IsCauSeq (padicNorm p) fun n => nthHom f r n", "theorem lift_spec (n : \u2115) : (toZModPow n).comp (lift f_compat) = f n", "theorem limNthHom_zero : limNthHom f_compat 0 = 0", "@[simp]\ntheorem mkUnits_eq {u : \u211a_[p]} (h : \u2016u\u2016 = 1) : ((mkUnits h : \u2124_[p]) : \u211a_[p]) = u", "@[simp]\ntheorem mk_zero {h} : (\u27e80, h\u27e9 : \u2124_[p]) = (0 : \u2124_[p])", "theorem norm_int_le_pow_iff_dvd {k : \u2124} {n : \u2115} :\n    \u2016(k : \u2124_[p])\u2016 \u2264 (p : \u211d) ^ (-n : \u2124) \u2194 (p ^ n : \u2124) \u2223 k", "@[simp]\ntheorem valuation_p_pow_mul (n : \u2115) (c : \u2124_[p]) (hc : c \u2260 0) :\n    ((p : \u2124_[p]) ^ n * c).valuation = n + c.valuation", "@[simp]\ntheorem valuation_zero : valuation (0 : \u2124_[p]) = 0", "theorem norm_equiv {f g : PadicSeq p} (hfg : f \u2248 g) : f.norm = g.norm", "@[simp]\ntheorem bind_map {\u03b3} (f : \u03b1 \u2192 \u03b2) (x) (g : \u03b2 \u2192 Part \u03b3) :\n    (map f x).bind g = x.bind fun y => g (f y)", "theorem bind_of_mem {o : Part \u03b1} {a : \u03b1} (h : a \u2208 o) (f : \u03b1 \u2192 Part \u03b2) : o.bind f = f a", "@[to_additive]\ntheorem div_mem_div [Div \u03b1] (a b : Part \u03b1) (ma mb : \u03b1) (ha : ma \u2208 a) (hb : mb \u2208 b) :\n    ma / mb \u2208 a / b", "theorem fix_eq : Part.fix f = f (Part.fix f)", "theorem fix_le {X : (a : _) \u2192 Part <| \u03b2 a} (hX : f X \u2264 X) : Part.fix f \u2264 X", "theorem map_map (g : \u03b2 \u2192 \u03b3) (f : \u03b1 \u2192 \u03b2) (o : Part \u03b1) : map g (map f o) = map (g \u2218 f) o", "@[simp]\ntheorem map_none (f : \u03b1 \u2192 \u03b2) : map f none = none", "@[simp]\ntheorem mem_map_iff (f : \u03b1 \u2192 \u03b2) {o : Part \u03b1} {b} : b \u2208 map f o \u2194 \u2203 a \u2208 o, f a = b", "@[simp]\ntheorem mem_mk_iff {p : Prop} {o : p \u2192 \u03b1} {a : \u03b1} : a \u2208 Part.mk p o \u2194 \u2203 h, o h = a", "@[to_additive]\ntheorem mul_mem_mul [Mul \u03b1] (a b : Part \u03b1) (ma mb : \u03b1) (ha : ma \u2208 a) (hb : mb \u2208 b) :\n    ma * mb \u2208 a * b", "@[simp]\ntheorem of_toOption (o : Part \u03b1) [Decidable o.Dom] : ofOption (toOption o) = o", "theorem one_def [One \u03b1] : (1 : Part \u03b1) = some 1", "theorem card_congr {\u03b1 : Type*} {\u03b2 : Type*} (f : \u03b1 \u2243 \u03b2) : PartENat.card \u03b1 = PartENat.card \u03b2", "theorem card_le_one_iff_subsingleton (\u03b1 : Type*) : card \u03b1 \u2264 1 \u2194 Subsingleton \u03b1", "@[simp]\ntheorem dom_some (x : \u2115) : (some x).Dom", "@[simp]\ntheorem dom_zero : (0 : PartENat).Dom", "lemma lt_coe_succ_iff_le {x : PartENat} {n : \u2115} (hx : x \u2260 \u22a4) : x < n.succ \u2194 x \u2264 n", "theorem lt_find (n : \u2115) (h : \u2200 m \u2264 n, \u00acP m) : (n : PartENat) < find P", "theorem lt_find_iff (n : \u2115) : (n : PartENat) < find P \u2194 \u2200 m \u2264 n, \u00acP m", "theorem copy_eq (e : PartialEquiv \u03b1 \u03b2) (f : \u03b1 \u2192 \u03b2) (hf : \u21d1e = f) (g : \u03b2 \u2192 \u03b1) (hg : \u21d1e.symm = g)\n    (s : Set \u03b1) (hs : e.source = s) (t : Set \u03b2) (ht : e.target = t) :\n    e.copy f hf g hg s hs t ht = e", "theorem image_symm_image_of_subset_target {s : Set \u03b2} (h : s \u2286 e.target) : e '' (e.symm '' s) = s", "protected theorem invOn : InvOn e.symm e e.source e.target", "protected theorem leftInvOn : LeftInvOn e.symm e e.source", "@[simp, mfld_simps]\ntheorem left_inv {x : \u03b1} (h : x \u2208 e.source) : e.symm (e x) = x", "@[simp, mfld_simps]\ntheorem pi_trans (ei : \u2200 i, PartialEquiv (\u03b1i i) (\u03b2i i)) (ei' : \u2200 i, PartialEquiv (\u03b2i i) (\u03b3i i)) :\n    (PartialEquiv.pi ei).trans (PartialEquiv.pi ei') = .pi fun i \u21a6 (ei i).trans (ei' i)", "theorem source_inter_preimage_inv_preimage (s : Set \u03b1) :\n    e.source \u2229 e \u207b\u00b9' (e.symm \u207b\u00b9' s) = e.source \u2229 s", "theorem symm_image_target_inter_eq' (s : Set \u03b2) : e.symm '' (e.target \u2229 s) = e.source \u2229 e \u207b\u00b9' s", "@[simp, mfld_simps]\ntheorem symm_source : e.symm.source = e.target", "@[simp, mfld_simps]\ntheorem symm_target : e.symm.target = e.source", "theorem trans_assoc (e'' : PartialEquiv \u03b3 \u03b4) : (e.trans e').trans e'' = e.trans (e'.trans e'')", "theorem trans_refl_restr (s : Set \u03b2) :\n    e.trans ((PartialEquiv.refl \u03b2).restr s) = e.restr (e \u207b\u00b9' s)", "theorem trans_symm_eq_symm_trans_symm : (e.trans e').symm = e'.symm.trans e.symm", "theorem iff_symm_preimage_eq : e.IsImage s t \u2194 e.target \u2229 e.symm \u207b\u00b9' s = e.target \u2229 t", "theorem leftInvOn_piecewise {e' : PartialHomeomorph X Y} [\u2200 i, Decidable (i \u2208 s)]\n    [\u2200 i, Decidable (i \u2208 t)] (h : e.IsImage s t) (h' : e'.IsImage s t) :\n    LeftInvOn (t.piecewise e.symm e'.symm) (s.piecewise e e') (s.ite e.source e'.source)", "protected theorem mapsTo (h : e.IsImage s t) : MapsTo e (e.source \u2229 s) (e.target \u2229 t)", "theorem comp_symm_deriv {x : M'} (hx : x \u2208 e.target) :\n    (mfderiv I I' e (e.symm x)).comp (mfderiv I' I e.symm x) =\n      ContinuousLinearMap.id \ud835\udd5c (TangentSpace I' x)", "theorem mdifferentiableAt_symm {x : M'} (hx : x \u2208 e.target) : MDifferentiableAt I' I e.symm x", "@[simp, mfld_simps]\ntheorem coe_trans_symm : ((e.trans e').symm : Z \u2192 X) = e.symm \u2218 e'.symm", "theorem continuousWithinAt_iff_continuousWithinAt_comp_left {f : Z \u2192 X} {s : Set Z} {x : Z}\n    (hx : f x \u2208 e.source) (h : f \u207b\u00b9' e.source \u2208 \ud835\udcdd[s] x) :\n    ContinuousWithinAt f s x \u2194 ContinuousWithinAt (e \u2218 f) s x", "theorem eqOnSource_refl : e \u2248 e", "theorem eventually_left_inverse {x} (hx : x \u2208 e.source) :\n    \u2200\u1da0 y in \ud835\udcdd x, e.symm (e y) = y", "theorem extend_coord_change_source_mem_nhdsWithin {x : E}\n    (hx : x \u2208 ((f.extend I).symm \u226b f'.extend I).source) :\n    ((f.extend I).symm \u226b f'.extend I).source \u2208 \ud835\udcdd[range I] x", "theorem extend_image_source_inter :\n    f.extend I '' (f.source \u2229 f'.source) = ((f.extend I).symm \u226b f'.extend I).source", "theorem extend_target_subset_range : (f.extend I).target \u2286 range I", "theorem image_eq_target_inter_inv_preimage {s : Set X} (h : s \u2286 e.source) :\n    e '' s = e.target \u2229 e.symm \u207b\u00b9' s", "theorem _root_.PartialHomeomorph.isLocalStructomorphWithinAt_source_iff {G : StructureGroupoid H}\n    [ClosedUnderRestriction G] (f : PartialHomeomorph H H) {x : H} :\n    G.IsLocalStructomorphWithinAt (\u21d1f) f.source x \u2194\n      x \u2208 f.source \u2192 \u2203 e : PartialHomeomorph H H,\n      e \u2208 G \u2227 e.source \u2286 f.source \u2227 EqOn f (\u21d1e) e.source \u2227 x \u2208 e.source", "theorem isOpen_extend_source : IsOpen (f.extend I).source", "theorem map_extend_symm_nhdsWithin_range {y : M} (hy : y \u2208 f.source) :\n    map (f.extend I).symm (\ud835\udcdd[range I] f.extend I y) = \ud835\udcdd y", "lemma map_source'' : e '' e.source \u2286 e.target", "protected theorem mapsTo : MapsTo e e.source e.target", "theorem ofSet_trans {s : Set X} (hs : IsOpen s) : (ofSet s hs).trans e = e.restr s", "theorem prod_eq_prod_of_nonempty'\n    {eX eX' : PartialHomeomorph X X'} {eY eY' : PartialHomeomorph Y Y'}\n    (h : (eX'.prod eY').source.Nonempty) : eX.prod eY = eX'.prod eY' \u2194 eX = eX' \u2227 eY = eY'", "@[simp, mfld_simps]\ntheorem prod_symm (eX : PartialHomeomorph X X') (eY : PartialHomeomorph Y Y') :\n    (eX.prod eY).symm = eX.symm.prod eY.symm", "@[simp, mfld_simps]\ntheorem refl_symm : (PartialHomeomorph.refl X).symm = PartialHomeomorph.refl X", "theorem self_trans_symm : e.trans e.symm \u2248 PartialHomeomorph.ofSet e.source e.open_source", "theorem singleton_hasGroupoid (h : e.source = Set.univ) (G : StructureGroupoid H)\n    [ClosedUnderRestriction G] : @HasGroupoid _ _ _ _ (e.singletonChartedSpace h) G", "theorem source_inter_preimage_target_inter (s : Set Y) :\n    e.source \u2229 e \u207b\u00b9' (e.target \u2229 s) = e.source \u2229 e \u207b\u00b9' s", "@[simp, mfld_simps]\ntheorem transHomeomorph_transHomeomorph (e : PartialHomeomorph X Y) (f' : Y \u2243\u209c Z) (f'' : Z \u2243\u209c Z') :\n    (e.transHomeomorph f').transHomeomorph f'' = e.transHomeomorph (f'.trans f'')", "theorem trans_apply {x : X} : (e.trans e') x = e' (e x)", "theorem sum_nonneg (x : X) : 0 \u2264 \u2211\u1da0 i, f i x", "theorem const' (s : Part \u03c3) : Partrec fun _ : \u03b1 => s", "theorem nat_iff {f : \u2115 \u2192. \u2115} : Partrec f \u2194 Nat.Partrec f", "theorem to\u2082 {f : \u03b1 \u00d7 \u03b2 \u2192. \u03c3} (hf : Partrec f) : Partrec\u2082 fun a b => f (a, b)", "nonrec theorem comp {f : \u03b2 \u2192 \u03b3 \u2192. \u03c3} {g : \u03b1 \u2192 \u03b2} {h : \u03b1 \u2192 \u03b3} (hf : Partrec\u2082 f) (hg : Computable g)\n    (hh : Computable h) : Partrec fun a => f (g a) (h a)", "theorem comp_pi_eq_pi_comp (\u03b3\u2080 : \u2200 i, Path.Homotopic.Quotient (as i) (bs i))\n    (\u03b3\u2081 : \u2200 i, Path.Homotopic.Quotient (bs i) (cs i)): pi \u03b3\u2080 \u2b1d pi \u03b3\u2081 = pi fun i => \u03b3\u2080 i \u2b1d \u03b3\u2081 i", "theorem reflTransSymmAux_mem_I (x : I \u00d7 I) : reflTransSymmAux x \u2208 I", "theorem transAssocReparamAux_zero : transAssocReparamAux 0 = 0", "theorem delayReflLeft_one (\u03b3 : Path x y) : delayReflLeft 1 \u03b3 = \u03b3", "@[simp]\ntheorem map_id (\u03b3 : Path x y) : \u03b3.map continuous_id = \u03b3", "@[simp]\ntheorem refl_range {a : X} : range (Path.refl a) = {a}", "@[simp]\ntheorem truncate_self {a b : X} (\u03b3 : Path a b) (t : \u211d) :\n    \u03b3.truncate t t = (Path.refl <| \u03b3.extend t).cast (by rw [min_self]) rfl", "@[simp]\ntheorem truncate_zero_one {a b : X} (\u03b3 : Path a b) :\n    \u03b3.truncate 0 1 = \u03b3.cast (by simp [zero_le_one, extend_zero]) (by simp)", "theorem d_pos {a : Solution\u2081 d} (h : IsFundamental a) : 0 < d", "@[simp]\ntheorem coe_mk (x y : \u2124) (prop : x ^ 2 - d * y ^ 2 = 1) : (\u2191(mk x y prop) : \u2124\u221ad) = \u27e8x, y\u27e9", "@[simp]\ntheorem x_mk (x y : \u2124) (prop : x ^ 2 - d * y ^ 2 = 1) : (mk x y prop).x = x", "theorem y_zpow_pos {a : Solution\u2081 d} (hax : 0 < a.x) (hay : 0 < a.y) {n : \u2124} (hn : 0 < n) :\n    0 < (a ^ n).y", "theorem eq_pellZd (b : \u2124\u221a(d a1)) (b1 : 1 \u2264 b) (hp : IsPell b) : \u2203 n, b = pellZd a1 n", "theorem eq_pell_lem : \u2200 (n) (b : \u2124\u221a(d a1)), 1 \u2264 b \u2192 IsPell b \u2192\n    b \u2264 pellZd a1 n \u2192 \u2203 n, b = pellZd a1 n", "theorem isPell_mul {b c : \u2124\u221ad} (hb : IsPell b) (hc : IsPell c) : IsPell (b * c)", "@[simp]\ntheorem pellZd_im (n : \u2115) : (pellZd a1 n).im = yn a1 n", "theorem pellZd_succ_succ (n) :\n    pellZd a1 (n + 2) + pellZd a1 n = (2 * a : \u2115) * pellZd a1 (n + 1)", "theorem x_sub_y_dvd_pow_lem (y2 y1 y0 yn1 yn0 xn1 xn0 ay a2 : \u2124) :\n    (a2 * yn1 - yn0) * ay + y2 - (a2 * xn1 - xn0) =\n      y2 - a2 * y1 + y0 + a2 * (yn1 * ay + y1 - xn1) - (yn0 * ay + y0 - xn0)", "theorem xn_modEq_x2n_sub {n j} (h : j \u2264 2 * n) : xn a1 (2 * n - j) + xn a1 j \u2261 0 [MOD xn a1 n]", "theorem xy_modEq_of_modEq {a b c} (a1 : 1 < a) (b1 : 1 < b) (h : a \u2261 b [MOD c]) :\n    \u2200 n, xn a1 n \u2261 xn b1 n [MOD c] \u2227 yn a1 n \u2261 yn b1 n [MOD c]", "theorem y_mul_dvd (n) : \u2200 k, yn a1 n \u2223 yn a1 (n * k)", "theorem yn_add (m n) : yn a1 (m + n) = xn a1 m * yn a1 n + yn a1 m * xn a1 n", "theorem separable_iff_squarefree {g : K[X]} : g.Separable \u2194 Squarefree g", "@[simp]\ntheorem apply_toDualLeft_symm_apply (f : Dual R N) (x : N) : p (p.toDualLeft.symm f) x = f x", "theorem liftAux_apply (x : L) (n : \u2115) (y : K) (h : i y = x ^ p ^ n) :\n    liftAux i j p x = (iterateFrobeniusEquiv M p n).symm (j y)", "theorem lift_comp_lift_apply_eq_self [PerfectRing L p] (x : L) :\n    lift j i p (lift i j p x) = x", "theorem frobenius_pthRoot : (frobenius _ p).comp (pthRoot R p) = RingHom.id _", "theorem hom_ext [PerfectRing R p] {S : Type u\u2082} [CommSemiring S] [CharP S p] {P : Type u\u2083}\n    [CommSemiring P] [CharP P p] [PerfectRing P p] (\u03c0 : P \u2192+* S) (m : PerfectionMap p \u03c0)\n    {f g : R \u2192+* P} (hfg : \u2200 x, \u03c0 (f x) = \u03c0 (g x)) : f = g", "theorem of : PerfectionMap p (Perfection.coeff R p 0)", "theorem Lex.noMaxOrder' [Preorder \u03b9] [\u2200 i, LT (\u03b2 i)] (i : \u03b9) [NoMaxOrder (\u03b2 i)] :\n    NoMaxOrder (Lex (\u2200 i, \u03b2 i))", "@[simp] lemma ceilDiv_apply (f : \u2200 i, \u03c0 i) (a : \u03b1) (i : \u03b9) : (f \u2308/\u2309 a) i = f i \u2308/\u2309 a := rfl", "theorem continuous_uncurry : Continuous <| monotoneUncurry \u03b1 \u03b2 \u03b3", "@[simp] lemma floorDiv_apply (f : \u2200 i, \u03c0 i) (a : \u03b1) (i : \u03b9) : (f \u230a/\u230b a) i = f i \u230a/\u230b a := rfl", "lemma floorDiv_def (f : \u2200 i, \u03c0 i) (a : \u03b1) : f \u230a/\u230b a = fun i \u21a6 f i \u230a/\u230b a := rfl", "@[to_additive]\ntheorem Pi.hasProd {f : \u03b9 \u2192 \u2200 x, \u03c0 x} {g : \u2200 x, \u03c0 x} :\n    HasProd f g \u2194 \u2200 x, HasProd (fun i \u21a6 f i x) (g x)", "lemma Pi.induced_precomp [TopologicalSpace Y] {\u03b9' : Type*} (\u03c6 : \u03b9' \u2192 \u03b9) :\n    induced (\u00b7 \u2218 \u03c6) Pi.topologicalSpace =\n    \u2a05 i', induced (eval (\u03c6 i')) \u2039TopologicalSpace Y\u203a", "theorem isAtom_iff {f : \u2200 i, \u03c0 i} [\u2200 i, PartialOrder (\u03c0 i)] [\u2200 i, OrderBot (\u03c0 i)] :\n    IsAtom f \u2194 \u2203 i, IsAtom (f i) \u2227 \u2200 j, j \u2260 i \u2192 f j = \u22a5", "theorem Pi.isPWO {\u03b1 : \u03b9 \u2192 Type*} [\u2200 i, LinearOrder (\u03b1 i)] [\u2200 i, IsWellOrder (\u03b1 i) (\u00b7 < \u00b7)]\n    [Finite \u03b9] (s : Set (\u2200 i, \u03b1 i)) : s.IsPWO", "@[simp]\ntheorem kstar_apply (a : \u2200 i, \u03c0 i) (i : \u03b9) : a\u2217 i = (a i)\u2217", "@[to_additive (attr := simp)] lemma leOnePart_apply (f : \u2200 i, \u03b1 i) (i : \u03b9) : f\u207b i = (f i)\u207b := rfl", "@[to_additive] lemma leOnePart_def (f : \u2200 i, \u03b1 i) : f\u207b = fun i \u21a6 (f i)\u207b := rfl", "@[to_additive (attr := simp)]\ntheorem mulSingle_eq_of_ne {i i' : I} (h : i' \u2260 i) (x : f i) : mulSingle i x i' = 1", "theorem nonUnitalRingHom_injective {\u03b3 : Type w} [Nonempty I]\n    [\u2200 i, NonUnitalNonAssocSemiring (f i)] [NonUnitalNonAssocSemiring \u03b3] (g : \u2200 i, \u03b3 \u2192\u2099+* f i)\n    (hg : \u2200 i, Function.Injective (g i)) : Function.Injective (Pi.nonUnitalRingHom g)", "@[simp]\ntheorem Pi.sdiff_apply [\u2200 i, SDiff (\u03c0 i)] (x y : \u2200 i, \u03c0 i) (i : \u03b9) :\n    (x \\ y) i = x i \\ y i", "lemma Pi.uniformContinuous_restrict (S : Set \u03b9) :\n    UniformContinuous (S.restrict : (\u2200 i : \u03b9, \u03b1 i) \u2192 (\u2200 i : S, \u03b1 i))", "lemma Pi.uniformSpace_comap_restrict (S : Set \u03b9) :\n    UniformSpace.comap (S.restrict) (Pi.uniformSpace (fun i : S \u21a6 \u03b1 i)) =\n    \u2a05 i \u2208 S, UniformSpace.comap (eval i) (U i)", "theorem Pi.uniformity :\n    \ud835\udce4 (\u2200 i, \u03b1 i) = \u2a05 i : \u03b9, (Filter.comap fun a => (a.1 i, a.2 i)) (\ud835\udce4 (\u03b1 i))", "nonrec theorem basis_toMatrix_basisFun_mul [Fintype \u03b9]\n    {\ud835\udd5c} [SeminormedCommRing \ud835\udd5c] (b : Basis \u03b9 \ud835\udd5c (PiLp p fun _ : \u03b9 => \ud835\udd5c))\n    (A : Matrix \u03b9 \u03b9 \ud835\udd5c) :\n    b.toMatrix (PiLp.basisFun _ _ _) * A =\n      Matrix.of fun i j => b.repr ((WithLp.equiv _ _).symm (A\u1d40 j)) i", "theorem dist_eq_iSup (f g : PiLp \u221e \u03b1) : dist f g = \u2a06 i, dist (f i) (g i)", "theorem dist_eq_sum {p : \u211d\u22650\u221e} (hp : 0 < p.toReal) (f g : PiLp p \u03b1) :\n    dist f g = (\u2211 i, dist (f i) (g i) ^ p.toReal) ^ (1 / p.toReal)", "@[simp]\ntheorem edist_equiv_symm_single_same (i : \u03b9) (b\u2081 b\u2082 : \u03b2 i) :\n    edist\n        ((WithLp.equiv p (\u2200 i, \u03b2 i)).symm (Pi.single i b\u2081))\n        ((WithLp.equiv p (\u2200 i, \u03b2 i)).symm (Pi.single i b\u2082)) =\n      edist b\u2081 b\u2082", "theorem norm_eq_ciSup (f : PiLp \u221e \u03b2) : \u2016f\u2016 = \u2a06 i, \u2016f i\u2016", "theorem norm_eq_sum (hp : 0 < p.toReal) (f : PiLp p \u03b2) :\n    \u2016f\u2016 = (\u2211 i, \u2016f i\u2016 ^ p.toReal) ^ (1 / p.toReal)", "theorem apply_firstDiff_ne {x y : \u2200 n, E n} (h : x \u2260 y) :\n    x (firstDiff x y) \u2260 y (firstDiff x y)", "theorem cylinder_eq_res (x : \u2115 \u2192 \u03b1) (n : \u2115) :\n    cylinder x n = { y | res y n = res x n }", "theorem dist_triangle_nonarch (x y z : \u2200 n, E n) : dist x z \u2264 max (dist x y) (dist y z)", "theorem exists_disjoint_cylinder {s : Set (\u2200 n, E n)} (hs : IsClosed s) {x : \u2200 n, E n}\n    (hx : x \u2209 s) : \u2203 n, Disjoint s (cylinder x n)", "theorem exists_retraction_of_isClosed {s : Set (\u2200 n, E n)} (hs : IsClosed s) (hne : s.Nonempty) :\n    \u2203 f : (\u2200 n, E n) \u2192 \u2200 n, E n, (\u2200 x \u2208 s, f x = x) \u2227 range f = s \u2227 Continuous f", "theorem exists_retraction_subtype_of_isClosed {s : Set (\u2200 n, E n)} (hs : IsClosed s)\n    (hne : s.Nonempty) :\n    \u2203 f : (\u2200 n, E n) \u2192 s, (\u2200 x : s, f x = x) \u2227 Surjective f \u2227 Continuous f", "theorem shortestPrefixDiff_pos {s : Set (\u2200 n, E n)} (hs : IsClosed s) (hne : s.Nonempty)\n    {x : \u2200 n, E n} (hx : x \u2209 s) : 0 < shortestPrefixDiff x s", "@[elab_as_elim]\nprotected theorem induction_on' {motive : (\u2a02[R] i, s i) \u2192 Prop} (z : \u2a02[R] i, s i)\n    (tprodCoeff : \u2200 (r : R) (f : \u03a0 i, s i), motive (tprodCoeff R r f))\n    (add : \u2200 x y, motive x \u2192 motive y \u2192 motive (x + y)) :\n    motive z", "theorem liftAux_tprod (\u03c6 : MultilinearMap R s E) (f : \u03a0 i, s i) : liftAux \u03c6 (tprod R f) = \u03c6 f", "@[simp]\ntheorem liftIsometry_tprodL :\n    liftIsometry \ud835\udd5c E _ (tprodL \ud835\udd5c) = ContinuousLinearMap.id \ud835\udd5c (\u2a02[\ud835\udd5c] i, E i)", "@[simp]\ntheorem lift_comp_reindex_symm (e : \u03b9 \u2243 \u03b9\u2082) (\u03c6 : MultilinearMap R s E) :\n    lift \u03c6 \u2218\u2097 (reindex R s e).symm = lift (domDomCongrLinearEquiv' R R s _ e \u03c6)", "@[simp]\ntheorem lift_reindex_symm\n    (e : \u03b9 \u2243 \u03b9\u2082) (\u03c6 : MultilinearMap R s E) (x : \u2a02[R] i, s (e.symm i)) :\n    lift \u03c6 (reindex R s e |>.symm x) = lift (domDomCongrLinearEquiv' R R s _ e \u03c6) x", "lemma lifts_zero : 0 \u2208 lifts (0 : \u2a02[R] i, s i)", "@[simp]\ntheorem mapL_apply (x : \u2a02[\ud835\udd5c] i, E i) : mapL f x = map (fun i \u21a6 (f i).toLinearMap) x", "@[simp]\ntheorem map_id : map (fun i \u21a6 (LinearMap.id : s i \u2192\u2097[R] s i)) = .id", "protected theorem map_smul [DecidableEq \u03b9] (i : \u03b9) (c : R) (u : s i \u2192\u2097[R] t i) :\n    map (update f i (c \u2022 u)) = c \u2022 map (update f i u)", "lemma mem_lifts_iff (x : \u2a02[R] i, s i) (p : FreeAddMonoid (R \u00d7 \u03a0 i, s i)) :\n    p \u2208 lifts x \u2194 List.sum (List.map (fun x \u21a6 x.1 \u2022 \u2a02\u209c[R] i, x.2 i) p) = x", "theorem norm_eval_le_projectiveSeminorm (x : \u2a02[\ud835\udd5c] i, E i) (G : Type*) [SeminormedAddCommGroup G]\n    [NormedSpace \ud835\udd5c G] (f : ContinuousMultilinearMap \ud835\udd5c E G) :\n    \u2016lift f.toMultilinearMap x\u2016 \u2264 projectiveSeminorm x * \u2016f\u2016", "theorem projectiveSeminormAux_add_le (p q : FreeAddMonoid (\ud835\udd5c \u00d7 \u03a0 i, E i)) :\n    projectiveSeminormAux (p + q) \u2264 projectiveSeminormAux p + projectiveSeminormAux q", "@[simp]\ntheorem reindex_symm (e : \u03b9 \u2243 \u03b9\u2082) :\n    (reindex R (fun _ \u21a6 M) e).symm = reindex R (fun _ \u21a6 M) e.symm", "@[simp]\ntheorem reindex_tprod (e : \u03b9 \u2243 \u03b9\u2082) (f : \u03a0 i, s i) :\n    reindex R s e (tprod R f) = tprod R fun i \u21a6 f (e.symm i)", "protected theorem nonempty_Icc : (Icc v.tMin v.tMax).Nonempty", "@[simp, norm_cast]\ntheorem toConvexCone_positive : \u2191(positive \ud835\udd5c E) = ConvexCone.positive \ud835\udd5c E", "@[simp]\ntheorem coe_sub (f g : Poly \u03b1) : \u21d1(f - g) = f - g", "@[simp]\ntheorem sub_apply (f g : Poly \u03b1) (x : \u03b1 \u2192 \u2115) : (f - g) x = f x - g x", "theorem C_add : C (a + b) = C a + C b", "theorem C_eq_algebraMap (r : R) : C r = algebraMap R R[X] r", "theorem C_leadingCoeff_mul_prod_multiset_X_sub_C (hroots : Multiset.card p.roots = p.natDegree) :\n    C p.leadingCoeff * (p.roots.map fun a => X - C a).prod = p", "theorem C_mul : C (a * b) = C a * C b", "@[simp]\ntheorem C_mul_monomial : C a * monomial n b = monomial n (a * b)", "theorem C_sub : C (a - b) = C a - C b", "@[simp]\ntheorem U_add_two : \u2200 n, U R (n + 2) = 2 * X * U R (n + 1) - U R n", "@[simp]\ntheorem U_one : U R 1 = 2 * X", "@[simp]\ntheorem aeval_T (x : A) (n : \u2124) : aeval x (T R n) = (T A n).eval x", "@[elab_as_elim]\nprotected theorem induct (motive : \u2124 \u2192 Prop)\n    (zero : motive 0)\n    (one : motive 1)\n    (add_two : \u2200 (n : \u2115), motive (\u2191n + 1) \u2192 motive \u2191n \u2192 motive (\u2191n + 2))\n    (neg_add_one : \u2200 (n : \u2115), motive (-\u2191n) \u2192 motive (-\u2191n + 1) \u2192 motive (-\u2191n - 1)) :\n    \u2200 (a : \u2124), motive a", "theorem one_sub_X_sq_mul_U_eq_pol_in_T (n : \u2124) :\n    (1 - X ^ 2) * U R n = X * T R (n + 1) - T R (n + 2)", "@[ext]\ntheorem ext {\u03c3 \u03c4 : p.Gal} (h : \u2200 x \u2208 p.rootSet p.SplittingField, \u03c3 x = \u03c4 x) : \u03c3 = \u03c4", "theorem galActionHom_bijective_of_prime_degree' {p : \u211a[X]} (p_irr : Irreducible p)\n    (p_deg : p.natDegree.Prime)\n    (p_roots1 : Fintype.card (p.rootSet \u211d) + 1 \u2264 Fintype.card (p.rootSet \u2102))\n    (p_roots2 : Fintype.card (p.rootSet \u2102) \u2264 Fintype.card (p.rootSet \u211d) + 3) :\n    Function.Bijective (galActionHom p \u2102)", "theorem galActionHom_injective [Fact (p.Splits (algebraMap F E))] :\n    Function.Injective (galActionHom p E)", "theorem IsPrimitive.dvd_iff_fraction_map_dvd_fraction_map {p q : R[X]} (hp : p.IsPrimitive)\n    (hq : q.IsPrimitive) : p \u2223 q \u2194 p.map (algebraMap R K) \u2223 q.map (algebraMap R K)", "theorem IsPrimitive.irreducible_of_irreducible_map_of_injective (h_irr : Irreducible (map \u03c6 f)) :\n    Irreducible f", "theorem IsRoot.of_map {R} [CommRing R] {f : R \u2192+* S} {x : R} {p : R[X]} (h : IsRoot (p.map f) (f x))\n    (hf : Function.Injective f) : IsRoot p x", "theorem adjoin_rootSet_eq_range [Algebra K F] (f : K[X]) [IsSplittingField K L f] (i : L \u2192\u2090[K] F) :\n    Algebra.adjoin K (rootSet f F) = i.range", "theorem irreducible_of_coprime' (hp : IsUnitTrinomial p)\n    (h : \u2200 z : \u2102, \u00ac(aeval z p = 0 \u2227 aeval z (mirror p) = 0)) : Irreducible p", "theorem map {A : Type v} [CommRing A] (\u03c6 : R \u2192+* A) : (f.map \u03c6).IsWeaklyEisensteinAt (\ud835\udcdf.map \u03c6)", "theorem Monic.isRegular {R : Type*} [Ring R] {p : R[X]} (hp : Monic p) : IsRegular p", "theorem _root_.Polynomial.Monic.leadingCoeff_not_mem (hf : f.Monic) (h : \ud835\udcdf \u2260 \u22a4) :\n    \u00acf.leadingCoeff \u2208 \ud835\udcdf", "theorem Monic.mul_left_ne_zero (hp : Monic p) {q : R[X]} (hq : q \u2260 0) : q * p \u2260 0", "theorem natSepDegree_eq_one_iff (q : \u2115) [ExpChar F q] (hm : f.Monic) :\n    f.natSepDegree = 1 \u2194 \u2203 (m n : \u2115) (y : F), m \u2260 0 \u2227 f = (X ^ q ^ n - C y) ^ m", "theorem Polynomial.Separable.map_irreducible_of_isPurelyInseparable {f : F[X]} (hsep : f.Separable)\n    (hirr : Irreducible f) [IsPurelyInseparable F E] : Irreducible (f.map (algebraMap F E))", "theorem Separable.unit_mul {f g : R[X]} (hf : IsUnit f) (hg : g.Separable) : (f * g).Separable", "theorem succ (n : \u2115) (f : K[X]) :\n    SplittingFieldAux (n + 1) f = SplittingFieldAux n f.removeFactor", "theorem X_eq_normalize : (X : Polynomial R) = normalize X", "theorem X_pow_sub_C_ne_zero {n : \u2115} (hn : 0 < n) (a : R) : (X : R[X]) ^ n - C a \u2260 0", "theorem X_pow_sub_one_separable_iff {n : \u2115} : (X ^ n - 1 : F[X]).Separable \u2194 (n : F) \u2260 0", "theorem addHom_ext {M : Type*} [AddMonoid M] {f g : R[X] \u2192+ M}\n    (h : \u2200 n a, f (monomial n a) = g (monomial n a)) : f = g", "@[simp]\ntheorem adjoin_X : Algebra.adjoin R ({X} : Set R[X]) = \u22a4", "@[simp]\ntheorem aevalTower_comp_C : (aevalTower g y : R[X] \u2192+* A').comp C = g", "theorem aeval_X_pow {n : \u2115} : aeval x ((X : R[X]) ^ n) = x ^ n", "lemma aeval_ofReal (p : \u211d[X]) (x : \u211d) : aeval (RCLike.ofReal x : K) p = eval x p", "@[simp]\ntheorem aroots_C [CommRing S] [IsDomain S] [Algebra T S] (a : T) : (C a).aroots S = 0", "@[simp]\ntheorem aroots_X [CommRing S] [IsDomain S] [Algebra T S] :\n    aroots (X : T[X]) S = {0}", "theorem as_sum_range' (p : R[X]) (n : \u2115) (w : p.natDegree < n) :\n    p = \u2211 i \u2208 range n, monomial i (coeff p i)", "theorem ascPochhammer_smeval_neg_eq_descPochhammer (r : R) (k : \u2115) :\n    (ascPochhammer \u2115 k).smeval (-r) = (-1)^k * (descPochhammer \u2124 k).smeval r", "theorem associated_of_dvd_of_natDegree_le {K} [Field K] {p q : K[X]} (hpq : p \u2223 q) (hq : q \u2260 0)\n    (h\u2081 : q.natDegree \u2264 p.natDegree) : Associated p q", "theorem card_nthRoots (n : \u2115) (a : R) : Multiset.card (nthRoots n a) \u2264 n", "theorem card_roots_toFinset_le_derivative (p : \u211d[X]) :\n    p.roots.toFinset.card \u2264 p.derivative.roots.toFinset.card + 1", "theorem card_support_eq_two :\n    f.support.card = 2 \u2194\n      \u2203 (k m : \u2115) (hkm : k < m) (x y : R) (hx : x \u2260 0) (hy : y \u2260 0),\n        f = C x * X ^ k + C y * X ^ m", "theorem cardinal_mk_le_max {R : Type u} [Semiring R] : #(R[X]) \u2264 max #R \u2135\u2080", "@[simp, norm_cast]\ntheorem coe_one : ((1 : R[X]) : PowerSeries R) = 1", "theorem coeff_C_mul_X (x : R) (n : \u2115) : coeff (C x * X : R[X]) n = if n = 1 then x else 0", "theorem coeff_C_mul_X_pow (x : R) (k n : \u2115) :\n    coeff (C x * X ^ k : R[X]) n = if n = k then x else 0", "@[simp]\ntheorem coeff_X_pow_mul (p : R[X]) (n d : \u2115) :\n    coeff (Polynomial.X ^ n * p) (d + n) = coeff p d", "theorem coeff_comp_degree_mul_degree (hqd0 : natDegree q \u2260 0) :\n    coeff (p.comp q) (natDegree p * natDegree q) =\n    leadingCoeff p * leadingCoeff q ^ natDegree p", "theorem coeff_divByMonic_X_sub_C (p : R[X]) (a : R) (n : \u2115) :\n    (p /\u2098 (X - C a)).coeff n = \u2211 i \u2208 Icc (n + 1) p.natDegree, a ^ (i - (n + 1)) * p.coeff i", "theorem coeff_eq_zero_of_lt_natTrailingDegree {p : R[X]} {n : \u2115} (h : n < p.natTrailingDegree) :\n    p.coeff n = 0", "theorem coeff_eq_zero_of_natDegree_lt {p : R[X]} {n : \u2115} (h : p.natDegree < n) :\n    p.coeff n = 0", "theorem coeff_hermite_succ_succ (n k : \u2115) : coeff (hermite (n + 1)) (k + 1) =\n    coeff (hermite n) k - (k + 2) * coeff (hermite n) (k + 2)", "theorem coeff_isUnit_isNilpotent_of_isUnit (hunit : IsUnit P) :\n    IsUnit (P.coeff 0) \u2227 (\u2200 i, i \u2260 0 \u2192 IsNilpotent (P.coeff i))", "@[simp]\ntheorem coeff_mul_C (p : R[X]) (n : \u2115) (a : R) : coeff (p * C a) n = coeff p n * a", "theorem coeff_mul_monomial (p : R[X]) (n d : \u2115) (r : R) :\n    coeff (p * monomial n r) (d + n) = coeff p d * r", "@[simp]\ntheorem coeff_restriction {p : R[X]} {n : \u2115} : \u2191(coeff (restriction p) n) = coeff p n", "theorem coeff_sum [Semiring S] (n : \u2115) (f : \u2115 \u2192 R \u2192 S[X]) :\n    coeff (p.sum f) n = p.sum fun a b => coeff (f a b) n", "theorem coeff_toSubring' {n : \u2115} : (coeff (toSubring p T hp) n).1 = coeff p n", "theorem coeff_update_apply (p : R[X]) (n : \u2115) (a : R) (i : \u2115) :\n    (p.update n a).coeff i = if i = n then a else p.coeff i", "@[simp]\ntheorem coeff_update_same (p : R[X]) (n : \u2115) (a : R) : (p.update n a).coeff n = a", "theorem coeff_zero_multiset_prod : t.prod.coeff 0 = (t.map fun f => coeff f 0).prod", "theorem comp_X_add_C_ne_zero_iff {p : R[X]} (t : R) :\n    p.comp (X + C t) \u2260 0 \u2194 p \u2260 0", "@[simp]\ntheorem content_C {r : R} : (C r).content = normalize r", "theorem content_eq_gcd_leadingCoeff_content_eraseLead (p : R[X]) :\n    p.content = GCDMonoid.gcd p.leadingCoeff (eraseLead p).content", "@[continuity, fun_prop]\nprotected theorem continuous_eval\u2082 [Semiring S] (p : S[X]) (f : S \u2192+* R) :\n    Continuous fun x => p.eval\u2082 f x", "theorem cyclotomic'.monic (n : \u2115) (R : Type*) [CommRing R] [IsDomain R] :\n    (cyclotomic' n R).Monic", "theorem cyclotomic_eq_prod_X_pow_sub_one_pow_moebius {n : \u2115} (R : Type*) [CommRing R]\n    [IsDomain R] : algebraMap _ (RatFunc R) (cyclotomic n R) =\n      \u220f i \u2208 n.divisorsAntidiagonal, algebraMap R[X] _ (X ^ i.snd - 1) ^ \u03bc i.fst", "theorem cyclotomic_irreducible_of_irreducible_pow {p : \u2115} (hp : Nat.Prime p) {R} [CommRing R]\n    [IsDomain R] {n : \u2115} (hn : n \u2260 0) (h : Irreducible (cyclotomic (p ^ n) R)) :\n    Irreducible (cyclotomic p R)", "theorem degree_add_eq_of_leadingCoeff_add_ne_zero (h : leadingCoeff p + leadingCoeff q \u2260 0) :\n    degree (p + q) = max p.degree q.degree", "theorem degree_add_eq_right_of_degree_lt (h : degree p < degree q) : degree (p + q) = degree q", "theorem degree_cyclotomic' {\u03b6 : R} {n : \u2115} (h : IsPrimitiveRoot \u03b6 n) :\n    (cyclotomic' n R).degree = Nat.totient n", "@[simp]\ntheorem degree_eq_bot : degree p = \u22a5 \u2194 p = 0", "theorem degree_eq_card_roots' {p : K[X]} {i : K \u2192+* L} (p_ne_zero : p.map i \u2260 0)\n    (hsplit : Splits i p) : (p.map i).degree = Multiset.card (p.map i).roots", "theorem degree_mono [Semiring S] {f : R[X]} {g : S[X]} (h : f.support \u2286 g.support) :\n    f.degree \u2264 g.degree", "theorem degree_mul_le_of_le {a b : WithBot \u2115} (hp : degree p \u2264 a) (hq : degree q \u2264 b) :\n    degree (p * q) \u2264 a + b", "theorem degree_pos_of_ne_zero_of_nonunit (hp0 : p \u2260 0) (hp : \u00acIsUnit p) : 0 < degree p", "theorem degree_pow_le_of_le {a : WithBot \u2115} (b : \u2115) (hp : degree p \u2264 a) :\n    degree (p ^ b) \u2264 b * a", "theorem degree_prod [Nontrivial R] : (\u220f i \u2208 s, f i).degree = \u2211 i \u2208 s, (f i).degree", "theorem degree_quadratic_lt_degree_C_mul_X_cb (ha : a \u2260 0) :\n    degree (C b * X ^ 2 + C c * X + C d) < degree (C a * X ^ 3)", "theorem degree_sub_lt (hd : degree p = degree q) (hp0 : p \u2260 0)\n    (hlc : leadingCoeff p = leadingCoeff q) : degree (p - q) < degree p", "theorem degree_sum_fin_lt {n : \u2115} (f : Fin n \u2192 R) :\n    degree (\u2211 i : Fin n, C (f i) * X ^ (i : \u2115)) < n", "protected theorem derivWithin (hxs : UniqueDiffWithinAt \ud835\udd5c s x) :\n    derivWithin (fun x => p.eval x) s x = p.derivative.eval x", "theorem derivative_X_add_C (c : R) : derivative (X + C c) = 1", "theorem derivative_comp (p q : R[X]) :\n    derivative (p.comp q) = derivative q * p.derivative.comp q", "@[simp]\ntheorem derivative_mul {f g : R[X]} : derivative (f * g) = derivative f * g + f * derivative g", "@[simp]\ntheorem derivative_one : derivative (1 : R[X]) = 0", "protected theorem differentiableAt : DifferentiableAt \ud835\udd5c (fun x => p.eval x) x", "theorem divByMonic_eq_div (p : R[X]) (hq : Monic q) : p /\u2098 q = p / q", "theorem divX_add : divX (p + q) = divX p + divX q", "theorem div_modByMonic_unique {f g} (q r : R[X]) (hg : Monic g)\n    (h : r + g * q = f \u2227 degree r < degree g) : f /\u2098 g = q \u2227 f %\u2098 g = r", "theorem div_tendsto_atTop_of_degree_gt' (hdeg : Q.degree < P.degree)\n    (hpos : 0 < P.leadingCoeff / Q.leadingCoeff) :\n    Tendsto (fun x => eval x P / eval x Q) atTop atTop", "theorem dvd_iff_content_dvd_content_and_primPart_dvd_primPart {p q : R[X]} (hq : q \u2260 0) :\n    p \u2223 q \u2194 p.content \u2223 q.content \u2227 p.primPart \u2223 q.primPart", "theorem eq_of_dvd_of_natDegree_le_of_leadingCoeff {p q : R[X]} (hpq : p \u2223 q)\n    (h\u2081 : q.natDegree \u2264 p.natDegree) (h\u2082 : p.leadingCoeff = q.leadingCoeff) :\n    p = q", "theorem eq_of_monic_of_associated (hp : p.Monic) (hq : q.Monic) (hpq : Associated p q) : p = q", "theorem eq_zero_of_degree_lt_of_eval_finset_eq_zero (degree_f_lt : f.degree < s.card)\n    (eval_f : \u2200 x \u2208 s, f.eval x = 0) : f = 0", "theorem eq_zero_of_hasseDeriv_eq_zero {R} [CommRing R] (f : R[X]) (r : R)\n    (h : \u2200 k, (hasseDeriv k f).eval r = 0) : f = 0", "theorem erase_mem_lifts {p : S[X]} (n : \u2115) (h : p \u2208 lifts f) : p.erase n \u2208 lifts f", "@[simp]\ntheorem eval_intCast {n : \u2124} {x : R} : (n : R[X]).eval x = n", "@[simp]\ntheorem eval\u2082_C : (C a).eval\u2082 f x = f a", "@[simp]\ntheorem eval\u2082_X : X.eval\u2082 f x = x", "@[simp]\ntheorem eval\u2082_at_natCast {S : Type*} [Semiring S] (f : R \u2192+* S) (n : \u2115) :\n    p.eval\u2082 f n = f (p.eval n)", "@[simp]\ntheorem eval\u2082_at_ofNat {S : Type*} [Semiring S] (f : R \u2192+* S) (n : \u2115) [n.AtLeastTwo] :\n    p.eval\u2082 f (no_index (OfNat.ofNat n)) = f (p.eval (OfNat.ofNat n))", "theorem eval\u2082_finset_prod (s : Finset \u03b9) (g : \u03b9 \u2192 R[X]) (x : S) :\n    (\u220f i \u2208 s, g i).eval\u2082 f x = \u220f i \u2208 s, (g i).eval\u2082 f x", "theorem eval\u2082_finset_sum (s : Finset \u03b9) (g : \u03b9 \u2192 R[X]) (x : S) :\n    (\u2211 i \u2208 s, g i).eval\u2082 f x = \u2211 i \u2208 s, (g i).eval\u2082 f x", "theorem eval\u2082_mul_eq_zero_of_left (q : R[X]) (hp : p.eval\u2082 f x = 0) : (p * q).eval\u2082 f x = 0", "theorem eval\u2082_pow (n : \u2115) : (p ^ n).eval\u2082 f x = p.eval\u2082 f x ^ n", "theorem eval\u2082_sum (p : T[X]) (g : \u2115 \u2192 T \u2192 R[X]) (x : S) :\n    (p.sum g).eval\u2082 f x = p.sum fun n a => (g n a).eval\u2082 f x", "theorem exists_approx_polynomial_aux [Ring Fq] {d : \u2115} {m : \u2115} (hm : Fintype.card Fq ^ d \u2264 m)\n    (b : Fq[X]) (A : Fin m.succ \u2192 Fq[X]) (hA : \u2200 i, degree (A i) < degree b) :\n    \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 degree (A i\u2081 - A i\u2080) < \u2191(natDegree b - d)", "theorem expand_contract [CharP R p] [NoZeroDivisors R] {f : R[X]} (hf : Polynomial.derivative f = 0)\n    (hp : p \u2260 0) : expand R p (contract p f) = f", "theorem expand_eq_sum {f : R[X]} : expand R p f = f.sum fun e a => C a * (X ^ p) ^ e", "theorem forall_eq_iff_forall_eq : (\u2200 f g : R[X], f = g) \u2194 \u2200 a b : R, a = b", "theorem hasseDeriv_C (r : R) (hk : 0 < k) : hasseDeriv k (C r) = 0", "theorem hermite_one : hermite 1 = X", "theorem image_rootSet [Algebra R K] [Algebra R L] {p : R[X]} (h : p.Splits (algebraMap R K))\n    (f : K \u2192\u2090[R] L) : f '' p.rootSet K = p.rootSet L", "theorem int_coeff_of_cyclotomic' {K : Type*} [CommRing K] [IsDomain K] {\u03b6 : K} {n : \u2115}\n    (h : IsPrimitiveRoot \u03b6 n) : \u2203 P : \u2124[X], map (Int.castRingHom K) P =\n      cyclotomic' n K \u2227 P.degree = (cyclotomic' n K).degree \u2227 P.Monic", "theorem int_cyclotomic_spec (n : \u2115) :\n    map (Int.castRingHom \u2102) (cyclotomic n \u2124) = cyclotomic' n \u2102 \u2227\n      (cyclotomic n \u2124).degree = (cyclotomic' n \u2102).degree \u2227 (cyclotomic n \u2124).Monic", "theorem integralNormalization_eval\u2082_eq_zero {p : R[X]} (f : R \u2192+* S) {z : S} (hz : eval\u2082 f z p = 0)\n    (inj : \u2200 x : R, f x = 0 \u2192 x = 0) :\n    eval\u2082 f (z * f p.leadingCoeff) (integralNormalization p) = 0", "theorem irreducible_iff_roots_eq_zero_of_degree_le_three\n    {p : K[X]} (hp2 : 2 \u2264 p.natDegree) (hp3 : p.natDegree \u2264 3) : Irreducible p \u2194 p.roots = 0", "theorem isCoprime_X_sub_C_of_isUnit_sub {R} [CommRing R] {a b : R} (h : IsUnit (a - b)) :\n    IsCoprime (X - C a) (X - C b)", "theorem isPrimitive_iff_isUnit_of_C_dvd {p : R[X]} : p.IsPrimitive \u2194 \u2200 r : R, C r \u2223 p \u2192 IsUnit r", "theorem isRoot_cyclotomic_iff [NeZero (n : R)] {\u03bc : R} :\n    IsRoot (cyclotomic n R) \u03bc \u2194 IsPrimitiveRoot \u03bc n", "theorem isRoot_gcd_iff_isRoot_left_right [DecidableEq R] {f g : R[X]} {\u03b1 : R} :\n    (EuclideanDomain.gcd f g).IsRoot \u03b1 \u2194 f.IsRoot \u03b1 \u2227 g.IsRoot \u03b1", "theorem isRoot_of_aeval_algebraMap_eq_zero [Algebra R S] {p : R[X]}\n    (inj : Function.Injective (algebraMap R S)) {r : R} (hr : aeval (algebraMap R S r) p = 0) :\n    p.IsRoot r", "theorem isUnitTrinomial_iff'' (h : p * p.mirror = q * q.mirror) :\n    p.IsUnitTrinomial \u2194 q.IsUnitTrinomial", "theorem isUnit_C {x : R} : IsUnit (C x) \u2194 IsUnit x", "@[simp] lemma isUnit_C_add_X_mul_iff :\n    IsUnit (C r + X * P) \u2194 IsUnit r \u2227 IsNilpotent P", "theorem isUnit_leadingCoeff_mul_left_eq_zero_iff (h : IsUnit p.leadingCoeff) {q : R[X]} :\n    q * p = 0 \u2194 q = 0", "theorem isUnit_map [Field k] (f : R \u2192+* k) : IsUnit (p.map f) \u2194 IsUnit p", "theorem isUnit_or_eq_zero_of_isUnit_integerNormalization_primPart {p : K[X]} (h0 : p \u2260 0)\n    (h : IsUnit (integerNormalization R\u2070 p).primPart) : IsUnit p", "@[simp]\ntheorem iterate_comp_eval\u2082 (k : \u2115) (t : S) :\n    eval\u2082 f t (p.comp^[k] q) = (fun x => eval\u2082 f x p)^[k] (eval\u2082 f t q)", "theorem iterate_derivative_X_sub_pow (n k : \u2115) (c : R) :\n    derivative^[k] ((X - C c) ^ n) = n.descFactorial k \u2022 (X - C c) ^ (n - k)", "theorem le_degree_of_mem_supp (a : \u2115) : a \u2208 p.support \u2192 \u2191a \u2264 degree p", "theorem le_natTrailingDegree (hp : p \u2260 0) (hn : \u2200 m < n, p.coeff m = 0) :\n    n \u2264 p.natTrailingDegree", "theorem leadingCoeff_add_of_degree_eq (h : degree p = degree q)\n    (hlc : leadingCoeff p + leadingCoeff q \u2260 0) :\n    leadingCoeff (p + q) = leadingCoeff p + leadingCoeff q", "theorem leadingCoeff_divByMonic_X_sub_C (p : R[X]) (hp : degree p \u2260 0) (a : R) :\n    leadingCoeff (p /\u2098 (X - C a)) = leadingCoeff p", "theorem leadingCoeff_divByMonic_of_monic {R : Type u} [CommRing R] {p q : R[X]} (hmonic : q.Monic)\n    (hdegree : q.degree \u2264 p.degree) : (p /\u2098 q).leadingCoeff = p.leadingCoeff", "theorem leadingCoeff_dvd_leadingCoeff {a p : R[X]} (hap : a \u2223 p) :\n    a.leadingCoeff \u2223 p.leadingCoeff", "@[simp]\ntheorem leadingCoeff_map [Semiring S] [Nontrivial S] (f : R \u2192+* S) :\n    leadingCoeff (p.map f) = f (leadingCoeff p)", "theorem leadingCoeff_mul_monic {p q : R[X]} (hq : Monic q) :\n    leadingCoeff (p * q) = leadingCoeff p", "theorem leadingCoeff_multiset_prod' (h : (t.map leadingCoeff).prod \u2260 0) :\n    t.prod.leadingCoeff = (t.map leadingCoeff).prod", "@[simp]\ntheorem leadingCoeff_quadratic (ha : a \u2260 0) : leadingCoeff (C a * X ^ 2 + C b * X + C c) = a", "theorem leadingCoeff_sub_of_degree_lt' (h : Polynomial.degree p < Polynomial.degree q) :\n    (p - q).leadingCoeff = -q.leadingCoeff", "theorem Polynomial.lift_of_splits {F K L : Type*} [Field F] [Field K] [Field L] [Algebra F K]\n    [Algebra F L] (s : Finset K) : (\u2200 x \u2208 s, IsIntegral F x \u2227\n      Splits (algebraMap F L) (minpoly F x)) \u2192 Nonempty (Algebra.adjoin F (s : Set K) \u2192\u2090[F] L)", "theorem linearIndependent_powers_iff_aeval (f : M \u2192\u2097[R] M) (v : M) :\n    (LinearIndependent R fun n : \u2115 => (f ^ n) v) \u2194 \u2200 p : R[X], aeval f p v = 0 \u2192 p = 0", "theorem lt_natDegree_of_mem_eraseLead_support {a : \u2115} (h : a \u2208 (eraseLead f).support) :\n    a < f.natDegree", "@[simp]\nprotected theorem map_add : (p + q).map f = p.map f + q.map f", "theorem map_contract {p : \u2115} (hp : p \u2260 0) {f : R \u2192+* S} {q : R[X]} :\n    (q.contract p).map f = (q.map f).contract p", "theorem map_injective (hf : Function.Injective f) : Function.Injective (map f)", "protected theorem map_list_prod (L : List R[X]) : L.prod.map f = (L.map <| map f).prod", "lemma map_mapRingHom_evalEval (x y : R) :\n    (p.map <| mapRingHom f).evalEval (f x) (f y) = f (p.evalEval x y)", "theorem map_mod [Field k] (f : R \u2192+* k) : (p % q).map f = p.map f % q.map f", "theorem map_monic_eq_zero_iff (hp : p.Monic) : p.map f = 0 \u2194 \u2200 x, f x = 0", "protected theorem map_multiset_prod (m : Multiset R[X]) : m.prod.map f = (m.map <| map f).prod", "@[simp]\ntheorem map_restriction {R : Type u} [CommRing R] (p : R[X]) :\n    p.restriction.map (algebraMap _ _) = p", "theorem map_roots_le [IsDomain A] [IsDomain B] {p : A[X]} {f : A \u2192+* B} (h : p.map f \u2260 0) :\n    p.roots.map f \u2264 (p.map f).roots", "@[simp]\nprotected theorem map_sub {S} [Ring S] (f : R \u2192+* S) : (p - q).map f = p.map f - q.map f", "theorem mem_degreeLT {n : \u2115} {f : R[X]} : f \u2208 degreeLT R n \u2194 degree f < n", "theorem mem_iff_annIdealGenerator_dvd {p : \ud835\udd5c[X]} {a : A} :\n    p \u2208 annIdeal \ud835\udd5c a \u2194 annIdealGenerator \ud835\udd5c a \u2223 p", "theorem mem_roots (hp : p \u2260 0) : a \u2208 p.roots \u2194 IsRoot p a", "theorem mirror_natDegree : p.mirror.natDegree = p.natDegree", "theorem mirror_natTrailingDegree : p.mirror.natTrailingDegree = p.natTrailingDegree", "@[simp]\ntheorem modByMonic_X_sub_C_eq_C_eval (p : R[X]) (a : R) : p %\u2098 (X - C a) = C (p.eval a)", "theorem modByMonic_add_div (p : R[X]) {q : R[X]} (hq : Monic q) : p %\u2098 q + q * (p /\u2098 q) = p", "@[simp]\ntheorem modByMonic_one (p : R[X]) : p %\u2098 1 = 0", "@[simp]\ntheorem monic_X_pow (n : \u2115) : Monic (X ^ n : R[X])", "theorem monomial_neg (n : \u2115) (a : R) : monomial n (-a) = -monomial n a", "theorem monomial_sub (n : \u2115) : monomial n (a - b) = monomial n a - monomial n b", "theorem multiplicity_X_sub_C_finite (a : R) (h0 : p \u2260 0) : multiplicity.Finite (X - C a) p", "theorem multiplicity_le_one_of_separable {p q : R[X]} (hq : \u00acIsUnit q) (hsep : Separable p) :\n    multiplicity q p \u2264 1", "theorem multiset_prod_X_sub_C_nextCoeff (t : Multiset R) :\n    nextCoeff (t.map fun x => X - C x).prod = -t.sum", "@[simp]\ntheorem natCast_comp {n : \u2115} : (n : R[X]).comp p = n", "theorem natDegree_X_add_C (x : R) : (X + C x).natDegree = 1", "@[simp] theorem natDegree_add_C {a : R} : (p + C a).natDegree = p.natDegree", "theorem natDegree_divX_eq_natDegree_tsub_one : p.divX.natDegree = p.natDegree - 1", "theorem natDegree_expand (p : \u2115) (f : R[X]) : (expand R p f).natDegree = f.natDegree * p", "theorem natDegree_mul_le {p q : R[X]} : natDegree (p * q) \u2264 natDegree p + natDegree q", "theorem natDegree_multiset_sum_le (l : Multiset S[X]) :\n    natDegree l.sum \u2264 (l.map natDegree).foldr max max_left_comm 0", "theorem natDegree_not_mem_eraseLead_support : f.natDegree \u2209 (eraseLead f).support", "theorem natDegree_quadratic (ha : a \u2260 0) : natDegree (C a * X ^ 2 + C b * X + C c) = 2", "theorem natDegree_sub_eq_right_of_natDegree_lt (h : natDegree p < natDegree q) :\n    natDegree (p - q) = natDegree q", "theorem natDegree_sub_le_iff_left (qn : q.natDegree \u2264 n) :\n    (p - q).natDegree \u2264 n \u2194 p.natDegree \u2264 n", "@[simp]\ntheorem natSepDegree_zero : (0 : F[X]).natSepDegree = 0", "@[simp]\ntheorem natTrailingDegree_natCast (n : \u2115) : natTrailingDegree (n : R[X]) = 0", "theorem ne_zero_of_degree_ge_degree (hpq : p.degree \u2264 q.degree) (hp : p \u2260 0) : q \u2260 0", "@[simp]\ntheorem nextCoeff_X_sub_C [Ring S] (c : S) : nextCoeff (X - C c) = -c", "lemma nextCoeff_eq_zero :\n    p.nextCoeff = 0 \u2194 p.natDegree = 0 \u2228 0 < p.natDegree \u2227 p.coeff (p.natDegree - 1) = 0", "lemma nnqsmul_eq_C_mul (q : \u211a\u22650) (f : R[X]) : q \u2022 f = Polynomial.C (q : R) * f", "theorem nodup_roots {p : R[X]} (hsep : Separable p) : p.roots.Nodup", "theorem normUnit_X : normUnit (X : Polynomial R) = 1", "theorem not_isUnit_X_pow_sub_one (R : Type*) [CommRing R] [Nontrivial R] (n : \u2115) :\n    \u00acIsUnit (X ^ n - 1 : R[X])", "theorem not_isUnit_of_natDegree_pos (p : R[X])\n    (hpl : 0 < p.natDegree) : \u00ac IsUnit p", "@[simp]\ntheorem not_monic_zero : \u00acMonic (0 : R[X])", "@[simp]\ntheorem ofFinsupp_add {a b} : (\u27e8a + b\u27e9 : R[X]) = \u27e8a\u27e9 + \u27e8b\u27e9", "theorem ofFinsupp_sum {\u03b9 : Type*} (s : Finset \u03b9) (f : \u03b9 \u2192 R[\u2115]) :\n    (\u27e8\u2211 i \u2208 s, f i\u27e9 : R[X]) = \u2211 i \u2208 s, \u27e8f i\u27e9", "@[simp]\ntheorem ofFinsupp_zero : (\u27e80\u27e9 : R[X]) = 0", "theorem opRingEquiv_symm_C_mul_X_pow (r : R\u1d50\u1d52\u1d56) (n : \u2115) :\n    (opRingEquiv R).symm (C r * X ^ n : R\u1d50\u1d52\u1d56[X]) = op (C (unop r) * X ^ n)", "@[simp]\ntheorem pow_comp {R : Type*} [CommSemiring R] (p q : R[X]) (n : \u2115) :\n    (p ^ n).comp q = p.comp q ^ n", "theorem pow_mul_divByMonic_rootMultiplicity_eq (p : R[X]) (a : R) :\n    (X - C a) ^ rootMultiplicity a p * (p /\u2098 (X - C a) ^ rootMultiplicity a p) = p", "theorem prod_cyclotomic_eq_X_pow_sub_one {n : \u2115} (hpos : 0 < n) (R : Type*) [CommRing R] :\n    \u220f i \u2208 Nat.divisors n, cyclotomic i R = X ^ n - 1", "theorem prod_multiset_X_sub_C_dvd (p : R[X]) : (p.roots.map fun a => X - C a).prod \u2223 p", "theorem prod_roots_eq_coeff_zero_of_monic_of_split {P : K[X]} (hmo : P.Monic)\n    (hP : P.Splits (RingHom.id K)) : coeff P 0 = (-1) ^ P.natDegree * P.roots.prod", "@[simp]\ntheorem revAt_invol {N i : \u2115} : (revAt N) (revAt N i) = i", "@[ext high]\ntheorem ringHom_ext' {S} [Semiring S] {f g : R[X] \u2192+* S} (h\u2081 : f.comp C = g.comp C)\n    (h\u2082 : f X = g X) : f = g", "@[simp]\ntheorem rootMultiplicity_eq_zero_iff {p : R[X]} {x : R} :\n    rootMultiplicity x p = 0 \u2194 IsRoot p x \u2192 p = 0", "theorem rootMultiplicity_mul' {p q : R[X]} {x : R}\n    (hpq : (p /\u2098 (X - C x) ^ p.rootMultiplicity x).eval x *\n      (q /\u2098 (X - C x) ^ q.rootMultiplicity x).eval x \u2260 0) :\n    rootMultiplicity x (p * q) = rootMultiplicity x p + rootMultiplicity x q", "theorem rootOfSplits'_eq_rootOfSplits {f : K[X]} (hf : f.Splits i) (hfd) :\n    rootOfSplits' i hf hfd = rootOfSplits i hf (f.degree_map i \u25b8 hfd)", "theorem rootSet_X_pow [CommRing S] [IsDomain S] [Algebra R S] {n : \u2115} (hn : n \u2260 0) :\n    (X ^ n : R[X]).rootSet S = {0}", "theorem root_mul_right_of_isRoot {p : R[X]} (q : R[X]) : IsRoot p a \u2192 IsRoot (p * q) a", "theorem roots.le_of_dvd (h : q \u2260 0) : p \u2223 q \u2192 roots p \u2264 roots q", "theorem scaleRoots_eval\u2082_eq_zero_of_eval\u2082_div_eq_zero {p : S[X]} {f : S \u2192+* K}\n    (hf : Function.Injective f) {r s : S} (hr : eval\u2082 f (f r / f s) p = 0)\n    (hs : s \u2208 nonZeroDivisors S) : eval\u2082 f (f r) (scaleRoots p s) = 0", "theorem separable_gcd_right {F : Type*} [Field F] {g : F[X]} (f : F[X]) (hg : g.Separable) :\n    (EuclideanDomain.gcd f g).Separable", "@[simp]\ntheorem smeval_C : (C r).smeval x = r \u2022 x ^ 0", "@[simp]\ntheorem smeval_X_pow {n : \u2115} :\n    (X ^ n : R[X]).smeval x = x ^ n", "@[simp]\ntheorem smeval_add : (p + q).smeval x = p.smeval x + q.smeval x", "theorem smeval_at_zero : p.smeval (0 : S) = (p.coeff 0) \u2022 (1 : S)", "theorem smul_X_eq_monomial {n} : a \u2022 X ^ n = monomial n (a : R)", "theorem smul_eq_C_mul (a : R) : a \u2022 p = C a * p", "theorem span_le_degreeLE_of_finite {s : Set R[X]} (s_fin : s.Finite) :\n    \u2203 n : \u2115, Submodule.span R s \u2264 degreeLE R n", "theorem splits_of_splits_mul' {f g : K[X]} (hfg : (f * g).map i \u2260 0) (h : Splits i (f * g)) :\n    Splits i f \u2227 Splits i g", "theorem sub_one_lt_natAbs_cyclotomic_eval {n : \u2115} {q : \u2115} (hn' : 1 < n) (hq : q \u2260 1) :\n    q - 1 < ((cyclotomic n \u2124).eval \u2191q).natAbs", "theorem sub_one_pow_totient_lt_cyclotomic_eval {n : \u2115} {q : \u211d} (hn' : 2 \u2264 n) (hq' : 1 < q) :\n    (q - 1) ^ totient n < (cyclotomic n \u211d).eval q", "theorem sum_fin [AddCommMonoid S] (f : \u2115 \u2192 R \u2192 S) (hf : \u2200 i, f i 0 = 0) {n : \u2115} {p : R[X]}\n    (hn : p.degree < n) : (\u2211 i : Fin n, f i (p.coeff i)) = p.sum f", "theorem support_X_empty (H : (1 : R) = 0) : (X : R[X]).support = \u2205", "theorem support_map_subset [Semiring R] [Semiring S] (f : R \u2192+* S) (p : R[X]) :\n    (map f p).support \u2286 p.support", "theorem support_update_zero (p : R[X]) (n : \u2115) : support (p.update n 0) = p.support.erase n", "theorem taylor_injective {R} [CommRing R] (r : R) : Function.Injective (taylor r)", "@[simp]\ntheorem toFinsupp_mul (a b : R[X]) : (a * b).toFinsupp = a.toFinsupp * b.toFinsupp", "@[simp]\ntheorem toFinsupp_pow (a : R[X]) (n : \u2115) : (a ^ n).toFinsupp = a.toFinsupp ^ n", "@[simp]\ntheorem trailingDegree_X : trailingDegree (X : R[X]) = 1", "theorem trinomial_leadingCoeff (hkm : k < m) (hmn : m < n) (hw : w \u2260 0) :\n    (trinomial k m n u v w).leadingCoeff = w", "theorem trinomial_middle_coeff (hkm : k < m) (hmn : m < n) :\n    (trinomial k m n u v w).coeff m = v", "theorem trinomial_mirror (hkm : k < m) (hmn : m < n) (hu : u \u2260 0) (hw : w \u2260 0) :\n    (trinomial k m n u v w).mirror = trinomial k (n - m + k) n w v u", "theorem zero_nmem_multiset_map_X_sub_C {\u03b1 : Type*} (m : Multiset \u03b1) (f : \u03b1 \u2192 R) :\n    (0 : R[X]) \u2209 m.map fun a => X - C (f a)", "theorem comp_eval (p : R[X]) (q : PolynomialModule R M) (r : R) :\n    eval r (comp p q) = eval (p.eval r) q", "theorem lsingle_apply (i : \u2115) (m : M) (n : \u2115) : lsingle R i m n = ite (i = n) m 0", "theorem add_one (n : PosNum) : n + 1 = succ n", "theorem natSize_to_nat (n) : natSize n = Nat.size n", "theorem to_nat_pos : \u2200 n : PosNum, 0 < (n : \u2115)", "lemma PosSMulReflectLE_iff_posSMulReflectLT : PosSMulReflectLE \u03b1 \u03b2 \u2194 PosSMulReflectLT \u03b1 \u03b2", "@[simp]\ntheorem val_one [Nontrivial R] : ((1 : { x : R // 0 < x }) : R) = 1", "theorem adjoin_gen_eq_top (B : PowerBasis R S) : adjoin R ({B.gen} : Set S) = \u22a4", "@[simp]\ntheorem coe_basis (pb : PowerBasis R S) : \u21d1pb.basis = fun i : Fin pb.dim => pb.gen ^ (i : \u2115)", "@[simp]\ntheorem equivOfMinpoly_symm (pb : PowerBasis A S) (pb' : PowerBasis A S')\n    (h : minpoly A pb.gen = minpoly A pb'.gen) :\n    (pb.equivOfMinpoly pb' h).symm = pb'.equivOfMinpoly pb h.symm", "theorem natDegree_minpolyGen [Nontrivial A] (pb : PowerBasis A S) :\n    natDegree (minpolyGen pb) = pb.dim", "@[simp]\ntheorem _root_.PowerBasis.ofGenMemAdjoin'_dim (B : PowerBasis R S) (hint : IsIntegral R x)\n    (hx : B.gen \u2208 adjoin R ({x} : Set S)) :\n    (B.ofGenMemAdjoin' hint hx).dim = (minpoly R x).natDegree", "theorem quotientEquivQuotientMinpolyMap_symm_apply_mk (pb : PowerBasis R S) (I : Ideal R)\n    (g : R[X]) :\n    (pb.quotientEquivQuotientMinpolyMap I).symm (Ideal.Quotient.mk (Ideal.span\n      ({(minpoly R pb.gen).map (Ideal.Quotient.mk I)} : Set (Polynomial (R \u29f8 I))))\n        (g.map (Ideal.Quotient.mk I))) = Ideal.Quotient.mk (I.map (algebraMap R S))\n          (aeval pb.gen g)", "theorem PowerBasis.trace_gen_eq_sum_roots [Nontrivial S] (pb : PowerBasis K S)\n    (hf : (minpoly K pb.gen).Splits (algebraMap K F)) :\n    algebraMap K F (trace K S pb.gen) = ((minpoly K pb.gen).aroots F).sum", "theorem C_eq_algebraMap {r : R} : C R r = (algebraMap R R\u27e6X\u27e7) r", "theorem X_pow_order_dvd (h : (order \u03c6).Dom) : X ^ (order \u03c6).get h \u2223 \u03c6", "@[simp]\ntheorem coeff_cos_bit1 : coeff A (bit1 n) (cos A) = 0", "@[simp]\ntheorem coeff_mul_X_pow (p : R\u27e6X\u27e7) (n d : \u2115) :\n    coeff R (d + n) (p * X ^ n) = coeff R d p", "theorem coeff_mul_X_pow' (p : R\u27e6X\u27e7) (n d : \u2115) :\n    coeff R d (p * X ^ n) = ite (n \u2264 d) (coeff R (d - n) p) 0", "theorem coeff_mul_one_sub_of_lt_order {R : Type*} [CommRing R] {\u03c6 \u03c8 : R\u27e6X\u27e7} (n : \u2115)\n    (h : \u2191n < \u03c8.order) : coeff R n (\u03c6 * (1 - \u03c8)) = coeff R n \u03c6", "@[simp]\ntheorem coeff_zero_X : coeff R 0 (X : R\u27e6X\u27e7) = 0", "@[simp]\ntheorem constantCoeff_smul {S : Type*} [Semiring S] [Module R S] (\u03c6 : PowerSeries S) (a : R) :\n    constantCoeff S (a \u2022 \u03c6) = a \u2022 constantCoeff S \u03c6", "theorem derivativeFun_coe (f : R[X]) : (f : R\u27e6X\u27e7).derivativeFun = derivative f", "theorem divided_by_X_pow_orderMul {f g : R\u27e6X\u27e7} (hf : f \u2260 0) (hg : g \u2260 0) :\n    divided_by_X_pow_order hf * divided_by_X_pow_order hg =\n      divided_by_X_pow_order (mul_ne_zero hf hg)", "theorem eq_inv_iff_mul_eq_one {\u03c6 \u03c8 : k\u27e6X\u27e7} (h : constantCoeff k \u03c8 \u2260 0) :\n    \u03c6 = \u03c8\u207b\u00b9 \u2194 \u03c6 * \u03c8 = 1", "@[simp]\ntheorem evalNegHom_X : evalNegHom (X : A\u27e6X\u27e7) = -X", "theorem ext_iff {\u03c6 \u03c8 : R\u27e6X\u27e7} : \u03c6 = \u03c8 \u2194 \u2200 n, coeff R n \u03c6 = coeff R n \u03c8", "theorem invOneSubPow_eq_inv_one_sub_pow :\n    invOneSubPow d = (Units.mkOfMulEqOne (1 - X) (mk 1 : S\u27e6X\u27e7)\n    <| Eq.trans (mul_comm _ _) mk_one_mul_one_sub_eq_one)\u207b\u00b9 ^ (d + 1)", "@[simp]\ntheorem map_X : map f X = X", "theorem natDegree_trunc_lt (f : R\u27e6X\u27e7) (n) : (trunc (n + 1) f).natDegree < n + 1", "theorem sub_const_eq_shift_mul_X (\u03c6 : R\u27e6X\u27e7) :\n    \u03c6 - C R (constantCoeff R \u03c6) = (PowerSeries.mk fun p => coeff R (p + 1) \u03c6) * X", "protected theorem subsingleton_iff : Subsingleton R\u27e6X\u27e7 \u2194 Subsingleton R", "@[simp]\ntheorem trunc_add (n) (\u03c6 \u03c8 : R\u27e6X\u27e7) : trunc n (\u03c6 + \u03c8) = trunc n \u03c6 + trunc n \u03c8", "theorem PreErgodic.ae_eq_const_of_ae_eq_comp (h : PreErgodic f \u03bc) (hgm : Measurable g)\n    (hg_eq : g \u2218 f = g) : \u2203 c, g =\u1d50[\u03bc] const \u03b1 c", "theorem congr_map {U V : Type*} [Quiver U] [Quiver V] (F : U \u2964q V) {X Y : U} {f g : X \u27f6 Y}\n    (h : f = g) : F.map f = F.map g", "@[simp]\ntheorem mapPath_comp {a b : V} (p : Path a b) :\n    \u2200 {c : V} (q : Path b c), F.mapPath (p.comp q) = (F.mapPath p).comp (F.mapPath q)", "theorem Prefunctor.pathStar_injective (h\u03c6 : \u2200 u, Injective (\u03c6.star u)) (u : U) :\n    Injective (\u03c6.pathStar u)", "protected theorem Prefunctor.symmetrifyCostar (u : U) :\n    \u03c6.symmetrify.costar u =\n      (Quiver.symmetrifyCostar _).symm \u2218\n        Sum.map (\u03c6.costar u) (\u03c6.star u) \u2218 Quiver.symmetrifyCostar u", "theorem to_group_eq_one_of_mem_closure : \u2200 x \u2208 Subgroup.normalClosure rels, F x = 1", "@[simp]\nlemma app_apply (d : M.Derivation' \u03c6') {X : D\u1d52\u1d56} (b : R.obj X) :\n    (d.app X).d b = d.d b := rfl", "lemma isSheaf_of_isLimit : Presheaf.IsSheaf J (c.pt.presheaf)", "@[simp, mfld_simps]\ntheorem coe_coe : \u21d1e.toPartialEquiv = e", "theorem proj_surjOn_baseSet [Nonempty F] : Set.SurjOn proj e.source e.baseSet", "theorem mem_range_encode : PrimrecPred (fun n => n \u2208 Set.range (encode : \u03b1 \u2192 \u2115))", "theorem Prime.dvd_prime_iff_associated [CancelCommMonoidWithZero \u03b1] {p q : \u03b1} (pp : Prime p)\n    (qp : Prime q) : p \u2223 q \u2194 Associated p q", "theorem Prime.dvd_prod_iff {p : M} {L : List M} (pp : Prime p) : p \u2223 L.prod \u2194 \u2203 a \u2208 L, p \u2223 a", "theorem Prime.left_dvd_or_dvd_right_of_dvd_mul [CancelCommMonoidWithZero \u03b1] {p : \u03b1} (hp : Prime p)\n    {a b : \u03b1} : a \u2223 p * b \u2192 p \u2223 a \u2228 a \u2223 b", "theorem Prime.not_dvd_finset_prod {S : Finset \u03b1} {p : M} (pp : Prime p) {g : \u03b1 \u2192 M}\n    (hS : \u2200 a \u2208 S, \u00acp \u2223 g a) : \u00acp \u2223 S.prod g", "theorem not_dvd_mul {a b : \u03b1} (ha : \u00ac p \u2223 a) (hb : \u00ac p \u2223 b) : \u00ac p \u2223 a * b", "theorem prod_inf (u v : PrimeMultiset) : (u \u2293 v).prod = PNat.gcd u.prod v.prod", "theorem prod_ofPNatMultiset (v : Multiset \u2115+) (h) : ((ofPNatMultiset v h).prod : \u2115+) = v.prod", "theorem prod_sup (u v : PrimeMultiset) : (u \u2294 v).prod = PNat.lcm u.prod v.prod", "@[simp]\ntheorem asIdeal_lt_asIdeal (x y : PrimeSpectrum R) : x.asIdeal < y.asIdeal \u2194 x < y", "@[simp]\ntheorem comap_asIdeal (y : PrimeSpectrum S) : (comap f y).asIdeal = Ideal.comap f y.asIdeal", "theorem comap_inducing_of_surjective (hf : Surjective f) : Inducing (comap f) where", "theorem isBasis_basic_opens : TopologicalSpace.Opens.IsBasis (Set.range (@basicOpen R _))", "theorem isOpen_basicOpen {a : R} : IsOpen (basicOpen a : Set (PrimeSpectrum R))", "theorem range_comap_of_surjective (hf : Surjective f) :\n    Set.range (comap f) = zeroLocus (ker f)", "theorem t1Space_iff_isField [IsDomain R] : T1Space (PrimeSpectrum R) \u2194 IsField R", "theorem vanishingIdeal_iUnion {\u03b9 : Sort*} (t : \u03b9 \u2192 Set (PrimeSpectrum R)) :\n    vanishingIdeal (\u22c3 i, t i) = \u2a05 i, vanishingIdeal (t i)", "theorem zeroLocus_inf (I J : Ideal R) :\n    zeroLocus ((I \u2293 J : Ideal R) : Set R) = zeroLocus I \u222a zeroLocus J", "theorem zeroLocus_subset_zeroLocus_singleton_iff (f g : R) :\n    zeroLocus ({f} : Set R) \u2286 zeroLocus {g} \u2194 g \u2208 (Ideal.span ({f} : Set R)).radical", "theorem dom_bool (f : Bool \u2192 \u03b1) : Primrec f", "theorem encode_iff {f : \u03b1 \u2192 \u03c3} : (Primrec fun a => encode (f a)) \u2194 Primrec f", "theorem nat_add : Primrec\u2082 ((\u00b7 + \u00b7) : \u2115 \u2192 \u2115 \u2192 \u2115)", "theorem nat_bodd : Primrec Nat.bodd", "theorem option_bind\u2081 {f : \u03b1 \u2192 Option \u03c3} (hf : Primrec f) : Primrec fun o => Option.bind o f", "theorem option_some : Primrec (@some \u03b1)", "theorem option_some_iff {f : \u03b1 \u2192 \u03c3} : (Primrec fun a => some (f a)) \u2194 Primrec f", "theorem subtype_val {p : \u03b1 \u2192 Prop} [DecidablePred p] {hp : PrimrecPred p} :\n    haveI := Primcodable.subtype hp\n    Primrec (@Subtype.val \u03b1 p)", "theorem succ : Primrec Nat.succ", "theorem vector_cons {n} : Primrec\u2082 (@Vector.cons \u03b1 n)", "theorem vector_head {n} : Primrec (@Vector.head \u03b1 n)", "theorem vector_tail {n} : Primrec (@Vector.tail \u03b1 n)", "theorem PrimrecPred.comp {p : \u03b2 \u2192 Prop} [DecidablePred p] {f : \u03b1 \u2192 \u03b2} :\n    PrimrecPred p \u2192 Primrec f \u2192 PrimrecPred fun a => p (f a)", "theorem PrimrecRel.of_eq {\u03b1 \u03b2} [Primcodable \u03b1] [Primcodable \u03b2] {r s : \u03b1 \u2192 \u03b2 \u2192 Prop}\n    [\u2200 a b, Decidable (r a b)] [\u2200 a b, Decidable (s a b)] (hr : PrimrecRel r)\n    (H : \u2200 a b, r a b \u2194 s a b) : PrimrecRel s", "theorem curry {f : \u03b1 \u00d7 \u03b2 \u2192 \u03c3} : Primrec\u2082 (Function.curry f) \u2194 Primrec f", "theorem coe_coe_fn' [IsTrans \u03b2 s] (f : r \u227ai s) : ((f : r \u227ci s) : \u03b1 \u2192 \u03b2) = f", "theorem down (f : r \u227ai s) : \u2200 {b : \u03b2}, s b f.top \u2194 \u2203 a, f a = b", "theorem CondIndep.condIndepSet_of_measurableSet {\u03bc : Measure \u03a9} [IsFiniteMeasure \u03bc]\n    (h_indep : CondIndep m' m\u2081 m\u2082 hm' \u03bc) {s t : Set \u03a9} (hs : MeasurableSet[m\u2081] s)\n    (ht : MeasurableSet[m\u2082] t) :\n    CondIndepSet m' hm' s t \u03bc", "protected theorem coe_nnreal_ennreal {f : \u03b1 \u2192 \u211d\u22650} {g : \u03b2 \u2192 \u211d\u22650} (h : IdentDistrib f g \u03bc \u03bd) :\n    IdentDistrib (fun x => (f x : \u211d\u22650\u221e)) (fun x => (g x : \u211d\u22650\u221e)) \u03bc \u03bd", "protected theorem norm [NormedAddCommGroup \u03b3] [BorelSpace \u03b3] (h : IdentDistrib f g \u03bc \u03bd) :\n    IdentDistrib (fun x => \u2016f x\u2016) (fun x => \u2016g x\u2016) \u03bc \u03bd", "theorem snorm_eq [NormedAddCommGroup \u03b3] [OpensMeasurableSpace \u03b3] (h : IdentDistrib f g \u03bc \u03bd)\n    (p : \u211d\u22650\u221e) : snorm f p \u03bc = snorm g p \u03bd", "theorem IndepFun.ae_eq {m\u03b2 : MeasurableSpace \u03b2} {m\u03b2' : MeasurableSpace \u03b2'}\n    {f' : \u03a9 \u2192 \u03b2} {g' : \u03a9 \u2192 \u03b2'} (hfg : IndepFun f g \u03bc)\n    (hf : f =\u1d50[\u03bc] f') (hg : g =\u1d50[\u03bc] g') : IndepFun f' g' \u03bc", "theorem IndepFun.cgf_add {X Y : \u03a9 \u2192 \u211d} (h_indep : IndepFun X Y \u03bc)\n    (h_int_X : Integrable (fun \u03c9 => exp (t * X \u03c9)) \u03bc)\n    (h_int_Y : Integrable (fun \u03c9 => exp (t * Y \u03c9)) \u03bc) :\n    cgf (X + Y) \u03bc t = cgf X \u03bc t + cgf Y \u03bc t", "lemma IsMeasurableRatCDF.stieltjesFunctionAux_def' (f : \u03b1 \u2192 \u211a \u2192 \u211d) (a : \u03b1) :\n    IsMeasurableRatCDF.stieltjesFunctionAux f a\n      = fun (t : \u211d) \u21a6 \u2a05 r : { r' : \u211a // t < r' }, f a r", "lemma IsRatCondKernelCDF.iInf_rat_gt_eq (hf : IsRatCondKernelCDF f \u03ba \u03bd) (a : \u03b1) :\n    \u2200\u1d50 b \u2202(\u03bd a), \u2200 q, \u2a05 r : Ioi q, f (a, b) r = f (a, b) q", "lemma IsRatCondKernelCDFAux.bddBelow_range (hf : IsRatCondKernelCDFAux f \u03ba \u03bd) (a : \u03b1) :\n    \u2200\u1d50 t \u2202(\u03bd a), \u2200 q : \u211a, BddBelow (range fun (r : Ioi q) \u21a6 f (a, t) r)", "theorem aestronglyMeasurable'_integral_condexpKernel [NormedSpace \u211d F]\n    (hf : AEStronglyMeasurable f \u03bc) :\n    AEStronglyMeasurable' m (fun \u03c9 => \u222b y, f y \u2202condexpKernel \u03bc m \u03c9) \u03bc", "theorem cgf_undef (hX : \u00acIntegrable (fun \u03c9 => exp (t * X \u03c9)) \u03bc) : cgf X \u03bc t = 0", "lemma compProd_toKernel [IsFiniteKernel \u03ba] [IsSFiniteKernel \u03bd] (hf : IsCondKernelCDF f \u03ba \u03bd) :\n    \u03bd \u2297\u2096 hf.toKernel f = \u03ba", "theorem condCount_singleton (\u03c9 : \u03a9) (t : Set \u03a9) [Decidable (\u03c9 \u2208 t)] :\n    condCount {\u03c9} t = if \u03c9 \u2208 t then 1 else 0", "theorem condDistrib_ae_eq_of_measure_eq_compProd (hX : Measurable X) (hY : Measurable Y)\n    (\u03ba : kernel \u03b2 \u03a9) [IsFiniteKernel \u03ba] (h\u03ba : \u03bc.map (fun x => (X x, Y x)) = \u03bc.map X \u2297\u2098 \u03ba) :\n    \u2200\u1d50 x \u2202\u03bc.map X, \u03ba x = condDistrib Y X \u03bc x", "lemma condDistrib_apply_of_ne_zero [MeasurableSingletonClass \u03b2]\n    (hY : Measurable Y) (x : \u03b2) (hX : \u03bc.map X {x} \u2260 0) (s : Set \u03a9) :\n    condDistrib Y X \u03bc x s = (\u03bc.map X {x})\u207b\u00b9 * \u03bc.map (fun a => (X a, Y a)) ({x} \u00d7\u02e2 s)", "theorem condIndepFun_iff_condexp_inter_preimage_eq_mul {m\u03b2 : MeasurableSpace \u03b2}\n    {m\u03b2' : MeasurableSpace \u03b2'} (hf : Measurable f) (hg : Measurable g) :\n    CondIndepFun m' hm' f g \u03bc \u2194\n      \u2200 s t, MeasurableSet s \u2192 MeasurableSet t\n        \u2192 (\u03bc\u27e6f \u207b\u00b9' s \u2229 g \u207b\u00b9' t | m'\u27e7) =\u1d50[\u03bc] fun \u03c9 \u21a6 (\u03bc\u27e6f \u207b\u00b9' s | m'\u27e7) \u03c9 * (\u03bc\u27e6g \u207b\u00b9' t | m'\u27e7) \u03c9", "lemma condIndepSet_iff (s t : Set \u03a9) (hs : MeasurableSet s) (ht : MeasurableSet t)\n    (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] :\n    CondIndepSet m' hm' s t \u03bc \u2194 (\u03bc\u27e6s \u2229 t | m'\u27e7) =\u1d50[\u03bc] (\u03bc\u27e6s | m'\u27e7) * (\u03bc\u27e6t | m'\u27e7)", "theorem condIndep_bot_left (m\u2081 : MeasurableSpace \u03a9) {m' : MeasurableSpace \u03a9}\n    {m\u03a9 : MeasurableSpace \u03a9} [StandardBorelSpace \u03a9] [Nonempty \u03a9]\n    {hm' : m' \u2264 m\u03a9} {\u03bc : Measure \u03a9} [IsFiniteMeasure \u03bc] :\n    CondIndep m' \u22a5 m\u2081 hm' \u03bc", "theorem condIndep_iSup_of_disjoint {m : \u03b9 \u2192 MeasurableSpace \u03a9}\n    (h_le : \u2200 i, m i \u2264 m\u03a9) (h_indep : iCondIndep m' hm' m \u03bc) {S T : Set \u03b9} (hST : Disjoint S T) :\n    CondIndep m' (\u2a06 i \u2208 S, m i) (\u2a06 i \u2208 T, m i) hm' \u03bc", "theorem condIndep_limsup_atBot_self [StandardBorelSpace \u03a9] [Nonempty \u03a9]\n    (hm : m \u2264 m0) [IsFiniteMeasure \u03bc]\n    (h_le : \u2200 n, s n \u2264 m0) (h_indep : iCondIndep m hm s \u03bc) :\n    CondIndep m (limsup s atBot) (limsup s atBot) hm \u03bc", "lemma condKernel_const [CountableOrCountablyGenerated \u03b1 \u03b2] (\u03c1 : Measure (\u03b2 \u00d7 \u03a9)) [IsFiniteMeasure \u03c1]\n    (a : \u03b1) :\n    (fun b \u21a6 kernel.condKernel (kernel.const \u03b1 \u03c1) (a, b)) =\u1d50[\u03c1.fst] \u03c1.condKernel", "theorem cond_apply' {t : Set \u03a9} (hA : MeasurableSet t) : \u03bc[t|s] = (\u03bc s)\u207b\u00b9 * \u03bc (s \u2229 t)", "lemma condexpKernel_ae_eq_condexp [IsFiniteMeasure \u03bc]\n    (hm : m \u2264 m\u03a9) {s : Set \u03a9} (hs : MeasurableSet s) :\n    (fun \u03c9 \u21a6 (condexpKernel \u03bc m \u03c9 s).toReal) =\u1d50[\u03bc] \u03bc\u27e6s | m\u27e7", "lemma condexpKernel_ae_eq_trim_condexp [IsFiniteMeasure \u03bc]\n    (hm : m \u2264 m\u03a9) {s : Set \u03a9} (hs : MeasurableSet s) :\n    (fun \u03c9 \u21a6 (condexpKernel \u03bc m \u03c9 s).toReal) =\u1d50[\u03bc.trim hm] \u03bc\u27e6s | m\u27e7", "theorem condexp_prod_ae_eq_integral_condDistrib [NormedSpace \u211d F] [CompleteSpace F]\n    (hX : Measurable X) (hY : AEMeasurable Y \u03bc) (hf : StronglyMeasurable f)\n    (hf_int : Integrable (fun a => f (X a, Y a)) \u03bc) :\n    \u03bc[fun a => f (X a, Y a)|m\u03b2.comap X] =\u1d50[\u03bc] fun a => \u222b y, f (X a, y) \u2202condDistrib Y X \u03bc (X a)", "lemma defaultRatCDF_le_one (q : \u211a) : defaultRatCDF q \u2264 1", "lemma defaultRatCDF_nonneg (q : \u211a) : 0 \u2264 defaultRatCDF q", "theorem eq_condKernel_of_measure_eq_compProd (\u03ba : kernel \u03b1 \u03a9) [IsFiniteKernel \u03ba]\n    (h\u03ba : \u03c1 = \u03c1.fst \u2297\u2098 \u03ba) :\n    \u2200\u1d50 x \u2202\u03c1.fst, \u03ba x = \u03c1.condKernel x", "theorem evariance_def' [@IsProbabilityMeasure \u03a9 _ \u2119] {X : \u03a9 \u2192 \u211d} (hX : AEStronglyMeasurable X \u2119) :\n    eVar[X] = (\u222b\u207b \u03c9, (\u2016X \u03c9\u2016\u208a ^ 2 :)) - ENNReal.ofReal (\ud835\udd3c[X] ^ 2)", "theorem evariance_mul (c : \u211d) (X : \u03a9 \u2192 \u211d) (\u03bc : Measure \u03a9) :\n    evariance (fun \u03c9 => c * X \u03c9) \u03bc = ENNReal.ofReal (c ^ 2) * evariance X \u03bc", "lemma gaussianPDFReal_mul {\u03bc : \u211d} {v : \u211d\u22650} {c : \u211d} (hc : c \u2260 0) (x : \u211d) :\n    gaussianPDFReal \u03bc v (c * x)\n      = |c\u207b\u00b9| * gaussianPDFReal (c\u207b\u00b9 * \u03bc) (\u27e8(c^2)\u207b\u00b9, inv_nonneg.mpr (sq_nonneg _)\u27e9 * v) x", "lemma gaussianPDFReal_sub {\u03bc : \u211d} {v : \u211d\u22650} (x y : \u211d) :\n    gaussianPDFReal \u03bc v (x - y) = gaussianPDFReal (\u03bc + y) v x", "lemma gaussianPDF_pos (\u03bc : \u211d) {v : \u211d\u22650} (hv : v \u2260 0) (x : \u211d) : 0 < gaussianPDF \u03bc v x", "@[simp]\nlemma gaussianPDF_zero_var (\u03bc : \u211d) : gaussianPDF \u03bc 0 = 0", "lemma geometricPMFReal_pos {n : \u2115} (hp_pos : 0 < p) (hp_lt_one : p < 1) :\n    0 < geometricPMFReal p n", "theorem iCondIndep.iCondIndepSets {m : \u03b9 \u2192 MeasurableSpace \u03a9}\n    {s : \u03b9 \u2192 Set (Set \u03a9)} (hms : \u2200 n, m n = generateFrom (s n))\n    (h_indep : iCondIndep m' hm' m \u03bc) :\n    iCondIndepSets m' hm' s \u03bc", "theorem iCondIndepSet.condIndep_generateFrom_le [LinearOrder \u03b9] {s : \u03b9 \u2192 Set \u03a9}\n    (hsm : \u2200 n, MeasurableSet (s n)) (hs : iCondIndepSet m' hm' s \u03bc) (i : \u03b9) {k : \u03b9} (hk : i < k) :\n    CondIndep m' (generateFrom {s k}) (generateFrom { t | \u2203 j \u2264 i, s j = t }) hm' \u03bc", "theorem iCondIndepSet.condIndep_generateFrom_lt [Preorder \u03b9] {s : \u03b9 \u2192 Set \u03a9}\n    (hsm : \u2200 n, MeasurableSet (s n)) (hs : iCondIndepSet m' hm' s \u03bc) (i : \u03b9) :\n    CondIndep m' (generateFrom {s i}) (generateFrom { t | \u2203 j < i, s j = t }) hm' \u03bc", "lemma iCondIndep_iff (m : \u03b9 \u2192 MeasurableSpace \u03a9) (hm : \u2200 i, m i \u2264 m\u03a9)\n    (\u03bc : @Measure \u03a9 m\u03a9) [IsFiniteMeasure \u03bc] :\n    iCondIndep m' hm' m \u03bc\n      \u2194 \u2200 (s : Finset \u03b9) {f : \u03b9 \u2192 Set \u03a9} (_H : \u2200 i, i \u2208 s \u2192 MeasurableSet[m i] (f i)),\n      \u03bc\u27e6\u22c2 i \u2208 s, f i | m'\u27e7 =\u1d50[\u03bc] \u220f i \u2208 s, (\u03bc\u27e6f i | m'\u27e7)", "theorem iIndepFun.mgf_sum [IsProbabilityMeasure \u03bc] {X : \u03b9 \u2192 \u03a9 \u2192 \u211d}\n    (h_indep : iIndepFun (fun i => inferInstance) X \u03bc) (h_meas : \u2200 i, Measurable (X i))\n    (s : Finset \u03b9) : mgf (\u2211 i \u2208 s, X i) \u03bc t = \u220f i \u2208 s, mgf (X i) \u03bc t", "theorem iIndepSet.indep_generateFrom_of_disjoint [IsProbabilityMeasure \u03bc] {s : \u03b9 \u2192 Set \u03a9}\n    (hsm : \u2200 n, MeasurableSet (s n)) (hs : iIndepSet s \u03bc) (S T : Set \u03b9) (hST : Disjoint S T) :\n    Indep (generateFrom { t | \u2203 n \u2208 S, s n = t }) (generateFrom { t | \u2203 k \u2208 T, s k = t }) \u03bc", "theorem indepSet_empty_left [IsProbabilityMeasure \u03bc] (s : Set \u03a9) : IndepSet \u2205 s \u03bc", "theorem indepSet_empty_right [IsProbabilityMeasure \u03bc] (s : Set \u03a9) : IndepSet s \u2205 \u03bc", "theorem indepSets_singleton_iff {s t : Set \u03a9} :\n    IndepSets {s} {t} \u03bc \u2194 \u03bc (s \u2229 t) = \u03bc s * \u03bc t", "theorem indep_biSup_compl (h_le : \u2200 n, s n \u2264 m0) (h_indep : iIndep s \u03bc) (t : Set \u03b9) :\n    Indep (\u2a06 n \u2208 t, s n) (\u2a06 n \u2208 t\u1d9c, s n) \u03bc", "theorem indep_iSup_limsup (h_le : \u2200 n, s n \u2264 m0) (h_indep : iIndep s \u03bc) (hf : \u2200 t, p t \u2192 t\u1d9c \u2208 f)\n    (hns : Directed (\u00b7 \u2264 \u00b7) ns) (hnsp : \u2200 a, p (ns a)) (hns_univ : \u2200 n, \u2203 a, n \u2208 ns a) :\n    Indep (\u2a06 n, s n) (limsup s f) \u03bc", "theorem indep_iSup_of_directed_le\n    [IsProbabilityMeasure \u03bc] (h_indep : \u2200 i, Indep (m i) m1 \u03bc)\n    (h_le : \u2200 i, m i \u2264 _m\u03a9) (h_le' : m1 \u2264 _m\u03a9) (hm : Directed (\u00b7 \u2264 \u00b7) m) :\n    Indep (\u2a06 i, m i) m1 \u03bc", "theorem integrable_toReal_condexpKernel {s : Set \u03a9} (hs : MeasurableSet s) :\n    Integrable (fun \u03c9 => (condexpKernel \u03bc m \u03c9 s).toReal) \u03bc", "lemma integral_gaussianPDFReal_eq_one (\u03bc : \u211d) {v : \u211d\u22650} (hv : v \u2260 0) :\n    \u222b x, gaussianPDFReal \u03bc v x = 1", "lemma integral_stieltjesOfMeasurableRat [IsFiniteKernel \u03ba] (hf : IsRatCondKernelCDF f \u03ba \u03bd)\n    (a : \u03b1) (x : \u211d) :\n    \u222b b, stieltjesOfMeasurableRat f hf.measurable (a, b) x \u2202(\u03bd a)\n      = (\u03ba a (univ \u00d7\u02e2 Iic x)).toReal", "lemma isRatStieltjesPoint_unit_prod_iff (f : \u03b1 \u2192 \u211a \u2192 \u211d) (a : \u03b1) :\n    IsRatStieltjesPoint (fun p : Unit \u00d7 \u03b1 \u21a6 f p.2) ((), a)\n      \u2194 IsRatStieltjesPoint f a", "lemma IsFiniteKernel.integrable (\u03bc : Measure \u03b1) [IsFiniteMeasure \u03bc]\n    (\u03ba : kernel \u03b1 \u03b2) [IsFiniteKernel \u03ba] {s : Set \u03b2} (hs : MeasurableSet s) :\n    Integrable (fun x => (\u03ba x s).toReal) \u03bc", "lemma compProd_add_right (\u03bc : kernel \u03b1 \u03b2) (\u03ba \u03b7 : kernel (\u03b1 \u00d7 \u03b2) \u03b3)\n    [IsSFiniteKernel \u03bc] [IsSFiniteKernel \u03ba] [IsSFiniteKernel \u03b7] :\n    \u03bc \u2297\u2096 (\u03ba + \u03b7) = \u03bc \u2297\u2096 \u03ba + \u03bc \u2297\u2096 \u03b7", "@[simp]\nlemma const_zero : kernel.const \u03b1 (0 : Measure \u03b2) = 0", "lemma density_fst_univ (\u03ba : kernel \u03b1 (\u03b3 \u00d7 \u03b2)) [IsFiniteKernel \u03ba] (a : \u03b1) :\n    \u2200\u1d50 x \u2202(fst \u03ba a), density \u03ba (fst \u03ba) a x univ = 1", "lemma density_nonneg (h\u03ba\u03bd : fst \u03ba \u2264 \u03bd) (a : \u03b1) (x : \u03b3) (s : Set \u03b2) :\n    0 \u2264 density \u03ba \u03bd a x s", "theorem deterministic_apply {f : \u03b1 \u2192 \u03b2} (hf : Measurable f) (a : \u03b1) :\n    deterministic f hf a = Measure.dirac (f a)", "theorem finset_sum_apply (I : Finset \u03b9) (\u03ba : \u03b9 \u2192 kernel \u03b1 \u03b2) (a : \u03b1) :\n    (\u2211 i \u2208 I, \u03ba i) a = \u2211 i \u2208 I, \u03ba i a", "theorem finset_sum_apply' (I : Finset \u03b9) (\u03ba : \u03b9 \u2192 kernel \u03b1 \u03b2) (a : \u03b1) (s : Set \u03b2) :\n    (\u2211 i \u2208 I, \u03ba i) a s = \u2211 i \u2208 I, \u03ba i a s", "@[simp] lemma fst_prod (\u03ba : kernel \u03b1 \u03b2) [IsSFiniteKernel \u03ba] (\u03b7 : kernel \u03b1 \u03b3) [IsMarkovKernel \u03b7] :\n    fst (\u03ba \u00d7\u2096 \u03b7) = \u03ba", "@[to_additive]\ntheorem iIndepFun.indepFun_prod_range_succ {f : \u2115 \u2192 \u03a9 \u2192 \u03b2}\n    (hf_Indep : iIndepFun (fun _ => m) f \u03ba \u03bc) (hf_meas : \u2200 i, Measurable (f i)) (n : \u2115) :\n    IndepFun (\u220f j \u2208 Finset.range n, f j) (f n) \u03ba \u03bc", "theorem iIndepFun_iff_measure_inter_preimage_eq_mul {\u03b9 : Type*} {\u03b2 : \u03b9 \u2192 Type*}\n    (m : \u2200 x, MeasurableSpace (\u03b2 x)) (f : \u2200 i, \u03a9 \u2192 \u03b2 i) :\n    iIndepFun m f \u03ba \u03bc \u2194\n      \u2200 (S : Finset \u03b9) {sets : \u2200 i : \u03b9, Set (\u03b2 i)} (_H : \u2200 i, i \u2208 S \u2192 MeasurableSet[m i] (sets i)),\n        \u2200\u1d50 a \u2202\u03bc, \u03ba a (\u22c2 i \u2208 S, (f i) \u207b\u00b9' (sets i)) = \u220f i \u2208 S, \u03ba a ((f i) \u207b\u00b9' (sets i))", "theorem iIndepSet_iff_meas_biInter {_m\u03a9 : MeasurableSpace \u03a9} {\u03ba : kernel \u03b1 \u03a9}\n    [IsMarkovKernel \u03ba] {\u03bc : Measure \u03b1} {f : \u03b9 \u2192 Set \u03a9} (hf : \u2200 i, MeasurableSet (f i)) :\n    iIndepSet f \u03ba \u03bc \u2194 \u2200 s, \u2200\u1d50 a \u2202\u03bc, \u03ba a (\u22c2 i \u2208 s, f i) = \u220f i \u2208 s, \u03ba a (f i)", "theorem iIndep_comap_mem_iff {f : \u03b9 \u2192 Set \u03a9} {_m\u03a9 : MeasurableSpace \u03a9}\n    {\u03ba : kernel \u03b1 \u03a9} {\u03bc : Measure \u03b1} :\n    iIndep (fun i => MeasurableSpace.comap (\u00b7 \u2208 f i) \u22a4) \u03ba \u03bc \u2194 iIndepSet f \u03ba \u03bc", "theorem kernel.indep_limsup_atTop_self (h_le : \u2200 n, s n \u2264 m0) (h_indep : iIndep s \u03ba \u03bc\u03b1) :\n    Indep (limsup s atTop) (limsup s atTop) \u03ba \u03bc\u03b1", "theorem kernel.integral_fn_integral_add \u2983f g : \u03b2 \u00d7 \u03b3 \u2192 E\u2984 (F : E \u2192 E')\n    (hf : Integrable f ((\u03ba \u2297\u2096 \u03b7) a)) (hg : Integrable g ((\u03ba \u2297\u2096 \u03b7) a)) :\n    \u222b x, F (\u222b y, f (x, y) + g (x, y) \u2202\u03b7 (a, x)) \u2202\u03ba a =\n      \u222b x, F (\u222b y, f (x, y) \u2202\u03b7 (a, x) + \u222b y, g (x, y) \u2202\u03b7 (a, x)) \u2202\u03ba a", "theorem integral_withDensity {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n    {f : \u03b2 \u2192 E} [IsSFiniteKernel \u03ba] {a : \u03b1} {g : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650}\n    (hg : Measurable (Function.uncurry g)) :\n    \u222b b, f b \u2202withDensity \u03ba (fun a b => g a b) a = \u222b b, g a b \u2022 f b \u2202\u03ba a", "lemma isCondKernelCDF_condKernelCDF (\u03ba : kernel \u03b1 (\u03b3 \u00d7 \u211d)) [IsFiniteKernel \u03ba] :\n    IsCondKernelCDF (condKernelCDF \u03ba) \u03ba (fst \u03ba)", "theorem isFiniteKernel_withDensity_of_bounded (\u03ba : kernel \u03b1 \u03b2) [IsFiniteKernel \u03ba] {B : \u211d\u22650\u221e}\n    (hB_top : B \u2260 \u221e) (hf_B : \u2200 a b, f a b \u2264 B) : IsFiniteKernel (withDensity \u03ba f)", "lemma lintegral_density (h\u03ba\u03bd : fst \u03ba \u2264 \u03bd) [IsFiniteKernel \u03bd]\n    (a : \u03b1) {s : Set \u03b2} (hs : MeasurableSet s) :\n    \u222b\u207b x, ENNReal.ofReal (density \u03ba \u03bd a x s) \u2202(\u03bd a) = \u03ba a (univ \u00d7\u02e2 s)", "theorem lintegral_prodMkRight (\u03ba : kernel \u03b1 \u03b2) (ca : \u03b1 \u00d7 \u03b3) (g : \u03b2 \u2192 \u211d\u22650\u221e) :\n    \u222b\u207b b, g b \u2202prodMkRight \u03b3 \u03ba ca = \u222b\u207b b, g b \u2202\u03ba ca.fst", "theorem measurable_compProdFun (\u03ba : kernel \u03b1 \u03b2) [IsSFiniteKernel \u03ba] (\u03b7 : kernel (\u03b1 \u00d7 \u03b2) \u03b3)\n    [IsSFiniteKernel \u03b7] (hs : MeasurableSet s) : Measurable fun a => compProdFun \u03ba \u03b7 a s", "theorem kernel.measure_zero_or_one_of_measurableSet_limsup_atBot (h_le : \u2200 n, s n \u2264 m0)\n    (h_indep : iIndep s \u03ba \u03bc\u03b1) {t : Set \u03a9} (ht_tail : MeasurableSet[limsup s atBot] t) :\n    \u2200\u1d50 a \u2202\u03bc\u03b1, \u03ba a t = 0 \u2228 \u03ba a t = 1", "theorem setLIntegral_compProd_univ_left (\u03ba : kernel \u03b1 \u03b2) [IsSFiniteKernel \u03ba] (\u03b7 : kernel (\u03b1 \u00d7 \u03b2) \u03b3)\n    [IsSFiniteKernel \u03b7] (a : \u03b1) {f : \u03b2 \u00d7 \u03b3 \u2192 \u211d\u22650\u221e} (hf : Measurable f) {t : Set \u03b3}\n    (ht : MeasurableSet t) :\n    \u222b\u207b z in Set.univ \u00d7\u02e2 t, f z \u2202(\u03ba \u2297\u2096 \u03b7) a = \u222b\u207b x, \u222b\u207b y in t, f (x, y) \u2202\u03b7 (a, x) \u2202\u03ba a", "lemma singularPart_of_subset_mutuallySingularSetSlice [IsFiniteKernel \u03ba]\n    [IsFiniteKernel \u03b7] {a : \u03b1} {s : Set \u03b3} (hsm : MeasurableSet s)\n    (hs : s \u2286 mutuallySingularSetSlice \u03ba \u03b7 a) :\n    singularPart \u03ba \u03b7 a s = \u03ba a s", "theorem sum_add [Countable \u03b9] (\u03ba \u03b7 : \u03b9 \u2192 kernel \u03b1 \u03b2) :\n    (kernel.sum fun n => \u03ba n + \u03b7 n) = kernel.sum \u03ba + kernel.sum \u03b7", "lemma tendsto_densityProcess_limitProcess (h\u03ba\u03bd : fst \u03ba \u2264 \u03bd)\n    [IsFiniteKernel \u03bd] (a : \u03b1) {s : Set \u03b2} (hs : MeasurableSet s) :\n    \u2200\u1d50 x \u2202(\u03bd a), Tendsto (fun n \u21a6 densityProcess \u03ba \u03bd n a x s) atTop\n      (\ud835\udcdd ((countableFiltration \u03b3).limitProcess\n      (fun n x \u21a6 densityProcess \u03ba \u03bd n a x s) (\u03bd a) x))", "lemma tendsto_snorm_one_restrict_densityProcess_limitProcess [IsFiniteKernel \u03bd]\n    (h\u03ba\u03bd : fst \u03ba \u2264 \u03bd) (a : \u03b1) {s : Set \u03b2} (hs : MeasurableSet s) (A : Set \u03b3) :\n    Tendsto (fun n \u21a6 snorm ((fun x \u21a6 densityProcess \u03ba \u03bd n a x s)\n      - (countableFiltration \u03b3).limitProcess (fun n x \u21a6 densityProcess \u03ba \u03bd n a x s) (\u03bd a))\n      1 ((\u03bd a).restrict A)) atTop (\ud835\udcdd 0)", "lemma withDensity_rnDeriv_eq_zero_iff_measure_eq_zero (\u03ba \u03b7 : kernel \u03b1 \u03b3)\n    [IsFiniteKernel \u03ba] [IsFiniteKernel \u03b7] (a : \u03b1) :\n    withDensity \u03b7 (rnDeriv \u03ba \u03b7) a = 0 \u2194 \u03ba a (mutuallySingularSetSlice \u03ba \u03b7 a)\u1d9c = 0", "lemma measure_cdf [IsProbabilityMeasure \u03bc] : (cdf \u03bc).measure = \u03bc", "theorem measure_ge_le_exp_cgf [IsFiniteMeasure \u03bc] (\u03b5 : \u211d) (ht : 0 \u2264 t)\n    (h_int : Integrable (fun \u03c9 => exp (t * X \u03c9)) \u03bc) :\n    (\u03bc {\u03c9 | \u03b5 \u2264 X \u03c9}).toReal \u2264 exp (-t * \u03b5 + cgf X \u03bc t)", "theorem mgf_const (c : \u211d) [IsProbabilityMeasure \u03bc] : mgf (fun _ => c) \u03bc t = exp (t * c)", "theorem mgf_nonneg : 0 \u2264 mgf X \u03bc t", "@[simp]\ntheorem mgf_zero_fun : mgf 0 \u03bc t = (\u03bc Set.univ).toReal", "lemma setIntegral_stieltjesOfMeasurableRat [IsFiniteKernel \u03ba] (hf : IsRatCondKernelCDF f \u03ba \u03bd)\n    (a : \u03b1) (x : \u211d) {s : Set \u03b2} (hs : MeasurableSet s) :\n    \u222b b in s, stieltjesOfMeasurableRat f hf.measurable (a, b) x \u2202(\u03bd a)\n      = (\u03ba a (s \u00d7\u02e2 Iic x)).toReal", "lemma setIntegral_stieltjesOfMeasurableRat_rat (hf : IsRatCondKernelCDF f \u03ba \u03bd) (a : \u03b1) (q : \u211a)\n    {s : Set \u03b2} (hs : MeasurableSet s) :\n    \u222b b in s, stieltjesOfMeasurableRat f hf.measurable (a, b) q \u2202(\u03bd a)\n      = (\u03ba a (s \u00d7\u02e2 Iic (q : \u211d))).toReal", "lemma stieltjesOfMeasurableRat_nonneg (hf : Measurable f) (a : \u03b1) (r : \u211d) :\n    0 \u2264 stieltjesOfMeasurableRat f hf a r := IsMeasurableRatCDF.stieltjesFunction_nonneg _ a r", "theorem strong_law_ae\n    (X : \u2115 \u2192 \u03a9 \u2192 E) (hint : Integrable (X 0))\n    (hindep : Pairwise (fun i j \u21a6 IndepFun (X i) (X j))) (hident : \u2200 i, IdentDistrib (X i) (X 0)) :\n    \u2200\u1d50 \u03c9, Tendsto (fun n : \u2115 \u21a6 (n : \u211d) \u207b\u00b9 \u2022 (\u2211 i \u2208 range n, X i \u03c9)) atTop (\ud835\udcdd \ud835\udd3c[X 0])", "theorem sum_variance_truncation_le {X : \u03a9 \u2192 \u211d} (hint : Integrable X) (hnonneg : 0 \u2264 X) (K : \u2115) :\n    \u2211 j \u2208 range K, ((j : \u211d) ^ 2)\u207b\u00b9 * \ud835\udd3c[truncation X j ^ 2] \u2264 2 * \ud835\udd3c[X]", "theorem tendsto_integral_truncation {f : \u03b1 \u2192 \u211d} (hf : Integrable f \u03bc) :\n    Tendsto (fun A => \u222b x, truncation f A x \u2202\u03bc) atTop (\ud835\udcdd (\u222b x, f x \u2202\u03bc))", "lemma tendsto_stieltjesOfMeasurableRat_atTop (hf : Measurable f) (a : \u03b1) :\n    Tendsto (stieltjesOfMeasurableRat f hf a) atTop (\ud835\udcdd 1)", "theorem comul_comp_snd :\n    comul \u2218\u2097 .snd R A B = TensorProduct.map (.snd R A B) (.snd R A B) \u2218\u2097 comul", "theorem Prod.edist_eq [PseudoEMetricSpace \u03b2] (x y : \u03b1 \u00d7 \u03b2) :\n    edist x y = max (edist x.1 y.1) (edist x.2 y.2)", "theorem fst_sSup [SupSet \u03b1] [SupSet \u03b2] (s : Set (\u03b1 \u00d7 \u03b2)) : (sSup s).fst = sSup (Prod.fst '' s)", "theorem gameAdd_le_lex : GameAdd r\u03b1 r\u03b2 \u2264 Prod.Lex r\u03b1 r\u03b2", "@[to_additive (attr := simp)]\ntheorem inv_mk [Inv G] [Inv H] (a : G) (b : H) : (a, b)\u207b\u00b9 = (a\u207b\u00b9, b\u207b\u00b9)", "@[simp]\ntheorem map_id : Prod.map (@id \u03b1) (@id \u03b2) = id", "@[simp]\ntheorem Prod.mk_dvd_mk {x\u2081 y\u2081 : G\u2081} {x\u2082 y\u2082 : G\u2082} :\n    (x\u2081, x\u2082) \u2223 (y\u2081, y\u2082) \u2194 x\u2081 \u2223 y\u2081 \u2227 x\u2082 \u2223 y\u2082", "@[to_additive existing (attr := simp) (reorder := 6 7) smul_fst]\ntheorem pow_fst (p : \u03b1 \u00d7 \u03b2) (c : E) : (p ^ c).fst = p.fst ^ c", "@[to_additive]\ntheorem smul_def (a : M) (x : \u03b1 \u00d7 \u03b2) : a \u2022 x = (a \u2022 x.1, a \u2022 x.2)", "@[to_additive (attr := simp)]\ntheorem smul_fst : (a \u2022 x).1 = a \u2022 x.1", "@[to_additive (attr := simp)]\ntheorem smul_mk (a : M) (b : \u03b1) (c : \u03b2) : a \u2022 (b, c) = (a \u2022 b, a \u2022 c)", "@[to_additive (attr := simp)]\ntheorem smul_swap : (a \u2022 x).swap = a \u2022 x.swap", "theorem swap_iInf [InfSet \u03b1] [InfSet \u03b2] (f : \u03b9 \u2192 \u03b1 \u00d7 \u03b2) : (iInf f).swap = \u2a05 i, (f i).swap", "@[simp]\ntheorem swap_swap_eq : swap \u2218 swap = @id (\u03b1 \u00d7 \u03b2)", "theorem \u03c9Sup_zip (c\u2080 : Chain \u03b1) (c\u2081 : Chain \u03b2) : \u03c9Sup (c\u2080.zip c\u2081) = (\u03c9Sup c\u2080, \u03c9Sup c\u2081)", "theorem isGood_mono {l : Products I} {o\u2081 o\u2082 : Ordinal} (h : o\u2081 \u2264 o\u2082)\n    (hl : l.isGood (\u03c0 C (ord I \u00b7 < o\u2081))) : l.isGood (\u03c0 C (ord I \u00b7 < o\u2082))", "theorem Products.isGood_nil : Products.isGood ({fun _ \u21a6 false} : Set (I \u2192 Bool)) Products.nil", "theorem prop_of_isGood  {l : Products I} (J : I \u2192 Prop) [\u2200 j, Decidable (J j)]\n    (h : l.isGood (\u03c0 C J)) : \u2200 a, a \u2208 l.val \u2192 J a", "theorem rel_head!_of_mem [Inhabited I] {i : I} {l : Products I} (hi : i \u2208 l.val) :\n    i \u2264 l.val.head!", "theorem isClosed_C1 : IsClosed (C1 C ho)", "theorem projRestricts_eq_id : ProjRestricts C (fun i (h : J i) \u21a6 h) = id", "theorem proj_prop_eq_self (hh : \u2200 i x, x \u2208 C \u2192 x i \u2260 false \u2192 J i) : \u03c0 C J = C", "theorem spanFinBasis.span : \u22a4 \u2264 Submodule.span \u2124 (Set.range (spanFinBasis C s))", "theorem succ_mono : CategoryTheory.Mono (ModuleCat.ofHom (\u03c0s C o))", "theorem swapTrue_eq_true (x : I \u2192 Bool) : SwapTrue o x (term I ho) = true", "@[simp]\ntheorem coe_id (X : Profinite) : (\ud835\udfd9 ((forget Profinite).obj X)) = id", "lemma projective_of_extrDisc {X : Profinite.{u}} (hX : ExtremallyDisconnected X) :\n    Projective X", "lemma pullback.hom_ext {Z : Profinite.{u}} (a b : Z \u27f6 pullback f g)\n    (hfst : a \u226b pullback.fst f g = b \u226b pullback.fst f g)\n    (hsnd : a \u226b pullback.snd f g = b \u226b pullback.snd f g) : a = b", "theorem basicOpen_mul (f g : A) : basicOpen \ud835\udc9c (f * g) = basicOpen \ud835\udc9c f \u2293 basicOpen \ud835\udc9c g", "theorem basicOpen_mul_le_left (f g : A) : basicOpen \ud835\udc9c (f * g) \u2264 basicOpen \ud835\udc9c f", "theorem mem_compl_zeroLocus_iff_not_mem {f : A} {I : ProjectiveSpectrum \ud835\udc9c} :\n    I \u2208 (zeroLocus \ud835\udc9c {f} : Set (ProjectiveSpectrum \ud835\udc9c))\u1d9c \u2194 f \u2209 I.asHomogeneousIdeal", "@[simp]\ntheorem vanishingIdeal_singleton (x : ProjectiveSpectrum \ud835\udc9c) :\n    vanishingIdeal ({x} : Set (ProjectiveSpectrum \ud835\udc9c)) = x.asHomogeneousIdeal", "theorem zeroLocus_iSup_homogeneousIdeal {\u03b3 : Sort*} (I : \u03b3 \u2192 HomogeneousIdeal \ud835\udc9c) :\n    zeroLocus _ ((\u2a06 i, I i : HomogeneousIdeal \ud835\udc9c) : Set A) = \u22c2 i, zeroLocus \ud835\udc9c (I i)", "theorem zeroLocus_mul_homogeneousIdeal (I J : HomogeneousIdeal \ud835\udc9c) :\n    zeroLocus \ud835\udc9c ((I * J : HomogeneousIdeal \ud835\udc9c) : Set A) = zeroLocus \ud835\udc9c I \u222a zeroLocus \ud835\udc9c J", "@[simp]\ntheorem zeroLocus_singleton_pow (f : A) (n : \u2115) (hn : 0 < n) :\n    zeroLocus \ud835\udc9c ({f ^ n} : Set A) = zeroLocus \ud835\udc9c {f}", "theorem zeroLocus_sup_homogeneousIdeal (I J : HomogeneousIdeal \ud835\udc9c) :\n    zeroLocus \ud835\udc9c ((I \u2294 J : HomogeneousIdeal \ud835\udc9c) : Set A) = zeroLocus _ I \u2229 zeroLocus _ J", "theorem span_le_subspace_iff {S : Set (\u2119 K V)} {W : Subspace K V} : span S \u2264 W \u2194 S \u2286 W", "@[simp]\ntheorem dependent_pair_iff_eq (u v : \u2119 K V) : Dependent ![u, v] \u2194 u = v", "@[elab_as_elim, cases_eliminator, induction_eliminator]\ntheorem ind {P : \u2119 K V \u2192 Prop} (h : \u2200 (v : V) (h : v \u2260 0), P (mk K v h)) : \u2200 p, P p", "theorem mk_eq_mk_iff' (v w : V) (hv : v \u2260 0) (hw : w \u2260 0) :\n    mk K v hv = mk K w hw \u2194 \u2203 a : K, a \u2022 w = v", "@[simp]\ntheorem dual_dual (K : ProperCone \u211d E) : K.dual.dual = K", "theorem Pullback.continuous_totalSpaceMk [\u2200 x, TopologicalSpace (E x)] [FiberBundle F E]\n    {f : B' \u2192 B} {x : B'} : Continuous (@TotalSpace.mk _ F (f *\u1d56 E) x)", "theorem eq : x * x + y * y = z * z", "theorem even_odd_of_coprime (hc : Int.gcd x y = 1) :\n    x % 2 = 0 \u2227 y % 2 = 1 \u2228 x % 2 = 1 \u2227 y % 2 = 0", "theorem Cofix.bisim' {\u03b1 : Type*} (Q : \u03b1 \u2192 Prop) (u v : \u03b1 \u2192 Cofix F)\n    (h : \u2200 x, Q x \u2192 \u2203 a f f', Cofix.dest (u x) = abs \u27e8a, f\u27e9 \u2227 Cofix.dest (v x) = abs \u27e8a, f'\u27e9 \u2227\n      \u2200 i, \u2203 x', Q x' \u2227 f i = u x' \u2227 f' i = v x') :\n    \u2200 x, Q x \u2192 u x = v x", "theorem comp_map {\u03b1 \u03b2 \u03b3 : Type _} (f : \u03b1 \u2192 \u03b2) (g : \u03b2 \u2192 \u03b3) (x : F \u03b1) :\n    (g \u2218 f) <$> x = g <$> f <$> x", "@[simp]\ntheorem coe_toLinearEquiv (f : Q\u2081.IsometryEquiv Q\u2082) : \u21d1(f : M\u2081 \u2243\u2097[R] M\u2082) = f", "theorem anisotropic_of_prod {R} [OrderedCommRing R] [Module R M\u2081] [Module R M\u2082]\n    {Q\u2081 : QuadraticForm R M\u2081} {Q\u2082 : QuadraticForm R M\u2082} (h : (Q\u2081.prod Q\u2082).Anisotropic) :\n    Q\u2081.Anisotropic \u2227 Q\u2082.Anisotropic", "@[simp]\ntheorem associated_isOrtho [Invertible (2 : R)] {x y : M} :\n    Q.associated.IsOrtho x y \u2194 Q.IsOrtho x y", "@[simp]\ntheorem associated_linMulLin (f g : M \u2192\u2097[R] R) :\n    associated (R := R) (linMulLin f g) =\n      \u215f (2 : R) \u2022 ((mul R R).compl\u2081\u2082 f g + (mul R R).compl\u2081\u2082 g f)", "theorem complex_equivalent {M : Type*} [AddCommGroup M] [Module \u2102 M] [FiniteDimensional \u2102 M]\n    (Q\u2081 Q\u2082 : QuadraticForm \u2102 M) (hQ\u2081 : (associated (R := \u2102) Q\u2081).SeparatingLeft)\n    (hQ\u2082 : (associated (R := \u2102) Q\u2082).SeparatingLeft) : Equivalent Q\u2081 Q\u2082", "@[ext]\ntheorem ext (H : \u2200 x : M, Q x = Q' x) : Q = Q'", "@[simp]\ntheorem isOrtho_polarBilin {x y : M} : Q.polarBilin.IsOrtho x y \u2194 IsOrtho Q x y", "theorem linMulLinSelfPosDef {R} [LinearOrderedCommRing R] [Module R M] (f : M \u2192\u2097[R] R)\n    (hf : LinearMap.ker f = \u22a5) : PosDef (linMulLin f f)", "@[simp]\ntheorem linMulLin_apply (f g : M \u2192\u2097[R] R) (x) : linMulLin f g x = f x * g x", "@[simp]\ntheorem map_neg (x : M) : Q (-x) = Q x", "theorem _root_.QuadraticForm.polarBilin_comp (Q : QuadraticForm R N) (f : M \u2192\u2097[R] N) :\n    polarBilin (Q.comp f) = compl\u2081\u2082 (polarBilin Q) f f", "@[simp]\ntheorem polar_smul_left (a : R) (x y : M) : polar Q (a \u2022 x) y = a * polar Q x y", "theorem posDef_of_nonneg {Q : QuadraticForm R\u2082 M} (h : \u2200 x, 0 \u2264 Q x) (h0 : Q.Anisotropic) :\n    PosDef Q", "theorem posDef_of_toMatrix' [DecidableEq n] {Q : QuadraticForm \u211d (n \u2192 \u211d)}\n    (hQ : Q.toMatrix'.PosDef) : Q.PosDef", "@[simp]\ntheorem sum_apply {\u03b9 : Type*} (Q : \u03b9 \u2192 QuadraticForm R M) (s : Finset \u03b9) (x : M) :\n    (\u2211 i \u2208 s, Q i) x = \u2211 i \u2208 s, Q i x", "@[simp]\ntheorem tensorDistrib_tmul (Q\u2081 : QuadraticForm A M\u2081) (Q\u2082 : QuadraticForm R M\u2082) (m\u2081 : M\u2081) (m\u2082 : M\u2082) :\n    tensorDistrib R A (Q\u2081 \u2297\u209c Q\u2082) (m\u2081 \u2297\u209c m\u2082) = Q\u2082 m\u2082 \u2022 Q\u2081 m\u2081", "@[simp]\ntheorem tmul_comp_tensorMap\n    {Q\u2081 : QuadraticForm R M\u2081} {Q\u2082 : QuadraticForm R M\u2082}\n    {Q\u2083 : QuadraticForm R M\u2083} {Q\u2084 : QuadraticForm R M\u2084}\n    (f : Q\u2081 \u2192q\u1d62 Q\u2082) (g : Q\u2083 \u2192q\u1d62 Q\u2084) :\n    (Q\u2082.tmul Q\u2084).comp (TensorProduct.map f.toLinearMap g.toLinearMap) = Q\u2081.tmul Q\u2083", "@[simp]\ntheorem toFun_eq_coe : Q.toFun = \u21d1Q", "theorem toQuadraticForm_associated : (associatedHom S Q).toQuadraticForm = Q", "theorem forget\u2082_map_associator_hom (X Y Z : QuadraticModuleCat.{u} R) :\n    (forget\u2082 (QuadraticModuleCat R) (ModuleCat R)).map (\u03b1_ X Y Z).hom =\n      (\u03b1_ X.toModuleCat Y.toModuleCat Z.toModuleCat).hom", "theorem ae_eq_const_of_ae_eq_comp_ae {g : \u03b1 \u2192 X} (h : QuasiErgodic f \u03bc)\n    (hgm : AEStronglyMeasurable g \u03bc) (hg_eq : g \u2218 f =\u1d50[\u03bc] g) : \u2203 c, g =\u1d50[\u03bc] const \u03b1 c", "protected lemma comp {R\u2081 R\u2082 R\u2083 A : Type*} [Semifield R\u2081] [Field R\u2082] [Field R\u2083]\n    [NonUnitalRing A] [Module R\u2081 A] [Module R\u2082 A] [Module R\u2083 A] [Algebra R\u2081 R\u2082] [Algebra R\u2082 R\u2083]\n    [Algebra R\u2081 R\u2083] [IsScalarTower R\u2081 R\u2082 R\u2083] [IsScalarTower R\u2082 R\u2083 A] [IsScalarTower R\u2083 A A]\n    [SMulCommClass R\u2083 A A] {a : A} {f : R\u2083 \u2192 R\u2082} {g : R\u2082 \u2192 R\u2081} {e : R\u2083 \u2192 R\u2081} (hfge : g \u2218 f = e)\n    (hf : QuasispectrumRestricts a f) (hg : QuasispectrumRestricts a g) :\n    QuasispectrumRestricts a e where\n  left_inv", "lemma nonUnitalStarAlgHom_id {a : A} {\u03c6 : C(\u03c3\u2099 S a, S)\u2080 \u2192\u22c6\u2099\u2090[S] A} {f : C(S, R)}\n    (h : QuasispectrumRestricts a f) (h_id : \u03c6 (.id rfl) = a) :\n    h.nonUnitalStarAlgHom \u03c6 (.id rfl) = a", "theorem subset_preimage : quasispectrum S a \u2286 f \u207b\u00b9' quasispectrum R a", "@[simp, norm_cast]\ntheorem coe_imI : (x : \u210d[R]).imI = 0", "@[ext]\ntheorem ext : a.re = b.re \u2192 a.imI = b.imI \u2192 a.imJ = b.imJ \u2192 a.imK = b.imK \u2192 a = b", "@[simp]\ntheorem fst_imJ_dualNumberEquiv_symm (d : DualNumber (Quaternion R)) :\n    (dualNumberEquiv.symm d).imJ.fst = d.fst.imJ", "@[simp, norm_cast] lemma imK_nnratCast (q : \u211a\u22650) : (q : \u210d[R]).imK = 0 := rfl", "theorem im_exp (q : \u210d[\u211d]) : (exp \u211d q).im = (exp \u211d q.re * (Real.sin \u2016q.im\u2016 / \u2016q.im\u2016)) \u2022 q.im", "@[simp]\ntheorem im_star : star a.im = -a.im", "theorem inner_def (a b : \u210d) : \u27eaa, b\u27eb = (a * star b).re", "@[simp]\ntheorem mul_re : (a * b).re = a.re * b.re - a.imI * b.imI - a.imJ * b.imJ - a.imK * b.imK", "@[simp, norm_cast]\ntheorem natCast_imJ (n : \u2115) : (n : \u210d[R]).imJ = 0", "@[simp] theorem neg_re : (-a).re = -a.re", "@[simp]\ntheorem snd_re_dualNumberEquiv_symm (d : DualNumber (Quaternion R)) :\n    (dualNumberEquiv.symm d).re.snd = d.snd.re", "@[simp] theorem star_imJ : (star a).imJ = -a.imJ", "@[simp] theorem star_re : (star a).re = a.re", "@[simp] nonrec theorem sub_self_re : a - \u2191a.re = a.im", "@[simp]\ntheorem j_mul_k : q.j * q.k = -c\u2082 \u2022 q.i", "theorem lift_mul (x y : \u210d[R,c\u2081,c\u2082]) : q.lift (x * y) = q.lift x * q.lift y", "@[simp, norm_cast]\ntheorem coe_add : ((x + y : R) : \u210d[R,c\u2081,c\u2082]) = x + y", "theorem coe_commutes : \u2191r * a = a * r", "@[simp, norm_cast]\ntheorem coe_imI : (x : \u210d[R,c\u2081,c\u2082]).imI = 0", "@[simp, norm_cast]\ntheorem coe_one : ((1 : R) : \u210d[R,c\u2081,c\u2082]) = 1", "theorem finrank_eq_four [StrongRankCondition R] : FiniteDimensional.finrank R \u210d[R,c\u2081,c\u2082] = 4", "@[simp, norm_cast]\ntheorem intCast_imI (z : \u2124) : (z : \u210d[R,c\u2081,c\u2082]).imI = 0", "@[simp]\ntheorem mul_imI : (a * b).imI = a.1 * b.2 + a.2 * b.1 - c\u2082 * a.3 * b.4 + c\u2082 * a.4 * b.3", "theorem mul_star_eq_coe : a * star a = (a * star a).re", "@[simp] theorem smul_re : (s \u2022 a).re = s \u2022 a.re", "theorem star_add_self' : star a + a = \u2191(2 * a.re)", "@[simp] theorem sub_imK : (a - b).imK = a.imK - b.imK", "@[simp]\ntheorem a_mul_a (i j : ZMod (2 * n)) : a i * a j = a (i + j)", "@[simp] theorem Quiver.Hom.unop_mk {X Y : C\u1d52\u1d56} (f : X \u27f6 Y) : Quiver.Hom.unop {unop := f} = f", "theorem Path.cast_cons {u v w u' w' : U} (p : Path u v) (e : v \u27f6 w) (hu : u = u') (hw : w = w') :\n    (p.cons e).cast hu hw = (p.cast hu rfl).cons (e.cast rfl hw)", "theorem comp_inj {p\u2081 p\u2082 : Path a b} {q\u2081 q\u2082 : Path b c} (hq : q\u2081.length = q\u2082.length) :\n    p\u2081.comp q\u2081 = p\u2082.comp q\u2082 \u2194 p\u2081 = p\u2082 \u2227 q\u2081 = q\u2082", "lemma ext {x y : SingleObj \u03b1} : x = y := Unit.ext x y", "theorem toPrefunctor_comp (f : \u03b1 \u2192 \u03b2) (g : \u03b2 \u2192 \u03b3) :\n    toPrefunctor (g \u2218 f) = toPrefunctor f \u22d9q toPrefunctor g", "@[simp]\nlemma map_comp (g : M' \u2192\u2097[R] M'') (f : M \u2192\u2097[R] M') :\n    map r (g \u2218\u2097 f) = map r g \u2218\u2097 map r f", "@[elab_as_elim]\nprotected theorem inductionOn\u2083' {p : Quotient s\u2081 \u2192 Quotient s\u2082 \u2192 Quotient s\u2083 \u2192 Prop}\n    (q\u2081 : Quotient s\u2081) (q\u2082 : Quotient s\u2082) (q\u2083 : Quotient s\u2083)\n    (h : \u2200 a\u2081 a\u2082 a\u2083, p (Quotient.mk'' a\u2081) (Quotient.mk'' a\u2082) (Quotient.mk'' a\u2083)) :\n    p q\u2081 q\u2082 q\u2083", "theorem Quotient.mk_eq_iff_out [s : Setoid \u03b1] {x : \u03b1} {y : Quotient s} :\n    \u27e6x\u27e7 = y \u2194 x \u2248 Quotient.out y", "@[simp]\ntheorem out_eq' (q : Quotient s\u2081) : Quotient.mk'' q.out' = q", "theorem surjective_Quotient_mk'' : Function.Surjective (Quotient.mk'' : \u03b1 \u2192 Quotient s\u2081)", "lemma QuotientAddGroup.integral_mul_eq_integral_automorphize_mul {K : Type*} [NormedField K]\n    [NormedSpace \u211d K] [\u03bc'.IsAddRightInvariant] {f : G' \u2192 K}\n    (f_\u2112_1 : Integrable f \u03bc') {g : G' \u29f8 \u0393' \u2192 K} (hg : AEStronglyMeasurable g \u03bc_\ud835\udcd5)\n    (g_\u2112_infinity : essSup (fun x \u21a6 (\u2016g x\u2016\u208a : \u211d\u22650\u221e)) \u03bc_\ud835\udcd5 \u2260 \u221e)\n    (F_ae_measurable : AEStronglyMeasurable (QuotientAddGroup.automorphize f) \u03bc_\ud835\udcd5)\n    (h\ud835\udcd5 : IsAddFundamentalDomain \u0393'.op \ud835\udcd5' \u03bc') :\n    \u222b x : G', g (x : G' \u29f8 \u0393') * (f x) \u2202\u03bc'\n      = \u222b x : G' \u29f8 \u0393', g x * (QuotientAddGroup.automorphize f x) \u2202\u03bc_\ud835\udcd5", "@[to_additive (attr := simp)]\ntheorem map_comp_map {I : Type*} [Group I] (M : Subgroup H) (O : Subgroup I) [M.Normal] [O.Normal]\n    (f : G \u2192* H) (g : H \u2192* I) (hf : N \u2264 Subgroup.comap f M) (hg : M \u2264 Subgroup.comap g O)\n    (hgf : N \u2264 Subgroup.comap (g.comp f) O", "@[to_additive (attr := measurability)]\ntheorem QuotientGroup.measurable_coe {G} [Group G] [MeasurableSpace G] {S : Subgroup G} :\n    Measurable ((\u2191) : G \u2192 G \u29f8 S)", "@[to_additive \"If the quotient by an additive subgroup gives a singleton then the additive subgroup\nis the whole additive group.\"]\ntheorem subgroup_eq_top_of_subsingleton (H : Subgroup G) (h : Subsingleton (G \u29f8 H)) : H = \u22a4", "@[simp, rclike_simps]\ntheorem I_to_real : IR = 0", "theorem abs_re_le_norm (z : K) : |re z| \u2264 \u2016z\u2016", "theorem conj_eq_iff_real {z : K} : conj z = z \u2194 \u2203 r : \u211d, z = (r : K)", "@[simp, rclike_simps]\ntheorem conj_im (z : K) : im (conj z) = -im z", "theorem div_re_ofReal {z : K} {r : \u211d} : re (z / r) = re z / r", "theorem ext_iff {z w : K} : z = w \u2194 re z = re w \u2227 im z = im w", "@[simp, rclike_simps]\ntheorem imCLM_apply : ((imCLM : K \u2192L[\u211d] \u211d) : K \u2192 \u211d) = im", "@[simp, rclike_simps, norm_cast]\ntheorem intCast_im (n : \u2124) : im (n : K) = 0", "@[simp, rclike_simps]\ntheorem inv_re (z : K) : re z\u207b\u00b9 = re z / normSq z", "theorem RCLike.nonUnitalContinuousFunctionalCalculus :\n    NonUnitalContinuousFunctionalCalculus \ud835\udd5c (p : A \u2192 Prop) where", "theorem ofReal_eq_zero {x : \u211d} : (x : K) = 0 \u2194 x = 0", "@[simp, rclike_simps, norm_cast]\ntheorem ofReal_ofNat (n : \u2115) [n.AtLeastTwo] :\n    ((no_index (OfNat.ofNat n) : \u211d) : K) = OfNat.ofNat n", "@[simp, norm_cast, rclike_simps]\ntheorem ofReal_re : \u2200 r : \u211d, re (r : K) = r", "@[simp, rclike_simps, norm_cast]\ntheorem ofReal_sum {\u03b1 : Type*} (s : Finset \u03b1) (f : \u03b1 \u2192 \u211d) :\n    ((\u2211 i \u2208 s, f i : \u211d) : K) = \u2211 i \u2208 s, (f i : K)", "@[simp, rclike_simps]\ntheorem one_re : re (1 : K) = 1", "theorem re_eq_self_of_le {a : K} (h : \u2016a\u2016 \u2264 re a) : (re a : K) = a", "@[symm]\ntheorem PreEnvelGroupRel.symm {R : Type u} [Rack R] {a b : PreEnvelGroup R} :\n    PreEnvelGroupRel R a b \u2192 PreEnvelGroupRel R b a", "@[simp]\ntheorem invAct_apply (x y : R) : (act' x)\u207b\u00b9 y = x \u25c3\u207b\u00b9 y", "theorem left_cancel_inv (x : R) {y y' : R} : x \u25c3\u207b\u00b9 y = x \u25c3\u207b\u00b9 y' \u2194 y = y'", "@[simp]\ntheorem self_invAct_invAct_eq {x y : R} : (x \u25c3\u207b\u00b9 x) \u25c3\u207b\u00b9 y = x \u25c3\u207b\u00b9 y", "protected theorem add_mul : (a + b) * c = a * c + b * c", "protected lemma add_nonneg : 0 \u2264 a \u2192 0 \u2264 b \u2192 0 \u2264 a + b", "lemma cast_def (q : \u211a) : (q : K) = q.num / q.den := DivisionRing.ratCast_def _", "@[mono]\ntheorem cast_mono : Monotone ((\u2191) : \u211a \u2192 K)", "@[simp, norm_cast]\ntheorem cast_natCast (n : \u2115) : ((n : \u211a) : \u03b1) = n", "theorem cast_pos_of_pos (hq : 0 < q) : (0 : K) < q", "@[norm_cast] lemma cast_sub_of_ne_zero (hp : (p.den : \u03b1) \u2260 0) (hq : (q.den : \u03b1) \u2260 0) :\n    \u2191(p - q) = (p - q : \u03b1)", "@[simp, norm_cast] lemma den_intCast (n : \u2124) : (n : \u211a).den = 1 := rfl", "lemma divInt_div_divInt_cancel_right {x : \u2124} (hx : x \u2260 0) (n d : \u2124) :\n    x /. n / (x /. d) = d /. n", "theorem divInt_self (a : Rat) : a.num /. a.den = a", "theorem fract_inv_num_lt_num_of_pos {q : \u211a} (q_pos : 0 < q) : (fract q\u207b\u00b9).num < q.num", "@[simp] theorem intCast_num (a : Int) : (a : Rat).num = a", "theorem inv_intCast_den_of_pos {a : \u2124} (ha0 : 0 < a) : ((a : \u211a)\u207b\u00b9.den : \u2124) = a", "theorem inv_natCast_den_of_pos {a : \u2115} (ha0 : 0 < a) : (a : \u211a)\u207b\u00b9.den = a", "theorem isSquare_iff {q : \u211a} : IsSquare q \u2194 IsSquare q.num \u2227 IsSquare q.den", "lemma mkRat_eq_divInt (n d) : mkRat n d = n /. d := rfl", "theorem mk_denom_ne_zero_of_ne_zero {q : \u211a} {n d : \u2124} (hq : q \u2260 0) (hqnd : q = n /. d) : d \u2260 0", "theorem mk_num_ne_zero_of_ne_zero {q : \u211a} {n d : \u2124} (hq : q \u2260 0) (hqnd : q = n /. d) : n \u2260 0", "protected theorem mul_comm (a b : Rat) : a * b = b * a", "theorem mul_den (q\u2081 q\u2082 : \u211a) :\n    (q\u2081 * q\u2082).den =\n      q\u2081.den * q\u2082.den / Nat.gcd (q\u2081.num * q\u2082.num).natAbs (q\u2081.den * q\u2082.den)", "theorem mul_den_dvd (q\u2081 q\u2082 : \u211a) : (q\u2081 * q\u2082).den \u2223 q\u2081.den * q\u2082.den", "protected lemma nonneg_antisymm : 0 \u2264 q \u2192 0 \u2264 -q \u2192 q = 0", "@[simp]\ntheorem Rat.not_irrational (q : \u211a) : \u00acIrrational q", "theorem num_den_mk {q : \u211a} {n d : \u2124} (hd : d \u2260 0) (qdf : q = n /. d) :\n    \u2203 c : \u2124, n = c * q.num \u2227 d = c * q.den", "@[simp] lemma num_nonpos {a : \u211a} : a.num \u2264 0 \u2194 a \u2264 0", "@[simp] lemma num_pos {a : \u211a} : 0 < a.num \u2194 0 < a := lt_iff_lt_of_le_iff_le num_nonpos", "@[simp]\ntheorem numberField_discr : discr \u211a = 1", "@[simp] theorem ofInt_ofNat : ofInt (OfNat.ofNat n) = OfNat.ofNat n", "@[simp] theorem ofNat_den : (OfNat.ofNat n : Rat).den = 1", "@[simp] theorem one_den : (1 : Rat).den = 1", "@[simp]\ntheorem preimage_cast_Ioi (q : \u211a) : (\u2191) \u207b\u00b9' Ioi (q : K) = Ioi q", "@[simp]\ntheorem preimage_cast_uIcc (p q : \u211a) : (\u2191) \u207b\u00b9' uIcc (p : K) q = uIcc p q", "@[simp] theorem zero_num : (0 : Rat).num = 0", "@[simp, norm_cast]\ntheorem coe_add : ((f + g : RatFunc F) : LaurentSeries F) = f + g", "theorem coe_def : (f : LaurentSeries F) = coeAlgHom F f", "theorem coe_mapRingHom_eq_coe_map [RingHomClass F R[X] S[X]] (\u03c6 : F) (h\u03c6 : R[X]\u2070 \u2264 S[X]\u2070.comap \u03c6) :\n    (mapRingHom \u03c6 h\u03c6 : RatFunc R \u2192 RatFunc S) = map \u03c6 h\u03c6", "@[simp]\ntheorem laurent_C (x : R) : laurent r (C x) = C x", "@[simp]\ntheorem mk_eq_div (p q : K[X]) : RatFunc.mk p q = algebraMap _ _ p / algebraMap _ _ q", "theorem monic_denom (x : RatFunc K) : (denom x).Monic", "theorem num_div_dvd (p : K[X]) {q : K[X]} (hq : q \u2260 0) :\n    num (algebraMap _ _ p / algebraMap _ _ q) \u2223 p", "theorem ofFractionRing_mul (p q : FractionRing K[X]) :\n    ofFractionRing (p * q) = ofFractionRing p * ofFractionRing q", "theorem ofFractionRing_neg (p : FractionRing K[X]) :\n    ofFractionRing (-p) = -ofFractionRing p", "theorem ofFractionRing_sub (p q : FractionRing K[X]) :\n    ofFractionRing (p - q) = ofFractionRing p - ofFractionRing q", "@[simp] theorem run_failure [Monad m] [Alternative m] (ctx : \u03c1) :\n    (failure : ReaderT \u03c1 m \u03b1).run ctx = failure", "theorem abs_sin_eq_of_two_zsmul_eq {\u03b8 \u03c8 : Angle} (h : (2 : \u2124) \u2022 \u03b8 = (2 : \u2124) \u2022 \u03c8) :\n    |sin \u03b8| = |sin \u03c8|", "theorem Real.Angle.coe_expMapCircle (\u03b8 : Real.Angle) :\n    (\u03b8.expMapCircle : \u2102) = \u03b8.cos + \u03b8.sin * I", "theorem coe_nsmul (n : \u2115) (x : \u211d) : \u2191(n \u2022 x : \u211d) = n \u2022 (\u2191x : Angle)", "theorem cos_antiperiodic : Function.Antiperiodic cos (\u03c0 : Angle)", "theorem cos_eq_zero_iff {\u03b8 : Angle} : cos \u03b8 = 0 \u2194 \u03b8 = (\u03c0 / 2 : \u211d) \u2228 \u03b8 = (-\u03c0 / 2 : \u211d)", "@[simp]\ntheorem cos_sq_add_sin_sq (\u03b8 : Real.Angle) : cos \u03b8 ^ 2 + sin \u03b8 ^ 2 = 1", "@[simp]\ntheorem Real.Angle.expMapCircle_coe (x : \u211d) : Real.Angle.expMapCircle x = _root_.expMapCircle x", "@[simp, norm_cast]\ntheorem intCast_mul_eq_zsmul (x : \u211d) (n : \u2124) : \u2191((n : \u211d) * x : \u211d) = n \u2022 (\u2191x : Angle)", "@[simp]\ntheorem sign_neg (\u03b8 : Angle) : (-\u03b8).sign = -\u03b8.sign", "@[simp]\ntheorem sin_coe (x : \u211d) : sin (x : Angle) = Real.sin x", "theorem tan_eq_of_two_zsmul_eq {\u03b8 \u03c8 : Angle} (h : (2 : \u2124) \u2022 \u03b8 = (2 : \u2124) \u2022 \u03c8) : tan \u03b8 = tan \u03c8", "theorem toReal_neg_iff_sign_neg {\u03b8 : Angle} : \u03b8.toReal < 0 \u2194 \u03b8.sign = -1", "@[simp]\ntheorem toReal_pi_div_two : ((\u03c0 / 2 : \u211d) : Angle).toReal = \u03c0 / 2", "theorem tendsto_logGammaSeq (hf_conv : ConvexOn \u211d (Ioi 0) f)\n    (hf_feq : \u2200 {y : \u211d}, 0 < y \u2192 f (y + 1) = f y + log y) (hx : 0 < x) :\n    Tendsto (logGammaSeq x) atTop (\ud835\udcdd <| f x - f 1)", "theorem Gamma_mul_Gamma_add_half (s : \u211d) :\n    Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : \u211d) ^ (1 - 2 * s) * \u221a\u03c0", "theorem Real.Gamma_one_half_eq : Real.Gamma (1 / 2) = \u221a\u03c0", "theorem arccos_neg (x : \u211d) : arccos (-x) = \u03c0 - arccos x", "theorem arccos_nonneg (x : \u211d) : 0 \u2264 arccos x", "@[simp]\ntheorem arcsin_eq_neg_pi_div_two {x : \u211d} : arcsin x = -(\u03c0 / 2) \u2194 x \u2264 -1", "@[simp]\ntheorem arcsin_lt_zero {x : \u211d} : arcsin x < 0 \u2194 x < 0", "@[simp]\ntheorem arcsin_nonneg {x : \u211d} : 0 \u2264 arcsin x \u2194 0 \u2264 x", "@[simp]\ntheorem arcsin_one : arcsin 1 = \u03c0 / 2", "theorem arctan_add {x y : \u211d} (h : x * y < 1) :\n    arctan x + arctan y = arctan ((x + y) / (1 - x * y))", "theorem _root_.Real.coe_toNNReal (r : \u211d) (hr : 0 \u2264 r) : (Real.toNNReal r : \u211d) = r", "lemma concaveOn_negMulLog : ConcaveOn \u211d (Set.Ici (0 : \u211d)) negMulLog", "theorem contDiff_cos {n} : ContDiff \u211d n cos", "theorem continuousAt_const_rpow {a b : \u211d} (h : a \u2260 0) : ContinuousAt (a ^ \u00b7) b", "@[fun_prop]\ntheorem continuousAt_rpow_const (x : \u211d) (q : \u211d) (h : x \u2260 0 \u2228 0 \u2264 q) :\n    ContinuousAt (fun x : \u211d => x ^ q) x", "theorem continuousAt_rpow_of_ne (p : \u211d \u00d7 \u211d) (hp : p.1 \u2260 0) :\n    ContinuousAt (fun p : \u211d \u00d7 \u211d => p.1 ^ p.2) p", "@[continuity]\ntheorem continuous_arccos : Continuous arccos", "@[continuity, fun_prop]\ntheorem continuous_sin : Continuous sin", "@[continuity, fun_prop]\ntheorem continuous_sinh : Continuous sinh", "@[simp]\ntheorem convergent_succ (\u03be : \u211d) (n : \u2115) :\n    \u03be.convergent (n + 1) = \u230a\u03be\u230b + ((fract \u03be)\u207b\u00b9.convergent n)\u207b\u00b9", "@[simp]\ntheorem cos_add_int_mul_two_pi (x : \u211d) (n : \u2124) : cos (x + n * (2 * \u03c0)) = cos x", "theorem cos_mem_Icc (x : \u211d) : cos x \u2208 Icc (-1 : \u211d) 1", "theorem cos_nonneg_of_neg_pi_div_two_le_of_le {x : \u211d} (hl : -(\u03c0 / 2) \u2264 x) (hu : x \u2264 \u03c0 / 2) :\n    0 \u2264 cos x", "theorem cos_sq' : cos x ^ 2 = 1 - sin x ^ 2", "theorem cos_sq_le_one : cos x ^ 2 \u2264 1", "@[simp]\ntheorem cosh_arsinh (x : \u211d) : cosh (arsinh x) = \u221a(1 + x ^ 2)", "@[simp]\ntheorem cosh_sub_sinh : cosh x - sinh x = exp (-x)", "nonrec theorem cosh_two_mul : cosh (2 * x) = cosh x ^ 2 + sinh x ^ 2", "@[simp]\ntheorem deriv_cos' : deriv cos = fun x => -sin x", "theorem differentiable_rpow_const {p : \u211d} (hp : 1 \u2264 p) : Differentiable \u211d fun x : \u211d => x ^ p", "lemma dist_le_of_mem_pi_Icc (hx : x \u2208 Icc x' y') (hy : y \u2208 Icc x' y') : dist x y \u2264 dist x' y'", "lemma Real.dist_right_le_of_mem_uIcc {x y z : \u211d} (h : y \u2208 uIcc x z) : dist y z \u2264 dist x z", "@[simp]\ntheorem ediam_Icc (a b : \u211d) : EMetric.diam (Icc a b) = ENNReal.ofReal (b - a)", "theorem exists_rat_pow_btwn_rat (hn : n \u2260 0) {x y : \u211a} (h : x < y) (hy : 0 < y) :\n    \u2203 q : \u211a, 0 < q \u2227 x < q ^ n \u2227 q ^ n < y", "theorem expNear_sub (n x r\u2081 r\u2082) : expNear n x r\u2081 -\n    expNear n x r\u2082 = x ^ n / n.factorial * (r\u2081 - r\u2082)", "lemma exp_abs_le (x : \u211d) : exp |x| \u2264 exp x + exp (-x)", "theorem exp_approx_start (x a b : \u211d) (h : |exp x - expNear 0 x a| \u2264 |x| ^ 0 / Nat.factorial 0 * b) :\n    |exp x - a| \u2264 b", "theorem exp_arsinh (x : \u211d) : exp (arsinh x) = x + \u221a(1 + x ^ 2)", "@[simp]\ntheorem exp_le_one_iff {x : \u211d} : exp x \u2264 1 \u2194 x \u2264 0", "@[simp]\ntheorem exp_lt_exp {x y : \u211d} : exp x < exp y \u2194 x < y", "theorem exp_neg_one_lt_d9 : exp (-1) < 0.3678794412", "lemma fderiv_fourierChar_neg_bilinear_left_apply (v y : V) (w : W) :\n    fderiv \u211d (fun v \u21a6 (\ud835\udc1e (-L v w) : \u2102)) v y = -2 * \u03c0 * I * L y w * \ud835\udc1e (-L v w)", "lemma fderiv_fourierChar_neg_bilinear_right_apply (v : V) (w y : W) :\n    fderiv \u211d (fun w \u21a6 (\ud835\udc1e (-L v w) : \u2102)) w y = -2 * \u03c0 * I * L v y * \ud835\udc1e (-L v w)", "lemma fourierIntegralInv_eq' (f : V \u2192 E) (w : V) :\n    \ud835\udcd5\u207b f w = \u222b v, Complex.exp ((\u2191(2 * \u03c0 * \u27eav, w\u27eb) * Complex.I)) \u2022 f v", "lemma fourierIntegral_eq (f : V \u2192 E) (w : V) :\n    \ud835\udcd5 f w = \u222b v, \ud835\udc1e (-\u27eav, w\u27eb) \u2022 f v := rfl", "theorem fourierIntegral_real_eq_integral_exp_smul (f : \u211d \u2192 E) (w : \u211d) :\n    \ud835\udcd5 f w = \u222b v : \u211d, Complex.exp (\u2191(-2 * \u03c0 * v * w) * Complex.I) \u2022 f v", "theorem geom_mean_le_arith_mean {\u03b9 : Type*} (s : Finset \u03b9) (w : \u03b9 \u2192 \u211d) (z : \u03b9 \u2192 \u211d)\n    (hw : \u2200 i \u2208 s, 0 \u2264 w i) (hw' : 0 < \u2211 i \u2208 s, w i) (hz : \u2200 i \u2208 s, 0 \u2264 z i) :\n    (\u220f i \u2208 s, z i ^ w i) ^ (\u2211 i \u2208 s, w i)\u207b\u00b9  \u2264  (\u2211 i \u2208 s, w i * z i) / (\u2211 i \u2208 s, w i)", "theorem hasStrictDerivAt_log (hx : x \u2260 0) : HasStrictDerivAt log x\u207b\u00b9 x", "theorem hasStrictDerivAt_rpow_const {x p : \u211d} (hx : x \u2260 0 \u2228 1 \u2264 p) :\n    HasStrictDerivAt (fun x => x ^ p) (p * x ^ (p - 1)) x", "theorem hasStrictDerivAt_tan {x : \u211d} (h : cos x \u2260 0) : HasStrictDerivAt tan (1 / cos x ^ 2) x", "theorem iInf_of_not_bddBelow {\u03b1 : Sort*} {f : \u03b1 \u2192 \u211d} (hf : \u00acBddBelow (Set.range f)) :\n    \u2a05 i, f i = 0", "theorem Real.iSup_mul_of_nonpos (ha : r \u2264 0) (f : \u03b9 \u2192 \u211d) : (\u2a06 i, f i) * r = \u2a05 i, f i * r", "theorem image_tan_Ioo : tan '' Ioo (-(\u03c0 / 2)) (\u03c0 / 2) = univ", "theorem injOn_cos : InjOn cos (Icc 0 \u03c0)", "theorem inv_rpow (hx : 0 \u2264 x) (y : \u211d) : x\u207b\u00b9 ^ y = (x ^ y)\u207b\u00b9", "theorem inv_sqrt_one_add_tan_sq {x : \u211d} (hx : 0 < cos x) : (\u221a(1 + tan x ^ 2))\u207b\u00b9 = cos x", "@[simp]\ntheorem isBigO_one_exp_comp {f : \u03b1 \u2192 \u211d} :\n    ((fun _ => 1 : \u03b1 \u2192 \u211d) =O[l] fun x => exp (f x)) \u2194 IsBoundedUnder (\u00b7 \u2265 \u00b7) l f", "theorem le_log_iff_exp_le (hy : 0 < y) : x \u2264 log y \u2194 exp x \u2264 y", "theorem le_logb_iff_rpow_le_of_base_lt_one (hy : 0 < y) : x \u2264 logb b y \u2194 y \u2264 b ^ x", "theorem le_sin_mul {x : \u211d} (hx : 0 \u2264 x) (hx' : x \u2264 1) : x \u2264 sin (\u03c0 / 2 * x)", "theorem le_tan {x : \u211d} (h1 : 0 \u2264 x) (h2 : x < \u03c0 / 2) : x \u2264 tan x", "@[gcongr]\nlemma log_le_log (hx : 0 < x) (hxy : x \u2264 y) : log x \u2264 log y", "theorem log_lt_sub_one_of_pos (hx1 : 0 < x) (hx2 : x \u2260 1) : log x < x - 1", "theorem log_nonpos (hx : 0 \u2264 x) (h'x : x \u2264 1) : log x \u2264 0", "theorem log_prod {\u03b1 : Type*} (s : Finset \u03b1) (f : \u03b1 \u2192 \u211d) (hf : \u2200 x \u2208 s, f x \u2260 0) :\n    log (\u220f i \u2208 s, f i) = \u2211 i \u2208 s, log (f i)", "@[simp]\ntheorem log_zero : log 0 = 0", "theorem logb_div_base {a b : \u211d} (h\u2081 : a \u2260 0) (h\u2082 : b \u2260 0) (c : \u211d) :\n    logb (a / b) c = ((logb a c)\u207b\u00b9 - (logb b c)\u207b\u00b9)\u207b\u00b9", "@[gcongr]\ntheorem logb_lt_logb (hx : 0 < x) (hxy : x < y) : logb b x < logb b y", "@[simp]\ntheorem logb_lt_logb_iff_of_base_lt_one (hx : 0 < x) (hy : 0 < y) :\n    logb b x < logb b y \u2194 y < x", "theorem logb_nonneg_iff_of_base_lt_one (hx : 0 < x) : 0 \u2264 logb b x \u2194 x \u2264 1", "theorem logb_pos_iff_of_base_lt_one (hx : 0 < x) : 0 < logb b x \u2194 x < 1", "theorem lt_logb_iff_rpow_lt (hy : 0 < y) : x < logb b y \u2194 b ^ x < y", "theorem lt_sInf_add_pos {s : Set \u211d} (h : s.Nonempty) {\u03b5 : \u211d} (h\u03b5 : 0 < \u03b5) :\n    \u2203 a \u2208 s, a < sInf s + \u03b5", "lemma lt_sin (hx : x < 0) : x < sin x", "theorem map_exp_nhds (x : \u211d) : map exp (\ud835\udcdd x) = \ud835\udcdd (exp x)", "theorem mk_one : mk 1 = 1", "theorem monotone_arcsin : Monotone arcsin", "theorem mul_le_sin {x : \u211d} (hx : 0 \u2264 x) (hx' : x \u2264 \u03c0 / 2) : 2 / \u03c0 * x \u2264 sin x", "lemma negMulLog_nonneg {x : \u211d} (h1 : 0 \u2264 x) (h2 : x \u2264 1) : 0 \u2264 negMulLog x", "theorem norm_rpow_of_nonneg {x y : \u211d} (hx_nonneg : 0 \u2264 x) : \u2016x ^ y\u2016 = \u2016x\u2016 ^ y", "theorem not_summable_natCast_inv : \u00acSummable (fun n => n\u207b\u00b9 : \u2115 \u2192 \u211d)", "theorem ofCauchy_zero : (\u27e80\u27e9 : \u211d) = 0", "theorem one_lt_rpow_iff_of_pos (hx : 0 < x) : 1 < x ^ y \u2194 1 < x \u2227 0 < y \u2228 x < 1 \u2227 y < 0", "theorem pow_sum_div_card_le_sum_pow {f : \u03b9 \u2192 \u211d} (n : \u2115) (hf : \u2200 a \u2208 s, 0 \u2264 f a) :\n    (\u2211 x \u2208 s, f x) ^ (n + 1) / (s.card : \u211d) ^ n \u2264 \u2211 x \u2208 s, f x ^ (n + 1)", "@[simp]\nlemma Real.rank_rat_real : Module.rank \u211a \u211d = continuum", "theorem real_sqrt_lt_nat_sqrt_succ {a : \u2115} : \u221a(a : \u211d) < Nat.sqrt a + 1", "theorem rpow_add (hx : 0 < x) (y z : \u211d) : x ^ (y + z) = x ^ y * x ^ z", "theorem rpow_def_of_nonpos {x : \u211d} (hx : x \u2264 0) (y : \u211d) :\n    x ^ y = if x = 0 then if y = 0 then 1 else 0 else exp (log x * y) * cos (y * \u03c0)", "theorem rpow_eq_zero_iff_of_nonneg (hx : 0 \u2264 x) : x ^ y = 0 \u2194 x = 0 \u2227 y \u2260 0", "lemma rpow_intCast_mul (hx : 0 \u2264 x) (n : \u2124) (z : \u211d) : x ^ (n * z) = (x ^ n) ^ z", "lemma rpow_inv_eq (hx : 0 \u2264 x) (hy : 0 \u2264 y) (hz : z \u2260 0) : x ^ z\u207b\u00b9 = y \u2194 x = y ^ z", "lemma rpow_inv_le_iff_of_pos (hx : 0 \u2264 x) (hy : 0 \u2264 y) (hz : 0 < z) : x ^ z\u207b\u00b9 \u2264 y \u2194 x \u2264 y ^ z", "theorem rpow_inv_lt_iff_of_neg (hx : 0 < x) (hy : 0 < y) (hz : z < 0) :\n    x ^ z\u207b\u00b9 < y \u2194 y ^ z < x", "@[gcongr]\ntheorem rpow_le_rpow {x y z : \u211d} (h : 0 \u2264 x) (h\u2081 : x \u2264 y) (h\u2082 : 0 \u2264 z) : x ^ z \u2264 y ^ z", "theorem rpow_le_rpow_of_exponent_ge (hx0 : 0 < x) (hx1 : x \u2264 1) (hyz : z \u2264 y) : x ^ y \u2264 x ^ z", "@[simp]\ntheorem rpow_lt_rpow_left_iff (hx : 1 < x) : x ^ y < x ^ z \u2194 y < z", "theorem rpow_neg_one (x : \u211d) : x ^ (-1 : \u211d) = x\u207b\u00b9", "theorem rpow_sub_one {x : \u211d} (hx : x \u2260 0) (y : \u211d) : x ^ (y - 1) = x ^ y / x", "theorem rpow_sum_of_pos {\u03b9 : Type*} {a : \u211d} (ha : 0 < a) (f : \u03b9 \u2192 \u211d) (s : Finset \u03b9) :\n    (a ^ \u2211 x \u2208 s, f x) = \u220f x \u2208 s, a ^ f x", "theorem Real.sSup_smul_of_nonpos (ha : a \u2264 0) (s : Set \u211d) : sSup (a \u2022 s) = a \u2022 sInf s", "@[simp]\ntheorem self_le_sinh_iff : x \u2264 sinh x \u2194 0 \u2264 x", "theorem sign_apply_eq (r : \u211d) : sign r = -1 \u2228 sign r = 0 \u2228 sign r = 1", "lemma sin_half_eq_sqrt {x : \u211d} (hl : 0 \u2264 x) (hr : x \u2264 2 * \u03c0) :\n    sin (x / 2) = sqrt ((1 - cos x) / 2)", "theorem sin_le_one : sin x \u2264 1", "theorem sin_lt_sin_of_lt_of_le_pi_div_two {x y : \u211d} (hx\u2081 : -(\u03c0 / 2) \u2264 x) (hy\u2082 : y \u2264 \u03c0 / 2)\n    (hxy : x < y) : sin x < sin y", "@[simp]\ntheorem sin_pi_div_eight : sin (\u03c0 / 8) = \u221a(2 - \u221a2) / 2", "@[simp]\ntheorem sin_pi_div_two : sin (\u03c0 / 2) = 1", "theorem sin_sub : sin (x - y) = sin x * cos y - cos x * sin y", "nonrec theorem sin_sub_sin : sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2)", "theorem sinh_sub : sinh (x - y) = sinh x * cosh y - cosh x * sinh y", "theorem sqrtTwoAddSeries_monotone_left {x y : \u211d} (h : x \u2264 y) :\n    \u2200 n : \u2115, sqrtTwoAddSeries x n \u2264 sqrtTwoAddSeries y n", "@[simp]\ntheorem sqrt_div {x : \u211d} (hx : 0 \u2264 x) (y : \u211d) : \u221a(x / y) = \u221ax / \u221ay", "theorem sqrt_eq_cases : \u221ax = y \u2194 y * y = x \u2227 0 \u2264 y \u2228 x < 0 \u2227 y = 0", "@[simp]\ntheorem sqrt_mul' (x) {y : \u211d} (hy : 0 \u2264 y) : \u221a(x * y) = \u221ax * \u221ay", "@[simp]\ntheorem sqrt_mul_self (h : 0 \u2264 x) : \u221a(x * x) = x", "@[simp]\ntheorem sqrt_one : \u221a1 = 1", "@[simp]\ntheorem sqrt_pos : 0 < \u221ax \u2194 0 < x", "theorem strictMonoOn_sin : StrictMonoOn sin (Icc (-(\u03c0 / 2)) (\u03c0 / 2))", "theorem sum_le_exp_of_nonneg {x : \u211d} (hx : 0 \u2264 x) (n : \u2115) : \u2211 i \u2208 range n, x ^ i / i ! \u2264 exp x", "theorem tan_lt_tan_of_lt_of_lt_pi_div_two {x y : \u211d} (hx\u2081 : -(\u03c0 / 2) < x) (hy\u2082 : y < \u03c0 / 2)\n    (hxy : x < y) : tan x < tan y", "theorem tendsto_comp_exp_atTop {f : \u211d \u2192 \u03b1} :\n    Tendsto (fun x => f (exp x)) atTop l \u2194 Tendsto f atTop l", "lemma tendsto_integral_gaussian_smul (hf : Integrable f) (h'f : Integrable (\ud835\udcd5 f)) (v : V) :\n    Tendsto (fun (c : \u211d) \u21a6\n      \u222b w : V, ((\u03c0 * c) ^ (finrank \u211d V / 2 : \u2102) * cexp (-\u03c0 ^ 2 * c * \u2016v - w\u2016 ^ 2)) \u2022 f w)\n    atTop (\ud835\udcdd (\ud835\udcd5\u207b (\ud835\udcd5 f) v))", "theorem toNNReal_add_le {r p : \u211d} : Real.toNNReal (r + p) \u2264 Real.toNNReal r + Real.toNNReal p", "theorem _root_.Real.toNNReal_div' {x y : \u211d} (hy : 0 \u2264 y) :\n    Real.toNNReal (x / y) = Real.toNNReal x / Real.toNNReal y", "@[simp]\ntheorem _root_.Real.toNNReal_ofNat (n : \u2115) [n.AtLeastTwo] :\n    Real.toNNReal (no_index (OfNat.ofNat n)) = OfNat.ofNat n", "theorem two_mul_arctan_sub_pi {x : \u211d} (h : x < -1) :\n    2 * arctan x = arctan (2 * x / (1 - x ^ 2)) - \u03c0", "theorem volume_pi_Ioo {a b : \u03b9 \u2192 \u211d} :\n    volume (pi univ fun i => Ioo (a i) (b i)) = \u220f i, ENNReal.ofReal (b i - a i)", "@[simp]\ntheorem one_def : (epsilon : RegularExpression \u03b1) = 1", "theorem codom_inv : r.inv.codom = r.dom", "theorem core_union (s t : Set \u03b2) : r.core s \u222a r.core t \u2286 r.core (s \u222a t)", "theorem image_univ : r.image Set.univ = r.codom", "@[simp]\ntheorem preimage_bot (s : Set \u03b2) : (\u22a5 : Rel \u03b1 \u03b2).preimage s = \u2205", "@[simp]\ntheorem preimage_inv (s : Set \u03b1) : r.inv.preimage s = r.image s", "@[simp]\ntheorem preimage_top {s : Set \u03b2} (h : Set.Nonempty s) :\n    (\u22a4 : Rel \u03b1 \u03b2).preimage s = Set.univ", "@[simp]\ntheorem coe_toRelHom {f : r \u21aar s} : ((f : r \u21aar s).toRelHom : \u03b1 \u2192 \u03b2) = f", "theorem injective (f : r \u21aar s) : Injective f", "protected theorem isIrrefl (f : r \u21aar s) [IsIrrefl \u03b2 s] : IsIrrefl \u03b1 r", "theorem ext_iff {f g : r \u2243r s} : f = g \u2194 \u2200 x, f x = g x", "protected theorem injective (e : r \u2243r s) : Injective e", "lemma append_apply_left (p q : RelSeries r) (connect : r p.last q.head)\n    (i : Fin (p.length + 1)) :\n    p.append q connect ((i.castAdd (q.length + 1)).cast (by dsimp; abel)) = p i", "@[simp] lemma last_snoc' (p : RelSeries r) (newLast : \u03b1) (rel : r p.last newLast) :\n    p.snoc newLast rel (Fin.last (p.length + 1)) = newLast := last_append _ _ _", "theorem head' (hab : r a b) (hbc : ReflTransGen r b c) : TransGen r a c", "theorem head'_iff : TransGen r a c \u2194 \u2203 b, r a b \u2227 ReflTransGen r b c", "theorem acc_of_singleton [IsIrrefl \u03b1 r] {s : Multiset \u03b1} (hs : \u2200 a \u2208 s, Acc (CutExpand r) {a}) :\n    Acc (CutExpand r) s", "theorem symmetric_join : Symmetric (Join r)", "theorem transGen_eq_self (trans : Transitive r) : TransGen r = r", "theorem transitive_transGen : Transitive (TransGen r)", "lemma left_unique_of_rel_eq {eq' : \u03b2 \u2192 \u03b2 \u2192 Prop} (he : (R \u21d2 (R \u21d2 Iff)) Eq eq') : LeftUnique R", "theorem trivial_def {V : Type u} [AddCommGroup V] [Module k V] (g : G) (v : V) :\n    (trivial k G V).\u03c1 g v = v", "@[simp]\ntheorem asModuleEquiv_symm_map_smul (r : k) (x : V) :\n    \u03c1.asModuleEquiv.symm (r \u2022 x) = algebraMap k (MonoidAlgebra k G) r \u2022 \u03c1.asModuleEquiv.symm x", "@[to_additive] lemma Right.min_le_max_of_mul_le_mul\n    [CovariantClass \u03b1 \u03b1 (\u00b7 * \u00b7) (\u00b7 \u2264 \u00b7)] [CovariantClass \u03b1 \u03b1 (swap (\u00b7 * \u00b7)) (\u00b7 < \u00b7)]\n    (h : a * b \u2264 c * d) : min a b \u2264 max c d", "@[to_additive \"Only assumes right strict covariance\"]\ntheorem Right.mul_lt_mul [CovariantClass \u03b1 \u03b1 (\u00b7 * \u00b7) (\u00b7 \u2264 \u00b7)]\n    [CovariantClass \u03b1 \u03b1 (swap (\u00b7 * \u00b7)) (\u00b7 < \u00b7)] {a b c d : \u03b1}\n    (h\u2081 : a < b) (h\u2082 : c < d) :\n    a * c < b * d", "@[to_additive \"Assumes right covariance.\nThe lemma assuming left covariance is `Left.add_neg'`.\"]\ntheorem Right.mul_lt_one' [CovariantClass \u03b1 \u03b1 (swap (\u00b7 * \u00b7)) (\u00b7 \u2264 \u00b7)] {a b : \u03b1} (ha : a < 1)\n    (hb : b < 1) :\n    a * b < 1", "theorem A_mem_nhdsWithin_Ioi {L : F} {r \u03b5 x : \u211d} (hx : x \u2208 A f L r \u03b5) : A f L r \u03b5 \u2208 \ud835\udcdd[>] x", "theorem B_mem_nhdsWithin_Ioi {K : Set F} {r s \u03b5 x : \u211d} (hx : x \u2208 B f K r s \u03b5) :\n    B f K r s \u03b5 \u2208 \ud835\udcdd[>] x", "theorem le_iff (hf : RightOrdContinuous f) (h : Injective f) {x y} : f x \u2264 f y \u2194 x \u2264 y", "theorem lift_unique [IsDirected \u03b9 (\u00b7 \u2264 \u00b7)] (F : DirectLimit G f \u2192+* P) (x) :\n    F x = lift G f P (fun i => F.comp <| of G f i) (fun i j hij x => by simp [of_f]) x", "theorem choose_natCast [NatPowAssoc R] (n k : \u2115) : choose (n : R) k = Nat.choose n k", "theorem closure_subset_iff {s t : Set R} (ht : IsSubring t) : closure s \u2286 t \u2194 s \u2286 t", "theorem factorial_nsmul_multichoose_eq_ascPochhammer (r : R) (n : \u2115) :\n    n.factorial \u2022 multichoose r n = (ascPochhammer \u2115 n).smeval r", "theorem Ring.inverse_add_inverse [Semiring \u03b1] {a b : \u03b1} (h : IsUnit a \u2194 IsUnit b) :\n    Ring.inverse a + Ring.inverse b = Ring.inverse a * (a + b) * Ring.inverse b", "@[simp]\ntheorem inverse_non_unit (x : M\u2080) (h : \u00acIsUnit x) : inverse x = 0", "theorem Ring.isArtinian_of_zero_eq_one {R} [Ring R] (h01 : (0 : R) = 1) : IsArtinianRing R", "theorem mul_inverse_cancel (x : M\u2080) (h : IsUnit x) : x * inverse x = 1", "lemma ext {X Y : RingCat} {f g : X \u27f6 Y} (w : \u2200 x : X, f x = g x) : f = g", "@[simp, norm_cast]\ntheorem coe_bot : \u21d1(\u22a5 : RingCon R) = Eq", "protected lemma multisetSum {\u03b9 S : Type*} [AddCommMonoid S] [Mul S] (t : RingCon S)\n    (s : Multiset \u03b9) {f g : \u03b9 \u2192 S} (h : \u2200 i \u2208 s, t (f i) (g i)) :\n    t (s.map f).sum (s.map g).sum", "lemma op_iff {c : RingCon R} {x y : R\u1d50\u1d52\u1d56} : c.op x y \u2194 c y.unop x.unop := Iff.rfl", "protected theorem refl (x) : c x x", "theorem ringConGen_mono {r s : R \u2192 R \u2192 Prop} (h : \u2200 x y, r x y \u2192 s x y) :\n    ringConGen r \u2264 ringConGen s", "@[simp]\ntheorem coe_prodComm : \u21d1(prodComm : R \u00d7 S \u2243+* S \u00d7 R) = Prod.swap", "theorem toRingHom_injective : Function.Injective (toRingHom : R \u2243+* S \u2192 R \u2192+* S)", "theorem of_comp_finiteType (f : A \u2192+* B) {g : B \u2192+* C} (hg : (g.comp f).FinitePresentation)\n    (hf : f.FiniteType) : g.FinitePresentation", "nonrec theorem RingHom.IsIntegral.tower_bot (hg : Function.Injective g)\n    (hfg : (g.comp f).IsIntegral) : f.IsIntegral", "theorem affineLocally_of_isOpenImmersion (hP : RingHom.PropertyIsLocal @P) {X Y : Scheme.{u}}\n    (f : X \u27f6 Y) [hf : IsOpenImmersion f] : affineLocally (@P) f", "theorem RingHom.PropertyIsLocal.ofLocalizationSpan (hP : RingHom.PropertyIsLocal @P) :\n    RingHom.OfLocalizationSpan @P", "theorem source_affine_openCover_iff {X Y : Scheme.{u}} (f : X \u27f6 Y) [IsAffine Y]\n    (\ud835\udcb0 : Scheme.OpenCover.{u} X) [\u2200 i, IsAffine (\ud835\udcb0.obj i)] :\n    sourceAffineLocally (@P) f \u2194 \u2200 i, P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)", "theorem RespectsIso.is_localization_away_iff (hP : RingHom.RespectsIso @P) {R S : Type u}\n    (R' S' : Type u) [CommRing R] [CommRing S] [CommRing R'] [CommRing S'] [Algebra R R']\n    [Algebra S S'] (f : R \u2192+* S) (r : R) [IsLocalization.Away r R'] [IsLocalization.Away (f r) S'] :\n    P (Localization.awayMap f r) \u2194 P (IsLocalization.Away.map R' S' f r)", "theorem RingHom.charZero_iff {\u03d5 : R \u2192+* S} (h\u03d5 : Function.Injective \u03d5) : CharZero R \u2194 CharZero S", "theorem eq_of_eqOn_set_top {f g : R \u2192+* S} (h : Set.EqOn f g (\u22a4 : Subring R)) : f = g", "theorem fieldRange_eq_map : f.fieldRange = Subfield.map f \u22a4", "@[simp]\ntheorem id_apply (x : \u03b1) : RingHom.id \u03b1 x = x", "@[simp]\ntheorem id_comp (f : \u03b1 \u2192+* \u03b2) : (id \u03b2).comp f = f", "@[simp]\ntheorem kerLift_mk (r : R) : kerLift f (Ideal.Quotient.mk (ker f) r) = f r", "@[simp]\ntheorem ker_coe_equiv (f : R \u2243+* S) : ker (f : R \u2192+* S) = \u22a5", "theorem RingHom.ker_isRadical_iff_reduced_of_surjective {S F} [CommSemiring R] [CommRing S]\n    [FunLike F R S] [RingHomClass F R S] {f : F} (hf : Function.Surjective f) :\n    (RingHom.ker f).IsRadical \u2194 IsReduced S", "theorem ker_ne_top [Nontrivial S] (f : F) : ker f \u2260 \u22a4", "theorem map_dotProduct [NonAssocSemiring R] [NonAssocSemiring S] (f : R \u2192+* S) (v w : n \u2192 R) :\n    f (v \u2b1d\u1d65 w) = f \u2218 v \u2b1d\u1d65 f \u2218 w", "theorem RingHom.map_frobenius : g (frobenius R p x) = frobenius S p (g x)", "protected theorem map_one (f : \u03b1 \u2192+* \u03b2) : f 1 = 1", "theorem map_one_ne_zero [Nontrivial \u03b2] : f 1 \u2260 0", "theorem prod_comp_prodMap (f : T \u2192+* R) (g : T \u2192+* S) (f' : R \u2192+* R') (g' : S \u2192+* S') :\n    (f'.prodMap g').comp (f.prod g) = (f'.comp f).prod (g'.comp g)", "@[simp]\ntheorem restrict_apply (f : R \u2192+* S) {s : \u03c3R} (x : s) : f.domRestrict s x = f x", "theorem sub_mem_ker_iff {x y} : x - y \u2208 ker f \u2194 f x = f y", "@[simp]\ntheorem comp_apply [RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083] {x : R\u2081} : \u03c3\u2082\u2083 (\u03c3\u2081\u2082 x) = \u03c3\u2081\u2083 x", "theorem symm (\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082) (\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081) [RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081] :\n    RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082", "lemma cons {r : R} {rs : List R} (h1 : IsSMulRegular M r)\n    (h2 : IsWeaklyRegular (QuotSMulTop r M) rs) : IsWeaklyRegular M (r :: rs)", "lemma injOn_dualMap_subtype_span_root_coroot [NoZeroSMulDivisors \u2124 M] :\n    InjOn ((span R (range P.root)).subtype.dualMap \u2218\u2097 P.toLin.flip) (range P.coroot)", "lemma ne_zero [CharZero R] : (P.root i : M) \u2260 0", "theorem comap {f : M \u2192\u2097[R] N} (hxy : f x \u2261 f y [SMOD V]) : x \u2261 y [SMOD V.comap f]", "@[simp]\ntheorem top : x \u2261 y [SMOD (\u22a4 : Submodule R M)]", "lemma SMulPosStrictMono.lift [SMulPosStrictMono \u03b1 \u03b3] : SMulPosStrictMono \u03b1 \u03b2 where\n  elim b hb a\u2081 a\u2082 ha", "lemma Quasicategory.hornFilling {S : SSet} [Quasicategory S] \u2983n : \u2115\u2984 \u2983i : Fin (n+1)\u2984\n    (h0 : 0 < i) (hn : i < Fin.last n)\n    (\u03c3\u2080 : \u039b[n, i] \u27f6 S) : \u2203 \u03c3 : \u0394[n] \u27f6 S, \u03c3\u2080 = hornInclusion n i \u226b \u03c3", "theorem exists_eq_smul_add (h : SameRay R v\u2081 v\u2082) :\n    \u2203 a b : R, 0 \u2264 a \u2227 0 \u2264 b \u2227 a + b = 1 \u2227 v\u2081 = a \u2022 (v\u2081 + v\u2082) \u2227 v\u2082 = b \u2022 (v\u2081 + v\u2082)", "theorem exists_nonneg_right (h : SameRay R x y) (hy : y \u2260 0) : \u2203 r : R, 0 \u2264 r \u2227 x = r \u2022 y", "theorem norm_add (h : SameRay \u211d x y) : \u2016x + y\u2016 = \u2016x\u2016 + \u2016y\u2016", "theorem norm_sub (h : SameRay \u211d x y) : \u2016x - y\u2016 = |\u2016x\u2016 - \u2016y\u2016|", "theorem sameRay_comm : SameRay R x y \u2194 SameRay R y x", "@[simp]\ntheorem sameRay_map_iff (e : M \u2243\u2097[R] N) : SameRay R (e x) (e y) \u2194 SameRay R x y", "lemma sameRay_nonneg_smul_right (v : M) (h : 0 \u2264 a) : SameRay R v (a \u2022 v)", "protected theorem map [Functor m] [LawfulFunctor m] {x : m \u03b1}\n    (hx : SatisfiesM p x) (hf : \u2200 {a}, p a \u2192 q (f a)) : SatisfiesM q (f <$> x)", "@[simp] theorem SatisfiesM_ExceptT_eq [Monad m] [LawfulMonad m] :\n    SatisfiesM (m := ExceptT \u03c1 m) (\u03b1 := \u03b1) p x \u2194 SatisfiesM (m := m) (\u2200 a, \u00b7 = .ok a \u2192 p a) x", "@[simp]\nlemma integralCLM_apply (f : \ud835\udce2(D, V)) : integralCLM \ud835\udd5c \u03bc f = \u222b x, f x \u2202\u03bc", "theorem norm_iteratedFDeriv_le_seminorm (f : \ud835\udce2(E, F)) (n : \u2115) (x\u2080 : E) :\n    \u2016iteratedFDeriv \u211d n f x\u2080\u2016 \u2264 (SchwartzMap.seminorm \ud835\udd5c 0 n) f", "theorem iso_isometry_of_normNoninc {V W : SemiNormedGrp} (i : V \u2245 W) (h1 : i.hom.NormNoninc)\n    (h2 : i.inv.NormNoninc) : Isometry i.hom", "@[simp (high)]\ntheorem coe_comp {M N K : SemiNormedGrp\u2081} (f : M \u27f6 N) (g : N \u27f6 K) :\n    (f \u226b g : M \u2192 K) = g \u2218 f", "@[simp]\ntheorem coe_of (V : Type u) [SeminormedAddCommGroup V] : (SemiNormedGrp\u2081.of V : Type u) = V", "@[to_additive (attr := simp)] lemma inv_right_iff : SemiconjBy a x\u207b\u00b9 y\u207b\u00b9 \u2194 SemiconjBy a x y", "@[to_additive (attr := simp)] lemma inv_symm_left_iff : SemiconjBy a\u207b\u00b9 y x \u2194 SemiconjBy a x y", "@[simp]\nlemma natCast_mul_left (h : SemiconjBy a x y) (n : \u2115) : SemiconjBy (n * a) x y", "theorem neg_one_left (x : R) : SemiconjBy (-1) x x", "theorem neg_one_right (a : R) : SemiconjBy a (-1) (-1)", "@[to_additive AddSemiconjBy.prod]\ntheorem SemiconjBy.prod {x y z : M \u00d7 N}\n    (hm : SemiconjBy x.1 y.1 z.1) (hn : SemiconjBy x.2 y.2 z.2) : SemiconjBy x y z", "@[to_additive (attr := simp)]\ntheorem units_inv_right_iff {a : M} {x y : M\u02e3} : SemiconjBy a \u2191x\u207b\u00b9 \u2191y\u207b\u00b9 \u2194 SemiconjBy a x y", "@[simp]\ntheorem map_right (g : N \u22ca[\u03c6] G) : (map f\u2081 f\u2082 h g).right = f\u2082 g.right", "@[simp]\ntheorem mk_eq_inl_mul_inr (g : G) (n : N) : (\u27e8n, g\u27e9 : N \u22ca[\u03c6] G) = inl n * inr g", "@[to_additive]\ntheorem _root_.Semigroup.mem_center_iff {z : M} :\n    z \u2208 Set.center M \u2194 \u2200 g, g * z = z * g", "@[simp]\ntheorem coe_forget_to_partOrd (X : SemilatInfCat) :\n    \u21a5((forget\u2082 SemilatInfCat PartOrd).obj X) = \u21a5X", "theorem SemilatticeInf.ext {\u03b1} {A B : SemilatticeInf \u03b1}\n    (H : \u2200 x y : \u03b1, (haveI := A; x \u2264 y) \u2194 x \u2264 y) :\n    A = B", "theorem nnnorm_map [SemilinearIsometryClass \ud835\udcd5 \u03c3\u2081\u2082 E E\u2082] (f : \ud835\udcd5) (x : E) : \u2016f x\u2016\u208a = \u2016x\u2016\u208a", "theorem ball_sup (p : Seminorm \ud835\udd5c E) (q : Seminorm \ud835\udd5c E) (e : E) (r : \u211d) :\n    ball (p \u2294 q) e r = ball p e r \u2229 ball q e r", "@[simp]\ntheorem ball_zero' (x : E) (hr : 0 < r) : ball (0 : Seminorm \ud835\udd5c E) x r = Set.univ", "theorem closedBall_smul_ball (p : Seminorm \ud835\udd5c E) {r\u2081 : \u211d} (hr\u2081 : r\u2081 \u2260 0) (r\u2082 : \u211d) :\n    Metric.closedBall (0 : \ud835\udd5c) r\u2081 \u2022 p.ball 0 r\u2082 \u2286 p.ball 0 (r\u2081 * r\u2082)", "theorem cont_normedSpace_to_withSeminorms (E) [SeminormedAddCommGroup E] [NormedSpace \ud835\udd5d E]\n    [TopologicalSpace F] {q : \u03b9 \u2192 Seminorm \ud835\udd5d\u2082 F} (hq : WithSeminorms q)\n    (f : E \u2192\u209b\u2097[\u03c4\u2081\u2082] F) (hf : \u2200 i : \u03b9, \u2203 C : \u211d\u22650, (q i).comp f \u2264 C \u2022 normSeminorm \ud835\udd5d E) :\n    Continuous f", "theorem continuous_of_continuous_comp {q : SeminormFamily \ud835\udd5d\u2082 F \u03b9'} [TopologicalSpace E]\n    [TopologicalAddGroup E] [TopologicalSpace F] (hq : WithSeminorms q)\n    (f : E \u2192\u209b\u2097[\u03c4\u2081\u2082] F) (hf : \u2200 i, Continuous ((q i).comp f)) : Continuous f", "theorem finset_sup_apply_le {p : \u03b9 \u2192 Seminorm \ud835\udd5c E} {s : Finset \u03b9} {x : E} {a : \u211d} (ha : 0 \u2264 a)\n    (h : \u2200 i, i \u2208 s \u2192 p i x \u2264 a) : s.sup p x \u2264 a", "theorem isBounded_const (\u03b9' : Type*) [Nonempty \u03b9'] {p : \u03b9 \u2192 Seminorm \ud835\udd5c E} {q : Seminorm \ud835\udd5c\u2082 F}\n    (f : E \u2192\u209b\u2097[\u03c3\u2081\u2082] F) :\n    IsBounded p (fun _ : \u03b9' => q) f \u2194 \u2203 (s : Finset \u03b9) (C : \u211d\u22650), q.comp f \u2264 C \u2022 s.sup p", "theorem isBounded_sup {p : \u03b9 \u2192 Seminorm \ud835\udd5c E} {q : \u03b9' \u2192 Seminorm \ud835\udd5c\u2082 F} {f : E \u2192\u209b\u2097[\u03c3\u2081\u2082] F}\n    (hf : IsBounded p q f) (s' : Finset \u03b9') :\n    \u2203 (C : \u211d\u22650) (s : Finset \u03b9), (s'.sup q).comp f \u2264 C \u2022 s.sup p", "theorem norm_sub_map_le_sub (p : Seminorm \ud835\udd5c E) (x y : E) : \u2016p x - p y\u2016 \u2264 p (x - y)", "theorem smul_le_smul {p q : Seminorm \ud835\udd5c E} {a b : \u211d\u22650} (hpq : p \u2264 q) (hab : a \u2264 b) :\n    a \u2022 p \u2264 b \u2022 q", "protected theorem uniformContinuous' [UniformSpace E] [UniformAddGroup E] [ContinuousConstSMul \ud835\udd5c E]\n    {p : Seminorm \ud835\udd5c E} {r : \u211d} (hp : p.closedBall 0 r \u2208 (\ud835\udcdd 0 : Filter E)) :\n    UniformContinuous p", "theorem basisSets_add (U) (hU : U \u2208 p.basisSets) :\n    \u2203 V \u2208 p.basisSets, V + V \u2286 U", "theorem basisSets_iff {U : Set E} :\n    U \u2208 p.basisSets \u2194 \u2203 (i : Finset \u03b9) (r : \u211d), 0 < r \u2227 U = ball (i.sup p) 0 r", "theorem basisSets_nonempty [Nonempty \u03b9] : p.basisSets.Nonempty", "@[simp]\ntheorem pure_le {a : \u03b1} {s : Semiquot \u03b1} : pure a \u2264 s \u2194 a \u2208 s", "@[ext]\ntheorem ext {\u03bc : YoungDiagram} {T T' : SemistandardYoungTableau \u03bc} (h : \u2200 i j, T i j = T' i j) :\n    T = T'", "@[symm]\ntheorem symm : SeparatedNhds s t \u2192 SeparatedNhds t s", "lemma exists_ne_zero {x : V} (hx : x \u2260 0) :\n    \u2203 f : V \u2192L[R] R, f x \u2260 0", "theorem comap_mk_nhdsSet : comap mk (\ud835\udcdd\u02e2 t) = \ud835\udcdd\u02e2 (mk \u207b\u00b9' t)", "theorem SeparationQuotient.completeSpace_iff :\n    CompleteSpace (SeparationQuotient \u03b1) \u2194 CompleteSpace \u03b1", "theorem continuous_mk : Continuous (mk : X \u2192 SeparationQuotient X)", "theorem map_mk {f : \u03b1 \u2192 \u03b2} (h : UniformContinuous f) (a : \u03b1) : map f (mk a) = mk (f a)", "theorem BijOn.exists_extend {t' : Set \u03b2} (h : BijOn f s t) (htt' : t \u2286 t') (ht' : t' \u2286 range f) :\n    \u2203 s', s \u2286 s' \u2227 BijOn f s' t'", "lemma BijOn.forall {p : \u03b2 \u2192 Prop} (hf : BijOn f s t) : (\u2200 b \u2208 t, p b) \u2194 \u2200 a \u2208 s, p (f a) where\n  mp h a ha := h _ $ hf.mapsTo ha\n  mpr h b hb", "theorem Countable.exists_eq_range {s : Set \u03b1} (hc : s.Countable) (hs : s.Nonempty) :\n    \u2203 f : \u2115 \u2192 \u03b1, s = range f", "theorem Set.Countable.isConnected_compl_of_one_lt_rank (h : 1 < Module.rank \u211d E) {s : Set E}\n    (hs : s.Countable) : IsConnected s\u1d9c", "theorem Set.Countable.isG\u03b4_compl {s : Set X} [T1Space X] (hs : s.Countable) : IsG\u03b4 s\u1d9c", "theorem Set.Countable.isLindelof_sUnion {S : Set (Set X)} (hf : S.Countable)\n    (hc : \u2200 s \u2208 S, IsLindelof s) : IsLindelof (\u22c3\u2080 S)", "theorem EqOn.piecewise_ite' {f f' g : \u03b1 \u2192 \u03b2} {t t'} (h : EqOn f g (t \u2229 s))\n    (h' : EqOn f' g (t' \u2229 s\u1d9c)) : EqOn (s.piecewise f f') g (s.ite t t')", "theorem coeSort_toFinset : \u21a5hs.toFinset = \u21a5s", "theorem Finite.diff {s : Set \u03b1} (hs : s.Finite) (t : Set \u03b1) : (s \\ t).Finite", "@[elab_as_elim]\ntheorem Finite.dinduction_on {C : \u2200 s : Set \u03b1, s.Finite \u2192 Prop} (s : Set \u03b1) (h : s.Finite)\n    (H0 : C \u2205 finite_empty)\n    (H1 : \u2200 {a s}, a \u2209 s \u2192 \u2200 h : Set.Finite s, C s h \u2192 C (insert a s) (h.insert a)) : C s h", "theorem Set.Finite.exists_ge [IsDirected \u03b1 (\u00b7 \u2265 \u00b7)] {s : Set \u03b1} (hs : s.Finite) :\n    \u2203 M, \u2200 i \u2208 s, M \u2264 i", "theorem Finite.exists_lt_map_eq_of_forall_mem [LinearOrder \u03b1] [Infinite \u03b1] {t : Set \u03b2} {f : \u03b1 \u2192 \u03b2}\n    (hf : \u2200 a, f a \u2208 t) (ht : t.Finite) : \u2203 a b, a < b \u2227 f a = f b", "theorem Finite.finite_subsets {\u03b1 : Type u} {a : Set \u03b1} (h : a.Finite) : { b | b \u2286 a }.Finite", "theorem Finite.iUnion {\u03b9 : Type*} {s : \u03b9 \u2192 Set \u03b1} {t : Set \u03b9} (ht : t.Finite)\n    (hs : \u2200 i \u2208 t, (s i).Finite) (he : \u2200 i, i \u2209 t \u2192 s i = \u2205) : (\u22c3 i, s i).Finite", "protected lemma Set.Finite.infClosure (hs : s.Finite) : (infClosure s).Finite", "theorem Set.Finite.interior_sInter {S : Set (Set X)} (hS : S.Finite) :\n    interior (\u22c2\u2080 S) = \u22c2 s \u2208 S, interior s", "lemma Set.Finite.latticeClosure (hs : s.Finite) : (latticeClosure s).Finite", "@[to_additive]\ntheorem Finite.mul : s.Finite \u2192 t.Finite \u2192 (s * t).Finite", "protected theorem Finite.ofFinset {p : Set \u03b1} (s : Finset \u03b1) (H : \u2200 x, x \u2208 s \u2194 x \u2208 p) : p.Finite", "lemma Finite.of_surjOn {s : Set \u03b1} {t : Set \u03b2} (f : \u03b1 \u2192 \u03b2) (hf : SurjOn f s t) (hs : s.Finite) :\n    t.Finite := (hs.image _).subset hf", "protected theorem Finite.prod (hs : s.Finite) (ht : t.Finite) : (s \u00d7\u02e2 t : Set (\u03b1 \u00d7 \u03b2)).Finite", "theorem Finite.sInter {\u03b1 : Type*} {s : Set (Set \u03b1)} {t : Set \u03b1} (ht : t \u2208 s) (hf : t.Finite) :\n    (\u22c2\u2080 s).Finite", "@[simp]\nprotected theorem toFinset_eq_univ [Fintype \u03b1] {h : s.Finite} :\n    h.toFinset = Finset.univ \u2194 s = univ", "@[simp high]\nprotected theorem toFinset_setOf [Fintype \u03b1] (p : \u03b1 \u2192 Prop) [DecidablePred p]\n    (h : { x | p x }.Finite) : h.toFinset = Finset.univ.filter p", "protected theorem Finite.upperClosure [LocallyFiniteOrderTop \u03b1] (hs : s.Finite) :\n    (upperClosure s : Set \u03b1).Finite", "@[simp, norm_cast]\ntheorem coe_pow (x : Icc (0 : \u03b1) 1) (n : \u2115) : \u2191(x ^ n) = ((x : \u03b1) ^ n)", "@[simp, norm_cast]\ntheorem coe_zero : \u2191(0 : Icc (0 : \u03b1) 1) = (0 : \u03b1)", "@[simp]\ntheorem mk_zero (h : (0 : \u03b1) \u2208 Icc (0 : \u03b1) 1) : (\u27e80, h\u27e9 : Icc (0 : \u03b1) 1) = 0", "theorem Icc_prod_eq (a b : \u03b1 \u00d7 \u03b2) : Icc a b = Icc a.1 b.1 \u00d7\u02e2 Icc a.2 b.2", "lemma Icc_subset_uIcc : Icc a b \u2286 [[a, b]] := Icc_subset_Icc inf_le_left le_sup_right", "theorem Icc_union_Icc' (h\u2081 : c \u2264 b) (h\u2082 : a \u2264 d) : Icc a b \u222a Icc c d = Icc (min a c) (max b d)", "@[simp]\ntheorem Icc_union_Ioo_eq_Ico (h\u2081 : a \u2264 b) (h\u2082 : b < c) : Icc a b \u222a Ioo b c = Ico a c", "theorem IciExtend_of_mem (f : Ici a \u2192 \u03b2) (hx : x \u2208 Ici a) : IciExtend f x = f \u27e8x, hx\u27e9", "@[simp]\ntheorem Ico_diff_Iio : Ico a b \\ Iio c = Ico (max a c) b", "theorem Ico_infinite : (Ico a b).Infinite", "theorem coe_biSup : (\u2191(\u2a06 i, \u2a06 (_ : p i), f i) : \u03b1) = \u2a06 i, \u2a06 (_ : p i), (f i : \u03b1)", "@[simp]\ntheorem coe_bot [Preorder \u03b1] [OrderBot \u03b1] : (\u22a5 : Iic a) = (\u22a5 : \u03b1)", "theorem Iic_iInf (f : \u03b9 \u2192 \u03b1) : Iic (\u2a05 i, f i) = \u22c2 i, Iic (f i)", "@[simp]\ntheorem Iic_prod_Iic (a : \u03b1) (b : \u03b2) : Iic a \u00d7\u02e2 Iic b = Iic (a, b)", "theorem Iic_union_Icc (h : min c d \u2264 b) : Iic b \u222a Icc c d = Iic (max b d)", "@[simp]\ntheorem Iic_union_Ioi : Iic a \u222a Ioi a = univ", "theorem Iio_inter_Ici : Iio a \u2229 Ici b = Ico b a", "theorem Iio_inter_Ioi : Iio a \u2229 Ioi b = Ioo b a", "@[simp]\ntheorem Iio_union_Icc_eq_Iic (h : a \u2264 b) : Iio a \u222a Icc a b = Iic b", "theorem Iio_union_Ico (h : min c d \u2264 b) : Iio b \u222a Ico c d = Iio (max b d)", "theorem Iio_union_Ioo (h : min c d < b) : Iio b \u222a Ioo c d = Iio (max b d)", "lemma _root_.Set.Infinite.card_eq_zero {s : Set \u03b1} (hs : s.Infinite) : Nat.card s = 0", "protected lemma Infinite.mono {s t : Set \u03b1} (h : s \u2286 t) : s.Infinite \u2192 t.Infinite", "theorem Infinite.nontrivial {s : Set \u03b1} (hs : s.Infinite) : s.Nontrivial", "theorem _root_.Set.Infinite.not_bddBelow {s : Set \u03b1} : s.Infinite \u2192 \u00acBddBelow s", "theorem InjOn.image_eq_image_iff (h : s.InjOn f) (h\u2081 : s\u2081 \u2286 s) (h\u2082 : s\u2082 \u2286 s) :\n    f '' s\u2081 = f '' s\u2082 \u2194 s\u2081 = s\u2082", "theorem InjOn.mono (h : s\u2081 \u2286 s\u2082) (ht : InjOn f s\u2082) : InjOn f s\u2081", "@[simp]\ntheorem mk_one [Nontrivial \u03b1] (h : (1 : \u03b1) \u2208 Ioc (0 : \u03b1) 1) : (\u27e81, h\u27e9 : Ioc (0 : \u03b1) 1) = 1", "@[simp]\ntheorem Ioc_diff_Ioo_same (h : a < b) : Ioc a b \\ Ioo a b = {b}", "@[simp]\ntheorem Ioc_union_Icc_eq_Ioc (h\u2081 : a < b) (h\u2082 : b \u2264 c) : Ioc a b \u222a Icc b c = Ioc a c", "@[simp]\ntheorem Ioc_union_Ioc_left : Ioc a c \u222a Ioc b c = Ioc (min a b) c", "@[simp]\ntheorem Ioc_union_Ioi_eq_Ioi (h : a \u2264 b) : Ioc a b \u222a Ioi b = Ioi a", "@[simp]\ntheorem Ioc_union_Ioo_eq_Ioo (h\u2081 : a \u2264 b) (h\u2082 : b < c) : Ioc a b \u222a Ioo b c = Ioo a c", "theorem lt_one (x : Ioo (0 : \u03b1) 1) : (x : \u03b1) < 1", "theorem one_minus_lt_one (x : Ioo (0 : \u03b2) 1) : 1 - (x : \u03b2) < 1", "theorem Ioo_def (a b : \u03b1) : { x | a < x \u2227 x < b } = Ioo a b", "theorem Ioo_inter_Iio : Ioo a b \u2229 Iio c = Ioo a (min b c)", "theorem Ioo_inter_Ioc_of_right_lt (h : b\u2082 < b\u2081) : Ioo a\u2081 b\u2081 \u2229 Ioc a\u2082 b\u2082 = Ioc (max a\u2081 a\u2082) b\u2082", "@[simp]\ntheorem Ioo_union_Ico_eq_Ioo (h\u2081 : a < b) (h\u2082 : b \u2264 c) : Ioo a b \u222a Ico b c = Ioo a c", "protected nonrec theorem IsPWO.union (hs : IsPWO s) (ht : IsPWO t) : IsPWO (s \u222a t)", "theorem IsWF.min_union (hs : s.IsWF) (hsn : s.Nonempty) (ht : t.IsWF) (htn : t.Nonempty) :\n    (hs.union ht).min (union_nonempty.2 (Or.intro_left _ hsn)) =\n      Min.min (hs.min hsn) (ht.min htn)", "theorem comp (hf' : LeftInvOn f' f s) (hg' : LeftInvOn g' g t) (hf : MapsTo f s t) :\n    LeftInvOn (f' \u2218 g') (g \u2218 f) s", "theorem eq (h : LeftInvOn f' f s) {x} (hx : x \u2208 s) : f' (f x) = x", "theorem mapsTo (h : LeftInvOn f' f s) (hf : SurjOn f s t) :\n    MapsTo f' t s", "theorem MapsTo.image_subset (h : MapsTo f s t) : f '' s \u2286 t", "theorem MapsTo.iterate_restrict {f : \u03b1 \u2192 \u03b1} {s : Set \u03b1} (h : MapsTo f s s) (n : \u2115) :\n    (h.restrict f s s)^[n] = (h.iterate n).restrict _ _ _", "theorem MapsTo.nonempty (h : MapsTo f s t) (hs : s.Nonempty) : t.Nonempty", "theorem MapsTo.piecewise_ite {s s\u2081 s\u2082 : Set \u03b1} {t t\u2081 t\u2082 : Set \u03b2} {f\u2081 f\u2082 : \u03b1 \u2192 \u03b2}\n    [\u2200 i, Decidable (i \u2208 s)] (h\u2081 : MapsTo f\u2081 (s\u2081 \u2229 s) (t\u2081 \u2229 t))\n    (h\u2082 : MapsTo f\u2082 (s\u2082 \u2229 s\u1d9c) (t\u2082 \u2229 t\u1d9c)) :\n    MapsTo (s.piecewise f\u2081 f\u2082) (s.ite s\u2081 s\u2082) (t.ite t\u2081 t\u2082)", "@[to_additive Set.AddAntidiagonal.eq_of_snd_eq_snd]\ntheorem eq_of_snd_eq_snd (h : (x : \u03b1 \u00d7 \u03b1).snd = (y : \u03b1 \u00d7 \u03b1).snd) : x = y", "theorem Nat.encard_range (k : \u2115) : {i | i < k}.encard = k", "theorem Nonempty.inr (ht : t.Nonempty) : (s \u222a t).Nonempty", "protected theorem Set.Nonempty.intrinsicInterior (hscv : Convex \u211d s) (hsne : s.Nonempty) :\n    (intrinsicInterior \u211d s).Nonempty", "theorem Nonempty.of_sUnion_eq_univ [Nonempty \u03b1] {s : Set (Set \u03b1)} (h : \u22c3\u2080s = univ) : s.Nonempty", "@[to_additive]\ntheorem Nonempty.subset_one_iff (h : s.Nonempty) : s \u2286 1 \u2194 s = 1", "theorem Nonempty.to_subtype : s.Nonempty \u2192 Nonempty (\u21a5s)", "theorem Nonempty.vsub : s.Nonempty \u2192 t.Nonempty \u2192 (s -\u1d65 t : Set \u03b1).Nonempty", "theorem Nontrivial.infsep_exists_of_finite [Finite s] (hs : s.Nontrivial) :\n    \u2203 x \u2208 s, \u2203 y \u2208 s, x \u2260 y \u2227 s.infsep = dist x y", "theorem Nontrivial.infsep_of_fintype [DecidableEq \u03b1] [Fintype s] (hs : s.Nontrivial) :\n    s.infsep = s.offDiag.toFinset.inf' (by simpa) (uncurry dist)", "protected theorem Nontrivial.nonempty (hs : s.Nontrivial) : s.Nonempty", "theorem Set.OrdConnected.measurableSet (h : OrdConnected s) : MeasurableSet s", "theorem Pairwise.mono' (H : r \u2264 p) (hr : s.Pairwise r) : s.Pairwise p", "theorem PairwiseDisjoint.biUnion {s : Set \u03b9'} {g : \u03b9' \u2192 Set \u03b9} {f : \u03b9 \u2192 \u03b1}\n    (hs : s.PairwiseDisjoint fun i' : \u03b9' => \u2a06 i \u2208 g i', f i)\n    (hg : \u2200 i \u2208 s, (g i).PairwiseDisjoint f) : (\u22c3 i \u2208 s, g i).PairwiseDisjoint f", "theorem PairwiseDisjoint.image_finset_of_le [DecidableEq \u03b9] {s : Finset \u03b9} {f : \u03b9 \u2192 \u03b1}\n    (hs : (s : Set \u03b9).PairwiseDisjoint f) {g : \u03b9 \u2192 \u03b9} (hf : \u2200 a, f (g a) \u2264 f a) :\n    (s.image g : Set \u03b9).PairwiseDisjoint f", "protected theorem PairwiseDisjoint.insert (hs : s.PairwiseDisjoint f) {i : \u03b9}\n    (h : \u2200 j \u2208 s, i \u2260 j \u2192 Disjoint (f i) (f j)) : (insert i s).PairwiseDisjoint f", "theorem PartiallyWellOrderedOn.image_of_monotone_on (hs : s.PartiallyWellOrderedOn r)\n    (hf : \u2200 a\u2081 \u2208 s, \u2200 a\u2082 \u2208 s, r a\u2081 a\u2082 \u2192 r' (f a\u2081) (f a\u2082)) : (f '' s).PartiallyWellOrderedOn r'", "protected theorem PartiallyWellOrderedOn.insert (h : PartiallyWellOrderedOn s r) (a : \u03b1) :\n    PartiallyWellOrderedOn (insert a s) r", "theorem Sized.empty_mem_iff (hA : A.Sized r) : \u2205 \u2208 A \u2194 A = {\u2205}", "theorem Subsingleton.equitableOn {s : Set \u03b1} (hs : s.Subsingleton) (f : \u03b1 \u2192 \u03b2) : s.EquitableOn f", "theorem Subsingleton.induction_on {p : Set \u03b1 \u2192 Prop} (hs : s.Subsingleton) (he : p \u2205)\n    (h\u2081 : \u2200 x, p {x}) : p s", "theorem SurjOn.rightInvOn_invFunOn [Nonempty \u03b1] (h : SurjOn f s t) :\n    RightInvOn (invFunOn f s) f t", "protected theorem Set.UniformEquicontinuousOn.closure {A : Set <| \u03b2 \u2192 \u03b1} {S : Set \u03b2}\n    (hA : A.UniformEquicontinuousOn S) : (closure A).UniformEquicontinuousOn S", "protected theorem Set.UniformEquicontinuousOn.uniformContinuousOn_of_mem {H : Set <| \u03b2 \u2192 \u03b1}\n    {S : Set \u03b2} (h : H.UniformEquicontinuousOn S) {f : \u03b2 \u2192 \u03b1} (hf : f \u2208 H) :\n    UniformContinuousOn f S", "theorem abs_sub_le_of_uIcc_subset_uIcc (h : [[c, d]] \u2286 [[a, b]]) : |d - c| \u2264 |b - a|", "@[simp]\ntheorem add_mem_centralizer [Distrib M] (ha : a \u2208 centralizer S) (hb : b \u2208 centralizer S) :\n    a + b \u2208 centralizer S", "theorem biInter_le_succ (u : \u2115 \u2192 Set \u03b1) (n : \u2115) : \u22c2 k \u2264 n + 1, u k = (\u22c2 k \u2264 n, u k) \u2229 u (n + 1)", "theorem biInter_le_succ' (u : \u2115 \u2192 Set \u03b1) (n : \u2115) : \u22c2 k \u2264 n + 1, u k = u 0 \u2229 \u22c2 k \u2264 n, u (k + 1)", "theorem biInter_union (s t : Set \u03b1) (u : \u03b1 \u2192 Set \u03b2) :\n    \u22c2 x \u2208 s \u222a t, u x = (\u22c2 x \u2208 s, u x) \u2229 \u22c2 x \u2208 t, u x", "theorem _root_.Set.biUnion_finsetSigma' (s : Finset \u03b9) (t : \u2200 i, Finset (\u03b1 i))\n    (f : \u2200 i, \u03b1 i \u2192 Set \u03b2) : \u22c3 i \u2208 s, \u22c3 j \u2208 t i, f i j = \u22c3 ij \u2208 s.sigma t, f ij.fst ij.snd", "theorem biUnion_sigma' (s : Set \u03b9) (t : \u2200 i, Set (\u03b1 i)) (f : \u2200 i, \u03b1 i \u2192 Set \u03b2) :\n    \u22c3 i \u2208 s, \u22c3 j \u2208 t i, f i j = \u22c3 ij \u2208 s.sigma t, f ij.fst ij.snd", "theorem biUnion_univ (s : \u03b1 \u2192 Set \u03b2) : \u22c3 x \u2208 @univ \u03b1, s x = \u22c3 x, s x", "theorem bijective_iff_bijOn_univ : Bijective f \u2194 BijOn f univ univ", "theorem bounded_le_inter_lt [LinearOrder \u03b1] (a : \u03b1) :\n    Bounded (\u00b7 \u2264 \u00b7) (s \u2229 { b | a < b }) \u2194 Bounded (\u00b7 \u2264 \u00b7) s", "theorem bounded_lt_Ioc [Preorder \u03b1] [NoMaxOrder \u03b1] (a b : \u03b1) : Bounded (\u00b7 < \u00b7) (Ioc a b)", "theorem bounded_lt_inter_not_lt [SemilatticeSup \u03b1] (a : \u03b1) :\n    Bounded (\u00b7 < \u00b7) (s \u2229 { b | \u00acb < a }) \u2194 Bounded (\u00b7 < \u00b7) s", "theorem card_le_card {s t : Set \u03b1} [Fintype s] [Fintype t] (hsub : s \u2286 t) :\n    Fintype.card s \u2264 Fintype.card t", "theorem card_range_of_injective [Fintype \u03b1] {f : \u03b1 \u2192 \u03b2} (hf : Injective f) [Fintype (range f)] :\n    Fintype.card (range f) = Fintype.card \u03b1", "theorem chainHeight_image (f : \u03b1 \u2192 \u03b2) (hf : \u2200 {x y}, x < y \u2194 f x < f y) (s : Set \u03b1) :\n    (f '' s).chainHeight = s.chainHeight", "@[simp]\ntheorem codRestrict_restrict (h : \u2200 x : s, f x \u2208 t) :\n    codRestrict (s.restrict f) t h = MapsTo.restrict f s t fun x hx => h \u27e8x, hx\u27e9", "@[simp]\ntheorem compl_Iic : (Iic a)\u1d9c = Ioi a", "@[simp]\ntheorem compl_inter_self (s : Set \u03b1) : s\u1d9c \u2229 s = \u2205", "theorem diagonal_eq_univ_iff : diagonal \u03b1 = univ \u2194 Subsingleton \u03b1", "theorem diff_subset_comm {s t u : Set \u03b1} : s \\ t \u2286 u \u2194 s \\ u \u2286 t", "theorem directedOn_sUnion {r} {S : Set (Set \u03b1)} (hd : DirectedOn (\u00b7 \u2286 \u00b7) S)\n    (h : \u2200 x \u2208 S, DirectedOn r x) : DirectedOn r (\u22c3\u2080 S)", "theorem disjoint_compl_right_iff_subset : Disjoint s t\u1d9c \u2194 s \u2286 t", "@[simp]\ntheorem disjoint_diagonal_offDiag : Disjoint (diagonal \u03b1) s.offDiag", "lemma disjoint_of_subset_right (h : t \u2286 u) (d : Disjoint s u) : Disjoint s t := d.mono_right h", "theorem disjoint_right : Disjoint s t \u2194 \u2200 \u2983a\u2984, a \u2208 t \u2192 a \u2209 s", "@[simp]\ntheorem disjoint_sUnion_right {s : Set \u03b1} {S : Set (Set \u03b1)} :\n    Disjoint s (\u22c3\u2080S) \u2194 \u2200 t \u2208 S, Disjoint s t", "@[to_additive (attr := simp)]\ntheorem div_empty : s / \u2205 = \u2205", "@[to_additive]\ntheorem div_iUnion (s : Set \u03b1) (t : \u03b9 \u2192 Set \u03b1) : (s / \u22c3 i, t i) = \u22c3 i, s / t i", "@[simp]\ntheorem dual_Ioc : Ioc (toDual a) (toDual b) = ofDual \u207b\u00b9' Ico b a", "theorem dual_ordConnectedSection (s : Set \u03b1) :\n    ordConnectedSection (ofDual \u207b\u00b9' s) = ofDual \u207b\u00b9' ordConnectedSection s", "@[simp]\ntheorem einfsep_empty : (\u2205 : Set \u03b1).einfsep = \u221e", "theorem einfsep_insert_le : (insert x s).einfsep \u2264 \u2a05 (y \u2208 s) (_ : x \u2260 y), edist x y", "theorem einfsep_pair_le_right (hxy : x \u2260 y) : ({x, y} : Set \u03b1).einfsep \u2264 edist y x", "theorem empty_card : Fintype.card (\u2205 : Set \u03b1) = 0", "theorem empty_definable_iff :\n    (\u2205 : Set M).Definable L s \u2194 \u2203 \u03c6 : L.Formula \u03b1, s = setOf \u03c6.Realize", "theorem encard_eq_coe_toFinset_card (s : Set \u03b1) [Fintype s] : encard s = s.toFinset.card", "theorem encard_le_one_iff : s.encard \u2264 1 \u2194 \u2200 a b, a \u2208 s \u2192 b \u2208 s \u2192 a = b", "theorem eqOn_refl (f : \u03b1 \u2192 \u03b2) (s : Set \u03b1) : EqOn f f s", "lemma eq_of_mem_uIoc_of_mem_uIoc : a \u2208 \u0399 b c \u2192 b \u2208 \u0399 a c \u2192 a = b", "theorem eq_of_nonempty_of_subsingleton {\u03b1} [Subsingleton \u03b1] (s t : Set \u03b1) [Nonempty s]\n    [Nonempty t] : s = t", "theorem eq_of_nonempty_of_subsingleton' {\u03b1} [Subsingleton \u03b1] {s : Set \u03b1} (t : Set \u03b1)\n    (hs : s.Nonempty) [Nonempty t] : s = t", "lemma eq_singleton_or_nontrivial (ha : a \u2208 s) : s = {a} \u2228 s.Nontrivial", "theorem equitableOn_iff_exists_eq_eq_add_one {s : Set \u03b1} {f : \u03b1 \u2192 \u2115} :\n    s.EquitableOn f \u2194 \u2203 b, \u2200 a \u2208 s, f a = b \u2228 f a = b + 1", "theorem equitableOn_iff_exists_le_le_add_one {s : Set \u03b1} {f : \u03b1 \u2192 \u2115} :\n    s.EquitableOn f \u2194 \u2203 b, \u2200 a \u2208 s, b \u2264 f a \u2227 f a \u2264 b + 1", "theorem exists_lower_bound_image [Nonempty \u03b1] [LinearOrder \u03b2] (s : Set \u03b1) (f : \u03b1 \u2192 \u03b2)\n    (h : s.Finite) : \u2203 a : \u03b1, \u2200 b \u2208 s, f a \u2264 f b", "lemma exists_ne_mem_inter_of_not_pairwiseDisjoint\n    {f : \u03b9 \u2192 Set \u03b1} (h : \u00ac s.PairwiseDisjoint f) :\n    \u2203 i \u2208 s, \u2203 j \u2208 s, i \u2260 j \u2227 \u2203 x : \u03b1, x \u2208 f i \u2229 f j", "theorem exists_prod_set {p : \u03b1 \u00d7 \u03b2 \u2192 Prop} : (\u2203 x \u2208 s \u00d7\u02e2 t, p x) \u2194 \u2203 x \u2208 s, \u2203 y \u2208 t, p (x, y)", "@[deprecated exists_subset_card_eq (since := \"2024-06-23\")]\ntheorem exists_smaller_set (s : Set \u03b1) (i : \u2115) (h\u2081 : i \u2264 s.ncard) :\n    \u2203 t : Set \u03b1, t \u2286 s \u2227 t.ncard = i", "theorem exists_subset_encard_eq {k : \u2115\u221e} (hk : k \u2264 s.encard) : \u2203 t, t \u2286 s \u2227 t.encard = k", "theorem finite_of_ncard_pos (hs : 0 < s.ncard) : s.Finite", "theorem forall_image2_iff {p : \u03b3 \u2192 Prop} :\n    (\u2200 z \u2208 image2 f s t, p z) \u2194 \u2200 x \u2208 s, \u2200 y \u2208 t, p (f x y)", "theorem iInter_and {p q : Prop} (s : p \u2227 q \u2192 Set \u03b1) : \u22c2 h, s h = \u22c2 (hp) (hq), s \u27e8hp, hq\u27e9", "theorem iInter_eq_iInter_finset (s : \u03b9 \u2192 Set \u03b1) : \u22c2 i, s i = \u22c2 t : Finset \u03b9, \u22c2 i \u2208 t, s i", "theorem iInter\u2082_mono {s t : \u2200 i, \u03ba i \u2192 Set \u03b1} (h : \u2200 i j, s i j \u2286 t i j) :\n    \u22c2 (i) (j), s i j \u2286 \u22c2 (i) (j), t i j", "theorem iInter\u2082_subset_of_subset {s : \u2200 i, \u03ba i \u2192 Set \u03b1} {t : Set \u03b1} (i : \u03b9) (j : \u03ba i)\n    (h : s i j \u2286 t) : \u22c2 (i) (j), s i j \u2286 t", "theorem iUnionLift_binary (dir : Directed (\u00b7 \u2264 \u00b7) S) (op : T \u2192 T \u2192 T) (opi : \u2200 i, S i \u2192 S i \u2192 S i)\n    (hopi :\n      \u2200 i x y,\n        Set.inclusion (show S i \u2286 T from hT'.symm \u25b8 Set.subset_iUnion S i) (opi i x y) =\n          op (Set.inclusion (show S i \u2286 T from hT'.symm \u25b8 Set.subset_iUnion S i) x)\n            (Set.inclusion (show S i \u2286 T from hT'.symm \u25b8 Set.subset_iUnion S i) y))\n    (op\u03b2 : \u03b2 \u2192 \u03b2 \u2192 \u03b2) (h : \u2200 (i) (x y : S i), f i (opi i x y) = op\u03b2 (f i x) (f i y)) (x y : T) :\n    iUnionLift S f hf T (le_of_eq hT') (op x y) =\n      op\u03b2 (iUnionLift S f hf T (le_of_eq hT') x) (iUnionLift S f hf T (le_of_eq hT') y)", "@[simp]\ntheorem iUnion_Icc_left (b : \u03b1) : \u22c3 a, Icc a b = Iic b", "@[simp]\ntheorem iUnion_Ico_left (b : \u03b1) : \u22c3 a, Ico a b = Iio b", "@[simp]\ntheorem iUnion_Ioc_left [NoMinOrder \u03b1] (b : \u03b1) : \u22c3 a, Ioc a b = Iic b", "@[simp]\ntheorem iUnion_Ioc_right (a : \u03b1) : \u22c3 b, Ioc a b = Ioi a", "theorem iUnion_diff (s : Set \u03b2) (t : \u03b9 \u2192 Set \u03b2) : (\u22c3 i, t i) \\ s = \u22c3 i, t i \\ s", "theorem iUnion_eq_compl_iInter_compl (s : \u03b9 \u2192 Set \u03b2) : \u22c3 i, s i = (\u22c2 i, (s i)\u1d9c)\u1d9c", "theorem iUnion_eq_iUnion_finset (s : \u03b9 \u2192 Set \u03b1) : \u22c3 i, s i = \u22c3 t : Finset \u03b9, \u22c3 i \u2208 t, s i", "theorem iUnion_eq_univ_iff {f : \u03b9 \u2192 Set \u03b1} : \u22c3 i, f i = univ \u2194 \u2200 x, \u2203 i, x \u2208 f i", "theorem iUnion_iInter_of_antitone {\u03b9 \u03b9' \u03b1 : Type*} [Finite \u03b9] [Preorder \u03b9']\n    [IsDirected \u03b9' (swap (\u00b7 \u2264 \u00b7))] [Nonempty \u03b9'] {s : \u03b9 \u2192 \u03b9' \u2192 Set \u03b1} (hs : \u2200 i, Antitone (s i)) :\n    \u22c3 j : \u03b9', \u22c2 i : \u03b9, s i j = \u22c2 i : \u03b9, \u22c3 j : \u03b9', s i j", "@[simp]\ntheorem iUnion_singleton_eq_range {\u03b1 \u03b2 : Type*} (f : \u03b1 \u2192 \u03b2) : \u22c3 x : \u03b1, {f x} = range f", "theorem iUnion_vsub (s : \u03b9 \u2192 Set \u03b2) (t : Set \u03b2) : (\u22c3 i, s i) -\u1d65 t = \u22c3 i, s i -\u1d65 t", "theorem iUnion\u2082_eq_univ_iff {s : \u2200 i, \u03ba i \u2192 Set \u03b1} :\n    \u22c3 (i) (j), s i j = univ \u2194 \u2200 a, \u2203 i j, a \u2208 s i j", "theorem iUnion\u2082_prod_const {s : \u2200 i, \u03ba i \u2192 Set \u03b1} {t : Set \u03b2} :\n    (\u22c3 (i) (j), s i j) \u00d7\u02e2 t = \u22c3 (i) (j), s i j \u00d7\u02e2 t", "theorem image2_iUnion_right (s : Set \u03b1) (t : \u03b9 \u2192 Set \u03b2) :\n    image2 f s (\u22c3 i, t i) = \u22c3 i, image2 f s (t i)", "theorem image2_iUnion\u2082_left (s : \u2200 i, \u03ba i \u2192 Set \u03b1) (t : Set \u03b2) :\n    image2 f (\u22c3 (i) (j), s i j) t = \u22c3 (i) (j), image2 f (s i j) t", "lemma image2_inter_left (hf : Injective2 f) :\n    image2 f (s \u2229 s') t = image2 f s t \u2229 image2 f s' t", "theorem image2_inter_subset_left : image2 f (s \u2229 s') t \u2286 image2 f s t \u2229 image2 f s' t", "theorem image2_inter_union_subset {f : \u03b1 \u2192 \u03b1 \u2192 \u03b2} {s t : Set \u03b1} (hf : \u2200 a b, f a b = f b a) :\n    image2 f (s \u2229 t) (s \u222a t) \u2286 image2 f s t", "@[simp]\ntheorem image2_singleton_right : image2 f s {b} = (fun a => f a b) '' s", "@[simp]\ntheorem image_add_const_Ioc : (fun x => x + a) '' Ioc b c = Ioc (b + a) (c + a)", "@[simp]\ntheorem image_const_add_Icc : (fun x => a + x) '' Icc b c = Icc (a + b) (a + c)", "@[simp]\ntheorem image_const_sub_Icc : (fun x => a - x) '' Icc b c = Icc (a - c) (a - b)", "@[simp]\ntheorem image_const_sub_Ico : (fun x => a - x) '' Ico b c = Ioc (a - c) (a - b)", "theorem image_eta (f : \u03b1 \u2192 \u03b2) : f '' s = (fun x => f x) '' s", "@[to_additive]\ntheorem image_inter_mulSupport_eq {s : Set \u03b2} {g : \u03b2 \u2192 \u03b1} :\n    g '' s \u2229 mulSupport f = g '' (s \u2229 mulSupport (f \u2218 g))", "@[to_additive]\ntheorem image_mulSingle_Ioc_right (i : \u03b9) (b : \u03b1 i) :\n    Pi.mulSingle i '' Ioc 1 b = Ioc 1 (Pi.mulSingle i b)", "theorem image_neg_Ico : Neg.neg '' Ico a b = Ioc (-b) (-a)", "theorem image_preimage_eq_range_inter {f : \u03b1 \u2192 \u03b2} {t : Set \u03b2} : f '' (f \u207b\u00b9' t) = range f \u2229 t", "@[simp]\ntheorem image_singleton {f : \u03b1 \u2192 \u03b2} {a : \u03b1} : f '' {a} = {f a}", "theorem image_subset_image_iff {f : \u03b1 \u2192 \u03b2} (hf : Injective f) : f '' s \u2286 f '' t \u2194 s \u2286 t", "@[simp]\nlemma image_subtype_val_Icc_Ici {a b : \u03b1} (c : Icc a b) : Subtype.val '' Ici c = Icc c.1 b", "@[simp]\nlemma image_subtype_val_Ico_Ioi {a b : \u03b1} (c : Ico a b) : Subtype.val '' Ioi c = Ioo c.1 b", "theorem image_subtype_val_Ico_subset (a b : {x // p x}) :\n    Subtype.val '' Ico a b \u2286 Ico a.val b.val", "@[simp]\nlemma image_subtype_val_Iio_Iio {a : \u03b1} (b : Iio a) : Subtype.val '' Iio b = Iio b.1", "@[simp]\nlemma image_subtype_val_Ioo {s : Set \u03b1} [OrdConnected s] (x y : s) :\n    Subtype.val '' Ioo x y = Ioo x.1 y", "theorem image_subtype_val_Ioo_subset (a b : {x // p x}) :\n    Subtype.val '' Ioo a b \u2286 Ioo a.val b.val", "theorem image_update_Ico_left (f : \u2200 i, \u03b1 i) (i : \u03b9) (a : \u03b1 i) :\n    update f i '' Ico a (f i) = Ico (update f i a) f", "theorem image_update_Ico_right (f : \u2200 i, \u03b1 i) (i : \u03b9) (b : \u03b1 i) :\n    update f i '' Ico (f i) b = Ico f (update f i b)", "theorem image_update_Ioc (f : \u2200 i, \u03b1 i) (i : \u03b9) (a b : \u03b1 i) :\n    update f i '' Ioc a b = Ioc (update f i a) (update f i b)", "lemma indicator_mul_right (s : Set \u03b9) (f g : \u03b9 \u2192 M\u2080) :\n    indicator s (fun j \u21a6 f j * g j) i = f i * indicator s g i", "lemma indicator_smul (s : Set \u03b1) (r : \u03b1 \u2192 R) (f : \u03b1 \u2192 M) :\n    indicator s (fun a \u21a6 r a \u2022 f a) = fun a \u21a6 r a \u2022 indicator s f a", "theorem infinite_coe_iff {s : Set \u03b1} : Infinite s \u2194 s.Infinite", "theorem infinite_image_iff {s : Set \u03b1} {f : \u03b1 \u2192 \u03b2} (hi : InjOn f s) :\n    (f '' s).Infinite \u2194 s.Infinite", "theorem infs_comm : s \u22bc t = t \u22bc s", "@[simp]\ntheorem infs_singleton : s \u22bc {b} = s.image fun a => a \u2293 b", "theorem infs_subset_right : s\u2081 \u2286 s\u2082 \u2192 s\u2081 \u22bc t \u2286 s\u2082 \u22bc t", "theorem infs_union_left : (s\u2081 \u222a s\u2082) \u22bc t = s\u2081 \u22bc t \u222a s\u2082 \u22bc t", "theorem infs_union_right : s \u22bc (t\u2081 \u222a t\u2082) = s \u22bc t\u2081 \u222a s \u22bc t\u2082", "theorem infsep_pos_iff_nontrivial_of_finite [Finite s] : 0 < s.infsep \u2194 s.Nontrivial", "@[simp]\ntheorem insert_nonempty (a : \u03b1) (s : Set \u03b1) : (insert a s).Nonempty", "@[simp] theorem insert_subset_insert_iff (ha : a \u2209 s) : insert a s \u2286 insert a t \u2194 s \u2286 t", "theorem inter_comm (a b : Set \u03b1) : a \u2229 b = b \u2229 a", "theorem inter_eq_iInter {s\u2081 s\u2082 : Set \u03b1} : s\u2081 \u2229 s\u2082 = \u22c2 b : Bool, cond b s\u2081 s\u2082", "theorem inter_insert_of_not_mem (h : a \u2209 s) : s \u2229 insert a t = s \u2229 t", "theorem inter_left_comm (s\u2081 s\u2082 s\u2083 : Set \u03b1) : s\u2081 \u2229 (s\u2082 \u2229 s\u2083) = s\u2082 \u2229 (s\u2081 \u2229 s\u2083)", "theorem inter_nonempty : (s \u2229 t).Nonempty \u2194 \u2203 x, x \u2208 s \u2227 x \u2208 t", "theorem inter_setOf_eq_sep (s : Set \u03b1) (p : \u03b1 \u2192 Prop) : s \u2229 {a | p a} = {a \u2208 s | p a}", "@[to_additive]\ntheorem inter_smul_union_subset_union : (s\u2081 \u2229 s\u2082) \u2022 (t\u2081 \u222a t\u2082) \u2286 s\u2081 \u2022 t\u2081 \u222a s\u2082 \u2022 t\u2082", "@[gcongr]\ntheorem inter_subset_inter_right {s t : Set \u03b1} (u : Set \u03b1) (H : s \u2286 t) : u \u2229 s \u2286 u \u2229 t", "@[simp, mfld_simps]\ntheorem inter_univ (a : Set \u03b1) : a \u2229 univ = a", "@[to_additive (attr := simp 500)]\ntheorem isUnit_iff : IsUnit s \u2194 \u2203 a, s = {a} \u2227 IsUnit a", "theorem isWF_iff_no_descending_seq :\n    IsWF s \u2194 \u2200 f : \u2115 \u2192 \u03b1, StrictAnti f \u2192 \u00ac\u2200 n, f (OrderDual.toDual n) \u2208 s", "@[simp]\ntheorem ite_same (t s : Set \u03b1) : t.ite s s = s", "lemma kernImage_compl {s : Set \u03b1} : kernImage f (s\u1d9c) = (f '' s)\u1d9c", "theorem le_einfsep_iff {d} :\n    d \u2264 s.einfsep \u2194 \u2200 x \u2208 s, \u2200 y \u2208 s, x \u2260 y \u2192 d \u2264 edist x y", "@[simp]\ntheorem mapsTo_union : MapsTo f (s\u2081 \u222a s\u2082) t \u2194 MapsTo f s\u2081 t \u2227 MapsTo f s\u2082 t", "theorem mapsTo_univ (f : \u03b1 \u2192 \u03b2) (s : Set \u03b1) : MapsTo f s univ", "@[deprecated mapsTo_range_iff (since := \"2023-12-25\")]\ntheorem maps_range_to (f : \u03b1 \u2192 \u03b2) (g : \u03b3 \u2192 \u03b1) (s : Set \u03b2) :\n    MapsTo f (range g) s \u2194 MapsTo (f \u2218 g) univ s", "theorem mem_Icc_of_Ioo (h : x \u2208 Ioo a b) : x \u2208 Icc a b", "theorem mem_diff_of_mem {s t : Set \u03b1} {x : \u03b1} (h1 : x \u2208 s) (h2 : x \u2209 t) : x \u2208 s \\ t", "theorem mem_dite_univ_left (p : Prop) [Decidable p] (t : \u00acp \u2192 Set \u03b1) (x : \u03b1) :\n    (x \u2208 if h : p then univ else t h) \u2194 \u2200 h : \u00acp, x \u2208 t h", "theorem mem_image2_of_mem (ha : a \u2208 s) (hb : b \u2208 t) : f a b \u2208 image2 f s t", "@[simp]\ntheorem mem_insert_iff {x a : \u03b1} {s : Set \u03b1} : x \u2208 insert a s \u2194 x = a \u2228 x \u2208 s", "theorem mem_of_mem_diff {s t : Set \u03b1} {x : \u03b1} (h : x \u2208 s \\ t) : x \u2208 s", "lemma mem_uIoc : a \u2208 \u0399 b c \u2194 b < a \u2227 a \u2264 c \u2228 c < a \u2227 a \u2264 b", "theorem mk_preimage_prod (f : \u03b3 \u2192 \u03b1) (g : \u03b3 \u2192 \u03b2) :\n    (fun x => (f x, g x)) \u207b\u00b9' s \u00d7\u02e2 t = f \u207b\u00b9' s \u2229 g \u207b\u00b9' t", "theorem mk_preimage_prod_left_eq_if [DecidablePred (\u00b7 \u2208 t)] :\n    (fun a => (a, b)) \u207b\u00b9' s \u00d7\u02e2 t = if b \u2208 t then s else \u2205", "theorem mk_preimage_prod_right_eq_if [DecidablePred (\u00b7 \u2208 s)] :\n    Prod.mk a \u207b\u00b9' s \u00d7\u02e2 t = if a \u2208 s then t else \u2205", "theorem monotone_powerset : Monotone (powerset : Set \u03b1 \u2192 Set (Set \u03b1))", "theorem monotone_projIic : Monotone (projIic a)", "@[to_additive (attr := simp)]\ntheorem mulIndicator_apply_eq_one : mulIndicator s f a = 1 \u2194 a \u2208 s \u2192 f a = 1", "@[to_additive]\ntheorem mulIndicator_const_preimage_eq_union (U : Set \u03b1) (s : Set M) (a : M) [Decidable (a \u2208 s)]\n    [Decidable ((1 : M) \u2208 s)] : (U.mulIndicator fun _ => a) \u207b\u00b9' s =\n      (if a \u2208 s then U else \u2205) \u222a if (1 : M) \u2208 s then U\u1d9c else \u2205", "@[to_additive]\nlemma mulIndicator_le_self (s : Set \u03b1) (f : \u03b1 \u2192 M) : mulIndicator s f \u2264 f", "@[to_additive]\ntheorem mulIndicator_mul' (s : Set \u03b1) (f g : \u03b1 \u2192 M) :\n    mulIndicator s (f * g) = mulIndicator s f * mulIndicator s g", "@[to_additive]\ntheorem mulIndicator_mul_eq_left {f g : \u03b1 \u2192 M} (h : Disjoint (mulSupport f) (mulSupport g)) :\n    (mulSupport f).mulIndicator (f * g) = f", "@[to_additive]\nprotected theorem mul_eq_one_iff : s * t = 1 \u2194 \u2203 a b, s = {a} \u2227 t = {b} \u2227 a * b = 1", "@[to_additive]\ntheorem mul_iUnion\u2082 (s : Set \u03b1) (t : \u2200 i, \u03ba i \u2192 Set \u03b1) :\n    (s * \u22c3 (i) (j), t i j) = \u22c3 (i) (j), s * t i j", "@[to_additive (attr := simp)]\ntheorem mul_singleton : s * {b} = (\u00b7 * b) '' s", "theorem ncard_diff_add_ncard_of_subset (h : s \u2286 t) (ht : t.Finite", "theorem ncard_eq_toFinset_card' (s : Set \u03b1) [Fintype s] :\n    s.ncard = s.toFinset.card", "theorem ncard_le_ncard_iff_ncard_diff_le_ncard_diff (hs : s.Finite", "theorem ncard_lt_ncard (h : s \u2282 t) (ht : t.Finite", "@[simp] theorem ncard_singleton (a : \u03b1) : ({a} : Set \u03b1).ncard = 1", "theorem nonempty_iInter\u2082 {s : \u2200 i, \u03ba i \u2192 Set \u03b1} :\n    (\u22c2 (i) (j), s i j).Nonempty \u2194 \u2203 a, \u2200 i j, a \u2208 s i j", "theorem nonempty_of_ssubset (ht : s \u2282 t) : (t \\ s).Nonempty", "theorem not_mem_Ico_of_ge (hb : b \u2264 c) : c \u2209 Ico a b", "theorem not_mem_Ioi_self : a \u2209 Ioi a", "theorem not_mem_of_not_mem_sUnion {x : \u03b1} {t : Set \u03b1} {S : Set (Set \u03b1)} (hx : x \u2209 \u22c3\u2080S)\n    (ht : t \u2208 S) : x \u2209 t", "lemma not_mem_uIoc : a \u2209 \u0399 b c \u2194 a \u2264 b \u2227 a \u2264 c \u2228 c < a \u2227 b < a", "@[simp]\ntheorem not_nonempty_empty : \u00ac(\u2205 : Set \u03b1).Nonempty", "lemma not_ordConnected_inter_Icc_iff (hx : x \u2208 s) (hy : y \u2208 s) :\n    \u00ac OrdConnected (s \u2229 Icc x y) \u2194 \u2203 z \u2209 s, z \u2208 Ioo x y", "@[simp]\ntheorem offDiag_empty : (\u2205 : Set \u03b1).offDiag = \u2205", "theorem offDiag_union (h : Disjoint s t) :\n    (s \u222a t).offDiag = s.offDiag \u222a t.offDiag \u222a s \u00d7\u02e2 t \u222a t \u00d7\u02e2 s", "@[simp] theorem one_le_encard_iff_nonempty : 1 \u2264 s.encard \u2194 s.Nonempty", "@[instance]\ntheorem ordConnected_image {E : Type*} [EquivLike E \u03b1 \u03b2] [OrderIsoClass E \u03b1 \u03b2] (e : E) {s : Set \u03b1}\n    [hs : OrdConnected s] : OrdConnected (e '' s)", "theorem pair_diff_left (hne : a \u2260 b) : ({a, b} : Set \u03b1) \\ {a} = {b}", "theorem pairwise_disjoint_Ioo_intCast : Pairwise (Disjoint on fun n : \u2124 => Ioo (n : \u03b1) (n + 1))", "theorem pairwise_eq_iff_exists_eq [Nonempty \u03b9] (s : Set \u03b1) (f : \u03b1 \u2192 \u03b9) :\n    (s.Pairwise fun x y => f x = f y) \u2194 \u2203 z, \u2200 x \u2208 s, f x = z", "theorem pi_nonempty_iff : (s.pi t).Nonempty \u2194 \u2200 i, \u2203 x, i \u2208 s \u2192 x \u2208 t i", "@[simp]\ntheorem pi_univ_Iic : (pi univ fun i \u21a6 Iic (x i)) = Iic x", "theorem pi_univ_Ioc_update_right {x y : \u2200 i, \u03b1 i} {i\u2080 : \u03b9} {m : \u03b1 i\u2080} (hm : m \u2264 y i\u2080) :\n    (pi univ fun i \u21a6 Ioc (x i) (update y i\u2080 m i)) =\n      { z | z i\u2080 \u2264 m } \u2229 pi univ fun i \u21a6 Ioc (x i) (y i)", "theorem piecewise_insert [DecidableEq \u03b1] (j : \u03b1) [\u2200 i, Decidable (i \u2208 insert j s)] :\n    (insert j s).piecewise f g = Function.update (s.piecewise f g) j (f j)", "theorem piecewise_mem_Icc {s : Set \u03b9} [\u2200 j, Decidable (j \u2208 s)] {f\u2081 f\u2082 g\u2081 g\u2082 : \u2200 i, \u03b1 i}\n    (h\u2081 : \u2200 i \u2208 s, f\u2081 i \u2208 Icc (g\u2081 i) (g\u2082 i)) (h\u2082 : \u2200 i \u2209 s, f\u2082 i \u2208 Icc (g\u2081 i) (g\u2082 i)) :\n    s.piecewise f\u2081 f\u2082 \u2208 Icc g\u2081 g\u2082", "@[simp]\ntheorem piecewise_univ [\u2200 i : \u03b1, Decidable (i \u2208 (Set.univ : Set \u03b1))] :\n    piecewise Set.univ f g = f", "@[to_additive]\ntheorem pow_subset_pow_of_one_mem (hs : (1 : \u03b1) \u2208 s) (hn : m \u2264 n) : s ^ m \u2286 s ^ n", "@[simp]\ntheorem powerset_mono : \ud835\udcab s \u2286 \ud835\udcab t \u2194 s \u2286 t", "@[simp]\ntheorem preimage_add_const_Ico : (fun x => x + a) \u207b\u00b9' Ico b c = Ico (b - a) (c - a)", "@[simp]\ntheorem preimage_add_const_uIcc : (fun x => x + a) \u207b\u00b9' [[b, c]] = [[b - a, c - a]]", "theorem preimage_compl_eq_image_compl [BooleanAlgebra \u03b1] (S : Set \u03b1) :\n    HasCompl.compl \u207b\u00b9' S = HasCompl.compl '' S", "@[simp]\ntheorem preimage_const_add_Ici : (fun x => a + x) \u207b\u00b9' Ici b = Ici (b - a)", "@[simp]\ntheorem preimage_const_mul_Icc_of_neg (a b : \u03b1) {c : \u03b1} (h : c < 0) :\n    (c * \u00b7) \u207b\u00b9' Icc a b = Icc (b / c) (a / c)", "@[simp]\ntheorem preimage_const_mul_Ioc_of_neg (a b : \u03b1) {c : \u03b1} (h : c < 0) :\n    (c * \u00b7) \u207b\u00b9' Ioc a b = Ico (b / c) (a / c)", "@[simp]\ntheorem preimage_const_sub_Ico : (fun x => a - x) \u207b\u00b9' Ico b c = Ioc (a - c) (a - b)", "theorem preimage_iInter\u2082 {f : \u03b1 \u2192 \u03b2} {s : \u2200 i, \u03ba i \u2192 Set \u03b2} :\n    (f \u207b\u00b9' \u22c2 (i) (j), s i j) = \u22c2 (i) (j), f \u207b\u00b9' s i j", "@[simp]\ntheorem preimage_inl_image_inr (s : Set \u03b2) : Sum.inl \u207b\u00b9' (@Sum.inr \u03b1 \u03b2 '' s) = \u2205", "@[simp]\ntheorem preimage_mul_const_Iio (a : \u03b1) {c : \u03b1} (h : 0 < c) :\n    (fun x => x * c) \u207b\u00b9' Iio a = Iio (a / c)", "@[simp]\ntheorem preimage_mul_const_uIcc (ha : a \u2260 0) (b c : \u03b1) :\n    (\u00b7 * a) \u207b\u00b9' [[b, c]] = [[b / a, c / a]]", "@[to_additive (attr := simp)]\ntheorem preimage_mul_left_one : (a * \u00b7) \u207b\u00b9' 1 = {a\u207b\u00b9}", "@[simp]\ntheorem preimage_neg_Ioo : -Ioo a b = Ioo (-b) (-a)", "theorem preimage_preimage {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} {s : Set \u03b3} :\n    f \u207b\u00b9' (g \u207b\u00b9' s) = (fun x => g (f x)) \u207b\u00b9' s", "theorem preimage_range_inter {f : \u03b1 \u2192 \u03b2} {s : Set \u03b2} : f \u207b\u00b9' (range f \u2229 s) = f \u207b\u00b9' s", "theorem preimage_subtype_coe_eq_compl {s u v : Set \u03b1} (hsuv : s \u2286 u \u222a v)\n    (H : s \u2229 (u \u2229 v) = \u2205) : ((\u2191) : s \u2192 \u03b1) \u207b\u00b9' u = ((\u2191) \u207b\u00b9' v)\u1d9c", "lemma preimage_val_sInter_eq_sInter : A \u2193\u2229 (\u22c2\u2080 S) = \u22c2\u2080 ((A \u2193\u2229 \u00b7) '' S)", "theorem prod_eq_biUnion_left : s \u00d7\u02e2 t = \u22c3 a \u2208 s, (fun b => (a, b)) '' t", "theorem prod_iInter {s : Set \u03b1} {t : \u03b9 \u2192 Set \u03b2} [h\u03b9 : Nonempty \u03b9] :\n    (s \u00d7\u02e2 \u22c2 i, t i) = \u22c2 i, s \u00d7\u02e2 t i", "theorem prod_iUnion {s : Set \u03b1} {t : \u03b9 \u2192 Set \u03b2} : (s \u00d7\u02e2 \u22c3 i, t i) = \u22c3 i, s \u00d7\u02e2 t i", "theorem prod_sInter {T : Set (Set \u03b2)} (hT : T.Nonempty) (s : Set \u03b1) :\n    s \u00d7\u02e2 \u22c2\u2080 T = \u22c2 t \u2208 T, s \u00d7\u02e2 t", "theorem prod_subset_prod_iff : s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205", "theorem prod_univ {s : Set \u03b1} : s \u00d7\u02e2 (univ : Set \u03b2) = Prod.fst \u207b\u00b9' s", "theorem projIcc_eq_left (h : a < b) : projIcc a b h.le x = \u27e8a, left_mem_Icc.mpr h.le\u27e9 \u2194 x \u2264 a", "@[simp]\ntheorem projIcc_left : projIcc a b h a = \u27e8a, left_mem_Icc.2 h\u27e9", "@[simp]\ntheorem range_diag : (range fun x => (x, x)) = diagonal \u03b1", "lemma range_enumerateCountable_of_mem {s : Set \u03b1} (h : s.Countable) {default : \u03b1}\n    (h_mem : default \u2208 s) :\n    range (enumerateCountable h default) = s", "theorem range_ite_subset' {p : Prop} [Decidable p] {f g : \u03b1 \u2192 \u03b2} :\n    range (if p then f else g) \u2286 range f \u222a range g", "@[simp]\ntheorem range_list_getD (d : \u03b1) : (range fun n : Nat => l[n]?.getD d) = insert d { x | x \u2208 l }", "theorem range_piecewise (f g : \u03b1 \u2192 \u03b2) : range (s.piecewise f g) = f '' s \u222a g '' s\u1d9c", "@[to_additive]\ntheorem range_smul_range {\u03b9 \u03ba : Type*} [SMul \u03b1 \u03b2] (b : \u03b9 \u2192 \u03b1) (c : \u03ba \u2192 \u03b2) :\n    range b \u2022 range c = range fun p : \u03b9 \u00d7 \u03ba \u21a6 b p.1 \u2022 c p.2", "theorem Set.restrictPreimage_closedEmbedding (s : Set \u03b2) (h : ClosedEmbedding f) :\n    ClosedEmbedding (s.restrictPreimage f)", "theorem Set.restrictPreimage_inducing (s : Set \u03b2) (h : Inducing f) :\n    Inducing (s.restrictPreimage f)", "@[simp]\ntheorem sInter_image (f : \u03b1 \u2192 Set \u03b2) (s : Set \u03b1) : \u22c2\u2080 (f '' s) = \u22c2 x \u2208 s, f x", "@[simp]\ntheorem sInter_range (f : \u03b9 \u2192 Set \u03b2) : \u22c2\u2080 range f = \u22c2 x, f x", "theorem sInter_subset_of_mem {S : Set (Set \u03b1)} {t : Set \u03b1} (tS : t \u2208 S) : \u22c2\u2080 S \u2286 t", "theorem sUnion_subset {S : Set (Set \u03b1)} {t : Set \u03b1} (h : \u2200 t' \u2208 S, t' \u2286 t) : \u22c3\u2080S \u2286 t", "@[simp]\ntheorem sep_eq_empty_iff_mem_false : { x \u2208 s | p x } = \u2205 \u2194 \u2200 x \u2208 s, \u00acp x", "@[simp]\ntheorem sep_inter : { x | (x \u2208 s \u2227 x \u2208 t) \u2227 p x } = { x \u2208 s | p x } \u2229 { x \u2208 t | p x }", "theorem sep_of_ncard_eq {a : \u03b1} {P : \u03b1 \u2192 Prop} (h : { x \u2208 s | P x }.ncard = s.ncard) (ha : a \u2208 s)\n    (hs : s.Finite", "@[simp]\ntheorem setOf_mem_eq {s : Set \u03b1} : { x | x \u2208 s } = s", "theorem sigmaToiUnion_surjective : Surjective (sigmaToiUnion t)", "theorem sigma_mono (hs : s\u2081 \u2286 s\u2082) (ht : \u2200 i, t\u2081 i \u2286 t\u2082 i) : s\u2081.sigma t\u2081 \u2286 s\u2082.sigma t\u2082", "@[simp]\ntheorem singleton_subset_iff {a : \u03b1} {s : Set \u03b1} : {a} \u2286 s \u2194 a \u2208 s", "@[simp 1100]\ntheorem singleton_unop_op (x : \u03b1\u1d52\u1d56) : ({unop x} : Set \u03b1).op = {x}", "@[simp] lemma sized_singleton : ({s} : Set (Finset \u03b1)).Sized r \u2194 s.card = r", "theorem _root_.Set.smul_mem_center (r : R) {a : A} (ha : a \u2208 Set.center A) :\n    r \u2022 a \u2208 Set.center A where", "@[simp]\ntheorem _root_.Set.smul_mem_centralizer {s : Set A} (r : R) {a : A} (ha : a \u2208 s.centralizer) :\n    r \u2022 a \u2208 s.centralizer", "@[simp]\ntheorem smul_mem_smul_set_iff\u2080 (ha : a \u2260 0) (A : Set \u03b2) (x : \u03b2) : a \u2022 x \u2208 a \u2022 A \u2194 x \u2208 A", "@[to_additive (attr := simp)]\ntheorem smul_set_eq_empty : a \u2022 s = \u2205 \u2194 s = \u2205", "@[to_additive]\ntheorem smul_set_subset_iff : a \u2022 s \u2286 t \u2194 \u2200 \u2983b\u2984, b \u2208 s \u2192 a \u2022 b \u2208 t", "theorem smul_zero_subset (s : Set \u03b1) : s \u2022 (0 : Set \u03b2) \u2286 0", "theorem ssubset_iff_insert {s t : Set \u03b1} : s \u2282 t \u2194 \u2203 a \u2209 s, insert a s \u2286 t", "theorem sub_mem_Ioc_iff_left : a - b \u2208 Set.Ioc c d \u2194 a \u2208 Set.Ioc (c + b) (d + b)", "@[to_additive subset_addCenter_add_units]\ntheorem subset_center_units [Monoid M] : ((\u2191) : M\u02e3 \u2192 M) \u207b\u00b9' center M \u2286 Set.center M\u02e3", "theorem subset_compl_comm : s \u2286 t\u1d9c \u2194 t \u2286 s\u1d9c", "theorem subset_compl_iff_disjoint_left : s \u2286 t\u1d9c \u2194 Disjoint t s", "theorem subset_compl_iff_disjoint_right : s \u2286 t\u1d9c \u2194 Disjoint s t", "theorem subset_eq_empty {s t : Set \u03b1} (h : t \u2286 s) (e : s = \u2205) : t = \u2205", "theorem subset_iInter {t : Set \u03b2} {s : \u03b9 \u2192 Set \u03b2} (h : \u2200 i, t \u2286 s i) : t \u2286 \u22c2 i, s i", "theorem subset_image_iff {t : Set \u03b2} :\n    t \u2286 f '' s \u2194 \u2203 u, u \u2286 s \u2227 f '' u = t", "@[simp]\ntheorem subset_insert (x : \u03b1) (s : Set \u03b1) : s \u2286 insert x s", "theorem subset_ite {t s s' u : Set \u03b1} : u \u2286 t.ite s s' \u2194 u \u2229 t \u2286 s \u2227 u \\ t \u2286 s'", "theorem subset_preimage_image (f : \u03b1 \u2192 \u03b2) (s : Set \u03b1) : s \u2286 f \u207b\u00b9' (f '' s)", "theorem subset_range_enumerate {s : Set \u03b1} (h : s.Countable) (default : \u03b1) :\n    s \u2286 range (enumerateCountable h default)", "theorem subset_sUnion_of_subset {s : Set \u03b1} (t : Set (Set \u03b1)) (u : Set \u03b1) (h\u2081 : s \u2286 u)\n    (h\u2082 : u \u2208 t) : s \u2286 \u22c3\u2080t", "theorem sups_right_comm : s \u22bb t \u22bb u = s \u22bb u \u22bb t", "theorem sups_subset_right : s\u2081 \u2286 s\u2082 \u2192 s\u2081 \u22bb t \u2286 s\u2082 \u22bb t", "@[simp]\ntheorem toFinset_eq_univ [Fintype \u03b1] [Fintype s] : s.toFinset = Finset.univ \u2194 s = univ", "@[to_additive (attr := simp)]\ntheorem toFinset_smul_set (a : \u03b1) (s : Set \u03b2) [Fintype s] [Fintype \u2191(a \u2022 s)] :\n    (a \u2022 s).toFinset = a \u2022 s.toFinset", "@[simp]\ntheorem top_eq_univ : (\u22a4 : Set \u03b1) = univ", "lemma uIoc_comm (a b : \u03b1) : \u0399 a b = \u0399 b a", "@[simp]\ntheorem union_subset_iff {s t u : Set \u03b1} : s \u222a t \u2286 u \u2194 s \u2286 u \u2227 t \u2286 u", "@[to_additive (attr := simp)]\ntheorem univ_mul (ht : t.Nonempty) : (univ : Set \u03b1) * t = univ", "@[simp]\ntheorem univ_nonempty : \u2200 [Nonempty \u03b1], (univ : Set \u03b1).Nonempty", "theorem univ_pi_eq_iInter (t : \u2200 i, Set (\u03c0 i)) : pi univ t = \u22c2 i, eval i \u207b\u00b9' t i", "@[simp]\ntheorem univ_union (s : Set \u03b1) : univ \u222a s = univ", "theorem update_image [DecidableEq \u03b9] (x : (i : \u03b9) \u2192 \u03b2 i) (i : \u03b9) (s : Set (\u03b2 i)) :\n    update x i '' s = Set.univ.pi (update (fun j \u21a6 {x j}) i s)", "theorem vsub_mem_vsub (hb : b \u2208 s) (hc : c \u2208 t) : b -\u1d65 c \u2208 s -\u1d65 t", "@[simp] theorem zero_mem_center : (0 : M\u2080) \u2208 center M\u2080 where", "@[simp] theorem zero_mem_centralizer : (0 : M\u2080) \u2208 centralizer s", "@[simp, norm_cast]\ntheorem coe_set_eq : (p : Set B) = q \u2194 p = q", "theorem list_prod_map_mem_graded {\u03b9'} (l : List \u03b9') (i : \u03b9' \u2192 \u03b9) (r : \u03b9' \u2192 R)\n    (h : \u2200 j \u2208 l, r j \u2208 A (i j)) : (l.map r).prod \u2208 A (l.map i).sum", "theorem up_le_up {s t : Set \u03b1} : Set.up s \u2264 Set.up t \u2194 s \u2286 t", "theorem PGame.equiv_iff_game_eq {x y : PGame} : x \u2248 y \u2194 (\u27e6x\u27e7 : Game) = \u27e6y\u27e7", "theorem equiv_iff_add_equiv_zero' (H : PGame) : (G \u2248 H) \u2194 (G + H \u2248 0)", "theorem forall_rightMoves_fuzzy_iff_equiv_zero : (\u2200 j, G.moveRight j \u2016 0) \u2194 (G \u2248 0)", "theorem lf_zero_iff {G : PGame} [G.Impartial] : G \u29cf 0 \u2194 0 \u29cf G", "theorem nonneg : \u00acG < 0", "theorem nonpos : \u00ac0 < G", "theorem lt_moveRight {x : PGame} (o : Numeric x) (j) : x < x.moveRight j", "private theorem add_le_add_right' : \u2200 {x y z : PGame}, x \u2264 y \u2192 x + z \u2264 y + z", "theorem add_lf_add_of_le_of_lf {w x y z : PGame} (hwx : w \u2264 x) (hyz : y \u29cf z) : w + y \u29cf x + z", "theorem add_lf_add_right {y z : PGame} (h : y \u29cf z) (x) : y + x \u29cf z + x", "@[simp]\ntheorem add_moveLeft_inr (x : PGame) {y : PGame} (i) :\n    (x + y).moveLeft (toLeftMovesAdd (Sum.inr i)) = x + y.moveLeft i", "lemma bddBelow_range_of_small {\u03b9 : Type*} [Small.{u} \u03b9] (f : \u03b9 \u2192 PGame.{u}) :\n    BddBelow (Set.range f)", "@[simp]\ntheorem birthday_one : birthday 1 = 1", "theorem equiv_nim_grundyValue : \u2200 (G : PGame.{u}) [G.Impartial], G \u2248 nim (grundyValue G)", "theorem fuzzy_congr_imp {x\u2081 y\u2081 x\u2082 y\u2082 : PGame} (hx : x\u2081 \u2248 x\u2082) (hy : y\u2081 \u2248 y\u2082) : x\u2081 \u2016 y\u2081 \u2192 x\u2082 \u2016 y\u2082", "@[trans]\ntheorem fuzzy_of_equiv_of_fuzzy {x y z : PGame} (h\u2081 : x \u2248 y) (h\u2082 : y \u2016 z) : x \u2016 z", "theorem grundyValue_eq_iff_equiv_nim {G : PGame} [G.Impartial] {o : Ordinal} :\n    grundyValue G = o \u2194 (G \u2248 nim o)", "@[simp]\ntheorem isOption_neg_neg {x y : PGame} : IsOption (-x) (-y) \u2194 IsOption x y", "theorem le_or_gf (x y : PGame) : x \u2264 y \u2228 y \u29cf x", "@[simp]\ntheorem leftMoves_mk {xl xr xL xR} : (\u27e8xl, xr, xL, xR\u27e9 : PGame).LeftMoves = xl", "@[trans]\ntheorem lf_of_lf_of_equiv {x y z : PGame} (h\u2081 : x \u29cf y) (h\u2082 : y \u2248 z) : x \u29cf z", "theorem lt_congr_imp {x\u2081 y\u2081 x\u2082 y\u2082 : PGame} (hx : x\u2081 \u2248 x\u2082) (hy : y\u2081 \u2248 y\u2082) (h : x\u2081 < y\u2081) : x\u2082 < y\u2082", "theorem lt_def {x y : PGame} (ox : x.Numeric) (oy : y.Numeric) :\n    x < y \u2194\n      (\u2203 i, (\u2200 i', x.moveLeft i' < y.moveLeft i) \u2227 \u2200 j, x < (y.moveLeft i).moveRight j) \u2228\n        \u2203 j, (\u2200 i, (x.moveRight j).moveLeft i < y) \u2227 \u2200 j', x.moveRight j < y.moveRight j'", "theorem lt_iff_le_and_lf {x y : PGame} : x < y \u2194 x \u2264 y \u2227 x \u29cf y", "theorem mk_lf_of_le {xl xr y j} (xL) {xR : xr \u2192 PGame} : xR j \u2264 y \u2192 mk xl xr xL xR \u29cf y", "@[simp]\ntheorem moveRight_neg' {x : PGame} (i) :\n    (-x).moveRight i = -x.moveLeft (toRightMovesNeg.symm i)", "theorem moveRight_neg_symm' {x : PGame} (i) :\n    x.moveRight i = -(-x).moveLeft (toLeftMovesNeg i)", "theorem moveRight_nim_hEq (o : Ordinal) :\n    have : IsWellOrder (Quotient.out o).\u03b1 (\u00b7 < \u00b7) := inferInstance\n    HEq (nim o).moveRight fun i : o.out.\u03b1 => nim (typein (\u00b7 < \u00b7) i)", "theorem mul_assoc_equiv (x y z : PGame) : x * y * z \u2248 x * (y * z)", "@[simp]\ntheorem mul_moveLeft_inr {x y : PGame} {i j} :\n    (x * y).moveLeft (toLeftMovesMul (Sum.inr (i, j))) =\n      x.moveRight i * y + x * y.moveRight j - x.moveRight i * y.moveRight j", "theorem neg_equiv_iff {x y : PGame} : (-x \u2248 y) \u2194 (x \u2248 -y)", "theorem neg_lf_iff {x y : PGame} : -y \u29cf x \u2194 -x \u29cf y", "theorem quot_mul_assoc (x y z : PGame) : (\u27e6x * y * z\u27e7 : Game) = \u27e6x * (y * z)\u27e7", "@[simp]\ntheorem quot_sub (a b : PGame) : \u27e6a - b\u27e7 = (\u27e6a\u27e7 : Game) - \u27e6b\u27e7", "@[simp]\ntheorem relabel_moveLeft' {x : PGame} {xl' xr'} (el : xl' \u2243 x.LeftMoves) (er : xr' \u2243 x.RightMoves)\n    (i : xl') : moveLeft (relabel el er) i = x.moveLeft (el i)", "theorem short_birthday (x : PGame.{u}) : [Short x] \u2192 x.birthday < Ordinal.omega", "theorem zero_le_add_left_neg (x : PGame) : 0 \u2264 -x + x", "theorem zero_lf_le {x : PGame} : 0 \u29cf x \u2194 \u2203 i, 0 \u2264 x.moveLeft i", "theorem eq_iff_rel_eq {r\u2081 r\u2082 : Setoid \u03b1} : r\u2081 = r\u2082 \u2194 r\u2081.Rel = r\u2082.Rel", "theorem eq_top_iff {s : Setoid \u03b1} : s = (\u22a4 : Setoid \u03b1) \u2194 \u2200 x y : \u03b1, s.Rel x y", "theorem eqvGen_eq (r : \u03b1 \u2192 \u03b1 \u2192 Prop) :\n    EqvGen.Setoid r = sInf { s : Setoid \u03b1 | \u2200 \u2983x y\u2984, r x y \u2192 s.Rel x y }", "theorem finite_classes_ker {\u03b1 \u03b2 : Type*} [Finite \u03b2] (f : \u03b1 \u2192 \u03b2) : (Setoid.ker f).classes.Finite", "theorem mapOfSurjective_eq_map (h : ker f \u2264 r) (hf : Surjective f) :\n    map r f = mapOfSurjective r f h hf", "theorem symm {a b : \u03b1} (hab : a \u2248 b) : b \u2248 a", "@[simp]\nlemma add_val {M N : SheafOfModules.{v} R} (f g : M \u27f6 N) :\n    (f + g).val = f.val + g.val := rfl", "@[ext]\ntheorem Shrink.ext {\u03b1 : Type v} [Small.{w} \u03b1] {x y : Shrink \u03b1}\n    (w : (equivShrink _).symm x = (equivShrink _).symm y) : x = y", "theorem find_mem {c : Set (PartialRefinement u s)} (i : \u03b9) (ne : c.Nonempty) : find c ne i \u2208 c", "theorem mem_find_carrier_iff {c : Set (PartialRefinement u s)} {i : \u03b9} (ne : c.Nonempty) :\n    i \u2208 (find c ne i).carrier \u2194 i \u2208 chainSupCarrier c", "@[simp]\ntheorem Ico_mk_mk : Ico (\u27e8i, a\u27e9 : Sigma \u03b1) \u27e8i, b\u27e9 = (Ico a b).map (Embedding.sigmaMk i)", "theorem card_Ioc : (Ioc a b).card = if h : a.1 = b.1 then (Ioc (h.rec a.2) b.2).card else 0", "@[to_additive (attr := simp)]\ntheorem curry_mul [\u2200 a b, Mul (\u03b3 a b)] (x y : (i : \u03a3 a, \u03b2 a) \u2192 \u03b3 i.1 i.2) :\n    Sigma.curry (x * y) = Sigma.curry x * Sigma.curry y", "protected theorem Sigma.eq {\u03b2 : \u03b1 \u2192 Type v} : \u2200 {p\u2081 p\u2082 : \u03a3 a, \u03b2 a} (h\u2081 : p\u2081.1 = p\u2082.1),\n    (Eq.recOn h\u2081 p\u2081.2 : \u03b2 p\u2082.1) = p\u2082.2 \u2192 p\u2081 = p\u2082", "theorem fst_injective_iff : Injective (fst : (\u03a3 a, \u03b2 a) \u2192 \u03b1) \u2194 \u2200 a, Subsingleton (\u03b2 a)", "theorem Sigma.isConnected_iff [\u2200 i, TopologicalSpace (\u03c0 i)] {s : Set (\u03a3i, \u03c0 i)} :\n    IsConnected s \u2194 \u2203 i t, IsConnected t \u2227 s = Sigma.mk i '' t", "theorem Sigma.nhds_mk (i : \u03b9) (x : \u03c3 i) : \ud835\udcdd (\u27e8i, x\u27e9 : Sigma \u03c3) = Filter.map (Sigma.mk i) (\ud835\udcdd x)", "@[to_additive]\ntheorem smul_def : a \u2022 x = x.map id fun _ => (a \u2022 \u00b7)", "@[simp, norm_cast] lemma coe_mul {\u03b1} [MulZeroOneClass \u03b1] [HasDistribNeg \u03b1] (a b : SignType) :\n    \u2191(a * b) = (a : \u03b1) * b", "private lemma mul_assoc : \u2200 (a b c : SignType), (a * b) * c = a * (b * c)", "@[simp]\ntheorem pos_eq_one : pos = 1", "theorem CliqueFree.anti (h : G \u2264 H) : H.CliqueFree n \u2192 G.CliqueFree n", "theorem Coloring.card_colorClasses_le [Fintype \u03b1] [Fintype C.colorClasses] :\n    Fintype.card C.colorClasses \u2264 Fintype.card \u03b1", "theorem Coloring.not_adj_of_mem_colorClass {c : \u03b1} {v w : V} (hv : v \u2208 C.colorClass c)\n    (hw : w \u2208 C.colorClass c) : \u00acG.Adj v w", "theorem hom_refl (C : G.ComponentCompl L) : C.hom (subset_refl L) = C", "theorem not_mem_of_mem {C : G.ComponentCompl K} {c : V} (cC : c \u2208 C) : c \u2209 K", "@[ext]\ntheorem ComponentCompl.supp_injective :\n    Function.Injective (ComponentCompl.supp : G.ComponentCompl K \u2192 Set V)", "theorem Connected.map {G : SimpleGraph V} {H : SimpleGraph V'} (f : G \u2192g H) (hf : Surjective f)\n    (hG : G.Connected) : H.Connected", "@[simp]\ntheorem iso_inv_image_comp_eq_iff_eq_map {C : G.ConnectedComponent} :\n    G.connectedComponentMk (\u03c6.symm v') = C \u2194 G'.connectedComponentMk v' = C.map \u03c6", "protected theorem sound {v w : V} :\n    G.Reachable v w \u2192 G.connectedComponentMk v = G.connectedComponentMk w", "@[ext]\ntheorem Dart.ext (d\u2081 d\u2082 : G.Dart) (h : d\u2081.toProd = d\u2082.toProd) : d\u2081 = d\u2082", "theorem FarFromTriangleFree.cliqueFinset_nonempty (hG : G.FarFromTriangleFree \u03b5) (h\u03b5 : 0 < \u03b5) :\n    (G.cliqueFinset 3).Nonempty", "protected theorem IsClique.map (h : G.IsClique s) {f : \u03b1 \u21aa \u03b2} : (G.map f).IsClique (f '' s)", "theorem IsNClique.mono (h : G \u2264 H) : G.IsNClique n s \u2192 H.IsNClique n s", "theorem IsNClique.not_cliqueFree (hG : G.IsNClique n s) : \u00acG.CliqueFree n", "theorem IsRegularOfDegree.degree_eq {d : \u2115} (h : G.IsRegularOfDegree d) (v : V) : G.degree v = d", "theorem IsSRGWith.top :\n    (\u22a4 : SimpleGraph V).IsSRGWith (Fintype.card V) (Fintype.card V - 1) (Fintype.card V - 2) \u03bc where", "theorem isEquipartition : h.finpartition.IsEquipartition", "lemma not_adj_iff_part_eq : \u00acG.Adj s t \u2194 h.finpartition.part s = h.finpartition.part t", "@[simp]\nlemma comap_apply (f : V \u2243 W) (G : SimpleGraph W) (v : V) :\n    SimpleGraph.Iso.comap f G v = f v := rfl", "theorem Iso.preconnected_iff {G : SimpleGraph V} {H : SimpleGraph V'} (e : G \u2243g H) :\n    G.Preconnected \u2194 H.Preconnected", "theorem Preconnected.set_univ_walk_nonempty (hconn : G.Preconnected) (u v : V) :\n    (Set.univ : Set (G.Walk u v)).Nonempty", "protected theorem Reachable.exists_walk_of_dist {u v : V} (hr : G.Reachable u v) :\n    \u2203 p : G.Walk u v, p.length = G.dist u v", "@[trans]\nprotected theorem Reachable.trans {u v w : V} (huv : G.Reachable u v) (hvw : G.Reachable v w) :\n    G.Reachable u w", "@[simp]\ntheorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) :\n    G.Adj u v", "@[simp] lemma coe_hom (x : Subgraph G) :\n    (x.hom : x.verts \u2192 V) = (fun (v : x.verts) => (v : V)) := rfl", "protected lemma connected_iff {H : G.Subgraph} :\n    H.Connected \u2194 H.Preconnected \u2227 H.verts.Nonempty", "@[simp]\ntheorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w \u2194 G'.Adj v w \u2227 \u00acs(v, w) \u2208 s", "@[simp]\ntheorem deleteEdges_spanningCoe_eq :\n    G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe", "@[mono]\ntheorem deleteVerts_mono {G' G'' : G.Subgraph} (h : G' \u2264 G'') :\n    G'.deleteVerts s \u2264 G''.deleteVerts s", "theorem deleteVerts_verts : (G'.deleteVerts s).verts = G'.verts \\ s", "@[simp]\ntheorem edgeSet_iSup (f : \u03b9 \u2192 G.Subgraph) :\n    (\u2a06 i, f i).edgeSet = \u22c3 i, (f i).edgeSet", "@[simp]\ntheorem edgeSet_top : (\u22a4 : Subgraph G).edgeSet = G.edgeSet", "@[simp]\ntheorem iInf_adj {f : \u03b9 \u2192 G.Subgraph} : (\u2a05 i, f i).Adj a b \u2194 (\u2200 i, (f i).Adj a b) \u2227 G.Adj a b", "protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj", "theorem map_monotone {G' : SimpleGraph W} (f : G \u2192g G') : Monotone (Subgraph.map f)", "theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v \u2286 G.neighborSet v", "@[simp]\ntheorem spanningCoe_top : (\u22a4 : Subgraph G).spanningCoe = G", "theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) :\n    G.subgraphOfAdj hvw = (\u22a4 : G.Subgraph).induce {v, w}", "lemma top_induce_pair_connected_of_adj {u v : V} (huv : G.Adj u v) :\n    ((\u22a4 : G.Subgraph).induce {u, v}).Connected", "@[simp] lemma in\u2080\u2082_iff : (graph t).Adj (in\u2080 a) (in\u2082 c) \u2194 \u2203 b, (a, b, c) \u2208 t", "@[simp] lemma in\u2081\u2080_iff : (graph t).Adj (in\u2081 b) (in\u2080 a) \u2194 \u2203 c, (a, b, c) \u2208 t", "lemma locallyLinear [ExplicitDisjoint t] [NoAccidental t] : (graph t).LocallyLinear", "lemma IsCycle.ne_bot : \u2200 {p : G.Walk u u}, p.IsCycle \u2192 G \u2260 \u22a5\n  | nil, hp => by cases hp.ne_nil rfl\n  | cons h _, hp => by rintro rfl; exact h", "protected lemma IsPath.cons {p : Walk G v w} (hp : p.IsPath) (hu : u \u2209 p.support) {h : G.Adj u v} :\n    (cons h p).IsPath", "theorem IsPath.of_append_left {u v w : V} {p : G.Walk u v} {q : G.Walk v w} :\n    (p.append q).IsPath \u2192 p.IsPath", "lemma IsPath.tail {p : G.Walk u v} (hp : p.IsPath) (hp' : \u00ac p.Nil) : (p.tail hp').IsPath", "theorem IsTrail.isEulerian_of_forall_mem {u v : V} {p : G.Walk u v} (h : p.IsTrail)\n    (hc : \u2200 e, e \u2208 G.edgeSet \u2192 e \u2208 p.edges) : p.IsEulerian", "theorem IsTrail.of_append_left {u v w : V} {p : G.Walk u v} {q : G.Walk v w}\n    (h : (p.append q).IsTrail) : p.IsTrail", "theorem IsTrail.of_cons {u v w : V} {h : G.Adj u v} {p : G.Walk v w} :\n    (cons h p).IsTrail \u2192 p.IsTrail", "theorem adj_getVert_succ {u v} (w : G.Walk u v) {i : \u2115} (hi : i < w.length) :\n    G.Adj (w.getVert i) (w.getVert (i + 1))", "theorem append_assoc {u v w x : V} (p : G.Walk u v) (q : G.Walk v w) (r : G.Walk w x) :\n    p.append (q.append r) = (p.append q).append r", "theorem append_concat {u v w x : V} (p : G.Walk u v) (q : G.Walk v w) (h : G.Adj w x) :\n    p.append (q.concat h) = (p.append q).concat h", "@[simp]\ntheorem append_nil {u v : V} (p : G.Walk u v) : p.append nil = p", "@[simp]\ntheorem cons_append {u v w x : V} (h : G.Adj u v) (p : G.Walk v w) (q : G.Walk w x) :\n    (cons h p).append q = cons h (p.append q)", "theorem darts_takeUntil_subset {u v w : V} (p : G.Walk v w) (h : u \u2208 p.support) :\n    (p.takeUntil u h).darts \u2286 p.darts", "@[simp]\ntheorem edges_append {u v w : V} (p : G.Walk u v) (p' : G.Walk v w) :\n    (p.append p').edges = p.edges ++ p'.edges", "lemma end_mem_verts_toSubgraph (p : G.Walk u v) : v \u2208 p.toSubgraph.verts", "theorem exists_boundary_dart {u v : V} (p : G.Walk u v) (S : Set V) (uS : u \u2208 S) (vS : v \u2209 S) :\n    \u2203 d : G.Dart, d \u2208 p.darts \u2227 d.fst \u2208 S \u2227 d.snd \u2209 S", "lemma isHamiltonianCycle_iff_isCycle_and_support_count_tail_eq_one :\n    p.IsHamiltonianCycle \u2194 p.IsCycle \u2227 \u2200 a, (support p).tail.count a = 1", "theorem length_bypass_le {u v : V} (p : G.Walk u v) : p.bypass.length \u2264 p.length", "@[simp]\ntheorem length_edges {u v : V} (p : G.Walk u v) : p.edges.length = p.length", "@[simp]\ntheorem length_eq_zero_iff {u : V} {p : G.Walk u u} : p.length = 0 \u2194 p = nil", "@[simp]\ntheorem length_support {u v : V} (p : G.Walk u v) : p.support.length = p.length + 1", "@[simp]\ntheorem mapLe_isPath {G G' : SimpleGraph V} (h : G \u2264 G') {u v : V} {p : G.Walk u v} :\n    (p.mapLe h).IsPath \u2194 p.IsPath", "theorem map_fst_darts {u v : V} (p : G.Walk u v) : p.darts.map (\u00b7.fst) = p.support.dropLast", "lemma nil_iff_eq_nil : \u2200 {p : G.Walk v v}, p.Nil \u2194 p = nil\n  | .nil | .cons _ _ => by simp", "@[simp]\ntheorem reverse_transfer (hp) :\n    (p.transfer H hp).reverse =\n      p.reverse.transfer H (by simp only [edges_reverse, List.mem_reverse]; exact hp)", "theorem rotate_edges {u v : V} (c : G.Walk v v) (h : u \u2208 c.support) :\n    (c.rotate h).edges ~r c.edges", "@[simp]\ntheorem start_mem_support {u v : V} (p : G.Walk u v) : u \u2208 p.support", "lemma _root_.SimpleGraph.Walk.toSubgraph_connected {u v : V} (p : G.Walk u v) :\n    p.toSubgraph.Connected", "lemma toSubgraph_le_induce_support (p : G.Walk u v) :\n    p.toSubgraph \u2264 (\u22a4 : G.Subgraph).induce {v | v \u2208 p.support}", "theorem adj_injective : Injective (Adj : SimpleGraph V \u2192 V \u2192 V \u2192 Prop)", "@[symm]\ntheorem adj_symm (h : G.Adj u v) : G.Adj v u", "theorem boxProd_neighborFinset (x : \u03b1 \u00d7 \u03b2)\n    [Fintype (G.neighborSet x.1)] [Fintype (H.neighborSet x.2)] [Fintype ((G \u25a1 H).neighborSet x)] :\n    (G \u25a1 H).neighborFinset x =\n      (G.neighborFinset x.1 \u00d7\u02e2 {x.2}).disjUnion ({x.1} \u00d7\u02e2 H.neighborFinset x.2)\n        (Finset.disjoint_product.mpr <| Or.inl <| neighborFinset_disjoint_singleton _ _)", "theorem card_edgeFinset_sup_edge [Fintype (edgeSet (G \u2294 edge s t))] (hn : \u00acG.Adj s t) (h : s \u2260 t) :\n    (G \u2294 edge s t).edgeFinset.card = G.edgeFinset.card + 1", "@[simp] lemma coe_recolorOfEmbedding (f : \u03b1 \u21aa \u03b2) :\n    \u21d1(G.recolorOfEmbedding f) = (Embedding.completeGraph f).toHom.comp := rfl", "theorem colorable_of_fintype (G : SimpleGraph V) [Fintype V] : G.Colorable (Fintype.card V)", "theorem commonNeighbors_top_eq {v w : V} :\n    (\u22a4 : SimpleGraph V).commonNeighbors v w = Set.univ \\ {v, w}", "theorem degMatrix_mulVec_apply [NonAssocSemiring R] (v : V) (vec : V \u2192 R) :\n    (G.degMatrix R *\u1d65 vec) v = G.degree v * vec v", "lemma deleteEdges_le (s : Set (Sym2 V)) : G.deleteEdges s \u2264 G := sdiff_le", "theorem deleteEdges_sdiff_eq_of_le {H : SimpleGraph V} (h : H \u2264 G) :\n    G.deleteEdges (G.edgeSet \\ H.edgeSet) = H", "theorem dotProduct_mulVec_adjMatrix [NonAssocSemiring \u03b1] (x y : V \u2192 \u03b1) :\n    x \u2b1d\u1d65 (G.adjMatrix \u03b1).mulVec y = \u2211 i : V, \u2211 j : V, if G.Adj i j then x i * y j else 0", "@[simp] lemma edgeSet_nonempty : G.edgeSet.Nonempty \u2194 G \u2260 \u22a5", "@[simp]\ntheorem edgeSet_top : (\u22a4 : SimpleGraph V).edgeSet = {e | \u00ace.IsDiag}", "lemma edge_adj (v w : V) : (edge s t).Adj v w \u2194 (v = s \u2227 w = t \u2228 v = t \u2227 w = s) \u2227 v \u2260 w", "lemma exists_girth_eq_length :\n    (\u2203 (a : \u03b1) (w : G.Walk a a), w.IsCycle \u2227 G.girth = w.length) \u2194 \u00ac G.IsAcyclic", "lemma farFromTriangleFree_of_disjoint_triangles (tris : Finset (Finset \u03b1))\n    (htris : tris \u2286 G.cliqueFinset 3)\n    (pd : (tris : Set (Finset \u03b1)).Pairwise fun x y \u21a6 (x \u2229 y : Set \u03b1).Subsingleton)\n    (tris_big : \u03b5 * (card \u03b1 ^ 2 : \u2115) \u2264 tris.card) :\n    G.FarFromTriangleFree \u03b5", "@[simp]\ntheorem SimpleGraph.fromRel_adj {V : Type u} (r : V \u2192 V \u2192 Prop) (v w : V) :\n    (SimpleGraph.fromRel r).Adj v w \u2194 v \u2260 w \u2227 (r v w \u2228 r w v)", "@[simp]\ntheorem iSup_adj {f : \u03b9 \u2192 SimpleGraph V} : (\u2a06 i, f i).Adj a b \u2194 \u2203 i, (f i).Adj a b", "theorem incMatrix_mul_transpose\n    [\u2200 a, Fintype (neighborSet G a)] [DecidableEq \u03b1] [DecidableRel G.Adj] :\n    G.incMatrix R * (G.incMatrix R)\u1d40 = fun a b =>\n      if a = b then (G.degree a : R) else if G.Adj a b then 1 else 0", "theorem incidenceSet_inter_incidenceSet_of_adj (h : G.Adj a b) :\n    G.incidenceSet a \u2229 G.incidenceSet b = {s(a, b)}", "theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) :\n    G.induce s = ((\u22a4 : G.Subgraph).induce s).coe", "lemma induce_sUnion_connected_of_pairwise_not_disjoint {S : Set (Set V)} (Sn : S.Nonempty)\n    (Snd : \u2200 {s t}, s \u2208 S \u2192 t \u2208 S \u2192 (s \u2229 t).Nonempty)\n    (Sc : \u2200 {s}, s \u2208 S \u2192 (G.induce s).Connected) :\n    (G.induce (\u22c3\u2080 S)).Connected", "theorem infinite_iff_in_eventualRange {K : (Finset V)\u1d52\u1d56} (C : G.componentComplFunctor.obj K) :\n    C.supp.Infinite \u2194 C \u2208 G.componentComplFunctor.eventualRange K", "@[simp]\ntheorem interedges_empty_left (t : Finset \u03b1) : G.interedges \u2205 t = \u2205", "@[simp]\ntheorem isNClique_bot_iff : (\u22a5 : SimpleGraph \u03b1).IsNClique n s \u2194 n \u2264 1 \u2227 s.card = n", "@[simp] lemma isNClique_empty : G.IsNClique n \u2205 \u2194 n = 0", "theorem isNClique_map_iff [DecidableEq \u03b2] (hn : 1 < n) {t : Finset \u03b2} {f : \u03b1 \u21aa \u03b2} :\n    (G.map f).IsNClique n t \u2194 \u2203 s : Finset \u03b1, G.IsNClique n s \u2227 s.map f = t", "theorem isTree_iff_existsUnique_path :\n    G.IsTree \u2194 Nonempty V \u2227 \u2200 v w : V, \u2203! p : G.Walk v w, p.IsPath", "theorem isTuranMaximal_of_iso (f : G \u2243g turanGraph n r) (hr : 0 < r) : G.IsTuranMaximal r", "theorem isUniform_comm : IsUniform G \u03b5 s t \u2194 IsUniform G \u03b5 t s", "@[simp] lemma isUniform_singleton : G.IsUniform \u03b5 {a} {b} \u2194 0 < \u03b5", "@[simp] lemma map_map (f : V \u21aa W) (g : W \u21aa X) : (G.map f).map g = G.map (f.trans g)", "theorem mem_commonNeighbors {u v w : V} : u \u2208 G.commonNeighbors v w \u2194 G.Adj v u \u2227 G.Adj w u", "theorem nonuniformWitnesses_spec (h : \u00acG.IsUniform \u03b5 s t) :\n    \u03b5 \u2264\n      |G.edgeDensity (G.nonuniformWitnesses \u03b5 s t).1 (G.nonuniformWitnesses \u03b5 s t).2 -\n          G.edgeDensity s t|", "theorem not_cliqueFree_card_of_top_embedding [Fintype \u03b1] (f : (\u22a4 : SimpleGraph \u03b1) \u21aag G) :\n    \u00acG.CliqueFree (card \u03b1)", "theorem not_mem_commonNeighbors_left (v w : V) : v \u2209 G.commonNeighbors v w", "theorem sdiff_compl_neighborFinset_inter_eq {v w : V} (h : G.Adj v w) :\n    ((G.neighborFinset v)\u1d9c \u2229 (G.neighborFinset w)\u1d9c) \\ ({w} \u222a {v}) =\n      (G.neighborFinset v)\u1d9c \u2229 (G.neighborFinset w)\u1d9c", "lemma triangleRemovalBound_mul_cube_lt (h\u03b5 : 0 < \u03b5) :\n    triangleRemovalBound \u03b5 * \u23084 / \u03b5\u2309\u208a ^ 3 < 1", "private lemma triangle_removal_aux (h\u03b5 : 0 < \u03b5) (hP\u2081 : P.IsEquipartition)\n    (hP\u2083 : P.parts.card \u2264 bound (\u03b5 / 8) \u23084/\u03b5\u2309\u208a)\n    (ht : t \u2208 (G.regularityReduced P (\u03b5/8) (\u03b5/4)).cliqueFinset 3) :\n    triangleRemovalBound \u03b5 * card \u03b1 ^ 3 \u2264 (G.cliqueFinset 3).card", "theorem turanGraph_cliqueFree : (turanGraph n r).CliqueFree (r + 1)", "@[reassoc (attr := simp 1100)]\ntheorem cofan_inj_naturality_symm {S\u2081 S\u2082 : Split C} (\u03a6 : S\u2081 \u27f6 S\u2082) {\u0394 : SimplexCategory\u1d52\u1d56}\n    (A : Splitting.IndexSet \u0394) :\n    (S\u2081.s.cofan \u0394).inj A \u226b \u03a6.F.app \u0394 = \u03a6.f A.1.unop.len \u226b (S\u2082.s.cofan \u0394).inj A", "theorem congr_f {S\u2081 S\u2082 : Split C} {\u03a6\u2081 \u03a6\u2082 : S\u2081 \u27f6 S\u2082} (h : \u03a6\u2081 = \u03a6\u2082) (n : \u2115) : \u03a6\u2081.f n = \u03a6\u2082.f n", "@[reassoc (attr := simp)]\ntheorem cofan_inj_comp_app (f : X \u27f6 Y) {\u0394 : SimplexCategory\u1d52\u1d56} (A : IndexSet \u0394) :\n    (s.cofan \u0394).inj A \u226b f.app \u0394 = s.\u03c6 f A.1.unop.len \u226b Y.map A.e.op", "@[simp]\ntheorem add_apply (f g : SlashInvariantForm \u0393 k) (z : \u210d) : (f + g) z = f z + g z", "lemma chooseNatLt_aux {lo hi : Nat} (a : Nat) (h : Nat.succ lo \u2264 a \u2227 a \u2264 hi) :\n    lo \u2264 Nat.pred a \u2227 Nat.pred a < hi", "protected nonrec theorem SmoothAt.coordChangeL\n    (hf : SmoothAt IM IB f x) (he : f x \u2208 e.baseSet) (he' : f x \u2208 e'.baseSet) :\n    SmoothAt IM \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F) (fun y \u21a6 (e.coordChangeL \ud835\udd5c e' (f y) : F \u2192L[\ud835\udd5c] F)) x", "@[to_additive]\nnonrec theorem SmoothAt.div {f g : M \u2192 G} {x\u2080 : M} (hf : SmoothAt I' I f x\u2080)\n    (hg : SmoothAt I' I g x\u2080) : SmoothAt I' I (fun x => f x / g x) x\u2080", "theorem SmoothAt.snd {f : N \u2192 M \u00d7 M'} {x : N} (hf : SmoothAt J (I.prod I') f x) :\n    SmoothAt J I' (fun x => (f x).2) x", "@[simp, nolint simpNF]\ntheorem isSubordinate_toBumpCovering {f : SmoothBumpCovering \u03b9 I M s} {U : M \u2192 Set M} :\n    (f.toBumpCovering.IsSubordinate fun i => U (f.c i)) \u2194 f.IsSubordinate U", "theorem eqOn_source : EqOn f (f.toContDiffBump \u2218 extChartAt I c) (chartAt H c).source", "theorem eventuallyEq_one_of_dist_lt (hs : x \u2208 (chartAt H c).source)\n    (hd : dist (extChartAt I c x) (extChartAt I c c) < f.rIn) : f =\u1da0[\ud835\udcdd x] 1", "theorem isOpen_support : IsOpen (support f)", "theorem mem_Icc : f x \u2208 Icc (0 : \u211d) 1", "@[simp]\ntheorem coe_smul {V : Type*} [NormedAddCommGroup V] [NormedSpace \ud835\udd5c V] (r : \ud835\udd5c)\n    (f : C^\u221e\u27eeI, N; \ud835\udcd8(\ud835\udd5c, V), V\u27ef) : \u21d1(r \u2022 f) = r \u2022 \u21d1f", "theorem SmoothOn.inv\u2080 (hf : SmoothOn I' I f s) (h0 : \u2200 x \u2208 s, f x \u2260 0) :\n    SmoothOn I' I (fun x => (f x)\u207b\u00b9) s", "theorem sum_finsupport_smul_eq_finsum {A : Type*} [AddCommGroup A] [Module \u211d A] (\u03c6 : \u03b9 \u2192 M \u2192 A) :\n    \u2211 i \u2208 \u03c1.finsupport x\u2080, \u03c1 i x\u2080 \u2022 \u03c6 i x\u2080 = \u2211\u1da0 i, \u03c1 i x\u2080 \u2022 \u03c6 i x\u2080", "nonrec theorem SmoothWithinAt.comp' {t : Set M'} {g : M' \u2192 M''} (x : M)\n    (hg : SmoothWithinAt I' I'' g t (f x)) (hf : SmoothWithinAt I I' f s x) :\n    SmoothWithinAt I I'' (g \u2218 f) (s \u2229 f \u207b\u00b9' t) x", "@[simp] lemma map_id : map (ContinuousMap.id \u03b1) = OrderHom.id := rfl", "@[simp] lemma ofEquiv_symm : (@ofEquiv \u03b1).symm = toEquiv := rfl", "@[to_additive]\ntheorem Specializes.const_smul {x y : \u03b1} (h : x \u2933 y) (c : M) : (c \u2022 x) \u2933 (c \u2022 y)", "theorem Specializes.fst {a b : X \u00d7 Y} (h : a \u2933 b) : a.1 \u2933 b.1", "theorem Specializes.joinedIn (h : x \u2933 y) (hx : x \u2208 F) (hy : y \u2208 F) : JoinedIn F x y", "theorem copy_eq (f : SpectralMap \u03b1 \u03b2) (f' : \u03b1 \u2192 \u03b2) (h : f' = f) : f.copy f' h = f", "@[ext]\ntheorem ext {f g : SpectralMap \u03b1 \u03b2} (h : \u2200 a, f a = g a) : f = g", "theorem algebraMap_image : algebraMap R S '' spectrum R a = spectrum S a", "theorem image : f '' spectrum S a = spectrum R a", "lemma spectralRadius_eq {\ud835\udd5c\u2081 \ud835\udd5c\u2082 A : Type*} [NormedField \ud835\udd5c\u2081] [NormedField \ud835\udd5c\u2082]\n    [NormedRing A] [NormedAlgebra \ud835\udd5c\u2081 A] [NormedAlgebra \ud835\udd5c\u2082 A] [NormedAlgebra \ud835\udd5c\u2081 \ud835\udd5c\u2082]\n    [IsScalarTower \ud835\udd5c\u2081 \ud835\udd5c\u2082 A] {f : \ud835\udd5c\u2082 \u2192 \ud835\udd5c\u2081} {a : A} (h : SpectrumRestricts a f) :\n    spectralRadius \ud835\udd5c\u2081 a = spectralRadius \ud835\udd5c\u2082 a", "theorem pow_dvd_of_squarefree_of_pow_succ_dvd_mul_left {k : \u2115}\n    (hy : Squarefree y) (hp : Prime p) (h : p ^ (k + 1) \u2223 x * y) :\n    p ^ k \u2223 x", "theorem fromStalk_to_skyscraper {\ud835\udcd5 : Presheaf C X} {c : C} (f : \ud835\udcd5.stalk p\u2080 \u27f6 c) :\n    fromStalk p\u2080 (toSkyscraperPresheaf _ f) = f", "@[simp]\ntheorem comp_apply (f : B \u2192\u22c6\u2090[R] C) (g : A \u2192\u22c6\u2090[R] B) (a : A) : comp f g a = f (g a)", "@[simp high]\nlemma realContinuousMapOfNNReal_apply_comp_toReal (\u03c6 : C(X, \u211d\u22650) \u2192\u22c6\u2090[\u211d\u22650] A)\n    (f : C(X, \u211d\u22650)) :\n    \u03c6.realContinuousMapOfNNReal ((ContinuousMap.mk toReal continuous_coe).comp f) = \u03c6 f", "@[simp]\ntheorem snd_prod (f : A \u2192\u22c6\u2090[R] B) (g : A \u2192\u22c6\u2090[R] C) : (snd R B C).comp (prod f g) = g", "@[simp]\ntheorem subtype_comp_codRestrict (f : A \u2192\u22c6\u2090[R] B) (S : StarSubalgebra R B)\n    (hf : \u2200 x : A, f x \u2208 S) : S.subtype.comp (f.codRestrict S hf) = f", "theorem self_mem_adjoin_singleton (x : A) : x \u2208 adjoin R ({x} : Set A)", "theorem StarConvex.add_left (hs : StarConvex \ud835\udd5c x s) (z : E) :\n    StarConvex \ud835\udd5c (z + x) ((fun x => z + x) '' s)", "theorem StarConvex.prod {y : F} {s : Set E} {t : Set F} (hs : StarConvex \ud835\udd5c x s)\n    (ht : StarConvex \ud835\udd5c y t) : StarConvex \ud835\udd5c (x, y) (s \u00d7\u02e2 t)", "private lemma mul_le_mul_of_nonneg_left {R : Type*} [CommSemiring R] [PartialOrder R]\n    [StarRing R] [StarOrderedRing R] {a b c : R} (hab : a \u2264 b) (hc : 0 \u2264 c) : c * a \u2264 c * b", "theorem nonneg_iff [NonUnitalSemiring R] [PartialOrder R] [StarRing R] [StarOrderedRing R] {x : R} :\n    0 \u2264 x \u2194 x \u2208 AddSubmonoid.closure (Set.range fun s : R => star s * s)", "theorem isClosed_topologicalClosure (s : StarSubalgebra R A) :\n    IsClosed (s.topologicalClosure : Set A)", "theorem mem_sup_right {S T : StarSubalgebra R A} : \u2200 {x : A}, x \u2208 T \u2192 x \u2208 S \u2294 T", "protected theorem smul_mem {x : A} (hx : x \u2208 S) (r : R) : r \u2022 x \u2208 S", "private theorem numElems_le_iff {start stop step i} (hstep : 0 < step) :\n    (stop - start + step - 1) / step \u2264 i \u2194 stop \u2264 start + step * i", "@[simp]\ntheorem id_leftLim (x : \u211d) : leftLim StieltjesFunction.id x = x", "theorem measure_Iic {l : \u211d} (hf : Tendsto f atBot (\ud835\udcdd l)) (x : \u211d) :\n    f.measure (Iic x) = ofReal (f x - l)", "@[simp]\ntheorem measure_Ioo {a b : \u211d} : f.measure (Ioo a b) = ofReal (leftLim f b - f a)", "lemma factorial_isEquivalent_stirling :\n    (fun n \u21a6 n ! : \u2115 \u2192 \u211d) ~[atTop] fun n \u21a6 Real.sqrt (2 * n * \u03c0) * (n / exp 1) ^ n", "theorem dropn_tail (s : Seq \u03b1) (n) : drop (tail s) n = drop s (n + 1)", "theorem eq_of_bisim (bisim : IsBisimulation R) {s\u2081 s\u2082} (r : s\u2081 ~ s\u2082) : s\u2081 = s\u2082", "@[simp]\ntheorem get?_cons_succ (a : \u03b1) (s : Seq \u03b1) (n : \u2115) : (cons a s).get? (n + 1) = s.get? n", "theorem join_cons_nil (a : \u03b1) (S) : join (cons (a, nil) S) = cons a (join S)", "@[simp]\ntheorem map_tail (f : \u03b1 \u2192 \u03b2) : \u2200 s, map f (tail s) = tail (map f s)", "@[simp]\ntheorem tail_nil : tail (nil : Seq \u03b1) = nil", "@[refl]\ntheorem Equiv.refl : \u2200 s : WSeq \u03b1, s ~\u02b7 s", "theorem LiftRel.refl (R : \u03b1 \u2192 \u03b1 \u2192 Prop) (H : Reflexive R) : Reflexive (LiftRel R)", "theorem cons_congr {s t : WSeq \u03b1} (a : \u03b1) (h : s ~\u02b7 t) : cons a s ~\u02b7 cons a t", "theorem destruct_map (f : \u03b1 \u2192 \u03b2) (s : WSeq \u03b1) :\n    destruct (map f s) = Computation.map (Option.map (Prod.map f (map f))) (destruct s)", "theorem exists_of_liftRel_right {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {s t} (H : LiftRel R s t) {b} (h : b \u2208 t) :\n    \u2203 a, a \u2208 s \u2227 R a b", "theorem get?_terminates_le {s : WSeq \u03b1} {m n} (h : m \u2264 n) :\n    Terminates (get? s n) \u2192 Terminates (get? s m)", "theorem head_some_of_head_tail_some {s : WSeq \u03b1} {a} (h : some a \u2208 head (tail s)) :\n    \u2203 a', some a' \u2208 head s", "@[simp]\ntheorem join_cons (s : WSeq \u03b1) (S) : join (cons s S) = think (append s (join S))", "theorem liftRel_dropn_destruct {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {s t} (H : LiftRel R s t) :\n    \u2200 n, Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct (drop s n)) (destruct (drop t n))", "@[simp]\ntheorem map_id (s : WSeq \u03b1) : map id s = s", "@[simp]\ntheorem map_nil (f : \u03b1 \u2192 \u03b2) : map f nil = nil", "theorem mem_of_mem_tail {s : WSeq \u03b1} {a} : a \u2208 tail s \u2192 a \u2208 s", "theorem not_mem_nil (a : \u03b1) : a \u2209 @nil \u03b1", "theorem productive_congr {s t : WSeq \u03b1} (h : s ~\u02b7 t) : Productive s \u2194 Productive t", "@[simp]\ntheorem tail_think (s : WSeq \u03b1) : tail (think s) = (tail s).think", "theorem think_congr {s t : WSeq \u03b1} (h : s ~\u02b7 t) : think s ~\u02b7 think t", "@[simp]\ntheorem toList_cons (a : \u03b1) (s) : toList (cons a s) = (List.cons a <$> toList s).think", "@[simp]\ntheorem toList_nil : toList (nil : WSeq \u03b1) = Computation.pure []", "@[simp]\ntheorem get_const (n : Nat) (a : \u03b1) : get (const a) n = a", "@[simp]\ntheorem get_map (n : Nat) (s : Stream' \u03b1) : get (map f s) n = f (get s n)", "theorem get_succ_iterate' (n : Nat) (f : \u03b1 \u2192 \u03b1) (a : \u03b1) :\n    get (iterate f a) (succ n) = f (get (iterate f a) n)", "@[simp]\ntheorem get_zip (n : Nat) (s\u2081 : Stream' \u03b1) (s\u2082 : Stream' \u03b2) :\n    get (zip f s\u2081 s\u2082) n = f (get s\u2081 n) (get s\u2082 n)", "theorem head_drop (a : Stream' \u03b1) (n : \u2115) : (a.drop n).head = a.get n", "theorem iterate_eq (f : \u03b1 \u2192 \u03b1) (a : \u03b1) : iterate f a = a::iterate f (f a)", "@[simp]\ntheorem mem_const (a : \u03b1) : a \u2208 const a", "theorem tails_eq_iterate (s : Stream' \u03b1) : tails s = iterate tail (tail s)", "theorem take_succ (n : Nat) (s : Stream' \u03b1) : take (succ n) s = head s::take n (tail s)", "theorem StrictAnti.injective (hf : StrictAnti f) : Injective f", "protected theorem StrictAnti.ite (hf : StrictAnti f) (hg : StrictAnti g) {p : \u03b1 \u2192 Prop}\n    [DecidablePred p] (hp : \u2200 \u2983x y\u2984, x < y \u2192 p y \u2192 p x) (hfg : \u2200 x, g x \u2264 f x) :\n    StrictAnti fun x \u21a6 if p x then f x else g x", "lemma StrictAntiOn.mapsTo_Iio (h : StrictAntiOn f (Iic b)) : MapsTo f (Iio b) (Ioi (f b))", "theorem StrictConcaveOn.add_const {\u03b3 : Type*} {f : E \u2192 \u03b3} [OrderedCancelAddCommMonoid \u03b3]\n    [Module \ud835\udd5c \u03b3] (hf : StrictConcaveOn \ud835\udd5c s f) (b : \u03b3) : StrictConcaveOn \ud835\udd5c s (f + fun _ => b)", "lemma StrictConcaveOn.eq_of_map_sum_eq (hf : StrictConcaveOn \ud835\udd5c s f) (h\u2080 : \u2200 i \u2208 t, 0 < w i)\n    (h\u2081 : \u2211 i \u2208 t, w i = 1) (hmem : \u2200 i \u2208 t, p i \u2208 s)\n    (h_eq : f (\u2211 i \u2208 t, w i \u2022 p i) \u2264 \u2211 i \u2208 t, w i \u2022 f (p i)) :\n    \u2200 \u2983j\u2984, j \u2208 t \u2192 \u2200 \u2983k\u2984, k \u2208 t \u2192 p j = p k", "lemma strictAntiOn_derivWithin (hfc : StrictConcaveOn \u211d S f) (hfd : DifferentiableOn \u211d f S) :\n    StrictAntiOn (derivWithin f S) S", "theorem StrictConvex.add (hs : StrictConvex \ud835\udd5c s) (ht : StrictConvex \ud835\udd5c t) :\n    StrictConvex \ud835\udd5c (s + t)", "theorem StrictConvex.add_smul_mem (hs : StrictConvex \ud835\udd5c s) (hx : x \u2208 s) (hxy : x + y \u2208 s)\n    (hy : y \u2260 0) {t : \ud835\udd5c} (ht\u2080 : 0 < t) (ht\u2081 : t < 1) : x + t \u2022 y \u2208 interior s", "theorem StrictConvexOn.add_const {\u03b3 : Type*} {f : E \u2192 \u03b3} [OrderedCancelAddCommMonoid \u03b3]\n    [Module \ud835\udd5c \u03b3] (hf : StrictConvexOn \ud835\udd5c s f) (b : \u03b3) : StrictConvexOn \ud835\udd5c s (f + fun _ => b)", "theorem StrictConvexOn.congr [SMul \ud835\udd5c \u03b2] (hf : StrictConvexOn \ud835\udd5c s f) (hfg : EqOn f g s) :\n    StrictConvexOn \ud835\udd5c s g", "theorem StrictConvexOn.secant_strict_mono (hf : StrictConvexOn \ud835\udd5c s f) {a x y : \ud835\udd5c} (ha : a \u2208 s)\n    (hx : x \u2208 s) (hy : y \u2208 s) (hxa : x \u2260 a) (hya : y \u2260 a) (hxy : x < y) :\n    (f x - f a) / (x - a) < (f y - f a) / (y - a)", "lemma slope_lt_of_hasDerivWithinAt_Iio (hfc : StrictConvexOn \u211d S f)\n    (hx : x \u2208 S) (hy : y \u2208 S) (hxy : x < y) (hf' : HasDerivWithinAt f f' (Iio y) y)  :\n    slope f x y < f'", "theorem StrictMono.embedding_of_ordConnected {\u03b1 \u03b2 : Type*} [LinearOrder \u03b1] [LinearOrder \u03b2]\n    [TopologicalSpace \u03b1] [h : OrderTopology \u03b1] [TopologicalSpace \u03b2] [OrderTopology \u03b2] {f : \u03b1 \u2192 \u03b2}\n    (hf : StrictMono f) (hc : OrdConnected (range f)) : Embedding f", "lemma StrictMono.image_Ico_subset (h : StrictMono f) : f '' Ico a b \u2286 Ico (f a) (f b)", "lemma StrictMono.image_Iio_subset (h : StrictMono f) : f '' Iio b \u2286 Iio (f b)", "nonrec theorem StrictMono.induced_topology_eq_preorder {\u03b1 \u03b2 : Type*} [LinearOrder \u03b1]\n    [LinearOrder \u03b2] [t : TopologicalSpace \u03b2] [OrderTopology \u03b2] {f : \u03b1 \u2192 \u03b2}\n    (hf : StrictMono f) (hc : OrdConnected (range f)) : t.induced f = Preorder.topology \u03b1", "lemma StrictMono.sign_comp {\u03b2 F : Type*} [Zero \u03b2] [Preorder \u03b2] [DecidableRel ((\u00b7 < \u00b7) : \u03b2 \u2192 \u03b2 \u2192 _)]\n    [FunLike F \u03b1 \u03b2] [ZeroHomClass F \u03b1 \u03b2] {f : F} (hf : StrictMono f) (a : \u03b1) :\n    sign (f a) = sign a", "theorem strictAnti_iterate_of_map_lt (hf : StrictMono f) (hx : f x < x) :\n    StrictAnti fun n => f^[n] x", "theorem StrictMonoOn.comp_strictAntiOn [Preorder \u03b1] [Preorder \u03b2] [Preorder \u03b3] {g : \u03b2 \u2192 \u03b3}\n    {f : \u03b1 \u2192 \u03b2} {s : Set \u03b1} {t : Set \u03b2} (hg : StrictMonoOn g t) (hf : StrictAntiOn f s)\n    (hs : Set.MapsTo f s t) : StrictAntiOn (g \u2218 f) s", "theorem _root_.StrictMonoOn.mono (h : StrictMonoOn f s) (h' : s\u2082 \u2286 s) : StrictMonoOn f s\u2082", "theorem next : \u2200 {it}, Valid it \u2192 it.hasNext \u2192 Valid it.next", "theorem prevn (h : Valid it) : \u2200 n, Valid (it.prevn n)", "theorem next : \u2200 {it}, ValidFor l (c :: r) it \u2192 ValidFor (c :: l) r it.next", "theorem out' : \u2200 {it}, ValidFor l r it \u2192 it = \u27e8\u27e8l.reverse ++ r\u27e9, \u27e8utf8Len l.reverse\u27e9\u27e9", "theorem setCurr' : \u2200 {it}, ValidFor l r it \u2192\n    ValidFor l (r.modifyHead fun _ => c) (it.setCurr c)", "@[simp] theorem forward_eq_nextn : forward = nextn", "theorem ext_iff {i\u2081 i\u2082 : Pos} : i\u2081 = i\u2082 \u2194 i\u2081.byteIdx = i\u2082.byteIdx", "private theorem zero_ne_addChar {i : Pos} {c : Char} : 0 \u2260 i + c", "@[simp] theorem data_drop (s : String) (n : Nat) : (s.drop n).1 = s.1.drop n", "theorem extract.go\u2081_append_right : \u2200 (s t : List Char) (i b : Nat) (e : Pos),\n    b = utf8Len s + i \u2192 go\u2081 (s ++ t) \u27e8i\u27e9 \u27e8b\u27e9 e = go\u2082 t \u27e8b\u27e9 e", "theorem extract_cons_addChar (c : Char) (cs : List Char) (b e : Pos) :\n    extract \u27e8c :: cs\u27e9 (b + c) (e + c) = extract \u27e8cs\u27e9 b e", "@[nolint unusedHavesSuffices] theorem foldlAux_of_valid (f : \u03b1 \u2192 Char \u2192 \u03b1) : \u2200 l m r a,\n    foldlAux f \u27e8l ++ m ++ r\u27e9 \u27e8utf8Len l + utf8Len m\u27e9 \u27e8utf8Len l\u27e9 a = m.foldl f a", "@[simp] lemma leftpad_length (n : \u2115) (c : Char) :\n    \u2200 (s : String), (leftpad n c s).length = max n s.length\n  | \u27e8s\u27e9 => by simp only [leftpad, String.length, List.leftpad_length]", "theorem offsetOfPos_of_valid (l r) : offsetOfPos \u27e8l ++ r\u27e9 \u27e8utf8Len l\u27e9 = l.length", "theorem utf8Len_le_of_infix (h : cs\u2081 <:+: cs\u2082) : utf8Len cs\u2081 \u2264 utf8Len cs\u2082", "theorem utf8PrevAux_of_valid {cs cs' : List Char} {c : Char} {i p : Nat}\n    (hp : i + (utf8Len cs + c.utf8Size) = p) :\n    utf8PrevAux (cs ++ c :: cs') \u27e8i\u27e9 \u27e8p\u27e9 = \u27e8i + utf8Len cs\u27e9", "theorem _root_.String.valid_toSubstring (s : String) : Valid s", "protected theorem StronglyMeasurableAtFilter.eventually (h : StronglyMeasurableAtFilter f l \u03bc) :\n    \u2200\u1da0 s in l.smallSets, AEStronglyMeasurable f (\u03bc.restrict s)", "theorem liftPropAt_congr_iff_of_eventuallyEq (h\u2081 : g' =\u1da0[\ud835\udcdd x] g) :\n    LiftPropAt P g' x \u2194 LiftPropAt P g x", "theorem liftPropWithinAt_indep_chart_source [HasGroupoid M G] (he : e \u2208 G.maximalAtlas M)\n    (xe : x \u2208 e.source) :\n    LiftPropWithinAt P g s x \u2194 LiftPropWithinAt P (g \u2218 e.symm) (e.symm \u207b\u00b9' s) (e x)", "theorem liftPropWithinAt_indep_chart_target [HasGroupoid M' G'] (hf : f \u2208 G'.maximalAtlas M')\n    (xf : g x \u2208 f.source) :\n    LiftPropWithinAt P g s x \u2194 ContinuousWithinAt g s x \u2227 LiftPropWithinAt P (f \u2218 g) s x", "theorem StructureGroupoid.compatible_of_mem_maximalAtlas {e e' : PartialHomeomorph M H}\n    (he : e \u2208 G.maximalAtlas M) (he' : e' \u2208 G.maximalAtlas M) : e.symm \u226b\u2095 e' \u2208 G", "lemma image_inclusion (s : SubMulAction M \u03b1) :\n    Set.range s.inclusion = s.carrier", "theorem apply_mul_add_le (k n r) : u (k * n + r) \u2264 k * u n + u r", "theorem tendsto_lim (hbdd : BddBelow (range fun n => u n / n)) :\n    Tendsto (fun n => u n / n) atTop (\ud835\udcdd h.lim)", "protected theorem add_mem {x y : A} (hx : x \u2208 S) (hy : y \u2208 S) : x + y \u2208 S", "@[simp]\ntheorem coe_pointwise_smul (m : R') (S : Subalgebra R A) : \u2191(m \u2022 S) = m \u2022 (S : Set A)", "theorem _root_.Subalgebra.fg_iff_finiteType (S : Subalgebra R A) : S.FG \u2194 Algebra.FiniteType R S", "theorem fg_of_fg_map (S : Subalgebra R A) (f : A \u2192\u2090[R] B) (hf : Function.Injective f)\n    (hs : (S.map f).FG) : S.FG", "theorem fg_top (S : Subalgebra R A) : (\u22a4 : Subalgebra R S).FG \u2194 S.FG", "@[deprecated Subalgebra.finite_sup (since := \"2024-04-11\")]\ntheorem Subalgebra.finiteDimensional_sup {K L : Type*} [Field K] [CommRing L] [Algebra K L]\n    (E1 E2 : Subalgebra K L) [FiniteDimensional K E1] [FiniteDimensional K E2] :\n    FiniteDimensional K (E1 \u2294 E2 : Subalgebra K L)", "theorem Subalgebra.finiteDimensional_toSubmodule {S : Subalgebra F E} :\n    FiniteDimensional F (Subalgebra.toSubmodule S) \u2194 FiniteDimensional F S", "@[simp]\ntheorem iSupLift_mk {i : \u03b9} (x : K i) (hx : (x : A) \u2208 T) :\n    iSupLift K dir f hf T hT \u27e8x, hx\u27e9 = f i x", "@[simp]\ntheorem inclusion_self {S : Subalgebra R A} : inclusion (le_refl S) = AlgHom.id R S", "theorem mem_center_iff {a : A} : a \u2208 center R A \u2194 \u2200 b : A, b * a = a * b", "@[simp]\ntheorem toSubsemiring_subtype : S.toSubsemiring.subtype = (S.val : S \u2192+* A)", "theorem _root_.Subalgebra.topologicalClosure_star_comm (s : Subalgebra R A) :\n    (star s).topologicalClosure = star s.topologicalClosure", "@[simp]\ntheorem unop_op (S : Subalgebra R A) : S.op.unop = S", "theorem unop_sSup (S : Set (Subalgebra R A\u1d50\u1d52\u1d56)) : (sSup S).unop = sSup (.op \u207b\u00b9' S)", "theorem smul_mem' (p : Submodule (A \u2297[R] B) M) (b : B) {m : M} (hm : m \u2208 p) : b \u2022 m \u2208 p", "theorem closure_preimage_le (f : K \u2192+* L) (s : Set L) : closure (f \u207b\u00b9' s) \u2264 (closure s).comap f", "@[simp, norm_cast]\ntheorem coe_div (x y : s) : (\u2191(x / y) : K) = \u2191x / \u2191y", "@[simp]\ntheorem coe_map : (s.map f : Set L) = f '' s", "@[simp]\ntheorem coe_toSubring (s : Subfield K) : (s.toSubring : Set K) = s", "@[simp]\ntheorem coe_top : ((\u22a4 : Subfield K) : Set K) = Set.univ", "theorem comap_comap (s : Subfield M) (g : L \u2192+* M) (f : K \u2192+* L) :\n    (s.comap g).comap f = s.comap (g.comp f)", "@[simp]\ntheorem sInf_toSubring (s : Set (Subfield K)) :\n    (sInf s).toSubring = \u2a05 t \u2208 s, Subfield.toSubring t", "protected theorem sub_mem {x y : K} : x \u2208 s \u2192 y \u2208 s \u2192 x - y \u2208 s", "@[aesop safe apply (rule_sets := [SetLike])]\nlemma ofScientific_mem (s : S) {b : Bool} {n m : \u2115} :\n    (OfScientific.ofScientific n b m : K) \u2208 s", "theorem IsComplement'.index_eq_card (h : IsComplement' H K) : K.index = Nat.card H", "@[to_additive]\ntheorem IsComplement'.symm (h : IsComplement' H K) : IsComplement' K H", "theorem equiv_fst_eq_mul_inv (g : G) : \u2191(hST.equiv g).fst = g * ((hST.equiv g).snd : G)\u207b\u00b9", "@[to_additive]\ntheorem IsComplement.existsUnique (h : IsComplement S T) (g : G) :\n    \u2203! x : S \u00d7 T, x.1.1 * x.2.1 = g", "theorem rightCosetEquivalence_equiv_snd (g : G) :\n    RightCosetEquivalence H g ((hHT.equiv g).snd : G)", "@[to_additive]\ntheorem inv_mul_toFun_mem (hS : S \u2208 Subgroup.leftTransversals (H : Set G)) (g : G) :\n    g\u207b\u00b9 * toFun hS g \u2208 H", "@[to_additive]\ntheorem Normal.map {H : Subgroup G} (h : H.Normal) (f : G \u2192* N) (hf : Function.Surjective f) :\n    (H.map f).Normal", "private theorem step1 (K : Subgroup G) (hK : K \u2294 N = \u22a4) : K = \u22a4", "private theorem step2 (K : Subgroup G) [K.Normal] (hK : K \u2264 N) : K = \u22a5 \u2228 K = N", "@[to_additive (attr := simp)]\ntheorem bot_subgroupOf : (\u22a5 : Subgroup G).subgroupOf H = \u22a5", "@[to_additive]\ntheorem centralizer_le {s t : Set G} (h : s \u2286 t) : centralizer t \u2264 centralizer s", "@[to_additive (attr := simp)]\ntheorem closure_eq_bot_iff : closure k = \u22a5 \u2194 k \u2286 {1}", "@[to_additive (attr := elab_as_elim)\n  \"An induction principle for additive closure membership. If `p` holds for `0` and all\n  elements of `k` and their negation, and is preserved under addition, then `p` holds for all\n  elements of the additive closure of `k`.\"]\ntheorem closure_induction'' {p : (g : G) \u2192 g \u2208 closure s \u2192 Prop}\n    (mem : \u2200 x (hx : x \u2208 s), p x (subset_closure hx))\n    (inv_mem : \u2200 x (hx : x \u2208 s), p x\u207b\u00b9 (inv_mem (subset_closure hx)))\n    (one : p 1 (one_mem _))\n    (mul : \u2200 x y hx hy, p x hx \u2192 p y hy \u2192 p (x * y) (mul_mem hx hy))\n    {x} (h : x \u2208 closure s) : p x h", "@[to_additive (attr := simp)]\ntheorem coeSubtype : \u21d1 H.subtype = ((\u2191) : H \u2192 G)", "@[to_additive (attr := simp, norm_cast)]\ntheorem coe_iInf {\u03b9 : Sort*} {S : \u03b9 \u2192 Subgroup G} : (\u2191(\u2a05 i, S i) : Set G) = \u22c2 i, S i", "@[to_additive (attr := simp)]\ntheorem coe_inclusion {H K : Subgroup G} {h : H \u2264 K} (a : H) : (inclusion h a : G) = a", "@[to_additive (attr := simp) \"If `x` is an element of a subgroup `s` of a seminormed group `E`, its\nnorm in `s` is equal to its norm in `E`.\"]\ntheorem coe_norm (x : s) : \u2016x\u2016 = \u2016(x : E)\u2016", "@[to_additive (attr := simp, norm_cast)]\ntheorem coe_one : ((1 : H) : G) = 1", "@[to_additive]\ntheorem comap_equiv_eq_map_symm (f : N \u2243* G) (K : Subgroup G) :\n    K.comap (G := N) f = K.map f.symm", "theorem commutator_commutator_eq_bot_of_rotate (h1 : \u2045\u2045H\u2082, H\u2083\u2046, H\u2081\u2046 = \u22a5) (h2 : \u2045\u2045H\u2083, H\u2081\u2046, H\u2082\u2046 = \u22a5) :\n    \u2045\u2045H\u2081, H\u2082\u2046, H\u2083\u2046 = \u22a5", "@[to_additive \"Elements of disjoint, normal subgroups commute.\"]\ntheorem commute_of_normal_of_disjoint (H\u2081 H\u2082 : Subgroup G) (hH\u2081 : H\u2081.Normal) (hH\u2082 : H\u2082.Normal)\n    (hdis : Disjoint H\u2081 H\u2082) (x y : G) (hx : x \u2208 H\u2081) (hy : y \u2208 H\u2082) : Commute x y", "@[to_additive \"An `AddSubgroup` is closed under subtraction.\"]\nprotected theorem div_mem {x y : G} (hx : x \u2208 H) (hy : y \u2208 H) : x / y \u2208 H", "theorem eq_one_of_smul_eq_one (hH : Nat.Coprime (Nat.card H) H.index) (\u03b1 : H.QuotientDiff)\n    (h : H) : h \u2022 \u03b1 = \u03b1 \u2192 h = 1", "@[to_additive]\nlemma exists_right_transversal (H : Subgroup G) (g : G) :\n    \u2203 S \u2208 rightTransversals (H : Set G), g \u2208 S", "@[to_additive (attr := elab_as_elim) \" An induction principle for elements of `\u2a06 i, S i`.\nIf `C` holds for `0` and all elements of `S i` for all `i`, and is preserved under addition,\nthen it holds for all elements of the supremum of `S`. \"]\ntheorem iSup_induction {\u03b9 : Sort*} (S : \u03b9 \u2192 Subgroup G) {C : G \u2192 Prop} {x : G} (hx : x \u2208 \u2a06 i, S i)\n    (mem : \u2200 (i), \u2200 x \u2208 S i, C x) (one : C 1) (mul : \u2200 x y, C x \u2192 C y \u2192 C (x * y)) : C x", "@[to_additive]\ntheorem independent_of_coprime_order [Finite \u03b9] [\u2200 i, Fintype (H i)]\n    (hcoprime : Pairwise fun i j => Nat.Coprime (Fintype.card (H i)) (Fintype.card (H j))) :\n    CompleteLattice.Independent H", "@[to_additive]\ntheorem index_comap {G' : Type*} [Group G'] (f : G' \u2192* G) :\n    (H.comap f).index = H.relindex f.range", "@[to_additive]\ntheorem index_iInf_le {\u03b9 : Type*} [Fintype \u03b9] (f : \u03b9 \u2192 Subgroup G) :\n    (\u2a05 i, f i).index \u2264 \u220f i, (f i).index", "@[to_additive]\ntheorem inf_subgroupOf_inf_normal_of_right (A B' B : Subgroup G) (hB : B' \u2264 B)\n    [hN : (B'.subgroupOf B).Normal] : ((A \u2293 B').subgroupOf (A \u2293 B)).Normal", "@[to_additive (attr := simp)]\ntheorem inv_subset_closure (S : Set G) : S\u207b\u00b9 \u2286 closure S", "lemma isCoatom_comap_of_surjective\n    {H : Type*} [Group H] {\u03c6 : G \u2192* H} (h\u03c6 : Function.Surjective \u03c6)\n    {M : Subgroup H} (hM : IsCoatom M) : IsCoatom (M.comap \u03c6)", "@[to_additive]\ntheorem isOpen_of_openSubgroup {U : OpenSubgroup G} (h : \u2191U \u2264 H) : IsOpen (H : Set G)", "@[to_additive \"Sum of a list of elements in an `AddSubgroup` is in the `AddSubgroup`.\"]\nprotected theorem list_prod_mem {l : List G} : (\u2200 x \u2208 l, x \u2208 K) \u2192 l.prod \u2208 K", "@[to_additive]\ntheorem map_inf_eq (H K : Subgroup G) (f : G \u2192* N) (hf : Function.Injective f) :\n    map f (H \u2293 K) = map f H \u2293 map f K", "@[to_additive]\ntheorem map_injective {f : G \u2192* N} (h : Function.Injective f) : Function.Injective (map f)", "@[to_additive (attr := simp, nolint simpNF)] theorem mem_carrier {s : Subgroup G} {x : G} : x \u2208 s.carrier \u2194 x \u2208 s", "@[to_additive]\ntheorem mem_leftTransversals_iff_existsUnique_quotient_mk''_eq :\n    S \u2208 leftTransversals (H : Set G) \u2194\n      \u2200 q : Quotient (QuotientGroup.leftRel H), \u2203! s : S, Quotient.mk'' s.1 = q", "@[to_additive]\ntheorem mem_sSup_of_mem {S : Set (Subgroup G)} {s : Subgroup G} (hs : s \u2208 S) :\n    \u2200 {x : G}, x \u2208 s \u2192 x \u2208 sSup S", "@[to_additive]\ntheorem mem_subgroupOf {H K : Subgroup G} {h : K} : h \u2208 H.subgroupOf K \u2194 (h : G) \u2208 H", "@[to_additive]\nlemma mem_units_iff_val_mem (H : Subgroup G) (x : G\u02e3): x \u2208 H.units \u2194 (x : G) \u2208 H", "@[to_additive]\ntheorem mul_injective_of_disjoint {H\u2081 H\u2082 : Subgroup G} (h : Disjoint H\u2081 H\u2082) :\n    Function.Injective (fun g => g.1 * g.2 : H\u2081 \u00d7 H\u2082 \u2192 G)", "@[to_additive]\nlemma ne_bot_iff_exists_ne_one {H : Subgroup G} : H \u2260 \u22a5 \u2194 \u2203 a : \u21a5H, a \u2260 1", "@[to_additive]\ntheorem nontrivial_iff_exists_ne_one (H : Subgroup G) : Nontrivial H \u2194 \u2203 x \u2208 H, x \u2260 (1 : G)", "@[to_additive]\ntheorem nontrivial_iff_ne_bot (H : Subgroup G) : Nontrivial H \u2194 H \u2260 \u22a5", "@[to_additive \"For finite index types, the `Subgroup.pi` is generated by the embeddings of the\n additive groups.\"]\ntheorem pi_le_iff [DecidableEq \u03b7] [Finite \u03b7] {H : \u2200 i, Subgroup (f i)} {J : Subgroup (\u2200 i, f i)} :\n    pi univ H \u2264 J \u2194 \u2200 i : \u03b7, map (MonoidHom.mulSingle f i) (H i) \u2264 J", "@[to_additive prod_le_iff]\ntheorem prod_le_iff {H : Subgroup G} {K : Subgroup N} {J : Subgroup (G \u00d7 N)} :\n    H.prod K \u2264 J \u2194 map (MonoidHom.inl G N) H \u2264 J \u2227 map (MonoidHom.inr G N) K \u2264 J", "@[to_additive\n  \"A subgroup `S` of an additive topological group `G` acts on `G` properly discontinuously\n  on the right, if it is discrete in the sense that `S \u2229 K` is finite for all compact `K`.\n  (See also `DiscreteTopology`.)\n\n  If `G` is Hausdorff, this can be combined with `t2Space_of_properlyDiscontinuousVAdd_of_t2Space`\n  to show that the quotient group `G \u29f8 S` is Hausdorff.\"]\ntheorem Subgroup.properlyDiscontinuousSMul_opposite_of_tendsto_cofinite (S : Subgroup G)\n    (hS : Tendsto S.subtype cofinite (cocompact G)) : ProperlyDiscontinuousSMul S.op G", "@[to_additive]\ntheorem relindex_bot_left_eq_card [Fintype H] : (\u22a5 : Subgroup G).relindex H = Fintype.card H", "@[to_additive]\ntheorem relindex_iInf_le {\u03b9 : Type*} [Fintype \u03b9] (f : \u03b9 \u2192 Subgroup G) :\n    (\u2a05 i, f i).relindex L \u2264 \u220f i, (f i).relindex L", "@[simp]\ntheorem smul_bot (a : \u03b1) : a \u2022 (\u22a5 : Subgroup G) = \u22a5", "@[simp]\ntheorem smul_mem_pointwise_smul_iff\u2080 {a : \u03b1} (ha : a \u2260 0) (S : Subgroup G) (x : G) :\n    a \u2022 x \u2208 a \u2022 S \u2194 x \u2208 S", "@[to_additive]\ntheorem Subgroup.topologicalClosure_minimal (s : Subgroup G) {t : Subgroup G} (h : s \u2264 t)\n    (ht : IsClosed (t : Set G)) : s.topologicalClosure \u2264 t", "@[to_additive]\nlemma unit_mem_of_mem_ofUnits (S : Subgroup M\u02e3) {x : M} {h\u2081 : IsUnit x}\n    (h\u2082 : x \u2208 S.ofUnits) : h\u2081.unit \u2208 S", "@[to_additive (attr := simp)] lemma unop_toSubmonoid (H : Subgroup G\u1d50\u1d52\u1d56) :\n    H.unop.toSubmonoid = H.toSubmonoid.unop", "@[to_additive (attr := simp 1100, norm_cast)]\ntheorem val_multiset_prod {G} [CommGroup G] (H : Subgroup G) (m : Multiset H) :\n    (m.prod : G) = (m.map Subtype.val).prod", "@[simp, norm_cast] lemma coe_eq_empty : L = (\u2205 : Set \u03b1) \u2194 L = \u22a5", "@[simp, norm_cast] lemma coe_inf' (L M : Sublattice \u03b1) : L \u2293 M = (L : Set \u03b1) \u2229 M := rfl", "@[simp, norm_cast] lemma coe_subtype (L : Sublattice \u03b1) : L.subtype = ((\u2191) : L \u2192 \u03b1) := rfl", "@[simp, norm_cast] lemma coe_top : (\u22a4 : Sublattice \u03b1) = (univ : Set \u03b1) := rfl", "@[simp] lemma comap_id (L : Sublattice \u03b1) : L.comap (LatticeHom.id _) = L := rfl", "lemma comap_inf (L M : Sublattice \u03b2) (f : LatticeHom \u03b1 \u03b2) :\n    (L \u2293 M).comap f = L.comap f \u2293 M.comap f := (gc_map_comap f).u_inf", "@[simp] lemma mem_sInf {S : Set (Sublattice \u03b1)} : a \u2208 sInf S \u2194 \u2200 L \u2208 S, a \u2208 L", "lemma subsingleton_iff : Subsingleton (Sublattice \u03b1) \u2194 IsEmpty \u03b1", "theorem FG.stabilizes_of_iSup_eq {M' : Submodule R M} (hM' : M'.FG) (N : \u2115 \u2192o Submodule R M)\n    (H : iSup N = M') : \u2203 n, M' = N n", "theorem IsOrtho.map (f : E \u2192\u2097\u1d62[\ud835\udd5c] F) {U V : Submodule \ud835\udd5c E} (h : U \u27c2 V) : U.map f \u27c2 V.map f", "@[simp]\ntheorem closedComplemented_bot : ClosedComplemented (\u22a5 : Submodule R M)", "@[elab_as_elim]\ntheorem closure_induction {p : M \u2192 Prop} (h : x \u2208 span R s) (zero : p 0)\n    (add : \u2200 x y, p x \u2192 p y \u2192 p (x + y)) (smul_mem : \u2200 r : R, \u2200 x \u2208 s, p (r \u2022 x)) : p x", "@[simp]\ntheorem coe_copy (S : Submodule R M) (s : Set M) (hs : s = \u2191S) : (S.copy s hs : Set M) = s", "protected theorem coe_neg (x : p) : ((-x : p) : M) = -x", "@[simp]\ntheorem _root_.Submodule.coe_subtypeL' (p : Submodule R\u2081 M\u2081) : \u21d1p.subtypeL = p.subtype", "@[simp]\ntheorem coe_toSubMulAction (p : Submodule R M) : (p.toSubMulAction : Set M) = p", "@[simp]\ntheorem colon_top {I : Ideal R} : I.colon \u22a4 = I", "theorem comap_equiv_eq_map_symm (e : M \u2243\u209b\u2097[\u03c4\u2081\u2082] M\u2082) (K : Submodule R\u2082 M\u2082) :\n    K.comap (e : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082) = K.map (e.symm : M\u2082 \u2192\u209b\u2097[\u03c4\u2082\u2081] M)", "@[simp]\ntheorem comap_fst : p.comap (fst R M M\u2082) = prod p \u22a4", "theorem comap_surjective_of_injective : Function.Surjective (comap f)", "theorem disjoint_def' {p p' : Submodule R M} :\n    Disjoint p p' \u2194 \u2200 x \u2208 p, \u2200 y \u2208 p', x = y \u2192 x = (0 : M)", "theorem disjoint_span_singleton' {K E : Type*} [DivisionRing K] [AddCommGroup E] [Module K E]\n    {p : Submodule K E} {x : E} (x0 : x \u2260 0) : Disjoint p (K \u2219 x) \u2194 x \u2209 p", "@[simp]\ntheorem dualCopairing_apply {W : Submodule R M} (\u03c6 : W.dualAnnihilator) (x : M) :\n    W.dualCopairing \u03c6 (Quotient.mk x) = \u03c6 x", "theorem dualRestrict_def (W : Submodule R M) : W.dualRestrict = W.subtype.dualMap", "theorem Submodule.exists_le_ker_of_lt_top (p : Submodule K V) (hp : p < \u22a4) :\n    \u2203 (f : V \u2192\u2097[K] K), f \u2260 0 \u2227 p \u2264 ker f", "theorem fg_iff_finiteDimensional (s : Submodule K V) : s.FG \u2194 FiniteDimensional K s", "theorem fg_of_fg_map_of_fg_inf_ker {R M P : Type*} [Ring R] [AddCommGroup M] [Module R M]\n    [AddCommGroup P] [Module R P] (f : M \u2192\u2097[R] P) {s : Submodule R M}\n    (hs1 : (s.map f).FG)\n    (hs2 : (s \u2293 LinearMap.ker f).FG) : s.FG", "theorem finite_span_isCompactElement (S : Set M) (h : S.Finite) :\n    CompleteLattice.IsCompactElement (span R S : Submodule R M)", "theorem Submodule.finrank_add_finrank_orthogonal [FiniteDimensional \ud835\udd5c E] (K : Submodule \ud835\udd5c E) :\n    finrank \ud835\udd5c K + finrank \ud835\udd5c K\u15ee = finrank \ud835\udd5c E", "theorem inclusion_apply (h : p \u2264 p') (x : p) : inclusion h x = \u27e8x, h x.2\u27e9", "theorem inf_iSup_genEigenspace [FiniteDimensional K V] (h : \u2200 x \u2208 p, f x \u2208 p) :\n    p \u2293 \u2a06 \u03bc, \u2a06 k, f.genEigenspace \u03bc k = \u2a06 \u03bc, \u2a06 k, p \u2293 f.genEigenspace \u03bc k", "theorem Submodule.isClosed_or_dense_of_isCoatom (s : Submodule R M) (hs : IsCoatom s) :\n    IsClosed (s : Set M) \u2228 Dense (s : Set M)", "theorem isOrtho_iff_inner_eq {U V : Submodule \ud835\udd5c E} : U \u27c2 V \u2194 \u2200 u \u2208 U, \u2200 v \u2208 V, \u27eau, v\u27eb = 0", "@[simp]\ntheorem isOrtho_sup_right {U V\u2081 V\u2082 : Submodule \ud835\udd5c E} : U \u27c2 V\u2081 \u2294 V\u2082 \u2194 U \u27c2 V\u2081 \u2227 U \u27c2 V\u2082", "theorem isTorsion'_iff_torsion'_eq_top : IsTorsion' M S \u2194 torsion' R M S = \u22a4", "@[simp]\ntheorem lTensorOne_one_tmul (n : N) : N.lTensorOne (1 \u2297\u209c[R] n) = n", "theorem le_comap_mkQ (p' : Submodule R (M \u29f8 p)) : p \u2264 comap p.mkQ p'", "theorem le_one_toAddSubmonoid : 1 \u2264 (1 : Submodule R A).toAddSubmonoid", "@[simp]\nprotected theorem map_div {B : Type*} [CommSemiring B] [Algebra R B] (I J : Submodule R A)\n    (h : A \u2243\u2090[R] B) : (I / J).map h.toLinearMap = I.map h.toLinearMap / J.map h.toLinearMap", "@[simp]\ntheorem map_mkQ_eq_top : map p.mkQ p' = \u22a4 \u2194 p \u2294 p' = \u22a4", "theorem map_op_mul :\n    map (\u2191(opLinearEquiv R : A \u2243\u2097[R] A\u1d50\u1d52\u1d56) : A \u2192\u2097[R] A\u1d50\u1d52\u1d56) (M * N) =\n      map (\u2191(opLinearEquiv R : A \u2243\u2097[R] A\u1d50\u1d52\u1d56) : A \u2192\u2097[R] A\u1d50\u1d52\u1d56) N *\n        map (\u2191(opLinearEquiv R : A \u2243\u2097[R] A\u1d50\u1d52\u1d56) : A \u2192\u2097[R] A\u1d50\u1d52\u1d56) M", "theorem map_span_le [RingHomSurjective \u03c3\u2081\u2082] (f : F) (s : Set M) (N : Submodule R\u2082 M\u2082) :\n    map f (span R s) \u2264 N \u2194 \u2200 m \u2208 s, f m \u2208 N", "theorem map_strictMono_of_injective : StrictMono (map f)", "theorem map_subtype_le (p' : Submodule R p) : map p.subtype p' \u2264 p", "@[simp]\ntheorem map_subtype_range_inclusion {p p' : Submodule R M} (h : p \u2264 p') :\n    map p'.subtype (range <| inclusion h) = p", "theorem map_subtype_top : map p.subtype (\u22a4 : Submodule R p) = p", "@[simp]\ntheorem map_zero : map (0 : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) p = \u22a5", "@[simp]\ntheorem map\u2082_bot_right (f : M \u2192\u2097[R] N \u2192\u2097[R] P) (p : Submodule R M) : map\u2082 f p \u22a5 = \u22a5", "theorem map\u2082_sup_right (f : M \u2192\u2097[R] N \u2192\u2097[R] P) (p : Submodule R M) (q\u2081 q\u2082 : Submodule R N) :\n    map\u2082 f p (q\u2081 \u2294 q\u2082) = map\u2082 f p q\u2081 \u2294 map\u2082 f p q\u2082", "@[simp]\ntheorem mem_bot {x : M} : x \u2208 (\u22a5 : Submodule R M) \u2194 x = 0", "@[simp]\ntheorem mem_dualCoannihilator {\u03a6 : Submodule R (Module.Dual R M)} (x : M) :\n    x \u2208 \u03a6.dualCoannihilator \u2194 \u2200 \u03c6 \u2208 \u03a6, (\u03c6 x : R) = 0", "@[simp]\ntheorem mem_one {x : A} : x \u2208 (1 : Submodule R A) \u2194 \u2203 y, algebraMap R A y = x", "theorem mem_orthogonal' (v : E) : v \u2208 K\u15ee \u2194 \u2200 u \u2208 K, \u27eav, u\u27eb = 0", "theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v \u2208 (\ud835\udd5c \u2219 u)\u15ee \u2194 \u27eav, u\u27eb = 0", "theorem Submodule.mem_sSup_iff_exists_finset {S : Set (Submodule R M)} {m : M} :\n    m \u2208 sSup S \u2194 \u2203 s : Finset (Submodule R M), \u2191s \u2286 S \u2227 m \u2208 \u2a06 i \u2208 s, i", "theorem mem_sSup_of_directed {s : Set (Submodule R M)} {z} (hs : s.Nonempty)\n    (hdir : DirectedOn (\u00b7 \u2264 \u00b7) s) : z \u2208 sSup s \u2194 \u2203 y \u2208 s, z \u2208 y", "@[simp]\ntheorem mk_quotientEquivOfIsCompl_apply (h : IsCompl p q) (x : E \u29f8 p) :\n    (Quotient.mk (quotientEquivOfIsCompl p q h x) : E \u29f8 p) = x", "theorem mulMap_one_left_eq : mulMap 1 N = N.subtype \u2218\u2097 N.lTensorOne.toLinearMap", "@[simp]\ntheorem mul_bot : M * \u22a5 = \u22a5", "@[elab_as_elim]\nprotected theorem mul_induction_on {C : A \u2192 Prop} {r : A} (hr : r \u2208 M * N)\n    (hm : \u2200 m \u2208 M, \u2200 n \u2208 N, C (m * n)) (ha : \u2200 x y, C x \u2192 C y \u2192 C (x + y)) : C r", "theorem mul_mem_mul (hm : m \u2208 M) (hn : n \u2208 N) : m * n \u2208 M * N", "theorem mul_mem_mul_rev (hm : m \u2208 M) (hn : n \u2208 N) : n * m \u2208 M * N", "theorem nontrivial_span_singleton {x : M} (h : x \u2260 0) : Nontrivial (R \u2219 x)", "theorem one_eq_range : (1 : Submodule R A) = LinearMap.range (Algebra.linearMap R A)", "theorem Submodule.orthogonal_orthogonal_eq_closure [CompleteSpace E] :\n    K\u15ee\u15ee = K.topologicalClosure", "@[simp]\ntheorem prodComm_trans_prodEquivOfIsCompl (h : IsCompl p q) :\n    LinearEquiv.prodComm R q p \u226a\u226b\u2097 prodEquivOfIsCompl p q h = prodEquivOfIsCompl q p h.symm", "@[simp]\ntheorem prodEquivOfIsCompl_symm_apply_snd_eq_zero (h : IsCompl p q) {x : E} :\n    ((prodEquivOfIsCompl p q h).symm x).2 = 0 \u2194 x \u2208 p", "@[simp]\ntheorem quotientEquivOfIsCompl_apply_mk_coe (h : IsCompl p q) (x : q) :\n    quotientEquivOfIsCompl p q h (Quotient.mk x) = x", "theorem range_liftQ [RingHomSurjective \u03c4\u2081\u2082] (f : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082) (h) :\n    range (p.liftQ f h) = range f", "theorem range_map_nonempty (N : Submodule R M) :\n    (Set.range (fun \u03d5 => Submodule.map \u03d5 N : (M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) \u2192 Submodule R\u2082 M\u2082)).Nonempty", "@[simp]\ntheorem range_snd : range (snd R M M\u2082) = \u22a4", "@[simp]\ntheorem restrictScalars_mem (V : Submodule R M) (m : M) : m \u2208 V.restrictScalars S \u2194 m \u2208 V", "@[simp]\ntheorem restrictScalars_top : restrictScalars S (\u22a4 : Submodule R M) = \u22a4", "theorem sInf_orthogonal (s : Set <| Submodule \ud835\udd5c E) : \u2a05 K \u2208 s, K\u15ee = (sSup s)\u15ee", "@[simp]\ntheorem smul_bot : I \u2022 (\u22a5 : Submodule R M) = \u22a5", "theorem smul_mem_pointwise_smul (m : M) (a : \u03b1) (S : Submodule R M) : m \u2208 S \u2192 a \u2022 m \u2208 a \u2022 S", "theorem smul_mem_smul {r} {n} (hr : r \u2208 I) (hn : n \u2208 N) : r \u2022 n \u2208 I \u2022 N", "theorem smul_mono (hij : I \u2264 J) (hnp : N \u2264 P) : I \u2022 N \u2264 J \u2022 P", "theorem smul_span (a : \u03b1) (s : Set M) : a \u2022 span R s = span R (a \u2022 s)", "theorem span_attach_biUnion [DecidableEq M] {\u03b1 : Type*} (s : Finset \u03b1) (f : s \u2192 Finset M) :\n    span R (s.attach.biUnion f : Set M) = \u2a06 x, span R (f x)", "theorem span_eq_span (hs : s \u2286 span R t) (ht : t \u2286 span R s) : span R s = span R t", "@[simp]\ntheorem span_insert_zero : span R (insert (0 : M) s) = span R s", "theorem span_mono (h : s \u2286 t) : span R s \u2264 span R t", "theorem span_smul_le (s : Set M) (r : R) : span R (r \u2022 s) \u2264 span R s", "theorem span_span : span R (span R s : Set M) = span R s", "theorem submodule_eq_sSup_le_nonzero_spans (p : Submodule R M) :\n    p = sSup { T : Submodule R M | \u2203 m \u2208 p, m \u2260 0 \u2227 T = span R {m} }", "theorem subtype_comp_inclusion (p q : Submodule R M) (h : p \u2264 q) :\n    q.subtype.comp (inclusion h) = p.subtype", "theorem sup_eq_sup_smul_of_le_smul_of_le_jacobson {I J : Ideal R} {N N' : Submodule R M}\n    (hN' : N'.FG) (hIJ : I \u2264 jacobson J) (hNN : N' \u2264 N \u2294 I \u2022 N') : N \u2294 N' = N \u2294 J \u2022 N'", "theorem Submodule.sup_orthogonal_of_completeSpace [HasOrthogonalProjection K] : K \u2294 K\u15ee = \u22a4", "@[simp]\ntheorem toConvexCone_le_iff {S T : Submodule \ud835\udd5c E} : S.toConvexCone \u2264 T.toConvexCone \u2194 S \u2264 T", "theorem toNonUnitalSubalgebra_mk (p : Submodule R A) hmul :\n    p.toNonUnitalSubalgebra hmul =\n      NonUnitalSubalgebra.mk \u27e8\u27e8\u27e8p, p.add_mem\u27e9, p.zero_mem\u27e9, hmul _ _\u27e9 p.smul_mem'", "@[mono]\ntheorem toSubMulAction_strictMono :\n    StrictMono (toSubMulAction : Submodule R M \u2192 SubMulAction R M)", "theorem toSubalgebra_mk (s : Submodule R A) (h1 hmul) :\n    s.toSubalgebra h1 hmul =\n      Subalgebra.mk \u27e8\u27e8\u27e8s, @hmul\u27e9, h1\u27e9, s.add_mem, s.zero_mem\u27e9\n        (by intro r; rw [Algebra.algebraMap_eq_smul_one]; apply s.smul_mem _ h1)", "lemma top_eq_ofList_cons_smul_iff :\n    (\u22a4 : Submodule R M) = Ideal.ofList (r :: rs) \u2022 \u22a4 \u2194\n      (\u22a4 : Submodule R (QuotSMulTop r M)) = Ideal.ofList rs \u2022 \u22a4", "theorem _root_.Submodule.topologicalClosure_map [RingHomSurjective \u03c3\u2081\u2082] [TopologicalSpace R\u2081]\n    [TopologicalSpace R\u2082] [ContinuousSMul R\u2081 M\u2081] [ContinuousAdd M\u2081] [ContinuousSMul R\u2082 M\u2082]\n    [ContinuousAdd M\u2082] (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (s : Submodule R\u2081 M\u2081) :\n    s.topologicalClosure.map (f : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) \u2264\n      (s.map (f : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082)).topologicalClosure", "lemma torsion_int {G} [AddCommGroup G] :\n    (torsion \u2124 G).toAddSubgroup = AddCommGroup.torsion G", "theorem wcovBy_span_singleton_sup (x : V) (p : Submodule K V) : WCovBy p ((K \u2219 x) \u2294 p)", "theorem nonarchimedean (hB : SubmodulesBasis B) : @NonarchimedeanAddGroup M _ hB.topology", "@[to_additive]\ntheorem lift_injective_iff :\n    Function.Injective (f.lift hg) \u2194 \u2200 x y, f.toMap x = f.toMap y \u2194 g x = g y", "@[to_additive\n    \"Given a Localization map `f : M \u2192+ N` for a Submonoid `S \u2286 M` and a map of\n`AddCommMonoid`s `g : M \u2192+ P` such that `g y` is invertible for all `y : S`, the homomorphism\ninduced from `N` to `P` maps `f x - f y` to `g x - g y` for all `x : M, y \u2208 S`.\"]\ntheorem lift_mk' (x y) : f.lift hg (f.mk' x y) = g x * (IsUnit.liftRight (g.restrict S) hg y)\u207b\u00b9", "@[to_additive\n    \"Given a Localization map `f : M \u2192+ N` for a Submonoid `S \u2286 M`, if an `AddCommMonoid` map\n`g : M \u2192+ P` induces a map `f.lift hg : N \u2192+ P` then for all\n`z : N, v w : P`, we have `f.lift hg z + w = v \u2194 g x + w = g y + v`, where `x : M, y \u2208 S` are such\nthat `z + f y = f x`.\"]\ntheorem lift_spec_mul (z w v) : f.lift hg z * w = v \u2194 g (f.sec z).1 * w = g (f.sec z).2 * v", "@[to_additive\n    \"If `AddCommMonoid` homs `g : M \u2192+ P, l : P \u2192+ A` induce maps of localizations, the composition\nof the induced maps equals the map of localizations induced by `l \u2218 g`.\"]\ntheorem map_map {A : Type*} [CommMonoid A] {U : Submonoid A} {R} [CommMonoid R]\n    (j : LocalizationMap U R) {l : P \u2192* A} (hl : \u2200 w : T, l w \u2208 U) (x) :\n    k.map hl j (f.map hy k x) = f.map (fun x \u21a6 show l.comp g x \u2208 U from hl \u27e8g x, hy x\u27e9) j x", "@[to_additive]\ntheorem mk'_eq_iff_eq' {x\u2081 x\u2082} {y\u2081 y\u2082 : S} :\n    f.mk' x\u2081 y\u2081 = f.mk' x\u2082 y\u2082 \u2194 f.toMap (x\u2081 * y\u2082) = f.toMap (x\u2082 * y\u2081)", "@[to_additive (attr := simp)]\ntheorem mulEquivOfLocalizations_apply {k : LocalizationMap S P} {x} :\n    f.mulEquivOfLocalizations k x = f.lift k.map_units x", "@[to_additive (attr := simp, nolint simpNF)]\ntheorem mulEquivOfMulEquiv_eq {k : LocalizationMap T Q} {j : M \u2243* P} (H : S.map j.toMonoidHom = T)\n    (x) :\n    f.mulEquivOfMulEquiv k H (f.toMap x) = k.toMap (j x)", "@[to_additive]\ntheorem sec_spec {f : LocalizationMap S N} (z : N) :\n    z * f.toMap (f.sec z).2 = f.toMap (f.sec z).1", "@[to_additive\n    \"Given a localization map `f : M \u2192+ N`, and `z w : N`, there exist `z' w' : M` and `d : S`\nsuch that `f z' - f d = z` and `f w' - f d = w`.\"]\ntheorem surj\u2082 (f : LocalizationMap S N) (z w : N) : \u2203 z' w' : M, \u2203 d : S,\n    (z * f.toMap d = f.toMap z') \u2227  (w * f.toMap d = f.toMap w')", "theorem leftCancelMulZero_of_le_isLeftRegular\n    (f : LocalizationWithZeroMap S N) [IsLeftCancelMulZero M]\n    (h : \u2200 \u2983x\u2984, x \u2208 S \u2192 IsLeftRegular x) : IsLeftCancelMulZero N", "@[to_additive (attr := elab_as_elim) \"A dependent version of `AddSubmonoid.closure_induction`. \"]\ntheorem closure_induction' (s : Set M) {p : \u2200 x, x \u2208 closure s \u2192 Prop}\n    (mem : \u2200 (x) (h : x \u2208 s), p x (subset_closure h)) (one : p 1 (one_mem _))\n    (mul : \u2200 x hx y hy, p x hx \u2192 p y hy \u2192 p (x * y) (mul_mem hx hy)) {x} (hx : x \u2208 closure s) :\n    p x hx", "@[to_additive]\ntheorem coe_center : \u2191(center M) = Set.center M", "@[to_additive (attr := simp)]\ntheorem coe_subtype : \u21d1S.subtype = Subtype.val", "@[to_additive]\ntheorem Submonoid.coe_topologicalClosure (s : Submonoid M) :\n    (s.topologicalClosure : Set M) = _root_.closure (s : Set M)", "@[to_additive]\ntheorem comap_injective_of_surjective : Function.Injective (comap f)", "@[to_additive]\ntheorem comap_strictMono_of_surjective : StrictMono (comap f)", "@[to_additive (attr := elab_as_elim)\n      \"If `s` is a dense set in an additive monoid `M`, `AddSubmonoid.closure s = \u22a4`, then in\n      order to prove that some predicate `p` holds for all `x : M` it suffices to verify `p x` for\n      `x \u2208 s`, verify `p 0`, and verify that `p x` and `p y` imply `p (x + y)`.\"]\ntheorem dense_induction {p : M \u2192 Prop} (x : M) {s : Set M} (hs : closure s = \u22a4)\n    (mem : \u2200 x \u2208 s, p x)\n    (one : p 1) (mul : \u2200 x y, p x \u2192 p y \u2192 p (x * y)) : p x", "@[to_additive]\nlemma inv_mem_units (S : Submonoid M) {x : M\u02e3} (h : x \u2208 S.units) : x\u207b\u00b9 \u2208 S.units := inv_mem h", "@[to_additive]\ntheorem le_op_iff {S\u2081 : Submonoid M\u1d50\u1d52\u1d56} {S\u2082 : Submonoid M} : S\u2081 \u2264 S\u2082.op \u2194 S\u2081.unop \u2264 S\u2082", "@[to_additive]\nlemma mem_units_of_val_mem_inv_val_mem (S : Submonoid M) {x : M\u02e3} (h\u2081 : (x : M) \u2208 S)\n    (h\u2082 : ((x\u207b\u00b9 : M\u02e3) : M) \u2208 S) : x \u2208 S.units := \u27e8h\u2081, h\u2082\u27e9", "@[to_additive]\ntheorem mrange_inr : mrange (inr M N) = prod \u22a5 \u22a4", "@[to_additive]\ntheorem mul_leftInvEquiv (x : S.leftInv) : (x : M) * S.leftInvEquiv hS x = 1", "@[to_additive\n      \"Sum of a multiset of elements in an `AddSubmonoid` of an `AddCommMonoid` is\n      in the `AddSubmonoid`.\"]\ntheorem multiset_prod_mem {M} [CommMonoid M] (S : Submonoid M) (m : Multiset M)\n    (hm : \u2200 a \u2208 m, a \u2208 S) : m.prod \u2208 S", "@[to_additive (attr := simp)]\ntheorem nontrivial_iff : Nontrivial (Submonoid M) \u2194 Nontrivial M", "@[to_additive (attr := simp)]\ntheorem op_top : (\u22a4 : Submonoid M).op = \u22a4", "@[to_additive]\ntheorem sup_eq_range (s t : Submonoid N) : s \u2294 t = mrange (s.subtype.coprod t.subtype)", "@[to_additive]\ntheorem Submonoid.top_closure_mul_self_eq (s : Submonoid M) :\n    _root_.closure (s : Set M) * _root_.closure s = _root_.closure s", "@[to_additive]\nlemma units_iInf\u2082 {\u03b9 : Sort*} {\u03ba : \u03b9 \u2192 Sort*} (f : (i : \u03b9) \u2192 \u03ba i \u2192 Submonoid M) :\n    (\u2a05 (i : \u03b9), \u2a05 (j : \u03ba i), f i j).units = \u2a05 (i : \u03b9), \u2a05 (j : \u03ba i), (f i j).units", "@[to_additive]\ntheorem _root_.SubmonoidClass.finsupp_prod_mem {S : Type*} [SetLike S N] [SubmonoidClass S N]\n    (s : S) (f : \u03b1 \u2192\u2080 M) (g : \u03b1 \u2192 M \u2192 N) (h : \u2200 c, f c \u2260 0 \u2192 g c (f c) \u2208 s) : f.prod g \u2208 s", "theorem center_le_centralizer (s) : center R \u2264 centralizer s", "theorem closure_eq (s : Subring R) : closure (s : Set R) = s", "@[simp, norm_cast]\ntheorem coe_add (x y : s) : (\u2191(x + y) : R) = \u2191x + \u2191y", "@[simp]\ntheorem coe_comap (s : Subring S) (f : R \u2192+* S) : (s.comap f : Set R) = f \u207b\u00b9' s", "protected theorem list_prod_mem {l : List R} : (\u2200 x \u2208 l, x \u2208 s) \u2192 l.prod \u2208 s", "theorem mem_center_iff {z : R} : z \u2208 center R \u2194 \u2200 g, g * z = z * g", "@[simp]\ntheorem mem_inf {p p' : Subring R} {x : R} : x \u2208 p \u2293 p' \u2194 x \u2208 p \u2227 x \u2208 p'", "@[simp]\ntheorem pointwise_smul_toAddSubgroup (m : M) (S : Subring R) :\n    (m \u2022 S).toAddSubgroup = m \u2022 S.toAddSubgroup", "protected theorem pow_mem {x : R} (hx : x \u2208 s) (n : \u2115) : x ^ n \u2208 s", "@[mono]\ntheorem prod_mono \u2983s\u2081 s\u2082 : Subring R\u2984 (hs : s\u2081 \u2264 s\u2082) \u2983t\u2081 t\u2082 : Subring S\u2984 (ht : t\u2081 \u2264 t\u2082) :\n    s\u2081.prod t\u2081 \u2264 s\u2082.prod t\u2082", "@[simp]\ntheorem range_subtype (s : Subring R) : s.subtype.range = s", "@[mono]\ntheorem toAddSubgroup_strictMono : StrictMono (toAddSubgroup : Subring R \u2192 AddSubgroup R)", "theorem unop_inf (S\u2081 S\u2082 : Subring R\u1d50\u1d52\u1d56) : (S\u2081 \u2293 S\u2082).unop = S\u2081.unop \u2293 S\u2082.unop", "@[to_additive (attr := simp)]\ntheorem centralizer_univ : centralizer Set.univ = center M", "@[to_additive (attr := simp, norm_cast)]\ntheorem coe_set_mk {s : Set M} (h_mul) : (mk s h_mul : Set M) = s", "@[to_additive (attr := simp)]\ntheorem comap_id (S : Subsemigroup P) : S.comap (MulHom.id _) = S", "@[to_additive]\ntheorem comap_inf (S T : Subsemigroup N) (f : M \u2192\u2099* N) : (S \u2293 T).comap f = S.comap f \u2293 T.comap f", "@[to_additive]\ntheorem iSup_eq_closure {\u03b9 : Sort*} (p : \u03b9 \u2192 Subsemigroup M) :\n    \u2a06 i, p i = Subsemigroup.closure (\u22c3 i, (p i : Set M))", "@[to_additive (attr := simp)]\ntheorem map_equiv_top (f : M \u2243* N) : (\u22a4 : Subsemigroup M).map (f : M \u2192\u2099* N) = \u22a4", "@[to_additive]\ntheorem map_inf_comap_of_surjective (S T : Subsemigroup N) :\n    (S.comap f \u2293 T.comap f).map f = S \u2293 T", "@[to_additive]\ntheorem map_map (g : N \u2192\u2099* P) (f : M \u2192\u2099* N) : (S.map f).map g = S.map (g.comp f)", "@[to_additive]\ntheorem mem_iSup {\u03b9 : Sort*} (p : \u03b9 \u2192 Subsemigroup M) {m : M} :\n    (m \u2208 \u2a06 i, p i) \u2194 \u2200 N, (\u2200 i, p i \u2264 N) \u2192 m \u2208 N", "@[to_additive (attr := simp)]\ntheorem mem_map {f : M \u2192\u2099* N} {S : Subsemigroup M} {y : N} : y \u2208 S.map f \u2194 \u2203 x \u2208 S, f x = y", "@[to_additive]\ntheorem mem_map_of_mem (f : M \u2192\u2099* N) {S : Subsemigroup M} {x : M} (hx : x \u2208 S) : f x \u2208 S.map f", "@[to_additive (attr := simp)]\ntheorem mk_le_mk {s t : Set M} (h_mul) (h_mul') : mk s h_mul \u2264 mk t h_mul' \u2194 s \u2286 t", "protected theorem add_mem {x y : R} : x \u2208 s \u2192 y \u2208 s \u2192 x + y \u2208 s", "@[simp]\ntheorem closure_empty : closure (\u2205 : Set R) = \u22a5", "@[simp]\ntheorem coe_pointwise_smul (m : M) (S : Subsemiring R) : \u2191(m \u2022 S) = m \u2022 (S : Set R)", "@[simp]\ntheorem coe_toSubmonoid (s : Subsemiring R) : (s.toSubmonoid : Set R) = s", "theorem gc_map_comap (f : R \u2192+* S) : GaloisConnection (map f) (comap f)", "theorem Subsemiring.le_topologicalClosure (s : Subsemiring \u03b1) : s \u2264 s.topologicalClosure", "theorem op_sSup (S : Set (Subsemiring R)) : (sSup S).op = sSup (.unop \u207b\u00b9' S)", "@[simp]\ntheorem op_unop (S : Subsemiring R\u1d50\u1d52\u1d56) : S.unop.op = S", "theorem pointwise_smul_subset_iff {a : M} {S T : Subsemiring R} : a \u2022 S \u2264 T \u2194 S \u2264 a\u207b\u00b9 \u2022 T", "@[simp]\ntheorem rangeS_subtype (s : Subsemiring R) : s.subtype.rangeS = s", "protected theorem Subsingleton.elim {\u03b1 : Sort u} [h : Subsingleton \u03b1] : (a b : \u03b1) \u2192 a = b", "@[nontriviality, measurability]\ntheorem Subsingleton.measurableSet [Subsingleton \u03b1] {s : Set \u03b1} : MeasurableSet s", "protected theorem monotone [Subsingleton \u03b1] (f : \u03b1 \u2192 \u03b2) : Monotone f", "@[elab_as_elim]\ntheorem set_cases {p : Set \u03b1 \u2192 Prop} (h0 : p \u2205) (h1 : p univ) (s) : p s", "theorem dualAnnihilator_inj {W W' : Subspace K V} :\n    W.dualAnnihilator = W'.dualAnnihilator \u2194 W = W'", "@[simp]\ntheorem quotAnnihilatorEquiv_apply (W : Subspace K V) (\u03c6 : Module.Dual K V) :\n    W.quotAnnihilatorEquiv (Submodule.Quotient.mk \u03c6) = W.dualRestrict \u03c6", "theorem extract : \u2200 {s}, Valid s \u2192 Valid \u27e8s.toString, b, e\u27e9 \u2192 Valid (s.extract b e)", "theorem nextn_stop : \u2200 {s}, ValidFor l m r s \u2192 \u2200 n, s.nextn n \u27e8utf8Len m\u27e9 = \u27e8utf8Len m\u27e9", "theorem prevn : \u2200 {s}, ValidFor l (m\u2081.reverse ++ m\u2082) r s \u2192\n    \u2200 n, s.prevn n \u27e8utf8Len m\u2081\u27e9 = \u27e8utf8Len (m\u2081.drop n)\u27e9", "@[simp, norm_cast]\ntheorem coe_le_coe [LE \u03b1] {p : \u03b1 \u2192 Prop} {x y : Subtype p} : (x : \u03b1) \u2264 y \u2194 x \u2264 y", "lemma coe_ne_coe {a b : Subtype p} : (a : \u03b1) \u2260 b \u2194 a \u2260 b := coe_injective.ne_iff", "theorem exists_set_subtype {t : Set \u03b1} (p : Set \u03b1 \u2192 Prop) :\n    (\u2203 s : Set t, p (((\u2191) : t \u2192 \u03b1) '' s)) \u2194 \u2203 s : Set \u03b1, s \u2286 t \u2227 p s", "theorem ext_val {a1 a2 : { x // p x }} : a1.1 = a2.1 \u2192 a1 = a2", "@[simp]\ntheorem image_preimage_coe (s t : Set \u03b1) : ((\u2191) : s \u2192 \u03b1) '' (((\u2191) : s \u2192 \u03b1) \u207b\u00b9' t) = s \u2229 t", "@[simp]\ntheorem mk_le_mk [LE \u03b1] {p : \u03b1 \u2192 Prop} {x y : \u03b1} {hx : p x} {hy : p y} :\n    (\u27e8x, hx\u27e9 : Subtype p) \u2264 \u27e8y, hy\u27e9 \u2194 x \u2264 y", "theorem val_injective : Injective (@val _ p)", "@[simp] theorem Ioo_inr_inl : Ioo (inr b\u2081) (inl a\u2082) = \u2205", "lemma Ioo_inr_inr :\n    Ioo (inr\u2097 b\u2081 : \u03b1 \u2295\u2097 \u03b2) (inr\u2097 b\u2082) = (Ioo b\u2081 b\u2082).map (Embedding.inr.trans toLex.toEmbedding)", "@[simp]\ntheorem inl_bot [LE \u03b1] [OrderBot \u03b1] [LE \u03b2] : toLex (inl \u22a5 : Sum \u03b1 \u03b2) = \u22a5", "theorem inl_strictMono : StrictMono (toLex \u2218 inl : \u03b1 \u2192 \u03b1 \u2295\u2097 \u03b2)", "@[simp]\ntheorem toLex_lt_toLex [LT \u03b1] [LT \u03b2] {a b : Sum \u03b1 \u03b2} :\n    toLex a < toLex b \u2194 Lex (\u00b7 < \u00b7) (\u00b7 < \u00b7) a b", "@[to_additive (attr := simp)]\ntheorem elim_one_one [One \u03b3] : Sum.elim (1 : \u03b1 \u2192 \u03b3) (1 : \u03b2 \u2192 \u03b3) = 1", "theorem eq_left_getLeft_of_isLeft : \u2200 {x : \u03b1 \u2295 \u03b2} (h : x.isLeft), x = inl (x.getLeft h)", "theorem eq_right_getRight_of_isRight : \u2200 {x : \u03b1 \u2295 \u03b2} (h : x.isRight), x = inr (x.getRight h)", "theorem inl_strictMono : StrictMono (inl : \u03b1 \u2192 Sum \u03b1 \u03b2)", "@[simp] theorem isLeft_inl : (inl x : \u03b1 \u2295 \u03b2).isLeft = true", "@[simp]\ntheorem map_bijective {f : \u03b1 \u2192 \u03b3} {g : \u03b2 \u2192 \u03b4} :\n    Bijective (Sum.map f g) \u2194 Bijective f \u2227 Bijective g", "theorem Summable.of_norm_bounded [CompleteSpace E] {f : \u03b9 \u2192 E} (g : \u03b9 \u2192 \u211d) (hg : Summable g)\n    (h : \u2200 i, \u2016f i\u2016 \u2264 g i) : Summable f", "@[simp] lemma coe_toBotHom (f : SupBotHom \u03b1 \u03b2) : \u21d1f.toBotHom = f := rfl", "@[simp] theorem dual_id : SupBotHom.dual (SupBotHom.id \u03b1) = InfTopHom.id _", "@[simp] theorem id_comp (f : SupBotHom \u03b1 \u03b2) : (SupBotHom.id \u03b2).comp f = f", "@[simp]\ntheorem symm_dual_id : SupHom.dual.symm (InfHom.id _) = SupHom.id \u03b1", "lemma P3_comm : P3 x\u2081 x\u2082 y\u2081 y\u2082 \u2194 P3 y\u2081 y\u2082 x\u2081 x\u2082", "lemma P4_neg_left : P4 x\u2081 x\u2082 y \u2194 P4 (-x\u2082) (-x\u2081) y", "lemma ih1 : IH1 x y", "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric", "@[simp]\ntheorem dyadicMap_apply (m : \u2124) (p : Submonoid.powers (2 : \u2124)) :\n    dyadicMap (IsLocalization.mk' (Localization (Submonoid.powers 2)) m p) =\n      m * powHalf (Submonoid.log p)", "@[simp]\ntheorem nsmul_pow_two_powHalf' (n k : \u2115) : 2 ^ n * powHalf (n + k) = powHalf k", "theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]\n    (P : Sylow p G) (x g : G) (hx : x \u2208 centralizer (P : Set G))\n    (hy : g\u207b\u00b9 * x * g \u2208 centralizer (P : Set G)) :\n    \u2203 n \u2208 (P : Subgroup G).normalizer, g\u207b\u00b9 * x * g = n\u207b\u00b9 * x * n", "theorem Sylow.pointwise_smul_def {\u03b1 : Type*} [Group \u03b1] [MulDistribMulAction \u03b1 G] {g : \u03b1}\n    {P : Sylow p G} : \u2191(g \u2022 P) = g \u2022 (P : Subgroup G)", "theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : \u2115} [hp : Fact p.Prime] (P : Sylow p G)\n    (hdvd : p ^ n \u2223 card G) : p ^ n \u2223 card P", "@[simp]\ntheorem coe_attach (s : Sym \u03b1 n) : (s.attach : Multiset { a // a \u2208 s }) =\n    Multiset.attach (s : Multiset \u03b1)", "theorem cons_swap (a b : \u03b1) (s : Sym \u03b1 n) : a ::\u209b b ::\u209b s = b ::\u209b a ::\u209b s", "@[ext] theorem ext {s\u2081 s\u2082 : Sym \u03b1 n} (h : (s\u2081 : Multiset \u03b1) = \u2191s\u2082) : s\u2081 = s\u2082", "@[simp]\ntheorem mem_map {n : \u2115} {f : \u03b1 \u2192 \u03b2} {b : \u03b2} {l : Sym \u03b1 n} :\n    b \u2208 Sym.map f l \u2194 \u2203 a, a \u2208 l \u2227 f a = b", "theorem IsDiag.mem_range_diag {z : Sym2 \u03b1} : IsDiag z \u2192 z \u2208 Set.range (@diag \u03b1)", "theorem card_image_offDiag (s : Finset \u03b1) :\n    (s.offDiag.image Sym2.mk).card = s.card.choose 2", "theorem congr_left {a b c : \u03b1} : s(b, a) = s(c, a) \u2194 b = c", "protected theorem \u00abforall\u00bb {\u03b1 : Sort _} {f : Sym2 \u03b1 \u2192 Prop} :\n    (\u2200 x : Sym2 \u03b1, f x) \u2194 \u2200 x y, f s(x, y)", "theorem rel_iff {x y z w : \u03b1} : Rel \u03b1 (x, y) (z, w) \u2194 x = z \u2227 y = w \u2228 x = w \u2227 y = z", "@[simp]\ntheorem unsym_neg [Neg \u03b1] (a : \u03b1\u02e2\u02b8\u1d50) : unsym (-a) = -unsym a", "@[simp]\ntheorem decode_inr (s : Sym \u03b1 n.succ) : decode (Sum.inr s) = s.map Embedding.some", "theorem SymmetricRel.inter {U V : Set (\u03b1 \u00d7 \u03b1)} (hU : SymmetricRel U) (hV : SymmetricRel V) :\n    SymmetricRel (U \u2229 V)", "theorem transpose_mem (hA : A \u2208 symplecticGroup l R) : A\u1d40 \u2208 symplecticGroup l R", "theorem edgeDensity_chunk_not_uniform [Nonempty \u03b1] (hP\u03b1 : P.parts.card * 16 ^ P.parts.card \u2264 card \u03b1)\n    (hP\u03b5 : \u2191100 \u2264 \u21914 ^ P.parts.card * \u03b5 ^ 5) (h\u03b5\u2081 : \u03b5 \u2264 1) {hU : U \u2208 P.parts} {hV : V \u2208 P.parts}\n    (hUVne : U \u2260 V) (hUV : \u00acG.IsUniform \u03b5 U V) :\n    (G.edgeDensity U V : \u211d) ^ 2 - \u03b5 ^ 5 / \u219125 + \u03b5 ^ 4 / \u21913 \u2264\n    (\u2211 ab \u2208 (chunk hP G \u03b5 hU).parts.product (chunk hP G \u03b5 hV).parts,\n      (G.edgeDensity ab.1 ab.2 : \u211d) ^ 2) / \u219116 ^ P.parts.card", "private theorem eps_le_card_star_div [Nonempty \u03b1] (hP\u03b1 : P.parts.card * 16 ^ P.parts.card \u2264 card \u03b1)\n    (hP\u03b5 : \u2191100 \u2264 \u21914 ^ P.parts.card * \u03b5 ^ 5) (h\u03b5\u2081 : \u03b5 \u2264 1) (hU : U \u2208 P.parts) (hV : V \u2208 P.parts)\n    (hUV : U \u2260 V) (hunif : \u00acG.IsUniform \u03b5 U V) :\n    \u21914 / \u21915 * \u03b5 \u2264 (star hP G \u03b5 hU V).card / \u21914 ^ P.parts.card", "private theorem m_add_one_div_m_le_one_add [Nonempty \u03b1]\n    (hP\u03b1 : P.parts.card * 16 ^ P.parts.card \u2264 card \u03b1) (hP\u03b5 : \u2191100 \u2264 \u21914 ^ P.parts.card * \u03b5 ^ 5)\n    (h\u03b5\u2081 : \u03b5 \u2264 1) : ((m + 1 : \u211d) / m) ^ 2 \u2264 \u21911 + \u03b5 ^ 5 / 49", "theorem m_le_card_of_mem_chunk_parts (hs : s \u2208 (chunk hP G \u03b5 hU).parts) : m \u2264 s.card", "private theorem one_sub_eps_mul_card_nonuniformWitness_le_card_star (hV : V \u2208 P.parts)\n    (hUV : U \u2260 V) (hunif : \u00acG.IsUniform \u03b5 U V) (hP\u03b5 : \u2191100 \u2264 \u21914 ^ P.parts.card * \u03b5 ^ 5)\n    (h\u03b5\u2081 : \u03b5 \u2264 1) :\n    (1 - \u03b5 / 10) * (G.nonuniformWitness \u03b5 U V).card \u2264 ((star hP G \u03b5 hU V).biUnion id).card", "lemma T2Space.isSeparatedMap [T2Space X] (f : X \u2192 Y) : IsSeparatedMap f := fun _ _ _ \u21a6 t2_separation", "private theorem T_pos : T > 0", "theorem int_not_isCoprime_helper (x y : \u2124) (d : \u2115) (hd : Int.gcd x y = d)\n    (h : Nat.beq d 1 = false) : \u00ac IsCoprime x y", "theorem isInt_isCoprime : {x y nx ny : \u2124} \u2192\n    IsInt x nx \u2192 IsInt y ny \u2192 IsCoprime nx ny \u2192 IsCoprime x y", "theorem nat_gcd_helper_2' (x y a b : \u2115) (h : x * a = y * b + 1) :\n    Nat.gcd x y = 1", "lemma CharP.intCast_eq_mod (R : Type _) [Ring R] (p : \u2115) [CharP R p] (k : \u2124) :\n    (k : R) = (k % p : \u2124)", "@[simp, mfld_simps]\ntheorem coe_chartAt_symm_fst (p : H \u00d7 E) (q : TM) :\n    ((chartAt (ModelProd H E) q).symm p).1 = ((chartAt H q.1).symm : H \u2192 M) p.1", "@[simp high, mfld_simps]\ntheorem symmL_model_space (b b' : F) :\n    (trivializationAt F (TangentSpace \ud835\udcd8(\ud835\udd5c, F)) b).symmL \ud835\udd5c b' = (1 : F \u2192L[\ud835\udd5c] F)", "theorem tangentMap_tangentBundle_pure (p : TangentBundle I M) :\n    tangentMap I I.tangent (zeroSection E (TangentSpace I)) p = \u27e8\u27e8p.proj, 0\u27e9, \u27e8p.2, 0\u27e9\u27e9", "@[simp, mfld_simps]\ntheorem trivializationAt_symmL {b\u2080 b : M}\n    (hb : b \u2208 (trivializationAt E (TangentSpace I) b\u2080).baseSet) :\n    (trivializationAt E (TangentSpace I) b\u2080).symmL \ud835\udd5c b =\n      (tangentBundleCore I M).coordChange (achart H b\u2080) (achart H b) b", "protected theorem TendstoLocallyUniformly.tendstoLocallyUniformlyOn\n    (h : TendstoLocallyUniformly F f p) : TendstoLocallyUniformlyOn F f p s", "theorem TendstoLocallyUniformlyOn.congr {G : \u03b9 \u2192 \u03b1 \u2192 \u03b2} (hf : TendstoLocallyUniformlyOn F f p s)\n    (hg : \u2200 n, s.EqOn (F n) (G n)) : TendstoLocallyUniformlyOn G f p s", "theorem TendstoLocallyUniformlyOn.tendsto_at (hf : TendstoLocallyUniformlyOn F f p s) {a : \u03b1}\n    (ha : a \u2208 s) : Tendsto (fun i => F i a) p (\ud835\udcdd (f a))", "@[to_additive]\ntheorem TendstoUniformly.mul (hf : TendstoUniformly f g l) (hf' : TendstoUniformly f' g' l) :\n    TendstoUniformly (f * f') (g * g') l", "theorem TendstoUniformly.tendsto_comp (h : TendstoUniformly F f p) (hf : ContinuousAt f x)\n    (hg : Tendsto g p (\ud835\udcdd x)) : Tendsto (fun n => F n (g n)) p (\ud835\udcdd (f x))", "@[simp]\ntheorem \u03b9_inj (x y : M) : \u03b9 R x = \u03b9 R y \u2194 x = y", "theorem algebraMap\u2080_mul_algebraMap\u2080 (r s : R) :\n    cast R M (add_zero _) (algebraMap\u2080 r \u209c* algebraMap\u2080 s) = algebraMap\u2080 (r * s)", "theorem gMul_def {i j} (a : \u2a02[R]^i M) (b : (\u2a02[R]^j) M) :\n    a \u209c* b = @mulEquiv R M _ _ _ i j (a \u2297\u209c b)", "@[simp]\ntheorem congr_one : congr (1 : M \u2243\u2097[A] M) (1 : N \u2243\u2097[R] N) = 1", "@[simp]\ntheorem dualDistrib_apply (f : Dual A M) (g : Dual R N) (m : M) (n : N) :\n    dualDistrib R A M N (f \u2297\u209c g) (m \u2297\u209c n) = g n \u2022 f m", "theorem liftLie_apply (f : M \u2192\u2097\u2045R,L\u2046 N \u2192\u2097[R] P) (m : M) (n : N) :\n    liftLie R L M N P f (m \u2297\u209c n) = f m n", "@[simp]\ntheorem directSum_lof_tmul_lof (i\u2081 : \u03b9\u2081) (m\u2081 : M\u2081 i\u2081) (i\u2082 : \u03b9\u2082) (m\u2082 : M\u2082 i\u2082) :\n    TensorProduct.directSum R S M\u2081 M\u2082 (DirectSum.lof S \u03b9\u2081 M\u2081 i\u2081 m\u2081 \u2297\u209c DirectSum.lof R \u03b9\u2082 M\u2082 i\u2082 m\u2082) =\n      DirectSum.lof S (\u03b9\u2081 \u00d7 \u03b9\u2082) (fun i => M\u2081 i.1 \u2297[R] M\u2082 i.2) (i\u2081, i\u2082) (m\u2081 \u2297\u209c m\u2082)", "@[simp]\ntheorem directSum_symm_lof_tmul (i\u2081 : \u03b9\u2081) (m\u2081 : M\u2081 i\u2081) (i\u2082 : \u03b9\u2082) (m\u2082 : M\u2082 i\u2082) :\n    (TensorProduct.directSum R S M\u2081 M\u2082).symm\n      (DirectSum.lof S (\u03b9\u2081 \u00d7 \u03b9\u2082) (fun i => M\u2081 i.1 \u2297[R] M\u2082 i.2) (i\u2081, i\u2082) (m\u2081 \u2297\u209c m\u2082)) =\n      (DirectSum.lof S \u03b9\u2081 M\u2081 i\u2081 m\u2081 \u2297\u209c DirectSum.lof R \u03b9\u2082 M\u2082 i\u2082 m\u2082)", "theorem exists_finite_submodule_of_finite (s : Set (M \u2297[R] N)) (hs : s.Finite) :\n    \u2203 (M' : Submodule R M) (N' : Submodule R N), Module.Finite R M' \u2227 Module.Finite R N' \u2227\n      s \u2286 LinearMap.range (mapIncl M' N')", "theorem ext' {g h : M \u2297[R] N \u2192\u2097[R] P} (H : \u2200 x y, g (x \u2297\u209c y) = h (x \u2297\u209c y)) : g = h", "theorem ext_threefold {g h : (M \u2297[R] N) \u2297[R] P \u2192\u2097[R] Q}\n    (H : \u2200 x y z, g (x \u2297\u209c y \u2297\u209c z) = h (x \u2297\u209c y \u2297\u209c z)) : g = h", "@[simp]\nlemma finsuppScalarLeft_apply_tmul_apply (p : \u03b9 \u2192\u2080 R) (n : N) (i : \u03b9) :\n    finsuppScalarLeft R N \u03b9 (p \u2297\u209c[R] n) i = p i \u2022 n", "theorem forall_vanishesTrivially_iff_forall_FG_rTensor_injective :\n    (\u2200 {\u03b9 : Type u} [Fintype \u03b9] {m : \u03b9 \u2192 M} {n : \u03b9 \u2192 N},\n      \u2211 i, m i \u2297\u209c n i = (0 : M \u2297[R] N) \u2192 VanishesTrivially R m n) \u2194\n    \u2200 (M' : Submodule R M) (_ : M'.FG), Injective (rTensor N M'.subtype)", "@[simp]\ntheorem lift.tmul (x y) : lift f (x \u2297\u209c y) = f x y", "theorem lift_comp_map (i : P \u2192\u2097[R] Q \u2192\u2097[R] Q') (f : M \u2192\u2097[R] P) (g : N \u2192\u2097[R] Q) :\n    (lift i).comp (map f g) = lift ((i.comp f).compl\u2082 g)", "theorem TensorProduct.map_ker :\n    ker (TensorProduct.map g g') = range (lTensor N f') \u2294 range (rTensor N' f)", "lemma map_map_assoc_symm (f : M \u2192\u2097[R] Q) (g : N \u2192\u2097[R] S) (h : P \u2192\u2097[R] T) (x : M \u2297[R] (N \u2297[R] P)) :\n    map (map f g) h ((TensorProduct.assoc R M N P).symm x) =\n      (TensorProduct.assoc R Q S T).symm (map f (map g h) x)", "theorem map\u2082_eq_range_lift_comp_mapIncl (f : P \u2192\u2097[R] Q \u2192\u2097[R] M)\n    (p : Submodule R P) (q : Submodule R Q) :\n    Submodule.map\u2082 f p q = LinearMap.range (lift f \u2218\u2097 mapIncl p q)", "@[simp] theorem prodLeft_tmul (m\u2081 : M\u2081) (m\u2082 : M\u2082) (m\u2083 : M\u2083) :\n    prodLeft R M\u2081 M\u2082 M\u2083 ((m\u2081, m\u2082) \u2297\u209c m\u2083) = (m\u2081 \u2297\u209c m\u2083, m\u2082 \u2297\u209c m\u2083)", "theorem span_tmul_eq_top : Submodule.span R { t : M \u2297[R] N | \u2203 m n, m \u2297\u209c n = t } = \u22a4", "theorem sum_tmul_eq_zero_of_vanishesTrivially (hmn : VanishesTrivially R m n) :\n    \u2211 i, m i \u2297\u209c n i = (0 : M \u2297[R] N)", "theorem tmul_of_gradedMul_of_tmul (j\u2081 i\u2082 : \u03b9)\n    (a\u2081 : \u2a01 i, \ud835\udc9c i) (b\u2081 : \u212c j\u2081) (a\u2082 : \ud835\udc9c i\u2082) (b\u2082 : \u2a01 i, \u212c i) :\n    gradedMul R \ud835\udc9c \u212c (a\u2081 \u2297\u209c lof R _ \u212c j\u2081 b\u2081) (lof R _ \ud835\udc9c i\u2082 a\u2082 \u2297\u209c b\u2082) =\n      (-1 : \u2124\u02e3)^(j\u2081 * i\u2082) \u2022 ((a\u2081 * lof R _ \ud835\udc9c _ a\u2082) \u2297\u209c (lof R _ \u212c _ b\u2081 * b\u2082))", "@[simp] lemma toDirectLimit_tmul_of\n    {i : \u03b9} (g : G i) (m : M) :\n    (toDirectLimit f M <| (of _ _ G f i g) \u2297\u209c m) = (of _ _ _ _ i (g \u2297\u209c m))", "theorem vanishesTrivially_iff_sum_tmul_eq_zero (hm : Submodule.span R (Set.range m) = \u22a4) :\n    VanishesTrivially R m n \u2194 \u2211 i, m i \u2297\u209c n i = (0 : M \u2297[R] N)", "@[to_additive]\ntheorem ThreeGPFree.smul_set (hs : ThreeGPFree s) : ThreeGPFree (a \u2022 s)", "lemma ThreeGPFree.smul_set\u2080 (hs : ThreeGPFree s) (ha : a \u2260 0) : ThreeGPFree (a \u2022 s)", "@[simp] theorem prod_get_snd : (prod a b).get.2 = b.get", "theorem range_fromOpenSubsetsGlue : Set.range (fromOpenSubsetsGlue U) = \u22c3 i, (U i : Set \u03b1)", "theorem rel_equiv : Equivalence D.Rel", "theorem PrelocalPredicate.sheafifyOf {T : X \u2192 Type v} {P : PrelocalPredicate T} {U : Opens X}\n    {f : \u2200 x : U, T x} (h : P.pred f) : P.sheafify.pred f", "@[simp]\ntheorem covering_presieve_eq_self {Y : Opens X} (R : Presieve Y) :\n    presieveOfCoveringAux (coveringOfPresieve Y R) Y = R", "@[ext]\nlemma ext {X : TopCat} {P Q : Presheaf C X} {f g : P \u27f6 Q}\n    (w : \u2200 U : Opens X, f.app (op U) = g.app (op U)) :\n    f = g", "theorem locally_surjective_iff_surjective_on_stalks (T : \u2131 \u27f6 \ud835\udca2) :\n    IsLocallySurjective T \u2194 \u2200 x : X, Function.Surjective ((stalkFunctor C x).map T)", "theorem toType_isSheaf (T : Type u) : (presheafToType X T).IsSheaf", "theorem epi_iff_surjective {X Y : TopCat.{u}} (f : X \u27f6 Y) : Epi f \u2194 Function.Surjective f", "theorem induced_of_isLimit {F : J \u2964 TopCat.{max v u}} (C : Cone F) (hC : IsLimit C) :\n    C.pt.str = \u2a05 j, (F.obj j).str.induced (C.\u03c0.app j)", "theorem isSheaf_of_isLimit [HasLimits C] {X : TopCat} (F : J \u2964 Presheaf.{v} C X)\n    (H : \u2200 j, (F.obj j).IsSheaf) {c : Cone F} (hc : IsLimit c) : c.pt.IsSheaf", "@[reassoc (attr := simp), elementwise]\ntheorem prodIsoProd_inv_fst (X Y : TopCat.{u}) :\n    (prodIsoProd X Y).inv \u226b Limits.prod.fst = prodFst", "theorem pullbackIsoProdSubtype_hom_apply {f : X \u27f6 Z} {g : Y \u27f6 Z}\n    (x : ConcreteCategory.forget.obj (pullback f g)) :\n    (pullbackIsoProdSubtype f g).hom x =\n      \u27e8\u27e8(pullback.fst : pullback f g \u27f6 _) x, (pullback.snd : pullback f g \u27f6 _) x\u27e9, by\n        simpa using ConcreteCategory.congr_hom pullback.condition x\u27e9", "theorem pullbackIsoProdSubtype_inv_fst_apply (f : X \u27f6 Z) (g : Y \u27f6 Z)\n    (x : { p : X \u00d7 Y // f p.1 = g p.2 }) :\n    (pullback.fst : pullback f g \u27f6 _) ((pullbackIsoProdSubtype f g).inv x) = (x : X \u00d7 Y).fst", "theorem pullbackIsoProdSubtype_inv_snd_apply (f : X \u27f6 Z) (g : Y \u27f6 Z)\n    (x : { p : X \u00d7 Y // f p.1 = g p.2 }) :\n    (pullback.snd : pullback f g \u27f6 _) ((pullbackIsoProdSubtype f g).inv x) = (x : X \u00d7 Y).snd", "lemma pullbackSnd_apply (f : X \u27f6 Z) (g : Y \u27f6 Z) (x) : pullbackSnd f g x = x.1.2 := rfl", "theorem pullback_map_openEmbedding_of_open_embeddings {W X Y Z S T : TopCat.{u}} (f\u2081 : W \u27f6 S)\n    (f\u2082 : X \u27f6 S) (g\u2081 : Y \u27f6 T) (g\u2082 : Z \u27f6 T) {i\u2081 : W \u27f6 Y} {i\u2082 : X \u27f6 Z} (H\u2081 : OpenEmbedding i\u2081)\n    (H\u2082 : OpenEmbedding i\u2082) (i\u2083 : S \u27f6 T) [H\u2083 : Mono i\u2083] (eq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081)\n    (eq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082) : OpenEmbedding (pullback.map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082)", "@[simp]\ntheorem coe_of (X : Type u) [CommRing X] [TopologicalSpace X] [TopologicalRing X] :\n    (of X : Type u) = X", "@[simp]\ntheorem cancel_left {g : TopHom \u03b2 \u03b3} {f\u2081 f\u2082 : TopHom \u03b1 \u03b2} (hg : Injective g) :\n    g.comp f\u2081 = g.comp f\u2082 \u2194 f\u2081 = f\u2082", "@[simp]\ntheorem symm_dual_id : TopHom.dual.symm (BotHom.id _) = TopHom.id \u03b1", "@[to_additive]\ntheorem TopologicalGroup.tendstoLocallyUniformlyOn_iff {\u03b9 \u03b1 : Type*} [TopologicalSpace \u03b1]\n    (F : \u03b9 \u2192 \u03b1 \u2192 G) (f : \u03b1 \u2192 G) (p : Filter \u03b9) (s : Set \u03b1) :\n    @TendstoLocallyUniformlyOn \u03b1 G \u03b9 (TopologicalGroup.toUniformSpace G) _ F f p s \u2194\n      \u2200 u \u2208 \ud835\udcdd (1 : G), \u2200 x \u2208 s, \u2203 t \u2208 \ud835\udcdd[s] x, \u2200\u1da0 i in p, \u2200 a \u2208 t, F i a / f a \u2208 u", "theorem TopologicalSemiring.toTopologicalRing [TopologicalSpace \u03b1] [NonAssocRing \u03b1]\n    (_ : TopologicalSemiring \u03b1) : TopologicalRing \u03b1 where", "@[simp] theorem coe_bot : (\u2191(\u22a5 : Clopens \u03b1) : Set \u03b1) = \u2205", "@[simp, norm_cast]\ntheorem coe_finset_inf (f : \u03b9 \u2192 Closeds \u03b1) (s : Finset \u03b9) :\n    (\u2191(s.inf f) : Set \u03b1) = s.inf ((\u2191) \u2218 f)", "nonrec theorem Closeds.compl_compl (s : Closeds \u03b1) : s.compl.compl = s", "@[simp] lemma mem_singleton [T1Space \u03b1] {a b : \u03b1} : a \u2208 singleton b \u2194 a = b := Iff.rfl", "theorem TopologicalSpace.IsTopologicalBasis.inseparable_iff {b : Set (Set X)}\n    (hb : IsTopologicalBasis b) {x y : X} : Inseparable x y \u2194 \u2200 s \u2208 b, (x \u2208 s \u2194 y \u2208 s)", "theorem IsTopologicalBasis.isOpen_iff {s : Set \u03b1} {b : Set (Set \u03b1)} (hb : IsTopologicalBasis b) :\n    IsOpen s \u2194 \u2200 a \u2208 s, \u2203 t \u2208 b, a \u2208 t \u2227 t \u2286 s", "lemma IsTopologicalBasis.subset_of_forall_subset {t : Set \u03b1} (hB : IsTopologicalBasis B)\n    (hs : IsOpen s) (h : \u2200 U \u2208 B, U \u2286 s \u2192 U \u2286 t) : s \u2286 t", "theorem NoetherianSpace.exists_open_ne_empty_le_irreducibleComponent [NoetherianSpace \u03b1]\n    (Z : Set \u03b1) (H : Z \u2208 irreducibleComponents \u03b1) :\n    \u2203 o : Set \u03b1, IsOpen o \u2227 o \u2260 \u2205 \u2227 o \u2264 Z", "@[simp]\ntheorem map_id_obj_unop (x : X) (U : (OpenNhds x)\u1d52\u1d56) : (map (\ud835\udfd9 X) x).obj (unop U) = unop U", "theorem dist_val_le_dist (x y : CompleteCopy s) : dist x.1 y.1 \u2264 dist x y", "@[simp, norm_cast]\ntheorem coe_sup (s t : Opens \u03b1) : (\u2191(s \u2294 t) : Set \u03b1) = \u2191s \u222a \u2191t", "theorem gc : GaloisConnection ((\u2191) : Opens \u03b1 \u2192 Set \u03b1) interior", "theorem isBasis_iff_cover {B : Set (Opens \u03b1)} :\n    IsBasis B \u2194 \u2200 U : Opens \u03b1, \u2203 Us, Us \u2286 B \u2227 U = sSup Us", "@[simp]\nlemma map_top (f : X \u27f6 Y) : (Opens.map f).obj \u22a4 = \u22a4 := rfl", "theorem not_nonempty_iff_eq_bot (U : Opens \u03b1) : \u00acSet.Nonempty (U : Set \u03b1) \u2194 U = \u22a5", "@[simp]\ntheorem openEmbedding_obj_top {X : TopCat} (U : Opens X) :\n    U.openEmbedding.isOpenMap.functor.obj \u22a4 = U", "theorem carrier_eq_coe (s : PositiveCompacts \u03b1) : s.carrier = s", "@[simp]\ntheorem coe_mk (s : Compacts \u03b1) (h) : (mk s h : Set \u03b1) = s", "@[simp]\ntheorem coe_prod (K : PositiveCompacts \u03b1) (L : PositiveCompacts \u03b2) :\n    (K.prod L : Set (\u03b1 \u00d7 \u03b2)) = (K : Set \u03b1) \u00d7\u02e2 (L : Set \u03b2)", "theorem map_comp (f : \u03b2 \u2192 \u03b3) (g : \u03b1 \u2192 \u03b2) (hf : Continuous f) (hg : Continuous g) (hf' : IsOpenMap f)\n    (hg' : IsOpenMap g) (K : PositiveCompacts \u03b1) :\n    K.map (f \u2218 g) (hf.comp hg) (hf'.comp hg') = (K.map g hg hg').map f hf hf'", "theorem TopologicalSpace.ext_iff_isClosed {t\u2081 t\u2082 : TopologicalSpace X} :\n    t\u2081 = t\u2082 \u2194 \u2200 s, IsClosed[t\u2081] s \u2194 IsClosed[t\u2082] s", "theorem TopologicalSpace.isOpen_top_iff {\u03b1} (U : Set \u03b1) : IsOpen[\u22a4] U \u2194 U = \u2205 \u2228 U = univ", "protected theorem le_def {\u03b1} {t s : TopologicalSpace \u03b1} : t \u2264 s \u2194 IsOpen[s] \u2264 IsOpen[t]", "theorem nhds_generateFrom {g : Set (Set \u03b1)} {a : \u03b1} :\n    @nhds \u03b1 (generateFrom g) a = \u2a05 s \u2208 { s | a \u2208 s \u2227 s \u2208 g }, \ud835\udcdf s", "lemma topology_eq : \u2039_\u203a = lower \u03b1 := topology_eq_lowerTopology", "lemma isClosed_iff_isUpper : IsClosed s \u2194 IsUpperSet s", "@[simp] lemma closure_singleton : closure {a} = Iic a := le_antisymm\n  (closure_minimal (by rw [singleton_subset_iff, mem_Iic]) isClosed_Iic) <| by\n    rw [\u2190 LowerSet.coe_Iic, \u2190 lowerClosure_singleton]\n    apply lowerClosure_subset_closure", "@[simp] lemma ofLawson_toLawson (a : \u03b1) : ofLawson (toLawson a) = a := rfl", "@[simp] lemma toLower_ofLower (a : WithLower \u03b1) : toLower (ofLower a) = a := rfl", "@[simp] lemma map_id : map (OrderHom.id : \u03b1 \u2192o \u03b1) = ContinuousMap.id _ := rfl", "@[simp] lemma ofLowerSet_toLowerSet (a : \u03b1) : ofLowerSet (toLowerSet a) = a := rfl", "@[simp] lemma toLowerSet_ofLowerSet (a : WithLowerSet \u03b1) : toLowerSet (ofLowerSet a) = a := rfl", "@[simp] lemma toUpperSet_ofUpperSet (a : WithUpperSet \u03b1) : toUpperSet (ofUpperSet a) = a := rfl", "@[simp] lemma toUpperSet_specializes_toUpperSet {a b : \u03b1} :\n    toUpperSet a \u2933 toUpperSet b \u2194 b \u2264 a", "lemma scottHausdorff_le_lawson : scottHausdorff \u03b1  \u2264 lawson \u03b1", "lemma scottHausdorff_le_scott : scottHausdorff \u03b1 \u2264 scott \u03b1 := le_sup_right", "theorem comp_sequence (x : t (F (G \u03b1))) :\n    sequence (Comp.mk <$> x) = Comp.mk (sequence <$> sequence x)", "theorem foldlm_toList {f : \u03b1 \u2192 \u03b2 \u2192 m \u03b1} {x : \u03b1} {xs : t \u03b2} :\n    foldlm f x xs = List.foldlM f x (toList xs)", "@[simp]\ntheorem foldr.ofFreeMonoid_comp_of (f : \u03b2 \u2192 \u03b1 \u2192 \u03b1) :\n    Foldr.ofFreeMonoid f \u2218 FreeMonoid.of = Foldr.mk \u2218 f", "@[simp]\ntheorem foldrm.ofFreeMonoid_comp_of {m} [Monad m] [LawfulMonad m] (f : \u03b2 \u2192 \u03b1 \u2192 m \u03b1) :\n    foldrM.ofFreeMonoid f \u2218 FreeMonoid.of = foldrM.mk \u2218 f", "theorem naturality' (\u03b7 : ApplicativeTransformation F G) (x : t (F \u03b1)) :\n    \u03b7 (sequence x) = sequence (@\u03b7 _ <$> x)", "theorem traverse_map' (g : \u03b1 \u2192 \u03b2) (h : \u03b2 \u2192 G \u03b3) :\n    traverse (h \u2218 g) = (traverse h \u2218 map g : t \u03b1 \u2192 G (t \u03b3))", "theorem height_le_numNodes : \u2200 x : Tree \u03b1, x.height \u2264 x.numNodes", "nonrec theorem continuous_snd : Continuous (snd : tsze R M \u2192 M)", "theorem exp_def_of_smul_comm (x : tsze R M) (hx : MulOpposite.op x.fst \u2022 x.snd = x.fst \u2022 x.snd) :\n    exp \ud835\udd5c x = inl (exp \ud835\udd5c x.fst) + inr (exp \ud835\udd5c x.fst \u2022 x.snd)", "@[simp]\ntheorem fst_inr [Zero R] (m : M) : (inr m : tsze R M).fst = 0", "@[simp]\ntheorem inl_pow [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction R\u1d50\u1d52\u1d56 M] (r : R)\n    (n : \u2115) : (inl r ^ n : tsze R M) = inl (r ^ n)", "@[simp]\ntheorem lift_comp_inrHom (f : R \u2192\u2090[S] A) (g : M \u2192\u2097[S] A)\n    (hg : \u2200 x y, g x * g y = 0)\n    (hfg : \u2200 r x, g (r \u2022> x) = f r * g x)\n    (hgf : \u2200 r x, g (x <\u2022 r) = g x * f r) :\n    (lift f g hg hfg hgf).toLinearMap.comp (inrHom R M |>.restrictScalars S) = g", "theorem mul_inl_eq_op_smul [Semiring R] [AddCommMonoid M] [Module R M] [Module R\u1d50\u1d52\u1d56 M]\n    (x : tsze R M) (r : R) :\n    x * inl r = x <\u2022 r", "protected theorem mul_inv_cancel {x : tsze R M} (hx : fst x \u2260 0) : x * x\u207b\u00b9 = 1", "protected theorem mul_inv_rev (a b : tsze R M) :\n    (a * b)\u207b\u00b9 = b\u207b\u00b9 * a\u207b\u00b9", "theorem nhds_def (x : tsze R M) : \ud835\udcdd x = (\ud835\udcdd x.fst).prod (\ud835\udcdd x.snd)", "@[simp]\ntheorem snd_mk (r : R) (m : M) : snd (r, m) = m", "theorem snd_pow_of_smul_comm [Monoid R] [AddMonoid M] [DistribMulAction R M]\n    [DistribMulAction R\u1d50\u1d52\u1d56 M] [SMulCommClass R R\u1d50\u1d52\u1d56 M] (x : tsze R M) (n : \u2115)\n    (h : x.snd <\u2022 x.fst = x.fst \u2022> x.snd) : snd (x ^ n) = n \u2022 x.fst ^ n.pred \u2022> x.snd", "theorem continuousAt_of_comp_left {X : Type*} [TopologicalSpace X] {f : X \u2192 Z} {x : X}\n    (e : Trivialization F proj) (hf_proj : ContinuousAt (proj \u2218 f) x) (he : proj (f x) \u2208 e.baseSet)\n    (hf : ContinuousAt (e \u2218 f) x) : ContinuousAt f x", "@[simp] theorem Trivialization.continuousLinearEquivAt_prod {e\u2081 : Trivialization F\u2081 (\u03c0 F\u2081 E\u2081)}\n    {e\u2082 : Trivialization F\u2082 (\u03c0 F\u2082 E\u2082)} [e\u2081.IsLinear \ud835\udd5c] [e\u2082.IsLinear \ud835\udd5c] {x : B}\n    (hx : x \u2208 (e\u2081.prod e\u2082).baseSet) :\n    (e\u2081.prod e\u2082).continuousLinearEquivAt \ud835\udd5c x hx =\n      (e\u2081.continuousLinearEquivAt \ud835\udd5c x hx.1).prod (e\u2082.continuousLinearEquivAt \ud835\udd5c x hx.2)", "protected theorem linear [AddCommMonoid F] [Module R F] [\u2200 x, AddCommMonoid (E x)]\n    [\u2200 x, Module R (E x)] [e.IsLinear R] {b : B} (hb : b \u2208 e.baseSet) :\n    IsLinearMap R fun y : E b => (e \u27e8b, y\u27e9).2", "theorem linearMapAt_def_of_mem (e : Trivialization F (\u03c0 F E)) [e.IsLinear R] {b : B}\n    (hb : b \u2208 e.baseSet) : e.linearMapAt R b = e.linearEquivAt R b hb", "theorem mem_target {x : B \u00d7 F} : x \u2208 e.target \u2194 x.1 \u2208 e.baseSet", "theorem mk_symm (e : Trivialization F (\u03c0 F E)) {b : B} (hb : b \u2208 e.baseSet) (y : F) :\n    TotalSpace.mk b (e.symm b y) = e.toPartialHomeomorph.symm (b, y)", "@[simp]\ntheorem preimageHomeomorph_apply {s : Set B} (hb : s \u2286 e.baseSet) (p : proj \u207b\u00b9' s) :\n    e.preimageHomeomorph hb p = (\u27e8proj p, p.2\u27e9, (e p).2)", "theorem prod_apply [e\u2081.IsLinear \ud835\udd5c] [e\u2082.IsLinear \ud835\udd5c] {x : B} (hx\u2081 : x \u2208 e\u2081.baseSet)\n    (hx\u2082 : x \u2208 e\u2082.baseSet) (v\u2081 : E\u2081 x) (v\u2082 : E\u2082 x) :\n    prod e\u2081 e\u2082 \u27e8x, (v\u2081, v\u2082)\u27e9 =\n      \u27e8x, e\u2081.continuousLinearEquivAt \ud835\udd5c x hx\u2081 v\u2081, e\u2082.continuousLinearEquivAt \ud835\udd5c x hx\u2082 v\u2082\u27e9", "theorem Trivialization.smoothOn_iff {f : M \u2192 TotalSpace F E} {s : Set M}\n    (he : MapsTo f s e.source) :\n    SmoothOn IM (IB.prod \ud835\udcd8(\ud835\udd5c, F)) f s \u2194\n      SmoothOn IM IB (fun x => (f x).proj) s \u2227 SmoothOn IM \ud835\udcd8(\ud835\udd5c, F) (fun x \u21a6 (e (f x)).2) s", "@[simp]\ntheorem add_eq_zero_iff {a b : Tropical (WithTop R)} : a + b = 0 \u2194 a = 0 \u2227 b = 0", "theorem rightInverse_trop : Function.RightInverse (trop : R \u2192 Tropical R) untrop", "@[simp]\ntheorem tropOrderIso_coe_fn [Preorder R] : (tropOrderIso : R \u2192 Tropical R) = trop", "theorem Trunc.finChoice_eq {\u03b9 : Type*} [DecidableEq \u03b9] [Fintype \u03b9] {\u03b1 : \u03b9 \u2192 Type*}\n    (f : \u2200 i, \u03b1 i) : (Trunc.finChoice fun i => Trunc.mk (f i)) = Trunc.mk f", "theorem ind {\u03b2 : Trunc \u03b1 \u2192 Prop} : (\u2200 a : \u03b1, \u03b2 (mk a)) \u2192 \u2200 q : Trunc \u03b1, \u03b2 q", "theorem card {R : Type*} [Fintype R] :\n    Fintype.card (TruncatedWittVector p n R) = Fintype.card R ^ n", "theorem out_injective : Injective (@out p n R _)", "theorem antitone_pair_of_not_sorted (h : f \u2260 f \u2218 sort f) : \u2203 i j, i < j \u2227 f j < f i", "theorem eq_sort_iff' : \u03c3 = sort f \u2194 StrictMono (\u03c3.trans <| graphEquiv\u2081 f)", "@[simp]\ntheorem graph.card (f : Fin n \u2192 \u03b1) : (graph f).card = n", "theorem graphEquiv\u2082_apply (f : Fin n \u2192 \u03b1) (i : Fin n) :\n    graphEquiv\u2082 f i = graphEquiv\u2081 f (sort f i)", "theorem lt_card_ge_iff_apply_ge_of_antitone [PartialOrder \u03b1] [DecidableRel (\u03b1 := \u03b1) LE.le]\n    {m : \u2115} (f : Fin m \u2192 \u03b1) (a : \u03b1) (h_sorted : Antitone f) (j : Fin m) :\n    j < Fintype.card {i // a \u2264 f i} \u2194 a \u2264 f j", "theorem unique_monotone [PartialOrder \u03b1] {f : Fin n \u2192 \u03b1} {\u03c3 \u03c4 : Equiv.Perm (Fin n)}\n    (hf\u03c3 : Monotone (f \u2218 \u03c3)) (hf\u03c4 : Monotone (f \u2218 \u03c4)) : f \u2218 \u03c3 = f \u2218 \u03c4", "theorem BlankRel.equivalence (\u0393) [Inhabited \u0393] : Equivalence (@BlankRel \u0393 _)", "theorem ListBlank.exists_cons {\u0393} [Inhabited \u0393] (l : ListBlank \u0393) :\n    \u2203 a l', l = ListBlank.cons a l'", "theorem K'.elim_rev (a b c d) : K'.elim a b c d K'.rev = b", "@[simp]\ntheorem K'.elim_update_stack {a b c d d'} :\n    update (K'.elim a b c d) stack d' = K'.elim a b c d'", "theorem copy_ok (q s a b c d) :\n    Reaches\u2081 (TM2.step tr) \u27e8some (\u039b'.copy q), s, K'.elim a b c d\u27e9\n      \u27e8some q, none, K'.elim (List.reverseAux b a) [] c (List.reverseAux b d)\u27e9", "theorem trNormal_respects (c k v s) :\n    \u2203 b\u2082,\n      TrCfg (stepNormal c k v) b\u2082 \u2227\n        Reaches\u2081 (TM2.step tr)\n          \u27e8some (trNormal c (trCont k)), s, K'.elim (trList v) [] [] (trContStack k)\u27e9 b\u2082", "theorem trStmts\u2081_trans {q q'} : q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q", "theorem tr_pred (q\u2081 q\u2082) : tr (\u039b'.pred q\u2081 q\u2082) = pop' main (branch (fun s => s = some \u0393'.bit0)\n    ((push rev fun _ => \u0393'.bit1) <| goto fun _ => \u039b'.pred q\u2081 q\u2082) <|\n    branch (fun s => natEnd s.iget) (goto fun _ => q\u2081)\n      (peek' main <|\n        branch (fun s => natEnd s.iget) (goto fun _ => unrev q\u2082)\n          ((push rev fun _ => \u0393'.bit0) <| goto fun _ => unrev q\u2082)))", "theorem tr_ret_halt : tr (\u039b'.ret Cont'.halt) = (load fun _ => none) halt", "theorem Reaches\u2080.single {\u03c3} {f : \u03c3 \u2192 Option \u03c3} {a b : \u03c3} (h : b \u2208 f a) : Reaches\u2080 f a b", "theorem Reaches\u2080.tail' {\u03c3} {f : \u03c3 \u2192 Option \u03c3} {a b c : \u03c3} (h : Reaches\u2080 f a b) (h\u2082 : c \u2208 f b) :\n    Reaches\u2081 f a c", "theorem Reaches\u2081.to\u2080 {\u03c3} {f : \u03c3 \u2192 Option \u03c3} {a b : \u03c3} (h : Reaches\u2081 f a b) : Reaches\u2080 f a b", "theorem supportsStmt_write {S : Finset \u039b'\u2081} {l : List Bool} {q : Stmt'\u2081} :\n    SupportsStmt S (write l q) = SupportsStmt S q", "theorem tr_supports {S : Finset \u039b} (ss : Supports M S) : Supports (tr enc dec M) (trSupp M S)", "theorem addBottom_modifyNth (f : (\u2200 k, Option (\u0393 k)) \u2192 \u2200 k, Option (\u0393 k))\n    (L : ListBlank (\u2200 k, Option (\u0393 k))) (n : \u2115) :\n    (addBottom L).modifyNth (fun a \u21a6 (a.1, f a.2)) n = addBottom (L.modifyNth f n)", "theorem addBottom_nth_snd (L : ListBlank (\u2200 k, Option (\u0393 k))) (n : \u2115) :\n    ((addBottom L).nth n).2 = L.nth n", "theorem Tape.map_mk\u2081 {\u0393 \u0393'} [Inhabited \u0393] [Inhabited \u0393'] (f : PointedMap \u0393 \u0393') (l : List \u0393) :\n    (Tape.mk\u2081 l).map f = Tape.mk\u2081 (l.map f)", "@[simp]\ntheorem Tape.mk'_nth_nat {\u0393} [Inhabited \u0393] (L R : ListBlank \u0393) (n : \u2115) :\n    (Tape.mk' L R).nth n = R.nth n", "@[simp]\ntheorem head_eval (v) : head.eval v = pure [v.headI]", "theorem proj_map_nth {\u03b9 : Type*} {\u0393 : \u03b9 \u2192 Type*} [\u2200 i, Inhabited (\u0393 i)] (i : \u03b9) (L n) :\n    (ListBlank.map (@proj \u03b9 \u0393 _ i) L).nth n = L.nth n i", "theorem reaches_total {\u03c3} {f : \u03c3 \u2192 Option \u03c3} {a b c} (hab : Reaches f a b) (hac : Reaches f a c) :\n    Reaches f b c \u2228 Reaches f c b", "theorem tr_eval_rev {\u03c3\u2081 \u03c3\u2082 f\u2081 f\u2082} {tr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop} (H : Respects f\u2081 f\u2082 tr) {a\u2081 b\u2082 a\u2082}\n    (aa : tr a\u2081 a\u2082) (ab : b\u2082 \u2208 eval f\u2082 a\u2082) : \u2203 b\u2081, tr b\u2081 b\u2082 \u2227 b\u2081 \u2208 eval f\u2081 a\u2081", "theorem tr_reaches_rev {\u03c3\u2081 \u03c3\u2082 f\u2081 f\u2082} {tr : \u03c3\u2081 \u2192 \u03c3\u2082 \u2192 Prop} (H : Respects f\u2081 f\u2082 tr) {a\u2081 a\u2082}\n    (aa : tr a\u2081 a\u2082) {b\u2082} (ab : Reaches f\u2082 a\u2082 b\u2082) :\n    \u2203 c\u2081 c\u2082, Reaches f\u2082 b\u2082 c\u2082 \u2227 tr c\u2081 c\u2082 \u2227 Reaches f\u2081 a\u2081 c\u2081", "@[simp]\ntheorem nonempty_two_pointing_iff : Nonempty (TwoPointing \u03b1) \u2194 Nontrivial \u03b1", "@[simp]\ntheorem sum_snd : (p.sum q).snd = Sum.inr q.snd", "@[to_additive]\ntheorem of_injective_mulHom (f : H \u2192\u2099* G) (hf : Function.Injective f)\n    (_ : TwoUniqueProds G) : TwoUniqueProds H", "@[to_additive]\ntheorem of_mulOpposite (h : TwoUniqueProds G\u1d50\u1d52\u1d56) : TwoUniqueProds G where", "@[simp]\ntheorem comp_id {\u03b1 \u03b2 : TypeVec n} (f : \u03b1 \u27f9 \u03b2) : f \u229a id = f", "@[simp]\ntheorem dropFun_from_append1_drop_last {\u03b1 : TypeVec (n + 1)} :\n    dropFun (@fromAppend1DropLast _ \u03b1) = id", "@[simp]\ntheorem lastFun_from_append1_drop_last {\u03b1 : TypeVec (n + 1)} :\n    lastFun (@fromAppend1DropLast _ \u03b1) = _root_.id", "@[simp]\ntheorem prod_snd_mk {\u03b1 \u03b2 : TypeVec n} (i : Fin2 n) (a : \u03b1 i) (b : \u03b2 i) :\n    TypeVec.prod.snd i (prod.mk i a b) = b", "@[simp] theorem UInt32.toUInt16_toNat (x : UInt32) : x.toUInt16.toNat = x.toNat % 2 ^ 16", "@[simp] theorem UInt32.val_ofNat (n) :\n    (no_index (OfNat.ofNat n) : UInt32).val = OfNat.ofNat n", "theorem UInt64.toNat_add (x y : UInt64) : (x + y).toNat = (x.toNat + y.toNat) % UInt64.size", "theorem UInt8.le_antisymm {x y : UInt8} (h1 : x \u2264 y) (h2 : y \u2264 x) : x = y", "theorem ULift.closedEmbedding_down [TopologicalSpace X] :\n    ClosedEmbedding (ULift.down : ULift.{v, u} X \u2192 X)", "@[simp]\ntheorem ULift.edist_up_up (x y : \u03b1) : edist (ULift.up x) (ULift.up y) = edist x y", "@[simp] lemma nnnorm_up (x : E) : \u2016ULift.up x\u2016\u208a = \u2016x\u2016\u208a := rfl", "@[to_additive (attr := simp)]\ntheorem one_down [One \u03b1] : (1 : ULift \u03b1).down = 1", "@[simp] theorem up_sup [Sup \u03b1] (a b : \u03b1) : up (a \u2294 b) = up a \u2294 up b", "theorem up_surjective : Surjective (@up \u03b1)", "theorem down_up {f : Type u\u2080 \u2192 Type u\u2081} {g : Type max u\u2080 v\u2080 \u2192 Type v\u2081} [ULiftable f g] {\u03b1}\n    (x : f \u03b1) : down (up x : g (ULift.{v\u2080} \u03b1)) = x", "theorem USize.toNat_div (x y : USize) : (x / y).toNat = x.toNat / y.toNat", "theorem disjoint_of_mem_compression_of_not_mem (h : a \u2208 \ud835\udcd2 u v s) (ha : a \u2209 s) : Disjoint v a", "theorem comap_inf_principal_neBot_of_image_mem (h : m '' s \u2208 g) : (Filter.comap m g \u2293 \ud835\udcdf s).NeBot", "@[to_additive]\ntheorem Ultrafilter.continuous_mul_left {M} [Semigroup M] (V : Ultrafilter M) :\n    Continuous (\u00b7 * V)", "@[ext]\ntheorem ext \u2983f g : Ultrafilter \u03b1\u2984 (h : \u2200 s, s \u2208 f \u2194 s \u2208 g) : f = g", "@[simp]\ntheorem frequently_iff_eventually : (\u2203\u1da0 x in f, p x) \u2194 \u2200\u1da0 x in f, p x", "theorem ofComapInfPrincipal_eq_of_map (h : m '' s \u2208 g) : (ofComapInfPrincipal h).map m = g", "theorem UniformCauchySeqOn.prod_map {\u03b9' \u03b1' \u03b2' : Type*} [UniformSpace \u03b2'] {F' : \u03b9' \u2192 \u03b1' \u2192 \u03b2'}\n    {p' : Filter \u03b9'} {s' : Set \u03b1'} (h : UniformCauchySeqOn F p s)\n    (h' : UniformCauchySeqOn F' p' s') :\n    UniformCauchySeqOn (fun i : \u03b9 \u00d7 \u03b9' => Prod.map (F i.1) (F' i.2)) (p \u00d7\u02e2 p') (s \u00d7\u02e2 s')", "theorem topologicalSpace_mono [TopologicalSpace F] [TopologicalAddGroup F] (h : \ud835\udd16\u2082 \u2286 \ud835\udd16\u2081) :\n    instTopologicalSpace \u03c3 F \ud835\udd16\u2081 \u2264 instTopologicalSpace \u03c3 F \ud835\udd16\u2082", "@[simp]\ntheorem apply_symm_apply (h : \u03b1 \u2243\u1d64 \u03b2) (x : \u03b2) : h (h.symm x) = x", "@[simp]\ntheorem coe_prodComm : \u21d1(prodComm \u03b1 \u03b2) = Prod.swap", "protected theorem gc : GaloisConnection lowerAdjoint fun \ud835\udcd5 => UniformFun.filter \u03b1 \u03b2 \ud835\udcd5", "protected theorem iInf_eq {u : \u03b9 \u2192 UniformSpace \u03b3} : \ud835\udcb0(\u03b1, \u03b3, (\u2a05 i, u i)) = \u2a05 i, \ud835\udcb0(\u03b1, \u03b3, u i)", "theorem isClosed_setOf_continuous [TopologicalSpace \u03b1] :\n    IsClosed {f : \u03b1 \u2192\u1d64 \u03b2 | Continuous (toFun f)}", "theorem UniformInducing.denseInducing {f : \u03b1 \u2192 \u03b2} (h : UniformInducing f) (hd : DenseRange f) :\n    DenseInducing f", "theorem UniformInducing.equicontinuousWithinAt_iff {F : \u03b9 \u2192 X \u2192 \u03b1} {S : Set X} {x\u2080 : X} {u : \u03b1 \u2192 \u03b2}\n    (hu : UniformInducing u) : EquicontinuousWithinAt F S x\u2080 \u2194\n      EquicontinuousWithinAt ((u \u2218 \u00b7) \u2218 F) S x\u2080", "theorem UniformInducing.uniformContinuous {f : \u03b1 \u2192 \u03b2} (hf : UniformInducing f) :\n    UniformContinuous f", "protected theorem isCountablyGenerated_uniformity [IsCountablyGenerated (\ud835\udce4 \u03b2)] {t : \u2115 \u2192 Set \u03b1}\n    (ht : \u2200 n, t n \u2208 \ud835\udd16) (hmono : Monotone t) (hex : \u2200 s \u2208 \ud835\udd16, \u2203 n, s \u2286 t n) :\n    IsCountablyGenerated (\ud835\udce4 (\u03b1 \u2192\u1d64[\ud835\udd16] \u03b2))", "@[to_additive (attr := simp)]\nlemma UniformOnFun.toFun_div [Div \u03b2] (f g : \u03b1 \u2192\u1d64[\ud835\udd16] \u03b2) :\n    toFun \ud835\udd16 (f / g) = toFun \ud835\udd16 f / toFun \ud835\udd16 g", "@[simp]\nlemma UniformOnFun.toFun_smul {M : Type*} [SMul M \u03b2] (c : M) (f : \u03b1 \u2192\u1d64[\ud835\udd16] \u03b2) :\n    toFun \ud835\udd16 (c \u2022 f) = c \u2022 toFun \ud835\udd16 f", "theorem uniformSpace_eq_iInf_precomp_of_cover {\u03b4 : \u03b9 \u2192 Type*} (\u03c6 : \u03a0 i, \u03b4 i \u2192 \u03b1)\n    (\ud835\udd17 : \u2200 i, Set (Set (\u03b4 i))) (h_image : \u2200 i, MapsTo (\u03c6 i '' \u00b7) (\ud835\udd17 i) \ud835\udd16)\n    (h_preimage : \u2200 i, MapsTo (\u03c6 i \u207b\u00b9' \u00b7) \ud835\udd16 (\ud835\udd17 i))\n    (h_cover : \u2200 S \u2208 \ud835\udd16, \u2203 I : Set \u03b9, I.Finite \u2227 S \u2286 \u22c3 i \u2208 I, range (\u03c6 i)) :\n    \ud835\udcb1(\u03b1, \u03b2, \ud835\udd16, _) = \u2a05 i, .comap (ofFun (\ud835\udd17 i) \u2218 (\u00b7 \u2218 \u03c6 i) \u2218 toFun \ud835\udd16) \ud835\udcb1(\u03b4 i, \u03b2, \ud835\udd17 i, _)", "theorem uniformSpace_eq_inf_precomp_of_cover {\u03b4\u2081 \u03b4\u2082 : Type*} (\u03c6\u2081 : \u03b4\u2081 \u2192 \u03b1) (\u03c6\u2082 : \u03b4\u2082 \u2192 \u03b1)\n    (\ud835\udd17\u2081 : Set (Set \u03b4\u2081)) (\ud835\udd17\u2082 : Set (Set \u03b4\u2082))\n    (h_image\u2081 : MapsTo (\u03c6\u2081 '' \u00b7) \ud835\udd17\u2081 \ud835\udd16) (h_image\u2082 : MapsTo (\u03c6\u2082 '' \u00b7) \ud835\udd17\u2082 \ud835\udd16)\n    (h_preimage\u2081 : MapsTo (\u03c6\u2081 \u207b\u00b9' \u00b7) \ud835\udd16 \ud835\udd17\u2081) (h_preimage\u2082 : MapsTo (\u03c6\u2082 \u207b\u00b9' \u00b7) \ud835\udd16 \ud835\udd17\u2082)\n    (h_cover : \u2200 S \u2208 \ud835\udd16, S \u2286 range \u03c6\u2081 \u222a range \u03c6\u2082) :\n    \ud835\udcb1(\u03b1, \u03b2, \ud835\udd16, _) =\n      .comap (ofFun \ud835\udd17\u2081 \u2218 (\u00b7 \u2218 \u03c6\u2081) \u2218 toFun \ud835\udd16) \ud835\udcb1(\u03b4\u2081, \u03b2, \ud835\udd17\u2081, _) \u2293\n      .comap (ofFun \ud835\udd17\u2082 \u2218 (\u00b7 \u2218 \u03c6\u2082) \u2218 toFun \ud835\udd16) \ud835\udcb1(\u03b4\u2082, \u03b2, \ud835\udd17\u2082, _)", "theorem Continuous.mul {\u03b2 : Type*} [TopologicalSpace \u03b2] {f g : \u03b2 \u2192 Completion \u03b1}\n    (hf : Continuous f) (hg : Continuous g) : Continuous fun b => f b * g b", "theorem coe_isometry : Isometry ((\u2191) : \u03b1 \u2192 Completion \u03b1)", "@[norm_cast]\ntheorem coe_one : ((1 : \u03b1) : Completion \u03b1) = 1", "theorem comap_coe_eq_uniformity :\n    ((\ud835\udce4 _).comap fun p : \u03b1 \u00d7 \u03b1 => ((p.1 : Completion \u03b1), (p.2 : Completion \u03b1))) = \ud835\udce4 \u03b1", "theorem extension_coe [T0Space \u03b2] (hf : UniformContinuous f) (a : \u03b1) :\n    (Completion.extension f) a = f a", "theorem UniformSpace.comap_inf {\u03b1 \u03b3} {u\u2081 u\u2082 : UniformSpace \u03b3} {f : \u03b1 \u2192 \u03b3} :\n    (u\u2081 \u2293 u\u2082).comap f = u\u2081.comap f \u2293 u\u2082.comap f", "theorem UniformSpace.compactSpace_iff_seqCompactSpace : CompactSpace X \u2194 SeqCompactSpace X", "theorem UniformSpace.hasBasis_nhds_prod (x y : \u03b1) :\n    HasBasis (\ud835\udcdd (x, y)) (fun s => s \u2208 \ud835\udce4 \u03b1 \u2227 SymmetricRel s) fun s => ball x s \u00d7\u02e2 ball y s", "theorem UniformSpace.mem_closure_iff_ball {s : Set \u03b1} {x} :\n    x \u2208 closure s \u2194 \u2200 {V}, V \u2208 \ud835\udce4 \u03b1 \u2192 (ball x V \u2229 s).Nonempty", "theorem UniformSpace.mem_closure_iff_symm_ball {s : Set \u03b1} {x} :\n    x \u2208 closure s \u2194 \u2200 {V}, V \u2208 \ud835\udce4 \u03b1 \u2192 SymmetricRel V \u2192 (s \u2229 ball x V).Nonempty", "protected theorem UniformSpace.metrizable_uniformity (X : Type*) [UniformSpace X]\n    [IsCountablyGenerated (\ud835\udce4 X)] : \u2203 I : PseudoMetricSpace X, I.toUniformSpace = \u2039_\u203a", "theorem UniformSpace.replaceTopology_eq {\u03b1 : Type*} [i : TopologicalSpace \u03b1] (u : UniformSpace \u03b1)\n    (h : i = u.toTopologicalSpace) : u.replaceTopology h = u", "theorem toTopologicalSpace_iInf {\u03b9 : Sort*} {u : \u03b9 \u2192 UniformSpace \u03b1} :\n    (iInf u).toTopologicalSpace = \u2a05 i, (u i).toTopologicalSpace", "theorem model' (self : UnionFind \u03b1) : \u2203 (m : UFModel self.arr.size), m.Models self.arr", "theorem eq_default (a : \u03b1) : a = default", "theorem UniqueDiffWithinAt.univ_pi (\u03b9 : Type*) [Finite \u03b9] (E : \u03b9 \u2192 Type*)\n    [\u2200 i, NormedAddCommGroup (E i)] [\u2200 i, NormedSpace \ud835\udd5c (E i)] (s : \u2200 i, Set (E i)) (x : \u2200 i, E i)\n    (h : \u2200 i, UniqueDiffWithinAt \ud835\udd5c (s i) (x i)) : UniqueDiffWithinAt \ud835\udd5c (Set.pi univ s) x", "theorem associated_iff_normalizedFactors_eq_normalizedFactors {x y : \u03b1} (hx : x \u2260 0) (hy : y \u2260 0) :\n    x ~\u1d64 y \u2194 normalizedFactors x = normalizedFactors y", "theorem count_normalizedFactors_eq' [DecidableEq R] {p x : R} (hp : p = 0 \u2228 Irreducible p)\n    (hnorm : normalize p = p) {n : \u2115} (hle : p ^ n \u2223 x) (hlt : \u00acp ^ (n + 1) \u2223 x) :\n    (normalizedFactors x).count p = n", "@[simp]\ntheorem factors_one : factors (1 : \u03b1) = 0", "theorem normalizedFactors_of_irreducible_pow {p : \u03b1} (hp : Irreducible p) (k : \u2115) :\n    normalizedFactors (p ^ k) = Multiset.replicate k (normalize p)", "@[simp]\ntheorem normalizedFactors_zero : normalizedFactors (0 : \u03b1) = 0", "theorem UniqueMDiffOn.tangentBundle_proj_preimage (hs : UniqueMDiffOn I s) :\n    UniqueMDiffOn I.tangent (\u03c0 E (TangentSpace I) \u207b\u00b9' s)", "@[to_additive \"`UniqueAdd` is preserved by inverse images under injective, additive maps.\"]\ntheorem mulHom_preimage (f : G \u2192\u2099* H) (hf : Function.Injective f) (a0 b0 : G) {A B : Finset H}\n    (u : UniqueMul A B (f a0) (f b0)) :\n    UniqueMul (A.preimage f hf.injOn) (B.preimage f hf.injOn) a0 b0", "@[to_additive] theorem toTwoUniqueProds_of_group {G}\n    [Group G] [UniqueProds G] : TwoUniqueProds G where", "lemma act_self_act_eq (x y : S) : x \u25c3 (x \u25c3 y) = x \u25c3 y", "lemma assoc (x y z : S) : (x \u25c3 y) \u25c3 z = x \u25c3 y \u25c3 z", "theorem antilipschitzWith_addEquiv :\n    AntilipschitzWith 2 (addEquiv \ud835\udd5c A)", "@[simp]\ntheorem fst_inl [Zero A] (r : R) : (inl r : Unitization R A).fst = r", "@[simp]\ntheorem inl_smul [Monoid S] [AddMonoid A] [SMul S R] [DistribMulAction S A] (s : S) (r : R) :\n    (inl (s \u2022 r) : Unitization R A) = s \u2022 inl r", "@[simp]\ntheorem inr_star [AddMonoid R] [StarAddMonoid R] [Star A] (a : A) :\n    \u2191(star a) = star (a : Unitization R A)", "theorem nnnorm_eq_sup (x : Unitization \ud835\udd5c A) :\n    \u2016x\u2016\u208a = \u2016x.fst\u2016\u208a \u2294 \u2016algebraMap \ud835\udd5c (A \u2192L[\ud835\udd5c] A) x.fst + mul \ud835\udd5c A x.snd\u2016\u208a", "theorem starLift_range_le\n    {f : A \u2192\u22c6\u2099\u2090[R] C} {S : StarSubalgebra R C} :\n    (starLift f).range \u2264 S \u2194 NonUnitalStarAlgHom.range f \u2264 S.toNonUnitalStarSubalgebra", "@[field_simps]\ntheorem add_divp (a b : \u03b1) (u : \u03b1\u02e3) : a + b /\u209a u = (a * u + b) /\u209a u", "@[simp]\ntheorem coe_map' {f : M \u2192 N} (hf : IsMonoidHom f) (x : M\u02e3) : \u2191((map' hf : M\u02e3 \u2192 N\u02e3) x) = f x", "@[fun_prop]\nlemma Units.continuousOn_zpow\u2080_spectrum (a : A\u02e3) (n : \u2124) :\n    ContinuousOn (\u00b7 ^ n) (spectrum R (a : A))", "@[field_simps]\ntheorem divp_add (a b : \u03b1) (u : \u03b1\u02e3) : a /\u209a u + b = (a + b * u) /\u209a u", "@[to_additive]\nprotected theorem inv_eq_of_mul_eq_one_left {a : \u03b1} (h : a * u = 1) : \u2191u\u207b\u00b9 = a", "@[to_additive]\nprotected theorem inv_eq_of_mul_eq_one_right {a : \u03b1} (h : \u2191u * a = 1) : \u2191u\u207b\u00b9 = a", "@[simp]\ntheorem Units.mem_posSubgroup {R : Type*} [LinearOrderedSemiring R] (u : R\u02e3) :\n    u \u2208 Units.posSubgroup R \u2194 (0 : R) < u", "theorem mul_left_dvd : \u2191u * a \u2223 b \u2194 a \u2223 b", "@[to_additive (attr := simp)]\ntheorem mul_right_inj (a : \u03b1\u02e3) {b c : \u03b1} : (a : \u03b1) * b = a * c \u2194 b = c", "@[simp]\ntheorem ne_zero [Nontrivial M\u2080] (u : M\u2080\u02e3) : (u : M\u2080) \u2260 0", "theorem Units.nnnorm_pos [Nontrivial \u03b1] (x : \u03b1\u02e3) : 0 < \u2016(x : \u03b1)\u2016\u208a", "@[simp high]\ntheorem SL_neg_smul (g : SL(2, \u2124)) (z : \u210d) : -g \u2022 z = g \u2022 z", "theorem dist_coe_le (z w : \u210d) : dist (z : \u2102) w \u2264 w.im * (Real.exp (dist z w) - 1)", "theorem im_smul (g : GL(2, \u211d)\u207a) (z : \u210d) : (g \u2022 z).im = (num g z / denom g z).im", "theorem modular_T_zpow_smul (z : \u210d) (n : \u2124) : ModularGroup.T ^ n \u2022 z = (n : \u211d) +\u1d65 z", "@[gcongr]\nlemma verticalStrip_anti_right (A) {B B'} (h : B' \u2264 B) : verticalStrip A B \u2286 verticalStrip A B'", "theorem zero_form_isBoundedAtImInfty {\u03b1 : Type*} [NormedField \u03b1] :\n    IsBoundedAtImInfty (0 : \u210d \u2192 \u03b1)", "theorem UpperSemicontinuous.add' {f g : \u03b1 \u2192 \u03b3} (hf : UpperSemicontinuous f)\n    (hg : UpperSemicontinuous g)\n    (hcont : \u2200 x, ContinuousAt (fun p : \u03b3 \u00d7 \u03b3 => p.1 + p.2) (f x, g x)) :\n    UpperSemicontinuous fun z => f z + g z", "theorem UpperSemicontinuousOn.add' {f g : \u03b1 \u2192 \u03b3} (hf : UpperSemicontinuousOn f s)\n    (hg : UpperSemicontinuousOn g s)\n    (hcont : \u2200 x \u2208 s, ContinuousAt (fun p : \u03b3 \u00d7 \u03b3 => p.1 + p.2) (f x, g x)) :\n    UpperSemicontinuousOn (fun z => f z + g z) s", "theorem UpperSemicontinuousOn.mono (h : UpperSemicontinuousOn f s) (hst : t \u2286 s) :\n    UpperSemicontinuousOn f t", "@[simp]\ntheorem coe_compl (s : UpperSet \u03b1) : (s.compl : Set \u03b1) = (\u2191s)\u1d9c", "@[norm_cast] theorem coe_iSup\u2082 (f : \u2200 i, \u03ba i \u2192 UpperSet \u03b1) :\n    (\u2191(\u2a06 (i) (j), f i j) : Set \u03b1) = \u22c2 (i) (j), f i j", "@[simp, norm_cast]\ntheorem coe_map : (map f s : Set \u03b2) = f '' s", "@[simp]\nprotected theorem compl_iInf (f : \u03b9 \u2192 UpperSet \u03b1) : (\u2a05 i, f i).compl = \u2a05 i, (f i).compl", "@[simp]\nprotected theorem compl_iSup (f : \u03b9 \u2192 UpperSet \u03b1) : (\u2a06 i, f i).compl = \u2a06 i, (f i).compl", "@[simp]\nprotected theorem compl_sInf (S : Set (UpperSet \u03b1)) : (sInf S).compl = \u2a05 s \u2208 S, UpperSet.compl s", "lemma erase_inf_Ici (ha : a \u2208 s) (has : \u2200 b \u2208 s, b \u2264 a \u2192 b = a) : s.erase a \u2293 Ici a = s", "@[simp] lemma lt_erase : s < s.erase a \u2194 a \u2208 s := le_erase.gt_iff_ne.trans erase_eq.not_left", "@[simp]\ntheorem mem_iSup_iff {f : \u03b9 \u2192 UpperSet \u03b1} : (a \u2208 \u2a06 i, f i) \u2194 \u2200 i, a \u2208 f i", "theorem prod_le_prod_iff : s\u2081 \u00d7\u02e2 t\u2081 \u2264 s\u2082 \u00d7\u02e2 t\u2082 \u2194 s\u2081 \u2264 s\u2082 \u2227 t\u2081 \u2264 t\u2082 \u2228 s\u2082 = \u22a4 \u2228 t\u2082 = \u22a4", "@[simp]\ntheorem prod_sup : s \u00d7\u02e2 (t\u2081 \u2294 t\u2082) = s \u00d7\u02e2 t\u2081 \u2294 s \u00d7\u02e2 t\u2082", "theorem continuous_lim (c : CU P) : Continuous c.lim", "theorem lim_eq_midpoint (c : CU P) (x : X) :\n    c.lim x = midpoint \u211d (c.left.lim x) (c.right.lim x)", "theorem le_of_dvd {x y : O} (h : x \u2223 y) : v (algebraMap O R y) \u2264 v (algebraMap O R x)", "@[trans]\ntheorem trans (h\u2081\u2082 : v\u2081.IsEquiv v\u2082) (h\u2082\u2083 : v\u2082.IsEquiv v\u2083) : v\u2081.IsEquiv v\u2083", "theorem isEquiv_iff_valuationSubring :\n    v\u2081.IsEquiv v\u2082 \u2194 v\u2081.valuationSubring = v\u2082.valuationSubring", "theorem isEquiv_of_val_le_one [LinearOrderedCommGroupWithZero \u0393\u2080]\n    [LinearOrderedCommGroupWithZero \u0393'\u2080] (v : Valuation K \u0393\u2080) (v' : Valuation K \u0393'\u2080)\n    (h : \u2200 {x : K}, v x \u2264 1 \u2194 v' x \u2264 1) : v.IsEquiv v'", "theorem isEquiv_valuation_valuationSubring : v.IsEquiv v.valuationSubring.valuation", "@[simp]\ntheorem map_add' : \u2200 x y, v (x + y) \u2264 v x \u2228 v (x + y) \u2264 v y", "@[simp]\ntheorem map_neg (x : R) : v (-x) = v x", "theorem map_one : v 1 = 1", "@[simp]\ntheorem mem_valuationSubring_iff (x : K) : x \u2208 v.valuationSubring \u2194 v x \u2264 1", "@[simp]\ntheorem onQuot_comap_eq {J : Ideal R} (hJ : J \u2264 supp v) :\n    (v.onQuot hJ).comap (Ideal.Quotient.mk J) = v", "theorem iff_isInteger_or_isInteger :\n    ValuationRing R \u2194 \u2200 x : K, IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9", "protected theorem tFAE (R : Type u) [CommRing R] [IsDomain R] :\n    List.TFAE\n      [ValuationRing R,\n        \u2200 x : FractionRing R, IsLocalization.IsInteger R x \u2228 IsLocalization.IsInteger R x\u207b\u00b9,\n        IsTotal R (\u00b7 \u2223 \u00b7), IsTotal (Ideal R) (\u00b7 \u2264 \u00b7), LocalRing R \u2227 IsBezout R]", "theorem unique_irreducible [ValuationRing R] \u2983p q : R\u2984 (hp : Irreducible p) (hq : Irreducible q) :\n    Associated p q", "@[simp]\ntheorem mapOfLE_valuation_apply (R S : ValuationSubring K) (h : R \u2264 S) (x : K) :\n    R.mapOfLE S h (R.valuation x) = S.valuation x", "@[simp]\ntheorem ofPrime_idealOfLE (R S : ValuationSubring K) (h : R \u2264 S) :\n    ofPrime R (idealOfLE R S h) = S", "@[simp]\ntheorem pointwise_smul_le_pointwise_smul_iff {g : G} {S T : ValuationSubring K} :\n    g \u2022 S \u2264 g \u2022 T \u2194 S \u2264 T", "theorem valuation_surjective : Function.Surjective A.valuation", "@[simp]\ntheorem get_cons_nil : \u2200 {ix : Fin 1} (x : \u03b1), get (x ::\u1d65 nil) ix = x", "@[simp]\ntheorem head_ofFn {n : \u2115} (f : Fin n.succ \u2192 \u03b1) : head (ofFn f) = f 0", "@[simp]\ntheorem inductionOn_cons {C : \u2200 {n : \u2115}, Vector \u03b1 n \u2192 Sort*} {n : \u2115} (x : \u03b1) (v : Vector \u03b1 n)\n    (nil : C nil) (cons : \u2200 {n : \u2115} {x : \u03b1} {w : Vector \u03b1 n}, C w \u2192 C (x ::\u1d65 w)) :\n    (x ::\u1d65 v).inductionOn nil cons = cons (v.inductionOn nil cons : C v)", "@[simp]\ntheorem mapAccumr_eq_map_of_unused_state (f : \u03b1 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b2) (s : \u03c3)\n    (h : \u2200 a s s', (f a s).snd = (f a s').snd) :\n    (mapAccumr f xs s).snd = (map (fun x => (f x s).snd) xs)", "@[simp]\ntheorem mapAccumr_snoc :\n    mapAccumr f (xs.snoc x) s\n    = let q := f x s\n      let r := mapAccumr f xs q.1\n      (r.1, r.2.snoc q.2)", "theorem mapAccumr\u2082_comm (f : \u03b1 \u2192 \u03b1 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b3) (comm : \u2200 a\u2081 a\u2082 s, f a\u2081 a\u2082 s = f a\u2082 a\u2081 s) :\n    mapAccumr\u2082 f xs ys s = mapAccumr\u2082 f ys xs s", "@[simp]\ntheorem mapAccumr\u2082_mapAccumr_left (f\u2081 : \u03b3 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b6) (f\u2082 : \u03b1 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3) :\n    (mapAccumr\u2082 f\u2081 (mapAccumr f\u2082 xs s\u2082).snd ys s\u2081)\n    = let m := (mapAccumr\u2082 (fun x y s =>\n          let r\u2082 := f\u2082 x s.snd\n          let r\u2081 := f\u2081 r\u2082.snd y s.fst\n          ((r\u2081.fst, r\u2082.fst), r\u2081.snd)\n        ) xs ys (s\u2081, s\u2082))\n      (m.fst.fst, m.snd)", "@[simp]\ntheorem mapAccumr\u2082_mapAccumr\u2082_left_right\n    (f\u2081 : \u03b3 \u2192 \u03b2 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03c6) (f\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3) :\n    (mapAccumr\u2082 f\u2081 (mapAccumr\u2082 f\u2082 xs ys s\u2082).snd ys s\u2081)\n    = let m := mapAccumr\u2082 (fun x y (s\u2081, s\u2082) =>\n                let r\u2082 := f\u2082 x y s\u2082\n                let r\u2081 := f\u2081 r\u2082.snd y s\u2081\n                ((r\u2081.fst, r\u2082.fst), r\u2081.snd)\n              )\n            xs ys (s\u2081, s\u2082)\n    (m.fst.fst, m.snd)", "@[simp]\ntheorem map\u2082_map_right (f\u2081 : \u03b1 \u2192 \u03b3 \u2192 \u03b6) (f\u2082 : \u03b2 \u2192 \u03b3) :\n    map\u2082 f\u2081 xs (map f\u2082 ys) = map\u2082 (fun x y => f\u2081 x (f\u2082 y)) xs ys", "theorem mem_map_succ_iff (b : \u03b2) (v : Vector \u03b1 (n + 1)) (f : \u03b1 \u2192 \u03b2) :\n    b \u2208 (v.map f).toList \u2194 f v.head = b \u2228 \u2203 a : \u03b1, a \u2208 v.tail.toList \u2227 f a = b", "theorem mem_succ_iff (v : Vector \u03b1 (n + 1)) : a \u2208 v.toList \u2194 a = v.head \u2228 a \u2208 v.tail.toList", "@[simp]\ntheorem replicate_succ (val : \u03b1) :\n    replicate (n+1) val = val ::\u1d65 (replicate n val)", "theorem reverse_get_zero {v : Vector \u03b1 (n + 1)} : v.reverse.head = v.last", "@[simp]\ntheorem scanl_val : \u2200 {v : Vector \u03b1 n}, (scanl f b v).val = List.scanl f b v.val", "@[simp]\ntheorem snoc_cons : (x ::\u1d65 xs).snoc y = x ::\u1d65 (xs.snoc y)", "@[simp]\ntheorem toList_length (v : Vector \u03b1 n) : (toList v).length = n", "@[simp]\ntheorem toList_map {\u03b2 : Type*} (v : Vector \u03b1 n) (f : \u03b1 \u2192 \u03b2) :\n    (v.map f).toList = v.toList.map f", "@[simp]\ntheorem toList_set (v : Vector \u03b1 n) (i : Fin n) (a : \u03b1) :\n    (v.set i a).toList = v.toList.set i a", "@[simp]\ntheorem append_add :\n    \u2200 {m} (v : Vector3 \u03b1 m) {n} (w : Vector3 \u03b1 n) (i : Fin2 n), (v +-+ w) (add i m) = w i", "@[simp, mfld_simps]\ntheorem localTriv_symmL {b : B} (hb : b \u2208 Z.baseSet i) :\n    (Z.localTriv i).symmL R b = Z.coordChange i (Z.indexAt b) b", "theorem fourierIntegral_continuous [FirstCountableTopology W] (he : Continuous e)\n    (hL : Continuous fun p : V \u00d7 W \u21a6 L p.1 p.2) {f : V \u2192 E} (hf : Integrable f \u03bc) :\n    Continuous (fourierIntegral e \u03bc L f)", "theorem fourierIntegral_continuousLinearMap_apply\n    {f : V \u2192 (F \u2192L[\u211d] E)} {a : F} {w : W} (he : Continuous e) (hf : Integrable f \u03bc) :\n    fourierIntegral e \u03bc L.toLinearMap\u2082 f w a =\n      fourierIntegral e \u03bc L.toLinearMap\u2082 (fun x \u21a6 f x a) w", "theorem index_countable [SecondCountableTopology \u03b1] : h.index.Countable", "theorem measure_diff_biUnion : \u03bc (s \\ \u22c3 p \u2208 h.index, h.covering p) = 0", "theorem measure_le_tsum [SecondCountableTopology \u03b1] : \u03bc s \u2264 \u2211' x : h.index, \u03bc (h.covering x)", "theorem ae_eventually_measure_zero_of_singular (h\u03c1 : \u03c1 \u27c2\u2098 \u03bc) :\n    \u2200\u1d50 x \u2202\u03bc, Tendsto (fun a => \u03c1 a / \u03bc a) (v.filterAt x) (\ud835\udcdd 0)", "theorem WCovBy.covBy_of_lt (h : a \u2a7f b) (h2 : a < b) : a \u22d6 b", "theorem cardinal_mk_le_of_le {\u03ba : Cardinal.{u}} (h\u03ba : (sum fun a : \u03b1 => \u03ba ^ #(\u03b2 a)) \u2264 \u03ba) :\n    #(WType \u03b2) \u2264 \u03ba", "theorem Wbtw.map {x y z : P} (h : Wbtw R x y z) (f : P \u2192\u1d43[R] P') : Wbtw R (f x) (f y) (f z)", "theorem _root_.Wbtw.oangle_sign_eq_of_ne_right {p\u2081 p\u2082 p\u2083 : P} (p\u2084 : P) (h : Wbtw \u211d p\u2081 p\u2082 p\u2083)\n    (hne : p\u2082 \u2260 p\u2083) : (\u2221 p\u2082 p\u2084 p\u2083).sign = (\u2221 p\u2081 p\u2084 p\u2083).sign", "theorem _root_.Wbtw.oangle\u2083\u2081\u2082_eq_zero {p\u2081 p\u2082 p\u2083 : P} (h : Wbtw \u211d p\u2081 p\u2082 p\u2083) : \u2221 p\u2083 p\u2081 p\u2082 = 0", "theorem Wbtw.trans_sbtw_right [NoZeroSMulDivisors R V] {w x y z : P} (h\u2081 : Wbtw R w x z)\n    (h\u2082 : Sbtw R x y z) : Sbtw R w y z", "theorem _root_.Wbtw.wOppSide\u2081\u2083 {s : AffineSubspace R P} {x y z : P} (h : Wbtw R x y z)\n    (hy : y \u2208 s) : s.WOppSide x z", "@[simp, norm_cast]\nprotected theorem coe_coe (\u03c6 : characterSpace \ud835\udd5c A) : \u21d1(\u03c6 : WeakDual \ud835\udd5c A) = (\u03c6 : A \u2192 \ud835\udd5c)", "theorem eval_continuous (y : E) : Continuous fun x : WeakDual \ud835\udd5c E => x y", "lemma hf_modif_FE (x : \u211d) (hx : 0 < x) :\n    P.f_modif (1 / x) = (P.\u03b5 * \u2191(x ^ P.k)) \u2022 P.g_modif x", "lemma hf_zero (P : WeakFEPair E) (r : \u211d) :\n    (fun x \u21a6 P.f x - (P.\u03b5 * \u2191(x ^ (-P.k))) \u2022 P.g\u2080) =O[\ud835\udcdd[>] 0] (\u00b7 ^ r)", "@[simp]\ntheorem coe_map (f : E \u2192L[\ud835\udd5c] F) : (WeakSpace.map f : E \u2192 F) = f", "lemma coe_norm_smul_basis (p q : R[X]) :\n    Algebra.norm R[X] (p \u2022 (1 : W.CoordinateRing) + q \u2022 mk W Y) =\n      mk W ((C p + C q * X) * (C p + C q * (-(Y : R[X][Y]) - C (C W.a\u2081 * X + C W.a\u2083))))", "lemma smul (x : R[X]) (y : W.CoordinateRing) : x \u2022 y = mk W (C x) * y", "lemma smul_basis_eq_zero {p q : R[X]} (hpq : p \u2022 (1 : W.CoordinateRing) + q \u2022 mk W Y = 0) :\n    p = 0 \u2227 q = 0", "@[simp]\nlemma add_of_Y_eq {x\u2081 x\u2082 y\u2081 y\u2082 : F} {h\u2081 : W.Nonsingular x\u2081 y\u2081} {h\u2082 : W.Nonsingular x\u2082 y\u2082}\n    (hx : x\u2081 = x\u2082) (hy : y\u2081 = W.negY x\u2082 y\u2082) : some h\u2081 + some h\u2082 = 0", "@[simp]\nlemma add_self_of_Y_eq {x\u2081 y\u2081 : F} {h\u2081 : W.Nonsingular x\u2081 y\u2081} (hy : y\u2081 = W.negY x\u2081 y\u2081) :\n    some h\u2081 + some h\u2081 = 0", "lemma toClass_eq_zero (P : W.Point) : toClass P = 0 \u2194 P = 0", "lemma zero_def : (zero : W.Point) = 0", "lemma Y_eq_of_Y_ne {x\u2081 x\u2082 y\u2081 y\u2082 : F} (h\u2081 : W.Equation x\u2081 y\u2081) (h\u2082 : W.Equation x\u2082 y\u2082) (hx : x\u2081 = x\u2082)\n    (hy : y\u2081 \u2260 W.negY x\u2082 y\u2082) : y\u2081 = y\u2082", "lemma map_addY (x\u2081 x\u2082 y\u2081 L : R) :\n    (W.map f).toAffine.addY (f x\u2081) (f x\u2082) (f y\u2081) (f L) = f (W.toAffine.addY x\u2081 x\u2082 y\u2081 L)", "lemma negY_negY (x y : R) : W.negY x (W.negY x y) = y", "lemma nonsingular_add {x\u2081 x\u2082 y\u2081 y\u2082 : F} (h\u2081 : W.Nonsingular x\u2081 y\u2081) (h\u2082 : W.Nonsingular x\u2082 y\u2082)\n    (hxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082) :\n    W.Nonsingular (W.addX x\u2081 x\u2082 <| W.slope x\u2081 x\u2082 y\u2081 y\u2082) (W.addY x\u2081 x\u2082 y\u2081 <| W.slope x\u2081 x\u2082 y\u2081 y\u2082)", "lemma polynomial_ne_zero [Nontrivial R] : W.polynomial \u2260 0", "lemma slope_of_Y_ne_eq_eval {x\u2081 x\u2082 y\u2081 y\u2082 : F} (hx : x\u2081 = x\u2082) (hy : y\u2081 \u2260 W.negY x\u2082 y\u2082) :\n    W.slope x\u2081 x\u2082 y\u2081 y\u2082 = -W.polynomialX.evalEval x\u2081 y\u2081 / W.polynomialY.evalEval x\u2081 y\u2081", "lemma add_def (P Q : W.Point) : P + Q = P.add Q", "lemma toAffineLift_add (P Q : W.Point) :\n    (P + Q).toAffineLift = P.toAffineLift + Q.toAffineLift", "lemma toAffineLift_zero : toAffineLift (0 : W.Point) = 0", "lemma X_ne_zero_of_Z_eq_zero [NoZeroDivisors R] {P : Fin 3 \u2192 R} (hP : W'.Nonsingular P)\n    (hPz : P z = 0) : P x \u2260 0", "lemma Y_eq_iff' {P Q : Fin 3 \u2192 F} (hPz : P z \u2260 0) (hQz : Q z \u2260 0) :\n    P y * Q z ^ 3 = W.negY Q * P z ^ 3 \u2194\n      P y / P z ^ 3 = W.toAffine.negY (Q x / Q z ^ 2) (Q y / Q z ^ 3)", "lemma Y_eq_negY_of_Y_eq [NoZeroDivisors R] {P Q : Fin 3 \u2192 R} (hQz : Q z \u2260 0)\n    (hx : P x * Q z ^ 2 = Q x * P z ^ 2) (hy : P y * Q z ^ 3 = Q y * P z ^ 3)\n    (hy' : P y * Q z ^ 3 = W'.negY Q * P z ^ 3) : P y = W'.negY P", "lemma addMap_of_Y_eq {P Q : Fin 3 \u2192 F} (hP : W.Nonsingular P) (hQ : W.Equation Q) (hPz : P z \u2260 0)\n    (hQz : Q z \u2260 0) (hx : P x * Q z ^ 2 = Q x * P z ^ 2)\n    (hy' : P y * Q z ^ 3 = W.negY Q * P z ^ 3) : W.addMap \u27e6P\u27e7 \u27e6Q\u27e7 = \u27e6![1, 1, 0]\u27e7", "lemma addY_neg {P : Fin 3 \u2192 R} (hP : W'.Equation P) : W'.addY P (W'.neg P) = -W'.dblZ P ^ 3", "lemma addY_self {P : Fin 3 \u2192 R} (hP : W'.Equation P) : W'.addY P P = 0", "lemma add_smul_of_equiv {P Q : Fin 3 \u2192 R} (h : P \u2248 Q) {u v : R} (hu : IsUnit u) (hv : IsUnit v) :\n    W'.add (u \u2022 P) (v \u2022 Q) = u ^ 4 \u2022 W'.add P Q", "lemma dblXYZ_of_Z_eq_zero {P : Fin 3 \u2192 R} (hP : W'.Equation P) (hPz : P z = 0) :\n    W'.dblXYZ P = P x ^ 2 \u2022 ![1, 1, 0]", "lemma dblX_of_Z_eq_zero {P : Fin 3 \u2192 R} (hP : W'.Equation P) (hPz : P z = 0) :\n    W'.dblX P = (P x ^ 2) ^ 2", "lemma equiv_some_of_Z_ne_zero {P : Fin 3 \u2192 F} (hPz : P z \u2260 0) :\n    P \u2248 ![P x / P z ^ 2, P y / P z ^ 3, 1]", "lemma isUnit_dblU_of_Y_eq {P Q : Fin 3 \u2192 F} (hP : W.Nonsingular P) (hPz : P z \u2260 0) (hQz : Q z \u2260 0)\n    (hx : P x * Q z ^ 2 = Q x * P z ^ 2) (hy : P y * Q z ^ 3 = Q y * P z ^ 3)\n    (hy' : P y * Q z ^ 3 = W.negY Q * P z ^ 3) : IsUnit (W.dblU P)", "@[simp] lemma map_negDblY : negDblY (W'.map f) (f \u2218 P) = f (W'.negDblY P)", "lemma map_polynomial : (W'.map f).toJacobian.polynomial = MvPolynomial.map f W'.polynomial", "lemma negDblY_smul (P : Fin 3 \u2192 R) (u : R) : W'.negDblY (u \u2022 P) = (u ^ 4) ^ 3 * W'.negDblY P", "lemma nonsingularLift_addMap {P Q : PointClass F} (hP : W.NonsingularLift P)\n    (hQ : W.NonsingularLift Q) : W.NonsingularLift <| W.addMap P Q", "lemma nonsingularLift_iff (P : Fin 3 \u2192 R) : W'.NonsingularLift \u27e6P\u27e7 \u2194 W'.Nonsingular P", "lemma nonsingularLift_zero [Nontrivial R] : W'.NonsingularLift \u27e6![1, 1, 0]\u27e7", "lemma nonsingular_iff_of_Y_eq_negY {P : Fin 3 \u2192 F} (hPz : P z \u2260 0) (hy : P y = W.negY P) :\n    W.Nonsingular P \u2194 W.Equation P \u2227 eval P W.polynomialX \u2260 0", "lemma nonsingular_smul (P : Fin 3 \u2192 R) {u : R} (hu : IsUnit u) :\n    W'.Nonsingular (u \u2022 P) \u2194 W'.Nonsingular P", "lemma nonsingular_some (X Y : R) : W'.Nonsingular ![X, Y, 1] \u2194 W'.toAffine.Nonsingular X Y", "@[simp]\nlemma nonsingularLift_iff (P : Fin 3 \u2192 R) : W.NonsingularLift \u27e6P\u27e7 \u2194 W.Nonsingular P", "lemma nonsingularLift_zero [Nontrivial R] : W.NonsingularLift \u27e6![0, 1, 0]\u27e7", "lemma nonsingular_iff_affine_of_Z_ne_zero {P : Fin 3 \u2192 F} (hPz : P z \u2260 0) :\n    W.Nonsingular P \u2194 W.toAffine.Nonsingular (P x / P z) (P y / P z)", "lemma nonsingular_some (X Y : R) : W.Nonsingular ![X, Y, 1] \u2194 W.toAffine.Nonsingular X Y", "lemma baseChange_\u03a8Sq (n : \u2124) : (W.baseChange B).\u03a8Sq n = ((W.baseChange A).\u03a8Sq n).map f", "lemma coeff_pre\u03a8'_ne_zero {n : \u2115} (h : (n : R) \u2260 0) :\n    (W.pre\u03a8' n).coeff ((n ^ 2 - if Even n then 4 else 1) / 2) \u2260 0", "@[simp]\nlemma coeff_\u03a6 (n : \u2124) : (W.\u03a6 n).coeff (n.natAbs ^ 2) = 1", "lemma coeff_\u03a6_ne_zero [Nontrivial R] (n : \u2124) : (W.\u03a6 n).coeff (n.natAbs ^ 2) \u2260 0", "lemma coeff_\u03a8Sq_ne_zero [NoZeroDivisors R] {n : \u2124} (h : (n : R) \u2260 0) :\n    (W.\u03a8Sq n).coeff (n.natAbs ^ 2 - 1) \u2260 0", "@[simp]\nlemma leadingCoeff_\u03a6 [Nontrivial R] (n : \u2124) : (W.\u03a6 n).leadingCoeff = 1", "@[simp]\nlemma map_b\u2088 : (W.map \u03c6).b\u2088 = \u03c6 W.b\u2088", "@[simp]\nlemma map_c\u2086 : (W.map \u03c6).c\u2086 = \u03c6 W.c\u2086", "lemma map_injective {\u03c6 : R \u2192+* A} (h\u03c6 : Function.Injective \u03c6) :\n    Function.Injective <| map (\u03c6 := \u03c6) := fun _ _ h => by\n  rcases mk.inj h with \u27e8_, _, _, _, _\u27e9\n  ext <;> apply_fun _ using h\u03c6 <;> assumption", "private lemma natDegree_coeff_\u03a8Sq_ofNat (n : \u2115) :\n    (W.\u03a8Sq n).natDegree \u2264 n ^ 2 - 1 \u2227 (W.\u03a8Sq n).coeff (n ^ 2 - 1) = (n ^ 2 : \u2124)", "lemma natDegree_pre\u03a8\u2084_le : W.pre\u03a8\u2084.natDegree \u2264 6", "@[simp]\nlemma natDegree_\u03a8\u2083 (h : (3 : R) \u2260 0) : W.\u03a8\u2083.natDegree = 4", "@[simp]\nlemma pre\u03a8'_zero : W.pre\u03a8' 0 = 0", "@[simp]\nlemma pre\u03a8_neg (n : \u2124) : W.pre\u03a8 (-n) = -W.pre\u03a8 n", "@[simp]\nlemma pre\u03a8_zero : W.pre\u03a8 0 = 0", "@[simp]\nlemma variableChange_c\u2084 : (W.variableChange C).c\u2084 = C.u\u207b\u00b9 ^ 4 * W.c\u2084", "lemma \u03a6_ne_zero [Nontrivial R] (n : \u2124) : W.\u03a6 n \u2260 0", "lemma \u03a8_even (m : \u2115) : W.\u03a8 (2 * (m + 3)) * W.\u03c8\u2082 =\n    W.\u03a8 (m + 2) ^ 2 * W.\u03a8 (m + 3) * W.\u03a8 (m + 5) - W.\u03a8 (m + 1) * W.\u03a8 (m + 3) * W.\u03a8 (m + 4) ^ 2", "@[simp]\nlemma \u03a8_zero : W.\u03a8 0 = 0", "@[simp]\nlemma \u03c6_one : W.\u03c6 1 = C X", "theorem exists_well_order_ge : \u2203 s, r \u2264 s \u2227 IsWellOrder \u03b1 s", "@[csimp] private theorem fixF_eq_fixFC : @fixF = @fixFC", "protected theorem lt_succ {r : \u03b1 \u2192 \u03b1 \u2192 Prop} (wf : WellFounded r) {x : \u03b1} (h : \u2203 y, r x y) :\n    r x (wf.succ x)", "theorem fix_eq {C : \u03b1 \u2192 Sort*} (F : \u2200 x : \u03b1, (\u2200 y : \u03b1, x < y \u2192 C y) \u2192 C x) :\n    \u2200 x, fix F x = F x fun y _ => fix F y", "theorem coe_strictMono : StrictMono (fun (a : \u03b1) => (a : WithBot \u03b1))", "lemma mul_eq_bot_iff : a * b = \u22a5 \u2194 a \u2260 0 \u2227 b = \u22a5 \u2228 a = \u22a5 \u2227 b \u2260 0", "theorem ne_bot_iff_exists {x : WithBot \u03b1} : x \u2260 \u22a5 \u2194 \u2203 a : \u03b1, \u2191a = x", "theorem ofDual_lt_iff {a : WithBot \u03b1\u1d52\u1d48} {b : WithTop \u03b1} :\n    WithBot.ofDual a < b \u2194 WithTop.toDual b < a", "@[simp]\ntheorem preimage_coe_Ioc : (some : \u03b1 \u2192 WithBot \u03b1) \u207b\u00b9' Ioc a b = Ioc a b", "@[simp]\ntheorem preimage_coe_Ioo_bot : (some : \u03b1 \u2192 WithBot \u03b1) \u207b\u00b9' Ioo \u22a5 a = Iio a", "@[simp]\ntheorem preimage_coe_bot : (some : \u03b1 \u2192 WithBot \u03b1) \u207b\u00b9' {\u22a5} = (\u2205 : Set \u03b1)", "theorem unbot'_le_iff {a : WithBot \u03b1} {b c : \u03b1} (h : a = \u22a5 \u2192 b \u2264 c) :\n    a.unbot' b \u2264 c \u2194 a \u2264 c", "theorem unbot_le_iff {a : WithBot \u03b1} (h : a \u2260 \u22a5) {b : \u03b1} :\n    unbot a h \u2264 b \u2194 a \u2264 (b : WithBot \u03b1)", "theorem prod_lipschitzWith_equiv [PseudoEMetricSpace \u03b1] [PseudoEMetricSpace \u03b2] :\n    LipschitzWith 1 (WithLp.equiv p (\u03b1 \u00d7 \u03b2))", "@[simp]\ntheorem prod_norm_eq_card (f : WithLp 0 (\u03b1 \u00d7 \u03b2)) :\n    \u2016f\u2016 = (if \u2016f.fst\u2016 = 0 then 0 else 1) + (if \u2016f.snd\u2016 = 0 then 0 else 1)", "theorem prod_norm_eq_of_nat [Norm \u03b1] [Norm \u03b2] (n : \u2115) (h : p = n) (f : WithLp p (\u03b1 \u00d7 \u03b2)) :\n    \u2016f\u2016 = (\u2016f.fst\u2016 ^ n + \u2016f.snd\u2016 ^ n) ^ (1 / (n : \u211d))", "@[to_additive (attr := simp)]\ntheorem map_id : map (MulHom.id \u03b1) = MonoidHom.id (WithOne \u03b1)", "theorem WithSeminorms.T1_of_separating (hp : WithSeminorms p)\n    (h : \u2200 x, x \u2260 0 \u2192 \u2203 i, p i x \u2260 0) : T1Space E", "theorem WithSeminorms.hasBasis (hp : WithSeminorms p) :\n    (\ud835\udcdd (0 : E)).HasBasis (fun s : Set E => s \u2208 p.basisSets) id", "theorem WithSeminorms.hasBasis_zero_ball (hp : WithSeminorms p) :\n    (\ud835\udcdd (0 : E)).HasBasis\n    (fun sr : Finset \u03b9 \u00d7 \u211d => 0 < sr.2) fun sr => (sr.1.sup p).ball 0 sr.2", "theorem WithSeminorms.mem_nhds_iff (hp : WithSeminorms p) (x : E) (U : Set E) :\n    U \u2208 \ud835\udcdd x \u2194 \u2203 s : Finset \u03b9, \u2203 r > 0, (s.sup p).ball x r \u2286 U", "theorem Ico_coe_coe : Ico (a : WithTop \u03b1) b = (Ico a b).map Embedding.some", "protected theorem add_le_add_iff_right [LE \u03b1] [CovariantClass \u03b1 \u03b1 (swap (\u00b7 + \u00b7)) (\u00b7 \u2264 \u00b7)]\n    [ContravariantClass \u03b1 \u03b1 (swap (\u00b7 + \u00b7)) (\u00b7 \u2264 \u00b7)] (ha : a \u2260 \u22a4) : b + a \u2264 c + a \u2194 b \u2264 c", "theorem coe_mono : Monotone (fun a : \u03b1 => (a : WithTop \u03b1))", "@[simp]\ntheorem WithTop.iInf_empty [IsEmpty \u03b9] [InfSet \u03b1] (f : \u03b9 \u2192 WithTop \u03b1) :\n    \u2a05 i, f i = \u22a4", "theorem isLUB_sSup (s : Set (WithTop \u03b1)) : IsLUB s (sSup s)", "theorem map_le_iff (f : \u03b1 \u2192 \u03b2) (a b : WithTop \u03b1)\n    (mono_iff : \u2200 {a b}, f a \u2264 f b \u2194 a \u2264 b) :\n    a.map f \u2264 b.map f \u2194 a \u2264 b", "theorem map_ofDual (f : \u03b1 \u2192 \u03b2) (a : WithBot \u03b1\u1d52\u1d48) : map f (WithBot.ofDual a) = a.map (ofDual \u2218 f)", "theorem mul_lt_top [LT \u03b1] {a b : WithTop \u03b1} (ha : a \u2260 \u22a4) (hb : b \u2260 \u22a4) : a * b < \u22a4", "theorem ofDual_le_iff {a : WithTop \u03b1\u1d52\u1d48} {b : WithBot \u03b1} :\n    WithTop.ofDual a \u2264 b \u2194 WithBot.toDual b \u2264 a", "@[to_additive (attr := simp, norm_cast)]\nlemma one_eq_coe : 1 = (a : WithTop \u03b1) \u2194 a = 1 := eq_comm.trans coe_eq_one", "@[to_additive (attr := simp, norm_cast) coe_nonneg]\ntheorem one_le_coe [LE \u03b1] {a : \u03b1} : 1 \u2264 (a : WithTop \u03b1) \u2194 1 \u2264 a", "lemma prod_lt_top [CommMonoidWithZero \u03b1] [NoZeroDivisors \u03b1] [Nontrivial \u03b1] [DecidableEq \u03b1] [LT \u03b1]\n    {s : Finset \u03b9} {f : \u03b9 \u2192 WithTop \u03b1} (h : \u2200 i \u2208 s, f i \u2260 \u22a4) : \u220f i \u2208 s, f i < \u22a4", "@[simp, deprecated coe_le_coe \"Don't mix Option and WithTop\" (since := \"2024-05-27\")]\ntheorem some_le_some : @LE.le (WithTop \u03b1) _ (Option.some a) (Option.some b) \u2194 a \u2264 b", "@[simp] lemma top_mul (hb : b \u2260 0) : \u22a4 * b = \u22a4", "lemma top_mul' : \u2200 (b : WithTop \u03b1), \u22a4 * b = if b = 0 then 0 else \u22a4\n  | (b : \u03b1) => if_congr coe_eq_zero.symm rfl rfl\n  | \u22a4 => (if_neg top_ne_zero).symm", "@[simp, norm_cast] lemma coe_le_one [One \u03b1] : (a : WithZero \u03b1) \u2264 1 \u2194 a \u2264 1 := coe_le_coe", "@[ext high]\ntheorem monoidWithZeroHom_ext \u2983f g : WithZero \u03b1 \u2192*\u2080 \u03b2\u2984\n    (h : f.toMonoidHom.comp coeMonoidHom = g.toMonoidHom.comp coeMonoidHom) :\n    f = g", "theorem Iio_mem_nhds (h : \u03b3\u2081 < \u03b3\u2082) : Iio \u03b3\u2082 \u2208 \ud835\udcdd \u03b3\u2081", "theorem isOpen_iff {s : Set \u0393\u2080} : IsOpen s \u2194 (0 : \u0393\u2080) \u2209 s \u2228 \u2203 \u03b3, \u03b3 \u2260 0 \u2227 Iio \u03b3 \u2286 s", "theorem solution_pow (a\u2081 a\u2082 : \ud835\udd4e k) : \u2203 x : k, x ^ (p - 1) = a\u2082.coeff 0 / a\u2081.coeff 0", "theorem aeval_verschiebung_poly' (x : \ud835\udd4e R) (n : \u2115) :\n    aeval x.coeff (verschiebungPoly n) = (verschiebungFun x).coeff n", "@[simp]\ntheorem bind\u2081_verschiebungPoly_wittPolynomial (n : \u2115) :\n    bind\u2081 verschiebungPoly (wittPolynomial p \u2124 n) =\n      if n = 0 then 0 else p * wittPolynomial p \u2124 (n - 1)", "theorem coeff_add_of_disjoint (x y : \ud835\udd4e R) (h : \u2200 n, x.coeff n = 0 \u2228 y.coeff n = 0) :\n    (x + y).coeff n = x.coeff n + y.coeff n", "theorem coeff_mk (x : \u2115 \u2192 R) : (mk p x).coeff = x", "theorem coeff_p_pow [CharP R p] (i : \u2115) : ((p : \ud835\udd4e R) ^ i).coeff i = 1", "@[simp]\ntheorem coeff_truncate (x : \ud835\udd4e R) (i : Fin n) : (truncate n x).coeff i = x.coeff i", "@[simp]\ntheorem constantCoeff_wittMul (n : \u2115) : constantCoeff (wittMul p n) = 0", "@[simp]\ntheorem constantCoeff_wittSub (n : \u2115) : constantCoeff (wittSub p n) = 0", "theorem ext_iff {x y : \ud835\udd4e R} : x = y \u2194 \u2200 n, x.coeff n = y.coeff n", "theorem frobeniusPoly_zmod (n : \u2115) :\n    MvPolynomial.map (Int.castRingHom (ZMod p)) (frobeniusPoly p n) = X n ^ p", "theorem frobenius_verschiebung (x : \ud835\udd4e R) : frobenius (verschiebung x) = x * p", "private theorem ghostFun_natCast (i : \u2115) : ghostFun (i : \ud835\udd4e R) = i", "private theorem ghostFun_sub : ghostFun (x - y) = ghostFun x - ghostFun y", "theorem init_pow (m : \u2115) (x : \ud835\udd4e R) (n : \u2115) : init n (x ^ m) = init n (init n x ^ m)", "theorem iterate_frobenius_coeff (x : \ud835\udd4e R) (i k : \u2115) :\n    (frobenius^[i] x).coeff k = x.coeff k ^ p ^ i", "theorem iterate_verschiebung_mul (x y : \ud835\udd4e R) (i j : \u2115) :\n    verschiebung^[i] x * verschiebung^[j] y =\n      verschiebung^[i + j] (frobenius^[j] x * frobenius^[i] y)", "theorem iterate_verschiebung_mul_coeff (x y : \ud835\udd4e R) (i j : \u2115) :\n    (verschiebung^[i] x * verschiebung^[j] y).coeff (i + j) =\n      x.coeff 0 ^ p ^ j * y.coeff 0 ^ p ^ i", "theorem mulN_coeff (n : \u2115) (x : \ud835\udd4e R) (k : \u2115) :\n    (x * n).coeff k = aeval x.coeff (wittMulN p n k)", "@[simp]\ntheorem truncate_lift (s : S) : WittVector.truncate n (lift _ f_compat s) = f n s", "@[simp]\ntheorem verschiebungFun_coeff_succ (x : \ud835\udd4e R) (n : \u2115) :\n    (verschiebungFun x).coeff n.succ = x.coeff n", "theorem verschiebung_shift (x : \ud835\udd4e R) (k : \u2115) (h : \u2200 i < k + 1, x.coeff i = 0) :\n    verschiebung (x.shift k.succ) = x.shift k", "theorem wittPolyProdRemainder_vars (n : \u2115) :\n    (wittPolyProdRemainder p n).vars \u2286 univ \u00d7\u02e2 range n", "@[simp]\ntheorem cells_bot : (\u22a5 : YoungDiagram).cells = \u2205", "theorem mk_mem_col_iff {\u03bc : YoungDiagram} {i j : \u2115} : (i, j) \u2208 \u03bc.col j \u2194 (i, j) \u2208 \u03bc", "@[simp]\ntheorem empty_isTransitive : IsTransitive \u2205", "theorem isTransitive_iff_subset_powerset : x.IsTransitive \u2194 x \u2286 powerset x", "@[simp]\ntheorem map_fval {f : ZFSet.{u} \u2192 ZFSet.{u}} [H : PSet.Definable 1 f] {x y : ZFSet.{u}}\n    (h : y \u2208 x) : (ZFSet.map f x \u2032 y : Class.{u}) = f y", "theorem regularity (x : ZFSet.{u}) (h : x \u2260 \u2205) : \u2203 y \u2208 x, x \u2229 y = \u2205", "@[simp]\ntheorem sInter_empty : \u22c2\u2080 (\u2205 : ZFSet) = \u2205", "@[simp]\ntheorem sUnion_empty : \u22c3\u2080 (\u2205 : ZFSet.{u}) = \u2205", "@[simp]\ntheorem toSet_pair (x y : ZFSet.{u}) : (pair x y).toSet = {{x}, {x, y}}", "theorem cast_cast_zmod_of_le {m n : \u2115} [hm : NeZero m] (h : m \u2264 n) (a : ZMod m) :\n    (cast (cast a : ZMod n) : ZMod m) = a", "theorem cast_mul (h : m \u2223 n) (a b : ZMod n) : (cast (a * b : ZMod n) : R) = cast a * cast b", "theorem cast_zmod_eq_zero_iff_of_le {m n : \u2115} [NeZero m] (h : m \u2264 n) (a : ZMod m) :\n    (cast a : ZMod n) = 0 \u2194 a = 0", "private theorem eisenstein_lemma_aux\u2081 (p : \u2115) [Fact p.Prime] [hp2 : Fact (p % 2 = 1)] {a : \u2115}\n    (hap : (a : ZMod p) \u2260 0) : ((\u2211 x \u2208 Ico 1 (p / 2).succ, a * x : \u2115) : ZMod 2) =\n    ((Ico 1 (p / 2).succ).filter fun x : \u2115 => p / 2 < (a * x : ZMod p).val).card +\n      \u2211 x \u2208 Ico 1 (p / 2).succ, x + (\u2211 x \u2208 Ico 1 (p / 2).succ, a * x / p : \u2115)", "theorem eq_zero_of_gcd_ne_one {a : \u2124} {p : \u2115} (pp : p.Prime) (h : a.gcd p \u2260 1) : (a : ZMod p) = 0", "@[simp]\nprotected theorem ZMod.exponent (n : \u2115) : AddMonoid.exponent (ZMod n) = n", "@[nolint unusedHavesSuffices]\ntheorem inv_zero : \u2200 n : \u2115, (0 : ZMod n)\u207b\u00b9 = 0", "@[simp]\ntheorem natCast_self' (n : \u2115) : (n + 1 : ZMod (n + 1)) = 0", "theorem ne_zero_of_gcd_eq_one {a : \u2124} {p : \u2115} (pp : p.Prime) (h : a.gcd p = 1) : (a : ZMod p) \u2260 0", "theorem neg_one_ne_one {n : \u2115} [Fact (2 < n)] : (-1 : ZMod n) \u2260 1", "theorem nonsquare_iff_jacobiSym_eq_neg_one {a : \u2124} {p : \u2115} [Fact p.Prime] :\n    J(a | p) = -1 \u2194 \u00acIsSquare (a : ZMod p)", "lemma pow_card_sub_one {p : \u2115} [Fact p.Prime] (a : ZMod p) :\n    a ^ (p - 1) = if a \u2260 0 then 1 else 0", "theorem ZMod.ringHom_eq_of_ker_eq {n : \u2115} {R : Type*} [CommRing R] (f g : R \u2192+* ZMod n)\n    (h : RingHom.ker f = RingHom.ker g) : f = g", "lemma unitsMap_def (hm : n \u2223 m) : unitsMap hm = Units.map (castHom hm (ZMod n)) := rfl", "@[simp]\ntheorem valMinAbs_def_zero (x : ZMod 0) : valMinAbs x = x", "theorem valMinAbs_spec {n : \u2115} [NeZero n] (x : ZMod n) (y : \u2124) :\n    x.valMinAbs = y \u2194 x = y \u2227 y * 2 \u2208 Set.Ioc (-n : \u2124) n", "theorem val_one (n : \u2115) [Fact (1 < n)] : (1 : ZMod n).val = 1", "theorem \u03c7\u2088'_int_eq_if_mod_eight (n : \u2124) :\n    \u03c7\u2088' n = if n % 2 = 0 then 0 else if n % 8 = 1 \u2228 n % 8 = 3 then 1 else -1", "@[simp]\ntheorem abs_to_nat : \u2200 n, (abs n : \u2115) = Int.natAbs n", "@[simp, norm_cast]\ntheorem cast_lt [LinearOrderedRing \u03b1] {m n : ZNum} : (m : \u03b1) < n \u2194 m < n", "@[simp]\ntheorem coe_copy (f : C\u2080(\u03b1, \u03b2)) (f' : \u03b1 \u2192 \u03b2) (h : f' = f) : \u21d1(f.copy f' h) = f'", "theorem copy_eq (f : C\u2080(\u03b1, \u03b2)) (f' : \u03b1 \u2192 \u03b2) (h : f' = f) : f.copy f' h = f", "theorem Zlattice.module_finite [IsZlattice K L] : Module.Finite \u2124 L", "theorem Zlattice.rank [hs : IsZlattice K L] : finrank \u2124 L = finrank K E", "@[simp]\ntheorem floor_eq_self_of_mem (m : E) (h : m \u2208 span \u2124 (Set.range b)) : (floor b m : E) = m", "@[simp]\ntheorem repr_ceil_apply (m : E) (i : \u03b9) : b.repr (ceil b m) i = \u2308b.repr m i\u2309", "theorem span_top : span K (span \u2124 (Set.range b) : Set E) = \u22a4", "theorem divides_sq_eq_zero {x y} (h : x * x = d * y * y) : x = 0 \u2227 y = 0", "theorem divides_sq_eq_zero_z {x y : \u2124} (h : x * x = d * y * y) : x = 0 \u2227 y = 0", "theorem intCast_dvd (z : \u2124) (a : \u2124\u221ad) : \u2191z \u2223 a \u2194 z \u2223 a.re \u2227 z \u2223 a.im", "theorem nonneg_muld {a : \u2124\u221ad} (ha : Nonneg a) : Nonneg (sqrtd * a)", "theorem norm_nonneg (hd : d \u2264 0) (n : \u2124\u221ad) : 0 \u2264 n.norm", "@[simp]\ntheorem ofNat_re (n : \u2115) [n.AtLeastTwo] : (no_index (OfNat.ofNat n) : \u2124\u221ad).re = n", "@[simp]\ntheorem one_re : (1 : \u2124\u221ad).re = 1", "@[simp]\ntheorem smuld_val (n x y : \u2124) : sqrtd * (n : \u2124\u221ad) * \u27e8x, y\u27e9 = \u27e8d * n * y, n * x\u27e9", "@[simp]\ntheorem abs_eq_self : |a| = a \u2194 0 \u2264 a", "@[simp]\nlemma _root_.absorbs_biUnion_finset {\u03b9 : Type*} {t : \u03b9 \u2192 Set \u03b1} {I : Finset \u03b9} :\n    Absorbs M s (\u22c3 i \u2208 I, t i) \u2194 \u2200 i \u2208 I, Absorbs M s (t i)", "@[simp]\nlemma absorbs_neg_neg {s t : Set E} : Absorbs M (-s) (-t) \u2194 Absorbs M s t", "theorem act_rel_of_rel_of_act_rel (ab : r a b) (rl : r (\u03bc m b) c) : r (\u03bc m a) c", "theorem add_eq_left_iff_le : a + b = a \u2194 b \u2264 a", "@[simp]\ntheorem add_halves' (a : R) : a / 2 + a / 2 = a", "theorem add_tsub_cancel_iff_le : a + (b - a) = b \u2194 a \u2264 b", "@[simp]\ntheorem add_tsub_cancel_of_le (h : a \u2264 b) : a + (b - a) = b", "theorem aeSeq_eq_fun_ae [Countable \u03b9] (hf : \u2200 i, AEMeasurable (f i) \u03bc)\n    (hp : \u2200\u1d50 x \u2202\u03bc, p x fun n => f n x) : \u2200\u1d50 a : \u03b1 \u2202\u03bc, \u2200 i : \u03b9, aeSeq hf p i a = f i a", "theorem ae_eq_const_or_norm_integral_lt_of_norm_le_const [StrictConvexSpace \u211d E] [IsFiniteMeasure \u03bc]\n    (h_le : \u2200\u1d50 x \u2202\u03bc, \u2016f x\u2016 \u2264 C) :\n    f =\u1d50[\u03bc] const \u03b1 (\u2a0d x, f x \u2202\u03bc) \u2228 \u2016\u222b x, f x \u2202\u03bc\u2016 < (\u03bc univ).toReal * C", "theorem aemeasurable_deriv [MeasurableSpace \ud835\udd5c] [OpensMeasurableSpace \ud835\udd5c] [MeasurableSpace F]\n    [BorelSpace F] (f : \ud835\udd5c \u2192 F) (\u03bc : Measure \ud835\udd5c) : AEMeasurable (deriv f) \u03bc", "theorem aemeasurable_indicator_iff\u2080 {s} (hs : NullMeasurableSet s \u03bc) :\n    AEMeasurable (indicator s f) \u03bc \u2194 AEMeasurable f (\u03bc.restrict s)", "theorem _root_.aestronglyMeasurable_const_smul_iff\u2080 {c : G\u2080} (hc : c \u2260 0) :\n    AEStronglyMeasurable (fun x => c \u2022 f x) \u03bc \u2194 AEStronglyMeasurable f \u03bc", "@[simp]\ntheorem _root_.aestronglyMeasurable_iUnion_iff [PseudoMetrizableSpace \u03b2] {s : \u03b9 \u2192 Set \u03b1} :\n    AEStronglyMeasurable f (\u03bc.restrict (\u22c3 i, s i)) \u2194\n      \u2200 i, AEStronglyMeasurable f (\u03bc.restrict (s i))", "theorem affineIndependent_iff_not_collinear_set {p\u2081 p\u2082 p\u2083 : P} :\n    AffineIndependent k ![p\u2081, p\u2082, p\u2083] \u2194 \u00acCollinear k ({p\u2081, p\u2082, p\u2083} : Set P)", "theorem affineIndependent_iff_of_fintype [Fintype \u03b9] (p : \u03b9 \u2192 P) :\n    AffineIndependent k p \u2194\n      \u2200 w : \u03b9 \u2192 k, \u2211 i, w i = 0 \u2192 Finset.univ.weightedVSub p w = (0 : V) \u2192 \u2200 i, w i = 0", "theorem affineIndependent_of_ne_of_mem_of_mem_of_not_mem {s : AffineSubspace k P} {p\u2081 p\u2082 p\u2083 : P}\n    (hp\u2081p\u2082 : p\u2081 \u2260 p\u2082) (hp\u2081 : p\u2081 \u2208 s) (hp\u2082 : p\u2082 \u2208 s) (hp\u2083 : p\u2083 \u2209 s) :\n    AffineIndependent k ![p\u2081, p\u2082, p\u2083]", "@[simp]\ntheorem affineSegment_image (f : P \u2192\u1d43[R] P') (x y : P) :\n    f '' affineSegment R x y = affineSegment R (f x) (f y)", "@[simp]\ntheorem affineSegment_same (x : P) : affineSegment R x x = {x}", "@[simp]\ntheorem affineSegment_vsub_const_image (x y p : P) :\n    (\u00b7 -\u1d65 p) '' affineSegment R x y = affineSegment R (x -\u1d65 p) (y -\u1d65 p)", "@[simp]\ntheorem algebraMap_star_comm (r : R) : algebraMap R A (star r) = star (algebraMap R A r)", "theorem algebraicIndependent_comp_subtype {s : Set \u03b9} :\n    AlgebraicIndependent R (x \u2218 (\u2191) : s \u2192 A) \u2194\n      \u2200 p \u2208 MvPolynomial.supported R s, aeval x p = 0 \u2192 p = 0", "theorem algebraicIndependent_sUnion_of_directed {s : Set (Set A)} (hsn : s.Nonempty)\n    (hs : DirectedOn (\u00b7 \u2286 \u00b7) s) (h : \u2200 a \u2208 s, AlgebraicIndependent R ((\u2191) : a \u2192 A)) :\n    AlgebraicIndependent R ((\u2191) : \u22c3\u2080 s \u2192 A)", "theorem isThreeCycle_isConj (h5 : 5 \u2264 Fintype.card \u03b1) {\u03c3 \u03c4 : alternatingGroup \u03b1}\n    (h\u03c3 : IsThreeCycle (\u03c3 : Perm \u03b1)) (h\u03c4 : IsThreeCycle (\u03c4 : Perm \u03b1)) : IsConj \u03c3 \u03c4", "theorem analyticAt_const {v : F} : AnalyticAt \ud835\udd5c (fun _ => v) x", "lemma analyticAt_mul (z : A \u00d7 A) : AnalyticAt \ud835\udd5c (fun x : A \u00d7 A \u21a6 x.1 * x.2) z", "theorem and_congr_left (h : c \u2192 (a \u2194 b)) : a \u2227 c \u2194 b \u2227 c", "@[simp] theorem and_imp : (a \u2227 b \u2192 c) \u2194 (a \u2192 b \u2192 c)", "@[simp]\ntheorem antisymmRel_iff_eq [IsRefl \u03b1 r] [IsAntisymm \u03b1 r] {a b : \u03b1} : AntisymmRel r a b \u2194 a = b", "theorem antisymm_iff [IsRefl \u03b1 r] [IsAntisymm \u03b1 r] {a b : \u03b1} : r a b \u2227 r b a \u2194 a = b", "@[elab_without_expected_type]\ntheorem antisymm_of (r : \u03b1 \u2192 \u03b1 \u2192 Prop) [IsAntisymm \u03b1 r] {a b : \u03b1} : r a b \u2192 r b a \u2192 a = b", "theorem antitoneOn_iff_forall_lt :\n    AntitoneOn f s \u2194 \u2200 \u2983a\u2984 (_ : a \u2208 s) \u2983b\u2984 (_ : b \u2208 s), a < b \u2192 f b \u2264 f a", "theorem antitoneOn_of_deriv_nonpos {D : Set \u211d} (hD : Convex \u211d D) {f : \u211d \u2192 \u211d}\n    (hf : ContinuousOn f D) (hf' : DifferentiableOn \u211d f (interior D))\n    (hf'_nonpos : \u2200 x \u2208 interior D, deriv f x \u2264 0) : AntitoneOn f D", "theorem antitone_iff_map_nonpos : Antitone (f : \u03b1 \u2192 \u03b2) \u2194 \u2200 a, 0 \u2264 a \u2192 f a \u2264 0", "theorem antitone_of_deriv_nonpos {f : \u211d \u2192 \u211d} (hf : Differentiable \u211d f) (hf' : \u2200 x, deriv f x \u2264 0) :\n    Antitone f", "@[to_additive (attr := simp)]\nlemma antivaryOn_inv_left : AntivaryOn f\u207b\u00b9 g s \u2194 MonovaryOn f g s", "theorem antivary_const_left (g : \u03b9 \u2192 \u03b2) (a : \u03b1) : Antivary (const \u03b9 a) g", "@[to_additive (attr := simp)] lemma antivary_inv_left : Antivary f\u207b\u00b9 g \u2194 Monovary f g", "@[to_additive (attr := simp)] lemma antivary_inv_right : Antivary f g\u207b\u00b9 \u2194 Monovary f g", "lemma antivary_inv\u2080 (hf : StrongLT 0 f) (hg : StrongLT 0 g) : Antivary f\u207b\u00b9 g\u207b\u00b9 \u2194 Antivary f g", "@[simp]\ntheorem apply_fdifferential (f : C^\u221e\u27eeI, M; I', M'\u27ef) {x : M} (v : PointDerivation I x)\n    (g : C^\u221e\u27eeI', M'; \ud835\udd5c\u27ef) : \ud835\udc85 f x v g = v (g.comp f)", "@[simp]\ntheorem approximatesLinearOn_empty (f : E \u2192 F) (f' : E \u2192L[\ud835\udd5c] F) (c : \u211d\u22650) :\n    ApproximatesLinearOn f f' \u2205 c", "theorem associatedPrimes.eq_empty_of_subsingleton [Subsingleton M] : associatedPrimes R M = \u2205", "theorem associated_isUnit_mul_left_iff {\u03b2 : Type*} [CommMonoid \u03b2] {u a b : \u03b2} (hu : IsUnit u) :\n    Associated (u * a) b \u2194 Associated a b", "theorem associated_mul_isUnit_right_iff {\u03b2 : Type*} [Monoid \u03b2] {a b u : \u03b2} (hu : IsUnit u) :\n    Associated a (b * u) \u2194 Associated a b", "@[simp]\ntheorem associated_mul_unit_right_iff {\u03b2 : Type*} [Monoid \u03b2] {a b : \u03b2} {u : Units \u03b2} :\n    Associated a (b * u) \u2194 Associated a b", "lemma autEquivZmod_symm_apply_intCast {\u03b6 : K} (h\u03b6 : IsPrimitiveRoot \u03b6 n) (m : \u2124) :\n    (autEquivZmod H L h\u03b6).symm (Multiplicative.ofAdd (m : ZMod n)) \u03b1 = \u03b6 ^ m \u2022 \u03b1", "private theorem aux : ((g x * f x ^ (g x - 1)) \u2022 (1 : \u2102 \u2192L[\u2102] \u2102).smulRight f' +\n    (f x ^ g x * log (f x)) \u2022 (1 : \u2102 \u2192L[\u2102] \u2102).smulRight g') 1 =\n      g x * f x ^ (g x - 1) * f' + f x ^ g x * log (f x) * g'", "private theorem aux1 {a b c : A} :\n    \u2045L a + L b + L c, L (a * a) + L (b * b) + L (c * c) +\n    2 \u2022 L (a * b) + 2 \u2022 L (c * a) + 2 \u2022 L (b * c)\u2046\n    =\n    \u2045L a, L (a * a)\u2046 + \u2045L a, L (b * b)\u2046 + \u2045L a, L (c * c)\u2046 +\n    \u2045L a, 2 \u2022 L (a * b)\u2046 + \u2045L a, 2 \u2022 L (c * a)\u2046 + \u2045L a, 2 \u2022 L (b * c)\u2046 +\n    (\u2045L b, L (a * a)\u2046 + \u2045L b, L (b * b)\u2046 + \u2045L b, L (c * c)\u2046 +\n    \u2045L b, 2 \u2022 L (a * b)\u2046 + \u2045L b, 2 \u2022 L (c * a)\u2046 + \u2045L b, 2 \u2022 L (b * c)\u2046) +\n    (\u2045L c, L (a * a)\u2046 + \u2045L c, L (b * b)\u2046 + \u2045L c, L (c * c)\u2046 +\n    \u2045L c, 2 \u2022 L (a * b)\u2046 + \u2045L c, 2 \u2022 L (c * a)\u2046 + \u2045L c, 2 \u2022 L (b * c)\u2046)", "theorem ball_add_closedBall (h\u03b5 : 0 < \u03b5) (h\u03b4 : 0 \u2264 \u03b4) (a b : E) :\n    ball a \u03b5 + closedBall b \u03b4 = ball (a + b) (\u03b5 + \u03b4)", "theorem banach_steinhaus_iSup_nnnorm {\u03b9 : Type*} [CompleteSpace E] {g : \u03b9 \u2192 E \u2192SL[\u03c3\u2081\u2082] F}\n    (h : \u2200 x, (\u2a06 i, \u2191\u2016g i x\u2016\u208a) < \u221e) : (\u2a06 i, \u2191\u2016g i\u2016\u208a) < \u221e", "theorem basis_toMatrix_basisFun_mul (b : Basis \u03b9 R (\u03b9 \u2192 R)) (A : Matrix \u03b9 \u03b9 R) :\n    b.toMatrix (Pi.basisFun R \u03b9) * A = of fun i j => b.repr (A\u1d40 j) i", "theorem bddAbove_Icc : BddAbove (Icc a b)", "lemma bddAbove_prod {s : Set (\u03b1 \u00d7 \u03b2)} :\n    BddAbove s \u2194 BddAbove (Prod.fst '' s) \u2227 BddAbove (Prod.snd '' s)", "theorem bddBelow_Ici : BddBelow (Ici a)", "theorem bddBelow_Ioo : BddBelow (Ioo a b)", "theorem bddBelow_bddAbove_iff_subset_Icc : BddBelow s \u2227 BddAbove s \u2194 \u2203 a b, s \u2286 Icc a b", "theorem bddLat_dual_comp_forget_to_bddOrd :\n    BddLat.dual \u22d9 forget\u2082 BddLat BddOrd =\n    forget\u2082 BddLat BddOrd \u22d9 BddOrd.dual", "@[simp]\ntheorem bernoulli'_one : bernoulli' 1 = 1 / 2", "theorem bernoulli'_spec (n : \u2115) :\n    (\u2211 k \u2208 range n.succ, (n.choose (n - k) : \u211a) / (n - k + 1) * bernoulli' k) = 1", "theorem bernoulliFourierCoeff_eq {k : \u2115} (hk : k \u2260 0) (n : \u2124) :\n    bernoulliFourierCoeff k n = -k ! / (2 * \u03c0 * I * n) ^ k", "theorem sum (n : \u2115) : (\u2211 \u03bd \u2208 Finset.range (n + 1), bernsteinPolynomial R n \u03bd) = 1", "theorem _root_.biInf_finsetSigma [CompleteLattice \u03b2] (s : Finset \u03b9) (t : \u2200 i, Finset (\u03b1 i))\n    (f : Sigma \u03b1 \u2192 \u03b2) : \u2a05 ij \u2208 s.sigma t, f ij = \u2a05 (i \u2208 s) (j \u2208 t i), f \u27e8i, j\u27e9", "theorem _root_.biSup_sigma' (s : Set \u03b9) (t : \u2200 i, Set (\u03b1 i)) (f : \u2200 i, \u03b1 i \u2192 \u03b2) :\n    \u2a06 (i \u2208 s) (j \u2208 t i), f i j = \u2a06 ij \u2208 s.sigma t, f ij.fst ij.snd", "theorem bihimp_assoc : a \u21d4 b \u21d4 c = a \u21d4 (b \u21d4 c)", "theorem bihimp_def [Inf \u03b1] [HImp \u03b1] (a b : \u03b1) : a \u21d4 b = (b \u21e8 a) \u2293 (a \u21e8 b)", "@[simp]\ntheorem bihimp_iff_iff {p q : Prop} : p \u21d4 q \u2194 (p \u2194 q)", "@[simp]\ntheorem bihimp_inf_sup : a \u21d4 b \u2293 (a \u2294 b) = a \u2293 b", "theorem birkhoffAverage_one (f : \u03b1 \u2192 \u03b1) (g : \u03b1 \u2192 M) (x : \u03b1) :\n    birkhoffAverage R f g 1 x = g x", "theorem birkhoffSum_one (f : \u03b1 \u2192 \u03b1) (g : \u03b1 \u2192 M) (x : \u03b1) : birkhoffSum f g 1 x = g x", "theorem birkhoffSum_succ (f : \u03b1 \u2192 \u03b1) (g : \u03b1 \u2192 M) (n : \u2115) (x : \u03b1) :\n    birkhoffSum f g (n + 1) x = birkhoffSum f g n x + g (f^[n] x)", "theorem bit1_injective : Function.Injective (bit1 : R \u2192 R)", "theorem borel_eq_top_of_countable [TopologicalSpace \u03b1] [T1Space \u03b1] [Countable \u03b1] : borel \u03b1 = \u22a4", "@[simp]\ntheorem bot_ne_top : (\u22a5 : \u03b1) \u2260 \u22a4", "@[reassoc (attr := simp)]\ntheorem boundariesToCycles'_naturality (i : \u03b9) :\n    boundariesMap f i \u226b C\u2082.boundariesToCycles' i =\n      C\u2081.boundariesToCycles' i \u226b cycles'Map f i", "theorem boundedSpace_induced_iff {\u03b1 \u03b2 : Type*} [Bornology \u03b2] {f : \u03b1 \u2192 \u03b2} :\n    @BoundedSpace \u03b1 (Bornology.induced f) \u2194 IsBounded (range f)", "theorem bounded_stdSimplex : IsBounded (stdSimplex \u211d \u03b9)", "theorem btw_rfl {a : \u03b1} : btw a a a", "theorem card_eq_pow_finrank [Fintype V] : Fintype.card V = q ^ FiniteDimensional.finrank K V", "theorem card_le_of_surjective' [RankCondition R] {\u03b1 \u03b2 : Type*} [Fintype \u03b1] [Fintype \u03b2]\n    (f : (\u03b1 \u2192\u2080 R) \u2192\u2097[R] \u03b2 \u2192\u2080 R) (i : Surjective f) : Fintype.card \u03b2 \u2264 Fintype.card \u03b1", "theorem cardinal_sInter_mem {S : Set (Set \u03b1)} [CardinalInterFilter l c] (hSc : #S < c) :\n    \u22c2\u2080 S \u2208 l \u2194 \u2200 s \u2208 S, s \u2208 l", "@[simp] theorem cast_cast : \u2200 (ha : \u03b1 = \u03b2) (hb : \u03b2 = \u03b3) (a : \u03b1),\n    cast hb (cast ha a) = cast (ha.trans hb) a", "theorem catalan_three : catalan 3 = 5", "@[to_additive]\ntheorem cauchySeq_finset_iff_nat_tprod_vanishing {f : \u2115 \u2192 G} :\n    (CauchySeq fun s : Finset \u2115 \u21a6 \u220f n \u2208 s, f n) \u2194\n      \u2200 e \u2208 \ud835\udcdd (1 : G), \u2203 N : \u2115, \u2200 t \u2286 {n | N \u2264 n}, (\u220f' n : t, f n) \u2208 e", "theorem cauchySeq_of_le_geometric : CauchySeq f", "theorem cauchySeq_of_le_tendsto_0 {s : \u03b2 \u2192 \u03b1} (b : \u03b2 \u2192 \u211d)\n    (h : \u2200 n m N : \u03b2, N \u2264 n \u2192 N \u2264 m \u2192 dist (s n) (s m) \u2264 b N) (h\u2080 : Tendsto b atTop (\ud835\udcdd 0)) :\n    CauchySeq s", "theorem cauchySeq_tendsto_of_complete [Preorder \u03b2] [CompleteSpace \u03b1] {u : \u03b2 \u2192 \u03b1}\n    (H : CauchySeq u) : \u2203 x, Tendsto u atTop (\ud835\udcdd x)", "theorem cauchy_iff_exists_le_nhds [CompleteSpace \u03b1] {l : Filter \u03b1} [NeBot l] :\n    Cauchy l \u2194 \u2203 x, l \u2264 \ud835\udcdd x", "theorem cauchy_map_of_uniformCauchySeqOn_fderiv {s : Set E} (hs : IsOpen s) (h's : IsPreconnected s)\n    (hf' : UniformCauchySeqOn f' l s) (hf : \u2200 n : \u03b9, \u2200 y : E, y \u2208 s \u2192 HasFDerivAt (f n) (f' n y) y)\n    {x\u2080 x : E} (hx\u2080 : x\u2080 \u2208 s) (hx : x \u2208 s) (hfg : Cauchy (map (fun n => f n x\u2080) l)) :\n    Cauchy (map (fun n => f n x) l)", "theorem _root_.cauchy_product (ha : IsCauSeq abs fun m \u21a6 \u2211 n \u2208 range m, abv (f n))\n    (hb : IsCauSeq abv fun m \u21a6 \u2211 n \u2208 range m, g n) (\u03b5 : \u03b1) (\u03b50 : 0 < \u03b5) :\n    \u2203 i : \u2115, \u2200 j \u2265 i,\n      abv ((\u2211 k \u2208 range j, f k) * \u2211 k \u2208 range j, g k -\n        \u2211 n \u2208 range j, \u2211 m \u2208 range (n + 1), f m * g (n - m)) < \u03b5", "@[simp] lemma ceilDiv_one [Nontrivial \u03b1] (b : \u03b2) : b \u2308/\u2309 (1 : \u03b1) = b", "lemma cfc_apply : cfc f a = cfcHom (a := a) ha \u27e8_, hf.restrict\u27e9", "lemma cfc_apply_pi {\u03b9 : Type*} (f : \u03b9 \u2192 R \u2192 R) (a : A) (ha : p a", "lemma cfc_comp_polynomial (q : R[X]) (f : R \u2192 R) (a : A)\n    (hf : ContinuousOn f (q.eval '' (spectrum R a))", "lemma cfc_nonneg_iff (f : R \u2192 R) (a : A) (hf : ContinuousOn f (spectrum R a)", "lemma cfc_nonpos_iff (f : R \u2192 R) (a : A) (hf : ContinuousOn f (spectrum R a)", "lemma cfc_smul {S : Type*} [SMul S R] [ContinuousConstSMul S R]\n    [SMulZeroClass S A] [IsScalarTower S R A] [IsScalarTower S R (R \u2192 R)]\n    (s : S) (f : R \u2192 R) (a : A) (hf : ContinuousOn f (spectrum R a)", "lemma cfc_star_id : cfc (star \u00b7 : R \u2192 R) a = star a", "lemma cfc\u2099_apply : cfc\u2099 f a = cfc\u2099Hom (a := a) ha \u27e8\u27e8_, hf.restrict\u27e9, hf0\u27e9", "lemma cfc\u2099_comp_star (hf : ContinuousOn f (star '' (\u03c3\u2099 R a))", "theorem charZero_of_inj_zero [AddGroupWithOne R] (H : \u2200 n : \u2115, (n : R) = 0 \u2192 n = 0) :\n    CharZero R", "theorem charpoly_leftMulMatrix {S : Type*} [Ring S] [Algebra R S] (h : PowerBasis R S) :\n    (leftMulMatrix h.basis h.gen).charpoly = minpoly R h.gen", "theorem chartAt_self_eq {H : Type*} [TopologicalSpace H] {x : H} :\n    chartAt H x = PartialHomeomorph.refl H", "theorem ciInf_le_of_le {f : \u03b9 \u2192 \u03b1} (H : BddBelow (range f)) (c : \u03b9) (h : f c \u2264 a) : iInf f \u2264 a", "@[simp]\ntheorem ciSup_of_empty [IsEmpty \u03b9] (f : \u03b9 \u2192 \u03b1) : \u2a06 i, f i = \u22a5", "theorem ciSup_subsingleton [Subsingleton \u03b9] (i : \u03b9) (s : \u03b9 \u2192 \u03b1) : \u2a06 i, s i = s i", "theorem circleIntegrable_iff [NormedSpace \u2102 E] {f : \u2102 \u2192 E} {c : \u2102} (R : \u211d) :\n    CircleIntegrable f c R \u2194 IntervalIntegrable (fun \u03b8 : \u211d =>\n      deriv (circleMap c R) \u03b8 \u2022 f (circleMap c R \u03b8)) volume 0 (2 * \u03c0)", "theorem integral_sub {f g : \u2102 \u2192 E} {c : \u2102} {R : \u211d} (hf : CircleIntegrable f c R)\n    (hg : CircleIntegrable g c R) :\n    (\u222e z in C(c, R), f z - g z) = (\u222e z in C(c, R), f z) - \u222e z in C(c, R), g z", "@[simp]\ntheorem circleMap_zero_radius (c : \u2102) : circleMap c 0 = const \u211d c", "theorem circle_def : \u2191circle = { z : \u2102 | abs z = 1 }", "@[to_additive]\ntheorem closedBall_one_div_singleton : closedBall 1 \u03b4 / {x} = closedBall x\u207b\u00b9 \u03b4", "@[to_additive]\ntheorem closedBall_one_mul_singleton : closedBall 1 \u03b4 * {x} = closedBall x \u03b4", "lemma closedBall_prod_same (x : \u03b1) (y : \u03b2) (r : \u211d) :\n    closedBall x r \u00d7\u02e2 closedBall y r = closedBall (x, y) r := ext fun z => by simp [Prod.dist_eq]", "theorem closedBall_sub_ball (h\u03b5 : 0 \u2264 \u03b5) (h\u03b4 : 0 < \u03b4) (a b : E) :\n    closedBall a \u03b5 - ball b \u03b4 = ball (a - b) (\u03b5 + \u03b4)", "theorem closedEmbedding_inl : ClosedEmbedding (inl : X \u2192 X \u2295 Y)", "theorem closedEmbedding_subtype_val (h : IsClosed { a | p a }) :\n    ClosedEmbedding ((\u2191) : Subtype p \u2192 X)", "theorem closedEmbedding_update {\u03b9 : Type*} {\u03b2 : \u03b9 \u2192 Type*}\n    [DecidableEq \u03b9] [(i : \u03b9) \u2192 TopologicalSpace (\u03b2 i)]\n    (x : (i : \u03b9) \u2192 \u03b2 i) (i : \u03b9) [(i : \u03b9) \u2192 T1Space (\u03b2 i)] :\n    ClosedEmbedding (update x i)", "@[simp]\ntheorem closure_Ioc {a b : \u03b1} (hab : a \u2260 b) : closure (Ioc a b) = Icc a b", "@[simp]\ntheorem closure_closure : closure (closure s) = closure s", "@[simp]\ntheorem closure_diff_intrinsicFrontier (s : Set P) :\n    closure s \\ intrinsicFrontier \ud835\udd5c s = intrinsicInterior \ud835\udd5c s", "@[simp]\ntheorem closure_empty_iff (s : Set X) : closure s = \u2205 \u2194 s = \u2205", "theorem closure_eq_cluster_pts : closure s = { a | ClusterPt a (\ud835\udcdf s) }", "lemma closure_iUnion (f : \u03b9 \u2192 Set \u03b1) : closure (\u22c3 i, f i) = \u22c3 i, closure (f i)", "theorem closure_image_closure (h : Continuous f) :\n    closure (f '' closure s) = closure (f '' s)", "theorem closure_induced {f : \u03b1 \u2192 \u03b2} {a : \u03b1} {s : Set \u03b1} :\n    a \u2208 @closure \u03b1 (t.induced f) s \u2194 f a \u2208 closure (f '' s)", "theorem closure_minimal (h\u2081 : s \u2286 t) (h\u2082 : IsClosed t) : closure s \u2286 t", "@[simp]\ntheorem closure_thickening (h\u03b4 : 0 < \u03b4) (s : Set E) :\n    closure (thickening \u03b4 s) = cthickening \u03b4 s", "@[simp]\ntheorem cmp_eq_gt_iff : cmp x y = Ordering.gt \u2194 y < x", "@[simp]\ntheorem cmp_ofDual [LT \u03b1] [@DecidableRel \u03b1 (\u00b7 < \u00b7)] (x y : \u03b1\u1d52\u1d48) :\n    cmp (ofDual x) (ofDual y) = cmp y x", "@[simp]\ntheorem cmp_swap [Preorder \u03b1] [@DecidableRel \u03b1 (\u00b7 < \u00b7)] (a b : \u03b1) : (cmp a b).swap = cmp b a", "theorem codisjoint_assoc : Codisjoint (a \u2294 b) c \u2194 Codisjoint a (b \u2294 c)", "theorem coe_algEquiv_lpBCF (f : lp (fun _ : \u03b1 \u21a6 A) \u221e) : (AlgEquiv.lpBCF \u03b1 A \ud835\udd5c f : \u03b1 \u2192 A) = f", "@[to_additive (attr := simp) coe_comp_nnnorm]\ntheorem coe_comp_nnnorm' : (toReal : \u211d\u22650 \u2192 \u211d) \u2218 (nnnorm : E \u2192 \u211d\u22650) = norm", "@[to_additive (attr := simp, norm_cast)]\nlemma coe_div_coe [SetLike S G] [DivisionMonoid G] [SubgroupClass S G] (H : S) :\n    H / H = (H : Set G)", "@[simp]\ntheorem coe_inv_circle (z : circle) : \u2191z\u207b\u00b9 = (z : \u2102)\u207b\u00b9", "theorem coe_iterateFrobenius : iterateFrobenius R p n = (frobenius R p)^[n]", "@[simp, norm_cast]\ntheorem coe_nnreal_ennreal_nndist (x y : \u03b1) : \u2191(nndist x y) = edist x y", "theorem collinear_iff_rank_le_one (s : Set P) :\n    Collinear k s \u2194 Module.rank k (vectorSpan k s) \u2264 1", "theorem collinear_insert_insert_insert_left_of_mem_affineSpan_pair {p\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P}\n    (h\u2081 : p\u2081 \u2208 line[k, p\u2084, p\u2085]) (h\u2082 : p\u2082 \u2208 line[k, p\u2084, p\u2085]) (h\u2083 : p\u2083 \u2208 line[k, p\u2084, p\u2085]) :\n    Collinear k ({p\u2081, p\u2082, p\u2083, p\u2084} : Set P)", "@[simp]\ntheorem commProb_eq_zero_of_infinite [Infinite M] : commProb M = 0", "theorem commutatorElement_def {G : Type*} [Group G] (g\u2081 g\u2082 : G) :\n    \u2045g\u2081, g\u2082\u2046 = g\u2081 * g\u2082 * g\u2081\u207b\u00b9 * g\u2082\u207b\u00b9", "@[simp]\ntheorem commute_lmul_sq_rmul (a : A) : Commute (L (a * a)) (R a)", "theorem comp_le_uniformity : ((\ud835\udce4 \u03b1).lift' fun s : Set (\u03b1 \u00d7 \u03b1) => s \u25cb s) \u2264 \ud835\udce4 \u03b1", "theorem comp_le_uniformity3 : ((\ud835\udce4 \u03b1).lift' fun s : Set (\u03b1 \u00d7 \u03b1) => s \u25cb (s \u25cb s)) \u2264 \ud835\udce4 \u03b1", "@[mono]\ntheorem compactCovering_subset \u2983m n : \u2115\u2984 (h : m \u2264 n) : compactCovering X m \u2286 compactCovering X n", "theorem compare_le_iff_le {a b : \u03b1} : (compare a b \u2260 .gt) \u2194 a \u2264 b", "@[simp]\ntheorem compl_bot : (\u22a5 : \u03b1)\u1d9c = \u22a4", "@[simp]\ntheorem compl_compl (x : \u03b1) : x\u1d9c\u1d9c = x", "theorem compl_sSup : (sSup s)\u1d9c = \u2a05 i \u2208 s, i\u1d9c", "theorem compl_sup_compl_le : a\u1d9c \u2294 b\u1d9c \u2264 (a \u2293 b)\u1d9c", "theorem compl_surjective : Function.Surjective (compl : \u03b1 \u2192 \u03b1)", "theorem concaveOn_of_slope_anti_adjacent (hs : Convex \ud835\udd5c s)\n    (hf :\n      \u2200 {x y z : \ud835\udd5c},\n        x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f z - f y) / (z - y) \u2264 (f y - f x) / (y - x)) :\n    ConcaveOn \ud835\udd5c s f", "theorem conductor_subset_adjoin : (conductor R x : Set S) \u2286 R<x>", "theorem conformalFactorAt_pos {f : E \u2192 F} {x : E} (h : ConformalAt f x) : 0 < conformalFactorAt h", "theorem congr {\u03b1 : Sort u} {\u03b2 : Sort v} {f\u2081 f\u2082 : \u03b1 \u2192 \u03b2} {a\u2081 a\u2082 : \u03b1} (h\u2081 : Eq f\u2081 f\u2082) (h\u2082 : Eq a\u2081 a\u2082) : Eq (f\u2081 a\u2081) (f\u2082 a\u2082)", "theorem conjugate_nonneg {a : R} (ha : 0 \u2264 a) (c : R) : 0 \u2264 star c * a * c", "theorem conjugate_pos {a : R} (ha : 0 < a) {c : R} (hc : IsRegular c) : 0 < star c * a * c", "theorem connectedComponentIn_eq {x y : \u03b1} {F : Set \u03b1} (h : y \u2208 connectedComponentIn F x) :\n    connectedComponentIn F x = connectedComponentIn F y", "theorem connectedComponentIn_eq_empty {F : Set \u03b1} {x : \u03b1} (h : x \u2209 F) :\n    connectedComponentIn F x = \u2205", "theorem connectedComponentIn_subset (F : Set \u03b1) (x : \u03b1) : connectedComponentIn F x \u2286 F", "theorem connectedComponents_preimage_singleton {x : \u03b1} :\n    (\u2191) \u207b\u00b9' ({\u2191x} : Set (ConnectedComponents \u03b1)) = connectedComponent x", "@[simp]\ntheorem constantCoeff_wittStructureRat_zero (\u03a6 : MvPolynomial idx \u211a) :\n    constantCoeff (wittStructureRat p \u03a6 0) = constantCoeff \u03a6", "theorem contDiffAt_const {c : F} : ContDiffAt \ud835\udd5c n (fun _ : E => c) x", "theorem contDiffAt_succ_iff_hasFDerivAt {n : \u2115} :\n    ContDiffAt \ud835\udd5c (n + 1 : \u2115) f x \u2194\n      \u2203 f' : E \u2192 E \u2192L[\ud835\udd5c] F, (\u2203 u \u2208 \ud835\udcdd x, \u2200 x \u2208 u, HasFDerivAt f (f' x) x) \u2227 ContDiffAt \ud835\udd5c n f' x", "theorem contDiffGroupoid_prod {I : ModelWithCorners \ud835\udd5c E H} {I' : ModelWithCorners \ud835\udd5c E' H'}\n    {e : PartialHomeomorph H H} {e' : PartialHomeomorph H' H'} (he : e \u2208 contDiffGroupoid \u22a4 I)\n    (he' : e' \u2208 contDiffGroupoid \u22a4 I') : e.prod e' \u2208 contDiffGroupoid \u22a4 (I.prod I')", "theorem contDiffOn_succ_iff_deriv_of_isOpen {n : \u2115} (hs : IsOpen s\u2082) :\n    ContDiffOn \ud835\udd5c (n + 1 : \u2115) f\u2082 s\u2082 \u2194 DifferentiableOn \ud835\udd5c f\u2082 s\u2082 \u2227 ContDiffOn \ud835\udd5c n (deriv f\u2082) s\u2082", "theorem contDiffOn_top_iff_fderivWithin (hs : UniqueDiffOn \ud835\udd5c s) :\n    ContDiffOn \ud835\udd5c \u221e f s \u2194\n      DifferentiableOn \ud835\udd5c f s \u2227 ContDiffOn \ud835\udd5c \u221e (fun y => fderivWithin \ud835\udd5c f s y) s", "theorem contDiffOn_top_iff_fderiv_of_isOpen (hs : IsOpen s) :\n    ContDiffOn \ud835\udd5c \u221e f s \u2194 DifferentiableOn \ud835\udd5c f s \u2227 ContDiffOn \ud835\udd5c \u221e (fun y => fderiv \ud835\udd5c f y) s", "theorem contDiffWithinAt_fst {s : Set (E \u00d7 F)} {p : E \u00d7 F} :\n    ContDiffWithinAt \ud835\udd5c n (Prod.fst : E \u00d7 F \u2192 E) s p", "theorem contDiff_euclidean {n : \u2115\u221e} : ContDiff \ud835\udd5c n f \u2194 \u2200 i, ContDiff \ud835\udd5c n fun x => f x i", "theorem contDiff_inner {n} : ContDiff \u211d n fun p : E \u00d7 E => \u27eap.1, p.2\u27eb", "theorem contDiff_mul : ContDiff \ud835\udd5c n fun p : \ud835\udd38 \u00d7 \ud835\udd38 => p.1 * p.2", "theorem contDiff_top : ContDiff \ud835\udd5c \u221e f \u2194 \u2200 n : \u2115, ContDiff \ud835\udd5c n f", "theorem contDiff_zero_fun : ContDiff \ud835\udd5c n fun _ : E => (0 : F)", "theorem contMDiffAt_extChartAt' {x' : M} (h : x' \u2208 (chartAt H x).source) :\n    ContMDiffAt I \ud835\udcd8(\ud835\udd5c, E) n (extChartAt I x) x'", "theorem contMDiffAt_fst {p : M \u00d7 N} : ContMDiffAt (I.prod J) I n Prod.fst p", "theorem contMDiffAt_prod_iff (f : M \u2192 M' \u00d7 N') {x : M} :\n    ContMDiffAt I (I'.prod J') n f x \u2194\n      ContMDiffAt I I' n (Prod.fst \u2218 f) x \u2227 ContMDiffAt I J' n (Prod.snd \u2218 f) x", "theorem contMDiffOn_chart : ContMDiffOn I I n (chartAt H x) (chartAt H x).source", "theorem contMDiffOn_congr (h\u2081 : \u2200 y \u2208 s, f\u2081 y = f y) :\n    ContMDiffOn I I' n f\u2081 s \u2194 ContMDiffOn I I' n f s", "@[to_additive]\ntheorem contMDiffOn_finset_prod' (h : \u2200 i \u2208 t, ContMDiffOn I' I n (f i) s) :\n    ContMDiffOn I' I n (\u220f i \u2208 t, f i) s", "theorem contMDiffOn_snd {s : Set (M \u00d7 N)} : ContMDiffOn (I.prod J) J n Prod.snd s", "theorem contMDiff_iff :\n    ContMDiff I I' n f \u2194\n      Continuous f \u2227\n        \u2200 (x : M) (y : M'),\n          ContDiffOn \ud835\udd5c n (extChartAt I' y \u2218 f \u2218 (extChartAt I x).symm)\n            ((extChartAt I x).target \u2229\n              (extChartAt I x).symm \u207b\u00b9' (f \u207b\u00b9' (extChartAt I' y).source))", "theorem contMDiff_prod_iff (f : M \u2192 M' \u00d7 N') :\n    ContMDiff I (I'.prod J') n f \u2194\n      ContMDiff I I' n (Prod.fst \u2218 f) \u2227 ContMDiff I J' n (Prod.snd \u2218 f)", "@[fun_prop]\ntheorem continuousAt_const : ContinuousAt (fun _ : X => y) x", "theorem continuousAt_cpow {p : \u2102 \u00d7 \u2102} (hp_fst : p.fst \u2208 slitPlane) :\n    ContinuousAt (fun x : \u2102 \u00d7 \u2102 => x.1 ^ x.2) p", "theorem continuousAt_gauge_zero (hs : s \u2208 \ud835\udcdd 0) : ContinuousAt (gauge s) 0", "theorem continuousAt_iff_continuous_left_right {a : \u03b1} {f : \u03b1 \u2192 \u03b2} :\n    ContinuousAt f a \u2194 ContinuousWithinAt f (Iic a) a \u2227 ContinuousWithinAt f (Ici a) a", "lemma continuousAt_jacobiTheta\u2082 (z : \u2102) {\u03c4 : \u2102} (h\u03c4 : 0 < im \u03c4) :\n    ContinuousAt (fun p : \u2102 \u00d7 \u2102 \u21a6 jacobiTheta\u2082 p.1 p.2) (z, \u03c4)", "theorem continuousAt_matrix_inv [Fintype n] [DecidableEq n] [CommRing R] [TopologicalRing R]\n    (A : Matrix n n R) (h : ContinuousAt Ring.inverse A.det) : ContinuousAt Inv.inv A", "theorem continuousAt_of_locally_lipschitz {x : \u03b1} {r : \u211d} (hr : 0 < r) (K : \u211d)\n    (h : \u2200 y, dist y x < r \u2192 dist (f y) (f x) \u2264 K * dist y x) : ContinuousAt f x", "theorem continuousLinearMapToWeakSpace_bijective :\n    Function.Bijective (continuousLinearMapToWeakSpace \ud835\udd5c E)", "theorem continuousOn_iff_isClosed {f : \u03b1 \u2192 \u03b2} {s : Set \u03b1} :\n    ContinuousOn f s \u2194 \u2200 t : Set \u03b2, IsClosed t \u2192 \u2203 u, IsClosed u \u2227 f \u207b\u00b9' t \u2229 s = u \u2229 s", "lemma continuousOn_integral_bilinear_of_locally_integrable_of_compact_support\n    [NormedSpace \ud835\udd5c E] (L : F \u2192L[\ud835\udd5c] G \u2192L[\ud835\udd5c] E)\n    {f : X \u2192 Y \u2192 G} {s : Set X} {k : Set Y} {g : Y \u2192 F}\n    (hk : IsCompact k) (hf : ContinuousOn f.uncurry (s \u00d7\u02e2 univ))\n    (hfs : \u2200 p, \u2200 x, p \u2208 s \u2192 x \u2209 k \u2192 f p x = 0) (hg : IntegrableOn g k \u03bc) :\n    ContinuousOn (fun x \u21a6 \u222b y, L (g y) (f x y) \u2202\u03bc) s", "lemma continuousOn_tangentCoordChange (x y : M) : ContinuousOn (tangentCoordChange I x y)\n    ((extChartAt I x).source \u2229 (extChartAt I y).source)", "theorem continuousWithinAt_const_smul_iff\u2080 (hc : c \u2260 0) :\n    ContinuousWithinAt (fun x => c \u2022 f x) s b \u2194 ContinuousWithinAt f s b", "@[simp]\ntheorem continuousWithinAt_diff_self {f : \u03b1 \u2192 \u03b2} {s : Set \u03b1} {x : \u03b1} :\n    ContinuousWithinAt f (s \\ {x}) x \u2194 ContinuousWithinAt f s x", "theorem continuous_def {_ : TopologicalSpace X} {_ : TopologicalSpace Y} {f : X \u2192 Y} :\n    Continuous f \u2194 \u2200 s, IsOpen s \u2192 IsOpen (f \u207b\u00b9' s)", "theorem continuous_iInf_dom {t\u2081 : \u03b9 \u2192 TopologicalSpace \u03b1} {t\u2082 : TopologicalSpace \u03b2} {i : \u03b9} :\n    Continuous[t\u2081 i, t\u2082] f \u2192 Continuous[iInf t\u2081, t\u2082] f", "theorem continuous_iSup_rng {t\u2081 : TopologicalSpace \u03b1} {t\u2082 : \u03b9 \u2192 TopologicalSpace \u03b2} {i : \u03b9}\n    (h : Continuous[t\u2081, t\u2082 i] f) : Continuous[t\u2081, iSup t\u2082] f", "theorem continuous_iff_ultrafilter :\n    Continuous f \u2194 \u2200 (x) (g : Ultrafilter X), \u2191g \u2264 \ud835\udcdd x \u2192 Tendsto f g (\ud835\udcdd (f x))", "theorem continuous_inclusion {s t : Set X} (h : s \u2286 t) : Continuous (inclusion h)", "theorem continuous_induced_rng {g : \u03b3 \u2192 \u03b1} {t\u2082 : TopologicalSpace \u03b2} {t\u2081 : TopologicalSpace \u03b3} :\n    Continuous[t\u2081, induced f t\u2082] g \u2194 Continuous[t\u2081, t\u2082] (f \u2218 g)", "@[continuity, fun_prop]\ntheorem continuous_isLeft : Continuous (isLeft : X \u2295 Y \u2192 Bool)", "@[continuity, fun_prop]\ntheorem continuous_isRight : Continuous (isRight : X \u2295 Y \u2192 Bool)", "theorem continuous_le_rng {t\u2081 : TopologicalSpace \u03b1} {t\u2082 t\u2083 : TopologicalSpace \u03b2} (h\u2081 : t\u2082 \u2264 t\u2083)\n    (h\u2082 : Continuous[t\u2081, t\u2082] f) : Continuous[t\u2081, t\u2083] f", "@[continuity]\ntheorem continuous_matrix [TopologicalSpace \u03b1] {f : \u03b1 \u2192 Matrix m n R}\n    (h : \u2200 i j, Continuous fun a => f a i j) : Continuous f", "@[to_additive continuous_of_continuousAt_zero\u2082]\ntheorem continuous_of_continuousAt_one\u2082 {H M : Type*} [CommMonoid M] [TopologicalSpace M]\n    [ContinuousMul M] [Group H] [TopologicalSpace H] [TopologicalGroup H] (f : G \u2192* H \u2192* M)\n    (hf : ContinuousAt (fun x : G \u00d7 H \u21a6 f x.1 x.2) (1, 1))\n    (hl : \u2200 x, ContinuousAt (f x) 1) (hr : \u2200 y, ContinuousAt (f \u00b7 y) 1) :\n    Continuous (fun x : G \u00d7 H \u21a6 f x.1 x.2)", "@[to_additive]\ntheorem continuous_of_mulTSupport [TopologicalSpace \u03b2] {f : \u03b1 \u2192 \u03b2}\n    (hf : \u2200 x \u2208 mulTSupport f, ContinuousAt f x) : Continuous f", "theorem continuous_of_uniform_approx_of_continuous\n    (L : \u2200 u \u2208 \ud835\udce4 \u03b2, \u2203 F, Continuous F \u2227 \u2200 y, (f y, F y) \u2208 u) : Continuous f", "theorem continuous_vsub : Continuous fun x : P \u00d7 P => x.1 -\u1d65 x.2", "theorem convexHull_sum {\u03b9} (s : Finset \u03b9) (t : \u03b9 \u2192 Set E) :\n    convexHull R (\u2211 i \u2208 s, t i) = \u2211 i \u2208 s, convexHull R (t i)", "theorem convexJoin_right_comm (s t u : Set E) :\n    convexJoin \ud835\udd5c (convexJoin \ud835\udd5c s t) u = convexJoin \ud835\udd5c (convexJoin \ud835\udd5c s u) t", "theorem convexOn_dist (z : E) (hs : Convex \u211d s) : ConvexOn \u211d s fun z' => dist z' z", "theorem convexOn_iff_convex_epigraph :\n    ConvexOn \ud835\udd5c s f \u2194 Convex \ud835\udd5c { p : E \u00d7 \u03b2 | p.1 \u2208 s \u2227 f p.1 \u2264 p.2 }", "theorem convexOn_rpow {p : \u211d} (hp : 1 \u2264 p) : ConvexOn \u211d (Ici 0) fun x : \u211d \u21a6 x ^ p", "theorem convexOn_univ_dist (z : E) : ConvexOn \u211d univ fun z' => dist z' z", "theorem convex_halfspace_im_lt (r : \u211d) : Convex \u211d { c : \u2102 | c.im < r }", "theorem convex_halfspace_re_lt (r : \u211d) : Convex \u211d { c : \u2102 | c.re < r }", "theorem convex_iff_div :\n    Convex \ud835\udd5c s \u2194 \u2200 \u2983x\u2984, x \u2208 s \u2192 \u2200 \u2983y\u2984, y \u2208 s \u2192\n      \u2200 \u2983a b : \ud835\udd5c\u2984, 0 \u2264 a \u2192 0 \u2264 b \u2192 0 < a + b \u2192 (a / (a + b)) \u2022 x + (b / (a + b)) \u2022 y \u2208 s", "theorem convex_openSegment (a b : E) : Convex \ud835\udd5c (openSegment \ud835\udd5c a b)", "theorem convex_stdSimplex : Convex \ud835\udd5c (stdSimplex \ud835\udd5c \u03b9)", "theorem countable_sInter_mem {S : Set (Set \u03b1)} (hSc : S.Countable) : \u22c2\u2080 S \u2208 l \u2194 \u2200 s \u2208 S, s \u2208 l", "theorem coverPreserving_opens_map : CoverPreserving (Opens.grothendieckTopology Y)\n    (Opens.grothendieckTopology X) (Opens.map f)", "theorem cpow_eq_nhds' {p : \u2102 \u00d7 \u2102} (hp_fst : p.fst \u2260 0) :\n    (fun x => x.1 ^ x.2) =\u1da0[\ud835\udcdd p] fun x => exp (log x.1 * x.2)", "theorem cross_apply (a b : Fin 3 \u2192 R) :\n    a \u00d7\u2083 b = ![a 1 * b 2 - a 2 * b 1, a 2 * b 0 - a 0 * b 2, a 0 * b 1 - a 1 * b 0]", "@[simp]\ntheorem csInf_Ici : sInf (Ici a) = a", "@[simp]\ntheorem csInf_Ioc [DenselyOrdered \u03b1] (h : a < b) : sInf (Ioc a b) = a", "@[simp]\ntheorem csInf_Ioo [DenselyOrdered \u03b1] (h : a < b) : sInf (Ioo a b) = a", "theorem csInf_le (h\u2081 : BddBelow s) (h\u2082 : a \u2208 s) : sInf s \u2264 a", "lemma csSup_mem_of_not_isSuccLimit'\n    (hbdd : BddAbove s) (hlim : \u00ac IsSuccLimit (sSup s)) :\n    sSup s \u2208 s", "lemma dNext_eq_zero (f : \u2200 i j, C.X i \u27f6 D.X j) (i : \u03b9) (hi : \u00ac c.Rel i (c.next i)) :\n    dNext i f = 0", "theorem denseRange_pure : DenseRange (pure : \u03b1 \u2192 Ultrafilter \u03b1)", "theorem dense_compl_singleton_iff_not_open :\n    Dense ({x}\u1d9c : Set X) \u2194 \u00acIsOpen ({x} : Set X)", "@[simp] theorem dense_discrete {s : Set \u03b1} : Dense s \u2194 s = univ", "theorem dense_iff_inter_open :\n    Dense s \u2194 \u2200 U, IsOpen U \u2192 U.Nonempty \u2192 (U \u2229 s).Nonempty", "theorem dense_pi {\u03b9 : Type*} {\u03b1 : \u03b9 \u2192 Type*} [\u2200 i, TopologicalSpace (\u03b1 i)] {s : \u2200 i, Set (\u03b1 i)}\n    (I : Set \u03b9) (hs : \u2200 i \u2208 I, Dense (s i)) : Dense (pi I s)", "theorem dense_sUnion_interior_of_closed {S : Set (Set X)} (hc : \u2200 s \u2208 S, IsClosed s)\n    (hS : S.Countable) (hU : \u22c3\u2080 S = univ) : Dense (\u22c3 s \u2208 S, interior s)", "theorem derivWithin_ccos (hf : DifferentiableWithinAt \u2102 f s x) (hxs : UniqueDiffWithinAt \u2102 s x) :\n    derivWithin (fun x => Complex.cos (f x)) s x = -Complex.sin (f x) * derivWithin f s x", "theorem derivWithin_const (hxs : UniqueDiffWithinAt \ud835\udd5c s x) : derivWithin (fun _ => c) s x = 0", "theorem derivWithin_pi (h : \u2200 i, DifferentiableWithinAt \ud835\udd5c (fun x => \u03c6 x i) s x)\n    (hs : UniqueDiffWithinAt \ud835\udd5c s x) :\n    derivWithin \u03c6 s x = fun i => derivWithin (fun x => \u03c6 x i) s x", "theorem derivWithin_pow' (hc : DifferentiableWithinAt \ud835\udd5c c s x) (hxs : UniqueDiffWithinAt \ud835\udd5c s x) :\n    derivWithin (fun x => c x ^ n) s x = (n : \ud835\udd5c) * c x ^ (n - 1) * derivWithin c s x", "theorem derivWithin_sum (hxs : UniqueDiffWithinAt \ud835\udd5c s x)\n    (h : \u2200 i \u2208 u, DifferentiableWithinAt \ud835\udd5c (A i) s x) :\n    derivWithin (fun y => \u2211 i \u2208 u, A i y) s x = \u2211 i \u2208 u, derivWithin (A i) s x", "theorem derivWithin_zero_of_isolated (h : \ud835\udcdd[s \\ {x}] x = \u22a5) : derivWithin f s x = 0", "@[simp]\ntheorem deriv_circleMap (c : \u2102) (R : \u211d) (\u03b8 : \u211d) : deriv (circleMap c R) \u03b8 = circleMap 0 R \u03b8 * I", "theorem deriv_circleMap_ne_zero {c : \u2102} {R : \u211d} {\u03b8 : \u211d} (hR : R \u2260 0) :\n    deriv (circleMap c R) \u03b8 \u2260 0", "theorem deriv_const_add (c : F) : deriv (fun y => c + f y) x = deriv f x", "@[simp]\ntheorem deriv_id' : deriv (@id \ud835\udd5c) = fun _ => 1", "theorem deriv_mem_iff {f : \ud835\udd5c \u2192 F} {s : Set F} {x : \ud835\udd5c} :\n    deriv f x \u2208 s \u2194\n      DifferentiableAt \ud835\udd5c f x \u2227 deriv f x \u2208 s \u2228 \u00acDifferentiableAt \ud835\udd5c f x \u2227 (0 : F) \u2208 s", "theorem deriv_pow : deriv (fun x : \ud835\udd5c => x ^ n) x = (n : \ud835\udd5c) * x ^ (n - 1)", "@[simp]\ntheorem deriv_pow' : (deriv fun x : \ud835\udd5c => x ^ n) = fun x => (n : \ud835\udd5c) * x ^ (n - 1)", "@[simp]\ntheorem deriv_sin (hc : DifferentiableAt \u211d f x) :\n    deriv (fun x => Real.sin (f x)) x = Real.cos (f x) * deriv f x", "theorem deriv_smul_const (hc : DifferentiableAt \ud835\udd5c c x) (f : F) :\n    deriv (fun y => c y \u2022 f) x = deriv c x \u2022 f", "theorem deriv_sqrt_mul_log' :\n    (deriv fun x => \u221ax * log x) = fun x => (2 + log x) / (2 * \u221ax)", "@[simp 1100]\ntheorem derivedSeries_one : derivedSeries G 1 = commutator G", "@[simp]\ntheorem descPochhammer_map (f : R \u2192+* T) (n : \u2115) :\n    (descPochhammer R n).map f = descPochhammer T n", "@[simp]\ntheorem descPochhammer_zero : descPochhammer R 0 = 1", "@[to_additive]\nprivate lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card \u2264 (y.1 * y.2).card)\n    (hadd : y.1.card + y.2.card \u2264 x.1.card + x.2.card) (hone : x.1.card < y.1.card) :\n    DevosMulRel x y", "theorem diffContOnCl_const {c : F} : DiffContOnCl \ud835\udd5c (fun _ : E => c) s", "theorem diff_mem_nhdsWithin_compl {x : \u03b1} {s : Set \u03b1} (hs : s \u2208 \ud835\udcdd x) (t : Set \u03b1) :\n    s \\ t \u2208 \ud835\udcdd[t\u1d9c] x", "lemma differentiableAt_comp_neg_iff {a : \ud835\udd5c} :\n    DifferentiableAt \ud835\udd5c f (-a) \u2194 DifferentiableAt \ud835\udd5c (fun x \u21a6 f (-x)) a", "@[simp, fun_prop]\ntheorem differentiableAt_const (c : F) : DifferentiableAt \ud835\udd5c (fun _ => c) x", "@[fun_prop]\ntheorem differentiableAt_fst : DifferentiableAt \ud835\udd5c Prod.fst p", "lemma differentiableAt_jacobiTheta\u2082_fst (z : \u2102) {\u03c4 : \u2102} (h\u03c4 : 0 < im \u03c4) :\n    DifferentiableAt \u2102 (jacobiTheta\u2082 \u00b7 \u03c4) z", "@[fun_prop]\ntheorem differentiableOn_id : DifferentiableOn \ud835\udd5c id s", "@[simp]\ntheorem differentiableOn_sub_const_iff (c : F) :\n    DifferentiableOn \ud835\udd5c (fun y => f y - c) s \u2194 DifferentiableOn \ud835\udd5c f s", "theorem differentiableWithinAt_dslope_of_ne (h : b \u2260 a) :\n    DifferentiableWithinAt \ud835\udd5c (dslope f a) s b \u2194 DifferentiableWithinAt \ud835\udd5c f s b", "@[fun_prop]\ntheorem differentiableWithinAt_inverse {x : R} (hx : IsUnit x) (s : Set R) :\n    DifferentiableWithinAt \ud835\udd5c (@Ring.inverse R _) s x", "@[simp]\ntheorem differentiable_const_add_iff (c : F) :\n    (Differentiable \ud835\udd5c fun y => c + f y) \u2194 Differentiable \ud835\udd5c f", "theorem differentiable_euclidean : Differentiable \ud835\udd5c f \u2194 \u2200 i, Differentiable \ud835\udd5c fun x => f x i", "lemma differentialIdeal_le_iff {I : Ideal B} (hI : I \u2260 \u22a5) [NoZeroSMulDivisors A B] :\n    differentIdeal A B \u2264 I \u2194 (((I\u207b\u00b9 : FractionalIdeal B\u2070 L) : Submodule B L).restrictScalars A).map\n      ((Algebra.trace K L).restrictScalars A) \u2264 1", "theorem directed_id [IsDirected \u03b1 r] : Directed r id", "theorem discreteTopology_iff_locallyInjective (y : Y) :\n    DiscreteTopology X \u2194 IsLocallyInjective fun _ : X \u21a6 y", "theorem discreteTopology_iff_nhds_ne [TopologicalSpace \u03b1] :\n    DiscreteTopology \u03b1 \u2194 \u2200 x : \u03b1, \ud835\udcdd[\u2260] x = \u22a5", "@[to_additive]\ntheorem discreteTopology_of_isOpen_singleton_one (h : IsOpen ({1} : Set G)) :\n    DiscreteTopology G", "theorem disjoint_comm : Disjoint a b \u2194 Disjoint b a", "theorem disjoint_compl_right_iff : Disjoint x y\u1d9c \u2194 x \u2264 y", "theorem disjoint_disjointed (f : \u2115 \u2192 \u03b1) : Pairwise (Disjoint on disjointed f)", "@[simp]\ntheorem disjoint_lift'_closure_nhds [T25Space X] {x y : X} :\n    Disjoint ((\ud835\udcdd x).lift' closure) ((\ud835\udcdd y).lift' closure) \u2194 x \u2260 y", "@[simp]\ntheorem disjoint_nhds_atTop_iff : Disjoint (\ud835\udcdd a) atTop \u2194 \u00acIsTop a", "theorem disjoint_sSup_iff {s : Set \u03b1} : Disjoint a (sSup s) \u2194 \u2200 b \u2208 s, Disjoint a b", "@[simp]\ntheorem disjoint_sup_left : Disjoint (a \u2294 b) c \u2194 Disjoint a c \u2227 Disjoint b c", "theorem disjointed_subset (f : \u2115 \u2192 Set \u03b1) (n : \u2115) : disjointed f n \u2286 f n", "@[simp]\ntheorem dist_center_homothety (p\u2081 p\u2082 : P) (c : \ud835\udd5c) :\n    dist p\u2081 (homothety p\u2081 c p\u2082) = \u2016c\u2016 * dist p\u2081 p\u2082", "theorem dist_dist_dist_le_right (x y z : \u03b1) : dist (dist x y) (dist x z) \u2264 dist y z", "theorem dist_le_Ico_sum_of_dist_le {f : \u2115 \u2192 \u03b1} {m n} (hmn : m \u2264 n) {d : \u2115 \u2192 \u211d}\n    (hd : \u2200 {k}, m \u2264 k \u2192 k < n \u2192 dist (f k) (f (k + 1)) \u2264 d k) :\n    dist (f m) (f n) \u2264 \u2211 i \u2208 Finset.Ico m n, d i", "theorem dist_le_range_sum_dist (f : \u2115 \u2192 \u03b1) (n : \u2115) :\n    dist (f 0) (f n) \u2264 \u2211 i \u2208 Finset.range n, dist (f i) (f (i + 1))", "theorem dist_le_tsum_dist_of_tendsto\u2080 (h : Summable fun n \u21a6 dist (f n) (f n.succ))\n    (ha : Tendsto f atTop (\ud835\udcdd a)) : dist (f 0) a \u2264 \u2211' n, dist (f n) (f n.succ)", "theorem dist_le_tsum_of_dist_le_of_tendsto\u2080 (d : \u2115 \u2192 \u211d) (hf : \u2200 n, dist (f n) (f n.succ) \u2264 d n)\n    (hd : Summable d) (ha : Tendsto f atTop (\ud835\udcdd a)) : dist (f 0) a \u2264 tsum d", "theorem dist_midpoint_midpoint_le (p\u2081 p\u2082 p\u2083 p\u2084 : V) :\n    dist (midpoint \u211d p\u2081 p\u2082) (midpoint \u211d p\u2083 p\u2084) \u2264 (dist p\u2081 p\u2083 + dist p\u2082 p\u2084) / 2", "lemma dist_mono_right_pi : MonotoneOn (dist x) (Ici x)", "@[to_additive (attr := simp)]\ntheorem dist_mul_left [PseudoMetricSpace M] [Mul M] [IsometricSMul M M] (a b c : M) :\n    dist (a * b) (a * c) = dist b c", "theorem dist_nndist (x y : \u03b1) : dist x y = nndist x y", "@[simp] theorem dist_ofMul (a b : X) : dist (ofMul a) (ofMul b) = dist a b", "lemma dist_pi_const_le (a b : \u03b1) : (dist (fun _ : \u03b2 => a) fun _ => b) \u2264 dist a b", "@[simp]\ntheorem dist_vadd_left (v : V) (x : P) : dist (v +\u1d65 x) x = \u2016v\u2016", "theorem dite_dite_comm {B : Q \u2192 \u03b1} {C : \u00acP \u2192 \u00acQ \u2192 \u03b1} (h : P \u2192 \u00acQ) :\n    (if p : P then A p else if q : Q then B q else C p q) =\n     if q : Q then B q else if p : P then A p else C p q", "@[simp] theorem dite_eq_ite [Decidable P] : (dite P (fun _ => a) fun _ => b) = ite P a b", "theorem dite_prop_iff_and {Q : P \u2192 Prop} {R : \u00acP \u2192 Prop} [Decidable P] :\n    dite P Q R \u2194 (\u2200 h, Q h) \u2227 (\u2200 h, R h)", "@[field_simps]\ntheorem div_add_div_same (a b c : \u03b1) : a / c + b / c = (a + b) / c", "@[to_additive]\ntheorem div_eq_of_eq_mul' {a b c : G} (h : a = b * c) : a / b = c", "theorem div_le_iff_of_neg' (hc : c < 0) : b / c \u2264 a \u2194 c * a \u2264 b", "theorem div_le_of_nonneg_of_le_mul (hb : 0 \u2264 b) (hc : 0 \u2264 c) (h : a \u2264 c * b) : a / b \u2264 c", "theorem div_le_one_iff : a / b \u2264 1 \u2194 0 < b \u2227 a \u2264 b \u2228 b = 0 \u2228 b < 0 \u2227 b \u2264 a", "@[to_additive (attr := simp)]\ntheorem div_left_inj : b / a = c / a \u2194 b = c", "@[to_additive]\ntheorem div_lt_iff_lt_mul : a / c < b \u2194 a < b * c", "theorem div_lt_one (hb : 0 < b) : a / b < 1 \u2194 a < b", "@[to_additive (attr := simp)]\ntheorem div_mul_cancel_left (a b : G) : a / (a * b) = b\u207b\u00b9", "lemma div_mul_cancel_of_imp (h : b = 0 \u2192 a = 0) : a / b * b = a", "@[to_additive]\ntheorem div_mul_div_comm : a / b * (c / d) = a * c / (b * d)", "@[to_additive]\ntheorem div_ne_one_of_ne : a \u2260 b \u2192 a / b \u2260 1", "theorem div_neg_iff : a / b < 0 \u2194 0 < a \u2227 b < 0 \u2228 a < 0 \u2227 0 < b", "lemma div_nonneg (ha : 0 \u2264 a) (hb : 0 \u2264 b) : 0 \u2264 a / b", "lemma div_sq_cancel (a b : G\u2080) : a ^ 2 * b / a = a * b", "theorem div_sub_div_same (a b c : K) : a / c - b / c = (a - b) / c", "theorem dslope_sub_smul [DecidableEq \ud835\udd5c] (f : \ud835\udd5c \u2192 E) (a : \ud835\udd5c) :\n    dslope (fun x => (x - a) \u2022 f x) a = update f a (deriv (fun x => (x - a) \u2022 f x) a)", "@[simp]\ntheorem dualTensorHom_apply (f : Module.Dual R M) (m : M) (n : N) :\n    dualTensorHom R M N (f \u2297\u209c n) m = f m \u2022 n", "theorem dvd_gcd_iff [GCDMonoid \u03b1] (a b c : \u03b1) : a \u2223 gcd b c \u2194 a \u2223 b \u2227 a \u2223 c", "theorem dvd_iff_exists_eq_mul_left : a \u2223 b \u2194 \u2203 c, b = c * a", "theorem dvd_mul_of_dvd_left (h : a \u2223 b) (c : \u03b1) : a \u2223 b * c", "theorem dvd_sub (h\u2081 : a \u2223 b) (h\u2082 : a \u2223 c) : a \u2223 b - c", "lemma dvd_zsmul_of_dvd [NonUnitalRing R] {x y : R} (z : \u2124) (h : x \u2223 y) : x \u2223 z \u2022 y", "theorem add_point (f : \u03b1 \u2192 E) {s : Set \u03b1} {x : \u03b1} (hx : x \u2208 s) (u : \u2115 \u2192 \u03b1) (hu : Monotone u)\n    (us : \u2200 i, u i \u2208 s) (n : \u2115) :\n    \u2203 (v : \u2115 \u2192 \u03b1) (m : \u2115), Monotone v \u2227 (\u2200 i, v i \u2208 s) \u2227 x \u2208 v '' Iio m \u2227\n      (\u2211 i \u2208 Finset.range n, edist (f (u (i + 1))) (f (u i))) \u2264\n        \u2211 j \u2208 Finset.range m, edist (f (v (j + 1))) (f (v j))", "theorem eq_zero_iff (f : \u03b1 \u2192 E) {s : Set \u03b1} :\n    eVariationOn f s = 0 \u2194 \u2200 x \u2208 s, \u2200 y \u2208 s, edist (f x) (f y) = 0", "theorem edist_le_of_edist_le_geometric_two_of_tendsto (n : \u2115) : edist (f n) a \u2264 2 * C / 2 ^ n", "theorem edist_le_pi_edist [\u2200 b, EDist (\u03c0 b)] (f g : \u2200 b, \u03c0 b) (b : \u03b2) :\n    edist (f b) (g b) \u2264 edist f g", "@[to_additive]\ntheorem edist_mulIndicator (s t : Set \u03b1) (f : \u03b1 \u2192 E) (x : \u03b1) :\n    edist (s.mulIndicator f x) (t.mulIndicator f x) = \u2016(s \u2206 t).mulIndicator f x\u2016\u208a", "theorem edist_naturalParameterization_eq_zero {f : \u03b1 \u2192 E} {s : Set \u03b1}\n    (hf : LocallyBoundedVariationOn f s) {a : \u03b1} (as : a \u2208 s) {b : \u03b1} (bs : b \u2208 s) :\n    edist (naturalParameterization f s a (variationOnFromTo f s a b)) (f b) = 0", "theorem edist_nndist (x y : \u03b1) : edist x y = nndist x y", "@[simp]\ntheorem edist_ofAdd (a b : X) : edist (ofAdd a) (ofAdd b) = edist a b", "@[simp]\ntheorem edist_ofDual (a b : X\u1d52\u1d48) : edist (ofDual a) (ofDual b) = edist a b", "theorem edist_triangle_left (x y z : \u03b1) : edist x y \u2264 edist z x + edist z y", "theorem edist_vsub_vsub_le (p\u2081 p\u2082 p\u2083 p\u2084 : P) :\n    edist (p\u2081 -\u1d65 p\u2082) (p\u2083 -\u1d65 p\u2084) \u2264 edist p\u2081 p\u2083 + edist p\u2082 p\u2084", "@[simp]\nlemma egauge_zero_zero : egauge \ud835\udd5c (0 : Set E) 0 = 0 := egauge_zero_right _ \u27e80, rfl\u27e9", "theorem embedding_prod_mk (x : X) : Embedding (Prod.mk x : Y \u2192 X \u00d7 Y)", "theorem eq_iff_atom_le_iff {a b : \u03b1} : a = b \u2194 \u2200 c, IsAtom c \u2192 (c \u2264 a \u2194 c \u2264 b)", "@[to_additive]\ntheorem eq_iff_eq_of_div_eq_div (H : a / b = c / d) : a = b \u2194 c = d", "lemma eq_iff_eq_of_lt_iff_lt_of_gt_iff_gt {x' y' : \u03b1}\n    (ltc : (x < y) \u2194 (x' < y')) (gtc : (y < x) \u2194 (y' < x')) :\n    x = y \u2194 x' = y'", "theorem eq_iff_le_not_lt [PartialOrder \u03b1] {a b : \u03b1} : a = b \u2194 a \u2264 b \u2227 \u00aca < b", "@[to_additive]\ntheorem eq_inv_mul_iff_mul_eq : a = b\u207b\u00b9 * c \u2194 b * a = c", "theorem eq_ite_iff : a = ite P b c \u2194 P \u2227 a = b \u2228 \u00acP \u2227 a = c", "theorem eq_max {a b c : \u03b1} (h\u2081 : a \u2264 c) (h\u2082 : b \u2264 c) (h\u2083 : \u2200 {d}, a \u2264 d \u2192 b \u2264 d \u2192 c \u2264 d) :\n    c = max a b", "theorem eq_of_forall_lt_rat_iff_lt (h : \u2200 q : \u211a, x < q \u2194 y < q) : x = y", "theorem eq_of_has_deriv_right_eq (derivf : \u2200 x \u2208 Ico a b, HasDerivWithinAt f (f' x) (Ici x) x)\n    (derivg : \u2200 x \u2208 Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b))\n    (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : \u2200 y \u2208 Icc a b, f y = g y", "theorem eq_of_heq {\u03b1 : Sort u} {a a' : \u03b1} (h : HEq a a') : Eq a a'", "theorem eq_of_le_of_not_lt [PartialOrder \u03b1] {a b : \u03b1} (hab : a \u2264 b) (hba : \u00aca < b) : a = b", "lemma eq_of_mul_inv_eq_one (h : a * b\u207b\u00b9 = 1) : a = b", "theorem eq_of_nndist_eq_zero {x y : \u03b3} : nndist x y = 0 \u2192 x = y", "@[to_additive]\ntheorem eq_one_div_of_mul_eq_one_left (h : b * a = 1) : b = 1 / a", "@[to_additive]\ntheorem eq_one_or_one_lt (a : \u03b1) : a = 1 \u2228 1 < a", "theorem eq_or_lt_of_le [PartialOrder \u03b1] {a b : \u03b1} (h : a \u2264 b) : a = b \u2228 a < b", "theorem eq_orthogonalProjection_of_eq_submodule {K' : Submodule \ud835\udd5c E} [HasOrthogonalProjection K']\n    (h : K = K') (u : E) : (orthogonalProjection K u : E) = (orthogonalProjection K' u : E)", "theorem eq_singleton_bot_of_sSup_eq_bot_of_nonempty {s : Set \u03b1} (h_sup : sSup s = \u22a5)\n    (hne : s.Nonempty) : s = {\u22a5}", "theorem eq_star_iff_eq_star [InvolutiveStar R] {r s : R} : r = star s \u2194 s = star r", "@[to_additive\n      \"If a function defined on a topological additive group has a support contained in a compact\n      set, then either the function is trivial or the group is locally compact.\"]\ntheorem eq_zero_or_locallyCompactSpace_of_support_subset_isCompact_of_group\n    [TopologicalSpace \u03b1] [Zero \u03b1] [T1Space \u03b1]\n    {f : G \u2192 \u03b1} {k : Set G} (hk : IsCompact k) (hf : support f \u2286 k) (h'f : Continuous f) :\n    f = 0 \u2228 LocallyCompactSpace G", "theorem equicontinuousWithinAt_finite [Finite \u03b9] {F : \u03b9 \u2192 X \u2192 \u03b1} {S : Set X} {x\u2080 : X} :\n    EquicontinuousWithinAt F S x\u2080 \u2194 \u2200 i, ContinuousWithinAt (F i) S x\u2080", "@[simp] lemma even_bit0 (a : \u03b1) : Even (bit0 a) := \u27e8a, rfl\u27e9", "theorem eventually_homothety_mem_of_mem_interior (x : Q) {s : Set Q} {y : Q} (hy : y \u2208 interior s) :\n    \u2200\u1da0 \u03b4 in \ud835\udcdd (1 : \ud835\udd5c), homothety x \u03b4 y \u2208 s", "theorem eventually_le_of_tendsto_lt {l : Filter \u03b3} {f : \u03b3 \u2192 \u03b1} {u v : \u03b1} (hv : v < u)\n    (h : Tendsto f l (\ud835\udcdd v)) : \u2200\u1da0 a in l, f a \u2264 u", "theorem existsUnique_sub_zsmul_mem_Ico {a : \u03b1} (ha : 0 < a) (b c : \u03b1) :\n    \u2203! m : \u2124, b - m \u2022 a \u2208 Set.Ico c (c + a)", "theorem exists_Ioc_subset_of_mem_nhds {a : \u03b1} {s : Set \u03b1} (hs : s \u2208 \ud835\udcdd a) (h : \u2203 l, l < a) :\n    \u2203 l < a, Ioc l a \u2286 s", "theorem exists_borelSpace_of_countablyGenerated_of_separatesPoints (\u03b1 : Type*)\n    [m : MeasurableSpace \u03b1] [CountablyGenerated \u03b1] [SeparatesPoints \u03b1] :\n    \u2203 \u03c4 : TopologicalSpace \u03b1, SecondCountableTopology \u03b1 \u2227 T4Space \u03b1 \u2227 BorelSpace \u03b1", "theorem exists_continuous_zero_one_of_isClosed [NormalSpace X]\n    {s t : Set X} (hs : IsClosed s) (ht : IsClosed t)\n    (hd : Disjoint s t) : \u2203 f : C(X, \u211d), EqOn f 0 s \u2227 EqOn f 1 t \u2227 \u2200 x, f x \u2208 Icc (0 : \u211d) 1", "theorem exists_embedding_euclidean_of_compact [T2Space M] [CompactSpace M] :\n    \u2203 (n : \u2115) (e : M \u2192 EuclideanSpace \u211d (Fin n)),\n      Smooth I (\ud835\udce1 n) e \u2227 ClosedEmbedding e \u2227 \u2200 x : M, Injective (mfderiv I (\ud835\udce1 n) e x)", "lemma exists_eq_ciInf_of_not_isPredLimit\n    (hf : BddBelow (Set.range f)) (hf' : \u00ac IsPredLimit (\u2a05 i, f i)) :\n    \u2203 i, f i = \u2a05 i, f i", "@[simp] theorem exists_eq_left : (\u2203 a, a = a' \u2227 p a) \u2194 p a'", "theorem exists_finset_piecewise_mem_of_mem_nhds [DecidableEq \u03b9] {s : Set (\u2200 a, \u03c0 a)} {x : \u2200 a, \u03c0 a}\n    (hs : s \u2208 \ud835\udcdd x) (y : \u2200 a, \u03c0 a) : \u2203 I : Finset \u03b9, I.piecewise x y \u2208 s", "theorem exists_forall_closed_ball_dist_add_le_two_mul_sub (h\u03b5 : 0 < \u03b5) (r : \u211d) :\n    \u2203 \u03b4, 0 < \u03b4 \u2227 \u2200 \u2983x : E\u2984, \u2016x\u2016 \u2264 r \u2192 \u2200 \u2983y\u2984, \u2016y\u2016 \u2264 r \u2192 \u03b5 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 * r - \u03b4", "lemma exists_linearIndependent_of_le_rank {n : \u2115} (hn : n \u2264 Module.rank R M) :\n    \u2203 f : Fin n \u2192 M, LinearIndependent R f", "theorem exists_linearIndependent_of_lt_rank [StrongRankCondition R]\n    {s : Set M} (hs : LinearIndependent (\u03b9 := s) R Subtype.val) :\n    \u2203 t, s \u2286 t \u2227 #t = Module.rank R M \u2227 LinearIndependent (\u03b9 := t) R Subtype.val", "theorem exists_mem_nhds_isCompact_isClosed (x : X) : \u2203 K \u2208 \ud835\udcdd x, IsCompact K \u2227 IsClosed K", "lemma exists_monotone_Icc_subset_open_cover_Icc {\u03b9} {a b : \u211d} (h : a \u2264 b) {c : \u03b9 \u2192 Set (Icc a b)}\n    (hc\u2081 : \u2200 i, IsOpen (c i)) (hc\u2082 : univ \u2286 \u22c3 i, c i) : \u2203 t : \u2115 \u2192 Icc a b, t 0 = a \u2227\n      Monotone t \u2227 (\u2203 m, \u2200 n \u2265 m, t n = b) \u2227 \u2200 n, \u2203 i, Icc (t n) (t (n + 1)) \u2286 c i", "@[to_additive exists_nhds_half_neg]\ntheorem exists_nhds_split_inv {s : Set G} (hs : s \u2208 \ud835\udcdd (1 : G)) :\n    \u2203 V \u2208 \ud835\udcdd (1 : G), \u2200 v \u2208 V, \u2200 w \u2208 V, v / w \u2208 s", "theorem exists_nontrivial_relation_sum_zero_of_not_affine_ind {t : Finset V}\n    (h : \u00acAffineIndependent k ((\u2191) : t \u2192 V)) :\n    \u2203 f : V \u2192 k, \u2211 e \u2208 t, f e \u2022 e = 0 \u2227 \u2211 e \u2208 t, f e = 0 \u2227 \u2203 x \u2208 t, f x \u2260 0", "lemma exists_of_schnirelmannDensity_eq_zero {\u03b5 : \u211d} (h\u03b5 : 0 < \u03b5) (hA : schnirelmannDensity A = 0) :\n    \u2203 n, 0 < n \u2227 ((Ioc 0 n).filter (\u00b7 \u2208 A)).card / n < \u03b5", "theorem exists_partition_approximatesLinearOn_of_hasFDerivWithinAt [SecondCountableTopology F]\n    (f : E \u2192 F) (s : Set E) (f' : E \u2192 E \u2192L[\u211d] F) (hf' : \u2200 x \u2208 s, HasFDerivWithinAt f (f' x) s x)\n    (r : (E \u2192L[\u211d] F) \u2192 \u211d\u22650) (rpos : \u2200 A, r A \u2260 0) :\n    \u2203 (t : \u2115 \u2192 Set E) (A : \u2115 \u2192 E \u2192L[\u211d] F),\n      Pairwise (Disjoint on t) \u2227\n        (\u2200 n, MeasurableSet (t n)) \u2227\n          (s \u2286 \u22c3 n, t n) \u2227\n            (\u2200 n, ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))) \u2227\n              (s.Nonempty \u2192 \u2200 n, \u2203 y \u2208 s, A n = f' y)", "@[congr]\ntheorem exists_prop_congr {p p' : Prop} {q q' : p \u2192 Prop} (hq : \u2200 h, q h \u2194 q' h) (hp : p \u2194 p') :\n    Exists q \u2194 \u2203 h : p', q' (hp.2 h)", "theorem exists_smooth_tsupport_subset {s : Set E} {x : E} (hs : s \u2208 \ud835\udcdd x) :\n    \u2203 f : E \u2192 \u211d,\n      tsupport f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 \u2227 f x = 1", "theorem exists_subset_iUnion_closure_subset (hs : IsClosed s) (uo : \u2200 i, IsOpen (u i))\n    (uf : \u2200 x \u2208 s, { i | x \u2208 u i }.Finite) (us : s \u2286 \u22c3 i, u i) :\n    \u2203 v : \u03b9 \u2192 Set X, s \u2286 iUnion v \u2227 (\u2200 i, IsOpen (v i)) \u2227 \u2200 i, closure (v i) \u2286 u i", "theorem exists_sum_eq_one_iff_pairwise_coprime' [Fintype I] [Nonempty I] [DecidableEq I] :\n    (\u2203 \u03bc : I \u2192 R, (\u2211 i : I, \u03bc i * \u220f j \u2208 {i}\u1d9c, s j) = 1) \u2194 Pairwise (IsCoprime on s)", "theorem expChar_of_injective_algebraMap {R A : Type*}\n    [CommSemiring R] [Semiring A] [Algebra R A] (h : Function.Injective (algebraMap R A))\n    (q : \u2115) [ExpChar R q] : ExpChar A q", "theorem exp_neg_mul_rpow_isLittleO_exp_neg {p b : \u211d} (hb : 0 < b) (hp : 1 < p) :\n    (fun x : \u211d => exp (- b * x ^ p)) =o[atTop] fun x : \u211d => exp (-x)", "theorem extChartAt_image_nhd_mem_nhds_of_boundaryless [I.Boundaryless]\n    {x : M} (hx : s \u2208 \ud835\udcdd x) : extChartAt I x '' s \u2208 \ud835\udcdd (extChartAt I x x)", "theorem extChartAt_target (x : M) :\n    (extChartAt I x).target = I.symm \u207b\u00b9' (chartAt H x).target \u2229 range I", "@[simp]\ntheorem extentClosure_iUnion (f : \u03b9 \u2192 Set \u03b2) :\n    extentClosure r (\u22c3 i, f i) = \u22c2 i, extentClosure r (f i)", "theorem extremePoints_convexHull_subset : (convexHull \ud835\udd5c A).extremePoints \ud835\udd5c \u2286 A", "@[simp]\ntheorem extremePoints_empty : (\u2205 : Set E).extremePoints \ud835\udd5c = \u2205", "@[simp]\ntheorem extremePoints_prod (s : Set E) (t : Set F) :\n    (s \u00d7\u02e2 t).extremePoints \ud835\udd5c = s.extremePoints \ud835\udd5c \u00d7\u02e2 t.extremePoints \ud835\udd5c", "theorem false_and_iff : False \u2227 p \u2194 False", "@[simp] theorem false_or (p : Prop) : (False \u2228 p) = p", "theorem fderiv.comp {g : F \u2192 G} (hg : DifferentiableAt \ud835\udd5c g (f x)) (hf : DifferentiableAt \ud835\udd5c f x) :\n    fderiv \ud835\udd5c (g \u2218 f) x = (fderiv \ud835\udd5c g (f x)).comp (fderiv \ud835\udd5c f x)", "theorem fderiv.comp_deriv (hl : DifferentiableAt \ud835\udd5c l (f x)) (hf : DifferentiableAt \ud835\udd5c f x) :\n    deriv (l \u2218 f) x = (fderiv \ud835\udd5c l (f x) : F \u2192 E) (deriv f x)", "theorem fderivWithin.comp {g : F \u2192 G} {t : Set F} (hg : DifferentiableWithinAt \ud835\udd5c g t (f x))\n    (hf : DifferentiableWithinAt \ud835\udd5c f s x) (h : MapsTo f s t) (hxs : UniqueDiffWithinAt \ud835\udd5c s x) :\n    fderivWithin \ud835\udd5c (g \u2218 f) s x = (fderivWithin \ud835\udd5c g t (f x)).comp (fderivWithin \ud835\udd5c f s x)", "theorem fderivWithin_const_apply (c : F) (hxs : UniqueDiffWithinAt \ud835\udd5c s x) :\n    fderivWithin \ud835\udd5c (fun _ => c) s x = 0", "theorem fderivWithin_eq_fderiv (hs : UniqueDiffWithinAt \ud835\udd5c s x) (h : DifferentiableAt \ud835\udd5c f x) :\n    fderivWithin \ud835\udd5c f s x = fderiv \ud835\udd5c f x", "theorem fderivWithin_eventually_congr_set (h : s =\u1da0[\ud835\udcdd x] t) :\n    fderivWithin \ud835\udd5c f s =\u1da0[\ud835\udcdd x] fderivWithin \ud835\udd5c f t", "theorem fderivWithin_fderivWithin {g : F \u2192 G} {f : E \u2192 F} {x : E} {y : F} {s : Set E} {t : Set F}\n    (hg : DifferentiableWithinAt \ud835\udd5c g t y) (hf : DifferentiableWithinAt \ud835\udd5c f s x) (h : MapsTo f s t)\n    (hxs : UniqueDiffWithinAt \ud835\udd5c s x) (hy : f x = y) (v : E) :\n    fderivWithin \ud835\udd5c g t y (fderivWithin \ud835\udd5c f s x v) = fderivWithin \ud835\udd5c (g \u2218 f) s x v", "theorem fderivWithin_fst {s : Set (E \u00d7 F)} (hs : UniqueDiffWithinAt \ud835\udd5c s p) :\n    fderivWithin \ud835\udd5c Prod.fst s p = fst \ud835\udd5c E F", "theorem fderivWithin_inter (ht : t \u2208 \ud835\udcdd x) : fderivWithin \ud835\udd5c f (s \u2229 t) x = fderivWithin \ud835\udd5c f s x", "theorem fderivWithin_mul_const (hxs : UniqueDiffWithinAt \ud835\udd5c s x)\n    (hc : DifferentiableWithinAt \ud835\udd5c c s x) (d : \ud835\udd38') :\n    fderivWithin \ud835\udd5c (fun y => c y * d) s x = d \u2022 fderivWithin \ud835\udd5c c s x", "theorem fderivWithin_of_mem (st : t \u2208 \ud835\udcdd[s] x) (ht : UniqueDiffWithinAt \ud835\udd5c s x)\n    (h : DifferentiableWithinAt \ud835\udd5c f t x) : fderivWithin \ud835\udd5c f s x = fderivWithin \ud835\udd5c f t x", "theorem fderiv_mem_iff {f : E \u2192 F} {s : Set (E \u2192L[\ud835\udd5c] F)} {x : E} : fderiv \ud835\udd5c f x \u2208 s \u2194\n    DifferentiableAt \ud835\udd5c f x \u2227 fderiv \ud835\udd5c f x \u2208 s \u2228 \u00acDifferentiableAt \ud835\udd5c f x \u2227 (0 : E \u2192L[\ud835\udd5c] F) \u2208 s", "@[simp]\ntheorem fderiv_sinh (hc : DifferentiableAt \u211d f x) :\n    fderiv \u211d (fun x => Real.sinh (f x)) x = Real.cosh (f x) \u2022 fderiv \u211d f x", "@[simp]\ntheorem fdifferential_comp (g : C^\u221e\u27eeI', M'; I'', M''\u27ef) (f : C^\u221e\u27eeI, M; I', M'\u27ef) (x : M) :\n    \ud835\udc85 (g.comp f) x = (\ud835\udc85 g (f x)).comp (\ud835\udc85 f x)", "lemma fermatLastTheoremFor_zero : FermatLastTheoremFor 0", "private lemma filter_collapse_eq (ha : a \u2209 s) (\ud835\udc9c : Finset (Finset \u03b1)) :\n    (\ud835\udc9c.filter fun t \u21a6 t.erase a = s) =\n      if s \u2208 \ud835\udc9c then\n        (if insert a s \u2208 \ud835\udc9c then {s, insert a s} else {s})\n      else\n        (if insert a s \u2208 \ud835\udc9c then {insert a s} else \u2205)", "@[simp]\ntheorem finAddFlip_apply_natAdd (k : Fin n) (m : \u2115) :\n    finAddFlip (Fin.natAdd m k) = Fin.castAdd m k", "lemma _root_.finCongr_symm_apply_coe (h : m = n) (k : Fin n) : ((finCongr h).symm k : \u2115) = k := rfl", "@[to_additive (attr := simp)]\nlemma finEquivPowers_apply (x : G) (hx) {n : Fin (orderOf x)} :\n    finEquivPowers x hx n = \u27e8x ^ (n : \u2115), n, rfl\u27e9 := rfl", "@[to_additive (attr := simp, nolint simpNF)]\nlemma finEquivZPowers_apply (hx) {n : Fin (orderOf x)} :\n    finEquivZPowers x hx n = \u27e8x ^ (n : \u2115), n, zpow_natCast x n\u27e9 := rfl", "@[simp]\ntheorem finSuccEquiv'_symm_none (i : Fin (n + 1)) : (finSuccEquiv' i).symm none = i", "theorem finite_ofLocalizationSpan : RingHom.OfLocalizationSpan @RingHom.Finite", "@[to_additive (attr := simp)]\nlemma finite_zpowers : (zpowers a : Set \u03b1).Finite \u2194 IsOfFinOrder a", "@[to_additive]\ntheorem finprod_curry\u2083 {\u03b3 : Type*} (f : \u03b1 \u00d7 \u03b2 \u00d7 \u03b3 \u2192 M) (h : (mulSupport f).Finite) :\n    \u220f\u1da0 abc, f abc = \u220f\u1da0 (a) (b) (c), f (a, b, c)", "@[to_additive\n      \"If the additive supports of `f` and `g` are finite, then the sum of `f i - g i`\n      equals the sum of `f i` minus the sum of `g i`.\"]\ntheorem finprod_div_distrib [DivisionCommMonoid G] {f g : \u03b1 \u2192 G} (hf : (mulSupport f).Finite)\n    (hg : (mulSupport g).Finite) : \u220f\u1da0 i, f i / g i = (\u220f\u1da0 i, f i) / \u220f\u1da0 i, g i", "@[to_additive]\ntheorem finprod_emb_domain' {f : \u03b1 \u2192 \u03b2} (hf : Injective f) [DecidablePred (\u00b7 \u2208 Set.range f)]\n    (g : \u03b1 \u2192 M) :\n    (\u220f\u1da0 b : \u03b2, if h : b \u2208 Set.range f then g (Classical.choose h) else 1) = \u220f\u1da0 a : \u03b1, g a", "@[to_additive]\ntheorem finprod_eq_dif {p : Prop} [Decidable p] (f : p \u2192 M) :\n    \u220f\u1da0 i, f i = if h : p then f h else 1", "@[to_additive]\ntheorem finprod_mem_def (s : Set \u03b1) (f : \u03b1 \u2192 M) : \u220f\u1da0 a \u2208 s, f a = \u220f\u1da0 a, mulIndicator s f a", "@[to_additive\n      \"Given a finite set `s` and an element `a \u2209 s`, the sum of `f i` over `i \u2208 insert a s`\n      equals `f a` plus the sum of `f i` over `i \u2208 s`.\"]\ntheorem finprod_mem_insert (f : \u03b1 \u2192 M) (h : a \u2209 s) (hs : s.Finite) :\n    \u220f\u1da0 i \u2208 insert a s, f i = f a * \u220f\u1da0 i \u2208 s, f i", "@[to_additive\n      \"A more general version of `finsum_mem_insert` that requires `s \u2229 support f` rather\n      than `s` to be finite.\"]\ntheorem finprod_mem_insert' (f : \u03b1 \u2192 M) (h : a \u2209 s) (hs : (s \u2229 mulSupport f).Finite) :\n    \u220f\u1da0 i \u2208 insert a s, f i = f a * \u220f\u1da0 i \u2208 s, f i", "@[to_additive \"The sum of `f i` over `i \u2208 {a}` equals `f a`.\"]\ntheorem finprod_mem_singleton : (\u220f\u1da0 i \u2208 ({a} : Set \u03b1), f i) = f a", "@[to_additive]\ntheorem finprod_mem_univ (f : \u03b1 \u2192 M) : \u220f\u1da0 i \u2208 @Set.univ \u03b1, f i = \u220f\u1da0 i : \u03b1, f i", "@[to_additive]\ntheorem finprod_of_infinite_mulSupport {f : \u03b1 \u2192 M} (hf : (mulSupport f).Infinite) :\n    \u220f\u1da0 i, f i = 1", "@[to_additive (attr := simp)]\ntheorem finprod_true (f : True \u2192 M) : \u220f\u1da0 i, f i = f trivial", "theorem finrank_eq_one_iff' [Module.Free K V] :\n    finrank K V = 1 \u2194 \u2203 v \u2260 0, \u2200 w : V, \u2203 c : K, c \u2022 v = w", "theorem finrank_quotient_span_eq_natDegree_norm [Algebra F S] [IsScalarTower F F[X] S]\n    (b : Basis \u03b9 F[X] S) {f : S} (hf : f \u2260 0) :\n    FiniteDimensional.finrank F (S \u29f8 span ({f} : Set S)) = (Algebra.norm F[X] f).natDegree", "theorem finrank_span_eq_card [Nontrivial R] {\u03b9 : Type*} [Fintype \u03b9] {b : \u03b9 \u2192 M}\n    (hb : LinearIndependent R b) :\n    finrank R (span R (Set.range b)) = Fintype.card \u03b9", "@[simp]\ntheorem finsuppTensorFinsupp'_single_tmul_single (a : \u03b9) (b : \u03ba) (r\u2081 r\u2082 : R) :\n    finsuppTensorFinsupp' R \u03b9 \u03ba (Finsupp.single a r\u2081 \u2297\u209c[R] Finsupp.single b r\u2082) =\n      Finsupp.single (a, b) (r\u2081 * r\u2082)", "theorem finsuppTensorFinsuppLid_self :\n    finsuppTensorFinsuppLid R R \u03b9 \u03ba = finsuppTensorFinsupp' R \u03b9 \u03ba", "@[simp]\ntheorem finsuppTensorFinsuppLid_single_tmul_single (a : \u03b9) (b : \u03ba) (r : R) (n : N) :\n    finsuppTensorFinsuppLid R N \u03b9 \u03ba (Finsupp.single a r \u2297\u209c[R] Finsupp.single b n) =\n      Finsupp.single (a, b) (r \u2022 n)", "@[simp]\ntheorem finsuppTensorFinsupp_apply (f : \u03b9 \u2192\u2080 M) (g : \u03ba \u2192\u2080 N) (i : \u03b9) (k : \u03ba) :\n    finsuppTensorFinsupp R S M N \u03b9 \u03ba (f \u2297\u209c g) (i, k) = f i \u2297\u209c g k", "theorem fixedPoints_subgroup_sup {P Q : Subgroup M} :\n    fixedPoints (\u21a5(P \u2294 Q)) \u03b1 = fixedPoints P \u03b1 \u2229 fixedPoints Q \u03b1", "theorem fixingSubmonoid_fixedPoints_gc :\n    GaloisConnection (OrderDual.toDual \u2218 fixingSubmonoid M)\n      ((fun P : Submonoid M => fixedPoints P \u03b1) \u2218 OrderDual.ofDual)", "lemma floorDiv_le_ceilDiv : b \u230a/\u230b a \u2264 b \u2308/\u2309 a", "theorem forall_apply_eq_imp_iff' {f : \u03b1 \u2192 \u03b2} {p : \u03b2 \u2192 Prop} :\n    (\u2200 a b, f a = b \u2192 p b) \u2194 \u2200 a, p (f a)", "theorem forall_imp_iff_exists_imp {\u03b1 : Sort*} {p : \u03b1 \u2192 Prop} {b : Prop} [ha : Nonempty \u03b1] :\n    (\u2200 x, p x) \u2192 b \u2194 \u2203 x, p x \u2192 b", "theorem forall\u2082_congr {p q : \u2200 a, \u03b2 a \u2192 Prop} (h : \u2200 a b, p a b \u2194 q a b) :\n    (\u2200 a b, p a b) \u2194 \u2200 a b, q a b", "private lemma four_functions_theorem_aux (h\u2081 : 0 \u2264 f\u2081) (h\u2082 : 0 \u2264 f\u2082) (h\u2083 : 0 \u2264 f\u2083) (h\u2084 : 0 \u2264 f\u2084)\n    (h : \u2200 s t, f\u2081 s * f\u2082 t \u2264 f\u2083 (s \u2229 t) * f\u2084 (s \u222a t)) (\ud835\udc9c \u212c : Finset (Finset \u03b1)) :\n    (\u2211 s \u2208 \ud835\udc9c, f\u2081 s) * \u2211 s \u2208 \u212c, f\u2082 s \u2264 (\u2211 s \u2208 \ud835\udc9c \u22bc \u212c, f\u2083 s) * \u2211 s \u2208 \ud835\udc9c \u22bb \u212c, f\u2084 s", "theorem fourier_coe_apply {n : \u2124} {x : \u211d} :\n    fourier n (x : AddCircle T) = Complex.exp (2 * \u03c0 * Complex.I * n * x / T)", "theorem frequently_nhdsWithin_iff {z : \u03b1} {s : Set \u03b1} {p : \u03b1 \u2192 Prop} :\n    (\u2203\u1da0 x in \ud835\udcdd[s] z, p x) \u2194 \u2203\u1da0 x in \ud835\udcdd z, p x \u2227 x \u2208 s", "theorem frobenius_natCast (n : \u2115) : frobenius R p n = n", "@[simp]\ntheorem frontier_Ico [NoMinOrder \u03b1] {a b : \u03b1} (h : a < b) : frontier (Ico a b) = {a, b}", "@[simp]\ntheorem frontier_Ioi' {a : \u03b1} (ha : (Ioi a).Nonempty) : frontier (Ioi a) = {a}", "theorem frontier_ball (x : E) {r : \u211d} (hr : r \u2260 0) :\n    frontier (ball x r) = sphere x r", "theorem frontier_inter_open_inter {s t : Set \u03b1} (ht : IsOpen t) :\n    frontier (s \u2229 t) \u2229 t = frontier s \u2229 t", "theorem frontier_inter_subset (s t : Set X) :\n    frontier (s \u2229 t) \u2286 frontier s \u2229 closure t \u222a closure s \u2229 frontier t", "theorem frontier_union_subset (s t : Set X) :\n    frontier (s \u222a t) \u2286 frontier s \u2229 closure t\u1d9c \u222a closure s\u1d9c \u2229 frontier t", "theorem gal_X_pow_sub_C_isSolvable_aux (n : \u2115) (a : F)\n    (h : (X ^ n - 1 : F[X]).Splits (RingHom.id F)) : IsSolvable (X ^ n - C a).Gal", "theorem gauge_add_le (hs : Convex \u211d s) (absorbs : Absorbent \u211d s) (x y : E) :\n    gauge s (x + y) \u2264 gauge s x + gauge s y", "theorem gauge_def : gauge s x = sInf ({ r \u2208 Set.Ioi (0 : \u211d) | x \u2208 r \u2022 s })", "theorem gauge_le_one_iff_mem_closure (hc : Convex \u211d s) (hs\u2080 : s \u2208 \ud835\udcdd 0) :\n    gauge s x \u2264 1 \u2194 x \u2208 closure s", "theorem gauge_mono (hs : Absorbent \u211d s) (h : s \u2286 t) : gauge t \u2264 gauge s", "lemma gc_floorDiv_mul (ha : 0 < a) : GaloisConnection (a * \u00b7) (\u00b7 \u230a/\u230b a) := gc_floorDiv_smul ha", "theorem gc_upperClosure_coe :\n    GaloisConnection (toDual \u2218 upperClosure : Set \u03b1 \u2192 (UpperSet \u03b1)\u1d52\u1d48) ((\u2191) \u2218 ofDual)", "theorem gcd_pow_left_dvd_pow_gcd [GCDMonoid \u03b1] {a b : \u03b1} {k : \u2115} : gcd (a ^ k) b \u2223 gcd a b ^ k", "theorem generateFrom_iUnion {f : \u03b9 \u2192 Set (Set \u03b1)} :\n    generateFrom (\u22c3 i, f i) = \u2a05 i, generateFrom (f i)", "theorem generateFrom_union (a\u2081 a\u2082 : Set (Set \u03b1)) :\n    generateFrom (a\u2081 \u222a a\u2082) = generateFrom a\u2081 \u2293 generateFrom a\u2082", "theorem geom_lt {u : \u2115 \u2192 \u211d} {c : \u211d} (hc : 0 \u2264 c) {n : \u2115} (hn : 0 < n)\n    (h : \u2200 k < n, c * u k < u (k + 1)) : c ^ n * u 0 < u n", "theorem geom_sum_Ico [DivisionRing \u03b1] {x : \u03b1} (hx : x \u2260 1) {m n : \u2115} (hmn : m \u2264 n) :\n    \u2211 i \u2208 Finset.Ico m n, x ^ i = (x ^ n - x ^ m) / (x - 1)", "theorem gold_pos : 0 < \u03c6", "@[simp]\ntheorem grade_toDual [GradeOrder \ud835\udd46 \u03b1] (a : \u03b1) : grade \ud835\udd46\u1d52\u1d48 (toDual a) = toDual (grade \ud835\udd46 a)", "theorem gronwallBound_continuous_\u03b5 (\u03b4 K x : \u211d) : Continuous fun \u03b5 => gronwallBound \u03b4 K \u03b5 x", "theorem dZero_ker_eq_invariants : LinearMap.ker (dZero A) = invariants A.\u03c1", "lemma groupCohomology\u03c0_comp_isoH0_hom  :\n    groupCohomology\u03c0 A 0 \u226b (isoH0 A).hom = (isoZeroCocycles A).hom", "theorem mem_twoCocycles_iff (f : G \u00d7 G \u2192 A) :\n    f \u2208 twoCocycles A \u2194 \u2200 g h j : G,\n      f (g * h, j) + f (g, h) =\n        A.\u03c1 g (f (h, j)) + f (g, h * j)", "@[simp] theorem oneCocycles_map_inv (f : oneCocycles A) (g : G) :\n    A.\u03c1 g (f.1 g\u207b\u00b9) = - f.1 g", "theorem ofMulAction_free :\n    Module.Free (MonoidAlgebra k G) (ofMulAction k G (Fin (n + 1) \u2192 G)).asModule", "lemma toCocycles_comp_isoOneCocycles_hom :\n    toCocycles A 0 1 \u226b (isoOneCocycles A).hom =\n      (zeroCochainsLequiv A).toModuleIso.hom \u226b\n        ModuleCat.ofHom (shortComplexH1 A).moduleCatToCycles", "lemma twoCocycles_\u03c1_map_inv_sub_map_inv (f : twoCocycles A) (g : G) :\n    A.\u03c1 g (f.1 (g\u207b\u00b9, g)) - f.1 (g, g\u207b\u00b9)\n      = f.1 (1, 1) - f.1 (g, 1)", "theorem hallMatchingsOn.nonempty {\u03b9 : Type u} {\u03b1 : Type v} [DecidableEq \u03b1] (t : \u03b9 \u2192 Finset \u03b1)\n    (h : \u2200 s : Finset \u03b9, s.card \u2264 (s.biUnion t).card) (\u03b9' : Finset \u03b9) :\n    Nonempty (hallMatchingsOn t \u03b9')", "theorem hammingNorm_ne_zero_iff {x : \u2200 i, \u03b2 i} : hammingNorm x \u2260 0 \u2194 x \u2260 0", "@[to_additive]\ntheorem _root_.hasCompactMulSupport_iff_eventuallyEq :\n    HasCompactMulSupport f \u2194 f =\u1da0[coclosedCompact \u03b1] 1", "theorem hasConstantSpeedOnWith_of_subsingleton (f : \u211d \u2192 E) {s : Set \u211d} (hs : s.Subsingleton)\n    (l : \u211d\u22650) : HasConstantSpeedOnWith f s l", "theorem hasDerivAt_bernoulliFun (k : \u2115) (x : \u211d) :\n    HasDerivAt (bernoulliFun k) (k * bernoulliFun (k - 1) x) x", "theorem hasDerivAt_exp_of_mem_ball [CharZero \ud835\udd42] {x : \ud835\udd42}\n    (hx : x \u2208 EMetric.ball (0 : \ud835\udd42) (expSeries \ud835\udd42 \ud835\udd42).radius) : HasDerivAt (exp \ud835\udd42) (exp \ud835\udd42 x) x", "theorem hasDerivAt_iff_isLittleO :\n    HasDerivAt f f' x \u2194 (fun x' : \ud835\udd5c => f x' - f x - (x' - x) \u2022 f') =o[\ud835\udcdd x] fun x' => x' - x", "theorem hasDerivAt_tsum_of_isPreconnected (hu : Summable u) (ht : IsOpen t)\n    (h't : IsPreconnected t) (hg : \u2200 n y, y \u2208 t \u2192 HasDerivAt (g n) (g' n y) y)\n    (hg' : \u2200 n y, y \u2208 t \u2192 \u2016g' n y\u2016 \u2264 u n) (hy\u2080 : y\u2080 \u2208 t) (hg0 : Summable fun n => g n y\u2080)\n    (hy : y \u2208 t) : HasDerivAt (fun z => \u2211' n, g n z) (\u2211' n, g' n y) y", "theorem hasDerivWithinAt_id : HasDerivWithinAt id 1 s x", "theorem hasDerivWithinAt_iff_tendsto_slope :\n    HasDerivWithinAt f f' s x \u2194 Tendsto (slope f x) (\ud835\udcdd[s \\ {x}] x) (\ud835\udcdd f')", "theorem hasDerivWithinAt_inter (h : t \u2208 \ud835\udcdd x) :\n    HasDerivWithinAt f f' (s \u2229 t) x \u2194 HasDerivWithinAt f f' s x", "@[fun_prop]\ntheorem hasFDerivAt_id (x : E) : HasFDerivAt id (id \ud835\udd5c E) x", "theorem hasFDerivAt_integral_of_dominated_of_fderiv_le {F' : H \u2192 \u03b1 \u2192 H \u2192L[\ud835\udd5c] E} (\u03b5_pos : 0 < \u03b5)\n    (hF_meas : \u2200\u1da0 x in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc) (hF_int : Integrable (F x\u2080) \u03bc)\n    (hF'_meas : AEStronglyMeasurable (F' x\u2080) \u03bc)\n    (h_bound : \u2200\u1d50 a \u2202\u03bc, \u2200 x \u2208 ball x\u2080 \u03b5, \u2016F' x a\u2016 \u2264 bound a)\n    (bound_integrable : Integrable (bound : \u03b1 \u2192 \u211d) \u03bc)\n    (h_diff : \u2200\u1d50 a \u2202\u03bc, \u2200 x \u2208 ball x\u2080 \u03b5, HasFDerivAt (F \u00b7 a) (F' x a) x) :\n    HasFDerivAt (fun x \u21a6 \u222b a, F x a \u2202\u03bc) (\u222b a, F' x\u2080 a \u2202\u03bc) x\u2080", "@[fun_prop]\ntheorem hasFDerivAt_single {i : \u03b9} (y : E i) :\n    HasFDerivAt (Pi.single i) (.pi (Pi.single i (.id \ud835\udd5c (E i)))) y", "theorem hasFDerivAt_stereoInvFunAux (v : E) :\n    HasFDerivAt (stereoInvFunAux v) (ContinuousLinearMap.id \u211d E) 0", "theorem hasFDerivWithinAt_inter' (h : t \u2208 \ud835\udcdd[s] x) :\n    HasFDerivWithinAt f f' (s \u2229 t) x \u2194 HasFDerivWithinAt f f' s x", "theorem hasFDerivWithinAt_inv (x_ne_zero : x \u2260 0) :\n    HasFDerivWithinAt (fun x => x\u207b\u00b9) (smulRight (1 : \ud835\udd5c \u2192L[\ud835\udd5c] \ud835\udd5c) (-(x ^ 2)\u207b\u00b9) : \ud835\udd5c \u2192L[\ud835\udd5c] \ud835\udd5c) s x", "theorem hasFDerivWithinAt_of_nmem_closure (h : x \u2209 closure s) : HasFDerivWithinAt f f' s x", "theorem hasFTaylorSeriesUpToOn_zero_iff :\n    HasFTaylorSeriesUpToOn 0 f p s \u2194 ContinuousOn f s \u2227 \u2200 x \u2208 s, (p x 0).uncurry0 = f x", "theorem hasGradientAt_iff_isLittleO_nhds_zero : HasGradientAt f f' x \u2194\n    (fun h => f (x + h) - f x - \u27eaf', h\u27eb) =o[\ud835\udcdd 0] fun h => h", "theorem hasGradientWithinAt_congr_set {s t : Set F} (h : s =\u1da0[\ud835\udcdd x] t) :\n    HasGradientWithinAt f f' s x \u2194 HasGradientWithinAt f f' t x", "theorem hasLineDerivAt_iff_tendsto_slope_zero :\n    HasLineDerivAt \ud835\udd5c f f' x v \u2194\n      Tendsto (fun (t : \ud835\udd5c) \u21a6 t\u207b\u00b9 \u2022 (f (x + t \u2022 v) - f x)) (\ud835\udcdd[\u2260] 0) (\ud835\udcdd f')", "theorem hasLineDerivWithinAt_smul_iff {c : \ud835\udd5c} (hc : c \u2260 0) :\n    HasLineDerivWithinAt \ud835\udd5c f (c \u2022 f') s x (c \u2022 v) \u2194 HasLineDerivWithinAt \ud835\udd5c f f' s x v", "@[simp] lemma hasLineDerivWithinAt_univ :\n    HasLineDerivWithinAt \ud835\udd5c f f' univ x v \u2194 HasLineDerivAt \ud835\udd5c f f' x v", "theorem hasMFDerivAt_snd (x : M \u00d7 M') :\n    HasMFDerivAt (I.prod I') I' Prod.snd x\n      (ContinuousLinearMap.snd \ud835\udd5c (TangentSpace I x.1) (TangentSpace I' x.2))", "theorem hasMFDerivWithinAt_inter (h : t \u2208 \ud835\udcdd x) :\n    HasMFDerivWithinAt I I' f (s \u2229 t) x f' \u2194 HasMFDerivWithinAt I I' f s x f'", "@[to_additive (attr := simp)] lemma hasProd_unique [Unique \u03b2] (f : \u03b2 \u2192 \u03b1) : HasProd f (f default)", "@[simp]\ntheorem hasStrictDerivAt_pi :\n    HasStrictDerivAt \u03c6 \u03c6' x \u2194 \u2200 i, HasStrictDerivAt (fun x => \u03c6 x i) (\u03c6' i) x", "theorem hasStrictFDerivAt_iff_hasStrictDerivAt {f' : \ud835\udd5c \u2192L[\ud835\udd5c] F} :\n    HasStrictFDerivAt f f' x \u2194 HasStrictDerivAt f (f' 1) x", "theorem hasSum_fourier_series_of_summable (h : Summable (fourierCoeff f)) :\n    HasSum (fun i => fourierCoeff f i \u2022 fourier i) f", "theorem hasSum_geometric_of_abs_lt_one {r : \u211d} (h : |r| < 1) :\n    HasSum (fun n : \u2115 \u21a6 r ^ n) (1 - r)\u207b\u00b9", "theorem hasSum_zeta_two : HasSum (fun n : \u2115 => (1 : \u211d) / (n : \u211d) ^ 2) (\u03c0 ^ 2 / 6)", "theorem hausdorffMeasure_of_dimH_lt {s : Set X} {d : \u211d\u22650} (h : dimH s < d) : \u03bcH[d] s = 0", "theorem hausdorffMeasure_of_lt_dimH {s : Set X} {d : \u211d\u22650} (h : \u2191d < dimH s) : \u03bcH[d] s = \u221e", "theorem himp_le_himp_himp_himp : b \u21e8 c \u2264 (a \u21e8 b) \u21e8 a \u21e8 c", "lemma himp_ne_right : x \u21e8 y \u2260 x \u2194 x \u2260 \u22a4 \u2228 y \u2260 \u22a4 := himp_eq_left.not.trans not_and_or", "@[simp]\ntheorem hnot_bot : \uffe2(\u22a5 : \u03b1) = \u22a4", "theorem hnot_hnot_le : \uffe2\uffe2a \u2264 a", "theorem hnot_hnot_sdiff_distrib (a b : \u03b1) : \uffe2\uffe2(a \\ b) = \uffe2\uffe2a \\ \uffe2\uffe2b", "lemma holley (h\u03bc\u2080 : 0 \u2264 \u03bc) (hf : 0 \u2264 f) (hg : 0 \u2264 g) (h\u03bc : Monotone \u03bc)\n    (hfg : \u2211 a, f a = \u2211 a, g a) (h : \u2200 a b, f a * g b \u2264 f (a \u2293 b) * g (a \u2294 b)) :\n    \u2211 a, \u03bc a * f a \u2264 \u2211 a, \u03bc a * g a", "theorem hom_coe_pow {F : Type*} [Monoid F] (c : F \u2192 M \u2192 M) (h1 : c 1 = id)\n    (hmul : \u2200 f g, c (f * g) = c f \u2218 c g) (f : F) : \u2200 n, c (f ^ n) = (c f)^[n]", "@[reassoc (attr := simp)]\ntheorem condition_\u03c0' : kernel.lift g f w \u226b \u03c0' f g w = 0", "theorem map_eq_lift_desc'_right (\u03b1 \u03b2 h) :\n    map w w' \u03b1 \u03b2 h =\n      homology'.lift f' g' _\n        (homology'.desc' f g _ (kernel.\u03b9 _ \u226b \u03b1.right \u226b cokernel.\u03c0 _)\n          (by\n            simp only [kernel.lift_\u03b9_assoc]\n            erw [\u2190 reassoc_of% \u03b1.w]\n            simp))\n        (by\n          ext\n          simp [h])", "@[reassoc (attr := simp)]\ntheorem \u03c0'_map (\u03b1 \u03b2 h) : \u03c0' _ _ _ \u226b map w w' \u03b1 \u03b2 h =\n    kernel.map _ _ \u03b1.right \u03b2.right (by simp [h, \u03b2.w.symm]) \u226b \u03c0' _ _ _", "@[simp]\nlemma homology'.\u03c0_map_apply [ConcreteCategory.{w} V] (p : \u03b1.right = \u03b2.left)\n    (x : (forget V).obj (Subobject.underlying.obj (kernelSubobject g))) :\n    homology'.map w w' \u03b1 \u03b2 p (homology'.\u03c0 f g w x) =\n      homology'.\u03c0 f' g' w' (kernelSubobjectMap \u03b2 x)", "theorem iInf_eq_iInf_finset' (s : \u03b9' \u2192 \u03b1) :\n    \u2a05 i, s i = \u2a05 t : Finset (PLift \u03b9'), \u2a05 i \u2208 t, s (PLift.down i)", "theorem iInf_iSup_ge_nat_add :\n    \u2200 (f : \u2115 \u2192 \u03b1) (k : \u2115), \u2a05 n, \u2a06 i \u2265 n, f (i + k) = \u2a05 n, \u2a06 i \u2265 n, f i", "theorem iInf_inf_eq : \u2a05 x, f x \u2293 g x = (\u2a05 x, f x) \u2293 \u2a05 x, g x", "theorem iInf_le_iInf_of_subset {f : \u03b2 \u2192 \u03b1} {s t : Set \u03b2} : s \u2286 t \u2192 \u2a05 x \u2208 t, f x \u2264 \u2a05 x \u2208 s, f x", "theorem iInf_nat_gt_zero_eq (f : \u2115 \u2192 \u03b1) : \u2a05 i > 0, f i = \u2a05 i, f (i + 1)", "theorem iInf_option (f : Option \u03b2 \u2192 \u03b1) : \u2a05 o, f o = f none \u2293 \u2a05 b, f (Option.some b)", "theorem iInf_pair {f : \u03b2 \u2192 \u03b1} {a b : \u03b2} : \u2a05 x \u2208 ({a, b} : Set \u03b2), f x = f a \u2293 f b", "lemma iInf_sigma' {\u03ba : \u03b2 \u2192 Type*} (f : \u2200 i, \u03ba i \u2192 \u03b1) :\n    (\u2a05 i, \u2a05 j, f i j) = \u2a05 x : \u03a3 i, \u03ba i, f x.1 x.2", "theorem iInf_subtype' {p : \u03b9 \u2192 Prop} {f : \u2200 i, p i \u2192 \u03b1} :\n    \u2a05 (i) (h : p i), f i h = \u2a05 x : Subtype p, f x x.property", "theorem iInf_sum {f : Sum \u03b2 \u03b3 \u2192 \u03b1} : \u2a05 x, f x = (\u2a05 i, f (Sum.inl i)) \u2293 \u2a05 j, f (Sum.inr j)", "theorem iInter_Iic_eq_empty_iff : \u22c2 i, Iic (f i) = \u2205 \u2194 \u00ac BddBelow (range f)", "theorem iSup_const_le : \u2a06 _ : \u03b9, a \u2264 a", "theorem iSup_extend_bot {e : \u03b9 \u2192 \u03b2} (he : Injective e) (f : \u03b9 \u2192 \u03b1) :\n    \u2a06 j, extend e f \u22a5 j = \u2a06 i, f i", "theorem iSup_nhds_le_uniformity : \u2a06 x : \u03b1, \ud835\udcdd (x, x) \u2264 \ud835\udce4 \u03b1", "theorem iSup_option (f : Option \u03b2 \u2192 \u03b1) : \u2a06 o, f o = f none \u2294 \u2a06 b, f (Option.some b)", "lemma iSup_sigma' {\u03ba : \u03b2 \u2192 Type*} (f : \u2200 i, \u03ba i \u2192 \u03b1) :\n    (\u2a06 i, \u2a06 j, f i j) = \u2a06 x : \u03a3 i, \u03ba i, f x.1 x.2", "theorem iSup_singleton {f : \u03b2 \u2192 \u03b1} {b : \u03b2} : \u2a06 x \u2208 (singleton b : Set \u03b2), f x = f b", "theorem iSup_split (f : \u03b2 \u2192 \u03b1) (p : \u03b2 \u2192 Prop) :\n    \u2a06 i, f i = (\u2a06 (i) (_ : p i), f i) \u2294 \u2a06 (i) (_ : \u00acp i), f i", "theorem iSup_subtype'' {\u03b9} (s : Set \u03b9) (f : \u03b9 \u2192 \u03b1) : \u2a06 i : s, f i = \u2a06 (t : \u03b9) (_ : t \u2208 s), f t", "theorem iSup_union {f : \u03b2 \u2192 \u03b1} {s t : Set \u03b2} :\n    \u2a06 x \u2208 s \u222a t, f x = (\u2a06 x \u2208 s, f x) \u2294 \u2a06 x \u2208 t, f x", "theorem iUnion_Ici_eq_Ioi_iInf {R : Type*} [CompleteLinearOrder R] {f : \u03b9 \u2192 R}\n    (no_least_elem : \u2a05 i, f i \u2209 range f) : \u22c3 i : \u03b9, Ici (f i) = Ioi (\u2a05 i, f i)", "theorem iUnion_Ioc_add_intCast : \u22c3 n : \u2124, Ioc (a + n) (a + n + 1) = Set.univ", "@[simp]\ntheorem id_compRel {r : Set (\u03b1 \u00d7 \u03b1)} : idRel \u25cb r = r", "theorem idealFactorsFunOfQuotHom_comp {f : R \u29f8 I \u2192+* A \u29f8 J} {g : A \u29f8 J \u2192+* B \u29f8 L}\n    (hf : Function.Surjective f) (hg : Function.Surjective g) :\n    (idealFactorsFunOfQuotHom hg).comp (idealFactorsFunOfQuotHom hf) =\n      idealFactorsFunOfQuotHom (show Function.Surjective (g.comp f) from hg.comp hf)", "theorem im_inner_eq_norm_sub_i_smul_mul_self_sub_norm_add_i_smul_mul_self_div_four (x y : E) :\n    im \u27eax, y\u27eb = (\u2016x - IK \u2022 y\u2016 * \u2016x - IK \u2022 y\u2016 - \u2016x + IK \u2022 y\u2016 * \u2016x + IK \u2022 y\u2016) / 4", "theorem image2_lowerBounds_lowerBounds_subset :\n    image2 f (lowerBounds s) (lowerBounds t) \u2286 lowerBounds (image2 f s t)", "theorem image2_upperBounds_lowerBounds_subset_lowerBounds_image2 :\n    image2 f (upperBounds s) (lowerBounds t) \u2286 lowerBounds (image2 f s t)", "theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : \u211d \u2192 \u211d} {a b : \u211d}\n    (hf : ContinuousOn f (Icc a b)) {B B' : \u211d \u2192 \u211d} (ha : f a \u2264 B a) (hB : ContinuousOn B (Icc a b))\n    (hB' : \u2200 x \u2208 Ico a b, HasDerivWithinAt B (B' x) (Ici x) x)\n    (bound : \u2200 x \u2208 Ico a b, \u2200 r, B' x < r \u2192 \u2203\u1da0 z in \ud835\udcdd[>] x, slope f x z < r) :\n    \u2200 \u2983x\u2984, x \u2208 Icc a b \u2192 f x \u2264 B x", "theorem image_parallelepiped (f : E \u2192\u2097[\u211d] F) (v : \u03b9 \u2192 E) :\n    f '' parallelepiped v = parallelepiped (f \u2218 v)", "@[simp]\ntheorem image_segment (f : E \u2192\u1d43[\ud835\udd5c] F) (a b : E) : f '' [a -[\ud835\udd5c] b] = [f a -[\ud835\udd5c] f b]", "@[simp] theorem image_smul_set :\n    h '' (c \u2022 s) = c \u2022 h '' s", "theorem imaginaryPart_smul (z : \u2102) (a : A) : \u2111 (z \u2022 a) = z.re \u2022 \u2111 a + z.im \u2022 \u211c a", "theorem imp_not_comm : (a \u2192 \u00acb) \u2194 (b \u2192 \u00aca)", "theorem indicator_ae_eq_zero_of_restrict_ae_eq_zero (hs : MeasurableSet s)\n    (hf : f =\u1d50[\u03bc.restrict s] 0) : s.indicator f =\u1d50[\u03bc] 0", "theorem induced_orderTopology {\u03b1 : Type u} {\u03b2 : Type v} [Preorder \u03b1] [ta : TopologicalSpace \u03b2]\n    [Preorder \u03b2] [OrderTopology \u03b2] (f : \u03b1 \u2192 \u03b2) (hf : \u2200 {x y}, f x < f y \u2194 x < y)\n    (H : \u2200 {x y}, x < y \u2192 \u2203 a, x < f a \u2227 f a < y) : @OrderTopology _ (induced f ta) _", "theorem induced_topology_eq_preorder [Preorder \u03b1] [Preorder \u03b2] [TopologicalSpace \u03b2]\n    [OrderTopology \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : \u2200 {x y}, f x < f y \u2194 x < y)\n    (H\u2081 : \u2200 {a b x}, b < f a \u2192 \u00ac(b < f x) \u2192 \u2203 y, y < a \u2227 b \u2264 f y)\n    (H\u2082 : \u2200 {a b x}, f a < b \u2192 \u00ac(f x < b) \u2192 \u2203 y, a < y \u2227 f y \u2264 b) :\n    induced f \u2039TopologicalSpace \u03b2\u203a = Preorder.topology \u03b1", "theorem inducing_id : Inducing (@id X)", "theorem inducing_induced (f : X \u2192 Y) : @Inducing X Y (TopologicalSpace.induced f \u2039_\u203a) _ f", "theorem infEdist_smul\u2080 {c : \ud835\udd5c} (hc : c \u2260 0) (s : Set E) (x : E) :\n    EMetric.infEdist (c \u2022 x) (c \u2022 s) = \u2016c\u2016\u208a \u2022 EMetric.infEdist x s", "theorem inf_congr_right (h1 : b \u2293 c \u2264 a) (h2 : a \u2293 c \u2264 b) : a \u2293 c = b \u2293 c", "theorem inf_covBy_of_covBy_sup_right : b \u22d6 a \u2294 b \u2192 a \u2293 b \u22d6 a", "lemma inf_eq_and_sup_eq_iff : a \u2293 b = c \u2227 a \u2294 b = c \u2194 a = c \u2227 b = c", "@[simp]\ntheorem inf_le_left : a \u2293 b \u2264 a", "theorem inf_le_of_right_le (h : b \u2264 c) : a \u2293 b \u2264 c", "theorem inf_le_sup : a \u2293 b \u2264 a \u2294 b", "theorem inf_lt_left_or_right (h : a \u2260 b) : a \u2293 b < a \u2228 a \u2293 b < b", "lemma inf_top_eq (a : \u03b1) : a \u2293 \u22a4 = a := inf_of_le_left le_top", "theorem inf_uniformity {u v : UniformSpace \u03b1} : \ud835\udce4[u \u2293 v] = \ud835\udce4[u] \u2293 \ud835\udce4[v]", "@[nolint checkType] theorem d_eq :\n    d n A =\n      (diagonalHomEquiv n A).toModuleIso.inv \u226b\n        (linearYonedaObjResolution A).d n (n + 1) \u226b\n          (diagonalHomEquiv (n + 1) A).toModuleIso.hom", "theorem injective_of_le_imp_le [PartialOrder \u03b1] [Preorder \u03b2] (f : \u03b1 \u2192 \u03b2)\n    (h : \u2200 {x y}, f x \u2264 f y \u2192 x \u2264 y) : Injective f", "theorem inner_self_eq_norm_sq_to_K (x : E) : \u27eax, x\u27eb = (\u2016x\u2016 : \ud835\udd5c) ^ 2", "theorem inner_smul_right (x y : E) (r : \ud835\udd5c) : \u27eax, r \u2022 y\u27eb = r * \u27eax, y\u27eb", "@[simp]\ntheorem inner_zero_left (x : E) : \u27ea0, x\u27eb = 0", "theorem inseparable_iff_ker_uniformity {x y : \u03b1} : Inseparable x y \u2194 (x, y) \u2208 (\ud835\udce4 \u03b1).ker", "theorem inseparable_of_nhdsWithin_eq (hx : x \u2208 s) (hy : y \u2208 s) (h : \ud835\udcdd[s] x = \ud835\udcdd[s] y) : x ~\u1d62 y", "theorem insert_mem_nhdsWithin_insert {a : \u03b1} {s t : Set \u03b1} (h : t \u2208 \ud835\udcdd[s] a) :\n    insert a t \u2208 \ud835\udcdd[insert a s] a", "theorem integrableOn_Ioi_cpow_iff {s : \u2102} {t : \u211d} (ht : 0 < t) :\n    IntegrableOn (fun x : \u211d \u21a6 (x : \u2102) ^ s) (Ioi t) \u2194 s.re < -1", "theorem integrableOn_peak_smul_of_integrableOn_of_tendsto\n    (hs : MeasurableSet s) (h'st : t \u2208 \ud835\udcdd[s] x\u2080)\n    (hl\u03c6 : \u2200 u : Set \u03b1, IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u))\n    (hi\u03c6 : Tendsto (fun i \u21a6 \u222b x in t, \u03c6 i x \u2202\u03bc) l (\ud835\udcdd 1))\n    (h'i\u03c6 : \u2200\u1da0 i in l, AEStronglyMeasurable (\u03c6 i) (\u03bc.restrict s))\n    (hmg : IntegrableOn g s \u03bc) (hcg : Tendsto g (\ud835\udcdd[s] x\u2080) (\ud835\udcdd a)) :\n    \u2200\u1da0 i in l, IntegrableOn (fun x => \u03c6 i x \u2022 g x) s \u03bc", "@[simp]\ntheorem integral_cos_sq : \u222b x in a..b, cos x ^ 2 = (cos b * sin b - cos a * sin a + b - a) / 2", "theorem integral_exp_neg_rpow {p : \u211d} (hp : 0 < p) :\n    \u222b x in Ioi (0:\u211d), exp (- x ^ p) = Gamma (1 / p + 1)", "theorem integral_gaussian (b : \u211d) : \u222b x : \u211d, exp (-b * x ^ 2) = \u221a(\u03c0 / b)", "theorem integral_gaussian_complex {b : \u2102} (hb : 0 < re b) :\n    \u222b x : \u211d, cexp (-b * (x : \u2102) ^ 2) = (\u03c0 / b) ^ (1 / 2 : \u2102)", "theorem integral_one_div_of_neg (ha : a < 0) (hb : b < 0) :\n    \u222b x : \u211d in a..b, 1 / x = log (b / a)", "theorem integral_sin_pow :\n    (\u222b x in a..b, sin x ^ (n + 2)) =\n      (sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b) / (n + 2) +\n        (n + 1) / (n + 2) * \u222b x in a..b, sin x ^ n", "@[simp]\ntheorem integral_sin_pow_three :\n    \u222b x in a..b, sin x ^ 3 = cos a - cos b - (cos a ^ 3 - cos b ^ 3) / 3", "@[simp]\ntheorem integral_sin_sq_mul_cos :\n    \u222b x in a..b, sin x ^ 2 * cos x = (sin b ^ 3 - sin a ^ 3) / 3", "@[simp]\ntheorem interior_Icc [NoMinOrder \u03b1] [NoMaxOrder \u03b1] {a b : \u03b1} : interior (Icc a b) = Ioo a b", "theorem interior_convexHull_nonempty_iff_affineSpan_eq_top [FiniteDimensional \u211d V] {s : Set V} :\n    (interior (convexHull \u211d s)).Nonempty \u2194 affineSpan \u211d s = \u22a4", "@[simp]\ntheorem interior_inter : interior (s \u2229 t) = interior s \u2229 interior t", "@[mono, gcongr]\ntheorem interior_mono (h : s \u2286 t) : interior s \u2286 interior t", "theorem interior_prod_eq (s : Set X) (t : Set Y) : interior (s \u00d7\u02e2 t) = interior s \u00d7\u02e2 interior t", "theorem intermediate_value_univ\u2082 [PreconnectedSpace X] {a b : X} {f g : X \u2192 \u03b1} (hf : Continuous f)\n    (hg : Continuous g) (ha : f a \u2264 g a) (hb : g b \u2264 f b) : \u2203 x, f x = g x", "theorem intervalIntegrable_iff' [NoAtoms \u03bc] :\n    IntervalIntegrable f \u03bc a b \u2194 IntegrableOn f (uIcc a b) \u03bc", "theorem continuousAt_of_dominated_interval {F : X \u2192 \u211d \u2192 E} {x\u2080 : X} {bound : \u211d \u2192 \u211d} {a b : \u211d}\n    (hF_meas : \u2200\u1da0 x in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) (\u03bc.restrict <| \u0399 a b))\n    (h_bound : \u2200\u1da0 x in \ud835\udcdd x\u2080, \u2200\u1d50 t \u2202\u03bc, t \u2208 \u0399 a b \u2192 \u2016F x t\u2016 \u2264 bound t)\n    (bound_integrable : IntervalIntegrable bound \u03bc a b)\n    (h_cont : \u2200\u1d50 t \u2202\u03bc, t \u2208 \u0399 a b \u2192 ContinuousAt (fun x => F x t) x\u2080) :\n    ContinuousAt (fun x => \u222b t in a..b, F x t \u2202\u03bc) x\u2080", "theorem continuousOn_primitive_Icc (h_int : IntegrableOn f (Icc a b) \u03bc) :\n    ContinuousOn (fun x => \u222b t in Icc a x, f t \u2202\u03bc) (Icc a b)", "theorem deriv_integral_left (hf : IntervalIntegrable f volume a b)\n    (hmeas : StronglyMeasurableAtFilter f (\ud835\udcdd a)) (hb : ContinuousAt f a) :\n    deriv (fun u => \u222b x in u..b, f x) a = -f a", "theorem integral_comp_add_right (d) : (\u222b x in a..b, f (x + d)) = \u222b x in a + d..b + d, f x", "theorem integral_deriv_mul_eq_sub_of_hasDerivWithinAt {u v u' v' : \u211d \u2192 A}\n    (hu : \u2200 x \u2208 [[a, b]], HasDerivWithinAt u (u' x) [[a, b]] x)\n    (hv : \u2200 x \u2208 [[a, b]], HasDerivWithinAt v (v' x) [[a, b]] x)\n    (hu' : IntervalIntegrable u' volume a b)\n    (hv' : IntervalIntegrable v' volume a b) :\n    \u222b x in a..b, u' x * v x + u x * v' x = u b * v b - u a * v a", "nonrec theorem integral_finset_sum {\u03b9} {s : Finset \u03b9} {f : \u03b9 \u2192 \u211d \u2192 E}\n    (h : \u2200 i \u2208 s, IntervalIntegrable (f i) \u03bc a b) :\n    \u222b x in a..b, \u2211 i \u2208 s, f i x \u2202\u03bc = \u2211 i \u2208 s, \u222b x in a..b, f i x \u2202\u03bc", "theorem integral_hasStrictDerivAt_of_tendsto_ae_right (hf : IntervalIntegrable f volume a b)\n    (hmeas : StronglyMeasurableAtFilter f (\ud835\udcdd b)) (hb : Tendsto f (\ud835\udcdd b \u2293 ae volume) (\ud835\udcdd c)) :\n    HasStrictDerivAt (fun u => \u222b x in a..u, f x) c b", "theorem integral_mono_interval {c d} (hca : c \u2264 a) (hab : a \u2264 b) (hbd : b \u2264 d)\n    (hf : 0 \u2264\u1d50[\u03bc.restrict (Ioc c d)] f) (hfi : IntervalIntegrable f \u03bc c d) :\n    (\u222b x in a..b, f x \u2202\u03bc) \u2264 \u222b x in c..d, f x \u2202\u03bc", "nonrec theorem integral_ofReal {a b : \u211d} {\u03bc : Measure \u211d} {f : \u211d \u2192 \u211d} :\n    (\u222b x in a..b, (f x : \u2102) \u2202\u03bc) = \u2191(\u222b x in a..b, f x \u2202\u03bc)", "theorem norm_integral_le_of_norm_le_const_ae {a b C : \u211d} {f : \u211d \u2192 E}\n    (h : \u2200\u1d50 x, x \u2208 \u0399 a b \u2192 \u2016f x\u2016 \u2264 C) : \u2016\u222b x in a..b, f x\u2016 \u2264 C * |b - a|", "theorem smul_integral_comp_mul_add (c d) :\n    (c \u2022 \u222b x in a..b, f (c * x + d)) = \u222b x in c * a + d..c * b + d, f x", "theorem sub_le_integral_of_hasDeriv_right_of_le_Ico (hab : a \u2264 b)\n    (hcont : ContinuousOn g (Icc a b)) (hderiv : \u2200 x \u2208 Ico a b, HasDerivWithinAt g (g' x) (Ioi x) x)\n    (\u03c6int : IntegrableOn \u03c6 (Icc a b)) (h\u03c6g : \u2200 x \u2208 Ico a b, g' x \u2264 \u03c6 x) :\n    g b - g a \u2264 \u222b y in a..b, \u03c6 y", "theorem intrinsicClosure_subset_affineSpan : intrinsicClosure \ud835\udd5c s \u2286 affineSpan \ud835\udd5c s", "@[simp]\ntheorem intrinsicInterior_union_intrinsicFrontier (s : Set P) :\n    intrinsicInterior \ud835\udd5c s \u222a intrinsicFrontier \ud835\udd5c s = intrinsicClosure \ud835\udd5c s", "theorem invOf_mul_self_assoc [Monoid \u03b1] (a b : \u03b1) [Invertible a] : \u215f a * (a * b) = b", "@[simp]\ntheorem invOf_nonneg [Invertible a] : 0 \u2264 \u215f a \u2194 0 \u2264 a", "theorem inv_eq_of_root_of_coeff_zero_ne_zero {x : L} {p : K[X]} (aeval_eq : aeval x p = 0)\n    (coeff_zero_ne : p.coeff 0 \u2260 0) : x\u207b\u00b9 = -(aeval x (divX p) / algebraMap _ _ (p.coeff 0))", "theorem inv_gold : \u03c6\u207b\u00b9 = -\u03c8", "@[to_additive] lemma inv_inf (a b : \u03b1) : (a \u2293 b)\u207b\u00b9 = a\u207b\u00b9 \u2294 b\u207b\u00b9 := (OrderIso.inv \u03b1).map_inf _ _", "@[to_additive (attr := simp)]\ntheorem inv_le_div_iff_le_mul : b\u207b\u00b9 \u2264 a / c \u2194 c \u2264 a * b", "@[to_additive neg_lt]\ntheorem inv_lt' : a\u207b\u00b9 < b \u2194 b\u207b\u00b9 < a", "@[to_additive] lemma inv_lt_of_mabs_lt (h : |a|\u2098 < b) : b\u207b\u00b9 < a := (mabs_lt.mp h).1", "@[to_additive (attr := simp)] lemma inv_mul_cancel_comm (a b : G) : a\u207b\u00b9 * b * a = b", "@[to_additive (attr := simp)]\ntheorem inv_mul_lt_iff_lt_mul : b\u207b\u00b9 * a < c \u2194 a < b * c", "@[simp]\ntheorem inv_neg' (a : \u03b1) : (-a)\u207b\u00b9 = -a\u207b\u00b9", "theorem inv_neg_one : (-1 : K)\u207b\u00b9 = -1", "theorem inv_pow_le_inv_pow_of_le (a1 : 1 \u2264 a) {m n : \u2115} (mn : m \u2264 n) : (a ^ n)\u207b\u00b9 \u2264 (a ^ m)\u207b\u00b9", "lemma inv_pow_sub_of_lt (a : G\u2080) (h : n < m) : a\u207b\u00b9 ^ (m - n) = (a ^ m)\u207b\u00b9 * a ^ n", "lemma inv_smul_le_iff_of_neg (h : a < 0) : a\u207b\u00b9 \u2022 b\u2081 \u2264 b\u2082 \u2194 a \u2022 b\u2082 \u2264 b\u2081", "theorem inv_sub_inv {a b : K} (ha : a \u2260 0) (hb : b \u2260 0) : a\u207b\u00b9 - b\u207b\u00b9 = (b - a) / (a * b)", "@[simp]\ntheorem irrational_int_mul_iff : Irrational (m * x) \u2194 m \u2260 0 \u2227 Irrational x", "theorem irrational_nrt_of_n_not_dvd_multiplicity {x : \u211d} (n : \u2115) {m : \u2124} (hm : m \u2260 0) (p : \u2115)\n    [hp : Fact p.Prime] (hxr : x ^ n = m)\n    (hv : (multiplicity (p : \u2124) m).get (finite_int_iff.2 \u27e8hp.1.ne_one, hm\u27e9) % n \u2260 0) :\n    Irrational x", "theorem irrational_sqrt_of_multiplicity_odd (m : \u2124) (hm : 0 < m) (p : \u2115) [hp : Fact p.Prime]\n    (Hpv :\n      (multiplicity (p : \u2124) m).get (finite_int_iff.2 \u27e8hp.1.ne_one, (ne_of_lt hm).symm\u27e9) % 2 = 1) :\n    Irrational (\u221am)", "@[deprecated irrational_sqrt_ratCast_iff (since := \"2024-06-16\")]\ntheorem irrational_sqrt_rat_iff (q : \u211a) :\n    Irrational (\u221aq) \u2194 Rat.sqrt q * Rat.sqrt q \u2260 q \u2227 0 \u2264 q", "theorem irreducibleComponent_mem_irreducibleComponents (x : X) :\n    irreducibleComponent x \u2208 irreducibleComponents X", "theorem irreducible_iff_prime [DecompositionMonoid \u03b1] {a : \u03b1} : Irreducible a \u2194 Prime a", "theorem irreducible_iff_prime_of_exists_unique_irreducible_factors [CancelCommMonoidWithZero \u03b1]\n    (eif : \u2200 a : \u03b1, a \u2260 0 \u2192 \u2203 f : Multiset \u03b1, (\u2200 b \u2208 f, Irreducible b) \u2227 f.prod ~\u1d64 a)\n    (uif :\n      \u2200 f g : Multiset \u03b1,\n        (\u2200 x \u2208 f, Irreducible x) \u2192\n          (\u2200 x \u2208 g, Irreducible x) \u2192 f.prod ~\u1d64 g.prod \u2192 Multiset.Rel Associated f g)\n    (p : \u03b1) : Irreducible p \u2194 Prime p", "theorem irreducible_or_factor {\u03b1} [Monoid \u03b1] (x : \u03b1) (h : \u00acIsUnit x) :\n    Irreducible x \u2228 \u2203 a b, \u00acIsUnit a \u2227 \u00acIsUnit b \u2227 a * b = x", "@[elab_without_expected_type]\ntheorem irrefl_of [IsIrrefl \u03b1 r] (a : \u03b1) : \u00aca \u227a a", "@[simp]\ntheorem isAdjointPair_toLinearMap\u2082 :\n    LinearMap.IsAdjointPair (Matrix.toLinearMap\u2082 b\u2081 b\u2081 J) (Matrix.toLinearMap\u2082 b\u2082 b\u2082 J')\n        (Matrix.toLin b\u2081 b\u2082 A) (Matrix.toLin b\u2082 b\u2081 A') \u2194\n      Matrix.IsAdjointPair J J' A A'", "theorem isAtomic_iff_forall_isAtomic_Iic [OrderBot \u03b1] :\n    IsAtomic \u03b1 \u2194 \u2200 x : \u03b1, IsAtomic (Set.Iic x)", "@[simp]\ntheorem isAtomistic_dual_iff_isCoatomistic : IsAtomistic \u03b1\u1d52\u1d48 \u2194 IsCoatomistic \u03b1", "@[simp]\ntheorem isBot_iff_eq_bot : IsBot a \u2194 a = \u22a5", "lemma isCancelMulZero_iff_isDomain_or_subsingleton [Semiring \u03b1] :\n    IsCancelMulZero \u03b1 \u2194 IsDomain \u03b1 \u2228 Subsingleton \u03b1", "theorem isClosedMap_const {X Y} [TopologicalSpace X] [TopologicalSpace Y] [T1Space Y] {y : Y} :\n    IsClosedMap (Function.const X y)", "theorem isClosedMap_fst_of_compactSpace [CompactSpace Y] : IsClosedMap (Prod.fst : X \u00d7 Y \u2192 X)", "theorem isClosed_and_discrete_iff {S : Set X} :\n    IsClosed S \u2227 DiscreteTopology S \u2194 \u2200 x, Disjoint (\ud835\udcdd[\u2260] x) (\ud835\udcdf S)", "theorem isClosed_iff_frequently : IsClosed s \u2194 \u2200 x, (\u2203\u1da0 y in \ud835\udcdd x, y \u2208 s) \u2192 x \u2208 s", "theorem isClosed_induced_iff [t : TopologicalSpace \u03b2] {s : Set \u03b1} {f : \u03b1 \u2192 \u03b2} :\n    IsClosed[t.induced f] s \u2194 \u2203 t, IsClosed t \u2227 f \u207b\u00b9' t = s", "lemma isClosed_nonneg : IsClosed {x : \u03b1 | 0 \u2264 x}", "theorem isClosed_sum_iff {s : Set (X \u2295 Y)} :\n    IsClosed s \u2194 IsClosed (inl \u207b\u00b9' s) \u2227 IsClosed (inr \u207b\u00b9' s)", "theorem isCompact_closure_singleton : IsCompact (closure {x})", "theorem isCompact_diagonal [CompactSpace X] : IsCompact (diagonal X)", "theorem isCompact_pi_infinite {s : \u2200 i, Set (X i)} :\n    (\u2200 i, IsCompact (s i)) \u2192 IsCompact { x : \u2200 i, X i | \u2200 i, x i \u2208 s i }", "theorem isConnected_iff_sUnion_disjoint_open {s : Set \u03b1} :\n    IsConnected s \u2194\n      \u2200 U : Finset (Set \u03b1), (\u2200 u v : Set \u03b1, u \u2208 U \u2192 v \u2208 U \u2192 (s \u2229 (u \u2229 v)).Nonempty \u2192 u = v) \u2192\n        (\u2200 u \u2208 U, IsOpen u) \u2192 (s \u2286 \u22c3\u2080 \u2191U) \u2192 \u2203 u \u2208 U, s \u2286 u", "theorem isCoprime_group_smul_right : IsCoprime y (x \u2022 z) \u2194 IsCoprime y z", "theorem isCoprime_mul_unit_left_left : IsCoprime (x * y) z \u2194 IsCoprime y z", "theorem isCoprime_self : IsCoprime x x \u2194 IsUnit x", "theorem isCoprime_zero_right : IsCoprime x 0 \u2194 IsUnit x", "lemma isCornerFree_iff (hAs : A \u2286 s \u00d7\u02e2 s) :\n    IsCornerFree A \u2194 \u2200 \u2983x\u2081\u2984, x\u2081 \u2208 s \u2192 \u2200 \u2983y\u2081\u2984, y\u2081 \u2208 s \u2192 \u2200 \u2983x\u2082\u2984, x\u2082 \u2208 s \u2192 \u2200 \u2983y\u2082\u2984, y\u2082 \u2208 s \u2192\n      IsCorner A x\u2081 y\u2081 x\u2082 y\u2082 \u2192 x\u2081 = x\u2082 where\n  mp hA _x\u2081 _ _y\u2081 _ _x\u2082 _ _y\u2082 _ hxy := hA hxy\n  mpr hA _x\u2081 _y\u2081 _x\u2082 _y\u2082 hxy := hA (hAs hxy.fst_fst_mem).1 (hAs hxy.fst_fst_mem).2\n    (hAs hxy.snd_fst_mem).1 (hAs hxy.fst_snd_mem).2 hxy", "theorem isCycle_finRotate {n : \u2115} : IsCycle (finRotate (n + 2))", "@[to_additive]\ntheorem isCyclic_of_orderOf_eq_card [Fintype \u03b1] (x : \u03b1) (hx : orderOf x = Fintype.card \u03b1) :\n    IsCyclic \u03b1", "@[simp]\ntheorem isEmpty_pprod : IsEmpty (PProd \u03b1 \u03b2) \u2194 IsEmpty \u03b1 \u2228 IsEmpty \u03b2", "@[simp]\ntheorem isEmpty_prod {\u03b1 \u03b2 : Type*} : IsEmpty (\u03b1 \u00d7 \u03b2) \u2194 IsEmpty \u03b1 \u2228 IsEmpty \u03b2", "@[simp]\ntheorem isEmpty_sigma {\u03b1} {E : \u03b1 \u2192 Type*} : IsEmpty (Sigma E) \u2194 \u2200 a, IsEmpty (E a)", "theorem isExtrFilter_const {b : \u03b2} : IsExtrFilter (fun _ => b) l a", "theorem isExtrFilter_dual_iff : IsExtrFilter (toDual \u2218 f) l a \u2194 IsExtrFilter f l a", "theorem isGLB_Ioi {a : \u03b3} : IsGLB (Ioi a) a", "theorem isGLB_Ioo {a b : \u03b3} (h : a < b) : IsGLB (Ioo a b) a", "lemma isGalois_of_isSplittingField_X_pow_sub_C : IsGalois K L", "theorem isGreatest_Icc (h : a \u2264 b) : IsGreatest (Icc a b) b", "theorem isGreatest_union_iff :\n    IsGreatest (s \u222a t) a \u2194\n      IsGreatest s a \u2227 a \u2208 upperBounds t \u2228 a \u2208 upperBounds s \u2227 IsGreatest t a", "lemma isIntegralCurveAt_comp_add {dt : \u211d} :\n    IsIntegralCurveAt \u03b3 v t\u2080 \u2194 IsIntegralCurveAt (\u03b3 \u2218 (\u00b7 + dt)) v (t\u2080 - dt)", "theorem isIntegral_iff_isIntegral_closure_finite {r : B} :\n    IsIntegral R r \u2194 \u2203 s : Set R, s.Finite \u2227 IsIntegral (Subring.closure s) r", "@[nolint unusedHavesSuffices] theorem isIntegral_localization' {R S : Type*} [CommRing R] [CommRing S] {f : R \u2192+* S}\n    (hf : f.IsIntegral) (M : Submonoid R) :\n    (map (Localization (M.map (f : R \u2192* S))) f\n          (M.le_comap_map : _ \u2264 Submonoid.comap (f : R \u2192* S) _) :\n        Localization M \u2192+* _).IsIntegral", "theorem isIntegrallyClosedIn_iff {R A : Type*} [CommRing R] [CommRing A] [Algebra R A] :\n    IsIntegrallyClosedIn R A \u2194\n      Function.Injective (algebraMap R A) \u2227\n        \u2200 {x : A}, IsIntegral R x \u2192 \u2203 y, algebraMap R A y = x", "theorem isIrreducible_singleton {x} : IsIrreducible ({x} : Set X)", "theorem isLUB_pair [SemilatticeSup \u03b3] {a b : \u03b3} : IsLUB {a, b} (a \u2294 b)", "theorem isLeast_csInf (hs : s.Nonempty) : IsLeast s (sInf s)", "theorem isLindelof_iUnion {\u03b9 : Sort*} {f : \u03b9 \u2192 Set X} [Countable \u03b9] (h : \u2200 i, IsLindelof (f i)) :\n    IsLindelof (\u22c3 i, f i)", "theorem isLittleO_exp_neg_mul_rpow_atTop {a : \u211d} (ha : 0 < a) (b : \u211d) :\n    IsLittleO atTop (fun x : \u211d => exp (-a * x)) fun x : \u211d => x ^ b", "theorem isLittleO_log_rpow_rpow_atTop {s : \u211d} (r : \u211d) (hs : 0 < s) :\n    (fun x => log x ^ r) =o[atTop] fun x => x ^ s", "theorem isLittleO_pow_const_const_pow_of_one_lt {R : Type*} [NormedRing R] (k : \u2115) {r : \u211d}\n    (hr : 1 < r) : (fun n \u21a6 (n : R) ^ k : \u2115 \u2192 R) =o[atTop] fun n \u21a6 r ^ n", "theorem isLocalMinOn_const {b : \u03b2} : IsLocalMinOn (fun _ => b) s a", "@[simp]\ntheorem isLowerSet_preimage_toDual_iff {s : Set \u03b1\u1d52\u1d48} : IsLowerSet (toDual \u207b\u00b9' s) \u2194 IsUpperSet s", "@[simp]\ntheorem isLowerSet_setOf : IsLowerSet { a | p a } \u2194 Antitone p", "theorem isMaxFilter_dual_iff : IsMaxFilter (toDual \u2218 f) l a \u2194 IsMinFilter f l a", "@[simp]\ntheorem isMax_top : IsMax (\u22a4 : \u03b1)", "theorem isMinFilter_dual_iff : IsMinFilter (toDual \u2218 f) l a \u2194 IsMaxFilter f l a", "@[to_additive]\nlemma isMulFreimanHom_const {b : \u03b2} (hb : b \u2208 B) : IsMulFreimanHom n A B fun _ \u21a6 b where\n  mapsTo _ _ := hb\n  map_prod_eq_map_prod s t _ _ hs ht _", "lemma isNilpotent_of_pos_nilpotencyClass (hx : 0 < nilpotencyClass x) :\n    IsNilpotent x", "theorem isNilpotent_pi_of_bounded_class [\u2200 i, IsNilpotent (Gs i)] (n : \u2115)\n    (h : \u2200 i, Group.nilpotencyClass (Gs i) \u2264 n) : IsNilpotent (\u2200 i, Gs i)", "lemma isNilpotent_sum {\u03b9 : Type*} {s : Finset \u03b9} {f : \u03b9 \u2192 R}\n    (hnp : \u2200 i \u2208 s, IsNilpotent (f i)) :\n    IsNilpotent (\u2211 i \u2208 s, f i)", "@[to_additive]\ntheorem isNormalSubgroup_of_commGroup [CommGroup G] {s : Set G} (hs : IsSubgroup s) :\n    IsNormalSubgroup s", "theorem isOpenMap_inl : IsOpenMap (@inl X Y)", "@[to_additive \"The quotient map by a group action is open, i.e. the quotient by a group\naction is an open quotient. \"]\ntheorem isOpenMap_quotient_mk'_mul [ContinuousConstSMul \u0393 T] :\n    letI := MulAction.orbitRel \u0393 T\n    IsOpenMap (Quotient.mk' : T \u2192 Quotient (MulAction.orbitRel \u0393 T))", "theorem isOpen_analyticAt : IsOpen { x | AnalyticAt \ud835\udd5c f x }", "theorem isOpen_extChartAt_source (x : M) : IsOpen (extChartAt I x).source", "theorem isOpen_iff_mem_nhds : IsOpen s \u2194 \u2200 x \u2208 s, s \u2208 \ud835\udcdd x", "theorem isOpen_iff_ultrafilter :\n    IsOpen s \u2194 \u2200 x \u2208 s, \u2200 (l : Ultrafilter X), \u2191l \u2264 \ud835\udcdd x \u2192 s \u2208 l", "theorem isOpen_pi_iff {s : Set (\u2200 a, \u03c0 a)} :\n    IsOpen s \u2194\n      \u2200 f, f \u2208 s \u2192 \u2203 (I : Finset \u03b9) (u : \u2200 a, Set (\u03c0 a)),\n        (\u2200 a, a \u2208 I \u2192 IsOpen (u a) \u2227 f a \u2208 u a) \u2227 (I : Set \u03b9).pi u \u2286 s", "theorem isOpen_pi_iff' [Finite \u03b9] {s : Set (\u2200 a, \u03c0 a)} :\n    IsOpen s \u2194\n      \u2200 f, f \u2208 s \u2192 \u2203 u : \u2200 a, Set (\u03c0 a), (\u2200 a, IsOpen (u a) \u2227 f a \u2208 u a) \u2227 univ.pi u \u2286 s", "theorem isPiSystem_Iic : IsPiSystem (range Iic : Set (Set \u03b1))", "theorem isPiSystem_image_Iic (s : Set \u03b1) : IsPiSystem (Iic '' s)", "theorem isPreconnected_Iic : IsPreconnected (Iic a)", "theorem isPreconnected_iff_ordConnected {s : Set \u03b1} : IsPreconnected s \u2194 OrdConnected s", "theorem isPurelyInseparable_iff : IsPurelyInseparable F E \u2194 \u2200 x : E,\n    IsIntegral F x \u2227 ((minpoly F x).Separable \u2192 x \u2208 (algebraMap F E).range)", "theorem isPurelyInseparable_of_finSepDegree_eq_one [Algebra.IsAlgebraic F E]\n    (hdeg : finSepDegree F E = 1) : IsPurelyInseparable F E", "theorem isRadical_iff_pow_one_lt [MonoidWithZero R] (k : \u2115) (hk : 1 < k) :\n    IsRadical y \u2194 \u2200 x, y \u2223 x ^ k \u2192 y \u2223 x", "theorem isRelPrime_of_no_nonunits_factors [MonoidWithZero \u03b1] {x y : \u03b1} (nonzero : \u00ac(x = 0 \u2227 y = 0))\n    (H : \u2200 z, \u00ac IsUnit z \u2192 z \u2260 0 \u2192 z \u2223 x \u2192 \u00acz \u2223 y) : IsRelPrime x y", "theorem isRelPrime_zero_right : IsRelPrime x 0 \u2194 IsUnit x", "theorem _root_.isRoot_of_unity_iff (h : 0 < n) (R : Type*) [CommRing R] [IsDomain R] {\u03b6 : R} :\n    \u03b6 ^ n = 1 \u2194 \u2203 i \u2208 n.divisors, (cyclotomic i R).IsRoot \u03b6", "lemma isSMulRegular_of_isSMulRegular_on_submodule_on_quotient\n    (h1 : IsSMulRegular N r) (h2 : IsSMulRegular (M \u29f8 N) r) : IsSMulRegular M r", "lemma isSMulRegular_on_submodule_iff_mem_imp_smul_eq_zero_imp_eq_zero :\n    IsSMulRegular N r \u2194 \u2200 x \u2208 N, r \u2022 x = 0 \u2192 x = 0", "theorem isSeparable_tower_bot_of_isSeparable [h : IsSeparable F E] : IsSeparable F K", "lemma isSigmaCompact_iUnion [Countable \u03b9] (s : \u03b9 \u2192 Set X)\n    (hcomp : \u2200 i, IsSigmaCompact (s i)) : IsSigmaCompact (\u22c3 i, s i)", "lemma isSigmaCompact_univ [h : SigmaCompactSpace X] : IsSigmaCompact (univ : Set X)", "theorem _root_.isSplittingField_iff_intermediateField {p : F[X]} :\n    p.IsSplittingField F E \u2194 p.Splits (algebraMap F E) \u2227 adjoin F (p.rootSet E) = \u22a4", "@[simp] lemma isSquare_ofAdd_iff {a : \u03b1} : IsSquare (Multiplicative.ofAdd a) \u2194 Even a := Iff.rfl", "@[simp] lemma isSublattice_empty : IsSublattice (\u2205 : Set \u03b1) := \u27e8supClosed_empty, infClosed_empty\u27e9", "@[simp] lemma isSublattice_singleton : IsSublattice ({a} : Set \u03b1)", "@[to_additive\n      \"The union of an indexed, directed, nonempty set of `AddSubmonoid`s of an `AddMonoid` `M`\n      is an `AddSubmonoid` of `M`. \"]\ntheorem isSubmonoid_iUnion_of_directed {\u03b9 : Type*} [h\u03b9 : Nonempty \u03b9] {s : \u03b9 \u2192 Set M}\n    (hs : \u2200 i, IsSubmonoid (s i)) (Directed : \u2200 i j, \u2203 k, s i \u2286 s k \u2227 s j \u2286 s k) :\n    IsSubmonoid (\u22c3 i, s i)", "@[simp]\ntheorem isTop_ofDual_iff {a : \u03b1\u1d52\u1d48} : IsTop (ofDual a) \u2194 IsBot a", "@[simp]\ntheorem isTop_top : IsTop (\u22a4 : \u03b1)", "theorem isTopologicalBasis_pi {\u03b9 : Type*} {X : \u03b9 \u2192 Type*} [\u2200 i, TopologicalSpace (X i)]\n    {T : \u2200 i, Set (Set (X i))} (cond : \u2200 i, IsTopologicalBasis (T i)) :\n    IsTopologicalBasis { S | \u2203 (U : \u2200 i, Set (X i)) (F : Finset \u03b9),\n      (\u2200 i, i \u2208 F \u2192 U i \u2208 T i) \u2227 S = (F : Set \u03b9).pi U }", "lemma isTotallyDisconnected_iff_lt {s : Set \u03b1} :\n    IsTotallyDisconnected s \u2194 \u2200 x \u2208 s, \u2200 y \u2208 s, x < y \u2192 \u2203 z \u2209 s, z \u2208 Ioo x y", "theorem isTotallyDisconnected_singleton {x} : IsTotallyDisconnected ({x} : Set \u03b1)", "@[simp]\ntheorem isUnit_gcd_one_right [GCDMonoid \u03b1] (a : \u03b1) : IsUnit (gcd a 1)", "@[to_additive]\ntheorem _root_.isUnit_map_of_leftInverse [MonoidHomClass F M N] [MonoidHomClass G N M]\n    {f : F} {x : M} (g : G) (hfg : Function.LeftInverse g f) :\n    IsUnit (f x) \u2194 IsUnit x", "@[simp]\ntheorem isUnit_of_map_unit (f : R \u2192+* S) [IsLocalRingHom f] (a) (h : IsUnit (f a)) : IsUnit a", "@[to_additive]\ntheorem isUnit_of_mul_isUnit_left [CommMonoid M] {x y : M} (hu : IsUnit (x * y)) : IsUnit x", "theorem isUpperSet_iInter {f : \u03b9 \u2192 Set \u03b1} (hf : \u2200 i, IsUpperSet (f i)) : IsUpperSet (\u22c2 i, f i)", "@[simp]\ntheorem isUpperSet_preimage_toDual_iff {s : Set \u03b1\u1d52\u1d48} : IsUpperSet (toDual \u207b\u00b9' s) \u2194 IsLowerSet s", "@[to_additive (attr := simp) smul_ite]\nlemma ite_pow (p : Prop) [Decidable p] (a b : \u03b1) (c : \u03b2) :\n    (if p then a else b) ^ c = if p then a ^ c else b ^ c := dite_pow _ _ _ _", "lemma ite_zero_smul (a : R) (b : M) : (if p then a else 0 : R) \u2022 b = if p then a \u2022 b else 0", "theorem iterateFrobeniusEquiv_add : iterateFrobeniusEquiv R p (m + n) =\n    (iterateFrobeniusEquiv R p n).trans (iterateFrobeniusEquiv R p m)", "theorem iterateFrobenius_mul_apply :\n    iterateFrobenius R p (m * n) x = (iterateFrobenius R p m)^[n] x", "@[simp]\ntheorem iterateFrobenius_zero : iterateFrobenius R p 0 = RingHom.id R", "@[to_additive (attr := simp)]\ntheorem iterate_map_inv {M F : Type*} [Group M] [FunLike F M M] [MonoidHomClass F M M]\n    (f : F) (n : \u2115) (x : M) :\n    f^[n] x\u207b\u00b9 = (f^[n] x)\u207b\u00b9", "theorem iteratedDerivWithin_neg :\n    iteratedDerivWithin n (-f) s x = -iteratedDerivWithin n f s x", "theorem iteratedDerivWithin_succ {x : \ud835\udd5c} (hxs : UniqueDiffWithinAt \ud835\udd5c s x) :\n    iteratedDerivWithin (n + 1) f s x = derivWithin (iteratedDerivWithin n f s) s x", "theorem iteratedDeriv_succ : iteratedDeriv (n + 1) f = deriv (iteratedDeriv n f)", "theorem iteratedFDerivWithin_eventually_congr_set (h : s =\u1da0[\ud835\udcdd x] t) (n : \u2115) :\n    iteratedFDerivWithin \ud835\udd5c n f s =\u1da0[\ud835\udcdd x] iteratedFDerivWithin \ud835\udd5c n f t", "@[simp]\ntheorem iteratedFDerivWithin_one_apply (h : UniqueDiffWithinAt \ud835\udd5c s x) (m : Fin 1 \u2192 E) :\n    iteratedFDerivWithin \ud835\udd5c 1 f s x m = fderivWithin \ud835\udd5c f s x (m 0)", "theorem iteratedFDerivWithin_succ_const (n : \u2115) (c : F) (hs : UniqueDiffOn \ud835\udd5c s) (hx : x \u2208 s) :\n    iteratedFDerivWithin \ud835\udd5c (n + 1) (fun _ : E \u21a6 c) s x = 0", "theorem iteratedFDerivWithin_univ {n : \u2115} :\n    iteratedFDerivWithin \ud835\udd5c n f univ = iteratedFDeriv \ud835\udd5c n f", "theorem eq_one_or_neg_one {a : \u2124} {b : \u2115} (h : a.gcd b = 1) : J(a | b) = 1 \u2228 J(a | b) = -1", "theorem mod_right (a : \u2124) {b : \u2115} (hb : Odd b) : J(a | b) = J(a | b % (4 * a.natAbs))", "@[simp]\ntheorem one_left (b : \u2115) : J(1 | b) = 1", "theorem quadratic_reciprocity_one_mod_four {a b : \u2115} (ha : a % 4 = 1) (hb : Odd b) :\n    J(a | b) = J(b | a)", "theorem zero_left {b : \u2115} (hb : 1 < b) : J(0 | b) = 0", "theorem jacobiTheta_eq_tsum_nat {\u03c4 : \u2102} (h\u03c4 : 0 < im \u03c4) :\n    jacobiTheta \u03c4 = \u21911 + \u21912 * \u2211' n : \u2115, cexp (\u03c0 * I * ((n : \u2102) + 1) ^ 2 * \u03c4)", "lemma jacobiTheta\u2082_add_left' (z \u03c4 : \u2102) :\n    jacobiTheta\u2082 (z + \u03c4) \u03c4 = cexp (-\u03c0 * I * (\u03c4 + 2 * z)) * jacobiTheta\u2082 z \u03c4", "lemma jacobiTheta\u2082_add_right (z \u03c4 : \u2102) : jacobiTheta\u2082 z (\u03c4 + 2) = jacobiTheta\u2082 z \u03c4", "theorem joinM_map_map {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2) (a : m (m \u03b1)) :\n    joinM (Functor.map f <$> a) = f <$> joinM a", "lemma latticeClosure_mono : Monotone (latticeClosure : Set \u03b1 \u2192 Set \u03b1) := latticeClosure.monotone", "@[to_additive (attr := simp)] lemma leOnePart_div_oneLePart (a : \u03b1) : a\u207b\u1d50 / a\u207a\u1d50 = a\u207b\u00b9", "@[to_additive (attr := simp)] lemma leOnePart_one : (1 : \u03b1)\u207b\u1d50 = 1", "theorem le_antisymm_iff {a b : \u03b1} : a = b \u2194 a \u2264 b \u2227 b \u2264 a", "@[simp] theorem le_compl_self : a \u2264 a\u1d9c \u2194 a = \u22a5", "theorem le_csInf_iff (hb : BddBelow s) (hs : s.Nonempty) : a \u2264 sInf s \u2194 \u2200 b \u2208 s, a \u2264 b", "theorem le_div_iff_of_neg (hc : c < 0) : a \u2264 b / c \u2194 b \u2264 a * c", "theorem le_gauge_of_subset_closedBall (hs : Absorbent \u211d s) (hr : 0 \u2264 r) (hsr : s \u2286 closedBall 0 r) :\n    \u2016x\u2016 / r \u2264 gauge s x", "theorem le_gronwallBound_of_liminf_deriv_right_le {f f' : \u211d \u2192 \u211d} {\u03b4 K \u03b5 : \u211d} {a b : \u211d}\n    (hf : ContinuousOn f (Icc a b))\n    (hf' : \u2200 x \u2208 Ico a b, \u2200 r, f' x < r \u2192 \u2203\u1da0 z in \ud835\udcdd[>] x, (z - x)\u207b\u00b9 * (f z - f x) < r)\n    (ha : f a \u2264 \u03b4) (bound : \u2200 x \u2208 Ico a b, f' x \u2264 K * f x + \u03b5) :\n    \u2200 x \u2208 Icc a b, f x \u2264 gronwallBound \u03b4 K \u03b5 (x - a)", "lemma le_iff_exists_nonneg_add (a b : \u03b1) : a \u2264 b \u2194 \u2203 c \u2265 0, b = a + c", "@[to_additive]\ntheorem le_iff_forall_one_lt_le_mul : a \u2264 b \u2194 \u2200 \u03b5, 1 < \u03b5 \u2192 a \u2264 b * \u03b5", "@[to_additive]\ntheorem le_iff_forall_one_lt_lt_mul : a \u2264 b \u2194 \u2200 \u03b5, 1 < \u03b5 \u2192 a < b * \u03b5", "@[simp]\ntheorem le_kstar : a \u2264 a\u2217", "@[to_additive le_max_of_two_nsmul_le_add]\ntheorem le_max_of_sq_le_mul {a b c : M} (h : a ^ 2 \u2264 b * c) : a \u2264 max b c", "theorem le_max_right (a b : \u03b1) : b \u2264 max a b", "@[simp]\nlemma le_mul_iff_one_le_right [PosMulMono \u03b1] [PosMulReflectLE \u03b1] (a0 : 0 < a) : a \u2264 a * b \u2194 1 \u2264 b", "@[to_additive (attr := simp)]\ntheorem le_mul_inv_iff_mul_le : c \u2264 a * b\u207b\u00b9 \u2194 c * b \u2264 a", "@[to_additive]\ntheorem le_mul_left (h : a \u2264 c) : a \u2264 b * c", "@[to_additive]\ntheorem le_mul_of_le_left : a \u2264 b \u2192 a \u2264 b * c", "theorem le_multiplicity_iff_replicate_subperm_factors {a b : \u2115} {n : \u2115} (ha : a.Prime)\n    (hb : b \u2260 0) :\n    \u2191n \u2264 multiplicity a b \u2194 replicate n a <+~ b.factors", "theorem le_nhds_iff {f} : f \u2264 \ud835\udcdd x \u2194 \u2200 s : Set X, x \u2208 s \u2192 IsOpen s \u2192 s \u2208 f", "theorem le_of_forall_ge_of_dense [LinearOrder \u03b1] [DenselyOrdered \u03b1] {a\u2081 a\u2082 : \u03b1}\n    (h : \u2200 a\u2083 < a\u2081, a\u2083 \u2264 a\u2082) : a\u2081 \u2264 a\u2082", "theorem le_of_max_le_left {a b c : \u03b1} (h : max a b \u2264 c) : a \u2264 c", "@[to_additive le_of_add_le_add_right]\ntheorem le_of_mul_le_mul_right' [i : ContravariantClass \u03b1 \u03b1 (swap (\u00b7 * \u00b7)) (\u00b7 \u2264 \u00b7)] {a b c : \u03b1}\n    (bc : b * a \u2264 c * a) :\n    b \u2264 c", "@[to_additive]\ntheorem le_of_mul_le_of_one_le_right [CovariantClass \u03b1 \u03b1 (swap (\u00b7 * \u00b7)) (\u00b7 \u2264 \u00b7)] {a b c : \u03b1}\n    (h : a * b \u2264 c) (hle : 1 \u2264 a) :\n    b \u2264 c", "lemma le_of_smul_le_smul_left [PosSMulReflectLE \u03b1 \u03b2] (h : a \u2022 b\u2081 \u2264 a \u2022 b\u2082) (ha : 0 < a) : b\u2081 \u2264 b\u2082", "lemma le_of_smul_le_smul_right [SMulPosReflectLE \u03b1 \u03b2] (h : a\u2081 \u2022 b \u2264 a\u2082 \u2022 b) (hb : 0 < b) :\n    a\u2081 \u2264 a\u2082 := SMulPosReflectLE.elim hb h", "theorem le_of_span_le_span [Nontrivial R] {s t u : Set M} (hl : LinearIndependent R ((\u2191) : u \u2192 M))\n    (hsu : s \u2286 u) (htu : t \u2286 u) (hst : span R s \u2264 span R t) : s \u2286 t", "@[to_additive]\ntheorem le_one_of_mul_le_left [ContravariantClass \u03b1 \u03b1 (swap (\u00b7 * \u00b7)) (\u00b7 \u2264 \u00b7)] {a b : \u03b1}\n    (h : a * b \u2264 b) :\n    a \u2264 1", "theorem le_padicValNat_iff_replicate_subperm_factors {a b : \u2115} {n : \u2115} (ha : a.Prime)\n    (hb : b \u2260 0) :\n    n \u2264 padicValNat a b \u2194 replicate n a <+~ b.factors", "theorem le_partialSups_of_le (f : \u2115 \u2192 \u03b1) {m n : \u2115} (h : m \u2264 n) : f m \u2264 partialSups f n", "theorem le_sInf_inter {s t : Set \u03b1} : sInf s \u2294 sInf t \u2264 sInf (s \u2229 t)", "theorem lebesgue_number_lemma_of_metric {s : Set \u03b1} {\u03b9 : Sort*} {c : \u03b9 \u2192 Set \u03b1} (hs : IsCompact s)\n    (hc\u2081 : \u2200 i, IsOpen (c i)) (hc\u2082 : s \u2286 \u22c3 i, c i) : \u2203 \u03b4 > 0, \u2200 x \u2208 s, \u2203 i, ball x \u03b4 \u2286 c i", "@[to_additive]\ntheorem leftInverse_mul_right_inv_mul (c : G) :\n    Function.LeftInverse (fun x \u21a6 c * x) fun x \u21a6 c\u207b\u00b9 * x", "@[simp]\ntheorem left_eq_sup : a = a \u2294 b \u2194 b \u2264 a", "theorem left_lt_lineMap_iff_lt (h : 0 < r) : a < lineMap a b r \u2194 a < b", "theorem left_subset_compRel {s t : Set (\u03b1 \u00d7 \u03b1)} (h : idRel \u2286 t) : s \u2286 s \u25cb t", "@[simp]\ntheorem at_one : legendreSym p 1 = 1", "@[simp]\ntheorem at_zero : legendreSym p 0 = 0", "theorem eq_one_iff' {a : \u2115} (ha0 : (a : ZMod p) \u2260 0) :\n    legendreSym p a = 1 \u2194 IsSquare (a : ZMod p)", "theorem eq_one_of_sq_sub_mul_sq_eq_zero {p : \u2115} [Fact p.Prime] {a : \u2124} (ha : (a : ZMod p) \u2260 0)\n    {x y : ZMod p} (hy : y \u2260 0) (hxy : x ^ 2 - a * y ^ 2 = 0) : legendreSym p a = 1", "protected theorem mod (a : \u2124) : legendreSym p a = legendreSym p (a % p)", "theorem quadratic_reciprocity' (hp : p \u2260 2) (hq : q \u2260 2) :\n    legendreSym q p = (-1) ^ (p / 2 * (q / 2)) * legendreSym p q", "theorem quadratic_reciprocity_one_mod_four (hp : p % 4 = 1) (hq : q \u2260 2) :\n    legendreSym q p = legendreSym p q", "theorem liftOfDerivationToSquareZero_mk_apply (d : Derivation R A I) (x : A) :\n    Ideal.Quotient.mk I (liftOfDerivationToSquareZero I hI d x) = algebraMap A (B \u29f8 I) x", "lemma limsup_const_sub (F : Filter \u03b9) [NeBot F] [AddCommSemigroup R] [Sub R] [ContinuousSub R]\n    [OrderedSub R] [CovariantClass R R (fun x y \u21a6 x + y) fun x y \u21a6 x \u2264 y] (f : \u03b9 \u2192 R) (c : R)\n    (bdd_above : F.IsBoundedUnder (\u00b7 \u2264 \u00b7) f) (bdd_below : F.IsBoundedUnder (\u00b7 \u2265 \u00b7) f) :\n    Filter.limsup (fun i \u21a6 c - f i) F = c - Filter.liminf f F", "theorem lineDerivWithin_congr_set (h : s =\u1da0[\ud835\udcdd x] t) :\n    lineDerivWithin \ud835\udd5c f s x v = lineDerivWithin \ud835\udd5c f t x v", "theorem lineDeriv_smul {c : \ud835\udd5c} : lineDeriv \ud835\udd5c f x (c \u2022 v) = c \u2022 lineDeriv \ud835\udd5c f x v", "theorem lineDifferentiableAt_smul_iff {c : \ud835\udd5c} (hc : c \u2260 0) :\n    LineDifferentiableAt \ud835\udd5c f x (c \u2022 v) \u2194 LineDifferentiableAt \ud835\udd5c f x v", "theorem lineMap_mono_left (ha : a \u2264 a') (hr : r \u2264 1) : lineMap a b r \u2264 lineMap a' b r", "theorem lineMap_strict_mono_left (ha : a < a') (hr : r < 1) : lineMap a b r < lineMap a' b r", "@[simp]\ntheorem linearEquiv_det_rotation (a : circle) : LinearEquiv.det (rotation a).toLinearEquiv = 1", "theorem linearIndependent_empty_type [IsEmpty \u03b9] : LinearIndependent R v", "theorem linearIndependent_equiv (e : \u03b9 \u2243 \u03b9') {f : \u03b9' \u2192 M} :\n    LinearIndependent R (f \u2218 e) \u2194 LinearIndependent R f", "theorem linearIndependent_fin_cons {n} {v : Fin n \u2192 V} :\n    LinearIndependent K (Fin.cons x v : Fin (n + 1) \u2192 V) \u2194\n      LinearIndependent K v \u2227 x \u2209 Submodule.span K (range v)", "theorem linearIndependent_iUnion_of_directed {\u03b7 : Type*} {s : \u03b7 \u2192 Set M} (hs : Directed (\u00b7 \u2286 \u00b7) s)\n    (h : \u2200 i, LinearIndependent R (fun x => x : s i \u2192 M)) :\n    LinearIndependent R (fun x => x : (\u22c3 i, s i) \u2192 M)", "theorem linearIndependent_iff'' :\n    LinearIndependent R v \u2194\n      \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), (\u2200 i \u2209 s, g i = 0) \u2192\n        \u2211 i \u2208 s, g i \u2022 v i = 0 \u2192 \u2200 i, g i = 0", "theorem linearIndependent_set_iff_affineIndependent_vadd_union_singleton {s : Set V}\n    (hs : \u2200 v \u2208 s, v \u2260 (0 : V)) (p\u2081 : P) : LinearIndependent k (fun v => v : s \u2192 V) \u2194\n    AffineIndependent k (fun p => p : ({p\u2081} \u222a (fun v => v +\u1d65 p\u2081) '' s : Set P) \u2192 P)", "theorem linearIndependent_unique_iff (v : \u03b9 \u2192 M) [Unique \u03b9] :\n    LinearIndependent R v \u2194 v default \u2260 0", "theorem lipschitzWith_iff_dist_le_mul [PseudoMetricSpace \u03b1] [PseudoMetricSpace \u03b2] {K : \u211d\u22650}\n    {f : \u03b1 \u2192 \u03b2} : LipschitzWith K f \u2194 \u2200 x y, dist (f x) (f y) \u2264 K * dist x y", "private theorem list_foldl' {f : \u03b1 \u2192 List \u03b2} {g : \u03b1 \u2192 \u03c3} {h : \u03b1 \u2192 \u03c3 \u00d7 \u03b2 \u2192 \u03c3}\n    (hf : haveI := prim H; Primrec f) (hg : Primrec g) (hh : haveI := prim H; Primrec\u2082 h) :\n    Primrec fun a => (f a).foldl (fun s b => h a (s, b)) (g a)", "theorem locallyConvexSpace_iff_zero : LocallyConvexSpace \ud835\udd5c E \u2194\n    (\ud835\udcdd 0 : Filter E).HasBasis (fun s : Set E => s \u2208 (\ud835\udcdd 0 : Filter E) \u2227 Convex \ud835\udd5c s) id", "@[to_additive (attr := simp)]\nlemma locallyLipschitz_inv_iff : LocallyLipschitz f\u207b\u00b9 \u2194 LocallyLipschitz f", "@[simp]\ntheorem lowerBounds_union : lowerBounds (s \u222a t) = lowerBounds s \u2229 lowerBounds t", "@[simp]\ntheorem lowerClosure_singleton (a : \u03b1) : lowerClosure ({a} : Set \u03b1) = LowerSet.Iic a", "theorem lowerSemicontinuousAt_biSup {p : \u03b9 \u2192 Prop} {f : \u2200 i, p i \u2192 \u03b1 \u2192 \u03b4}\n    (h : \u2200 i hi, LowerSemicontinuousAt (f i hi) x) :\n    LowerSemicontinuousAt (fun x' => \u2a06 (i) (hi), f i hi x') x", "theorem lowerSemicontinuousWithinAt_tsum {f : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e}\n    (h : \u2200 i, LowerSemicontinuousWithinAt (f i) s x) :\n    LowerSemicontinuousWithinAt (fun x' => \u2211' i, f i x') s x", "theorem lowerSemicontinuous_iff_isOpen_preimage :\n    LowerSemicontinuous f \u2194 \u2200 y, IsOpen (f \u207b\u00b9' Ioi y)", "@[simp]\ntheorem coeFn_smul (c : \ud835\udd5c) (f : lp E p) : \u21d1(c \u2022 f) = c \u2022 \u21d1f", "@[ext]\ntheorem ext {f g : lp E p} (h : (f : \u2200 i, E i) = g) : f = g", "theorem inner_single_right (i : \u03b9) (a : G i) (f : lp G 2) : \u27eaf, lp.single 2 i a\u27eb = \u27eaf i, a\u27eb", "theorem norm_eq_tsum_rpow (hp : 0 < p.toReal) (f : lp E p) :\n    \u2016f\u2016 = (\u2211' i, \u2016f i\u2016 ^ p.toReal) ^ (1 / p.toReal)", "@[simp]\nlemma ltByCases_gt (h : y < x) {h\u2081 : x < y \u2192 P} {h\u2082 : x = y \u2192 P} {h\u2083 : y < x \u2192 P} :\n    ltByCases x y h\u2081 h\u2082 h\u2083 = h\u2083 h := (dif_neg h.not_lt).trans (dif_pos h)", "theorem lt_add_of_tsub_lt_left (h : a - b < c) : a < b + c", "theorem lt_imp_lt_of_le_imp_le {\u03b2} [LinearOrder \u03b1] [Preorder \u03b2] {a b : \u03b1} {c d : \u03b2}\n    (H : a \u2264 b \u2192 c \u2264 d) (h : d < c) : b < a", "@[simp] protected theorem lt_irrefl (a : $typeName) : \u00ac a < a", "theorem lt_mem_nhds {a b : \u03b1} (h : a < b) : \u2200\u1da0 x in \ud835\udcdd b, a < x", "@[simp]\ntheorem lt_mul_iff_one_lt_right [PosMulStrictMono \u03b1] [PosMulReflectLT \u03b1] (a0 : 0 < a) :\n    a < a * b \u2194 1 < b", "theorem lt_mul_of_one_lt_right [PosMulStrictMono \u03b1] (ha : 0 < a) (h : 1 < b) : a < a * b", "theorem lt_of_lt_of_le' : b < c \u2192 a \u2264 b \u2192 a < c", "theorem lt_of_mul_lt_mul_of_le\u2080 (h : a * b < c * d) (hc : 0 < c) (hh : c \u2264 a) : b < d", "theorem lt_of_mul_lt_of_one_le_of_nonneg_left [PosMulMono \u03b1] (h : a * b < c) (hle : 1 \u2264 b)\n    (ha : 0 \u2264 a) : a < c", "theorem lt_of_mul_self_lt_mul_self (hb : 0 \u2264 b) : a * a < b * b \u2192 a < b", "theorem lt_of_one_div_lt_one_div (ha : 0 < a) (h : 1 / a < 1 / b) : b < a", "@[to_additive]\ntheorem lt_one_of_mul_lt_right [ContravariantClass \u03b1 \u03b1 (\u00b7 * \u00b7) (\u00b7 < \u00b7)] {a b : \u03b1} (h : a * b < a) :\n    b < 1", "theorem lt_or_eq_of_le {a b : \u03b1} : a \u2264 b \u2192 a < b \u2228 a = b", "theorem lt_tsub_of_add_lt_left : a + c < b \u2192 c < b - a", "@[to_additive] lemma mabs_eq_mabs : |a|\u2098 = |b|\u2098 \u2194 a = b \u2228 a = b\u207b\u00b9", "@[to_additive abs_add_eq_two_nsmul_posPart]\nlemma mabs_mul_eq_oneLePart_sq (a : \u03b1) : |a|\u2098 * a = a\u207a\u1d50 ^ 2", "@[simp]\ntheorem manyOneReducible_toNat {p : Set \u03b1} : p \u2264\u2080 toNat p", "theorem mapClusterPt_iff {\u03b9 : Type*} (x : X) (F : Filter \u03b9) (u : \u03b9 \u2192 X) :\n    MapClusterPt x F u \u2194 \u2200 s \u2208 \ud835\udcdd x, \u2203\u1da0 a in F, u a \u2208 s", "theorem mapClusterPt_iff_ultrafilter {\u03b9 : Type*} (x : X) (F : Filter \u03b9) (u : \u03b9 \u2192 X) :\n    MapClusterPt x F u \u2194 \u2203 U : Ultrafilter \u03b9, U \u2264 F \u2227 Tendsto u U (\ud835\udcdd x)", "theorem mapClusterPt_of_comp {F : Filter \u03b1} {\u03c6 : \u03b2 \u2192 \u03b1} {p : Filter \u03b2}\n    {u : \u03b1 \u2192 X} [NeBot p] (h : Tendsto \u03c6 p F) (H : Tendsto (u \u2218 \u03c6) p (\ud835\udcdd x)) :\n    MapClusterPt x F u", "@[simp]\ntheorem map_bihimp (a b : \u03b1) : f (a \u21d4 b) = f a \u21d4 f b", "theorem map_bind (x : m \u03b1) {g : \u03b1 \u2192 m \u03b2} {f : \u03b2 \u2192 \u03b3} :\n    f <$> (x >>= g) = x >>= fun a => f <$> g a", "theorem map_coe_atBot_of_Ioo_subset (ha : s \u2286 Ioi a) (hs : \u2200 b' > a, \u2203 b > a, Ioo a b \u2286 s) :\n    map ((\u2191) : s \u2192 \u03b1) atBot = \ud835\udcdd[>] a", "theorem map_comap_subtype_coe {m0 : MeasurableSpace \u03b1} {s : Set \u03b1} (hs : MeasurableSet s)\n    (\u03bc : Measure \u03b1) : (comap (\u2191) \u03bc).map ((\u2191) : s \u2192 \u03b1) = \u03bc.restrict s", "theorem map_commutatorElement : (f \u2045g\u2081, g\u2082\u2046 : G') = \u2045f g\u2081, f g\u2082\u2046", "@[to_additive (attr := simp)]\nlemma map_comp_zpow' [DivInvMonoid G] [DivInvMonoid H] [MonoidHomClass F G H] (f : F)\n    (hf : \u2200 x : G, f x\u207b\u00b9 = (f x)\u207b\u00b9) (g : \u03b9 \u2192 G) (n : \u2124) : f \u2218 (g ^ n) = f \u2218 g ^ n", "theorem map_continuousAt (f : F) (a : \u03b1) : ContinuousAt f a", "theorem map_derivedSeries_le_derivedSeries (n : \u2115) :\n    (derivedSeries G n).map f \u2264 derivedSeries G' n", "theorem map_extChartAt_nhdsWithin' {x y : M} (hy : y \u2208 (extChartAt I x).source) :\n    map (extChartAt I x) (\ud835\udcdd[s] y) = \ud835\udcdd[(extChartAt I x).symm \u207b\u00b9' s \u2229 range I] extChartAt I x y", "theorem map_extChartAt_nhdsWithin_eq_image' {x y : M} (hy : y \u2208 (extChartAt I x).source) :\n    map (extChartAt I x) (\ud835\udcdd[s] y) =\n      \ud835\udcdd[extChartAt I x '' ((extChartAt I x).source \u2229 s)] extChartAt I x y", "theorem map_extChartAt_symm_nhdsWithin' {x y : M} (hy : y \u2208 (extChartAt I x).source) :\n    map (extChartAt I x).symm (\ud835\udcdd[(extChartAt I x).symm \u207b\u00b9' s \u2229 range I] extChartAt I x y) =\n      \ud835\udcdd[s] y", "@[to_additive]\ntheorem map_finsupp_prod [Zero M] [CommMonoid N] [CommMonoid P] {H : Type*}\n    [FunLike H N P] [MonoidHomClass H N P]\n    (h : H) (f : \u03b1 \u2192\u2080 M) (g : \u03b1 \u2192 M \u2192 N) : h (f.prod g) = f.prod fun a b => h (g a b)", "theorem map_iSup\u2082 [SupSet \u03b1] [SupSet \u03b2] [sSupHomClass F \u03b1 \u03b2] (f : F) (g : \u2200 i, \u03ba i \u2192 \u03b1) :\n    f (\u2a06 (i) (j), g i j) = \u2a06 (i) (j), f (g i j)", "@[simp]\ntheorem map_inv_le_iff (f : F) {a : \u03b1} {b : \u03b2} : EquivLike.inv f b \u2264 a \u2194 b \u2264 f a", "theorem map_le_lineMap_iff_slope_le_slope_right (h : 0 < (1 - r) * (b - a)) :\n    f c \u2264 lineMap (f a) (f b) r \u2194 slope f a b \u2264 slope f c b", "@[to_additive]\ntheorem map_list_prod {F : Type*} [FunLike F M N] [MonoidHomClass F M N] (f : F) (l : List M) :\n    f l.prod = (l.map f).prod", "@[to_additive]\ntheorem map_mul_eq_one [MonoidHomClass F M N] (f : F) {a b : M} (h : a * b = 1) :\n    f a * f b = 1", "theorem map_mul_left_nhds\u2080 (ha : a \u2260 0) (b : G\u2080) : map (a * \u00b7) (\ud835\udcdd b) = \ud835\udcdd (a * b)", "@[to_additive]\ntheorem map_ne_one_iff {R S F : Type*} [One R] [One S] [FunLike F R S] [OneHomClass F R S] (f : F)\n    (hf : Function.Injective f) {x : R} : f x \u2260 1 \u2194 x \u2260 1", "@[simp]\ntheorem map_snd_nhds (x : X \u00d7 Y) : map Prod.snd (\ud835\udcdd x) = \ud835\udcdd x.2", "theorem map_uniformity_set_coe {s : Set \u03b1} [UniformSpace \u03b1] :\n    map (Prod.map (\u2191) (\u2191)) (\ud835\udce4 s) = \ud835\udce4 \u03b1 \u2293 \ud835\udcdf (s \u00d7\u02e2 s)", "@[simp]\nlemma matPolyEquiv_map_smul (p : R[X]) (M : Matrix n n R[X]) :\n    matPolyEquiv (p \u2022 M) = p.map (algebraMap _ _) * matPolyEquiv M", "@[simp]\ntheorem matrixEquivTensor_apply (M : Matrix n n A) :\n    matrixEquivTensor R A n M = \u2211 p : n \u00d7 n, M p.1 p.2 \u2297\u209c stdBasisMatrix p.1 p.2 1", "theorem max_assoc (a b c : \u03b1) : max (max a b) c = max a (max b c)", "@[to_additive max_sub_sub_right]\ntheorem max_div_div_right' (a b c : \u03b1) : max (a / c) (b / c) = max a b / c", "theorem max_lt {a b c : \u03b1} (h\u2081 : a < c) (h\u2082 : b < c) : max a b < c", "@[to_additive max_add_add_le_max_add_max]\ntheorem max_mul_mul_le_max_mul_max' : max (a * b) (c * d) \u2264 max a c * max b d", "theorem mdifferentiableAt_fst {x : M \u00d7 M'} : MDifferentiableAt (I.prod I') I Prod.fst x", "theorem mdifferentiableWithinAt_fst {s : Set (M \u00d7 M')} {x : M \u00d7 M'} :\n    MDifferentiableWithinAt (I.prod I') I Prod.fst s x", "theorem mdifferentiableWithinAt_inter' (ht : t \u2208 \ud835\udcdd[s] x) :\n    MDifferentiableWithinAt I I' f (s \u2229 t) x \u2194 MDifferentiableWithinAt I I' f s x", "theorem mdifferentiable_iff_differentiable :\n    MDifferentiable \ud835\udcd8(\ud835\udd5c, E) \ud835\udcd8(\ud835\udd5c, E') f \u2194 Differentiable \ud835\udd5c f", "theorem mdifferentiable_of_mem_atlas (h : e \u2208 atlas H M) : e.MDifferentiable I I", "@[simp, measurability]\ntheorem measurableSet_Ioc : MeasurableSet (Ioc a b)", "@[measurability]\ntheorem measurableSet_closedBall : MeasurableSet (Metric.closedBall x \u03b5)", "@[measurability]\ntheorem measurableSet_eball : MeasurableSet (EMetric.ball x \u03b5)", "@[measurability]\ntheorem measurableSet_eq {a : \u03b1} : MeasurableSet { x | x = a }", "@[measurability]\nlemma measurableSet_eq_fun' {\u03b2 : Type*} [CanonicallyOrderedAddCommMonoid \u03b2] [Sub \u03b2] [OrderedSub \u03b2]\n    {_ : MeasurableSpace \u03b2} [MeasurableSub\u2082 \u03b2] [MeasurableSingletonClass \u03b2]\n    {f g : \u03b1 \u2192 \u03b2} (hf : Measurable f) (hg : Measurable g) :\n    MeasurableSet {x | f x = g x}", "theorem measurableSet_of_differentiableAt_with_param (hf : Continuous f.uncurry) :\n    MeasurableSet {p : \u03b1 \u00d7 E | DifferentiableAt \ud835\udd5c (f p.1) p.2}", "theorem measurableSet_prod {s : Set \u03b1} {t : Set \u03b2} :\n    MeasurableSet (s \u00d7\u02e2 t) \u2194 MeasurableSet s \u2227 MeasurableSet t \u2228 s = \u2205 \u2228 t = \u2205", "theorem measurableSet_region_between_co (hf : Measurable f) (hg : Measurable g)\n    (hs : MeasurableSet s) :\n    MeasurableSet { p : \u03b1 \u00d7 \u211d | p.fst \u2208 s \u2227 p.snd \u2208 Ico (f p.fst) (g p.fst) }", "theorem measurableSet_region_between_oc (hf : Measurable f) (hg : Measurable g)\n    (hs : MeasurableSet s) :\n    MeasurableSet { p : \u03b1 \u00d7 \u211d | p.fst \u2208 s \u2227 p.snd \u2208 Ioc (f p.fst) (g p.fst) }", "@[measurability]\ntheorem measurable_ereal_toReal : Measurable EReal.toReal", "@[measurability]\ntheorem measurable_iSup {\u03b9} [Countable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} (hf : \u2200 i, Measurable (f i)) :\n    Measurable (fun b \u21a6 \u2a06 i, f i b)", "@[measurability]\ntheorem measurable_id {_ : MeasurableSpace \u03b1} : Measurable (@id \u03b1)", "@[to_additive (attr := simp)]\ntheorem measurable_inv_iff {G : Type*} [Group G] [MeasurableSpace G] [MeasurableInv G]\n    {f : \u03b1 \u2192 G} : (Measurable fun x => (f x)\u207b\u00b9) \u2194 Measurable f", "@[measurability]\ntheorem measurable_limsup {f : \u2115 \u2192 \u03b4 \u2192 \u03b1} (hf : \u2200 i, Measurable (f i)) :\n    Measurable fun x => limsup (fun i => f i x) atTop", "theorem measurable_measure_prod_mk_left_finite [IsFiniteMeasure \u03bd] {s : Set (\u03b1 \u00d7 \u03b2)}\n    (hs : MeasurableSet s) : Measurable fun x => \u03bd (Prod.mk x \u207b\u00b9' s)", "theorem measurable_of_isClosed {f : \u03b4 \u2192 \u03b3} (hf : \u2200 s, IsClosed s \u2192 MeasurableSet (f \u207b\u00b9' s)) :\n    Measurable f", "theorem measurable_of_restrict_of_restrict_compl {f : \u03b1 \u2192 \u03b2} {s : Set \u03b1} (hs : MeasurableSet s)\n    (h\u2081 : Measurable (s.restrict f)) (h\u2082 : Measurable (s\u1d9c.restrict f)) : Measurable f", "theorem measurable_sSup {\u03b9} {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} {s : Set \u03b9} (hs : s.Countable)\n    (hf : \u2200 i \u2208 s, Measurable (f i)) :\n    Measurable fun x => sSup ((fun i => f i x) '' s)", "@[measurability]\ntheorem measurable_swap : Measurable (Prod.swap : \u03b1 \u00d7 \u03b2 \u2192 \u03b2 \u00d7 \u03b1)", "theorem measure_Ioo_lt_top : \u03bc (Ioo a b) < \u221e", "theorem mellin_const_smul (f : \u211d \u2192 E) (s : \u2102) {\ud835\udd5c : Type*} [NontriviallyNormedField \ud835\udd5c]\n    [NormedSpace \ud835\udd5c E] [SMulCommClass \u2102 \ud835\udd5c E] (c : \ud835\udd5c) :\n    mellin (fun t => c \u2022 f t) s = c \u2022 mellin f s", "theorem mem_adjoin_of_dvd_coeff_of_dvd_aeval {A B : Type*} [CommSemiring A] [CommRing B]\n    [Algebra A B] [NoZeroSMulDivisors A B] {Q : A[X]} {p : A} {x z : B} (hp : p \u2260 0)\n    (hQ : \u2200 i \u2208 range (Q.natDegree + 1), p \u2223 Q.coeff i) (hz : aeval x Q = p \u2022 z) :\n    z \u2208 adjoin A ({x} : Set B)", "theorem mem_affineSpan {p : P} {s : Set P} (hp : p \u2208 s) : p \u2208 affineSpan k s", "@[to_additive] theorem mem_ball_one_iff : a \u2208 ball (1 : E) r \u2194 \u2016a\u2016 < r", "@[to_additive mem_closedBall_iff_norm']\ntheorem mem_closedBall_iff_norm''' : b \u2208 closedBall a r \u2194 \u2016a / b\u2016 \u2264 r", "@[to_additive]\ntheorem mem_closure_iff_nhds_one {x : G} {s : Set G} :\n    x \u2208 closure s \u2194 \u2200 U \u2208 (\ud835\udcdd 1 : Filter G), \u2203 y \u2208 s, y / x \u2208 U", "theorem mem_closure_iff_seq_limit [FrechetUrysohnSpace X] {s : Set X} {a : X} :\n    a \u2208 closure s \u2194 \u2203 x : \u2115 \u2192 X, (\u2200 n : \u2115, x n \u2208 s) \u2227 Tendsto x atTop (\ud835\udcdd a)", "theorem mem_closure_iff_ultrafilter :\n    x \u2208 closure s \u2194 \u2203 u : Ultrafilter X, s \u2208 u \u2227 \u2191u \u2264 \ud835\udcdd x", "theorem mem_conjugatesOf_self {a : \u03b1} : a \u2208 conjugatesOf a", "theorem mem_lowerBounds_image2 (ha : a \u2208 lowerBounds s) (hb : b \u2208 lowerBounds t) :\n    f a b \u2208 lowerBounds (image2 f s t)", "theorem mem_nhdsWithin {t : Set \u03b1} {a : \u03b1} {s : Set \u03b1} :\n    t \u2208 \ud835\udcdd[s] a \u2194 \u2203 u, IsOpen u \u2227 a \u2208 u \u2227 u \u2229 s \u2286 t", "theorem mem_nhdsWithin_iff_eventually {s t : Set \u03b1} {x : \u03b1} :\n    t \u2208 \ud835\udcdd[s] x \u2194 \u2200\u1da0 y in \ud835\udcdd x, y \u2208 s \u2192 y \u2208 t", "theorem mem_nhdsWithin_of_mem_nhds {s t : Set \u03b1} {a : \u03b1} (h : s \u2208 \ud835\udcdd a) : s \u2208 \ud835\udcdd[t] a", "theorem mem_nhdsWithin_subtype {s : Set \u03b1} {a : { x // x \u2208 s }} {t u : Set { x // x \u2208 s }} :\n    t \u2208 \ud835\udcdd[u] a \u2194 t \u2208 comap ((\u2191) : s \u2192 \u03b1) (\ud835\udcdd[(\u2191) '' u] a)", "theorem mem_nhds_induced [T : TopologicalSpace \u03b1] (f : \u03b2 \u2192 \u03b1) (a : \u03b2) (s : Set \u03b2) :\n    s \u2208 @nhds \u03b2 (TopologicalSpace.induced f T) a \u2194 \u2203 u \u2208 \ud835\udcdd (f a), f \u207b\u00b9' u \u2286 s", "@[simp]\ntheorem mem_nonunits_iff [Monoid \u03b1] : a \u2208 nonunits \u03b1 \u2194 \u00acIsUnit a", "lemma mem_of_mem_measurableAtom {x y : \u03b2} (h : y \u2208 measurableAtom x) {s : Set \u03b2}\n    (hs : MeasurableSet s) (hxs : x \u2208 s) : y \u2208 s", "theorem mem_omegaLimit_singleton_iff_map_cluster_point (x : \u03b1) (y : \u03b2) :\n    y \u2208 \u03c9 f \u03d5 {x} \u2194 MapClusterPt y f fun t \u21a6 \u03d5 t x", "theorem mem_openSegment_of_ne_left_right (hx : x \u2260 z) (hy : y \u2260 z) (hz : z \u2208 [x -[\ud835\udd5c] y]) :\n    z \u2208 openSegment \ud835\udd5c x y", "theorem mem_perfectClosure_iff {x : E} :\n    x \u2208 perfectClosure F E \u2194 \u2203 n : \u2115, x ^ (ringExpChar F) ^ n \u2208 (algebraMap F E).range", "theorem mem_permsOfList_iff {l : List \u03b1} {f : Perm \u03b1} :\n    f \u2208 permsOfList l \u2194 \u2200 {x}, f x \u2260 x \u2192 x \u2208 l", "theorem mem_residual_iff {s : Set X} :\n    s \u2208 residual X \u2194\n      \u2203 S : Set (Set X), (\u2200 t \u2208 S, IsOpen t) \u2227 (\u2200 t \u2208 S, Dense t) \u2227 S.Countable \u2227 \u22c2\u2080 S \u2286 s", "@[to_additive mem_rightAddCoset_rightAddCoset]\ntheorem mem_rightCoset_rightCoset {a : \u03b1} (ha : op a \u2022 (s : Set \u03b1) = s) : a \u2208 s", "@[simp]\ntheorem mem_skewAdjointMatricesSubmodule :\n    A\u2081 \u2208 skewAdjointMatricesSubmodule J \u2194 J.IsSkewAdjoint A\u2081", "theorem mem_span_finset {s : Finset M} {x : M} :\n    x \u2208 span R (\u2191s : Set M) \u2194 \u2203 f : M \u2192 R, \u2211 i \u2208 s, f i \u2022 i = x", "theorem mem_tangentConeAt_of_pow_smul {r : \ud835\udd5c} (hr\u2080 : r \u2260 0) (hr : \u2016r\u2016 < 1)\n    (hs : \u2200\u1da0 n : \u2115 in atTop, x + r ^ n \u2022 y \u2208 s) : y \u2208 tangentConeAt \ud835\udd5c s x", "theorem mem\u2113p_zero {f : \u2200 i, E i} (hf : Set.Finite { i | f i \u2260 0 }) : Mem\u2113p f 0", "theorem mfderiv_prod_left {x\u2080 : M} {y\u2080 : M'} :\n    mfderiv I (I.prod I') (fun x => (x, y\u2080)) x\u2080 =\n      ContinuousLinearMap.inl \ud835\udd5c (TangentSpace I x\u2080) (TangentSpace I' y\u2080)", "theorem midpoint_eq_iff {x y z : P} : midpoint R x y = z \u2194 pointReflection R z x = y", "@[simp]\ntheorem midpoint_pointReflection_right (x y : P) :\n    midpoint R y (Equiv.pointReflection x y) = x", "theorem min_abs_abs_le_abs_min : min |a| |b| \u2264 |min a b|", "theorem min_def (a b : \u03b1) : min a b = if a \u2264 b then a else b", "@[simp]\ntheorem min_eq_left_iff : min a b = a \u2194 a \u2264 b", "theorem min_le_left (a b : \u03b1) : min a b \u2264 a", "theorem min_lt_min (h\u2081 : a < c) (h\u2082 : b < d) : min a b < min c d", "theorem min_lt_of_left_lt (h : a < c) : min a b < c", "@[to_additive min_add_min_le_min_add_add]\ntheorem min_mul_min_le_min_mul_mul' : min a c * min b d \u2264 min (a * b) (c * d)", "theorem minimal_nonempty_open_subsingleton [T0Space X] {s : Set X} (hs : IsOpen s)\n    (hmin : \u2200 t, t \u2286 s \u2192 t.Nonempty \u2192 IsOpen t \u2192 t = s) : s.Subsingleton", "theorem minimals_antichain : IsAntichain r (minimals r s)", "theorem minimals_of_symm [IsSymm \u03b1 r] : minimals r s = s", "theorem minimals_union : minimals r (s \u222a t) \u2286 minimals r s \u222a minimals r t", "theorem ToAdjoin.injective (hx : IsIntegral R x) : Function.Injective (Minpoly.toAdjoin R x)", "theorem eq_X_sub_C' (a : A) : minpoly A a = X - C a", "theorem eq_of_irreducible [Nontrivial B] {p : A[X]} (hp1 : Irreducible p)\n    (hp2 : Polynomial.aeval x p = 0) : p * C p.leadingCoeff\u207b\u00b9 = minpoly A x", "theorem eq_of_irreducible_of_monic [Nontrivial B] {p : A[X]} (hp1 : Irreducible p)\n    (hp2 : Polynomial.aeval x p = 0) (hp3 : p.Monic) : p = minpoly A x", "theorem minpoly.map_eq_of_separable_of_isPurelyInseparable (x : K)\n    (hsep : (minpoly F x).Separable) [IsPurelyInseparable F E] :\n    (minpoly F x).map (algebraMap F E) = minpoly E x", "lemma minpolyDiv_spec :\n    minpolyDiv R x * (X - C x) = (minpoly R x).map (algebraMap R S)", "lemma minpoly_algHom_toLinearMap (\u03c3 : L \u2192\u2090[K] L) (h\u03c3 : IsOfFinOrder \u03c3) :\n    minpoly K \u03c3.toLinearMap = X ^ (orderOf \u03c3) - C 1", "theorem mk_mem_nonZeroDivisors_associates : Associates.mk a \u2208 (Associates M\u2080)\u2070 \u2194 a \u2208 M\u2080\u2070", "@[simp, mfld_simps]\ntheorem modelWithCorners_prod_coe_symm (I : ModelWithCorners \ud835\udd5c E H)\n    (I' : ModelWithCorners \ud835\udd5c E' H') :\n    ((I.prod I').symm : _ \u00d7 _ \u2192 _ \u00d7 _) = Prod.map I.symm I'.symm", "lemma monotoneOn_of_hasDerivWithinAt_nonneg {D : Set \u211d} (hD : Convex \u211d D) {f f' : \u211d \u2192 \u211d}\n    (hf : ContinuousOn f D) (hf' : \u2200 x \u2208 interior D, HasDerivWithinAt f (f' x) (interior D) x)\n    (hf'\u2080 : \u2200 x \u2208 interior D, 0 \u2264 f' x) : MonotoneOn f D", "theorem monotone_app (f : \u03b2 \u2192 \u03b1 \u2192 \u03b3) (b : \u03b2) (hf : Monotone fun a b \u21a6 f b a) : Monotone (f b)", "theorem monotone_nat_of_le_succ {f : \u2115 \u2192 \u03b1} (hf : \u2200 n, f n \u2264 f (n + 1)) : Monotone f", "theorem monotone_sInf_of_monotone {s : Set (\u03b1 \u2192 \u03b2)} (m_s : \u2200 f \u2208 s, Monotone f) :\n    Monotone (sInf s)", "@[simp]\ntheorem monotone_vecEmpty : Monotone ![a]", "lemma monovaryOn_iff_forall_mul_nonneg :\n    MonovaryOn f g s \u2194 \u2200 \u2983i\u2984, i \u2208 s \u2192 \u2200 \u2983j\u2984, j \u2208 s \u2192 0 \u2264 (f j - f i) * (g j - g i)", "lemma monovaryOn_iff_mul_rearrangement :\n    MonovaryOn f g s \u2194\n      \u2200 \u2983i\u2984, i \u2208 s \u2192 \u2200 \u2983j\u2984, j \u2208 s \u2192 f i * g j + f j * g i \u2264 f i * g i + f j * g j", "theorem monovary_comm : Monovary f g \u2194 Monovary g f", "@[simp]\ntheorem monovary_id_iff : Monovary f id \u2194 Monotone f", "@[simp] lemma monovary_inv_right\u2080 (hg : StrongLT 0 g) : Monovary f g\u207b\u00b9 \u2194 Antivary f g", "theorem mt {a b : Prop} (h\u2081 : a \u2192 b) (h\u2082 : \u00acb) : \u00aca", "@[to_additive (attr := simp)]\nlemma mulEquivOfOrderOfEq_apply_gen : mulEquivOfOrderOfEq hg hg' h g = g'", "@[to_additive]\ntheorem mulIndicator_eventuallyLE_mulIndicator (h : f \u2264\u1da0[l \u2293 \ud835\udcdf s] g) :\n    mulIndicator s f \u2264\u1da0[l] mulIndicator s g", "theorem mulPosStrictMono_iff_mulPosReflectLE : MulPosStrictMono \u03b1 \u2194 MulPosReflectLE \u03b1", "@[to_additive]\ntheorem mulRothNumber_le : mulRothNumber s \u2264 s.card", "@[to_additive (attr := simp)]\ntheorem mulRothNumber_map_mul_left :\n    mulRothNumber (s.map <| mulLeftEmbedding a) = mulRothNumber s", "@[to_additive]\ntheorem mulTSupport_eq_empty_iff {f : X \u2192 \u03b1} : mulTSupport f = \u2205 \u2194 f = 1", "@[to_additive]\ntheorem mul_div (a b c : G) : a * (b / c) = a * b / c", "@[to_additive (attr := simp)]\ntheorem mul_div_div_cancel (a b c : G) : a * b / (a / c) = b * c", "theorem mul_dvd_mul_right (h : a \u2223 b) (c : \u03b1) : a * c \u2223 b * c", "@[to_additive]\ntheorem mul_eq_of_eq_div' (h : b = c / a) : a * b = c", "@[to_additive]\ntheorem mul_eq_one_iff_inv_eq : a * b = 1 \u2194 a\u207b\u00b9 = b", "@[to_additive]\ntheorem mul_inv_le_one_iff : b * a\u207b\u00b9 \u2264 1 \u2194 b \u2264 a", "@[to_additive (attr := simp) neg_add_rev]\ntheorem mul_inv_rev (a b : G) : (a * b)\u207b\u00b9 = b\u207b\u00b9 * a\u207b\u00b9", "@[gcongr]\ntheorem mul_le_mul [PosMulMono \u03b1] [MulPosMono \u03b1] (h\u2081 : a \u2264 b) (h\u2082 : c \u2264 d) (c0 : 0 \u2264 c)\n    (b0 : 0 \u2264 b) : a * c \u2264 b * d", "theorem mul_le_mul_of_nonpos_right (h : b \u2264 a) (hc : c \u2264 0) : a * c \u2264 b * c", "theorem mul_le_mul_right\u2080 (hc : c \u2260 0) : a * c \u2264 b * c \u2194 a \u2264 b", "theorem mul_le_of_le_of_le_one_of_nonneg [PosMulMono \u03b1] (h : b \u2264 c) (ha : a \u2264 1) (hb : 0 \u2264 b) :\n    b * a \u2264 c", "@[to_additive]\ntheorem mul_le_of_mul_le_left [CovariantClass \u03b1 \u03b1 (\u00b7 * \u00b7) (\u00b7 \u2264 \u00b7)] {a b c d : \u03b1} (h : a * b \u2264 c)\n    (hle : d \u2264 b) :\n    a * d \u2264 c", "theorem mul_le_one\u2080 (ha : a \u2264 1) (hb : b \u2264 1) : a * b \u2264 1", "@[to_additive]\ntheorem mul_left_surjective (a : G) : Surjective (a * \u00b7)", "theorem mul_lt_mul (hac : a < c) (hbd : b \u2264 d) (hb : 0 < b) (hc : 0 \u2264 c) : a * b < c * d", "theorem mul_lt_mul_of_le_of_lt' [PosMulMono \u03b1] [MulPosStrictMono \u03b1] (h\u2081 : a < b) (h\u2082 : c \u2264 d)\n    (b0 : 0 \u2264 b) (c0 : 0 < c) : a * c < b * d", "theorem mul_lt_of_le_of_lt_one_of_pos [PosMulStrictMono \u03b1] (bc : b \u2264 c) (ha : a < 1) (b0 : 0 < b) :\n    b * a < c", "@[to_additive]\ntheorem mul_lt_of_lt_one_of_le [CovariantClass \u03b1 \u03b1 (swap (\u00b7 * \u00b7)) (\u00b7 < \u00b7)] {a b c : \u03b1} (ha : a < 1)\n    (hbc : b \u2264 c) :\n    a * b < c", "theorem mul_lt_of_one_lt_right (ha : a < 0) (h : 1 < b) : a * b < a", "@[to_additive]\ntheorem mul_mem_cancel_right {x y : G} (h : x \u2208 H) : y * x \u2208 H \u2194 y \u2208 H", "theorem mul_min_of_nonneg (b c : \u03b1) (ha : 0 \u2264 a) : a * min b c = min (a * b) (a * c)", "theorem mul_nonpos_of_nonneg_of_nonpos [PosMulMono \u03b1] (ha : 0 \u2264 a) (hb : b \u2264 0) : a * b \u2264 0", "@[to_additive]\nlemma mul_not_mem_of_exponent_two (h : Monoid.exponent G = 2) {x y : G}\n    (hx : x \u2260 1) (hy : y \u2260 1) (hxy : x \u2260 y) : x * y \u2209 ({x, y, 1} : Set G)", "@[to_additive]\nlemma mul_not_mem_of_orderOf_eq_two {x y : G} (hx : orderOf x = 2)\n    (hy : orderOf y = 2) (hxy : x \u2260 y) : x * y \u2209 ({x, y, 1} : Set G)", "@[to_additive (attr := simp)]\ntheorem mul_one : \u2200 a : M, a * 1 = a", "@[simp]\ntheorem mul_one_add_self : a * (1 + a) = 0", "@[to_additive (attr := simp)]\ntheorem mul_right_iterate : (\u00b7 * a)^[n] = (\u00b7 * a ^ n)", "@[to_additive]\ntheorem mul_right_iterate_apply_one : (\u00b7 * a)^[n] 1 = a ^ n", "theorem mul_right_surjective\u2080 {a : G\u2080} (h : a \u2260 0) : Surjective fun g => g * a", "@[to_additive (attr := simp)]\nlemma mul_smul_one {M N} [MulOneClass N] [SMul M N] [SMulCommClass M N N] (x : M) (y : N) :\n    y * x \u2022 (1 : N) = x \u2022 y", "theorem mul_sub_mul_div_mul_nonpos_iff (hc : c \u2260 0) (hd : d \u2260 0) :\n    (a * d - b * c) / (c * d) \u2264 0 \u2194 a / c \u2264 b / d", "@[to_additive zsmul_add] lemma mul_zpow : \u2200 n : \u2124, (a * b) ^ n = a ^ n * b ^ n\n  | (n : \u2115) => by simp_rw [zpow_natCast, mul_pow]\n  | .negSucc n => by simp_rw [zpow_negSucc, \u2190 inv_pow, mul_inv, mul_pow]", "@[simp] lemma multiplesAddHom_apply (x : \u03b1) (n : \u2115) : multiplesAddHom \u03b1 x n = n \u2022 x := rfl", "@[to_additive additive_of_symmetric_of_isTotal]\nlemma multiplicative_of_symmetric_of_isTotal\n    (hsymm : Symmetric p) (hf_swap : \u2200 {a b}, p a b \u2192 f a b * f b a = 1)\n    (hmul : \u2200 {a b c}, r a b \u2192 r b c \u2192 p a b \u2192 p b c \u2192 p a c \u2192 f a c = f a b * f b c)\n    {a b c : \u03b1} (pab : p a b) (pbc : p b c) (pac : p a c) : f a c = f a b * f b c", "theorem eq_of_associated_left {a b c : \u03b1} (h : Associated a b) :\n    multiplicity b c = multiplicity a c", "theorem eq_of_associated_right {a b c : \u03b1} (h : Associated b c) :\n    multiplicity a b = multiplicity a c", "theorem finite_int_iff {a b : \u2124} : Finite a b \u2194 a.natAbs \u2260 1 \u2227 b \u2260 0", "theorem min_le_multiplicity_add {p a b : \u03b1} :\n    min (multiplicity p a) (multiplicity p b) \u2264 multiplicity p (a + b)", "theorem multiplicity_add_of_gt {p a b : \u03b1} (h : multiplicity p b < multiplicity p a) :\n    multiplicity p (a + b) = multiplicity p b", "theorem multiplicity_le_multiplicity_of_dvd_right {a b c : \u03b1} (h : b \u2223 c) :\n    multiplicity a b \u2264 multiplicity a c", "theorem multiplicity_map_eq {F : Type*} [EquivLike F \u03b1 \u03b2] [MulEquivClass F \u03b1 \u03b2]\n    (f : F) {a b : \u03b1} : multiplicity (f a) (f b) = multiplicity a b", "theorem pow_sub_pow_of_prime {p : R} (hp : Prime p) {x y : R} (hxy : p \u2223 x - y) (hx : \u00acp \u2223 x)\n    {n : \u2115} (hn : \u00acp \u2223 n) : multiplicity p (x ^ n - y ^ n) = multiplicity p (x - y)", "theorem multiplicity_normalizedFactorsEquivSpanNormalizedFactors_eq_multiplicity {r d : R}\n    (hr : r \u2260 0) (hd : d \u2208 normalizedFactors r) :\n    multiplicity d r =\n      multiplicity (normalizedFactorsEquivSpanNormalizedFactors hr \u27e8d, hd\u27e9 : Ideal R)\n        (Ideal.span {r})", "theorem nat_abs_sum_le {\u03b9 : Type*} (s : Finset \u03b9) (f : \u03b9 \u2192 \u2124) :\n    (\u2211 i \u2208 s, f i).natAbs \u2264 \u2211 i \u2208 s, (f i).natAbs", "theorem nat_sub_dvd_pow_sub_pow (x y n : \u2115) : x - y \u2223 x ^ n - y ^ n", "private theorem ncs_tendsto_lim :\n    Tendsto (fun i => \u2016F.derivative.eval (ncs i)\u2016) atTop (\ud835\udcdd \u2016F.derivative.eval ncs.lim\u2016)", "theorem ne_iff_lt_or_gt {a b : \u03b1} : a \u2260 b \u2194 a < b \u2228 a > b", "lemma ne_of_eq_of_ne {\u03b1 : Sort*} {a b c : \u03b1} (h\u2081 : a = b) (h\u2082 : b \u2260 c) : a \u2260 c := h\u2081.symm \u25b8 h\u2082", "lemma ne_of_ne_of_eq' (hab : (a : \u03b1) \u2260 c) (hbc : a = b) : b \u2260 c := hbc \u25b8 hab", "theorem ne_zero_of_dvd_ne_zero {p q : \u03b1} (h\u2081 : q \u2260 0) (h\u2082 : p \u2223 q) : p \u2260 0", "lemma ne_zero_of_eq_one [One \u03b1] [NeZero (1 : \u03b1)] {a : \u03b1} (h : a = 1) : a \u2260 0 := h \u25b8 one_ne_zero", "theorem ne_zero_or_ne_zero_of_nat_coprime {A : Type u} [CommRing A] [Nontrivial A] {a b : \u2115}\n    (h : Nat.Coprime a b) : (a : A) \u2260 0 \u2228 (b : A) \u2260 0", "@[field_simps]\ntheorem neg_div' (a b : K) : -(b / a) = -b / a", "theorem neg_of_mul_neg_left (h : a * b < 0) (hb : 0 \u2264 b) : a < 0", "theorem neg_of_mul_pos_left [PosMulMono \u03b1] [MulPosMono \u03b1] (h : 0 < a * b) (ha : b \u2264 0) : a < 0", "lemma neg_one_pow_eq_one_iff_even {R : Type*} [Monoid R] [HasDistribNeg R] {n : \u2115}\n    (h : (-1 : R) \u2260 1) : (-1 : R) ^ n = 1 \u2194 Even n where\n  mp h' := of_not_not fun hn \u21a6 h <| (Odd.neg_one_pow <| odd_iff_not_even.mpr hn).symm.trans h'\n  mpr := Even.neg_one_pow", "lemma neg_one_pow_eq_or : \u2200 n : \u2115, (-1 : R) ^ n = 1 \u2228 (-1 : R) ^ n = -1\n  | 0 => Or.inl (pow_zero _)\n  | n + 1 => (neg_one_pow_eq_or n).symm.imp\n    (fun h \u21a6 by rw [pow_succ, h, neg_one_mul, neg_neg])\n    (fun h \u21a6 by rw [pow_succ, h, one_mul])", "lemma neg_one_sq : (-1 : R) ^ 2 = 1", "lemma neg_pow (a : R) (n : \u2115) : (-a) ^ n = (-1) ^ n * a ^ n", "@[simp]\nlemma neg_pow_bit1 (a : R) (n : \u2115) : (-a) ^ bit1 n = -a ^ bit1 n", "theorem nhdsSet_Ioc (h : a < b) : \ud835\udcdd\u02e2 (Ioc a b) = \ud835\udcdd b \u2294 \ud835\udcdf (Ioo a b)", "theorem nhdsSet_diagonal (X) [TopologicalSpace (X \u00d7 X)] :\n    \ud835\udcdd\u02e2 (diagonal X) = \u2a06 (x : X), \ud835\udcdd (x, x)", "theorem nhdsSet_diagonal_eq_uniformity [CompactSpace \u03b1] : \ud835\udcdd\u02e2 (diagonal \u03b1) = \ud835\udce4 \u03b1", "@[simp]\ntheorem nhdsSet_univ : \ud835\udcdd\u02e2 (univ : Set X) = \u22a4", "@[simp]\ntheorem nhdsWithin_Icc_eq_nhdsWithin_Iic (h : a < b) : \ud835\udcdd[Icc a b] b = \ud835\udcdd[\u2264] b", "theorem nhdsWithin_eq_map_subtype_coe {s : Set \u03b1} {a : \u03b1} (h : a \u2208 s) :\n    \ud835\udcdd[s] a = map ((\u2191) : s \u2192 \u03b1) (\ud835\udcdd \u27e8a, h\u27e9)", "@[simp]\ntheorem nhdsWithin_insert (a : \u03b1) (s : Set \u03b1) : \ud835\udcdd[insert a s] a = pure a \u2294 \ud835\udcdd[s] a", "lemma nhdsWithin_neBot : (\ud835\udcdd[s] x).NeBot \u2194 \u2200 \u2983t\u2984, t \u2208 \ud835\udcdd x \u2192 (t \u2229 s).Nonempty", "theorem nhdsWithin_prod_eq (x : X) (y : Y) (s : Set X) (t : Set Y) :\n    \ud835\udcdd[s \u00d7\u02e2 t] (x, y) = \ud835\udcdd[s] x \u00d7\u02e2 \ud835\udcdd[t] y", "theorem nhdsWithin_restrict {a : \u03b1} (s : Set \u03b1) {t : Set \u03b1} (h\u2080 : a \u2208 t) (h\u2081 : IsOpen t) :\n    \ud835\udcdd[s] a = \ud835\udcdd[s \u2229 t] a", "theorem nhdsWithin_restrict' {a : \u03b1} (s : Set \u03b1) {t : Set \u03b1} (h : t \u2208 \ud835\udcdd a) : \ud835\udcdd[s] a = \ud835\udcdd[s \u2229 t] a", "theorem nhdsWithin_subtype_eq_bot_iff {s t : Set X} {x : s} :\n    \ud835\udcdd[((\u2191) : s \u2192 X) \u207b\u00b9' t] x = \u22a5 \u2194 \ud835\udcdd[t] (x : X) \u2293 \ud835\udcdf s = \u22a5", "theorem nhds_basis_Ioo_pos [NoMaxOrder \u03b1] (a : \u03b1) :\n    (\ud835\udcdd a).HasBasis (fun \u03b5 : \u03b1 => (0 : \u03b1) < \u03b5) fun \u03b5 => Ioo (a - \u03b5) (a + \u03b5)", "theorem nhds_basis_Ioo_pos_of_pos [NoMaxOrder \u03b1] {a : \u03b1} (ha : 0 < a) :\n    (\ud835\udcdd a).HasBasis (fun \u03b5 : \u03b1 => (0 : \u03b1) < \u03b5 \u2227 \u03b5 \u2264 a) fun \u03b5 => Ioo (a - \u03b5) (a + \u03b5)", "theorem nhds_eq_comap_uniformity {x : \u03b1} : \ud835\udcdd x = (\ud835\udce4 \u03b1).comap (Prod.mk x)", "theorem nhds_eq_nhds_emetric_ball (a x : \u03b2) (r : \u211d\u22650\u221e) (h : x \u2208 ball a r) :\n    \ud835\udcdd x = map ((\u2191) : ball a r \u2192 \u03b2) (\ud835\udcdd \u27e8x, h\u27e9)", "@[simp]\ntheorem nhds_false : \ud835\udcdd False = \u22a4", "theorem nhds_injective [T0Space X] : Injective (\ud835\udcdd : X \u2192 Filter X)", "theorem nhds_of_nhdsWithin_of_nhds {s t : Set \u03b1} {a : \u03b1} (h1 : s \u2208 \ud835\udcdd a) (h2 : t \u2208 \ud835\udcdd[s] a) :\n    t \u2208 \ud835\udcdd a", "theorem nhds_pi {a : \u2200 i, \u03c0 i} : \ud835\udcdd a = pi fun i => \ud835\udcdd (a i)", "theorem nhds_toAdd (x : Multiplicative X) : \ud835\udcdd (toAdd x) = map toAdd (\ud835\udcdd x)", "theorem nhds_top {a : \u03b1} : @nhds \u03b1 \u22a4 a = \u22a4", "theorem nndist_edist (x y : \u03b1) : nndist x y = (edist x y).toNNReal", "@[simp]\ntheorem nndist_homothety_center (p\u2081 p\u2082 : P) (c : \ud835\udd5c) :\n    nndist (homothety p\u2081 c p\u2082) p\u2081 = \u2016c\u2016\u208a * nndist p\u2081 p\u2082", "@[simp] theorem nndist_toDual (a b : X) : nndist (toDual a) (toDual b) = nndist a b", "theorem noncompactSpace_of_neBot (_ : NeBot (Filter.cocompact X)) : NoncompactSpace X", "theorem nonempty_linearEquiv_of_lift_rank_eq\n    (cnd : Cardinal.lift.{v'} (Module.rank R M) = Cardinal.lift.{v} (Module.rank R M')) :\n    Nonempty (M \u2243\u2097[R] M')", "theorem nonempty_measurable_superset (s : Set \u03b1) : Nonempty { t // s \u2286 t \u2227 MeasurableSet t }", "theorem nonempty_sections_of_finite_cofiltered_system.init {J : Type u} [SmallCategory J]\n    [IsCofilteredOrEmpty J] (F : J \u2964 Type u) [hf : \u2200 j, Finite (F.obj j)]\n    [hne : \u2200 j, Nonempty (F.obj j)] : F.sections.Nonempty", "theorem nonpos_of_mul_nonpos_right (h : a * b \u2264 0) (ha : 0 < a) : b \u2264 0", "theorem norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero (x y : E) (h : \u27eax, y\u27eb = 0) :\n    \u2016x + y\u2016 * \u2016x + y\u2016 = \u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016", "theorem norm_fderiv_iteratedFDeriv {n : \u2115} :\n    \u2016fderiv \ud835\udd5c (iteratedFDeriv \ud835\udd5c n f) x\u2016 = \u2016iteratedFDeriv \ud835\udd5c (n + 1) f x\u2016", "lemma norm_fst_le (x : E \u00d7 F) : \u2016x.1\u2016 \u2264 \u2016x\u2016 := le_max_left _ _", "theorem norm_image_sub_le_of_norm_deriv_le_segment {C : \u211d} (hf : DifferentiableOn \u211d f (Icc a b))\n    (bound : \u2200 x \u2208 Ico a b, \u2016derivWithin f (Icc a b) x\u2016 \u2264 C) :\n    \u2200 x \u2208 Icc a b, \u2016f x - f a\u2016 \u2264 C * (x - a)", "theorem norm_inner_div_norm_mul_norm_eq_one_iff (x y : E) :\n    \u2016\u27eax, y\u27eb / (\u2016x\u2016 * \u2016y\u2016)\u2016 = 1 \u2194 x \u2260 0 \u2227 \u2203 r : \ud835\udd5c, r \u2260 0 \u2227 y = r \u2022 x", "theorem norm_iteratedFDerivWithin_clm_apply_const {f : E \u2192 F \u2192L[\ud835\udd5c] G} {c : F} {s : Set E} {x : E}\n    {N : \u2115\u221e} {n : \u2115} (hf : ContDiffOn \ud835\udd5c N f s) (hs : UniqueDiffOn \ud835\udd5c s) (hx : x \u2208 s) (hn : \u2191n \u2264 N) :\n    \u2016iteratedFDerivWithin \ud835\udd5c n (fun y : E => (f y) c) s x\u2016 \u2264\n      \u2016c\u2016 * \u2016iteratedFDerivWithin \ud835\udd5c n f s x\u2016", "theorem norm_iteratedFDerivWithin_comp_le {g : F \u2192 G} {f : E \u2192 F} {n : \u2115} {s : Set E} {t : Set F}\n    {x : E} {N : \u2115\u221e} (hg : ContDiffOn \ud835\udd5c N g t) (hf : ContDiffOn \ud835\udd5c N f s) (hn : (n : \u2115\u221e) \u2264 N)\n    (ht : UniqueDiffOn \ud835\udd5c t) (hs : UniqueDiffOn \ud835\udd5c s) (hst : MapsTo f s t) (hx : x \u2208 s) {C : \u211d}\n    {D : \u211d} (hC : \u2200 i, i \u2264 n \u2192 \u2016iteratedFDerivWithin \ud835\udd5c i g t (f x)\u2016 \u2264 C)\n    (hD : \u2200 i, 1 \u2264 i \u2192 i \u2264 n \u2192 \u2016iteratedFDerivWithin \ud835\udd5c i f s x\u2016 \u2264 D ^ i) :\n    \u2016iteratedFDerivWithin \ud835\udd5c n (g \u2218 f) s x\u2016 \u2264 n ! * C * D ^ n", "theorem norm_iteratedFDerivWithin_eq_norm_iteratedDerivWithin :\n    \u2016iteratedFDerivWithin \ud835\udd5c n f s x\u2016 = \u2016iteratedDerivWithin n f s x\u2016", "theorem norm_iteratedFDeriv_fderiv {n : \u2115} :\n    \u2016iteratedFDeriv \ud835\udd5c n (fderiv \ud835\udd5c f) x\u2016 = \u2016iteratedFDeriv \ud835\udd5c (n + 1) f x\u2016", "theorem norm_le_gronwallBound_of_norm_deriv_right_le {f f' : \u211d \u2192 E} {\u03b4 K \u03b5 : \u211d} {a b : \u211d}\n    (hf : ContinuousOn f (Icc a b)) (hf' : \u2200 x \u2208 Ico a b, HasDerivWithinAt f (f' x) (Ici x) x)\n    (ha : \u2016f a\u2016 \u2264 \u03b4) (bound : \u2200 x \u2208 Ico a b, \u2016f' x\u2016 \u2264 K * \u2016f x\u2016 + \u03b5) :\n    \u2200 x \u2208 Icc a b, \u2016f x\u2016 \u2264 gronwallBound \u03b4 K \u03b5 (x - a)", "@[simp]\ntheorem norm_mul (a b : \u03b1) : \u2016a * b\u2016 = \u2016a\u2016 * \u2016b\u2016", "theorem norm_sum_le {\u03b9 E} [SeminormedAddCommGroup E] (s : Finset \u03b9) (f : \u03b9 \u2192 E) :\n    \u2016\u2211 i \u2208 s, f i\u2016 \u2264 \u2211 i \u2208 s, \u2016f i\u2016", "theorem norm_sup_le_add (x y : \u03b1) : \u2016x \u2294 y\u2016 \u2264 \u2016x\u2016 + \u2016y\u2016", "theorem normal_exists_closure_subset [NormalSpace X] {s t : Set X} (hs : IsClosed s) (ht : IsOpen t)\n    (hst : s \u2286 t) : \u2203 u, IsOpen u \u2227 s \u2286 u \u2227 closure u \u2286 t", "theorem normalizeScaleRoots_coeff_mul_leadingCoeff_pow (i : \u2115) (hp : 1 \u2264 natDegree p) :\n    (normalizeScaleRoots p).coeff i * p.leadingCoeff ^ i =\n      p.coeff i * p.leadingCoeff ^ (p.natDegree - 1)", "theorem normalize_eq (x : \u03b1) : normalize x = x", "theorem not_and_or : \u00ac(a \u2227 b) \u2194 \u00aca \u2228 \u00acb", "theorem not_covBy [DenselyOrdered \u03b1] : \u00aca \u22d6 b", "theorem not_covBy_of_lt_of_lt (h\u2081 : a < b) (h\u2082 : b < c) : \u00aca \u22d6 c", "@[simp] theorem not_false_eq_true : (\u00ac False) = True", "theorem not_forall_not : (\u00ac\u2200 x, \u00acp x) \u2194 \u2203 x, p x", "theorem not_iff_not : (\u00aca \u2194 \u00acb) \u2194 (a \u2194 b)", "theorem not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter_aux\n    [CompleteSpace E] {f : \u211d \u2192 E} {g : \u211d \u2192 F}\n    {k : Set \u211d} (l : Filter \u211d) [NeBot l] [TendstoIxxClass Icc l l]\n    (hl : k \u2208 l) (hd : \u2200\u1da0 x in l, DifferentiableAt \u211d f x) (hf : Tendsto (fun x => \u2016f x\u2016) l atTop)\n    (hfg : deriv f =O[l] g) : \u00acIntegrableOn g k", "@[to_additive]\nlemma not_isCyclic_iff_exponent_eq_prime [Group \u03b1] {p : \u2115} (hp : p.Prime)\n    (h\u03b1 : Nat.card \u03b1 = p ^ 2) : \u00ac IsCyclic \u03b1 \u2194 Monoid.exponent \u03b1 = p", "theorem not_isRightRegular_zero_iff : \u00acIsRightRegular (0 : R) \u2194 Nontrivial R", "@[simp]\ntheorem not_isTop [NoTopOrder \u03b1] (a : \u03b1) : \u00acIsTop a", "@[to_additive]\nlemma not_mulDissociated_iff_exists_disjoint :\n    \u00ac MulDissociated s \u2194\n      \u2203 t u : Finset \u03b1, \u2191t \u2286 s \u2227 \u2191u \u2286 s \u2227 Disjoint t u \u2227 t \u2260 u \u2227 \u220f a in t, a = \u220f a in u, a", "theorem not_small_ordinal : \u00acSmall.{u} Ordinal.{max u v}", "theorem npow_mul_comm (m n : \u2115) (x : M) :\n    x ^ m * x ^ n = x ^ n * x ^ m", "@[simp] lemma odd_add_one_self : Odd (a + 1 + a)", "@[simp] lemma ofAdd_smul [VAdd \u03b1 \u03b2] (a : \u03b1) (b : \u03b2) : ofAdd a \u2022 b = a +\u1d65 b := rfl", "@[simp]\ntheorem ofDual_sInf (s : Set \u03b1\u1d52\u1d48) : ofDual (sInf s) = sSup (toDual \u207b\u00b9' s)", "@[to_additive (attr := simp)]\ntheorem ofLex_one [One \u03b1] : (ofLex 1 : \u03b1) = 1", "@[aesop safe apply (rule_sets := [SetLike])]\nlemma ofNat_mem [AddSubmonoidWithOneClass S R] (s : S) (n : \u2115) [n.AtLeastTwo] :\n    no_index (OfNat.ofNat n) \u2208 s", "theorem of_not_not {a : Prop} : \u00ac\u00aca \u2192 a", "theorem omegaLimit_def : \u03c9 f \u03d5 s = \u22c2 u \u2208 f, closure (image2 \u03d5 u s)", "theorem omegaLimit_iInter (p : \u03b9 \u2192 Set \u03b1) : \u03c9 f \u03d5 (\u22c2 i, p i) \u2286 \u22c2 i, \u03c9 f \u03d5 (p i)", "theorem omegaLimit_mono_right {s\u2081 s\u2082 : Set \u03b1} (hs : s\u2081 \u2286 s\u2082) : \u03c9 f \u03d5 s\u2081 \u2286 \u03c9 f \u03d5 s\u2082", "theorem one_add_div (h : b \u2260 0) : 1 + a / b = (b + a) / b", "theorem one_add_one_eq_two [AddMonoidWithOne R] : 1 + 1 = (2 : R)", "theorem one_div_le_of_neg (ha : a < 0) (hb : b < 0) : 1 / a \u2264 b \u2194 1 / b \u2264 a", "theorem one_div_lt_one_div_of_neg_of_lt (hb : b < 0) (h : a < b) : 1 / b < 1 / a", "theorem one_div_ne_zero {a : G\u2080} (h : a \u2260 0) : 1 / a \u2260 0", "@[to_additive]\ntheorem one_div_one : (1 : G) / 1 = 1", "lemma one_div_pos : 0 < 1 / a \u2194 0 < a := inv_eq_one_div a \u25b8 inv_pos", "@[to_additive zsmul_zero_sub]\nlemma one_div_zpow (a : \u03b1) (n : \u2124) : (1 / a) ^ n = 1 / a ^ n", "lemma one_half_le_sum_primes_ge_one_div (k : \u2115) :\n    1 / 2 \u2264 \u2211 p \u2208 (4 ^ (k.primesBelow.card + 1)).succ.primesBelow \\ k.primesBelow,\n      (1 / p : \u211d)", "theorem one_half_lt_one : (1 / 2 : \u03b1) < 1", "@[to_additive]\ntheorem one_le_of_le_mul_right [ContravariantClass \u03b1 \u03b1 (\u00b7 * \u00b7) (\u00b7 \u2264 \u00b7)] {a b : \u03b1} (h : a \u2264 a * b) :\n    1 \u2264 b", "@[to_additive dite_pos]\nlemma one_lt_dite [LT \u03b1] (ha : \u2200 h, 1 < a h) (hb : \u2200 h, 1 < b h) : 1 < dite p a b", "theorem one_lt_inv (h\u2081 : 0 < a) (h\u2082 : a < 1) : 1 < a\u207b\u00b9", "theorem one_lt_natDegree_of_irrational_root (hx : Irrational x) (p_nonzero : p \u2260 0)\n    (x_is_root : aeval x p = 0) : 1 < p.natDegree", "theorem openSegment_eq_image (x y : E) :\n    openSegment \ud835\udd5c x y = (fun \u03b8 : \ud835\udd5c => (1 - \u03b8) \u2022 x + \u03b8 \u2022 y) '' Ioo (0 : \ud835\udd5c) 1", "theorem openSegment_eq_image' (x y : E) :\n    openSegment \ud835\udd5c x y = (fun \u03b8 : \ud835\udd5c => x + \u03b8 \u2022 (y - x)) '' Ioo (0 : \ud835\udd5c) 1", "@[simp]\ntheorem openSegment_same (x : E) : openSegment \ud835\udd5c x x = {x}", "lemma opensMeasurableSpace_iff_forall_measurableSet\n    [TopologicalSpace \u03b1] [MeasurableSpace \u03b1] :\n    OpensMeasurableSpace \u03b1 \u2194  (\u2200 (s : Set \u03b1), IsOpen s \u2192 MeasurableSet s)", "theorem or_assoc : (a \u2228 b) \u2228 c \u2194 a \u2228 (b \u2228 c)", "theorem or_congr (h\u2081 : a \u2194 c) (h\u2082 : b \u2194 d) : (a \u2228 b) \u2194 (c \u2228 d)", "theorem or_iff_right (ha : \u00aca) : a \u2228 b \u2194 b", "@[simp] theorem or_self (p : Prop) : (p \u2228 p) = p", "@[to_additive]\ntheorem orderOf_dvd_natCard {G : Type*} [Group G] (x : G) : orderOf x \u2223 Nat.card G", "@[to_additive]\nlemma orderOf_generator_eq_natCard (h : \u2200 x, x \u2208 Subgroup.zpowers a) : orderOf a = Nat.card \u03b1", "@[to_additive (attr := simp)]\ntheorem orderOf_one : orderOf (1 : G) = 1", "theorem orthogonalProjectionFn_mem (v : E) : orthogonalProjectionFn K v \u2208 K", "theorem orthogonalProjection_eq_linear_proj [HasOrthogonalProjection K] (x : E) :\n    orthogonalProjection K x =\n      K.linearProjOfIsCompl _ Submodule.isCompl_orthogonal_of_completeSpace x", "theorem orthogonalProjection_orthogonalProjection_of_le {U V : Submodule \ud835\udd5c E}\n    [HasOrthogonalProjection U] [HasOrthogonalProjection V] (h : U \u2264 V) (x : E) :\n    orthogonalProjection U (orthogonalProjection V x) = orthogonalProjection U x", "theorem orthonormal_empty : Orthonormal \ud835\udd5c (fun x => x : (\u2205 : Set E) \u2192 E)", "theorem orthonormal_subtype_iff_ite [DecidableEq E] {s : Set E} :\n    Orthonormal \ud835\udd5c (Subtype.val : s \u2192 E) \u2194 \u2200 v \u2208 s, \u2200 w \u2208 s, \u27eav, w\u27eb = if v = w then 1 else 0", "theorem pNilradical_prime {R : Type*} [CommSemiring R] {p : \u2115} (hp : p.Prime) :\n    pNilradical R p = nilradical R", "theorem add_eq_max_of_ne {q r : \u211a} (hne : padicNorm p q \u2260 padicNorm p r) :\n    padicNorm p (q + r) = max (padicNorm p q) (padicNorm p r)", "theorem int_eq_one_iff (m : \u2124) : padicNorm p m = 1 \u2194 \u00ac(p : \u2124) \u2223 m", "protected theorem of_int (z : \u2124) : padicNorm p z \u2264 1", "theorem sum_lt {\u03b1 : Type*} {F : \u03b1 \u2192 \u211a} {t : \u211a} {s : Finset \u03b1} :\n    s.Nonempty \u2192 (\u2200 i \u2208 s, padicNorm p (F i) < t) \u2192 padicNorm p (\u2211 i \u2208 s, F i) < t", "theorem zero_of_padicNorm_eq_zero {q : \u211a} (h : padicNorm p q = 0) : q = 0", "theorem pow_add_pow (hxy : p \u2223 x + y) (hx : \u00acp \u2223 x) {n : \u2115} (hn : Odd n) :\n    padicValNat p (x ^ n + y ^ n) = padicValNat p (x + y) + padicValNat p n", "theorem padicValNat_def' {n : \u2115} (hp : p \u2260 1) (hn : 0 < n) :\n    \u2191(padicValNat p n) = multiplicity p n", "theorem padicValNat_factorial_mul (n : \u2115) [hp : Fact p.Prime] :\n    padicValNat p (p * n) ! = padicValNat p n ! + n", "@[simp]\ntheorem partialSups_succ (f : \u2115 \u2192 \u03b1) (n : \u2115) :\n    partialSups f (n + 1) = partialSups f n \u2294 f (n + 1)", "theorem pathConnectedSpace_iff_univ : PathConnectedSpace X \u2194 IsPathConnected (univ : Set X)", "theorem pcontinuous_iff' {f : X \u2192. Y} :\n    PContinuous f \u2194 \u2200 {x y} (h : y \u2208 f x), PTendsto' f (\ud835\udcdd x) (\ud835\udcdd y)", "theorem perfectField_iff_splits_of_natSepDegree_eq_one (F : Type*) [Field F] :\n    PerfectField F \u2194 \u2200 f : F[X], f.natSepDegree = 1 \u2192 f.Splits (RingHom.id F)", "theorem pi_Ico_mem_nhds' (ha : \u2200 i, a' i < x' i) (hb : \u2200 i, x' i < b' i) : Ico a' b' \u2208 \ud835\udcdd x'", "theorem pi_Iio_mem_nhds' (ha : \u2200 i, x' i < a' i) : Iio a' \u2208 \ud835\udcdd x'", "theorem pi_Ioc_mem_nhds' (ha : \u2200 i, a' i < x' i) (hb : \u2200 i, x' i < b' i) : Ioc a' b' \u2208 \ud835\udcdd x'", "theorem pi_dvd_iff {x y : \u2200 i, G i} : x \u2223 y \u2194 \u2200 i, x i \u2223 y i", "theorem pi_generateFrom_eq {\u03c0 : \u03b9 \u2192 Type*} {g : \u2200 a, Set (Set (\u03c0 a))} :\n    (@Pi.topologicalSpace \u03b9 \u03c0 fun a => generateFrom (g a)) =\n      generateFrom\n        { t | \u2203 (s : \u2200 a, Set (\u03c0 a)) (i : Finset \u03b9), (\u2200 a \u2208 i, s a \u2208 g a) \u2227 t = pi (\u2191i) s }", "theorem piiUnionInter_singleton (\u03c0 : \u03b9 \u2192 Set (Set \u03b1)) (i : \u03b9) :\n    piiUnionInter \u03c0 {i} = \u03c0 i \u222a {univ}", "theorem polynomialFunctions.eq_adjoin_X (s : Set R) :\n    polynomialFunctions s = Algebra.adjoin R {toContinuousMapOnAlgHom s X}", "lemma posSMulStrictMono_iff_PosSMulReflectLE : PosSMulStrictMono \u03b1 \u03b2 \u2194 PosSMulReflectLE \u03b1 \u03b2", "theorem posTangentConeAt_mono : Monotone fun s => posTangentConeAt s a", "@[deprecated Even.pow_pos_iff (since := \"2024-04-06\")]\nlemma pow_bit0_pos_iff (a : R) {n : \u2115} (hn : n \u2260 0) : 0 < a ^ bit0 n \u2194 a \u2260 0", "theorem pow_inv_comm\u2080 (a : G\u2080) (m n : \u2115) : a\u207b\u00b9 ^ m * a ^ n = a ^ n * a\u207b\u00b9 ^ m", "@[to_additive (attr := mono, gcongr) nsmul_lt_nsmul_right]\nlemma pow_lt_pow_left' (hn : n \u2260 0) {a b : M} (hab : a < b) : a ^ n < b ^ n", "@[to_additive (attr := simp) mod_card_nsmul]\nlemma pow_mod_card (a : G) (n : \u2115) : a ^ (n % card G) = a ^ n", "@[to_additive (attr := simp) mod_addOrderOf_nsmul]\nlemma pow_mod_orderOf (x : G) (n : \u2115) : x ^ (n % orderOf x) = x ^ n", "@[to_additive] lemma pow_mul_comm (a : M) (m n : \u2115) : a ^ m * a ^ n = a ^ n * a ^ m", "@[simp]\ntheorem pow_pos (H : 0 < a) : \u2200 n : \u2115, 0 < a ^ n", "theorem pow_right_mono (h : 1 \u2264 a) : Monotone (a ^ \u00b7)", "theorem pow_right_strictAnti (h\u2080 : 0 < a) (h\u2081 : a < 1) : StrictAnti (a ^ \u00b7)", "@[to_additive add_sub_one_nsmul]\nlemma pow_sub_one_mul (hn : n \u2260 0) (a : M) : a ^ (n - 1) * a = a ^ n", "theorem preStoneCechExtend_extends : preStoneCechExtend hg \u2218 preStoneCechUnit = g", "lemma pred_max (a b : \u03b1) : pred (max a b) = max (pred a) (pred b) := pred_mono.map_max", "theorem preimage_smul_set\u209b\u2097\n    {G : Type*} [FunLike G R S] [MonoidHomClass G R S] (\u03c3 : G)\n    [MulAction R M] [MulAction S N]\n    {F : Type*} [FunLike F M N] [MulActionSemiHomClass F \u03c3 M N] (h : F)\n    {c : R} (hc : IsUnit c) (t : Set N) :\n    h \u207b\u00b9' (\u03c3 c \u2022 t) = c \u2022 h \u207b\u00b9' t", "@[simp 1100]\ntheorem prevD_comp_right (f : \u2200 i j, C.X i \u27f6 D.X j) (g : D \u27f6 E) (j : \u03b9) :\n    (prevD j fun i j => f i j \u226b g.f j) = prevD j f \u226b g.f j", "theorem prime_dvd_char_iff_dvd_card {R : Type*} [CommRing R] [Fintype R] (p : \u2115) [Fact p.Prime] :\n    p \u2223 ringChar R \u2194 p \u2223 Fintype.card R", "theorem prod_eq_generateFrom :\n    instTopologicalSpaceProd =\n      generateFrom { g | \u2203 (s : Set X) (t : Set Y), IsOpen s \u2227 IsOpen t \u2227 g = s \u00d7\u02e2 t }", "@[to_additive\n      \"Sum of elements in an `AddSubmonoid` of an `AddCommMonoid` indexed by a `Finset`\n      is in the `AddSubmonoid`.\"]\ntheorem prod_mem {M : Type*} [CommMonoid M] [SetLike B M] [SubmonoidClass B M] {\u03b9 : Type*}\n    {t : Finset \u03b9} {f : \u03b9 \u2192 M} (h : \u2200 c \u2208 t, f c \u2208 S) : (\u220f c \u2208 t, f c) \u2208 S", "@[to_additive]\ntheorem prod_mul_tprod_subtype_compl [T2Space \u03b1] {f : \u03b2 \u2192 \u03b1} (hf : Multipliable f) (s : Finset \u03b2) :\n    (\u220f x \u2208 s, f x) * \u220f' x : { x // x \u2209 s }, f x = \u220f' x, f x", "@[simp]\ntheorem projIcc_eq_one {x : \u211d} : projIcc (0 : \u211d) 1 zero_le_one x = 1 \u2194 1 \u2264 x", "theorem pure_le_nhds : pure \u2264 (\ud835\udcdd : X \u2192 Filter X)", "theorem quasiIso'_of_comp_left (f : C \u27f6 D) [QuasiIso' f] (g : D \u27f6 E) [QuasiIso' (f \u226b g)] :\n    QuasiIso' g", "theorem quasiIso'_of_comp_right (f : C \u27f6 D) (g : D \u27f6 E) [QuasiIso' g] [QuasiIso' (f \u226b g)] :\n    QuasiIso' f", "lemma quasispectrum.mem_of_not_quasiregular (a : A) {r : R\u02e3}\n    (hr : \u00ac IsQuasiregular (-(r\u207b\u00b9 \u2022 a))) : (r : R) \u2208 quasispectrum R a", "lemma rTensor.inverse_of_rightInverse_apply\n    {h : P \u2192 N} (hgh : Function.RightInverse h g) (y : N \u2297[R] Q) :\n    (rTensor.inverse_of_rightInverse Q hfg hgh) ((rTensor Q g) y) =\n      Submodule.Quotient.mk (p := LinearMap.range (rTensor Q f)) y", "@[simp] lemma range_two_mul_add_one (\u03b1 : Type*) [Semiring \u03b1] :\n    Set.range (fun x : \u03b1 \u21a6 2 * x + 1) = {a | Odd a}", "theorem rank_eq_one_iff [Module.Free K V] :\n    Module.rank K V = 1 \u2194 \u2203 v\u2080 : V, v\u2080 \u2260 0 \u2227 \u2200 v, \u2203 r : K, r \u2022 v\u2080 = v", "@[simp]\ntheorem rank_prod : Module.rank R (M \u00d7 M') =\n    Cardinal.lift.{v'} (Module.rank R M) + Cardinal.lift.{v, v'} (Module.rank R M')", "@[simp]\ntheorem rank_self : Module.rank R R = 1", "theorem rank_span_of_finset (s : Finset M) : Module.rank R (span R (s : Set M)) < \u2135\u2080", "theorem rank_submodule_le_one_iff (s : Submodule K V) [Module.Free K s] :\n    Module.rank K s \u2264 1 \u2194 \u2203 v\u2080 \u2208 s, s \u2264 K \u2219 v\u2080", "theorem rank_tensorProduct' :\n    Module.rank R (M \u2297[S] M\u2081) = Module.rank R M * Module.rank S M\u2081", "theorem realPart_add_I_smul_imaginaryPart (a : A) : (\u211c a : A) + I \u2022 (\u2111 a : A) = a", "theorem real_inner_add_add_self (x y : F) :\n    \u27eax + y, x + y\u27eb_\u211d = \u27eax, x\u27eb_\u211d + 2 * \u27eax, y\u27eb_\u211d + \u27eay, y\u27eb_\u211d", "theorem rec_heq_of_heq {C : \u03b1 \u2192 Sort*} {x : C a} {y : \u03b2} (e : a = b) (h : HEq x y) :\n    HEq (e \u25b8 x) y", "theorem rec_subsingleton {p : Prop} [h : Decidable p] {h\u2081 : p \u2192 Sort u} {h\u2082 : \u00acp \u2192 Sort u}\n    [h\u2083 : \u2200 h : p, Subsingleton (h\u2081 h)] [h\u2084 : \u2200 h : \u00acp, Subsingleton (h\u2082 h)] :\n    Subsingleton (Decidable.recOn h h\u2082 h\u2081)", "theorem reflTransGen_of_pred_of_ge (r : \u03b1 \u2192 \u03b1 \u2192 Prop) {n m : \u03b1} (h : \u2200 i \u2208 Ioc m n, r i (pred i))\n    (hnm : m \u2264 n) : ReflTransGen r n m", "@[simp]\ntheorem reflection_bot : reflection (\u22a5 : Submodule \ud835\udd5c E) = LinearIsometryEquiv.neg \ud835\udd5c", "@[to_additive \"If a function is countably sub-additive then it is sub-additive on countable types\"]\ntheorem rel_iSup_tprod [CompleteLattice \u03b1] (m : \u03b1 \u2192 M) (m0 : m \u22a5 = 1) (R : M \u2192 M \u2192 Prop)\n    (m_iSup : \u2200 s : \u2115 \u2192 \u03b1, R (m (\u2a06 i, s i)) (\u220f' i, m (s i))) (s : \u03b2 \u2192 \u03b1) :\n    R (m (\u2a06 b : \u03b2, s b)) (\u220f' b : \u03b2, m (s b))", "lemma rescale_to_shell_semi_normed {c : \ud835\udd5c} (hc : 1 < \u2016c\u2016) {\u03b5 : \u211d} (\u03b5pos : 0 < \u03b5)\n    {x : E} (hx : \u2016x\u2016 \u2260 0) :\n    \u2203d:\ud835\udd5c, d \u2260 0 \u2227 \u2016d \u2022 x\u2016 < \u03b5 \u2227 (\u03b5/\u2016c\u2016 \u2264 \u2016d \u2022 x\u2016) \u2227 (\u2016d\u2016\u207b\u00b9 \u2264 \u03b5\u207b\u00b9 * \u2016c\u2016 * \u2016x\u2016)", "lemma riemannZeta_def_of_ne_zero {s : \u2102} (hs : s \u2260 0) :\n    riemannZeta s = completedRiemannZeta s / Gamma\u211d s", "theorem riemannZeta_eulerProduct (hs : 1 < s.re) :\n    Tendsto (fun n : \u2115 \u21a6 \u220f p \u2208 primesBelow n, (1 - (p : \u2102) ^ (-s))\u207b\u00b9) atTop\n      (\ud835\udcdd (riemannZeta s))", "lemma _root_.riemannZeta_one_ne_zero : riemannZeta 1 \u2260 0", "theorem riesz_lemma_of_norm_lt {c : \ud835\udd5c} (hc : 1 < \u2016c\u2016) {R : \u211d} (hR : \u2016c\u2016 < R) {F : Subspace \ud835\udd5c E}\n    (hFc : IsClosed (F : Set E)) (hF : \u2203 x : E, x \u2209 F) :\n    \u2203 x\u2080 : E, \u2016x\u2080\u2016 \u2264 R \u2227 \u2200 y \u2208 F, 1 \u2264 \u2016x\u2080 - y\u2016", "@[simp]\ntheorem right_eq_midpoint_iff {x y : P} : y = midpoint R x y \u2194 x = y", "theorem ringExpChar.eq (q : \u2115) [h : ExpChar R q] : ringExpChar R = q", "lemma root_X_pow_sub_C_eq_zero_iff {n : \u2115} {a : K} (H : Irreducible (X ^ n - C a)) :\n    (AdjoinRoot.root (X ^ n - C a)) = 0 \u2194 a = 0", "theorem rootsOfUnity.integer_power_of_ringEquiv (g : L \u2243+* L) :\n    \u2203 m : \u2124, \u2200 t : rootsOfUnity n L, g (t : L\u02e3) = (t ^ m : L\u02e3)", "theorem roots_mem_integralClosure {f : R[X]} (hf : f.Monic) {a : S}\n    (ha : a \u2208 f.aroots S) : a \u2208 integralClosure R S", "@[simp]\ntheorem rotation_apply (a : circle) (z : \u2102) : rotation a z = a * z", "@[simp]\ntheorem round_add_one (a : \u03b1) : round (a + 1) = round a + 1", "@[simp]\ntheorem round_eq_zero_iff {x : \u03b1} : round x = 0 \u2194 x \u2208 Ico (-(1 / 2)) ((1 : \u03b1) / 2)", "@[simp]\ntheorem round_nat_add (x : \u03b1) (y : \u2115) : round ((y : \u03b1) + x) = y + round x", "@[simp]\ntheorem round_two_inv : round (2\u207b\u00b9 : \u03b1) = 1", "@[simp]\ntheorem coe_comp (f : sInfHom \u03b2 \u03b3) (g : sInfHom \u03b1 \u03b2) : \u21d1(f.comp g) = f \u2218 g", "@[simp]\ntheorem comp_id (f : sInfHom \u03b1 \u03b2) : f.comp (sInfHom.id \u03b1) = f", "@[simp]\ntheorem id_comp (f : sInfHom \u03b1 \u03b2) : (sInfHom.id \u03b2).comp f = f", "theorem sInf_image {s : Set \u03b2} {f : \u03b2 \u2192 \u03b1} : sInf (f '' s) = \u2a05 a \u2208 s, f a", "theorem sInf_le : a \u2208 s \u2192 sInf s \u2264 a", "lemma sInf_mem_of_not_isPredLimit (hlim : \u00ac IsPredLimit (sInf s)) :\n    sInf s \u2208 s", "@[simp]\ntheorem symm_dual_comp (g : sInfHom \u03b2\u1d52\u1d48 \u03b3\u1d52\u1d48) (f : sInfHom \u03b1\u1d52\u1d48 \u03b2\u1d52\u1d48) :\n    sSupHom.dual.symm (g.comp f) = (sSupHom.dual.symm g).comp (sSupHom.dual.symm f)", "theorem sSup_image2_eq_sSup_sSup (h\u2081 : \u2200 b, GaloisConnection (swap l b) (u\u2081 b))\n    (h\u2082 : \u2200 a, GaloisConnection (l a) (u\u2082 a)) : sSup (image2 l s t) = l (sSup s) (sSup t)", "theorem sameRay_iff_of_norm_eq (h : \u2016x\u2016 = \u2016y\u2016) : SameRay \u211d x y \u2194 x = y", "theorem sameRay_neg_smul_left_iff_of_ne {v : M} {r : R} (hv : v \u2260 0) (hr : r \u2260 0) :\n    SameRay R (r \u2022 v) (-v) \u2194 r < 0", "theorem sbtw_of_btw_not_btw {a b c : \u03b1} (habc : btw a b c) (hcba : \u00acbtw c b a) : sbtw a b c", "theorem sbtw_pointReflection_of_ne {x y : P} (h : x \u2260 y) : Sbtw R y x (pointReflection R x y)", "lemma schnirelmannDensity_le_of_not_mem {k : \u2115} (hk : k \u2209 A) :\n    schnirelmannDensity A \u2264 1 - (k\u207b\u00b9 : \u211d)", "lemma schnirelmannDensity_setOf_modeq_one {m : \u2115} :\n    schnirelmannDensity {n | n \u2261 1 [MOD m]} = (m\u207b\u00b9 : \u211d)", "@[simp] lemma sdiff_eq_left : x \\ y = x \u2194 Disjoint x y", "@[simp] lemma sdiff_eq_right : x \\ y = y \u2194 x = \u22a5 \u2227 y = \u22a5", "@[simp]\ntheorem sdiff_inf_self_left (a b : \u03b1) : a \\ (a \u2293 b) = a \\ b", "theorem sdiff_le_comm : a \\ b \u2264 c \u2194 a \\ c \u2264 b", "theorem sdiff_right_comm (a b c : \u03b1) : (a \\ b) \\ c = (a \\ c) \\ b", "@[simp]\ntheorem sdiff_self : a \\ a = \u22a5", "theorem segment_eq_image (x y : E) :\n    [x -[\ud835\udd5c] y] = (fun \u03b8 : \ud835\udd5c => (1 - \u03b8) \u2022 x + \u03b8 \u2022 y) '' Icc (0 : \ud835\udd5c) 1", "theorem segment_eq_image_lineMap (x y : E) : [x -[\ud835\udd5c] y] =\n    AffineMap.lineMap x y '' Icc (0 : \ud835\udd5c) 1", "theorem segment_subset_iff :\n    [x -[\ud835\udd5c] y] \u2286 s \u2194 \u2200 a b : \ud835\udd5c, 0 \u2264 a \u2192 0 \u2264 b \u2192 a + b = 1 \u2192 a \u2022 x + b \u2022 y \u2208 s", "theorem segment_symm (x y : E) : [x -[\ud835\udd5c] y] = [y -[\ud835\udd5c] x]", "@[simp, norm_cast]\ntheorem val_pow (x : selfAdjoint R) (n : \u2115) : \u2191(x ^ n) = (x : R) ^ n", "@[simp, norm_cast] lemma val_qsmul (q : \u211a) (x : selfAdjoint R) : \u2191(q \u2022 x) = q \u2022 (x : R) := rfl", "@[simp, norm_cast] lemma val_ratCast (q : \u211a) : (q : selfAdjoint R) = (q : R) := rfl", "theorem selfZPow_of_neg {n : \u2124} (hn : n < 0) :\n    selfZPow x B n = mk' _ (1 : R) (Submonoid.pow x n.natAbs)", "@[simp]\ntheorem semiconjBy_star_star_star {x y z : R} :\n    SemiconjBy (star x) (star z) (star y) \u2194 SemiconjBy x y z", "theorem adjoin_eq_of_isAlgebraic [Algebra.IsAlgebraic F E] :\n    adjoin E (separableClosure F K) = separableClosure E K", "lemma adjoin_eq_of_isAlgebraic_of_isSeparable [Algebra.IsAlgebraic F E]\n    [IsSeparable E K] : adjoin E (separableClosure F K : Set K) = \u22a4 := top_unique fun x _ \u21a6 by\n  set S := separableClosure F K\n  set L := adjoin E (S : Set K)\n  have := isSeparable_tower_top_of_isSeparable E L K\n  let i : S \u2192+* L := Subsemiring.inclusion fun x hx \u21a6 subset_adjoin E (S : Set K) hx\n  let _ : Algebra S L := i.toAlgebra\n  let _ : SMul S L := Algebra.toSMul\n  have : IsScalarTower S L K := IsScalarTower.of_algebraMap_eq (congrFun rfl)\n  have : Algebra.IsAlgebraic F K := Algebra.IsAlgebraic.trans (L := E)\n  have : IsPurelyInseparable S K := separableClosure.isPurelyInseparable F K\n  have := IsPurelyInseparable.tower_top S L K\n  obtain \u27e8y, rfl\u27e9 := IsPurelyInseparable.surjective_algebraMap_of_isSeparable L K x\n  exact y.2", "theorem seqClosure_eq_closure [FrechetUrysohnSpace X] (s : Set X) : seqClosure s = closure s", "theorem set_smul_mem_nhds_smul {c : G\u2080} {s : Set \u03b1} {x : \u03b1} (hs : s \u2208 \ud835\udcdd x) (hc : c \u2260 0) :\n    c \u2022 s \u2208 \ud835\udcdd (c \u2022 x : \u03b1)", "theorem set_smul_mem_nhds_zero_iff {s : Set \u03b1} {c : G\u2080} (hc : c \u2260 0) :\n    c \u2022 s \u2208 \ud835\udcdd (0 : \u03b1) \u2194 s \u2208 \ud835\udcdd (0 : \u03b1)", "@[simp]\ntheorem sigmaFinsuppEquivDFinsupp_smul {R} [Monoid R] [AddMonoid N] [DistribMulAction R N] (r : R)\n    (f : (\u03a3 i, \u03b7 i) \u2192\u2080 N) :\n    sigmaFinsuppEquivDFinsupp (r \u2022 f) = r \u2022 sigmaFinsuppEquivDFinsupp f", "theorem sign_eq_neg_one_iff : sign a = -1 \u2194 a < 0", "@[simp]\ntheorem sign_finRotate (n : \u2115) : Perm.sign (finRotate (n + 1)) = (-1) ^ n", "theorem simple_iff_isSimpleModule' (M : ModuleCat R) : Simple M \u2194 IsSimpleModule R M", "@[to_additive]\ntheorem single_le_finprod {M : Type*} [OrderedCommMonoid M] (i : \u03b1) {f : \u03b1 \u2192 M}\n    (hf : (mulSupport f).Finite) (h : \u2200 j, 1 \u2264 f j) : f i \u2264 \u220f\u1da0 j, f j", "theorem singleton_mem_nhdsWithin_of_mem_discrete {s : Set X} [DiscreteTopology s] {x : X}\n    (hx : x \u2208 s) : {x} \u2208 \ud835\udcdd[s] x", "@[to_additive (attr := simp 1100)]\ntheorem singleton_mul_closedBall : {x} * closedBall y \u03b4 = closedBall (x * y) \u03b4", "theorem singletons_open_iff_discrete {X : Type*} [TopologicalSpace X] :\n    (\u2200 a : X, IsOpen ({a} : Set X)) \u2194 DiscreteTopology X", "@[aesop safe apply (rule_sets := [SetLike])]\ntheorem smul_mem [Monoid R] [DistribMulAction R A] [StarModule R A] (r : R) {x : A}\n    (h : x \u2208 skewAdjoint A) : r \u2022 x \u2208 skewAdjoint A", "theorem skewAdjointMatricesLieSubalgebraEquiv_apply (P : Matrix n n R) (h : Invertible P)\n    (A : skewAdjointMatricesLieSubalgebra J) :\n    \u2191(skewAdjointMatricesLieSubalgebraEquiv J P h A) = P\u207b\u00b9 * (A : Matrix n n R) * P", "theorem small_congr {\u03b1 : Type*} {\u03b2 : Type*} (e : \u03b1 \u2243 \u03b2) : Small.{w} \u03b1 \u2194 Small.{w} \u03b2", "theorem small_of_surjective {\u03b1 : Type v} {\u03b2 : Type w} [Small.{u} \u03b1] {f : \u03b1 \u2192 \u03b2}\n    (hf : Function.Surjective f) : Small.{u} \u03b2", "theorem smoothAt_fst {p : M \u00d7 N} : SmoothAt (I.prod J) I Prod.fst p", "theorem smoothAt_hom_bundle (f : M \u2192 LE\u2081E\u2082) {x\u2080 : M} :\n    SmoothAt IM (IB.prod \ud835\udcd8(\ud835\udd5c, F\u2081 \u2192L[\ud835\udd5c] F\u2082)) f x\u2080 \u2194\n      SmoothAt IM IB (fun x => (f x).1) x\u2080 \u2227\n        SmoothAt IM \ud835\udcd8(\ud835\udd5c, F\u2081 \u2192L[\ud835\udd5c] F\u2082)\n          (fun x => inCoordinates F\u2081 E\u2081 F\u2082 E\u2082 (f x\u2080).1 (f x).1 (f x\u2080).1 (f x).1 (f x).2) x\u2080", "theorem smoothWithinAt_pi_space :\n    SmoothWithinAt I \ud835\udcd8(\ud835\udd5c, \u2200 i, Fi i) \u03c6 s x \u2194\n      \u2200 i, SmoothWithinAt I \ud835\udcd8(\ud835\udd5c, Fi i) (fun x => \u03c6 x i) s x", "@[to_additive]\ntheorem smooth_mul : Smooth (I.prod I) I fun p : G \u00d7 G => p.1 * p.2", "@[simp]\ntheorem smulAddHom_apply (r : R) (x : M) : smulAddHom R M r x = r \u2022 x", "lemma smul_add_smul_lt_smul_add_smul [PosSMulStrictMono \u03b1 \u03b2]\n    {b\u2081 b\u2082 : \u03b1} {a d : \u03b2} (hab : b\u2081 < b\u2082) (hcd : a < d) :\n    b\u2081 \u2022 d + b\u2082 \u2022 a < b\u2081 \u2022 a + b\u2082 \u2022 d", "lemma smul_eq_zero_of_left (h : a = 0) (b : M) : a \u2022 b = 0 := h.symm \u25b8 zero_smul _ b", "lemma smul_lt_of_lt_one_left [SMulPosStrictMono \u03b1 \u03b2] (hb : 0 < b) (h : a < 1) : a \u2022 b < b", "lemma smul_lt_smul' [PosSMulStrictMono \u03b1 \u03b2] [SMulPosStrictMono \u03b1 \u03b2] (ha : a\u2081 < a\u2082) (hb : b\u2081 < b\u2082)\n    (h\u2082 : 0 < a\u2082) (h\u2081 : 0 < b\u2081) : a\u2081 \u2022 b\u2081 < a\u2082 \u2022 b\u2082", "@[simp]\nlemma smul_lt_smul_iff_of_pos_left [PosSMulStrictMono \u03b1 \u03b2] [PosSMulReflectLT \u03b1 \u03b2] (ha : 0 < a) :\n    a \u2022 b\u2081 < a \u2022 b\u2082 \u2194 b\u2081 < b\u2082", "lemma smul_max_of_nonpos (ha : a \u2264 0) (b\u2081 b\u2082 : \u03b2) : a \u2022 max b\u2081 b\u2082 = min (a \u2022 b\u2081) (a \u2022 b\u2082)", "@[to_additive] lemma smul_mul_smul [Mul \u03b1] (r s : M) (x y : \u03b1) [IsScalarTower M \u03b1 \u03b1] [SMulCommClass M \u03b1 \u03b1] :\n    r \u2022 x * s \u2022 y = (r * s) \u2022 (x * y)", "theorem smul_ne_zero_iff_ne (a : \u03b1) {x : \u03b2} : a \u2022 x \u2260 0 \u2194 x \u2260 0", "@[simp]\ntheorem smul_neg (r : M) (x : A) : r \u2022 -x = -(r \u2022 x)", "lemma smul_neg_iff_of_pos_left [PosSMulStrictMono \u03b1 \u03b2] [PosSMulReflectLT \u03b1 \u03b2] (ha : 0 < a) :\n    a \u2022 b < 0 \u2194 b < 0", "lemma smul_nonpos_iff_neg_imp_nonneg : a \u2022 b \u2264 0 \u2194 (a < 0 \u2192 0 \u2264 b) \u2227 (0 < b \u2192 a \u2264 0)", "lemma smul_nonpos_of_nonpos_of_nonneg [SMulPosMono \u03b1 \u03b2] (ha : a \u2264 0) (hb : 0 \u2264 b) : a \u2022 b \u2264 0", "lemma smul_pow (r : M) (x : N) : \u2200 n, (r \u2022 x) ^ n = r ^ n \u2022 x ^ n\n  | 0 => by simp\n  | n + 1 => by rw [pow_succ', smul_pow _ _ n, smul_mul_smul, \u2190 pow_succ', \u2190 pow_succ']", "@[to_additive \"Consider a sigma-compact additive group acting continuously and transitively on a\nBaire space. Then the orbit map is open around zero. It follows in\n`isOpenMap_vadd_of_sigmaCompact` that it is open around any point.\"]\ntheorem smul_singleton_mem_nhds_of_sigmaCompact\n    {U : Set G} (hU : U \u2208 \ud835\udcdd 1) (x : X) : U \u2022 {x} \u2208 \ud835\udcdd x", "@[simp]\ntheorem snd_himp [HImp \u03b1] [HImp \u03b2] (a b : \u03b1 \u00d7 \u03b2) : (a \u21e8 b).2 = a.2 \u21e8 b.2", "lemma span_eq_iUnion_nat (s : Set M) :\n    (Submodule.span R s : Set M) = \u22c3 (n : \u2115),\n      (fun (f : Fin n \u2192 (R \u00d7 M)) \u21a6 \u2211 i, (f i).1 \u2022 (f i).2) '' ({f | \u2200 i, (f i).2 \u2208 s})", "theorem span_eq_top_localization_localization {v : Set A} (hv : span R v = \u22a4) :\n    span R\u209b (algebraMap A A\u209b '' v) = \u22a4", "theorem span_le_span_iff [Nontrivial R] {s t u : Set M} (hl : LinearIndependent R ((\u2191) : u \u2192 M))\n    (hsu : s \u2286 u) (htu : t \u2286 u) : span R s \u2264 span R t \u2194 s \u2286 t", "lemma spec_cfc\u2099Aux (f : C(\u03c3\u2099 \ud835\udd5c a, \ud835\udd5c)\u2080) : \u03c3 \ud835\udd5c (cfc\u2099Aux hp\u2081 a ha f) = Set.range f", "theorem spectrum.gelfandTransform_eq (a : A) :\n    spectrum \u2102 (gelfandTransform \u2102 A a) = spectrum \u2102 a", "theorem hasFPowerSeriesOnBall_inverse_one_sub_smul [CompleteSpace A] (a : A) :\n    HasFPowerSeriesOnBall (fun z : \ud835\udd5c => Ring.inverse (1 - z \u2022 a))\n      (fun n => ContinuousMultilinearMap.mkPiRing \ud835\udd5c (Fin n) (a ^ n)) 0 \u2016a\u2016\u208a\u207b\u00b9", "theorem inv_mem_resolventSet {r : R\u02e3} {a : A\u02e3} (h : (r : R) \u2208 resolventSet R (a : A)) :\n    (\u2191r\u207b\u00b9 : R) \u2208 resolventSet R (\u2191a\u207b\u00b9 : A)", "theorem limsup_pow_nnnorm_pow_one_div_le_spectralRadius (a : A) :\n    limsup (fun n : \u2115 => (\u2016a ^ n\u2016\u208a : \u211d\u22650\u221e) ^ (1 / n : \u211d)) atTop \u2264 spectralRadius \u2102 a", "protected theorem nonempty : (spectrum \u2102 a).Nonempty", "theorem pow_image_subset (a : A) (n : \u2115) : (fun x => x ^ n) '' \u03c3 a \u2286 \u03c3 (a ^ n)", "@[simp]\nlemma spectrum.units_conjugate' {a : A} {u : A\u02e3} :\n    spectrum R (u\u207b\u00b9 * a * u) = spectrum R a", "theorem zero_not_mem_iff {a : A} : (0 : R) \u2209 \u03c3 a \u2194 IsUnit a", "theorem sphere_ext_iff (u v : sphere (0 : E) 1) : u = v \u2194 \u27ea(u : E), v\u27eb_\u211d = 1", "lemma sphere_prod (x : \u03b1 \u00d7 \u03b2) (r : \u211d) :\n    sphere x r = sphere x.1 r \u00d7\u02e2 closedBall x.2 r \u222a closedBall x.1 r \u00d7\u02e2 sphere x.2 r", "@[simp]\ntheorem sq_eq_sq {a b : R} (ha : 0 \u2264 a) (hb : 0 \u2264 b) : a ^ 2 = b ^ 2 \u2194 a = b", "@[simp] lemma sq_le_one_iff_abs_le_one (a : \u03b1) : a ^ 2 \u2264 1 \u2194 |a| \u2264 1", "theorem sqrt_one_add_norm_sq_le (x : E) : \u221a((1 : \u211d) + \u2016x\u2016 ^ 2) \u2264 1 + \u2016x\u2016", "@[to_additive \"Special case of the sandwich theorem: if the norm of `f` is bounded by a real\nfunction `a` which tends to `0`, then `f` tends to `0`.\"]\ntheorem squeeze_one_norm {f : \u03b1 \u2192 E} {a : \u03b1 \u2192 \u211d} {t\u2080 : Filter \u03b1} (h : \u2200 n, \u2016f n\u2016 \u2264 a n) :\n    Tendsto a t\u2080 (\ud835\udcdd 0) \u2192 Tendsto f t\u2080 (\ud835\udcdd 1)", "theorem starConvex_pi {\u03b9 : Type*} {E : \u03b9 \u2192 Type*} [\u2200 i, AddCommMonoid (E i)] [\u2200 i, SMul \ud835\udd5c (E i)]\n    {x : \u2200 i, E i} {s : Set \u03b9} {t : \u2200 i, Set (E i)} (ht : \u2200 \u2983i\u2984, i \u2208 s \u2192 StarConvex \ud835\udd5c (x i) (t i)) :\n    StarConvex \ud835\udd5c x (s.pi t)", "theorem starConvex_sInter {S : Set (Set E)} (h : \u2200 s \u2208 S, StarConvex \ud835\udd5c x s) :\n    StarConvex \ud835\udd5c x (\u22c2\u2080 S)", "theorem starRingEnd_self_apply (x : R) : starRingEnd R (starRingEnd R x) = x", "@[simp]\ntheorem star_intCast_smul [Ring R] [AddCommGroup M] [Module R M] [StarAddMonoid M] (n : \u2124)\n    (x : M) : star ((n : R) \u2022 x) = (n : R) \u2022 star x", "@[simp]\nlemma star_le_star_iff {x y : R} : star x \u2264 star y \u2194 x \u2264 y", "@[nontriviality] lemma stdSimplex_of_subsingleton [Subsingleton \ud835\udd5c] : stdSimplex \ud835\udd5c \u03b9 = univ", "@[simp]\ntheorem stereographic'_source {n : \u2115} [Fact (finrank \u211d E = n + 1)] (v : sphere (0 : E) 1) :\n    (stereographic' n v).source = {v}\u1d9c", "theorem strictAntiOn_Iic_of_succ_lt [SuccOrder \u03b1] [IsSuccArchimedean \u03b1] {n : \u03b1}\n    (h\u03c8 : \u2200 m, m < n \u2192 \u03c8 (succ m) < \u03c8 m) : StrictAntiOn \u03c8 (Set.Iic n)", "theorem strictAnti_nat_of_succ_lt {f : \u2115 \u2192 \u03b1} (hf : \u2200 n, f (n + 1) < f n) : StrictAnti f", "theorem strictAnti_of_deriv_neg {f : \u211d \u2192 \u211d} (hf' : \u2200 x, deriv f x < 0) : StrictAnti f", "@[simp]\ntheorem strictAnti_vecEmpty : StrictAnti (vecCons a vecEmpty)", "theorem strictConcaveOn_of_slope_strict_anti_adjacent (hs : Convex \ud835\udd5c s)\n    (hf :\n      \u2200 {x y z : \ud835\udd5c},\n        x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f z - f y) / (z - y) < (f y - f x) / (y - x)) :\n    StrictConcaveOn \ud835\udd5c s f", "theorem strictConvexOn_of_slope_strict_mono_adjacent (hs : Convex \ud835\udd5c s)\n    (hf :\n      \u2200 {x y z : \ud835\udd5c},\n        x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) < (f z - f y) / (z - y)) :\n    StrictConvexOn \ud835\udd5c s f", "theorem strictMono_of_odd_strictMonoOn_nonneg {f : G \u2192 H} (h\u2081 : \u2200 x, f (-x) = -f x)\n    (h\u2082 : StrictMonoOn f (Ici 0)) : StrictMono f", "theorem stronglyMeasurable_deriv_with_param [LocallyCompactSpace \ud835\udd5c] [MeasurableSpace \ud835\udd5c]\n    [OpensMeasurableSpace \ud835\udd5c] [h : SecondCountableTopologyEither \u03b1 F]\n    {f : \u03b1 \u2192 \ud835\udd5c \u2192 F} (hf : Continuous f.uncurry) :\n    StronglyMeasurable (fun (p : \u03b1 \u00d7 \ud835\udd5c) \u21a6 deriv (f p.1) p.2)", "@[field_simps]\ntheorem sub_div' (a b c : K) (hc : c \u2260 0) : b - a / c = (b * c - a) / c", "theorem sub_lt_of_abs_sub_lt_left (h : |a - b| < c) : b - c < a", "theorem sub_pow_expChar_pow_of_commute [Ring R] {q : \u2115} [hR : ExpChar R q]\n    {n : \u2115} (x y : R) (h : Commute x y) : (x - y) ^ q ^ n = x ^ q ^ n - y ^ q ^ n", "theorem sub_self_div_two (a : \u03b1) : a - a / 2 = a / 2", "theorem sub_smul_slope_vadd (f : k \u2192 PE) (a b : k) : (b - a) \u2022 slope f a b +\u1d65 f a = f b", "theorem subalgebra_top_finrank_eq_submodule_top_finrank :\n    finrank F (\u22a4 : Subalgebra F E) = finrank F (\u22a4 : Submodule F E)", "theorem subset_convexJoin_right (h : s.Nonempty) : t \u2286 convexJoin \ud835\udd5c s t", "theorem subset_sInf_of_within [Inhabited s] {t : Set s}\n    (h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((\u2191) '' t : Set \u03b1) \u2208 s) :\n    sInf ((\u2191) '' t : Set \u03b1) = (@sInf s _ t : \u03b1)", "theorem suffixLevenshtein_length (xs : List \u03b1) (ys : List \u03b2) :\n    (suffixLevenshtein C xs ys).1.length = xs.length + 1", "theorem sum_div_card_sq_le_sum_sq_div_card :\n    ((\u2211 i \u2208 s, f i) / s.card) ^ 2 \u2264 (\u2211 i \u2208 s, f i ^ 2) / s.card", "theorem summable_iff_cauchySeq_finset_and_tsum_mem (f : \u03b2 \u2192 \u03b1) :\n    Summable f \u2194 CauchySeq (fun s : Finset \u03b2 \u21a6 \u2211 b in s, f b) \u2227\n      \u2211' i, toCompl (f i) \u2208 Set.range toCompl", "lemma summable_jacobiTheta\u2082_term_fderiv_iff (z \u03c4 : \u2102) :\n    Summable (jacobiTheta\u2082_term_fderiv \u00b7 z \u03c4) \u2194 0 < im \u03c4", "lemma summable_jacobiTheta\u2082_term_iff (z \u03c4 : \u2102) : Summable (jacobiTheta\u2082_term \u00b7 z \u03c4) \u2194 0 < im \u03c4", "theorem summable_matrix_blockDiagonal [DecidableEq p] {f : X \u2192 p \u2192 Matrix m n R} :\n    (Summable fun x => blockDiagonal (f x)) \u2194 Summable f", "theorem summable_of_summable_hasFDerivAt (hu : Summable u)\n    (hf : \u2200 n x, HasFDerivAt (f n) (f' n x) x) (hf' : \u2200 n x, \u2016f' n x\u2016 \u2264 u n)\n    (hf0 : Summable fun n => f n x\u2080) (x : E) : Summable fun n => f n x", "theorem sup_comm (a b : \u03b1) : a \u2294 b = b \u2294 a", "@[simp]\ntheorem sup_eq_left : a \u2294 b = a \u2194 b \u2264 a", "@[simp]\ntheorem sup_eq_right : a \u2294 b = b \u2194 a \u2264 b", "theorem sup_eq_sdiff_sup_sdiff_sup_inf : x \u2294 y = x \\ y \u2294 y \\ x \u2294 x \u2293 y", "@[simp]\ntheorem sup_himp_self_left (a b : \u03b1) : a \u2294 b \u21e8 a = b \u21e8 a", "theorem sup_le : a \u2264 c \u2192 b \u2264 c \u2192 a \u2294 b \u2264 c", "@[gcongr]\ntheorem sup_le_sup_right (h\u2081 : a \u2264 b) (c) : a \u2294 c \u2264 b \u2294 c", "theorem sup_right_idem (a b : \u03b1) : a \u2294 b \u2294 b = a \u2294 b", "theorem sup_sdiff_injOn [GeneralizedBooleanAlgebra \u03b1] (u v : \u03b1) :\n    { x | Disjoint u x \u2227 v \u2264 x }.InjOn fun x => (x \u2294 u) \\ v", "theorem surjective_algebraMap_of_linearMap (hb : Function.Surjective b) :\n    Function.Surjective (algebraMap F E)", "theorem symmDiff_comm : a \u2206 b = b \u2206 a", "theorem symmDiff_le_iff {a b c : \u03b1} : a \u2206 b \u2264 c \u2194 a \u2264 b \u2294 c \u2227 b \u2264 a \u2294 c", "@[simp]\ntheorem symmDiff_left_inj : a \u2206 b = c \u2206 b \u2194 a = c", "theorem symmDiff_right_injective (a : \u03b1) : Injective (a \u2206 \u00b7)", "theorem symmDiff_right_surjective (a : \u03b1) : Surjective (a \u2206 \u00b7)", "@[simp]\ntheorem symmDiff_sdiff_eq_sup : a \u2206 (b \\ a) = a \u2294 b", "@[simp]\ntheorem symmDiff_sdiff_inf : a \u2206 b \\ (a \u2293 b) = a \u2206 b", "@[simp]\ntheorem symmDiff_self : a \u2206 a = \u22a5", "theorem symmDiff_symmDiff_left :\n    a \u2206 b \u2206 c = a \\ (b \u2294 c) \u2294 b \\ (a \u2294 c) \u2294 c \\ (a \u2294 b) \u2294 a \u2293 b \u2293 c", "theorem t0Space_iff_uniformity' :\n    T0Space \u03b1 \u2194 Pairwise fun x y \u21a6 \u2203 r \u2208 \ud835\udce4 \u03b1, (x, y) \u2209 r", "theorem t1Space_TFAE (X : Type u) [TopologicalSpace X] :\n    List.TFAE [T1Space X,\n      \u2200 x, IsClosed ({ x } : Set X),\n      \u2200 x, IsOpen ({ x }\u1d9c : Set X),\n      Continuous (@CofiniteTopology.of X),\n      \u2200 \u2983x y : X\u2984, x \u2260 y \u2192 {y}\u1d9c \u2208 \ud835\udcdd x,\n      \u2200 \u2983x y : X\u2984, x \u2260 y \u2192 \u2203 s \u2208 \ud835\udcdd x, y \u2209 s,\n      \u2200 \u2983x y : X\u2984, x \u2260 y \u2192 \u2203 U : Set X, IsOpen U \u2227 x \u2208 U \u2227 y \u2209 U,\n      \u2200 \u2983x y : X\u2984, x \u2260 y \u2192 Disjoint (\ud835\udcdd x) (pure y),\n      \u2200 \u2983x y : X\u2984, x \u2260 y \u2192 Disjoint (pure x) (\ud835\udcdd y),\n      \u2200 \u2983x y : X\u2984, x \u2933 y \u2192 x = y]", "lemma unique_lift {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] [T2Space Y]\n    {f : X \u2192 Y} (hf : Continuous f) {g : t2Quotient X \u2192 Y} (hfg : g \u2218 mk = f) :\n    g = lift hf", "@[simp, mfld_simps]\ntheorem tangentBundleModelSpaceHomeomorph_coe :\n    (tangentBundleModelSpaceHomeomorph H I : TangentBundle I H \u2192 ModelProd H E) =\n      TotalSpace.toProd H E", "theorem tangentConeAt.lim_zero {\u03b1 : Type*} (l : Filter \u03b1) {c : \u03b1 \u2192 \ud835\udd5c} {d : \u03b1 \u2192 E}\n    (hc : Tendsto (fun n => \u2016c n\u2016) l atTop) (hd : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd y)) :\n    Tendsto d l (\ud835\udcdd 0)", "theorem tangentMapWithin_congr (h : \u2200 x \u2208 s, f x = f\u2081 x) (p : TangentBundle I M) (hp : p.1 \u2208 s)\n    (hs : UniqueMDiffWithinAt I s p.1) :\n    tangentMapWithin I I' f s p = tangentMapWithin I I' f\u2081 s p", "@[simp]\ntheorem taylorWithinEval_self (f : \u211d \u2192 E) (n : \u2115) (s : Set \u211d) (x\u2080 : \u211d) :\n    taylorWithinEval f n s x\u2080 x\u2080 = f x\u2080", "@[simp]\ntheorem taylor_within_zero_eval (f : \u211d \u2192 E) (s : Set \u211d) (x\u2080 x : \u211d) :\n    taylorWithinEval f 0 s x\u2080 x = f x\u2080", "@[simp]\ntheorem tendsto_Ioo_atBot {f : \u03b2 \u2192 Ioo a b} :\n    Tendsto f l atBot \u2194 Tendsto (fun x => (f x : \u03b1)) l (\ud835\udcdd[>] a)", "theorem tendsto_add_one_pow_atTop_atTop_of_pos [LinearOrderedSemiring \u03b1] [Archimedean \u03b1] {r : \u03b1}\n    (h : 0 < r) : Tendsto (fun n : \u2115 \u21a6 (r + 1) ^ n) atTop atTop", "theorem tendsto_atBot_isGLB (h_mono : Monotone f) (ha : IsGLB (Set.range f) a) :\n    Tendsto f atBot (\ud835\udcdd a)", "theorem tendsto_atTop_ciInf (h_anti : Antitone f) (hbdd : BddBelow <| range f) :\n    Tendsto f atTop (\ud835\udcdd (\u2a05 i, f i))", "theorem tendsto_ceil_left' (n : \u2124) :\n    Tendsto (fun x => ceil x : \u03b1 \u2192 \u03b1) (\ud835\udcdd[\u2264] n) (\ud835\udcdd n)", "@[to_additive]\ntheorem tendsto_div_comap_self (x\u2080 : \u03b1) :\n    Tendsto (fun t : \u03b2 \u00d7 \u03b2 => t.2 / t.1) ((comap fun p : \u03b2 \u00d7 \u03b2 => (e p.1, e p.2)) <| \ud835\udcdd (x\u2080, x\u2080))\n      (\ud835\udcdd 1)", "theorem tendsto_floor_left' (n : \u2124) :\n    Tendsto (fun x => floor x : \u03b1 \u2192 \u03b1) (\ud835\udcdd[<] n) (\ud835\udcdd (n - 1))", "theorem tendsto_floor_left_pure_sub_one (n : \u2124) :\n    Tendsto (floor : \u03b1 \u2192 \u2124) (\ud835\udcdd[<] n) (pure (n - 1))", "theorem tendsto_floor_right' (n : \u2124) : Tendsto (fun x => floor x : \u03b1 \u2192 \u03b1) (\ud835\udcdd[\u2265] n) (\ud835\udcdd n)", "theorem tendsto_fract_left' [OrderClosedTopology \u03b1] [TopologicalAddGroup \u03b1] (n : \u2124) :\n    Tendsto (fract : \u03b1 \u2192 \u03b1) (\ud835\udcdd[<] n) (\ud835\udcdd 1)", "theorem tendsto_iff_forall_eval_tendsto_topDualPairing {l : Filter \u03b1} {f : \u03b1 \u2192 WeakDual \ud835\udd5c E}\n    {x : WeakDual \ud835\udd5c E} :\n    Tendsto f l (\ud835\udcdd x) \u2194\n      \u2200 y, Tendsto (fun i => topDualPairing \ud835\udd5c E (f i) y) l (\ud835\udcdd (topDualPairing \ud835\udd5c E x y))", "lemma tendsto_indicator_const_apply_iff_eventually' (b : \u03b2)\n    (nhd_b : {0}\u1d9c \u2208 \ud835\udcdd b) (nhd_o : {b}\u1d9c \u2208 \ud835\udcdd 0) (x : \u03b1) :\n    Tendsto (fun i \u21a6 (As i).indicator (fun (_ : \u03b1) \u21a6 b) x) L (\ud835\udcdd (A.indicator (fun (_ : \u03b1) \u21a6 b) x))\n      \u2194 \u2200\u1da0 i in L, (x \u2208 As i \u2194 x \u2208 A)", "@[to_additive]\ntheorem tendsto_multiset_prod {f : \u03b9 \u2192 \u03b1 \u2192 M} {x : Filter \u03b1} {a : \u03b9 \u2192 M} (s : Multiset \u03b9) :\n    (\u2200 i \u2208 s, Tendsto (f i) x (\ud835\udcdd (a i))) \u2192\n      Tendsto (fun b => (s.map fun c => f c b).prod) x (\ud835\udcdd (s.map a).prod)", "theorem tendsto_natCast_atTop_iff [StrictOrderedSemiring R] [Archimedean R] {f : \u03b1 \u2192 \u2115}\n    {l : Filter \u03b1} : Tendsto (fun n => (f n : R)) l atTop \u2194 Tendsto f l atTop", "theorem tendsto_nat_ceil_div_atTop : Tendsto (fun x \u21a6 (\u2308x\u2309\u208a : R) / x) atTop (\ud835\udcdd 1)", "theorem tendsto_nat_floor_mul_div_atTop {a : R} (ha : 0 \u2264 a) :\n    Tendsto (fun x \u21a6 (\u230aa * x\u230b\u208a : R) / x) atTop (\ud835\udcdd a)", "@[simp]\ntheorem tendsto_nhdsWithin_range {a : \u03b1} {l : Filter \u03b2} {f : \u03b2 \u2192 \u03b1} :\n    Tendsto f l (\ud835\udcdd[range f] a) \u2194 Tendsto f l (\ud835\udcdd a)", "theorem tendsto_nhds_bot_mono [TopologicalSpace \u03b2] [Preorder \u03b2] [OrderBot \u03b2] [OrderTopology \u03b2]\n    {l : Filter \u03b1} {f g : \u03b1 \u2192 \u03b2} (hf : Tendsto f l (\ud835\udcdd \u22a5)) (hg : g \u2264\u1da0[l] f) : Tendsto g l (\ud835\udcdd \u22a5)", "@[to_additive tendsto_norm_cobounded_atTop]\nlemma tendsto_norm_cobounded_atTop' : Tendsto norm (cobounded E) atTop", "theorem tendsto_of_monotone {\u03b9 \u03b1 : Type*} [Preorder \u03b9] [TopologicalSpace \u03b1]\n    [ConditionallyCompleteLinearOrder \u03b1] [OrderTopology \u03b1] {f : \u03b9 \u2192 \u03b1} (h_mono : Monotone f) :\n    Tendsto f atTop atTop \u2228 \u2203 l, Tendsto f atTop (\ud835\udcdd l)", "theorem tendsto_of_tendsto_of_tendsto_of_le_of_le' {f g h : \u03b2 \u2192 \u03b1} {b : Filter \u03b2} {a : \u03b1}\n    (hg : Tendsto g b (\ud835\udcdd a)) (hh : Tendsto h b (\ud835\udcdd a)) (hgf : \u2200\u1da0 b in b, g b \u2264 f b)\n    (hfh : \u2200\u1da0 b in b, f b \u2264 h b) : Tendsto f b (\ud835\udcdd a)", "theorem tendsto_pow_atTop_nhdsWithin_zero_of_lt_one {\ud835\udd5c : Type*} [LinearOrderedField \ud835\udd5c]\n    [Archimedean \ud835\udd5c] [TopologicalSpace \ud835\udd5c] [OrderTopology \ud835\udd5c] {r : \ud835\udd5c} (h\u2081 : 0 < r) (h\u2082 : r < 1) :\n    Tendsto (fun n : \u2115 \u21a6 r ^ n) atTop (\ud835\udcdd[>] 0)", "theorem tendsto_pow_const_mul_const_pow_of_abs_lt_one (k : \u2115) {r : \u211d} (hr : |r| < 1) :\n    Tendsto (fun n \u21a6 (n : \u211d) ^ k * r ^ n : \u2115 \u2192 \u211d) atTop (\ud835\udcdd 0)", "theorem tendsto_rpow_abs_mul_exp_neg_mul_sq_cocompact {a : \u211d} (ha : 0 < a) (s : \u211d) :\n    Tendsto (fun x : \u211d => |x| ^ s * rexp (-a * x ^ 2)) (cocompact \u211d) (\ud835\udcdd 0)", "private theorem tendsto_zero_of_norm_tendsto_zero : Tendsto (fun i => F.eval (ncs i)) atTop (\ud835\udcdd 0)", "theorem thickenedIndicatorAux_le_one (\u03b4 : \u211d) (E : Set \u03b1) (x : \u03b1) :\n    thickenedIndicatorAux \u03b4 E x \u2264 1", "theorem thickenedIndicatorAux_lt_top {\u03b4 : \u211d} {E : Set \u03b1} {x : \u03b1} :\n    thickenedIndicatorAux \u03b4 E x < \u221e", "theorem threeGPFree_smul_set\u2080 (ha : a \u2260 0) : ThreeGPFree (a \u2022 s) \u2194 ThreeGPFree s", "@[simp]\ntheorem toIcoDiv_add_right (a b : \u03b1) : toIcoDiv hp a (b + p) = toIcoDiv hp a b + 1", "@[simp]\ntheorem toIcoDiv_apply_left (a : \u03b1) : toIcoDiv hp a a = 0", "theorem toIcoDiv_apply_right (a : \u03b1) : toIcoDiv hp a (a + p) = 1", "theorem toIcoDiv_neg (a b : \u03b1) : toIcoDiv hp a (-b) = -(toIocDiv hp (-a) b + 1)", "theorem toIcoMod_eq_toIcoMod : toIcoMod hp a b = toIcoMod hp a c \u2194 \u2203 n : \u2124, c - b = n \u2022 p", "theorem toIcoMod_le_toIocMod (a b : \u03b1) : toIcoMod hp a b \u2264 toIocMod hp a b", "theorem toIcoMod_mem_Ico (a b : \u03b1) : toIcoMod hp a b \u2208 Set.Ico a (a + p)", "@[simp]\ntheorem toIocDiv_add_zsmul (a b : \u03b1) (m : \u2124) : toIocDiv hp a (b + m \u2022 p) = toIocDiv hp a b + m", "theorem toIocDiv_eq_neg_floor (a b : \u03b1) : toIocDiv hp a b = -\u230a(a + p - b) / p\u230b", "theorem toIocDiv_sub_eq_toIocDiv_add' (a b c : \u03b1) :\n    toIocDiv hp (a - c) b = toIocDiv hp a (b + c)", "theorem toIocMod_eq_sub (a b : \u03b1) : toIocMod hp a b = toIocMod hp 0 (b - a) + a", "@[simp]\ntheorem toIocMod_sub_zsmul' (a b : \u03b1) (m : \u2124) :\n    toIocMod hp (a - m \u2022 p) b = toIocMod hp a b - m \u2022 p", "@[simp]\ntheorem toIocMod_toIocMod (a\u2081 a\u2082 b : \u03b1) : toIocMod hp a\u2081 (toIocMod hp a\u2082 b) = toIocMod hp a\u2081 b", "@[to_additive (attr := simp)]\ntheorem toLex_inv [Inv \u03b1] (a : \u03b1) : toLex a\u207b\u00b9 = (toLex a)\u207b\u00b9", "@[simp]\ntheorem toMul_sub [Div \u03b1] (x y : Additive \u03b1) : toMul (x - y) = toMul x / toMul y", "@[to_additive toReal_coe_nnnorm \"The non negative norm seen as an `ENNReal` and\nthen as a `Real` is equal to the norm.\"]\ntheorem toReal_coe_nnnorm' (a : E) : (\u2016a\u2016\u208a : \u211d\u22650\u221e).toReal = \u2016a\u2016", "theorem toRingHom_injective [MulSemiringAction M R] [FaithfulSMul M R] :\n    Function.Injective (MulSemiringAction.toRingHom M R)", "theorem toTopologicalSpace_prod {\u03b1} {\u03b2} [u : UniformSpace \u03b1] [v : UniformSpace \u03b2] :\n    @UniformSpace.toTopologicalSpace (\u03b1 \u00d7 \u03b2) instUniformSpaceProd =\n      @instTopologicalSpaceProd \u03b1 \u03b2 u.toTopologicalSpace v.toTopologicalSpace", "@[to_additive (attr := simp)]\nlemma toUnits_val_apply {G : Type*} [Group G] (x : G\u02e3) : toUnits (x : G) = x", "theorem toZ_iterate_succ_le (n : \u2115) : toZ i0 (succ^[n] i0) \u2264 n", "theorem toZ_mono {i j : \u03b9} (h_le : i \u2264 j) : toZ i0 i \u2264 toZ i0 j", "theorem toZ_of_lt (hi : i < i0) : toZ i0 i = -Nat.find (exists_pred_iterate_of_le hi.le)", "lemma top_inf_eq (a : \u03b1) : \u22a4 \u2293 a = a := inf_of_le_right le_top", "theorem top_mem_upperBounds [OrderTop \u03b1] (s : Set \u03b1) : \u22a4 \u2208 upperBounds s", "@[simp]\ntheorem top_sdiff' (a : \u03b1) : \u22a4 \\ a = \uffe2a", "@[simp]\ntheorem torusIntegral_dim0 (f : \u2102\u2070 \u2192 E) (c : \u2102\u2070) (R : \u211d\u2070) : (\u222f x in T(c, R), f x) = f c", "theorem torusMap_eq_center_iff {c : \u2102\u207f} {R : \u211d\u207f} {\u03b8 : \u211d\u207f} : torusMap c R \u03b8 = c \u2194 R = 0", "theorem totallyBounded_subset {s\u2081 s\u2082 : Set \u03b1} (hs : s\u2081 \u2286 s\u2082) (h : TotallyBounded s\u2082) :\n    TotallyBounded s\u2081", "theorem totallySeparatedSpace_of_t1_of_basis_clopen [T1Space X]\n    (h : IsTopologicalBasis { s : Set X | IsClopen s }) : TotallySeparatedSpace X", "@[to_additive]\ntheorem tprod_image {g : \u03b3 \u2192 \u03b2} (f : \u03b2 \u2192 \u03b1) {s : Set \u03b3} (hg : Set.InjOn g s) :\n    \u220f' x : g '' s, f x = \u220f' x : s, f (g x)", "@[to_additive]\ntheorem tprod_le_tprod (h : \u2200 i, f i \u2264 g i) (hf : Multipliable f) (hg : Multipliable g) :\n    \u220f' i, f i \u2264 \u220f' i, g i", "@[to_additive]\ntheorem tprod_ne_one_iff (hf : Multipliable f) : \u220f' i, f i \u2260 1 \u2194 \u2203 x, f x \u2260 1", "@[to_additive tsum_setProd_singleton_right]\nlemma tprod_setProd_singleton_right (s : Set \u03b2) (c : \u03b3) (f : \u03b2 \u00d7 \u03b3 \u2192 \u03b1) :\n    (\u220f' x : s \u00d7\u02e2 {c}, f x) = \u220f' b : s, f (b, c)", "@[to_additive (attr := simp, nolint simpNF)]\ntheorem tprod_singleton (b : \u03b2) (f : \u03b2 \u2192 \u03b1) : \u220f' x : ({b} : Set \u03b2), f x = f b", "@[to_additive]\nlemma tprod_subtype_le {\u03ba \u03b3 : Type*} [OrderedCommGroup \u03b3] [UniformSpace \u03b3] [UniformGroup \u03b3]\n    [OrderClosedTopology \u03b3] [CompleteSpace \u03b3] (f : \u03ba \u2192 \u03b3) (\u03b2 : Set \u03ba) (h : \u2200 a : \u03ba, 1 \u2264 f a)\n    (hf : Multipliable f) : (\u220f' (b : \u03b2), f b) \u2264 (\u220f' (a : \u03ba), f a)", "@[to_additive]\ntheorem tprod_subtype_mulSupport (f : \u03b2 \u2192 \u03b1) : \u220f' x : mulSupport f, f x = \u220f' x, f x", "lemma transGen_wcovBy_of_le [Preorder \u03b1] [LocallyFiniteOrder \u03b1] {x y : \u03b1} (hxy : x \u2264 y) :\n    TransGen (\u00b7 \u2a7f \u00b7) x y", "@[elab_without_expected_type]\ntheorem trichotomous_of [IsTrichotomous \u03b1 r] : \u2200 a b : \u03b1, a \u227a b \u2228 a = b \u2228 b \u227a a", "theorem true_imp_iff (\u03b1 : Prop) : (True \u2192 \u03b1) \u2194 \u03b1", "theorem tsub_add_tsub_comm (hba : b \u2264 a) (hdc : d \u2264 c) : a - b + (c - d) = a + c - (b + d)", "theorem tsub_eq_zero_iff_le : a - b = 0 \u2194 a \u2264 b", "@[simp]\ntheorem tsub_le_iff_right [LE \u03b1] [Add \u03b1] [Sub \u03b1] [OrderedSub \u03b1] {a b c : \u03b1} :\n    a - b \u2264 c \u2194 a \u2264 c + b", "@[gcongr] theorem tsub_le_tsub_right (h : a \u2264 b) (c : \u03b1) : a - c \u2264 b - c", "theorem tsub_min : a - min a b = a - b", "theorem tsub_tsub_cancel_of_le (h : a \u2264 b) : b - (b - a) = a", "lemma tsub_tsub_eq_min (a b : \u03b1) : a - (a - b) = min a b", "theorem tsub_tsub_le : b - (b - a) \u2264 a", "lemma tsum_const_smul' {\u03b3 : Type*} [Group \u03b3] [DistribMulAction \u03b3 \u03b1] [ContinuousConstSMul \u03b3 \u03b1]\n    [T2Space \u03b1] (g : \u03b3) : \u2211' (i : \u03b2), g \u2022 f i = g \u2022 \u2211' (i : \u03b2), f i", "theorem tsupport_mul_subset_right {\u03b1 : Type*} [MulZeroClass \u03b1] {f g : X \u2192 \u03b1} :\n    (tsupport fun x => f x * g x) \u2286 tsupport g", "theorem ultrafilter_isOpen_basic (s : Set \u03b1) : IsOpen { u : Ultrafilter \u03b1 | s \u2208 u }", "theorem uniformCauchySeqOnFilter_of_fderiv (hf' : UniformCauchySeqOnFilter f' l (\ud835\udcdd x))\n    (hf : \u2200\u1da0 n : \u03b9 \u00d7 E in l \u00d7\u02e2 \ud835\udcdd x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2)\n    (hfg : Cauchy (map (fun n => f n x) l)) : UniformCauchySeqOnFilter f l (\ud835\udcdd x)", "theorem uniformCauchySeqOn_iff_uniformCauchySeqOnFilter :\n    UniformCauchySeqOn F p s \u2194 UniformCauchySeqOnFilter F p (\ud835\udcdf s)", "theorem uniformContinuousOn_univ [UniformSpace \u03b2] {f : \u03b1 \u2192 \u03b2} :\n    UniformContinuousOn f univ \u2194 UniformContinuous f", "theorem uniformContinuous_of_const [UniformSpace \u03b2] {c : \u03b1 \u2192 \u03b2} (h : \u2200 a b, c a = c b) :\n    UniformContinuous c", "@[to_additive]\ntheorem uniformContinuous_of_tendsto_one {hom : Type*} [UniformSpace \u03b2] [Group \u03b2] [UniformGroup \u03b2]\n    [FunLike hom \u03b1 \u03b2] [MonoidHomClass hom \u03b1 \u03b2] {f : hom} (h : Tendsto f (\ud835\udcdd 1) (\ud835\udcdd 1)) :\n    UniformContinuous f", "theorem uniformContinuous_subtype_val {p : \u03b1 \u2192 Prop} [UniformSpace \u03b1] :\n    UniformContinuous (Subtype.val : { a : \u03b1 // p a } \u2192 \u03b1)", "theorem uniformEmbedding_subtype_val {p : \u03b1 \u2192 Prop} :\n    UniformEmbedding (Subtype.val : Subtype p \u2192 \u03b1)", "@[to_additive]\ntheorem uniformEmbedding_translate_mul (a : \u03b1) : UniformEmbedding fun x : \u03b1 => x * a", "@[simp]\ntheorem uniformSpace_comap_id {\u03b1 : Type*} : UniformSpace.comap (id : \u03b1 \u2192 \u03b1) = id", "theorem uniformity_basis_edist_nnreal_le :\n    (\ud835\udce4 \u03b1).HasBasis (fun \u03b5 : \u211d\u22650 => 0 < \u03b5) fun \u03b5 => { p : \u03b1 \u00d7 \u03b1 | edist p.1 p.2 \u2264 \u03b5 }", "theorem uniformity_hasBasis_closure : HasBasis (\ud835\udce4 \u03b1) (fun V : Set (\u03b1 \u00d7 \u03b1) => V \u2208 \ud835\udce4 \u03b1) closure", "theorem uniformity_pseudoedist : \ud835\udce4 \u03b1 = \u2a05 \u03b5 > 0, \ud835\udcdf { p : \u03b1 \u00d7 \u03b1 | edist p.1 p.2 < \u03b5 }", "theorem uniqueDiffOn_Ici (a : \u211d) : UniqueDiffOn \u211d (Ici a)", "theorem uniqueDiffOn_empty : UniqueDiffOn \ud835\udd5c (\u2205 : Set E)", "theorem uniqueDiffWithinAt_convex {s : Set G} (conv : Convex \u211d s) (hs : (interior s).Nonempty)\n    {x : G} (hx : x \u2208 closure s) : UniqueDiffWithinAt \u211d s x", "@[simp]\ntheorem uniqueElim_const {\u03b2 : Sort*} {_ : Unique \u03b9} (x : \u03b2) (i : \u03b9) :\n    uniqueElim (\u03b1 := fun _ \u21a6 \u03b2) x i = x", "theorem unique_unit_speed {\u03c6 : \u211d \u2192 \u211d} (\u03c6m : MonotoneOn \u03c6 s) (hf\u03c6 : HasUnitSpeedOn (f \u2218 \u03c6) s)\n    (hf : HasUnitSpeedOn f (\u03c6 '' s)) \u2983x : \u211d\u2984 (xs : x \u2208 s) : EqOn \u03c6 (fun y => y - x + \u03c6 x) s", "theorem half_le_symm_iff (t : I) : 1 / 2 \u2264 (\u03c3 t : \u211d) \u2194 (t : \u211d) \u2264 1 / 2", "theorem one_mem : (1 : \u211d) \u2208 I", "theorem coe_star_mul_self (U : unitary R) : (star U : R) * U = 1", "@[simp]\nlemma spectrum.unitary_conjugate {a : A} {u : unitary A} :\n    spectrum R (u * a * (star u : A)) = spectrum R a", "@[simp]\nlemma unop_nonZeroSMulDivisors_mulOpposite_eq_nonZeroDivisors :\n    (R\u1d50\u1d52\u1d56 \u2070[R]).unop = R\u2070 := rfl", "theorem upperBounds_mono_mem \u2983a b\u2984 (hab : a \u2264 b) : a \u2208 upperBounds s \u2192 b \u2208 upperBounds s", "@[simp]\ntheorem upperCentralSeries_nilpotencyClass : upperCentralSeries G (Group.nilpotencyClass G) = \u22a4", "@[simp]\ntheorem upperCentralSeries_one : upperCentralSeries G 1 = center G", "@[to_additive (attr := simp)]\ntheorem upperClosure_mul_distrib : upperClosure (s * t) = upperClosure s * upperClosure t", "@[to_additive (attr := simp)]\ntheorem upperClosure_one : upperClosure (1 : Set \u03b1) = 1", "theorem upperSemicontinuousAt_ciInf {f : \u03b9 \u2192 \u03b1 \u2192 \u03b4'}\n    (bdd : \u2200\u1da0 y in \ud835\udcdd x, BddBelow (range fun i => f i y)) (h : \u2200 i, UpperSemicontinuousAt (f i) x) :\n    UpperSemicontinuousAt (fun x' => \u2a05 i, f i x') x", "theorem upperSemicontinuousOn_biInf {p : \u03b9 \u2192 Prop} {f : \u2200 i, p i \u2192 \u03b1 \u2192 \u03b4}\n    (h : \u2200 i hi, UpperSemicontinuousOn (f i hi) s) :\n    UpperSemicontinuousOn (fun x' => \u2a05 (i) (hi), f i hi x') s", "theorem upperSemicontinuousWithinAt_univ_iff :\n    UpperSemicontinuousWithinAt f univ x \u2194 UpperSemicontinuousAt f x", "theorem upperSemicontinuous_iInf {f : \u03b9 \u2192 \u03b1 \u2192 \u03b4} (h : \u2200 i, UpperSemicontinuous (f i)) :\n    UpperSemicontinuous fun x' => \u2a05 i, f i x'", "theorem upperSemicontinuous_iff_limsup_le {f : \u03b1 \u2192 \u03b3} :\n    UpperSemicontinuous f \u2194 \u2200 x, limsup f (\ud835\udcdd x) \u2264 f x", "@[norm_cast] lemma uzpow_natCast (u : \u2124\u02e3) (n : \u2115) : u ^ (n : R) = u ^ n", "lemma uzpow_neg (s : \u2124\u02e3) (x : R) : s ^ (-x) = (s ^ x)\u207b\u00b9", "theorem vadd_right_mem_affineSpan_pair {p\u2081 p\u2082 : P} {v : V} :\n    v +\u1d65 p\u2082 \u2208 line[k, p\u2081, p\u2082] \u2194 \u2203 r : k, r \u2022 (p\u2081 -\u1d65 p\u2082) = v", "theorem wbtw_pointReflection (x y : P) : Wbtw R y x (pointReflection R x y)", "theorem wellFounded_gt_exact_sequence {\u03b2 \u03b3 : Type*} [Preorder \u03b2] [PartialOrder \u03b3]\n    (h\u2081 : WellFounded ((\u00b7 > \u00b7) : \u03b2 \u2192 \u03b2 \u2192 Prop)) (h\u2082 : WellFounded ((\u00b7 > \u00b7) : \u03b3 \u2192 \u03b3 \u2192 Prop)) (K : \u03b1)\n    (f\u2081 : \u03b2 \u2192 \u03b1) (f\u2082 : \u03b1 \u2192 \u03b2) (g\u2081 : \u03b3 \u2192 \u03b1) (g\u2082 : \u03b1 \u2192 \u03b3) (gci : GaloisCoinsertion f\u2081 f\u2082)\n    (gi : GaloisInsertion g\u2082 g\u2081) (hf : \u2200 a, f\u2081 (f\u2082 a) = a \u2293 K) (hg : \u2200 a, g\u2081 (g\u2082 a) = a \u2294 K) :\n    WellFounded ((\u00b7 > \u00b7) : \u03b1 \u2192 \u03b1 \u2192 Prop)", "theorem wellFounded_iff_wellFounded_subrel {\u03b2 : Type*} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} [IsTrans \u03b2 s] :\n    WellFounded s \u2194 \u2200 b, WellFounded (Subrel s { b' | s b' b })", "theorem wellFounded_of_isEmpty {\u03b1} [IsEmpty \u03b1] (r : \u03b1 \u2192 \u03b1 \u2192 Prop) : WellFounded r", "theorem with_gaugeSeminormFamily : WithSeminorms (gaugeSeminormFamily \ud835\udd5c E)", "theorem wittStructureInt_prop (\u03a6 : MvPolynomial idx \u2124) (n) :\n    bind\u2081 (wittStructureInt p \u03a6) (wittPolynomial p \u2124 n) =\n      bind\u2081 (fun i => rename (Prod.mk i) (W_ \u2124 n)) \u03a6", "theorem wittStructureInt_rename {\u03c3 : Type*} (\u03a6 : MvPolynomial idx \u2124) (f : idx \u2192 \u03c3) (n : \u2115) :\n    wittStructureInt p (rename f \u03a6) n = rename (Prod.map f id) (wittStructureInt p \u03a6 n)", "theorem wittStructureRat_rec_aux (\u03a6 : MvPolynomial idx \u211a) (n : \u2115) :\n    wittStructureRat p \u03a6 n * C ((p : \u211a) ^ n) =\n      bind\u2081 (fun b => rename (fun i => (b, i)) (W_ \u211a n)) \u03a6 -\n        \u2211 i \u2208 range n, C ((p : \u211a) ^ i) * wittStructureRat p \u03a6 i ^ p ^ (n - i)", "@[simp] theorem xor_false : Xor' False = id", "@[simp] theorem xor_self (a : Prop) : Xor' a a = False", "@[simp]\ntheorem zero_dvd_iff : 0 \u2223 a \u2194 a = 0", "@[simp]\ntheorem zero_eq_dist {x y : \u03b3} : 0 = dist x y \u2194 x = y", "@[simp] lemma zero_le_one [Zero \u03b1] [One \u03b1] [LE \u03b1] [ZeroLEOneClass \u03b1] : (0 : \u03b1) \u2264 1", "theorem zero_lt_iff : 0 < a \u2194 a \u2260 0", "theorem zero_mem_nonunits [Semiring \u03b1] : 0 \u2208 nonunits \u03b1 \u2194 (0 : \u03b1) \u2260 1", "theorem zero_mem\u2113p : Mem\u2113p (0 : \u2200 i, E i) p", "lemma zero_pow_eq (n : \u2115) : (0 : M\u2080) ^ n = if n = 0 then 1 else 0", "@[simp]\nlemma zero_pow_eq_zero [Nontrivial M\u2080] : (0 : M\u2080) ^ n = 0 \u2194 n \u2260 0", "theorem zero_pow_le_one : \u2200 n : \u2115, (0 : R) ^ n \u2264 1", "@[simp]\ntheorem zero_smul (m : M) : (0 : R) \u2022 m = 0", "theorem zeta_nat_eq_tsum_of_gt_one {k : \u2115} (hk : 1 < k) :\n    riemannZeta k = \u2211' n : \u2115, 1 / (n : \u2102) ^ k", "theorem zorn_nonempty_partialOrder\u2080 (s : Set \u03b1)\n    (ih : \u2200 c \u2286 s, IsChain (\u00b7 \u2264 \u00b7) c \u2192 \u2200 y \u2208 c, \u2203 ub \u2208 s, \u2200 z \u2208 c, z \u2264 ub) (x : \u03b1)\n    (hxs : x \u2208 s) : \u2203 m \u2208 s, x \u2264 m \u2227 \u2200 z \u2208 s, m \u2264 z \u2192 z = m", "theorem zorn_preorder (h : \u2200 c : Set \u03b1, IsChain (\u00b7 \u2264 \u00b7) c \u2192 BddAbove c) :\n    \u2203 m : \u03b1, \u2200 a, m \u2264 a \u2192 a \u2264 m", "@[to_additive (attr := simp) abs_zsmul_eq_zero]\nlemma zpow_abs_eq_one (a : G) (n : \u2124) : a ^ |n| = 1 \u2194 a ^ n = 1", "@[to_additive]\ntheorem zpow_eq_one_iff_modEq {n : \u2124} : x ^ n = 1 \u2194 n \u2261 0 [ZMOD orderOf x]", "@[simp]\ntheorem zpow_le_iff_le (hx : 1 < a) : a ^ m \u2264 a ^ n \u2194 m \u2264 n", "@[to_additive (attr := aesop safe apply (rule_sets := [SetLike]))]\ntheorem zpow_mem {x : M} (hx : x \u2208 K) : \u2200 n : \u2124, x ^ n \u2208 K", "@[to_additive (attr := simp) mod_card_zsmul]\ntheorem zpow_mod_card (a : G) (n : \u2124) : a ^ (n % Fintype.card G : \u2124) = a ^ n", "@[to_additive (attr := simp) one_zsmul]\nlemma zpow_one (a : G) : a ^ (1 : \u2124) = a", "lemma zpow_one_add\u2080 (h : a \u2260 0) (i : \u2124) : a ^ (1 + i) = a * a ^ i", "lemma zpow_pos_of_pos (ha : 0 < a) : \u2200 n : \u2124, 0 < a ^ n\n  | (n : \u2115) => by rw [zpow_natCast]; exact pow_pos ha _\n  | -(n + 1 : \u2115) => by rw [zpow_neg, inv_pos, zpow_natCast]; exact pow_pos ha _", "@[to_additive \"Any non-identity element of a finite group of prime order generates the group.\"]\ntheorem zpowers_eq_top_of_prime_card {G : Type*} [Group G] {_ : Fintype G} {p : \u2115}\n    [hp : Fact p.Prime] (h : Fintype.card G = p) {g : G} (hg : g \u2260 1) : zpowers g = \u22a4", "@[simp]\ntheorem evalFrom_append_singleton (S : Set \u03c3) (x : List \u03b1) (a : \u03b1) :\n    M.evalFrom S (x ++ [a]) = M.stepSet (M.evalFrom S x) a"], "paths": ["Mathlib/NumberTheory/ADEInequality.lean", "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean", "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "Mathlib/MeasureTheory/Order/Lattice.lean", 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"Mathlib/MeasureTheory/Function/LocallyIntegrable.lean", "Mathlib/Topology/Semicontinuous.lean", "Mathlib/Topology/Instances/Matrix.lean", "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "Mathlib/Topology/Constructions.lean", "Mathlib/LinearAlgebra/AffineSpace/ContinuousAffineEquiv.lean", "Mathlib/LinearAlgebra/AffineSpace/ContinuousAffineEquiv.lean", "Mathlib/LinearAlgebra/AffineSpace/ContinuousAffineEquiv.lean", "Mathlib/LinearAlgebra/AffineSpace/ContinuousAffineEquiv.lean", "Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean", "Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean", "Mathlib/Topology/Algebra/ContinuousAffineMap.lean", "Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean", "Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean", "Mathlib/Topology/Algebra/Module/Alternating/Basic.lean", "Mathlib/Topology/Algebra/Module/Alternating/Basic.lean", "Mathlib/Topology/Algebra/Module/Alternating/Basic.lean", "Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean", "Mathlib/Topology/Order/Lattice.lean", "Mathlib/Topology/Algebra/Monoid.lean", "Mathlib/Topology/Constructions.lean", "Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean", "Mathlib/Topology/Algebra/Module/Basic.lean", "Mathlib/Topology/Algebra/Module/StrongTopology.lean", "Mathlib/Topology/Algebra/Module/Basic.lean", "Mathlib/Geometry/Manifold/Diffeomorph.lean", "Mathlib/Analysis/Calculus/FDeriv/Equiv.lean", "Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean", "Mathlib/Analysis/Calculus/FDeriv/Equiv.lean", "Mathlib/Topology/Algebra/Module/Basic.lean", "Mathlib/Analysis/Calculus/FDeriv/Equiv.lean", "Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean", "Mathlib/Analysis/InnerProductSpace/Adjoint.lean", "Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean", "Mathlib/Analysis/NormedSpace/Banach.lean", "Mathlib/Topology/Algebra/Module/Basic.lean", "Mathlib/Topology/Algebra/Module/Basic.lean", "Mathlib/Analysis/NormedSpace/OperatorNorm/Basic.lean", "Mathlib/Topology/Algebra/Module/Basic.lean", "Mathlib/Topology/ContinuousFunction/Compact.lean", "Mathlib/Analysis/Analytic/CPolynomial.lean", "Mathlib/Geometry/Manifold/ContMDiff/NormedSpace.lean", "Mathlib/Analysis/Calculus/FDeriv/Linear.lean", "Mathlib/Analysis/NormedSpace/Complemented.lean", "Mathlib/Topology/Algebra/Module/Basic.lean", "Mathlib/Topology/Algebra/Module/Basic.lean", "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean", "Mathlib/Analysis/Distribution/SchwartzSpace.lean", "Mathlib/Analysis/InnerProductSpace/Rayleigh.lean", "Mathlib/Analysis/InnerProductSpace/Adjoint.lean", "Mathlib/Topology/Algebra/Module/Basic.lean", "Mathlib/Analysis/InnerProductSpace/Adjoint.lean", "Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean", "Mathlib/Analysis/NormedSpace/OperatorNorm/Mul.lean", "Mathlib/Analysis/NormedSpace/OperatorNorm/NNNorm.lean", "Mathlib/Analysis/NormedSpace/OperatorNorm/NNNorm.lean", 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"Mathlib/Topology/ContinuousFunction/Ordered.lean", "Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean", "Mathlib/Topology/ContinuousFunction/Basic.lean", "Mathlib/Topology/ContinuousFunction/Algebra.lean", "Mathlib/Topology/ContinuousFunction/Ideals.lean", "Mathlib/Topology/ContinuousFunction/Ideals.lean", "Mathlib/Topology/ContinuousFunction/Algebra.lean", "Mathlib/Topology/ContinuousFunction/Algebra.lean", "Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean", "Mathlib/Topology/ContinuousFunction/Compact.lean", "Mathlib/MeasureTheory/Function/LpSpace.lean", "Mathlib/Topology/ContinuousFunction/Algebra.lean", "Mathlib/Topology/ContinuousFunction/ContinuousMapZero.lean", "Mathlib/Analysis/Calculus/FDeriv/Analytic.lean", "Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean", "Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean", "Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean", "Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean", "Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean", "Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean", "Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean", "Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean", "Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean", "Mathlib/Analysis/NormedSpace/Multilinear/Curry.lean", "Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean", "Mathlib/Topology/ContinuousOn.lean", "Mathlib/Topology/Algebra/GroupWithZero.lean", "Mathlib/Topology/Algebra/Order/Compact.lean", "Mathlib/Topology/Algebra/Order/Compact.lean", "Mathlib/Topology/ContinuousOn.lean", "Mathlib/Topology/Algebra/Order/Compact.lean", "Mathlib/Topology/Algebra/GroupWithZero.lean", "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "Mathlib/Topology/ContinuousOn.lean", "Mathlib/Topology/Order/IntermediateValue.lean", "Mathlib/Analysis/Normed/Group/AddTorsor.lean", "Mathlib/Topology/Hom/Open.lean", "Mathlib/Topology/Hom/Open.lean", "Mathlib/Topology/Order/Hom/Basic.lean", "Mathlib/Topology/Order/Hom/Basic.lean", "Mathlib/Topology/Algebra/FilterBasis.lean", "Mathlib/Topology/CompactOpen.lean", "Mathlib/Analysis/InnerProductSpace/Basic.lean", "Mathlib/Analysis/Normed/Group/Basic.lean", "Mathlib/Topology/MetricSpace/Contracting.lean", "Mathlib/Topology/MetricSpace/Contracting.lean", "Mathlib/Analysis/Convex/Basic.lean", "Mathlib/Analysis/Convex/Basic.lean", "Mathlib/Analysis/Calculus/MeanValue.lean", "Mathlib/Analysis/Convex/Combination.lean", "Mathlib/Analysis/Calculus/MeanValue.lean", "Mathlib/Analysis/Convex/Basic.lean", "Mathlib/Analysis/Convex/Segment.lean", "Mathlib/Analysis/Convex/Topology.lean", "Mathlib/Analysis/Convex/Basic.lean", "Mathlib/Analysis/Convex/Cone/Basic.lean", "Mathlib/Analysis/Convex/Body.lean", "Mathlib/Analysis/Convex/Cone/Basic.lean", "Mathlib/Analysis/Convex/Cone/Basic.lean", "Mathlib/Analysis/Convex/Cone/Closure.lean", "Mathlib/Analysis/Convex/Cone/Basic.lean", "Mathlib/Analysis/Convex/Cone/Basic.lean", 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"Mathlib/Analysis/NormedSpace/Star/Basic.lean", "Mathlib/Algebra/CubicDiscriminant.lean", "Mathlib/Algebra/CubicDiscriminant.lean", "Mathlib/Algebra/CubicDiscriminant.lean", "Mathlib/Algebra/CubicDiscriminant.lean", "Mathlib/Algebra/CubicDiscriminant.lean", "Mathlib/Data/List/Cycle.lean", "Mathlib/GroupTheory/Perm/Cycle/Concrete.lean", "Mathlib/Data/List/Cycle.lean", "Mathlib/Data/List/Cycle.lean", "Mathlib/Data/List/Cycle.lean", "Mathlib/Data/List/Cycle.lean", "Mathlib/Data/List/Cycle.lean", "Mathlib/Data/List/Cycle.lean", "Mathlib/Data/List/Cycle.lean", "Mathlib/Data/List/Cycle.lean", "Mathlib/Computability/DFA.lean", "Mathlib/Data/DFinsupp/WellFounded.lean", "Mathlib/Data/DFinsupp/Order.lean", "Mathlib/Data/DFinsupp/Order.lean", "Mathlib/Data/DFinsupp/Basic.lean", "Mathlib/Data/DFinsupp/Basic.lean", "Mathlib/Data/DFinsupp/Basic.lean", "Mathlib/Data/DFinsupp/Basic.lean", "Mathlib/Data/DFinsupp/Basic.lean", "Mathlib/Data/DFinsupp/Basic.lean", "Mathlib/Data/DFinsupp/Basic.lean", 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"Mathlib/Order/Basic.lean", "Mathlib/Init/Order/Defs.lean", "Mathlib/Logic/Basic.lean", "Mathlib/RingTheory/DedekindDomain/FiniteAdeleRing.lean", "Mathlib/Topology/Separation.lean", "Mathlib/Topology/Algebra/ConstMulAction.lean", "Mathlib/Topology/DenseEmbedding.lean", "Mathlib/Topology/DenseEmbedding.lean", "Mathlib/Topology/Basic.lean", "Mathlib/Topology/Constructions.lean", "Mathlib/Topology/Algebra/Module/Basic.lean", "Mathlib/Logic/Equiv/List.lean", "Mathlib/RingTheory/Derivation/Basic.lean", "Mathlib/RingTheory/Derivation/Basic.lean", "Mathlib/RingTheory/Derivation/Basic.lean", "Mathlib/Algebra/Homology/DerivedCategory/HomologySequence.lean", "Mathlib/Analysis/Calculus/DiffContOnCl.lean", "Mathlib/Geometry/Manifold/Diffeomorph.lean", "Mathlib/Geometry/Manifold/Diffeomorph.lean", "Mathlib/Geometry/Manifold/LocalDiffeomorph.lean", "Mathlib/Analysis/Calculus/FDeriv/Comp.lean", "Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean", "Mathlib/Analysis/InnerProductSpace/Calculus.lean", "Mathlib/Analysis/Calculus/FDeriv/Mul.lean", "Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean", "Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean", "Mathlib/Analysis/Calculus/Deriv/Mul.lean", "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean", "Mathlib/Analysis/Calculus/DiffContOnCl.lean", "Mathlib/Analysis/Calculus/Deriv/Mul.lean", "Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean", "Mathlib/Analysis/SpecialFunctions/Trigonometric/ArctanDeriv.lean", "Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean", "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean", "Mathlib/Analysis/Calculus/Deriv/ZPow.lean", "Mathlib/GroupTheory/SpecificGroups/Dihedral.lean", "Mathlib/GroupTheory/CommutingProbability.lean", "Mathlib/GroupTheory/SpecificGroups/Dihedral.lean", "Mathlib/Topology/MetricSpace/Dilation.lean", 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"Mathlib/GroupTheory/GroupAction/DomAct/Basic.lean", "Mathlib/GroupTheory/GroupAction/DomAct/Basic.lean", "Mathlib/GroupTheory/GroupAction/DomAct/ActionHom.lean", "Mathlib/Algebra/Module/LinearMap/Basic.lean", "Mathlib/GroupTheory/DoubleCoset.lean", "Mathlib/GroupTheory/DoubleCoset.lean", "Mathlib/GroupTheory/DoubleCoset.lean", "Mathlib/Analysis/NormedSpace/Star/Multiplier.lean", "Mathlib/Analysis/NormedSpace/Star/Multiplier.lean", "Mathlib/Analysis/NormedSpace/Star/Multiplier.lean", "Mathlib/Analysis/NormedSpace/Star/Multiplier.lean", "Mathlib/RingTheory/Ideal/QuotientOperations.lean", "Mathlib/RingTheory/Ideal/QuotientOperations.lean", "Mathlib/Combinatorics/SetFamily/Compression/Down.lean", "Mathlib/Combinatorics/SetFamily/Compression/Down.lean", "Mathlib/Algebra/DualNumber.lean", "Mathlib/Algebra/DualNumber.lean", "Mathlib/Algebra/DualNumber.lean", "Mathlib/Algebra/Divisibility/Basic.lean", "Mathlib/Algebra/Divisibility/Basic.lean", "Mathlib/Topology/EMetricSpace/Basic.lean", "Mathlib/Topology/EMetricSpace/Basic.lean", "Mathlib/Topology/MetricSpace/PartitionOfUnity.lean", "Mathlib/Topology/MetricSpace/PartitionOfUnity.lean", "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "Mathlib/Topology/EMetricSpace/Basic.lean", "Mathlib/Topology/EMetricSpace/Basic.lean", "Mathlib/Topology/EMetricSpace/Basic.lean", "Mathlib/Topology/EMetricSpace/Basic.lean", "Mathlib/Topology/EMetricSpace/Basic.lean", "Mathlib/Topology/EMetricSpace/Basic.lean", "Mathlib/Data/ENNReal/Inv.lean", "Mathlib/Data/ENNReal/Operations.lean", "Mathlib/MeasureTheory/Function/AEMeasurableOrder.lean", "Mathlib/Topology/Instances/ENNReal.lean", "Mathlib/Data/ENNReal/Basic.lean", "Mathlib/Data/ENNReal/Inv.lean", "Mathlib/Data/ENNReal/Basic.lean", "Mathlib/Data/ENNReal/Basic.lean", "Mathlib/Data/ENNReal/Basic.lean", "Mathlib/Data/ENNReal/Basic.lean", "Mathlib/Data/ENNReal/Basic.lean", "Mathlib/Topology/Instances/ENNReal.lean", "Mathlib/Topology/Instances/ENNReal.lean", 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