leandojo-lean4-formal-informal-strings / random /test_formal_informal_metadata.json
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{
"total_valid_proofs": 2000,
"total_no_tactics": 1015,
"deliminter": "**",
"no_tactics": [
"LaurentPolynomial.isUnit_T",
"PartENat.toWithTop_some",
"Finset.coe_smul",
"GenLoop.continuous_fromLoop",
"Int.neg_eq_neg",
"Polynomial.degree_sum_fin_lt",
"SimpleGraph.Walk.edges_bypass_subset",
"PrimeSpectrum.closedEmbedding_comap_of_surjective",
"Filter.tendsto_fst",
"IsMaxOn.norm_add_self",
"Rat.cast_sub",
"Set.finite_le_nat",
"Finset.eq_univ_of_forall",
"RingEquiv.trans_apply",
"CategoryTheory.ConcreteCategory.surjective_of_epi_of_preservesPushout",
"AlgEquiv.ext_iff",
"RingCon.coe_nsmul",
"MvPolynomial.coeToMvPowerSeries.ringHom_apply",
"FreimanHom.comp_id",
"ContinuousAt.vsub",
"Multiset.ndinsert_zero",
"Rat.mkRat_num_den",
"MeasureTheory.AEStronglyMeasurable.restrict",
"LinearMap.SeparatingLeft.toMatrix\u2082",
"Cardinal.pow_cast_right",
"t1Space_iff_continuous_cofinite_of",
"Antitone.pairwise_disjoint_on_Ioc_pred",
"continuous_quotient_mk'",
"Finset.smul_finset_union",
"Subsingleton.antitone'",
"LocalHomeomorph.symm_image_target_inter_eq",
"DoubleCentralizer.toProdMulOpposite_injective",
"Submodule.mk_quotientEquivOfIsCompl_apply",
"Filter.prod_mono_left",
"Seminorm.ext",
"List.isPrefixOf_cons\u2082",
"Cardinal.toNat_eq_iff",
"Polynomial.continuousWithinAt_aeval",
"Directed.mono",
"NonUnitalSubsemiring.mk'_toAddSubmonoid",
"Filter.inf_principal_neBot_iff",
"Filter.eventually_of_mem",
"Relation.TransGen.head",
"MvPolynomial.eval\u2082_zero_apply",
"LowerSet.coe_div",
"Finset.mem_uIcc_of_le",
"ofAdd_smul",
"IsMonoidHom.id",
"deriv_sqrt",
"IsROrC.conj_bit0",
"Order.pred_succ_of_not_isMax",
"TwoPointing.sum_fst",
"SmoothBumpFunction.hasCompactSupport",
"UpperHalfPlane.im_ne_zero",
"Filter.eventuallyEq_bind",
"Mathlib.Meta.Positivity.pow_zero_pos",
"CategoryTheory.Subgroupoid.isThin_iff",
"CategoryTheory.ActionCategory.\u03c0_map",
"coe_basisOfLinearIndependentOfCardEqFinrank",
"fderivWithin_pi",
"HasDerivWithinAt.const_sub",
"WithBot.toDual_lt_iff",
"le_iSup'",
"LocalHomeomorph.secondCountableTopology_source",
"IsLowerSet.le_card_inter_finset",
"supIrred_ofDual",
"ContinuousMultilinearMap.nnnorm_ofSubsingleton",
"Hyperreal.coe_ne_zero",
"SetTheory.PGame.zero_lt_one",
"IsCompact.isLUB_sSup",
"IsBoundedBilinearMap.differentiableOn",
"tendsto_atTop_of_eventually_const",
"Basis.algebraMap_injective",
"ContinuousAt.exp",
"StarSubalgebra.map_mono",
"tsum_sigma'",
"norm_le_norm_add_const_of_dist_le'",
"Equiv.sumArrowEquivProdArrow_apply_fst",
"Set.iUnion_vsub_left_image",
"CategoryTheory.Subobject.underlyingIso_hom_comp_eq_mk",
"Real.volume_pi_Ioo",
"BoundedLatticeHom.coe_id",
"OrderIso.map_csInf",
"SetSemiring.add_def",
"Metric.isOpen_thickening",
"IsClosed.compl_mem_nhds",
"isOpen_Ioo",
"HasFDerivAt.csin",
"ContinuousMap.gen_inter",
"NonUnitalSubring.mem_carrier",
"CircleDeg1Lift.coe_mul",
"WellFoundedLT.fix_eq",
"IsCyclotomicExtension.finiteDimensional",
"Multiset.count_zero",
"Polynomial.eval\u2082_finset_prod",
"lt_or_lt_iff_ne",
"Equiv.coe_vaddConst_symm",
"AEMeasurable.const_inf",
"Ordinal.nfp_eq_self",
"Nat.Odd.of_mul_right",
"Set.mapsTo_iInter_iInter",
"Urysohns.CU.left_U_subset_right_C",
"FirstOrder.Language.BoundedFormula.IsAtomic.relabel",
"abelianizationCongr_of",
"MonoidHom.ker_restrict",
"MeasureTheory.measure_toMeasurable_union",
"ContinuousWithinAt.mem_closure",
"IsLocalMaxOn.sup",
"Rep.MonoidalCategory.braiding_inv_apply",
"MultilinearMap.map_update_zero",
"commutatorElement_self",
"LinearMap.fst_surjective",
"LinearMap.IsRefl.flip_isRefl_iff",
"Equiv.Perm.VectorsProdEqOne.zero_eq",
"IsCompact.isGreatest_sSup",
"Set.image_add_const_Icc",
"Colex.eq_iff",
"Ordinal.typein_enum",
"QuadraticForm.complex_equivalent",
"affineSpan_le",
"MeasureTheory.integral_iUnion_ae",
"ContinuousWithinAt.comp'",
"RePred.of_eq",
"Bool.decide_false_iff",
"Metric.self_subset_cthickening",
"MvPolynomial.aeval_def",
"lt_of_eq_of_lt",
"WithBot.add_coe_eq_bot_iff",
"Set.iUnion_or",
"Covby.image",
"Subalgebra.restrictScalars_toSubmodule",
"Asymptotics.isBigO_pure",
"reflexive_oneOneReducible",
"nhds_right'_le_nhds_ne",
"Algebra.trace_apply",
"Btw.btw.antisymm",
"CategoryTheory.mop_comp",
"iInf_congr_Prop",
"Monotone.strictMono_of_injective",
"Convex.image_deriv",
"PrimeSpectrum.vanishingIdeal_union",
"Lists'.mem_of_subset",
"Finite.card_pos",
"IsROrC.re_add_im",
"Set.eq_or_ssubset_of_subset",
"AffineIsometryEquiv.coe_mul",
"Part.get_mem",
"TensorProduct.AlgebraTensorModule.tensorTensorTensorComm_tmul",
"Real.Angle.continuous_sin",
"Int.card_Icc",
"IsLocalization.lift_id",
"FractionalIdeal.div_zero",
"summable_sum",
"FormalMultilinearSeries.norm_mul_pow_le_of_lt_radius",
"Num.succ_to_nat",
"UpperSet.coe_subset_coe",
"Complementeds.coe_bot",
"AEMeasurable.div_const",
"FirstOrder.Language.Equiv.coe_toEmbedding",
"Matrix.isUnit_of_left_inverse",
"selfZpow_of_nonneg",
"AList.keys_erase",
"CategoryTheory.Functor.const.unop_functor_op_obj_map",
"UniformFun.hasBasis_nhds",
"ContinuousMap.HomotopyEquiv.continuous",
"ONote.opow_def",
"AlgEquiv.coe_algHom",
"SimpleGraph.singleton_disjoint_neighborFinset",
"edist_ofMul",
"Std.RBNode.insert_toList_zoom_nil",
"Mathlib.Tactic.ComputeDegree.natDegree_C_le",
"mulIndicator_eventuallyLE_mulIndicator",
"lt_inv_of_neg",
"PerfectClosure.induction_on",
"ComplexShape.down_mk",
"inv_add_inv",
"InfHom.ext",
"OrderEmbedding.isCoatom_of_map_top_of_image",
"Std.RBNode.cmpEq.RootOrdered_congr",
"nndist_midpoint_midpoint_le",
"EuclideanGeometry.oangle_eq_zero_iff_oangle_rev_eq_zero",
"Ordinal.mod_mod",
"Matrix.submatrix_mul_equiv",
"LinearMap.map_neg",
"Std.RBNode.Path.Ordered.insert",
"Finset.Subset.trans",
"Finset.pi_const_singleton",
"IsGalois.integral",
"MeasureTheory.Lp.stronglyMeasurable",
"NonUnitalStarSubalgebra.toNonUnitalSubalgebra_inj",
"RelEmbedding.codRestrict_apply",
"FiniteDimensional.nontrivial_of_finrank_pos",
"ContDiff.rpow_const_of_ne",
"transGen_of_succ_of_ne",
"Set.preimage_singleton_eq_empty",
"lcm_dvd_lcm_mul_left",
"Bornology.IsBounded.smul\u2080",
"selfAdjoint.val_pow",
"NatOrdinal.toOrdinal_symm_eq",
"Isometry.uniformContinuous",
"ContinuousLinearEquiv.continuousWithinAt",
"continuousAt_id",
"Disjoint.exists_mem_filter_basis",
"Nat.ArithmeticFunction.vonMangoldt_eq_zero_iff",
"List.IsPrefix.trans",
"coe_inv_unitSphere",
"Metric.preimage_mul_left_closedBall",
"Subalgebra.center_toSubsemiring",
"Algebra.surjective_algebraMap_iff",
"closure_diff_intrinsicFrontier",
"CliffordAlgebra.reverse_mem_evenOdd_iff",
"Submodule.orthogonal_gc",
"List.destutter_pair",
"Int.ofNat_mul_negSucc",
"codisjoint_map_orderIso_iff",
"Set.preimage_pi",
"Ordinal.toNatOrdinal_zero",
"AlternatingMap.map_eq_zero_of_eq",
"MeasureTheory.integrableOn_condexpL2_of_measure_ne_top",
"mul_lt_mul\u2080",
"Filter.BoundedAtFilter.smul",
"Subsemiring.le_pointwise_smul_iff\u2080",
"Submodule.restrictScalars_top",
"Set.eq_empty_of_isEmpty",
"CategoryTheory.Sieve.pullback_pushforward_le",
"Path.coe_mk",
"Matrix.trace_fin_two",
"mul_one_eq_id",
"Iic_mem_nhds",
"Filter.eventually_ne_atBot",
"intermediate_value_Ioc",
"ContinuousOrderHom.id_comp",
"MeasureTheory.Measure.IsComplete.out",
"LocalHomeomorph.preimage_interior",
"ModelWithCorners.coe_extChartAt_transDiffeomorph",
"LipschitzWith.mul_end",
"Ordnode.Valid'.node",
"HasStrictFDerivAt.cos",
"ProbabilityTheory.kernel.prodMkLeft_apply'",
"Filter.isAtom_pure",
"nhds_bind_nhdsWithin",
"EuclideanGeometry.cospherical_empty",
"Real.contDiffAt_rpow_const_of_ne",
"Polynomial.rootMultiplicity_zero",
"Set.vsub_subset_vsub_left",
"IsBoundedBilinearMap.map_sub_left",
"Set.iUnion_ge_eq_iUnion_nat_add",
"Con.toSetoid_inj",
"AlgHom.toLieHom_comp",
"IsLocalizedModule.fromLocalizedModule'_mk",
"OptionT.goto_mkLabel",
"Complementeds.mk_top",
"MeasureTheory.SimpleFunc.inf_apply",
"Sylow.coe_comapOfKerIsPGroup",
"Smooth.snd",
"CategoryTheory.Functor.essImage.ofNatIso",
"Continuous.nndist",
"dense_univ",
"CategoryTheory.Abelian.comp_epiDesc",
"NonUnitalSubsemiring.gc_map_comap",
"CategoryTheory.Limits.Multiequalizer.condition",
"OrderIso.image_symm_image",
"List.Sublist.length_le",
"ContinuousConstSMul.secondCountableTopology",
"Finset.inr_mem_disjSum",
"UniformSpace.Completion.uniformEmbedding_coe",
"Matrix.kroneckerMap_reindex_right",
"Iff.not_left",
"MvPolynomial.monomial_sum_one",
"Absorbs.add",
"measurable_smul_const",
"MeasureTheory.mem_lpMeas_indicatorConstLp",
"Filter.NeBot.zero_div_nonneg",
"mem_integralClosure_iff_mem_FG",
"CategoryTheory.NatTrans.naturality_app",
"ContinuousLinearEquiv.mdifferentiableOn",
"nndist_triangle",
"VectorPrebundle.smoothCoordChange_apply",
"Filter.comap_cofinite_le",
"Metric.closedBall_subset_closedBall'",
"AddValuation.ext_iff",
"UniformEmbedding.comp",
"LinearMap.map_smul\u2082",
"Symmetric.swap_eq",
"IsMetricSeparated.union_right",
"FirstOrder.Language.Substructure.le_comap_of_map_le",
"Nat.mul_lt_mul'",
"AddMonoidHom.extension_coe",
"IsROrC.algebraMap_eq_ofReal",
"List.foldr_nil",
"SymAlg.mul_def",
"inf_right_comm",
"inv_ball",
"Antitone.le_leftLim",
"Prod.fst_comp_toSigma",
"Ordinal.enumOrd_def_nonempty",
"IsMin.Iio_eq",
"IsLocalMinOn.min",
"DifferentiableOn.analyticOn",
"TopCat.Presheaf.isSheaf_iff_isSheafEqualizerProducts",
"Option.to_list_none",
"GeneralizedContinuedFraction.convergent_eq_conts_a_div_conts_b",
"NNReal.summable_mk",
"SemidirectProduct.inv_right",
"Finset.compl_eq_univ_iff",
"Matrix.blockDiagonal_apply_ne",
"Basis.card_le_card_of_submodule",
"Set.mem_sigma_iff",
"Vector.mapAccumr_cons",
"Order.Cofinal.le_above",
"Subsemigroup.map_strictMono_of_injective",
"csSup_Iic",
"Monoid.coe_one",
"ZMod.injective_valMinAbs",
"intervalIntegral.integral_nonneg_of_ae",
"measurable_mem",
"Submodule.toAddSubmonoid_toNatSubmodule",
"dNext_comp_left",
"Cardinal.lift_mk_shrink",
"IsIntegral.sum",
"CategoryTheory.Presheaf.isSheaf_of_isTerminal",
"NonemptyInterval.ext_iff",
"ProjectiveSpectrum.mem_coe_basicOpen",
"ModuleCat.piIsoPi_inv_kernel_\u03b9",
"Function.Injective.hasLeftInverse",
"Subsemigroup.mem_closure",
"WithBot.Icc_bot_coe",
"Std.RBNode.Path.Balanced.insert",
"CategoryTheory.Preadditive.hasCoequalizers_of_hasCokernels",
"Filter.Tendsto.mono_right",
"Filter.Tendsto.nndist",
"ContinuousMap.congr_fun",
"isCompactOperator_iff_isCompact_closure_image_closedBall",
"Relation.TransGen.head'",
"CategoryTheory.Limits.coequalizer.condition",
"Set.Ioc_subset_Ioo_union_Icc",
"Subgroup.mem_closure",
"isOpen_extChartAt_source",
"AddMonoidAlgebra.le_infDegree_add",
"DFinsupp.sumAddHom_zero",
"Filter.tendsto_add_atTop_nat",
"PicardLindelof.FunSpace.uniformInducing_toContinuousMap",
"MulAction.orbit_smul_subset",
"max_mul_mul_le_max_mul_max'",
"Submodule.span_insert_eq_span",
"SimpleGraph.mem_cliqueFinset_iff",
"HasDerivAt.arsinh",
"Matrix.UnitaryGroup.toLin'_one",
"Codisjoint.himp_le_of_right_le",
"bit0_eq_zero",
"Fin.pos",
"Std.RBNode.insert_toList_zoom_node",
"IntermediateField.mem_mk",
"MeasureTheory.measurableEmbedding_of_fderivWithin",
"InitialSeg.trans_apply",
"List.rel_sections",
"Submodule.mul_comm",
"AlternatingMap.curryLeft_smul",
"bddAbove_smul_iff_of_pos",
"OrderDual.Preorder.dual_dual",
"Int.not_irrational",
"left_mem_affineSpan_pair",
"AddSubgroup.mem_smul_pointwise_iff_exists",
"CategoryTheory.Idempotents.app_p_comm",
"SetLike.exists",
"Fintype.exists_le_card_fiber_of_mul_le_card",
"Filter.blimsup_eq",
"Continuous.inv\u2080",
"Continuous.exp",
"MeasureTheory.quasiMeasurePreserving_div",
"isConnected_Ico",
"Subtype.coe_sup",
"SimpleGraph.swap_mem_interedges_iff",
"IsROrC.I_mul_I",
"MeasureTheory.SimpleFunc.mem\u2112p_iff",
"ENNReal.inv_limsup",
"Algebra.TensorProduct.map_comp",
"div_mem_comm_iff",
"Set.exists_image_iff",
"Nat.digits_zero_zero",
"NonUnitalAlgebra.inf_toNonUnitalSubsemiring",
"iInf_ne_top_subtype",
"Complex.zero_lt_real",
"AddSubgroup.pointwise_smul_le_pointwise_smul_iff",
"RingHom.ker_ne_top",
"MulAction.Supports.mono",
"Nat.Prime.ne_one",
"QuotientGroup.homQuotientZPowOfHom_id",
"continuousWithinAt_const",
"derivWithin_pow'",
"Set.Nonempty.inv",
"Finset.singleton_one",
"Equiv.Perm.ofSubtype_apply_coe",
"LieEquiv.refl_symm",
"Function.one_lt_const",
"measure_Ico_lt_top",
"CategoryTheory.Functor.initial_natIso_iff",
"ENNReal.top_ne_zero",
"frontier_Iic_subset",
"OrderIso.ext",
"Filter.comap_embedding_atBot",
"AlgEquiv.restrictScalars_apply",
"IsUnit.mul_left_eq_zero",
"smul_vsub_rev_vadd_mem_affineSpan_pair",
"Filter.EventuallyEq.differentiableAt_iff",
"nonempty_measurable_superset",
"Set.iInter_plift_down",
"SimpleGraph.edgeDensity_le_one",
"Finsupp.not_mem_graph_snd_zero",
"normUnit_eq_one",
"not_imp_eq_of_eq_false_right",
"AlgHom.one_apply",
"iInf_iInf_eq_right",
"Finset.Nonempty.sym",
"AddCommGroupCat.asHom_apply",
"PSet.Equiv.symm",
"SeparatedNhds.preimage",
"CliffordAlgebraQuaternion.toQuaternion_ofQuaternion",
"Finsupp.prod_antidiagonal_swap",
"LocalHomeomorph.image_source_eq_target",
"Std.BinomialHeap.Imp.Heap.WF.merge",
"lowerClosure_eq_bot_iff",
"compl_sup_compl_le",
"Stream'.Seq.mem_cons_of_mem",
"measurableEmbedding_const_smul\u2080",
"iff_true_iff",
"AlgebraicGeometry.quasiCompact_stableUnderBaseChange",
"EsakiaHom.ext",
"NonUnitalSubalgebra.coe_map",
"CategoryTheory.finitaryExtensive_of_preserves_and_reflects",
"Finset.min_insert",
"Set.ordConnected_uIoc",
"Asymptotics.IsBigO.pow",
"Monotone.map_csInf",
"GenLoop.continuous_toLoop",
"Quotient.exact'",
"eq_of_abs_sub_nonpos",
"FormalMultilinearSeries.apply_order_ne_zero'",
"IsHausdorff.haus",
"ProbabilityTheory.IndepSets.union_iff",
"FirstOrder.Language.Substructure.map_inf_comap_of_surjective",
"FractionalIdeal.one_mem_one",
"isCompl_toDual_iff",
"smul_inv''",
"Cardinal.toNat_apply_of_lt_aleph0",
"Metric.equicontinuous_of_continuity_modulus",
"StarSubalgebra.centralizer_le",
"List.Sorted.get_strictMono",
"ContractingWith.efixedPoint_mem'",
"MeasureTheory.Integrable.norm",
"Sum.Lex.inl_lt_inl_iff",
"MeasureTheory.ProbabilityMeasure.tendsto_nhds_iff_toFiniteMeasure_tendsto_nhds",
"List.Sublist.count_le",
"min_one",
"Finset.eq_of_mem_slice",
"MvPolynomial.IsWeightedHomogeneous.add",
"Int.ofNat_eq_zero",
"contMDiffAt_const",
"Subring.prod_mono",
"Set.diff_diff_cancel_left",
"Set.mapsTo_prod_map_diagonal",
"VectorAllP.imp",
"Equiv.Perm.support_refl",
"IsLocallyConstant.iff_isOpen_fiber",
"Polynomial.fderivWithin_aeval",
"CategoryTheory.IsFilteredOrEmpty.of_isRightAdjoint",
"MonotoneOn.Ico",
"QuadraticForm.linMulLin_comp",
"LocallyConstant.coe_desc",
"IsLprojection.le_def",
"CategoryTheory.Faithful.div_faithful",
"HNNExtension.NormalWord.consRecOn_ofGroup",
"isG\u03b4_empty",
"ack_le_iff_right",
"Tuple.antitone_pair_of_not_sorted",
"Matrix.isDiag_iff_diagonal_diag",
"AddValuation.map_add",
"CoheytingHom.coe_id",
"CategoryTheory.Limits.image.isoStrongEpiMono_inv_comp_mono",
"Prod.uIcc_eq",
"Finsupp.mapRange.linearEquiv_apply",
"DifferentiableOn.hasFDerivAt",
"decidableEq_inr_neg",
"DifferentiableWithinAt.sub_const",
"AEMeasurable.indicator",
"MeasureTheory.AEStronglyMeasurable.indicator",
"span_gramSchmidt_Iio",
"Set.bounded_gt_Ioo",
"Finset.ssubset_of_ssubset_of_subset",
"Set.Infinite.of_smul_set",
"DifferentiableAt.sinh",
"Set.Subsingleton.coe_sort",
"ContMDiffAt.smul",
"MulAction.fixed_eq_iInter_fixedBy",
"upperSemicontinuousOn_const",
"Finset.singleton_ne_empty",
"List.some_nthLe_eq",
"UpperHalfPlane.continuous_re",
"Subalgebra.coe_sub",
"Embedding.isSeparable_preimage",
"Complex.continuousOn_tan",
"Sum.Lex.inl_le_inr",
"WithTop.forall",
"Set.Finite.einfsep_pos",
"Ideal.mem_span_singleton'",
"MeasureTheory.Ioo_ae_eq_Ioc'",
"CategoryTheory.Equivalence.functor_map_inj_iff",
"HasFPowerSeriesOnBall.add",
"CategoryTheory.IsDetector.isSeparator",
"Set.mem_sep_iff",
"List.mem_keys_of_mem_keys_kerase",
"IsCompactOperator.clm_comp",
"EuclideanHalfSpace.ext",
"NonUnitalStarAlgebra.map_top",
"Set.finite_range_findGreatest",
"Filter.Tendsto.zpow",
"MeasurableEmbedding.comap_eq",
"HasFPowerSeriesOnBall.analyticAt",
"CauSeq.lim_le",
"Int.div_mul_cancel",
"Complex.nnnorm_int",
"Subsemigroup.map_map",
"HasFDerivWithinAt.mono_of_mem",
"Set.Icc_subset_Ioi_iff",
"Algebra.toSubsemiring_eq_top",
"Set.mem_union_left",
"LinearPMap.adjoint_apply_eq",
"not_eq_of_eq_true",
"elementalStarAlgebra.isClosed",
"Matrix.toLin'_apply",
"Ordinal.blsub_id",
"measurable_quotient_mk'",
"EuclideanGeometry.left_ne_right_of_oangle_eq_pi_div_two",
"isClopen_compl_iff",
"LieSubalgebra.mem_mk_iff",
"Ici_toDual",
"antivaryOn_toDual_right",
"Asymptotics.SuperpolynomialDecay.mul_param_zpow",
"Cardinal.mk_out",
"Module.Ray.someRayVector_ray",
"OrderMonoidHom.toMonoidHom_eq_coe",
"Fin.forall_fin_succ_pi",
"apply_wcovby_apply_iff",
"Subtype.mk_le_mk",
"StarSubalgebra.range_le",
"ContinuousOrderHom.coe_id",
"IntervalIntegrable.aestronglyMeasurable",
"AntisymmRel.trans_wcovby",
"Continuous.prod_mk",
"HasStrictFDerivAt.eq_implicitFunctionOfComplemented",
"Matrix.of_add_of",
"ContinuousMap.one_comp",
"CategoryTheory.Limits.IsZero.eq_of_tgt",
"\u03b5NFA.start_one",
"CategoryTheory.Limits.WalkingPair.equivBool_symm_apply_false",
"Matrix.conjTranspose_list_prod",
"Nat.sqrt_add_eq'",
"SetTheory.PGame.moveRight_nim'",
"Turing.Tape.write_nth",
"Hyperreal.Infinite.ne_zero",
"SimpleGraph.Subgraph.neighborSet_sup",
"CategoryTheory.isIso_of_epi_of_nonzero",
"MeasureTheory.Measure.coe_add",
"Option.Mem.leftUnique",
"Quot.liftOn\u2082_mk",
"CochainComplex.mkHom_f_1",
"div_neg_of_pos_of_neg",
"RingHom.Finite.of_surjective",
"Finset.ne_empty_of_mem",
"FractionalIdeal.map_comp",
"CircleDeg1Lift.monotone",
"Finset.disjiUnion_cons",
"CategoryTheory.op_zsmul",
"Ordinal.nat_cast_succ",
"IsROrC.conj_re",
"Set.div_eq_empty",
"MvPolynomial.counit_surjective",
"CauSeq.Completion.mk_mul",
"Submodule.quotEquivOfEq_mk",
"Continuous.Icc_extend'",
"Function.Injective.sum_elim",
"RingHom.map_list_prod",
"ENNReal.essSup_const_mul",
"Std.RBSet.mkRBSet_eq",
"Ideal.IsPrime.radical_le_iff",
"NonUnitalStarSubalgebra.mem_toNonUnitalSubalgebra",
"Set.image_iInter_subset",
"pathComponent_symm",
"MeasureTheory.ProbabilityMeasure.toFiniteMeasure_embedding",
"Hindman.exists_FP_of_finite_cover",
"Prefunctor.pathStar_apply",
"Set.Nonempty.of_mul_left",
"Nat.bit0_lt_bit1",
"Hyperreal.InfinitePos.pos",
"CliffordAlgebra.EquivEven.Q'_apply",
"SemiNormedGroupCat.explicitCokernelDesc_zero",
"StrictMono.orderIsoOfSurjective_symm_apply_self",
"OrderIso.toFun_eq_coe",
"LinearMap.mulRight_toAddMonoidHom",
"MulAut.mul_apply",
"WithBot.le_of_add_le_add_right",
"ExistsContDiffBumpBase.u_smooth",
"StrictMonoOn.comp_strictAntiOn",
"CentroidHom.toAddMonoidHom_eq_coe",
"SetSemiring.up_down",
"IsCoatom.le_iff",
"IsCompact.totallyBounded",
"isBoundedBilinearMap_smulRight",
"IsMax.Ici_eq",
"Finset.forall_of_forall_insert",
"MonotoneOn.quasiconvexOn",
"Subfield.closure_le",
"NumberField.Units.dirichletUnitTheorem.seq_norm_ne_zero",
"Ultrafilter.em",
"AlternatingMap.curryLeft_add",
"pi_nnnorm_const'",
"Int.nonneg_of_pos",
"contDiffWithinAt_inter",
"Finsupp.support_neg",
"Abelianization.lift.unique",
"Unitization.inr_zero",
"ENNReal.coe_lt_top",
"SmoothBumpFunction.nhds_basis_support",
"Set.center_eq_univ",
"WithTop.toDual_symm_apply",
"Finset.div_inter_subset",
"LinearIsometryEquiv.norm_map",
"BotHom.coe_comp",
"isOpen_uniformity",
"SemidirectProduct.inr_injective",
"Nat.cast_min",
"Derivation.sub_apply",
"Complex.normSq_eq_zero",
"ContMDiffOn.congr",
"Fin.prod_univ_zero",
"Order.bot_lt_succ",
"IsScalarTower.coe_toAlgHom",
"parallelogram_law_with_nnnorm",
"Rat.zero_den",
"Finset.codisjoint_inf_right",
"Finsupp.comapSMul_single",
"FirstOrder.Language.Equiv.toEmbedding_toHom",
"Commute.multiset_sum_left",
"Finset.monotone_filter_left",
"DFinsupp.not_mem_neLocus",
"isClopen_biInter_finset",
"continuous_empty_function",
"Set.Ioo_subset_Ico_self",
"ConvexCone.Salient.anti",
"Submodule.map_smul''",
"IsROrC.continuous_normSq",
"Set.sInf_eq_sInter",
"ContDiffWithinAt.cos",
"Polynomial.degree_lt_wf",
"PSet.equiv_iff_mem",
"CategoryTheory.ihom.coev_ev",
"CategoryTheory.Limits.Bicone.toCone_\u03c0_app_mk",
"Real.hasDerivAt_arcsin",
"SetLike.homogeneous_coe",
"Field.subset_closure",
"lowerSemicontinuous_iff_isOpen_preimage",
"HasDerivAt.tendsto_punctured_nhds",
"CategoryTheory.Subgroupoid.map_comap_le",
"Multiset.of_mem_filter",
"MeasureTheory.AEDisjoint.of_null_right",
"extChartAt_preimage_mem_nhdsWithin",
"ApplicativeTransformation.preserves_pure",
"Set.Subsingleton.measurableSet",
"Set.Bounded.mono",
"MulPosMono.toMulPosStrictMono",
"inf_le_bihimp",
"IsometryEquiv.symm_trans_apply",
"Real.pi_div_two_eq_arcsin",
"deriv_sq_norm_ne_zero",
"Filter.eventuallyEq_principal",
"ContinuousLinearMap.coe_restrictScalars",
"Finset.fold_singleton",
"IsCoatom.lt_iff",
"ContinuousOrderHom.comp_id",
"Matrix.fromBlocks_apply\u2081\u2081",
"IsClopen.isClosed",
"LieModuleHom.add_apply",
"Finset.Icc_subset_Icc_left",
"Complex.I_re",
"frobenius_sub",
"LinearMap.IsAlt.self_eq_zero",
"CliffordAlgebra.EquivEven.reverse_e0",
"NonUnitalAlgHom.subtype_comp_codRestrict",
"star_intCast",
"Filter.exists_eventuallyEq_const_of_forall_separating",
"Int.eq_of_associated_of_nonneg",
"Finset.mem_union_left",
"Fin.val_two",
"lowerBounds_mono",
"HolderOnWith.ediam_image_inter_le_of_le",
"edist_dist",
"StrictConcaveOn.inf",
"CategoryTheory.Limits.BinaryBicone.fstKernelFork_\u03b9",
"TensorProduct.lid_tmul",
"MeasurableSpace.generateFrom_sup_generateFrom",
"ContinuousAt.rpow_const",
"Subalgebra.coe_restrictScalars",
"TrivSqZeroExt.inl_add",
"Equiv.Perm.VectorsProdEqOne.rotate_zero",
"ENNReal.continuous_nnreal_sub",
"Finset.mul_univ_of_one_mem",
"PowerSeries.coe_X",
"Set.image_subtype_val_Ioo_subset",
"Metric.isometry_inl",
"Ideal.le_map_of_comap_le_of_surjective",
"embedding_of_embedding_compose",
"LieSubalgebra.span_empty",
"hammingNorm_lt_one",
"SubMulAction.mem_carrier",
"Monoid.End.coe_pow",
"IsROrC.I_mul_I_of_nonzero",
"MeasureTheory.Integrable.integral_condDistrib",
"MeasureTheory.StronglyMeasurable.div",
"pow_ne_zero",
"MeasurableEquiv.coe_smul\u2080",
"Even.mul_left",
"BoxIntegral.Prepartition.le_biUnionIndex",
"MeasureTheory.Submartingale.sum_mul_sub'",
"iteratedFDeriv_add_apply'",
"QuotientGroup.homQuotientZPowOfHom_comp_of_rightInverse",
"SimpleGraph.Reachable.trans",
"zero_smul",
"RingEquiv.toMonoidHom_commutes",
"le_of_neg_of_one_div_le_one_div",
"CategoryTheory.IsCofilteredOrEmpty.of_initial",
"IsLeast.isGLB",
"EMetric.diam_eq_zero_iff",
"Antitone.vecCons",
"Set.image3_congr'",
"LatticeOrderedGroup.m_neg_part_def",
"MeasureTheory.SimpleFunc.inv_apply",
"LinearMap.range_toAddSubmonoid",
"LinearMap.detAux_id",
"min_lt_iff",
"Submodule.mem_div_iff_forall_mul_mem",
"ModuleCat.monoidalClosed_curry",
"List.Forall\u2082.mp",
"CategoryTheory.\u03bc_inv_naturality",
"Asymptotics.IsLittleO.add",
"CategoryTheory.FreeBicategory.mk_id",
"Set.diff_univ",
"ContinuousMultilinearMap.coe_restrictScalars",
"prod_mk_mem_compRel",
"Real.Angle.cos_coe",
"nhds_eq_nhds_emetric_ball",
"Std.RBNode.Stream.toList_nil",
"Set.ordConnected_biInter",
"VectorBundleCore.mem_localTriv_source",
"MeasureTheory.Measure.measure_Ioo_eq_zero",
"mul_left_cancel''",
"AbstractCompletion.continuous_map",
"LipschitzOnWith.edist_le_mul_of_le",
"ContinuousOrderHom.coe_copy",
"DiffContOnCl.const_smul",
"Asymptotics.superpolynomialDecay_iff_abs_tendsto_zero",
"IsGenericPoint.specializes_iff_mem",
"CauSeq.mul_pos",
"ContinuousMultilinearMap.bounds_bddBelow",
"MeasureTheory.Measure.lintegral_bind",
"PrimeMultiset.coeNat_injective",
"List.argmin_singleton",
"Subsemiring.center_eq_top",
"IsLUB.exists_between_sub_self",
"WithLp.equiv_symm_single",
"AntitoneOn.convex_le",
"inducing_subtype_val",
"Subgroup.mem_iSup_of_mem",
"NonUnitalStarSubalgebra.coe_center",
"Set.image2_subset_iff",
"MvPolynomial.le_weightedTotalDegree",
"EReal.coe_sub",
"Finset.inf'_insert",
"MeasureTheory.LocallyIntegrableOn.add",
"Polynomial.SplittingField.splits",
"monotone_of_deriv_nonneg",
"List.dProdIndex_nil",
"CategoryTheory.MonoidalClosed.homEquiv_symm_apply_eq",
"NormedAddGroupHom.smul_apply",
"MeasureTheory.L1.SimpleFunc.setToL1SCLM_mono",
"MeasureTheory.map_mul_right_eq_self",
"ENNReal.mul_left_mono",
"UpperSet.prod_sup",
"selfAdjoint.val_one",
"Filter.map\u2082_map_left_anticomm",
"Multiset.monotone_filter_left",
"ContinuousLinearMap.map_smul_of_tower",
"Disjoint.dual",
"Subsemigroup.le_comap_of_map_le",
"AbstractCompletion.extend_comp_coe",
"WithTop.ofDual_apply_top",
"Pi.default_def",
"List.singleton_eq",
"CocompactMap.coe_copy",
"Subgroup.map_inf_le",
"Filter.HasBasis.clusterPt_iff",
"ENNReal.inv_liminf",
"Algebra.TensorProduct.productMap_left",
"IsPreconnected.intermediate_value",
"RingEquiv.toNonUnitalRingHom_refl",
"Submodule.sInf_coe",
"Orthonormal.toSubtypeRange",
"LocalHomeomorph.contDiff_unitBallBall_symm",
"SupHom.dual_id",
"TypeVec.lastFun_subtypeVal",
"isMinFilter_dual_iff",
"CategoryTheory.GrothendieckTopology.Subpresheaf.sheafify_sheafify",
"CategoryTheory.Preadditive.forkOfKernelFork_\u03b9",
"USize.modn_toNat",
"SetTheory.PGame.numeric_zero",
"Function.minimalPeriod_id",
"CliffordAlgebraComplex.Q_apply",
"IsTorsion.quotient_iff",
"Subring.toSubsemiring_injective",
"Monoid.CoprodI.NeWord.singleton_last",
"Ideal.isPrime_of_irreducible_absNorm",
"MeasureTheory.MeasurePreserving.prod",
"Equiv.eq_symm_comp",
"Units.isSMulRegular",
"Finite.one_lt_card",
"IsClosed.upperSemicontinuousOn_indicator",
"Subsingleton.strictMono",
"MvPolynomial.vars_one",
"CategoryTheory.Limits.Multicoequalizer.multicofork_\u03b9_app_right",
"mul_eq_zero_comm",
"NonUnitalSubsemiring.mem_closure_iff",
"LocalHomeomorph.EqOnSource.eqOn",
"Function.is_periodic_id",
"LinearMap.mulRight_apply",
"List.Chain'.take",
"Set.symmDiff_def",
"Smooth.smul",
"SimpleGraph.Walk.darts_cons",
"IsROrC.conj_I",
"ContinuousAlternatingMap.map_piecewise_add",
"SlashInvariantForm.coe_zero",
"MeasureTheory.AEEqFun.lintegral_mk",
"Set.inter_compl_self",
"cauchySeq_tendsto_of_complete",
"Cubic.ne_zero_of_b_ne_zero",
"wellFounded_antisymmetrization_iff",
"isIrreducible_singleton",
"Subgroup.prod_mono_right",
"Quaternion.mul_coe_eq_smul",
"tendsto_pow_atTop_nhds_0_of_abs_lt_1",
"ConjAct.toConjAct_mul",
"Function.Commute.comp_right",
"Filter.Germ.le_def",
"Set.eqOn_comp_right_iff",
"Monovary.sum_comp_perm_smul_le_sum_smul",
"MvPolynomial.constantCoeff_smul",
"AlgHom.range_comp",
"CategoryTheory.Functor.const.opObjUnop_hom_app",
"GroupFilterBasis.prod_subset_self",
"MeasureTheory.SimpleFunc.bind_apply",
"TopHom.ext",
"Function.Surjective.of_comp_iff",
"SetTheory.PGame.Impartial.equiv_zero_iff_le",
"Nat.div2_zero",
"Sum.getLeft?_inl",
"AlternatingMap.domDomCongr_perm",
"Filter.coext",
"Matrix.conjTranspose_list_sum",
"coeFn_fourierLp",
"Filter.prod_mono_right",
"BoxIntegral.Prepartition.iUnion_biUnionTagged",
"NNRat.toNNRat_mono",
"uniformContinuous_toMul",
"contDiffOn_succ_of_fderiv_apply",
"MeasureTheory.AEStronglyMeasurable.inv",
"RingHom.asBoolAlg_id",
"BoundedContinuousFunction.dist_toContinuousMap",
"HilbertBasis.tsum_inner_mul_inner",
"tendsto_ceil_atTop",
"SimpleGraph.Walk.length_concat",
"MeasureTheory.condexpL1_congr_ae",
"IsBaseChange.inductionOn",
"FirstOrder.Language.Substructure.cg_closure",
"ModelWithCorners.right_inv",
"Cardinal.toNat_lt_of_lt_of_lt_aleph0",
"isOpen_gt'",
"Disjoint.inf_right",
"Filter.mem_inf_iff",
"LocalHomeomorph.refl_prod_refl",
"Cardinal.mk_preimage_of_injective_of_subset_range_lift",
"PerfectClosure.liftOn_mk",
"Set.iInter\u2082_mono'",
"LipschitzWith.continuous_compLp",
"LinearEquivClass.range",
"continuous_const",
"DFinsupp.neLocus_add_left",
"Order.Ideal.carrier_eq_coe",
"Traversable.foldlm_map",
"Subsemiring.range_fst",
"Set.add_mem_Ioc_iff_left",
"MeasureTheory.ae_eq_univ",
"AddSubgroup.pointwise_smul_le_iff\u2080",
"total_of",
"ModularGroup.lcRow0_apply",
"Monotone.map_sSup_of_continuousAt'",
"StrictMono.minimal_of_minimal_image",
"ContinuousMap.HomotopyEquiv.toFun_eq_coe",
"PicardLindelof.continuousOn",
"Finset.prod_range_add_div_prod_range",
"NonUnitalStarSubalgebra.coe_centralizer",
"LocalEquiv.prod_target",
"Set.preimage_comp_eq",
"LinearPMap.map_smul",
"Set.biUnion_lt_succ",
"Antivary.antivaryOn",
"convex_Ioi",
"Int.eq_of_mul_eq_one",
"Setoid.mkClasses_classes",
"Nat.recDiagAux_succ_succ",
"WithTop.coe_mono",
"Function.update_inv",
"Set.Icc_union_Ico_eq_Ico",
"balanced_iInter",
"FirstOrder.Language.BoundedFormula.realize_rel",
"Set.biInter_eq_iInter",
"Set.Icc.coe_pow",
"Int.covby_iff_succ_eq",
"FreeLieAlgebra.liftAux_map_mul",
"Con.comap_rel",
"Equiv.emptySum_apply_inr",
"hasBasis_nhdsSet_Ici_Ioi",
"Set.image_mulSingle_Ioc_left",
"LocalEquiv.prod_source",
"isTotallySeparated_singleton",
"Filter.tendsto_atTop_add_nonneg_right",
"QuasiconcaveOn.inf",
"isOpenMap_ofDual",
"finprod_false",
"Subsemigroup.coe_bot",
"bihimp_right_surjective",
"PrimeSpectrum.comap_singleton_isClosed_of_isIntegral",
"Finset.insert_subset_insert",
"MeasureTheory.aestronglyMeasurable_one",
"Subring.pointwise_smul_subset_iff",
"IsOpenMap.nhds_le",
"Set.nontrivial_iff_exists_lt",
"Monotone.Ici",
"RingHom.congr_fun",
"iInf_mono",
"PerfectionMap.hom_ext",
"Set.wellFoundedOn_singleton",
"coe_mul_unitClosedBall",
"upperClosure_empty",
"BoundedContinuousFunction.dist_mkOfCompact",
"CategoryTheory.Groupoid.isIsomorphic_iff_nonempty_hom",
"OpenEmbedding.comp",
"LinearEquiv.ofLinear_symm_apply",
"himp_le_himp_right",
"LipschitzWith.ae_differentiableAt_real",
"forall_of_ball",
"ContinuousLinearMap.adjointAux_apply",
"Set.isSimpleOrder_Iic_iff_isAtom",
"SetTheory.PGame.leftMoves_add",
"Function.not_lt_argmin",
"isLeast_univ",
"MeasureTheory.Adapted.progMeasurable_of_discrete",
"WellFounded.rank_lt_of_rel",
"Finset.erase_mem_shadow",
"Subalgebra.mem_centralizer_iff",
"Filter.le_vsub_iff",
"Set.Ioc.coe_pow",
"CategoryTheory.Monoidal.rightUnitor_inv_app",
"AddMonoidHom.mul_apply",
"AlgebraicGeometry.PresheafedSpace.stalkMap_germ'",
"ContinuousLinearEquiv.equivOfInverse_apply",
"CategoryTheory.yonedaEquiv_apply",
"Or.symm",
"ContDiff.restrict_scalars",
"Ordinal.nat_cast_pos",
"QuaternionAlgebra.nat_cast_re",
"PartitionOfUnity.le_one",
"LocalHomeomorph.image_open_of_open'",
"MonoidHomClass.continuous_of_bound",
"antitone_const",
"tsum_mul_tsum",
"Polynomial.degree_one",
"Zsqrtd.coe_int_sub",
"MeasureTheory.Measure.MutuallySingular.smul_nnreal",
"compl_singleton_mem_nhds_iff",
"DualNumber.snd_eps",
"FixedPoints.minpoly_eq_minpoly",
"Ordnode.Valid.merge",
"Set.mapsTo_empty",
"Set.EqOn.comp_right",
"Option.map_id''",
"MeasureTheory.measure_mono_null",
"MonoidHom.map_finprod_of_injective",
"Set.Ioc.coe_mul",
"tendsto_ceil_left_pure_ceil",
"CliffordAlgebra.\u03b9_comp_lift",
"lineMap_le_left_iff_le",
"Set.iInter_comm",
"Associates.out_injective",
"ContinuousLinearMap.op_norm_le_of_shell",
"PLift.up_surjective",
"SetTheory.Game.PGame.equiv_iff_game_eq",
"Set.mk_preimage_sigma_eq_empty",
"Dense.exists_lt",
"ContDiffAt.inversion",
"TypeVec.dropFun_appendFun",
"Set.image2_vsub",
"zpow_lt_zpow",
"MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonpos'",
"nndist_lineMap_right",
"Action.sum_hom",
"Rack.op_act_op_eq"
]
}