task stringclasses 1
value | query stringlengths 12 5.5k | positives listlengths 1 42 | candidates listlengths 100 141 |
|---|---|---|---|
t1 | alpha : Type u_1 lt : alpha -> alpha -> Bool x : alpha self : BinaryHeap alpha lt e : self.max = some x h : ~0 < self.arr.size |- False | [
"BinaryHeap.max",
"Array.get"
] | [
"BinaryHeap.max",
"Array.get",
"Real.Angle.coe_neg",
"Set.mem_singleton_iff",
"CategoryTheory.SimplicialObject",
"Matrix.mulVec",
"h2U.interior_eq",
"Word.mem_smul_iff",
"Scheme.Spec_map_presheaf_map_eqToHom",
"Nat.cast_sum",
"Upto.GT",
"Polynomial.eval_C",
"LieSubmodule.isCompl_iff_coe_toSu... |
t1 | alpha : Type u_1 le : alpha -> alpha -> Bool s' s : Heap alpha |- tail? le s = some s' -> s'.NoSibling | [
"Heap.tail"
] | [
"Heap.tail",
"X.fiber",
"Option.some_inj",
"IntFractPair.coe_of_rat_eq",
"List.map_singleton",
"Algebra.comap_top",
"Metric.continuous_iff'",
"Finset.erase_eq_of_not_mem",
"Limits.biproduct.components",
"Products.eval_eq",
"Ideal.map_sup",
"Opens.functor_obj_map_obj",
"Set.smul_set_compl",
... |
t1 | alpha : Type u_1 le : alpha -> alpha -> Bool s : Heap alpha |- (tail le s).NoSibling | [
"Heap.tail"
] | [
"Heap.tail",
"List.forIn'.loop",
"LinearEquiv.range",
"Submodule.mem_iSup_of_chain",
"R.eventually_log_b_mul_pos",
"Set.image2_iUnion_right",
"Bool.and_false",
"S.exact_iff_epi_toCycles",
"Set.inv_Ioo_0_left",
"Colimit.w_apply",
"I.add_mem",
"Submodule.coe_toAddSubmonoid",
"ContinuousMultili... |
t1 | alpha : Type u_1 le : alpha -> alpha -> Bool s' s : Heap alpha h : s.NoSibling |- tail? le s = some s' -> s.size = s'.size + 1 | [
"Heap.tail"
] | [
"Heap.tail",
"Vector.toList_map",
"List.forall₂_and_left",
"Commute.zero_left",
"Nat.factors_eq",
"Int.dvd_trans",
"hf_meas.ae_eq_mk",
"Disjoint.mul_eq_one_iff",
"Nat.cast_ite",
"S.p_fromOpcycles",
"NonemptyFinLinOrd.epi_iff_surjective",
"Nat.add_mul_mod_self_left",
"Set.image_compl_eq",
"... |
t1 | alpha : Type u_1 le : alpha -> alpha -> Bool s : Heap alpha h : s.NoSibling |- (tail le s).size = s.size - 1 | [
"Heap.tail"
] | [
"Heap.tail",
"R.prop.commute",
"NNReal.coe_two",
"U.coe_inner",
"D.t'_iij",
"Equalizer.Presieve.Arrows.forkMap",
"hP.inf_normalizer_sylow",
"Cardinal.sum_const",
"LinearIsometryEquiv.nnnorm_map",
"Finset.sum_pi_single'",
"CategoryTheory.epi_iff_surjective",
"Nat.bit1_div_two",
"Algebra.adjoi... |
t1 | alpha : Type u_1 le : alpha -> alpha -> Bool s : Heap alpha h : s.NoSibling |- ((tail? le s).getD nil).size = s.size - 1 | [
"Heap.size_tail"
] | [
"Heap.size_tail",
"ENNReal.inv_lt_inv",
"Cochain.zero_comp",
"Category.id_comp",
"W₂.Q",
"M.toBlock",
"Basis.equivFun_apply",
"hZ₁₂.inter_eq",
"LieIdeal.mem_killingCompl",
"LinearMap.map_coe_ker",
"Exact.lift",
"colimit.comp_coconePointUniqueUpToIso_hom_assoc",
"hinj.pairwiseDisjoint_image",... |
t1 | alpha : Type u_1 le : alpha -> alpha -> Bool s : Heap alpha h : s.NoSibling tl : Heap alpha eq : tail? le s = some tl |- ((some tl).getD nil).size = s.size - 1 | [
"Heap.size_tail"
] | [
"Heap.size_tail",
"Nat.add_one_le_iff",
"pi_pos.ne",
"Set.indicator_univ",
"Fintype.piFinset_univ",
"Set.not_mem_of_mem_diff",
"Nat.abs_cast",
"PiNat.dist_nonneg",
"EqOn.image_eq_self",
"List.sym",
"Submodule.coe_mem",
"ProbabilityTheory.setIntegral_compProd",
"hσ.coe_stabilzer_mk",
"ContD... |
t1 | alpha : Type u_1 le : alpha -> alpha -> Bool s' s : Heap alpha |- tail? le s = some s' -> s'.size < s.size | [
"Heap.tail"
] | [
"Heap.tail",
"Fin.mul_comm",
"Set.coe_toFinset",
"mk_image_le.trans",
"S.toCycles",
"MorphismProperty.LeftFraction.map_comp_map_s_assoc",
"N.index_mul_card",
"Subgroup.commutator_eq_bot_iff_le_centralizer",
"Prod.tendsto_iff",
"Finsupp.singleAddHom_apply",
"H.invApp",
"Finset.noncommProd_eq_pr... |
t1 | alpha : Type u_1 s : Heap alpha le : alpha -> alpha -> Bool tl : Heap alpha hwf : WF le s |- Heap.tail? le s = some tl -> WF le tl | [
"Heap.tail"
] | [
"Heap.tail",
"nposℝ.ne'",
"SheafedSpace.forgetToPresheafedSpace",
"Substring.drop",
"ZeroHom.toFun_eq_coe",
"M.dest_eq_dest'",
"Units.val_oneSub",
"LieHom.map_add",
"Ordering.Compares",
"Measure.fst_map_prod_mk₀",
"CharP.ofCharZero",
"Closeds.compl_bijective.injective.eq_iff",
"Complex.isPri... |
t1 | alpha : Type u_1 s : Heap alpha le : alpha -> alpha -> Bool hwf : WF le s |- WF le (Heap.tail le s) | [
"Heap.tail"
] | [
"Heap.tail",
"Matrix.toLinAlgEquiv_symm",
"Metric.tendsto_nhds",
"LinearMap.stdBasis_ne",
"Algebra.id.map_eq_self",
"Finset.iSup_coe",
"Filter.eventually_ge_atTop",
"Ideal.span_singleton_one",
"Set.image_add_const_uIcc",
"List.toFinmap'",
"LieSubmodule.coe_toEnd",
"Multiset.filter_add",
"Add... |
t1 | sep : String escape : Bool m n : Name |- m.beq n = true <-> m = n | [
"Name.beq",
"And.comm"
] | [
"Name.beq",
"And.comm",
"Pi.single_star",
"AssocList.forIn",
"_root_.Equiv.symm_apply_apply",
"ContinuousLinearMap.smulRightL",
"B.dualBasis",
"List.isEmpty",
"CochainComplex.single₀_map_f_zero",
"AlgEquiv.symm_toRingEquiv",
"LinearMap.BilinForm.flip_apply",
"NeZero.ne'",
"D.t_inv_apply",
... |
t1 | a b : Nat |- ^(a ^ b) = ^a ^ b | [
"Nat.pow_succ",
"Int.ofNat_mul"
] | [
"Nat.pow_succ",
"Int.ofNat_mul",
"hP.basicOpen_iff_localization",
"Over.comp_left",
"Real.Angle.toReal_zero",
"Localization.isoOfHom_hom",
"KaehlerDifferential.kerTotal",
"Finset.cons_eq_insert",
"NormedAddGroupHom.completion_coe_to_fun",
"List.map_eq_map",
"inst₁.natCast_zero",
"Pi.single_eq_... |
t1 | case succ a b : Nat ih : ^(a ^ b) = ^a ^ b |- ^(a ^ (b + 1)) = ^a ^ (b + 1) | [
"Nat.pow_succ",
"Int.ofNat_mul"
] | [
"Nat.pow_succ",
"Int.ofNat_mul",
"Metric.mem_ball",
"LowerSet.coe_iInf",
"StarAlgebra.adjoin_le_iff",
"Polynomial.aeval_def",
"NonUnitalStarAlgebra.adjoin_eq",
"Equiv.symm_apply_eq",
"Nat.eq_zero_of_zero_dvd",
"BinaryCofan.mk_inl",
"_root_.disjoint_comm",
"AffineScheme.ofHom",
"Localization.... |
t1 | a : Int b : Nat h : 0 < a |- 0 < a ^ b | [
"Int.eq_natAbs_of_zero_le",
"Int.le_of_lt",
"Int.ofNat_zero",
"Int.ofNat_pow",
"Int.ofNat_lt"
] | [
"Int.eq_natAbs_of_zero_le",
"Int.le_of_lt",
"Int.ofNat_zero",
"Int.ofNat_pow",
"Int.ofNat_lt",
"ENNReal.toReal_nat",
"Fin.extractNth",
"Set.toFinset_union",
"Pi.inv_def",
"Submonoid.LocalizationMap.mk'_self",
"Order.lt_succ_iff",
"QuotientGroup.mk_prod",
"hB1_has.tsum_eq",
"Pos.ext_iff",
... |
t1 | a : Nat |- 0 < ^a <-> 0 < a | [
"Int.ofNat_zero",
"Int.ofNat_lt"
] | [
"Int.ofNat_zero",
"Int.ofNat_lt",
"Real.arccos_eq_zero",
"yzero.le",
"AlgHom.comp_toLinearMap",
"HomogeneousIdeal.mem_iff",
"AddMonoidHom.id",
"Real.logb",
"Ideal.span_eq_bot",
"Function.minimalPeriod_eq_minimalPeriod_iff",
"Antitone.map_sSup_of_continuousAt'",
"B.orthogonal_top",
"Nat.surje... |
t1 | x y : Nat |- ^(x <<< y) = ^x * ^(2 ^ y) | [
"Nat.shiftLeft_eq"
] | [
"Nat.shiftLeft_eq",
"PartENat.natCast_ne_top",
"Finset.untrop_sum'",
"Finset.sup_eq_union",
"FiniteDimensional.finrank",
"Polynomial.coeff_C_mul_X_pow",
"I.closure.eq_top_iff_one",
"BitVec.toInt",
"SheafedSpace.forget",
"RingEquiv.ofHomInv_symm_apply",
"Tape.map_move",
"ENNReal.mul_inv",
"EN... |
t1 | x y : Nat |- ^(x >>> y) = ^x / ^(2 ^ y) | [
"Nat.shiftRight_eq_div_pow",
"Int.ofNat_ediv"
] | [
"Nat.shiftRight_eq_div_pow",
"Int.ofNat_ediv",
"encode_injective.extend_apply",
"Subtype.coe_image_of_subset",
"Cardinal.one_toNat",
"Walk.isTrail_def",
"Fin.cases_succ'",
"Measure.sum_apply_of_countable",
"evaln.eq_8",
"ZeroMemClass.zero_def",
"pi_pos.ne'",
"Real.tsum_eq_tsum_fourierIntegral_... |
t1 | case neg a b : Nat h : ~b <= a |- b <= a /\ ^(a - b) = ^a - ^b \/ a < b /\ ^(a - b) = 0 | [
"Int.ofNat_sub_eq_zero"
] | [
"Int.ofNat_sub_eq_zero",
"Ordinal.typein_enum",
"biprod.inl_snd_assoc",
"LinearMap.isNilpotent_mulRight_iff",
"Finite.injective_iff_surjective",
"Rat.ofScientific_false_def",
"Real.exp_sub",
"TensorProduct.add_tmul",
"hF.sub",
"T.A₁_sa",
"Limits.image.fac",
"RingHom.congr_fun",
"cofinite.bli... |
t1 | case neg.h a b : Nat h : ~b <= a t : a < b |- a < b /\ ^(a - b) = 0 | [
"Int.ofNat_sub_eq_zero"
] | [
"Int.ofNat_sub_eq_zero",
"Set.indicator_apply_eq_zero.mpr",
"M.Indep",
"QuotientAddGroup.mk_zero",
"ENNReal.rpow_two",
"zero_eq_ofReal.mpr",
"disjoint_iff.mp",
"Or.inr",
"LinearMap.trace_mul_comm",
"Subsemiring.coe_toSubmonoid",
"List.take_length",
"Perm.smul_def",
"WalkingMultispan.Hom.snd"... |
t1 | a b : Nat |- ^(min a b) = min ^a ^b | [
"Nat.min_def",
"Int.min_def",
"Int.ofNat_le"
] | [
"Nat.min_def",
"Int.min_def",
"Int.ofNat_le",
"L.counit_app_functor",
"PrimeMultiset.prod_ofPrime",
"Cofork.condition",
"Array.ugetElem_eq_getElem",
"Seminorm.closedBall_zero_eq_preimage_closedBall",
"J₁.pullback_close",
"Equiv.inv_def",
"Set.empty_inter",
"Set.covBy_iff_exists_insert",
"mea... |
t1 | a b : Nat |- ^(max a b) = max ^a ^b | [
"Nat.max_def",
"Int.max_def",
"Int.ofNat_le"
] | [
"Nat.max_def",
"Int.max_def",
"Int.ofNat_le",
"LinearEquiv.symm_apply_eq",
"PartialEquiv.image_source_inter_eq'",
"FactorSet.coe_add",
"sub_eq_iff_eq_add'.mp",
"LieModuleHom.map_top",
"Int.bmod_pos",
"LinearMap.mem_ker",
"RingHom.coe_mk",
"IsPushout.isVanKampen_iff",
"Stream'.nil_append_stre... |
t1 | a : Int |- ^a.natAbs = if 0 <= a then a else -a | [
"Int.natAbs"
] | [
"Int.natAbs",
"Measure.smul_apply",
"SatelliteConfig.centerAndRescale",
"ShortComplex.p_fromOpcycles",
"UFModel.push",
"Continuous.fst",
"HomologicalComplex₂.shiftFunctor₁₂CommIso",
"hm₁₂.trans",
"Polynomial.comp_assoc",
"IsStoppingTime.measurableSpace_min",
"List.zipWith_nil_left",
"Functor.m... |
t1 | case h_1 n+ : Int n : Nat |- ^n = if 0 <= Int.ofNat n then Int.ofNat n else -Int.ofNat n case h_2 n+ : Int n : Nat |- ^n.succ = if 0 <= Int.negSucc n then Int.negSucc n else -Int.negSucc n | [
"Int.ofNat_eq_coe",
"Int.ofNat_nonneg"
] | [
"Int.ofNat_eq_coe",
"Int.ofNat_nonneg",
"List.applyId_eq_self",
"Int.smul_one_eq_cast",
"ShortComplex.homologyπ_naturality",
"LieModule.mem_maxTrivSubmodule",
"Measure.compProd",
"hI.basis_inter_ground.inter_eq_of_subset_indep",
"I.splitLower_ne_splitUpper",
"foldlm.ofFreeMonoid_comp_of",
"Finsu... |
t1 | case h_1 n+ : Int n : Nat |- ^n = if 0 <= Int.ofNat n then Int.ofNat n else -Int.ofNat n | [
"Int.ofNat_eq_coe"
] | [
"Int.ofNat_eq_coe",
"NNReal.coe_rpow",
"Filter.mem_comap",
"Set.singleton_mul_singleton",
"Matrix.vecMulLinear_apply",
"MonoidAlgebra.lift_single",
"FiniteField.isSquare_iff",
"Polynomial.C_mul'",
"filterMap.go",
"iᶜ.inter_subset_right",
"ENNReal.essSup_indicator_eq_essSup_restrict",
"AdjoinRo... |
t1 | case h_1 n+ : Int n : Nat |- ^n = if 0 <= ^n then ^n else -^n | [
"Int.ofNat_nonneg"
] | [
"Int.ofNat_nonneg",
"Rat.smul_def",
"zero_lt_one.not_le",
"Finset.compl_eq_univ_sdiff",
"Dart.edge",
"hnodup.mem_erase_iff",
"Measure.restrict_inter_add_diff₀",
"zero_lt_two.le",
"Nat.cast_le_one",
"Pi.inf_apply",
"Valuation.map_zero",
"S.X₂.pOpcycles",
"Nat.range_succ",
"Theory.model_iff"... |
t1 | case pos a : Int h : 0 <= a |- 0 <= a /\ ^a.natAbs = a \/ a < 0 /\ ^a.natAbs = -a case neg a : Int h : ~0 <= a |- 0 <= a /\ ^a.natAbs = a \/ a < 0 /\ ^a.natAbs = -a | [
"Int.natAbs_of_nonneg"
] | [
"Int.natAbs_of_nonneg",
"NNReal.coe_max",
"Real.log_neg_eq_log",
"ModelWithCorners.symm_comp_self",
"hA.spectral_theorem",
"IsSemisimple.sSup_atoms_eq_top",
"Pi.uniformSpace_comap_restrict_sUnion",
"SeminormFamily.basisSets_iff",
"Nat.mul_ne_zero_iff",
"Array.mkEmpty",
"Classical.imp_iff_right_i... |
t1 | case neg a : Int h : ~0 <= a |- 0 <= a /\ ^a.natAbs = a \/ a < 0 /\ ^a.natAbs = -a | [
"Int.not_le",
"Int.ofNat_natAbs_of_nonpos",
"Int.le_of_lt"
] | [
"Int.not_le",
"Int.ofNat_natAbs_of_nonpos",
"Int.le_of_lt",
"Subgroup.inv_mem_iff",
"LinearMap.restrictScalars_apply",
"Mon_.one_mul",
"Int.sign_neg",
"Filter.mem_sets",
"Path.reverse_comp",
"ZeroHom.map_zero",
"Metric.nhds_basis_ball.isVonNBounded_iff",
"Pi.mulSingle_mul",
"legendreSym.mul"... |
t1 | case pos.h a : Int h : 0 <= a |- 0 <= a /\ ^a.natAbs = a | [
"Int.natAbs_of_nonneg"
] | [
"Int.natAbs_of_nonneg",
"Monoid.pow_exponent_eq_one",
"MvPFunctor.wDest'_wMk",
"Real.sin_add_pi_div_two",
"NormalWord.prod",
"Nat.Partition.ext_iff",
"Equiv.plift.surjective.iInf_comp",
"Finset.prod_apply",
"_root_.zero_add",
"Prod.map_comp_map",
"Equiv.permCongr_apply",
"LinearMap.map_zero",
... |
t1 | case neg.h a : Int h : ~0 <= a |- a < 0 /\ ^a.natAbs = -a | [
"Int.not_le"
] | [
"Int.not_le",
"PartENat.top_add",
"Opens.map_coe",
"Lp.simpleFunc.coeFn_zero",
"Set.BijOn.image_eq",
"Real.nnnorm_of_nonneg",
"Associated.comm",
"NNReal.sub_def",
"List.length_eq_zero",
"LSeries.delta",
"Equiv.prod_assoc_preimage",
"FormalMultilinearSeries.sum",
"List.sum_map_hom",
"hf_f_L... |
t1 | case neg.h a : Int h : a < 0 |- a < 0 /\ ^a.natAbs = -a | [
"Int.ofNat_natAbs_of_nonpos",
"Int.le_of_lt"
] | [
"Int.ofNat_natAbs_of_nonpos",
"Int.le_of_lt",
"ConjClasses.mem_carrier_iff_mk_eq",
"NNReal.sqrt_le_iff_le_sq",
"Ideal.map_quotient_self",
"Nat.card_congr",
"MorphismProperty.LeftFraction.Localization.map_eq_iff",
"hβ.measurable_eq",
"Finset.prod_Ico_id_eq_factorial",
"sub_eq_zero.mp",
"Ordering.... |
t1 | a b c : Int |- a + b <= c <-> a <= c - b | [
"Int.add_zero",
"Int.sub_self",
"Int.sub_eq_add_neg",
"Int.add_assoc",
"Int.neg_neg",
"Int.add_le_add_iff_right"
] | [
"Int.add_zero",
"Int.sub_self",
"Int.sub_eq_add_neg",
"Int.add_assoc",
"Int.neg_neg",
"Int.add_le_add_iff_right",
"Finset.two_lt_card_iff",
"Bool.ite_eq_false_distrib",
"_root_.abs_of_pos",
"Set.iUnion_inter",
"TopCat.GlueData.isOpen_iff",
"Subalgebra.topologicalClosure_coe",
"PrimeMultiset.... |
t1 | a b c : Int |- a <= b + c <-> a - c <= b | [
"Int.neg_neg",
"Int.sub_eq_add_neg"
] | [
"Int.neg_neg",
"Int.sub_eq_add_neg",
"DFunLike.ext'_iff",
"ContinuousLinearMap.coe_sum'",
"NNReal.coe_lt_coe.symm",
"LieSubmodule.ucs_add",
"Substring.next",
"Nat.irreducible_iff_prime",
"one_lt_two.trans_le",
"Iterator.extract",
"QuotientGroup.ker_mk'",
"DMatrix.sub_apply",
"Set.prod_range_... |
t1 | a b : Int |- a + b <= 0 <-> a <= -b | [
"Int.zero_sub"
] | [
"Int.zero_sub",
"Finset.coe_sup'",
"List.map_replicate",
"Pi.mulSingle_mul",
"RingHom.ext",
"Fin.coe_pred",
"Module.mapEvalEquiv_symm_apply",
"Real.cos_zero",
"splitOnP.go",
"Polynomial.coeff_C_mul",
"Bool.cond_decide",
"OpenCover.fromGlued",
"LinearMap.smulRight_apply",
"Real.tsum_eq_tsum... |
t1 | a b : Int |- a + b <= 0 <-> b <= -a | [
"Int.add_comm"
] | [
"Int.add_comm",
"LocallyRingedSpace.IsOpenImmersion.ofRestrict_invApp",
"Multiset.id_traverse",
"Quot.liftOn_mk",
"hG.liftPropAt_iff_comp_inclusion",
"Associates.mk_zero",
"biprod.snd",
"hG.is_local",
"cofinite.bliminf_set_eq",
"IsUnit.isUnit_iff_mulRight_bijective",
"Set.nonempty_Ioo",
"Metri... |
t1 | a b : Int |- 0 <= a + b <-> -b <= a | [
"Int.zero_sub"
] | [
"Int.zero_sub",
"Equiv.ulift.coinduced_symm",
"Set.inter_eq_self_of_subset_right",
"Measure.mkMetric_toOuterMeasure",
"X.basicOpen_le",
"Sieve.generateSingleton_eq",
"_root_.div_mul_div_comm",
"Quotient.out_eq",
"FractionalIdeal.mem_coe",
"SetCoe.exists",
"IsometryEquiv.subLeft_symm_apply",
"S... |
t1 | a b : Int |- 0 <= a + b <-> -a <= b | [
"Int.add_comm"
] | [
"Int.add_comm",
"Composition.ofFn_blocksFun",
"Finsupp.total",
"Path.Homotopic.comp_lift",
"InnerProductGeometry.sin_angle_sub_mul_norm_of_inner_eq_zero",
"biprod.inr_snd_assoc",
"LinearMap.IsOrtho",
"LinearMap.range_eq_bot",
"Cubic.coeff_eq_c",
"Finset.prod_map",
"Subalgebra.coe_algebraMap",
... |
t1 | case mk.mk n val+_1 : Nat isLt+_1 : val+_1 < n val+ : Nat isLt+ : val+ < n |- <val+_1, isLt+_1> /= <val+, isLt+> <-> <val+_1, isLt+_1> < <val+, isLt+> \/ <val+_1, isLt+_1> > <val+, isLt+> | [
"Fin.mk.injEq",
"Nat.ne_iff_lt_or_gt"
] | [
"Fin.mk.injEq",
"Nat.ne_iff_lt_or_gt",
"iotaTR.go",
"OpenEmbedding.map_nhds_eq",
"RingedSpace.mem_basicOpen",
"Option.some_ne_none",
"S.v₀₁.comm₂₃",
"MulEquiv.symm_symm",
"AlgHom.map_sub",
"LeftHomologyData.leftHomologyπ_comp_leftHomologyIso_hom",
"Thunk.pure",
"Nat.add_lt_add_iff_left",
"S.... |
t1 | alpha : Type u_1 r : alpha -> alpha -> Prop inst+ : DecidableEq alpha beta : Type u_2 s : beta -> beta -> Prop p q : alpha * beta w : ~Prod.Lex r s p q |- ~r p.fst q.fst /\ (p.fst /= q.fst \/ ~s p.snd q.snd) | [
"Prod.lex_def",
"Decidable.not_and_iff_or_not_not"
] | [
"Prod.lex_def",
"Decidable.not_and_iff_or_not_not",
"Real.log_le_log_iff",
"Metric.isCompact_iff_isClosed_bounded",
"TotalSpace.ext_iff",
"Multiset.coe_mk",
"Bifunctor.map_id",
"Scheme.Hom.naturality_assoc",
"Asymptotics.IsBigOWith",
"Subtype.coe_injective",
"Finpartition.not_isEquipartition",
... |
t1 | case isTrue.h_2 m : Type u_1 -> Type u_2 beta : Type u_1 alpha : Type u_3 inst+ : Monad m f : beta -> alpha -> m beta arr : Array alpha j : Nat b : beta h+ : j < arr.size i+ i'+ : Nat H : arr.size <= i'+.succ + j |- (let_fun this := ...; do let __do_lift <- f b arr[j] foldlM.loop f arr arr.size ... i'+ (j + 1) __do_lif... | [
"Nat.succ_add",
"foldlM_eq_foldlM_data.aux",
"List.get_drop_eq_drop"
] | [
"Nat.succ_add",
"foldlM_eq_foldlM_data.aux",
"List.get_drop_eq_drop",
"Algebra.leftMulMatrix_complex",
"CompleteLattice.independent_iff_supIndep_of_injOn",
"Int.pow_succ",
"Ideal.ofList_cons",
"_root_.dist_comm",
"IsModularLattice.inf_sup_inf_assoc",
"h_mf.mul",
"MvPolynomial.eval₂_eq'",
"Equi... |
t1 | case isFalse m : Type u_1 -> Type u_2 beta : Type u_1 alpha : Type u_3 inst+ : Monad m f : beta -> alpha -> m beta arr : Array alpha i j : Nat H : arr.size <= i + j b : beta h+ : ~j < arr.size |- pure b = List.foldlM f b (List.drop j arr.data) | [
"List.drop_length_le",
"Nat.ge_of_not_lt"
] | [
"List.drop_length_le",
"Nat.ge_of_not_lt",
"Subgroup.index_eq_card",
"PosNum.one.succ",
"Cofan.mk",
"homotopyCofiber.inrX_sndX_assoc",
"LinearMap.det_eq_det_toMatrix_of_finset",
"LieIdeal.killingForm_eq",
"hC.tsum_eq",
"RingHom.injective",
"X.V",
"LinearIsometry.norm_map",
"ENNReal.rpow_eq_t... |
t1 | case isTrue.h_2 m : Type u_1 -> Type u_2 beta : Type u_1 alpha : Type u_3 inst+ : Monad m f : beta -> alpha -> m beta arr : Array alpha j : Nat b : beta h+ : j < arr.size i+ i : Nat H : arr.size <= i.succ + j |- (let_fun this := ...; do let __do_lift <- f b arr[j] foldlM.loop f arr arr.size ... i (j + 1) __do_lift) = L... | [
"Nat.succ_add"
] | [
"Nat.succ_add",
"Finset.sum_fn",
"ContinuousMap.toFun_eq_coe",
"S₁.leftHomologyData.homologyIso.hom",
"RingOfIntegers.algebraMap_norm_algebraMap",
"WithBot.coe_eq_coe",
"Nat.prime_iff_card_units",
"Nat.lt_iff_le_and_ne.trans",
"WithTop.coe_untop",
"Fintype.card_eq_zero_iff",
"Multiset.cons_eq_co... |
t1 | case isTrue.h_2 m : Type u_1 -> Type u_2 beta : Type u_1 alpha : Type u_3 inst+ : Monad m f : beta -> alpha -> m beta arr : Array alpha j : Nat b : beta h+ : j < arr.size i+ i : Nat H : arr.size <= (i + j).succ |- (let_fun this := ...; do let __do_lift <- f b arr[j] foldlM.loop f arr arr.size ... i (j + 1) __do_lift) =... | [
"foldlM_eq_foldlM_data.aux"
] | [
"foldlM_eq_foldlM_data.aux",
"B'.map_sub₂",
"SimpleFunc.integral_eq_integral",
"Fix.dest",
"EuclideanSpace.ball_zero_eq",
"Finset.prod_eq_single",
"Complex.add_re",
"skewAdjoint.negISMul_apply_coe",
"MvQPF.liftP_iff",
"IsLocalization.AtPrime.to_map_mem_maximal_iff",
"Polynomial.toContinuousMap",... |
t1 | case isTrue.h_2 m : Type u_1 -> Type u_2 beta : Type u_1 alpha : Type u_3 inst+ : Monad m f : beta -> alpha -> m beta arr : Array alpha j : Nat b : beta h+ : j < arr.size i+ i : Nat H : arr.size <= (i + j).succ |- (do let __do_lift <- f b arr[j] List.foldlM f __do_lift (List.drop (j + 1) arr.data)) = List.foldlM f b (L... | [
"List.get_drop_eq_drop"
] | [
"List.get_drop_eq_drop",
"Finite.toFinset_univ",
"hP.respectsIso.cancel_right_isIso",
"ArithmeticFunction.mul_apply",
"biprod.lift_fst",
"List.countP_go_eq_add",
"Module.Free.chooseBasis",
"hI.right",
"Nat.card_congr",
"MeasurableSet.univ.nullMeasurableSet",
"IsPrimitiveRoot.coe_units_iff",
"R... |
t1 | case isFalse m : Type u_1 -> Type u_2 beta : Type u_1 alpha : Type u_3 inst+ : Monad m f : beta -> alpha -> m beta arr : Array alpha i j : Nat H : arr.size <= i + j b : beta h+ : ~j < arr.size |- pure b = List.foldlM f b (List.drop j arr.data) | [
"List.drop_length_le",
"Nat.ge_of_not_lt"
] | [
"List.drop_length_le",
"Nat.ge_of_not_lt",
"OreLocalization.nonempty_oreSet_iff_of_noZeroDivisors",
"Measure.count_apply_lt_top",
"Option.join",
"LinearMap.restrictScalars_apply",
"CommShift.isoAdd_hom_app",
"hYX.trans",
"List.exists_iff_exists_tuple",
"Subgroup.comap_map_eq",
"hfinj.eq_iff",
... |
t1 | m : Type u_1 -> Type u_2 beta : Type u_1 alpha : Type u_3 inst+ : Monad m f : beta -> alpha -> m beta init : beta arr : Array alpha |- foldlM f init arr 0 = List.foldlM f init arr.data | [
"foldlM_eq_foldlM_data.aux"
] | [
"foldlM_eq_foldlM_data.aux",
"Submodule.mem_annihilator",
"ENNReal.tsum_comm",
"QuadraticForm.ext",
"Q.natDegree",
"Measure.restrict_zero",
"Pi.cons",
"Function.injective_id.injOn",
"Set.iUnion_prod",
"J.toPlus_plusLift",
"IsOfFinAddOrder.addOrderOf_nsmul",
"H.mul_mem_cancel_left",
"Sym.map"... |
t1 | m : Type u_1 -> Type u_2 alpha : Type u_3 beta : Type u_1 inst+ : Monad m f : alpha -> beta -> m beta arr : Array alpha init : beta i : Nat h : i <= arr.size |- List.foldlM (fun x y => f y x) init (List.take i arr.data).reverse = if (i == 0) = true then pure init else match i, h with | 0, x => pure init | i.succ, h => ... | [
"List.foldlM",
"List.take",
"List.take_concat_get"
] | [
"List.foldlM",
"List.take",
"List.take_concat_get",
"Set.setOf_eq_eq_singleton'",
"Fin.eq_zero",
"Nat.gcd_mul_right",
"Memℒp.coeFn_toLp",
"HomologicalComplex.homologyFunctor_obj",
"Functor.id_obj",
"Karoubi.decompId_i_toKaroubi",
"Nat.factorial_mul_ascFactorial",
"Finset.sum_nsmul",
"Homolog... |
t1 | m : Type u_1 -> Type u_2 alpha : Type u_3 beta : Type u_1 inst+ : Monad m f : alpha -> beta -> m beta arr : Array alpha init : beta i : Nat h : 0 <= arr.size |- List.foldlM (fun x y => f y x) init (List.take 0 arr.data).reverse = if (0 == 0) = true then pure init else match 0, h with | 0, x => pure init | i.succ, h => ... | [
"List.foldlM",
"List.take"
] | [
"List.foldlM",
"List.take",
"LHom.coe_substructureReduct",
"Complex.coe_basisOneI",
"Fin.sum_cons",
"Measure.count_apply_finite",
"Sieve.pullback_top",
"Fin.addNat_one",
"Cardinal.isPrimePow_iff",
"Filter.map_top",
"NonUnitalStarAlgebra.mem_bot",
"Num.zneg_toZNum",
"Finset.centerMass_insert"... |
t1 | m : Type u_1 -> Type u_2 alpha : Type u_3 beta : Type u_1 inst+ : Monad m f : alpha -> beta -> m beta arr : Array alpha init : beta i+ i : Nat h : i + 1 <= arr.size |- List.foldlM (fun x y => f y x) init (List.take (i + 1) arr.data).reverse = if (i + 1 == 0) = true then pure init else match i + 1, h with | 0, x => pure... | [
"List.take_concat_get"
] | [
"List.take_concat_get",
"LSeries.convolution_def",
"Set.disjoint_iff",
"MulAut.conj_symm_apply",
"Filter.prod_mem_prod",
"Fin.sum_univ_eight",
"Int.floor_eq_iff",
"Finset.mem_coe.mpr",
"GradedMonoid.list_prod_map_eq_dProd",
"reverse.loop",
"List.foldr_fixed'",
"LinearMap.rank",
"WithLawson.i... |
t1 | m : Type u_1 -> Type u_2 alpha : Type u_3 beta : Type u_1 inst+ : Monad m f : alpha -> beta -> m beta init : beta arr+ : Array alpha this : arr+ = #[] \/ 0 < arr+.size arr : Array alpha h : 0 < arr.size |- foldrM f init arr arr.size = List.foldlM (fun x y => f y x) init arr.data.reverse | [
"foldrM_eq_reverse_foldlM_data.aux",
"List.take_length"
] | [
"foldrM_eq_reverse_foldlM_data.aux",
"List.take_length",
"Set.inclusion_eq_id",
"Set.setOf_subset_setOf",
"Opens.openEmbedding",
"Set.subset_union_left",
"RCLike.inner_apply",
"CategoryTheory.epi_iff_surjective",
"Measure.prod",
"Unitization.quasispectrum_eq_spectrum_inr'",
"AlgHom.map_pow",
"... |
t1 | m : Type u_1 -> Type u_2 alpha : Type u_3 beta : Type u_1 inst+ : Monad m f : alpha -> beta -> m beta init : beta arr : Array alpha |- foldrM f init arr arr.size = List.foldrM f init arr.data | [
"List.foldlM_reverse"
] | [
"List.foldlM_reverse",
"QuadraticForm.zero_apply",
"Finset.sum_range_add",
"DFinsupp.card_Icc",
"Subalgebra.zero_mem",
"kernel.comap_apply",
"Finset.card_eq_two",
"FiberBundle.chartedSpace'_chartAt",
"G.id_mem_maximalAtlas",
"HomologyData.comm_assoc",
"Ordinal.succ_ne_zero",
"Set.image_diff",
... |
t1 | alpha : Type u_1 arr : Array alpha a : alpha |- (arr.push a).data = arr.data ++ [a] | [
"List.concat_eq_append"
] | [
"List.concat_eq_append",
"List.reverseAux_cons",
"Iso.trans_assoc",
"MulOpposite.unop_op",
"Hom.appLE",
"Int.natAbs_ne_zero.mpr",
"Submodule.sup_orthogonal_inf_of_completeSpace",
"LinearMap.ringLmapEquivSelf_symm_apply",
"ClosedEmbedding.closed_iff_image_closed",
"ENNReal.rpow_natCast",
"LinearM... |
t1 | alpha : Type u_1 arr : Array alpha |- arr.toListRev = arr.data.reverse | [
"List.foldr_reverse",
"List.foldr_self"
] | [
"List.foldr_reverse",
"List.foldr_self",
"Subtype.coe_injective.mem_set_image",
"rootsOfUnity.coe_pow",
"WithBot.preimage_coe_Ioi",
"MonoidHom.snd_comp_prod",
"Int.gcd_eq_natAbs",
"AddMonoidHom.coe_comp",
"SimpleFunc.restrict_apply",
"Pi.div_def",
"Nat.card_submonoidPowers",
"StarAlgebra.adjoi... |
t1 | alpha : Type ?u.22043 a : Array alpha x : alpha i : Nat h : i < a.size |- i < (a.push x).size | [
"Nat.lt_succ_of_le",
"Nat.le_of_lt"
] | [
"Nat.lt_succ_of_le",
"Nat.le_of_lt",
"MapsTo.val_restrict_apply",
"Term.relabel_comp_relabel",
"Constraint.sat",
"Fin.range_succAbove",
"Set.preimage_iInter",
"LinearOrder.min_def",
"Measure.ae_le_iff_absolutelyContinuous",
"Metric.eball_top_eq_univ",
"FiniteField.isSquare_neg_two_iff",
"Finsu... |
t1 | alpha : Type u_1 a : Array alpha x : alpha i : Nat h : i < a.size |- let_fun this := ...; (a.push x)[i] = a[i] | [
"List.concat_eq_append",
"List.getElem_append_left"
] | [
"List.concat_eq_append",
"List.getElem_append_left",
"Pi.le_def",
"Nat.one_mem_divisors",
"Finset.mem_insertNone",
"BinaryBiproduct.bicone_inr",
"FractionalIdeal.coeIdeal_mul",
"Polynomial.cyclotomic_three",
"NatIso.isIso_map_iff",
"IsUnit.unit_spec",
"Finset.mem_preimage",
"Multiset.toFinsupp... |
t1 | alpha : Type u_1 a : Array alpha x : alpha |- (a.push x)[a.size] = x | [
"List.concat_eq_append"
] | [
"List.concat_eq_append",
"Option.map₂_none_right",
"IsCoprime.mul_right_iff",
"H.adj_sub",
"NNReal.coe_sSup",
"intervalIntegral.integral_of_le",
"kernel.fst_apply",
"Int.lcm",
"SplittingField.adjoin_rootSet",
"Fin.partialProd_succ'",
"Matrix.charpoly",
"Rat.natCast_div",
"LieModuleHom.range_... |
t1 | alpha : Type u_1 a : Array alpha x : alpha |- (a.data ++ [x])[a.size] = x | [
"List.getElem_append_right",
"Nat.zero_lt_one"
] | [
"List.getElem_append_right",
"Nat.zero_lt_one",
"DFinsupp.single_eq_pi_single",
"Exact.lift",
"isMultiplicative_sigma.multiplicative_factorization",
"Multiset.coe_nodup",
"NNReal.rpow_le_one_of_one_le_of_nonpos",
"SpecialLinearGroup.coe_GLPos_coe_GL_coe_matrix",
"P.isoRightDerivedToHomotopyCategoryO... |
t1 | case neg alpha : Type u_1 a : Array alpha x : alpha i : Nat h : i < (a.push x).size h' : ~i < a.size |- (a.push x)[i] = if h : i < a.size then a[i] else x | [
"Nat.le_antisymm",
"Nat.le_of_lt_succ",
"Nat.ge_of_not_lt"
] | [
"Nat.le_antisymm",
"Nat.le_of_lt_succ",
"Nat.ge_of_not_lt",
"Measure.measurePreserving_neg",
"inv_eq_zero.mpr",
"Function.uncurry",
"Finsupp.mapRange.addEquiv_toEquiv",
"ENNReal.mul_iInf_of_ne",
"G.edgeFinset",
"Exact.lift",
"Matrix.det_vandermonde_sub",
"CochainComplex.homologyFunctor0Single₀... |
t1 | case neg alpha : Type u_1 a : Array alpha x : alpha i : Nat h+ : i < (a.push x).size h' : ~i < a.size h : i < a.size + 1 |- (a.push x)[i] = if h : i < a.size then a[i] else x | [
"Nat.le_antisymm",
"Nat.le_of_lt_succ",
"Nat.ge_of_not_lt"
] | [
"Nat.le_antisymm",
"Nat.le_of_lt_succ",
"Nat.ge_of_not_lt",
"Submodule.iSup_eq_span",
"Iso.hom_inv_id_app_assoc",
"Subtype.preimage_coe_nonempty",
"Set.indicator_eq_zero_iff_not_mem",
"Nat.not_succ_le_self",
"SheafedSpace.comp_c_app'",
"Set.union_right_comm",
"Vector.tail_val",
"IntFractPair.s... |
t1 | case isTrue m : Type u_1 -> Type u_2 alpha : Type u_3 beta : Type u_1 inst+_1 : Monad m inst+ : LawfulMonad m f : alpha -> m beta arr : Array alpha i : Nat r : Array beta h+ : i < arr.size |- (do let __do_lift <- f arr[i] mapM.map f arr (i + 1) (r.push __do_lift)) = List.foldlM (fun bs a => bs.push <$> f a) r (List.dro... | [
"List.get_drop_eq_drop",
"List.foldlM_cons"
] | [
"List.get_drop_eq_drop",
"List.foldlM_cons",
"LocallyConstant.coe_mk",
"List.pmap_eq_map_attach",
"Chain.count_not",
"LinearMap.funLeft",
"Nat.not_prime_one",
"eX.hom",
"MonoidHom.coe_snd",
"Bool.not_eq_true'",
"Real.pi_pos.le",
"Presieve.functorPullback_mem",
"K.flip.D₁_shape",
"List.mem_... |
t1 | case isFalse m : Type u_1 -> Type u_2 alpha : Type u_3 beta : Type u_1 inst+_1 : Monad m inst+ : LawfulMonad m f : alpha -> m beta arr : Array alpha i : Nat r : Array beta h+ : ~i < arr.size |- pure r = List.foldlM (fun bs a => bs.push <$> f a) r (List.drop i arr.data) | [
"List.drop_length_le",
"Nat.ge_of_not_lt"
] | [
"List.drop_length_le",
"Nat.ge_of_not_lt",
"Int.cast_prod",
"LinearIsometryEquiv.piLpCongrLeft_apply",
"Algebra.isAlgebraic_iff_isIntegral",
"Units.val_mk",
"Finset.coe_mul",
"Iterator.next",
"Submodule.fst_inf_snd",
"MvPFunctor.wRec_eq",
"Nat.coprime_zero_right",
"MvPolynomial.aeval_C",
"Ad... |
t1 | case isTrue m : Type u_1 -> Type u_2 alpha : Type u_3 beta : Type u_1 inst+_1 : Monad m inst+ : LawfulMonad m f : alpha -> m beta arr : Array alpha i : Nat r : Array beta h+ : i < arr.size |- (do let __do_lift <- f arr[i] mapM.map f arr (i + 1) (r.push __do_lift)) = List.foldlM (fun bs a => bs.push <$> f a) r (List.dro... | [
"List.get_drop_eq_drop"
] | [
"List.get_drop_eq_drop",
"F.apply_one",
"LieModuleHom.map_sub",
"Vector.head_cons",
"Nat.le_add_right",
"Fin.mul_one",
"Associated.comm",
"Bool.and_iff_right_iff_imp",
"Dual.transpose_apply",
"LinearPMap.map_add",
"rangeRestrict.map_inv",
"Set.pairwise_singleton",
"Nat.sub_eq_zero_iff_le",
... |
t1 | case isTrue m : Type u_1 -> Type u_2 alpha : Type u_3 beta : Type u_1 inst+_1 : Monad m inst+ : LawfulMonad m f : alpha -> m beta arr : Array alpha i : Nat r : Array beta h+ : i < arr.size |- (do let __do_lift <- f arr[i] mapM.map f arr (i + 1) (r.push __do_lift)) = List.foldlM (fun bs a => bs.push <$> f a) r (arr.data... | [
"List.foldlM_cons"
] | [
"List.foldlM_cons",
"Multiset.replicate_one",
"Ideal.span_singleton_le_span_singleton",
"Hinj.image_iInter_eq",
"X.carrier",
"Cardinal.le_def",
"add_eq_zero_iff.mp",
"Nat.lor",
"reverse.loop",
"List.foldlM",
"Polynomial.degree_le_zero_iff",
"Finset.insert_subset_iff",
"L.functor.map_comp",
... |
t1 | case isFalse m : Type u_1 -> Type u_2 alpha : Type u_3 beta : Type u_1 inst+_1 : Monad m inst+ : LawfulMonad m f : alpha -> m beta arr : Array alpha i : Nat r : Array beta h+ : ~i < arr.size |- pure r = List.foldlM (fun bs a => bs.push <$> f a) r (List.drop i arr.data) | [
"List.drop_length_le",
"Nat.ge_of_not_lt"
] | [
"List.drop_length_le",
"Nat.ge_of_not_lt",
"P.hε",
"circleIntegral.integral_congr",
"LinearEquiv.coe_toLinearMap",
"List.Perm.prod_eq",
"Cocones.precompose_obj_pt",
"hI.hom_ext",
"IsFractionRing.fieldEquivOfRingEquiv",
"multiplicity.Int.natCast_multiplicity",
"ZMod.expand_card",
"Prop.exists_i... |
t1 | alpha : Type u_1 beta : Type u_2 l : List alpha acc : Array beta F : Array beta -> alpha -> Array beta G : alpha -> List beta H : forall (acc : Array beta) (a : alpha), (F acc a).data = acc.data ++ G a |- (List.foldl F acc l).data = acc.data ++ l.bind G | [
"List.bind"
] | [
"List.bind",
"QuotientAddGroup.eq",
"Gr.map_comp",
"ENNReal.tsum_add_one_eq_top",
"MeasurableSet.subtype_image",
"Quot.rec",
"NonUnitalStarSubalgebra.mem_comap",
"NormedAddCommGroup.nhds_zero_basis_norm_lt.eventually_iff",
"NNRat.cast_pow",
"SimpleFunc.map_apply",
"ONote.mulNat_eq_mul",
"IsFix... |
t1 | m : Type -> Type u_1 alpha : Type u_2 inst+ : Monad m p : alpha -> m Bool as : Array alpha start stop : Nat |- anyM p as start stop = anyM.loop p as (min stop as.size) ... start | [
"Nat.min_def"
] | [
"Nat.min_def",
"WeierstrassCurve.VariableChange.id",
"G.le_card_nonuniformWitness",
"J.toPlus_plusLift",
"Nonempty.forall",
"Complex.arg_ofReal_of_neg",
"Equiv.Perm.sign_swap",
"counit.app",
"hcard.ge",
"M.vecMul_transpose",
"Finset.sum_add_sum_compl",
"Matrix.one_apply_ne'",
"ONote.repr_ofN... |
t1 | m : Type -> Type u_1 alpha : Type u_2 inst+ : Monad m p : alpha -> m Bool as : Array alpha start stop : Nat h : min stop as.size <= start |- anyM p as start stop = pure false | [
"anyM.loop",
"Nat.not_lt"
] | [
"anyM.loop",
"Nat.not_lt",
"Bornology.ext_iff_isBounded",
"Nat.Coprime.gcd_eq_one",
"LieSubalgebra.coe_submodule_le_coe_submodule",
"Finset.iInter_mem_sets",
"Matrix.one_apply",
"Matrix.eq_zero_of_mulVec_eq_zero",
"Submodule.restrictScalars_top",
"WittVector.ext_iff",
"LinearEquiv.map_smulₛₗ",
... |
t1 | alpha : Type u_1 a : Array alpha i : Fin a.size v : alpha j : Nat pj : j < (a.set i v).size h : ^i /= j |- (a.set i v)[j] = a[j] | [
"List.getElem_set_ne"
] | [
"List.getElem_set_ne",
"FiniteDimensional.finBasis",
"Set.image2_swap",
"OrderDual.toDual_le_toDual",
"LieSubmodule.sup_coe_toSubmodule",
"OuterMeasure.trim_eq_iInf",
"Option.guard",
"PartialEquiv.symm_symm",
"Prod.braiding_inverse_obj",
"Function.Embedding.nonempty_iff_card_le",
"Nat.max_le",
... |
t1 | alpha : Type u_1 a : Array alpha i : Fin a.size j : Nat v : alpha ne : ^i /= j |- (a.set i v)[j]? = a[j]? | [
"Nat.ge_of_not_lt"
] | [
"Nat.ge_of_not_lt",
"Subalgebra.coe_toSubsemiring",
"Subgroup.toSubmonoid_le",
"Finset.left_mem_Icc.mpr",
"StarSubalgebra.mem_toSubalgebra",
"UniformSpace.mem_nhds_iff",
"ENat.toNat_eq_zero",
"Box.coe_subset_coe",
"TensorProduct.curry_apply",
"Splitting.IndexSet.id_fst",
"LinearIsometryEquiv.ofE... |
t1 | alpha : Type u_1 a : Array alpha i : Nat v d : alpha |- (a.setD i v)[i]?.getD d = if i < a.size then v else d | [
"Nat.not_lt_of_le"
] | [
"Nat.not_lt_of_le",
"Set.mapsTo_inter",
"Set.nonempty_Ioc",
"hP.respectsIso.cancel_right_isIso",
"Nat.lt_irrefl",
"WithBot.image_coe_Ico",
"SplittingField.adjoin_rootSet",
"List.filterMap_nil",
"Int.div_nonneg_iff_of_pos",
"Substring.extract",
"Multiset.mem_union",
"Finset.univ_sum_single",
... |
t1 | alpha : Type u_1 n : Nat f : Fin n -> alpha i : Nat acc : Array alpha |- (ofFn.go f i acc).size = acc.size + (n - i) | [
"Nat.add_assoc"
] | [
"Nat.add_assoc",
"List.mem_iff_getElem",
"Equiv.left_inv",
"F.asEquivalence.counitIso.symm",
"Int.units_mul_self",
"BoundedContinuousFunction.coe_one",
"Nat.cast_zero",
"Sphere.secondInter_eq_self_iff",
"Matrix.add_mul",
"Eq.refl",
"fund_dom_s.measure_zero_of_invariant",
"Bool.false_xor",
"A... |
t1 | alpha : Type u_1 n : Nat f : Fin n -> alpha i : Nat acc : Array alpha hin : i < n this : 1 + (n - (i + 1)) = n - i |- (if h : i < n then ofFn.go f (i + 1) (acc.push (f <i, h>)) else acc).size = acc.size + (n - i) | [
"Nat.add_assoc"
] | [
"Nat.add_assoc",
"Finset.subset_union_right",
"Int.fract_eq_fract",
"WithBot.coe_unbot",
"homology'.map_desc",
"FormalMultilinearSeries.partialSum",
"Array.mkEmpty",
"Category.id_comp",
"S.exact_iff_kernel_ι_comp_cokernel_π_zero",
"Finset.toFinset_coe",
"ContinuousMap.coe_neg",
"IsStoppingTime... |
t1 | n : Nat alpha : Type u_1 f : Fin n -> alpha i : Nat acc : Array alpha k : Nat hki : k < n hin : i <= n hi : i = acc.size hacc : forall (j : Nat) (hj : j < acc.size), acc[j] = f <j, ...> |- (if h : i < n then ofFn.go f (i + 1) (acc.push (f <i, h>)) else acc)[k] = f <k, hki> | [
"Nat.lt_of_lt_of_le",
"Nat.le_of_not_lt"
] | [
"Nat.lt_of_lt_of_le",
"Nat.le_of_not_lt",
"HomotopyCategory.isoOfHomotopyEquiv",
"ModuleCat.mono_iff_injective",
"Stream'.corec'",
"Sym.map",
"Finsupp.prod_add_index'",
"LieModule.ker",
"Memℒp.integrable",
"Int.not_lt.mpr",
"Finset.not_nonempty_iff_eq_empty",
"LinearEquiv.dualMap_apply",
"Fi... |
End of preview. Expand in Data Studio
PSEB — Proof-State Embedding Benchmark
A kernel-free benchmark of eight retrieval tasks over Lean 4 proof states, built from 303,870 steps in 64,031 Mathlib theorems. Labels are mined from proof traces already checked by the Lean kernel, so no task here requires a Lean install or a kernel call to construct or score.
See the paper, "Which Tactics Deserve a Kernel Call? Selection and Benchmarking for Lean 4 Neural Theorem Proving," for task definitions, label-quality measurements against gold annotations, and baseline results.
Tasks
| Config | Rows | Labels |
|---|---|---|
t1 |
5,695 | premises named in tactic [...] brackets; gold-checked |
t2 |
30,856 | tactic used at this canonical state (multi-positive) |
t3 |
30,856 | tactic for this (before, after) transition |
t4 |
3,516 | duplicate-state sanity check (identical canonical hash) |
t5 |
55 | duplicate-statement sanity check |
t6 |
154,280 | applicability: 1 observed tactic + 4 sampled negatives |
t7 |
30,358 | renaming sensitivity: canonical renamed variant vs. hard/random negatives |
t8 |
6,418 | subject area, from file path (e.g. Mathlib/Algebra → Algebra) |
Split: file-level (every step of a theorem stays on one side), seed 0, 6,422 test theorems / 30,856 test steps out of 64,031 theorems total.
Usage
from datasets import load_dataset
t1 = load_dataset("stillrolli/PSEB", "t1")
print(t1["train"][0])
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