ProofWalaDataset / lean /eval /leandojo-eval.yaml
amitayusht's picture
Added raw dataset
db04d2b
name: leandojo-eval
num_files: 729
language: LEAN4
few_shot_data_path_for_retrieval: null
few_shot_metadata_filename_for_retrieval: null
dfs_data_path_for_retrieval: null
dfs_metadata_filename_for_retrieval: local.meta.json
theorem_cnt: 1035
datasets:
- project: <path-to-repo>/mathlib4
files:
- path: Mathlib/Data/Set/Pointwise/Basic.lean
theorems:
- Set.div_subset_range
- Set.pow_mem_pow
- path: Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
theorems:
- Complex.abs_cpow_eq_rpow_re_of_nonneg
- Real.rpow_div_two_eq_sqrt
- path: Mathlib/Order/Filter/AtTopBot.lean
theorems:
- Filter.exists_eventually_atTop
- Filter.tendsto_const_mul_atTop_of_pos
- path: Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean
theorems:
- CategoryTheory.ShortComplex.HasRightHomology.hasKernel
- path: Mathlib/RingTheory/WittVector/Verschiebung.lean
theorems:
- WittVector.verschiebungFun_coeff
- path: Mathlib/Order/WellFounded.lean
theorems:
- WellFounded.eq_strictMono_iff_eq_range_aux
- path: Mathlib/Algebra/Order/Group/MinMax.lean
theorems:
- max_one_div_max_inv_one_eq_self
- path: Mathlib/Data/List/Intervals.lean
theorems:
- List.Ico.filter_lt_of_succ_bot
- List.Ico.length
- path: Mathlib/Data/Seq/Computation.lean
theorems:
- Computation.results_of_terminates'
- path: Mathlib/Probability/ProbabilityMassFunction/Basic.lean
theorems:
- PMF.toMeasure_injective
- path: Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean
theorems:
- expNegInvGlue.hasDerivAt_polynomial_eval_inv_mul
- expNegInvGlue.pos_of_pos
- path: Mathlib/SetTheory/Game/PGame.lean
theorems:
- SetTheory.PGame.add_le_add_right'
- SetTheory.PGame.isOption_neg
- SetTheory.PGame.moveLeft_neg_symm
- SetTheory.PGame.bddAbove_range_of_small
- path: Mathlib/Data/List/Range.lean
theorems:
- List.unzip_enum_eq_prod
- List.pairwise_lt_finRange
- List.chain'_range_succ
- path: Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean
theorems:
- "CliffordAlgebra.\u03B9_range_map_reverse"
- CliffordAlgebra.submodule_comap_pow_reverse
- path: Mathlib/Data/Nat/Order/Basic.lean
theorems:
- Nat.div_eq_self
- Nat.add_sub_one_le_mul
- path: Mathlib/Topology/Compactness/Compact.lean
theorems:
- IsCompact.nhdsSet_prod_eq_biSup
- IsCompact.union
- IsCompact.inter_right
- path: .lake/packages/std/Std/Data/List/Lemmas.lean
theorems:
- List.diff_cons_right
- List.length_insert_of_not_mem
- List.forall_mem_append
- List.map_subset
- List.take_take
- List.exists_of_replaceF
- List.drop_sizeOf_le
- List.suffix_of_suffix_length_le
- List.append_ne_nil_of_left_ne_nil
- List.findIdx?_eq_some_iff
- List.erase_of_not_mem
- List.drop_eq_nil_iff_le
- List.take_eq_take
- path: Mathlib/NumberTheory/ADEInequality.lean
theorems:
- ADEInequality.admissible_of_one_lt_sumInv_aux'
- path: Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean
theorems:
- InnerProductGeometry.norm_div_sin_angle_add_of_inner_eq_zero
- EuclideanGeometry.angle_eq_arcsin_of_angle_eq_pi_div_two
- InnerProductGeometry.angle_sub_eq_arccos_of_inner_eq_zero
- InnerProductGeometry.norm_div_sin_angle_sub_of_inner_eq_zero
- path: Mathlib/FieldTheory/SeparableDegree.lean
theorems:
- IntermediateField.finSepDegree_bot
- path: Mathlib/NumberTheory/FLT/Four.lean
theorems:
- Fermat42.exists_pos_odd_minimal
- path: Mathlib/Data/Matrix/Basis.lean
theorems:
- Matrix.StdBasisMatrix.mul_of_ne
- path: Mathlib/Data/Matrix/Block.lean
theorems:
- Matrix.toBlock_diagonal_disjoint
- path: Mathlib/Data/Polynomial/AlgebraMap.lean
theorems:
- Polynomial.aeval_apply_smul_mem_of_le_comap'
- path: Mathlib/Algebra/Order/Sub/Canonical.lean
theorems:
- tsub_tsub_eq_add_tsub_of_le
- path: Mathlib/Data/Nat/Parity.lean
theorems:
- Nat.even_add
- Odd.ne_two_of_dvd_nat
- pow_eq_neg_one_iff
- path: Mathlib/ModelTheory/Basic.lean
theorems:
- FirstOrder.Language.Embedding.coe_injective
- path: Mathlib/Topology/Covering.lean
theorems:
- isCoveringMap_iff_isCoveringMapOn_univ
- path: Mathlib/Data/Int/Cast/Lemmas.lean
theorems:
- Int.cast_nonneg
- Int.cast_le_neg_one_of_neg
- path: Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean
theorems:
- NonUnitalSubalgebra.toSubalgebra_toNonUnitalSubalgebra
- path: Mathlib/Data/Polynomial/FieldDivision.lean
theorems:
- Polynomial.mod_eq_self_iff
- Polynomial.rootSet_prod
- Polynomial.monic_normalize
- Polynomial.one_lt_rootMultiplicity_iff_isRoot_iterate_derivative
- path: Mathlib/MeasureTheory/Measure/WithDensity.lean
theorems:
- MeasureTheory.withDensity_eq_zero
- MeasureTheory.withDensity_add_measure
- MeasureTheory.restrict_withDensity
- path: Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean
theorems:
- HasSum.sigma
- path: Mathlib/Data/Set/Lattice.lean
theorems:
- Set.biUnion_iUnion
- Set.iUnion_of_singleton_coe
- path: Mathlib/GroupTheory/Perm/Fin.lean
theorems:
- Fin.cycleRange_apply
- path: Mathlib/Data/Set/Basic.lean
theorems:
- Set.ite_right
- Set.ite_compl
- path: Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean
theorems:
- CategoryTheory.Limits.HasZeroObject.zeroIsoIsTerminal_hom
- CategoryTheory.Limits.HasZeroObject.zeroIsoIsInitial_inv
- "CategoryTheory.Limits.image.\u03B9_zero'"
- "CategoryTheory.Limits.image_\u03B9_comp_eq_zero"
- CategoryTheory.Limits.HasZeroObject.zeroIsoTerminal_inv
- path: Mathlib/Analysis/Seminorm.lean
theorems:
- Seminorm.smul_inf
- Seminorm.bddBelow_range_add
- Seminorm.comp_smul
- absorbent_ball_zero
- path: Mathlib/Analysis/Calculus/FDeriv/Basic.lean
theorems:
- DifferentiableWithinAt.hasFDerivWithinAt
- fderivWithin_zero_of_isolated
- HasFDerivWithinAt.lim
- path: Mathlib/Data/Complex/ExponentialBounds.lean
theorems:
- Real.exp_neg_one_gt_d9
- path: Mathlib/SetTheory/Cardinal/Ordinal.lean
theorems:
- Cardinal.ciSup_mul_ciSup
- Cardinal.mk_cardinal
- path: Mathlib/Geometry/Euclidean/Circumcenter.lean
theorems:
- Affine.Simplex.dist_circumcenter_sq_eq_sq_sub_circumradius
- path: Mathlib/Order/Filter/Archimedean.lean
theorems:
- Filter.Tendsto.atBot_zsmul_const
- Filter.Tendsto.atTop_mul_const'
- Filter.Eventually.int_cast_atBot
- path: Mathlib/Data/Polynomial/Eval.lean
theorems:
- "Polynomial.eval\u2082_monomial"
- Polynomial.coeff_comp_degree_mul_degree
- Polynomial.eval_ofNat
- path: Mathlib/Data/PNat/Factors.lean
theorems:
- PNat.factorMultiset_gcd
- path: Mathlib/MeasureTheory/MeasurableSpace/Defs.lean
theorems:
- MeasurableSpace.generateFrom_insert_univ
- MeasurableSet.univ
- path: Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean
theorems:
- CategoryTheory.Limits.KernelFork.app_one
- "CategoryTheory.Limits.CokernelCofork.IsColimit.isIso_\u03C0"
- CategoryTheory.Limits.cokernel.map_desc
- path: Mathlib/SetTheory/Surreal/Dyadic.lean
theorems:
- Surreal.dyadicMap_apply_pow'
- path: Mathlib/Order/LocallyFinite.lean
theorems:
- Multiset.mem_Ioo
- path: Mathlib/Condensed/Equivalence.lean
theorems:
- Condensed.StoneanProfinite.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily
- path: Mathlib/Analysis/NormedSpace/lpSpace.lean
theorems:
- lp.norm_le_of_tendsto
- "Mem\u2113p.infty_mul"
- path: Mathlib/GroupTheory/Subgroup/Pointwise.lean
theorems:
- Subgroup.sup_eq_closure_mul
- Subgroup.conj_smul_le_of_le
- path: Mathlib/Combinatorics/SimpleGraph/Ends/Defs.lean
theorems:
- SimpleGraph.ComponentCompl.exists_adj_boundary_pair
- path: Mathlib/Algebra/Module/Zlattice.lean
theorems:
- Zspan.repr_floor_apply
- path: Mathlib/Analysis/InnerProductSpace/Symmetric.lean
theorems:
- LinearMap.IsSymmetric.conj_inner_sym
- path: Mathlib/Tactic/Abel.lean
theorems:
- Mathlib.Tactic.Abel.term_add_termg
- path: Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
theorems:
- PartialHomeomorph.contDiffOn_extend_coord_change
- path: Mathlib/Data/Fintype/Basic.lean
theorems:
- Set.toFinset_image
- path: Mathlib/Order/Filter/Basic.lean
theorems:
- Filter.frequently_and_distrib_left
- Filter.iInf_principal_finite
- Filter.iSup_join
- Filter.sup_join
- path: Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean
theorems:
- Orientation.two_zsmul_oangle_smul_right_of_ne_zero
- Orientation.oangle_self
- path: Mathlib/Algebra/BigOperators/Intervals.lean
theorems:
- Finset.prod_Ioc_mul_eq_prod_Icc
- path: Mathlib/Topology/Compactification/OnePoint.lean
theorems:
- OnePoint.ne_infty_iff_exists
- OnePoint.continuousAt_infty
- OnePoint.inseparable_iff
- path: Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean
theorems:
- UniformOnFun.topologicalSpace_eq
- UniformOnFun.tendsto_iff_tendstoUniformlyOn
- path: Mathlib/LinearAlgebra/Matrix/Determinant.lean
theorems:
- Matrix.det_eq_of_forall_col_eq_smul_add_pred
- path: Mathlib/Topology/Algebra/Group/Basic.lean
theorems:
- QuotientGroup.isOpenMap_coe
- topologicalGroup_iInf
- tendsto_inv_nhdsWithin_Iic
- path: Mathlib/CategoryTheory/Limits/Preserves/Shapes/Biproducts.lean
theorems:
- CategoryTheory.Limits.biprod.mapBiprod_hom_desc
- path: Mathlib/Data/Set/Pairwise/Basic.lean
theorems:
- Set.pairwise_insert
- path: Mathlib/Topology/MetricSpace/Perfect.lean
theorems:
- Perfect.small_diam_aux
- path: Mathlib/Topology/PartialHomeomorph.lean
theorems:
- PartialHomeomorph.subtypeRestr_symm_eqOn
- PartialHomeomorph.ofSet_trans'
- PartialHomeomorph.IsImage.interior
- path: Mathlib/Analysis/InnerProductSpace/Basic.lean
theorems:
- inner_add_right
- Orthonormal.sum_inner_products_le
- path: Mathlib/Order/SuccPred/Limit.lean
theorems:
- Order.mem_range_pred_of_not_isPredLimit
- path: Mathlib/Data/Polynomial/Module/Basic.lean
theorems:
- PolynomialModule.comp_smul
- path: Mathlib/Data/Int/Bitwise.lean
theorems:
- Int.bit_val
- Int.testBit_bit_zero
- path: Mathlib/Probability/Kernel/Disintegration.lean
theorems:
- ProbabilityTheory.set_integral_condKernel_univ_right
- path: Mathlib/FieldTheory/RatFunc.lean
theorems:
- RatFunc.num_mul_eq_mul_denom_iff
- RatFunc.induction_on
- RatFunc.ofFractionRing_sub
- path: Mathlib/Data/Finset/Card.lean
theorems:
- Finset.le_card_of_inj_on_range
- path: Mathlib/Topology/Order/LowerUpperTopology.lean
theorems:
- Topology.IsLower.isClosed_upperClosure
- path: Mathlib/Analysis/Calculus/SmoothSeries.lean
theorems:
- differentiable_tsum
- path: Mathlib/GroupTheory/Index.lean
theorems:
- Subgroup.relindex_self
- Subgroup.relindex_bot_right
- path: Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean
theorems:
- UpperHalfPlane.isometry_pos_mul
- path: Mathlib/Data/Polynomial/EraseLead.lean
theorems:
- Polynomial.eraseLead_C_mul_X_pow
- path: Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean
theorems:
- Asymptotics.isLittleO_pow_pow_atTop_of_lt
- path: Mathlib/Data/Finsupp/Basic.lean
theorems:
- Finsupp.equivMapDomain_trans
- path: Mathlib/Order/SupIndep.lean
theorems:
- Finset.supIndep_univ_bool
- CompleteLattice.independent_iff_supIndep_univ
- path: Mathlib/RingTheory/PowerBasis.lean
theorems:
- PowerBasis.natDegree_minpoly
- path: Mathlib/CategoryTheory/Subobject/Basic.lean
theorems:
- CategoryTheory.Subobject.ofMkLEMk_refl
- path: Mathlib/Order/SupClosed.lean
theorems:
- infClosed_empty
- path: Mathlib/Algebra/Ring/Hom/Defs.lean
theorems:
- RingHom.map_ite_one_zero
- path: Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean
theorems:
- InnerProductGeometry.cos_angle_mul_norm_mul_norm
- path: Mathlib/MeasureTheory/PiSystem.lean
theorems:
- generateFrom_piiUnionInter_le
- MeasurableSpace.DynkinSystem.ext
- path: Mathlib/RingTheory/Localization/InvSubmonoid.lean
theorems:
- IsLocalization.finiteType_of_monoid_fg
- path: Mathlib/Analysis/Calculus/ContDiff/Bounds.lean
theorems:
- norm_iteratedFDeriv_clm_apply
- norm_iteratedFDeriv_clm_apply_const
- path: Mathlib/LinearAlgebra/Matrix/Adjugate.lean
theorems:
- Matrix.adjugate_diagonal
- Matrix.adjugate_transpose
- path: Mathlib/RingTheory/WittVector/Teichmuller.lean
theorems:
- WittVector.map_teichmullerFun
- path: Mathlib/Geometry/Manifold/Instances/Sphere.lean
theorems:
- contMDiff_coe_sphere
- stereographic_apply_neg
- path: Mathlib/CategoryTheory/Abelian/RightDerived.lean
theorems:
- CategoryTheory.InjectiveResolution.toRightDerivedZero'_comp_iCycles
- path: Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean
theorems:
- MvQPF.wrepr_equiv
- path: Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
theorems:
- CategoryTheory.Limits.biprod.braid_natural
- path: Mathlib/RingTheory/DedekindDomain/Ideal.lean
theorems:
- FractionalIdeal.coe_ideal_span_singleton_inv_mul
- multiplicity_normalizedFactorsEquivSpanNormalizedFactors_eq_multiplicity
- Ideal.prime_span_singleton_iff
- path: Mathlib/Analysis/Convex/StrictConvexSpace.lean
theorems:
- combo_mem_ball_of_ne
- path: Mathlib/Deprecated/Subgroup.lean
theorems:
- IsGroupHom.injective_of_trivial_ker
- Group.normalClosure_subset
- path: Mathlib/LinearAlgebra/Lagrange.lean
theorems:
- Lagrange.eval_basis_self
- Lagrange.eval_nodal_at_node
- path: Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean
theorems:
- NonUnitalAlgHom.coe_range
- path: Mathlib/Data/Fintype/Perm.lean
theorems:
- mem_perms_of_finset_iff
- path: Mathlib/GroupTheory/Subgroup/Basic.lean
theorems:
- Subgroup.eq_bot_of_subsingleton
- Subgroup.Normal.mem_comm
- CommGroup.center_eq_top
- path: Mathlib/Algebra/Module/LocalizedModule.lean
theorems:
- IsLocalizedModule.iso_symm_apply_aux
- IsLocalizedModule.mk'_sub
- path: Mathlib/GroupTheory/Coset.lean
theorems:
- Subgroup.card_comap_dvd_of_injective
- path: Mathlib/Data/Real/NNReal.lean
theorems:
- NNReal.mul_eq_mul_left
- Real.toNNReal_pos
- NNReal.sSup_of_not_bddAbove
- path: Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean
theorems:
- CochainComplex.HomComplex.Cochain.leftUnshift_smul
- CochainComplex.HomComplex.Cochain.shift_v'
- path: Mathlib/Algebra/EuclideanDomain/Defs.lean
theorems:
- EuclideanDomain.lt_one
- path: Mathlib/Combinatorics/Hall/Basic.lean
theorems:
- Fintype.all_card_le_filter_rel_iff_exists_injective
- path: Mathlib/Analysis/InnerProductSpace/Rayleigh.lean
theorems:
- IsSelfAdjoint.hasEigenvector_of_isMaxOn
- path: Mathlib/Order/Bounds/Basic.lean
theorems:
- IsGreatest.insert
- path: Mathlib/Tactic/Ring/RingNF.lean
theorems:
- Mathlib.Tactic.RingNF.mul_neg
- path: Mathlib/Data/Matrix/Basic.lean
theorems:
- Matrix.dotProduct_pUnit
- Matrix.mulVec_zero
- Matrix.mul_neg
- path: Mathlib/GroupTheory/Sylow.lean
theorems:
- IsPGroup.sylow_mem_fixedPoints_iff
- path: Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean
theorems:
- gramSchmidt_orthonormal'
- gramSchmidt_triangular
- path: Mathlib/Analysis/NormedSpace/OperatorNorm/Basic.lean
theorems:
- ContinuousLinearMap.le_of_opNorm_le_of_le
- path: Mathlib/Algebra/GroupWithZero/Basic.lean
theorems:
- "mul_eq_right\u2080"
- path: Mathlib/Data/Finsupp/Multiset.lean
theorems:
- Finsupp.toMultiset_sup
- path: Mathlib/Data/Polynomial/RingDivision.lean
theorems:
- Polynomial.roots_C
- Polynomial.root_mul
- Polynomial.not_dvd_of_natDegree_lt
- Polynomial.zero_of_eval_zero
- Polynomial.roots_pow
- path: Mathlib/CategoryTheory/Limits/Shapes/Images.lean
theorems:
- CategoryTheory.Limits.image.factorThruImage_preComp
- path: Mathlib/MeasureTheory/Integral/IntegrableOn.lean
theorems:
- MeasureTheory.hasFiniteIntegral_restrict_of_bounded
- integrableOn_Ico_iff_integrableOn_Ioo
- path: Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean
theorems:
- Gamma_is_cong_sub
- Gamma1_in_Gamma0
- path: Mathlib/Order/Monotone/Odd.lean
theorems:
- monotone_of_odd_of_monotoneOn_nonneg
- path: Mathlib/Data/Nat/Log.lean
theorems:
- Nat.log_le_clog
- path: Mathlib/Data/Set/Sigma.lean
theorems:
- Set.range_sigmaMk
- path: Mathlib/Algebra/Order/Ring/Abs.lean
theorems:
- abs_lt_of_sq_lt_sq
- path: Mathlib/Algebra/Order/Field/Basic.lean
theorems:
- one_le_one_div
- inv_le_one
- inv_le_inv
- path: Mathlib/MeasureTheory/OuterMeasure/Basic.lean
theorems:
- MeasureTheory.OuterMeasure.restrict_mono
- MeasureTheory.OuterMeasure.sUnion_null_iff
- MeasureTheory.OuterMeasure.isCaratheodory_iUnion_lt
- MeasureTheory.OuterMeasure.ext_nonempty
- MeasureTheory.OuterMeasure.iSup_apply
- MeasureTheory.OuterMeasure.union_null
- path: Mathlib/Combinatorics/SimpleGraph/Prod.lean
theorems:
- SimpleGraph.Connected.boxProd
- path: Mathlib/Logic/Equiv/PartialEquiv.lean
theorems:
- PartialEquiv.symm_symm
- PartialEquiv.prod_symm
- PartialEquiv.IsImage.leftInvOn_piecewise
- path: Mathlib/Topology/MetricSpace/IsometricSMul.lean
theorems:
- nndist_div_left
- path: Mathlib/Topology/Category/Profinite/Nobeling.lean
theorems:
- Profinite.NobelingProof.contained_eq_proj
- path: .lake/packages/std/Std/Data/AssocList.lean
theorems:
- Std.AssocList.length_toList
- path: Mathlib/Analysis/Calculus/ContDiff/Basic.lean
theorems:
- ContinuousLinearEquiv.iteratedFDerivWithin_comp_right
- ContDiff.pow
- contDiffOn_top_iff_derivWithin
- path: Mathlib/InformationTheory/Hamming.lean
theorems:
- hammingDist_lt_one
- path: Mathlib/Data/Nat/Count.lean
theorems:
- Nat.count_lt_count_succ_iff
- path: Mathlib/Analysis/BoundedVariation.lean
theorems:
- LocallyBoundedVariationOn.ae_differentiableWithinAt
- variationOnFromTo.eq_left_iff
- path: Mathlib/Data/List/NodupEquivFin.lean
theorems:
- List.sublist_iff_exists_fin_orderEmbedding_get_eq
- path: Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
theorems:
- Real.Angle.neg_coe_abs_toReal_of_sign_nonpos
- path: Mathlib/Probability/Kernel/MeasurableIntegral.lean
theorems:
- MeasureTheory.StronglyMeasurable.integral_kernel_prod_left''
- Measurable.lintegral_kernel_prod_right'
- path: Mathlib/NumberTheory/Padics/PadicVal.lean
theorems:
- one_le_padicValNat_of_dvd
- path: Mathlib/Topology/Connected/PathConnected.lean
theorems:
- PathConnectedSpace.exists_path_through_family'
- path: Mathlib/Algebra/BigOperators/Finprod.lean
theorems:
- finprod_mem_insert'
- finprod_mem_eq_one_of_infinite
- finprod_eq_dif
- path: Mathlib/MeasureTheory/Measure/Restrict.lean
theorems:
- MeasureTheory.Measure.restrict_toMeasurable
- MeasureTheory.Measure.restrict_sUnion_congr
- path: Mathlib/Topology/LocallyConstant/Basic.lean
theorems:
- LocallyConstant.piecewise_apply_right
- path: Mathlib/CategoryTheory/MorphismProperty.lean
theorems:
- CategoryTheory.MorphismProperty.RespectsIso.isoClosure_eq
- path: Mathlib/Logic/Function/Basic.lean
theorems:
- Function.surjective_of_right_cancellable_Prop
- path: Mathlib/SetTheory/Ordinal/NaturalOps.lean
theorems:
- Ordinal.nadd_nat
- path: Mathlib/MeasureTheory/Integral/Bochner.lean
theorems:
- MeasureTheory.SimpleFunc.integral_eq_sum_filter
- MeasureTheory.integral_undef
- MeasureTheory.ae_le_trim_of_stronglyMeasurable
- path: Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean
theorems:
- MeasureTheory.LpAddConst_of_one_le
- path: Mathlib/NumberTheory/Padics/PadicNumbers.lean
theorems:
- padicNormE.eq_of_norm_add_lt_right
- Padic.valuation_one
- PadicSeq.norm_nonarchimedean_aux
- Padic.padicNormE_lim_le
- path: Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean
theorems:
- CategoryTheory.tensorLeftHomEquiv_whiskerLeft_comp_evaluation
- path: Mathlib/Data/Fin/Basic.lean
theorems:
- Fin.val_bit1
- Fin.le_pred_iff
- Fin.rev_castPred
- path: Mathlib/RingTheory/Polynomial/Bernstein.lean
theorems:
- bernsteinPolynomial.sum_mul_smul
- path: Mathlib/Analysis/Convex/Body.lean
theorems:
- ConvexBody.hausdorffEdist_ne_top
- path: Mathlib/Analysis/Matrix.lean
theorems:
- Matrix.linfty_opNorm_def
- path: Mathlib/Algebra/Homology/Single.lean
theorems:
- "ChainComplex.toSingle\u2080Equiv_symm_apply_f_zero"
- path: Mathlib/Data/MvPolynomial/Basic.lean
theorems:
- MvPolynomial.prod_X_pow_eq_monomial
- MvPolynomial.induction_on_monomial
- "MvPolynomial.eval_eval\u2082"
- path: Mathlib/Analysis/Asymptotics/Asymptotics.lean
theorems:
- Asymptotics.isLittleO_pow_sub_sub
- Asymptotics.isLittleO_insert
- path: Mathlib/Analysis/NormedSpace/Pointwise.lean
theorems:
- cthickening_cthickening
- path: Mathlib/LinearAlgebra/BilinearForm/Basic.lean
theorems:
- BilinForm.flip_flip_aux
- path: Mathlib/Topology/MetricSpace/Contracting.lean
theorems:
- ContractingWith.isFixedPt_fixedPoint_iterate
- path: Mathlib/Data/Int/CharZero.lean
theorems:
- Int.cast_eq_one
- path: Mathlib/Topology/IndicatorConstPointwise.lean
theorems:
- tendsto_indicator_const_apply_iff_eventually
- path: Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean
theorems:
- Polynomial.IsWeaklyEisensteinAt.map
- path: Mathlib/Topology/Algebra/Constructions.lean
theorems:
- Units.topology_eq_inf
- path: Mathlib/RingTheory/TensorProduct/Basic.lean
theorems:
- Algebra.TensorProduct.map_comp_includeLeft
- path: Mathlib/Data/QPF/Univariate/Basic.lean
theorems:
- QPF.Cofix.dest_corec
- QPF.Fix.rec_eq
- QPF.Fix.ind
- path: Mathlib/Algebra/Group/Hom/Defs.lean
theorems:
- map_comp_div'
- path: Mathlib/Topology/Instances/NNReal.lean
theorems:
- NNReal.summable_nat_add_iff
- NNReal.tendsto_tsum_compl_atTop_zero
- path: Mathlib/Algebra/GroupPower/Ring.lean
theorems:
- add_sq
- path: Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
theorems:
- Measurable.ennreal_tsum
- path: Mathlib/Analysis/Convex/Side.lean
theorems:
- AffineSubspace.sSameSide_vadd_right_iff
- AffineSubspace.wOppSide_iff_exists_right
- path: Mathlib/Data/NNRat/Lemmas.lean
theorems:
- Rat.toNNRat_div
- Rat.toNNRat_inv
- path: Mathlib/Analysis/Normed/Group/Basic.lean
theorems:
- dist_prod_prod_le_of_le
- path: Mathlib/Algebra/Homology/Exact.lean
theorems:
- CategoryTheory.exact_of_image_eq_kernel
- path: Mathlib/CategoryTheory/Preadditive/Generator.lean
theorems:
- CategoryTheory.isCoseparator_iff_faithful_preadditiveYoneda
- path: Mathlib/Data/List/Sigma.lean
theorems:
- List.mem_keys_kinsert
- path: Mathlib/Analysis/BoxIntegral/Partition/Split.lean
theorems:
- BoxIntegral.Box.coe_splitUpper
- BoxIntegral.Prepartition.IsPartition.exists_splitMany_le
- BoxIntegral.Box.splitLower_eq_self
- path: Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean
theorems:
- Orientation.tan_oangle_add_right_smul_rotation_pi_div_two
- Orientation.oangle_add_right_eq_arccos_of_oangle_eq_pi_div_two
- path: Mathlib/MeasureTheory/Integral/Lebesgue.lean
theorems:
- MeasureTheory.lintegral_iSup
- ENNReal.count_const_le_le_of_tsum_le
- MeasureTheory.lintegral_lintegral_mul
- path: Mathlib/GroupTheory/PushoutI.lean
theorems:
- Monoid.PushoutI.NormalWord.eq_one_of_smul_normalized
- path: Mathlib/Order/LiminfLimsup.lean
theorems:
- Filter.blimsup_mono
- Filter.isCobounded_principal
- Filter.limsup_congr
- path: Mathlib/Data/Fin/OrderHom.lean
theorems:
- Fin.predAbove_rev_left
- path: Mathlib/Algebra/Algebra/Spectrum.lean
theorems:
- spectrum.add_mem_add_iff
- spectrum.unit_smul_eq_smul
- spectrum.map_star
- path: Mathlib/Data/Polynomial/Derivative.lean
theorems:
- Polynomial.derivative_C_mul_X_sq
- path: Mathlib/CategoryTheory/Subobject/Comma.lean
theorems:
- CategoryTheory.CostructuredArrow.unop_left_comp_underlyingIso_hom_unop
- path: Mathlib/MeasureTheory/Measure/Dirac.lean
theorems:
- MeasureTheory.Measure.tsum_indicator_apply_singleton
- path: Mathlib/Order/CompactlyGenerated/Basic.lean
theorems:
- DirectedOn.disjoint_sSup_left
- path: Mathlib/Data/Polynomial/Inductions.lean
theorems:
- Polynomial.divX_X_pow
- path: Mathlib/Data/Real/Basic.lean
theorems:
- Real.cauchy_sub
- path: .lake/packages/std/Std/Data/Nat/Gcd.lean
theorems:
- Nat.Coprime.gcd_mul_left_cancel_right
- path: Mathlib/Topology/NoetherianSpace.lean
theorems:
- TopologicalSpace.NoetherianSpace.exists_finset_irreducible
- path: Mathlib/NumberTheory/Dioph.lean
theorems:
- Dioph.ext
- path: Mathlib/LinearAlgebra/Matrix/ToLin.lean
theorems:
- Matrix.toLin'_mul_apply
- path: .lake/packages/std/Std/Data/Nat/Lemmas.lean
theorems:
- Nat.compare_eq_eq
- path: Mathlib/Analysis/Convex/Gauge.lean
theorems:
- gauge_neg
- continuousAt_gauge
- path: Mathlib/Data/Multiset/Sum.lean
theorems:
- Multiset.mem_disjSum
- path: Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
theorems:
- WithTop.iInf_empty
- ciSup_unique
- path: Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean
theorems:
- CochainComplex.mappingCone.inr_snd
- CochainComplex.mappingCone.inl_v_descCochain_v
- path: Mathlib/Data/Set/Intervals/OrdConnected.lean
theorems:
- OrderEmbedding.image_Ioo
- path: Mathlib/Data/Set/Card.lean
theorems:
- Set.encard_union_le
- Set.Nat.encard_range
- Set.one_lt_ncard_iff
- Set.eq_of_subset_of_ncard_le
- path: Mathlib/Data/Part.lean
theorems:
- Part.bind_some_eq_map
- Part.mem_restrict
- Part.sdiff_get_eq
- Part.bind_none
- path: Mathlib/ModelTheory/Types.lean
theorems:
- FirstOrder.Language.Theory.CompleteType.setOf_subset_eq_univ_iff
- path: Mathlib/Data/List/Rotate.lean
theorems:
- List.rotate'_cons_succ
- List.rotate'_length_mul
- List.rotate_rotate
- path: Mathlib/Analysis/Convex/StoneSeparation.lean
theorems:
- exists_convex_convex_compl_subset
- path: Mathlib/LinearAlgebra/Orientation.lean
theorems:
- Basis.orientation_unitsSMul
- path: Mathlib/GroupTheory/CoprodI.lean
theorems:
- Monoid.CoprodI.lift_word_prod_nontrivial_of_not_empty
- path: Mathlib/Algebra/GCDMonoid/Basic.lean
theorems:
- gcd_mul_dvd_mul_gcd
- path: Mathlib/Data/Nat/Bitwise.lean
theorems:
- Nat.xor_self
- path: Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
theorems:
- MeasureTheory.StronglyMeasurable.norm_approxBounded_le
- MeasureTheory.StronglyMeasurable.tendsto_approxBounded_ae
- MeasureTheory.StronglyMeasurable.piecewise
- List.aestronglyMeasurable_prod'
- path: Mathlib/Algebra/Order/Monoid/Lemmas.lean
theorems:
- lt_one_of_mul_lt_right
- path: Mathlib/Algebra/Order/Monovary.lean
theorems:
- monovaryOn_iff_mul_rearrangement
- path: Mathlib/Order/UpperLower/Basic.lean
theorems:
- isLowerSet_iff_Iic_subset
- UpperSet.codisjoint_coe
- path: Mathlib/LinearAlgebra/Vandermonde.lean
theorems:
- Matrix.det_vandermonde
- path: Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
theorems:
- SimpleGraph.Subgraph.deleteVerts_le
- SimpleGraph.Subgraph.coe_degree
- path: Mathlib/Algebra/Group/Basic.lean
theorems:
- div_mul_mul_cancel
- path: Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean
theorems:
- LinearIsometry.norm_toContinuousLinearMap_comp
- path: Mathlib/Data/Set/Intervals/Group.lean
theorems:
- Set.pairwise_disjoint_Ioo_add_int_cast
- path: .lake/packages/std/Std/Data/RBMap/WF.lean
theorems:
- Std.RBNode.RedRed.balance2
- path: Mathlib/CategoryTheory/Comma/Presheaf.lean
theorems:
- "CategoryTheory.OverPresheafAux.MakesOverArrow.map\u2081"
- CategoryTheory.OverPresheafAux.counitForward_val_snd
- path: Mathlib/Topology/VectorBundle/Basic.lean
theorems:
- VectorBundleCore.coordChange_linear_comp
- path: Mathlib/Topology/NhdsSet.lean
theorems:
- nhdsSet_diagonal
- path: Mathlib/Data/Nat/Pairing.lean
theorems:
- Nat.unpair_left_le
- path: Mathlib/RingTheory/MvPolynomial/Homogeneous.lean
theorems:
- MvPolynomial.homogeneousComponent_zero
- path: Mathlib/Geometry/Manifold/Sheaf/LocallyRingedSpace.lean
theorems:
- smoothSheafCommRing.isUnit_stalk_iff
- path: Mathlib/Topology/Algebra/UniformGroup.lean
theorems:
- UniformGroup.uniformContinuous_iff_open_ker
- path: Mathlib/Algebra/Lie/Killing.lean
theorems:
- LieAlgebra.IsKilling.ker_restrictBilinear_eq_bot_of_isCartanSubalgebra
- path: Mathlib/Geometry/Euclidean/Sphere/Power.lean
theorems:
- EuclideanGeometry.mul_dist_eq_mul_dist_of_cospherical_of_angle_eq_pi
- path: Mathlib/Geometry/Manifold/ChartedSpace.lean
theorems:
- TopologicalSpace.Opens.chartAt_inclusion_symm_eventuallyEq
- path: Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
theorems:
- MeasureTheory.integrable_of_intervalIntegral_norm_bounded
- path: Mathlib/GroupTheory/Complement.lean
theorems:
- Subgroup.IsComplement.equiv_snd_eq_iff_rightCosetEquivalence
- path: Mathlib/RingTheory/Localization/Basic.lean
theorems:
- IsLocalization.map_eq_zero_iff
- IsLocalization.algebraMap_eq_map_map_submonoid
- path: Mathlib/Data/MvPolynomial/Degrees.lean
theorems:
- MvPolynomial.degrees_rename
- MvPolynomial.totalDegree_zero
- MvPolynomial.degrees_monomial_eq
- path: Mathlib/LinearAlgebra/CliffordAlgebra/Fold.lean
theorems:
- "CliffordAlgebra.foldl_prod_map_\u03B9"
- path: Mathlib/Topology/MetricSpace/Bounded.lean
theorems:
- Metric.dist_le_diam_of_mem'
- path: Mathlib/Analysis/NormedSpace/Banach.lean
theorems:
- LinearMap.continuous_of_seq_closed_graph
- ContinuousLinearMap.range_eq_map_coprodSubtypeLEquivOfIsCompl
- path: Mathlib/Data/List/Basic.lean
theorems:
- List.head_mem
- List.getLastI_eq_getLast?
- List.get_set_of_ne
- List.map_bijective_iff
- List.disjoint_map
- List.zipWith_flip
- path: Mathlib/GroupTheory/FiniteAbelian.lean
theorems:
- AddCommGroup.equiv_directSum_zmod_of_finite
- path: Mathlib/Data/Bool/Count.lean
theorems:
- List.Chain'.length_div_two_le_count_bool
- path: Mathlib/Analysis/SpecialFunctions/Gaussian.lean
theorems:
- GaussianFourier.integrable_cexp_neg_mul_sq_norm_add
- norm_cexp_neg_mul_sq
- integrable_mul_cexp_neg_mul_sq
- path: Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean
theorems:
- SimpleGraph.incMatrix_apply_eq_zero_iff
- SimpleGraph.incMatrix_apply'
- path: Mathlib/NumberTheory/Cyclotomic/CyclotomicCharacter.lean
theorems:
- ModularCyclotomicCharacter.id
- path: Mathlib/Data/Polynomial/Degree/Definitions.lean
theorems:
- Polynomial.eq_X_add_C_of_degree_eq_one
- Polynomial.ne_zero_of_natDegree_gt
- Polynomial.degree_linear
- path: Mathlib/Analysis/InnerProductSpace/Adjoint.lean
theorems:
- LinearIsometryEquiv.adjoint_eq_symm
- path: Mathlib/Topology/ContinuousFunction/Bounded.lean
theorems:
- BoundedContinuousFunction.add_norm_nonneg
- BoundedContinuousFunction.abs_diff_coe_le_dist
- path: Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
theorems:
- CategoryTheory.Limits.pullback_symmetry_hom_of_epi_eq
- path: Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
theorems:
- CircleDeg1Lift.translationNumber_one
- CircleDeg1Lift.translationNumber_mul_of_commute
- path: Mathlib/Combinatorics/SimpleGraph/Basic.lean
theorems:
- SimpleGraph.not_mem_neighborSet_self
- SimpleGraph.mem_incidence_iff_neighbor
- SimpleGraph.disjoint_edgeSet
- path: Mathlib/Analysis/Convex/Combination.lean
theorems:
- Set.Finite.convexHull_eq_image
- path: Mathlib/LinearAlgebra/Matrix/Circulant.lean
theorems:
- Matrix.circulant_mul_comm
- path: Mathlib/NumberTheory/Divisors.lean
theorems:
- Nat.image_div_divisors_eq_divisors
- Nat.swap_mem_divisorsAntidiagonal
- path: Mathlib/Topology/MetricSpace/Basic.lean
theorems:
- dist_pos
- MetricSpace.replaceUniformity_eq
- path: Mathlib/CategoryTheory/Adjunction/AdjointFunctorTheorems.lean
theorems:
- CategoryTheory.solutionSetCondition_of_isRightAdjoint
- path: Mathlib/Topology/UniformSpace/Completion.lean
theorems:
- CauchyFilter.uniformInducing_pureCauchy
- path: Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean
theorems:
- Set.self_mem_ordConnectedComponent
- path: Mathlib/Data/Nat/ModEq.lean
theorems:
- Nat.add_mod_add_of_le_add_mod
- Nat.odd_mod_four_iff
- path: Mathlib/Topology/UniformSpace/Equicontinuity.lean
theorems:
- Filter.HasBasis.uniformEquicontinuousOn_iff_left
- path: Mathlib/Algebra/GroupWithZero/NeZero.lean
theorems:
- inv_mul_cancel
- path: Mathlib/Data/Fin/Tuple/Curry.lean
theorems:
- Function.FromTypes.curry_uncurry
- path: Mathlib/Order/JordanHolder.lean
theorems:
- JordanHolderLattice.second_iso_of_eq
- path: Mathlib/Data/Vector/Snoc.lean
theorems:
- Vector.map_snoc
- path: Mathlib/Topology/ContinuousOn.lean
theorems:
- continuous_mulIndicator
- nhdsWithin_pi_eq'
- path: Mathlib/NumberTheory/SumPrimeReciprocals.lean
theorems:
- one_half_le_sum_primes_ge_one_div
- path: Mathlib/Data/Polynomial/Basic.lean
theorems:
- Polynomial.coeff_update_apply
- Polynomial.support_monomial
- Polynomial.toFinsupp_add
- path: Mathlib/Analysis/Calculus/LineDeriv/Basic.lean
theorems:
- hasLineDerivAt_iff_tendsto_slope_zero
- HasLineDerivAt.of_comp
- path: Mathlib/Topology/Baire/Lemmas.lean
theorems:
- "IsG\u03B4.dense_biUnion_interior_of_closed"
- "dense_biInter_of_G\u03B4"
- path: Mathlib/LinearAlgebra/Span.lean
theorems:
- Submodule.iSup_eq_span
- Submodule.mem_sSup
- Submodule.sup_span
- path: Mathlib/Algebra/BigOperators/Fin.lean
theorems:
- Fin.partialProd_succ'
- path: Mathlib/CategoryTheory/Abelian/ProjectiveResolution.lean
theorems:
- CategoryTheory.ProjectiveResolution.iso_inv_naturality
- path: Mathlib/Algebra/Order/Ring/Defs.lean
theorems:
- le_mul_of_le_one_right
- one_le_bit1
- path: Mathlib/RingTheory/Valuation/Basic.lean
theorems:
- Valuation.one_lt_val_iff
- Valuation.map_sub_le
- path: Mathlib/Order/GaloisConnection.lean
theorems:
- "GaloisConnection.l_iSup\u2082"
- path: Mathlib/GroupTheory/Transfer.lean
theorems:
- MonoidHom.transferSylow_eq_pow_aux
- path: Mathlib/LinearAlgebra/Basis.lean
theorems:
- Basis.constr_apply
- path: Mathlib/Computability/Primrec.lean
theorems:
- Nat.Primrec.sub
- Primrec.bind_decode_iff
- Primrec.const
- path: Mathlib/Analysis/Subadditive.lean
theorems:
- Subadditive.apply_mul_add_le
- path: Mathlib/RingTheory/UniqueFactorizationDomain.lean
theorems:
- Associates.map_subtype_coe_factors'
- UniqueFactorizationMonoid.multiplicity_eq_count_normalizedFactors
- Associates.FactorSet.prod_eq_zero_iff
- UniqueFactorizationMonoid.dvdNotUnit_iff_normalizedFactors_lt_normalizedFactors
- path: Mathlib/Data/Seq/WSeq.lean
theorems:
- Stream'.WSeq.LiftRel.refl
- path: Mathlib/Combinatorics/Composition.lean
theorems:
- List.map_length_splitWrtCompositionAux
- CompositionAsSet.mem_boundaries_iff_exists_blocks_sum_take_eq
- path: Mathlib/Order/WellFoundedSet.lean
theorems:
- Set.PartiallyWellOrderedOn.wellFoundedOn
- path: Mathlib/Analysis/Calculus/FDeriv/Analytic.lean
theorems:
- ContinuousMultilinearMap.cPolynomialAt
- path: Mathlib/LinearAlgebra/BilinearForm/Properties.lean
theorems:
- BilinForm.dualBasis_dualBasis
- path: Mathlib/Data/Set/Defs.lean
theorems:
- Set.mem_univ_pi
- path: Mathlib/Data/Rat/Defs.lean
theorems:
- Rat.coe_nat_eq_divInt
- Rat.zero_add
- path: .lake/packages/std/Std/Data/Array/Lemmas.lean
theorems:
- Array.get?_push_lt
- Array.contains_def
- path: Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean
theorems:
- FreeAbelianGroup.support_nsmul
- path: Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean
theorems:
- CliffordAlgebra.changeForm_changeForm
- CliffordAlgebra.changeForm.associated_neg_proof
- path: Mathlib/Data/List/OfFn.lean
theorems:
- List.last_ofFn
- path: Mathlib/Algebra/Ring/Defs.lean
theorems:
- ite_sub_ite
- path: Mathlib/Analysis/Convex/Between.lean
theorems:
- wbtw_iff_right_eq_or_left_mem_image_Ici
- path: Mathlib/Analysis/Convex/Star.lean
theorems:
- StarConvex.preimage_add_left
- path: Mathlib/Analysis/InnerProductSpace/LaxMilgram.lean
theorems:
- IsCoercive.antilipschitz
- IsCoercive.bounded_below
- path: Mathlib/Logic/Basic.lean
theorems:
- xor_false
- path: Mathlib/Analysis/Calculus/LHopital.lean
theorems:
- HasDerivAt.lhopital_zero_atTop_on_Ioi
- deriv.lhopital_zero_nhds
- path: Mathlib/Data/Polynomial/Reverse.lean
theorems:
- Polynomial.reverse_neg
- Polynomial.reverse_mul
- path: Mathlib/Algebra/Homology/HomotopyCofiber.lean
theorems:
- "HomologicalComplex.cylinder.map_\u03B9\u2080_eq_map_\u03B9\u2081"
- path: Mathlib/Probability/Independence/Kernel.lean
theorems:
- ProbabilityTheory.kernel.iIndep_comap_mem_iff
- path: Mathlib/Data/Real/Pointwise.lean
theorems:
- Real.sSup_smul_of_nonpos
- path: Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean
theorems:
- Orientation.det_rotation
- path: Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
theorems:
- affineSpan_le_toAffineSubspace_span
- vectorSpan_eq_span_vsub_set_left
- AffineSubspace.spanPoints_subset_coe_of_subset_coe
- path: Mathlib/MeasureTheory/Group/Convolution.lean
theorems:
- MeasureTheory.Measure.mconv_add
- path: Mathlib/Algebra/BigOperators/Basic.lean
theorems:
- Multiset.prod_sum
- path: Mathlib/LinearAlgebra/LinearIndependent.lean
theorems:
- LinearIndependent.maximal_iff
- LinearIndependent.disjoint_span_image
- path: Mathlib/RingTheory/Polynomial/GaussLemma.lean
theorems:
- Polynomial.Monic.irreducible_iff_irreducible_map_fraction_map
- path: Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean
theorems:
- CategoryTheory.Limits.pullbackDiagonalMapIdIso_inv_snd_snd
- CategoryTheory.Limits.diagonalObjPullbackFstIso_hom_snd
- path: Mathlib/Data/Polynomial/Expand.lean
theorems:
- Polynomial.contract_expand
- Polynomial.natDegree_expand
- path: Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean
theorems:
- TopCat.pullback_fst_image_snd_preimage
- path: Mathlib/Data/PFunctor/Univariate/Basic.lean
theorems:
- PFunctor.liftp_iff'
- PFunctor.liftr_iff
- path: .lake/packages/std/Std/Data/Int/DivMod.lean
theorems:
- Int.fdiv_self
- path: Mathlib/Data/String/Lemmas.lean
theorems:
- String.leftpad_suffix
- String.leftpad_length
- path: Mathlib/RingTheory/MvPowerSeries/Basic.lean
theorems:
- MvPowerSeries.monomial_mul_monomial
- MvPowerSeries.algebraMap_apply
- MvPowerSeries.commute_monomial
- path: Mathlib/Data/Nat/Size.lean
theorems:
- Nat.size_pow
- path: Mathlib/AlgebraicTopology/DoldKan/Decomposition.lean
theorems:
- "AlgebraicTopology.DoldKan.MorphComponents.preComp_\u03C6"
- path: Mathlib/Data/Finset/Fold.lean
theorems:
- Finset.fold_image
- path: Mathlib/Combinatorics/Partition.lean
theorems:
- Nat.Partition.partition_one_parts
- path: Mathlib/Combinatorics/Derangements/Basic.lean
theorems:
- derangements.Equiv.RemoveNone.fiber_none
- path: Mathlib/Computability/TuringMachine.lean
theorems:
- Turing.TM2to1.addBottom_modifyNth
- "Turing.eval_maximal\u2081"
- path: Mathlib/Logic/Equiv/Basic.lean
theorems:
- Equiv.sumCongr_refl
- path: Mathlib/CategoryTheory/Category/Basic.lean
theorems:
- CategoryTheory.eq_whisker
- path: Mathlib/CategoryTheory/EqToHom.lean
theorems:
- CategoryTheory.eqToIso_map_trans
- path: Mathlib/Data/Real/GoldenRatio.lean
theorems:
- geom_gold_isSol_fibRec
- goldConj_neg
- path: Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
theorems:
- Complex.cos_pi_div_two
- Real.sin_lt_sin_of_lt_of_le_pi_div_two
- path: Mathlib/MeasureTheory/Measure/EverywherePos.lean
theorems:
- MeasureTheory.Measure.measure_eq_zero_of_subset_diff_everywherePosSubset
- path: Mathlib/GroupTheory/OrderOfElement.lean
theorems:
- Commute.orderOf_mul_dvd_lcm
- isOfFinOrder_iff_zpow_eq_one
- orderOf_eq_prime
- path: Mathlib/Algebra/Order/Monoid/WithTop.lean
theorems:
- WithTop.map_add
- path: Mathlib/Combinatorics/Additive/SalemSpencer.lean
theorems:
- MulSalemSpencer.mul_right
- path: Mathlib/Probability/ProbabilityMassFunction/Constructions.lean
theorems:
- PMF.map_comp
- path: Mathlib/Data/Polynomial/Coeff.lean
theorems:
- Polynomial.coeff_mul_C
- path: Mathlib/Data/Complex/Exponential.lean
theorems:
- Real.exp_approx_start
- Real.quadratic_le_exp_of_nonneg
- Complex.sin_zero
- Complex.cos_two_mul
- path: Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean
theorems:
- Path.Homotopy.continuous_transReflReparamAux
- path: Mathlib/RingTheory/WittVector/Truncated.lean
theorems:
- TruncatedWittVector.card
- path: Mathlib/Analysis/ODE/Gronwall.lean
theorems:
- ODE_solution_unique_of_eventually
- path: Mathlib/Topology/Connected/Basic.lean
theorems:
- "PreconnectedSpace.induction\u2082'"
- IsPreconnected.prod
- IsPreconnected.union'
- connectedComponentIn_eq
- path: Mathlib/Data/Option/NAry.lean
theorems:
- "Option.map\u2082_assoc"
- "Option.map\u2082_map_left_anticomm"
- path: Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean
theorems:
- Real.surjOn_tan
- path: Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
theorems:
- NNReal.rpow_neg_one
- ENNReal.top_rpow_of_neg
- path: Mathlib/Probability/Kernel/Basic.lean
theorems:
- ProbabilityTheory.kernel.sum_add
- ProbabilityTheory.kernel.integral_piecewise
- ProbabilityTheory.kernel.integral_deterministic
- ProbabilityTheory.kernel.restrict_apply'
- path: Mathlib/RingTheory/Ideal/Operations.lean
theorems:
- Submodule.mem_smul_span
- Ideal.IsPrime.multiset_prod_le
- path: Mathlib/RingTheory/Etale.lean
theorems:
- Algebra.FormallyUnramified.localization_map
- path: Mathlib/RingTheory/Ideal/QuotientOperations.lean
theorems:
- Ideal.mem_quotient_iff_mem
- Ideal.fst_comp_quotientInfEquivQuotientProd
- path: Mathlib/CategoryTheory/EffectiveEpi/Preserves.lean
theorems:
- CategoryTheory.effectiveEpiFamilyStructOfEquivalence_aux
- path: Mathlib/Logic/Relation.lean
theorems:
- Relation.TransGen.head_induction_on
- Relation.map_map
- Relation.reflTransGen_eq_reflGen
- Relation.ReflTransGen.trans_induction_on
- path: Mathlib/RingTheory/Polynomial/Vieta.lean
theorems:
- MvPolynomial.prod_C_add_X_eq_sum_esymm
- path: Mathlib/Algebra/Periodic.lean
theorems:
- Function.Periodic.const_inv_smul
- path: Mathlib/Algebra/Lie/Solvable.lean
theorems:
- LieAlgebra.derivedSeriesOfIdeal_le
- path: Mathlib/Algebra/BigOperators/Module.lean
theorems:
- Finset.sum_range_by_parts
- path: Mathlib/CategoryTheory/Sites/Sheaf.lean
theorems:
- CategoryTheory.Presheaf.isSheaf_iff_multiequalizer
- path: Mathlib/SetTheory/Cardinal/Basic.lean
theorems:
- Cardinal.one_lt_iff_nontrivial
- Cardinal.self_le_power
- Cardinal.mk_subset_ge_of_subset_image_lift
- path: Mathlib/Topology/Basic.lean
theorems:
- closure_diff_frontier
- interior_eq_empty_iff_dense_compl
- path: Mathlib/Data/Finsupp/ToDFinsupp.lean
theorems:
- finsuppLequivDFinsupp_apply_apply
- path: Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean
theorems:
- one_lt_mul_iff
- path: Mathlib/NumberTheory/RamificationInertia.lean
theorems:
- Ideal.ramificationIdx_lt
- path: Mathlib/Analysis/Analytic/Meromorphic.lean
theorems:
- MeromorphicAt.order_eq_top_iff
- path: Mathlib/Algebra/Order/ToIntervalMod.lean
theorems:
- toIcoMod_sub_zsmul
- toIocMod_eq_iff
- path: Mathlib/Computability/Language.lean
theorems:
- Language.mul_self_kstar_comm
- path: Mathlib/MeasureTheory/MeasurableSpace/Basic.lean
theorems:
- MeasurableSpace.comap_compl
- MeasurableEmbedding.iff_comap_eq
- measurableSet_tendsto
- path: Mathlib/Analysis/Convex/GaugeRescale.lean
theorems:
- image_gaugeRescaleHomeomorph_interior
- path: Mathlib/LinearAlgebra/Basis/VectorSpace.lean
theorems:
- LinearMap.exists_rightInverse_of_surjective
- Submodule.exists_le_ker_of_lt_top
- path: Mathlib/AlgebraicTopology/DoldKan/PInfty.lean
theorems:
- AlgebraicTopology.DoldKan.PInfty_comp_QInfty
- AlgebraicTopology.DoldKan.QInfty_f_comp_PInfty_f
- AlgebraicTopology.DoldKan.QInfty_idem
- path: Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean
theorems:
- MeasureTheory.condexp_condexp_of_le
- path: Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
theorems:
- intervalIntegral.integral_comp_smul_deriv''
- path: Mathlib/Algebra/ModEq.lean
theorems:
- AddCommGroup.intCast_modEq_intCast'
- path: Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean
theorems:
- CategoryTheory.Limits.Multiequalizer.hom_ext
- path: Mathlib/Analysis/SpecialFunctions/Log/Basic.lean
theorems:
- Mathlib.Meta.Positivity.log_pos_of_isRat_neg
- Real.tendsto_log_atTop
- path: Mathlib/Data/QPF/Multivariate/Basic.lean
theorems:
- MvQPF.liftR_iff
- path: Mathlib/Algebra/Homology/HomotopyCategory/DegreewiseSplit.lean
theorems:
- HomotopyCategory.distinguished_iff_iso_trianglehOfDegreewiseSplit
- path: Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean
theorems:
- midpoint_sub_add
- path: Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean
theorems:
- Module.End.isRoot_of_hasEigenvalue
- path: Mathlib/Data/Finmap.lean
theorems:
- Finmap.mem_singleton
- path: Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean
theorems:
- Real.Gamma_one
- Complex.Gamma_ofReal
- path: Mathlib/Data/Rat/Floor.lean
theorems:
- Rat.floor_def'
- path: Mathlib/Algebra/Homology/Homotopy.lean
theorems:
- Homotopy.comp_nullHomotopicMap
- path: Mathlib/LinearAlgebra/Matrix/Symmetric.lean
theorems:
- Matrix.IsSymm.fromBlocks
- path: Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean
theorems:
- "jacobiTheta\u2082_add_left"
- path: Mathlib/Data/PFunctor/Univariate/M.lean
theorems:
- PFunctor.M.cases_mk
- path: Mathlib/Geometry/RingedSpace/PresheafedSpace.lean
theorems:
- AlgebraicGeometry.PresheafedSpace.Hom.ext
- path: Mathlib/Topology/Algebra/Order/Compact.lean
theorems:
- IsCompact.exists_isLeast
- path: Mathlib/Topology/UniformSpace/UniformConvergence.lean
theorems:
- TendstoUniformly.prod_map
- path: Mathlib/Data/Set/Pointwise/ListOfFn.lean
theorems:
- Set.mem_pow
- path: Mathlib/RingTheory/Derivation/Basic.lean
theorems:
- Derivation.eqOn_adjoin
- path: Mathlib/RingTheory/FractionalIdeal/Basic.lean
theorems:
- FractionalIdeal.coeIdeal_mul
- path: Mathlib/CategoryTheory/Abelian/LeftDerived.lean
theorems:
- CategoryTheory.ProjectiveResolution.isoLeftDerivedObj_hom_naturality
- path: Mathlib/Combinatorics/Young/YoungDiagram.lean
theorems:
- YoungDiagram.exists_not_mem_row
- path: Mathlib/CategoryTheory/Monoidal/Preadditive.lean
theorems:
- CategoryTheory.leftDistributor_assoc
- path: Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
theorems:
- AkraBazziRecurrence.GrowsPolynomially.neg_iff
- path: Mathlib/LinearAlgebra/Finsupp.lean
theorems:
- Finsupp.totalOn_range
- path: Mathlib/Data/IsROrC/Basic.lean
theorems:
- IsROrC.exists_norm_eq_mul_self
- IsROrC.div_re
- IsROrC.nonneg_iff
- IsROrC.conj_smul
- IsROrC.one_re
- path: Mathlib/Data/Nat/Defs.lean
theorems:
- Nat.mul_div_cancel_left'
- Nat.div_le_div_right
- Nat.div_lt_one_iff
- path: Mathlib/Order/CompleteBooleanAlgebra.lean
theorems:
- "disjoint_iSup\u2082_iff"
- path: Mathlib/Algebra/CharP/MixedCharZero.lean
theorems:
- MixedCharZero.reduce_to_maximal_ideal
- path: Mathlib/Data/Set/Pointwise/BigOperators.lean
theorems:
- Set.multiset_prod_subset_multiset_prod
- path: Mathlib/Probability/StrongLaw.lean
theorems:
- ProbabilityTheory.abs_truncation_le_abs_self
- path: Mathlib/Combinatorics/SetFamily/LYM.lean
theorems:
- Finset.mem_falling
- path: Mathlib/NumberTheory/Rayleigh.lean
theorems:
- compl_beattySeq
- path: Mathlib/Init/Data/Int/CompLemmas.lean
theorems:
- Int.natAbs_bit0_step
- path: Mathlib/Algebra/Star/StarAlgHom.lean
theorems:
- NonUnitalStarAlgHom.mk_coe
- path: Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.lean
theorems:
- AlgebraicClosure.Step.isIntegral
- path: Mathlib/NumberTheory/NumberField/Embeddings.lean
theorems:
- NumberField.InfinitePlace.isUnramified_mk_iff_forall_isConj
- NumberField.InfinitePlace.card_eq_card_isUnramifiedIn
- NumberField.InfinitePlace.isReal_of_mk_isReal
- NumberField.InfinitePlace.ComplexEmbedding.exists_comp_symm_eq_of_comp_eq
- path: Mathlib/Topology/Maps.lean
theorems:
- OpenEmbedding.open_iff_image_open
- path: Mathlib/Topology/Order/Basic.lean
theorems:
- eventually_abs_sub_lt
- path: Mathlib/Data/Finset/Image.lean
theorems:
- Finset.image_biUnion
- path: Mathlib/Topology/Constructions.lean
theorems:
- continuous_sigma_iff
- path: Mathlib/RingTheory/Adjoin/Basic.lean
theorems:
- Algebra.adjoin_adjoin_coe_preimage
- path: Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean
theorems:
- CategoryTheory.Subgroupoid.generated_le_generatedNormal
- path: Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean
theorems:
- Finset.sups_aux
- Finset.truncatedInf_union
- path: Mathlib/Topology/MetricSpace/Thickening.lean
theorems:
- Metric.ediam_cthickening_le
- path: Mathlib/LinearAlgebra/Eigenspace/Basic.lean
theorems:
- Module.End.iSup_generalizedEigenspace_inf_le_add
- Module.End.hasGeneralizedEigenvalue_of_hasEigenvalue
- path: Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean
theorems:
- AlgebraicTopology.alternatingFaceMapComplex_obj_d
- path: Mathlib/Topology/Order/MonotoneConvergence.lean
theorems:
- tendsto_atBot_isLUB
- path: Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean
theorems:
- MultilinearMap.bound_of_shell
- path: Mathlib/MeasureTheory/Integral/SetToL1.lean
theorems:
- MeasureTheory.tendsto_setToFun_of_L1
- path: Mathlib/Data/Matrix/Notation.lean
theorems:
- Matrix.tail_transpose
- path: Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean
theorems:
- Basis.addHaar_eq_iff
- path: Mathlib/Data/Finset/LocallyFinite/Basic.lean
theorems:
- Finset.Ioc_subset_Ioo_right
- Finset.map_add_left_Ioc
- Finset.Ioc_insert_left
- path: Mathlib/Data/Sign.lean
theorems:
- SignType.pos_iff
- SignType.mul_assoc
- abs_mul_sign
- SignType.le_trans
- path: Mathlib/Data/Real/Sqrt.lean
theorems:
- Real.sqrt_one_add_le
- path: Mathlib/GroupTheory/FixedPointFree.lean
theorems:
- MonoidHom.FixedPointFree.prod_pow_eq_one
- path: Mathlib/CategoryTheory/Whiskering.lean
theorems:
- CategoryTheory.Functor.pentagon
- path: Mathlib/NumberTheory/NumberField/CanonicalEmbedding.lean
theorems:
- NumberField.canonicalEmbedding.nnnorm_eq
- NumberField.mixedEmbedding.finrank
- path: Mathlib/Data/Nat/Sqrt.lean
theorems:
- Nat.sqrt_le_add
- Nat.le_three_of_sqrt_eq_one
- path: Mathlib/Topology/MetricSpace/Isometry.lean
theorems:
- IsometryEquiv.ediam_preimage
- path: Mathlib/Topology/Homotopy/Contractible.lean
theorems:
- ContinuousMap.nullhomotopic_of_constant
- path: Mathlib/AlgebraicTopology/DoldKan/SplitSimplicialObject.lean
theorems:
- "SimplicialObject.Splitting.cofan_inj_\u03C0Summand_eq_id"
- path: Mathlib/GroupTheory/GroupAction/Hom.lean
theorems:
- DistribMulActionHom.ext_ring
- path: Mathlib/Algebra/MonoidAlgebra/Support.lean
theorems:
- MonoidAlgebra.mem_span_support
- path: Mathlib/SetTheory/Lists.lean
theorems:
- Lists'.to_ofList
- Lists'.ofList_subset
- path: Mathlib/Data/Nat/Prime.lean
theorems:
- Nat.prime_mul_iff
- Nat.minFac_le_div
- path: Mathlib/Analysis/InnerProductSpace/EuclideanDist.lean
theorems:
- Euclidean.closure_ball
- path: Mathlib/Topology/MetricSpace/PseudoMetric.lean
theorems:
- edist_le_ofReal
- nndist_dist
- dist_dist_dist_le_right
- Metric.eventually_isCompact_closedBall
- path: Mathlib/Tactic/NormNum/OfScientific.lean
theorems:
- Mathlib.Meta.NormNum.isRat_ofScientific_of_true
- path: Mathlib/SetTheory/Ordinal/Arithmetic.lean
theorems:
- Ordinal.bsup_eq_blsub_iff_succ
- Ordinal.IsNormal.blsub_eq
- Ordinal.enum_succ_eq_top
- Ordinal.sup_typein_succ
- path: Mathlib/RingTheory/RootsOfUnity/Complex.lean
theorems:
- Complex.isPrimitiveRoot_exp
- path: Mathlib/Analysis/InnerProductSpace/Projection.lean
theorems:
- smul_orthogonalProjection_singleton
- path: Mathlib/Analysis/ConstantSpeed.lean
theorems:
- hasConstantSpeedOnWith_of_subsingleton
- path: Mathlib/CategoryTheory/Monoidal/Category.lean
theorems:
- CategoryTheory.MonoidalCategory.comp_tensor_id
- CategoryTheory.MonoidalCategory.associator_naturality_right
- path: Mathlib/Tactic/NormNum/LegendreSymbol.lean
theorems:
- "Mathlib.Meta.NormNum.jacobiSymNat.qr\u2083"
- path: Mathlib/LinearAlgebra/RootSystem/Basic.lean
theorems:
- RootPairing.isReduced_iff
- path: Mathlib/NumberTheory/ZetaFunction.lean
theorems:
- riemannZeta_one_sub
- path: Mathlib/Analysis/Convex/Normed.lean
theorems:
- Convex.thickening
- path: Mathlib/Algebra/Free.lean
theorems:
- FreeMagma.length_toFreeSemigroup
- path: Mathlib/Algebra/Ring/CentroidHom.lean
theorems:
- CentroidHom.centroid_eq_centralizer_mulLeftRight
- path: Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean
theorems:
- AffineMap.image_vsub_image
- path: Mathlib/Topology/Partial.lean
theorems:
- pcontinuous_iff'
- path: Mathlib/MeasureTheory/Constructions/Cylinders.lean
theorems:
- MeasureTheory.disjoint_cylinder_iff
- path: Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean
theorems:
- WeierstrassCurve.Jacobian.equation_zero'
- WeierstrassCurve.Jacobian.equation_some
- path: Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean
theorems:
- GeneralizedContinuedFraction.IntFractPair.succ_nth_stream_eq_none_iff
- path: Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean
theorems:
- MeasureTheory.ProbabilityMeasure.continuous_map
- MeasureTheory.ProbabilityMeasure.ennreal_coeFn_eq_coeFn_toMeasure
- path: Mathlib/LinearAlgebra/Matrix/SchurComplement.lean
theorems:
- Matrix.det_one_sub_mul_comm
- path: Mathlib/Order/Directed.lean
theorems:
- directedOn_image
- path: .lake/packages/std/Std/Data/List/Pairwise.lean
theorems:
- List.Pairwise.imp_of_mem
- path: Mathlib/LinearAlgebra/Matrix/ZPow.lean
theorems:
- Matrix.pow_inv_comm'
- path: Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean
theorems:
- mfderiv_sub
- path: Mathlib/Data/Multiset/Basic.lean
theorems:
- Multiset.filter_attach'
- Multiset.union_le
- Multiset.one_le_count_iff_mem
- Multiset.pairwise_coe_iff
- Multiset.lt_replicate_succ
- path: Mathlib/LinearAlgebra/Matrix/Transvection.lean
theorems:
- Matrix.transvection_mul_apply_of_ne
- path: Mathlib/Geometry/Manifold/ContMDiff/Basic.lean
theorems:
- contMDiff_inclusion
- path: Mathlib/Algebra/GeomSum.lean
theorems:
- geom_sum_pos'
- path: Mathlib/Data/Vector/Basic.lean
theorems:
- Vector.reverse_get_zero
- path: Mathlib/LinearAlgebra/Matrix/PosDef.lean
theorems:
- Matrix.PosSemidef.one
- path: Mathlib/MeasureTheory/Measure/MeasureSpace.lean
theorems:
- "MeasureTheory.measure_sUnion\u2080"
- MeasureTheory.Measure.measure_le_sInf
- MeasureTheory.sum_measure_preimage_singleton
- path: Mathlib/MeasureTheory/Measure/Hausdorff.lean
theorems:
- MeasureTheory.Measure.mkMetric_le_liminf_sum
- path: Mathlib/MeasureTheory/Measure/GiryMonad.lean
theorems:
- MeasureTheory.Measure.join_map_join
- MeasureTheory.Measure.measurable_of_measurable_coe
- path: Mathlib/Data/Nat/Factorial/Cast.lean
theorems:
- Nat.cast_descFactorial_two
- path: Mathlib/MeasureTheory/Measure/Typeclasses.lean
theorems:
- "MeasureTheory.Measure.countable_meas_pos_of_disjoint_iUnion\u2080"
- path: Mathlib/Data/Nat/Factorial/DoubleFactorial.lean
theorems:
- Nat.doubleFactorial_two_mul
- path: Mathlib/Data/Finset/Basic.lean
theorems:
- Finset.singleton_inter_of_not_mem
- Finset.filter_inter
- Finset.erase_insert
- path: Mathlib/Geometry/RingedSpace/LocallyRingedSpace/HasColimits.lean
theorems:
- AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
- path: .lake/packages/std/Std/Data/Rat/Lemmas.lean
theorems:
- Rat.mkRat_mul_mkRat
- Rat.maybeNormalize_eq
- path: Mathlib/LinearAlgebra/Determinant.lean
theorems:
- Basis.det_unitsSMul
- path: Mathlib/AlgebraicTopology/DoldKan/Homotopies.lean
theorems:
- "AlgebraicTopology.DoldKan.h\u03C3'_eq"
- path: Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean
theorems:
- Complex.cpow_neg
- path: Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean
theorems:
- Real.arccos_neg_one
- path: Mathlib/GroupTheory/Submonoid/Operations.lean
theorems:
- Submonoid.mrange_inl_sup_mrange_inr
- Submonoid.equivMapOfInjective_coe_mulEquiv
- path: Mathlib/Analysis/InnerProductSpace/LinearPMap.lean
theorems:
- LinearPMap.adjoint_apply_of_not_dense
- path: Mathlib/Algebra/Group/Units.lean
theorems:
- Units.isUnit_units_mul
- path: Mathlib/RingTheory/Polynomial/IntegralNormalization.lean
theorems:
- Polynomial.integralNormalization_support
- path: Mathlib/MeasureTheory/Function/LocallyIntegrable.lean
theorems:
- MeasureTheory.IntegrableOn.continuousOn_mul_of_subset
- path: Mathlib/Data/Finset/NAry.lean
theorems:
- "Finset.image\u2082_left_comm"
- path: Mathlib/Analysis/Calculus/FDeriv/RestrictScalars.lean
theorems:
- hasFDerivAt_of_restrictScalars
- path: Mathlib/Algebra/Order/Chebyshev.lean
theorems:
- sum_div_card_sq_le_sum_sq_div_card
- path: Mathlib/NumberTheory/Padics/RingHoms.lean
theorems:
- PadicInt.denseRange_int_cast
- path: Mathlib/Analysis/Calculus/Deriv/Add.lean
theorems:
- deriv_sub_const
- path: Mathlib/LinearAlgebra/Dual.lean
theorems:
- TensorProduct.dualDistribInvOfBasis_apply
- Subspace.finrank_add_finrank_dualCoannihilator_eq
- LinearMap.range_dualMap_le_dualAnnihilator_ker
- Submodule.iSup_dualAnnihilator_le_iInf
- path: Mathlib/Analysis/NormedSpace/MStructure.lean
theorems:
- IsLprojection.compl_mul
- path: .lake/packages/std/Std/Data/PairingHeap.lean
theorems:
- Std.PairingHeapImp.Heap.noSibling_merge
- path: Mathlib/RingTheory/RootsOfUnity/Basic.lean
theorems:
- IsPrimitiveRoot.of_map_of_injective
- path: Mathlib/Data/Finset/Pointwise.lean
theorems:
- Finset.preimage_mul_right_one
- Finset.op_smul_finset_smul_eq_smul_smul_finset
- Finset.inv_smul_mem_iff
- path: Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean
theorems:
- Matrix.Represents.one
- path: Mathlib/RingTheory/Noetherian.lean
theorems:
- isNoetherian_of_fg_of_noetherian
- path: Mathlib/Probability/Independence/ZeroOne.lean
theorems:
- ProbabilityTheory.measure_eq_zero_or_one_of_indepSet_self
- ProbabilityTheory.kernel.measure_eq_zero_or_one_of_indepSet_self
- path: Mathlib/AlgebraicGeometry/ProjectiveSpectrum/StructureSheaf.lean
theorems:
- AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.SectionSubring.addMem'
- path: Mathlib/Topology/Homeomorph.lean
theorems:
- Homeomorph.symm_map_nhds_eq
- path: Mathlib/ModelTheory/Substructures.lean
theorems:
- FirstOrder.Language.Substructure.comap_id
- path: Mathlib/Topology/AlexandrovDiscrete.lean
theorems:
- isOpen_iff_forall_specializes
- path: Mathlib/CategoryTheory/Products/Bifunctor.lean
theorems:
- CategoryTheory.Bifunctor.diagonal'
- path: Mathlib/Data/Nat/Digits.lean
theorems:
- Nat.toDigitsCore_lens_eq
- path: Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
theorems:
- SimpleGraph.Walk.edges_copy
- SimpleGraph.Walk.support_concat
- path: Mathlib/RingTheory/Polynomial/Dickson.lean
theorems:
- Polynomial.map_dickson
- path: Mathlib/ModelTheory/Semantics.lean
theorems:
- FirstOrder.Language.BoundedFormula.realize_toPrenex
- path: Mathlib/Analysis/SpecificLimits/Normed.lean
theorems:
- isLittleO_pow_const_mul_const_pow_const_pow_of_norm_lt
- path: Mathlib/Analysis/Normed/Group/Pointwise.lean
theorems:
- singleton_div_closedBall
- path: Mathlib/Analysis/MeanInequalitiesPow.lean
theorems:
- ENNReal.rpow_add_rpow_le_add
- path: Mathlib/Computability/TMToPartrec.lean
theorems:
- Turing.PartrecToTM2.codeSupp_case
- Turing.PartrecToTM2.trNormal_supports
- Turing.PartrecToTM2.K'.elim_update_main
- Turing.PartrecToTM2.trNormal_respects
- path: Mathlib/RingTheory/Coprime/Basic.lean
theorems:
- IsCoprime.of_mul_right_right
- path: Mathlib/Data/Sym/Basic.lean
theorems:
- Sym.map_id'
- path: Mathlib/Geometry/Manifold/LocalDiffeomorph.lean
theorems:
- IsLocalDiffeomorphAt.contMDiffAt
- path: Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean
theorems:
- MeasureTheory.snorm_le_snorm_mul_snorm_of_nnnorm
- path: Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean
theorems:
- MeasureTheory.Measure.inv_rnDeriv_aux
- path: Mathlib/Order/CompleteLattice.lean
theorems:
- iSup_split_single
- path: Mathlib/Topology/UniformSpace/Basic.lean
theorems:
- uniformity_lift_le_swap
- UniformSpace.toTopologicalSpace_iInf
- Disjoint.exists_uniform_thickening
- path: Mathlib/Data/Nat/Interval.lean
theorems:
- Nat.Ico_insert_succ_left
- path: Mathlib/Algebra/Order/Rearrangement.lean
theorems:
- Monovary.sum_smul_comp_perm_eq_sum_smul_iff
- path: Mathlib/Data/List/AList.lean
theorems:
- AList.lookup_insert
- path: Mathlib/LinearAlgebra/SesquilinearForm.lean
theorems:
- LinearMap.IsRefl.nondegenerate_of_separatingRight
- path: Mathlib/NumberTheory/DiophantineApproximation.lean
theorems:
- Real.exists_rat_eq_convergent
- path: Mathlib/RingTheory/Jacobson.lean
theorems:
- Ideal.Polynomial.jacobson_bot_of_integral_localization
- path: Mathlib/Data/Vector3.lean
theorems:
- Vector3.append_left
- path: Mathlib/Topology/Homotopy/Product.lean
theorems:
- Path.Homotopic.comp_pi_eq_pi_comp
- path: Mathlib/AlgebraicGeometry/Gluing.lean
theorems:
- "AlgebraicGeometry.Scheme.GlueData.\u03B9_isoCarrier_inv"
- path: Mathlib/Algebra/Homology/ShortComplex/ConcreteCategory.lean
theorems:
- CategoryTheory.Preadditive.epi_iff_surjective'
- path: Mathlib/Data/Polynomial/Splits.lean
theorems:
- Polynomial.splits_of_splits_mul'
- path: Mathlib/Data/ENNReal/Real.lean
theorems:
- ENNReal.ofReal_sub
- ENNReal.mul_iInf
- path: Mathlib/Order/BooleanAlgebra.lean
theorems:
- inf_sdiff_eq_bot_iff
- path: Mathlib/Data/List/Zip.lean
theorems:
- List.revzip_map_fst
- List.map_prod_right_eq_zip
- path: Mathlib/Analysis/Normed/Group/Hom.lean
theorems:
- NormedAddGroupHom.mem_ker
- path: Mathlib/RingTheory/NonUnitalSubring/Basic.lean
theorems:
- "NonUnitalSubring.closure_induction\u2082"
- path: Mathlib/MeasureTheory/Group/Measure.lean
theorems:
- MeasureTheory.Measure.isHaarMeasure_map
- path: Mathlib/CategoryTheory/Pi/Basic.lean
theorems:
- CategoryTheory.isIso_pi_iff
- path: Mathlib/Logic/Equiv/List.lean
theorems:
- Denumerable.list_ofNat_succ
- path: Mathlib/AlgebraicGeometry/Morphisms/FiniteType.lean
theorems:
- AlgebraicGeometry.locallyOfFiniteType_eq
- path: Mathlib/MeasureTheory/Covering/Vitali.lean
theorems:
- Vitali.exists_disjoint_subfamily_covering_enlargment_closedBall
- path: Mathlib/CategoryTheory/Abelian/FunctorCategory.lean
theorems:
- CategoryTheory.Abelian.FunctorCategory.coimageImageComparison_app'
- path: Mathlib/CategoryTheory/Preadditive/OfBiproducts.lean
theorems:
- CategoryTheory.SemiadditiveOfBinaryBiproducts.isUnital_leftAdd
- path: Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
theorems:
- Complex.abs_mul_cos_add_sin_mul_I
- path: Mathlib/RingTheory/Localization/Module.lean
theorems:
- LinearIndependent.of_isLocalizedModule
- path: Mathlib/Data/Real/EReal.lean
theorems:
- EReal.exists_between_coe_real
- EReal.sub_lt_sub_of_lt_of_le
- path: Mathlib/Topology/Gluing.lean
theorems:
- "TopCat.GlueData.eqvGen_of_\u03C0_eq"
- path: Mathlib/MeasureTheory/Measure/Sub.lean
theorems:
- MeasureTheory.Measure.sub_apply
- path: Mathlib/MeasureTheory/Covering/VitaliFamily.lean
theorems:
- VitaliFamily.mem_filterAt_iff
- path: Mathlib/LinearAlgebra/Matrix/Nondegenerate.lean
theorems:
- Matrix.nondegenerate_of_det_ne_zero
- path: Mathlib/NumberTheory/LegendreSymbol/MulCharacter.lean
theorems:
- MulChar.pow_apply'
- path: Mathlib/Order/ModularLattice.lean
theorems:
- wellFounded_lt_exact_sequence
- path: Mathlib/Data/Sym/Sym2.lean
theorems:
- Sym2.other_spec
- path: Mathlib/RingTheory/DedekindDomain/Different.lean
theorems:
- Submodule.le_traceDual
- aeval_derivative_mem_differentIdeal
- path: Mathlib/Analysis/NormedSpace/CompactOperator.lean
theorems:
- IsCompactOperator.continuous
- path: Mathlib/Algebra/Order/CompleteField.lean
theorems:
- LinearOrderedField.cutMap_bddAbove
- path: Mathlib/Analysis/Convolution.lean
theorems:
- BddAbove.continuous_convolution_left_of_integrable
- HasCompactSupport.convolution_integrand_bound_left
- path: Mathlib/MeasureTheory/Measure/Stieltjes.lean
theorems:
- StieltjesFunction.measure_Ioo
- path: Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
theorems:
- Set.Finite.summable
- path: Mathlib/Order/Filter/Cofinite.lean
theorems:
- "Filter.coprod\u1D62_cofinite"
- path: Mathlib/CategoryTheory/Monoidal/End.lean
theorems:
- "CategoryTheory.\u03BC_inv_naturality\u2097"
- path: Mathlib/Probability/Kernel/MeasureCompProd.lean
theorems:
- MeasureTheory.Measure.dirac_compProd_apply
- path: Mathlib/Data/List/Sublists.lean
theorems:
- List.sublists'_eq_sublists
- path: Mathlib/Algebra/Divisibility/Units.lean
theorems:
- isRelPrime_mul_unit_right_right
- path: Mathlib/Data/Int/Units.lean
theorems:
- Int.isUnit_iff
- path: Mathlib/Data/Num/Lemmas.lean
theorems:
- PosNum.to_nat_inj
- path: Mathlib/RingTheory/PrincipalIdealDomain.lean
theorems:
- gcd_isUnit_iff
- path: Mathlib/GroupTheory/Perm/Cycle/Type.lean
theorems:
- Equiv.Perm.cycleType_eq
- path: Mathlib/Data/Int/Parity.lean
theorems:
- Int.add_one_ediv_two_mul_two_of_odd
- Int.ne_of_odd_add
- path: Mathlib/Combinatorics/Additive/Behrend.lean
theorems:
- Behrend.map_succ
- path: Mathlib/Data/Set/Intervals/Monoid.lean
theorems:
- Set.image_const_add_Ioi
- path: Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean
theorems:
- MeasureTheory.snorm_condexpL2_le
- path: Mathlib/LinearAlgebra/AffineSpace/Combination.lean
theorems:
- Finset.affineCombination_eq_of_weightedVSub_eq_zero_of_eq_neg_one
- path: Mathlib/Data/Nat/Choose/Basic.lean
theorems:
- Nat.choose_one_right
- path: Mathlib/Order/Disjoint.lean
theorems:
- Disjoint.left_le_of_le_sup_left
- path: Mathlib/Algebra/Category/Ring/Constructions.lean
theorems:
- CommRingCat.pushoutCocone_inl
- path: Mathlib/Data/List/Indexes.lean
theorems:
- List.foldlIdx_eq_foldlIdxSpec
- path: Mathlib/Analysis/InnerProductSpace/Orientation.lean
theorems:
- Orientation.abs_volumeForm_apply_of_pairwise_orthogonal
- path: Mathlib/GroupTheory/GroupAction/Basic.lean
theorems:
- MulAction.disjoint_image_image_iff
- path: Mathlib/Analysis/LocallyConvex/WithSeminorms.lean
theorems:
- WithSeminorms.partial_sups
- path: Mathlib/Algebra/GroupPower/Order.lean
theorems:
- zero_pow_le_one
- path: Mathlib/Data/Ordmap/Ordset.lean
theorems:
- Ordnode.balanceL_eq_balance
- Ordnode.Raised.add_left
- path: Mathlib/Data/List/Nodup.lean
theorems:
- List.Nodup.erase_get
- path: Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean
theorems:
- integrableOn_exp_Iic
- path: Mathlib/Analysis/Calculus/InverseFunctionTheorem/ApproximatesLinearOn.lean
theorems:
- ApproximatesLinearOn.open_image
- path: Mathlib/RingTheory/Polynomial/Basic.lean
theorems:
- MvPolynomial.prime_rename_iff
- path: Mathlib/Topology/Order/NhdsSet.lean
theorems:
- nhdsSet_Ici
- path: Mathlib/NumberTheory/Multiplicity.lean
theorems:
- padicValNat.pow_add_pow
- path: Mathlib/Analysis/Calculus/Deriv/Basic.lean
theorems:
- DifferentiableAt.derivWithin
- path: Mathlib/RingTheory/RootsOfUnity/Minpoly.lean
theorems:
- IsPrimitiveRoot.minpoly_eq_pow_coprime
- path: Mathlib/Data/List/MinMax.lean
theorems:
- List.getElem_le_maximum_of_length_pos
- path: Mathlib/Analysis/Calculus/ContDiff/Defs.lean
theorems:
- iteratedFDerivWithin_succ_apply_right
- contDiffWithinAt_succ_iff_hasFDerivWithinAt'
- path: Mathlib/SetTheory/Cardinal/Cofinality.lean
theorems:
- Cardinal.nfpBFamily_lt_ord_of_isRegular
- path: Mathlib/Geometry/Manifold/MFDeriv/Basic.lean
theorems:
- mfderiv_comp_of_eq
- path: Mathlib/Algebra/Quaternion.lean
theorems:
- Quaternion.self_mul_star
- path: Mathlib/Logic/Denumerable.lean
theorems:
- Nat.Subtype.ofNat_surjective_aux
- path: Mathlib/LinearAlgebra/QuadraticForm/TensorProduct.lean
theorems:
- QuadraticForm.associated_baseChange
- path: Mathlib/Init/Data/Nat/Bitwise.lean
theorems:
- Nat.bodd_add
- path: Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean
theorems:
- hasDerivAt_sqrt_mul_log
- path: Mathlib/RingTheory/PowerSeries/Order.lean
theorems:
- PowerSeries.coeff_of_lt_order
- path: Mathlib/MeasureTheory/Function/LpSpace.lean
theorems:
- MeasureTheory.Lp.dist_def
- path: Mathlib/Geometry/Manifold/VectorBundle/Basic.lean
theorems:
- Bundle.contMDiffAt_totalSpace
- path: Mathlib/RepresentationTheory/Rep.lean
theorems:
- Rep.ihom_ev_app_hom
- path: Mathlib/Data/Int/GCD.lean
theorems:
- Int.ne_zero_of_gcd
- path: Mathlib/CategoryTheory/Yoneda.lean
theorems:
- CategoryTheory.isIso_of_yoneda_map_bijective
- path: Mathlib/CategoryTheory/Shift/Pullback.lean
theorems:
- CategoryTheory.pullbackShiftFunctorZero_hom_app
- path: Mathlib/Computability/PartrecCode.lean
theorems:
- Nat.Partrec.Code.eval_prec_zero
- path: Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean
theorems:
- CategoryTheory.ShortComplex.LeftHomologyMapData.cyclesMap_eq
- CategoryTheory.ShortComplex.f'_cyclesMap'
- path: Mathlib/RingTheory/WittVector/Identities.lean
theorems:
- WittVector.coeff_p_pow_eq_zero
- path: Mathlib/Topology/Instances/ENNReal.lean
theorems:
- ENNReal.tendsto_nhds_top_iff_nnreal
- path: Mathlib/Computability/Halting.lean
theorems:
- ComputablePred.computable_iff_re_compl_re
- path: Mathlib/Data/Fin/Tuple/Basic.lean
theorems:
- Fin.snoc_init_self
- Fin.snoc_zero
- path: Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean
theorems:
- "AlgebraicGeometry.\u0393Spec.right_triangle"
- path: Mathlib/GroupTheory/Perm/Cycle/Basic.lean
theorems:
- Equiv.Perm.isCycleOn_singleton
- path: Mathlib/RingTheory/Ideal/LocalRing.lean
theorems:
- LocalRing.isLocalRingHom_residue
- path: Mathlib/Data/PFunctor/Multivariate/Basic.lean
theorems:
- MvPFunctor.comp.get_mk
- path: .lake/packages/std/Std/Data/Array/Monadic.lean
theorems:
- Array.SatisfiesM_anyM
- path: Mathlib/CategoryTheory/SmallObject/Construction.lean
theorems:
- "CategoryTheory.SmallObject.\u03B9FunctorObj_\u03C0FunctorObj"
- path: Mathlib/GroupTheory/Subsemigroup/Center.lean
theorems:
- Set.neg_mem_center
- path: Mathlib/Topology/Category/TopCat/Opens.lean
theorems:
- TopologicalSpace.Opens.map_eq
- path: Mathlib/Data/Set/Prod.lean
theorems:
- Set.univ_prod_univ
- path: Mathlib/LinearAlgebra/Semisimple.lean
theorems:
- Module.End.isSemisimple_neg
- path: Mathlib/SetTheory/Ordinal/Exponential.lean
theorems:
- Ordinal.log_opow_mul_add
- path: Mathlib/Topology/MetricSpace/Lipschitz.lean
theorems:
- LipschitzWith.dist
- path: Mathlib/RingTheory/FractionalIdeal/Operations.lean
theorems:
- FractionalIdeal.spanSingleton_eq_spanSingleton
- path: Mathlib/LinearAlgebra/PiTensorProduct.lean
theorems:
- PiTensorProduct.induction_on'
- path: Mathlib/Data/DFinsupp/Basic.lean
theorems:
- DFinsupp.extendWith_zero
- path: Mathlib/Deprecated/Submonoid.lean
theorems:
- Submonoid.isSubmonoid
- path: Mathlib/LinearAlgebra/LinearPMap.lean
theorems:
- LinearPMap.mem_graph_snd_inj
- path: Mathlib/Topology/Algebra/GroupCompletion.lean
theorems:
- AddMonoidHom.completion_add
- path: Mathlib/Topology/Compactness/LocallyCompact.lean
theorems:
- exists_mem_nhdsSet_isCompact_mapsTo
- path: Mathlib/Algebra/Lie/IdealOperations.lean
theorems:
- LieSubmodule.lie_sup
- path: Mathlib/CategoryTheory/Limits/MonoCoprod.lean
theorems:
- CategoryTheory.Limits.MonoCoprod.monoCoprod_of_preservesCoprod_of_reflectsMono
- path: Mathlib/Topology/Algebra/Ring/Ideal.lean
theorems:
- QuotientRing.quotientMap_coe_coe
- path: Mathlib/LinearAlgebra/AffineSpace/Ordered.lean
theorems:
- left_lt_lineMap_iff_lt
- path: Mathlib/Analysis/MellinInversion.lean
theorems:
- rexp_neg_image_aux
- path: Mathlib/Algebra/Lie/Nilpotent.lean
theorems:
- LieSubmodule.lowerCentralSeries_eq_lcs_comap
- LieModule.isTrivial_of_nilpotencyLength_le_one
- path: Mathlib/LinearAlgebra/Prod.lean
theorems:
- Submodule.prod_comap_inl
- LinearMap.tailing_sup_tunnel_succ_le_tunnel
- path: Mathlib/FieldTheory/Adjoin.lean
theorems:
- IntermediateField.adjoin_simple_toSubalgebra_of_integral
- path: Mathlib/Analysis/Convex/Strict.lean
theorems:
- DirectedOn.strictConvex_sUnion
- path: Mathlib/Data/List/BigOperators/Basic.lean
theorems:
- List.prod_drop_succ
- path: Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean
theorems:
- ProjectiveSpectrum.vanishingIdeal_iUnion
- path: Mathlib/Data/Multiset/Powerset.lean
theorems:
- Multiset.powersetCard_zero_left
- path: Mathlib/Algebra/Ring/BooleanRing.lean
theorems:
- ofBoolAlg_symmDiff
- path: Mathlib/MeasureTheory/Measure/VectorMeasure.lean
theorems:
- MeasureTheory.VectorMeasure.nonpos_of_restrict_le_zero
- path: Mathlib/Analysis/Convex/Segment.lean
theorems:
- openSegment_eq_Ioo'
- path: Mathlib/Control/Fold.lean
theorems:
- Traversable.toList_spec
- path: Mathlib/Analysis/SpecialFunctions/Log/Base.lean
theorems:
- Real.logb_inv
- path: Mathlib/FieldTheory/Finite/Basic.lean
theorems:
- ZMod.pow_card_sub_one_eq_one
- path: Mathlib/Probability/Martingale/Upcrossing.lean
theorems:
- MeasureTheory.upperCrossingTime_le
- path: Mathlib/Topology/Separation.lean
theorems:
- Filter.coclosedCompact_eq_cocompact
- path: Mathlib/AlgebraicTopology/FundamentalGroupoid/InducedMaps.lean
theorems:
- ContinuousMap.Homotopy.apply_zero_path
- path: Mathlib/Order/OmegaCompletePartialOrder.lean
theorems:
- OmegaCompletePartialOrder.ContinuousHom.map_continuous'
- path: Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean
theorems:
- Algebra.TensorProduct.rTensor_ker
- path: Mathlib/Init/Data/Bool/Lemmas.lean
theorems:
- Bool.and_eq_false_eq_eq_false_or_eq_false
- path: Mathlib/Order/Filter/NAry.lean
theorems:
- "Filter.map_map\u2082"
- path: Mathlib/LinearAlgebra/Matrix/BilinearForm.lean
theorems:
- Matrix.toBilin_comp
- path: Mathlib/Order/InitialSeg.lean
theorems:
- InitialSeg.acc
- path: Mathlib/Analysis/Calculus/TangentCone.lean
theorems:
- uniqueDiffWithinAt_of_mem_nhds
- path: Mathlib/Algebra/Star/Subalgebra.lean
theorems:
- StarSubalgebra.sInf_toSubalgebra
- path: Mathlib/MeasureTheory/Decomposition/Lebesgue.lean
theorems:
- MeasureTheory.Measure.rnDeriv_smul_left'
- path: Mathlib/Data/Finsupp/Defs.lean
theorems:
- Finsupp.support_single_disjoint
- Finsupp.erase_single_ne
- path: Mathlib/LinearAlgebra/AffineSpace/Basis.lean
theorems:
- AffineBasis.affineCombination_coord_eq_self
- path: Mathlib/Order/CompleteLatticeIntervals.lean
theorems:
- Set.Iic.coe_iInf
- path: Mathlib/Algebra/Order/Group/PosPart.lean
theorems:
- leOnePart_one
- path: Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean
theorems:
- HomologicalComplex.d_pOpcycles
- path: Mathlib/RingTheory/ClassGroup.lean
theorems:
- ClassGroup.mk_eq_one_of_coe_ideal
- path: Mathlib/Analysis/InnerProductSpace/PiL2.lean
theorems:
- OrthonormalBasis.sum_repr_symm
- path: Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean
theorems:
- "Wbtw.oangle\u2083\u2081\u2082_eq_zero"
- path: Mathlib/Order/Heyting/Basic.lean
theorems:
- Disjoint.sup_sdiff_cancel_left
- sup_sdiff_distrib
- path: Mathlib/Algebra/Lie/Matrix.lean
theorems:
- Matrix.lieConj_symm_apply
- path: Mathlib/RingTheory/Discriminant.lean
theorems:
- Algebra.discr_of_matrix_mulVec
- path: Mathlib/Probability/Kernel/Condexp.lean
theorems:
- MeasureTheory.AEStronglyMeasurable.comp_snd_map_prod_id
- path: Mathlib/Data/ENNReal/Inv.lean
theorems:
- ENNReal.mul_lt_of_lt_div
- path: Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean
theorems:
- CochainComplex.HomComplex.Cochain.ofHom_injective
- path: Mathlib/RingTheory/Localization/Away/Basic.lean
theorems:
- selfZPow_pow_sub
- path: Mathlib/Order/Hom/Lattice.lean
theorems:
- LatticeHom.withTopWithBot_id
- path: Mathlib/RingTheory/JacobsonIdeal.lean
theorems:
- Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
- path: Mathlib/Data/Nat/GCD/Basic.lean
theorems:
- Nat.gcd_add_self_right
- path: .lake/packages/std/Std/Data/String/Lemmas.lean
theorems:
- String.Iterator.forward_eq_nextn
- path: Mathlib/Algebra/Order/Ring/Lemmas.lean
theorems:
- mul_eq_mul_iff_eq_and_eq_of_pos'
- path: Mathlib/Data/List/Perm.lean
theorems:
- List.length_permutations'Aux
- path: Mathlib/RingTheory/FreeCommRing.lean
theorems:
- FreeCommRing.isSupported_of
- path: Mathlib/Data/ZMod/Basic.lean
theorems:
- ZMod.nontrivial_iff
- ZMod.nat_cast_self'
- path: Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean
theorems:
- ModuleCat.MonoidalCategory.associator_naturality
- path: Mathlib/Data/Set/Intervals/WithBotTop.lean
theorems:
- WithTop.preimage_coe_Ioc
- WithTop.preimage_coe_Iio_top
- path: Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean
theorems:
- CliffordAlgebra.induction
- path: Mathlib/Order/Filter/Lift.lean
theorems:
- Filter.lift_iInf_of_map_univ
- path: Mathlib/Topology/UrysohnsBounded.lean
theorems:
- exists_bounded_mem_Icc_of_closed_of_le
- path: Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean
theorems:
- GaussianInt.toComplex_re
- path: Mathlib/Algebra/GroupPower/Basic.lean
theorems:
- pow_bit0'
- path: Mathlib/Algebra/Lie/Submodule.lean
theorems:
- LieSubmodule.iInf_coe
- path: Mathlib/RingTheory/ChainOfDivisors.lean
theorems:
- coe_factor_orderIso_map_eq_one_iff
- path: Mathlib/MeasureTheory/Decomposition/Jordan.lean
theorems:
- MeasureTheory.SignedMeasure.absolutelyContinuous_ennreal_iff
- path: Mathlib/Analysis/Complex/Liouville.lean
theorems:
- Complex.liouville_theorem_aux
- path: Mathlib/Analysis/InnerProductSpace/Positive.lean
theorems:
- ContinuousLinearMap.isPositive_zero
- path: Mathlib/Data/Set/Image.lean
theorems:
- Set.range_subtype_map
- path: Mathlib/Combinatorics/Additive/ETransform.lean
theorems:
- Finset.MulETransform.card
- path: Mathlib/Algebra/GroupWithZero/Units/Basic.lean
theorems:
- Units.mk0_one
- path: Mathlib/AlgebraicTopology/DoldKan/Degeneracies.lean
theorems:
- "AlgebraicTopology.DoldKan.\u03C3_comp_PInfty"
- path: Mathlib/Algebra/Order/Sub/Defs.lean
theorems:
- AddLECancellable.lt_add_of_tsub_lt_right
- path: Mathlib/Topology/Category/TopCat/Limits/Products.lean
theorems:
- TopCat.prodIsoProd_hom_apply
- path: Mathlib/RingTheory/IntegralDomain.lean
theorems:
- sum_hom_units_eq_zero
- path: Mathlib/Algebra/Order/Group/Defs.lean
theorems:
- div_le_div_iff_right
- path: Mathlib/Computability/Partrec.lean
theorems:
- Nat.rfindOpt_mono
- path: Mathlib/RingTheory/Subsemiring/Basic.lean
theorems:
- Subsemiring.mem_sSup_of_directedOn
- path: Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean
theorems:
- quadraticChar_eq_pow_of_char_ne_two'
- path: Mathlib/Probability/IdentDistrib.lean
theorems:
- ProbabilityTheory.IdentDistrib.ae_snd
- path: Mathlib/Analysis/NormedSpace/Star/ContinuousFunctionalCalculus.lean
theorems:
- elementalStarAlgebra.isUnit_of_isUnit_of_isStarNormal
- path: Mathlib/Topology/MetricSpace/Infsep.lean
theorems:
- Set.infsep_pair_le_toReal_inf
- path: Mathlib/RingTheory/WittVector/Basic.lean
theorems:
- WittVector.ghostFun_sub
- path: Mathlib/Data/Nat/Choose/Multinomial.lean
theorems:
- Nat.multinomial_nil
- path: Mathlib/RingTheory/Filtration.lean
theorems:
- Ideal.isIdempotentElem_iff_eq_bot_or_top_of_localRing
- path: Mathlib/FieldTheory/Laurent.lean
theorems:
- RatFunc.laurentAux_algebraMap
- path: Mathlib/Probability/Kernel/CondDistrib.lean
theorems:
- MeasureTheory.Integrable.integral_norm_condDistrib_map
- path: Mathlib/MeasureTheory/Constructions/Prod/Integral.lean
theorems:
- measurableSet_integrable
- path: Mathlib/Data/Finset/Sigma.lean
theorems:
- Finset.sigma_nonempty
- path: Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean
theorems:
- "CategoryTheory.Limits.\u03B9_colimitConstInitial_hom"
- path: Mathlib/Topology/MetricSpace/HausdorffDistance.lean
theorems:
- Metric.infDist_image
- path: Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean
theorems:
- "CategoryTheory.Pretriangulated.Triangle.mor\u2083_eq_zero_iff_mono\u2081"
- path: Mathlib/LinearAlgebra/Contraction.lean
theorems:
- comp_dualTensorHom
- path: Mathlib/Order/Filter/ModEq.lean
theorems:
- Nat.frequently_modEq
- path: Mathlib/Data/Set/Pointwise/Interval.lean
theorems:
- Set.preimage_neg_Ioc
- path: Mathlib/Data/PNat/Basic.lean
theorems:
- PNat.sub_coe
- path: Mathlib/MeasureTheory/Measure/Haar/Basic.lean
theorems:
- MeasureTheory.Measure.haar.index_elim
- path: Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
theorems:
- Matrix.det_smul_inv_vecMul_eq_cramer_transpose
- path: Mathlib/NumberTheory/LucasLehmer.lean
theorems:
- "LucasLehmer.X.\u03C9_mul_\u03C9b"
- path: Mathlib/Analysis/BoxIntegral/Partition/Basic.lean
theorems:
- BoxIntegral.Prepartition.ofWithBot_le
- path: Mathlib/RepresentationTheory/Basic.lean
theorems:
- Representation.asAlgebraHom_single
- path: Mathlib/LinearAlgebra/Dimension/DivisionRing.lean
theorems:
- rank_dual_eq_card_dual_of_aleph0_le_rank'
- path: Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean
theorems:
- CategoryTheory.Limits.parallelPair_functor_obj
- path: Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean
theorems:
- WeierstrassCurve.Affine.CoordinateRing.mk_XYIdeal'_mul_mk_XYIdeal'
- path: Mathlib/RingTheory/PowerSeries/Trunc.lean
theorems:
- PowerSeries.trunc_coe_eq_self
- path: Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean
theorems:
- Real.negMulLog_one
- path: Mathlib/Data/Polynomial/Degree/TrailingDegree.lean
theorems:
- Polynomial.nextCoeffUp_C_eq_zero
- path: Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean
theorems:
- MeasureTheory.snorm_one_condexp_le_snorm
- path: Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean
theorems:
- "TopCat.Presheaf.SheafConditionEqualizerProducts.res_\u03C0"