Documentation Verification Report

Nat

šŸ“ Source: Mathlib/Order/Nat.lean

Statistics

MetricCount
DefinitionsinstOrderBot
1
Theoremsbot_eq_zero, instNoMaxOrder, isLeast_find, isLeast_natFind
4
Total5

Nat

Definitions

NameCategoryTheorems
instOrderBot šŸ“–CompOp
185 mathmath: SchwartzMap.norm_toLp_le_seminorm, MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt, Fin.attachFin_Iic, ProbabilityTheory.Kernel.partialTraj_succ_of_le, MeasureTheory.preimage_spanningSetsIndex_singleton, ProbabilityTheory.Kernel.condExp_traj', largeSchroder_succ, ProbabilityTheory.Kernel.map_frestrictLe_trajMeasure_compProd_eq_map_trajMeasure, MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt, ProbabilityTheory.Kernel.trajContent_cylinder, Filter.Tendsto.partialSups_apply, disjointed_zero, MeasureTheory.StronglyAdapted.isStoppingTime_upperCrossingTime, ProbabilityTheory.Kernel.partialTraj_comp_partialTraj', partialSups_eq_sup'_range, MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part, MeasureTheory.lowerCrossingTime_lt_upperCrossingTime, MeasureTheory.isProjectiveLimit_nat_iff, biUnion_range_succ_disjointed, Polynomial.length_coeffList_eq_withBotSucc_degree, ProbabilityTheory.Kernel.traj_map_updateFinset, MvPolynomial.degreeOf_sum_le, ProbabilityTheory.Kernel.map_partialTraj_succ_self, MeasureTheory.upcrossingsBefore_zero', MvPolynomial.totalDegree_eq, MeasureTheory.Submartingale.stoppedValue_leastGE, MeasureTheory.spanningSetsIndex_eq_iff, ProbabilityTheory.Kernel.partialTraj_comp_partialTraj'', MeasureTheory.upperCrossingTime_lt_of_le_upcrossingsBefore, ProbabilityTheory.Kernel.le_lmarginalPartialTraj_succ, range_succ_eq_Iic, MeasureTheory.integral_mul_upcrossingsBefore_le_integral, MeasureTheory.upperCrossingTime_bound_eq, frestrictLe_iterateInduction, ProbabilityTheory.Kernel.integral_traj_partialTraj', Finset.biUnion_slice, ContinuousWithinAt.partialSups_apply, ProbabilityTheory.Kernel.aestronglyMeasurable_traj, MeasureTheory.crossing_pos_eq, Fin.finsetImage_val_Iio, Polynomial.coeffList_eraseLead, partialSups_eq_sup_range, MeasureTheory.stoppedValue_lowerCrossingTime, Polynomial.natDegree_sum_eq_of_disjoint, MeasureTheory.lowerCrossingTime_stabilize, Finset.subset_range_sup_succ, ProbabilityTheory.Kernel.partialTraj_compProd_traj, MeasureTheory.upperCrossingTime_lt_lowerCrossingTime, MeasureTheory.Submartingale.eLpNorm_stoppedAbove_le, MeasureTheory.lowerCrossingTime_lt_of_lt_upcrossingsBefore, ProbabilityTheory.Kernel.partialTraj_succ_eq_comp, Finset.sum_card_slice, ProbabilityTheory.Kernel.partialTraj_le_def, ProbabilityTheory.Kernel.condDistrib_trajMeasure, MeasureTheory.mul_integral_upcrossingsBefore_le_integral_pos_part_aux, MeasureTheory.StronglyAdapted.measurable_upcrossings, MeasureTheory.lowerCrossingTime_stabilize', ProbabilityTheory.Kernel.traj_comp_partialTraj, MeasureTheory.inducedFamily_Iic, MeasureTheory.Measure.pi_prod_map_IicProdIoc, card_Iic, MeasureTheory.measurableCylinders_nat, partialSups_eq_sUnion_image, ProbabilityTheory.Kernel.measurable_trajFun, withBotSucc_zero, ProbabilityTheory.Kernel.partialTraj_map_frestrictLeā‚‚_apply, MeasureTheory.Submartingale.eLpNorm_stoppedAbove_le', nilpotencyClass_pi, ProbabilityTheory.Kernel.lintegral_traj, ProbabilityTheory.Kernel.instIsSFiniteKernelForallValNatMemFinsetIicPartialTraj, ProbabilityTheory.Kernel.partialTraj_succ_map_frestrictLeā‚‚, MeasureTheory.StronglyAdapted.isStoppingTime_crossing, MvPolynomial.weightedTotalDegree_one, MeasureTheory.stoppedValue_upperCrossingTime, ProbabilityTheory.Kernel.condExp_traj, MeasureTheory.partialTraj_const, ProbabilityTheory.Kernel.partialTraj_le, ProbabilityTheory.Kernel.partialTraj_succ_self, MeasureTheory.isPredictable_of_measurable_add_one, WittVector.dvd_sub_sum_teichmuller_iterateFrobeniusEquiv_coeff, SimpleGraph.extremalNumber_of_fintypeCard_eq, MeasureTheory.upperCrossingTime_lt_bddAbove, MeasureTheory.upcrossings_eq_top_of_frequently_lt, ProbabilityTheory.Kernel.isProjectiveLimit_trajFun, MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt, withBotSucc_one, ContinuousWithinAt.partialSups, ProbabilityTheory.Kernel.traj_map_frestrictLe_of_le, MeasureTheory.upperCrossingTime_stabilize, MeasureTheory.upperCrossingTime_lt_nonempty, ProbabilityTheory.Kernel.instIsFiniteKernelForallValNatMemFinsetIicPartialTrajOfHAddOfNat, ContinuousOn.partialSups, MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing, MeasureTheory.Submartingale.stoppedValue_leastGE_eLpNorm_le, MeasureTheory.upcrossingsBefore_le, SchwartzMap.eLpNorm_le_seminorm, ProbabilityTheory.Kernel.partialTraj_comp_partialTraj, MeasureTheory.NullMeasurableSet.disjointed, ContinuousAt.partialSups, ProbabilityTheory.Kernel.instIsMarkovKernelForallValNatMemFinsetIicForallTraj, partialSups_zero, MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part, MeasureTheory.mem_disjointed_spanningSetsIndex, ProbabilityTheory.Kernel.integral_traj_partialTraj, MvPolynomial.degreeOf_eq_sup, ProbabilityTheory.Kernel.instIsZeroOrMarkovKernelForallValNatMemFinsetIicPartialTrajOfHAddOfNat, MeasureTheory.StronglyAdapted.measurable_upcrossingsBefore, MeasureTheory.le_sub_of_le_upcrossingsBefore, MeasureTheory.norm_stoppedValue_leastGE_le, ProbabilityTheory.Kernel.isProjectiveMeasureFamily_partialTraj, disjointedRec_zero, MeasureTheory.upcrossingsBefore_eq_sum, DependsOn.dependsOn_lmarginalPartialTraj, MeasureTheory.Submartingale.stoppedAbove, ProbabilityTheory.Kernel.partialTraj_self, ProbabilityTheory.Kernel.traj_apply, Multiset.count_finset_sup, MvPolynomial.weightedTotalDegree_piSingle, Fin.attachFin_Iio, sup_divisors_id, MeasureTheory.upcrossingsBefore_zero, cast_finsetSup, Set.partialSups_eq_accumulate, ProbabilityTheory.Kernel.integrable_traj, MeasurableSet.disjointed, MeasureTheory.upperCrossingTime_eq_of_bound_le, MvPowerSeries.totalDegree_truncFinset, ProbabilityTheory.Kernel.traj_map_frestrictLe, MeasureTheory.upperCrossingTime_stabilize', ProbabilityTheory.Kernel.integral_traj, MeasureTheory.isPredictable_iff_measurable_add_one, ProbabilityTheory.Kernel.traj_map_frestrictLe_apply, MeasureTheory.measurableSet_predictable_singleton_prod, MeasureTheory.partialTraj_const_restrictā‚‚, MeasureTheory.Submartingale.upcrossings_ae_lt_top', Fin.map_valEmbedding_univ, Continuous.partialSups_apply, ProbabilityTheory.Kernel.map_traj_succ_self, MvPolynomial.totalDegree_finset_sum, ContinuousOn.partialSups_apply, Fin.finsetImage_val_Iic, Finset.card_shatterer_le_sum_vcDim, ProbabilityTheory.Kernel.setIntegral_traj_partialTraj, MeasureTheory.StronglyAdapted.isStoppingTime_lowerCrossingTime, ProbabilityTheory.Kernel.partialTraj_map_frestrictLeā‚‚, card_Iio, MeasureTheory.StronglyAdapted.integrable_upcrossingsBefore, Continuous.partialSups, MvPolynomial.weightedTotalDegree_rename_of_injective, MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt, Polynomial.natDegree_sum_eq_of_linearIndepOn, Polynomial.withBotSucc_degree_eq_natDegree_add_one, MeasureTheory.upperCrossingTime_lt_succ, partialSups_eq_biUnion_range, ProbabilityTheory.Kernel.lintegral_trajā‚€, MeasureTheory.stoppedAbove_le, preimage_find_eq_disjointed, ProbabilityTheory.Kernel.trajContent_eq_lmarginalPartialTraj, Fin.map_valEmbedding_Iio, ProbabilityTheory.Kernel.lmarginalPartialTraj_succ, MeasureTheory.upcrossingsBefore_mono, MeasureTheory.mul_upcrossingsBefore_le, Polynomial.supDegree_eq_natDegree, ProbabilityTheory.Kernel.partialTraj_zero, MeasureTheory.upcrossings_lt_top_iff, MeasureTheory.Measure.FiniteSpanningSetsIn.disjointed_set_eq, MeasureTheory.upcrossingsBefore_bot, MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt, ProbabilityTheory.Kernel.lmarginalPartialTraj_eq_lintegral_map, MeasureTheory.upcrossingsBefore_pos_eq, Iio_eq_range, SchwartzMap.one_add_le_sup_seminorm_apply, ProbabilityTheory.Kernel.partialTraj_compProd_eq_map_traj, ProbabilityTheory.Kernel.setIntegral_traj_partialTraj', MeasureTheory.exists_upperCrossingTime_eq, Filter.Tendsto.partialSups, bot_eq_zero, MeasureTheory.sum_restrict_disjointed_spanningSets, ContinuousAt.partialSups_apply, Fin.map_valEmbedding_Iic, MeasureTheory.Submartingale.stoppedValue_leastGE_eLpNorm_le', MeasureTheory.isProjectiveLimit_nat_iff', ProbabilityTheory.Kernel.partialTraj_eq_prod, ProbabilityTheory.Kernel.instIsMarkovKernelForallValNatMemFinsetIicPartialTrajOfHAddOfNat, ProbabilityTheory.Kernel.traj_eq_prod

Theorems

NameKindAssumesProvesValidatesDepends On
bot_eq_zero šŸ“–mathematical—Bot.bot
OrderBot.toBot
instOrderBot
——
instNoMaxOrder šŸ“–mathematical—NoMaxOrder——
isLeast_find šŸ“–mathematical—IsLeast
setOf
find
—find_spec
find_min'

Set.Nonempty

Theorems

NameKindAssumesProvesValidatesDepends On
isLeast_natFind šŸ“–mathematicalSet.NonemptyIsLeast
Nat.find
Set
Set.instMembership
—Nat.isLeast_find

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