Documentation Verification Report

Nat

πŸ“ Source: Mathlib/Order/Interval/Finset/Nat.lean

Statistics

MetricCount
DefinitionsinstLocallyFiniteOrder, instUniqueSubtypeMemFinsetIicOfNat
2
Theoremsrange_add_eq_union, range_eq_Ico, range_image_pred_top_sub, toFinset_range, toFinset_range'_1, toFinset_range'_1_1, Icc_eq_range', Ico_eq_range', Ico_image_const_sub_eq_Ico, Ico_pred_singleton, Ico_succ_left_eq_erase_Ico, Ico_succ_right_eq_insert_Ico, Ico_succ_singleton, Ico_zero_eq_range, Iio_eq_range, Ioc_eq_range', Ioc_succ_singleton, Ioo_eq_range', card_Icc, card_Ico, card_Iic, card_Iio, card_Ioc, card_Ioo, card_uIcc, cauchy_induction, cauchy_induction', cauchy_induction_mul, cauchy_induction_two_mul, decreasing_induction_of_infinite, decreasing_induction_of_not_bddAbove, image_Ico_mod, image_sub_const_Ico, mem_Ioc_succ, mem_Ioc_succ', mod_injOn_Ico, multiset_Ico_map_mod, pow_imp_self_of_one_lt, range_eq_Icc_zero_sub_one, range_succ_eq_Icc_zero, range_succ_eq_Iic, strong_decreasing_induction, uIcc_eq_range'
43
Total45

Finset

Theorems

NameKindAssumesProvesValidatesDepends On
range_add_eq_union πŸ“–mathematicalβ€”range
Finset
instUnion
map
addLeftEmbedding
AddLeftCancelSemigroup.toIsLeftCancelAdd
AddLeftCancelMonoid.toAddLeftCancelSemigroup
AddCancelCommMonoid.toAddLeftCancelMonoid
Nat.instAddCancelCommMonoid
β€”AddLeftCancelSemigroup.toIsLeftCancelAdd
range_eq_Ico
map_eq_image
ext
Nat.Ico_zero_eq_range
Ico_union_Ico_eq_Ico
range_eq_Ico πŸ“–mathematicalβ€”range
Ico
Nat.instPreorder
Nat.instLocallyFiniteOrder
β€”Nat.Ico_zero_eq_range
range_image_pred_top_sub πŸ“–mathematicalβ€”image
range
β€”range_zero
image_empty
range_eq_Ico
Nat.Ico_image_const_sub_eq_Ico

List

Theorems

NameKindAssumesProvesValidatesDepends On
toFinset_range πŸ“–mathematicalβ€”toFinset
Finset.range
β€”Finset.ext
mem_toFinset
Finset.mem_range
toFinset_range'_1 πŸ“–mathematicalβ€”toFinset
Finset.Ico
Nat.instPreorder
Nat.instLocallyFiniteOrder
β€”Finset.ext
mem_toFinset
Finset.mem_Ico
toFinset_range'_1_1 πŸ“–mathematicalβ€”toFinset
Finset.Icc
Nat.instPreorder
Nat.instLocallyFiniteOrder
β€”Finset.ext
mem_toFinset
add_comm
Finset.mem_Icc

Nat

Definitions

NameCategoryTheorems
instLocallyFiniteOrder πŸ“–CompOp
284 mathmath: SchwartzMap.norm_toLp_le_seminorm, Finset.sum_Ico_sub_bot, eVariationOn.sum_le_of_monotoneOn_Icc, Finset.sum_Ico_Ico_comm', FormalMultilinearSeries.comp_partialSum, Finset.sum_Ico_eq_add_neg, PNat.map_subtype_embedding_Ico, Ioc_filter_modEq_cast, Fin.attachFin_Iic, Ioc_succ_singleton, ProbabilityTheory.Kernel.partialTraj_succ_of_le, MeasureTheory.preimage_spanningSetsIndex_singleton, Ico_succ_left_eq_erase_Ico, largeSchroder_succ, PNat.Icc_eq_finset_subtype, Finset.sum_range_add_sum_Ico, image_Ico_mod, PNat.Ioo_eq_finset_subtype, ArithmeticFunction.sum_Ioc_mul_zeta_eq_sum, Filter.Tendsto.partialSups_apply, Fin.finsetImage_val_Ioi, Finset.prod_Ico_consecutive, add_pow_prime_pow_eq, image_sub_const_Ico, disjointed_zero, Fin.attachFin_Ioo_eq_Ioi, edist_le_Ico_sum_of_edist_le, Finset.prod_range_mul_prod_Ico, partialSups_eq_sup'_range, schnirelmannDensity_le_iff_forall, Finset.sum_Ico_sub, Fin.finsetImage_val_Ioo, Finset.sum_condensed_le, le_schnirelmannDensity_iff, Multiset.Icc_eq, MeasureTheory.isProjectiveLimit_nat_iff, biUnion_range_succ_disjointed, Ico_image_const_sub_eq_Ico, List.toFinset_range'_1, Partition.toFinsuppAntidiag_mem_finsuppAntidiag, Finset.prod_Ico_reflect, factorization_factorial, Prime.emultiplicity_factorial, Ico_pow_dvd_eq_Ico_of_lt, Fin.map_valEmbedding_uIcc, exists_of_schnirelmannDensity_eq_zero, Chebyshev.psi_eq_sum_theta, MeasureTheory.spanningSetsIndex_eq_iff, Chebyshev.psi_sub_theta_eq_sum_not_prime, Finset.le_sum_condensed', SummationFilter.conditional_filter_eq_map_range, range_succ_eq_Iic, factorization_eq_card_pow_dvd_of_lt, AkraBazziRecurrence.sumTransform_def, Finset.prod_Icc_succ_top, uIcc_eq_range', List.toFinset_range'_1_1, Finset.sum_Ico_eq_sub, frestrictLe_iterateInduction, Fin.attachFin_Icc, MonotoneOn.integral_le_sum_Ico, Finset.le_sum_schlomilch', MeasureTheory.Measure.map_piSingleton, integral_le_sum_mul_Ico_of_antitone_monotone, PNat.Ico_eq_finset_subtype, Fin.attachFin_Ico_eq_Ici, primorial_add, sum_mul_eq_sub_integral_mul', Finset.biUnion_slice, ContinuousWithinAt.partialSups_apply, Commute.add_pow_prime_pow_eq, MeasureTheory.Measure.pi_prod_map_IocProdIoc, Fin.finsetImage_val_Iio, Finset.sum_schlomilch_le, Ioc_filter_dvd_card_eq_div, Chebyshev.sum_PrimePow_eq_sum_sum, partialSups_eq_sup_range, Prime.emultiplicity_choose', Ioc_eq_range', range_eq_Icc_zero_sub_one, ZMod.sum_mul_div_add_sum_mul_div_eq_mul, padicValNat_choose', Prime.pow_dvd_factorial_iff, sum_Ico_le_integral_of_le, ProbabilityTheory.Kernel.partialTraj_succ_eq_comp, Finset.sum_card_slice, ProbabilityTheory.Kernel.partialTraj_le_def, geom_sum_Ico_mul, geom_sum_Ico_mul_neg, Finset.sum_Ico_succ_top, PNat.map_subtype_embedding_Icc, intervalIntegral.sum_integral_adjacent_intervals_Ico, AntitoneOn.sum_le_integral_Ico, Prime.emultiplicity_choose, Ioc_filter_modEq_card, PNat.uIcc_eq_finset_subtype, card_Ioo, MeasureTheory.inducedFamily_Iic, Fin.finsetImage_val_Ioc, MeasureTheory.Measure.pi_prod_map_IicProdIoc, card_Iic, MeasureTheory.measurableCylinders_nat, partialSups_eq_sUnion_image, Fin.map_valEmbedding_Ici, MonotoneOn.sum_le_integral_Ico, ProbabilityTheory.Kernel.instIsSFiniteKernelForallValNatMemFinsetIicPartialTraj, sum_mul_Ico_le_integral_of_monotone_antitone, Multiset.uIcc_eq, sum_Ioo_inv_sq_le, sum_mul_eq_sub_integral_mulβ‚€', MeasureTheory.partialTraj_const, Ico_filter_modEq_card, ProbabilityTheory.Kernel.partialTraj_le, sum_Ico_pow, Finset.sum_Ico_by_parts, ProbabilityTheory.Kernel.partialTraj_succ_self, WittVector.dvd_sub_sum_teichmuller_iterateFrobeniusEquiv_coeff, prod_Icc_factorial, card_Icc, Fin.finsetImage_val_Ico, Fin.attachFin_uIcc, PNat.map_subtype_embedding_Ioo, multiplicity_choose_aux, Commute.geom_sumβ‚‚_Ico, AntitoneOn.integral_le_sum_Ico, ArithmeticFunction.sum_Ioc_zeta, ArithmeticFunction.sum_Ioc_mul_eq_sum_prod_filter, Finset.prod_Ico_id_eq_factorial, Finset.sum_Ico_Ico_comm, Fin.finsetImage_val_Ici, PNat.Ioc_eq_finset_subtype, mem_Ioc_succ', Ico_filter_modEq_cast, sum_Icc_choose, card_Ico, Fin.map_valEmbedding_Icc, Ico_pred_singleton, Fin.attachFin_Ico, factorization_choose', sum_mul_eq_sub_integral_mul₁, Chebyshev.psi_eq_theta_add_sum_theta, ContinuousWithinAt.partialSups, ZMod.gauss_lemma, schnirelmannDensity_le_div, add_pow_prime_eq, ContinuousOn.partialSups, Polynomial.Chebyshev.isLocalExtr_T_real_iff, Icc_factorization_eq_pow_dvd, Finset.prod_eq_prod_Ico_succ_bot, SchwartzMap.eLpNorm_le_seminorm, Finset.sum_Ioc_succ_top, ProbabilityTheory.Kernel.partialTraj_comp_partialTraj, Finset.sum_eq_sum_Ico_succ_bot, MeasureTheory.NullMeasurableSet.disjointed, ContinuousAt.partialSups, gaussSum_pow_eq_prod_jacobiSum_aux, Finset.prod_Ico_eq_div, partialSups_zero, MeasureTheory.mem_disjointed_spanningSetsIndex, PNat.map_subtype_embedding_uIcc, PowerSeries.trunc_apply, Fin.attachFin_Ioo, Finset.sum_Ico_eq_sum_range, Ico_eq_range', Fin.map_valEmbedding_Ioi, padicValNat_choose, Fin.finsetImage_val_Icc, Chebyshev.theta_eq_sum_Icc, smallSchroder_succ, padicValNat_factorial, mul_geom_sumβ‚‚_Ico, Fin.map_valEmbedding_Ioo, disjointedRec_zero, MeasureTheory.upcrossingsBefore_eq_sum, MeasureTheory.stoppedValue_sub_eq_sum, Finset.prod_Ico_succ_top, Ico_filter_coprime_le, Finset.sum_Ico_succ_sub_top, ZMod.Ico_map_valMinAbs_natAbs_eq_Ico_map_id, ProbabilityTheory.Kernel.partialTraj_self, Ico_succ_right_eq_insert_Ico, Fin.attachFin_Iio, divisorsAntidiagonal_eq_prod_filter_of_le, locallyIntegrableOn_mul_sum_Icc, Fin.map_valEmbedding_Ico, add_pow_prime_pow_eq', Finset.sum_Ico_consecutive, filter_coprime_Ico_eq_totient, Ico_zero_eq_range, Finset.prod_Ico_div, Finset.sum_Icc_succ_top, Set.partialSups_eq_accumulate, FormalMultilinearSeries.radius_right_inv_pos_of_radius_pos_aux1, Finset.prod_Ioc_consecutive, range_succ_eq_Icc_zero, ZMod.gauss_lemma_aux, MeasurableSet.disjointed, Ico_filter_pow_dvd_eq, Chebyshev.psi_eq_sum_Icc, card_Ioc, harmonic_eq_sum_Icc, gaussSum_pow_eq_prod_jacobiSum, add_pow_prime_eq', factorization_eq_card_pow_dvd, integrableOn_mul_sum_Icc, MeasureTheory.partialTraj_const_restrictβ‚‚, integral_le_sum_Ico_of_le, geom_sum_Ico', Fin.map_valEmbedding_univ, Continuous.partialSups_apply, Finset.range_eq_Ico, schnirelmannDensity_lt_iff, ContinuousOn.partialSups_apply, Fin.finsetImage_val_Iic, Ico_succ_singleton, Finset.sum_Ico_reflect, Finset.card_shatterer_le_sum_vcDim, Finset.prod_Ico_succ_div_top, sum_Ioc_inv_sq_le_sub, Finset.prod_Ico_div_bot, Commute.mul_geom_sumβ‚‚_Ico, Icc_eq_range', Polynomial.Monic.irreducible_iff_natDegree', Fin.finsetImage_val_uIcc, Commute.add_pow_prime_eq, ZMod.eisenstein_lemma, Finset.sum_Ioc_by_parts, Finset.prod_Ioc_succ_top, card_Iio, ArithmeticFunction.sum_Ioc_sigma0_eq_sum_div, multiset_Ico_map_mod, sum_mul_eq_sub_sub_integral_mul', FormalMultilinearSeries.radius_rightInv_pos_of_radius_pos_aux2, Continuous.partialSups, Finset.prod_Ico_eq_prod_range, sum_mul_eq_sub_integral_mul, partialSups_eq_biUnion_range, addRothNumber_Ico, factorization_choose, Finset.sum_schlomilch_le', preimage_find_eq_disjointed, dist_le_Ico_sum_of_dist_le, edist_le_Ico_sum_edist, ZMod.prod_Ico_one_prime, ArithmeticFunction.sum_Ioc_mul_eq_sum_sum, Commute.add_pow_prime_eq', Fin.map_valEmbedding_Iio, geom_sum_Ico, geom_sumβ‚‚_Ico, PNat.map_subtype_embedding_Ioc, filter_multiset_Ico_card_eq_of_periodic, sum_mul_eq_sub_integral_mulβ‚€, emultiplicity_eq_card_pow_dvd, ProbabilityTheory.Kernel.partialTraj_zero, MeasureTheory.Measure.FiniteSpanningSetsIn.disjointed_set_eq, Polynomial.coeff_divByMonic_X_sub_C, sum_mul_eq_sub_sub_integral_mul, ZMod.eisenstein_lemma_aux, mod_injOn_Ico, Iio_eq_range, mem_Ioc_succ, SchwartzMap.one_add_le_sup_seminorm_apply, Commute.geom_sumβ‚‚_Ico_mul, dist_le_Ico_sum_dist, Filter.Tendsto.partialSups, Commute.add_pow_prime_pow_eq', ZMod.div_eq_filter_card, geom_sum_Ico_le_of_lt_one, MeasureTheory.sum_restrict_disjointed_spanningSets, Finset.sum_condensed_le', schnirelmannDensity_mul_le_card_filter, Finset.prod_range_eq_mul_Ico, geom_sum_Ico_le, ContinuousAt.partialSups_apply, Finset.sum_Ioc_consecutive, Ioo_eq_range', card_uIcc, Fin.map_valEmbedding_Iic, Finset.prod_Ico_eq_mul_inv, MeasureTheory.isProjectiveLimit_nat_iff', Finset.sum_range_eq_add_Ico, Fin.map_valEmbedding_Ioc, Fin.attachFin_Ioc, filter_Ico_card_eq_of_periodic
instUniqueSubtypeMemFinsetIicOfNat πŸ“–CompOpβ€”

Theorems

NameKindAssumesProvesValidatesDepends On
Icc_eq_range' πŸ“–mathematicalβ€”Finset.Icc
instPreorder
instLocallyFiniteOrder
Multiset.ofList
β€”β€”
Ico_eq_range' πŸ“–mathematicalβ€”Finset.Ico
instPreorder
instLocallyFiniteOrder
Multiset.ofList
β€”β€”
Ico_image_const_sub_eq_Ico πŸ“–mathematicalβ€”Finset.image
Finset.Ico
instPreorder
instLocallyFiniteOrder
β€”Finset.ext
Ico_pred_singleton πŸ“–mathematicalβ€”Finset.Ico
instPreorder
instLocallyFiniteOrder
Finset
Finset.instSingleton
β€”Finset.ext
Finset.mem_Ico
Finset.mem_singleton
Ico_succ_left_eq_erase_Ico πŸ“–mathematicalβ€”Finset.Ico
instPreorder
instLocallyFiniteOrder
Finset.erase
β€”Finset.ext
Ico_succ_right_eq_insert_Ico πŸ“–mathematicalβ€”Finset.Ico
instPreorder
instLocallyFiniteOrder
Finset
Finset.instInsert
β€”Finset.ext
Ico_succ_singleton πŸ“–mathematicalβ€”Finset.Ico
instPreorder
instLocallyFiniteOrder
Finset
Finset.instSingleton
β€”β€”
Ico_zero_eq_range πŸ“–mathematicalβ€”Finset.Ico
instPreorder
instLocallyFiniteOrder
Finset.range
β€”bot_eq_zero
Finset.Iio_eq_Ico
Iio_eq_range
Iio_eq_range πŸ“–mathematicalβ€”Finset.Iio
instPreorder
LocallyFiniteOrder.toLocallyFiniteOrderBot
instOrderBot
instLocallyFiniteOrder
Finset.range
β€”β€”
Ioc_eq_range' πŸ“–mathematicalβ€”Finset.Ioc
instPreorder
instLocallyFiniteOrder
Multiset.ofList
β€”β€”
Ioc_succ_singleton πŸ“–mathematicalβ€”Finset.Ioc
instPreorder
instLocallyFiniteOrder
Finset
Finset.instSingleton
β€”β€”
Ioo_eq_range' πŸ“–mathematicalβ€”Finset.Ioo
instPreorder
instLocallyFiniteOrder
Multiset.ofList
β€”β€”
card_Icc πŸ“–mathematicalβ€”Finset.card
Finset.Icc
instPreorder
instLocallyFiniteOrder
β€”β€”
card_Ico πŸ“–mathematicalβ€”Finset.card
Finset.Ico
instPreorder
instLocallyFiniteOrder
β€”β€”
card_Iic πŸ“–mathematicalβ€”Finset.card
Finset.Iic
instPreorder
LocallyFiniteOrder.toLocallyFiniteOrderBot
instOrderBot
instLocallyFiniteOrder
β€”Finset.Iic_eq_Icc
card_Icc
bot_eq_zero
card_Iio πŸ“–mathematicalβ€”Finset.card
Finset.Iio
instPreorder
LocallyFiniteOrder.toLocallyFiniteOrderBot
instOrderBot
instLocallyFiniteOrder
β€”Finset.Iio_eq_Ico
card_Ico
bot_eq_zero
card_Ioc πŸ“–mathematicalβ€”Finset.card
Finset.Ioc
instPreorder
instLocallyFiniteOrder
β€”β€”
card_Ioo πŸ“–mathematicalβ€”Finset.card
Finset.Ioo
instPreorder
instLocallyFiniteOrder
β€”β€”
card_uIcc πŸ“–mathematicalβ€”Finset.card
Finset.uIcc
DistribLattice.toLattice
instDistribLatticeNat
instLocallyFiniteOrder
β€”card_Icc
cauchy_induction πŸ“–β€”β€”β€”β€”cauchy_induction'
cauchy_induction' πŸ“–β€”β€”β€”β€”decreasing_induction_of_infinite
Set.Finite.exists_maximal
instIsTransLe
le_of_not_gt
not_lt_iff_le_imp_ge
LT.lt.not_ge
LT.lt.le
cauchy_induction_mul πŸ“–β€”β€”β€”β€”cauchy_induction
one_mul
LT.lt.trans_le
cauchy_induction_two_mul πŸ“–β€”β€”β€”β€”cauchy_induction_mul
decreasing_induction_of_infinite πŸ“–β€”Set.Infinite
setOf
β€”β€”decreasing_induction_of_not_bddAbove
BddAbove.finite
decreasing_induction_of_not_bddAbove πŸ“–β€”BddAbove
setOf
β€”β€”not_bddAbove_iff
LT.lt.le
image_Ico_mod πŸ“–mathematicalβ€”Finset.image
Finset.Ico
instPreorder
instLocallyFiniteOrder
Finset.range
β€”eq_or_ne
Finset.range_zero
add_zero
Finset.Ico_self
Finset.image_empty
Finset.ext
Ne.bot_lt
lt_or_ge
add_comm
mul_one
add_assoc
LE.le.trans
Eq.le
zero_add
LT.lt.le
lt_of_lt_of_le
le_refl
image_sub_const_Ico πŸ“–mathematicalβ€”Finset.image
Finset.Ico
instPreorder
instLocallyFiniteOrder
β€”Finset.ext
mem_Ioc_succ πŸ“–mathematicalβ€”Finset
Finset.instMembership
Finset.Ioc
instPreorder
instLocallyFiniteOrder
β€”Ioc_succ_singleton
mem_Ioc_succ' πŸ“–mathematicalβ€”Finset
SetLike.instMembership
Finset.instSetLike
Finset.Ioc
instPreorder
instLocallyFiniteOrder
Finset.instMembership
Preorder.toLT
Preorder.toLE
Finset.mem_Ioc
β€”Finset.mem_Ioc
mem_Ioc_succ
mod_injOn_Ico πŸ“–mathematicalβ€”Set.InjOn
SetLike.coe
Finset
Finset.instSetLike
Finset.Ico
instPreorder
instLocallyFiniteOrder
β€”zero_add
Ico_zero_eq_range
Ico_succ_left_eq_erase_Ico
Ico_succ_right_eq_insert_Ico
add_zero
Finset.Ico_eq_empty_of_le
Finset.instLawfulSingleton
Finset.erase_singleton
Finset.coe_empty
Finset.coe_Ico
multiset_Ico_map_mod πŸ“–mathematicalβ€”Multiset.map
Multiset.Ico
instPreorder
instLocallyFiniteOrder
Multiset.range
β€”Multiset.Nodup.dedup
Multiset.nodup_map_iff_inj_on
Finset.nodup
mod_injOn_Ico
image_Ico_mod
pow_imp_self_of_one_lt πŸ“–β€”MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
Monoid.toNatPow
β€”β€”cauchy_induction_mul
pow_succ
pow_succ'
pow_one
pow_mul
range_eq_Icc_zero_sub_one πŸ“–mathematicalβ€”Finset.range
Finset.Icc
instPreorder
instLocallyFiniteOrder
β€”β€”
range_succ_eq_Icc_zero πŸ“–mathematicalβ€”Finset.range
Finset.Icc
instPreorder
instLocallyFiniteOrder
β€”range_eq_Icc_zero_sub_one
range_succ_eq_Iic πŸ“–mathematicalβ€”Finset.range
Finset.Iic
instPreorder
LocallyFiniteOrder.toLocallyFiniteOrderBot
instOrderBot
instLocallyFiniteOrder
β€”range_succ_eq_Icc_zero
strong_decreasing_induction πŸ“–β€”β€”β€”β€”decreasing_induction_of_not_bddAbove
LE.le.eq_or_lt
le_rfl
uIcc_eq_range' πŸ“–mathematicalβ€”Finset.uIcc
DistribLattice.toLattice
instDistribLatticeNat
instLocallyFiniteOrder
Multiset.ofList
β€”β€”

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