Documentation Verification Report

Defs

πŸ“ Source: Mathlib/Analysis/Asymptotics/Defs.lean

Statistics

MetricCount
DefinitionsIsBigO, IsBigOWith, IsEquivalent, IsLittleO, IsTheta, transEventuallyEqIsBigO, transEventuallyEqIsLittleO, transIsBigOEventuallyEq, transIsBigOIsBigO, transIsBigOIsLittleO, transIsLittleOEventuallyEq, transIsLittleOIsBigO, transIsLittleOIsLittleO, «term_=O[_]_», «term_=o[_]_», «term_=Θ[_]_», «term_~[_]_»
17
Theoremsabs_abs, abs_left, abs_right, add, add_iff_left, add_iff_right, add_isLittleO, antisymm, comp_fst, comp_neg_int, comp_snd, comp_tendsto, congr', congr_left, congr_of_sub, congr_right, const_mul_left, const_mul_right, const_mul_right', eq_zero_imp, eventually_mul_div_cancel, exists_mem_basis, exists_nonneg, exists_pos, fiberwise_left, fiberwise_right, inv_rev, isBigOWith, mono, mul, mul_isLittleO, neg_left, neg_right, norm_left, norm_norm, norm_right, not_isLittleO, of_abs_abs, of_abs_left, of_abs_right, of_bound, of_bound', of_const_mul_right, of_neg_left, of_neg_right, of_norm_eventuallyLE, of_norm_le, of_norm_left, of_norm_norm, of_norm_right, pow, prod_left, prod_left_fst, prod_left_snd, prod_rightl, prod_rightr, sub, sub_iff_left, sub_iff_right, sum, sup, trans, trans_eventuallyEq, trans_isLittleO, trans_le, triangle, abs_abs, abs_left, abs_right, add, add_isLittleO, comp_tendsto, congr', congr_const, congr_left, congr_right, const_mul_left, const_mul_right, const_mul_right', eq_zero_imp, eventually_mul_div_cancel, exists_nonneg, exists_pos, insert, inv_rev, isBigO, mono, mul, neg_left, neg_right, norm_left, norm_norm, norm_right, of_abs_abs, of_abs_left, of_abs_right, of_bound, of_const_mul_right, of_neg_left, of_neg_right, of_norm_left, of_norm_norm, of_norm_right, of_pow, pow, pow', prod_left, prod_left_fst, prod_left_same, prod_left_snd, prod_rightl, prod_rightr, sub, sub_isLittleO, sum, sup, sup', trans, trans_isLittleO, trans_le, triangle, weaken, IsBigOWith_def, IsBigO_def, abs_abs, abs_left, abs_right, add, add_add, add_iff_left, add_iff_right, add_isBigO, add_isBigOWith, comp_fst, comp_snd, comp_tendsto, congr', congr_left, congr_of_sub, congr_right, const_mul_left, const_mul_right, const_mul_right', def, def', eventuallyLE, eventually_mul_div_cancel, forall_isBigOWith, insert, inv_rev, isBigO, isBigOWith, mono, mul, mul_isBigO, neg_left, neg_right, norm_left, norm_norm, norm_right, not_isBigO, of_abs_abs, of_abs_left, of_abs_right, of_bound, of_const_mul_right, of_isBigOWith, of_neg_left, of_neg_right, of_norm_left, of_norm_norm, of_norm_right, of_pow, pow, prod_left, prod_left_fst, prod_left_snd, prod_rightl, prod_rightr, sub, sub_iff_left, sub_iff_right, sum, sup, trans, trans_eventuallyEq, trans_isBigO, trans_isBigOWith, trans_le, triangle, IsLittleO_def, isBigO, isBigO_symm, isBigOWith_abs_abs, isBigOWith_abs_left, isBigOWith_abs_right, isBigOWith_bot, isBigOWith_comm, isBigOWith_congr, isBigOWith_const_const, isBigOWith_const_mul_self, isBigOWith_fst_prod, isBigOWith_iff, isBigOWith_insert, isBigOWith_inv, isBigOWith_map, isBigOWith_neg_left, isBigOWith_neg_right, isBigOWith_norm_left, isBigOWith_norm_norm, isBigOWith_norm_right, isBigOWith_of_le, isBigOWith_of_le', isBigOWith_prod_left, isBigOWith_pure, isBigOWith_refl, isBigOWith_self_const_mul, isBigOWith_self_const_mul', isBigOWith_snd_prod, isBigOWith_zero, isBigOWith_zero', isBigOWith_zero_right_iff, isBigO_abs_abs, isBigO_abs_left, isBigO_abs_right, isBigO_bot, isBigO_comm, isBigO_congr, isBigO_const_const, isBigO_const_const_iff, isBigO_const_mul_left_iff, isBigO_const_mul_left_iff', isBigO_const_mul_right_iff, isBigO_const_mul_right_iff', isBigO_const_mul_self, isBigO_fst_prod, isBigO_fst_prod', isBigO_iff, isBigO_iff', isBigO_iff'', isBigO_iff_eventually, isBigO_iff_eventually_isBigOWith, isBigO_iff_isBigOWith, isBigO_map, isBigO_neg_left, isBigO_neg_right, isBigO_norm_left, isBigO_norm_norm, isBigO_norm_right, isBigO_of_le, isBigO_of_le', isBigO_of_subsingleton, isBigO_prod_left, isBigO_pure, isBigO_refl, isBigO_refl_left, isBigO_self_const_mul, isBigO_self_const_mul', isBigO_snd_prod, isBigO_snd_prod', isBigO_sup, isBigO_zero, isBigO_zero_right_iff, isLittleO_abs_abs, isLittleO_abs_left, isLittleO_abs_right, isLittleO_bot, isLittleO_comm, isLittleO_congr, isLittleO_const_mul_left_iff, isLittleO_const_mul_left_iff', isLittleO_const_mul_right_iff, isLittleO_const_mul_right_iff', isLittleO_iff, isLittleO_iff_forall_isBigOWith, isLittleO_iff_nat_mul_le, isLittleO_iff_nat_mul_le', isLittleO_iff_nat_mul_le_aux, isLittleO_insert, isLittleO_irrefl, isLittleO_irrefl', isLittleO_map, isLittleO_neg_left, isLittleO_neg_right, isLittleO_norm_left, isLittleO_norm_norm, isLittleO_norm_right, isLittleO_of_subsingleton, isLittleO_prod_left, isLittleO_refl_left, isLittleO_sup, isLittleO_zero, isLittleO_zero_right_iff, isBigO, trans_isBigO, isBigO, trans_isBigO, trans_isLittleO
298
Total315

Asymptotics

Definitions

NameCategoryTheorems
IsBigO πŸ“–MathDef
223 mathmath: HasStrictFDerivAt.isBigO_sub, CuspFormClass.qExpansion_isBigO, isBigO_const_const_iff, HurwitzZeta.isBigO_atTop_evenKernel_sub, isBigO_norm_norm, isBigO_neg_left, Real.isBigO_log_mul_const_log_atTop, AkraBazziRecurrence.T_isBigO_smoothingFn_mul_asympBound, ContinuousLinearMap.isBigO_comp, IsBigOTVS.isBigO, IsBigO.of_norm_eventuallyLE, EisensteinSeries.linear_inv_isBigO_left, isBigO_zero_right_iff, IsBigO.rpow_rpow_nhdsGE_zero_of_le, NormedRing.inverse_one_sub_norm, isBigO_const_smul_right, isBigO_nat_atTop_induction, isBigO_abs_abs, Filter.Tendsto.isBigO_one, isBigO_prod_left, Filter.IsBoundedUnder.isBigO_const, IsBigO.of_norm_le, Filter.EventuallyEq.isBigO, NormedRing.inverse_add_norm_diff_nth_order, isBigO_zero, ContDiffPointwiseHolderAt.isBigO, AkraBazziRecurrence.isBigO_apply_r_sub_b, Complex.IsExpCmpFilter.isBigO_im_pow_re, isBigO_const_mul_left_iff', ContinuousOn.isBigO_principal, WeakFEPair.hf_zero', isBigO_pi, HurwitzKernelBounds.isBigO_atTop_F_int_one, isBigO_iff_div_isBoundedUnder, HasFPowerSeriesWithinOnBall.isBigO_image_sub_image_sub_deriv_principal, ContinuousLinearEquiv.isBigO_comp, isBigO_norm_right, IsBoundedBilinearMap.isBigO_comp, isBigO_abs_left, isBigO_neg_right, IsEquivalent.isBigO_symm, isBigO_fst_prod, IsBoundedBilinearMap.isBigO', isBigO_iff, isBigO_map, ContinuousLinearEquiv.isBigO_sub, Function.hasTemperateGrowth_iff_isBigO, HasDerivAtFilter.isBigO_sub_rev, IsTheta.isBigO_congr_left, Chebyshev.integral_one_div_log_sq_isBigO, AkraBazziRecurrence.isBigO_symm_asympBound, ValueDistribution.isBigO_characteristic_sub_characteristic_shift, ContinuousLinearMap.isBigO_id, Real.isBigO_one_exp_comp, isBigO_top, Real.isBigO_exp_comp_one, ContinuousOn.isBigO_rev_principal, isBigOTVS_iff_isBigO, isBigO_nat_atTop_iff, StrongFEPair.hf_zero', isBigO_one_iff, isBigO_congr, isBigO_mul_iff_isBigO_div, isBigO_fst_prod', UpperHalfPlane.IsZeroAtImInfty.petersson_exp_decay_left, Complex.isBigO_re_sub_re, OpenPartialHomeomorph.isBigO_congr, Complex.isBigO_cpow_rpow, Real.isBigO_log_const_mul_log_atTop, HurwitzKernelBounds.isBigO_atTop_F_nat_one, isBigO_const_one, IsBoundedLinearMap.isBigO_comp, isBigO_completion_right, Polynomial.isBigO_atBot_of_degree_le, contDiffPointwiseHolderAt_iff, HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub, isBigO_riemannZeta_sub_one_div, Real.isBigO_logb_mul_const_log_atTop, isBigO_const_iff, ModularFormClass.exp_decay_sub_atImInfty, Chebyshev.primeCounting_sub_theta_div_log_isBigO, UpperHalfPlane.IsZeroAtImInfty.exp_decay_atImInfty, Real.isBigO_logb_log, Complex.log_sub_self_isBigO, Polynomial.isBigO_of_degree_le, isBigO_const_const, isBigO_at_im_infty_jacobiTheta_sub_one, HurwitzKernelBounds.isBigO_atTop_F_nat_zero_sub, isBigO_const_mul_right_iff, IsBigO_def, HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal, IsEquivalent.isBigO, HasFDerivWithinAt.isBigO_sub, WeakFEPair.hg_top, HasDerivAtFilter.isBigO_sub, HurwitzZeta.isBigO_atTop_oddKernel, NormedRing.inverse_add_norm, isBigO_atTop_iff_eventually_exists, DifferentiableAt.isBigO_sub, isBigO_atTop_iff_eventually_exists_pos, Tactic.ComputeAsymptotics.isBigO_of_div_tendsto_atBot, HasGradientAtFilter.isBigO_sub, WeakFEPair.hf_zero, EisensteinSeries.linear_inv_isBigO_right_add, ContinuousLinearEquiv.isBigO_sub_rev, isBigO_of_le', NormedRing.inverse_add_norm_diff_second_order, ValueDistribution.isBigO_characteristic_sub_characteristic_inv, isBigO_of_div_tendsto_nhds_of_ne_zero, Homeomorph.isBigO_congr, isBigO_const_mul_left_iff, isBigO_snd_prod', IsLittleO.isBigO, EisensteinSeries.isBigO_linear_add_const_vec, isBigO_norm_left, IsTheta.isBigO_symm, Tactic.ComputeAsymptotics.isBigO_of_div_tendsto_atTop, Complex.isBigO_im_sub_im, isBigO_const_of_tendsto, isBigO_snd_prod, superpolynomialDecay_iff_isBigO, IsLittleO.right_isBigO_sub, isBigO_sup, Complex.isBigO_ofReal_left, isBigO_of_div_tendsto_nhds, isBigO_one_nat_atTop_iff, isBigO_bot, isBigO_const_mul_right_iff', IsTheta.isBigO, isBigO_cofinite_iff, Complex.exp_sub_sum_range_isBigO_pow, Filter.Eventually.isBigO, isBigO_atTop_natCast_rpow_of_tendsto_div_rpow, HasFDerivAt.isBigO_sub, HasFPowerSeriesWithinAt.isBigO_sub_partialSum_pow, HasFPowerSeriesWithinAt.isBigO_image_sub_norm_mul_norm_sub, isBigO_principal, IsBoundedLinearMap.isBigO_sub, isBigO_self_const_mul', isBigO_const_left_iff_pos_le_norm, ValueDistribution.abs_characteristic_sub_characteristic_shift_eqO, IsLittleO.not_isBigO, IsBigO.rpow_rpow_nhdsGE_zero_of_le_of_imp, isBigO_iff_isBoundedUnder_le_div, isBigO_comm, Real.isBigO_exp_comp_exp_comp, isBigO_deriv_ofReal_cpow_const_atTop, isBigO_refl_left, isBigO_iff_exists_eq_mul, AkraBazziRecurrence.isBigO_asympBound, SchwartzMap.isBigO_cocompact_zpow_neg_nat, isBigOTVS.isBigO, LSeriesSummable.isBigO_rpow, ContinuousLinearEquiv.isBigO_comp_rev, ContinuousLinearMap.isBigO_sub, IsBigO.of_bound', isBigO_const_mul_self, IsBigOWith.isBigO, ModularFormClass.qExpansion_isBigO, isBigO_deriv_rpow_const_atTop, NormedRing.inverse_add_norm_diff_first_order, isBigO_self_const_mul, Chebyshev.integral_theta_div_log_sq_isBigO, IsBoundedBilinearMap.isBigO, CuspFormClass.exp_decay_atImInfty', DifferentiableWithinAt.isBigO_sub, Complex.log_sub_logTaylor_isBigO, HurwitzKernelBounds.isBigO_exp_neg_mul_of_le, UpperHalfPlane.IsZeroAtImInfty.exp_decay_atImInfty', Complex.isBigO_ofReal_right, Filter.IsBoundedUnder.isBigO_one, HasFPowerSeriesAt.isBigO_sub_partialSum_pow, StrongFEPair.hf_top', isBigO_iff_eventually_isBigOWith, DirichletCharacter.LFunctionTrivChar_isBigO_near_one_horizontal, IsBigO.of_bound, IsLittleO.right_isBigO_add', isBigO_iff_isBigOWith, isBigO_iff'', IsBigO.id_rpow_of_le_one, isBigO_const_smul_left, isEquivalent_zero_iff_isBigO_zero, HasFDerivAtFilter.isBigO_sub, isBigO_nat_atTop_induction_of_eventually_pos, isBigO_of_le, Real.isBigO_logb_const_mul_log_atTop, IsBigO.const_smul_self, isBigO_iff_eventually, CuspFormClass.exp_decay_atImInfty, TFAE_exists_lt_isLittleO_pow, SchwartzMap.isBigO_cocompact_zpow, HurwitzZeta.isBigO_atTop_cosKernel_sub, DirichletCharacter.LFunction_isBigO_horizontal, ruzsaSzemerediNumberNat_asymptotic_lower_bound, isBigO_pure, IsBigO.pow_of_le_right, IsBoundedLinearMap.isBigO_id, isBigO_of_subsingleton, UpperHalfPlane.IsZeroAtImInfty.petersson_exp_decay_right, spectrum.resolvent_isBigO_inv, isBigO_completion_left, isBigO_const_of_ne, ContDiffPointwiseHolderAt.zero_order_iff, Real.exp_sub_sum_range_isBigO_pow, ModularFormClass.exp_decay_sub_atImInfty', Function.HasTemperateGrowth.isBigO_uniform, IsTheta.isBigO_congr_right, Function.HasTemperateGrowth.isBigO, HurwitzZeta.isBigO_atTop_sinKernel, isBigO_iff', WeakFEPair.hf_top, Function.Periodic.exp_decay_sub_of_bounded_at_inf, SchwartzMap.isBigO_cocompact_rpow, HurwitzKernelBounds.isBigO_atTop_F_int_zero_sub, IsLittleO.right_isBigO_add, isBigO_pow_pow_of_le_left, Function.Periodic.exp_decay_of_zero_at_inf, AkraBazziRecurrence.smoothingFn_mul_asympBound_isBigO_T, isBigO_refl, isBigO_abs_right, EisensteinSeries.linear_inv_isBigO_right, Polynomial.isBigO_atTop_of_degree_le, isBigO_one_nhds_ne_iff
IsBigOWith πŸ“–MathDef
59 mathmath: isBigOWith_pi, ContinuousLinearMap.isBigOWith_sub, isBigOWith_neg_right, isBigOWith_mul_iff_isBigOWith_div, isBigOWith_norm_right, ContinuousLinearMap.isBigOWith_id, IsBigOWith.const_smul_self, isBigOWith_zero_right_iff, IsBigOWith_def, isBigOWith_refl, isBigOWith_comm, isBigOWith_self_const_mul', IsLittleO.forall_isBigOWith, isBigOWith_of_le', Homeomorph.isBigOWith_congr, isBigOWith_norm_norm, isBigOWith_abs_left, isBigOWith_map, Tactic.ComputeAsymptotics.isBigOWith_of_tendsto_bot, isBigOWith_zero, isBigOWith_self_const_mul, IsLittleO.isBigOWith, isBigOWith_of_le, isBigOWith_iff, isBigOWith_bot, ContinuousOn.isBigOWith_rev_principal, IsBigO_def, isBigOWith_abs_abs, isBigOWith_insert, isBigOWith_of_eq_mul, isBigOWith_norm_left, isBigOWith_iff_exists_eq_mul, isBigOWith_congr, ContinuousLinearMap.isBigOWith_comp, isBigOWith_top, ContinuousOn.isBigOWith_principal, IsBigO.exists_nonneg, isBigOWith_zero', isBigOWith_prod_left, isBigOWith_neg_left, isBigOWith_of_div_tendsto_nhds, isBigOWith_snd_prod, Tactic.ComputeAsymptotics.isBigOWith_of_tendsto_top, isBigOWith_principal, isBigOWith_const_mul_self, isBigOWith_const_const, IsBigO.isBigOWith, OpenPartialHomeomorph.isBigOWith_congr, isBigO_iff_eventually_isBigOWith, isBigO_iff_isBigOWith, isBigOWith_abs_right, IsLittleO.def', isBigOWith_inv, IsBigO.exists_pos, IsBigOWith.of_bound, isLittleO_iff_forall_isBigOWith, isBigOWith_fst_prod, isBigOWith_pure, isBigOWith_const_one
IsEquivalent πŸ“–MathDef
33 mathmath: isEquivalent_descFactorial, Polynomial.isEquivalent_atBot_div, Stirling.factorial_isEquivalent_stirling, Polynomial.isEquivalent_atTop_lead, isEquivalent_choose, isEquivalent_zero_iff_eventually_zero, isEquivalent_nat_ceil, Real.isEquivalent_sin, isEquivalent_iff_exists_eq_mul, Complex.isEquivalent_sinh, Polynomial.isEquivalent_atTop_div, Polynomial.isEquivalent_atBot_lead, isEquivalent_of_tendsto_one, HasStrictFDerivAt.isEquivalent_sub, AkraBazziRecurrence.isEquivalent_one_sub_smoothingFn_one, isEquivalent_nat_floor, AkraBazziRecurrence.isEquivalent_smoothingFn_sub_self, IsEquivalent.refl, AkraBazziRecurrence.isEquivalent_deriv_rpow_p_mul_one_add_smoothingFn, AkraBazziRecurrence.isEquivalent_one_add_smoothingFn_one, isEquivalent_iff_tendsto_one, HasDerivAtFilter.isEquivalent_sub, Filter.EventuallyEq.isEquivalent, SimpleGraph.isEquivalent_extremalNumber, isEquivalent_zero_iff_isBigO_zero, isEquivalent_map, HasFDerivAtFilter.isEquivalent_sub, isEquivalent_const_iff_tendsto, isEquivalent_of_tendsto_one', IsLittleO.isEquivalent, AkraBazziRecurrence.isEquivalent_deriv_rpow_p_mul_one_sub_smoothingFn, Complex.isEquivalent_sin, Real.isEquivalent_sinh
IsLittleO πŸ“–MathDef
179 mathmath: isLittleO_congr, Real.isLittleO_logb_id_atTop, IsLittleO.of_bound, isLittleO_completion_left, isLittleO_completion_right, isLittleO_prod_left, isLittleO_const_left, isLittleO_exp_neg_mul_sq_cocompact, isLittleO_const_iff, intervalIntegral.measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_right, isLittleO_abs_abs, AkraBazziRecurrence.isLittleO_deriv_one_add_smoothingFn, isLittleO_of_tendsto', isLittleOTVS.isLittleO, isLittleO_comm, hasDerivWithinAt_iff_isLittleO, isLittleO_log_rpow_rpow_atTop, HasFPowerSeriesWithinAt.hasStrictFDerivWithinAt, OpenPartialHomeomorph.isLittleO_congr, isLittleO_const_id_atTop, hasLineDerivAt_iff_isLittleO_nhds_zero, hasDerivAt_iff_isLittleO, isLittleO_irrefl, Real.isLittleO_log_id_atTop, intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_ge', isLittleO_const_mul_right_iff, Polynomial.isLittleO_atTop_of_degree_lt, isLittleO_insert, isLittleO_const_iff_isLittleO_one, isLittleO_pow_pow, isLittleO_iff_nat_mul_le, isLittleO_bot, Real.exp_sub_sum_range_succ_isLittleO_pow, hasFDerivAtFilter_iff_isLittleO, isLittleO_zero, hasGradientAtFilter_iff_isLittleO, hasGradientAt_iff_isLittleO, IsLittleO.of_isBigOWith, isLittleO_of_tendsto, isLittleO_norm_right, isLittleO_const_id_cobounded, hasStrictFDerivAt_iff_isLittleO, Complex.IsExpCmpFilter.isLittleO_exp_cpow, hasGradientAt_iff_isLittleO_nhds_zero, IsTheta.isLittleO_congr_left, isLittleO_pow_id, isLittleO_irrefl', HasStrictFDerivAt.isLittleO, isLittleO_neg_right, isLittleO_exp_mul_rpow_of_lt, Real.isLittleO_pow_exp_atTop, Convex.taylor_approx_two_segment, isLittleO_const_smul_left, IsLittleO.of_tendsto_div_atBot, isLittleO_rpow_exp_pos_mul_atTop, isLittleO_sum_range_of_tendsto_zero, isLittleO_pow_sub_pow_sub, isLittleO_const_smul_right, isLittleO_id_one, Convex.isLittleO_alternate_sum_square, isLittleO_log_rpow_nhdsGT_zero, isLittleO_log_rpow_atTop, isLittleO_iff_tendsto', isLittleO_exp_neg_mul_rpow_atTop, continuousAt_iff_isLittleO, isLittleO_const_id_atBot, isLittleO_refl_left, hasDerivAtFilter_iff_isLittleO, FormalMultilinearSeries.isLittleO_one_of_lt_radius, isLittleO_principal, isLittleO_one_left_iff, intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae, isLittleO_pow_pow_of_abs_lt_left, ContinuousWithinAt.integral_sub_linear_isLittleO_ae, isLittleO_abs_log_rpow_rpow_nhdsGT_zero, superpolynomialDecay_iff_isLittleO, rothNumberNat_isLittleO_id, isLittleOTVS_iff_isLittleO, isLittleO_norm_pow_norm_pow, HasFDerivAtFilter.isLittleO, intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae', intervalIntegral.integral_sub_linear_isLittleO_of_tendsto_ae, rpow_mul_exp_neg_mul_sq_isLittleO_exp_neg, Real.isLittleO_const_log_atTop, Real.isLittleO_exp_comp_exp_comp, isLittleO_rpow_exp_atTop, isLittleO_zpow_exp_pos_mul_atTop, isLittleO_const_left_of_ne, isLittleO_norm_left, cexp_neg_quadratic_isLittleO_abs_rpow_cocompact, intervalIntegral.measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_left, intervalIntegral.measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae, Complex.exp_sub_sum_range_succ_isLittleO_pow, isLittleO_const_mul_left_iff, Complex.IsExpCmpFilter.isLittleO_log_norm_re, taylor_isLittleO, isLittleO_map, Complex.IsExpCmpFilter.isLittleO_zpow_mul_exp, isLittleO_zero_right_iff, hasDerivAt_iff_isLittleO_nhds_zero, isLittleO_iff_exists_eq_mul, isLittleO_pure, isLittleO_of_subsingleton, FormalMultilinearSeries.isLittleO_of_lt_radius, Complex.IsExpCmpFilter.isLittleO_im_pow_exp_re, isLittleO_neg_left, isLittleO_abs_right, intervalIntegral.integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae, exp_neg_mul_rpow_isLittleO_exp_neg, ContinuousOn.integral_sub_linear_isLittleO_ae, exp_neg_mul_sq_isLittleO_exp_neg, AkraBazziRecurrence.isLittleO_smoothingFn_one, AkraBazziRecurrence.dist_r_b, isLittleO_pow_exp_pos_mul_atTop, isLittleO_pow_pow_of_lt_left, isLittleO_coe_const_pow_of_one_lt, isLittleO_const_mul_left_iff', rexp_neg_quadratic_isLittleO_rpow_atTop, ProbabilityTheory.strong_law_aux4, cexp_neg_quadratic_isLittleO_rpow_atTop, IsLittleO_def, isLittleO_norm_pow_id, isLittleO_iff_nat_mul_le_aux, Filter.Tendsto.integral_sub_linear_isLittleO_ae, Complex.IsExpCmpFilter.isLittleO_log_re_re, isLittleO_one_iff, Chebyshev.integral_theta_div_log_sq_isLittleO, isLittleO_sup, Tactic.ComputeAsymptotics.WellFormedBasis.tail_isLittleO_head, isLittleO_id_const, AkraBazziRecurrence.isLittleO_deriv_one_sub_smoothingFn, isLittleO_iff_tendsto, intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_le, Real.Gamma_integrand_isLittleO, ProbabilityTheory.strong_law_aux2, Complex.IsExpCmpFilter.isLittleO_cpow_exp, isLittleO_iff, ProbabilityTheory.strong_law_aux3, isLittleO_norm_norm, intervalIntegral.integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_left, isLittleO_top, ProbabilityTheory.strong_law_aux5, Polynomial.isLittleO_atBot_of_degree_lt, isLittleO_const_const_iff, Complex.IsExpCmpFilter.isLittleO_pow_mul_exp, AkraBazziRecurrence.isLittleO_deriv_smoothingFn, ContinuousAt.integral_sub_linear_isLittleO_ae, EisensteinSeries.isLittleO_const_left_of_properSpace_of_discreteTopology, TFAE_exists_lt_isLittleO_pow, IsLittleO.of_tendsto_div_atTop, hasGradientWithinAt_iff_isLittleO, isLittleO_pow_const_const_pow_of_one_lt, Complex.isLittleO_ofReal_right, Real.isLittleO_pow_log_id_atTop, isLittleO_pow_sub_sub, IsEquivalent.isLittleO, Real.isLittleO_const_logb_atTop, isLittleO_pow_const_mul_const_pow_const_pow_of_norm_lt, intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_le', IsBigO.not_isLittleO, isLittleO_mul_iff_isLittleO_div, Complex.IsExpCmpFilter.isLittleO_cpow_mul_exp, isLittleO_const_mul_right_iff', intervalIntegral.integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_right, isLittleO_iff_forall_isBigOWith, Real.isLittleO_pow_logb_id_atTop, Homeomorph.isLittleO_congr, hasFDerivAt_iff_isLittleO_nhds_zero, isLittleO_pow_pow_atTop_of_lt, Complex.isLittleO_ofReal_left, Filter.IsBoundedUnder.isLittleO_sub_self_inv, Real.isLittleO_one_exp_comp, IsTheta.isLittleO_congr_right, isLittleO_iff_nat_mul_le', isLittleO_abs_left, intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_ge, AkraBazziRecurrence.isLittleO_self_div_log_id, rpow_mul_exp_neg_mul_rpow_isLittleO_exp_neg, isLittleO_pi
IsTheta πŸ“–MathDef
54 mathmath: isTheta_completion_right, Complex.isTheta_exp_arg_mul_im, Int.cast_complex_isTheta_cast_real, Complex.isTheta_cpow_rpow, isTheta_const_const_iff, Filter.EventuallyEq.isTheta, IsLittleO.right_isTheta_add, AkraBazziRecurrence.isTheta_deriv_rpow_p_mul_one_sub_smoothingFn, isTheta_inv, Real.isTheta_exp_comp_one, IsBigO.antisymm, isTheta_choose, IsEquivalent.isTheta_symm, AkraBazziRecurrence.isTheta_smoothingFn_sub_self, Complex.isTheta_ofReal, IsTheta.of_norm_eventuallyEq, IsTheta.of_norm_eventuallyEq_norm, ContinuousOn.isTheta_principal, isTheta_comm, isTheta_const_smul_left, AkraBazziRecurrence.isTheta_deriv_rpow_p_mul_one_add_smoothingFn, HasFDerivAtFilter.isTheta_sub, isTheta_norm_right, isTheta_deriv_rpow_const_atTop, isTheta_sup, isTheta_rfl, isTheta_const_mul_right, Complex.isTheta_ofReal_right, isTheta_of_div_tendsto_nhds_ne_zero, isTheta_refl, EisensteinSeries.linear_isTheta_left, Complex.isTheta_cpow_const_rpow, Real.isTheta_exp_comp_exp_comp, EisensteinSeries.linear_isTheta_right_add, isTheta_norm_left, isTheta_completion_left, isTheta_const_const, isTheta_zero_left, HasStrictFDerivAt.isTheta_sub, isTheta_zero_right, IsEquivalent.isTheta, isTheta_bot, EisensteinSeries.linear_isTheta_right, AkraBazziRecurrence.isTheta_asympBound, Complex.isTheta_ofReal_left, isTheta_deriv_ofReal_cpow_const_atTop, IsThetaTVS.isTheta, isTheta_const_mul_left, isTheta_const_smul_right, HasDerivAtFilter.isTheta_sub, Complex.IsExpCmpFilter.isTheta_cpow_exp_re_mul_log, EisensteinSeries.vec_add_const_isTheta, isThetaTVS_iff_isTheta, IsLittleO.right_isTheta_add'
transEventuallyEqIsBigO πŸ“–CompOpβ€”
transEventuallyEqIsLittleO πŸ“–CompOpβ€”
transIsBigOEventuallyEq πŸ“–CompOpβ€”
transIsBigOIsBigO πŸ“–CompOpβ€”
transIsBigOIsLittleO πŸ“–CompOpβ€”
transIsLittleOEventuallyEq πŸ“–CompOpβ€”
transIsLittleOIsBigO πŸ“–CompOpβ€”
transIsLittleOIsLittleO πŸ“–CompOpβ€”
Β«term_=O[_]_Β» πŸ“–CompOpβ€”
Β«term_=o[_]_Β» πŸ“–CompOpβ€”
Β«term_=Θ[_]_Β» πŸ“–CompOpβ€”
Β«term_~[_]_Β» πŸ“–CompOpβ€”

Theorems

NameKindAssumesProvesValidatesDepends On
IsBigOWith_def πŸ“–mathematicalβ€”IsBigOWith
Filter.Eventually
Real
Real.instLE
Norm.norm
Real.instMul
β€”β€”
IsBigO_def πŸ“–mathematicalβ€”IsBigO
Real
IsBigOWith
β€”β€”
IsLittleO_def πŸ“–mathematicalβ€”IsLittleOβ€”β€”
isBigOWith_abs_abs πŸ“–mathematicalβ€”IsBigOWith
Real
Real.norm
abs
Real.lattice
Real.instAddGroup
β€”isBigOWith_abs_left
isBigOWith_abs_right
isBigOWith_abs_left πŸ“–mathematicalβ€”IsBigOWith
Real
Real.norm
abs
Real.lattice
Real.instAddGroup
β€”isBigOWith_norm_left
isBigOWith_abs_right πŸ“–mathematicalβ€”IsBigOWith
Real
Real.norm
abs
Real.lattice
Real.instAddGroup
β€”isBigOWith_norm_right
isBigOWith_bot πŸ“–mathematicalβ€”IsBigOWith
Bot.bot
Filter
Filter.instBot
β€”IsBigOWith.of_bound
isBigOWith_comm πŸ“–mathematicalβ€”IsBigOWith
SeminormedAddCommGroup.toNorm
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
SeminormedAddGroup.toAddGroup
SeminormedAddCommGroup.toSeminormedAddGroup
β€”IsBigOWith.symm
isBigOWith_congr πŸ“–mathematicalFilter.EventuallyEqIsBigOWithβ€”IsBigOWith_def
Filter.eventually_congr
Filter.mp_mem
Filter.univ_mem'
isBigOWith_const_const πŸ“–mathematicalβ€”IsBigOWith
NormedAddCommGroup.toNorm
Real
DivInvMonoid.toDiv
Real.instDivInvMonoid
Norm.norm
β€”IsBigOWith_def
Filter.univ_mem'
Set.mem_setOf
div_mul_cancelβ‚€
norm_ne_zero_iff
le_refl
isBigOWith_const_mul_self πŸ“–mathematicalβ€”IsBigOWith
SeminormedRing.toNorm
Norm.norm
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
SeminormedRing.toRing
β€”isBigOWith_of_le'
norm_mul_le
isBigOWith_fst_prod πŸ“–mathematicalβ€”IsBigOWith
SeminormedAddCommGroup.toNorm
Prod.toNorm
Real
Real.instOne
β€”isBigOWith_of_le
le_max_left
isBigOWith_iff πŸ“–mathematicalβ€”IsBigOWith
Filter.Eventually
Real
Real.instLE
Norm.norm
Real.instMul
β€”IsBigOWith_def
isBigOWith_insert πŸ“–mathematicalReal
Real.instLE
Norm.norm
Real.instMul
IsBigOWith
nhdsWithin
Set
Set.instInsert
β€”IsBigOWith_def
nhdsWithin_insert
isBigOWith_inv πŸ“–mathematicalReal
Real.instLT
Real.instZero
IsBigOWith
Real.instInv
Filter.Eventually
Real.instLE
Real.instMul
Norm.norm
β€”IsBigOWith_def
le_div_iffβ‚€'
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
Real.instIsStrictOrderedRing
isBigOWith_map πŸ“–mathematicalβ€”IsBigOWith
Filter.map
β€”IsBigOWith_def
Filter.eventually_map
isBigOWith_neg_left πŸ“–mathematicalβ€”IsBigOWith
SeminormedAddCommGroup.toNorm
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”IsBigOWith_def
norm_neg
isBigOWith_neg_right πŸ“–mathematicalβ€”IsBigOWith
SeminormedAddCommGroup.toNorm
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”IsBigOWith_def
norm_neg
isBigOWith_norm_left πŸ“–mathematicalβ€”IsBigOWith
Real
Real.norm
Norm.norm
SeminormedAddCommGroup.toNorm
β€”IsBigOWith_def
norm_norm
isBigOWith_norm_norm πŸ“–mathematicalβ€”IsBigOWith
Real
Real.norm
Norm.norm
SeminormedAddCommGroup.toNorm
β€”isBigOWith_norm_left
isBigOWith_norm_right
isBigOWith_norm_right πŸ“–mathematicalβ€”IsBigOWith
Real
Real.norm
Norm.norm
SeminormedAddCommGroup.toNorm
β€”IsBigOWith_def
norm_norm
isBigOWith_of_le πŸ“–mathematicalReal
Real.instLE
Norm.norm
IsBigOWith
Real.instOne
β€”isBigOWith_of_le'
one_mul
isBigOWith_of_le' πŸ“–mathematicalReal
Real.instLE
Norm.norm
Real.instMul
IsBigOWithβ€”IsBigOWith.of_bound
Filter.univ_mem'
isBigOWith_prod_left πŸ“–mathematicalβ€”IsBigOWith
Prod.toNorm
SeminormedAddCommGroup.toNorm
β€”IsBigOWith.prod_left_fst
IsBigOWith.prod_left_snd
IsBigOWith.prod_left_same
isBigOWith_pure πŸ“–mathematicalβ€”IsBigOWith
Filter
Filter.instPure
Real
Real.instLE
Norm.norm
Real.instMul
β€”isBigOWith_iff
isBigOWith_refl πŸ“–mathematicalβ€”IsBigOWith
Real
Real.instOne
β€”isBigOWith_of_le
le_rfl
isBigOWith_self_const_mul πŸ“–mathematicalβ€”IsBigOWith
NormedRing.toNorm
Real
Real.instInv
Norm.norm
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
NormedRing.toRing
β€”IsBigOWith_def
norm_mul
inv_mul_cancel_leftβ‚€
norm_ne_zero_iff
isBigOWith_self_const_mul' πŸ“–mathematicalβ€”IsBigOWith
SeminormedRing.toNorm
Norm.norm
Units.val
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Units
Units.instInv
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
β€”IsBigOWith.congr_left
isBigOWith_const_mul_self
Units.inv_mul_cancel_left
isBigOWith_snd_prod πŸ“–mathematicalβ€”IsBigOWith
SeminormedAddCommGroup.toNorm
Prod.toNorm
Real
Real.instOne
β€”isBigOWith_of_le
le_max_right
isBigOWith_zero πŸ“–mathematicalReal
Real.instLE
Real.instZero
IsBigOWith
SeminormedAddCommGroup.toNorm
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”IsBigOWith.of_bound
Filter.univ_mem'
norm_zero
mul_nonneg
IsOrderedRing.toPosMulMono
Real.instIsOrderedRing
norm_nonneg
isBigOWith_zero' πŸ“–mathematicalβ€”IsBigOWith
SeminormedAddCommGroup.toNorm
Real
Real.instZero
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”IsBigOWith.of_bound
Filter.univ_mem'
norm_zero
MulZeroClass.zero_mul
isBigOWith_zero_right_iff πŸ“–mathematicalβ€”IsBigOWith
NormedAddCommGroup.toNorm
SeminormedAddCommGroup.toNorm
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
Filter.EventuallyEq
Pi.instZero
NormedAddCommGroup.toAddCommGroup
β€”IsBigOWith_def
norm_zero
MulZeroClass.mul_zero
isBigO_abs_abs πŸ“–mathematicalβ€”IsBigO
Real
Real.norm
abs
Real.lattice
Real.instAddGroup
β€”isBigO_abs_left
isBigO_abs_right
isBigO_abs_left πŸ“–mathematicalβ€”IsBigO
Real
Real.norm
abs
Real.lattice
Real.instAddGroup
β€”isBigO_norm_left
isBigO_abs_right πŸ“–mathematicalβ€”IsBigO
Real
Real.norm
abs
Real.lattice
Real.instAddGroup
β€”isBigO_norm_right
isBigO_bot πŸ“–mathematicalβ€”IsBigO
Bot.bot
Filter
Filter.instBot
β€”IsBigOWith.isBigO
isBigOWith_bot
isBigO_comm πŸ“–mathematicalβ€”IsBigO
SeminormedAddCommGroup.toNorm
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
SeminormedAddGroup.toAddGroup
SeminormedAddCommGroup.toSeminormedAddGroup
β€”IsBigO.symm
isBigO_congr πŸ“–mathematicalFilter.EventuallyEqIsBigOβ€”IsBigO_def
isBigOWith_congr
isBigO_const_const πŸ“–mathematicalβ€”IsBigO
NormedAddCommGroup.toNorm
β€”IsBigOWith.isBigO
isBigOWith_const_const
isBigO_const_const_iff πŸ“–mathematicalβ€”IsBigO
NormedAddCommGroup.toNorm
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
NormedAddCommGroup.toAddCommGroup
β€”eq_or_ne
isBigO_const_const
instIsEmptyFalse
isBigO_const_mul_left_iff πŸ“–mathematicalβ€”IsBigO
NormedRing.toNorm
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
NormedRing.toRing
β€”IsBigO.trans
isBigO_self_const_mul
isBigO_const_mul_self
isBigO_const_mul_left_iff' πŸ“–mathematicalIsUnit
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
IsBigO
SeminormedRing.toNorm
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
β€”IsBigO.trans
isBigO_self_const_mul'
IsBigO.const_mul_left
isBigO_const_mul_right_iff πŸ“–mathematicalβ€”IsBigO
NormedRing.toNorm
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
NormedRing.toRing
β€”IsBigO.of_const_mul_right
IsBigO.const_mul_right
isBigO_const_mul_right_iff' πŸ“–mathematicalIsUnit
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
IsBigO
SeminormedRing.toNorm
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
β€”IsBigO.of_const_mul_right
IsBigO.const_mul_right'
isBigO_const_mul_self πŸ“–mathematicalβ€”IsBigO
SeminormedRing.toNorm
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
SeminormedRing.toRing
β€”IsBigOWith.isBigO
isBigOWith_const_mul_self
isBigO_fst_prod πŸ“–mathematicalβ€”IsBigO
SeminormedAddCommGroup.toNorm
Prod.toNorm
β€”IsBigOWith.isBigO
isBigOWith_fst_prod
isBigO_fst_prod' πŸ“–mathematicalβ€”IsBigO
SeminormedAddCommGroup.toNorm
Prod.toNorm
β€”IsBigO_def
IsBigOWith_def
isBigO_fst_prod
isBigO_iff πŸ“–mathematicalβ€”IsBigO
Filter.Eventually
Real
Real.instLE
Norm.norm
Real.instMul
β€”IsBigO_def
IsBigOWith_def
isBigO_iff' πŸ“–mathematicalβ€”IsBigO
SeminormedAddGroup.toNorm
Real
Real.instLT
Real.instZero
Filter.Eventually
Real.instLE
Norm.norm
Real.instMul
β€”isBigO_iff
LT.lt.trans_le
zero_lt_one
Real.instZeroLEOneClass
NeZero.charZero_one
IsStrictOrderedRing.toCharZero
Real.instIsStrictOrderedRing
le_max_right
Filter.mp_mem
Filter.univ_mem'
LE.le.trans
mul_le_mul_of_nonneg_right
IsOrderedRing.toMulPosMono
Real.instIsOrderedRing
le_max_left
norm_nonneg
isBigO_iff'' πŸ“–mathematicalβ€”IsBigO
SeminormedAddGroup.toNorm
Real
Real.instLT
Real.instZero
Filter.Eventually
Real.instLE
Real.instMul
Norm.norm
β€”isBigO_iff'
inv_pos_of_pos
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
Real.instIsStrictOrderedRing
Filter.mp_mem
Filter.univ_mem'
inv_mul_le_iffβ‚€
inv_inv
isBigO_iff_eventually πŸ“–mathematicalβ€”IsBigO
SeminormedAddCommGroup.toNorm
Filter.Eventually
Real
Real.instLE
Norm.norm
Real.instMul
Filter.atTop
Real.instPreorder
β€”isBigO_iff_eventually_isBigOWith
IsBigOWith_def
isBigO_iff_eventually_isBigOWith πŸ“–mathematicalβ€”IsBigO
SeminormedAddCommGroup.toNorm
Filter.Eventually
Real
IsBigOWith
Filter.atTop
Real.instPreorder
β€”isBigO_iff_isBigOWith
Filter.mem_atTop_sets
instIsDirectedOrder
Real.instIsOrderedRing
Real.instArchimedean
IsBigOWith.weaken
Filter.Eventually.exists
Filter.atTop_neBot
isBigO_iff_isBigOWith πŸ“–mathematicalβ€”IsBigO
IsBigOWith
β€”IsBigO_def
isBigO_map πŸ“–mathematicalβ€”IsBigO
Filter.map
β€”IsBigO_def
isBigO_neg_left πŸ“–mathematicalβ€”IsBigO
SeminormedAddCommGroup.toNorm
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”IsBigO_def
isBigOWith_neg_left
isBigO_neg_right πŸ“–mathematicalβ€”IsBigO
SeminormedAddCommGroup.toNorm
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”IsBigO_def
isBigOWith_neg_right
isBigO_norm_left πŸ“–mathematicalβ€”IsBigO
Real
Real.norm
Norm.norm
SeminormedAddCommGroup.toNorm
β€”IsBigO_def
isBigOWith_norm_left
isBigO_norm_norm πŸ“–mathematicalβ€”IsBigO
Real
Real.norm
Norm.norm
SeminormedAddCommGroup.toNorm
β€”isBigO_norm_left
isBigO_norm_right
isBigO_norm_right πŸ“–mathematicalβ€”IsBigO
Real
Real.norm
Norm.norm
SeminormedAddCommGroup.toNorm
β€”IsBigO_def
isBigOWith_norm_right
isBigO_of_le πŸ“–mathematicalReal
Real.instLE
Norm.norm
IsBigOβ€”IsBigOWith.isBigO
isBigOWith_of_le
isBigO_of_le' πŸ“–mathematicalReal
Real.instLE
Norm.norm
Real.instMul
IsBigOβ€”IsBigOWith.isBigO
isBigOWith_of_le'
isBigO_of_subsingleton πŸ“–mathematicalβ€”IsBigO
SeminormedAddCommGroup.toNorm
β€”IsLittleO.isBigO
isLittleO_of_subsingleton
isBigO_prod_left πŸ“–mathematicalβ€”IsBigO
Prod.toNorm
SeminormedAddCommGroup.toNorm
β€”IsBigO.prod_left_fst
IsBigO.prod_left_snd
IsBigO.prod_left
isBigO_pure πŸ“–mathematicalβ€”IsBigO
NormedAddCommGroup.toNorm
Filter
Filter.instPure
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
NormedAddCommGroup.toAddCommGroup
β€”isBigO_congr
isBigO_const_const_iff
Filter.pure_neBot
isBigO_refl πŸ“–mathematicalβ€”IsBigOβ€”IsBigOWith.isBigO
isBigOWith_refl
isBigO_refl_left πŸ“–mathematicalβ€”IsBigO
SeminormedAddCommGroup.toNorm
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
SeminormedAddGroup.toAddGroup
SeminormedAddCommGroup.toSeminormedAddGroup
β€”IsBigO.congr_left
isBigO_zero
sub_self
isBigO_self_const_mul πŸ“–mathematicalβ€”IsBigO
NormedRing.toNorm
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
NormedRing.toRing
β€”IsBigOWith.isBigO
isBigOWith_self_const_mul
isBigO_self_const_mul' πŸ“–mathematicalIsUnit
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
IsBigO
SeminormedRing.toNorm
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
β€”IsBigOWith.isBigO
isBigOWith_self_const_mul'
isBigO_snd_prod πŸ“–mathematicalβ€”IsBigO
SeminormedAddCommGroup.toNorm
Prod.toNorm
β€”IsBigOWith.isBigO
isBigOWith_snd_prod
isBigO_snd_prod' πŸ“–mathematicalβ€”IsBigO
SeminormedAddCommGroup.toNorm
Prod.toNorm
β€”IsBigO_def
IsBigOWith_def
isBigO_snd_prod
isBigO_sup πŸ“–mathematicalβ€”IsBigO
SeminormedAddCommGroup.toNorm
Filter
Filter.instSup
β€”IsBigO.mono
le_sup_left
le_sup_right
IsBigO.sup
isBigO_zero πŸ“–mathematicalβ€”IsBigO
SeminormedAddCommGroup.toNorm
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”isBigO_iff_isBigOWith
isBigOWith_zero'
isBigO_zero_right_iff πŸ“–mathematicalβ€”IsBigO
NormedAddCommGroup.toNorm
SeminormedAddCommGroup.toNorm
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
Filter.EventuallyEq
Pi.instZero
NormedAddCommGroup.toAddCommGroup
β€”IsBigO.isBigOWith
isBigOWith_zero_right_iff
IsBigOWith.isBigO
isLittleO_abs_abs πŸ“–mathematicalβ€”IsLittleO
Real
Real.norm
abs
Real.lattice
Real.instAddGroup
β€”isLittleO_abs_left
isLittleO_abs_right
isLittleO_abs_left πŸ“–mathematicalβ€”IsLittleO
Real
Real.norm
abs
Real.lattice
Real.instAddGroup
β€”isLittleO_norm_left
isLittleO_abs_right πŸ“–mathematicalβ€”IsLittleO
Real
Real.norm
abs
Real.lattice
Real.instAddGroup
β€”isLittleO_norm_right
isLittleO_bot πŸ“–mathematicalβ€”IsLittleO
Bot.bot
Filter
Filter.instBot
β€”IsLittleO.of_isBigOWith
isBigOWith_bot
isLittleO_comm πŸ“–mathematicalβ€”IsLittleO
SeminormedAddCommGroup.toNorm
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
SeminormedAddGroup.toAddGroup
SeminormedAddCommGroup.toSeminormedAddGroup
β€”IsLittleO.symm
isLittleO_congr πŸ“–mathematicalFilter.EventuallyEqIsLittleOβ€”IsLittleO_def
isBigOWith_congr
isLittleO_const_mul_left_iff πŸ“–mathematicalβ€”IsLittleO
NormedRing.toNorm
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
NormedRing.toRing
β€”IsBigO.trans_isLittleO
isBigO_self_const_mul
isBigO_const_mul_self
isLittleO_const_mul_left_iff' πŸ“–mathematicalIsUnit
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
IsLittleO
SeminormedRing.toNorm
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
β€”IsBigO.trans_isLittleO
isBigO_self_const_mul'
IsLittleO.const_mul_left
isLittleO_const_mul_right_iff πŸ“–mathematicalβ€”IsLittleO
NormedRing.toNorm
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
NormedRing.toRing
β€”IsLittleO.of_const_mul_right
IsLittleO.trans_isBigO
isBigO_self_const_mul
isLittleO_const_mul_right_iff' πŸ“–mathematicalIsUnit
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
IsLittleO
SeminormedRing.toNorm
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
β€”IsLittleO.of_const_mul_right
IsLittleO.const_mul_right'
isLittleO_iff πŸ“–mathematicalβ€”IsLittleO
Filter.Eventually
Real
Real.instLE
Norm.norm
Real.instMul
β€”IsLittleO_def
IsBigOWith_def
isLittleO_iff_forall_isBigOWith πŸ“–mathematicalβ€”IsLittleO
IsBigOWith
β€”IsLittleO_def
isLittleO_iff_nat_mul_le πŸ“–mathematicalβ€”IsLittleO
SeminormedAddCommGroup.toNorm
Filter.Eventually
Real
Real.instLE
Real.instMul
Real.instNatCast
Norm.norm
β€”isLittleO_iff_nat_mul_le_aux
norm_nonneg
isLittleO_iff_nat_mul_le' πŸ“–mathematicalβ€”IsLittleO
SeminormedAddCommGroup.toNorm
Filter.Eventually
Real
Real.instLE
Real.instMul
Real.instNatCast
Norm.norm
β€”isLittleO_iff_nat_mul_le_aux
norm_nonneg
isLittleO_iff_nat_mul_le_aux πŸ“–mathematicalReal
Real.instLE
Real.instZero
Norm.norm
IsLittleO
Filter.Eventually
Real.instMul
Real.instNatCast
β€”Filter.Eventually.mono
IsLittleO.def
one_pos
Real.instZeroLEOneClass
NeZero.charZero_one
IsStrictOrderedRing.toCharZero
Real.instIsStrictOrderedRing
Nat.cast_zero
MulZeroClass.zero_mul
LE.le.trans
one_mul
Nat.cast_pos
Real.instIsOrderedRing
Real.instNontrivial
isBigOWith_inv
IsLittleO.def'
inv_pos
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
isLittleO_iff
exists_nat_gt
Real.instArchimedean
LT.lt.trans
IsBigOWith.bound
mul_le_mul_of_nonneg_right
IsOrderedRing.toMulPosMono
inv_le_of_inv_leβ‚€
MulPosReflectLE.toMulPosReflectLT
MulPosStrictMono.toMulPosReflectLE
IsStrictOrderedRing.toMulPosStrictMono
LT.lt.le
nonneg_of_mul_nonneg_right
isLittleO_insert πŸ“–mathematicalNegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
IsLittleO
SeminormedAddCommGroup.toNorm
nhdsWithin
Set
Set.instInsert
β€”IsLittleO_def
isBigOWith_insert
norm_zero
mul_nonneg
IsOrderedRing.toPosMulMono
Real.instIsOrderedRing
LT.lt.le
norm_nonneg
isLittleO_irrefl πŸ“–mathematicalFilter.Frequently
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
NormedAddCommGroup.toAddCommGroup
IsLittleO
NormedAddCommGroup.toNorm
β€”isLittleO_irrefl'
Filter.Frequently.mono
norm_ne_zero_iff
isLittleO_irrefl' πŸ“–mathematicalFilter.Frequently
Real
Norm.norm
SeminormedAddCommGroup.toNorm
Real.instZero
IsLittleOβ€”Nat.instAtLeastTwoHAddOfNat
Filter.Frequently.exists
Filter.Eventually.and_frequently
IsLittleO.bound
one_half_pos
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
Real.instIsStrictOrderedRing
LT.lt.not_ge
half_lt_self
lt_of_le_of_ne
norm_nonneg
div_eq_inv_mul
one_div
isLittleO_map πŸ“–mathematicalβ€”IsLittleO
Filter.map
β€”IsLittleO_def
isLittleO_neg_left πŸ“–mathematicalβ€”IsLittleO
SeminormedAddCommGroup.toNorm
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”IsLittleO_def
isBigOWith_neg_left
isLittleO_neg_right πŸ“–mathematicalβ€”IsLittleO
SeminormedAddCommGroup.toNorm
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”IsLittleO_def
isBigOWith_neg_right
isLittleO_norm_left πŸ“–mathematicalβ€”IsLittleO
Real
Real.norm
Norm.norm
SeminormedAddCommGroup.toNorm
β€”IsLittleO_def
isBigOWith_norm_left
isLittleO_norm_norm πŸ“–mathematicalβ€”IsLittleO
Real
Real.norm
Norm.norm
SeminormedAddCommGroup.toNorm
β€”isLittleO_norm_left
isLittleO_norm_right
isLittleO_norm_right πŸ“–mathematicalβ€”IsLittleO
Real
Real.norm
Norm.norm
SeminormedAddCommGroup.toNorm
β€”IsLittleO_def
isBigOWith_norm_right
isLittleO_of_subsingleton πŸ“–mathematicalβ€”IsLittleO
SeminormedAddCommGroup.toNorm
β€”IsLittleO.of_bound
norm_zero
mul_nonneg
IsOrderedRing.toPosMulMono
Real.instIsOrderedRing
LT.lt.le
isLittleO_prod_left πŸ“–mathematicalβ€”IsLittleO
Prod.toNorm
SeminormedAddCommGroup.toNorm
β€”IsLittleO.prod_left_fst
IsLittleO.prod_left_snd
IsLittleO.prod_left
isLittleO_refl_left πŸ“–mathematicalβ€”IsLittleO
SeminormedAddCommGroup.toNorm
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
SeminormedAddGroup.toAddGroup
SeminormedAddCommGroup.toSeminormedAddGroup
β€”IsLittleO.congr_left
isLittleO_zero
sub_self
isLittleO_sup πŸ“–mathematicalβ€”IsLittleO
Filter
Filter.instSup
β€”IsLittleO.mono
le_sup_left
le_sup_right
IsLittleO.sup
isLittleO_zero πŸ“–mathematicalβ€”IsLittleO
SeminormedAddCommGroup.toNorm
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”IsLittleO.of_bound
Filter.univ_mem'
norm_zero
mul_nonneg
IsOrderedRing.toPosMulMono
Real.instIsOrderedRing
LT.lt.le
norm_nonneg
isLittleO_zero_right_iff πŸ“–mathematicalβ€”IsLittleO
NormedAddCommGroup.toNorm
SeminormedAddCommGroup.toNorm
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
Filter.EventuallyEq
Pi.instZero
NormedAddCommGroup.toAddCommGroup
β€”isBigO_zero_right_iff
IsLittleO.isBigO
IsLittleO.of_isBigOWith
isBigOWith_zero_right_iff

Asymptotics.IsBigO

Theorems

NameKindAssumesProvesValidatesDepends On
abs_abs πŸ“–mathematicalAsymptotics.IsBigO
Real
Real.norm
abs
Real.lattice
Real.instAddGroup
β€”Asymptotics.isBigO_abs_abs
abs_left πŸ“–mathematicalAsymptotics.IsBigO
Real
Real.norm
abs
Real.lattice
Real.instAddGroup
β€”Asymptotics.isBigO_abs_left
abs_right πŸ“–mathematicalAsymptotics.IsBigO
Real
Real.norm
abs
Real.lattice
Real.instAddGroup
β€”Asymptotics.isBigO_abs_right
add πŸ“–mathematicalAsymptotics.IsBigO
SeminormedAddCommGroup.toNorm
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
AddCommGroup.toAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”isBigOWith
Asymptotics.IsBigOWith.isBigO
Asymptotics.IsBigOWith.add
add_iff_left πŸ“–mathematicalAsymptotics.IsBigO
SeminormedAddCommGroup.toNorm
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
AddCommGroup.toAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”congr
sub
add_sub_cancel_right
add
add_iff_right πŸ“–mathematicalAsymptotics.IsBigO
SeminormedAddCommGroup.toNorm
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
AddCommGroup.toAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”congr
sub
eq_sub_of_add_eq'
add
add_isLittleO πŸ“–mathematicalAsymptotics.IsBigO
SeminormedAddCommGroup.toNorm
Asymptotics.IsLittleO
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
AddCommGroup.toAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”add
Asymptotics.IsLittleO.isBigO
antisymm πŸ“–mathematicalAsymptotics.IsBigOAsymptotics.IsThetaβ€”β€”
comp_fst πŸ“–mathematicalAsymptotics.IsBigOSProd.sprod
Filter
Filter.instSProd
β€”Filter.eventually_true
comp_neg_int πŸ“–β€”Asymptotics.IsBigO
Filter.cofinite
β€”β€”Equiv.neg_apply
comp_tendsto
Function.Injective.tendsto_cofinite
Equiv.injective
comp_snd πŸ“–mathematicalAsymptotics.IsBigOSProd.sprod
Filter
Filter.instSProd
β€”Filter.eventually_true
comp_tendsto πŸ“–β€”Asymptotics.IsBigO
Filter.Tendsto
β€”β€”Asymptotics.isBigO_iff_isBigOWith
Asymptotics.IsBigOWith.comp_tendsto
isBigOWith
congr' πŸ“–β€”Asymptotics.IsBigO
Filter.EventuallyEq
β€”β€”Asymptotics.isBigO_congr
congr_left πŸ“–β€”Asymptotics.IsBigOβ€”β€”congr
congr_of_sub πŸ“–β€”Asymptotics.IsBigO
SeminormedAddCommGroup.toNorm
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
SeminormedAddGroup.toAddGroup
SeminormedAddCommGroup.toSeminormedAddGroup
β€”β€”congr_left
sub
sub_sub_cancel
add
sub_add_cancel
congr_right πŸ“–β€”Asymptotics.IsBigOβ€”β€”congr
const_mul_left πŸ“–mathematicalAsymptotics.IsBigO
SeminormedRing.toNorm
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
SeminormedRing.toRing
β€”isBigOWith
Asymptotics.IsBigOWith.isBigO
Asymptotics.IsBigOWith.const_mul_left
const_mul_right πŸ“–mathematicalAsymptotics.IsBigO
NormedRing.toNorm
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
NormedRing.toRing
β€”exists_nonneg
Asymptotics.IsBigOWith.isBigO
Asymptotics.IsBigOWith.const_mul_right
const_mul_right' πŸ“–mathematicalIsUnit
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Asymptotics.IsBigO
SeminormedRing.toNorm
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
β€”trans
Asymptotics.isBigO_self_const_mul'
eq_zero_imp πŸ“–mathematicalAsymptotics.IsBigO
NormedAddCommGroup.toNorm
Filter.Eventuallyβ€”isBigOWith
Asymptotics.IsBigOWith.eq_zero_imp
eventually_mul_div_cancel πŸ“–mathematicalAsymptotics.IsBigO
NormedDivisionRing.toNorm
Filter.EventuallyEq
Pi.instMul
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
NormedRing.toRing
NormedDivisionRing.toNormedRing
Pi.instDiv
DivInvMonoid.toDiv
DivisionRing.toDivInvMonoid
NormedDivisionRing.toDivisionRing
β€”isBigOWith
Asymptotics.IsBigOWith.eventually_mul_div_cancel
exists_mem_basis πŸ“–mathematicalAsymptotics.IsBigO
SeminormedAddCommGroup.toNorm
Filter.HasBasis
Real
Real.instLT
Real.instZero
Real.instLE
Norm.norm
Real.instMul
β€”exists_pos
Filter.HasBasis.eventually_iff
exists_nonneg πŸ“–mathematicalAsymptotics.IsBigO
SeminormedAddCommGroup.toNorm
Real
Real.instLE
Real.instZero
Asymptotics.IsBigOWith
β€”isBigOWith
Asymptotics.IsBigOWith.exists_nonneg
exists_pos πŸ“–mathematicalAsymptotics.IsBigO
SeminormedAddCommGroup.toNorm
Real
Real.instLT
Real.instZero
Asymptotics.IsBigOWith
β€”isBigOWith
Asymptotics.IsBigOWith.exists_pos
fiberwise_left πŸ“–mathematicalAsymptotics.IsBigO
SProd.sprod
Filter
Filter.instSProd
Filter.Eventuallyβ€”Filter.mem_of_superset
fiberwise_right πŸ“–mathematicalAsymptotics.IsBigO
SProd.sprod
Filter
Filter.instSProd
Filter.Eventuallyβ€”Filter.mem_of_superset
inv_rev πŸ“–mathematicalAsymptotics.IsBigO
NormedDivisionRing.toNorm
Filter.Eventually
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
DivisionSemiring.toGroupWithZero
DivisionRing.toDivisionSemiring
NormedDivisionRing.toDivisionRing
β€”isBigOWith
Asymptotics.IsBigOWith.isBigO
Asymptotics.IsBigOWith.inv_rev
isBigOWith πŸ“–mathematicalAsymptotics.IsBigOAsymptotics.IsBigOWithβ€”Asymptotics.isBigO_iff_isBigOWith
mono πŸ“–β€”Asymptotics.IsBigO
Filter
Preorder.toLE
PartialOrder.toPreorder
Filter.instPartialOrder
β€”β€”Asymptotics.isBigO_iff_isBigOWith
Asymptotics.IsBigOWith.mono
isBigOWith
mul πŸ“–mathematicalAsymptotics.IsBigO
SeminormedRing.toNorm
NormedRing.toNorm
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
SeminormedRing.toRing
NormedRing.toRing
β€”isBigOWith
Asymptotics.IsBigOWith.isBigO
Asymptotics.IsBigOWith.mul
mul_isLittleO πŸ“–mathematicalAsymptotics.IsBigO
SeminormedRing.toNorm
NormedRing.toNorm
Asymptotics.IsLittleO
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
SeminormedRing.toRing
NormedRing.toRing
β€”Asymptotics.IsLittleO_def
exists_pos
Asymptotics.IsBigOWith.congr_const
Asymptotics.IsBigOWith.mul
div_pos
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
Real.instIsStrictOrderedRing
mul_div_cancelβ‚€
ne_of_gt
neg_left πŸ“–mathematicalAsymptotics.IsBigO
SeminormedAddCommGroup.toNorm
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”Asymptotics.isBigO_neg_left
neg_right πŸ“–mathematicalAsymptotics.IsBigO
SeminormedAddCommGroup.toNorm
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”Asymptotics.isBigO_neg_right
norm_left πŸ“–mathematicalAsymptotics.IsBigO
SeminormedAddCommGroup.toNorm
Real
Real.norm
Norm.norm
β€”Asymptotics.isBigO_norm_left
norm_norm πŸ“–mathematicalAsymptotics.IsBigO
SeminormedAddCommGroup.toNorm
Real
Real.norm
Norm.norm
β€”Asymptotics.isBigO_norm_norm
norm_right πŸ“–mathematicalAsymptotics.IsBigO
SeminormedAddCommGroup.toNorm
Real
Real.norm
Norm.norm
β€”Asymptotics.isBigO_norm_right
not_isLittleO πŸ“–mathematicalAsymptotics.IsBigO
NormedAddCommGroup.toNorm
SeminormedAddCommGroup.toNorm
Filter.Frequently
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
NormedAddCommGroup.toAddCommGroup
Asymptotics.IsLittleOβ€”Asymptotics.isLittleO_irrefl
trans_isLittleO
of_abs_abs πŸ“–β€”Asymptotics.IsBigO
Real
Real.norm
abs
Real.lattice
Real.instAddGroup
β€”β€”Asymptotics.isBigO_abs_abs
of_abs_left πŸ“–β€”Asymptotics.IsBigO
Real
Real.norm
abs
Real.lattice
Real.instAddGroup
β€”β€”Asymptotics.isBigO_abs_left
of_abs_right πŸ“–β€”Asymptotics.IsBigO
Real
Real.norm
abs
Real.lattice
Real.instAddGroup
β€”β€”Asymptotics.isBigO_abs_right
of_bound πŸ“–mathematicalFilter.Eventually
Real
Real.instLE
Norm.norm
Real.instMul
Asymptotics.IsBigOβ€”Asymptotics.isBigO_iff
of_bound' πŸ“–mathematicalFilter.Eventually
Real
Real.instLE
Norm.norm
Asymptotics.IsBigOβ€”of_bound
one_mul
of_const_mul_right πŸ“–β€”Asymptotics.IsBigO
SeminormedRing.toNorm
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
SeminormedRing.toRing
β€”β€”exists_nonneg
Asymptotics.IsBigOWith.isBigO
Asymptotics.IsBigOWith.of_const_mul_right
of_neg_left πŸ“–β€”Asymptotics.IsBigO
SeminormedAddCommGroup.toNorm
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”β€”Asymptotics.isBigO_neg_left
of_neg_right πŸ“–β€”Asymptotics.IsBigO
SeminormedAddCommGroup.toNorm
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”β€”Asymptotics.isBigO_neg_right
of_norm_eventuallyLE πŸ“–mathematicalFilter.EventuallyLE
Real
Real.instLE
Norm.norm
Asymptotics.IsBigO
Real.norm
β€”of_bound'
Filter.Eventually.mono
LE.le.trans
le_abs_self
of_norm_le πŸ“–mathematicalReal
Real.instLE
Norm.norm
Asymptotics.IsBigO
Real.norm
β€”of_norm_eventuallyLE
Filter.Eventually.of_forall
of_norm_left πŸ“–β€”Asymptotics.IsBigO
Real
Real.norm
Norm.norm
SeminormedAddCommGroup.toNorm
β€”β€”Asymptotics.isBigO_norm_left
of_norm_norm πŸ“–β€”Asymptotics.IsBigO
Real
Real.norm
Norm.norm
SeminormedAddCommGroup.toNorm
β€”β€”Asymptotics.isBigO_norm_norm
of_norm_right πŸ“–β€”Asymptotics.IsBigO
Real
Real.norm
Norm.norm
SeminormedAddCommGroup.toNorm
β€”β€”Asymptotics.isBigO_norm_right
pow πŸ“–mathematicalAsymptotics.IsBigO
SeminormedRing.toNorm
NormedRing.toNorm
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
NormedRing.toRing
β€”isBigOWith
Asymptotics.isBigO_iff_isBigOWith
Asymptotics.IsBigOWith.pow'
prod_left πŸ“–mathematicalAsymptotics.IsBigO
SeminormedAddCommGroup.toNorm
Prod.toNormβ€”isBigOWith
Asymptotics.IsBigOWith.isBigO
Asymptotics.IsBigOWith.prod_left
prod_left_fst πŸ“–β€”Asymptotics.IsBigO
Prod.toNorm
SeminormedAddCommGroup.toNorm
β€”β€”trans
Asymptotics.isBigO_fst_prod
prod_left_snd πŸ“–β€”Asymptotics.IsBigO
Prod.toNorm
SeminormedAddCommGroup.toNorm
β€”β€”trans
Asymptotics.isBigO_snd_prod
prod_rightl πŸ“–mathematicalAsymptotics.IsBigO
SeminormedAddCommGroup.toNorm
Prod.toNormβ€”exists_nonneg
Asymptotics.IsBigOWith.isBigO
Asymptotics.IsBigOWith.prod_rightl
prod_rightr πŸ“–mathematicalAsymptotics.IsBigO
SeminormedAddCommGroup.toNorm
Prod.toNormβ€”exists_nonneg
Asymptotics.IsBigOWith.isBigO
Asymptotics.IsBigOWith.prod_rightr
sub πŸ“–mathematicalAsymptotics.IsBigO
SeminormedAddCommGroup.toNorm
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
SeminormedAddGroup.toAddGroup
SeminormedAddCommGroup.toSeminormedAddGroup
β€”sub_eq_add_neg
add
neg_left
sub_iff_left πŸ“–mathematicalAsymptotics.IsBigO
SeminormedAddCommGroup.toNorm
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
SeminormedAddGroup.toAddGroup
SeminormedAddCommGroup.toSeminormedAddGroup
β€”congr
add
sub_add_cancel
sub
sub_iff_right πŸ“–mathematicalAsymptotics.IsBigO
SeminormedAddCommGroup.toNorm
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
SeminormedAddGroup.toAddGroup
SeminormedAddCommGroup.toSeminormedAddGroup
β€”congr
sub
sub_sub_self
sum πŸ“–mathematicalAsymptotics.IsBigO
SeminormedAddCommGroup.toNorm
Finset.sum
AddCommGroup.toAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”Asymptotics.IsBigO_def
Asymptotics.IsBigOWith.sum
Function.sometimes_spec
sup πŸ“–mathematicalAsymptotics.IsBigO
SeminormedAddCommGroup.toNorm
Filter
Filter.instSup
β€”isBigOWith
Asymptotics.IsBigOWith.isBigO
Asymptotics.IsBigOWith.sup'
trans πŸ“–β€”Asymptotics.IsBigO
SeminormedAddCommGroup.toNorm
β€”β€”exists_nonneg
isBigOWith
Asymptotics.IsBigOWith.isBigO
Asymptotics.IsBigOWith.trans
trans_eventuallyEq πŸ“–β€”Asymptotics.IsBigO
Filter.EventuallyEq
β€”β€”congr'
Filter.EventuallyEq.rfl
trans_isLittleO πŸ“–β€”Asymptotics.IsBigO
SeminormedAddCommGroup.toNorm
Asymptotics.IsLittleO
β€”β€”exists_pos
Asymptotics.IsBigOWith.trans_isLittleO
trans_le πŸ“–β€”Asymptotics.IsBigO
SeminormedAddCommGroup.toNorm
Real
Real.instLE
Norm.norm
β€”β€”trans
Asymptotics.isBigO_of_le
triangle πŸ“–β€”Asymptotics.IsBigO
SeminormedAddCommGroup.toNorm
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
SeminormedAddGroup.toAddGroup
SeminormedAddCommGroup.toSeminormedAddGroup
β€”β€”congr_left
add
sub_add_sub_cancel

Asymptotics.IsBigOWith

Theorems

NameKindAssumesProvesValidatesDepends On
abs_abs πŸ“–mathematicalAsymptotics.IsBigOWith
Real
Real.norm
abs
Real.lattice
Real.instAddGroup
β€”Asymptotics.isBigOWith_abs_abs
abs_left πŸ“–mathematicalAsymptotics.IsBigOWith
Real
Real.norm
abs
Real.lattice
Real.instAddGroup
β€”Asymptotics.isBigOWith_abs_left
abs_right πŸ“–mathematicalAsymptotics.IsBigOWith
Real
Real.norm
abs
Real.lattice
Real.instAddGroup
β€”Asymptotics.isBigOWith_abs_right
add πŸ“–mathematicalAsymptotics.IsBigOWith
SeminormedAddCommGroup.toNorm
Real
Real.instAdd
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
AddCommGroup.toAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”Asymptotics.IsBigOWith_def
Filter.mp_mem
Filter.univ_mem'
norm_add_le_of_le
add_mul
Distrib.rightDistribClass
add_isLittleO πŸ“–mathematicalAsymptotics.IsBigOWith
SeminormedAddCommGroup.toNorm
Asymptotics.IsLittleO
Real
Real.instLT
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
AddCommGroup.toAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”congr_const
add
Asymptotics.IsLittleO.forall_isBigOWith
sub_pos
IsRightCancelAdd.addRightStrictMono_of_addRightMono
AddRightCancelSemigroup.toIsRightCancelAdd
covariant_swap_add_of_covariant_add
IsOrderedAddMonoid.toAddLeftMono
Real.instIsOrderedAddMonoid
add_sub_cancel
comp_tendsto πŸ“–β€”Asymptotics.IsBigOWith
Filter.Tendsto
β€”β€”of_bound
bound
congr' πŸ“–β€”Asymptotics.IsBigOWith
Filter.EventuallyEq
β€”β€”Asymptotics.isBigOWith_congr
congr_const πŸ“–β€”Asymptotics.IsBigOWithβ€”β€”congr
congr_left πŸ“–β€”Asymptotics.IsBigOWithβ€”β€”congr
congr_right πŸ“–β€”Asymptotics.IsBigOWithβ€”β€”congr
const_mul_left πŸ“–mathematicalAsymptotics.IsBigOWith
SeminormedRing.toNorm
Real
Real.instMul
Norm.norm
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
SeminormedRing.toRing
β€”trans
Asymptotics.isBigOWith_const_mul_self
norm_nonneg
const_mul_right πŸ“–mathematicalReal
Real.instLE
Real.instZero
Asymptotics.IsBigOWith
NormedRing.toNorm
Real.instMul
Real.instInv
Norm.norm
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
NormedRing.toRing
β€”trans
Asymptotics.isBigOWith_self_const_mul
const_mul_right' πŸ“–mathematicalReal
Real.instLE
Real.instZero
Asymptotics.IsBigOWith
SeminormedRing.toNorm
Real.instMul
Norm.norm
Units.val
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Units
Units.instInv
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
β€”trans
Asymptotics.isBigOWith_self_const_mul'
eq_zero_imp πŸ“–mathematicalAsymptotics.IsBigOWith
NormedAddCommGroup.toNorm
Filter.Eventuallyβ€”Filter.Eventually.mono
bound
norm_le_zero_iff
norm_zero
MulZeroClass.mul_zero
eventually_mul_div_cancel πŸ“–mathematicalAsymptotics.IsBigOWith
NormedDivisionRing.toNorm
Filter.EventuallyEq
Pi.instMul
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
NormedRing.toRing
NormedDivisionRing.toNormedRing
Pi.instDiv
DivInvMonoid.toDiv
DivisionRing.toDivInvMonoid
NormedDivisionRing.toDivisionRing
β€”Filter.Eventually.mono
bound
div_mul_cancel_of_imp
norm_zero
MulZeroClass.mul_zero
exists_nonneg πŸ“–mathematicalAsymptotics.IsBigOWith
SeminormedAddCommGroup.toNorm
Real
Real.instLE
Real.instZero
β€”exists_pos
le_of_lt
exists_pos πŸ“–mathematicalAsymptotics.IsBigOWith
SeminormedAddCommGroup.toNorm
Real
Real.instLT
Real.instZero
β€”lt_of_lt_of_le
zero_lt_one
Real.instZeroLEOneClass
NeZero.charZero_one
IsStrictOrderedRing.toCharZero
Real.instIsStrictOrderedRing
le_max_right
weaken
le_max_left
insert πŸ“–mathematicalAsymptotics.IsBigOWith
nhdsWithin
Real
Real.instLE
Norm.norm
Real.instMul
Set
Set.instInsert
β€”Asymptotics.isBigOWith_insert
inv_rev πŸ“–mathematicalAsymptotics.IsBigOWith
NormedDivisionRing.toNorm
Filter.Eventually
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
DivisionSemiring.toGroupWithZero
DivisionRing.toDivisionSemiring
NormedDivisionRing.toDivisionRing
β€”of_bound
Filter.Eventually.mp
bound
Filter.Eventually.mono
eq_or_ne
inv_zero
norm_zero
MulZeroClass.mul_zero
pos_of_mul_pos_left
MulPosReflectLE.toMulPosReflectLT
MulPosStrictMono.toMulPosReflectLE
IsStrictOrderedRing.toMulPosStrictMono
Real.instIsStrictOrderedRing
LT.lt.trans_le
norm_pos_iff
norm_nonneg
inv_antiβ‚€
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
norm_inv
mul_inv
div_le_iffβ‚€
isBigO πŸ“–mathematicalAsymptotics.IsBigOWithAsymptotics.IsBigOβ€”Asymptotics.IsBigO_def
mono πŸ“–β€”Asymptotics.IsBigOWith
Filter
Preorder.toLE
PartialOrder.toPreorder
Filter.instPartialOrder
β€”β€”of_bound
bound
mul πŸ“–mathematicalAsymptotics.IsBigOWith
SeminormedRing.toNorm
NormedRing.toNorm
Real
Real.instMul
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
SeminormedRing.toRing
NormedRing.toRing
β€”Asymptotics.IsBigOWith_def
Filter.mp_mem
Filter.univ_mem'
le_trans
norm_mul_le
norm_mul
mul_mul_mul_comm
mul_le_mul
IsOrderedRing.toPosMulMono
Real.instIsOrderedRing
IsOrderedRing.toMulPosMono
norm_nonneg
neg_left πŸ“–mathematicalAsymptotics.IsBigOWith
SeminormedAddCommGroup.toNorm
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”Asymptotics.isBigOWith_neg_left
neg_right πŸ“–mathematicalAsymptotics.IsBigOWith
SeminormedAddCommGroup.toNorm
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”Asymptotics.isBigOWith_neg_right
norm_left πŸ“–mathematicalAsymptotics.IsBigOWith
SeminormedAddCommGroup.toNorm
Real
Real.norm
Norm.norm
β€”Asymptotics.isBigOWith_norm_left
norm_norm πŸ“–mathematicalAsymptotics.IsBigOWith
SeminormedAddCommGroup.toNorm
Real
Real.norm
Norm.norm
β€”Asymptotics.isBigOWith_norm_norm
norm_right πŸ“–mathematicalAsymptotics.IsBigOWith
SeminormedAddCommGroup.toNorm
Real
Real.norm
Norm.norm
β€”Asymptotics.isBigOWith_norm_right
of_abs_abs πŸ“–β€”Asymptotics.IsBigOWith
Real
Real.norm
abs
Real.lattice
Real.instAddGroup
β€”β€”Asymptotics.isBigOWith_abs_abs
of_abs_left πŸ“–β€”Asymptotics.IsBigOWith
Real
Real.norm
abs
Real.lattice
Real.instAddGroup
β€”β€”Asymptotics.isBigOWith_abs_left
of_abs_right πŸ“–β€”Asymptotics.IsBigOWith
Real
Real.norm
abs
Real.lattice
Real.instAddGroup
β€”β€”Asymptotics.isBigOWith_abs_right
of_bound πŸ“–mathematicalFilter.Eventually
Real
Real.instLE
Norm.norm
Real.instMul
Asymptotics.IsBigOWithβ€”Asymptotics.isBigOWith_iff
of_const_mul_right πŸ“–mathematicalReal
Real.instLE
Real.instZero
Asymptotics.IsBigOWith
SeminormedRing.toNorm
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
SeminormedRing.toRing
Real.instMul
Norm.norm
β€”trans
Asymptotics.isBigOWith_const_mul_self
of_neg_left πŸ“–β€”Asymptotics.IsBigOWith
SeminormedAddCommGroup.toNorm
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”β€”Asymptotics.isBigOWith_neg_left
of_neg_right πŸ“–β€”Asymptotics.IsBigOWith
SeminormedAddCommGroup.toNorm
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”β€”Asymptotics.isBigOWith_neg_right
of_norm_left πŸ“–β€”Asymptotics.IsBigOWith
Real
Real.norm
Norm.norm
SeminormedAddCommGroup.toNorm
β€”β€”Asymptotics.isBigOWith_norm_left
of_norm_norm πŸ“–β€”Asymptotics.IsBigOWith
Real
Real.norm
Norm.norm
SeminormedAddCommGroup.toNorm
β€”β€”Asymptotics.isBigOWith_norm_norm
of_norm_right πŸ“–β€”Asymptotics.IsBigOWith
Real
Real.norm
Norm.norm
SeminormedAddCommGroup.toNorm
β€”β€”Asymptotics.isBigOWith_norm_right
of_pow πŸ“–β€”Asymptotics.IsBigOWith
NormedRing.toNorm
SeminormedRing.toNorm
Pi.instPow
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
NormedRing.toRing
SeminormedRing.toRing
Real
Real.instLE
Real.instMonoid
Real.instZero
β€”β€”of_bound
Filter.Eventually.mono
bound
weaken
le_of_pow_le_pow_leftβ‚€
IsStrictOrderedRing.toPosMulStrictMono
Real.instIsStrictOrderedRing
IsOrderedRing.toMulPosMono
Real.instIsOrderedRing
mul_nonneg
IsOrderedRing.toPosMulMono
norm_nonneg
norm_pow
mul_le_mul_of_nonneg_left
norm_pow_le'
Ne.bot_lt
pow_nonneg
IsOrderedRing.toZeroLEOneClass
mul_pow
pow πŸ“–mathematicalAsymptotics.IsBigOWith
SeminormedRing.toNorm
NormedRing.toNorm
Real
Monoid.toNatPow
Real.instMonoid
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
NormedRing.toRing
β€”pow_zero
NormOneClass.norm_one
pow'
pow' πŸ“–mathematicalAsymptotics.IsBigOWith
SeminormedRing.toNorm
NormedRing.toNorm
Real
Norm.norm
AddMonoidWithOne.toOne
AddGroupWithOne.toAddMonoidWithOne
Ring.toAddGroupWithOne
SeminormedRing.toRing
Monoid.toNatPow
Real.instMonoid
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
NormedRing.toRing
β€”NormOneClass.nontrivial
pow_zero
NormOneClass.norm_one
div_one
Asymptotics.isBigOWith_const_const
one_ne_zero'
NeZero.one
zero_add
pow_one
pow_succ
mul
prod_left πŸ“–mathematicalAsymptotics.IsBigOWith
SeminormedAddCommGroup.toNorm
Prod.toNorm
Real
Real.instMax
β€”prod_left_same
weaken
le_max_left
le_max_right
prod_left_fst πŸ“–β€”Asymptotics.IsBigOWith
Prod.toNorm
SeminormedAddCommGroup.toNorm
β€”β€”congr_const
trans
Asymptotics.isBigOWith_fst_prod
zero_le_one
Real.instZeroLEOneClass
one_mul
prod_left_same πŸ“–mathematicalAsymptotics.IsBigOWith
SeminormedAddCommGroup.toNorm
Prod.toNormβ€”Asymptotics.isBigOWith_iff
Filter.mp_mem
Filter.univ_mem'
max_le
prod_left_snd πŸ“–β€”Asymptotics.IsBigOWith
Prod.toNorm
SeminormedAddCommGroup.toNorm
β€”β€”congr_const
trans
Asymptotics.isBigOWith_snd_prod
zero_le_one
Real.instZeroLEOneClass
one_mul
prod_rightl πŸ“–mathematicalAsymptotics.IsBigOWith
SeminormedAddCommGroup.toNorm
Real
Real.instLE
Real.instZero
Prod.toNormβ€”congr_const
trans
Asymptotics.isBigOWith_fst_prod
mul_one
prod_rightr πŸ“–mathematicalAsymptotics.IsBigOWith
SeminormedAddCommGroup.toNorm
Real
Real.instLE
Real.instZero
Prod.toNormβ€”congr_const
trans
Asymptotics.isBigOWith_snd_prod
mul_one
sub πŸ“–mathematicalAsymptotics.IsBigOWith
SeminormedAddCommGroup.toNorm
Real
Real.instAdd
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
SeminormedAddGroup.toAddGroup
SeminormedAddCommGroup.toSeminormedAddGroup
β€”sub_eq_add_neg
add
neg_left
sub_isLittleO πŸ“–mathematicalAsymptotics.IsBigOWith
SeminormedAddCommGroup.toNorm
Asymptotics.IsLittleO
Real
Real.instLT
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
SeminormedAddGroup.toAddGroup
SeminormedAddCommGroup.toSeminormedAddGroup
β€”sub_eq_add_neg
add_isLittleO
Asymptotics.IsLittleO.neg_left
sum πŸ“–mathematicalAsymptotics.IsBigOWith
SeminormedAddCommGroup.toNorm
Finset.sum
Real
Real.instAddCommMonoid
AddCommGroup.toAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”Finset.cons_induction
Finset.sum_cons
add
sup πŸ“–mathematicalAsymptotics.IsBigOWithFilter
Filter.instSup
β€”of_bound
Filter.mem_sup
bound
sup' πŸ“–mathematicalAsymptotics.IsBigOWith
SeminormedAddCommGroup.toNorm
Real
Real.instMax
Filter
Filter.instSup
β€”of_bound
Filter.mem_sup
bound
weaken
le_max_left
le_max_right
trans πŸ“–mathematicalAsymptotics.IsBigOWith
Real
Real.instLE
Real.instZero
Real.instMulβ€”Asymptotics.IsBigOWith_def
Filter.mp_mem
Filter.univ_mem'
mul_le_mul_of_nonneg_left
IsOrderedRing.toPosMulMono
Real.instIsOrderedRing
mul_assoc
trans_isLittleO πŸ“–β€”Asymptotics.IsBigOWith
Asymptotics.IsLittleO
Real
Real.instLT
Real.instZero
β€”β€”Asymptotics.IsLittleO_def
div_pos
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
Real.instIsStrictOrderedRing
congr_const
trans
LT.lt.le
mul_div_cancelβ‚€
LT.lt.ne'
trans_le πŸ“–β€”Asymptotics.IsBigOWith
Real
Real.instLE
Norm.norm
Real.instZero
β€”β€”congr_const
trans
Asymptotics.isBigOWith_of_le
mul_one
triangle πŸ“–mathematicalAsymptotics.IsBigOWith
SeminormedAddCommGroup.toNorm
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
SeminormedAddGroup.toAddGroup
SeminormedAddCommGroup.toSeminormedAddGroup
Real
Real.instAdd
β€”congr_left
add
sub_add_sub_cancel
weaken πŸ“–β€”Asymptotics.IsBigOWith
SeminormedAddCommGroup.toNorm
Real
Real.instLE
β€”β€”of_bound
Filter.mem_of_superset
bound
mul_le_mul_of_nonneg_right
IsOrderedRing.toMulPosMono
Real.instIsOrderedRing
norm_nonneg

Asymptotics.IsLittleO

Theorems

NameKindAssumesProvesValidatesDepends On
abs_abs πŸ“–mathematicalAsymptotics.IsLittleO
Real
Real.norm
abs
Real.lattice
Real.instAddGroup
β€”Asymptotics.isLittleO_abs_abs
abs_left πŸ“–mathematicalAsymptotics.IsLittleO
Real
Real.norm
abs
Real.lattice
Real.instAddGroup
β€”Asymptotics.isLittleO_abs_left
abs_right πŸ“–mathematicalAsymptotics.IsLittleO
Real
Real.norm
abs
Real.lattice
Real.instAddGroup
β€”Asymptotics.isLittleO_abs_right
add πŸ“–mathematicalAsymptotics.IsLittleO
SeminormedAddCommGroup.toNorm
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
AddCommGroup.toAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”of_isBigOWith
Asymptotics.IsBigOWith.congr_const
Nat.instAtLeastTwoHAddOfNat
Asymptotics.IsBigOWith.add
forall_isBigOWith
half_pos
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
Real.instIsStrictOrderedRing
add_halves
CharZero.NeZero.two
IsStrictOrderedRing.toCharZero
add_add πŸ“–mathematicalAsymptotics.IsLittleO
SeminormedAddCommGroup.toNorm
Real
Real.norm
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
AddCommGroup.toAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
Real.instAdd
Norm.norm
β€”add
trans_le
abs_of_nonneg
IsOrderedAddMonoid.toAddLeftMono
Real.instIsOrderedAddMonoid
IsLeftCancelAdd.addLeftReflectLE_of_addLeftReflectLT
AddLeftCancelSemigroup.toIsLeftCancelAdd
contravariant_lt_of_covariant_le
covariant_swap_add_of_covariant_add
IsRightCancelAdd.addRightReflectLE_of_addRightReflectLT
AddRightCancelSemigroup.toIsRightCancelAdd
contravariant_swap_add_of_contravariant_add
add_iff_left πŸ“–mathematicalAsymptotics.IsLittleO
SeminormedAddCommGroup.toNorm
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
AddCommGroup.toAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”congr
sub
add_sub_cancel_right
add
add_iff_right πŸ“–mathematicalAsymptotics.IsLittleO
SeminormedAddCommGroup.toNorm
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
AddCommGroup.toAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”congr
sub
eq_sub_of_add_eq'
add
add_isBigO πŸ“–mathematicalAsymptotics.IsLittleO
SeminormedAddCommGroup.toNorm
Asymptotics.IsBigO
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
AddCommGroup.toAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”Asymptotics.IsBigO.add
isBigO
add_isBigOWith πŸ“–mathematicalAsymptotics.IsLittleO
SeminormedAddCommGroup.toNorm
Asymptotics.IsBigOWith
Real
Real.instLT
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
AddCommGroup.toAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”Asymptotics.IsBigOWith.congr_left
Asymptotics.IsBigOWith.add_isLittleO
add_comm
comp_fst πŸ“–mathematicalAsymptotics.IsLittleOSProd.sprod
Filter
Filter.instSProd
β€”Filter.eventually_true
comp_snd πŸ“–mathematicalAsymptotics.IsLittleOSProd.sprod
Filter
Filter.instSProd
β€”Filter.eventually_true
comp_tendsto πŸ“–β€”Asymptotics.IsLittleO
Filter.Tendsto
β€”β€”of_isBigOWith
Asymptotics.IsBigOWith.comp_tendsto
forall_isBigOWith
congr' πŸ“–β€”Asymptotics.IsLittleO
Filter.EventuallyEq
β€”β€”Asymptotics.isLittleO_congr
congr_left πŸ“–β€”Asymptotics.IsLittleOβ€”β€”congr
congr_of_sub πŸ“–β€”Asymptotics.IsLittleO
SeminormedAddCommGroup.toNorm
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
SeminormedAddGroup.toAddGroup
SeminormedAddCommGroup.toSeminormedAddGroup
β€”β€”congr_left
sub
sub_sub_cancel
add
sub_add_cancel
congr_right πŸ“–β€”Asymptotics.IsLittleOβ€”β€”congr
const_mul_left πŸ“–mathematicalAsymptotics.IsLittleO
SeminormedRing.toNorm
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
SeminormedRing.toRing
β€”Asymptotics.IsBigO.trans_isLittleO
Asymptotics.isBigO_const_mul_self
const_mul_right πŸ“–mathematicalAsymptotics.IsLittleO
NormedRing.toNorm
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
NormedRing.toRing
β€”trans_isBigO
Asymptotics.isBigO_self_const_mul
const_mul_right' πŸ“–mathematicalIsUnit
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
Asymptotics.IsLittleO
SeminormedRing.toNorm
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
β€”trans_isBigO
Asymptotics.isBigO_self_const_mul'
def πŸ“–mathematicalAsymptotics.IsLittleO
Real
Real.instLT
Real.instZero
Filter.Eventually
Real.instLE
Norm.norm
Real.instMul
β€”Asymptotics.isLittleO_iff
def' πŸ“–mathematicalAsymptotics.IsLittleO
Real
Real.instLT
Real.instZero
Asymptotics.IsBigOWithβ€”Asymptotics.isBigOWith_iff
Asymptotics.isLittleO_iff
eventuallyLE πŸ“–mathematicalAsymptotics.IsLittleOFilter.Eventually
Real
Real.instLE
Norm.norm
β€”one_mul
def
zero_lt_one
Real.instZeroLEOneClass
NeZero.charZero_one
IsStrictOrderedRing.toCharZero
Real.instIsStrictOrderedRing
eventually_mul_div_cancel πŸ“–mathematicalAsymptotics.IsLittleO
NormedDivisionRing.toNorm
Filter.EventuallyEq
Pi.instMul
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
NormedRing.toRing
NormedDivisionRing.toNormedRing
Pi.instDiv
DivInvMonoid.toDiv
DivisionRing.toDivInvMonoid
NormedDivisionRing.toDivisionRing
β€”Asymptotics.IsBigOWith.eventually_mul_div_cancel
forall_isBigOWith
zero_lt_one
Real.instZeroLEOneClass
NeZero.charZero_one
IsStrictOrderedRing.toCharZero
Real.instIsStrictOrderedRing
forall_isBigOWith πŸ“–mathematicalAsymptotics.IsLittleO
Real
Real.instLT
Real.instZero
Asymptotics.IsBigOWithβ€”Asymptotics.isLittleO_iff_forall_isBigOWith
insert πŸ“–mathematicalAsymptotics.IsLittleO
SeminormedAddCommGroup.toNorm
nhdsWithin
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
Set
Set.instInsert
β€”Asymptotics.isLittleO_insert
inv_rev πŸ“–mathematicalAsymptotics.IsLittleO
NormedDivisionRing.toNorm
Filter.Eventually
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
GroupWithZero.toDivisionMonoid
DivisionSemiring.toGroupWithZero
DivisionRing.toDivisionSemiring
NormedDivisionRing.toDivisionRing
β€”of_isBigOWith
Asymptotics.IsBigOWith.inv_rev
def'
isBigO πŸ“–mathematicalAsymptotics.IsLittleOAsymptotics.IsBigOβ€”Asymptotics.IsBigOWith.isBigO
isBigOWith
isBigOWith πŸ“–mathematicalAsymptotics.IsLittleOAsymptotics.IsBigOWith
Real
Real.instOne
β€”def'
zero_lt_one
Real.instZeroLEOneClass
NeZero.charZero_one
IsStrictOrderedRing.toCharZero
Real.instIsStrictOrderedRing
mono πŸ“–β€”Asymptotics.IsLittleO
Filter
Preorder.toLE
PartialOrder.toPreorder
Filter.instPartialOrder
β€”β€”of_isBigOWith
Asymptotics.IsBigOWith.mono
forall_isBigOWith
mul πŸ“–mathematicalAsymptotics.IsLittleO
SeminormedRing.toNorm
NormedRing.toNorm
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
SeminormedRing.toRing
NormedRing.toRing
β€”mul_isBigO
isBigO
mul_isBigO πŸ“–mathematicalAsymptotics.IsLittleO
SeminormedRing.toNorm
NormedRing.toNorm
Asymptotics.IsBigO
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
SeminormedRing.toRing
NormedRing.toRing
β€”Asymptotics.IsLittleO_def
Asymptotics.IsBigO.exists_pos
Asymptotics.IsBigOWith.congr_const
Asymptotics.IsBigOWith.mul
div_pos
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
Real.instIsStrictOrderedRing
div_mul_cancelβ‚€
ne_of_gt
neg_left πŸ“–mathematicalAsymptotics.IsLittleO
SeminormedAddCommGroup.toNorm
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”Asymptotics.isLittleO_neg_left
neg_right πŸ“–mathematicalAsymptotics.IsLittleO
SeminormedAddCommGroup.toNorm
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”Asymptotics.isLittleO_neg_right
norm_left πŸ“–mathematicalAsymptotics.IsLittleO
SeminormedAddCommGroup.toNorm
Real
Real.norm
Norm.norm
β€”Asymptotics.isLittleO_norm_left
norm_norm πŸ“–mathematicalAsymptotics.IsLittleO
SeminormedAddCommGroup.toNorm
Real
Real.norm
Norm.norm
β€”Asymptotics.isLittleO_norm_norm
norm_right πŸ“–mathematicalAsymptotics.IsLittleO
SeminormedAddCommGroup.toNorm
Real
Real.norm
Norm.norm
β€”Asymptotics.isLittleO_norm_right
not_isBigO πŸ“–mathematicalAsymptotics.IsLittleO
NormedAddCommGroup.toNorm
SeminormedAddCommGroup.toNorm
Filter.Frequently
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
NormedAddCommGroup.toAddCommGroup
Asymptotics.IsBigOβ€”Asymptotics.isLittleO_irrefl
trans_isBigO
of_abs_abs πŸ“–β€”Asymptotics.IsLittleO
Real
Real.norm
abs
Real.lattice
Real.instAddGroup
β€”β€”Asymptotics.isLittleO_abs_abs
of_abs_left πŸ“–β€”Asymptotics.IsLittleO
Real
Real.norm
abs
Real.lattice
Real.instAddGroup
β€”β€”Asymptotics.isLittleO_abs_left
of_abs_right πŸ“–β€”Asymptotics.IsLittleO
Real
Real.norm
abs
Real.lattice
Real.instAddGroup
β€”β€”Asymptotics.isLittleO_abs_right
of_bound πŸ“–mathematicalFilter.Eventually
Real
Real.instLE
Norm.norm
Real.instMul
Asymptotics.IsLittleOβ€”Asymptotics.isLittleO_iff
of_const_mul_right πŸ“–β€”Asymptotics.IsLittleO
SeminormedRing.toNorm
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
SeminormedRing.toRing
β€”β€”trans_isBigO
Asymptotics.isBigO_const_mul_self
of_isBigOWith πŸ“–mathematicalAsymptotics.IsBigOWithAsymptotics.IsLittleOβ€”Asymptotics.isLittleO_iff_forall_isBigOWith
of_neg_left πŸ“–β€”Asymptotics.IsLittleO
SeminormedAddCommGroup.toNorm
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”β€”Asymptotics.isLittleO_neg_left
of_neg_right πŸ“–β€”Asymptotics.IsLittleO
SeminormedAddCommGroup.toNorm
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”β€”Asymptotics.isLittleO_neg_right
of_norm_left πŸ“–β€”Asymptotics.IsLittleO
Real
Real.norm
Norm.norm
SeminormedAddCommGroup.toNorm
β€”β€”Asymptotics.isLittleO_norm_left
of_norm_norm πŸ“–β€”Asymptotics.IsLittleO
Real
Real.norm
Norm.norm
SeminormedAddCommGroup.toNorm
β€”β€”Asymptotics.isLittleO_norm_norm
of_norm_right πŸ“–β€”Asymptotics.IsLittleO
Real
Real.norm
Norm.norm
SeminormedAddCommGroup.toNorm
β€”β€”Asymptotics.isLittleO_norm_right
of_pow πŸ“–β€”Asymptotics.IsLittleO
NormedRing.toNorm
SeminormedRing.toNorm
Pi.instPow
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
NormedRing.toRing
SeminormedRing.toRing
β€”β€”of_isBigOWith
Asymptotics.IsBigOWith.of_pow
def'
pow_pos
IsStrictOrderedRing.toPosMulStrictMono
Real.instIsStrictOrderedRing
Real.instZeroLEOneClass
le_rfl
LT.lt.le
pow πŸ“–mathematicalAsymptotics.IsLittleO
SeminormedRing.toNorm
NormedRing.toNorm
Monoid.toNatPow
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SeminormedRing.toRing
NormedRing.toRing
β€”LT.lt.ne'
pow_one
pow_succ
mul
prod_left πŸ“–mathematicalAsymptotics.IsLittleO
SeminormedAddCommGroup.toNorm
Prod.toNormβ€”of_isBigOWith
Asymptotics.IsBigOWith.prod_left_same
forall_isBigOWith
prod_left_fst πŸ“–β€”Asymptotics.IsLittleO
Prod.toNorm
SeminormedAddCommGroup.toNorm
β€”β€”Asymptotics.IsBigO.trans_isLittleO
Asymptotics.isBigO_fst_prod
prod_left_snd πŸ“–β€”Asymptotics.IsLittleO
Prod.toNorm
SeminormedAddCommGroup.toNorm
β€”β€”Asymptotics.IsBigO.trans_isLittleO
Asymptotics.isBigO_snd_prod
prod_rightl πŸ“–mathematicalAsymptotics.IsLittleO
SeminormedAddCommGroup.toNorm
Prod.toNormβ€”of_isBigOWith
Asymptotics.IsBigOWith.prod_rightl
forall_isBigOWith
LT.lt.le
prod_rightr πŸ“–mathematicalAsymptotics.IsLittleO
SeminormedAddCommGroup.toNorm
Prod.toNormβ€”of_isBigOWith
Asymptotics.IsBigOWith.prod_rightr
forall_isBigOWith
LT.lt.le
sub πŸ“–mathematicalAsymptotics.IsLittleO
SeminormedAddCommGroup.toNorm
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
SeminormedAddGroup.toAddGroup
SeminormedAddCommGroup.toSeminormedAddGroup
β€”sub_eq_add_neg
add
neg_left
sub_iff_left πŸ“–mathematicalAsymptotics.IsLittleO
SeminormedAddCommGroup.toNorm
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
SeminormedAddGroup.toAddGroup
SeminormedAddCommGroup.toSeminormedAddGroup
β€”congr
add
sub_add_cancel
sub
sub_iff_right πŸ“–mathematicalAsymptotics.IsLittleO
SeminormedAddCommGroup.toNorm
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
SeminormedAddGroup.toAddGroup
SeminormedAddCommGroup.toSeminormedAddGroup
β€”congr
sub
sub_sub_self
sum πŸ“–mathematicalAsymptotics.IsLittleO
SeminormedAddCommGroup.toNorm
Finset.sum
AddCommGroup.toAddCommMonoid
SeminormedAddCommGroup.toAddCommGroup
β€”Finset.sum_induction
add
Asymptotics.isLittleO_zero
sup πŸ“–mathematicalAsymptotics.IsLittleOFilter
Filter.instSup
β€”of_isBigOWith
Asymptotics.IsBigOWith.sup
forall_isBigOWith
trans πŸ“–β€”Asymptotics.IsLittleOβ€”β€”trans_isBigOWith
isBigOWith
one_pos
Real.instZeroLEOneClass
NeZero.charZero_one
IsStrictOrderedRing.toCharZero
Real.instIsStrictOrderedRing
trans_eventuallyEq πŸ“–β€”Asymptotics.IsLittleO
Filter.EventuallyEq
β€”β€”congr'
Filter.EventuallyEq.rfl
trans_isBigO πŸ“–β€”Asymptotics.IsLittleO
Asymptotics.IsBigO
SeminormedAddCommGroup.toNorm
β€”β€”Asymptotics.IsBigO.exists_pos
trans_isBigOWith
trans_isBigOWith πŸ“–β€”Asymptotics.IsLittleO
Asymptotics.IsBigOWith
Real
Real.instLT
Real.instZero
β€”β€”Asymptotics.IsLittleO_def
div_pos
PosMulReflectLE.toPosMulReflectLT
PosMulStrictMono.toPosMulReflectLE
IsStrictOrderedRing.toPosMulStrictMono
Real.instIsStrictOrderedRing
Asymptotics.IsBigOWith.congr_const
Asymptotics.IsBigOWith.trans
LT.lt.le
div_mul_cancelβ‚€
LT.lt.ne'
trans_le πŸ“–β€”Asymptotics.IsLittleO
Real
Real.instLE
Norm.norm
β€”β€”trans_isBigOWith
Asymptotics.isBigOWith_of_le
zero_lt_one
Real.instZeroLEOneClass
NeZero.charZero_one
IsStrictOrderedRing.toCharZero
Real.instIsStrictOrderedRing
triangle πŸ“–β€”Asymptotics.IsLittleO
SeminormedAddCommGroup.toNorm
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
SeminormedAddGroup.toAddGroup
SeminormedAddCommGroup.toSeminormedAddGroup
β€”β€”congr_left
add
sub_add_sub_cancel

Asymptotics.IsTheta

Theorems

NameKindAssumesProvesValidatesDepends On
isBigO πŸ“–mathematicalAsymptotics.IsThetaAsymptotics.IsBigOβ€”β€”
isBigO_symm πŸ“–mathematicalAsymptotics.IsThetaAsymptotics.IsBigOβ€”β€”

Filter.Eventually

Theorems

NameKindAssumesProvesValidatesDepends On
isBigO πŸ“–mathematicalFilter.Eventually
Real
Real.instLE
Norm.norm
Asymptotics.IsBigO
Real.norm
β€”Asymptotics.IsBigO.of_norm_eventuallyLE
trans_isBigO πŸ“–β€”Filter.Eventually
Real
Real.instLE
Norm.norm
SeminormedAddCommGroup.toNorm
Asymptotics.IsBigO
β€”β€”Asymptotics.IsBigO.trans
Asymptotics.IsBigO.of_bound'

Filter.EventuallyEq

Theorems

NameKindAssumesProvesValidatesDepends On
isBigO πŸ“–mathematicalFilter.EventuallyEqAsymptotics.IsBigOβ€”trans_isBigO
Asymptotics.isBigO_refl
trans_isBigO πŸ“–β€”Filter.EventuallyEq
Asymptotics.IsBigO
β€”β€”Asymptotics.IsBigO.congr'
symm
rfl
trans_isLittleO πŸ“–β€”Filter.EventuallyEq
Asymptotics.IsLittleO
β€”β€”Asymptotics.IsLittleO.congr'
symm
rfl

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