rw obtain rw exact intro obtain rw refine rwa


ext refine obtain rw rw intro obtain obtain refine rw specialize rw refine exact refine exact


simp obtain obtain replace clear let obtain apply obtain exact have rw exact have let have have calc conv_rhs refine simp intro by_cases have rw exact rw exact calc apply apply simpa rw calc simp inv_one,


rw simp obtain simp obtain let refine rw exact exact exact simp rw change rw rw exact intro apply rw


classical induction empty insert intro have have rw by_cases have have simp simp rw exact rw refine exact intro exact exact


have by_contra rw exact have rw refine rwa rwa


classical refine intro by_cases rw exact convert simp grind simp intro simp cases


classical refine intro obtain exact rintro let have rw sum_ite_of_true, intros simp_all refine rintro by_cases simp simp_rw affineCombination_eq_linear_combination add_smul, rw simp intro simp intro simp rintro exact


refine intro use simp rintro rw refine rintro rw rcases apply_rules simp intro rw rcases rintro exact


constructor simp rintro refine simpa rintro exact


classical refine intro rw refine intros split_ifs exacts rw rintro rw refine rintro apply_rules simp rintro exact


classical refine rw rintro simp refine rw exact apply simp exact exact


rw obtain obtain have refine ext simp_rw simp_rw


rw ext refine rintro by_cases rw exact rw have rw exact obtain refine intro specialize rw exact


classical refine simp_rw rintro let have replace hs.eq_zero_of_sum_eq_zero_subtype simp have refine simp exact all_goals rwa rintro simp_intro simp


classical cases wlog rw refine split_ifs exact simp subst simp choose refine mem_convexHull_of_exists_fintype rw exact ext calc rw simp_rw congr exact congr calc rw mul_one] simp


obtain let have hSclos.closure_subset_iff.2 have fun_prop simpa


obtain exact obtain have refine by_contra simpa hv


RieszExtension.step

obtain obtain set set suffices simpa refine rcases rw exact rintro have rw replace rwa refine refine replace rw exact rintro rcases rcases simp rw rcases have rwa one_smul, replace rwa mul_assoc, subst simp apply exact have replace rwa


set have intro clear have have refine rintro have directedOn_image.2 rcases have convert exact obtain refine contrapose have let rcases exact


let smul_mem calc add_mem set have simpa have rintro refine simp obtain replace refine simp calc


obtain have use constructor rintro specialize rwa one_smul, have hinner rw have rw contrapose rwa rw one_smul] calc


set have obtain simp have have have let replace have mem_ball_iff_norm.2 calc let let have have calc x rw calc ε have simp field_simp linear_combination rw gcongr exact exact refine simp_rw exact


rw simp forall_apply_eq_imp_iff₂] obtain rw exact


refine rintro refine have simp filter_upwards refine rw calc have simp not_false_eq_true, cancel_denoms rwa


tfae_have tfae_have obtain exact tfae_have tfae_have have have exact obtain set refine simp obtain refine have simp sub_smul] let have have calc f hf.le_max_of_mem_segment tfae_have tfae_have h, obtain obtain simp obtain exists_lipschitzOnWith_of_isBounded isBounded_iff_forall_norm_le.2 exact tfae_finish


by_cases apply rcases obtain apply exact exact intro rw apply exact obtain apply refine exact intro rw apply exact refine simp have apply exact rw exact linarith


have refine rw exact rintro convert dsimp rw exact


have rw obtain simp_rw simp have have suffices apply rintro obtain exact


intro rcases simp_rw refine csInf_le_of_le have rw obtain rw obtain exact rintro rw exact


intro rcases simp_rw refine le_csSup_of_le have rw obtain rw obtain exact rintro exact


have rw obtain rw refine rw obtain exact rintro refine rw obtain exact rintro exact


intro rcases suffices simp or_false] exact exact simp suffices simp or_false] exact rw exact


classical constructor rintro rw simp exact rcases simp intro refine ext specialize rw exact rintro exact


rw intro rcases exact obtain grw exact apply intro rcases exact


rw rintro rcases simp refine rw refine simp


refine rcases simp rw enorm_inv refine rw


refine refine have choose obtain have rcases obtain use simp rcases exact obtain by_cases exact have have have rcases refine simp refine rw intro exact


letI rcases rw apply simpa simpa rcases have simp refine simp rcases calc egauge


obtain refine rw rintro rw have contradiction exact


IsExposed.inter

rintro obtain obtain refine rintro exact rintro refine exact exact


IsExposed.sInter

classical induction empty insert rw obtain rw exact exact


IsExposed.isExtreme

refine obtain have have have refine rw exact


rintro dsimp rw rcases exact have have rcases rcases suffices calc ya


intro let have have have have simpa have rw exact have refine exact simpa


ext refine rintro ext exact exact rintro refine rw exact rintro refine rw exact


classical ext simp refine rintro rw rintro obtain rw exact rintro obtain rintro ext exact


rw refine constructor rw rintro exacts intro rcases exacts


refine and_congr_right obtain rw subst simp_rw obtain rw subst simp_rw exact


rintro refine calc f add_lt_add_of_lt_of_le


constructor rintro refine simpa rintro refine rw simp_rw exact


constructor rintro refine simp exact rintro refine rw simp_rw exact


ext simp_rw refine have rw obtain suffices rw refine rw exact linarith


refine obtain obtain calc gauge gauge_le_of_mem rw exact


rw obtain refine rintro refine have refine rw exact dsimp rw


obtain rw rw congr ext simp_rw constructor rintro simp_rw rw have refine rwa inv_inv] rintro rw rw have refine rw exact


obtain rw ext rw congr ext simp_rw constructor rintro simp_rw refine rwa rintro rw refine rw


rw obtain rw rw congr ext refine rw exact


intro have rw have filter_upwards exact rcases exact


have filter_upwards apply rw exact refine exact


have refine rw have inter_mem filter_upwards constructor rw calc gauge calc gauge


ext obtain rw by_contra have have refine rw exact refine rintro rw rw exact have refine rw exact


rcases simp rw div_eq_inv_mul] simp_rw exact


rcases rw apply rw exact suffices refine exact filter_upwards rw refine exact


have refine rcases rw nth_rewrite simp exact exact


rsuffices refine simp_rw rcases rcases set apply simp refine calc simp_rw calc simp_rw


constructor rintro rw exact rintro suffices rw exact exact rintro exact


refine rw exact by_contra refine rw exact


constructor rintro refine rw exact intro rw exact


rw constructor rintro exact rintro by_contra exact


refine rwa have rintro rw refine rw exact rw simp_rw obtain rw refine suffices refine rwa exact


borelize rcases haveI obtain rcases exact rw have filter_upwards apply exact set have tendsto_integral_approxOn_of_measurable refine exact simp_rw rw finiteness simp intro apply exact


refine refine simp by_cases rw by_cases rw rw have rw rw


have hs.convex.set_average_mem refine rintro exact hs.openSegment_subset


have hg.convexOn.set_average_mem_epigraph hgi.integrableOn refine rintro rcases rw average_pair simp rw calc g hg.2 gcongr exacts


rcases have simp exact by_cases simp rcases simp haveI replace simpa


obtain simp haveI let have rw φ.injective.preimage_image]


ext simp refine rintro obtain exact rintro obtain exact by_contra obtain exact


Set.Nonempty.intrinsicInterior

haveI obtain let rw exact hscv.affine_preimage


classical obtain have let have have have simp clear_value subst simp have have let have calc f congrm match_scalars gcongr


constructor obtain simp intro have intro apply calc p intro have intro simp rw


set obtain have refine rw centerMass_filter_ne_zero] exact rw exact


Convex.convexJoin

simp rintro hpq rcases rcases refine linear_combination rw module


let rsuffices obtain refine rwa by_contra obtain obtain have rw exact refine obtain exact refine sInter_subset_of_mem haveI rw refine obtain exacts


apply by_contra obtain obtain have obtain obtain linarith hy.1.2


rintro have have obtain refine have rw have exact


hyperplane, rcases refine exact frontier_subset_closure.trans rw rcases suffices suffices let have refine rw exact refine set refine have suffices rw exact intro replace suffices rwa set have refine rw have refine refine nhdsWithin_le_nhds simp


ConvexOn.smul'

refine dsimp refine add_nonneg calc exact simp simp rw smul_eq_mul, add_comm


ConcaveOn.smul'

refine dsimp refine calc a simp_rw simp exact simp rw smul_eq_mul, add_comm


choose obtain mem_iUnion.2 refine hf.continuous_finsum_smul exact


refine obtain have rw exact rw obtain rw have trans exact rw gcongr exact rw have trans exact rw gcongr exact rw obtain rw have trans exact rw gcongr exact rw have trans exact rw gcongr exact have rw exact


rw push_neg obtain let let let have simpa have rcases exact sum_pos have centerMass_of_sum_add_sum_eq_zero simpa refine exact rw refine centerMass_mem_convexHull_of_nonpos linarith exact


classical obtain exact generalize rw induction base succ hk let apply exact intro exact simp lia dimension have rw exact simp lia lies obtain use apply intro let hull suffices by_cases exact apply rwa show intro rw rintro rw apply apply simp suffices simp constructor exact assumption apply


classical rcases simp have apply exact have rw apply exact intro exact intro simpa


apply intro have by_cases exact have have apply simp rwa obtain apply exact


classical rw refine helly_theorem_compact intro let rw apply simp apply assumption


apply intro obtain exists_superset_subset_encard_eq apply have let rw apply simpa rwa


simp true_and] rintro refine rintro rw rintro rw exact


intro rw rcases rcases have have rw have obtain simp


apply simp suffices simp nontriviality rw aesop


constructor rintro refine rw rintro rw exact


constructor rintro use simp rintro refine rw


constructor rintro use rw rintro have refine rw


rcases simp rintro rcases right left have refine simp left rfl right right have simp refine simp sub_sub_sub_cancel_right]


rintro obtain rw exact rw rw refine rw rw sub_mul,


refine obtain obtain rw rw exact exact rintro obtain rw rwa exact


refine rcases rw rcases rcases refine simp_rw exact


constructor rintro refine rwa rintro refine rwa


rcases refine rcases simp rcases simp refine rw module


constructor rintro obtain rw rw exact exact


constructor rintro rw rw exact refine rw exact refine rw vsub_sub_vsub_cancel_right] rintro exact exact


constructor rintro rw rw exact refine rw exact refine rw linear_combination rintro exact exact


rw constructor rintro exact refine rwa rintro exact refine rwa


AffineSubspace.WOppSide.trans

rcases rw rcases rw refine refine rw exact


constructor rintro rw by_contra rw exact rintro exact exact


refine rcases rw rw exact rw rw exact refine have rw linear_combination rw vadd_eq_add, exact exact


refine rcases rw have rintro simp have rw s.direction.smul_mem_iff rwa


ext simp_rw constructor rw rintro rw exact rw refine simp refine rw rintro exact


ext simp_rw constructor rw rintro rw exact rw exact refine rw rintro exact


ext simp_rw constructor rw rintro rw exact rw refine simp refine rw neg_vsub_eq_vsub_rev, rintro exact


ext simp_rw constructor rw rintro rw exact rw exact refine rw neg_vsub_eq_vsub_rev, rintro exact


obtain haveI by_cases simp have exact rw refine convert


rcases rw simp exact by_cases simp exact exact


have refine obtain have refine rw let have have refine simp left rcases rw exact rw positivity


let have have have have rwa refine h'.sOppSide_of_notMem_of_mem simp rw simp


refine have refine have rcases exact exfalso rcases rw exact rw exact


refine have refine have rcases exfalso exact exact rcases rw exact rw exact


refine by_cases simp by_cases simp exact by_cases exact by_cases exact simp exact


refine by_cases simp by_cases simp exact by_cases exact by_cases exact simp rcases rw exact rw exact


refine classical have by_contra rwa convexHull_empty]


refine push_neg obtain exact rintro have rw exact


have rw have have have simp ring_nf field_simp linarith


have have rw have have have simp ring_nf field_simp linarith


have have have have have have simp ring_nf field_simp linear_combination field


rcases simp rcases rcases convert convert rw exact


ConvexOn.strictMonoOn

intro have intro refine rw exact rcases exact refine apply rw exact rw exact


apply rintro trans have have rw calc exp have rw calc exp gcongr linarith


apply intro have trans have rw have have contrapose rw simp calc log have rw have have contrapose rw simp calc y⁻¹


have rcases rwa have rcases exact have have rwa rw apply rcases rw convert rw rw gcongr exact rw convert rw rw gcongr exact


rcases rwa have have rw rcases exact exact have have rwa rw apply rcases rw convert rw rw gcongr exact rw convert rw rw gcongr exact


rcases simp rcases simp by_cases simp exact


apply intro have have trans have rw sub_lt_comm, rpow_sub neg_sub, apply rw exact rw exact have rw lt_sub_iff_add_lt', mul_div_assoc, div_self apply rw exact rw exact


rcases induction zero succ rw rw mul_assoc] refine rcases have convert convert exact


apply exact intro rw rw refine norm_cast refine rw norm_cast fin_cases


simp rcases rw refine refine rw have convert mul_ne_zero nth_rw field


apply exact rw exact div_neg_of_neg_of_pos


have have let have refine exact have refine simp exact


refine let let let let have have exact_mod_cast


refine intro obtain rw rwa obtain rw rwa exact


refine intro obtain rw rwa obtain rw rwa obtain rwa exact


refine rw contrapose refine calc b rw gcongr


Set.OrdConnected.starConvex

intro obtain refine calc x calc a refine calc y calc a


classical rintro refine rw exact simp exact


rw obtain obtain rw rw refine obtain obtain rw rw exact obtain have refine positivity positivity rw classical let let have rintro fin_cases exact exact exact have trans simp linear_combination have convert rw trans module congr simp


let obtain zorn_subset_nonempty s obtain refine suffices obtain obtain refine not_disjoint_segment_convexHull_triple simp rintro by_contra suffices rw exact rw refine rwa


classical obtain simp_rw split_ifs exact exact refine unfold exact


classical induction empty insert intro have have by_cases have have exfalso rcases exact exact rw by_cases simp grind by_cases have refine have grw simp exact simp by_cases grind exfalso obtain have have have grind exact strictConvex_iff_div.1 hzi


have rw rcases rw exact rw exact rw rcases simp_rw exact


rcases exact by_cases simp have have rw exact by_cases simp have have rw exact


have rw dist_eq_norm_vsub', obtain obtain simp rw obtain rw have rw rw


refine rw rcases use rwa sub_eq_iff_eq_add.2


refine simp mem_sphere_zero_iff_norm] intro rcases exact


simp rcases rw have have combo_mem_ball_of_ne rwa smul_inv_smul₀ div_lt_one


Convex.strictConvex'

refine intro by_cases exact by_cases exact rcases refine exacts


refine intro rcases have obtain apply nth_rw exact


intro have refine rw homothety_eq_lineMap] exact


constructor rintro have rw exact rintro rcases rw exact exact


rcases rw exact replace conv rw exact diff_singleton_eventually_mem_nhds_left


rw intro obtain obtain obtain obtain hV₁) use simp calc convexHull


have obtain set refine obtain rw exact obtain rw exact have have rintro rw have rintro nth_rw rwa inv_mul_cancel₀ set set have calc ε gcongr exact exact have rw exact calc have rw exact gcongr exacts suffices exact


obtain exact obtain refine rw have simp_rw div_le_div_iff_of_pos_right exact


classical obtain convert rw rw sum_eq_zero_iff_of_nonneg refine rw rintro replace replace have have refine hw sbtw_lineMap_iff.2 hx).symm, have refine refine calc have rw simp_rw exact simp_rw match_scalars refine intro positivity rw exact exact


classical obtain rw exact subset_convexHull


let have have let have obtain refine rintro replace replace rw exact


calc push_cast refine gcongr exact suffices span simp affineSpan_insert_zero]


refine apply_rules have refine refine simp simp


BddAbove.convolutionExistsAt'

rw rw set have filter_upwards refine apply_rules refine rwa have rw refine rw exact


HasCompactSupport.convolutionExistsAt

let let apply isClosed_closure.measurableSet have apply exact convert v.measurableEmbedding).1 ext simp


BddAbove.convolutionExistsAt

refine simp_rw have hmg.mono_ac apply exact


intro by_contra apply simp_rw rw convert ext rcases rw rw exact


have by_cases have rw specialize rw rw rw simp_rw simp_rw


have refine swap rw refine rintro refine rwa have intro have rw specialize rw refine exact rw simp_rw rfl simp_rw simp_rw rw refine rw refine have intro exact refine rw


simp_rw rw simp_rw intro have obtain dsimp have have filter_upwards have have intro refine exact have refine ring


let have refine simpa have apply simpa have intro refine have have rw filter_upwards rintro exact obtain have obtain have apply simp obtain obtain obtain refine exact exact obtain refine have by_cases have apply refine simp rw simpa have rwa have rw simpa have filter_upwards rintro refine apply exact have have apply have have apply have apply intro simpa set have obtain obtain obtain rcases refine exact letI let have filter_upwards rw have apply rw apply rw refine apply rw exact simp apply intro exact rw exact have rw exact have dist filter_upwards apply have apply apply rw exact have have have ext rw exact exact exact


everything let borelize let let let let let let let let let let let let have have have refine intro simp exact exact apply apply rintro simpa hp have apply apply rintro simpa have apply rintro simp simp rw rfl exact simp_rw exact


have have have have let apply obtain obtain exists_seq_strictAnti_tendsto let obtain have intro rcases exists_contMDiff_support_eq_eq_one_iff exact choose have rw let have have rw exact have intro filter_upwards rw refine have have rw exact exact have simp have filter_upwards by_cases have simp simp_rw apply suffices obtain rw simpa filter_upwards rw contrapose exact exact have refine apply simpa simpa


have rw let change haveI refine exact specialize h rw rw simp_rw exact rw apply rintro


have apply intro simp rw exact exact exact filter_upwards simpa


have congrm by_cases simp simp by_cases simp push_neg simp


induction zero rfl succ have have calc congr exact


obtain simp simp_rw refine refine rw exact


let have refine suffices simp_rw refine rw exact


rw simp norm_cast rw push_cast rw have grw gcongr move_mul gcongr grw apply exact


SchwartzMap.bilinLeftCLM

refine mkCLM rintro rcases use intro have simp_rw have grw rw move_mul simp_rw gcongr rw have simp_rw rw refine simp move_mul gcongr swap norm_cast exact specialize grw move_mul gcongr rw change refine rw move_mul gcongr simp


SchwartzMap.compCLM

refine rintro rcases rcases have refine specialize rw exact let use intro let have rw gcongr rw refine nth_rw gcongr apply simp have have rw change exact have refine rw have gcongr exact refine simp have have pow_le_pow_left₀ grw have rw ring rw have rw grw


SchwartzMap.compCLMOfAntilipschitz

refine calc rw apply gcongr exact gcongr exact simp gcongr exact


SchwartzMap.integralCLM

refine rcases let use refine have intro rw simpa apply unfold rw apply


intro refine obtain use have filter_upwards simp simp eLpNorm_congr_ae grw simp


SchwartzMap.fourierTransformCLM

refine intro simp rw exact exact intro simp intro exact rintro refine apply le_top simp gcongr calc f gcongr simp apply simp rw gcongr apply have simp apply apply have simp lia apply


choose use intro refine rw gcongr simp exact


rcases set have simp_rw exact rcases simp exact


Function.HasTemperateGrowth.comp'

refine obtain obtain have set use intro have intro calc rw simp apply intro grw simp_rw move_mul gcongr simp grind have intro calc apply grind calc norm_iteratedFDeriv_comp_le


Function.HasTemperateGrowth.add

rw refine obtain obtain use rw have exact


Function.HasTemperateGrowth.sum

classical induction empty insert obtain simpa


rw constructor apply intro rcases rcases use have refine simp_rw refine


open set have rintro simp exact have contDiffOn_fun_id.rpow_const_of_ne have apply intro obtain have have have field_simp grind use intro have rw iteratedDeriv_eq_iterate, gcongr have apply have have simp apply by_cases have rw exact have calc the rw congr simpa simp rw gcongr exact rw norm_num gcongr apply gcongr rw norm_num gcongr grind


have have rw simp ring rw rcases gcongr apply apply positivity linarith simp linarith calc x rw simp apply exact exact


rw apply filter_upwards exact filter_upwards exact


cases top coe cases inl inr have have rcases let have use suffices rw rw simp refine rw exact refine rw gcongr simp


rw intro rw apply exact intro have have calc


apply refine intro refine ext1 rintro refine ext1 exact


convert span_fourier_closure_eq_top rw unfold rw simp


rw intro rw simp_rw split_ifs simp_rw have rw have convert


ext rw congr apply simp filter_upwards rw


simp_rw have apply_mod_cast simp have have rw simp rwa exact


haveI rw have convert simp nth_rw rw simp apply filter_upwards simp


rw have simp_rw conv rw have conv rw have rw congr conv_lhs rw ring ring


apply refine ext simp rintro refine ext simp


classical intro rw obtain refine dsimp rw contrapose exact


rw intro simp split_ifs simpa integral_const rw obtain apply simpa


contrapose obtain obtain use simp rw simp grind


simp obtain let let refine ext1 simp star_sum, rw simp


intro obtain contrapose exact use simp exists_exists_and_eq_and, exact


apply rw constructor filter_upwards exact have apply filter_upwards convert ext simp_rw rw congr convert rfl


obtain classical have let have circleEquivComplex.injective.comp mkZModAux_injective.comp exact


refine rintro exact classical by_contra let have have simp if_neg exact


have have simp_rw exact refine have simp_rw one_smul] exact


apply exact exact exact filter_upwards fun_prop


rw have apply change have refine exact have hg.1.comp_fst.prodMk have apply filter_upwards simp convert simp


by_cases calc congr apply exact simp congr apply apply exact


by_cases calc congr apply exact congr congr rw simp smul_eq_mul, simp congr apply have exact


have convert ext simp ofRealCLM_apply, ring


let let let have hf have have refine refine fun_prop have filter_upwards rw exact have have ae_of_all exact


ext let 𝐞 suffices rw simpa coe_smul, have rw simp have convert simp exact exact exact exact


thanks thanks iterated argues multilinear the simp_rw let compContinuousLinearMapLRight have have apply simp apply simp gcongr refine exact have rw apply simp gcongr exact exact have n.descFactorial apply rcases rw ring simp have rw norm_smul simp abs_of_nonneg gcongr apply calc k.choose k.choose gcongr exact congr ring gcongr rw calc n.descFactorial exact simp


rw congr rw intro simpa intro have μ apply simp exact apply filter_upwards refine apply intro positivity simpa


rw apply have simp exact rw apply filter_upwards exact filter_upwards apply intro apply intro positivity simp exact


have have simp rwa rcases simp apply conv_rhs gcongr exact rcases simp positivity rw apply conv_rhs simp gcongr exact


have let have suffices simpa convert hf apply rw map_smul] exact rw convert erw simp_rw neg_mul, rfl


ext have simpa have simp rw simp conj_trivial,


ext have convert simp have rw fourier_continuousMultilinearMap_apply congr suffices simp


ext have intro rw exact change simp_rw simp


have have have simpa convert rw congr simp ring simp have apply filter_upwards have simpa apply filter_upwards congr rw congr simp simp


let have have intro simp_rw have intro have simpa calc fourierCoeff simp_rw simp_rw refine convert exact simp simp_rw suffices apply convert simp_rw simp_rw exact funext neK rw congr rw congr push_cast ring


let have convert exact convert simpa zero_add] simp_rw


have rw obtain rw apply linarith obtain simp obtain refine rw have rw rw refine have rw convert apply exact


have convert ext1 have have simp rw refine rw rintro rw refine rw exact


obtain rw have intro rw rintro refine simp_rw exact simp_rw constructor refine simp_rw refine simp_rw


refine suffices simp_rw exact let obtain let have suffices rw exact simp_rw obtain have simpa let replace refine rw exact obtain refine have contrapose exact have rw div_div, have rw this, refine simp_rw have refine dsimp simp rw rw refine rw refine rw refine refine exact exact rw have simp_rw simp_rw refine rw div_div] exact have have rw refine rw mul_lt_mul_iff_right₀ refine rw exact


by_cases convert apply rwa refine obtain refine rw have refine simp_rw integral_sub exact replace rw refine simp


rw simp_rw rw apply let have ext1 congr rw rw exact


let have borelize let let have convert w suffices simpa simp


ext1 simp rw simp have split_ifs simp exact have simp


innermost but additional and calc simp_rw congr push_cast ring rw simp change simp refine exact lmarginal_mono simp set have hf.lmarginal have let have calc clear_value congr simp_rw ring_nf have intro rw exact simp_rw rw exact gcongr apply exact intros exact simp ring exact exact congr rw refine have simp exact rw rw ring exact exact congr ring congr rw congr rw rfl rw simp exact


rw intro convert rw rw rfl rw exact refine gcongr suffices rw norm_cast gcongr have rwa


have have have have have have rw convert field_simp simp


classical we have have linarith calc congr rw field_simp congr simp_rw norm_cast apply fun_prop gcongr calc apply exact exact gcongr exact calc rw


an and set let have have let have have let have have have have simp_rw have rw simp let have have have calc refine borelize exact congr apply exact gcongr rw apply rw fun_prop rw congr rw fun_prop rw rw exact


have rw eLpNorm_nnreal_pow_eq_lintegral exact


by_cases simp set let have have rwa exact_mod_cast have have have have have rcases convert suffices rw simp norm_cast simp_rw simp field_simp simp let have have have have have simp_rw rw simp let have have rwa exact have rw simp have rw have simp have by_cases rw have refine rw exact have have have let have have set have calc rw simp rw gcongr simp_rw exact dsimp fun_prop gcongr convert simp borelize fun_prop fun_prop ring calc eLpNorm rwa suffices rw congr norm_cast simp_rw refold_let rw left positivity


finite-dimensional let let let let let let have have have have calc rw calc eLpNorm apply exact exact simp_rw push_cast simp_rw


by_cases simp let have rw simp gcongr have have norm_cast rwa dsimp have simp gcongr positivity positivity set let calc eLpNorm convert rw simp rw norm_cast rw simpa have apply exact rel simp_rw


hofer

by_contra have intro rw positivity replace intro have simpa haveI choose let have exact hF have intro induction hz hi have rw let range calc d rw simp field_simp have refine exact exact obtain have refine simpa obtain have let suffices have calc apply exact have exact


refine refine rw exact nth_rw refine rw exact


rw apply all_goals refine simp rintro simp_all


refine calc rw refine have calc re calc simp_rw


refine suffices rw simp_rw have calc exact


rw intro have rw symm convert rw intro rw exact simpa


tfae_have refine have rw simp letI erw rw rw rwa tfae_have tfae_have rintro tfae_have tfae_finish


constructor intro have have refine simpa rintro simp exact


have rw constructor have rw simp conv_lhs field


constructor intro have have refine exact rintro exact


let calc congr rw simp simp tauto simp_rw simp rw


let map_zero map_add have rw change exact


constructor rintro refine simp rintro obtain refine simp ext1 exact


suffices rw rw linarith refine by_cases simp obtain positivity have convert field_simp rw rw ring field_simp rw ring


let suffices rw simp have ext simp change conv_lhs rw exact rw refine have have simp exact exact


apply refine intro simp rw conv_rhs exact


simp apply exacts toEuclidean.injective.ne


suffices rcases exact rw exact clear intro revert apply intro simp rw by_cases simp rw rwa intro simp right rcases rw exact exact


rw simp set convert exact apply rintro have have simp


intro rw simp_rw refine let exact termination_by


ext rw trans congr apply intro rw suffices apply rw exact rw rintro apply exact simp


by_contra have rw apply intro simp apply rw exact have rw simpa apply simp


unfold constructor simp intro simp inv_eq_zero, repeat exact


refine span_eq_span simp by_cases simp refine exact


rw suffices apply rw exact rw rintro by_cases rw rw have rintro exact exact


have funext simp rw linarith exact rw apply by_cases exact rw exact analyticAt_harmonicAt_log_normSq


intro obtain have simp apply exact exact


rcases refine intro by_cases refine calc have simp


haveI rw rw intro rcases obtain rw apply calc positivity exact


constructor intro ext rw simp simp rintro simp


have simp have calc a have calc b ring_nf have have refine simp_rw exact


by_contra rw have rw intro rw refine simp exact simp


simp forall_exists_index, constructor rintro rw specialize dsimp simp exact intro refine intro simp


ext simp forall_apply_eq_imp_iff] constructor rintro rw intro simp_rw have apply use rintro rw use apply rintro rw


apply rw apply intro simp zero_sub, apply rintro exact


that to apply exact clear rw refine simpa have simp


by_cases simp have rw calc ext rw ring_nf refine convert simp apply convert simp_rw ring_nf


simp rw by_cases simp have have have convert linear_combination have convert linear_combination rw ring


unfold have have have have have have have have apply_fun simp abel_nf linear_combination


by_cases rw simpa intro have rw have have rw linear_combination


constructor intro dsimp congr intro rw simp


cases classical have simp_rw simp_rw rw exact


rcases classical have simp_rw let simp_rw apply convert exact


rcases refine simp simp let have have symm rw rw


cases refine rw b.abs_det_adjustToOrientation]


rcases refine haveI have let have rw apply intro positivity intro convert simp


rcases refine haveI have let have rw by_cases obtain rw congr ext have simp rw field positivity


rcases refine let have have rw exact rw rw simp


rcases refine have convert symm rwa rw ext simp


refine have comap_map_eq_of_injective simpa


constructor intro split_ifs simp exact intro constructor intro have have have rwa intro simpa


classical rw rintro obtain have rw exact


have classical have intro simp norm_cast suffices rw simp rw simp simp sub_right_inj] simp


use apply intro simp refine use rintro exact


let let refine suffices intro simp convert intro trans simp congr


calc gcongr exact simp congr rw cases simp exact


classical ext obtain simp obtain simp simp


obtain rw have let let have refine simpa simpa simp


have have obtain have refine exact have have rw exact obtain use intro simp


LinearIsometry.extend

let let have have calc finrank simp exact let haveI haveI let let let have intro have simp have have have simp apply apply rw simp rw simp exact norm_map


refine specialize have conv_lhs rw exact simp_rw sq_le_sq, obtain simpa


refine rw simp_rw by_cases simp apply exact


rw have have orthogonalProjectionFn_inner_eq_zero have have rw rwa


by_cases swap rw exact let let have ext simp reflection_mem_subspace_eq_self, rw one_mul,


induction zero refine have rwa symm ext have simpa succ let have by_cases obtain exact haveI obtain have intro rw exact have intro exact hv let let let have intro change rw have intro apply rw refine rw exact have apply have rw exact reflection_reflection have change have finrank_lt_finrank_of_lt have have lia obtain refine rw have dsimp rwa


rw intro rcases apply intro rcases exact


induction mem hx refine rw exact zero add simp_rw exact


induction zero of x simp_rw obtain rw rw exact add simp_rw exact


rw constructor contrapose rintro let have have have intro have have simpa have contradiction refine rintro rcases simp exact have intro exact rintro rcases rintro have rw simpa rintro rcases simpa have intro apply simpa exact simp rintro refine intro refine intro have obtain rw simp exact


rw rw have constructor refine convert rintro rw


letI constructor intro let let let have have have have calc simp rw apply exacts have rw simp rw rw simp change rw have rw have ring rw exact by_cases rw have have linarith exact have by_contra rw let have calc θ have calc linarith intro apply apply intro apply have calc rw refine exact have rw change apply use rintro exact


letI let constructor intro have intro apply simp symm calc im intro have intro rw exact exact


set suffices apply simpa have have apply rw rw exact convert simp abel


ext rw let have constructor rintro exact exact


ext constructor obtain intro have have simpa simp intro rw exact


refine cases let have suffices simpa intro obtain have rw exact rw obtain refine rw refine change exact


have rw infer_instance convert rw trivial


ext constructor rintro let have refine simp rw exact rintro exact


have convert ext simp obtain refine apply simp simp change apply simpa


obtain by_cases simp by_cases have refine apply simpa have set convert simpa


convert have have rw refine intro dsimp have simp rw gcongr exact


haveI let obtain have have obtain have have have have simpa exact


refine exact simp suffices replace exact


rintro by_cases simp have rw refine simpa


suffices simpa congr_arg intro have simp


let let simp_rw change have have simp_rw exact have exact have rw exact simpa


suffices simpa congr_arg intro simp_rw apply intro rw


rw intro contrapose rw rw set exact isUnit_of_forall_le_norm_inner_map calc rw one_apply, mul_comm] simpa


classical calc simp congr congr apply simp


constructor rintro simpa rintro by_cases subst have subst exact simpa exact


rw let have have have intro dsimp split_ifs have have simpa convert refine simp refine simp simp simp


classical simp constructor intro obtain use intro rw refine have simp have rw exact have linarith intro have obtain use intro refine have rw have convert have rw simp exact have convert have rw simp exact linarith


OrthogonalFamily.independent

classical apply refine rw intro rw ext suffices calc simpa


constructor intro apply intro rw rw simp simp rw simp rw norm_num ring


LinearMap.IsSymmetric.continuous

refine rw have intro rw refine simp_rw rw refine rw exact


classical have conv_lhs simp conv_lhs simp simp


in obtain obtain obtain obtain have rw simpa have rw


by_cases simp apply intro fin_cases simp simp


by_cases simp constructor let let suffices rw simp inner_add_right, zero_add, exact rintro have have have exact intro obtain simp and_true] positivity


trans apply simp rw linear_combination simp rw linear_combination norm_cast


rw linear_combination positivity positivity


refine simp_rw convert ext split_ifs subst simp


classical refine rintro refine intro rw refine intro refine exact apply refine rintro use exact exact


Submodule.isHilbertSumOrthogonal

have intro cases first exact assumption refine refine rw refine exact


classical suffices b.repr.symm.toContinuousLinearEquiv rw have exact ext apply letI have lp.single_smul rw rw exact


classical rw rintro refine intro simp refine rintro refine exact


ext simp conv conv rw simp simpa fun_prop


refine rintro refine exact hs_balanced.interior rintro exact


intro obtain apply have have refine rw ring_nf


refine refine refine convert rw refine rw rcases rw refine balanced_iInter₂ convex_iInter₂ have have rw exact


rintro obtain simp_rw rw intro have rw exact have rwa


IsClosed.balancedCore

by_cases rw refine refine have rw exact have contrapose exact rw exact


obtain have continuous_smul.tendsto simpa h.basis_left obtain have refine rw rintro refine rw exact


obtain rw simp calc a smul_set_mono hs rw exact


rw by_contra have filter_upwards rw push_neg rcases rcases refine rw exact rcases refine filter_upwards


Bornology.IsVonNBounded.closure

intro rcases specialize filter_upwards grw apply grw


TotallyBounded.isVonNBounded

letI rw intro have have simp_rw rcases rcases refine rw have refine exact haveI exact


intro refine refine rw mul_div_cancel_right₀


refine rcases rcases specialize rw exact


rw constructor intro rcases rcases exact rintro exact


obtain have rcases simp have refine intro use exact intro simpa simp_rw by_contra rcases choose have apply intro simpa have specialize simp have have have exact this.congr simp refine filter_upwards


ext constructor intro obtain contrapose rw simpa intro simp


obtain rcases exact obtain rcases exact exact


let have exists_extension_of_le_sublinear obtain have rw have refine refine φ.continuous_of_nonzero_on_open change rw specialize linarith rintro obtain rw simp obtain exact rw exact


obtain exact obtain exact let let have have have intro rw exact obtain have intro have simp linarith refine rw refine exact rintro simp exact


obtain exact obtain exact obtain refine rw refine rintro simp_rw exact


obtain exact obtain exact obtain obtain obtain have exact hf₂


let constructor exact continuous_zero intro refine simp_rw have simp_rw simpa


letI letI letI suffices simp_rw rw have have topologicalAddGroup_iInf have have simp_rw conv conv rw nhds_iInf, rw


letI let have have simp_rw comap_norm_nhds_zero rfl rw constructor rw rcases exact exact


refine le_antisymm obtain have obtain let cont have have refine rintro simpa exact


rw closedConvexHull_eq_closure_convexHull] congr refine


suffices have rw simpa he'.continuousOn.image_closure exact


have simp have have simpa ext conv_rhs simp refine simpa simpa


classical rcases rcases use refine rw ball_finset_sup_eq_iInter exact


rcases rw by_cases rw use exact refine simp preimage_const_of_mem,


refine rw intro refine rw exact rw rw rintro rcases rw exact


obtain clear refine classical induction empty insert simp exact


classical obtain exact choose use have intro refine exact refine simp_rw refine rw


rw constructor intro simp specialize rcases obtain specialize rw refine intro specialize exact intro rcases rw rcases simp_rw refine exact


rw constructor intro convert rw intro by_cases choose have rcases exact refine refine simp exact simp exact


simp_rw congrm constructor rintro use grind rintro use grind


we refine apply exact apply exact rintro apply ext refine boundedess obtain have have rwa apply apply change apply simpa fact which for rcases suffices rcases rcases simp simp have have rcases ext simp simpa rw obtain have have rw exact apply simpa


have refine rcases rw refine change simp_rw exact


refine intro refine rw rfl intro refine rw rfl


refine intro refine rw rfl intro refine rw exact


have rcases rcases have refine by_cases rw exact refine refine rwa


rcases rcases clear let let have rcases exact


WithSeminorms.toLocallyConvexSpace

have apply rw exact intro change simp_rw rcases exact


rw simp constructor rintro rw intro ext simpa


ext by_cases simp dotProduct_comm] rfl simp rw rfl


simp_rw congr refine obtain rw rw exact refine rw


simp_rw calc refine gcongr exact simp_rw rfl


rw rw refine cases simp classical let have specialize rw replace dsimp simp_rw exact


simp_rw let rw rw dsimp simp_rw suffices intro exact


simp_rw rw gcongr rw simpa nnnorm_inner_le_nnnorm


classical refine obtain simp_rw star_star, rw


rw simp hA.isHermitian.det_eq_prod_eigenvalues, grind


refine simp rintro exact_mod_cast rw intro simpa


refine obtain simp_rw dotProduct_star_self_pos_iff] contrapose rw hv,


classical have contrapose have rwa map_zero] obtain exact


by_cases rcases rw exact rw exact have simp convert rcases simp exact rcases simp rw


convert rw refine rw simp_rw exact simp_rw exact


by_cases rcases rw constructor intro rw intro apply apply exact intro convert exact rw exact have have simp convert apply simp constructor rw apply intro simp rw apply intro simp


have have by_contra simp rw simp simp rwa simp_all


constructor intro by_contra rw exact intro rw apply rintro have by_contra apply apply simp


have geom_mean_le_arith_mean_weighted have have norm_num rw apply have rw gcongr exact rw


have simp set nth_rw simp_rw convert rw refine rw exact simp_rw exact


by_cases refine repeat rcases simp rw coe_rpow_of_nonneg coe_add, exact


obtain exact obtain calc congr rw let let suffices simp_rw rwa exact refine simp_rw simp_rw


obtain simp calc congr rw simp have have simp


have intro refine rw exact rw exact have refine rintro exact have exact


rcases simp let have have have rw ring simpa


constructor use obtain simp have have refine simp simp div_le_iff₀ rw simpa rintro apply simpa


rcases simp have have simp rcases rw exact


have have intro rw refine refine exacts have refine rintro exact have refine rw exact


have obtain have have refine simpa simpa


simp_rw rw gcongr exact any_goals exact exact


lift lift lift lift beta_reduce norm_cast obtain refine norm_cast


by_cases replace simpa sum_eq_zero_iff_of_nonneg] have simp by_cases rcases replace simpa not_or] have simp convert refine simp


obtain simp have have by_cases replace simpa have simp by_cases rcases replace simpa have coe_le_coe.2 rw simp_rw sum_congr simp convert obtain simp rw simpa convert exact


rcases simp let have have have rw ring simpa rpow_one,


by_cases rcases have replace simpa have push_cast convert refine simp


have by_cases simp have have have rw have have nth_rw rwa mul_inv_cancel₀


have have have have simp refine rw grind intro have haveI intro rw exact have intro specialize rwa simp_rw refine have rw refine refine exact rwa


have by_cases rw rw exact obtain lift lift simpa


refine rw refine dsimp rw cpow_add mul_div_cancel₀


obtain simp_rw obtain have refine refine refine exact have refine have rw norm_of_nonneg gcongr exact refine exact


obtain obtain have rw min_eq_left rw refine refine exact


have refine simp simp_rw refine simp_rw congr abel


set set obtain obtain obtain exact let have refine refine refine exact have have apply refine hfc.continuousOn_smul exact refine exact have refine simp_rw gcongr rw rcases refine rw rw have rwa refine rw rw have rwa have simp_rw suffices refine all_goals intro obtain obtain refine simp_rw refine refine exact refine simp refine simp have refine have have convert rw ring exact have simpa


have have have simp_rw smul_eq_mul, rw refine rw ofReal_one,


obtain rcases left filter_upwards rw exact right filter_upwards exact


MeromorphicAt.add

rcases rcases refine have simp_rw rw exact


MeromorphicAt.smul

rcases rcases refine convert simp module


MeromorphicAt.prod

classical induction empty rw exact insert rw exact


MeromorphicAt.sum

classical induction empty simp exact insert rw exact


MeromorphicAt.congr

rcases refine have refine rw filter_upwards rcases simp rw


MeromorphicAt.inv

rcases by_cases refine rw filter_upwards rw obtain have refine filter_upwards rcases simp replace simp rw eq_inv_smul_iff₀ simp


obtain have filter_upwards simp have apply filter_upwards simp


obtain apply refine simp rw apply intro rw have apply intro simp


refine filter_upwards match_scalars simp refine exact


obtain by_cases refine filter_upwards grind obtain obtain set have refine filter_upwards simp rw


intro apply simp_rw filter_upwards simp tauto


intro apply simp_rw have apply use filter_upwards simp tauto


Meromorphic.measurable

set have have have have exact h₃.restrict.measurable


ext by_cases simp congr apply simp_rw filter_upwards simp tauto simp


by_cases ext by_cases simp congr apply simp_rw have apply use filter_upwards simp tauto simp simp


classical ext simp ite_eq_right_iff, tauto


intro by_cases simp simp by_cases simp rw apply


classical ext by_cases simp by_cases simp simp


ext by_cases suffices simpa by_cases suffices calc suffices rwa calc meromorphicOrderAt simp


ext rw simp intro contrapose simp_all simp


Function.FactorizedRational.analyticAt

apply intro by_cases apply rwa apply rwa


by_cases simp have rw have rw congr have rw intro by_cases simp_all


classical by_cases intro rw right use simp rw rw exact


classical rw use simp filter_upwards simp


intro contrapose simp by_cases rw simp simp_all


have rw finprod_eq_prod_of_mulSupport_subset apply intro rw by_cases rw simp_all grind


classical rw have intro contrapose simp_all rw intro congr apply by_contra simp_all


classical rw have intro by_cases rw simp_all simp_all rw have intro contrapose simp_all rw apply intro rw by_cases simp rw


let have let have refine rw divisor_smul neg_add_cancel] intro simpa intro rw meromorphicOrderAt_congr meromorphicOrderAt_smul simp lift rw simp filter_upwards self_mem_codiscreteWithin unfold simp toMeromorphicNFAt_eq_self.2 rw rwa


have ext rw simp constructor obtain have simp rcases assumption simp linarith simp simp linarith simp_all by_cases rw rw filter_upwards simp filter_upwards rw rw finsum_eq_sum_of_support_subset have intro rw rw apply rwa simp rw simp


have rw t₀.meromorphicTrailingCoeffAt_smul simp


constructor rintro simp have apply fun_prop have rw use exact by_cases left rw apply right use simp rintro by_cases rw tauto right use have rw obtain use simpa right lift obtain use filter_upwards intro by_cases simp rw apply by_contra simp exact


constructor rw rcases exact by_contra exact rw simp


constructor intro have rw rwa intro rcases have tauto have by_contra have simp tauto simp apply use simp exact


constructor intro have by_cases rw have have exact apply let rw let rw simp rw exact


constructor rintro exact exact rintro exact exact


rcases left filter_upwards simp_all right use constructor simp filter_upwards intro simp rw


toMeromorphicNFAt

by_cases classical apply by_cases rw exact exact exact


by_cases by_cases have apply exact simp apply rw exact lift obtain right use apply simp split_ifs obtain apply apply filter_upwards simp_all simp simp exact


MeromorphicNFAt.inv

rcases left filter_upwards simp right use filter_upwards simp ring


constructor intro rw have simp assumption by_cases simp exact simp


ext constructor simp_all simp constructor exact rw have simp simp_all


toMeromorphicNFOn

by_cases intro by_cases exact exact exact


constructor rw apply ext by_cases simp rw simp


by_cases simp contrapose exact simp by_cases simp rw filter_upwards rwa obtain simp contrapose rw rw apply filter_upwards rw


simp by_cases rw rw refine apply rw apply rw filter_upwards rwa exact obtain rw obtain replace rw filter_upwards rwa add_sub_assoc, exact


have simp obtain have rcases have simp apply have refine have simpa filter_upwards simpa exact apply filter_upwards


have cases top apply filter_upwards coe rcases lift have have simp rw have apply filter_upwards


rcases simp simp intro rcases apply rw exact apply exact


rcases simp simp intro rcases obtain apply exact apply rw exact


by_cases have contrapose exact simp by_cases rw rw exact obtain obtain rw use exact


MeromorphicAt.analyticAt

cases top it have apply rw filter_upwards coe would shows continuity rcases have apply exact lift have apply filter_upwards exact


cases top simp filter_upwards coe cases top simp filter_upwards coe rw obtain obtain use filter_upwards simp


induction simp rw exact apply hn simp cases aesop norm_cast simp ring


by_cases simp rw exact apply by_cases simp rw filter_upwards intro simp obtain rw use constructor simp_all filter_upwards intro rw congr rw


by_cases have contrapose simpa simp by_cases rw rw filter_upwards simp lift apply obtain use rw filter_upwards intro simp ring


by_cases rw filter_upwards simp by_cases simp rw filter_upwards simp lift lift obtain obtain let let have apply apply apply have filter_upwards simp_all have have rw exact


by_cases have contrapose simpa simp by_cases rw filter_upwards simp lift lift obtain obtain rw refine apply apply simpa filter_upwards simp_all


constructor rw intro rcases rw tauto obtain use constructor intro push by_cases rwa rw apply rw use mem_singleton_iff.not.2 exact apply intro conv arg rw simp rw obtain use simp intro simp by_cases rw tauto use constructor exact apply exact


constructor intro have rcases intro have rw tauto obtain have rw tauto intro obtain use


and analytic. have have exact filter_upwards exact


have rw simp refine filter_upwards filter_upwards


rw intro rw rcases filter_upwards suffices obtain exact rw obtain use exact filter_upwards simp not_exists, rcases simp exact


rcases rw rw rw exact simp have rw set rw obtain have have rw rw meromorphicOrderAt_zpow, fun_prop


have rw fun_prop by_cases simp apply rw apply unfold simp obtain apply rw filter_upwards rw rwa simp_all


have rw fun_prop have simp use exact simp_all


cases have simp haveI have ext simp interior_Ici,


obtain obtain obtain let have refine simp rw abs_div, exact have div_ne_zero classical let refine intro rintro by_cases simp exact rw


refine obtain suffices rw refine lift exact


have rw map_prod_eq_map₂, change simp_rw exact


refine intro have simp_rw choose have rw refine rw let have rw have rw exact


have have rw rw have have have field_simp exact


classical wlog obtain use simpa subst obtain exists_affineBasis_of_finiteDimensional set have simpa b.centroid_mem_interior_convexHull set have simpa obtain set have have positivity have set have have simp simpa refine simpa calc convexHull


classical rcases rcases rcases rcases hs₂.elim_finite_subcover_image lift refine all_goals grw intros gcongr simpa grw


set set haveI have refine rintro calc dist rw set set have intro dsimp rw have rintro constructor simp set have apply rintro simp exact replace refine exact


set have have have simp rwa pointReflection_self,


rw rcases apply cases nth_rw apply cases rw apply cases rw nth_rw apply cases


let simp intro split_ifs simp simp simp simp, simp intro split_ifs simp simp simp simp⟩ simp_rw convert simp split_ifs simp simp


refine rcases refine ext simpa rcases refine ext simpa


Affine.Simplex.Regular.equilateral

refine have by_cases rw rcases nth_rw simp_rw simp convert rw rcases nth_rw simp_rw simp convert exact rw simp


refine h.dist_ne h.dist_ne intro simp simp fin_cases simp simp


refine simp refine intro have clear decide rcases rcases rfl exact exact rw rw rw


by_cases obtain rw simp_rw have rintro exact rw


have apply simp_rw div_self apply apply exact simp


classical induction empty insert ih rw ih refine exact


intro apply have refine simpa have exact


refine simp_rw let suffices convert simp congr ext rw exact have exact


suffices refine rw intro filter_upwards simp refine suffices refine filter_upwards exact exact


induction zero succ obtain obtain have simpa mul_le_mul


set have refine norm_eq_of_isMinOn_of_forall_le dsimp rw show simp le_div_iff₀ obtain sub_dvd_pow_sub_pow grw rw apply_fun simp rw exact


obtain set rcases exact by_contra have rw replace rw norm_cast rw linarith


induction zero succ rw obtain obtain rw exact mul_le_mul and is


set have positivity refine have simp rw let have have have nth_rewrite exact have grind obtain rw grw rw apply_fun rw rw exact


simp_rw refine rw refine rw obtain suffices refine filter_upwards rw refine specialize simp gcongr simp exact suffices refine filter_upwards calc simp simp_all rw exact


obtain rcases rw exact simp refine simpa tendsto_norm_cobounded_atTop.comp


obtain suffices by_contra set have rw convert have simp exact


rw have letI calc k⁻¹ congr rw push_cast rfl rw norm_cast ring_nf


rw obtain simp have have let calc k⁻¹ congr rw norm_cast rw congr simp_rw ring


replace replace refine convert ext rw convert ext rw


rw nontriviality have letI rw have simpa


refine iSup₂_le have simpa subset_polynomial_aeval have simpa coe_mono have convert all_goals rw positivity rw


refine by_cases simp simp conv_rhs rw obtain refine refine simp refine norm_cast grw


by_cases simp let suffices rw refine rwa lt_inv_iff_lt_inv]


let have rw let have refine filter_upwards rw gcongr exact exact have simpa have simpa have rw convert simp exact rw exact


have rw exact suffices refine all_goals apply rw apply suffices rw apply simpa


have intro by_contra rw rw obtain rw apply rw refine calc rw apply simp


suffices rw have have rw diff_eq_empty.mpr exact apply suffices disjoint_of_subset_left rw apply rw apply refine exact


suffices rw simp refine obtain have simpa


rw have apply all_goals refine exact have subst exact


have intro rw exact rw refine rw obtain obtain lift rw simp replace simpa peel rw linarith


rw refine filter_upwards apply gcongr apply gcongr simpa exact nonunits.isClosed.mem_of_tendsto


rw intro induction rw simp obtain simp rw rw


rw refine rw refine rw exact nontriviality rw calc simpa norm_le_add_norm_add


refine rw refine rw exact nontriviality calc rw


constructor contrapose simp have simp_rw obtain wlog rcases rw replace rw exact obtain rw specialize simp


rw intro rcases simp have refine calc dist rw gcongr


rw intro rcases refine tendsto_inf.mpr filter_upwards have have simp linarith simp_all


by_cases exact simp_rw refine contrapose refine simp


constructor exact rcases rwa refine obtain have obtain have have intro simp rw exact obtain refine have contrapose simp simp


rcases rcases have have convert ext simp norm_inv, have exact


refine rcases simpa have simpa div_mul_cancel₀


refine rcases exact suffices rw apply simp


suffices rcases simp have rw rw clear intro simp refine rw rw constructor simpa simpa simpa eq_neg_add_iff_add_eq,


refine suffices rcases rw exact rw rw exact clear intro rcases simp suffices rw obtain replace constructor rwa rwa


rcases simp ext simp refine rintro rw inv_mul_cancel_left₀ exact


rcases rcases simp simp exact simp suffices simp intro simp rcases simp simp rintro obtain rcases rcases have linarith have linarith simp linarith rcases have linarith have linarith


obtain replace norm_cast exact conv_lhs rw contrapose simpa


rw refine rw exact norm_mul_sub_norm_div_le_two_mul


obtain rcases rw intro rcases filter_upwards refine rcases exact ε₀ calc


have rw specialize rcases use intro suffices apply simp


rw intro rcases rcases apply use intro simp apply simp


apply intro obtain simp let have ext exact obtain refine refine rw have simpa


refine obtain refine simpa obtain refine simpa


obtain obtain haveI simp_rw exact set have have simpa obtain lim_z.cauchySeq.subseq_mem set have set refine rintro change apply apply intro obtain apply


rintro by_cases rw use simp the of set have obtain controlled_sum_of_mem_closure have choose set have apply rintro calc mul_comm obtain refine have ext simp definition rw exact suffices intro have have calc calc have calc simp calc rw gcongr


refine filter_upwards have rw apply apply exact


classical refine have simp obtain rw obtain refine have apply apply simp aesop rw apply apply apply simp apply have simp_all


refine rintro rw intro rcases rcases obtain obtain have refine have calc calc


refine rcases classical refine have intro simp exact calc


cases inl rcases simpa inr rw rintro exact


fconstructor apply rintro change simp exact


constructor intro rw have rw simp rw simp intro ext simpa


rcases simpa rcases simp have have have filter_upwards exact rw intro let obtain rw classical have calc simpa have rw filter_upwards rw refine simpa rw refine refine exact refine exact


simp induction singleton cons simp refine exact exact


inhabit induction empty cons obtain by_cases refine rw exact rcases exact refine simp rw exact


rcases simp by_cases rw by_cases exact rw have refine exact exact


induction empty cons rcases simp specialize obtain suffices rw apply


let rw suffices intro dsimp rw intro simp


obtain simp_rw have intro rw refine simp rw exact


have rw specialize rcases use intro suffices apply simp_all


rw intro rw rw rcases rcases use intro aesop


rcases simpa have rw calc edist simp gcongr exact


intro rcases simp iSup_le_iff] exact have have have rw simp calc gcongr exact simp have rw


haveI induction top simp_rw refine obtain rw rw exact rw coe have rw intro rw


rcases simp have rw constructor calc edist simp gcongr simp calc edist simp gcongr simp


intro rcases simp have have have rw simp calc simp


by_cases f use simpa have have rw exact set have have rw obtain refine hε₂⟩ intro rw exact


have have obtain obtain refine have grw eLpNorm_sub_le hg₂,


intro refine obtain have use use rw le_ofReal_iff_toReal_le convert apply gcongr exact


rcases apply have exact apply have simpa exact


apply use apply intro exact apply use rintro exact


rcases apply simp apply simp exact apply simp


Memℓp.neg

rcases apply simp apply simpa apply simpa


rcases exact apply obtain use rintro by_cases simp exact apply have intro have simp exact exact apply obtain use rintro have simpa apply have refine have simpa exact change intro have simp contrapose exact


Memℓp.add

rcases apply refine simp contrapose rintro simp apply obtain obtain refine rintro exact apply let refine intro refine dsimp split_ifs simpa let simp simpa


haveI revert refine simp intro simp exact


rcases apply refine exact obtain refine rintro dsimp refine gcongr exact apply dsimp have simp_rw refine intro gcongr apply


rcases simp rcases rw simp inhabit exact rw refine exact


refine rcases ext have have tauto rcases simp have ext have simpa have have rwa have rw ext have exact


rcases simp cases simp apply simpa suffices exact apply simpa


rcases exact cases simp have have simp_rw refine have simp_rw intro exact letI have suffices clear_value rw have have simp_rw refine dsimp rw gcongr apply


haveI induction top simp refine obtain rw rw exact rw coe have rw exact intro rw


refine let have intro suffices simpa this simp have intro simp simp have rwa


have have have rw intro refine intro rw rw simp have simpa rw have exact


have rw intro rw intro refine rintro have exact


have have let have refine intro have exact refine refine intro refine gcongr


obtain have rcases apply exact have have apply exact


obtain rcases apply use rintro exact apply exact


rw intro have refine rintro refine rw intro exact


rcases use rintro calc g gcongr exact exact


LipschitzOnWith.coordinate

simp_rw constructor intro calc dist intro apply positivity intro apply


refine obtain obtain simp simp have rwa le_inv_smul_iff_of_pos


obtain obtain have filter_upwards simp intro obtain have calc gcongr exacts have apply simpa


obtain rw exact refine refine refine cases top exact coe r have refine have rw exact rw obtain refine infEDist_lt_iff.2 rw


rw rintro wlog replace rw exact specialize simp simpa have lift have replace rw homeomorphSphereProd_symm_apply_coe] filter_upwards rintro rw apply exacts exact simp


have rcases rw exact rw refine ext obtain rw mul_lt_iff_lt_one_left


but refine le_antisymm mul_le_of_forall_lt_of_nonneg calc a


NormedSpace.noncompactSpace

by_cases letI exact rcases suffices refine simp rw rwa


obtain obtain refine simp simp forall_apply_eq_imp_iff₂] have obtain have have refine calc gcongr simpa refine simpa


have obtain refine intro rcases exact let let have simp module have simp module have obtain have apply intro apply have rw apply apply apply have apply intro apply have rw apply rw apply apply convert simp obtain let simp have apply rw convert exact have apply rw convert exact exact


is rcases simpa let have intro apply apply simpa have have have apply rintro have simp intro refine intro simp simp exact simp rwa


suffices rw refine subset_antisymm have have obtain rw exact by_cases simp_all by_cases rwa set have simp grw have apply simp_all have rwa mul_one,


by_cases simp exact have have intro rw apply have rwa


apply intro rw apply intro rw rw apply rw exact rw exact


obtain obtain exact isBounded_closedBall.subset


change by_cases rcases haveI classical simp_rw refine exact rw simpa


let let have have obtain refine have apply rw exact intro have simp


rw rintro rcases have filter_upwards apply exact


LinearIndependent.eventually

cases classical simp rcases have tendsto_finset_sum simp refine replace rw push_cast exact rw refine simp refine rw gcongr exact


classical rcases exact simp_rw let cases haveI convert_to exact


set suffices intro obtain let obtain v.exists_opNorm_le have have have intro have simp intro have obtain rw specialize simpa exact choose use replace have rw specialize rwa choose set change use intro calc dist


let have haveI have have contrapose have ext simp rw exact obtain have intro rw simpa exact


have constructor intro rw simpa apply rintro exact


by_contra obtain exists_seq_norm_le_one_le_norm_sub have obtain let have intro simp calc gcongr exact apply obtain h.tendsto_subseq have obtain apply calc conv_lhs simp gcongr apply exact


rcases rcases have have simpa have exact


obtain rw exact rw rcases have exact refine rw rw


refine suffices obtain set have refine simpa clear intro have refine rw refine exact exact


rcases apply calc apply filter_upwards simp simpa convert isBigO_norm_right.mpr simp


rcases exists_extension_of_le_sublinear rw exact set refine apply refine dsimp rw apply


letI letI letI letI let reCLM.comp obtain refine have rw simp refine calc exact


by_cases obtain obtain refine simp exact


IsLprojection.commute

have refine rw have calc rw rw have simpa rw nth_rewrite rw rw refine rwa have have calc rwa change rw


classical rw have inseparable_pi.2 simpa


cases refine obtain simp obtain simp rcases have refine refine refine rw gcongr apply


have induction empty insert have have have rw apply gcongr by_cases rw simp by_cases calc rw exact simp convert simp


classical have intro calc gcongr rcases simp exact simp rw simp calc f.norm_image_sub_le_of_bound rw ring


let have replace refine refine continuousAt_of_locally_lipschitz rw have calc f.norm_image_sub_le_of_bound


refine rcases have simp use suffices simpa intro refine opNorm_le_bound f.1.bound_of_shell_of_continuous calc rcases rcases refine simp replace replace calc


refine refine rw gcongr exact obtain simp rw intro rw refine rw


simp obtain suffices refine exact refine simpa filter_upwards


simp by_cases simp haveI haveI let have simp filter_upwards have multipliable_one_add_of_summable convert simp exact


existsi rw simp intro rw intro simp exact


property not through the immediately. to it of set set set letI letI set have change simp exact set have have induction smul_tprod simp add suffices change rw exact have injectiveSeminorm simp refine rw existsi rfl refine simp rw exact


rw refine existsi simp existsi ext simp rw intro simp obtain rw exact


rw intro rw simp refine rw intro simp refine rw exact


PiTensorProduct.projectiveSeminorm

refine refine refine rfl letI exact intro letI exact le_ciInf_add_ciInf ciInf_le_of_le intro letI rw refine intro rw exact ciInf_le_of_le


letI rw unfold refine intro rw conv_lhs refine simp rw refine simp map_smul, intro rw exact


by_cases rw simp exact set have apply rw exact have have rw rw simp rw div_mul_eq_mul_div, apply exact apply


refine have convert rw simp rintro rw mul_one rw replace apply


rcases rw refine intro rcases exact set suffices have interior_mono simp intro rw simpa


rcases wlog rw rw exact rw rw abs_of_nonneg


classical obtain let have have let have simp norm_num have have obtain have by_contra have simp exact refine have calc r rw exact


have have rw simpa rcases have obtain rescale_to_shell refine set have calc simp ring


use constructor exact intro contrapose exact


have have metrizable_of_isCompact have continuous_iff_seqContinuous.mp simpa


have refine rcases rcases refine refine rwa have nonempty_interior_of_iUnion_of_closed simp rcases rcases refine rcases simp have rcases let have have rcases rcases rw rw have simp exact rcases rcases rw rw let have calc congr simp abel have calc rwa rw have calc gcongr simpa exact exact


obtain choose let have intro rw exact refine have intro induction zero succ rw apply rw gcongr let have apply calc have refine exact have let have calc have intro induction zero succ have have simp have refine rw simp refine rw refine have rw exact


ContinuousLinearMap.isOpenMap

intro rcases refine rcases rcases refine rcases have rw have calc dist apply rwa exact


choose use toFun nnnorm bound right_inv exact


refine rintro refine have ext exact rw exact


refine rw simp_rw intro rw apply simp_all


refine rcases refine suffices simpa intro refine exact rcases rcases refine simp rw exact


refine refine calc obtain simp rw refine rw calc


IsBoundedBilinearMap.continuous

refine suffices simpa apply rw refine refine exact exact rw refine apply rw exact


ContinuousLinearEquiv.isOpen

rw refine let have convert ext constructor rintro exact rintro use ext simp


IsCompactOperator.continuous

letI haveI refine rw rcases rcases rcases have suffices refine have rwa rw refine rw exact


refine obtain obtain let have obtain refine have have exact simpa


refine have choose cases let compContinuousLinearMapL have refine exact obtain refine have simpa


rcases suffices tendsto_iff_norm_sub_tendsto_zero.2 intro have refine le_of_tendsto exact


have refine have isClosed_Iic.preimage refine exact


given a by_cases by_cases exact simpa simp_rw eq_comp_toContinuousLinearMap_symm] obtain obtain obtain simp_rw smulRight_apply, exact set have have simp congr have have have have apply obtain simp simpa set set set continuous_toFun continuous_invFun change suffices simp exact


refine cases inl inr hN have obtain simp exact rw calc


have rintro exact refine rw gcongr exacts


obtain use intro refine calc exact rw exact


rcases simp lift rcases refine by_cases rw obtain exact have rw rcases simp calc


cases haveI simp refine convert simp convert ext simp haveI simp


rw refine rw calc a gcongr exact rw exact


HasSolidNorm.continuousInf

refine have norm_inf_sub_inf_le_add_norm refine convert simp


cases obtain obtain rw have refine rw exact obtain refine rintro refine simp_rw exact


refine grw apply simp ring obtain refine rw rw replace replace dsimp rw linarith


cases refine choose obtain obtain tendsto_subseq_of_bounded exact


cases refine choose haveI obtain obtain tendsto_subseq_of_bounded exact


classical suffices let apply rw have rw convert simp


simp refine rw have simp change have have linarith


obtain use constructor intro obtain exists_mem_eq_sup conv_lhs rw simp_rw have obtain simp apply obtain obtain exact apply le_trans rw exact mul_le_mul have have rw simp exact


by_cases rw rw obtain obtain have rw apply have simp exact simp rw


obtain have nth_rewrite exact tendsto_const_nhds apply simp use intro have have rw apply rw apply rw exact


obtain obtain have have rw ext exact le_antisymm


ext set set have have let have have have obtain obtain rw


refine obtain exfalso refine simpa refine simpa


RingSeminorm.isBoundedUnder

have intro rw grw apply cases inl inr use refine exact


rcases simp change have specialize rw replace rcases linarith rw linarith positivity


by_cases specialize rw have rw rw conv_lhs exact


apply by_cases rw intro rw exact intro rw apply rw by_cases have simp_rw exact rw by_cases rw exact rw apply rw


simp_rw apply refine by_cases rw rw rw have use rintro by_cases simp simp exact


refine suffices exact le_trans by_cases simp rw exact


refine rw suffices rcases exact exact by_cases simp rw div_le_div_iff_of_pos_right exact


simp_rw apply refine by_cases rw rw have use rintro by_cases simp exact rw mul_div_assoc, exact convert by_cases rw have apply specialize rw exact rw


refine rw ext simp exact rw seminormFromBounded_eq_zero_iff exact


intro simp nth_rw rw have apply cases inl have rw conv_rhs exact inr hlt have rw exact rw mul_assoc


have apply rintro rw apply have simp ext ring_nf rw simpa


rw have tendsto_mod_div_atTop_nhds_zero_nat apply have rw exact rw apply rw exact simp_rw exact


have have apply rw exact rw have le_antisymm simpa


let have rw intro obtain obtain ε have have rw set have rw exact obtain use intro specialize rw exact use intro have rw have rw apply refine suffices by_cases submultiplicativity simp nth_rw have apply rw zero_rpow mul_rpow exact have simp nth_rw have μ apply rw exact map_pow_le_pow have add_pos_of_nonneg_of_pos have have rw rpow_mul nth_rw have nth_rw nth_rw have div_pos rw rpow_mul exact have by_cases simpa exact rpow_le_rpow have have rw nth_rw rw exact have have mul_pos calc μ μ


apply simp use rintro simp rw conv_rhs have apply rw exact exact


have by_cases use simp rintro apply rw exact use simp rintro apply rw conv_rhs rw exact


by_cases use simp exact rpow_le_one use simp use intro nth_rw apply rw exact


simp rw intro simp forall_exists_index] by_cases apply intro exact rpow_le_one have nth_rw convert simp rw rw apply intro have simp obtain exact


by_cases rw calc limsup limsup apply intro simp obtain use intro apply rw exact use simp forall_exists_index] exact have have μ have apply exact simp_rw exact exact tendsto_natCast_atTop_iff.mp hψ_lim h_eq


intro have let set have have have limit obtain rw set have apply simp use intro have cast_ne_zero.mpr rw have have have limsup_mu_le have limsup_mu_le have limsup smoothingFun have limsup limsup limsup limsup_mul_le have apply le_trans apply intro rw have rw split_ifs rw apply le_trans rw add_sub_cancel_left, rw apply le_trans rw add_sub_cancel_left, apply have μ smoothingFun hxy obtain apply le_trans apply simpa


intro have apply have have simp rw mul_assoc, simpa


apply simp use intro simp by_cases have apply apply rw rw have have rw mul_one_div_cancel


have rw exact apply simp use intro have simp rw rpow_natCast,


rw unfold simp suffices rw apply ext by_cases rw rpow_one] rw apply split_ifs exact exact


refine rw ext rw split_ifs rw by_cases have have rw exact have rw specialize simp rw exact exact


rw refine obtain simp rw have contrapose simp exact apply rw split_ifs apply rw exact exact


by_cases rw by_contra have have rw cast_sub have rw have simp exact rw exact have have have by_cases rw exact have rw this, gcongr exact set obtain obtain rw exact exact have have rw max_eq_left_of_lt exact exact exact


have apply split_ifs exacts apply apply by_cases have rw exact simp apply by_cases rw have rw one_div_mul_cancel have rw have rw rw have rcases rw rw rw obtain card hf_na.multiset_powerset_image_add apply have le_prod_of_submultiplicative_of_nonneg apply have have have obtain exact toFinset_nonempty.mp obtain have set have have intro obtain exact have simpa have use intro obtain simp rw split_ifs exact exact exact simp


simp have minpoly rw minpoly.algebraMap_eq simp_rw


classical have rw simp refine le_antisymm norm_root_le_spectralValue set have obtain have rw simp rw apply intro split_ifs obtain minpoly.exists_algEquiv_of_root rw apply exact


ext have set have have have rw simp assumptions


set haveI set have rw spectralNorm.eq_of_normalClosure exact


set haveI set have have algebraMap k rw spectralNorm.eq_of_normalClosure rw apply


intro set haveI set set have rw spectralNorm.eq_of_normalClosure spectralNorm.eq_of_normalClosure apply


set f smul x simp rw congr rw rfl mul_le have have rw


by_cases simp have have set have rw exact


have trans rw rw exact rw apply ext rw split_ifs have intro obtain rw rwa rw intro obtain rw exact


set have have have rw map_one, simp exact_mod_cast


by_cases subst simp convert rw simp convert simp ring


intro unfold split_ifs gcongr have gcongr


have intro apply rwa exact intro grw gcongr exact intro change convert simp exact


simp have apply intro have have rw refine rw


have intro replace push constructor rw exact have have have suffices rw convert simp apply intro convert simp intro convert simp


ext obtain use rw refine exact exact exact exact exact


ext constructor rintro exact rintro exact


rw intro apply gcongr exact have positivity


obtain have have positivity refine exact


nth_rw rw apply intro rw apply rw gcongr exact


rw integral_eq_sub_of_hasDeriv_right intro apply exact


by_cases have induction zero simp exact succ simp rw refine apply intro exact rw intro rw exact


by_cases set have have intro have rw exact rw cases inl intro rw exfalso exact inr rw intro rw exact


obtain obtain set have have rw calc gcongr rw apply apply exact gcongr rw exact let have refine half_pos apply rw mul_comm,


obtain refine apply hasDerivWithinAt_picard_Icc intro nth_rw rw


have choose set refine rw beta_reduce rw rw beta_reduce rw apply hasDerivWithinAt_picard_Icc intro nth_rw rw obtain refine simp_rw rw have apply exact


obtain choose refine rw refine rw exact grind


let have refine refine refine rw have intro have simp obtain obtain simp have intro grw by_cases exact obtain use have obtain obtain by_cases use intro obtain simp exact rw use use intro obtain grw gcongr


induction zero succ ihn suffices rw exact exacts have convert simp


induction zero succ ihn suffices rw exacts add_le_add_left have hf convert simp


rw gcongr suffices refine exact have intro rw gcongr exact exact convert simp


simp constructor replace have obtain simpa replace have simpa


have intro simp convert summable_schlomilch_iff simp


lift simp have have simp simp_rw exact_mod_cast


have intro simp convert summable_schlomilch_iff_of_nonneg two_ne_zero simp


rcases rw simp_rw rpow_mul nth_rw rw rpow_lt_one_iff simp intro positivity intro gcongr suffices have simpa intro obtain apply rw simpa


refine simp intro rw grw push_cast have field_simp linarith


suffices simp_rw refine filter_upwards have exfalso revert positivity simp_rw refine have simpa have refine filter_upwards simp apply filter_upwards rw rw


refine have by_cases simp exact rw exact refine have simp exact


refine by_cases simp rcases rw exact exact have rw rcases exact rw exact


have simp exact rw refine apply apply lia


by_cases simpa have have rcases exact exact


rw simp apply_fun simp suffices rcases simp apply intro simp rcases apply simp rw calc gcongr congr have field gcongr apply contrapose simp apply exact simp apply have calc rw simpa gcongr apply gcongr rw field_simp apply simp field


obtain exists_monic_irreducible_factor rw have exact


obtain exists_monic_irreducible_factor refine have exact


obtain all_goals have obtain rw exact rw exact exact


by_cases simpa rw simp rw congr rw apply rw apply simp apply apply rintro contrapose simp_all


by_cases simp have rw simp mahlerMeasure_def_of_ne_zero gcongr apply rw use constructor rw apply simp rw apply injOn_circleMap_of_abs_sub_le simp intro gcongr rw apply simp


by_cases simp rcases simp rw coeff_eq_esymm_roots_of_card norm_pow, esymm, apply simp_rw let have rw calc gcongr exact simp_rw norm_cast exact gcongr exact exact hx.1 calc gcongr rw simp_rw exact rw congr norm_cast simp congr exact


simp_rw constructor rintro exact rintro exact


ext simp toContinuousMapₐ_apply, constructor intro use ext simp exact intro have simp have ext simp use refine ext simp


have apply norm_num induction zero let have intro cases rw contradiction simp let have intro cases rw simp apply refine have rw convert norm_num rwa convert rw apply refine intro cases contradiction simp exact succ rw rw gcongr exact


intro classical contrapose have let have intro apply exact cases apply refine apply rw apply exact apply refine apply rw apply exact


apply rw apply apply rw convert rw exact simp norm_num


refine rw intro refine have convert exact rw grw have convert simp rw refine refine rw exact


refine have refine rw replace have refine rw


refine rcases simp simp contrapose rw exact


set set have obtain exact have obtain exact intro constructor exact replace exact


rcases rw cases inl convert exact inr convert exact exact


obtain simp rify convert simp_rw rw have field simp_rw refine simp


have induction zero succ ih rw mul_inv_cancel₀ set norm_cast rw


have rw simp simp rw rw tsum_geometric_of_lt_one push_cast have field rw apply


n obtain generalize rify refine simp fun_prop exact simp apply calc gcongr linarith


have rw focus apply apply all_goals refine rw


have rw calc π sub_lt_comm.1 calc π refine ring simp rw rw all_goals


refine have have rw div_le_div_iff₀ exact


rw let let let let let let let let let have have have have have have have have have convert simp have convert simp ring have have convert simp ring have convert_to simp_rw rw hu₁_eval_one, convert_to rw congr dsimp ring rw mul_sub, have cases zero succ simp_rw have rw congr simp_rw ring all_goals


rcases rcases exact exact refine conv rw simp refine have rw exact


rw refine intro simp rw refine rw constructor


apply rw by_contra obtain have have rwa have have have filter_upwards rw exact obtain have obtain is_integer have rw have nth_rw simp linear_combination have norm_cast lia


obtain apply exact apply exact have have have apply nth_rw exact rw use exact replace rw replace apply rw use intro rw rw have rw simp_rw norm_cast rwa


refine have rw refine have intro rw one_mul] congr simp_rw refine refine exact


rcases rw exact intro rw refine rwa mul_assoc, rw


refine apply rw suffices rcases exact rw simpa


have rcases have have refine show show rw exact show rw mul_inv_rev, le_div_iff₀ exact show have rw exact


wlog specialize rwa have rw tsub_tsub, rw show gcongr


wlog simpa rw calc gcongr rw gcongr nth_rw rw


suffices obtain exact exact intro induction zero succ


suffices obtain exact exact intro induction zero succ rw exact


refine tendsto_nhds_unique conv enter rw exact


rw set set apply have have nth_rw gcongr


bernstein.variance

convert simp field_simp rw field simp field


letI have suffices rw nhds_basis_uniformity rintro filter_upwards exact intro obtain have simpa isCompact_range have le_trans whenever obtain have rw exact this tendsto_gauge_nhds_zero positivity have filter_upwards set calc gauge simp simp calc gcongr refine simpa rw calc simp gcongr apply gcongr conv_lhs gcongr simpa simp


rw suffices by_cases have rw exact nlinarith have wlog simp nlinarith nlinarith


refine contrapose obtain exact obtain exact exact


refine rintro simp wlog have rw exact obtain exact have calc strictConcaveOn_log_Ioi.2


refine rw simp fun_prop rw simp fun_prop


apply have filter_upwards gcongr linarith apply


apply have apply have have intro simp_all convert exact have filter_upwards gcongr exact


have have ext ring rw apply exact apply simp filter_upwards intros apply simp_all linarith simp_all linarith


have have rw exact apply simp filter_upwards intros apply simp_all linarith simp_all linarith


simp by_cases p p obtain suffices refine rw rw simp have have simp ring apply simp exact filter_upwards have rw simp_all cases intro have cases


intro apply exact intro have have have simp have linarith simp rw simp rw apply simp_all have have convert simp linarith exact


intro apply exact intro have have have simp rw simp rw apply simp_all linarith have ring_nf simp have have have calc rwa nlinarith exact


induction zero succ simp_rw refine have rw simpa


AnalyticWithinAt.cpow

have filter_upwards intro simp apply exact


unfold rw exp_zero, have contrapose rw exact have contrapose rw exact have rw exact rw show


refine rw norm_cast rcases left rw rw norm_cast change rwa exact


have unfold rw conv_lhs enter rw rw show all_goals rwa rwa


have have rw rw have have simp_rw exact have have rw rw exact


have simp_rw replace dsimp simp_rw refine dsimp convert rw zero_mul, show congr rw neg_neg,


rw split_ifs rw field_simp simp field rw field_simp simp field rw field_simp simp field


simp simp mul_div_cancel_left₀ by_cases rw exacts rw rcases replace have rw refine have rw exact replace have have rw exact


have rcases simp rcases rw rw have push_cast rwa


by_cases simp constructor intro rw simp obtain rintro rw simp


by_cases constructor intro rw simp obtain rintro rw


by_cases constructor intro rw simp obtain rintro rw simp


simp_rw rcases simp simp simp simp simp simp simp simp simp


rw constructor rintro refine convert ring rw exp_mul_I_periodic.sub_zsmul_eq] rintro exact


rcases simp simp rcases simp rw exacts simp rw exact


rcases simp simp rcases simp rw exact simp rw exacts


rcases simp rw rcases rw simp simp rw simp simp


rcases rw rw rcases rw two_mul, exact rw zero_add] rw


suffices h_forall_nhds.mono refine exact


suffices refine have filter_upwards rw convert continuous_norm.continuousWithinAt π simp lift simpa


have have filter_upwards rw refine refine continuous_norm.continuousWithinAt tendsto_const_nhds lift simpa rw


by_cases exact rw refine suffices rwa have rw exact rw replace push_neg refine rw exact


intro induction induction simp_rw obtain rw use field_simp linarith


have let refine refine convert e.isQuotientMap.comp isQuotientMap_quotient_mk ext convert ext simp


rw obtain simp norm_cast simp abs_of_pos simp norm_cast simp


constructor intro rcases use rw have rw rintro exact


refine refine rw exact rw refine by_cases rcases simp exact simp


convert tendsto_const_nhds) simp lift simpa


refine refine refine exact have refine exact


induction zero succ rw simp refine simp have simp_rw convert simp convert ring


convert have intro rw simp exact simp_rw simp


have let let have intro rw exact hasDerivAt_log_sub_logTaylor have simp exact have convert simp simp unfold simp simp_rw intervalIntegral.integral_const_mul, pow_succ', gcongr calc refine filter_upwards rw gcongr exact rw simp


rw use filter_upwards rw have apply rw gcongr


apply have rw linarith linarith have gcongr rw linarith rw gcongr simp linarith


refine refine intro simp rcases simp conv gcongr norm_cast rw conv rw refine have have rw cases zero succ refine rw gcongr exact linarith exact refine convert exact


of fact monotone let let let have refine refine rw intro by_cases have exact simp eventually have filter_upwards have intro simp rw cfc_sub refine gcongr grind


by_cases simp rw refine simp obtain simp have simp


rw refine rw rw have conv_lhs refine rw rw simpa


refine rw apply_rules all_goals refine first apply apply


by_cases simp refine obtain simp simp exact


by_cases obtain all_goals simp rw congr ext simp simp


by_cases simp cases inl inr have have rw


unfold cases inl inr refine rw gcongr linarith


by_cases have calc simp rw simp push have simp


let refine simp refine refine exact exact intro simp


refine refine refine exact refine apply rw rw linarith exact refine rw exact


refine refine exact refine refine apply rw refine linarith refine rw exact


obtain simp suffices rwa exact simpa simpa simpa simp rw exact have simp integral_smul_const] rw


let property rw exact refine unfold rw exact simp_rw


have have intro simp positivity calc refine refine refine refine have fun_prop congr refine exact


obtain refine nontriviality have let let have let have refine grind have filter_upwards intro have unfold rw refine exact have rw exact have filter_upwards simp refine exact calc a rw exact


intro calc rw gcongr unfold simp refine all_goals


obtain have refine refine filter_upwards exact


intro by_cases have have rw obtain exact simp_all simp_all have simp


let change cases inl simp_rw exact inr simp intro by_cases have simp have simp


refine obtain have simp norm_cast aesop lift lift simp exact


rw intro obtain refine rw refine obtain refine intros obtain simp contrapose exact


have have calc rw simpa simpa gcongr rw exact congr rw ring gcongr refine simp linarith


refine rintro split_ifs simpa have have simp rw refine norm_cast gcongr refine norm_cast linarith exact exact


refine refine split_ifs simp refine have simpa refine intro simp rw congr split_ifs simp have hl₁ obtain simpa rw simp_rw rw simp field_simp intros split_ifs simp have hl₁ obtain simpa exact


apply refine exact refine rintro exact refine intro simp rw simpa simp eq_sub_iff_add_eq] exact simp rw congr ring


have by_cases have dsimp rw simp ring have simp one_div] rw simp ring


let add_mem zero_mem neg_mem have rw rintro ext exact ext exact exact


delta cases zero succ split_ifs trans rw simp rw simp_rw congr split_ifs simp dsimp exact have simp


obtain obtain refine have simpa simpa let rw apply simpa dsimp have have simpa have refine intro split_ifs simp have obtain simpa obtain simp calc simp gcongr norm_cast exact


trans delta congr split_ifs simp simpa trans simp simp_rw apply exact


constructor apply convert simp simp intro simpa exact intro replace have simp convert rw smul_eq_mul]


trans rw refine have suffices by_cases simpa fun_prop simp filter_upwards have simp norm_num


filter_upwards by_cases simp replace simp simp field


refine suffices rw rw simpa rw refine by_cases simpa by_cases simp by_cases exact rw simp_rw simp iteratedDeriv_const, iteratedDeriv_weierstrassPExcept_self] obtain simp simp


by_cases lift have rw convert ext simp have have have fun_prop apply filter_upwards simp_all


have have rw exact have intro have rw exact calc ring


rcases simpa refine rw refine convert simp


UniformContinuousOn.cexp

have rw intro simp have have rw obtain refine intro have rw have rw ring_nf rw have simp apply simp


refine have have obtain eventually_atTop.1 simp refine rw have rw calc x


have intro convert ext simp rcases exact convert ext field_simp rw rw


rw have refine filter_upwards simp exact


have rw suffices exp refine rw exact have filter_upwards rw ring


have rw suffices apply rw have rw exact exact have rw exact have this.eventually filter_upwards have rw mul_sub_left_distrib,


rw constructor rw refine refine exact refine refine intro exact


constructor refine apply apply intro have exact rw refine apply filter_upwards rw simp


rw refine dsimp rw cpow_def_of_ne_zero conj_ofReal,


have rw conv_rhs refine intro conv_rhs rw simp


have ring rw constructor refine apply apply intro refine exact rw simp_rw refine rw filter_upwards rw simp exact


rw have x intro have have have simpa exact simpa have have have apply_fun rw intervalIntegral.integral_neg, rw have ring rw rw rw contrapose


suffices refine apply have apply intro rw ring_nf refine suffices simpa rw have refine intro rw exact


induction zero simp dsimp rw simp succ dsimp have rw rw rw


rcases simp dsimp rw rintro rw exact dsimp have have rw rw rw rw


have induction zero succ rw refine rw have simp refine rw convert by_cases apply exact_mod_cast rw lia linarith


induction zero succ have rw apply have rw contrapose exact


suffices intro induction zero succ intro rw dsimp rw suffices rw


rw have rw intro rw exact rw rw exact refine exact exact


suffices apply swap exact rw exact intro induction zero intro refine rwa succ intro have apply intro specialize contrapose rw rw push_cast ring rw linarith rw exact simpa


apply refine rwa apply intro rw apply norm_cast exact


have rw have rw rw intervalIntegral.integral_comp_div simp_rw refine rw congr rw rw mul_cpow_ofReal_nonneg push_cast rw


have integral_posConvolution simp_rw have rw conv_int, refine rw congr push_cast suffices rw


let have have have refine refine rw refine rw exact have x intro have have simp exact rw have have swap simp_rw have convert ring convert ring have rw have have simp contrapose have simp contrapose rw intervalIntegral.integral_const_mul] rw convert apply convert apply convert


induction zero rw simp succ have swap rw swap rw swap rw div_div] congr push_cast


have conv_rhs rw have intervalIntegral.integral_comp_div dsimp rw intervalIntegral.integral_const_mul] swap simp_rw refine push_cast have have conv_lhs simp have rw rw


suffices rw apply rw rcases exact refine rw intro induction zero rw rw refine refine exact rwa succ rw rw have pick_goal conv rwa simp_rw exact


have rw have rw rw add_comm have intro push_cast have field simp_rw rw mul_comm exact


have by_cases rw rw obtain rw rw cases rw rw refine have convert convert have convert rw convert ext1


by_cases have conv_lhs rw rw refine specialize contrapose rwa have rw intro apply_fun rw exact have rw exact


suffices exact convert ext1 dsimp push_cast rw


suffices convert rw rw have refine refine exact have refine refine apply exact refine exact have rw exact exact refine refine refine rw cpow_neg,


let have have have positivity have mul_nonneg have have intro dsimp rw congr field have intro have have rw memLp_one_iff_integrable] apply refine dsimp rw congr exact refine refine exact continuousAt_rpow_const rw convert refine dsimp have have ring rw ring rw congr


refine have subst simp_rw push rw gcongr exact all_goals


refine have simp rw conv_rhs congr exact


induction zero succ have rw rw abel linarith


dsimp conv_rhs rw rw rw have congr push_cast abel rw ring


suffices refine rcases rwa convert abel rw intro induction zero rw exact succ intro rw rw specialize have logGammaSeq refine have rw rw ring replace have ext1 dsimp rw rw convert ring have rw rw ring


suffices log_injOn_pos intro have rw exact intro rw ring


have have swap rw rw le_sub_iff_add_le', refine rw refine rw


convert convexOn_Gamma.strictMonoOn symm rw exact


intro have have have have rw log_mul ring_nf


refine convert simpa ext1 simp rw refine convert convexOn_log_Gamma.comp_affineMap change rw div_self ext1 change rw simpa


have refine rw exact eventually_of_mem have rw refine exact convert rw ring_nf


have rw intro specialize simp_all have intro specialize simp_all rw


by_cases rw div_mul_cancel₀ rw have rcases rwa rw convert push_cast ring rw simp_rw generalize field


rw convert refine exact rw simp_rw simpa simp_rw refine rw exact


induction zero refine rw succ dsimp specialize have have intro contrapose rw simpa refine rw exact simpa


let have apply let have rw refine have exact rw exact apply intro apply


induction rw exact contrapose rw suffices suffices this.comp rw have exact


by_cases obtain rw deriv_zero_of_not_differentiableAt deriv_zero_of_not_differentiableAt mul_zero] rwa suffices rw exact refine simpa


have y| intro rw gcongr gcongr refine rw gcongr rwa apply rw conv_lhs conv_rhs conv congr rw rw intro have rcases rw rw exact rw rw exact rw refine rw simpa


refine rw simp_rw suffices exact simp_rw apply


rw simp convert simp ofReal_sum, norm_cast rw simp exact


rw simp convert simp norm_cast left rw exact simp simp_rw simp congr norm_cast rw exact


simp real_inner_comm convert congr simp ring have simp ring


rw suffices refine refine rw rw field apply refine exact


obtain have rw constructor rw refine refine exact intro rw refine exact have intro refine exact refine integrable_of_isBigO_exp_neg rw exact convert rw simp_rw convert rw


have suffices rw refine rw mul_inv_cancel_left₀] all_goals refine rw exact


rw refine rw simp apply apply exact have filter_upwards have rw apply simpa


refine by_contra have apply simp exact simpa


have have intro convert field have refine simp_rw exact convert integral_Ioi_of_hasDerivAt_of_tendsto simp reduceCtorEq]


of calc rw congr ext1 rw rw simp rw congr rw congr conv_rhs ring simp integral_mul_cexp_neg_mul_sq rw simp ring


rcases rw exact simpa refine rw sq_sqrt convert rw


let obtain have refine intro rw gcongr exact exact continuousAt_of_dominated


have apply exact refine continuousOn_of_forall_continuousAt rw exact have intro simp simp_rw coe_algebraMap] conv_rhs congr rw rw rw convert rw rw dsimp rw conv_lhs rw norm_num rw exact


have have rw suffices rw rwa have intro have simpa have have simp_rw refine apply exact


rcases rw exact rwa rw convert simp rw simp exact


rw convert rw refine dsimp have rw norm_num rw rw simp norm_num rw


rw constructor refine simp refine simp refine refine simp_rw


let have refine exact exact have have rw exact have rw refine contrapose rw have convert rwa have rw have intro simp_rw conv refine rwa convert simp rw simp


have intro convert simp left have rw exact exact


refine contrapose intro have have apply filter_upwards have simp rw exact exact


intro rcases have rw exact have rw exact


have intro convert simp have apply simpa convert simp


refine suffices refine dsimp rw refine apply_fun rw exact simp_rw refine rw refine simp_rw


suffices exact have intro rw have intro convert simp apply intro refine intro exact rw have rw refine rw simp rpow_def_of_neg


refine swap rw apply contrapose intro have have have apply filter_upwards simp rwa exact have rwa simp


suffices exact have intro rw rw apply rw intro rw dsimp rw exact refine refine rw refine rw exact intro exact rw have rw refine dsimp have rw simp


refine swap rw exact have apply intro simp rwa


apply intro apply rw apply apply exact intro norm_num simpa intro norm_num exact apply apply apply simpa


by_cases obtain hf.extract_zeros_poles have rw apply apply intro apply rw apply simp apply apply simp_all rw push_neg have apply filter_upwards simp left by_contra simp_all rw apply tauto


by_cases obtain hf.extract_zeros_poles have apply apply apply intro apply apply apply apply apply apply apply apply apply intro simp intro rw apply rw push_neg have filter_upwards simp left by_contra simp_all apply


rw have rcases apply_fun rw exact rw by_cases apply refine contrapose replace suffices rw integral_symm, ring intro apply refine rwa refine rcases rw rw contrapose exact


have cases left right have integral_cpow apply_fun simp_rw rw refine rcases exact exact rw simp_rw


rcases calc simp refine rw exact calc simp refine rw exact


have intro conv rw apply simpa rw ring fun_prop


rw intervalIntegrable_log')] abel exact intro simpa simpa exact


rcases conv rw integral_log_from_zero_of_pos ring simp exact


apply swap apply intro have have have have simp convert conv_rhs rw


have apply intro have convert simp have intro convert simp convert ring exact apply refine apply intro norm_cast exact


have intro norm_cast rw exact rw convert rw congr rw simp_rw push_cast rw exact


let have have simpa have simpa have calc rw all_goals


induction zero succ rw norm_cast field_simp simp


let have have simpa have have calc rw all_goals


calc apply apply have apply simp have apply simp filter_upwards simp_all mul_eq_zero, ring rw intervalIntegral.integral_const] simp simpa apply simpa exact simp simp smul_eq_mul, linarith


calc circleAverage apply intro simp_all congr have calc rw sub_zero, intro have calc apply rw by_contra rwa


calc have rw simp apply apply have have simp forall_const] filter_upwards simp not_false_eq_true, rw rw intervalIntegral.integral_const_mul] simp ring


simp right obtain calc simp have have simp_all apply intro simp rw congr calc rw norm_num rw simp rw ring nth_rw ring


rw intro apply rw by_contra simp_all linarith


calc circleAverage rw congr ext congr rw field apply have apply intro use simp apply intro simp_all rw filter_upwards rw simp rw circleAverage_add circleAverage_const] simp congr rcases simp rw


have have calc gcongr rw exact simp refine refine rw apply rwa


have have have rw have congr ext simp exact rw set set μ calc have refine rw exact rw exact have simp rw exact


constructor apply have lintegral_enorm_of_nonneg rw exact


rw intro use intro simp intro rw tauto exact


obtain rw by_cases have apply rw refine exact rw exact rw cases simp_all simp_all


obtain rw by_cases have apply rw exact rw refine exact rw cases simp_all simp_all


wlog exact wlog exact refine filter_upwards rw exact


refine rintro cases inl exact not_tendsto_nhds_of_tendsto_atTop inr exact


suffices intro have refine rw exact intro induction zero succ


cases inl have exact inr cases inl inr have rw exact


constructor intro rcases refine rw exact exact exact rintro


have replace replace rw have refine rw apply rw exact rw exact


induction nil cons ih simp_all sum_cons] have rw


have tendsto_const_nhds.add rw refine filter_upwards rw


refine have apply convert rw exact rcases have apply apply apply convert ext simp exact


and let let have have convert calc simp ring congr rw have calc have have gcongr exact have refine exact simp simp simpa


refine simp have congr simp have have simp ring


rw show rw simp refine suffices simp refine exact show refine calc have rw gcongr exact


set have ext dsimp rw push_cast ring_nf rw have convert ring_nf intro rw dsimp rw


have rw linarith linarith exact convert have have rw congr simp ring field_simp positivity simp


rcases simp rw rcases rw simp rw simp exact rw rcases simp simp rw_mod_cast exact rw positivity


rcases rcases simp rw have have simp norm_cast exact simp rcases rw rw have have simp norm_cast exact


intro induction rw exact induction simp simp_rw exact simp exact


induction simp induction simp simp rw simp rw simp induction simp simp rw simp simp


simp smul_eq_mul, have refine simp refine refine exact


suffices not_continuousAt_of_tendsto have exact


rcases simpa rcases simpa rcases simpa simp_all


simp intro have have have have rw calc log


simp intro have have nth_rw nth_rw rw log_rpow mul_le_mul_iff_right₀ refine simp convert rw simp field positivity simp convert rw simp field positivity gcongr


rw intro obtain exact rw simp_rw nth_rewrite have rw simp refine simpa convert simp


simp by_cases rw exact suffices refine rw filter_upwards


simp by_cases simp by_cases simp rw ring


constructor unfold intro simp exact intro have have apply simp_all


intro simp by_cases tauto right have simp linarith


classical induction empty insert calc log⁺


by_cases simp obtain calc log⁺ apply simp apply apply gcongr rw apply


have calc simp congr rw gcongr refine simp simp grind


refine intro rw intro let have have refine grind have refine grind have have have have have filter_upwards intro have calc gcongr refine grind have rw calc gcongr refine grind


classical induction empty insert rw show refine generalize rw transitivity grw rw exact


suffices this.imp suffices let have exact


classical refine obtain obtain simp let refine rw show apply refine refine exact refine simp intro exact


suffices refine simp constructor simpa simpa


rw have exp rw set have rw exact apply simp refine exact


apply intro obtain apply exact use rw use intro exact exact


apply apply exact obtain use rw use intro have apply exact apply intro simp refine mul_le_of_le_one_right apply rw refine apply apply apply rw exact apply intro apply simpa


have apply apply exact have rw exact exact


intro obtain rcases have rcases simp exact replace obtain rcases have exact specialize replace have have have rw have exact refine calc apply


let let have filter_upwards have have apply rw suffices filter_upwards have rw have simp rfl exact simp simp exact


constructor refine exact rintro exact refine exact


refine rw simp rcases simp all_goals obtain exact obtain exact


simp calc simp repeat all_goals any_goals apply push_neg exact exact


convert suffices simp_rw simp convert exact have simp_rw apply convert simp


induction zero succ intro rw simp_rw apply apply intro rw apply exact intro rwa


have intro simp exact have apply rw intro have simp simp exact


symm calc rw polarCoord.symm_image_target_eq_source] rw simp_rw exacts refine rw


rcases simp_rw rw refine rw refine rw linarith rw exact refine rw filter_upwards simp simp_rw refine exact


refine simpa tendsto_log_atTop) apply intro simp rw field_simp


obtain refine filter_upwards gcongr simpa


rw intro obtain have filter_upwards by_cases simp lift simp rw exact


apply refine rw exact refine refine simp


have simp have simp intro simp refine nth_rewrite refine refine push simp


constructor intro simp grind rintro exact exact


rcases simp rcases rw rcases rw have have rw cpow_def_of_ne_zero


suffices this.mono dsimp rw refine change rw exact


rw refine exact continuous_snd.continuousAt


rw cases inl rw refine refine exact inr rw refine refine exact


obtain dsimp obtain exact have tendsto_exp_atBot.comp have squeeze_zero_norm have rw exact simpa zero_rpow


have rw refine simp refine convert simp simp rcases refine refine rw exact


rcases have refine rwa have rw apply tauto have exact suffices refine exact have apply refine rwa fun_prop


have ext simp exact_mod_cast rw refine apply simpa fun_prop


by_cases rw convert simp lift rw convert ext1 simp


refine rcases exact simp exact obtain have simp_rw exact