ext dsimp rw congr apply rw coe_support_inter]


rintro obtain refine simp intro have rwa


refine intro simp apply rintro obtain simp refine apply rw simp simpa


cases have simp suffices ext obtain simpa intro simp


have have ext have let have rw simp simp simpa


constructor intro rw pullback.lift_fst exact intro have simp rw exact


letI let letI have simp refine simp simp


intro apply intro obtain refine have apply simp simp rw have simp exact simp simp convert rw simp eqToHom_op, exact


letI refine intro simp homOfLE_app, convert_to rw exact x simp have rw rw simp rfl have simp_rw simp simp conv_lhs rw


have have rw have simp rw rw simp_rw congr rw simp


intro obtain obtain change have have rwa obtain exists_basicOpen_le_affine_inter have b subst refine change rwa obtain obtain simp simp obtain simp


refine obtain change rw exact intro obtain obtain rw exact


have simp replace simp eqToHom_op, simp_rw simp homOfLE_leOfHom] convert exact


let have rw simp eqToIso.hom, pullback.condition, rw exact rw refine


have rw f.ker.subschemeι_app, simp rw exact


rw constructor suffices ext refine fun_prop rintro have obtain by_contra exact rw rw intro simpa intro have rw refine simp simp


obtain convert refine obtain simp apply intro exact


have refine simpa simp apply rw intro have obtain use simp


let let refine let convert intro cases simpa intro


refine apply rintro obtain rw simp simp cases rintro exact


obtain have rw rw letI have infer_instance have rw rw letI have infer_instance


have isOpenImmersion_sigmaDesc convert ext simp have infer_instance


let have have have hc.conePointUniqueUpToIso exact


have refine createsColimitOfFullyFaithfulOfIso prop let colimit preservesColimitIso let rw intro simpa


rw let preservesColimitIso rw simp exact


let let let let change have congr_arg simp


rw refine obtain let refine simpa germ_ext obtain refine obtain refine simp smul_eq_iff_of_mem simpa


constructor intro fapply convert rw ext1 refine simp


rw intro obtain rw apply intro rw refine rw exact rw exact


have have obtain rw simp apply exact constructor refine change simp exact rw rw exact simp refine convert exact


refine have have let convert apply intro obtain replace exact replace exact rintro obtain simpa simpa


apply intro by_cases obtain obtain obtain obtain obtain exact obtain refine convert simp simpa obtain rw refine convert exact


simp refine obtain use have rwa refine change rw exact


haveI rw simp intro constructor wlog intro rw intro have rw hQ.respectsIso.cancel_right_isIso] apply rw exact intro have rw exact intro rw rw exact


have have rw refine let rw hQi.cancel_right_isIso have refine have simp convert exact have simp convert exact


let rw apply rw intro have change simp let simp rw exact apply dsimp rw simp


choose apply rintro simp exact intro exact


refine rintro rw convert simp rw convert simp


refine choose rw intro rw exact rw rintro simp use


apply introv rw exact introv rintro haveI rw exact


letI rw classical intro induction basicOpen haveI have exact openCover apply intro obtain exact hU


letI apply rw intro rw exact rintro let apply rintro have refine exact rw exact


letI rw intro let have simp rw apply exact


refine intro simp_rw rw refine simp_rw exact


let suffices rwa isIso_iff_of_reflects_iso rw simpa


let let let refine rw obtain obtain let have change rw refine simp have rw simp rw simp have obtain exact have choose refine rw simp apply obtain apply rw exact


constructor intro obtain let refine exact simp change simp simp rintro infer_instance


delta apply introv rw apply introv rw apply


letI let have apply rintro dsimp convert simp ext1 simp convert ext1


letI rw rintro rw convert apply infer_instance infer_instance


let pullback.map refine change apply rw apply exact change refine rw exact


refine intro refine apply intro rw exact apply intro rw intro dsimp rw exact


apply intro simp_rw exact intro apply rw exact


have letI apply intro apply intro have obtain exact


letI apply intro rw exact intro have obtain rw simpa


apply introv wlog rw intro let let pullbackLeftPullbackSndIso pullback.congrHom pullback.congrHom have e.hom refine exact change rw exact obtain exact


obtain obtain algebraize replace hf rw refine rwa


apply intro wlog have obtain refine rw exact rw exact obtain obtain exact


apply introv wlog rw intro have pullback.fst pullback.map exact obtain apply simp


apply introv have convert exact wlog rw have pullback.fst pullback.map refine rwa obtain apply simp


wlog change rw simp_rw rw refine trans exact exact


wlog rw have have have exact obtain wlog rw have refine cases exact have obtain have obtain rw have letI letI have change exact


constructor exact have inferInstance intro simpa


wlog refine have have convert refine apply rw change rw simp obtain wlog have let rw refine have convert rw obtain obtain rw exact


wlog rw intro have have have let have exact obtain wlog have rw intro have let exact obtain obtain have algebraize have exact


rw refine wlog let rw intro rw apply refine rw exact obtain wlog have let let refine rw exact obtain obtain refine exact apply rwa


apply apply apply intro apply algebraize have exact


refine rw exact refine let let pullback.map have rw exact rwa


apply rw exact intro have have of_pullback_fst_of_descendsAlong have of_pullback_fst_of_descendsAlong have of_pullback_fst_of_descendsAlong have have isAffine_of_isAffineHom wlog rw have pullback.fst pullback.map refine change rwa obtain obtain refine of_pullback_fst_Spec_of_codescendsAlong intro rwa simp_rw


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


wlog let have let have have have of_postcomp have rw have infer_instance have have have exact


wlog rw intro have exact obtain wlog rw intro have exact obtain obtain rw algebraize let have have rwa obtain refine obtain obtain refine rw exact


wlog let let apply intro rw have comp_mem have comp_mem exact obtain obtain rw algebraize rw cancel_right_of_respectsIso infer_instance


suffices isImmersion_eq_inf apply refine intro change change simpa intro change change simp refine rw exact simp_rw exact


let let have top_le_iff.mp have have rwa


refine rw ext simp dsimp rw rw infer_instance


revert rw apply intro wlog rw exact obtain wlog intro have rw rw exact obtain obtain rw exact


refine clear wlog rw rw rw exact obtain wlog have rw exact obtain obtain rw apply algebraize exact


constructor intro obtain refine let have have have let have have rw simp rintro letI have exact


refine intro convert ext exact intro obtain rw infer_instance


refine have constructor let simp have exact


constructor intro convert rw delta exact exact


classical refine obtain let rw exact haveI refine rw


letI apply apply intro let have have exact


refine wlog change rw intro have refine exact obtain clear intro replace wlog obtain have have simp_rw exact obtain obtain exact


refine have have rw rw change rw obtain rw use


rw constructor intro intro suffices intro intro refine simp intro change rw apply clear refine simp intro change rw apply apply


delta rw constructor intro haveI haveI let have rw erw rw exact introv let have rw simp_rw exact


let have apply rintro refine exact pullback.map exact


obtain apply change simp exact change simp exact use intro rw simp rw


constructor rintro simp exact intro obtain refine simpa intro rw simp_rw generalize intro obtain refine simpa


refine rw apply exact refine exact exact simp


erw rw have congr_arg simp rw hP'.cancel_right_isIso] exact


apply introv have inferInstanceAs rw exact introv rw exact


constructor exact intro rw intro have simp rw simpa rw simp apply exact introv apply intro simp_rw have hs rwa h₁.isLocalization_away_iff]


obtain have have simpa obtain haveI haveI haveI have rw apply simp have f.appLE f.appLE simp congr apply refine rw apply rw exact


rw intro let induction basicOpen simp_rw have exact openCover apply hs rintro refine rw exact hU specialize rw exact


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


wlog rw intro have rw exact wlog rw intro have rw exact wlog rw intro have rw exact rw exact


apply letI apply intro rw wlog rw intro simp apply rw exact


wlog obtain let rw intro let have let simpa have simp rw refine rwa obtain use apply apply simp conv_rhs ext have simp obtain wlog rw intro rw exact rw exact


have refine obtain obtain simp_rw obtain apply have have obtain refine rw haveI exact obtain use simp haveI toMorphismProperty_respectsIso_iff.mp exact


haveI haveI apply infer_instance rfl intro exact rw haveI apply


have wlog rw intro refine exact wlog rw intro refine intro rw exact wlog rw rw refine rw apply use obtain obtain rw apply intro specialize rwa


wlog obtain obtain rw exact obtain wlog have refine rwa use rw have rwa obtain obtain rw rw apply


wlog rw intro have exact have rw infer_instance


rw rintro simp have hf let obtain obtain let obtain refine simp_rw change simp rw simp


rintro simp let obtain obtain let obtain refine rw simp openCoverOfBase_X, simp_rw congr apply


let have rw exact have rw intro rw infer_instance


let have intro have have obtain obtain algebraize rw convert ext1 obtain simp ext simp erw rfl simp erw rfl let let let let have simp intro simp obtain have rwa obtain hsx.symm obtain hsy.symm obtain obtain refine let refine fun_prop rintro obtain hx₁.symm obtain hx₂.symm obtain refine simp simp dsimp rw erw congr apply intro have convert apply


have rw refine intro obtain obtain exact


apply introv rw intro simp obtain rw exact introv simp_rw intro have obtain rw have have simp rwa obtain rw simp apply_fun simp use


classical let let let let let let let let have simp exact let let let have have intro suffices exact obtain let let let have show have suffices rw contrapose obtain obtain suffices intro rw cases


apply intro refine rw rfl rw universallyInjective_eq] apply intro obtain obtain rw


change wlog rw intro dsimp refine exact obtain wlog rw intro refine exact obtain obtain algebraize convert rwa apply


AlgebraicGeometry.Flat.generalizingMap

have change rw intro algebraize apply convert exact


apply intro let let let have intro apply let exact let use have intro rw iInf_subtype', apply intro congr rw exact


refine induction basicOpen have exact openCover apply intro have have have exact hU


constructor intro let have apply apply exact intro let apply rw intro apply apply exact


apply intro rw apply apply convert apply exact exact


apply intro have apply rw apply apply convert apply exact rw exact rw exact


change id wlog exact this wlog exact obtain obtain obtain have algebraize simp_all


constructor intro apply exact intro convert exact constructor rw apply intro apply convert have apply


constructor intro apply exact intro convert exact exact


apply let rw suffices intro have rw apply let convert apply


intro let let let let have have let dsimp rw exact isLocalization_basicOpen_of_qcqs change letI have have simp have have exact


rw trans congr delta generalize_proofs exact


refine let simp rw letI convert_to rw simp have rw exact rw exact inferInstanceAs


refine colimit.desc ι.app ι.naturality algebraize have simp exact intros dsimp rw congr ext apply dsimp simp


refine letI refine dsimp simp change simp


refine dsimp rw congr ext dsimp rw simp_rw


apply let have have have have let have refine eq_of_SpecMap_comp_eq_of_isAffineOpen simp simp have simp have simp have have have trans simp rw have f.normalization.presheaf.map rw simp simp rw have f.normalization.presheaf.map rw simp


have rw have exact inferInstanceAs simp infer_instance


delta refine infer_instance refine dsimp infer_instance


ext by_cases obtain simp simp constructor rintro rintro obtain rw cases


obtain obtain obtain hV'.elim_finite_subcover exact replace have rw trans simp rw exact have top_le_iff.mp simpa


have apply simpa exact have have rw infer_instance obtain obtain convert_to infer_instance


rintro have isIso_of_comp_eq_sigmaSpec rw


intro let have refine rintro exact choose have pointsPi_surjective_of_isAffine choose simp_rw let let invFun right_inv map_mul map_add refine ext have rfl rw ι_sigmaSpec


classical refine by_contra simp refine rwa


classical apply intro convert_to rw sub_zero] clear have induction mem obtain obtain rw by_cases rw exact rw exact algebraMap convert rw exact add rw exact mul hy convert rw ring


ext apply ext obtain simp erw rw refine rw use simp ring


ext1 trans rw simp rw opensRange_awayι simp letI letI exact


simp have rw congr simp subst rw basicOpenIsoSpecAway_inv_homOfLE_assoc simp congr ext simp simp congr apply ext simp


refine rintro let pullback rw trans toBasicOpenOfGlobalSections simpa homOfLE_toBasicOpenOfGlobalSections_ι simpa


apply intro obtain rw simp fromOfGlobalSections, simp simp rw rw refine change simp exact intro let obtain simpa rw simp simp


intro obtain by_cases use have exact have apply convert infer_instance exact exact let num convert rw cases contradiction simp den den_mem let num den den_mem use ext simp y0] rw use simp cases contradiction simp ring


AlgebraicGeometry.Proj.isSeparated

refine intro dsimp refine let pullback refine simp simp let pullbackAwayιIso rw let have convert rw apply simp erw congr ext exact x simp erw congr ext exact x


classical cases let valuation have let have have have refine have simp have simp letI have have isUnit_iff_ne_zero.mpr hKmax.ne let have use rintro rw have induction zero add simp exact smul hx1 simp refine rw awayMap_fromZeroRingHom change simp exists_eq, mem obtain simp have convert simp cases cases simp suffices convert rw congr ext simp use simp cases cases rw rw div_le_iff₀ one_mul, calc simp congr apply simp simp rw use simp simp_rw mul_assoc pow_mul, congr simp cases cases simp ext ring exacts rw simp_rw cases cases simp simp congr rw


trans q.asIdeal refine convert apply simp simp


obtain have have exact convert apply have rw


refine change simp_rw let rotate_left rw apply rw rw apply rw convert_to refine exacts rw change dsimp apply intro change rw split_ifs exact all_goals congr dsimp simp_rw dsimp simp_rw


revert refine rw rintro simp_rw let split_ifs convert_to dsimp rw simp_rw congr apply rw mem_tac simp simpa rw mem_tac simp simp_rw


letI constructor rintro obtain refine simp intro obtain refine val_injective convert rw simp rw intro obtain obtain obtain apply_fun simp obtain obtain refine apply_fun conv_rhs rwa coe_decompose_mul_add_of_right_mem, exacts refine val_injective simp mul_assoc] congr rw mul_assoc,


obtain change homogeneousLocalizationToStalk obtain simp rw refine change apply ext rw simp


apply rintro exact rintro rw obtain refine rw exact


refine apply intro change rw change exact


constructor rintro obtain rw obtain rw replace rw


constructor intro rw exact isReduced_of_injective


apply intro have exact isReduced_of_injective


apply intro obtain let let refine obtain apply apply


apply intro change rw induction h₁ obtain specialize rw specialize rw exact h₂ refine rintro haveI specialize rw have rw apply_fun rw exact h₃ rw replace rw rw


constructor intro rcases have haveI infer_instance haveI infer_instance


by_contra replace toNonempty simp_rw rcases rw haveI haveI haveI let have exact ext constructor rintro rcases exacts simp


constructor intro haveI have refine simp_rw by_contra obtain replace rw refine convert exact exact


constructor intro rw have apply exact exact


rw intro rw rw revert contrapose simp_rw apply simpa


constructor rintro simp obtain obtain rintro simpa


refine rw contrapose rintro obtain exact


ext constructor rintro constructor use rw simp use rw simp intro let obtain let obtain let obtain use rw simp


refine rw rw exact obtain obtain use simpa


fapply exact pullback.map intro rw simp_rw congr apply rw congr rw rw simp_rw exact


apply intro dsimp trans rw simp_rw congr rw apply simp_rw simp_rw p2] simp rw simp_rw


fconstructor exact pullback.lift exact apply simpa simp_rw apply simp_rw simp


fapply pullbackSymmetry intro dsimp apply


apply intro haveI refine inferInstance simp simp


apply pullback.map pullback.condition.symm) intro rw ext simp rfl


have suffices have let have simpa have rw simpa convert aesop exact


refine refine exact pullback.congrHom pullbackDiagonalMapIso have pullback.snd pullback.snd rw apply trans pullback.snd simp symm trans pullback.snd simp rw simp simp dsimp apply simp simp apply simp simp apply


refine obtain simp_rw refine rw convert exact


suffices have choose exact refine obtain refine suffices rw have exact


have have have have have refine change rw refine let pullbackRestrictIsoRestrict refine simp dsimp simp simp_rw congr rw simp simp simp


constructor rintro simp obtain obtain rintro congr simpa


dsimp have rw rfl rw simp congr rw simp rw


have have have spread_out_unique_of_isGermInjective obtain refine rw simp convert rw g.fromSpecStalkOfMem_restrict simpa


constructor intro obtain rw intro obtain obtain exact


dsimp refine generalize_proofs have exact


ext refine obtain obtain simp obtain exists_preimage_of_isPullback apply simp rw simp use simp rintro use rw simp


refine intro apply rw rintro use rw simp intro rw simp intro simpa


constructor rintro simp obtain obtain rintro simpa


rw apply intro simp convert refine exact


refine have rw rw erw rw erw ext1 dsimp rw exact


simp intro obtain assumption have contradiction


apply intro rw rintro obtain let intro apply exact refine rintro dsimp rw intro apply intro rw exact


refine intro exact intro simp infer_instance


apply rw yonedaEquiv_symm_app_apply] apply


ext simp constructor rintro exact rintro have refine ext simpa


ext rw constructor intro obtain rw letI letI isOver_map use rwa rintro use rwa


ext simp mem_overGrothendieckTopology] constructor intro have rw exact refine use rintro let have rwa rintro use rintro exact


rw refine rintro intro obtain refine use choose let f mem₀ rw refine letI isOver_map refine use rintro exact


obtain constructor rintro subst obtain refine rw exact rintro refine exact


rw refine simp constructor simp rintro exact rintro refine


constructor rintro let have let have rw exact obtain refine simp simp refine change rw exact refine change rw exact


haveI haveI let let change apply rintro obtain dsimp replace have obtain obtain suffices rw rw exact refine congr change congr simp refine rintro simpa


obtain obtain change replace e set replace rw clear_value obtain refine rw iterate erw rw let rw dsimp rw congr


apply intro change exact apply intro rw toPushforwardStalk] rw obtain obtain apply_fun simp have have this obtain refine rw exact


refine obtain obtain obtain obtain obtain refine replace replace simpa


refine obtain refine rw exact rw intro obtain x rw swap exact erw obtain obtain refine exact


suffices refine exact refine let have choose obtain have refine rw refine rw intro rw refine rw exact


obtain have obtain refine rw congr have suffices f.appLE rw simp congr apply rintro have rw simp simp


letI obtain choose have rw exact refine letI letI letI let let change rw rintro refine exact


obtain obtain let have rw exact let refine iV'V.trans ext change simp rfl ext apply change rw ext change rw rfl


obtain have obtain have rw simp obtain exists_lift_of_germInjective refine rw simp rw hW.isoSpec_hom,


obtain transitivity delta rw erw rw delta rw erw rw


ext constructor rintro exact rintro have rw exact


obtain have rw rw germ_stalkClosedPointTo] simpa


constructor rintro obtain obtain rintro simp


intro obtain obtain simp rw let zero_mem add_mem smul_mem suffices obtain exact rw intro contrapose rw have dsimp rw obtain exact


obtain obtain obtain replace rw obtain have have use rw swap intro rw exact apply rw


have intro let let let homOfLE apply rw simp trans on_goal convert convert all_goals ring exacts have intro obtain use rw dsimp convert let have let have refine simpa intro have obtain simp convert simp ring intro rw swap exact apply dsimp rw ring


convert apply intro rw congr change rw exact


rintro rcases refine rintro specialize refine dsimp erw rfl


intro let obtain let have rw simp_rw rfl obtain dsimp refine simp_rw exact rw simp have comap_closedPoint rw


rintro haveI haveI letI replace let obtain let let have have let have simp stalkClosedPointIso_inv, change simp simp rw erw rw simp have let have have rw have hαβ congr_arg refine simp simp rw simp simp rw simp rw


constructor intro let i₁ i₂ commSq obtain refine apply simpa simp rw rw


ext constructor intro apply apply rwa apply


intro constructor rintro ext dsimp have let let have have rw exact suffices rw conv_lhs conv_rhs simp suffices refine exact constructor let pullback.lift apply simpa simpa have have rw let let have change rw exact rw exact rw exact


dsimp simp let let rw and let apply intro simp_rw lia rintro rw exact simpa rintro simp_rw refine rintro simp simpa simp rintro dsimp simp congr simp apply rw simpa


apply intro ext simp intro apply rintro dsimp simp rfl ext rfl


dsimp apply face we zero intro simp rw swap intro rw factorThru_arrow kernelSubobject_arrow_comp, rw dsimp cases


ext1 dsimp simp equivalenceCounitIso_hom_app] simp rw congr simp simp simp simp id_comp]


induction zero succ by_cases rw congr ext simp_rw lia obtain rw symm conv_rhs let rw congr have simp rfl ext simp_rw lia


induction zero succ by_cases rw replace rcases fin_cases dsimp rw change simp rw dsimp rw simp one_mul, rw simp rw have dsimp rw add_zero, intro simp obtain rw have ext simp have hq simp lia) rw comp_zero] simp lia


rw cases exfalso apply intro fin_cases fin_cases rfl obtain rw


CategoryTheory.Idempotents.DoldKan.hε

dsimp ext1 rw ext rw dsimp toKaroubiEquivalence, rw


have rw hσ'_eq simp simp rw swap rintro suffices simp convert dsimp lia rw swap rintro rw intro replace lia dsimp rw lia simp dsimp lia simp conv_lhs congr rw rw dsimp to have intro rw apply rw exact rw simp rw apply rintro simp have rw dsimp lia generalize_proofs rw neg_zsmul] congr ring


simp rw by_cases rw simp eqToHom_refl, rw simp comp_neg, intro rw simp dsimp lia simp


intro dsimp simp by_cases rw obtain rw rcases simp rfl by_cases simp rfl have have rw swap rw simpa obtain rw swap rw dsimp lia simp rw swap ext cases dsimp dsimp lia simp


constructor rintro by_contra exact rintro exact


by_cases subst simp by_cases subst simp obtain obtain have rw by_contra simp by_contra simp lia by_cases by_cases rw simp simp


simp have subst congr exact dsimp congr exact


dsimp change rw dsimp rw dsimp rw dsimp rw rfl


ext dsimp simp conv_rhs congr apply intro rw apply


ext dsimp simp convert apply intro rw erw


unfold rw rcases rw simp erw rw neg_smul, δ_comp_σ_self rw


have subst simp unfold split_ifs rw simp


induction obtain rw exact simp rcases change subst obtain rw have by_contra exact exact simp clear subst obtain eq_comp_δ_of_not_surjective rw grind obtain eq_comp_δ_of_not_surjective rw grind by_cases subst rw simp erw simp lia simp erw by_contra exact


induction induction dsimp by_cases subst simp dsimp simp by_cases have subst simp dsimp rw simp rw rotate_left intro rw rw exact zsmul_zero] simp simp rfl rw swap by_contra exact rw exact assumption


ext have have dsimp rw dsimp rw dsimp rw simp simp


have intro have have dsimp rw rw infer_instance exact apply


refine refine ext rfl dsimp simp rintro ext have dsimp simp ext dsimp simp app_p_comp, app_comp_p,


let let let let let let let let change have have have simp congr have dsimp rw rw simp let have simp dsimp rw


induction zero simp apply succ hq simp hq by_cases exact obtain have simp have v simp rfl) simp simp


apply intro dsimp rw rw change rw have dsimp lia


constructor intro rcases dsimp rw rw have dsimp rw rw intro simp intro rw intro by_cases dsimp subst rw simp


dsimp split_ifs constructor apply rw apply constructor apply apply


ext simp split_ifs iterate linarith linarith grind


between set set set set refine all_goals


rw erw dsimp constructor slice_lhs rw slice_lhs rfl slice_lhs rw slice_lhs rfl


rw constructor intro exact intro refine rw replace have calc


rw refine lift rcases h toFun source target exact rcases exact map_zero_left map_one_left prop


have have rw infer_instance have have have have r have simpa


rintro obtain let rw suffices simp have simp infer_instance) simp


let let let let let let let let let let have have let have exact


constructor intro obtain obtain simp obtain have rw exact h h₀ have rwa h₁ have rwa intro obtain have exact


let have rw convert ext obtain simp convert ext obtain simp


obtain obtain dsimp refine rw apply simp rw simpa


revert suffices intro have rw infer_instance


rintro let rw suffices have rw infer_instance) simp


obtain let let i₀ i₁ π weakEquivalence_π have rw infer_instance have exact


obtain obtain have exact h h₀ have simp simp h₁ have simp simp


revert suffices from intro have rw infer_instance


obtain let let p₀ p₁ ι weakEquivalence_ι have rw infer_instance have exact


SSet.Quasicategory.hornFilling

cases zero succ cases zero simp simp succ


SSet.StrictSegal.quasicategory

apply intro use intro apply ext dsimp rw let obtain rw dsimp apply apply dsimp rw stdSimplex.yonedaEquiv_map, rfl rw dsimp apply apply dsimp rw stdSimplex.yonedaEquiv_map, rfl obtain obtain fin_cases the it let horn.primitiveTriangle of have ext dsimp rw apply apply ext fin_cases rw simp rw apply apply ext dsimp rw dsimp rw simp rw fin_cases


refine have c.isColimit₁.hom_ext rw exact


constructor rintro exact rintro rw obtain let let let let exact π cofan₁ cofan₂ isColimit₁ isColimit₂ hm isPushout


refine dsimp induction isMin obtain simpa succ apply simpa intro simpa isSuccLimit exact


ext dsimp rcases rcases rcases split_ifs


ext dsimp rcases rw rcases rwa rw exact rw have rw rwa


ext dsimp rcases rw rcases rw rw exact rw rwa rw have rw rwa


ext dsimp cases last cast cases zero succ rcases simp_rw rw rcases rwa rw exact simp_rw rw


ext fin_cases subst simp rfl simp subst rw rfl


have refine ext refine ext refine ext refine ext rw exact


use ext simp by_cases rw dsimp rw rw exact simp let have rw rw by_cases rw refine congr dsimp rw exact dsimp rw simp lia


simp have rcases by_cases exact refine lia rcases use apply apply exact rw exact


use ext suffices dsimp exact intro cases


obtain eq_σ_comp_of_not_injective rw grind use haveI rw infer_instance haveI rw


obtain eq_comp_δ_of_not_surjective rw grind use haveI rw infer_instance haveI rw


apply intro apply apply exact rintro exact exact


suffices rw rw apply intro induction of rcases simpa simpa id comp_of rcases simpa simpa


suffices rw top_le_iff] rw apply intro induction of rcases simpa simpa id of_comp rcases simpa simpa


induction of id comp_of rcases rw exact exact


obtain rw obtain obtain rw obtain exact exact exact obtain rw obtain rw exact


induction zero cases of id comp_of obtain obtain simpa simp succ cases of id comp_of obtain obtain simpa obtain exact obtain exact


induction obtain exact obtain cases of rcases obtain exact comp_of cases zero cases exact succ rcases obtain exact obtain exact


induction nil obtain simp cons dsimp obtain simp


induction nil obtain simp cons simp subst set have simp split_ifs rfl simp simp rfl have have simpa toSimplexCategory_map_σ, simplicialEvalσ,


induction nil cons simp split_ifs simp have simp have grind have convert grind


induction id use rfl of cases use simp cases obtain obtain use subst_vars have standardσ simp_all


ext simp induction induction dsimp generalize induction zero obtain obtain obtain ext apply apply succ by_cases obtain exact obtain replace rw exact by_cases obtain exact obtain replace rw exact rw rw obtain obtain apply


have constructor intro have apply rintro apply by_cases trans apply trans apply apply trans apply trans apply apply


simp constructor intro by_contra exact intro by_contra obtain simp exact


simp intro constructor intro by_contra exact intro obtain simp by_contra exact apply


obtain obtain rw rw obtain rw rw constructor refine simp exact intro exact rw rw intro exact rw rw intro apply simp


obtain by_cases rw obtain simp constructor obtain rwa simpa intro rwa obtain simp obtain rw map'_eq_castSucc_iff] simp intro by_cases simpa rwa simp


obtain by_cases rw simp constructor rw intro by_cases rwa rw exact rw simp constructor simp intro by_cases rw exact rwa simp


constructor intro dsimp rw rfl intro rcases simp subst refine simp exact


rw constructor intro rw intro rw ext exact


constructor intro dsimp subst dsimp infer_instance intro rw exact


SimplicialObject.Split.Hom.ext

rcases rcases have ext apply subst simp apply intro dsimp rw


SSet.S.IsUniquelyCodimOneFace.iff

constructor rintro obtain exact rintro refine obtain obtain rfl


let let let have obtain refine simpa rwa choose apply intro rw


obtain revert induction zero rintro obtain have mkOfLe rw let have dsimp rw simp refine unfold rw congr succ intro let strArrowMk₂ let let let let let let have subst exact have change have apply apply intro dsimp rw have rw rfl have rw rfl rw dsimp


constructor rintro rw have exact rintro exact


ext constructor intro rw obtain obtain rw have lia) aesop intro simp obtain apply


induction zero succ by_cases exact simp obtain obtain exact


obtain induction rw by_contra refine rw exact


ext suffices obtain dsimp simpa obtain let have dsimp split_ifs rw apply have rintro by_contra suffices simpa exact


rw constructor rintro exact obtain rintro refine have simpa


constructor aesop rintro induction obtain simp subst rw exact


constructor intro ext simp intro simp intro simp ext simpa


have let obtain by_cases obtain exact rw simp exact refine ext simp intro have dsimp lia


obtain induction let have obtain have exact exact


obtain have exact finite_of_hasDimensionLT


obtain have refine have rw simp exact infer_instance


refine refine intro obtain simp obtain rw exact


rw refine simp intro exact simp intro exact


ext simp have have simp obtain obtain exact


ext constructor rintro exact rintro exact


ext simp let have have simp have lia rw obtain simp refine fin_cases exact


SSet.horn.primitiveTriangle

refine simp simp simp have obtain by_cases subst have rw simp or_self, simp lia have rw simp and_false, obtain rw obtain simp refine fin_cases exact exact exact


ext constructor rintro rw exact rintro exact


let refine rw apply convert_to exact exact have have have have simp exact congr_arg


obtain obtain obtain obtain obtain obtain convert


apply ext fin_cases dsimp rw dsimp rw dsimp rw simp


apply ext fin_cases dsimp rw simp dsimp rw dsimp rw


refine obtain fin_cases ext ext fin_cases ext ext ext lia obtain fin_cases ext fin_cases ext ext lia


have rw exact exact isIso_app_coconePt_of_preservesColimit


constructor rintro simp intro refine ext rw by_cases obtain rw simp grind


obtain simpa rw simp apply intro have grind


obtain exact obtain obtain obtain obtain obtain obtain simp


refine apply exact rintro dsimp infer_instance simp


SSet.Finite.instIsFinitelyPresentable

obtain obtain have have apply isCardinalPresentable_of_isColimit exact rintro


constructor intro obtain obtain infer_instance intro obtain obtain simp


rw constructor rintro exact rintro exact


apply conv_lhs simp rintro rw obtain exact exact simp constructor exact intro apply


dsimp apply simp refine by_cases exact exact simp exact


apply rintro induction rw let monotone refine ext simpa simp


ext induction obtain have dsimp simpa congr_arg exact


apply ext dsimp rw spineToSimplex_arrow,


SSet.StrictSegalCore.spineToSimplexAux

induction zero succ refine rw exact ext obtain dsimp rw simpa


refine rcases simp_all rw mul_inv_lt_iff₀ exact


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


classical let have simpa have simpa simp_rw choose choose have let refine exact simpa


classical let have have simpa have have map_inv₀ simpa have have simpa simp_rw choose choose choose have let refine exact le_max_iff.2 exact exact


classical have let v revert refine let exact rcases let let exact have let let ih let ih rcases let refine rcases rcases refine rcases let refine rcases


by_contra wlog apply simpa wlog apply simpa wlog exact have have rw div_lt_div_iff₀ let rw rw log_lt_log_iff map_pow, rw log_lt_log_iff map_pow, exact


refine by_cases let refine rcases rcases rw rpow_inv_log exact


rw obtain obtain refine rw refine rw rpow_lt_rpow_iff simpa


HasFPowerSeriesWithinOnBall.congr

refine intro convert simp rcases simpa apply refine simpa


HasFPowerSeriesWithinAt.congr

rcases obtain let refine have apply intro exact


rcases rcases refine have refine apply simp rcases simp apply simp and_true] exact


have have have intro have exact have simpa


have apply suffices simpa apply obtain simp simpa have have obtain have apply exact have have exact have suffices have filter_upwards have filter_upwards rintro simp have calc rw congr rw exact apply apply calc congr simp abel norm_add₃_le gcongr exact simp


obtain p.norm_mul_pow_le_mul_pow_of_lt_radius refine have rw exact have have refine simpa rw have suffices refine have gcongr apply_rules apply intro calc


obtain hf.uniform_geometric_approx refine have have gcongr exact


rcases rcases obtain hf.uniform_geometric_approx refine replace filter_upwards simpa


lift rcases simp obtain p.norm_mul_pow_le_mul_pow_of_lt_radius simp set have intro have rw exact set have convert rw rw set have calc simpa sub_sub_sub_cancel_right] rw ring have gcongr apply apply have apply simp have rw exact exact suffices simp_rw exact


obtain hf.uniform_geometric_approx refine have tendsto_const_nhds.mul rw refine rw exact


intro rcases have refine rw exact apply apply simpa


refine simp rintro refine simp obtain have simp apply convert simp refine simp exact simp refine simpa


suffices convert norm_num apply intro apply rw simpa rw exact simp ext rw let suffices specialize rw symm simpa induction zero succ ih have iteratedDerivWithin derivWithin ext rw rw have intro rw apply intro simp rw show congr simp apply simpa


have rw have have simp convert ext simp simp intro simp


have rw convert ext simp simp norm_cast simp intro have have simp


apply intro apply exact intro rw rw intro


intro rw hf.eq_partialSum simp congr apply simp


let p.changeOriginSeriesTerm have apply apply simp_rw exact refine simp dsimp rw have simp_rw have have rw rw rw refine rw have refine refine rw simp_rw exact


rw dsimp have intro convert_to ext1 rw rw exact refine simp exact


have have have refine rwa have refine exact set p.changeOriginSeriesTerm have refine rintro dsimp exact have refine dsimp refine have rw refine simp apply refine refine simp refine refine rw simp_rw exact


ext refine dsimp congr convert obtain have congr simp


ext by_cases rw let simp change let suffices suffices exact exact simp let change suffices apply rwa


ext1 dsimp rw show ext rw apply intros rw refine rw change intro obtain obtain let have ext rw apply dsimp rw simp


ext1 obtain ext simp congr exact dsimp rw show ext rw dsimp refine ext change b intro have have have lia ext rw simp


giving fact rcases rcases simp obtain obtain rcases exact let have set have have rintro have calc mul_le_mul have calc simp calc grw simp simp refine simp_rw refine have intro convert simp refine convert ext1 rw simp


refine le_radius_of_bound_nnreal calc rw exact


rcases refine simp simp exact dsimp rw simp conv_rhs


apply rintro rw simp intro simp rintro obtain have rwa congr funext calc blocks_fun intro apply simpa assumption


apply intro have have simp_all rintro simp obtain refine lt_of_le_of_lt apply simp


suffices rw apply rintro apply intros rw


HasFPowerSeriesWithinAt.comp

of rcases rcases rcases obtain have apply exact have rcases exact let have simp radius refine refine have have apply have simpa we the along but have apply have rw refine symm rw have exact have have apply simpa simpa simpa have have haveI apply simp rintro calc apply rw gcongr rw have rwa tendsto_nhds_of_cauchySeq_of_subseq have apply intro simp exact rw exact


HasFiniteFPowerSeriesAt.comp

rcases refine apply rintro ext simp rcases simp obtain contrapose rw rcases have apply have grind rw exact calc apply exact apply simp


refine rcases rcases rintro obtain obtain rfl


refine rcases rcases dsimp obtain ext1 have map map simp change ofFn simpa ext1 rfl simp ext1 ext1 exact


induction zero change induction zero succ have have have rw have take take rw apply rw take_append_drop congr rw succ have have rw have simp exact have have rw simp rw


ext let r let r suffices simpa compAlongComposition_apply, variables rw one apply rintro dsimp apply intro apply intro apply intro refine dsimp rw


apply refine rw have constructor all_goals apply refine rw refine rw apply apply refine rw have refine rw refine rw refine apply


apply obtain apply apply gcongr rw exact


apply have choose obtain apply rcases exact simp rw intro exact


AnalyticWithinAt.pow

induction zero simp apply succ simp exact


constructor exact exact intro simp simp exact


rcases convert let let let have have apply simp filter_upwards simp have rw simp simp rw congr simp apply apply apply simp exact exact


constructor exact intro rw exact filter_upwards intro rw simp multiplication


classical induction empty simp exact insert simp_rw simp exact


have rw simp rcases rw have rw refine calc


have rcases rw lift have norm_cast rw refine rw exact


constructor intro obtain p.norm_mul_pow_le_mul_pow_of_lt_radius refine specialize rwa exact


obtain simp rw refine rw obtain refine calc gcongr exact simp rw rw positivity


have u.symm.toContinuousLinearMap simp rwa


simp congr ext apply apply intro use apply intro simp intro rcases simp right rw gcongr apply intro use apply intro simp intro cases inl rw cases inr right rw specialize rw rwa


ext1 induction simp simp n simp refine rcases ext simp


ext classical simp n have refine by_cases simp simp have have p p.leftInv apply exact have simp convert simp congr ext congr ext simp simp


classical have refine by_cases simp have rw simp have have p apply simp simp


ext simp congr ext exact simp n have simp


obtain le_mul_pow_of_radius_pos let obtain have tendsto_const_nhds.mul have simp have have have rcases exact let have apply simp rw exact intro have have calc r calc radius_rightInv_pos_of_radius_pos_aux2 exact gcongr simp positivity linarith gcongr let suffices apply simpa apply filter_upwards calc single_le_sum


rcases rcases let have filter_upwards have have have apply simp rintro calc apply apply rw gcongr rw have rwa apply simpa have have simpa apply filter_upwards rintro congr rw congr exact


have have apply exact have exact filter_upwards h''y have apply rintro simp rcases obtain simpa refine have have apply apply convert exact apply filter_upwards have have exact exact


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


obtain simpa by_cases have exact refine have have convert match_scalars simp simp


have have simp refine by_cases convert have suffices simpa simpa


have filter_upwards have obtain convert simp exact


wlog exact let let contrapose have apply rw filter_upwards rw smul_right_inj exact rw rw conv exact


constructor rintro contrapose apply rw refine filter_upwards rwa exact intro rcases exact hp.iterate_dslope_fslope_ne_zero


by_cases simp obtain by_cases simpa apply have have cases inl inr


have filter_upwards tauto contrapose rw use rw apply intro rw tauto


simp_rw intro apply rw apply intro push push_neg tauto


rw let change rw simp rw congr congr have have exact rcases simp congr symm simp


have rw exact have apply simpa let have apply exact let have apply intro rw simp simp intro simp intro have simp have have have nth_rewrite rw have apply simp apply have iteratedFDerivWithin_eq_iteratedFDeriv simp let have rw exact


have rw apply exact exact exact exact exact


have rw apply exact apply rw exact exact exact


rcases rw have iteratedFDerivWithin_inter_open rw exact


intro refine simp_rw push_neg obtain refine simp exact


refine rw by_cases simp apply refine refine refine simp_all simp_rw exact filter_upwards simp gcongr exact


refine refine rw have contrapose apply simp_rw exact


refine by_cases apply simp refine apply simp exact simp_all filter_upwards simp gcongr exact


suffices refine rw have contrapose refine contrapose apply apply filter_upwards rw cases filter_upwards simp rw convert rw aesop simp_all refine rw simp


rcases simp_rw by_cases apply refine filter_upwards simp apply simp obtain obtain refine nontriviality simp contrapose exact simp_rw apply refine filter_upwards apply simp_all have have convert simp simp_rw refine filter_upwards simp


unfold split_ifs simp contrapose rw exact rw refine rw


by_cases rw constructor intro simpa exact simp


unfold split_ifs simpa let obtain rw refine filter_upwards contrapose have rw simp filter_upwards z rw rw


by_cases simp by_cases simp obtain obtain rw refine simp tauto obtain obtain exact


generalize cases top suffices simp obtain filter_upwards simp coe r have intro rw grind obtain rw congr rw obtain simp obtain have intro rw deriv_fun_smul simp rw have rw refine simpa simpa simp_rw fun_prop rwa rw exact


induction zero succ by_cases simpa have simpa simp


by_cases have rw rw by_cases simpa rw grind by_cases rw rw rw obtain obtain rw rw rcases rw rcases refine filter_upwards simp


constructor rw intro rcases rw tauto obtain use constructor intro push by_cases rwa rw exact exact apply intro conv arg rw simp rw obtain use simp grind


have obtain suffices simpa isClopen_iff.1


have intro have conv_lhs rw mul_inv_cancel₀ apply have obtain apply infer_param rw refine push_cast exact refine filter_upwards simpa


obtain obtain obtain clear rcases exact norm_eq_zero.mp simpa ht refine replace refine have obtain have rw exact have calc simpa simp ring have inv_mul_cancel_right₀ calc simpa congr_arg rw simp rw simpa


refine have funext refine have simp exact replace simp exact


that let suffices have intro simpa mem_of_mem_nhds contained on it rintro rcases obtain let have have have simp apply apply apply exact have filter_upwards have have have convert ext exact exact


let set have have apply filter_upwards simp intro have exact


rcases rcases exact r_pos hasSum intro simp simp have rcases simp exact simp apply simp


intro rcases rcases rcases exact r_le hasSum intro simp apply simp rcases simp simp apply simp simp


refine intro rcases rcases exact r_le hasSum intro simp refine apply apply simp


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


constructor intro rcases refine refine exact intro simp rcases refine refine intro refine simp exact exact


classical simp refine rintro rcases let refine have simpa intro by_cases have simpa simp apply filter_upwards rintro exact


obtain let refine exact intro have exact


simp constructor have simp exact have rwa


rw constructor conv exact convert conv exact convert


suffices rw exact by_contra replace simpa


constructor intro have simp have convert simp simp exact


Asymptotics.IsEquivalent.smul

rcases have convert ext rw refine rcases rw rw rw simp intro specialize specialize refine have calc calc


refine rw obtain refine filter_upwards apply gcongr exact rw obtain exact


refine rw obtain refine filter_upwards rwa rw obtain refine filter_upwards rwa


constructor rintro refine rw refine rwa have exact refine rcases have refine refine refine exact


apply rcases simp apply exact exact exact


rw apply refine rw exact refine exact sup_le


refine simp have apply apply exact apply exact


refine simp obtain use filter_upwards rcases apply exact


constructor intro rcases refine exact rintro refine rw exact


Asymptotics.IsLittleO.listProd

induction nil cons ihL simp cases inl inr


constructor intro use refine simpa rintro exact


constructor rintro rcases rcases exact rintro exact


refine obtain simp refine by_cases simp exact


obtain refine filter_upwards by_cases simp exact


rw obtain obtain rw rw obtain let let have rw intro calc gcongr exact refine filter_upwards induction ind refine by_cases have grw grind grind


refine rw replace have refine le_mul_of_one_le_right obtain rw exact


obtain apply refine lift rw obtain refine rw exact


have rw exact apply refine have filter_upwards replace rw exact


have refine refine have rw exact have filter_upwards replace rw exact


simp obtain have rcases rw exact rw exact


by_cases rw have refine have have rcases rw apply obtain have have replace filter_upwards rw exact suffices rw replace refine obtain obtain refine rw apply gcongr rw rw rw have exact


have hv.symm.lt_of_le by_cases rw apply intro apply obtain have have have replace filter_upwards exact suffices rw replace refine obtain obtain refine rw apply gcongr rw rw rw have exact


have rw exact have by_cases rw by_cases replace obtain exact have apply exact rw apply apply exact replace apply rw exact rw exact


have rw exact have by_cases rw by_cases have apply apply exact rw apply apply exact replace apply rw exact rw exact


Asymptotics.IsEquivalent.rpow

obtain rw have rw exact use conv filter_upwards simp


Asymptotics.IsEquivalent.log

have have rw have exact calc isLittleO_one_left_iff


have have apply intro obtain simpa have apply exact filter_upwards calc gcongr exact rw exact simp ring


rw have apply filter_upwards have simp rw filter_upwards have rw


refine by_cases unfold lift simpa have have exact


refine have refine refine have rw exact suffices refine refine simp


refine have have isLittleO_of_tendsto have simpa refine simp


Asymptotics.IsBigOTVS.trans

refine obtain obtain refine filter_upwards


rw refine rintro exact refine simp gcongr


rw intro rcases rcases have refine refine simp grw simp


constructor intro rcases use filter_upwards refine apply grw exact


Asymptotics.IsLittleOTVS.prodMk

rw rintro rcases refine filter_upwards simp


Asymptotics.IsLittleOTVS.pi

have rw simp rintro have rcases refine refine simpa


Asymptotics.IsBigOTVS.pi

have rw simp rintro have rcases use refine simpa


constructor intro rw rw rintro rcases lift norm_cast rcases obtain apply refine fun_prop first apply positivity) filter_upwards refine calc rw gcongr simpa simp intro refine rcases replace filter_upwards calc egauge apply simpa


replace rw rintro obtain apply refine refine suffices simpa use filter_upwards calc egauge simpa apply refine simp rwa


rcases have simp refine rcases lift filter_upwards suffices rw rw exact_mod_cast calc gcongr exacts filter_upwards lift calc egauge simp


rcases constructor rw intro rcases lift norm_cast refine suffices simp exact calc rw have rintro lift lift refine calc egauge gcongr simp exact simp_all ac_rfl


have have apply exact filter_upwards exact


let suffices filter_upwards have rw exact apply exact


BoxIntegral.HasIntegral.sum

classical induction empty insert simp exact


by_cases refine refine simp abs_of_nonneg gcongr exact rw exact


refine le_of_forall_pos_le_add set have have set have πi.IsPartition dist intro have have set have rcases have exact have exact exact same choose have biUnionTagged_memBaseSet have have rw calc dist dist dist dist_triangle


two sufficiently refine rcases rcases have obtain exact rcases have let let have rcases exact obtain rw have have intro suffices simpa rcases an fully refine simp have let let let classical let have have lt_top_iff_ne_top.1 have iUnion_subset_iff.2 rw apply le_trans the have intro rw rw gcongr obtain have have apply refine exact exact h₂.isSubordinate.infPrepartition rw simpa refine rw trans linarith simp_rw refine simp_rw rw refine refine exact have intro rw gcongr apply apply simp_rw suffices linarith rw refine trans simp exact le_of_eq measure_biUnion_finset apply simp_rw exact


that that need box refine intro simp choose choose have rcases simp rcases classical set refine simp rintro rw sum_filter_add_sum_filter_not sum_filter_add_sum_filter_not refine rcases simp same specialize have rw refine convert refine rw refine rintro rw refine refine exact rw exact calc dist dist_sum_sum_le_of_le rw refine convert gcongr exact apply


induction change simp induction change simp change simp


simp congr have by_contra have rw exact have simp_rw


set i.insertNth volume integrate required have have have have set change clear_value convert_to congr have simp have replace intro refine rw exact rw exact calc integral f rw integral_sub simp refine rw gcongr exact rw ac_rfl


box-additive have by_cases exacts set integral. set set change refine exact exact intro have δ refine rcases filter_upwards have rw calc dist have refine simpa rcases refine have have have have have simp rw refine simp_rw refine have calc dist calc prod_le_prod refine refine rw refine calc le_abs_self intro rcases rcases refine simp refine exact rw exact


BoxIntegral.hasIntegralIndicatorConst

refine lift have have obtain obtain have rcases exact choose have obtain exact choose classical set refine rw hence dsimp simp gcongr set change have simp refine simpa refine refine refine refine exact have refine refine rintro refine simp rcases refine contrapose refine simpa


measure refine lift rcases haveI change set have have refine rcases simp refine exact simp choose have obtain exact choose refine rw grw apply rintro dsimp have rw rw gcongr exact refine rw let change have simp refine simp rintro exact clear_value lift grw


borelize rcases haveI rw refine clear set have have have intro rw edist_nndist, exact refine lift obtain have refine exact have have exact choose rcases the of at set refine integral integral refine rw refine the rw refine rw gcongr rw to to rw grw refine have have obtain rfl have convert refine simp refine rw exact than refine have have have rw integral_biUnion_finset integral_biUnion_finset refine rw refine exact


obtain convert refine constructor let have refine repeat apply refine rw refine exact have have isFiniteMeasure_of_le obtain refine use


classical refine replace induction empty insert rw refine by_cases rw exact rw intro have rcases rw exact


rintro suffices choose exact intro simp rcases have have have rw exact have exact


constructor refine rcases rcases rwa rintro simp rcases exact


ext simp constructor rintro refine rwa rintro refine rwa


classical simp refine refine intro rw exact


refine refine exact simp rintro obtain exact


constructor rw exact rw rcases rcases rcases exact


refine rw exact rintro rcases have rcases exact


classical refine rcases set have refine suffices have ext fconstructor rintro exacts rintro by_cases exacts have simpa


simp set refine refine refine rw refine refine congr exact exact


classical rw ext simp forall_update_iff and_forall_ne exact


simp rcases refine rcases have rw exact rcases have rw exact


refine refine refine simp rintro refine exact rw rcases refine exact


refine choose choose have have intro rcases refine simp refine refine rcases exact refine intro rw refine distortion_single simp refine simp


have choose refine isSubordinate_biUnionTagged.2 rw exact


have obtain let have let refine suffices intro simp_rw exact


refine refine exact exact exact lt_top_iff_ne_top.mp measure_lt_top_of_subset rw


classical unfold have apply rw exact rw simp_rw mul_zero, refine rw refine simp_rw tag_index_eq_self_of_mem_smul_span rw exact rw refine rw exact rw rw tag_index_eq_self_of_mem_smul_span


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


intro simp rw rw rw have cfc rw cfc_comp exact rw gcongr rw exact


intro calc rw refine intro have simp rw refine intro have simp


have ext simp rw constructor have simp simp field_simp simp


let let suffices rintro simp refine simpa exact exact rintro calc a conv_lhs refine exact rw fun_prop have refine exact fun_prop have exact


nth_rw rw simp have have rw refine refine trans simpa gcongr exact


refine rw intro lift rw refine rintro simp rw have rw nnreal_cfcₙ_eq_cfc_inr rw rw rw generalize set have rw fun_prop have have have refine suffices rw refine calc y gcongr refine have rw cfc_nnreal_le_iff exact exact rw cfc_id congr rw cfc_tsub refine simp


classical rw intro ext rw toCLM_apply_single_apply] simp


classical suffices obtain simp exact rw apply apply simp


refine obtain rw positivity refine simpa


refine calc rw refine simp apply gcongr apply gcongr apply rw gcongr exact gcongr exact


let let have simp exact rw have rw exact


refine exact exact simpa rintro have rw obtain exact


have refine change exact trans congrm exact


rw refine gcongr all_goals rw first rfl assumption


rw nontriviality intro by_cases simp exact lift rw refine simpa


set have simp set have calc simp simp have simp rw rintro positivity have rw refine exact exact rw refine simp rw cfcₙ_eq_cfc have exact


Commute.cfcHom

open induction const r conv enter equals rw exact id star_id add mul frequently hf rw apply exact


Commute.cfcₙHom

open induction zero smul id star_id add mul frequently hf rw apply exact


open obtain simp have by_cases let rw conv enter rw rw apply rw intro simp peel simp_all simpa


open obtain simp have have tendsto_nhds_unique h_tendsto.tendsto_at by_cases let have rw conv enter rw rw apply rw change rw intro simp peel simp_all simpa


rw simp toFun_ofFun, refine all_goals exacts


have rw refine rw apply apply rw exact have simp convert aesop


refine have exact rintro rw nth_rw rw refine obtain exact


use constructor simpa and_imp, rw nth_rw apply rw apply simp


obtain rw simpa refine simp rintro exact


let map_smul map_add map_mul map_zero map_star let suffices refine refine exact simp congr ext simp


by_cases rw exact simp obtain rwa rwa rwa


by_cases rw congr exact simp obtain simp rw exact rw exact


by_cases obtain rw congr simp obtain simp rw exact rw exact


by_cases obtain simpa simp obtain simp simp


intro simp rw simp rw ext constructor rintro exact exact rintro have simp_rw obtain exact apply push rw


by_cases exact refine all_goals exact exact


suffices rw convert ext simp congr exact simp and_congr_right refine all_goals


ext constructor rintro simp exact rintro simp exact


rw refine rintro simp obtain refine rwa exact exact


rw refine rintro simp obtain simpa refine rwa exact exact exact


by_cases rw apply lift simp have refine rw simpa hg'.comp rw


by_cases obtain rw apply lift simp simp obtain have refine rw simpa hg'.comp rw rw


let suffices refine exact simp congr ext simp


by_cases rw congr exact obtain simp rw exact


by_cases obtain rw congr obtain simp rw exact


have continuousOn_singleton rw congr ext apply simp_all


refine self-adjoint, roots have unfold rw simp simpa


refine hard nontrivial let star_mem rintro use ext1 dsimp simp suffices have le_antisymm refine h of contrapose exact


have refine by_contra obtain exists_continuous_zero_one_of_isClosed simpa hence suffices rw exact zero_ne_one calc calc φ


have apply intro rw obtain rw apply rw use simp have simp norm_cast have rw rw norm_cast have have simp rw exact


set rw refine refine refine simp calc refine calc rw simp


with_reducible_and_instances rfl reducible whereas


rcases simp have calc simp mul_sub, abel gcongr calc rw positivity specialize simp rwa


have intro have have f.opNNNorm_le_bound convert simp simp simp have intro calc have intro calc exact


parts suffices obtain refine obtain conv_lhs simp apply simp apply apply simp gcongr exact by_contra have obtain set have refine simp simpa choose simp obtain have refine have simp obtain tendsto_pow_atTop_atTop_of_one_lt apply trans have rw simpa have refine gcongr exact


rw intro obtain exact suffices rw refine norm_cast exact simp apply norm_cast positivity


use rw constructor intro fin_cases all_goals constructor simp cfc_tac exact exact exact exact nth_rw simp module


nontriviality refine simpa rw have rw have simpa


refine have funext rw nnnorm_star_mul_self, have rw convert rw


let have let have simpa exact simpa neg_eq_zero]


suffices rw rw convert exact refine nonempty have exact apply apply all_goals refine rw simp_all


refine have have have rw spectrum.zero_notMem_iff] have have exact exact exact


have suffices refine simp_rw exact intro suffices rwa exact simpa


obtain exfalso rw have have have simpa have have simpa exact


nontriviality rw constructor intro have norm_num contrapose obtain spectrum.isCompact rw obtain intro replace have rw exact


nontriviality apply simp rw x.2.spectrumRestricts.algebraMap_image] simp refine cases inl inr rintro exact


nontriviality ext have rw norm_sq_expUnitary_sub_one calc gcongr exact simp rw refine rwa conv_rhs refine rw obtain simp replace simp rw


ext have continuousOn_arg.mono rw conv_rhs refine have have simpa


have rw simp_rw gcongr exact rw rw trans gcongr exact exact


rw simp rw intro obtain have have apply apply apply rintro simp refine intro simp trans exact refine simpa exact refine apply norm_num exact


constructor revert simp_rw refine refine intro rw obtain refine simp rintro induction nil cons


obtain simp have rw rwa sq_le_one_iff₀ rw cfc_real_eq_complex cfc_const_one rw intro rw exact


let let obtain exact have extract_lets use constructor conv_lhs simp module intro fin_cases all_goals


applying shuffling refine simp rw refine rintro simp rw refine calc nth_rewrite gcongr exact le_csSup refine rintro refine gcongr rw refine simpa exact simp mul_add, simp


filter_upwards simp have tendsto_nhdsWithin_iff.2 have simp apply filter_upwards apply rw simp filter_upwards change simp intro simp simpa filter_upwards rw exact


obtain let rOut rIn_pos rIn_lt_rOut let have intro have rwa have rw exact refine refine have refine exact apply exact rintro exact apply apply exact


nonzero countably tending derivative summability rcases exact simp let obtain have refine intro rcases let have exact simp_rw apply rintro exact obtain apply rcases simp exact let have have rw obtain grind have have have obtain have intro have intro have apply apply apply rcases exact choose let have have intro refine apply apply simp lia refine calc rw exact rw choose have refine rw simpa refine apply intro simp contrapose have intro contrapose exact simp intro obtain have exact apply refine contDiff_tsum_of_eventually intro simp exact rintro refine have simp apply intro apply simpa


have obtain have refine exact intro push constructor linarith linarith refine apply have apply rwa linarith have have rw simpa simp simp intro


have have norm_cast exact simp support_const


change have have apply simp linarith intro simp intro linarith have have rw simp


change have intro simp intro have apply rw linarith exact have have rw simp


have apply convolution_mono_right_of_nonneg refine exact have simp exact


simp refine have have have exact set have have intro simp simp not_forall, constructor apply simp field_simp linarith have rw field field_simp linarith rw apply rw field_simp nlinarith apply apply exact


have have refine rintro simp right contrapose have simp rw exact exact apply norm_cast refine ne_of_gt apply simp_rw apply exact intro apply exact


refine borelize have exact mem_Icc push split_ifs refine simp symmetric split_ifs simp rfl smooth suffices apply rintro simp apply apply refine rintro apply dsimp linarith apply refine rintro apply dsimp linarith rintro have have simp eq_one have simp apply simp calc support have have simp refine field


have apply rw addHaar_real_closedBall gcongr exact


have have gcongr exact exact apply rw exact exact


intro rw simp intro have rcases rcases refine rw exact


constructor rintro rwa intro by_cases exact nontriviality exact


ω obtain refine change apply exact intro rcases exact


induction zero simp coe_coe] succ rw have fderivWithin fderivWithin fderivWithin_congr simp_rw rw simp rw


ω obtain refine refine exact intro change apply exact exact intro rcases refine refine exact


induction zero simp succ simp iteratedFDerivWithin_succ_apply_left] have fderivWithin fderivWithin fderivWithin_congr rw simp rw


HasFTaylorSeriesUpToOn.prodMk

set constructor intro intro convert intro exact


ContDiffWithinAt.prodMk

ω obtain obtain refine change apply exact intro rcases rcases exact


ContDiffWithinAt.comp

ω have obtain obtain let have have refine apply apply apply simp apply exact have have have simpa exact intro rcases rcases let have have refine apply apply apply simp apply exact


obtain x have have have hf.eventually_hasFTaylorSeriesUpToOn have hf_tendsto.image_smallSets.eventually rcases exact exact iteratedFDerivWithin_inter_open


induction zero succ replace have have simp rw rw rw rw simp rw


have refine simp_rw subset_preimage_image] obtain refine refine have refine refine rw exact intro have refine exact exact x₀


ContDiffWithinAt.fderivWithin''

have intro obtain refine filter_upwards exact ω obtain obtain refine filter_upwards exact rw exact


the and the original universe. induction zero simp apply succ have have n.choose calc n.choose n.choose congr apply rw have n.choose calc n.choose n.choose congr apply rw have iteratedFDerivWithin iteratedFDerivWithin apply exact rw have have rw apply simp_rw congr exact n).symm


already to let let let let have have have have let let let let have rfl have ext1 simp have refine refine simp calc let have let have have have isoE.symm.contDiff.comp_contDiffOn have isoF.symm.contDiff.comp_contDiffOn have intro rw rw rwa have intro rw rw rwa have rw rw rw rwa have simp exact


induction empty cases zero succ insert conv simp refine rw rw rw simp refine intro specialize grw rw refine simp intro refine rw have rw suffices rw ring refine intro have rw


derivatives controlled on induction hz simpa comp_apply] hi have have have have simpa have intro simp apply apply grw apply intro have rw exact intro exact hD have intro have rw apply simp apply calc x‖ congr refine apply exact exact have apply apply simp have apply simp exact gcongr exact exact congr simp ring congr exact rw congr rw exact gcongr apply simpa simp


already to let let have have let let let have have simpa have simp have have isoG.symm.contDiff.comp_contDiffOn have rw simp_rw have rw rw rw exact simp_rw have have ext simp rw rw exact


CPolynomialOn.contDiffOn

let suffices suffices have exact have exact


have constructor intro rcases refine rintro exact apply refine convert have exact rw refine rintro change apply exact rintro rw rcases refine apply apply exact rw refine change rw convert rw ext change congr norm_num convert ext change rw ext change p p rw intro simp simp apply exact i simp apply exact


refine obtain obtain rw refine intro apply refine refine exact exact rw rintro exact


ContDiffWithinAt.contDiffOn'

rcases obtain rcases rw refine suffices intro refine simp ω rcases rcases rw exact


constructor intro rcases rcases rw have rw exact intro rw rcases have exact


constructor intro obtain refine simpa simp exact rintro rw have exact


ContDiffOn.ftaylorSeriesWithin

constructor intro simp intro have rcases rw rcases rw have change rw exact rw have rintro change rw exact exact intro apply intro rcases rcases rw rw refine have rintro change rw exact exact


rcases have rw exact refine rw refine exact


intro rw refine constructor intro simp intro convert intro exact


suffices intro refine rw constructor intro simp intro exact intro exact intro rw exact


rcases rw intro apply rw exact intro rw exact


refine refine rintro exact have rcases rcases rw have rw apply have rw refine have rwa ω


rw simp constructor rintro rcases refine refine intro refine exact rintro refine exact


rw rcases rw apply exact exact apply simpa exact


ω constructor intro use rw refine rw exact rintro exact constructor intro use rw exact rintro exact


constructor intro apply intro constructor exact intro apply intro apply


have suffices x from rw have convert ext change congr rw rfl


constructor exact intro constructor exact intro by_cases exact have rw exact intro by_cases apply have rw exact


constructor intro rfl intro have rw exact change rw convert ext change rw intro suffices refine rw exact


constructor intro refine exact rintro constructor exact intro rcases exact have rw exact have rw convert ext change rw intro rcases have exact refine refine rw exact


induction zero rw simp succ let have iteratedFDerivWithin ext simp calc rw simp


induction zero succ ext rw simp congr rw


constructor intro constructor exact intro rw exact intro rw exact intro constructor exact intro rw exact intro rw exact


induction zero rw exact succ rw exact closure_minimal


induction zero intro ext1 simp succ intro rw dsimp congr rw apply filter_upwards exact


rintro intro simp_all have subst simp_all ext exact mk.inj_iff.mp for


rintro have contrapose have apply subst simpa


have rw rw rcases rw have apply simpa simpa apply have range simp convert simp apply exact


intro rcases have rw simp have refine simp simp exact have rw simp exact


ext by_cases rw simp congr rw simp exact simp congr apply have have exact


simp refine refine refine have have have have rw have simpa have have simpa


apply let change apply apply exact exact


ext simp compAlongOrderFinpartition_apply] congr ext exact


ext simp simp compAlongOrderedFinpartitionL_apply, congr ext rcases simp apply intro a simp apply simp


HasFTaylorSeriesUpToOn.comp

is multilinear thanks needs classical constructor intro simp intro have let change have have have hf.fderivWithin have hg.fderivWithin have hf.hasFDerivWithinAt convert simp have s suffices rw convert ext simp rw exact intro apply let change apply have exact apply have exact


set set refine exact ext1 exact intro exact intro exact


set refine ω choose refine set change apply exact intro choose exact


rw intro dsimp split_ifs subst exact exact


have obtain have have have have rw exact hut


induction zero succ calc iteratedFDerivWithin fderivWithin iteratedFDerivWithin_succ_apply_left rw rw


ContDiffWithinAt.sum

classical induction empty insert simp exact


nontriviality let let have rw have have refine convert simp


ω apply exact induction zero apply exact succ obtain apply contDiffAt_succ_iff_hasFDerivAt.2 have refine have have rw exact obtain use refine exact exact intro obtain have rw exact convert simp have rw exact have refine norm_cast exact exact top


ω obtain refine change apply exact intro rcases exact


rcases rcases replace clear rw rcases rw exact


induction zero filter_upwards ext simp succ have have nth_rw apply filter_upwards h₁a rw ext simp rw simp apply rw simp have simpa apply


apply rintro rcases have rcases exact have rcases exact


suffices refine refine rintro simp simp ring refine rw exact


let have have convert ring have have convert ring let have intro exact rcases exact


obtain exists_ratio_hasDerivAt_eq_ratio_slope use rwa


replace refine filter_upwards have exact by_cases intro have rw simp intro replace rw have rw rw obtain refine simp exact have rw specialize filter_upwards have filter_upwards have refine have exact have intro refine have refine simp ext exact rw exact have intro by_cases rw exact exact obtain specialize replace have rwa rw linarith simp


let have rw specialize rw rw obtain refine simp suffices refine rw have have exact have refine refine exact intro have refine simp fun_prop intro simp exact


intro have have obtain exists_deriv_eq_slope have exact


intro rcases rw have have obtain exists_deriv_eq_slope have exact


set set have have have have refine exact exact rcases rw refine simpa


by_cases exact rw rcases simp refine simpa


rintro by_cases simpa by_cases rw exact have rw exact have apply rw exact rw apply filter_upwards simp refine apply apply exact apply suffices apply apply rw exact


have have have apply filter_upwards simp exact


have have have mem_of_superset have rw filter_upwards simp exact


have lift simp convert rw norm_cast rcases have have simp convert rw congr abel simp exact


induction zero succ rw ext simp by_cases simp have have ext field_simp rw simp field_simp ring_nf


obtain obtain exact isLocalMax_of_deriv_Ioo


obtain obtain exact isLocalMin_of_deriv_Ioo


rw refine have filter_upwards rw obtain rw rw


have have replace exact isLocalMax_of_deriv


by_cases exact have contrapose exact simp


intro have rcases exact apply suffices rw exact


rcases convert simp have apply rw exact simpa


HasFPowerSeriesOnBall.fderiv

refine refine continuousMultilinearCurryFin1 simpa dsimp rw simpa


refine refine continuousMultilinearCurryFin1 apply simpa dsimp rw simpa simpa


AnalyticOnNhd.iteratedFDeriv

induction zero rw exact succ rw convert exact simp


rcases rcases refine suffices exact exact


of the the let let let rw have a.comp_hasFPowerSeriesWithinOnBall have have apply exact rw convert ext simp rfl


AnalyticOn.iteratedFDerivWithin

induction zero rw exact succ rw apply apply


HasFiniteFPowerSeriesOnBall.fderiv'

obtain rw refine rw simp exact apply change rw exact rw exact


ext rw cases have simp_rw rw intro rw rw rw simp_rw refine exact use intro have obtain exact rw dif_pos simp_rw congr grind


let have convert ext suffices congr congr rcases suffices congr congr rcases


rcases have rintro simp simp have simp_rw simp


classical constructor simp rintro suffices curryLeft have x apply exact rwa ext simp rw congr congr by_cases simp split_ifs simp obtain rcases simpa simp split_ifs congr exfalso apply simp_rw have contrapose rw exact simp rw contrapose rw exact rintro apply exact


induction zero succ intro rw rw iteratedFDeriv_succ_apply_right] rfl


rcases simp_rw apply simp refine simp_rw


IsBoundedBilinearMap.hasStrictFDerivAt

simp simp set calc ext rcases simp h.map_sub_left, abel refine suffices simpa refine exact


cases constructor rw refine simp apply simp exact


refine rw have rw rfl have rw rw apply intro simpa


have refine clear refine intro simp refine rintro simp


replace replace have f refine simp refine clear refine rintro simp refine rintro simp


rcases rw have isBigO_iff.2 have simpa exact


intro rcases apply rw refine simpa exact exact


classical by_cases rw rw suffices of proof. intro obtain simpa set suffices mem_nhdsWithin_iff.2 suffices rw intro apply have exact have rintro have have have intro have have have rw exact rw exact simpa rintro have intro exact refine all_goals have obtain apply simp rw have simp exact apply rw exact


setting we obtain let have have have have have have rw intro by_cases rw have exact have simp exact have rw exact exact


setting apply obtain let have have have have have have rw intro by_cases rw have exact have simp exact have rw exact exact


have have apply have apply apply intro exact have have apply have apply apply intro exact simpa


rw rintro obtain have refine have intro exact


let obtain eventually_nhds_iff_ball.1 refine have refine calc


refine intro rw calc apply apply simp simp apply simp simp


intro rw intro have rcases obtain simp refine refine exact


have rcases intro have intro have rcases refine simp rcases exact such choose have n intro let have have have have exact have have have exact have have have exact calc abel ring is let have rw intro obtain refine rw calc field obtain cauchySeq_tendsto_of_isComplete have intro apply rw exact have simp some this the intro have obtain exists_pow_lt_of_lt_one rw refine by_cases simp have have have le_trans obtain exists_nat_pow_near_of_lt_one have have rw lia set have have rw have apply simp positivity simpa have calc gcongr calc rw exact


rcases obtain have filter_upwards use have apply linarith intro exact


have simp_rw rcases refine have refine calc gcongr rw rw


have intro have intro have rcases refine simp rcases exact such choose have intro let have have have have exact have have have exact have have have exact calc abel ring is let have rw intro obtain refine rw calc ring obtain cauchySeq_tendsto_of_isComplete have intro apply rw exact have simp some this the intro obtain filter_upwards rcases simp have have have le_trans obtain exists_nat_pow_near_of_lt_one have have rw lia set have have rw have calc apply simp positivity simp simpa calc simp ring rw exact


borelize apply obtain suffices rintro suffices apply calc apply simpa rw exact


have simp apply rintro rcases rcases obtain have have rcases simp refine have exact obtain obtain have have rintro simp simp obtain obtain generalized_tube_lemma refine rintro exact have filter_upwards rintro simp refine have have calc abel gcongr rw change apply exact simp rw change apply exact simp simp


have simp rw simp simp refine refine refine refine refine refine have exact


borelize rcases have apply have rintro have rw apply have rintro exact exact exact exact


obtain intro have have simp_rw have rw exact exact


rcases apply exact obtain suffices rw exact rw by_contra exact exact


have hasStrictFDerivAt_id have conv convert rw ext simp rw


have hasFDerivAt_id have conv convert simp rw ext simp rw


obtain suffices rw by_contra exact exact


rw have simp_rw rw conv_lhs rw simp exact exact exact


cases simp_rw by_cases obtain simp exact rw rw simp exact


set have ext dsimp split_ifs subst simp simp rw have convert rw


have have nhdsWithin_le_iff.2 filter_upwards exact have apply exact rw apply exact


have have have convert module have convert module have convert module have convert module have convert module have convert module have convert module have have convert ext simp abel


have convert ext simp abel have have have apply filter_upwards intro match_scalars filter_upwards simp simpa


let let let let let let have filter_upwards let map_add map_smul map_add map_smul let refine simp exact have simp exact change exact


ContDiffAt.isSymmSndFDerivAt

by_cases intro apply obtain rcases filter_upwards have have exact exact have apply exact exact simp apply simpa


continuity symmetric. obtain rcases simp have obtain have have filter_upwards have apply exact have apply apply exact intro rw exact have have apply intro apply filter_upwards change simp_all have have apply filter_upwards exact have have exact have apply filter_upwards intro exact


have rw φ.toOpenPartialHomeomorph.leftInvOn conv_rhs congr ext suffices x φ.leftFun intro rw exact


convert swap ext simp swap ext simp simp


have have rw apply apply simp_rw intro rw congr ext rw rfl


intro rcases refine have simp rw have have have linarith is the to bound control. set set have simp have intro rw exact f'symm.bound have intro set calc dist simp gcongr simpa ring have dist w apply rw calc f'symm.nnnorm gcongr rw positivity rw gcongr exact field bound have have dist dist f'symm.nnnorm induction zero succ rw have dist f'symm.nnnorm calc dist f'symm.nnnorm f'symm.nnnorm gcongr exact exact replace simp ring refine calc dist gcongr apply ring have refine calc dist gcongr exact obtain have refine have simp exact have have rw refine simpa exact


rcases exact simp intro rcases refine exact


ApproximatesLinearOn.surjective

rcases haveI exact apply have let have have refine refine exact refine exact


AnalyticOn.hasFPowerSeriesOnSubball

rw intro set let have have replace apply apply symm apply have apply unfold rw obtain exact


AnalyticAt.hasFPowerSeriesAt

obtain convert obtain ext have simp smul_eq_mul, simp mul_eq_mul_left_iff] left rw norm_cast positivity


rw simp simp simp iteratedFDerivWithin_one_apply have congr


induction zero succ have have s.EqOn have hf.differentiableOn_iteratedDerivWithin have rw apply exact exact exact rw iteratedDerivWithin_succ, derivWithin_const_mul


induction zero succ obtain rw rw simp nth_rw have simp abel all_goals filter_upwards rw all_goals


induction zero succ obtain simp simp refine have rw cases obtain simp show show show ring


induction zero succ ih have rw deriv_comp_neg


classical induction empty insert rw simp rw


have comp have comp rw rw have lhopital_zero_right_on_Ioo intro apply rwa simp exact have unfold simpa


obtain have have have comp have comp have lhopital_zero_right_on_Ioo intro refine exact refine filter_upwards simp rw exact have unfold simpa


have comp have comp rw rw have lhopital_zero_atTop_on_Ioi intro apply rwa simp exact have unfold simpa


rw rcases rcases rcases let have rw rcases refine first exact exact exact


rw rcases rcases rcases let have rw rcases have refine apply_assumption first exact exact


obtain refine contrapose simp refine simpa ext have have simpa


letI replace rcases rcases refine simpa ext


apply have have filter_upwards simp simpa


HasLineDerivWithinAt.smul

simp let let have have have simp convert ext simp


by_cases simp rcases have have simp have let let obtain have have obtain apply simpa simp exact let suffices have have simpa have apply simpa simpa simpa intro have specialize rw convert simp simp intro have specialize rw convert simp simp


constructor intro rcases have apply filter_upwards refine simpa intro apply rw simpa


rcases rw exact have refine rw obtain refine rwa


induction empty cons rw simp_rw rw exact simpa exact exact


rcases rcases simp rw logDeriv_apply] simp


rcases simpa constructor refine have intro rw simp deriv_comp ring have hderiv.trans simp grind letI obtain grind rintro simp mul_div_mul_left


change set have have simp exact apply rintro rcases refine have have exact rcases specialize have obtain hf'.and_eventually refine have rwa sub_le_sub_iff_right]


have apply rwa exact intro exact intro rw exact exact intro have convert


let have have intro simp let have intro simpa convert simp


simp exact hi constant_of_has_deriv_right_zero simpa


letI letI set have have simpa have le_of_opNorm_le simpa


obtain mem_nhdsWithin_iff.1 rw refine exact


refine obtain refine have simpa intro simpa


obtain exact refine have have by_contra obtain have obtain hs exact


letI let suffices exact convex_univ.eqOn_of_fderivWithin_eq


rw intro simp_rw simp_rw have have filter_upwards refine gcongr exact filter_upwards convert simp simp exact


rw refine rcases refine rintro rw have rw exact le_of_lt letI refine exact


have have apply exact simp apply filter_upwards field_simp ring have rw apply have filter_upwards simp


to and argument filter_upwards rnDeriv_lt_top have apply filter_upwards rintro simp rw exact have apply filter_upwards rintro simp rw neg_sub, exact have apply apply apply have tendsto_id.add simpa filter_upwards rintro rw nlinarith have apply filter_upwards rintro refine apply have linarith filter_upwards rintro refine simpa f.mono.leftLim_le rw exact


differentiable values scale filter_upwards have rw apply rw exact rw have have apply apply apply have tendsto_id.add simpa filter_upwards rintro rw nlinarith apply filter_upwards rw gcongr exact filter_upwards rw gcongr exact have have apply apply apply have tendsto_id.add simpa filter_upwards rintro rw rw nlinarith apply filter_upwards rintro rw apply exact filter_upwards rintro rw have apply exact rw exact


almost apply intro obtain filter_upwards intro apply have filter_upwards exact


obtain have have set have have replace h_lipsch.mono have have simp refine rw rw exact exact have have apply rintro exact b_int.mono refine by_cases rcases have convert have simp exact have have apply intro rw exacts rw show apply filter_upwards apply exact refine apply on_goal show replace calc rw simp apply_rules first rfl positivity exact apply intro suffices rw have ext rw rwa


obtain eventually_nhds_iff_ball.mp choose replace h_lip.mono replace apply


letI rcases have have have apply rintro refine rw exact apply diff_x₀).2


let suffices ae_mem_of_ae_add_linearMap_mem intro have filter_upwards have have simp convert simp


apply filter_upwards apply apply apply apply exact filter_upwards filter_upwards calc exact filter_upwards exact


let have apply filter_upwards apply apply apply apply exact filter_upwards have filter_upwards by_cases calc have rw contrapose exact have rw contrapose apply simp exact simp exact rw exact filter_upwards exact


the conclude apply ae_eq_of_integral_contDiff_smul_eq simp_rw have rw suffices dsimp simp_rw suffices obtain simp_rw simp_rw simp simp exact suffices simp_rw intro let change refine exact exact


have have have filter_upwards let refine have simpa


rw intro obtain obtain have apply obtain exact exact have apply apply exact obtain rcases exact apply rcases simp obtain refine simp have have obtain have calc gcongr exact rw gcongr exact conv_rhs apply simpa gcongr


have have apply exact filter_upwards exact


have suffices filter_upwards have rw exact apply exact


classical have intro apply exact refine exact


letI let exact hasFDerivAt_tsum_of_isPreconnected


by_cases rcases intro exact have rw exact


induction zero ext1 simp_rw exact succ have have simp_rw rw ext1 exact intro simpa


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


classical refine let have haveI rw intro apply simp exact eventually_cofinite.2 let have ext1 refine refine filter_upwards simpa rw apply have refine rintro simp not_and, exact


simp refine gcongr rw suffices refine exact


intro rcases have have simp simpa have simpa


suffices rw rw exact rw refine exact simp


rw refine refine gcongr apply refine rw filter_upwards simp exact


refine apply exact simpa apply refine have simp


rw intro rcases apply rw intro rcases refine rw intro rcases rw intro rcases


classical simp refine simp gcongr refine gcongr exact


rw rintro rcases refine mem_tangentConeAt_of_seq simpa


of a the obtain exists_seq_strictAnti_tendsto have choose choose let have let have choose obtain apply exact simp refine push contrapose simp norm_pos_iff, contrapose simp apply refine exact exact exact


rcases suffices apply exact rw replace apply rintro refine exact


constructor intro simp rcases simp exact rintro exact


refine mem_tangentConeAt_of_add_smul_mem hs simp


constructor rcases rw intro obtain have rw rcases use simp_all choose refine simp exact tendsto_cd rw refine rw suffices rw refine grw simp norm_num1 rw rintro refine replace have replace simp refine rw simpa rintro exact


simp_rw refine refine rw simp refine simp


replace convert rw exact have have rw exact convert field_simp module


have induction zero simp one_smul] simp rw exact succ simp_rw simp have have specialize convert exact


change simp_rw have apply rw apply intro rw congr simp


induction zero simp rw exact succ rcases simp replace simp convert f simp intro refine exact


rcases exists_ratio_hasDerivAt_eq_ratio_slope use simp rw rw simp


have have intro refine rw rw exact have mul_ne_zero rcases use simp rw simp


have have refine norm_cast simp obtain use rw iteratedFDerivWithin_eq_iteratedFDeriv exact


have have hasDerivAt_id rcases use rw simp


rcases rw simp have hf.differentiableOn_iteratedDerivWithin have rintro rw gcongr rw gcongr exact exact hC have intro have exact have simp refine this.trans_eq rw ring


rcases refine have simp let use intro rw refine exact


letI letI rw suffices have rw exact constructor rw intro have obtain obtain let have have hR have intro rw exact have intro exact refine eventually_prod_iff.mpr simp refine exact refine obtain exact eventually_prod_iff.mpr eventually_prod_iff.mpr


letI letI have rcases simp eventually_const, rw suffices have rw refine filter_upwards constructor rw intro obtain simp_rw exact apply intro simp_rw have refine have mul_lt_mul exact refine obtain rw refine intro rw exact


have let suffices have have rw intro rcases exact have suffices rintro obtain obtain have exact


let refine rw rw have rw rw intro obtain specialize have obtain obtain rw refine simp rw refine by_cases have rw simp refine


letI rw suffices rw intro specialize rw simp exact conv congr ext rw rw have ext simp abel simp_rw have rw refine have rw rw intro apply intro rw convert module rw intro rw refine have rw refine rw simp rw exact refine have rw exact have rw exact refine intro simp_rw by_cases have rw simp rw exact


have have have eventually_of_mem refine simpa


rw intro obtain apply intro refine simp refine intro simp rw gcongr


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


have have eventually_prod_iff.2 refine simpa


simp have have this.differentiableWithinAt have fderivWithin refine exact rw rw rw rw simp abel


classical rcases let have have rw exact have exact let have filter_upwards simpa have have have apply have have convert simp refine have simp have fderivWithin have rw simpa have rw simp


rcases simp have apply exact exact apply filter_upwards have simp


by_cases simp rcases have have fderivWithin apply filter_upwards have fderivWithin apply filter_upwards have simp rw simp exact exact exact exact exact exact


refine have calc simp_rw refine rw exact calc


let have simp_rw range_eq_Ico, conv enter rw suffices simp_rw replace conv enter rw enter rw rwa rw apply simpa conv enter rw rw apply simp


rcases simp_rw let let have rw simp intro obtain hm let use intro have simp_rw obtain clear replace suffices calc calc gcongr simp_rw have calc gcongr gcongr have calc gcongr have rw exact gcongr exact rw convert linarith


have replace rw push_cast replace rw rw convert simp_rw norm_cast


set have refine rintro have dist_pos.2 suffices refine have hd.circleIntegral_sub_inv_smul simp suffices rwa have refine show refine exact show rintro rw exact show rw exact


intro rw rcases set have suffices have have refine simpa exact


set have suffices have simpa have set have have have exact


obtain isCompact_closedBall suffices apply obtain exact


refine have have have have filter_upwards


have obtain hc.exists_isMaxOn rw rcases have rcases refine refine rw exact


rw rcases rcases set have have replace have rcases calc


wlog have rwa rintro simp subst rw obtain rw norm_cast rw refine rw calc rw calc convert ring


simp_rw contrapose refine rintro have simpa


set have have simpa have calc simp gcongr linarith simpa


set have have simp set set have rintro simp set set rw have have have simp_rw refine w.re rw simpa


have set have have have have have have calc have rw circleIntegral.integral_sub, exacts


have suffices lift have have refine rwa have refine intro rw simp refine mem_closure_iff_nhds.2 set have rcases obtain refine have rw exact exact


constructor intro filter_upwards apply intro rcases have refine intro exact exact


have refine intro simp have nth_rw rw simp field ext1 simp


rcases obtain rcases refine ext1 simp refine ext1 simp


constructor exact rintro refine contrapose simp have ext1 simp starRingEnd_self_apply] rw refine contrapose simp


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


rw have simp have simp repeat have simp match_scalars simp ring


rw have have have let convert exact ext refine this.isQuotientCoveringMap_of_subgroup simp


rw refine convert rcases exact dsimp induction zero succ rw simp simp


have intro refine rw refine have have rw simp rw simp simp_rw apply simp exact


rw refine simp_rw change rw calc refine rw rw simp_rw gcongr rw exact gcongr exact


rw refine simp_rw change let have rw calc gcongr exact simp rw gcongr simp_rw rw trans linarith simp positivity linarith linarith


rw refine simp_rw calc rw refine congr simp gcongr exact


rw refine simp_rw change rw calc refine rw rw simp_rw gcongr rw simp congr refine grind intro simpa intro simp simp refine rw exact rw simp gcongr refine exact


have calc calc exp rw repeat norm_num nlinarith


apply apply apply left rw simp apply apply rw exact


simp rw rcases use simp intro specialize hB simp gcongr by_cases ε sSupNormIm apply simp rw simp apply simp by_cases ε sSupNormIm apply simp rw simp


simp rcases simp inv_mul_le_iff₀ rw apply simp rw simp mul_one] rw apply simp


apply use rw refine obtain simp use rw use rw apply intro norm_num exact


by_cases rw rcases simp left rw simp simp right rw simp replace exact


by_cases sSupNormIm sSupNormIm eta_expand rcases rw apply intro eta_expand simp_rw refine apply left refine apply left


simp mul_zero, simp obtain simp rw nth_rewrite have gcongr


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


simp_rw have ext simp constructor intro obtain use simp intro obtain use constructor norm_cast rw simp rw simp rw


simp_rw have ext simp constructor intro obtain use simp add_sub_cancel, intro obtain use constructor norm_cast rw simp add_zero] rw div_self rw simp rw


apply simp rw convert simp simp_rw have intro simp repeat simp simpa simpa apply simp rw norm_cpow_eq_rpow_re_of_nonneg apply apply nth_rw exact right apply nth_rw exact right simpa simpa


have by_cases rw left exact rcases rw simp_rw simp rw rw simp apply apply apply specialize simp exact apply rw apply simpa use simp apply rw apply simpa use simp


simp simp sub_zero, obtain simp rw nth_rewrite gcongr simp exact


unfold apply apply apply exact rw intro exact


have rw have refine rw repeat simp mul_sub] ring_nf rw simp rw congr norm_cast apply have simp simp


by_cases simp let have obtain let have have have use rw intro apply exact exact exact intro have have rw fderivWithin_eq_fderiv fderiv_comp ext nth_rw rw simp simp_all simp assumption


obtain rw obtain have have h₂F rw apply simp exact


set set have rintro rw grind suffices exact


refine set set set set set set set set have have have have have rw exact exact exact exact have rw exact exact exact simpa have have have have simpa grind up


suffices calc refine exact intervalIntegrable_const).symm have rw exact rw intro replace simp replace rw obtain exact filter_upwards calc


have apply_fun simp rw rw simp rw rw simp simp rw


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


have calc circleAverage rw intro contrapose aesop rw intro apply apply apply apply intro simp_rw congr rw apply simp_all rw intro aesop


have simpa have have rw simp calc rw exact


set have e.hasFDerivAt.comp_hasDerivAt calc rw exact norm_deriv_le_aux


have obtain obtain apply simpa calc obtain convert exact


have obtain exact have intro simp simp exact have simp refine simp


have refine rwa simp_rw congr simpa circleIntegral.integral_sub


TendstoUniformlyOn.cderiv

rcases simp have rw rintro filter_upwards simp_rw have h have have simpa


have filter_upwards have have apply filter_upwards exact


TendstoLocallyUniformlyOn.deriv

rw rcases simp rintro obtain refine exact


calc circleAverage rw match_scalars simp


the at hence rintro have have continuous_norm.comp_continuousOn have have linarith have obtain exists_isLocalMin_mem_ball refine have refine have replace have have have exact


is every provides refine refine have have obtain simpa nhds_basis_closedBall.mem_iff.mp replace let have have have have h4 have have continuous_norm.comp_continuousOn obtain refine exact


to constant any one-dimensional let let obtain have refine exact exact by_cases left refine refine replace let have have have specialize apply simpa have replace simpa coe_smul] right simp obtain specialize have rw refine have simpa


by_cases obtain exact simp refine rintro exact


let have simpa have let have have have have rw rw exact


obtain replace simp obtain have have use simp intro simpa have rw exact


have c₁ rcases refine refine exacts le_max_left


have c₁ filter_upwards simp gcongr rcases refine refine exacts le_max_left


rw rcases rcases wlog refine exact exact refine rw obtain have have rw have rcases obtain simpa have set set exp for suffices refine apply simp filter_upwards obtain refine mul_neg_of_neg_of_pos replace rw rw gcongr simpa sub_zero] have refine exacts mul_nonpos_of_nonpos_of_nonneg particular, obtain refine rcases simp suffices filter_upwards calc rw gcongr exacts refine suffices rw refine simpa refine exact have have replace convert rw frontier_Ioo by_cases rw exact replace have exact rw exact


suffices have refine horizontal_strip rcases refine have refine simpa simpa all_goals


rcases exact obtain refine rw exact change have rw have have constructor refine rw rcases refine rw refine have refine simp refine rw positivity simp filter_upwards rw gcongr intro rw exact intro rw exact


obtain simp change have simpa rcases refine quadrant_I simpa hO.comp_tendsto rw exact


obtain simp change have intro simpa refine rcases refine simpa rw exact rw exact


obtain simp change have simpa refine rcases refine simpa rw exact rw exact


the revert have rcases rcases refine quadrant_IV exact refine quadrant_I exact hO.mono obtain have refine rwa by_cases refine rcases have suffices simpa rw exact rcases simpa replace rintro simpa have apply intro rw rw contrapose calc x₀ abs_re_le_norm refine simpa


suffices refine exact filter_upwards set replace have refine rcases refine refine rw refine exact simp_rw have exact rw exact


by_cases haveI simp_rw have rw let on_goal on_goal change have have intro simp_rw constructor rintro exact rintro have ext rfl rw exact have intro rw exact have intro simp_rw constructor rintro exact rintro exact rw apply simp_rw tauto rw intro rw tauto


classical have simp_rw rw exact exact let refine apply rwa rw simpa prime_degree_dvd_card exact rw apply rw exact


have by_cases simp simp apply intro apply suffices rintro simp_all rw apply rwa


set set suffices calc z rw exact simp by_cases simp rw calc simp refine have rw rintro simp_all simp rw simp


refine suffices rcases replace rw replace rw rwa intro rw rw obtain rw refine obtain rw obtain rw


have rw have rw refine rw norm_cast positivity


have cases zero succ specialize rw rwa exact


convert apply_le_of_iteratedDeriv_nonneg simp simp simpa


have have h.hasStrictFDerivAt.restrictScalars have simpa


have have h.hasFDerivAt.restrictScalars have simpa


rcases lift replace refine rcases exacts exact


refine rintro rcases refine refine exact simpa


set suffices rw have rwa refine rw have have simpa


have have rw congr trans have refine exact have refine exact have have simpa simp refine simp not_false_iff,


have wlog suffices refine refine fun_prop rw filter_upwards apply exact exact rwa exact exact clear rcases rcases simp set set have refine apply exact refine fun_prop refine simp refine rw simp suffices have simpa refine grw intro have have rw simpa exact


have have rcases trans have rcases set have intro simpa have refine simpa have g.differentiable.comp_differentiableOn have simp refine rw refine exact simpa have rw simpa


rcases simpa rw div_eq_inv_mul, exact simpa


set set rintro have have ext simp rw rintro exact have norm_dslope_le_div_of_mapsTo_ball have isMaxOn_iff.mpr have have simpa norm_eqOn_of_isPreconnected_of_isMaxOn have simpa norm_eqOn_of_isPreconnected_of_isMaxOn h_z₀ have eq_of_norm_eq_of_norm_add_eq simp


obtain obtain exact lift have rw exact have simp convert simpa


Set.instTietzeExtensionUnitClosedBall

have have let classical suffices refine by_cases simpa simp by_cases ext simp refine replace simp refine simp not_lt, exact


rw have ext convert rw rfl exact contrapose rw rw filter_upwards simpa


letI rcases trans exacts have have simp rw


have rw dist_of_re_eq, nth_rw simp congr field


ext simp generalize have have rw exact rw div_add conv_rhs rw congr simp ring


have induction replace refine ext1 suffices simpa replace replace replace grind


let use rw refine have norm_cast rw simp have have rw apply


split_ifs exact change simp rintro exact


calc characteristic rw rw unfold have simp


have have apply apply rw simp apply apply apply intro simp by_cases simpa simp convert rw abel_nf simp


intro simp_all gcongr let have repeat gcongr by_cases simp have rw by_cases rw gcongr exact simp apply apply simp_all intro aesop intro simp inv_eq_zero, intro have rw rwa simp_all linarith exact


ext by_cases simp congr exact simp congr apply filter_upwards


classical induction empty insert rw calc logCounting


calc proximity intro apply simp_rw apply apply fun_prop apply exact ext apply exact exact


refine rcases exact rw exact rintro cases cases rfl


constructor rintro rw rcases rw exact rw have simp_all rw rintro rw


HasConstantSpeedOnWith.union

rw rintro have ext rintro exact exact rintro rw have ext rintro exacts rintro exacts rw have simp rw ring_nf exacts cases simp exact have ext rintro exact exact rintro rw


rcases rcases rw exact rintro rw rintro rw sup_of_le_right


dsimp simp constructor by_contra simp_rw push_neg obtain rcases exact rw exact rintro simpa


rw convert have have obtain apply have exact simp rfl


intro obtain rw rwa rw calc a add_lt_add_of_lt_of_le


rw rintro refine linear_combination module


rintro obtain obtain refine apply_fun simpa


rcases use simp replace refine match_scalars


simp intro obtain obtain convert rw rw rw exact


refine constructor rintro refine rwa rwa


rcases rcases exfalso exact exfalso exact exact


refine rcases refine rw neg_vsub_eq_vsub_rev simp


constructor rintro rcases rcases rw rw smul_eq_zero] rcases nth_rw rw have refine rw exact simp rw rw rintro exact


rw rcases rw change rw rw sbtw_lineMap_iff, intro have refine have rwa


have rcases simp nth_rw simp rw congr convert exact lia convert exact lia


ext constructor rintro have rw ext fin_cases rw exact rintro rw affineCombination_mem_interior_iff intro fin_cases


rcases simp_rw rcases rcases refine simp ring_nf


cases inl subst rw inr have let have simp hxz.ne]


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


refine obtain rw rcases exact simp rcases rw refine hne.symm, rw nth_rw rw exact


rcases rcases refine div_le_one_of_le₀ simp vadd_right_cancel_iff, nth_rw rw by_cases rw rw suffices exact rw ring_nf


Wbtw.collinear

rw refine intro simp_rw rcases refine simp rcases exact refine simp


rw rcases simp_rw have have rcases rcases rcases rcases rw rw exact simp exact simp rcases refine simp rw rw exact


refine rcases rw simp rw simp refine div_le_one_of_le₀ have rw simp convert field


rw induction nil cons rw refine rcases refine clear induction nil cons rw refine refine simp rw grind rw exact grind have simp rw refine rintro cases nil cons rw rcases cases nil cons simpa rintro simp


List.SortedLE.wbtw

induction nil cons rw refine clear induction nil cons rw refine simp_rw exact simp


refine rw exact suffices rcases exact induction nil cons by_cases refine simp rw replace rcases simp cases nil cons by_cases exact simp rw have rcases refine simp forall_apply_eq_imp_iff₂, intro have nlinarith simp forall_apply_eq_imp_iff₂] refine intro have nlinarith refine gcongr linarith simp lineMap_lineMap_left, ring_nf simp


refine rw rcases grind exact rw rcases refine rw have exact cases nil cons simp cases nil cons simp rw refine simp


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


rw let have have have have suffices rw trans refine exact exact obtain rw refine simp rintro simpa


rcases exact set clear_value induction rcases use simp simp obtain obtain rw let have let have simp exact have rw simp smul_eq_mul, refine split exact exact have rw gcongr rw simp_rw refine simp rintro dsimp simp obtain linarith obtain refine simp sub_add_cancel, intro split


rcases exact obtain refine inhabit have simp rw simpa


refine rw exact rintro refine suffices obtain simp_all intro have image_openSegment by_contra have rw apply specialize aesop