Add lean source and binary files (part 39)

lean/OAI/Analysis/Triangular/Certificate/ResidualMinus07.lean

@@ -0,0 +1,42 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+open AtomicTriangular.Interval
+
+def residualMinus07 : Fin 20 → RI :=
+  ![⟨-3157620568658079683423400785178708576437630537324, 3157620567147838372710022693341604137238876750298⟩,
+    ⟨-49772070815213840130010346671498595834630117349677, 49772070799941209397105677271559994706875600665318⟩,
+    ⟨-7869107327615297337406780794448719126743758485296, 7869107324528045727471467744139532083242242288117⟩,
+    ⟨-46096706185658971634781747452368879214737938817836, 46096706173428800069489099881352431936113111884332⟩,
+    ⟨-2181669866813058846664176424987457814157101991540, 2181669867619818211515982651705843933183437264520⟩,
+    ⟨-11146819775589694022058852300221826678024936363728, 11146819771836308580850341083974351935709612947142⟩,
+    ⟨-9974682380421860182744454092815900137452148542, 9974682378249761522103001866817914413310680018⟩,
+    ⟨-22110638999179604808313533739102288250387606525, 22110639008296641426265835809855013557188017303⟩,
+    ⟨-648000357210315253857863031068565598748411959, 648000357074449388739060951799198014012621066⟩,
+    ⟨-5446332793760267886457023265283891852847368982, 5446332796844752470225982111182428897424810543⟩,
+    ⟨-690434751528493312262491926213726817539213144, 690434751297913656774839453834896254482233789⟩,
+    ⟨-4929150393744273770104199162697222899713143603, 4929150392230774874011927914792824950867175242⟩,
+    ⟨-339665876635886425607236878288044048596788290, 339665876762516586929744524642657540437819500⟩,
+    ⟨-961486693582799036172022331279628525051362203, 961486693288421471176215140778723644389620955⟩,
+    ⟨-366835808700844938057701862963579124623339, 366835808028043517026078131119028981293479⟩,
+    ⟨-1612437591759009256591968655271653370280209, 1612437594408680260509369334933796243926791⟩,
+    ⟨-14960759715328725404925234476523927459046, 14960759690919172554845270149418495043541⟩,
+    ⟨-178531621305756946062660516029191786704171, 178531620928914083210669342713969119747881⟩,
+    ⟨-5975531974443267939240577383084444505467, 5975531954246401456978538823671068977357⟩,
+    ⟨-9163655230693914134399711113310242932322, 9163655217674493798098365187151924816116⟩]
+
+theorem residualMinus07_exact (i : Fin 20) :
+    residual 1 i 7 = residualMinus07 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualMinus07_bound :
+    matrixColBound (residual 1) 7 ≤ minusEpsilonNumerator := by
+  unfold matrixColBound
+  simp_rw [residualMinus07_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/ResidualMinus08.lean

@@ -0,0 +1,42 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+open AtomicTriangular.Interval
+
+def residualMinus08 : Fin 20 → RI :=
+  ![⟨-466186447597031595086393570979840600757693156347, 466186447832573562971869681421791418949411040411⟩,
+    ⟨-9088545400193004890166758288788459212980568564140, 9088545402617528540709426499515227243897368140026⟩,
+    ⟨-1081655000816702997641086016224466245509924686941, 1081655001303903966840614068414199618232645693410⟩,
+    ⟨-7000677208445894378719824445992239562986186740590, 7000677210354411631515719122712100663057895147823⟩,
+    ⟨-304051232180957215585034377490998718791017428737, 304051232054513111944129025336795335014822756332⟩,
+    ⟨-1630577713169044111427875890798900402168950855061, 1630577713758788814994350762120649315981311044905⟩,
+    ⟨-954252827540109875912113712971027942700737528, 954252827879774756131342158753722466507148212⟩,
+    ⟨-3175051292959262051676220778098851151637962186, 3175051290432340487260621653863847413951690925⟩,
+    ⟨-46062823983344508327261134453399492777239400, 46062824003953235926050749231335154371286639⟩,
+    ⟨-505235817148290814769422087651599765285132769, 505235816663742172113903399281096859694827270⟩,
+    ⟨-64050302977213304844069057341895663977976902, 64050303013251876599083470832072566492992997⟩,
+    ⟨-367997665864501357678425998979742241893109089, 367997666101230776537225181939291287644587281⟩,
+    ⟨-27075748681149866657412208436594448644880654, 27075748661315717641853004618616404191867101⟩,
+    ⟨-96173390114633828002799090827263499663913074, 96173390160624940730677929916677370515846360⟩,
+    ⟨-26777248107353351532953875965749950677391, 26777248213157864258641444603575689636562⟩,
+    ⟨-120485134482004640631339489493193214176674, 120485134065358652941541987683684007117045⟩,
+    ⟨-1178254674940212717024425893605408831013, 1178254678762558061136790954232783398573⟩,
+    ⟨-12911030530501239721928246580403753887798, 12911030589770054755366798323681097043547⟩,
+    ⟨-423119765009371300622483500198669225308, 423119768186884929803054319638286376830⟩,
+    ⟨-684485248176631613697289972389555239108, 684485250223381475899327723751925244470⟩]
+
+theorem residualMinus08_exact (i : Fin 20) :
+    residual 1 i 8 = residualMinus08 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualMinus08_bound :
+    matrixColBound (residual 1) 8 ≤ minusEpsilonNumerator := by
+  unfold matrixColBound
+  simp_rw [residualMinus08_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/ResidualMinus09.lean

@@ -0,0 +1,42 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+open AtomicTriangular.Interval
+
+def residualMinus09 : Fin 20 → RI :=
+  ![⟨-2169289073847007109526087649525465120869084761319, 2169289073511133393053805874201915621246605631216⟩,
+    ⟨-37038924809312660272616158221463637430751759091479, 37038924805582146827255686255060083558283517173210⟩,
+    ⟨-5110030878824669121164767476049279187109800020180, 5110030877990359413264646506965797313813892185988⟩,
+    ⟨-35847300057933561216447207722424106318144140530243, 35847300055844401816985331737604389153139016637951⟩,
+    ⟨-1433922906266199729636405812033406246438734191788, 1433922906453815784067448413046081904749384005499⟩,
+    ⟨-10018479986227287372751856915221922082727483667250, 10018479985417637517075816045054222427425276087505⟩,
+    ⟨-7983111599756541680186057387669817207095166851, 7983111599259076043563962310387008675071976223⟩,
+    ⟨-15193960186813038494749732775030716151200319620, 15193960193589173196666176270069761516096051238⟩,
+    ⟨-585342407961489708613038990555279149050053091, 585342407978908699959938993252251773632188564⟩,
+    ⟨-4401019241713823697249607141016766834182351668, 4401019242689434979959325362901207904529546933⟩,
+    ⟨-549682359233528548511341548683474788513870513, 549682359153348764866623030361019976533284818⟩,
+    ⟨-4213017131189701125779816212741753384512686212, 4213017130669554666135650031331425146306087540⟩,
+    ⟨-280006303489438040579629807199229187571466201, 280006303542402222250121020010222626375232045⟩,
+    ⟨-773016483323729163310280908828318293157012315, 773016483255671161558488662818905485119212246⟩,
+    ⟨-312193366374852016305349736106648536830460, 312193365872934820827797027836698561875409⟩,
+    ⟨-1347035671903097875535369367852867960614680, 1347035673845563724259772836333085112046541⟩,
+    ⟨-12342943246968530622286213743838295033007, 12342943227989657119643242243386886513889⟩,
+    ⟨-150836048336293647446878646057788564908997, 150836048052671462163512834468865841639996⟩,
+    ⟨-5020047535736230777942081943996761651186, 5020047520019844188263875877459343108145⟩,
+    ⟨-7769914682496645570571293271961213509499, 7769914673254839964881916052744302610980⟩]
+
+theorem residualMinus09_exact (i : Fin 20) :
+    residual 1 i 9 = residualMinus09 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualMinus09_bound :
+    matrixColBound (residual 1) 9 ≤ minusEpsilonNumerator := by
+  unfold matrixColBound
+  simp_rw [residualMinus09_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/ResidualMinus10.lean

@@ -0,0 +1,42 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+open AtomicTriangular.Interval
+
+def residualMinus10 : Fin 20 → RI :=
+  ![⟨-311732259349534652323544615465387877665746863673, 311732259416803228204501262631083341732317927566⟩,
+    ⟨-7426382752726593666963516028712624377537854886568, 7426382753498726290629289752139014804315588085263⟩,
+    ⟨-492801777754173620655373684060139088769158260503, 492801777916257209951149205401958615261611265833⟩,
+    ⟨-6545342616899392344901672116376646744341721800527, 6545342617326650553145103101991694982109562456143⟩,
+    ⟨-135827352356179105437443185863252619145709235349, 135827352319241278838518730992890124219596441845⟩,
+    ⟨-1650312175898434019835658234890753500498216307941, 1650312176078260617928510470066653673869871719549⟩,
+    ⟨-633656906938548046595723906235307153635880529, 633656907008613598888952130972099251322132292⟩,
+    ⟨-1542639696499063643918680050820392541513790163, 1542639695286630242607602809955528915526837990⟩,
+    ⟨-28852285995536049741388038153357947808695645, 28852285990041660803285193444751067830378394⟩,
+    ⟨-325432527351301599822095279225111708432940912, 325432527069513414266023350020234799808099817⟩,
+    ⟨-37189291768729971838392137323050962003783950, 37189291782548440771344562006744578795990176⟩,
+    ⟨-218163718262777228331882623991841318235092134, 218163718352312572169016296082950584995204014⟩,
+    ⟨-15584274236802321738945237260997647297047922, 15584274228019274349598224783315320888345001⟩,
+    ⟨-56265668407873856537098777306997635872747909, 56265668421043155754612495993343358689602258⟩,
+    ⟨-12832903885646248297875307176797320484967, 12832903961873288601842231444136721941586⟩,
+    ⟨-66009471930933687187584631487948479553170, 66009471635026115586498082188462401093035⟩,
+    ⟨-655409911422708415908608634051310763355, 655409914316737017322202278451866764079⟩,
+    ⟨-6381180642468163390784404156992963162933, 6381180685527114491189240848486634758751⟩,
+    ⟨-215588801389022148365925763403638352220, 215588803766616221300301040520630507071⟩,
+    ⟨-365490981592819141219931496487657182089, 365490983005790290573415501962597385893⟩]
+
+theorem residualMinus10_exact (i : Fin 20) :
+    residual 1 i 10 = residualMinus10 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualMinus10_bound :
+    matrixColBound (residual 1) 10 ≤ minusEpsilonNumerator := by
+  unfold matrixColBound
+  simp_rw [residualMinus10_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/ResidualMinus11.lean

@@ -0,0 +1,42 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+open AtomicTriangular.Interval
+
+def residualMinus11 : Fin 20 → RI :=
+  ![⟨-1499161722939384424047785808931054028872615760638, 1499161722723945487311126531245867901555814700335⟩,
+    ⟨-27620677993560420903757083973332580207843786206978, 27620677991100650172788264991618187213835893985240⟩,
+    ⟨-2995146283431618962676499427283911920183951657810, 2995146282887213460705994277609363538975239486873⟩,
+    ⟨-28072211747288683133574555016203155016246390898445, 28072211746083420296937004916102469768644792116641⟩,
+    ⟨-840263414650933146996797389843585902061503114948, 840263414769311937300792067649661103555344534658⟩,
+    ⟨-9549795080071298297058218737453020944235702948808, 9549795079494434469749723267801280128731549469277⟩,
+    ⟨-5769945944549654088570143669706213986244240189, 5769945944350133515535819358826257886287050504⟩,
+    ⟨-9270436657188234364679871958335058809115225780, 9270436661945193228946657436474251068993957998⟩,
+    ⟨-504191251946650773803162890339136257437110579, 504191251978869045684364372374009877262100762⟩,
+    ⟨-3320248530363282011460570612862910827557973312, 3320248531513124518442039056697319212997316724⟩,
+    ⟨-411196023278063433467267704746964672897329916, 411196023228196066742303066540684710928220897⟩,
+    ⟨-3562014736287058261844291875794522470340403162, 3562014735964017830830576161336069201016733911⟩,
+    ⟨-222707807278825574848517345858123826011780599, 222707807312076340278928078277334051068924149⟩,
+    ⟨-634012761420396678490582585282103992978348041, 634012761378367981788797932362909288145150453⟩,
+    ⟨-264548763124365643689292730030742288004718, 264548762804880827833866447177146634101798⟩,
+    ⟨-1104482409609496849984852346924568216587442, 1104482410847094238765429967316308422456585⟩,
+    ⟨-9891581304676936235878088198726277003098, 9891581292688834741283210576784883866968⟩,
+    ⟨-126923384821327068362798093062332776092036, 126923384640592038553108568230418377296793⟩,
+    ⟨-4183824299460406223701582378027886097898, 4183824289435646825498092622186972826324⟩,
+    ⟨-6616174737960037753731034238706605084421, 6616174732076565107511503936455055308295⟩]
+
+theorem residualMinus11_exact (i : Fin 20) :
+    residual 1 i 11 = residualMinus11 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualMinus11_bound :
+    matrixColBound (residual 1) 11 ≤ minusEpsilonNumerator := by
+  unfold matrixColBound
+  simp_rw [residualMinus11_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/ResidualMinus12.lean

@@ -0,0 +1,42 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+open AtomicTriangular.Interval
+
+def residualMinus12 : Fin 20 → RI :=
+  ![⟨-288566724023513704542903551155454085399389205233, 288566724097904069567941613595850856322264284370⟩,
+    ⟨-9278001795307357536228964551514263002440702084557, 9278001796199667272540708951056024649880556143291⟩,
+    ⟨-412460571801176188977211090166441888532562164192, 412460571984649461670503049581738343925214210203⟩,
+    ⟨-6891144083776167891352972860775121736019981013125, 6891144084257469972867499966257407235373051348859⟩,
+    ⟨-111153681014090960670997359963711806496871074925, 111153680972672667118861183327508802405654429995⟩,
+    ⟨-1638433108960364730401413751112946472480568862499, 1638433109164484657596841775690525690660087415656⟩,
+    ⟨-574117086787789933744250940474420475815024929, 574117086868899683068315276980265908068948244⟩,
+    ⟨-1240916498316728127870046099980593495858146909, 1240916496983164879953058939443660807956976422⟩,
+    ⟨-30224560562853567302330643089531521572773497, 30224560557154999992085431057628192916347486⟩,
+    ⟨-295972237855540670850568615621344605470391169, 295972237545843614260322058760414617180673009⟩,
+    ⟨-32601467020976690250596633049304043172638432, 32601467036333423424050130851228379212709400⟩,
+    ⟨-201009854177334475188032505347171797308855535, 201009854276702538053967272093140555016779305⟩,
+    ⟨-13970084435717386861169601839476173124730029, 13970084426017926047929606272276725280764205⟩,
+    ⟨-49268784431197008410111179609598019126848032, 49268784445938660893837712149429979550901322⟩,
+    ⟨-11218673140795097189446965075222332886926, 11218673223924823164156617172013173611268⟩,
+    ⟨-58919553611737081175453805804441452816919, 58919553288911302566511736000600332332651⟩,
+    ⟨-582400901079546018950100763622405420019, 582400904211283527270465851965859007600⟩,
+    ⟨-5494254215005459228233451866861885049246, 5494254261957058416045084001256204096241⟩,
+    ⟨-189053819622995509253005307866723761533, 189053822214031266018931608104321511920⟩,
+    ⟨-322110124553071592497242442480320956926, 322110126095434189738409387273119937011⟩]
+
+theorem residualMinus12_exact (i : Fin 20) :
+    residual 1 i 12 = residualMinus12 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualMinus12_bound :
+    matrixColBound (residual 1) 12 ≤ minusEpsilonNumerator := by
+  unfold matrixColBound
+  simp_rw [residualMinus12_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/ResidualMinus13.lean

@@ -0,0 +1,42 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+open AtomicTriangular.Interval
+
+def residualMinus13 : Fin 20 → RI :=
+  ![⟨-1331971280484960455182492116096415766916063042350, 1331971280261720445118141580348104447053100773249⟩,
+    ⟨-28359319642251920066305342347831397477587956476830, 28359319639686126205582346855854326540926238755743⟩,
+    ⟨-2544843519285840230979541399983371149564728636517, 2544843518725448644305879958347575089183262620945⟩,
+    ⟨-28289186117527353765929226091616158948506610653781, 28289186116264007125488869982674189596641474849681⟩,
+    ⟨-717454467313428164641272718079032827903997395336, 717454467436291781178274949077909535710871390009⟩,
+    ⟨-9891619058402064423898457833138661023928453290450, 9891619057800235065166542735769470110100643210450⟩,
+    ⟨-5166665344005271693880472984340572472596793399, 5166665343802824005923521960375884849439556666⟩,
+    ⟨-8221848616363160902048184839574169895986969985, 8221848621169770477069462878994371688805865436⟩,
+    ⟨-474651291037405587435654640548578219016790880, 474651291070012088479951987144917831473733080⟩,
+    ⟨-3056446263650095320698891933890937734352717309, 3056446264798813801092999239148951229085226903⟩,
+    ⟨-376364168600210124564653558477546749788309134, 376364168549345304585922385194570735194600046⟩,
+    ⟨-3395377853435274362225827075141745238640681735, 3395377853108259260939011388168718029042171390⟩,
+    ⟨-207963779172768573555363149966024355391448732, 207963779206191778781344969390211630437201601⟩,
+    ⟨-617812284505252197939303113141297108598751459, 617812284461670247801669900475915289184956058⟩,
+    ⟨-252728107816773915319829579598741891877122, 252728107500170754247095038848477111426395⟩,
+    ⟨-1044832907961646410183852971342982521690579, 1044832909188396391646885831421625903158329⟩,
+    ⟨-9274269892217387279059183307519742716280, 9274269880201261906712599771914769193461⟩,
+    ⟨-121206223029102666915216403429981793938802, 121206222850044424072826747751145534356100⟩,
+    ⟨-3982239991630905023528186293931821238640, 3982239981705351410137859383892842299248⟩,
+    ⟨-6361176507986002066988116980941339682754, 6361176502150937144881282558974371738347⟩]
+
+theorem residualMinus13_exact (i : Fin 20) :
+    residual 1 i 13 = residualMinus13 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualMinus13_bound :
+    matrixColBound (residual 1) 13 ≤ minusEpsilonNumerator := by
+  unfold matrixColBound
+  simp_rw [residualMinus13_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/ResidualMinus14.lean

@@ -0,0 +1,42 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+open AtomicTriangular.Interval
+
+def residualMinus14 : Fin 20 → RI :=
+  ![⟨-247556158302465533771562210405367340944406794254, 247556158404016874693621842037993697730544326584⟩,
+    ⟨-21923827087159982400727297724214136553562604517179, 21923827088697607277160649040486933973950439897406⟩,
+    ⟨-279494893091494351158902448042922767776374433230, 279494893386855521203119257071320564356146906514⟩,
+    ⟨-6230086602921028431317507085500338390932810579313, 6230086603649418643980465480132387290310027809812⟩,
+    ⟨-69965413372042257664592531567254892379408382755, 69965413310459250595901367516177695155726475293⟩,
+    ⟨-1096685955546625254579016798415001975052404861796, 1096685955862814896161813663698931879378122243633⟩,
+    ⟨-431666362203779603845534598483951351082637379, 431666362359900796058963865745857181485906853⟩,
+    ⟨-1153528821528840496008861727817793310777070776, 1153528819704532965670183939230919741042392407⟩,
+    ⟨-32856301810204011659525978600200564757069817, 32856301806492011223961129923324725714785625⟩,
+    ⟨-222115893073188329848002376011067619931111751, 222115892611919627327111072477073507262815213⟩,
+    ⟨-21906405010197432825583687508027002758546787, 21906405032428533279084134973421232167612584⟩,
+    ⟨-167690879727524129127714849384346635765074326, 167690879880883148932135930628422184166076229⟩,
+    ⟨-10626028687804978324008288163008892945310624, 10626028672827722206952510451853327710990685⟩,
+    ⟨-32334078969019661302404945894611325534612027, 32334078990864210665582106130101911263796290⟩,
+    ⟨-9466136519594927590234418369896540872867, 9466136647503502575957131296838241758806⟩,
+    ⟨-45904063180808560353374129291124893853444, 45904062683759593774060046166263195139506⟩,
+    ⟨-436630766085986346846396385225243724864, 436630770402800119789150953299557491737⟩,
+    ⟨-4323482716522298311420786686434504925040, 4323482788804088428964577138376089525721⟩,
+    ⟨-145350471836420799855422769961305720104, 145350475825238111823397033782478724069⟩,
+    ⟨-241355351638978460927501795040687897390, 241355354014682894480102108730335942738⟩]
+
+theorem residualMinus14_exact (i : Fin 20) :
+    residual 1 i 14 = residualMinus14 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualMinus14_bound :
+    matrixColBound (residual 1) 14 ≤ minusEpsilonNumerator := by
+  unfold matrixColBound
+  simp_rw [residualMinus14_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/ResidualMinus15.lean

@@ -0,0 +1,42 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+open AtomicTriangular.Interval
+
+def residualMinus15 : Fin 20 → RI :=
+  ![⟨-928025442227252497860487955624248545816788327333, 928025441847042406892866721239813687817735684449⟩,
+    ⟨-48012064721804046265409711897706691580967462689674, 48012064717214660847973386655671296151542515698551⟩,
+    ⟨-1536000934176458285203538083886280679168251577290, 1536000933231267840951655067998248164021566786412⟩,
+    ⟨-36095121340959358267369299365138351228921545975120, 36095121338531441370654375286005557725097548501723⟩,
+    ⟨-454653290366583892509877431522276244430952797119, 454653290579827395343631023789872047741567311190⟩,
+    ⟨-11092948501230734750713523922898938298839679454907, 11092948500173043127086147269017378222878667968224⟩,
+    ⟨-3670357850417135807429247197834459976503562850, 3670357850024748346566832615814897204853588747⟩,
+    ⟨-6918782686980057798471785154856650412028130564, 6918782693856339290925107026257515201539821564⟩,
+    ⟨-379251896108419661533290277002651083278407667, 379251896142123229005103513781692556909815568⟩,
+    ⟨-2446018086572645138593977640071010400400925505, 2446018088171037697772746690373859358914850161⟩,
+    ⟨-295470082324017766152599224698063391805220080, 295470082244437962711426528678339381827947982⟩,
+    ⟨-2959795861219192655748006507138874155166368864, 2959795860708511540002007044860521975686076959⟩,
+    ⟨-171454051472666839118658471094933204265397216, 171454051522627031164306029161093367145365682⟩,
+    ⟨-576289793218934915062434856824519591486654124, 576289793143178289413882965095895135030201238⟩,
+    ⟨-222265067060679137022982602490643254227834, 222265066630422265009005015328318278726637⟩,
+    ⟨-898053873637287442789982780225202304445441, 898053875305783962097635653567913519489342⟩,
+    ⟨-7764758204540088501820354082379927678797, 7764758188103500934975471378801012715179⟩,
+    ⟨-107135938483589900855527060436056540805133, 107135938240584677806964554101232948668904⟩,
+    ⟨-3490513017678892330590919793770093537112, 3490513004265963117238278067877882031585⟩,
+    ⟨-5766440436137915525419143985492948137425, 5766440428158606771183408555546728884674⟩]
+
+theorem residualMinus15_exact (i : Fin 20) :
+    residual 1 i 15 = residualMinus15 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualMinus15_bound :
+    matrixColBound (residual 1) 15 ≤ minusEpsilonNumerator := by
+  unfold matrixColBound
+  simp_rw [residualMinus15_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/ResidualMinus16.lean

@@ -0,0 +1,42 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+open AtomicTriangular.Interval
+
+def residualMinus16 : Fin 20 → RI :=
+  ![⟨-232057973008361956602470799932350520761943884392, 232057973165245592531169292252488283272777394386⟩,
+    ⟨-21785100793909859661795759242340269466116510030603, 21785100795893300792298396996166156002853560276920⟩,
+    ⟨-240096152885917883587604231610954465229563099470, 240096153209691018518123377595291266268335198760⟩,
+    ⟨-10264521508913461449248177377914049652140452647153, 10264521509818770758076887348815334884600907278686⟩,
+    ⟨-64688659989498069184619992707633030359592305839, 64688659904739125471849942276440553019574149099⟩,
+    ⟨-1932467272923700831408256524656904161182172087733, 1932467273366464926561603769770299701065207212489⟩,
+    ⟨-335129556502163233920460965354497669847511128, 335129556499229326527848254725889731012972194⟩,
+    ⟨-1268344654791973527725563514147916665260088973, 1268344654283085632095372249224898871950822447⟩,
+    ⟨-32535652043243729929130034152896666248472497, 32535652021203028180009298590897510926873568⟩,
+    ⟨-175702353403770583455381107174517360993596658, 175702353016474527270240952831328807720666027⟩,
+    ⟨-14610312772284449474363706749845308360638083, 14610312801510282196199890244989991503317360⟩,
+    ⟨-150070765342329953093887952481603066112815771, 150070765484259700536836456381590566469780625⟩,
+    ⟨-8329623202783746653580209656842038626682189, 8329623190556458410039946116516004762596892⟩,
+    ⟨-34331614488508384238699947710546071008958886, 34331614518416997693575721187559000912723595⟩,
+    ⟨-9107961505358687786359685546828124004206, 9107961578336986677595804465204887146015⟩,
+    ⟨-38831579726763274136383579157062904951730, 38831579441693640247900867366931431139034⟩,
+    ⟨-341606913906298903920057509406292561924, 341606919323760738592023439969538593616⟩,
+    ⟨-3970741248530293155069738447708809333294, 3970741289191557385155893351151686485638⟩,
+    ⟨-128593643499635842042885501237827114144, 128593645691219528520732758104164396950⟩,
+    ⟨-197226111419863466831304759737693984235, 197226112800355069336059040945172858191⟩]
+
+theorem residualMinus16_exact (i : Fin 20) :
+    residual 1 i 16 = residualMinus16 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualMinus16_bound :
+    matrixColBound (residual 1) 16 ≤ minusEpsilonNumerator := by
+  unfold matrixColBound
+  simp_rw [residualMinus16_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/ResidualMinus17.lean

@@ -0,0 +1,42 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+open AtomicTriangular.Interval
+
+def residualMinus17 : Fin 20 → RI :=
+  ![⟨-628583204442600019453960262434072595030093840475, 628583203009071404585187615236294231171922108374⟩,
+    ⟨-72214238414024397617100237629395841750980364773725, 72214238398571608805050156235797649408329453963476⟩,
+    ⟨-904108911302979180711112741794420710997441379091, 904108908282466945009620910643641401270376125003⟩,
+    ⟨-38134995445225372970724353249127774581178366340102, 38134995433473237967253630215597860249500489620506⟩,
+    ⟨-307469306745476440643419416197834858980166294448, 307469307527544000945058599470440028105596438581⟩,
+    ⟨-10306014406888168706823255024480626128252993647231, 10306014403169476556506388340564215545065778834691⟩,
+    ⟨-2658779293941691545889224838322813422812540924, 2658779291874901799279066147104910175062190741⟩,
+    ⟨-7279734361172596844184681217964704646540724710, 7279734376368578498883672532395108687395182233⟩,
+    ⟨-288088696856445817516812135681206369339277887, 288088696729349194584215774397473932043758516⟩,
+    ⟨-2082478699540540849878586492912055567062404426, 2082478702313048441140889043442458844097919808⟩,
+    ⟨-251103398948006278840312151014225766536077420, 251103398731257344942025734800302818634481217⟩,
+    ⟨-2617033119695501075335979534777314512847796984, 2617033118287947571164354863240278208640405210⟩,
+    ⟨-147550079394780146235837039155113485205233588, 147550079510249450010168816476730959627327620⟩,
+    ⟨-533900017949367779323238100048545866730803578, 533900017665872710784031670729340000366053559⟩,
+    ⟨-199185557660512562707509664895673550385137, 199185557103648230964233492334111362216717⟩,
+    ⟨-799802147926273241915658798746567759858844, 799802150127826681676281404794391426027956⟩,
+    ⟨-6797453681437306084625971071810431143205, 6797453660559350485010797343873588733547⟩,
+    ⟨-97334396250977995683166108955320523247767, 97334395939873144038221106757419224653615⟩,
+    ⟨-3159681784590397006925675499821243624077, 3159681768053362219588123460864404003840⟩,
+    ⟨-5380370661489624654999194532989494477541, 5380370650597352785510234082331268534260⟩]
+
+theorem residualMinus17_exact (i : Fin 20) :
+    residual 1 i 17 = residualMinus17 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualMinus17_bound :
+    matrixColBound (residual 1) 17 ≤ minusEpsilonNumerator := by
+  unfold matrixColBound
+  simp_rw [residualMinus17_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/ResidualMinus18.lean

@@ -0,0 +1,42 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+open AtomicTriangular.Interval
+
+def residualMinus18 : Fin 20 → RI :=
+  ![⟨-227003766965379207440778828329925515567918867977, 227003767152049119040489503072667050926233838365⟩,
+    ⟨-35943684552321810819476423854795777526817133064588, 35943684554992019352596014939770637813496396297106⟩,
+    ⟨-248013229590498281787362878238528256269139747555, 248013229990900817431867442867701694051726262586⟩,
+    ⟨-9774227619595350572137298188523893550595666841096, 9774227620616430548777773077767305458120334929521⟩,
+    ⟨-69381660199848796314764098746071775970610692752, 69381660096237396864608577631069344655997487163⟩,
+    ⟨-2225139509935356448386434210430871159078500422302, 2225139510496931442220468646342530039882957361813⟩,
+    ⟨-290971075924087371549453405616066941874668478, 290971075862808711245158130636729519172315304⟩,
+    ⟨-1328080099908007196259753582738992077675057560, 1328080099378435993102626497801804402898171969⟩,
+    ⟨-31287968867138216538845839242842183018638676, 31287968834165363410859064695299772283450894⟩,
+    ⟨-152421666124836568535106798415762081188281891, 152421665676893076893356281875431889172153252⟩,
+    ⟨-17934386895543905016095166492805424418962804, 17934386931689576019215895359546289514262461⟩,
+    ⟨-148439736903686557753047379836746802150311325, 148439737066840854912380896019574195849173440⟩,
+    ⟨-7209458702468249507242925550558546237999833, 7209458688670117349932752490747121268120144⟩,
+    ⟨-38803676523868559869157889290977264148296859, 38803676560119383452758527756813620858172360⟩,
+    ⟨-9168304470892709939278571550801497320641, 9168304547214509159196741689857442419565⟩,
+    ⟨-36176372249921384317564471344560195592349, 36176371951550697800009854299152167051636⟩,
+    ⟨-299339618357817567137770285796936302973, 299339625252034878681197141855142608504⟩,
+    ⟨-3986104502145811224173701238897707513777, 3986104544445986888376471448471076284101⟩,
+    ⟨-125702590751402363373574262188374801381, 125702593015279116465502992152355338460⟩,
+    ⟨-195917459569464566614034035895207513141, 195917461016898032300820419646660142962⟩]
+
+theorem residualMinus18_exact (i : Fin 20) :
+    residual 1 i 18 = residualMinus18 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualMinus18_bound :
+    matrixColBound (residual 1) 18 ≤ minusEpsilonNumerator := by
+  unfold matrixColBound
+  simp_rw [residualMinus18_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/ResidualMinus19.lean

@@ -0,0 +1,42 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+open AtomicTriangular.Interval
+
+def residualMinus19 : Fin 20 → RI :=
+  ![⟨-566519433582462804153332765699741596498535857526, 566519434011580098898545487122614791375671717367⟩,
+    ⟨-79996113554000789379777363323374230194606204557857, 79996113560787327518376645201007970776830570269241⟩,
+    ⟨-605005737661219458145152987551928051667287432035, 605005738499449447273231744067270946340497683273⟩,
+    ⟨-41026153499812257743043011330893372996042250806498, 41026153503490641838283641202813369215527953640587⟩,
+    ⟨-247203647868927912541647244379502949143617798754, 247203647666457478993235033238055613863223710766⟩,
+    ⟨-9942672948687214951393958973439729256125206754113, 9942672950010040856466001548339329582300379720293⟩,
+    ⟨-2178443340032940192225251097479685839537305605, 2178443340732270352835890929053869786165343474⟩,
+    ⟨-7855329329141386682856809601202369140699325704, 7855329332245064811834178383113240593827339719⟩,
+    ⟨-232616022674104653918663370377806631852934400, 232616022685818949550148610401228771034378983⟩,
+    ⟨-1927558528078343376263127302175346750905043361, 1927558529416886058546669859932610354672386712⟩,
+    ⟨-236778628924817293598207932978858611323606759, 236778628925054670215516588010437419915716201⟩,
+    ⟨-2414127876059068718303926814363719786113315722, 2414127876035877291347752519033144138290783316⟩,
+    ⟨-136890090524086669693809605547736180578463874, 136890090543360843394659226524069291724568521⟩,
+    ⟨-496468858073592624009394640905012120370460838, 496468858137568579265731593311120065921015143⟩,
+    ⟨-186035495002095402822860370522034667237518, 186035494481819359955607742306035071295437⟩,
+    ⟨-751904482216434201551483288046230542496741, 751904484202519882680581620896085426959491⟩,
+    ⟨-6374031897643410567289061715393332843946, 6374031878917807088580401273158611623927⟩,
+    ⟨-92026209746314020609365615738354992865714, 92026209449682789385688092601102507192539⟩,
+    ⟨-2990222170028562650849755451975493403430, 2990222153137618289177066496165564021822⟩,
+    ⟨-5163927134007994876234457423589534601821, 5163927124835429711461170897562425035068⟩]
+
+theorem residualMinus19_exact (i : Fin 20) :
+    residual 1 i 19 = residualMinus19 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualMinus19_bound :
+    matrixColBound (residual 1) 19 ≤ minusEpsilonNumerator := by
+  unfold matrixColBound
+  simp_rw [residualMinus19_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/ResidualPlus00.lean

@@ -0,0 +1,45 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+open scoped BigOperators
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+open AtomicTriangular.Interval
+
+def residualPlus00 : Fin 20 → RI := ![
+  ⟨-17721807902065955573555783544366620496685934826027, 17721808185191316491784444716543907493762438434827⟩,
+  ⟨-318291302511429929739027852752616907143079095460548, 318291303168094893187053128601030380532151493694842⟩,
+  ⟨-45957774167818342930592894247393201108774192487753, 45957774816553097225001063111036199259589578643718⟩,
+  ⟨-163453903439758543120297501033937849330676149734628, 163453903839841282611341355366919215547809408697224⟩,
+  ⟨-10187257796279646953571868664241151441255971794600, 10187257582925834597506940970532708505275464819507⟩,
+  ⟨-55438712421812397529205045057861705787384272560621, 55438712460816356028706683363760852489902139898836⟩,
+  ⟨-29432691778457254282181440309887620232528902698, 29432695752555171402167984984271767234235357809⟩,
+  ⟨-99140127584091246002328607609914167563607268015, 99140116434775486670170110898241108236456112760⟩,
+  ⟨-1527565155462629061691297034695559562313703937, 1527565493702015020140141084569439092198173439⟩,
+  ⟨-17804775708672544031628737283835324953432308446, 17804771926766109471098276766344776551088738350⟩,
+  ⟨-2330447537963181771950230058904889338058419974, 2330447771848237039225823227587912063030197522⟩,
+  ⟨-15392966788634492795702159685617109961083164214, 15392967879571496333165021915685732418745055625⟩,
+  ⟨-1091701209662712316311370411124635571089306184, 1091701074792143056175058252207789476799129218⟩,
+  ⟨-3479319654592559199247849123390418607599924355, 3479319756692468583223247066773363026002641300⟩,
+  ⟨-1142946334472924166855042070483257983563889, 1142948005539661527038199989069403281562202⟩,
+  ⟨-5184399701545829712366985153031313643507250, 5184393227468135290967612953458500005129632⟩,
+  ⟨-48610407112347111770262564783610120320646, 48610493251631122562819537280213954366063⟩,
+  ⟨-568776996010930569830095626870114678586547, 568777952447054168434061553024822678525112⟩,
+  ⟨-19148641485099896750633714150113667631597, 19148695164808266101246736657371906931608⟩,
+  ⟨-29494052725515066441912776119301595340508, 29494083501983313294635279871442327471045⟩]
+
+theorem residualPlus00_exact (i : Fin 20) :
+    residual 0 i 0 = residualPlus00 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualPlus00_bound :
+    matrixColBound (residual 0) 0 ≤ plusEpsilonNumerator := by
+  change (∑ i : Fin 20, ((residual 0 i 0).abs).hi) ≤ plusEpsilonNumerator
+  simp only [residualPlus00_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/ResidualPlus01.lean

@@ -0,0 +1,45 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+open scoped BigOperators
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+open AtomicTriangular.Interval
+
+def residualPlus01 : Fin 20 → RI := ![
+  ⟨-28687778583439044304703993844185607673016656346423, 28687778058861223412230942896381146988429170859325⟩,
+  ⟨-571955143534314258002551832651339972704456801650961, 571955142326716142139511068895239172614530571516326⟩,
+  ⟨-85895466681710881440068459063459951773051905346426, 85895465479661244035331038922002980433185969991479⟩,
+  ⟨-288345214934755546940396831666224034731324173617393, 288345214204190231745243313591992483282307873128636⟩,
+  ⟨-19024510948517214597141375619744921178484128800564, 19024511343696396479258622398159973423888000555383⟩,
+  ⟨-101784228742121595103329742317513825565979587551675, 101784228667640939828301421094970490138697934475923⟩,
+  ⟨-54147032746905724535011487203188964627334887560, 54147025389869425881081394571531245489709147718⟩,
+  ⟨-184513648368569150271431647045819102580490204406, 184513669009493031472099900046447050843443856026⟩,
+  ⟨-2784487153879534687167467721380031295329912431, 2784486527739108671054659219271807322538292255⟩,
+  ⟨-33019627129764268852738788536890099922735152564, 33019634130781763587238853093167658669892918229⟩,
+  ⟨-4342801095438446772694233677541428986833311766, 4342800662446281879979512303038933974449453693⟩,
+  ⟨-28590412975731496765325482208345446987834427146, 28590410955984804158662420226233226142814758850⟩,
+  ⟨-2031310035879975970829336090881573270780307412, 2031310285559988988136610757736318927880254343⟩,
+  ⟨-6478418776248415998570966552094741379022712456, 6478418587173724626969901794458166473784442642⟩,
+  ⟨-2126627098592905730736964118076749951128255, 2126624005283110101285678439807393261960373⟩,
+  ⟨-9652506033575175679601445432053824820511938, 9652518017754463742473376213795503507492568⟩,
+  ⟨-90545668144684678361276035943325926369669, 90545508692467485223316793859973235624291⟩,
+  ⟨-1059514012613564082971226191867002732141818, 1059512242157301145070109389299087677059951⟩,
+  ⟨-35679014402015183284455136415274140322927, 35678915035915944252591687383867088153111⟩,
+  ⟨-54967554499254188856890470465534210516791, 54967497528778040756321999775379749329925⟩]
+
+theorem residualPlus01_exact (i : Fin 20) :
+    residual 0 i 1 = residualPlus01 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualPlus01_bound :
+    matrixColBound (residual 0) 1 ≤ plusEpsilonNumerator := by
+  change (∑ i : Fin 20, ((residual 0 i 1).abs).hi) ≤ plusEpsilonNumerator
+  simp only [residualPlus01_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/ResidualPlus02.lean

@@ -0,0 +1,45 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+open scoped BigOperators
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+open AtomicTriangular.Interval
+
+def residualPlus02 : Fin 20 → RI := ![
+  ⟨-3297407235746062622699773988374924055264748582359, 3297407258626130212764634450769295955469980858056⟩,
+  ⟨-50790253888813663913432143944813122865942547533395, 50790253964705306663981140296983307993772929222061⟩,
+  ⟨-8111570717299919486682955996300668270881192121890, 8111570769530521011263810393100974643737507248180⟩,
+  ⟨-30185851587076786032236617256806352310914325490097, 30185851651891178371299224261199194963651991794808⟩,
+  ⟨-1709025133961621665686680221880006140406067559059, 1709025116244887624028662211291845509221246527306⟩,
+  ⟨-9762524021839495647413679522538734688216897116912, 9762523986948004402431689100208668271064055862362⟩,
+  ⟨-4824415757738721351484611231527203766679718815, 4824416104610504309879404025489340357491690043⟩,
+  ⟨-16257284941697363493719458122945214040397881860, 16257283990732695312847203255826245642792371406⟩,
+  ⟨-274242282831484530251257465993102106771441522, 274242312793404380565125825699996908911496752⟩,
+  ⟨-2802790428483084837219714103101288702048381472, 2802790099791000528939835526390751018369430660⟩,
+  ⟨-362254903374724870040330713343045879439216048, 362254923532131891199246504508551502161532382⟩,
+  ⟨-2440451274348247492783099919622569598543690388, 2440451368565684533044876342860684334906828868⟩,
+  ⟨-169477828473180639286003229170366276002233803, 169477816802472029516404465692153998618017020⟩,
+  ⟨-549974664210910577277838736838330039021851171, 549974672768547420738764106822081079974233496⟩,
+  ⟨-178960177845924776759590843221906571437762, 178960323250245450896656674328581198147224⟩,
+  ⟨-805022355153028384386806518604269083009826, 805021792290218063251231634240186139636980⟩,
+  ⟨-7502794592298814809121031583296538351282, 7502802078023901306630331440813439953935⟩,
+  ⟨-88320537486850034983746998460357900800837, 88320620629014606342644496177685860910249⟩,
+  ⟨-2964391319844454767805390051892975616608, 2964395985592122687124448864210756099865⟩,
+  ⟨-4562464723039717315531678240767280426945, 4562467395818530201849072650009686475007⟩]
+
+theorem residualPlus02_exact (i : Fin 20) :
+    residual 0 i 2 = residualPlus02 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualPlus02_bound :
+    matrixColBound (residual 0) 2 ≤ plusEpsilonNumerator := by
+  change (∑ i : Fin 20, ((residual 0 i 2).abs).hi) ≤ plusEpsilonNumerator
+  simp only [residualPlus02_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/ResidualPlus03.lean

@@ -0,0 +1,45 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+open scoped BigOperators
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+open AtomicTriangular.Interval
+
+def residualPlus03 : Fin 20 → RI := ![
+  ⟨-1642571459251064309761222693825333426220417051819, 1642571443887569381813131260304307379682012479191⟩,
+  ⟨-24897836834895894013834214121699263884017543902806, 24897836828246120827558947953498716498307148481560⟩,
+  ⟨-4887573392007044654493362911508537737263100808622, 4887573355457936149180915610764583256856690571361⟩,
+  ⟨-21932371381029471889508684357267231300237914504851, 21932371318957146484742580207972649258511344813689⟩,
+  ⟨-1220973741417007490468115773975469952842060610363, 1220973753152170612115890001017128914378259523857⟩,
+  ⟨-5148699181019622769603748422435548811545196402089, 5148699195431560694120939516491577232838645513193⟩,
+  ⟨-4067234430490256232151154338395761762403913983, 4067234220536037325719912362802557858421698465⟩,
+  ⟨-11557744610547215900464664235283246441713360515, 11557745187525321289875257729681718531515841174⟩,
+  ⟨-225420885612027911449260511921929816863677667, 225420867572253271215208531364101912806743180⟩,
+  ⟨-2307453490312314536269617890312068327436709903, 2307453688011395853350068310818127631466644991⟩,
+  ⟨-298261346591933806618236774497500808808135167, 298261334331302670672008220286520288680285870⟩,
+  ⟨-2003088300142918780134022923962808790464284774, 2003088242303296031594753926896519530624172080⟩,
+  ⟨-141514359899700521695831440956547930702756722, 141514366969870406631962573156047764920565894⟩,
+  ⟨-427984935511000712851531458597354316402040891, 427984930015113880602079457318660402848522875⟩,
+  ⟨-148636210488155680453190803659432860687108, 148636123852050716600617791681036948709668⟩,
+  ⟨-668220003944774769337149029684669918291953, 668220339381910252498532703482537610528154⟩,
+  ⟨-6261165784037546603707669235510918517492, 6261161325653082018783989289245109912180⟩,
+  ⟨-73323891405774106705756164062847984175832, 73323841885131737148140943963106804955952⟩,
+  ⟨-2466132304678826544331630843135168898138, 2466129527283283864141515930068057642134⟩,
+  ⟨-3771643654977463834871749036869232268948, 3771642061910712999794443425053635225989⟩]
+
+theorem residualPlus03_exact (i : Fin 20) :
+    residual 0 i 3 = residualPlus03 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualPlus03_bound :
+    matrixColBound (residual 0) 3 ≤ plusEpsilonNumerator := by
+  change (∑ i : Fin 20, ((residual 0 i 3).abs).hi) ≤ plusEpsilonNumerator
+  simp only [residualPlus03_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/ResidualPlus04.lean

@@ -0,0 +1,45 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+open scoped BigOperators
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+open AtomicTriangular.Interval
+
+def residualPlus04 : Fin 20 → RI := ![
+  ⟨-1100055363513728839378935082609480002701627706110, 1100055375062888837350454691291616560278558410614⟩,
+  ⟨-19864936007604032457864253049961379842814036139530, 19864935997106157816119831382966663304402237838858⟩,
+  ⟨-3554849109938581352192742993762644058687687653852, 3554849137280776442289946073435800712869648829270⟩,
+  ⟨-9887680489293945703058051348505246236753266784698, 9887680513062350508230352568212977537117511928316⟩,
+  ⟨-707466655999453461427751945031138641960968877758, 707466647365360972272766492012477896327733953684⟩,
+  ⟨-4285965969804878342157280641946781556672339818686, 4285965956700835544447593287382429173204847905487⟩,
+  ⟨-1514775331767376402863710908313778028887458912, 1514775480235313327359247776295138500343988713⟩,
+  ⟨-6360421511680613056691396399073021756862822126, 6360421103943939405367855331400272395949993414⟩,
+  ⟨-98326438932182823229246645942792581533195135, 98326451654103367519690876116077529991637150⟩,
+  ⟨-885981634989636103596408085201154627811131205, 885981496560892886750572281739911604383353105⟩,
+  ⟨-118995703430252152000717585917599924506733664, 118995712085854711861240320654339771725947513⟩,
+  ⟨-798691871836933163806039996322699888611074634, 798691912869209887060612027798017445394682807⟩,
+  ⟨-53075513626982815904620583020719075878849943, 53075508664918349971357857356095056115721314⟩,
+  ⟨-190471987554513000483003219283188460941900912, 190471991539276318213284392608042629835528197⟩,
+  ⟨-57378695668252623261859582660651282026272, 57378755994855755031594315311740010536736⟩,
+  ⟨-254961005408457553809784983581325002669950, 254960771867954589814561865025150278571663⟩,
+  ⟨-2382206760142912862478627078452592689291, 2382209862867853583020384296839602791095⟩,
+  ⟨-28107969912968413149310401825314951944870, 28108004377504186824433286461426519087336⟩,
+  ⟨-937676871942400123299923649311091628636, 937678804184469981723304786542750001645⟩,
+  ⟨-1455720532086626728212977478511584883776, 1455721640880309788001108021283908820218⟩]
+
+theorem residualPlus04_exact (i : Fin 20) :
+    residual 0 i 4 = residualPlus04 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualPlus04_bound :
+    matrixColBound (residual 0) 4 ≤ plusEpsilonNumerator := by
+  change (∑ i : Fin 20, ((residual 0 i 4).abs).hi) ≤ plusEpsilonNumerator
+  simp only [residualPlus04_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/ResidualPlus05.lean

@@ -0,0 +1,45 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+open scoped BigOperators
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+open AtomicTriangular.Interval
+
+def residualPlus05 : Fin 20 → RI := ![
+  ⟨-1965110368241573381383939024881512617429894142209, 1965110391384685322203080468344448841594581211827⟩,
+  ⟨-34294305199172318334395752362923867730525591180559, 34294305230092414485609128100525230495469079861811⟩,
+  ⟨-6651734247521384893055186453042606401285570047579, 6651734300687445727655266915113138722348751805594⟩,
+  ⟨-34876946178924976656444555442805570889986819941422, 34876946224942414463836391498285856649961354383377⟩,
+  ⟨-1831931431718657800307616919937565854297091422386, 1831931414072347058224938955280978941063939092218⟩,
+  ⟨-6543507021438592383940095016528643914394658066507, 6543506973724422789670999088630911310888565126792⟩,
+  ⟨-6944199300857440777889819564145794303320032281, 6944199630186985261212656696432921158849312161⟩,
+  ⟨-17667297078969064360103358351880779484433813110, 17667296181765446382962303626381699006215556958⟩,
+  ⟨-401432157160577211949244952067437725577431410, 401432185598069599279844604439177103170009992⟩,
+  ⟨-3840886983587748795998925876536581417788204753, 3840886675473297057980443758734217525937728155⟩,
+  ⟨-493636257662325129658446121175837499672562300, 493636276722903720743172434243270313936060707⟩,
+  ⟨-3359099687774230416015944162112866810303509959, 3359099777519495580492657390020780057599122539⟩,
+  ⟨-236854412087003652962180175422001266888406681, 236854401118545588662133660522372419493492285⟩,
+  ⟨-691610039916963600094497616211852460926883700, 691610048275297722735164100453760486539923228⟩,
+  ⟨-250057208651949667082787202993210143964116, 250057343989215725757845709511900093743937⟩,
+  ⟨-1116948735660879871596628261169327660934582, 1116948211939335862583518472350099694527354⟩,
+  ⟨-10452747279700707187215189051397841275734, 10452754240268362525616982471859345454420⟩,
+  ⟨-122782278414975960036232524178414148753462, 122782355735019631202895330732569811160794⟩,
+  ⟨-4126503544359321756587748011386823729883, 4126507881132024025040769462355555765511⟩,
+  ⟨-6298270747798941180289545720163238389724, 6298273233250741300072325714654126688240⟩]
+
+theorem residualPlus05_exact (i : Fin 20) :
+    residual 0 i 5 = residualPlus05 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualPlus05_bound :
+    matrixColBound (residual 0) 5 ≤ plusEpsilonNumerator := by
+  change (∑ i : Fin 20, ((residual 0 i 5).abs).hi) ≤ plusEpsilonNumerator
+  simp only [residualPlus05_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/ResidualPlus06.lean

@@ -0,0 +1,45 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+open scoped BigOperators
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+open AtomicTriangular.Interval
+
+def residualPlus06 : Fin 20 → RI := ![
+  ⟨-383243203487448504963768174695517186892859608901, 383243211044966084192919980745900195060776081711⟩,
+  ⟨-7185506364147700712906670822935041926990155067403, 7185506380086039285876825517678033034629177506023⟩,
+  ⟨-1056202899256588859762616913513321902873468122555, 1056202916467315225870564018725708103516110418998⟩,
+  ⟨-5812397907788388440738165803211151486160998413029, 5812397915999101164563258914551028063514381272808⟩,
+  ⟨-291156216157643949598115741793536671347341887349, 291156210494158999530002874157180904224869003609⟩,
+  ⟨-1415141805999633892548069789680233582405760232339, 1415141808552932225680035679731816780679674450798⟩,
+  ⟨-552376716557480007032250009528066098218105868, 552376821115501220265636049175044300923341554⟩,
+  ⟨-2799752720574402875684171685334711464776803159, 2799752425834149533639339968797655636127902307⟩,
+  ⟨-41169989666708147830859386792567783122065194, 41169998547894574167659638159289678021668550⟩,
+  ⟨-322949137707230429099269736428255944803773238, 322949038171944284338126026536270260546944680⟩,
+  ⟨-42345987939453363785548325391850195980459187, 42345994103130979811438061948111777974539419⟩,
+  ⟨-249761146037945510747260193556455022178283085, 249761174839488254834736690125011367411528389⟩,
+  ⟨-16507742580349349596001273637017180163623426, 16507739028440453639482559223105792619891694⟩,
+  ⟨-69376632852119493032838488914505737525484363, 69376635561293802496176520221600799650671016⟩,
+  ⟨-17278199788771282225778756708158010250933, 17278243746962966090650046247173938757235⟩,
+  ⟨-73157767226331600583430083788161833153654, 73157596903408794256999507592575788111461⟩,
+  ⟨-730184443095850797038402212900927837159, 730186709202876303605530508339093882521⟩,
+  ⟨-8096111471168391457592512768406991504356, 8096136633509250284487632810489445135071⟩,
+  ⟨-260150231865041065003177024067823706527, 260151644090246599989379328527207146334⟩,
+  ⟨-428570784826705111227161928024649632857, 428571594612866197769461530237595135493⟩]
+
+theorem residualPlus06_exact (i : Fin 20) :
+    residual 0 i 6 = residualPlus06 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualPlus06_bound :
+    matrixColBound (residual 0) 6 ≤ plusEpsilonNumerator := by
+  change (∑ i : Fin 20, ((residual 0 i 6).abs).hi) ≤ plusEpsilonNumerator
+  simp only [residualPlus06_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/ResidualPlus07.lean

@@ -0,0 +1,45 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+open scoped BigOperators
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+open AtomicTriangular.Interval
+
+def residualPlus07 : Fin 20 → RI := ![
+  ⟨-1476401450062268960191229083903871085779666054504, 1476401467461328792582646337438049772389988005745⟩,
+  ⟨-27398134642094839275970273140404794126202253904543, 27398134666352689721333172474916009804081235563566⟩,
+  ⟨-3805007448864831961649930166397190472968225383169, 3805007488839403832252758574989556618361182844447⟩,
+  ⟨-28690240522418407332252760404132432376318606141052, 28690240559830027425572703504127165119632845056882⟩,
+  ⟨-1236541102956134073565815017057564696095132757489, 1236541089666375649947461555569556620117746241166⟩,
+  ⟨-6581362405134075937764895071987909815899055498374, 6581362368595714355892598559953770892924896339891⟩,
+  ⟨-7066579932762450670963043997155934472911540014, 7066580181293980633973657936629325888074383492⟩,
+  ⟨-13226630265801894808027077946469657547147970941, 13226629588961324445070537376301153634683314009⟩,
+  ⟨-486357127218643582196105950736756479625328766, 486357148684208351117931594991916271118412409⟩,
+  ⟨-3789533686860343633929078401323711458540039273, 3789533454332413934511027224510252675678981289⟩,
+  ⟨-476763378952780145260846412247621258882108626, 476763393334093251582783781848974596914232387⟩,
+  ⟨-3495124008732250061392675462867105495917417269, 3495124076438679868973134653275205483729906690⟩,
+  ⟨-239032775019228954415586242985306003052169803, 239032766742711422871883797014725974109060149⟩,
+  ⟨-647603610024973044498553436375036889642168312, 647603616324516589113921470804400081102758924⟩,
+  ⟨-261153368430600929450534801134952347720461, 261153470581137754142986489067814910129588⟩,
+  ⟨-1136715057853472133090050395387980681850224, 1136714662561743765157812966598784116627261⟩,
+  ⟨-10508410974721728035230019659314465566806, 10508416228391166631885687802939763204076⟩,
+  ⟨-126395697478808381628867596748863211361921, 126395755838353398589777515734904441710230⟩,
+  ⟨-4223234090654924016698692874594024965230, 4223237363977534545768978403830101716248⟩,
+  ⟨-6478930129362424440777824625825887097890, 6478932005288905078886421937040895092065⟩]
+
+theorem residualPlus07_exact (i : Fin 20) :
+    residual 0 i 7 = residualPlus07 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualPlus07_bound :
+    matrixColBound (residual 0) 7 ≤ plusEpsilonNumerator := by
+  change (∑ i : Fin 20, ((residual 0 i 7).abs).hi) ≤ plusEpsilonNumerator
+  simp only [residualPlus07_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/ResidualPlus08.lean

@@ -0,0 +1,45 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+open scoped BigOperators
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+open AtomicTriangular.Interval
+
+def residualPlus08 : Fin 20 → RI := ![
+  ⟨-217486972653906511339291843808021261346650910252, 217486969522051342776860773273607866737165131020⟩,
+  ⟨-5693601521527496746324285634924038518389122377110, 5693601515903410273193725597064057625186715280067⟩,
+  ⟨-459856991572431162390622870933461373987169051312, 459856984462967345739837185921909755514800059192⟩,
+  ⟨-4227271258937786618863135378230808201536835410171, 4227271253938416985080124063126602755785676750310⟩,
+  ⟨-155296672204479478104674074444817814214788833763, 155296674591091252441074464313205453566531980853⟩,
+  ⟨-940595606727513976713649535918466120220588288888, 940595613083471943074987994311239354649245613129⟩,
+  ⟨-474702450789367069893195157061834806468871817, 474702405663890192177439087108946610870691531⟩,
+  ⟨-1769459264918755692936340154579656850229992635, 1769459387594722091651641867621909520019108951⟩,
+  ⟨-19027874287224288422438025817648140393294300, 19027870391241253325610931659626039087549142⟩,
+  ⟨-234185296337154934894474936447508612821379549, 234185338630534141650380692126917493137184271⟩,
+  ⟨-29148088657133662666461089201946247092225767, 29148086044874288083773013414128045059139337⟩,
+  ⟨-132632732232685467613509437044969777246218613, 132632720002083869799138803210898235697176023⟩,
+  ⟨-10559753411820727934726500928313126132863749, 10559754915720456018788546429491639921828591⟩,
+  ⟨-45408109262128701683550553614740416824070651, 45408108130043899068572329026662968709764085⟩,
+  ⟨-9399127368372771542972145393818263938212, 9399108762199564963023303978062865542876⟩,
+  ⟨-42439840907559804465520182505549755968105, 42439912911755254382550328127259657481159⟩,
+  ⟨-448080816145397081695359824517777170856, 448079858873264754070742811237777867762⟩,
+  ⟨-4350710920591338192795723330289418874493, 4350700288964454893746029725416915216302⟩,
+  ⟨-135460756165072368863883083262023871019, 135460159774107639353246155997800899328⟩,
+  ⟨-244087818591875066764394909855001795847, 244087476851547732431603139803088161978⟩]
+
+theorem residualPlus08_exact (i : Fin 20) :
+    residual 0 i 8 = residualPlus08 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualPlus08_bound :
+    matrixColBound (residual 0) 8 ≤ plusEpsilonNumerator := by
+  change (∑ i : Fin 20, ((residual 0 i 8).abs).hi) ≤ plusEpsilonNumerator
+  simp only [residualPlus08_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/ResidualPlus09.lean

@@ -0,0 +1,45 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+open scoped BigOperators
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+open AtomicTriangular.Interval
+
+def residualPlus09 : Fin 20 → RI := ![
+  ⟨-1158131167177430320448501256405473288313835944252, 1158131181599960263635520937997935305971009018259⟩,
+  ⟨-24087227238281450349213252393291764529925547552101, 24087227260667857899592331104484034452960338666078⟩,
+  ⟨-2762860483678995419486961620876215973380435280784, 2762860516727264464340622634399012474554214456921⟩,
+  ⟨-25755208631930160762622735722896990795407425157573, 25755208661133185235740288992028866175758100398290⟩,
+  ⟨-892188722621374317942170531193268401932275971805, 892188711601148615804467557428925166415693297006⟩,
+  ⟨-7362416670937795522268635052053431254787895878463, 7362416640123777006750390969971282050684078800793⟩,
+  ⟨-6329881583402177422072704164766336738911869141, 6329881790552734670205462096418723451206067631⟩,
+  ⟨-10112653197612517980711085921147657636949118123, 10112652633580757046173934618016360674378451095⟩,
+  ⟨-493236546783873099385187142610033517507546742, 493236564681778506135769630588751243589420232⟩,
+  ⟨-3461578957139926858814107459442471257384891792, 3461578763256835291162565012313258705977824329⟩,
+  ⟨-428678044925473994640708416292696350874634187, 428678056908503340113483093031715327240738957⟩,
+  ⟨-3400880053368951339517730848141606703387041739, 3400880109755987977862198788653333114002836646⟩,
+  ⟨-223078961599725106598135697089058410894760141, 223078954701613161645013801270536866177373752⟩,
+  ⟨-594795879515691276111752087948264768701044029, 594795884748244973121538376091974605389995717⟩,
+  ⟨-252425536943455990472716877253890827131811, 252425622163185802770239540624993966582877⟩,
+  ⟨-1077988759225774206768613800951223698949001, 1077988429455370309860416801312967063083596⟩,
+  ⟨-9825240882693479340883204834564116648311, 9825245265692067791339358742903984978677⟩,
+  ⟨-121362700631940535138229547741576264466616, 121362749319680877608098693973377278254010⟩,
+  ⟨-4029569754794476459446429839604467684569, 4029572485725338502267018325569744540246⟩,
+  ⟨-6252520632146494840811755767099832574224, 6252522197114064965257321414206225236318⟩]
+
+theorem residualPlus09_exact (i : Fin 20) :
+    residual 0 i 9 = residualPlus09 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualPlus09_bound :
+    matrixColBound (residual 0) 9 ≤ plusEpsilonNumerator := by
+  change (∑ i : Fin 20, ((residual 0 i 9).abs).hi) ≤ plusEpsilonNumerator
+  simp only [residualPlus09_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/ResidualPlus10.lean

@@ -0,0 +1,45 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+open scoped BigOperators
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+open AtomicTriangular.Interval
+
+def residualPlus10 : Fin 20 → RI := ![
+  ⟨-214544260691037519092034541525758925752934434743, 214544257835396441489380594317612079281479170037⟩,
+  ⟨-6193756035069058409100972526412805129587696663624, 6193756028289591622055166662602559610418772091271⟩,
+  ⟨-270813600464984670712670318354602125005303040973, 270813594058826196915944090091263009239926724659⟩,
+  ⟨-5577732957381857485045782239197643209860731749507, 5577732953706296273124873780172164494312513574220⟩,
+  ⟨-84341677724130853194582832951426085904505555284, 84341679894369753729768318119880030942602317618⟩,
+  ⟨-1396945258759350874908226115302112074674322223256, 1396945262981828838149215539034985496992508598060⟩,
+  ⟨-474619334341909804447186622295138058172931641, 474619292829879255478703230508241539782818870⟩,
+  ⟨-1058937298827019518504833668604700758456658942, 1058937412438037074727888209747785253442208357⟩,
+  ⟨-19940358718492069984840059571785787427700012, 19940355147256922569164845430053299155043950⟩,
+  ⟨-235157483369122498446317572815869154593794091, 235157522415984890522845353582191813843668192⟩,
+  ⟨-25572591835553429438630568038179417437616194, 25572589424957946572947207986183719454453764⟩,
+  ⟨-139976284420887684664916714073381202069617513, 139976273175889262803820488640895714586114244⟩,
+  ⟨-10108926107013526652302212849725128764759817, 10108927495056072364567897118185329384672810⟩,
+  ⟨-39181744567019420937209000891743074663300416, 39181743529594976982026616686768533303243752⟩,
+  ⟨-7077866870481299589143460362085509147352, 7077849649804824146224689185335310761369⟩,
+  ⟨-40125565167268519069888994048129386543716, 40125631825742834383777633866049230948602⟩,
+  ⟨-413269222803625029518630008072423943647, 413268336440170371382284942409108577205⟩,
+  ⟨-3544519760644792152605347797528216613413, 3544509916235658216196908222972139282220⟩,
+  ⟨-120277952222589431937441146829505769666, 120277399875050155774967147387819874859⟩,
+  ⟨-219452962241016694854637569415630583805, 219452645756945262572518670451649341165⟩]
+
+theorem residualPlus10_exact (i : Fin 20) :
+    residual 0 i 10 = residualPlus10 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualPlus10_bound :
+    matrixColBound (residual 0) 10 ≤ plusEpsilonNumerator := by
+  change (∑ i : Fin 20, ((residual 0 i 10).abs).hi) ≤ plusEpsilonNumerator
+  simp only [residualPlus10_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/ResidualPlus11.lean

@@ -0,0 +1,45 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+open scoped BigOperators
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+open AtomicTriangular.Interval
+
+def residualPlus11 : Fin 20 → RI := ![
+  ⟨-939781633570194275503234717577707247402806106488, 939781644665924667608631431435382887669969998117⟩,
+  ⟨-20505161460501689844528795459042290989423125524211, 20505161480387339560114081592411815063707157855067⟩,
+  ⟨-1715284715415380426049796668376613165288341815608, 1715284740618668959821477594624040390388622662317⟩,
+  ⟨-22522879703797515501854327468596358863212194284801, 22522879723660091169494158524626328230537272095831⟩,
+  ⟨-545413643939785775516662716619082999460198928617, 545413635458612530257941055017445663570500686879⟩,
+  ⟨-8080416329057546018643952965441188580325129314711, 8080416303935989445897316135208963368011000421112⟩,
+  ⟨-4862589360767141182356800301058140487240583968, 4862589521557928144161382122191250087678652227⟩,
+  ⟨-6501329328232723787276076097474568550088162817, 6501328892706804159683227487448120791654141373⟩,
+  ⟨-453066950679478908311300054987699215638422750, 453066964584909836232980821656051344208636324⟩,
+  ⟨-2805228702534489944567031917888296165563206295, 2805228552012128413397837384634108599349127520⟩,
+  ⟨-344947798425985401152715388347734419116304120, 344947807718203267968417210207169521252044227⟩,
+  ⟨-3115265267365409516111959878090527274745613008, 3115265310890006784795740695969637271684295580⟩,
+  ⟨-191476135761511546663173035230470923476703852, 191476130410334842877454338123397735776812970⟩,
+  ⟨-536347659496722995945843254339478575554719835, 536347663515392979053007867777606677674882088⟩,
+  ⟨-231685090083278386272481896725060313756035, 231685156294974152673382231325412448640121⟩,
+  ⟨-956769050938433842522543198799016661633394, 956768794734987296584666718803970456147552⟩,
+  ⟨-8510538361674312795299445345211242514061, 8510541766996049413603889450565997769870⟩,
+  ⟨-110730358792832648141069867301581239099743, 110730396620376015276432555996554441396167⟩,
+  ⟨-3639881146400753938434877487825326681283, 3639883268261337518066854553480695340059⟩,
+  ⟨-5782428358059059385180329379232372002685, 5782429573821239424395367432806761912229⟩]
+
+theorem residualPlus11_exact (i : Fin 20) :
+    residual 0 i 11 = residualPlus11 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualPlus11_bound :
+    matrixColBound (residual 0) 11 ≤ plusEpsilonNumerator := by
+  change (∑ i : Fin 20, ((residual 0 i 11).abs).hi) ≤ plusEpsilonNumerator
+  simp only [residualPlus11_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/ResidualPlus12.lean

@@ -0,0 +1,45 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+open scoped BigOperators
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+open AtomicTriangular.Interval
+
+def residualPlus12 : Fin 20 → RI := ![
+  ⟨-210450151893874879886445719840635267084179365078, 210450148238809494137342475574195425017101789391⟩,
+  ⟨-8301179141617323028384424750551307940891333213160, 8301179132340674334530166969631445843369797200803⟩,
+  ⟨-237141132368278717036049573743872824014459696148, 237141124160864768509632800654502247121999211826⟩,
+  ⟨-6122438940892740469738580660644021543424124016725, 6122438937146263093401751724962106164766576086690⟩,
+  ⟨-70490755694483206229251770833154148562824062314, 70490758469161351812404696908377297521822611278⟩,
+  ⟨-1437804308668194006538710092900405304412902454976, 1437804313765504801932547395127268781247670484332⟩,
+  ⟨-448483370967978629888072745341512334232663377, 448483317687869275482914782135266146462114117⟩,
+  ⟨-858978589007434790459741002423660451842874017, 858978735578647785052075869755806107477360038⟩,
+  ⟨-23182543696687259251872658528893174087547510, 23182539115254080910410324911296799302518272⟩,
+  ⟨-224714902080680737575229243095615788668909000, 224714952241706694283099908596725524759464009⟩,
+  ⟨-23436976422028124536254105077159300757801073, 23436973326547754497013486450676524293229579⟩,
+  ⟨-139314480981386001433943080821578391761214265, 139314466496023570523770800247899263483459808⟩,
+  ⟨-9650537022678562835524325837192016663973950, 9650538805070727006836549283688904504194944⟩,
+  ⟨-35787809048802279915547127745800379008037643, 35787807714570274469193474857382134168765200⟩,
+  ⟨-6678973381071106702503466046369391508331, 6678951247408664239401010521847680192831⟩,
+  ⟨-38501582808111730617781991762706248186885, 38501668487429640799471535383006773968057⟩,
+  ⟨-391391368692462117241475909992897051707, 391390229140206093301474863380798966737⟩,
+  ⟨-3256866403397995132411605812708850772601, 3256853749217078104206763452653865536686⟩,
+  ⟨-113880081926487634725642225516814745348, 113879371891178261579259557505808313909⟩,
+  ⟨-206929786288720346952954984672446400737, 206929379468154529051549248500566171576⟩]
+
+theorem residualPlus12_exact (i : Fin 20) :
+    residual 0 i 12 = residualPlus12 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualPlus12_bound :
+    matrixColBound (residual 0) 12 ≤ plusEpsilonNumerator := by
+  change (∑ i : Fin 20, ((residual 0 i 12).abs).hi) ≤ plusEpsilonNumerator
+  simp only [residualPlus12_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/ResidualPlus13.lean

@@ -0,0 +1,45 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+open scoped BigOperators
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+open AtomicTriangular.Interval
+
+def residualPlus13 : Fin 20 → RI := ![
+  ⟨-861248641284464651129281221823415333879910028270, 861248652750447245427662964277991026788310402385⟩,
+  ⟨-22361370694536838231959029175514465680313734629447, 22361370716833271946084888510194571897947257489029⟩,
+  ⟨-1464176216671115861074099128186691213973407471226, 1464176242750378179850962193622152056615501807608⟩,
+  ⟨-23623862338061988413134816065811864932881873261798, 23623862356475523341667320472857769048067283820686⟩,
+  ⟨-468759365925129418706510371110302101651544570431, 468759357159026170164987307093646867098565944572⟩,
+  ⟨-8655958155876706272482512778882114595758724927892, 8655958129594581108362152558520443080765278629089⟩,
+  ⟨-4404441190088299869231749926005594397496334037, 4404441357093394155495742853483668077435646226⟩,
+  ⟨-5887296490813242709481288517419296934228057138, 5887296036799193923824883233409487556932045575⟩,
+  ⟨-431849022013122256335384551436438078927138182, 431849036460875981368297288435943137307939153⟩,
+  ⟨-2623719432521844886432191688855565124304025422, 2623719276130082101230604958377553887009890406⟩,
+  ⟨-320682716448481614822506104819567770788720479, 320682726096876579186198755984589530104081197⟩,
+  ⟨-3020522538665538844974718601121637993187774030, 3020522584014400971244941485419890958142999972⟩,
+  ⟨-181737785738893900054448917406774793449726011, 181737780181309750119486747981499170775095548⟩,
+  ⟨-535732244545094976187527805747802729074211123, 535732248721795625204888667947835154154971089⟩,
+  ⟨-225154609764882870229900800285321433369502, 225154678594335284281442453142494431189197⟩,
+  ⟨-920829440050220844715984152658752179623716, 920829173721191724904069590350850278306608⟩,
+  ⟨-8114580413846383413793560096341367107473, 8114583953864963906524720604667393472556⟩,
+  ⟨-107616703753271554905505590513237893335923, 107616743077068814673017543203432343888075⟩,
+  ⟨-3525689424139172505178981470593508971135, 3525691629997496575455400546215532907822⟩,
+  ⟨-5661522386623356999764030292743443710999, 5661523650426629547674511921697733380478⟩]
+
+theorem residualPlus13_exact (i : Fin 20) :
+    residual 0 i 13 = residualPlus13 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualPlus13_bound :
+    matrixColBound (residual 0) 13 ≤ plusEpsilonNumerator := by
+  change (∑ i : Fin 20, ((residual 0 i 13).abs).hi) ≤ plusEpsilonNumerator
+  simp only [residualPlus13_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/ResidualPlus14.lean

@@ -0,0 +1,45 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+open scoped BigOperators
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+open AtomicTriangular.Interval
+
+def residualPlus14 : Fin 20 → RI := ![
+  ⟨-199608528033815933371916145177801170991918060216, 199608519810843696538265178817502406209105605825⟩,
+  ⟨-21349131384365850432849691272551054948249881026154, 21349131352489502504616104868465715666066168312917⟩,
+  ⟨-177455037834431923987459438668992047664413816210, 177455019480231359428037569866562478778694354501⟩,
+  ⟨-5773496604588831668554025524105755259133612262681, 5773496580622354426012417085492885895762126779057⟩,
+  ⟨-46199511034695740329926637169072921771918517295, 46199517411277186171324698671235899969915437786⟩,
+  ⟨-979997781052722604605759788477864850715913733465, 979997792022646313767303572416122715734718995689⟩,
+  ⟨-358199259704745596394937124360168357627207521, 358199133253287292205655901686837584893604335⟩,
+  ⟨-930270858134038493396575688646956400800606213, 930271207209945359938893066047687368116222485⟩,
+  ⟨-28790204107609648706896856191258540141715582, 28790193248346231621678856365610058661785435⟩,
+  ⟨-180423456273411620428203465250230096050703729, 180423575849870513860193421228031428690452415⟩,
+  ⟨-16551404904253067799875340259807963276416523, 16551397546773587747005991496915130081797981⟩,
+  ⟨-131725830766888178195444495969658506917028521, 131725796498708612047781387060592151315341064⟩,
+  ⟨-8102075147474099123985670014406256163178291, 8102079389310188515637004516183519060522997⟩,
+  ⟨-24461114015752724609550228025614250852442176, 24461110903391533932262296271097160247428504⟩,
+  ⟨-6819667103876980512239534416416764669644, 6819614129130050959858470460339619487842⟩,
+  ⟨-33982815486266133015086585590850393956778, 33983020575195129530764550147334223170450⟩,
+  ⟨-325042546421138151485760670158681787880, 325039818140857616010633283563140291785⟩,
+  ⟨-3018191593671423927143635083045936199855, 3018161295275195469286902225236114891207⟩,
+  ⟨-101476821439889357259021701575345817119, 101475120914281770814867048751253263457⟩,
+  ⟨-174157842107777341338804404554407717978, 174156868092233162811504727028628628771⟩]
+
+theorem residualPlus14_exact (i : Fin 20) :
+    residual 0 i 14 = residualPlus14 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualPlus14_bound :
+    matrixColBound (residual 0) 14 ≤ plusEpsilonNumerator := by
+  change (∑ i : Fin 20, ((residual 0 i 14).abs).hi) ≤ plusEpsilonNumerator
+  simp only [residualPlus14_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/ResidualPlus15.lean

@@ -0,0 +1,45 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+open scoped BigOperators
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+open AtomicTriangular.Interval
+
+def residualPlus15 : Fin 20 → RI := ![
+  ⟨-650945724157420636897084609190134538339093092807, 650945744441334392902781474328956593372675379454⟩,
+  ⟨-44333367588574892926154996131912978713724064095442, 44333367636469854895800890586050178501103545981383⟩,
+  ⟨-871788917397750377162587514213058452358532517783, 871788963144724290441052343236943631704379378240⟩,
+  ⟨-33264749741931278871743858241580274630145455750120, 33264749761642624666460200470660801796454262304491⟩,
+  ⟨-300972322183246881038314157993018257667908749933, 300972306716436462709834751887631448907971910260⟩,
+  ⟨-10338176094403358774886476591335887128915460377452, 10338176050533133527979968339970025920279406174522⟩,
+  ⟨-3197957702971965659052486302956719281248160228, 3197958000777957357932464516446643440226703477⟩,
+  ⟨-5473284291259768386433895345761471550002396284, 5473283479694551931594138956253599243652615448⟩,
+  ⟨-352892988985007670734382742910097938376050085, 352893014731039361960048193514260362188555019⟩,
+  ⟨-2176693294538397286740827065795831066594060289, 2176693015101789025473476179104709983800965858⟩,
+  ⟨-260744166334536613070248820098211467247353053, 260744183552811814258939838832616867294396863⟩,
+  ⟨-2726427490720481816127017115113500756843069942, 2726427571495230817024506111635838055906134759⟩,
+  ⟨-155094264143585294002609858852301614878331960, 155094254220670958750873397800888313821587787⟩,
+  ⟨-525212168139309480140431951888317302164628527, 525212175542933980252225054204220979715708407⟩,
+  ⟨-205071173801817536615983507525073158660714, 205071296984002882985567375078351128266990⟩,
+  ⟨-820687752426240290598345654822792327800796, 820687275748332670576869439396928848287178⟩,
+  ⟨-7040914664105178820659581835982201545317, 7040921001141533527574707187836509224352⟩,
+  ⟨-98655276920283411142450478332858881104877, 98655347311084257346430225439829570258245⟩,
+  ⟨-3205520919987791510401192352777788152699, 3205524869031559082540598386900049151151⟩,
+  ⟨-5329659317943845271204905571344001576113, 5329661580204930778689577336368444871689⟩]
+
+theorem residualPlus15_exact (i : Fin 20) :
+    residual 0 i 15 = residualPlus15 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualPlus15_bound :
+    matrixColBound (residual 0) 15 ≤ plusEpsilonNumerator := by
+  change (∑ i : Fin 20, ((residual 0 i 15).abs).hi) ≤ plusEpsilonNumerator
+  simp only [residualPlus15_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/ResidualPlus16.lean

@@ -0,0 +1,45 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+open scoped BigOperators
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+open AtomicTriangular.Interval
+
+def residualPlus16 : Fin 20 → RI := ![
+  ⟨-196632328840299715513559273988508299593483218553, 196632320925339248802178865165964829984541780015⟩,
+  ⟨-21379204697098745568431502121152043363881626298879, 21379204665462404174299209399992725766245329919617⟩,
+  ⟨-169113588510528404364569040743096925759506552914, 169113570863920129584128622305722039108811393397⟩,
+  ⟨-9936648995195672126052667412291078798938230898638, 9936648970823439901668919766513307857326207785223⟩,
+  ⟨-47969931473210791271599833793043215339284768801, 47969937629914652539833760144879588797436596986⟩,
+  ⟨-1851250091684918153793740342797437842363870978058, 1851250103783419242461009626173725844665499503747⟩,
+  ⟨-282849902544592576994226851633498828177828264, 282849779775684579761663472749768633436625622⟩,
+  ⟨-1110970131427486115241821650998992595518628676, 1110970469545527373259983904917049102925722906⟩,
+  ⟨-29682413917275673246411470078410252466784858, 29682403359658452897526241046162526914628359⟩,
+  ⟨-146025921577585333727044109186023830330343888, 146026037593294043023995575157657276241164216⟩,
+  ⟨-10805385747805501047040478096254588443662576, 10805378614417897015959964339222356732790401⟩,
+  ⟨-124566030243159992474618563946261772142319423, 124565997005346805964497027311624686630593353⟩,
+  ⟨-6534032617744747038249999086558754328657658, 6534036731285134373231206441241499669461617⟩,
+  ⟨-28754228116333976964613174451127603063539839, 28754225107462652028881216272349157405371916⟩,
+  ⟨-7229809635387566922626560742332692457271, 7229758221595573493535164021485976359754⟩,
+  ⟨-30358512729393205902447687210427427143265, 30358711757318795603075897681650614705321⟩,
+  ⟨-262240034362717903161009512754226712858, 262237387413588569562636377667499457561⟩,
+  ⟨-3043666041081785590597826302969204870874, 3043636638553152995960440056138873065199⟩,
+  ⟨-97417267356623877935549611034860411989, 97415617135639466483750105864205881623⟩,
+  ⟨-149500978153163195019959228508390477295, 149500033036781259087763453108018137326⟩]
+
+theorem residualPlus16_exact (i : Fin 20) :
+    residual 0 i 16 = residualPlus16 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualPlus16_bound :
+    matrixColBound (residual 0) 16 ≤ plusEpsilonNumerator := by
+  change (∑ i : Fin 20, ((residual 0 i 16).abs).hi) ≤ plusEpsilonNumerator
+  simp only [residualPlus16_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/ResidualPlus17.lean

@@ -0,0 +1,45 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+open scoped BigOperators
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+open AtomicTriangular.Interval
+
+def residualPlus17 : Fin 20 → RI := ![
+  ⟨-511327553712155001004396215953603078307398982512, 511327488136553935769225289005960897759466327528⟩,
+  ⟨-70654946519948628898459161525565852202067376771678, 70654946490476749800972707099699785180746412209788⟩,
+  ⟨-591424167685222724475466961218111710812562484132, 591424013512504598273800089470329855191022314454⟩,
+  ⟨-36644879669228121499468484213172709654136165151830, 36644879536485893529370352658558530171246717244881⟩,
+  ⟨-229360566575130870467163132806897865489917478361, 229360615436273129862425171669233705516563590209⟩,
+  ⟨-9974664789612955204672686068375731760356075439839, 9974664747258581858906627504706358445078931592489⟩,
+  ⟨-2393983488228821081659647182082179120946479120, 2393982634990408037121507486815531635911098033⟩,
+  ⟨-6544209998384289506140162395578549640271704416, 6544212405764347647671791388655162499140074089⟩,
+  ⟨-272923899867968212322668152336825463319948129, 272923827689077876534970711352467122767607105⟩,
+  ⟨-1935084370136860827566624930229735116872474333, 1935085179409251383921550851774367568015079116⟩,
+  ⟨-232173421016292031527550079300858706127172796, 232173370533535171867352336500503370494536471⟩,
+  ⟨-2488764384330120197358308807228927252536098623, 2488764147538688200740301650421042575228082194⟩,
+  ⟨-138551300135803708386429615442046793381664042, 138551329151686381541875947910689893013337474⟩,
+  ⟨-506733813429239640978928742737178359106384613, 506733790519489023652407276991901969909258545⟩,
+  ⟨-189704625214292076775873249634148392497921, 189704269865489038149886237963972516251131⟩,
+  ⟨-757336513298779935913880275870046371321366, 757337890385937459040402319421930121988666⟩,
+  ⟨-6400225346292757599006999496258123532552, 6400207029446233201813599506354929658991⟩,
+  ⟨-92674729762211817985681615759335031346518, 92674526387904830118416019152799303317337⟩,
+  ⟨-3003134662837911109103561563728261285628, 3003123252479673076672714691875246009629⟩,
+  ⟨-5141136205013378119074305275804509335188, 5141129656673190984961979022098299532404⟩]
+
+theorem residualPlus17_exact (i : Fin 20) :
+    residual 0 i 17 = residualPlus17 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualPlus17_bound :
+    matrixColBound (residual 0) 17 ≤ plusEpsilonNumerator := by
+  change (∑ i : Fin 20, ((residual 0 i 17).abs).hi) ≤ plusEpsilonNumerator
+  simp only [residualPlus17_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/ResidualPlus18.lean

@@ -0,0 +1,45 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+open scoped BigOperators
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+open AtomicTriangular.Interval
+
+def residualPlus18 : Fin 20 → RI := ![
+  ⟨-195921953636906696882221757636576840832850049080, 195921940246287321994059657553150870777295815238⟩,
+  ⟨-35600788570703554232001701747006678084061379656211, 35600788517277988690664212642036780447515180051999⟩,
+  ⟨-188913151200233984432255315052478318918126303654, 188913120752673442800772473537558860136636955713⟩,
+  ⟨-9492539474917390159683765032748697912392455371772, 9492539435757579749501107160237396852760664556893⟩,
+  ⟨-55281060226258717139847252544413722655495586794, 55281070633613002196012860909163943165194825154⟩,
+  ⟨-2157470936263636057436138580472363388757426261968, 2157470954314130029300893758941020504868830107862⟩,
+  ⟨-246249934720966749465736541887698970973250022, 246249727601847711700438373049347967472905460⟩,
+  ⟨-1195021536383465838330454797555881613500591506, 1195022107857415491051829112332532456691965823⟩,
+  ⟨-28876904220294159541772771807469424580599478, 28876886425519676406837854662582455683340506⟩,
+  ⟨-127032938251816391808512781757327883730173679, 127033134125147222768137497414028988496752644⟩,
+  ⟨-14684292511653168737048836233970993280463827, 14684280464468107980629866521664157800889143⟩,
+  ⟨-126686920257163886074946143806249094798200528, 126686864145347021844100403030031655563659703⟩,
+  ⟨-5673375181552336684674230846297302903063651, 5673382127150448959981255027551877128711799⟩,
+  ⟨-34056409082588028333968436325866248783866903, 34056403996290917499404113217094758988497417⟩,
+  ⟨-7564936186906068311740565830792394054449, 7564849380864349180591092461419947214647⟩,
+  ⟨-28933908160359222933114311685862720659980, 28934244214979366794329457988852549614746⟩,
+  ⟨-231457911662925222646828491935688425217, 231453441398056520116824742256614403803⟩,
+  ⟨-3194132235142025609520374422178014825662, 3194082587827666796445917407610421811899⟩,
+  ⟨-99057783037068224734884672310228891431, 99054996508557712632629858567802372262⟩,
+  ⟨-155149034898261118315622556427653542844, 155147438898976887103928064862501975948⟩]
+
+theorem residualPlus18_exact (i : Fin 20) :
+    residual 0 i 18 = residualPlus18 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualPlus18_bound :
+    matrixColBound (residual 0) 18 ≤ plusEpsilonNumerator := by
+  change (∑ i : Fin 20, ((residual 0 i 18).abs).hi) ≤ plusEpsilonNumerator
+  simp only [residualPlus18_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/ResidualPlus19.lean

@@ -0,0 +1,45 @@
+import OAI.Analysis.Triangular.Certificate.AtomicResidual
+
+namespace OAI
+
+open scoped BigOperators
+
+namespace AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+open AtomicTriangular.Interval
+
+def residualPlus19 : Fin 20 → RI := ![
+  ⟨-619886860193411409473982091756940193385456292726, 619886830792804078399996579713598607948238639409⟩,
+  ⟨-81152422545433073515540742815682974668983629219913, 81152422424069584278133567718816075590852480067953⟩,
+  ⟨-814883809835990753552194005120340839496819141874, 814883743353012758510408852357154903849036457430⟩,
+  ⟨-41809199868340075473716280781825235074104098782299, 41809199777176385989282509022841528820136307814037⟩,
+  ⟨-298045129286504856782002354025270698267105871261, 298045152200224365949068396126337386827937859325⟩,
+  ⟨-10166769794437190668124483828959175089869453102357, 10166769841062936781673315230675051654496135149960⟩,
+  ⟨-2337400933284266379118765047324034482128219308, 2337400474509916073402011797398459507262765977⟩,
+  ⟨-8342238477303014165594760995985377283573954584, 8342239735776499459490016878290467369449309052⟩,
+  ⟨-240882081248925795074059314470565237457023736, 240882041594040534903465909254617655391187471⟩,
+  ⟨-2021810676045790820199973695142844602901616644, 2021811111646850172287962029409611144002470145⟩,
+  ⟨-249138400077472398172384678167162691284656036, 249138373429971938297131820926504057384580602⟩,
+  ⟨-2495756676024715427351503187997328634634740151, 2495756551769920092548766514658407978053538138⟩,
+  ⟨-142718715008438609903177647022612775877835995, 142718730452194113376652112651289598387898652⟩,
+  ⟨-514349876406939755929993631870879671423367483, 514349865234419135134652535371761538538137940⟩,
+  ⟨-192131767592121969032774450995366816457563, 192131574460395468175426945684689537680632⟩,
+  ⟨-779498274697651406076858651896476264208173, 779499022308123043587105327685203088719900⟩,
+  ⟨-6633205655477896379331057230564473036855, 6633195711084109889706339106179893399317⟩,
+  ⟨-95055521647285779410994992703198397456872, 95055411199697415545509590371219211681557⟩,
+  ⟨-3092282266458174211204531785906692991273, 3092276067508684309909150393439480084061⟩,
+  ⟨-5320448442262813711463942121995825969838, 5320444892113181216112631604089798917805⟩]
+
+theorem residualPlus19_exact (i : Fin 20) :
+    residual 0 i 19 = residualPlus19 i := by
+  fin_cases i <;> decide +kernel
+
+theorem residualPlus19_bound :
+    matrixColBound (residual 0) 19 ≤ plusEpsilonNumerator := by
+  change (∑ i : Fin 20, ((residual 0 i 19).abs).hi) ≤ plusEpsilonNumerator
+  simp only [residualPlus19_exact]
+  decide +kernel
+
+end AtomicTriangular.FiniteCertificate.AtomicMatrix
+
+end OAI

lean/OAI/Analysis/Triangular/Certificate/TargetInterval.lean

@@ -0,0 +1,343 @@
+import OAI.Analysis.Triangular.Envelope.TargetEnvelope
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators
+namespace AtomicTriangular.FiniteCertificate
+open Construction AtomicTriangular.Interval
+def targetZBox : RI := (RI.point 391).divNat 1000
+lemma targetZBox_mem : targetZBox.Mem Z := by
+  simpa [Z,targetZBox] using RI.mem_divNat (RI.mem_point 391) (by norm_num : 0<1000)
+def targetRateBox (q : Fin 2) : RI := if q.val=0 then RI.point 0 else piBox.mul ((RI.point 1).divNat 5)
+lemma targetRateBox_mem (q : Fin 2) : (targetRateBox q).Mem (Real.pi*(damping q-h)) := by
+  fin_cases q
+  · simpa [targetRateBox,damping] using ri0_mem
+  · convert RI.mem_mul piBox_mem (RI.mem_divNat (RI.mem_point 1) (by norm_num : 0<5)) using 1 <;> norm_num [targetRateBox,damping,H,h]
+def targetExpInput (q : Fin 2) (n : ℤ) : RI :=
+  ((targetRateBox q).mul (RI.point n)).add ((RI.point (1-n)).mul targetZBox.invPos)
+lemma targetExpInput_mem (q : Fin 2) (n : ℤ) :
+    (targetExpInput q n).Mem (Real.pi*(damping q-h)*n+(1-n)/Z) := by
+  apply RI.mem_add (RI.mem_mul (targetRateBox_mem q) (RI.mem_point n))
+  rw [div_eq_mul_inv]
+  exact RI.mem_mul (by simpa using RI.mem_point (1-n))
+    (RI.mem_invPos targetZBox_mem (by decide +kernel))
+def targetEnvBox (q : Fin 2) (qb db eb : RI) : RI :=
+  ((QBox1.mul qb.invPos).mul (((RI.point 1).add targetZBox).add
+    (targetZBox.mul ((targetRateBox q).sub db).abs))).mul eb
+lemma targetEnvBox_mem (q : Fin 2) (n : ℤ) {qb db eb : RI}
+    (hq : qb.Mem (Q n)) (hd : db.Mem (D n))
+    (he : eb.Mem (Real.exp (Real.pi*(damping q-h)*n+(1-n)/Z))) (hp : 0<qb.lo) :
+    (targetEnvBox q qb db eb).Mem (targetEnvelope (damping q) n) := by
+  unfold targetEnvelope
+  rw [div_eq_mul_inv]
+  exact RI.mem_mul (RI.mem_mul (RI.mem_mul QBox1_mem (RI.mem_invPos hq hp))
+    (RI.mem_add (RI.mem_add ri1_mem targetZBox_mem)
+      (RI.mem_mul targetZBox_mem (RI.mem_abs (RI.mem_sub (targetRateBox_mem q) hd))))) he
+
+def targetExpf9 : RI := ⟨371583854405964955268998186837214768409740484407723760,371583854405964955268998186837214768409740484407730647⟩
+lemma targetExpf9_calc : (CI.ofReal (targetExpInput 1 9)).exp.re=targetExpf9 := by decide +kernel
+lemma targetExpf9_mem : targetExpf9.Mem (Real.exp (Real.pi*(damping 1-h)*(9:ℤ)+(1-(9:ℤ))/Z)) := by
+  rw [← targetExpf9_calc]
+  exact ri_exp_mem (targetExpInput_mem 1 9) (by decide +kernel)
+def targetEnvf9 : RI := ⟨79929592308069886255923802412064702646456854029325769,79929592308069886255923802412064702646459470384903306⟩
+lemma targetEnvf9_calc : targetEnvBox 1 QBox9 DBox9 targetExpf9=targetEnvf9 := by decide +kernel
+lemma targetEnvf9_mem : targetEnvf9.Mem (targetEnvelope (damping 1) 9) := by
+  rw [← targetEnvf9_calc]
+  apply targetEnvBox_mem 1 9 _ _ targetExpf9_mem (by decide +kernel)
+  · exact QBox9_mem
+  · exact DBox9_mem
+def targetExpf12 : RI := ⟨1138937577370360867550942274425471604845725879364144,1138937577370360867550942274425471604845725879364176⟩
+lemma targetExpf12_calc : (CI.ofReal (targetExpInput 1 12)).exp.re=targetExpf12 := by decide +kernel
+lemma targetExpf12_mem : targetExpf12.Mem (Real.exp (Real.pi*(damping 1-h)*(12:ℤ)+(1-(12:ℤ))/Z)) := by
+  rw [← targetExpf12_calc]
+  exact ri_exp_mem (targetExpInput_mem 1 12) (by decide +kernel)
+def targetEnvf12 : RI := ⟨737900748185172127488690301649742871122126409324216,737900748185172127488690301649742871122155359330918⟩
+lemma targetEnvf12_calc : targetEnvBox 1 QBox12 DBox12 targetExpf12=targetEnvf12 := by decide +kernel
+lemma targetEnvf12_mem : targetEnvf12.Mem (targetEnvelope (damping 1) 12) := by
+  rw [← targetEnvf12_calc]
+  apply targetEnvBox_mem 1 12 _ _ targetExpf12_mem (by decide +kernel)
+  · exact QBox12_mem
+  · exact DBox12_mem
+def targetExpf13 : RI := ⟨165442703236073607826867801824880264180323233409359,165442703236073607826867801824880264180323233409366⟩
+lemma targetExpf13_calc : (CI.ofReal (targetExpInput 1 13)).exp.re=targetExpf13 := by decide +kernel
+lemma targetExpf13_mem : targetExpf13.Mem (Real.exp (Real.pi*(damping 1-h)*(13:ℤ)+(1-(13:ℤ))/Z)) := by
+  rw [← targetExpf13_calc]
+  exact ri_exp_mem (targetExpInput_mem 1 13) (by decide +kernel)
+def targetEnvf13 : RI := ⟨72358775014941788641752679186749217513948768859638,72358775014941788641752679186749217513951534988133⟩
+lemma targetEnvf13_calc : targetEnvBox 1 QBox13 DBox13 targetExpf13=targetEnvf13 := by decide +kernel
+lemma targetEnvf13_mem : targetEnvf13.Mem (targetEnvelope (damping 1) 13) := by
+  rw [← targetEnvf13_calc]
+  apply targetEnvBox_mem 1 13 _ _ targetExpf13_mem (by decide +kernel)
+  · exact QBox13_mem
+  · exact DBox13_mem
+def targetExpf16 : RI := ⟨507096606548016980840453496425200306337042956607,507096606548016980840453496425200306337042956610⟩
+lemma targetExpf16_calc : (CI.ofReal (targetExpInput 1 16)).exp.re=targetExpf16 := by decide +kernel
+lemma targetExpf16_mem : targetExpf16.Mem (Real.exp (Real.pi*(damping 1-h)*(16:ℤ)+(1-(16:ℤ))/Z)) := by
+  rw [← targetExpf16_calc]
+  exact ri_exp_mem (targetExpInput_mem 1 16) (by decide +kernel)
+def targetEnvf16 : RI := ⟨14984081603111771939722335761054167345044608929,14984081603111771939722335761054167345045051454⟩
+lemma targetEnvf16_calc : targetEnvBox 1 QBox16 DBox16 targetExpf16=targetEnvf16 := by decide +kernel
+lemma targetEnvf16_mem : targetEnvf16.Mem (targetEnvelope (damping 1) 16) := by
+  rw [← targetEnvf16_calc]
+  apply targetEnvBox_mem 1 16 _ _ targetExpf16_mem (by decide +kernel)
+  · exact QBox16_mem
+  · exact DBox16_mem
+def targetExpf19 : RI := ⟨1554296220641329958753034193228726425123681830,1554296220641329958753034193228726425123681833⟩
+lemma targetExpf19_calc : (CI.ofReal (targetExpInput 1 19)).exp.re=targetExpf19 := by decide +kernel
+lemma targetExpf19_mem : targetExpf19.Mem (Real.exp (Real.pi*(damping 1-h)*(19:ℤ)+(1-(19:ℤ))/Z)) := by
+  rw [← targetExpf19_calc]
+  exact ri_exp_mem (targetExpInput_mem 1 19) (by decide +kernel)
+def targetEnvf19 : RI := ⟨42209558519464300166257608381105721464756918,42209558519464300166257608381105721464758160⟩
+lemma targetEnvf19_calc : targetEnvBox 1 QBox19 DBox19 targetExpf19=targetEnvf19 := by decide +kernel
+lemma targetEnvf19_mem : targetEnvf19.Mem (targetEnvelope (damping 1) 19) := by
+  rw [← targetEnvf19_calc]
+  apply targetEnvBox_mem 1 19 _ _ targetExpf19_mem (by decide +kernel)
+  · exact QBox19_mem
+  · exact DBox19_mem
+def targetExpf21 : RI := ⟨32796627117040229186147692193117594753946052,32796627117040229186147692193117594753946055⟩
+lemma targetExpf21_calc : (CI.ofReal (targetExpInput 1 21)).exp.re=targetExpf21 := by decide +kernel
+lemma targetExpf21_mem : targetExpf21.Mem (Real.exp (Real.pi*(damping 1-h)*(21:ℤ)+(1-(21:ℤ))/Z)) := by
+  rw [← targetExpf21_calc]
+  exact ri_exp_mem (targetExpInput_mem 1 21) (by decide +kernel)
+def targetEnvf21 : RI := ⟨490708736100926712270907069267985577233720,490708736100926712270907069267985577233735⟩
+lemma targetEnvf21_calc : targetEnvBox 1 QBox21 DBox21 targetExpf21=targetEnvf21 := by decide +kernel
+lemma targetEnvf21_mem : targetEnvf21.Mem (targetEnvelope (damping 1) 21) := by
+  rw [← targetEnvf21_calc]
+  apply targetEnvBox_mem 1 21 _ _ targetExpf21_mem (by decide +kernel)
+  · exact QBox21_mem
+  · exact DBox21_mem
+def targetEnvTablef : Fin 6 → RI := ![targetEnvf9,targetEnvf12,targetEnvf13,targetEnvf16,targetEnvf19,targetEnvf21]
+lemma targetEnvTablef_mem (i : Fin 6) : (targetEnvTablef i).Mem (targetEnvelope (damping 1) (finiteNode ⟨i.val+4,by omega⟩)) := by
+  fin_cases i
+  · exact targetEnvf9_mem
+  · exact targetEnvf12_mem
+  · exact targetEnvf13_mem
+  · exact targetEnvf16_mem
+  · exact targetEnvf19_mem
+  · exact targetEnvf21_mem
+def targetExpt36 : RI := ⟨1332093987218979628888,1332093987218979628891⟩
+lemma targetExpt36_calc : (CI.ofReal (targetExpInput 0 36)).exp.re=targetExpt36 := by decide +kernel
+lemma targetExpt36_mem : targetExpt36.Mem (Real.exp (Real.pi*(damping 0-h)*(36:ℤ)+(1-(36:ℤ))/Z)) := by
+  rw [← targetExpt36_calc]
+  exact ri_exp_mem (targetExpInput_mem 0 36) (by decide +kernel)
+def targetEnvt36 : RI := ⟨983699026673584249414,983699026673584249417⟩
+lemma targetEnvt36_calc : targetEnvBox 0 QBox0 DBox0 targetExpt36=targetEnvt36 := by decide +kernel
+lemma targetEnvt36_mem : targetEnvt36.Mem (targetEnvelope (damping 0) 36) := by
+  rw [← targetEnvt36_calc]
+  apply targetEnvBox_mem 0 36 _ _ targetExpt36_mem (by decide +kernel)
+  · rw [Q_emod 36]; norm_num; exact QBox0_mem
+  · rw [D_emod 36]; norm_num; exact DBox0_mem
+def targetExpt37 : RI := ⟨103230324563422963144,103230324563422963147⟩
+lemma targetExpt37_calc : (CI.ofReal (targetExpInput 0 37)).exp.re=targetExpt37 := by decide +kernel
+lemma targetExpt37_mem : targetExpt37.Mem (Real.exp (Real.pi*(damping 0-h)*(37:ℤ)+(1-(37:ℤ))/Z)) := by
+  rw [← targetExpt37_calc]
+  exact ri_exp_mem (targetExpInput_mem 0 37) (by decide +kernel)
+def targetEnvt37 : RI := ⟨156853136021545377530,156853136021545377536⟩
+lemma targetEnvt37_calc : targetEnvBox 0 QBox1 DBox1 targetExpt37=targetEnvt37 := by decide +kernel
+lemma targetEnvt37_mem : targetEnvt37.Mem (targetEnvelope (damping 0) 37) := by
+  rw [← targetEnvt37_calc]
+  apply targetEnvBox_mem 0 37 _ _ targetExpt37_mem (by decide +kernel)
+  · rw [Q_emod 37]; norm_num; exact QBox1_mem
+  · rw [D_emod 37]; norm_num; exact DBox1_mem
+def targetExpt39 : RI := ⟨619943545016735189,619943545016735192⟩
+lemma targetExpt39_calc : (CI.ofReal (targetExpInput 0 39)).exp.re=targetExpt39 := by decide +kernel
+lemma targetExpt39_mem : targetExpt39.Mem (Real.exp (Real.pi*(damping 0-h)*(39:ℤ)+(1-(39:ℤ))/Z)) := by
+  rw [← targetExpt39_calc]
+  exact ri_exp_mem (targetExpInput_mem 0 39) (by decide +kernel)
+def targetEnvt39 : RI := ⟨2314918595180738609,2314918595180738621⟩
+lemma targetEnvt39_calc : targetEnvBox 0 QBox3 DBox3 targetExpt39=targetEnvt39 := by decide +kernel
+lemma targetEnvt39_mem : targetEnvt39.Mem (targetEnvelope (damping 0) 39) := by
+  rw [← targetEnvt39_calc]
+  apply targetEnvBox_mem 0 39 _ _ targetExpt39_mem (by decide +kernel)
+  · rw [Q_emod 39]; norm_num; exact QBox3_mem
+  · rw [D_emod 39]; norm_num; exact DBox3_mem
+def targetExpt40 : RI := ⟨48042385880506405,48042385880506408⟩
+lemma targetExpt40_calc : (CI.ofReal (targetExpInput 0 40)).exp.re=targetExpt40 := by decide +kernel
+lemma targetExpt40_mem : targetExpt40.Mem (Real.exp (Real.pi*(damping 0-h)*(40:ℤ)+(1-(40:ℤ))/Z)) := by
+  rw [← targetExpt40_calc]
+  exact ri_exp_mem (targetExpInput_mem 0 40) (by decide +kernel)
+def targetEnvt40 : RI := ⟨140114137776113727,140114137776113737⟩
+lemma targetEnvt40_calc : targetEnvBox 0 QBox4 DBox4 targetExpt40=targetEnvt40 := by decide +kernel
+lemma targetEnvt40_mem : targetEnvt40.Mem (targetEnvelope (damping 0) 40) := by
+  rw [← targetEnvt40_calc]
+  apply targetEnvBox_mem 0 40 _ _ targetExpt40_mem (by decide +kernel)
+  · rw [Q_emod 40]; norm_num; exact QBox4_mem
+  · rw [D_emod 40]; norm_num; exact DBox4_mem
+def targetExpt43 : RI := ⟨22358457661085,22358457661088⟩
+lemma targetExpt43_calc : (CI.ofReal (targetExpInput 0 43)).exp.re=targetExpt43 := by decide +kernel
+lemma targetExpt43_mem : targetExpt43.Mem (Real.exp (Real.pi*(damping 0-h)*(43:ℤ)+(1-(43:ℤ))/Z)) := by
+  rw [← targetExpt43_calc]
+  exact ri_exp_mem (targetExpInput_mem 0 43) (by decide +kernel)
+def targetEnvt43 : RI := ⟨6589731472279,6589731472281⟩
+lemma targetEnvt43_calc : targetEnvBox 0 QBox7 DBox7 targetExpt43=targetEnvt43 := by decide +kernel
+lemma targetEnvt43_mem : targetEnvt43.Mem (targetEnvelope (damping 0) 43) := by
+  rw [← targetEnvt43_calc]
+  apply targetEnvBox_mem 0 43 _ _ targetExpt43_mem (by decide +kernel)
+  · rw [Q_emod 43]; norm_num; exact QBox7_mem
+  · rw [D_emod 43]; norm_num; exact DBox7_mem
+def targetExpt45 : RI := ⟨134272381318,134272381321⟩
+lemma targetExpt45_calc : (CI.ofReal (targetExpInput 0 45)).exp.re=targetExpt45 := by decide +kernel
+lemma targetExpt45_mem : targetExpt45.Mem (Real.exp (Real.pi*(damping 0-h)*(45:ℤ)+(1-(45:ℤ))/Z)) := by
+  rw [← targetExpt45_calc]
+  exact ri_exp_mem (targetExpInput_mem 0 45) (by decide +kernel)
+def targetEnvt45 : RI := ⟨31298807609,31298807611⟩
+lemma targetEnvt45_calc : targetEnvBox 0 QBox9 DBox9 targetExpt45=targetEnvt45 := by decide +kernel
+lemma targetEnvt45_mem : targetEnvt45.Mem (targetEnvelope (damping 0) 45) := by
+  rw [← targetEnvt45_calc]
+  apply targetEnvBox_mem 0 45 _ _ targetExpt45_mem (by decide +kernel)
+  · rw [Q_emod 45]; norm_num; exact QBox9_mem
+  · rw [D_emod 45]; norm_num; exact DBox9_mem
+def targetExpt48 : RI := ⟨62489055,62489058⟩
+lemma targetExpt48_calc : (CI.ofReal (targetExpInput 0 48)).exp.re=targetExpt48 := by decide +kernel
+lemma targetExpt48_mem : targetExpt48.Mem (Real.exp (Real.pi*(damping 0-h)*(48:ℤ)+(1-(48:ℤ))/Z)) := by
+  rw [← targetExpt48_calc]
+  exact ri_exp_mem (targetExpInput_mem 0 48) (by decide +kernel)
+def targetEnvt48 : RI := ⟨35940479,35940482⟩
+lemma targetEnvt48_calc : targetEnvBox 0 QBox12 DBox12 targetExpt48=targetEnvt48 := by decide +kernel
+lemma targetEnvt48_mem : targetEnvt48.Mem (targetEnvelope (damping 0) 48) := by
+  rw [← targetEnvt48_calc]
+  apply targetEnvBox_mem 0 48 _ _ targetExpt48_mem (by decide +kernel)
+  · rw [Q_emod 48]; norm_num; exact QBox12_mem
+  · rw [D_emod 48]; norm_num; exact DBox12_mem
+def targetExpt49 : RI := ⟨4842574,4842577⟩
+lemma targetExpt49_calc : (CI.ofReal (targetExpInput 0 49)).exp.re=targetExpt49 := by decide +kernel
+lemma targetExpt49_mem : targetExpt49.Mem (Real.exp (Real.pi*(damping 0-h)*(49:ℤ)+(1-(49:ℤ))/Z)) := by
+  rw [← targetExpt49_calc]
+  exact ri_exp_mem (targetExpInput_mem 0 49) (by decide +kernel)
+def targetEnvt49 : RI := ⟨2393126,2393128⟩
+lemma targetEnvt49_calc : targetEnvBox 0 QBox13 DBox13 targetExpt49=targetEnvt49 := by decide +kernel
+lemma targetEnvt49_mem : targetEnvt49.Mem (targetEnvelope (damping 0) 49) := by
+  rw [← targetEnvt49_calc]
+  apply targetEnvBox_mem 0 49 _ _ targetExpt49_mem (by decide +kernel)
+  · rw [Q_emod 49]; norm_num; exact QBox13_mem
+  · rw [D_emod 49]; norm_num; exact DBox13_mem
+def targetExpt52 : RI := ⟨2252,2255⟩
+lemma targetExpt52_calc : (CI.ofReal (targetExpInput 0 52)).exp.re=targetExpt52 := by decide +kernel
+lemma targetExpt52_mem : targetExpt52.Mem (Real.exp (Real.pi*(damping 0-h)*(52:ℤ)+(1-(52:ℤ))/Z)) := by
+  rw [← targetExpt52_calc]
+  exact ri_exp_mem (targetExpInput_mem 0 52) (by decide +kernel)
+def targetEnvt52 : RI := ⟨67,68⟩
+lemma targetEnvt52_calc : targetEnvBox 0 QBox16 DBox16 targetExpt52=targetEnvt52 := by decide +kernel
+lemma targetEnvt52_mem : targetEnvt52.Mem (targetEnvelope (damping 0) 52) := by
+  rw [← targetEnvt52_calc]
+  apply targetEnvBox_mem 0 52 _ _ targetExpt52_mem (by decide +kernel)
+  · rw [Q_emod 52]; norm_num; exact QBox16_mem
+  · rw [D_emod 52]; norm_num; exact DBox16_mem
+def targetExpt55 : RI := ⟨0,3⟩
+lemma targetExpt55_calc : (CI.ofReal (targetExpInput 0 55)).exp.re=targetExpt55 := by decide +kernel
+lemma targetExpt55_mem : targetExpt55.Mem (Real.exp (Real.pi*(damping 0-h)*(55:ℤ)+(1-(55:ℤ))/Z)) := by
+  rw [← targetExpt55_calc]
+  exact ri_exp_mem (targetExpInput_mem 0 55) (by decide +kernel)
+def targetEnvt55 : RI := ⟨0,1⟩
+lemma targetEnvt55_calc : targetEnvBox 0 QBox19 DBox19 targetExpt55=targetEnvt55 := by decide +kernel
+lemma targetEnvt55_mem : targetEnvt55.Mem (targetEnvelope (damping 0) 55) := by
+  rw [← targetEnvt55_calc]
+  apply targetEnvBox_mem 0 55 _ _ targetExpt55_mem (by decide +kernel)
+  · rw [Q_emod 55]; norm_num; exact QBox19_mem
+  · rw [D_emod 55]; norm_num; exact DBox19_mem
+def targetExpt57 : RI := ⟨-1,2⟩
+lemma targetExpt57_calc : (CI.ofReal (targetExpInput 0 57)).exp.re=targetExpt57 := by decide +kernel
+lemma targetExpt57_mem : targetExpt57.Mem (Real.exp (Real.pi*(damping 0-h)*(57:ℤ)+(1-(57:ℤ))/Z)) := by
+  rw [← targetExpt57_calc]
+  exact ri_exp_mem (targetExpInput_mem 0 57) (by decide +kernel)
+def targetEnvt57 : RI := ⟨-1,1⟩
+lemma targetEnvt57_calc : targetEnvBox 0 QBox21 DBox21 targetExpt57=targetEnvt57 := by decide +kernel
+lemma targetEnvt57_mem : targetEnvt57.Mem (targetEnvelope (damping 0) 57) := by
+  rw [← targetEnvt57_calc]
+  apply targetEnvBox_mem 0 57 _ _ targetExpt57_mem (by decide +kernel)
+  · rw [Q_emod 57]; norm_num; exact QBox21_mem
+  · rw [D_emod 57]; norm_num; exact DBox21_mem
+def targetExpt25 : RI := ⟨2200585237517198393137309427282307,2200585237517198393137309427282310⟩
+lemma targetExpt25_calc : (CI.ofReal (targetExpInput 0 25)).exp.re=targetExpt25 := by decide +kernel
+lemma targetExpt25_mem : targetExpt25.Mem (Real.exp (Real.pi*(damping 0-h)*(25:ℤ)+(1-(25:ℤ))/Z)) := by
+  rw [← targetExpt25_calc]
+  exact ri_exp_mem (targetExpInput_mem 0 25) (by decide +kernel)
+def targetEnvt25 : RI := ⟨87131799812251848004403109991140,87131799812251848004403109991141⟩
+lemma targetEnvt25_calc : targetEnvBox 0 QBox25 DBox25 targetExpt25=targetEnvt25 := by decide +kernel
+lemma targetEnvt25_mem : targetEnvt25.Mem (targetEnvelope (damping 0) 25) := by
+  rw [← targetEnvt25_calc]
+  apply targetEnvBox_mem 0 25 _ _ targetExpt25_mem (by decide +kernel)
+  · exact QBox25_mem
+  · exact DBox25_mem
+def targetExpt27 : RI := ⟨13215483134703708916534863346746,13215483134703708916534863346749⟩
+lemma targetExpt27_calc : (CI.ofReal (targetExpInput 0 27)).exp.re=targetExpt27 := by decide +kernel
+lemma targetExpt27_mem : targetExpt27.Mem (Real.exp (Real.pi*(damping 0-h)*(27:ℤ)+(1-(27:ℤ))/Z)) := by
+  rw [← targetExpt27_calc]
+  exact ri_exp_mem (targetExpInput_mem 0 27) (by decide +kernel)
+def targetEnvt27 : RI := ⟨3697163849264717858191507441874,3697163849264717858191507441876⟩
+lemma targetEnvt27_calc : targetEnvBox 0 QBox27 DBox27 targetExpt27=targetEnvt27 := by decide +kernel
+lemma targetEnvt27_mem : targetEnvt27.Mem (targetEnvelope (damping 0) 27) := by
+  rw [← targetEnvt27_calc]
+  apply targetEnvBox_mem 0 27 _ _ targetExpt27_mem (by decide +kernel)
+  · exact QBox27_mem
+  · exact DBox27_mem
+def targetExpt28 : RI := ⟨1024130899431529660013217530607,1024130899431529660013217530610⟩
+lemma targetExpt28_calc : (CI.ofReal (targetExpInput 0 28)).exp.re=targetExpt28 := by decide +kernel
+lemma targetExpt28_mem : targetExpt28.Mem (Real.exp (Real.pi*(damping 0-h)*(28:ℤ)+(1-(28:ℤ))/Z)) := by
+  rw [← targetExpt28_calc]
+  exact ri_exp_mem (targetExpInput_mem 0 28) (by decide +kernel)
+def targetEnvt28 : RI := ⟨238163983735114174003852346393,238163983735114174003852346395⟩
+lemma targetEnvt28_calc : targetEnvBox 0 QBox28 DBox28 targetExpt28=targetEnvt28 := by decide +kernel
+lemma targetEnvt28_mem : targetEnvt28.Mem (targetEnvelope (damping 0) 28) := by
+  rw [← targetEnvt28_calc]
+  apply targetEnvBox_mem 0 28 _ _ targetExpt28_mem (by decide +kernel)
+  · exact QBox28_mem
+  · exact DBox28_mem
+def targetExpt31 : RI := ⟨476620528616191269558476256,476620528616191269558476259⟩
+lemma targetExpt31_calc : (CI.ofReal (targetExpInput 0 31)).exp.re=targetExpt31 := by decide +kernel
+lemma targetExpt31_mem : targetExpt31.Mem (Real.exp (Real.pi*(damping 0-h)*(31:ℤ)+(1-(31:ℤ))/Z)) := by
+  rw [← targetExpt31_calc]
+  exact ri_exp_mem (targetExpInput_mem 0 31) (by decide +kernel)
+def targetEnvt31 : RI := ⟨6138805489362733871230068,6138805489362733871230069⟩
+lemma targetEnvt31_calc : targetEnvBox 0 QBox31 DBox31 targetExpt31=targetEnvt31 := by decide +kernel
+lemma targetEnvt31_mem : targetEnvt31.Mem (targetEnvelope (damping 0) 31) := by
+  rw [← targetEnvt31_calc]
+  apply targetEnvBox_mem 0 31 _ _ targetExpt31_mem (by decide +kernel)
+  · exact QBox31_mem
+  · exact DBox31_mem
+def targetEnvTablet : Fin 15 → RI := ![targetEnvt36,targetEnvt37,targetEnvt39,targetEnvt40,targetEnvt43,targetEnvt45,targetEnvt48,targetEnvt49,targetEnvt52,targetEnvt55,targetEnvt57,targetEnvt25,targetEnvt27,targetEnvt28,targetEnvt31]
+lemma targetEnvTablet_mem (i : Fin 15) : (targetEnvTablet i).Mem (targetEnvelope (damping 0) (tailBase i)) := by
+  fin_cases i
+  · exact targetEnvt36_mem
+  · exact targetEnvt37_mem
+  · exact targetEnvt39_mem
+  · exact targetEnvt40_mem
+  · exact targetEnvt43_mem
+  · exact targetEnvt45_mem
+  · exact targetEnvt48_mem
+  · exact targetEnvt49_mem
+  · exact targetEnvt52_mem
+  · exact targetEnvt55_mem
+  · exact targetEnvt57_mem
+  · exact targetEnvt25_mem
+  · exact targetEnvt27_mem
+  · exact targetEnvt28_mem
+  · exact targetEnvt31_mem
+def finiteTargetBox : RI := ⟨80739866858051870539391411993765471239633981294109190,80739866858051870539391411993765471239636629366265706⟩
+lemma finiteTargetBox_calc : RI.sum targetEnvTablef=finiteTargetBox := by decide +kernel
+lemma finiteTargetEnvelope_bound : finiteTargetEnvelope < (81/10^9:ℝ) := by
+  have hh := RI.mem_sum targetEnvTablef_mem
+  rw [finiteTargetBox_calc] at hh
+  have hb := ri_mem_lt hh 81 (10^9) (by norm_num) (by decide +kernel)
+  simpa [finiteTargetEnvelope,tailTargetEnvelope,damping,div_eq_mul_inv,show (10:ℝ)^9=1000000000 by norm_num,show (10:ℝ)^30=1000000000000000000000000000000 by norm_num] using hb
+def targetRatioBox : RI := ⟨103230324563422963144,103230324563422963147⟩
+lemma targetRatioBox_calc : (CI.ofReal ((RI.point (-36)).mul targetZBox.invPos)).exp.re=targetRatioBox := by decide +kernel
+lemma targetRatioBox_mem : targetRatioBox.Mem (Real.exp (-36/Z)) := by
+  rw [←targetRatioBox_calc]
+  apply ri_exp_mem
+  · rw [div_eq_mul_inv]
+    exact RI.mem_mul (by simpa using RI.mem_point (-36)) (RI.mem_invPos targetZBox_mem (by decide +kernel))
+  · decide +kernel
+def tailTargetBox : RI := ⟨91067133785200176601381496102314,91067133785200176601381496102365⟩
+lemma tailTargetBox_calc : (RI.sum targetEnvTablet).mul (((RI.point 1).sub targetRatioBox).invPos)=tailTargetBox := by decide +kernel
+lemma tailTargetEnvelope_bound : tailTargetEnvelope < (92/10^30:ℝ) := by
+  have hh := RI.mem_mul (RI.mem_sum targetEnvTablet_mem)
+    (RI.mem_invPos (RI.mem_sub ri1_mem targetRatioBox_mem) (by decide +kernel))
+  rw [tailTargetBox_calc] at hh
+  have hb := ri_mem_lt hh 92 (10^30) (by norm_num) (by decide +kernel)
+  simpa [finiteTargetEnvelope,tailTargetEnvelope,damping,div_eq_mul_inv,show (10:ℝ)^9=1000000000 by norm_num,show (10:ℝ)^30=1000000000000000000000000000000 by norm_num] using hb
+end AtomicTriangular.FiniteCertificate
+end
+
+end OAI

lean/OAI/Analysis/Triangular/Construction/ActualCorrection.lean

@@ -0,0 +1,137 @@
+import OAI.Analysis.Triangular.Construction.ConcreteSchur
+import OAI.Analysis.Triangular.Certificate.ParamActual
+import OAI.Analysis.Triangular.Certificate.TargetInterval
+import OAI.Analysis.Triangular.Certificate.ParamNorm
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators lp
+namespace AtomicTriangular.Construction
+open AtomicTriangular.L1Kernel AtomicTriangular.FiniteCertificate
+
+def finiteResidual (z : ℝ) : FiniteCoefficients :=
+  column (fun i (_ : Unit) => if i.val<8 then 0 else
+    gaussianTarget H z (finiteNode ⟨i.val/2,by omega⟩) ⟨i.val%2,by omega⟩)
+    (fun _ => (hasSum_fintype _).summable) ()
+lemma finiteResidual_norm {z : ℝ} (hz : 0<z) (hzB : z≤Z) :
+    ‖finiteResidual z‖ ≤ finiteResidualBound := by
+  rw [l1_norm,tsum_fintype]
+  change (∑ i : Fin 20, |if i.val<8 then 0 else
+    gaussianTarget H z (finiteNode ⟨i.val/2,by omega⟩) ⟨i.val%2,by omega⟩|) ≤ _
+  have he : (∑ i : Fin 20, |if i.val<8 then 0 else
+      gaussianTarget H z (finiteNode ⟨i.val/2,by omega⟩) ⟨i.val%2,by omega⟩|) =
+      ∑ n : Fin 10, if n.val<4 then 0 else ∑ d : Fin 2, |gaussianTarget H z (finiteNode n) d| := by
+    rw [show (fun i : Fin 20 => |if i.val<8 then 0 else
+      gaussianTarget H z (finiteNode ⟨i.val/2,by omega⟩) ⟨i.val%2,by omega⟩|) =
+      (fun i : Fin 20 => (fun n : Fin 10 => fun d : Fin 2 =>
+        if n.val<4 then (0:ℝ) else |gaussianTarget H z (finiteNode n) d|)
+        ⟨i.val/2,by omega⟩ ⟨i.val%2,by omega⟩) by
+        funext i; by_cases hi : i.val<8 <;> simp [hi,show i.val/2<4 ↔ i.val<8 by omega]]
+    rw [sum_fin20_pairs (fun n d => if n.val<4 then (0:ℝ) else |gaussianTarget H z (finiteNode n) d|)]
+    apply Finset.sum_congr rfl
+    intro n _; by_cases hn : n.val<4 <;> simp [hn]
+  rw [he]
+  have hb : (∑ n : Fin 10, if n.val<4 then 0 else ∑ d : Fin 2, |gaussianTarget H z (finiteNode n) d|) ≤
+      ∑ n : Fin 10, if n.val<4 then 0 else targetEnvelope H (finiteNode n) := by
+    apply Finset.sum_le_sum
+    intro n _
+    split_ifs with hn
+    · rfl
+    · exact gaussianTarget_pair_bound hz hzB H
+        ⟨finiteNode n, by fin_cases n <;> decide +kernel⟩ (by fin_cases n <;> norm_num [finiteNode,residue])
+  have he2 : (∑ n : Fin 10, if n.val<4 then 0 else targetEnvelope H (finiteNode n)) = finiteTargetEnvelope := by
+    simp [finiteTargetEnvelope,Fin.sum_univ_succ]
+  exact hb.trans ((he2 ▸ finiteTargetEnvelope_bound.le).trans (by norm_num [finiteResidualBound]))
+
+def finiteApprox (side : Fin 2) (e : Fin 8 → ℝ) : TrigonometricCoefficients :=
+  column (fun i (_ : Unit) => ∑ j : Fin 8, Acoefficient side ⟨i.val,by omega⟩ j*e j)
+    (fun _ => (hasSum_fintype _).summable) ()
+lemma finiteApprox_norm (side : Fin 2) (e : Fin 8 → ℝ) (he : ParameterPolytope e) :
+    ‖finiteApprox side e‖ < 3 := by
+  rw [l1_norm,tsum_fintype]
+  have hb : (∑ i : Fin 22, |∑ j : Fin 8, Acoefficient side i.castSucc j*e j|) ≤
+      ∑ i : Fin 23, |∑ j : Fin 8, Acoefficient side i j*e j| := by
+    conv_rhs => rw [Fin.sum_univ_castSucc]
+    exact le_add_of_nonneg_right (abs_nonneg _)
+  change (∑ i : Fin 22, |∑ j : Fin 8, Acoefficient side i.castSucc j*e j|) < 3
+  refine hb.trans_lt ?_
+  fin_cases side
+  · exact (A1_norm_bound e he).trans (by norm_num)
+  · exact (A2_norm_bound e he).trans (by norm_num)
+
+def approximationTailResidual (e : Fin 2) (z : ℝ) (hz : 0<z) (hzB : z≤Z) : TailCoefficients :=
+  tailTarget z hz hzB +
+  (if e.val=0 then CallOperator (finiteApprox 0 (actualParameter z)+finiteApprox 1 (actualParameter z)) -
+      AallOperator (finiteApprox 0 (actualParameter z)+finiteApprox 1 (actualParameter z))
+    else CallOperator (finiteApprox 1 (actualParameter z)-finiteApprox 0 (actualParameter z)) -
+      AallOperator (finiteApprox 0 (actualParameter z)-finiteApprox 1 (actualParameter z)) +
+      (12/1000:ℝ) • tailFixedColumn 22)
+lemma approximationTailResidual_norm (e : Fin 2) {z : ℝ} (hz : 0<z) (hzB : z≤Z) :
+    ‖approximationTailResidual e z hz hzB‖ ≤ tailResidualBound e := by
+  have h1 := (finiteApprox_norm 0 (actualParameter z) (actualParameter_mem hz hzB)).le
+  have h2 := (finiteApprox_norm 1 (actualParameter z) (actualParameter_mem hz hzB)).le
+  have ht : ‖tailTarget z hz hzB‖ ≤ (1/10^26:ℝ) :=
+    (tailTarget_norm hz hzB).trans (tailTargetEnvelope_bound.le.trans (by norm_num))
+  have hp : ‖finiteApprox 0 (actualParameter z)+finiteApprox 1 (actualParameter z)‖ ≤ 6 :=
+    (norm_add_le _ _).trans (by linarith)
+  have hm : ‖finiteApprox 0 (actualParameter z)-finiteApprox 1 (actualParameter z)‖ ≤ 6 :=
+    (norm_sub_le _ _).trans (by linarith)
+  have hm' : ‖finiteApprox 1 (actualParameter z)-finiteApprox 0 (actualParameter z)‖ ≤ 6 :=
+    (norm_sub_le _ _).trans (by linarith)
+  have hc (a b : TrigonometricCoefficients) (ha : ‖a‖≤6) (hb : ‖b‖≤6) :
+      ‖CallOperator a-AallOperator b‖ ≤ 6*beta := by
+    calc
+      _ ≤ ‖CallOperator a‖+‖AallOperator b‖ := norm_sub_le _ _
+      _ ≤ ‖CallOperator‖*‖a‖+‖AallOperator‖*‖b‖ :=
+        add_le_add (ContinuousLinearMap.le_opNorm _ _) (ContinuousLinearMap.le_opNorm _ _)
+      _ ≤ ‖CallOperator‖*6+‖AallOperator‖*6 := by gcongr
+      _ ≤ 6*beta := by nlinarith [AC_all_bound]
+  have hv : ‖(12/1000:ℝ) • tailFixedColumn 22‖ ≤ (12/1000:ℝ)*(13/10^8) := by
+    rw [norm_smul,Real.norm_eq_abs,abs_of_pos (by norm_num : (0:ℝ)<12/1000)]
+    exact mul_le_mul_of_nonneg_left (tailFixedColumn_norm_constant.trans (by norm_num)) (by norm_num)
+  fin_cases e <;> simp only [approximationTailResidual,tailResidualBound,↓reduceIte,Nat.one_ne_zero]
+  · change ‖tailTarget z hz hzB + _‖ ≤ 1/10^26+6*beta+0
+    exact (norm_add_le _ _).trans (by linarith [hc _ _ hp hp])
+  · change ‖tailTarget z hz hzB + (_+_)‖ ≤ 1/10^26+6*beta+(12/1000)*(13/10^8)
+    have hh := (norm_add_le _ _).trans (add_le_add ht
+      ((norm_add_le _ _).trans (add_le_add (hc _ _ hm' hm) hv)))
+    simpa only [add_assoc] using hh
+
+def actualTailSplit (e : Fin 2) (z : ℝ) (hz : 0<z) (hzB : z≤Z) : TailCoefficients :=
+  correctionTail e (finiteResidual z) (approximationTailResidual e z hz hzB)
+def actualFiniteSplit (e : Fin 2) (z : ℝ) (hz : 0<z) (hzB : z≤Z) : FiniteCoefficients :=
+  correctionFinite e (finiteResidual z) (approximationTailResidual e z hz hzB)
+lemma actualTailSplit_norm (e : Fin 2) {z : ℝ} (hz : 0<z) (hzB : z≤Z) :
+    ‖actualTailSplit e z hz hzB‖ ≤ tailErrorBound e :=
+  correctionTail_small _ _ _ (finiteResidual_norm hz hzB) (approximationTailResidual_norm e hz hzB)
+lemma actualFiniteSplit_norm (e : Fin 2) {z : ℝ} (hz : 0<z) (hzB : z≤Z) :
+    ‖actualFiniteSplit e z hz hzB‖ ≤ finiteErrorBound e :=
+  correctionFinite_small _ _ _ (finiteResidual_norm hz hzB) (approximationTailResidual_norm e hz hzB)
+
+def actualTail (side : Fin 2) (z : ℝ) (hz : 0<z) (hzB : z≤Z) : TailCoefficients :=
+  (1/2:ℝ) • (actualTailSplit 0 z hz hzB + (if side.val=0 then actualTailSplit 1 z hz hzB else -actualTailSplit 1 z hz hzB))
+def actualFiniteError (side : Fin 2) (z : ℝ) (hz : 0<z) (hzB : z≤Z) : FiniteCoefficients :=
+  (1/2:ℝ) • (actualFiniteSplit 0 z hz hzB + (if side.val=0 then actualFiniteSplit 1 z hz hzB else -actualFiniteSplit 1 z hz hzB))
+lemma actualTail_norm (side : Fin 2) {z : ℝ} (hz : 0<z) (hzB : z≤Z) :
+    ‖actualTail side z hz hzB‖ < (3/10^9:ℝ) := by
+  have hp := actualTailSplit_norm 0 hz hzB
+  have hm := actualTailSplit_norm 1 hz hzB
+  unfold actualTail
+  rw [norm_smul,Real.norm_eq_abs,abs_of_pos (by norm_num : (0:ℝ)<1/2)]
+  have hh := norm_add_le (actualTailSplit 0 z hz hzB)
+    (if side.val=0 then actualTailSplit 1 z hz hzB else -actualTailSplit 1 z hz hzB)
+  split_ifs at hh ⊢ <;> (try simp only [norm_neg] at hh) <;> norm_num [tailErrorBound] at hp hm <;> linarith
+lemma actualFiniteError_norm (side : Fin 2) {z : ℝ} (hz : 0<z) (hzB : z≤Z) :
+    ‖actualFiniteError side z hz hzB‖ < (1/10^5:ℝ) := by
+  have hp := actualFiniteSplit_norm 0 hz hzB
+  have hm := actualFiniteSplit_norm 1 hz hzB
+  unfold actualFiniteError
+  rw [norm_smul,Real.norm_eq_abs,abs_of_pos (by norm_num : (0:ℝ)<1/2)]
+  have hh := norm_add_le (actualFiniteSplit 0 z hz hzB)
+    (if side.val=0 then actualFiniteSplit 1 z hz hzB else -actualFiniteSplit 1 z hz hzB)
+  split_ifs at hh ⊢ <;> (try simp only [norm_neg] at hh) <;> norm_num [finiteErrorBound] at hp hm <;> linarith
+end AtomicTriangular.Construction
+end
+
+end OAI

lean/OAI/Analysis/Triangular/Construction/ActualInput.lean

@@ -0,0 +1,86 @@
+import OAI.Analysis.Triangular.Construction.ActualCorrection
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators lp
+namespace AtomicTriangular.Construction
+open AtomicTriangular.L1Kernel AtomicTriangular.MatrixBounds
+
+lemma tail_pairs_summable (w : TailCoefficients) :
+    Summable (fun n => |w (n,0)|+|w (n,1)|) := by
+  have hh := (l1_summable w).prod
+  simpa [tsum_fintype,Fin.sum_univ_succ] using hh
+lemma tail_pairs_norm (w : TailCoefficients) :
+    (∑' n, (|w (n,0)| + |w (n,1)|))=‖w‖ := by
+  rw [l1_norm]
+  symm
+  calc
+    (∑' p, |w p|) = ∑' n : TailNode, ∑' d : Fin 2, |w (n,d)| := (l1_summable w).tsum_prod
+    _ = _ := by simp only [tsum_fintype,Fin.sum_univ_succ,Finset.univ_eq_empty,Finset.sum_empty,add_zero]
+                rfl
+
+def coefficientInput (C : ℝ) (a : TrigonometricCoefficients) (w : TailCoefficients) : InputCoefficients where
+  C := C
+  finiteC j := a ⟨2*j.val,by omega⟩
+  finiteD j := a ⟨2*j.val+1,by omega⟩
+  tailC n := w (n,0)
+  tailD n := w (n,1)
+  summable := tail_pairs_summable w
+
+def extendFiniteError (u : FiniteCoefficients) : TrigonometricCoefficients :=
+  column (fun i (_ : Unit) => if hi : i.val<2 then 0 else u ⟨i.val-2,by omega⟩)
+    (fun _ => (hasSum_fintype _).summable) ()
+lemma extendFiniteError_norm (u : FiniteCoefficients) : ‖extendFiniteError u‖=‖u‖ := by
+  rw [l1_norm,l1_norm,tsum_fintype,tsum_fintype]
+  change (∑ i : Fin 22, |if hi : i.val<2 then 0 else u ⟨i.val-2,by omega⟩|)=_
+  rw [Fin.sum_univ_succ,Fin.sum_univ_succ]
+  simp only [Fin.val_zero,Fin.val_succ,show 0<2 by omega,show 0+1<2 by omega,↓reduceDIte,abs_zero,zero_add]
+  apply Finset.sum_congr rfl
+  intro i _
+  simp only [show ¬(i.val+1+1<2) by omega,↓reduceDIte]
+  congr 2
+
+def actualTrigonometric (side : Fin 2) (z : ℝ) (hz : 0<z) (hzB : z≤Z) : TrigonometricCoefficients :=
+  finiteApprox side (actualParameter z)+extendFiniteError (actualFiniteError side z hz hzB)
+def actualInput (side : Fin 2) (z : ℝ) (hz : 0<z) (hzB : z≤Z) : InputCoefficients :=
+  coefficientInput (if side.val=0 then -(3/500) else 3/500)
+    (actualTrigonometric side z hz hzB) (actualTail side z hz hzB)
+lemma actualTrigonometric_norm (side : Fin 2) {z : ℝ} (hz : 0<z) (hzB : z≤Z) :
+    ‖actualTrigonometric side z hz hzB‖ < (300001/100000:ℝ) := by
+  have h1 := finiteApprox_norm side (actualParameter z) (actualParameter_mem hz hzB)
+  have h2 := actualFiniteError_norm side hz hzB
+  have hh := norm_add_le (finiteApprox side (actualParameter z)) (extendFiniteError (actualFiniteError side z hz hzB))
+  rw [extendFiniteError_norm] at hh
+  change ‖finiteApprox side (actualParameter z)+_‖ < _
+  linarith
+
+lemma correctionFinite_plus (r : FiniteCoefficients) (s : TailCoefficients) :
+    correctionFinite 0 r s = Roperator (correctionFinite 0 r s)+Boperator (correctionTail 0 r s)+r := by
+  have hh := congrArg (fun T : FiniteCoefficients →L[ℝ] FiniteCoefficients => T (r+Boperator (correctionTail 0 r s))) Lplus_left_inverse
+  change correctionFinite 0 r s-Roperator (correctionFinite 0 r s)=r+Boperator (correctionTail 0 r s) at hh
+  exact sub_eq_iff_eq_add.mp hh |>.trans (by abel)
+lemma correctionFinite_minus (r : FiniteCoefficients) (s : TailCoefficients) :
+    correctionFinite 1 r s = -Roperator (correctionFinite 1 r s)-Boperator (correctionTail 1 r s)+r := by
+  have hh := congrArg (fun T : FiniteCoefficients →L[ℝ] FiniteCoefficients => T (r-Boperator (correctionTail 1 r s))) Lminus_left_inverse
+  change correctionFinite 1 r s+Roperator (correctionFinite 1 r s)=r-Boperator (correctionTail 1 r s) at hh
+  exact eq_sub_of_add_eq hh |>.trans (by abel)
+lemma actualFiniteError_equations {z : ℝ} (hz : 0<z) (hzB : z≤Z) :
+    actualFiniteError 0 z hz hzB = Roperator (actualFiniteError 1 z hz hzB)+Boperator (actualTail 1 z hz hzB)+finiteResidual z ∧
+    actualFiniteError 1 z hz hzB = Roperator (actualFiniteError 0 z hz hzB)+Boperator (actualTail 0 z hz hzB) := by
+  have hp := correctionFinite_plus (finiteResidual z) (approximationTailResidual 0 z hz hzB)
+  have hm := correctionFinite_minus (finiteResidual z) (approximationTailResidual 1 z hz hzB)
+  change actualFiniteSplit 0 z hz hzB=Roperator (actualFiniteSplit 0 z hz hzB)+Boperator (actualTailSplit 0 z hz hzB)+finiteResidual z at hp
+  change actualFiniteSplit 1 z hz hzB= -Roperator (actualFiniteSplit 1 z hz hzB)-Boperator (actualTailSplit 1 z hz hzB)+finiteResidual z at hm
+  constructor <;> ext i
+  all_goals
+    have hp' := congrArg (fun u : FiniteCoefficients => u i) hp
+    have hm' := congrArg (fun u : FiniteCoefficients => u i) hm
+    simp only [actualFiniteError,actualTail,Fin.val_zero,Fin.val_one,Nat.one_ne_zero,↓reduceIte,
+      map_smul,map_add,map_neg,lp.coeFn_add,lp.coeFn_sub,lp.coeFn_neg,lp.coeFn_smul,
+      Pi.add_apply,Pi.sub_apply,Pi.neg_apply,Pi.smul_apply,smul_eq_mul] at hp' hm' ⊢
+    linarith
+end AtomicTriangular.Construction
+end
+
+end OAI

lean/OAI/Analysis/Triangular/Construction/ActualInterpolation.lean

@@ -0,0 +1,94 @@
+import OAI.Analysis.Triangular.Construction.FiniteInterpolation
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators lp
+open Complex Matrix
+namespace AtomicTriangular.Construction
+lemma otherSide_zero : otherSide 0=1 := rfl
+lemma otherSide_one : otherSide 1=0 := rfl
+def actualJ (side : Fin 2) (z : ℝ) (hz : 0<z) (hzB : z≤Z) : ℝ → ℝ :=
+  pairJ (actualInput side z hz hzB) (actualInput (otherSide side) z hz hzB)
+def gaussianG (z s : ℝ) : ℝ := Q 1*z*Real.exp ((1-s)/z)
+lemma gaussianG_deriv {z : ℝ} (hz : 0<z) (s : ℝ) :
+    HasDerivAt (gaussianG z) (-Q 1*Real.exp ((1-s)/z)) s := by
+  have hh := ((((hasDerivAt_const s (1:ℝ)).sub (hasDerivAt_id s)).div_const z).exp.const_mul (Q 1*z))
+  convert hh using 1 <;> first | rfl | (simp only [Pi.sub_apply,id_eq]; field_simp; ring)
+lemma negativeJet_gaussian {z : ℝ} (hz : 0<z) (v : ℝ) (n : ℤ) (d : Fin 2) :
+    (negativeJet v n d (gaussianG z n:ℂ) ((deriv (gaussianG z) (n:ℝ):ℝ):ℂ)).re = -gaussianTarget v z n d := by
+  rw [negativeJet_re,(gaussianG_deriv hz n).deriv]
+  simp only [ofReal_re,gaussianG,gaussianTarget,Real.exp_add]
+  split_ifs <;> ring
+lemma negativeJet_injective (v : ℝ) (n : Node) (a b c d : ℂ)
+    (h0 : (negativeJet v n.val 0 a b).re=(negativeJet v n.val 0 c d).re)
+    (h1 : (negativeJet v n.val 1 a b).re=(negativeJet v n.val 1 c d).re) :
+    a.re=c.re ∧ b.re=d.re := by
+  simp only [negativeJet_re,Fin.val_zero,Fin.val_one,Nat.one_ne_zero,↓reduceIte] at h0 h1
+  have hp : Real.exp (Real.pi*(v-h)*n.val)/Q n.val≠0 := div_ne_zero (Real.exp_ne_zero _) (Q_node_ne n)
+  have ha : a.re=c.re := by
+    have hh := mul_left_cancel₀ hp h0
+    linarith
+  have hb := mul_left_cancel₀ hp h1
+  constructor
+  · exact ha
+  · rw [ha] at hb
+    linarith
+lemma finiteNode_surj {n : ℤ} (hn : n∈FiniteNodes) : ∃ i : Fin 10,finiteNode i=n := by
+  have hall : ∀ n∈FiniteNodes,∃ i : Fin 10,finiteNode i=n := by decide +kernel
+  exact hall n hn
+lemma rowNode_cover (n : Node) (hn : n.val∈FiniteNodes) (d : Fin 2) :
+    ∃ i : Fin 20,rowNode i=n ∧ rowDerivative i=d := by
+  obtain ⟨j,hj⟩ := finiteNode_surj hn
+  refine ⟨⟨2*j.val+d.val,by omega⟩,?_,?_⟩
+  · apply Subtype.ext
+    change finiteNode ⟨(2*j.val+d.val)/2,by omega⟩=n.val
+    convert hj using 1
+    congr 1
+    apply Fin.ext
+    dsimp
+    omega
+  · apply Fin.ext
+    change (2*j.val+d.val)%2=d.val
+    omega
+lemma actual_node_negative {z : ℝ} (hz : 0<z) (hzB : z≤Z) (n : Node) :
+    ∃ v : ℝ,∀ d : Fin 2,
+      (negativeJet v n.val d
+        (pairJJet (actualInput 0 z hz hzB) (actualInput 1 z hz hzB) 0 n.val)
+        (pairJJet (actualInput 0 z hz hzB) (actualInput 1 z hz hzB) 1 n.val)).re= -gaussianTarget v z n.val d ∧
+      (negativeJet v n.val d
+        (pairJJet (actualInput 1 z hz hzB) (actualInput 0 z hz hzB) 0 n.val)
+        (pairJJet (actualInput 1 z hz hzB) (actualInput 0 z hz hzB) 1 n.val)).re=0 := by
+  by_cases hn : n.val∈FiniteNodes
+  · refine ⟨H,?_⟩
+    intro d
+    obtain ⟨i,hi,hd⟩ := rowNode_cover n hn d
+    have hh := actual_finite_negative hz hzB i
+    simpa only [hi,hd] using hh
+  · refine ⟨h,?_⟩
+    intro d
+    exact actual_tail_negative hz hzB (⟨n,hn⟩,d)
+lemma actual_interpolation {z : ℝ} (hz : 0<z) (hzB : z≤Z) (n : Node) :
+    actualJ 0 z hz hzB n.val=gaussianG z n.val ∧ deriv (actualJ 0 z hz hzB) n.val=deriv (gaussianG z) n.val ∧
+    actualJ 1 z hz hzB n.val=0 ∧ deriv (actualJ 1 z hz hzB) n.val=0 := by
+  obtain ⟨v,hv⟩ := actual_node_negative hz hzB n
+  have h0 := negativeJet_injective v n
+    (pairJJet (actualInput 0 z hz hzB) (actualInput 1 z hz hzB) 0 n.val)
+    (pairJJet (actualInput 0 z hz hzB) (actualInput 1 z hz hzB) 1 n.val)
+    (gaussianG z n.val:ℂ) ((deriv (gaussianG z) (n.val:ℝ):ℝ):ℂ)
+    ((hv 0).1.trans (negativeJet_gaussian hz v n.val 0).symm)
+    ((hv 1).1.trans (negativeJet_gaussian hz v n.val 1).symm)
+  have hn0 (d : Fin 2) : negativeJet v n.val d 0 0=0 := by simp [negativeJet]
+  have h1 := negativeJet_injective v n
+    (pairJJet (actualInput 1 z hz hzB) (actualInput 0 z hz hzB) 0 n.val)
+    (pairJJet (actualInput 1 z hz hzB) (actualInput 0 z hz hzB) 1 n.val) 0 0
+    (by simpa only [hn0,zero_re] using (hv 0).2) (by simpa only [hn0,zero_re] using (hv 1).2)
+  have hd0 := ((hasDerivAt_pairJJet (actualInput 0 z hz hzB) (actualInput 1 z hz hzB) 0 n.val).real_of_complex).deriv
+  have hd1 := ((hasDerivAt_pairJJet (actualInput 1 z hz hzB) (actualInput 0 z hz hzB) 0 n.val).real_of_complex).deriv
+  change deriv (actualJ 0 z hz hzB) n.val=(pairJJet _ _ 1 n.val).re at hd0
+  change deriv (actualJ 1 z hz hzB) n.val=(pairJJet _ _ 1 n.val).re at hd1
+  exact ⟨h0.1,hd0.trans h0.2,h1.1,hd1.trans h1.2⟩
+end AtomicTriangular.Construction
+end
+
+end OAI

lean/OAI/Analysis/Triangular/Construction/ActualTail.lean

@@ -0,0 +1,75 @@
+import OAI.Analysis.Triangular.Construction.ActualInput
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators lp
+namespace AtomicTriangular.Construction
+open AtomicTriangular.L1Kernel AtomicTriangular.FiniteCertificate
+lemma actualTail_equations {z : ℝ} (hz : 0<z) (hzB : z≤Z) :
+    actualTail 0 z hz hzB = Doperator (actualTail 1 z hz hzB)+Coperator (actualFiniteError 1 z hz hzB)-
+      Aoperator (actualFiniteError 0 z hz hzB)+tailTarget z hz hzB+
+      CallOperator (finiteApprox 1 (actualParameter z))-AallOperator (finiteApprox 0 (actualParameter z))+
+      (3/500:ℝ) • tailFixedColumn 22 ∧
+    actualTail 1 z hz hzB = Doperator (actualTail 0 z hz hzB)+Coperator (actualFiniteError 0 z hz hzB)-
+      Aoperator (actualFiniteError 1 z hz hzB)+
+      CallOperator (finiteApprox 0 (actualParameter z))-AallOperator (finiteApprox 1 (actualParameter z))-
+      (3/500:ℝ) • tailFixedColumn 22 := by
+  have hp := correctionTail_equation 0 (finiteResidual z) (approximationTailResidual 0 z hz hzB)
+  have hm := correctionTail_equation 1 (finiteResidual z) (approximationTailResidual 1 z hz hzB)
+  change actualTailSplit 0 z hz hzB=Doperator (actualTailSplit 0 z hz hzB)+
+    (Coperator-Aoperator) (actualFiniteSplit 0 z hz hzB)+approximationTailResidual 0 z hz hzB at hp
+  change actualTailSplit 1 z hz hzB=(-Doperator) (actualTailSplit 1 z hz hzB)+
+    (-Coperator-Aoperator) (actualFiniteSplit 1 z hz hzB)+approximationTailResidual 1 z hz hzB at hm
+  constructor <;> ext i
+  all_goals
+    have hp' := congrArg (fun u : TailCoefficients => u i) hp
+    have hm' := congrArg (fun u : TailCoefficients => u i) hm
+    simp only [actualFiniteError,actualTail,approximationTailResidual,
+      Fin.val_zero,Fin.val_one,Nat.one_ne_zero,↓reduceIte,
+      map_smul,map_add,map_sub,map_neg,sub_apply,neg_apply,
+      lp.coeFn_add,lp.coeFn_sub,lp.coeFn_neg,lp.coeFn_smul,
+      Pi.add_apply,Pi.sub_apply,Pi.neg_apply,Pi.smul_apply,smul_eq_mul] at hp' hm' ⊢
+    linarith
+lemma CallOperator_extend (u : FiniteCoefficients) :
+    CallOperator (extendFiniteError u)=Coperator u := by
+  ext p
+  unfold CallOperator Coperator
+  erw [operator_apply,operator_apply,tsum_fintype,tsum_fintype]
+  change (∑ i : Fin 22, (if hi : i.val<2 then 0 else u ⟨i.val-2,by omega⟩)*CallKernel p i)=_
+  rw [Fin.sum_univ_succ,Fin.sum_univ_succ]
+  simp only [Fin.val_zero,Fin.val_succ,show 0<2 by omega,show 0+1<2 by omega,↓reduceDIte,zero_mul,zero_add]
+  apply Finset.sum_congr rfl
+  intro i _
+  simp only [show ¬(i.val+1+1<2) by omega,↓reduceDIte]
+  congr 1
+lemma AallOperator_extend (u : FiniteCoefficients) :
+    AallOperator (extendFiniteError u)=Aoperator u := by
+  ext p
+  unfold AallOperator Aoperator
+  erw [operator_apply,operator_apply,tsum_fintype,tsum_fintype]
+  change (∑ i : Fin 22, (if hi : i.val<2 then 0 else u ⟨i.val-2,by omega⟩)*AallKernel p i)=_
+  rw [Fin.sum_univ_succ,Fin.sum_univ_succ]
+  simp only [Fin.val_zero,Fin.val_succ,show 0<2 by omega,show 0+1<2 by omega,↓reduceDIte,zero_mul,zero_add]
+  apply Finset.sum_congr rfl
+  intro i _
+  simp only [show ¬(i.val+1+1<2) by omega,↓reduceDIte]
+  congr 1
+  unfold AallKernel Akernel
+  congr 1 <;> apply Fin.ext <;> dsimp <;> omega
+lemma actualTail_full_equations {z : ℝ} (hz : 0<z) (hzB : z≤Z) :
+    actualTail 0 z hz hzB = Doperator (actualTail 1 z hz hzB)+
+      CallOperator (actualTrigonometric 1 z hz hzB)-AallOperator (actualTrigonometric 0 z hz hzB)+
+      tailTarget z hz hzB+(3/500:ℝ) • tailFixedColumn 22 ∧
+    actualTail 1 z hz hzB = Doperator (actualTail 0 z hz hzB)+
+      CallOperator (actualTrigonometric 0 z hz hzB)-AallOperator (actualTrigonometric 1 z hz hzB)-
+      (3/500:ℝ) • tailFixedColumn 22 := by
+  rcases actualTail_equations hz hzB with ⟨h0,h1⟩
+  simp only [actualTrigonometric,map_add,CallOperator_extend,AallOperator_extend]
+  constructor
+  · rw [h0]; abel
+  · rw [h1]; abel
+end AtomicTriangular.Construction
+end
+
+end OAI

lean/OAI/Analysis/Triangular/Construction/AtomicCurvature.lean

@@ -0,0 +1,48 @@
+import OAI.Analysis.Triangular.Basic
+
+namespace OAI
+
+noncomputable section
+namespace AtomicTriangular.Envelope
+open Real Set
+
+lemma linear_sqrt_exp_hasDerivAt {a b δ x : ℝ} (hx : 0<x) :
+    HasDerivAt (fun y : ℝ => (a+b*y)*sqrt y*exp (-δ*y))
+      ((b*sqrt x+(a+b*x)/(2*sqrt x)-δ*((a+b*x)*sqrt x))*exp (-δ*x)) x := by
+  have hd := (((hasDerivAt_const x a).add ((hasDerivAt_id x).const_mul b)).mul
+    (hasDerivAt_sqrt (ne_of_gt hx))).mul
+    (((hasDerivAt_id x).const_mul (-δ)).exp)
+  convert hd using 1 <;> first | rfl | (dsimp; ring)
+
+lemma linear_sqrt_exp_antitone {a b δ x y : ℝ} (hx : 0<x) (hxy : x≤y)
+    (ha : 0≤a) (hb : 0≤b) (hδ : 3/2≤δ*x) :
+    (a+b*y)*sqrt y*exp (-δ*y) ≤ (a+b*x)*sqrt x*exp (-δ*x) := by
+  have hδ0 : 0<δ := by nlinarith
+  have hanti : AntitoneOn (fun u : ℝ => (a+b*u)*sqrt u*exp (-δ*u)) (Ici x) := by
+    apply antitoneOn_of_deriv_nonpos (convex_Ici x)
+    · exact (by fun_prop : ContinuousOn (fun u : ℝ => (a+b*u)*sqrt u*exp (-δ*u)) (Ici x))
+    · intro u hu
+      exact (linear_sqrt_exp_hasDerivAt (a:=a) (b:=b) (δ:=δ) (hx.trans_le (mem_Ici.mp (interior_subset hu)))).differentiableAt.differentiableWithinAt
+    · intro u hu
+      have hux : x≤u := mem_Ici.mp (interior_subset hu)
+      have hu0 := hx.trans_le hux
+      rw [(linear_sqrt_exp_hasDerivAt (a:=a) (b:=b) (δ:=δ) hu0).deriv]
+      apply mul_nonpos_of_nonpos_of_nonneg _ (exp_pos _).le
+      have hs := sq_sqrt hu0.le
+      have hsp : 0<sqrt u := sqrt_pos.mpr hu0
+      have hδu : 3/2≤δ*u := hδ.trans (mul_le_mul_of_nonneg_left hux hδ0.le)
+      apply nonpos_of_mul_nonpos_left (b:=2*sqrt u) _ (by positivity)
+      have he : (b*sqrt u+(a+b*u)/(2*sqrt u)-δ*((a+b*u)*sqrt u))*(2*sqrt u) =
+          a+3*b*u-2*δ*u*(a+b*u) := by
+        field_simp
+        rw [hs]
+        ring
+      rw [he]
+      have h1 := mul_nonneg (show 0≤2*δ*u-1 by linarith) ha
+      have h2 := mul_nonneg (show 0≤2*δ*u-3 by linarith) (mul_nonneg hb hu0.le)
+      nlinarith
+  exact hanti (mem_Ici.mpr le_rfl) (mem_Ici.mpr hxy) hxy
+end AtomicTriangular.Envelope
+end
+
+end OAI

lean/OAI/Analysis/Triangular/Construction/BlockOperators.lean

@@ -0,0 +1,83 @@
+import OAI.Analysis.Triangular.Envelope.KernelBounds
+import OAI.Analysis.Triangular.Construction.L1Matrix
+import OAI.Analysis.Triangular.Certificate.Inverse
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators lp
+namespace AtomicTriangular.Construction
+open AtomicTriangular.L1Kernel AtomicTriangular.MatrixBounds
+abbrev FiniteCoefficients := ℓ¹(Fin 20,ℝ)
+abbrev TailCoefficients := ℓ¹(TailNode × Fin 2,ℝ)
+lemma sum_fin20_pairs (g : Fin 10 → Fin 2 → ℝ) :
+    (∑ i : Fin 20, g ⟨i.val/2,by omega⟩ ⟨i.val%2,by omega⟩) =
+      ∑ n : Fin 10, ∑ d : Fin 2, g n d := by
+  simpa only [finProdFinEquiv_symm_apply,Fintype.sum_prod_type,Fin.divNat,Fin.modNat] using
+    ((finProdFinEquiv (m:=10) (n:=2)).symm.sum_comp (fun p => g p.1 p.2))
+def Bkernel (i : Fin 20) (p : TailNode × Fin 2) : ℝ :=
+  tailJet h H (finiteNode ⟨i.val/2,by omega⟩) p.1.val.val p.2 ⟨i.val%2,by omega⟩
+lemma Bkernel_bound (p : TailNode × Fin 2) : (∑' i, |Bkernel i p|) ≤ 335 := by
+  rw [tsum_fintype]
+  change (∑ i : Fin 20, |tailJet h H (finiteNode ⟨i.val/2,by omega⟩) p.1.val.val p.2 ⟨i.val%2,by omega⟩|) ≤ 335
+  exact (sum_fin20_pairs (fun n d => |tailJet h H (finiteNode n) p.1.val.val p.2 d|)).trans_le (finite_tail_column_bound _ _).le
+def Boperator : TailCoefficients →L[ℝ] FiniteCoefficients :=
+  operator Bkernel (fun _ => (hasSum_fintype _).summable) 335 Bkernel_bound
+lemma Boperator_norm : ‖Boperator‖ ≤ 335 := operator_norm _ _ _ (by norm_num) _
+def Ckernel (p : TailNode × Fin 2) (i : Fin 20) : ℝ :=
+  jet H h p.1.val.val p.2 (finiteColumn ⟨i.val+2,by omega⟩)
+lemma Ckernel_summable (i : Fin 20) : Summable (fun p => |Ckernel p i|) :=
+  (tail_atom_column_bounds 1 0 _).1
+lemma Ckernel_bound (i : Fin 20) : (∑' p, |Ckernel p i|) ≤ 7/10^15 :=
+  (tail_finite_column_bound ⟨i.val+2,by omega⟩).le
+def Coperator : FiniteCoefficients →L[ℝ] TailCoefficients :=
+  operator Ckernel Ckernel_summable (7/10^15) Ckernel_bound
+lemma Coperator_norm : ‖Coperator‖ ≤ 7/10^15 := operator_norm _ _ _ (by norm_num) _
+def Dkernel (p q : TailNode × Fin 2) : ℝ :=
+  tailJet h h p.1.val.val q.1.val.val q.2 p.2
+lemma Dkernel_summable (q : TailNode × Fin 2) : Summable (fun p => |Dkernel p q|) :=
+  (tail_tail_column_bounds 0 0 q.1.val.val q.2).1
+lemma Dkernel_bound (q : TailNode × Fin 2) : (∑' p, |Dkernel p q|) ≤ 23/10^9 :=
+  (tail_tail_column_bound q.1.val.val q.2).le
+def Doperator : TailCoefficients →L[ℝ] TailCoefficients :=
+  operator Dkernel Dkernel_summable (23/10^9) Dkernel_bound
+lemma Doperator_norm : ‖Doperator‖ ≤ 23/10^9 := operator_norm _ _ _ (by norm_num) _
+def tailFixedColumn (c : Fin 23) : TailCoefficients :=
+  column (fun p (_ : Unit) => jet (if c.val=22 then h else H) h p.1.val.val p.2 (finiteColumn c))
+    (fun _ => by
+      by_cases hc : c.val=22
+      · have hh := (tail_atom_column_bounds 0 0 (finiteColumn c)).1
+        simpa [hc,damping] using hh
+      · have hh := (tail_atom_column_bounds 1 0 (finiteColumn c)).1
+        simpa [hc,damping] using hh) ()
+lemma tailFixedColumn_apply (c : Fin 23) (p : TailNode × Fin 2) :
+    tailFixedColumn c p = jet (if c.val=22 then h else H) h p.1.val.val p.2 (finiteColumn c) := rfl
+lemma tailFixedColumn_norm_finite (c : Fin 22) : ‖tailFixedColumn ⟨c.val,by omega⟩‖ ≤ 7/10^15 := by
+  rw [l1_norm]
+  have hc : c.val≠22 := by omega
+  simp only [tailFixedColumn_apply,hc,↓reduceIte]
+  exact (tail_finite_column_bound c).le
+lemma tailFixedColumn_norm_constant : ‖tailFixedColumn 22‖ ≤ 127/10^9 := by
+  rw [l1_norm]
+  simp only [tailFixedColumn_apply,Fin.isValue]
+  have hh := tail_constant_column_bound
+  convert hh.le using 1
+  rfl
+
+def Roperator : FiniteCoefficients →L[ℝ] FiniteCoefficients := l1Matrix R
+def Lplus : FiniteCoefficients →L[ℝ] FiniteCoefficients := l1Matrix (1-R)⁻¹
+def Lminus : FiniteCoefficients →L[ℝ] FiniteCoefficients := l1Matrix (1+R)⁻¹
+lemma Lplus_norm : ‖Lplus‖ ≤ 17 :=
+  ((l1Matrix_norm _).trans_lt FiniteCertificate.minusInverse_bound).le.trans (by norm_num)
+lemma Lminus_norm : ‖Lminus‖ ≤ 5 :=
+  ((l1Matrix_norm _).trans_lt FiniteCertificate.plusInverse_bound).le.trans (by norm_num)
+lemma Lplus_left_inverse : (1-Roperator).comp Lplus=1 := by
+  have hh := (l1Matrix_inverse (1-R) FiniteCertificate.minusBlock_isUnit).1
+  simpa [Lplus,l1Matrix_sub,l1Matrix_one,Roperator] using hh
+lemma Lminus_left_inverse : (1+Roperator).comp Lminus=1 := by
+  have hh := (l1Matrix_inverse (1+R) FiniteCertificate.plusBlock_isUnit).1
+  simpa [Lminus,l1Matrix_add,l1Matrix_one,Roperator] using hh
+end AtomicTriangular.Construction
+end
+
+end OAI

lean/OAI/Analysis/Triangular/Construction/CoefficientAddition.lean

@@ -0,0 +1,35 @@
+import OAI.Analysis.Triangular.Construction.ActualInterpolation
+import OAI.Analysis.Triangular.Construction.FinitePairBridge
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators lp
+open Complex Matrix
+namespace AtomicTriangular.Construction
+lemma inputJJet_split (C : ℝ) (a da : TrigonometricCoefficients) (w : TailCoefficients) (br : Fin 2) (k : ℕ) (s : ℂ) :
+    inputJJet (coefficientInput C (a+da) w) br k s=
+      inputJJet (coefficientInput C a 0) br k s+inputJJet (coefficientInput 0 da w) br k s := by
+  have hzero (T : C(Spectrum,ℂ) →L[ℂ] ℂ) : (0:ℂ) • T=0 := by
+    ext f
+    exact zero_mul _
+  simp only [hzero,inputJJet,tailSpectral_coefficient,finiteSpectral_coefficient_apply,
+    lp.coeFn_add,lp.coeFn_zero,Pi.zero_apply,add_zero,tsum_zero,Pi.add_apply,ofReal_add,ofReal_zero,zero_add,_root_.add_apply,
+    _root_.smul_apply,smul_eq_mul,add_mul,Finset.sum_add_distrib]
+  ring
+lemma gamma_actual_constant (side : Fin 2) (z : ℝ) :
+    (gamma side *ᵥ actualParameter z) 2=if side.val=0 then -(3/500:ℝ) else 3/500 := by
+  rw [gamma_actual]
+  rfl
+lemma actualJ_split (side : Fin 2) {z : ℝ} (hz : 0<z) (hzB : z≤Z) :
+    actualJ side z hz hzB = finiteJ side (actualParameter z)+
+      pairJ (coefficientInput 0 (extendFiniteError (actualFiniteError side z hz hzB)) (actualTail side z hz hzB))
+        (coefficientInput 0 (extendFiniteError (actualFiniteError (otherSide side) z hz hzB)) (actualTail (otherSide side) z hz hzB)) := by
+  funext s
+  simp only [Pi.add_apply,actualJ,actualInput,actualTrigonometric,finiteJ,finiteJJet_pair,pairJ,pairJJet,inputJJet_split,gamma_actual_constant]
+  simp only [add_re]
+  ring
+end AtomicTriangular.Construction
+end
+
+end OAI

lean/OAI/Analysis/Triangular/Construction/ConcreteSchur.lean

@@ -0,0 +1,121 @@
+import OAI.Analysis.Triangular.Construction.RepetitionOperators
+import OAI.Analysis.Triangular.Construction.Schur
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators lp
+namespace AtomicTriangular.Construction
+open AtomicTriangular.Schur
+
+def splitL (e : Fin 2) : FiniteCoefficients →L[ℝ] FiniteCoefficients :=
+  if e.val=0 then Lplus else Lminus
+def splitB (e : Fin 2) : TailCoefficients →L[ℝ] FiniteCoefficients :=
+  if e.val=0 then Boperator else -Boperator
+def splitK (e : Fin 2) : FiniteCoefficients →L[ℝ] TailCoefficients :=
+  if e.val=0 then Coperator-Aoperator else -Coperator-Aoperator
+def splitD (e : Fin 2) : TailCoefficients →L[ℝ] TailCoefficients :=
+  if e.val=0 then Doperator else -Doperator
+def splitN (e : Fin 2) : ℝ := if e.val=0 then 17 else 5
+def splitQ (e : Fin 2) : ℝ := delta+340*beta*splitN e
+lemma splitL_norm (e : Fin 2) : ‖splitL e‖ ≤ splitN e := by
+  fin_cases e
+  · exact Lplus_norm
+  · exact Lminus_norm
+lemma splitB_norm (e : Fin 2) : ‖splitB e‖ ≤ 340 := by
+  have hh := Boperator_norm
+  fin_cases e <;> simp only [splitB,↓reduceIte,Nat.one_ne_zero,norm_neg] <;> linarith
+lemma splitK_norm (e : Fin 2) : ‖splitK e‖ ≤ beta := by
+  have ha := AC_bound
+  fin_cases e
+  · exact (norm_sub_le _ _).trans (by linarith)
+  · exact (norm_sub_le _ _).trans (by simpa only [norm_neg] using (show ‖Coperator‖+‖Aoperator‖≤beta by linarith))
+lemma splitD_norm (e : Fin 2) : ‖splitD e‖ ≤ delta := by
+  have hh := Doperator_norm
+  fin_cases e <;> simp only [splitD,↓reduceIte,Nat.one_ne_zero,norm_neg] <;>
+    norm_num [delta] at * <;> linarith
+lemma splitN_nonneg (e : Fin 2) : 0 ≤ splitN e := by fin_cases e <;> norm_num [splitN]
+lemma splitQ_nonneg (e : Fin 2) : 0 ≤ splitQ e := by fin_cases e <;> norm_num [splitQ,splitN,beta,delta]
+lemma splitQ_lt (e : Fin 2) : splitQ e<1 := by fin_cases e <;> norm_num [splitQ,splitN,beta,delta]
+lemma splitOperator_norm (e : Fin 2) :
+    ‖tailOperator (splitL e) (splitB e) (splitK e) (splitD e)‖ ≤ splitQ e := by
+  have hb : 0≤beta := by norm_num [beta]
+  have hN := splitN_nonneg e
+  calc
+    _ ≤ ‖splitD e‖+‖(splitK e).comp ((splitL e).comp (splitB e))‖ := norm_add_le _ _
+    _ ≤ delta+‖splitK e‖*(‖splitL e‖*‖splitB e‖) :=
+      add_le_add (splitD_norm e) ((ContinuousLinearMap.opNorm_comp_le _ _).trans
+        (mul_le_mul_of_nonneg_left (ContinuousLinearMap.opNorm_comp_le _ _) (norm_nonneg _)))
+    _ ≤ delta+beta*(splitN e*340) := by
+      gcongr
+      · exact splitK_norm e
+      · exact splitL_norm e
+      · exact splitB_norm e
+    _ = splitQ e := by unfold splitQ; ring
+
+def correctionTail (e : Fin 2) (r : FiniteCoefficients) (s : TailCoefficients) : TailCoefficients :=
+  tailSolution (splitL e) (splitB e) (splitK e) (splitD e) r s
+def correctionFinite (e : Fin 2) (r : FiniteCoefficients) (s : TailCoefficients) : FiniteCoefficients :=
+  AtomicTriangular.Schur.finiteSolution (splitL e) (splitB e) (splitK e) (splitD e) r s
+lemma correctionTail_equation (e : Fin 2) (r : FiniteCoefficients) (s : TailCoefficients) :
+    correctionTail e r s = splitD e (correctionTail e r s) +
+      splitK e (correctionFinite e r s)+s :=
+  solution_equations _ _ _ _ _ _ ((splitOperator_norm e).trans_lt (splitQ_lt e))
+lemma correctionTail_norm (e : Fin 2) (r : FiniteCoefficients) (s : TailCoefficients) :
+    ‖correctionTail e r s‖ ≤ (‖s‖+beta*splitN e*‖r‖)/(1-splitQ e) := by
+  have hn := neumann_norm (tailOperator (splitL e) (splitB e) (splitK e) (splitD e))
+    (splitQ_nonneg e) (splitQ_lt e) (splitOperator_norm e)
+  have hq : 0≤1/(1-splitQ e) := by exact div_nonneg (by norm_num) (sub_pos.mpr (splitQ_lt e)).le
+  have hbeta : 0≤beta := by norm_num [beta]
+  have hN := splitN_nonneg e
+  calc
+    _ ≤ ‖neumann (tailOperator (splitL e) (splitB e) (splitK e) (splitD e))‖*
+        ‖s+splitK e (splitL e r)‖ := ContinuousLinearMap.le_opNorm _ _
+    _ ≤ (1/(1-splitQ e))*(‖s‖+‖splitK e‖*(‖splitL e‖*‖r‖)) := by
+      apply mul_le_mul hn _ (norm_nonneg _) hq
+      exact (norm_add_le _ _).trans (add_le_add_right ((ContinuousLinearMap.le_opNorm _ _).trans
+        (mul_le_mul_of_nonneg_left (ContinuousLinearMap.le_opNorm _ _) (norm_nonneg _))) _)
+    _ ≤ (1/(1-splitQ e))*(‖s‖+beta*(splitN e*‖r‖)) := by
+      gcongr
+      · exact splitK_norm e
+      · exact splitL_norm e
+    _ = _ := by ring
+lemma correctionFinite_norm (e : Fin 2) (r : FiniteCoefficients) (s : TailCoefficients) :
+    ‖correctionFinite e r s‖ ≤ splitN e*(‖r‖+340*‖correctionTail e r s‖) := by
+  have hN := splitN_nonneg e
+  calc
+    _ ≤ ‖splitL e‖*‖r+splitB e (correctionTail e r s)‖ := ContinuousLinearMap.le_opNorm _ _
+    _ ≤ splitN e*(‖r‖+‖splitB e‖*‖correctionTail e r s‖) :=
+      mul_le_mul (splitL_norm e) ((norm_add_le _ _).trans
+        (add_le_add_right (ContinuousLinearMap.le_opNorm _ _) _)) (norm_nonneg _) (splitN_nonneg e)
+    _ ≤ _ := by gcongr; exact splitB_norm e
+
+def finiteResidualBound : ℝ := 82/10^9
+def tailResidualBound (e : Fin 2) : ℝ := 1/10^26+6*beta+(if e.val=0 then 0 else (12/1000)*(13/10^8))
+def tailErrorBound (e : Fin 2) : ℝ := if e.val=0 then 1801/10^12 else 3361/10^12
+def finiteErrorBound (e : Fin 2) : ℝ := if e.val=0 then 11804/10^9 else 6124/10^9
+lemma correctionTail_small (e : Fin 2) (r : FiniteCoefficients) (s : TailCoefficients)
+    (hr : ‖r‖≤finiteResidualBound) (hs : ‖s‖≤tailResidualBound e) :
+    ‖correctionTail e r s‖ ≤ tailErrorBound e := by
+  have hh := correctionTail_norm e r s
+  have hb : (‖s‖+beta*splitN e*‖r‖)/(1-splitQ e) ≤
+      (tailResidualBound e+beta*splitN e*finiteResidualBound)/(1-splitQ e) := by
+    apply div_le_div_of_nonneg_right _ (sub_pos.mpr (splitQ_lt e)).le
+    apply add_le_add hs
+    exact mul_le_mul_of_nonneg_left hr (mul_nonneg (by norm_num [beta]) (splitN_nonneg e))
+  refine (hh.trans hb).trans ?_
+  fin_cases e <;> norm_num [tailResidualBound,tailErrorBound,finiteResidualBound,splitQ,splitN,beta,delta]
+lemma correctionFinite_small (e : Fin 2) (r : FiniteCoefficients) (s : TailCoefficients)
+    (hr : ‖r‖≤finiteResidualBound) (hs : ‖s‖≤tailResidualBound e) :
+    ‖correctionFinite e r s‖ ≤ finiteErrorBound e := by
+  have hN := splitN_nonneg e
+  have hh := correctionFinite_norm e r s
+  have hb := correctionTail_small e r s hr hs
+  calc
+    _ ≤ splitN e*(‖r‖+340*‖correctionTail e r s‖) := hh
+    _ ≤ splitN e*(finiteResidualBound+340*tailErrorBound e) := by gcongr
+    _ ≤ _ := by fin_cases e <;> norm_num [splitN,finiteResidualBound,tailErrorBound,finiteErrorBound]
+end AtomicTriangular.Construction
+end
+
+end OAI

lean/OAI/Analysis/Triangular/Construction/CurvatureAnalytic.lean

@@ -0,0 +1,98 @@
+import OAI.Analysis.Triangular.Envelope.TailEnvelopeSeries
+import OAI.Analysis.Triangular.Construction.CurvatureBoxes
+import OAI.Analysis.Triangular.Construction.AtomicCurvature
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators
+namespace AtomicTriangular.Construction
+open Complex
+
+lemma taylorPhase_re_nonpos (q br : Fin 2) {t : ℝ} (ht : |t|≤5/6) :
+    (taylorPhase q br t).re≤0 := by
+  fin_cases br
+  · simp only [taylorPhase,↓reduceIte,sub_re,mul_re,mul_im,I_re,I_im,
+      ofReal_re,ofReal_im,mul_zero,zero_mul,sub_zero,zero_sub]
+    have hh : h≤damping q := by fin_cases q <;> norm_num [damping,h,H]
+    exact neg_nonpos.mpr (mul_nonneg Real.pi_pos.le (sub_nonneg.mpr hh))
+  · change (I*Real.pi*zeta (damping q) h t).re≤0
+    rw [wavePhase_re]
+    have hh := (frequencyDecay_pos q 0 ht).le
+    change 0≤Real.pi*(damping q*Xwave (damping q) t-h) at hh
+    nlinarith
+
+lemma curvatureWave_nonneg (q br : Fin 2) (s t : ℝ) : 0≤curvatureWave q br s t := by
+  unfold curvatureWave; positivity
+lemma curvatureWave_antitone (q br : Fin 2) {s u t : ℝ} (ht : |t|≤5/6) (hsu : s≤ u) :
+    curvatureWave q br u t≤curvatureWave q br s t := by
+  unfold curvatureWave
+  apply mul_le_mul_of_nonneg_left _ (by positivity)
+  exact Real.exp_le_exp.mpr (mul_le_mul_of_nonpos_left hsu (taylorPhase_re_nonpos q br ht))
+lemma curvatureWave_direct (q : Fin 2) (s t : ℝ) :
+    curvatureWave q 0 s t = Real.pi^2*(t^2+(damping q-h)^2)*Real.exp (-Real.pi*(damping q-h)*s) := by
+  unfold curvatureWave
+  rw [←Complex.normSq_eq_norm_sq]
+  simp only [taylorPhase,taylorAmplitude,Fin.val_zero,↓reduceIte,norm_one,mul_one,
+    Complex.normSq_apply,sub_re,sub_im,mul_re,mul_im,ofReal_re,ofReal_im,I_re,I_im,
+    zero_mul,mul_zero,one_mul,sub_zero,zero_sub,zero_add]
+  ring_nf
+lemma curvatureWave_transformed (q : Fin 2) (s t : ℝ) :
+    curvatureWave q 1 s t = Real.pi^2*(h^2+(4/3-2*damping q*h)*Xwave (damping q) t)*
+      Real.sqrt (Xwave (damping q) t)*Real.exp (-Real.pi*(damping q*Xwave (damping q) t-h)*s) := by
+  simp only [curvatureWave,taylorPhase,taylorAmplitude,Fin.val_one,Nat.one_ne_zero,↓reduceIte]
+  rw [norm_wavePhase (damping_pos q),norm_lambda (damping_pos q),wavePhase_re,mul_pow,
+    Real.sq_sqrt]
+  rw [←normSq_zeta (damping_pos q)]
+  exact normSq_nonneg _
+lemma curvatureWave_even (q br : Fin 2) (s t : ℝ) :
+    curvatureWave q br s (-t)=curvatureWave q br s t := by
+  fin_cases br
+  · simp [curvatureWave_direct]
+  · simp [curvatureWave_transformed,Xwave,waveDen]
+lemma curvatureWave_abs (q br : Fin 2) (s t : ℝ) :
+    curvatureWave q br s |t|=curvatureWave q br s t := by
+  rcases le_total 0 t with ht|ht
+  · rw [abs_of_nonneg ht]
+  · rw [abs_of_nonpos ht,curvatureWave_even]
+lemma curvature_tail_direct {s t : ℝ} (ht : |t|≤5/6) :
+    curvatureWave 0 0 s t≤curvatureWave 0 0 s (5/6) := by
+  rw [curvatureWave_direct,curvatureWave_direct]
+  norm_num [damping]
+  have hh : t^2≤(5/6:ℝ)^2 := by simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg t) (by norm_num)).mpr ht
+  nlinarith [mul_nonneg (sq_nonneg Real.pi) (sub_nonneg.mpr hh)]
+lemma curvature_tail_zero {t : ℝ} (_ht : |t|≤5/6) :
+    curvatureWave 0 1 0 t≤curvatureWave 0 1 0 0 := by
+  rw [curvatureWave_transformed,curvatureWave_transformed]
+  simp only [mul_zero,Real.exp_zero,mul_one]
+  have hx : Xwave (damping 0) t≤Xwave (damping 0) 0 := by
+    rw [←show Xwave (damping 0) |t|=Xwave (damping 0) t by simp [Xwave,waveDen]]
+    exact Xwave_antitone (damping_pos 0) le_rfl (abs_nonneg t)
+  have h1 : 0≤4/3-2*damping 0*h := by norm_num [damping,h]
+  apply mul_le_mul (mul_le_mul_of_nonneg_left (add_le_add_right (mul_le_mul_of_nonneg_left hx h1) _) (sq_nonneg _)) (Real.sqrt_le_sqrt hx) (Real.sqrt_nonneg _)
+  exact mul_nonneg (sq_nonneg _) (add_nonneg (sq_nonneg _) (mul_nonneg h1 (Xwave_pos (damping_pos 0) 0).le))
+lemma curvature_tail_positive {s t : ℝ} (hs : 1 ≤ s) (ht : |t| ≤ 5/6) :
+    curvatureWave 0 1 s t≤curvatureWave 0 1 s (5/6) := by
+  rw [←curvatureWave_abs 0 1 s t]
+  rw [curvatureWave_transformed,curvatureWave_transformed]
+  have hx := Xwave_pos (damping_pos 0) (5/6)
+  have hxy := Xwave_antitone (damping_pos 0) (abs_nonneg t) ht
+  have hbase : (3/2:ℝ)≤Real.pi*h*Xwave h (5/6) := by
+    have hp := Real.pi_gt_three
+    norm_num [Xwave,waveDen,h] at *
+    linarith
+  have hb : (3/2:ℝ)≤(Real.pi*damping 0*s)*Xwave (damping 0) (5/6) := by
+    change (3/2:ℝ)≤(Real.pi*h*s)*Xwave h (5/6)
+    have hh : 0 ≤ h := by norm_num [h]
+    nlinarith [mul_nonneg (sub_nonneg.mpr hs) (mul_nonneg Real.pi_pos.le (mul_nonneg hh hx.le))]
+  have hh := Envelope.linear_sqrt_exp_antitone hx hxy (sq_nonneg h)
+    (show 0≤4/3-2*damping 0*h by norm_num [damping,h]) hb
+  have he (x : ℝ) : Real.exp (-Real.pi*(damping 0*x-h)*s) =
+      Real.exp (Real.pi*h*s)*Real.exp (-(Real.pi*damping 0*s)*x) := by
+    rw [←Real.exp_add]; congr 1; ring
+  rw [he,he]
+  convert mul_le_mul_of_nonneg_left hh (show 0≤Real.pi^2*Real.exp (Real.pi*h*s) by positivity) using 1 <;> first | rfl | ring
+end AtomicTriangular.Construction
+end
+
+end OAI

lean/OAI/Analysis/Triangular/Construction/CurvatureBoxes.lean

@@ -0,0 +1,54 @@
+import OAI.Analysis.Triangular.Envelope.EnvelopeBoxes
+import OAI.Analysis.Triangular.Envelope.EnvelopeWaves
+import OAI.Analysis.Triangular.Taylor.TaylorPhase
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators
+namespace AtomicTriangular.Construction
+open Complex
+
+def curvatureWave (q br : Fin 2) (s t : ℝ) : ℝ :=
+  ‖taylorPhase q br t‖^2 * ‖taylorAmplitude q br t‖ * Real.exp ((taylorPhase q br t).re*s)
+def curvaturePoint : Fin 5 → ℝ := ![0,1,2,29/2,23]
+def rawCurvatureColumn (br : Fin 2) (s : ℝ) (c : Fin 22) : ℝ :=
+  ∑ j : Fin 31, curvatureWave 1 br s (frequency j)*‖finiteColumn ⟨c.val,by omega⟩ j‖
+def rawConstantCurvature (s : ℝ) : ℝ :=
+  (∑ j : Fin 31, curvatureWave 0 1 s (frequency j)*‖fourierCoeff j‖)/2
+def rawTailCurvature (i : Fin 5) : ℝ :=
+  60*(curvatureWave 0 0 (curvaturePoint i) (frequency 30)+
+    curvatureWave 0 1 (curvaturePoint i) (frequency (if i.val=0 then 15 else 30)))
+end AtomicTriangular.Construction
+namespace AtomicTriangular.FiniteCertificate
+open Construction AtomicTriangular.Interval Complex
+def curvaturePointBox : Fin 5 → RI :=
+  ![RI.point 0,RI.point 1,RI.point 2,(RI.point 29).divNat 2,RI.point 23]
+lemma curvaturePointBox_mem (i : Fin 5) : (curvaturePointBox i).Mem (curvaturePoint i) := by
+  fin_cases i <;> norm_num [curvaturePointBox,curvaturePoint,RI.Mem,RI.point,RI.divNat,scale]
+def amplitudeNormBox (q br : Fin 2) (j : Fin 31) : RI :=
+  if br.val=0 then RI.point 1 else if q.val=0 then lambdaNorm0 j else lambdaNorm1 j
+lemma amplitudeNormBox_mem (q br : Fin 2) (j : Fin 31) :
+    (amplitudeNormBox q br j).Mem ‖taylorAmplitude q br (frequency j)‖ := by
+  fin_cases br
+  · simpa [amplitudeNormBox,taylorAmplitude] using ri1_mem
+  · fin_cases q
+    · simpa [amplitudeNormBox,taylorAmplitude] using lambdaNorm0_mem j
+    · simpa [amplitudeNormBox,taylorAmplitude] using lambdaNorm1_mem j
+
+def curvatureExpInput (q br : Fin 2) (i : Fin 5) (j : Fin 31) : CI :=
+  CI.ofReal ((taylorPhaseTable q br j).re.mul (curvaturePointBox i))
+def curvatureBox (q br : Fin 2) (i : Fin 5) (j : Fin 31) : RI :=
+  ((taylorPhaseTable q br j).normSq.mul (amplitudeNormBox q br j)).mul
+    (curvatureExpInput q br i j).exp.re
+lemma curvatureBox_mem (q br : Fin 2) (i : Fin 5) (j : Fin 31)
+    (hp : (curvatureExpInput q br i j).normSq.hi ≤ scale*1024^2) :
+    (curvatureBox q br i j).Mem (curvatureWave q br (curvaturePoint i) (frequency j)) := by
+  have hs := CI.mem_normSq (taylorPhaseTable_mem q br j)
+  rw [Complex.normSq_eq_norm_sq] at hs
+  exact RI.mem_mul (RI.mem_mul hs (amplitudeNormBox_mem q br j))
+    (ri_exp_mem (RI.mem_mul (taylorPhaseTable_mem q br j).1 (curvaturePointBox_mem i)) hp)
+end AtomicTriangular.FiniteCertificate
+end
+
+end OAI

lean/OAI/Analysis/Triangular/Construction/CurvatureEnvelope.lean

@@ -0,0 +1,101 @@
+import OAI.Analysis.Triangular.Construction.CurvatureAnalytic
+import OAI.Analysis.Triangular.Taylor.CoefficientSpectral
+import OAI.Analysis.Triangular.Construction.CurvatureSums
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators lp
+namespace AtomicTriangular.Construction
+open Complex AtomicTriangular.FiniteCertificate
+
+def finiteCurvatureCap (br : Fin 2) : Fin 5 → ℝ :=
+  if br.val=0 then ![828/10,44164/1000,23561/1000,9147/1000000,43834/1000000000]
+  else ![291294/100,45877/1000,3893/1000,635/10000000000,167/100000000000000]
+def finiteCurvatureEnvelope : Fin 5 → ℝ := ![1510,46,14,465/100000,23/1000000]
+def tailCurvatureEnvelope : Fin 5 → ℝ := ![26500,1160,800,800,800]
+lemma finiteCurvatureCap_bound (br : Fin 2) (i : Fin 5) (c : Fin 22) :
+    rawCurvatureColumn br (curvaturePoint i) c≤finiteCurvatureCap br i := by
+  fin_cases br <;> fin_cases i
+  all_goals first
+    | exact (curvatureColumn0_0_bound c).le | exact (curvatureColumn0_1_bound c).le
+    | exact (curvatureColumn0_2_bound c).le | exact (curvatureColumn0_3_bound c).le
+    | exact (curvatureColumn0_4_bound c).le | exact (curvatureColumn1_0_bound c).le
+    | exact (curvatureColumn1_1_bound c).le | exact (curvatureColumn1_2_bound c).le
+    | exact (curvatureColumn1_3_bound c).le | exact (curvatureColumn1_4_bound c).le
+lemma finiteCurvatureCap_sum (i : Fin 5) :
+    (finiteCurvatureCap 0 i+finiteCurvatureCap 1 i)/2≤finiteCurvatureEnvelope i := by
+  fin_cases i <;> norm_num [finiteCurvatureCap,finiteCurvatureEnvelope]
+lemma finiteCurvatureCap_nonneg (br : Fin 2) (i : Fin 5) : 0≤finiteCurvatureCap br i := by
+  fin_cases br <;> fin_cases i <;> norm_num [finiteCurvatureCap]
+lemma rawCurvatureColumn_antitone (br : Fin 2) (c : Fin 22) {u s : ℝ} (hus : u≤ s) :
+    rawCurvatureColumn br s c≤rawCurvatureColumn br u c := by
+  apply Finset.sum_le_sum
+  intro j _
+  exact mul_le_mul_of_nonneg_right (curvatureWave_antitone 1 br (abs_le.mpr (frequency_mem j)) hus) (norm_nonneg _)
+lemma finiteSpectral_curvature (C : ℝ) (a : TrigonometricCoefficients) (w : TailCoefficients)
+    (br : Fin 2) (i : Fin 5) {s : ℝ} (hs : curvaturePoint i≤ s) :
+    ‖finiteSpectral (coefficientInput C a w) (waveFunctionJet 1 br 2 s)‖≤‖a‖*finiteCurvatureCap br i := by
+  apply finiteSpectral_wave_bound
+  intro c
+  exact (rawCurvatureColumn_antitone br c hs).trans (finiteCurvatureCap_bound br i c)
+def tailCurvatureCap (br : Fin 2) (i : Fin 5) : ℝ :=
+  curvatureWave 0 br (curvaturePoint i) (if br.val=1 ∧ i.val=0 then 0 else 5/6)
+lemma tailCurvatureCap_nonneg (br : Fin 2) (i : Fin 5) : 0≤tailCurvatureCap br i := curvatureWave_nonneg _ _ _ _
+lemma tailCurvatureCap_bound (br : Fin 2) (i : Fin 5) {s t : ℝ}
+    (hs : curvaturePoint i≤ s) (ht : |t|≤5/6) : curvatureWave 0 br s t≤tailCurvatureCap br i := by
+  apply (curvatureWave_antitone 0 br ht hs).trans
+  fin_cases br
+  · exact curvature_tail_direct ht
+  · by_cases hi : i=0
+    · subst i
+      exact curvature_tail_zero ht
+    · have hu : 1≤curvaturePoint i := by fin_cases i <;> norm_num [curvaturePoint] at *
+      have hval : ¬i.val=0 := by simpa only [Fin.val_eq_zero_iff] using hi
+      simp only [tailCurvatureCap,true_and,hval,↓reduceIte]
+      convert curvature_tail_positive hu ht using 1 <;> rfl
+lemma tailCurvatureCap_sum (i : Fin 5) :
+    (119/2)*(tailCurvatureCap 0 i+tailCurvatureCap 1 i)≤tailCurvatureEnvelope i := by
+  have he : 60*(tailCurvatureCap 0 i+tailCurvatureCap 1 i)=rawTailCurvature i := by
+    fin_cases i <;> norm_num [tailCurvatureCap,rawTailCurvature,frequency]
+  have hb : rawTailCurvature i<tailCurvatureEnvelope i := by
+    fin_cases i
+    · exact curvatureTail0_bound
+    · exact curvatureTail1_bound
+    · exact curvatureTail2_bound
+    · exact curvatureTail3_bound
+    · exact curvatureTail4_bound
+  nlinarith [tailCurvatureCap_nonneg 0 i,tailCurvatureCap_nonneg 1 i]
+lemma tailSpectral_curvature (a : TrigonometricCoefficients) (w : TailCoefficients)
+    (br : Fin 2) (i : Fin 5) {s : ℝ} (hs : curvaturePoint i≤ s) :
+    ‖tailSpectral (coefficientInput 0 a w) (waveFunctionJet 0 br 2 s)‖≤119*‖w‖*tailCurvatureCap br i := by
+  exact ((tailSpectral _).le_opNorm _).trans (mul_le_mul (tailSpectral_zero_constant_norm a w)
+    (waveFunctionJet_uniform _ _ _ (tailCurvatureCap_nonneg br i) (fun t ht => tailCurvatureCap_bound br i hs ht))
+    (norm_nonneg _) (by positivity))
+lemma pairJ_curvature (a b : TrigonometricCoefficients) (w v : TailCoefficients)
+    (i : Fin 5) {s aL aT : ℝ} (hs : curvaturePoint i≤ s)
+    (ha : ‖a‖≤aL) (hb : ‖b‖≤aL) (hw : ‖w‖≤aT) (hv : ‖v‖≤aT) :
+    |iteratedDeriv 2 (pairJ (coefficientInput 0 a w) (coefficientInput 0 b v)) s|/2≤
+      aL*finiteCurvatureEnvelope i+aT*tailCurvatureEnvelope i := by
+  rw [←pairJJet_iterated]
+  have hTa := tailSpectral_curvature a w 0 i hs
+  have hTb := tailSpectral_curvature b v 1 i hs
+  have hLa := finiteSpectral_curvature 0 a w 0 i hs
+  have hLb := finiteSpectral_curvature 0 b v 1 i hs
+  have hh := Complex.abs_re_le_norm (pairJJet (coefficientInput 0 a w) (coefficientInput 0 b v) 2 s)
+  have hN := norm_add_le (inputJJet (coefficientInput 0 a w) 0 2 s) (inputJJet (coefficientInput 0 b v) 1 2 s)
+  have hNa := norm_add_le (tailSpectral (coefficientInput 0 a w) (waveFunctionJet 0 0 2 s)) (finiteSpectral (coefficientInput 0 a w) (waveFunctionJet 1 0 2 s))
+  have hNb := norm_add_le (tailSpectral (coefficientInput 0 b v) (waveFunctionJet 0 1 2 s)) (finiteSpectral (coefficientInput 0 b v) (waveFunctionJet 1 1 2 s))
+  have hL0 := mul_le_mul_of_nonneg_right ha (finiteCurvatureCap_nonneg 0 i)
+  have hL1 := mul_le_mul_of_nonneg_right hb (finiteCurvatureCap_nonneg 1 i)
+  have hT0 := mul_le_mul_of_nonneg_right hw (tailCurvatureCap_nonneg 0 i)
+  have hT1 := mul_le_mul_of_nonneg_right hv (tailCurvatureCap_nonneg 1 i)
+  have hLf := mul_le_mul_of_nonneg_left (finiteCurvatureCap_sum i) ((norm_nonneg a).trans ha)
+  have hTf := mul_le_mul_of_nonneg_left (tailCurvatureCap_sum i) ((norm_nonneg w).trans hw)
+  change |(pairJJet _ _ 2 s).re|/2≤_
+  dsimp only [pairJJet,inputJJet] at hh hN ⊢
+  nlinarith
+end AtomicTriangular.Construction
+end
+
+end OAI

lean/OAI/Analysis/Triangular/Construction/CurvatureSums.lean

@@ -0,0 +1,177 @@
+import OAI.Analysis.Triangular.Envelope.EnvelopeSumsBase
+import OAI.Analysis.Triangular.Envelope.EnvelopeNorms
+import OAI.Analysis.Triangular.Taylor.CurvatureData00
+import OAI.Analysis.Triangular.Taylor.CurvatureData01
+import OAI.Analysis.Triangular.Taylor.CurvatureData10
+import OAI.Analysis.Triangular.Taylor.CurvatureData11
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators
+namespace AtomicTriangular.FiniteCertificate
+open Construction AtomicTriangular.Interval Complex
+def curvatureColumn0_0 : Fin 22 → RI := ![⟨18931278953418004980981475350303064450074551871382092339685998,18931278953418004980981475350323566839439921866716315086729279⟩,⟨53153971995326934122877094040140299390063339738708527238055811,53153971995326934122877094040140300227771194920246741417165810⟩,⟨10906661929182473281713140014972139345824732340463494747135871,10906661929182473281713140014992642614594582274364751588478823⟩,⟨39910794816684720571431237362312035443007220174642251180355223,39910794816684720571431237362312062710999669093301784031260291⟩,⟨5171459074464329276346267760914009460734455361639903528531148,5171459074464329276346267760934549495431818609173876445333941⟩,⟨25552119883320652976287576564643637183688623268688467964614394,25552119883320652976287576564645303496653581345673454036276311⟩,⟨4050667990041770235350762754727619171544511661115989064320886,4050667990041770235350762754748197851927653664513364582633457⟩,⟨21189146532965609575648057106675042960720483923249955867507375,21189146532965609575648057106678708303162073897146027459888648⟩,⟨3170024831191457863357988051835930047935532821096119276246995,3170024831191457863357988051856441383748919647718543468991344⟩,⟨13641100870844811559736685323914012826011918252890656293489020,13641100870844811559736685323914344708045339139810406967306356⟩,⟨4257338934769909478475186436649126153387702488665189820909304,4257338934769909478475186436669628541156838573647247232029314⟩,⟨12056225255145991332415226171747301433641240891321531621118492,12056225255145991332415226171759402395082195433248858729169900⟩,⟨4687058798308615317454560150776022606226564183537041405362170,4687058798308615317454560150796563640182762162300082072790461⟩,⟨13405653299561784449405535281100747885199663356148886958002555,13405653299561784449405535281123551230330284418991753439916629⟩,⟨6959348639004267529004408373125516740656320733551809531984855,6959348639004267529004408373146099324023897202292721578977076⟩,⟨17421475348538079960123039275314841127751640054818818610532594,17421475348538079960123039275318249088860636594975958561765259⟩,⟨23692305685851956133903664854954503341660492352698432530051452,23692305685851956133903664854975013940435526148636252024482656⟩,⟨38953628097574911294595585002706736098602568869154574558311259,38953628097574911294595585002707065091648242008295457224700586⟩,⟨31898183105943367001537947368084484460791891565102351882507821,31898183105943367001537947368104987724493390620932233311848909⟩,⟨51870935959559777069906569196424487810808541920297387509966171,51870935959559777069906569196424515586567374225863168686094767⟩,⟨48697204576559363243147058658536891341040415083104777792831871,48697204576559363243147058658557431788360499096354193678391084⟩,⟨82782070552052637321534860770614371769314130420709126926616388,82782070552052637321534860770616022460080603773879098429307209⟩]
+lemma curvatureColumn0_0_calc (c : Fin 22) : RI.sum (fun j : Fin 31 => (curvatureTable10 0 j).mul (columnNormTable ⟨c.val,by omega⟩ j))=curvatureColumn0_0 c := by
+  revert c
+  decide +kernel
+lemma curvatureColumn0_0_bound (c : Fin 22) : rawCurvatureColumn 0 (curvaturePoint 0) c < (828/10:ℝ) := by
+  have hh := RI.mem_sum (fun j : Fin 31 => RI.mem_mul (curvatureTable10_mem 0 j) (columnNormTable_mem ⟨c.val,by omega⟩ j))
+  rw [curvatureColumn0_0_calc] at hh
+  have hb := ri_mem_lt hh 828 10 (by norm_num) (by clear hh; revert c; decide +kernel)
+  simpa only [rawCurvatureColumn, Nat.cast_ofNat, Nat.cast_one, Int.cast_ofNat, Int.cast_one] using hb
+def curvatureColumn0_1 : Fin 22 → RI := ![⟨10099611870772150463903374712483665635789245849855202842367989,10099611870772150463903374712494603416354583624502812411580649⟩,⟨28357011053696926667774517557794339456067415376376488197676815,28357011053696926667774517557794339902974579929197734449562675⟩,⟨5818574252775567222942745269580739056234852583591372264104688,5818574252775567222942745269591677305952007657629504776169146⟩,⟨21291933740681829691328782995547616246799309597217530103449399,21291933740681829691328782995547630793948549057451026330799143⟩,⟨2758911829791738608239320561600807893057901176074696414532469,2758911829791738608239320561611765756959542521675353728166545⟩,⟨13631751659883759092995686955172465915957457774875153957929387,13631751659883759092995686955173354874080293615789516463073547⟩,⟨2160983133651220036444169192958627165103911722834548554054316,2160983133651220036444169192969605646018688083496744741393808⟩,⟨11304157335721491759955944093328278841884413009650144925714390,11304157335721491759955944093330234258426772048441823867550833⟩,⟨1691170495903727679459132502248422186353122759603429941932935,1691170495903727679459132502259364739741935079361335141774572⟩,⟨7277364863968184715208718587326380089037083832518719351890280,7277364863968184715208718587326557144149560975213330037116524⟩,⟨2271239621438229950229096209008056180750654575953111224462865,2271239621438229950229096209018993960464420568832041196120037⟩,⟨6431852597132184159883505885306866794481647895840513502877020,6431852597132184159883505885313322513301149780560738702960088⟩,⟨2500490051141423509539923377665623967568102649324399930145950,2500490051141423509539923377676582364562432241608678774382479⟩,⟨7151756388612366122191593453940482907416203538514365994444500,7151756388612366122191593453952648220480430173853894047880101⟩,⟨3712729620659854111980538528224259005982073891756769470626533,3712729620659854111980538528235239569092565753624268995197334⟩,⟨9294149627682292963261072081765979914098420476064636780504649,9294149627682292963261072081767798020764971759535602859757906⟩,⟨12639562933892051862322384183918321501192700581010544162522880,12639562933892051862322384183929263661380328881497886900055192⟩,⟨20781296694848003320827528692249623672431249942091575745532252,20781296694848003320827528692249799186303168353237939261578176⟩,⟨17017300814464205803313101021243084884391469225703580586630825,17017300814464205803313101021254023131404719464819520450176593⟩,⟨27672526608174409576854076499561611981512702306951422823717722,27672526608174409576854076499561626799549260360498829007157527⟩,⟨25979378711021591704856352723377551477205137972313807230429251,25979378711021591704856352723388509561236086960033372331191720⟩,⟨44163248795383593382531747115579451958004733308925423515917645,44163248795383593382531747115580332581870720888206040328333088⟩]
+lemma curvatureColumn0_1_calc (c : Fin 22) : RI.sum (fun j : Fin 31 => (curvatureTable10 1 j).mul (columnNormTable ⟨c.val,by omega⟩ j))=curvatureColumn0_1 c := by
+  revert c
+  decide +kernel
+lemma curvatureColumn0_1_bound (c : Fin 22) : rawCurvatureColumn 0 (curvaturePoint 1) c < (44164/1000:ℝ) := by
+  have hh := RI.mem_sum (fun j : Fin 31 => RI.mem_mul (curvatureTable10_mem 1 j) (columnNormTable_mem ⟨c.val,by omega⟩ j))
+  rw [curvatureColumn0_1_calc] at hh
+  have hb := ri_mem_lt hh 44164 1000 (by norm_num) (by clear hh; revert c; decide +kernel)
+  simpa only [rawCurvatureColumn, Nat.cast_ofNat, Nat.cast_one, Int.cast_ofNat, Int.cast_one] using hb
+def curvatureColumn0_2 : Fin 22 → RI := ![⟨5388022657699280723992928518703689853912306170355691151179656,5388022657699280723992928518709525029586881587892792393350225⟩,⟨15128127696086087801175958661940776654546401796890999599715860,15128127696086087801175958661940776892966051909113200959158862⟩,⟨3104140070985079840845152154853575630971934357239231202673152,3104140070985079840845152154859411056933417217750209259490534⟩,⟨11358993086954602748008476581125002234807521578523896409320255,11358993086954602748008476581125009995538400155558348581034202⟩,⟨1471846605564257395081805206638268607525111838522796424106333,1471846605564257395081805206644114497420434588781346007761935⟩,⟨7272377171259364815861742713032459091351210499307157502840454,7272377171259364815861742713032933339923222122558104468780125⟩,⟨1152858766851659826275096276718949803634436956611335792515198,1152858766851659826275096276724806692460741106257247489727253⟩,⟨6030633318427550231777347479019357138944372090288755007368684,6030633318427550231777347479020400330382843179334746574002767⟩,⟨902219319569274129967815879512436833262938955427714915564785,902219319569274129967815879518274555181998922842527972394853⟩,⟨3882387489451853147551613139111923731711241451225965570614188,3882387489451853147551613139112018188505214802659639766951459⟩,⟨1211679290051561284278905489992744524226615678470282613227501,1211679290051561284278905489998579699446887691724553031900304⟩,⟨3431316764223403684798503424303957699265276521760357150378044,3431316764223403684798503424307401748374913492783792025167282⟩,⟨1333981664175733172318528326744139214539098581783744102213078,1333981664175733172318528326749985388833021959390152817733372⟩,⟨3815376863709413600140459324842183747818274092671935569273080,3815376863709413600140459324848673797462434020322302830814200⟩,⟨1980697038063221469717550014902855785352557875954285418934202,1980697038063221469717550014908713785005479566412123520719711⟩,⟨4958318143187314475468451981984269122153397637516668401251673,4958318143187314475468451981985239060408336090717510995126622⟩,⟨6743056301827935224821194428173157978848228739054049593026270,6743056301827935224821194428178995490999139629266608563778145⟩,⟨11086574304132314519338071208534244642352470379693483100597473,11086574304132314519338071208534338276912960141248090157397440⟩,⟨9078527327031585772273737145670387083129124243923266340359134,9078527327031585772273737145676222507648106075559611179076075���,⟨14762963395862727902459620085164850163883472798653845072446523,14762963395862727902459620085164858069129509872822862689545707⟩,⟨13859689156275755482504897634461963172242957290387694426265182,13859689156275755482504897634467809179574644167846491792560468⟩,⟨23560567296230658394109238891965789521227584001231156296564775,23560567296230658394109238891966259323572818982428585417944630⟩]
+lemma curvatureColumn0_2_calc (c : Fin 22) : RI.sum (fun j : Fin 31 => (curvatureTable10 2 j).mul (columnNormTable ⟨c.val,by omega⟩ j))=curvatureColumn0_2 c := by
+  revert c
+  decide +kernel
+lemma curvatureColumn0_2_bound (c : Fin 22) : rawCurvatureColumn 0 (curvaturePoint 2) c < (23561/1000:ℝ) := by
+  have hh := RI.mem_sum (fun j : Fin 31 => RI.mem_mul (curvatureTable10_mem 2 j) (columnNormTable_mem ⟨c.val,by omega⟩ j))
+  rw [curvatureColumn0_2_calc] at hh
+  have hb := ri_mem_lt hh 23561 1000 (by norm_num) (by clear hh; revert c; decide +kernel)
+  simpa only [rawCurvatureColumn, Nat.cast_ofNat, Nat.cast_one, Int.cast_ofNat, Int.cast_one] using hb
+def curvatureColumn0_3 : Fin 22 → RI := ![⟨2091647658548870926330018214280994892214515139338851753764,2091647658548870926330018214283260126106860845953738992876⟩,⟨5872787641033867980711936500282481093497733108333233962585,5872787641033867980711936500282481186053005160996636198832⟩,⟨1205037120993867603897779706893130424988403495650970066041,1205037120993867603897779706895395756042928572550291454102⟩,⟨4409597509737765697906659839950682466633488578083523576662,4409597509737765697906659839950685479374080455075924764114⟩,⟨571375567968780098227917713716909732156901575153899657133,571375567968780098227917713719179125344068974963075092240⟩,⟨2823160118046758664730596647312655018685039329687982609447,2823160118046758664730596647312839123500151935556640113929⟩,⟨447543466966875163370650848946382253174806035040012606238,447543466966875163370650848948655916182220183462770293605⟩,⟨2341111175921081332373917898546971726297348474838822973918,2341111175921081332373917898547376696556071923644011793963⟩,⟨350244430501419210616447220271366824830985345142027093916,350244430501419210616447220273633047183597934876856375940⟩,⟨1507155262290403790066985026813606471333709694888275825490,1507155262290403790066985026813643139763762800379987538553⟩,⟨470377782529725594198326207045435885661878919346804872921,470377782529725594198326207047701119377862588864218847388⟩,⟨1332048161559149678646172881045321130630094056979195446432,1332048161559149678646172881046658121528936255766367826143⟩,⟨517855956012579153152552834511696593119886476319760742169,517855956012579153152552834513966096711501831954206943724⟩,⟨1481141522680051318584080496194004004907455800084452718512,1481141522680051318584080496196523462972962453195682250967⟩,⟨768912936184398651913159612937001947738672563825116449685,768912936184398651913159612939276041972538648723671050620⟩,⟨1924834989273530006491791254861456813570808933294285808264,1924834989273530006491791254861833346708987177431344188831⟩,⟨2617676060628176192874849746171282511020214816525811920232,2617676060628176192874849746173548651940159852703249422122⟩,⟨4303843665436323494715746401391359818295440747540283382303,4303843665436323494715746401391396167531821147574834259507⟩,⟨3524313395290362780236926789070842418914673968534176848970,3524313395290362780236926789073107749409215524356768037853⟩,⟨5731029689727484104467202143541648399032303728226466574941,5731029689727484104467202143541651467874143149790977209748⟩,⟨5380375735895307936724282246038064726512455166350564778448,5380375735895307936724282246040334165288795377849356977632⟩,⟨9146287710728929951189203722741628289142951515661081138483,9146287710728929951189203722741810667918584044106235693263⟩]
+lemma curvatureColumn0_3_calc (c : Fin 22) : RI.sum (fun j : Fin 31 => (curvatureTable10 3 j).mul (columnNormTable ⟨c.val,by omega⟩ j))=curvatureColumn0_3 c := by
+  revert c
+  decide +kernel
+lemma curvatureColumn0_3_bound (c : Fin 22) : rawCurvatureColumn 0 (curvaturePoint 3) c < (9147/1000000:ℝ) := by
+  have hh := RI.mem_sum (fun j : Fin 31 => RI.mem_mul (curvatureTable10_mem 3 j) (columnNormTable_mem ⟨c.val,by omega⟩ j))
+  rw [curvatureColumn0_3_calc] at hh
+  have hb := ri_mem_lt hh 9147 1000000 (by norm_num) (by clear hh; revert c; decide +kernel)
+  simpa only [rawCurvatureColumn, Nat.cast_ofNat, Nat.cast_one, Int.cast_ofNat, Int.cast_one] using hb
+def curvatureColumn0_4 : Fin 22 → RI := ![⟨10024177015858215932838726634059815047242158049720294777,10024177015858215932838726634070671132599740912213449237⟩,⟨28145210140751058852955064512256060187280944403205323059,28145210140751058852955064512256060630850121561125855721⟩,⟨5775114829762064553976883764877752675987806184314541072,5775114829762064553976883764888609226993071759303002440⟩,⟨21132902487489458734796150046167424676159298654241028494,21132902487489458734796150046167439114654650585811609064⟩,⟨2738305284088436636902884596714567876179200572905427944,2738305284088436636902884596725443894868950250172836543⟩,⟨13529934954267377840065211580614063243345078502912595110,13529934954267377840065211580614945561764539727076042995⟩,⟨2144842567930056691188415032373355095314565267559747502,2144842567930056691188415032384251577027235831032241914⟩,⟨11219725628894004985686161663108534629526006612793405232,11219725628894004985686161663110475440896894058765107360⟩,⟨1678539000493253950567320195242269218480893705563733732,1678539000493253950567320195253130041013329073139321291⟩,⟨7223009610549200161218045656700245845209505632184731036,7223009610549200161218045656700421577882378918845907884⟩,⟨2254275540688374991510978476415391350211019404151217776,2254275540688374991510978476426247434723390946507774482⟩,⟨6383812546316312047511384681867524806917531847696840749,6383812546316312047511384681873932307427385259798533953⟩,⟨2481813679550604242826044836430667603005191773836409409,2481813679550604242826044836441544150805915340036229658⟩,⟨7098339315515127168140112725370001549063106791808789962,7098339315515127168140112725382075998375565076464260701⟩,⟨3684998929237973657362595100066738325102547644164597486,3684998929237973657362595100077636873458837874763549514⟩,⟨9224730838358163157933307657323308293140631775954084777,9224730838358163157933307657325112820214782458814339403⟩,⟨12545156969752679841302531312306172378460356244500977460,12545156969752679841302531312317032810728460478640538029⟩,⟨20626079432921735331321394208831988439722456585052487348,20626079432921735331321394208832162642666427174379248531⟩,⟨16890197155987898025755310864236240080038523712906256343,16890197155987898025755310864247096628360080146781195673⟩,⟨27465838167420494263614374387691835736186262493467847646,27465838167420494263614374387691850443545652734161949846⟩,⟨25785336535055254940617645130668766206884107787483524150,25785336535055254940617645130679642444059000697707450799⟩,⟨43833389756438856736036931141894651599373411099242844407,43833389756438856736036931141895525645785293064228745766⟩]
+lemma curvatureColumn0_4_calc (c : Fin 22) : RI.sum (fun j : Fin 31 => (curvatureTable10 4 j).mul (columnNormTable ⟨c.val,by omega⟩ j))=curvatureColumn0_4 c := by
+  revert c
+  decide +kernel
+lemma curvatureColumn0_4_bound (c : Fin 22) : rawCurvatureColumn 0 (curvaturePoint 4) c < (43834/1000000000:ℝ) := by
+  have hh := RI.mem_sum (fun j : Fin 31 => RI.mem_mul (curvatureTable10_mem 4 j) (columnNormTable_mem ⟨c.val,by omega⟩ j))
+  rw [curvatureColumn0_4_calc] at hh
+  have hb := ri_mem_lt hh 43834 1000000000 (by norm_num) (by clear hh; revert c; decide +kernel)
+  simpa only [rawCurvatureColumn, Nat.cast_ofNat, Nat.cast_one, Int.cast_ofNat, Int.cast_one] using hb
+def curvatureColumn1_0 : Fin 22 → RI := ![⟨1058499932694297960295845242156766039153190046624018212610135487,1058499932694297960295845242156812199561029336678790007456361467⟩,⟨2422019594396479840112235111927886942388328603172501572816687649,2422019594396479840112235111927886944751436743301852524841070739⟩,⟨512929371103136489128038858862251381154513195020104627419606229,512929371103136489128038858862297544629981831238889059496542592⟩,⟨1710812219114451314420354774967307022862246709922899168329053899,1710812219114451314420354774967307095886959757785903360867337506⟩,⟨220722556479815099731798455180928928745767189634487947797423821,220722556479815099731798455180975212849025500603782749698527938⟩,⟨1050380114163500856414032728698156134501460941336323789223313591,1050380114163500856414032728698160464539801088034831272687315696⟩,⟨167428954823840485917511441512051139687057474093271616617038055,167428954823840485917511441512097549064080690776007140342770918⟩,⟨861852346726326464114200873982381134783905630825990199679877536,861852346726326464114200873982390622202968045439660740437014328⟩,⟨123690710020062520758016987182361092613217718621934961529529800,123690710020062520758016987182407283336368586252934200296459631⟩,⟨511216114745745943443653892216227294351336465403333715248075398,511216114745745943443653892216228169103115356815094201791204539⟩,⟨170832868374401166609870222505323671970080505807538944366000695,170832868374401166609870222505369832370816429133219135812587836⟩,⟨384627149369005958233521415715264632052800161851321640645078603,384627149369005958233521415715481063304381741551419365986192192⟩,⟨142949747028953210297425020919416258288498325617985070199562411,142949747028953210297425020919462546620603105033455495586625873⟩,⟨396398463401248617919267216535216163376367963464099827127769974,396398463401248617919267216535395265894679537601909413880945906⟩,⟨238059293790868663995364503248530352563243939025592309015755308,238059293790868663995364503248576775791311955261183134267174086⟩,⟨483099045015476558457052171035507155017706393628534625003319415,483099045015476558457052171035515818611231742698425938372298945⟩,⟨1260577198657405506011828913972117831340287470875556470158080162,1260577198657405506011828913972164018894536393113441234236523849⟩,⟨1053084952382937612174359381168037551795251616276329462120548551,1053084952382937612174359381168038419169821377691563374149584073⟩,⟨1510326873385029286950778038402781511134585271232170315685743424,1510326873385029286950778038402827674581303317840960817374782937⟩,⟨1478079488516475082264873232116025962363823329170199354291498963,1478079488516475082264873232116026037256637681071698570074246760⟩,⟨2808301589493704590979127227184162915950583115106120455280573563,2808301589493704590979127227184209202178538149831867257756356932⟩,⟨2912933586518956541139929332473683756538058684744762080888513837,2912933586518956541139929332473688025683657170033607150130266758⟩]
+lemma curvatureColumn1_0_calc (c : Fin 22) : RI.sum (fun j : Fin 31 => (curvatureTable11 0 j).mul (columnNormTable ⟨c.val,by omega⟩ j))=curvatureColumn1_0 c := by
+  revert c
+  decide +kernel
+lemma curvatureColumn1_0_bound (c : Fin 22) : rawCurvatureColumn 1 (curvaturePoint 0) c < (291294/100:ℝ) := by
+  have hh := RI.mem_sum (fun j : Fin 31 => RI.mem_mul (curvatureTable11_mem 0 j) (columnNormTable_mem ⟨c.val,by omega⟩ j))
+  rw [curvatureColumn1_0_calc] at hh
+  have hb := ri_mem_lt hh 291294 100 (by norm_num) (by clear hh; revert c; decide +kernel)
+  simpa only [rawCurvatureColumn, Nat.cast_ofNat, Nat.cast_one, Int.cast_ofNat, Int.cast_one] using hb
+def curvatureColumn1_1 : Fin 22 → RI := ![⟨9270259362346716978466799944457304084114037333341404923620760,9270259362346716978466799944470852947668652837833837981935356⟩,⟨27872111694533796162667262297548217267220090721459286275602402,27872111694533796162667262297548217830568099466141286890817868⟩,⟨5666026977098111673216153849428611006268414737730139484890449,5666026977098111673216153849442160475650652811769389304088025⟩,⟨21314071875525581465196204539713715341700720345180361152283212,21314071875525581465196204539713733634682358176092258693154921⟩,⟨2765544057976893594882064717794268832948198932882768124481396,2765544057976893594882064717807843592246589575847879741117312⟩,⟨13803961780347957739080071808413588040745118084177704522445195,13803961780347957739080071808414704262611245108059010194152248⟩,⟨2182069542415650330586295068199032939551792961385605379304877,2182069542415650330586295068212634258235120460331981285581498⟩,⟨11478120585351167333785359299521166686954476456570090729873530,11478120585351167333785359299523621344848954597040371561352305⟩,⟨1740537632190099241930054432317353484763512262818070517075116,1740537632190099241930054432330908508128523074694959054728486⟩,⟨7545201734234663671084394356751160887545154111332571527361903,7545201734234663671084394356751383367971029787866359146796613⟩,⟨2307435542176419854279345516466552719764801004753261267134835,2307435542176419854279345516480101582207319844307205873149155⟩,⟨6860901960868398845947545357672339167202965457021621918194102,6860901960868398845947545357680378068660545574164344317543727⟩,⟨2776039856083173213030980089771677863514643166940299869715061,2776039856083173213030980089785253319804845666905399907903065⟩,⟨7867199364802920160885877260151455413588226676621315798706886,7867199364802920160885877260166669893165748086822032339287282⟩,⟨4005911433086196820315416305739695047758724459661875034240667,4005911433086196820315416305753299096944557151534940341603772⟩,⟨10385980574123612329244027411133114197399764373032731384686132,10385980574123612329244027411135391843431099203513751257034385⟩,⟨11834881908768397279563847846071356820849847179573371909860792,11834881908768397279563847846084911330341523761192402365734668⟩,⟨22744854138567463076442858393702819738451661324648470087714999,22744854138567463076442858393703040231975389397319063148514865⟩,⟨16632110560540465780749888314401211898461731510228923054606859,16632110560540465780749888314414761364330992276476380650878282⟩,⟨30839664347490436475292992345200376683394552663207878087124915,30839664347490436475292992345200395330947461948655006260875933⟩,⟨23563627475565198277634050028845244599404887934752247079436683,23563627475565198277634050028858819646623892715617886980414484⟩,⟨45876472112082918662366993203297603245388858626401184781017299,45876472112082918662366993203298708541513383365798219266894991⟩]
+lemma curvatureColumn1_1_calc (c : Fin 22) : RI.sum (fun j : Fin 31 => (curvatureTable11 1 j).mul (columnNormTable ⟨c.val,by omega⟩ j))=curvatureColumn1_1 c := by
+  revert c
+  decide +kernel
+lemma curvatureColumn1_1_bound (c : Fin 22) : rawCurvatureColumn 1 (curvaturePoint 1) c < (45877/1000:ℝ) := by
+  have hh := RI.mem_sum (fun j : Fin 31 => RI.mem_mul (curvatureTable11_mem 1 j) (columnNormTable_mem ⟨c.val,by omega⟩ j))
+  rw [curvatureColumn1_1_calc] at hh
+  have hb := ri_mem_lt hh 45877 1000 (by norm_num) (by clear hh; revert c; decide +kernel)
+  simpa only [rawCurvatureColumn, Nat.cast_ofNat, Nat.cast_one, Int.cast_ofNat, Int.cast_one] using hb
+def curvatureColumn1_2 : Fin 22 → RI := ![⟨492737994145033258700847028046463667588492418873980399900044,492737994145033258700847028050440491097563524774734674439459⟩,⟨2088225632928948965862684975488287924817843951890290961125088,2088225632928948965862684975488288068920447118821604914009135⟩,⟨373259149351668988056358864045218250189980081629738817969576,373259149351668988056358864049195200175335144260500511298646⟩,⟨1730740367376542488918229535524054157407011376294068374878621,1730740367376542488918229535524058969784406138827643301889767⟩,⟨219272662534707682167763991420799780154700399892488090903117,219272662534707682167763991424782227867682253103174660269307⟩,⟨1204796664817511404320493407604269445398193685512890150121326,1204796664817511404320493407604567767730513142964726992135074⟩,⟨178686680211071685234108861684963464727159670710887448489495,178686680211071685234108861688951745768145011599439292576608⟩,⟨1021065582736533232735111218482377889096478055674461397327563,1021065582736533232735111218483035447415013645602860319770280⟩,⟨150101691034635769719078825107773978472035350010418773478875,150101691034635769719078825111752112037074136280076271457321⟩,⟨768863569952289287926862957712573757772030933970583739067370,768863569952289287926862957712632760930932724388281382835873⟩,⟨198469835165894648430327472150392663053309590747421413645226,198469835165894648430327472154369486381987855089091999344079⟩,⟨834494366484370953353125088747187130385162605937530597745682,834494366484370953353125088747950906379540149131122186007742⟩,⟨294188263747599027157576910856834958495359851064983130884618,294188263747599027157576910860817524454698224763826032828141⟩,⟨1135548606651355606860482908266551594232553299527499636403108,1135548606651355606860482908268017408079263282959352781623739⟩,⟨371385030865852095875809226481846514902983741735185942181989,371385030865852095875809226485835314301186787736098384090502⟩,⟨1411179907701577172759281232326094485496625845649448157584513,1411179907701577172759281232326713078263324436152987205493851⟩,⟨689198484301338120805034581534468972446045660363658863837789,689198484301338120805034581538447019802160348578856082547394⟩,⟨2432278457694674313559771427280971762529699907023985993746813,2432278457694674313559771427281030285293109522087730778178041⟩,⟨977312728288406147189024143737030731493125833943428704124525,977312728288406147189024143741007681040864560130667115369395⟩,⟨3334579168264614880614730288335750974819194761209570545796425,3334579168264614880614730288335755853863555589647297285521655⟩,⟨1126968346473033480137737149160140766837091109860588827712525,1126968346473033480137737149164123253683200381973453232401409⟩,⟨3892280461323035124905217620209048740852949672501748608844940,3892280461323035124905217620209345115188615995110685068087506⟩]
+lemma curvatureColumn1_2_calc (c : Fin 22) : RI.sum (fun j : Fin 31 => (curvatureTable11 2 j).mul (columnNormTable ⟨c.val,by omega⟩ j))=curvatureColumn1_2 c := by
+  revert c
+  decide +kernel
+lemma curvatureColumn1_2_bound (c : Fin 22) : rawCurvatureColumn 1 (curvaturePoint 2) c < (3893/1000:ℝ) := by
+  have hh := RI.mem_sum (fun j : Fin 31 => RI.mem_mul (curvatureTable11_mem 2 j) (columnNormTable_mem ⟨c.val,by omega⟩ j))
+  rw [curvatureColumn1_2_calc] at hh
+  have hb := ri_mem_lt hh 3893 1000 (by norm_num) (by clear hh; revert c; decide +kernel)
+  simpa only [rawCurvatureColumn, Nat.cast_ofNat, Nat.cast_one, Int.cast_ofNat, Int.cast_one] using hb
+def curvatureColumn1_3 : Fin 22 → RI := ![⟨828291193688351065221289482776556229364831097352885,828291193688351065221289483657518019560837852255545⟩,⟨59023234651583563077564286501807644410579526107038576,59023234651583563077564286501807665488871694472325990⟩,⟨821514986992793647703932763469964337554816767102428,821514986992793647703932764350927302192507881171724⟩,⟨58300047068333113679285032194627224059617204093666593,58300047068333113679285032194627975218168093954641754⟩,⟨807245979127971556650196989563876474730587402626466,807245979127971556650196990444896750215062510251413⟩,⟨56781954717707114801063844314479127287888655370811338,56781954717707114801063844314527773178852725300839792⟩,⟨799861065510841784000286263802705631310823796875988,799861065510841784000286264683790173817278747241084⟩,⟨55999929334559295782400124046843776375377214548104542,55999929334559295782400124046951904918051598273509148⟩,⟨792381090351517351169531645997916081537869248844767,792381090351517351169531646878890698658342313713051⟩,⟨55211609170930812194673482619002051087927615099543873,55211609170930812194673482619011555412401876707758614⟩,⟨807244573466176795237307810854725234698106138372522,807244573466176795237307811735687024824275615472456⟩,⟨56781046956026617019652650653619876835845309135386360,56781046956026617019652650653619898030134226043802547⟩,⟨828279377991535332502114094166439089828785638184964,828279377991535332502114095047459417961883713062077⟩,⟨59021115987766726911233527608358573531825890873866951,59021115987766726911233527608406973829222237240058202⟩,⟨834747110639239283341368853634373515552796919651387,834747110639239283341368854515458352251344366484857⟩,⟨59712676867531556757295528327508957891604456608782133,59712676867531556757295528327616671392308448075366674⟩,⟨851782656720845330124515675910226934324330289719864,851782656720845330124515676791201516279189139624652⟩,⟨61538913962159813190420892026634397545673897368232435,61538913962159813190420892026643896711483493235059028⟩,⟨864304828848271133703899648529360152712070406524289,864304828848271133703899649410323117460706971138485⟩,⟨62884815823075131562678055504605269022076259280912375,62884815823075131562678055504606020288339035320627235⟩,⟨869652553692509682954881910877302387656899972160165,869652553692509682954881911758322677636330280389834⟩,⟨63460083219929588203904981288693286943491044026426131,63460083219929588203904981288741943466566456123485412⟩]
+lemma curvatureColumn1_3_calc (c : Fin 22) : RI.sum (fun j : Fin 31 => (curvatureTable11 3 j).mul (columnNormTable ⟨c.val,by omega⟩ j))=curvatureColumn1_3 c := by
+  revert c
+  decide +kernel
+lemma curvatureColumn1_3_bound (c : Fin 22) : rawCurvatureColumn 1 (curvaturePoint 3) c < (635/10000000000:ℝ) := by
+  have hh := RI.mem_sum (fun j : Fin 31 => RI.mem_mul (curvatureTable11_mem 3 j) (columnNormTable_mem ⟨c.val,by omega⟩ j))
+  rw [curvatureColumn1_3_calc] at hh
+  have hb := ri_mem_lt hh 635 10000000000 (by norm_num) (by clear hh; revert c; decide +kernel)
+  simpa only [rawCurvatureColumn, Nat.cast_ofNat, Nat.cast_one, Int.cast_ofNat, Int.cast_one] using hb
+def curvatureColumn1_4 : Fin 22 → RI := ![⟨3363994217262576426026759788738909011532115958,3363994217262576426026759815032114576133903227⟩,⟨1641702348658942796387296095868126236366780573291,1641702348658942796387296095868126853003078038708⟩,⟨3360874979691564559664658327453458167727166099,3360874979691564559664658353746668684385148024⟩,⟨1638724646252702774593086094723728011882128575789,1638724646252702774593086094723750040045954225543⟩,⟨3354275192893685854228300473131200167139716013,3354275192893685854228300499424653286477127664⟩,⟨1632427446509710339480629826955186881717697983701,1632427446509710339480629826956616894846367448539⟩,⟨3350844778800741105768102958990001255712974844,3350844778800741105768102985283727182575505922⟩,⟨1629156424562103617443300916909936099642520059103,1629156424562103617443300916913116308433823852781⟩,⟨3347361753625311826816310743325360026955249978,3347361753625311826816310769618619775086224861⟩,⟨1625837344319338801695032959870056516559762570220,1625837344319338801695032959870335895248514068416⟩,⟨3354275189524729012367889053063324012034852024,3354275189524729012367889079356529576602238880⟩,⟨1632427411850071157167235552141188504637379464678,1632427411850071157167235552141189121056405117038⟩,⟨3363994188668989003144924388286232301292948893,3363994188668989003144924414579685457019503825⟩,⟨1641702267225306201235397266577579811427347654033,1641702267225306201235397266579008841985455793190⟩,⟨3366960955963685075437729729998795678679450118,3366960955963685075437729756292521778924211533⟩,⟨1644535214707477572708230739910382443526742864052,1644535214707477572708230739913561092095918525234⟩,⟨3374726216469945217393809000283666505918872620,3374726216469945217393809026576926241310360899⟩,⟨1651953715875160642804910734195658411112439473246,1651953715875160642804910734195937786880845754239⟩,⟨3380389537948522881303619586036679798009370040,3380389537948522881303619612329890315157378183⟩,⟨1657367270612848520494907948264097688418356103821,1657367270612848520494907948264119718701097622593⟩,⟨3382797814785111546996181070700560350180601040,3382797814785111546996181096994013528823613237⟩,⟨1659670116207951544873702557793619978360077817422,1659670116207951544873702557795050333224277373680⟩]
+lemma curvatureColumn1_4_calc (c : Fin 22) : RI.sum (fun j : Fin 31 => (curvatureTable11 4 j).mul (columnNormTable ⟨c.val,by omega⟩ j))=curvatureColumn1_4 c := by
+  revert c
+  decide +kernel
+lemma curvatureColumn1_4_bound (c : Fin 22) : rawCurvatureColumn 1 (curvaturePoint 4) c < (167/100000000000000:ℝ) := by
+  have hh := RI.mem_sum (fun j : Fin 31 => RI.mem_mul (curvatureTable11_mem 4 j) (columnNormTable_mem ⟨c.val,by omega⟩ j))
+  rw [curvatureColumn1_4_calc] at hh
+  have hb := ri_mem_lt hh 167 100000000000000 (by norm_num) (by clear hh; revert c; decide +kernel)
+  simpa only [rawCurvatureColumn, Nat.cast_ofNat, Nat.cast_one, Int.cast_ofNat, Int.cast_one] using hb
+def curvatureConstant0 : RI := ⟨13522464111263058769274694040018732303032654392195743820811661829,13522464111263058769274694040018732303032665260832714006879471591⟩
+lemma curvatureConstant0_calc : (RI.sum (fun j : Fin 31 => (curvatureTable01 0 j).mul (columnNorm22 j))).divNat 2=curvatureConstant0 := by decide +kernel
+lemma curvatureConstant0_bound : rawConstantCurvature (curvaturePoint 0) < (13530/1:ℝ) := by
+  have hh := RI.mem_divNat (RI.mem_sum (fun j : Fin 31 => RI.mem_mul (curvatureTable01_mem 0 j) (fourierNorm_mem j))) (by norm_num : 0<2)
+  rw [curvatureConstant0_calc] at hh
+  have hb := ri_mem_lt hh 13530 1 (by norm_num) (by decide +kernel)
+  simpa only [rawConstantCurvature, Nat.cast_ofNat, Nat.cast_one, Int.cast_ofNat, Int.cast_one] using hb
+def curvatureTail0 : RI := ⟨26209350125794453268038359194214686316500054332137775901261727180,26209350125794453268038359194214686316500054332137942758578330681⟩
+lemma curvatureTail0_calc : ((curvatureTable00 0 30).add (curvatureTable01 0 15)).mul (RI.point 60)=curvatureTail0 := by decide +kernel
+lemma curvatureTail0_bound : rawTailCurvature 0 < (26500:ℝ) := by
+  have hh := RI.mem_mul (RI.mem_add (curvatureTable00_mem 0 30) (curvatureTable01_mem 0 15)) (RI.mem_point 60)
+  rw [curvatureTail0_calc] at hh
+  have hh' := ri_mem_lt hh 26500 1 (by norm_num) (by decide +kernel)
+  simpa [rawTailCurvature,mul_comm] using hh'
+def curvatureConstant1 : RI := ⟨150371258648440095142820936910505736961983543519393796861914761,150371258648440095142820936910505736961983548419330132820503909⟩
+lemma curvatureConstant1_calc : (RI.sum (fun j : Fin 31 => (curvatureTable01 1 j).mul (columnNorm22 j))).divNat 2=curvatureConstant1 := by decide +kernel
+lemma curvatureConstant1_bound : rawConstantCurvature (curvaturePoint 1) < (151/1:ℝ) := by
+  have hh := RI.mem_divNat (RI.mem_sum (fun j : Fin 31 => RI.mem_mul (curvatureTable01_mem 1 j) (fourierNorm_mem j))) (by norm_num : 0<2)
+  rw [curvatureConstant1_calc] at hh
+  have hb := ri_mem_lt hh 151 1 (by norm_num) (by decide +kernel)
+  simpa only [rawConstantCurvature, Nat.cast_ofNat, Nat.cast_one, Int.cast_ofNat, Int.cast_one] using hb
+def curvatureTail1 : RI := ⟨1146570904437445124234805277954807889641704200443903319238089820,1146570904437445124234805277954807889641704200443912235954319601⟩
+lemma curvatureTail1_calc : ((curvatureTable00 1 30).add (curvatureTable01 1 30)).mul (RI.point 60)=curvatureTail1 := by decide +kernel
+lemma curvatureTail1_bound : rawTailCurvature 1 < (1160:ℝ) := by
+  have hh := RI.mem_mul (RI.mem_add (curvatureTable00_mem 1 30) (curvatureTable01_mem 1 30)) (RI.mem_point 60)
+  rw [curvatureTail1_calc] at hh
+  have hh' := ri_mem_lt hh 1160 1 (by norm_num) (by decide +kernel)
+  simpa [rawTailCurvature,mul_comm] using hh'
+def curvatureConstant2 : RI := ⟨30877143611153424423463150047039229594938245849736446159092976,30877143611153424423463150047039229594938245932926941674785634⟩
+lemma curvatureConstant2_calc : (RI.sum (fun j : Fin 31 => (curvatureTable01 2 j).mul (columnNorm22 j))).divNat 2=curvatureConstant2 := by decide +kernel
+lemma curvatureConstant2_bound : rawConstantCurvature (curvaturePoint 2) < (31/1:ℝ) := by
+  have hh := RI.mem_divNat (RI.mem_sum (fun j : Fin 31 => RI.mem_mul (curvatureTable01_mem 2 j) (fourierNorm_mem j))) (by norm_num : 0<2)
+  rw [curvatureConstant2_calc] at hh
+  have hb := ri_mem_lt hh 31 1 (by norm_num) (by decide +kernel)
+  simpa only [rawConstantCurvature, Nat.cast_ofNat, Nat.cast_one, Int.cast_ofNat, Int.cast_one] using hb
+def curvatureTail2 : RI := ⟨780053907590041165674185831867232483829602501942465790620589260,780053907590041165674185831867232483829602501942472381224322021⟩
+lemma curvatureTail2_calc : ((curvatureTable00 2 30).add (curvatureTable01 2 30)).mul (RI.point 60)=curvatureTail2 := by decide +kernel
+lemma curvatureTail2_bound : rawTailCurvature 2 < (800:ℝ) := by
+  have hh := RI.mem_mul (RI.mem_add (curvatureTable00_mem 2 30) (curvatureTable01_mem 2 30)) (RI.mem_point 60)
+  rw [curvatureTail2_calc] at hh
+  have hh' := ri_mem_lt hh 800 1 (by norm_num) (by decide +kernel)
+  simpa [rawTailCurvature,mul_comm] using hh'
+def curvatureConstant3 : RI := ⟨1215198976953092218551294390379033113285331535323125796493,1215198976953092218551294390379033113285331535325211480446⟩
+lemma curvatureConstant3_calc : (RI.sum (fun j : Fin 31 => (curvatureTable01 3 j).mul (columnNorm22 j))).divNat 2=curvatureConstant3 := by decide +kernel
+lemma curvatureConstant3_bound : rawConstantCurvature (curvaturePoint 3) < (122/100000:ℝ) := by
+  have hh := RI.mem_divNat (RI.mem_sum (fun j : Fin 31 => RI.mem_mul (curvatureTable01_mem 3 j) (fourierNorm_mem j))) (by norm_num : 0<2)
+  rw [curvatureConstant3_calc] at hh
+  have hb := ri_mem_lt hh 122 100000 (by norm_num) (by decide +kernel)
+  simpa only [rawConstantCurvature, Nat.cast_ofNat, Nat.cast_one, Int.cast_ofNat, Int.cast_one] using hb
+def curvatureTail3 : RI := ⟨411299725916832894558639495174632402962049280878910989332766820,411299725916832894558639495174632402962049280878913609952289521⟩
+lemma curvatureTail3_calc : ((curvatureTable00 3 30).add (curvatureTable01 3 30)).mul (RI.point 60)=curvatureTail3 := by decide +kernel
+lemma curvatureTail3_bound : rawTailCurvature 3 < (800:ℝ) := by
+  have hh := RI.mem_mul (RI.mem_add (curvatureTable00_mem 3 30) (curvatureTable01_mem 3 30)) (RI.mem_point 60)
+  rw [curvatureTail3_calc] at hh
+  have hh' := ri_mem_lt hh 800 1 (by norm_num) (by decide +kernel)
+  simpa [rawTailCurvature,mul_comm] using hh'
+def curvatureConstant4 : RI := ⟨3178971454699645734361456476976127512300929112829300349,3178971454699645734361456476976127512300929112832218312⟩
+lemma curvatureConstant4_calc : (RI.sum (fun j : Fin 31 => (curvatureTable01 4 j).mul (columnNorm22 j))).divNat 2=curvatureConstant4 := by decide +kernel
+lemma curvatureConstant4_bound : rawConstantCurvature (curvaturePoint 4) < (4/1000000:ℝ) := by
+  have hh := RI.mem_divNat (RI.mem_sum (fun j : Fin 31 => RI.mem_mul (curvatureTable01_mem 4 j) (fourierNorm_mem j))) (by norm_num : 0<2)
+  rw [curvatureConstant4_calc] at hh
+  have hb := ri_mem_lt hh 4 1000000 (by norm_num) (by decide +kernel)
+  simpa only [rawConstantCurvature, Nat.cast_ofNat, Nat.cast_one, Int.cast_ofNat, Int.cast_one] using hb
+def curvatureTail4 : RI := ⟨411233704517733732772216847552037848865353700964006719747166500,411233704517733732772216847552037848865353700964009337805933001⟩
+lemma curvatureTail4_calc : ((curvatureTable00 4 30).add (curvatureTable01 4 30)).mul (RI.point 60)=curvatureTail4 := by decide +kernel
+lemma curvatureTail4_bound : rawTailCurvature 4 < (800:ℝ) := by
+  have hh := RI.mem_mul (RI.mem_add (curvatureTable00_mem 4 30) (curvatureTable01_mem 4 30)) (RI.mem_point 60)
+  rw [curvatureTail4_calc] at hh
+  have hh' := ri_mem_lt hh 800 1 (by norm_num) (by decide +kernel)
+  simpa [rawTailCurvature,mul_comm] using hh'
+end AtomicTriangular.FiniteCertificate
+end
+
+end OAI

lean/OAI/Analysis/Triangular/Construction/DirectCoordinates.lean

@@ -0,0 +1,93 @@
+import OAI.Analysis.Triangular.Construction.NormalizedJets
+import OAI.Analysis.Triangular.Construction.DirectWaveJets
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators lp
+open Complex
+namespace AtomicTriangular.Construction
+lemma negativeJet_re (v : ℝ) (n : ℤ) (d : Fin 2) (a b : ℂ) :
+    (negativeJet v n d a b).re=
+      Real.exp (Real.pi*(v-h)*n)/(Q n)*
+        (if d.val=0 then -a.re else (D n-Real.pi*(v-h))*a.re-b.re) := by
+  have he : exp (Real.pi*(v-h)*(n:ℂ))=(Real.exp (Real.pi*(v-h)*n):ℂ) := by
+    rw [ofReal_exp]; congr 1; push_cast; rfl
+  unfold negativeJet
+  rw [he,←ofReal_inv]
+  split_ifs <;> simp only [mul_re,mul_im,ofReal_re,ofReal_im,neg_re,sub_re,sub_im,
+    zero_mul,mul_zero,sub_zero,add_zero]
+  all_goals ring
+lemma damped_wave_re (T : C(Spectrum,ℂ) →L[ℂ] ℂ) (q : Fin 2) (n : ℤ) :
+    (T (waveFunctionJet q 0 0 n)).re=Real.exp (-Real.pi*(damping q-h)*n)*(T (waveFunctionJet 0 0 0 n)).re ∧
+    (T (waveFunctionJet q 0 1 n)).re=Real.exp (-Real.pi*(damping q-h)*n)*
+      ((T (waveFunctionJet 0 0 1 n)).re-Real.pi*(damping q-h)*(T (waveFunctionJet 0 0 0 n)).re) := by
+  have he : exp (-(Real.pi:ℂ)*(damping q-h)*(n:ℂ))=(Real.exp (-Real.pi*(damping q-h)*n):ℂ) := by
+    rw [ofReal_exp]; congr 1; push_cast; rfl
+  constructor
+  · rw [waveFunctionJet_direct_zero,map_smul,←waveFunctionJet_zero_phase,he]
+    simp only [smul_eq_mul,mul_re,ofReal_re,ofReal_im,zero_mul,sub_zero]
+  · rw [waveFunctionJet_direct_one,map_smul,map_sub,map_smul,←waveFunctionJet_zero_phase,he]
+    simp only [smul_eq_mul,mul_re,ofReal_re,ofReal_im,zero_mul,sub_zero,sub_re,mul_im,sub_im,mul_zero,add_zero]
+lemma negativeJet_direct_re (T : C(Spectrum,ℂ) →L[ℂ] ℂ) (q : Fin 2) (v : ℝ) (n : ℤ) (d : Fin 2) :
+    (negativeJet v n d (T (waveFunctionJet q 0 0 n)) (T (waveFunctionJet q 0 1 n))).re=
+      Real.exp (Real.pi*(v-damping q)*n)/(Q n)*
+        (if d.val=0 then -(T (waveFunctionJet 0 0 0 n)).re else
+          (D n-Real.pi*(v-damping q))*(T (waveFunctionJet 0 0 0 n)).re-(T (waveFunctionJet 0 0 1 n)).re) := by
+  rw [negativeJet_re,(damped_wave_re T q n).1,(damped_wave_re T q n).2]
+  have he : Real.exp (Real.pi*(v-h)*n)*Real.exp (-Real.pi*(damping q-h)*n)=
+      Real.exp (Real.pi*(v-damping q)*n) := by rw [←Real.exp_add]; congr 1; ring
+  split_ifs
+  · linear_combination (-(T (waveFunctionJet 0 0 0 n)).re/(Q n))*he
+  · linear_combination (((D n-Real.pi*(v-damping q))*(T (waveFunctionJet 0 0 0 n)).re-(T (waveFunctionJet 0 0 1 n)).re)/(Q n))*he
+
+def directJetFunction (q : Fin 2) (v : ℝ) (n : ℤ) (d : Fin 2) : C(Spectrum,ℂ) :=
+  (exp (Real.pi*(v-h)*(n:ℂ))*(Q n:ℂ)⁻¹) •
+    (if d.val=0 then -waveFunctionJet q 0 0 n else
+      ((D n:ℂ)-Real.pi*(v-h)) • waveFunctionJet q 0 0 n-waveFunctionJet q 0 1 n)
+lemma negativeJet_direct_linear (T : C(Spectrum,ℂ) →L[ℂ] ℂ) (q : Fin 2) (v : ℝ) (n : ℤ) (d : Fin 2) :
+    negativeJet v n d (T (waveFunctionJet q 0 0 n)) (T (waveFunctionJet q 0 1 n))=
+      T (directJetFunction q v n d) := by
+  unfold directJetFunction negativeJet
+  split_ifs <;> simp only [map_smul,map_neg,map_sub,smul_eq_mul]
+lemma Q_node_ne (n : Node) : Q n.val≠0 :=
+  (lt_trans (by norm_num) (FiniteCertificate.Q_node_bounds n.val n.property.2).1).ne'
+lemma spectralP_directJet (q : Fin 2) (v : ℝ) (n : Node) (d : Fin 2) :
+    (spectralPCLM (directJetFunction q v n.val d)).re=0 := by
+  rw [←negativeJet_direct_linear,negativeJet_direct_re,(spectralP_direct _).1,(spectralP_direct _).2,P_node,deriv_P_node]
+  split_ifs <;> simp
+lemma spectralC_directJet (n m : Node) (d : Fin 2) :
+    (spectralCCLM m (directJetFunction 0 h n.val d)).re=
+      if n.val=m.val ∧ d.val=0 then -1 else 0 := by
+  rw [←negativeJet_direct_linear,negativeJet_direct_re,(spectralC_direct m _).1,(spectralC_direct m _).2]
+  simp only [damping,Fin.val_zero,↓reduceIte,sub_self,mul_zero,zero_mul,Real.exp_zero,sub_zero]
+  by_cases hm : n=m
+  · subst m
+    rw [cBasis_at_node,cBasis_deriv_re]
+    by_cases hd : d.val=0
+    · simp only [hd,↓reduceIte,and_self]
+      field_simp [Q_node_ne n]
+    · simp only [hd,↓reduceIte,and_false]
+      ring
+  · have hmn : n.val≠m.val := fun he => hm (Subtype.ext he)
+    rw [cBasis_other_node hmn,deriv_cBasis_other_node hmn]
+    simp [hmn]
+lemma spectralD_directJet (n m : Node) (d : Fin 2) :
+    (spectralDCLM m (directJetFunction 0 h n.val d)).re=
+      if n.val=m.val ∧ d.val=1 then -1 else 0 := by
+  rw [←negativeJet_direct_linear,negativeJet_direct_re,(spectralD_direct m _).1,(spectralD_direct m _).2]
+  simp only [damping,Fin.val_zero,↓reduceIte,sub_self,mul_zero,zero_mul,Real.exp_zero,sub_zero]
+  by_cases hm : n=m
+  · subst m
+    have hd0 : dBasis n n.val=0 := by rw [dBasis_cBasis]; simp
+    rw [hd0,deriv_dBasis_at_node,cBasis_at_node]
+    fin_cases d <;> simp only [Nat.one_ne_zero,↓reduceIte,zero_re,mul_zero,zero_sub,neg_zero,mul_zero,and_true]
+    · norm_num
+    · field_simp [Q_node_ne n]
+  · have hmn : n.val≠m.val := fun he => hm (Subtype.ext he)
+    rw [dBasis_other_node hmn,deriv_dBasis_other_node hmn]
+    simp [hmn]
+end AtomicTriangular.Construction
+end
+
+end OAI