Leo-III---1.8.0.THM-Ref.s

%------------------------------------------------------------------------------
% File     : Leo-III---1.8.0
% Problem  : ITP234_1 : TPTP v9.3.1. Released v8.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p  --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39

% Computer : n004.cluster.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory   : 8046.5625MB
% OS       : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit  : 300s
% DateTime : Sun Sep 27 07:37:04 AM UTC 2026

% Result   : Theorem 22.61s 8.75s
% Output   : Refutation 23.81s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    3
%            Number of leaves      :  310
% Syntax   : Number of formulae    :  622 ( 404 unt;   0 typ;   0 def)
%            Number of atoms       : 1135 ( 960 equ;   0 cnn)
%            Maximal formula atoms :   10 (   1 avg)
%            Number of connectives : 8108 ( 164   ~;  40   |; 131   &;7431   @)
%                                         (  49 <=>; 293  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   16 (   5 avg)
%            Number of types       :  463 ( 462 usr)
%            Number of type conns  :    0 (   0   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   78 (  76 usr;  35 con; 0-4 aty)
%            Number of variables   : 1327 (   0   ^;1290   !;  37   ?;1327   :)

% Comments : 
%------------------------------------------------------------------------------
thf(fun_fu4272674133525273539t_bool_type,type,
    fun_fu4272674133525273539t_bool: $tType ).

thf(fun_fu7272995576155203506omplex_type,type,
    fun_fu7272995576155203506omplex: $tType ).

thf(fun_fu5286008751397070186omplex_type,type,
    fun_fu5286008751397070186omplex: $tType ).

thf(fun_fu8372565977779661586at_nat_type,type,
    fun_fu8372565977779661586at_nat: $tType ).

thf(fun_fu6906207628721142290at_int_type,type,
    fun_fu6906207628721142290at_int: $tType ).

thf(fun_fu5983729185816342290at_int_type,type,
    fun_fu5983729185816342290at_int: $tType ).

thf(fun_fu3241646296630235056omplex_type,type,
    fun_fu3241646296630235056omplex: $tType ).

thf(fun_fu5276431085538847372omplex_type,type,
    fun_fu5276431085538847372omplex: $tType ).

thf(fun_fu2295278630349710275d_enat_type,type,
    fun_fu2295278630349710275d_enat: $tType ).

thf(fun_fu2544567023866575296at_nat_type,type,
    fun_fu2544567023866575296at_nat: $tType ).

thf(fun_fu3445227072600493688at_int_type,type,
    fun_fu3445227072600493688at_int: $tType ).

thf(fun_fu6293522500203156436nt_int_type,type,
    fun_fu6293522500203156436nt_int: $tType ).

thf(fun_fu4637029388598527722ex_nat_type,type,
    fun_fu4637029388598527722ex_nat: $tType ).

thf(fun_fu1666499901826068458ex_int_type,type,
    fun_fu1666499901826068458ex_int: $tType ).

thf(fun_fu2836861327644265266ex_int_type,type,
    fun_fu2836861327644265266ex_int: $tType ).

thf(fun_fu8778301040783821591l_bool_type,type,
    fun_fu8778301040783821591l_bool: $tType ).

thf(fun_fu1817986637694864396at_nat_type,type,
    fun_fu1817986637694864396at_nat: $tType ).

thf(fun_fu3438847080485518604at_int_type,type,
    fun_fu3438847080485518604at_int: $tType ).

thf(fun_fu6400036368084001456nt_int_type,type,
    fun_fu6400036368084001456nt_int: $tType ).

thf(fun_fu4671764723984660721t_bool_type,type,
    fun_fu4671764723984660721t_bool: $tType ).

thf(fun_fu5040975675184911655d_enat_type,type,
    fun_fu5040975675184911655d_enat: $tType ).

thf(set_Pr1542805901266377927at_nat_type,type,
    set_Pr1542805901266377927at_nat: $tType ).

thf(fun_Pr2477239776880934807at_nat_type,type,
    fun_Pr2477239776880934807at_nat: $tType ).

thf(produc9217583074051984497at_nat_type,type,
    produc9217583074051984497at_nat: $tType ).

thf(fun_fu6463170612142064108al_nat_type,type,
    fun_fu6463170612142064108al_nat: $tType ).

thf(fun_fu6487107696295728876al_int_type,type,
    fun_fu6487107696295728876al_int: $tType ).

thf(fun_fu4310402290150085684al_int_type,type,
    fun_fu4310402290150085684al_int: $tType ).

thf(fun_fu3370445115705697296at_nat_type,type,
    fun_fu3370445115705697296at_nat: $tType ).

thf(fun_fu1164691120062057232at_int_type,type,
    fun_fu1164691120062057232at_int: $tType ).

thf(fun_fu5129330665597888180nt_int_type,type,
    fun_fu5129330665597888180nt_int: $tType ).

thf(fun_fu2946920718754786320at_int_type,type,
    fun_fu2946920718754786320at_int: $tType ).

thf(fun_fu6165624312919156332nt_int_type,type,
    fun_fu6165624312919156332nt_int: $tType ).

thf(fun_fu5002972218919249680at_nat_type,type,
    fun_fu5002972218919249680at_nat: $tType ).

thf(fun_fu8221675813083619692nt_nat_type,type,
    fun_fu8221675813083619692nt_nat: $tType ).

thf(fun_Pr817965127378462947t_bool_type,type,
    fun_Pr817965127378462947t_bool: $tType ).

thf(fun_fu3686806697881729723t_real_type,type,
    fun_fu3686806697881729723t_real: $tType ).

thf(produc6392793444374437607at_nat_type,type,
    produc6392793444374437607at_nat: $tType ).

thf(fun_fu5527893311006036234t_bool_type,type,
    fun_fu5527893311006036234t_bool: $tType ).

thf(fun_fu2303348052976528778t_bool_type,type,
    fun_fu2303348052976528778t_bool: $tType ).

thf(fun_fu2784792389871991434t_bool_type,type,
    fun_fu2784792389871991434t_bool: $tType ).

thf(fun_fu5303113179588540421d_enat_type,type,
    fun_fu5303113179588540421d_enat: $tType ).

thf(fun_fu8366861321737653343at_nat_type,type,
    fun_fu8366861321737653343at_nat: $tType ).

thf(fun_fu2469135700067826619at_int_type,type,
    fun_fu2469135700067826619at_int: $tType ).

thf(fun_fu3064074204415222357nt_int_type,type,
    fun_fu3064074204415222357nt_int: $tType ).

thf(fun_na1668100129766719005at_nat_type,type,
    fun_na1668100129766719005at_nat: $tType ).

thf(fun_fu3612569900928539882l_real_type,type,
    fun_fu3612569900928539882l_real: $tType ).

thf(fun_fu4101317034166783747x_bool_type,type,
    fun_fu4101317034166783747x_bool: $tType ).

thf(fun_fu8829865716563604795l_bool_type,type,
    fun_fu8829865716563604795l_bool: $tType ).

thf(fun_fu1584491925252458057omplex_type,type,
    fun_fu1584491925252458057omplex: $tType ).

thf(fun_fu709861064203714431x_real_type,type,
    fun_fu709861064203714431x_real: $tType ).

thf(fun_fu5995210064384465734x_bool_type,type,
    fun_fu5995210064384465734x_bool: $tType ).

thf(fun_li8232715659020003717t_bool_type,type,
    fun_li8232715659020003717t_bool: $tType ).

thf(fun_fu674922162107492667t_bool_type,type,
    fun_fu674922162107492667t_bool: $tType ).

thf(fun_fu2429753552268211515t_bool_type,type,
    fun_fu2429753552268211515t_bool: $tType ).

thf(fun_fu8451253408283314807t_bool_type,type,
    fun_fu8451253408283314807t_bool: $tType ).

thf(fun_fu8444152097983751227t_bool_type,type,
    fun_fu8444152097983751227t_bool: $tType ).

thf(fun_fu7830736900494809023t_bool_type,type,
    fun_fu7830736900494809023t_bool: $tType ).

thf(list_P8469869581646625389at_nat_type,type,
    list_P8469869581646625389at_nat: $tType ).

thf(fun_fu6981909852427741054x_bool_type,type,
    fun_fu6981909852427741054x_bool: $tType ).

thf(fun_fu5639431007873792710x_bool_type,type,
    fun_fu5639431007873792710x_bool: $tType ).

thf(set_Pr8693737435421807431at_nat_type,type,
    set_Pr8693737435421807431at_nat: $tType ).

thf(fun_fu3593947265352647275omplex_type,type,
    fun_fu3593947265352647275omplex: $tType ).

thf(fun_fu2513798379308225899ex_nat_type,type,
    fun_fu2513798379308225899ex_nat: $tType ).

thf(fun_fu4692242605864086655ex_int_type,type,
    fun_fu4692242605864086655ex_int: $tType ).

thf(fun_fu4189668387880940419x_bool_type,type,
    fun_fu4189668387880940419x_bool: $tType ).

thf(fun_fu6440467031773438595l_bool_type,type,
    fun_fu6440467031773438595l_bool: $tType ).

thf(fun_fu813382667476268716l_bool_type,type,
    fun_fu813382667476268716l_bool: $tType ).

thf(fun_fu8281188428872102787x_bool_type,type,
    fun_fu8281188428872102787x_bool: $tType ).

thf(fun_fu2548079958729413763t_bool_type,type,
    fun_fu2548079958729413763t_bool: $tType ).

thf(fun_fu1598310740840436355x_bool_type,type,
    fun_fu1598310740840436355x_bool: $tType ).

thf(fun_fu5088574307552523139t_bool_type,type,
    fun_fu5088574307552523139t_bool: $tType ).

thf(fun_fu8510821397121033091x_bool_type,type,
    fun_fu8510821397121033091x_bool: $tType ).

thf(fun_fu4844541694308389507t_bool_type,type,
    fun_fu4844541694308389507t_bool: $tType ).

thf(fun_fu7919257609373989805t_bool_type,type,
    fun_fu7919257609373989805t_bool: $tType ).

thf(fun_Pr248683899324676269t_bool_type,type,
    fun_Pr248683899324676269t_bool: $tType ).

thf(fun_Pr3099820731930151597t_bool_type,type,
    fun_Pr3099820731930151597t_bool: $tType ).

thf(fun_na8713714802964096592at_nat_type,type,
    fun_na8713714802964096592at_nat: $tType ).

thf(fun_fu9132278742410140537l_real_type,type,
    fun_fu9132278742410140537l_real: $tType ).

thf(fun_fu2568719150003003596l_bool_type,type,
    fun_fu2568719150003003596l_bool: $tType ).

thf(set_Pr1916528119006554503T_VEBT_type,type,
    set_Pr1916528119006554503T_VEBT: $tType ).

thf(produc859450856879609959at_nat_type,type,
    produc859450856879609959at_nat: $tType ).

thf(fun_fu6810213084838398785t_real_type,type,
    fun_fu6810213084838398785t_real: $tType ).

thf(fun_co3732688428756473980omplex_type,type,
    fun_co3732688428756473980omplex: $tType ).

thf(fun_fu5569630428773330436l_bool_type,type,
    fun_fu5569630428773330436l_bool: $tType ).

thf(fun_fu946007832778367820l_bool_type,type,
    fun_fu946007832778367820l_bool: $tType ).

thf(fun_fu9073671928554219267t_bool_type,type,
    fun_fu9073671928554219267t_bool: $tType ).

thf(fun_fu4470565429337120259l_bool_type,type,
    fun_fu4470565429337120259l_bool: $tType ).

thf(fun_fu3139726878529933315t_bool_type,type,
    fun_fu3139726878529933315t_bool: $tType ).

thf(fun_fu66453571216240387l_bool_type,type,
    fun_fu66453571216240387l_bool: $tType ).

thf(fun_fu3768230810189712741al_nat_type,type,
    fun_fu3768230810189712741al_nat: $tType ).

thf(fun_fu5645595436833112697al_int_type,type,
    fun_fu5645595436833112697al_int: $tType ).

thf(fun_fu1620311477532340197t_real_type,type,
    fun_fu1620311477532340197t_real: $tType ).

thf(fun_fu8639250699203096928t_bool_type,type,
    fun_fu8639250699203096928t_bool: $tType ).

thf(fun_fu3401574108658714444t_bool_type,type,
    fun_fu3401574108658714444t_bool: $tType ).

thf(fun_fu7872074469501006851d_enat_type,type,
    fun_fu7872074469501006851d_enat: $tType ).

thf(fun_fu5397991209436124507omplex_type,type,
    fun_fu5397991209436124507omplex: $tType ).

thf(fun_fu2509202510690934885at_nat_type,type,
    fun_fu2509202510690934885at_nat: $tType ).

thf(fun_fu19119365759208129at_int_type,type,
    fun_fu19119365759208129at_int: $tType ).

thf(fun_fu8400218827278457117at_int_type,type,
    fun_fu8400218827278457117at_int: $tType ).

thf(fun_fu4970875099359191043t_bool_type,type,
    fun_fu4970875099359191043t_bool: $tType ).

thf(fun_fu3078129256884107011t_bool_type,type,
    fun_fu3078129256884107011t_bool: $tType ).

thf(fun_fu7165551060251750659t_bool_type,type,
    fun_fu7165551060251750659t_bool: $tType ).

thf(fun_fu7719728765329718531t_bool_type,type,
    fun_fu7719728765329718531t_bool: $tType ).

thf(fun_fu2583778531842586371t_bool_type,type,
    fun_fu2583778531842586371t_bool: $tType ).

thf(fun_fu8885438963528759omplex_type,type,
    fun_fu8885438963528759omplex: $tType ).

thf(fun_fu9131882981303784088t_bool_type,type,
    fun_fu9131882981303784088t_bool: $tType ).

thf(fun_fu4664640653530332640t_bool_type,type,
    fun_fu4664640653530332640t_bool: $tType ).

thf(fun_fu6323702069054234244t_bool_type,type,
    fun_fu6323702069054234244t_bool: $tType ).

thf(fun_fu1856459741280782796t_bool_type,type,
    fun_fu1856459741280782796t_bool: $tType ).

thf(fun_fu5935951186182374353at_nat_type,type,
    fun_fu5935951186182374353at_nat: $tType ).

thf(fun_fu960360787657196261at_int_type,type,
    fun_fu960360787657196261at_int: $tType ).

thf(fun_fu4902120768161409089at_int_type,type,
    fun_fu4902120768161409089at_int: $tType ).

thf(fun_fu6652553491520710942omplex_type,type,
    fun_fu6652553491520710942omplex: $tType ).

thf(fun_fu973771059875708133nt_nat_type,type,
    fun_fu973771059875708133nt_nat: $tType ).

thf(fun_co3194742168270691230omplex_type,type,
    fun_co3194742168270691230omplex: $tType ).

thf(fun_fu3811875341156932185omplex_type,type,
    fun_fu3811875341156932185omplex: $tType ).

thf(fun_fu2868497722653781188t_bool_type,type,
    fun_fu2868497722653781188t_bool: $tType ).

thf(fun_se2100927751953696370t_real_type,type,
    fun_se2100927751953696370t_real: $tType ).

thf(fun_na4786964484885125043t_bool_type,type,
    fun_na4786964484885125043t_bool: $tType ).

thf(fun_na4143055713577223196at_nat_type,type,
    fun_na4143055713577223196at_nat: $tType ).

thf(fun_fu875838123782038019d_enat_type,type,
    fun_fu875838123782038019d_enat: $tType ).

thf(fun_fu2357351296865450298t_real_type,type,
    fun_fu2357351296865450298t_real: $tType ).

thf(fun_fu3728411652088879705omplex_type,type,
    fun_fu3728411652088879705omplex: $tType ).

thf(fun_fu2009456248099990617ex_nat_type,type,
    fun_fu2009456248099990617ex_nat: $tType ).

thf(fun_fu8742745140023039669ex_int_type,type,
    fun_fu8742745140023039669ex_int: $tType ).

thf(fun_fu7570473958957503833ex_int_type,type,
    fun_fu7570473958957503833ex_int: $tType ).

thf(produc9211091688327510695T_VEBT_type,type,
    produc9211091688327510695T_VEBT: $tType ).

thf(fun_fu4528473944208963392omplex_type,type,
    fun_fu4528473944208963392omplex: $tType ).

thf(fun_fu8303687057036190147t_bool_type,type,
    fun_fu8303687057036190147t_bool: $tType ).

thf(fun_fu893568184458893719at_nat_type,type,
    fun_fu893568184458893719at_nat: $tType ).

thf(fun_fu7626857076381942771at_int_type,type,
    fun_fu7626857076381942771at_int: $tType ).

thf(fun_fu3628248698684322547nt_int_type,type,
    fun_fu3628248698684322547nt_int: $tType ).

thf(fun_Pr2118491707810114563at_nat_type,type,
    fun_Pr2118491707810114563at_nat: $tType ).

thf(fun_fu5354524256550068955t_real_type,type,
    fun_fu5354524256550068955t_real: $tType ).

thf(fun_fu3477889739401271834t_real_type,type,
    fun_fu3477889739401271834t_real: $tType ).

thf(fun_fu7495562154832025027t_bool_type,type,
    fun_fu7495562154832025027t_bool: $tType ).

thf(fun_fu2279060208127989371at_nat_type,type,
    fun_fu2279060208127989371at_nat: $tType ).

thf(fun_fu9012349100051038423at_int_type,type,
    fun_fu9012349100051038423at_int: $tType ).

thf(fun_fu5013740722353418199nt_int_type,type,
    fun_fu5013740722353418199nt_int: $tType ).

thf(fun_fu1590586450949517859at_nat_type,type,
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thf(fun_fu3492966305048685787at_int_type,type,
    fun_fu3492966305048685787at_int: $tType ).

thf(fun_fu5956788528830221615omplex_type,type,
    fun_fu5956788528830221615omplex: $tType ).

thf(fun_re6135230694098607860t_real_type,type,
    fun_re6135230694098607860t_real: $tType ).

thf(fun_fu5589980185390252312t_real_type,type,
    fun_fu5589980185390252312t_real: $tType ).

thf(fun_fu8064956510386623256t_real_type,type,
    fun_fu8064956510386623256t_real: $tType ).

thf(fun_fu7336479014630192387t_real_type,type,
    fun_fu7336479014630192387t_real: $tType ).

thf(fun_re7819166435457029784t_real_type,type,
    fun_re7819166435457029784t_real: $tType ).

thf(fun_fu6241273384353077465al_nat_type,type,
    fun_fu6241273384353077465al_nat: $tType ).

thf(fun_fu3751190239421350709al_int_type,type,
    fun_fu3751190239421350709al_int: $tType ).

thf(fun_fu2654107258777352153al_int_type,type,
    fun_fu2654107258777352153al_int: $tType ).

thf(fun_fu3738011059562840620t_bool_type,type,
    fun_fu3738011059562840620t_bool: $tType ).

thf(fun_fu4696490002264930058t_real_type,type,
    fun_fu4696490002264930058t_real: $tType ).

thf(fun_se7035039998181355731t_bool_type,type,
    fun_se7035039998181355731t_bool: $tType ).

thf(fun_fu2115079063411676732t_real_type,type,
    fun_fu2115079063411676732t_real: $tType ).

thf(fun_fu4540635375566031512t_real_type,type,
    fun_fu4540635375566031512t_real: $tType ).

thf(fun_fu163975356400620604t_real_type,type,
    fun_fu163975356400620604t_real: $tType ).

thf(fun_fu739766798173130341t_bool_type,type,
    fun_fu739766798173130341t_bool: $tType ).

thf(fun_fu8831944113610316029at_nat_type,type,
    fun_fu8831944113610316029at_nat: $tType ).

thf(fun_fu6341860968678589273at_int_type,type,
    fun_fu6341860968678589273at_int: $tType ).

thf(fun_fu2343252590980969049nt_int_type,type,
    fun_fu2343252590980969049nt_int: $tType ).

thf(fun_fu4185298608880980944t_bool_type,type,
    fun_fu4185298608880980944t_bool: $tType ).

thf(fun_fu3901247534510871084t_bool_type,type,
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thf(fun_fu5513983380133311534at_nat_type,type,
    fun_fu5513983380133311534at_nat: $tType ).

thf(fun_fu3023900235201584778at_int_type,type,
    fun_fu3023900235201584778at_int: $tType ).

thf(fun_VE7346385682144900493T_bool_type,type,
    fun_VE7346385682144900493T_bool: $tType ).

thf(fun_fu2955637221277575299d_enat_type,type,
    fun_fu2955637221277575299d_enat: $tType ).

thf(fun_fu5112335056569181187at_nat_type,type,
    fun_fu5112335056569181187at_nat: $tType ).

thf(fun_fu6685514224203069699at_int_type,type,
    fun_fu6685514224203069699at_int: $tType ).

thf(fun_Ex8979681104671372885d_enat_type,type,
    fun_Ex8979681104671372885d_enat: $tType ).

thf(fun_fu1288400193039323353at_int_type,type,
    fun_fu1288400193039323353at_int: $tType ).

thf(fun_fu3475321767732345141nt_int_type,type,
    fun_fu3475321767732345141nt_int: $tType ).

thf(fun_fu228726614528254077at_nat_type,type,
    fun_fu228726614528254077at_nat: $tType ).

thf(fun_fu2415648189221275865nt_nat_type,type,
    fun_fu2415648189221275865nt_nat: $tType ).

thf(fun_fu248857536038387758omplex_type,type,
    fun_fu248857536038387758omplex: $tType ).

thf(fun_se1925730536474940935t_bool_type,type,
    fun_se1925730536474940935t_bool: $tType ).

thf(fun_fu4413992040155564420at_nat_type,type,
    fun_fu4413992040155564420at_nat: $tType ).

thf(fun_fu1567874110255095356at_int_type,type,
    fun_fu1567874110255095356at_int: $tType ).

thf(fun_fu5519318738670804632at_int_type,type,
    fun_fu5519318738670804632at_int: $tType ).

thf(fun_in281396392327853336at_int_type,type,
    fun_in281396392327853336at_int: $tType ).

thf(fun_co6147893885205795665omplex_type,type,
    fun_co6147893885205795665omplex: $tType ).

thf(set_Pr3451248702717554689st_nat_type,type,
    set_Pr3451248702717554689st_nat: $tType ).

thf(set_Pr765067013931698361st_int_type,type,
    set_Pr765067013931698361st_int: $tType ).

thf(list_P7413028617227757229T_VEBT_type,type,
    list_P7413028617227757229T_VEBT: $tType ).

thf(set_Pr6192946355708809607T_VEBT_type,type,
    set_Pr6192946355708809607T_VEBT: $tType ).

thf(fun_fu8384527901749275179omplex_type,type,
    fun_fu8384527901749275179omplex: $tType ).

thf(fun_fu7714627026455246123x_real_type,type,
    fun_fu7714627026455246123x_real: $tType ).

thf(fun_Pr5932607294251420486t_bool_type,type,
    fun_Pr5932607294251420486t_bool: $tType ).

thf(fun_na4827981024643048934t_bool_type,type,
    fun_na4827981024643048934t_bool: $tType ).

thf(fun_fu6356453956411242500x_bool_type,type,
    fun_fu6356453956411242500x_bool: $tType ).

thf(fun_fu1911030855436031788x_real_type,type,
    fun_fu1911030855436031788x_real: $tType ).

thf(fun_se6358542376187484365t_bool_type,type,
    fun_se6358542376187484365t_bool: $tType ).

thf(fun_na4138050610385705875at_nat_type,type,
    fun_na4138050610385705875at_nat: $tType ).

thf(fun_in1696471567231710611nt_int_type,type,
    fun_in1696471567231710611nt_int: $tType ).

thf(fun_fu5204410621998794179x_bool_type,type,
    fun_fu5204410621998794179x_bool: $tType ).

thf(fun_VE491066615515474744t_bool_type,type,
    fun_VE491066615515474744t_bool: $tType ).

thf(fun_fu5450758583235971936x_bool_type,type,
    fun_fu5450758583235971936x_bool: $tType ).

thf(fun_fu5688335724886546948x_bool_type,type,
    fun_fu5688335724886546948x_bool: $tType ).

thf(fun_fu2104585616785278288ex_nat_type,type,
    fun_fu2104585616785278288ex_nat: $tType ).

thf(fun_fu8837874508708327340ex_int_type,type,
    fun_fu8837874508708327340ex_int: $tType ).

thf(fun_fu40286998686754419omplex_type,type,
    fun_fu40286998686754419omplex: $tType ).

thf(fun_fu705315015666092147ex_nat_type,type,
    fun_fu705315015666092147ex_nat: $tType ).

thf(fun_fu4213752561377164075ex_int_type,type,
    fun_fu4213752561377164075ex_int: $tType ).

thf(fun_na6632006986671646835omplex_type,type,
    fun_na6632006986671646835omplex: $tType ).

thf(fun_co6883864892131739507omplex_type,type,
    fun_co6883864892131739507omplex: $tType ).

thf(produc7248412053542808358at_nat_type,type,
    produc7248412053542808358at_nat: $tType ).

thf(fun_fu1393723446858488622omplex_type,type,
    fun_fu1393723446858488622omplex: $tType ).

thf(fun_li8598321315282354825t_bool_type,type,
    fun_li8598321315282354825t_bool: $tType ).

thf(fun_na2508794161012263405T_bool_type,type,
    fun_na2508794161012263405T_bool: $tType ).

thf(fun_fu7905158015251895726omplex_type,type,
    fun_fu7905158015251895726omplex: $tType ).

thf(fun_se707272970763642173t_bool_type,type,
    fun_se707272970763642173t_bool: $tType ).

thf(fun_fu3077872570260440615l_real_type,type,
    fun_fu3077872570260440615l_real: $tType ).

thf(fun_na4695037785009495611t_bool_type,type,
    fun_na4695037785009495611t_bool: $tType ).

thf(fun_fu7376312551830202376l_bool_type,type,
    fun_fu7376312551830202376l_bool: $tType ).

thf(fun_fu1760894646116856364l_real_type,type,
    fun_fu1760894646116856364l_real: $tType ).

thf(produc1828647624359046049st_nat_type,type,
    produc1828647624359046049st_nat: $tType ).

thf(produc1186641810826059865st_int_type,type,
    produc1186641810826059865st_int: $tType ).

thf(fun_fu5874800227564645059l_bool_type,type,
    fun_fu5874800227564645059l_bool: $tType ).

thf(fun_re8542384036533109323t_real_type,type,
    fun_re8542384036533109323t_real: $tType ).

thf(fun_re4112884445705685835l_real_type,type,
    fun_re4112884445705685835l_real: $tType ).

thf(list_P7037539587688870467BT_nat_type,type,
    list_P7037539587688870467BT_nat: $tType ).

thf(list_P4547456442757143711BT_int_type,type,
    list_P4547456442757143711BT_int: $tType ).

thf(list_P5647936690300460905T_VEBT_type,type,
    list_P5647936690300460905T_VEBT: $tType ).

thf(list_P7524865323317820941T_VEBT_type,type,
    list_P7524865323317820941T_VEBT: $tType ).

thf(fun_fu1424522630487402084l_bool_type,type,
    fun_fu1424522630487402084l_bool: $tType ).

thf(fun_fu5258761731371428616l_bool_type,type,
    fun_fu5258761731371428616l_bool: $tType ).

thf(fun_fu4405974721632067664al_nat_type,type,
    fun_fu4405974721632067664al_nat: $tType ).

thf(fun_fu1915891576700340908al_int_type,type,
    fun_fu1915891576700340908al_int: $tType ).

thf(produc8243902056947475879T_VEBT_type,type,
    produc8243902056947475879T_VEBT: $tType ).

thf(fun_fu3214020956574444655al_nat_type,type,
    fun_fu3214020956574444655al_nat: $tType ).

thf(fun_fu3737244727311182119al_int_type,type,
    fun_fu3737244727311182119al_int: $tType ).

thf(fun_fu6899955116716589935t_real_type,type,
    fun_fu6899955116716589935t_real: $tType ).

thf(fun_fu121957152452960295t_real_type,type,
    fun_fu121957152452960295t_real: $tType ).

thf(fun_re4756235295228751087t_real_type,type,
    fun_re4756235295228751087t_real: $tType ).

thf(fun_na5225841814499866095t_real_type,type,
    fun_na5225841814499866095t_real: $tType ).

thf(set_Pr7556676689462069481BT_nat_type,type,
    set_Pr7556676689462069481BT_nat: $tType ).

thf(fun_fu7195134416904984768t_bool_type,type,
    fun_fu7195134416904984768t_bool: $tType ).

thf(fun_fu549321512867696136t_bool_type,type,
    fun_fu549321512867696136t_bool: $tType ).

thf(fun_fu6185597869953674668t_real_type,type,
    fun_fu6185597869953674668t_real: $tType ).

thf(fun_na376774557589183419x_bool_type,type,
    fun_na376774557589183419x_bool: $tType ).

thf(fun_Ex8598408993869023341t_bool_type,type,
    fun_Ex8598408993869023341t_bool: $tType ).

thf(fun_fu3889889255038636099t_bool_type,type,
    fun_fu3889889255038636099t_bool: $tType ).

thf(fun_fu7822069302422884163t_bool_type,type,
    fun_fu7822069302422884163t_bool: $tType ).

thf(fun_na7718645245721260691t_real_type,type,
    fun_na7718645245721260691t_real: $tType ).

thf(fun_na1154345429118027792d_enat_type,type,
    fun_na1154345429118027792d_enat: $tType ).

thf(fun_Ex2441982661509597004d_enat_type,type,
    fun_Ex2441982661509597004d_enat: $tType ).

thf(fun_fu8422113094057456412t_bool_type,type,
    fun_fu8422113094057456412t_bool: $tType ).

thf(fun_fu5590568769550944704t_bool_type,type,
    fun_fu5590568769550944704t_bool: $tType ).

thf(fun_fu6629554324982842668at_int_type,type,
    fun_fu6629554324982842668at_int: $tType ).

thf(fun_fu5706916984221235812t_bool_type,type,
    fun_fu5706916984221235812t_bool: $tType ).

thf(fun_fu2875372659714724104t_bool_type,type,
    fun_fu2875372659714724104t_bool: $tType ).

thf(fun_fu9051645886418017232nt_nat_type,type,
    fun_fu9051645886418017232nt_nat: $tType ).

thf(fun_na3408937105440547634t_bool_type,type,
    fun_na3408937105440547634t_bool: $tType ).

thf(fun_fu7002938529518989751at_nat_type,type,
    fun_fu7002938529518989751at_nat: $tType ).

thf(fun_fu8464300013002972271at_int_type,type,
    fun_fu8464300013002972271at_int: $tType ).

thf(fun_fu8905809029841901679nt_nat_type,type,
    fun_fu8905809029841901679nt_nat: $tType ).

thf(fun_fu1143798476471108391nt_int_type,type,
    fun_fu1143798476471108391nt_int: $tType ).

thf(fun_na3244735409550470327at_nat_type,type,
    fun_na3244735409550470327at_nat: $tType ).

thf(fun_na7654559027617327379at_int_type,type,
    fun_na7654559027617327379at_int: $tType ).

thf(fun_na2382631619178260847at_int_type,type,
    fun_na2382631619178260847at_int: $tType ).

thf(fun_in6793278964266839919at_int_type,type,
    fun_in6793278964266839919at_int: $tType ).

thf(fun_in1521351555827773387at_int_type,type,
    fun_in1521351555827773387at_int: $tType ).

thf(fun_se2197967325210717499l_bool_type,type,
    fun_se2197967325210717499l_bool: $tType ).

thf(fun_Pr672294278902479040t_real_type,type,
    fun_Pr672294278902479040t_real: $tType ).

thf(fun_Pr7757163693017731481t_bool_type,type,
    fun_Pr7757163693017731481t_bool: $tType ).

thf(fun_Pr803113274115125473t_bool_type,type,
    fun_Pr803113274115125473t_bool: $tType ).

thf(fun_VE2066785035971159626l_bool_type,type,
    fun_VE2066785035971159626l_bool: $tType ).

thf(fun_Pr2602117396471556196at_nat_type,type,
    fun_Pr2602117396471556196at_nat: $tType ).

thf(fun_Pr7647638413817135296at_int_type,type,
    fun_Pr7647638413817135296at_int: $tType ).

thf(fun_na5614638261385347865t_bool_type,type,
    fun_na5614638261385347865t_bool: $tType ).

thf(fun_na7919044170365968317t_bool_type,type,
    fun_na7919044170365968317t_bool: $tType ).

thf(fun_na1476630412504255114at_nat_type,type,
    fun_na1476630412504255114at_nat: $tType ).

thf(fun_in4514472496968388910at_nat_type,type,
    fun_in4514472496968388910at_nat: $tType ).

thf(fun_in4211310926729683942nt_int_type,type,
    fun_in4211310926729683942nt_int: $tType ).

thf(fun_VE3763956218935105199t_bool_type,type,
    fun_VE3763956218935105199t_bool: $tType ).

thf(fun_re7208826350674946562omplex_type,type,
    fun_re7208826350674946562omplex: $tType ).

thf(fun_na2351039247863542899T_bool_type,type,
    fun_na2351039247863542899T_bool: $tType ).

thf(fun_na819858167508000762st_nat_type,type,
    fun_na819858167508000762st_nat: $tType ).

thf(fun_co286204877037288194omplex_type,type,
    fun_co286204877037288194omplex: $tType ).

thf(fun_se680526843285281907t_bool_type,type,
    fun_se680526843285281907t_bool: $tType ).

thf(fun_se6612071603986899387t_bool_type,type,
    fun_se6612071603986899387t_bool: $tType ).

thf(fun_na5042103099923536755t_bool_type,type,
    fun_na5042103099923536755t_bool: $tType ).

thf(fun_co1378794544173401837x_bool_type,type,
    fun_co1378794544173401837x_bool: $tType ).

thf(option4927543243414619207at_nat_type,type,
    option4927543243414619207at_nat: $tType ).

thf(filter1242075044329608583at_nat_type,type,
    filter1242075044329608583at_nat: $tType ).

thf(fun_re7357811994743134493t_bool_type,type,
    fun_re7357811994743134493t_bool: $tType ).

thf(fun_na1591161823130732966omplex_type,type,
    fun_na1591161823130732966omplex: $tType ).

thf(fun_Ex4578939836232520097l_bool_type,type,
    fun_Ex4578939836232520097l_bool: $tType ).

thf(fun_co3258106519541737638omplex_type,type,
    fun_co3258106519541737638omplex: $tType ).

thf(fun_na2978037997607195054t_real_type,type,
    fun_na2978037997607195054t_real: $tType ).

thf(fun_na938974098958486209t_bool_type,type,
    fun_na938974098958486209t_bool: $tType ).

thf(fun_in2147091231886866205t_bool_type,type,
    fun_in2147091231886866205t_bool: $tType ).

thf(fun_Ex7682446853002821117t_bool_type,type,
    fun_Ex7682446853002821117t_bool: $tType ).

thf(fun_Ex7222163901356174497t_bool_type,type,
    fun_Ex7222163901356174497t_bool: $tType ).

thf(fun_na7542918316964655096d_enat_type,type,
    fun_na7542918316964655096d_enat: $tType ).

thf(fun_na40478436225343698at_nat_type,type,
    fun_na40478436225343698at_nat: $tType ).

thf(fun_na6773767328148392750at_int_type,type,
    fun_na6773767328148392750at_int: $tType ).

thf(fun_in5181886200629118930at_int_type,type,
    fun_in5181886200629118930at_int: $tType ).

thf(fun_Ex7123704652682714774at_nat_type,type,
    fun_Ex7123704652682714774at_nat: $tType ).

thf(fun_Ex4633621507750988018at_int_type,type,
    fun_Ex4633621507750988018at_int: $tType ).

thf(fun_Ex6820543082444009806nt_int_type,type,
    fun_Ex6820543082444009806nt_int: $tType ).

thf(list_P6011104703257516679at_nat_type,type,
    list_P6011104703257516679at_nat: $tType ).

thf(list_P3521021558325789923at_int_type,type,
    list_P3521021558325789923at_int: $tType ).

thf(list_P8198026277950538467nt_nat_type,type,
    list_P8198026277950538467nt_nat: $tType ).

thf(list_P5707943133018811711nt_int_type,type,
    list_P5707943133018811711nt_int: $tType ).

thf(fun_li6006399982103069984T_bool_type,type,
    fun_li6006399982103069984T_bool: $tType ).

thf(produc9072475918466114483BT_nat_type,type,
    produc9072475918466114483BT_nat: $tType ).

thf(produc4894624898956917775BT_int_type,type,
    produc4894624898956917775BT_int: $tType ).

thf(produc8025551001238799321T_VEBT_type,type,
    produc8025551001238799321T_VEBT: $tType ).

thf(produc1531783533982839933T_VEBT_type,type,
    produc1531783533982839933T_VEBT: $tType ).

thf(fun_re3010880487650781423x_bool_type,type,
    fun_re3010880487650781423x_bool: $tType ).

thf(fun_li1940556759920501533BT_nat_type,type,
    fun_li1940556759920501533BT_nat: $tType ).

thf(fun_co2286642472385812463l_bool_type,type,
    fun_co2286642472385812463l_bool: $tType ).

thf(set_Pr1261947904930325089at_nat_type,type,
    set_Pr1261947904930325089at_nat: $tType ).

thf(set_Pr958786334691620121nt_int_type,type,
    set_Pr958786334691620121nt_int: $tType ).

thf(fun_se448528567104302406T_bool_type,type,
    fun_se448528567104302406T_bool: $tType ).

thf(fun_na5723442673449079972x_real_type,type,
    fun_na5723442673449079972x_real: $tType ).

thf(fun_set_set_nat_bool_type,type,
    fun_set_set_nat_bool: $tType ).

thf(fun_na5941084329111710283x_bool_type,type,
    fun_na5941084329111710283x_bool: $tType ).

thf(fun_in5750378413879446255x_bool_type,type,
    fun_in5750378413879446255x_bool: $tType ).

thf(fun_co2841405903858895691t_bool_type,type,
    fun_co2841405903858895691t_bool: $tType ).

thf(fun_co2381122952212249071t_bool_type,type,
    fun_co2381122952212249071t_bool: $tType ).

thf(fun_na1142814060743658952omplex_type,type,
    fun_na1142814060743658952omplex: $tType ).

thf(fun_na8457291176974909128ex_nat_type,type,
    fun_na8457291176974909128ex_nat: $tType ).

thf(fun_na5967208032043182372ex_int_type,type,
    fun_na5967208032043182372ex_int: $tType ).

thf(fun_li6133858669135478382t_bool_type,type,
    fun_li6133858669135478382t_bool: $tType ).

thf(fun_in3657390099950231424ex_int_type,type,
    fun_in3657390099950231424ex_int: $tType ).

thf(fun_co636942507509095880at_nat_type,type,
    fun_co636942507509095880at_nat: $tType ).

thf(fun_co7370231399432144932at_int_type,type,
    fun_co7370231399432144932at_int: $tType ).

thf(fun_co333780937270390912nt_int_type,type,
    fun_co333780937270390912nt_int: $tType ).

thf(fun_se2015444840911768724t_bool_type,type,
    fun_se2015444840911768724t_bool: $tType ).

thf(fun_re1255401033344146673l_bool_type,type,
    fun_re1255401033344146673l_bool: $tType ).

thf(fun_re4238438383274954274t_real_type,type,
    fun_re4238438383274954274t_real: $tType ).

thf(fun_na6724076748692969250l_real_type,type,
    fun_na6724076748692969250l_real: $tType ).

thf(fun_re9121486324101951821t_bool_type,type,
    fun_re9121486324101951821t_bool: $tType ).

thf(fun_re8661203372455305201t_bool_type,type,
    fun_re8661203372455305201t_bool: $tType ).

thf(fun_na6801867027684862029l_bool_type,type,
    fun_na6801867027684862029l_bool: $tType ).

thf(fun_li356976171784545970x_bool_type,type,
    fun_li356976171784545970x_bool: $tType ).

thf(fun_in3970322703178350321l_bool_type,type,
    fun_in3970322703178350321l_bool: $tType ).

thf(fun_bo1841576699660343469l_bool_type,type,
    fun_bo1841576699660343469l_bool: $tType ).

thf(set_list_VEBT_VEBT_type,type,
    set_list_VEBT_VEBT: $tType ).

thf(fun_fu7168253649841883962l_bool_type,type,
    fun_fu7168253649841883962l_bool: $tType ).

thf(fun_real_fun_nat_nat_type,type,
    fun_real_fun_nat_nat: $tType ).

thf(fun_real_fun_nat_int_type,type,
    fun_real_fun_nat_int: $tType ).

thf(fun_real_fun_int_int_type,type,
    fun_real_fun_int_int: $tType ).

thf(fun_nat_fun_real_nat_type,type,
    fun_nat_fun_real_nat: $tType ).

thf(fun_nat_fun_real_int_type,type,
    fun_nat_fun_real_int: $tType ).

thf(fun_nat_fun_nat_real_type,type,
    fun_nat_fun_nat_real: $tType ).

thf(fun_nat_fun_int_real_type,type,
    fun_nat_fun_int_real: $tType ).

thf(fun_int_fun_real_int_type,type,
    fun_int_fun_real_int: $tType ).

thf(fun_Ex5852502975177633731d_enat_type,type,
    fun_Ex5852502975177633731d_enat: $tType ).

thf(fun_fun_int_int_bool_type,type,
    fun_fun_int_int_bool: $tType ).

thf(fun_set_complex_bool_type,type,
    fun_set_complex_bool: $tType ).

thf(fun_nat_fun_nat_bool_type,type,
    fun_nat_fun_nat_bool: $tType ).

thf(fun_nat_fun_int_bool_type,type,
    fun_nat_fun_int_bool: $tType ).

thf(fun_int_fun_nat_bool_type,type,
    fun_int_fun_nat_bool: $tType ).

thf(fun_int_fun_int_bool_type,type,
    fun_int_fun_int_bool: $tType ).

thf(fun_nat_fun_nat_nat_type,type,
    fun_nat_fun_nat_nat: $tType ).

thf(fun_nat_fun_nat_int_type,type,
    fun_nat_fun_nat_int: $tType ).

thf(fun_nat_fun_int_nat_type,type,
    fun_nat_fun_int_nat: $tType ).

thf(fun_nat_fun_int_int_type,type,
    fun_nat_fun_int_int: $tType ).

thf(fun_int_fun_nat_nat_type,type,
    fun_int_fun_nat_nat: $tType ).

thf(fun_int_fun_nat_int_type,type,
    fun_int_fun_nat_int: $tType ).

thf(fun_int_fun_int_int_type,type,
    fun_int_fun_int_int: $tType ).

thf(set_li5464603477888414924d_enat_type,type,
    set_li5464603477888414924d_enat: $tType ).

thf(fun_num_option_num_type,type,
    fun_num_option_num: $tType ).

thf(fun_Ex3741802288643809319omplex_type,type,
    fun_Ex3741802288643809319omplex: $tType ).

thf(fun_co6496365337113880337d_enat_type,type,
    fun_co6496365337113880337d_enat: $tType ).

thf(set_se7270636423289371942d_enat_type,type,
    set_se7270636423289371942d_enat: $tType ).

thf(fun_set_real_bool_type,type,
    fun_set_real_bool: $tType ).

thf(fun_list_nat_bool_type,type,
    fun_list_nat_bool: $tType ).

thf(fun_list_int_bool_type,type,
    fun_list_int_bool: $tType ).

thf(product_prod_nat_nat_type,type,
    product_prod_nat_nat: $tType ).

thf(product_prod_nat_int_type,type,
    product_prod_nat_int: $tType ).

thf(product_prod_int_nat_type,type,
    product_prod_int_nat: $tType ).

thf(product_prod_int_int_type,type,
    product_prod_int_int: $tType ).

thf(fun_VEBT_VEBT_bool_type,type,
    fun_VEBT_VEBT_bool: $tType ).

thf(fun_set_nat_real_type,type,
    fun_set_nat_real: $tType ).

thf(fun_real_set_nat_type,type,
    fun_real_set_nat: $tType ).

thf(fun_nat_list_nat_type,type,
    fun_nat_list_nat: $tType ).

thf(fun_list_nat_nat_type,type,
    fun_list_nat_nat: $tType ).

thf(fun_list_int_nat_type,type,
    fun_list_int_nat: $tType ).

thf(set_list_complex_type,type,
    set_list_complex: $tType ).

thf(fun_VEBT_VEBT_nat_type,type,
    fun_VEBT_VEBT_nat: $tType ).

thf(fun_set_nat_bool_type,type,
    fun_set_nat_bool: $tType ).

thf(fun_set_int_bool_type,type,
    fun_set_int_bool: $tType ).

thf(fun_set_nat_nat_type,type,
    fun_set_nat_nat: $tType ).

thf(fun_set_nat_int_type,type,
    fun_set_nat_int: $tType ).

thf(fun_nat_set_nat_type,type,
    fun_nat_set_nat: $tType ).

thf(fun_nat_set_int_type,type,
    fun_nat_set_int: $tType ).

thf(fun_complex_complex_type,type,
    fun_complex_complex: $tType ).

thf(set_set_complex_type,type,
    set_set_complex: $tType ).

thf(fun_re6065469302365115155d_enat_type,type,
    fun_re6065469302365115155d_enat: $tType ).

thf(fun_Ex1451620480498224165t_real_type,type,
    fun_Ex1451620480498224165t_real: $tType ).

thf(option_VEBT_VEBT_type,type,
    option_VEBT_VEBT: $tType ).

thf(fun_Ex8563670519792538868t_bool_type,type,
    fun_Ex8563670519792538868t_bool: $tType ).

thf(option_set_nat_type,type,
    option_set_nat: $tType ).

thf(fun_na8797796813480665199d_enat_type,type,
    fun_na8797796813480665199d_enat: $tType ).

thf(fun_in5914218760682845459d_enat_type,type,
    fun_in5914218760682845459d_enat: $tType ).

thf(fun_Ex7660298398204468809at_nat_type,type,
    fun_Ex7660298398204468809at_nat: $tType ).

thf(fun_Ex3482447378695272101at_int_type,type,
    fun_Ex3482447378695272101at_int: $tType ).

thf(option_Extended_enat_type,type,
    option_Extended_enat: $tType ).

thf(list_VEBT_VEBT_type,type,
    list_VEBT_VEBT: $tType ).

thf(fun_real_complex_type,type,
    fun_real_complex: $tType ).

thf(fun_complex_real_type,type,
    fun_complex_real: $tType ).

thf(set_list_nat_type,type,
    set_list_nat: $tType ).

thf(set_list_int_type,type,
    set_list_int: $tType ).

thf(list_set_nat_type,type,
    list_set_nat: $tType ).

thf(fun_complex_bool_type,type,
    fun_complex_bool: $tType ).

thf(set_VEBT_VEBT_type,type,
    set_VEBT_VEBT: $tType ).

thf(fun_nat_complex_type,type,
    fun_nat_complex: $tType ).

thf(fun_int_complex_type,type,
    fun_int_complex: $tType ).

thf(fun_complex_nat_type,type,
    fun_complex_nat: $tType ).

thf(fun_complex_int_type,type,
    fun_complex_int: $tType ).

thf(set_set_nat_type,type,
    set_set_nat: $tType ).

thf(set_set_int_type,type,
    set_set_int: $tType ).

thf(list_Extended_enat_type,type,
    list_Extended_enat: $tType ).

thf(set_Product_unit_type,type,
    set_Product_unit: $tType ).

thf(set_Extended_enat_type,type,
    set_Extended_enat: $tType ).

thf(fun_char_nat_type,type,
    fun_char_nat: $tType ).

thf(fun_real_real_type,type,
    fun_real_real: $tType ).

thf(fun_nat_char_type,type,
    fun_nat_char: $tType ).

thf(fun_real_bool_type,type,
    fun_real_bool: $tType ).

thf(fun_real_nat_type,type,
    fun_real_nat: $tType ).

thf(fun_real_int_type,type,
    fun_real_int: $tType ).

thf(fun_nat_real_type,type,
    fun_nat_real: $tType ).

thf(fun_int_real_type,type,
    fun_int_real: $tType ).

thf(fun_bool_bool_type,type,
    fun_bool_bool: $tType ).

thf(list_complex_type,type,
    list_complex: $tType ).

thf(fun_num_bool_type,type,
    fun_num_bool: $tType ).

thf(fun_nat_bool_type,type,
    fun_nat_bool: $tType ).

thf(fun_int_bool_type,type,
    fun_int_bool: $tType ).

thf(fun_num_num_type,type,
    fun_num_num: $tType ).

thf(fun_num_nat_type,type,
    fun_num_nat: $tType ).

thf(fun_nat_nat_type,type,
    fun_nat_nat: $tType ).

thf(fun_nat_int_type,type,
    fun_nat_int: $tType ).

thf(fun_int_nat_type,type,
    fun_int_nat: $tType ).

thf(fun_int_int_type,type,
    fun_int_int: $tType ).

thf(set_complex_type,type,
    set_complex: $tType ).

thf(option_real_type,type,
    option_real: $tType ).

thf(filter_real_type,type,
    filter_real: $tType ).

thf(option_num_type,type,
    option_num: $tType ).

thf(option_nat_type,type,
    option_nat: $tType ).

thf(option_int_type,type,
    option_int: $tType ).

thf(filter_nat_type,type,
    filter_nat: $tType ).

thf(set_char_type,type,
    set_char: $tType ).

thf(list_real_type,type,
    list_real: $tType ).

thf(set_real_type,type,
    set_real: $tType ).

thf(list_num_type,type,
    list_num: $tType ).

thf(list_nat_type,type,
    list_nat: $tType ).

thf(list_int_type,type,
    list_int: $tType ).

thf(vEBT_VEBT_type,type,
    vEBT_VEBT: $tType ).

thf(set_num_type,type,
    set_num: $tType ).

thf(set_nat_type,type,
    set_nat: $tType ).

thf(set_int_type,type,
    set_int: $tType ).

thf(extended_enat_type,type,
    extended_enat: $tType ).

thf(complex_type,type,
    complex: $tType ).

thf(char_type,type,
    char: $tType ).

thf(real_type,type,
    real: $tType ).

thf(bool_type,type,
    bool: $tType ).

thf(num_type,type,
    num: $tType ).

thf(nat_type,type,
    nat: $tType ).

thf(int_type,type,
    int: $tType ).

thf(aTP_Lamm_aaa_decl,type,
    aTP_Lamm_aaa: fun_nat_fun_nat_nat ).

thf(aTP_Lamm_aas_decl,type,
    aTP_Lamm_aas: fun_nat_fun_nat_nat ).

thf(aTP_Lamm_aba_decl,type,
    aTP_Lamm_aba: fun_nat_bool > fun_nat_bool ).

thf(aTP_Lamm_abl_decl,type,
    aTP_Lamm_abl: fun_nat_fun_nat_nat ).

thf(aTP_Lamm_cd_decl,type,
    aTP_Lamm_cd: fun_nat_nat ).

thf(aTP_Lamm_ko_decl,type,
    aTP_Lamm_ko: nat > fun_nat_fun_nat_nat ).

thf(aTP_Lamm_la_decl,type,
    aTP_Lamm_la: fun_nat_nat > fun_nat_fun_nat_nat ).

thf(aTP_Lamm_mw_decl,type,
    aTP_Lamm_mw: fun_nat_nat > fun_nat_fun_nat_nat ).

thf(aTP_Lamm_py_decl,type,
    aTP_Lamm_py: fun_nat_nat > fun_nat_fun_nat_nat ).

thf(aTP_Lamm_rp_decl,type,
    aTP_Lamm_rp: fun_nat_nat > fun_nat_fun_nat_nat ).

thf(aTP_Lamm_tl_decl,type,
    aTP_Lamm_tl: fun_nat_fun_nat_nat ).

thf(aTP_Lamm_tp_decl,type,
    aTP_Lamm_tp: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

thf(aTP_Lamm_tz_decl,type,
    aTP_Lamm_tz: fun_nat_fun_nat_nat ).

thf(aTP_Lamm_ua_decl,type,
    aTP_Lamm_ua: fun_nat_fun_nat_nat ).

thf(aTP_Lamm_ue_decl,type,
    aTP_Lamm_ue: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

thf(aTP_Lamm_ul_decl,type,
    aTP_Lamm_ul: nat > fun_nat_fun_nat_nat ).

thf(aTP_Lamm_zl_decl,type,
    aTP_Lamm_zl: nat > fun_na5042103099923536755t_bool ).

thf(aTP_Lamm_zy_decl,type,
    aTP_Lamm_zy: fun_nat_fun_nat_nat ).

thf(binomial_decl,type,
    binomial: fun_nat_fun_nat_nat ).

thf(cOMBC_771164880915498576T_bool_decl,type,
    cOMBC_771164880915498576T_bool: fun_VE7346385682144900493T_bool > set_VEBT_VEBT > fun_VEBT_VEBT_bool ).

thf(finite_card_set_nat_decl,type,
    finite_card_set_nat: set_set_nat > nat ).

thf(minus_minus_nat_decl,type,
    minus_minus_nat: fun_nat_fun_nat_nat ).

thf(one_one_nat_decl,type,
    one_one_nat: nat ).

thf(plus_plus_nat_decl,type,
    plus_plus_nat: fun_nat_fun_nat_nat ).

thf(times_times_nat_decl,type,
    times_times_nat: fun_nat_fun_nat_nat ).

thf(zero_zero_nat_decl,type,
    zero_zero_nat: nat ).

thf(if_VEBT_VEBT_decl,type,
    if_VEBT_VEBT: bool > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).

thf(drop_nat_decl,type,
    drop_nat: nat > list_nat > list_nat ).

thf(find_VEBT_VEBT_decl,type,
    find_VEBT_VEBT: fun_VEBT_VEBT_bool > list_VEBT_VEBT > option_VEBT_VEBT ).

thf(linord738340561235409698at_nat_decl,type,
    linord738340561235409698at_nat: fun_nat_nat > list_nat > list_nat ).

thf(map_nat_nat_decl,type,
    map_nat_nat: fun_nat_nat > list_nat > list_nat ).

thf(set_VEBT_VEBT2_decl,type,
    set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).

thf(tl_nat_decl,type,
    tl_nat: list_nat > list_nat ).

thf(list_u1324408373059187874T_VEBT_decl,type,
    list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).

thf(nth_VEBT_VEBT_decl,type,
    nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).

thf(replicate_VEBT_VEBT_decl,type,
    replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).

thf(upt_decl,type,
    upt: nat > nat > list_nat ).

thf(suc_decl,type,
    suc: fun_nat_nat ).

thf(compow_fun_nat_nat_decl,type,
    compow_fun_nat_nat: nat > fun_nat_nat > fun_nat_nat ).

thf(nat_triangle_decl,type,
    nat_triangle: fun_nat_nat ).

thf(none_VEBT_VEBT_decl,type,
    none_VEBT_VEBT: option_VEBT_VEBT ).

thf(ord_Least_nat_decl,type,
    ord_Least_nat: fun_nat_bool > nat ).

thf(ord_le4337996190870823476T_VEBT_decl,type,
    ord_le4337996190870823476T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o ).

thf(ord_max_nat_decl,type,
    ord_max_nat: fun_nat_fun_nat_nat ).

thf(power_power_nat_decl,type,
    power_power_nat: fun_nat_fun_nat_nat ).

thf(divide_divide_nat_decl,type,
    divide_divide_nat: fun_nat_fun_nat_nat ).

thf(modulo_modulo_nat_decl,type,
    modulo_modulo_nat: fun_nat_fun_nat_nat ).

thf(collect_set_nat_decl,type,
    collect_set_nat: fun_set_nat_bool > set_set_nat ).

thf(vEBT_Node_decl,type,
    vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).

thf(vEBT_V8194947554948674370ptions_decl,type,
    vEBT_V8194947554948674370ptions: fun_VE3763956218935105199t_bool ).

thf(vEBT_VEBT_high_decl,type,
    vEBT_VEBT_high: fun_nat_fun_nat_nat ).

thf(vEBT_V5917875025757280293ildren_decl,type,
    vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > fun_nat_bool ).

thf(vEBT_VEBT_low_decl,type,
    vEBT_VEBT_low: fun_nat_fun_nat_nat ).

thf(vEBT_invar_vebt_decl,type,
    vEBT_invar_vebt: vEBT_VEBT > fun_nat_bool ).

thf(vEBT_vebt_insert_decl,type,
    vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).

thf(vEBT_VEBT_bit_concat_decl,type,
    vEBT_VEBT_bit_concat: nat > fun_nat_fun_nat_nat ).

thf(vEBT_VEBT_minNull_decl,type,
    vEBT_VEBT_minNull: fun_VEBT_VEBT_bool ).

thf(vEBT_vebt_member_decl,type,
    vEBT_vebt_member: vEBT_VEBT > fun_nat_bool ).

thf(aa_nat_bool_decl,type,
    aa_nat_bool: fun_nat_bool > nat > bool ).

thf(aa_nat_nat_decl,type,
    aa_nat_nat: fun_nat_nat > nat > nat ).

thf(aa_nat_fun_nat_nat_decl,type,
    aa_nat_fun_nat_nat: fun_nat_fun_nat_nat > nat > fun_nat_nat ).

thf(aa_nat8857836526276733660t_bool_decl,type,
    aa_nat8857836526276733660t_bool: fun_na5042103099923536755t_bool > nat > fun_set_nat_bool ).

thf(aa_set8260762808118238015T_bool_decl,type,
    aa_set8260762808118238015T_bool: fun_se448528567104302406T_bool > set_VEBT_VEBT > bool ).

thf(aa_VEBT_VEBT_bool_decl,type,
    aa_VEBT_VEBT_bool: fun_VEBT_VEBT_bool > vEBT_VEBT > bool ).

thf(aa_VEB4832260192069861672t_bool_decl,type,
    aa_VEB4832260192069861672t_bool: fun_VE3763956218935105199t_bool > vEBT_VEBT > fun_nat_bool ).

thf(aa_VEB289770264548726718T_bool_decl,type,
    aa_VEB289770264548726718T_bool: fun_VE7346385682144900493T_bool > vEBT_VEBT > fun_se448528567104302406T_bool ).

thf(member_VEBT_VEBT_decl,type,
    member_VEBT_VEBT: fun_VE7346385682144900493T_bool ).

thf(pp_decl,type,
    pp: bool > $o ).

thf(deg_decl,type,
    deg: nat ).

thf(i_decl,type,
    i: nat ).

thf(m_decl,type,
    m: nat ).

thf(ma_decl,type,
    ma: nat ).

thf(mi_decl,type,
    mi: nat ).

thf(na_decl,type,
    na: nat ).

thf(summary_decl,type,
    summary: vEBT_VEBT ).

thf(treeList_decl,type,
    treeList: list_VEBT_VEBT ).

thf(215,axiom,
    ! [A: nat,B: nat] :
      ( A
      = ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( B @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3347_mod__div__decomp) ).

thf(1322,plain,
    ! [A: nat,B: nat] :
      ( A
      = ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( B @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).

thf(289,axiom,
    ( ( na @ ( suc @ aa_nat_nat ) )
    = m ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_35__C8_C) ).

thf(1586,plain,
    ( ( na @ ( suc @ aa_nat_nat ) )
    = m ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[289]) ).

thf(220,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( B @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3351_mod__div__mult__eq) ).

thf(1337,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( B @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).

thf(239,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ upt ) @ ( aTP_Lamm_cd @ linord738340561235409698at_nat ) )
      = ( B @ ( A @ upt ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8000_sort__upt) ).

thf(1411,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ upt ) @ ( aTP_Lamm_cd @ linord738340561235409698at_nat ) )
      = ( B @ ( A @ upt ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).

thf(170,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3689_left__diff__distrib_H) ).

thf(1105,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).

thf(107,axiom,
    ! [A: nat] :
      ( ( A @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2891_mod__self) ).

thf(846,plain,
    ! [A: nat] :
      ( ( A @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).

thf(47,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( suc @ aa_nat_nat ) @ ( A @ ( suc @ aa_nat_nat ) @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1479_max__Suc__Suc) ).

thf(555,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( suc @ aa_nat_nat ) @ ( A @ ( suc @ aa_nat_nat ) @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).

thf(277,axiom,
    ! [A: nat] :
      ( ( A @ ( aTP_Lamm_cd @ aa_nat_nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8838_ATP_Olambda__719) ).

thf(1549,plain,
    ! [A: nat] :
      ( ( A @ ( aTP_Lamm_cd @ aa_nat_nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[277]) ).

thf(62,axiom,
    ! [A: nat,B: nat] :
      ( ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
       != zero_zero_nat )
     => ( ( B != zero_zero_nat )
        & ( A != zero_zero_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2409_mult__not__zero) ).

thf(628,plain,
    ! [A: nat,B: nat] :
      ( ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
       != zero_zero_nat )
     => ( ( B != zero_zero_nat )
        & ( A != zero_zero_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).

thf(149,axiom,
    ! [A: fun_nat_nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ aTP_Lamm_mw @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( suc @ aa_nat_nat ) @ ( B @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8636_ATP_Olambda__517) ).

thf(1031,plain,
    ! [A: fun_nat_nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ aTP_Lamm_mw @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( suc @ aa_nat_nat ) @ ( B @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).

thf(15,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1507_max__add__distrib__right) ).

thf(408,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).

thf(186,axiom,
    ! [A: nat,B: nat] :
      ( ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) )
    <=> ( ( B
          = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) )
        & ( A
          = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2342_mult__eq__1__iff) ).

thf(1211,plain,
    ! [A: nat,B: nat] :
      ( ( ( ( B
            = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) )
          & ( A
            = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) ) )
       => ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) ) )
      & ( ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) )
       => ( ( B
            = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) )
          & ( A
            = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).

thf(293,axiom,
    ! [A: nat,B: list_VEBT_VEBT,C: nat] :
      ( ( C @ ( B @ ( A @ vEBT_V5917875025757280293ildren ) @ aa_nat_bool ) @ pp )
    <=> ( A @ ( C @ ( vEBT_VEBT_low @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( C @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ nth_VEBT_VEBT ) @ ( vEBT_V8194947554948674370ptions @ aa_VEB4832260192069861672t_bool ) @ aa_nat_bool ) @ pp ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_21_in__children__def) ).

thf(1596,plain,
    ! [A: nat,B: list_VEBT_VEBT,C: nat] :
      ( ( ( A @ ( C @ ( vEBT_VEBT_low @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( C @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ nth_VEBT_VEBT ) @ ( vEBT_V8194947554948674370ptions @ aa_VEB4832260192069861672t_bool ) @ aa_nat_bool ) @ pp )
       => ( C @ ( B @ ( A @ vEBT_V5917875025757280293ildren ) @ aa_nat_bool ) @ pp ) )
      & ( ( C @ ( B @ ( A @ vEBT_V5917875025757280293ildren ) @ aa_nat_bool ) @ pp )
       => ( A @ ( C @ ( vEBT_VEBT_low @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( C @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ nth_VEBT_VEBT ) @ ( vEBT_V8194947554948674370ptions @ aa_VEB4832260192069861672t_bool ) @ aa_nat_bool ) @ pp ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[293]) ).

thf(76,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2955_bits__mod__div__trivial) ).

thf(683,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).

thf(161,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( B @ ( C @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3737_diff__cancel2) ).

thf(1068,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( B @ ( C @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).

thf(298,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( A
        = ( C @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
     => ( ( A @ ( D @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( C @ ( D @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_917_group__cancel_Oadd2) ).

thf(1618,plain,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( A
        = ( C @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
     => ( ( A @ ( D @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( C @ ( D @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[298]) ).

thf(116,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
    <=> ( B = zero_zero_nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1392_add__0__iff) ).

thf(873,plain,
    ! [A: nat,B: nat] :
      ( ( ( B = zero_zero_nat )
       => ( A
          = ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) )
      & ( ( A
          = ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
       => ( B = zero_zero_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).

thf(291,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ upt ) @ ( A @ drop_nat ) )
      = ( C @ ( A @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ upt ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_7961_drop__upt) ).

thf(1591,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ upt ) @ ( A @ drop_nat ) )
      = ( C @ ( A @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ upt ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[291]) ).

thf(247,axiom,
    m @ ( summary @ vEBT_invar_vebt @ aa_nat_bool ) @ pp,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_26__C1_C) ).

thf(1437,plain,
    m @ ( summary @ vEBT_invar_vebt @ aa_nat_bool ) @ pp,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).

thf(152,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( D @ ( B @ ( C @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( D @ ( B @ ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2432_combine__common__factor) ).

thf(1041,plain,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( D @ ( B @ ( C @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( D @ ( B @ ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).

thf(180,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2437_distrib__right) ).

thf(1173,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).

thf(244,axiom,
    ! [A: fun_nat_bool,B: nat,C: nat] :
      ( ( B @ ( A @ aa_nat_bool ) @ pp )
     => ( ! [D: nat] :
            ( ( D @ ( suc @ aa_nat_nat ) @ ( A @ aa_nat_bool ) @ pp )
           => ( D @ ( A @ aa_nat_bool ) @ pp ) )
       => ( C @ ( B @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ aa_nat_bool ) @ pp ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3724_zero__induct__lemma) ).

thf(1426,plain,
    ! [A: fun_nat_bool,B: nat,C: nat] :
      ( ( B @ ( A @ aa_nat_bool ) @ pp )
     => ( ! [D: nat] :
            ( ( D @ ( suc @ aa_nat_nat ) @ ( A @ aa_nat_bool ) @ pp )
           => ( D @ ( A @ aa_nat_bool ) @ pp ) )
       => ( C @ ( B @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ aa_nat_bool ) @ pp ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).

thf(173,axiom,
    ! [A: nat] :
      ( ( A @ ( suc @ aa_nat_nat ) )
      = ( A @ ( one_one_nat @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2167_Suc__eq__plus1__left) ).

thf(1137,plain,
    ! [A: nat] :
      ( ( A @ ( suc @ aa_nat_nat ) )
      = ( A @ ( one_one_nat @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).

thf(211,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( B @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3866_minus__div__mult__eq__mod) ).

thf(1310,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( B @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).

thf(35,axiom,
    ! [A: nat] :
      ( ( A @ ( zero_zero_nat @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_987_comm__monoid__add__class_Oadd__0) ).

thf(500,plain,
    ! [A: nat] :
      ( ( A @ ( zero_zero_nat @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).

thf(99,axiom,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( A @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1483_max__nat_Oright__neutral) ).

thf(797,plain,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( A @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).

thf(191,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( B @ ( C @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( C @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2959_mod__mult__self3) ).

thf(1233,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( B @ ( C @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( C @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).

thf(88,axiom,
    ! [A: nat] :
      ( ( A @ ( zero_zero_nat @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1870_div__0) ).

thf(747,plain,
    ! [A: nat] :
      ( ( A @ ( zero_zero_nat @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).

thf(252,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( suc @ aa_nat_nat ) @ ( A @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( B @ ( A @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2603_power__Suc2) ).

thf(1458,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( suc @ aa_nat_nat ) @ ( A @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( B @ ( A @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).

thf(286,axiom,
    ! [A: nat,B: nat,C: list_VEBT_VEBT,D: vEBT_VEBT] :
      ( ( A != B )
     => ( ( B @ ( D @ ( A @ ( C @ list_u1324408373059187874T_VEBT ) ) @ nth_VEBT_VEBT ) )
        = ( B @ ( C @ nth_VEBT_VEBT ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_10_nth__list__update__neq) ).

thf(1576,plain,
    ! [A: nat,B: nat,C: list_VEBT_VEBT,D: vEBT_VEBT] :
      ( ( A != B )
     => ( ( B @ ( D @ ( A @ ( C @ list_u1324408373059187874T_VEBT ) ) @ nth_VEBT_VEBT ) )
        = ( B @ ( C @ nth_VEBT_VEBT ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[286]) ).

thf(296,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( aTP_Lamm_tl @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( B @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8274_ATP_Olambda__155) ).

thf(1613,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( aTP_Lamm_tl @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( B @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[296]) ).

thf(308,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2380_mult_Ocommute) ).

thf(1645,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[308]) ).

thf(103,axiom,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2893_mod__by__0) ).

thf(829,plain,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).

thf(14,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ~ ! [B: nat] :
            ( A
           != ( B @ ( suc @ aa_nat_nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_147_old_Onat_Oexhaust) ).

thf(404,plain,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ~ ! [B: nat] :
            ( A
           != ( B @ ( suc @ aa_nat_nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).

thf(96,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( A @ ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( A @ ( B @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2362_div__mult__self2) ).

thf(782,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( A @ ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( A @ ( B @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).

thf(249,axiom,
    ! [A: nat] :
      ( ( one_one_nat @ ( A @ ( suc @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3633_diff__Suc__1) ).

thf(1441,plain,
    ! [A: nat] :
      ( ( one_one_nat @ ( A @ ( suc @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).

thf(269,axiom,
    ~ ? [A: nat] : ( A @ ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( treeList @ nth_VEBT_VEBT ) @ vEBT_vebt_member @ aa_nat_bool ) @ pp ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_4__C161_C) ).

thf(1518,plain,
    ~ ? [A: nat] : ( A @ ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( treeList @ nth_VEBT_VEBT ) @ vEBT_vebt_member @ aa_nat_bool ) @ pp ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).

thf(200,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( B @ ( C @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( C @ ( suc @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2978_Suc__mod__mult__self3) ).

thf(1265,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( B @ ( C @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( C @ ( suc @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).

thf(299,axiom,
    ( i
    = ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_0_True) ).

thf(1623,plain,
    ( i
    = ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[299]) ).

thf(260,axiom,
    ! [A: fun_nat_bool,B: nat] :
      ( ( B @ ( A @ aTP_Lamm_aba @ aa_nat_bool ) @ pp )
    <=> ( B @ ( suc @ aa_nat_nat ) @ ( A @ aa_nat_bool ) @ pp ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8293_ATP_Olambda__174) ).

thf(1482,plain,
    ! [A: fun_nat_bool,B: nat] :
      ( ( ( B @ ( suc @ aa_nat_nat ) @ ( A @ aa_nat_bool ) @ pp )
       => ( B @ ( A @ aTP_Lamm_aba @ aa_nat_bool ) @ pp ) )
      & ( ( B @ ( A @ aTP_Lamm_aba @ aa_nat_bool ) @ pp )
       => ( B @ ( suc @ aa_nat_nat ) @ ( A @ aa_nat_bool ) @ pp ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).

thf(301,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( B @ ( C @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
     => ( A = C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_940_add__right__imp__eq) ).

thf(1628,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( B @ ( C @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
     => ( A = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[301]) ).

thf(157,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2442_distrib__left) ).

thf(1055,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).

thf(83,axiom,
    ! [A: nat,B: nat] :
      ( ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = zero_zero_nat )
    <=> ( ( B = zero_zero_nat )
        & ( A = zero_zero_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_93_add__is__0) ).

thf(719,plain,
    ! [A: nat,B: nat] :
      ( ( ( ( B = zero_zero_nat )
          & ( A = zero_zero_nat ) )
       => ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = zero_zero_nat ) )
      & ( ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = zero_zero_nat )
       => ( ( B = zero_zero_nat )
          & ( A = zero_zero_nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).

thf(106,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ( A @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( A @ ( C @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
      <=> ( B = C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2391_mult__right__cancel) ).

thf(838,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ( B = C )
         => ( ( A @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
            = ( A @ ( C @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) )
        & ( ( ( A @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
            = ( A @ ( C @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
         => ( B = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).

thf(267,axiom,
    ! [A: fun_nat_fun_nat_nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ aTP_Lamm_tp @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( C @ ( A @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8449_ATP_Olambda__330) ).

thf(1511,plain,
    ! [A: fun_nat_fun_nat_nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ aTP_Lamm_tp @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( C @ ( A @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).

thf(212,axiom,
    ! [A: nat] :
      ( ( A @ ( suc @ aa_nat_nat ) @ ( zero_zero_nat @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_6681_binomial__0__Suc) ).

thf(1313,plain,
    ! [A: nat] :
      ( ( A @ ( suc @ aa_nat_nat ) @ ( zero_zero_nat @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).

thf(251,axiom,
    ! [A: nat] :
      ( ( suc @ ( A @ compow_fun_nat_nat ) )
      = ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_7741_Suc__funpow) ).

thf(1455,plain,
    ! [A: nat] :
      ( ( suc @ ( A @ compow_fun_nat_nat ) )
      = ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).

thf(184,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2472_add__mult__distrib) ).

thf(1191,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).

thf(127,axiom,
    ! [A: nat] :
      ( ( A @ ( zero_zero_nat @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3543_zero__diff) ).

thf(949,plain,
    ! [A: nat] :
      ( ( A @ ( zero_zero_nat @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).

thf(268,axiom,
    ! [A: nat,B: nat,C: list_VEBT_VEBT,D: vEBT_VEBT,E: vEBT_VEBT] :
      ( ( A != B )
     => ( ( E @ ( B @ ( D @ ( A @ ( C @ list_u1324408373059187874T_VEBT ) ) @ list_u1324408373059187874T_VEBT ) ) )
        = ( D @ ( A @ ( E @ ( B @ ( C @ list_u1324408373059187874T_VEBT ) ) @ list_u1324408373059187874T_VEBT ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_18_list__update__swap) ).

thf(1514,plain,
    ! [A: nat,B: nat,C: list_VEBT_VEBT,D: vEBT_VEBT,E: vEBT_VEBT] :
      ( ( A != B )
     => ( ( E @ ( B @ ( D @ ( A @ ( C @ list_u1324408373059187874T_VEBT ) ) @ list_u1324408373059187874T_VEBT ) ) )
        = ( D @ ( A @ ( E @ ( B @ ( C @ list_u1324408373059187874T_VEBT ) ) @ list_u1324408373059187874T_VEBT ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).

thf(214,axiom,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( suc @ aa_nat_nat ) @ ( A @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_6682_binomial__1) ).

thf(1319,plain,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( suc @ aa_nat_nat ) @ ( A @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).

thf(259,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3740_diff__mult__distrib2) ).

thf(1479,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).

thf(285,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( C = D )
        & ( A = B ) )
     => ( ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( D @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_909_add__mono__thms__linordered__semiring_I4_J) ).

thf(1572,plain,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( C = D )
        & ( A = B ) )
     => ( ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( D @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[285]) ).

thf(117,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
    <=> ( B = zero_zero_nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_879_add__cancel__right__right) ).

thf(881,plain,
    ! [A: nat,B: nat] :
      ( ( ( B = zero_zero_nat )
       => ( A
          = ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) )
      & ( ( A
          = ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
       => ( B = zero_zero_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).

thf(18,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( A @ ( A @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2310_nonzero__mult__div__cancel__right) ).

thf(416,plain,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( A @ ( A @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).

thf(26,axiom,
    ! [A: vEBT_VEBT] :
      ~ ( zero_zero_nat @ ( A @ vEBT_invar_vebt @ aa_nat_bool ) @ pp ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_61_valid__tree__deg__neq__0) ).

thf(448,plain,
    ! [A: vEBT_VEBT] :
      ~ ( zero_zero_nat @ ( A @ vEBT_invar_vebt @ aa_nat_bool ) @ pp ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).

thf(263,axiom,
    ! [A: vEBT_VEBT,B: nat,C: nat] :
      ( ( B @ ( A @ vEBT_invar_vebt @ aa_nat_bool ) @ pp )
     => ( ( C @ ( A @ ( vEBT_V8194947554948674370ptions @ aa_VEB4832260192069861672t_bool ) @ aa_nat_bool ) @ pp )
      <=> ( C @ ( A @ vEBT_vebt_member @ aa_nat_bool ) @ pp ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_27_both__member__options__equiv__member) ).

thf(1494,plain,
    ! [A: vEBT_VEBT,B: nat,C: nat] :
      ( ( B @ ( A @ vEBT_invar_vebt @ aa_nat_bool ) @ pp )
     => ( ( ( C @ ( A @ vEBT_vebt_member @ aa_nat_bool ) @ pp )
         => ( C @ ( A @ ( vEBT_V8194947554948674370ptions @ aa_VEB4832260192069861672t_bool ) @ aa_nat_bool ) @ pp ) )
        & ( ( C @ ( A @ ( vEBT_V8194947554948674370ptions @ aa_VEB4832260192069861672t_bool ) @ aa_nat_bool ) @ pp )
         => ( C @ ( A @ vEBT_vebt_member @ aa_nat_bool ) @ pp ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).

thf(30,axiom,
    ! [A: nat] :
      ( ( A @ ( one_one_nat @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2473_nat__mult__1) ).

thf(475,plain,
    ! [A: nat] :
      ( ( A @ ( one_one_nat @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).

thf(55,axiom,
    ! [A: nat] :
      ( ( one_one_nat @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2269_mult_Oright__neutral) ).

thf(591,plain,
    ! [A: nat] :
      ( ( one_one_nat @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).

thf(303,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( aTP_Lamm_abl @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8268_ATP_Olambda__149) ).

thf(1632,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( aTP_Lamm_abl @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[303]) ).

thf(206,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ aTP_Lamm_ko @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( C @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8628_ATP_Olambda__509) ).

thf(1283,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ aTP_Lamm_ko @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( C @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).

thf(102,axiom,
    ! [A: nat,B: nat] :
      ( ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = B )
    <=> ( A = zero_zero_nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_867_add__cancel__left__left) ).

thf(821,plain,
    ! [A: nat,B: nat] :
      ( ( ( A = zero_zero_nat )
       => ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = B ) )
      & ( ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = B )
       => ( A = zero_zero_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).

thf(219,axiom,
    ! [A: fun_VE7346385682144900493T_bool,B: set_VEBT_VEBT,C: vEBT_VEBT] :
      ( ( C @ ( B @ ( A @ cOMBC_771164880915498576T_bool ) @ aa_VEBT_VEBT_bool ) )
      = ( B @ ( C @ ( A @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_COMBC_1_1_COMBC_001t__VEBT____Definitions__OVEBT_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J_001t__HOL__Obool_U) ).

thf(1334,plain,
    ! [A: fun_VE7346385682144900493T_bool,B: set_VEBT_VEBT,C: vEBT_VEBT] :
      ( ( C @ ( B @ ( A @ cOMBC_771164880915498576T_bool ) @ aa_VEBT_VEBT_bool ) )
      = ( B @ ( C @ ( A @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).

thf(66,axiom,
    ! [A: nat] :
      ( ( A @ ( zero_zero_nat @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_148_plus__nat_Oadd__0) ).

thf(646,plain,
    ! [A: nat] :
      ( ( A @ ( zero_zero_nat @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).

thf(135,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3860_minus__mult__div__eq__mod) ).

thf(981,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).

thf(172,axiom,
    ! [A: list_VEBT_VEBT,B: list_VEBT_VEBT,C: fun_VEBT_VEBT_bool,D: fun_VEBT_VEBT_bool] :
      ( ( A = B )
     => ( ! [E: vEBT_VEBT] :
            ( ( B @ set_VEBT_VEBT2 @ ( E @ ( member_VEBT_VEBT @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) @ pp )
           => ( ( E @ ( C @ aa_VEBT_VEBT_bool ) @ pp )
            <=> ( E @ ( D @ aa_VEBT_VEBT_bool ) @ pp ) ) )
       => ( ( A @ ( C @ find_VEBT_VEBT ) )
          = ( B @ ( D @ find_VEBT_VEBT ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1794_find__cong) ).

thf(1119,plain,
    ! [A: list_VEBT_VEBT,B: list_VEBT_VEBT,C: fun_VEBT_VEBT_bool,D: fun_VEBT_VEBT_bool] :
      ( ( A = B )
     => ( ! [E: vEBT_VEBT] :
            ( ( B @ set_VEBT_VEBT2 @ ( E @ ( member_VEBT_VEBT @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) @ pp )
           => ( ( ( E @ ( D @ aa_VEBT_VEBT_bool ) @ pp )
               => ( E @ ( C @ aa_VEBT_VEBT_bool ) @ pp ) )
              & ( ( E @ ( C @ aa_VEBT_VEBT_bool ) @ pp )
               => ( E @ ( D @ aa_VEBT_VEBT_bool ) @ pp ) ) ) )
       => ( ( A @ ( C @ find_VEBT_VEBT ) )
          = ( B @ ( D @ find_VEBT_VEBT ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).

thf(16,axiom,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1396_verit__sum__simplify) ).

thf(411,plain,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).

thf(237,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( A @ ( B @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( B @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3343_cancel__div__mod__rules_I2_J) ).

thf(1405,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( A @ ( B @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( B @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).

thf(270,axiom,
    ! [A: list_VEBT_VEBT,B: nat,C: vEBT_VEBT,D: vEBT_VEBT] :
      ( ( D @ ( B @ ( C @ ( B @ ( A @ list_u1324408373059187874T_VEBT ) ) @ list_u1324408373059187874T_VEBT ) ) )
      = ( D @ ( B @ ( A @ list_u1324408373059187874T_VEBT ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_14_list__update__overwrite) ).

thf(1521,plain,
    ! [A: list_VEBT_VEBT,B: nat,C: vEBT_VEBT,D: vEBT_VEBT] :
      ( ( D @ ( B @ ( C @ ( B @ ( A @ list_u1324408373059187874T_VEBT ) ) @ list_u1324408373059187874T_VEBT ) ) )
      = ( D @ ( B @ ( A @ list_u1324408373059187874T_VEBT ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).

thf(1,conjecture,
    ( ( i @ ( na @ ( mi @ ( vEBT_VEBT_low @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( treeList @ nth_VEBT_VEBT ) @ vEBT_vebt_insert ) @ ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( treeList @ list_u1324408373059187874T_VEBT ) ) @ nth_VEBT_VEBT ) )
    = ( na @ ( mi @ ( vEBT_VEBT_low @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( treeList @ nth_VEBT_VEBT ) @ vEBT_vebt_insert ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_0) ).

thf(2,negated_conjecture,
    ( ( i @ ( na @ ( mi @ ( vEBT_VEBT_low @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( treeList @ nth_VEBT_VEBT ) @ vEBT_vebt_insert ) @ ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( treeList @ list_u1324408373059187874T_VEBT ) ) @ nth_VEBT_VEBT ) )
   != ( na @ ( mi @ ( vEBT_VEBT_low @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( treeList @ nth_VEBT_VEBT ) @ vEBT_vebt_insert ) ) ),
    inference(neg_conjecture,[status(cth)],[1]) ).

thf(312,plain,
    ( ( i @ ( na @ ( mi @ ( vEBT_VEBT_low @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( treeList @ nth_VEBT_VEBT ) @ vEBT_vebt_insert ) @ ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( treeList @ list_u1324408373059187874T_VEBT ) ) @ nth_VEBT_VEBT ) )
   != ( na @ ( mi @ ( vEBT_VEBT_low @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( treeList @ nth_VEBT_VEBT ) @ vEBT_vebt_insert ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(282,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( C @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3656_diff__right__commute) ).

thf(1564,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( C @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[282]) ).

thf(194,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( C @ ( B @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3122_mod__add__right__eq) ).

thf(1242,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( C @ ( B @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).

thf(108,axiom,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( A @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = one_one_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2160_power__0) ).

thf(849,plain,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( A @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = one_one_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).

thf(310,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2375_mult_Oassoc) ).

thf(1651,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[310]) ).

thf(122,axiom,
    ! [A: nat] :
      ( ( ( A != zero_zero_nat )
       => ( ( A @ ( zero_zero_nat @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = zero_zero_nat ) )
      & ( ( A = zero_zero_nat )
       => ( ( A @ ( zero_zero_nat @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = one_one_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2180_power__0__left) ).

thf(920,plain,
    ! [A: nat] :
      ( ( ( A != zero_zero_nat )
       => ( ( A @ ( zero_zero_nat @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = zero_zero_nat ) )
      & ( ( A = zero_zero_nat )
       => ( ( A @ ( zero_zero_nat @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = one_one_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).

thf(11,axiom,
    ! [A: nat,B: nat] :
      ( ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = zero_zero_nat )
    <=> ( ( B = zero_zero_nat )
        & ( A = zero_zero_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_883_add__eq__0__iff__both__eq__0) ).

thf(380,plain,
    ! [A: nat,B: nat] :
      ( ( ( ( B = zero_zero_nat )
          & ( A = zero_zero_nat ) )
       => ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = zero_zero_nat ) )
      & ( ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = zero_zero_nat )
       => ( ( B = zero_zero_nat )
          & ( A = zero_zero_nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).

thf(256,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( B @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3397_mod__mult2__eq) ).

thf(1470,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( B @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).

thf(154,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3739_diff__add__inverse2) ).

thf(1047,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).

thf(248,axiom,
    ( ( mi = ma )
   => ! [A: vEBT_VEBT] :
        ( ( treeList @ set_VEBT_VEBT2 @ ( A @ ( member_VEBT_VEBT @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) @ pp )
       => ~ ? [B: nat] : ( B @ ( A @ ( vEBT_V8194947554948674370ptions @ aa_VEB4832260192069861672t_bool ) @ aa_nat_bool ) @ pp ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_29__C5_C) ).

thf(1438,plain,
    ( ( mi = ma )
   => ! [A: vEBT_VEBT] :
        ( ( treeList @ set_VEBT_VEBT2 @ ( A @ ( member_VEBT_VEBT @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) @ pp )
       => ~ ? [B: nat] : ( B @ ( A @ ( vEBT_V8194947554948674370ptions @ aa_VEB4832260192069861672t_bool ) @ aa_nat_bool ) @ pp ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).

thf(71,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
     => ( ( A = zero_zero_nat )
        | ( B = one_one_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2620_mult__eq__self__implies__10) ).

thf(661,plain,
    ! [A: nat,B: nat] :
      ( ( A
        = ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
     => ( ( A = zero_zero_nat )
        | ( B = one_one_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).

thf(203,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3565_add__diff__cancel__right_H) ).

thf(1274,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).

thf(167,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2471_add__mult__distrib2) ).

thf(1091,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).

thf(222,axiom,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( suc @ aa_nat_nat ) @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2967_mod__by__Suc__0) ).

thf(1343,plain,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( suc @ aa_nat_nat ) @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).

thf(95,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B != zero_zero_nat )
       => ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
         != zero_zero_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2399_no__zero__divisors) ).

thf(778,plain,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B != zero_zero_nat )
       => ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
         != zero_zero_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).

thf(128,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( A @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
    <=> ( B = zero_zero_nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_875_add__cancel__right__left) ).

thf(952,plain,
    ! [A: nat,B: nat] :
      ( ( ( B = zero_zero_nat )
       => ( A
          = ( A @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) )
      & ( ( A
          = ( A @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
       => ( B = zero_zero_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).

thf(112,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2480_nat__mult__max__right) ).

thf(862,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).

thf(284,axiom,
    ( deg
    = ( m @ ( na @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_57__C3_C) ).

thf(1570,plain,
    ( deg
    = ( m @ ( na @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[284]) ).

thf(57,axiom,
    ! [A: nat] :
      ( ( A @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3571_diff__self__eq__0) ).

thf(598,plain,
    ! [A: nat] :
      ( ( A @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).

thf(22,axiom,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1864_div__by__0) ).

thf(429,plain,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).

thf(84,axiom,
    ! [A: nat] :
      ( ( A @ ( suc @ aa_nat_nat ) )
     != zero_zero_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_138_Suc__neq__Zero) ).

thf(729,plain,
    ! [A: nat] :
      ( ( A @ ( suc @ aa_nat_nat ) )
     != zero_zero_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).

thf(226,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2447_comm__semiring__class_Odistrib) ).

thf(1355,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).

thf(287,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2385_mult_Oleft__commute) ).

thf(1580,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[287]) ).

thf(243,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3817_diff__add__0) ).

thf(1423,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).

thf(294,axiom,
    ( ~ ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( treeList @ nth_VEBT_VEBT ) @ ( vEBT_VEBT_minNull @ aa_VEBT_VEBT_bool ) @ pp )
   => ( summary
      = ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( summary @ vEBT_vebt_insert ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_15_nsprop) ).

thf(1602,plain,
    ( ~ ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( treeList @ nth_VEBT_VEBT ) @ ( vEBT_VEBT_minNull @ aa_VEBT_VEBT_bool ) @ pp )
   => ( summary
      = ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( summary @ vEBT_vebt_insert ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[294]) ).

thf(52,axiom,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_992_add_Ocomm__neutral) ).

thf(583,plain,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).

thf(118,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
    <=> ( ( A = zero_zero_nat )
        | ( B = C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2288_mult__cancel1) ).

thf(889,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( ( A = zero_zero_nat )
          | ( B = C ) )
       => ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) )
      & ( ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
       => ( ( A = zero_zero_nat )
          | ( B = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).

thf(177,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( B @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3345_cancel__div__mod__rules_I1_J) ).

thf(1159,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( B @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).

thf(163,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ upt ) @ ( suc @ map_nat_nat ) )
      = ( B @ ( suc @ aa_nat_nat ) @ ( A @ ( suc @ aa_nat_nat ) @ upt ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_7983_map__Suc__upt) ).

thf(1079,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ upt ) @ ( suc @ map_nat_nat ) )
      = ( B @ ( suc @ aa_nat_nat ) @ ( A @ ( suc @ aa_nat_nat ) @ upt ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).

thf(46,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B @ ( A @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
       != zero_zero_nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1921_power__not__zero) ).

thf(551,plain,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B @ ( A @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
       != zero_zero_nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).

thf(233,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( suc @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_146_add__Suc) ).

thf(1392,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( suc @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).

thf(133,axiom,
    ! [A: nat] :
      ( zero_zero_nat
     != ( A @ ( suc @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_153_nat_Odistinct_I1_J) ).

thf(974,plain,
    ! [A: nat] :
      ( zero_zero_nat
     != ( A @ ( suc @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).

thf(143,axiom,
    ~ ( i @ ( summary @ ( vEBT_V8194947554948674370ptions @ aa_VEB4832260192069861672t_bool ) @ aa_nat_bool ) @ pp ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_33__C163_C) ).

thf(1012,plain,
    ~ ( i @ ( summary @ ( vEBT_V8194947554948674370ptions @ aa_VEB4832260192069861672t_bool ) @ aa_nat_bool ) @ pp ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).

thf(109,axiom,
    ! [A: nat] :
      ( ( A @ ( zero_zero_nat @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1484_max__0L) ).

thf(852,plain,
    ! [A: nat] :
      ( ( A @ ( zero_zero_nat @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).

thf(279,axiom,
    ! [A: fun_nat_fun_nat_nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ aTP_Lamm_ue @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( A @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8435_ATP_Olambda__316) ).

thf(1555,plain,
    ! [A: fun_nat_fun_nat_nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ aTP_Lamm_ue @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( A @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[279]) ).

thf(236,axiom,
    ! [A: nat,B: nat] :
      ( ( A @ ( suc @ aa_nat_nat ) @ ( B @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( suc @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( one_one_nat @ ( B @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_6739_binomial__absorption) ).

thf(1402,plain,
    ! [A: nat,B: nat] :
      ( ( A @ ( suc @ aa_nat_nat ) @ ( B @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( suc @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( one_one_nat @ ( B @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).

thf(217,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( suc @ aa_nat_nat ) @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_91_add__Suc__right) ).

thf(1328,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( suc @ aa_nat_nat ) @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).

thf(189,axiom,
    ! [A: nat,B: nat] :
      ( ( A @ ( suc @ aa_nat_nat ) @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( suc @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( suc @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_6723_Suc__times__binomial__add) ).

thf(1227,plain,
    ! [A: nat,B: nat] :
      ( ( A @ ( suc @ aa_nat_nat ) @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( suc @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( suc @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).

thf(141,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( B @ ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2961_mod__mult__self2) ).

thf(1006,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( B @ ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).

thf(59,axiom,
    ! [A: nat,B: nat] :
      ( ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = zero_zero_nat )
    <=> ( ( B = zero_zero_nat )
        | ( A = zero_zero_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2254_mult__eq__0__iff) ).

thf(610,plain,
    ! [A: nat,B: nat] :
      ( ( ( ( B = zero_zero_nat )
          | ( A = zero_zero_nat ) )
       => ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = zero_zero_nat ) )
      & ( ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = zero_zero_nat )
       => ( ( B = zero_zero_nat )
          | ( A = zero_zero_nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).

thf(19,axiom,
    ! [A: nat] :
      ( ( A @ ( zero_zero_nat @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2001_bits__div__0) ).

thf(420,plain,
    ! [A: nat] :
      ( ( A @ ( zero_zero_nat @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).

thf(138,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( D @ ( B @ ( C @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( D @ ( B @ ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2828_left__add__mult__distrib) ).

thf(989,plain,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( D @ ( B @ ( C @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( D @ ( B @ ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).

thf(198,axiom,
    ! [A: vEBT_VEBT,B: nat,C: nat] :
      ( ( B @ ( A @ vEBT_invar_vebt @ aa_nat_bool ) @ pp )
     => ( ( C @ ( A @ ( vEBT_V8194947554948674370ptions @ aa_VEB4832260192069861672t_bool ) @ aa_nat_bool ) @ pp )
       => ( C @ ( A @ vEBT_vebt_member @ aa_nat_bool ) @ pp ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_28_valid__member__both__member__options) ).

thf(1259,plain,
    ! [A: vEBT_VEBT,B: nat,C: nat] :
      ( ( B @ ( A @ vEBT_invar_vebt @ aa_nat_bool ) @ pp )
     => ( ( C @ ( A @ ( vEBT_V8194947554948674370ptions @ aa_VEB4832260192069861672t_bool ) @ aa_nat_bool ) @ pp )
       => ( C @ ( A @ vEBT_vebt_member @ aa_nat_bool ) @ pp ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).

thf(4,axiom,
    ! [A: vEBT_VEBT,B: nat,C: vEBT_VEBT] :
      ( ( C @ ( B @ replicate_VEBT_VEBT ) @ set_VEBT_VEBT2 @ ( A @ ( member_VEBT_VEBT @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) @ pp )
    <=> ( ( B != zero_zero_nat )
        & ( A = C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1735_in__set__replicate) ).

thf(317,plain,
    ! [A: vEBT_VEBT,B: nat,C: vEBT_VEBT] :
      ( ( ( ( B != zero_zero_nat )
          & ( A = C ) )
       => ( C @ ( B @ replicate_VEBT_VEBT ) @ set_VEBT_VEBT2 @ ( A @ ( member_VEBT_VEBT @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) @ pp ) )
      & ( ( C @ ( B @ replicate_VEBT_VEBT ) @ set_VEBT_VEBT2 @ ( A @ ( member_VEBT_VEBT @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) @ pp )
       => ( ( B != zero_zero_nat )
          & ( A = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).

thf(228,axiom,
    ! [A: nat,B: nat] :
      ( ( ( zero_zero_nat @ ( suc @ aa_nat_nat ) )
        = ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
    <=> ( ( ( B
            = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) )
          & ( A = zero_zero_nat ) )
        | ( ( B = zero_zero_nat )
          & ( A
            = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_142_one__is__add) ).

thf(1361,plain,
    ! [A: nat,B: nat] :
      ( ( ( ( ( B
              = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) )
            & ( A = zero_zero_nat ) )
          | ( ( B = zero_zero_nat )
            & ( A
              = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) ) ) )
       => ( ( zero_zero_nat @ ( suc @ aa_nat_nat ) )
          = ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) )
      & ( ( ( zero_zero_nat @ ( suc @ aa_nat_nat ) )
          = ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
       => ( ( ( B
              = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) )
            & ( A = zero_zero_nat ) )
          | ( ( B = zero_zero_nat )
            & ( A
              = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).

thf(253,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( C @ ( B @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( B @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3357_div__mult1__eq) ).

thf(1461,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( C @ ( B @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( B @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).

thf(77,axiom,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2259_mult__zero__right) ).

thf(686,plain,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).

thf(65,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( ( A
         != ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) )
       => ~ ! [B: nat] :
              ( A
             != ( B @ ( suc @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1390_vebt__buildup_Ocases) ).

thf(642,plain,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( ( A
         != ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) )
       => ~ ! [B: nat] :
              ( A
             != ( B @ ( suc @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).

thf(240,axiom,
    ! [A: nat,B: nat] :
      ( ( A @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( B @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2903_mod__add__self1) ).

thf(1414,plain,
    ! [A: nat,B: nat] :
      ( ( A @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( B @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).

thf(132,axiom,
    ! [A: nat] :
      ( zero_zero_nat
     != ( A @ ( suc @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_136_Zero__not__Suc) ).

thf(970,plain,
    ! [A: nat] :
      ( zero_zero_nat
     != ( A @ ( suc @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).

thf(44,axiom,
    ( ( one_one_nat @ ( zero_zero_nat @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
    = one_one_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2034_max__0__1_I1_J) ).

thf(547,plain,
    ( ( one_one_nat @ ( zero_zero_nat @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
    = one_one_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).

thf(5,axiom,
    ! [A: nat,B: vEBT_VEBT,C: nat,D: vEBT_VEBT] :
      ( ( ( B @ ( A @ replicate_VEBT_VEBT ) )
        = ( D @ ( C @ replicate_VEBT_VEBT ) ) )
    <=> ( ( ( A != zero_zero_nat )
         => ( B = D ) )
        & ( A = C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1706_replicate__eq__replicate) ).

thf(328,plain,
    ! [A: nat,B: vEBT_VEBT,C: nat,D: vEBT_VEBT] :
      ( ( ( ( ( A != zero_zero_nat )
           => ( B = D ) )
          & ( A = C ) )
       => ( ( B @ ( A @ replicate_VEBT_VEBT ) )
          = ( D @ ( C @ replicate_VEBT_VEBT ) ) ) )
      & ( ( ( B @ ( A @ replicate_VEBT_VEBT ) )
          = ( D @ ( C @ replicate_VEBT_VEBT ) ) )
       => ( ( ( A != zero_zero_nat )
           => ( B = D ) )
          & ( A = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).

thf(283,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( aTP_Lamm_aas @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8265_ATP_Olambda__146) ).

thf(1567,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( aTP_Lamm_aas @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[283]) ).

thf(148,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3862_minus__mod__eq__mult__div) ).

thf(1028,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).

thf(164,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A @ ( C @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( C @ ( suc @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2977_Suc__mod__mult__self4) ).

thf(1082,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( A @ ( C @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( C @ ( suc @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).

thf(290,axiom,
    ! [A: fun_nat_nat,B: nat] :
      ( ( B @ ( A @ aTP_Lamm_py @ aa_nat_fun_nat_nat ) )
      = ( B @ ( A @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8340_ATP_Olambda__221) ).

thf(1588,plain,
    ! [A: fun_nat_nat,B: nat] :
      ( ( B @ ( A @ aTP_Lamm_py @ aa_nat_fun_nat_nat ) )
      = ( B @ ( A @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[290]) ).

thf(79,axiom,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( suc @ aa_nat_nat ) @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1995_div__by__Suc__0) ).

thf(701,plain,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( suc @ aa_nat_nat ) @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).

thf(209,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( suc @ aa_nat_nat ) @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( one_one_nat @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3826_diff__Suc__eq__diff__pred) ).

thf(1297,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( suc @ aa_nat_nat ) @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( one_one_nat @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).

thf(273,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
    <=> ( B = C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_844_add__left__cancel) ).

thf(1530,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( B = C )
       => ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) )
      & ( ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
       => ( B = C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[273]) ).

thf(69,axiom,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2003_bits__div__by__0) ).

thf(654,plain,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).

thf(278,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( aTP_Lamm_aaa @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( B @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8259_ATP_Olambda__140) ).

thf(1552,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( aTP_Lamm_aaa @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( B @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).

thf(155,axiom,
    ! [A: vEBT_VEBT] :
      ( ( treeList @ set_VEBT_VEBT2 @ ( A @ ( member_VEBT_VEBT @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) @ pp )
     => ( na @ ( A @ vEBT_invar_vebt @ aa_nat_bool ) @ pp ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_32__C0_C) ).

thf(1050,plain,
    ! [A: vEBT_VEBT] :
      ( ( treeList @ set_VEBT_VEBT2 @ ( A @ ( member_VEBT_VEBT @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) @ pp )
     => ( na @ ( A @ vEBT_invar_vebt @ aa_nat_bool ) @ pp ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).

thf(36,axiom,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_860_add_Oright__neutral) ).

thf(503,plain,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).

thf(136,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( suc @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2618_mult__Suc) ).

thf(984,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( suc @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).

thf(121,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( A != zero_zero_nat )
       => ( ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( C @ ( B @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) )
      & ( ( A = zero_zero_nat )
       => ( ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = zero_zero_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2325_div__mult__mult1__if) ).

thf(911,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( A != zero_zero_nat )
       => ( ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( C @ ( B @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) )
      & ( ( A = zero_zero_nat )
       => ( ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = zero_zero_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).

thf(246,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( B @ ( C @ ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3124_mod__add__left__eq) ).

thf(1434,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( B @ ( C @ ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).

thf(6,axiom,
    ! [A: fun_nat_bool,B: nat,C: fun_nat_bool,D: nat] :
      ( ( B @ ( A @ aa_nat_bool ) @ pp )
     => ( ( D @ ( C @ aa_nat_bool ) @ pp )
       => ( ~ ( zero_zero_nat @ ( A @ aa_nat_bool ) @ pp )
         => ( ! [E: nat] :
                ( ( E @ ( suc @ aa_nat_nat ) @ ( A @ aa_nat_bool ) @ pp )
              <=> ( E @ ( C @ aa_nat_bool ) @ pp ) )
           => ( ( A @ ord_Least_nat )
              = ( C @ ord_Least_nat @ ( suc @ aa_nat_nat ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8015_Least__Suc2) ).

thf(343,plain,
    ! [A: fun_nat_bool,B: nat,C: fun_nat_bool,D: nat] :
      ( ( B @ ( A @ aa_nat_bool ) @ pp )
     => ( ( D @ ( C @ aa_nat_bool ) @ pp )
       => ( ~ ( zero_zero_nat @ ( A @ aa_nat_bool ) @ pp )
         => ( ! [E: nat] :
                ( ( ( E @ ( C @ aa_nat_bool ) @ pp )
                 => ( E @ ( suc @ aa_nat_nat ) @ ( A @ aa_nat_bool ) @ pp ) )
                & ( ( E @ ( suc @ aa_nat_nat ) @ ( A @ aa_nat_bool ) @ pp )
                 => ( E @ ( C @ aa_nat_bool ) @ pp ) ) )
           => ( ( A @ ord_Least_nat )
              = ( C @ ord_Least_nat @ ( suc @ aa_nat_nat ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).

thf(70,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( A @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2318_nonzero__mult__div__cancel__left) ).

thf(657,plain,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( A @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).

thf(179,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2963_mod__mult__self1) ).

thf(1170,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).

thf(207,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) )
         != B )
       => ( ( B @ ( A @ ( suc @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) ) ) )
      & ( ( ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) )
          = B )
       => ( ( B @ ( A @ ( suc @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = zero_zero_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3274_mod__Suc) ).

thf(1286,plain,
    ! [A: nat,B: nat] :
      ( ( ( ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) )
         != B )
       => ( ( B @ ( A @ ( suc @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) ) ) )
      & ( ( ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) )
          = B )
       => ( ( B @ ( A @ ( suc @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = zero_zero_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).

thf(75,axiom,
    ! [A: nat] :
      ( ( one_one_nat = A )
    <=> ( A = one_one_nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2089_one__reorient) ).

thf(674,plain,
    ! [A: nat] :
      ( ( ( A = one_one_nat )
       => ( one_one_nat = A ) )
      & ( ( one_one_nat = A )
       => ( A = one_one_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).

thf(297,axiom,
    ! [A: vEBT_VEBT] :
      ( ( na @ ( mi @ ( vEBT_VEBT_low @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( treeList @ nth_VEBT_VEBT ) @ vEBT_vebt_insert ) @ ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( treeList @ list_u1324408373059187874T_VEBT ) ) @ set_VEBT_VEBT2 @ ( A @ ( member_VEBT_VEBT @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) @ pp )
     => ( na @ ( A @ vEBT_invar_vebt @ aa_nat_bool ) @ pp ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_16__C11_C) ).

thf(1616,plain,
    ! [A: vEBT_VEBT] :
      ( ( na @ ( mi @ ( vEBT_VEBT_low @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( treeList @ nth_VEBT_VEBT ) @ vEBT_vebt_insert ) @ ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( treeList @ list_u1324408373059187874T_VEBT ) ) @ set_VEBT_VEBT2 @ ( A @ ( member_VEBT_VEBT @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) @ pp )
     => ( na @ ( A @ vEBT_invar_vebt @ aa_nat_bool ) @ pp ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[297]) ).

thf(48,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat
        = ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
    <=> ( ( B = zero_zero_nat )
        & ( A = zero_zero_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_885_zero__eq__add__iff__both__eq__0) ).

thf(558,plain,
    ! [A: nat,B: nat] :
      ( ( ( ( B = zero_zero_nat )
          & ( A = zero_zero_nat ) )
       => ( zero_zero_nat
          = ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) )
      & ( ( zero_zero_nat
          = ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
       => ( ( B = zero_zero_nat )
          & ( A = zero_zero_nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).

thf(162,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( B @ ( C @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
    <=> ? [D: nat,E: nat] :
          ( ( D @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( E @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3315_nat__mod__eq__iff) ).

thf(1071,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ? [D: nat,E: nat] :
            ( ( D @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
            = ( E @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
       => ( ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( B @ ( C @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) )
      & ( ( ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( B @ ( C @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
       => ? [D: nat,E: nat] :
            ( ( D @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
            = ( E @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).

thf(98,axiom,
    ! [A: nat] :
      ( ( one_one_nat @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2006_div__by__1) ).

thf(794,plain,
    ! [A: nat] :
      ( ( one_one_nat @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).

thf(53,axiom,
    ! [A: option4927543243414619207at_nat,B: list_VEBT_VEBT,C: vEBT_VEBT,D: nat] :
      ( ( D @ ( C @ ( B @ ( zero_zero_nat @ ( A @ vEBT_Node ) ) ) @ vEBT_vebt_insert ) )
      = ( C @ ( B @ ( zero_zero_nat @ ( A @ vEBT_Node ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1648_vebt__insert_Osimps_I2_J) ).

thf(586,plain,
    ! [A: option4927543243414619207at_nat,B: list_VEBT_VEBT,C: vEBT_VEBT,D: nat] :
      ( ( D @ ( C @ ( B @ ( zero_zero_nat @ ( A @ vEBT_Node ) ) ) @ vEBT_vebt_insert ) )
      = ( C @ ( B @ ( zero_zero_nat @ ( A @ vEBT_Node ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).

thf(309,axiom,
    ~ ? [A: nat] : ( A @ ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( treeList @ nth_VEBT_VEBT ) @ ( vEBT_V8194947554948674370ptions @ aa_VEB4832260192069861672t_bool ) @ aa_nat_bool ) @ pp ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_11__C162_C) ).

thf(1648,plain,
    ~ ? [A: nat] : ( A @ ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( treeList @ nth_VEBT_VEBT ) @ ( vEBT_V8194947554948674370ptions @ aa_VEB4832260192069861672t_bool ) @ aa_nat_bool ) @ pp ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[309]) ).

thf(89,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( B @ ( suc @ aa_nat_nat ) ) )
     => ( A != zero_zero_nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_150_nat_OdiscI) ).

thf(750,plain,
    ! [A: nat,B: nat] :
      ( ( A
        = ( B @ ( suc @ aa_nat_nat ) ) )
     => ( A != zero_zero_nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).

thf(197,axiom,
    ! [A: list_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( ( B @ ( A @ set_VEBT_VEBT2 @ ord_le4337996190870823476T_VEBT ) )
    <=> ! [C: vEBT_VEBT] :
          ( ( A @ set_VEBT_VEBT2 @ ( C @ ( member_VEBT_VEBT @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) @ pp )
         => ( B @ ( C @ ( member_VEBT_VEBT @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) @ pp ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_224_subset__code_I1_J) ).

thf(1251,plain,
    ! [A: list_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( ( ! [C: vEBT_VEBT] :
            ( ( A @ set_VEBT_VEBT2 @ ( C @ ( member_VEBT_VEBT @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) @ pp )
           => ( B @ ( C @ ( member_VEBT_VEBT @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) @ pp ) )
       => ( B @ ( A @ set_VEBT_VEBT2 @ ord_le4337996190870823476T_VEBT ) ) )
      & ( ( B @ ( A @ set_VEBT_VEBT2 @ ord_le4337996190870823476T_VEBT ) )
       => ! [C: vEBT_VEBT] :
            ( ( A @ set_VEBT_VEBT2 @ ( C @ ( member_VEBT_VEBT @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) @ pp )
           => ( B @ ( C @ ( member_VEBT_VEBT @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) @ pp ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).

thf(78,axiom,
    ! [A: nat,B: vEBT_VEBT,C: fun_VEBT_VEBT_bool] :
      ( ! [D: vEBT_VEBT] :
          ( ( B @ ( A @ replicate_VEBT_VEBT ) @ set_VEBT_VEBT2 @ ( D @ ( member_VEBT_VEBT @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) @ pp )
         => ( D @ ( C @ aa_VEBT_VEBT_bool ) @ pp ) )
    <=> ( ( A = zero_zero_nat )
        | ( B @ ( C @ aa_VEBT_VEBT_bool ) @ pp ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1741_Ball__set__replicate) ).

thf(689,plain,
    ! [A: nat,B: vEBT_VEBT,C: fun_VEBT_VEBT_bool] :
      ( ( ( ( A = zero_zero_nat )
          | ( B @ ( C @ aa_VEBT_VEBT_bool ) @ pp ) )
       => ! [D: vEBT_VEBT] :
            ( ( B @ ( A @ replicate_VEBT_VEBT ) @ set_VEBT_VEBT2 @ ( D @ ( member_VEBT_VEBT @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) @ pp )
           => ( D @ ( C @ aa_VEBT_VEBT_bool ) @ pp ) ) )
      & ( ! [D: vEBT_VEBT] :
            ( ( B @ ( A @ replicate_VEBT_VEBT ) @ set_VEBT_VEBT2 @ ( D @ ( member_VEBT_VEBT @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) @ pp )
           => ( D @ ( C @ aa_VEBT_VEBT_bool ) @ pp ) )
       => ( ( A = zero_zero_nat )
          | ( B @ ( C @ aa_VEBT_VEBT_bool ) @ pp ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).

thf(58,axiom,
    ! [A: nat] :
      ( ( zero_zero_nat = A )
    <=> ( A = zero_zero_nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_900_zero__reorient) ).

thf(601,plain,
    ! [A: nat] :
      ( ( ( A = zero_zero_nat )
       => ( zero_zero_nat = A ) )
      & ( ( zero_zero_nat = A )
       => ( A = zero_zero_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).

thf(302,axiom,
    na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( treeList @ nth_VEBT_VEBT ) @ ( vEBT_VEBT_minNull @ aa_VEBT_VEBT_bool ) @ pp,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3_tc) ).

thf(1631,plain,
    na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( treeList @ nth_VEBT_VEBT ) @ ( vEBT_VEBT_minNull @ aa_VEBT_VEBT_bool ) @ pp,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[302]) ).

thf(85,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1504_max__add__distrib__left) ).

thf(733,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).

thf(20,axiom,
    ! [A: nat] :
      ( ( A @ ( zero_zero_nat @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2465_mult__0) ).

thf(423,plain,
    ! [A: nat] :
      ( ( A @ ( zero_zero_nat @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).

thf(232,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3741_diff__mult__distrib) ).

thf(1389,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).

thf(261,axiom,
    ! [A: nat,B: nat] :
      ( ( A @ ( B @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( B @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = B ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3355_mult__div__mod__eq) ).

thf(1488,plain,
    ! [A: nat,B: nat] :
      ( ( A @ ( B @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( B @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).

thf(25,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2953_mod__div__trivial) ).

thf(445,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).

thf(202,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3353_mod__mult__div__eq) ).

thf(1271,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).

thf(81,axiom,
    ! [A: nat] :
      ( ( one_one_nat @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2474_nat__mult__1__right) ).

thf(713,plain,
    ! [A: nat] :
      ( ( one_one_nat @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).

thf(304,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2370_ab__semigroup__mult__class_Omult__ac_I1_J) ).

thf(1635,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[304]) ).

thf(158,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A
        = ( C @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
     => ( ( A @ ( suc @ aa_nat_nat ) )
        = ( C @ ( suc @ aa_nat_nat ) @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_149_nat__arith_Osuc1) ).

thf(1058,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( A
        = ( C @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
     => ( ( A @ ( suc @ aa_nat_nat ) )
        = ( C @ ( suc @ aa_nat_nat ) @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).

thf(193,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ upt ) @ tl_nat )
      = ( B @ ( A @ ( suc @ aa_nat_nat ) @ upt ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_7959_tl__upt) ).

thf(1239,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ upt ) @ tl_nat )
      = ( B @ ( A @ ( suc @ aa_nat_nat ) @ upt ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).

thf(80,axiom,
    ! [A: nat,B: nat] :
      ( ( ( A != zero_zero_nat )
       => ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( B @ ( one_one_nat @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) ) ) )
      & ( ( A = zero_zero_nat )
       => ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3893_add__eq__if) ).

thf(704,plain,
    ! [A: nat,B: nat] :
      ( ( ( A != zero_zero_nat )
       => ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( B @ ( one_one_nat @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) ) ) )
      & ( ( A = zero_zero_nat )
       => ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).

thf(254,axiom,
    ! [A: nat,B: nat] :
      ( ( A @ ( suc @ aa_nat_nat ) @ ( B @ ( suc @ aa_nat_nat ) @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( suc @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( B @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( suc @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_6715_Suc__times__binomial) ).

thf(1464,plain,
    ! [A: nat,B: nat] :
      ( ( A @ ( suc @ aa_nat_nat ) @ ( B @ ( suc @ aa_nat_nat ) @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( suc @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( B @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( suc @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).

thf(305,axiom,
    m @ ( summary @ ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( summary @ vEBT_vebt_insert ) @ ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( treeList @ nth_VEBT_VEBT ) @ ( vEBT_VEBT_minNull @ aa_VEBT_VEBT_bool ) @ if_VEBT_VEBT ) ) @ vEBT_invar_vebt @ aa_nat_bool ) @ pp,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_23__C12_C) ).

thf(1638,plain,
    m @ ( summary @ ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( summary @ vEBT_vebt_insert ) @ ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( treeList @ nth_VEBT_VEBT ) @ ( vEBT_VEBT_minNull @ aa_VEBT_VEBT_bool ) @ if_VEBT_VEBT ) ) @ vEBT_invar_vebt @ aa_nat_bool ) @ pp,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[305]) ).

thf(188,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A @ ( C @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( C @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2957_mod__mult__self4) ).

thf(1224,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( A @ ( C @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( C @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).

thf(185,axiom,
    ! [A: nat,B: nat] :
      ( ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) )
    <=> ( ( ( B
            = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) )
          & ( A = zero_zero_nat ) )
        | ( ( B = zero_zero_nat )
          & ( A
            = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_145_add__is__1) ).

thf(1194,plain,
    ! [A: nat,B: nat] :
      ( ( ( ( ( B
              = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) )
            & ( A = zero_zero_nat ) )
          | ( ( B = zero_zero_nat )
            & ( A
              = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) ) ) )
       => ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) ) )
      & ( ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) )
       => ( ( ( B
              = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) )
            & ( A = zero_zero_nat ) )
          | ( ( B = zero_zero_nat )
            & ( A
              = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).

thf(119,axiom,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3545_diff__zero) ).

thf(900,plain,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).

thf(238,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( suc @ aa_nat_nat ) @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2345_mult__Suc__right) ).

thf(1408,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( suc @ aa_nat_nat ) @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).

thf(271,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_905_ab__semigroup__add__class_Oadd__ac_I1_J) ).

thf(1524,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).

thf(86,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
      <=> ( B = C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2395_mult__left__cancel) ).

thf(736,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ( B = C )
         => ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
            = ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) )
        & ( ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
            = ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
         => ( B = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).

thf(224,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( C @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_6718_choose__mult__lemma) ).

thf(1349,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( C @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).

thf(27,axiom,
    zero_zero_nat != one_one_nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2096_zero__neq__one) ).

thf(451,plain,
    zero_zero_nat != one_one_nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).

thf(67,axiom,
    ! [A: nat] :
      ( ( A @ ( suc @ aa_nat_nat ) @ ( zero_zero_nat @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1874_power__0__Suc) ).

thf(649,plain,
    ! [A: nat] :
      ( ( A @ ( suc @ aa_nat_nat ) @ ( zero_zero_nat @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).

thf(101,axiom,
    ! [A: nat,B: nat] :
      ( ( one_one_nat
        = ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
    <=> ( ( B = one_one_nat )
        & ( A = one_one_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2291_nat__1__eq__mult__iff) ).

thf(811,plain,
    ! [A: nat,B: nat] :
      ( ( ( ( B = one_one_nat )
          & ( A = one_one_nat ) )
       => ( one_one_nat
          = ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) )
      & ( ( one_one_nat
          = ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
       => ( ( B = one_one_nat )
          & ( A = one_one_nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).

thf(33,axiom,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( suc @ aa_nat_nat ) @ ( A @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1883_power__Suc0__right) ).

thf(494,plain,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( suc @ aa_nat_nat ) @ ( A @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).

thf(91,axiom,
    ! [A: nat,B: nat] :
      ( ( ( A != zero_zero_nat )
       => ( ( A @ ( B @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( one_one_nat @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) )
      & ( ( A = zero_zero_nat )
       => ( ( A @ ( B @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = one_one_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3904_power__eq__if) ).

thf(758,plain,
    ! [A: nat,B: nat] :
      ( ( ( A != zero_zero_nat )
       => ( ( A @ ( B @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( one_one_nat @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) )
      & ( ( A = zero_zero_nat )
       => ( ( A @ ( B @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = one_one_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).

thf(9,axiom,
    ! [A: nat] :
      ( ( one_one_nat @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2939_bits__mod__by__1) ).

thf(367,plain,
    ! [A: nat] :
      ( ( one_one_nat @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).

thf(56,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( A @ ( C @ ( A @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( A @ ( C @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2360_div__mult__self3) ).

thf(594,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( A @ ( C @ ( A @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( A @ ( C @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).

thf(178,axiom,
    ! [A: nat,B: nat] :
      ( ( ( A @ ( suc @ aa_nat_nat ) )
        = ( B @ ( suc @ aa_nat_nat ) ) )
    <=> ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_63_old_Onat_Oinject) ).

thf(1162,plain,
    ! [A: nat,B: nat] :
      ( ( ( A = B )
       => ( ( A @ ( suc @ aa_nat_nat ) )
          = ( B @ ( suc @ aa_nat_nat ) ) ) )
      & ( ( ( A @ ( suc @ aa_nat_nat ) )
          = ( B @ ( suc @ aa_nat_nat ) ) )
       => ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).

thf(60,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ? [B: nat] :
          ( A
          = ( B @ ( suc @ aa_nat_nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_133_not0__implies__Suc) ).

thf(621,plain,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ? [B: nat] :
          ( A
          = ( B @ ( suc @ aa_nat_nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).

thf(216,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3736_Nat_Odiff__cancel) ).

thf(1325,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).

thf(73,axiom,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_92_Nat_Oadd__0__right) ).

thf(668,plain,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).

thf(230,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( A @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2609_power__add) ).

thf(1381,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( A @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).

thf(124,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
    <=> ( ( B = C )
        | ( A = zero_zero_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2250_mult__cancel__left) ).

thf(932,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( ( B = C )
          | ( A = zero_zero_nat ) )
       => ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) )
      & ( ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
       => ( ( B = C )
          | ( A = zero_zero_nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).

thf(125,axiom,
    ! [A: nat] :
      ( ( A @ ( zero_zero_nat @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_887_add__0) ).

thf(943,plain,
    ! [A: nat] :
      ( ( A @ ( zero_zero_nat @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).

thf(146,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( suc @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3138_mod__Suc__eq) ).

thf(1022,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( suc @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).

thf(39,axiom,
    ! [A: nat,B: nat] :
      ( ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = zero_zero_nat )
    <=> ( ( B = zero_zero_nat )
        | ( A = zero_zero_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2290_mult__is__0) ).

thf(521,plain,
    ! [A: nat,B: nat] :
      ( ( ( ( B = zero_zero_nat )
          | ( A = zero_zero_nat ) )
       => ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = zero_zero_nat ) )
      & ( ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = zero_zero_nat )
       => ( ( B = zero_zero_nat )
          | ( A = zero_zero_nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).

thf(292,axiom,
    ( ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( na @ ( mi @ ( vEBT_VEBT_low @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( treeList @ nth_VEBT_VEBT ) @ vEBT_vebt_insert ) @ ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( treeList @ list_u1324408373059187874T_VEBT ) ) @ nth_VEBT_VEBT ) )
    = ( na @ ( mi @ ( vEBT_VEBT_low @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( treeList @ nth_VEBT_VEBT ) @ vEBT_vebt_insert ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1_insprop) ).

thf(1594,plain,
    ( ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( na @ ( mi @ ( vEBT_VEBT_low @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( treeList @ nth_VEBT_VEBT ) @ vEBT_vebt_insert ) @ ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( treeList @ list_u1324408373059187874T_VEBT ) ) @ nth_VEBT_VEBT ) )
    = ( na @ ( mi @ ( vEBT_VEBT_low @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( na @ ( mi @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( treeList @ nth_VEBT_VEBT ) @ vEBT_vebt_insert ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[292]) ).

thf(114,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( ( A @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = one_one_nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2012_div__self) ).

thf(868,plain,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( ( A @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = one_one_nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).

thf(181,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = C )
     => ( A
        = ( B @ ( C @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3713_add__implies__diff) ).

thf(1176,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = C )
     => ( A
        = ( B @ ( C @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).

thf(169,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( B @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3834_nat__minus__add__max) ).

thf(1102,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( B @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).

thf(195,axiom,
    ! [A: nat] :
      ( ( A @ ( suc @ aa_nat_nat ) @ ( nat_triangle @ aa_nat_nat ) )
      = ( A @ ( suc @ aa_nat_nat ) @ ( A @ ( nat_triangle @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1438_triangle__Suc) ).

thf(1245,plain,
    ! [A: nat] :
      ( ( A @ ( suc @ aa_nat_nat ) @ ( nat_triangle @ aa_nat_nat ) )
      = ( A @ ( suc @ aa_nat_nat ) @ ( A @ ( nat_triangle @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).

thf(63,axiom,
    ! [A: nat] :
      ( ( A @ ( one_one_nat @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2274_mult__1) ).

thf(636,plain,
    ! [A: nat] :
      ( ( A @ ( one_one_nat @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).

thf(153,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( suc @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( suc @ aa_nat_nat ) @ ( B @ ( suc @ aa_nat_nat ) @ ( A @ ( suc @ aa_nat_nat ) @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_6716_Suc__times__binomial__eq) ).

thf(1044,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( suc @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( suc @ aa_nat_nat ) @ ( B @ ( suc @ aa_nat_nat ) @ ( A @ ( suc @ aa_nat_nat ) @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).

thf(150,axiom,
    ! [A: nat] :
      ( ! [B: nat] :
          ( A
         != ( B @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
     => ~ ! [B: nat] :
            ( A
           != ( B @ ( suc @ aa_nat_nat ) @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_60_even__odd__cases) ).

thf(1034,plain,
    ! [A: nat] :
      ( ! [B: nat] :
          ( A
         != ( B @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
     => ~ ! [B: nat] :
            ( A
           != ( B @ ( suc @ aa_nat_nat ) @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).

thf(288,axiom,
    ! [A: list_VEBT_VEBT,B: nat] :
      ( ( B @ ( A @ nth_VEBT_VEBT ) @ ( B @ ( A @ list_u1324408373059187874T_VEBT ) ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_7_list__update__id) ).

thf(1583,plain,
    ! [A: list_VEBT_VEBT,B: nat] :
      ( ( B @ ( A @ nth_VEBT_VEBT ) @ ( B @ ( A @ list_u1324408373059187874T_VEBT ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[288]) ).

thf(21,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1510_nat__add__max__right) ).

thf(426,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).

thf(54,axiom,
    ( ( zero_zero_nat @ ( one_one_nat @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
    = one_one_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2030_max__0__1_I2_J) ).

thf(589,plain,
    ( ( zero_zero_nat @ ( one_one_nat @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
    = one_one_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).

thf(255,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( B @ ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( suc @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2979_Suc__mod__mult__self2) ).

thf(1467,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( B @ ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( suc @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).

thf(242,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( suc @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3137_mod__Suc__Suc__eq) ).

thf(1420,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( suc @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).

thf(204,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3573_diff__diff__left) ).

thf(1277,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).

thf(221,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( suc @ aa_nat_nat ) @ ( A @ ( suc @ aa_nat_nat ) @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( suc @ aa_nat_nat ) @ ( B @ ( A @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( suc @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_6728_binomial__Suc__Suc__eq__times) ).

thf(1340,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( suc @ aa_nat_nat ) @ ( A @ ( suc @ aa_nat_nat ) @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( suc @ aa_nat_nat ) @ ( B @ ( A @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( suc @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).

thf(175,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( B @ ( B @ ( C @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3128_mod__add__eq) ).

thf(1145,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( B @ ( B @ ( C @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).

thf(145,axiom,
    ! [A: nat,B: nat,C: nat,D: nat,E: nat] :
      ( ( ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( B @ ( C @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
     => ( ( ( B @ ( D @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( B @ ( E @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
       => ( ( B @ ( D @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( B @ ( E @ ( C @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3126_mod__add__cong) ).

thf(1018,plain,
    ! [A: nat,B: nat,C: nat,D: nat,E: nat] :
      ( ( ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( B @ ( C @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
     => ( ( ( B @ ( D @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( B @ ( E @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
       => ( ( B @ ( D @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( B @ ( E @ ( C @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).

thf(140,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2901_mod__add__self2) ).

thf(1003,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).

thf(147,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( suc @ aa_nat_nat ) @ ( A @ ( suc @ aa_nat_nat ) @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( suc @ aa_nat_nat ) @ ( A @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_6690_binomial__Suc__Suc) ).

thf(1025,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( suc @ aa_nat_nat ) @ ( A @ ( suc @ aa_nat_nat ) @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( suc @ aa_nat_nat ) @ ( A @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).

thf(17,axiom,
    ( ( one_one_nat @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) )
    = suc ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2166_plus__1__eq__Suc) ).

thf(414,plain,
    ( ( one_one_nat @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) )
    = suc ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).

thf(295,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( B @ ( C @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
    <=> ( A = C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_847_add__right__cancel) ).

thf(1605,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( A = C )
       => ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( B @ ( C @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) )
      & ( ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( B @ ( C @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
       => ( A = C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[295]) ).

thf(190,axiom,
    ! [A: nat] :
      ( ( A @ ( A @ ( suc @ aa_nat_nat ) @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( suc @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_6667_binomial__Suc__n) ).

thf(1230,plain,
    ! [A: nat] :
      ( ( A @ ( A @ ( suc @ aa_nat_nat ) @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( suc @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).

thf(51,axiom,
    ! [A: nat] :
      ( ( A @ ( zero_zero_nat @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2897_bits__mod__0) ).

thf(580,plain,
    ! [A: nat] :
      ( ( A @ ( zero_zero_nat @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).

thf(110,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ~ ! [B: nat] :
            ( A
           != ( B @ ( suc @ aa_nat_nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1389_list__decode_Ocases) ).

thf(855,plain,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ~ ! [B: nat] :
            ( A
           != ( B @ ( suc @ aa_nat_nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).

thf(210,axiom,
    ! [A: nat,B: nat] :
      ( ( ( zero_zero_nat @ ( suc @ aa_nat_nat ) )
        = ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
    <=> ( ( B
          = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) )
        & ( A
          = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2341_one__eq__mult__iff) ).

thf(1300,plain,
    ! [A: nat,B: nat] :
      ( ( ( ( B
            = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) )
          & ( A
            = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) ) )
       => ( ( zero_zero_nat @ ( suc @ aa_nat_nat ) )
          = ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) )
      & ( ( ( zero_zero_nat @ ( suc @ aa_nat_nat ) )
          = ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
       => ( ( B
            = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) )
          & ( A
            = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).

thf(183,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( zero_zero_nat @ upt ) @ ( A @ ( aTP_Lamm_abl @ aa_nat_fun_nat_nat ) @ map_nat_nat ) )
      = ( A @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ upt ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_7984_map__add__upt) ).

thf(1188,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( zero_zero_nat @ upt ) @ ( A @ ( aTP_Lamm_abl @ aa_nat_fun_nat_nat ) @ map_nat_nat ) )
      = ( A @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ upt ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).

thf(199,axiom,
    ! [A: nat,B: nat] :
      ( A
      = ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( B @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3396_div__mod__decomp) ).

thf(1262,plain,
    ! [A: nat,B: nat] :
      ( A
      = ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( B @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).

thf(165,axiom,
    ! [A: vEBT_VEBT,B: nat] :
      ( ( A @ ( vEBT_VEBT_minNull @ aa_VEBT_VEBT_bool ) @ pp )
     => ~ ( B @ ( A @ vEBT_vebt_member @ aa_nat_bool ) @ pp ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_25_min__Null__member) ).

thf(1085,plain,
    ! [A: vEBT_VEBT,B: nat] :
      ( ( A @ ( vEBT_VEBT_minNull @ aa_VEBT_VEBT_bool ) @ pp )
     => ~ ( B @ ( A @ vEBT_vebt_member @ aa_nat_bool ) @ pp ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).

thf(274,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( A
        = ( C @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
     => ( ( D @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( D @ ( C @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_913_group__cancel_Oadd1) ).

thf(1538,plain,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( A
        = ( C @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
     => ( ( D @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( D @ ( C @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).

thf(151,axiom,
    ! [A: nat,B: nat] :
      ( ( A @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = B ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3738_diff__add__inverse) ).

thf(1038,plain,
    ! [A: nat,B: nat] :
      ( ( A @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).

thf(10,axiom,
    ! [A: nat,B: nat] :
      ( ( ( B @ ( A @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = zero_zero_nat )
    <=> ( ( B = zero_zero_nat )
        & ( A = zero_zero_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1480_max__nat_Oeq__neutr__iff) ).

thf(370,plain,
    ! [A: nat,B: nat] :
      ( ( ( ( B = zero_zero_nat )
          & ( A = zero_zero_nat ) )
       => ( ( B @ ( A @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = zero_zero_nat ) )
      & ( ( ( B @ ( A @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = zero_zero_nat )
       => ( ( B = zero_zero_nat )
          & ( A = zero_zero_nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).

thf(227,axiom,
    ! [A: nat] :
      ( ( A @ ( suc @ aa_nat_nat ) )
      = ( one_one_nat @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2165_Suc__eq__plus1) ).

thf(1358,plain,
    ! [A: nat] :
      ( ( A @ ( suc @ aa_nat_nat ) )
      = ( one_one_nat @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).

thf(87,axiom,
    ! [A: option4927543243414619207at_nat,B: nat,C: list_VEBT_VEBT,D: vEBT_VEBT,E: nat] :
      ( ( E @ ( D @ ( C @ ( B @ ( A @ vEBT_Node ) ) ) @ vEBT_invar_vebt @ aa_nat_bool ) @ pp )
     => ( B = E ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1466_deg__deg__n) ).

thf(744,plain,
    ! [A: option4927543243414619207at_nat,B: nat,C: list_VEBT_VEBT,D: vEBT_VEBT,E: nat] :
      ( ( E @ ( D @ ( C @ ( B @ ( A @ vEBT_Node ) ) ) @ vEBT_invar_vebt @ aa_nat_bool ) @ pp )
     => ( B = E ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).

thf(275,axiom,
    ! [A: fun_nat_nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ aTP_Lamm_la @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( C @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8641_ATP_Olambda__522) ).

thf(1543,plain,
    ! [A: fun_nat_nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ aTP_Lamm_la @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( C @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[275]) ).

thf(187,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3685_right__diff__distrib_H) ).

thf(1221,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).

thf(156,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3716_diff__diff__eq) ).

thf(1052,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).

thf(3,axiom,
    ! [A: fun_nat_bool] :
      ( ( zero_zero_nat @ ( A @ aa_nat_bool ) @ pp )
     => ( ( A @ ord_Least_nat )
        = zero_zero_nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8014_Least__eq__0) ).

thf(314,plain,
    ! [A: fun_nat_bool] :
      ( ( zero_zero_nat @ ( A @ aa_nat_bool ) @ pp )
     => ( ( A @ ord_Least_nat )
        = zero_zero_nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).

thf(50,axiom,
    ! [A: nat,B: nat] :
      ( ( ( A != zero_zero_nat )
       => ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( B @ ( one_one_nat @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) )
      & ( ( A = zero_zero_nat )
       => ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = zero_zero_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3896_mult__eq__if) ).

thf(571,plain,
    ! [A: nat,B: nat] :
      ( ( ( A != zero_zero_nat )
       => ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( B @ ( one_one_nat @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) )
      & ( ( A = zero_zero_nat )
       => ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = zero_zero_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).

thf(34,axiom,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( A @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = one_one_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_6691_binomial__n__0) ).

thf(497,plain,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( A @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = one_one_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).

thf(111,axiom,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3725_minus__nat_Odiff__0) ).

thf(859,plain,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).

thf(168,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( B @ ( A @ ( suc @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( C @ ( A @ ( suc @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
    <=> ( B = C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2464_Suc__mult__cancel1) ).

thf(1094,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( B = C )
       => ( ( B @ ( A @ ( suc @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( C @ ( A @ ( suc @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) )
      & ( ( ( B @ ( A @ ( suc @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( C @ ( A @ ( suc @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
       => ( B = C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).

thf(134,axiom,
    ! [A: nat] :
      ( ( A @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3549_cancel__comm__monoid__add__class_Odiff__cancel) ).

thf(978,plain,
    ! [A: nat] :
      ( ( A @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).

thf(174,axiom,
    ! [A: nat,B: nat,C: nat,D: nat,E: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ( D != E )
          & ( B = C ) )
       => ( ( D @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
         != ( E @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2577_add__scale__eq__noteq) ).

thf(1140,plain,
    ! [A: nat,B: nat,C: nat,D: nat,E: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ( D != E )
          & ( B = C ) )
       => ( ( D @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
         != ( E @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).

thf(29,axiom,
    ! [A: nat,B: nat] :
      ( ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = one_one_nat )
    <=> ( ( B = one_one_nat )
        & ( A = one_one_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2292_nat__mult__eq__1__iff) ).

thf(465,plain,
    ! [A: nat,B: nat] :
      ( ( ( ( B = one_one_nat )
          & ( A = one_one_nat ) )
       => ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = one_one_nat ) )
      & ( ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = one_one_nat )
       => ( ( B = one_one_nat )
          & ( A = one_one_nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).

thf(213,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( suc @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( suc @ aa_nat_nat ) @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_135_add__Suc__shift) ).

thf(1316,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( suc @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( suc @ aa_nat_nat ) @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).

thf(182,axiom,
    ! [A: nat,B: nat] :
      ( ( ( A @ ( suc @ aa_nat_nat ) )
        = ( B @ ( suc @ aa_nat_nat ) ) )
    <=> ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_64_nat_Oinject) ).

thf(1180,plain,
    ! [A: nat,B: nat] :
      ( ( ( A = B )
       => ( ( A @ ( suc @ aa_nat_nat ) )
          = ( B @ ( suc @ aa_nat_nat ) ) ) )
      & ( ( ( A @ ( suc @ aa_nat_nat ) )
          = ( B @ ( suc @ aa_nat_nat ) ) )
       => ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).

thf(8,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( A @ ( A @ ( C @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( A @ ( B @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2364_div__mult__self1) ).

thf(363,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( A @ ( A @ ( C @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( A @ ( B @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).

thf(42,axiom,
    ! [A: fun_nat_bool,B: nat] :
      ( ( B @ ( A @ aa_nat_bool ) @ pp )
     => ( ~ ( zero_zero_nat @ ( A @ aa_nat_bool ) @ pp )
       => ( ( A @ ord_Least_nat )
          = ( A @ aTP_Lamm_aba @ ord_Least_nat @ ( suc @ aa_nat_nat ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8016_Least__Suc) ).

thf(539,plain,
    ! [A: fun_nat_bool,B: nat] :
      ( ( B @ ( A @ aa_nat_bool ) @ pp )
     => ( ~ ( zero_zero_nat @ ( A @ aa_nat_bool ) @ pp )
       => ( ( A @ ord_Least_nat )
          = ( A @ aTP_Lamm_aba @ ord_Least_nat @ ( suc @ aa_nat_nat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).

thf(74,axiom,
    ! [A: nat,B: nat] :
      ( ( ( B @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = zero_zero_nat )
     => ( ( ( A @ ( B @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = zero_zero_nat )
       => ( A = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3726_diffs0__imp__equal) ).

thf(671,plain,
    ! [A: nat,B: nat] :
      ( ( ( B @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = zero_zero_nat )
     => ( ( ( A @ ( B @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = zero_zero_nat )
       => ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).

thf(120,axiom,
    ! [A: nat,B: nat] :
      ( ( ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = A )
    <=> ( ( B @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = zero_zero_nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3233_mod__eq__self__iff__div__eq__0) ).

thf(903,plain,
    ! [A: nat,B: nat] :
      ( ( ( ( B @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = zero_zero_nat )
       => ( ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = A ) )
      & ( ( ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = A )
       => ( ( B @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = zero_zero_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).

thf(311,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
     => ( B = C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_937_add__left__imp__eq) ).

thf(1654,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
     => ( B = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[311]) ).

thf(93,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( A @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( one_one_nat @ ( A @ ( B @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2105_div__add__self1) ).

thf(771,plain,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( A @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( one_one_nat @ ( A @ ( B @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).

thf(171,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( C != D )
        & ( A != B ) )
    <=> ( ( D @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
       != ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( D @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2424_crossproduct__noteq) ).

thf(1108,plain,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( ( D @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
         != ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( D @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
       => ( ( C != D )
          & ( A != B ) ) )
      & ( ( ( C != D )
          & ( A != B ) )
       => ( ( D @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
         != ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( D @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).

thf(258,axiom,
    ! [A: vEBT_VEBT,B: nat] :
      ( ( B @ ( suc @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) @ ( A @ vEBT_invar_vebt @ aa_nat_bool ) @ pp )
     => ? [C: option4927543243414619207at_nat,D: list_VEBT_VEBT,E: vEBT_VEBT] :
          ( A
          = ( E @ ( D @ ( B @ ( suc @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) @ ( C @ vEBT_Node ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1467_deg__SUcn__Node) ).

thf(1476,plain,
    ! [A: vEBT_VEBT,B: nat] :
      ( ( B @ ( suc @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) @ ( A @ vEBT_invar_vebt @ aa_nat_bool ) @ pp )
     => ? [C: option4927543243414619207at_nat,D: list_VEBT_VEBT,E: vEBT_VEBT] :
          ( A
          = ( E @ ( D @ ( B @ ( suc @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) @ ( C @ vEBT_Node ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).

thf(306,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_929_add_Ocommute) ).

thf(1639,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[306]) ).

thf(126,axiom,
    ! [A: nat] :
      ( ( A @ ( zero_zero_nat @ ( suc @ aa_nat_nat ) @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1888_power__Suc__0) ).

thf(946,plain,
    ! [A: nat] :
      ( ( A @ ( zero_zero_nat @ ( suc @ aa_nat_nat ) @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).

thf(68,axiom,
    ( aTP_Lamm_cd
    = ( one_one_nat @ ( times_times_nat @ aa_nat_fun_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3102_lambda__one) ).

thf(652,plain,
    ( aTP_Lamm_cd
    = ( one_one_nat @ ( times_times_nat @ aa_nat_fun_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).

thf(12,axiom,
    ! [A: nat,B: nat] :
      ( ( ( B @ ( A @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) )
    <=> ( ( A
          = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) )
        | ( B = zero_zero_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1887_nat__power__eq__Suc__0__iff) ).

thf(390,plain,
    ! [A: nat,B: nat] :
      ( ( ( ( A
            = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) )
          | ( B = zero_zero_nat ) )
       => ( ( B @ ( A @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) ) )
      & ( ( ( B @ ( A @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) )
       => ( ( A
            = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) )
          | ( B = zero_zero_nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(13,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1509_nat__add__max__left) ).

thf(401,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(97,axiom,
    ! [A: nat,B: nat] :
      ( ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = A )
    <=> ( B = zero_zero_nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_871_add__cancel__left__right) ).

thf(786,plain,
    ! [A: nat,B: nat] :
      ( ( ( B = zero_zero_nat )
       => ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = A ) )
      & ( ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = A )
       => ( B = zero_zero_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).

thf(280,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( B @ ( A @ ( vEBT_VEBT_low @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ vEBT_VEBT_bit_concat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2_bit__split__inv) ).

thf(1558,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( B @ ( A @ ( vEBT_VEBT_low @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( vEBT_VEBT_high @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ vEBT_VEBT_bit_concat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[280]) ).

thf(130,axiom,
    ! [A: nat] :
      ( zero_zero_nat
     != ( A @ ( suc @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_137_Zero__neq__Suc) ).

thf(963,plain,
    ! [A: nat] :
      ( zero_zero_nat
     != ( A @ ( suc @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).

thf(38,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( A @ ( C @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( C @ ( B @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2327_div__mult__mult2) ).

thf(517,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( A @ ( C @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( C @ ( B @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).

thf(159,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( suc @ aa_nat_nat ) @ ( A @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2598_power__Suc) ).

thf(1062,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( suc @ aa_nat_nat ) @ ( A @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( power_power_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).

thf(266,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ aTP_Lamm_zl @ aa_nat8857836526276733660t_bool ) @ collect_set_nat @ finite_card_set_nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_7690_binomial__def) ).

thf(1508,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ aTP_Lamm_zl @ aa_nat8857836526276733660t_bool ) @ collect_set_nat @ finite_card_set_nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).

thf(262,axiom,
    ! [A: nat,B: nat] :
      ( ( ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = zero_zero_nat )
     => ? [C: nat] :
          ( A
          = ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3302_mod__eq__0D) ).

thf(1491,plain,
    ! [A: nat,B: nat] :
      ( ( ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = zero_zero_nat )
     => ? [C: nat] :
          ( A
          = ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).

thf(49,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2479_nat__mult__max__left) ).

thf(568,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).

thf(208,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3615_diff__add__zero) ).

thf(1294,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).

thf(123,axiom,
    ! [A: nat] :
      ( ( A @ ( zero_zero_nat @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1481_max__nat_Oleft__neutral) ).

thf(929,plain,
    ! [A: nat] :
      ( ( A @ ( zero_zero_nat @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).

thf(31,axiom,
    ! [A: nat,B: vEBT_VEBT,C: fun_VEBT_VEBT_bool] :
      ( ? [D: vEBT_VEBT] :
          ( ( D @ ( C @ aa_VEBT_VEBT_bool ) @ pp )
          & ( B @ ( A @ replicate_VEBT_VEBT ) @ set_VEBT_VEBT2 @ ( D @ ( member_VEBT_VEBT @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) @ pp ) )
    <=> ( ( A != zero_zero_nat )
        & ( B @ ( C @ aa_VEBT_VEBT_bool ) @ pp ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1738_Bex__set__replicate) ).

thf(478,plain,
    ! [A: nat,B: vEBT_VEBT,C: fun_VEBT_VEBT_bool] :
      ( ( ( ( A != zero_zero_nat )
          & ( B @ ( C @ aa_VEBT_VEBT_bool ) @ pp ) )
       => ? [D: vEBT_VEBT] :
            ( ( D @ ( C @ aa_VEBT_VEBT_bool ) @ pp )
            & ( B @ ( A @ replicate_VEBT_VEBT ) @ set_VEBT_VEBT2 @ ( D @ ( member_VEBT_VEBT @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) @ pp ) ) )
      & ( ? [D: vEBT_VEBT] :
            ( ( D @ ( C @ aa_VEBT_VEBT_bool ) @ pp )
            & ( B @ ( A @ replicate_VEBT_VEBT ) @ set_VEBT_VEBT2 @ ( D @ ( member_VEBT_VEBT @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) @ pp ) )
       => ( ( A != zero_zero_nat )
          & ( B @ ( C @ aa_VEBT_VEBT_bool ) @ pp ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).

thf(104,axiom,
    ! [A: nat] :
      ( ( A @ ( one_one_nat @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2414_comm__monoid__mult__class_Omult__1) ).

thf(832,plain,
    ! [A: nat] :
      ( ( A @ ( one_one_nat @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).

thf(41,axiom,
    ! [A: nat,B: nat] :
      ( ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = zero_zero_nat )
     => ( ( B = zero_zero_nat )
        | ( A = zero_zero_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2404_divisors__zero) ).

thf(536,plain,
    ! [A: nat,B: nat] :
      ( ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = zero_zero_nat )
     => ( ( B = zero_zero_nat )
        | ( A = zero_zero_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).

thf(45,axiom,
    ( ( zero_zero_nat @ ( nat_triangle @ aa_nat_nat ) )
    = zero_zero_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1486_triangle__0) ).

thf(549,plain,
    ( ( zero_zero_nat @ ( nat_triangle @ aa_nat_nat ) )
    = zero_zero_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).

thf(250,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( D @ ( C @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( B @ ( C @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( D @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
    <=> ( ( B = D )
        | ( A = C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2428_crossproduct__eq) ).

thf(1444,plain,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( ( B = D )
          | ( A = C ) )
       => ( ( D @ ( C @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( B @ ( C @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( D @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) )
      & ( ( ( D @ ( C @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( B @ ( C @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( D @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
       => ( ( B = D )
          | ( A = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).

thf(82,axiom,
    ! [A: vEBT_VEBT] :
      ~ ( zero_zero_nat @ ( A @ vEBT_invar_vebt @ aa_nat_bool ) @ pp ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_62_valid__0__not) ).

thf(716,plain,
    ! [A: vEBT_VEBT] :
      ~ ( zero_zero_nat @ ( A @ vEBT_invar_vebt @ aa_nat_bool ) @ pp ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).

thf(100,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( B @ ( C @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
    <=> ( ( A = C )
        | ( B = zero_zero_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2246_mult__cancel__right) ).

thf(800,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( ( A = C )
          | ( B = zero_zero_nat ) )
       => ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( B @ ( C @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) )
      & ( ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( B @ ( C @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
       => ( ( A = C )
          | ( B = zero_zero_nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).

thf(307,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( aTP_Lamm_zy @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( B @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8266_ATP_Olambda__147) ).

thf(1642,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( aTP_Lamm_zy @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( B @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[307]) ).

thf(205,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( suc @ aa_nat_nat ) @ ( B @ ( A @ ( suc @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3569_Suc__diff__diff) ).

thf(1280,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( suc @ aa_nat_nat ) @ ( B @ ( A @ ( suc @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).

thf(23,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat
        = ( B @ ( A @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
    <=> ( ( B = zero_zero_nat )
        & ( A = zero_zero_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1482_max__nat_Oneutr__eq__iff) ).

thf(432,plain,
    ! [A: nat,B: nat] :
      ( ( ( ( B = zero_zero_nat )
          & ( A = zero_zero_nat ) )
       => ( zero_zero_nat
          = ( B @ ( A @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) )
      & ( ( zero_zero_nat
          = ( B @ ( A @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
       => ( ( B = zero_zero_nat )
          & ( A = zero_zero_nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).

thf(131,axiom,
    ! [A: nat] :
      ( ( A @ ( zero_zero_nat @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2264_mult__zero__left) ).

thf(967,plain,
    ! [A: nat] :
      ( ( A @ ( zero_zero_nat @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).

thf(105,axiom,
    ! [A: nat] :
      ( ( one_one_nat @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2937_mod__by__1) ).

thf(835,plain,
    ! [A: nat] :
      ( ( one_one_nat @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).

thf(276,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_933_add_Oleft__commute) ).

thf(1546,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[276]) ).

thf(196,axiom,
    ! [A: list_VEBT_VEBT,B: set_VEBT_VEBT,C: vEBT_VEBT,D: nat] :
      ( ( B @ ( A @ set_VEBT_VEBT2 @ ord_le4337996190870823476T_VEBT ) )
     => ( ( B @ ( C @ ( member_VEBT_VEBT @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) @ pp )
       => ( B @ ( C @ ( D @ ( A @ list_u1324408373059187874T_VEBT ) ) @ set_VEBT_VEBT2 @ ord_le4337996190870823476T_VEBT ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_268_set__update__subsetI) ).

thf(1248,plain,
    ! [A: list_VEBT_VEBT,B: set_VEBT_VEBT,C: vEBT_VEBT,D: nat] :
      ( ( B @ ( A @ set_VEBT_VEBT2 @ ord_le4337996190870823476T_VEBT ) )
     => ( ( B @ ( C @ ( member_VEBT_VEBT @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) @ pp )
       => ( B @ ( C @ ( D @ ( A @ list_u1324408373059187874T_VEBT ) ) @ set_VEBT_VEBT2 @ ord_le4337996190870823476T_VEBT ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).

thf(113,axiom,
    ! [A: option4927543243414619207at_nat,B: list_VEBT_VEBT,C: vEBT_VEBT,D: nat] :
      ( ( D @ ( C @ ( B @ ( zero_zero_nat @ ( suc @ aa_nat_nat ) @ ( A @ vEBT_Node ) ) ) @ vEBT_vebt_insert ) )
      = ( C @ ( B @ ( zero_zero_nat @ ( suc @ aa_nat_nat ) @ ( A @ vEBT_Node ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1587_vebt__insert_Osimps_I3_J) ).

thf(865,plain,
    ! [A: option4927543243414619207at_nat,B: list_VEBT_VEBT,C: vEBT_VEBT,D: nat] :
      ( ( D @ ( C @ ( B @ ( zero_zero_nat @ ( suc @ aa_nat_nat ) @ ( A @ vEBT_Node ) ) ) @ vEBT_vebt_insert ) )
      = ( C @ ( B @ ( zero_zero_nat @ ( suc @ aa_nat_nat ) @ ( A @ vEBT_Node ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).

thf(234,axiom,
    ! [A: nat] :
      ( A
     != ( A @ ( suc @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_140_n__not__Suc__n) ).

thf(1395,plain,
    ! [A: nat] :
      ( A
     != ( A @ ( suc @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).

thf(300,axiom,
    ! [A: fun_nat_nat,B: nat] :
      ( ( B @ ( A @ aTP_Lamm_rp @ aa_nat_fun_nat_nat ) )
      = ( B @ ( A @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8350_ATP_Olambda__231) ).

thf(1625,plain,
    ! [A: fun_nat_nat,B: nat] :
      ( ( B @ ( A @ aTP_Lamm_rp @ aa_nat_fun_nat_nat ) )
      = ( B @ ( A @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[300]) ).

thf(272,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_921_add_Oassoc) ).

thf(1527,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).

thf(176,axiom,
    ! [A: fun_VEBT_VEBT_bool,B: list_VEBT_VEBT] :
      ( ( none_VEBT_VEBT
        = ( B @ ( A @ find_VEBT_VEBT ) ) )
    <=> ~ ? [C: vEBT_VEBT] :
            ( ( C @ ( A @ aa_VEBT_VEBT_bool ) @ pp )
            & ( B @ set_VEBT_VEBT2 @ ( C @ ( member_VEBT_VEBT @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) @ pp ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1767_find__None__iff2) ).

thf(1148,plain,
    ! [A: fun_VEBT_VEBT_bool,B: list_VEBT_VEBT] :
      ( ( ~ ? [C: vEBT_VEBT] :
              ( ( C @ ( A @ aa_VEBT_VEBT_bool ) @ pp )
              & ( B @ set_VEBT_VEBT2 @ ( C @ ( member_VEBT_VEBT @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) @ pp ) )
       => ( none_VEBT_VEBT
          = ( B @ ( A @ find_VEBT_VEBT ) ) ) )
      & ( ( none_VEBT_VEBT
          = ( B @ ( A @ find_VEBT_VEBT ) ) )
       => ~ ? [C: vEBT_VEBT] :
              ( ( C @ ( A @ aa_VEBT_VEBT_bool ) @ pp )
              & ( B @ set_VEBT_VEBT2 @ ( C @ ( member_VEBT_VEBT @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) @ pp ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).

thf(37,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( B @ ( C @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
    <=> ( ( B = zero_zero_nat )
        | ( A = C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2287_mult__cancel2) ).

thf(506,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( ( B = zero_zero_nat )
          | ( A = C ) )
       => ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( B @ ( C @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) )
      & ( ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( B @ ( C @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
       => ( ( B = zero_zero_nat )
          | ( A = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).

thf(281,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( C @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3655_diff__commute) ).

thf(1561,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( C @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[281]) ).

thf(92,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( C @ ( B @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2329_div__mult__mult1) ).

thf(767,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( C @ ( B @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).

thf(72,axiom,
    ! [A: nat] :
      ( ( A @ ( suc @ aa_nat_nat ) )
     != zero_zero_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_152_old_Onat_Odistinct_I2_J) ).

thf(664,plain,
    ! [A: nat] :
      ( ( A @ ( suc @ aa_nat_nat ) )
     != zero_zero_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).

thf(223,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( B @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3864_minus__mod__eq__div__mult) ).

thf(1346,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( B @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).

thf(129,axiom,
    ! [A: nat,B: nat] :
      ( ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = A )
     => ( B = zero_zero_nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_134_add__eq__self__zero) ).

thf(960,plain,
    ! [A: nat,B: nat] :
      ( ( ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = A )
     => ( B = zero_zero_nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).

thf(257,axiom,
    ! [A: nat,B: nat] :
      ( ( A @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = B ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3559_add__diff__cancel__left_H) ).

thf(1473,plain,
    ! [A: nat,B: nat] :
      ( ( A @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).

thf(40,axiom,
    ! [A: nat] :
      ( zero_zero_nat
     != ( A @ ( suc @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_151_old_Onat_Odistinct_I1_J) ).

thf(532,plain,
    ! [A: nat] :
      ( zero_zero_nat
     != ( A @ ( suc @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).

thf(160,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2935_mod__mult__self2__is__0) ).

thf(1065,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).

thf(61,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( A @ ( A @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( one_one_nat @ ( A @ ( B @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2107_div__add__self2) ).

thf(624,plain,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( A @ ( A @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( one_one_nat @ ( A @ ( B @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).

thf(225,axiom,
    ! [A: nat,B: nat] :
      ( ( ( A @ ( suc @ aa_nat_nat ) )
        = ( B @ ( suc @ aa_nat_nat ) ) )
     => ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_144_Suc__inject) ).

thf(1352,plain,
    ! [A: nat,B: nat] :
      ( ( ( A @ ( suc @ aa_nat_nat ) )
        = ( B @ ( suc @ aa_nat_nat ) ) )
     => ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).

thf(201,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( suc @ aa_nat_nat ) @ ( A @ ( suc @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3568_diff__Suc__Suc) ).

thf(1268,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( suc @ aa_nat_nat ) @ ( A @ ( suc @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).

thf(231,axiom,
    ! [A: fun_nat_bool,B: nat] :
      ( ( B @ ( A @ aa_nat_bool ) @ pp )
     => ( ! [C: nat] :
            ( ( C @ ( suc @ aa_nat_nat ) @ ( A @ aa_nat_bool ) @ pp )
           => ( C @ ( A @ aa_nat_bool ) @ pp ) )
       => ( zero_zero_nat @ ( A @ aa_nat_bool ) @ pp ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_139_zero__induct) ).

thf(1384,plain,
    ! [A: fun_nat_bool,B: nat] :
      ( ( B @ ( A @ aa_nat_bool ) @ pp )
     => ( ! [C: nat] :
            ( ( C @ ( suc @ aa_nat_nat ) @ ( A @ aa_nat_bool ) @ pp )
           => ( C @ ( A @ aa_nat_bool ) @ pp ) )
       => ( zero_zero_nat @ ( A @ aa_nat_bool ) @ pp ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).

thf(229,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( A @ ( suc @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2980_Suc__mod__mult__self1) ).

thf(1378,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( C @ ( B @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( suc @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( A @ ( suc @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).

thf(139,axiom,
    ! [A: fun_VEBT_VEBT_bool,B: list_VEBT_VEBT] :
      ( ( ( B @ ( A @ find_VEBT_VEBT ) )
        = none_VEBT_VEBT )
    <=> ~ ? [C: vEBT_VEBT] :
            ( ( C @ ( A @ aa_VEBT_VEBT_bool ) @ pp )
            & ( B @ set_VEBT_VEBT2 @ ( C @ ( member_VEBT_VEBT @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) @ pp ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1759_find__None__iff) ).

thf(992,plain,
    ! [A: fun_VEBT_VEBT_bool,B: list_VEBT_VEBT] :
      ( ( ~ ? [C: vEBT_VEBT] :
              ( ( C @ ( A @ aa_VEBT_VEBT_bool ) @ pp )
              & ( B @ set_VEBT_VEBT2 @ ( C @ ( member_VEBT_VEBT @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) @ pp ) )
       => ( ( B @ ( A @ find_VEBT_VEBT ) )
          = none_VEBT_VEBT ) )
      & ( ( ( B @ ( A @ find_VEBT_VEBT ) )
          = none_VEBT_VEBT )
       => ~ ? [C: vEBT_VEBT] :
              ( ( C @ ( A @ aa_VEBT_VEBT_bool ) @ pp )
              & ( B @ set_VEBT_VEBT2 @ ( C @ ( member_VEBT_VEBT @ aa_VEB289770264548726718T_bool ) @ aa_set8260762808118238015T_bool ) @ pp ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).

thf(264,axiom,
    ! [A: fun_nat_bool,B: nat] :
      ( ( zero_zero_nat @ ( A @ aa_nat_bool ) @ pp )
     => ( ! [C: nat] :
            ( ( C @ ( A @ aa_nat_bool ) @ pp )
           => ( C @ ( suc @ aa_nat_nat ) @ ( A @ aa_nat_bool ) @ pp ) )
       => ( B @ ( A @ aa_nat_bool ) @ pp ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_143_nat__induct) ).

thf(1500,plain,
    ! [A: fun_nat_bool,B: nat] :
      ( ( zero_zero_nat @ ( A @ aa_nat_bool ) @ pp )
     => ( ! [C: nat] :
            ( ( C @ ( A @ aa_nat_bool ) @ pp )
           => ( C @ ( suc @ aa_nat_nat ) @ ( A @ aa_nat_bool ) @ pp ) )
       => ( B @ ( A @ aa_nat_bool ) @ pp ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).

thf(245,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( B @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3349_div__mult__mod__eq) ).

thf(1431,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( B @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).

thf(24,axiom,
    ! [A: nat] :
      ( ( A @ ( zero_zero_nat @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2895_mod__0) ).

thf(442,plain,
    ! [A: nat] :
      ( ( A @ ( zero_zero_nat @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).

thf(115,axiom,
    ( one_one_nat
    = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2164_One__nat__def) ).

thf(871,plain,
    ( one_one_nat
    = ( zero_zero_nat @ ( suc @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).

thf(192,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( aTP_Lamm_ua @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8176_ATP_Olambda__57) ).

thf(1236,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( aTP_Lamm_ua @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( A @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).

thf(137,axiom,
    ! [A: vEBT_VEBT] :
      ( ~ ( A @ ( vEBT_VEBT_minNull @ aa_VEBT_VEBT_bool ) @ pp )
     => ? [B: nat] : ( B @ ( A @ ( vEBT_V8194947554948674370ptions @ aa_VEB4832260192069861672t_bool ) @ aa_nat_bool ) @ pp ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_24_not__min__Null__member) ).

thf(987,plain,
    ! [A: vEBT_VEBT] :
      ( ~ ( A @ ( vEBT_VEBT_minNull @ aa_VEBT_VEBT_bool ) @ pp )
     => ? [B: nat] : ( B @ ( A @ ( vEBT_V8194947554948674370ptions @ aa_VEB4832260192069861672t_bool ) @ aa_nat_bool ) @ pp ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).

thf(265,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( C @ ( B @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( B @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3236_div__add1__eq) ).

thf(1505,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( C @ ( B @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( A @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( B @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( A @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).

thf(235,axiom,
    ! [A: nat,B: nat] :
      ( ( A @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2933_mod__mult__self1__is__0) ).

thf(1399,plain,
    ! [A: nat,B: nat] :
      ( ( A @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( modulo_modulo_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).

thf(218,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3556_add__diff__cancel__left) ).

thf(1331,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( B @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).

thf(7,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( A != zero_zero_nat )
       => ( ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( C @ ( B @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) )
      & ( ( A = zero_zero_nat )
       => ( ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = zero_zero_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2815_nat__mult__div__cancel__disj) ).

thf(354,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( A != zero_zero_nat )
       => ( ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( C @ ( B @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) )
      & ( ( A = zero_zero_nat )
       => ( ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = zero_zero_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).

thf(142,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ aTP_Lamm_ul @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( B @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8498_ATP_Olambda__379) ).

thf(1009,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( C @ ( B @ ( A @ aTP_Lamm_ul @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( C @ ( B @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).

thf(43,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( A @ ( C @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( A @ ( C @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2358_div__mult__self4) ).

thf(543,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( A @ ( C @ ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( A @ ( C @ ( divide_divide_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).

thf(94,axiom,
    ! [A: nat] :
      ( ( one_one_nat @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2419_mult_Ocomm__neutral) ).

thf(775,plain,
    ! [A: nat] :
      ( ( one_one_nat @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).

thf(64,axiom,
    ! [A: nat] :
      ( ( A @ ( zero_zero_nat @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3570_diff__0__eq__0) ).

thf(639,plain,
    ! [A: nat] :
      ( ( A @ ( zero_zero_nat @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).

thf(90,axiom,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( A @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1485_max__0R) ).

thf(755,plain,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( A @ ( ord_max_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).

thf(28,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
        = ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
    <=> ( ( B = C )
        | ( A = zero_zero_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2827_nat__mult__eq__cancel__disj) ).

thf(454,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( ( B = C )
          | ( A = zero_zero_nat ) )
       => ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ) )
      & ( ( ( B @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
          = ( C @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) )
       => ( ( B = C )
          | ( A = zero_zero_nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).

thf(32,axiom,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2289_mult__0__right) ).

thf(491,plain,
    ! [A: nat] :
      ( ( zero_zero_nat @ ( A @ ( times_times_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = zero_zero_nat ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).

thf(166,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( B @ ( C @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3562_add__diff__cancel__right) ).

thf(1088,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( B @ ( C @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( C @ ( A @ ( minus_minus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).

thf(144,axiom,
    ! [A: fun_nat_bool] :
      ( ~ ( zero_zero_nat @ ( A @ aa_nat_bool ) @ pp )
     => ( ? [B: nat] : ( B @ ( A @ aa_nat_bool ) @ pp )
       => ? [B: nat] :
            ( ( B @ ( suc @ aa_nat_nat ) @ ( A @ aa_nat_bool ) @ pp )
            & ~ ( B @ ( A @ aa_nat_bool ) @ pp ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1391_exists__least__lemma) ).

thf(1014,plain,
    ! [A: fun_nat_bool] :
      ( ~ ( zero_zero_nat @ ( A @ aa_nat_bool ) @ pp )
     => ( ? [B: nat] : ( B @ ( A @ aa_nat_bool ) @ pp )
       => ? [B: nat] :
            ( ( B @ ( suc @ aa_nat_nat ) @ ( A @ aa_nat_bool ) @ pp )
            & ~ ( B @ ( A @ aa_nat_bool ) @ pp ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).

thf(241,axiom,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( aTP_Lamm_tz @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8177_ATP_Olambda__58) ).

thf(1417,plain,
    ! [A: nat,B: nat] :
      ( ( B @ ( A @ ( aTP_Lamm_tz @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) )
      = ( B @ ( B @ ( A @ ( plus_plus_nat @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) @ ( binomial @ aa_nat_fun_nat_nat ) @ aa_nat_nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).

thf(1667,plain,
    $false,
    inference(e,[status(thm)],[1322,1586,1337,1411,1105,846,555,1549,628,1031,408,1211,1596,683,1068,1618,873,1591,1437,1041,1173,1426,1137,1310,500,797,1233,747,1458,1576,1613,1645,829,404,782,1441,1518,1265,1623,1482,1628,1055,719,838,1511,1313,1455,1191,949,1514,1319,1479,1572,881,416,448,1494,475,591,1632,1283,821,1334,646,981,1119,411,1405,1521,312,1564,1242,849,1651,920,380,1470,1047,1438,661,1274,1091,1343,778,952,862,1570,598,429,729,1355,1580,1423,1602,583,889,1159,1079,551,1392,974,1012,852,1555,1402,1328,1227,1006,610,420,989,1259,317,1361,1461,686,642,1414,970,547,328,1567,1028,1082,1588,701,1297,1530,654,1552,1050,503,984,911,1434,343,657,1170,1286,674,1616,558,1071,794,586,1648,750,1251,689,601,1631,733,423,1389,1488,445,1271,713,1635,1058,1239,704,1464,1638,1224,1194,900,1408,1524,736,1349,451,649,811,494,758,367,594,1162,621,1325,668,1381,932,943,1022,521,1594,868,1176,1102,1245,636,1044,1034,1583,426,589,1467,1420,1277,1340,1145,1018,1003,1025,414,1605,1230,580,855,1300,1188,1262,1085,1538,1038,370,1358,744,1543,1221,1052,314,571,497,859,1094,978,1140,465,1316,1180,363,539,671,903,1654,771,1108,1476,1639,946,652,390,401,786,1558,963,517,1062,1508,1491,568,1294,929,478,832,536,549,1444,716,800,1642,1280,432,967,835,1546,1248,865,1395,1625,1527,1148,506,1561,767,664,1346,960,1473,532,1065,624,1352,1268,1384,1378,992,1500,1431,442,871,1236,987,1505,1399,1331,354,1009,543,775,639,755,454,491,1088,1014,1417]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03  % Problem  : ITP234_1 : TPTP v9.3.1. Released v8.0.0.
% 0.00/0.08  % Command  : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p  --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% 0.13/0.38  % Computer : n004.cluster.edu
% 0.13/0.38  % Model    : x86_64 x86_64
% 0.13/0.38  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.38  % Memory   : 8046.5625MB
% 0.13/0.38  % OS       : Linux 6.8.0-71-generic
% 0.13/0.39  % CPULimit : 300
% 0.13/0.39  % WCLimit  : 300
% 0.13/0.39  % DateTime : Fri Sep 25 18:22:51 UTC 2026
% 0.13/0.39  % CPUTime  : 
% 0.13/0.39  Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p  --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% 0.87/0.98  % [INFO] 	 Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ... 
% 5.75/2.47  % [INFO] 	 Parsing done (1487ms). 
% 5.75/2.49  % [INFO] 	 Running in sequential loop mode. 
% 6.46/2.85  % [INFO] 	 eprover registered as external prover. 
% 6.46/2.86  % [INFO] 	 Scanning for conjecture ... 
% 10.51/3.97  % [INFO] 	 Found a conjecture (or negated_conjecture) and 8910 axioms. Running axiom selection ... 
% 12.82/5.25  % [INFO] 	 Axiom selection finished. Selected 309 axioms (removed 8601 axioms). 
% 13.57/5.52  % [INFO] 	 Problem is typed first-order (TPTP TFF). 
% 13.57/5.54  % [INFO] 	 Type checking passed. 
% 13.57/5.55  % [CONFIG] 	 Using configuration: timeout(300) with strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>.  Searching for refutation ... 
% 22.61/8.74  % External prover 'e' found a proof!
% 22.61/8.74  % [INFO] 	 Killing All external provers ... 
% 22.61/8.74  % Time passed: 8210ms (effective reasoning time: 6250ms)
% 22.61/8.74  % Solved by strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>
% 22.61/8.75  % Axioms used in derivation (309): fact_8265_ATP_Olambda__146, fact_3571_diff__self__eq__0, fact_6716_Suc__times__binomial__eq, fact_33__C163_C, fact_2577_add__scale__eq__noteq, fact_2375_mult_Oassoc, fact_3860_minus__mult__div__eq__mod, fact_2953_mod__div__trivial, fact_940_add__right__imp__eq, fact_3124_mod__add__left__eq, fact_2414_comm__monoid__mult__class_Omult__1, fact_3864_minus__mod__eq__div__mult, fact_6739_binomial__absorption, fact_1874_power__0__Suc, fact_2935_mod__mult__self2__is__0, fact_8449_ATP_Olambda__330, fact_8641_ATP_Olambda__522, fact_867_add__cancel__left__left, fact_3355_mult__div__mod__eq, fact_2618_mult__Suc, fact_2370_ab__semigroup__mult__class_Omult__ac_I1_J, fact_2291_nat__1__eq__mult__iff, fact_3741_diff__mult__distrib, fact_875_add__cancel__right__left, fact_3345_cancel__div__mod__rules_I1_J, fact_3236_div__add1__eq, fact_151_old_Onat_Odistinct_I1_J, fact_2464_Suc__mult__cancel1, fact_8015_Least__Suc2, fact_3726_diffs0__imp__equal, fact_6691_binomial__n__0, fact_1507_max__add__distrib__right, fact_1390_vebt__buildup_Ocases, fact_1481_max__nat_Oleft__neutral, fact_3543_zero__diff, fact_2603_power__Suc2, fact_1392_add__0__iff, fact_2254_mult__eq__0__iff, fact_2903_mod__add__self1, fact_1995_div__by__Suc__0, fact_2978_Suc__mod__mult__self3, fact_2246_mult__cancel__right, fact_8628_ATP_Olambda__509, fact_2_bit__split__inv, fact_2980_Suc__mod__mult__self1, fact_3102_lambda__one, fact_1870_div__0, fact_6728_binomial__Suc__Suc__eq__times, fact_3_tc, fact_2287_mult__cancel2, fact_7690_binomial__def, fact_2096_zero__neq__one, fact_1794_find__cong, fact_933_add_Oleft__commute, fact_8350_ATP_Olambda__231, fact_1483_max__nat_Oright__neutral, fact_150_nat_OdiscI, fact_2290_mult__is__0, fact_2395_mult__left__cancel, fact_6718_choose__mult__lemma, fact_2358_div__mult__self4, fact_8435_ATP_Olambda__316, fact_992_add_Ocomm__neutral, fact_3862_minus__mod__eq__mult__div, fact_2967_mod__by__Suc__0, fact_3397_mod__mult2__eq, 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fact_2003_bits__div__by__0, fact_3834_nat__minus__add__max, fact_2399_no__zero__divisors, fact_879_add__cancel__right__right, fact_29__C5_C, fact_1466_deg__deg__n, fact_2264_mult__zero__left, fact_3559_add__diff__cancel__left_H, fact_3573_diff__diff__left, fact_1391_exists__least__lemma, fact_2089_one__reorient, fact_14_list__update__overwrite, fact_3549_cancel__comm__monoid__add__class_Odiff__cancel, fact_1_insprop, fact_8838_ATP_Olambda__719, fact_3545_diff__zero, fact_2957_mod__mult__self4, fact_2107_div__add__self2, fact_16__C11_C, fact_148_plus__nat_Oadd__0, fact_3343_cancel__div__mod__rules_I2_J, fact_147_old_Onat_Oexhaust, fact_2250_mult__cancel__left, fact_8177_ATP_Olambda__58, fact_2961_mod__mult__self2, fact_1864_div__by__0, fact_883_add__eq__0__iff__both__eq__0, fact_134_add__eq__self__zero, fact_2955_bits__mod__div__trivial, help_COMBC_1_1_COMBC_001t__VEBT____Definitions__OVEBT_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J_001t__HOL__Obool_U, fact_3713_add__implies__diff, fact_2325_div__mult__mult1__if, fact_25_min__Null__member, fact_0_True, fact_2006_div__by__1, fact_1396_verit__sum__simplify, fact_8498_ATP_Olambda__379, fact_3904_power__eq__if, fact_7984_map__add__upt, fact_2480_nat__mult__max__right, fact_3347_mod__div__decomp, fact_2166_plus__1__eq__Suc, fact_6667_binomial__Suc__n, fact_3357_div__mult1__eq, fact_2939_bits__mod__by__1, fact_133_not0__implies__Suc, fact_909_add__mono__thms__linordered__semiring_I4_J, fact_2385_mult_Oleft__commute, fact_10_nth__list__update__neq, fact_26__C1_C, fact_2959_mod__mult__self3, fact_3656_diff__right__commute, fact_1482_max__nat_Oneutr__eq__iff, fact_268_set__update__subsetI, fact_7983_map__Suc__upt, fact_2465_mult__0, fact_3736_Nat_Odiff__cancel, fact_8293_ATP_Olambda__174, fact_3353_mod__mult__div__eq, fact_2259_mult__zero__right, fact_3866_minus__div__mult__eq__mod, fact_3570_diff__0__eq__0, fact_152_old_Onat_Odistinct_I2_J, fact_224_subset__code_I1_J, fact_2620_mult__eq__self__implies__10
% 22.61/8.75  % No. of inferences in proof: 622
% 22.61/8.75  % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 8210 ms resp. 6250 ms w/o parsing
% 23.81/9.07  % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 23.81/9.08  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------