%------------------------------------------------------------------------------
% File     : ITP294_3 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Example 00111_003949
% Version  : [Des22] axioms.
% English  :

% Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
%          : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source   : [Des22]
% Names    : 0097_VEBT_Example_00111_003949 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 4624 (1621 unt;1288 typ;   0 def)
%            Number of atoms       : 7557 (2957 equ)
%            Maximal formula atoms :   28 (   2 avg)
%            Number of connectives : 4576 ( 577   ~; 105   |; 396   &)
%                                         ( 635 <=>;2863  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   23 (   4 avg)
%            Maximal term depth    :   35 (   2 avg)
%            Number of FOOLs       :  551 ( 222 fml; 329 var)
%            Number of X terms     :  221 (   0  []; 181 ite;  40 let)
%            Number of types       :  257 ( 256 usr)
%            Number of type conns  : 1264 ( 794   >; 470   *;   0   +;   0  <<)
%            Number of predicates  :   72 (  69 usr;   2 prp; 0-3 aty)
%            Number of functors    :  974 ( 974 usr; 249 con; 0-7 aty)
%            Number of variables   : 7535 (7329   !; 206   ?;7535   :)
% SPC      : TX0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-18 23:52:31.477
%------------------------------------------------------------------------------
% Could-be-implicit typings (256)
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tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    fun_nat_fun_int_o: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    fun_int_fun_nat_o: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    fun_int_fun_int_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    fun_o_fun_nat_nat: $tType ).

tff(ty_n_t__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J,type,
    fun_uint32_uint32: $tType ).

tff(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    list_list_nat: $tType ).

tff(ty_n_t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
    list_list_int: $tType ).

tff(ty_n_t__fun_It__List__Olist_It__Real__Oreal_J_M_Eo_J,type,
    fun_list_real_o: $tType ).

tff(ty_n_t__Heap____Time____Monad__OHeap_It__Nat__Onat_J,type,
    heap_Time_Heap_nat: $tType ).

tff(ty_n_t__Heap____Time____Monad__OHeap_It__Int__Oint_J,type,
    heap_Time_Heap_int: $tType ).

tff(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    set_list_nat: $tType ).

tff(ty_n_t__fun_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
    fun_list_nat_o: $tType ).

tff(ty_n_t__fun_It__List__Olist_It__Int__Oint_J_M_Eo_J,type,
    fun_list_int_o: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J,type,
    fun_nat_complex: $tType ).

tff(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    set_set_nat: $tType ).

tff(ty_n_t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    set_Code_integer: $tType ).

tff(ty_n_t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    fun_set_nat_o: $tType ).

tff(ty_n_t__fun_It__Code____Numeral__Ointeger_M_Eo_J,type,
    fun_Code_integer_o: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_I_Eo_M_Eo_J_J,type,
    fun_nat_fun_o_o: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__fun_I_Eo_M_Eo_J_J,type,
    fun_int_fun_o_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    fun_o_fun_nat_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    fun_o_fun_int_o: $tType ).

tff(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
    set_Product_unit: $tType ).

tff(ty_n_t__Set__Oset_It__Numeral____Type__Onum1_J,type,
    set_Numeral_num1: $tType ).

tff(ty_n_t__fun_It__String__Ochar_Mt__Nat__Onat_J,type,
    fun_char_nat: $tType ).

tff(ty_n_t__fun_It__Real__Oreal_Mt__Real__Oreal_J,type,
    fun_real_real: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__String__Ochar_J,type,
    fun_nat_char: $tType ).

tff(ty_n_t__fun_It__Extended____Nat__Oenat_M_Eo_J,type,
    fun_Extended_enat_o: $tType ).

tff(ty_n_t__List__Olist_It__List__Olist_I_Eo_J_J,type,
    list_list_o: $tType ).

tff(ty_n_t__itself_It__Numeral____Type__Onum1_J,type,
    itself_Numeral_num1: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Real__Oreal_J,type,
    fun_nat_real: $tType ).

tff(ty_n_t__Heap____Time____Monad__OHeap_I_Eo_J,type,
    heap_Time_Heap_o: $tType ).

tff(ty_n_t__fun_It__Num__Onum_Mt__Int__Oint_J,type,
    fun_num_int: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    fun_nat_nat: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
    fun_nat_int: $tType ).

tff(ty_n_t__fun_It__List__Olist_I_Eo_J_M_Eo_J,type,
    fun_list_o_o: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Nat__Onat_J,type,
    fun_int_nat: $tType ).

tff(ty_n_t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
    fun_int_int: $tType ).

tff(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
    set_complex: $tType ).

tff(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
    filter_real: $tType ).

tff(ty_n_t__fun_It__Complex__Ocomplex_M_Eo_J,type,
    fun_complex_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__Assertions__Oassn_J,type,
    fun_o_assn: $tType ).

tff(ty_n_t__itself_It__Enum__Ofinite____3_J,type,
    itself_finite_3: $tType ).

tff(ty_n_t__itself_It__Enum__Ofinite____2_J,type,
    itself_finite_2: $tType ).

tff(ty_n_t__itself_It__Enum__Ofinite____1_J,type,
    itself_finite_1: $tType ).

tff(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
    option_nat: $tType ).

tff(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
    filter_nat: $tType ).

tff(ty_n_t__Filter__Ofilter_It__Int__Oint_J,type,
    filter_int: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_M_Eo_J_J,type,
    fun_o_fun_o_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__Uint32__Ouint32_J,type,
    fun_o_uint32: $tType ).

tff(ty_n_t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_VEBTi: $tType ).

tff(ty_n_t__Set__Oset_It__String__Ochar_J,type,
    set_char: $tType ).

tff(ty_n_t__List__Olist_It__Real__Oreal_J,type,
    list_real: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__String__Ochar_J,type,
    fun_o_char: $tType ).

tff(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
    set_real: $tType ).

tff(ty_n_t__List__Olist_It__Nat__Onat_J,type,
    list_nat: $tType ).

tff(ty_n_t__List__Olist_It__Int__Oint_J,type,
    list_int: $tType ).

tff(ty_n_t__fun_It__Real__Oreal_M_Eo_J,type,
    fun_real_o: $tType ).

tff(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
    set_nat: $tType ).

tff(ty_n_t__Set__Oset_It__Int__Oint_J,type,
    set_int: $tType ).

tff(ty_n_t__Code____Numeral__Ointeger,type,
    code_integer: $tType ).

tff(ty_n_t__fun_It__Num__Onum_M_Eo_J,type,
    fun_num_o: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_M_Eo_J,type,
    fun_nat_o: $tType ).

tff(ty_n_t__fun_It__Int__Oint_M_Eo_J,type,
    fun_int_o: $tType ).

tff(ty_n_t__fun_I_Eo_Mt__Int__Oint_J,type,
    fun_o_int: $tType ).

tff(ty_n_t__Product____Type__Ounit,type,
    product_unit: $tType ).

tff(ty_n_t__Extended____Nat__Oenat,type,
    extended_enat: $tType ).

tff(ty_n_t__List__Olist_I_Eo_J,type,
    list_o: $tType ).

tff(ty_n_t__Complex__Ocomplex,type,
    complex: $tType ).

tff(ty_n_t__Assertions__Oassn,type,
    assn: $tType ).

tff(ty_n_t__Set__Oset_I_Eo_J,type,
    set_o: $tType ).

tff(ty_n_t__fun_I_Eo_M_Eo_J,type,
    fun_o_o: $tType ).

tff(ty_n_t__Uint32__Ouint32,type,
    uint32: $tType ).

tff(ty_n_t__String__Ochar,type,
    char: $tType ).

tff(ty_n_t__Real__Oreal,type,
    real: $tType ).

tff(ty_n_t__Rat__Orat,type,
    rat: $tType ).

tff(ty_n_t__Num__Onum,type,
    num: $tType ).

tff(ty_n_t__Nat__Onat,type,
    nat: $tType ).

tff(ty_n_t__Int__Oint,type,
    int: $tType ).

% Explicit typings (1032)
tff(sy_c_ATP_058Lamm__a____,type,
    aTP_Lamm_a: fun_op1688020532487384469ap_nat ).

tff(sy_c_ATP_058Lamm__aa____,type,
    aTP_Lamm_aa: fun_na3533371890617757541ap_nat ).

tff(sy_c_ATP_058Lamm__ab____,type,
    aTP_Lamm_ab: set_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__ac____,type,
    aTP_Lamm_ac: set_wo3913738467083021356l_num1 > fun_wo7859709073469921915num1_o ).

tff(sy_c_ATP_058Lamm__ad____,type,
    aTP_Lamm_ad: set_complex > fun_complex_o ).

tff(sy_c_ATP_058Lamm__ae____,type,
    aTP_Lamm_ae: set_list_nat > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__af____,type,
    aTP_Lamm_af: set_set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: set_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: set_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: fun_op1688020532487384469ap_nat > fun_fu1289858818758400691ap_nat ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: fun_VE6418762861507722832st_nat > fun_fu9081731133224488430st_nat ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: fun_op4683986414774905957on_nat > fun_fu6764168361233458435ap_nat ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: fun_VE4765431424076933413_VEBTi > fun_fu8993078228354513475st_nat ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: int > fun_li3813585616024374629st_int ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: fun_na1043288745686030785ap_int > fun_li2063098588606496750st_int ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: $o > fun_li4296730278180249433list_o ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: fun_na2343073368905879023Heap_o > fun_li440184404239406540list_o ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: fun_in1280865887336605797ap_int > fun_li8089996857832718994st_int ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: fun_in6294517997321588299Heap_o > fun_li3447736378634091120list_o ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: fun_o_1668507913897003235ap_int > fun_li3655962233941422746st_int ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: fun_o_8325599348076409421Heap_o > fun_li3441379867450936952list_o ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: nat > fun_li4609997795046405733st_nat ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: fun_in3770949032268332553ap_nat > fun_li1515057313908316562st_nat ).

tff(sy_c_ATP_058Lamm__aw____,type,
    aTP_Lamm_aw: fun_o_4158591058828729991ap_nat > fun_li6304394726871796122st_nat ).

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: fun_na3533371890617757541ap_nat > fun_li4711531081536870126st_nat ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: heap_Time_Heap_nat > fun_He5398114094724351478ap_nat ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: nat > fun_se292914136812062355o_assn ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: nat > fun_se7717841832296886164t_assn ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: fun_na2426279534609329326_VEBTi ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: nat > fun_se7717841832296886164t_assn ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: vEBT_VEBTi > fun_na3533371890617757541ap_nat ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: list_nat > fun_VE6418762861507722832st_nat ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: list_nat > fun_li2555252496525066953st_nat ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: fun_nat_real ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: fun_nat_real > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: fun_nat_real ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: fun_real_o ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: fun_real_o ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: real > fun_int_o ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: fun_Pr6626665806141890376t_assn ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: complex > fun_real_o ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: nat > fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: nat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: fun_nat_real ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: fun_int_int ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: complex > fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: fun_int_o ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: fun_nat_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: int > fun_int_int ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: list_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: list_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: list_nat > fun_li6410458269556549730t_assn ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: fun_real_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: real > fun_real_real ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: fun_real_real > fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: nat > fun_fu3612569900928539882l_real ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: fun_real_real ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: fun_nat_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: fun_re4238438383274954274t_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: fun_re4238438383274954274t_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: fun_real_real ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: fun_nat_real ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: fun_nat_real ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: fun_nat_real ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: fun_nat_real > fun_fu6119738925743952495t_real ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: real > fun_real_real ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: nat > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: fun_real_real ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: real > fun_real_real ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: fun_real_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: fun_real_real > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: fun_nat_o ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: fun_real_o > fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: fun_real_o > fun_real_o ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: real > fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: fun_real_o > fun_real_o ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: fun_nat_complex > fun_nat_real ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: fun_nat_complex > fun_nat_real ).

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: fun_o_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: code_integer > fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: fun_Co1092983284983208416er_int ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: fun_Co8521658056209057742er_num ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: fun_Co3583066429914935172er_nat ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: fun_int_int ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: fun_int_int ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: code_integer > fun_Co3544761417380347684eger_o ).

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: $o > fun_o_5488095450495718970o_char ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: $o > fun_o_6360995121869435791o_char ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: $o > fun_o_2683029531876921444o_char ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: $o > fun_o_3037882179896959673o_char ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: $o > fun_o_335877185394996366o_char ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: $o > fun_o_2016307092959984611o_char ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: $o > fun_Co681828205884031672o_char ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: fun_Co681828205884031672o_char ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: fun_nat_o ).

tff(sy_c_ATP_058Lamm__hy____,type,
    aTP_Lamm_hy: fun_o_fun_nat_o ).

tff(sy_c_ATP_058Lamm__hz____,type,
    aTP_Lamm_hz: nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
    archim2889992004027027881ng_rat: rat > int ).

tff(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
    archim7802044766580827645g_real: real > int ).

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
    archim3151403230148437115or_rat: rat > int ).

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
    archim6058952711729229775r_real: real > int ).

tff(sy_c_Assertions_Opure__assn,type,
    pure_assn: $o > assn ).

tff(sy_c_Binomial_Obinomial,type,
    binomial: nat > fun_nat_nat ).

tff(sy_c_Bit__Comprehension_Obit__comprehension__class_Oset__bits_001t__Int__Oint,type,
    bit_bi6516823479961619367ts_int: fun_nat_o > int ).

tff(sy_c_Bit__Comprehension_Obit__comprehension__class_Oset__bits_001t__Uint32__Ouint32,type,
    bit_bi705532357378895591uint32: fun_nat_o > uint32 ).

tff(sy_c_Bit__Comprehension_Obit__comprehension__class_Oset__bits_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    bit_bi5746210779246519537l_num1: fun_nat_o > word_N3645301735248828278l_num1 ).

tff(sy_c_Bit__Comprehension_Owf__set__bits__int,type,
    bit_wf_set_bits_int: fun_nat_o > $o ).

tff(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: fun_Pr9018117161326028078_int_o ).

tff(sy_c_Bit__Operations_Oconcat__bit,type,
    bit_concat_bit: ( nat * int ) > fun_int_int ).

tff(sy_c_Bit__Operations_Oor__not__num__neg,type,
    bit_or_not_num_neg: ( num * num ) > num ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
    bit_ri7919022796975470100ot_int: int > int ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Uint32__Ouint32,type,
    bit_ri5940046801117472212uint32: uint32 > uint32 ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    bit_ri6656324446590079838l_num1: word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
    bit_ri631733984087533419it_int: ( nat * int ) > int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Int__Oint,type,
    bit_se725231765392027082nd_int: ( int * int ) > int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
    bit_se727722235901077358nd_nat: nat > fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Uint32__Ouint32,type,
    bit_se6294004230839889034uint32: ( uint32 * uint32 ) > uint32 ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    bit_se6221597996545983252l_num1: ( word_N3645301735248828278l_num1 * word_N3645301735248828278l_num1 ) > word_N3645301735248828278l_num1 ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Code____Numeral__Ointeger,type,
    bit_se3928097537394005634nteger: ( nat * code_integer ) > code_integer ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Int__Oint,type,
    bit_se8568078237143864401it_int: ( nat * int ) > int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
    bit_se8570568707652914677it_nat: nat > fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Uint32__Ouint32,type,
    bit_se3964402333458159761uint32: fun_na3857091055356329634uint32 ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    bit_se5176125413884933531l_num1: ( nat * word_N3645301735248828278l_num1 ) > word_N3645301735248828278l_num1 ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Int__Oint,type,
    bit_se2159334234014336723it_int: ( nat * int ) > int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
    bit_se2161824704523386999it_nat: nat > fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Uint32__Ouint32,type,
    bit_se7025624438249859091uint32: ( nat * uint32 ) > uint32 ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    bit_se4491814353640558621l_num1: ( nat * word_N3645301735248828278l_num1 ) > word_N3645301735248828278l_num1 ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
    bit_se2000444600071755411sk_int: fun_nat_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
    bit_se2002935070580805687sk_nat: fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Uint32__Ouint32,type,
    bit_se2524848472035186643uint32: nat > uint32 ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    bit_se6623363288439087069l_num1: nat > word_N3645301735248828278l_num1 ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Code____Numeral__Ointeger,type,
    bit_se1080825931792720795nteger: ( code_integer * code_integer ) > code_integer ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Int__Oint,type,
    bit_se1409905431419307370or_int: ( int * int ) > int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
    bit_se1412395901928357646or_nat: nat > fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Uint32__Ouint32,type,
    bit_se2966626333419230250uint32: ( uint32 * uint32 ) > uint32 ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    bit_se2579674332251232948l_num1: ( word_N3645301735248828278l_num1 * word_N3645301735248828278l_num1 ) > word_N3645301735248828278l_num1 ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Code____Numeral__Ointeger,type,
    bit_se7788150548672797655nteger: ( nat * code_integer ) > code_integer ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Int__Oint,type,
    bit_se545348938243370406it_int: ( nat * int ) > int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Nat__Onat,type,
    bit_se547839408752420682it_nat: nat > fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Uint32__Ouint32,type,
    bit_se5742574853984576102uint32: fun_na3857091055356329634uint32 ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    bit_se837345729053750000l_num1: ( nat * word_N3645301735248828278l_num1 ) > word_N3645301735248828278l_num1 ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
    bit_se7879613467334960850it_int: ( nat * int ) > int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Nat__Onat,type,
    bit_se7882103937844011126it_nat: nat > fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Uint32__Ouint32,type,
    bit_se6647067497041451410uint32: fun_na3857091055356329634uint32 ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    bit_se4894374433684937756l_num1: ( nat * word_N3645301735248828278l_num1 ) > word_N3645301735248828278l_num1 ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Int__Oint,type,
    bit_se2923211474154528505it_int: ( nat * int ) > int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Nat__Onat,type,
    bit_se2925701944663578781it_nat: nat > fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Uint32__Ouint32,type,
    bit_se4189161883738761017uint32: ( nat * uint32 ) > uint32 ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    bit_se6195711425208868931l_num1: ( nat * word_N3645301735248828278l_num1 ) > word_N3645301735248828278l_num1 ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
    bit_se4203085406695923979it_int: ( nat * int ) > int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
    bit_se4205575877204974255it_nat: nat > fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Uint32__Ouint32,type,
    bit_se4315839071623982667uint32: fun_na3857091055356329634uint32 ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    bit_se5331074070815623765l_num1: ( nat * word_N3645301735248828278l_num1 ) > word_N3645301735248828278l_num1 ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Int__Oint,type,
    bit_se6526347334894502574or_int: ( int * int ) > int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Nat__Onat,type,
    bit_se6528837805403552850or_nat: nat > fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Uint32__Ouint32,type,
    bit_se8572831644974689134uint32: ( uint32 * uint32 ) > uint32 ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    bit_se4696820032942467064l_num1: ( word_N3645301735248828278l_num1 * word_N3645301735248828278l_num1 ) > word_N3645301735248828278l_num1 ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Int__Oint,type,
    bit_se1146084159140164899it_int: fun_int_fun_nat_o ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
    bit_se1148574629649215175it_nat: fun_nat_fun_nat_o ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Uint32__Ouint32,type,
    bit_se5367290876889521763uint32: fun_uint32_fun_nat_o ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    bit_se6859397288646540909l_num1: word_N3645301735248828278l_num1 > fun_nat_o ).

tff(sy_c_Bit__Shifts__Infix__Syntax_Osemiring__bit__operations__class_Oshiftl_001t__Nat__Onat,type,
    bit_Sh3965577149348748681tl_nat: nat > fun_nat_nat ).

tff(sy_c_Bits__Integer_OBit__integer,type,
    bits_Bit_integer: ( code_integer * $o ) > code_integer ).

tff(sy_c_Bits__Integer_Obin__last__integer,type,
    bits_b8758750999018896077nteger: code_integer > $o ).

tff(sy_c_Bits__Integer_Obin__rest__integer,type,
    bits_b2549910563261871055nteger: fun_Co8499238673658658103nteger ).

tff(sy_c_Code__Numeral_Obit__cut__integer,type,
    code_bit_cut_integer: code_integer > produc6271795597528267376eger_o ).

tff(sy_c_Code__Numeral_Odivmod__abs,type,
    code_divmod_abs: ( code_integer * code_integer ) > produc8923325533196201883nteger ).

tff(sy_c_Code__Numeral_Odivmod__integer,type,
    code_divmod_integer: ( code_integer * code_integer ) > produc8923325533196201883nteger ).

tff(sy_c_Code__Numeral_Odup,type,
    code_dup: fun_Co8499238673658658103nteger ).

tff(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
    code_int_of_integer: fun_Code_integer_int ).

tff(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
    code_integer_of_int: fun_int_Code_integer ).

tff(sy_c_Code__Numeral_Ointeger__of__nat,type,
    code_integer_of_nat: nat > code_integer ).

tff(sy_c_Code__Numeral_Ointeger__of__num,type,
    code_integer_of_num: num > code_integer ).

tff(sy_c_Code__Numeral_Onat__of__integer,type,
    code_nat_of_integer: code_integer > nat ).

tff(sy_c_Code__Numeral_Onum__of__integer,type,
    code_num_of_integer: code_integer > num ).

tff(sy_c_Code__Target__Int_Onegative,type,
    code_Target_negative: fun_num_int ).

tff(sy_c_Code__Target__Int_Opositive,type,
    code_Target_positive: fun_num_int ).

tff(sy_c_Code__Target__Nat_Oint__of__nat,type,
    code_T6385005292777649522of_nat: fun_nat_int ).

tff(sy_c_Code__Target__Word__Base_Oset__bits__aux_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J,type,
    code_T8486762787009228264l_num1: ( fun_nat_o * nat * word_N3645301735248828278l_num1 ) > word_N3645301735248828278l_num1 ).

tff(sy_c_Complex_OArg,type,
    arg: complex > real ).

tff(sy_c_Complex_Ocis,type,
    cis: real > complex ).

tff(sy_c_Complex_Ocnj,type,
    cnj: complex > complex ).

tff(sy_c_Complex_Ocomplex_OComplex,type,
    complex2: ( real * real ) > complex ).

tff(sy_c_Complex_Ocomplex_OIm,type,
    im: complex > real ).

tff(sy_c_Complex_Ocomplex_ORe,type,
    re: complex > real ).

tff(sy_c_Complex_Ocsqrt,type,
    csqrt: complex > complex ).

tff(sy_c_Complex_Oimaginary__unit,type,
    imaginary_unit: complex ).

tff(sy_c_Deriv_Odifferentiable_001t__Real__Oreal_001t__Real__Oreal,type,
    differ6690327859849518006l_real: ( fun_real_real * filter_real ) > $o ).

tff(sy_c_Deriv_Ohas__field__derivative_001t__Real__Oreal,type,
    has_fi5821293074295781190e_real: ( fun_real_real * real * filter_real ) > $o ).

tff(sy_c_Divides_Oadjust__div,type,
    adjust_div: product_prod_int_int > int ).

tff(sy_c_Divides_Oadjust__mod,type,
    adjust_mod: ( int * int ) > int ).

tff(sy_c_Divides_Odivmod__nat,type,
    divmod_nat: ( nat * nat ) > product_prod_nat_nat ).

tff(sy_c_Divides_Oeucl__rel__int,type,
    eucl_rel_int: ( int * int * product_prod_int_int ) > $o ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Int__Oint,type,
    unique5052692396658037445od_int: ( num * num ) > product_prod_int_int ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Nat__Onat,type,
    unique5055182867167087721od_nat: ( num * num ) > product_prod_nat_nat ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Code____Numeral__Ointeger,type,
    unique4921790084139445826nteger: ( num * produc8923325533196201883nteger ) > produc8923325533196201883nteger ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Int__Oint,type,
    unique5024387138958732305ep_int: ( num * product_prod_int_int ) > product_prod_int_int ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Nat__Onat,type,
    unique5026877609467782581ep_nat: ( num * product_prod_nat_nat ) > product_prod_nat_nat ).

tff(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
    semiri1408675320244567234ct_nat: fun_nat_nat ).

tff(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
    semiri2265585572941072030t_real: fun_nat_real ).

tff(sy_c_Fields_Oinverse__class_Oinverse_001t__Complex__Ocomplex,type,
    invers8013647133539491842omplex: complex > complex ).

tff(sy_c_Fields_Oinverse__class_Oinverse_001t__Rat__Orat,type,
    inverse_inverse_rat: rat > rat ).

tff(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
    inverse_inverse_real: fun_real_real ).

tff(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
    at_bot_real: filter_real ).

tff(sy_c_Filter_Oat__top_001t__Int__Oint,type,
    at_top_int: filter_int ).

tff(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
    at_top_nat: filter_nat ).

tff(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
    at_top_real: filter_real ).

tff(sy_c_Filter_Oeventually_001t__Nat__Onat,type,
    eventually_nat: ( fun_nat_o * filter_nat ) > $o ).

tff(sy_c_Filter_Oeventually_001t__Real__Oreal,type,
    eventually_real: ( fun_real_o * filter_real ) > $o ).

tff(sy_c_Filter_Ofilterlim_001t__Int__Oint_001t__Nat__Onat,type,
    filterlim_int_nat: ( fun_int_nat * filter_nat * filter_int ) > $o ).

tff(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Int__Oint,type,
    filterlim_nat_int: ( fun_nat_int * filter_int * filter_nat ) > $o ).

tff(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
    filterlim_nat_nat: ( fun_nat_nat * filter_nat * filter_nat ) > $o ).

tff(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
    filterlim_nat_real: ( fun_nat_real * filter_real * filter_nat ) > $o ).

tff(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
    filterlim_real_real: ( fun_real_real * filter_real * filter_real ) > $o ).

tff(sy_c_Finite__Set_Ocard_001_Eo,type,
    finite_card_o: set_o > nat ).

tff(sy_c_Finite__Set_Ocard_001t__Complex__Ocomplex,type,
    finite_card_complex: set_complex > nat ).

tff(sy_c_Finite__Set_Ocard_001t__Int__Oint,type,
    finite_card_int: set_int > nat ).

tff(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
    finite_card_list_nat: set_list_nat > nat ).

tff(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
    finite_card_nat: set_nat > nat ).

tff(sy_c_Finite__Set_Ocard_001t__Numeral____Type__Onum1,type,
    finite6454714172617411597l_num1: set_Numeral_num1 > nat ).

tff(sy_c_Finite__Set_Ocard_001t__Product____Type__Ounit,type,
    finite410649719033368117t_unit: set_Product_unit > nat ).

tff(sy_c_Finite__Set_Ocard_001t__String__Ochar,type,
    finite_card_char: set_char > nat ).

tff(sy_c_Finite__Set_Ofinite_001t__Code____Numeral__Ointeger,type,
    finite6017078050557962740nteger: set_Code_integer > $o ).

tff(sy_c_Finite__Set_Ofinite_001t__Complex__Ocomplex,type,
    finite3207457112153483333omplex: set_complex > $o ).

tff(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
    finite_finite_int: set_int > $o ).

tff(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
    finite_finite_nat: fun_set_nat_o ).

tff(sy_c_Fun_Obij__betw_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    bij_be1856998921033663316omplex: ( fun_complex_complex * set_complex * set_complex ) > $o ).

tff(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Complex__Ocomplex,type,
    bij_betw_nat_complex: ( fun_nat_complex * set_nat * set_complex ) > $o ).

tff(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
    bij_betw_nat_nat: ( fun_nat_nat * set_nat ) > fun_set_nat_o ).

tff(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
    comp_C7848664941760151970nteger: ( fun_Co1561183069634336733nteger * fun_Co8499238673658658103nteger * code_integer ) > fun_Pr5876161306168176259nteger ).

tff(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Int__Oint_001t__Num__Onum,type,
    comp_int_int_num: ( fun_int_int * fun_num_int ) > fun_num_int ).

tff(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Nat__Onat_001t__Int__Oint,type,
    comp_int_nat_int: ( fun_int_nat * fun_int_int ) > fun_int_nat ).

tff(sy_c_Fun_Ocomp_001t__Nat__Onat_001_Eo_001t__Nat__Onat,type,
    comp_nat_o_nat: ( fun_nat_o * fun_nat_nat ) > fun_nat_o ).

tff(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Real__Oreal_001t__Nat__Onat,type,
    comp_nat_real_nat: ( fun_nat_real * fun_nat_nat ) > fun_nat_real ).

tff(sy_c_Fun_Ocomp_001t__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
    comp_f6739083541385432444nteger: ( fun_fu4800041728187187127nteger * fun_Co1981900195452921617nteger ) > fun_Co1561183069634336733nteger ).

tff(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
    inj_on_nat_nat: fun_nat_nat > fun_set_nat_o ).

tff(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__String__Ochar,type,
    inj_on_nat_char: fun_nat_char > fun_set_nat_o ).

tff(sy_c_Fun_Oinj__on_001t__Real__Oreal_001t__Real__Oreal,type,
    inj_on_real_real: ( fun_real_real * set_real ) > $o ).

tff(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
    inj_on_set_nat_nat: ( fun_set_nat_nat * set_set_nat ) > $o ).

tff(sy_c_Fun_Omap__fun_001t__Code____Numeral__Ointeger_001t__Int__Oint_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
    map_fu2599414010547811884nteger: ( fun_Code_integer_int * fun_int_Code_integer * fun_int_int ) > fun_Co8499238673658658103nteger ).

tff(sy_c_Fun_Othe__inv__into_001t__Real__Oreal_001t__Real__Oreal,type,
    the_in5290026491893676941l_real: ( set_real * fun_real_real ) > fun_real_real ).

tff(sy_c_GCD_Obezw,type,
    bezw: ( nat * nat ) > product_prod_int_int ).

tff(sy_c_GCD_Obezw__rel,type,
    bezw_rel: fun_Pr2028404973800837934_nat_o ).

tff(sy_c_GCD_Ogcd__class_Ogcd_001t__Int__Oint,type,
    gcd_gcd_int: ( int * int ) > int ).

tff(sy_c_GCD_Ogcd__class_Ogcd_001t__Nat__Onat,type,
    gcd_gcd_nat: nat > fun_nat_nat ).

tff(sy_c_Generic__set__bit_Oset__bit__class_Oset__bit_001t__Int__Oint,type,
    generi8991105624351003935it_int: ( int * nat * $o ) > int ).

tff(sy_c_Generic__set__bit_Oset__bit__class_Oset__bit_001t__Uint32__Ouint32,type,
    generi1993664874377053279uint32: fun_ui928617633769132301uint32 ).

tff(sy_c_Generic__set__bit_Oset__bit__class_Oset__bit_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    generi5268133209446125161l_num1: ( word_N3645301735248828278l_num1 * nat * $o ) > word_N3645301735248828278l_num1 ).

tff(sy_c_Groups_Oabs__class_Oabs_001t__Code____Numeral__Ointeger,type,
    abs_abs_Code_integer: code_integer > code_integer ).

tff(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
    abs_abs_int: fun_int_int ).

tff(sy_c_Groups_Oabs__class_Oabs_001t__Rat__Orat,type,
    abs_abs_rat: rat > rat ).

tff(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
    abs_abs_real: fun_real_real ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Code____Numeral__Ointeger,type,
    minus_8373710615458151222nteger: ( code_integer * code_integer ) > code_integer ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex,type,
    minus_minus_complex: ( complex * complex ) > complex ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Nat__Oenat,type,
    minus_3235023915231533773d_enat: ( extended_enat * extended_enat ) > extended_enat ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
    minus_minus_int: ( int * int ) > int ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
    minus_minus_nat: nat > fun_nat_nat ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Rat__Orat,type,
    minus_minus_rat: ( rat * rat ) > rat ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
    minus_minus_real: real > fun_real_real ).

tff(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
    minus_minus_set_nat: ( set_nat * set_nat ) > set_nat ).

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tff(sy_c_Groups_Ozero__class_Ozero_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
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tff(sy_c_HOL_OThe_001t__Int__Oint,type,
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tff(sy_c_HOL_Oundefined_001t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_Mt__fun_It__Code____Numeral__Ointeger_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_J,type,
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tff(sy_c_Heap__Time__Monad_Obind_001_Eo_001t__List__Olist_I_Eo_J,type,
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tff(sy_c_Heap__Time__Monad_Obind_001t__Option__Ooption_It__Nat__Onat_J_001t__Nat__Onat,type,
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tff(sy_c_Hoare__Triple_Ohoare__triple_001_Eo,type,
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tff(sy_c_Hoare__Triple_Ohoare__triple_001t__Product____Type__Ounit,type,
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tff(sy_c_Int_Oint__ge__less__than2,type,
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tff(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
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    distinct_nat: fun_list_nat_o ).

tff(sy_c_List_Ofoldr_001_Eo_001t__Nat__Onat,type,
    foldr_o_nat: ( fun_o_fun_nat_nat * list_o ) > fun_nat_nat ).

tff(sy_c_List_Ofoldr_001t__Nat__Onat_001t__Nat__Onat,type,
    foldr_nat_nat: ( fun_nat_fun_nat_nat * list_nat ) > fun_nat_nat ).

tff(sy_c_List_Ofoldr_001t__Real__Oreal_001t__Real__Oreal,type,
    foldr_real_real: ( fun_re7681423839302404990l_real * list_real ) > fun_real_real ).

tff(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
    linord2614967742042102400et_nat: set_nat > list_nat ).

tff(sy_c_List_Olist_OCons_001_Eo,type,
    cons_o: ( $o * list_o ) > list_o ).

tff(sy_c_List_Olist_OCons_001t__Int__Oint,type,
    cons_int: ( int * list_int ) > list_int ).

tff(sy_c_List_Olist_OCons_001t__List__Olist_I_Eo_J,type,
    cons_list_o: ( list_o * list_list_o ) > list_list_o ).

tff(sy_c_List_Olist_OCons_001t__List__Olist_It__Int__Oint_J,type,
    cons_list_int: ( list_int * list_list_int ) > list_list_int ).

tff(sy_c_List_Olist_OCons_001t__List__Olist_It__Nat__Onat_J,type,
    cons_list_nat: ( list_nat * list_list_nat ) > list_list_nat ).

tff(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
    cons_nat: ( nat * list_nat ) > list_nat ).

tff(sy_c_List_Olist_OCons_001t__Real__Oreal,type,
    cons_real: ( real * list_real ) > list_real ).

tff(sy_c_List_Olist_ONil_001_Eo,type,
    nil_o: list_o ).

tff(sy_c_List_Olist_ONil_001t__Int__Oint,type,
    nil_int: list_int ).

tff(sy_c_List_Olist_ONil_001t__List__Olist_I_Eo_J,type,
    nil_list_o: list_list_o ).

tff(sy_c_List_Olist_ONil_001t__List__Olist_It__Int__Oint_J,type,
    nil_list_int: list_list_int ).

tff(sy_c_List_Olist_ONil_001t__List__Olist_It__Nat__Onat_J,type,
    nil_list_nat: list_list_nat ).

tff(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
    nil_nat: list_nat ).

tff(sy_c_List_Olist_ONil_001t__Real__Oreal,type,
    nil_real: list_real ).

tff(sy_c_List_Olist_Omap_001t__Nat__Onat_001_Eo,type,
    map_nat_o: ( fun_nat_o * list_nat ) > list_o ).

tff(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
    map_nat_nat: ( fun_nat_nat * list_nat ) > list_nat ).

tff(sy_c_List_Olist_Oset_001t__Int__Oint,type,
    set_int2: list_int > set_int ).

tff(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
    set_nat2: list_nat > set_nat ).

tff(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
    set_real2: list_real > set_real ).

tff(sy_c_List_Onth_001t__Int__Oint,type,
    nth_int: list_int > fun_nat_int ).

tff(sy_c_List_Onth_001t__Nat__Onat,type,
    nth_nat: list_nat > fun_nat_nat ).

tff(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
    sorted_wrt_int: fun_int_fun_int_o > fun_list_int_o ).

tff(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
    sorted_wrt_nat: fun_nat_fun_nat_o > fun_list_nat_o ).

tff(sy_c_List_Oupt,type,
    upt: ( nat * nat ) > list_nat ).

tff(sy_c_List_Oupto,type,
    upto: ( int * int ) > list_int ).

tff(sy_c_List_Oupto__aux,type,
    upto_aux: ( int * int * list_int ) > list_int ).

tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun_Pr9018117161326028078_int_o ).

tff(sy_c_Most__significant__bit_Omsb__class_Omsb_001t__Int__Oint,type,
    most_s5051101344085556sb_int: fun_int_o ).

tff(sy_c_Most__significant__bit_Omsb__class_Omsb_001t__Uint32__Ouint32,type,
    most_s9063628576841037300uint32: uint32 > $o ).

tff(sy_c_Most__significant__bit_Omsb__class_Omsb_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    most_s3860955602682368894l_num1: fun_wo7859709073469921915num1_o ).

tff(sy_c_Nat_OSuc,type,
    suc: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
    semiri4939895301339042750nteger: fun_nat_Code_integer ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
    semiri8010041392384452111omplex: fun_nat_complex ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
    semiri1314217659103216013at_int: fun_nat_int ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
    semiri1316708129612266289at_nat: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
    semiri681578069525770553at_rat: nat > rat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
    semiri5074537144036343181t_real: fun_nat_real ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Uint32__Ouint32,type,
    semiri2565882477558803405uint32: nat > uint32 ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    semiri8819519690708144855l_num1: nat > word_N3645301735248828278l_num1 ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
    size_size_list_o: list_o > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
    size_size_list_int: list_int > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
    size_size_list_nat: list_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
    size_size_list_real: list_real > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
    size_size_num: num > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
    size_size_char: char > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Uint32__Ouint32,type,
    size_size_uint32: uint32 > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    size_s8261804613246490634l_num1: word_N3645301735248828278l_num1 > nat ).

tff(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: nat > set_nat ).

tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: fun_set_nat_nat ).

tff(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: fun_nat_nat ).

tff(sy_c_NthRoot_Oroot,type,
    root: fun_na6724076748692969250l_real ).

tff(sy_c_NthRoot_Osqrt,type,
    sqrt: fun_real_real ).

tff(sy_c_Num_OBitM,type,
    bitM: num > num ).

tff(sy_c_Num_Oinc,type,
    inc: num > num ).

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
    neg_numeral_dbl_int: int > int ).

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
    neg_numeral_dbl_rat: rat > rat ).

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
    neg_numeral_dbl_real: fun_real_real ).

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Uint32__Ouint32,type,
    neg_nu5314729912787363643uint32: uint32 > uint32 ).

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    neg_nu7865238048354675525l_num1: word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
    neg_nu5851722552734809277nc_int: int > int ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
    neg_nu5219082963157363817nc_rat: rat > rat ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
    neg_nu8295874005876285629c_real: fun_real_real ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Uint32__Ouint32,type,
    neg_nu4269007558841261821uint32: uint32 > uint32 ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    neg_nu8115118780965096967l_num1: word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ).

tff(sy_c_Num_Onum_OBit0,type,
    bit0: num > num ).

tff(sy_c_Num_Onum_OBit1,type,
    bit1: num > num ).

tff(sy_c_Num_Onum_OOne,type,
    one: num ).

tff(sy_c_Num_Onum_Osize__num,type,
    size_num: num > nat ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
    numera6620942414471956472nteger: num > code_integer ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex,type,
    numera6690914467698888265omplex: num > complex ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
    numera1916890842035813515d_enat: num > extended_enat ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
    numeral_numeral_int: fun_num_int ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
    numeral_numeral_nat: num > nat ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
    numeral_numeral_rat: num > rat ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
    numeral_numeral_real: num > real ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Uint32__Ouint32,type,
    numera9087168376688890119uint32: num > uint32 ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    numera7442385471795722001l_num1: num > word_N3645301735248828278l_num1 ).

tff(sy_c_Num_Opow,type,
    pow: ( num * num ) > num ).

tff(sy_c_Num_Opred__numeral,type,
    pred_numeral: num > nat ).

tff(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
    some_nat: nat > option_nat ).

tff(sy_c_Option_Ooption_Othe_001t__Nat__Onat,type,
    the_nat: option_nat > nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
    bot_bo4199563552545308370d_enat: extended_enat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
    bot_bot_nat: nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
    bot_bot_set_int: set_int ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Ointeger,type,
    ord_le6747313008572928689nteger: ( code_integer * code_integer ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
    ord_le72135733267957522d_enat: extended_enat > fun_Extended_enat_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
    ord_less_int: fun_int_fun_int_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
    ord_less_nat: fun_nat_fun_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
    ord_less_num: num > fun_num_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
    ord_less_rat: ( rat * rat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
    ord_less_real: fun_real_fun_real_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__String__Ochar,type,
    ord_less_char: ( char * char ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Uint32__Ouint32,type,
    ord_less_uint32: ( uint32 * uint32 ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    ord_le750835935415966154l_num1: word_N3645301735248828278l_num1 > fun_wo7859709073469921915num1_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Ointeger,type,
    ord_le3102999989581377725nteger: ( code_integer * code_integer ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
    ord_le2932123472753598470d_enat: extended_enat > fun_Extended_enat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    ord_le2510731241096832064er_nat: ( filter_nat * filter_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
    ord_less_eq_int: fun_int_fun_int_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
    ord_less_eq_nat: fun_nat_fun_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
    ord_less_eq_num: num > fun_num_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
    ord_less_eq_rat: ( rat * rat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
    ord_less_eq_real: fun_real_fun_real_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_eq_set_nat: set_nat > fun_set_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__String__Ochar,type,
    ord_less_eq_char: ( char * char ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Uint32__Ouint32,type,
    ord_less_eq_uint32: ( uint32 * uint32 ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    ord_le3335648743751981014l_num1: word_N3645301735248828278l_num1 > fun_wo7859709073469921915num1_o ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Extended____Nat__Oenat,type,
    ord_ma741700101516333627d_enat: fun_Ex8341929632020906428d_enat ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
    ord_max_int: fun_int_fun_int_int ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
    ord_max_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Orderings_Oord__class_Omin_001t__Extended____Nat__Oenat,type,
    ord_mi8085742599997312461d_enat: ( extended_enat * extended_enat ) > extended_enat ).

tff(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
    ord_min_nat: nat > fun_nat_nat ).

tff(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
    order_Greatest_nat: fun_nat_o > nat ).

tff(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Real__Oreal,type,
    order_9091379641038594480t_real: fun_nat_real > $o ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Assertions__Oassn,type,
    top_top_assn: assn ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
    top_top_set_o: set_o ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
    top_top_set_int: set_int ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Numeral____Type__Onum1_J,type,
    top_to3689904429138878997l_num1: set_Numeral_num1 ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    top_to1996260823553986621t_unit: set_Product_unit ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
    top_top_set_real: set_real ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
    top_top_set_char: set_char ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J_J,type,
    top_to6090768113777838812l_num1: set_wo3913738467083021356l_num1 ).

tff(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
    power_8256067586552552935nteger: ( code_integer * nat ) > code_integer ).

tff(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
    power_power_complex: complex > fun_nat_complex ).

tff(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
    power_power_int: int > fun_nat_int ).

tff(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
    power_power_nat: nat > fun_nat_nat ).

tff(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
    power_power_real: fun_re4238438383274954274t_real ).

tff(sy_c_Power_Opower__class_Opower_001t__Uint32__Ouint32,type,
    power_power_uint32: ( uint32 * nat ) > uint32 ).

tff(sy_c_Power_Opower__class_Opower_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    power_2184487114949457152l_num1: ( word_N3645301735248828278l_num1 * nat ) > word_N3645301735248828278l_num1 ).

tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001_Eo,type,
    produc6677183202524767010eger_o: ( code_integer * $o ) > produc6271795597528267376eger_o ).

tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    produc1086072967326762835nteger: ( code_integer * code_integer ) > produc8923325533196201883nteger ).

tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
    product_Pair_int_int: int > fun_in4211310926729683942nt_int ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
    product_Pair_nat_nat: nat > fun_na1476630412504255114at_nat ).

tff(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
    product_Pair_num_num: ( num * num ) > product_prod_num_num ).

tff(sy_c_Product__Type_OPair_001t__Uint32__Ouint32_001t__Uint32__Ouint32,type,
    produc1400373151660368625uint32: ( uint32 * uint32 ) > produc827990862158126777uint32 ).

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    produc6499014454317279255nteger: fun_fu4800041728187187127nteger ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001_Eo_001t__String__Ochar,type,
    produc4188289175737317920o_char: ( fun_Co681828205884031672o_char * produc6271795597528267376eger_o ) > char ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
    produc1553301316500091796er_int: ( fun_Co1092983284983208416er_int * produc8923325533196201883nteger ) > int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
    produc1555791787009142072er_nat: ( fun_Co3583066429914935172er_nat * produc8923325533196201883nteger ) > nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
    produc7336495610019696514er_num: ( fun_Co8521658056209057742er_num * produc8923325533196201883nteger ) > num ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
    produc9125791028180074456eger_o: ( fun_Co3544761417380347684eger_o * produc8923325533196201883nteger ) > produc6271795597528267376eger_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc6916734918728496179nteger: ( fun_Co2470855966376689427nteger * produc8923325533196201883nteger ) > produc8923325533196201883nteger ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
    produc4947309494688390418_int_o: fun_int_fun_int_o > fun_Pr3238374608701176162_int_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    produc8211389475949308722nt_int: ( fun_int_fun_int_int * product_prod_int_int ) > int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc4245557441103728435nt_int: ( fun_in1696471567231710611nt_int * product_prod_int_int ) > product_prod_int_int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc2626176000494625587at_nat: ( fun_na4138050610385705875at_nat * product_prod_nat_nat ) > product_prod_nat_nat ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
    product_fst_int_int: product_prod_int_int > int ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
    product_fst_nat_nat: product_prod_nat_nat > nat ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__Uint32__Ouint32_001t__Uint32__Ouint32,type,
    produc9004433772639906525uint32: produc827990862158126777uint32 > uint32 ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
    product_snd_nat_nat: product_prod_nat_nat > nat ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__Uint32__Ouint32_001t__Uint32__Ouint32,type,
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tff(sy_c_Pure_Otype_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J,type,
    type_N8448461349408098053l_num1: itself8794530163899892676l_num1 ).

tff(sy_c_Rat_OFrct,type,
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tff(sy_c_Rat_Onormalize,type,
    normalize: product_prod_int_int > product_prod_int_int ).

tff(sy_c_Rat_Oof__int,type,
    of_int: int > rat ).

tff(sy_c_Rat_Oquotient__of,type,
    quotient_of: rat > product_prod_int_int ).

tff(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
    real_V2521375963428798218omplex: set_complex ).

tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
    real_V1022390504157884413omplex: complex > real ).

tff(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
    real_V4546457046886955230omplex: real > complex ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
    divide6298287555418463151nteger: ( code_integer * code_integer ) > code_integer ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
    divide1717551699836669952omplex: ( complex * complex ) > complex ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
    divide_divide_int: fun_int_fun_int_int ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
    divide_divide_nat: nat > fun_nat_nat ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
    divide_divide_rat: ( rat * rat ) > rat ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
    divide_divide_real: real > fun_real_real ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Uint32__Ouint32,type,
    divide_divide_uint32: fun_ui2637848667930195262uint32 ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
    dvd_dvd_int: fun_int_fun_int_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
    dvd_dvd_nat: fun_nat_fun_nat_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Uint32__Ouint32,type,
    dvd_dvd_uint32: ( uint32 * uint32 ) > $o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
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tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
    modulo_modulo_int: ( int * int ) > int ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
    modulo_modulo_nat: nat > fun_nat_nat ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Uint32__Ouint32,type,
    modulo_modulo_uint32: fun_ui2637848667930195262uint32 ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
    zero_n2684676970156552555ol_int: fun_o_int ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Uint32__Ouint32,type,
    zero_n412250872926760619uint32: $o > uint32 ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    zero_n2087535428495186613l_num1: $o > word_N3645301735248828278l_num1 ).

tff(sy_c_Series_Osuminf_001t__Real__Oreal,type,
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tff(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
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tff(sy_c_Series_Osummable_001t__Real__Oreal,type,
    summable_real: fun_nat_real > $o ).

tff(sy_c_Series_Osums_001t__Complex__Ocomplex,type,
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tff(sy_c_Series_Osums_001t__Real__Oreal,type,
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tff(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
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tff(sy_c_Set_OCollect_001t__Int__Oint,type,
    collect_int: fun_int_o > set_int ).

tff(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
    collect_list_nat: fun_list_nat_o > set_list_nat ).

tff(sy_c_Set_OCollect_001t__Nat__Onat,type,
    collect_nat: fun_nat_o > set_nat ).

tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(sy_c_Set_OCollect_001t__Real__Oreal,type,
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tff(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
    collect_set_nat: fun_set_nat_o > set_set_nat ).

tff(sy_c_Set_OCollect_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
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tff(sy_c_Set_Oimage_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    image_4470545334726330049nteger: ( fun_Co8499238673658658103nteger * set_Code_integer ) > set_Code_integer ).

tff(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
    image_int_int: ( fun_int_int * set_int ) > set_int ).

tff(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
    image_int_nat: ( fun_int_nat * set_int ) > set_nat ).

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
    image_1215581382706833972nteger: ( fun_nat_Code_integer * set_nat ) > set_Code_integer ).

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
    image_nat_int: ( fun_nat_int * set_nat ) > set_int ).

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
    image_nat_nat: ( fun_nat_nat * set_nat ) > set_nat ).

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__String__Ochar,type,
    image_nat_char: ( fun_nat_char * set_nat ) > set_char ).

tff(sy_c_Set_Oimage_001t__String__Ochar_001t__Nat__Onat,type,
    image_char_nat: ( fun_char_nat * set_char ) > set_nat ).

tff(sy_c_Set_Oinsert_001t__Int__Oint,type,
    insert_int: ( int * set_int ) > set_int ).

tff(sy_c_Set_Oinsert_001t__Nat__Onat,type,
    insert_nat: ( nat * set_nat ) > set_nat ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Nat__Onat,type,
    set_fo2584398358068434914at_nat: ( fun_nat_fun_nat_nat * nat * nat ) > fun_nat_nat ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
    set_or1266510415728281911st_int: ( int * int ) > set_int ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
    set_or1269000886237332187st_nat: ( nat * nat ) > set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
    set_or4665077453230672383an_nat: ( nat * nat ) > set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
    set_ord_atMost_nat: nat > set_nat ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
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tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
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tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
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tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
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tff(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
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tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
    set_ord_lessThan_nat: nat > set_nat ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
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tff(sy_c_Signed__Division_Osigned__division__class_Osigned__divide_001t__Int__Oint,type,
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tff(sy_c_Signed__Division_Osigned__division__class_Osigned__divide_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
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tff(sy_c_Signed__Division_Osigned__division__class_Osigned__modulo_001t__Int__Oint,type,
    signed6292675348222524329lo_int: ( int * int ) > int ).

tff(sy_c_Simple__TBOUND__Cond_Ocond__TBOUND_001_Eo,type,
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tff(sy_c_Simple__TBOUND__Cond_Ocond__TBOUND_001t__List__Olist_It__Nat__Onat_J,type,
    simple8287821899582105911st_nat: ( assn * heap_T290393402774840812st_nat ) > fun_nat_o ).

tff(sy_c_Simple__TBOUND__Cond_Ocond__TBOUND_001t__Nat__Onat,type,
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tff(sy_c_Simple__TBOUND__Cond_Ocond__TBOUND_001t__Option__Ooption_It__Nat__Onat_J,type,
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tff(sy_c_Simple__TBOUND__Cond_Ocond__TBOUND_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_String_Oascii__of,type,
    ascii_of: char > char ).

tff(sy_c_String_Ochar_OChar,type,
    char2: ( $o * $o * $o * $o * $o * $o * $o ) > fun_o_char ).

tff(sy_c_String_Ochar__of__integer,type,
    char_of_integer: code_integer > char ).

tff(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
    comm_s629917340098488124ar_nat: fun_char_nat ).

tff(sy_c_String_Ointeger__of__char,type,
    integer_of_char: char > code_integer ).

tff(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
    unique3096191561947761185of_nat: fun_nat_char ).

tff(sy_c_Time__Reasoning_OTBOUND_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_Time__Reasoning_Ohtt_001_Eo,type,
    time_htt_o: ( assn * heap_Time_Heap_o * fun_o_assn ) > fun_nat_o ).

tff(sy_c_Time__Reasoning_Ohtt_001t__Option__Ooption_It__Nat__Onat_J,type,
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tff(sy_c_Time__Reasoning_Ohtt_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
    topolo4422821103128117721l_real: ( filter_real * fun_real_real ) > $o ).

tff(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
    topolo6980174941875973593q_real: fun_nat_real > $o ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
    topolo2177554685111907308n_real: ( real * set_real ) > filter_real ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
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tff(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Complex__Ocomplex,type,
    topolo6517432010174082258omplex: fun_nat_complex > $o ).

tff(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
    topolo4055970368930404560y_real: fun_nat_real > $o ).

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun_real_real ).

tff(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
    arcosh_real: fun_real_real ).

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: fun_real_real ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: fun_real_real ).

tff(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
    arsinh_real: fun_real_real ).

tff(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
    artanh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
    cos_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
    cosh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
    cot_real: fun_real_real ).

tff(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
    exp_complex: complex > complex ).

tff(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
    exp_real: fun_real_real ).

tff(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
    ln_ln_real: fun_real_real ).

tff(sy_c_Transcendental_Olog,type,
    log: real > fun_real_real ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

tff(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
    powr_real: real > fun_real_real ).

tff(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
    sin_real: fun_real_real ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
    sinh_real: fun_real_real ).

tff(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
    tan_real: fun_real_real ).

tff(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
    tanh_real: fun_real_real ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Enum__Ofinite____1,type,
    type_l31302759751748491nite_1: itself_finite_1 > nat ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Enum__Ofinite____2,type,
    type_l31302759751748492nite_2: itself_finite_2 > nat ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Enum__Ofinite____3,type,
    type_l31302759751748493nite_3: itself_finite_3 > nat ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J,type,
    type_l796852477590012082l_num1: itself8794530163899892676l_num1 > nat ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Numeral____Type__Onum1,type,
    type_l4264026598287037465l_num1: itself_Numeral_num1 > nat ).

tff(sy_c_Uint32_ORep__uint32_H,type,
    rep_uint32: fun_ui3911142380820847071l_num1 ).

tff(sy_c_Uint32_Odiv0__uint32,type,
    div0_uint32: uint32 > uint32 ).

tff(sy_c_Uint32_Ointeger__of__uint32,type,
    integer_of_uint32: fun_ui1560626348547333958nteger ).

tff(sy_c_Uint32_Ointeger__of__uint32__signed,type,
    intege5370686899274169573signed: uint32 > code_integer ).

tff(sy_c_Uint32_Omod0__uint32,type,
    mod0_uint32: uint32 > uint32 ).

tff(sy_c_Uint32_Oset__bits__aux__uint32,type,
    set_bits_aux_uint32: ( fun_nat_o * nat * uint32 ) > uint32 ).

tff(sy_c_Uint32_Osigned__drop__bit__uint32,type,
    signed489701013188660438uint32: fun_na3857091055356329634uint32 ).

tff(sy_c_Uint32_Ouint32_ORep__uint32,type,
    rep_uint322: fun_ui3911142380820847071l_num1 ).

tff(sy_c_Uint32_Ouint32__div,type,
    uint32_div: ( uint32 * uint32 ) > uint32 ).

tff(sy_c_Uint32_Ouint32__divmod,type,
    uint32_divmod: ( uint32 * uint32 ) > produc827990862158126777uint32 ).

tff(sy_c_Uint32_Ouint32__mod,type,
    uint32_mod: ( uint32 * uint32 ) > uint32 ).

tff(sy_c_Uint32_Ouint32__sdiv,type,
    uint32_sdiv: ( uint32 * uint32 ) > uint32 ).

tff(sy_c_Uint32_Ouint32__set__bit,type,
    uint32_set_bit: ( uint32 * code_integer * $o ) > uint32 ).

tff(sy_c_Uint32_Ouint32__shiftl,type,
    uint32_shiftl: ( uint32 * code_integer ) > uint32 ).

tff(sy_c_Uint32_Ouint32__shiftr,type,
    uint32_shiftr: ( uint32 * code_integer ) > uint32 ).

tff(sy_c_Uint32_Ouint32__sshiftr,type,
    uint32_sshiftr: ( uint32 * code_integer ) > uint32 ).

tff(sy_c_Uint32_Ouint32__test__bit,type,
    uint32_test_bit: ( uint32 * code_integer ) > $o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi,type,
    vEBT_V441764108873111860ildupi: fun_nat_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H,type,
    vEBT_V9176841429113362141ildupi: fun_nat_int ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H__rel,type,
    vEBT_V3352910403632780892pi_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi__rel,type,
    vEBT_V2957053500504383685pi_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb,type,
    vEBT_VEBT_Tb: fun_nat_int ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H,type,
    vEBT_VEBT_Tb2: fun_nat_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H__rel,type,
    vEBT_VEBT_Tb_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb__rel,type,
    vEBT_VEBT_Tb_rel2: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__buildupi,type,
    vEBT_vebt_buildupi: nat > heap_T8145700208782473153_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__inserti,type,
    vEBT_vebt_inserti: ( vEBT_VEBTi * nat ) > heap_T8145700208782473153_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__memberi,type,
    vEBT_vebt_memberi: vEBT_VEBTi > fun_na2343073368905879023Heap_o ).

tff(sy_c_VEBT__DelImperative_Ovebt__deletei,type,
    vEBT_vebt_deletei: ( vEBT_VEBTi * nat ) > heap_T8145700208782473153_VEBTi ).

tff(sy_c_VEBT__Example_Otest,type,
    vEBT_test: ( nat * list_nat ) > fun_li4609997795046405733st_nat ).

tff(sy_c_VEBT__Example__Setup_OmIf_001t__Nat__Onat,type,
    vEBT_Example_mIf_nat: ( heap_Time_Heap_o * heap_Time_Heap_nat * heap_Time_Heap_nat ) > heap_Time_Heap_nat ).

tff(sy_c_VEBT__Example__Setup_Omfold_001t__Nat__Onat_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_E6105538542217078229_VEBTi: ( fun_na2426279534609329326_VEBTi * list_nat ) > fun_VE4765431424076933413_VEBTi ).

tff(sy_c_VEBT__Example__Setup_Ommap_001_Eo_001_Eo,type,
    vEBT_E3732904050032093615ap_o_o: ( fun_o_8325599348076409421Heap_o * list_o ) > heap_T844314716496656296list_o ).

tff(sy_c_VEBT__Example__Setup_Ommap_001_Eo_001t__Int__Oint,type,
    vEBT_E7578125389175783381_o_int: ( fun_o_1668507913897003235ap_int * list_o ) > heap_T7023682294697889864st_int ).

tff(sy_c_VEBT__Example__Setup_Ommap_001_Eo_001t__Nat__Onat,type,
    vEBT_E7580615859684833657_o_nat: ( fun_o_4158591058828729991ap_nat * list_o ) > heap_T290393402774840812st_nat ).

tff(sy_c_VEBT__Example__Setup_Ommap_001t__Int__Oint_001_Eo,type,
    vEBT_E8862244875360331003_int_o: ( fun_in6294517997321588299Heap_o * list_int ) > heap_T844314716496656296list_o ).

tff(sy_c_VEBT__Example__Setup_Ommap_001t__Int__Oint_001t__Int__Oint,type,
    vEBT_E5210436028176674697nt_int: ( fun_in1280865887336605797ap_int * list_int ) > heap_T7023682294697889864st_int ).

tff(sy_c_VEBT__Example__Setup_Ommap_001t__Int__Oint_001t__Nat__Onat,type,
    vEBT_E5212926498685724973nt_nat: ( fun_in3770949032268332553ap_nat * list_int ) > heap_T290393402774840812st_nat ).

tff(sy_c_VEBT__Example__Setup_Ommap_001t__Nat__Onat_001_Eo,type,
    vEBT_E1459915183938243159_nat_o: ( fun_na2343073368905879023Heap_o * list_nat ) > heap_T844314716496656296list_o ).

tff(sy_c_VEBT__Example__Setup_Ommap_001t__Nat__Onat_001t__Int__Oint,type,
    vEBT_E4211082316275620141at_int: ( fun_na1043288745686030785ap_int * list_nat ) > heap_T7023682294697889864st_int ).

tff(sy_c_VEBT__Example__Setup_Ommap_001t__Nat__Onat_001t__Nat__Onat,type,
    vEBT_E4213572786784670417at_nat: fun_na3533371890617757541ap_nat > fun_li4609997795046405733st_nat ).

tff(sy_c_VEBT__Intf__Imperative_Ovebt__assn,type,
    vEBT_Intf_vebt_assn: ( nat * set_nat ) > fun_VEBT_VEBTi_assn ).

tff(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
    vEBT_VEBT_bit_concat: ( nat * nat ) > fun_nat_nat ).

tff(sy_c_VEBT__Pred_Ois__pred__in__set,type,
    vEBT_is_pred_in_set: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d,type,
    vEBT_V8646137997579335489_i_l_d: fun_nat_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p,type,
    vEBT_V8346862874174094_d_u_p: fun_nat_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p__rel,type,
    vEBT_V1247956027447740395_p_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d__rel,type,
    vEBT_V5144397997797733112_d_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__SuccPredImperative_Ovebt__predi,type,
    vEBT_vebt_predi: ( vEBT_VEBTi * nat ) > heap_T2636463487746394924on_nat ).

tff(sy_c_VEBT__SuccPredImperative_Ovebt__succi,type,
    vEBT_vebt_succi: ( vEBT_VEBTi * nat ) > heap_T2636463487746394924on_nat ).

tff(sy_c_VEBT__Succ_Ois__succ__in__set,type,
    vEBT_is_succ_in_set: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
    accp_nat: fun_nat_fun_nat_o > fun_nat_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    accp_P1096762738010456898nt_int: ( fun_Pr9018117161326028078_int_o * product_prod_int_int ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    accp_P4275260045618599050at_nat: ( fun_Pr2028404973800837934_nat_o * product_prod_nat_nat ) > $o ).

tff(sy_c_Word_Osemiring__1__class_Ounsigned_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_001t__Code____Numeral__Ointeger,type,
    semiri4716579097644248933nteger: word_N3645301735248828278l_num1 > code_integer ).

tff(sy_c_Word_Osemiring__1__class_Ounsigned_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_001t__Int__Oint,type,
    semiri7338730514057886004m1_int: word_N3645301735248828278l_num1 > int ).

tff(sy_c_Word_Osigned__drop__bit_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J,type,
    signed5000768011106662067l_num1: ( nat * word_N3645301735248828278l_num1 ) > word_N3645301735248828278l_num1 ).

tff(sy_c_aa_001_Eo_001_Eo,type,
    aa_o_o: ( fun_o_o * $o ) > $o ).

tff(sy_c_aa_001_Eo_001t__Assertions__Oassn,type,
    aa_o_assn: ( fun_o_assn * $o ) > assn ).

tff(sy_c_aa_001_Eo_001t__Heap____Time____Monad__OHeap_I_Eo_J,type,
    aa_o_H7399033255965945470Heap_o: ( fun_o_8325599348076409421Heap_o * $o ) > heap_Time_Heap_o ).

tff(sy_c_aa_001_Eo_001t__Heap____Time____Monad__OHeap_It__Int__Oint_J,type,
    aa_o_H4561945182497200476ap_int: ( fun_o_1668507913897003235ap_int * $o ) > heap_Time_Heap_int ).

tff(sy_c_aa_001_Eo_001t__Heap____Time____Monad__OHeap_It__List__Olist_I_Eo_J_J,type,
    aa_o_H3316518667072757508list_o: ( fun_o_8145193230459182547list_o * $o ) > heap_T844314716496656296list_o ).

tff(sy_c_aa_001_Eo_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J,type,
    aa_o_H8739796202006397184ap_nat: ( fun_o_4158591058828729991ap_nat * $o ) > heap_Time_Heap_nat ).

tff(sy_c_aa_001_Eo_001t__Int__Oint,type,
    aa_o_int: ( fun_o_int * $o ) > int ).

tff(sy_c_aa_001_Eo_001t__String__Ochar,type,
    aa_o_char: ( fun_o_char * $o ) > char ).

tff(sy_c_aa_001_Eo_001t__Uint32__Ouint32,type,
    aa_o_uint32: ( fun_o_uint32 * $o ) > uint32 ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_M_Eo_J,type,
    aa_o_fun_o_o: ( fun_o_fun_o_o * $o ) > fun_o_o ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__Code____Numeral__Ointeger_Mt__fun_I_Eo_Mt__String__Ochar_J_J_J_J_J_J_J,type,
    aa_o_f3038258172971468011o_char: ( fun_o_5488095450495718970o_char * $o ) > fun_o_6360995121869435791o_char ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__Code____Numeral__Ointeger_Mt__fun_I_Eo_Mt__String__Ochar_J_J_J_J_J_J,type,
    aa_o_f1143307120959153672o_char: ( fun_o_6360995121869435791o_char * $o ) > fun_o_2683029531876921444o_char ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__Code____Numeral__Ointeger_Mt__fun_I_Eo_Mt__String__Ochar_J_J_J_J_J,type,
    aa_o_f2932186792858379797o_char: ( fun_o_2683029531876921444o_char * $o ) > fun_o_3037882179896959673o_char ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_I_Eo_Mt__fun_It__Code____Numeral__Ointeger_Mt__fun_I_Eo_Mt__String__Ochar_J_J_J_J,type,
    aa_o_f4183413333637126194o_char: ( fun_o_3037882179896959673o_char * $o ) > fun_o_335877185394996366o_char ).

tff(sy_c_aa_001_Eo_001t__fun_I_Eo_Mt__fun_It__Code____Numeral__Ointeger_Mt__fun_I_Eo_Mt__String__Ochar_J_J_J,type,
    aa_o_f9003794812476387647o_char: ( fun_o_335877185394996366o_char * $o ) > fun_o_2016307092959984611o_char ).

tff(sy_c_aa_001_Eo_001t__fun_It__Code____Numeral__Ointeger_Mt__fun_I_Eo_Mt__String__Ochar_J_J,type,
    aa_o_f2742674633914528860o_char: ( fun_o_2016307092959984611o_char * $o ) > fun_Co681828205884031672o_char ).

tff(sy_c_aa_001_Eo_001t__fun_It__Int__Oint_M_Eo_J,type,
    aa_o_fun_int_o: ( fun_o_fun_int_o * $o ) > fun_int_o ).

tff(sy_c_aa_001_Eo_001t__fun_It__Nat__Onat_M_Eo_J,type,
    aa_o_fun_nat_o: ( fun_o_fun_nat_o * $o ) > fun_nat_o ).

tff(sy_c_aa_001_Eo_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    aa_o_fun_nat_nat: ( fun_o_fun_nat_nat * $o ) > fun_nat_nat ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001_Eo,type,
    aa_Code_integer_o: ( fun_Code_integer_o * code_integer ) > $o ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    aa_Cod1783974652533396728nteger: ( fun_Co8499238673658658103nteger * code_integer ) > code_integer ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
    aa_Code_integer_int: ( fun_Code_integer_int * code_integer ) > int ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
    aa_Code_integer_nat: ( fun_Code_integer_nat * code_integer ) > nat ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
    aa_Code_integer_num: ( fun_Code_integer_num * code_integer ) > num ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
    aa_Cod3168588076859124107eger_o: ( fun_Co1766037875893669450eger_o * code_integer ) > produc6271795597528267376eger_o ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    aa_Cod6977894654539884864nteger: ( fun_Co6597186368970283863nteger * code_integer ) > produc8923325533196201883nteger ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Uint32__Ouint32,type,
    aa_Cod8401701657498852871uint32: ( fun_Co1554742298434871494uint32 * code_integer ) > uint32 ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_I_Eo_Mt__String__Ochar_J,type,
    aa_Cod7842134615204088097o_char: ( fun_Co681828205884031672o_char * code_integer ) > fun_o_char ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_I_Eo_Mt__Uint32__Ouint32_J,type,
    aa_Cod7056071456392079164uint32: ( fun_Co8473515836739849555uint32 * code_integer ) > fun_o_uint32 ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    aa_Cod4161879620478718930nteger: ( fun_Co1981900195452921617nteger * code_integer ) > fun_Co8499238673658658103nteger ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Int__Oint_J,type,
    aa_Cod1242632294114111009er_int: ( fun_Co1092983284983208416er_int * code_integer ) > fun_Code_integer_int ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Nat__Onat_J,type,
    aa_Cod5420483313623307717er_nat: ( fun_Co3583066429914935172er_nat * code_integer ) > fun_Code_integer_nat ).

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_It__Code____Numeral__Ointeger_Mt__Num__Onum_J,type,
    aa_Cod6128727250572145423er_num: ( fun_Co8521658056209057742er_num * code_integer ) > fun_Code_integer_num ).

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tff(sy_c_aa_001t__fun_It__Option__Ooption_It__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__Nat__Onat_J_J_001t__fun_It__Option__Ooption_It__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__Nat__Onat_J_J,type,
    aa_fun7982982223196242428ap_nat: ( fun_fu6764168361233458435ap_nat * fun_op1688020532487384469ap_nat ) > fun_op1688020532487384469ap_nat ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J_001t__fun_It__Real__Oreal_M_Eo_J,type,
    aa_fun2895569901277098738real_o: ( fun_fu3022329567491847689real_o * fun_real_real ) > fun_real_o ).

tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J,type,
    aa_fun6946558556880649704l_real: ( fun_fu3077872570260440615l_real * fun_real_real ) > fun_real_real ).

tff(sy_c_aa_001t__fun_It__Uint32__Ouint32_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Uint32__Ouint32_Mt__Code____Numeral__Ointeger_J,type,
    aa_fun3533256999457416010nteger: ( fun_fu5166283944260750601nteger * fun_ui1560626348547333958nteger ) > fun_ui1560626348547333958nteger ).

tff(sy_c_aa_001t__fun_It__Uint32__Ouint32_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__fun_It__Uint32__Ouint32_Mt__fun_It__Code____Numeral__Ointeger_M_Eo_J_J,type,
    aa_fun5262225984791465967eger_o: ( fun_fu3466578569608634614eger_o * fun_uint32_fun_nat_o ) > fun_ui2235522947382672961eger_o ).

tff(sy_c_aa_001t__fun_It__Uint32__Ouint32_Mt__fun_It__Nat__Onat_Mt__fun_I_Eo_Mt__Uint32__Ouint32_J_J_J_001t__fun_It__Uint32__Ouint32_Mt__fun_It__Code____Numeral__Ointeger_Mt__fun_I_Eo_Mt__Uint32__Ouint32_J_J_J,type,
    aa_fun8001877339868165167uint32: ( fun_fu8139443716560175030uint32 * fun_ui928617633769132301uint32 ) > fun_ui4964537544116621120uint32 ).

tff(sy_c_aa_001t__fun_It__Uint32__Ouint32_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_001t__fun_It__Uint32__Ouint32_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J,type,
    aa_fun8139942314387009082uint32: ( fun_fu6749172196426063865uint32 * fun_ui2637848667930195262uint32 ) > fun_ui2637848667930195262uint32 ).

tff(sy_c_aa_001t__fun_It__VEBT____BuildupMemImp__OVEBTi_Mt__Heap____Time____Monad__OHeap_It__List__Olist_It__Nat__Onat_J_J_J_001t__fun_It__VEBT____BuildupMemImp__OVEBTi_Mt__Heap____Time____Monad__OHeap_It__List__Olist_It__Nat__Onat_J_J_J,type,
    aa_fun7495903629695767740st_nat: ( fun_fu8993078228354513475st_nat * fun_VE6418762861507722832st_nat ) > fun_VE6418762861507722832st_nat ).

tff(sy_c_fChoice_001t__Real__Oreal,type,
    fChoice_real: fun_real_o > real ).

tff(sy_c_member_001t__Complex__Ocomplex,type,
    member_complex: ( complex * set_complex ) > $o ).

tff(sy_c_member_001t__Int__Oint,type,
    member_int: ( int * set_int ) > $o ).

tff(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
    member_list_nat: ( list_nat * set_list_nat ) > $o ).

tff(sy_c_member_001t__Nat__Onat,type,
    member_nat: nat > fun_set_nat_o ).

tff(sy_c_member_001t__Real__Oreal,type,
    member_real: ( real * set_real ) > $o ).

tff(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
    member_set_nat: ( set_nat * set_set_nat ) > $o ).

tff(sy_c_member_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    member890509984587605005l_num1: ( word_N3645301735248828278l_num1 * set_wo3913738467083021356l_num1 ) > $o ).

tff(sy_v_S,type,
    s: set_nat ).

tff(sy_v_a____,type,
    a: nat ).

tff(sy_v_b,type,
    b: list_nat > fun_nat_nat ).

tff(sy_v_n,type,
    n: nat ).

tff(sy_v_ti,type,
    ti: vEBT_VEBTi ).

tff(sy_v_ysa____,type,
    ysa: list_nat ).

% Relevant facts (3333)
tff(fact_0_ceiling__add__numeral,axiom,
    ! [X: real,V: num] : archim7802044766580827645g_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),numeral_numeral_real(V))) = plus_plus_int(archim7802044766580827645g_real(X),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_1_ceiling__add__numeral,axiom,
    ! [X: rat,V: num] : archim2889992004027027881ng_rat(plus_plus_rat(X,numeral_numeral_rat(V))) = plus_plus_int(archim2889992004027027881ng_rat(X),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_2_ceiling__le__numeral,axiom,
    ! [X: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim7802044766580827645g_real(X)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),numeral_numeral_real(V)) ) ).

% ceiling_le_numeral
tff(fact_3_ceiling__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(X)),aa_num_int(numeral_numeral_int,V))
    <=> ord_less_eq_rat(X,numeral_numeral_rat(V)) ) ).

% ceiling_le_numeral
tff(fact_4_of__nat__mult,axiom,
    ! [M: nat,Na: nat] : semiri681578069525770553at_rat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = times_times_rat(semiri681578069525770553at_rat(M),semiri681578069525770553at_rat(Na)) ).

% of_nat_mult
tff(fact_5_of__nat__mult,axiom,
    ! [M: nat,Na: nat] : semiri2565882477558803405uint32(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = times_times_uint32(semiri2565882477558803405uint32(M),semiri2565882477558803405uint32(Na)) ).

% of_nat_mult
tff(fact_6_of__nat__mult,axiom,
    ! [M: nat,Na: nat] : semiri8819519690708144855l_num1(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = times_7065122842183080059l_num1(semiri8819519690708144855l_num1(M),semiri8819519690708144855l_num1(Na)) ).

% of_nat_mult
tff(fact_7_of__nat__mult,axiom,
    ! [M: nat,Na: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Na)) ).

% of_nat_mult
tff(fact_8_of__nat__mult,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ).

% of_nat_mult
tff(fact_9_of__nat__mult,axiom,
    ! [M: nat,Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = times_times_int(aa_nat_int(semiri1314217659103216013at_int,M),aa_nat_int(semiri1314217659103216013at_int,Na)) ).

% of_nat_mult
tff(fact_10_of__nat__add,axiom,
    ! [M: nat,Na: nat] : semiri681578069525770553at_rat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = plus_plus_rat(semiri681578069525770553at_rat(M),semiri681578069525770553at_rat(Na)) ).

% of_nat_add
tff(fact_11_of__nat__add,axiom,
    ! [M: nat,Na: nat] : semiri2565882477558803405uint32(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = plus_plus_uint32(semiri2565882477558803405uint32(M),semiri2565882477558803405uint32(Na)) ).

% of_nat_add
tff(fact_12_of__nat__add,axiom,
    ! [M: nat,Na: nat] : semiri8819519690708144855l_num1(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = plus_p361126936061061375l_num1(semiri8819519690708144855l_num1(M),semiri8819519690708144855l_num1(Na)) ).

% of_nat_add
tff(fact_13_of__nat__add,axiom,
    ! [M: nat,Na: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Na)) ).

% of_nat_add
tff(fact_14_of__nat__add,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ).

% of_nat_add
tff(fact_15_of__nat__add,axiom,
    ! [M: nat,Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,M),aa_nat_int(semiri1314217659103216013at_int,Na)) ).

% of_nat_add
tff(fact_16_of__nat__le__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% of_nat_le_iff
tff(fact_17_of__nat__le__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ord_less_eq_rat(semiri681578069525770553at_rat(M),semiri681578069525770553at_rat(Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% of_nat_le_iff
tff(fact_18_of__nat__le__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% of_nat_le_iff
tff(fact_19_of__nat__le__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% of_nat_le_iff
tff(fact_20_of__nat__numeral,axiom,
    ! [Na: num] : semiri681578069525770553at_rat(numeral_numeral_nat(Na)) = numeral_numeral_rat(Na) ).

% of_nat_numeral
tff(fact_21_of__nat__numeral,axiom,
    ! [Na: num] : semiri2565882477558803405uint32(numeral_numeral_nat(Na)) = numera9087168376688890119uint32(Na) ).

% of_nat_numeral
tff(fact_22_of__nat__numeral,axiom,
    ! [Na: num] : semiri8819519690708144855l_num1(numeral_numeral_nat(Na)) = numera7442385471795722001l_num1(Na) ).

% of_nat_numeral
tff(fact_23_of__nat__numeral,axiom,
    ! [Na: num] : aa_nat_real(semiri5074537144036343181t_real,numeral_numeral_nat(Na)) = numeral_numeral_real(Na) ).

% of_nat_numeral
tff(fact_24_of__nat__numeral,axiom,
    ! [Na: num] : aa_nat_nat(semiri1316708129612266289at_nat,numeral_numeral_nat(Na)) = numeral_numeral_nat(Na) ).

% of_nat_numeral
tff(fact_25_of__nat__numeral,axiom,
    ! [Na: num] : aa_nat_int(semiri1314217659103216013at_int,numeral_numeral_nat(Na)) = aa_num_int(numeral_numeral_int,Na) ).

% of_nat_numeral
tff(fact_26_distrib__left__numeral,axiom,
    ! [V: num,Ba: uint32,C: uint32] : times_times_uint32(numera9087168376688890119uint32(V),plus_plus_uint32(Ba,C)) = plus_plus_uint32(times_times_uint32(numera9087168376688890119uint32(V),Ba),times_times_uint32(numera9087168376688890119uint32(V),C)) ).

% distrib_left_numeral
tff(fact_27_distrib__left__numeral,axiom,
    ! [V: num,Ba: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] : times_7065122842183080059l_num1(numera7442385471795722001l_num1(V),plus_p361126936061061375l_num1(Ba,C)) = plus_p361126936061061375l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(V),Ba),times_7065122842183080059l_num1(numera7442385471795722001l_num1(V),C)) ).

% distrib_left_numeral
tff(fact_28_distrib__left__numeral,axiom,
    ! [V: num,Ba: real,C: real] : aa_real_real(times_times_real(numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(numeral_numeral_real(V)),Ba)),aa_real_real(times_times_real(numeral_numeral_real(V)),C)) ).

% distrib_left_numeral
tff(fact_29_distrib__left__numeral,axiom,
    ! [V: num,Ba: rat,C: rat] : times_times_rat(numeral_numeral_rat(V),plus_plus_rat(Ba,C)) = plus_plus_rat(times_times_rat(numeral_numeral_rat(V),Ba),times_times_rat(numeral_numeral_rat(V),C)) ).

% distrib_left_numeral
tff(fact_30_distrib__left__numeral,axiom,
    ! [V: num,Ba: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(V)),Ba)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(V)),C)) ).

% distrib_left_numeral
tff(fact_31_distrib__left__numeral,axiom,
    ! [V: num,Ba: int,C: int] : times_times_int(aa_num_int(numeral_numeral_int,V),plus_plus_int(Ba,C)) = plus_plus_int(times_times_int(aa_num_int(numeral_numeral_int,V),Ba),times_times_int(aa_num_int(numeral_numeral_int,V),C)) ).

% distrib_left_numeral
tff(fact_32_distrib__right__numeral,axiom,
    ! [Aa: uint32,Ba: uint32,V: num] : times_times_uint32(plus_plus_uint32(Aa,Ba),numera9087168376688890119uint32(V)) = plus_plus_uint32(times_times_uint32(Aa,numera9087168376688890119uint32(V)),times_times_uint32(Ba,numera9087168376688890119uint32(V))) ).

% distrib_right_numeral
tff(fact_33_distrib__right__numeral,axiom,
    ! [Aa: word_N3645301735248828278l_num1,Ba: word_N3645301735248828278l_num1,V: num] : times_7065122842183080059l_num1(plus_p361126936061061375l_num1(Aa,Ba),numera7442385471795722001l_num1(V)) = plus_p361126936061061375l_num1(times_7065122842183080059l_num1(Aa,numera7442385471795722001l_num1(V)),times_7065122842183080059l_num1(Ba,numera7442385471795722001l_num1(V))) ).

% distrib_right_numeral
tff(fact_34_distrib__right__numeral,axiom,
    ! [Aa: real,Ba: real,V: num] : aa_real_real(times_times_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)),numeral_numeral_real(V)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(Aa),numeral_numeral_real(V))),aa_real_real(times_times_real(Ba),numeral_numeral_real(V))) ).

% distrib_right_numeral
tff(fact_35_distrib__right__numeral,axiom,
    ! [Aa: rat,Ba: rat,V: num] : times_times_rat(plus_plus_rat(Aa,Ba),numeral_numeral_rat(V)) = plus_plus_rat(times_times_rat(Aa,numeral_numeral_rat(V)),times_times_rat(Ba,numeral_numeral_rat(V))) ).

% distrib_right_numeral
tff(fact_36_distrib__right__numeral,axiom,
    ! [Aa: nat,Ba: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba)),numeral_numeral_nat(V)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),numeral_numeral_nat(V))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),numeral_numeral_nat(V))) ).

% distrib_right_numeral
tff(fact_37_distrib__right__numeral,axiom,
    ! [Aa: int,Ba: int,V: num] : times_times_int(plus_plus_int(Aa,Ba),aa_num_int(numeral_numeral_int,V)) = plus_plus_int(times_times_int(Aa,aa_num_int(numeral_numeral_int,V)),times_times_int(Ba,aa_num_int(numeral_numeral_int,V))) ).

% distrib_right_numeral
tff(fact_38_pred__time__pure,axiom,
    ! [Na: nat,Sa: set_nat,Tia: vEBT_VEBTi,Aa: nat] : aa_nat_o(simple9182585030592575607on_nat(aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Na,Sa),Tia),vEBT_vebt_predi(Tia,Aa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit1(bit1(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit1(bit1(one)))),aa_int_nat(nat2,archim7802044766580827645g_real(aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Na))))))) ).

% pred_time_pure
tff(fact_39_member__time__pure,axiom,
    ! [Na: nat,Sa: set_nat,Tia: vEBT_VEBTi,Aa: nat] : aa_nat_o(simple83843512214091265OUND_o(aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Na,Sa),Tia),aa_nat2910048432523430448Heap_o(vEBT_vebt_memberi(Tia),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit1(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit1(bit0(one)))),aa_int_nat(nat2,archim7802044766580827645g_real(aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Na))))))) ).

% member_time_pure
tff(fact_40_ceiling__of__nat,axiom,
    ! [Na: nat] : archim7802044766580827645g_real(aa_nat_real(semiri5074537144036343181t_real,Na)) = aa_nat_int(semiri1314217659103216013at_int,Na) ).

% ceiling_of_nat
tff(fact_41_ceiling__of__nat,axiom,
    ! [Na: nat] : archim2889992004027027881ng_rat(semiri681578069525770553at_rat(Na)) = aa_nat_int(semiri1314217659103216013at_int,Na) ).

% ceiling_of_nat
tff(fact_42_ceiling__numeral,axiom,
    ! [V: num] : archim7802044766580827645g_real(numeral_numeral_real(V)) = aa_num_int(numeral_numeral_int,V) ).

% ceiling_numeral
tff(fact_43_ceiling__numeral,axiom,
    ! [V: num] : archim2889992004027027881ng_rat(numeral_numeral_rat(V)) = aa_num_int(numeral_numeral_int,V) ).

% ceiling_numeral
tff(fact_44_numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( numeral_numeral_real(M) = numeral_numeral_real(Na) )
    <=> ( M = Na ) ) ).

% numeral_eq_iff
tff(fact_45_numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( numeral_numeral_rat(M) = numeral_numeral_rat(Na) )
    <=> ( M = Na ) ) ).

% numeral_eq_iff
tff(fact_46_numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( numeral_numeral_nat(M) = numeral_numeral_nat(Na) )
    <=> ( M = Na ) ) ).

% numeral_eq_iff
tff(fact_47_numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( aa_num_int(numeral_numeral_int,M) = aa_num_int(numeral_numeral_int,Na) )
    <=> ( M = Na ) ) ).

% numeral_eq_iff
tff(fact_48_of__nat__eq__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(semiri5074537144036343181t_real,Na) )
    <=> ( M = Na ) ) ).

% of_nat_eq_iff
tff(fact_49_of__nat__eq__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M) = aa_nat_nat(semiri1316708129612266289at_nat,Na) )
    <=> ( M = Na ) ) ).

% of_nat_eq_iff
tff(fact_50_of__nat__eq__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_nat_int(semiri1314217659103216013at_int,Na) )
    <=> ( M = Na ) ) ).

% of_nat_eq_iff
tff(fact_51_of__nat__eq__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( semiri681578069525770553at_rat(M) = semiri681578069525770553at_rat(Na) )
    <=> ( M = Na ) ) ).

% of_nat_eq_iff
tff(fact_52_numeral__le__iff,axiom,
    ! [M: num,Na: num] :
      ( ord_less_eq_rat(numeral_numeral_rat(M),numeral_numeral_rat(Na))
    <=> aa_num_o(ord_less_eq_num(M),Na) ) ).

% numeral_le_iff
tff(fact_53_numeral__le__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(M)),numeral_numeral_nat(Na))
    <=> aa_num_o(ord_less_eq_num(M),Na) ) ).

% numeral_le_iff
tff(fact_54_numeral__le__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na))
    <=> aa_num_o(ord_less_eq_num(M),Na) ) ).

% numeral_le_iff
tff(fact_55_numeral__le__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(M)),numeral_numeral_real(Na))
    <=> aa_num_o(ord_less_eq_num(M),Na) ) ).

% numeral_le_iff
tff(fact_56_add__numeral__left,axiom,
    ! [V: num,W: num,Z: word_N3645301735248828278l_num1] : plus_p361126936061061375l_num1(numera7442385471795722001l_num1(V),plus_p361126936061061375l_num1(numera7442385471795722001l_num1(W),Z)) = plus_p361126936061061375l_num1(numera7442385471795722001l_num1(plus_plus_num(V,W)),Z) ).

% add_numeral_left
tff(fact_57_add__numeral__left,axiom,
    ! [V: num,W: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(W)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(plus_plus_num(V,W))),Z) ).

% add_numeral_left
tff(fact_58_add__numeral__left,axiom,
    ! [V: num,W: num,Z: rat] : plus_plus_rat(numeral_numeral_rat(V),plus_plus_rat(numeral_numeral_rat(W),Z)) = plus_plus_rat(numeral_numeral_rat(plus_plus_num(V,W)),Z) ).

% add_numeral_left
tff(fact_59_add__numeral__left,axiom,
    ! [V: num,W: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(W)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(plus_plus_num(V,W))),Z) ).

% add_numeral_left
tff(fact_60_add__numeral__left,axiom,
    ! [V: num,W: num,Z: int] : plus_plus_int(aa_num_int(numeral_numeral_int,V),plus_plus_int(aa_num_int(numeral_numeral_int,W),Z)) = plus_plus_int(aa_num_int(numeral_numeral_int,plus_plus_num(V,W)),Z) ).

% add_numeral_left
tff(fact_61_add__numeral__left,axiom,
    ! [V: num,W: num,Z: uint32] : plus_plus_uint32(numera9087168376688890119uint32(V),plus_plus_uint32(numera9087168376688890119uint32(W),Z)) = plus_plus_uint32(numera9087168376688890119uint32(plus_plus_num(V,W)),Z) ).

% add_numeral_left
tff(fact_62_numeral__plus__numeral,axiom,
    ! [M: num,Na: num] : plus_p361126936061061375l_num1(numera7442385471795722001l_num1(M),numera7442385471795722001l_num1(Na)) = numera7442385471795722001l_num1(plus_plus_num(M,Na)) ).

% numeral_plus_numeral
tff(fact_63_numeral__plus__numeral,axiom,
    ! [M: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(M)),numeral_numeral_real(Na)) = numeral_numeral_real(plus_plus_num(M,Na)) ).

% numeral_plus_numeral
tff(fact_64_numeral__plus__numeral,axiom,
    ! [M: num,Na: num] : plus_plus_rat(numeral_numeral_rat(M),numeral_numeral_rat(Na)) = numeral_numeral_rat(plus_plus_num(M,Na)) ).

% numeral_plus_numeral
tff(fact_65_numeral__plus__numeral,axiom,
    ! [M: num,Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(M)),numeral_numeral_nat(Na)) = numeral_numeral_nat(plus_plus_num(M,Na)) ).

% numeral_plus_numeral
tff(fact_66_numeral__plus__numeral,axiom,
    ! [M: num,Na: num] : plus_plus_int(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,Na)) = aa_num_int(numeral_numeral_int,plus_plus_num(M,Na)) ).

% numeral_plus_numeral
tff(fact_67_numeral__plus__numeral,axiom,
    ! [M: num,Na: num] : plus_plus_uint32(numera9087168376688890119uint32(M),numera9087168376688890119uint32(Na)) = numera9087168376688890119uint32(plus_plus_num(M,Na)) ).

% numeral_plus_numeral
tff(fact_68_num__double,axiom,
    ! [Na: num] : times_times_num(bit0(one),Na) = bit0(Na) ).

% num_double
tff(fact_69_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: word_N3645301735248828278l_num1] : times_7065122842183080059l_num1(numera7442385471795722001l_num1(V),times_7065122842183080059l_num1(numera7442385471795722001l_num1(W),Z)) = times_7065122842183080059l_num1(numera7442385471795722001l_num1(times_times_num(V,W)),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_70_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: real] : aa_real_real(times_times_real(numeral_numeral_real(V)),aa_real_real(times_times_real(numeral_numeral_real(W)),Z)) = aa_real_real(times_times_real(numeral_numeral_real(times_times_num(V,W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_71_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: rat] : times_times_rat(numeral_numeral_rat(V),times_times_rat(numeral_numeral_rat(W),Z)) = times_times_rat(numeral_numeral_rat(times_times_num(V,W)),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_72_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(V)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(W)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(times_times_num(V,W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_73_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: int] : times_times_int(aa_num_int(numeral_numeral_int,V),times_times_int(aa_num_int(numeral_numeral_int,W),Z)) = times_times_int(aa_num_int(numeral_numeral_int,times_times_num(V,W)),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_74_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: uint32] : times_times_uint32(numera9087168376688890119uint32(V),times_times_uint32(numera9087168376688890119uint32(W),Z)) = times_times_uint32(numera9087168376688890119uint32(times_times_num(V,W)),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_75_numeral__times__numeral,axiom,
    ! [M: num,Na: num] : times_7065122842183080059l_num1(numera7442385471795722001l_num1(M),numera7442385471795722001l_num1(Na)) = numera7442385471795722001l_num1(times_times_num(M,Na)) ).

% numeral_times_numeral
tff(fact_76_numeral__times__numeral,axiom,
    ! [M: num,Na: num] : aa_real_real(times_times_real(numeral_numeral_real(M)),numeral_numeral_real(Na)) = numeral_numeral_real(times_times_num(M,Na)) ).

% numeral_times_numeral
tff(fact_77_numeral__times__numeral,axiom,
    ! [M: num,Na: num] : times_times_rat(numeral_numeral_rat(M),numeral_numeral_rat(Na)) = numeral_numeral_rat(times_times_num(M,Na)) ).

% numeral_times_numeral
tff(fact_78_numeral__times__numeral,axiom,
    ! [M: num,Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(M)),numeral_numeral_nat(Na)) = numeral_numeral_nat(times_times_num(M,Na)) ).

% numeral_times_numeral
tff(fact_79_numeral__times__numeral,axiom,
    ! [M: num,Na: num] : times_times_int(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,Na)) = aa_num_int(numeral_numeral_int,times_times_num(M,Na)) ).

% numeral_times_numeral
tff(fact_80_numeral__times__numeral,axiom,
    ! [M: num,Na: num] : times_times_uint32(numera9087168376688890119uint32(M),numera9087168376688890119uint32(Na)) = numera9087168376688890119uint32(times_times_num(M,Na)) ).

% numeral_times_numeral
tff(fact_81_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% nat_add_left_cancel_le
tff(fact_82_add__One__commute,axiom,
    ! [Na: num] : plus_plus_num(one,Na) = plus_plus_num(Na,one) ).

% add_One_commute
tff(fact_83_le__num__One__iff,axiom,
    ! [X: num] :
      ( aa_num_o(ord_less_eq_num(X),one)
    <=> ( X = one ) ) ).

% le_num_One_iff
tff(fact_84_ceiling__add__le,axiom,
    ! [X: real,Y: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim7802044766580827645g_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),plus_plus_int(archim7802044766580827645g_real(X),archim7802044766580827645g_real(Y))) ).

% ceiling_add_le
tff(fact_85_ceiling__add__le,axiom,
    ! [X: rat,Y: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(plus_plus_rat(X,Y))),plus_plus_int(archim2889992004027027881ng_rat(X),archim2889992004027027881ng_rat(Y))) ).

% ceiling_add_le
tff(fact_86_is__num__normalize_I1_J,axiom,
    ! [Aa: real,Ba: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),C)) ).

% is_num_normalize(1)
tff(fact_87_is__num__normalize_I1_J,axiom,
    ! [Aa: rat,Ba: rat,C: rat] : plus_plus_rat(plus_plus_rat(Aa,Ba),C) = plus_plus_rat(Aa,plus_plus_rat(Ba,C)) ).

% is_num_normalize(1)
tff(fact_88_is__num__normalize_I1_J,axiom,
    ! [Aa: int,Ba: int,C: int] : plus_plus_int(plus_plus_int(Aa,Ba),C) = plus_plus_int(Aa,plus_plus_int(Ba,C)) ).

% is_num_normalize(1)
tff(fact_89_is__num__normalize_I1_J,axiom,
    ! [Aa: word_N3645301735248828278l_num1,Ba: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] : plus_p361126936061061375l_num1(plus_p361126936061061375l_num1(Aa,Ba),C) = plus_p361126936061061375l_num1(Aa,plus_p361126936061061375l_num1(Ba,C)) ).

% is_num_normalize(1)
tff(fact_90_is__num__normalize_I1_J,axiom,
    ! [Aa: uint32,Ba: uint32,C: uint32] : plus_plus_uint32(plus_plus_uint32(Aa,Ba),C) = plus_plus_uint32(Aa,plus_plus_uint32(Ba,C)) ).

% is_num_normalize(1)
tff(fact_91_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun_nat_o,K: nat,Ba: nat] :
      ( aa_nat_o(P,K)
     => ( ! [Y2: nat] :
            ( aa_nat_o(P,Y2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),Ba) )
       => ? [X2: nat] :
            ( aa_nat_o(P,X2)
            & ! [Y3: nat] :
                ( aa_nat_o(P,Y3)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),X2) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_92_nat__le__linear,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M) ) ).

% nat_le_linear
tff(fact_93_le__antisym,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( M = Na ) ) ) ).

% le_antisym
tff(fact_94_eq__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( ( M = Na )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% eq_imp_le
tff(fact_95_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K) ) ) ).

% le_trans
tff(fact_96_le__refl,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Na) ).

% le_refl
tff(fact_97_real__arch__simple,axiom,
    ! [X: rat] :
    ? [N: nat] : ord_less_eq_rat(X,semiri681578069525770553at_rat(N)) ).

% real_arch_simple
tff(fact_98_real__arch__simple,axiom,
    ! [X: real] :
    ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% real_arch_simple
tff(fact_99_mult__of__nat__commute,axiom,
    ! [X: nat,Y: uint32] : times_times_uint32(semiri2565882477558803405uint32(X),Y) = times_times_uint32(Y,semiri2565882477558803405uint32(X)) ).

% mult_of_nat_commute
tff(fact_100_mult__of__nat__commute,axiom,
    ! [X: nat,Y: word_N3645301735248828278l_num1] : times_7065122842183080059l_num1(semiri8819519690708144855l_num1(X),Y) = times_7065122842183080059l_num1(Y,semiri8819519690708144855l_num1(X)) ).

% mult_of_nat_commute
tff(fact_101_mult__of__nat__commute,axiom,
    ! [X: nat,Y: real] : aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,X)),Y) = aa_real_real(times_times_real(Y),aa_nat_real(semiri5074537144036343181t_real,X)) ).

% mult_of_nat_commute
tff(fact_102_mult__of__nat__commute,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(semiri1316708129612266289at_nat,X)) ).

% mult_of_nat_commute
tff(fact_103_mult__of__nat__commute,axiom,
    ! [X: nat,Y: int] : times_times_int(aa_nat_int(semiri1314217659103216013at_int,X),Y) = times_times_int(Y,aa_nat_int(semiri1314217659103216013at_int,X)) ).

% mult_of_nat_commute
tff(fact_104_mult__of__nat__commute,axiom,
    ! [X: nat,Y: rat] : times_times_rat(semiri681578069525770553at_rat(X),Y) = times_times_rat(Y,semiri681578069525770553at_rat(X)) ).

% mult_of_nat_commute
tff(fact_105_nat__le__iff__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
    <=> ? [K2: nat] : Na = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2) ) ).

% nat_le_iff_add
tff(fact_106_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J)) ) ).

% trans_le_add2
tff(fact_107_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M)) ) ).

% trans_le_add1
tff(fact_108_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ) ).

% add_le_mono1
tff(fact_109_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_le_mono
tff(fact_110_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
     => ? [N: nat] : L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N) ) ).

% le_Suc_ex
tff(fact_111_add__leD2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na) ) ).

% add_leD2
tff(fact_112_add__leD1,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% add_leD1
tff(fact_113_le__add2,axiom,
    ! [Na: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) ).

% le_add2
tff(fact_114_le__add1,axiom,
    ! [Na: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)) ).

% le_add1
tff(fact_115_add__leE,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),Na)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na) ) ) ).

% add_leE
tff(fact_116_mem__Collect__eq,axiom,
    ! [Aa: real,P: fun_real_o] :
      ( member_real(Aa,collect_real(P))
    <=> aa_real_o(P,Aa) ) ).

% mem_Collect_eq
tff(fact_117_mem__Collect__eq,axiom,
    ! [Aa: word_N3645301735248828278l_num1,P: fun_wo7859709073469921915num1_o] :
      ( member890509984587605005l_num1(Aa,collec7814023847061821259l_num1(P))
    <=> aa_wor4452366487969921764num1_o(P,Aa) ) ).

% mem_Collect_eq
tff(fact_118_mem__Collect__eq,axiom,
    ! [Aa: complex,P: fun_complex_o] :
      ( member_complex(Aa,collect_complex(P))
    <=> aa_complex_o(P,Aa) ) ).

% mem_Collect_eq
tff(fact_119_mem__Collect__eq,axiom,
    ! [Aa: list_nat,P: fun_list_nat_o] :
      ( member_list_nat(Aa,collect_list_nat(P))
    <=> aa_list_nat_o(P,Aa) ) ).

% mem_Collect_eq
tff(fact_120_mem__Collect__eq,axiom,
    ! [Aa: set_nat,P: fun_set_nat_o] :
      ( member_set_nat(Aa,collect_set_nat(P))
    <=> aa_set_nat_o(P,Aa) ) ).

% mem_Collect_eq
tff(fact_121_mem__Collect__eq,axiom,
    ! [Aa: nat,P: fun_nat_o] :
      ( aa_set_nat_o(member_nat(Aa),collect_nat(P))
    <=> aa_nat_o(P,Aa) ) ).

% mem_Collect_eq
tff(fact_122_mem__Collect__eq,axiom,
    ! [Aa: int,P: fun_int_o] :
      ( member_int(Aa,collect_int(P))
    <=> aa_int_o(P,Aa) ) ).

% mem_Collect_eq
tff(fact_123_Collect__mem__eq,axiom,
    ! [A: set_real] : collect_real(aTP_Lamm_ab(A)) = A ).

% Collect_mem_eq
tff(fact_124_Collect__mem__eq,axiom,
    ! [A: set_wo3913738467083021356l_num1] : collec7814023847061821259l_num1(aTP_Lamm_ac(A)) = A ).

% Collect_mem_eq
tff(fact_125_Collect__mem__eq,axiom,
    ! [A: set_complex] : collect_complex(aTP_Lamm_ad(A)) = A ).

% Collect_mem_eq
tff(fact_126_Collect__mem__eq,axiom,
    ! [A: set_list_nat] : collect_list_nat(aTP_Lamm_ae(A)) = A ).

% Collect_mem_eq
tff(fact_127_Collect__mem__eq,axiom,
    ! [A: set_set_nat] : collect_set_nat(aTP_Lamm_af(A)) = A ).

% Collect_mem_eq
tff(fact_128_Collect__mem__eq,axiom,
    ! [A: set_nat] : collect_nat(aTP_Lamm_ag(A)) = A ).

% Collect_mem_eq
tff(fact_129_Collect__mem__eq,axiom,
    ! [A: set_int] : collect_int(aTP_Lamm_ah(A)) = A ).

% Collect_mem_eq
tff(fact_130_Collect__cong,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( ! [X2: complex] :
          ( aa_complex_o(P,X2)
        <=> aa_complex_o(Q,X2) )
     => ( collect_complex(P) = collect_complex(Q) ) ) ).

% Collect_cong
tff(fact_131_Collect__cong,axiom,
    ! [P: fun_list_nat_o,Q: fun_list_nat_o] :
      ( ! [X2: list_nat] :
          ( aa_list_nat_o(P,X2)
        <=> aa_list_nat_o(Q,X2) )
     => ( collect_list_nat(P) = collect_list_nat(Q) ) ) ).

% Collect_cong
tff(fact_132_Collect__cong,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ! [X2: set_nat] :
          ( aa_set_nat_o(P,X2)
        <=> aa_set_nat_o(Q,X2) )
     => ( collect_set_nat(P) = collect_set_nat(Q) ) ) ).

% Collect_cong
tff(fact_133_Collect__cong,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ! [X2: nat] :
          ( aa_nat_o(P,X2)
        <=> aa_nat_o(Q,X2) )
     => ( collect_nat(P) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_134_Collect__cong,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( aa_int_o(P,X2)
        <=> aa_int_o(Q,X2) )
     => ( collect_int(P) = collect_int(Q) ) ) ).

% Collect_cong
tff(fact_135_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)) ).

% add_mult_distrib2
tff(fact_136_add__mult__distrib,axiom,
    ! [M: nat,Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K)) ).

% add_mult_distrib
tff(fact_137_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J)) ) ).

% mult_le_mono2
tff(fact_138_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K)) ) ).

% mult_le_mono1
tff(fact_139_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),L)) ) ) ).

% mult_le_mono
tff(fact_140_le__square,axiom,
    ! [M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M)) ).

% le_square
tff(fact_141_le__cube,axiom,
    ! [M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M))) ).

% le_cube
tff(fact_142_ceiling__mono,axiom,
    ! [Y: rat,X: rat] :
      ( ord_less_eq_rat(Y,X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(Y)),archim2889992004027027881ng_rat(X)) ) ).

% ceiling_mono
tff(fact_143_ceiling__mono,axiom,
    ! [Y: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim7802044766580827645g_real(Y)),archim7802044766580827645g_real(X)) ) ).

% ceiling_mono
tff(fact_144_numeral__Bit0,axiom,
    ! [Na: num] : numera7442385471795722001l_num1(bit0(Na)) = plus_p361126936061061375l_num1(numera7442385471795722001l_num1(Na),numera7442385471795722001l_num1(Na)) ).

% numeral_Bit0
tff(fact_145_numeral__Bit0,axiom,
    ! [Na: num] : numeral_numeral_real(bit0(Na)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(Na)),numeral_numeral_real(Na)) ).

% numeral_Bit0
tff(fact_146_numeral__Bit0,axiom,
    ! [Na: num] : numeral_numeral_rat(bit0(Na)) = plus_plus_rat(numeral_numeral_rat(Na),numeral_numeral_rat(Na)) ).

% numeral_Bit0
tff(fact_147_numeral__Bit0,axiom,
    ! [Na: num] : numeral_numeral_nat(bit0(Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(Na)),numeral_numeral_nat(Na)) ).

% numeral_Bit0
tff(fact_148_numeral__Bit0,axiom,
    ! [Na: num] : aa_num_int(numeral_numeral_int,bit0(Na)) = plus_plus_int(aa_num_int(numeral_numeral_int,Na),aa_num_int(numeral_numeral_int,Na)) ).

% numeral_Bit0
tff(fact_149_numeral__Bit0,axiom,
    ! [Na: num] : numera9087168376688890119uint32(bit0(Na)) = plus_plus_uint32(numera9087168376688890119uint32(Na),numera9087168376688890119uint32(Na)) ).

% numeral_Bit0
tff(fact_150_mult__numeral__1__right,axiom,
    ! [Aa: word_N3645301735248828278l_num1] : times_7065122842183080059l_num1(Aa,numera7442385471795722001l_num1(one)) = Aa ).

% mult_numeral_1_right
tff(fact_151_mult__numeral__1__right,axiom,
    ! [Aa: real] : aa_real_real(times_times_real(Aa),numeral_numeral_real(one)) = Aa ).

% mult_numeral_1_right
tff(fact_152_mult__numeral__1__right,axiom,
    ! [Aa: rat] : times_times_rat(Aa,numeral_numeral_rat(one)) = Aa ).

% mult_numeral_1_right
tff(fact_153_mult__numeral__1__right,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),numeral_numeral_nat(one)) = Aa ).

% mult_numeral_1_right
tff(fact_154_mult__numeral__1__right,axiom,
    ! [Aa: int] : times_times_int(Aa,aa_num_int(numeral_numeral_int,one)) = Aa ).

% mult_numeral_1_right
tff(fact_155_mult__numeral__1__right,axiom,
    ! [Aa: uint32] : times_times_uint32(Aa,numera9087168376688890119uint32(one)) = Aa ).

% mult_numeral_1_right
tff(fact_156_mult__numeral__1,axiom,
    ! [Aa: word_N3645301735248828278l_num1] : times_7065122842183080059l_num1(numera7442385471795722001l_num1(one),Aa) = Aa ).

% mult_numeral_1
tff(fact_157_mult__numeral__1,axiom,
    ! [Aa: real] : aa_real_real(times_times_real(numeral_numeral_real(one)),Aa) = Aa ).

% mult_numeral_1
tff(fact_158_mult__numeral__1,axiom,
    ! [Aa: rat] : times_times_rat(numeral_numeral_rat(one),Aa) = Aa ).

% mult_numeral_1
tff(fact_159_mult__numeral__1,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(one)),Aa) = Aa ).

% mult_numeral_1
tff(fact_160_mult__numeral__1,axiom,
    ! [Aa: int] : times_times_int(aa_num_int(numeral_numeral_int,one),Aa) = Aa ).

% mult_numeral_1
tff(fact_161_mult__numeral__1,axiom,
    ! [Aa: uint32] : times_times_uint32(numera9087168376688890119uint32(one),Aa) = Aa ).

% mult_numeral_1
tff(fact_162_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one )
     => ( ! [X22: num] : Y != bit0(X22)
       => ~ ! [X3: num] : Y != bit1(X3) ) ) ).

% num.exhaust
tff(fact_163_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ord_less_eq_rat(semiri681578069525770553at_rat(I),semiri681578069525770553at_rat(J)) ) ).

% of_nat_mono
tff(fact_164_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,I)),aa_nat_nat(semiri1316708129612266289at_nat,J)) ) ).

% of_nat_mono
tff(fact_165_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,I)),aa_nat_int(semiri1314217659103216013at_int,J)) ) ).

% of_nat_mono
tff(fact_166_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,I)),aa_nat_real(semiri5074537144036343181t_real,J)) ) ).

% of_nat_mono
tff(fact_167_numeral__code_I2_J,axiom,
    ! [Na: num] :
      numera7442385471795722001l_num1(bit0(Na)) = $let(
        m: word_N3645301735248828278l_num1,
        m:= numera7442385471795722001l_num1(Na),
        plus_p361126936061061375l_num1(m,m) ) ).

% numeral_code(2)
tff(fact_168_numeral__code_I2_J,axiom,
    ! [Na: num] :
      numeral_numeral_real(bit0(Na)) = $let(
        m: real,
        m:= numeral_numeral_real(Na),
        aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m),m) ) ).

% numeral_code(2)
tff(fact_169_numeral__code_I2_J,axiom,
    ! [Na: num] :
      numeral_numeral_rat(bit0(Na)) = $let(
        m: rat,
        m:= numeral_numeral_rat(Na),
        plus_plus_rat(m,m) ) ).

% numeral_code(2)
tff(fact_170_numeral__code_I2_J,axiom,
    ! [Na: num] :
      numeral_numeral_nat(bit0(Na)) = $let(
        m: nat,
        m:= numeral_numeral_nat(Na),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m),m) ) ).

% numeral_code(2)
tff(fact_171_numeral__code_I2_J,axiom,
    ! [Na: num] :
      aa_num_int(numeral_numeral_int,bit0(Na)) = $let(
        m: int,
        m:= aa_num_int(numeral_numeral_int,Na),
        plus_plus_int(m,m) ) ).

% numeral_code(2)
tff(fact_172_numeral__code_I2_J,axiom,
    ! [Na: num] :
      numera9087168376688890119uint32(bit0(Na)) = $let(
        m: uint32,
        m:= numera9087168376688890119uint32(Na),
        plus_plus_uint32(m,m) ) ).

% numeral_code(2)
tff(fact_173_of__nat__ceiling,axiom,
    ! [R: rat] : ord_less_eq_rat(R,semiri681578069525770553at_rat(aa_int_nat(nat2,archim2889992004027027881ng_rat(R)))) ).

% of_nat_ceiling
tff(fact_174_of__nat__ceiling,axiom,
    ! [R: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,archim7802044766580827645g_real(R)))) ).

% of_nat_ceiling
tff(fact_175_left__add__twice,axiom,
    ! [Aa: word_N3645301735248828278l_num1,Ba: word_N3645301735248828278l_num1] : plus_p361126936061061375l_num1(Aa,plus_p361126936061061375l_num1(Aa,Ba)) = plus_p361126936061061375l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(bit0(one)),Aa),Ba) ).

% left_add_twice
tff(fact_176_left__add__twice,axiom,
    ! [Aa: real,Ba: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),Aa)),Ba) ).

% left_add_twice
tff(fact_177_left__add__twice,axiom,
    ! [Aa: rat,Ba: rat] : plus_plus_rat(Aa,plus_plus_rat(Aa,Ba)) = plus_plus_rat(times_times_rat(numeral_numeral_rat(bit0(one)),Aa),Ba) ).

% left_add_twice
tff(fact_178_left__add__twice,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Aa)),Ba) ).

% left_add_twice
tff(fact_179_left__add__twice,axiom,
    ! [Aa: int,Ba: int] : plus_plus_int(Aa,plus_plus_int(Aa,Ba)) = plus_plus_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),Aa),Ba) ).

% left_add_twice
tff(fact_180_left__add__twice,axiom,
    ! [Aa: uint32,Ba: uint32] : plus_plus_uint32(Aa,plus_plus_uint32(Aa,Ba)) = plus_plus_uint32(times_times_uint32(numera9087168376688890119uint32(bit0(one)),Aa),Ba) ).

% left_add_twice
tff(fact_181_mult__2__right,axiom,
    ! [Z: word_N3645301735248828278l_num1] : times_7065122842183080059l_num1(Z,numera7442385471795722001l_num1(bit0(one))) = plus_p361126936061061375l_num1(Z,Z) ).

% mult_2_right
tff(fact_182_mult__2__right,axiom,
    ! [Z: real] : aa_real_real(times_times_real(Z),numeral_numeral_real(bit0(one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2_right
tff(fact_183_mult__2__right,axiom,
    ! [Z: rat] : times_times_rat(Z,numeral_numeral_rat(bit0(one))) = plus_plus_rat(Z,Z) ).

% mult_2_right
tff(fact_184_mult__2__right,axiom,
    ! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),numeral_numeral_nat(bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).

% mult_2_right
tff(fact_185_mult__2__right,axiom,
    ! [Z: int] : times_times_int(Z,aa_num_int(numeral_numeral_int,bit0(one))) = plus_plus_int(Z,Z) ).

% mult_2_right
tff(fact_186_mult__2__right,axiom,
    ! [Z: uint32] : times_times_uint32(Z,numera9087168376688890119uint32(bit0(one))) = plus_plus_uint32(Z,Z) ).

% mult_2_right
tff(fact_187_mult__2,axiom,
    ! [Z: word_N3645301735248828278l_num1] : times_7065122842183080059l_num1(numera7442385471795722001l_num1(bit0(one)),Z) = plus_p361126936061061375l_num1(Z,Z) ).

% mult_2
tff(fact_188_mult__2,axiom,
    ! [Z: real] : aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),Z) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2
tff(fact_189_mult__2,axiom,
    ! [Z: rat] : times_times_rat(numeral_numeral_rat(bit0(one)),Z) = plus_plus_rat(Z,Z) ).

% mult_2
tff(fact_190_mult__2,axiom,
    ! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Z) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).

% mult_2
tff(fact_191_mult__2,axiom,
    ! [Z: int] : times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),Z) = plus_plus_int(Z,Z) ).

% mult_2
tff(fact_192_mult__2,axiom,
    ! [Z: uint32] : times_times_uint32(numera9087168376688890119uint32(bit0(one)),Z) = plus_plus_uint32(Z,Z) ).

% mult_2
tff(fact_193_nat__ceiling__le__eq,axiom,
    ! [X: real,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,archim7802044766580827645g_real(X))),Aa)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,Aa)) ) ).

% nat_ceiling_le_eq
tff(fact_194_numeral__le__real__of__nat__iff,axiom,
    ! [Na: num,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(Na)),aa_nat_real(semiri5074537144036343181t_real,M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(Na)),M) ) ).

% numeral_le_real_of_nat_iff
tff(fact_195_nat__numeral,axiom,
    ! [K: num] : aa_int_nat(nat2,aa_num_int(numeral_numeral_int,K)) = numeral_numeral_nat(K) ).

% nat_numeral
tff(fact_196_int__eq__iff__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_num_int(numeral_numeral_int,V) )
    <=> ( M = numeral_numeral_nat(V) ) ) ).

% int_eq_iff_numeral
tff(fact_197_semiring__norm_I3_J,axiom,
    ! [Na: num] : plus_plus_num(one,bit0(Na)) = bit1(Na) ).

% semiring_norm(3)
tff(fact_198_semiring__norm_I4_J,axiom,
    ! [Na: num] : plus_plus_num(one,bit1(Na)) = bit0(plus_plus_num(Na,one)) ).

% semiring_norm(4)
tff(fact_199_semiring__norm_I5_J,axiom,
    ! [M: num] : plus_plus_num(bit0(M),one) = bit1(M) ).

% semiring_norm(5)
tff(fact_200_semiring__norm_I8_J,axiom,
    ! [M: num] : plus_plus_num(bit1(M),one) = bit0(plus_plus_num(M,one)) ).

% semiring_norm(8)
tff(fact_201_semiring__norm_I10_J,axiom,
    ! [M: num,Na: num] : plus_plus_num(bit1(M),bit1(Na)) = bit0(plus_plus_num(plus_plus_num(M,Na),one)) ).

% semiring_norm(10)
tff(fact_202_nat__int,axiom,
    ! [Na: nat] : aa_int_nat(nat2,aa_nat_int(semiri1314217659103216013at_int,Na)) = Na ).

% nat_int
tff(fact_203_enat__ord__number_I1_J,axiom,
    ! [M: num,Na: num] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(M)),numeral_numeral_nat(Na)) ) ).

% enat_ord_number(1)
tff(fact_204_semiring__norm_I70_J,axiom,
    ! [M: num] : ~ aa_num_o(ord_less_eq_num(bit1(M)),one) ).

% semiring_norm(70)
tff(fact_205_semiring__norm_I87_J,axiom,
    ! [M: num,Na: num] :
      ( ( bit0(M) = bit0(Na) )
    <=> ( M = Na ) ) ).

% semiring_norm(87)
tff(fact_206_semiring__norm_I90_J,axiom,
    ! [M: num,Na: num] :
      ( ( bit1(M) = bit1(Na) )
    <=> ( M = Na ) ) ).

% semiring_norm(90)
tff(fact_207_semiring__norm_I85_J,axiom,
    ! [M: num] : bit0(M) != one ).

% semiring_norm(85)
tff(fact_208_semiring__norm_I83_J,axiom,
    ! [Na: num] : one != bit0(Na) ).

% semiring_norm(83)
tff(fact_209_semiring__norm_I89_J,axiom,
    ! [M: num,Na: num] : bit1(M) != bit0(Na) ).

% semiring_norm(89)
tff(fact_210_semiring__norm_I88_J,axiom,
    ! [M: num,Na: num] : bit0(M) != bit1(Na) ).

% semiring_norm(88)
tff(fact_211_semiring__norm_I86_J,axiom,
    ! [M: num] : bit1(M) != one ).

% semiring_norm(86)
tff(fact_212_semiring__norm_I84_J,axiom,
    ! [Na: num] : one != bit1(Na) ).

% semiring_norm(84)
tff(fact_213_semiring__norm_I6_J,axiom,
    ! [M: num,Na: num] : plus_plus_num(bit0(M),bit0(Na)) = bit0(plus_plus_num(M,Na)) ).

% semiring_norm(6)
tff(fact_214_semiring__norm_I13_J,axiom,
    ! [M: num,Na: num] : times_times_num(bit0(M),bit0(Na)) = bit0(bit0(times_times_num(M,Na))) ).

% semiring_norm(13)
tff(fact_215_semiring__norm_I12_J,axiom,
    ! [Na: num] : times_times_num(one,Na) = Na ).

% semiring_norm(12)
tff(fact_216_semiring__norm_I11_J,axiom,
    ! [M: num] : times_times_num(M,one) = M ).

% semiring_norm(11)
tff(fact_217_semiring__norm_I71_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(ord_less_eq_num(bit0(M)),bit0(Na))
    <=> aa_num_o(ord_less_eq_num(M),Na) ) ).

% semiring_norm(71)
tff(fact_218_semiring__norm_I68_J,axiom,
    ! [Na: num] : aa_num_o(ord_less_eq_num(one),Na) ).

% semiring_norm(68)
tff(fact_219_semiring__norm_I73_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(ord_less_eq_num(bit1(M)),bit1(Na))
    <=> aa_num_o(ord_less_eq_num(M),Na) ) ).

% semiring_norm(73)
tff(fact_220_semiring__norm_I2_J,axiom,
    plus_plus_num(one,one) = bit0(one) ).

% semiring_norm(2)
tff(fact_221_semiring__norm_I9_J,axiom,
    ! [M: num,Na: num] : plus_plus_num(bit1(M),bit0(Na)) = bit1(plus_plus_num(M,Na)) ).

% semiring_norm(9)
tff(fact_222_semiring__norm_I7_J,axiom,
    ! [M: num,Na: num] : plus_plus_num(bit0(M),bit1(Na)) = bit1(plus_plus_num(M,Na)) ).

% semiring_norm(7)
tff(fact_223_semiring__norm_I15_J,axiom,
    ! [M: num,Na: num] : times_times_num(bit1(M),bit0(Na)) = bit0(times_times_num(bit1(M),Na)) ).

% semiring_norm(15)
tff(fact_224_semiring__norm_I14_J,axiom,
    ! [M: num,Na: num] : times_times_num(bit0(M),bit1(Na)) = bit0(times_times_num(M,bit1(Na))) ).

% semiring_norm(14)
tff(fact_225_semiring__norm_I69_J,axiom,
    ! [M: num] : ~ aa_num_o(ord_less_eq_num(bit0(M)),one) ).

% semiring_norm(69)
tff(fact_226_semiring__norm_I72_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(ord_less_eq_num(bit0(M)),bit1(Na))
    <=> aa_num_o(ord_less_eq_num(M),Na) ) ).

% semiring_norm(72)
tff(fact_227_semiring__norm_I16_J,axiom,
    ! [M: num,Na: num] : times_times_num(bit1(M),bit1(Na)) = bit1(plus_plus_num(plus_plus_num(M,Na),bit0(times_times_num(M,Na)))) ).

% semiring_norm(16)
tff(fact_228_complete__real,axiom,
    ! [Sa: set_real] :
      ( ? [X4: real] : member_real(X4,Sa)
     => ( ? [Z2: real] :
          ! [X2: real] :
            ( member_real(X2,Sa)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Z2) )
       => ? [Y2: real] :
            ( ! [X4: real] :
                ( member_real(X4,Sa)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Y2) )
            & ! [Z2: real] :
                ( ! [X2: real] :
                    ( member_real(X2,Sa)
                   => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Z2) )
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),Z2) ) ) ) ) ).

% complete_real
tff(fact_229_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W: int] : times_times_int(plus_plus_int(Z1,Z22),W) = plus_plus_int(times_times_int(Z1,W),times_times_int(Z22,W)) ).

% int_distrib(1)
tff(fact_230_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z22: int] : times_times_int(W,plus_plus_int(Z1,Z22)) = plus_plus_int(times_times_int(W,Z1),times_times_int(W,Z22)) ).

% int_distrib(2)
tff(fact_231_int__int__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_nat_int(semiri1314217659103216013at_int,Na) )
    <=> ( M = Na ) ) ).

% int_int_eq
tff(fact_232_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),U)),K) ).

% left_add_mult_distrib
tff(fact_233_zle__int,axiom,
    ! [M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% zle_int
tff(fact_234_zadd__int__left,axiom,
    ! [M: nat,Na: nat,Z: int] : plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,M),plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,Na),Z)) = plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)),Z) ).

% zadd_int_left
tff(fact_235_nat__mono,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ).

% nat_mono
tff(fact_236_zle__iff__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z)
    <=> ? [N2: nat] : Z = plus_plus_int(W,aa_nat_int(semiri1314217659103216013at_int,N2)) ) ).

% zle_iff_zadd
tff(fact_237_real__nat__ceiling__ge,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,archim7802044766580827645g_real(X)))) ).

% real_nat_ceiling_ge
tff(fact_238_nat__le__iff,axiom,
    ! [X: int,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,X)),Na)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_nat_int(semiri1314217659103216013at_int,Na)) ) ).

% nat_le_iff
tff(fact_239_nat__int__add,axiom,
    ! [Aa: nat,Ba: nat] : aa_int_nat(nat2,plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,Aa),aa_nat_int(semiri1314217659103216013at_int,Ba))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba) ).

% nat_int_add
tff(fact_240_add__le__cancel__left,axiom,
    ! [C: rat,Aa: rat,Ba: rat] :
      ( ord_less_eq_rat(plus_plus_rat(C,Aa),plus_plus_rat(C,Ba))
    <=> ord_less_eq_rat(Aa,Ba) ) ).

% add_le_cancel_left
tff(fact_241_add__le__cancel__left,axiom,
    ! [C: nat,Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Ba))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba) ) ).

% add_le_cancel_left
tff(fact_242_add__le__cancel__left,axiom,
    ! [C: int,Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,plus_plus_int(C,Aa)),plus_plus_int(C,Ba))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba) ) ).

% add_le_cancel_left
tff(fact_243_add__le__cancel__left,axiom,
    ! [C: real,Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),Ba))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba) ) ).

% add_le_cancel_left
tff(fact_244_add__le__cancel__right,axiom,
    ! [Aa: rat,C: rat,Ba: rat] :
      ( ord_less_eq_rat(plus_plus_rat(Aa,C),plus_plus_rat(Ba,C))
    <=> ord_less_eq_rat(Aa,Ba) ) ).

% add_le_cancel_right
tff(fact_245_add__le__cancel__right,axiom,
    ! [Aa: nat,C: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),C))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba) ) ).

% add_le_cancel_right
tff(fact_246_add__le__cancel__right,axiom,
    ! [Aa: int,C: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,plus_plus_int(Aa,C)),plus_plus_int(Ba,C))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba) ) ).

% add_le_cancel_right
tff(fact_247_add__le__cancel__right,axiom,
    ! [Aa: real,C: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba) ) ).

% add_le_cancel_right
tff(fact_248__092_060open_062_092_060section_062_Avebt__assn_An_AS_Ati_A_092_060section_062_ATBOUND_Areturn_A_091_093_Ab_A_091_093_An_092_060close_062,axiom,
    aa_nat_o(simple8287821899582105911st_nat(aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(n,s),ti),aa_lis1877347954636394190st_nat(heap_T3164785745270913723st_nat,nil_nat)),aa_nat_nat(b(nil_nat),n)) ).

% \<open>\<section> vebt_assn n S ti \<section> TBOUND return [] b [] n\<close>
tff(fact_249_nat__plus__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),Xa) = aa_int_nat(nat2,plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,X4),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_plus_as_int
tff(fact_250_Heap__Time__Monad_Obind__bind,axiom,
    ! [F: heap_T2636463487746394924on_nat,G: fun_op1688020532487384469ap_nat,K: fun_na3533371890617757541ap_nat] : heap_T7049098217575491753at_nat(heap_T2919350798565477881at_nat(F,G),K) = heap_T2919350798565477881at_nat(F,aa_fun34237272164541356ap_nat(aTP_Lamm_ai(G),K)) ).

% Heap_Time_Monad.bind_bind
tff(fact_251_Heap__Time__Monad_Obind__bind,axiom,
    ! [F: heap_T8145700208782473153_VEBTi,G: fun_VE6418762861507722832st_nat,K: fun_li4609997795046405733st_nat] : heap_T7479857938255823433st_nat(heap_T3876627890315925662st_nat(F,G),K) = heap_T3876627890315925662st_nat(F,aa_fun1096608658413694055st_nat(aTP_Lamm_aj(G),K)) ).

% Heap_Time_Monad.bind_bind
tff(fact_252_Heap__Time__Monad_Obind__bind,axiom,
    ! [F: heap_T2636463487746394924on_nat,G: fun_op4683986414774905957on_nat,K: fun_op1688020532487384469ap_nat] : heap_T2919350798565477881at_nat(heap_T3669509953089699273on_nat(F,G),K) = heap_T2919350798565477881at_nat(F,aa_fun7982982223196242428ap_nat(aTP_Lamm_ak(G),K)) ).

% Heap_Time_Monad.bind_bind
tff(fact_253_Heap__Time__Monad_Obind__bind,axiom,
    ! [F: heap_T8145700208782473153_VEBTi,G: fun_VE4765431424076933413_VEBTi,K: fun_VE6418762861507722832st_nat] : heap_T3876627890315925662st_nat(heap_T1006145433769338483_VEBTi(F,G),K) = heap_T3876627890315925662st_nat(F,aa_fun7495903629695767740st_nat(aTP_Lamm_al(G),K)) ).

% Heap_Time_Monad.bind_bind
tff(fact_254_nat__times__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X4),Xa) = aa_int_nat(nat2,times_times_int(aa_nat_int(semiri1314217659103216013at_int,X4),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_times_as_int
tff(fact_255_nat__leq__as__int,axiom,
    ! [X4: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Xa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa)) ) ).

% nat_leq_as_int
tff(fact_256_b__def,axiom,
    ! [Ysb: list_nat,Na: nat] : aa_nat_nat(b(Ysb),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Ysb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit1(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit1(bit0(one)))),aa_int_nat(nat2,archim7802044766580827645g_real(aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Na))))))),numeral_numeral_nat(bit1(bit0(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit1(bit1(one)))),aa_int_nat(nat2,archim7802044766580827645g_real(aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Na)))))))) ).

% b_def
tff(fact_257_int__plus,axiom,
    ! [Na: nat,M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)) = plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,Na),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% int_plus
tff(fact_258_int__ops_I5_J,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba)) = plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,Aa),aa_nat_int(semiri1314217659103216013at_int,Ba)) ).

% int_ops(5)
tff(fact_259_nat__int__comparison_I3_J,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Aa)),aa_nat_int(semiri1314217659103216013at_int,Ba)) ) ).

% nat_int_comparison(3)
tff(fact_260_nat__numeral__as__int,axiom,
    ! [X4: num] : numeral_numeral_nat(X4) = aa_int_nat(nat2,aa_num_int(numeral_numeral_int,X4)) ).

% nat_numeral_as_int
tff(fact_261_verit__eq__simplify_I8_J,axiom,
    ! [X23: num,Y22: num] :
      ( ( bit0(X23) = bit0(Y22) )
    <=> ( X23 = Y22 ) ) ).

% verit_eq_simplify(8)
tff(fact_262_add__right__cancel,axiom,
    ! [Ba: real,Aa: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),Aa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),Aa) )
    <=> ( Ba = C ) ) ).

% add_right_cancel
tff(fact_263_add__right__cancel,axiom,
    ! [Ba: rat,Aa: rat,C: rat] :
      ( ( plus_plus_rat(Ba,Aa) = plus_plus_rat(C,Aa) )
    <=> ( Ba = C ) ) ).

% add_right_cancel
tff(fact_264_add__right__cancel,axiom,
    ! [Ba: nat,Aa: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Aa) )
    <=> ( Ba = C ) ) ).

% add_right_cancel
tff(fact_265_add__right__cancel,axiom,
    ! [Ba: int,Aa: int,C: int] :
      ( ( plus_plus_int(Ba,Aa) = plus_plus_int(C,Aa) )
    <=> ( Ba = C ) ) ).

% add_right_cancel
tff(fact_266_add__right__cancel,axiom,
    ! [Ba: word_N3645301735248828278l_num1,Aa: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1(Ba,Aa) = plus_p361126936061061375l_num1(C,Aa) )
    <=> ( Ba = C ) ) ).

% add_right_cancel
tff(fact_267_add__right__cancel,axiom,
    ! [Ba: uint32,Aa: uint32,C: uint32] :
      ( ( plus_plus_uint32(Ba,Aa) = plus_plus_uint32(C,Aa) )
    <=> ( Ba = C ) ) ).

% add_right_cancel
tff(fact_268_add__left__cancel,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C) )
    <=> ( Ba = C ) ) ).

% add_left_cancel
tff(fact_269_add__left__cancel,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( ( plus_plus_rat(Aa,Ba) = plus_plus_rat(Aa,C) )
    <=> ( Ba = C ) ) ).

% add_left_cancel
tff(fact_270_add__left__cancel,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C) )
    <=> ( Ba = C ) ) ).

% add_left_cancel
tff(fact_271_add__left__cancel,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( ( plus_plus_int(Aa,Ba) = plus_plus_int(Aa,C) )
    <=> ( Ba = C ) ) ).

% add_left_cancel
tff(fact_272_add__left__cancel,axiom,
    ! [Aa: word_N3645301735248828278l_num1,Ba: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1(Aa,Ba) = plus_p361126936061061375l_num1(Aa,C) )
    <=> ( Ba = C ) ) ).

% add_left_cancel
tff(fact_273_add__left__cancel,axiom,
    ! [Aa: uint32,Ba: uint32,C: uint32] :
      ( ( plus_plus_uint32(Aa,Ba) = plus_plus_uint32(Aa,C) )
    <=> ( Ba = C ) ) ).

% add_left_cancel
tff(fact_274_verit__eq__simplify_I9_J,axiom,
    ! [X32: num,Y32: num] :
      ( ( bit1(X32) = bit1(Y32) )
    <=> ( X32 = Y32 ) ) ).

% verit_eq_simplify(9)
tff(fact_275_mult__1,axiom,
    ! [Aa: real] : aa_real_real(times_times_real(one_one_real),Aa) = Aa ).

% mult_1
tff(fact_276_mult__1,axiom,
    ! [Aa: rat] : times_times_rat(one_one_rat,Aa) = Aa ).

% mult_1
tff(fact_277_mult__1,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),Aa) = Aa ).

% mult_1
tff(fact_278_mult__1,axiom,
    ! [Aa: int] : times_times_int(one_one_int,Aa) = Aa ).

% mult_1
tff(fact_279_mult__1,axiom,
    ! [Aa: word_N3645301735248828278l_num1] : times_7065122842183080059l_num1(one_on7727431528512463931l_num1,Aa) = Aa ).

% mult_1
tff(fact_280_mult__1,axiom,
    ! [Aa: uint32] : times_times_uint32(one_one_uint32,Aa) = Aa ).

% mult_1
tff(fact_281_mult_Oright__neutral,axiom,
    ! [Aa: real] : aa_real_real(times_times_real(Aa),one_one_real) = Aa ).

% mult.right_neutral
tff(fact_282_mult_Oright__neutral,axiom,
    ! [Aa: rat] : times_times_rat(Aa,one_one_rat) = Aa ).

% mult.right_neutral
tff(fact_283_mult_Oright__neutral,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),one_one_nat) = Aa ).

% mult.right_neutral
tff(fact_284_mult_Oright__neutral,axiom,
    ! [Aa: int] : times_times_int(Aa,one_one_int) = Aa ).

% mult.right_neutral
tff(fact_285_mult_Oright__neutral,axiom,
    ! [Aa: word_N3645301735248828278l_num1] : times_7065122842183080059l_num1(Aa,one_on7727431528512463931l_num1) = Aa ).

% mult.right_neutral
tff(fact_286_mult_Oright__neutral,axiom,
    ! [Aa: uint32] : times_times_uint32(Aa,one_one_uint32) = Aa ).

% mult.right_neutral
tff(fact_287_nat__mult__eq__1__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) = one_one_nat )
    <=> ( ( M = one_one_nat )
        & ( Na = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_288_nat__1__eq__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) )
    <=> ( ( M = one_one_nat )
        & ( Na = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_289_ceiling__one,axiom,
    archim7802044766580827645g_real(one_one_real) = one_one_int ).

% ceiling_one
tff(fact_290_ceiling__one,axiom,
    archim2889992004027027881ng_rat(one_one_rat) = one_one_int ).

% ceiling_one
tff(fact_291_one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( one_one_real = numeral_numeral_real(Na) )
    <=> ( one = Na ) ) ).

% one_eq_numeral_iff
tff(fact_292_one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( one_one_rat = numeral_numeral_rat(Na) )
    <=> ( one = Na ) ) ).

% one_eq_numeral_iff
tff(fact_293_one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( one_one_nat = numeral_numeral_nat(Na) )
    <=> ( one = Na ) ) ).

% one_eq_numeral_iff
tff(fact_294_one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( one_one_int = aa_num_int(numeral_numeral_int,Na) )
    <=> ( one = Na ) ) ).

% one_eq_numeral_iff
tff(fact_295_numeral__eq__one__iff,axiom,
    ! [Na: num] :
      ( ( numeral_numeral_real(Na) = one_one_real )
    <=> ( Na = one ) ) ).

% numeral_eq_one_iff
tff(fact_296_numeral__eq__one__iff,axiom,
    ! [Na: num] :
      ( ( numeral_numeral_rat(Na) = one_one_rat )
    <=> ( Na = one ) ) ).

% numeral_eq_one_iff
tff(fact_297_numeral__eq__one__iff,axiom,
    ! [Na: num] :
      ( ( numeral_numeral_nat(Na) = one_one_nat )
    <=> ( Na = one ) ) ).

% numeral_eq_one_iff
tff(fact_298_numeral__eq__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_num_int(numeral_numeral_int,Na) = one_one_int )
    <=> ( Na = one ) ) ).

% numeral_eq_one_iff
tff(fact_299_of__nat__1,axiom,
    semiri2565882477558803405uint32(one_one_nat) = one_one_uint32 ).

% of_nat_1
tff(fact_300_of__nat__1,axiom,
    semiri8819519690708144855l_num1(one_one_nat) = one_on7727431528512463931l_num1 ).

% of_nat_1
tff(fact_301_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_302_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_303_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_304_of__nat__1,axiom,
    semiri681578069525770553at_rat(one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_305_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_real = aa_nat_real(semiri5074537144036343181t_real,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_306_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_307_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_308_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_rat = semiri681578069525770553at_rat(Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_309_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Na) = one_one_real )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_310_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Na) = one_one_nat )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_311_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Na) = one_one_int )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_312_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( semiri681578069525770553at_rat(Na) = one_one_rat )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_313_ceiling__le__one,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(X)),one_one_int)
    <=> ord_less_eq_rat(X,one_one_rat) ) ).

% ceiling_le_one
tff(fact_314_ceiling__le__one,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim7802044766580827645g_real(X)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real) ) ).

% ceiling_le_one
tff(fact_315_ceiling__add__one,axiom,
    ! [X: real] : archim7802044766580827645g_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)) = plus_plus_int(archim7802044766580827645g_real(X),one_one_int) ).

% ceiling_add_one
tff(fact_316_ceiling__add__one,axiom,
    ! [X: rat] : archim2889992004027027881ng_rat(plus_plus_rat(X,one_one_rat)) = plus_plus_int(archim2889992004027027881ng_rat(X),one_one_int) ).

% ceiling_add_one
tff(fact_317_one__add__one,axiom,
    plus_p361126936061061375l_num1(one_on7727431528512463931l_num1,one_on7727431528512463931l_num1) = numera7442385471795722001l_num1(bit0(one)) ).

% one_add_one
tff(fact_318_one__add__one,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real) = numeral_numeral_real(bit0(one)) ).

% one_add_one
tff(fact_319_one__add__one,axiom,
    plus_plus_rat(one_one_rat,one_one_rat) = numeral_numeral_rat(bit0(one)) ).

% one_add_one
tff(fact_320_one__add__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = numeral_numeral_nat(bit0(one)) ).

% one_add_one
tff(fact_321_one__add__one,axiom,
    plus_plus_int(one_one_int,one_one_int) = aa_num_int(numeral_numeral_int,bit0(one)) ).

% one_add_one
tff(fact_322_one__add__one,axiom,
    plus_plus_uint32(one_one_uint32,one_one_uint32) = numera9087168376688890119uint32(bit0(one)) ).

% one_add_one
tff(fact_323_one__plus__numeral,axiom,
    ! [Na: num] : plus_p361126936061061375l_num1(one_on7727431528512463931l_num1,numera7442385471795722001l_num1(Na)) = numera7442385471795722001l_num1(plus_plus_num(one,Na)) ).

% one_plus_numeral
tff(fact_324_one__plus__numeral,axiom,
    ! [Na: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),numeral_numeral_real(Na)) = numeral_numeral_real(plus_plus_num(one,Na)) ).

% one_plus_numeral
tff(fact_325_one__plus__numeral,axiom,
    ! [Na: num] : plus_plus_rat(one_one_rat,numeral_numeral_rat(Na)) = numeral_numeral_rat(plus_plus_num(one,Na)) ).

% one_plus_numeral
tff(fact_326_one__plus__numeral,axiom,
    ! [Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),numeral_numeral_nat(Na)) = numeral_numeral_nat(plus_plus_num(one,Na)) ).

% one_plus_numeral
tff(fact_327_one__plus__numeral,axiom,
    ! [Na: num] : plus_plus_int(one_one_int,aa_num_int(numeral_numeral_int,Na)) = aa_num_int(numeral_numeral_int,plus_plus_num(one,Na)) ).

% one_plus_numeral
tff(fact_328_one__plus__numeral,axiom,
    ! [Na: num] : plus_plus_uint32(one_one_uint32,numera9087168376688890119uint32(Na)) = numera9087168376688890119uint32(plus_plus_num(one,Na)) ).

% one_plus_numeral
tff(fact_329_numeral__plus__one,axiom,
    ! [Na: num] : plus_p361126936061061375l_num1(numera7442385471795722001l_num1(Na),one_on7727431528512463931l_num1) = numera7442385471795722001l_num1(plus_plus_num(Na,one)) ).

% numeral_plus_one
tff(fact_330_numeral__plus__one,axiom,
    ! [Na: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(Na)),one_one_real) = numeral_numeral_real(plus_plus_num(Na,one)) ).

% numeral_plus_one
tff(fact_331_numeral__plus__one,axiom,
    ! [Na: num] : plus_plus_rat(numeral_numeral_rat(Na),one_one_rat) = numeral_numeral_rat(plus_plus_num(Na,one)) ).

% numeral_plus_one
tff(fact_332_numeral__plus__one,axiom,
    ! [Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(Na)),one_one_nat) = numeral_numeral_nat(plus_plus_num(Na,one)) ).

% numeral_plus_one
tff(fact_333_numeral__plus__one,axiom,
    ! [Na: num] : plus_plus_int(aa_num_int(numeral_numeral_int,Na),one_one_int) = aa_num_int(numeral_numeral_int,plus_plus_num(Na,one)) ).

% numeral_plus_one
tff(fact_334_numeral__plus__one,axiom,
    ! [Na: num] : plus_plus_uint32(numera9087168376688890119uint32(Na),one_one_uint32) = numera9087168376688890119uint32(plus_plus_num(Na,one)) ).

% numeral_plus_one
tff(fact_335_numeral__le__one__iff,axiom,
    ! [Na: num] :
      ( ord_less_eq_rat(numeral_numeral_rat(Na),one_one_rat)
    <=> aa_num_o(ord_less_eq_num(Na),one) ) ).

% numeral_le_one_iff
tff(fact_336_numeral__le__one__iff,axiom,
    ! [Na: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(Na)),one_one_nat)
    <=> aa_num_o(ord_less_eq_num(Na),one) ) ).

% numeral_le_one_iff
tff(fact_337_numeral__le__one__iff,axiom,
    ! [Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Na)),one_one_int)
    <=> aa_num_o(ord_less_eq_num(Na),one) ) ).

% numeral_le_one_iff
tff(fact_338_numeral__le__one__iff,axiom,
    ! [Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(Na)),one_one_real)
    <=> aa_num_o(ord_less_eq_num(Na),one) ) ).

% numeral_le_one_iff
tff(fact_339_one__reorient,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ( one_on7727431528512463931l_num1 = X )
    <=> ( X = one_on7727431528512463931l_num1 ) ) ).

% one_reorient
tff(fact_340_one__reorient,axiom,
    ! [X: real] :
      ( ( one_one_real = X )
    <=> ( X = one_one_real ) ) ).

% one_reorient
tff(fact_341_one__reorient,axiom,
    ! [X: rat] :
      ( ( one_one_rat = X )
    <=> ( X = one_one_rat ) ) ).

% one_reorient
tff(fact_342_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
    <=> ( X = one_one_nat ) ) ).

% one_reorient
tff(fact_343_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
    <=> ( X = one_one_int ) ) ).

% one_reorient
tff(fact_344_size__neq__size__imp__neq,axiom,
    ! [X: list_real,Y: list_real] :
      ( ( size_size_list_real(X) != size_size_list_real(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_345_size__neq__size__imp__neq,axiom,
    ! [X: list_o,Y: list_o] :
      ( ( size_size_list_o(X) != size_size_list_o(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_346_size__neq__size__imp__neq,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( size_size_list_nat(X) != size_size_list_nat(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_347_size__neq__size__imp__neq,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( size_size_list_int(X) != size_size_list_int(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_348_size__neq__size__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( ( size_size_num(X) != size_size_num(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_349_mult_Ocomm__neutral,axiom,
    ! [Aa: real] : aa_real_real(times_times_real(Aa),one_one_real) = Aa ).

% mult.comm_neutral
tff(fact_350_mult_Ocomm__neutral,axiom,
    ! [Aa: rat] : times_times_rat(Aa,one_one_rat) = Aa ).

% mult.comm_neutral
tff(fact_351_mult_Ocomm__neutral,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),one_one_nat) = Aa ).

% mult.comm_neutral
tff(fact_352_mult_Ocomm__neutral,axiom,
    ! [Aa: int] : times_times_int(Aa,one_one_int) = Aa ).

% mult.comm_neutral
tff(fact_353_mult_Ocomm__neutral,axiom,
    ! [Aa: word_N3645301735248828278l_num1] : times_7065122842183080059l_num1(Aa,one_on7727431528512463931l_num1) = Aa ).

% mult.comm_neutral
tff(fact_354_mult_Ocomm__neutral,axiom,
    ! [Aa: uint32] : times_times_uint32(Aa,one_one_uint32) = Aa ).

% mult.comm_neutral
tff(fact_355_comm__monoid__mult__class_Omult__1,axiom,
    ! [Aa: real] : aa_real_real(times_times_real(one_one_real),Aa) = Aa ).

% comm_monoid_mult_class.mult_1
tff(fact_356_comm__monoid__mult__class_Omult__1,axiom,
    ! [Aa: rat] : times_times_rat(one_one_rat,Aa) = Aa ).

% comm_monoid_mult_class.mult_1
tff(fact_357_comm__monoid__mult__class_Omult__1,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),Aa) = Aa ).

% comm_monoid_mult_class.mult_1
tff(fact_358_comm__monoid__mult__class_Omult__1,axiom,
    ! [Aa: int] : times_times_int(one_one_int,Aa) = Aa ).

% comm_monoid_mult_class.mult_1
tff(fact_359_comm__monoid__mult__class_Omult__1,axiom,
    ! [Aa: word_N3645301735248828278l_num1] : times_7065122842183080059l_num1(one_on7727431528512463931l_num1,Aa) = Aa ).

% comm_monoid_mult_class.mult_1
tff(fact_360_comm__monoid__mult__class_Omult__1,axiom,
    ! [Aa: uint32] : times_times_uint32(one_one_uint32,Aa) = Aa ).

% comm_monoid_mult_class.mult_1
tff(fact_361_le__numeral__extra_I4_J,axiom,
    ord_less_eq_rat(one_one_rat,one_one_rat) ).

% le_numeral_extra(4)
tff(fact_362_le__numeral__extra_I4_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),one_one_nat) ).

% le_numeral_extra(4)
tff(fact_363_le__numeral__extra_I4_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),one_one_int) ).

% le_numeral_extra(4)
tff(fact_364_le__numeral__extra_I4_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),one_one_real) ).

% le_numeral_extra(4)
tff(fact_365_nat__mult__1__right,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),one_one_nat) = Na ).

% nat_mult_1_right
tff(fact_366_nat__mult__1,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),Na) = Na ).

% nat_mult_1
tff(fact_367_one__le__numeral,axiom,
    ! [Na: num] : ord_less_eq_rat(one_one_rat,numeral_numeral_rat(Na)) ).

% one_le_numeral
tff(fact_368_one__le__numeral,axiom,
    ! [Na: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),numeral_numeral_nat(Na)) ).

% one_le_numeral
tff(fact_369_one__le__numeral,axiom,
    ! [Na: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_num_int(numeral_numeral_int,Na)) ).

% one_le_numeral
tff(fact_370_one__le__numeral,axiom,
    ! [Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),numeral_numeral_real(Na)) ).

% one_le_numeral
tff(fact_371_numeral__One,axiom,
    numera7442385471795722001l_num1(one) = one_on7727431528512463931l_num1 ).

% numeral_One
tff(fact_372_numeral__One,axiom,
    numeral_numeral_real(one) = one_one_real ).

% numeral_One
tff(fact_373_numeral__One,axiom,
    numeral_numeral_rat(one) = one_one_rat ).

% numeral_One
tff(fact_374_numeral__One,axiom,
    numeral_numeral_nat(one) = one_one_nat ).

% numeral_One
tff(fact_375_numeral__One,axiom,
    aa_num_int(numeral_numeral_int,one) = one_one_int ).

% numeral_One
tff(fact_376_numeral__One,axiom,
    numera9087168376688890119uint32(one) = one_one_uint32 ).

% numeral_One
tff(fact_377_one__plus__numeral__commute,axiom,
    ! [X: num] : plus_p361126936061061375l_num1(one_on7727431528512463931l_num1,numera7442385471795722001l_num1(X)) = plus_p361126936061061375l_num1(numera7442385471795722001l_num1(X),one_on7727431528512463931l_num1) ).

% one_plus_numeral_commute
tff(fact_378_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),numeral_numeral_real(X)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(X)),one_one_real) ).

% one_plus_numeral_commute
tff(fact_379_one__plus__numeral__commute,axiom,
    ! [X: num] : plus_plus_rat(one_one_rat,numeral_numeral_rat(X)) = plus_plus_rat(numeral_numeral_rat(X),one_one_rat) ).

% one_plus_numeral_commute
tff(fact_380_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),numeral_numeral_nat(X)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(X)),one_one_nat) ).

% one_plus_numeral_commute
tff(fact_381_one__plus__numeral__commute,axiom,
    ! [X: num] : plus_plus_int(one_one_int,aa_num_int(numeral_numeral_int,X)) = plus_plus_int(aa_num_int(numeral_numeral_int,X),one_one_int) ).

% one_plus_numeral_commute
tff(fact_382_one__plus__numeral__commute,axiom,
    ! [X: num] : plus_plus_uint32(one_one_uint32,numera9087168376688890119uint32(X)) = plus_plus_uint32(numera9087168376688890119uint32(X),one_one_uint32) ).

% one_plus_numeral_commute
tff(fact_383_numerals_I1_J,axiom,
    numeral_numeral_nat(one) = one_one_nat ).

% numerals(1)
tff(fact_384_verit__la__disequality,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( Aa = Ba )
      | ~ ord_less_eq_rat(Aa,Ba)
      | ~ ord_less_eq_rat(Ba,Aa) ) ).

% verit_la_disequality
tff(fact_385_verit__la__disequality,axiom,
    ! [Aa: num,Ba: num] :
      ( ( Aa = Ba )
      | ~ aa_num_o(ord_less_eq_num(Aa),Ba)
      | ~ aa_num_o(ord_less_eq_num(Ba),Aa) ) ).

% verit_la_disequality
tff(fact_386_verit__la__disequality,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( Aa = Ba )
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),Aa) ) ).

% verit_la_disequality
tff(fact_387_verit__la__disequality,axiom,
    ! [Aa: int,Ba: int] :
      ( ( Aa = Ba )
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa) ) ).

% verit_la_disequality
tff(fact_388_verit__la__disequality,axiom,
    ! [Aa: real,Ba: real] :
      ( ( Aa = Ba )
      | ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
      | ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),Aa) ) ).

% verit_la_disequality
tff(fact_389_verit__comp__simplify1_I2_J,axiom,
    ! [Aa: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(Aa),Aa) ).

% verit_comp_simplify1(2)
tff(fact_390_verit__comp__simplify1_I2_J,axiom,
    ! [Aa: rat] : ord_less_eq_rat(Aa,Aa) ).

% verit_comp_simplify1(2)
tff(fact_391_verit__comp__simplify1_I2_J,axiom,
    ! [Aa: num] : aa_num_o(ord_less_eq_num(Aa),Aa) ).

% verit_comp_simplify1(2)
tff(fact_392_verit__comp__simplify1_I2_J,axiom,
    ! [Aa: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Aa) ).

% verit_comp_simplify1(2)
tff(fact_393_verit__comp__simplify1_I2_J,axiom,
    ! [Aa: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Aa) ).

% verit_comp_simplify1(2)
tff(fact_394_verit__comp__simplify1_I2_J,axiom,
    ! [Aa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Aa) ).

% verit_comp_simplify1(2)
tff(fact_395_add__right__imp__eq,axiom,
    ! [Ba: real,Aa: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),Aa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),Aa) )
     => ( Ba = C ) ) ).

% add_right_imp_eq
tff(fact_396_add__right__imp__eq,axiom,
    ! [Ba: rat,Aa: rat,C: rat] :
      ( ( plus_plus_rat(Ba,Aa) = plus_plus_rat(C,Aa) )
     => ( Ba = C ) ) ).

% add_right_imp_eq
tff(fact_397_add__right__imp__eq,axiom,
    ! [Ba: nat,Aa: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Aa) )
     => ( Ba = C ) ) ).

% add_right_imp_eq
tff(fact_398_add__right__imp__eq,axiom,
    ! [Ba: int,Aa: int,C: int] :
      ( ( plus_plus_int(Ba,Aa) = plus_plus_int(C,Aa) )
     => ( Ba = C ) ) ).

% add_right_imp_eq
tff(fact_399_add__right__imp__eq,axiom,
    ! [Ba: word_N3645301735248828278l_num1,Aa: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1(Ba,Aa) = plus_p361126936061061375l_num1(C,Aa) )
     => ( Ba = C ) ) ).

% add_right_imp_eq
tff(fact_400_add__right__imp__eq,axiom,
    ! [Ba: uint32,Aa: uint32,C: uint32] :
      ( ( plus_plus_uint32(Ba,Aa) = plus_plus_uint32(C,Aa) )
     => ( Ba = C ) ) ).

% add_right_imp_eq
tff(fact_401_add__left__imp__eq,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C) )
     => ( Ba = C ) ) ).

% add_left_imp_eq
tff(fact_402_add__left__imp__eq,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( ( plus_plus_rat(Aa,Ba) = plus_plus_rat(Aa,C) )
     => ( Ba = C ) ) ).

% add_left_imp_eq
tff(fact_403_add__left__imp__eq,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C) )
     => ( Ba = C ) ) ).

% add_left_imp_eq
tff(fact_404_add__left__imp__eq,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( ( plus_plus_int(Aa,Ba) = plus_plus_int(Aa,C) )
     => ( Ba = C ) ) ).

% add_left_imp_eq
tff(fact_405_add__left__imp__eq,axiom,
    ! [Aa: word_N3645301735248828278l_num1,Ba: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1(Aa,Ba) = plus_p361126936061061375l_num1(Aa,C) )
     => ( Ba = C ) ) ).

% add_left_imp_eq
tff(fact_406_add__left__imp__eq,axiom,
    ! [Aa: uint32,Ba: uint32,C: uint32] :
      ( ( plus_plus_uint32(Aa,Ba) = plus_plus_uint32(Aa,C) )
     => ( Ba = C ) ) ).

% add_left_imp_eq
tff(fact_407_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [Ba: real,Aa: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_408_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [Ba: rat,Aa: rat,C: rat] : plus_plus_rat(Ba,plus_plus_rat(Aa,C)) = plus_plus_rat(Aa,plus_plus_rat(Ba,C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_409_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [Ba: nat,Aa: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_410_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [Ba: int,Aa: int,C: int] : plus_plus_int(Ba,plus_plus_int(Aa,C)) = plus_plus_int(Aa,plus_plus_int(Ba,C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_411_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [Ba: word_N3645301735248828278l_num1,Aa: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] : plus_p361126936061061375l_num1(Ba,plus_p361126936061061375l_num1(Aa,C)) = plus_p361126936061061375l_num1(Aa,plus_p361126936061061375l_num1(Ba,C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_412_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [Ba: uint32,Aa: uint32,C: uint32] : plus_plus_uint32(Ba,plus_plus_uint32(Aa,C)) = plus_plus_uint32(Aa,plus_plus_uint32(Ba,C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_413_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [Aa: real,Ba: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),Aa) ).

% ab_semigroup_add_class.add.commute
tff(fact_414_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [Aa: rat,Ba: rat] : plus_plus_rat(Aa,Ba) = plus_plus_rat(Ba,Aa) ).

% ab_semigroup_add_class.add.commute
tff(fact_415_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),Aa) ).

% ab_semigroup_add_class.add.commute
tff(fact_416_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [Aa: int,Ba: int] : plus_plus_int(Aa,Ba) = plus_plus_int(Ba,Aa) ).

% ab_semigroup_add_class.add.commute
tff(fact_417_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [Aa: word_N3645301735248828278l_num1,Ba: word_N3645301735248828278l_num1] : plus_p361126936061061375l_num1(Aa,Ba) = plus_p361126936061061375l_num1(Ba,Aa) ).

% ab_semigroup_add_class.add.commute
tff(fact_418_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [Aa: uint32,Ba: uint32] : plus_plus_uint32(Aa,Ba) = plus_plus_uint32(Ba,Aa) ).

% ab_semigroup_add_class.add.commute
tff(fact_419_add_Oright__cancel,axiom,
    ! [Ba: real,Aa: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),Aa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),Aa) )
    <=> ( Ba = C ) ) ).

% add.right_cancel
tff(fact_420_add_Oright__cancel,axiom,
    ! [Ba: rat,Aa: rat,C: rat] :
      ( ( plus_plus_rat(Ba,Aa) = plus_plus_rat(C,Aa) )
    <=> ( Ba = C ) ) ).

% add.right_cancel
tff(fact_421_add_Oright__cancel,axiom,
    ! [Ba: int,Aa: int,C: int] :
      ( ( plus_plus_int(Ba,Aa) = plus_plus_int(C,Aa) )
    <=> ( Ba = C ) ) ).

% add.right_cancel
tff(fact_422_add_Oright__cancel,axiom,
    ! [Ba: word_N3645301735248828278l_num1,Aa: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1(Ba,Aa) = plus_p361126936061061375l_num1(C,Aa) )
    <=> ( Ba = C ) ) ).

% add.right_cancel
tff(fact_423_add_Oright__cancel,axiom,
    ! [Ba: uint32,Aa: uint32,C: uint32] :
      ( ( plus_plus_uint32(Ba,Aa) = plus_plus_uint32(C,Aa) )
    <=> ( Ba = C ) ) ).

% add.right_cancel
tff(fact_424_add_Oleft__cancel,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C) )
    <=> ( Ba = C ) ) ).

% add.left_cancel
tff(fact_425_add_Oleft__cancel,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( ( plus_plus_rat(Aa,Ba) = plus_plus_rat(Aa,C) )
    <=> ( Ba = C ) ) ).

% add.left_cancel
tff(fact_426_add_Oleft__cancel,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( ( plus_plus_int(Aa,Ba) = plus_plus_int(Aa,C) )
    <=> ( Ba = C ) ) ).

% add.left_cancel
tff(fact_427_add_Oleft__cancel,axiom,
    ! [Aa: word_N3645301735248828278l_num1,Ba: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1(Aa,Ba) = plus_p361126936061061375l_num1(Aa,C) )
    <=> ( Ba = C ) ) ).

% add.left_cancel
tff(fact_428_add_Oleft__cancel,axiom,
    ! [Aa: uint32,Ba: uint32,C: uint32] :
      ( ( plus_plus_uint32(Aa,Ba) = plus_plus_uint32(Aa,C) )
    <=> ( Ba = C ) ) ).

% add.left_cancel
tff(fact_429_add_Oassoc,axiom,
    ! [Aa: real,Ba: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),C)) ).

% add.assoc
tff(fact_430_add_Oassoc,axiom,
    ! [Aa: rat,Ba: rat,C: rat] : plus_plus_rat(plus_plus_rat(Aa,Ba),C) = plus_plus_rat(Aa,plus_plus_rat(Ba,C)) ).

% add.assoc
tff(fact_431_add_Oassoc,axiom,
    ! [Aa: nat,Ba: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),C)) ).

% add.assoc
tff(fact_432_add_Oassoc,axiom,
    ! [Aa: int,Ba: int,C: int] : plus_plus_int(plus_plus_int(Aa,Ba),C) = plus_plus_int(Aa,plus_plus_int(Ba,C)) ).

% add.assoc
tff(fact_433_add_Oassoc,axiom,
    ! [Aa: word_N3645301735248828278l_num1,Ba: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] : plus_p361126936061061375l_num1(plus_p361126936061061375l_num1(Aa,Ba),C) = plus_p361126936061061375l_num1(Aa,plus_p361126936061061375l_num1(Ba,C)) ).

% add.assoc
tff(fact_434_add_Oassoc,axiom,
    ! [Aa: uint32,Ba: uint32,C: uint32] : plus_plus_uint32(plus_plus_uint32(Aa,Ba),C) = plus_plus_uint32(Aa,plus_plus_uint32(Ba,C)) ).

% add.assoc
tff(fact_435_group__cancel_Oadd2,axiom,
    ! [B: real,K: real,Ba: real,Aa: real] :
      ( ( B = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),Ba) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)) ) ) ).

% group_cancel.add2
tff(fact_436_group__cancel_Oadd2,axiom,
    ! [B: rat,K: rat,Ba: rat,Aa: rat] :
      ( ( B = plus_plus_rat(K,Ba) )
     => ( plus_plus_rat(Aa,B) = plus_plus_rat(K,plus_plus_rat(Aa,Ba)) ) ) ).

% group_cancel.add2
tff(fact_437_group__cancel_Oadd2,axiom,
    ! [B: nat,K: nat,Ba: nat,Aa: nat] :
      ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Ba) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba)) ) ) ).

% group_cancel.add2
tff(fact_438_group__cancel_Oadd2,axiom,
    ! [B: int,K: int,Ba: int,Aa: int] :
      ( ( B = plus_plus_int(K,Ba) )
     => ( plus_plus_int(Aa,B) = plus_plus_int(K,plus_plus_int(Aa,Ba)) ) ) ).

% group_cancel.add2
tff(fact_439_group__cancel_Oadd2,axiom,
    ! [B: word_N3645301735248828278l_num1,K: word_N3645301735248828278l_num1,Ba: word_N3645301735248828278l_num1,Aa: word_N3645301735248828278l_num1] :
      ( ( B = plus_p361126936061061375l_num1(K,Ba) )
     => ( plus_p361126936061061375l_num1(Aa,B) = plus_p361126936061061375l_num1(K,plus_p361126936061061375l_num1(Aa,Ba)) ) ) ).

% group_cancel.add2
tff(fact_440_group__cancel_Oadd2,axiom,
    ! [B: uint32,K: uint32,Ba: uint32,Aa: uint32] :
      ( ( B = plus_plus_uint32(K,Ba) )
     => ( plus_plus_uint32(Aa,B) = plus_plus_uint32(K,plus_plus_uint32(Aa,Ba)) ) ) ).

% group_cancel.add2
tff(fact_441_group__cancel_Oadd1,axiom,
    ! [A: real,K: real,Aa: real,Ba: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),Aa) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),Ba) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)) ) ) ).

% group_cancel.add1
tff(fact_442_group__cancel_Oadd1,axiom,
    ! [A: rat,K: rat,Aa: rat,Ba: rat] :
      ( ( A = plus_plus_rat(K,Aa) )
     => ( plus_plus_rat(A,Ba) = plus_plus_rat(K,plus_plus_rat(Aa,Ba)) ) ) ).

% group_cancel.add1
tff(fact_443_group__cancel_Oadd1,axiom,
    ! [A: nat,K: nat,Aa: nat,Ba: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Aa) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),Ba) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba)) ) ) ).

% group_cancel.add1
tff(fact_444_group__cancel_Oadd1,axiom,
    ! [A: int,K: int,Aa: int,Ba: int] :
      ( ( A = plus_plus_int(K,Aa) )
     => ( plus_plus_int(A,Ba) = plus_plus_int(K,plus_plus_int(Aa,Ba)) ) ) ).

% group_cancel.add1
tff(fact_445_group__cancel_Oadd1,axiom,
    ! [A: word_N3645301735248828278l_num1,K: word_N3645301735248828278l_num1,Aa: word_N3645301735248828278l_num1,Ba: word_N3645301735248828278l_num1] :
      ( ( A = plus_p361126936061061375l_num1(K,Aa) )
     => ( plus_p361126936061061375l_num1(A,Ba) = plus_p361126936061061375l_num1(K,plus_p361126936061061375l_num1(Aa,Ba)) ) ) ).

% group_cancel.add1
tff(fact_446_group__cancel_Oadd1,axiom,
    ! [A: uint32,K: uint32,Aa: uint32,Ba: uint32] :
      ( ( A = plus_plus_uint32(K,Aa) )
     => ( plus_plus_uint32(A,Ba) = plus_plus_uint32(K,plus_plus_uint32(Aa,Ba)) ) ) ).

% group_cancel.add1
tff(fact_447_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_448_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( plus_plus_rat(I,K) = plus_plus_rat(J,L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_449_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_450_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( plus_plus_int(I,K) = plus_plus_int(J,L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_451_numeral__Bit1,axiom,
    ! [Na: num] : numera7442385471795722001l_num1(bit1(Na)) = plus_p361126936061061375l_num1(plus_p361126936061061375l_num1(numera7442385471795722001l_num1(Na),numera7442385471795722001l_num1(Na)),one_on7727431528512463931l_num1) ).

% numeral_Bit1
tff(fact_452_numeral__Bit1,axiom,
    ! [Na: num] : numeral_numeral_real(bit1(Na)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(Na)),numeral_numeral_real(Na))),one_one_real) ).

% numeral_Bit1
tff(fact_453_numeral__Bit1,axiom,
    ! [Na: num] : numeral_numeral_rat(bit1(Na)) = plus_plus_rat(plus_plus_rat(numeral_numeral_rat(Na),numeral_numeral_rat(Na)),one_one_rat) ).

% numeral_Bit1
tff(fact_454_numeral__Bit1,axiom,
    ! [Na: num] : numeral_numeral_nat(bit1(Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(Na)),numeral_numeral_nat(Na))),one_one_nat) ).

% numeral_Bit1
tff(fact_455_numeral__Bit1,axiom,
    ! [Na: num] : aa_num_int(numeral_numeral_int,bit1(Na)) = plus_plus_int(plus_plus_int(aa_num_int(numeral_numeral_int,Na),aa_num_int(numeral_numeral_int,Na)),one_one_int) ).

% numeral_Bit1
tff(fact_456_numeral__Bit1,axiom,
    ! [Na: num] : numera9087168376688890119uint32(bit1(Na)) = plus_plus_uint32(plus_plus_uint32(numera9087168376688890119uint32(Na),numera9087168376688890119uint32(Na)),one_one_uint32) ).

% numeral_Bit1
tff(fact_457_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [Ba: real,Aa: real,C: real] : aa_real_real(times_times_real(Ba),aa_real_real(times_times_real(Aa),C)) = aa_real_real(times_times_real(Aa),aa_real_real(times_times_real(Ba),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_458_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [Ba: rat,Aa: rat,C: rat] : times_times_rat(Ba,times_times_rat(Aa,C)) = times_times_rat(Aa,times_times_rat(Ba,C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_459_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [Ba: nat,Aa: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_460_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [Ba: int,Aa: int,C: int] : times_times_int(Ba,times_times_int(Aa,C)) = times_times_int(Aa,times_times_int(Ba,C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_461_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [Ba: word_N3645301735248828278l_num1,Aa: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] : times_7065122842183080059l_num1(Ba,times_7065122842183080059l_num1(Aa,C)) = times_7065122842183080059l_num1(Aa,times_7065122842183080059l_num1(Ba,C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_462_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [Ba: uint32,Aa: uint32,C: uint32] : times_times_uint32(Ba,times_times_uint32(Aa,C)) = times_times_uint32(Aa,times_times_uint32(Ba,C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_463_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [Aa: real,Ba: real] : aa_real_real(times_times_real(Aa),Ba) = aa_real_real(times_times_real(Ba),Aa) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_464_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [Aa: rat,Ba: rat] : times_times_rat(Aa,Ba) = times_times_rat(Ba,Aa) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_465_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Ba) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),Aa) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_466_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [Aa: int,Ba: int] : times_times_int(Aa,Ba) = times_times_int(Ba,Aa) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_467_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [Aa: word_N3645301735248828278l_num1,Ba: word_N3645301735248828278l_num1] : times_7065122842183080059l_num1(Aa,Ba) = times_7065122842183080059l_num1(Ba,Aa) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_468_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [Aa: uint32,Ba: uint32] : times_times_uint32(Aa,Ba) = times_times_uint32(Ba,Aa) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_469_mult_Oassoc,axiom,
    ! [Aa: real,Ba: real,C: real] : aa_real_real(times_times_real(aa_real_real(times_times_real(Aa),Ba)),C) = aa_real_real(times_times_real(Aa),aa_real_real(times_times_real(Ba),C)) ).

% mult.assoc
tff(fact_470_mult_Oassoc,axiom,
    ! [Aa: rat,Ba: rat,C: rat] : times_times_rat(times_times_rat(Aa,Ba),C) = times_times_rat(Aa,times_times_rat(Ba,C)) ).

% mult.assoc
tff(fact_471_mult_Oassoc,axiom,
    ! [Aa: nat,Ba: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Ba)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),C)) ).

% mult.assoc
tff(fact_472_mult_Oassoc,axiom,
    ! [Aa: int,Ba: int,C: int] : times_times_int(times_times_int(Aa,Ba),C) = times_times_int(Aa,times_times_int(Ba,C)) ).

% mult.assoc
tff(fact_473_mult_Oassoc,axiom,
    ! [Aa: word_N3645301735248828278l_num1,Ba: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] : times_7065122842183080059l_num1(times_7065122842183080059l_num1(Aa,Ba),C) = times_7065122842183080059l_num1(Aa,times_7065122842183080059l_num1(Ba,C)) ).

% mult.assoc
tff(fact_474_mult_Oassoc,axiom,
    ! [Aa: uint32,Ba: uint32,C: uint32] : times_times_uint32(times_times_uint32(Aa,Ba),C) = times_times_uint32(Aa,times_times_uint32(Ba,C)) ).

% mult.assoc
tff(fact_475_fold__if__return,axiom,
    ! [Ba: $o,C: nat,D: nat] :
      $ite((Ba),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,C),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,D)) = aa_nat5821310369566504398ap_nat(heap_Time_return_nat,
        $ite((Ba),C,D)) ).

% fold_if_return
tff(fact_476_fold__if__return,axiom,
    ! [Ba: $o,C: list_nat,D: list_nat] :
      $ite((Ba),aa_lis1877347954636394190st_nat(heap_T3164785745270913723st_nat,C),aa_lis1877347954636394190st_nat(heap_T3164785745270913723st_nat,D)) = aa_lis1877347954636394190st_nat(heap_T3164785745270913723st_nat,
        $ite((Ba),C,D)) ).

% fold_if_return
tff(fact_477_verit__la__generic,axiom,
    ! [Aa: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),X)
      | ( Aa = X )
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Aa) ) ).

% verit_la_generic
tff(fact_478_nat__int__comparison_I1_J,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( Aa = Ba )
    <=> ( aa_nat_int(semiri1314217659103216013at_int,Aa) = aa_nat_int(semiri1314217659103216013at_int,Ba) ) ) ).

% nat_int_comparison(1)
tff(fact_479_int__if,axiom,
    ! [P: $o,Aa: nat,Ba: nat] :
      aa_nat_int(semiri1314217659103216013at_int,
        $ite((P),Aa,Ba)) = $ite((P),aa_nat_int(semiri1314217659103216013at_int,Aa),aa_nat_int(semiri1314217659103216013at_int,Ba)) ).

% int_if
tff(fact_480_numeral__code_I3_J,axiom,
    ! [Na: num] :
      numera7442385471795722001l_num1(bit1(Na)) = $let(
        m: word_N3645301735248828278l_num1,
        m:= numera7442385471795722001l_num1(Na),
        plus_p361126936061061375l_num1(plus_p361126936061061375l_num1(m,m),one_on7727431528512463931l_num1) ) ).

% numeral_code(3)
tff(fact_481_numeral__code_I3_J,axiom,
    ! [Na: num] :
      numeral_numeral_real(bit1(Na)) = $let(
        m: real,
        m:= numeral_numeral_real(Na),
        aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m),m)),one_one_real) ) ).

% numeral_code(3)
tff(fact_482_numeral__code_I3_J,axiom,
    ! [Na: num] :
      numeral_numeral_rat(bit1(Na)) = $let(
        m: rat,
        m:= numeral_numeral_rat(Na),
        plus_plus_rat(plus_plus_rat(m,m),one_one_rat) ) ).

% numeral_code(3)
tff(fact_483_numeral__code_I3_J,axiom,
    ! [Na: num] :
      numeral_numeral_nat(bit1(Na)) = $let(
        m: nat,
        m:= numeral_numeral_nat(Na),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m),m)),one_one_nat) ) ).

% numeral_code(3)
tff(fact_484_numeral__code_I3_J,axiom,
    ! [Na: num] :
      aa_num_int(numeral_numeral_int,bit1(Na)) = $let(
        m: int,
        m:= aa_num_int(numeral_numeral_int,Na),
        plus_plus_int(plus_plus_int(m,m),one_one_int) ) ).

% numeral_code(3)
tff(fact_485_numeral__code_I3_J,axiom,
    ! [Na: num] :
      numera9087168376688890119uint32(bit1(Na)) = $let(
        m: uint32,
        m:= numera9087168376688890119uint32(Na),
        plus_plus_uint32(plus_plus_uint32(m,m),one_one_uint32) ) ).

% numeral_code(3)
tff(fact_486_nat__1__add__1,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = numeral_numeral_nat(bit0(one)) ).

% nat_1_add_1
tff(fact_487_distrib__if__bind,axiom,
    ! [Ba: $o,C: heap_T2636463487746394924on_nat,D: heap_T2636463487746394924on_nat,F: fun_op1688020532487384469ap_nat] :
      heap_T2919350798565477881at_nat($ite((Ba),C,D),F) = $ite((Ba),heap_T2919350798565477881at_nat(C,F),heap_T2919350798565477881at_nat(D,F)) ).

% distrib_if_bind
tff(fact_488_distrib__if__bind,axiom,
    ! [Ba: $o,C: heap_T8145700208782473153_VEBTi,D: heap_T8145700208782473153_VEBTi,F: fun_VE6418762861507722832st_nat] :
      heap_T3876627890315925662st_nat($ite((Ba),C,D),F) = $ite((Ba),heap_T3876627890315925662st_nat(C,F),heap_T3876627890315925662st_nat(D,F)) ).

% distrib_if_bind
tff(fact_489_add__le__imp__le__right,axiom,
    ! [Aa: rat,C: rat,Ba: rat] :
      ( ord_less_eq_rat(plus_plus_rat(Aa,C),plus_plus_rat(Ba,C))
     => ord_less_eq_rat(Aa,Ba) ) ).

% add_le_imp_le_right
tff(fact_490_add__le__imp__le__right,axiom,
    ! [Aa: nat,C: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),C))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba) ) ).

% add_le_imp_le_right
tff(fact_491_add__le__imp__le__right,axiom,
    ! [Aa: int,C: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,plus_plus_int(Aa,C)),plus_plus_int(Ba,C))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba) ) ).

% add_le_imp_le_right
tff(fact_492_add__le__imp__le__right,axiom,
    ! [Aa: real,C: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),C))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba) ) ).

% add_le_imp_le_right
tff(fact_493_add__le__imp__le__left,axiom,
    ! [C: rat,Aa: rat,Ba: rat] :
      ( ord_less_eq_rat(plus_plus_rat(C,Aa),plus_plus_rat(C,Ba))
     => ord_less_eq_rat(Aa,Ba) ) ).

% add_le_imp_le_left
tff(fact_494_add__le__imp__le__left,axiom,
    ! [C: nat,Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Ba))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba) ) ).

% add_le_imp_le_left
tff(fact_495_add__le__imp__le__left,axiom,
    ! [C: int,Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,plus_plus_int(C,Aa)),plus_plus_int(C,Ba))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba) ) ).

% add_le_imp_le_left
tff(fact_496_add__le__imp__le__left,axiom,
    ! [C: real,Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),Ba))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba) ) ).

% add_le_imp_le_left
tff(fact_497_le__iff__add,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
    <=> ? [C2: nat] : Ba = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C2) ) ).

% le_iff_add
tff(fact_498_add__right__mono,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( ord_less_eq_rat(Aa,Ba)
     => ord_less_eq_rat(plus_plus_rat(Aa,C),plus_plus_rat(Ba,C)) ) ).

% add_right_mono
tff(fact_499_add__right__mono,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),C)) ) ).

% add_right_mono
tff(fact_500_add__right__mono,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,plus_plus_int(Aa,C)),plus_plus_int(Ba,C)) ) ).

% add_right_mono
tff(fact_501_add__right__mono,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),C)) ) ).

% add_right_mono
tff(fact_502_less__eqE,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ~ ! [C3: nat] : Ba != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C3) ) ).

% less_eqE
tff(fact_503_add__left__mono,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( ord_less_eq_rat(Aa,Ba)
     => ord_less_eq_rat(plus_plus_rat(C,Aa),plus_plus_rat(C,Ba)) ) ).

% add_left_mono
tff(fact_504_add__left__mono,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Ba)) ) ).

% add_left_mono
tff(fact_505_add__left__mono,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,plus_plus_int(C,Aa)),plus_plus_int(C,Ba)) ) ).

% add_left_mono
tff(fact_506_add__left__mono,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),Ba)) ) ).

% add_left_mono
tff(fact_507_add__mono,axiom,
    ! [Aa: rat,Ba: rat,C: rat,D: rat] :
      ( ord_less_eq_rat(Aa,Ba)
     => ( ord_less_eq_rat(C,D)
       => ord_less_eq_rat(plus_plus_rat(Aa,C),plus_plus_rat(Ba,D)) ) ) ).

% add_mono
tff(fact_508_add__mono,axiom,
    ! [Aa: nat,Ba: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),D)) ) ) ).

% add_mono
tff(fact_509_add__mono,axiom,
    ! [Aa: int,Ba: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,plus_plus_int(Aa,C)),plus_plus_int(Ba,D)) ) ) ).

% add_mono
tff(fact_510_add__mono,axiom,
    ! [Aa: real,Ba: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),D)) ) ) ).

% add_mono
tff(fact_511_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ord_less_eq_rat(I,J)
        & ord_less_eq_rat(K,L) )
     => ord_less_eq_rat(plus_plus_rat(I,K),plus_plus_rat(J,L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_512_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_513_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,plus_plus_int(I,K)),plus_plus_int(J,L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_514_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_515_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ord_less_eq_rat(K,L) )
     => ord_less_eq_rat(plus_plus_rat(I,K),plus_plus_rat(J,L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_516_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_517_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,plus_plus_int(I,K)),plus_plus_int(J,L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_518_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_519_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ord_less_eq_rat(I,J)
        & ( K = L ) )
     => ord_less_eq_rat(plus_plus_rat(I,K),plus_plus_rat(J,L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_520_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & ( K = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_521_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & ( K = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,plus_plus_int(I,K)),plus_plus_int(J,L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_522_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & ( K = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_523_verit__eq__simplify_I10_J,axiom,
    ! [X23: num] : one != bit0(X23) ).

% verit_eq_simplify(10)
tff(fact_524_verit__eq__simplify_I14_J,axiom,
    ! [X23: num,X32: num] : bit0(X23) != bit1(X32) ).

% verit_eq_simplify(14)
tff(fact_525_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one != bit1(X32) ).

% verit_eq_simplify(12)
tff(fact_526_int__ops_I7_J,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Ba)) = times_times_int(aa_nat_int(semiri1314217659103216013at_int,Aa),aa_nat_int(semiri1314217659103216013at_int,Ba)) ).

% int_ops(7)
tff(fact_527_int__ops_I3_J,axiom,
    ! [Na: num] : aa_nat_int(semiri1314217659103216013at_int,numeral_numeral_nat(Na)) = aa_num_int(numeral_numeral_int,Na) ).

% int_ops(3)
tff(fact_528_list__decomp__2,axiom,
    ! [L: list_real] :
      ( ( size_size_list_real(L) = numeral_numeral_nat(bit0(one)) )
     => ? [A2: real,B2: real] : L = cons_real(A2,cons_real(B2,nil_real)) ) ).

% list_decomp_2
tff(fact_529_list__decomp__2,axiom,
    ! [L: list_o] :
      ( ( size_size_list_o(L) = numeral_numeral_nat(bit0(one)) )
     => ? [A2: $o,B2: $o] : L = cons_o((A2),cons_o((B2),nil_o)) ) ).

% list_decomp_2
tff(fact_530_list__decomp__2,axiom,
    ! [L: list_nat] :
      ( ( size_size_list_nat(L) = numeral_numeral_nat(bit0(one)) )
     => ? [A2: nat,B2: nat] : L = cons_nat(A2,cons_nat(B2,nil_nat)) ) ).

% list_decomp_2
tff(fact_531_list__decomp__2,axiom,
    ! [L: list_int] :
      ( ( size_size_list_int(L) = numeral_numeral_nat(bit0(one)) )
     => ? [A2: int,B2: int] : L = cons_int(A2,cons_int(B2,nil_int)) ) ).

% list_decomp_2
tff(fact_532_mmap_Oelims,axiom,
    ! [X: fun_na1043288745686030785ap_int,Xa2: list_nat,Y: heap_T7023682294697889864st_int] :
      ( ( vEBT_E4211082316275620141at_int(X,Xa2) = Y )
     => ( ( ( Xa2 = nil_nat )
         => ( Y != heap_T8210306762616492823st_int(nil_int) ) )
       => ~ ! [X2: nat,Xs: list_nat] :
              ( ( Xa2 = cons_nat(X2,Xs) )
             => ( Y != heap_T5172143623661458417st_int(aa_nat1643459350057307690ap_int(X,X2),aa_lis7701896087915484375st_int(aTP_Lamm_an(X),Xs)) ) ) ) ) ).

% mmap.elims
tff(fact_533_mmap_Oelims,axiom,
    ! [X: fun_na2343073368905879023Heap_o,Xa2: list_nat,Y: heap_T844314716496656296list_o] :
      ( ( vEBT_E1459915183938243159_nat_o(X,Xa2) = Y )
     => ( ( ( Xa2 = nil_nat )
         => ( Y != heap_T8290161705520373763list_o(nil_o) ) )
       => ~ ! [X2: nat,Xs: list_nat] :
              ( ( Xa2 = cons_nat(X2,Xs) )
             => ( Y != heap_T3790336413049173261list_o(aa_nat2910048432523430448Heap_o(X,X2),aa_lis540506999666758581list_o(aTP_Lamm_ap(X),Xs)) ) ) ) ) ).

% mmap.elims
tff(fact_534_mmap_Oelims,axiom,
    ! [X: fun_in1280865887336605797ap_int,Xa2: list_int,Y: heap_T7023682294697889864st_int] :
      ( ( vEBT_E5210436028176674697nt_int(X,Xa2) = Y )
     => ( ( ( Xa2 = nil_int )
         => ( Y != heap_T8210306762616492823st_int(nil_int) ) )
       => ~ ! [X2: int,Xs: list_int] :
              ( ( Xa2 = cons_int(X2,Xs) )
             => ( Y != heap_T5172143623661458417st_int(aa_int1881036491707882702ap_int(X,X2),aa_lis4505422320286930811st_int(aTP_Lamm_aq(X),Xs)) ) ) ) ) ).

% mmap.elims
tff(fact_535_mmap_Oelims,axiom,
    ! [X: fun_in6294517997321588299Heap_o,Xa2: list_int,Y: heap_T844314716496656296list_o] :
      ( ( vEBT_E8862244875360331003_int_o(X,Xa2) = Y )
     => ( ( ( Xa2 = nil_int )
         => ( Y != heap_T8290161705520373763list_o(nil_o) ) )
       => ~ ! [X2: int,Xs: list_int] :
              ( ( Xa2 = cons_int(X2,Xs) )
             => ( Y != heap_T3790336413049173261list_o(aa_int6861493060939139724Heap_o(X,X2),aa_lis3548058974061443161list_o(aTP_Lamm_ar(X),Xs)) ) ) ) ) ).

% mmap.elims
tff(fact_536_mmap_Oelims,axiom,
    ! [X: fun_o_1668507913897003235ap_int,Xa2: list_o,Y: heap_T7023682294697889864st_int] :
      ( ( vEBT_E7578125389175783381_o_int(X,Xa2) = Y )
     => ( ( ( Xa2 = nil_o )
         => ( Y != heap_T8210306762616492823st_int(nil_int) ) )
       => ~ ! [X2: $o,Xs: list_o] :
              ( ( Xa2 = cons_o((X2),Xs) )
             => ( Y != heap_T5172143623661458417st_int(aa_o_H4561945182497200476ap_int(X,(X2)),aa_lis8355408446276760467st_int(aTP_Lamm_as(X),Xs)) ) ) ) ) ).

% mmap.elims
tff(fact_537_mmap_Oelims,axiom,
    ! [X: fun_o_8325599348076409421Heap_o,Xa2: list_o,Y: heap_T844314716496656296list_o] :
      ( ( vEBT_E3732904050032093615ap_o_o(X,Xa2) = Y )
     => ( ( ( Xa2 = nil_o )
         => ( Y != heap_T8290161705520373763list_o(nil_o) ) )
       => ~ ! [X2: $o,Xs: list_o] :
              ( ( Xa2 = cons_o((X2),Xs) )
             => ( Y != heap_T3790336413049173261list_o(aa_o_H7399033255965945470Heap_o(X,(X2)),aa_lis4141098589498572913list_o(aTP_Lamm_at(X),Xs)) ) ) ) ) ).

% mmap.elims
tff(fact_538_mmap_Oelims,axiom,
    ! [X: fun_in3770949032268332553ap_nat,Xa2: list_int,Y: heap_T290393402774840812st_nat] :
      ( ( vEBT_E5212926498685724973nt_nat(X,Xa2) = Y )
     => ( ( ( Xa2 = nil_int )
         => ( Y != aa_lis1877347954636394190st_nat(heap_T3164785745270913723st_nat,nil_nat) ) )
       => ~ ! [X2: int,Xs: list_int] :
              ( ( Xa2 = cons_int(X2,Xs) )
             => ( Y != heap_T3341184992175080249st_nat(aa_int6058887511217079410ap_nat(X,X2),aa_lis1507422941235949179st_nat(aTP_Lamm_av(X),Xs)) ) ) ) ) ).

% mmap.elims
tff(fact_539_mmap_Oelims,axiom,
    ! [X: fun_o_4158591058828729991ap_nat,Xa2: list_o,Y: heap_T290393402774840812st_nat] :
      ( ( vEBT_E7580615859684833657_o_nat(X,Xa2) = Y )
     => ( ( ( Xa2 = nil_o )
         => ( Y != aa_lis1877347954636394190st_nat(heap_T3164785745270913723st_nat,nil_nat) ) )
       => ~ ! [X2: $o,Xs: list_o] :
              ( ( Xa2 = cons_o((X2),Xs) )
             => ( Y != heap_T3341184992175080249st_nat(aa_o_H8739796202006397184ap_nat(X,(X2)),aa_lis5357409067225778835st_nat(aTP_Lamm_aw(X),Xs)) ) ) ) ) ).

% mmap.elims
tff(fact_540_mmap_Oelims,axiom,
    ! [X: fun_na3533371890617757541ap_nat,Xa2: list_nat,Y: heap_T290393402774840812st_nat] :
      ( ( aa_lis1877347954636394190st_nat(vEBT_E4213572786784670417at_nat(X),Xa2) = Y )
     => ( ( ( Xa2 = nil_nat )
         => ( Y != aa_lis1877347954636394190st_nat(heap_T3164785745270913723st_nat,nil_nat) ) )
       => ~ ! [X2: nat,Xs: list_nat] :
              ( ( Xa2 = cons_nat(X2,Xs) )
             => ( Y != heap_T3341184992175080249st_nat(aa_nat5821310369566504398ap_nat(X,X2),aa_lis4703896708864502743st_nat(aTP_Lamm_ax(X),Xs)) ) ) ) ) ).

% mmap.elims
tff(fact_541_mmap_Osimps_I2_J,axiom,
    ! [F: fun_na1043288745686030785ap_int,X: nat,Xs2: list_nat] : vEBT_E4211082316275620141at_int(F,cons_nat(X,Xs2)) = heap_T5172143623661458417st_int(aa_nat1643459350057307690ap_int(F,X),aa_lis7701896087915484375st_int(aTP_Lamm_an(F),Xs2)) ).

% mmap.simps(2)
tff(fact_542_mmap_Osimps_I2_J,axiom,
    ! [F: fun_na2343073368905879023Heap_o,X: nat,Xs2: list_nat] : vEBT_E1459915183938243159_nat_o(F,cons_nat(X,Xs2)) = heap_T3790336413049173261list_o(aa_nat2910048432523430448Heap_o(F,X),aa_lis540506999666758581list_o(aTP_Lamm_ap(F),Xs2)) ).

% mmap.simps(2)
tff(fact_543_mmap_Osimps_I2_J,axiom,
    ! [F: fun_in1280865887336605797ap_int,X: int,Xs2: list_int] : vEBT_E5210436028176674697nt_int(F,cons_int(X,Xs2)) = heap_T5172143623661458417st_int(aa_int1881036491707882702ap_int(F,X),aa_lis4505422320286930811st_int(aTP_Lamm_aq(F),Xs2)) ).

% mmap.simps(2)
tff(fact_544_mmap_Osimps_I2_J,axiom,
    ! [F: fun_in6294517997321588299Heap_o,X: int,Xs2: list_int] : vEBT_E8862244875360331003_int_o(F,cons_int(X,Xs2)) = heap_T3790336413049173261list_o(aa_int6861493060939139724Heap_o(F,X),aa_lis3548058974061443161list_o(aTP_Lamm_ar(F),Xs2)) ).

% mmap.simps(2)
tff(fact_545_mmap_Osimps_I2_J,axiom,
    ! [F: fun_o_1668507913897003235ap_int,X: $o,Xs2: list_o] : vEBT_E7578125389175783381_o_int(F,cons_o((X),Xs2)) = heap_T5172143623661458417st_int(aa_o_H4561945182497200476ap_int(F,(X)),aa_lis8355408446276760467st_int(aTP_Lamm_as(F),Xs2)) ).

% mmap.simps(2)
tff(fact_546_mmap_Osimps_I2_J,axiom,
    ! [F: fun_o_8325599348076409421Heap_o,X: $o,Xs2: list_o] : vEBT_E3732904050032093615ap_o_o(F,cons_o((X),Xs2)) = heap_T3790336413049173261list_o(aa_o_H7399033255965945470Heap_o(F,(X)),aa_lis4141098589498572913list_o(aTP_Lamm_at(F),Xs2)) ).

% mmap.simps(2)
tff(fact_547_mmap_Osimps_I2_J,axiom,
    ! [F: fun_in3770949032268332553ap_nat,X: int,Xs2: list_int] : vEBT_E5212926498685724973nt_nat(F,cons_int(X,Xs2)) = heap_T3341184992175080249st_nat(aa_int6058887511217079410ap_nat(F,X),aa_lis1507422941235949179st_nat(aTP_Lamm_av(F),Xs2)) ).

% mmap.simps(2)
tff(fact_548_mmap_Osimps_I2_J,axiom,
    ! [F: fun_o_4158591058828729991ap_nat,X: $o,Xs2: list_o] : vEBT_E7580615859684833657_o_nat(F,cons_o((X),Xs2)) = heap_T3341184992175080249st_nat(aa_o_H8739796202006397184ap_nat(F,(X)),aa_lis5357409067225778835st_nat(aTP_Lamm_aw(F),Xs2)) ).

% mmap.simps(2)
tff(fact_549_mmap_Osimps_I2_J,axiom,
    ! [F: fun_na3533371890617757541ap_nat,X: nat,Xs2: list_nat] : aa_lis1877347954636394190st_nat(vEBT_E4213572786784670417at_nat(F),cons_nat(X,Xs2)) = heap_T3341184992175080249st_nat(aa_nat5821310369566504398ap_nat(F,X),aa_lis4703896708864502743st_nat(aTP_Lamm_ax(F),Xs2)) ).

% mmap.simps(2)
tff(fact_550_list__decomp__1,axiom,
    ! [L: list_real] :
      ( ( size_size_list_real(L) = one_one_nat )
     => ? [A2: real] : L = cons_real(A2,nil_real) ) ).

% list_decomp_1
tff(fact_551_list__decomp__1,axiom,
    ! [L: list_o] :
      ( ( size_size_list_o(L) = one_one_nat )
     => ? [A2: $o] : L = cons_o((A2),nil_o) ) ).

% list_decomp_1
tff(fact_552_list__decomp__1,axiom,
    ! [L: list_nat] :
      ( ( size_size_list_nat(L) = one_one_nat )
     => ? [A2: nat] : L = cons_nat(A2,nil_nat) ) ).

% list_decomp_1
tff(fact_553_list__decomp__1,axiom,
    ! [L: list_int] :
      ( ( size_size_list_int(L) = one_one_nat )
     => ? [A2: int] : L = cons_int(A2,nil_int) ) ).

% list_decomp_1
tff(fact_554_length__compl__induct,axiom,
    ! [P: fun_list_real_o,L: list_real] :
      ( aa_list_real_o(P,nil_real)
     => ( ! [E: real,L2: list_real] :
            ( ! [Ll: list_real] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_real(Ll)),size_size_list_real(L2))
               => aa_list_real_o(P,Ll) )
           => aa_list_real_o(P,cons_real(E,L2)) )
       => aa_list_real_o(P,L) ) ) ).

% length_compl_induct
tff(fact_555_length__compl__induct,axiom,
    ! [P: fun_list_o_o,L: list_o] :
      ( aa_list_o_o(P,nil_o)
     => ( ! [E: $o,L2: list_o] :
            ( ! [Ll: list_o] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Ll)),size_size_list_o(L2))
               => aa_list_o_o(P,Ll) )
           => aa_list_o_o(P,cons_o((E),L2)) )
       => aa_list_o_o(P,L) ) ) ).

% length_compl_induct
tff(fact_556_length__compl__induct,axiom,
    ! [P: fun_list_nat_o,L: list_nat] :
      ( aa_list_nat_o(P,nil_nat)
     => ( ! [E: nat,L2: list_nat] :
            ( ! [Ll: list_nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Ll)),size_size_list_nat(L2))
               => aa_list_nat_o(P,Ll) )
           => aa_list_nat_o(P,cons_nat(E,L2)) )
       => aa_list_nat_o(P,L) ) ) ).

% length_compl_induct
tff(fact_557_length__compl__induct,axiom,
    ! [P: fun_list_int_o,L: list_int] :
      ( aa_list_int_o(P,nil_int)
     => ( ! [E: int,L2: list_int] :
            ( ! [Ll: list_int] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Ll)),size_size_list_int(L2))
               => aa_list_int_o(P,Ll) )
           => aa_list_int_o(P,cons_int(E,L2)) )
       => aa_list_int_o(P,L) ) ) ).

% length_compl_induct
tff(fact_558_mmap_Osimps_I1_J,axiom,
    ! [F: fun_na1043288745686030785ap_int] : vEBT_E4211082316275620141at_int(F,nil_nat) = heap_T8210306762616492823st_int(nil_int) ).

% mmap.simps(1)
tff(fact_559_mmap_Osimps_I1_J,axiom,
    ! [F: fun_na2343073368905879023Heap_o] : vEBT_E1459915183938243159_nat_o(F,nil_nat) = heap_T8290161705520373763list_o(nil_o) ).

% mmap.simps(1)
tff(fact_560_mmap_Osimps_I1_J,axiom,
    ! [F: fun_in1280865887336605797ap_int] : vEBT_E5210436028176674697nt_int(F,nil_int) = heap_T8210306762616492823st_int(nil_int) ).

% mmap.simps(1)
tff(fact_561_mmap_Osimps_I1_J,axiom,
    ! [F: fun_in6294517997321588299Heap_o] : vEBT_E8862244875360331003_int_o(F,nil_int) = heap_T8290161705520373763list_o(nil_o) ).

% mmap.simps(1)
tff(fact_562_mmap_Osimps_I1_J,axiom,
    ! [F: fun_o_1668507913897003235ap_int] : vEBT_E7578125389175783381_o_int(F,nil_o) = heap_T8210306762616492823st_int(nil_int) ).

% mmap.simps(1)
tff(fact_563_mmap_Osimps_I1_J,axiom,
    ! [F: fun_o_8325599348076409421Heap_o] : vEBT_E3732904050032093615ap_o_o(F,nil_o) = heap_T8290161705520373763list_o(nil_o) ).

% mmap.simps(1)
tff(fact_564_mmap_Osimps_I1_J,axiom,
    ! [F: fun_in3770949032268332553ap_nat] : vEBT_E5212926498685724973nt_nat(F,nil_int) = aa_lis1877347954636394190st_nat(heap_T3164785745270913723st_nat,nil_nat) ).

% mmap.simps(1)
tff(fact_565_mmap_Osimps_I1_J,axiom,
    ! [F: fun_o_4158591058828729991ap_nat] : vEBT_E7580615859684833657_o_nat(F,nil_o) = aa_lis1877347954636394190st_nat(heap_T3164785745270913723st_nat,nil_nat) ).

% mmap.simps(1)
tff(fact_566_mmap_Osimps_I1_J,axiom,
    ! [F: fun_na3533371890617757541ap_nat] : aa_lis1877347954636394190st_nat(vEBT_E4213572786784670417at_nat(F),nil_nat) = aa_lis1877347954636394190st_nat(heap_T3164785745270913723st_nat,nil_nat) ).

% mmap.simps(1)
tff(fact_567_cond__TBOUND__return,axiom,
    ! [P: assn,X: nat] : aa_nat_o(simple7468303322168157607ND_nat(P,aa_nat5821310369566504398ap_nat(heap_Time_return_nat,X)),one_one_nat) ).

% cond_TBOUND_return
tff(fact_568_cond__TBOUND__return,axiom,
    ! [P: assn,X: list_nat] : aa_nat_o(simple8287821899582105911st_nat(P,aa_lis1877347954636394190st_nat(heap_T3164785745270913723st_nat,X)),one_one_nat) ).

% cond_TBOUND_return
tff(fact_569_cond__TBOUND__return,axiom,
    ! [P: assn,X: option_nat] : aa_nat_o(simple9182585030592575607on_nat(P,aa_opt9163293676379396302on_nat(heap_T3487192422709364219on_nat,X)),one_one_nat) ).

% cond_TBOUND_return
tff(fact_570_cond__TBOUND__return,axiom,
    ! [P: assn,X: $o] : aa_nat_o(simple83843512214091265OUND_o(P,aa_o_H7399033255965945470Heap_o(heap_Time_return_o,(X))),one_one_nat) ).

% cond_TBOUND_return
tff(fact_571_cond__TBOUND__return,axiom,
    ! [P: assn,X: vEBT_VEBTi] : aa_nat_o(simple9179169597231168140_VEBTi(P,aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,X)),one_one_nat) ).

% cond_TBOUND_return
tff(fact_572_dbl__inc__simps_I3_J,axiom,
    neg_nu8115118780965096967l_num1(one_on7727431528512463931l_num1) = numera7442385471795722001l_num1(bit1(one)) ).

% dbl_inc_simps(3)
tff(fact_573_dbl__inc__simps_I3_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,one_one_real) = numeral_numeral_real(bit1(one)) ).

% dbl_inc_simps(3)
tff(fact_574_dbl__inc__simps_I3_J,axiom,
    neg_nu5219082963157363817nc_rat(one_one_rat) = numeral_numeral_rat(bit1(one)) ).

% dbl_inc_simps(3)
tff(fact_575_dbl__inc__simps_I3_J,axiom,
    neg_nu5851722552734809277nc_int(one_one_int) = aa_num_int(numeral_numeral_int,bit1(one)) ).

% dbl_inc_simps(3)
tff(fact_576_dbl__inc__simps_I3_J,axiom,
    neg_nu4269007558841261821uint32(one_one_uint32) = numera9087168376688890119uint32(bit1(one)) ).

% dbl_inc_simps(3)
tff(fact_577_dbl__simps_I3_J,axiom,
    neg_nu7865238048354675525l_num1(one_on7727431528512463931l_num1) = numera7442385471795722001l_num1(bit0(one)) ).

% dbl_simps(3)
tff(fact_578_dbl__simps_I3_J,axiom,
    aa_real_real(neg_numeral_dbl_real,one_one_real) = numeral_numeral_real(bit0(one)) ).

% dbl_simps(3)
tff(fact_579_dbl__simps_I3_J,axiom,
    neg_numeral_dbl_rat(one_one_rat) = numeral_numeral_rat(bit0(one)) ).

% dbl_simps(3)
tff(fact_580_dbl__simps_I3_J,axiom,
    neg_numeral_dbl_int(one_one_int) = aa_num_int(numeral_numeral_int,bit0(one)) ).

% dbl_simps(3)
tff(fact_581_dbl__simps_I3_J,axiom,
    neg_nu5314729912787363643uint32(one_one_uint32) = numera9087168376688890119uint32(bit0(one)) ).

% dbl_simps(3)
tff(fact_582_mlex__leI,axiom,
    ! [Aa: nat,A3: nat,Ba: nat,B3: nat,N3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),B3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),N3)),Ba)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A3),N3)),B3)) ) ) ).

% mlex_leI
tff(fact_583_impossible__Cons,axiom,
    ! [Xs2: list_real,Ysb: list_real,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_real(Xs2)),size_size_list_real(Ysb))
     => ( Xs2 != cons_real(X,Ysb) ) ) ).

% impossible_Cons
tff(fact_584_impossible__Cons,axiom,
    ! [Xs2: list_o,Ysb: list_o,X: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Xs2)),size_size_list_o(Ysb))
     => ( Xs2 != cons_o((X),Ysb) ) ) ).

% impossible_Cons
tff(fact_585_impossible__Cons,axiom,
    ! [Xs2: list_nat,Ysb: list_nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Xs2)),size_size_list_nat(Ysb))
     => ( Xs2 != cons_nat(X,Ysb) ) ) ).

% impossible_Cons
tff(fact_586_impossible__Cons,axiom,
    ! [Xs2: list_int,Ysb: list_int,X: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Xs2)),size_size_list_int(Ysb))
     => ( Xs2 != cons_int(X,Ysb) ) ) ).

% impossible_Cons
tff(fact_587_list__induct2,axiom,
    ! [Xs2: list_real,Ysb: list_real,P: fun_li4346391027978563666real_o] :
      ( ( size_size_list_real(Xs2) = size_size_list_real(Ysb) )
     => ( aa_list_real_o(aa_lis6564241592085543739real_o(P,nil_real),nil_real)
       => ( ! [X2: real,Xs: list_real,Y2: real,Ys: list_real] :
              ( ( size_size_list_real(Xs) = size_size_list_real(Ys) )
             => ( aa_list_real_o(aa_lis6564241592085543739real_o(P,Xs),Ys)
               => aa_list_real_o(aa_lis6564241592085543739real_o(P,cons_real(X2,Xs)),cons_real(Y2,Ys)) ) )
         => aa_list_real_o(aa_lis6564241592085543739real_o(P,Xs2),Ysb) ) ) ) ).

% list_induct2
tff(fact_588_list__induct2,axiom,
    ! [Xs2: list_real,Ysb: list_o,P: fun_li8656390842604288118st_o_o] :
      ( ( size_size_list_real(Xs2) = size_size_list_o(Ysb) )
     => ( aa_list_o_o(aa_lis8056432215191216183st_o_o(P,nil_real),nil_o)
       => ( ! [X2: real,Xs: list_real,Y2: $o,Ys: list_o] :
              ( ( size_size_list_real(Xs) = size_size_list_o(Ys) )
             => ( aa_list_o_o(aa_lis8056432215191216183st_o_o(P,Xs),Ys)
               => aa_list_o_o(aa_lis8056432215191216183st_o_o(P,cons_real(X2,Xs)),cons_o((Y2),Ys)) ) )
         => aa_list_o_o(aa_lis8056432215191216183st_o_o(P,Xs2),Ysb) ) ) ) ).

% list_induct2
tff(fact_589_list__induct2,axiom,
    ! [Xs2: list_real,Ysb: list_nat,P: fun_li6994085003866780206_nat_o] :
      ( ( size_size_list_real(Xs2) = size_size_list_nat(Ysb) )
     => ( aa_list_nat_o(aa_lis8968956283152119063_nat_o(P,nil_real),nil_nat)
       => ( ! [X2: real,Xs: list_real,Y2: nat,Ys: list_nat] :
              ( ( size_size_list_real(Xs) = size_size_list_nat(Ys) )
             => ( aa_list_nat_o(aa_lis8968956283152119063_nat_o(P,Xs),Ys)
               => aa_list_nat_o(aa_lis8968956283152119063_nat_o(P,cons_real(X2,Xs)),cons_nat(Y2,Ys)) ) )
         => aa_list_nat_o(aa_lis8968956283152119063_nat_o(P,Xs2),Ysb) ) ) ) ).

% list_induct2
tff(fact_590_list__induct2,axiom,
    ! [Xs2: list_real,Ysb: list_int,P: fun_li7066035496089741522_int_o] :
      ( ( size_size_list_real(Xs2) = size_size_list_int(Ysb) )
     => ( aa_list_int_o(aa_lis6723811315481673147_int_o(P,nil_real),nil_int)
       => ( ! [X2: real,Xs: list_real,Y2: int,Ys: list_int] :
              ( ( size_size_list_real(Xs) = size_size_list_int(Ys) )
             => ( aa_list_int_o(aa_lis6723811315481673147_int_o(P,Xs),Ys)
               => aa_list_int_o(aa_lis6723811315481673147_int_o(P,cons_real(X2,Xs)),cons_int(Y2,Ys)) ) )
         => aa_list_int_o(aa_lis6723811315481673147_int_o(P,Xs2),Ysb) ) ) ) ).

% list_induct2
tff(fact_591_list__induct2,axiom,
    ! [Xs2: list_o,Ysb: list_real,P: fun_li5080777396398478298real_o] :
      ( ( size_size_list_o(Xs2) = size_size_list_real(Ysb) )
     => ( aa_list_real_o(aa_lis4736231737363803859real_o(P,nil_o),nil_real)
       => ( ! [X2: $o,Xs: list_o,Y2: real,Ys: list_real] :
              ( ( size_size_list_o(Xs) = size_size_list_real(Ys) )
             => ( aa_list_real_o(aa_lis4736231737363803859real_o(P,Xs),Ys)
               => aa_list_real_o(aa_lis4736231737363803859real_o(P,cons_o((X2),Xs)),cons_real(Y2,Ys)) ) )
         => aa_list_real_o(aa_lis4736231737363803859real_o(P,Xs2),Ysb) ) ) ) ).

% list_induct2
tff(fact_592_list__induct2,axiom,
    ! [Xs2: list_o,Ysb: list_o,P: fun_li4634889159120903918st_o_o] :
      ( ( size_size_list_o(Xs2) = size_size_list_o(Ysb) )
     => ( aa_list_o_o(aa_lis8221177634450426527st_o_o(P,nil_o),nil_o)
       => ( ! [X2: $o,Xs: list_o,Y2: $o,Ys: list_o] :
              ( ( size_size_list_o(Xs) = size_size_list_o(Ys) )
             => ( aa_list_o_o(aa_lis8221177634450426527st_o_o(P,Xs),Ys)
               => aa_list_o_o(aa_lis8221177634450426527st_o_o(P,cons_o((X2),Xs)),cons_o((Y2),Ys)) ) )
         => aa_list_o_o(aa_lis8221177634450426527st_o_o(P,Xs2),Ysb) ) ) ) ).

% list_induct2
tff(fact_593_list__induct2,axiom,
    ! [Xs2: list_o,Ysb: list_nat,P: fun_li6400911775203056054_nat_o] :
      ( ( size_size_list_o(Xs2) = size_size_list_nat(Ysb) )
     => ( aa_list_nat_o(aa_lis3624166925546173615_nat_o(P,nil_o),nil_nat)
       => ( ! [X2: $o,Xs: list_o,Y2: nat,Ys: list_nat] :
              ( ( size_size_list_o(Xs) = size_size_list_nat(Ys) )
             => ( aa_list_nat_o(aa_lis3624166925546173615_nat_o(P,Xs),Ys)
               => aa_list_nat_o(aa_lis3624166925546173615_nat_o(P,cons_o((X2),Xs)),cons_nat(Y2,Ys)) ) )
         => aa_list_nat_o(aa_lis3624166925546173615_nat_o(P,Xs2),Ysb) ) ) ) ).

% list_induct2
tff(fact_594_list__induct2,axiom,
    ! [Xs2: list_o,Ysb: list_int,P: fun_li6472862267426017370_int_o] :
      ( ( size_size_list_o(Xs2) = size_size_list_int(Ysb) )
     => ( aa_list_int_o(aa_lis1379021957875727699_int_o(P,nil_o),nil_int)
       => ( ! [X2: $o,Xs: list_o,Y2: int,Ys: list_int] :
              ( ( size_size_list_o(Xs) = size_size_list_int(Ys) )
             => ( aa_list_int_o(aa_lis1379021957875727699_int_o(P,Xs),Ys)
               => aa_list_int_o(aa_lis1379021957875727699_int_o(P,cons_o((X2),Xs)),cons_int(Y2,Ys)) ) )
         => aa_list_int_o(aa_lis1379021957875727699_int_o(P,Xs2),Ysb) ) ) ) ).

% list_induct2
tff(fact_595_list__induct2,axiom,
    ! [Xs2: list_nat,Ysb: list_real,P: fun_li7467302237394473774real_o] :
      ( ( size_size_list_nat(Xs2) = size_size_list_real(Ysb) )
     => ( aa_list_real_o(aa_lis5411116565844171287real_o(P,nil_nat),nil_real)
       => ( ! [X2: nat,Xs: list_nat,Y2: real,Ys: list_real] :
              ( ( size_size_list_nat(Xs) = size_size_list_real(Ys) )
             => ( aa_list_real_o(aa_lis5411116565844171287real_o(P,Xs),Ys)
               => aa_list_real_o(aa_lis5411116565844171287real_o(P,cons_nat(X2,Xs)),cons_real(Y2,Ys)) ) )
         => aa_list_real_o(aa_lis5411116565844171287real_o(P,Xs2),Ysb) ) ) ) ).

% list_induct2
tff(fact_596_list__induct2,axiom,
    ! [Xs2: list_nat,Ysb: list_o,P: fun_li920423630202665114st_o_o] :
      ( ( size_size_list_nat(Xs2) = size_size_list_o(Ysb) )
     => ( aa_list_o_o(aa_lis3080580898186519643st_o_o(P,nil_nat),nil_o)
       => ( ! [X2: nat,Xs: list_nat,Y2: $o,Ys: list_o] :
              ( ( size_size_list_nat(Xs) = size_size_list_o(Ys) )
             => ( aa_list_o_o(aa_lis3080580898186519643st_o_o(P,Xs),Ys)
               => aa_list_o_o(aa_lis3080580898186519643st_o_o(P,cons_nat(X2,Xs)),cons_o((Y2),Ys)) ) )
         => aa_list_o_o(aa_lis3080580898186519643st_o_o(P,Xs2),Ysb) ) ) ) ).

% list_induct2
tff(fact_597_list_Oinject,axiom,
    ! [X21: nat,X222: list_nat,Y21: nat,Y222: list_nat] :
      ( ( cons_nat(X21,X222) = cons_nat(Y21,Y222) )
    <=> ( ( X21 = Y21 )
        & ( X222 = Y222 ) ) ) ).

% list.inject
tff(fact_598_list_Oinject,axiom,
    ! [X21: int,X222: list_int,Y21: int,Y222: list_int] :
      ( ( cons_int(X21,X222) = cons_int(Y21,Y222) )
    <=> ( ( X21 = Y21 )
        & ( X222 = Y222 ) ) ) ).

% list.inject
tff(fact_599_list_Oinject,axiom,
    ! [X21: $o,X222: list_o,Y21: $o,Y222: list_o] :
      ( ( cons_o((X21),X222) = cons_o((Y21),Y222) )
    <=> ( ( (X21)
        <=> (Y21) )
        & ( X222 = Y222 ) ) ) ).

% list.inject
tff(fact_600_dbl__simps_I5_J,axiom,
    ! [K: num] : neg_nu7865238048354675525l_num1(numera7442385471795722001l_num1(K)) = numera7442385471795722001l_num1(bit0(K)) ).

% dbl_simps(5)
tff(fact_601_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_numeral_dbl_real,numeral_numeral_real(K)) = numeral_numeral_real(bit0(K)) ).

% dbl_simps(5)
tff(fact_602_dbl__simps_I5_J,axiom,
    ! [K: num] : neg_numeral_dbl_rat(numeral_numeral_rat(K)) = numeral_numeral_rat(bit0(K)) ).

% dbl_simps(5)
tff(fact_603_dbl__simps_I5_J,axiom,
    ! [K: num] : neg_numeral_dbl_int(aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,bit0(K)) ).

% dbl_simps(5)
tff(fact_604_dbl__simps_I5_J,axiom,
    ! [K: num] : neg_nu5314729912787363643uint32(numera9087168376688890119uint32(K)) = numera9087168376688890119uint32(bit0(K)) ).

% dbl_simps(5)
tff(fact_605_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : neg_nu8115118780965096967l_num1(numera7442385471795722001l_num1(K)) = numera7442385471795722001l_num1(bit1(K)) ).

% dbl_inc_simps(5)
tff(fact_606_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_nu8295874005876285629c_real,numeral_numeral_real(K)) = numeral_numeral_real(bit1(K)) ).

% dbl_inc_simps(5)
tff(fact_607_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : neg_nu5219082963157363817nc_rat(numeral_numeral_rat(K)) = numeral_numeral_rat(bit1(K)) ).

% dbl_inc_simps(5)
tff(fact_608_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : neg_nu5851722552734809277nc_int(aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,bit1(K)) ).

% dbl_inc_simps(5)
tff(fact_609_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : neg_nu4269007558841261821uint32(numera9087168376688890119uint32(K)) = numera9087168376688890119uint32(bit1(K)) ).

% dbl_inc_simps(5)
tff(fact_610_dbl__def,axiom,
    ! [X: real] : aa_real_real(neg_numeral_dbl_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),X) ).

% dbl_def
tff(fact_611_dbl__def,axiom,
    ! [X: rat] : neg_numeral_dbl_rat(X) = plus_plus_rat(X,X) ).

% dbl_def
tff(fact_612_dbl__def,axiom,
    ! [X: int] : neg_numeral_dbl_int(X) = plus_plus_int(X,X) ).

% dbl_def
tff(fact_613_dbl__def,axiom,
    ! [X: word_N3645301735248828278l_num1] : neg_nu7865238048354675525l_num1(X) = plus_p361126936061061375l_num1(X,X) ).

% dbl_def
tff(fact_614_dbl__def,axiom,
    ! [X: uint32] : neg_nu5314729912787363643uint32(X) = plus_plus_uint32(X,X) ).

% dbl_def
tff(fact_615_int__ops_I2_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% int_ops(2)
tff(fact_616_nat__one__as__int,axiom,
    one_one_nat = aa_int_nat(nat2,one_one_int) ).

% nat_one_as_int
tff(fact_617_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I)
     => ( aa_int_o(P,K)
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I2)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,plus_plus_int(I2,one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_ge_induct
tff(fact_618_ord__eq__le__eq__trans,axiom,
    ! [Aa: set_nat,Ba: set_nat,C: set_nat,D: set_nat] :
      ( ( Aa = Ba )
     => ( aa_set_nat_o(ord_less_eq_set_nat(Ba),C)
       => ( ( C = D )
         => aa_set_nat_o(ord_less_eq_set_nat(Aa),D) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_619_ord__eq__le__eq__trans,axiom,
    ! [Aa: rat,Ba: rat,C: rat,D: rat] :
      ( ( Aa = Ba )
     => ( ord_less_eq_rat(Ba,C)
       => ( ( C = D )
         => ord_less_eq_rat(Aa,D) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_620_ord__eq__le__eq__trans,axiom,
    ! [Aa: num,Ba: num,C: num,D: num] :
      ( ( Aa = Ba )
     => ( aa_num_o(ord_less_eq_num(Ba),C)
       => ( ( C = D )
         => aa_num_o(ord_less_eq_num(Aa),D) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_621_ord__eq__le__eq__trans,axiom,
    ! [Aa: nat,Ba: nat,C: nat,D: nat] :
      ( ( Aa = Ba )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),C)
       => ( ( C = D )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),D) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_622_ord__eq__le__eq__trans,axiom,
    ! [Aa: int,Ba: int,C: int,D: int] :
      ( ( Aa = Ba )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),C)
       => ( ( C = D )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),D) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_623_ord__eq__le__eq__trans,axiom,
    ! [Aa: real,Ba: real,C: real,D: real] :
      ( ( Aa = Ba )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),C)
       => ( ( C = D )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),D) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_624_not__Cons__self2,axiom,
    ! [X: nat,Xs2: list_nat] : cons_nat(X,Xs2) != Xs2 ).

% not_Cons_self2
tff(fact_625_not__Cons__self2,axiom,
    ! [X: int,Xs2: list_int] : cons_int(X,Xs2) != Xs2 ).

% not_Cons_self2
tff(fact_626_not__Cons__self2,axiom,
    ! [X: $o,Xs2: list_o] : cons_o((X),Xs2) != Xs2 ).

% not_Cons_self2
tff(fact_627_list__tail__coinc,axiom,
    ! [N1: nat,R1: list_nat,N22: nat,R2: list_nat] :
      ( ( cons_nat(N1,R1) = cons_nat(N22,R2) )
     => ( ( N1 = N22 )
        & ( R1 = R2 ) ) ) ).

% list_tail_coinc
tff(fact_628_list__tail__coinc,axiom,
    ! [N1: int,R1: list_int,N22: int,R2: list_int] :
      ( ( cons_int(N1,R1) = cons_int(N22,R2) )
     => ( ( N1 = N22 )
        & ( R1 = R2 ) ) ) ).

% list_tail_coinc
tff(fact_629_list__tail__coinc,axiom,
    ! [N1: $o,R1: list_o,N22: $o,R2: list_o] :
      ( ( cons_o((N1),R1) = cons_o((N22),R2) )
     => ( ( (N1)
        <=> (N22) )
        & ( R1 = R2 ) ) ) ).

% list_tail_coinc
tff(fact_630_neq__if__length__neq,axiom,
    ! [Xs2: list_real,Ysb: list_real] :
      ( ( size_size_list_real(Xs2) != size_size_list_real(Ysb) )
     => ( Xs2 != Ysb ) ) ).

% neq_if_length_neq
tff(fact_631_neq__if__length__neq,axiom,
    ! [Xs2: list_o,Ysb: list_o] :
      ( ( size_size_list_o(Xs2) != size_size_list_o(Ysb) )
     => ( Xs2 != Ysb ) ) ).

% neq_if_length_neq
tff(fact_632_neq__if__length__neq,axiom,
    ! [Xs2: list_nat,Ysb: list_nat] :
      ( ( size_size_list_nat(Xs2) != size_size_list_nat(Ysb) )
     => ( Xs2 != Ysb ) ) ).

% neq_if_length_neq
tff(fact_633_neq__if__length__neq,axiom,
    ! [Xs2: list_int,Ysb: list_int] :
      ( ( size_size_list_int(Xs2) != size_size_list_int(Ysb) )
     => ( Xs2 != Ysb ) ) ).

% neq_if_length_neq
tff(fact_634_Ex__list__of__length,axiom,
    ! [Na: nat] :
    ? [Xs: list_real] : size_size_list_real(Xs) = Na ).

% Ex_list_of_length
tff(fact_635_Ex__list__of__length,axiom,
    ! [Na: nat] :
    ? [Xs: list_o] : size_size_list_o(Xs) = Na ).

% Ex_list_of_length
tff(fact_636_Ex__list__of__length,axiom,
    ! [Na: nat] :
    ? [Xs: list_nat] : size_size_list_nat(Xs) = Na ).

% Ex_list_of_length
tff(fact_637_Ex__list__of__length,axiom,
    ! [Na: nat] :
    ? [Xs: list_int] : size_size_list_int(Xs) = Na ).

% Ex_list_of_length
tff(fact_638_dbl__inc__def,axiom,
    ! [X: real] : aa_real_real(neg_nu8295874005876285629c_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),X)),one_one_real) ).

% dbl_inc_def
tff(fact_639_dbl__inc__def,axiom,
    ! [X: rat] : neg_nu5219082963157363817nc_rat(X) = plus_plus_rat(plus_plus_rat(X,X),one_one_rat) ).

% dbl_inc_def
tff(fact_640_dbl__inc__def,axiom,
    ! [X: int] : neg_nu5851722552734809277nc_int(X) = plus_plus_int(plus_plus_int(X,X),one_one_int) ).

% dbl_inc_def
tff(fact_641_dbl__inc__def,axiom,
    ! [X: word_N3645301735248828278l_num1] : neg_nu8115118780965096967l_num1(X) = plus_p361126936061061375l_num1(plus_p361126936061061375l_num1(X,X),one_on7727431528512463931l_num1) ).

% dbl_inc_def
tff(fact_642_dbl__inc__def,axiom,
    ! [X: uint32] : neg_nu4269007558841261821uint32(X) = plus_plus_uint32(plus_plus_uint32(X,X),one_one_uint32) ).

% dbl_inc_def
tff(fact_643_list_Odistinct_I1_J,axiom,
    ! [X21: nat,X222: list_nat] : nil_nat != cons_nat(X21,X222) ).

% list.distinct(1)
tff(fact_644_list_Odistinct_I1_J,axiom,
    ! [X21: int,X222: list_int] : nil_int != cons_int(X21,X222) ).

% list.distinct(1)
tff(fact_645_list_Odistinct_I1_J,axiom,
    ! [X21: $o,X222: list_o] : nil_o != cons_o((X21),X222) ).

% list.distinct(1)
tff(fact_646_list_OdiscI,axiom,
    ! [List: list_nat,X21: nat,X222: list_nat] :
      ( ( List = cons_nat(X21,X222) )
     => ( List != nil_nat ) ) ).

% list.discI
tff(fact_647_list_OdiscI,axiom,
    ! [List: list_int,X21: int,X222: list_int] :
      ( ( List = cons_int(X21,X222) )
     => ( List != nil_int ) ) ).

% list.discI
tff(fact_648_list_OdiscI,axiom,
    ! [List: list_o,X21: $o,X222: list_o] :
      ( ( List = cons_o((X21),X222) )
     => ( List != nil_o ) ) ).

% list.discI
tff(fact_649_list_Oexhaust,axiom,
    ! [Y: list_nat] :
      ( ( Y != nil_nat )
     => ~ ! [X212: nat,X223: list_nat] : Y != cons_nat(X212,X223) ) ).

% list.exhaust
tff(fact_650_list_Oexhaust,axiom,
    ! [Y: list_int] :
      ( ( Y != nil_int )
     => ~ ! [X212: int,X223: list_int] : Y != cons_int(X212,X223) ) ).

% list.exhaust
tff(fact_651_list_Oexhaust,axiom,
    ! [Y: list_o] :
      ( ( Y != nil_o )
     => ~ ! [X212: $o,X223: list_o] : Y != cons_o((X212),X223) ) ).

% list.exhaust
tff(fact_652_min__list_Ocases,axiom,
    ! [X: list_nat] :
      ( ! [X2: nat,Xs: list_nat] : X != cons_nat(X2,Xs)
     => ( X = nil_nat ) ) ).

% min_list.cases
tff(fact_653_min__list_Ocases,axiom,
    ! [X: list_int] :
      ( ! [X2: int,Xs: list_int] : X != cons_int(X2,Xs)
     => ( X = nil_int ) ) ).

% min_list.cases
tff(fact_654_min__list_Ocases,axiom,
    ! [X: list_o] :
      ( ! [X2: $o,Xs: list_o] : X != cons_o((X2),Xs)
     => ( X = nil_o ) ) ).

% min_list.cases
tff(fact_655_remdups__adj_Ocases,axiom,
    ! [X: list_nat] :
      ( ( X != nil_nat )
     => ( ! [X2: nat] : X != cons_nat(X2,nil_nat)
       => ~ ! [X2: nat,Y2: nat,Xs: list_nat] : X != cons_nat(X2,cons_nat(Y2,Xs)) ) ) ).

% remdups_adj.cases
tff(fact_656_remdups__adj_Ocases,axiom,
    ! [X: list_int] :
      ( ( X != nil_int )
     => ( ! [X2: int] : X != cons_int(X2,nil_int)
       => ~ ! [X2: int,Y2: int,Xs: list_int] : X != cons_int(X2,cons_int(Y2,Xs)) ) ) ).

% remdups_adj.cases
tff(fact_657_remdups__adj_Ocases,axiom,
    ! [X: list_o] :
      ( ( X != nil_o )
     => ( ! [X2: $o] : X != cons_o((X2),nil_o)
       => ~ ! [X2: $o,Y2: $o,Xs: list_o] : X != cons_o((X2),cons_o((Y2),Xs)) ) ) ).

% remdups_adj.cases
tff(fact_658_neq__NilE,axiom,
    ! [L: list_nat] :
      ( ( L != nil_nat )
     => ~ ! [X2: nat,Xs: list_nat] : L != cons_nat(X2,Xs) ) ).

% neq_NilE
tff(fact_659_neq__NilE,axiom,
    ! [L: list_int] :
      ( ( L != nil_int )
     => ~ ! [X2: int,Xs: list_int] : L != cons_int(X2,Xs) ) ).

% neq_NilE
tff(fact_660_neq__NilE,axiom,
    ! [L: list_o] :
      ( ( L != nil_o )
     => ~ ! [X2: $o,Xs: list_o] : L != cons_o((X2),Xs) ) ).

% neq_NilE
tff(fact_661_neq__Nil__conv,axiom,
    ! [Xs2: list_nat] :
      ( ( Xs2 != nil_nat )
    <=> ? [Y4: nat,Ys2: list_nat] : Xs2 = cons_nat(Y4,Ys2) ) ).

% neq_Nil_conv
tff(fact_662_neq__Nil__conv,axiom,
    ! [Xs2: list_int] :
      ( ( Xs2 != nil_int )
    <=> ? [Y4: int,Ys2: list_int] : Xs2 = cons_int(Y4,Ys2) ) ).

% neq_Nil_conv
tff(fact_663_neq__Nil__conv,axiom,
    ! [Xs2: list_o] :
      ( ( Xs2 != nil_o )
    <=> ? [Y4: $o,Ys2: list_o] : Xs2 = cons_o((Y4),Ys2) ) ).

% neq_Nil_conv
tff(fact_664_list__induct2_H,axiom,
    ! [P: fun_li1025265955262268170_nat_o,Xs2: list_nat,Ysb: list_nat] :
      ( aa_list_nat_o(aa_lis557685189327716851_nat_o(P,nil_nat),nil_nat)
     => ( ! [X2: nat,Xs: list_nat] : aa_list_nat_o(aa_lis557685189327716851_nat_o(P,cons_nat(X2,Xs)),nil_nat)
       => ( ! [Y2: nat,Ys: list_nat] : aa_list_nat_o(aa_lis557685189327716851_nat_o(P,nil_nat),cons_nat(Y2,Ys))
         => ( ! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
                ( aa_list_nat_o(aa_lis557685189327716851_nat_o(P,Xs),Ys)
               => aa_list_nat_o(aa_lis557685189327716851_nat_o(P,cons_nat(X2,Xs)),cons_nat(Y2,Ys)) )
           => aa_list_nat_o(aa_lis557685189327716851_nat_o(P,Xs2),Ysb) ) ) ) ) ).

% list_induct2'
tff(fact_665_list__induct2_H,axiom,
    ! [P: fun_li1097216447485229486_int_o,Xs2: list_nat,Ysb: list_int] :
      ( aa_list_int_o(aa_lis7535912258512046743_int_o(P,nil_nat),nil_int)
     => ( ! [X2: nat,Xs: list_nat] : aa_list_int_o(aa_lis7535912258512046743_int_o(P,cons_nat(X2,Xs)),nil_int)
       => ( ! [Y2: int,Ys: list_int] : aa_list_int_o(aa_lis7535912258512046743_int_o(P,nil_nat),cons_int(Y2,Ys))
         => ( ! [X2: nat,Xs: list_nat,Y2: int,Ys: list_int] :
                ( aa_list_int_o(aa_lis7535912258512046743_int_o(P,Xs),Ys)
               => aa_list_int_o(aa_lis7535912258512046743_int_o(P,cons_nat(X2,Xs)),cons_int(Y2,Ys)) )
           => aa_list_int_o(aa_lis7535912258512046743_int_o(P,Xs2),Ysb) ) ) ) ) ).

% list_induct2'
tff(fact_666_list__induct2_H,axiom,
    ! [P: fun_li920423630202665114st_o_o,Xs2: list_nat,Ysb: list_o] :
      ( aa_list_o_o(aa_lis3080580898186519643st_o_o(P,nil_nat),nil_o)
     => ( ! [X2: nat,Xs: list_nat] : aa_list_o_o(aa_lis3080580898186519643st_o_o(P,cons_nat(X2,Xs)),nil_o)
       => ( ! [Y2: $o,Ys: list_o] : aa_list_o_o(aa_lis3080580898186519643st_o_o(P,nil_nat),cons_o((Y2),Ys))
         => ( ! [X2: nat,Xs: list_nat,Y2: $o,Ys: list_o] :
                ( aa_list_o_o(aa_lis3080580898186519643st_o_o(P,Xs),Ys)
               => aa_list_o_o(aa_lis3080580898186519643st_o_o(P,cons_nat(X2,Xs)),cons_o((Y2),Ys)) )
           => aa_list_o_o(aa_lis3080580898186519643st_o_o(P,Xs2),Ysb) ) ) ) ) ).

% list_induct2'
tff(fact_667_list__induct2_H,axiom,
    ! [P: fun_li1262843096912843182_nat_o,Xs2: list_int,Ysb: list_nat] :
      ( aa_list_nat_o(aa_lis795262330978291863_nat_o(P,nil_int),nil_nat)
     => ( ! [X2: int,Xs: list_int] : aa_list_nat_o(aa_lis795262330978291863_nat_o(P,cons_int(X2,Xs)),nil_nat)
       => ( ! [Y2: nat,Ys: list_nat] : aa_list_nat_o(aa_lis795262330978291863_nat_o(P,nil_int),cons_nat(Y2,Ys))
         => ( ! [X2: int,Xs: list_int,Y2: nat,Ys: list_nat] :
                ( aa_list_nat_o(aa_lis795262330978291863_nat_o(P,Xs),Ys)
               => aa_list_nat_o(aa_lis795262330978291863_nat_o(P,cons_int(X2,Xs)),cons_nat(Y2,Ys)) )
           => aa_list_nat_o(aa_lis795262330978291863_nat_o(P,Xs2),Ysb) ) ) ) ) ).

% list_induct2'
tff(fact_668_list__induct2_H,axiom,
    ! [P: fun_li1334793589135804498_int_o,Xs2: list_int,Ysb: list_int] :
      ( aa_list_int_o(aa_lis7773489400162621755_int_o(P,nil_int),nil_int)
     => ( ! [X2: int,Xs: list_int] : aa_list_int_o(aa_lis7773489400162621755_int_o(P,cons_int(X2,Xs)),nil_int)
       => ( ! [Y2: int,Ys: list_int] : aa_list_int_o(aa_lis7773489400162621755_int_o(P,nil_int),cons_int(Y2,Ys))
         => ( ! [X2: int,Xs: list_int,Y2: int,Ys: list_int] :
                ( aa_list_int_o(aa_lis7773489400162621755_int_o(P,Xs),Ys)
               => aa_list_int_o(aa_lis7773489400162621755_int_o(P,cons_int(X2,Xs)),cons_int(Y2,Ys)) )
           => aa_list_int_o(aa_lis7773489400162621755_int_o(P,Xs2),Ysb) ) ) ) ) ).

% list_induct2'
tff(fact_669_list__induct2_H,axiom,
    ! [P: fun_li4871868258618374390st_o_o,Xs2: list_int,Ysb: list_o] :
      ( aa_list_o_o(aa_lis7032025526602228919st_o_o(P,nil_int),nil_o)
     => ( ! [X2: int,Xs: list_int] : aa_list_o_o(aa_lis7032025526602228919st_o_o(P,cons_int(X2,Xs)),nil_o)
       => ( ! [Y2: $o,Ys: list_o] : aa_list_o_o(aa_lis7032025526602228919st_o_o(P,nil_int),cons_o((Y2),Ys))
         => ( ! [X2: int,Xs: list_int,Y2: $o,Ys: list_o] :
                ( aa_list_o_o(aa_lis7032025526602228919st_o_o(P,Xs),Ys)
               => aa_list_o_o(aa_lis7032025526602228919st_o_o(P,cons_int(X2,Xs)),cons_o((Y2),Ys)) )
           => aa_list_o_o(aa_lis7032025526602228919st_o_o(P,Xs2),Ysb) ) ) ) ) ).

% list_induct2'
tff(fact_670_list__induct2_H,axiom,
    ! [P: fun_li6400911775203056054_nat_o,Xs2: list_o,Ysb: list_nat] :
      ( aa_list_nat_o(aa_lis3624166925546173615_nat_o(P,nil_o),nil_nat)
     => ( ! [X2: $o,Xs: list_o] : aa_list_nat_o(aa_lis3624166925546173615_nat_o(P,cons_o((X2),Xs)),nil_nat)
       => ( ! [Y2: nat,Ys: list_nat] : aa_list_nat_o(aa_lis3624166925546173615_nat_o(P,nil_o),cons_nat(Y2,Ys))
         => ( ! [X2: $o,Xs: list_o,Y2: nat,Ys: list_nat] :
                ( aa_list_nat_o(aa_lis3624166925546173615_nat_o(P,Xs),Ys)
               => aa_list_nat_o(aa_lis3624166925546173615_nat_o(P,cons_o((X2),Xs)),cons_nat(Y2,Ys)) )
           => aa_list_nat_o(aa_lis3624166925546173615_nat_o(P,Xs2),Ysb) ) ) ) ) ).

% list_induct2'
tff(fact_671_list__induct2_H,axiom,
    ! [P: fun_li6472862267426017370_int_o,Xs2: list_o,Ysb: list_int] :
      ( aa_list_int_o(aa_lis1379021957875727699_int_o(P,nil_o),nil_int)
     => ( ! [X2: $o,Xs: list_o] : aa_list_int_o(aa_lis1379021957875727699_int_o(P,cons_o((X2),Xs)),nil_int)
       => ( ! [Y2: int,Ys: list_int] : aa_list_int_o(aa_lis1379021957875727699_int_o(P,nil_o),cons_int(Y2,Ys))
         => ( ! [X2: $o,Xs: list_o,Y2: int,Ys: list_int] :
                ( aa_list_int_o(aa_lis1379021957875727699_int_o(P,Xs),Ys)
               => aa_list_int_o(aa_lis1379021957875727699_int_o(P,cons_o((X2),Xs)),cons_int(Y2,Ys)) )
           => aa_list_int_o(aa_lis1379021957875727699_int_o(P,Xs2),Ysb) ) ) ) ) ).

% list_induct2'
tff(fact_672_list__induct2_H,axiom,
    ! [P: fun_li4634889159120903918st_o_o,Xs2: list_o,Ysb: list_o] :
      ( aa_list_o_o(aa_lis8221177634450426527st_o_o(P,nil_o),nil_o)
     => ( ! [X2: $o,Xs: list_o] : aa_list_o_o(aa_lis8221177634450426527st_o_o(P,cons_o((X2),Xs)),nil_o)
       => ( ! [Y2: $o,Ys: list_o] : aa_list_o_o(aa_lis8221177634450426527st_o_o(P,nil_o),cons_o((Y2),Ys))
         => ( ! [X2: $o,Xs: list_o,Y2: $o,Ys: list_o] :
                ( aa_list_o_o(aa_lis8221177634450426527st_o_o(P,Xs),Ys)
               => aa_list_o_o(aa_lis8221177634450426527st_o_o(P,cons_o((X2),Xs)),cons_o((Y2),Ys)) )
           => aa_list_o_o(aa_lis8221177634450426527st_o_o(P,Xs2),Ysb) ) ) ) ) ).

% list_induct2'
tff(fact_673_list__2pre__induct,axiom,
    ! [P: fun_li1025265955262268170_nat_o,W1: list_nat,W2: list_nat] :
      ( aa_list_nat_o(aa_lis557685189327716851_nat_o(P,nil_nat),nil_nat)
     => ( ! [E: nat,W12: list_nat,W22: list_nat] :
            ( aa_list_nat_o(aa_lis557685189327716851_nat_o(P,W12),W22)
           => aa_list_nat_o(aa_lis557685189327716851_nat_o(P,cons_nat(E,W12)),W22) )
       => ( ! [E: nat,W13: list_nat,W23: list_nat] :
              ( aa_list_nat_o(aa_lis557685189327716851_nat_o(P,W13),W23)
             => aa_list_nat_o(aa_lis557685189327716851_nat_o(P,W13),cons_nat(E,W23)) )
         => aa_list_nat_o(aa_lis557685189327716851_nat_o(P,W1),W2) ) ) ) ).

% list_2pre_induct
tff(fact_674_list__2pre__induct,axiom,
    ! [P: fun_li1097216447485229486_int_o,W1: list_nat,W2: list_int] :
      ( aa_list_int_o(aa_lis7535912258512046743_int_o(P,nil_nat),nil_int)
     => ( ! [E: nat,W12: list_nat,W22: list_int] :
            ( aa_list_int_o(aa_lis7535912258512046743_int_o(P,W12),W22)
           => aa_list_int_o(aa_lis7535912258512046743_int_o(P,cons_nat(E,W12)),W22) )
       => ( ! [E: int,W13: list_nat,W23: list_int] :
              ( aa_list_int_o(aa_lis7535912258512046743_int_o(P,W13),W23)
             => aa_list_int_o(aa_lis7535912258512046743_int_o(P,W13),cons_int(E,W23)) )
         => aa_list_int_o(aa_lis7535912258512046743_int_o(P,W1),W2) ) ) ) ).

% list_2pre_induct
tff(fact_675_list__2pre__induct,axiom,
    ! [P: fun_li920423630202665114st_o_o,W1: list_nat,W2: list_o] :
      ( aa_list_o_o(aa_lis3080580898186519643st_o_o(P,nil_nat),nil_o)
     => ( ! [E: nat,W12: list_nat,W22: list_o] :
            ( aa_list_o_o(aa_lis3080580898186519643st_o_o(P,W12),W22)
           => aa_list_o_o(aa_lis3080580898186519643st_o_o(P,cons_nat(E,W12)),W22) )
       => ( ! [E: $o,W13: list_nat,W23: list_o] :
              ( aa_list_o_o(aa_lis3080580898186519643st_o_o(P,W13),W23)
             => aa_list_o_o(aa_lis3080580898186519643st_o_o(P,W13),cons_o((E),W23)) )
         => aa_list_o_o(aa_lis3080580898186519643st_o_o(P,W1),W2) ) ) ) ).

% list_2pre_induct
tff(fact_676_list__2pre__induct,axiom,
    ! [P: fun_li1262843096912843182_nat_o,W1: list_int,W2: list_nat] :
      ( aa_list_nat_o(aa_lis795262330978291863_nat_o(P,nil_int),nil_nat)
     => ( ! [E: int,W12: list_int,W22: list_nat] :
            ( aa_list_nat_o(aa_lis795262330978291863_nat_o(P,W12),W22)
           => aa_list_nat_o(aa_lis795262330978291863_nat_o(P,cons_int(E,W12)),W22) )
       => ( ! [E: nat,W13: list_int,W23: list_nat] :
              ( aa_list_nat_o(aa_lis795262330978291863_nat_o(P,W13),W23)
             => aa_list_nat_o(aa_lis795262330978291863_nat_o(P,W13),cons_nat(E,W23)) )
         => aa_list_nat_o(aa_lis795262330978291863_nat_o(P,W1),W2) ) ) ) ).

% list_2pre_induct
tff(fact_677_list__2pre__induct,axiom,
    ! [P: fun_li1334793589135804498_int_o,W1: list_int,W2: list_int] :
      ( aa_list_int_o(aa_lis7773489400162621755_int_o(P,nil_int),nil_int)
     => ( ! [E: int,W12: list_int,W22: list_int] :
            ( aa_list_int_o(aa_lis7773489400162621755_int_o(P,W12),W22)
           => aa_list_int_o(aa_lis7773489400162621755_int_o(P,cons_int(E,W12)),W22) )
       => ( ! [E: int,W13: list_int,W23: list_int] :
              ( aa_list_int_o(aa_lis7773489400162621755_int_o(P,W13),W23)
             => aa_list_int_o(aa_lis7773489400162621755_int_o(P,W13),cons_int(E,W23)) )
         => aa_list_int_o(aa_lis7773489400162621755_int_o(P,W1),W2) ) ) ) ).

% list_2pre_induct
tff(fact_678_list__2pre__induct,axiom,
    ! [P: fun_li4871868258618374390st_o_o,W1: list_int,W2: list_o] :
      ( aa_list_o_o(aa_lis7032025526602228919st_o_o(P,nil_int),nil_o)
     => ( ! [E: int,W12: list_int,W22: list_o] :
            ( aa_list_o_o(aa_lis7032025526602228919st_o_o(P,W12),W22)
           => aa_list_o_o(aa_lis7032025526602228919st_o_o(P,cons_int(E,W12)),W22) )
       => ( ! [E: $o,W13: list_int,W23: list_o] :
              ( aa_list_o_o(aa_lis7032025526602228919st_o_o(P,W13),W23)
             => aa_list_o_o(aa_lis7032025526602228919st_o_o(P,W13),cons_o((E),W23)) )
         => aa_list_o_o(aa_lis7032025526602228919st_o_o(P,W1),W2) ) ) ) ).

% list_2pre_induct
tff(fact_679_list__2pre__induct,axiom,
    ! [P: fun_li6400911775203056054_nat_o,W1: list_o,W2: list_nat] :
      ( aa_list_nat_o(aa_lis3624166925546173615_nat_o(P,nil_o),nil_nat)
     => ( ! [E: $o,W12: list_o,W22: list_nat] :
            ( aa_list_nat_o(aa_lis3624166925546173615_nat_o(P,W12),W22)
           => aa_list_nat_o(aa_lis3624166925546173615_nat_o(P,cons_o((E),W12)),W22) )
       => ( ! [E: nat,W13: list_o,W23: list_nat] :
              ( aa_list_nat_o(aa_lis3624166925546173615_nat_o(P,W13),W23)
             => aa_list_nat_o(aa_lis3624166925546173615_nat_o(P,W13),cons_nat(E,W23)) )
         => aa_list_nat_o(aa_lis3624166925546173615_nat_o(P,W1),W2) ) ) ) ).

% list_2pre_induct
tff(fact_680_list__2pre__induct,axiom,
    ! [P: fun_li6472862267426017370_int_o,W1: list_o,W2: list_int] :
      ( aa_list_int_o(aa_lis1379021957875727699_int_o(P,nil_o),nil_int)
     => ( ! [E: $o,W12: list_o,W22: list_int] :
            ( aa_list_int_o(aa_lis1379021957875727699_int_o(P,W12),W22)
           => aa_list_int_o(aa_lis1379021957875727699_int_o(P,cons_o((E),W12)),W22) )
       => ( ! [E: int,W13: list_o,W23: list_int] :
              ( aa_list_int_o(aa_lis1379021957875727699_int_o(P,W13),W23)
             => aa_list_int_o(aa_lis1379021957875727699_int_o(P,W13),cons_int(E,W23)) )
         => aa_list_int_o(aa_lis1379021957875727699_int_o(P,W1),W2) ) ) ) ).

% list_2pre_induct
tff(fact_681_list__2pre__induct,axiom,
    ! [P: fun_li4634889159120903918st_o_o,W1: list_o,W2: list_o] :
      ( aa_list_o_o(aa_lis8221177634450426527st_o_o(P,nil_o),nil_o)
     => ( ! [E: $o,W12: list_o,W22: list_o] :
            ( aa_list_o_o(aa_lis8221177634450426527st_o_o(P,W12),W22)
           => aa_list_o_o(aa_lis8221177634450426527st_o_o(P,cons_o((E),W12)),W22) )
       => ( ! [E: $o,W13: list_o,W23: list_o] :
              ( aa_list_o_o(aa_lis8221177634450426527st_o_o(P,W13),W23)
             => aa_list_o_o(aa_lis8221177634450426527st_o_o(P,W13),cons_o((E),W23)) )
         => aa_list_o_o(aa_lis8221177634450426527st_o_o(P,W1),W2) ) ) ) ).

% list_2pre_induct
tff(fact_682_list__induct__first2,axiom,
    ! [P: fun_list_nat_o,Xs2: list_nat] :
      ( aa_list_nat_o(P,nil_nat)
     => ( ! [X2: nat] : aa_list_nat_o(P,cons_nat(X2,nil_nat))
       => ( ! [X1: nat,X22: nat,Xs: list_nat] :
              ( aa_list_nat_o(P,Xs)
             => aa_list_nat_o(P,cons_nat(X1,cons_nat(X22,Xs))) )
         => aa_list_nat_o(P,Xs2) ) ) ) ).

% list_induct_first2
tff(fact_683_list__induct__first2,axiom,
    ! [P: fun_list_int_o,Xs2: list_int] :
      ( aa_list_int_o(P,nil_int)
     => ( ! [X2: int] : aa_list_int_o(P,cons_int(X2,nil_int))
       => ( ! [X1: int,X22: int,Xs: list_int] :
              ( aa_list_int_o(P,Xs)
             => aa_list_int_o(P,cons_int(X1,cons_int(X22,Xs))) )
         => aa_list_int_o(P,Xs2) ) ) ) ).

% list_induct_first2
tff(fact_684_list__induct__first2,axiom,
    ! [P: fun_list_o_o,Xs2: list_o] :
      ( aa_list_o_o(P,nil_o)
     => ( ! [X2: $o] : aa_list_o_o(P,cons_o((X2),nil_o))
       => ( ! [X1: $o,X22: $o,Xs: list_o] :
              ( aa_list_o_o(P,Xs)
             => aa_list_o_o(P,cons_o((X1),cons_o((X22),Xs))) )
         => aa_list_o_o(P,Xs2) ) ) ) ).

% list_induct_first2
tff(fact_685_list__nonempty__induct,axiom,
    ! [Xs2: list_nat,P: fun_list_nat_o] :
      ( ( Xs2 != nil_nat )
     => ( ! [X2: nat] : aa_list_nat_o(P,cons_nat(X2,nil_nat))
       => ( ! [X2: nat,Xs: list_nat] :
              ( ( Xs != nil_nat )
             => ( aa_list_nat_o(P,Xs)
               => aa_list_nat_o(P,cons_nat(X2,Xs)) ) )
         => aa_list_nat_o(P,Xs2) ) ) ) ).

% list_nonempty_induct
tff(fact_686_list__nonempty__induct,axiom,
    ! [Xs2: list_int,P: fun_list_int_o] :
      ( ( Xs2 != nil_int )
     => ( ! [X2: int] : aa_list_int_o(P,cons_int(X2,nil_int))
       => ( ! [X2: int,Xs: list_int] :
              ( ( Xs != nil_int )
             => ( aa_list_int_o(P,Xs)
               => aa_list_int_o(P,cons_int(X2,Xs)) ) )
         => aa_list_int_o(P,Xs2) ) ) ) ).

% list_nonempty_induct
tff(fact_687_list__nonempty__induct,axiom,
    ! [Xs2: list_o,P: fun_list_o_o] :
      ( ( Xs2 != nil_o )
     => ( ! [X2: $o] : aa_list_o_o(P,cons_o((X2),nil_o))
       => ( ! [X2: $o,Xs: list_o] :
              ( ( Xs != nil_o )
             => ( aa_list_o_o(P,Xs)
               => aa_list_o_o(P,cons_o((X2),Xs)) ) )
         => aa_list_o_o(P,Xs2) ) ) ) ).

% list_nonempty_induct
tff(fact_688_mergesort__by__rel__merge__induct,axiom,
    ! [P: fun_li1025265955262268170_nat_o,R3: fun_nat_fun_nat_o,Xs2: list_nat,Ysb: list_nat] :
      ( ! [Xs: list_nat] : aa_list_nat_o(aa_lis557685189327716851_nat_o(P,Xs),nil_nat)
     => ( ! [Ys: list_nat] : aa_list_nat_o(aa_lis557685189327716851_nat_o(P,nil_nat),Ys)
       => ( ! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R3,X2),Y2)
             => ( aa_list_nat_o(aa_lis557685189327716851_nat_o(P,Xs),cons_nat(Y2,Ys))
               => aa_list_nat_o(aa_lis557685189327716851_nat_o(P,cons_nat(X2,Xs)),cons_nat(Y2,Ys)) ) )
         => ( ! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
                ( ~ aa_nat_o(aa_nat_fun_nat_o(R3,X2),Y2)
               => ( aa_list_nat_o(aa_lis557685189327716851_nat_o(P,cons_nat(X2,Xs)),Ys)
                 => aa_list_nat_o(aa_lis557685189327716851_nat_o(P,cons_nat(X2,Xs)),cons_nat(Y2,Ys)) ) )
           => aa_list_nat_o(aa_lis557685189327716851_nat_o(P,Xs2),Ysb) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_689_mergesort__by__rel__merge__induct,axiom,
    ! [P: fun_li1262843096912843182_nat_o,R3: fun_int_fun_nat_o,Xs2: list_int,Ysb: list_nat] :
      ( ! [Xs: list_int] : aa_list_nat_o(aa_lis795262330978291863_nat_o(P,Xs),nil_nat)
     => ( ! [Ys: list_nat] : aa_list_nat_o(aa_lis795262330978291863_nat_o(P,nil_int),Ys)
       => ( ! [X2: int,Xs: list_int,Y2: nat,Ys: list_nat] :
              ( aa_nat_o(aa_int_fun_nat_o(R3,X2),Y2)
             => ( aa_list_nat_o(aa_lis795262330978291863_nat_o(P,Xs),cons_nat(Y2,Ys))
               => aa_list_nat_o(aa_lis795262330978291863_nat_o(P,cons_int(X2,Xs)),cons_nat(Y2,Ys)) ) )
         => ( ! [X2: int,Xs: list_int,Y2: nat,Ys: list_nat] :
                ( ~ aa_nat_o(aa_int_fun_nat_o(R3,X2),Y2)
               => ( aa_list_nat_o(aa_lis795262330978291863_nat_o(P,cons_int(X2,Xs)),Ys)
                 => aa_list_nat_o(aa_lis795262330978291863_nat_o(P,cons_int(X2,Xs)),cons_nat(Y2,Ys)) ) )
           => aa_list_nat_o(aa_lis795262330978291863_nat_o(P,Xs2),Ysb) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_690_mergesort__by__rel__merge__induct,axiom,
    ! [P: fun_li6400911775203056054_nat_o,R3: fun_o_fun_nat_o,Xs2: list_o,Ysb: list_nat] :
      ( ! [Xs: list_o] : aa_list_nat_o(aa_lis3624166925546173615_nat_o(P,Xs),nil_nat)
     => ( ! [Ys: list_nat] : aa_list_nat_o(aa_lis3624166925546173615_nat_o(P,nil_o),Ys)
       => ( ! [X2: $o,Xs: list_o,Y2: nat,Ys: list_nat] :
              ( aa_nat_o(aa_o_fun_nat_o(R3,(X2)),Y2)
             => ( aa_list_nat_o(aa_lis3624166925546173615_nat_o(P,Xs),cons_nat(Y2,Ys))
               => aa_list_nat_o(aa_lis3624166925546173615_nat_o(P,cons_o((X2),Xs)),cons_nat(Y2,Ys)) ) )
         => ( ! [X2: $o,Xs: list_o,Y2: nat,Ys: list_nat] :
                ( ~ aa_nat_o(aa_o_fun_nat_o(R3,(X2)),Y2)
               => ( aa_list_nat_o(aa_lis3624166925546173615_nat_o(P,cons_o((X2),Xs)),Ys)
                 => aa_list_nat_o(aa_lis3624166925546173615_nat_o(P,cons_o((X2),Xs)),cons_nat(Y2,Ys)) ) )
           => aa_list_nat_o(aa_lis3624166925546173615_nat_o(P,Xs2),Ysb) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_691_mergesort__by__rel__merge__induct,axiom,
    ! [P: fun_li1097216447485229486_int_o,R3: fun_nat_fun_int_o,Xs2: list_nat,Ysb: list_int] :
      ( ! [Xs: list_nat] : aa_list_int_o(aa_lis7535912258512046743_int_o(P,Xs),nil_int)
     => ( ! [Ys: list_int] : aa_list_int_o(aa_lis7535912258512046743_int_o(P,nil_nat),Ys)
       => ( ! [X2: nat,Xs: list_nat,Y2: int,Ys: list_int] :
              ( aa_int_o(aa_nat_fun_int_o(R3,X2),Y2)
             => ( aa_list_int_o(aa_lis7535912258512046743_int_o(P,Xs),cons_int(Y2,Ys))
               => aa_list_int_o(aa_lis7535912258512046743_int_o(P,cons_nat(X2,Xs)),cons_int(Y2,Ys)) ) )
         => ( ! [X2: nat,Xs: list_nat,Y2: int,Ys: list_int] :
                ( ~ aa_int_o(aa_nat_fun_int_o(R3,X2),Y2)
               => ( aa_list_int_o(aa_lis7535912258512046743_int_o(P,cons_nat(X2,Xs)),Ys)
                 => aa_list_int_o(aa_lis7535912258512046743_int_o(P,cons_nat(X2,Xs)),cons_int(Y2,Ys)) ) )
           => aa_list_int_o(aa_lis7535912258512046743_int_o(P,Xs2),Ysb) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_692_mergesort__by__rel__merge__induct,axiom,
    ! [P: fun_li1334793589135804498_int_o,R3: fun_int_fun_int_o,Xs2: list_int,Ysb: list_int] :
      ( ! [Xs: list_int] : aa_list_int_o(aa_lis7773489400162621755_int_o(P,Xs),nil_int)
     => ( ! [Ys: list_int] : aa_list_int_o(aa_lis7773489400162621755_int_o(P,nil_int),Ys)
       => ( ! [X2: int,Xs: list_int,Y2: int,Ys: list_int] :
              ( aa_int_o(aa_int_fun_int_o(R3,X2),Y2)
             => ( aa_list_int_o(aa_lis7773489400162621755_int_o(P,Xs),cons_int(Y2,Ys))
               => aa_list_int_o(aa_lis7773489400162621755_int_o(P,cons_int(X2,Xs)),cons_int(Y2,Ys)) ) )
         => ( ! [X2: int,Xs: list_int,Y2: int,Ys: list_int] :
                ( ~ aa_int_o(aa_int_fun_int_o(R3,X2),Y2)
               => ( aa_list_int_o(aa_lis7773489400162621755_int_o(P,cons_int(X2,Xs)),Ys)
                 => aa_list_int_o(aa_lis7773489400162621755_int_o(P,cons_int(X2,Xs)),cons_int(Y2,Ys)) ) )
           => aa_list_int_o(aa_lis7773489400162621755_int_o(P,Xs2),Ysb) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_693_mergesort__by__rel__merge__induct,axiom,
    ! [P: fun_li6472862267426017370_int_o,R3: fun_o_fun_int_o,Xs2: list_o,Ysb: list_int] :
      ( ! [Xs: list_o] : aa_list_int_o(aa_lis1379021957875727699_int_o(P,Xs),nil_int)
     => ( ! [Ys: list_int] : aa_list_int_o(aa_lis1379021957875727699_int_o(P,nil_o),Ys)
       => ( ! [X2: $o,Xs: list_o,Y2: int,Ys: list_int] :
              ( aa_int_o(aa_o_fun_int_o(R3,(X2)),Y2)
             => ( aa_list_int_o(aa_lis1379021957875727699_int_o(P,Xs),cons_int(Y2,Ys))
               => aa_list_int_o(aa_lis1379021957875727699_int_o(P,cons_o((X2),Xs)),cons_int(Y2,Ys)) ) )
         => ( ! [X2: $o,Xs: list_o,Y2: int,Ys: list_int] :
                ( ~ aa_int_o(aa_o_fun_int_o(R3,(X2)),Y2)
               => ( aa_list_int_o(aa_lis1379021957875727699_int_o(P,cons_o((X2),Xs)),Ys)
                 => aa_list_int_o(aa_lis1379021957875727699_int_o(P,cons_o((X2),Xs)),cons_int(Y2,Ys)) ) )
           => aa_list_int_o(aa_lis1379021957875727699_int_o(P,Xs2),Ysb) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_694_mergesort__by__rel__merge__induct,axiom,
    ! [P: fun_li920423630202665114st_o_o,R3: fun_nat_fun_o_o,Xs2: list_nat,Ysb: list_o] :
      ( ! [Xs: list_nat] : aa_list_o_o(aa_lis3080580898186519643st_o_o(P,Xs),nil_o)
     => ( ! [Ys: list_o] : aa_list_o_o(aa_lis3080580898186519643st_o_o(P,nil_nat),Ys)
       => ( ! [X2: nat,Xs: list_nat,Y2: $o,Ys: list_o] :
              ( aa_o_o(aa_nat_fun_o_o(R3,X2),(Y2))
             => ( aa_list_o_o(aa_lis3080580898186519643st_o_o(P,Xs),cons_o((Y2),Ys))
               => aa_list_o_o(aa_lis3080580898186519643st_o_o(P,cons_nat(X2,Xs)),cons_o((Y2),Ys)) ) )
         => ( ! [X2: nat,Xs: list_nat,Y2: $o,Ys: list_o] :
                ( ~ aa_o_o(aa_nat_fun_o_o(R3,X2),(Y2))
               => ( aa_list_o_o(aa_lis3080580898186519643st_o_o(P,cons_nat(X2,Xs)),Ys)
                 => aa_list_o_o(aa_lis3080580898186519643st_o_o(P,cons_nat(X2,Xs)),cons_o((Y2),Ys)) ) )
           => aa_list_o_o(aa_lis3080580898186519643st_o_o(P,Xs2),Ysb) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_695_mergesort__by__rel__merge__induct,axiom,
    ! [P: fun_li4871868258618374390st_o_o,R3: fun_int_fun_o_o,Xs2: list_int,Ysb: list_o] :
      ( ! [Xs: list_int] : aa_list_o_o(aa_lis7032025526602228919st_o_o(P,Xs),nil_o)
     => ( ! [Ys: list_o] : aa_list_o_o(aa_lis7032025526602228919st_o_o(P,nil_int),Ys)
       => ( ! [X2: int,Xs: list_int,Y2: $o,Ys: list_o] :
              ( aa_o_o(aa_int_fun_o_o(R3,X2),(Y2))
             => ( aa_list_o_o(aa_lis7032025526602228919st_o_o(P,Xs),cons_o((Y2),Ys))
               => aa_list_o_o(aa_lis7032025526602228919st_o_o(P,cons_int(X2,Xs)),cons_o((Y2),Ys)) ) )
         => ( ! [X2: int,Xs: list_int,Y2: $o,Ys: list_o] :
                ( ~ aa_o_o(aa_int_fun_o_o(R3,X2),(Y2))
               => ( aa_list_o_o(aa_lis7032025526602228919st_o_o(P,cons_int(X2,Xs)),Ys)
                 => aa_list_o_o(aa_lis7032025526602228919st_o_o(P,cons_int(X2,Xs)),cons_o((Y2),Ys)) ) )
           => aa_list_o_o(aa_lis7032025526602228919st_o_o(P,Xs2),Ysb) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_696_mergesort__by__rel__merge__induct,axiom,
    ! [P: fun_li4634889159120903918st_o_o,R3: fun_o_fun_o_o,Xs2: list_o,Ysb: list_o] :
      ( ! [Xs: list_o] : aa_list_o_o(aa_lis8221177634450426527st_o_o(P,Xs),nil_o)
     => ( ! [Ys: list_o] : aa_list_o_o(aa_lis8221177634450426527st_o_o(P,nil_o),Ys)
       => ( ! [X2: $o,Xs: list_o,Y2: $o,Ys: list_o] :
              ( aa_o_o(aa_o_fun_o_o(R3,(X2)),(Y2))
             => ( aa_list_o_o(aa_lis8221177634450426527st_o_o(P,Xs),cons_o((Y2),Ys))
               => aa_list_o_o(aa_lis8221177634450426527st_o_o(P,cons_o((X2),Xs)),cons_o((Y2),Ys)) ) )
         => ( ! [X2: $o,Xs: list_o,Y2: $o,Ys: list_o] :
                ( ~ aa_o_o(aa_o_fun_o_o(R3,(X2)),(Y2))
               => ( aa_list_o_o(aa_lis8221177634450426527st_o_o(P,cons_o((X2),Xs)),Ys)
                 => aa_list_o_o(aa_lis8221177634450426527st_o_o(P,cons_o((X2),Xs)),cons_o((Y2),Ys)) ) )
           => aa_list_o_o(aa_lis8221177634450426527st_o_o(P,Xs2),Ysb) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_697_transpose_Ocases,axiom,
    ! [X: list_list_nat] :
      ( ( X != nil_list_nat )
     => ( ! [Xss: list_list_nat] : X != cons_list_nat(nil_nat,Xss)
       => ~ ! [X2: nat,Xs: list_nat,Xss: list_list_nat] : X != cons_list_nat(cons_nat(X2,Xs),Xss) ) ) ).

% transpose.cases
tff(fact_698_transpose_Ocases,axiom,
    ! [X: list_list_int] :
      ( ( X != nil_list_int )
     => ( ! [Xss: list_list_int] : X != cons_list_int(nil_int,Xss)
       => ~ ! [X2: int,Xs: list_int,Xss: list_list_int] : X != cons_list_int(cons_int(X2,Xs),Xss) ) ) ).

% transpose.cases
tff(fact_699_transpose_Ocases,axiom,
    ! [X: list_list_o] :
      ( ( X != nil_list_o )
     => ( ! [Xss: list_list_o] : X != cons_list_o(nil_o,Xss)
       => ~ ! [X2: $o,Xs: list_o,Xss: list_list_o] : X != cons_list_o(cons_o((X2),Xs),Xss) ) ) ).

% transpose.cases
tff(fact_700_mIf__def,axiom,
    ! [Ba: heap_Time_Heap_o,T: heap_Time_Heap_nat,E2: heap_Time_Heap_nat] : vEBT_Example_mIf_nat(Ba,T,E2) = heap_Time_bind_o_nat(Ba,aa_Hea3464058753474827431ap_nat(aTP_Lamm_ay(T),E2)) ).

% mIf_def
tff(fact_701_cond__TBOUND__mono,axiom,
    ! [P: assn,C: heap_T290393402774840812st_nat,Ba: nat,B3: nat] :
      ( aa_nat_o(simple8287821899582105911st_nat(P,C),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),B3)
       => aa_nat_o(simple8287821899582105911st_nat(P,C),B3) ) ) ).

% cond_TBOUND_mono
tff(fact_702_cond__TBOUND__mono,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Ba: nat,B3: nat] :
      ( aa_nat_o(simple9182585030592575607on_nat(P,C),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),B3)
       => aa_nat_o(simple9182585030592575607on_nat(P,C),B3) ) ) ).

% cond_TBOUND_mono
tff(fact_703_cond__TBOUND__mono,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Ba: nat,B3: nat] :
      ( aa_nat_o(simple83843512214091265OUND_o(P,C),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),B3)
       => aa_nat_o(simple83843512214091265OUND_o(P,C),B3) ) ) ).

% cond_TBOUND_mono
tff(fact_704_cond__TBOUND__mono,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Ba: nat,B3: nat] :
      ( aa_nat_o(simple9179169597231168140_VEBTi(P,C),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),B3)
       => aa_nat_o(simple9179169597231168140_VEBTi(P,C),B3) ) ) ).

% cond_TBOUND_mono
tff(fact_705_list__induct4,axiom,
    ! [Xs2: list_nat,Ysb: list_o,Zs: list_real,Ws: list_real,P: fun_li2803985326509624148real_o] :
      ( ( size_size_list_nat(Xs2) = size_size_list_o(Ysb) )
     => ( ( size_size_list_o(Ysb) = size_size_list_real(Zs) )
       => ( ( size_size_list_real(Zs) = size_size_list_real(Ws) )
         => ( aa_list_real_o(aa_lis6564241592085543739real_o(aa_lis3441229985154202864real_o(aa_lis7152931814914681877real_o(P,nil_nat),nil_o),nil_real),nil_real)
           => ( ! [X2: nat,Xs: list_nat,Y2: $o,Ys: list_o,Z3: real,Zs2: list_real,W3: real,Ws2: list_real] :
                  ( ( size_size_list_nat(Xs) = size_size_list_o(Ys) )
                 => ( ( size_size_list_o(Ys) = size_size_list_real(Zs2) )
                   => ( ( size_size_list_real(Zs2) = size_size_list_real(Ws2) )
                     => ( aa_list_real_o(aa_lis6564241592085543739real_o(aa_lis3441229985154202864real_o(aa_lis7152931814914681877real_o(P,Xs),Ys),Zs2),Ws2)
                       => aa_list_real_o(aa_lis6564241592085543739real_o(aa_lis3441229985154202864real_o(aa_lis7152931814914681877real_o(P,cons_nat(X2,Xs)),cons_o((Y2),Ys)),cons_real(Z3,Zs2)),cons_real(W3,Ws2)) ) ) ) )
             => aa_list_real_o(aa_lis6564241592085543739real_o(aa_lis3441229985154202864real_o(aa_lis7152931814914681877real_o(P,Xs2),Ysb),Zs),Ws) ) ) ) ) ) ).

% list_induct4
tff(fact_706_list__induct4,axiom,
    ! [Xs2: list_nat,Ysb: list_o,Zs: list_real,Ws: list_o,P: fun_li418994105990606880st_o_o] :
      ( ( size_size_list_nat(Xs2) = size_size_list_o(Ysb) )
     => ( ( size_size_list_o(Ysb) = size_size_list_real(Zs) )
       => ( ( size_size_list_real(Zs) = size_size_list_o(Ws) )
         => ( aa_list_o_o(aa_lis8056432215191216183st_o_o(aa_lis6899351466067655000st_o_o(aa_lis3174993183282415113st_o_o(P,nil_nat),nil_o),nil_real),nil_o)
           => ( ! [X2: nat,Xs: list_nat,Y2: $o,Ys: list_o,Z3: real,Zs2: list_real,W3: $o,Ws2: list_o] :
                  ( ( size_size_list_nat(Xs) = size_size_list_o(Ys) )
                 => ( ( size_size_list_o(Ys) = size_size_list_real(Zs2) )
                   => ( ( size_size_list_real(Zs2) = size_size_list_o(Ws2) )
                     => ( aa_list_o_o(aa_lis8056432215191216183st_o_o(aa_lis6899351466067655000st_o_o(aa_lis3174993183282415113st_o_o(P,Xs),Ys),Zs2),Ws2)
                       => aa_list_o_o(aa_lis8056432215191216183st_o_o(aa_lis6899351466067655000st_o_o(aa_lis3174993183282415113st_o_o(P,cons_nat(X2,Xs)),cons_o((Y2),Ys)),cons_real(Z3,Zs2)),cons_o((W3),Ws2)) ) ) ) )
             => aa_list_o_o(aa_lis8056432215191216183st_o_o(aa_lis6899351466067655000st_o_o(aa_lis3174993183282415113st_o_o(P,Xs2),Ysb),Zs),Ws) ) ) ) ) ) ).

% list_induct4
tff(fact_707_list__induct4,axiom,
    ! [Xs2: list_nat,Ysb: list_o,Zs: list_real,Ws: list_nat,P: fun_li2505530234024511280_nat_o] :
      ( ( size_size_list_nat(Xs2) = size_size_list_o(Ysb) )
     => ( ( size_size_list_o(Ysb) = size_size_list_real(Zs) )
       => ( ( size_size_list_real(Zs) = size_size_list_nat(Ws) )
         => ( aa_list_nat_o(aa_lis8968956283152119063_nat_o(aa_lis6616997229935472076_nat_o(aa_lis2286876058617023473_nat_o(P,nil_nat),nil_o),nil_real),nil_nat)
           => ( ! [X2: nat,Xs: list_nat,Y2: $o,Ys: list_o,Z3: real,Zs2: list_real,W3: nat,Ws2: list_nat] :
                  ( ( size_size_list_nat(Xs) = size_size_list_o(Ys) )
                 => ( ( size_size_list_o(Ys) = size_size_list_real(Zs2) )
                   => ( ( size_size_list_real(Zs2) = size_size_list_nat(Ws2) )
                     => ( aa_list_nat_o(aa_lis8968956283152119063_nat_o(aa_lis6616997229935472076_nat_o(aa_lis2286876058617023473_nat_o(P,Xs),Ys),Zs2),Ws2)
                       => aa_list_nat_o(aa_lis8968956283152119063_nat_o(aa_lis6616997229935472076_nat_o(aa_lis2286876058617023473_nat_o(P,cons_nat(X2,Xs)),cons_o((Y2),Ys)),cons_real(Z3,Zs2)),cons_nat(W3,Ws2)) ) ) ) )
             => aa_list_nat_o(aa_lis8968956283152119063_nat_o(aa_lis6616997229935472076_nat_o(aa_lis2286876058617023473_nat_o(P,Xs2),Ysb),Zs),Ws) ) ) ) ) ) ).

% list_induct4
tff(fact_708_list__induct4,axiom,
    ! [Xs2: list_nat,Ysb: list_o,Zs: list_real,Ws: list_int,P: fun_li3745517388504521172_int_o] :
      ( ( size_size_list_nat(Xs2) = size_size_list_o(Ysb) )
     => ( ( size_size_list_o(Ysb) = size_size_list_real(Zs) )
       => ( ( size_size_list_real(Zs) = size_size_list_int(Ws) )
         => ( aa_list_int_o(aa_lis6723811315481673147_int_o(aa_lis6688947722158433392_int_o(aa_lis1421439516909678741_int_o(P,nil_nat),nil_o),nil_real),nil_int)
           => ( ! [X2: nat,Xs: list_nat,Y2: $o,Ys: list_o,Z3: real,Zs2: list_real,W3: int,Ws2: list_int] :
                  ( ( size_size_list_nat(Xs) = size_size_list_o(Ys) )
                 => ( ( size_size_list_o(Ys) = size_size_list_real(Zs2) )
                   => ( ( size_size_list_real(Zs2) = size_size_list_int(Ws2) )
                     => ( aa_list_int_o(aa_lis6723811315481673147_int_o(aa_lis6688947722158433392_int_o(aa_lis1421439516909678741_int_o(P,Xs),Ys),Zs2),Ws2)
                       => aa_list_int_o(aa_lis6723811315481673147_int_o(aa_lis6688947722158433392_int_o(aa_lis1421439516909678741_int_o(P,cons_nat(X2,Xs)),cons_o((Y2),Ys)),cons_real(Z3,Zs2)),cons_int(W3,Ws2)) ) ) ) )
             => aa_list_int_o(aa_lis6723811315481673147_int_o(aa_lis6688947722158433392_int_o(aa_lis1421439516909678741_int_o(P,Xs2),Ysb),Zs),Ws) ) ) ) ) ) ).

% list_induct4
tff(fact_709_list__induct4,axiom,
    ! [Xs2: list_nat,Ysb: list_o,Zs: list_o,Ws: list_real,P: fun_li3450975573230380932real_o] :
      ( ( size_size_list_nat(Xs2) = size_size_list_o(Ysb) )
     => ( ( size_size_list_o(Ysb) = size_size_list_o(Zs) )
       => ( ( size_size_list_o(Zs) = size_size_list_real(Ws) )
         => ( aa_list_real_o(aa_lis4736231737363803859real_o(aa_lis3323738019861845180real_o(aa_lis4150974259312799597real_o(P,nil_nat),nil_o),nil_o),nil_real)
           => ( ! [X2: nat,Xs: list_nat,Y2: $o,Ys: list_o,Z3: $o,Zs2: list_o,W3: real,Ws2: list_real] :
                  ( ( size_size_list_nat(Xs) = size_size_list_o(Ys) )
                 => ( ( size_size_list_o(Ys) = size_size_list_o(Zs2) )
                   => ( ( size_size_list_o(Zs2) = size_size_list_real(Ws2) )
                     => ( aa_list_real_o(aa_lis4736231737363803859real_o(aa_lis3323738019861845180real_o(aa_lis4150974259312799597real_o(P,Xs),Ys),Zs2),Ws2)
                       => aa_list_real_o(aa_lis4736231737363803859real_o(aa_lis3323738019861845180real_o(aa_lis4150974259312799597real_o(P,cons_nat(X2,Xs)),cons_o((Y2),Ys)),cons_o((Z3),Zs2)),cons_real(W3,Ws2)) ) ) ) )
             => aa_list_real_o(aa_lis4736231737363803859real_o(aa_lis3323738019861845180real_o(aa_lis4150974259312799597real_o(P,Xs2),Ysb),Zs),Ws) ) ) ) ) ) ).

% list_induct4
tff(fact_710_list__induct4,axiom,
    ! [Xs2: list_nat,Ysb: list_o,Zs: list_o,Ws: list_o,P: fun_li452254204978466800st_o_o] :
      ( ( size_size_list_nat(Xs2) = size_size_list_o(Ysb) )
     => ( ( size_size_list_o(Ysb) = size_size_list_o(Zs) )
       => ( ( size_size_list_o(Zs) = size_size_list_o(Ws) )
         => ( aa_list_o_o(aa_lis8221177634450426527st_o_o(aa_lis735068535524438156st_o_o(aa_lis6671440321569411249st_o_o(P,nil_nat),nil_o),nil_o),nil_o)
           => ( ! [X2: nat,Xs: list_nat,Y2: $o,Ys: list_o,Z3: $o,Zs2: list_o,W3: $o,Ws2: list_o] :
                  ( ( size_size_list_nat(Xs) = size_size_list_o(Ys) )
                 => ( ( size_size_list_o(Ys) = size_size_list_o(Zs2) )
                   => ( ( size_size_list_o(Zs2) = size_size_list_o(Ws2) )
                     => ( aa_list_o_o(aa_lis8221177634450426527st_o_o(aa_lis735068535524438156st_o_o(aa_lis6671440321569411249st_o_o(P,Xs),Ys),Zs2),Ws2)
                       => aa_list_o_o(aa_lis8221177634450426527st_o_o(aa_lis735068535524438156st_o_o(aa_lis6671440321569411249st_o_o(P,cons_nat(X2,Xs)),cons_o((Y2),Ys)),cons_o((Z3),Zs2)),cons_o((W3),Ws2)) ) ) ) )
             => aa_list_o_o(aa_lis8221177634450426527st_o_o(aa_lis735068535524438156st_o_o(aa_lis6671440321569411249st_o_o(P,Xs2),Ysb),Zs),Ws) ) ) ) ) ) ).

% list_induct4
tff(fact_711_list__induct4,axiom,
    ! [Xs2: list_nat,Ysb: list_o,Zs: list_o,Ws: list_nat,P: fun_li2490675048253111136_nat_o] :
      ( ( size_size_list_nat(Xs2) = size_size_list_o(Ysb) )
     => ( ( size_size_list_o(Ysb) = size_size_list_o(Zs) )
       => ( ( size_size_list_o(Zs) = size_size_list_nat(Ws) )
         => ( aa_list_nat_o(aa_lis3624166925546173615_nat_o(aa_lis212630285342190488_nat_o(aa_lis6408773780859353929_nat_o(P,nil_nat),nil_o),nil_o),nil_nat)
           => ( ! [X2: nat,Xs: list_nat,Y2: $o,Ys: list_o,Z3: $o,Zs2: list_o,W3: nat,Ws2: list_nat] :
                  ( ( size_size_list_nat(Xs) = size_size_list_o(Ys) )
                 => ( ( size_size_list_o(Ys) = size_size_list_o(Zs2) )
                   => ( ( size_size_list_o(Zs2) = size_size_list_nat(Ws2) )
                     => ( aa_list_nat_o(aa_lis3624166925546173615_nat_o(aa_lis212630285342190488_nat_o(aa_lis6408773780859353929_nat_o(P,Xs),Ys),Zs2),Ws2)
                       => aa_list_nat_o(aa_lis3624166925546173615_nat_o(aa_lis212630285342190488_nat_o(aa_lis6408773780859353929_nat_o(P,cons_nat(X2,Xs)),cons_o((Y2),Ys)),cons_o((Z3),Zs2)),cons_nat(W3,Ws2)) ) ) ) )
             => aa_list_nat_o(aa_lis3624166925546173615_nat_o(aa_lis212630285342190488_nat_o(aa_lis6408773780859353929_nat_o(P,Xs2),Ysb),Zs),Ws) ) ) ) ) ) ).

% list_induct4
tff(fact_712_list__induct4,axiom,
    ! [Xs2: list_nat,Ysb: list_o,Zs: list_o,Ws: list_int,P: fun_li3730662202733121028_int_o] :
      ( ( size_size_list_nat(Xs2) = size_size_list_o(Ysb) )
     => ( ( size_size_list_o(Ysb) = size_size_list_o(Zs) )
       => ( ( size_size_list_o(Zs) = size_size_list_int(Ws) )
         => ( aa_list_int_o(aa_lis1379021957875727699_int_o(aa_lis284580777565151804_int_o(aa_lis5543337239152009197_int_o(P,nil_nat),nil_o),nil_o),nil_int)
           => ( ! [X2: nat,Xs: list_nat,Y2: $o,Ys: list_o,Z3: $o,Zs2: list_o,W3: int,Ws2: list_int] :
                  ( ( size_size_list_nat(Xs) = size_size_list_o(Ys) )
                 => ( ( size_size_list_o(Ys) = size_size_list_o(Zs2) )
                   => ( ( size_size_list_o(Zs2) = size_size_list_int(Ws2) )
                     => ( aa_list_int_o(aa_lis1379021957875727699_int_o(aa_lis284580777565151804_int_o(aa_lis5543337239152009197_int_o(P,Xs),Ys),Zs2),Ws2)
                       => aa_list_int_o(aa_lis1379021957875727699_int_o(aa_lis284580777565151804_int_o(aa_lis5543337239152009197_int_o(P,cons_nat(X2,Xs)),cons_o((Y2),Ys)),cons_o((Z3),Zs2)),cons_int(W3,Ws2)) ) ) ) )
             => aa_list_int_o(aa_lis1379021957875727699_int_o(aa_lis284580777565151804_int_o(aa_lis5543337239152009197_int_o(P,Xs2),Ysb),Zs),Ws) ) ) ) ) ) ).

% list_induct4
tff(fact_713_list__induct4,axiom,
    ! [Xs2: list_nat,Ysb: list_o,Zs: list_nat,Ws: list_real,P: fun_li8882586394930325552real_o] :
      ( ( size_size_list_nat(Xs2) = size_size_list_o(Ysb) )
     => ( ( size_size_list_o(Ysb) = size_size_list_nat(Zs) )
       => ( ( size_size_list_nat(Zs) = size_size_list_real(Ws) )
         => ( aa_list_real_o(aa_lis5411116565844171287real_o(aa_lis7090214463463165644real_o(aa_lis7090452776162487537real_o(P,nil_nat),nil_o),nil_nat),nil_real)
           => ( ! [X2: nat,Xs: list_nat,Y2: $o,Ys: list_o,Z3: nat,Zs2: list_nat,W3: real,Ws2: list_real] :
                  ( ( size_size_list_nat(Xs) = size_size_list_o(Ys) )
                 => ( ( size_size_list_o(Ys) = size_size_list_nat(Zs2) )
                   => ( ( size_size_list_nat(Zs2) = size_size_list_real(Ws2) )
                     => ( aa_list_real_o(aa_lis5411116565844171287real_o(aa_lis7090214463463165644real_o(aa_lis7090452776162487537real_o(P,Xs),Ys),Zs2),Ws2)
                       => aa_list_real_o(aa_lis5411116565844171287real_o(aa_lis7090214463463165644real_o(aa_lis7090452776162487537real_o(P,cons_nat(X2,Xs)),cons_o((Y2),Ys)),cons_nat(Z3,Zs2)),cons_real(W3,Ws2)) ) ) ) )
             => aa_list_real_o(aa_lis5411116565844171287real_o(aa_lis7090214463463165644real_o(aa_lis7090452776162487537real_o(P,Xs2),Ysb),Zs),Ws) ) ) ) ) ) ).

% list_induct4
tff(fact_714_list__induct4,axiom,
    ! [Xs2: list_nat,Ysb: list_o,Zs: list_nat,Ws: list_o,P: fun_li2959864849711494724st_o_o] :
      ( ( size_size_list_nat(Xs2) = size_size_list_o(Ysb) )
     => ( ( size_size_list_o(Ysb) = size_size_list_nat(Zs) )
       => ( ( size_size_list_nat(Zs) = size_size_list_o(Ws) )
         => ( aa_list_o_o(aa_lis3080580898186519643st_o_o(aa_lis3955514177196575356st_o_o(aa_lis1347999119914788909st_o_o(P,nil_nat),nil_o),nil_nat),nil_o)
           => ( ! [X2: nat,Xs: list_nat,Y2: $o,Ys: list_o,Z3: nat,Zs2: list_nat,W3: $o,Ws2: list_o] :
                  ( ( size_size_list_nat(Xs) = size_size_list_o(Ys) )
                 => ( ( size_size_list_o(Ys) = size_size_list_nat(Zs2) )
                   => ( ( size_size_list_nat(Zs2) = size_size_list_o(Ws2) )
                     => ( aa_list_o_o(aa_lis3080580898186519643st_o_o(aa_lis3955514177196575356st_o_o(aa_lis1347999119914788909st_o_o(P,Xs),Ys),Zs2),Ws2)
                       => aa_list_o_o(aa_lis3080580898186519643st_o_o(aa_lis3955514177196575356st_o_o(aa_lis1347999119914788909st_o_o(P,cons_nat(X2,Xs)),cons_o((Y2),Ys)),cons_nat(Z3,Zs2)),cons_o((W3),Ws2)) ) ) ) )
             => aa_list_o_o(aa_lis3080580898186519643st_o_o(aa_lis3955514177196575356st_o_o(aa_lis1347999119914788909st_o_o(P,Xs2),Ysb),Zs),Ws) ) ) ) ) ) ).

% list_induct4
tff(fact_715_uint32_Osize__eq,axiom,
    ! [P2: uint32] : size_size_uint32(P2) = numeral_numeral_nat(bit0(bit0(bit0(bit0(bit0(one)))))) ).

% uint32.size_eq
tff(fact_716_list__encode_Ocases,axiom,
    ! [X: list_nat] :
      ( ( X != nil_nat )
     => ~ ! [X2: nat,Xs: list_nat] : X != cons_nat(X2,Xs) ) ).

% list_encode.cases
tff(fact_717_plus__and__or,axiom,
    ! [X: int,Y: int] : plus_plus_int(bit_se725231765392027082nd_int(X,Y),bit_se1409905431419307370or_int(X,Y)) = plus_plus_int(X,Y) ).

% plus_and_or
tff(fact_718_or__nat__def,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(bit_se1412395901928357646or_nat(M),Na) = aa_int_nat(nat2,bit_se1409905431419307370or_int(aa_nat_int(semiri1314217659103216013at_int,M),aa_nat_int(semiri1314217659103216013at_int,Na))) ).

% or_nat_def
tff(fact_719_xor__nat__def,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(bit_se6528837805403552850or_nat(M),Na) = aa_int_nat(nat2,bit_se6526347334894502574or_int(aa_nat_int(semiri1314217659103216013at_int,M),aa_nat_int(semiri1314217659103216013at_int,Na))) ).

% xor_nat_def
tff(fact_720_and__nat__def,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(bit_se727722235901077358nd_nat(M),Na) = aa_int_nat(nat2,bit_se725231765392027082nd_int(aa_nat_int(semiri1314217659103216013at_int,M),aa_nat_int(semiri1314217659103216013at_int,Na))) ).

% and_nat_def
tff(fact_721_or__not__numerals_I8_J,axiom,
    ! [M: num,Na: num] : bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,bit1(M)),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,bit0(Na)))) = plus_plus_int(one_one_int,times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,Na))))) ).

% or_not_numerals(8)
tff(fact_722_or__not__numerals_I9_J,axiom,
    ! [M: num,Na: num] : bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,bit1(M)),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,bit1(Na)))) = plus_plus_int(one_one_int,times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,Na))))) ).

% or_not_numerals(9)
tff(fact_723_and__not__numerals_I8_J,axiom,
    ! [M: num,Na: num] : bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,bit1(M)),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,bit0(Na)))) = plus_plus_int(one_one_int,times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,Na))))) ).

% and_not_numerals(8)
tff(fact_724_vebt__memberi__rule,axiom,
    ! [Na: nat,S: set_nat,Tia: vEBT_VEBTi,X: nat] : aa_nat_o(time_htt_o(aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Na,S),Tia),aa_nat2910048432523430448Heap_o(vEBT_vebt_memberi(Tia),X),aa_nat_fun_o_assn(aa_VEB8176250900481067561o_assn(aa_set2974300032405964540o_assn(aTP_Lamm_az(Na),S),Tia),X)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit1(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit1(bit0(one)))),aa_int_nat(nat2,archim7802044766580827645g_real(aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Na))))))) ).

% vebt_memberi_rule
tff(fact_725_le__log2__of__power,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na)),M)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,M))) ) ).

% le_log2_of_power
tff(fact_726_negative__zle,axiom,
    ! [Na: nat,M: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na))),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% negative_zle
tff(fact_727_and__minus__numerals_I6_J,axiom,
    ! [Na: num] : bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit1(Na))),one_one_int) = one_one_int ).

% and_minus_numerals(6)
tff(fact_728_and__minus__numerals_I2_J,axiom,
    ! [Na: num] : bit_se725231765392027082nd_int(one_one_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit1(Na)))) = one_one_int ).

% and_minus_numerals(2)
tff(fact_729_or__minus__numerals_I6_J,axiom,
    ! [Na: num] : bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit1(Na))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit1(Na))) ).

% or_minus_numerals(6)
tff(fact_730_or__minus__numerals_I2_J,axiom,
    ! [Na: num] : bit_se1409905431419307370or_int(one_one_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit1(Na)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit1(Na))) ).

% or_minus_numerals(2)
tff(fact_731_or__int__def,axiom,
    ! [K: int,L: int] : bit_se1409905431419307370or_int(K,L) = bit_ri7919022796975470100ot_int(bit_se725231765392027082nd_int(bit_ri7919022796975470100ot_int(K),bit_ri7919022796975470100ot_int(L))) ).

% or_int_def
tff(fact_732_xor__int__def,axiom,
    ! [K: int,L: int] : bit_se6526347334894502574or_int(K,L) = bit_se1409905431419307370or_int(bit_se725231765392027082nd_int(K,bit_ri7919022796975470100ot_int(L)),bit_se725231765392027082nd_int(bit_ri7919022796975470100ot_int(K),L)) ).

% xor_int_def
tff(fact_733_real__minus__mult__self__le,axiom,
    ! [U: real,X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(U),U))),aa_real_real(times_times_real(X),X)) ).

% real_minus_mult_self_le
tff(fact_734_int__cases2,axiom,
    ! [Z: int] :
      ( ! [N: nat] : Z != aa_nat_int(semiri1314217659103216013at_int,N)
     => ~ ! [N: nat] : Z != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) ) ).

% int_cases2
tff(fact_735_zmult__eq__1__iff,axiom,
    ! [M: int,Na: int] :
      ( ( times_times_int(M,Na) = one_one_int )
    <=> ( ( ( M = one_one_int )
          & ( Na = one_one_int ) )
        | ( ( M = aa_int_int(uminus_uminus_int,one_one_int) )
          & ( Na = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_736_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,Na: int] :
      ( ( times_times_int(M,Na) = one_one_int )
     => ( ( M = one_one_int )
        | ( M = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_737_and__not__numerals_I4_J,axiom,
    ! [M: num] : bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,bit0(M)),bit_ri7919022796975470100ot_int(one_one_int)) = aa_num_int(numeral_numeral_int,bit0(M)) ).

% and_not_numerals(4)
tff(fact_738_and__not__numerals_I2_J,axiom,
    ! [Na: num] : bit_se725231765392027082nd_int(one_one_int,bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,bit0(Na)))) = one_one_int ).

% and_not_numerals(2)
tff(fact_739_or__not__numerals_I4_J,axiom,
    ! [M: num] : bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,bit0(M)),bit_ri7919022796975470100ot_int(one_one_int)) = bit_ri7919022796975470100ot_int(one_one_int) ).

% or_not_numerals(4)
tff(fact_740_or__not__numerals_I2_J,axiom,
    ! [Na: num] : bit_se1409905431419307370or_int(one_one_int,bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,bit0(Na)))) = bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,bit0(Na))) ).

% or_not_numerals(2)
tff(fact_741_and__not__numerals_I5_J,axiom,
    ! [M: num,Na: num] : bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,bit0(M)),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,bit0(Na)))) = times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,Na)))) ).

% and_not_numerals(5)
tff(fact_742_and__not__numerals_I7_J,axiom,
    ! [M: num] : bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,bit1(M)),bit_ri7919022796975470100ot_int(one_one_int)) = aa_num_int(numeral_numeral_int,bit0(M)) ).

% and_not_numerals(7)
tff(fact_743_or__not__numerals_I3_J,axiom,
    ! [Na: num] : bit_se1409905431419307370or_int(one_one_int,bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,bit1(Na)))) = bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,bit0(Na))) ).

% or_not_numerals(3)
tff(fact_744_log2__of__power__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ( M = aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na) )
     => ( aa_nat_real(semiri5074537144036343181t_real,Na) = aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,M)) ) ) ).

% log2_of_power_eq
tff(fact_745_and__not__numerals_I6_J,axiom,
    ! [M: num,Na: num] : bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,bit0(M)),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,bit1(Na)))) = times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,Na)))) ).

% and_not_numerals(6)
tff(fact_746_and__not__numerals_I9_J,axiom,
    ! [M: num,Na: num] : bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,bit1(M)),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,bit1(Na)))) = times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,Na)))) ).

% and_not_numerals(9)
tff(fact_747_or__not__numerals_I6_J,axiom,
    ! [M: num,Na: num] : bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,bit0(M)),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,bit1(Na)))) = times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,Na)))) ).

% or_not_numerals(6)
tff(fact_748_or__not__numerals_I5_J,axiom,
    ! [M: num,Na: num] : bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,bit0(M)),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,bit0(Na)))) = plus_plus_int(one_one_int,times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,Na))))) ).

% or_not_numerals(5)
tff(fact_749_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,Na: nat] :
      ( ( aa_int_nat(nat2,Y) = aa_nat_nat(power_power_nat(numeral_numeral_nat(X)),Na) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_750_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X: num,Na: nat,Y: int] :
      ( ( aa_nat_nat(power_power_nat(numeral_numeral_nat(X)),Na) = aa_int_nat(nat2,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na) = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_751_nat__le__numeral__power__cancel__iff,axiom,
    ! [Aa: int,X: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Aa)),aa_nat_nat(power_power_nat(numeral_numeral_nat(X)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_752_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,Na: nat,Aa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(power_power_nat(numeral_numeral_nat(X)),Na)),aa_int_nat(nat2,Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)),Aa) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_753_Bernoulli__inequality,axiom,
    ! [X: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),X))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),Na)) ) ).

% Bernoulli_inequality
tff(fact_754_realpow__square__minus__le,axiom,
    ! [U: real,X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,U),numeral_numeral_nat(bit0(one))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))) ).

% realpow_square_minus_le
tff(fact_755_two__realpow__ge__one,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),Na)) ).

% two_realpow_ge_one
tff(fact_756_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(power_power_nat(K),M)) ) ).

% self_le_ge2_pow
tff(fact_757_power2__nat__le__eq__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(power_power_nat(M),numeral_numeral_nat(bit0(one)))),aa_nat_nat(power_power_nat(Na),numeral_numeral_nat(bit0(one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% power2_nat_le_eq_le
tff(fact_758_power2__nat__le__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(power_power_nat(M),numeral_numeral_nat(bit0(one)))),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% power2_nat_le_imp_le
tff(fact_759_L2__set__mult__ineq__lemma,axiom,
    ! [Aa: real,C: real,Ba: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_real_real(times_times_real(Aa),C))),aa_real_real(times_times_real(Ba),D))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),numeral_numeral_nat(bit0(one))))),aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,C),numeral_numeral_nat(bit0(one)))))) ).

% L2_set_mult_ineq_lemma
tff(fact_760_VEBT__internal_Otwo__realpow__ge__two,axiom,
    ! [Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Na))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(bit0(one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),Na)) ) ).

% VEBT_internal.two_realpow_ge_two
tff(fact_761_four__x__squared,axiom,
    ! [X: real] : aa_real_real(times_times_real(numeral_numeral_real(bit0(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),X)),numeral_numeral_nat(bit0(one))) ).

% four_x_squared
tff(fact_762_ceiling__log__nat__eq__if,axiom,
    ! [Ba: nat,Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(power_power_nat(Ba),Na)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_nat_nat(power_power_nat(Ba),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Ba)
         => ( archim7802044766580827645g_real(aa_real_real(log(aa_nat_real(semiri5074537144036343181t_real,Ba)),aa_nat_real(semiri5074537144036343181t_real,K))) = plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,Na),one_one_int) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_763_VEBT__internal_Obit__concat__def,axiom,
    ! [H: nat,L: nat,D: nat] : aa_nat_nat(vEBT_VEBT_bit_concat(H,L),D) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),D))),L) ).

% VEBT_internal.bit_concat_def
tff(fact_764_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% nat_add_left_cancel_less
tff(fact_765_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X) ) ) ).

% linorder_neqE_nat
tff(fact_766_infinite__descent,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( ! [N: nat] :
          ( ~ aa_nat_o(P,N)
         => ? [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
              & ~ aa_nat_o(P,M2) ) )
     => aa_nat_o(P,Na) ) ).

% infinite_descent
tff(fact_767_nat__less__induct,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
             => aa_nat_o(P,M2) )
         => aa_nat_o(P,N) )
     => aa_nat_o(P,Na) ) ).

% nat_less_induct
tff(fact_768_less__irrefl__nat,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Na) ).

% less_irrefl_nat
tff(fact_769_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,S),T)
     => ( S != T ) ) ).

% less_not_refl3
tff(fact_770_less__not__refl2,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
     => ( M != Na ) ) ).

% less_not_refl2
tff(fact_771_less__not__refl,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Na) ).

% less_not_refl
tff(fact_772_nat__neq__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( M != Na )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ).

% nat_neq_iff
tff(fact_773_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% less_add_eq_less
tff(fact_774_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J)) ) ).

% trans_less_add2
tff(fact_775_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M)) ) ).

% trans_less_add1
tff(fact_776_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ) ).

% add_less_mono1
tff(fact_777_not__add__less2,axiom,
    ! [J: nat,I: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),I) ).

% not_add_less2
tff(fact_778_not__add__less1,axiom,
    ! [I: nat,J: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),I) ).

% not_add_less1
tff(fact_779_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_less_mono
tff(fact_780_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K) ) ).

% add_lessD1
tff(fact_781_less__mono__imp__le__mono,axiom,
    ! [F: fun_nat_nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,I2)),aa_nat_nat(F,J2)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,I)),aa_nat_nat(F,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_782_le__neq__implies__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( ( M != Na )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% le_neq_implies_less
tff(fact_783_less__or__eq__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
        | ( M = Na ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% less_or_eq_imp_le
tff(fact_784_le__eq__less__or__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
        | ( M = Na ) ) ) ).

% le_eq_less_or_eq
tff(fact_785_less__imp__le__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% less_imp_le_nat
tff(fact_786_nat__less__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
        & ( M != Na ) ) ) ).

% nat_less_le
tff(fact_787_exists__leI,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ( ! [N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N4),Na)
           => ~ aa_nat_o(P,N4) )
       => aa_nat_o(P,Na) )
     => ? [N5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),Na)
          & aa_nat_o(P,N5) ) ) ).

% exists_leI
tff(fact_788_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,M: nat,K: nat] :
      ( ! [M3: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,M3)),aa_nat_nat(F,N)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(F,M)),K)),aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K))) ) ).

% mono_nat_linear_lb
tff(fact_789_mlex__snd__decrI,axiom,
    ! [Aa: nat,A3: nat,Ba: nat,B3: nat,N3: nat] :
      ( ( Aa = A3 )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),B3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),N3)),Ba)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A3),N3)),B3)) ) ) ).

% mlex_snd_decrI
tff(fact_790_mlex__fst__decrI,axiom,
    ! [Aa: nat,A3: nat,Ba: nat,N3: nat,B3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),N3)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B3),N3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),N3)),Ba)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A3),N3)),B3)) ) ) ) ).

% mlex_fst_decrI
tff(fact_791_mlex__bound,axiom,
    ! [Aa: nat,A: nat,Ba: nat,N3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),N3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),N3)),Ba)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N3)) ) ) ).

% mlex_bound
tff(fact_792_nat__less__real__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Na)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,M)) ) ).

% nat_less_real_le
tff(fact_793_ex__power__ivl2,axiom,
    ! [Ba: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),K)
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(power_power_nat(Ba),N)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_nat_nat(power_power_nat(Ba),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) ) ) ).

% ex_power_ivl2
tff(fact_794_ex__power__ivl1,axiom,
    ! [Ba: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),K)
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(power_power_nat(Ba),N)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(power_power_nat(Ba),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) ) ) ).

% ex_power_ivl1
tff(fact_795_power__2__mult__step__le,axiom,
    ! [N6: nat,Na: nat,K3: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),N6)),K3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na)),K))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),N6)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K3),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na)),K)) ) ) ).

% power_2_mult_step_le
tff(fact_796_nat__add__offset__less,axiom,
    ! [Y: nat,Na: nat,X: nat,M: nat,Sz: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),M))
       => ( ( Sz = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na))),Y)),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Sz)) ) ) ) ).

% nat_add_offset_less
tff(fact_797_insert__time__pure,axiom,
    ! [Aa: nat,Na: nat,Sa: set_nat,Tia: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na))
     => aa_nat_o(simple9179169597231168140_VEBTi(aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Na,Sa),Tia),vEBT_vebt_inserti(Tia,Aa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit1(bit0(bit1(one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit1(bit0(bit1(one))))),aa_int_nat(nat2,archim7802044766580827645g_real(aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Na))))))) ) ).

% insert_time_pure
tff(fact_798_n__less__equal__power__2,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na)) ).

% n_less_equal_power_2
tff(fact_799_ceiling__log__nat__eq__powr__iff,axiom,
    ! [Ba: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( ( archim7802044766580827645g_real(aa_real_real(log(aa_nat_real(semiri5074537144036343181t_real,Ba)),aa_nat_real(semiri5074537144036343181t_real,K))) = plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,Na),one_one_int) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(power_power_nat(Ba),Na)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_nat_nat(power_power_nat(Ba),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_800_log2__of__power__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ).

% log2_of_power_le
tff(fact_801_vebt__predi__rule,axiom,
    ! [Na: nat,S: set_nat,Tia: vEBT_VEBTi,X: nat,Y: nat] : aa_nat_o(time_htt_option_nat(aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Na,S),Tia),vEBT_vebt_predi(Tia,X),aa_nat435388790147128531t_assn(aa_nat336663792692944416t_assn(aa_VEB746558900233519186t_assn(aa_set2727930972950555477t_assn(aTP_Lamm_ba(Na),S),Tia),X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit1(bit1(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit1(bit1(one)))),aa_int_nat(nat2,archim7802044766580827645g_real(aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Na))))))) ).

% vebt_predi_rule
tff(fact_802_bot__nat__0_Onot__eq__extremum,axiom,
    ! [Aa: nat] :
      ( ( Aa != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Aa) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_803_neq0__conv,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% neq0_conv
tff(fact_804_less__nat__zero__code,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),zero_zero_nat) ).

% less_nat_zero_code
tff(fact_805_add__is__0,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        & ( Na = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_806_Nat_Oadd__0__right,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),zero_zero_nat) = M ).

% Nat.add_0_right
tff(fact_807_bot__nat__0_Oextremum,axiom,
    ! [Aa: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa) ).

% bot_nat_0.extremum
tff(fact_808_le0,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Na) ).

% le0
tff(fact_809_mult__is__0,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        | ( Na = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_810_mult__0__right,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),zero_zero_nat) = zero_zero_nat ).

% mult_0_right
tff(fact_811_mult__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na) )
    <=> ( ( M = Na )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_812_mult__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K) )
    <=> ( ( M = Na )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_813_semiring__norm_I78_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(ord_less_num(bit0(M)),bit0(Na))
    <=> aa_num_o(ord_less_num(M),Na) ) ).

% semiring_norm(78)
tff(fact_814_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ aa_num_o(ord_less_num(M),one) ).

% semiring_norm(75)
tff(fact_815_semiring__norm_I80_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(ord_less_num(bit1(M)),bit1(Na))
    <=> aa_num_o(ord_less_num(M),Na) ) ).

% semiring_norm(80)
tff(fact_816_add__gr__0,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% add_gr_0
tff(fact_817_less__one,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),one_one_nat)
    <=> ( Na = zero_zero_nat ) ) ).

% less_one
tff(fact_818_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_819_nat__0__less__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% nat_0_less_mult_iff
tff(fact_820_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% mult_less_cancel2
tff(fact_821_negative__eq__positive,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na)) = aa_nat_int(semiri1314217659103216013at_int,M) )
    <=> ( ( Na = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% negative_eq_positive
tff(fact_822_semiring__norm_I76_J,axiom,
    ! [Na: num] : aa_num_o(ord_less_num(one),bit0(Na)) ).

% semiring_norm(76)
tff(fact_823_semiring__norm_I81_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(ord_less_num(bit1(M)),bit0(Na))
    <=> aa_num_o(ord_less_num(M),Na) ) ).

% semiring_norm(81)
tff(fact_824_semiring__norm_I77_J,axiom,
    ! [Na: num] : aa_num_o(ord_less_num(one),bit1(Na)) ).

% semiring_norm(77)
tff(fact_825_enat__ord__number_I2_J,axiom,
    ! [M: num,Na: num] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,numeral_numeral_nat(M)),numeral_numeral_nat(Na)) ) ).

% enat_ord_number(2)
tff(fact_826_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_827_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% mult_le_cancel2
tff(fact_828_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(W)),aa_nat_real(semiri5074537144036343181t_real,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,numeral_numeral_nat(W)),Na) ) ).

% numeral_less_real_of_nat_iff
tff(fact_829_real__of__nat__less__numeral__iff,axiom,
    ! [Na: nat,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),numeral_numeral_real(W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),numeral_numeral_nat(W)) ) ).

% real_of_nat_less_numeral_iff
tff(fact_830_nat__neg__numeral,axiom,
    ! [K: num] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = zero_zero_nat ).

% nat_neg_numeral
tff(fact_831_nat__zminus__int,axiom,
    ! [Na: nat] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na))) = zero_zero_nat ).

% nat_zminus_int
tff(fact_832_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),plus_plus_int(Z,one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).

% zle_add1_eq_le
tff(fact_833_semiring__norm_I79_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(ord_less_num(bit0(M)),bit1(Na))
    <=> aa_num_o(ord_less_eq_num(M),Na) ) ).

% semiring_norm(79)
tff(fact_834_semiring__norm_I74_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(ord_less_eq_num(bit1(M)),bit0(Na))
    <=> aa_num_o(ord_less_num(M),Na) ) ).

% semiring_norm(74)
tff(fact_835_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,Na: nat,Aa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(power_power_nat(numeral_numeral_nat(X)),Na)),aa_int_nat(nat2,Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)),Aa) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_836_nat__less__numeral__power__cancel__iff,axiom,
    ! [Aa: int,X: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,Aa)),aa_nat_nat(power_power_nat(numeral_numeral_nat(X)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),Na)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_837_less__eq__real__def,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
        | ( X = Y ) ) ) ).

% less_eq_real_def
tff(fact_838_num_Osize_I4_J,axiom,
    size_size_num(one) = zero_zero_nat ).

% num.size(4)
tff(fact_839_bot__nat__0_Oextremum__strict,axiom,
    ! [Aa: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),zero_zero_nat) ).

% bot_nat_0.extremum_strict
tff(fact_840_gr0I,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% gr0I
tff(fact_841_not__gr0,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
    <=> ( Na = zero_zero_nat ) ) ).

% not_gr0
tff(fact_842_not__less0,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),zero_zero_nat) ).

% not_less0
tff(fact_843_less__zeroE,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),zero_zero_nat) ).

% less_zeroE
tff(fact_844_gr__implies__not0,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( Na != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_845_infinite__descent0,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( ~ aa_nat_o(P,N)
             => ? [M2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
                  & ~ aa_nat_o(P,M2) ) ) )
       => aa_nat_o(P,Na) ) ) ).

% infinite_descent0
tff(fact_846_plus__nat_Oadd__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),Na) = Na ).

% plus_nat.add_0
tff(fact_847_add__eq__self__zero,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na) = M )
     => ( Na = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_848_less__eq__nat_Osimps_I1_J,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Na) ).

% less_eq_nat.simps(1)
tff(fact_849_bot__nat__0_Oextremum__unique,axiom,
    ! [Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),zero_zero_nat)
    <=> ( Aa = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
tff(fact_850_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),zero_zero_nat)
     => ( Aa = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_851_le__0__eq,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),zero_zero_nat)
    <=> ( Na = zero_zero_nat ) ) ).

% le_0_eq
tff(fact_852_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na) )
    <=> ( ( K = zero_zero_nat )
        | ( M = Na ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_853_mult__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),Na) = zero_zero_nat ).

% mult_0
tff(fact_854_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,times_times_int(aa_nat_int(semiri1314217659103216013at_int,K),I)),times_times_int(aa_nat_int(semiri1314217659103216013at_int,K),J)) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_855_nat__int__comparison_I2_J,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Aa)),aa_nat_int(semiri1314217659103216013at_int,Ba)) ) ).

% nat_int_comparison(2)
tff(fact_856_int__gr__induct,axiom,
    ! [K: int,I: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I)
     => ( aa_int_o(P,plus_plus_int(K,one_one_int))
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I2)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,plus_plus_int(I2,one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_gr_induct
tff(fact_857_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),plus_plus_int(Z,one_one_int))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
        | ( W = Z ) ) ) ).

% zless_add1_eq
tff(fact_858_not__int__zless__negative,axiom,
    ! [Na: nat,M: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M))) ).

% not_int_zless_negative
tff(fact_859_real__arch__pow,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) ) ).

% real_arch_pow
tff(fact_860_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ? [K4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K4)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K4) = J ) ) ) ).

% less_imp_add_positive
tff(fact_861_ex__least__nat__le,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( aa_nat_o(P,Na)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ? [K4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K4),Na)
            & ! [I3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K4)
               => ~ aa_nat_o(P,I3) )
            & aa_nat_o(P,K4) ) ) ) ).

% ex_least_nat_le
tff(fact_862_nat__geq__1__eq__neqz,axiom,
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),X)
    <=> ( X != zero_zero_nat ) ) ).

% nat_geq_1_eq_neqz
tff(fact_863_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na) )
      <=> ( M = Na ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_864_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% nat_mult_less_cancel1
tff(fact_865_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J)) ) ) ).

% mult_less_mono2
tff(fact_866_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K)) ) ) ).

% mult_less_mono1
tff(fact_867_mult__eq__self__implies__10,axiom,
    ! [M: nat,Na: nat] :
      ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) )
     => ( ( Na = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_868_nat__less__as__int,axiom,
    ! [X4: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Xa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa)) ) ).

% nat_less_as_int
tff(fact_869_log__of__power__less,axiom,
    ! [M: nat,Ba: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),Na))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ba)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(Ba),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ) ).

% log_of_power_less
tff(fact_870_log__of__power__le,axiom,
    ! [M: nat,Ba: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),Na))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ba)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(Ba),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ) ).

% log_of_power_le
tff(fact_871_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),Z) ) ).

% zless_nat_eq_int_zless
tff(fact_872_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,plus_plus_int(W,one_one_int)),Z) ) ).

% zless_imp_add1_zle
tff(fact_873_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,plus_plus_int(W,one_one_int)),Z)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% add1_zle_eq
tff(fact_874_bounded__Max__nat,axiom,
    ! [P: fun_nat_o,X: nat,M4: nat] :
      ( aa_nat_o(P,X)
     => ( ! [X2: nat] :
            ( aa_nat_o(P,X2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),M4) )
       => ~ ! [M3: nat] :
              ( aa_nat_o(P,M3)
             => ~ ! [X4: nat] :
                    ( aa_nat_o(P,X4)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),M3) ) ) ) ) ).

% bounded_Max_nat
tff(fact_875_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% nat_mult_le_cancel1
tff(fact_876_int__cases4,axiom,
    ! [M: int] :
      ( ! [N: nat] : M != aa_nat_int(semiri1314217659103216013at_int,N)
     => ~ ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( M != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) ) ) ) ).

% int_cases4
tff(fact_877_int__zle__neg,axiom,
    ! [Na: nat,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M)))
    <=> ( ( Na = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
tff(fact_878_log2__of__power__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ).

% log2_of_power_less
tff(fact_879_nat__le__real__less,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),one_one_real)) ) ).

% nat_le_real_less
tff(fact_880_pos2,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),numeral_numeral_nat(bit0(one))) ).

% pos2
tff(fact_881_log__of__power__eq,axiom,
    ! [M: nat,Ba: real,Na: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),Na) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ba)
       => ( aa_nat_real(semiri5074537144036343181t_real,Na) = aa_real_real(log(Ba),aa_nat_real(semiri5074537144036343181t_real,M)) ) ) ) ).

% log_of_power_eq
tff(fact_882_less__log__of__power,axiom,
    ! [Ba: real,Na: nat,M: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),Na)),M)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ba)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(log(Ba),M)) ) ) ).

% less_log_of_power
tff(fact_883_le__log__of__power,axiom,
    ! [Ba: real,Na: nat,M: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),Na)),M)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ba)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(log(Ba),M)) ) ) ).

% le_log_of_power
tff(fact_884_less__log2__of__power,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na)),M)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,M))) ) ).

% less_log2_of_power
tff(fact_885_nat__induct2,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [N: nat] :
              ( aa_nat_o(P,N)
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),numeral_numeral_nat(bit0(one)))) )
         => aa_nat_o(P,Na) ) ) ) ).

% nat_induct2
tff(fact_886_TBOUND__mfold,axiom,
    ! [Xs2: list_nat,Na: nat,Sa: set_nat,Tia: vEBT_VEBTi] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),set_nat2(Xs2))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na)) )
     => aa_nat_o(simple9179169597231168140_VEBTi(aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Na,Sa),Tia),aa_VEB836174949890266638_VEBTi(vEBT_E6105538542217078229_VEBTi(aTP_Lamm_bb,Xs2),Tia)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit1(bit0(bit1(one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit1(bit0(bit1(one))))),aa_int_nat(nat2,archim7802044766580827645g_real(aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Na)))))))),one_one_nat)) ) ).

% TBOUND_mfold
tff(fact_887_vebt__succi__rule,axiom,
    ! [Na: nat,S: set_nat,Tia: vEBT_VEBTi,X: nat,Y: nat] : aa_nat_o(time_htt_option_nat(aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Na,S),Tia),vEBT_vebt_succi(Tia,X),aa_nat435388790147128531t_assn(aa_nat336663792692944416t_assn(aa_VEB746558900233519186t_assn(aa_set2727930972950555477t_assn(aTP_Lamm_bc(Na),S),Tia),X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit1(bit1(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit1(bit1(one)))),aa_int_nat(nat2,archim7802044766580827645g_real(aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Na))))))) ).

% vebt_succi_rule
tff(fact_888_test__def,axiom,
    ! [Na: nat,Xs2: list_nat,Ysb: list_nat] : aa_lis1877347954636394190st_nat(vEBT_test(Na,Xs2),Ysb) = heap_T3876627890315925662st_nat(vEBT_vebt_buildupi(Na),aa_lis4214751270122123570st_nat(aTP_Lamm_bf(Xs2),Ysb)) ).

% test_def
tff(fact_889_i0__less,axiom,
    ! [Na: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),Na)
    <=> ( Na != zero_z5237406670263579293d_enat ) ) ).

% i0_less
tff(fact_890_real__add__minus__iff,axiom,
    ! [X: real,Aa: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(uminus_uminus_real,Aa)) = zero_zero_real )
    <=> ( X = Aa ) ) ).

% real_add_minus_iff
tff(fact_891_log__one,axiom,
    ! [Aa: real] : aa_real_real(log(Aa),one_one_real) = zero_zero_real ).

% log_one
tff(fact_892_log__eq__one,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( ( Aa != one_one_real )
       => ( aa_real_real(log(Aa),Aa) = one_one_real ) ) ) ).

% log_eq_one
tff(fact_893_log__less__cancel__iff,axiom,
    ! [Aa: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(Aa),X)),aa_real_real(log(Aa),Y))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_894_log__less__one__cancel__iff,axiom,
    ! [Aa: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(Aa),X)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Aa) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_895_one__less__log__cancel__iff,axiom,
    ! [Aa: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(log(Aa),X))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),X) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_896_log__less__zero__cancel__iff,axiom,
    ! [Aa: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(Aa),X)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_897_zero__less__log__cancel__iff,axiom,
    ! [Aa: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(log(Aa),X))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_898_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),bit_se725231765392027082nd_int(K,L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% and_nonnegative_int_iff
tff(fact_899_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,bit_se725231765392027082nd_int(K,L)),zero_zero_int)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% and_negative_int_iff
tff(fact_900_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),bit_se1409905431419307370or_int(K,L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% or_nonnegative_int_iff
tff(fact_901_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,bit_se1409905431419307370or_int(K,L)),zero_zero_int)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% or_negative_int_iff
tff(fact_902_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),bit_se6526347334894502574or_int(K,L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% xor_nonnegative_int_iff
tff(fact_903_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,bit_se6526347334894502574or_int(K,L)),zero_zero_int)
    <=> ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        <=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% xor_negative_int_iff
tff(fact_904_nat__le__0,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int)
     => ( aa_int_nat(nat2,Z) = zero_zero_nat ) ) ).

% nat_le_0
tff(fact_905_nat__0__iff,axiom,
    ! [I: int] :
      ( ( aa_int_nat(nat2,I) = zero_zero_nat )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int) ) ).

% nat_0_iff
tff(fact_906_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ) ).

% zless_nat_conj
tff(fact_907_zero__le__log__cancel__iff,axiom,
    ! [Aa: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(log(Aa),X))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_908_log__le__zero__cancel__iff,axiom,
    ! [Aa: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(Aa),X)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_909_one__le__log__cancel__iff,axiom,
    ! [Aa: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(log(Aa),X))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_910_log__le__one__cancel__iff,axiom,
    ! [Aa: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(Aa),X)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Aa) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_911_log__le__cancel__iff,axiom,
    ! [Aa: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(Aa),X)),aa_real_real(log(Aa),Y))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_912_int__nat__eq,axiom,
    ! [Z: int] :
      aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z),Z,zero_zero_int) ).

% int_nat_eq
tff(fact_913_not__negative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,bit_ri7919022796975470100ot_int(K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% not_negative_int_iff
tff(fact_914_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),bit_ri7919022796975470100ot_int(K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% not_nonnegative_int_iff
tff(fact_915_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% zero_less_nat_eq
tff(fact_916_log__pow__cancel,axiom,
    ! [Aa: real,Ba: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( ( Aa != one_one_real )
       => ( aa_real_real(log(Aa),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Ba)) = aa_nat_real(semiri5074537144036343181t_real,Ba) ) ) ) ).

% log_pow_cancel
tff(fact_917_and__minus__numerals_I1_J,axiom,
    ! [Na: num] : bit_se725231765392027082nd_int(one_one_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(Na)))) = zero_zero_int ).

% and_minus_numerals(1)
tff(fact_918_and__minus__numerals_I5_J,axiom,
    ! [Na: num] : bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(Na))),one_one_int) = zero_zero_int ).

% and_minus_numerals(5)
tff(fact_919_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,Na: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),times_7803423173614009249d_enat(M,Na))
    <=> ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),M)
        & aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),Na) ) ) ).

% enat_0_less_mult_iff
tff(fact_920_enat__less__induct,axiom,
    ! [P: fun_Extended_enat_o,Na: extended_enat] :
      ( ! [N: extended_enat] :
          ( ! [M2: extended_enat] :
              ( aa_Extended_enat_o(ord_le72135733267957522d_enat(M2),N)
             => aa_Extended_enat_o(P,M2) )
         => aa_Extended_enat_o(P,N) )
     => aa_Extended_enat_o(P,Na) ) ).

% enat_less_induct
tff(fact_921_not__iless0,axiom,
    ! [Na: extended_enat] : ~ aa_Extended_enat_o(ord_le72135733267957522d_enat(Na),zero_z5237406670263579293d_enat) ).

% not_iless0
tff(fact_922_less__eq__int__code_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),zero_zero_int) ).

% less_eq_int_code(1)
tff(fact_923_less__int__code_I1_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),zero_zero_int) ).

% less_int_code(1)
tff(fact_924_uminus__int__code_I1_J,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% uminus_int_code(1)
tff(fact_925_plus__int__code_I2_J,axiom,
    ! [L: int] : plus_plus_int(zero_zero_int,L) = L ).

% plus_int_code(2)
tff(fact_926_plus__int__code_I1_J,axiom,
    ! [K: int] : plus_plus_int(K,zero_zero_int) = K ).

% plus_int_code(1)
tff(fact_927_times__int__code_I2_J,axiom,
    ! [L: int] : times_times_int(zero_zero_int,L) = zero_zero_int ).

% times_int_code(2)
tff(fact_928_times__int__code_I1_J,axiom,
    ! [K: int] : times_times_int(K,zero_zero_int) = zero_zero_int ).

% times_int_code(1)
tff(fact_929_ile0__eq,axiom,
    ! [Na: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Na),zero_z5237406670263579293d_enat)
    <=> ( Na = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
tff(fact_930_i0__lb,axiom,
    ! [Na: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(zero_z5237406670263579293d_enat),Na) ).

% i0_lb
tff(fact_931_int__ops_I1_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% int_ops(1)
tff(fact_932_nat__zero__as__int,axiom,
    zero_zero_nat = aa_int_nat(nat2,zero_zero_int) ).

% nat_zero_as_int
tff(fact_933_odd__nonzero,axiom,
    ! [Z: int] : plus_plus_int(plus_plus_int(one_one_int,Z),Z) != zero_zero_int ).

% odd_nonzero
tff(fact_934_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,times_times_int(K,I)),times_times_int(K,J)) ) ) ).

% zmult_zless_mono2
tff(fact_935_nonneg__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ~ ! [N: nat] : K != aa_nat_int(semiri1314217659103216013at_int,N) ) ).

% nonneg_int_cases
tff(fact_936_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ? [N: nat] : K = aa_nat_int(semiri1314217659103216013at_int,N) ) ).

% zero_le_imp_eq_int
tff(fact_937_real__arch__pow__inv,axiom,
    ! [Y: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)),Y) ) ) ).

% real_arch_pow_inv
tff(fact_938_ex__nat,axiom,
    ! [P: fun_nat_o] :
      ( ? [X_1: nat] : aa_nat_o(P,X_1)
    <=> ? [X5: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X5)
          & aa_nat_o(P,aa_int_nat(nat2,X5)) ) ) ).

% ex_nat
tff(fact_939_all__nat,axiom,
    ! [P: fun_nat_o] :
      ( ! [X_1: nat] : aa_nat_o(P,X_1)
    <=> ! [X5: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X5)
         => aa_nat_o(P,aa_int_nat(nat2,X5)) ) ) ).

% all_nat
tff(fact_940_eq__nat__nat__iff,axiom,
    ! [Z: int,Z4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z4)
       => ( ( aa_int_nat(nat2,Z) = aa_int_nat(nat2,Z4) )
        <=> ( Z = Z4 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_941_reals__Archimedean3,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ! [Y3: real] :
        ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),X)) ) ).

% reals_Archimedean3
tff(fact_942_real__add__less__0__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(uminus_uminus_real,X)) ) ).

% real_add_less_0_iff
tff(fact_943_real__0__less__add__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,X)),Y) ) ).

% real_0_less_add_iff
tff(fact_944_AND__lower,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),bit_se725231765392027082nd_int(X,Y)) ) ).

% AND_lower
tff(fact_945_AND__upper1,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,bit_se725231765392027082nd_int(X,Y)),X) ) ).

% AND_upper1
tff(fact_946_AND__upper2,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,bit_se725231765392027082nd_int(X,Y)),Y) ) ).

% AND_upper2
tff(fact_947_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,bit_se725231765392027082nd_int(Y,Ya)),Z) ) ) ).

% AND_upper1'
tff(fact_948_AND__upper2_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,bit_se725231765392027082nd_int(X,Y)),Z) ) ) ).

% AND_upper2'
tff(fact_949_OR__lower,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),bit_se1409905431419307370or_int(X,Y)) ) ) ).

% OR_lower
tff(fact_950_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),bit_se1409905431419307370or_int(K,L)) ) ).

% or_greater_eq
tff(fact_951_XOR__lower,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),bit_se6526347334894502574or_int(X,Y)) ) ) ).

% XOR_lower
tff(fact_952_nat__mono__iff,axiom,
    ! [Z: int,W: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ) ).

% nat_mono_iff
tff(fact_953_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% int_one_le_iff_zero_less
tff(fact_954_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,plus_plus_int(plus_plus_int(one_one_int,Z),Z)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% odd_less_0_iff
tff(fact_955_pos__zmult__eq__1__iff,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M)
     => ( ( times_times_int(M,Na) = one_one_int )
      <=> ( ( M = one_one_int )
          & ( Na = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_956_negative__zle__0,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na))),zero_zero_int) ).

% negative_zle_0
tff(fact_957_nonpos__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ~ ! [N: nat] : K != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) ) ).

% nonpos_int_cases
tff(fact_958_real__add__le__0__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(uminus_uminus_real,X)) ) ).

% real_add_le_0_iff
tff(fact_959_real__0__le__add__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,X)),Y) ) ).

% real_0_le_add_iff
tff(fact_960_nat__0__le,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = Z ) ) ).

% nat_0_le
tff(fact_961_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = Z )
    <=> ( ( M = aa_int_nat(nat2,Z) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ) ).

% int_eq_iff
tff(fact_962_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,bit_se725231765392027082nd_int(X,Y)),Z) ) ) ).

% AND_upper2''
tff(fact_963_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,bit_se725231765392027082nd_int(Y,Ya)),Z) ) ) ).

% AND_upper1''
tff(fact_964_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,bit_se725231765392027082nd_int(K,L)),K) ) ).

% and_less_eq
tff(fact_965_and__not__numerals_I1_J,axiom,
    bit_se725231765392027082nd_int(one_one_int,bit_ri7919022796975470100ot_int(one_one_int)) = zero_zero_int ).

% and_not_numerals(1)
tff(fact_966_or__not__numerals_I1_J,axiom,
    bit_se1409905431419307370or_int(one_one_int,bit_ri7919022796975470100ot_int(one_one_int)) = bit_ri7919022796975470100ot_int(zero_zero_int) ).

% or_not_numerals(1)
tff(fact_967_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ? [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
          & ( K = aa_nat_int(semiri1314217659103216013at_int,N) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_968_pos__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ~ ! [N: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% pos_int_cases
tff(fact_969_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) )
       => ~ ! [N: nat] :
              ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) )
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ) ).

% int_cases3
tff(fact_970_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ) ).

% nat_less_eq_zless
tff(fact_971_nat__eq__iff,axiom,
    ! [W: int,M: nat] :
      ( ( aa_int_nat(nat2,W) = M )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W),W = aa_nat_int(semiri1314217659103216013at_int,M),M = zero_zero_nat) ) ).

% nat_eq_iff
tff(fact_972_nat__eq__iff2,axiom,
    ! [M: nat,W: int] :
      ( ( M = aa_int_nat(nat2,W) )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W),W = aa_nat_int(semiri1314217659103216013at_int,M),M = zero_zero_nat) ) ).

% nat_eq_iff2
tff(fact_973_split__nat,axiom,
    ! [P: fun_nat_o,I: int] :
      ( aa_nat_o(P,aa_int_nat(nat2,I))
    <=> ( ! [N2: nat] :
            ( ( I = aa_nat_int(semiri1314217659103216013at_int,N2) )
           => aa_nat_o(P,N2) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),zero_zero_int)
         => aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_nat
tff(fact_974_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),W)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ) ).

% nat_le_eq_zle
tff(fact_975_nat__add__distrib,axiom,
    ! [Z: int,Z4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z4)
       => ( aa_int_nat(nat2,plus_plus_int(Z,Z4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z4)) ) ) ) ).

% nat_add_distrib
tff(fact_976_le__imp__0__less,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),plus_plus_int(one_one_int,Z)) ) ).

% le_imp_0_less
tff(fact_977_pos__mult__pos__ge,axiom,
    ! [X: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,times_times_int(Na,one_one_int)),times_times_int(Na,X)) ) ) ).

% pos_mult_pos_ge
tff(fact_978_le__nat__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_int_nat(nat2,K))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Na)),K) ) ) ).

% le_nat_iff
tff(fact_979_q__pos__lemma,axiom,
    ! [B3: int,Q2: int,R4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),plus_plus_int(times_times_int(B3,Q2),R4))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B3)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Q2) ) ) ) ).

% q_pos_lemma
tff(fact_980_zdiv__mono2__lemma,axiom,
    ! [Ba: int,Q3: int,R: int,B3: int,Q2: int,R4: int] :
      ( ( plus_plus_int(times_times_int(Ba,Q3),R) = plus_plus_int(times_times_int(B3,Q2),R4) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),plus_plus_int(times_times_int(B3,Q2),R4))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B3)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B3),Ba)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q3),Q2) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_981_zdiv__mono2__neg__lemma,axiom,
    ! [Ba: int,Q3: int,R: int,B3: int,Q2: int,R4: int] :
      ( ( plus_plus_int(times_times_int(Ba,Q3),R) = plus_plus_int(times_times_int(B3,Q2),R4) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,plus_plus_int(times_times_int(B3,Q2),R4)),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R),Ba)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R4)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B3),Ba)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q2),Q3) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_982_unique__quotient__lemma,axiom,
    ! [Ba: int,Q2: int,R4: int,Q3: int,R: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,plus_plus_int(times_times_int(Ba,Q2),R4)),plus_plus_int(times_times_int(Ba,Q3),R))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),Ba)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R),Ba)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q2),Q3) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_983_unique__quotient__lemma__neg,axiom,
    ! [Ba: int,Q2: int,R4: int,Q3: int,R: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,plus_plus_int(times_times_int(Ba,Q2),R4)),plus_plus_int(times_times_int(Ba,Q3),R))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),R)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),R4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q3),Q2) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_984_nat__mult__distrib,axiom,
    ! [Z: int,Z4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,times_times_int(Z,Z4)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z4)) ) ) ).

% nat_mult_distrib
tff(fact_985_nat__power__eq,axiom,
    ! [Z: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,aa_nat_int(power_power_int(Z),Na)) = aa_nat_nat(power_power_nat(aa_int_nat(nat2,Z)),Na) ) ) ).

% nat_power_eq
tff(fact_986_log__mult,axiom,
    ! [Aa: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( ( Aa != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
           => ( aa_real_real(log(Aa),aa_real_real(times_times_real(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(log(Aa),X)),aa_real_real(log(Aa),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_987_log__nat__power,axiom,
    ! [X: real,Ba: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(log(Ba),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na)) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(log(Ba),X)) ) ) ).

% log_nat_power
tff(fact_988_not__exp__less__eq__0__int,axiom,
    ! [Na: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na)),zero_zero_int) ).

% not_exp_less_eq_0_int
tff(fact_989_neg__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => ~ ! [N: nat] :
            ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% neg_int_cases
tff(fact_990_real__archimedian__rdiv__eq__0,axiom,
    ! [X: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => ( ! [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,M3)),X)),C) )
         => ( X = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_991_nat__less__iff,axiom,
    ! [W: int,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),M)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_nat_int(semiri1314217659103216013at_int,M)) ) ) ).

% nat_less_iff
tff(fact_992_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int)
     => ( aa_int_nat(nat2,times_times_int(Z,Z4)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z))),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z4))) ) ) ).

% nat_mult_distrib_neg
tff(fact_993_and__not__numerals_I3_J,axiom,
    ! [Na: num] : bit_se725231765392027082nd_int(one_one_int,bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,bit1(Na)))) = zero_zero_int ).

% and_not_numerals(3)
tff(fact_994_or__not__numerals_I7_J,axiom,
    ! [M: num] : bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,bit1(M)),bit_ri7919022796975470100ot_int(one_one_int)) = bit_ri7919022796975470100ot_int(zero_zero_int) ).

% or_not_numerals(7)
tff(fact_995_int__bit__induct,axiom,
    ! [P: fun_int_o,K: int] :
      ( aa_int_o(P,zero_zero_int)
     => ( aa_int_o(P,aa_int_int(uminus_uminus_int,one_one_int))
       => ( ! [K4: int] :
              ( aa_int_o(P,K4)
             => ( ( K4 != zero_zero_int )
               => aa_int_o(P,times_times_int(K4,aa_num_int(numeral_numeral_int,bit0(one)))) ) )
         => ( ! [K4: int] :
                ( aa_int_o(P,K4)
               => ( ( K4 != aa_int_int(uminus_uminus_int,one_one_int) )
                 => aa_int_o(P,plus_plus_int(one_one_int,times_times_int(K4,aa_num_int(numeral_numeral_int,bit0(one))))) ) )
           => aa_int_o(P,K) ) ) ) ) ).

% int_bit_induct
tff(fact_996_OR__upper,axiom,
    ! [X: int,Na: nat,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,bit_se1409905431419307370or_int(X,Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na)) ) ) ) ).

% OR_upper
tff(fact_997_XOR__upper,axiom,
    ! [X: int,Na: nat,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,bit_se6526347334894502574or_int(X,Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na)) ) ) ) ).

% XOR_upper
tff(fact_998_linear__plus__1__le__power,axiom,
    ! [X: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),X)),one_one_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)),Na)) ) ).

% linear_plus_1_le_power
tff(fact_999_incr__mult__lemma,axiom,
    ! [D: int,P: fun_int_o,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X2: int] :
            ( aa_int_o(P,X2)
           => aa_int_o(P,plus_plus_int(X2,D)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
         => ! [X4: int] :
              ( aa_int_o(P,X4)
             => aa_int_o(P,plus_plus_int(X4,times_times_int(K,D))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_1000_int__not__code_I1_J,axiom,
    bit_ri7919022796975470100ot_int(zero_zero_int) = aa_int_int(uminus_uminus_int,one_one_int) ).

% int_not_code(1)
tff(fact_1001_vebt__buildupi__rule,axiom,
    ! [Na: nat] : aa_nat_o(time_htt_VEBT_VEBTi(pure_assn(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)),vEBT_vebt_buildupi(Na),vEBT_Intf_vebt_assn(Na,bot_bot_set_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(bit1(bit0(one))))),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na))) ).

% vebt_buildupi_rule
tff(fact_1002_is__succ__in__set__def,axiom,
    ! [Xs2: set_nat,X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(Xs2),X),Y)
    <=> ( aa_set_nat_o(member_nat(Y),Xs2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
        & ! [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),Xs2)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),X5)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X5) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_1003_is__pred__in__set__def,axiom,
    ! [Xs2: set_nat,X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(Xs2),X),Y)
    <=> ( aa_set_nat_o(member_nat(Y),Xs2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),X)
        & ! [X5: nat] :
            ( aa_set_nat_o(member_nat(X5),Xs2)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X5),X)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),Y) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_1004_floor__log__nat__eq__powr__iff,axiom,
    ! [Ba: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( ( archim6058952711729229775r_real(aa_real_real(log(aa_nat_real(semiri5074537144036343181t_real,Ba)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,Na) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(power_power_nat(Ba),Na)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(power_power_nat(Ba),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_1005_iadd__is__0,axiom,
    ! [M: extended_enat,Na: extended_enat] :
      ( ( plus_p3455044024723400733d_enat(M,Na) = zero_z5237406670263579293d_enat )
    <=> ( ( M = zero_z5237406670263579293d_enat )
        & ( Na = zero_z5237406670263579293d_enat ) ) ) ).

% iadd_is_0
tff(fact_1006_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
tff(fact_1007_imult__is__0,axiom,
    ! [M: extended_enat,Na: extended_enat] :
      ( ( times_7803423173614009249d_enat(M,Na) = zero_z5237406670263579293d_enat )
    <=> ( ( M = zero_z5237406670263579293d_enat )
        | ( Na = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
tff(fact_1008_nat__floor__neg,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
     => ( aa_int_nat(nat2,archim6058952711729229775r_real(X)) = zero_zero_nat ) ) ).

% nat_floor_neg
tff(fact_1009_le__nat__floor,axiom,
    ! [X: nat,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),Aa)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_int_nat(nat2,archim6058952711729229775r_real(Aa))) ) ).

% le_nat_floor
tff(fact_1010_conj__le__cong,axiom,
    ! [X: int,X6: int,P: $o,P3: $o] :
      ( ( X = X6 )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X6)
         => ( (P)
          <=> (P3) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
            & (P) )
        <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X6)
            & (P3) ) ) ) ) ).

% conj_le_cong
tff(fact_1011_imp__le__cong,axiom,
    ! [X: int,X6: int,P: $o,P3: $o] :
      ( ( X = X6 )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X6)
         => ( (P)
          <=> (P3) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
           => (P) )
        <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X6)
           => (P3) ) ) ) ) ).

% imp_le_cong
tff(fact_1012_floor__log__nat__eq__if,axiom,
    ! [Ba: nat,Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(power_power_nat(Ba),Na)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(power_power_nat(Ba),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Ba)
         => ( archim6058952711729229775r_real(aa_real_real(log(aa_nat_real(semiri5074537144036343181t_real,Ba)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,Na) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_1013_ceiling__log__eq__powr__iff,axiom,
    ! [X: real,Ba: real,K: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ba)
       => ( ( archim7802044766580827645g_real(aa_real_real(log(Ba),X)) = plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,K),one_one_int) )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(Ba),aa_nat_real(semiri5074537144036343181t_real,K))),X)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(powr_real(Ba),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)))) ) ) ) ) ).

% ceiling_log_eq_powr_iff
tff(fact_1014_floor__log2__div2,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Na)
     => ( archim6058952711729229775r_real(aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Na))) = plus_plus_int(archim6058952711729229775r_real(aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one)))))),one_one_int) ) ) ).

% floor_log2_div2
tff(fact_1015_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      aa_nat_nat(divide_divide_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)) = $ite(K = zero_zero_nat,zero_zero_nat,aa_nat_nat(divide_divide_nat(M),Na)) ).

% nat_mult_div_cancel_disj
tff(fact_1016_powr__nonneg__iff,axiom,
    ! [Aa: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(Aa),X)),zero_zero_real)
    <=> ( Aa = zero_zero_real ) ) ).

% powr_nonneg_iff
tff(fact_1017_powr__one__gt__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(powr_real(X),one_one_real) = X )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X) ) ).

% powr_one_gt_zero_iff
tff(fact_1018_powr__one,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_real(powr_real(X),one_one_real) = X ) ) ).

% powr_one
tff(fact_1019_powr__le__cancel__iff,axiom,
    ! [X: real,Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(X),Aa)),aa_real_real(powr_real(X),Ba))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba) ) ) ).

% powr_le_cancel_iff
tff(fact_1020_numeral__powr__numeral__real,axiom,
    ! [M: num,Na: num] : aa_real_real(powr_real(numeral_numeral_real(M)),numeral_numeral_real(Na)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(M)),numeral_numeral_nat(Na)) ).

% numeral_powr_numeral_real
tff(fact_1021_powr__log__cancel,axiom,
    ! [Aa: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( ( Aa != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( aa_real_real(powr_real(Aa),aa_real_real(log(Aa),X)) = X ) ) ) ) ).

% powr_log_cancel
tff(fact_1022_log__powr__cancel,axiom,
    ! [Aa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( ( Aa != one_one_real )
       => ( aa_real_real(log(Aa),aa_real_real(powr_real(Aa),Y)) = Y ) ) ) ).

% log_powr_cancel
tff(fact_1023_powr__numeral,axiom,
    ! [X: real,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_real(powr_real(X),numeral_numeral_real(Na)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(Na)) ) ) ).

% powr_numeral
tff(fact_1024_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_1025_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
tff(fact_1026_powr__mono2,axiom,
    ! [Aa: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(X),Aa)),aa_real_real(powr_real(Y),Aa)) ) ) ) ).

% powr_mono2
tff(fact_1027_powr__ge__pzero,axiom,
    ! [X: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(powr_real(X),Y)) ).

% powr_ge_pzero
tff(fact_1028_powr__mono,axiom,
    ! [Aa: real,Ba: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(X),Aa)),aa_real_real(powr_real(X),Ba)) ) ) ).

% powr_mono
tff(fact_1029_powr__mono2_H,axiom,
    ! [Aa: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(Y),Aa)),aa_real_real(powr_real(X),Aa)) ) ) ) ).

% powr_mono2'
tff(fact_1030_powr__less__mono2,axiom,
    ! [Aa: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(X),Aa)),aa_real_real(powr_real(Y),Aa)) ) ) ) ).

% powr_less_mono2
tff(fact_1031_ge__one__powr__ge__zero,axiom,
    ! [X: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(powr_real(X),Aa)) ) ) ).

% ge_one_powr_ge_zero
tff(fact_1032_powr__mono__both,axiom,
    ! [Aa: real,Ba: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(X),Aa)),aa_real_real(powr_real(Y),Ba)) ) ) ) ) ).

% powr_mono_both
tff(fact_1033_powr__le1,axiom,
    ! [Aa: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(X),Aa)),one_one_real) ) ) ) ).

% powr_le1
tff(fact_1034_powr__mult,axiom,
    ! [X: real,Y: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_real(powr_real(aa_real_real(times_times_real(X),Y)),Aa) = aa_real_real(times_times_real(aa_real_real(powr_real(X),Aa)),aa_real_real(powr_real(Y),Aa)) ) ) ) ).

% powr_mult
tff(fact_1035_log__powr,axiom,
    ! [X: real,Ba: real,Y: real] :
      ( ( X != zero_zero_real )
     => ( aa_real_real(log(Ba),aa_real_real(powr_real(X),Y)) = aa_real_real(times_times_real(Y),aa_real_real(log(Ba),X)) ) ) ).

% log_powr
tff(fact_1036_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_nat(divide_divide_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)) = aa_nat_nat(divide_divide_nat(M),Na) ) ) ).

% nat_mult_div_cancel1
tff(fact_1037_powr__realpow,axiom,
    ! [X: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(powr_real(X),aa_nat_real(semiri5074537144036343181t_real,Na)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na) ) ) ).

% powr_realpow
tff(fact_1038_powr__less__iff,axiom,
    ! [Ba: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(Ba),Y)),X)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(log(Ba),X)) ) ) ) ).

% powr_less_iff
tff(fact_1039_less__powr__iff,axiom,
    ! [Ba: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(powr_real(Ba),Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(Ba),X)),Y) ) ) ) ).

% less_powr_iff
tff(fact_1040_log__less__iff,axiom,
    ! [Ba: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(Ba),X)),Y)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(powr_real(Ba),Y)) ) ) ) ).

% log_less_iff
tff(fact_1041_less__log__iff,axiom,
    ! [Ba: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(log(Ba),X))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(Ba),Y)),X) ) ) ) ).

% less_log_iff
tff(fact_1042_msrevs_I1_J,axiom,
    ! [Na: nat,K: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(divide_divide_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)),M)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(divide_divide_nat(M),Na)),K) ) ) ).

% msrevs(1)
tff(fact_1043_powr__mult__base,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_real(times_times_real(X),aa_real_real(powr_real(X),Y)) = aa_real_real(powr_real(X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Y)) ) ) ).

% powr_mult_base
tff(fact_1044_powr__le__iff,axiom,
    ! [Ba: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(Ba),Y)),X)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(log(Ba),X)) ) ) ) ).

% powr_le_iff
tff(fact_1045_le__powr__iff,axiom,
    ! [Ba: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(powr_real(Ba),Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(Ba),X)),Y) ) ) ) ).

% le_powr_iff
tff(fact_1046_log__le__iff,axiom,
    ! [Ba: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(Ba),X)),Y)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(powr_real(Ba),Y)) ) ) ) ).

% log_le_iff
tff(fact_1047_le__log__iff,axiom,
    ! [Ba: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(log(Ba),X))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(Ba),Y)),X) ) ) ) ).

% le_log_iff
tff(fact_1048_log__add__eq__powr,axiom,
    ! [Ba: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ba)
     => ( ( Ba != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(log(Ba),X)),Y) = aa_real_real(log(Ba),aa_real_real(times_times_real(X),aa_real_real(powr_real(Ba),Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_1049_add__log__eq__powr,axiom,
    ! [Ba: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ba)
     => ( ( Ba != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),aa_real_real(log(Ba),X)) = aa_real_real(log(Ba),aa_real_real(times_times_real(aa_real_real(powr_real(Ba),Y)),X)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_1050_add__self__div__2,axiom,
    ! [M: nat] : aa_nat_nat(divide_divide_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),M)),numeral_numeral_nat(bit0(one))) = M ).

% add_self_div_2
tff(fact_1051_div__mult__self__is__m,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(divide_divide_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)),Na) = M ) ) ).

% div_mult_self_is_m
tff(fact_1052_div__mult__self1__is__m,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(divide_divide_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),M)),Na) = M ) ) ).

% div_mult_self1_is_m
tff(fact_1053_div__eq__dividend__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( ( aa_nat_nat(divide_divide_nat(M),Na) = M )
      <=> ( Na = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_1054_zdiv__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,bit0(V))),aa_num_int(numeral_numeral_int,bit0(W))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit0
tff(fact_1055_zdiv__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,bit1(V))),aa_num_int(numeral_numeral_int,bit0(W))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit1
tff(fact_1056_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),L)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
tff(fact_1057_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
tff(fact_1058_floor__divide__eq__div__numeral,axiom,
    ! [Aa: num,Ba: num] : archim6058952711729229775r_real(aa_real_real(divide_divide_real(numeral_numeral_real(Aa)),numeral_numeral_real(Ba))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,Aa)),aa_num_int(numeral_numeral_int,Ba)) ).

% floor_divide_eq_div_numeral
tff(fact_1059_ceiling__divide__eq__div__numeral,axiom,
    ! [Aa: num,Ba: num] : archim7802044766580827645g_real(aa_real_real(divide_divide_real(numeral_numeral_real(Aa)),numeral_numeral_real(Ba))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Aa))),aa_num_int(numeral_numeral_int,Ba))) ).

% ceiling_divide_eq_div_numeral
tff(fact_1060_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one))))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% half_nonnegative_int_iff
tff(fact_1061_half__negative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one)))),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% half_negative_int_iff
tff(fact_1062_floor__one__divide__eq__div__numeral,axiom,
    ! [Ba: num] : archim6058952711729229775r_real(aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(Ba))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,Ba)) ).

% floor_one_divide_eq_div_numeral
tff(fact_1063_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [Aa: num,Ba: num] : archim7802044766580827645g_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(numeral_numeral_real(Aa)),numeral_numeral_real(Ba)))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,Aa)),aa_num_int(numeral_numeral_int,Ba))) ).

% ceiling_minus_divide_eq_div_numeral
tff(fact_1064_floor__minus__divide__eq__div__numeral,axiom,
    ! [Aa: num,Ba: num] : archim6058952711729229775r_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(numeral_numeral_real(Aa)),numeral_numeral_real(Ba)))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Aa))),aa_num_int(numeral_numeral_int,Ba)) ).

% floor_minus_divide_eq_div_numeral
tff(fact_1065_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [Ba: num] : archim6058952711729229775r_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(Ba)))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Ba)) ).

% floor_minus_one_divide_eq_div_numeral
tff(fact_1066_zdiv__zmult2__eq,axiom,
    ! [C: int,Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),times_times_int(Ba,C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)),C) ) ) ).

% zdiv_zmult2_eq
tff(fact_1067_zdiv__int,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(divide_divide_nat(Aa),Ba)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,Aa)),aa_nat_int(semiri1314217659103216013at_int,Ba)) ).

% zdiv_int
tff(fact_1068_nat__div__distrib_H,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)) = aa_nat_nat(divide_divide_nat(aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib'
tff(fact_1069_nat__div__distrib,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)) = aa_nat_nat(divide_divide_nat(aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib
tff(fact_1070_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba))
      <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_1071_pos__imp__zdiv__nonneg__iff,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_1072_neg__imp__zdiv__nonneg__iff,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_1073_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,I),K))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_1074_div__nonpos__pos__le0,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)),zero_zero_int) ) ) ).

% div_nonpos_pos_le0
tff(fact_1075_div__nonneg__neg__le0,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)),zero_zero_int) ) ) ).

% div_nonneg_neg_le0
tff(fact_1076_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L)) ) ) ).

% div_positive_int
tff(fact_1077_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L))
    <=> ( ( K = zero_zero_int )
        | ( L = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ) ).

% div_int_pos_iff
tff(fact_1078_zdiv__mono2__neg,axiom,
    ! [Aa: int,B3: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B3),Ba)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),B3)),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_1079_zdiv__mono1__neg,axiom,
    ! [Aa: int,A3: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),A3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),Ba)),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)) ) ) ).

% zdiv_mono1_neg
tff(fact_1080_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,I),K) = zero_zero_int )
    <=> ( ( K = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,I),K) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K),I) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_1081_zdiv__mono2,axiom,
    ! [Aa: int,B3: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B3),Ba)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),B3)) ) ) ) ).

% zdiv_mono2
tff(fact_1082_zdiv__mono1,axiom,
    ! [Aa: int,A3: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),A3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),Ba)) ) ) ).

% zdiv_mono1
tff(fact_1083_int__div__less__self,axiom,
    ! [X: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),K)),X) ) ) ).

% int_div_less_self
tff(fact_1084_real__of__nat__div4,axiom,
    ! [Na: nat,X: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(divide_divide_nat(Na),X))),aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(semiri5074537144036343181t_real,X))) ).

% real_of_nat_div4
tff(fact_1085_powr__divide,axiom,
    ! [X: real,Y: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_real(powr_real(aa_real_real(divide_divide_real(X),Y)),Aa) = aa_real_real(divide_divide_real(aa_real_real(powr_real(X),Aa)),aa_real_real(powr_real(Y),Aa)) ) ) ) ).

% powr_divide
tff(fact_1086_log__base__powr,axiom,
    ! [Aa: real,Ba: real,X: real] :
      ( ( Aa != zero_zero_real )
     => ( aa_real_real(log(aa_real_real(powr_real(Aa),Ba)),X) = aa_real_real(divide_divide_real(aa_real_real(log(Aa),X)),Ba) ) ) ).

% log_base_powr
tff(fact_1087_nat__div__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa_nat_nat(divide_divide_nat(X4),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_div_as_int
tff(fact_1088_not__int__div__2,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,bit_ri7919022796975470100ot_int(K)),aa_num_int(numeral_numeral_int,bit0(one))) = bit_ri7919022796975470100ot_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one)))) ).

% not_int_div_2
tff(fact_1089_verit__less__mono__div__int2,axiom,
    ! [A: int,B: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,Na))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),Na)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),Na)) ) ) ).

% verit_less_mono_div_int2
tff(fact_1090_div__eq__minus1,axiom,
    ! [Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),Ba) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% div_eq_minus1
tff(fact_1091_log__base__change,axiom,
    ! [Aa: real,Ba: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( ( Aa != one_one_real )
       => ( aa_real_real(log(Ba),X) = aa_real_real(divide_divide_real(aa_real_real(log(Aa),X)),aa_real_real(log(Aa),Ba)) ) ) ) ).

% log_base_change
tff(fact_1092_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,plus_plus_int(K,L)),zero_zero_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% div_pos_neg_trivial
tff(fact_1093_int__div__pos__eq,axiom,
    ! [Aa: int,Ba: int,Q3: int,R: int] :
      ( ( Aa = plus_plus_int(times_times_int(Ba,Q3),R) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R),Ba)
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba) = Q3 ) ) ) ) ).

% int_div_pos_eq
tff(fact_1094_int__div__neg__eq,axiom,
    ! [Aa: int,Ba: int,Q3: int,R: int] :
      ( ( Aa = plus_plus_int(times_times_int(Ba,Q3),R) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),R)
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba) = Q3 ) ) ) ) ).

% int_div_neg_eq
tff(fact_1095_split__zdiv,axiom,
    ! [P: fun_int_o,Na: int,K: int] :
      ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),K))
    <=> ( ( ( K = zero_zero_int )
         => aa_int_o(P,zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
         => ! [I4: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
                & ( Na = plus_plus_int(times_times_int(K,I4),J3) ) )
             => aa_int_o(P,I4) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I4: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
                & ( Na = plus_plus_int(times_times_int(K,I4),J3) ) )
             => aa_int_o(P,I4) ) ) ) ) ).

% split_zdiv
tff(fact_1096_log__base__pow,axiom,
    ! [Aa: real,Na: nat,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( aa_real_real(log(aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na)),X) = aa_real_real(divide_divide_real(aa_real_real(log(Aa),X)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ).

% log_base_pow
tff(fact_1097_minus__1__div__exp__eq__int,axiom,
    ! [Na: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_exp_eq_int
tff(fact_1098_div__le__mono,axiom,
    ! [M: nat,Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(divide_divide_nat(M),K)),aa_nat_nat(divide_divide_nat(Na),K)) ) ).

% div_le_mono
tff(fact_1099_div__le__dividend,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(divide_divide_nat(M),Na)),M) ).

% div_le_dividend
tff(fact_1100_div__mult2__eq,axiom,
    ! [M: nat,Na: nat,Q3: nat] : aa_nat_nat(divide_divide_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q3)) = aa_nat_nat(divide_divide_nat(aa_nat_nat(divide_divide_nat(M),Na)),Q3) ).

% div_mult2_eq
tff(fact_1101_pos__zdiv__mult__2,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,plus_plus_int(one_one_int,times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),Ba))),times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),Aa)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,Ba),Aa) ) ) ).

% pos_zdiv_mult_2
tff(fact_1102_neg__zdiv__mult__2,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,plus_plus_int(one_one_int,times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),Ba))),times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),Aa)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,plus_plus_int(Ba,one_one_int)),Aa) ) ) ).

% neg_zdiv_mult_2
tff(fact_1103_powr__neg__numeral,axiom,
    ! [X: real,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(powr_real(X),aa_real_real(uminus_uminus_real,numeral_numeral_real(Na))) = aa_real_real(divide_divide_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(Na))) ) ) ).

% powr_neg_numeral
tff(fact_1104_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(divide_divide_nat(M),Na)),I) ) ).

% less_mult_imp_div_less
tff(fact_1105_div__times__less__eq__dividend,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(divide_divide_nat(M),Na)),Na)),M) ).

% div_times_less_eq_dividend
tff(fact_1106_times__div__less__eq__dividend,axiom,
    ! [Na: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(divide_divide_nat(M),Na))),M) ).

% times_div_less_eq_dividend
tff(fact_1107_div__le__mono2,axiom,
    ! [M: nat,Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(divide_divide_nat(K),Na)),aa_nat_nat(divide_divide_nat(K),M)) ) ) ).

% div_le_mono2
tff(fact_1108_div__greater__zero__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(divide_divide_nat(M),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% div_greater_zero_iff
tff(fact_1109_div__less__dividend,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(divide_divide_nat(M),Na)),M) ) ) ).

% div_less_dividend
tff(fact_1110_div__less__iff__less__mult,axiom,
    ! [Q3: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(divide_divide_nat(M),Q3)),Na)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q3)) ) ) ).

% div_less_iff_less_mult
tff(fact_1111_split__div,axiom,
    ! [P: fun_nat_o,M: nat,Na: nat] :
      ( aa_nat_o(P,aa_nat_nat(divide_divide_nat(M),Na))
    <=> ( ( ( Na = zero_zero_nat )
         => aa_nat_o(P,zero_zero_nat) )
        & ( ( Na != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Na)
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),I4)),J3) )
               => aa_nat_o(P,I4) ) ) ) ) ) ).

% split_div
tff(fact_1112_dividend__less__div__times,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(divide_divide_nat(M),Na)),Na))) ) ).

% dividend_less_div_times
tff(fact_1113_dividend__less__times__div,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(divide_divide_nat(M),Na)))) ) ).

% dividend_less_times_div
tff(fact_1114_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q3: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(divide_divide_nat(Na),Q3))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)),Na) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_1115_int__div__minus__is__minus1,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),zero_zero_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba) = aa_int_int(uminus_uminus_int,Aa) )
      <=> ( Ba = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% int_div_minus_is_minus1
tff(fact_1116_int__div__same__is__1,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba) = Aa )
      <=> ( Ba = one_one_int ) ) ) ).

% int_div_same_is_1
tff(fact_1117_two__pow__div__gt__le,axiom,
    ! [V: nat,Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,V),aa_nat_nat(divide_divide_nat(aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na)),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),M)))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% two_pow_div_gt_le
tff(fact_1118_z1pdiv2,axiom,
    ! [Ba: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,plus_plus_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),Ba),one_one_int)),aa_num_int(numeral_numeral_int,bit0(one))) = Ba ).

% z1pdiv2
tff(fact_1119_axxdiv2,axiom,
    ! [X: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,plus_plus_int(plus_plus_int(one_one_int,X),X)),aa_num_int(numeral_numeral_int,bit0(one))) = X )
      & ( aa_int_int(aa_int_fun_int_int(divide_divide_int,plus_plus_int(plus_plus_int(zero_zero_int,X),X)),aa_num_int(numeral_numeral_int,bit0(one))) = X ) ) ).

% axxdiv2
tff(fact_1120_VEBT__internal_Omulcomm,axiom,
    ! [I: nat,Va: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(divide_divide_nat(Va),numeral_numeral_nat(bit0(one)))))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(divide_divide_nat(Va),numeral_numeral_nat(bit0(one)))))),I) ).

% VEBT_internal.mulcomm
tff(fact_1121_VEBT__internal_Opow__sum,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_nat(divide_divide_nat(aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba))),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Aa)) = aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Ba) ).

% VEBT_internal.pow_sum
tff(fact_1122_td__gal,axiom,
    ! [C: nat,Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),C)),Aa)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),aa_nat_nat(divide_divide_nat(Aa),C)) ) ) ).

% td_gal
tff(fact_1123_div__mult__le,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(divide_divide_nat(Aa),Ba)),Ba)),Aa) ).

% div_mult_le
tff(fact_1124_zdiv__le__dividend,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)),Aa) ) ) ).

% zdiv_le_dividend
tff(fact_1125_zdiv__mult__self,axiom,
    ! [M: int,Aa: int,Na: int] :
      ( ( M != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,plus_plus_int(Aa,times_times_int(M,Na))),M) = plus_plus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),M),Na) ) ) ).

% zdiv_mult_self
tff(fact_1126_td__gal__lt,axiom,
    ! [C: nat,Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(divide_divide_nat(Aa),C)),Ba) ) ) ).

% td_gal_lt
tff(fact_1127_ceiling__log2__div2,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Na)
     => ( archim7802044766580827645g_real(aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Na))) = plus_plus_int(archim7802044766580827645g_real(aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(divide_divide_nat(aa_nat_nat(minus_minus_nat(Na),one_one_nat)),numeral_numeral_nat(bit0(one)))),one_one_nat)))),one_one_int) ) ) ).

% ceiling_log2_div2
tff(fact_1128_central__binomial__lower__bound,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(bit0(one)))),Na)),aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Na)))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(binomial(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Na)),Na))) ) ).

% central_binomial_lower_bound
tff(fact_1129_and__int__unfold,axiom,
    ! [K: int,L: int] :
      bit_se725231765392027082nd_int(K,L) = $ite(
        ( ( K = zero_zero_int )
        | ( L = zero_zero_int ) ),
        zero_zero_int,
        $ite(
          K = aa_int_int(uminus_uminus_int,one_one_int),
          L,
          $ite(L = aa_int_int(uminus_uminus_int,one_one_int),K,plus_plus_int(times_times_int(modulo_modulo_int(K,aa_num_int(numeral_numeral_int,bit0(one))),modulo_modulo_int(L,aa_num_int(numeral_numeral_int,bit0(one)))),times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),bit_se725231765392027082nd_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,bit0(one))))))) ) ) ).

% and_int_unfold
tff(fact_1130_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),one_one_nat)
    <=> ( M = one_one_nat ) ) ).

% nat_dvd_1_iff_1
tff(fact_1131_int__dvd__int__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na) ) ).

% int_dvd_int_iff
tff(fact_1132_diff__0__eq__0,axiom,
    ! [Na: nat] : aa_nat_nat(minus_minus_nat(zero_zero_nat),Na) = zero_zero_nat ).

% diff_0_eq_0
tff(fact_1133_diff__self__eq__0,axiom,
    ! [M: nat] : aa_nat_nat(minus_minus_nat(M),M) = zero_zero_nat ).

% diff_self_eq_0
tff(fact_1134_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
    <=> ( ( K = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_1135_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] : aa_nat_nat(minus_minus_nat(aa_nat_nat(minus_minus_nat(I),J)),K) = aa_nat_nat(minus_minus_nat(I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ).

% diff_diff_left
tff(fact_1136_diff__diff__cancel,axiom,
    ! [I: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),Na)
     => ( aa_nat_nat(minus_minus_nat(Na),aa_nat_nat(minus_minus_nat(Na),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_1137_unset__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),bit_se4203085406695923979it_int(Na,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_1138_set__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),bit_se7879613467334960850it_int(Na,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% set_bit_nonnegative_int_iff
tff(fact_1139_unset__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,bit_se4203085406695923979it_int(Na,K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% unset_bit_negative_int_iff
tff(fact_1140_set__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,bit_se7879613467334960850it_int(Na,K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% set_bit_negative_int_iff
tff(fact_1141_zero__less__diff,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(minus_minus_nat(Na),M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% zero_less_diff
tff(fact_1142_diff__is__0__eq_H,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_nat(minus_minus_nat(M),Na) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_1143_diff__is__0__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(minus_minus_nat(M),Na) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% diff_is_0_eq
tff(fact_1144_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(minus_minus_nat(J),K)) = aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_1145_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(minus_minus_nat(J),K)),I) = aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_1146_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(minus_minus_nat(I),aa_nat_nat(minus_minus_nat(J),K)) = aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_1147_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),L)
       => ( modulo_modulo_int(K,L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_1148_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K)
       => ( modulo_modulo_int(K,L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_1149_zmod__numeral__Bit0,axiom,
    ! [V: num,W: num] : modulo_modulo_int(aa_num_int(numeral_numeral_int,bit0(V)),aa_num_int(numeral_numeral_int,bit0(W))) = times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),modulo_modulo_int(aa_num_int(numeral_numeral_int,V),aa_num_int(numeral_numeral_int,W))) ).

% zmod_numeral_Bit0
tff(fact_1150_even__diff__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(minus_minus_nat(M),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) ) ) ).

% even_diff_nat
tff(fact_1151_odd__two__times__div__two__nat,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one)))) = aa_nat_nat(minus_minus_nat(Na),one_one_nat) ) ) ).

% odd_two_times_div_two_nat
tff(fact_1152_one__mod__exp__eq__one,axiom,
    ! [Na: nat] : modulo_modulo_int(one_one_int,times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na))) = one_one_int ).

% one_mod_exp_eq_one
tff(fact_1153_zmod__numeral__Bit1,axiom,
    ! [V: num,W: num] : modulo_modulo_int(aa_num_int(numeral_numeral_int,bit1(V)),aa_num_int(numeral_numeral_int,bit0(W))) = plus_plus_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),modulo_modulo_int(aa_num_int(numeral_numeral_int,V),aa_num_int(numeral_numeral_int,W))),one_one_int) ).

% zmod_numeral_Bit1
tff(fact_1154_dvd__minus__self,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(minus_minus_nat(Na),M))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na) ) ) ).

% dvd_minus_self
tff(fact_1155_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] : aa_nat_nat(minus_minus_nat(aa_nat_nat(minus_minus_nat(I),J)),K) = aa_nat_nat(minus_minus_nat(aa_nat_nat(minus_minus_nat(I),K)),J) ).

% diff_commute
tff(fact_1156_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Na)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(minus_minus_nat(M),Na)) ) ) ).

% dvd_diff_nat
tff(fact_1157_dvd__diffD,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(minus_minus_nat(M),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M) ) ) ) ).

% dvd_diffD
tff(fact_1158_dvd__diffD1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(minus_minus_nat(M),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Na) ) ) ) ).

% dvd_diffD1
tff(fact_1159_less__eq__dvd__minus,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(minus_minus_nat(Na),M)) ) ) ).

% less_eq_dvd_minus
tff(fact_1160_minus__nat_Odiff__0,axiom,
    ! [M: nat] : aa_nat_nat(minus_minus_nat(M),zero_zero_nat) = M ).

% minus_nat.diff_0
tff(fact_1161_diffs0__imp__equal,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(minus_minus_nat(M),Na) = zero_zero_nat )
     => ( ( aa_nat_nat(minus_minus_nat(Na),M) = zero_zero_nat )
       => ( M = Na ) ) ) ).

% diffs0_imp_equal
tff(fact_1162_mod__plus__cong,axiom,
    ! [Ba: int,B3: int,X: int,X6: int,Y: int,Y5: int,Z4: int] :
      ( ( Ba = B3 )
     => ( ( modulo_modulo_int(X,B3) = modulo_modulo_int(X6,B3) )
       => ( ( modulo_modulo_int(Y,B3) = modulo_modulo_int(Y5,B3) )
         => ( ( plus_plus_int(X6,Y5) = Z4 )
           => ( modulo_modulo_int(plus_plus_int(X,Y),Ba) = modulo_modulo_int(Z4,B3) ) ) ) ) ) ).

% mod_plus_cong
tff(fact_1163_zmod__helper,axiom,
    ! [Na: int,M: int,K: int,Aa: int] :
      ( ( modulo_modulo_int(Na,M) = K )
     => ( modulo_modulo_int(plus_plus_int(Na,Aa),M) = modulo_modulo_int(plus_plus_int(K,Aa),M) ) ) ).

% zmod_helper
tff(fact_1164_diff__less__mono2,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(minus_minus_nat(L),Na)),aa_nat_nat(minus_minus_nat(L),M)) ) ) ).

% diff_less_mono2
tff(fact_1165_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(minus_minus_nat(J),Na)),K) ) ).

% less_imp_diff_less
tff(fact_1166_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,Na: nat] : aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na)) = aa_nat_nat(minus_minus_nat(M),Na) ).

% Nat.diff_cancel
tff(fact_1167_diff__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] : aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K)) = aa_nat_nat(minus_minus_nat(M),Na) ).

% diff_cancel2
tff(fact_1168_diff__add__inverse,axiom,
    ! [Na: nat,M: nat] : aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)),Na) = M ).

% diff_add_inverse
tff(fact_1169_diff__add__inverse2,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)),Na) = M ).

% diff_add_inverse2
tff(fact_1170_eq__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => ( ( aa_nat_nat(minus_minus_nat(M),K) = aa_nat_nat(minus_minus_nat(Na),K) )
        <=> ( M = Na ) ) ) ) ).

% eq_diff_iff
tff(fact_1171_le__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(minus_minus_nat(M),K)),aa_nat_nat(minus_minus_nat(Na),K))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ) ).

% le_diff_iff
tff(fact_1172_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => ( aa_nat_nat(minus_minus_nat(aa_nat_nat(minus_minus_nat(M),K)),aa_nat_nat(minus_minus_nat(Na),K)) = aa_nat_nat(minus_minus_nat(M),Na) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_1173_diff__le__mono,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(minus_minus_nat(M),L)),aa_nat_nat(minus_minus_nat(Na),L)) ) ).

% diff_le_mono
tff(fact_1174_diff__le__self,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(minus_minus_nat(M),Na)),M) ).

% diff_le_self
tff(fact_1175_le__diff__iff_H,axiom,
    ! [Aa: nat,C: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),C)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(minus_minus_nat(C),Aa)),aa_nat_nat(minus_minus_nat(C),Ba))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),Aa) ) ) ) ).

% le_diff_iff'
tff(fact_1176_diff__le__mono2,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(minus_minus_nat(L),Na)),aa_nat_nat(minus_minus_nat(L),M)) ) ).

% diff_le_mono2
tff(fact_1177_diff__mult__distrib,axiom,
    ! [M: nat,Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(M),Na)),K) = aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K)) ).

% diff_mult_distrib
tff(fact_1178_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(minus_minus_nat(M),Na)) = aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)) ).

% diff_mult_distrib2
tff(fact_1179_unset__bit__nat__def,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(bit_se4205575877204974255it_nat(M),Na) = aa_int_nat(nat2,bit_se4203085406695923979it_int(M,aa_nat_int(semiri1314217659103216013at_int,Na))) ).

% unset_bit_nat_def
tff(fact_1180_unset__bit__less__eq,axiom,
    ! [Na: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,bit_se4203085406695923979it_int(Na,K)),K) ).

% unset_bit_less_eq
tff(fact_1181_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),modulo_modulo_int(K,L))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( ( L = zero_zero_int )
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L) ) ) ).

% mod_int_pos_iff
tff(fact_1182_set__bit__greater__eq,axiom,
    ! [K: int,Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),bit_se7879613467334960850it_int(Na,K)) ).

% set_bit_greater_eq
tff(fact_1183_dvd__minus__add,axiom,
    ! [Q3: nat,Na: nat,R: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q3),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),M))
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(minus_minus_nat(Na),Q3))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),M)),Q3))) ) ) ) ).

% dvd_minus_add
tff(fact_1184_nat__dvd__not__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
       => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),M) ) ) ).

% nat_dvd_not_less
tff(fact_1185_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),M)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,modulo_modulo_int(M,K)),M) ) ).

% zmod_le_nonneg_dividend
tff(fact_1186_zmod__eq__0__iff,axiom,
    ! [M: int,D: int] :
      ( ( modulo_modulo_int(M,D) = zero_zero_int )
    <=> ? [Q4: int] : M = times_times_int(D,Q4) ) ).

% zmod_eq_0_iff
tff(fact_1187_zmod__eq__0D,axiom,
    ! [M: int,D: int] :
      ( ( modulo_modulo_int(M,D) = zero_zero_int )
     => ? [Q5: int] : M = times_times_int(D,Q5) ) ).

% zmod_eq_0D
tff(fact_1188_diff__less,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(minus_minus_nat(M),Na)),M) ) ) ).

% diff_less
tff(fact_1189_zdvd__antisym__nonneg,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Na)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),Na)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Na),M)
           => ( M = Na ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_1190_diff__add__0,axiom,
    ! [Na: nat,M: nat] : aa_nat_nat(minus_minus_nat(Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)) = zero_zero_nat ).

% diff_add_0
tff(fact_1191_zdvd__not__zless,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,M),Na)
       => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Na),M) ) ) ).

% zdvd_not_zless
tff(fact_1192_add__diff__inverse__nat,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(minus_minus_nat(M),Na)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_1193_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(minus_minus_nat(J),K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ).

% less_diff_conv
tff(fact_1194_diff__less__mono,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),Aa)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(minus_minus_nat(Aa),C)),aa_nat_nat(minus_minus_nat(Ba),C)) ) ) ).

% diff_less_mono
tff(fact_1195_less__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(minus_minus_nat(M),K)),aa_nat_nat(minus_minus_nat(Na),K))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ) ).

% less_diff_iff
tff(fact_1196_zdvd__mono,axiom,
    ! [K: int,M: int,T: int] :
      ( ( K != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),T)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,times_times_int(K,M)),times_times_int(K,T)) ) ) ).

% zdvd_mono
tff(fact_1197_zdvd__mult__cancel,axiom,
    ! [K: int,M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,times_times_int(K,M)),times_times_int(K,Na))
     => ( ( K != zero_zero_int )
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),Na) ) ) ).

% zdvd_mult_cancel
tff(fact_1198_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(minus_minus_nat(J),K)),I)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)) ) ).

% le_diff_conv
tff(fact_1199_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(minus_minus_nat(J),K))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_1200_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(minus_minus_nat(J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_1201_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(minus_minus_nat(J),K)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_1202_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( ( aa_nat_nat(minus_minus_nat(J),I) = K )
      <=> ( J = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_1203_zdvd__reduce,axiom,
    ! [K: int,Na: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),plus_plus_int(Na,times_times_int(K,M)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Na) ) ).

% zdvd_reduce
tff(fact_1204_zdvd__period,axiom,
    ! [Aa: int,D: int,X: int,T: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),D)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),plus_plus_int(X,T))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),plus_plus_int(plus_plus_int(X,times_times_int(C,D)),T)) ) ) ).

% zdvd_period
tff(fact_1205_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(nat2,Z)),M)
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z),aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),aa_nat_int(semiri1314217659103216013at_int,M)),M = zero_zero_nat) ) ).

% nat_dvd_iff
tff(fact_1206_dvd__imp__le,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na) ) ) ).

% dvd_imp_le
tff(fact_1207_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_1208_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na) ) ) ).

% dvd_mult_cancel
tff(fact_1209_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),modulo_modulo_int(K,L)) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_1210_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,modulo_modulo_int(K,L)),zero_zero_int) ) ).

% neg_mod_sign
tff(fact_1211_int__mod__ge,axiom,
    ! [Aa: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),modulo_modulo_int(Aa,Na)) ) ) ).

% int_mod_ge
tff(fact_1212_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( modulo_modulo_int(I,K) = I )
    <=> ( ( K = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,I),K) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K),I) ) ) ) ).

% zmod_trivial_iff
tff(fact_1213_pos__mod__conj,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),modulo_modulo_int(Aa,Ba))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,modulo_modulo_int(Aa,Ba)),Ba) ) ) ).

% pos_mod_conj
tff(fact_1214_neg__mod__conj,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,modulo_modulo_int(Aa,Ba)),zero_zero_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),modulo_modulo_int(Aa,Ba)) ) ) ).

% neg_mod_conj
tff(fact_1215_int__mod__lem,axiom,
    ! [Na: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Na)
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ba)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),Na) )
      <=> ( modulo_modulo_int(Ba,Na) = Ba ) ) ) ).

% int_mod_lem
tff(fact_1216_int__mod__eq,axiom,
    ! [Ba: int,Na: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ba)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),Na)
       => ( ( modulo_modulo_int(Aa,Na) = modulo_modulo_int(Ba,Na) )
         => ( modulo_modulo_int(Aa,Na) = Ba ) ) ) ) ).

% int_mod_eq
tff(fact_1217_nat__diff__split__asm,axiom,
    ! [P: fun_nat_o,Aa: nat,Ba: nat] :
      ( aa_nat_o(P,aa_nat_nat(minus_minus_nat(Aa),Ba))
    <=> ~ ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
            & ~ aa_nat_o(P,zero_zero_nat) )
          | ? [D2: nat] :
              ( ( Aa = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),D2) )
              & ~ aa_nat_o(P,D2) ) ) ) ).

% nat_diff_split_asm
tff(fact_1218_nat__diff__split,axiom,
    ! [P: fun_nat_o,Aa: nat,Ba: nat] :
      ( aa_nat_o(P,aa_nat_nat(minus_minus_nat(Aa),Ba))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
         => aa_nat_o(P,zero_zero_nat) )
        & ! [D2: nat] :
            ( ( Aa = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),D2) )
           => aa_nat_o(P,D2) ) ) ) ).

% nat_diff_split
tff(fact_1219_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(minus_minus_nat(J),K)),I)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)) ) ) ).

% less_diff_conv2
tff(fact_1220_nonneg__mod__div,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ba)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),modulo_modulo_int(Aa,Ba))
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)) ) ) ) ).

% nonneg_mod_div
tff(fact_1221_zdiv__mono__strict,axiom,
    ! [A: int,B: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Na)
       => ( ( modulo_modulo_int(A,Na) = zero_zero_int )
         => ( ( modulo_modulo_int(B,Na) = zero_zero_int )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),Na)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),Na)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_1222_zdvd__imp__le,axiom,
    ! [Z: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Na) ) ) ).

% zdvd_imp_le
tff(fact_1223_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(I),J)),U)),M) = Na ) ) ) ).

% nat_eq_add_iff1
tff(fact_1224_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na) )
      <=> ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(J),I)),U)),Na) ) ) ) ).

% nat_eq_add_iff2
tff(fact_1225_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(I),J)),U)),M)),Na) ) ) ).

% nat_le_add_iff1
tff(fact_1226_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(J),I)),U)),Na)) ) ) ).

% nat_le_add_iff2
tff(fact_1227_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na)) = aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(I),J)),U)),M)),Na) ) ) ).

% nat_diff_add_eq1
tff(fact_1228_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na)) = aa_nat_nat(minus_minus_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(J),I)),U)),Na)) ) ) ).

% nat_diff_add_eq2
tff(fact_1229_div__mod__decomp__int,axiom,
    ! [A: int,Na: int] : A = plus_plus_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),Na),Na),modulo_modulo_int(A,Na)) ).

% div_mod_decomp_int
tff(fact_1230_real__of__nat__div,axiom,
    ! [D: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),Na)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(divide_divide_nat(Na),D)) = aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(semiri5074537144036343181t_real,D)) ) ) ).

% real_of_nat_div
tff(fact_1231_emep1,axiom,
    ! [Na: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),Na)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),D)
         => ( modulo_modulo_int(plus_plus_int(Na,one_one_int),D) = plus_plus_int(modulo_modulo_int(Na,D),one_one_int) ) ) ) ) ).

% emep1
tff(fact_1232_eme1p,axiom,
    ! [Na: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),Na)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),D)
         => ( modulo_modulo_int(plus_plus_int(one_one_int,Na),D) = plus_plus_int(one_one_int,modulo_modulo_int(Na,D)) ) ) ) ) ).

% eme1p
tff(fact_1233_even__nat__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_int_nat(nat2,K))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),K) ) ) ).

% even_nat_iff
tff(fact_1234_pos__mod__bound2,axiom,
    ! [Aa: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,modulo_modulo_int(Aa,aa_num_int(numeral_numeral_int,bit0(one)))),aa_num_int(numeral_numeral_int,bit0(one))) ).

% pos_mod_bound2
tff(fact_1235_div2__even__ext__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(divide_divide_nat(X),numeral_numeral_nat(bit0(one))) = aa_nat_nat(divide_divide_nat(Y),numeral_numeral_nat(bit0(one))) )
     => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),X)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Y) )
       => ( X = Y ) ) ) ).

% div2_even_ext_nat
tff(fact_1236_dvd__mult__cancel2,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),M)),M)
      <=> ( Na = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_1237_dvd__mult__cancel1,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)),M)
      <=> ( Na = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_1238_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(power_power_nat(I),M)),aa_nat_nat(power_power_nat(I),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),I)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% power_dvd_imp_le
tff(fact_1239_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,plus_plus_int(K,L)),zero_zero_int)
       => ( modulo_modulo_int(K,L) = plus_plus_int(K,L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_1240_int__mod__ge_H,axiom,
    ! [Ba: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,plus_plus_int(Ba,Na)),modulo_modulo_int(Ba,Na)) ) ) ).

% int_mod_ge'
tff(fact_1241_even__and__iff__int,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),bit_se725231765392027082nd_int(K,L))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),K)
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),L) ) ) ).

% even_and_iff_int
tff(fact_1242_even__not__iff__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),bit_ri7919022796975470100ot_int(K))
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),K) ) ).

% even_not_iff_int
tff(fact_1243_mult__eq__if,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) = $ite(M = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(M),one_one_nat)),Na))) ).

% mult_eq_if
tff(fact_1244_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(I),J)),U)),M)),Na) ) ) ).

% nat_less_add_iff1
tff(fact_1245_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(J),I)),U)),Na)) ) ) ).

% nat_less_add_iff2
tff(fact_1246_nat__mult__power__less__eq,axiom,
    ! [Ba: nat,Aa: nat,Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),aa_nat_nat(power_power_nat(Ba),Na))),aa_nat_nat(power_power_nat(Ba),M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),aa_nat_nat(power_power_nat(Ba),aa_nat_nat(minus_minus_nat(M),Na))) ) ) ).

% nat_mult_power_less_eq
tff(fact_1247_binomial__maximum,axiom,
    ! [Na: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(binomial(Na),K)),aa_nat_nat(binomial(Na),aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one))))) ).

% binomial_maximum
tff(fact_1248_binomial__antimono,axiom,
    ! [K: nat,K3: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),K3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one)))),K)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),Na)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(binomial(Na),K3)),aa_nat_nat(binomial(Na),K)) ) ) ) ).

% binomial_antimono
tff(fact_1249_binomial__mono,axiom,
    ! [K: nat,K3: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),K3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),K3)),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(binomial(Na),K)),aa_nat_nat(binomial(Na),K3)) ) ) ).

% binomial_mono
tff(fact_1250_binomial__maximum_H,axiom,
    ! [Na: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(binomial(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Na)),K)),aa_nat_nat(binomial(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Na)),Na)) ).

% binomial_maximum'
tff(fact_1251_pos__mod__sign2,axiom,
    ! [Aa: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),modulo_modulo_int(Aa,aa_num_int(numeral_numeral_int,bit0(one)))) ).

% pos_mod_sign2
tff(fact_1252_mod__2__neq__1__eq__eq__0,axiom,
    ! [K: int] :
      ( ( modulo_modulo_int(K,aa_num_int(numeral_numeral_int,bit0(one))) != one_one_int )
    <=> ( modulo_modulo_int(K,aa_num_int(numeral_numeral_int,bit0(one))) = zero_zero_int ) ) ).

% mod_2_neq_1_eq_eq_0
tff(fact_1253_nmod2,axiom,
    ! [Na: int] :
      ( ( modulo_modulo_int(Na,aa_num_int(numeral_numeral_int,bit0(one))) = zero_zero_int )
      | ( modulo_modulo_int(Na,aa_num_int(numeral_numeral_int,bit0(one))) = one_one_int ) ) ).

% nmod2
tff(fact_1254_odd__pos,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% odd_pos
tff(fact_1255_mod__exp__less__eq__exp,axiom,
    ! [Aa: int,Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,modulo_modulo_int(Aa,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na)) ).

% mod_exp_less_eq_exp
tff(fact_1256_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(power_power_nat(K),M)),aa_nat_nat(power_power_nat(K),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% dvd_power_iff_le
tff(fact_1257_mod__power__lem,axiom,
    ! [Aa: int,Na: nat,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Aa)
     => ( modulo_modulo_int(aa_nat_int(power_power_int(Aa),Na),aa_nat_int(power_power_int(Aa),M)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na),zero_zero_int,aa_nat_int(power_power_int(Aa),Na)) ) ) ).

% mod_power_lem
tff(fact_1258_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(minus_minus_nat(M),Na)),aa_nat_nat(minus_minus_nat(aa_nat_nat(power_power_nat(K),M)),aa_nat_nat(power_power_nat(K),Na))) ) ).

% diff_le_diff_pow
tff(fact_1259_int__mod__pos__eq,axiom,
    ! [Aa: int,Ba: int,Q3: int,R: int] :
      ( ( Aa = plus_plus_int(times_times_int(Ba,Q3),R) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R),Ba)
         => ( modulo_modulo_int(Aa,Ba) = R ) ) ) ) ).

% int_mod_pos_eq
tff(fact_1260_int__mod__neg__eq,axiom,
    ! [Aa: int,Ba: int,Q3: int,R: int] :
      ( ( Aa = plus_plus_int(times_times_int(Ba,Q3),R) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),R)
         => ( modulo_modulo_int(Aa,Ba) = R ) ) ) ) ).

% int_mod_neg_eq
tff(fact_1261_split__zmod,axiom,
    ! [P: fun_int_o,Na: int,K: int] :
      ( aa_int_o(P,modulo_modulo_int(Na,K))
    <=> ( ( ( K = zero_zero_int )
         => aa_int_o(P,Na) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
         => ! [I4: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
                & ( Na = plus_plus_int(times_times_int(K,I4),J3) ) )
             => aa_int_o(P,J3) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I4: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
                & ( Na = plus_plus_int(times_times_int(K,I4),J3) ) )
             => aa_int_o(P,J3) ) ) ) ) ).

% split_zmod
tff(fact_1262_zmod__zmult2__eq,axiom,
    ! [C: int,Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( modulo_modulo_int(Aa,times_times_int(Ba,C)) = plus_plus_int(times_times_int(Ba,modulo_modulo_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba),C)),modulo_modulo_int(Aa,Ba)) ) ) ).

% zmod_zmult2_eq
tff(fact_1263_power__sub,axiom,
    ! [Na: nat,M: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Aa)
       => ( aa_nat_nat(power_power_nat(Aa),aa_nat_nat(minus_minus_nat(M),Na)) = aa_nat_nat(divide_divide_nat(aa_nat_nat(power_power_nat(Aa),M)),aa_nat_nat(power_power_nat(Aa),Na)) ) ) ) ).

% power_sub
tff(fact_1264_binomial__strict__mono,axiom,
    ! [K: nat,K3: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),K3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),K3)),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(binomial(Na),K)),aa_nat_nat(binomial(Na),K3)) ) ) ).

% binomial_strict_mono
tff(fact_1265_binomial__strict__antimono,axiom,
    ! [K: nat,K3: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),K3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),K))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),Na)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(binomial(Na),K3)),aa_nat_nat(binomial(Na),K)) ) ) ) ).

% binomial_strict_antimono
tff(fact_1266_axxmod2,axiom,
    ! [X: int] :
      ( ( modulo_modulo_int(plus_plus_int(plus_plus_int(one_one_int,X),X),aa_num_int(numeral_numeral_int,bit0(one))) = one_one_int )
      & ( modulo_modulo_int(plus_plus_int(plus_plus_int(zero_zero_int,X),X),aa_num_int(numeral_numeral_int,bit0(one))) = zero_zero_int ) ) ).

% axxmod2
tff(fact_1267_z1pmod2,axiom,
    ! [Ba: int] : modulo_modulo_int(plus_plus_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),Ba),one_one_int),aa_num_int(numeral_numeral_int,bit0(one))) = one_one_int ).

% z1pmod2
tff(fact_1268_nat__power__less__diff,axiom,
    ! [Na: nat,Q3: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na)),Q3)),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),M))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Q3),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(minus_minus_nat(M),Na))) ) ).

% nat_power_less_diff
tff(fact_1269_power__minus__is__div,axiom,
    ! [Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),Aa)
     => ( aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(minus_minus_nat(Aa),Ba)) = aa_nat_nat(divide_divide_nat(aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Aa)),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Ba)) ) ) ).

% power_minus_is_div
tff(fact_1270_nat__le__power__trans,axiom,
    ! [Na: nat,M: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(minus_minus_nat(M),K)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),K)),Na)),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),M)) ) ) ).

% nat_le_power_trans
tff(fact_1271_verit__le__mono__div__int,axiom,
    ! [A: int,B: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,
              plus_plus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),Na),
                $ite(modulo_modulo_int(B,Na) = zero_zero_int,one_one_int,zero_zero_int))),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),Na)) ) ) ).

% verit_le_mono_div_int
tff(fact_1272_split__pos__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_o,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),K)),modulo_modulo_int(Na,K))
      <=> ! [I4: int,J3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
              & ( Na = plus_plus_int(times_times_int(K,I4),J3) ) )
           => aa_int_o(aa_int_fun_int_o(P,I4),J3) ) ) ) ).

% split_pos_lemma
tff(fact_1273_split__neg__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_o,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),K)),modulo_modulo_int(Na,K))
      <=> ! [I4: int,J3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
              & ( Na = plus_plus_int(times_times_int(K,I4),J3) ) )
           => aa_int_o(aa_int_fun_int_o(P,I4),J3) ) ) ) ).

% split_neg_lemma
tff(fact_1274_p1mod22k_H,axiom,
    ! [Ba: int,Na: nat] : modulo_modulo_int(plus_plus_int(one_one_int,times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),Ba)),times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na))) = plus_plus_int(one_one_int,times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),modulo_modulo_int(Ba,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na)))) ).

% p1mod22k'
tff(fact_1275_p1mod22k,axiom,
    ! [Ba: int,Na: nat] : modulo_modulo_int(plus_plus_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),Ba),one_one_int),times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na))) = plus_plus_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),modulo_modulo_int(Ba,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na))),one_one_int) ).

% p1mod22k
tff(fact_1276_less__two__pow__divD,axiom,
    ! [X: nat,Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(divide_divide_nat(aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na)),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),M)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(minus_minus_nat(Na),M))) ) ) ).

% less_two_pow_divD
tff(fact_1277_less__two__pow__divI,axiom,
    ! [X: nat,Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(minus_minus_nat(Na),M)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(divide_divide_nat(aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na)),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),M))) ) ) ).

% less_two_pow_divI
tff(fact_1278_nat__less__power__trans,axiom,
    ! [Na: nat,M: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(minus_minus_nat(M),K)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),K)),Na)),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),M)) ) ) ).

% nat_less_power_trans
tff(fact_1279_pos__zmod__mult__2,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( modulo_modulo_int(plus_plus_int(one_one_int,times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),Ba)),times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),Aa)) = plus_plus_int(one_one_int,times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),modulo_modulo_int(Ba,Aa))) ) ) ).

% pos_zmod_mult_2
tff(fact_1280_sb__dec__lem_H,axiom,
    ! [K: nat,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K)),Aa)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,modulo_modulo_int(plus_plus_int(Aa,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K)),times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K)))),plus_plus_int(aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K)),Aa)) ) ).

% sb_dec_lem'
tff(fact_1281_small__powers__of__2,axiom,
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit1(one))),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(minus_minus_nat(X),one_one_nat))) ) ).

% small_powers_of_2
tff(fact_1282_sb__dec__lem,axiom,
    ! [K: nat,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),plus_plus_int(aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K)),Aa))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,modulo_modulo_int(plus_plus_int(Aa,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K)),times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K)))),plus_plus_int(aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K)),Aa)) ) ).

% sb_dec_lem
tff(fact_1283_Bernoulli__inequality__even,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),X))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),Na)) ) ).

% Bernoulli_inequality_even
tff(fact_1284_zero__less__binomial__iff,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(binomial(Na),K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na) ) ).

% zero_less_binomial_iff
tff(fact_1285_choose__two,axiom,
    ! [Na: nat] : aa_nat_nat(binomial(Na),numeral_numeral_nat(bit0(one))) = aa_nat_nat(divide_divide_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(minus_minus_nat(Na),one_one_nat))),numeral_numeral_nat(bit0(one))) ).

% choose_two
tff(fact_1286_binomial__n__0,axiom,
    ! [Na: nat] : aa_nat_nat(binomial(Na),zero_zero_nat) = one_one_nat ).

% binomial_n_0
tff(fact_1287_times__binomial__minus1__eq,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(binomial(Na),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(binomial(aa_nat_nat(minus_minus_nat(Na),one_one_nat)),aa_nat_nat(minus_minus_nat(K),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_1288_choose__reduce__nat,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( aa_nat_nat(binomial(Na),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(binomial(aa_nat_nat(minus_minus_nat(Na),one_one_nat)),aa_nat_nat(minus_minus_nat(K),one_one_nat))),aa_nat_nat(binomial(aa_nat_nat(minus_minus_nat(Na),one_one_nat)),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_1289_binomial__le__pow2,axiom,
    ! [Na: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(binomial(Na),K)),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na)) ).

% binomial_le_pow2
tff(fact_1290_idiff__0__right,axiom,
    ! [Na: extended_enat] : minus_3235023915231533773d_enat(Na,zero_z5237406670263579293d_enat) = Na ).

% idiff_0_right
tff(fact_1291_idiff__0,axiom,
    ! [Na: extended_enat] : minus_3235023915231533773d_enat(zero_z5237406670263579293d_enat,Na) = zero_z5237406670263579293d_enat ).

% idiff_0
tff(fact_1292_binomial__n__n,axiom,
    ! [Na: nat] : aa_nat_nat(binomial(Na),Na) = one_one_nat ).

% binomial_n_n
tff(fact_1293_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),minus_minus_int(Z,one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% zle_diff1_eq
tff(fact_1294_diff__nat__numeral,axiom,
    ! [V: num,V2: num] : aa_nat_nat(minus_minus_nat(numeral_numeral_nat(V)),numeral_numeral_nat(V2)) = aa_int_nat(nat2,minus_minus_int(aa_num_int(numeral_numeral_int,V),aa_num_int(numeral_numeral_int,V2))) ).

% diff_nat_numeral
tff(fact_1295_add__self__mod__2,axiom,
    ! [M: nat] : aa_nat_nat(modulo_modulo_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),M)),numeral_numeral_nat(bit0(one))) = zero_zero_nat ).

% add_self_mod_2
tff(fact_1296_mod2__gr__0,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(modulo_modulo_nat(M),numeral_numeral_nat(bit0(one))))
    <=> ( aa_nat_nat(modulo_modulo_nat(M),numeral_numeral_nat(bit0(one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_1297_nat__numeral__diff__1,axiom,
    ! [V: num] : aa_nat_nat(minus_minus_nat(numeral_numeral_nat(V)),one_one_nat) = aa_int_nat(nat2,minus_minus_int(aa_num_int(numeral_numeral_int,V),one_one_int)) ).

% nat_numeral_diff_1
tff(fact_1298_and__minus__minus__numerals,axiom,
    ! [M: num,Na: num] : bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = bit_ri7919022796975470100ot_int(bit_se1409905431419307370or_int(minus_minus_int(aa_num_int(numeral_numeral_int,M),one_one_int),minus_minus_int(aa_num_int(numeral_numeral_int,Na),one_one_int))) ).

% and_minus_minus_numerals
tff(fact_1299_or__minus__minus__numerals,axiom,
    ! [M: num,Na: num] : bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = bit_ri7919022796975470100ot_int(bit_se725231765392027082nd_int(minus_minus_int(aa_num_int(numeral_numeral_int,M),one_one_int),minus_minus_int(aa_num_int(numeral_numeral_int,Na),one_one_int))) ).

% or_minus_minus_numerals
tff(fact_1300_dvd__antisym,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),M)
       => ( M = Na ) ) ) ).

% dvd_antisym
tff(fact_1301_zdvd__zdiffD,axiom,
    ! [K: int,M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),minus_minus_int(M,Na))
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Na)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),M) ) ) ).

% zdvd_zdiffD
tff(fact_1302_mod__plus__right,axiom,
    ! [Aa: nat,X: nat,M: nat,Ba: nat] :
      ( ( aa_nat_nat(modulo_modulo_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),X)),M) = aa_nat_nat(modulo_modulo_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),X)),M) )
    <=> ( aa_nat_nat(modulo_modulo_nat(Aa),M) = aa_nat_nat(modulo_modulo_nat(Ba),M) ) ) ).

% mod_plus_right
tff(fact_1303_mod__less__eq__dividend,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(modulo_modulo_nat(M),Na)),M) ).

% mod_less_eq_dividend
tff(fact_1304_minus__int__code_I1_J,axiom,
    ! [K: int] : minus_minus_int(K,zero_zero_int) = K ).

% minus_int_code(1)
tff(fact_1305_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z22: int] : times_times_int(W,minus_minus_int(Z1,Z22)) = minus_minus_int(times_times_int(W,Z1),times_times_int(W,Z22)) ).

% int_distrib(4)
tff(fact_1306_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W: int] : times_times_int(minus_minus_int(Z1,Z22),W) = minus_minus_int(times_times_int(Z1,W),times_times_int(Z22,W)) ).

% int_distrib(3)
tff(fact_1307_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M3: nat,N: nat] : Z != minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,M3),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% int_diff_cases
tff(fact_1308_Word_Omod__minus__cong,axiom,
    ! [Ba: int,B3: int,X: int,X6: int,Y: int,Y5: int,Z4: int] :
      ( ( Ba = B3 )
     => ( ( modulo_modulo_int(X,B3) = modulo_modulo_int(X6,B3) )
       => ( ( modulo_modulo_int(Y,B3) = modulo_modulo_int(Y5,B3) )
         => ( ( minus_minus_int(X6,Y5) = Z4 )
           => ( modulo_modulo_int(minus_minus_int(X,Y),Ba) = modulo_modulo_int(Z4,B3) ) ) ) ) ) ).

% Word.mod_minus_cong
tff(fact_1309_word__rot__lem,axiom,
    ! [L: nat,K: nat,D: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,L),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,D),aa_nat_nat(modulo_modulo_nat(K),L)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),L)
       => ( aa_nat_nat(modulo_modulo_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,D),Na)),L) = Na ) ) ) ).

% word_rot_lem
tff(fact_1310_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( aa_nat_nat(modulo_modulo_nat(M),D) = zero_zero_nat )
     => ? [Q5: nat] : M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q5) ) ).

% mod_eq_0D
tff(fact_1311_nat__minus__mod__plus__right,axiom,
    ! [Na: nat,X: nat,M: nat] : aa_nat_nat(modulo_modulo_nat(aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),X)),aa_nat_nat(modulo_modulo_nat(Na),M))),M) = aa_nat_nat(modulo_modulo_nat(X),M) ).

% nat_minus_mod_plus_right
tff(fact_1312_le__mod__geq,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_nat(modulo_modulo_nat(M),Na) = aa_nat_nat(modulo_modulo_nat(aa_nat_nat(minus_minus_nat(M),Na)),Na) ) ) ).

% le_mod_geq
tff(fact_1313_mod__eq__dvd__iff__nat,axiom,
    ! [Na: nat,M: nat,Q3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( ( aa_nat_nat(modulo_modulo_nat(M),Q3) = aa_nat_nat(modulo_modulo_nat(Na),Q3) )
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Q3),aa_nat_nat(minus_minus_nat(M),Na)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_1314_nat__mod__eq__iff,axiom,
    ! [X: nat,Na: nat,Y: nat] :
      ( ( aa_nat_nat(modulo_modulo_nat(X),Na) = aa_nat_nat(modulo_modulo_nat(Y),Na) )
    <=> ? [Q1: nat,Q22: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q1)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_1315_msrevs_I2_J,axiom,
    ! [K: nat,Na: nat,M: nat] : aa_nat_nat(modulo_modulo_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)),M)),Na) = aa_nat_nat(modulo_modulo_nat(M),Na) ).

% msrevs(2)
tff(fact_1316_minus__int__code_I2_J,axiom,
    ! [L: int] : minus_minus_int(zero_zero_int,L) = aa_int_int(uminus_uminus_int,L) ).

% minus_int_code(2)
tff(fact_1317_zmod__int,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(modulo_modulo_nat(Aa),Ba)) = modulo_modulo_int(aa_nat_int(semiri1314217659103216013at_int,Aa),aa_nat_int(semiri1314217659103216013at_int,Ba)) ).

% zmod_int
tff(fact_1318_int__le__induct,axiom,
    ! [I: int,K: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),K)
     => ( aa_int_o(P,K)
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),K)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,minus_minus_int(I2,one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_le_induct
tff(fact_1319_int__less__induct,axiom,
    ! [I: int,K: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),K)
     => ( aa_int_o(P,minus_minus_int(K,one_one_int))
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),K)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,minus_minus_int(I2,one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_less_induct
tff(fact_1320_minus__real__def,axiom,
    ! [X: real,Y: real] : aa_real_real(minus_minus_real(X),Y) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(uminus_uminus_real,Y)) ).

% minus_real_def
tff(fact_1321_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,X: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Z),Y)
     => ( plus_p3455044024723400733d_enat(X,minus_3235023915231533773d_enat(Y,Z)) = minus_3235023915231533773d_enat(plus_p3455044024723400733d_enat(X,Y),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_1322_even__even__mod__4__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(modulo_modulo_nat(Na),numeral_numeral_nat(bit0(bit0(one))))) ) ).

% even_even_mod_4_iff
tff(fact_1323_mod__le__divisor,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(modulo_modulo_nat(M),Na)),Na) ) ).

% mod_le_divisor
tff(fact_1324_div__less__mono,axiom,
    ! [A: nat,B: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( ( aa_nat_nat(modulo_modulo_nat(A),Na) = zero_zero_nat )
         => ( ( aa_nat_nat(modulo_modulo_nat(B),Na) = zero_zero_nat )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(divide_divide_nat(A),Na)),aa_nat_nat(divide_divide_nat(B),Na)) ) ) ) ) ).

% div_less_mono
tff(fact_1325_mod__nat__add,axiom,
    ! [X: nat,Z: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z)
       => ( aa_nat_nat(modulo_modulo_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),Z) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y),aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),Z)) ) ) ) ).

% mod_nat_add
tff(fact_1326_mod__nat__eqI,axiom,
    ! [R: nat,Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,R),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R),M)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),aa_nat_nat(minus_minus_nat(M),R))
         => ( aa_nat_nat(modulo_modulo_nat(M),Na) = R ) ) ) ) ).

% mod_nat_eqI
tff(fact_1327_diff__mod__le,axiom,
    ! [Aa: nat,D: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),D)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(minus_minus_nat(Aa),aa_nat_nat(modulo_modulo_nat(Aa),Ba))),aa_nat_nat(minus_minus_nat(D),Ba)) ) ) ).

% diff_mod_le
tff(fact_1328_nat__mod__eq__lemma,axiom,
    ! [X: nat,Na: nat,Y: nat] :
      ( ( aa_nat_nat(modulo_modulo_nat(X),Na) = aa_nat_nat(modulo_modulo_nat(Y),Na) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
       => ? [Q5: nat] : X = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q5)) ) ) ).

% nat_mod_eq_lemma
tff(fact_1329_mod__eq__nat2E,axiom,
    ! [M: nat,Q3: nat,Na: nat] :
      ( ( aa_nat_nat(modulo_modulo_nat(M),Q3) = aa_nat_nat(modulo_modulo_nat(Na),Q3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => ~ ! [S2: nat] : Na != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q3),S2)) ) ) ).

% mod_eq_nat2E
tff(fact_1330_mod__eq__nat1E,axiom,
    ! [M: nat,Q3: nat,Na: nat] :
      ( ( aa_nat_nat(modulo_modulo_nat(M),Q3) = aa_nat_nat(modulo_modulo_nat(Na),Q3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ~ ! [S2: nat] : M != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q3),S2)) ) ) ).

% mod_eq_nat1E
tff(fact_1331_div__mod__decomp,axiom,
    ! [A: nat,Na: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(divide_divide_nat(A),Na)),Na)),aa_nat_nat(modulo_modulo_nat(A),Na)) ).

% div_mod_decomp
tff(fact_1332_mod__mult2__eq,axiom,
    ! [M: nat,Na: nat,Q3: nat] : aa_nat_nat(modulo_modulo_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(modulo_modulo_nat(aa_nat_nat(divide_divide_nat(M),Na)),Q3))),aa_nat_nat(modulo_modulo_nat(M),Na)) ).

% mod_mult2_eq
tff(fact_1333_modulo__nat__def,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(modulo_modulo_nat(M),Na) = aa_nat_nat(minus_minus_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(divide_divide_nat(M),Na)),Na)) ).

% modulo_nat_def
tff(fact_1334_Bolzano,axiom,
    ! [Aa: real,Ba: real,P: fun_real_fun_real_o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => ( ! [A2: real,B2: real,C3: real] :
            ( aa_real_o(aa_real_fun_real_o(P,A2),B2)
           => ( aa_real_o(aa_real_fun_real_o(P,B2),C3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),C3)
                 => aa_real_o(aa_real_fun_real_o(P,A2),C3) ) ) ) )
       => ( ! [X2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Ba)
               => ? [D3: real] :
                    ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
                    & ! [A2: real,B2: real] :
                        ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),X2)
                          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B2)
                          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(B2),A2)),D3) )
                       => aa_real_o(aa_real_fun_real_o(P,A2),B2) ) ) ) )
         => aa_real_o(aa_real_fun_real_o(P,Aa),Ba) ) ) ) ).

% Bolzano
tff(fact_1335_minusinfinity,axiom,
    ! [D: int,P1: fun_int_o,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X2: int,K4: int] :
            ( aa_int_o(P1,X2)
          <=> aa_int_o(P1,minus_minus_int(X2,times_times_int(K4,D))) )
       => ( ? [Z2: int] :
            ! [X2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z2)
             => ( aa_int_o(P,X2)
              <=> aa_int_o(P1,X2) ) )
         => ( ? [X_12: int] : aa_int_o(P1,X_12)
           => ? [X_13: int] : aa_int_o(P,X_13) ) ) ) ) ).

% minusinfinity
tff(fact_1336_plusinfinity,axiom,
    ! [D: int,P3: fun_int_o,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X2: int,K4: int] :
            ( aa_int_o(P3,X2)
          <=> aa_int_o(P3,minus_minus_int(X2,times_times_int(K4,D))) )
       => ( ? [Z2: int] :
            ! [X2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X2)
             => ( aa_int_o(P,X2)
              <=> aa_int_o(P3,X2) ) )
         => ( ? [X_12: int] : aa_int_o(P3,X_12)
           => ? [X_13: int] : aa_int_o(P,X_13) ) ) ) ) ).

% plusinfinity
tff(fact_1337_less__1__helper,axiom,
    ! [Na: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Na),M)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,minus_minus_int(Na,one_one_int)),M) ) ).

% less_1_helper
tff(fact_1338_int__induct,axiom,
    ! [P: fun_int_o,K: int,I: int] :
      ( aa_int_o(P,K)
     => ( ! [I2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I2)
           => ( aa_int_o(P,I2)
             => aa_int_o(P,plus_plus_int(I2,one_one_int)) ) )
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),K)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,minus_minus_int(I2,one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_induct
tff(fact_1339_int__mod__le_H,axiom,
    ! [Ba: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),minus_minus_int(Ba,Na))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,modulo_modulo_int(Ba,Na)),minus_minus_int(Ba,Na)) ) ).

% int_mod_le'
tff(fact_1340_int__minus,axiom,
    ! [Na: nat,M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(minus_minus_nat(Na),M)) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,Na),aa_nat_int(semiri1314217659103216013at_int,M)))) ).

% int_minus
tff(fact_1341_int__div__sub__1,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),M)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,minus_minus_int(Na,one_one_int)),M) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),Na),minus_minus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),M),one_one_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),M)) ) ) ).

% int_div_sub_1
tff(fact_1342_mod__div__equality__div__eq,axiom,
    ! [Aa: int,Ba: int] : times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba),Ba) = minus_minus_int(Aa,modulo_modulo_int(Aa,Ba)) ).

% mod_div_equality_div_eq
tff(fact_1343_not__int__def,axiom,
    ! [K: int] : bit_ri7919022796975470100ot_int(K) = minus_minus_int(aa_int_int(uminus_uminus_int,K),one_one_int) ).

% not_int_def
tff(fact_1344_nat__mod__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa_nat_nat(modulo_modulo_nat(X4),Xa) = aa_int_nat(nat2,modulo_modulo_int(aa_nat_int(semiri1314217659103216013at_int,X4),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_mod_as_int
tff(fact_1345_nat__minus__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa_nat_nat(minus_minus_nat(X4),Xa) = aa_int_nat(nat2,minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,X4),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_minus_as_int
tff(fact_1346_mod__lemma,axiom,
    ! [C: nat,R: nat,Ba: nat,Q3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,R),Ba)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),aa_nat_nat(modulo_modulo_nat(Q3),C))),R)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),C)) ) ) ).

% mod_lemma
tff(fact_1347_split__mod,axiom,
    ! [P: fun_nat_o,M: nat,Na: nat] :
      ( aa_nat_o(P,aa_nat_nat(modulo_modulo_nat(M),Na))
    <=> ( ( ( Na = zero_zero_nat )
         => aa_nat_o(P,M) )
        & ( ( Na != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Na)
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),I4)),J3) )
               => aa_nat_o(P,J3) ) ) ) ) ) ).

% split_mod
tff(fact_1348_even__diff__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),minus_minus_int(K,L))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),plus_plus_int(K,L)) ) ).

% even_diff_iff
tff(fact_1349_nat__mod__distrib,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_nat(nat2,modulo_modulo_int(X,Y)) = aa_nat_nat(modulo_modulo_nat(aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_1350_real__of__nat__div__aux,axiom,
    ! [X: nat,D: nat] : aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(semiri5074537144036343181t_real,D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(divide_divide_nat(X),D))),aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(modulo_modulo_nat(X),D))),aa_nat_real(semiri5074537144036343181t_real,D))) ).

% real_of_nat_div_aux
tff(fact_1351_int__ops_I6_J,axiom,
    ! [Aa: nat,Ba: nat] :
      aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(minus_minus_nat(Aa),Ba)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Aa)),aa_nat_int(semiri1314217659103216013at_int,Ba)),zero_zero_int,minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,Aa),aa_nat_int(semiri1314217659103216013at_int,Ba))) ).

% int_ops(6)
tff(fact_1352_nat__diff__distrib,axiom,
    ! [Z4: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z4),Z)
       => ( aa_int_nat(nat2,minus_minus_int(Z,Z4)) = aa_nat_nat(minus_minus_nat(aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z4)) ) ) ) ).

% nat_diff_distrib
tff(fact_1353_nat__diff__distrib_H,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_nat(nat2,minus_minus_int(X,Y)) = aa_nat_nat(minus_minus_nat(aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_1354_decr__mult__lemma,axiom,
    ! [D: int,P: fun_int_o,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X2: int] :
            ( aa_int_o(P,X2)
           => aa_int_o(P,minus_minus_int(X2,D)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
         => ! [X4: int] :
              ( aa_int_o(P,X4)
             => aa_int_o(P,minus_minus_int(X4,times_times_int(K,D))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_1355_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
       => ( modulo_modulo_int(K,L) = modulo_modulo_int(minus_minus_int(K,L),L) ) ) ) ).

% mod_pos_geq
tff(fact_1356_zdiff__int__split,axiom,
    ! [P: fun_int_o,X: nat,Y: nat] :
      ( aa_int_o(P,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(minus_minus_nat(X),Y)))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
         => aa_int_o(P,minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,X),aa_nat_int(semiri1314217659103216013at_int,Y))) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y)
         => aa_int_o(P,zero_zero_int) ) ) ) ).

% zdiff_int_split
tff(fact_1357_real__of__nat__div2,axiom,
    ! [Na: nat,X: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(semiri5074537144036343181t_real,X))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(divide_divide_nat(Na),X)))) ).

% real_of_nat_div2
tff(fact_1358_real__of__nat__div3,axiom,
    ! [Na: nat,X: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(semiri5074537144036343181t_real,X))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(divide_divide_nat(Na),X)))),one_one_real) ).

% real_of_nat_div3
tff(fact_1359_log__divide,axiom,
    ! [Aa: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( ( Aa != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
           => ( aa_real_real(log(Aa),aa_real_real(divide_divide_real(X),Y)) = aa_real_real(minus_minus_real(aa_real_real(log(Aa),X)),aa_real_real(log(Aa),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_1360_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L)
     => ( modulo_modulo_int(aa_int_int(uminus_uminus_int,K),L) = minus_minus_int(minus_minus_int(L,one_one_int),modulo_modulo_int(minus_minus_int(K,one_one_int),L)) ) ) ).

% minus_mod_int_eq
tff(fact_1361_zmod__minus1,axiom,
    ! [Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
     => ( modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int),Ba) = minus_minus_int(Ba,one_one_int) ) ) ).

% zmod_minus1
tff(fact_1362_mod__sub__if__z,axiom,
    ! [X: int,Z: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
             => ( modulo_modulo_int(minus_minus_int(X,Y),Z) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X),minus_minus_int(X,Y),plus_plus_int(minus_minus_int(X,Y),Z)) ) ) ) ) ) ) ).

% mod_sub_if_z
tff(fact_1363_mod__add__if__z,axiom,
    ! [X: int,Z: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
             => ( modulo_modulo_int(plus_plus_int(X,Y),Z) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,plus_plus_int(X,Y)),Z),plus_plus_int(X,Y),minus_minus_int(plus_plus_int(X,Y),Z)) ) ) ) ) ) ) ).

% mod_add_if_z
tff(fact_1364_zdiv__zminus1__eq__if,axiom,
    ! [Ba: int,Aa: int] :
      ( ( Ba != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,Aa)),Ba) = $ite(modulo_modulo_int(Aa,Ba) = zero_zero_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)),minus_minus_int(aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)),one_one_int)) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_1365_zdiv__zminus2__eq__if,axiom,
    ! [Ba: int,Aa: int] :
      ( ( Ba != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_int_int(uminus_uminus_int,Ba)) = $ite(modulo_modulo_int(Aa,Ba) = zero_zero_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)),minus_minus_int(aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)),one_one_int)) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_1366_diff__nat__eq__if,axiom,
    ! [Z: int,Z4: int] :
      aa_nat_nat(minus_minus_nat(aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z4)) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),zero_zero_int),
        aa_int_nat(nat2,Z),
        $let(
          d: int,
          d:= minus_minus_int(Z,Z4),
          $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,d),zero_zero_int),zero_zero_nat,aa_int_nat(nat2,d)) ) ) ).

% diff_nat_eq_if
tff(fact_1367_power__mod__div,axiom,
    ! [X: nat,Na: nat,M: nat] : aa_nat_nat(divide_divide_nat(aa_nat_nat(modulo_modulo_nat(X),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na))),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),M)) = aa_nat_nat(modulo_modulo_nat(aa_nat_nat(divide_divide_nat(X),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),M))),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(minus_minus_nat(Na),M))) ).

% power_mod_div
tff(fact_1368_verit__le__mono__div,axiom,
    ! [A: nat,B: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(divide_divide_nat(A),Na)),
                $ite(aa_nat_nat(modulo_modulo_nat(B),Na) = zero_zero_nat,one_one_nat,zero_zero_nat))),aa_nat_nat(divide_divide_nat(B),Na)) ) ) ).

% verit_le_mono_div
tff(fact_1369_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = plus_plus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,minus_minus_int(K,L)),L),one_one_int) ) ) ) ).

% div_pos_geq
tff(fact_1370_minus__log__eq__powr,axiom,
    ! [Ba: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ba)
     => ( ( Ba != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( aa_real_real(minus_minus_real(Y),aa_real_real(log(Ba),X)) = aa_real_real(log(Ba),aa_real_real(divide_divide_real(aa_real_real(powr_real(Ba),Y)),X)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_1371_m1mod2k,axiom,
    ! [Na: nat] : modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na)) = minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na),one_one_int) ).

% m1mod2k
tff(fact_1372_choose__one,axiom,
    ! [Na: nat] : aa_nat_nat(binomial(Na),one_one_nat) = Na ).

% choose_one
tff(fact_1373_log__minus__eq__powr,axiom,
    ! [Ba: real,X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ba)
     => ( ( Ba != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( aa_real_real(minus_minus_real(aa_real_real(log(Ba),X)),Y) = aa_real_real(log(Ba),aa_real_real(times_times_real(X),aa_real_real(powr_real(Ba),aa_real_real(uminus_uminus_real,Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_1374_xor__nat__unfold,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(bit_se6528837805403552850or_nat(M),Na) = $ite(
        M = zero_zero_nat,
        Na,
        $ite(Na = zero_zero_nat,M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(modulo_modulo_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(modulo_modulo_nat(M),numeral_numeral_nat(bit0(one)))),aa_nat_nat(modulo_modulo_nat(Na),numeral_numeral_nat(bit0(one))))),numeral_numeral_nat(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(bit_se6528837805403552850or_nat(aa_nat_nat(divide_divide_nat(M),numeral_numeral_nat(bit0(one)))),aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one))))))) ) ).

% xor_nat_unfold
tff(fact_1375_and__nat__unfold,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(bit_se727722235901077358nd_nat(M),Na) = $ite(
        ( ( M = zero_zero_nat )
        | ( Na = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(modulo_modulo_nat(M),numeral_numeral_nat(bit0(one)))),aa_nat_nat(modulo_modulo_nat(Na),numeral_numeral_nat(bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(bit_se727722235901077358nd_nat(aa_nat_nat(divide_divide_nat(M),numeral_numeral_nat(bit0(one)))),aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one)))))) ) ).

% and_nat_unfold
tff(fact_1376_m1mod22k,axiom,
    ! [Na: nat] : modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int),times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na))) = minus_minus_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na)),one_one_int) ).

% m1mod22k
tff(fact_1377_neg__zmod__mult__2,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int)
     => ( modulo_modulo_int(plus_plus_int(one_one_int,times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),Ba)),times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),Aa)) = minus_minus_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),modulo_modulo_int(plus_plus_int(Ba,one_one_int),Aa)),one_one_int) ) ) ).

% neg_zmod_mult_2
tff(fact_1378_binomial__symmetric,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_nat(binomial(Na),K) = aa_nat_nat(binomial(Na),aa_nat_nat(minus_minus_nat(Na),K)) ) ) ).

% binomial_symmetric
tff(fact_1379_choose__mult__lemma,axiom,
    ! [M: nat,R: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(binomial(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R)),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K))),aa_nat_nat(binomial(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(binomial(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R)),K)),K)),aa_nat_nat(binomial(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R)),M)) ).

% choose_mult_lemma
tff(fact_1380_binomial__le__pow,axiom,
    ! [R: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(binomial(Na),R)),aa_nat_nat(power_power_nat(Na),R)) ) ).

% binomial_le_pow
tff(fact_1381_zero__less__binomial,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(binomial(Na),K)) ) ).

% zero_less_binomial
tff(fact_1382_choose__mult,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(binomial(Na),M)),aa_nat_nat(binomial(M),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(binomial(Na),K)),aa_nat_nat(binomial(aa_nat_nat(minus_minus_nat(Na),K)),aa_nat_nat(minus_minus_nat(M),K))) ) ) ) ).

% choose_mult
tff(fact_1383_binomial__absorb__comp,axiom,
    ! [Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(Na),K)),aa_nat_nat(binomial(Na),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(binomial(aa_nat_nat(minus_minus_nat(Na),one_one_nat)),K)) ).

% binomial_absorb_comp
tff(fact_1384_real__average__minus__second,axiom,
    ! [Ba: real,Aa: real] : aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),Aa)),numeral_numeral_real(bit0(one)))),Aa) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(Ba),Aa)),numeral_numeral_real(bit0(one))) ).

% real_average_minus_second
tff(fact_1385_real__average__minus__first,axiom,
    ! [Aa: real,Ba: real] : aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)),numeral_numeral_real(bit0(one)))),Aa) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(Ba),Aa)),numeral_numeral_real(bit0(one))) ).

% real_average_minus_first
tff(fact_1386_mod__exhaust__less__4,axiom,
    ! [M: nat] :
      ( ( aa_nat_nat(modulo_modulo_nat(M),numeral_numeral_nat(bit0(bit0(one)))) = zero_zero_nat )
      | ( aa_nat_nat(modulo_modulo_nat(M),numeral_numeral_nat(bit0(bit0(one)))) = one_one_nat )
      | ( aa_nat_nat(modulo_modulo_nat(M),numeral_numeral_nat(bit0(bit0(one)))) = numeral_numeral_nat(bit0(one)) )
      | ( aa_nat_nat(modulo_modulo_nat(M),numeral_numeral_nat(bit0(bit0(one)))) = numeral_numeral_nat(bit1(one)) ) ) ).

% mod_exhaust_less_4
tff(fact_1387_bezout__add__strong__nat,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( Aa != zero_zero_nat )
     => ? [D4: nat,X2: nat,Y2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D4),Aa)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D4),Ba)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),Y2)),D4) ) ) ) ).

% bezout_add_strong_nat
tff(fact_1388_signed__take__bit__minus,axiom,
    ! [Na: nat,K: int] : bit_ri631733984087533419it_int(Na,aa_int_int(uminus_uminus_int,bit_ri631733984087533419it_int(Na,K))) = bit_ri631733984087533419it_int(Na,aa_int_int(uminus_uminus_int,K)) ).

% signed_take_bit_minus
tff(fact_1389_signed__take__bit__add,axiom,
    ! [Na: nat,K: int,L: int] : bit_ri631733984087533419it_int(Na,plus_plus_int(bit_ri631733984087533419it_int(Na,K),bit_ri631733984087533419it_int(Na,L))) = bit_ri631733984087533419it_int(Na,plus_plus_int(K,L)) ).

% signed_take_bit_add
tff(fact_1390_signed__take__bit__mult,axiom,
    ! [Na: nat,K: int,L: int] : bit_ri631733984087533419it_int(Na,times_times_int(bit_ri631733984087533419it_int(Na,K),bit_ri631733984087533419it_int(Na,L))) = bit_ri631733984087533419it_int(Na,times_times_int(K,L)) ).

% signed_take_bit_mult
tff(fact_1391_signed__take__bit__diff,axiom,
    ! [Na: nat,K: int,L: int] : bit_ri631733984087533419it_int(Na,minus_minus_int(bit_ri631733984087533419it_int(Na,K),bit_ri631733984087533419it_int(Na,L))) = bit_ri631733984087533419it_int(Na,minus_minus_int(K,L)) ).

% signed_take_bit_diff
tff(fact_1392_eq__diff__eq_H,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( X = aa_real_real(minus_minus_real(Y),Z) )
    <=> ( Y = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Z) ) ) ).

% eq_diff_eq'
tff(fact_1393_signed__take__bit__int__less__exp,axiom,
    ! [Na: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,bit_ri631733984087533419it_int(Na,K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na)) ).

% signed_take_bit_int_less_exp
tff(fact_1394_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),bit_ri631733984087533419it_int(Na,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_1395_signed__take__bit__int__less__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,bit_ri631733984087533419it_int(Na,K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na)),K) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_1396_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [Na: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na))),bit_ri631733984087533419it_int(Na,K)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_1397_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,bit_ri631733984087533419it_int(Na,K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na))),K) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_1398_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),bit_ri631733984087533419it_int(Na,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_1399_signed__take__bit__int__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( bit_ri631733984087533419it_int(Na,K) = K )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na))),K)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_1400_signed__take__bit__int__eq__self,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na))),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na))
       => ( bit_ri631733984087533419it_int(Na,K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_1401_Euclid__induct,axiom,
    ! [P: fun_nat_fun_nat_o,Aa: nat,Ba: nat] :
      ( ! [A2: nat,B2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(P,A2),B2)
        <=> aa_nat_o(aa_nat_fun_nat_o(P,B2),A2) )
     => ( ! [A2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A2),zero_zero_nat)
       => ( ! [A2: nat,B2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,A2),B2)
             => aa_nat_o(aa_nat_fun_nat_o(P,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,Aa),Ba) ) ) ) ).

% Euclid_induct
tff(fact_1402_bezout__add__nat,axiom,
    ! [Aa: nat,Ba: nat] :
    ? [D4: nat,X2: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D4),Aa)
      & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D4),Ba)
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),Y2)),D4) )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Y2)),D4) ) ) ) ).

% bezout_add_nat
tff(fact_1403_bezout__lemma__nat,axiom,
    ! [D: nat,Aa: nat,Ba: nat,X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),Ba)
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),Y)),D) )
            | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Y)),D) ) )
         => ? [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),Aa)
              & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba))
              & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba)),Y2)),D) )
                | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba)),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Y2)),D) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_1404_bezout1__nat,axiom,
    ! [Aa: nat,Ba: nat] :
    ? [D4: nat,X2: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D4),Aa)
      & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D4),Ba)
      & ( ( aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),Y2)) = D4 )
        | ( aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Y2)) = D4 ) ) ) ).

% bezout1_nat
tff(fact_1405_xor__int__unfold,axiom,
    ! [K: int,L: int] :
      bit_se6526347334894502574or_int(K,L) = $ite(
        K = aa_int_int(uminus_uminus_int,one_one_int),
        bit_ri7919022796975470100ot_int(L),
        $ite(
          L = aa_int_int(uminus_uminus_int,one_one_int),
          bit_ri7919022796975470100ot_int(K),
          $ite(
            K = zero_zero_int,
            L,
            $ite(L = zero_zero_int,K,plus_plus_int(aa_int_int(abs_abs_int,minus_minus_int(modulo_modulo_int(K,aa_num_int(numeral_numeral_int,bit0(one))),modulo_modulo_int(L,aa_num_int(numeral_numeral_int,bit0(one))))),times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),bit_se6526347334894502574or_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,bit0(one))))))) ) ) ) ).

% xor_int_unfold
tff(fact_1406_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : bit_ri631733984087533419it_int(numeral_numeral_nat(L),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit1(K)))) = plus_plus_int(times_times_int(bit_ri631733984087533419it_int(pred_numeral(L),minus_minus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)),one_one_int)),aa_num_int(numeral_numeral_int,bit0(one))),one_one_int) ).

% signed_take_bit_numeral_minus_bit1
tff(fact_1407_signed__take__bit__Suc__minus__bit1,axiom,
    ! [Na: nat,K: num] : bit_ri631733984087533419it_int(aa_nat_nat(suc,Na),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit1(K)))) = plus_plus_int(times_times_int(bit_ri631733984087533419it_int(Na,minus_minus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)),one_one_int)),aa_num_int(numeral_numeral_int,bit0(one))),one_one_int) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_1408_ln__one__minus__pos__lower__bound,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,X)),aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))))),aa_real_real(ln_ln_real,aa_real_real(minus_minus_real(one_one_real),X))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_1409_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( aa_nat_nat(suc,Nat) = aa_nat_nat(suc,Nat2) )
    <=> ( Nat = Nat2 ) ) ).

% old.nat.inject
tff(fact_1410_nat_Oinject,axiom,
    ! [X23: nat,Y22: nat] :
      ( ( aa_nat_nat(suc,X23) = aa_nat_nat(suc,Y22) )
    <=> ( X23 = Y22 ) ) ).

% nat.inject
tff(fact_1411_Suc__less__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% Suc_less_eq
tff(fact_1412_Suc__mono,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Na)) ) ).

% Suc_mono
tff(fact_1413_lessI,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,Na)) ).

% lessI
tff(fact_1414_add__Suc__right,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) ).

% add_Suc_right
tff(fact_1415_Suc__le__mono,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),aa_nat_nat(suc,M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M) ) ).

% Suc_le_mono
tff(fact_1416_Suc__diff__diff,axiom,
    ! [M: nat,Na: nat,K: nat] : aa_nat_nat(minus_minus_nat(aa_nat_nat(minus_minus_nat(aa_nat_nat(suc,M)),Na)),aa_nat_nat(suc,K)) = aa_nat_nat(minus_minus_nat(aa_nat_nat(minus_minus_nat(M),Na)),K) ).

% Suc_diff_diff
tff(fact_1417_diff__Suc__Suc,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(minus_minus_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,Na)) = aa_nat_nat(minus_minus_nat(M),Na) ).

% diff_Suc_Suc
tff(fact_1418_zero__less__Suc,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,Na)) ).

% zero_less_Suc
tff(fact_1419_less__Suc0,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,zero_zero_nat))
    <=> ( Na = zero_zero_nat ) ) ).

% less_Suc0
tff(fact_1420_mult__eq__1__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_1421_one__eq__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_1422_diff__Suc__1,axiom,
    ! [Na: nat] : aa_nat_nat(minus_minus_nat(aa_nat_nat(suc,Na)),one_one_nat) = Na ).

% diff_Suc_1
tff(fact_1423_mult__Suc__right,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) ).

% mult_Suc_right
tff(fact_1424_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(suc,zero_zero_nat))
    <=> ( M = aa_nat_nat(suc,zero_zero_nat) ) ) ).

% dvd_1_iff_1
tff(fact_1425_dvd__1__left,axiom,
    ! [K: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(suc,zero_zero_nat)),K) ).

% dvd_1_left
tff(fact_1426_binomial__Suc__Suc,axiom,
    ! [Na: nat,K: nat] : aa_nat_nat(binomial(aa_nat_nat(suc,Na)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(binomial(Na),K)),aa_nat_nat(binomial(Na),aa_nat_nat(suc,K))) ).

% binomial_Suc_Suc
tff(fact_1427_pred__numeral__simps_I1_J,axiom,
    pred_numeral(one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_1428_eq__numeral__Suc,axiom,
    ! [K: num,Na: nat] :
      ( ( numeral_numeral_nat(K) = aa_nat_nat(suc,Na) )
    <=> ( pred_numeral(K) = Na ) ) ).

% eq_numeral_Suc
tff(fact_1429_Suc__eq__numeral,axiom,
    ! [Na: nat,K: num] :
      ( ( aa_nat_nat(suc,Na) = numeral_numeral_nat(K) )
    <=> ( Na = pred_numeral(K) ) ) ).

% Suc_eq_numeral
tff(fact_1430_zdvd1__eq,axiom,
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),one_one_int)
    <=> ( aa_int_int(abs_abs_int,X) = one_one_int ) ) ).

% zdvd1_eq
tff(fact_1431_Suc__pred,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(suc,aa_nat_nat(minus_minus_nat(Na),aa_nat_nat(suc,zero_zero_nat))) = Na ) ) ).

% Suc_pred
tff(fact_1432_one__le__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),Na) ) ) ).

% one_le_mult_iff
tff(fact_1433_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(minus_minus_nat(aa_nat_nat(suc,aa_nat_nat(minus_minus_nat(J),K))),I) = aa_nat_nat(minus_minus_nat(aa_nat_nat(suc,J)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_1434_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(minus_minus_nat(I),aa_nat_nat(suc,aa_nat_nat(minus_minus_nat(J),K))) = aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_1435_Suc__diff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),M)
       => ( aa_nat_nat(suc,aa_nat_nat(minus_minus_nat(Na),M)) = aa_nat_nat(minus_minus_nat(Na),aa_nat_nat(minus_minus_nat(M),one_one_nat)) ) ) ) ).

% Suc_diff
tff(fact_1436_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_1437_Suc__mod__mult__self4,axiom,
    ! [Na: nat,K: nat,M: nat] : aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K)),M))),Na) = aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,M)),Na) ).

% Suc_mod_mult_self4
tff(fact_1438_Suc__mod__mult__self3,axiom,
    ! [K: nat,Na: nat,M: nat] : aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)),M))),Na) = aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,M)),Na) ).

% Suc_mod_mult_self3
tff(fact_1439_Suc__mod__mult__self2,axiom,
    ! [M: nat,Na: nat,K: nat] : aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K)))),Na) = aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,M)),Na) ).

% Suc_mod_mult_self2
tff(fact_1440_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,Na: nat] : aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)))),Na) = aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,M)),Na) ).

% Suc_mod_mult_self1
tff(fact_1441_Suc__numeral,axiom,
    ! [Na: num] : aa_nat_nat(suc,numeral_numeral_nat(Na)) = numeral_numeral_nat(plus_plus_num(Na,one)) ).

% Suc_numeral
tff(fact_1442_negative__zless,axiom,
    ! [Na: nat,M: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Na)))),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% negative_zless
tff(fact_1443_pred__numeral__simps_I3_J,axiom,
    ! [K: num] : pred_numeral(bit1(K)) = numeral_numeral_nat(bit0(K)) ).

% pred_numeral_simps(3)
tff(fact_1444_less__numeral__Suc,axiom,
    ! [K: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,numeral_numeral_nat(K)),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pred_numeral(K)),Na) ) ).

% less_numeral_Suc
tff(fact_1445_less__Suc__numeral,axiom,
    ! [Na: nat,K: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),numeral_numeral_nat(K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),pred_numeral(K)) ) ).

% less_Suc_numeral
tff(fact_1446_le__numeral__Suc,axiom,
    ! [K: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(K)),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,pred_numeral(K)),Na) ) ).

% le_numeral_Suc
tff(fact_1447_le__Suc__numeral,axiom,
    ! [Na: nat,K: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),numeral_numeral_nat(K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),pred_numeral(K)) ) ).

% le_Suc_numeral
tff(fact_1448_diff__Suc__numeral,axiom,
    ! [Na: nat,K: num] : aa_nat_nat(minus_minus_nat(aa_nat_nat(suc,Na)),numeral_numeral_nat(K)) = aa_nat_nat(minus_minus_nat(Na),pred_numeral(K)) ).

% diff_Suc_numeral
tff(fact_1449_diff__numeral__Suc,axiom,
    ! [K: num,Na: nat] : aa_nat_nat(minus_minus_nat(numeral_numeral_nat(K)),aa_nat_nat(suc,Na)) = aa_nat_nat(minus_minus_nat(pred_numeral(K)),Na) ).

% diff_numeral_Suc
tff(fact_1450_ln__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ) ) ).

% ln_le_cancel_iff
tff(fact_1451_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Z)),one_one_int)
    <=> ( Z = zero_zero_int ) ) ).

% zabs_less_one_iff
tff(fact_1452_add__2__eq__Suc_H,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),numeral_numeral_nat(bit0(one))) = aa_nat_nat(suc,aa_nat_nat(suc,Na)) ).

% add_2_eq_Suc'
tff(fact_1453_add__2__eq__Suc,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(one))),Na) = aa_nat_nat(suc,aa_nat_nat(suc,Na)) ).

% add_2_eq_Suc
tff(fact_1454_Suc__1,axiom,
    aa_nat_nat(suc,one_one_nat) = numeral_numeral_nat(bit0(one)) ).

% Suc_1
tff(fact_1455_div2__Suc__Suc,axiom,
    ! [M: nat] : aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,M))),numeral_numeral_nat(bit0(one))) = aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(M),numeral_numeral_nat(bit0(one)))) ).

% div2_Suc_Suc
tff(fact_1456_even__Suc__Suc__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Na)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na) ) ).

% even_Suc_Suc_iff
tff(fact_1457_even__Suc,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,Na))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na) ) ).

% even_Suc
tff(fact_1458_mod2__Suc__Suc,axiom,
    ! [M: nat] : aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,aa_nat_nat(suc,M))),numeral_numeral_nat(bit0(one))) = aa_nat_nat(modulo_modulo_nat(M),numeral_numeral_nat(bit0(one))) ).

% mod2_Suc_Suc
tff(fact_1459_Suc__diff__1,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(suc,aa_nat_nat(minus_minus_nat(Na),one_one_nat)) = Na ) ) ).

% Suc_diff_1
tff(fact_1460_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,Na: nat] :
      ( ( numeral_numeral_nat(K) != one_one_nat )
     => ( aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(K)),Na))),numeral_numeral_nat(K)) = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_1461_ln__ge__zero__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X) ) ) ).

% ln_ge_zero_iff
tff(fact_1462_ln__le__zero__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,X)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real) ) ) ).

% ln_le_zero_iff
tff(fact_1463_and__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(bit_se727722235901077358nd_nat(aa_nat_nat(suc,zero_zero_nat)),numeral_numeral_nat(bit0(Y))) = zero_zero_nat ).

% and_nat_numerals(1)
tff(fact_1464_and__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(bit_se727722235901077358nd_nat(numeral_numeral_nat(bit0(X))),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% and_nat_numerals(3)
tff(fact_1465_or__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(bit_se1412395901928357646or_nat(aa_nat_nat(suc,zero_zero_nat)),numeral_numeral_nat(bit1(Y))) = numeral_numeral_nat(bit1(Y)) ).

% or_nat_numerals(2)
tff(fact_1466_or__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(bit_se1412395901928357646or_nat(numeral_numeral_nat(bit1(X))),aa_nat_nat(suc,zero_zero_nat)) = numeral_numeral_nat(bit1(X)) ).

% or_nat_numerals(4)
tff(fact_1467_nat__abs__dvd__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),Na)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_nat_int(semiri1314217659103216013at_int,Na)) ) ).

% nat_abs_dvd_iff
tff(fact_1468_dvd__nat__abs__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),aa_int_nat(nat2,aa_int_int(abs_abs_int,K)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,Na)),K) ) ).

% dvd_nat_abs_iff
tff(fact_1469_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V: num] : aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),numeral_numeral_nat(V)) = aa_nat_nat(divide_divide_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit1(one))),M)),numeral_numeral_nat(V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_1470_div__Suc__eq__div__add3,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(divide_divide_nat(M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Na)))) = aa_nat_nat(divide_divide_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit1(one))),Na)) ).

% div_Suc_eq_div_add3
tff(fact_1471_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(modulo_modulo_nat(Na),numeral_numeral_nat(bit0(one))) != aa_nat_nat(suc,zero_zero_nat) )
    <=> ( aa_nat_nat(modulo_modulo_nat(Na),numeral_numeral_nat(bit0(one))) = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_1472_even__Suc__div__two,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na)
     => ( aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,Na)),numeral_numeral_nat(bit0(one))) = aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one))) ) ) ).

% even_Suc_div_two
tff(fact_1473_odd__Suc__div__two,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na)
     => ( aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,Na)),numeral_numeral_nat(bit0(one))) = aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one)))) ) ) ).

% odd_Suc_div_two
tff(fact_1474_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V: num] : aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),numeral_numeral_nat(V)) = aa_nat_nat(modulo_modulo_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit1(one))),M)),numeral_numeral_nat(V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_1475_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(modulo_modulo_nat(M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Na)))) = aa_nat_nat(modulo_modulo_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit1(one))),Na)) ).

% mod_Suc_eq_mod_add3
tff(fact_1476_signed__take__bit__Suc__bit0,axiom,
    ! [Na: nat,K: num] : bit_ri631733984087533419it_int(aa_nat_nat(suc,Na),aa_num_int(numeral_numeral_int,bit0(K))) = times_times_int(bit_ri631733984087533419it_int(Na,aa_num_int(numeral_numeral_int,K)),aa_num_int(numeral_numeral_int,bit0(one))) ).

% signed_take_bit_Suc_bit0
tff(fact_1477_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% one_less_nat_eq
tff(fact_1478_xor__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(bit_se6528837805403552850or_nat(aa_nat_nat(suc,zero_zero_nat)),numeral_numeral_nat(bit0(Y))) = numeral_numeral_nat(bit1(Y)) ).

% xor_nat_numerals(1)
tff(fact_1479_xor__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(bit_se6528837805403552850or_nat(aa_nat_nat(suc,zero_zero_nat)),numeral_numeral_nat(bit1(Y))) = numeral_numeral_nat(bit0(Y)) ).

% xor_nat_numerals(2)
tff(fact_1480_xor__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(bit_se6528837805403552850or_nat(numeral_numeral_nat(bit0(X))),aa_nat_nat(suc,zero_zero_nat)) = numeral_numeral_nat(bit1(X)) ).

% xor_nat_numerals(3)
tff(fact_1481_xor__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(bit_se6528837805403552850or_nat(numeral_numeral_nat(bit1(X))),aa_nat_nat(suc,zero_zero_nat)) = numeral_numeral_nat(bit0(X)) ).

% xor_nat_numerals(4)
tff(fact_1482_or__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(bit_se1412395901928357646or_nat(aa_nat_nat(suc,zero_zero_nat)),numeral_numeral_nat(bit0(Y))) = numeral_numeral_nat(bit1(Y)) ).

% or_nat_numerals(1)
tff(fact_1483_or__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(bit_se1412395901928357646or_nat(numeral_numeral_nat(bit0(X))),aa_nat_nat(suc,zero_zero_nat)) = numeral_numeral_nat(bit1(X)) ).

% or_nat_numerals(3)
tff(fact_1484_and__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(bit_se727722235901077358nd_nat(numeral_numeral_nat(bit1(X))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% and_nat_numerals(4)
tff(fact_1485_and__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(bit_se727722235901077358nd_nat(aa_nat_nat(suc,zero_zero_nat)),numeral_numeral_nat(bit1(Y))) = one_one_nat ).

% and_nat_numerals(2)
tff(fact_1486_odd__Suc__minus__one,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na)
     => ( aa_nat_nat(suc,aa_nat_nat(minus_minus_nat(Na),aa_nat_nat(suc,zero_zero_nat))) = Na ) ) ).

% odd_Suc_minus_one
tff(fact_1487_signed__take__bit__Suc__minus__bit0,axiom,
    ! [Na: nat,K: num] : bit_ri631733984087533419it_int(aa_nat_nat(suc,Na),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(K)))) = times_times_int(bit_ri631733984087533419it_int(Na,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,bit0(one))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_1488_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : bit_ri631733984087533419it_int(numeral_numeral_nat(L),aa_num_int(numeral_numeral_int,bit0(K))) = times_times_int(bit_ri631733984087533419it_int(pred_numeral(L),aa_num_int(numeral_numeral_int,K)),aa_num_int(numeral_numeral_int,bit0(one))) ).

% signed_take_bit_numeral_bit0
tff(fact_1489_and__Suc__0__eq,axiom,
    ! [Na: nat] : aa_nat_nat(bit_se727722235901077358nd_nat(Na),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(modulo_modulo_nat(Na),numeral_numeral_nat(bit0(one))) ).

% and_Suc_0_eq
tff(fact_1490_Suc__0__and__eq,axiom,
    ! [Na: nat] : aa_nat_nat(bit_se727722235901077358nd_nat(aa_nat_nat(suc,zero_zero_nat)),Na) = aa_nat_nat(modulo_modulo_nat(Na),numeral_numeral_nat(bit0(one))) ).

% Suc_0_and_eq
tff(fact_1491_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : bit_ri631733984087533419it_int(numeral_numeral_nat(L),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(K)))) = times_times_int(bit_ri631733984087533419it_int(pred_numeral(L),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,bit0(one))) ).

% signed_take_bit_numeral_minus_bit0
tff(fact_1492_signed__take__bit__Suc__bit1,axiom,
    ! [Na: nat,K: num] : bit_ri631733984087533419it_int(aa_nat_nat(suc,Na),aa_num_int(numeral_numeral_int,bit1(K))) = plus_plus_int(times_times_int(bit_ri631733984087533419it_int(Na,aa_num_int(numeral_numeral_int,K)),aa_num_int(numeral_numeral_int,bit0(one))),one_one_int) ).

% signed_take_bit_Suc_bit1
tff(fact_1493_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : bit_ri631733984087533419it_int(numeral_numeral_nat(L),aa_num_int(numeral_numeral_int,bit1(K))) = plus_plus_int(times_times_int(bit_ri631733984087533419it_int(pred_numeral(L),aa_num_int(numeral_numeral_int,K)),aa_num_int(numeral_numeral_int,bit0(one))),one_one_int) ).

% signed_take_bit_numeral_bit1
tff(fact_1494_numeral__eq__Suc,axiom,
    ! [K: num] : numeral_numeral_nat(K) = aa_nat_nat(suc,pred_numeral(K)) ).

% numeral_eq_Suc
tff(fact_1495_n__not__Suc__n,axiom,
    ! [Na: nat] : Na != aa_nat_nat(suc,Na) ).

% n_not_Suc_n
tff(fact_1496_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(suc,X) = aa_nat_nat(suc,Y) )
     => ( X = Y ) ) ).

% Suc_inject
tff(fact_1497_exists__least__lemma,axiom,
    ! [P: fun_nat_o] :
      ( ~ aa_nat_o(P,zero_zero_nat)
     => ( ? [X_12: nat] : aa_nat_o(P,X_12)
       => ? [N: nat] :
            ( ~ aa_nat_o(P,N)
            & aa_nat_o(P,aa_nat_nat(suc,N)) ) ) ) ).

% exists_least_lemma
tff(fact_1498_not0__implies__Suc,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
     => ? [M3: nat] : Na = aa_nat_nat(suc,M3) ) ).

% not0_implies_Suc
tff(fact_1499_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_not_Suc
tff(fact_1500_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_neq_Suc
tff(fact_1501_Suc__neq__Zero,axiom,
    ! [M: nat] : aa_nat_nat(suc,M) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_1502_zero__induct,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( aa_nat_o(P,K)
     => ( ! [N: nat] :
            ( aa_nat_o(P,aa_nat_nat(suc,N))
           => aa_nat_o(P,N) )
       => aa_nat_o(P,zero_zero_nat) ) ) ).

% zero_induct
tff(fact_1503_diff__induct,axiom,
    ! [P: fun_nat_fun_nat_o,M: nat,Na: nat] :
      ( ! [X2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,X2),zero_zero_nat)
     => ( ! [Y2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,zero_zero_nat),aa_nat_nat(suc,Y2))
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(P,aa_nat_nat(suc,X2)),aa_nat_nat(suc,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,M),Na) ) ) ) ).

% diff_induct
tff(fact_1504_nat__induct,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(P,N)
           => aa_nat_o(P,aa_nat_nat(suc,N)) )
       => aa_nat_o(P,Na) ) ) ).

% nat_induct
tff(fact_1505_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] : Y != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_1506_nat_OdiscI,axiom,
    ! [Nat: nat,X23: nat] :
      ( ( Nat = aa_nat_nat(suc,X23) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_1507_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_1508_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_1509_nat_Odistinct_I1_J,axiom,
    ! [X23: nat] : zero_zero_nat != aa_nat_nat(suc,X23) ).

% nat.distinct(1)
tff(fact_1510_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ~ ! [N: nat] : X != aa_nat_nat(suc,N) ) ).

% list_decode.cases
tff(fact_1511_zdvd__antisym__abs,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),Ba)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),Aa)
       => ( aa_int_int(abs_abs_int,Aa) = aa_int_int(abs_abs_int,Ba) ) ) ) ).

% zdvd_antisym_abs
tff(fact_1512_not__less__less__Suc__eq,axiom,
    ! [Na: nat,M: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,M))
      <=> ( Na = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_1513_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( ! [I2: nat] :
            ( ( J = aa_nat_nat(suc,I2) )
           => aa_nat_o(P,I2) )
       => ( ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
             => ( aa_nat_o(P,aa_nat_nat(suc,I2))
               => aa_nat_o(P,I2) ) )
         => aa_nat_o(P,I) ) ) ) ).

% strict_inc_induct
tff(fact_1514_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( ! [I2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I2),aa_nat_nat(suc,I2))
       => ( ! [I2: nat,J2: nat,K4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),K4)
               => ( aa_nat_o(aa_nat_fun_nat_o(P,I2),J2)
                 => ( aa_nat_o(aa_nat_fun_nat_o(P,J2),K4)
                   => aa_nat_o(aa_nat_fun_nat_o(P,I2),K4) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(P,I),J) ) ) ) ).

% less_Suc_induct
tff(fact_1515_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),K) ) ) ).

% less_trans_Suc
tff(fact_1516_Suc__less__SucD,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% Suc_less_SucD
tff(fact_1517_less__antisym,axiom,
    ! [Na: nat,M: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,M))
       => ( M = Na ) ) ) ).

% less_antisym
tff(fact_1518_Suc__less__eq2,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),M)
    <=> ? [M5: nat] :
          ( ( M = aa_nat_nat(suc,M5) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M5) ) ) ).

% Suc_less_eq2
tff(fact_1519_Nat_OAll__less__Suc,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,Na))
         => aa_nat_o(P,I4) )
    <=> ( aa_nat_o(P,Na)
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Na)
           => aa_nat_o(P,I4) ) ) ) ).

% Nat.All_less_Suc
tff(fact_1520_not__less__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,M)) ) ).

% not_less_eq
tff(fact_1521_less__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
        | ( M = Na ) ) ) ).

% less_Suc_eq
tff(fact_1522_Ex__less__Suc,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,Na))
          & aa_nat_o(P,I4) )
    <=> ( aa_nat_o(P,Na)
        | ? [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Na)
            & aa_nat_o(P,I4) ) ) ) ).

% Ex_less_Suc
tff(fact_1523_less__SucI,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Na)) ) ).

% less_SucI
tff(fact_1524_less__SucE,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Na))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
       => ( M = Na ) ) ) ).

% less_SucE
tff(fact_1525_Suc__lessI,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( ( aa_nat_nat(suc,M) != Na )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),Na) ) ) ).

% Suc_lessI
tff(fact_1526_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),K)
     => ~ ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
           => ( K != aa_nat_nat(suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_1527_Suc__lessD,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% Suc_lessD
tff(fact_1528_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K)
     => ( ( K != aa_nat_nat(suc,I) )
       => ~ ! [J2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
             => ( K != aa_nat_nat(suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_1529_VEBT__internal_Oeven__odd__cases,axiom,
    ! [X: nat] :
      ( ! [N: nat] : X != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N)
     => ~ ! [N: nat] : X != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(suc,N)) ) ).

% VEBT_internal.even_odd_cases
tff(fact_1530_add__Suc__shift,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,Na)) ).

% add_Suc_shift
tff(fact_1531_add__Suc,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) ).

% add_Suc
tff(fact_1532_nat__arith_Osuc1,axiom,
    ! [A: nat,K: nat,Aa: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Aa) )
     => ( aa_nat_nat(suc,A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(suc,Aa)) ) ) ).

% nat_arith.suc1
tff(fact_1533_transitive__stepwise__le,axiom,
    ! [M: nat,Na: nat,R3: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( ! [X2: nat] : aa_nat_o(aa_nat_fun_nat_o(R3,X2),X2)
       => ( ! [X2: nat,Y2: nat,Z3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R3,X2),Y2)
             => ( aa_nat_o(aa_nat_fun_nat_o(R3,Y2),Z3)
               => aa_nat_o(aa_nat_fun_nat_o(R3,X2),Z3) ) )
         => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(R3,N),aa_nat_nat(suc,N))
           => aa_nat_o(aa_nat_fun_nat_o(R3,M),Na) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_1534_nat__induct__at__least,axiom,
    ! [M: nat,Na: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_o(P,M)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
             => ( aa_nat_o(P,N)
               => aa_nat_o(P,aa_nat_nat(suc,N)) ) )
         => aa_nat_o(P,Na) ) ) ) ).

% nat_induct_at_least
tff(fact_1535_full__nat__induct,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M2)),N)
             => aa_nat_o(P,M2) )
         => aa_nat_o(P,N) )
     => aa_nat_o(P,Na) ) ).

% full_nat_induct
tff(fact_1536_not__less__eq__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),M) ) ).

% not_less_eq_eq
tff(fact_1537_Suc__n__not__le__n,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),Na) ).

% Suc_n_not_le_n
tff(fact_1538_le__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
        | ( M = aa_nat_nat(suc,Na) ) ) ) ).

% le_Suc_eq
tff(fact_1539_Suc__le__D,axiom,
    ! [Na: nat,M6: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),M6)
     => ? [M3: nat] : M6 = aa_nat_nat(suc,M3) ) ).

% Suc_le_D
tff(fact_1540_le__SucI,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na)) ) ).

% le_SucI
tff(fact_1541_le__SucE,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => ( M = aa_nat_nat(suc,Na) ) ) ) ).

% le_SucE
tff(fact_1542_Suc__leD,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% Suc_leD
tff(fact_1543_zero__induct__lemma,axiom,
    ! [P: fun_nat_o,K: nat,I: nat] :
      ( aa_nat_o(P,K)
     => ( ! [N: nat] :
            ( aa_nat_o(P,aa_nat_nat(suc,N))
           => aa_nat_o(P,N) )
       => aa_nat_o(P,aa_nat_nat(minus_minus_nat(K),I)) ) ) ).

% zero_induct_lemma
tff(fact_1544_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Na) )
    <=> ( M = Na ) ) ).

% Suc_mult_cancel1
tff(fact_1545_log__def,axiom,
    ! [Aa: real,X: real] : aa_real_real(log(Aa),X) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Aa)) ).

% log_def
tff(fact_1546_abs__zmult__eq__1,axiom,
    ! [M: int,Na: int] :
      ( ( aa_int_int(abs_abs_int,times_times_int(M,Na)) = one_one_int )
     => ( aa_int_int(abs_abs_int,M) = one_one_int ) ) ).

% abs_zmult_eq_1
tff(fact_1547_less__Suc__eq__0__disj,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Na))
    <=> ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M = aa_nat_nat(suc,J3) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Na) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_1548_gr0__implies__Suc,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ? [M3: nat] : Na = aa_nat_nat(suc,M3) ) ).

% gr0_implies_Suc
tff(fact_1549_All__less__Suc2,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,Na))
         => aa_nat_o(P,I4) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Na)
           => aa_nat_o(P,aa_nat_nat(suc,I4)) ) ) ) ).

% All_less_Suc2
tff(fact_1550_gr0__conv__Suc,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
    <=> ? [M7: nat] : Na = aa_nat_nat(suc,M7) ) ).

% gr0_conv_Suc
tff(fact_1551_Ex__less__Suc2,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,Na))
          & aa_nat_o(P,I4) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        | ? [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Na)
            & aa_nat_o(P,aa_nat_nat(suc,I4)) ) ) ) ).

% Ex_less_Suc2
tff(fact_1552_add__is__1,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( Na = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_1553_one__is__add,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( Na = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_1554_nat__compl__induct,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( ! [Nn: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nn),N)
               => aa_nat_o(P,Nn) )
           => aa_nat_o(P,aa_nat_nat(suc,N)) )
       => aa_nat_o(P,Na) ) ) ).

% nat_compl_induct
tff(fact_1555_nat__compl__induct_H,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( ! [Nn: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nn),N)
               => aa_nat_o(P,Nn) )
           => aa_nat_o(P,aa_nat_nat(suc,N)) )
       => aa_nat_o(P,Na) ) ) ).

% nat_compl_induct'
tff(fact_1556_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_1557_less__natE,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ~ ! [Q5: nat] : Na != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q5)) ) ).

% less_natE
tff(fact_1558_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M))) ).

% less_add_Suc1
tff(fact_1559_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),I))) ).

% less_add_Suc2
tff(fact_1560_less__iff__Suc__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
    <=> ? [K2: nat] : Na = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2)) ) ).

% less_iff_Suc_add
tff(fact_1561_less__imp__Suc__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ? [K4: nat] : Na = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K4)) ) ).

% less_imp_Suc_add
tff(fact_1562_Suc__leI,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Na) ) ).

% Suc_leI
tff(fact_1563_Suc__le__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Na)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% Suc_le_eq
tff(fact_1564_dec__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(P,I)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),N)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),J)
               => ( aa_nat_o(P,N)
                 => aa_nat_o(P,aa_nat_nat(suc,N)) ) ) )
         => aa_nat_o(P,J) ) ) ) ).

% dec_induct
tff(fact_1565_inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(P,J)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),N)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),J)
               => ( aa_nat_o(P,aa_nat_nat(suc,N))
                 => aa_nat_o(P,N) ) ) )
         => aa_nat_o(P,I) ) ) ) ).

% inc_induct
tff(fact_1566_Suc__le__lessD,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% Suc_le_lessD
tff(fact_1567_le__less__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,M))
      <=> ( Na = M ) ) ) ).

% le_less_Suc_eq
tff(fact_1568_less__Suc__eq__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% less_Suc_eq_le
tff(fact_1569_less__eq__Suc__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),M) ) ).

% less_eq_Suc_le
tff(fact_1570_le__imp__less__Suc,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Na)) ) ).

% le_imp_less_Suc
tff(fact_1571_nat__in__between__eq_I2_J,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),aa_nat_nat(suc,Aa)) )
    <=> ( Ba = Aa ) ) ).

% nat_in_between_eq(2)
tff(fact_1572_nat__in__between__eq_I1_J,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),aa_nat_nat(suc,Aa)) )
    <=> ( Ba = aa_nat_nat(suc,Aa) ) ) ).

% nat_in_between_eq(1)
tff(fact_1573_Suc__to__right,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa_nat_nat(suc,Na) = M )
     => ( Na = aa_nat_nat(minus_minus_nat(M),aa_nat_nat(suc,zero_zero_nat)) ) ) ).

% Suc_to_right
tff(fact_1574_diff__less__Suc,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(minus_minus_nat(M),Na)),aa_nat_nat(suc,M)) ).

% diff_less_Suc
tff(fact_1575_Suc__diff__Suc,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
     => ( aa_nat_nat(suc,aa_nat_nat(minus_minus_nat(M),aa_nat_nat(suc,Na))) = aa_nat_nat(minus_minus_nat(M),Na) ) ) ).

% Suc_diff_Suc
tff(fact_1576_Suc__eq__plus1,axiom,
    ! [Na: nat] : aa_nat_nat(suc,Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat) ).

% Suc_eq_plus1
tff(fact_1577_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_1578_Suc__eq__plus1__left,axiom,
    ! [Na: nat] : aa_nat_nat(suc,Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),Na) ).

% Suc_eq_plus1_left
tff(fact_1579_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% Suc_mult_less_cancel1
tff(fact_1580_Suc__div__le__mono,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(divide_divide_nat(M),Na)),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,M)),Na)) ).

% Suc_div_le_mono
tff(fact_1581_Suc__diff__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_nat(minus_minus_nat(aa_nat_nat(suc,M)),Na) = aa_nat_nat(suc,aa_nat_nat(minus_minus_nat(M),Na)) ) ) ).

% Suc_diff_le
tff(fact_1582_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(minus_minus_nat(M),aa_nat_nat(suc,Na)) = aa_nat_nat(minus_minus_nat(aa_nat_nat(minus_minus_nat(M),one_one_nat)),Na) ).

% diff_Suc_eq_diff_pred
tff(fact_1583_mult__Suc,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,M)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) ).

% mult_Suc
tff(fact_1584_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% Suc_mult_le_cancel1
tff(fact_1585_mod__Suc__le__divisor,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(modulo_modulo_nat(M),aa_nat_nat(suc,Na))),Na) ).

% mod_Suc_le_divisor
tff(fact_1586_int__of__nat__induct,axiom,
    ! [P: fun_int_o,Z: int] :
      ( ! [N: nat] : aa_int_o(P,aa_nat_int(semiri1314217659103216013at_int,N))
     => ( ! [N: nat] : aa_int_o(P,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))))
       => aa_int_o(P,Z) ) ) ).

% int_of_nat_induct
tff(fact_1587_int__cases,axiom,
    ! [Z: int] :
      ( ! [N: nat] : Z != aa_nat_int(semiri1314217659103216013at_int,N)
     => ~ ! [N: nat] : Z != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))) ) ).

% int_cases
tff(fact_1588_Suc__times__binomial__eq,axiom,
    ! [Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Na)),aa_nat_nat(binomial(Na),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(binomial(aa_nat_nat(suc,Na)),aa_nat_nat(suc,K))),aa_nat_nat(suc,K)) ).

% Suc_times_binomial_eq
tff(fact_1589_Suc__times__binomial,axiom,
    ! [K: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(binomial(aa_nat_nat(suc,Na)),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Na)),aa_nat_nat(binomial(Na),K)) ).

% Suc_times_binomial
tff(fact_1590_nat__intermed__int__val,axiom,
    ! [M: nat,Na: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),I2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Na) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,minus_minus_int(aa_nat_int(F,aa_nat_nat(suc,I2)),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,M)),K)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Na))
           => ? [I2: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),I2)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Na)
                & ( aa_nat_int(F,I2) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_1591_ln__bound,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,X)),X) ) ).

% ln_bound
tff(fact_1592_ln__ge__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X)) ) ).

% ln_ge_zero
tff(fact_1593_nat__ivt__aux,axiom,
    ! [Na: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Na)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,minus_minus_int(aa_nat_int(F,aa_nat_nat(suc,I2)),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Na))
         => ? [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Na)
              & ( aa_nat_int(F,I2) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_1594_zabs__def,axiom,
    ! [I: int] :
      aa_int_int(abs_abs_int,I) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,I),zero_zero_int),aa_int_int(uminus_uminus_int,I),I) ).

% zabs_def
tff(fact_1595_nat__abs__mult__distrib,axiom,
    ! [W: int,Z: int] : aa_int_nat(nat2,aa_int_int(abs_abs_int,times_times_int(W,Z))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,W))),aa_int_nat(nat2,aa_int_int(abs_abs_int,Z))) ).

% nat_abs_mult_distrib
tff(fact_1596_dvd__imp__le__int,axiom,
    ! [I: int,D: int] :
      ( ( I != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),I)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,D)),aa_int_int(abs_abs_int,I)) ) ) ).

% dvd_imp_le_int
tff(fact_1597_numeral__1__eq__Suc__0,axiom,
    numeral_numeral_nat(one) = aa_nat_nat(suc,zero_zero_nat) ).

% numeral_1_eq_Suc_0
tff(fact_1598_eval__nat__numeral_I3_J,axiom,
    ! [Na: num] : numeral_numeral_nat(bit1(Na)) = aa_nat_nat(suc,numeral_numeral_nat(bit0(Na))) ).

% eval_nat_numeral(3)
tff(fact_1599_ex__least__nat__less,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( aa_nat_o(P,Na)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ? [K4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K4),Na)
            & ! [I3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),K4)
               => ~ aa_nat_o(P,I3) )
            & aa_nat_o(P,aa_nat_nat(suc,K4)) ) ) ) ).

% ex_least_nat_less
tff(fact_1600_nat__induct__non__zero,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => ( aa_nat_o(P,N)
               => aa_nat_o(P,aa_nat_nat(suc,N)) ) )
         => aa_nat_o(P,Na) ) ) ) ).

% nat_induct_non_zero
tff(fact_1601_diff__Suc__less,axiom,
    ! [Na: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(minus_minus_nat(Na),aa_nat_nat(suc,I))),Na) ) ).

% diff_Suc_less
tff(fact_1602_one__less__mult,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) ) ) ).

% one_less_mult
tff(fact_1603_n__less__m__mult__n,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) ) ) ).

% n_less_m_mult_n
tff(fact_1604_n__less__n__mult__m,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),M)) ) ) ).

% n_less_n_mult_m
tff(fact_1605_pred__numeral__def,axiom,
    ! [K: num] : pred_numeral(K) = aa_nat_nat(minus_minus_nat(numeral_numeral_nat(K)),one_one_nat) ).

% pred_numeral_def
tff(fact_1606_nat__one__le__power,axiom,
    ! [I: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),I)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(power_power_nat(I),Na)) ) ).

% nat_one_le_power
tff(fact_1607_int__ops_I4_J,axiom,
    ! [Aa: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Aa)) = plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,Aa),one_one_int) ).

% int_ops(4)
tff(fact_1608_int__Suc,axiom,
    ! [Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Na)) = plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,Na),one_one_int) ).

% int_Suc
tff(fact_1609_zless__iff__Suc__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
    <=> ? [N2: nat] : Z = plus_plus_int(W,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))) ) ).

% zless_iff_Suc_zadd
tff(fact_1610_Suc__times__binomial__add,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Aa)),aa_nat_nat(binomial(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba))),aa_nat_nat(suc,Aa))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Ba)),aa_nat_nat(binomial(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba))),Aa)) ).

% Suc_times_binomial_add
tff(fact_1611_binomial__Suc__Suc__eq__times,axiom,
    ! [Na: nat,K: nat] : aa_nat_nat(binomial(aa_nat_nat(suc,Na)),aa_nat_nat(suc,K)) = aa_nat_nat(divide_divide_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Na)),aa_nat_nat(binomial(Na),K))),aa_nat_nat(suc,K)) ).

% binomial_Suc_Suc_eq_times
tff(fact_1612_ln__ge__zero__imp__ge__one,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_1613_ln__add__one__self__le__self,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X))),X) ) ).

% ln_add_one_self_le_self
tff(fact_1614_ln__mult,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( aa_real_real(ln_ln_real,aa_real_real(times_times_real(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_mult
tff(fact_1615_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,plus_plus_int(K,L)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% nat_abs_triangle_ineq
tff(fact_1616_zdvd__mult__cancel1,axiom,
    ! [M: int,Na: int] :
      ( ( M != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,times_times_int(M,Na)),M)
      <=> ( aa_int_int(abs_abs_int,Na) = one_one_int ) ) ) ).

% zdvd_mult_cancel1
tff(fact_1617_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(divide_divide_nat(aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% div_abs_eq_div_nat
tff(fact_1618_numeral__2__eq__2,axiom,
    numeral_numeral_nat(bit0(one)) = aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat)) ).

% numeral_2_eq_2
tff(fact_1619_mod__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : modulo_modulo_int(aa_int_int(abs_abs_int,K),aa_int_int(abs_abs_int,L)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(modulo_modulo_nat(aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% mod_abs_eq_div_nat
tff(fact_1620_numeral__3__eq__3,axiom,
    numeral_numeral_nat(bit1(one)) = aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))) ).

% numeral_3_eq_3
tff(fact_1621_double__not__eq__Suc__double,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),M) != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Na)) ).

% double_not_eq_Suc_double
tff(fact_1622_Suc__double__not__eq__double,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),M)) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Na) ).

% Suc_double_not_eq_double
tff(fact_1623_Suc3__eq__add__3,axiom,
    ! [Na: nat] : aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit1(one))),Na) ).

% Suc3_eq_add_3
tff(fact_1624_nz__le__conv__less,axiom,
    ! [K: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(minus_minus_nat(K),aa_nat_nat(suc,zero_zero_nat))),M) ) ) ).

% nz_le_conv_less
tff(fact_1625_Suc__diff__eq__diff__pred,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(minus_minus_nat(aa_nat_nat(suc,M)),Na) = aa_nat_nat(minus_minus_nat(M),aa_nat_nat(minus_minus_nat(Na),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_1626_Suc__pred_H,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( Na = aa_nat_nat(suc,aa_nat_nat(minus_minus_nat(Na),one_one_nat)) ) ) ).

% Suc_pred'
tff(fact_1627_add__eq__if,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na) = $ite(M = zero_zero_nat,Na,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(minus_minus_nat(M),one_one_nat)),Na))) ).

% add_eq_if
tff(fact_1628_Suc__n__minus__m__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),M)
       => ( aa_nat_nat(suc,aa_nat_nat(minus_minus_nat(Na),M)) = aa_nat_nat(minus_minus_nat(Na),aa_nat_nat(minus_minus_nat(M),one_one_nat)) ) ) ) ).

% Suc_n_minus_m_eq
tff(fact_1629_div__nat__eqI,axiom,
    ! [Na: nat,Q3: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q3)),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(suc,Q3)))
       => ( aa_nat_nat(divide_divide_nat(M),Na) = Q3 ) ) ) ).

% div_nat_eqI
tff(fact_1630_Suc__nat__number__of__add,axiom,
    ! [V: num,Na: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(V)),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(plus_plus_num(V,one))),Na) ).

% Suc_nat_number_of_add
tff(fact_1631_not__zle__0__negative,axiom,
    ! [Na: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Na)))) ).

% not_zle_0_negative
tff(fact_1632_negD,axiom,
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),zero_zero_int)
     => ? [N: nat] : X = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))) ) ).

% negD
tff(fact_1633_negative__zless__0,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Na)))),zero_zero_int) ).

% negative_zless_0
tff(fact_1634_binomial__absorption,axiom,
    ! [K: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(binomial(Na),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(binomial(aa_nat_nat(minus_minus_nat(Na),one_one_nat)),K)) ).

% binomial_absorption
tff(fact_1635_Suc__as__int,axiom,
    ! [X4: nat] : aa_nat_nat(suc,X4) = aa_int_nat(nat2,plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,X4),one_one_int)) ).

% Suc_as_int
tff(fact_1636_ln__2__less__1,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,numeral_numeral_real(bit0(one)))),one_one_real) ).

% ln_2_less_1
tff(fact_1637_floor__eq3,axiom,
    ! [Na: nat,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na)))
       => ( aa_int_nat(nat2,archim6058952711729229775r_real(X)) = Na ) ) ) ).

% floor_eq3
tff(fact_1638_ln__le__minus__one,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,X)),aa_real_real(minus_minus_real(X),one_one_real)) ) ).

% ln_le_minus_one
tff(fact_1639_ln__diff__le,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y))),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(X),Y)),Y)) ) ) ).

% ln_diff_le
tff(fact_1640_ln__add__one__self__le__self2,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X))),X) ) ).

% ln_add_one_self_le_self2
tff(fact_1641_ln__realpow,axiom,
    ! [X: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(ln_ln_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na)) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(ln_ln_real,X)) ) ) ).

% ln_realpow
tff(fact_1642_num_Osize_I5_J,axiom,
    ! [X23: num] : size_size_num(bit0(X23)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_num(X23)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(5)
tff(fact_1643_num_Osize_I6_J,axiom,
    ! [X32: num] : size_size_num(bit1(X32)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_num(X32)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(6)
tff(fact_1644_even__abs__add__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),plus_plus_int(aa_int_int(abs_abs_int,K),L))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),plus_plus_int(K,L)) ) ).

% even_abs_add_iff
tff(fact_1645_even__add__abs__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),plus_plus_int(K,aa_int_int(abs_abs_int,L)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),plus_plus_int(K,L)) ) ).

% even_add_abs_iff
tff(fact_1646_nat__abs__int__diff,axiom,
    ! [Aa: nat,Ba: nat] :
      aa_int_nat(nat2,aa_int_int(abs_abs_int,minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,Aa),aa_nat_int(semiri1314217659103216013at_int,Ba)))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba),aa_nat_nat(minus_minus_nat(Ba),Aa),aa_nat_nat(minus_minus_nat(Aa),Ba)) ).

% nat_abs_int_diff
tff(fact_1647_less__2__cases__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),numeral_numeral_nat(bit0(one)))
    <=> ( ( Na = zero_zero_nat )
        | ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_1648_less__2__cases,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),numeral_numeral_nat(bit0(one)))
     => ( ( Na = zero_zero_nat )
        | ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_1649_Suc__div__eq__add3__div,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),Na) = aa_nat_nat(divide_divide_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit1(one))),M)),Na) ).

% Suc_div_eq_add3_div
tff(fact_1650_nat__2,axiom,
    aa_int_nat(nat2,aa_num_int(numeral_numeral_int,bit0(one))) = aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat)) ).

% nat_2
tff(fact_1651_le__div__geq,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( aa_nat_nat(divide_divide_nat(M),Na) = aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(aa_nat_nat(minus_minus_nat(M),Na)),Na)) ) ) ) ).

% le_div_geq
tff(fact_1652_split__div_H,axiom,
    ! [P: fun_nat_o,M: nat,Na: nat] :
      ( aa_nat_o(P,aa_nat_nat(divide_divide_nat(M),Na))
    <=> ( ( ( Na = zero_zero_nat )
          & aa_nat_o(P,zero_zero_nat) )
        | ? [Q4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q4)),M)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(suc,Q4)))
            & aa_nat_o(P,Q4) ) ) ) ).

% split_div'
tff(fact_1653_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),Na) = aa_nat_nat(modulo_modulo_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit1(one))),M)),Na) ).

% Suc_mod_eq_add3_mod
tff(fact_1654_Suc__times__mod__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
     => ( aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na))),M) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_1655_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_nat(suc,aa_int_nat(nat2,Z)) = aa_int_nat(nat2,plus_plus_int(one_one_int,Z)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_1656_ln__one__minus__pos__upper__bound,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(minus_minus_real(one_one_real),X))),aa_real_real(uminus_uminus_real,X)) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_1657_ln__powr__bound,axiom,
    ! [X: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,X)),aa_real_real(divide_divide_real(aa_real_real(powr_real(X),Aa)),Aa)) ) ) ).

% ln_powr_bound
tff(fact_1658_ln__powr__bound2,axiom,
    ! [X: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(aa_real_real(ln_ln_real,X)),Aa)),aa_real_real(times_times_real(aa_real_real(powr_real(Aa),Aa)),X)) ) ) ).

% ln_powr_bound2
tff(fact_1659_log__eq__div__ln__mult__log,axiom,
    ! [Aa: real,Ba: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( ( Aa != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ba)
         => ( ( Ba != one_one_real )
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
             => ( aa_real_real(log(Aa),X) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,Ba)),aa_real_real(ln_ln_real,Aa))),aa_real_real(log(Ba),X)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_1660_floor__eq4,axiom,
    ! [Na: nat,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Na)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na)))
       => ( aa_int_nat(nat2,archim6058952711729229775r_real(X)) = Na ) ) ) ).

% floor_eq4
tff(fact_1661_incr__lemma,axiom,
    ! [D: int,Z: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Z),plus_plus_int(X,times_times_int(plus_plus_int(aa_int_int(abs_abs_int,minus_minus_int(X,Z)),one_one_int),D))) ) ).

% incr_lemma
tff(fact_1662_decr__lemma,axiom,
    ! [D: int,X: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,minus_minus_int(X,times_times_int(plus_plus_int(aa_int_int(abs_abs_int,minus_minus_int(X,Z)),one_one_int),D))),Z) ) ).

% decr_lemma
tff(fact_1663_Suc__n__div__2__gt__zero,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,Na)),numeral_numeral_nat(bit0(one)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_1664_div__2__gt__zero,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one)))) ) ).

% div_2_gt_zero
tff(fact_1665_nat__bit__induct,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(P,N)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),N)) ) )
       => ( ! [N: nat] :
              ( aa_nat_o(P,N)
             => aa_nat_o(P,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),N))) )
         => aa_nat_o(P,Na) ) ) ) ).

% nat_bit_induct
tff(fact_1666_binomial__less__binomial__Suc,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one))))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(binomial(Na),K)),aa_nat_nat(binomial(Na),aa_nat_nat(suc,K))) ) ).

% binomial_less_binomial_Suc
tff(fact_1667_central__binomial__odd,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na)
     => ( aa_nat_nat(binomial(Na),aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one))))) = aa_nat_nat(binomial(Na),aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one)))) ) ) ).

% central_binomial_odd
tff(fact_1668_signed__take__bit__int__less__eq,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na)),K)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,bit_ri631733984087533419it_int(Na,K)),minus_minus_int(K,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,Na)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_1669_binomial__addition__formula,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(binomial(Na),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(binomial(aa_nat_nat(minus_minus_nat(Na),one_one_nat)),aa_nat_nat(suc,K))),aa_nat_nat(binomial(aa_nat_nat(minus_minus_nat(Na),one_one_nat)),K)) ) ) ).

% binomial_addition_formula
tff(fact_1670_nat0__intermed__int__val,axiom,
    ! [Na: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Na)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,minus_minus_int(aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),one_one_nat)),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Na))
         => ? [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Na)
              & ( aa_nat_int(F,I2) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_1671_nat__div__eq__Suc__0__iff,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa_nat_nat(divide_divide_nat(Na),M) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),M)) ) ) ).

% nat_div_eq_Suc_0_iff
tff(fact_1672_ln__one__plus__pos__lower__bound,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(X),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one))))),aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_1673_even__mod__4__div__2,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(modulo_modulo_nat(Na),numeral_numeral_nat(bit0(bit0(one)))) = aa_nat_nat(suc,zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(divide_divide_nat(aa_nat_nat(minus_minus_nat(Na),aa_nat_nat(suc,zero_zero_nat))),numeral_numeral_nat(bit0(one)))) ) ).

% even_mod_4_div_2
tff(fact_1674_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na)))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,plus_plus_int(K,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,Na)))),bit_ri631733984087533419it_int(Na,K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_1675_odd__mod__4__div__2,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(modulo_modulo_nat(Na),numeral_numeral_nat(bit0(bit0(one)))) = numeral_numeral_nat(bit1(one)) )
     => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(divide_divide_nat(aa_nat_nat(minus_minus_nat(Na),aa_nat_nat(suc,zero_zero_nat))),numeral_numeral_nat(bit0(one)))) ) ).

% odd_mod_4_div_2
tff(fact_1676_sb__inc__lem,axiom,
    ! [Aa: int,K: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,plus_plus_int(Aa,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K))),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,plus_plus_int(plus_plus_int(Aa,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,K)))),modulo_modulo_int(plus_plus_int(Aa,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,K)))) ) ).

% sb_inc_lem
tff(fact_1677_sb__inc__lem_H,axiom,
    ! [Aa: int,K: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K)))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,plus_plus_int(plus_plus_int(Aa,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,K)))),modulo_modulo_int(plus_plus_int(Aa,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,K)))) ) ).

% sb_inc_lem'
tff(fact_1678_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Oelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V8646137997579335489_i_l_d,X) = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != numeral_numeral_nat(bit0(bit0(one))) ) )
       => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != numeral_numeral_nat(bit0(bit0(one))) ) )
         => ~ ! [Va2: nat] :
                ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
               => ( Y != $ite(
                      aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                        $let(
                          half: nat,
                          half:= aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one))),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(bit1(bit0(one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),half)),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) )),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one))),
                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(bit0(bit1(one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,half))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.elims
tff(fact_1679_mask__nat__positive__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% mask_nat_positive_iff
tff(fact_1680_square__powr__half,axiom,
    ! [X: real] : aa_real_real(powr_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one)))) = aa_real_real(abs_abs_real,X) ).

% square_powr_half
tff(fact_1681_nat__mask__eq,axiom,
    ! [Na: nat] : aa_int_nat(nat2,aa_nat_int(bit_se2000444600071755411sk_int,Na)) = aa_nat_nat(bit_se2002935070580805687sk_nat,Na) ).

% nat_mask_eq
tff(fact_1682_less__eq__mask,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(bit_se2002935070580805687sk_nat,Na)) ).

% less_eq_mask
tff(fact_1683_mask__nonnegative__int,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(bit_se2000444600071755411sk_int,Na)) ).

% mask_nonnegative_int
tff(fact_1684_not__mask__negative__int,axiom,
    ! [Na: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(bit_se2000444600071755411sk_int,Na)),zero_zero_int) ).

% not_mask_negative_int
tff(fact_1685_abs__real__def,axiom,
    ! [Aa: real] :
      aa_real_real(abs_abs_real,Aa) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real),aa_real_real(uminus_uminus_real,Aa),Aa) ).

% abs_real_def
tff(fact_1686_sin__bound__lemma,axiom,
    ! [X: real,Y: real,U: real,V: real] :
      ( ( X = Y )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,U)),V)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),U)),Y))),V) ) ) ).

% sin_bound_lemma
tff(fact_1687_less__mask,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(bit_se2002935070580805687sk_nat,Na)) ) ).

% less_mask
tff(fact_1688_lemma__interval,axiom,
    ! [Aa: real,X: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Ba)
       => ? [D4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
            & ! [Y3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X),Y3))),D4)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Y3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),Ba) ) ) ) ) ) ).

% lemma_interval
tff(fact_1689_Suc__mask__eq__exp,axiom,
    ! [Na: nat] : aa_nat_nat(suc,aa_nat_nat(bit_se2002935070580805687sk_nat,Na)) = aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na) ).

% Suc_mask_eq_exp
tff(fact_1690_mask__nat__less__exp,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,Na)),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na)) ).

% mask_nat_less_exp
tff(fact_1691_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I1_J,axiom,
    aa_nat_nat(vEBT_V8646137997579335489_i_l_d,zero_zero_nat) = numeral_numeral_nat(bit0(bit0(one))) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(1)
tff(fact_1692_mask__half__int,axiom,
    ! [Na: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(bit_se2000444600071755411sk_int,Na)),aa_num_int(numeral_numeral_int,bit0(one))) = aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(minus_minus_nat(Na),one_one_nat)) ).

% mask_half_int
tff(fact_1693_mask__int__def,axiom,
    ! [Na: nat] : aa_nat_int(bit_se2000444600071755411sk_int,Na) = minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na),one_one_int) ).

% mask_int_def
tff(fact_1694_mask__nat__def,axiom,
    ! [Na: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,Na) = aa_nat_nat(minus_minus_nat(aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na)),one_one_nat) ).

% mask_nat_def
tff(fact_1695_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I2_J,axiom,
    aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,zero_zero_nat)) = numeral_numeral_nat(bit0(bit0(one))) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(2)
tff(fact_1696_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X))),X))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_1697_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X))),X))),aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_1698_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X))),X))),aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_1699_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [Va: nat] :
      aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,aa_nat_nat(suc,Va))) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
          $let(
            half: nat,
            half:= aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one))),
            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(bit1(bit0(one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),half)),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) )),
        $let(
          half: nat,
          half:= aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one))),
          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(bit0(bit1(one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,half))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(3)
tff(fact_1700_VEBT__internal_OT__vebt__buildupi_H_Oelims,axiom,
    ! [X: nat,Y: int] :
      ( ( aa_nat_int(vEBT_V9176841429113362141ildupi,X) = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != one_one_int ) )
       => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != one_one_int ) )
         => ~ ! [N: nat] :
                ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
               => ( Y != $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N),plus_plus_int(aa_num_int(numeral_numeral_int,bit1(one)),plus_plus_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one))))),plus_plus_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(bit0(one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one))))),times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),times_times_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one))))))))),plus_plus_int(aa_num_int(numeral_numeral_int,bit1(one)),plus_plus_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one)))))),plus_plus_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(bit0(bit0(one)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one))))),times_times_int(aa_num_int(numeral_numeral_int,bit0(bit0(one))),times_times_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one)))))))))) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.elims
tff(fact_1701_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Oelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V8346862874174094_d_u_p,X) = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != numeral_numeral_nat(bit1(one)) ) )
       => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != numeral_numeral_nat(bit1(one)) ) )
         => ~ ! [Va2: nat] :
                ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
               => ( Y != $ite(
                      aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                        $let(
                          half: nat,
                          half:= aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one))),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit1(bit0(bit0(one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),half)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) )),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one))),
                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit1(bit1(bit0(one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,half))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.elims
tff(fact_1702_VEBT__internal_OTb_Oelims,axiom,
    ! [X: nat,Y: int] :
      ( ( aa_nat_int(vEBT_VEBT_Tb,X) = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != aa_num_int(numeral_numeral_int,bit1(one)) ) )
       => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_num_int(numeral_numeral_int,bit1(one)) ) )
         => ~ ! [N: nat] :
                ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
               => ( Y != $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N),plus_plus_int(plus_plus_int(aa_num_int(numeral_numeral_int,bit1(bit0(one))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one)))))),times_times_int(aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one))))))),plus_plus_int(plus_plus_int(aa_num_int(numeral_numeral_int,bit1(bit0(one))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one))))))),times_times_int(aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one))))))))) ) ) ) ) ) ).

% VEBT_internal.Tb.elims
tff(fact_1703_monoseq__arctan__series,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => topolo6980174941875973593q_real(aa_real_fun_nat_real(aTP_Lamm_bg,X)) ) ).

% monoseq_arctan_series
tff(fact_1704_complex__mod__minus__le__complex__mod,axiom,
    ! [X: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,real_V1022390504157884413omplex(X))),real_V1022390504157884413omplex(X)) ).

% complex_mod_minus_le_complex_mod
tff(fact_1705_complex__mod__triangle__ineq2,axiom,
    ! [Ba: complex,Aa: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(real_V1022390504157884413omplex(plus_plus_complex(Ba,Aa))),real_V1022390504157884413omplex(Ba))),real_V1022390504157884413omplex(Aa)) ).

% complex_mod_triangle_ineq2
tff(fact_1706_VEBT__internal_OT__vebt__buildupi_H_Osimps_I1_J,axiom,
    aa_nat_int(vEBT_V9176841429113362141ildupi,zero_zero_nat) = one_one_int ).

% VEBT_internal.T_vebt_buildupi'.simps(1)
tff(fact_1707_VEBT__internal_OTb__T__vebt__buildupi_H,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(vEBT_V9176841429113362141ildupi,Na)),minus_minus_int(aa_nat_int(vEBT_VEBT_Tb,Na),aa_num_int(numeral_numeral_int,bit0(one)))) ).

% VEBT_internal.Tb_T_vebt_buildupi'
tff(fact_1708_VEBT__internal_OT__vebt__buildupi_H_Osimps_I2_J,axiom,
    aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% VEBT_internal.T_vebt_buildupi'.simps(2)
tff(fact_1709_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I1_J,axiom,
    aa_nat_nat(vEBT_V8346862874174094_d_u_p,zero_zero_nat) = numeral_numeral_nat(bit1(one)) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(1)
tff(fact_1710_VEBT__internal_OTb_Osimps_I1_J,axiom,
    aa_nat_int(vEBT_VEBT_Tb,zero_zero_nat) = aa_num_int(numeral_numeral_int,bit1(one)) ).

% VEBT_internal.Tb.simps(1)
tff(fact_1711_monoseq__realpow,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
       => topolo6980174941875973593q_real(aa_real_fun_nat_real(power_power_real,X)) ) ) ).

% monoseq_realpow
tff(fact_1712_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I2_J,axiom,
    aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,zero_zero_nat)) = numeral_numeral_nat(bit1(one)) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(2)
tff(fact_1713_VEBT__internal_OTb_Osimps_I2_J,axiom,
    aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,zero_zero_nat)) = aa_num_int(numeral_numeral_int,bit1(one)) ).

% VEBT_internal.Tb.simps(2)
tff(fact_1714_VEBT__internal_OTb_Osimps_I3_J,axiom,
    ! [Na: nat] :
      aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na),plus_plus_int(plus_plus_int(aa_num_int(numeral_numeral_int,bit1(bit0(one))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one)))))),times_times_int(aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one))))))),plus_plus_int(plus_plus_int(aa_num_int(numeral_numeral_int,bit1(bit0(one))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one))))))),times_times_int(aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one))))))))) ).

% VEBT_internal.Tb.simps(3)
tff(fact_1715_VEBT__internal_OT__vebt__buildupi_H_Osimps_I3_J,axiom,
    ! [Na: nat] :
      aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na),plus_plus_int(aa_num_int(numeral_numeral_int,bit1(one)),plus_plus_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one))))),plus_plus_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(bit0(one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one))))),times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),times_times_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one))))))))),plus_plus_int(aa_num_int(numeral_numeral_int,bit1(one)),plus_plus_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one)))))),plus_plus_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(bit0(bit0(one)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one))))),times_times_int(aa_num_int(numeral_numeral_int,bit0(bit0(one))),times_times_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one)))))))))) ).

% VEBT_internal.T_vebt_buildupi'.simps(3)
tff(fact_1716_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I3_J,axiom,
    ! [Va: nat] :
      aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,aa_nat_nat(suc,Va))) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
          $let(
            half: nat,
            half:= aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one))),
            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit1(bit0(bit0(one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),half)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) )),
        $let(
          half: nat,
          half:= aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va))),numeral_numeral_nat(bit0(one))),
          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit1(bit1(bit0(one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,half))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(3)
tff(fact_1717_vebt__inst_Otime__vebt__buildup,axiom,
    ! [U: nat,Na: nat] :
      ( ( U = aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(bit1(bit0(bit1(one)))))),U)) ) ).

% vebt_inst.time_vebt_buildup
tff(fact_1718_VEBT__internal_OTb_H_Oelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_VEBT_Tb2,X) = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != numeral_numeral_nat(bit1(one)) ) )
       => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != numeral_numeral_nat(bit1(one)) ) )
         => ~ ! [N: nat] :
                ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
               => ( Y != $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit1(bit0(one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit1(bit0(one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one))))))))) ) ) ) ) ) ).

% VEBT_internal.Tb'.elims
tff(fact_1719_ln__series,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),numeral_numeral_real(bit0(one)))
       => ( aa_real_real(ln_ln_real,X) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_bh,X)) ) ) ) ).

% ln_series
tff(fact_1720_VEBT__internal_OTb__Tb_H,axiom,
    ! [T: nat] : aa_nat_int(vEBT_VEBT_Tb,T) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(vEBT_VEBT_Tb2,T)) ).

% VEBT_internal.Tb_Tb'
tff(fact_1721_VEBT__internal_OTb_H_Osimps_I1_J,axiom,
    aa_nat_nat(vEBT_VEBT_Tb2,zero_zero_nat) = numeral_numeral_nat(bit1(one)) ).

% VEBT_internal.Tb'.simps(1)
tff(fact_1722_VEBT__internal_OTb_H_Osimps_I2_J,axiom,
    aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,zero_zero_nat)) = numeral_numeral_nat(bit1(one)) ).

% VEBT_internal.Tb'.simps(2)
tff(fact_1723_VEBT__internal_OTb_H_Osimps_I3_J,axiom,
    ! [Na: nat] :
      aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit1(bit0(one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit1(bit0(one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one))))))))) ).

% VEBT_internal.Tb'.simps(3)
tff(fact_1724_arctan__series,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ( aa_real_real(arctan,X) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_bi,X)) ) ) ).

% arctan_series
tff(fact_1725_pi__series,axiom,
    aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(bit0(one)))) = suminf_real(aTP_Lamm_bj) ).

% pi_series
tff(fact_1726_summable__arctan__series,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_bi,X)) ) ).

% summable_arctan_series
tff(fact_1727_arctan__le__zero__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,X)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real) ) ).

% arctan_le_zero_iff
tff(fact_1728_zero__le__arctan__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arctan,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X) ) ).

% zero_le_arctan_iff
tff(fact_1729_summable__rabs__cancel,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_bk(F))
     => summable_real(F) ) ).

% summable_rabs_cancel
tff(fact_1730_arctan__le__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ).

% arctan_le_iff
tff(fact_1731_arctan__monotone_H,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y)) ) ).

% arctan_monotone'
tff(fact_1732_pi__ge__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),pi) ).

% pi_ge_zero
tff(fact_1733_summable__rabs__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N7: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(F,N))),aa_nat_real(G,N)) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_bk(F)) ) ) ).

% summable_rabs_comparison_test
tff(fact_1734_summable__rabs,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_bk(F))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,suminf_real(F))),suminf_real(aTP_Lamm_bk(F))) ) ).

% summable_rabs
tff(fact_1735_arctan__ubound,axiom,
    ! [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one)))) ).

% arctan_ubound
tff(fact_1736_arctan__one,axiom,
    aa_real_real(arctan,one_one_real) = aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(bit0(one)))) ).

% arctan_one
tff(fact_1737_arctan__lbound,axiom,
    ! [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),aa_real_real(arctan,Y)) ).

% arctan_lbound
tff(fact_1738_arctan__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),aa_real_real(arctan,Y))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one)))) ) ).

% arctan_bounded
tff(fact_1739_machin__Euler,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(numeral_numeral_real(bit1(bit0(one)))),aa_real_real(arctan,aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit1(bit1(one))))))),aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_real_real(arctan,aa_real_real(divide_divide_real(numeral_numeral_real(bit1(one))),numeral_numeral_real(bit1(bit1(bit1(bit1(bit0(bit0(one))))))))))) = aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(bit0(one)))) ).

% machin_Euler
tff(fact_1740_machin,axiom,
    aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(bit0(one)))) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(bit0(one)))),aa_real_real(arctan,aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit1(bit0(one))))))),aa_real_real(arctan,aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit1(bit1(bit1(bit1(bit0(bit1(bit1(one))))))))))) ).

% machin
tff(fact_1741_pi__less__4,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,pi),numeral_numeral_real(bit0(bit0(one)))) ).

% pi_less_4
tff(fact_1742_pi__ge__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(bit0(one))),pi) ).

% pi_ge_two
tff(fact_1743_pi__half__neq__two,axiom,
    aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))) != numeral_numeral_real(bit0(one)) ).

% pi_half_neq_two
tff(fact_1744_summable__power__series,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,I2)),one_one_real)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I2))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Z)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),one_one_real)
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_bl(F),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_1745_pi__half__neq__zero,axiom,
    aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))) != zero_zero_real ).

% pi_half_neq_zero
tff(fact_1746_pi__half__less__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one)))),numeral_numeral_real(bit0(one))) ).

% pi_half_less_two
tff(fact_1747_pi__half__le__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one)))),numeral_numeral_real(bit0(one))) ).

% pi_half_le_two
tff(fact_1748_pi__half__gt__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one)))) ).

% pi_half_gt_zero
tff(fact_1749_pi__half__ge__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one)))) ).

% pi_half_ge_zero
tff(fact_1750_m2pi__less__pi,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),pi))),pi) ).

% m2pi_less_pi
tff(fact_1751_minus__pi__half__less__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),zero_zero_real) ).

% minus_pi_half_less_zero
tff(fact_1752_arctan__add,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y)) = aa_real_real(arctan,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_real_real(minus_minus_real(one_one_real),aa_real_real(times_times_real(X),Y)))) ) ) ) ).

% arctan_add
tff(fact_1753_arctan__double,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ( aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_real_real(arctan,X)) = aa_real_real(arctan,aa_real_real(divide_divide_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),X)),aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))))) ) ) ).

% arctan_double
tff(fact_1754_sin__cos__npi,axiom,
    ! [Na: nat] : aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Na)))),pi)),numeral_numeral_real(bit0(one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Na) ).

% sin_cos_npi
tff(fact_1755_tanh__ln__real,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(tanh_real,aa_real_real(ln_ln_real,X)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))),one_one_real)) ) ) ).

% tanh_ln_real
tff(fact_1756_VEBT__internal_OT__vebt__buildupi_Oelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V441764108873111860ildupi,X) = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_nat_nat(suc,zero_zero_nat) ) )
         => ~ ! [N: nat] :
                ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
               => ( Y != $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(bit0(one)))),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one))))))))))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(bit0(bit0(one))))),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one)))))))))))) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.elims
tff(fact_1757_cos__pi__eq__zero,axiom,
    ! [M: nat] : aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(times_times_real(pi),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),M))))),numeral_numeral_real(bit0(one)))) = zero_zero_real ).

% cos_pi_eq_zero
tff(fact_1758_flip__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),bit_se2159334234014336723it_int(Na,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_1759_flip__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,bit_se2159334234014336723it_int(Na,K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% flip_bit_negative_int_iff
tff(fact_1760_tanh__real__le__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tanh_real,X)),aa_real_real(tanh_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ).

% tanh_real_le_iff
tff(fact_1761_tanh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tanh_real,X)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real) ) ).

% tanh_real_nonpos_iff
tff(fact_1762_tanh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(tanh_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X) ) ).

% tanh_real_nonneg_iff
tff(fact_1763_cos__periodic__pi,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,X)) ).

% cos_periodic_pi
tff(fact_1764_cos__periodic__pi2,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,pi),X)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,X)) ).

% cos_periodic_pi2
tff(fact_1765_sin__periodic__pi2,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,pi),X)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X)) ).

% sin_periodic_pi2
tff(fact_1766_sin__periodic__pi,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X)) ).

% sin_periodic_pi
tff(fact_1767_sin__npi2,axiom,
    ! [Na: nat] : aa_real_real(sin_real,aa_real_real(times_times_real(pi),aa_nat_real(semiri5074537144036343181t_real,Na))) = zero_zero_real ).

% sin_npi2
tff(fact_1768_sin__npi,axiom,
    ! [Na: nat] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),pi)) = zero_zero_real ).

% sin_npi
tff(fact_1769_cos__pi__half,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one)))) = zero_zero_real ).

% cos_pi_half
tff(fact_1770_sin__two__pi,axiom,
    aa_real_real(sin_real,aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),pi)) = zero_zero_real ).

% sin_two_pi
tff(fact_1771_cos__two__pi,axiom,
    aa_real_real(cos_real,aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),pi)) = one_one_real ).

% cos_two_pi
tff(fact_1772_sin__pi__half,axiom,
    aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one)))) = one_one_real ).

% sin_pi_half
tff(fact_1773_cos__periodic,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),pi))) = aa_real_real(cos_real,X) ).

% cos_periodic
tff(fact_1774_sin__periodic,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),pi))) = aa_real_real(sin_real,X) ).

% sin_periodic
tff(fact_1775_cos__2pi__minus,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(minus_minus_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),pi)),X)) = aa_real_real(cos_real,X) ).

% cos_2pi_minus
tff(fact_1776_cos__npi,axiom,
    ! [Na: nat] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),pi)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Na) ).

% cos_npi
tff(fact_1777_cos__npi2,axiom,
    ! [Na: nat] : aa_real_real(cos_real,aa_real_real(times_times_real(pi),aa_nat_real(semiri5074537144036343181t_real,Na))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Na) ).

% cos_npi2
tff(fact_1778_sin__2npi,axiom,
    ! [Na: nat] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)) = zero_zero_real ).

% sin_2npi
tff(fact_1779_cos__2npi,axiom,
    ! [Na: nat] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)) = one_one_real ).

% cos_2npi
tff(fact_1780_sin__2pi__minus,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(minus_minus_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),pi)),X)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X)) ).

% sin_2pi_minus
tff(fact_1781_cos__3over2__pi,axiom,
    aa_real_real(cos_real,aa_real_real(times_times_real(aa_real_real(divide_divide_real(numeral_numeral_real(bit1(one))),numeral_numeral_real(bit0(one)))),pi)) = zero_zero_real ).

% cos_3over2_pi
tff(fact_1782_sin__3over2__pi,axiom,
    aa_real_real(sin_real,aa_real_real(times_times_real(aa_real_real(divide_divide_real(numeral_numeral_real(bit1(one))),numeral_numeral_real(bit0(one)))),pi)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% sin_3over2_pi
tff(fact_1783_sincos__principal__value,axiom,
    ! [X: real] :
    ? [Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Y2)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),pi)
      & ( aa_real_real(sin_real,Y2) = aa_real_real(sin_real,X) )
      & ( aa_real_real(cos_real,Y2) = aa_real_real(cos_real,X) ) ) ).

% sincos_principal_value
tff(fact_1784_sin__x__le__x,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,X)),X) ) ).

% sin_x_le_x
tff(fact_1785_sin__le__one,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,X)),one_one_real) ).

% sin_le_one
tff(fact_1786_cos__le__one,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,X)),one_one_real) ).

% cos_le_one
tff(fact_1787_abs__sin__x__le__abs__x,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,X))),aa_real_real(abs_abs_real,X)) ).

% abs_sin_x_le_abs_x
tff(fact_1788_sin__cos__le1,axiom,
    ! [X: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_real_real(sin_real,X)),aa_real_real(sin_real,Y))),aa_real_real(times_times_real(aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))))),one_one_real) ).

% sin_cos_le1
tff(fact_1789_sin__x__ge__neg__x,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,X)),aa_real_real(sin_real,X)) ) ).

% sin_x_ge_neg_x
tff(fact_1790_sin__ge__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),pi)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,X)) ) ) ).

% sin_ge_zero
tff(fact_1791_sin__ge__minus__one,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(sin_real,X)) ).

% sin_ge_minus_one
tff(fact_1792_cos__monotone__0__pi__le,axiom,
    ! [Y: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),pi)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y)) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_1793_cos__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),pi)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),X) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_1794_cos__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),pi)
           => ( ( aa_real_real(cos_real,X) = aa_real_real(cos_real,Y) )
             => ( X = Y ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_1795_cos__ge__minus__one,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(cos_real,X)) ).

% cos_ge_minus_one
tff(fact_1796_abs__sin__le__one,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,X))),one_one_real) ).

% abs_sin_le_one
tff(fact_1797_abs__cos__le__one,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(cos_real,X))),one_one_real) ).

% abs_cos_le_one
tff(fact_1798_VEBT__internal_OTbuildupi__buildupi_H,axiom,
    ! [Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(vEBT_V441764108873111860ildupi,Na)) = aa_nat_int(vEBT_V9176841429113362141ildupi,Na) ).

% VEBT_internal.Tbuildupi_buildupi'
tff(fact_1799_cos__two__neq__zero,axiom,
    aa_real_real(cos_real,numeral_numeral_real(bit0(one))) != zero_zero_real ).

% cos_two_neq_zero
tff(fact_1800_cos__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),pi)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_1801_cos__monotone__0__pi,axiom,
    ! [Y: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),pi)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y)) ) ) ) ).

% cos_monotone_0_pi
tff(fact_1802_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,X)) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_1803_sincos__total__pi,axiom,
    ! [Y: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one)))) = one_one_real )
       => ? [T2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T2)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T2),pi)
            & ( X = aa_real_real(cos_real,T2) )
            & ( Y = aa_real_real(sin_real,T2) ) ) ) ) ).

% sincos_total_pi
tff(fact_1804_sin__expansion__lemma,axiom,
    ! [X: real,M: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),pi)),numeral_numeral_real(bit0(one))))) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,M)),pi)),numeral_numeral_real(bit0(one))))) ).

% sin_expansion_lemma
tff(fact_1805_cos__expansion__lemma,axiom,
    ! [X: real,M: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),pi)),numeral_numeral_real(bit0(one))))) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,M)),pi)),numeral_numeral_real(bit0(one)))))) ).

% cos_expansion_lemma
tff(fact_1806_sin__gt__zero__02,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),numeral_numeral_real(bit0(one)))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,X)) ) ) ).

% sin_gt_zero_02
tff(fact_1807_cos__two__less__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,numeral_numeral_real(bit0(one)))),zero_zero_real) ).

% cos_two_less_zero
tff(fact_1808_cos__two__le__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,numeral_numeral_real(bit0(one)))),zero_zero_real) ).

% cos_two_le_zero
tff(fact_1809_cos__is__zero,axiom,
    ? [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X2)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),numeral_numeral_real(bit0(one)))
      & ( aa_real_real(cos_real,X2) = zero_zero_real )
      & ! [Y3: real] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y3)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),numeral_numeral_real(bit0(one)))
            & ( aa_real_real(cos_real,Y3) = zero_zero_real ) )
         => ( Y3 = X2 ) ) ) ).

% cos_is_zero
tff(fact_1810_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,X)) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_1811_cos__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ? [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X2)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),pi)
            & ( aa_real_real(cos_real,X2) = Y )
            & ! [Y3: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),pi)
                  & ( aa_real_real(cos_real,Y3) = Y ) )
               => ( Y3 = X2 ) ) ) ) ) ).

% cos_total
tff(fact_1812_sincos__total__pi__half,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one)))) = one_one_real )
         => ? [T2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T2)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T2),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
              & ( X = aa_real_real(cos_real,T2) )
              & ( Y = aa_real_real(sin_real,T2) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_1813_sincos__total__2pi__le,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one)))) = one_one_real )
     => ? [T2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T2)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T2),aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),pi))
          & ( X = aa_real_real(cos_real,T2) )
          & ( Y = aa_real_real(sin_real,T2) ) ) ) ).

% sincos_total_2pi_le
tff(fact_1814_sincos__total__2pi,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one)))) = one_one_real )
     => ~ ! [T2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T2),aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),pi))
             => ( ( X = aa_real_real(cos_real,T2) )
               => ( Y != aa_real_real(sin_real,T2) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_1815_sin__pi__divide__n__ge__0,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_nat_real(semiri5074537144036343181t_real,Na)))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_1816_sin__gt__zero2,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,X)) ) ) ).

% sin_gt_zero2
tff(fact_1817_sin__lt__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,pi),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),pi))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,X)),zero_zero_real) ) ) ).

% sin_lt_zero
tff(fact_1818_sin__30,axiom,
    aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(bit1(one))))) = aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one))) ).

% sin_30
tff(fact_1819_cos__double__less__one,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),numeral_numeral_real(bit0(one)))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),X))),one_one_real) ) ) ).

% cos_double_less_one
tff(fact_1820_cos__gt__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cos_real,X)) ) ) ).

% cos_gt_zero
tff(fact_1821_sin__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
           => ( ( aa_real_real(sin_real,X) = aa_real_real(sin_real,Y) )
             => ( X = Y ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_1822_sin__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_1823_sin__monotone__2pi__le,axiom,
    ! [Y: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,X)) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_1824_cos__60,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit1(one)))) = aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one))) ).

% cos_60
tff(fact_1825_sin__le__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,pi),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),pi))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,X)),zero_zero_real) ) ) ).

% sin_le_zero
tff(fact_1826_sin__less__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(bit0(one)))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,X)),zero_zero_real) ) ) ).

% sin_less_zero
tff(fact_1827_sin__monotone__2pi,axiom,
    ! [Y: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,X)) ) ) ) ).

% sin_monotone_2pi
tff(fact_1828_sin__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_1829_sin__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ? [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),X2)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
            & ( aa_real_real(sin_real,X2) = Y )
            & ! [Y3: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),Y3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
                  & ( aa_real_real(sin_real,Y3) = Y ) )
               => ( Y3 = X2 ) ) ) ) ) ).

% sin_total
tff(fact_1830_cos__gt__zero__pi,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cos_real,X)) ) ) ).

% cos_gt_zero_pi
tff(fact_1831_cos__ge__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(cos_real,X)) ) ) ).

% cos_ge_zero
tff(fact_1832_cos__one__2pi,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = one_one_real )
    <=> ( ? [X5: nat] : X = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,X5)),numeral_numeral_real(bit0(one)))),pi)
        | ? [X5: nat] : X = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,X5)),numeral_numeral_real(bit0(one)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_1833_VEBT__internal_OTb__T__vebt__buildupi_H_H,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,Na)),aa_nat_nat(minus_minus_nat(aa_nat_nat(vEBT_VEBT_Tb2,Na)),numeral_numeral_nat(bit0(one)))) ).

% VEBT_internal.Tb_T_vebt_buildupi''
tff(fact_1834_sin__pi__divide__n__gt__0,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_nat_real(semiri5074537144036343181t_real,Na)))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_1835_VEBT__internal_OT__vebt__buildupi__univ,axiom,
    ! [U: nat,Na: nat] :
      ( ( U = aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(bit1(bit0(one))))),U)) ) ).

% VEBT_internal.T_vebt_buildupi_univ
tff(fact_1836_VEBT__internal_OTb__T__vebt__buildupi,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(vEBT_V441764108873111860ildupi,Na))),minus_minus_int(aa_nat_int(vEBT_VEBT_Tb,Na),aa_num_int(numeral_numeral_int,bit0(one)))) ).

% VEBT_internal.Tb_T_vebt_buildupi
tff(fact_1837_sin__zero__lemma,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( ( aa_real_real(sin_real,X) = zero_zero_real )
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N)
            & ( X = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_1838_sin__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
    <=> ( ? [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N2)
            & ( X = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N2)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one)))) ) )
        | ? [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N2)
            & ( X = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N2)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_1839_cos__zero__lemma,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( ( aa_real_real(cos_real,X) = zero_zero_real )
       => ? [N: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N)
            & ( X = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_1840_cos__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = zero_zero_real )
    <=> ( ? [N2: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N2)
            & ( X = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N2)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one)))) ) )
        | ? [N2: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N2)
            & ( X = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N2)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_1841_VEBT__internal_OT__vebt__buildupi_Osimps_I3_J,axiom,
    ! [Na: nat] :
      aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(bit0(one)))),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one))))))))))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(bit0(bit0(one))))),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one)))))))))))) ).

% VEBT_internal.T_vebt_buildupi.simps(3)
tff(fact_1842_triangle__def,axiom,
    ! [Na: nat] : aa_nat_nat(nat_triangle,Na) = aa_nat_nat(divide_divide_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(suc,Na))),numeral_numeral_nat(bit0(one))) ).

% triangle_def
tff(fact_1843_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( real_V1022390504157884413omplex(Z) = one_one_real )
     => ~ ! [T2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T2),aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),pi))
             => ( Z != complex2(aa_real_real(cos_real,T2),aa_real_real(sin_real,T2)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_1844_summable__complex__of__real,axiom,
    ! [F: fun_nat_real] :
      ( summable_complex(aTP_Lamm_bm(F))
    <=> summable_real(F) ) ).

% summable_complex_of_real
tff(fact_1845_tan__periodic__pi,axiom,
    ! [X: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),pi)) = aa_real_real(tan_real,X) ).

% tan_periodic_pi
tff(fact_1846_triangle__0,axiom,
    aa_nat_nat(nat_triangle,zero_zero_nat) = zero_zero_nat ).

% triangle_0
tff(fact_1847_triangle__Suc,axiom,
    ! [Na: nat] : aa_nat_nat(nat_triangle,aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,Na)),aa_nat_nat(suc,Na)) ).

% triangle_Suc
tff(fact_1848_tan__npi,axiom,
    ! [Na: nat] : aa_real_real(tan_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),pi)) = zero_zero_real ).

% tan_npi
tff(fact_1849_tan__periodic__n,axiom,
    ! [X: real,Na: num] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(times_times_real(numeral_numeral_real(Na)),pi))) = aa_real_real(tan_real,X) ).

% tan_periodic_n
tff(fact_1850_tan__periodic__nat,axiom,
    ! [X: real,Na: nat] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),pi))) = aa_real_real(tan_real,X) ).

% tan_periodic_nat
tff(fact_1851_tan__periodic,axiom,
    ! [X: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),pi))) = aa_real_real(tan_real,X) ).

% tan_periodic
tff(fact_1852_Complex__add__complex__of__real,axiom,
    ! [X: real,Y: real,R: real] : plus_plus_complex(complex2(X,Y),real_V4546457046886955230omplex(R)) = complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),R),Y) ).

% Complex_add_complex_of_real
tff(fact_1853_complex__of__real__add__Complex,axiom,
    ! [R: real,X: real,Y: real] : plus_plus_complex(real_V4546457046886955230omplex(R),complex2(X,Y)) = complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R),X),Y) ).

% complex_of_real_add_Complex
tff(fact_1854_Complex__eq__numeral,axiom,
    ! [Aa: real,Ba: real,W: num] :
      ( ( complex2(Aa,Ba) = numera6690914467698888265omplex(W) )
    <=> ( ( Aa = numeral_numeral_real(W) )
        & ( Ba = zero_zero_real ) ) ) ).

% Complex_eq_numeral
tff(fact_1855_complex__add,axiom,
    ! [Aa: real,Ba: real,C: real,D: real] : plus_plus_complex(complex2(Aa,Ba),complex2(C,D)) = complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),D)) ).

% complex_add
tff(fact_1856_Complex__eq__neg__numeral,axiom,
    ! [Aa: real,Ba: real,W: num] :
      ( ( complex2(Aa,Ba) = uminus1482373934393186551omplex(numera6690914467698888265omplex(W)) )
    <=> ( ( Aa = aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) )
        & ( Ba = zero_zero_real ) ) ) ).

% Complex_eq_neg_numeral
tff(fact_1857_complex__mult,axiom,
    ! [Aa: real,Ba: real,C: real,D: real] : aa_complex_complex(times_times_complex(complex2(Aa,Ba)),complex2(C,D)) = complex2(aa_real_real(minus_minus_real(aa_real_real(times_times_real(Aa),C)),aa_real_real(times_times_real(Ba),D)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(Aa),D)),aa_real_real(times_times_real(Ba),C))) ).

% complex_mult
tff(fact_1858_tan__45,axiom,
    aa_real_real(tan_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(bit0(one))))) = one_one_real ).

% tan_45
tff(fact_1859_power__half__series,axiom,
    aa_real_o(sums_real(aTP_Lamm_bn),one_one_real) ).

% power_half_series
tff(fact_1860_tan__gt__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(tan_real,X)) ) ) ).

% tan_gt_zero
tff(fact_1861_lemma__tan__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ? [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(tan_real,X2)) ) ) ).

% lemma_tan_total
tff(fact_1862_tan__total,axiom,
    ! [Y: real] :
    ? [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),X2)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
      & ( aa_real_real(tan_real,X2) = Y )
      & ! [Y3: real] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),Y3)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
            & ( aa_real_real(tan_real,Y3) = Y ) )
         => ( Y3 = X2 ) ) ) ).

% tan_total
tff(fact_1863_tan__monotone,axiom,
    ! [Y: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,X)) ) ) ) ).

% tan_monotone
tff(fact_1864_tan__monotone_H,axiom,
    ! [Y: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),X)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,X)) ) ) ) ) ) ).

% tan_monotone'
tff(fact_1865_tan__mono__lt__eq,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_1866_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),X2)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
      & ( aa_real_real(tan_real,X2) = Y ) ) ).

% lemma_tan_total1
tff(fact_1867_tan__minus__45,axiom,
    aa_real_real(tan_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(bit0(one)))))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% tan_minus_45
tff(fact_1868_tan__inverse,axiom,
    ! [Y: real] : aa_real_real(divide_divide_real(one_one_real),aa_real_real(tan_real,Y)) = aa_real_real(tan_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one)))),Y)) ).

% tan_inverse
tff(fact_1869_sums__if_H,axiom,
    ! [G: fun_nat_real,X: real] :
      ( aa_real_o(sums_real(G),X)
     => aa_real_o(sums_real(aTP_Lamm_bo(G)),X) ) ).

% sums_if'
tff(fact_1870_sums__if,axiom,
    ! [G: fun_nat_real,X: real,F: fun_nat_real,Y: real] :
      ( aa_real_o(sums_real(G),X)
     => ( aa_real_o(sums_real(F),Y)
       => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_bp(G),F)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) ) ) ).

% sums_if
tff(fact_1871_tan__total__pos,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ? [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
          & ( aa_real_real(tan_real,X2) = Y ) ) ) ).

% tan_total_pos
tff(fact_1872_tan__pos__pi2__le,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(tan_real,X)) ) ) ).

% tan_pos_pi2_le
tff(fact_1873_tan__less__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(bit0(one)))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,X)),zero_zero_real) ) ) ).

% tan_less_zero
tff(fact_1874_tan__mono__le,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y)) ) ) ) ).

% tan_mono_le
tff(fact_1875_tan__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_1876_tan__bound__pi2,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(bit0(one)))))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(tan_real,X))),one_one_real) ) ).

% tan_bound_pi2
tff(fact_1877_arctan,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),aa_real_real(arctan,Y))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
      & ( aa_real_real(tan_real,aa_real_real(arctan,Y)) = Y ) ) ).

% arctan
tff(fact_1878_arctan__tan,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
       => ( aa_real_real(arctan,aa_real_real(tan_real,X)) = X ) ) ) ).

% arctan_tan
tff(fact_1879_arctan__unique,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
       => ( ( aa_real_real(tan_real,X) = Y )
         => ( aa_real_real(arctan,Y) = X ) ) ) ) ).

% arctan_unique
tff(fact_1880_tan__total__pi4,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ? [Z3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(bit0(one)))))),Z3)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(bit0(one)))))
          & ( aa_real_real(tan_real,Z3) = X ) ) ) ).

% tan_total_pi4
tff(fact_1881_sin__paired,axiom,
    ! [X: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_bq,X)),aa_real_real(sin_real,X)) ).

% sin_paired
tff(fact_1882_sin__tan,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
     => ( aa_real_real(sin_real,X) = aa_real_real(divide_divide_real(aa_real_real(tan_real,X)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,X)),numeral_numeral_nat(bit0(one)))))) ) ) ).

% sin_tan
tff(fact_1883_cos__tan,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
     => ( aa_real_real(cos_real,X) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,X)),numeral_numeral_nat(bit0(one)))))) ) ) ).

% cos_tan
tff(fact_1884_real__sqrt__le__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ).

% real_sqrt_le_iff
tff(fact_1885_real__sqrt__le__0__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,X)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real) ) ).

% real_sqrt_le_0_iff
tff(fact_1886_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y) ) ).

% real_sqrt_ge_0_iff
tff(fact_1887_real__sqrt__le__1__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,X)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real) ) ).

% real_sqrt_le_1_iff
tff(fact_1888_real__sqrt__ge__1__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y) ) ).

% real_sqrt_ge_1_iff
tff(fact_1889_real__sqrt__four,axiom,
    aa_real_real(sqrt,numeral_numeral_real(bit0(bit0(one)))) = numeral_numeral_real(bit0(one)) ).

% real_sqrt_four
tff(fact_1890_real__sqrt__abs,axiom,
    ! [X: real] : aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))) = aa_real_real(abs_abs_real,X) ).

% real_sqrt_abs
tff(fact_1891_real__sqrt__pow2__iff,axiom,
    ! [X: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,X)),numeral_numeral_nat(bit0(one))) = X )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X) ) ).

% real_sqrt_pow2_iff
tff(fact_1892_real__sqrt__pow2,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,X)),numeral_numeral_nat(bit0(one))) = X ) ) ).

% real_sqrt_pow2
tff(fact_1893_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X: real,Y: real,Xa2: real,Ya: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,aa_real_real(times_times_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa2),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),numeral_numeral_nat(bit0(one))))))),numeral_numeral_nat(bit0(one))) = aa_real_real(times_times_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa2),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),numeral_numeral_nat(bit0(one))))) ).

% real_sqrt_sum_squares_mult_squared_eq
tff(fact_1894_real__sqrt__le__mono,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y)) ) ).

% real_sqrt_le_mono
tff(fact_1895_real__sqrt__ge__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,X)) ) ).

% real_sqrt_ge_zero
tff(fact_1896_real__sqrt__eq__zero__cancel,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( ( aa_real_real(sqrt,X) = zero_zero_real )
       => ( X = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_1897_real__sqrt__ge__one,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(sqrt,X)) ) ).

% real_sqrt_ge_one
tff(fact_1898_real__div__sqrt,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_real(divide_divide_real(X),aa_real_real(sqrt,X)) = aa_real_real(sqrt,X) ) ) ).

% real_div_sqrt
tff(fact_1899_sqrt__add__le__add__sqrt,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_1900_le__real__sqrt__sumsq,axiom,
    ! [X: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(X),X)),aa_real_real(times_times_real(Y),Y)))) ).

% le_real_sqrt_sumsq
tff(fact_1901_sqrt2__less__2,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,numeral_numeral_real(bit0(one)))),numeral_numeral_real(bit0(one))) ).

% sqrt2_less_2
tff(fact_1902_arsinh__real__def,axiom,
    ! [X: real] : aa_real_real(arsinh_real,X) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))),one_one_real)))) ).

% arsinh_real_def
tff(fact_1903_real__less__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(sqrt,Y)) ) ).

% real_less_rsqrt
tff(fact_1904_real__le__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(sqrt,Y)) ) ).

% real_le_rsqrt
tff(fact_1905_sqrt__le__D,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,X)),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one)))) ) ).

% sqrt_le_D
tff(fact_1906_tan__60,axiom,
    aa_real_real(tan_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit1(one)))) = aa_real_real(sqrt,numeral_numeral_real(bit1(one))) ).

% tan_60
tff(fact_1907_square__fact__le__2__fact,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_nat_real(semiri2265585572941072030t_real,Na)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Na))) ).

% square_fact_le_2_fact
tff(fact_1908_real__sqrt__unique,axiom,
    ! [Y: real,X: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one))) = X )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_real(sqrt,X) = Y ) ) ) ).

% real_sqrt_unique
tff(fact_1909_real__le__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,X)),Y) ) ) ) ).

% real_le_lsqrt
tff(fact_1910_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),U)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(U),aa_real_real(sqrt,numeral_numeral_real(bit0(one))))),U) ) ).

% lemma_real_divide_sqrt_less
tff(fact_1911_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one))))) = Y )
     => ( X = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
tff(fact_1912_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one))))) = X )
     => ( Y = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
tff(fact_1913_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [Aa: real,C: real,Ba: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C)),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),D)),numeral_numeral_nat(bit0(one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),numeral_numeral_nat(bit0(one)))))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,C),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),numeral_numeral_nat(bit0(one))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_1914_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one)))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_1915_real__sqrt__sum__squares__ge1,axiom,
    ! [X: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one)))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_1916_sqrt__ge__absD,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(sqrt,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))),Y) ) ).

% sqrt_ge_absD
tff(fact_1917_cos__45,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(bit0(one))))) = aa_real_real(divide_divide_real(aa_real_real(sqrt,numeral_numeral_real(bit0(one)))),numeral_numeral_real(bit0(one))) ).

% cos_45
tff(fact_1918_sin__45,axiom,
    aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(bit0(one))))) = aa_real_real(divide_divide_real(aa_real_real(sqrt,numeral_numeral_real(bit0(one)))),numeral_numeral_real(bit0(one))) ).

% sin_45
tff(fact_1919_real__less__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,X)),Y) ) ) ) ).

% real_less_lsqrt
tff(fact_1920_sqrt__sum__squares__le__sum,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_1921_sqrt__even__pow2,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na)
     => ( aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),Na)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(bit0(one))),aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one)))) ) ) ).

% sqrt_even_pow2
tff(fact_1922_tan__30,axiom,
    aa_real_real(tan_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(bit1(one))))) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,numeral_numeral_real(bit1(one)))) ).

% tan_30
tff(fact_1923_real__sqrt__ge__abs1,axiom,
    ! [X: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one)))))) ).

% real_sqrt_ge_abs1
tff(fact_1924_real__sqrt__ge__abs2,axiom,
    ! [Y: real,X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one)))))) ).

% real_sqrt_ge_abs2
tff(fact_1925_sqrt__sum__squares__le__sum__abs,axiom,
    ! [X: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,X)),aa_real_real(abs_abs_real,Y))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_1926_ln__sqrt,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(ln_ln_real,aa_real_real(sqrt,X)) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,X)),numeral_numeral_real(bit0(one))) ) ) ).

% ln_sqrt
tff(fact_1927_cos__30,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(bit1(one))))) = aa_real_real(divide_divide_real(aa_real_real(sqrt,numeral_numeral_real(bit1(one)))),numeral_numeral_real(bit0(one))) ).

% cos_30
tff(fact_1928_sin__60,axiom,
    aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit1(one)))) = aa_real_real(divide_divide_real(aa_real_real(sqrt,numeral_numeral_real(bit1(one)))),numeral_numeral_real(bit0(one))) ).

% sin_60
tff(fact_1929_complex__norm,axiom,
    ! [X: real,Y: real] : real_V1022390504157884413omplex(complex2(X,Y)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one))))) ).

% complex_norm
tff(fact_1930_real__sqrt__power__even,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,X)),Na) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_1931_arsinh__real__aux,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))),one_one_real)))) ).

% arsinh_real_aux
tff(fact_1932_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X: real,Y: real,Xa2: real,Ya: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,aa_real_real(times_times_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa2),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),numeral_numeral_nat(bit0(one))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_1933_arith__geo__mean__sqrt,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(times_times_real(X),Y))),aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),numeral_numeral_real(bit0(one)))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_1934_powr__half__sqrt,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_real(powr_real(X),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one)))) = aa_real_real(sqrt,X) ) ) ).

% powr_half_sqrt
tff(fact_1935_cos__x__y__le__one,axiom,
    ! [X: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(X),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one)))))))),one_one_real) ).

% cos_x_y_le_one
tff(fact_1936_real__sqrt__sum__squares__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(divide_divide_real(U),aa_real_real(sqrt,numeral_numeral_real(bit0(one)))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Y)),aa_real_real(divide_divide_real(U),aa_real_real(sqrt,numeral_numeral_real(bit0(one)))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one)))))),U) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_1937_arcosh__real__def,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X)
     => ( aa_real_real(arcosh_real,X) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(sqrt,aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))),one_one_real)))) ) ) ).

% arcosh_real_def
tff(fact_1938_cos__arctan,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(arctan,X)) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))))) ).

% cos_arctan
tff(fact_1939_sin__arctan,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(arctan,X)) = aa_real_real(divide_divide_real(X),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))))) ).

% sin_arctan
tff(fact_1940_sqrt__sum__squares__half__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(U),numeral_numeral_real(bit0(one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(divide_divide_real(U),numeral_numeral_real(bit0(one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one)))))),U) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_1941_sin__cos__sqrt,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,X))
     => ( aa_real_real(sin_real,X) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,X)),numeral_numeral_nat(bit0(one))))) ) ) ).

% sin_cos_sqrt
tff(fact_1942_arctan__half,axiom,
    ! [X: real] : aa_real_real(arctan,X) = aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),aa_real_real(arctan,aa_real_real(divide_divide_real(X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one))))))))) ).

% arctan_half
tff(fact_1943_cos__paired,axiom,
    ! [X: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_br,X)),aa_real_real(cos_real,X)) ).

% cos_paired
tff(fact_1944_sin__coeff__def,axiom,
    ! [X4: nat] :
      aa_nat_real(sin_coeff,X4) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),X4),zero_zero_real,aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(divide_divide_nat(aa_nat_nat(minus_minus_nat(X4),aa_nat_nat(suc,zero_zero_nat))),numeral_numeral_nat(bit0(one))))),aa_nat_real(semiri2265585572941072030t_real,X4))) ).

% sin_coeff_def
tff(fact_1945_cos__coeff__def,axiom,
    ! [X4: nat] :
      aa_nat_real(cos_coeff,X4) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),X4),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(divide_divide_nat(X4),numeral_numeral_nat(bit0(one))))),aa_nat_real(semiri2265585572941072030t_real,X4)),zero_zero_real) ).

% cos_coeff_def
tff(fact_1946_cos__arcsin,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arcsin,X)) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one))))) ) ) ) ).

% cos_arcsin
tff(fact_1947_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
     => ( aa_real_real(sin_real,aa_real_real(arccos,Y)) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),numeral_numeral_nat(bit0(one))))) ) ) ).

% sin_arccos_abs
tff(fact_1948_sin__arccos,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arccos,X)) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one))))) ) ) ) ).

% sin_arccos
tff(fact_1949_cos__arccos,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ) ).

% cos_arccos
tff(fact_1950_sin__arcsin,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ).

% sin_arcsin
tff(fact_1951_arccos__0,axiom,
    aa_real_real(arccos,zero_zero_real) = aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))) ).

% arccos_0
tff(fact_1952_arcsin__1,axiom,
    aa_real_real(arcsin,one_one_real) = aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))) ).

% arcsin_1
tff(fact_1953_arcsin__minus__1,axiom,
    aa_real_real(arcsin,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one)))) ).

% arcsin_minus_1
tff(fact_1954_fact__mono__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ) ).

% fact_mono_nat
tff(fact_1955_fact__ge__self,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ).

% fact_ge_self
tff(fact_1956_sin__coeff__Suc,axiom,
    ! [Na: nat] : aa_nat_real(sin_coeff,aa_nat_nat(suc,Na)) = aa_real_real(divide_divide_real(aa_nat_real(cos_coeff,Na)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na))) ).

% sin_coeff_Suc
tff(fact_1957_fact__ge__Suc__0__nat,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ).

% fact_ge_Suc_0_nat
tff(fact_1958_dvd__fact,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ) ) ).

% dvd_fact
tff(fact_1959_cos__coeff__Suc,axiom,
    ! [Na: nat] : aa_nat_real(cos_coeff,aa_nat_nat(suc,Na)) = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,aa_nat_real(sin_coeff,Na))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na))) ).

% cos_coeff_Suc
tff(fact_1960_arccos__le__arccos,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),aa_real_real(arccos,X)) ) ) ) ).

% arccos_le_arccos
tff(fact_1961_arccos__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real) )
     => ( ( aa_real_real(arccos,X) = aa_real_real(arccos,Y) )
      <=> ( X = Y ) ) ) ).

% arccos_eq_iff
tff(fact_1962_arccos__le__mono,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,X)),aa_real_real(arccos,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),X) ) ) ) ).

% arccos_le_mono
tff(fact_1963_arcsin__minus,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
       => ( aa_real_real(arcsin,aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(arcsin,X)) ) ) ) ).

% arcsin_minus
tff(fact_1964_arcsin__le__arcsin,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y)) ) ) ) ).

% arcsin_le_arcsin
tff(fact_1965_arcsin__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( ( aa_real_real(arcsin,X) = aa_real_real(arcsin,Y) )
        <=> ( X = Y ) ) ) ) ).

% arcsin_eq_iff
tff(fact_1966_arcsin__le__mono,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ) ) ).

% arcsin_le_mono
tff(fact_1967_fact__diff__Suc,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,M))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(minus_minus_nat(aa_nat_nat(suc,M)),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(minus_minus_nat(aa_nat_nat(suc,M)),Na)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(minus_minus_nat(M),Na))) ) ) ).

% fact_diff_Suc
tff(fact_1968_fact__div__fact__le__pow,axiom,
    ! [R: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(divide_divide_nat(aa_nat_nat(semiri1408675320244567234ct_nat,Na)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(minus_minus_nat(Na),R)))),aa_nat_nat(power_power_nat(Na),R)) ) ).

% fact_div_fact_le_pow
tff(fact_1969_binomial__fact__lemma,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(minus_minus_nat(Na),K)))),aa_nat_nat(binomial(Na),K)) = aa_nat_nat(semiri1408675320244567234ct_nat,Na) ) ) ).

% binomial_fact_lemma
tff(fact_1970_arccos__lbound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y)) ) ) ).

% arccos_lbound
tff(fact_1971_arccos__less__arccos,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Y)),aa_real_real(arccos,X)) ) ) ) ).

% arccos_less_arccos
tff(fact_1972_arccos__less__mono,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,X)),aa_real_real(arccos,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X) ) ) ) ).

% arccos_less_mono
tff(fact_1973_arccos__ubound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),pi) ) ) ).

% arccos_ubound
tff(fact_1974_arccos__cos,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),pi)
       => ( aa_real_real(arccos,aa_real_real(cos_real,X)) = X ) ) ) ).

% arccos_cos
tff(fact_1975_arcsin__less__arcsin,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y)) ) ) ) ).

% arcsin_less_arcsin
tff(fact_1976_arcsin__less__mono,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ) ) ).

% arcsin_less_mono
tff(fact_1977_cos__arccos__abs,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
     => ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ).

% cos_arccos_abs
tff(fact_1978_binomial__altdef__nat,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_nat(binomial(Na),K) = aa_nat_nat(divide_divide_nat(aa_nat_nat(semiri1408675320244567234ct_nat,Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(minus_minus_nat(Na),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_1979_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Theta)),pi)
     => ( aa_real_real(arccos,aa_real_real(cos_real,Theta)) = aa_real_real(abs_abs_real,Theta) ) ) ).

% arccos_cos_eq_abs
tff(fact_1980_arccos__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),pi) ) ) ) ).

% arccos_bounded
tff(fact_1981_arccos__cos2,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),X)
       => ( aa_real_real(arccos,aa_real_real(cos_real,X)) = aa_real_real(uminus_uminus_real,X) ) ) ) ).

% arccos_cos2
tff(fact_1982_arccos__minus,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
       => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,X)) = aa_real_real(minus_minus_real(pi),aa_real_real(arccos,X)) ) ) ) ).

% arccos_minus
tff(fact_1983_arccos,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),pi)
          & ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ) ) ).

% arccos
tff(fact_1984_arccos__minus__abs,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,X)) = aa_real_real(minus_minus_real(pi),aa_real_real(arccos,X)) ) ) ).

% arccos_minus_abs
tff(fact_1985_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one)))) ) ) ).

% arccos_le_pi2
tff(fact_1986_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Y)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one)))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_1987_arcsin__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one)))) ) ) ) ).

% arcsin_bounded
tff(fact_1988_arcsin__ubound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one)))) ) ) ).

% arcsin_ubound
tff(fact_1989_arcsin__lbound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),aa_real_real(arcsin,Y)) ) ) ).

% arcsin_lbound
tff(fact_1990_arcsin__sin,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
       => ( aa_real_real(arcsin,aa_real_real(sin_real,X)) = X ) ) ) ).

% arcsin_sin
tff(fact_1991_binomial__code,axiom,
    ! [Na: nat,K: nat] :
      aa_nat_nat(binomial(Na),K) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K),
        zero_zero_nat,
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),K)),aa_nat_nat(binomial(Na),aa_nat_nat(minus_minus_nat(Na),K)),aa_nat_nat(divide_divide_nat(aa_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(minus_minus_nat(Na),K)),one_one_nat),Na),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K))) ) ).

% binomial_code
tff(fact_1992_arcsin,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ) ).

% arcsin
tff(fact_1993_arcsin__pi,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),pi)
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ) ).

% arcsin_pi
tff(fact_1994_arcsin__le__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(bit0(one)))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,X)),Y)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(sin_real,Y)) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_1995_le__arcsin__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(bit0(one)))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(arcsin,X))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Y)),X) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_1996_cos__npi__int,axiom,
    ! [Na: int] :
      aa_real_real(cos_real,aa_real_real(times_times_real(pi),ring_1_of_int_real(Na))) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),Na),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ).

% cos_npi_int
tff(fact_1997_cot__less__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(bit0(one)))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cot_real,X)),zero_zero_real) ) ) ).

% cot_less_zero
tff(fact_1998_or__int__unfold,axiom,
    ! [K: int,L: int] :
      bit_se1409905431419307370or_int(K,L) = $ite(
        ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
        | ( L = aa_int_int(uminus_uminus_int,one_one_int) ) ),
        aa_int_int(uminus_uminus_int,one_one_int),
        $ite(
          K = zero_zero_int,
          L,
          $ite(L = zero_zero_int,K,plus_plus_int(aa_int_int(aa_int_fun_int_int(ord_max_int,modulo_modulo_int(K,aa_num_int(numeral_numeral_int,bit0(one)))),modulo_modulo_int(L,aa_num_int(numeral_numeral_int,bit0(one)))),times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),bit_se1409905431419307370or_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,bit0(one))))))) ) ) ).

% or_int_unfold
tff(fact_1999_tan__periodic__int,axiom,
    ! [X: real,I: int] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(times_times_real(ring_1_of_int_real(I)),pi))) = aa_real_real(tan_real,X) ).

% tan_periodic_int
tff(fact_2000_cot__npi,axiom,
    ! [Na: nat] : aa_real_real(cot_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),pi)) = zero_zero_real ).

% cot_npi
tff(fact_2001_cot__periodic,axiom,
    ! [X: real] : aa_real_real(cot_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),pi))) = aa_real_real(cot_real,X) ).

% cot_periodic
tff(fact_2002_sin__int__2pin,axiom,
    ! [Na: int] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),pi)),ring_1_of_int_real(Na))) = zero_zero_real ).

% sin_int_2pin
tff(fact_2003_cos__int__2pin,axiom,
    ! [Na: int] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),pi)),ring_1_of_int_real(Na))) = one_one_real ).

% cos_int_2pin
tff(fact_2004_real__of__int__div4,axiom,
    ! [Na: int,X: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,ring_1_of_int_real(aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),X))),aa_real_real(divide_divide_real(ring_1_of_int_real(Na)),ring_1_of_int_real(X))) ).

% real_of_int_div4
tff(fact_2005_real__of__int__div,axiom,
    ! [D: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),Na)
     => ( ring_1_of_int_real(aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),D)) = aa_real_real(divide_divide_real(ring_1_of_int_real(Na)),ring_1_of_int_real(D)) ) ) ).

% real_of_int_div
tff(fact_2006_int__le__real__less,axiom,
    ! [Na: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Na),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,ring_1_of_int_real(Na)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,ring_1_of_int_real(M)),one_one_real)) ) ).

% int_le_real_less
tff(fact_2007_int__less__real__le,axiom,
    ! [Na: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Na),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,ring_1_of_int_real(Na)),one_one_real)),ring_1_of_int_real(M)) ) ).

% int_less_real_le
tff(fact_2008_real__of__int__div__aux,axiom,
    ! [X: int,D: int] : aa_real_real(divide_divide_real(ring_1_of_int_real(X)),ring_1_of_int_real(D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,ring_1_of_int_real(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),D))),aa_real_real(divide_divide_real(ring_1_of_int_real(modulo_modulo_int(X,D))),ring_1_of_int_real(D))) ).

% real_of_int_div_aux
tff(fact_2009_real__of__int__floor__add__one__gt,axiom,
    ! [R: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,R),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,ring_1_of_int_real(archim6058952711729229775r_real(R))),one_one_real)) ).

% real_of_int_floor_add_one_gt
tff(fact_2010_floor__eq,axiom,
    ! [Na: int,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,ring_1_of_int_real(Na)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,ring_1_of_int_real(Na)),one_one_real))
       => ( archim6058952711729229775r_real(X) = Na ) ) ) ).

% floor_eq
tff(fact_2011_real__of__int__floor__add__one__ge,axiom,
    ! [R: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,ring_1_of_int_real(archim6058952711729229775r_real(R))),one_one_real)) ).

% real_of_int_floor_add_one_ge
tff(fact_2012_real__of__int__floor__gt__diff__one,axiom,
    ! [R: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(R),one_one_real)),ring_1_of_int_real(archim6058952711729229775r_real(R))) ).

% real_of_int_floor_gt_diff_one
tff(fact_2013_real__of__int__floor__ge__diff__one,axiom,
    ! [R: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(R),one_one_real)),ring_1_of_int_real(archim6058952711729229775r_real(R))) ).

% real_of_int_floor_ge_diff_one
tff(fact_2014_real__of__int__div2,axiom,
    ! [Na: int,X: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(ring_1_of_int_real(Na)),ring_1_of_int_real(X))),ring_1_of_int_real(aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),X)))) ).

% real_of_int_div2
tff(fact_2015_real__of__int__div3,axiom,
    ! [Na: int,X: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(ring_1_of_int_real(Na)),ring_1_of_int_real(X))),ring_1_of_int_real(aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),X)))),one_one_real) ).

% real_of_int_div3
tff(fact_2016_floor__eq2,axiom,
    ! [Na: int,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,ring_1_of_int_real(Na)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,ring_1_of_int_real(Na)),one_one_real))
       => ( archim6058952711729229775r_real(X) = Na ) ) ) ).

% floor_eq2
tff(fact_2017_floor__divide__real__eq__div,axiom,
    ! [Ba: int,Aa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ba)
     => ( archim6058952711729229775r_real(aa_real_real(divide_divide_real(Aa),ring_1_of_int_real(Ba))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,archim6058952711729229775r_real(Aa)),Ba) ) ) ).

% floor_divide_real_eq_div
tff(fact_2018_cos__one__2pi__int,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = one_one_real )
    <=> ? [X5: int] : X = aa_real_real(times_times_real(aa_real_real(times_times_real(ring_1_of_int_real(X5)),numeral_numeral_real(bit0(one)))),pi) ) ).

% cos_one_2pi_int
tff(fact_2019_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K4: int] : aa_real_real(arccos,aa_real_real(cos_real,Theta)) != aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Theta),aa_real_real(times_times_real(ring_1_of_int_real(K4)),aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),pi)))) ).

% arccos_cos_eq_abs_2pi
tff(fact_2020_floor__log__eq__powr__iff,axiom,
    ! [X: real,Ba: real,K: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ba)
       => ( ( archim6058952711729229775r_real(aa_real_real(log(Ba),X)) = K )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(Ba),ring_1_of_int_real(K))),X)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(powr_real(Ba),ring_1_of_int_real(plus_plus_int(K,one_one_int)))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_2021_cot__gt__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cot_real,X)) ) ) ).

% cot_gt_zero
tff(fact_2022_tan__cot_H,axiom,
    ! [X: real] : aa_real_real(tan_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one)))),X)) = aa_real_real(cot_real,X) ).

% tan_cot'
tff(fact_2023_cos__zero__iff__int,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = zero_zero_real )
    <=> ? [I4: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),I4)
          & ( X = aa_real_real(times_times_real(ring_1_of_int_real(I4)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_2024_sin__zero__iff__int,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
    <=> ? [I4: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),I4)
          & ( X = aa_real_real(times_times_real(ring_1_of_int_real(I4)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_2025_powr__int,axiom,
    ! [X: real,I: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(powr_real(X),ring_1_of_int_real(I)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_int_nat(nat2,I)),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,I))))) ) ) ).

% powr_int
tff(fact_2026_VEBT__internal_Olog__ceil__idem,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X)
     => ( archim7802044766580827645g_real(aa_real_real(log(numeral_numeral_real(bit0(one))),X)) = archim7802044766580827645g_real(aa_real_real(log(numeral_numeral_real(bit0(one))),ring_1_of_int_real(archim7802044766580827645g_real(X)))) ) ) ).

% VEBT_internal.log_ceil_idem
tff(fact_2027_max__enat__simps_I2_J,axiom,
    ! [Q3: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Q3),zero_z5237406670263579293d_enat) = Q3 ).

% max_enat_simps(2)
tff(fact_2028_max__enat__simps_I3_J,axiom,
    ! [Q3: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,zero_z5237406670263579293d_enat),Q3) = Q3 ).

% max_enat_simps(3)
tff(fact_2029_max__Suc__Suc,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),Na)) ).

% max_Suc_Suc
tff(fact_2030_max__0R,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Na),zero_zero_nat) = Na ).

% max_0R
tff(fact_2031_max__0L,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),Na) = Na ).

% max_0L
tff(fact_2032_max__nat_Oright__neutral,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),zero_zero_nat) = Aa ).

% max_nat.right_neutral
tff(fact_2033_max__nat_Oneutr__eq__iff,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),Ba) )
    <=> ( ( Aa = zero_zero_nat )
        & ( Ba = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_2034_max__nat_Oleft__neutral,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),Aa) = Aa ).

% max_nat.left_neutral
tff(fact_2035_max__nat_Oeq__neutr__iff,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),Ba) = zero_zero_nat )
    <=> ( ( Aa = zero_zero_nat )
        & ( Ba = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_2036_max__Suc__numeral,axiom,
    ! [Na: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Na)),numeral_numeral_nat(K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Na),pred_numeral(K))) ).

% max_Suc_numeral
tff(fact_2037_max__numeral__Suc,axiom,
    ! [K: num,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,numeral_numeral_nat(K)),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,pred_numeral(K)),Na)) ).

% max_numeral_Suc
tff(fact_2038_nat__mult__max__left,axiom,
    ! [M: nat,Na: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),Na)),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q3)) ).

% nat_mult_max_left
tff(fact_2039_nat__mult__max__right,axiom,
    ! [M: nat,Na: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Na),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)) ).

% nat_mult_max_right
tff(fact_2040_nat__add__max__left,axiom,
    ! [M: nat,Na: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),Na)),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),Q3)) ).

% nat_add_max_left
tff(fact_2041_nat__add__max__right,axiom,
    ! [M: nat,Na: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Na),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q3)) ).

% nat_add_max_right
tff(fact_2042_nat__minus__add__max,axiom,
    ! [Na: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(minus_minus_nat(Na),M)),M) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Na),M) ).

% nat_minus_add_max
tff(fact_2043_or__nat__unfold,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(bit_se1412395901928357646or_nat(M),Na) = $ite(
        M = zero_zero_nat,
        Na,
        $ite(Na = zero_zero_nat,M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(modulo_modulo_nat(M),numeral_numeral_nat(bit0(one)))),aa_nat_nat(modulo_modulo_nat(Na),numeral_numeral_nat(bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(bit_se1412395901928357646or_nat(aa_nat_nat(divide_divide_nat(M),numeral_numeral_nat(bit0(one)))),aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one))))))) ) ).

% or_nat_unfold
tff(fact_2044_log__base__10__eq1,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(log(numeral_numeral_real(bit0(bit1(bit0(one))))),X) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,numeral_numeral_real(bit0(bit1(bit0(one))))))),aa_real_real(ln_ln_real,X)) ) ) ).

% log_base_10_eq1
tff(fact_2045_exp__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ).

% exp_le_cancel_iff
tff(fact_2046_one__le__exp__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(exp_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X) ) ).

% one_le_exp_iff
tff(fact_2047_exp__le__one__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,X)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real) ) ).

% exp_le_one_iff
tff(fact_2048_exp__ge__zero,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(exp_real,X)) ).

% exp_ge_zero
tff(fact_2049_not__exp__le__zero,axiom,
    ! [X: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,X)),zero_zero_real) ).

% not_exp_le_zero
tff(fact_2050_exp__ge__add__one__self,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),aa_real_real(exp_real,X)) ).

% exp_ge_add_one_self
tff(fact_2051_log__ln,axiom,
    ! [X: real] : aa_real_real(ln_ln_real,X) = aa_real_real(log(aa_real_real(exp_real,one_one_real)),X) ).

% log_ln
tff(fact_2052_exp__ge__add__one__self__aux,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),aa_real_real(exp_real,X)) ) ).

% exp_ge_add_one_self_aux
tff(fact_2053_lemma__exp__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y)
     => ? [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X2)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),aa_real_real(minus_minus_real(Y),one_one_real))
          & ( aa_real_real(exp_real,X2) = Y ) ) ) ).

% lemma_exp_total
tff(fact_2054_ln__ge__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(ln_ln_real,X))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Y)),X) ) ) ).

% ln_ge_iff
tff(fact_2055_ln__x__over__x__mono,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,Y)),Y)),aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,X)),X)) ) ) ).

% ln_x_over_x_mono
tff(fact_2056_exp__le,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),numeral_numeral_real(bit1(one))) ).

% exp_le
tff(fact_2057_exp__half__le2,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one))))),numeral_numeral_real(bit0(one))) ).

% exp_half_le2
tff(fact_2058_exp__bound,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,X)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one))))) ) ) ).

% exp_bound
tff(fact_2059_real__exp__bound__lemma,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,X)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),X))) ) ) ).

% real_exp_bound_lemma
tff(fact_2060_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [Na: nat,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Na))),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(divide_divide_real(X),aa_nat_real(semiri5074537144036343181t_real,Na)))),Na)),aa_real_real(exp_real,X)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_2061_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(one_one_real),aa_real_real(divide_divide_real(X),aa_nat_real(semiri5074537144036343181t_real,Na)))),Na)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_2062_exp__lower__Taylor__quadratic,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))),numeral_numeral_real(bit0(one))))),aa_real_real(exp_real,X)) ) ).

% exp_lower_Taylor_quadratic
tff(fact_2063_log__base__10__eq2,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(log(numeral_numeral_real(bit0(bit1(bit0(one))))),X) = aa_real_real(times_times_real(aa_real_real(log(numeral_numeral_real(bit0(bit1(bit0(one))))),aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,X)) ) ) ).

% log_base_10_eq2
tff(fact_2064_tanh__real__altdef,axiom,
    ! [X: real] : aa_real_real(tanh_real,X) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(one_one_real),aa_real_real(exp_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(bit0(one)))),X)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(exp_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(bit0(one)))),X)))) ).

% tanh_real_altdef
tff(fact_2065_set__decode__0,axiom,
    ! [X: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),nat_set_decode(X))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),X) ) ).

% set_decode_0
tff(fact_2066_nat__of__bool,axiom,
    ! [P: $o] : aa_int_nat(nat2,aa_o_int(zero_n2684676970156552555ol_int,(P))) = zero_n2687167440665602831ol_nat((P)) ).

% nat_of_bool
tff(fact_2067_set__decode__zero,axiom,
    nat_set_decode(zero_zero_nat) = bot_bot_set_nat ).

% set_decode_zero
tff(fact_2068_set__decode__Suc,axiom,
    ! [Na: nat,X: nat] :
      ( aa_set_nat_o(member_nat(aa_nat_nat(suc,Na)),nat_set_decode(X))
    <=> aa_set_nat_o(member_nat(Na),nat_set_decode(aa_nat_nat(divide_divide_nat(X),numeral_numeral_nat(bit0(one))))) ) ).

% set_decode_Suc
tff(fact_2069_subset__decode__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(nat_set_decode(M)),nat_set_decode(Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% subset_decode_imp_le
tff(fact_2070_Suc__0__or__eq,axiom,
    ! [Na: nat] : aa_nat_nat(bit_se1412395901928357646or_nat(aa_nat_nat(suc,zero_zero_nat)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na))) ).

% Suc_0_or_eq
tff(fact_2071_or__Suc__0__eq,axiom,
    ! [Na: nat] : aa_nat_nat(bit_se1412395901928357646or_nat(Na),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na))) ).

% or_Suc_0_eq
tff(fact_2072_or__nat__rec,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(bit_se1412395901928357646or_nat(M),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          zero_n2687167440665602831ol_nat(( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),M)
            | ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(bit_se1412395901928357646or_nat(aa_nat_nat(divide_divide_nat(M),numeral_numeral_nat(bit0(one)))),aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one)))))) ).

% or_nat_rec
tff(fact_2073_xor__nat__rec,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(bit_se6528837805403552850or_nat(M),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),M) != ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(bit_se6528837805403552850or_nat(aa_nat_nat(divide_divide_nat(M),numeral_numeral_nat(bit0(one)))),aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one)))))) ).

% xor_nat_rec
tff(fact_2074_and__nat__rec,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(bit_se727722235901077358nd_nat(M),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          zero_n2687167440665602831ol_nat(( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),M)
            & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(bit_se727722235901077358nd_nat(aa_nat_nat(divide_divide_nat(M),numeral_numeral_nat(bit0(one)))),aa_nat_nat(divide_divide_nat(Na),numeral_numeral_nat(bit0(one)))))) ).

% and_nat_rec
tff(fact_2075_and__int__rec,axiom,
    ! [K: int,L: int] :
      bit_se725231765392027082nd_int(K,L) = plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),K)
          & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),L) )),times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),bit_se725231765392027082nd_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,bit0(one)))))) ).

% and_int_rec
tff(fact_2076_not__int__rec,axiom,
    ! [K: int] : bit_ri7919022796975470100ot_int(K) = plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),K)),times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),bit_ri7919022796975470100ot_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one)))))) ).

% not_int_rec
tff(fact_2077_or__int__rec,axiom,
    ! [K: int,L: int] :
      bit_se1409905431419307370or_int(K,L) = plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),K)
          | ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),L) )),times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),bit_se1409905431419307370or_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,bit0(one)))))) ).

% or_int_rec
tff(fact_2078_xor__int__rec,axiom,
    ! [K: int,L: int] :
      bit_se6526347334894502574or_int(K,L) = plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
          ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),K) != ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),L)),times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),bit_se6526347334894502574or_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,bit0(one)))))) ).

% xor_int_rec
tff(fact_2079_Suc__0__xor__eq,axiom,
    ! [Na: nat] : aa_nat_nat(bit_se6528837805403552850or_nat(aa_nat_nat(suc,zero_zero_nat)),Na) = aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na)))),zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na))) ).

% Suc_0_xor_eq
tff(fact_2080_xor__Suc__0__eq,axiom,
    ! [Na: nat] : aa_nat_nat(bit_se6528837805403552850or_nat(Na),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na)))),zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na))) ).

% xor_Suc_0_eq
tff(fact_2081_set__decode__def,axiom,
    ! [X: nat] : nat_set_decode(X) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bs,X)) ).

% set_decode_def
tff(fact_2082_and__int_Oelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( bit_se725231765392027082nd_int(X,Xa2) = Y )
     => ( Y = $ite(
            ( member_int(X,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & member_int(Xa2,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
            aa_int_int(uminus_uminus_int,
              aa_o_int(zero_n2684676970156552555ol_int,
                ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),X)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),Xa2) ))),
            plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
                ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),X)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),Xa2) )),times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),bit_se725231765392027082nd_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa2),aa_num_int(numeral_numeral_int,bit0(one)))))) ) ) ) ).

% and_int.elims
tff(fact_2083_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      bit_se725231765392027082nd_int(K,L) = $ite(
        ( member_int(K,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
        & member_int(L,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
        aa_int_int(uminus_uminus_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),K)
            & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),L) ))),
        plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),K)
            & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),L) )),times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),bit_se725231765392027082nd_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,bit0(one)))))) ) ).

% and_int.simps
tff(fact_2084_pred__numeral__inc,axiom,
    ! [K: num] : pred_numeral(inc(K)) = numeral_numeral_nat(K) ).

% pred_numeral_inc
tff(fact_2085_take__bit__of__Suc__0,axiom,
    ! [Na: nat] : aa_nat_nat(bit_se2925701944663578781it_nat(Na),aa_nat_nat(suc,zero_zero_nat)) = zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% take_bit_of_Suc_0
tff(fact_2086_take__bit__minus,axiom,
    ! [Na: nat,K: int] : bit_se2923211474154528505it_int(Na,aa_int_int(uminus_uminus_int,bit_se2923211474154528505it_int(Na,K))) = bit_se2923211474154528505it_int(Na,aa_int_int(uminus_uminus_int,K)) ).

% take_bit_minus
tff(fact_2087_take__bit__mult,axiom,
    ! [Na: nat,K: int,L: int] : bit_se2923211474154528505it_int(Na,times_times_int(bit_se2923211474154528505it_int(Na,K),bit_se2923211474154528505it_int(Na,L))) = bit_se2923211474154528505it_int(Na,times_times_int(K,L)) ).

% take_bit_mult
tff(fact_2088_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,Na: nat,Q3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(bit_se2925701944663578781it_nat(M),Q3)),aa_nat_nat(bit_se2925701944663578781it_nat(Na),Q3)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_2089_take__bit__nat__less__eq__self,axiom,
    ! [Na: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(bit_se2925701944663578781it_nat(Na),M)),M) ).

% take_bit_nat_less_eq_self
tff(fact_2090_take__bit__diff,axiom,
    ! [Na: nat,K: int,L: int] : bit_se2923211474154528505it_int(Na,minus_minus_int(bit_se2923211474154528505it_int(Na,K),bit_se2923211474154528505it_int(Na,L))) = bit_se2923211474154528505it_int(Na,minus_minus_int(K,L)) ).

% take_bit_diff
tff(fact_2091_num__induct,axiom,
    ! [P: fun_num_o,X: num] :
      ( aa_num_o(P,one)
     => ( ! [X2: num] :
            ( aa_num_o(P,X2)
           => aa_num_o(P,inc(X2)) )
       => aa_num_o(P,X) ) ) ).

% num_induct
tff(fact_2092_nat__take__bit__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_nat(nat2,bit_se2923211474154528505it_int(Na,K)) = aa_nat_nat(bit_se2925701944663578781it_nat(Na),aa_int_nat(nat2,K)) ) ) ).

% nat_take_bit_eq
tff(fact_2093_take__bit__nat__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_nat_nat(bit_se2925701944663578781it_nat(Na),aa_int_nat(nat2,K)) = aa_int_nat(nat2,bit_se2923211474154528505it_int(Na,K)) ) ) ).

% take_bit_nat_eq
tff(fact_2094_add__inc,axiom,
    ! [X: num,Y: num] : plus_plus_num(X,inc(Y)) = inc(plus_plus_num(X,Y)) ).

% add_inc
tff(fact_2095_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,Na: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,bit_se2923211474154528505it_int(M,K)),bit_se2923211474154528505it_int(Na,K)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_2096_take__bit__int__less__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,bit_se2923211474154528505it_int(Na,K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_2097_take__bit__nonnegative,axiom,
    ! [Na: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),bit_se2923211474154528505it_int(Na,K)) ).

% take_bit_nonnegative
tff(fact_2098_take__bit__int__greater__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),bit_se2923211474154528505it_int(Na,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% take_bit_int_greater_self_iff
tff(fact_2099_not__take__bit__negative,axiom,
    ! [Na: nat,K: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,bit_se2923211474154528505it_int(Na,K)),zero_zero_int) ).

% not_take_bit_negative
tff(fact_2100_inc_Osimps_I1_J,axiom,
    inc(one) = bit0(one) ).

% inc.simps(1)
tff(fact_2101_inc_Osimps_I2_J,axiom,
    ! [X: num] : inc(bit0(X)) = bit1(X) ).

% inc.simps(2)
tff(fact_2102_inc_Osimps_I3_J,axiom,
    ! [X: num] : inc(bit1(X)) = bit0(inc(X)) ).

% inc.simps(3)
tff(fact_2103_add__One,axiom,
    ! [X: num] : plus_plus_num(X,one) = inc(X) ).

% add_One
tff(fact_2104_mult__inc,axiom,
    ! [X: num,Y: num] : times_times_num(X,inc(Y)) = plus_plus_num(times_times_num(X,Y),X) ).

% mult_inc
tff(fact_2105_take__bit__decr__eq,axiom,
    ! [Na: nat,K: int] :
      ( ( bit_se2923211474154528505it_int(Na,K) != zero_zero_int )
     => ( bit_se2923211474154528505it_int(Na,minus_minus_int(K,one_one_int)) = minus_minus_int(bit_se2923211474154528505it_int(Na,K),one_one_int) ) ) ).

% take_bit_decr_eq
tff(fact_2106_take__bit__eq__mask__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( bit_se2923211474154528505it_int(Na,K) = aa_nat_int(bit_se2000444600071755411sk_int,Na) )
    <=> ( bit_se2923211474154528505it_int(Na,plus_plus_int(K,one_one_int)) = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
tff(fact_2107_take__bit__nat__eq__self__iff,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa_nat_nat(bit_se2925701944663578781it_nat(Na),M) = M )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_2108_take__bit__nat__less__exp,axiom,
    ! [Na: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(bit_se2925701944663578781it_nat(Na),M)),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na)) ).

% take_bit_nat_less_exp
tff(fact_2109_take__bit__nat__eq__self,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na))
     => ( aa_nat_nat(bit_se2925701944663578781it_nat(Na),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_2110_take__bit__nat__def,axiom,
    ! [Na: nat,M: nat] : aa_nat_nat(bit_se2925701944663578781it_nat(Na),M) = aa_nat_nat(modulo_modulo_nat(M),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na)) ).

% take_bit_nat_def
tff(fact_2111_take__bit__Suc__minus__bit1,axiom,
    ! [Na: nat,K: num] : bit_se2923211474154528505it_int(aa_nat_nat(suc,Na),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit1(K)))) = plus_plus_int(times_times_int(bit_se2923211474154528505it_int(Na,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K)))),aa_num_int(numeral_numeral_int,bit0(one))),one_one_int) ).

% take_bit_Suc_minus_bit1
tff(fact_2112_take__bit__int__less__exp,axiom,
    ! [Na: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,bit_se2923211474154528505it_int(Na,K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na)) ).

% take_bit_int_less_exp
tff(fact_2113_take__bit__int__def,axiom,
    ! [Na: nat,K: int] : bit_se2923211474154528505it_int(Na,K) = modulo_modulo_int(K,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na)) ).

% take_bit_int_def
tff(fact_2114_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : bit_se2923211474154528505it_int(numeral_numeral_nat(L),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit1(K)))) = plus_plus_int(times_times_int(bit_se2923211474154528505it_int(pred_numeral(L),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K)))),aa_num_int(numeral_numeral_int,bit0(one))),one_one_int) ).

% take_bit_numeral_minus_bit1
tff(fact_2115_take__bit__nat__less__self__iff,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(bit_se2925701944663578781it_nat(Na),M)),M)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na)),M) ) ).

% take_bit_nat_less_self_iff
tff(fact_2116_take__bit__Suc__minus__bit0,axiom,
    ! [Na: nat,K: num] : bit_se2923211474154528505it_int(aa_nat_nat(suc,Na),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(K)))) = times_times_int(bit_se2923211474154528505it_int(Na,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,bit0(one))) ).

% take_bit_Suc_minus_bit0
tff(fact_2117_take__bit__int__less__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,bit_se2923211474154528505it_int(Na,K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na)),K) ) ).

% take_bit_int_less_self_iff
tff(fact_2118_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),bit_se2923211474154528505it_int(Na,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_2119_take__bit__int__eq__self,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na))
       => ( bit_se2923211474154528505it_int(Na,K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_2120_take__bit__int__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( bit_se2923211474154528505it_int(Na,K) = K )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_2121_take__bit__incr__eq,axiom,
    ! [Na: nat,K: int] :
      ( ( bit_se2923211474154528505it_int(Na,K) != minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na),one_one_int) )
     => ( bit_se2923211474154528505it_int(Na,plus_plus_int(K,one_one_int)) = plus_plus_int(one_one_int,bit_se2923211474154528505it_int(Na,K)) ) ) ).

% take_bit_incr_eq
tff(fact_2122_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : bit_se2923211474154528505it_int(numeral_numeral_nat(L),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(K)))) = times_times_int(bit_se2923211474154528505it_int(pred_numeral(L),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,bit0(one))) ).

% take_bit_numeral_minus_bit0
tff(fact_2123_take__bit__int__less__eq,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,bit_se2923211474154528505it_int(Na,K)),minus_minus_int(K,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na))) ) ) ).

% take_bit_int_less_eq
tff(fact_2124_take__bit__int__greater__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,plus_plus_int(K,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na))),bit_se2923211474154528505it_int(Na,K)) ) ).

% take_bit_int_greater_eq
tff(fact_2125_signed__take__bit__eq__take__bit__shift,axiom,
    ! [Na: nat,K: int] : bit_ri631733984087533419it_int(Na,K) = minus_minus_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Na),plus_plus_int(K,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_2126_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [Na: nat,K: int] :
      ( ( bit_se2923211474154528505it_int(Na,K) = aa_nat_int(bit_se2000444600071755411sk_int,Na) )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na)),plus_plus_int(K,one_one_int)) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_2127_take__bit__minus__small__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na))
       => ( bit_se2923211474154528505it_int(Na,aa_int_int(uminus_uminus_int,K)) = minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_2128_set__decode__plus__power__2,axiom,
    ! [Na: nat,Z: nat] :
      ( ~ aa_set_nat_o(member_nat(Na),nat_set_decode(Z))
     => ( nat_set_decode(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na)),Z)) = insert_nat(Na,nat_set_decode(Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_2129_vebt__deletei__rule,axiom,
    ! [Na: nat,S: set_nat,Tia: vEBT_VEBTi,X: nat] : aa_nat_o(time_htt_VEBT_VEBTi(aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Na,S),Tia),vEBT_vebt_deletei(Tia,X),vEBT_Intf_vebt_assn(Na,minus_minus_set_nat(S,insert_nat(X,bot_bot_set_nat)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(bit0(bit1(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(bit0(bit1(bit0(one)))))),aa_int_nat(nat2,archim7802044766580827645g_real(aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Na))))))) ).

% vebt_deletei_rule
tff(fact_2130_vebt__inserti__rule,axiom,
    ! [X: nat,Na: nat,S: set_nat,Tia: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na))
     => aa_nat_o(time_htt_VEBT_VEBTi(aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Na,S),Tia),vEBT_vebt_inserti(Tia,X),vEBT_Intf_vebt_assn(Na,sup_sup_set_nat(S,insert_nat(X,bot_bot_set_nat)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit1(bit0(bit1(one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit1(bit0(bit1(one))))),aa_int_nat(nat2,archim7802044766580827645g_real(aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,Na))))))) ) ).

% vebt_inserti_rule
tff(fact_2131_arctan__def,axiom,
    ! [Y: real] : aa_real_real(arctan,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_bt,Y)) ).

% arctan_def
tff(fact_2132_modulo__int__def,axiom,
    ! [K: int,L: int] :
      modulo_modulo_int(K,L) = $ite(
        L = zero_zero_int,
        K,
        $ite(sgn_sgn_int(K) = sgn_sgn_int(L),times_times_int(sgn_sgn_int(L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(modulo_modulo_nat(aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))),times_times_int(sgn_sgn_int(L),minus_minus_int(times_times_int(aa_int_int(abs_abs_int,L),aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(modulo_modulo_nat(aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))))) ) ).

% modulo_int_def
tff(fact_2133_sgn__mult__dvd__iff,axiom,
    ! [R: int,L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,times_times_int(sgn_sgn_int(R),L)),K)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        & ( ( R = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_2134_mult__sgn__dvd__iff,axiom,
    ! [L: int,R: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,times_times_int(L,sgn_sgn_int(R))),K)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        & ( ( R = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_2135_dvd__sgn__mult__iff,axiom,
    ! [L: int,R: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),times_times_int(sgn_sgn_int(R),K))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( R = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_2136_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),times_times_int(K,sgn_sgn_int(R)))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( R = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_2137_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N: nat,L2: int] : K != times_times_int(sgn_sgn_int(L2),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% int_sgnE
tff(fact_2138_ln__real__def,axiom,
    ! [X: real] : aa_real_real(ln_ln_real,X) = the_real(aa_real_fun_real_o(aTP_Lamm_bu,X)) ).

% ln_real_def
tff(fact_2139_ln__neg__is__const,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
     => ( aa_real_real(ln_ln_real,X) = the_real(aTP_Lamm_bv) ) ) ).

% ln_neg_is_const
tff(fact_2140_zsgn__def,axiom,
    ! [I: int] :
      sgn_sgn_int(I) = $ite(
        I = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),I),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ) ).

% zsgn_def
tff(fact_2141_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L: int] :
      ( ( V != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,times_times_int(sgn_sgn_int(V),aa_int_int(abs_abs_int,K))),times_times_int(sgn_sgn_int(V),aa_int_int(abs_abs_int,L))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_2142_div__dvd__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = times_times_int(times_times_int(sgn_sgn_int(K),sgn_sgn_int(L)),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L))) ) ) ).

% div_dvd_sgn_abs
tff(fact_2143_arccos__def,axiom,
    ! [Y: real] : aa_real_real(arccos,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_bw,Y)) ).

% arccos_def
tff(fact_2144_pi__half,axiom,
    aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))) = the_real(aTP_Lamm_bx) ).

% pi_half
tff(fact_2145_pi__def,axiom,
    pi = aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),the_real(aTP_Lamm_bx)) ).

% pi_def
tff(fact_2146_divide__int__unfold,axiom,
    ! [K: int,M: nat,L: int,Na: nat] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,times_times_int(sgn_sgn_int(K),aa_nat_int(semiri1314217659103216013at_int,M))),times_times_int(sgn_sgn_int(L),aa_nat_int(semiri1314217659103216013at_int,Na))) = $ite(
        ( ( sgn_sgn_int(L) = zero_zero_int )
        | ( sgn_sgn_int(K) = zero_zero_int )
        | ( Na = zero_zero_nat ) ),
        zero_zero_int,
        $ite(sgn_sgn_int(K) = sgn_sgn_int(L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(divide_divide_nat(M),Na)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(divide_divide_nat(M),Na)),zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),M)))))) ) ).

% divide_int_unfold
tff(fact_2147_modulo__int__unfold,axiom,
    ! [K: int,M: nat,L: int,Na: nat] :
      modulo_modulo_int(times_times_int(sgn_sgn_int(K),aa_nat_int(semiri1314217659103216013at_int,M)),times_times_int(sgn_sgn_int(L),aa_nat_int(semiri1314217659103216013at_int,Na))) = $ite(
        ( ( sgn_sgn_int(L) = zero_zero_int )
        | ( sgn_sgn_int(K) = zero_zero_int )
        | ( Na = zero_zero_nat ) ),
        times_times_int(sgn_sgn_int(K),aa_nat_int(semiri1314217659103216013at_int,M)),
        $ite(sgn_sgn_int(K) = sgn_sgn_int(L),times_times_int(sgn_sgn_int(L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(modulo_modulo_nat(M),Na))),times_times_int(sgn_sgn_int(L),minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),M)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(modulo_modulo_nat(M),Na))))) ) ).

% modulo_int_unfold
tff(fact_2148_divide__int__def,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = $ite(
        L = zero_zero_int,
        zero_zero_int,
        $ite(sgn_sgn_int(K) = sgn_sgn_int(L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(divide_divide_nat(aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(divide_divide_nat(aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),zero_n2687167440665602831ol_nat(~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)))))) ) ).

% divide_int_def
tff(fact_2149_arcsin__def,axiom,
    ! [Y: real] : aa_real_real(arcsin,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_by,Y)) ).

% arcsin_def
tff(fact_2150_sgn__div__eq__sgn__mult,axiom,
    ! [Aa: int,Ba: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba) != zero_zero_int )
     => ( sgn_sgn_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)) = sgn_sgn_int(times_times_int(Aa,Ba)) ) ) ).

% sgn_div_eq_sgn_mult
tff(fact_2151_zero__le__sgn__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sgn_sgn_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X) ) ).

% zero_le_sgn_iff
tff(fact_2152_sgn__le__0__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sgn_sgn_real,X)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real) ) ).

% sgn_le_0_iff
tff(fact_2153_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_2154_sup__enat__def,axiom,
    sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).

% sup_enat_def
tff(fact_2155_sup__int__def,axiom,
    sup_sup_int = ord_max_int ).

% sup_int_def
tff(fact_2156_sgn__real__def,axiom,
    ! [Aa: real] :
      aa_real_real(sgn_sgn_real,Aa) = $ite(
        Aa = zero_zero_real,
        zero_zero_real,
        $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% sgn_real_def
tff(fact_2157_floor__real__def,axiom,
    ! [X: real] : archim6058952711729229775r_real(X) = the_int(aTP_Lamm_bz(X)) ).

% floor_real_def
tff(fact_2158_arctan__inverse,axiom,
    ! [X: real] :
      ( ( X != zero_zero_real )
     => ( aa_real_real(arctan,aa_real_real(divide_divide_real(one_one_real),X)) = aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,X)),pi)),numeral_numeral_real(bit0(one)))),aa_real_real(arctan,X)) ) ) ).

% arctan_inverse
tff(fact_2159_fold__list__rl,axiom,
    ! [Xs2: list_nat,Na: nat,S: set_nat,T: vEBT_VEBTi] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),set_nat2(Xs2))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na)) )
     => hoare_1429296392585015714_VEBTi(aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Na,S),T),aa_VEB836174949890266638_VEBTi(vEBT_E6105538542217078229_VEBTi(aTP_Lamm_bb,Xs2),T),vEBT_Intf_vebt_assn(Na,sup_sup_set_nat(S,set_nat2(Xs2)))) ) ).

% fold_list_rl
tff(fact_2160_signed__take__bit__eq__take__bit__minus,axiom,
    ! [Na: nat,K: int] : bit_ri631733984087533419it_int(Na,K) = minus_minus_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Na),K),times_times_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,Na)),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_2161_vebt__heap__rules_I3_J,axiom,
    ! [X: nat,Na: nat,S: set_nat,Tia: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na))
     => hoare_1429296392585015714_VEBTi(aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Na,S),Tia),vEBT_vebt_inserti(Tia,X),vEBT_Intf_vebt_assn(Na,sup_sup_set_nat(S,insert_nat(X,bot_bot_set_nat)))) ) ).

% vebt_heap_rules(3)
tff(fact_2162_num_Osize__gen_I3_J,axiom,
    ! [X32: num] : size_num(bit1(X32)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_num(X32)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(3)
tff(fact_2163_signed__take__bit__nonnegative__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),bit_ri631733984087533419it_int(Na,K))
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_2164_signed__take__bit__negative__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,bit_ri631733984087533419it_int(Na,K)),zero_zero_int)
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na) ) ).

% signed_take_bit_negative_iff
tff(fact_2165_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,Na: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(W)))),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Na) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_2166_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,Na: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit1(W)))),aa_nat_nat(suc,Na))
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W)),Na) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_2167_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,Na: num] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(W)))),numeral_numeral_nat(Na))
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),pred_numeral(Na)) ) ).

% bit_minus_numeral_int(1)
tff(fact_2168_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,Na: num] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit1(W)))),numeral_numeral_nat(Na))
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W)),pred_numeral(Na)) ) ).

% bit_minus_numeral_int(2)
tff(fact_2169_bin__nth__minus__Bit0,axiom,
    ! [Na: nat,W: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,bit0(W))),Na)
      <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W)),aa_nat_nat(minus_minus_nat(Na),one_one_nat)) ) ) ).

% bin_nth_minus_Bit0
tff(fact_2170_bin__nth__minus__Bit1,axiom,
    ! [Na: nat,W: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,bit1(W))),Na)
      <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W)),aa_nat_nat(minus_minus_nat(Na),one_one_nat)) ) ) ).

% bin_nth_minus_Bit1
tff(fact_2171_bit__xor__int__iff,axiom,
    ! [K: int,L: int,Na: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,bit_se6526347334894502574or_int(K,L)),Na)
    <=> ~ ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na)
        <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,L),Na) ) ) ).

% bit_xor_int_iff
tff(fact_2172_bit__not__int__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,bit_ri7919022796975470100ot_int(K)),Na)
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na) ) ).

% bit_not_int_iff
tff(fact_2173_bit__or__int__iff,axiom,
    ! [K: int,L: int,Na: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,bit_se1409905431419307370or_int(K,L)),Na)
    <=> ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na)
        | aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,L),Na) ) ) ).

% bit_or_int_iff
tff(fact_2174_bit__and__int__iff,axiom,
    ! [K: int,L: int,Na: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,bit_se725231765392027082nd_int(K,L)),Na)
    <=> ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na)
        & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,L),Na) ) ) ).

% bit_and_int_iff
tff(fact_2175_bit__not__int__iff_H,axiom,
    ! [K: int,Na: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,minus_minus_int(aa_int_int(uminus_uminus_int,K),one_one_int)),Na)
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na) ) ).

% bit_not_int_iff'
tff(fact_2176_bit__imp__take__bit__positive,axiom,
    ! [Na: nat,M: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
     => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),bit_se2923211474154528505it_int(M,K)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_2177_bit__minus__int__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,K)),Na)
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,bit_ri7919022796975470100ot_int(minus_minus_int(K,one_one_int))),Na) ) ).

% bit_minus_int_iff
tff(fact_2178_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2)
             => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),M2)
              <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N) ) )
         => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),aa_nat_nat(minus_minus_nat(N),one_one_nat))
              <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N) ) ) ) ).

% int_bit_bound
tff(fact_2179_num_Osize__gen_I1_J,axiom,
    size_num(one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_2180_vebt__buildupi__rule__basic,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => hoare_1429296392585015714_VEBTi(one_one_assn,vEBT_vebt_buildupi(Na),vEBT_Intf_vebt_assn(Na,bot_bot_set_nat)) ) ).

% vebt_buildupi_rule_basic
tff(fact_2181_vebt__heap__rules_I8_J,axiom,
    ! [Na: nat,S: set_nat,Tia: vEBT_VEBTi,X: nat] : hoare_1429296392585015714_VEBTi(aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Na,S),Tia),vEBT_vebt_deletei(Tia,X),vEBT_Intf_vebt_assn(Na,minus_minus_set_nat(S,insert_nat(X,bot_bot_set_nat)))) ).

% vebt_heap_rules(8)
tff(fact_2182_bit__int__def,axiom,
    ! [K: int,Na: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na))) ) ).

% bit_int_def
tff(fact_2183_vebt__heap__rules_I1_J,axiom,
    ! [Na: nat] : hoare_1429296392585015714_VEBTi(pure_assn(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)),vEBT_vebt_buildupi(Na),vEBT_Intf_vebt_assn(Na,bot_bot_set_nat)) ).

% vebt_heap_rules(1)
tff(fact_2184_Bit__Operations_Oset__bit__eq,axiom,
    ! [Na: nat,K: int] : bit_se7879613467334960850it_int(Na,K) = plus_plus_int(K,times_times_int(aa_o_int(zero_n2684676970156552555ol_int,~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na))) ).

% Bit_Operations.set_bit_eq
tff(fact_2185_unset__bit__eq,axiom,
    ! [Na: nat,K: int] : bit_se4203085406695923979it_int(Na,K) = minus_minus_int(K,times_times_int(aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na))) ).

% unset_bit_eq
tff(fact_2186_take__bit__Suc__from__most,axiom,
    ! [Na: nat,K: int] : bit_se2923211474154528505it_int(aa_nat_nat(suc,Na),K) = plus_plus_int(times_times_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na))),bit_se2923211474154528505it_int(Na,K)) ).

% take_bit_Suc_from_most
tff(fact_2187_num_Osize__gen_I2_J,axiom,
    ! [X23: num] : size_num(bit0(X23)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_num(X23)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(2)
tff(fact_2188_floor__rat__def,axiom,
    ! [X: rat] : archim3151403230148437115or_rat(X) = the_int(aTP_Lamm_ca(X)) ).

% floor_rat_def
tff(fact_2189_wait__rule,axiom,
    ! [Na: nat] : hoare_8945653483474564448t_unit(one_one_assn,heap_Time_wait(Na),aTP_Lamm_cb) ).

% wait_rule
tff(fact_2190_obtain__pos__sum,axiom,
    ! [R: rat] :
      ( ord_less_rat(zero_zero_rat,R)
     => ~ ! [S2: rat] :
            ( ord_less_rat(zero_zero_rat,S2)
           => ! [T2: rat] :
                ( ord_less_rat(zero_zero_rat,T2)
               => ( R != plus_plus_rat(S2,T2) ) ) ) ) ).

% obtain_pos_sum
tff(fact_2191_not__bit__Suc__0__Suc,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(suc,Na)) ).

% not_bit_Suc_0_Suc
tff(fact_2192_bit__Suc__0__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
    <=> ( Na = zero_zero_nat ) ) ).

% bit_Suc_0_iff
tff(fact_2193_not__bit__Suc__0__numeral,axiom,
    ! [Na: num] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),numeral_numeral_nat(Na)) ).

% not_bit_Suc_0_numeral
tff(fact_2194_bit__nat__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_int_nat(nat2,K)),Na)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na) ) ) ).

% bit_nat_iff
tff(fact_2195_vebt__heap__rules_I2_J,axiom,
    ! [Na: nat,S: set_nat,Tia: vEBT_VEBTi,X: nat] : hoare_hoare_triple_o(aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Na,S),Tia),aa_nat2910048432523430448Heap_o(vEBT_vebt_memberi(Tia),X),aa_nat_fun_o_assn(aa_VEB8176250900481067561o_assn(aa_set2974300032405964540o_assn(aTP_Lamm_az(Na),S),Tia),X)) ).

% vebt_heap_rules(2)
tff(fact_2196_bit__nat__def,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,M),Na)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(divide_divide_nat(M),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na))) ) ).

% bit_nat_def
tff(fact_2197_vebt__heap__rules_I7_J,axiom,
    ! [Na: nat,S: set_nat,Tia: vEBT_VEBTi,X: nat,Y: nat] : hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Na,S),Tia),vEBT_vebt_predi(Tia,X),aa_nat435388790147128531t_assn(aa_nat336663792692944416t_assn(aa_VEB746558900233519186t_assn(aa_set2727930972950555477t_assn(aTP_Lamm_ba(Na),S),Tia),X),Y)) ).

% vebt_heap_rules(7)
tff(fact_2198_vebt__heap__rules_I6_J,axiom,
    ! [Na: nat,S: set_nat,Tia: vEBT_VEBTi,X: nat,Y: nat] : hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Na,S),Tia),vEBT_vebt_succi(Tia,X),aa_nat435388790147128531t_assn(aa_nat336663792692944416t_assn(aa_VEB746558900233519186t_assn(aa_set2727930972950555477t_assn(aTP_Lamm_bc(Na),S),Tia),X),Y)) ).

% vebt_heap_rules(6)
tff(fact_2199_VEBT__internal_OT__vebt__buildupi_H_Opelims,axiom,
    ! [X: nat,Y: int] :
      ( ( aa_nat_int(vEBT_V9176841429113362141ildupi,X) = Y )
     => ( aa_nat_o(accp_nat(vEBT_V3352910403632780892pi_rel),X)
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = one_one_int )
             => ~ aa_nat_o(accp_nat(vEBT_V3352910403632780892pi_rel),zero_zero_nat) ) )
         => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = one_one_int )
               => ~ aa_nat_o(accp_nat(vEBT_V3352910403632780892pi_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [N: nat] :
                  ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                 => ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N),plus_plus_int(aa_num_int(numeral_numeral_int,bit1(one)),plus_plus_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one))))),plus_plus_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(bit0(one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one))))),times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),times_times_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one))))))))),plus_plus_int(aa_num_int(numeral_numeral_int,bit1(one)),plus_plus_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one)))))),plus_plus_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(bit0(bit0(one)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one))))),times_times_int(aa_num_int(numeral_numeral_int,bit0(bit0(one))),times_times_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one)))))))))) )
                   => ~ aa_nat_o(accp_nat(vEBT_V3352910403632780892pi_rel),aa_nat_nat(suc,aa_nat_nat(suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.pelims
tff(fact_2200_diff__rat__def,axiom,
    ! [Q3: rat,R: rat] : minus_minus_rat(Q3,R) = plus_plus_rat(Q3,uminus_uminus_rat(R)) ).

% diff_rat_def
tff(fact_2201_pow_Osimps_I1_J,axiom,
    ! [X: num] : pow(X,one) = X ).

% pow.simps(1)
tff(fact_2202_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Opelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V8646137997579335489_i_l_d,X) = Y )
     => ( aa_nat_o(accp_nat(vEBT_V5144397997797733112_d_rel),X)
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = numeral_numeral_nat(bit0(bit0(one))) )
             => ~ aa_nat_o(accp_nat(vEBT_V5144397997797733112_d_rel),zero_zero_nat) ) )
         => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = numeral_numeral_nat(bit0(bit0(one))) )
               => ~ aa_nat_o(accp_nat(vEBT_V5144397997797733112_d_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va2: nat] :
                  ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                 => ( ( Y = $ite(
                          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                            $let(
                              half: nat,
                              half:= aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one))),
                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(bit1(bit0(one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),half)),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) )),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one))),
                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(bit0(bit1(one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,half))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) ) ) )
                   => ~ aa_nat_o(accp_nat(vEBT_V5144397997797733112_d_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.pelims
tff(fact_2203_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Opelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V8346862874174094_d_u_p,X) = Y )
     => ( aa_nat_o(accp_nat(vEBT_V1247956027447740395_p_rel),X)
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = numeral_numeral_nat(bit1(one)) )
             => ~ aa_nat_o(accp_nat(vEBT_V1247956027447740395_p_rel),zero_zero_nat) ) )
         => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = numeral_numeral_nat(bit1(one)) )
               => ~ aa_nat_o(accp_nat(vEBT_V1247956027447740395_p_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va2: nat] :
                  ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                 => ( ( Y = $ite(
                          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                            $let(
                              half: nat,
                              half:= aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one))),
                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit1(bit0(bit0(one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),half)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) )),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,aa_nat_nat(suc,Va2))),numeral_numeral_nat(bit0(one))),
                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit1(bit1(bit0(one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,half))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) ) ) )
                   => ~ aa_nat_o(accp_nat(vEBT_V1247956027447740395_p_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.pelims
tff(fact_2204_VEBT__internal_OTb_Opelims,axiom,
    ! [X: nat,Y: int] :
      ( ( aa_nat_int(vEBT_VEBT_Tb,X) = Y )
     => ( aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel2),X)
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = aa_num_int(numeral_numeral_int,bit1(one)) )
             => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel2),zero_zero_nat) ) )
         => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_num_int(numeral_numeral_int,bit1(one)) )
               => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel2),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [N: nat] :
                  ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                 => ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N),plus_plus_int(plus_plus_int(aa_num_int(numeral_numeral_int,bit1(bit0(one))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one)))))),times_times_int(aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one))))))),plus_plus_int(plus_plus_int(aa_num_int(numeral_numeral_int,bit1(bit0(one))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one))))))),times_times_int(aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one))))))))) )
                   => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel2),aa_nat_nat(suc,aa_nat_nat(suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.Tb.pelims
tff(fact_2205_VEBT__internal_OTb_H_Opelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_VEBT_Tb2,X) = Y )
     => ( aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel),X)
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = numeral_numeral_nat(bit1(one)) )
             => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel),zero_zero_nat) ) )
         => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = numeral_numeral_nat(bit1(one)) )
               => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [N: nat] :
                  ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                 => ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit1(bit0(one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit1(bit0(one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one))))))))) )
                   => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel),aa_nat_nat(suc,aa_nat_nat(suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.pelims
tff(fact_2206_VEBT__internal_OT__vebt__buildupi_Opelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V441764108873111860ildupi,X) = Y )
     => ( aa_nat_o(accp_nat(vEBT_V2957053500504383685pi_rel),X)
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = aa_nat_nat(suc,zero_zero_nat) )
             => ~ aa_nat_o(accp_nat(vEBT_V2957053500504383685pi_rel),zero_zero_nat) ) )
         => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_nat_nat(suc,zero_zero_nat) )
               => ~ aa_nat_o(accp_nat(vEBT_V2957053500504383685pi_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [N: nat] :
                  ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                 => ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(bit0(one)))),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one))))))))))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(bit0(bit0(one))))),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one)))))),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(divide_divide_nat(N),numeral_numeral_nat(bit0(one)))))))))))) )
                   => ~ aa_nat_o(accp_nat(vEBT_V2957053500504383685pi_rel),aa_nat_nat(suc,aa_nat_nat(suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.pelims
tff(fact_2207_VEBT__internal_OTBOUND__buildupi,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(time_T5737551269749752165_VEBTi(vEBT_vebt_buildupi(Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(bit1(bit0(one))))),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na))) ) ).

% VEBT_internal.TBOUND_buildupi
tff(fact_2208_powr__real__of__int,axiom,
    ! [X: real,Na: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(powr_real(X),ring_1_of_int_real(Na)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Na),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_int_nat(nat2,Na)),aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Na))))) ) ) ).

% powr_real_of_int
tff(fact_2209_exp__two__pi__i_H,axiom,
    exp_complex(aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(times_times_complex(real_V4546457046886955230omplex(pi)),numera6690914467698888265omplex(bit0(one))))) = one_one_complex ).

% exp_two_pi_i'
tff(fact_2210_power2__i,axiom,
    aa_nat_complex(power_power_complex(imaginary_unit),numeral_numeral_nat(bit0(one))) = uminus1482373934393186551omplex(one_one_complex) ).

% power2_i
tff(fact_2211_i__even__power,axiom,
    ! [Na: nat] : aa_nat_complex(power_power_complex(imaginary_unit),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),numeral_numeral_nat(bit0(one)))) = aa_nat_complex(power_power_complex(uminus1482373934393186551omplex(one_one_complex)),Na) ).

% i_even_power
tff(fact_2212_exp__two__pi__i,axiom,
    exp_complex(aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(bit0(one))),real_V4546457046886955230omplex(pi))),imaginary_unit)) = one_one_complex ).

% exp_two_pi_i
tff(fact_2213_divide__real__def,axiom,
    ! [X: real,Y: real] : aa_real_real(divide_divide_real(X),Y) = aa_real_real(times_times_real(X),aa_real_real(inverse_inverse_real,Y)) ).

% divide_real_def
tff(fact_2214_inverse__powr,axiom,
    ! [Y: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_real(powr_real(aa_real_real(inverse_inverse_real,Y)),Aa) = aa_real_real(inverse_inverse_real,aa_real_real(powr_real(Y),Aa)) ) ) ).

% inverse_powr
tff(fact_2215_forall__pos__mono__1,axiom,
    ! [P: fun_real_o,E2: real] :
      ( ! [D4: real,E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D4),E)
         => ( aa_real_o(P,D4)
           => aa_real_o(P,E) ) )
     => ( ! [N: nat] : aa_real_o(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
         => aa_real_o(P,E2) ) ) ) ).

% forall_pos_mono_1
tff(fact_2216_forall__pos__mono,axiom,
    ! [P: fun_real_o,E2: real] :
      ( ! [D4: real,E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D4),E)
         => ( aa_real_o(P,D4)
           => aa_real_o(P,E) ) )
     => ( ! [N: nat] :
            ( ( N != zero_zero_nat )
           => aa_real_o(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N))) )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
         => aa_real_o(P,E2) ) ) ) ).

% forall_pos_mono
tff(fact_2217_real__arch__inverse,axiom,
    ! [E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
    <=> ? [N2: nat] :
          ( ( N2 != zero_zero_nat )
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N2)))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N2))),E2) ) ) ).

% real_arch_inverse
tff(fact_2218_sqrt__divide__self__eq,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_real(divide_divide_real(aa_real_real(sqrt,X)),X) = aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X)) ) ) ).

% sqrt_divide_self_eq
tff(fact_2219_log__inverse,axiom,
    ! [Aa: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( ( Aa != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( aa_real_real(log(Aa),aa_real_real(inverse_inverse_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(log(Aa),X)) ) ) ) ) ).

% log_inverse
tff(fact_2220_Complex__eq,axiom,
    ! [Aa: real,Ba: real] : complex2(Aa,Ba) = plus_plus_complex(real_V4546457046886955230omplex(Aa),aa_complex_complex(times_times_complex(imaginary_unit),real_V4546457046886955230omplex(Ba))) ).

% Complex_eq
tff(fact_2221_exp__plus__inverse__exp,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,X)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,X)))) ).

% exp_plus_inverse_exp
tff(fact_2222_complex__split__polar,axiom,
    ! [Z: complex] :
    ? [R5: real,A2: real] : Z = aa_complex_complex(times_times_complex(real_V4546457046886955230omplex(R5)),plus_plus_complex(real_V4546457046886955230omplex(aa_real_real(cos_real,A2)),aa_complex_complex(times_times_complex(imaginary_unit),real_V4546457046886955230omplex(aa_real_real(sin_real,A2))))) ).

% complex_split_polar
tff(fact_2223_plus__inverse__ge__2,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(bit0(one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(inverse_inverse_real,X))) ) ).

% plus_inverse_ge_2
tff(fact_2224_real__inv__sqrt__pow2,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),numeral_numeral_nat(bit0(one))) = aa_real_real(inverse_inverse_real,X) ) ) ).

% real_inv_sqrt_pow2
tff(fact_2225_tan__cot,axiom,
    ! [X: real] : aa_real_real(tan_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one)))),X)) = aa_real_real(inverse_inverse_real,aa_real_real(tan_real,X)) ).

% tan_cot
tff(fact_2226_real__le__x__sinh,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(exp_real,X)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,X)))),numeral_numeral_real(bit0(one)))) ) ).

% real_le_x_sinh
tff(fact_2227_real__le__abs__sinh,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(exp_real,X)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,X)))),numeral_numeral_real(bit0(one))))) ).

% real_le_abs_sinh
tff(fact_2228_cmod__unit__one,axiom,
    ! [Aa: real] : real_V1022390504157884413omplex(plus_plus_complex(real_V4546457046886955230omplex(aa_real_real(cos_real,Aa)),aa_complex_complex(times_times_complex(imaginary_unit),real_V4546457046886955230omplex(aa_real_real(sin_real,Aa))))) = one_one_real ).

% cmod_unit_one
tff(fact_2229_cmod__complex__polar,axiom,
    ! [R: real,Aa: real] : real_V1022390504157884413omplex(aa_complex_complex(times_times_complex(real_V4546457046886955230omplex(R)),plus_plus_complex(real_V4546457046886955230omplex(aa_real_real(cos_real,Aa)),aa_complex_complex(times_times_complex(imaginary_unit),real_V4546457046886955230omplex(aa_real_real(sin_real,Aa)))))) = aa_real_real(abs_abs_real,R) ).

% cmod_complex_polar
tff(fact_2230_Arg__minus__ii,axiom,
    arg(uminus1482373934393186551omplex(imaginary_unit)) = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(bit0(one))) ).

% Arg_minus_ii
tff(fact_2231_csqrt__ii,axiom,
    csqrt(imaginary_unit) = divide1717551699836669952omplex(plus_plus_complex(one_one_complex,imaginary_unit),real_V4546457046886955230omplex(aa_real_real(sqrt,numeral_numeral_real(bit0(one))))) ).

% csqrt_ii
tff(fact_2232_Arg__ii,axiom,
    arg(imaginary_unit) = aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))) ).

% Arg_ii
tff(fact_2233_cis__minus__pi__half,axiom,
    cis(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))) = uminus1482373934393186551omplex(imaginary_unit) ).

% cis_minus_pi_half
tff(fact_2234_power2__csqrt,axiom,
    ! [Z: complex] : aa_nat_complex(power_power_complex(csqrt(Z)),numeral_numeral_nat(bit0(one))) = Z ).

% power2_csqrt
tff(fact_2235_cis__pi__half,axiom,
    cis(aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one)))) = imaginary_unit ).

% cis_pi_half
tff(fact_2236_cis__2pi,axiom,
    cis(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),pi)) = one_one_complex ).

% cis_2pi
tff(fact_2237_DeMoivre,axiom,
    ! [Aa: real,Na: nat] : aa_nat_complex(power_power_complex(cis(Aa)),Na) = cis(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),Aa)) ).

% DeMoivre
tff(fact_2238_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( sgn_sgn_complex(Z) = cis(arg(Z)) )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),arg(Z))
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,arg(Z)),pi) ) ) ).

% Arg_correct
tff(fact_2239_cis__Arg__unique,axiom,
    ! [Z: complex,X: real] :
      ( ( sgn_sgn_complex(Z) = cis(X) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),pi)
         => ( arg(Z) = X ) ) ) ) ).

% cis_Arg_unique
tff(fact_2240_cis__mult,axiom,
    ! [Aa: real,Ba: real] : aa_complex_complex(times_times_complex(cis(Aa)),cis(Ba)) = cis(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)) ).

% cis_mult
tff(fact_2241_of__real__sqrt,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( real_V4546457046886955230omplex(aa_real_real(sqrt,X)) = csqrt(real_V4546457046886955230omplex(X)) ) ) ).

% of_real_sqrt
tff(fact_2242_Arg__bounded,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),arg(Z))
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,arg(Z)),pi) ) ).

% Arg_bounded
tff(fact_2243_complex__inverse,axiom,
    ! [Aa: real,Ba: real] : invers8013647133539491842omplex(complex2(Aa,Ba)) = complex2(aa_real_real(divide_divide_real(Aa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),numeral_numeral_nat(bit0(one))))),aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,Ba)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),numeral_numeral_nat(bit0(one)))))) ).

% complex_inverse
tff(fact_2244_sinh__ln__real,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(sinh_real,aa_real_real(ln_ln_real,X)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(X),aa_real_real(inverse_inverse_real,X))),numeral_numeral_real(bit0(one))) ) ) ).

% sinh_ln_real
tff(fact_2245_cis__multiple__2pi,axiom,
    ! [Na: real] :
      ( member_real(Na,ring_1_Ints_real)
     => ( cis(aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),pi)),Na)) = one_one_complex ) ) ).

% cis_multiple_2pi
tff(fact_2246_sinh__real__le__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,X)),aa_real_real(sinh_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ).

% sinh_real_le_iff
tff(fact_2247_sinh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,X)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real) ) ).

% sinh_real_nonpos_iff
tff(fact_2248_sinh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sinh_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X) ) ).

% sinh_real_nonneg_iff
tff(fact_2249_sin__integer__2pi,axiom,
    ! [Na: real] :
      ( member_real(Na,ring_1_Ints_real)
     => ( aa_real_real(sin_real,aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),pi)),Na)) = zero_zero_real ) ) ).

% sin_integer_2pi
tff(fact_2250_cos__integer__2pi,axiom,
    ! [Na: real] :
      ( member_real(Na,ring_1_Ints_real)
     => ( aa_real_real(cos_real,aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),pi)),Na)) = one_one_real ) ) ).

% cos_integer_2pi
tff(fact_2251_Arg__def,axiom,
    ! [Z: complex] :
      arg(Z) = $ite(Z = zero_zero_complex,zero_zero_real,fChoice_real(aTP_Lamm_cc(Z))) ).

% Arg_def
tff(fact_2252_sum__choose__upper,axiom,
    ! [M: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cd(M)),set_ord_atMost_nat(Na)) = aa_nat_nat(binomial(aa_nat_nat(suc,Na)),aa_nat_nat(suc,M)) ).

% sum_choose_upper
tff(fact_2253_choose__rising__sum_I2_J,axiom,
    ! [Na: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ce(Na)),set_ord_atMost_nat(M)) = aa_nat_nat(binomial(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)),one_one_nat)),M) ).

% choose_rising_sum(2)
tff(fact_2254_choose__rising__sum_I1_J,axiom,
    ! [Na: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ce(Na)),set_ord_atMost_nat(M)) = aa_nat_nat(binomial(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat)) ).

% choose_rising_sum(1)
tff(fact_2255_sum__choose__lower,axiom,
    ! [R: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cf(R)),set_ord_atMost_nat(Na)) = aa_nat_nat(binomial(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,R),Na))),Na) ).

% sum_choose_lower
tff(fact_2256_sum__choose__diagonal,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_cg(M),Na)),set_ord_atMost_nat(M)) = aa_nat_nat(binomial(aa_nat_nat(suc,Na)),M) ) ) ).

% sum_choose_diagonal
tff(fact_2257_vandermonde,axiom,
    ! [M: nat,Na: nat,R: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ch(M),Na),R)),set_ord_atMost_nat(R)) = aa_nat_nat(binomial(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)),R) ).

% vandermonde
tff(fact_2258_cosh__real__nonpos__le__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),X) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_2259_cosh__real__nonneg__le__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_2260_cosh__real__nonneg,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(cosh_real,X)) ).

% cosh_real_nonneg
tff(fact_2261_cosh__real__ge__1,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(cosh_real,X)) ).

% cosh_real_ge_1
tff(fact_2262_sinh__le__cosh__real,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,X)),aa_real_real(cosh_real,X)) ).

% sinh_le_cosh_real
tff(fact_2263_choose__row__sum,axiom,
    ! [Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(binomial(Na)),set_ord_atMost_nat(Na)) = aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na) ).

% choose_row_sum
tff(fact_2264_binomial,axiom,
    ! [Aa: nat,Ba: nat,Na: nat] : aa_nat_nat(power_power_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba)),Na) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ci(Aa),Ba),Na)),set_ord_atMost_nat(Na)) ).

% binomial
tff(fact_2265_polynomial__product__nat,axiom,
    ! [M: nat,Aa: fun_nat_nat,Na: nat,Ba: fun_nat_nat,X: nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),I2)
         => ( aa_nat_nat(Aa,I2) = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),J2)
           => ( aa_nat_nat(Ba,J2) = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_cj(Aa),X)),set_ord_atMost_nat(M))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_cj(Ba),X)),set_ord_atMost_nat(Na))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_cl(Aa),Ba),X)),set_ord_atMost_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))) ) ) ) ).

% polynomial_product_nat
tff(fact_2266_choose__square__sum,axiom,
    ! [Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cm(Na)),set_ord_atMost_nat(Na)) = aa_nat_nat(binomial(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Na)),Na) ).

% choose_square_sum
tff(fact_2267_cosh__real__nonpos__less__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_2268_cosh__real__nonneg__less__iff,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_2269_cosh__real__strict__mono,axiom,
    ! [X: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y)) ) ) ).

% cosh_real_strict_mono
tff(fact_2270_atMost__nat__numeral,axiom,
    ! [K: num] : set_ord_atMost_nat(numeral_numeral_nat(K)) = insert_nat(numeral_numeral_nat(K),set_ord_atMost_nat(pred_numeral(K))) ).

% atMost_nat_numeral
tff(fact_2271_arcosh__cosh__real,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_real(arcosh_real,aa_real_real(cosh_real,X)) = X ) ) ).

% arcosh_cosh_real
tff(fact_2272_binomial__r__part__sum,axiom,
    ! [M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(binomial(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),M)),one_one_nat))),set_ord_atMost_nat(M)) = aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),M)) ).

% binomial_r_part_sum
tff(fact_2273_choose__linear__sum,axiom,
    ! [Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cn(Na)),set_ord_atMost_nat(Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(minus_minus_nat(Na),one_one_nat))) ).

% choose_linear_sum
tff(fact_2274_mask__eq__sum__exp__nat,axiom,
    ! [Na: nat] : aa_nat_nat(minus_minus_nat(aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na)),aa_nat_nat(suc,zero_zero_nat)) = aa_set_nat_nat(groups3542108847815614940at_nat(power_power_nat(numeral_numeral_nat(bit0(one)))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_co,Na))) ).

% mask_eq_sum_exp_nat
tff(fact_2275_cosh__ln__real,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(cosh_real,aa_real_real(ln_ln_real,X)) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(inverse_inverse_real,X))),numeral_numeral_real(bit0(one))) ) ) ).

% cosh_ln_real
tff(fact_2276_sum__nth__roots,axiom,
    ! [Na: nat,C: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Na)
     => ( groups7754918857620584856omplex(aTP_Lamm_cp,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_cq(Na),C))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_2277_sum__roots__unity,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Na)
     => ( groups7754918857620584856omplex(aTP_Lamm_cp,collect_complex(aTP_Lamm_cr(Na))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_2278_Maclaurin__cos__expansion2,axiom,
    ! [X: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ? [T2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T2)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T2),X)
            & ( aa_real_real(cos_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_cs,X),set_ord_lessThan_nat(Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T2),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one)))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_2279_Maclaurin__minus__cos__expansion,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
       => ? [T2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),T2)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T2),zero_zero_real)
            & ( aa_real_real(cos_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_cs,X),set_ord_lessThan_nat(Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T2),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one)))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_2280_Maclaurin__sin__expansion3,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ? [T2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T2)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T2),X)
            & ( aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ct,X),set_ord_lessThan_nat(Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T2),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one)))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_2281_sumr__cos__zero__one,axiom,
    ! [Na: nat] : groups6591440286371151544t_real(aTP_Lamm_cu,set_ord_lessThan_nat(aa_nat_nat(suc,Na))) = one_one_real ).

% sumr_cos_zero_one
tff(fact_2282_lessThan__nat__numeral,axiom,
    ! [K: num] : set_ord_lessThan_nat(numeral_numeral_nat(K)) = insert_nat(pred_numeral(K),set_ord_lessThan_nat(pred_numeral(K))) ).

% lessThan_nat_numeral
tff(fact_2283_Maclaurin__lemma,axiom,
    ! [H: real,F: fun_real_real,J: fun_nat_real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ? [B4: real] : aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_cv(H),J),set_ord_lessThan_nat(Na))),aa_real_real(times_times_real(B4),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Na)),aa_nat_real(semiri2265585572941072030t_real,Na)))) ) ).

% Maclaurin_lemma
tff(fact_2284_sum__split__even__odd,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,Na: nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_cw(F),G),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Na))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aTP_Lamm_cx(F),set_ord_lessThan_nat(Na))),groups6591440286371151544t_real(aTP_Lamm_cy(G),set_ord_lessThan_nat(Na))) ).

% sum_split_even_odd
tff(fact_2285_Maclaurin__exp__le,axiom,
    ! [X: real,Na: nat] :
    ? [T2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X))
      & ( aa_real_real(exp_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_cz,X),set_ord_lessThan_nat(Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(exp_real,T2)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na))) ) ) ).

% Maclaurin_exp_le
tff(fact_2286_Maclaurin__sin__bound,axiom,
    ! [X: real,Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(sin_real,X)),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ct,X),set_ord_lessThan_nat(Na))))),aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,X)),Na))) ).

% Maclaurin_sin_bound
tff(fact_2287_sum__pos__lt__pair,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( ! [D4: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D4)))),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D4)),one_one_nat)))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,groups6591440286371151544t_real(F,set_ord_lessThan_nat(K))),suminf_real(F)) ) ) ).

% sum_pos_lt_pair
tff(fact_2288_Maclaurin__exp__lt,axiom,
    ! [X: real,Na: nat] :
      ( ( X != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ? [T2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T2))
            & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X))
            & ( aa_real_real(exp_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_cz,X),set_ord_lessThan_nat(Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(exp_real,T2)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_2289_Maclaurin__sin__expansion,axiom,
    ! [X: real,Na: nat] :
    ? [T2: real] : aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ct,X),set_ord_lessThan_nat(Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T2),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one)))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na))) ).

% Maclaurin_sin_expansion
tff(fact_2290_Maclaurin__sin__expansion2,axiom,
    ! [X: real,Na: nat] :
    ? [T2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X))
      & ( aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ct,X),set_ord_lessThan_nat(Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T2),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one)))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_2291_Maclaurin__cos__expansion,axiom,
    ! [X: real,Na: nat] :
    ? [T2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X))
      & ( aa_real_real(cos_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_cs,X),set_ord_lessThan_nat(Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T2),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one)))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_2292_Maclaurin__sin__expansion4,axiom,
    ! [X: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ? [T2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T2)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T2),X)
          & ( aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ct,X),set_ord_lessThan_nat(Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T2),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one)))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_2293_bij__betw__roots__unity,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => bij_betw_nat_complex(aTP_Lamm_da(Na),set_ord_lessThan_nat(Na),collect_complex(aTP_Lamm_cr(Na))) ) ).

% bij_betw_roots_unity
tff(fact_2294_all__nat__less,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ! [M7: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M7),Na)
         => aa_nat_o(P,M7) )
    <=> ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),set_or1269000886237332187st_nat(zero_zero_nat,Na))
         => aa_nat_o(P,X5) ) ) ).

% all_nat_less
tff(fact_2295_ex__nat__less,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ? [M7: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M7),Na)
          & aa_nat_o(P,M7) )
    <=> ? [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),set_or1269000886237332187st_nat(zero_zero_nat,Na))
          & aa_nat_o(P,X5) ) ) ).

% ex_nat_less
tff(fact_2296_atLeastAtMost__insertL,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( insert_nat(M,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),Na)) = set_or1269000886237332187st_nat(M,Na) ) ) ).

% atLeastAtMost_insertL
tff(fact_2297_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Na)) = insert_nat(aa_nat_nat(suc,Na),set_or1269000886237332187st_nat(M,Na)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_2298_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( set_or1269000886237332187st_nat(M,Na) = insert_nat(M,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),Na)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_2299_gauss__sum__nat,axiom,
    ! [Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_db),set_or1269000886237332187st_nat(zero_zero_nat,Na)) = aa_nat_nat(divide_divide_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(suc,Na))),numeral_numeral_nat(bit0(one))) ).

% gauss_sum_nat
tff(fact_2300_arith__series__nat,axiom,
    ! [Aa: nat,D: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_dc(Aa),D)),set_or1269000886237332187st_nat(zero_zero_nat,Na)) = aa_nat_nat(divide_divide_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Na)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),D)))),numeral_numeral_nat(bit0(one))) ).

% arith_series_nat
tff(fact_2301_Sum__Icc__nat,axiom,
    ! [M: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_db),set_or1269000886237332187st_nat(M,Na)) = aa_nat_nat(divide_divide_nat(aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(minus_minus_nat(M),one_one_nat)))),numeral_numeral_nat(bit0(one))) ).

% Sum_Icc_nat
tff(fact_2302_Sum__Icc__int,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),Na)
     => ( groups4538972089207619220nt_int(aTP_Lamm_dd,set_or1266510415728281911st_int(M,Na)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,minus_minus_int(times_times_int(Na,plus_plus_int(Na,one_one_int)),times_times_int(M,minus_minus_int(M,one_one_int)))),aa_num_int(numeral_numeral_int,bit0(one))) ) ) ).

% Sum_Icc_int
tff(fact_2303_bset_I1_J,axiom,
    ! [D5: int,B: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D5))
             => ! [Xb: int] :
                  ( member_int(Xb,B)
                 => ( X2 != plus_plus_int(Xb,Xa) ) ) )
         => ( aa_int_o(P,X2)
           => aa_int_o(P,minus_minus_int(X2,D5)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb: int] :
                    ( member_int(Xb,B)
                   => ( X2 != plus_plus_int(Xb,Xa) ) ) )
           => ( aa_int_o(Q,X2)
             => aa_int_o(Q,minus_minus_int(X2,D5)) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb2: int] :
                    ( member_int(Xb2,B)
                   => ( X4 != plus_plus_int(Xb2,Xa3) ) ) )
           => ( ( aa_int_o(P,X4)
                & aa_int_o(Q,X4) )
             => ( aa_int_o(P,minus_minus_int(X4,D5))
                & aa_int_o(Q,minus_minus_int(X4,D5)) ) ) ) ) ) ).

% bset(1)
tff(fact_2304_bset_I2_J,axiom,
    ! [D5: int,B: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D5))
             => ! [Xb: int] :
                  ( member_int(Xb,B)
                 => ( X2 != plus_plus_int(Xb,Xa) ) ) )
         => ( aa_int_o(P,X2)
           => aa_int_o(P,minus_minus_int(X2,D5)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb: int] :
                    ( member_int(Xb,B)
                   => ( X2 != plus_plus_int(Xb,Xa) ) ) )
           => ( aa_int_o(Q,X2)
             => aa_int_o(Q,minus_minus_int(X2,D5)) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb2: int] :
                    ( member_int(Xb2,B)
                   => ( X4 != plus_plus_int(Xb2,Xa3) ) ) )
           => ( ( aa_int_o(P,X4)
                | aa_int_o(Q,X4) )
             => ( aa_int_o(P,minus_minus_int(X4,D5))
                | aa_int_o(Q,minus_minus_int(X4,D5)) ) ) ) ) ) ).

% bset(2)
tff(fact_2305_aset_I1_J,axiom,
    ! [D5: int,A: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D5))
             => ! [Xb: int] :
                  ( member_int(Xb,A)
                 => ( X2 != minus_minus_int(Xb,Xa) ) ) )
         => ( aa_int_o(P,X2)
           => aa_int_o(P,plus_plus_int(X2,D5)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb: int] :
                    ( member_int(Xb,A)
                   => ( X2 != minus_minus_int(Xb,Xa) ) ) )
           => ( aa_int_o(Q,X2)
             => aa_int_o(Q,plus_plus_int(X2,D5)) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb2: int] :
                    ( member_int(Xb2,A)
                   => ( X4 != minus_minus_int(Xb2,Xa3) ) ) )
           => ( ( aa_int_o(P,X4)
                & aa_int_o(Q,X4) )
             => ( aa_int_o(P,plus_plus_int(X4,D5))
                & aa_int_o(Q,plus_plus_int(X4,D5)) ) ) ) ) ) ).

% aset(1)
tff(fact_2306_aset_I2_J,axiom,
    ! [D5: int,A: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D5))
             => ! [Xb: int] :
                  ( member_int(Xb,A)
                 => ( X2 != minus_minus_int(Xb,Xa) ) ) )
         => ( aa_int_o(P,X2)
           => aa_int_o(P,plus_plus_int(X2,D5)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb: int] :
                    ( member_int(Xb,A)
                   => ( X2 != minus_minus_int(Xb,Xa) ) ) )
           => ( aa_int_o(Q,X2)
             => aa_int_o(Q,plus_plus_int(X2,D5)) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb2: int] :
                    ( member_int(Xb2,A)
                   => ( X4 != minus_minus_int(Xb2,Xa3) ) ) )
           => ( ( aa_int_o(P,X4)
                | aa_int_o(Q,X4) )
             => ( aa_int_o(P,plus_plus_int(X4,D5))
                | aa_int_o(Q,plus_plus_int(X4,D5)) ) ) ) ) ) ).

% aset(2)
tff(fact_2307_aset_I10_J,axiom,
    ! [D: int,D5: int,A: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D5)
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
             => ! [Xb2: int] :
                  ( member_int(Xb2,A)
                 => ( X4 != minus_minus_int(Xb2,Xa3) ) ) )
         => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),plus_plus_int(X4,T))
           => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),plus_plus_int(plus_plus_int(X4,D5),T)) ) ) ) ).

% aset(10)
tff(fact_2308_aset_I9_J,axiom,
    ! [D: int,D5: int,A: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D5)
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
             => ! [Xb2: int] :
                  ( member_int(Xb2,A)
                 => ( X4 != minus_minus_int(Xb2,Xa3) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),plus_plus_int(X4,T))
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),plus_plus_int(plus_plus_int(X4,D5),T)) ) ) ) ).

% aset(9)
tff(fact_2309_bset_I10_J,axiom,
    ! [D: int,D5: int,B: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D5)
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
             => ! [Xb2: int] :
                  ( member_int(Xb2,B)
                 => ( X4 != plus_plus_int(Xb2,Xa3) ) ) )
         => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),plus_plus_int(X4,T))
           => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),plus_plus_int(minus_minus_int(X4,D5),T)) ) ) ) ).

% bset(10)
tff(fact_2310_bset_I9_J,axiom,
    ! [D: int,D5: int,B: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D5)
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
             => ! [Xb2: int] :
                  ( member_int(Xb2,B)
                 => ( X4 != plus_plus_int(Xb2,Xa3) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),plus_plus_int(X4,T))
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),plus_plus_int(minus_minus_int(X4,D5),T)) ) ) ) ).

% bset(9)
tff(fact_2311_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),plus_plus_int(one_one_int,Na))
     => ( set_or1266510415728281911st_int(M,plus_plus_int(one_one_int,Na)) = insert_int(plus_plus_int(one_one_int,Na),set_or1266510415728281911st_int(M,Na)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_2312_bset_I3_J,axiom,
    ! [D5: int,T: int,B: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ( member_int(minus_minus_int(T,one_one_int),B)
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb2: int] :
                    ( member_int(Xb2,B)
                   => ( X4 != plus_plus_int(Xb2,Xa3) ) ) )
           => ( ( X4 = T )
             => ( minus_minus_int(X4,D5) = T ) ) ) ) ) ).

% bset(3)
tff(fact_2313_bset_I4_J,axiom,
    ! [D5: int,T: int,B: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ( member_int(T,B)
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb2: int] :
                    ( member_int(Xb2,B)
                   => ( X4 != plus_plus_int(Xb2,Xa3) ) ) )
           => ( ( X4 != T )
             => ( minus_minus_int(X4,D5) != T ) ) ) ) ) ).

% bset(4)
tff(fact_2314_bset_I5_J,axiom,
    ! [D5: int,B: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
             => ! [Xb2: int] :
                  ( member_int(Xb2,B)
                 => ( X4 != plus_plus_int(Xb2,Xa3) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),T)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,minus_minus_int(X4,D5)),T) ) ) ) ).

% bset(5)
tff(fact_2315_bset_I7_J,axiom,
    ! [D5: int,T: int,B: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ( member_int(T,B)
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb2: int] :
                    ( member_int(Xb2,B)
                   => ( X4 != plus_plus_int(Xb2,Xa3) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,T),X4)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,T),minus_minus_int(X4,D5)) ) ) ) ) ).

% bset(7)
tff(fact_2316_aset_I3_J,axiom,
    ! [D5: int,T: int,A: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ( member_int(plus_plus_int(T,one_one_int),A)
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb2: int] :
                    ( member_int(Xb2,A)
                   => ( X4 != minus_minus_int(Xb2,Xa3) ) ) )
           => ( ( X4 = T )
             => ( plus_plus_int(X4,D5) = T ) ) ) ) ) ).

% aset(3)
tff(fact_2317_aset_I4_J,axiom,
    ! [D5: int,T: int,A: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ( member_int(T,A)
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb2: int] :
                    ( member_int(Xb2,A)
                   => ( X4 != minus_minus_int(Xb2,Xa3) ) ) )
           => ( ( X4 != T )
             => ( plus_plus_int(X4,D5) != T ) ) ) ) ) ).

% aset(4)
tff(fact_2318_aset_I5_J,axiom,
    ! [D5: int,T: int,A: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ( member_int(T,A)
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb2: int] :
                    ( member_int(Xb2,A)
                   => ( X4 != minus_minus_int(Xb2,Xa3) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),T)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,plus_plus_int(X4,D5)),T) ) ) ) ) ).

% aset(5)
tff(fact_2319_aset_I7_J,axiom,
    ! [D5: int,A: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
             => ! [Xb2: int] :
                  ( member_int(Xb2,A)
                 => ( X4 != minus_minus_int(Xb2,Xa3) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,T),X4)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,T),plus_plus_int(X4,D5)) ) ) ) ).

% aset(7)
tff(fact_2320_periodic__finite__ex,axiom,
    ! [D: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X2: int,K4: int] :
            ( aa_int_o(P,X2)
          <=> aa_int_o(P,minus_minus_int(X2,times_times_int(K4,D))) )
       => ( ? [X_1: int] : aa_int_o(P,X_1)
        <=> ? [X5: int] :
              ( member_int(X5,set_or1266510415728281911st_int(one_one_int,D))
              & aa_int_o(P,X5) ) ) ) ) ).

% periodic_finite_ex
tff(fact_2321_fact__eq__fact__times,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),groups708209901874060359at_nat(aTP_Lamm_db,set_or1269000886237332187st_nat(aa_nat_nat(suc,Na),M))) ) ) ).

% fact_eq_fact_times
tff(fact_2322_simp__from__to,axiom,
    ! [I: int,J: int] :
      set_or1266510415728281911st_int(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I),bot_bot_set_int,insert_int(I,set_or1266510415728281911st_int(plus_plus_int(I,one_one_int),J))) ).

% simp_from_to
tff(fact_2323_bset_I6_J,axiom,
    ! [D5: int,B: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
             => ! [Xb2: int] :
                  ( member_int(Xb2,B)
                 => ( X4 != plus_plus_int(Xb2,Xa3) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),T)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,minus_minus_int(X4,D5)),T) ) ) ) ).

% bset(6)
tff(fact_2324_bset_I8_J,axiom,
    ! [D5: int,T: int,B: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ( member_int(minus_minus_int(T,one_one_int),B)
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb2: int] :
                    ( member_int(Xb2,B)
                   => ( X4 != plus_plus_int(Xb2,Xa3) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),X4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),minus_minus_int(X4,D5)) ) ) ) ) ).

% bset(8)
tff(fact_2325_aset_I6_J,axiom,
    ! [D5: int,T: int,A: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ( member_int(plus_plus_int(T,one_one_int),A)
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
               => ! [Xb2: int] :
                    ( member_int(Xb2,A)
                   => ( X4 != minus_minus_int(Xb2,Xa3) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),T)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,plus_plus_int(X4,D5)),T) ) ) ) ) ).

% aset(6)
tff(fact_2326_aset_I8_J,axiom,
    ! [D5: int,A: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D5))
             => ! [Xb2: int] :
                  ( member_int(Xb2,A)
                 => ( X4 != minus_minus_int(Xb2,Xa3) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),X4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),plus_plus_int(X4,D5)) ) ) ) ).

% aset(8)
tff(fact_2327_cppi,axiom,
    ! [D5: int,P: fun_int_o,P3: fun_int_o,A: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ( ? [Z2: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X2)
           => ( aa_int_o(P,X2)
            <=> aa_int_o(P3,X2) ) )
       => ( ! [X2: int] :
              ( ! [Xa: int] :
                  ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D5))
                 => ! [Xb: int] :
                      ( member_int(Xb,A)
                     => ( X2 != minus_minus_int(Xb,Xa) ) ) )
             => ( aa_int_o(P,X2)
               => aa_int_o(P,plus_plus_int(X2,D5)) ) )
         => ( ! [X2: int,K4: int] :
                ( aa_int_o(P3,X2)
              <=> aa_int_o(P3,minus_minus_int(X2,times_times_int(K4,D5))) )
           => ( ? [X_1: int] : aa_int_o(P,X_1)
            <=> ( ? [X5: int] :
                    ( member_int(X5,set_or1266510415728281911st_int(one_one_int,D5))
                    & aa_int_o(P3,X5) )
                | ? [X5: int] :
                    ( member_int(X5,set_or1266510415728281911st_int(one_one_int,D5))
                    & ? [Xa4: int] :
                        ( member_int(Xa4,A)
                        & aa_int_o(P,minus_minus_int(Xa4,X5)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_2328_cpmi,axiom,
    ! [D5: int,P: fun_int_o,P3: fun_int_o,B: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D5)
     => ( ? [Z2: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z2)
           => ( aa_int_o(P,X2)
            <=> aa_int_o(P3,X2) ) )
       => ( ! [X2: int] :
              ( ! [Xa: int] :
                  ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D5))
                 => ! [Xb: int] :
                      ( member_int(Xb,B)
                     => ( X2 != plus_plus_int(Xb,Xa) ) ) )
             => ( aa_int_o(P,X2)
               => aa_int_o(P,minus_minus_int(X2,D5)) ) )
         => ( ! [X2: int,K4: int] :
                ( aa_int_o(P3,X2)
              <=> aa_int_o(P3,minus_minus_int(X2,times_times_int(K4,D5))) )
           => ( ? [X_1: int] : aa_int_o(P,X_1)
            <=> ( ? [X5: int] :
                    ( member_int(X5,set_or1266510415728281911st_int(one_one_int,D5))
                    & aa_int_o(P3,X5) )
                | ? [X5: int] :
                    ( member_int(X5,set_or1266510415728281911st_int(one_one_int,D5))
                    & ? [Xa4: int] :
                        ( member_int(Xa4,B)
                        & aa_int_o(P,plus_plus_int(Xa4,X5)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_2329_fact__div__fact,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_nat(divide_divide_nat(aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) = groups708209901874060359at_nat(aTP_Lamm_db,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat),M)) ) ) ).

% fact_div_fact
tff(fact_2330_bij__betw__nth__root__unity,axiom,
    ! [C: complex,Na: nat] :
      ( ( C != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => bij_be1856998921033663316omplex(times_times_complex(aa_complex_complex(times_times_complex(real_V4546457046886955230omplex(aa_real_real(aa_nat_fun_real_real(root,Na),real_V1022390504157884413omplex(C)))),cis(aa_real_real(divide_divide_real(arg(C)),aa_nat_real(semiri5074537144036343181t_real,Na))))),collect_complex(aTP_Lamm_cr(Na)),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_de(C),Na))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_2331_drop__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),bit_se8568078237143864401it_int(Na,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_2332_drop__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,bit_se8568078237143864401it_int(Na,K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% drop_bit_negative_int_iff
tff(fact_2333_drop__bit__minus__one,axiom,
    ! [Na: nat] : bit_se8568078237143864401it_int(Na,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% drop_bit_minus_one
tff(fact_2334_drop__bit__Suc__minus__bit0,axiom,
    ! [Na: nat,K: num] : bit_se8568078237143864401it_int(aa_nat_nat(suc,Na),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(K)))) = bit_se8568078237143864401it_int(Na,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ).

% drop_bit_Suc_minus_bit0
tff(fact_2335_real__root__le__iff,axiom,
    ! [Na: nat,X: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) ) ) ).

% real_root_le_iff
tff(fact_2336_drop__bit__of__Suc__0,axiom,
    ! [Na: nat] : aa_nat_nat(bit_se8570568707652914677it_nat(Na),aa_nat_nat(suc,zero_zero_nat)) = zero_n2687167440665602831ol_nat(Na = zero_zero_nat) ).

% drop_bit_of_Suc_0
tff(fact_2337_real__root__ge__0__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y) ) ) ).

% real_root_ge_0_iff
tff(fact_2338_real__root__le__0__iff,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real) ) ) ).

% real_root_le_0_iff
tff(fact_2339_real__root__ge__1__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y) ) ) ).

% real_root_ge_1_iff
tff(fact_2340_real__root__le__1__iff,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real) ) ) ).

% real_root_le_1_iff
tff(fact_2341_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : bit_se8568078237143864401it_int(numeral_numeral_nat(L),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(K)))) = bit_se8568078237143864401it_int(pred_numeral(L),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ).

% drop_bit_numeral_minus_bit0
tff(fact_2342_drop__bit__Suc__minus__bit1,axiom,
    ! [Na: nat,K: num] : bit_se8568078237143864401it_int(aa_nat_nat(suc,Na),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit1(K)))) = bit_se8568078237143864401it_int(Na,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_2343_real__root__pow__pos2,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),Na) = X ) ) ) ).

% real_root_pow_pos2
tff(fact_2344_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : bit_se8568078237143864401it_int(numeral_numeral_nat(L),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit1(K)))) = bit_se8568078237143864401it_int(pred_numeral(L),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K)))) ).

% drop_bit_numeral_minus_bit1
tff(fact_2345_drop__bit__nat__eq,axiom,
    ! [Na: nat,K: int] : aa_nat_nat(bit_se8570568707652914677it_nat(Na),aa_int_nat(nat2,K)) = aa_int_nat(nat2,bit_se8568078237143864401it_int(Na,K)) ).

% drop_bit_nat_eq
tff(fact_2346_real__root__mult__exp,axiom,
    ! [M: nat,Na: nat,X: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)),X) = aa_real_real(aa_nat_fun_real_real(root,M),aa_real_real(aa_nat_fun_real_real(root,Na),X)) ).

% real_root_mult_exp
tff(fact_2347_real__root__pos__pos__le,axiom,
    ! [X: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Na),X)) ) ).

% real_root_pos_pos_le
tff(fact_2348_real__root__le__mono,axiom,
    ! [Na: nat,X: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_real_real(aa_nat_fun_real_real(root,Na),Y)) ) ) ).

% real_root_le_mono
tff(fact_2349_prod__int__eq,axiom,
    ! [I: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(I,J)) = groups1705073143266064639nt_int(aTP_Lamm_dd,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I),aa_nat_int(semiri1314217659103216013at_int,J))) ).

% prod_int_eq
tff(fact_2350_sqrt__def,axiom,
    sqrt = aa_nat_fun_real_real(root,numeral_numeral_nat(bit0(one))) ).

% sqrt_def
tff(fact_2351_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J))) = groups1705073143266064639nt_int(aTP_Lamm_dd,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)))) ).

% prod_int_plus_eq
tff(fact_2352_real__root__pos__pos,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Na),X)) ) ) ).

% real_root_pos_pos
tff(fact_2353_real__root__decreasing,axiom,
    ! [Na: nat,N3: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N3),X)),aa_real_real(aa_nat_fun_real_real(root,Na),X)) ) ) ) ).

% real_root_decreasing
tff(fact_2354_odd__real__root__pow,axiom,
    ! [Na: nat,X: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),Na) = X ) ) ).

% odd_real_root_pow
tff(fact_2355_odd__real__root__unique,axiom,
    ! [Na: nat,Y: real,X: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na)
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Na) = X )
       => ( aa_real_real(aa_nat_fun_real_real(root,Na),X) = Y ) ) ) ).

% odd_real_root_unique
tff(fact_2356_odd__real__root__power__cancel,axiom,
    ! [Na: nat,X: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na)
     => ( aa_real_real(aa_nat_fun_real_real(root,Na),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na)) = X ) ) ).

% odd_real_root_power_cancel
tff(fact_2357_real__root__pos__unique,axiom,
    ! [Na: nat,Y: real,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Na) = X )
         => ( aa_real_real(aa_nat_fun_real_real(root,Na),X) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_2358_real__root__power__cancel,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
       => ( aa_real_real(aa_nat_fun_real_real(root,Na),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na)) = X ) ) ) ).

% real_root_power_cancel
tff(fact_2359_bin__rest__code,axiom,
    ! [I: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,I),aa_num_int(numeral_numeral_int,bit0(one))) = bit_se8568078237143864401it_int(one_one_nat,I) ).

% bin_rest_code
tff(fact_2360_drop__bit__int__def,axiom,
    ! [Na: nat,K: int] : bit_se8568078237143864401it_int(Na,K) = aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na)) ).

% drop_bit_int_def
tff(fact_2361_drop__bit__nat__def,axiom,
    ! [Na: nat,M: nat] : aa_nat_nat(bit_se8570568707652914677it_nat(Na),M) = aa_nat_nat(divide_divide_nat(M),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na)) ).

% drop_bit_nat_def
tff(fact_2362_real__root__increasing,axiom,
    ! [Na: nat,N3: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_real_real(aa_nat_fun_real_real(root,N3),X)) ) ) ) ) ).

% real_root_increasing
tff(fact_2363_log__root,axiom,
    ! [Na: nat,Aa: real,Ba: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
       => ( aa_real_real(log(Ba),aa_real_real(aa_nat_fun_real_real(root,Na),Aa)) = aa_real_real(divide_divide_real(aa_real_real(log(Ba),Aa)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ) ).

% log_root
tff(fact_2364_log__base__root,axiom,
    ! [Na: nat,Ba: real,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ba)
       => ( aa_real_real(log(aa_real_real(aa_nat_fun_real_real(root,Na),Ba)),X) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(log(Ba),X)) ) ) ) ).

% log_base_root
tff(fact_2365_ln__root,axiom,
    ! [Na: nat,Ba: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ba)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_nat_fun_real_real(root,Na),Ba)) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,Ba)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ) ).

% ln_root
tff(fact_2366_root__powr__inverse,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_real(aa_nat_fun_real_real(root,Na),X) = aa_real_real(powr_real(X),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,Na))) ) ) ) ).

% root_powr_inverse
tff(fact_2367_upto__aux__rec,axiom,
    ! [I: int,J: int,Js: list_int] :
      upto_aux(I,J,Js) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I),Js,upto_aux(I,minus_minus_int(J,one_one_int),cons_int(J,Js))) ).

% upto_aux_rec
tff(fact_2368_div__half__nat,axiom,
    ! [Y: nat,X: nat] :
      ( ( Y != zero_zero_nat )
     => ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(divide_divide_nat(X),Y)),aa_nat_nat(modulo_modulo_nat(X),Y)) = $let(
            q: nat,
            q:= aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(divide_divide_nat(aa_nat_nat(divide_divide_nat(X),numeral_numeral_nat(bit0(one)))),Y)),
            $let(
              r: nat,
              r:= aa_nat_nat(minus_minus_nat(X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,q),Y)),
              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),r),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,q),one_one_nat)),aa_nat_nat(minus_minus_nat(r),Y)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(q),r)) ) ) ) ) ).

% div_half_nat
tff(fact_2369_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod_nat_nat] :
      ~ ! [K4: nat,M3: nat] : X != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(K4),M3) ).

% prod_decode_aux.cases
tff(fact_2370_int__lsb__numeral_I2_J,axiom,
    aa_int_o(least_4859182151741483524sb_int,one_one_int) ).

% int_lsb_numeral(2)
tff(fact_2371_int__lsb__numeral_I6_J,axiom,
    ! [W: num] : ~ aa_int_o(least_4859182151741483524sb_int,aa_num_int(numeral_numeral_int,bit0(W))) ).

% int_lsb_numeral(6)
tff(fact_2372_int__lsb__numeral_I3_J,axiom,
    aa_int_o(least_4859182151741483524sb_int,aa_num_int(numeral_numeral_int,one)) ).

% int_lsb_numeral(3)
tff(fact_2373_int__lsb__numeral_I7_J,axiom,
    ! [W: num] : aa_int_o(least_4859182151741483524sb_int,aa_num_int(numeral_numeral_int,bit1(W))) ).

% int_lsb_numeral(7)
tff(fact_2374_int__lsb__numeral_I4_J,axiom,
    aa_int_o(least_4859182151741483524sb_int,aa_int_int(uminus_uminus_int,one_one_int)) ).

% int_lsb_numeral(4)
tff(fact_2375_int__lsb__numeral_I8_J,axiom,
    ! [W: num] : ~ aa_int_o(least_4859182151741483524sb_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(W)))) ).

% int_lsb_numeral(8)
tff(fact_2376_int__lsb__numeral_I5_J,axiom,
    aa_int_o(least_4859182151741483524sb_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))) ).

% int_lsb_numeral(5)
tff(fact_2377_int__lsb__numeral_I9_J,axiom,
    ! [W: num] : aa_int_o(least_4859182151741483524sb_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit1(W)))) ).

% int_lsb_numeral(9)
tff(fact_2378_xor__num_Ocases,axiom,
    ! [X: product_prod_num_num] :
      ( ( X != product_Pair_num_num(one,one) )
     => ( ! [N: num] : X != product_Pair_num_num(one,bit0(N))
       => ( ! [N: num] : X != product_Pair_num_num(one,bit1(N))
         => ( ! [M3: num] : X != product_Pair_num_num(bit0(M3),one)
           => ( ! [M3: num,N: num] : X != product_Pair_num_num(bit0(M3),bit0(N))
             => ( ! [M3: num,N: num] : X != product_Pair_num_num(bit0(M3),bit1(N))
               => ( ! [M3: num] : X != product_Pair_num_num(bit1(M3),one)
                 => ( ! [M3: num,N: num] : X != product_Pair_num_num(bit1(M3),bit0(N))
                   => ~ ! [M3: num,N: num] : X != product_Pair_num_num(bit1(M3),bit1(N)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_2379_bin__last__conv__lsb,axiom,
    aTP_Lamm_df = least_4859182151741483524sb_int ).

% bin_last_conv_lsb
tff(fact_2380_neg__eucl__rel__int__mult__2,axiom,
    ! [Ba: int,Aa: int,Q3: int,R: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),zero_zero_int)
     => ( eucl_rel_int(plus_plus_int(Aa,one_one_int),Ba,aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),R))
       => eucl_rel_int(plus_plus_int(one_one_int,times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),Aa)),times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),Ba),aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),minus_minus_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),R),one_one_int))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_2381_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q3: int] :
      ( ( L != zero_zero_int )
     => ( ( K = times_times_int(Q3,L) )
       => eucl_rel_int(K,L,aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),zero_zero_int)) ) ) ).

% eucl_rel_int_dividesI
tff(fact_2382_divmod__int__def,axiom,
    ! [M: num,Na: num] : unique5052692396658037445od_int(M,Na) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na))),modulo_modulo_int(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,Na))) ).

% divmod_int_def
tff(fact_2383_divmod_H__nat__def,axiom,
    ! [M: num,Na: num] : unique5055182867167087721od_nat(M,Na) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(divide_divide_nat(numeral_numeral_nat(M)),numeral_numeral_nat(Na))),aa_nat_nat(modulo_modulo_nat(numeral_numeral_nat(M)),numeral_numeral_nat(Na))) ).

% divmod'_nat_def
tff(fact_2384_zminus1__lemma,axiom,
    ! [Aa: int,Ba: int,Q3: int,R: int] :
      ( eucl_rel_int(Aa,Ba,aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),R))
     => ( ( Ba != zero_zero_int )
       => eucl_rel_int(aa_int_int(uminus_uminus_int,Aa),Ba,
            aa_int7462565939943104719nt_int(product_Pair_int_int($ite(R = zero_zero_int,aa_int_int(uminus_uminus_int,Q3),minus_minus_int(aa_int_int(uminus_uminus_int,Q3),one_one_int))),
              $ite(R = zero_zero_int,zero_zero_int,minus_minus_int(Ba,R)))) ) ) ).

% zminus1_lemma
tff(fact_2385_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q3: int,R: int] :
      ( eucl_rel_int(K,L,aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),R))
    <=> ( ( K = plus_plus_int(times_times_int(L,Q3),R) )
        & $ite(
            aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L),
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R)
            & aa_int_o(aa_int_fun_int_o(ord_less_int,R),L) ),
            $ite(
              aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int),
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),R)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R),zero_zero_int) ),
              Q3 = zero_zero_int ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_2386_eucl__rel__int__remainderI,axiom,
    ! [R: int,L: int,K: int,Q3: int] :
      ( ( sgn_sgn_int(R) = sgn_sgn_int(L) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R)),aa_int_int(abs_abs_int,L))
       => ( ( K = plus_plus_int(times_times_int(Q3,L),R) )
         => eucl_rel_int(K,L,aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),R)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_2387_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( eucl_rel_int(A1,A22,A32)
     => ( ( ( A22 = zero_zero_int )
         => ( A32 != aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),A1) ) )
       => ( ! [Q5: int] :
              ( ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q5),zero_zero_int) )
             => ( ( A22 != zero_zero_int )
               => ( A1 != times_times_int(Q5,A22) ) ) )
         => ~ ! [R5: int,Q5: int] :
                ( ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q5),R5) )
               => ( ( sgn_sgn_int(R5) = sgn_sgn_int(A22) )
                 => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R5)),aa_int_int(abs_abs_int,A22))
                   => ( A1 != plus_plus_int(times_times_int(Q5,A22),R5) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_2388_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( eucl_rel_int(A1,A22,A32)
    <=> ( ? [K2: int] :
            ( ( A1 = K2 )
            & ( A22 = zero_zero_int )
            & ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),K2) ) )
        | ? [L3: int,K2: int,Q4: int] :
            ( ( A1 = K2 )
            & ( A22 = L3 )
            & ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),zero_zero_int) )
            & ( L3 != zero_zero_int )
            & ( K2 = times_times_int(Q4,L3) ) )
        | ? [R6: int,L3: int,K2: int,Q4: int] :
            ( ( A1 = K2 )
            & ( A22 = L3 )
            & ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),R6) )
            & ( sgn_sgn_int(R6) = sgn_sgn_int(L3) )
            & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R6)),aa_int_int(abs_abs_int,L3))
            & ( K2 = plus_plus_int(times_times_int(Q4,L3),R6) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_2389_pos__eucl__rel__int__mult__2,axiom,
    ! [Ba: int,Aa: int,Q3: int,R: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ba)
     => ( eucl_rel_int(Aa,Ba,aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),R))
       => eucl_rel_int(plus_plus_int(one_one_int,times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),Aa)),times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),Ba),aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),plus_plus_int(one_one_int,times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),R)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_2390_one__div__minus__numeral,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,Na))) ).

% one_div_minus_numeral
tff(fact_2391_minus__one__div__numeral,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Na)) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,Na))) ).

% minus_one_div_numeral
tff(fact_2392_Divides_Oadjust__div__eq,axiom,
    ! [Q3: int,R: int] : adjust_div(aa_int7462565939943104719nt_int(product_Pair_int_int(Q3),R)) = plus_plus_int(Q3,aa_o_int(zero_n2684676970156552555ol_int,R != zero_zero_int)) ).

% Divides.adjust_div_eq
tff(fact_2393_numeral__div__minus__numeral,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(M,Na))) ).

% numeral_div_minus_numeral
tff(fact_2394_minus__numeral__div__numeral,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Na)) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(M,Na))) ).

% minus_numeral_div_numeral
tff(fact_2395_and__int_Opelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( bit_se725231765392027082nd_int(X,Xa2) = Y )
     => ( accp_P1096762738010456898nt_int(bit_and_int_rel,aa_int7462565939943104719nt_int(product_Pair_int_int(X),Xa2))
       => ~ ( ( Y = $ite(
                  ( member_int(X,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                  & member_int(Xa2,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
                  aa_int_int(uminus_uminus_int,
                    aa_o_int(zero_n2684676970156552555ol_int,
                      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),X)
                      & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),Xa2) ))),
                  plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
                      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),X)
                      & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),Xa2) )),times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),bit_se725231765392027082nd_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa2),aa_num_int(numeral_numeral_int,bit0(one)))))) ) )
           => ~ accp_P1096762738010456898nt_int(bit_and_int_rel,aa_int7462565939943104719nt_int(product_Pair_int_int(X),Xa2)) ) ) ) ).

% and_int.pelims
tff(fact_2396_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( accp_P1096762738010456898nt_int(bit_and_int_rel,aa_int7462565939943104719nt_int(product_Pair_int_int(K),L))
     => ( bit_se725231765392027082nd_int(K,L) = $ite(
            ( member_int(K,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & member_int(L,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
            aa_int_int(uminus_uminus_int,
              aa_o_int(zero_n2684676970156552555ol_int,
                ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),K)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),L) ))),
            plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
                ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),K)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),L) )),times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),bit_se725231765392027082nd_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,bit0(one)))))) ) ) ) ).

% and_int.psimps
tff(fact_2397_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_o] :
      ( accp_P1096762738010456898nt_int(bit_and_int_rel,aa_int7462565939943104719nt_int(product_Pair_int_int(A0),A1))
     => ( ! [K4: int,L2: int] :
            ( accp_P1096762738010456898nt_int(bit_and_int_rel,aa_int7462565939943104719nt_int(product_Pair_int_int(K4),L2))
           => ( ( ~ ( member_int(K4,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                    & member_int(L2,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
               => aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,K4),aa_num_int(numeral_numeral_int,bit0(one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L2),aa_num_int(numeral_numeral_int,bit0(one)))) )
             => aa_int_o(aa_int_fun_int_o(P,K4),L2) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_2398_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_o] :
      ( accp_P1096762738010456898nt_int(upto_rel,aa_int7462565939943104719nt_int(product_Pair_int_int(A0),A1))
     => ( ! [I2: int,J2: int] :
            ( accp_P1096762738010456898nt_int(upto_rel,aa_int7462565939943104719nt_int(product_Pair_int_int(I2),J2))
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J2)
               => aa_int_o(aa_int_fun_int_o(P,plus_plus_int(I2,one_one_int)),J2) )
             => aa_int_o(aa_int_fun_int_o(P,I2),J2) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% upto.pinduct
tff(fact_2399_push__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),bit_se545348938243370406it_int(Na,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% push_bit_nonnegative_int_iff
tff(fact_2400_push__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,bit_se545348938243370406it_int(Na,K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% push_bit_negative_int_iff
tff(fact_2401_push__bit__of__Suc__0,axiom,
    ! [Na: nat] : aa_nat_nat(bit_se547839408752420682it_nat(Na),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na) ).

% push_bit_of_Suc_0
tff(fact_2402_push__bit__nat__eq,axiom,
    ! [Na: nat,K: int] : aa_nat_nat(bit_se547839408752420682it_nat(Na),aa_int_nat(nat2,K)) = aa_int_nat(nat2,bit_se545348938243370406it_int(Na,K)) ).

% push_bit_nat_eq
tff(fact_2403_drop__bit__push__bit__int,axiom,
    ! [M: nat,Na: nat,K: int] : bit_se8568078237143864401it_int(M,bit_se545348938243370406it_int(Na,K)) = bit_se8568078237143864401it_int(aa_nat_nat(minus_minus_nat(M),Na),bit_se545348938243370406it_int(aa_nat_nat(minus_minus_nat(Na),M),K)) ).

% drop_bit_push_bit_int
tff(fact_2404_set__bit__nat__def,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(bit_se7882103937844011126it_nat(M),Na) = aa_nat_nat(bit_se1412395901928357646or_nat(Na),aa_nat_nat(bit_se547839408752420682it_nat(M),one_one_nat)) ).

% set_bit_nat_def
tff(fact_2405_flip__bit__nat__def,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(bit_se2161824704523386999it_nat(M),Na) = aa_nat_nat(bit_se6528837805403552850or_nat(Na),aa_nat_nat(bit_se547839408752420682it_nat(M),one_one_nat)) ).

% flip_bit_nat_def
tff(fact_2406_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,Na: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,bit_se545348938243370406it_int(M,K)),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
        & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),aa_nat_nat(minus_minus_nat(Na),M)) ) ) ).

% bit_push_bit_iff_int
tff(fact_2407_Bit__Operations_Oset__bit__int__def,axiom,
    ! [Na: nat,K: int] : bit_se7879613467334960850it_int(Na,K) = bit_se1409905431419307370or_int(K,bit_se545348938243370406it_int(Na,one_one_int)) ).

% Bit_Operations.set_bit_int_def
tff(fact_2408_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q3: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(bit_se547839408752420682it_nat(M),Q3)),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Q3),aa_nat_nat(minus_minus_nat(Na),M)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_2409_flip__bit__int__def,axiom,
    ! [Na: nat,K: int] : bit_se2159334234014336723it_int(Na,K) = bit_se6526347334894502574or_int(K,bit_se545348938243370406it_int(Na,one_one_int)) ).

% flip_bit_int_def
tff(fact_2410_unset__bit__int__def,axiom,
    ! [Na: nat,K: int] : bit_se4203085406695923979it_int(Na,K) = bit_se725231765392027082nd_int(K,bit_ri7919022796975470100ot_int(bit_se545348938243370406it_int(Na,one_one_int))) ).

% unset_bit_int_def
tff(fact_2411_push__bit__int__def,axiom,
    ! [Na: nat,K: int] : bit_se545348938243370406it_int(Na,K) = times_times_int(K,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na)) ).

% push_bit_int_def
tff(fact_2412_push__bit__nat__def,axiom,
    ! [Na: nat,M: nat] : aa_nat_nat(bit_se547839408752420682it_nat(Na),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na)) ).

% push_bit_nat_def
tff(fact_2413_push__bit__minus__one,axiom,
    ! [Na: nat] : bit_se545348938243370406it_int(Na,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),Na)) ).

% push_bit_minus_one
tff(fact_2414_divmod__BitM__2__eq,axiom,
    ! [M: num] : unique5052692396658037445od_int(bitM(M),bit0(one)) = aa_int7462565939943104719nt_int(product_Pair_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,M),one_one_int)),one_one_int) ).

% divmod_BitM_2_eq
tff(fact_2415_pred__numeral__simps_I2_J,axiom,
    ! [K: num] : pred_numeral(bit0(K)) = numeral_numeral_nat(bitM(K)) ).

% pred_numeral_simps(2)
tff(fact_2416_semiring__norm_I26_J,axiom,
    bitM(one) = one ).

% semiring_norm(26)
tff(fact_2417_semiring__norm_I28_J,axiom,
    ! [Na: num] : bitM(bit1(Na)) = bit1(bit0(Na)) ).

% semiring_norm(28)
tff(fact_2418_semiring__norm_I27_J,axiom,
    ! [Na: num] : bitM(bit0(Na)) = bit1(bitM(Na)) ).

% semiring_norm(27)
tff(fact_2419_inc__BitM__eq,axiom,
    ! [Na: num] : inc(bitM(Na)) = bit0(Na) ).

% inc_BitM_eq
tff(fact_2420_BitM__inc__eq,axiom,
    ! [Na: num] : bitM(inc(Na)) = bit1(Na) ).

% BitM_inc_eq
tff(fact_2421_eval__nat__numeral_I2_J,axiom,
    ! [Na: num] : numeral_numeral_nat(bit0(Na)) = aa_nat_nat(suc,numeral_numeral_nat(bitM(Na))) ).

% eval_nat_numeral(2)
tff(fact_2422_BitM__plus__one,axiom,
    ! [Na: num] : plus_plus_num(bitM(Na),one) = bit0(Na) ).

% BitM_plus_one
tff(fact_2423_one__plus__BitM,axiom,
    ! [Na: num] : plus_plus_num(one,bitM(Na)) = bit0(Na) ).

% one_plus_BitM
tff(fact_2424_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = produc2626176000494625587at_nat(aTP_Lamm_dg(L),Qr) ).

% divmod_step_nat_def
tff(fact_2425_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_dh(L),Qr) ).

% divmod_step_int_def
tff(fact_2426_divmod__nat__if,axiom,
    ! [M: nat,Na: nat] :
      divmod_nat(M,Na) = $ite(
        ( ( Na = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ),
        aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),M),
        produc2626176000494625587at_nat(aTP_Lamm_di,divmod_nat(aa_nat_nat(minus_minus_nat(M),Na),Na)) ) ).

% divmod_nat_if
tff(fact_2427_or__minus__numerals_I1_J,axiom,
    ! [Na: num] : bit_se1409905431419307370or_int(one_one_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(Na)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit_or_not_num_neg(one,bitM(Na)))) ).

% or_minus_numerals(1)
tff(fact_2428_or__minus__numerals_I8_J,axiom,
    ! [Na: num,M: num] : bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit1(Na))),aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit_or_not_num_neg(M,bit0(Na)))) ).

% or_minus_numerals(8)
tff(fact_2429_or__minus__numerals_I4_J,axiom,
    ! [M: num,Na: num] : bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit1(Na)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit_or_not_num_neg(M,bit0(Na)))) ).

% or_minus_numerals(4)
tff(fact_2430_or__minus__numerals_I7_J,axiom,
    ! [Na: num,M: num] : bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(Na))),aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit_or_not_num_neg(M,bitM(Na)))) ).

% or_minus_numerals(7)
tff(fact_2431_or__minus__numerals_I3_J,axiom,
    ! [M: num,Na: num] : bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(Na)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit_or_not_num_neg(M,bitM(Na)))) ).

% or_minus_numerals(3)
tff(fact_2432_or__minus__numerals_I5_J,axiom,
    ! [Na: num] : bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit0(Na))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit_or_not_num_neg(one,bitM(Na)))) ).

% or_minus_numerals(5)
tff(fact_2433_or__not__num__neg_Osimps_I1_J,axiom,
    bit_or_not_num_neg(one,one) = one ).

% or_not_num_neg.simps(1)
tff(fact_2434_or__not__num__neg_Osimps_I4_J,axiom,
    ! [Na: num] : bit_or_not_num_neg(bit0(Na),one) = bit0(one) ).

% or_not_num_neg.simps(4)
tff(fact_2435_or__not__num__neg_Osimps_I6_J,axiom,
    ! [Na: num,M: num] : bit_or_not_num_neg(bit0(Na),bit1(M)) = bit0(bit_or_not_num_neg(Na,M)) ).

% or_not_num_neg.simps(6)
tff(fact_2436_or__not__num__neg_Osimps_I3_J,axiom,
    ! [M: num] : bit_or_not_num_neg(one,bit1(M)) = bit1(M) ).

% or_not_num_neg.simps(3)
tff(fact_2437_or__not__num__neg_Osimps_I7_J,axiom,
    ! [Na: num] : bit_or_not_num_neg(bit1(Na),one) = one ).

% or_not_num_neg.simps(7)
tff(fact_2438_or__not__num__neg_Osimps_I5_J,axiom,
    ! [Na: num,M: num] : bit_or_not_num_neg(bit0(Na),bit0(M)) = bitM(bit_or_not_num_neg(Na,M)) ).

% or_not_num_neg.simps(5)
tff(fact_2439_or__not__num__neg_Osimps_I9_J,axiom,
    ! [Na: num,M: num] : bit_or_not_num_neg(bit1(Na),bit1(M)) = bitM(bit_or_not_num_neg(Na,M)) ).

% or_not_num_neg.simps(9)
tff(fact_2440_or__not__num__neg_Osimps_I2_J,axiom,
    ! [M: num] : bit_or_not_num_neg(one,bit0(M)) = bit1(M) ).

% or_not_num_neg.simps(2)
tff(fact_2441_or__not__num__neg_Osimps_I8_J,axiom,
    ! [Na: num,M: num] : bit_or_not_num_neg(bit1(Na),bit0(M)) = bitM(bit_or_not_num_neg(Na,M)) ).

% or_not_num_neg.simps(8)
tff(fact_2442_or__not__num__neg_Oelims,axiom,
    ! [X: num,Xa2: num,Y: num] :
      ( ( bit_or_not_num_neg(X,Xa2) = Y )
     => ( ( ( X = one )
         => ( ( Xa2 = one )
           => ( Y != one ) ) )
       => ( ( ( X = one )
           => ! [M3: num] :
                ( ( Xa2 = bit0(M3) )
               => ( Y != bit1(M3) ) ) )
         => ( ( ( X = one )
             => ! [M3: num] :
                  ( ( Xa2 = bit1(M3) )
                 => ( Y != bit1(M3) ) ) )
           => ( ( ? [N: num] : X = bit0(N)
               => ( ( Xa2 = one )
                 => ( Y != bit0(one) ) ) )
             => ( ! [N: num] :
                    ( ( X = bit0(N) )
                   => ! [M3: num] :
                        ( ( Xa2 = bit0(M3) )
                       => ( Y != bitM(bit_or_not_num_neg(N,M3)) ) ) )
               => ( ! [N: num] :
                      ( ( X = bit0(N) )
                     => ! [M3: num] :
                          ( ( Xa2 = bit1(M3) )
                         => ( Y != bit0(bit_or_not_num_neg(N,M3)) ) ) )
                 => ( ( ? [N: num] : X = bit1(N)
                     => ( ( Xa2 = one )
                       => ( Y != one ) ) )
                   => ( ! [N: num] :
                          ( ( X = bit1(N) )
                         => ! [M3: num] :
                              ( ( Xa2 = bit0(M3) )
                             => ( Y != bitM(bit_or_not_num_neg(N,M3)) ) ) )
                     => ~ ! [N: num] :
                            ( ( X = bit1(N) )
                           => ! [M3: num] :
                                ( ( Xa2 = bit1(M3) )
                               => ( Y != bitM(bit_or_not_num_neg(N,M3)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
tff(fact_2443_numeral__or__not__num__eq,axiom,
    ! [M: num,Na: num] : aa_num_int(numeral_numeral_int,bit_or_not_num_neg(M,Na)) = aa_int_int(uminus_uminus_int,bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,Na)))) ).

% numeral_or_not_num_eq
tff(fact_2444_int__numeral__not__or__num__neg,axiom,
    ! [M: num,Na: num] : bit_se1409905431419307370or_int(bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit_or_not_num_neg(Na,M))) ).

% int_numeral_not_or_num_neg
tff(fact_2445_int__numeral__or__not__num__neg,axiom,
    ! [M: num,Na: num] : bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M),bit_ri7919022796975470100ot_int(aa_num_int(numeral_numeral_int,Na))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,bit_or_not_num_neg(M,Na))) ).

% int_numeral_or_not_num_neg
tff(fact_2446_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod_int_int] : adjust_div(Qr) = produc8211389475949308722nt_int(aTP_Lamm_dj,Qr) ).

% Divides.adjust_div_def
tff(fact_2447_int__ge__less__than2__def,axiom,
    ! [D: int] : int_ge_less_than2(D) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_dk(D))) ).

% int_ge_less_than2_def
tff(fact_2448_int__ge__less__than__def,axiom,
    ! [D: int] : int_ge_less_than(D) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_dl(D))) ).

% int_ge_less_than_def
tff(fact_2449_rat__inverse__code,axiom,
    ! [P2: rat] : quotient_of(inverse_inverse_rat(P2)) = produc4245557441103728435nt_int(aTP_Lamm_dm,quotient_of(P2)) ).

% rat_inverse_code
tff(fact_2450_upto_Opelims,axiom,
    ! [X: int,Xa2: int,Y: list_int] :
      ( ( upto(X,Xa2) = Y )
     => ( accp_P1096762738010456898nt_int(upto_rel,aa_int7462565939943104719nt_int(product_Pair_int_int(X),Xa2))
       => ~ ( ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa2),cons_int(X,upto(plus_plus_int(X,one_one_int),Xa2)),nil_int) )
           => ~ accp_P1096762738010456898nt_int(upto_rel,aa_int7462565939943104719nt_int(product_Pair_int_int(X),Xa2)) ) ) ) ).

% upto.pelims
tff(fact_2451_rat__one__code,axiom,
    quotient_of(one_one_rat) = aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),one_one_int) ).

% rat_one_code
tff(fact_2452_upto__empty,axiom,
    ! [J: int,I: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,J),I)
     => ( upto(I,J) = nil_int ) ) ).

% upto_empty
tff(fact_2453_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil_int = upto(I,J) )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I) ) ).

% upto_Nil2
tff(fact_2454_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( upto(I,J) = nil_int )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I) ) ).

% upto_Nil
tff(fact_2455_upto__single,axiom,
    ! [I: int] : upto(I,I) = cons_int(I,nil_int) ).

% upto_single
tff(fact_2456_quotient__of__number_I3_J,axiom,
    ! [K: num] : quotient_of(numeral_numeral_rat(K)) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_num_int(numeral_numeral_int,K)),one_one_int) ).

% quotient_of_number(3)
tff(fact_2457_length__upto,axiom,
    ! [I: int,J: int] : size_size_list_int(upto(I,J)) = aa_int_nat(nat2,plus_plus_int(minus_minus_int(J,I),one_one_int)) ).

% length_upto
tff(fact_2458_rat__zero__code,axiom,
    quotient_of(zero_zero_rat) = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int) ).

% rat_zero_code
tff(fact_2459_quotient__of__number_I4_J,axiom,
    quotient_of(uminus_uminus_rat(one_one_rat)) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% quotient_of_number(4)
tff(fact_2460_quotient__of__number_I5_J,axiom,
    ! [K: num] : quotient_of(uminus_uminus_rat(numeral_numeral_rat(K))) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int) ).

% quotient_of_number(5)
tff(fact_2461_upto__rec__numeral_I1_J,axiom,
    ! [M: num,Na: num] :
      upto(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,Na)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na)),cons_int(aa_num_int(numeral_numeral_int,M),upto(plus_plus_int(aa_num_int(numeral_numeral_int,M),one_one_int),aa_num_int(numeral_numeral_int,Na))),nil_int) ).

% upto_rec_numeral(1)
tff(fact_2462_upto__rec__numeral_I4_J,axiom,
    ! [M: num,Na: num] :
      upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))),cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),upto(plus_plus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na)))),nil_int) ).

% upto_rec_numeral(4)
tff(fact_2463_upto__rec__numeral_I3_J,axiom,
    ! [M: num,Na: num] :
      upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Na)),cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),upto(plus_plus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),one_one_int),aa_num_int(numeral_numeral_int,Na))),nil_int) ).

% upto_rec_numeral(3)
tff(fact_2464_upto__rec__numeral_I2_J,axiom,
    ! [M: num,Na: num] :
      upto(aa_num_int(numeral_numeral_int,M),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))),cons_int(aa_num_int(numeral_numeral_int,M),upto(plus_plus_int(aa_num_int(numeral_numeral_int,M),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na)))),nil_int) ).

% upto_rec_numeral(2)
tff(fact_2465_atLeastAtMost__upto,axiom,
    ! [I: int,J: int] : set_or1266510415728281911st_int(I,J) = set_int2(upto(I,J)) ).

% atLeastAtMost_upto
tff(fact_2466_upto__code,axiom,
    ! [I: int,J: int] : upto(I,J) = upto_aux(I,J,nil_int) ).

% upto_code
tff(fact_2467_rat__abs__code,axiom,
    ! [P2: rat] : quotient_of(abs_abs_rat(P2)) = produc4245557441103728435nt_int(aTP_Lamm_dn,quotient_of(P2)) ).

% rat_abs_code
tff(fact_2468_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( upto(I,J) = cons_int(I,upto(plus_plus_int(I,one_one_int),J)) ) ) ).

% upto_rec1
tff(fact_2469_upto_Osimps,axiom,
    ! [I: int,J: int] :
      upto(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J),cons_int(I,upto(plus_plus_int(I,one_one_int),J)),nil_int) ).

% upto.simps
tff(fact_2470_upto_Oelims,axiom,
    ! [X: int,Xa2: int,Y: list_int] :
      ( ( upto(X,Xa2) = Y )
     => ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa2),cons_int(X,upto(plus_plus_int(X,one_one_int),Xa2)),nil_int) ) ) ).

% upto.elims
tff(fact_2471_rat__uminus__code,axiom,
    ! [P2: rat] : quotient_of(uminus_uminus_rat(P2)) = produc4245557441103728435nt_int(aTP_Lamm_do,quotient_of(P2)) ).

% rat_uminus_code
tff(fact_2472_rat__less__code,axiom,
    ! [P2: rat,Q3: rat] :
      ( ord_less_rat(P2,Q3)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_dq(Q3)),quotient_of(P2)) ) ).

% rat_less_code
tff(fact_2473_rat__floor__code,axiom,
    ! [P2: rat] : archim3151403230148437115or_rat(P2) = produc8211389475949308722nt_int(divide_divide_int,quotient_of(P2)) ).

% rat_floor_code
tff(fact_2474_rat__less__eq__code,axiom,
    ! [P2: rat,Q3: rat] :
      ( ord_less_eq_rat(P2,Q3)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_ds(Q3)),quotient_of(P2)) ) ).

% rat_less_eq_code
tff(fact_2475_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( accp_P1096762738010456898nt_int(upto_rel,aa_int7462565939943104719nt_int(product_Pair_int_int(I),J))
     => ( upto(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J),cons_int(I,upto(plus_plus_int(I,one_one_int),J)),nil_int) ) ) ).

% upto.psimps
tff(fact_2476_quotient__of__int,axiom,
    ! [Aa: int] : quotient_of(of_int(Aa)) = aa_int7462565939943104719nt_int(product_Pair_int_int(Aa),one_one_int) ).

% quotient_of_int
tff(fact_2477_rat__minus__code,axiom,
    ! [P2: rat,Q3: rat] : quotient_of(minus_minus_rat(P2,Q3)) = produc4245557441103728435nt_int(aTP_Lamm_du(Q3),quotient_of(P2)) ).

% rat_minus_code
tff(fact_2478_rat__plus__code,axiom,
    ! [P2: rat,Q3: rat] : quotient_of(plus_plus_rat(P2,Q3)) = produc4245557441103728435nt_int(aTP_Lamm_dw(Q3),quotient_of(P2)) ).

% rat_plus_code
tff(fact_2479_normalize__denom__zero,axiom,
    ! [P2: int] : normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(P2),zero_zero_int)) = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int) ).

% normalize_denom_zero
tff(fact_2480_normalize__crossproduct,axiom,
    ! [Q3: int,S: int,P2: int,R: int] :
      ( ( Q3 != zero_zero_int )
     => ( ( S != zero_zero_int )
       => ( ( normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(P2),Q3)) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(R),S)) )
         => ( times_times_int(P2,S) = times_times_int(R,Q3) ) ) ) ) ).

% normalize_crossproduct
tff(fact_2481_rat__divide__code,axiom,
    ! [P2: rat,Q3: rat] : quotient_of(divide_divide_rat(P2,Q3)) = produc4245557441103728435nt_int(aTP_Lamm_dy(Q3),quotient_of(P2)) ).

% rat_divide_code
tff(fact_2482_rat__times__code,axiom,
    ! [P2: rat,Q3: rat] : quotient_of(times_times_rat(P2,Q3)) = produc4245557441103728435nt_int(aTP_Lamm_ea(Q3),quotient_of(P2)) ).

% rat_times_code
tff(fact_2483_Frct__code__post_I5_J,axiom,
    ! [K: num] : frct(aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),aa_num_int(numeral_numeral_int,K))) = divide_divide_rat(one_one_rat,numeral_numeral_rat(K)) ).

% Frct_code_post(5)
tff(fact_2484_concat__bit__Suc,axiom,
    ! [Na: nat,K: int,L: int] : aa_int_int(bit_concat_bit(aa_nat_nat(suc,Na),K),L) = plus_plus_int(modulo_modulo_int(K,aa_num_int(numeral_numeral_int,bit0(one))),times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),aa_int_int(bit_concat_bit(Na,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one)))),L))) ).

% concat_bit_Suc
tff(fact_2485_concat__bit__0,axiom,
    ! [K: int,L: int] : aa_int_int(bit_concat_bit(zero_zero_nat,K),L) = L ).

% concat_bit_0
tff(fact_2486_concat__bit__of__zero__2,axiom,
    ! [Na: nat,K: int] : aa_int_int(bit_concat_bit(Na,K),zero_zero_int) = bit_se2923211474154528505it_int(Na,K) ).

% concat_bit_of_zero_2
tff(fact_2487_concat__bit__nonnegative__iff,axiom,
    ! [Na: nat,K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_concat_bit(Na,K),L))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ).

% concat_bit_nonnegative_iff
tff(fact_2488_concat__bit__negative__iff,axiom,
    ! [Na: nat,K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_concat_bit(Na,K),L)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ).

% concat_bit_negative_iff
tff(fact_2489_concat__bit__of__zero__1,axiom,
    ! [Na: nat,L: int] : aa_int_int(bit_concat_bit(Na,zero_zero_int),L) = bit_se545348938243370406it_int(Na,L) ).

% concat_bit_of_zero_1
tff(fact_2490_concat__bit__take__bit__eq,axiom,
    ! [Na: nat,Ba: int] : bit_concat_bit(Na,bit_se2923211474154528505it_int(Na,Ba)) = bit_concat_bit(Na,Ba) ).

% concat_bit_take_bit_eq
tff(fact_2491_concat__bit__eq__iff,axiom,
    ! [Na: nat,K: int,L: int,R: int,S: int] :
      ( ( aa_int_int(bit_concat_bit(Na,K),L) = aa_int_int(bit_concat_bit(Na,R),S) )
    <=> ( ( bit_se2923211474154528505it_int(Na,K) = bit_se2923211474154528505it_int(Na,R) )
        & ( L = S ) ) ) ).

% concat_bit_eq_iff
tff(fact_2492_concat__bit__assoc,axiom,
    ! [Na: nat,K: int,M: nat,L: int,R: int] : aa_int_int(bit_concat_bit(Na,K),aa_int_int(bit_concat_bit(M,L),R)) = aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na),aa_int_int(bit_concat_bit(Na,K),L)),R) ).

% concat_bit_assoc
tff(fact_2493_concat__bit__eq,axiom,
    ! [Na: nat,K: int,L: int] : aa_int_int(bit_concat_bit(Na,K),L) = plus_plus_int(bit_se2923211474154528505it_int(Na,K),bit_se545348938243370406it_int(Na,L)) ).

% concat_bit_eq
tff(fact_2494_concat__bit__def,axiom,
    ! [Na: nat,K: int,L: int] : aa_int_int(bit_concat_bit(Na,K),L) = bit_se1409905431419307370or_int(bit_se2923211474154528505it_int(Na,K),bit_se545348938243370406it_int(Na,L)) ).

% concat_bit_def
tff(fact_2495_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,Na: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(bit_concat_bit(M,K),L)),Na)
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
          & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
          & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,L),aa_nat_nat(minus_minus_nat(Na),M)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_2496_signed__take__bit__eq__concat__bit,axiom,
    ! [Na: nat,K: int] : bit_ri631733984087533419it_int(Na,K) = aa_int_int(bit_concat_bit(Na,K),aa_int_int(uminus_uminus_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na)))) ).

% signed_take_bit_eq_concat_bit
tff(fact_2497_Frct__code__post_I3_J,axiom,
    frct(aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),one_one_int)) = one_one_rat ).

% Frct_code_post(3)
tff(fact_2498_Frct__code__post_I6_J,axiom,
    ! [K: num,L: num] : frct(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_num_int(numeral_numeral_int,K)),aa_num_int(numeral_numeral_int,L))) = divide_divide_rat(numeral_numeral_rat(K),numeral_numeral_rat(L)) ).

% Frct_code_post(6)
tff(fact_2499_Frct__code__post_I4_J,axiom,
    ! [K: num] : frct(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_num_int(numeral_numeral_int,K)),one_one_int)) = numeral_numeral_rat(K) ).

% Frct_code_post(4)
tff(fact_2500_int__set__bit__conv__ops,axiom,
    ! [I: int,Na: nat,Ba: $o] :
      generi8991105624351003935it_int(I,Na,(Ba)) = $ite((Ba),bit_se1409905431419307370or_int(I,bit_se545348938243370406it_int(Na,one_one_int)),bit_se725231765392027082nd_int(I,bit_ri7919022796975470100ot_int(bit_se545348938243370406it_int(Na,one_one_int)))) ).

% int_set_bit_conv_ops
tff(fact_2501_set__encode__def,axiom,
    nat_set_encode = groups3542108847815614940at_nat(power_power_nat(numeral_numeral_nat(bit0(one)))) ).

% set_encode_def
tff(fact_2502_set__decode__inverse,axiom,
    ! [Na: nat] : aa_set_nat_nat(nat_set_encode,nat_set_decode(Na)) = Na ).

% set_decode_inverse
tff(fact_2503_set__encode__empty,axiom,
    aa_set_nat_nat(nat_set_encode,bot_bot_set_nat) = zero_zero_nat ).

% set_encode_empty
tff(fact_2504_int__set__bit__True__conv__OR,axiom,
    ! [I: int,Na: nat] : generi8991105624351003935it_int(I,Na,$true) = bit_se1409905431419307370or_int(I,bit_se545348938243370406it_int(Na,one_one_int)) ).

% int_set_bit_True_conv_OR
tff(fact_2505_int__set__bit__False__conv__NAND,axiom,
    ! [I: int,Na: nat] : generi8991105624351003935it_int(I,Na,$false) = bit_se725231765392027082nd_int(I,bit_ri7919022796975470100ot_int(bit_se545348938243370406it_int(Na,one_one_int))) ).

% int_set_bit_False_conv_NAND
tff(fact_2506_set__encode__insert,axiom,
    ! [A: set_nat,Na: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ~ aa_set_nat_o(member_nat(Na),A)
       => ( aa_set_nat_nat(nat_set_encode,insert_nat(Na,A)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na)),aa_set_nat_nat(nat_set_encode,A)) ) ) ) ).

% set_encode_insert
tff(fact_2507_int__sdiv__simps_I1_J,axiom,
    ! [Aa: int] : signed6714573509424544716de_int(Aa,one_one_int) = Aa ).

% int_sdiv_simps(1)
tff(fact_2508_int__sdiv__same__is__1,axiom,
    ! [Aa: int,Ba: int] :
      ( ( Aa != zero_zero_int )
     => ( ( signed6714573509424544716de_int(Aa,Ba) = Aa )
      <=> ( Ba = one_one_int ) ) ) ).

% int_sdiv_same_is_1
tff(fact_2509_int__sdiv__simps_I3_J,axiom,
    ! [Aa: int] : signed6714573509424544716de_int(Aa,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,Aa) ).

% int_sdiv_simps(3)
tff(fact_2510_sdiv__int__numeral__numeral,axiom,
    ! [M: num,Na: num] : signed6714573509424544716de_int(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,Na)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na)) ).

% sdiv_int_numeral_numeral
tff(fact_2511_set__encode__inverse,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( nat_set_decode(aa_set_nat_nat(nat_set_encode,A)) = A ) ) ).

% set_encode_inverse
tff(fact_2512_int__sdiv__negated__is__minus1,axiom,
    ! [Aa: int,Ba: int] :
      ( ( Aa != zero_zero_int )
     => ( ( signed6714573509424544716de_int(Aa,Ba) = aa_int_int(uminus_uminus_int,Aa) )
      <=> ( Ba = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% int_sdiv_negated_is_minus1
tff(fact_2513_set__encode__eq,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(finite_finite_nat,B)
       => ( ( aa_set_nat_nat(nat_set_encode,A) = aa_set_nat_nat(nat_set_encode,B) )
        <=> ( A = B ) ) ) ) ).

% set_encode_eq
tff(fact_2514_finite__set__decode,axiom,
    ! [Na: nat] : aa_set_nat_o(finite_finite_nat,nat_set_decode(Na)) ).

% finite_set_decode
tff(fact_2515_finite__M__bounded__by__nat,axiom,
    ! [P: fun_nat_o,I: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_eb(P),I))) ).

% finite_M_bounded_by_nat
tff(fact_2516_finite__nat__set__iff__bounded__le,axiom,
    ! [N3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N3)
    <=> ? [M7: nat] :
        ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),N3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X5),M7) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_2517_finite__less__ub,axiom,
    ! [F: fun_nat_nat,U: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),aa_nat_nat(F,N))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ec(F),U))) ) ).

% finite_less_ub
tff(fact_2518_set__encode__inf,axiom,
    ! [A: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_nat(nat_set_encode,A) = zero_zero_nat ) ) ).

% set_encode_inf
tff(fact_2519_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ed,M))) ) ).

% finite_divisors_nat
tff(fact_2520_sgn__sdiv__eq__sgn__mult,axiom,
    ! [Aa: int,Ba: int] :
      ( ( signed6714573509424544716de_int(Aa,Ba) != zero_zero_int )
     => ( sgn_sgn_int(signed6714573509424544716de_int(Aa,Ba)) = sgn_sgn_int(times_times_int(Aa,Ba)) ) ) ).

% sgn_sdiv_eq_sgn_mult
tff(fact_2521_signed__divide__int__def,axiom,
    ! [K: int,L: int] : signed6714573509424544716de_int(K,L) = times_times_int(times_times_int(sgn_sgn_int(K),sgn_sgn_int(L)),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L))) ).

% signed_divide_int_def
tff(fact_2522_even__set__encode__iff,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_set_nat_nat(nat_set_encode,A))
      <=> ~ aa_set_nat_o(member_nat(zero_zero_nat),A) ) ) ).

% even_set_encode_iff
tff(fact_2523_finite__Collect__le__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ee,K))) ).

% finite_Collect_le_nat
tff(fact_2524_finite__Collect__less__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_co,K))) ).

% finite_Collect_less_nat
tff(fact_2525_finite__interval__int1,axiom,
    ! [Aa: int,Ba: int] : finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_ef(Aa),Ba))) ).

% finite_interval_int1
tff(fact_2526_finite__interval__int4,axiom,
    ! [Aa: int,Ba: int] : finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_eg(Aa),Ba))) ).

% finite_interval_int4
tff(fact_2527_finite__interval__int2,axiom,
    ! [Aa: int,Ba: int] : finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_eh(Aa),Ba))) ).

% finite_interval_int2
tff(fact_2528_finite__interval__int3,axiom,
    ! [Aa: int,Ba: int] : finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_ei(Aa),Ba))) ).

% finite_interval_int3
tff(fact_2529_finite__nth__roots,axiom,
    ! [Na: nat,C: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => finite3207457112153483333omplex(collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_cq(Na),C))) ) ).

% finite_nth_roots
tff(fact_2530_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero_int )
     => finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_ej,I))) ) ).

% finite_divisors_int
tff(fact_2531_infinite__int__iff__unbounded__le,axiom,
    ! [Sa: set_int] :
      ( ~ finite_finite_int(Sa)
    <=> ! [M7: int] :
        ? [N2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M7),aa_int_int(abs_abs_int,N2))
          & member_int(N2,Sa) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_2532_Cauchy__iff2,axiom,
    ! [X7: fun_nat_real] :
      ( topolo4055970368930404560y_real(X7)
    <=> ! [J3: nat] :
        ? [M8: nat] :
        ! [M7: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M8),M7)
         => ! [N2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M8),N2)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_nat_real(X7,M7)),aa_nat_real(X7,N2)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J3)))) ) ) ) ).

% Cauchy_iff2
tff(fact_2533_int__of__nat__def,axiom,
    code_T6385005292777649522of_nat = semiri1314217659103216013at_int ).

% int_of_nat_def
tff(fact_2534_infinite__nat__iff__unbounded__le,axiom,
    ! [Sa: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,Sa)
    <=> ! [M7: nat] :
        ? [N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M7),N2)
          & aa_set_nat_o(member_nat(N2),Sa) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_2535_Sum__Ico__nat,axiom,
    ! [M: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_db),set_or4665077453230672383an_nat(M,Na)) = aa_nat_nat(divide_divide_nat(aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(minus_minus_nat(Na),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(minus_minus_nat(M),one_one_nat)))),numeral_numeral_nat(bit0(one))) ).

% Sum_Ico_nat
tff(fact_2536_sum__power2,axiom,
    ! [K: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(power_power_nat(numeral_numeral_nat(bit0(one)))),set_or4665077453230672383an_nat(zero_zero_nat,K)) = aa_nat_nat(minus_minus_nat(aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),K)),one_one_nat) ).

% sum_power2
tff(fact_2537_smod__int__range,axiom,
    ! [Ba: int,Aa: int] :
      ( ( Ba != zero_zero_int )
     => member_int(signed6292675348222524329lo_int(Aa,Ba),set_or1266510415728281911st_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_int_int(abs_abs_int,Ba)),one_one_int),minus_minus_int(aa_int_int(abs_abs_int,Ba),one_one_int))) ) ).

% smod_int_range
tff(fact_2538_smod__int__numeral__numeral,axiom,
    ! [M: num,Na: num] : signed6292675348222524329lo_int(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,Na)) = modulo_modulo_int(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,Na)) ).

% smod_int_numeral_numeral
tff(fact_2539_ex__nat__less__eq,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ? [M7: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M7),Na)
          & aa_nat_o(P,M7) )
    <=> ? [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),set_or4665077453230672383an_nat(zero_zero_nat,Na))
          & aa_nat_o(P,X5) ) ) ).

% ex_nat_less_eq
tff(fact_2540_all__nat__less__eq,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ! [M7: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M7),Na)
         => aa_nat_o(P,M7) )
    <=> ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),set_or4665077453230672383an_nat(zero_zero_nat,Na))
         => aa_nat_o(P,X5) ) ) ).

% all_nat_less_eq
tff(fact_2541_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( set_or4665077453230672383an_nat(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = sup_sup_set_nat(set_or4665077453230672383an_nat(I,J),set_or4665077453230672383an_nat(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_2542_smod__int__compares_I1_J,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,signed6292675348222524329lo_int(Aa,Ba)),Ba) ) ) ).

% smod_int_compares(1)
tff(fact_2543_smod__int__compares_I2_J,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),signed6292675348222524329lo_int(Aa,Ba)) ) ) ).

% smod_int_compares(2)
tff(fact_2544_smod__int__compares_I4_J,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,signed6292675348222524329lo_int(Aa,Ba)),zero_zero_int) ) ) ).

% smod_int_compares(4)
tff(fact_2545_smod__int__compares_I6_J,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),signed6292675348222524329lo_int(Aa,Ba)) ) ) ).

% smod_int_compares(6)
tff(fact_2546_smod__int__compares_I7_J,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,signed6292675348222524329lo_int(Aa,Ba)),zero_zero_int) ) ) ).

% smod_int_compares(7)
tff(fact_2547_smod__int__compares_I8_J,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),signed6292675348222524329lo_int(Aa,Ba)) ) ) ).

% smod_int_compares(8)
tff(fact_2548_smod__mod__positive,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ba)
       => ( signed6292675348222524329lo_int(Aa,Ba) = modulo_modulo_int(Aa,Ba) ) ) ) ).

% smod_mod_positive
tff(fact_2549_signed__modulo__int__def,axiom,
    ! [K: int,L: int] : signed6292675348222524329lo_int(K,L) = minus_minus_int(K,times_times_int(signed6714573509424544716de_int(K,L),L)) ).

% signed_modulo_int_def
tff(fact_2550_atLeastLessThanSuc,axiom,
    ! [M: nat,Na: nat] :
      set_or4665077453230672383an_nat(M,aa_nat_nat(suc,Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na),insert_nat(Na,set_or4665077453230672383an_nat(M,Na)),bot_bot_set_nat) ).

% atLeastLessThanSuc
tff(fact_2551_smod__int__compares_I3_J,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,Ba)),signed6292675348222524329lo_int(Aa,Ba)) ) ) ).

% smod_int_compares(3)
tff(fact_2552_smod__int__compares_I5_J,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,signed6292675348222524329lo_int(Aa,Ba)),aa_int_int(uminus_uminus_int,Ba)) ) ) ).

% smod_int_compares(5)
tff(fact_2553_smod__int__alt__def,axiom,
    ! [Aa: int,Ba: int] : signed6292675348222524329lo_int(Aa,Ba) = times_times_int(sgn_sgn_int(Aa),modulo_modulo_int(aa_int_int(abs_abs_int,Aa),aa_int_int(abs_abs_int,Ba))) ).

% smod_int_alt_def
tff(fact_2554_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      set_or4665077453230672383an_nat(M,numeral_numeral_nat(K)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),pred_numeral(K)),insert_nat(pred_numeral(K),set_or4665077453230672383an_nat(M,pred_numeral(K))),bot_bot_set_nat) ).

% atLeastLessThan_nat_numeral
tff(fact_2555_Chebyshev__sum__upper__nat,axiom,
    ! [Na: nat,Aa: fun_nat_nat,Ba: fun_nat_nat] :
      ( ! [I2: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Na)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Aa,I2)),aa_nat_nat(Aa,J2)) ) )
     => ( ! [I2: nat,J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Na)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Ba,J2)),aa_nat_nat(Ba,I2)) ) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ek(Aa),Ba)),set_or4665077453230672383an_nat(zero_zero_nat,Na)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Aa),set_or4665077453230672383an_nat(zero_zero_nat,Na))),aa_set_nat_nat(groups3542108847815614940at_nat(Ba),set_or4665077453230672383an_nat(zero_zero_nat,Na)))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_2556_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : set_or4662586982721622107an_int(L,plus_plus_int(U,one_one_int)) = set_or1266510415728281911st_int(L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
tff(fact_2557_atLeastLessThan__upto,axiom,
    ! [I: int,J: int] : set_or4662586982721622107an_int(I,J) = set_int2(upto(I,minus_minus_int(J,one_one_int))) ).

% atLeastLessThan_upto
tff(fact_2558_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list_o] : aa_int_o(aa_int_fun_int_o(ord_less_int,groups9116527308978886569_o_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,bit0(one)),Bs)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,bit0(one))),size_size_list_o(Bs))) ).

% horner_sum_of_bool_2_less
tff(fact_2559_uint32_Osize__eq__length,axiom,
    numeral_numeral_nat(bit0(bit0(bit0(bit0(bit0(one)))))) = type_l796852477590012082l_num1(type_N8448461349408098053l_num1) ).

% uint32.size_eq_length
tff(fact_2560_len__of__finite__1__def,axiom,
    ! [X: itself_finite_1] : type_l31302759751748491nite_1(X) = one_one_nat ).

% len_of_finite_1_def
tff(fact_2561_len__num1,axiom,
    ! [Uu: itself_Numeral_num1] : type_l4264026598287037465l_num1(Uu) = one_one_nat ).

% len_num1
tff(fact_2562_len__of__finite__3__def,axiom,
    ! [X: itself_finite_3] : type_l31302759751748493nite_3(X) = numeral_numeral_nat(bit0(bit0(one))) ).

% len_of_finite_3_def
tff(fact_2563_len__of__finite__2__def,axiom,
    ! [X: itself_finite_2] : type_l31302759751748492nite_2(X) = numeral_numeral_nat(bit0(one)) ).

% len_of_finite_2_def
tff(fact_2564_bij__betw__Suc,axiom,
    ! [M4: set_nat,N3: set_nat] :
      ( aa_set_nat_o(bij_betw_nat_nat(suc,M4),N3)
    <=> ( image_nat_nat(suc,M4) = N3 ) ) ).

% bij_betw_Suc
tff(fact_2565_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,plus_plus_int(I,aa_nat_int(semiri1314217659103216013at_int,K))),J)
     => ( aa_nat_int(nth_int(upto(I,J)),K) = plus_plus_int(I,aa_nat_int(semiri1314217659103216013at_int,K)) ) ) ).

% nth_upto
tff(fact_2566_image__int__atLeastAtMost,axiom,
    ! [Aa: nat,Ba: nat] : image_nat_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(Aa,Ba)) = set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,Aa),aa_nat_int(semiri1314217659103216013at_int,Ba)) ).

% image_int_atLeastAtMost
tff(fact_2567_image__int__atLeastLessThan,axiom,
    ! [Aa: nat,Ba: nat] : image_nat_int(semiri1314217659103216013at_int,set_or4665077453230672383an_nat(Aa,Ba)) = set_or4662586982721622107an_int(aa_nat_int(semiri1314217659103216013at_int,Aa),aa_nat_int(semiri1314217659103216013at_int,Ba)) ).

% image_int_atLeastLessThan
tff(fact_2568_image__Suc__lessThan,axiom,
    ! [Na: nat] : image_nat_nat(suc,set_ord_lessThan_nat(Na)) = set_or1269000886237332187st_nat(one_one_nat,Na) ).

% image_Suc_lessThan
tff(fact_2569_image__Suc__atMost,axiom,
    ! [Na: nat] : image_nat_nat(suc,set_ord_atMost_nat(Na)) = set_or1269000886237332187st_nat(one_one_nat,aa_nat_nat(suc,Na)) ).

% image_Suc_atMost
tff(fact_2570_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(aTP_Lamm_el(L),set_or4662586982721622107an_int(zero_zero_int,minus_minus_int(U,L))) = set_or4662586982721622107an_int(L,U) ).

% image_add_int_atLeastLessThan
tff(fact_2571_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
     => ( set_or4662586982721622107an_int(zero_zero_int,U) = image_nat_int(semiri1314217659103216013at_int,set_ord_lessThan_nat(aa_int_nat(nat2,U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_2572_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,X: nat,Y: nat] :
      image_nat_nat(aTP_Lamm_em(C),set_or4665077453230672383an_nat(X,Y)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Y),
        set_or4665077453230672383an_nat(aa_nat_nat(minus_minus_nat(X),C),aa_nat_nat(minus_minus_nat(Y),C)),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y),insert_nat(zero_zero_nat,bot_bot_set_nat),bot_bot_set_nat) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_2573_min__Suc__Suc,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(ord_min_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(ord_min_nat(M),Na)) ).

% min_Suc_Suc
tff(fact_2574_min__0R,axiom,
    ! [Na: nat] : aa_nat_nat(ord_min_nat(Na),zero_zero_nat) = zero_zero_nat ).

% min_0R
tff(fact_2575_min__0L,axiom,
    ! [Na: nat] : aa_nat_nat(ord_min_nat(zero_zero_nat),Na) = zero_zero_nat ).

% min_0L
tff(fact_2576_min__Suc__gt_I2_J,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
     => ( aa_nat_nat(ord_min_nat(Ba),aa_nat_nat(suc,Aa)) = aa_nat_nat(suc,Aa) ) ) ).

% min_Suc_gt(2)
tff(fact_2577_min__Suc__gt_I1_J,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
     => ( aa_nat_nat(ord_min_nat(aa_nat_nat(suc,Aa)),Ba) = aa_nat_nat(suc,Aa) ) ) ).

% min_Suc_gt(1)
tff(fact_2578_min__pm,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(ord_min_nat(Aa),Ba)),aa_nat_nat(minus_minus_nat(Aa),Ba)) = Aa ).

% min_pm
tff(fact_2579_min__pm1,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(minus_minus_nat(Aa),Ba)),aa_nat_nat(ord_min_nat(Aa),Ba)) = Aa ).

% min_pm1
tff(fact_2580_rev__min__pm,axiom,
    ! [Ba: nat,Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(ord_min_nat(Ba),Aa)),aa_nat_nat(minus_minus_nat(Aa),Ba)) = Aa ).

% rev_min_pm
tff(fact_2581_rev__min__pm1,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(minus_minus_nat(Aa),Ba)),aa_nat_nat(ord_min_nat(Ba),Aa)) = Aa ).

% rev_min_pm1
tff(fact_2582_min__numeral__Suc,axiom,
    ! [K: num,Na: nat] : aa_nat_nat(ord_min_nat(numeral_numeral_nat(K)),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(ord_min_nat(pred_numeral(K)),Na)) ).

% min_numeral_Suc
tff(fact_2583_min__Suc__numeral,axiom,
    ! [Na: nat,K: num] : aa_nat_nat(ord_min_nat(aa_nat_nat(suc,Na)),numeral_numeral_nat(K)) = aa_nat_nat(suc,aa_nat_nat(ord_min_nat(Na),pred_numeral(K))) ).

% min_Suc_numeral
tff(fact_2584_nat__mult__min__right,axiom,
    ! [M: nat,Na: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(ord_min_nat(Na),Q3)) = aa_nat_nat(ord_min_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)) ).

% nat_mult_min_right
tff(fact_2585_nat__mult__min__left,axiom,
    ! [M: nat,Na: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(ord_min_nat(M),Na)),Q3) = aa_nat_nat(ord_min_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q3)) ).

% nat_mult_min_left
tff(fact_2586_min__diff,axiom,
    ! [M: nat,I: nat,Na: nat] : aa_nat_nat(ord_min_nat(aa_nat_nat(minus_minus_nat(M),I)),aa_nat_nat(minus_minus_nat(Na),I)) = aa_nat_nat(minus_minus_nat(aa_nat_nat(ord_min_nat(M),Na)),I) ).

% min_diff
tff(fact_2587_concat__bit__assoc__sym,axiom,
    ! [M: nat,Na: nat,K: int,L: int,R: int] : aa_int_int(bit_concat_bit(M,aa_int_int(bit_concat_bit(Na,K),L)),R) = aa_int_int(bit_concat_bit(aa_nat_nat(ord_min_nat(M),Na),K),aa_int_int(bit_concat_bit(aa_nat_nat(minus_minus_nat(M),Na),L),R)) ).

% concat_bit_assoc_sym
tff(fact_2588_take__bit__concat__bit__eq,axiom,
    ! [M: nat,Na: nat,K: int,L: int] : bit_se2923211474154528505it_int(M,aa_int_int(bit_concat_bit(Na,K),L)) = aa_int_int(bit_concat_bit(aa_nat_nat(ord_min_nat(M),Na),K),bit_se2923211474154528505it_int(aa_nat_nat(minus_minus_nat(M),Na),L)) ).

% take_bit_concat_bit_eq
tff(fact_2589_mod__mod__power,axiom,
    ! [K: nat,M: nat,Na: nat] : aa_nat_nat(modulo_modulo_nat(aa_nat_nat(modulo_modulo_nat(K),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),M))),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na)) = aa_nat_nat(modulo_modulo_nat(K),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),aa_nat_nat(ord_min_nat(M),Na))) ).

% mod_mod_power
tff(fact_2590_min__enat__simps_I2_J,axiom,
    ! [Q3: extended_enat] : ord_mi8085742599997312461d_enat(Q3,zero_z5237406670263579293d_enat) = zero_z5237406670263579293d_enat ).

% min_enat_simps(2)
tff(fact_2591_min__enat__simps_I3_J,axiom,
    ! [Q3: extended_enat] : ord_mi8085742599997312461d_enat(zero_z5237406670263579293d_enat,Q3) = zero_z5237406670263579293d_enat ).

% min_enat_simps(3)
tff(fact_2592_infinite__UNIV__int,axiom,
    ~ finite_finite_int(top_top_set_int) ).

% infinite_UNIV_int
tff(fact_2593_int__in__range__abs,axiom,
    ! [Na: nat] : member_int(aa_nat_int(semiri1314217659103216013at_int,Na),image_int_int(abs_abs_int,top_top_set_int)) ).

% int_in_range_abs
tff(fact_2594_range__mod,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( image_nat_nat(aTP_Lamm_en(Na),top_top_set_nat) = set_or4665077453230672383an_nat(zero_zero_nat,Na) ) ) ).

% range_mod
tff(fact_2595_root__def,axiom,
    ! [Na: nat,X: real] :
      aa_real_real(aa_nat_fun_real_real(root,Na),X) = $ite(Na = zero_zero_nat,zero_zero_real,aa_real_real(the_in5290026491893676941l_real(top_top_set_real,aa_nat_fun_real_real(aTP_Lamm_eo,Na)),X)) ).

% root_def
tff(fact_2596_test__hoare,axiom,
    ! [Xs2: list_nat,Na: nat,Ysb: list_nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),set_nat2(Xs2))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => hoare_7964568885773372237st_nat(one_one_assn,aa_lis1877347954636394190st_nat(vEBT_test(Na,Xs2),Ysb),aa_lis5667661702080215587t_assn(aTP_Lamm_er(Xs2),Ysb)) ) ) ).

% test_hoare
tff(fact_2597_GreatestI__ex__nat,axiom,
    ! [P: fun_nat_o,Ba: nat] :
      ( ? [X_12: nat] : aa_nat_o(P,X_12)
     => ( ! [Y2: nat] :
            ( aa_nat_o(P,Y2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),Ba) )
       => aa_nat_o(P,order_Greatest_nat(P)) ) ) ).

% GreatestI_ex_nat
tff(fact_2598_Greatest__le__nat,axiom,
    ! [P: fun_nat_o,K: nat,Ba: nat] :
      ( aa_nat_o(P,K)
     => ( ! [Y2: nat] :
            ( aa_nat_o(P,Y2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),Ba) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),order_Greatest_nat(P)) ) ) ).

% Greatest_le_nat
tff(fact_2599_GreatestI__nat,axiom,
    ! [P: fun_nat_o,K: nat,Ba: nat] :
      ( aa_nat_o(P,K)
     => ( ! [Y2: nat] :
            ( aa_nat_o(P,Y2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),Ba) )
       => aa_nat_o(P,order_Greatest_nat(P)) ) ) ).

% GreatestI_nat
tff(fact_2600_shiftl__Suc__0,axiom,
    ! [Na: nat] : aa_nat_nat(bit_Sh3965577149348748681tl_nat(aa_nat_nat(suc,zero_zero_nat)),Na) = aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Na) ).

% shiftl_Suc_0
tff(fact_2601_inj__Suc,axiom,
    ! [N3: set_nat] : aa_set_nat_o(inj_on_nat_nat(suc),N3) ).

% inj_Suc
tff(fact_2602_inj__on__diff__nat,axiom,
    ! [N3: set_nat,K: nat] :
      ( ! [N: nat] :
          ( aa_set_nat_o(member_nat(N),N3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N) )
     => aa_set_nat_o(inj_on_nat_nat(aTP_Lamm_em(K)),N3) ) ).

% inj_on_diff_nat
tff(fact_2603_inj__on__set__encode,axiom,
    inj_on_set_nat_nat(nat_set_encode,collect_set_nat(finite_finite_nat)) ).

% inj_on_set_encode
tff(fact_2604_inj__sgn__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => inj_on_real_real(aa_nat_fun_real_real(aTP_Lamm_eo,Na),top_top_set_real) ) ).

% inj_sgn_power
tff(fact_2605_DERIV__real__root__generic,axiom,
    ! [Na: nat,X: real,D5: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( X != zero_zero_real )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
             => ( D5 = aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_nat_nat(minus_minus_nat(Na),aa_nat_nat(suc,zero_zero_nat))))) ) ) )
         => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
               => ( D5 = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_nat_nat(minus_minus_nat(Na),aa_nat_nat(suc,zero_zero_nat)))))) ) ) )
           => ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na)
               => ( D5 = aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_nat_nat(minus_minus_nat(Na),aa_nat_nat(suc,zero_zero_nat))))) ) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Na),D5,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_2606_DERIV__mirror,axiom,
    ! [F: fun_real_real,Y: real,X: real] :
      ( has_fi5821293074295781190e_real(F,Y,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,X),top_top_set_real))
    <=> has_fi5821293074295781190e_real(aTP_Lamm_es(F),aa_real_real(uminus_uminus_real,Y),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_mirror
tff(fact_2607_DERIV__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,X: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
            & ! [H2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H2),D4)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H2))),aa_real_real(F,X)) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_2608_DERIV__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,X: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
            & ! [H2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H2),D4)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H2))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_2609_deriv__nonneg__imp__mono,axiom,
    ! [Aa: real,Ba: real,G: fun_real_real,G2: fun_real_real] :
      ( ! [X2: real] :
          ( member_real(X2,set_or1222579329274155063t_real(Aa,Ba))
         => has_fi5821293074295781190e_real(G,aa_real_real(G2,X2),topolo2177554685111907308n_real(X2,top_top_set_real)) )
     => ( ! [X2: real] :
            ( member_real(X2,set_or1222579329274155063t_real(Aa,Ba))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(G2,X2)) )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(G,Aa)),aa_real_real(G,Ba)) ) ) ) ).

% deriv_nonneg_imp_mono
tff(fact_2610_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [Aa: real,Ba: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Ba)
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y3) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Aa)),aa_real_real(F,Ba)) ) ) ).

% DERIV_nonneg_imp_nondecreasing
tff(fact_2611_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [Aa: real,Ba: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Ba)
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),zero_zero_real) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Ba)),aa_real_real(F,Aa)) ) ) ).

% DERIV_nonpos_imp_nonincreasing
tff(fact_2612_DERIV__pos__imp__increasing,axiom,
    ! [Aa: real,Ba: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Ba)
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y3) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Aa)),aa_real_real(F,Ba)) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_2613_DERIV__neg__imp__decreasing,axiom,
    ! [Aa: real,Ba: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Ba)
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),zero_zero_real) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Ba)),aa_real_real(F,Aa)) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_2614_has__real__derivative__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,X: real,Sa: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,Sa))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
            & ! [H2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H2)
               => ( member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H2),Sa)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H2),D4)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H2))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_2615_has__real__derivative__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,X: real,Sa: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,Sa))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
            & ! [H2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H2)
               => ( member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H2),Sa)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H2),D4)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H2))),aa_real_real(F,X)) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_2616_MVT2,axiom,
    ! [Aa: real,Ba: real,F: fun_real_real,F2: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Ba)
             => has_fi5821293074295781190e_real(F,aa_real_real(F2,X2),topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
       => ? [Z3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Z3)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),Ba)
            & ( aa_real_real(minus_minus_real(aa_real_real(F,Ba)),aa_real_real(F,Aa)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(Ba),Aa)),aa_real_real(F2,Z3)) ) ) ) ) ).

% MVT2
tff(fact_2617_DERIV__const__average,axiom,
    ! [Aa: real,Ba: real,V: fun_real_real,K: real] :
      ( ( Aa != Ba )
     => ( ! [X2: real] : has_fi5821293074295781190e_real(V,K,topolo2177554685111907308n_real(X2,top_top_set_real))
       => ( aa_real_real(V,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)),numeral_numeral_real(bit0(one)))) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(V,Aa)),aa_real_real(V,Ba))),numeral_numeral_real(bit0(one))) ) ) ) ).

% DERIV_const_average
tff(fact_2618_DERIV__local__min,axiom,
    ! [F: fun_real_real,L: real,X: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X),Y2))),D)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X)),aa_real_real(F,Y2)) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
tff(fact_2619_DERIV__local__max,axiom,
    ! [F: fun_real_real,L: real,X: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X),Y2))),D)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Y2)),aa_real_real(F,X)) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
tff(fact_2620_DERIV__pow,axiom,
    ! [Na: nat,X: real,S: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_et,Na),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(minus_minus_nat(Na),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(X,S)) ).

% DERIV_pow
tff(fact_2621_DERIV__fun__pow,axiom,
    ! [G: fun_real_real,M: real,X: real,Na: nat] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(X,top_top_set_real))
     => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_eu(G),Na),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(G,X)),aa_nat_nat(minus_minus_nat(Na),one_one_nat)))),M),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_fun_pow
tff(fact_2622_has__real__derivative__powr,axiom,
    ! [Z: real,R: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => has_fi5821293074295781190e_real(aTP_Lamm_ev(R),aa_real_real(times_times_real(R),aa_real_real(powr_real(Z),aa_real_real(minus_minus_real(R),one_one_real))),topolo2177554685111907308n_real(Z,top_top_set_real)) ) ).

% has_real_derivative_powr
tff(fact_2623_DERIV__fun__powr,axiom,
    ! [G: fun_real_real,M: real,X: real,R: real] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,X))
       => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_ew(G),R),aa_real_real(times_times_real(aa_real_real(times_times_real(R),aa_real_real(powr_real(aa_real_real(G,X)),aa_real_real(minus_minus_real(R),aa_nat_real(semiri5074537144036343181t_real,one_one_nat))))),M),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_fun_powr
tff(fact_2624_DERIV__log,axiom,
    ! [X: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => has_fi5821293074295781190e_real(log(Ba),aa_real_real(divide_divide_real(one_one_real),aa_real_real(times_times_real(aa_real_real(ln_ln_real,Ba)),X)),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_log
tff(fact_2625_DERIV__powr,axiom,
    ! [G: fun_real_real,M: real,X: real,F: fun_real_real,R: real] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,X))
       => ( has_fi5821293074295781190e_real(F,R,topolo2177554685111907308n_real(X,top_top_set_real))
         => has_fi5821293074295781190e_real(aa_fun6946558556880649704l_real(aTP_Lamm_ex(G),F),aa_real_real(times_times_real(aa_real_real(powr_real(aa_real_real(G,X)),aa_real_real(F,X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(R),aa_real_real(ln_ln_real,aa_real_real(G,X)))),aa_real_real(divide_divide_real(aa_real_real(times_times_real(M),aa_real_real(F,X))),aa_real_real(G,X)))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_powr
tff(fact_2626_artanh__real__has__field__derivative,axiom,
    ! [X: real,A: set_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => has_fi5821293074295781190e_real(artanh_real,aa_real_real(divide_divide_real(one_one_real),aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one))))),topolo2177554685111907308n_real(X,A)) ) ).

% artanh_real_has_field_derivative
tff(fact_2627_DERIV__real__sqrt,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => has_fi5821293074295781190e_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),numeral_numeral_real(bit0(one))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_real_sqrt
tff(fact_2628_DERIV__arctan,axiom,
    ! [X: real] : has_fi5821293074295781190e_real(arctan,aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one))))),topolo2177554685111907308n_real(X,top_top_set_real)) ).

% DERIV_arctan
tff(fact_2629_arsinh__real__has__field__derivative,axiom,
    ! [X: real,A: set_real] : has_fi5821293074295781190e_real(arsinh_real,aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))),one_one_real))),topolo2177554685111907308n_real(X,A)) ).

% arsinh_real_has_field_derivative
tff(fact_2630_DERIV__real__sqrt__generic,axiom,
    ! [X: real,D5: real] :
      ( ( X != zero_zero_real )
     => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( D5 = aa_real_real(divide_divide_real(aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),numeral_numeral_real(bit0(one))) ) )
       => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
           => ( D5 = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X)))),numeral_numeral_real(bit0(one))) ) )
         => has_fi5821293074295781190e_real(sqrt,D5,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_2631_arcosh__real__has__field__derivative,axiom,
    ! [X: real,A: set_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => has_fi5821293074295781190e_real(arcosh_real,aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))),one_one_real))),topolo2177554685111907308n_real(X,A)) ) ).

% arcosh_real_has_field_derivative
tff(fact_2632_DERIV__real__root,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Na),aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_nat_nat(minus_minus_nat(Na),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_real_root
tff(fact_2633_DERIV__arccos,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => has_fi5821293074295781190e_real(arccos,aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one))))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_arccos
tff(fact_2634_DERIV__arcsin,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => has_fi5821293074295781190e_real(arcsin,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),numeral_numeral_nat(bit0(one)))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_arcsin
tff(fact_2635_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,X: real,Na: nat] :
      ( ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
        & ! [M3: nat,X2: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),X2),topolo2177554685111907308n_real(X2,top_top_set_real)) )
     => ? [T2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X))
          & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ey(Diff),X),set_ord_lessThan_nat(Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Na),T2)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_2636_Maclaurin__all__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,X: real,Na: nat] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M3: nat,X2: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),X2),topolo2177554685111907308n_real(X2,top_top_set_real))
       => ? [T2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X))
            & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ey(Diff),X),set_ord_lessThan_nat(Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Na),T2)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_2637_DERIV__odd__real__root,axiom,
    ! [Na: nat,X: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na)
     => ( ( X != zero_zero_real )
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Na),aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_nat_nat(minus_minus_nat(Na),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_odd_real_root
tff(fact_2638_Maclaurin__minus,axiom,
    ! [H: real,Na: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),zero_zero_real)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M3: nat,T2: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Na)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,H),T2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T2),zero_zero_real) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
           => ? [T2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),T2)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T2),zero_zero_real)
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_ez(H),Diff),set_ord_lessThan_nat(Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Na),T2)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Na))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_2639_Maclaurin2,axiom,
    ! [H: real,Diff: fun_na6724076748692969250l_real,F: fun_real_real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M3: nat,T2: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Na)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T2)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T2),H) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
         => ? [T2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T2)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T2),H)
              & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_ez(H),Diff),set_ord_lessThan_nat(Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Na),T2)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Na))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_2640_Maclaurin,axiom,
    ! [H: real,Na: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M3: nat,T2: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Na)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T2),H) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
           => ? [T2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T2)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T2),H)
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_ez(H),Diff),set_ord_lessThan_nat(Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Na),T2)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Na))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_2641_Maclaurin__all__lt,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Na: nat,X: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( ( X != zero_zero_real )
         => ( ! [M3: nat,X2: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),X2),topolo2177554685111907308n_real(X2,top_top_set_real))
           => ? [T2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T2))
                & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X))
                & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ey(Diff),X),set_ord_lessThan_nat(Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Na),T2)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_2642_Maclaurin__bi__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Na: nat,X: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M3: nat,T2: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Na)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X)) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
       => ? [T2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,X))
            & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ey(Diff),X),set_ord_lessThan_nat(Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Na),T2)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_2643_Taylor__down,axiom,
    ! [Na: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,Aa: real,Ba: real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M3: nat,T2: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Na)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),T2)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T2),Ba) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),Ba)
             => ? [T2: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),T2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T2),C)
                  & ( aa_real_real(F,Aa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_fa(Diff),Aa),C),set_ord_lessThan_nat(Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Na),T2)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Aa),C)),Na))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_2644_Taylor__up,axiom,
    ! [Na: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,Aa: real,Ba: real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M3: nat,T2: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Na)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),T2)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T2),Ba) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),Ba)
             => ? [T2: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T2),Ba)
                  & ( aa_real_real(F,Ba) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_fa(Diff),Ba),C),set_ord_lessThan_nat(Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Na),T2)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Ba),C)),Na))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_2645_Taylor,axiom,
    ! [Na: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,Aa: real,Ba: real,C: real,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M3: nat,T2: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Na)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),T2)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T2),Ba) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),Ba)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Ba)
                 => ( ( X != C )
                   => ? [T2: real] :
                        ( $ite(
                            aa_real_o(aa_real_fun_real_o(ord_less_real,X),C),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),T2)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T2),C) ),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T2)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T2),X) ) )
                        & ( aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_fb(Diff),C),X),set_ord_lessThan_nat(Na))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Na),T2)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(X),C)),Na))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_2646_Maclaurin__lemma2,axiom,
    ! [Na: nat,H: real,Diff: fun_na6724076748692969250l_real,K: nat,B: real] :
      ( ! [M3: nat,T2: real] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Na)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T2)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T2),H) )
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
     => ( ( Na = aa_nat_nat(suc,K) )
       => ! [M2: nat,T3: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T3)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),H) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_fd(Na),Diff),B),M2),aa_real_real(minus_minus_real(aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T3)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_fe(Diff),M2),T3),set_ord_lessThan_nat(aa_nat_nat(minus_minus_nat(Na),aa_nat_nat(suc,M2))))),aa_real_real(times_times_real(B),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,T3),aa_nat_nat(minus_minus_nat(Na),aa_nat_nat(suc,M2)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(minus_minus_nat(Na),aa_nat_nat(suc,M2))))))),topolo2177554685111907308n_real(T3,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_2647_DERIV__arctan__series,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => has_fi5821293074295781190e_real(aTP_Lamm_ff,suminf_real(aa_real_fun_nat_real(aTP_Lamm_fg,X)),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_2648_DERIV__even__real__root,axiom,
    ! [Na: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Na),aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),X)),aa_nat_nat(minus_minus_nat(Na),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_even_real_root
tff(fact_2649_DERIV__power__series_H,axiom,
    ! [R3: real,F: fun_nat_real,X0: real] :
      ( ! [X2: real] :
          ( member_real(X2,set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R3),R3))
         => summable_real(aa_real_fun_nat_real(aTP_Lamm_fh(F),X2)) )
     => ( member_real(X0,set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R3),R3))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
         => has_fi5821293074295781190e_real(aTP_Lamm_fj(F),suminf_real(aa_real_fun_nat_real(aTP_Lamm_fh(F),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_2650_DERIV__series_H,axiom,
    ! [F: fun_re4238438383274954274t_real,F2: fun_re4238438383274954274t_real,X0: real,Aa: real,Ba: real,L4: fun_nat_real] :
      ( ! [N: nat] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_fk(F),N),aa_nat_real(aa_real_fun_nat_real(F2,X0),N),topolo2177554685111907308n_real(X0,top_top_set_real))
     => ( ! [X2: real] :
            ( member_real(X2,set_or1633881224788618240n_real(Aa,Ba))
           => summable_real(aa_real_fun_nat_real(F,X2)) )
       => ( member_real(X0,set_or1633881224788618240n_real(Aa,Ba))
         => ( summable_real(aa_real_fun_nat_real(F2,X0))
           => ( summable_real(L4)
             => ( ! [N: nat,X2: real,Y2: real] :
                    ( member_real(X2,set_or1633881224788618240n_real(Aa,Ba))
                   => ( member_real(Y2,set_or1633881224788618240n_real(Aa,Ba))
                     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(F,X2),N)),aa_nat_real(aa_real_fun_nat_real(F,Y2),N)))),aa_real_real(times_times_real(aa_nat_real(L4,N)),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X2),Y2)))) ) )
               => has_fi5821293074295781190e_real(aTP_Lamm_fl(F),suminf_real(aa_real_fun_nat_real(F2,X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_2651_isCont__Lb__Ub,axiom,
    ! [Aa: real,Ba: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => ( ! [X2: real] :
            ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X2)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Ba) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X2,top_top_set_real),F) )
       => ? [L5: real,M9: real] :
            ( ! [X4: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Ba) )
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L5),aa_real_real(F,X4))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X4)),M9) ) )
            & ! [Y3: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L5),Y3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),M9) )
               => ? [X2: real] :
                    ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X2)
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Ba)
                    & ( aa_real_real(F,X2) = Y3 ) ) ) ) ) ) ).

% isCont_Lb_Ub
tff(fact_2652_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : set_or4662586982721622107an_int(plus_plus_int(L,one_one_int),U) = set_or5832277885323065728an_int(L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
tff(fact_2653_isCont__inverse__function2,axiom,
    ! [Aa: real,X: real,Ba: real,G: fun_real_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Ba)
       => ( ! [Z3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Z3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z3),Ba)
               => ( aa_real_real(G,aa_real_real(F,Z3)) = Z3 ) ) )
         => ( ! [Z3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Z3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z3),Ba)
                 => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),F) ) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,X),top_top_set_real),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_2654_LIM__cos__div__sin,axiom,
    filterlim_real_real(aTP_Lamm_fm,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))),top_top_set_real)) ).

% LIM_cos_div_sin
tff(fact_2655_LIM__less__bound,axiom,
    ! [Ba: real,X: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),X)
     => ( ! [X2: real] :
            ( member_real(X2,set_or1633881224788618240n_real(Ba,X))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X2)) )
       => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),F)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X)) ) ) ) ).

% LIM_less_bound
tff(fact_2656_greaterThanLessThan__upto,axiom,
    ! [I: int,J: int] : set_or5832277885323065728an_int(I,J) = set_int2(upto(plus_plus_int(I,one_one_int),minus_minus_int(J,one_one_int))) ).

% greaterThanLessThan_upto
tff(fact_2657_isCont__inverse__function,axiom,
    ! [D: real,X: real,G: fun_real_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
     => ( ! [Z3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Z3),X))),D)
           => ( aa_real_real(G,aa_real_real(F,Z3)) = Z3 ) )
       => ( ! [Z3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Z3),X))),D)
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),F) )
         => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,X),top_top_set_real),G) ) ) ) ).

% isCont_inverse_function
tff(fact_2658_GMVT_H,axiom,
    ! [Aa: real,Ba: real,F: fun_real_real,G: fun_real_real,G2: fun_real_real,F2: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( ! [Z3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Z3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z3),Ba)
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),F) ) )
       => ( ! [Z3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Z3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z3),Ba)
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),G) ) )
         => ( ! [Z3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Z3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),Ba)
                 => has_fi5821293074295781190e_real(G,aa_real_real(G2,Z3),topolo2177554685111907308n_real(Z3,top_top_set_real)) ) )
           => ( ! [Z3: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Z3)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),Ba)
                   => has_fi5821293074295781190e_real(F,aa_real_real(F2,Z3),topolo2177554685111907308n_real(Z3,top_top_set_real)) ) )
             => ? [C3: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),C3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,C3),Ba)
                  & ( aa_real_real(times_times_real(aa_real_real(minus_minus_real(aa_real_real(F,Ba)),aa_real_real(F,Aa))),aa_real_real(G2,C3)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(aa_real_real(G,Ba)),aa_real_real(G,Aa))),aa_real_real(F2,C3)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_2659_summable__Leibniz_I2_J,axiom,
    ! [Aa: fun_nat_real] :
      ( filterlim_nat_real(Aa,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(Aa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(Aa,zero_zero_nat))
         => ! [N8: nat] : member_real(suminf_real(aTP_Lamm_fn(Aa)),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_fn(Aa),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),N8))),groups6591440286371151544t_real(aTP_Lamm_fn(Aa),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),N8)),one_one_nat))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_2660_summable__Leibniz_I3_J,axiom,
    ! [Aa: fun_nat_real] :
      ( filterlim_nat_real(Aa,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(Aa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(Aa,zero_zero_nat)),zero_zero_real)
         => ! [N8: nat] : member_real(suminf_real(aTP_Lamm_fn(Aa)),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_fn(Aa),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),N8)),one_one_nat))),groups6591440286371151544t_real(aTP_Lamm_fn(Aa),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),N8))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_2661_mult__nat__left__at__top,axiom,
    ! [C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
     => filterlim_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),at_top_nat,at_top_nat) ) ).

% mult_nat_left_at_top
tff(fact_2662_mult__nat__right__at__top,axiom,
    ! [C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
     => filterlim_nat_nat(aTP_Lamm_fo(C),at_top_nat,at_top_nat) ) ).

% mult_nat_right_at_top
tff(fact_2663_monoseq__convergent,axiom,
    ! [X7: fun_nat_real,B: real] :
      ( topolo6980174941875973593q_real(X7)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(X7,I2))),B)
       => ~ ! [L5: real] : ~ filterlim_nat_real(X7,topolo2815343760600316023s_real(L5),at_top_nat) ) ) ).

% monoseq_convergent
tff(fact_2664_LIMSEQ__root,axiom,
    filterlim_nat_real(aTP_Lamm_fp,topolo2815343760600316023s_real(one_one_real),at_top_nat) ).

% LIMSEQ_root
tff(fact_2665_nested__sequence__unique,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(G,aa_nat_nat(suc,N))),aa_nat_real(G,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(G,N))
         => ( filterlim_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_fq(F),G),topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
           => ? [L2: real] :
                ( ! [N8: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N8)),L2)
                & filterlim_nat_real(F,topolo2815343760600316023s_real(L2),at_top_nat)
                & ! [N8: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L2),aa_nat_real(G,N8))
                & filterlim_nat_real(G,topolo2815343760600316023s_real(L2),at_top_nat) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_2666_LIMSEQ__inverse__zero,axiom,
    ! [X7: fun_nat_real] :
      ( ! [R5: real] :
        ? [N7: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,R5),aa_nat_real(X7,N)) )
     => filterlim_nat_real(aTP_Lamm_fr(X7),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_2667_lim__inverse__n_H,axiom,
    filterlim_nat_real(aTP_Lamm_fs,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% lim_inverse_n'
tff(fact_2668_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim_nat_real(aTP_Lamm_ft,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_2669_LIMSEQ__root__const,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_fu,C),topolo2815343760600316023s_real(one_one_real),at_top_nat) ) ).

% LIMSEQ_root_const
tff(fact_2670_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_fv,R),topolo2815343760600316023s_real(R),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_2671_increasing__LIMSEQ,axiom,
    ! [F: fun_nat_real,L: real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),L)
       => ( ! [E: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
             => ? [N8: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,N8)),E)) )
         => filterlim_nat_real(F,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).

% increasing_LIMSEQ
tff(fact_2672_LIMSEQ__realpow__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_2673_LIMSEQ__divide__realpow__zero,axiom,
    ! [X: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_fw(X),Aa),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_2674_LIMSEQ__abs__realpow__zero,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,C)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,C)),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero
tff(fact_2675_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,C)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,C),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_2676_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_fx,X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_2677_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_fy,R),topolo2815343760600316023s_real(R),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_2678_tendsto__exp__limit__sequentially,axiom,
    ! [X: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_fz,X),topolo2815343760600316023s_real(aa_real_real(exp_real,X)),at_top_nat) ).

% tendsto_exp_limit_sequentially
tff(fact_2679_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_ga,R),topolo2815343760600316023s_real(R),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_2680_summable__Leibniz_I1_J,axiom,
    ! [Aa: fun_nat_real] :
      ( filterlim_nat_real(Aa,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(Aa)
       => summable_real(aTP_Lamm_fn(Aa)) ) ) ).

% summable_Leibniz(1)
tff(fact_2681_summable,axiom,
    ! [Aa: fun_nat_real] :
      ( filterlim_nat_real(Aa,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(Aa,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(Aa,aa_nat_nat(suc,N))),aa_nat_real(Aa,N))
         => summable_real(aTP_Lamm_fn(Aa)) ) ) ) ).

% summable
tff(fact_2682_cos__diff__limit__1,axiom,
    ! [Theta: fun_nat_real,Theta2: real] :
      ( filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_gb(Theta),Theta2),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ~ ! [K4: fun_nat_int] : ~ filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_gc(Theta),K4),topolo2815343760600316023s_real(Theta2),at_top_nat) ) ).

% cos_diff_limit_1
tff(fact_2683_cos__limit__1,axiom,
    ! [Theta: fun_nat_real] :
      ( filterlim_nat_real(aTP_Lamm_gd(Theta),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ? [K4: fun_nat_int] : filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_gc(Theta),K4),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% cos_limit_1
tff(fact_2684_summable__Leibniz_I4_J,axiom,
    ! [Aa: fun_nat_real] :
      ( filterlim_nat_real(Aa,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(Aa)
       => filterlim_nat_real(aTP_Lamm_ge(Aa),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_fn(Aa))),at_top_nat) ) ) ).

% summable_Leibniz(4)
tff(fact_2685_zeroseq__arctan__series,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_bg,X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% zeroseq_arctan_series
tff(fact_2686_summable__Leibniz_H_I2_J,axiom,
    ! [Aa: fun_nat_real,Na: nat] :
      ( filterlim_nat_real(Aa,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(Aa,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(Aa,aa_nat_nat(suc,N))),aa_nat_real(Aa,N))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_fn(Aa),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Na)))),suminf_real(aTP_Lamm_fn(Aa))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_2687_summable__Leibniz_H_I3_J,axiom,
    ! [Aa: fun_nat_real] :
      ( filterlim_nat_real(Aa,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(Aa,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(Aa,aa_nat_nat(suc,N))),aa_nat_real(Aa,N))
         => filterlim_nat_real(aTP_Lamm_ge(Aa),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_fn(Aa))),at_top_nat) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_2688_sums__alternating__upper__lower,axiom,
    ! [Aa: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(Aa,aa_nat_nat(suc,N))),aa_nat_real(Aa,N))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(Aa,N))
       => ( filterlim_nat_real(Aa,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
         => ? [L2: real] :
              ( ! [N8: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_fn(Aa),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),N8)))),L2)
              & filterlim_nat_real(aTP_Lamm_ge(Aa),topolo2815343760600316023s_real(L2),at_top_nat)
              & ! [N8: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L2),groups6591440286371151544t_real(aTP_Lamm_fn(Aa),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),N8)),one_one_nat))))
              & filterlim_nat_real(aTP_Lamm_gf(Aa),topolo2815343760600316023s_real(L2),at_top_nat) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_2689_summable__Leibniz_I5_J,axiom,
    ! [Aa: fun_nat_real] :
      ( filterlim_nat_real(Aa,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(Aa)
       => filterlim_nat_real(aTP_Lamm_gf(Aa),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_fn(Aa))),at_top_nat) ) ) ).

% summable_Leibniz(5)
tff(fact_2690_summable__Leibniz_H_I4_J,axiom,
    ! [Aa: fun_nat_real,Na: nat] :
      ( filterlim_nat_real(Aa,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(Aa,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(Aa,aa_nat_nat(suc,N))),aa_nat_real(Aa,N))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(aTP_Lamm_fn(Aa))),groups6591440286371151544t_real(aTP_Lamm_fn(Aa),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Na)),one_one_nat)))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_2691_summable__Leibniz_H_I5_J,axiom,
    ! [Aa: fun_nat_real] :
      ( filterlim_nat_real(Aa,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(Aa,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(Aa,aa_nat_nat(suc,N))),aa_nat_real(Aa,N))
         => filterlim_nat_real(aTP_Lamm_gf(Aa),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_fn(Aa))),at_top_nat) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_2692_tendsto__exp__limit__at__right,axiom,
    ! [X: real] : filterlim_real_real(aTP_Lamm_gg(X),topolo2815343760600316023s_real(aa_real_real(exp_real,X)),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% tendsto_exp_limit_at_right
tff(fact_2693_tendsto__arctan__at__bot,axiom,
    filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),at_bot_real) ).

% tendsto_arctan_at_bot
tff(fact_2694_ln__at__0,axiom,
    filterlim_real_real(ln_ln_real,at_bot_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% ln_at_0
tff(fact_2695_filterlim__tan__at__right,axiom,
    filterlim_real_real(tan_real,at_bot_real,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one)))),set_or5849166863359141190n_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))))) ).

% filterlim_tan_at_right
tff(fact_2696_filterlim__inverse__at__bot__neg,axiom,
    filterlim_real_real(inverse_inverse_real,at_bot_real,topolo2177554685111907308n_real(zero_zero_real,set_or5984915006950818249n_real(zero_zero_real))) ).

% filterlim_inverse_at_bot_neg
tff(fact_2697_log__inj,axiom,
    ! [Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ba)
     => inj_on_real_real(log(Ba),set_or5849166863359141190n_real(zero_zero_real)) ) ).

% log_inj
tff(fact_2698_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [Ba: real,F: fun_real_real,Flim: real] :
      ( ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Ba)
         => ? [Y3: real] :
              ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X2,top_top_set_real))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y3) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_bot_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Flim),aa_real_real(F,Ba)) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_2699_filterlim__pow__at__bot__odd,axiom,
    ! [Na: nat,F: fun_real_real,F3: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( filterlim_real_real(F,at_bot_real,F3)
       => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gh(Na),F),at_bot_real,F3) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_2700_filterlim__pow__at__bot__even,axiom,
    ! [Na: nat,F: fun_real_real,F3: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( filterlim_real_real(F,at_bot_real,F3)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Na)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gh(Na),F),at_top_real,F3) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_2701_exp__at__top,axiom,
    filterlim_real_real(exp_real,at_top_real,at_top_real) ).

% exp_at_top
tff(fact_2702_sqrt__at__top,axiom,
    filterlim_real_real(sqrt,at_top_real,at_top_real) ).

% sqrt_at_top
tff(fact_2703_ln__at__top,axiom,
    filterlim_real_real(ln_ln_real,at_top_real,at_top_real) ).

% ln_at_top
tff(fact_2704_filterlim__real__sequentially,axiom,
    filterlim_nat_real(semiri5074537144036343181t_real,at_top_real,at_top_nat) ).

% filterlim_real_sequentially
tff(fact_2705_filterlim__uminus__at__top__at__bot,axiom,
    filterlim_real_real(uminus_uminus_real,at_top_real,at_bot_real) ).

% filterlim_uminus_at_top_at_bot
tff(fact_2706_filterlim__uminus__at__bot__at__top,axiom,
    filterlim_real_real(uminus_uminus_real,at_bot_real,at_top_real) ).

% filterlim_uminus_at_bot_at_top
tff(fact_2707_filterlim__real__at__infinity__sequentially,axiom,
    filterlim_nat_real(semiri5074537144036343181t_real,at_infinity_real,at_top_nat) ).

% filterlim_real_at_infinity_sequentially
tff(fact_2708_ln__x__over__x__tendsto__0,axiom,
    filterlim_real_real(aTP_Lamm_gi,topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% ln_x_over_x_tendsto_0
tff(fact_2709_filterlim__inverse__at__top__right,axiom,
    filterlim_real_real(inverse_inverse_real,at_top_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% filterlim_inverse_at_top_right
tff(fact_2710_filterlim__inverse__at__right__top,axiom,
    filterlim_real_real(inverse_inverse_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)),at_top_real) ).

% filterlim_inverse_at_right_top
tff(fact_2711_tendsto__power__div__exp__0,axiom,
    ! [K: nat] : filterlim_real_real(aa_nat_fun_real_real(aTP_Lamm_gj,K),topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% tendsto_power_div_exp_0
tff(fact_2712_tendsto__exp__limit__at__top,axiom,
    ! [X: real] : filterlim_real_real(aTP_Lamm_gk(X),topolo2815343760600316023s_real(aa_real_real(exp_real,X)),at_top_real) ).

% tendsto_exp_limit_at_top
tff(fact_2713_filterlim__tan__at__left,axiom,
    filterlim_real_real(tan_real,at_top_real,topolo2177554685111907308n_real(aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))),set_or5984915006950818249n_real(aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one)))))) ).

% filterlim_tan_at_left
tff(fact_2714_tendsto__arctan__at__top,axiom,
    filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one)))),at_top_real) ).

% tendsto_arctan_at_top
tff(fact_2715_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [Ba: real,F: fun_real_real,Flim: real] :
      ( ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),X2)
         => ? [Y3: real] :
              ( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X2,top_top_set_real))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),zero_zero_real) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_top_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Flim),aa_real_real(F,Ba)) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_2716_lhopital__left__at__top,axiom,
    ! [G: fun_real_real,X: real,G2: fun_real_real,F: fun_real_real,F2: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
     => ( eventually_real(aTP_Lamm_gl(G2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_gm(F),F2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_gm(G),G2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gn(G2),F2),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gn(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X))) ) ) ) ) ) ).

% lhopital_left_at_top
tff(fact_2717_eventually__sequentially__Suc,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(aTP_Lamm_go(P),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_Suc
tff(fact_2718_eventually__sequentially__seg,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_gp(P),K),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_seg
tff(fact_2719_sequentially__offset,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( eventually_nat(P,at_top_nat)
     => eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_gp(P),K),at_top_nat) ) ).

% sequentially_offset
tff(fact_2720_eventually__sequentially,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(P,at_top_nat)
    <=> ? [N9: nat] :
        ! [N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N9),N2)
         => aa_nat_o(P,N2) ) ) ).

% eventually_sequentially
tff(fact_2721_eventually__sequentiallyI,axiom,
    ! [C: nat,P: fun_nat_o] :
      ( ! [X2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),X2)
         => aa_nat_o(P,X2) )
     => eventually_nat(P,at_top_nat) ) ).

% eventually_sequentiallyI
tff(fact_2722_le__sequentially,axiom,
    ! [F3: filter_nat] :
      ( ord_le2510731241096832064er_nat(F3,at_top_nat)
    <=> ! [N9: nat] : eventually_nat(aa_nat_fun_nat_o(ord_less_eq_nat,N9),F3) ) ).

% le_sequentially
tff(fact_2723_eventually__False__sequentially,axiom,
    ~ eventually_nat(aTP_Lamm_gq,at_top_nat) ).

% eventually_False_sequentially
tff(fact_2724_eventually__at__right__to__0,axiom,
    ! [P: fun_real_o,Aa: real] :
      ( eventually_real(P,topolo2177554685111907308n_real(Aa,set_or5849166863359141190n_real(Aa)))
    <=> eventually_real(aa_real_fun_real_o(aTP_Lamm_gr(P),Aa),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ).

% eventually_at_right_to_0
tff(fact_2725_eventually__at__left__to__right,axiom,
    ! [P: fun_real_o,Aa: real] :
      ( eventually_real(P,topolo2177554685111907308n_real(Aa,set_or5984915006950818249n_real(Aa)))
    <=> eventually_real(aTP_Lamm_gs(P),topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,Aa),set_or5849166863359141190n_real(aa_real_real(uminus_uminus_real,Aa)))) ) ).

% eventually_at_left_to_right
tff(fact_2726_eventually__at__right__real,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => eventually_real(aa_real_fun_real_o(aTP_Lamm_gt(Aa),Ba),topolo2177554685111907308n_real(Aa,set_or5849166863359141190n_real(Aa))) ) ).

% eventually_at_right_real
tff(fact_2727_eventually__at__left__real,axiom,
    ! [Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),Aa)
     => eventually_real(aa_real_fun_real_o(aTP_Lamm_gt(Ba),Aa),topolo2177554685111907308n_real(Aa,set_or5984915006950818249n_real(Aa))) ) ).

% eventually_at_left_real
tff(fact_2728_eventually__at__top__to__right,axiom,
    ! [P: fun_real_o] :
      ( eventually_real(P,at_top_real)
    <=> eventually_real(aTP_Lamm_gu(P),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ).

% eventually_at_top_to_right
tff(fact_2729_eventually__at__right__to__top,axiom,
    ! [P: fun_real_o] :
      ( eventually_real(P,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
    <=> eventually_real(aTP_Lamm_gu(P),at_top_real) ) ).

% eventually_at_right_to_top
tff(fact_2730_lhopital__at__top__at__top,axiom,
    ! [F: fun_real_real,Aa: real,G: fun_real_real,F2: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(Aa,top_top_set_real))
     => ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(Aa,top_top_set_real))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_gm(F),F2),topolo2177554685111907308n_real(Aa,top_top_set_real))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_gm(G),G2),topolo2177554685111907308n_real(Aa,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gv(F2),G2),at_top_real,topolo2177554685111907308n_real(Aa,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gv(F),G),at_top_real,topolo2177554685111907308n_real(Aa,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top_at_top
tff(fact_2731_lhopital,axiom,
    ! [F: fun_real_real,X: real,G: fun_real_real,G2: fun_real_real,F2: fun_real_real,F3: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X,top_top_set_real))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X,top_top_set_real))
       => ( eventually_real(aTP_Lamm_gl(G),topolo2177554685111907308n_real(X,top_top_set_real))
         => ( eventually_real(aTP_Lamm_gl(G2),topolo2177554685111907308n_real(X,top_top_set_real))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_gm(F),F2),topolo2177554685111907308n_real(X,top_top_set_real))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_gm(G),G2),topolo2177554685111907308n_real(X,top_top_set_real))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gn(G2),F2),F3,topolo2177554685111907308n_real(X,top_top_set_real))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gv(F),G),F3,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ) ) ).

% lhopital
tff(fact_2732_lhopital__right__at__top__at__top,axiom,
    ! [F: fun_real_real,Aa: real,G: fun_real_real,F2: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(Aa,set_or5849166863359141190n_real(Aa)))
     => ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(Aa,set_or5849166863359141190n_real(Aa)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_gm(F),F2),topolo2177554685111907308n_real(Aa,set_or5849166863359141190n_real(Aa)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_gm(G),G2),topolo2177554685111907308n_real(Aa,set_or5849166863359141190n_real(Aa)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gv(F2),G2),at_top_real,topolo2177554685111907308n_real(Aa,set_or5849166863359141190n_real(Aa)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gv(F),G),at_top_real,topolo2177554685111907308n_real(Aa,set_or5849166863359141190n_real(Aa))) ) ) ) ) ) ).

% lhopital_right_at_top_at_top
tff(fact_2733_lhopital__at__top__at__bot,axiom,
    ! [F: fun_real_real,Aa: real,G: fun_real_real,F2: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(Aa,top_top_set_real))
     => ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(Aa,top_top_set_real))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_gm(F),F2),topolo2177554685111907308n_real(Aa,top_top_set_real))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_gm(G),G2),topolo2177554685111907308n_real(Aa,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gv(F2),G2),at_bot_real,topolo2177554685111907308n_real(Aa,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gv(F),G),at_bot_real,topolo2177554685111907308n_real(Aa,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top_at_bot
tff(fact_2734_lhopital__left__at__top__at__top,axiom,
    ! [F: fun_real_real,Aa: real,G: fun_real_real,F2: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(Aa,set_or5984915006950818249n_real(Aa)))
     => ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(Aa,set_or5984915006950818249n_real(Aa)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_gm(F),F2),topolo2177554685111907308n_real(Aa,set_or5984915006950818249n_real(Aa)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_gm(G),G2),topolo2177554685111907308n_real(Aa,set_or5984915006950818249n_real(Aa)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gv(F2),G2),at_top_real,topolo2177554685111907308n_real(Aa,set_or5984915006950818249n_real(Aa)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gv(F),G),at_top_real,topolo2177554685111907308n_real(Aa,set_or5984915006950818249n_real(Aa))) ) ) ) ) ) ).

% lhopital_left_at_top_at_top
tff(fact_2735_lhospital__at__top__at__top,axiom,
    ! [G: fun_real_real,G2: fun_real_real,F: fun_real_real,F2: fun_real_real,X: real] :
      ( filterlim_real_real(G,at_top_real,at_top_real)
     => ( eventually_real(aTP_Lamm_gl(G2),at_top_real)
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_gm(F),F2),at_top_real)
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_gm(G),G2),at_top_real)
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gn(G2),F2),topolo2815343760600316023s_real(X),at_top_real)
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gn(G),F),topolo2815343760600316023s_real(X),at_top_real) ) ) ) ) ) ).

% lhospital_at_top_at_top
tff(fact_2736_lhopital__at__top,axiom,
    ! [G: fun_real_real,X: real,G2: fun_real_real,F: fun_real_real,F2: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( eventually_real(aTP_Lamm_gl(G2),topolo2177554685111907308n_real(X,top_top_set_real))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_gm(F),F2),topolo2177554685111907308n_real(X,top_top_set_real))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_gm(G),G2),topolo2177554685111907308n_real(X,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gn(G2),F2),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gn(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top
tff(fact_2737_lhopital__right,axiom,
    ! [F: fun_real_real,X: real,G: fun_real_real,G2: fun_real_real,F2: fun_real_real,F3: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
       => ( eventually_real(aTP_Lamm_gl(G),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
         => ( eventually_real(aTP_Lamm_gl(G2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_gm(F),F2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_gm(G),G2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gn(G2),F2),F3,topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gv(F),G),F3,topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X))) ) ) ) ) ) ) ) ).

% lhopital_right
tff(fact_2738_lhopital__right__0,axiom,
    ! [F0: fun_real_real,G0: fun_real_real,G2: fun_real_real,F2: fun_real_real,F3: filter_real] :
      ( filterlim_real_real(F0,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
     => ( filterlim_real_real(G0,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
       => ( eventually_real(aTP_Lamm_gl(G0),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
         => ( eventually_real(aTP_Lamm_gl(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_gm(F0),F2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_gm(G0),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gn(G2),F2),F3,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gv(F0),G0),F3,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ) ) ).

% lhopital_right_0
tff(fact_2739_lhopital__left,axiom,
    ! [F: fun_real_real,X: real,G: fun_real_real,G2: fun_real_real,F2: fun_real_real,F3: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
       => ( eventually_real(aTP_Lamm_gl(G),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
         => ( eventually_real(aTP_Lamm_gl(G2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_gm(F),F2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_gm(G),G2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gn(G2),F2),F3,topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gv(F),G),F3,topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X))) ) ) ) ) ) ) ) ).

% lhopital_left
tff(fact_2740_lhopital__right__at__top__at__bot,axiom,
    ! [F: fun_real_real,Aa: real,G: fun_real_real,F2: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(Aa,set_or5849166863359141190n_real(Aa)))
     => ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(Aa,set_or5849166863359141190n_real(Aa)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_gm(F),F2),topolo2177554685111907308n_real(Aa,set_or5849166863359141190n_real(Aa)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_gm(G),G2),topolo2177554685111907308n_real(Aa,set_or5849166863359141190n_real(Aa)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gv(F2),G2),at_bot_real,topolo2177554685111907308n_real(Aa,set_or5849166863359141190n_real(Aa)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gv(F),G),at_bot_real,topolo2177554685111907308n_real(Aa,set_or5849166863359141190n_real(Aa))) ) ) ) ) ) ).

% lhopital_right_at_top_at_bot
tff(fact_2741_lhopital__left__at__top__at__bot,axiom,
    ! [F: fun_real_real,Aa: real,G: fun_real_real,F2: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(Aa,set_or5984915006950818249n_real(Aa)))
     => ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(Aa,set_or5984915006950818249n_real(Aa)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_gm(F),F2),topolo2177554685111907308n_real(Aa,set_or5984915006950818249n_real(Aa)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_gm(G),G2),topolo2177554685111907308n_real(Aa,set_or5984915006950818249n_real(Aa)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gv(F2),G2),at_bot_real,topolo2177554685111907308n_real(Aa,set_or5984915006950818249n_real(Aa)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gv(F),G),at_bot_real,topolo2177554685111907308n_real(Aa,set_or5984915006950818249n_real(Aa))) ) ) ) ) ) ).

% lhopital_left_at_top_at_bot
tff(fact_2742_lhopital__right__at__top,axiom,
    ! [G: fun_real_real,X: real,G2: fun_real_real,F: fun_real_real,F2: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
     => ( eventually_real(aTP_Lamm_gl(G2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_gm(F),F2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_gm(G),G2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gn(G2),F2),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gn(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X))) ) ) ) ) ) ).

% lhopital_right_at_top
tff(fact_2743_lhopital__right__0__at__top,axiom,
    ! [G: fun_real_real,G2: fun_real_real,F: fun_real_real,F2: fun_real_real,X: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
     => ( eventually_real(aTP_Lamm_gl(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_gm(F),F2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_gm(G),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gn(G2),F2),topolo2815343760600316023s_real(X),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gn(G),F),topolo2815343760600316023s_real(X),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ).

% lhopital_right_0_at_top
tff(fact_2744_filterlim__int__sequentially,axiom,
    filterlim_nat_int(semiri1314217659103216013at_int,at_top_int,at_top_nat) ).

% filterlim_int_sequentially
tff(fact_2745_filterlim__nat__sequentially,axiom,
    filterlim_int_nat(nat2,at_top_nat,at_top_int) ).

% filterlim_nat_sequentially
tff(fact_2746_Bseq__realpow,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
       => bfun_nat_real(aa_real_fun_nat_real(power_power_real,X),at_top_nat) ) ) ).

% Bseq_realpow
tff(fact_2747_decseq__bounded,axiom,
    ! [X7: fun_nat_real,B: real] :
      ( order_9091379641038594480t_real(X7)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_nat_real(X7,I2))
       => bfun_nat_real(X7,at_top_nat) ) ) ).

% decseq_bounded
tff(fact_2748_decseq__convergent,axiom,
    ! [X7: fun_nat_real,B: real] :
      ( order_9091379641038594480t_real(X7)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_nat_real(X7,I2))
       => ~ ! [L5: real] :
              ( filterlim_nat_real(X7,topolo2815343760600316023s_real(L5),at_top_nat)
             => ~ ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L5),aa_nat_real(X7,I3)) ) ) ) ).

% decseq_convergent
tff(fact_2749_GMVT,axiom,
    ! [Aa: real,Ba: real,F: fun_real_real,G: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( ! [X2: real] :
            ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X2)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Ba) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X2,top_top_set_real),F) )
       => ( ! [X2: real] :
              ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),X2)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Ba) )
             => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X2,top_top_set_real)) )
         => ( ! [X2: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Ba) )
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X2,top_top_set_real),G) )
           => ( ! [X2: real] :
                  ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),X2)
                    & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Ba) )
                 => differ6690327859849518006l_real(G,topolo2177554685111907308n_real(X2,top_top_set_real)) )
             => ? [G_c: real,F_c: real,C3: real] :
                  ( has_fi5821293074295781190e_real(G,G_c,topolo2177554685111907308n_real(C3,top_top_set_real))
                  & has_fi5821293074295781190e_real(F,F_c,topolo2177554685111907308n_real(C3,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),C3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,C3),Ba)
                  & ( aa_real_real(times_times_real(aa_real_real(minus_minus_real(aa_real_real(F,Ba)),aa_real_real(F,Aa))),G_c) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(aa_real_real(G,Ba)),aa_real_real(G,Aa))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_2750_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : set_or1266510415728281911st_int(plus_plus_int(L,one_one_int),U) = set_or6656581121297822940st_int(L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
tff(fact_2751_greaterThanAtMost__upto,axiom,
    ! [I: int,J: int] : set_or6656581121297822940st_int(I,J) = set_int2(upto(plus_plus_int(I,one_one_int),J)) ).

% greaterThanAtMost_upto
tff(fact_2752_csqrt_Osimps_I1_J,axiom,
    ! [Z: complex] : re(csqrt(Z)) = aa_real_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,real_V1022390504157884413omplex(Z)),re(Z))),numeral_numeral_real(bit0(one)))) ).

% csqrt.simps(1)
tff(fact_2753_complex__Re__of__nat,axiom,
    ! [Na: nat] : re(aa_nat_complex(semiri8010041392384452111omplex,Na)) = aa_nat_real(semiri5074537144036343181t_real,Na) ).

% complex_Re_of_nat
tff(fact_2754_complex__Re__numeral,axiom,
    ! [V: num] : re(numera6690914467698888265omplex(V)) = numeral_numeral_real(V) ).

% complex_Re_numeral
tff(fact_2755_Re__divide__of__nat,axiom,
    ! [Z: complex,Na: nat] : re(divide1717551699836669952omplex(Z,aa_nat_complex(semiri8010041392384452111omplex,Na))) = aa_real_real(divide_divide_real(re(Z)),aa_nat_real(semiri5074537144036343181t_real,Na)) ).

% Re_divide_of_nat
tff(fact_2756_Re__divide__numeral,axiom,
    ! [Z: complex,W: num] : re(divide1717551699836669952omplex(Z,numera6690914467698888265omplex(W))) = aa_real_real(divide_divide_real(re(Z)),numeral_numeral_real(W)) ).

% Re_divide_numeral
tff(fact_2757_complex__Re__le__cmod,axiom,
    ! [X: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,re(X)),real_V1022390504157884413omplex(X)) ).

% complex_Re_le_cmod
tff(fact_2758_plus__complex_Osimps_I1_J,axiom,
    ! [X: complex,Y: complex] : re(plus_plus_complex(X,Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,re(X)),re(Y)) ).

% plus_complex.simps(1)
tff(fact_2759_Cauchy__Re,axiom,
    ! [X7: fun_nat_complex] :
      ( topolo6517432010174082258omplex(X7)
     => topolo4055970368930404560y_real(aTP_Lamm_gw(X7)) ) ).

% Cauchy_Re
tff(fact_2760_sums__Re,axiom,
    ! [X7: fun_nat_complex,Aa: complex] :
      ( aa_complex_o(sums_complex(X7),Aa)
     => aa_real_o(sums_real(aTP_Lamm_gw(X7)),re(Aa)) ) ).

% sums_Re
tff(fact_2761_summable__Re,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(F)
     => summable_real(aTP_Lamm_gw(F)) ) ).

% summable_Re
tff(fact_2762_abs__Re__le__cmod,axiom,
    ! [X: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,re(X))),real_V1022390504157884413omplex(X)) ).

% abs_Re_le_cmod
tff(fact_2763_Re__csqrt,axiom,
    ! [Z: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),re(csqrt(Z))) ).

% Re_csqrt
tff(fact_2764_cmod__plus__Re__le__0__iff,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,real_V1022390504157884413omplex(Z)),re(Z))),zero_zero_real)
    <=> ( re(Z) = aa_real_real(uminus_uminus_real,real_V1022390504157884413omplex(Z)) ) ) ).

% cmod_plus_Re_le_0_iff
tff(fact_2765_cos__n__Re__cis__pow__n,axiom,
    ! [Na: nat,Aa: real] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),Aa)) = re(aa_nat_complex(power_power_complex(cis(Aa)),Na)) ).

% cos_n_Re_cis_pow_n
tff(fact_2766_csqrt_Ocode,axiom,
    ! [Z: complex] :
      csqrt(Z) = complex2(aa_real_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,real_V1022390504157884413omplex(Z)),re(Z))),numeral_numeral_real(bit0(one)))),
        aa_real_real(times_times_real($ite(im(Z) = zero_zero_real,one_one_real,aa_real_real(sgn_sgn_real,im(Z)))),aa_real_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(real_V1022390504157884413omplex(Z)),re(Z))),numeral_numeral_real(bit0(one)))))) ).

% csqrt.code
tff(fact_2767_csqrt_Osimps_I2_J,axiom,
    ! [Z: complex] :
      im(csqrt(Z)) = aa_real_real(times_times_real($ite(im(Z) = zero_zero_real,one_one_real,aa_real_real(sgn_sgn_real,im(Z)))),aa_real_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(real_V1022390504157884413omplex(Z)),re(Z))),numeral_numeral_real(bit0(one))))) ).

% csqrt.simps(2)
tff(fact_2768_Im__divide__numeral,axiom,
    ! [Z: complex,W: num] : im(divide1717551699836669952omplex(Z,numera6690914467698888265omplex(W))) = aa_real_real(divide_divide_real(im(Z)),numeral_numeral_real(W)) ).

% Im_divide_numeral
tff(fact_2769_Im__divide__of__nat,axiom,
    ! [Z: complex,Na: nat] : im(divide1717551699836669952omplex(Z,aa_nat_complex(semiri8010041392384452111omplex,Na))) = aa_real_real(divide_divide_real(im(Z)),aa_nat_real(semiri5074537144036343181t_real,Na)) ).

% Im_divide_of_nat
tff(fact_2770_csqrt__of__real__nonneg,axiom,
    ! [X: complex] :
      ( ( im(X) = zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),re(X))
       => ( csqrt(X) = real_V4546457046886955230omplex(aa_real_real(sqrt,re(X))) ) ) ) ).

% csqrt_of_real_nonneg
tff(fact_2771_csqrt__minus,axiom,
    ! [X: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,im(X)),zero_zero_real)
        | ( ( im(X) = zero_zero_real )
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),re(X)) ) )
     => ( csqrt(uminus1482373934393186551omplex(X)) = aa_complex_complex(times_times_complex(imaginary_unit),csqrt(X)) ) ) ).

% csqrt_minus
tff(fact_2772_csqrt__of__real__nonpos,axiom,
    ! [X: complex] :
      ( ( im(X) = zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,re(X)),zero_zero_real)
       => ( csqrt(X) = aa_complex_complex(times_times_complex(imaginary_unit),real_V4546457046886955230omplex(aa_real_real(sqrt,aa_real_real(abs_abs_real,re(X))))) ) ) ) ).

% csqrt_of_real_nonpos
tff(fact_2773_sums__complex__iff,axiom,
    ! [F: fun_nat_complex,X: complex] :
      ( aa_complex_o(sums_complex(F),X)
    <=> ( aa_real_o(sums_real(aTP_Lamm_gw(F)),re(X))
        & aa_real_o(sums_real(aTP_Lamm_gx(F)),im(X)) ) ) ).

% sums_complex_iff
tff(fact_2774_sums__Im,axiom,
    ! [X7: fun_nat_complex,Aa: complex] :
      ( aa_complex_o(sums_complex(X7),Aa)
     => aa_real_o(sums_real(aTP_Lamm_gx(X7)),im(Aa)) ) ).

% sums_Im
tff(fact_2775_Cauchy__Im,axiom,
    ! [X7: fun_nat_complex] :
      ( topolo6517432010174082258omplex(X7)
     => topolo4055970368930404560y_real(aTP_Lamm_gx(X7)) ) ).

% Cauchy_Im
tff(fact_2776_plus__complex_Osimps_I2_J,axiom,
    ! [X: complex,Y: complex] : im(plus_plus_complex(X,Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,im(X)),im(Y)) ).

% plus_complex.simps(2)
tff(fact_2777_summable__Im,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(F)
     => summable_real(aTP_Lamm_gx(F)) ) ).

% summable_Im
tff(fact_2778_abs__Im__le__cmod,axiom,
    ! [X: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,im(X))),real_V1022390504157884413omplex(X)) ).

% abs_Im_le_cmod
tff(fact_2779_summable__complex__iff,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(F)
    <=> ( summable_real(aTP_Lamm_gw(F))
        & summable_real(aTP_Lamm_gx(F)) ) ) ).

% summable_complex_iff
tff(fact_2780_times__complex_Osimps_I2_J,axiom,
    ! [X: complex,Y: complex] : im(aa_complex_complex(times_times_complex(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(re(X)),im(Y))),aa_real_real(times_times_real(im(X)),re(Y))) ).

% times_complex.simps(2)
tff(fact_2781_cmod__Re__le__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( im(X) = im(Y) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,real_V1022390504157884413omplex(X)),real_V1022390504157884413omplex(Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,re(X))),aa_real_real(abs_abs_real,re(Y))) ) ) ).

% cmod_Re_le_iff
tff(fact_2782_cmod__Im__le__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( re(X) = re(Y) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,real_V1022390504157884413omplex(X)),real_V1022390504157884413omplex(Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,im(X))),aa_real_real(abs_abs_real,im(Y))) ) ) ).

% cmod_Im_le_iff
tff(fact_2783_plus__complex_Ocode,axiom,
    ! [X: complex,Y: complex] : plus_plus_complex(X,Y) = complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,re(X)),re(Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,im(X)),im(Y))) ).

% plus_complex.code
tff(fact_2784_csqrt__principal,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),re(csqrt(Z)))
      | ( ( re(csqrt(Z)) = zero_zero_real )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),im(csqrt(Z))) ) ) ).

% csqrt_principal
tff(fact_2785_cmod__le,axiom,
    ! [Z: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,real_V1022390504157884413omplex(Z)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,re(Z))),aa_real_real(abs_abs_real,im(Z)))) ).

% cmod_le
tff(fact_2786_sin__n__Im__cis__pow__n,axiom,
    ! [Na: nat,Aa: real] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Na)),Aa)) = im(aa_nat_complex(power_power_complex(cis(Aa)),Na)) ).

% sin_n_Im_cis_pow_n
tff(fact_2787_complex__eq,axiom,
    ! [Aa: complex] : Aa = plus_plus_complex(real_V4546457046886955230omplex(re(Aa)),aa_complex_complex(times_times_complex(imaginary_unit),real_V4546457046886955230omplex(im(Aa)))) ).

% complex_eq
tff(fact_2788_times__complex_Ocode,axiom,
    ! [X: complex,Y: complex] : aa_complex_complex(times_times_complex(X),Y) = complex2(aa_real_real(minus_minus_real(aa_real_real(times_times_real(re(X)),re(Y))),aa_real_real(times_times_real(im(X)),im(Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(re(X)),im(Y))),aa_real_real(times_times_real(im(X)),re(Y)))) ).

% times_complex.code
tff(fact_2789_cmod__power2,axiom,
    ! [Z: complex] : aa_nat_real(aa_real_fun_nat_real(power_power_real,real_V1022390504157884413omplex(Z)),numeral_numeral_nat(bit0(one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,re(Z)),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,im(Z)),numeral_numeral_nat(bit0(one)))) ).

% cmod_power2
tff(fact_2790_Im__power2,axiom,
    ! [X: complex] : im(aa_nat_complex(power_power_complex(X),numeral_numeral_nat(bit0(one)))) = aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),re(X))),im(X)) ).

% Im_power2
tff(fact_2791_Re__power2,axiom,
    ! [X: complex] : re(aa_nat_complex(power_power_complex(X),numeral_numeral_nat(bit0(one)))) = aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,re(X)),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,im(X)),numeral_numeral_nat(bit0(one)))) ).

% Re_power2
tff(fact_2792_complex__eq__0,axiom,
    ! [Z: complex] :
      ( ( Z = zero_zero_complex )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,re(Z)),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,im(Z)),numeral_numeral_nat(bit0(one)))) = zero_zero_real ) ) ).

% complex_eq_0
tff(fact_2793_norm__complex__def,axiom,
    ! [Z: complex] : real_V1022390504157884413omplex(Z) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,re(Z)),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,im(Z)),numeral_numeral_nat(bit0(one))))) ).

% norm_complex_def
tff(fact_2794_inverse__complex_Osimps_I1_J,axiom,
    ! [X: complex] : re(invers8013647133539491842omplex(X)) = aa_real_real(divide_divide_real(re(X)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,re(X)),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,im(X)),numeral_numeral_nat(bit0(one))))) ).

% inverse_complex.simps(1)
tff(fact_2795_complex__neq__0,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,re(Z)),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,im(Z)),numeral_numeral_nat(bit0(one))))) ) ).

% complex_neq_0
tff(fact_2796_Re__divide,axiom,
    ! [X: complex,Y: complex] : re(divide1717551699836669952omplex(X,Y)) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(re(X)),re(Y))),aa_real_real(times_times_real(im(X)),im(Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,re(Y)),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,im(Y)),numeral_numeral_nat(bit0(one))))) ).

% Re_divide
tff(fact_2797_csqrt__square,axiom,
    ! [Ba: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),re(Ba))
        | ( ( re(Ba) = zero_zero_real )
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),im(Ba)) ) )
     => ( csqrt(aa_nat_complex(power_power_complex(Ba),numeral_numeral_nat(bit0(one)))) = Ba ) ) ).

% csqrt_square
tff(fact_2798_csqrt__unique,axiom,
    ! [W: complex,Z: complex] :
      ( ( aa_nat_complex(power_power_complex(W),numeral_numeral_nat(bit0(one))) = Z )
     => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),re(W))
          | ( ( re(W) = zero_zero_real )
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),im(W)) ) )
       => ( csqrt(Z) = W ) ) ) ).

% csqrt_unique
tff(fact_2799_inverse__complex_Osimps_I2_J,axiom,
    ! [X: complex] : im(invers8013647133539491842omplex(X)) = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,im(X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,re(X)),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,im(X)),numeral_numeral_nat(bit0(one))))) ).

% inverse_complex.simps(2)
tff(fact_2800_Im__divide,axiom,
    ! [X: complex,Y: complex] : im(divide1717551699836669952omplex(X,Y)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(im(X)),re(Y))),aa_real_real(times_times_real(re(X)),im(Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,re(Y)),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,im(Y)),numeral_numeral_nat(bit0(one))))) ).

% Im_divide
tff(fact_2801_complex__abs__le__norm,axiom,
    ! [Z: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,re(Z))),aa_real_real(abs_abs_real,im(Z)))),aa_real_real(times_times_real(aa_real_real(sqrt,numeral_numeral_real(bit0(one)))),real_V1022390504157884413omplex(Z))) ).

% complex_abs_le_norm
tff(fact_2802_complex__unit__circle,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(re(Z)),real_V1022390504157884413omplex(Z))),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(im(Z)),real_V1022390504157884413omplex(Z))),numeral_numeral_nat(bit0(one)))) = one_one_real ) ) ).

% complex_unit_circle
tff(fact_2803_inverse__complex_Ocode,axiom,
    ! [X: complex] : invers8013647133539491842omplex(X) = complex2(aa_real_real(divide_divide_real(re(X)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,re(X)),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,im(X)),numeral_numeral_nat(bit0(one))))),aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,im(X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,re(X)),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,im(X)),numeral_numeral_nat(bit0(one)))))) ).

% inverse_complex.code
tff(fact_2804_Complex__divide,axiom,
    ! [X: complex,Y: complex] : divide1717551699836669952omplex(X,Y) = complex2(aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(re(X)),re(Y))),aa_real_real(times_times_real(im(X)),im(Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,re(Y)),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,im(Y)),numeral_numeral_nat(bit0(one))))),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(im(X)),re(Y))),aa_real_real(times_times_real(re(X)),im(Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,re(Y)),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,im(Y)),numeral_numeral_nat(bit0(one)))))) ).

% Complex_divide
tff(fact_2805_Im__Reals__divide,axiom,
    ! [R: complex,Z: complex] :
      ( member_complex(R,real_V2521375963428798218omplex)
     => ( im(divide1717551699836669952omplex(R,Z)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,re(R))),im(Z))),aa_nat_real(aa_real_fun_nat_real(power_power_real,real_V1022390504157884413omplex(Z)),numeral_numeral_nat(bit0(one)))) ) ) ).

% Im_Reals_divide
tff(fact_2806_Re__Reals__divide,axiom,
    ! [R: complex,Z: complex] :
      ( member_complex(R,real_V2521375963428798218omplex)
     => ( re(divide1717551699836669952omplex(R,Z)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(re(R)),re(Z))),aa_nat_real(aa_real_fun_nat_real(power_power_real,real_V1022390504157884413omplex(Z)),numeral_numeral_nat(bit0(one)))) ) ) ).

% Re_Reals_divide
tff(fact_2807_complex__mult__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(Z),cnj(Z)) = real_V4546457046886955230omplex(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,re(Z)),numeral_numeral_nat(bit0(one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,im(Z)),numeral_numeral_nat(bit0(one))))) ).

% complex_mult_cnj
tff(fact_2808_card__UNIV__unit,axiom,
    finite410649719033368117t_unit(top_to1996260823553986621t_unit) = one_one_nat ).

% card_UNIV_unit
tff(fact_2809_card__Collect__less__nat,axiom,
    ! [Na: nat] : finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_co,Na))) = Na ).

% card_Collect_less_nat
tff(fact_2810_card__Collect__le__nat,axiom,
    ! [Na: nat] : finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ee,Na))) = aa_nat_nat(suc,Na) ).

% card_Collect_le_nat
tff(fact_2811_complex__cnj__add,axiom,
    ! [X: complex,Y: complex] : cnj(plus_plus_complex(X,Y)) = plus_plus_complex(cnj(X),cnj(Y)) ).

% complex_cnj_add
tff(fact_2812_card__UNIV__bool,axiom,
    finite_card_o(top_top_set_o) = numeral_numeral_nat(bit0(one)) ).

% card_UNIV_bool
tff(fact_2813_card__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or4662586982721622107an_int(L,U)) = aa_int_nat(nat2,minus_minus_int(U,L)) ).

% card_atLeastLessThan_int
tff(fact_2814_card__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or6656581121297822940st_int(L,U)) = aa_int_nat(nat2,minus_minus_int(U,L)) ).

% card_greaterThanAtMost_int
tff(fact_2815_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or1266510415728281911st_int(L,U)) = aa_int_nat(nat2,plus_plus_int(minus_minus_int(U,L),one_one_int)) ).

% card_atLeastAtMost_int
tff(fact_2816_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or5832277885323065728an_int(L,U)) = aa_int_nat(nat2,minus_minus_int(U,plus_plus_int(L,one_one_int))) ).

% card_greaterThanLessThan_int
tff(fact_2817_sums__cnj,axiom,
    ! [F: fun_nat_complex,L: complex] :
      ( aa_complex_o(sums_complex(aTP_Lamm_gy(F)),cnj(L))
    <=> aa_complex_o(sums_complex(F),L) ) ).

% sums_cnj
tff(fact_2818_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] : finite_card_int(set_or4662586982721622107an_int(zero_zero_int,U)) = aa_int_nat(nat2,U) ).

% card_atLeastZeroLessThan_int
tff(fact_2819_card__less__Suc2,axiom,
    ! [M4: set_nat,I: nat] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),M4)
     => ( finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_gz(M4),I))) = finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ha(M4),I))) ) ) ).

% card_less_Suc2
tff(fact_2820_card__less__Suc,axiom,
    ! [M4: set_nat,I: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M4)
     => ( aa_nat_nat(suc,finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_gz(M4),I)))) = finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ha(M4),I))) ) ) ).

% card_less_Suc
tff(fact_2821_card__less,axiom,
    ! [M4: set_nat,I: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M4)
     => ( finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ha(M4),I))) != zero_zero_nat ) ) ).

% card_less
tff(fact_2822_subset__card__intvl__is__intvl,axiom,
    ! [A: set_nat,K: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),set_or4665077453230672383an_nat(K,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),finite_card_nat(A))))
     => ( A = set_or4665077453230672383an_nat(K,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),finite_card_nat(A))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_2823_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N3: set_nat,Na: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N3),set_or4665077453230672383an_nat(zero_zero_nat,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_nat(N3)),Na) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_2824_card__sum__le__nat__sum,axiom,
    ! [Sa: set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_db),set_or4665077453230672383an_nat(zero_zero_nat,finite_card_nat(Sa)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_db),Sa)) ).

% card_sum_le_nat_sum
tff(fact_2825_card__nth__roots,axiom,
    ! [C: complex,Na: nat] :
      ( ( C != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( finite_card_complex(collect_complex(aa_nat_fun_complex_o(aTP_Lamm_de(C),Na))) = Na ) ) ) ).

% card_nth_roots
tff(fact_2826_card__roots__unity__eq,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( finite_card_complex(collect_complex(aTP_Lamm_cr(Na))) = Na ) ) ).

% card_roots_unity_eq
tff(fact_2827_Re__complex__div__ge__0,axiom,
    ! [Aa: complex,Ba: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),re(divide1717551699836669952omplex(Aa,Ba)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),re(aa_complex_complex(times_times_complex(Aa),cnj(Ba)))) ) ).

% Re_complex_div_ge_0
tff(fact_2828_Re__complex__div__le__0,axiom,
    ! [Aa: complex,Ba: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,re(divide1717551699836669952omplex(Aa,Ba))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,re(aa_complex_complex(times_times_complex(Aa),cnj(Ba)))),zero_zero_real) ) ).

% Re_complex_div_le_0
tff(fact_2829_Im__complex__div__ge__0,axiom,
    ! [Aa: complex,Ba: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),im(divide1717551699836669952omplex(Aa,Ba)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),im(aa_complex_complex(times_times_complex(Aa),cnj(Ba)))) ) ).

% Im_complex_div_ge_0
tff(fact_2830_Im__complex__div__le__0,axiom,
    ! [Aa: complex,Ba: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,im(divide1717551699836669952omplex(Aa,Ba))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,im(aa_complex_complex(times_times_complex(Aa),cnj(Ba)))),zero_zero_real) ) ).

% Im_complex_div_le_0
tff(fact_2831_complex__mod__mult__cnj,axiom,
    ! [Z: complex] : real_V1022390504157884413omplex(aa_complex_complex(times_times_complex(Z),cnj(Z))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,real_V1022390504157884413omplex(Z)),numeral_numeral_nat(bit0(one))) ).

% complex_mod_mult_cnj
tff(fact_2832_complex__norm__square,axiom,
    ! [Z: complex] : real_V4546457046886955230omplex(aa_nat_real(aa_real_fun_nat_real(power_power_real,real_V1022390504157884413omplex(Z)),numeral_numeral_nat(bit0(one)))) = aa_complex_complex(times_times_complex(Z),cnj(Z)) ).

% complex_norm_square
tff(fact_2833_complex__add__cnj,axiom,
    ! [Z: complex] : plus_plus_complex(Z,cnj(Z)) = real_V4546457046886955230omplex(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),re(Z))) ).

% complex_add_cnj
tff(fact_2834_complex__diff__cnj,axiom,
    ! [Z: complex] : minus_minus_complex(Z,cnj(Z)) = aa_complex_complex(times_times_complex(real_V4546457046886955230omplex(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),im(Z)))),imaginary_unit) ).

% complex_diff_cnj
tff(fact_2835_complex__div__cnj,axiom,
    ! [Aa: complex,Ba: complex] : divide1717551699836669952omplex(Aa,Ba) = divide1717551699836669952omplex(aa_complex_complex(times_times_complex(Aa),cnj(Ba)),real_V4546457046886955230omplex(aa_nat_real(aa_real_fun_nat_real(power_power_real,real_V1022390504157884413omplex(Ba)),numeral_numeral_nat(bit0(one))))) ).

% complex_div_cnj
tff(fact_2836_cnj__add__mult__eq__Re,axiom,
    ! [Z: complex,W: complex] : plus_plus_complex(aa_complex_complex(times_times_complex(Z),cnj(W)),aa_complex_complex(times_times_complex(cnj(Z)),W)) = real_V4546457046886955230omplex(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),re(aa_complex_complex(times_times_complex(Z),cnj(W))))) ).

% cnj_add_mult_eq_Re
tff(fact_2837_card__num1,axiom,
    finite6454714172617411597l_num1(top_to3689904429138878997l_num1) = one_one_nat ).

% card_num1
tff(fact_2838_distinct__upto,axiom,
    ! [I: int,J: int] : aa_list_int_o(distinct_int,upto(I,J)) ).

% distinct_upto
tff(fact_2839_card__UNIV__char,axiom,
    finite_card_char(top_top_set_char) = numeral_numeral_nat(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one))))))))) ).

% card_UNIV_char
tff(fact_2840_UNIV__char__of__nat,axiom,
    top_top_set_char = image_nat_char(unique3096191561947761185of_nat,set_or4665077453230672383an_nat(zero_zero_nat,numeral_numeral_nat(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one))))))))))) ).

% UNIV_char_of_nat
tff(fact_2841_inj__on__char__of__nat,axiom,
    aa_set_nat_o(inj_on_nat_char(unique3096191561947761185of_nat),set_or4665077453230672383an_nat(zero_zero_nat,numeral_numeral_nat(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one))))))))))) ).

% inj_on_char_of_nat
tff(fact_2842_char_Osize_I2_J,axiom,
    ! [X12: $o,X23: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X8: $o] : size_size_char(aa_o_char(char2((X12),(X23),(X32),(X42),(X52),(X62),(X72)),(X8))) = zero_zero_nat ).

% char.size(2)
tff(fact_2843_nat__of__char__less__256,axiom,
    ! [C: char] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_char_nat(comm_s629917340098488124ar_nat,C)),numeral_numeral_nat(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one)))))))))) ).

% nat_of_char_less_256
tff(fact_2844_range__nat__of__char,axiom,
    image_char_nat(comm_s629917340098488124ar_nat,top_top_set_char) = set_or4665077453230672383an_nat(zero_zero_nat,numeral_numeral_nat(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one)))))))))) ).

% range_nat_of_char
tff(fact_2845_integer__of__char__code,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B5: $o,B6: $o,B7: $o] : integer_of_char(aa_o_char(char2((B0),(B1),(B22),(B32),(B42),(B5),(B6)),(B7))) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger((B7))),numera6620942414471956472nteger(bit0(one))),zero_n356916108424825756nteger((B6)))),numera6620942414471956472nteger(bit0(one))),zero_n356916108424825756nteger((B5)))),numera6620942414471956472nteger(bit0(one))),zero_n356916108424825756nteger((B42)))),numera6620942414471956472nteger(bit0(one))),zero_n356916108424825756nteger((B32)))),numera6620942414471956472nteger(bit0(one))),zero_n356916108424825756nteger((B22)))),numera6620942414471956472nteger(bit0(one))),zero_n356916108424825756nteger((B1)))),numera6620942414471956472nteger(bit0(one))),zero_n356916108424825756nteger((B0))) ).

% integer_of_char_code
tff(fact_2846_less__eq__char__simp,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B5: $o,B6: $o,B7: $o,C0: $o,C1: $o,C22: $o,C32: $o,C4: $o,C5: $o,C6: $o,C7: $o] :
      ( ord_less_eq_char(aa_o_char(char2((B0),(B1),(B22),(B32),(B42),(B5),(B6)),(B7)),aa_o_char(char2((C0),(C1),(C22),(C32),(C4),(C5),(C6)),(C7)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(foldr_o_nat(aTP_Lamm_hb,cons_o((B0),cons_o((B1),cons_o((B22),cons_o((B32),cons_o((B42),cons_o((B5),cons_o((B6),cons_o((B7),nil_o))))))))),zero_zero_nat)),aa_nat_nat(foldr_o_nat(aTP_Lamm_hb,cons_o((C0),cons_o((C1),cons_o((C22),cons_o((C32),cons_o((C4),cons_o((C5),cons_o((C6),cons_o((C7),nil_o))))))))),zero_zero_nat)) ) ).

% less_eq_char_simp
tff(fact_2847_less__char__simp,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B5: $o,B6: $o,B7: $o,C0: $o,C1: $o,C22: $o,C32: $o,C4: $o,C5: $o,C6: $o,C7: $o] :
      ( ord_less_char(aa_o_char(char2((B0),(B1),(B22),(B32),(B42),(B5),(B6)),(B7)),aa_o_char(char2((C0),(C1),(C22),(C32),(C4),(C5),(C6)),(C7)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(foldr_o_nat(aTP_Lamm_hb,cons_o((B0),cons_o((B1),cons_o((B22),cons_o((B32),cons_o((B42),cons_o((B5),cons_o((B6),cons_o((B7),nil_o))))))))),zero_zero_nat)),aa_nat_nat(foldr_o_nat(aTP_Lamm_hb,cons_o((C0),cons_o((C1),cons_o((C22),cons_o((C32),cons_o((C4),cons_o((C5),cons_o((C6),cons_o((C7),nil_o))))))))),zero_zero_nat)) ) ).

% less_char_simp
tff(fact_2848_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or8404916559141939852nteger(L,plus_p5714425477246183910nteger(U,one_one_Code_integer)) = set_or189985376899183464nteger(L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_integer
tff(fact_2849_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or8404916559141939852nteger(plus_p5714425477246183910nteger(L,one_one_Code_integer),U) = set_or4266950643985792945nteger(L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_integer
tff(fact_2850_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or189985376899183464nteger(plus_p5714425477246183910nteger(L,one_one_Code_integer),U) = set_or2715278749043346189nteger(L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_integer
tff(fact_2851_image__add__integer__atLeastLessThan,axiom,
    ! [L: code_integer,U: code_integer] : image_4470545334726330049nteger(aTP_Lamm_hc(L),set_or8404916559141939852nteger(zero_z3403309356797280102nteger,minus_8373710615458151222nteger(U,L))) = set_or8404916559141939852nteger(L,U) ).

% image_add_integer_atLeastLessThan
tff(fact_2852_less__eq__char__def,axiom,
    ! [C1: char,C22: char] :
      ( ord_less_eq_char(C1,C22)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_char_nat(comm_s629917340098488124ar_nat,C1)),aa_char_nat(comm_s629917340098488124ar_nat,C22)) ) ).

% less_eq_char_def
tff(fact_2853_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_hd(L),Qr) ).

% divmod_step_integer_def
tff(fact_2854_shiftl__integer__conv__mult__pow2,axiom,
    ! [Na: nat,X: code_integer] : bit_se7788150548672797655nteger(Na,X) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),Na)) ).

% shiftl_integer_conv_mult_pow2
tff(fact_2855_finite__atLeastLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : finite6017078050557962740nteger(set_or8404916559141939852nteger(L,U)) ).

% finite_atLeastLessThan_integer
tff(fact_2856_finite__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : finite6017078050557962740nteger(set_or189985376899183464nteger(L,U)) ).

% finite_atLeastAtMost_integer
tff(fact_2857_finite__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : finite6017078050557962740nteger(set_or2715278749043346189nteger(L,U)) ).

% finite_greaterThanAtMost_integer
tff(fact_2858_finite__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : finite6017078050557962740nteger(set_or4266950643985792945nteger(L,U)) ).

% finite_greaterThanLessThan_integer
tff(fact_2859_VEBT__internal_Ofoldr__one,axiom,
    ! [D: nat,Ysb: list_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D),aa_nat_nat(foldr_nat_nat(plus_plus_nat,Ysb),D)) ).

% VEBT_internal.foldr_one
tff(fact_2860_VEBT__internal_Ofoldr0,axiom,
    ! [Xs2: list_real,C: real,D: real] : aa_real_real(foldr_real_real(plus_plus_real,Xs2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(foldr_real_real(plus_plus_real,Xs2),D)),C) ).

% VEBT_internal.foldr0
tff(fact_2861_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : finite6017078050557962740nteger(set_or8404916559141939852nteger(zero_z3403309356797280102nteger,U)) ).

% finite_atLeastZeroLessThan_integer
tff(fact_2862_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
tff(fact_2863_one__integer_Orsp,axiom,
    one_one_int = one_one_int ).

% one_integer.rsp
tff(fact_2864_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] : plus_p5714425477246183910nteger(zero_z3403309356797280102nteger,L) = L ).

% plus_integer_code(2)
tff(fact_2865_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] : plus_p5714425477246183910nteger(K,zero_z3403309356797280102nteger) = K ).

% plus_integer_code(1)
tff(fact_2866_VEBT__internal_Ofoldr__mono,axiom,
    ! [Xs2: list_nat,Ysb: list_nat,C: nat,D: nat] :
      ( ( size_size_list_nat(Xs2) = size_size_list_nat(Ysb) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(Xs2))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(nth_nat(Xs2),I2)),aa_nat_nat(nth_nat(Ysb),I2)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,Xs2),C)),size_size_list_nat(Ysb))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,Ysb),D)) ) ) ) ).

% VEBT_internal.foldr_mono
tff(fact_2867_VEBT__internal_Ofoldr__same,axiom,
    ! [Xs2: list_real,Y: real] :
      ( ! [X2: real,Y2: real] :
          ( member_real(X2,set_real2(Xs2))
         => ( member_real(Y2,set_real2(Xs2))
           => ( X2 = Y2 ) ) )
     => ( ! [X2: real] :
            ( member_real(X2,set_real2(Xs2))
           => ( X2 = Y ) )
       => ( aa_real_real(foldr_real_real(plus_plus_real,Xs2),zero_zero_real) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,size_size_list_real(Xs2))),Y) ) ) ) ).

% VEBT_internal.foldr_same
tff(fact_2868_VEBT__internal_Ofoldr__same__int,axiom,
    ! [Xs2: list_nat,Y: nat] :
      ( ! [X2: nat,Y2: nat] :
          ( aa_set_nat_o(member_nat(X2),set_nat2(Xs2))
         => ( aa_set_nat_o(member_nat(Y2),set_nat2(Xs2))
           => ( X2 = Y2 ) ) )
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),set_nat2(Xs2))
           => ( X2 = Y ) )
       => ( aa_nat_nat(foldr_nat_nat(plus_plus_nat,Xs2),zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs2)),Y) ) ) ) ).

% VEBT_internal.foldr_same_int
tff(fact_2869_VEBT__internal_Ofoldr__zero,axiom,
    ! [Xs2: list_nat,D: nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(Xs2))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(nth_nat(Xs2),I2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Xs2)),aa_nat_nat(minus_minus_nat(aa_nat_nat(foldr_nat_nat(plus_plus_nat,Xs2),D)),D)) ) ).

% VEBT_internal.foldr_zero
tff(fact_2870_shiftr__integer__conv__div__pow2,axiom,
    ! [Na: nat,X: code_integer] : bit_se3928097537394005634nteger(Na,X) = divide6298287555418463151nteger(X,power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),Na)) ).

% shiftr_integer_conv_div_pow2
tff(fact_2871_integer__of__int__code,axiom,
    ! [K: int] :
      aa_int_Code_integer(code_integer_of_int,K) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
        aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(code_integer_of_int,aa_int_int(uminus_uminus_int,K))),
        $ite(
          K = zero_zero_int,
          zero_z3403309356797280102nteger,
          $let(
            l: code_integer,
            l:= aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,bit0(one))))),
            $ite(modulo_modulo_int(K,aa_num_int(numeral_numeral_int,bit0(one))) = zero_zero_int,l,plus_p5714425477246183910nteger(l,one_one_Code_integer)) ) ) ) ).

% integer_of_int_code
tff(fact_2872_String_Ochar__of__ascii__of,axiom,
    ! [C: char] : aa_char_nat(comm_s629917340098488124ar_nat,ascii_of(C)) = aa_nat_nat(bit_se2925701944663578781it_nat(numeral_numeral_nat(bit1(bit1(one)))),aa_char_nat(comm_s629917340098488124ar_nat,C)) ).

% String.char_of_ascii_of
tff(fact_2873_less__eq__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( ord_le3102999989581377725nteger(aa_int_Code_integer(code_integer_of_int,Xa2),aa_int_Code_integer(code_integer_of_int,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa2),X) ) ).

% less_eq_integer.abs_eq
tff(fact_2874_times__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(code_integer_of_int,Xa2)),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,times_times_int(Xa2,X)) ).

% times_integer.abs_eq
tff(fact_2875_plus__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] : plus_p5714425477246183910nteger(aa_int_Code_integer(code_integer_of_int,Xa2),aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,plus_plus_int(Xa2,X)) ).

% plus_integer.abs_eq
tff(fact_2876_one__integer__def,axiom,
    one_one_Code_integer = aa_int_Code_integer(code_integer_of_int,one_one_int) ).

% one_integer_def
tff(fact_2877_Bit__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: $o] : bits_Bit_integer(aa_int_Code_integer(code_integer_of_int,Xa2),(X)) = aa_int_Code_integer(code_integer_of_int,plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,(X)),times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),Xa2))) ).

% Bit_integer.abs_eq
tff(fact_2878_dup__1,axiom,
    aa_Cod1783974652533396728nteger(code_dup,one_one_Code_integer) = numera6620942414471956472nteger(bit0(one)) ).

% dup_1
tff(fact_2879_dup_Oabs__eq,axiom,
    ! [X: int] : aa_Cod1783974652533396728nteger(code_dup,aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,plus_plus_int(X,X)) ).

% dup.abs_eq
tff(fact_2880_Bit__integer__code_I1_J,axiom,
    ! [I: code_integer] : bits_Bit_integer(I,$false) = bit_se7788150548672797655nteger(one_one_nat,I) ).

% Bit_integer_code(1)
tff(fact_2881_Bit__integer__code_I2_J,axiom,
    ! [I: code_integer] : bits_Bit_integer(I,$true) = plus_p5714425477246183910nteger(bit_se7788150548672797655nteger(one_one_nat,I),one_one_Code_integer) ).

% Bit_integer_code(2)
tff(fact_2882_bin__rest__integer_Oabs__eq,axiom,
    ! [X: int] : aa_Cod1783974652533396728nteger(bits_b2549910563261871055nteger,aa_int_Code_integer(code_integer_of_int,X)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),aa_num_int(numeral_numeral_int,bit0(one)))) ).

% bin_rest_integer.abs_eq
tff(fact_2883_bin__last__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( bits_b8758750999018896077nteger(aa_int_Code_integer(code_integer_of_int,X))
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),X) ) ).

% bin_last_integer.abs_eq
tff(fact_2884_bin__last__integer__nbe,axiom,
    ! [I: code_integer] :
      ( bits_b8758750999018896077nteger(I)
    <=> ( modulo364778990260209775nteger(I,numera6620942414471956472nteger(bit0(one))) != zero_z3403309356797280102nteger ) ) ).

% bin_last_integer_nbe
tff(fact_2885_bin__rest__integer__code,axiom,
    ! [I: code_integer] : aa_Cod1783974652533396728nteger(bits_b2549910563261871055nteger,I) = divide6298287555418463151nteger(I,numera6620942414471956472nteger(bit0(one))) ).

% bin_rest_integer_code
tff(fact_2886_int__of__integer__code,axiom,
    ! [K: code_integer] :
      aa_Code_integer_int(code_int_of_integer,K) = $ite(
        ord_le6747313008572928689nteger(K,zero_z3403309356797280102nteger),
        aa_int_int(uminus_uminus_int,aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K))),
        $ite(K = zero_z3403309356797280102nteger,zero_zero_int,produc1553301316500091796er_int(aTP_Lamm_he,code_divmod_integer(K,numera6620942414471956472nteger(bit0(one))))) ) ).

% int_of_integer_code
tff(fact_2887_integer__of__num_I3_J,axiom,
    ! [Na: num] :
      code_integer_of_num(bit1(Na)) = $let(
        k: code_integer,
        k:= code_integer_of_num(Na),
        plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(k,k),one_one_Code_integer) ) ).

% integer_of_num(3)
tff(fact_2888_int__of__integer__of__nat,axiom,
    ! [Na: nat] : aa_Code_integer_int(code_int_of_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) = aa_nat_int(semiri1314217659103216013at_int,Na) ).

% int_of_integer_of_nat
tff(fact_2889_int__of__integer__numeral,axiom,
    ! [K: num] : aa_Code_integer_int(code_int_of_integer,numera6620942414471956472nteger(K)) = aa_num_int(numeral_numeral_int,K) ).

% int_of_integer_numeral
tff(fact_2890_one__integer_Orep__eq,axiom,
    aa_Code_integer_int(code_int_of_integer,one_one_Code_integer) = one_one_int ).

% one_integer.rep_eq
tff(fact_2891_plus__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] : aa_Code_integer_int(code_int_of_integer,plus_p5714425477246183910nteger(X,Xa2)) = plus_plus_int(aa_Code_integer_int(code_int_of_integer,X),aa_Code_integer_int(code_int_of_integer,Xa2)) ).

% plus_integer.rep_eq
tff(fact_2892_times__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Xa2)) = times_times_int(aa_Code_integer_int(code_int_of_integer,X),aa_Code_integer_int(code_int_of_integer,Xa2)) ).

% times_integer.rep_eq
tff(fact_2893_int__of__integer__less__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,Y))
    <=> ord_le6747313008572928689nteger(X,Y) ) ).

% int_of_integer_less_iff
tff(fact_2894_integer__less__eq__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ord_le3102999989581377725nteger(K,L)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Code_integer_int(code_int_of_integer,K)),aa_Code_integer_int(code_int_of_integer,L)) ) ).

% integer_less_eq_iff
tff(fact_2895_less__eq__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( ord_le3102999989581377725nteger(X,Xa2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Code_integer_int(code_int_of_integer,X)),aa_Code_integer_int(code_int_of_integer,Xa2)) ) ).

% less_eq_integer.rep_eq
tff(fact_2896_dup_Orep__eq,axiom,
    ! [X: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(code_dup,X)) = plus_plus_int(aa_Code_integer_int(code_int_of_integer,X),aa_Code_integer_int(code_int_of_integer,X)) ).

% dup.rep_eq
tff(fact_2897_integer__of__num__triv_I1_J,axiom,
    code_integer_of_num(one) = one_one_Code_integer ).

% integer_of_num_triv(1)
tff(fact_2898_bin__last__integer_Orep__eq,axiom,
    ! [X: code_integer] :
      ( bits_b8758750999018896077nteger(X)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),aa_Code_integer_int(code_int_of_integer,X)) ) ).

% bin_last_integer.rep_eq
tff(fact_2899_bin__rest__integer_Orep__eq,axiom,
    ! [X: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(bits_b2549910563261871055nteger,X)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Code_integer_int(code_int_of_integer,X)),aa_num_int(numeral_numeral_int,bit0(one))) ).

% bin_rest_integer.rep_eq
tff(fact_2900_integer__of__num_I2_J,axiom,
    ! [Na: num] :
      code_integer_of_num(bit0(Na)) = $let(
        k: code_integer,
        k:= code_integer_of_num(Na),
        plus_p5714425477246183910nteger(k,k) ) ).

% integer_of_num(2)
tff(fact_2901_integer__of__num__triv_I2_J,axiom,
    code_integer_of_num(bit0(one)) = numera6620942414471956472nteger(bit0(one)) ).

% integer_of_num_triv(2)
tff(fact_2902_Bit__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: $o] : aa_Code_integer_int(code_int_of_integer,bits_Bit_integer(X,(Xa2))) = plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,(Xa2)),times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),aa_Code_integer_int(code_int_of_integer,X))) ).

% Bit_integer.rep_eq
tff(fact_2903_num__of__integer__code,axiom,
    ! [K: code_integer] :
      code_num_of_integer(K) = $ite(ord_le3102999989581377725nteger(K,one_one_Code_integer),one,produc7336495610019696514er_num(aTP_Lamm_hf,code_divmod_integer(K,numera6620942414471956472nteger(bit0(one))))) ).

% num_of_integer_code
tff(fact_2904_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      code_nat_of_integer(K) = $ite(ord_le3102999989581377725nteger(K,zero_z3403309356797280102nteger),zero_zero_nat,produc1555791787009142072er_nat(aTP_Lamm_hg,code_divmod_integer(K,numera6620942414471956472nteger(bit0(one))))) ).

% nat_of_integer_code
tff(fact_2905_nat__of__integer__numeral,axiom,
    ! [Na: num] : code_nat_of_integer(numera6620942414471956472nteger(Na)) = numeral_numeral_nat(Na) ).

% nat_of_integer_numeral
tff(fact_2906_nat__of__integer__code__post_I3_J,axiom,
    ! [K: num] : code_nat_of_integer(numera6620942414471956472nteger(K)) = numeral_numeral_nat(K) ).

% nat_of_integer_code_post(3)
tff(fact_2907_nat__of__integer__1,axiom,
    code_nat_of_integer(one_one_Code_integer) = one_one_nat ).

% nat_of_integer_1
tff(fact_2908_nat__of__integer_Orep__eq,axiom,
    ! [X: code_integer] : code_nat_of_integer(X) = aa_int_nat(nat2,aa_Code_integer_int(code_int_of_integer,X)) ).

% nat_of_integer.rep_eq
tff(fact_2909_nat__of__integer_Oabs__eq,axiom,
    ! [X: int] : code_nat_of_integer(aa_int_Code_integer(code_integer_of_int,X)) = aa_int_nat(nat2,X) ).

% nat_of_integer.abs_eq
tff(fact_2910_nat__of__integer__less__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ord_le3102999989581377725nteger(zero_z3403309356797280102nteger,X)
     => ( ord_le3102999989581377725nteger(zero_z3403309356797280102nteger,Y)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,code_nat_of_integer(X)),code_nat_of_integer(Y))
        <=> ord_le6747313008572928689nteger(X,Y) ) ) ) ).

% nat_of_integer_less_iff
tff(fact_2911_image__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] :
      ( ord_le3102999989581377725nteger(zero_z3403309356797280102nteger,U)
     => ( set_or8404916559141939852nteger(zero_z3403309356797280102nteger,U) = image_1215581382706833972nteger(semiri4939895301339042750nteger,set_ord_lessThan_nat(code_nat_of_integer(U))) ) ) ).

% image_atLeastZeroLessThan_integer
tff(fact_2912_Code__Target__Int_Opositive__def,axiom,
    code_Target_positive = numeral_numeral_int ).

% Code_Target_Int.positive_def
tff(fact_2913_bin__rest__integer__def,axiom,
    bits_b2549910563261871055nteger = map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int,aTP_Lamm_hh) ).

% bin_rest_integer_def
tff(fact_2914_Code__Numeral_Odup__def,axiom,
    code_dup = map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int,aTP_Lamm_hi) ).

% Code_Numeral.dup_def
tff(fact_2915_bit__cut__integer__def,axiom,
    ! [K: code_integer] : code_bit_cut_integer(K) = produc6677183202524767010eger_o(divide6298287555418463151nteger(K,numera6620942414471956472nteger(bit0(one))),~ dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one)),K)) ).

% bit_cut_integer_def
tff(fact_2916_bit__cut__integer__code,axiom,
    ! [K: code_integer] :
      code_bit_cut_integer(K) = $ite(K = zero_z3403309356797280102nteger,produc6677183202524767010eger_o(zero_z3403309356797280102nteger,$false),produc9125791028180074456eger_o(aTP_Lamm_hj(K),code_divmod_abs(K,numera6620942414471956472nteger(bit0(one))))) ).

% bit_cut_integer_code
tff(fact_2917_char__of__integer__code,axiom,
    ! [K: code_integer] : char_of_integer(K) = produc4188289175737317920o_char(aTP_Lamm_hr,code_bit_cut_integer(K)) ).

% char_of_integer_code
tff(fact_2918_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_z3403309356797280102nteger,
        produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
        $ite(
          ord_le6747313008572928689nteger(zero_z3403309356797280102nteger,L),
          $ite(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger,K),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_hs(L),code_divmod_abs(K,L))),
          $ite(
            L = zero_z3403309356797280102nteger,
            produc1086072967326762835nteger(zero_z3403309356797280102nteger,K),
            aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,uminus1351360451143612070nteger),
              $ite(ord_le6747313008572928689nteger(K,zero_z3403309356797280102nteger),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_ht(L),code_divmod_abs(K,L)))) ) ) ) ).

% divmod_integer_code
tff(fact_2919_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [Na: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(minus_minus_nat(J),I))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(set_or6659071591806873216st_nat(I,J))),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Na)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_2920_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),J)
     => ( linord2614967742042102400et_nat(set_or5834768355832116004an_nat(I,J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(set_or5834768355832116004an_nat(aa_nat_nat(suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_2921_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,I)),J)
     => ( linord2614967742042102400et_nat(set_or6659071591806873216st_nat(I,J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(set_or6659071591806873216st_nat(aa_nat_nat(suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_2922_card__length__sum__list__rec,axiom,
    ! [M: nat,N3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),M)
     => ( finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_hu(M),N3))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_hv(M),N3)))),finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_hw(M),N3)))) ) ) ).

% card_length_sum_list_rec
tff(fact_2923_card__length__sum__list,axiom,
    ! [M: nat,N3: nat] : finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_hu(M),N3))) = aa_nat_nat(binomial(aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),M)),one_one_nat)),N3) ).

% card_length_sum_list
tff(fact_2924_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [Na: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(minus_minus_nat(J),aa_nat_nat(suc,I)))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(set_or5834768355832116004an_nat(I,J))),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Na)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_2925_Suc__0__div__numeral,axiom,
    ! [K: num] : aa_nat_nat(divide_divide_nat(aa_nat_nat(suc,zero_zero_nat)),numeral_numeral_nat(K)) = product_fst_nat_nat(unique5055182867167087721od_nat(one,K)) ).

% Suc_0_div_numeral
tff(fact_2926_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I: nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_nat),Ns)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Ns))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(nth_nat(Ns),I)) ) ) ).

% sorted_wrt_less_idx
tff(fact_2927_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : aa_list_int_o(sorted_wrt_int(ord_less_int),upto(I,J)) ).

% sorted_wrt_upto
tff(fact_2928_sorted__upto,axiom,
    ! [M: int,Na: int] : aa_list_int_o(sorted_wrt_int(ord_less_eq_int),upto(M,Na)) ).

% sorted_upto
tff(fact_2929_rat__sgn__code,axiom,
    ! [P2: rat] : quotient_of(sgn_sgn_rat(P2)) = aa_int7462565939943104719nt_int(product_Pair_int_int(sgn_sgn_int(product_fst_int_int(quotient_of(P2)))),one_one_int) ).

% rat_sgn_code
tff(fact_2930_Suc__0__mod__numeral,axiom,
    ! [K: num] : aa_nat_nat(modulo_modulo_nat(aa_nat_nat(suc,zero_zero_nat)),numeral_numeral_nat(K)) = product_snd_nat_nat(unique5055182867167087721od_nat(one,K)) ).

% Suc_0_mod_numeral
tff(fact_2931_minus__one__mod__numeral,axiom,
    ! [Na: num] : modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int),aa_num_int(numeral_numeral_int,Na)) = adjust_mod(aa_num_int(numeral_numeral_int,Na),product_snd_int_int(unique5052692396658037445od_int(one,Na))) ).

% minus_one_mod_numeral
tff(fact_2932_one__mod__minus__numeral,axiom,
    ! [Na: num] : modulo_modulo_int(one_one_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = aa_int_int(uminus_uminus_int,adjust_mod(aa_num_int(numeral_numeral_int,Na),product_snd_int_int(unique5052692396658037445od_int(one,Na)))) ).

% one_mod_minus_numeral
tff(fact_2933_numeral__mod__minus__numeral,axiom,
    ! [M: num,Na: num] : modulo_modulo_int(aa_num_int(numeral_numeral_int,M),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = aa_int_int(uminus_uminus_int,adjust_mod(aa_num_int(numeral_numeral_int,Na),product_snd_int_int(unique5052692396658037445od_int(M,Na)))) ).

% numeral_mod_minus_numeral
tff(fact_2934_minus__numeral__mod__numeral,axiom,
    ! [M: num,Na: num] : modulo_modulo_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na)) = adjust_mod(aa_num_int(numeral_numeral_int,Na),product_snd_int_int(unique5052692396658037445od_int(M,Na))) ).

% minus_numeral_mod_numeral
tff(fact_2935_bezw_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( bezw(X,Xa2) = Y )
     => ( Y = $ite(Xa2 = zero_zero_nat,aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(product_Pair_int_int(product_snd_int_int(bezw(Xa2,aa_nat_nat(modulo_modulo_nat(X),Xa2)))),minus_minus_int(product_fst_int_int(bezw(Xa2,aa_nat_nat(modulo_modulo_nat(X),Xa2))),times_times_int(product_snd_int_int(bezw(Xa2,aa_nat_nat(modulo_modulo_nat(X),Xa2))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(divide_divide_nat(X),Xa2)))))) ) ) ).

% bezw.elims
tff(fact_2936_bezw_Osimps,axiom,
    ! [X: nat,Y: nat] :
      bezw(X,Y) = $ite(Y = zero_zero_nat,aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(product_Pair_int_int(product_snd_int_int(bezw(Y,aa_nat_nat(modulo_modulo_nat(X),Y)))),minus_minus_int(product_fst_int_int(bezw(Y,aa_nat_nat(modulo_modulo_nat(X),Y))),times_times_int(product_snd_int_int(bezw(Y,aa_nat_nat(modulo_modulo_nat(X),Y))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(divide_divide_nat(X),Y)))))) ).

% bezw.simps
tff(fact_2937_bezw__0,axiom,
    ! [X: nat] : bezw(X,zero_zero_nat) = aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),zero_zero_int) ).

% bezw_0
tff(fact_2938_bezw__non__0,axiom,
    ! [Y: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Y)
     => ( bezw(X,Y) = aa_int7462565939943104719nt_int(product_Pair_int_int(product_snd_int_int(bezw(Y,aa_nat_nat(modulo_modulo_nat(X),Y)))),minus_minus_int(product_fst_int_int(bezw(Y,aa_nat_nat(modulo_modulo_nat(X),Y))),times_times_int(product_snd_int_int(bezw(Y,aa_nat_nat(modulo_modulo_nat(X),Y))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(divide_divide_nat(X),Y))))) ) ) ).

% bezw_non_0
tff(fact_2939_bezw_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( bezw(X,Xa2) = Y )
     => ( accp_P4275260045618599050at_nat(bezw_rel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2))
       => ~ ( ( Y = $ite(Xa2 = zero_zero_nat,aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(product_Pair_int_int(product_snd_int_int(bezw(Xa2,aa_nat_nat(modulo_modulo_nat(X),Xa2)))),minus_minus_int(product_fst_int_int(bezw(Xa2,aa_nat_nat(modulo_modulo_nat(X),Xa2))),times_times_int(product_snd_int_int(bezw(Xa2,aa_nat_nat(modulo_modulo_nat(X),Xa2))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(divide_divide_nat(X),Xa2)))))) )
           => ~ accp_P4275260045618599050at_nat(bezw_rel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2)) ) ) ) ).

% bezw.pelims
tff(fact_2940_normalize__def,axiom,
    ! [P2: product_prod_int_int] :
      normalize(P2) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),product_snd_int_int(P2)),
        $let(
          a2: int,
          a2:= gcd_gcd_int(product_fst_int_int(P2),product_snd_int_int(P2)),
          aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,product_fst_int_int(P2)),a2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,product_snd_int_int(P2)),a2)) ),
        $ite(
          product_snd_int_int(P2) = zero_zero_int,
          aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int),
          $let(
            a2: int,
            a2:= aa_int_int(uminus_uminus_int,gcd_gcd_int(product_fst_int_int(P2),product_snd_int_int(P2))),
            aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,product_fst_int_int(P2)),a2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,product_snd_int_int(P2)),a2)) ) ) ) ).

% normalize_def
tff(fact_2941_gcd__1__int,axiom,
    ! [M: int] : gcd_gcd_int(M,one_one_int) = one_one_int ).

% gcd_1_int
tff(fact_2942_gcd__neg__numeral__2__int,axiom,
    ! [X: int,Na: num] : gcd_gcd_int(X,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = gcd_gcd_int(X,aa_num_int(numeral_numeral_int,Na)) ).

% gcd_neg_numeral_2_int
tff(fact_2943_gcd__neg__numeral__1__int,axiom,
    ! [Na: num,X: int] : gcd_gcd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na)),X) = gcd_gcd_int(aa_num_int(numeral_numeral_int,Na),X) ).

% gcd_neg_numeral_1_int
tff(fact_2944_gcd__unique__int,axiom,
    ! [D: int,Aa: int,Ba: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),D)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),Aa)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),Ba)
        & ! [E3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),Aa)
              & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),Ba) )
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),D) ) )
    <=> ( D = gcd_gcd_int(Aa,Ba) ) ) ).

% gcd_unique_int
tff(fact_2945_gcd__ge__0__int,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),gcd_gcd_int(X,Y)) ).

% gcd_ge_0_int
tff(fact_2946_gcd__le1__int,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,gcd_gcd_int(Aa,Ba)),Aa) ) ).

% gcd_le1_int
tff(fact_2947_gcd__le2__int,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,gcd_gcd_int(Aa,Ba)),Ba) ) ).

% gcd_le2_int
tff(fact_2948_gcd__mult__distrib__int,axiom,
    ! [K: int,M: int,Na: int] : times_times_int(aa_int_int(abs_abs_int,K),gcd_gcd_int(M,Na)) = gcd_gcd_int(times_times_int(K,M),times_times_int(K,Na)) ).

% gcd_mult_distrib_int
tff(fact_2949_bezout__int,axiom,
    ! [X: int,Y: int] :
    ? [U2: int,V3: int] : plus_plus_int(times_times_int(U2,X),times_times_int(V3,Y)) = gcd_gcd_int(X,Y) ).

% bezout_int
tff(fact_2950_gcd__cases__int,axiom,
    ! [X: int,Y: int,P: fun_int_o] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => aa_int_o(P,gcd_gcd_int(X,Y)) ) )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
           => aa_int_o(P,gcd_gcd_int(X,aa_int_int(uminus_uminus_int,Y))) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),zero_zero_int)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
             => aa_int_o(P,gcd_gcd_int(aa_int_int(uminus_uminus_int,X),Y)) ) )
         => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),zero_zero_int)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
               => aa_int_o(P,gcd_gcd_int(aa_int_int(uminus_uminus_int,X),aa_int_int(uminus_uminus_int,Y))) ) )
           => aa_int_o(P,gcd_gcd_int(X,Y)) ) ) ) ) ).

% gcd_cases_int
tff(fact_2951_uint32__shiftl__def,axiom,
    ! [X: uint32,Na: code_integer] :
      uint32_shiftl(X,Na) = $ite(
        ( ord_le6747313008572928689nteger(Na,zero_z3403309356797280102nteger)
        | ord_le3102999989581377725nteger(numera6620942414471956472nteger(bit0(bit0(bit0(bit0(bit0(one)))))),Na) ),
        aa_Cod8401701657498852871uint32(aa_uin6552852722274082928uint32(aa_fun8838986389337097743uint32(undefi4150998591671624934uint32,bit_se5742574853984576102uint32),X),Na),
        aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se5742574853984576102uint32,code_nat_of_integer(Na)),X) ) ).

% uint32_shiftl_def
tff(fact_2952_uint32__shiftr__def,axiom,
    ! [X: uint32,Na: code_integer] :
      uint32_shiftr(X,Na) = $ite(
        ( ord_le6747313008572928689nteger(Na,zero_z3403309356797280102nteger)
        | ord_le3102999989581377725nteger(numera6620942414471956472nteger(bit0(bit0(bit0(bit0(bit0(one)))))),Na) ),
        aa_Cod8401701657498852871uint32(aa_uin6552852722274082928uint32(aa_fun8838986389337097743uint32(undefi4150998591671624934uint32,bit_se3964402333458159761uint32),X),Na),
        aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se3964402333458159761uint32,code_nat_of_integer(Na)),X) ) ).

% uint32_shiftr_def
tff(fact_2953_gcd__1__nat,axiom,
    ! [M: nat] : aa_nat_nat(gcd_gcd_nat(M),one_one_nat) = one_one_nat ).

% gcd_1_nat
tff(fact_2954_gcd__int__int__eq,axiom,
    ! [M: nat,Na: nat] : gcd_gcd_int(aa_nat_int(semiri1314217659103216013at_int,M),aa_nat_int(semiri1314217659103216013at_int,Na)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(gcd_gcd_nat(M),Na)) ).

% gcd_int_int_eq
tff(fact_2955_gcd__nat__abs__right__eq,axiom,
    ! [Na: nat,K: int] : aa_nat_nat(gcd_gcd_nat(Na),aa_int_nat(nat2,aa_int_int(abs_abs_int,K))) = aa_int_nat(nat2,gcd_gcd_int(aa_nat_int(semiri1314217659103216013at_int,Na),K)) ).

% gcd_nat_abs_right_eq
tff(fact_2956_gcd__nat__abs__left__eq,axiom,
    ! [K: int,Na: nat] : aa_nat_nat(gcd_gcd_nat(aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),Na) = aa_int_nat(nat2,gcd_gcd_int(K,aa_nat_int(semiri1314217659103216013at_int,Na))) ).

% gcd_nat_abs_left_eq
tff(fact_2957_bezout__gcd__nat_H,axiom,
    ! [Ba: nat,Aa: nat] :
    ? [X2: nat,Y2: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),Y2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),X2))
        & ( aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),Y2)) = aa_nat_nat(gcd_gcd_nat(Aa),Ba) ) )
      | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Y2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),X2))
        & ( aa_nat_nat(minus_minus_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Y2)) = aa_nat_nat(gcd_gcd_nat(Aa),Ba) ) ) ) ).

% bezout_gcd_nat'
tff(fact_2958_gcd__diff2__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_nat(gcd_gcd_nat(aa_nat_nat(minus_minus_nat(Na),M)),Na) = aa_nat_nat(gcd_gcd_nat(M),Na) ) ) ).

% gcd_diff2_nat
tff(fact_2959_gcd__diff1__nat,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_nat(gcd_gcd_nat(aa_nat_nat(minus_minus_nat(M),Na)),Na) = aa_nat_nat(gcd_gcd_nat(M),Na) ) ) ).

% gcd_diff1_nat
tff(fact_2960_bezout__nat,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( Aa != zero_zero_nat )
     => ? [X2: nat,Y2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),Y2)),aa_nat_nat(gcd_gcd_nat(Aa),Ba)) ) ).

% bezout_nat
tff(fact_2961_gcd__le1__nat,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( Aa != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(gcd_gcd_nat(Aa),Ba)),Aa) ) ).

% gcd_le1_nat
tff(fact_2962_gcd__le2__nat,axiom,
    ! [Ba: nat,Aa: nat] :
      ( ( Ba != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(gcd_gcd_nat(Aa),Ba)),Ba) ) ).

% gcd_le2_nat
tff(fact_2963_gcd__mult__distrib__nat,axiom,
    ! [K: nat,M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(gcd_gcd_nat(M),Na)) = aa_nat_nat(gcd_gcd_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)) ).

% gcd_mult_distrib_nat
tff(fact_2964_gcd__int__def,axiom,
    ! [X: int,Y: int] : gcd_gcd_int(X,Y) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(gcd_gcd_nat(aa_int_nat(nat2,aa_int_int(abs_abs_int,X))),aa_int_nat(nat2,aa_int_int(abs_abs_int,Y)))) ).

% gcd_int_def
tff(fact_2965_bezw__aux,axiom,
    ! [X: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(gcd_gcd_nat(X),Y)) = plus_plus_int(times_times_int(product_fst_int_int(bezw(X,Y)),aa_nat_int(semiri1314217659103216013at_int,X)),times_times_int(product_snd_int_int(bezw(X,Y)),aa_nat_int(semiri1314217659103216013at_int,Y))) ).

% bezw_aux
tff(fact_2966_uint32__set__bit__def,axiom,
    ! [X: uint32,Na: code_integer,Ba: $o] :
      uint32_set_bit(X,Na,(Ba)) = $ite(
        ( ord_le6747313008572928689nteger(Na,zero_z3403309356797280102nteger)
        | ord_le6747313008572928689nteger(numera6620942414471956472nteger(bit1(bit1(bit1(bit1(one))))),Na) ),
        aa_o_uint32(aa_Cod7056071456392079164uint32(aa_uin937759467983514409uint32(aa_fun8001877339868165167uint32(undefi8935238026407452494uint32,generi1993664874377053279uint32),X),Na),(Ba)),
        aa_o_uint32(aa_nat_fun_o_uint32(aa_uin7041821913289200502uint32(generi1993664874377053279uint32,X),code_nat_of_integer(Na)),(Ba)) ) ).

% uint32_set_bit_def
tff(fact_2967_uint32__test__bit__def,axiom,
    ! [X: uint32,Na: code_integer] :
      ( uint32_test_bit(X,Na)
    <=> $ite(
          ( ord_le6747313008572928689nteger(Na,zero_z3403309356797280102nteger)
          | ord_le6747313008572928689nteger(numera6620942414471956472nteger(bit1(bit1(bit1(bit1(one))))),Na) ),
          aa_Code_integer_o(aa_uin5206268876086501546eger_o(aa_fun5262225984791465967eger_o(undefi9030187962712549518eger_o,bit_se5367290876889521763uint32),X),Na),
          aa_nat_o(aa_uint32_fun_nat_o(bit_se5367290876889521763uint32,X),code_nat_of_integer(Na)) ) ) ).

% uint32_test_bit_def
tff(fact_2968_set__bit__uint32__eq,axiom,
    ! [Aa: uint32,Na: nat,Ba: $o] :
      aa_o_uint32(aa_nat_fun_o_uint32(aa_uin7041821913289200502uint32(generi1993664874377053279uint32,Aa),Na),(Ba)) = aa_uint32_uint32(aa_nat4578243230388868707uint32($ite((Ba),bit_se6647067497041451410uint32,bit_se4315839071623982667uint32),Na),Aa) ).

% set_bit_uint32_eq
tff(fact_2969_uint32__shiftr__code,axiom,
    ! [W: uint32,Na: code_integer] :
      aa_uin171991283339207200l_num1(rep_uint322,uint32_shiftr(W,Na)) = $ite(
        ( ord_le6747313008572928689nteger(Na,zero_z3403309356797280102nteger)
        | ord_le3102999989581377725nteger(numera6620942414471956472nteger(bit0(bit0(bit0(bit0(bit0(one)))))),Na) ),
        aa_uin171991283339207200l_num1(rep_uint322,aa_Cod8401701657498852871uint32(aa_uin6552852722274082928uint32(aa_fun8838986389337097743uint32(undefi4150998591671624934uint32,bit_se3964402333458159761uint32),W),Na)),
        bit_se5176125413884933531l_num1(code_nat_of_integer(Na),aa_uin171991283339207200l_num1(rep_uint322,W)) ) ).

% uint32_shiftr_code
tff(fact_2970_uint32_Oword__of__numeral,axiom,
    ! [Na: num] : aa_uin171991283339207200l_num1(rep_uint322,numera9087168376688890119uint32(Na)) = numera7442385471795722001l_num1(Na) ).

% uint32.word_of_numeral
tff(fact_2971_uint32_Oword__of__generic__set__bit,axiom,
    ! [P2: uint32,Na: nat,Ba: $o] : aa_uin171991283339207200l_num1(rep_uint322,aa_o_uint32(aa_nat_fun_o_uint32(aa_uin7041821913289200502uint32(generi1993664874377053279uint32,P2),Na),(Ba))) = generi5268133209446125161l_num1(aa_uin171991283339207200l_num1(rep_uint322,P2),Na,(Ba)) ).

% uint32.word_of_generic_set_bit
tff(fact_2972_uint32_Obit__eq__word__of,axiom,
    ! [P2: uint32] : aa_uint32_fun_nat_o(bit_se5367290876889521763uint32,P2) = bit_se6859397288646540909l_num1(aa_uin171991283339207200l_num1(rep_uint322,P2)) ).

% uint32.bit_eq_word_of
tff(fact_2973_bit__uint32_Orep__eq,axiom,
    ! [X: uint32] : aa_uint32_fun_nat_o(bit_se5367290876889521763uint32,X) = bit_se6859397288646540909l_num1(aa_uin171991283339207200l_num1(rep_uint322,X)) ).

% bit_uint32.rep_eq
tff(fact_2974_uint32_Oword__of__nat,axiom,
    ! [Na: nat] : aa_uin171991283339207200l_num1(rep_uint322,semiri2565882477558803405uint32(Na)) = semiri8819519690708144855l_num1(Na) ).

% uint32.word_of_nat
tff(fact_2975_uint32_Oword__of__minus,axiom,
    ! [P2: uint32] : aa_uin171991283339207200l_num1(rep_uint322,uminus_uminus_uint32(P2)) = uminus8244633308260627903l_num1(aa_uin171991283339207200l_num1(rep_uint322,P2)) ).

% uint32.word_of_minus
tff(fact_2976_uminus__uint32_Orep__eq,axiom,
    ! [X: uint32] : aa_uin171991283339207200l_num1(rep_uint322,uminus_uminus_uint32(X)) = uminus8244633308260627903l_num1(aa_uin171991283339207200l_num1(rep_uint322,X)) ).

% uminus_uint32.rep_eq
tff(fact_2977_divide__uint32_Orep__eq,axiom,
    ! [X: uint32,Xa2: uint32] : aa_uin171991283339207200l_num1(rep_uint322,aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,X),Xa2)) = divide1791077408188789448l_num1(aa_uin171991283339207200l_num1(rep_uint322,X),aa_uin171991283339207200l_num1(rep_uint322,Xa2)) ).

% divide_uint32.rep_eq
tff(fact_2978_uint32_Oword__of__div,axiom,
    ! [P2: uint32,Q3: uint32] : aa_uin171991283339207200l_num1(rep_uint322,aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,P2),Q3)) = divide1791077408188789448l_num1(aa_uin171991283339207200l_num1(rep_uint322,P2),aa_uin171991283339207200l_num1(rep_uint322,Q3)) ).

% uint32.word_of_div
tff(fact_2979_uint32_Oword__of__mult,axiom,
    ! [P2: uint32,Q3: uint32] : aa_uin171991283339207200l_num1(rep_uint322,times_times_uint32(P2,Q3)) = times_7065122842183080059l_num1(aa_uin171991283339207200l_num1(rep_uint322,P2),aa_uin171991283339207200l_num1(rep_uint322,Q3)) ).

% uint32.word_of_mult
tff(fact_2980_times__uint32_Orep__eq,axiom,
    ! [X: uint32,Xa2: uint32] : aa_uin171991283339207200l_num1(rep_uint322,times_times_uint32(X,Xa2)) = times_7065122842183080059l_num1(aa_uin171991283339207200l_num1(rep_uint322,X),aa_uin171991283339207200l_num1(rep_uint322,Xa2)) ).

% times_uint32.rep_eq
tff(fact_2981_plus__uint32_Orep__eq,axiom,
    ! [X: uint32,Xa2: uint32] : aa_uin171991283339207200l_num1(rep_uint322,plus_plus_uint32(X,Xa2)) = plus_p361126936061061375l_num1(aa_uin171991283339207200l_num1(rep_uint322,X),aa_uin171991283339207200l_num1(rep_uint322,Xa2)) ).

% plus_uint32.rep_eq
tff(fact_2982_uint32_Oword__of__add,axiom,
    ! [P2: uint32,Q3: uint32] : aa_uin171991283339207200l_num1(rep_uint322,plus_plus_uint32(P2,Q3)) = plus_p361126936061061375l_num1(aa_uin171991283339207200l_num1(rep_uint322,P2),aa_uin171991283339207200l_num1(rep_uint322,Q3)) ).

% uint32.word_of_add
tff(fact_2983_uint32_Oword__of__bool,axiom,
    ! [Na: $o] : aa_uin171991283339207200l_num1(rep_uint322,zero_n412250872926760619uint32((Na))) = zero_n2087535428495186613l_num1((Na)) ).

% uint32.word_of_bool
tff(fact_2984_or__uint32_Orep__eq,axiom,
    ! [X: uint32,Xa2: uint32] : aa_uin171991283339207200l_num1(rep_uint322,bit_se2966626333419230250uint32(X,Xa2)) = bit_se2579674332251232948l_num1(aa_uin171991283339207200l_num1(rep_uint322,X),aa_uin171991283339207200l_num1(rep_uint322,Xa2)) ).

% or_uint32.rep_eq
tff(fact_2985_and__uint32_Orep__eq,axiom,
    ! [X: uint32,Xa2: uint32] : aa_uin171991283339207200l_num1(rep_uint322,bit_se6294004230839889034uint32(X,Xa2)) = bit_se6221597996545983252l_num1(aa_uin171991283339207200l_num1(rep_uint322,X),aa_uin171991283339207200l_num1(rep_uint322,Xa2)) ).

% and_uint32.rep_eq
tff(fact_2986_uint32_Oword__of__or,axiom,
    ! [P2: uint32,Q3: uint32] : aa_uin171991283339207200l_num1(rep_uint322,bit_se2966626333419230250uint32(P2,Q3)) = bit_se2579674332251232948l_num1(aa_uin171991283339207200l_num1(rep_uint322,P2),aa_uin171991283339207200l_num1(rep_uint322,Q3)) ).

% uint32.word_of_or
tff(fact_2987_xor__uint32_Orep__eq,axiom,
    ! [X: uint32,Xa2: uint32] : aa_uin171991283339207200l_num1(rep_uint322,bit_se8572831644974689134uint32(X,Xa2)) = bit_se4696820032942467064l_num1(aa_uin171991283339207200l_num1(rep_uint322,X),aa_uin171991283339207200l_num1(rep_uint322,Xa2)) ).

% xor_uint32.rep_eq
tff(fact_2988_uint32_Oword__of__and,axiom,
    ! [P2: uint32,Q3: uint32] : aa_uin171991283339207200l_num1(rep_uint322,bit_se6294004230839889034uint32(P2,Q3)) = bit_se6221597996545983252l_num1(aa_uin171991283339207200l_num1(rep_uint322,P2),aa_uin171991283339207200l_num1(rep_uint322,Q3)) ).

% uint32.word_of_and
tff(fact_2989_uint32_Oword__of__xor,axiom,
    ! [P2: uint32,Q3: uint32] : aa_uin171991283339207200l_num1(rep_uint322,bit_se8572831644974689134uint32(P2,Q3)) = bit_se4696820032942467064l_num1(aa_uin171991283339207200l_num1(rep_uint322,P2),aa_uin171991283339207200l_num1(rep_uint322,Q3)) ).

% uint32.word_of_xor
tff(fact_2990_Rep__uint32__inject,axiom,
    ! [X: uint32,Y: uint32] :
      ( ( aa_uin171991283339207200l_num1(rep_uint322,X) = aa_uin171991283339207200l_num1(rep_uint322,Y) )
    <=> ( X = Y ) ) ).

% Rep_uint32_inject
tff(fact_2991_uint32_Oeq__iff__word__of,axiom,
    ! [P2: uint32,Q3: uint32] :
      ( ( P2 = Q3 )
    <=> ( aa_uin171991283339207200l_num1(rep_uint322,P2) = aa_uin171991283339207200l_num1(rep_uint322,Q3) ) ) ).

% uint32.eq_iff_word_of
tff(fact_2992_uint32_Oword__of__eqI,axiom,
    ! [P2: uint32,Q3: uint32] :
      ( ( aa_uin171991283339207200l_num1(rep_uint322,P2) = aa_uin171991283339207200l_num1(rep_uint322,Q3) )
     => ( P2 = Q3 ) ) ).

% uint32.word_of_eqI
tff(fact_2993_not__uint32_Orep__eq,axiom,
    ! [X: uint32] : aa_uin171991283339207200l_num1(rep_uint322,bit_ri5940046801117472212uint32(X)) = bit_ri6656324446590079838l_num1(aa_uin171991283339207200l_num1(rep_uint322,X)) ).

% not_uint32.rep_eq
tff(fact_2994_uint32_Oword__of__not,axiom,
    ! [P2: uint32] : aa_uin171991283339207200l_num1(rep_uint322,bit_ri5940046801117472212uint32(P2)) = bit_ri6656324446590079838l_num1(aa_uin171991283339207200l_num1(rep_uint322,P2)) ).

% uint32.word_of_not
tff(fact_2995_mask__uint32_Orep__eq,axiom,
    ! [X: nat] : aa_uin171991283339207200l_num1(rep_uint322,bit_se2524848472035186643uint32(X)) = bit_se6623363288439087069l_num1(X) ).

% mask_uint32.rep_eq
tff(fact_2996_uint32_Oword__of__mask,axiom,
    ! [Na: nat] : aa_uin171991283339207200l_num1(rep_uint322,bit_se2524848472035186643uint32(Na)) = bit_se6623363288439087069l_num1(Na) ).

% uint32.word_of_mask
tff(fact_2997_uint32_Oword__of__take__bit,axiom,
    ! [Na: nat,P2: uint32] : aa_uin171991283339207200l_num1(rep_uint322,bit_se4189161883738761017uint32(Na,P2)) = bit_se6195711425208868931l_num1(Na,aa_uin171991283339207200l_num1(rep_uint322,P2)) ).

% uint32.word_of_take_bit
tff(fact_2998_take__bit__uint32_Orep__eq,axiom,
    ! [X: nat,Xa2: uint32] : aa_uin171991283339207200l_num1(rep_uint322,bit_se4189161883738761017uint32(X,Xa2)) = bit_se6195711425208868931l_num1(X,aa_uin171991283339207200l_num1(rep_uint322,Xa2)) ).

% take_bit_uint32.rep_eq
tff(fact_2999_lsb__uint32_Orep__eq,axiom,
    ! [X: uint32] :
      ( least_6765572793814168004uint32(X)
    <=> aa_wor4452366487969921764num1_o(least_8743040477163521870l_num1,aa_uin171991283339207200l_num1(rep_uint322,X)) ) ).

% lsb_uint32.rep_eq
tff(fact_3000_uint32_Olsb__iff__word__of,axiom,
    ! [P2: uint32] :
      ( least_6765572793814168004uint32(P2)
    <=> aa_wor4452366487969921764num1_o(least_8743040477163521870l_num1,aa_uin171991283339207200l_num1(rep_uint322,P2)) ) ).

% uint32.lsb_iff_word_of
tff(fact_3001_uint32_Oword__of__flip__bit,axiom,
    ! [Na: nat,P2: uint32] : aa_uin171991283339207200l_num1(rep_uint322,bit_se7025624438249859091uint32(Na,P2)) = bit_se4491814353640558621l_num1(Na,aa_uin171991283339207200l_num1(rep_uint322,P2)) ).

% uint32.word_of_flip_bit
tff(fact_3002_flip__bit__uint32_Orep__eq,axiom,
    ! [X: nat,Xa2: uint32] : aa_uin171991283339207200l_num1(rep_uint322,bit_se7025624438249859091uint32(X,Xa2)) = bit_se4491814353640558621l_num1(X,aa_uin171991283339207200l_num1(rep_uint322,Xa2)) ).

% flip_bit_uint32.rep_eq
tff(fact_3003_uint32_Oword__of__set__bit,axiom,
    ! [Na: nat,P2: uint32] : aa_uin171991283339207200l_num1(rep_uint322,aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se6647067497041451410uint32,Na),P2)) = bit_se4894374433684937756l_num1(Na,aa_uin171991283339207200l_num1(rep_uint322,P2)) ).

% uint32.word_of_set_bit
tff(fact_3004_set__bit__uint32_Orep__eq,axiom,
    ! [X: nat,Xa2: uint32] : aa_uin171991283339207200l_num1(rep_uint322,aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se6647067497041451410uint32,X),Xa2)) = bit_se4894374433684937756l_num1(X,aa_uin171991283339207200l_num1(rep_uint322,Xa2)) ).

% set_bit_uint32.rep_eq
tff(fact_3005_unset__bit__uint32_Orep__eq,axiom,
    ! [X: nat,Xa2: uint32] : aa_uin171991283339207200l_num1(rep_uint322,aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se4315839071623982667uint32,X),Xa2)) = bit_se5331074070815623765l_num1(X,aa_uin171991283339207200l_num1(rep_uint322,Xa2)) ).

% unset_bit_uint32.rep_eq
tff(fact_3006_uint32_Oword__of__unset__bit,axiom,
    ! [Na: nat,P2: uint32] : aa_uin171991283339207200l_num1(rep_uint322,aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se4315839071623982667uint32,Na),P2)) = bit_se5331074070815623765l_num1(Na,aa_uin171991283339207200l_num1(rep_uint322,P2)) ).

% uint32.word_of_unset_bit
tff(fact_3007_Rep__uint32,axiom,
    ! [X: uint32] : member890509984587605005l_num1(aa_uin171991283339207200l_num1(rep_uint322,X),top_to6090768113777838812l_num1) ).

% Rep_uint32
tff(fact_3008_Rep__uint32__cases,axiom,
    ! [Y: word_N3645301735248828278l_num1] :
      ( member890509984587605005l_num1(Y,top_to6090768113777838812l_num1)
     => ~ ! [X2: uint32] : Y != aa_uin171991283339207200l_num1(rep_uint322,X2) ) ).

% Rep_uint32_cases
tff(fact_3009_Rep__uint32__induct,axiom,
    ! [Y: word_N3645301735248828278l_num1,P: fun_wo7859709073469921915num1_o] :
      ( member890509984587605005l_num1(Y,top_to6090768113777838812l_num1)
     => ( ! [X2: uint32] : aa_wor4452366487969921764num1_o(P,aa_uin171991283339207200l_num1(rep_uint322,X2))
       => aa_wor4452366487969921764num1_o(P,Y) ) ) ).

% Rep_uint32_induct
tff(fact_3010_zero__uint32_Orep__eq,axiom,
    aa_uin171991283339207200l_num1(rep_uint322,zero_zero_uint32) = zero_z3563351764282998399l_num1 ).

% zero_uint32.rep_eq
tff(fact_3011_less__uint32_Orep__eq,axiom,
    ! [X: uint32,Xa2: uint32] :
      ( ord_less_uint32(X,Xa2)
    <=> aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(aa_uin171991283339207200l_num1(rep_uint322,X)),aa_uin171991283339207200l_num1(rep_uint322,Xa2)) ) ).

% less_uint32.rep_eq
tff(fact_3012_uint32_Oless__iff__word__of,axiom,
    ! [P2: uint32,Q3: uint32] :
      ( ord_less_uint32(P2,Q3)
    <=> aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(aa_uin171991283339207200l_num1(rep_uint322,P2)),aa_uin171991283339207200l_num1(rep_uint322,Q3)) ) ).

% uint32.less_iff_word_of
tff(fact_3013_uint32_Oword__of__power,axiom,
    ! [P2: uint32,Na: nat] : aa_uin171991283339207200l_num1(rep_uint322,power_power_uint32(P2,Na)) = power_2184487114949457152l_num1(aa_uin171991283339207200l_num1(rep_uint322,P2),Na) ).

% uint32.word_of_power
tff(fact_3014_one__uint32_Orep__eq,axiom,
    aa_uin171991283339207200l_num1(rep_uint322,one_one_uint32) = one_on7727431528512463931l_num1 ).

% one_uint32.rep_eq
tff(fact_3015_less__eq__uint32_Orep__eq,axiom,
    ! [X: uint32,Xa2: uint32] :
      ( ord_less_eq_uint32(X,Xa2)
    <=> aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(aa_uin171991283339207200l_num1(rep_uint322,X)),aa_uin171991283339207200l_num1(rep_uint322,Xa2)) ) ).

% less_eq_uint32.rep_eq
tff(fact_3016_uint32_Oless__eq__iff__word__of,axiom,
    ! [P2: uint32,Q3: uint32] :
      ( ord_less_eq_uint32(P2,Q3)
    <=> aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(aa_uin171991283339207200l_num1(rep_uint322,P2)),aa_uin171991283339207200l_num1(rep_uint322,Q3)) ) ).

% uint32.less_eq_iff_word_of
tff(fact_3017_modulo__uint32_Orep__eq,axiom,
    ! [X: uint32,Xa2: uint32] : aa_uin171991283339207200l_num1(rep_uint322,aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,X),Xa2)) = modulo1504961113040953224l_num1(aa_uin171991283339207200l_num1(rep_uint322,X),aa_uin171991283339207200l_num1(rep_uint322,Xa2)) ).

% modulo_uint32.rep_eq
tff(fact_3018_uint32_Oword__of__mod,axiom,
    ! [P2: uint32,Q3: uint32] : aa_uin171991283339207200l_num1(rep_uint322,aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,P2),Q3)) = modulo1504961113040953224l_num1(aa_uin171991283339207200l_num1(rep_uint322,P2),aa_uin171991283339207200l_num1(rep_uint322,Q3)) ).

% uint32.word_of_mod
tff(fact_3019_uint32_Oeven__iff__word__of,axiom,
    ! [P2: uint32] :
      ( dvd_dvd_uint32(numera9087168376688890119uint32(bit0(one)),P2)
    <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(bit0(one))),aa_uin171991283339207200l_num1(rep_uint322,P2)) ) ).

% uint32.even_iff_word_of
tff(fact_3020_uint32_Oword__of__diff,axiom,
    ! [P2: uint32,Q3: uint32] : aa_uin171991283339207200l_num1(rep_uint322,minus_minus_uint32(P2,Q3)) = minus_4019991460397169231l_num1(aa_uin171991283339207200l_num1(rep_uint322,P2),aa_uin171991283339207200l_num1(rep_uint322,Q3)) ).

% uint32.word_of_diff
tff(fact_3021_minus__uint32_Orep__eq,axiom,
    ! [X: uint32,Xa2: uint32] : aa_uin171991283339207200l_num1(rep_uint322,minus_minus_uint32(X,Xa2)) = minus_4019991460397169231l_num1(aa_uin171991283339207200l_num1(rep_uint322,X),aa_uin171991283339207200l_num1(rep_uint322,Xa2)) ).

% minus_uint32.rep_eq
tff(fact_3022_size__uint32_Orep__eq,axiom,
    ! [X: uint32] : size_size_uint32(X) = size_s8261804613246490634l_num1(aa_uin171991283339207200l_num1(rep_uint322,X)) ).

% size_uint32.rep_eq
tff(fact_3023_uint32_Osize__eq__word__of,axiom,
    ! [P2: uint32] : size_size_uint32(P2) = size_s8261804613246490634l_num1(aa_uin171991283339207200l_num1(rep_uint322,P2)) ).

% uint32.size_eq_word_of
tff(fact_3024_uint32_Oword__of__push__bit,axiom,
    ! [Na: nat,P2: uint32] : aa_uin171991283339207200l_num1(rep_uint322,aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se5742574853984576102uint32,Na),P2)) = bit_se837345729053750000l_num1(Na,aa_uin171991283339207200l_num1(rep_uint322,P2)) ).

% uint32.word_of_push_bit
tff(fact_3025_push__bit__uint32_Orep__eq,axiom,
    ! [X: nat,Xa2: uint32] : aa_uin171991283339207200l_num1(rep_uint322,aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se5742574853984576102uint32,X),Xa2)) = bit_se837345729053750000l_num1(X,aa_uin171991283339207200l_num1(rep_uint322,Xa2)) ).

% push_bit_uint32.rep_eq
tff(fact_3026_drop__bit__uint32_Orep__eq,axiom,
    ! [X: nat,Xa2: uint32] : aa_uin171991283339207200l_num1(rep_uint322,aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se3964402333458159761uint32,X),Xa2)) = bit_se5176125413884933531l_num1(X,aa_uin171991283339207200l_num1(rep_uint322,Xa2)) ).

% drop_bit_uint32.rep_eq
tff(fact_3027_uint32_Oword__of__drop__bit,axiom,
    ! [Na: nat,P2: uint32] : aa_uin171991283339207200l_num1(rep_uint322,aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se3964402333458159761uint32,Na),P2)) = bit_se5176125413884933531l_num1(Na,aa_uin171991283339207200l_num1(rep_uint322,P2)) ).

% uint32.word_of_drop_bit
tff(fact_3028_uint32__test__bit__code,axiom,
    ! [W: uint32,Na: code_integer] :
      ( uint32_test_bit(W,Na)
    <=> $ite(
          ( ord_le6747313008572928689nteger(Na,zero_z3403309356797280102nteger)
          | ord_le6747313008572928689nteger(numera6620942414471956472nteger(bit1(bit1(bit1(bit1(one))))),Na) ),
          aa_Code_integer_o(aa_uin5206268876086501546eger_o(aa_fun5262225984791465967eger_o(undefi9030187962712549518eger_o,bit_se5367290876889521763uint32),W),Na),
          aa_nat_o(bit_se6859397288646540909l_num1(aa_uin171991283339207200l_num1(rep_uint322,W)),code_nat_of_integer(Na)) ) ) ).

% uint32_test_bit_code
tff(fact_3029_uint32__set__bit__code,axiom,
    ! [W: uint32,Na: code_integer,Ba: $o] :
      aa_uin171991283339207200l_num1(rep_uint322,uint32_set_bit(W,Na,(Ba))) = $ite(
        ( ord_le6747313008572928689nteger(Na,zero_z3403309356797280102nteger)
        | ord_le6747313008572928689nteger(numera6620942414471956472nteger(bit1(bit1(bit1(bit1(one))))),Na) ),
        aa_uin171991283339207200l_num1(rep_uint322,aa_o_uint32(aa_Cod7056071456392079164uint32(aa_uin937759467983514409uint32(aa_fun8001877339868165167uint32(undefi8935238026407452494uint32,generi1993664874377053279uint32),W),Na),(Ba))),
        generi5268133209446125161l_num1(aa_uin171991283339207200l_num1(rep_uint322,W),code_nat_of_integer(Na),(Ba)) ) ).

% uint32_set_bit_code
tff(fact_3030_uint32__shiftl__code,axiom,
    ! [W: uint32,Na: code_integer] :
      aa_uin171991283339207200l_num1(rep_uint322,uint32_shiftl(W,Na)) = $ite(
        ( ord_le6747313008572928689nteger(Na,zero_z3403309356797280102nteger)
        | ord_le3102999989581377725nteger(numera6620942414471956472nteger(bit0(bit0(bit0(bit0(bit0(one)))))),Na) ),
        aa_uin171991283339207200l_num1(rep_uint322,aa_Cod8401701657498852871uint32(aa_uin6552852722274082928uint32(aa_fun8838986389337097743uint32(undefi4150998591671624934uint32,bit_se5742574853984576102uint32),W),Na)),
        bit_se837345729053750000l_num1(code_nat_of_integer(Na),aa_uin171991283339207200l_num1(rep_uint322,W)) ) ).

% uint32_shiftl_code
tff(fact_3031_set__bit__uint32__code,axiom,
    ! [X: uint32,Na: nat,Ba: $o] :
      aa_o_uint32(aa_nat_fun_o_uint32(aa_uin7041821913289200502uint32(generi1993664874377053279uint32,X),Na),(Ba)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),numeral_numeral_nat(bit0(bit0(bit0(bit0(bit0(one))))))),uint32_set_bit(X,code_integer_of_nat(Na),(Ba)),X) ).

% set_bit_uint32_code
tff(fact_3032_shiftr__uint32__code,axiom,
    ! [Na: nat,X: uint32] :
      aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se3964402333458159761uint32,Na),X) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),numeral_numeral_nat(bit0(bit0(bit0(bit0(bit0(one))))))),uint32_shiftr(X,code_integer_of_nat(Na)),zero_zero_uint32) ).

% shiftr_uint32_code
tff(fact_3033_integer__of__nat_Orep__eq,axiom,
    ! [X: nat] : aa_Code_integer_int(code_int_of_integer,code_integer_of_nat(X)) = aa_nat_int(semiri1314217659103216013at_int,X) ).

% integer_of_nat.rep_eq
tff(fact_3034_int__of__integer__integer__of__nat,axiom,
    ! [Na: nat] : aa_Code_integer_int(code_int_of_integer,code_integer_of_nat(Na)) = aa_nat_int(semiri1314217659103216013at_int,Na) ).

% int_of_integer_integer_of_nat
tff(fact_3035_integer__of__nat__numeral,axiom,
    ! [Na: num] : code_integer_of_nat(numeral_numeral_nat(Na)) = numera6620942414471956472nteger(Na) ).

% integer_of_nat_numeral
tff(fact_3036_integer__of__nat__1,axiom,
    code_integer_of_nat(one_one_nat) = one_one_Code_integer ).

% integer_of_nat_1
tff(fact_3037_one__uint32_Orsp,axiom,
    one_on7727431528512463931l_num1 = one_on7727431528512463931l_num1 ).

% one_uint32.rsp
tff(fact_3038_zero__uint32_Orsp,axiom,
    zero_z3563351764282998399l_num1 = zero_z3563351764282998399l_num1 ).

% zero_uint32.rsp
tff(fact_3039_integer__of__nat_Oabs__eq,axiom,
    ! [X: nat] : code_integer_of_nat(X) = aa_int_Code_integer(code_integer_of_int,aa_nat_int(semiri1314217659103216013at_int,X)) ).

% integer_of_nat.abs_eq
tff(fact_3040_test__bit__uint32__code,axiom,
    ! [X: uint32,Na: nat] :
      ( aa_nat_o(aa_uint32_fun_nat_o(bit_se5367290876889521763uint32,X),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),numeral_numeral_nat(bit0(bit0(bit0(bit0(bit0(one)))))))
        & uint32_test_bit(X,code_integer_of_nat(Na)) ) ) ).

% test_bit_uint32_code
tff(fact_3041_shiftl__uint32__code,axiom,
    ! [Na: nat,X: uint32] :
      aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se5742574853984576102uint32,Na),X) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),numeral_numeral_nat(bit0(bit0(bit0(bit0(bit0(one))))))),uint32_shiftl(X,code_integer_of_nat(Na)),zero_zero_uint32) ).

% shiftl_uint32_code
tff(fact_3042_uint32_Oset__bits__aux__code,axiom,
    ! [F: fun_nat_o,Na: nat,W: uint32] :
      set_bits_aux_uint32(F,Na,W) = $ite(
        Na = zero_zero_nat,
        W,
        $let(
          n2: nat,
          n2:= aa_nat_nat(minus_minus_nat(Na),one_one_nat),
          set_bits_aux_uint32(F,n2,
            bit_se2966626333419230250uint32(aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se5742574853984576102uint32,one_one_nat),W),
              $ite(aa_nat_o(F,n2),one_one_uint32,zero_zero_uint32))) ) ) ).

% uint32.set_bits_aux_code
tff(fact_3043_uint32__divmod__code,axiom,
    ! [X: uint32,Y: uint32] :
      uint32_divmod(X,Y) = $ite(
        ord_less_eq_uint32(numera9087168376688890119uint32(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one)))))))))))))))))))))))))))))))),Y),
        $ite(ord_less_uint32(X,Y),produc1400373151660368625uint32(zero_zero_uint32,X),produc1400373151660368625uint32(one_one_uint32,minus_minus_uint32(X,Y))),
        $ite(
          Y = zero_zero_uint32,
          produc1400373151660368625uint32(div0_uint32(X),mod0_uint32(X)),
          $let(
            q: uint32,
            q:= aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se5742574853984576102uint32,one_one_nat),uint32_sdiv(aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se3964402333458159761uint32,one_one_nat),X),Y)),
            $let(
              r: uint32,
              r:= minus_minus_uint32(X,times_times_uint32(q,Y)),
              $ite(ord_less_eq_uint32(Y,r),produc1400373151660368625uint32(plus_plus_uint32(q,one_one_uint32),minus_minus_uint32(r,Y)),produc1400373151660368625uint32(q,r)) ) ) ) ) ).

% uint32_divmod_code
tff(fact_3044_uint32_Oword__of__set__bits__aux,axiom,
    ! [P: fun_nat_o,Na: nat,P2: uint32] : aa_uin171991283339207200l_num1(rep_uint322,set_bits_aux_uint32(P,Na,P2)) = code_T8486762787009228264l_num1(P,Na,aa_uin171991283339207200l_num1(rep_uint322,P2)) ).

% uint32.word_of_set_bits_aux
tff(fact_3045_set__bits__aux__uint32_Orep__eq,axiom,
    ! [X: fun_nat_o,Xa2: nat,Xb3: uint32] : aa_uin171991283339207200l_num1(rep_uint322,set_bits_aux_uint32(X,Xa2,Xb3)) = code_T8486762787009228264l_num1(X,Xa2,aa_uin171991283339207200l_num1(rep_uint322,Xb3)) ).

% set_bits_aux_uint32.rep_eq
tff(fact_3046_mod0__uint32__def,axiom,
    ! [X: uint32] : mod0_uint32(X) = aa_uint32_uint32(aa_uin3778660748068847871uint32(aa_fun8139942314387009082uint32(undefi5883245521155754897uint32,modulo_modulo_uint32),X),zero_zero_uint32) ).

% mod0_uint32_def
tff(fact_3047_div0__uint32__def,axiom,
    ! [X: uint32] : div0_uint32(X) = aa_uint32_uint32(aa_uin3778660748068847871uint32(aa_fun8139942314387009082uint32(undefi5883245521155754897uint32,divide_divide_uint32),X),zero_zero_uint32) ).

% div0_uint32_def
tff(fact_3048_uint32__divmod__def,axiom,
    ! [X: uint32,Y: uint32] :
      uint32_divmod(X,Y) = $ite(Y = zero_zero_uint32,produc1400373151660368625uint32(aa_uint32_uint32(aa_uin3778660748068847871uint32(aa_fun8139942314387009082uint32(undefi5883245521155754897uint32,divide_divide_uint32),X),zero_zero_uint32),aa_uint32_uint32(aa_uin3778660748068847871uint32(aa_fun8139942314387009082uint32(undefi5883245521155754897uint32,modulo_modulo_uint32),X),zero_zero_uint32)),produc1400373151660368625uint32(aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,X),Y),aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,X),Y))) ).

% uint32_divmod_def
tff(fact_3049_uint32__sdiv__code,axiom,
    ! [X: uint32,Y: uint32] :
      aa_uin171991283339207200l_num1(rep_uint322,uint32_sdiv(X,Y)) = $ite(Y = zero_zero_uint32,aa_uin171991283339207200l_num1(rep_uint322,aa_uint32_uint32(aa_uin3778660748068847871uint32(aa_fun8139942314387009082uint32(undefi5883245521155754897uint32,divide_divide_uint32),X),zero_zero_uint32)),signed6753297604338940182l_num1(aa_uin171991283339207200l_num1(rep_uint322,X),aa_uin171991283339207200l_num1(rep_uint322,Y))) ).

% uint32_sdiv_code
tff(fact_3050_sshiftr__uint32__code,axiom,
    ! [Na: nat,X: uint32] :
      aa_uint32_uint32(aa_nat4578243230388868707uint32(signed489701013188660438uint32,Na),X) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),numeral_numeral_nat(bit0(bit0(bit0(bit0(bit0(one))))))),
        uint32_sshiftr(X,code_integer_of_nat(Na)),
        $ite(aa_nat_o(aa_uint32_fun_nat_o(bit_se5367290876889521763uint32,X),numeral_numeral_nat(bit1(bit1(bit1(bit1(one)))))),uminus_uminus_uint32(one_one_uint32),zero_zero_uint32) ) ).

% sshiftr_uint32_code
tff(fact_3051_div__uint32__code,axiom,
    ! [X: uint32,Y: uint32] :
      aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,X),Y) = $ite(Y = zero_zero_uint32,zero_zero_uint32,uint32_div(X,Y)) ).

% div_uint32_code
tff(fact_3052_signed__drop__bit__uint32_Orep__eq,axiom,
    ! [X: nat,Xa2: uint32] : aa_uin171991283339207200l_num1(rep_uint322,aa_uint32_uint32(aa_nat4578243230388868707uint32(signed489701013188660438uint32,X),Xa2)) = signed5000768011106662067l_num1(X,aa_uin171991283339207200l_num1(rep_uint322,Xa2)) ).

% signed_drop_bit_uint32.rep_eq
tff(fact_3053_uint32_Oword__of__signed__drop__bit,axiom,
    ! [Na: nat,P2: uint32] : aa_uin171991283339207200l_num1(rep_uint322,aa_uint32_uint32(aa_nat4578243230388868707uint32(signed489701013188660438uint32,Na),P2)) = signed5000768011106662067l_num1(Na,aa_uin171991283339207200l_num1(rep_uint322,P2)) ).

% uint32.word_of_signed_drop_bit
tff(fact_3054_uint32__div__def,axiom,
    ! [X: uint32,Y: uint32] : uint32_div(X,Y) = produc9004433772639906525uint32(uint32_divmod(X,Y)) ).

% uint32_div_def
tff(fact_3055_uint32__sshiftr__def,axiom,
    ! [X: uint32,Na: code_integer] :
      uint32_sshiftr(X,Na) = $ite(
        ( ord_le6747313008572928689nteger(Na,zero_z3403309356797280102nteger)
        | ord_le3102999989581377725nteger(numera6620942414471956472nteger(bit0(bit0(bit0(bit0(bit0(one)))))),Na) ),
        aa_uint32_uint32(aa_Cod3565511682404152816uint32(aa_fun7349710212561521167uint32(undefi424500106705823974uint32,signed489701013188660438uint32),Na),X),
        aa_uint32_uint32(aa_nat4578243230388868707uint32(signed489701013188660438uint32,code_nat_of_integer(Na)),X) ) ).

% uint32_sshiftr_def
tff(fact_3056_uint32__sshiftr__code,axiom,
    ! [W: uint32,Na: code_integer] :
      aa_uin171991283339207200l_num1(rep_uint322,uint32_sshiftr(W,Na)) = $ite(
        ( ord_le6747313008572928689nteger(Na,zero_z3403309356797280102nteger)
        | ord_le3102999989581377725nteger(numera6620942414471956472nteger(bit0(bit0(bit0(bit0(bit0(one)))))),Na) ),
        aa_uin171991283339207200l_num1(rep_uint322,aa_uint32_uint32(aa_Cod3565511682404152816uint32(aa_fun7349710212561521167uint32(undefi424500106705823974uint32,signed489701013188660438uint32),Na),W)),
        signed5000768011106662067l_num1(code_nat_of_integer(Na),aa_uin171991283339207200l_num1(rep_uint322,W)) ) ).

% uint32_sshiftr_code
tff(fact_3057_mod__uint32__code,axiom,
    ! [X: uint32,Y: uint32] :
      aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,X),Y) = $ite(Y = zero_zero_uint32,X,uint32_mod(X,Y)) ).

% mod_uint32_code
tff(fact_3058_uint32__mod__def,axiom,
    ! [X: uint32,Y: uint32] : uint32_mod(X,Y) = produc1510406741064981791uint32(uint32_divmod(X,Y)) ).

% uint32_mod_def
tff(fact_3059_integer__of__uint32__code,axiom,
    ! [Na: uint32] :
      aa_uin2106300854182906631nteger(integer_of_uint32,Na) = $ite(aa_nat_o(aa_uint32_fun_nat_o(bit_se5367290876889521763uint32,Na),numeral_numeral_nat(bit1(bit1(bit1(bit1(one)))))),bit_se1080825931792720795nteger(intege5370686899274169573signed(bit_se6294004230839889034uint32(Na,numera9087168376688890119uint32(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(one))))))))))))))))))))))))))))))))),numera6620942414471956472nteger(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one))))))))))))))))))))))))))))))))),intege5370686899274169573signed(Na)) ).

% integer_of_uint32_code
tff(fact_3060_uint32_Oset__bits__code,axiom,
    ! [P: fun_nat_o] : bit_bi705532357378895591uint32(P) = set_bits_aux_uint32(P,numeral_numeral_nat(bit0(bit0(bit0(bit0(bit0(one)))))),zero_zero_uint32) ).

% uint32.set_bits_code
tff(fact_3061_uint32_Ointeger__of__eq__word__of,axiom,
    ! [P2: uint32] : aa_uin2106300854182906631nteger(integer_of_uint32,P2) = semiri4716579097644248933nteger(aa_uin171991283339207200l_num1(rep_uint322,P2)) ).

% uint32.integer_of_eq_word_of
tff(fact_3062_integer__of__uint32_Orep__eq,axiom,
    ! [X: uint32] : aa_Code_integer_int(code_int_of_integer,aa_uin2106300854182906631nteger(integer_of_uint32,X)) = semiri7338730514057886004m1_int(aa_uin171991283339207200l_num1(rep_uint322,X)) ).

% integer_of_uint32.rep_eq
tff(fact_3063_integer__of__uint32__signed__def,axiom,
    ! [Na: uint32] :
      intege5370686899274169573signed(Na) = $ite(aa_nat_o(aa_uint32_fun_nat_o(bit_se5367290876889521763uint32,Na),numeral_numeral_nat(bit1(bit1(bit1(bit1(one)))))),aa_uin2106300854182906631nteger(aa_fun3533256999457416010nteger(undefi4551866269130008737nteger,integer_of_uint32),Na),aa_uin2106300854182906631nteger(integer_of_uint32,Na)) ).

% integer_of_uint32_signed_def
tff(fact_3064_integer__of__uint32__signed__code,axiom,
    ! [Na: uint32] :
      intege5370686899274169573signed(Na) = $ite(aa_nat_o(aa_uint32_fun_nat_o(bit_se5367290876889521763uint32,Na),numeral_numeral_nat(bit1(bit1(bit1(bit1(one)))))),aa_uin2106300854182906631nteger(aa_fun3533256999457416010nteger(undefi4551866269130008737nteger,integer_of_uint32),Na),aa_int_Code_integer(code_integer_of_int,semiri7338730514057886004m1_int(aa_uin171991283339207200l_num1(rep_uint32,Na)))) ).

% integer_of_uint32_signed_code
tff(fact_3065_int__set__bits__K__False,axiom,
    bit_bi6516823479961619367ts_int(aTP_Lamm_gq) = zero_zero_int ).

% int_set_bits_K_False
tff(fact_3066_int__set__bits__K__True,axiom,
    bit_bi6516823479961619367ts_int(aTP_Lamm_hx) = aa_int_int(uminus_uminus_int,one_one_int) ).

% int_set_bits_K_True
tff(fact_3067_uint32_Oword__of__set__bits,axiom,
    ! [P: fun_nat_o] : aa_uin171991283339207200l_num1(rep_uint322,bit_bi705532357378895591uint32(P)) = bit_bi5746210779246519537l_num1(P) ).

% uint32.word_of_set_bits
tff(fact_3068_set__bits__uint32_Orep__eq,axiom,
    ! [X: fun_nat_o] : aa_uin171991283339207200l_num1(rep_uint322,bit_bi705532357378895591uint32(X)) = bit_bi5746210779246519537l_num1(X) ).

% set_bits_uint32.rep_eq
tff(fact_3069_Rep__uint32_H__code,axiom,
    ! [X: uint32] : aa_uin171991283339207200l_num1(rep_uint32,X) = bit_bi5746210779246519537l_num1(aa_uint32_fun_nat_o(bit_se5367290876889521763uint32,X)) ).

% Rep_uint32'_code
tff(fact_3070_Rep__uint32_H__def,axiom,
    rep_uint32 = rep_uint322 ).

% Rep_uint32'_def
tff(fact_3071_msb__numeral_I1_J,axiom,
    ! [Na: num] : ~ aa_int_o(most_s5051101344085556sb_int,aa_num_int(numeral_numeral_int,Na)) ).

% msb_numeral(1)
tff(fact_3072_msb__1,axiom,
    ~ aa_int_o(most_s5051101344085556sb_int,one_one_int) ).

% msb_1
tff(fact_3073_msb__numeral_I2_J,axiom,
    ! [Na: num] : aa_int_o(most_s5051101344085556sb_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) ).

% msb_numeral(2)
tff(fact_3074_msb__bin__rest,axiom,
    ! [X: int] :
      ( aa_int_o(most_s5051101344085556sb_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),aa_num_int(numeral_numeral_int,bit0(one))))
    <=> aa_int_o(most_s5051101344085556sb_int,X) ) ).

% msb_bin_rest
tff(fact_3075_msb__uint32_Orep__eq,axiom,
    ! [X: uint32] :
      ( most_s9063628576841037300uint32(X)
    <=> aa_wor4452366487969921764num1_o(most_s3860955602682368894l_num1,aa_uin171991283339207200l_num1(rep_uint322,X)) ) ).

% msb_uint32.rep_eq
tff(fact_3076_uint32_Omsb__iff__word__of,axiom,
    ! [P2: uint32] :
      ( most_s9063628576841037300uint32(P2)
    <=> aa_wor4452366487969921764num1_o(most_s3860955602682368894l_num1,aa_uin171991283339207200l_num1(rep_uint322,P2)) ) ).

% uint32.msb_iff_word_of
tff(fact_3077_uint32__msb__test__bit,axiom,
    ! [X: uint32] :
      ( most_s9063628576841037300uint32(X)
    <=> aa_nat_o(aa_uint32_fun_nat_o(bit_se5367290876889521763uint32,X),numeral_numeral_nat(bit1(bit1(bit1(bit1(one)))))) ) ).

% uint32_msb_test_bit
tff(fact_3078_msb__uint32__code,axiom,
    ! [X: uint32] :
      ( most_s9063628576841037300uint32(X)
    <=> uint32_test_bit(X,numera6620942414471956472nteger(bit1(bit1(bit1(bit1(one)))))) ) ).

% msb_uint32_code
tff(fact_3079_bin__last__set__bits,axiom,
    ! [F: fun_nat_o] :
      ( bit_wf_set_bits_int(F)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),bit_bi6516823479961619367ts_int(F))
      <=> aa_nat_o(F,zero_zero_nat) ) ) ).

% bin_last_set_bits
tff(fact_3080_wf__set__bits__int__Suc,axiom,
    ! [F: fun_nat_o] :
      ( bit_wf_set_bits_int(aTP_Lamm_go(F))
    <=> bit_wf_set_bits_int(F) ) ).

% wf_set_bits_int_Suc
tff(fact_3081_wf__set__bits__int__const,axiom,
    ! [Ba: $o] : bit_wf_set_bits_int(aa_o_fun_nat_o(aTP_Lamm_hy,(Ba))) ).

% wf_set_bits_int_const
tff(fact_3082_wf__set__bits__int__simps,axiom,
    ! [F: fun_nat_o] :
      ( bit_wf_set_bits_int(F)
    <=> ? [N2: nat] :
          ( ! [N10: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N10)
             => ~ aa_nat_o(F,N10) )
          | ! [N10: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N10)
             => aa_nat_o(F,N10) ) ) ) ).

% wf_set_bits_int_simps
tff(fact_3083_zeros,axiom,
    ! [Na: nat,F: fun_nat_o] :
      ( ! [N5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N5)
         => ~ aa_nat_o(F,N5) )
     => bit_wf_set_bits_int(F) ) ).

% zeros
tff(fact_3084_wf__set__bits__int_Osimps,axiom,
    ! [F: fun_nat_o] :
      ( bit_wf_set_bits_int(F)
    <=> ( ? [N2: nat] :
          ! [N10: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N10)
           => ~ aa_nat_o(F,N10) )
        | ? [N2: nat] :
          ! [N10: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N10)
           => aa_nat_o(F,N10) ) ) ) ).

% wf_set_bits_int.simps
tff(fact_3085_wf__set__bits__int_Ocases,axiom,
    ! [F: fun_nat_o] :
      ( bit_wf_set_bits_int(F)
     => ( ! [N: nat] :
            ~ ! [N4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N4)
               => ~ aa_nat_o(F,N4) )
       => ~ ! [N: nat] :
              ~ ! [N4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N4)
                 => aa_nat_o(F,N4) ) ) ) ).

% wf_set_bits_int.cases
tff(fact_3086_ones,axiom,
    ! [Na: nat,F: fun_nat_o] :
      ( ! [N5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N5)
         => aa_nat_o(F,N5) )
     => bit_wf_set_bits_int(F) ) ).

% ones
tff(fact_3087_int__set__bits__unfold__BIT,axiom,
    ! [F: fun_nat_o] :
      ( bit_wf_set_bits_int(F)
     => ( bit_bi6516823479961619367ts_int(F) = plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(F,zero_zero_nat)),times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),bit_bi6516823479961619367ts_int(comp_nat_o_nat(F,suc)))) ) ) ).

% int_set_bits_unfold_BIT
tff(fact_3088_upt__0__eq__Nil__conv,axiom,
    ! [J: nat] :
      ( ( upt(zero_zero_nat,J) = nil_nat )
    <=> ( J = zero_zero_nat ) ) ).

% upt_0_eq_Nil_conv
tff(fact_3089_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( upt(I,J) = nil_nat ) ) ).

% upt_conv_Nil
tff(fact_3090_length__upt,axiom,
    ! [I: nat,J: nat] : size_size_list_nat(upt(I,J)) = aa_nat_nat(minus_minus_nat(J),I) ).

% length_upt
tff(fact_3091_sorted__list__of__set__range,axiom,
    ! [M: nat,Na: nat] : linord2614967742042102400et_nat(set_or4665077453230672383an_nat(M,Na)) = upt(M,Na) ).

% sorted_list_of_set_range
tff(fact_3092_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil_nat )
    <=> ( ( J = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I) ) ) ).

% upt_eq_Nil_conv
tff(fact_3093_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J)
     => ( aa_nat_nat(nth_nat(upt(I,J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) ) ) ).

% nth_upt
tff(fact_3094_sum__list__upt,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( groups4561878855575611511st_nat(upt(M,Na)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_db),set_or4665077453230672383an_nat(M,Na)) ) ) ).

% sum_list_upt
tff(fact_3095_upt__rec__numeral,axiom,
    ! [M: num,Na: num] :
      upt(numeral_numeral_nat(M),numeral_numeral_nat(Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,numeral_numeral_nat(M)),numeral_numeral_nat(Na)),cons_nat(numeral_numeral_nat(M),upt(aa_nat_nat(suc,numeral_numeral_nat(M)),numeral_numeral_nat(Na))),nil_nat) ).

% upt_rec_numeral
tff(fact_3096_bin__rest__set__bits,axiom,
    ! [F: fun_nat_o] :
      ( bit_wf_set_bits_int(F)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,bit_bi6516823479961619367ts_int(F)),aa_num_int(numeral_numeral_int,bit0(one))) = bit_bi6516823479961619367ts_int(comp_nat_o_nat(F,suc)) ) ) ).

% bin_rest_set_bits
tff(fact_3097_map__Suc__upt,axiom,
    ! [M: nat,Na: nat] : map_nat_nat(suc,upt(M,Na)) = upt(aa_nat_nat(suc,M),aa_nat_nat(suc,Na)) ).

% map_Suc_upt
tff(fact_3098_upt__conv__Cons__Cons,axiom,
    ! [M: nat,Na: nat,Ns: list_nat,Q3: nat] :
      ( ( cons_nat(M,cons_nat(Na,Ns)) = upt(M,Q3) )
    <=> ( cons_nat(Na,Ns) = upt(aa_nat_nat(suc,M),Q3) ) ) ).

% upt_conv_Cons_Cons
tff(fact_3099_distinct__upt,axiom,
    ! [I: nat,J: nat] : aa_list_nat_o(distinct_nat,upt(I,J)) ).

% distinct_upt
tff(fact_3100_atLeastAtMost__upt,axiom,
    ! [Na: nat,M: nat] : set_or1269000886237332187st_nat(Na,M) = set_nat2(upt(Na,aa_nat_nat(suc,M))) ).

% atLeastAtMost_upt
tff(fact_3101_atLeastLessThan__upt,axiom,
    ! [I: nat,J: nat] : set_or4665077453230672383an_nat(I,J) = set_nat2(upt(I,J)) ).

% atLeastLessThan_upt
tff(fact_3102_greaterThanAtMost__upt,axiom,
    ! [Na: nat,M: nat] : set_or6659071591806873216st_nat(Na,M) = set_nat2(upt(aa_nat_nat(suc,Na),aa_nat_nat(suc,M))) ).

% greaterThanAtMost_upt
tff(fact_3103_map__bit__range__eq__if__take__bit__eq,axiom,
    ! [Na: nat,K: int,L: int] :
      ( ( bit_se2923211474154528505it_int(Na,K) = bit_se2923211474154528505it_int(Na,L) )
     => ( map_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),upt(zero_zero_nat,Na)) = map_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,L),upt(zero_zero_nat,Na)) ) ) ).

% map_bit_range_eq_if_take_bit_eq
tff(fact_3104_atLeast__upt,axiom,
    ! [Na: nat] : set_ord_lessThan_nat(Na) = set_nat2(upt(zero_zero_nat,Na)) ).

% atLeast_upt
tff(fact_3105_sorted__upt,axiom,
    ! [M: nat,Na: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),upt(M,Na)) ).

% sorted_upt
tff(fact_3106_sorted__wrt__upt,axiom,
    ! [M: nat,Na: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_nat),upt(M,Na)) ).

% sorted_wrt_upt
tff(fact_3107_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero_nat) = nil_nat ).

% upt_0
tff(fact_3108_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( upt(I,J) = cons_nat(I,upt(aa_nat_nat(suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_3109_map__add__upt_H,axiom,
    ! [Ofs: nat,Aa: nat,Ba: nat] : map_nat_nat(aTP_Lamm_hz(Ofs),upt(Aa,Ba)) = upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ofs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),Ofs)) ).

% map_add_upt'
tff(fact_3110_greaterThanLessThan__upt,axiom,
    ! [Na: nat,M: nat] : set_or5834768355832116004an_nat(Na,M) = set_nat2(upt(aa_nat_nat(suc,Na),M)) ).

% greaterThanLessThan_upt
tff(fact_3111_map__add__upt,axiom,
    ! [Na: nat,M: nat] : map_nat_nat(aTP_Lamm_hz(Na),upt(zero_zero_nat,M)) = upt(Na,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) ).

% map_add_upt
tff(fact_3112_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X: nat,Xs2: list_nat] :
      ( ( upt(I,J) = cons_nat(X,Xs2) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & ( I = X )
        & ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat),J) = Xs2 ) ) ) ).

% upt_eq_Cons_conv
tff(fact_3113_atMost__upto,axiom,
    ! [Na: nat] : set_ord_atMost_nat(Na) = set_nat2(upt(zero_zero_nat,aa_nat_nat(suc,Na))) ).

% atMost_upto
tff(fact_3114_upt__rec,axiom,
    ! [I: nat,J: nat] :
      upt(I,J) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J),cons_nat(I,upt(aa_nat_nat(suc,I),J)),nil_nat) ).

% upt_rec
tff(fact_3115_map__decr__upt,axiom,
    ! [M: nat,Na: nat] : map_nat_nat(aTP_Lamm_ia,upt(aa_nat_nat(suc,M),aa_nat_nat(suc,Na))) = upt(M,Na) ).

% map_decr_upt
tff(fact_3116_infinite__int__iff__infinite__nat__abs,axiom,
    ! [Sa: set_int] :
      ( ~ finite_finite_int(Sa)
    <=> ~ aa_set_nat_o(finite_finite_nat,image_int_nat(comp_int_nat_int(nat2,abs_abs_int),Sa)) ) ).

% infinite_int_iff_infinite_nat_abs
tff(fact_3117_summable__reindex,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( aa_set_nat_o(inj_on_nat_nat(G),top_top_set_nat)
       => ( ! [X2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X2))
         => summable_real(comp_nat_real_nat(F,G)) ) ) ) ).

% summable_reindex
tff(fact_3118_divmod__integer__eq__cases,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_z3403309356797280102nteger,
        produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
        $ite(
          L = zero_z3403309356797280102nteger,
          produc1086072967326762835nteger(zero_z3403309356797280102nteger,K),
          aa_Pro4693618923521437820nteger(comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(produc6499014454317279255nteger,times_3573771949741848930nteger),sgn_sgn_Code_integer,L),
            $ite(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,L),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_ib(L),code_divmod_abs(K,L)))) ) ) ).

% divmod_integer_eq_cases
tff(fact_3119_suminf__reindex__mono,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( aa_set_nat_o(inj_on_nat_nat(G),top_top_set_nat)
       => ( ! [X2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X2))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(comp_nat_real_nat(F,G))),suminf_real(F)) ) ) ) ).

% suminf_reindex_mono
tff(fact_3120_suminf__reindex,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( aa_set_nat_o(inj_on_nat_nat(G),top_top_set_nat)
       => ( ! [X2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X2))
         => ( ! [X2: nat] :
                ( ~ aa_set_nat_o(member_nat(X2),image_nat_nat(G,top_top_set_nat))
               => ( aa_nat_real(F,X2) = zero_zero_real ) )
           => ( suminf_real(comp_nat_real_nat(F,G)) = suminf_real(F) ) ) ) ) ) ).

% suminf_reindex
tff(fact_3121_Code__Target__Int_Onegative__def,axiom,
    code_Target_negative = comp_int_int_num(uminus_uminus_int,numeral_numeral_int) ).

% Code_Target_Int.negative_def
tff(fact_3122_ATP_Olambda__1,axiom,
    ! [Uu: nat] : aa_nat5821310369566504398ap_nat(aTP_Lamm_aa,Uu) = vEBT_Example_mIf_nat(aa_nat2910048432523430448Heap_o(vEBT_vebt_memberi(ti),Uu),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uu),heap_T2919350798565477881at_nat(vEBT_vebt_predi(ti,Uu),aTP_Lamm_a)) ).

% ATP.lambda_1
tff(fact_3123_ATP_Olambda__2,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_bj,Uu) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uu)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),numeral_numeral_nat(bit0(one)))),one_one_nat))) ).

% ATP.lambda_2
tff(fact_3124_ATP_Olambda__3,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_bn,Uu) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(bit0(one)))),aa_nat_nat(suc,Uu)) ).

% ATP.lambda_3
tff(fact_3125_ATP_Olambda__4,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_bx,Uu)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Uu)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uu),numeral_numeral_real(bit0(one)))
        & ( aa_real_real(cos_real,Uu) = zero_zero_real ) ) ) ).

% ATP.lambda_4
tff(fact_3126_ATP_Olambda__5,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_cu,Uu) = aa_real_real(times_times_real(aa_nat_real(cos_coeff,Uu)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uu)) ).

% ATP.lambda_5
tff(fact_3127_ATP_Olambda__6,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_fm,Uu) = aa_real_real(divide_divide_real(aa_real_real(cos_real,Uu)),aa_real_real(sin_real,Uu)) ).

% ATP.lambda_6
tff(fact_3128_ATP_Olambda__7,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_gi,Uu) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,Uu)),Uu) ).

% ATP.lambda_7
tff(fact_3129_ATP_Olambda__8,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_hh,Uu) = aa_int_int(aa_int_fun_int_int(divide_divide_int,Uu),aa_num_int(numeral_numeral_int,bit0(one))) ).

% ATP.lambda_8
tff(fact_3130_ATP_Olambda__9,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_fp,Uu) = aa_real_real(aa_nat_fun_real_real(root,Uu),aa_nat_real(semiri5074537144036343181t_real,Uu)) ).

% ATP.lambda_9
tff(fact_3131_ATP_Olambda__10,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_ia,Uu) = aa_nat_nat(minus_minus_nat(Uu),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_10
tff(fact_3132_ATP_Olambda__11,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_hi,Uu) = plus_plus_int(Uu,Uu) ).

% ATP.lambda_11
tff(fact_3133_ATP_Olambda__12,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_fs,Uu) = aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,Uu)) ).

% ATP.lambda_12
tff(fact_3134_ATP_Olambda__13,axiom,
    ! [Uu: int] :
      ( aa_int_o(aTP_Lamm_df,Uu)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,bit0(one))),Uu) ) ).

% ATP.lambda_13
tff(fact_3135_ATP_Olambda__14,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_ff,Uu) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_bi,Uu)) ).

% ATP.lambda_14
tff(fact_3136_ATP_Olambda__15,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_ft,Uu) = aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uu))) ).

% ATP.lambda_15
tff(fact_3137_ATP_Olambda__16,axiom,
    ! [Uu: option_nat] : aa_opt1867772761376694270ap_nat(aTP_Lamm_a,Uu) = aa_nat5821310369566504398ap_nat(heap_Time_return_nat,the_nat(Uu)) ).

% ATP.lambda_16
tff(fact_3138_ATP_Olambda__17,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_do,Uu) = product_Pair_int_int(aa_int_int(uminus_uminus_int,Uu)) ).

% ATP.lambda_17
tff(fact_3139_ATP_Olambda__18,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_dn,Uu) = product_Pair_int_int(aa_int_int(abs_abs_int,Uu)) ).

% ATP.lambda_18
tff(fact_3140_ATP_Olambda__19,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_di,Uu) = product_Pair_nat_nat(aa_nat_nat(suc,Uu)) ).

% ATP.lambda_19
tff(fact_3141_ATP_Olambda__20,axiom,
    ! [Uu: fun_nat_real,Uua: nat] :
      aa_nat_real(aTP_Lamm_bo(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Uua),zero_zero_real,aa_nat_real(Uu,aa_nat_nat(divide_divide_nat(aa_nat_nat(minus_minus_nat(Uua),one_one_nat)),numeral_numeral_nat(bit0(one))))) ).

% ATP.lambda_20
tff(fact_3142_ATP_Olambda__21,axiom,
    ! [Uu: vEBT_VEBTi,Uua: nat] : aa_nat5821310369566504398ap_nat(aTP_Lamm_bd(Uu),Uua) = vEBT_Example_mIf_nat(aa_nat2910048432523430448Heap_o(vEBT_vebt_memberi(Uu),Uua),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uua),heap_T2919350798565477881at_nat(vEBT_vebt_predi(Uu,Uua),aTP_Lamm_a)) ).

% ATP.lambda_21
tff(fact_3143_ATP_Olambda__22,axiom,
    ! [Uu: int,Uua: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_dm,Uu),Uua) = $ite(Uu = zero_zero_int,aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(product_Pair_int_int(times_times_int(sgn_sgn_int(Uu),Uua)),aa_int_int(abs_abs_int,Uu))) ).

% ATP.lambda_22
tff(fact_3144_ATP_Olambda__23,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_bh,Uu),Uua) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Uu),one_one_real)),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_23
tff(fact_3145_ATP_Olambda__24,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_bq,Uu),Uua) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua)),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua)),one_one_nat))) ).

% ATP.lambda_24
tff(fact_3146_ATP_Olambda__25,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_br,Uu),Uua) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua))) ).

% ATP.lambda_25
tff(fact_3147_ATP_Olambda__26,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_by,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
        & ( aa_real_real(sin_real,Uua) = Uu ) ) ) ).

% ATP.lambda_26
tff(fact_3148_ATP_Olambda__27,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_bt,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Uua),aa_real_real(divide_divide_real(pi),numeral_numeral_real(bit0(one))))
        & ( aa_real_real(tan_real,Uua) = Uu ) ) ) ).

% ATP.lambda_27
tff(fact_3149_ATP_Olambda__28,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_he,Uu),Uua) = $let(
        l2: int,
        l2:= times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),aa_Code_integer_int(code_int_of_integer,Uu)),
        $ite(Uua = zero_z3403309356797280102nteger,l2,plus_plus_int(l2,one_one_int)) ) ).

% ATP.lambda_28
tff(fact_3150_ATP_Olambda__29,axiom,
    ! [Uu: complex,Uua: real] :
      ( aa_real_o(aTP_Lamm_cc(Uu),Uua)
    <=> ( ( sgn_sgn_complex(Uu) = cis(Uua) )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),pi) ) ) ).

% ATP.lambda_29
tff(fact_3151_ATP_Olambda__30,axiom,
    ! [Uu: real,Uua: int] :
      ( aa_int_o(aTP_Lamm_bz(Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,ring_1_of_int_real(Uua)),Uu)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Uu),ring_1_of_int_real(plus_plus_int(Uua,one_one_int))) ) ) ).

% ATP.lambda_30
tff(fact_3152_ATP_Olambda__31,axiom,
    ! [Uu: rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_ca(Uu),Uua)
    <=> ( ord_less_eq_rat(ring_1_of_int_rat(Uua),Uu)
        & ord_less_rat(Uu,ring_1_of_int_rat(plus_plus_int(Uua,one_one_int))) ) ) ).

% ATP.lambda_31
tff(fact_3153_ATP_Olambda__32,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_bi,Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),numeral_numeral_nat(bit0(one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),numeral_numeral_nat(bit0(one)))),one_one_nat)))) ).

% ATP.lambda_32
tff(fact_3154_ATP_Olambda__33,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fg,Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),numeral_numeral_nat(bit0(one))))) ).

% ATP.lambda_33
tff(fact_3155_ATP_Olambda__34,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_fn(Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_34
tff(fact_3156_ATP_Olambda__35,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aTP_Lamm_cm(Uu),Uua) = aa_nat_nat(power_power_nat(aa_nat_nat(binomial(Uu),Uua)),numeral_numeral_nat(bit0(one))) ).

% ATP.lambda_35
tff(fact_3157_ATP_Olambda__36,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_cz,Uu),Uua) = aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uua)) ).

% ATP.lambda_36
tff(fact_3158_ATP_Olambda__37,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_gj,Uu),Uua) = aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu)),aa_real_real(exp_real,Uua)) ).

% ATP.lambda_37
tff(fact_3159_ATP_Olambda__38,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aTP_Lamm_ce(Uu),Uua) = aa_nat_nat(binomial(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)),Uu) ).

% ATP.lambda_38
tff(fact_3160_ATP_Olambda__39,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aTP_Lamm_cf(Uu),Uua) = aa_nat_nat(binomial(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)),Uua) ).

% ATP.lambda_39
tff(fact_3161_ATP_Olambda__40,axiom,
    ! [Uu: nat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_cr(Uu),Uua)
    <=> ( aa_nat_complex(power_power_complex(Uua),Uu) = one_one_complex ) ) ).

% ATP.lambda_40
tff(fact_3162_ATP_Olambda__41,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fz,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(divide_divide_real(Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)))),Uua) ).

% ATP.lambda_41
tff(fact_3163_ATP_Olambda__42,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_gg(Uu),Uua) = aa_real_real(powr_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(times_times_real(Uu),Uua))),aa_real_real(divide_divide_real(one_one_real),Uua)) ).

% ATP.lambda_42
tff(fact_3164_ATP_Olambda__43,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_gk(Uu),Uua) = aa_real_real(powr_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(divide_divide_real(Uu),Uua))),Uua) ).

% ATP.lambda_43
tff(fact_3165_ATP_Olambda__44,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_bg,Uu),Uua) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),numeral_numeral_nat(bit0(one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),numeral_numeral_nat(bit0(one)))),one_one_nat))) ).

% ATP.lambda_44
tff(fact_3166_ATP_Olambda__45,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_bw,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),pi)
        & ( aa_real_real(cos_real,Uua) = Uu ) ) ) ).

% ATP.lambda_45
tff(fact_3167_ATP_Olambda__46,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_gf(Uu),Uua) = groups6591440286371151544t_real(aTP_Lamm_fn(Uu),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua)),one_one_nat))) ).

% ATP.lambda_46
tff(fact_3168_ATP_Olambda__47,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ge(Uu),Uua) = groups6591440286371151544t_real(aTP_Lamm_fn(Uu),set_ord_lessThan_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua))) ).

% ATP.lambda_47
tff(fact_3169_ATP_Olambda__48,axiom,
    ! [Uu: code_integer,Uua: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aTP_Lamm_hr,Uu),(Uua)) = produc4188289175737317920o_char(aTP_Lamm_hq((Uua)),code_bit_cut_integer(Uu)) ).

% ATP.lambda_48
tff(fact_3170_ATP_Olambda__49,axiom,
    ! [Uu: list_nat,Uua: vEBT_VEBTi] : aa_VEB6430634201118579385st_nat(aTP_Lamm_be(Uu),Uua) = aa_lis1877347954636394190st_nat(vEBT_E4213572786784670417at_nat(aTP_Lamm_bd(Uua)),Uu) ).

% ATP.lambda_49
tff(fact_3171_ATP_Olambda__50,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_eo,Uu),Uua) = aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Uua)),Uu)) ).

% ATP.lambda_50
tff(fact_3172_ATP_Olambda__51,axiom,
    ! [Uu: $o,Uua: nat] : aa_nat_nat(aa_o_fun_nat_nat(aTP_Lamm_hb,(Uu)),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat((Uu))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),numeral_numeral_nat(bit0(one)))) ).

% ATP.lambda_51
tff(fact_3173_ATP_Olambda__52,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ct,Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(sin_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_52
tff(fact_3174_ATP_Olambda__53,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_cs,Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(cos_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_53
tff(fact_3175_ATP_Olambda__54,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_hf,Uu),Uua) = $let(
        l2: num,
        l2:= code_num_of_integer(Uu),
        $let(
          l3: num,
          l3:= plus_plus_num(l2,l2),
          $ite(Uua = zero_z3403309356797280102nteger,l3,plus_plus_num(l3,one)) ) ) ).

% ATP.lambda_54
tff(fact_3176_ATP_Olambda__55,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_hg,Uu),Uua) = $let(
        l2: nat,
        l2:= code_nat_of_integer(Uu),
        $let(
          l3: nat,
          l3:= aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,l2),l2),
          $ite(Uua = zero_z3403309356797280102nteger,l3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,l3),one_one_nat)) ) ) ).

% ATP.lambda_55
tff(fact_3177_ATP_Olambda__56,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_bu,Uu),Uua)
    <=> ( aa_real_real(exp_real,Uua) = Uu ) ) ).

% ATP.lambda_56
tff(fact_3178_ATP_Olambda__57,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aTP_Lamm_cn(Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_nat_nat(binomial(Uu),Uua)) ).

% ATP.lambda_57
tff(fact_3179_ATP_Olambda__58,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ga,Uu),Uua) = aa_real_real(times_times_real(Uu),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))))) ).

% ATP.lambda_58
tff(fact_3180_ATP_Olambda__59,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_dj,Uu),Uua) = plus_plus_int(Uu,aa_o_int(zero_n2684676970156552555ol_int,Uua != zero_zero_int)) ).

% ATP.lambda_59
tff(fact_3181_ATP_Olambda__60,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fy,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))))) ).

% ATP.lambda_60
tff(fact_3182_ATP_Olambda__61,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fv,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))) ).

% ATP.lambda_61
tff(fact_3183_ATP_Olambda__62,axiom,
    ! [Uu: nat,Uua: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_nat5661399468481021847_VEBTi(aTP_Lamm_bb,Uu),Uua) = vEBT_vebt_inserti(Uua,Uu) ).

% ATP.lambda_62
tff(fact_3184_ATP_Olambda__63,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ee,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uu) ) ).

% ATP.lambda_63
tff(fact_3185_ATP_Olambda__64,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aTP_Lamm_en(Uu),Uua) = aa_nat_nat(modulo_modulo_nat(Uua),Uu) ).

% ATP.lambda_64
tff(fact_3186_ATP_Olambda__65,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_co,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu) ) ).

% ATP.lambda_65
tff(fact_3187_ATP_Olambda__66,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aTP_Lamm_fo(Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu) ).

% ATP.lambda_66
tff(fact_3188_ATP_Olambda__67,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aTP_Lamm_em(Uu),Uua) = aa_nat_nat(minus_minus_nat(Uua),Uu) ).

% ATP.lambda_67
tff(fact_3189_ATP_Olambda__68,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_et,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) ).

% ATP.lambda_68
tff(fact_3190_ATP_Olambda__69,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_hc(Uu),Uua) = plus_p5714425477246183910nteger(Uua,Uu) ).

% ATP.lambda_69
tff(fact_3191_ATP_Olambda__70,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aTP_Lamm_hz(Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu) ).

% ATP.lambda_70
tff(fact_3192_ATP_Olambda__71,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aTP_Lamm_el(Uu),Uua) = plus_plus_int(Uua,Uu) ).

% ATP.lambda_71
tff(fact_3193_ATP_Olambda__72,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_ev(Uu),Uua) = aa_real_real(powr_real(Uua),Uu) ).

% ATP.lambda_72
tff(fact_3194_ATP_Olambda__73,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ed,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),Uu) ) ).

% ATP.lambda_73
tff(fact_3195_ATP_Olambda__74,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ej,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),Uu) ) ).

% ATP.lambda_74
tff(fact_3196_ATP_Olambda__75,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aTP_Lamm_cd(Uu),Uua) = aa_nat_nat(binomial(Uua),Uu) ).

% ATP.lambda_75
tff(fact_3197_ATP_Olambda__76,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fu,Uu),Uua) = aa_real_real(aa_nat_fun_real_real(root,Uua),Uu) ).

% ATP.lambda_76
tff(fact_3198_ATP_Olambda__77,axiom,
    ! [Uu: set_complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_ad(Uu),Uua)
    <=> member_complex(Uua,Uu) ) ).

% ATP.lambda_77
tff(fact_3199_ATP_Olambda__78,axiom,
    ! [Uu: set_wo3913738467083021356l_num1,Uua: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(aTP_Lamm_ac(Uu),Uua)
    <=> member890509984587605005l_num1(Uua,Uu) ) ).

% ATP.lambda_78
tff(fact_3200_ATP_Olambda__79,axiom,
    ! [Uu: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_ab(Uu),Uua)
    <=> member_real(Uua,Uu) ) ).

% ATP.lambda_79
tff(fact_3201_ATP_Olambda__80,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_ae(Uu),Uua)
    <=> member_list_nat(Uua,Uu) ) ).

% ATP.lambda_80
tff(fact_3202_ATP_Olambda__81,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_af(Uu),Uua)
    <=> member_set_nat(Uua,Uu) ) ).

% ATP.lambda_81
tff(fact_3203_ATP_Olambda__82,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ag(Uu),Uua)
    <=> aa_set_nat_o(member_nat(Uua),Uu) ) ).

% ATP.lambda_82
tff(fact_3204_ATP_Olambda__83,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_ah(Uu),Uua)
    <=> member_int(Uua,Uu) ) ).

% ATP.lambda_83
tff(fact_3205_ATP_Olambda__84,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_cy(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua)),one_one_nat)) ).

% ATP.lambda_84
tff(fact_3206_ATP_Olambda__85,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_cx(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua)) ).

% ATP.lambda_85
tff(fact_3207_ATP_Olambda__86,axiom,
    ! [Uu: fun_real_o,Uua: real] :
      ( aa_real_o(aTP_Lamm_gu(Uu),Uua)
    <=> aa_real_o(Uu,aa_real_real(inverse_inverse_real,Uua)) ) ).

% ATP.lambda_86
tff(fact_3208_ATP_Olambda__87,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_es(Uu),Uua) = aa_real_real(Uu,aa_real_real(uminus_uminus_real,Uua)) ).

% ATP.lambda_87
tff(fact_3209_ATP_Olambda__88,axiom,
    ! [Uu: fun_real_o,Uua: real] :
      ( aa_real_o(aTP_Lamm_gs(Uu),Uua)
    <=> aa_real_o(Uu,aa_real_real(uminus_uminus_real,Uua)) ) ).

% ATP.lambda_88
tff(fact_3210_ATP_Olambda__89,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_go(Uu),Uua)
    <=> aa_nat_o(Uu,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_89
tff(fact_3211_ATP_Olambda__90,axiom,
    ! [Uu: list_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_eq(Uu),Uua) = order_Greatest_nat(aa_nat_fun_nat_o(aTP_Lamm_ep(Uu),Uua)) ).

% ATP.lambda_90
tff(fact_3212_ATP_Olambda__91,axiom,
    ! [Uu: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_fj(Uu),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_fi(Uu),Uua)) ).

% ATP.lambda_91
tff(fact_3213_ATP_Olambda__92,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_da(Uu),Uua) = cis(aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),pi)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,Uu))) ).

% ATP.lambda_92
tff(fact_3214_ATP_Olambda__93,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_gl(Uu),Uua)
    <=> ( aa_real_real(Uu,Uua) != zero_zero_real ) ) ).

% ATP.lambda_93
tff(fact_3215_ATP_Olambda__94,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bs,Uu),Uua)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(divide_divide_nat(Uu),aa_nat_nat(power_power_nat(numeral_numeral_nat(bit0(one))),Uua))) ) ).

% ATP.lambda_94
tff(fact_3216_ATP_Olambda__95,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fx,Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_95
tff(fact_3217_ATP_Olambda__96,axiom,
    ! [Uu: $o,Uua: list_o] : aa_lis8745123556223272074list_o(aTP_Lamm_ao((Uu)),Uua) = heap_T8290161705520373763list_o(cons_o((Uu),Uua)) ).

% ATP.lambda_96
tff(fact_3218_ATP_Olambda__97,axiom,
    ! [Uu: nat,Uua: list_nat] : aa_lis1877347954636394190st_nat(aTP_Lamm_au(Uu),Uua) = aa_lis1877347954636394190st_nat(heap_T3164785745270913723st_nat,cons_nat(Uu,Uua)) ).

% ATP.lambda_97
tff(fact_3219_ATP_Olambda__98,axiom,
    ! [Uu: int,Uua: list_int] : aa_lis3404401049470555086st_int(aTP_Lamm_am(Uu),Uua) = heap_T8210306762616492823st_int(cons_int(Uu,Uua)) ).

% ATP.lambda_98
tff(fact_3220_ATP_Olambda__99,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_fr(Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_99
tff(fact_3221_ATP_Olambda__100,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_complex(aTP_Lamm_bm(Uu),Uua) = real_V4546457046886955230omplex(aa_nat_real(Uu,Uua)) ).

% ATP.lambda_100
tff(fact_3222_ATP_Olambda__101,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_bk(Uu),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_101
tff(fact_3223_ATP_Olambda__102,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_gd(Uu),Uua) = aa_real_real(cos_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_102
tff(fact_3224_ATP_Olambda__103,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_real(aTP_Lamm_gw(Uu),Uua) = re(aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_103
tff(fact_3225_ATP_Olambda__104,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_real(aTP_Lamm_gx(Uu),Uua) = im(aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_104
tff(fact_3226_ATP_Olambda__105,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: real] : aa_real_real(aTP_Lamm_fl(Uu),Uua) = suminf_real(aa_real_fun_nat_real(Uu,Uua)) ).

% ATP.lambda_105
tff(fact_3227_ATP_Olambda__106,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_gy(Uu),Uua) = cnj(aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_106
tff(fact_3228_ATP_Olambda__107,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_bp(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Uub),aa_nat_real(Uua,aa_nat_nat(divide_divide_nat(Uub),numeral_numeral_nat(bit0(one)))),aa_nat_real(Uu,aa_nat_nat(divide_divide_nat(aa_nat_nat(minus_minus_nat(Uub),one_one_nat)),numeral_numeral_nat(bit0(one))))) ).

% ATP.lambda_107
tff(fact_3229_ATP_Olambda__108,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_cw(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),Uub),aa_nat_real(Uu,Uub),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_108
tff(fact_3230_ATP_Olambda__109,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_hd(Uu),Uua),Uub) = $ite(ord_le3102999989581377725nteger(numera6620942414471956472nteger(Uu),Uub),produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uub,numera6620942414471956472nteger(Uu))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),Uua),Uub)) ).

% ATP.lambda_109
tff(fact_3231_ATP_Olambda__110,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_dg(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(Uu)),Uub),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua)),one_one_nat)),aa_nat_nat(minus_minus_nat(Uub),numeral_numeral_nat(Uu))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua)),Uub)) ).

% ATP.lambda_110
tff(fact_3232_ATP_Olambda__111,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_dh(Uu),Uua),Uub) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Uu)),Uub),aa_int7462565939943104719nt_int(product_Pair_int_int(plus_plus_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),Uua),one_one_int)),minus_minus_int(Uub,aa_num_int(numeral_numeral_int,Uu))),aa_int7462565939943104719nt_int(product_Pair_int_int(times_times_int(aa_num_int(numeral_numeral_int,bit0(one)),Uua)),Uub)) ).

% ATP.lambda_111
tff(fact_3233_ATP_Olambda__112,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_ht(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uu),Uub))) ).

% ATP.lambda_112
tff(fact_3234_ATP_Olambda__113,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_ib(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(abs_abs_Code_integer(Uu),Uub))) ).

% ATP.lambda_113
tff(fact_3235_ATP_Olambda__114,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_hs(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uu,Uub))) ).

% ATP.lambda_114
tff(fact_3236_ATP_Olambda__115,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_gm(Uu),Uua),Uub)
    <=> has_fi5821293074295781190e_real(Uu,aa_real_real(Uua,Uub),topolo2177554685111907308n_real(Uub,top_top_set_real)) ) ).

% ATP.lambda_115
tff(fact_3237_ATP_Olambda__116,axiom,
    ! [Uu: heap_Time_Heap_nat,Uua: heap_Time_Heap_nat,Uub: $o] :
      aa_o_H8739796202006397184ap_nat(aa_Hea3464058753474827431ap_nat(aTP_Lamm_ay(Uu),Uua),(Uub)) = $ite((Uub),Uu,Uua) ).

% ATP.lambda_116
tff(fact_3238_ATP_Olambda__117,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_fk(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_117
tff(fact_3239_ATP_Olambda__118,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod3168588076859124107eger_o(aa_Cod1096751537565085541eger_o(aTP_Lamm_hj(Uu),Uua),Uub) = produc6677183202524767010eger_o($ite(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger,Uu),Uua,minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),Uub)),Uub = one_one_Code_integer) ).

% ATP.lambda_118
tff(fact_3240_ATP_Olambda__119,axiom,
    ! [Uu: list_nat,Uua: list_nat,Uub: vEBT_VEBTi] : aa_VEB6430634201118579385st_nat(aa_lis4214751270122123570st_nat(aTP_Lamm_bf(Uu),Uua),Uub) = heap_T3876627890315925662st_nat(aa_VEB836174949890266638_VEBTi(vEBT_E6105538542217078229_VEBTi(aTP_Lamm_bb,cons_nat(zero_zero_nat,Uu)),Uub),aTP_Lamm_be(Uua)) ).

% ATP.lambda_119
tff(fact_3241_ATP_Olambda__120,axiom,
    ! [Uu: $o,Uua: code_integer,Uub: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aTP_Lamm_hq((Uu)),Uua),(Uub)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aTP_Lamm_hp((Uu)),(Uub)),code_bit_cut_integer(Uua)) ).

% ATP.lambda_120
tff(fact_3242_ATP_Olambda__121,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ea(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_dz(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_121
tff(fact_3243_ATP_Olambda__122,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_dy(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_dx(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_122
tff(fact_3244_ATP_Olambda__123,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_dw(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_dv(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_123
tff(fact_3245_ATP_Olambda__124,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_du(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_dt(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_124
tff(fact_3246_ATP_Olambda__125,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ds(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_dr(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_125
tff(fact_3247_ATP_Olambda__126,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_dq(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_dp(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_126
tff(fact_3248_ATP_Olambda__127,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ey(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_127
tff(fact_3249_ATP_Olambda__128,axiom,
    ! [Uu: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_ez(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uua,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_128
tff(fact_3250_ATP_Olambda__129,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fh(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uub)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_129
tff(fact_3251_ATP_Olambda__130,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_cv(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_nat_real(Uua,Uub)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_130
tff(fact_3252_ATP_Olambda__131,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_hv(Uu),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = aa_nat_nat(minus_minus_nat(Uu),one_one_nat) )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_131
tff(fact_3253_ATP_Olambda__132,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_hw(Uu),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = Uu )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4561878855575611511st_nat(Uub)),one_one_nat) = Uua ) ) ) ).

% ATP.lambda_132
tff(fact_3254_ATP_Olambda__133,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_hu(Uu),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = Uu )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_133
tff(fact_3255_ATP_Olambda__134,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_gz(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(aa_nat_nat(suc,Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_134
tff(fact_3256_ATP_Olambda__135,axiom,
    ! [Uu: list_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ep(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),insert_nat(zero_zero_nat,set_nat2(Uu)))
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uub),Uua) ) ) ).

% ATP.lambda_135
tff(fact_3257_ATP_Olambda__136,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ha(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),aa_nat_nat(suc,Uua)) ) ) ).

% ATP.lambda_136
tff(fact_3258_ATP_Olambda__137,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_cg(Uu),Uua),Uub) = aa_nat_nat(binomial(aa_nat_nat(minus_minus_nat(Uua),Uub)),aa_nat_nat(minus_minus_nat(Uu),Uub)) ).

% ATP.lambda_137
tff(fact_3259_ATP_Olambda__138,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_dl(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uua)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uub) ) ) ).

% ATP.lambda_138
tff(fact_3260_ATP_Olambda__139,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ef(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uub),Uua) ) ) ).

% ATP.lambda_139
tff(fact_3261_ATP_Olambda__140,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_dk(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uub) ) ) ).

% ATP.lambda_140
tff(fact_3262_ATP_Olambda__141,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_eh(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uub),Uua) ) ) ).

% ATP.lambda_141
tff(fact_3263_ATP_Olambda__142,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ei(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uub),Uua) ) ) ).

% ATP.lambda_142
tff(fact_3264_ATP_Olambda__143,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_eg(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uub),Uua) ) ) ).

% ATP.lambda_143
tff(fact_3265_ATP_Olambda__144,axiom,
    ! [Uu: fun_o_4158591058828729991ap_nat,Uua: list_o,Uub: nat] : aa_nat5122079553830896478st_nat(aa_lis5357409067225778835st_nat(aTP_Lamm_aw(Uu),Uua),Uub) = heap_T7479857938255823433st_nat(vEBT_E7580615859684833657_o_nat(Uu,Uua),aTP_Lamm_au(Uub)) ).

% ATP.lambda_144
tff(fact_3266_ATP_Olambda__145,axiom,
    ! [Uu: fun_na3533371890617757541ap_nat,Uua: list_nat,Uub: nat] : aa_nat5122079553830896478st_nat(aa_lis4703896708864502743st_nat(aTP_Lamm_ax(Uu),Uua),Uub) = heap_T7479857938255823433st_nat(aa_lis1877347954636394190st_nat(vEBT_E4213572786784670417at_nat(Uu),Uua),aTP_Lamm_au(Uub)) ).

% ATP.lambda_145
tff(fact_3267_ATP_Olambda__146,axiom,
    ! [Uu: fun_in3770949032268332553ap_nat,Uua: list_int,Uub: nat] : aa_nat5122079553830896478st_nat(aa_lis1507422941235949179st_nat(aTP_Lamm_av(Uu),Uua),Uub) = heap_T7479857938255823433st_nat(vEBT_E5212926498685724973nt_nat(Uu,Uua),aTP_Lamm_au(Uub)) ).

% ATP.lambda_146
tff(fact_3268_ATP_Olambda__147,axiom,
    ! [Uu: fun_o_8325599348076409421Heap_o,Uua: list_o,Uub: $o] : aa_o_H3316518667072757508list_o(aa_lis4141098589498572913list_o(aTP_Lamm_at(Uu),Uua),(Uub)) = heap_T3493894765904877959list_o(vEBT_E3732904050032093615ap_o_o(Uu,Uua),aTP_Lamm_ao((Uub))) ).

% ATP.lambda_147
tff(fact_3269_ATP_Olambda__148,axiom,
    ! [Uu: fun_na2343073368905879023Heap_o,Uua: list_nat,Uub: $o] : aa_o_H3316518667072757508list_o(aa_lis540506999666758581list_o(aTP_Lamm_ap(Uu),Uua),(Uub)) = heap_T3493894765904877959list_o(vEBT_E1459915183938243159_nat_o(Uu,Uua),aTP_Lamm_ao((Uub))) ).

% ATP.lambda_148
tff(fact_3270_ATP_Olambda__149,axiom,
    ! [Uu: fun_in6294517997321588299Heap_o,Uua: list_int,Uub: $o] : aa_o_H3316518667072757508list_o(aa_lis3548058974061443161list_o(aTP_Lamm_ar(Uu),Uua),(Uub)) = heap_T3493894765904877959list_o(vEBT_E8862244875360331003_int_o(Uu,Uua),aTP_Lamm_ao((Uub))) ).

% ATP.lambda_149
tff(fact_3271_ATP_Olambda__150,axiom,
    ! [Uu: fun_o_1668507913897003235ap_int,Uua: list_o,Uub: int] : aa_int1220172169883294302st_int(aa_lis8355408446276760467st_int(aTP_Lamm_as(Uu),Uua),Uub) = heap_T5150084250145367297st_int(vEBT_E7578125389175783381_o_int(Uu,Uua),aTP_Lamm_am(Uub)) ).

% ATP.lambda_150
tff(fact_3272_ATP_Olambda__151,axiom,
    ! [Uu: fun_na1043288745686030785ap_int,Uua: list_nat,Uub: int] : aa_int1220172169883294302st_int(aa_lis7701896087915484375st_int(aTP_Lamm_an(Uu),Uua),Uub) = heap_T5150084250145367297st_int(vEBT_E4211082316275620141at_int(Uu,Uua),aTP_Lamm_am(Uub)) ).

% ATP.lambda_151
tff(fact_3273_ATP_Olambda__152,axiom,
    ! [Uu: fun_in1280865887336605797ap_int,Uua: list_int,Uub: int] : aa_int1220172169883294302st_int(aa_lis4505422320286930811st_int(aTP_Lamm_aq(Uu),Uua),Uub) = heap_T5150084250145367297st_int(vEBT_E5210436028176674697nt_int(Uu,Uua),aTP_Lamm_am(Uub)) ).

% ATP.lambda_152
tff(fact_3274_ATP_Olambda__153,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_cq(Uu),Uua),Uub)
    <=> ( aa_nat_complex(power_power_complex(Uub),Uu) = Uua ) ) ).

% ATP.lambda_153
tff(fact_3275_ATP_Olambda__154,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_de(Uu),Uua),Uub)
    <=> ( aa_nat_complex(power_power_complex(Uub),Uua) = Uu ) ) ).

% ATP.lambda_154
tff(fact_3276_ATP_Olambda__155,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_int,Uub: nat] : aa_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_gc(Uu),Uua),Uub) = aa_real_real(minus_minus_real(aa_nat_real(Uu,Uub)),aa_real_real(times_times_real(ring_1_of_int_real(aa_nat_int(Uua,Uub))),aa_real_real(times_times_real(numeral_numeral_real(bit0(one))),pi))) ).

% ATP.lambda_155
tff(fact_3277_ATP_Olambda__156,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fi(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_156
tff(fact_3278_ATP_Olambda__157,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_bl(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_157
tff(fact_3279_ATP_Olambda__158,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_cj(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(power_power_nat(Uua),Uub)) ).

% ATP.lambda_158
tff(fact_3280_ATP_Olambda__159,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_eb(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_159
tff(fact_3281_ATP_Olambda__160,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gv(Uu),Uua),Uub) = aa_real_real(divide_divide_real(aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_160
tff(fact_3282_ATP_Olambda__161,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gn(Uu),Uua),Uub) = aa_real_real(divide_divide_real(aa_real_real(Uua,Uub)),aa_real_real(Uu,Uub)) ).

% ATP.lambda_161
tff(fact_3283_ATP_Olambda__162,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ek(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_162
tff(fact_3284_ATP_Olambda__163,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_fq(Uu),Uua),Uub) = aa_real_real(minus_minus_real(aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_163
tff(fact_3285_ATP_Olambda__164,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ex(Uu),Uua),Uub) = aa_real_real(powr_real(aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_164
tff(fact_3286_ATP_Olambda__165,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ec(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_165
tff(fact_3287_ATP_Olambda__166,axiom,
    ! [Uu: fun_real_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_eu(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_166
tff(fact_3288_ATP_Olambda__167,axiom,
    ! [Uu: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_gh(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uua,Uub)),Uu) ).

% ATP.lambda_167
tff(fact_3289_ATP_Olambda__168,axiom,
    ! [Uu: fun_VE4765431424076933413_VEBTi,Uua: fun_VE6418762861507722832st_nat,Uub: vEBT_VEBTi] : aa_VEB6430634201118579385st_nat(aa_fun7495903629695767740st_nat(aTP_Lamm_al(Uu),Uua),Uub) = heap_T3876627890315925662st_nat(aa_VEB836174949890266638_VEBTi(Uu,Uub),Uua) ).

% ATP.lambda_168
tff(fact_3290_ATP_Olambda__169,axiom,
    ! [Uu: fun_VE6418762861507722832st_nat,Uua: fun_li4609997795046405733st_nat,Uub: vEBT_VEBTi] : aa_VEB6430634201118579385st_nat(aa_fun1096608658413694055st_nat(aTP_Lamm_aj(Uu),Uua),Uub) = heap_T7479857938255823433st_nat(aa_VEB6430634201118579385st_nat(Uu,Uub),Uua) ).

% ATP.lambda_169
tff(fact_3291_ATP_Olambda__170,axiom,
    ! [Uu: fun_op4683986414774905957on_nat,Uua: fun_op1688020532487384469ap_nat,Uub: option_nat] : aa_opt1867772761376694270ap_nat(aa_fun7982982223196242428ap_nat(aTP_Lamm_ak(Uu),Uua),Uub) = heap_T2919350798565477881at_nat(aa_opt9163293676379396302on_nat(Uu,Uub),Uua) ).

% ATP.lambda_170
tff(fact_3292_ATP_Olambda__171,axiom,
    ! [Uu: fun_op1688020532487384469ap_nat,Uua: fun_na3533371890617757541ap_nat,Uub: option_nat] : aa_opt1867772761376694270ap_nat(aa_fun34237272164541356ap_nat(aTP_Lamm_ai(Uu),Uua),Uub) = heap_T7049098217575491753at_nat(aa_opt1867772761376694270ap_nat(Uu,Uub),Uua) ).

% ATP.lambda_171
tff(fact_3293_ATP_Olambda__172,axiom,
    ! [Uu: fun_real_real,Uua: real,Uub: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_ew(Uu),Uua),Uub) = aa_real_real(powr_real(aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_172
tff(fact_3294_ATP_Olambda__173,axiom,
    ! [Uu: list_nat,Uua: list_nat,Uub: list_nat] : aa_list_nat_assn(aa_lis5667661702080215587t_assn(aTP_Lamm_er(Uu),Uua),Uub) = times_times_assn(pure_assn(Uub = map_nat_nat(aTP_Lamm_eq(Uu),Uua)),top_top_assn) ).

% ATP.lambda_173
tff(fact_3295_ATP_Olambda__174,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fw(Uu),Uua),Uub) = aa_real_real(divide_divide_real(Uua),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_174
tff(fact_3296_ATP_Olambda__175,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_dc(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)) ).

% ATP.lambda_175
tff(fact_3297_ATP_Olambda__176,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_gt(Uu),Uua),Uub)
    <=> member_real(Uub,set_or1633881224788618240n_real(Uu,Uua)) ) ).

% ATP.lambda_176
tff(fact_3298_ATP_Olambda__177,axiom,
    ! [Uu: fun_real_o,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_gr(Uu),Uua),Uub)
    <=> aa_real_o(Uu,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uub),Uua)) ) ).

% ATP.lambda_177
tff(fact_3299_ATP_Olambda__178,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_gp(Uu),Uua),Uub)
    <=> aa_nat_o(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_178
tff(fact_3300_ATP_Olambda__179,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_gb(Uu),Uua),Uub) = aa_real_real(cos_real,aa_real_real(minus_minus_real(aa_nat_real(Uu,Uub)),Uua)) ).

% ATP.lambda_179
tff(fact_3301_ATP_Olambda__180,axiom,
    ! [Uu: $o,Uua: $o,Uub: code_integer,Uuc: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aTP_Lamm_hp((Uu)),(Uua)),Uub),(Uuc)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aTP_Lamm_ho((Uu)),(Uua)),(Uuc)),code_bit_cut_integer(Uub)) ).

% ATP.lambda_180
tff(fact_3302_ATP_Olambda__181,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_cl(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_ck(Uu),Uua),Uuc)),set_ord_atMost_nat(Uuc))),aa_nat_nat(power_power_nat(Uub),Uuc)) ).

% ATP.lambda_181
tff(fact_3303_ATP_Olambda__182,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_fe(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Uua)),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_182
tff(fact_3304_ATP_Olambda__183,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_fc(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_183
tff(fact_3305_ATP_Olambda__184,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_fb(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Uub),Uua)),Uuc)) ).

% ATP.lambda_184
tff(fact_3306_ATP_Olambda__185,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_fa(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uub)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Uua),Uub)),Uuc)) ).

% ATP.lambda_185
tff(fact_3307_ATP_Olambda__186,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ci(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(binomial(Uub),Uuc))),aa_nat_nat(power_power_nat(Uu),Uuc))),aa_nat_nat(power_power_nat(Uua),aa_nat_nat(minus_minus_nat(Uub),Uuc))) ).

% ATP.lambda_186
tff(fact_3308_ATP_Olambda__187,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ch(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(binomial(Uu),Uuc)),aa_nat_nat(binomial(Uua),aa_nat_nat(minus_minus_nat(Uub),Uuc))) ).

% ATP.lambda_187
tff(fact_3309_ATP_Olambda__188,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_dr(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,times_times_int(Uu,Uuc)),times_times_int(Uua,Uub)) ) ).

% ATP.lambda_188
tff(fact_3310_ATP_Olambda__189,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_dp(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,times_times_int(Uu,Uuc)),times_times_int(Uua,Uub)) ) ).

% ATP.lambda_189
tff(fact_3311_ATP_Olambda__190,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_ck(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uuc)),aa_nat_nat(Uua,aa_nat_nat(minus_minus_nat(Uub),Uuc))) ).

% ATP.lambda_190
tff(fact_3312_ATP_Olambda__191,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_dt(Uu),Uua),Uub),Uuc) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(minus_minus_int(times_times_int(Uu,Uuc),times_times_int(Uub,Uua))),times_times_int(Uua,Uuc))) ).

% ATP.lambda_191
tff(fact_3313_ATP_Olambda__192,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_dv(Uu),Uua),Uub),Uuc) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(plus_plus_int(times_times_int(Uu,Uuc),times_times_int(Uub,Uua))),times_times_int(Uua,Uuc))) ).

% ATP.lambda_192
tff(fact_3314_ATP_Olambda__193,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_dz(Uu),Uua),Uub),Uuc) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(times_times_int(Uu,Uub)),times_times_int(Uua,Uuc))) ).

% ATP.lambda_193
tff(fact_3315_ATP_Olambda__194,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_dx(Uu),Uua),Uub),Uuc) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(times_times_int(Uu,Uuc)),times_times_int(Uua,Uub))) ).

% ATP.lambda_194
tff(fact_3316_ATP_Olambda__195,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: code_integer,Uud: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aTP_Lamm_ho((Uu)),(Uua)),(Uub)),Uuc),(Uud)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aTP_Lamm_hn((Uu)),(Uua)),(Uub)),(Uud)),code_bit_cut_integer(Uuc)) ).

% ATP.lambda_195
tff(fact_3317_ATP_Olambda__196,axiom,
    ! [Uu: nat,Uua: set_nat,Uub: vEBT_VEBTi,Uuc: nat,Uud: $o] : aa_o_assn(aa_nat_fun_o_assn(aa_VEB8176250900481067561o_assn(aa_set2974300032405964540o_assn(aTP_Lamm_az(Uu),Uua),Uub),Uuc),(Uud)) = times_times_assn(aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Uu,Uua),Uub),pure_assn((Uud) = aa_set_nat_o(member_nat(Uuc),Uua))) ).

% ATP.lambda_196
tff(fact_3318_ATP_Olambda__197,axiom,
    ! [Uu: nat,Uua: fun_na6724076748692969250l_real,Uub: real,Uuc: nat,Uud: real] : aa_real_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_fd(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(minus_minus_real(aa_real_real(aa_nat_fun_real_real(Uua,Uuc),Uud)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_fc(Uua),Uuc),Uud),set_ord_lessThan_nat(aa_nat_nat(minus_minus_nat(Uu),Uuc)))),aa_real_real(times_times_real(Uub),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Uud),aa_nat_nat(minus_minus_nat(Uu),Uuc))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(minus_minus_nat(Uu),Uuc)))))) ).

% ATP.lambda_197
tff(fact_3319_ATP_Olambda__198,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: code_integer,Uue: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aTP_Lamm_hn((Uu)),(Uua)),(Uub)),(Uuc)),Uud),(Uue)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aTP_Lamm_hm((Uu)),(Uua)),(Uub)),(Uuc)),(Uue)),code_bit_cut_integer(Uud)) ).

% ATP.lambda_198
tff(fact_3320_ATP_Olambda__199,axiom,
    ! [Uu: nat,Uua: set_nat,Uub: vEBT_VEBTi,Uuc: nat,Uud: nat,Uue: option_nat] :
      aa_option_nat_assn(aa_nat435388790147128531t_assn(aa_nat336663792692944416t_assn(aa_VEB746558900233519186t_assn(aa_set2727930972950555477t_assn(aTP_Lamm_bc(Uu),Uua),Uub),Uuc),Uud),Uue) = times_times_assn(aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Uu,Uua),Uub),
        pure_assn(( ( Uue = some_nat(Uud) )
          = aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(Uua),Uuc),Uud) ))) ).

% ATP.lambda_199
tff(fact_3321_ATP_Olambda__200,axiom,
    ! [Uu: nat,Uua: set_nat,Uub: vEBT_VEBTi,Uuc: nat,Uud: nat,Uue: option_nat] :
      aa_option_nat_assn(aa_nat435388790147128531t_assn(aa_nat336663792692944416t_assn(aa_VEB746558900233519186t_assn(aa_set2727930972950555477t_assn(aTP_Lamm_ba(Uu),Uua),Uub),Uuc),Uud),Uue) = times_times_assn(aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(Uu,Uua),Uub),
        pure_assn(( ( Uue = some_nat(Uud) )
          = aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(Uua),Uuc),Uud) ))) ).

% ATP.lambda_200
tff(fact_3322_ATP_Olambda__201,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: code_integer,Uuf: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aTP_Lamm_hm((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),Uue),(Uuf)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aa_o_f1143307120959153672o_char(aTP_Lamm_hl((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uuf)),code_bit_cut_integer(Uue)) ).

% ATP.lambda_201
tff(fact_3323_ATP_Olambda__202,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: $o,Uug: code_integer] : aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aa_o_f1143307120959153672o_char(aa_o_f3038258172971468011o_char(aTP_Lamm_hk((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uuf)),Uug) = char2((Uu),(Uua),(Uub),(Uuc),(Uud),(Uue),(Uuf)) ).

% ATP.lambda_202
tff(fact_3324_ATP_Olambda__203,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: code_integer,Uug: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aa_o_f1143307120959153672o_char(aTP_Lamm_hl((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),Uuf),(Uug)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aa_o_f1143307120959153672o_char(aa_o_f3038258172971468011o_char(aTP_Lamm_hk((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uug)),code_bit_cut_integer(Uuf)) ).

% ATP.lambda_203
tff(fact_3325_ATP_Olambda__204,axiom,
    ! [Uu: $o,Uua: nat] :
      ( aa_nat_o(aa_o_fun_nat_o(aTP_Lamm_hy,(Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_204
tff(fact_3326_ATP_Olambda__205,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_cp,Uu) = Uu ).

% ATP.lambda_205
tff(fact_3327_ATP_Olambda__206,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_db,Uu) = Uu ).

% ATP.lambda_206
tff(fact_3328_ATP_Olambda__207,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_dd,Uu) = Uu ).

% ATP.lambda_207
tff(fact_3329_ATP_Olambda__208,axiom,
    ! [Uu: product_unit] : aa_Product_unit_assn(aTP_Lamm_cb,Uu) = one_one_assn ).

% ATP.lambda_208
tff(fact_3330_ATP_Olambda__209,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_bv,Uu)
    <=> $false ) ).

% ATP.lambda_209
tff(fact_3331_ATP_Olambda__210,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_gq,Uu)
    <=> $false ) ).

% ATP.lambda_210
tff(fact_3332_ATP_Olambda__211,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_hx,Uu)
    <=> $true ) ).

% ATP.lambda_211

% Helper facts (1)
tff(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: fun_real_o] :
      ( aa_real_o(P,fChoice_real(P))
      = ( ? [X9: real] : aa_real_o(P,X9) ) ) ).

% Conjectures (2)
tff(conj_0,hypothesis,
    aa_nat_o(simple8287821899582105911st_nat(aa_VEBT_VEBTi_assn(vEBT_Intf_vebt_assn(n,s),ti),aa_lis1877347954636394190st_nat(vEBT_E4213572786784670417at_nat(aTP_Lamm_aa),ysa)),aa_nat_nat(b(ysa),n)) ).

tff(conj_1,conjecture,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(bit0(bit1(bit1(one))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(bit0(bit1(one))))),aa_int_nat(nat2,archim7802044766580827645g_real(aa_real_real(log(numeral_numeral_real(bit0(one))),aa_nat_real(semiri5074537144036343181t_real,n)))))),aa_nat_nat(b(ysa),n)))),aa_nat_nat(b(cons_nat(a,ysa)),n)) ).

%------------------------------------------------------------------------------