TPTP Problem File: ITP250_1.p
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%------------------------------------------------------------------------------
% File : ITP250_1 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem VEBT_Bounds 00802_043357
% 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 : 0071_VEBT_Bounds_00802_043357 [Des22]
% Status : Theorem
% Rating : 1.00 v8.1.0
% Syntax : Number of formulae : 13028 (4308 unt;2209 typ; 0 def)
% Number of atoms : 26420 (10356 equ)
% Maximal formula atoms : 73 ( 2 avg)
% Number of connectives : 18409 (2808 ~; 503 |;1795 &)
% (2720 <=>;10583 =>; 0 <=; 0 <~>)
% Maximal formula depth : 35 ( 5 avg)
% Maximal term depth : 23 ( 2 avg)
% Number of types : 404 ( 403 usr)
% Number of type conns : 2010 (1338 >; 672 *; 0 +; 0 <<)
% Number of predicates : 61 ( 58 usr; 2 prp; 0-4 aty)
% Number of functors : 1748 (1748 usr; 468 con; 0-8 aty)
% Number of variables : 25716 (24917 !; 799 ?;25716 :)
% SPC : TF0_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 02:50:13.032
%------------------------------------------------------------------------------
% Could-be-implicit typings (403)
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tff(ty_n_t__fun_It__Extended____Nat__Oenat_Mt__HOL__Obool_J,type,
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tff(ty_n_t__fun_It__HOL__Obool_Mt__HOL__Obool_J,type,
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tff(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
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tff(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
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tff(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
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tff(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
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tff(ty_n_t__Set__Oset_It__String__Ochar_J,type,
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tff(ty_n_t__List__Olist_It__Real__Oreal_J,type,
list_real: $tType ).
tff(ty_n_t__List__Olist_It__HOL__Obool_J,type,
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tff(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
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tff(ty_n_t__List__Olist_It__Num__Onum_J,type,
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tff(ty_n_t__List__Olist_It__Nat__Onat_J,type,
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tff(ty_n_t__Set__Oset_It__Num__Onum_J,type,
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tff(ty_n_t__Code____Numeral__Ointeger,type,
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tff(ty_n_t__Extended____Nat__Oenat,type,
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% Explicit typings (1806)
tff(sy_c_ATP_058Lamm__a____,type,
aTP_Lamm_a: set_Pr1261947904930325089at_nat > fun_Pr7757163693017731481t_bool ).
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aTP_Lamm_ae: set_int > fun_int_bool ).
tff(sy_c_ATP_058Lamm__af____,type,
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tff(sy_c_ATP_058Lamm__ag____,type,
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tff(sy_c_ATP_058Lamm__ah____,type,
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aTP_Lamm_ak: fun_int_fun_int_int ).
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aTP_Lamm_au: set_int > fun_na7919044170365968317t_bool ).
tff(sy_c_ATP_058Lamm__av____,type,
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tff(sy_c_ATP_058Lamm__ax____,type,
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aTP_Lamm_bb: fun_complex_complex ).
tff(sy_c_ATP_058Lamm__bc____,type,
aTP_Lamm_bc: fun_real_real ).
tff(sy_c_ATP_058Lamm__bd____,type,
aTP_Lamm_bd: fun_rat_rat ).
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aTP_Lamm_be: fun_nat_nat ).
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tff(sy_c_ATP_058Lamm__bg____,type,
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tff(sy_c_ATP_058Lamm__bh____,type,
aTP_Lamm_bh: fun_nat_fun_nat_bool ).
tff(sy_c_ATP_058Lamm__bi____,type,
aTP_Lamm_bi: nat > fun_real_bool ).
tff(sy_c_ATP_058Lamm__bj____,type,
aTP_Lamm_bj: nat > fun_complex_bool ).
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aTP_Lamm_bk: int > fun_int_fun_int_bool ).
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aTP_Lamm_bl: int > fun_int_fun_int_bool ).
tff(sy_c_ATP_058Lamm__bm____,type,
aTP_Lamm_bm: int > fun_int_fun_int_bool ).
tff(sy_c_ATP_058Lamm__bn____,type,
aTP_Lamm_bn: set_real > fun_fu914841811607101702l_bool ).
tff(sy_c_ATP_058Lamm__bo____,type,
aTP_Lamm_bo: set_real > fun_fu7100649987335273288l_bool ).
tff(sy_c_ATP_058Lamm__bp____,type,
aTP_Lamm_bp: set_nat > fun_fu4346542433904371390t_bool ).
tff(sy_c_ATP_058Lamm__bq____,type,
aTP_Lamm_bq: set_nat > fun_fu8382797250367883996t_bool ).
tff(sy_c_ATP_058Lamm__br____,type,
aTP_Lamm_br: set_int > fun_fu7152481525987436038t_bool ).
tff(sy_c_ATP_058Lamm__bs____,type,
aTP_Lamm_bs: set_int > fun_fu5129045444369776328t_bool ).
tff(sy_c_ATP_058Lamm__bt____,type,
aTP_Lamm_bt: set_complex > fun_fu5519326811792979714x_bool ).
tff(sy_c_ATP_058Lamm__bu____,type,
aTP_Lamm_bu: set_complex > fun_fu3191020621175015362x_bool ).
tff(sy_c_ATP_058Lamm__bv____,type,
aTP_Lamm_bv: set_real > fun_fu7376312551830202376l_bool ).
tff(sy_c_ATP_058Lamm__bw____,type,
aTP_Lamm_bw: set_real > fun_fu2568719150003003596l_bool ).
tff(sy_c_ATP_058Lamm__bx____,type,
aTP_Lamm_bx: set_nat > fun_fu7195134416904984768t_bool ).
tff(sy_c_ATP_058Lamm__by____,type,
aTP_Lamm_by: set_nat > fun_fu8639250699203096928t_bool ).
tff(sy_c_ATP_058Lamm__bz____,type,
aTP_Lamm_bz: set_int > fun_fu549321512867696136t_bool ).
tff(sy_c_ATP_058Lamm__ca____,type,
aTP_Lamm_ca: set_int > fun_fu3401574108658714444t_bool ).
tff(sy_c_ATP_058Lamm__cb____,type,
aTP_Lamm_cb: set_complex > fun_fu6356453956411242500x_bool ).
tff(sy_c_ATP_058Lamm__cc____,type,
aTP_Lamm_cc: set_complex > fun_fu5995210064384465734x_bool ).
tff(sy_c_ATP_058Lamm__cd____,type,
aTP_Lamm_cd: set_real > fun_fu2277326744902863452l_bool ).
tff(sy_c_ATP_058Lamm__ce____,type,
aTP_Lamm_ce: set_real > fun_fu6750725090257165812l_bool ).
tff(sy_c_ATP_058Lamm__cf____,type,
aTP_Lamm_cf: set_nat > fun_fu3052598110916186900t_bool ).
tff(sy_c_ATP_058Lamm__cg____,type,
aTP_Lamm_cg: set_nat > fun_fu2971308621445215880t_bool ).
tff(sy_c_ATP_058Lamm__ch____,type,
aTP_Lamm_ch: fun_complex_complex ).
tff(sy_c_ATP_058Lamm__ci____,type,
aTP_Lamm_ci: fun_real_real ).
tff(sy_c_ATP_058Lamm__cj____,type,
aTP_Lamm_cj: fun_rat_rat ).
tff(sy_c_ATP_058Lamm__ck____,type,
aTP_Lamm_ck: fun_nat_nat ).
tff(sy_c_ATP_058Lamm__cl____,type,
aTP_Lamm_cl: fun_int_int ).
tff(sy_c_ATP_058Lamm__cm____,type,
aTP_Lamm_cm: set_real > fun_fu914841811607101702l_bool ).
tff(sy_c_ATP_058Lamm__cn____,type,
aTP_Lamm_cn: set_real > fun_fu7100649987335273288l_bool ).
tff(sy_c_ATP_058Lamm__co____,type,
aTP_Lamm_co: set_nat > fun_fu4346542433904371390t_bool ).
tff(sy_c_ATP_058Lamm__cp____,type,
aTP_Lamm_cp: set_nat > fun_fu8382797250367883996t_bool ).
tff(sy_c_ATP_058Lamm__cq____,type,
aTP_Lamm_cq: set_int > fun_fu7152481525987436038t_bool ).
tff(sy_c_ATP_058Lamm__cr____,type,
aTP_Lamm_cr: set_int > fun_fu5129045444369776328t_bool ).
tff(sy_c_ATP_058Lamm__cs____,type,
aTP_Lamm_cs: set_complex > fun_fu5519326811792979714x_bool ).
tff(sy_c_ATP_058Lamm__ct____,type,
aTP_Lamm_ct: set_complex > fun_fu3191020621175015362x_bool ).
tff(sy_c_ATP_058Lamm__cu____,type,
aTP_Lamm_cu: set_real > fun_fu7376312551830202376l_bool ).
tff(sy_c_ATP_058Lamm__cv____,type,
aTP_Lamm_cv: set_real > fun_fu2568719150003003596l_bool ).
tff(sy_c_ATP_058Lamm__cw____,type,
aTP_Lamm_cw: set_nat > fun_fu7195134416904984768t_bool ).
tff(sy_c_ATP_058Lamm__cx____,type,
aTP_Lamm_cx: set_nat > fun_fu8639250699203096928t_bool ).
tff(sy_c_ATP_058Lamm__cy____,type,
aTP_Lamm_cy: set_int > fun_fu549321512867696136t_bool ).
tff(sy_c_ATP_058Lamm__cz____,type,
aTP_Lamm_cz: set_int > fun_fu3401574108658714444t_bool ).
tff(sy_c_ATP_058Lamm__da____,type,
aTP_Lamm_da: set_complex > fun_fu6356453956411242500x_bool ).
tff(sy_c_ATP_058Lamm__db____,type,
aTP_Lamm_db: set_complex > fun_fu5995210064384465734x_bool ).
tff(sy_c_ATP_058Lamm__dc____,type,
aTP_Lamm_dc: set_real > fun_fu2277326744902863452l_bool ).
tff(sy_c_ATP_058Lamm__dd____,type,
aTP_Lamm_dd: set_real > fun_fu6750725090257165812l_bool ).
tff(sy_c_ATP_058Lamm__de____,type,
aTP_Lamm_de: set_nat > fun_fu3052598110916186900t_bool ).
tff(sy_c_ATP_058Lamm__df____,type,
aTP_Lamm_df: set_nat > fun_fu2971308621445215880t_bool ).
tff(sy_c_ATP_058Lamm__dg____,type,
aTP_Lamm_dg: nat > fun_co1378794544173401837x_bool ).
tff(sy_c_ATP_058Lamm__dh____,type,
aTP_Lamm_dh: fun_nat_bool > fun_nat_fun_nat_bool ).
tff(sy_c_ATP_058Lamm__di____,type,
aTP_Lamm_di: fun_nat_nat > fun_nat_fun_nat_bool ).
tff(sy_c_ATP_058Lamm__dj____,type,
aTP_Lamm_dj: fun_in1696471567231710611nt_int ).
tff(sy_c_ATP_058Lamm__dk____,type,
aTP_Lamm_dk: fun_na4138050610385705875at_nat ).
tff(sy_c_ATP_058Lamm__dl____,type,
aTP_Lamm_dl: fun_Co2470855966376689427nteger ).
tff(sy_c_ATP_058Lamm__dm____,type,
aTP_Lamm_dm: fun_in1696471567231710611nt_int ).
tff(sy_c_ATP_058Lamm__dn____,type,
aTP_Lamm_dn: fun_na4138050610385705875at_nat ).
tff(sy_c_ATP_058Lamm__do____,type,
aTP_Lamm_do: fun_Co2470855966376689427nteger ).
tff(sy_c_ATP_058Lamm__dp____,type,
aTP_Lamm_dp: fun_co1378794544173401837x_bool ).
tff(sy_c_ATP_058Lamm__dq____,type,
aTP_Lamm_dq: fun_re1255401033344146673l_bool ).
tff(sy_c_ATP_058Lamm__dr____,type,
aTP_Lamm_dr: fun_int_fun_int_bool ).
tff(sy_c_ATP_058Lamm__ds____,type,
aTP_Lamm_ds: fun_Co8310303586393106575r_bool ).
tff(sy_c_ATP_058Lamm__dt____,type,
aTP_Lamm_dt: fun_nat_fun_nat_bool ).
tff(sy_c_ATP_058Lamm__du____,type,
aTP_Lamm_du: fun_Pr2477239776880934807at_nat > fun_na1668100129766719005at_nat ).
tff(sy_c_ATP_058Lamm__dv____,type,
aTP_Lamm_dv: fun_Pr248683899324676269t_bool > fun_na4786964484885125043t_bool ).
tff(sy_c_ATP_058Lamm__dw____,type,
aTP_Lamm_dw: fun_Pr2478283820680522243nt_int > fun_in1696471567231710611nt_int ).
tff(sy_c_ATP_058Lamm__dx____,type,
aTP_Lamm_dx: fun_Pr803113274115125473t_bool > fun_int_fun_int_bool ).
tff(sy_c_ATP_058Lamm__dy____,type,
aTP_Lamm_dy: fun_Pr982841511892225912nt_int > fun_int_fun_int_int ).
tff(sy_c_ATP_058Lamm__dz____,type,
aTP_Lamm_dz: num > fun_na4138050610385705875at_nat ).
tff(sy_c_ATP_058Lamm__ea____,type,
aTP_Lamm_ea: num > fun_in1696471567231710611nt_int ).
tff(sy_c_ATP_058Lamm__eb____,type,
aTP_Lamm_eb: num > fun_Co2470855966376689427nteger ).
tff(sy_c_ATP_058Lamm__ec____,type,
aTP_Lamm_ec: fun_na4138050610385705875at_nat ).
tff(sy_c_ATP_058Lamm__ed____,type,
aTP_Lamm_ed: fun_int_fun_int_int ).
tff(sy_c_ATP_058Lamm__ee____,type,
aTP_Lamm_ee: int > fun_int_fun_int_bool ).
tff(sy_c_ATP_058Lamm__ef____,type,
aTP_Lamm_ef: int > fun_int_fun_int_bool ).
tff(sy_c_ATP_058Lamm__eg____,type,
aTP_Lamm_eg: fun_nat_fun_nat_bool ).
tff(sy_c_ATP_058Lamm__eh____,type,
aTP_Lamm_eh: fun_int_int > fun_int_int ).
tff(sy_c_ATP_058Lamm__ei____,type,
aTP_Lamm_ei: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__ej____,type,
aTP_Lamm_ej: int > fun_fu1143798476471108391nt_int ).
tff(sy_c_ATP_058Lamm__ek____,type,
aTP_Lamm_ek: complex > fun_fu5956788528830221615omplex ).
tff(sy_c_ATP_058Lamm__el____,type,
aTP_Lamm_el: nat > fun_fu7002938529518989751at_nat ).
tff(sy_c_ATP_058Lamm__em____,type,
aTP_Lamm_em: real > fun_fu6899955116716589935t_real ).
tff(sy_c_ATP_058Lamm__en____,type,
aTP_Lamm_en: fun_int_int > fun_int_fun_int_int ).
tff(sy_c_ATP_058Lamm__eo____,type,
aTP_Lamm_eo: fun_complex_complex > fun_co7673397065115134084omplex ).
tff(sy_c_ATP_058Lamm__ep____,type,
aTP_Lamm_ep: fun_nat_nat > fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__eq____,type,
aTP_Lamm_eq: fun_nat_real > fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__er____,type,
aTP_Lamm_er: fun_int_int > fun_fu923337204662071120nt_int ).
tff(sy_c_ATP_058Lamm__es____,type,
aTP_Lamm_es: fun_int_int > fun_fu3739269683040691718nt_int ).
tff(sy_c_ATP_058Lamm__et____,type,
aTP_Lamm_et: fun_complex_complex > fun_fu8998703528969179354omplex ).
tff(sy_c_ATP_058Lamm__eu____,type,
aTP_Lamm_eu: fun_complex_complex > fun_fu8084223252616112272omplex ).
tff(sy_c_ATP_058Lamm__ev____,type,
aTP_Lamm_ev: fun_nat_nat > fun_fu4413992040155564420at_nat ).
tff(sy_c_ATP_058Lamm__ew____,type,
aTP_Lamm_ew: fun_nat_nat > fun_fu2454420574033104442at_nat ).
tff(sy_c_ATP_058Lamm__ex____,type,
aTP_Lamm_ex: fun_nat_real > fun_fu163975356400620604t_real ).
tff(sy_c_ATP_058Lamm__ey____,type,
aTP_Lamm_ey: fun_nat_real > fun_fu3306995184869731826t_real ).
tff(sy_c_ATP_058Lamm__ez____,type,
aTP_Lamm_ez: fun_int_int > fun_fu1143798476471108391nt_int ).
tff(sy_c_ATP_058Lamm__fa____,type,
aTP_Lamm_fa: fun_complex_complex > fun_fu5956788528830221615omplex ).
tff(sy_c_ATP_058Lamm__fb____,type,
aTP_Lamm_fb: fun_nat_nat > fun_fu7002938529518989751at_nat ).
tff(sy_c_ATP_058Lamm__fc____,type,
aTP_Lamm_fc: fun_nat_real > fun_fu6899955116716589935t_real ).
tff(sy_c_ATP_058Lamm__fd____,type,
aTP_Lamm_fd: fun_int_int > fun_fu1143798476471108391nt_int ).
tff(sy_c_ATP_058Lamm__fe____,type,
aTP_Lamm_fe: fun_complex_complex > fun_fu5956788528830221615omplex ).
tff(sy_c_ATP_058Lamm__ff____,type,
aTP_Lamm_ff: fun_nat_real > fun_fu6899955116716589935t_real ).
tff(sy_c_ATP_058Lamm__fg____,type,
aTP_Lamm_fg: fun_complex_complex > fun_co7673397065115134084omplex ).
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_int_int > fun_int_fun_int_int ).
tff(sy_c_ATP_058Lamm__fj____,type,
aTP_Lamm_fj: fun_nat_nat > fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__fk____,type,
aTP_Lamm_fk: fun_real_complex > fun_real_bool ).
tff(sy_c_ATP_058Lamm__fl____,type,
aTP_Lamm_fl: fun_int_complex > fun_int_bool ).
tff(sy_c_ATP_058Lamm__fm____,type,
aTP_Lamm_fm: fun_real_real > fun_real_bool ).
tff(sy_c_ATP_058Lamm__fn____,type,
aTP_Lamm_fn: fun_int_real > fun_int_bool ).
tff(sy_c_ATP_058Lamm__fo____,type,
aTP_Lamm_fo: fun_complex_real > fun_complex_bool ).
tff(sy_c_ATP_058Lamm__fp____,type,
aTP_Lamm_fp: fun_real_rat > fun_real_bool ).
tff(sy_c_ATP_058Lamm__fq____,type,
aTP_Lamm_fq: fun_int_rat > fun_int_bool ).
tff(sy_c_ATP_058Lamm__fr____,type,
aTP_Lamm_fr: fun_complex_rat > fun_complex_bool ).
tff(sy_c_ATP_058Lamm__fs____,type,
aTP_Lamm_fs: fun_real_nat > fun_real_bool ).
tff(sy_c_ATP_058Lamm__ft____,type,
aTP_Lamm_ft: fun_int_nat > fun_int_bool ).
tff(sy_c_ATP_058Lamm__fu____,type,
aTP_Lamm_fu: fun_co8153928105989009288nteger > fun_fu1400560744564786701nteger ).
tff(sy_c_ATP_058Lamm__fv____,type,
aTP_Lamm_fv: fun_re6978948838062957830nteger > fun_fu8643405727515436425nteger ).
tff(sy_c_ATP_058Lamm__fw____,type,
aTP_Lamm_fw: fun_nat_Code_integer > fun_fu2499394402704830801nteger ).
tff(sy_c_ATP_058Lamm__fx____,type,
aTP_Lamm_fx: fun_int_Code_integer > fun_fu3628599891403802633nteger ).
tff(sy_c_ATP_058Lamm__fy____,type,
aTP_Lamm_fy: fun_complex_real > fun_fu7714627026455246123x_real ).
tff(sy_c_ATP_058Lamm__fz____,type,
aTP_Lamm_fz: fun_real_real > fun_fu3077872570260440615l_real ).
tff(sy_c_ATP_058Lamm__ga____,type,
aTP_Lamm_ga: fun_int_real > fun_fu121957152452960295t_real ).
tff(sy_c_ATP_058Lamm__gb____,type,
aTP_Lamm_gb: fun_complex_rat > fun_fu2990765671291504771ex_rat ).
tff(sy_c_ATP_058Lamm__gc____,type,
aTP_Lamm_gc: fun_real_rat > fun_fu2598743018846156927al_rat ).
tff(sy_c_ATP_058Lamm__gd____,type,
aTP_Lamm_gd: fun_nat_rat > fun_fu8328862064278946247at_rat ).
tff(sy_c_ATP_058Lamm__ge____,type,
aTP_Lamm_ge: fun_nat_complex > fun_nat_complex ).
tff(sy_c_ATP_058Lamm__gf____,type,
aTP_Lamm_gf: fun_nat_rat > fun_nat_rat ).
tff(sy_c_ATP_058Lamm__gg____,type,
aTP_Lamm_gg: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__gh____,type,
aTP_Lamm_gh: fun_nat_complex > fun_fu40286998686754419omplex ).
tff(sy_c_ATP_058Lamm__gi____,type,
aTP_Lamm_gi: fun_nat_rat > fun_fu8328862064278946247at_rat ).
tff(sy_c_ATP_058Lamm__gj____,type,
aTP_Lamm_gj: fun_nat_real > fun_fu6899955116716589935t_real ).
tff(sy_c_ATP_058Lamm__gk____,type,
aTP_Lamm_gk: fun_Pr2602117396471556196at_nat > fun_fu7732904369943091395at_nat ).
tff(sy_c_ATP_058Lamm__gl____,type,
aTP_Lamm_gl: fun_complex_nat > fun_fu705315015666092147ex_nat ).
tff(sy_c_ATP_058Lamm__gm____,type,
aTP_Lamm_gm: fun_real_nat > fun_fu3214020956574444655al_nat ).
tff(sy_c_ATP_058Lamm__gn____,type,
aTP_Lamm_gn: fun_int_nat > fun_fu8905809029841901679nt_nat ).
tff(sy_c_ATP_058Lamm__go____,type,
aTP_Lamm_go: fun_nat_nat > fun_fu7002938529518989751at_nat ).
tff(sy_c_ATP_058Lamm__gp____,type,
aTP_Lamm_gp: fun_nat_nat > fun_nat_nat ).
tff(sy_c_ATP_058Lamm__gq____,type,
aTP_Lamm_gq: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__gr____,type,
aTP_Lamm_gr: fun_nat_nat > fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__gs____,type,
aTP_Lamm_gs: fun_nat_real > fun_nat_fun_nat_real ).
tff(sy_c_ATP_058Lamm__gt____,type,
aTP_Lamm_gt: complex > fun_na1142814060743658952omplex ).
tff(sy_c_ATP_058Lamm__gu____,type,
aTP_Lamm_gu: rat > fun_nat_fun_nat_rat ).
tff(sy_c_ATP_058Lamm__gv____,type,
aTP_Lamm_gv: int > fun_nat_fun_nat_int ).
tff(sy_c_ATP_058Lamm__gw____,type,
aTP_Lamm_gw: real > fun_nat_fun_nat_real ).
tff(sy_c_ATP_058Lamm__gx____,type,
aTP_Lamm_gx: fun_nat_nat > fun_na3244735409550470327at_nat ).
tff(sy_c_ATP_058Lamm__gy____,type,
aTP_Lamm_gy: fun_nat_real > fun_na7718645245721260691t_real ).
tff(sy_c_ATP_058Lamm__gz____,type,
aTP_Lamm_gz: fun_nat_rat > fun_nat_rat ).
tff(sy_c_ATP_058Lamm__ha____,type,
aTP_Lamm_ha: fun_nat_int > fun_nat_int ).
tff(sy_c_ATP_058Lamm__hb____,type,
aTP_Lamm_hb: fun_nat_rat > fun_nat_rat ).
tff(sy_c_ATP_058Lamm__hc____,type,
aTP_Lamm_hc: fun_nat_int > fun_nat_int ).
tff(sy_c_ATP_058Lamm__hd____,type,
aTP_Lamm_hd: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__he____,type,
aTP_Lamm_he: fun_nat_complex > fun_nat_complex ).
tff(sy_c_ATP_058Lamm__hf____,type,
aTP_Lamm_hf: fun_nat_rat > fun_nat_rat ).
tff(sy_c_ATP_058Lamm__hg____,type,
aTP_Lamm_hg: fun_nat_int > fun_nat_int ).
tff(sy_c_ATP_058Lamm__hh____,type,
aTP_Lamm_hh: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__hi____,type,
aTP_Lamm_hi: fun_nat_rat > fun_nat_rat ).
tff(sy_c_ATP_058Lamm__hj____,type,
aTP_Lamm_hj: fun_nat_int > fun_nat_int ).
tff(sy_c_ATP_058Lamm__hk____,type,
aTP_Lamm_hk: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__hl____,type,
aTP_Lamm_hl: fun_nat_rat > fun_nat_rat ).
tff(sy_c_ATP_058Lamm__hm____,type,
aTP_Lamm_hm: fun_nat_int > fun_nat_int ).
tff(sy_c_ATP_058Lamm__hn____,type,
aTP_Lamm_hn: fun_nat_nat > fun_nat_nat ).
tff(sy_c_ATP_058Lamm__ho____,type,
aTP_Lamm_ho: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__hp____,type,
aTP_Lamm_hp: nat > fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__hq____,type,
aTP_Lamm_hq: int > fun_int_fun_nat_int ).
tff(sy_c_ATP_058Lamm__hr____,type,
aTP_Lamm_hr: complex > fun_co3258106519541737638omplex ).
tff(sy_c_ATP_058Lamm__hs____,type,
aTP_Lamm_hs: rat > fun_rat_fun_nat_rat ).
tff(sy_c_ATP_058Lamm__ht____,type,
aTP_Lamm_ht: nat > fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__hu____,type,
aTP_Lamm_hu: real > fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__hv____,type,
aTP_Lamm_hv: fun_nat_fun_nat_int ).
tff(sy_c_ATP_058Lamm__hw____,type,
aTP_Lamm_hw: fun_nat_fun_nat_real ).
tff(sy_c_ATP_058Lamm__hx____,type,
aTP_Lamm_hx: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__hy____,type,
aTP_Lamm_hy: fun_na1142814060743658952omplex ).
tff(sy_c_ATP_058Lamm__hz____,type,
aTP_Lamm_hz: fun_nat_fun_nat_rat ).
tff(sy_c_ATP_058Lamm__ia____,type,
aTP_Lamm_ia: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__ib____,type,
aTP_Lamm_ib: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__ic____,type,
aTP_Lamm_ic: complex > fun_co6883864892131739507omplex ).
tff(sy_c_ATP_058Lamm__id____,type,
aTP_Lamm_id: complex > fun_co6883864892131739507omplex ).
tff(sy_c_ATP_058Lamm__ie____,type,
aTP_Lamm_ie: rat > fun_ra5513060666784058567at_rat ).
tff(sy_c_ATP_058Lamm__if____,type,
aTP_Lamm_if: rat > fun_ra5513060666784058567at_rat ).
tff(sy_c_ATP_058Lamm__ig____,type,
aTP_Lamm_ig: real > fun_re4756235295228751087t_real ).
tff(sy_c_ATP_058Lamm__ih____,type,
aTP_Lamm_ih: real > fun_re4756235295228751087t_real ).
tff(sy_c_ATP_058Lamm__ii____,type,
aTP_Lamm_ii: fun_rat_fun_rat_bool ).
tff(sy_c_ATP_058Lamm__ij____,type,
aTP_Lamm_ij: fun_num_fun_num_bool ).
tff(sy_c_ATP_058Lamm__ik____,type,
aTP_Lamm_ik: fun_int_fun_int_bool ).
tff(sy_c_ATP_058Lamm__il____,type,
aTP_Lamm_il: fun_re1255401033344146673l_bool ).
tff(sy_c_ATP_058Lamm__im____,type,
aTP_Lamm_im: fun_nat_nat > fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__in____,type,
aTP_Lamm_in: fun_nat_real > fun_nat_fun_nat_real ).
tff(sy_c_ATP_058Lamm__io____,type,
aTP_Lamm_io: fun_nat_rat > fun_nat_rat ).
tff(sy_c_ATP_058Lamm__ip____,type,
aTP_Lamm_ip: fun_nat_int > fun_nat_int ).
tff(sy_c_ATP_058Lamm__iq____,type,
aTP_Lamm_iq: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__ir____,type,
aTP_Lamm_ir: fun_nat_int > fun_int_fun_nat_int ).
tff(sy_c_ATP_058Lamm__is____,type,
aTP_Lamm_is: fun_nat_complex > fun_co3258106519541737638omplex ).
tff(sy_c_ATP_058Lamm__it____,type,
aTP_Lamm_it: fun_nat_rat > fun_rat_fun_nat_rat ).
tff(sy_c_ATP_058Lamm__iu____,type,
aTP_Lamm_iu: fun_nat_real > fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__iv____,type,
aTP_Lamm_iv: complex > fun_co6883864892131739507omplex ).
tff(sy_c_ATP_058Lamm__iw____,type,
aTP_Lamm_iw: complex > fun_co6883864892131739507omplex ).
tff(sy_c_ATP_058Lamm__ix____,type,
aTP_Lamm_ix: rat > fun_ra5513060666784058567at_rat ).
tff(sy_c_ATP_058Lamm__iy____,type,
aTP_Lamm_iy: rat > fun_ra5513060666784058567at_rat ).
tff(sy_c_ATP_058Lamm__iz____,type,
aTP_Lamm_iz: int > fun_in6793278964266839919at_int ).
tff(sy_c_ATP_058Lamm__ja____,type,
aTP_Lamm_ja: int > fun_in6793278964266839919at_int ).
tff(sy_c_ATP_058Lamm__jb____,type,
aTP_Lamm_jb: real > fun_re4756235295228751087t_real ).
tff(sy_c_ATP_058Lamm__jc____,type,
aTP_Lamm_jc: real > fun_re4756235295228751087t_real ).
tff(sy_c_ATP_058Lamm__jd____,type,
aTP_Lamm_jd: complex > fun_na6632006986671646835omplex ).
tff(sy_c_ATP_058Lamm__je____,type,
aTP_Lamm_je: rat > fun_na1148497095348402375at_rat ).
tff(sy_c_ATP_058Lamm__jf____,type,
aTP_Lamm_jf: int > fun_na2382631619178260847at_int ).
tff(sy_c_ATP_058Lamm__jg____,type,
aTP_Lamm_jg: real > fun_na5225841814499866095t_real ).
tff(sy_c_ATP_058Lamm__jh____,type,
aTP_Lamm_jh: complex > fun_na6632006986671646835omplex ).
tff(sy_c_ATP_058Lamm__ji____,type,
aTP_Lamm_ji: rat > fun_na1148497095348402375at_rat ).
tff(sy_c_ATP_058Lamm__jj____,type,
aTP_Lamm_jj: int > fun_na2382631619178260847at_int ).
tff(sy_c_ATP_058Lamm__jk____,type,
aTP_Lamm_jk: real > fun_na5225841814499866095t_real ).
tff(sy_c_ATP_058Lamm__jl____,type,
aTP_Lamm_jl: complex > fun_na1142814060743658952omplex ).
tff(sy_c_ATP_058Lamm__jm____,type,
aTP_Lamm_jm: rat > fun_nat_fun_nat_rat ).
tff(sy_c_ATP_058Lamm__jn____,type,
aTP_Lamm_jn: int > fun_nat_fun_nat_int ).
tff(sy_c_ATP_058Lamm__jo____,type,
aTP_Lamm_jo: real > fun_nat_fun_nat_real ).
tff(sy_c_ATP_058Lamm__jp____,type,
aTP_Lamm_jp: fun_nat_real > fun_fu6899955116716589935t_real ).
tff(sy_c_ATP_058Lamm__jq____,type,
aTP_Lamm_jq: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__jr____,type,
aTP_Lamm_jr: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__js____,type,
aTP_Lamm_js: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__jt____,type,
aTP_Lamm_jt: fun_nat_complex > fun_nat_real ).
tff(sy_c_ATP_058Lamm__ju____,type,
aTP_Lamm_ju: fun_complex_complex > fun_complex_real ).
tff(sy_c_ATP_058Lamm__jv____,type,
aTP_Lamm_jv: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__jw____,type,
aTP_Lamm_jw: fun_nat_real ).
tff(sy_c_ATP_058Lamm__jx____,type,
aTP_Lamm_jx: fun_nat_real ).
tff(sy_c_ATP_058Lamm__jy____,type,
aTP_Lamm_jy: complex > fun_fu40286998686754419omplex ).
tff(sy_c_ATP_058Lamm__jz____,type,
aTP_Lamm_jz: fun_nat_complex > fun_co3258106519541737638omplex ).
tff(sy_c_ATP_058Lamm__ka____,type,
aTP_Lamm_ka: fun_nat_int > fun_fu8464300013002972271at_int ).
tff(sy_c_ATP_058Lamm__kb____,type,
aTP_Lamm_kb: fun_nat_real > fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__kc____,type,
aTP_Lamm_kc: fun_nat_real > fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__kd____,type,
aTP_Lamm_kd: fun_nat_real > fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__ke____,type,
aTP_Lamm_ke: fun_nat_complex > fun_co3258106519541737638omplex ).
tff(sy_c_ATP_058Lamm__kf____,type,
aTP_Lamm_kf: fun_nat_complex > fun_co8378349014452122532t_real ).
tff(sy_c_ATP_058Lamm__kg____,type,
aTP_Lamm_kg: fun_nat_int > fun_nat_int ).
tff(sy_c_ATP_058Lamm__kh____,type,
aTP_Lamm_kh: fun_nat_complex > fun_co3258106519541737638omplex ).
tff(sy_c_ATP_058Lamm__ki____,type,
aTP_Lamm_ki: fun_nat_real > fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__kj____,type,
aTP_Lamm_kj: fun_nat_complex > fun_na6632006986671646835omplex ).
tff(sy_c_ATP_058Lamm__kk____,type,
aTP_Lamm_kk: fun_nat_real > fun_na5225841814499866095t_real ).
tff(sy_c_ATP_058Lamm__kl____,type,
aTP_Lamm_kl: fun_nat_complex > fun_na1142814060743658952omplex ).
tff(sy_c_ATP_058Lamm__km____,type,
aTP_Lamm_km: real > fun_fu7631928937237476593t_real ).
tff(sy_c_ATP_058Lamm__kn____,type,
aTP_Lamm_kn: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__ko____,type,
aTP_Lamm_ko: vEBT_VEBT > fun_fu4141963935195952137T_real ).
tff(sy_c_ATP_058Lamm__kp____,type,
aTP_Lamm_kp: complex > fun_fu709861064203714431x_real ).
tff(sy_c_ATP_058Lamm__kq____,type,
aTP_Lamm_kq: vEBT_VEBT > fun_fu4463953228700081077BT_rat ).
tff(sy_c_ATP_058Lamm__kr____,type,
aTP_Lamm_kr: complex > fun_fu6638204120336560003ex_rat ).
tff(sy_c_ATP_058Lamm__ks____,type,
aTP_Lamm_ks: vEBT_VEBT > fun_fu5586652545743564717BT_nat ).
tff(sy_c_ATP_058Lamm__kt____,type,
aTP_Lamm_kt: complex > fun_fu2513798379308225899ex_nat ).
tff(sy_c_ATP_058Lamm__ku____,type,
aTP_Lamm_ku: vEBT_VEBT > fun_fu7059288016784409353BT_int ).
tff(sy_c_ATP_058Lamm__kv____,type,
aTP_Lamm_kv: complex > fun_fu4692242605864086655ex_int ).
tff(sy_c_ATP_058Lamm__kw____,type,
aTP_Lamm_kw: int > fun_fu4328765270694171897t_real ).
tff(sy_c_ATP_058Lamm__kx____,type,
aTP_Lamm_kx: int > fun_fu1617438229336688381nt_rat ).
tff(sy_c_ATP_058Lamm__ky____,type,
aTP_Lamm_ky: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__kz____,type,
aTP_Lamm_kz: real > fun_fu3477889739401271834t_real ).
tff(sy_c_ATP_058Lamm__la____,type,
aTP_Lamm_la: real > fun_fu5589980185390252312t_real ).
tff(sy_c_ATP_058Lamm__lb____,type,
aTP_Lamm_lb: real > fun_fu4806068833880117316t_real ).
tff(sy_c_ATP_058Lamm__lc____,type,
aTP_Lamm_lc: real > fun_fu2115079063411676732t_real ).
tff(sy_c_ATP_058Lamm__ld____,type,
aTP_Lamm_ld: real > fun_fu4540635375566031512t_real ).
tff(sy_c_ATP_058Lamm__le____,type,
aTP_Lamm_le: real > fun_fu6899955116716589935t_real ).
tff(sy_c_ATP_058Lamm__lf____,type,
aTP_Lamm_lf: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__lg____,type,
aTP_Lamm_lg: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__lh____,type,
aTP_Lamm_lh: complex > fun_fu40286998686754419omplex ).
tff(sy_c_ATP_058Lamm__li____,type,
aTP_Lamm_li: real > fun_fu6899955116716589935t_real ).
tff(sy_c_ATP_058Lamm__lj____,type,
aTP_Lamm_lj: fun_nat_complex > fun_nat_complex ).
tff(sy_c_ATP_058Lamm__lk____,type,
aTP_Lamm_lk: nat > fun_fu40286998686754419omplex ).
tff(sy_c_ATP_058Lamm__ll____,type,
aTP_Lamm_ll: nat > fun_fu6899955116716589935t_real ).
tff(sy_c_ATP_058Lamm__lm____,type,
aTP_Lamm_lm: nat > fun_co3258106519541737638omplex ).
tff(sy_c_ATP_058Lamm__ln____,type,
aTP_Lamm_ln: nat > fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__lo____,type,
aTP_Lamm_lo: nat > fun_int_fun_nat_int ).
tff(sy_c_ATP_058Lamm__lp____,type,
aTP_Lamm_lp: fun_nat_real > fun_fu6899955116716589935t_real ).
tff(sy_c_ATP_058Lamm__lq____,type,
aTP_Lamm_lq: fun_nat_real > fun_se2100927751953696370t_real ).
tff(sy_c_ATP_058Lamm__lr____,type,
aTP_Lamm_lr: fun_nat_real ).
tff(sy_c_ATP_058Lamm__ls____,type,
aTP_Lamm_ls: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__lt____,type,
aTP_Lamm_lt: fun_nat_real > fun_fu6899955116716589935t_real ).
tff(sy_c_ATP_058Lamm__lu____,type,
aTP_Lamm_lu: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__lv____,type,
aTP_Lamm_lv: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__lw____,type,
aTP_Lamm_lw: complex > fun_nat_complex ).
tff(sy_c_ATP_058Lamm__lx____,type,
aTP_Lamm_lx: fun_nat_real > fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__ly____,type,
aTP_Lamm_ly: fun_nat_real > fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__lz____,type,
aTP_Lamm_lz: fun_nat_complex > fun_co3258106519541737638omplex ).
tff(sy_c_ATP_058Lamm__ma____,type,
aTP_Lamm_ma: fun_nat_complex > fun_co3258106519541737638omplex ).
tff(sy_c_ATP_058Lamm__mb____,type,
aTP_Lamm_mb: real > fun_fu6899955116716589935t_real ).
tff(sy_c_ATP_058Lamm__mc____,type,
aTP_Lamm_mc: complex > fun_fu40286998686754419omplex ).
tff(sy_c_ATP_058Lamm__md____,type,
aTP_Lamm_md: fun_int_int ).
tff(sy_c_ATP_058Lamm__me____,type,
aTP_Lamm_me: fun_real_real ).
tff(sy_c_ATP_058Lamm__mf____,type,
aTP_Lamm_mf: fun_nat_nat ).
tff(sy_c_ATP_058Lamm__mg____,type,
aTP_Lamm_mg: fun_complex_complex ).
tff(sy_c_ATP_058Lamm__mh____,type,
aTP_Lamm_mh: fun_rat_rat ).
tff(sy_c_ATP_058Lamm__mi____,type,
aTP_Lamm_mi: complex > fun_nat_complex ).
tff(sy_c_ATP_058Lamm__mj____,type,
aTP_Lamm_mj: rat > fun_nat_rat ).
tff(sy_c_ATP_058Lamm__mk____,type,
aTP_Lamm_mk: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__ml____,type,
aTP_Lamm_ml: nat > fun_nat_complex ).
tff(sy_c_ATP_058Lamm__mm____,type,
aTP_Lamm_mm: nat > fun_nat_rat ).
tff(sy_c_ATP_058Lamm__mn____,type,
aTP_Lamm_mn: fun_nat_fun_nat_real ).
tff(sy_c_ATP_058Lamm__mo____,type,
aTP_Lamm_mo: complex > fun_na1591161823130732966omplex ).
tff(sy_c_ATP_058Lamm__mp____,type,
aTP_Lamm_mp: rat > fun_nat_fun_rat_rat ).
tff(sy_c_ATP_058Lamm__mq____,type,
aTP_Lamm_mq: real > fun_na6724076748692969250l_real ).
tff(sy_c_ATP_058Lamm__mr____,type,
aTP_Lamm_mr: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__ms____,type,
aTP_Lamm_ms: complex > fun_nat_complex ).
tff(sy_c_ATP_058Lamm__mt____,type,
aTP_Lamm_mt: rat > fun_nat_rat ).
tff(sy_c_ATP_058Lamm__mu____,type,
aTP_Lamm_mu: int > fun_nat_fun_int_int ).
tff(sy_c_ATP_058Lamm__mv____,type,
aTP_Lamm_mv: real > fun_na6724076748692969250l_real ).
tff(sy_c_ATP_058Lamm__mw____,type,
aTP_Lamm_mw: complex > fun_na1591161823130732966omplex ).
tff(sy_c_ATP_058Lamm__mx____,type,
aTP_Lamm_mx: rat > fun_nat_fun_rat_rat ).
tff(sy_c_ATP_058Lamm__my____,type,
aTP_Lamm_my: nat > fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__mz____,type,
aTP_Lamm_mz: fun_nat_nat ).
tff(sy_c_ATP_058Lamm__na____,type,
aTP_Lamm_na: fun_nat_int ).
tff(sy_c_ATP_058Lamm__nb____,type,
aTP_Lamm_nb: fun_int_int ).
tff(sy_c_ATP_058Lamm__nc____,type,
aTP_Lamm_nc: real > fun_fu8384527901749275179omplex ).
tff(sy_c_ATP_058Lamm__nd____,type,
aTP_Lamm_nd: nat > fun_fu40286998686754419omplex ).
tff(sy_c_ATP_058Lamm__ne____,type,
aTP_Lamm_ne: int > fun_fu3613808716698379563omplex ).
tff(sy_c_ATP_058Lamm__nf____,type,
aTP_Lamm_nf: complex > fun_fu5956788528830221615omplex ).
tff(sy_c_ATP_058Lamm__ng____,type,
aTP_Lamm_ng: real > fun_fu3077872570260440615l_real ).
tff(sy_c_ATP_058Lamm__nh____,type,
aTP_Lamm_nh: nat > fun_fu6899955116716589935t_real ).
tff(sy_c_ATP_058Lamm__ni____,type,
aTP_Lamm_ni: int > fun_fu121957152452960295t_real ).
tff(sy_c_ATP_058Lamm__nj____,type,
aTP_Lamm_nj: complex > fun_fu7714627026455246123x_real ).
tff(sy_c_ATP_058Lamm__nk____,type,
aTP_Lamm_nk: real > fun_fu2598743018846156927al_rat ).
tff(sy_c_ATP_058Lamm__nl____,type,
aTP_Lamm_nl: nat > fun_fu8328862064278946247at_rat ).
tff(sy_c_ATP_058Lamm__nm____,type,
aTP_Lamm_nm: real > fun_fu8384527901749275179omplex ).
tff(sy_c_ATP_058Lamm__nn____,type,
aTP_Lamm_nn: nat > fun_fu40286998686754419omplex ).
tff(sy_c_ATP_058Lamm__no____,type,
aTP_Lamm_no: int > fun_fu3613808716698379563omplex ).
tff(sy_c_ATP_058Lamm__np____,type,
aTP_Lamm_np: complex > fun_fu5956788528830221615omplex ).
tff(sy_c_ATP_058Lamm__nq____,type,
aTP_Lamm_nq: real > fun_fu3077872570260440615l_real ).
tff(sy_c_ATP_058Lamm__nr____,type,
aTP_Lamm_nr: nat > fun_fu6899955116716589935t_real ).
tff(sy_c_ATP_058Lamm__ns____,type,
aTP_Lamm_ns: int > fun_fu121957152452960295t_real ).
tff(sy_c_ATP_058Lamm__nt____,type,
aTP_Lamm_nt: complex > fun_fu7714627026455246123x_real ).
tff(sy_c_ATP_058Lamm__nu____,type,
aTP_Lamm_nu: real > fun_fu2598743018846156927al_rat ).
tff(sy_c_ATP_058Lamm__nv____,type,
aTP_Lamm_nv: nat > fun_fu8328862064278946247at_rat ).
tff(sy_c_ATP_058Lamm__nw____,type,
aTP_Lamm_nw: fun_nat_nat > fun_fu7002938529518989751at_nat ).
tff(sy_c_ATP_058Lamm__nx____,type,
aTP_Lamm_nx: fun_nat_int > fun_fu8464300013002972271at_int ).
tff(sy_c_ATP_058Lamm__ny____,type,
aTP_Lamm_ny: fun_int_int > fun_fu1143798476471108391nt_int ).
tff(sy_c_ATP_058Lamm__nz____,type,
aTP_Lamm_nz: fun_nat_nat > fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__oa____,type,
aTP_Lamm_oa: fun_nat_int > fun_nat_fun_nat_int ).
tff(sy_c_ATP_058Lamm__ob____,type,
aTP_Lamm_ob: fun_int_int > fun_nat_fun_int_int ).
tff(sy_c_ATP_058Lamm__oc____,type,
aTP_Lamm_oc: fun_nat_int > fun_int_fun_nat_int ).
tff(sy_c_ATP_058Lamm__od____,type,
aTP_Lamm_od: fun_nat_fun_nat_nat > fun_nat_nat ).
tff(sy_c_ATP_058Lamm__oe____,type,
aTP_Lamm_oe: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__of____,type,
aTP_Lamm_of: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__og____,type,
aTP_Lamm_og: fun_nat_fun_nat_int > fun_nat_int ).
tff(sy_c_ATP_058Lamm__oh____,type,
aTP_Lamm_oh: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).
tff(sy_c_ATP_058Lamm__oi____,type,
aTP_Lamm_oi: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).
tff(sy_c_ATP_058Lamm__oj____,type,
aTP_Lamm_oj: fun_re1255401033344146673l_bool ).
tff(sy_c_ATP_058Lamm__ok____,type,
aTP_Lamm_ok: set_nat > fun_set_nat_bool ).
tff(sy_c_ATP_058Lamm__ol____,type,
aTP_Lamm_ol: fun_rat_fun_rat_bool ).
tff(sy_c_ATP_058Lamm__om____,type,
aTP_Lamm_om: fun_num_fun_num_bool ).
tff(sy_c_ATP_058Lamm__on____,type,
aTP_Lamm_on: fun_int_fun_int_bool ).
tff(sy_c_ATP_058Lamm__oo____,type,
aTP_Lamm_oo: fun_nat_nat > fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__op____,type,
aTP_Lamm_op: fun_nat_nat > fun_fu4413992040155564420at_nat ).
tff(sy_c_ATP_058Lamm__oq____,type,
aTP_Lamm_oq: fun_nat_nat > fun_fu4413992040155564420at_nat ).
tff(sy_c_ATP_058Lamm__or____,type,
aTP_Lamm_or: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__os____,type,
aTP_Lamm_os: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__ot____,type,
aTP_Lamm_ot: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__ou____,type,
aTP_Lamm_ou: nat > fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__ov____,type,
aTP_Lamm_ov: nat > fun_na3244735409550470327at_nat ).
tff(sy_c_ATP_058Lamm__ow____,type,
aTP_Lamm_ow: nat > fun_na3244735409550470327at_nat ).
tff(sy_c_ATP_058Lamm__ox____,type,
aTP_Lamm_ox: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__oy____,type,
aTP_Lamm_oy: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__oz____,type,
aTP_Lamm_oz: nat > fun_nat_complex ).
tff(sy_c_ATP_058Lamm__pa____,type,
aTP_Lamm_pa: fun_re1255401033344146673l_bool ).
tff(sy_c_ATP_058Lamm__pb____,type,
aTP_Lamm_pb: fun_real_bool ).
tff(sy_c_ATP_058Lamm__pc____,type,
aTP_Lamm_pc: fun_re1255401033344146673l_bool ).
tff(sy_c_ATP_058Lamm__pd____,type,
aTP_Lamm_pd: fun_real_bool ).
tff(sy_c_ATP_058Lamm__pe____,type,
aTP_Lamm_pe: fun_re1255401033344146673l_bool ).
tff(sy_c_ATP_058Lamm__pf____,type,
aTP_Lamm_pf: complex > fun_na5941084329111710283x_bool ).
tff(sy_c_ATP_058Lamm__pg____,type,
aTP_Lamm_pg: complex > fun_real_bool ).
tff(sy_c_ATP_058Lamm__ph____,type,
aTP_Lamm_ph: nat > fun_nat_fun_nat_bool ).
tff(sy_c_ATP_058Lamm__pi____,type,
aTP_Lamm_pi: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__pj____,type,
aTP_Lamm_pj: fun_int_fun_int_int ).
tff(sy_c_ATP_058Lamm__pk____,type,
aTP_Lamm_pk: set_nat > fun_nat_fun_nat_bool ).
tff(sy_c_ATP_058Lamm__pl____,type,
aTP_Lamm_pl: set_nat > fun_nat_fun_nat_bool ).
tff(sy_c_ATP_058Lamm__pm____,type,
aTP_Lamm_pm: real > fun_int_bool ).
tff(sy_c_ATP_058Lamm__pn____,type,
aTP_Lamm_pn: rat > fun_int_bool ).
tff(sy_c_ATP_058Lamm__po____,type,
aTP_Lamm_po: fun_Co1092983284983208416er_int ).
tff(sy_c_ATP_058Lamm__pp____,type,
aTP_Lamm_pp: fun_Co8521658056209057742er_num ).
tff(sy_c_ATP_058Lamm__pq____,type,
aTP_Lamm_pq: code_integer > fun_Co3025358997185726883r_bool ).
tff(sy_c_ATP_058Lamm__pr____,type,
aTP_Lamm_pr: fun_Co3583066429914935172er_nat ).
tff(sy_c_ATP_058Lamm__ps____,type,
aTP_Lamm_ps: code_integer > fun_Co2470855966376689427nteger ).
tff(sy_c_ATP_058Lamm__pt____,type,
aTP_Lamm_pt: code_integer > fun_Co2470855966376689427nteger ).
tff(sy_c_ATP_058Lamm__pu____,type,
aTP_Lamm_pu: int > fun_in2204242111582549184nt_int ).
tff(sy_c_ATP_058Lamm__pv____,type,
aTP_Lamm_pv: rat > fun_in1696471567231710611nt_int ).
tff(sy_c_ATP_058Lamm__pw____,type,
aTP_Lamm_pw: fun_nat_bool ).
tff(sy_c_ATP_058Lamm__px____,type,
aTP_Lamm_px: fun_nat_bool ).
tff(sy_c_ATP_058Lamm__py____,type,
aTP_Lamm_py: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__pz____,type,
aTP_Lamm_pz: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__qa____,type,
aTP_Lamm_qa: int > fun_in6148020733644367646t_bool ).
tff(sy_c_ATP_058Lamm__qb____,type,
aTP_Lamm_qb: rat > fun_int_fun_int_bool ).
tff(sy_c_ATP_058Lamm__qc____,type,
aTP_Lamm_qc: int > fun_in2204242111582549184nt_int ).
tff(sy_c_ATP_058Lamm__qd____,type,
aTP_Lamm_qd: rat > fun_in1696471567231710611nt_int ).
tff(sy_c_ATP_058Lamm__qe____,type,
aTP_Lamm_qe: code_integer > fun_Co2470855966376689427nteger ).
tff(sy_c_ATP_058Lamm__qf____,type,
aTP_Lamm_qf: nat > fun_nat_fun_nat_bool ).
tff(sy_c_ATP_058Lamm__qg____,type,
aTP_Lamm_qg: nat > fun_na4143055713577223196at_nat ).
tff(sy_c_ATP_058Lamm__qh____,type,
aTP_Lamm_qh: fun_na1668100129766719005at_nat ).
tff(sy_c_ATP_058Lamm__qi____,type,
aTP_Lamm_qi: fun_na4138050610385705875at_nat ).
tff(sy_c_ATP_058Lamm__qj____,type,
aTP_Lamm_qj: nat > fun_na3408937105440547634t_bool ).
tff(sy_c_ATP_058Lamm__qk____,type,
aTP_Lamm_qk: fun_na4786964484885125043t_bool ).
tff(sy_c_ATP_058Lamm__ql____,type,
aTP_Lamm_ql: nat > fun_na3408937105440547634t_bool ).
tff(sy_c_ATP_058Lamm__qm____,type,
aTP_Lamm_qm: fun_na4786964484885125043t_bool ).
tff(sy_c_ATP_058Lamm__qn____,type,
aTP_Lamm_qn: nat > fun_na4143055713577223196at_nat ).
tff(sy_c_ATP_058Lamm__qo____,type,
aTP_Lamm_qo: fun_na1668100129766719005at_nat ).
tff(sy_c_ATP_058Lamm__qp____,type,
aTP_Lamm_qp: nat > fun_na4143055713577223196at_nat ).
tff(sy_c_ATP_058Lamm__qq____,type,
aTP_Lamm_qq: fun_na1668100129766719005at_nat ).
tff(sy_c_ATP_058Lamm__qr____,type,
aTP_Lamm_qr: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__qs____,type,
aTP_Lamm_qs: num > fun_num_int ).
tff(sy_c_ATP_058Lamm__qt____,type,
aTP_Lamm_qt: fun_nat_option_num ).
tff(sy_c_ATP_058Lamm__qu____,type,
aTP_Lamm_qu: fun_num_option_num ).
tff(sy_c_ATP_058Lamm__qv____,type,
aTP_Lamm_qv: fun_num_option_num ).
tff(sy_c_ATP_058Lamm__qw____,type,
aTP_Lamm_qw: num > fun_nat_option_num ).
tff(sy_c_ATP_058Lamm__qx____,type,
aTP_Lamm_qx: num > fun_nat_option_num ).
tff(sy_c_ATP_058Lamm__qy____,type,
aTP_Lamm_qy: nat > fun_num_option_num ).
tff(sy_c_ATP_058Lamm__qz____,type,
aTP_Lamm_qz: nat > fun_num_option_num ).
tff(sy_c_ATP_058Lamm__ra____,type,
aTP_Lamm_ra: num > fun_nat_option_num ).
tff(sy_c_ATP_058Lamm__rb____,type,
aTP_Lamm_rb: fun_na7595719129879163854on_num ).
tff(sy_c_ATP_058Lamm__rc____,type,
aTP_Lamm_rc: fun_Pr803113274115125473t_bool ).
tff(sy_c_ATP_058Lamm__rd____,type,
aTP_Lamm_rd: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__re____,type,
aTP_Lamm_re: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__rf____,type,
aTP_Lamm_rf: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__rg____,type,
aTP_Lamm_rg: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__rh____,type,
aTP_Lamm_rh: fun_na6724076748692969250l_real ).
tff(sy_c_ATP_058Lamm__ri____,type,
aTP_Lamm_ri: fun_na6724076748692969250l_real ).
tff(sy_c_ATP_058Lamm__rj____,type,
aTP_Lamm_rj: fun_real_real > fun_na6724076748692969250l_real ).
tff(sy_c_ATP_058Lamm__rk____,type,
aTP_Lamm_rk: fun_re7681423839302404990l_real ).
tff(sy_c_ATP_058Lamm__rl____,type,
aTP_Lamm_rl: fun_real_real > fun_re7681423839302404990l_real ).
tff(sy_c_ATP_058Lamm__rm____,type,
aTP_Lamm_rm: fun_real_real > fun_fu3077872570260440615l_real ).
tff(sy_c_ATP_058Lamm__rn____,type,
aTP_Lamm_rn: fun_re4238438383274954274t_real > fun_na6724076748692969250l_real ).
tff(sy_c_ATP_058Lamm__ro____,type,
aTP_Lamm_ro: fun_re4238438383274954274t_real > fun_real_real ).
tff(sy_c_ATP_058Lamm__rp____,type,
aTP_Lamm_rp: fun_nat_real > fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__rq____,type,
aTP_Lamm_rq: fun_nat_real > fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__rr____,type,
aTP_Lamm_rr: fun_nat_real > fun_real_real ).
tff(sy_c_ATP_058Lamm__rs____,type,
aTP_Lamm_rs: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__rt____,type,
aTP_Lamm_rt: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).
tff(sy_c_ATP_058Lamm__ru____,type,
aTP_Lamm_ru: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).
tff(sy_c_ATP_058Lamm__rv____,type,
aTP_Lamm_rv: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).
tff(sy_c_ATP_058Lamm__rw____,type,
aTP_Lamm_rw: nat > fun_fu3612569900928539882l_real ).
tff(sy_c_ATP_058Lamm__rx____,type,
aTP_Lamm_rx: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).
tff(sy_c_ATP_058Lamm__ry____,type,
aTP_Lamm_ry: fun_real_real ).
tff(sy_c_ATP_058Lamm__rz____,type,
aTP_Lamm_rz: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__sa____,type,
aTP_Lamm_sa: fun_real_real ).
tff(sy_c_ATP_058Lamm__sb____,type,
aTP_Lamm_sb: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__sc____,type,
aTP_Lamm_sc: fun_nat_real ).
tff(sy_c_ATP_058Lamm__sd____,type,
aTP_Lamm_sd: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__se____,type,
aTP_Lamm_se: fun_nat_real ).
tff(sy_c_ATP_058Lamm__sf____,type,
aTP_Lamm_sf: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__sg____,type,
aTP_Lamm_sg: fun_nat_real ).
tff(sy_c_ATP_058Lamm__sh____,type,
aTP_Lamm_sh: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__si____,type,
aTP_Lamm_si: real > fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__sj____,type,
aTP_Lamm_sj: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__sk____,type,
aTP_Lamm_sk: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__sl____,type,
aTP_Lamm_sl: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__sm____,type,
aTP_Lamm_sm: fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__sn____,type,
aTP_Lamm_sn: fun_nat_real > fun_re4238438383274954274t_real ).
tff(sy_c_ATP_058Lamm__so____,type,
aTP_Lamm_so: fun_nat_real > fun_fu6119738925743952495t_real ).
tff(sy_c_ATP_058Lamm__sp____,type,
aTP_Lamm_sp: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__sq____,type,
aTP_Lamm_sq: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__sr____,type,
aTP_Lamm_sr: fun_nat_real > fun_nat_real ).
tff(sy_c_ATP_058Lamm__ss____,type,
aTP_Lamm_ss: fun_re7681423839302404990l_real ).
tff(sy_c_ATP_058Lamm__st____,type,
aTP_Lamm_st: nat > fun_fu3077872570260440615l_real ).
tff(sy_c_ATP_058Lamm__su____,type,
aTP_Lamm_su: fun_real_real ).
tff(sy_c_ATP_058Lamm__sv____,type,
aTP_Lamm_sv: fun_na6724076748692969250l_real ).
tff(sy_c_ATP_058Lamm__sw____,type,
aTP_Lamm_sw: fun_re7681423839302404990l_real ).
tff(sy_c_ATP_058Lamm__sx____,type,
aTP_Lamm_sx: fun_real_real > fun_real_bool ).
tff(sy_c_ATP_058Lamm__sy____,type,
aTP_Lamm_sy: fun_real_real > fun_fu7376312551830202376l_bool ).
tff(sy_c_ATP_058Lamm__sz____,type,
aTP_Lamm_sz: fun_real_real > fun_fu3077872570260440615l_real ).
tff(sy_c_ATP_058Lamm__ta____,type,
aTP_Lamm_ta: real > fun_re1255401033344146673l_bool ).
tff(sy_c_ATP_058Lamm__tb____,type,
aTP_Lamm_tb: fun_real_bool > fun_re1255401033344146673l_bool ).
tff(sy_c_ATP_058Lamm__tc____,type,
aTP_Lamm_tc: fun_real_real > fun_fu3077872570260440615l_real ).
tff(sy_c_ATP_058Lamm__td____,type,
aTP_Lamm_td: fun_nat_bool > fun_nat_bool ).
tff(sy_c_ATP_058Lamm__te____,type,
aTP_Lamm_te: fun_nat_bool > fun_nat_fun_nat_bool ).
tff(sy_c_ATP_058Lamm__tf____,type,
aTP_Lamm_tf: nat > fun_li8232715659020003717t_bool ).
tff(sy_c_ATP_058Lamm__tg____,type,
aTP_Lamm_tg: fun_real_real > fun_real_real ).
tff(sy_c_ATP_058Lamm__th____,type,
aTP_Lamm_th: fun_real_real > fun_real_real ).
tff(sy_c_ATP_058Lamm__ti____,type,
aTP_Lamm_ti: real > fun_re1255401033344146673l_bool ).
tff(sy_c_ATP_058Lamm__tj____,type,
aTP_Lamm_tj: fun_re5512370375968561900l_real ).
tff(sy_c_ATP_058Lamm__tk____,type,
aTP_Lamm_tk: real > fun_co1378794544173401837x_bool ).
tff(sy_c_ATP_058Lamm__tl____,type,
aTP_Lamm_tl: fun_re3844104906058862192omplex ).
tff(sy_c_ATP_058Lamm__tm____,type,
aTP_Lamm_tm: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__tn____,type,
aTP_Lamm_tn: fun_nat_real > fun_fu3106432010075488275t_real ).
tff(sy_c_ATP_058Lamm__to____,type,
aTP_Lamm_to: fun_re7681423839302404990l_real ).
tff(sy_c_ATP_058Lamm__tp____,type,
aTP_Lamm_tp: fun_re7681423839302404990l_real ).
tff(sy_c_ATP_058Lamm__tq____,type,
aTP_Lamm_tq: fun_nat_fun_nat_bool ).
tff(sy_c_ATP_058Lamm__tr____,type,
aTP_Lamm_tr: fun_real_bool ).
tff(sy_c_ATP_058Lamm__ts____,type,
aTP_Lamm_ts: fun_real_bool ).
tff(sy_c_ATP_058Lamm__tt____,type,
aTP_Lamm_tt: fun_nat_fun_nat_nat ).
tff(sy_c_ATP_058Lamm__tu____,type,
aTP_Lamm_tu: fun_nat_nat ).
tff(sy_c_ATP_058Lamm__tv____,type,
aTP_Lamm_tv: nat > fun_na5614638261385347865t_bool ).
tff(sy_c_ATP_058Lamm__tw____,type,
aTP_Lamm_tw: nat > fun_na5614638261385347865t_bool ).
tff(sy_c_ATP_058Lamm__tx____,type,
aTP_Lamm_tx: nat > fun_na5614638261385347865t_bool ).
tff(sy_c_ATP_058Lamm__ty____,type,
aTP_Lamm_ty: rat > fun_Pr803113274115125473t_bool ).
tff(sy_c_ATP_058Lamm__tz____,type,
aTP_Lamm_tz: fun_fu6579618355643354601at_rat ).
tff(sy_c_ATP_058Lamm__ua____,type,
aTP_Lamm_ua: rat > fun_nat_rat ).
tff(sy_c_ATP_058Lamm__ub____,type,
aTP_Lamm_ub: fun_nat_rat > fun_fu8328862064278946247at_rat ).
tff(sy_c_ATP_058Lamm__uc____,type,
aTP_Lamm_uc: fun_fu8328862064278946247at_rat ).
tff(sy_c_ATP_058Lamm__ud____,type,
aTP_Lamm_ud: fun_fu6579618355643354601at_rat ).
tff(sy_c_ATP_058Lamm__ue____,type,
aTP_Lamm_ue: nat > fun_nat_fun_nat_bool ).
tff(sy_c_ATP_058Lamm__uf____,type,
aTP_Lamm_uf: nat > fun_nat_set_nat ).
tff(sy_c_ATP_058Lamm__ug____,type,
aTP_Lamm_ug: nat > fun_nat_set_nat ).
tff(sy_c_ATP_058Lamm__uh____,type,
aTP_Lamm_uh: nat > fun_nat_set_nat ).
tff(sy_c_ATP_058Lamm__ui____,type,
aTP_Lamm_ui: nat > fun_nat_fun_nat_bool ).
tff(sy_c_ATP_058Lamm__uj____,type,
aTP_Lamm_uj: nat > fun_nat_set_nat ).
tff(sy_c_ATP_058Lamm__uk____,type,
aTP_Lamm_uk: fun_nat_rat > fun_nat_rat ).
tff(sy_c_ATP_058Lamm__ul____,type,
aTP_Lamm_ul: fun_nat_rat > fun_fu8328862064278946247at_rat ).
tff(sy_c_ATP_058Lamm__um____,type,
aTP_Lamm_um: nat > fun_nat_set_nat ).
tff(sy_c_ATP_058Lamm__un____,type,
aTP_Lamm_un: fun_nat_rat ).
tff(sy_c_ATP_058Lamm__uo____,type,
aTP_Lamm_uo: fun_nat_rat ).
tff(sy_c_ATP_058Lamm__up____,type,
aTP_Lamm_up: nat > fun_nat_rat ).
tff(sy_c_ATP_058Lamm__uq____,type,
aTP_Lamm_uq: int > fun_nat_rat ).
tff(sy_c_ATP_058Lamm__ur____,type,
aTP_Lamm_ur: fun_fun_nat_rat_bool ).
tff(sy_c_ATP_058Lamm__us____,type,
aTP_Lamm_us: fun_fu8328862064278946247at_rat ).
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: fun_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: fun_real_int ).
tff(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
archim7778729529865785530nd_rat: rat > int ).
tff(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
archim8280529875227126926d_real: fun_real_int ).
tff(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat ).
tff(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat ).
tff(sy_c_BNF__Def_Orel__fun_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J_001t__HOL__Obool_001t__HOL__Obool,type,
bNF_re2717234250851183977l_bool: ( fun_fu2636726645394866687t_bool * fun_bo1841576699660343469l_bool * fun_fun_nat_rat_bool * fun_fun_nat_rat_bool ) > $o ).
tff(sy_c_Binomial_Obinomial,type,
binomial: fun_nat_fun_nat_nat ).
tff(sy_c_Binomial_Ogbinomial_001t__Complex__Ocomplex,type,
gbinomial_complex: complex > fun_nat_complex ).
tff(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
gbinomial_int: int > fun_nat_int ).
tff(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
gbinomial_nat: fun_nat_fun_nat_nat ).
tff(sy_c_Binomial_Ogbinomial_001t__Rat__Orat,type,
gbinomial_rat: rat > fun_nat_rat ).
tff(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
gbinomial_real: fun_re4238438383274954274t_real ).
tff(sy_c_Bit__Operations_Oand__int__rel,type,
bit_and_int_rel: fun_Pr3099820731930151597t_bool ).
tff(sy_c_Bit__Operations_Oand__not__num,type,
bit_and_not_num: ( num * num ) > option_num ).
tff(sy_c_Bit__Operations_Oand__not__num__rel,type,
bit_and_not_num_rel: fun_Pr5837708615769641901m_bool ).
tff(sy_c_Bit__Operations_Oconcat__bit,type,
bit_concat_bit: nat > fun_int_fun_int_int ).
tff(sy_c_Bit__Operations_Oor__not__num__neg,type,
bit_or_not_num_neg: fun_num_fun_num_num ).
tff(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
bit_or3848514188828904588eg_rel: fun_Pr5837708615769641901m_bool ).
tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
bit_ri7919022796975470100ot_int: fun_int_int ).
tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Code____Numeral__Ointeger,type,
bit_ri6519982836138164636nteger: ( nat * code_integer ) > code_integer ).
tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
bit_ri631733984087533419it_int: nat > fun_int_int ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Code____Numeral__Ointeger,type,
bit_se3949692690581998587nteger: ( code_integer * code_integer ) > code_integer ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Int__Oint,type,
bit_se725231765392027082nd_int: fun_int_fun_int_int ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
bit_se727722235901077358nd_nat: fun_nat_fun_nat_nat ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Int__Oint,type,
bit_se8568078237143864401it_int: nat > fun_int_int ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
bit_se8570568707652914677it_nat: fun_nat_fun_nat_nat ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Code____Numeral__Ointeger,type,
bit_se1345352211410354436nteger: ( nat * code_integer ) > code_integer ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Int__Oint,type,
bit_se2159334234014336723it_int: nat > fun_int_int ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
bit_se2161824704523386999it_nat: fun_nat_fun_nat_nat ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Code____Numeral__Ointeger,type,
bit_se2119862282449309892nteger: fun_nat_Code_integer ).
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_Oor_001t__Int__Oint,type,
bit_se1409905431419307370or_int: fun_int_fun_int_int ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
bit_se1412395901928357646or_nat: fun_nat_fun_nat_nat ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Int__Oint,type,
bit_se545348938243370406it_int: nat > fun_int_int ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Nat__Onat,type,
bit_se547839408752420682it_nat: fun_nat_fun_nat_nat ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Code____Numeral__Ointeger,type,
bit_se2793503036327961859nteger: ( nat * code_integer ) > code_integer ).
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tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Num__Onum,type,
set_ord_atMost_num: num > set_num ).
tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Rat__Orat,type,
set_ord_atMost_rat: rat > set_rat ).
tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Real__Oreal,type,
set_ord_atMost_real: real > set_real ).
tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4236626031148496127et_nat: set_nat > set_set_nat ).
tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
set_or6656581121297822940st_int: ( int * int ) > set_int ).
tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
set_or6659071591806873216st_nat: nat > fun_nat_set_nat ).
tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
set_or5832277885323065728an_int: ( int * int ) > set_int ).
tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
set_or5834768355832116004an_nat: nat > fun_nat_set_nat ).
tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
set_or1633881224788618240n_real: ( real * real ) > set_real ).
tff(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
set_or1210151606488870762an_nat: fun_nat_set_nat ).
tff(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
set_or5849166863359141190n_real: real > set_real ).
tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
set_ord_lessThan_int: int > set_int ).
tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: fun_nat_set_nat ).
tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Num__Onum,type,
set_ord_lessThan_num: num > set_num ).
tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Rat__Orat,type,
set_ord_lessThan_rat: rat > set_rat ).
tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
set_or5984915006950818249n_real: real > set_real ).
tff(sy_c_String_Oascii__of,type,
ascii_of: char > char ).
tff(sy_c_String_Ochar_OChar,type,
char2: ( bool * bool * bool * bool * bool * bool * bool * bool ) > 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_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
topolo4422821103128117721l_real: ( filter_real * fun_real_real ) > $o ).
tff(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
topolo5044208981011980120l_real: ( set_real * fun_real_real ) > $o ).
tff(sy_c_Topological__Spaces_Omonoseq_001t__Int__Oint,type,
topolo4899668324122417113eq_int: fun_nat_int > $o ).
tff(sy_c_Topological__Spaces_Omonoseq_001t__Nat__Onat,type,
topolo4902158794631467389eq_nat: fun_nat_nat > $o ).
tff(sy_c_Topological__Spaces_Omonoseq_001t__Num__Onum,type,
topolo1459490580787246023eq_num: fun_nat_num > $o ).
tff(sy_c_Topological__Spaces_Omonoseq_001t__Rat__Orat,type,
topolo4267028734544971653eq_rat: fun_nat_rat > $o ).
tff(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
topolo6980174941875973593q_real: fun_nat_real > $o ).
tff(sy_c_Topological__Spaces_Omonoseq_001t__Set__Oset_It__Nat__Onat_J,type,
topolo7278393974255667507et_nat: fun_nat_set_nat > $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,
topolo2815343760600316023s_real: real > filter_real ).
tff(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
topolo4055970368930404560y_real: fun_nat_real > $o ).
tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
topolo896644834953643431omplex: filter6041513312241820739omplex ).
tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
topolo1511823702728130853y_real: filter2146258269922977983l_real ).
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__Complex__Ocomplex,type,
cos_complex: fun_complex_complex ).
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_Odiffs_001t__Complex__Ocomplex,type,
diffs_complex: fun_nat_complex > fun_nat_complex ).
tff(sy_c_Transcendental_Odiffs_001t__Int__Oint,type,
diffs_int: fun_nat_int > fun_nat_int ).
tff(sy_c_Transcendental_Odiffs_001t__Rat__Orat,type,
diffs_rat: fun_nat_rat > fun_nat_rat ).
tff(sy_c_Transcendental_Odiffs_001t__Real__Oreal,type,
diffs_real: fun_nat_real > fun_nat_real ).
tff(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
exp_complex: fun_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: fun_re7681423839302404990l_real ).
tff(sy_c_Transcendental_Opi,type,
pi: real ).
tff(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
powr_real: fun_re7681423839302404990l_real ).
tff(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
sin_complex: fun_complex_complex ).
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__Complex__Ocomplex,type,
tan_complex: fun_complex_complex ).
tff(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
tan_real: fun_real_real ).
tff(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
tanh_complex: fun_complex_complex ).
tff(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
tanh_real: fun_real_real ).
tff(sy_c_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
tff(sy_c_Transitive__Closure_Otrancl_001t__Nat__Onat,type,
transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t,type,
vEBT_T_i_n_s_e_r_t: vEBT_VEBT > fun_nat_nat ).
tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H,type,
vEBT_T_i_n_s_e_r_t2: vEBT_VEBT > fun_nat_nat ).
tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H__rel,type,
vEBT_T5076183648494686801_t_rel: fun_Pr817965127378462947t_bool ).
tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t__rel,type,
vEBT_T9217963907923527482_t_rel: fun_Pr817965127378462947t_bool ).
tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t,type,
vEBT_T_m_a_x_t: fun_VEBT_VEBT_nat ).
tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t__rel,type,
vEBT_T_m_a_x_t_rel: fun_VE5877677516038742509T_bool ).
tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r,type,
vEBT_T_m_e_m_b_e_r: vEBT_VEBT > fun_nat_nat ).
tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H,type,
vEBT_T_m_e_m_b_e_r2: vEBT_VEBT > fun_nat_nat ).
tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H__rel,type,
vEBT_T8099345112685741742_r_rel: fun_Pr817965127378462947t_bool ).
tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r__rel,type,
vEBT_T5837161174952499735_r_rel: fun_Pr817965127378462947t_bool ).
tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l,type,
vEBT_T_m_i_n_N_u_l_l: fun_VEBT_VEBT_nat ).
tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l__rel,type,
vEBT_T5462971552011256508_l_rel: fun_VE5877677516038742509T_bool ).
tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t,type,
vEBT_T_m_i_n_t: fun_VEBT_VEBT_nat ).
tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t__rel,type,
vEBT_T_m_i_n_t_rel: fun_VE5877677516038742509T_bool ).
tff(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
vEBT_Leaf: ( bool * bool ) > vEBT_VEBT ).
tff(sy_c_VEBT__Definitions_OVEBT_ONode,type,
vEBT_Node: ( option4927543243414619207at_nat * nat * list_VEBT_VEBT * vEBT_VEBT ) > vEBT_VEBT ).
tff(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
vEBT_size_VEBT: fun_VEBT_VEBT_nat ).
tff(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
vEBT_V8194947554948674370ptions: fun_VE3763956218935105199t_bool ).
tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
vEBT_VEBT_high: fun_nat_fun_nat_nat ).
tff(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
vEBT_V5917875025757280293ildren: ( nat * list_VEBT_VEBT ) > fun_nat_bool ).
tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
vEBT_VEBT_low: fun_nat_fun_nat_nat ).
tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
vEBT_VEBT_membermima: vEBT_VEBT > fun_nat_bool ).
tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
vEBT_V4351362008482014158ma_rel: fun_Pr817965127378462947t_bool ).
tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
vEBT_V5719532721284313246member: vEBT_VEBT > fun_nat_bool ).
tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
vEBT_V5765760719290551771er_rel: fun_Pr817965127378462947t_bool ).
tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
vEBT_VEBT_valid: vEBT_VEBT > fun_nat_bool ).
tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
vEBT_VEBT_valid_rel: fun_Pr817965127378462947t_bool ).
tff(sy_c_VEBT__Definitions_Oinvar__vebt,type,
vEBT_invar_vebt: vEBT_VEBT > fun_nat_bool ).
tff(sy_c_VEBT__Definitions_Oset__vebt,type,
vEBT_set_vebt: vEBT_VEBT > set_nat ).
tff(sy_c_VEBT__Definitions_Ovebt__buildup,type,
vEBT_vebt_buildup: nat > vEBT_VEBT ).
tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
vEBT_v4011308405150292612up_rel: fun_nat_fun_nat_bool ).
tff(sy_c_VEBT__Height_OVEBT__internal_Oheight,type,
vEBT_VEBT_height: fun_VEBT_VEBT_nat ).
tff(sy_c_VEBT__Height_OVEBT__internal_Oheight__rel,type,
vEBT_VEBT_height_rel: fun_VE5877677516038742509T_bool ).
tff(sy_c_VEBT__Insert_Ovebt__insert,type,
vEBT_vebt_insert: ( vEBT_VEBT * nat ) > vEBT_VEBT ).
tff(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
vEBT_vebt_insert_rel: fun_Pr817965127378462947t_bool ).
tff(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
vEBT_VEBT_bit_concat: nat > fun_nat_fun_nat_nat ).
tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
vEBT_VEBT_minNull: fun_VEBT_VEBT_bool ).
tff(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
vEBT_V6963167321098673237ll_rel: fun_VE5877677516038742509T_bool ).
tff(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).
tff(sy_c_VEBT__Member_Ovebt__member,type,
vEBT_vebt_member: vEBT_VEBT > fun_nat_bool ).
tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
vEBT_vebt_member_rel: fun_Pr817965127378462947t_bool ).
tff(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
vEBT_VEBT_add: fun_op7210469962420818650on_nat ).
tff(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
vEBT_VEBT_greater: fun_op7792301606653096457t_bool ).
tff(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
vEBT_VEBT_less: fun_op7792301606653096457t_bool ).
tff(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
vEBT_VEBT_lesseq: fun_op7792301606653096457t_bool ).
tff(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
vEBT_VEBT_max_in_set: set_nat > fun_nat_bool ).
tff(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
vEBT_VEBT_min_in_set: set_nat > fun_nat_bool ).
tff(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
vEBT_VEBT_mul: fun_op7210469962420818650on_nat ).
tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Nat__Onat,type,
vEBT_V4262088993061758097ft_nat: fun_nat_fun_nat_nat > fun_op7210469962420818650on_nat ).
tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Num__Onum,type,
vEBT_V819420779217536731ft_num: ( fun_num_fun_num_num * option_num * option_num ) > option_num ).
tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
vEBT_V1502963449132264192at_nat: ( fun_Pr2477239776880934807at_nat * option4927543243414619207at_nat * option4927543243414619207at_nat ) > option4927543243414619207at_nat ).
tff(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
vEBT_VEBT_power: fun_op7210469962420818650on_nat ).
tff(sy_c_VEBT__MinMax_Ovebt__maxt,type,
vEBT_vebt_maxt: vEBT_VEBT > option_nat ).
tff(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
vEBT_vebt_maxt_rel: fun_VE5877677516038742509T_bool ).
tff(sy_c_VEBT__MinMax_Ovebt__mint,type,
vEBT_vebt_mint: vEBT_VEBT > option_nat ).
tff(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
vEBT_vebt_mint_rel: fun_VE5877677516038742509T_bool ).
tff(sy_c_VEBT__Pred_Ois__pred__in__set,type,
vEBT_is_pred_in_set: set_nat > fun_nat_fun_nat_bool ).
tff(sy_c_VEBT__Pred_Ovebt__pred,type,
vEBT_vebt_pred: ( vEBT_VEBT * nat ) > option_nat ).
tff(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
vEBT_vebt_pred_rel: fun_Pr817965127378462947t_bool ).
tff(sy_c_VEBT__Succ_Ois__succ__in__set,type,
vEBT_is_succ_in_set: set_nat > fun_nat_fun_nat_bool ).
tff(sy_c_VEBT__Succ_Ovebt__succ,type,
vEBT_vebt_succ: ( vEBT_VEBT * nat ) > option_nat ).
tff(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
vEBT_vebt_succ_rel: fun_Pr817965127378462947t_bool ).
tff(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
accp_list_nat: fun_li8598321315282354825t_bool > fun_list_nat_bool ).
tff(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
accp_nat: fun_nat_fun_nat_bool > fun_nat_bool ).
tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
accp_P1096762738010456898nt_int: fun_Pr3099820731930151597t_bool > fun_Pr803113274115125473t_bool ).
tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
accp_P4275260045618599050at_nat: fun_Pr248683899324676269t_bool > fun_Pr7757163693017731481t_bool ).
tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
accp_P3113834385874906142um_num: fun_Pr5837708615769641901m_bool > fun_Pr6287530515171585797m_bool ).
tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
accp_P2887432264394892906BT_nat: ( fun_Pr817965127378462947t_bool * produc9072475918466114483BT_nat ) > $o ).
tff(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
accp_VEBT_VEBT: fun_VE5877677516038742509T_bool > fun_VEBT_VEBT_bool ).
tff(sy_c_Wellfounded_Opred__nat,type,
pred_nat: set_Pr1261947904930325089at_nat ).
tff(sy_c_Wellfounded_Owf_001t__Nat__Onat,type,
wf_nat: fun_se8074476271188067385t_bool ).
tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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aa_fun1349230436907296553t_bool: ( fun_fu7195134416904984768t_bool * fun_nat_real ) > fun_nat_bool ).
tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J,type,
aa_fun1839206346889086768t_real: ( fun_fu6899955116716589935t_real * fun_nat_real ) > fun_nat_real ).
tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
aa_fun3023522663955195517t_real: ( fun_fu163975356400620604t_real * fun_nat_real ) > fun_nat_fun_nat_real ).
tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J_001t__fun_It__Set__Oset_It__Nat__Onat_J_Mt__fun_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
aa_fun1400677770452585267t_real: ( fun_fu3306995184869731826t_real * fun_nat_real ) > fun_se1654316830156123260t_real ).
tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J_001t__fun_It__fun_It__Nat__Onat_Mt__Real__Oreal_J_Mt__fun_It__Nat__Onat_Mt__HOL__Obool_J_J,type,
aa_fun849216055071118793t_bool: ( fun_fu8639250699203096928t_bool * fun_nat_real ) > fun_fu7195134416904984768t_bool ).
tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Complex__Ocomplex_Mt__Real__Oreal_J_J_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J,type,
aa_fun5746040925223427547t_real: ( fun_fu3477889739401271834t_real * fun_na5723442673449079972x_real ) > fun_nat_real ).
tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Int__Oint_Mt__Real__Oreal_J_J_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J,type,
aa_fun241848180832411225t_real: ( fun_fu4540635375566031512t_real * fun_nat_fun_int_real ) > fun_nat_real ).
tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Real__Oreal_J_J_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J,type,
aa_fun7039663905532832253t_real: ( fun_fu2115079063411676732t_real * fun_nat_fun_nat_real ) > fun_nat_real ).
tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Rat__Orat_Mt__Real__Oreal_J_J_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J,type,
aa_fun507281639146497029t_real: ( fun_fu4806068833880117316t_real * fun_nat_fun_rat_real ) > fun_nat_real ).
tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Real__Oreal_Mt__Real__Oreal_J_J_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J,type,
aa_fun1113820866546176217t_real: ( fun_fu5589980185390252312t_real * fun_na6724076748692969250l_real ) > fun_nat_real ).
tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__fun_It__Real__Oreal_Mt__Real__Oreal_J_J_001t__fun_It__Real__Oreal_Mt__fun_It__Nat__Onat_Mt__fun_It__Real__Oreal_Mt__Real__Oreal_J_J_J,type,
aa_fun2567993205388704171l_real: ( fun_fu3612569900928539882l_real * fun_na6724076748692969250l_real ) > fun_re4112884445705685835l_real ).
tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__HOL__Obool_J_001t__HOL__Obool,type,
aa_fun8663595511526100224l_bool: ( fun_fu5641360819561228055l_bool * fun_Pr7757163693017731481t_bool ) > bool ).
tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
aa_fun2351638177499875132at_nat: ( fun_fu7732904369943091395at_nat * fun_Pr2602117396471556196at_nat ) > fun_Pr2602117396471556196at_nat ).
tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__HOL__Obool,type,
aa_fun4519828053337421206t_bool: ( fun_fu7919257609373989805t_bool * fun_Pr2118491707810114563at_nat ) > bool ).
tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
aa_fun8399641135911395862nt_int: ( fun_fu3064074204415222357nt_int * fun_Pr2118491707810114563at_nat ) > fun_int_int ).
tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Code____Numeral__Ointeger_J_001t__fun_It__Real__Oreal_Mt__Code____Numeral__Ointeger_J,type,
aa_fun7923231501384339402nteger: ( fun_fu8643405727515436425nteger * fun_re6978948838062957830nteger ) > fun_re6978948838062957830nteger ).
tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Complex__Ocomplex_J_001t__fun_It__Real__Oreal_Mt__Complex__Ocomplex_J,type,
aa_fun2233226938935547116omplex: ( fun_fu8384527901749275179omplex * fun_real_complex ) > fun_real_complex ).
tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Complex__Ocomplex_J_001t__fun_It__Real__Oreal_Mt__HOL__Obool_J,type,
aa_fun3813300650643916655l_bool: ( fun_fu914841811607101702l_bool * fun_real_complex ) > fun_real_bool ).
tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Complex__Ocomplex_J_001t__fun_It__fun_It__Real__Oreal_Mt__Complex__Ocomplex_J_Mt__fun_It__Real__Oreal_Mt__HOL__Obool_J_J,type,
aa_fun5520078544946397361l_bool: ( fun_fu7100649987335273288l_bool * fun_real_complex ) > fun_fu914841811607101702l_bool ).
tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__HOL__Obool_J_001t__HOL__Obool,type,
aa_fun3352386287069783311l_bool: ( fun_fu7511176317168184086l_bool * fun_real_bool ) > bool ).
tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Nat__Onat_J_001t__fun_It__Real__Oreal_Mt__Nat__Onat_J,type,
aa_fun211602392931233840al_nat: ( fun_fu3214020956574444655al_nat * fun_real_nat ) > fun_real_nat ).
tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Rat__Orat_J_001t__fun_It__Real__Oreal_Mt__HOL__Obool_J,type,
aa_fun8987621170725249221l_bool: ( fun_fu2277326744902863452l_bool * fun_real_rat ) > fun_real_bool ).
tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Rat__Orat_J_001t__fun_It__Real__Oreal_Mt__Rat__Orat_J,type,
aa_fun2468082234287226944al_rat: ( fun_fu2598743018846156927al_rat * fun_real_rat ) > fun_real_rat ).
tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Rat__Orat_J_001t__fun_It__fun_It__Real__Oreal_Mt__Rat__Orat_J_Mt__fun_It__Real__Oreal_Mt__HOL__Obool_J_J,type,
aa_fun8530242357055030877l_bool: ( fun_fu6750725090257165812l_bool * fun_real_rat ) > fun_fu2277326744902863452l_bool ).
tff(sy_c_aa_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J_001t__fun_It__Real__Oreal_Mt__HOL__Obool_J,type,
aa_fun9137271176455094897l_bool: ( fun_fu7376312551830202376l_bool * fun_real_real ) > fun_real_bool ).
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__Real__Oreal_Mt__Real__Oreal_J_001t__fun_It__fun_It__Real__Oreal_Mt__Real__Oreal_J_Mt__fun_It__Real__Oreal_Mt__HOL__Obool_J_J,type,
aa_fun334843040499603765l_bool: ( fun_fu2568719150003003596l_bool * fun_real_real ) > fun_fu7376312551830202376l_bool ).
tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
aa_fun7144222995531728252BT_int: ( fun_fu5786399312697532803BT_int * fun_VEBT_VEBT_int ) > fun_VEBT_VEBT_int ).
tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_Mt__fun_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
aa_fun7485955581636121530BT_int: ( fun_fu7059288016784409353BT_int * fun_VEBT_VEBT_int ) > fun_fu5786399312697532803BT_int ).
tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
aa_fun8284231930576791164BT_nat: ( fun_fu5238640373165125763BT_nat * fun_VEBT_VEBT_nat ) > fun_VEBT_VEBT_nat ).
tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_Mt__fun_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
aa_fun8778301808485728350BT_nat: ( fun_fu5586652545743564717BT_nat * fun_VEBT_VEBT_nat ) > fun_fu5238640373165125763BT_nat ).
tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Rat__Orat_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Rat__Orat_J,type,
aa_fun8105832842879916156BT_rat: ( fun_fu2188483506383969923BT_rat * fun_VEBT_VEBT_rat ) > fun_VEBT_VEBT_rat ).
tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Rat__Orat_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__Rat__Orat_J_Mt__fun_It__VEBT____Definitions__OVEBT_Mt__Rat__Orat_J_J,type,
aa_fun2693410532875676774BT_rat: ( fun_fu4463953228700081077BT_rat * fun_VEBT_VEBT_rat ) > fun_fu2188483506383969923BT_rat ).
tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
aa_fun1825138017709814396T_real: ( fun_fu8198533418994508931T_real * fun_VEBT_VEBT_real ) > fun_VEBT_VEBT_real ).
tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_Mt__fun_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_J,type,
aa_fun910392650362481850T_real: ( fun_fu4141963935195952137T_real * fun_VEBT_VEBT_real ) > fun_fu8198533418994508931T_real ).
tff(sy_c_aa_001t__fun_It__fun_It__Nat__Onat_Mt__Rat__Orat_J_Mt__fun_It__Nat__Onat_Mt__Rat__Orat_J_J_001t__fun_It__Real__Oreal_Mt__Real__Oreal_J,type,
aa_fun8686829112900759002l_real: ( fun_fu3711856879280950809l_real * fun_fu8328862064278946247at_rat ) > fun_real_real ).
tff(sy_c_fAll_001t__VEBT____Definitions__OVEBT,type,
fAll_VEBT_VEBT: fun_VEBT_VEBT_bool > bool ).
tff(sy_c_fChoice_001t__Real__Oreal,type,
fChoice_real: fun_real_bool > real ).
tff(sy_c_fEx_001t__Nat__Onat,type,
fEx_nat: fun_fu7168253649841883962l_bool ).
tff(sy_c_fEx_001t__Real__Oreal,type,
fEx_real: fun_fu7511176317168184086l_bool ).
tff(sy_c_fFalse,type,
fFalse: bool ).
tff(sy_c_fNot,type,
fNot: fun_bool_bool ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_fconj,type,
fconj: bool > fun_bool_bool ).
tff(sy_c_fdisj,type,
fdisj: bool > fun_bool_bool ).
tff(sy_c_fequal_001t__Code____Numeral__Ointeger,type,
fequal_Code_integer: fun_Co8310303586393106575r_bool ).
tff(sy_c_fequal_001t__Complex__Ocomplex,type,
fequal_complex: fun_co1378794544173401837x_bool ).
tff(sy_c_fequal_001t__HOL__Obool,type,
fequal_bool: fun_bo1841576699660343469l_bool ).
tff(sy_c_fequal_001t__Int__Oint,type,
fequal_int: fun_int_fun_int_bool ).
tff(sy_c_fequal_001t__Nat__Onat,type,
fequal_nat: fun_nat_fun_nat_bool ).
tff(sy_c_fequal_001t__Option__Ooption_It__Nat__Onat_J,type,
fequal_option_nat: fun_op7792301606653096457t_bool ).
tff(sy_c_fequal_001t__Real__Oreal,type,
fequal_real: fun_re1255401033344146673l_bool ).
tff(sy_c_fequal_001t__VEBT____Definitions__OVEBT,type,
fequal_VEBT_VEBT: vEBT_VEBT > fun_VEBT_VEBT_bool ).
tff(sy_c_fimplies,type,
fimplies: fun_bo1841576699660343469l_bool ).
tff(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > fun_set_complex_bool ).
tff(sy_c_member_001t__HOL__Obool,type,
member_bool: ( bool * set_bool ) > $o ).
tff(sy_c_member_001t__Int__Oint,type,
member_int: int > fun_set_int_bool ).
tff(sy_c_member_001t__List__Olist_It__HOL__Obool_J,type,
member_list_bool: ( list_bool * set_list_bool ) > $o ).
tff(sy_c_member_001t__List__Olist_It__Int__Oint_J,type,
member_list_int: ( list_int * set_list_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__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
member2936631157270082147T_VEBT: ( list_VEBT_VEBT * set_list_VEBT_VEBT ) > $o ).
tff(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > fun_set_nat_bool ).
tff(sy_c_member_001t__Num__Onum,type,
member_num: num > fun_set_num_bool ).
tff(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > fun_se8074476271188067385t_bool ).
tff(sy_c_member_001t__Rat__Orat,type,
member_rat: rat > fun_set_rat_bool ).
tff(sy_c_member_001t__Real__Oreal,type,
member_real: real > fun_set_real_bool ).
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__VEBT____Definitions__OVEBT,type,
member_VEBT_VEBT: fun_VE7346385682144900493T_bool ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_deg____,type,
deg: nat ).
tff(sy_v_m____,type,
m: nat ).
tff(sy_v_ma____,type,
ma: nat ).
tff(sy_v_mi____,type,
mi: nat ).
tff(sy_v_na____,type,
na: nat ).
tff(sy_v_summary____,type,
summary: vEBT_VEBT ).
tff(sy_v_treeList____,type,
treeList: list_VEBT_VEBT ).
tff(sy_v_xa____,type,
xa: nat ).
% Relevant facts (10752)
tff(fact_0_False,axiom,
~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),mi)) ).
% False
tff(fact_1__C5_Ohyps_C_I7_J,axiom,
pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,mi),ma)) ).
% "5.hyps"(7)
tff(fact_2_max__in__set__def,axiom,
! [Xs: set_nat,X: nat] :
( pp(aa_nat_bool(vEBT_VEBT_max_in_set(Xs),X))
<=> ( pp(aa_set_nat_bool(member_nat(X),Xs))
& ! [X2: nat] :
( pp(aa_set_nat_bool(member_nat(X2),Xs))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),X)) ) ) ) ).
% max_in_set_def
tff(fact_3_min__in__set__def,axiom,
! [Xs: set_nat,X: nat] :
( pp(aa_nat_bool(vEBT_VEBT_min_in_set(Xs),X))
<=> ( pp(aa_set_nat_bool(member_nat(X),Xs))
& ! [X2: nat] :
( pp(aa_set_nat_bool(member_nat(X2),Xs))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),X2)) ) ) ) ).
% min_in_set_def
tff(fact_4__C5_OIH_C_I2_J,axiom,
! [X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t(summary),X)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,summary))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one)))))))) ).
% "5.IH"(2)
tff(fact_5_height__compose__summary,axiom,
! [Summary: vEBT_VEBT,Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Summary))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,vEBT_Node(Info,Deg,TreeList,Summary)))) ).
% height_compose_summary
tff(fact_6__092_060open_0622_A_092_060le_062_Adeg_092_060close_062,axiom,
pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg)) ).
% \<open>2 \<le> deg\<close>
tff(fact_7__092_060open_062T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_Ax_A_092_060le_062_A23_A_L_AT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Asummary_A_Ihigh_A_Iif_Ax_A_060_Ami_Athen_Ami_Aelse_Ax_J_A_Ideg_Adiv_A2_J_J_092_060close_062,axiom,
pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary)),xa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_T_i_n_s_e_r_t(summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),mi)),mi),xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ).
% \<open>T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t (Node (Some (mi, ma)) deg treeList summary) x \<le> 23 + T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t summary (high (if x < mi then mi else x) (deg div 2))\<close>
tff(fact_8__C5_Ohyps_C_I8_J,axiom,
pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg))) ).
% "5.hyps"(8)
tff(fact_9_insert__simp__mima,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X = Mi )
| ( X = Ma ) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
=> ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary) ) ) ) ).
% insert_simp_mima
tff(fact_10_one__add__one,axiom,
aa_complex_complex(plus_plus_complex(one_one_complex),one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).
% one_add_one
tff(fact_11_one__add__one,axiom,
aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real) = numeral_numeral_real(aa_num_num(bit0,one)) ).
% one_add_one
tff(fact_12_one__add__one,axiom,
plus_plus_rat(one_one_rat,one_one_rat) = numeral_numeral_rat(aa_num_num(bit0,one)) ).
% one_add_one
tff(fact_13_one__add__one,axiom,
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).
% one_add_one
tff(fact_14_one__add__one,axiom,
aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).
% one_add_one
tff(fact_15_numeral__plus__one,axiom,
! [N: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),one_one_complex) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).
% numeral_plus_one
tff(fact_16_numeral__plus__one,axiom,
! [N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(N)),one_one_real) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).
% numeral_plus_one
tff(fact_17_numeral__plus__one,axiom,
! [N: num] : plus_plus_rat(numeral_numeral_rat(N),one_one_rat) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).
% numeral_plus_one
tff(fact_18_numeral__plus__one,axiom,
! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).
% numeral_plus_one
tff(fact_19_numeral__plus__one,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).
% numeral_plus_one
tff(fact_20_one__plus__numeral,axiom,
! [N: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(N)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).
% one_plus_numeral
tff(fact_21_one__plus__numeral,axiom,
! [N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),numeral_numeral_real(N)) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).
% one_plus_numeral
tff(fact_22_one__plus__numeral,axiom,
! [N: num] : plus_plus_rat(one_one_rat,numeral_numeral_rat(N)) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).
% one_plus_numeral
tff(fact_23_one__plus__numeral,axiom,
! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).
% one_plus_numeral
tff(fact_24_one__plus__numeral,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).
% one_plus_numeral
tff(fact_25_numeral__le__one__iff,axiom,
! [N: num] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(N)),one_one_real))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).
% numeral_le_one_iff
tff(fact_26_numeral__le__one__iff,axiom,
! [N: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,numeral_numeral_rat(N)),one_one_rat))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).
% numeral_le_one_iff
tff(fact_27_numeral__le__one__iff,axiom,
! [N: num] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N)),one_one_nat))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).
% numeral_le_one_iff
tff(fact_28_numeral__le__one__iff,axiom,
! [N: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),one_one_int))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),one)) ) ).
% numeral_le_one_iff
tff(fact_29_numeral__eq__one__iff,axiom,
! [N: num] :
( ( numera6690914467698888265omplex(N) = one_one_complex )
<=> ( N = one ) ) ).
% numeral_eq_one_iff
tff(fact_30_numeral__eq__one__iff,axiom,
! [N: num] :
( ( numeral_numeral_real(N) = one_one_real )
<=> ( N = one ) ) ).
% numeral_eq_one_iff
tff(fact_31_numeral__eq__one__iff,axiom,
! [N: num] :
( ( numeral_numeral_rat(N) = one_one_rat )
<=> ( N = one ) ) ).
% numeral_eq_one_iff
tff(fact_32_numeral__eq__one__iff,axiom,
! [N: num] :
( ( aa_num_nat(numeral_numeral_nat,N) = one_one_nat )
<=> ( N = one ) ) ).
% numeral_eq_one_iff
tff(fact_33_numeral__eq__one__iff,axiom,
! [N: num] :
( ( aa_num_int(numeral_numeral_int,N) = one_one_int )
<=> ( N = one ) ) ).
% numeral_eq_one_iff
tff(fact_34_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_complex = numera6690914467698888265omplex(N) )
<=> ( one = N ) ) ).
% one_eq_numeral_iff
tff(fact_35_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_real = numeral_numeral_real(N) )
<=> ( one = N ) ) ).
% one_eq_numeral_iff
tff(fact_36_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_rat = numeral_numeral_rat(N) )
<=> ( one = N ) ) ).
% one_eq_numeral_iff
tff(fact_37_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat = aa_num_nat(numeral_numeral_nat,N) )
<=> ( one = N ) ) ).
% one_eq_numeral_iff
tff(fact_38_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_int = aa_num_int(numeral_numeral_int,N) )
<=> ( one = N ) ) ).
% one_eq_numeral_iff
tff(fact_39_distrib__left__numeral,axiom,
! [V: num,B: complex,C: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(plus_plus_complex(B),C)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),B)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),C)) ).
% distrib_left_numeral
tff(fact_40_distrib__left__numeral,axiom,
! [V: num,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),C)) ).
% distrib_left_numeral
tff(fact_41_distrib__left__numeral,axiom,
! [V: num,B: rat,C: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),plus_plus_rat(B,C)) = plus_plus_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),B),aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),C)) ).
% distrib_left_numeral
tff(fact_42_distrib__left__numeral,axiom,
! [V: num,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),C)) ).
% distrib_left_numeral
tff(fact_43_distrib__left__numeral,axiom,
! [V: num,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),C)) ).
% distrib_left_numeral
tff(fact_44_distrib__right__numeral,axiom,
! [A: complex,B: complex,V: num] : aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B)),numera6690914467698888265omplex(V)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(B),numera6690914467698888265omplex(V))) ).
% distrib_right_numeral
tff(fact_45_distrib__right__numeral,axiom,
! [A: real,B: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),numeral_numeral_real(V)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),numeral_numeral_real(V))) ).
% distrib_right_numeral
tff(fact_46_distrib__right__numeral,axiom,
! [A: rat,B: rat,V: num] : aa_rat_rat(times_times_rat(plus_plus_rat(A,B)),numeral_numeral_rat(V)) = plus_plus_rat(aa_rat_rat(times_times_rat(A),numeral_numeral_rat(V)),aa_rat_rat(times_times_rat(B),numeral_numeral_rat(V))) ).
% distrib_right_numeral
tff(fact_47_distrib__right__numeral,axiom,
! [A: nat,B: 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,A),B)),aa_num_nat(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,A),aa_num_nat(numeral_numeral_nat,V))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_num_nat(numeral_numeral_nat,V))) ).
% distrib_right_numeral
tff(fact_48_distrib__right__numeral,axiom,
! [A: int,B: int,V: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_num_int(numeral_numeral_int,V))) ).
% distrib_right_numeral
tff(fact_49_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) )
<=> ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
tff(fact_50_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = one_one_nat )
<=> ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
tff(fact_51_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( numera6690914467698888265omplex(M) = numera6690914467698888265omplex(N) )
<=> ( M = N ) ) ).
% numeral_eq_iff
tff(fact_52_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( numeral_numeral_real(M) = numeral_numeral_real(N) )
<=> ( M = N ) ) ).
% numeral_eq_iff
tff(fact_53_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( numeral_numeral_rat(M) = numeral_numeral_rat(N) )
<=> ( M = N ) ) ).
% numeral_eq_iff
tff(fact_54_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( aa_num_nat(numeral_numeral_nat,M) = aa_num_nat(numeral_numeral_nat,N) )
<=> ( M = N ) ) ).
% numeral_eq_iff
tff(fact_55_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( aa_num_int(numeral_numeral_int,M) = aa_num_int(numeral_numeral_int,N) )
<=> ( M = N ) ) ).
% numeral_eq_iff
tff(fact_56_pow__sum,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ).
% pow_sum
tff(fact_57_high__def,axiom,
! [X: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).
% high_def
tff(fact_58_high__bound__aux,axiom,
! [Ma: nat,N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))) ) ).
% high_bound_aux
tff(fact_59_numeral__less__iff,axiom,
! [M: num,N: num] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(M)),numeral_numeral_real(N)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).
% numeral_less_iff
tff(fact_60_numeral__less__iff,axiom,
! [M: num,N: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,numeral_numeral_rat(M)),numeral_numeral_rat(N)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).
% numeral_less_iff
tff(fact_61_numeral__less__iff,axiom,
! [M: num,N: num] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).
% numeral_less_iff
tff(fact_62_numeral__less__iff,axiom,
! [M: num,N: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).
% numeral_less_iff
tff(fact_63_high__inv,axiom,
! [X: nat,N: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),X)),N) = Y ) ) ).
% high_inv
tff(fact_64_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),Z)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).
% mult_numeral_left_semiring_numeral
tff(fact_65_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).
% mult_numeral_left_semiring_numeral
tff(fact_66_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),Z)) = aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).
% mult_numeral_left_semiring_numeral
tff(fact_67_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,W)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).
% mult_numeral_left_semiring_numeral
tff(fact_68_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W)),Z)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).
% mult_numeral_left_semiring_numeral
tff(fact_69_numeral__times__numeral,axiom,
! [M: num,N: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(M)),numera6690914467698888265omplex(N)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).
% numeral_times_numeral
tff(fact_70_numeral__times__numeral,axiom,
! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(M)),numeral_numeral_real(N)) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).
% numeral_times_numeral
tff(fact_71_numeral__times__numeral,axiom,
! [M: num,N: num] : aa_rat_rat(times_times_rat(numeral_numeral_rat(M)),numeral_numeral_rat(N)) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).
% numeral_times_numeral
tff(fact_72_numeral__times__numeral,axiom,
! [M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).
% numeral_times_numeral
tff(fact_73_numeral__times__numeral,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).
% numeral_times_numeral
tff(fact_74_num__double,axiom,
! [N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,one)),N) = aa_num_num(bit0,N) ).
% num_double
tff(fact_75_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% nat_add_left_cancel_less
tff(fact_76_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% nat_add_left_cancel_le
tff(fact_77_bit__concat__def,axiom,
! [H: nat,L: nat,D: nat] : aa_nat_nat(aa_nat_fun_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(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),D))),L) ).
% bit_concat_def
tff(fact_78_add__numeral__left,axiom,
! [V: num,W: num,Z: complex] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(V)),aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(W)),Z)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).
% add_numeral_left
tff(fact_79_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(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).
% add_numeral_left
tff(fact_80_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(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)),Z) ).
% add_numeral_left
tff(fact_81_add__numeral__left,axiom,
! [V: num,W: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,W)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).
% add_numeral_left
tff(fact_82_add__numeral__left,axiom,
! [V: num,W: num,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,W)),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).
% add_numeral_left
tff(fact_83_numeral__plus__numeral,axiom,
! [M: num,N: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(M)),numera6690914467698888265omplex(N)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).
% numeral_plus_numeral
tff(fact_84_numeral__plus__numeral,axiom,
! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(M)),numeral_numeral_real(N)) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).
% numeral_plus_numeral
tff(fact_85_numeral__plus__numeral,axiom,
! [M: num,N: num] : plus_plus_rat(numeral_numeral_rat(M),numeral_numeral_rat(N)) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).
% numeral_plus_numeral
tff(fact_86_numeral__plus__numeral,axiom,
! [M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).
% numeral_plus_numeral
tff(fact_87_numeral__plus__numeral,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).
% numeral_plus_numeral
tff(fact_88_numeral__le__iff,axiom,
! [M: num,N: num] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(M)),numeral_numeral_real(N)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).
% numeral_le_iff
tff(fact_89_numeral__le__iff,axiom,
! [M: num,N: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,numeral_numeral_rat(M)),numeral_numeral_rat(N)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).
% numeral_le_iff
tff(fact_90_numeral__le__iff,axiom,
! [M: num,N: num] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).
% numeral_le_iff
tff(fact_91_numeral__le__iff,axiom,
! [M: num,N: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).
% numeral_le_iff
tff(fact_92_one__less__numeral__iff,axiom,
! [N: num] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),numeral_numeral_real(N)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).
% one_less_numeral_iff
tff(fact_93_one__less__numeral__iff,axiom,
! [N: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),numeral_numeral_rat(N)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).
% one_less_numeral_iff
tff(fact_94_one__less__numeral__iff,axiom,
! [N: num] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).
% one_less_numeral_iff
tff(fact_95_one__less__numeral__iff,axiom,
! [N: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_num_int(numeral_numeral_int,N)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),N)) ) ).
% one_less_numeral_iff
tff(fact_96_le__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W: num] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W))))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W))),B)) ) ).
% le_divide_eq_numeral1(1)
tff(fact_97_le__divide__eq__numeral1_I1_J,axiom,
! [A: rat,B: rat,W: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),divide_divide_rat(B,numeral_numeral_rat(W))))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W))),B)) ) ).
% le_divide_eq_numeral1(1)
tff(fact_98_divide__le__eq__numeral1_I1_J,axiom,
! [B: real,W: num,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W))),A))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W)))) ) ).
% divide_le_eq_numeral1(1)
tff(fact_99_divide__le__eq__numeral1_I1_J,axiom,
! [B: rat,W: num,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(B,numeral_numeral_rat(W))),A))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W)))) ) ).
% divide_le_eq_numeral1(1)
tff(fact_100_less__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W: num] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W))))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W))),B)) ) ).
% less_divide_eq_numeral1(1)
tff(fact_101_less__divide__eq__numeral1_I1_J,axiom,
! [A: rat,B: rat,W: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),divide_divide_rat(B,numeral_numeral_rat(W))))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W))),B)) ) ).
% less_divide_eq_numeral1(1)
tff(fact_102_divide__less__eq__numeral1_I1_J,axiom,
! [B: real,W: num,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W))),A))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W)))) ) ).
% divide_less_eq_numeral1(1)
tff(fact_103_divide__less__eq__numeral1_I1_J,axiom,
! [B: rat,W: num,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(B,numeral_numeral_rat(W))),A))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W)))) ) ).
% divide_less_eq_numeral1(1)
tff(fact_104_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
| pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).
% nat_neq_iff
tff(fact_105_less__not__refl,axiom,
! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N)) ).
% less_not_refl
tff(fact_106_less__not__refl2,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
=> ( M != N ) ) ).
% less_not_refl2
tff(fact_107_less__not__refl3,axiom,
! [S: nat,T: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,S),T))
=> ( S != T ) ) ).
% less_not_refl3
tff(fact_108_mem__Collect__eq,axiom,
! [A: product_prod_nat_nat,P: fun_Pr7757163693017731481t_bool] :
( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),collec3392354462482085612at_nat(P)))
<=> pp(aa_Pro2232559814647311250t_bool(P,A)) ) ).
% mem_Collect_eq
tff(fact_109_mem__Collect__eq,axiom,
! [A: complex,P: fun_complex_bool] :
( pp(aa_set_complex_bool(member_complex(A),collect_complex(P)))
<=> pp(aa_complex_bool(P,A)) ) ).
% mem_Collect_eq
tff(fact_110_mem__Collect__eq,axiom,
! [A: real,P: fun_real_bool] :
( pp(aa_set_real_bool(member_real(A),collect_real(P)))
<=> pp(aa_real_bool(P,A)) ) ).
% mem_Collect_eq
tff(fact_111_mem__Collect__eq,axiom,
! [A: list_nat,P: fun_list_nat_bool] :
( member_list_nat(A,collect_list_nat(P))
<=> pp(aa_list_nat_bool(P,A)) ) ).
% mem_Collect_eq
tff(fact_112_mem__Collect__eq,axiom,
! [A: nat,P: fun_nat_bool] :
( pp(aa_set_nat_bool(member_nat(A),collect_nat(P)))
<=> pp(aa_nat_bool(P,A)) ) ).
% mem_Collect_eq
tff(fact_113_mem__Collect__eq,axiom,
! [A: int,P: fun_int_bool] :
( pp(aa_set_int_bool(member_int(A),collect_int(P)))
<=> pp(aa_int_bool(P,A)) ) ).
% mem_Collect_eq
tff(fact_114_Collect__mem__eq,axiom,
! [A2: set_Pr1261947904930325089at_nat] : collec3392354462482085612at_nat(aTP_Lamm_a(A2)) = A2 ).
% Collect_mem_eq
tff(fact_115_Collect__mem__eq,axiom,
! [A2: set_complex] : collect_complex(aTP_Lamm_aa(A2)) = A2 ).
% Collect_mem_eq
tff(fact_116_Collect__mem__eq,axiom,
! [A2: set_real] : collect_real(aTP_Lamm_ab(A2)) = A2 ).
% Collect_mem_eq
tff(fact_117_Collect__mem__eq,axiom,
! [A2: set_list_nat] : collect_list_nat(aTP_Lamm_ac(A2)) = A2 ).
% Collect_mem_eq
tff(fact_118_Collect__mem__eq,axiom,
! [A2: set_nat] : collect_nat(aTP_Lamm_ad(A2)) = A2 ).
% Collect_mem_eq
tff(fact_119_Collect__mem__eq,axiom,
! [A2: set_int] : collect_int(aTP_Lamm_ae(A2)) = A2 ).
% Collect_mem_eq
tff(fact_120_Collect__cong,axiom,
! [P: fun_complex_bool,Q: fun_complex_bool] :
( ! [X3: complex] :
( pp(aa_complex_bool(P,X3))
<=> pp(aa_complex_bool(Q,X3)) )
=> ( collect_complex(P) = collect_complex(Q) ) ) ).
% Collect_cong
tff(fact_121_Collect__cong,axiom,
! [P: fun_real_bool,Q: fun_real_bool] :
( ! [X3: real] :
( pp(aa_real_bool(P,X3))
<=> pp(aa_real_bool(Q,X3)) )
=> ( collect_real(P) = collect_real(Q) ) ) ).
% Collect_cong
tff(fact_122_Collect__cong,axiom,
! [P: fun_list_nat_bool,Q: fun_list_nat_bool] :
( ! [X3: list_nat] :
( pp(aa_list_nat_bool(P,X3))
<=> pp(aa_list_nat_bool(Q,X3)) )
=> ( collect_list_nat(P) = collect_list_nat(Q) ) ) ).
% Collect_cong
tff(fact_123_Collect__cong,axiom,
! [P: fun_nat_bool,Q: fun_nat_bool] :
( ! [X3: nat] :
( pp(aa_nat_bool(P,X3))
<=> pp(aa_nat_bool(Q,X3)) )
=> ( collect_nat(P) = collect_nat(Q) ) ) ).
% Collect_cong
tff(fact_124_Collect__cong,axiom,
! [P: fun_int_bool,Q: fun_int_bool] :
( ! [X3: int] :
( pp(aa_int_bool(P,X3))
<=> pp(aa_int_bool(Q,X3)) )
=> ( collect_int(P) = collect_int(Q) ) ) ).
% Collect_cong
tff(fact_125_less__irrefl__nat,axiom,
! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N)) ).
% less_irrefl_nat
tff(fact_126_nat__less__induct,axiom,
! [P: fun_nat_bool,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N2))
=> pp(aa_nat_bool(P,M2)) )
=> pp(aa_nat_bool(P,N2)) )
=> pp(aa_nat_bool(P,N)) ) ).
% nat_less_induct
tff(fact_127_infinite__descent,axiom,
! [P: fun_nat_bool,N: nat] :
( ! [N2: nat] :
( ~ pp(aa_nat_bool(P,N2))
=> ? [M2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N2))
& ~ pp(aa_nat_bool(P,M2)) ) )
=> pp(aa_nat_bool(P,N)) ) ).
% infinite_descent
tff(fact_128_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).
% linorder_neqE_nat
tff(fact_129_add__One__commute,axiom,
! [N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N) = aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one) ).
% add_One_commute
tff(fact_130_le__num__One__iff,axiom,
! [X: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),one))
<=> ( X = one ) ) ).
% le_num_One_iff
tff(fact_131_less__numeral__extra_I4_J,axiom,
~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),one_one_real)) ).
% less_numeral_extra(4)
tff(fact_132_less__numeral__extra_I4_J,axiom,
~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),one_one_rat)) ).
% less_numeral_extra(4)
tff(fact_133_less__numeral__extra_I4_J,axiom,
~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),one_one_nat)) ).
% less_numeral_extra(4)
tff(fact_134_less__numeral__extra_I4_J,axiom,
~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),one_one_int)) ).
% less_numeral_extra(4)
tff(fact_135_less__mono__imp__le__mono,axiom,
! [F: fun_nat_nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,I2)),aa_nat_nat(F,J2))) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,I)),aa_nat_nat(F,J))) ) ) ).
% less_mono_imp_le_mono
tff(fact_136_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( ( M != N )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).
% le_neq_implies_less
tff(fact_137_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
| ( M = N ) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% less_or_eq_imp_le
tff(fact_138_le__eq__less__or__eq,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
| ( M = N ) ) ) ).
% le_eq_less_or_eq
tff(fact_139_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% less_imp_le_nat
tff(fact_140_nat__less__le,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
& ( M != N ) ) ) ).
% nat_less_le
tff(fact_141_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).
% less_add_eq_less
tff(fact_142_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J))) ) ).
% trans_less_add2
tff(fact_143_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M))) ) ).
% trans_less_add1
tff(fact_144_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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_145_not__add__less2,axiom,
! [J: nat,I: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),I)) ).
% not_add_less2
tff(fact_146_not__add__less1,axiom,
! [I: nat,J: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),I)) ).
% not_add_less1
tff(fact_147_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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_148_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K)) ) ).
% add_lessD1
tff(fact_149_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).
% is_num_normalize(1)
tff(fact_150_is__num__normalize_I1_J,axiom,
! [A: rat,B: rat,C: rat] : plus_plus_rat(plus_plus_rat(A,B),C) = plus_plus_rat(A,plus_plus_rat(B,C)) ).
% is_num_normalize(1)
tff(fact_151_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).
% is_num_normalize(1)
tff(fact_152_Nat_Oex__has__greatest__nat,axiom,
! [P: fun_nat_bool,K: nat,B: nat] :
( pp(aa_nat_bool(P,K))
=> ( ! [Y2: nat] :
( pp(aa_nat_bool(P,Y2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y2),B)) )
=> ? [X3: nat] :
( pp(aa_nat_bool(P,X3))
& ! [Y3: nat] :
( pp(aa_nat_bool(P,Y3))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y3),X3)) ) ) ) ) ).
% Nat.ex_has_greatest_nat
tff(fact_153_nat__le__linear,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
| pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ).
% nat_le_linear
tff(fact_154_le__antisym,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( M = N ) ) ) ).
% le_antisym
tff(fact_155_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% eq_imp_le
tff(fact_156_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),K))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K)) ) ) ).
% le_trans
tff(fact_157_le__refl,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N)) ).
% le_refl
tff(fact_158_divide__numeral__1,axiom,
! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),numera6690914467698888265omplex(one)) = A ).
% divide_numeral_1
tff(fact_159_divide__numeral__1,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),numeral_numeral_real(one)) = A ).
% divide_numeral_1
tff(fact_160_divide__numeral__1,axiom,
! [A: rat] : divide_divide_rat(A,numeral_numeral_rat(one)) = A ).
% divide_numeral_1
tff(fact_161_not__numeral__less__one,axiom,
! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(N)),one_one_real)) ).
% not_numeral_less_one
tff(fact_162_not__numeral__less__one,axiom,
! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,numeral_numeral_rat(N)),one_one_rat)) ).
% not_numeral_less_one
tff(fact_163_not__numeral__less__one,axiom,
! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,N)),one_one_nat)) ).
% not_numeral_less_one
tff(fact_164_not__numeral__less__one,axiom,
! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),one_one_int)) ).
% not_numeral_less_one
tff(fact_165_mono__nat__linear__lb,axiom,
! [F: fun_nat_nat,M: nat,K: nat] :
( ! [M3: nat,N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,M3)),aa_nat_nat(F,N2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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_166_le__numeral__extra_I4_J,axiom,
pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),one_one_real)) ).
% le_numeral_extra(4)
tff(fact_167_le__numeral__extra_I4_J,axiom,
pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),one_one_rat)) ).
% le_numeral_extra(4)
tff(fact_168_le__numeral__extra_I4_J,axiom,
pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),one_one_nat)) ).
% le_numeral_extra(4)
tff(fact_169_le__numeral__extra_I4_J,axiom,
pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),one_one_int)) ).
% le_numeral_extra(4)
tff(fact_170_nat__le__iff__add,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
<=> ? [K2: nat] : N = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2) ) ).
% nat_le_iff_add
tff(fact_171_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J))) ) ).
% trans_le_add2
tff(fact_172_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M))) ) ).
% trans_le_add1
tff(fact_173_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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_174_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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_175_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L))
=> ? [N2: nat] : L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N2) ) ).
% le_Suc_ex
tff(fact_176_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ).
% add_leD2
tff(fact_177_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% add_leD1
tff(fact_178_le__add2,axiom,
! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ).
% le_add2
tff(fact_179_le__add1,axiom,
! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))) ).
% le_add1
tff(fact_180_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
=> ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ) ).
% add_leE
tff(fact_181_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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_182_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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_183_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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_184_le__square,axiom,
! [M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M))) ).
% le_square
tff(fact_185_le__cube,axiom,
! [M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(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_186_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: 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),N)) = 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),N)) ).
% add_mult_distrib2
tff(fact_187_add__mult__distrib,axiom,
! [M: nat,N: 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),N)),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,N),K)) ).
% add_mult_distrib
tff(fact_188_nat__mult__1__right,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),one_one_nat) = N ).
% nat_mult_1_right
tff(fact_189_nat__mult__1,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),N) = N ).
% nat_mult_1
tff(fact_190_one__le__numeral,axiom,
! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),numeral_numeral_real(N))) ).
% one_le_numeral
tff(fact_191_one__le__numeral,axiom,
! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),numeral_numeral_rat(N))) ).
% one_le_numeral
tff(fact_192_one__le__numeral,axiom,
! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N))) ).
% one_le_numeral
tff(fact_193_one__le__numeral,axiom,
! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_num_int(numeral_numeral_int,N))) ).
% one_le_numeral
tff(fact_194_mult__numeral__1__right,axiom,
! [A: complex] : aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(one)) = A ).
% mult_numeral_1_right
tff(fact_195_mult__numeral__1__right,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(one)) = A ).
% mult_numeral_1_right
tff(fact_196_mult__numeral__1__right,axiom,
! [A: rat] : aa_rat_rat(times_times_rat(A),numeral_numeral_rat(one)) = A ).
% mult_numeral_1_right
tff(fact_197_mult__numeral__1__right,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_num_nat(numeral_numeral_nat,one)) = A ).
% mult_numeral_1_right
tff(fact_198_mult__numeral__1__right,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,one)) = A ).
% mult_numeral_1_right
tff(fact_199_mult__numeral__1,axiom,
! [A: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(one)),A) = A ).
% mult_numeral_1
tff(fact_200_mult__numeral__1,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(one)),A) = A ).
% mult_numeral_1
tff(fact_201_mult__numeral__1,axiom,
! [A: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(one)),A) = A ).
% mult_numeral_1
tff(fact_202_mult__numeral__1,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,one)),A) = A ).
% mult_numeral_1
tff(fact_203_mult__numeral__1,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,one)),A) = A ).
% mult_numeral_1
tff(fact_204_numeral__Bit0,axiom,
! [N: num] : numera6690914467698888265omplex(aa_num_num(bit0,N)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),numera6690914467698888265omplex(N)) ).
% numeral_Bit0
tff(fact_205_numeral__Bit0,axiom,
! [N: num] : numeral_numeral_real(aa_num_num(bit0,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(N)),numeral_numeral_real(N)) ).
% numeral_Bit0
tff(fact_206_numeral__Bit0,axiom,
! [N: num] : numeral_numeral_rat(aa_num_num(bit0,N)) = plus_plus_rat(numeral_numeral_rat(N),numeral_numeral_rat(N)) ).
% numeral_Bit0
tff(fact_207_numeral__Bit0,axiom,
! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N)) ).
% numeral_Bit0
tff(fact_208_numeral__Bit0,axiom,
! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N)) ).
% numeral_Bit0
tff(fact_209_one__plus__numeral__commute,axiom,
! [X: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(X)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(X)),one_one_complex) ).
% one_plus_numeral_commute
tff(fact_210_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_211_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_212_one__plus__numeral__commute,axiom,
! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,X)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,X)),one_one_nat) ).
% one_plus_numeral_commute
tff(fact_213_one__plus__numeral__commute,axiom,
! [X: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,X)),one_one_int) ).
% one_plus_numeral_commute
tff(fact_214_numeral__One,axiom,
numera6690914467698888265omplex(one) = one_one_complex ).
% numeral_One
tff(fact_215_numeral__One,axiom,
numeral_numeral_real(one) = one_one_real ).
% numeral_One
tff(fact_216_numeral__One,axiom,
numeral_numeral_rat(one) = one_one_rat ).
% numeral_One
tff(fact_217_numeral__One,axiom,
aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).
% numeral_One
tff(fact_218_numeral__One,axiom,
aa_num_int(numeral_numeral_int,one) = one_one_int ).
% numeral_One
tff(fact_219_num_Oexhaust,axiom,
! [Y: num] :
( ( Y != one )
=> ( ! [X22: num] : Y != aa_num_num(bit0,X22)
=> ~ ! [X32: num] : Y != aa_num_num(bit1,X32) ) ) ).
% num.exhaust
tff(fact_220_numerals_I1_J,axiom,
aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).
% numerals(1)
tff(fact_221_numeral__Bit1,axiom,
! [N: num] : numera6690914467698888265omplex(aa_num_num(bit1,N)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),numera6690914467698888265omplex(N))),one_one_complex) ).
% numeral_Bit1
tff(fact_222_numeral__Bit1,axiom,
! [N: num] : numeral_numeral_real(aa_num_num(bit1,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(N)),numeral_numeral_real(N))),one_one_real) ).
% numeral_Bit1
tff(fact_223_numeral__Bit1,axiom,
! [N: num] : numeral_numeral_rat(aa_num_num(bit1,N)) = plus_plus_rat(plus_plus_rat(numeral_numeral_rat(N),numeral_numeral_rat(N)),one_one_rat) ).
% numeral_Bit1
tff(fact_224_numeral__Bit1,axiom,
! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N))),one_one_nat) ).
% numeral_Bit1
tff(fact_225_numeral__Bit1,axiom,
! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N))),one_one_int) ).
% numeral_Bit1
tff(fact_226_left__add__twice,axiom,
! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),A)),B) ).
% left_add_twice
tff(fact_227_left__add__twice,axiom,
! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),A)),B) ).
% left_add_twice
tff(fact_228_left__add__twice,axiom,
! [A: rat,B: rat] : plus_plus_rat(A,plus_plus_rat(A,B)) = plus_plus_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),A),B) ).
% left_add_twice
tff(fact_229_left__add__twice,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),B) ).
% left_add_twice
tff(fact_230_left__add__twice,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),B) ).
% left_add_twice
tff(fact_231_mult__2__right,axiom,
! [Z: complex] : aa_complex_complex(times_times_complex(Z),numera6690914467698888265omplex(aa_num_num(bit0,one))) = aa_complex_complex(plus_plus_complex(Z),Z) ).
% mult_2_right
tff(fact_232_mult__2__right,axiom,
! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),numeral_numeral_real(aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).
% mult_2_right
tff(fact_233_mult__2__right,axiom,
! [Z: rat] : aa_rat_rat(times_times_rat(Z),numeral_numeral_rat(aa_num_num(bit0,one))) = plus_plus_rat(Z,Z) ).
% mult_2_right
tff(fact_234_mult__2__right,axiom,
! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).
% mult_2_right
tff(fact_235_mult__2__right,axiom,
! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Z),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z) ).
% mult_2_right
tff(fact_236_mult__2,axiom,
! [Z: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Z) = aa_complex_complex(plus_plus_complex(Z),Z) ).
% mult_2
tff(fact_237_mult__2,axiom,
! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),Z) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).
% mult_2
tff(fact_238_mult__2,axiom,
! [Z: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),Z) = plus_plus_rat(Z,Z) ).
% mult_2
tff(fact_239_mult__2,axiom,
! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Z) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).
% mult_2
tff(fact_240_mult__2,axiom,
! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Z) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z) ).
% mult_2
tff(fact_241_nat__1__add__1,axiom,
aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).
% nat_1_add_1
tff(fact_242_add__self__div__2,axiom,
! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),M)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = M ).
% add_self_div_2
tff(fact_243_power__increasing__iff,axiom,
! [B: real,X: nat,Y: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Y)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).
% power_increasing_iff
tff(fact_244_power__increasing__iff,axiom,
! [B: rat,X: nat,Y: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),X)),aa_nat_rat(power_power_rat(B),Y)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).
% power_increasing_iff
tff(fact_245_power__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).
% power_increasing_iff
tff(fact_246_power__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),X)),aa_nat_int(power_power_int(B),Y)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).
% power_increasing_iff
tff(fact_247_semiring__norm_I3_J,axiom,
! [N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bit0,N)) = aa_num_num(bit1,N) ).
% semiring_norm(3)
tff(fact_248_semiring__norm_I4_J,axiom,
! [N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bit1,N)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).
% semiring_norm(4)
tff(fact_249_semiring__norm_I5_J,axiom,
! [M: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,M)),one) = aa_num_num(bit1,M) ).
% semiring_norm(5)
tff(fact_250_semiring__norm_I8_J,axiom,
! [M: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,M)),one) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one)) ).
% semiring_norm(8)
tff(fact_251_semiring__norm_I10_J,axiom,
! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,M)),aa_num_num(bit1,N)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)),one)) ).
% semiring_norm(10)
tff(fact_252_power__add__numeral2,axiom,
! [A: complex,M: num,N: num,B: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,M))),aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,N))),B)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)))),B) ).
% power_add_numeral2
tff(fact_253_power__add__numeral2,axiom,
! [A: real,M: num,N: num,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,M))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,N))),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)))),B) ).
% power_add_numeral2
tff(fact_254_power__add__numeral2,axiom,
! [A: rat,M: num,N: num,B: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,M))),aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,N))),B)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)))),B) ).
% power_add_numeral2
tff(fact_255_power__add__numeral2,axiom,
! [A: nat,M: num,N: num,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,N))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)))),B) ).
% power_add_numeral2
tff(fact_256_power__add__numeral2,axiom,
! [A: int,M: num,N: num,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,N))),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)))),B) ).
% power_add_numeral2
tff(fact_257_power__add__numeral,axiom,
! [A: complex,M: num,N: num] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).
% power_add_numeral
tff(fact_258_power__add__numeral,axiom,
! [A: real,M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).
% power_add_numeral
tff(fact_259_power__add__numeral,axiom,
! [A: rat,M: num,N: num] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).
% power_add_numeral
tff(fact_260_power__add__numeral,axiom,
! [A: nat,M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).
% power_add_numeral
tff(fact_261_power__add__numeral,axiom,
! [A: int,M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,M))),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).
% power_add_numeral
tff(fact_262_power__strict__increasing__iff,axiom,
! [B: real,X: nat,Y: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Y)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).
% power_strict_increasing_iff
tff(fact_263_power__strict__increasing__iff,axiom,
! [B: rat,X: nat,Y: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(B),X)),aa_nat_rat(power_power_rat(B),Y)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).
% power_strict_increasing_iff
tff(fact_264_power__strict__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).
% power_strict_increasing_iff
tff(fact_265_power__strict__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),X)),aa_nat_int(power_power_int(B),Y)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).
% power_strict_increasing_iff
tff(fact_266_semiring__norm_I70_J,axiom,
! [M: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(bit1,M)),one)) ).
% semiring_norm(70)
tff(fact_267_semiring__norm_I72_J,axiom,
! [M: num,N: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(bit0,M)),aa_num_num(bit1,N)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).
% semiring_norm(72)
tff(fact_268_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( aa_num_num(bit0,M) = aa_num_num(bit0,N) )
<=> ( M = N ) ) ).
% semiring_norm(87)
tff(fact_269_semiring__norm_I90_J,axiom,
! [M: num,N: num] :
( ( aa_num_num(bit1,M) = aa_num_num(bit1,N) )
<=> ( M = N ) ) ).
% semiring_norm(90)
tff(fact_270_semiring__norm_I85_J,axiom,
! [M: num] : aa_num_num(bit0,M) != one ).
% semiring_norm(85)
tff(fact_271_semiring__norm_I83_J,axiom,
! [N: num] : one != aa_num_num(bit0,N) ).
% semiring_norm(83)
tff(fact_272_power__one,axiom,
! [N: nat] : aa_nat_rat(power_power_rat(one_one_rat),N) = one_one_rat ).
% power_one
tff(fact_273_power__one,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),N) = one_one_nat ).
% power_one
tff(fact_274_power__one,axiom,
! [N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,one_one_real),N) = one_one_real ).
% power_one
tff(fact_275_power__one,axiom,
! [N: nat] : aa_nat_complex(power_power_complex(one_one_complex),N) = one_one_complex ).
% power_one
tff(fact_276_power__one,axiom,
! [N: nat] : aa_nat_int(power_power_int(one_one_int),N) = one_one_int ).
% power_one
tff(fact_277_semiring__norm_I89_J,axiom,
! [M: num,N: num] : aa_num_num(bit1,M) != aa_num_num(bit0,N) ).
% semiring_norm(89)
tff(fact_278_semiring__norm_I88_J,axiom,
! [M: num,N: num] : aa_num_num(bit0,M) != aa_num_num(bit1,N) ).
% semiring_norm(88)
tff(fact_279_semiring__norm_I86_J,axiom,
! [M: num] : aa_num_num(bit1,M) != one ).
% semiring_norm(86)
tff(fact_280_semiring__norm_I84_J,axiom,
! [N: num] : one != aa_num_num(bit1,N) ).
% semiring_norm(84)
tff(fact_281_power__one__right,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),one_one_nat) = A ).
% power_one_right
tff(fact_282_power__one__right,axiom,
! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),one_one_nat) = A ).
% power_one_right
tff(fact_283_power__one__right,axiom,
! [A: complex] : aa_nat_complex(power_power_complex(A),one_one_nat) = A ).
% power_one_right
tff(fact_284_power__one__right,axiom,
! [A: int] : aa_nat_int(power_power_int(A),one_one_nat) = A ).
% power_one_right
tff(fact_285_semiring__norm_I6_J,axiom,
! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,M)),aa_num_num(bit0,N)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).
% semiring_norm(6)
tff(fact_286_semiring__norm_I13_J,axiom,
! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,M)),aa_num_num(bit0,N)) = aa_num_num(bit0,aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).
% semiring_norm(13)
tff(fact_287_semiring__norm_I12_J,axiom,
! [N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,one),N) = N ).
% semiring_norm(12)
tff(fact_288_semiring__norm_I11_J,axiom,
! [M: num] : aa_num_num(aa_num_fun_num_num(times_times_num,M),one) = M ).
% semiring_norm(11)
tff(fact_289_semiring__norm_I78_J,axiom,
! [M: num,N: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(bit0,M)),aa_num_num(bit0,N)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).
% semiring_norm(78)
tff(fact_290_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(bit0,M)),aa_num_num(bit0,N)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).
% semiring_norm(71)
tff(fact_291_semiring__norm_I75_J,axiom,
! [M: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),one)) ).
% semiring_norm(75)
tff(fact_292_semiring__norm_I68_J,axiom,
! [N: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,one),N)) ).
% semiring_norm(68)
tff(fact_293_semiring__norm_I80_J,axiom,
! [M: num,N: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(bit1,M)),aa_num_num(bit1,N)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).
% semiring_norm(80)
tff(fact_294_semiring__norm_I73_J,axiom,
! [M: num,N: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(bit1,M)),aa_num_num(bit1,N)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).
% semiring_norm(73)
tff(fact_295_power__inject__exp,axiom,
! [A: real,M: nat,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
=> ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) )
<=> ( M = N ) ) ) ).
% power_inject_exp
tff(fact_296_power__inject__exp,axiom,
! [A: rat,M: nat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
=> ( ( aa_nat_rat(power_power_rat(A),M) = aa_nat_rat(power_power_rat(A),N) )
<=> ( M = N ) ) ) ).
% power_inject_exp
tff(fact_297_power__inject__exp,axiom,
! [A: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) )
<=> ( M = N ) ) ) ).
% power_inject_exp
tff(fact_298_power__inject__exp,axiom,
! [A: int,M: nat,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
=> ( ( aa_nat_int(power_power_int(A),M) = aa_nat_int(power_power_int(A),N) )
<=> ( M = N ) ) ) ).
% power_inject_exp
tff(fact_299_semiring__norm_I2_J,axiom,
aa_num_num(aa_num_fun_num_num(plus_plus_num,one),one) = aa_num_num(bit0,one) ).
% semiring_norm(2)
tff(fact_300_semiring__norm_I9_J,axiom,
! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,M)),aa_num_num(bit0,N)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).
% semiring_norm(9)
tff(fact_301_semiring__norm_I7_J,axiom,
! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,M)),aa_num_num(bit1,N)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).
% semiring_norm(7)
tff(fact_302_power__mult__numeral,axiom,
! [A: nat,M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).
% power_mult_numeral
tff(fact_303_power__mult__numeral,axiom,
! [A: real,M: num,N: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).
% power_mult_numeral
tff(fact_304_power__mult__numeral,axiom,
! [A: complex,M: num,N: num] : aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,N)) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).
% power_mult_numeral
tff(fact_305_power__mult__numeral,axiom,
! [A: int,M: num,N: num] : aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,N)) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).
% power_mult_numeral
tff(fact_306_semiring__norm_I15_J,axiom,
! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,M)),aa_num_num(bit0,N)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,M)),N)) ).
% semiring_norm(15)
tff(fact_307_semiring__norm_I14_J,axiom,
! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,M)),aa_num_num(bit1,N)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,M),aa_num_num(bit1,N))) ).
% semiring_norm(14)
tff(fact_308_semiring__norm_I76_J,axiom,
! [N: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),aa_num_num(bit0,N))) ).
% semiring_norm(76)
tff(fact_309_semiring__norm_I69_J,axiom,
! [M: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(bit0,M)),one)) ).
% semiring_norm(69)
tff(fact_310_semiring__norm_I81_J,axiom,
! [M: num,N: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(bit1,M)),aa_num_num(bit0,N)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).
% semiring_norm(81)
tff(fact_311_semiring__norm_I77_J,axiom,
! [N: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,one),aa_num_num(bit1,N))) ).
% semiring_norm(77)
tff(fact_312_semiring__norm_I16_J,axiom,
! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,M)),aa_num_num(bit1,N)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)),aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,M),N)))) ).
% semiring_norm(16)
tff(fact_313_semiring__norm_I79_J,axiom,
! [M: num,N: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(bit0,M)),aa_num_num(bit1,N)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N)) ) ).
% semiring_norm(79)
tff(fact_314_semiring__norm_I74_J,axiom,
! [M: num,N: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(bit1,M)),aa_num_num(bit0,N)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).
% semiring_norm(74)
tff(fact_315_div__mult2__numeral__eq,axiom,
! [A: nat,K: num,L: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,L)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,K),L))) ).
% div_mult2_numeral_eq
tff(fact_316_div__mult2__numeral__eq,axiom,
! [A: int,K: num,L: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,L)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,K),L))) ).
% div_mult2_numeral_eq
tff(fact_317_power__commutes,axiom,
! [A: complex,N: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),N)),A) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(A),N)) ).
% power_commutes
tff(fact_318_power__commutes,axiom,
! [A: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ).
% power_commutes
tff(fact_319_power__commutes,axiom,
! [A: rat,N: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),N)),A) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),N)) ).
% power_commutes
tff(fact_320_power__commutes,axiom,
! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ).
% power_commutes
tff(fact_321_power__commutes,axiom,
! [A: int,N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),N)),A) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N)) ).
% power_commutes
tff(fact_322_power__mult__distrib,axiom,
! [A: complex,B: complex,N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(times_times_complex(A),B)),N) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),N)),aa_nat_complex(power_power_complex(B),N)) ).
% power_mult_distrib
tff(fact_323_power__mult__distrib,axiom,
! [A: real,B: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),N) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)) ).
% power_mult_distrib
tff(fact_324_power__mult__distrib,axiom,
! [A: rat,B: rat,N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(times_times_rat(A),B)),N) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N)) ).
% power_mult_distrib
tff(fact_325_power__mult__distrib,axiom,
! [A: nat,B: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)) ).
% power_mult_distrib
tff(fact_326_power__mult__distrib,axiom,
! [A: int,B: int,N: nat] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N)) ).
% power_mult_distrib
tff(fact_327_power__commuting__commutes,axiom,
! [X: complex,Y: complex,N: nat] :
( ( aa_complex_complex(times_times_complex(X),Y) = aa_complex_complex(times_times_complex(Y),X) )
=> ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),N)),Y) = aa_complex_complex(times_times_complex(Y),aa_nat_complex(power_power_complex(X),N)) ) ) ).
% power_commuting_commutes
tff(fact_328_power__commuting__commutes,axiom,
! [X: real,Y: real,N: nat] :
( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),X) )
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) ) ) ).
% power_commuting_commutes
tff(fact_329_power__commuting__commutes,axiom,
! [X: rat,Y: rat,N: nat] :
( ( aa_rat_rat(times_times_rat(X),Y) = aa_rat_rat(times_times_rat(Y),X) )
=> ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),N)),Y) = aa_rat_rat(times_times_rat(Y),aa_nat_rat(power_power_rat(X),N)) ) ) ).
% power_commuting_commutes
tff(fact_330_power__commuting__commutes,axiom,
! [X: nat,Y: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),X) )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)) ) ) ).
% power_commuting_commutes
tff(fact_331_power__commuting__commutes,axiom,
! [X: int,Y: int,N: nat] :
( ( aa_int_int(aa_int_fun_int_int(times_times_int,X),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),X) )
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X),N)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(power_power_int(X),N)) ) ) ).
% power_commuting_commutes
tff(fact_332_power__divide,axiom,
! [A: complex,B: complex,N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(divide1717551699836669952omplex(A),B)),N) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),N)),aa_nat_complex(power_power_complex(B),N)) ).
% power_divide
tff(fact_333_power__divide,axiom,
! [A: real,B: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),N) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)) ).
% power_divide
tff(fact_334_power__divide,axiom,
! [A: rat,B: rat,N: nat] : aa_nat_rat(power_power_rat(divide_divide_rat(A,B)),N) = divide_divide_rat(aa_nat_rat(power_power_rat(A),N),aa_nat_rat(power_power_rat(B),N)) ).
% power_divide
tff(fact_335_power__mult,axiom,
! [A: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),N) ).
% power_mult
tff(fact_336_power__mult,axiom,
! [A: real,M: nat,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),N) ).
% power_mult
tff(fact_337_power__mult,axiom,
! [A: complex,M: nat,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),M)),N) ).
% power_mult
tff(fact_338_power__mult,axiom,
! [A: int,M: nat,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),M)),N) ).
% power_mult
tff(fact_339_div__le__mono,axiom,
! [M: nat,N: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),K))) ) ).
% div_le_mono
tff(fact_340_div__le__dividend,axiom,
! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),M)) ).
% div_le_dividend
tff(fact_341_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_342_div__mult2__eq,axiom,
! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),Q2) ).
% div_mult2_eq
tff(fact_343_one__le__power,axiom,
! [A: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ).
% one_le_power
tff(fact_344_one__le__power,axiom,
! [A: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_nat_rat(power_power_rat(A),N))) ) ).
% one_le_power
tff(fact_345_one__le__power,axiom,
! [A: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).
% one_le_power
tff(fact_346_one__le__power,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_nat_int(power_power_int(A),N))) ) ).
% one_le_power
tff(fact_347_left__right__inverse__power,axiom,
! [X: complex,Y: complex,N: nat] :
( ( aa_complex_complex(times_times_complex(X),Y) = one_one_complex )
=> ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),N)),aa_nat_complex(power_power_complex(Y),N)) = one_one_complex ) ) ).
% left_right_inverse_power
tff(fact_348_left__right__inverse__power,axiom,
! [X: real,Y: real,N: nat] :
( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y) = one_one_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),N)) = one_one_real ) ) ).
% left_right_inverse_power
tff(fact_349_left__right__inverse__power,axiom,
! [X: rat,Y: rat,N: nat] :
( ( aa_rat_rat(times_times_rat(X),Y) = one_one_rat )
=> ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),N)),aa_nat_rat(power_power_rat(Y),N)) = one_one_rat ) ) ).
% left_right_inverse_power
tff(fact_350_left__right__inverse__power,axiom,
! [X: nat,Y: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y) = one_one_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),N)) = one_one_nat ) ) ).
% left_right_inverse_power
tff(fact_351_left__right__inverse__power,axiom,
! [X: int,Y: int,N: nat] :
( ( aa_int_int(aa_int_fun_int_int(times_times_int,X),Y) = one_one_int )
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X),N)),aa_nat_int(power_power_int(Y),N)) = one_one_int ) ) ).
% left_right_inverse_power
tff(fact_352_power__one__over,axiom,
! [A: complex,N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A)),N) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_nat_complex(power_power_complex(A),N)) ).
% power_one_over
tff(fact_353_power__one__over,axiom,
! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),N) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ).
% power_one_over
tff(fact_354_power__one__over,axiom,
! [A: rat,N: nat] : aa_nat_rat(power_power_rat(divide_divide_rat(one_one_rat,A)),N) = divide_divide_rat(one_one_rat,aa_nat_rat(power_power_rat(A),N)) ).
% power_one_over
tff(fact_355_power__add,axiom,
! [A: complex,M: nat,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),M)),aa_nat_complex(power_power_complex(A),N)) ).
% power_add
tff(fact_356_power__add,axiom,
! [A: real,M: nat,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ).
% power_add
tff(fact_357_power__add,axiom,
! [A: rat,M: nat,N: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N)) ).
% power_add
tff(fact_358_power__add,axiom,
! [A: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ).
% power_add
tff(fact_359_power__add,axiom,
! [A: int,M: nat,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)) ).
% power_add
tff(fact_360_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),I)) ) ).
% less_mult_imp_div_less
tff(fact_361_div__times__less__eq__dividend,axiom,
! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),N)),M)) ).
% div_times_less_eq_dividend
tff(fact_362_times__div__less__eq__dividend,axiom,
! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N))),M)) ).
% times_div_less_eq_dividend
tff(fact_363_power__gt1__lemma,axiom,
! [A: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)))) ) ).
% power_gt1_lemma
tff(fact_364_power__gt1__lemma,axiom,
! [A: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),N)))) ) ).
% power_gt1_lemma
tff(fact_365_power__gt1__lemma,axiom,
! [A: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))) ) ).
% power_gt1_lemma
tff(fact_366_power__gt1__lemma,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N)))) ) ).
% power_gt1_lemma
tff(fact_367_power__less__power__Suc,axiom,
! [A: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)))) ) ).
% power_less_power_Suc
tff(fact_368_power__less__power__Suc,axiom,
! [A: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),N)))) ) ).
% power_less_power_Suc
tff(fact_369_power__less__power__Suc,axiom,
! [A: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))) ) ).
% power_less_power_Suc
tff(fact_370_power__less__power__Suc,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N)))) ) ).
% power_less_power_Suc
tff(fact_371_numeral__Bit0__div__2,axiom,
! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,N) ).
% numeral_Bit0_div_2
tff(fact_372_numeral__Bit0__div__2,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,N) ).
% numeral_Bit0_div_2
tff(fact_373_power__strict__increasing,axiom,
! [N: nat,N3: nat,A: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N3))) ) ) ).
% power_strict_increasing
tff(fact_374_power__strict__increasing,axiom,
! [N: nat,N3: nat,A: rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(A),N3))) ) ) ).
% power_strict_increasing
tff(fact_375_power__strict__increasing,axiom,
! [N: nat,N3: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N3))) ) ) ).
% power_strict_increasing
tff(fact_376_power__strict__increasing,axiom,
! [N: nat,N3: nat,A: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(A),N3))) ) ) ).
% power_strict_increasing
tff(fact_377_power__less__imp__less__exp,axiom,
! [A: real,M: nat,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).
% power_less_imp_less_exp
tff(fact_378_power__less__imp__less__exp,axiom,
! [A: rat,M: nat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).
% power_less_imp_less_exp
tff(fact_379_power__less__imp__less__exp,axiom,
! [A: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).
% power_less_imp_less_exp
tff(fact_380_power__less__imp__less__exp,axiom,
! [A: int,M: nat,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).
% power_less_imp_less_exp
tff(fact_381_power__increasing,axiom,
! [N: nat,N3: nat,A: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N3))) ) ) ).
% power_increasing
tff(fact_382_power__increasing,axiom,
! [N: nat,N3: nat,A: rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(A),N3))) ) ) ).
% power_increasing
tff(fact_383_power__increasing,axiom,
! [N: nat,N3: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N3))) ) ) ).
% power_increasing
tff(fact_384_power__increasing,axiom,
! [N: nat,N3: nat,A: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(A),N3))) ) ) ).
% power_increasing
tff(fact_385_power__le__imp__le__exp,axiom,
! [A: real,M: nat,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).
% power_le_imp_le_exp
tff(fact_386_power__le__imp__le__exp,axiom,
! [A: rat,M: nat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),M)),aa_nat_rat(power_power_rat(A),N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).
% power_le_imp_le_exp
tff(fact_387_power__le__imp__le__exp,axiom,
! [A: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).
% power_le_imp_le_exp
tff(fact_388_power__le__imp__le__exp,axiom,
! [A: int,M: nat,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).
% power_le_imp_le_exp
tff(fact_389_power4__eq__xxxx,axiom,
! [X: complex] : aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(X),X)),X)),X) ).
% power4_eq_xxxx
tff(fact_390_power4__eq__xxxx,axiom,
! [X: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),X)),X) ).
% power4_eq_xxxx
tff(fact_391_power4__eq__xxxx,axiom,
! [X: rat] : aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(X),X)),X)),X) ).
% power4_eq_xxxx
tff(fact_392_power4__eq__xxxx,axiom,
! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = 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(aa_nat_fun_nat_nat(times_times_nat,X),X)),X)),X) ).
% power4_eq_xxxx
tff(fact_393_power4__eq__xxxx,axiom,
! [X: int] : aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),X)),X) ).
% power4_eq_xxxx
tff(fact_394_power2__eq__square,axiom,
! [A: complex] : aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(times_times_complex(A),A) ).
% power2_eq_square
tff(fact_395_power2__eq__square,axiom,
! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A) ).
% power2_eq_square
tff(fact_396_power2__eq__square,axiom,
! [A: rat] : aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(times_times_rat(A),A) ).
% power2_eq_square
tff(fact_397_power2__eq__square,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),A) ).
% power2_eq_square
tff(fact_398_power2__eq__square,axiom,
! [A: int] : aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,A),A) ).
% power2_eq_square
tff(fact_399_one__power2,axiom,
aa_nat_rat(power_power_rat(one_one_rat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat ).
% one_power2
tff(fact_400_one__power2,axiom,
aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).
% one_power2
tff(fact_401_one__power2,axiom,
aa_nat_real(aa_real_fun_nat_real(power_power_real,one_one_real),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real ).
% one_power2
tff(fact_402_one__power2,axiom,
aa_nat_complex(power_power_complex(one_one_complex),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex ).
% one_power2
tff(fact_403_one__power2,axiom,
aa_nat_int(power_power_int(one_one_int),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int ).
% one_power2
tff(fact_404_numeral__Bit1__div__2,axiom,
! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,N) ).
% numeral_Bit1_div_2
tff(fact_405_numeral__Bit1__div__2,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,N) ).
% numeral_Bit1_div_2
tff(fact_406_power3__eq__cube,axiom,
! [A: complex] : aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(A),A)),A) ).
% power3_eq_cube
tff(fact_407_power3__eq__cube,axiom,
! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)),A) ).
% power3_eq_cube
tff(fact_408_power3__eq__cube,axiom,
! [A: rat] : aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A),A)),A) ).
% power3_eq_cube
tff(fact_409_power3__eq__cube,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),A)),A) ).
% power3_eq_cube
tff(fact_410_power3__eq__cube,axiom,
! [A: int] : aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),A)),A) ).
% power3_eq_cube
tff(fact_411_power__even__eq,axiom,
! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% power_even_eq
tff(fact_412_power__even__eq,axiom,
! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% power_even_eq
tff(fact_413_power__even__eq,axiom,
! [A: complex,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% power_even_eq
tff(fact_414_power__even__eq,axiom,
! [A: int,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% power_even_eq
tff(fact_415_less__exp,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).
% less_exp
tff(fact_416_self__le__ge2__pow,axiom,
! [K: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M))) ) ).
% self_le_ge2_pow
tff(fact_417_power2__nat__le__eq__le,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% power2_nat_le_eq_le
tff(fact_418_power2__nat__le__imp__le,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% power2_nat_le_imp_le
tff(fact_419_power2__sum,axiom,
! [X: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)),Y)) ).
% power2_sum
tff(fact_420_power2__sum,axiom,
! [X: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),X)),Y)) ).
% power2_sum
tff(fact_421_power2__sum,axiom,
! [X: rat,Y: rat] : aa_nat_rat(power_power_rat(plus_plus_rat(X,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = plus_plus_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),X)),Y)) ).
% power2_sum
tff(fact_422_power2__sum,axiom,
! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = 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(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X)),Y)) ).
% power2_sum
tff(fact_423_power2__sum,axiom,
! [X: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X)),Y)) ).
% power2_sum
tff(fact_424_ex__power__ivl1,axiom,
! [B: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),K))
=> ? [N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2)),K))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)))) ) ) ) ).
% ex_power_ivl1
tff(fact_425_ex__power__ivl2,axiom,
! [B: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
=> ? [N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N2)),K))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat)))) ) ) ) ).
% ex_power_ivl2
tff(fact_426_sum__squares__bound,axiom,
! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),X)),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).
% sum_squares_bound
tff(fact_427_sum__squares__bound,axiom,
! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),X)),Y)),plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).
% sum_squares_bound
tff(fact_428_div__exp__eq,axiom,
! [A: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ).
% div_exp_eq
tff(fact_429_div__exp__eq,axiom,
! [A: int,M: nat,N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ).
% div_exp_eq
tff(fact_430_field__less__half__sum,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),numeral_numeral_real(aa_num_num(bit0,one))))) ) ).
% field_less_half_sum
tff(fact_431_field__less__half__sum,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),divide_divide_rat(plus_plus_rat(X,Y),numeral_numeral_rat(aa_num_num(bit0,one))))) ) ).
% field_less_half_sum
tff(fact_432__092_060open_062T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_Ax_A_092_060le_062_A23_A_L_A_Iif_AminNull_A_ItreeList_A_B_Ahigh_A_Iif_Ax_A_060_Ami_Athen_Ami_Aelse_Ax_J_A_Ideg_Adiv_A2_J_J_Athen_AT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Asummary_A_Ihigh_A_Iif_Ax_A_060_Ami_Athen_Ami_Aelse_Ax_J_A_Ideg_Adiv_A2_J_J_Aelse_A1_J_092_060close_062,axiom,
pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary)),xa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),mi)),mi),xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(vEBT_T_i_n_s_e_r_t(summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),mi)),mi),xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat)))) ).
% \<open>T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t (Node (Some (mi, ma)) deg treeList summary) x \<le> 23 + (if minNull (treeList ! high (if x < mi then mi else x) (deg div 2)) then T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t summary (high (if x < mi then mi else x) (deg div 2)) else 1)\<close>
tff(fact_433_mi__ma__2__deg,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),Ma))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))) ) ) ).
% mi_ma_2_deg
tff(fact_434_True,axiom,
pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),mi)),mi),xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).
% True
tff(fact_435_enat__ord__number_I1_J,axiom,
! [M: num,N: num] :
( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))) ) ).
% enat_ord_number(1)
tff(fact_436_enat__ord__number_I2_J,axiom,
! [M: num,N: num] :
( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))) ) ).
% enat_ord_number(2)
tff(fact_437_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,aa_num_num(bit1,V))),aa_num_int(numeral_numeral_int,aa_num_num(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_438_div__by__1,axiom,
! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),one_one_complex) = A ).
% div_by_1
tff(fact_439_div__by__1,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),one_one_real) = A ).
% div_by_1
tff(fact_440_div__by__1,axiom,
! [A: rat] : divide_divide_rat(A,one_one_rat) = A ).
% div_by_1
tff(fact_441_div__by__1,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),one_one_nat) = A ).
% div_by_1
tff(fact_442_div__by__1,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),one_one_int) = A ).
% div_by_1
tff(fact_443_bits__div__by__1,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),one_one_nat) = A ).
% bits_div_by_1
tff(fact_444_bits__div__by__1,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),one_one_int) = A ).
% bits_div_by_1
tff(fact_445_times__divide__eq__left,axiom,
! [B: complex,C: complex,A: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(B),C)),A) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(B),A)),C) ).
% times_divide_eq_left
tff(fact_446_times__divide__eq__left,axiom,
! [B: real,C: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),C) ).
% times_divide_eq_left
tff(fact_447_times__divide__eq__left,axiom,
! [B: rat,C: rat,A: rat] : aa_rat_rat(times_times_rat(divide_divide_rat(B,C)),A) = divide_divide_rat(aa_rat_rat(times_times_rat(B),A),C) ).
% times_divide_eq_left
tff(fact_448_deg__deg__n,axiom,
! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary)),N))
=> ( Deg = N ) ) ).
% deg_deg_n
tff(fact_449__C5_Ohyps_C_I1_J,axiom,
pp(aa_nat_bool(vEBT_invar_vebt(summary),m)) ).
% "5.hyps"(1)
tff(fact_450_times__divide__eq__right,axiom,
! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B)),C) ).
% times_divide_eq_right
tff(fact_451_times__divide__eq__right,axiom,
! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C) ).
% times_divide_eq_right
tff(fact_452_times__divide__eq__right,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),divide_divide_rat(B,C)) = divide_divide_rat(aa_rat_rat(times_times_rat(A),B),C) ).
% times_divide_eq_right
tff(fact_453_divide__divide__eq__right,axiom,
! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(divide1717551699836669952omplex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),B) ).
% divide_divide_eq_right
tff(fact_454_divide__divide__eq__right,axiom,
! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B) ).
% divide_divide_eq_right
tff(fact_455_divide__divide__eq__right,axiom,
! [A: rat,B: rat,C: rat] : divide_divide_rat(A,divide_divide_rat(B,C)) = divide_divide_rat(aa_rat_rat(times_times_rat(A),C),B) ).
% divide_divide_eq_right
tff(fact_456_divide__divide__eq__left,axiom,
! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(A),B)),C) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(B),C)) ).
% divide_divide_eq_left
tff(fact_457_divide__divide__eq__left,axiom,
! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).
% divide_divide_eq_left
tff(fact_458_divide__divide__eq__left,axiom,
! [A: rat,B: rat,C: rat] : divide_divide_rat(divide_divide_rat(A,B),C) = divide_divide_rat(A,aa_rat_rat(times_times_rat(B),C)) ).
% divide_divide_eq_left
tff(fact_459_two__powr__height__bound__deg,axiom,
! [T: vEBT_VEBT,N: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ).
% two_powr_height_bound_deg
tff(fact_460_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,aa_num_num(bit0,V))),aa_num_int(numeral_numeral_int,aa_num_num(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_461_enat__less__induct,axiom,
! [P: fun_Ex8563670519792538868t_bool,N: extended_enat] :
( ! [N2: extended_enat] :
( ! [M2: extended_enat] :
( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(M2),N2))
=> pp(aa_Ext2772979874139974381t_bool(P,M2)) )
=> pp(aa_Ext2772979874139974381t_bool(P,N2)) )
=> pp(aa_Ext2772979874139974381t_bool(P,N)) ) ).
% enat_less_induct
tff(fact_462_insertsimp,axiom,
! [T: vEBT_VEBT,N: nat,L: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t(T),L)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))) ) ) ).
% insertsimp
tff(fact_463_linordered__field__no__lb,axiom,
! [X4: real] :
? [Y2: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y2),X4)) ).
% linordered_field_no_lb
tff(fact_464_linordered__field__no__lb,axiom,
! [X4: rat] :
? [Y2: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y2),X4)) ).
% linordered_field_no_lb
tff(fact_465_linordered__field__no__ub,axiom,
! [X4: real] :
? [X_1: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),X_1)) ).
% linordered_field_no_ub
tff(fact_466_linordered__field__no__ub,axiom,
! [X4: rat] :
? [X_1: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),X_1)) ).
% linordered_field_no_ub
tff(fact_467_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ).
% linorder_neqE_linordered_idom
tff(fact_468_linorder__neqE__linordered__idom,axiom,
! [X: rat,Y: rat] :
( ( X != Y )
=> ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),X)) ) ) ).
% linorder_neqE_linordered_idom
tff(fact_469_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).
% linorder_neqE_linordered_idom
tff(fact_470_height__node,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)))) ) ).
% height_node
tff(fact_471_combine__common__factor,axiom,
! [A: real,E: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),E)),C) ).
% combine_common_factor
tff(fact_472_combine__common__factor,axiom,
! [A: rat,E: rat,B: rat,C: rat] : plus_plus_rat(aa_rat_rat(times_times_rat(A),E),plus_plus_rat(aa_rat_rat(times_times_rat(B),E),C)) = plus_plus_rat(aa_rat_rat(times_times_rat(plus_plus_rat(A,B)),E),C) ).
% combine_common_factor
tff(fact_473_combine__common__factor,axiom,
! [A: nat,E: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),E)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),E)),C)) = 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,A),B)),E)),C) ).
% combine_common_factor
tff(fact_474_combine__common__factor,axiom,
! [A: int,E: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),E)),C) ).
% combine_common_factor
tff(fact_475_distrib__right,axiom,
! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).
% distrib_right
tff(fact_476_distrib__right,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(plus_plus_rat(A,B)),C) = plus_plus_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(B),C)) ).
% distrib_right
tff(fact_477_distrib__right,axiom,
! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).
% distrib_right
tff(fact_478_distrib__right,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).
% distrib_right
tff(fact_479_distrib__left,axiom,
! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).
% distrib_left
tff(fact_480_distrib__left,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),plus_plus_rat(B,C)) = plus_plus_rat(aa_rat_rat(times_times_rat(A),B),aa_rat_rat(times_times_rat(A),C)) ).
% distrib_left
tff(fact_481_distrib__left,axiom,
! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).
% distrib_left
tff(fact_482_distrib__left,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).
% distrib_left
tff(fact_483_comm__semiring__class_Odistrib,axiom,
! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).
% comm_semiring_class.distrib
tff(fact_484_comm__semiring__class_Odistrib,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(plus_plus_rat(A,B)),C) = plus_plus_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(B),C)) ).
% comm_semiring_class.distrib
tff(fact_485_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).
% comm_semiring_class.distrib
tff(fact_486_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).
% comm_semiring_class.distrib
tff(fact_487_ring__class_Oring__distribs_I1_J,axiom,
! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).
% ring_class.ring_distribs(1)
tff(fact_488_ring__class_Oring__distribs_I1_J,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),plus_plus_rat(B,C)) = plus_plus_rat(aa_rat_rat(times_times_rat(A),B),aa_rat_rat(times_times_rat(A),C)) ).
% ring_class.ring_distribs(1)
tff(fact_489_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).
% ring_class.ring_distribs(1)
tff(fact_490_ring__class_Oring__distribs_I2_J,axiom,
! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).
% ring_class.ring_distribs(2)
tff(fact_491_ring__class_Oring__distribs_I2_J,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(plus_plus_rat(A,B)),C) = plus_plus_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(B),C)) ).
% ring_class.ring_distribs(2)
tff(fact_492_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).
% ring_class.ring_distribs(2)
tff(fact_493_two__realpow__ge__one,axiom,
! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),N))) ).
% two_realpow_ge_one
tff(fact_494_divide__divide__eq__left_H,axiom,
! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(A),B)),C) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(C),B)) ).
% divide_divide_eq_left'
tff(fact_495_divide__divide__eq__left_H,axiom,
! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) ).
% divide_divide_eq_left'
tff(fact_496_divide__divide__eq__left_H,axiom,
! [A: rat,B: rat,C: rat] : divide_divide_rat(divide_divide_rat(A,B),C) = divide_divide_rat(A,aa_rat_rat(times_times_rat(C),B)) ).
% divide_divide_eq_left'
tff(fact_497_divide__divide__times__eq,axiom,
! [X: complex,Y: complex,Z: complex,W: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(Z),W)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(X),W)),aa_complex_complex(times_times_complex(Y),Z)) ).
% divide_divide_times_eq
tff(fact_498_divide__divide__times__eq,axiom,
! [X: real,Y: real,Z: real,W: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Z),W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),W)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ).
% divide_divide_times_eq
tff(fact_499_divide__divide__times__eq,axiom,
! [X: rat,Y: rat,Z: rat,W: rat] : divide_divide_rat(divide_divide_rat(X,Y),divide_divide_rat(Z,W)) = divide_divide_rat(aa_rat_rat(times_times_rat(X),W),aa_rat_rat(times_times_rat(Y),Z)) ).
% divide_divide_times_eq
tff(fact_500_times__divide__times__eq,axiom,
! [X: complex,Y: complex,Z: complex,W: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(Z),W)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(X),Z)),aa_complex_complex(times_times_complex(Y),W)) ).
% times_divide_times_eq
tff(fact_501_times__divide__times__eq,axiom,
! [X: real,Y: real,Z: real,W: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Z),W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),W)) ).
% times_divide_times_eq
tff(fact_502_times__divide__times__eq,axiom,
! [X: rat,Y: rat,Z: rat,W: rat] : aa_rat_rat(times_times_rat(divide_divide_rat(X,Y)),divide_divide_rat(Z,W)) = divide_divide_rat(aa_rat_rat(times_times_rat(X),Z),aa_rat_rat(times_times_rat(Y),W)) ).
% times_divide_times_eq
tff(fact_503_add__divide__distrib,axiom,
! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),B)),C) = aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),C)),aa_complex_complex(divide1717551699836669952omplex(B),C)) ).
% add_divide_distrib
tff(fact_504_add__divide__distrib,axiom,
! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ).
% add_divide_distrib
tff(fact_505_add__divide__distrib,axiom,
! [A: rat,B: rat,C: rat] : divide_divide_rat(plus_plus_rat(A,B),C) = plus_plus_rat(divide_divide_rat(A,C),divide_divide_rat(B,C)) ).
% add_divide_distrib
tff(fact_506_less__1__mult,axiom,
! [M: real,N: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),M))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),N))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,M),N))) ) ) ).
% less_1_mult
tff(fact_507_less__1__mult,axiom,
! [M: rat,N: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),M))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),N))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(M),N))) ) ) ).
% less_1_mult
tff(fact_508_less__1__mult,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))) ) ) ).
% less_1_mult
tff(fact_509_less__1__mult,axiom,
! [M: int,N: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),M))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,M),N))) ) ) ).
% less_1_mult
tff(fact_510_add__mono1,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),one_one_real))) ) ).
% add_mono1
tff(fact_511_add__mono1,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(A,one_one_rat)),plus_plus_rat(B,one_one_rat))) ) ).
% add_mono1
tff(fact_512_add__mono1,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),one_one_nat))) ) ).
% add_mono1
tff(fact_513_add__mono1,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int))) ) ).
% add_mono1
tff(fact_514_less__add__one,axiom,
! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real))) ).
% less_add_one
tff(fact_515_less__add__one,axiom,
! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),plus_plus_rat(A,one_one_rat))) ).
% less_add_one
tff(fact_516_less__add__one,axiom,
! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat))) ).
% less_add_one
tff(fact_517_less__add__one,axiom,
! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int))) ).
% less_add_one
tff(fact_518_L2__set__mult__ineq__lemma,axiom,
! [A: real,C: real,B: real,D: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,C),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).
% L2_set_mult_ineq_lemma
tff(fact_519_four__x__squared,axiom,
! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% four_x_squared
tff(fact_520_discrete,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B)) ) ).
% discrete
tff(fact_521_discrete,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),B)) ) ).
% discrete
tff(fact_522_less__half__sum,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real)))) ) ).
% less_half_sum
tff(fact_523_less__half__sum,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),divide_divide_rat(plus_plus_rat(A,B),plus_plus_rat(one_one_rat,one_one_rat)))) ) ).
% less_half_sum
tff(fact_524_gt__half__sum,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))),B)) ) ).
% gt_half_sum
tff(fact_525_gt__half__sum,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(plus_plus_rat(A,B),plus_plus_rat(one_one_rat,one_one_rat))),B)) ) ).
% gt_half_sum
tff(fact_526_field__sum__of__halves,axiom,
! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),numeral_numeral_real(aa_num_num(bit0,one)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),numeral_numeral_real(aa_num_num(bit0,one)))) = X ).
% field_sum_of_halves
tff(fact_527_field__sum__of__halves,axiom,
! [X: rat] : plus_plus_rat(divide_divide_rat(X,numeral_numeral_rat(aa_num_num(bit0,one))),divide_divide_rat(X,numeral_numeral_rat(aa_num_num(bit0,one)))) = X ).
% field_sum_of_halves
tff(fact_528__C2_C,axiom,
pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary)),xa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one)))))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),mi)),mi),xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),mi)),mi),xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(vEBT_T_i_n_s_e_r_t(summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),mi)),mi),xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat)))) ).
% "2"
tff(fact_529__092_060open_062T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_A_ItreeList_A_B_Ahigh_A_Iif_Ax_A_060_Ami_Athen_Ami_Aelse_Ax_J_A_Ideg_Adiv_A2_J_J_A_Ilow_A_Iif_Ax_A_060_Ami_Athen_Ami_Aelse_Ax_J_A_Ideg_Adiv_A2_J_J_A_092_060le_062_A3_092_060close_062,axiom,
pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),mi)),mi),xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),mi)),mi),xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))) ).
% \<open>T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t (treeList ! high (if x < mi then mi else x) (deg div 2)) (low (if x < mi then mi else x) (deg div 2)) \<le> 3\<close>
tff(fact_530_post__member__pre__member,axiom,
! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
=> ( pp(aa_nat_bool(vEBT_vebt_member(vEBT_vebt_insert(T,X)),Y))
=> ( pp(aa_nat_bool(vEBT_vebt_member(T),Y))
| ( X = Y ) ) ) ) ) ) ).
% post_member_pre_member
tff(fact_531__C5_OIH_C_I1_J,axiom,
! [X4: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(treeList)))
=> ( pp(aa_nat_bool(vEBT_invar_vebt(X4),na))
& ! [Xa: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t(X4),Xa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,X4))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one)))))))) ) ) ).
% "5.IH"(1)
tff(fact_532__C1_C,axiom,
aa_nat_nat(vEBT_T_i_n_s_e_r_t(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary)),xa) = 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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one)))))),aa_nat_nat(vEBT_T_i_n_s_e_r_t(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),mi)),mi),xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),mi)),mi),xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),mi)),mi),xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),mi)),mi),xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(vEBT_T_i_n_s_e_r_t(summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),mi)),mi),xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat)) ).
% "1"
tff(fact_533_member__bound,axiom,
! [Tree: vEBT_VEBT,X: nat,N: nat] :
( pp(aa_nat_bool(vEBT_vebt_member(Tree),X))
=> ( pp(aa_nat_bool(vEBT_invar_vebt(Tree),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ) ).
% member_bound
tff(fact_534_mult__1,axiom,
! [A: complex] : aa_complex_complex(times_times_complex(one_one_complex),A) = A ).
% mult_1
tff(fact_535_mult__1,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ).
% mult_1
tff(fact_536_mult__1,axiom,
! [A: rat] : aa_rat_rat(times_times_rat(one_one_rat),A) = A ).
% mult_1
tff(fact_537_mult__1,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),A) = A ).
% mult_1
tff(fact_538_mult__1,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),A) = A ).
% mult_1
tff(fact_539_mult_Oright__neutral,axiom,
! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).
% mult.right_neutral
tff(fact_540_mult_Oright__neutral,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),one_one_real) = A ).
% mult.right_neutral
tff(fact_541_mult_Oright__neutral,axiom,
! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).
% mult.right_neutral
tff(fact_542_mult_Oright__neutral,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),one_one_nat) = A ).
% mult.right_neutral
tff(fact_543_mult_Oright__neutral,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),one_one_int) = A ).
% mult.right_neutral
tff(fact_544_add__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).
% add_less_cancel_right
tff(fact_545_add__less__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(A,C)),plus_plus_rat(B,C)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).
% add_less_cancel_right
tff(fact_546_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).
% add_less_cancel_right
tff(fact_547_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).
% add_less_cancel_right
tff(fact_548_add__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).
% add_less_cancel_left
tff(fact_549_add__less__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(C,A)),plus_plus_rat(C,B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).
% add_less_cancel_left
tff(fact_550_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).
% add_less_cancel_left
tff(fact_551_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).
% add_less_cancel_left
tff(fact_552_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).
% add_le_cancel_right
tff(fact_553_add__le__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(A,C)),plus_plus_rat(B,C)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).
% add_le_cancel_right
tff(fact_554_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).
% add_le_cancel_right
tff(fact_555_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).
% add_le_cancel_right
tff(fact_556_min__Null__member,axiom,
! [T: vEBT_VEBT,X: nat] :
( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
=> ~ pp(aa_nat_bool(vEBT_vebt_member(T),X)) ) ).
% min_Null_member
tff(fact_557__C5_Ohyps_C_I4_J,axiom,
deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,na),m) ).
% "5.hyps"(4)
tff(fact_558_bit__split__inv,axiom,
! [X: nat,D: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_bit_concat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),D)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),D)),D) = X ).
% bit_split_inv
tff(fact_559_add__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
<=> ( B = C ) ) ).
% add_left_cancel
tff(fact_560_add__left__cancel,axiom,
! [A: rat,B: rat,C: rat] :
( ( plus_plus_rat(A,B) = plus_plus_rat(A,C) )
<=> ( B = C ) ) ).
% add_left_cancel
tff(fact_561_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
<=> ( B = C ) ) ).
% add_left_cancel
tff(fact_562_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
<=> ( B = C ) ) ).
% add_left_cancel
tff(fact_563_add__right__cancel,axiom,
! [B: real,A: real,C: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
<=> ( B = C ) ) ).
% add_right_cancel
tff(fact_564_add__right__cancel,axiom,
! [B: rat,A: rat,C: rat] :
( ( plus_plus_rat(B,A) = plus_plus_rat(C,A) )
<=> ( B = C ) ) ).
% add_right_cancel
tff(fact_565_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
<=> ( B = C ) ) ).
% add_right_cancel
tff(fact_566_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
<=> ( B = C ) ) ).
% add_right_cancel
tff(fact_567_real__divide__square__eq,axiom,
! [R: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),R)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),R) ).
% real_divide_square_eq
tff(fact_568_height__compose__child,axiom,
! [T: vEBT_VEBT,TreeList: list_VEBT_VEBT,Info: option4927543243414619207at_nat,Deg: nat,Summary: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,T),set_VEBT_VEBT2(TreeList)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,vEBT_Node(Info,Deg,TreeList,Summary)))) ) ).
% height_compose_child
tff(fact_569_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).
% add_le_cancel_left
tff(fact_570_add__le__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(C,A)),plus_plus_rat(C,B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).
% add_le_cancel_left
tff(fact_571_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).
% add_le_cancel_left
tff(fact_572_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).
% add_le_cancel_left
tff(fact_573_low__inv,axiom,
! [X: nat,N: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),X)),N) = X ) ) ).
% low_inv
tff(fact_574_member__correct,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( pp(aa_nat_bool(vEBT_vebt_member(T),X))
<=> pp(aa_set_nat_bool(member_nat(X),vEBT_set_vebt(T))) ) ) ).
% member_correct
tff(fact_575__C5_Ohyps_C_I3_J,axiom,
m = aa_nat_nat(suc,na) ).
% "5.hyps"(3)
tff(fact_576__C5_Ohyps_C_I2_J,axiom,
size_s6755466524823107622T_VEBT(treeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) ).
% "5.hyps"(2)
tff(fact_577__C5_Ohyps_C_I6_J,axiom,
( ( mi = ma )
=> ! [X4: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(treeList)))
=> ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_12)) ) ) ).
% "5.hyps"(6)
tff(fact_578__C5_Ohyps_C_I5_J,axiom,
! [I3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m)))
=> ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList,I3)),X_13))
<=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,summary),I3)) ) ) ).
% "5.hyps"(5)
tff(fact_579__C5_Ohyps_C_I9_J,axiom,
( ( mi != ma )
=> ! [I3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m)))
=> ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ma),na) = I3 )
=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,ma),na))) )
& ! [X4: nat] :
( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),na) = I3 )
& pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(treeList,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),na))) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,mi),X4))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),ma)) ) ) ) ) ) ).
% "5.hyps"(9)
tff(fact_580_real__arch__pow,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
=> ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N2))) ) ).
% real_arch_pow
tff(fact_581_less__eq__real__def,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
<=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
| ( X = Y ) ) ) ).
% less_eq_real_def
tff(fact_582_complete__real,axiom,
! [S2: set_real] :
( ? [X4: real] : pp(aa_set_real_bool(member_real(X4),S2))
=> ( ? [Z2: real] :
! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),S2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Z2)) )
=> ? [Y2: real] :
( ! [X4: real] :
( pp(aa_set_real_bool(member_real(X4),S2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),Y2)) )
& ! [Z2: real] :
( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),S2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Z2)) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y2),Z2)) ) ) ) ) ).
% complete_real
tff(fact_583_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).
% ab_semigroup_mult_class.mult_ac(1)
tff(fact_584_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A),B)),C) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C)) ).
% ab_semigroup_mult_class.mult_ac(1)
tff(fact_585_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: 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,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).
% ab_semigroup_mult_class.mult_ac(1)
tff(fact_586_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).
% ab_semigroup_mult_class.mult_ac(1)
tff(fact_587_mult_Oassoc,axiom,
! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).
% mult.assoc
tff(fact_588_mult_Oassoc,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A),B)),C) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C)) ).
% mult.assoc
tff(fact_589_mult_Oassoc,axiom,
! [A: nat,B: 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,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).
% mult.assoc
tff(fact_590_mult_Oassoc,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).
% mult.assoc
tff(fact_591_mult_Ocommute,axiom,
! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A) ).
% mult.commute
tff(fact_592_mult_Ocommute,axiom,
! [A: rat,B: rat] : aa_rat_rat(times_times_rat(A),B) = aa_rat_rat(times_times_rat(B),A) ).
% mult.commute
tff(fact_593_mult_Ocommute,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A) ).
% mult.commute
tff(fact_594_mult_Ocommute,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = aa_int_int(aa_int_fun_int_int(times_times_int,B),A) ).
% mult.commute
tff(fact_595_mult_Oleft__commute,axiom,
! [B: real,A: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).
% mult.left_commute
tff(fact_596_mult_Oleft__commute,axiom,
! [B: rat,A: rat,C: rat] : aa_rat_rat(times_times_rat(B),aa_rat_rat(times_times_rat(A),C)) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C)) ).
% mult.left_commute
tff(fact_597_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).
% mult.left_commute
tff(fact_598_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).
% mult.left_commute
tff(fact_599_one__reorient,axiom,
! [X: complex] :
( ( one_one_complex = X )
<=> ( X = one_one_complex ) ) ).
% one_reorient
tff(fact_600_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
<=> ( X = one_one_real ) ) ).
% one_reorient
tff(fact_601_one__reorient,axiom,
! [X: rat] :
( ( one_one_rat = X )
<=> ( X = one_one_rat ) ) ).
% one_reorient
tff(fact_602_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
<=> ( X = one_one_nat ) ) ).
% one_reorient
tff(fact_603_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
<=> ( X = one_one_int ) ) ).
% one_reorient
tff(fact_604_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).
% ab_semigroup_add_class.add_ac(1)
tff(fact_605_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: rat,B: rat,C: rat] : plus_plus_rat(plus_plus_rat(A,B),C) = plus_plus_rat(A,plus_plus_rat(B,C)) ).
% ab_semigroup_add_class.add_ac(1)
tff(fact_606_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: 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,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).
% ab_semigroup_add_class.add_ac(1)
tff(fact_607_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).
% ab_semigroup_add_class.add_ac(1)
tff(fact_608_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_609_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_610_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_611_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L) ) ) ).
% add_mono_thms_linordered_semiring(4)
tff(fact_612_group__cancel_Oadd1,axiom,
! [A2: real,K: real,A: real,B: real] :
( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).
% group_cancel.add1
tff(fact_613_group__cancel_Oadd1,axiom,
! [A2: rat,K: rat,A: rat,B: rat] :
( ( A2 = plus_plus_rat(K,A) )
=> ( plus_plus_rat(A2,B) = plus_plus_rat(K,plus_plus_rat(A,B)) ) ) ).
% group_cancel.add1
tff(fact_614_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).
% group_cancel.add1
tff(fact_615_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).
% group_cancel.add1
tff(fact_616_group__cancel_Oadd2,axiom,
! [B2: real,K: real,B: real,A: real] :
( ( B2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B) )
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ) ) ).
% group_cancel.add2
tff(fact_617_group__cancel_Oadd2,axiom,
! [B2: rat,K: rat,B: rat,A: rat] :
( ( B2 = plus_plus_rat(K,B) )
=> ( plus_plus_rat(A,B2) = plus_plus_rat(K,plus_plus_rat(A,B)) ) ) ).
% group_cancel.add2
tff(fact_618_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),B) )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).
% group_cancel.add2
tff(fact_619_group__cancel_Oadd2,axiom,
! [B2: int,K: int,B: int,A: int] :
( ( B2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B) )
=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).
% group_cancel.add2
tff(fact_620_add_Oassoc,axiom,
! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).
% add.assoc
tff(fact_621_add_Oassoc,axiom,
! [A: rat,B: rat,C: rat] : plus_plus_rat(plus_plus_rat(A,B),C) = plus_plus_rat(A,plus_plus_rat(B,C)) ).
% add.assoc
tff(fact_622_add_Oassoc,axiom,
! [A: nat,B: 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,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).
% add.assoc
tff(fact_623_add_Oassoc,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).
% add.assoc
tff(fact_624_add_Oleft__cancel,axiom,
! [A: real,B: real,C: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
<=> ( B = C ) ) ).
% add.left_cancel
tff(fact_625_add_Oleft__cancel,axiom,
! [A: rat,B: rat,C: rat] :
( ( plus_plus_rat(A,B) = plus_plus_rat(A,C) )
<=> ( B = C ) ) ).
% add.left_cancel
tff(fact_626_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
<=> ( B = C ) ) ).
% add.left_cancel
tff(fact_627_add_Oright__cancel,axiom,
! [B: real,A: real,C: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
<=> ( B = C ) ) ).
% add.right_cancel
tff(fact_628_add_Oright__cancel,axiom,
! [B: rat,A: rat,C: rat] :
( ( plus_plus_rat(B,A) = plus_plus_rat(C,A) )
<=> ( B = C ) ) ).
% add.right_cancel
tff(fact_629_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
<=> ( B = C ) ) ).
% add.right_cancel
tff(fact_630_add_Ocommute,axiom,
! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) ).
% add.commute
tff(fact_631_add_Ocommute,axiom,
! [A: rat,B: rat] : plus_plus_rat(A,B) = plus_plus_rat(B,A) ).
% add.commute
tff(fact_632_add_Ocommute,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) ).
% add.commute
tff(fact_633_add_Ocommute,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) ).
% add.commute
tff(fact_634_add_Oleft__commute,axiom,
! [B: real,A: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).
% add.left_commute
tff(fact_635_add_Oleft__commute,axiom,
! [B: rat,A: rat,C: rat] : plus_plus_rat(B,plus_plus_rat(A,C)) = plus_plus_rat(A,plus_plus_rat(B,C)) ).
% add.left_commute
tff(fact_636_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).
% add.left_commute
tff(fact_637_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).
% add.left_commute
tff(fact_638_add__left__imp__eq,axiom,
! [A: real,B: real,C: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
=> ( B = C ) ) ).
% add_left_imp_eq
tff(fact_639_add__left__imp__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( plus_plus_rat(A,B) = plus_plus_rat(A,C) )
=> ( B = C ) ) ).
% add_left_imp_eq
tff(fact_640_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
=> ( B = C ) ) ).
% add_left_imp_eq
tff(fact_641_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
=> ( B = C ) ) ).
% add_left_imp_eq
tff(fact_642_add__right__imp__eq,axiom,
! [B: real,A: real,C: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
=> ( B = C ) ) ).
% add_right_imp_eq
tff(fact_643_add__right__imp__eq,axiom,
! [B: rat,A: rat,C: rat] :
( ( plus_plus_rat(B,A) = plus_plus_rat(C,A) )
=> ( B = C ) ) ).
% add_right_imp_eq
tff(fact_644_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
=> ( B = C ) ) ).
% add_right_imp_eq
tff(fact_645_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
=> ( B = C ) ) ).
% add_right_imp_eq
tff(fact_646_minNull__bound,axiom,
! [T: vEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,T)),one_one_nat)) ).
% minNull_bound
tff(fact_647_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I5_J,axiom,
! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,vEBT_Node(some_P7363390416028606310at_nat(Uz),Va,Vb,Vc)) = one_one_nat ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(5)
tff(fact_648_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),J))
& ( K = L ) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_649_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),J))
& ( K = L ) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(I,K)),plus_plus_rat(J,L))) ) ).
% add_mono_thms_linordered_semiring(3)
tff(fact_650_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
& ( K = L ) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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_651_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
& ( K = L ) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).
% add_mono_thms_linordered_semiring(3)
tff(fact_652_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K),L)) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_653_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( I = J )
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K),L)) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(I,K)),plus_plus_rat(J,L))) ) ).
% add_mono_thms_linordered_semiring(2)
tff(fact_654_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L)) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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_655_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),L)) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).
% add_mono_thms_linordered_semiring(2)
tff(fact_656_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),J))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K),L)) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_657_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),J))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K),L)) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(I,K)),plus_plus_rat(J,L))) ) ).
% add_mono_thms_linordered_semiring(1)
tff(fact_658_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L)) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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_659_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),L)) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).
% add_mono_thms_linordered_semiring(1)
tff(fact_660_add__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D))) ) ) ).
% add_mono
tff(fact_661_add__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(A,C)),plus_plus_rat(B,D))) ) ) ).
% add_mono
tff(fact_662_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).
% add_mono
tff(fact_663_add__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D))) ) ) ).
% add_mono
tff(fact_664_add__left__mono,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))) ) ).
% add_left_mono
tff(fact_665_add__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(C,A)),plus_plus_rat(C,B))) ) ).
% add_left_mono
tff(fact_666_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))) ) ).
% add_left_mono
tff(fact_667_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))) ) ).
% add_left_mono
tff(fact_668_less__eqE,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ~ ! [C2: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) ) ).
% less_eqE
tff(fact_669_add__right__mono,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))) ) ).
% add_right_mono
tff(fact_670_add__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(A,C)),plus_plus_rat(B,C))) ) ).
% add_right_mono
tff(fact_671_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))) ) ).
% add_right_mono
tff(fact_672_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))) ) ).
% add_right_mono
tff(fact_673_le__iff__add,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
<=> ? [C3: nat] : B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) ) ).
% le_iff_add
tff(fact_674_add__le__imp__le__left,axiom,
! [C: real,A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).
% add_le_imp_le_left
tff(fact_675_add__le__imp__le__left,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(C,A)),plus_plus_rat(C,B)))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).
% add_le_imp_le_left
tff(fact_676_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).
% add_le_imp_le_left
tff(fact_677_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).
% add_le_imp_le_left
tff(fact_678_add__le__imp__le__right,axiom,
! [A: real,C: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).
% add_le_imp_le_right
tff(fact_679_add__le__imp__le__right,axiom,
! [A: rat,C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(A,C)),plus_plus_rat(B,C)))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).
% add_le_imp_le_right
tff(fact_680_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).
% add_le_imp_le_right
tff(fact_681_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).
% add_le_imp_le_right
tff(fact_682_add__mono__thms__linordered__field_I5_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),J))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K),L)) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_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_field(5)
tff(fact_683_add__mono__thms__linordered__field_I5_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),J))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K),L)) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(I,K)),plus_plus_rat(J,L))) ) ).
% add_mono_thms_linordered_field(5)
tff(fact_684_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L)) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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_mono_thms_linordered_field(5)
tff(fact_685_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L)) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).
% add_mono_thms_linordered_field(5)
tff(fact_686_add__mono__thms__linordered__field_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K),L)) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_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_field(2)
tff(fact_687_add__mono__thms__linordered__field_I2_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( ( I = J )
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K),L)) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(I,K)),plus_plus_rat(J,L))) ) ).
% add_mono_thms_linordered_field(2)
tff(fact_688_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L)) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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_mono_thms_linordered_field(2)
tff(fact_689_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L)) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).
% add_mono_thms_linordered_field(2)
tff(fact_690_add__mono__thms__linordered__field_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),J))
& ( K = L ) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_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_field(1)
tff(fact_691_add__mono__thms__linordered__field_I1_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),J))
& ( K = L ) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(I,K)),plus_plus_rat(J,L))) ) ).
% add_mono_thms_linordered_field(1)
tff(fact_692_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
& ( K = L ) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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_mono_thms_linordered_field(1)
tff(fact_693_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
& ( K = L ) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).
% add_mono_thms_linordered_field(1)
tff(fact_694_add__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D))) ) ) ).
% add_strict_mono
tff(fact_695_add__strict__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(A,C)),plus_plus_rat(B,D))) ) ) ).
% add_strict_mono
tff(fact_696_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).
% add_strict_mono
tff(fact_697_add__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D))) ) ) ).
% add_strict_mono
tff(fact_698_add__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))) ) ).
% add_strict_left_mono
tff(fact_699_add__strict__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(C,A)),plus_plus_rat(C,B))) ) ).
% add_strict_left_mono
tff(fact_700_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))) ) ).
% add_strict_left_mono
tff(fact_701_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))) ) ).
% add_strict_left_mono
tff(fact_702_add__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))) ) ).
% add_strict_right_mono
tff(fact_703_add__strict__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(A,C)),plus_plus_rat(B,C))) ) ).
% add_strict_right_mono
tff(fact_704_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))) ) ).
% add_strict_right_mono
tff(fact_705_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))) ) ).
% add_strict_right_mono
tff(fact_706_add__less__imp__less__left,axiom,
! [C: real,A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).
% add_less_imp_less_left
tff(fact_707_add__less__imp__less__left,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(C,A)),plus_plus_rat(C,B)))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).
% add_less_imp_less_left
tff(fact_708_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).
% add_less_imp_less_left
tff(fact_709_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).
% add_less_imp_less_left
tff(fact_710_add__less__imp__less__right,axiom,
! [A: real,C: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).
% add_less_imp_less_right
tff(fact_711_add__less__imp__less__right,axiom,
! [A: rat,C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(A,C)),plus_plus_rat(B,C)))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).
% add_less_imp_less_right
tff(fact_712_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ).
% add_less_imp_less_right
tff(fact_713_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).
% add_less_imp_less_right
tff(fact_714_comm__monoid__mult__class_Omult__1,axiom,
! [A: complex] : aa_complex_complex(times_times_complex(one_one_complex),A) = A ).
% comm_monoid_mult_class.mult_1
tff(fact_715_comm__monoid__mult__class_Omult__1,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ).
% comm_monoid_mult_class.mult_1
tff(fact_716_comm__monoid__mult__class_Omult__1,axiom,
! [A: rat] : aa_rat_rat(times_times_rat(one_one_rat),A) = A ).
% comm_monoid_mult_class.mult_1
tff(fact_717_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),A) = A ).
% comm_monoid_mult_class.mult_1
tff(fact_718_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),A) = A ).
% comm_monoid_mult_class.mult_1
tff(fact_719_mult_Ocomm__neutral,axiom,
! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).
% mult.comm_neutral
tff(fact_720_mult_Ocomm__neutral,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),one_one_real) = A ).
% mult.comm_neutral
tff(fact_721_mult_Ocomm__neutral,axiom,
! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).
% mult.comm_neutral
tff(fact_722_mult_Ocomm__neutral,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),one_one_nat) = A ).
% mult.comm_neutral
tff(fact_723_mult_Ocomm__neutral,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),one_one_int) = A ).
% mult.comm_neutral
tff(fact_724_add__mono__thms__linordered__field_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),J))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,K),L)) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_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_field(4)
tff(fact_725_add__mono__thms__linordered__field_I4_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),J))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,K),L)) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(I,K)),plus_plus_rat(J,L))) ) ).
% add_mono_thms_linordered_field(4)
tff(fact_726_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),L)) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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_mono_thms_linordered_field(4)
tff(fact_727_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L)) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).
% add_mono_thms_linordered_field(4)
tff(fact_728_add__mono__thms__linordered__field_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),J))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,K),L)) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_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_field(3)
tff(fact_729_add__mono__thms__linordered__field_I3_J,axiom,
! [I: rat,J: rat,K: rat,L: rat] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),J))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,K),L)) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(I,K)),plus_plus_rat(J,L))) ) ).
% add_mono_thms_linordered_field(3)
tff(fact_730_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),L)) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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_mono_thms_linordered_field(3)
tff(fact_731_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),L)) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).
% add_mono_thms_linordered_field(3)
tff(fact_732_add__le__less__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D))) ) ) ).
% add_le_less_mono
tff(fact_733_add__le__less__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(A,C)),plus_plus_rat(B,D))) ) ) ).
% add_le_less_mono
tff(fact_734_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).
% add_le_less_mono
tff(fact_735_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D))) ) ) ).
% add_le_less_mono
tff(fact_736_add__less__le__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D))) ) ) ).
% add_less_le_mono
tff(fact_737_add__less__le__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(A,C)),plus_plus_rat(B,D))) ) ) ).
% add_less_le_mono
tff(fact_738_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D))) ) ) ).
% add_less_le_mono
tff(fact_739_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D))) ) ) ).
% add_less_le_mono
tff(fact_740__C0_C,axiom,
aa_nat_nat(vEBT_T_i_n_s_e_r_t(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary)),xa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),mi)),mi),xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(treeList))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,xa),mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,xa),ma))))),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(vEBT_T_i_n_s_e_r_t(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),mi)),mi),xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),mi)),mi),xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),mi)),mi),xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),mi)),mi),xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(vEBT_T_i_n_s_e_r_t(summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,xa),mi)),mi),xa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat)) ).
% "0"
tff(fact_741_member__inv,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),X))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
& ( ( X = Mi )
| ( X = Ma )
| ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Ma))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
& pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ).
% member_inv
tff(fact_742_both__member__options__from__complete__tree__to__child,axiom,
! [Deg: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),Deg))
=> ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),X))
=> ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
| ( X = Mi )
| ( X = Ma ) ) ) ) ).
% both_member_options_from_complete_tree_to_child
tff(fact_743_both__member__options__ding,axiom,
! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary)),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)))
=> ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_Node(Info,Deg,TreeList,Summary)),X)) ) ) ) ).
% both_member_options_ding
tff(fact_744_set__n__deg__not__0,axiom,
! [TreeList: list_VEBT_VEBT,N: nat,M: nat] :
( ! [X3: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList)))
=> pp(aa_nat_bool(vEBT_invar_vebt(X3),N)) )
=> ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N)) ) ) ).
% set_n_deg_not_0
tff(fact_745_power__minus__is__div,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) ) ) ).
% power_minus_is_div
tff(fact_746_valid__insert__both__member__options__add,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_vebt_insert(T,X)),X)) ) ) ).
% valid_insert_both_member_options_add
tff(fact_747_valid__insert__both__member__options__pres,axiom,
! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
=> ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_vebt_insert(T,Y)),X)) ) ) ) ) ).
% valid_insert_both_member_options_pres
tff(fact_748_member__bound__height,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_T_m_e_m_b_e_r(T),X)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))) ) ).
% member_bound_height
tff(fact_749_low__def,axiom,
! [X: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).
% low_def
tff(fact_750_VEBT__internal_OminNull_Osimps_I5_J,axiom,
! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Node(some_P7363390416028606310at_nat(Uz),Va,Vb,Vc))) ).
% VEBT_internal.minNull.simps(5)
tff(fact_751_even__odd__cases,axiom,
! [X: nat] :
( ! [N2: nat] : X != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N2)
=> ~ ! [N2: nat] : X != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(suc,N2)) ) ).
% even_odd_cases
tff(fact_752_not__min__Null__member,axiom,
! [T: vEBT_VEBT] :
( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
=> ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X_1)) ) ).
% not_min_Null_member
tff(fact_753_deg__SUcn__Node,axiom,
! [Tree: vEBT_VEBT,N: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(Tree),aa_nat_nat(suc,aa_nat_nat(suc,N))))
=> ? [Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] : Tree = vEBT_Node(Info2,aa_nat_nat(suc,aa_nat_nat(suc,N)),TreeList2,S3) ) ).
% deg_SUcn_Node
tff(fact_754_both__member__options__equiv__member,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
<=> pp(aa_nat_bool(vEBT_vebt_member(T),X)) ) ) ).
% both_member_options_equiv_member
tff(fact_755_valid__member__both__member__options,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
=> pp(aa_nat_bool(vEBT_vebt_member(T),X)) ) ) ).
% valid_member_both_member_options
tff(fact_756_inthall,axiom,
! [Xs: list_P6011104703257516679at_nat,P: fun_Pr7757163693017731481t_bool,N: nat] :
( ! [X3: product_prod_nat_nat] :
( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),set_Pr5648618587558075414at_nat(Xs)))
=> pp(aa_Pro2232559814647311250t_bool(P,X3)) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s5460976970255530739at_nat(Xs)))
=> pp(aa_Pro2232559814647311250t_bool(P,nth_Pr7617993195940197384at_nat(Xs,N))) ) ) ).
% inthall
tff(fact_757_inthall,axiom,
! [Xs: list_complex,P: fun_complex_bool,N: nat] :
( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),set_complex2(Xs)))
=> pp(aa_complex_bool(P,X3)) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3451745648224563538omplex(Xs)))
=> pp(aa_complex_bool(P,aa_nat_complex(nth_complex(Xs),N))) ) ) ).
% inthall
tff(fact_758_inthall,axiom,
! [Xs: list_real,P: fun_real_bool,N: nat] :
( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),set_real2(Xs)))
=> pp(aa_real_bool(P,X3)) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
=> pp(aa_real_bool(P,aa_nat_real(nth_real(Xs),N))) ) ) ).
% inthall
tff(fact_759_inthall,axiom,
! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool,N: nat] :
( ! [X3: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(Xs)))
=> pp(aa_VEBT_VEBT_bool(P,X3)) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
=> pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,N))) ) ) ).
% inthall
tff(fact_760_inthall,axiom,
! [Xs: list_bool,P: fun_bool_bool,N: nat] :
( ! [X3: bool] :
( member_bool(X3,set_bool2(Xs))
=> pp(aa_bool_bool(P,X3)) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
=> pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),N))) ) ) ).
% inthall
tff(fact_761_inthall,axiom,
! [Xs: list_nat,P: fun_nat_bool,N: nat] :
( ! [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),set_nat2(Xs)))
=> pp(aa_nat_bool(P,X3)) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_nat(Xs)))
=> pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),N))) ) ) ).
% inthall
tff(fact_762_inthall,axiom,
! [Xs: list_int,P: fun_int_bool,N: nat] :
( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),set_int2(Xs)))
=> pp(aa_int_bool(P,X3)) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
=> pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),N))) ) ) ).
% inthall
tff(fact_763_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_764_nat_Oinject,axiom,
! [X23: nat,Y22: nat] :
( ( aa_nat_nat(suc,X23) = aa_nat_nat(suc,Y22) )
<=> ( X23 = Y22 ) ) ).
% nat.inject
tff(fact_765_mod__mod__trivial,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).
% mod_mod_trivial
tff(fact_766_mod__mod__trivial,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).
% mod_mod_trivial
tff(fact_767_mod__mod__trivial,axiom,
! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(modulo364778990260209775nteger(A,B),B) = modulo364778990260209775nteger(A,B) ).
% mod_mod_trivial
tff(fact_768_mi__eq__ma__no__ch,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),Deg))
=> ( ( Mi = Ma )
=> ( ! [X4: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList)))
=> ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_12)) )
& ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_12)) ) ) ) ).
% mi_eq_ma_no_ch
tff(fact_769_add__diff__cancel__right_H,axiom,
! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B) = A ).
% add_diff_cancel_right'
tff(fact_770_add__diff__cancel__right_H,axiom,
! [A: rat,B: rat] : minus_minus_rat(plus_plus_rat(A,B),B) = A ).
% add_diff_cancel_right'
tff(fact_771_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = A ).
% add_diff_cancel_right'
tff(fact_772_add__diff__cancel__right_H,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = A ).
% add_diff_cancel_right'
tff(fact_773_add__diff__cancel__right,axiom,
! [A: real,C: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ).
% add_diff_cancel_right
tff(fact_774_add__diff__cancel__right,axiom,
! [A: rat,C: rat,B: rat] : minus_minus_rat(plus_plus_rat(A,C),plus_plus_rat(B,C)) = minus_minus_rat(A,B) ).
% add_diff_cancel_right
tff(fact_775_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).
% add_diff_cancel_right
tff(fact_776_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).
% add_diff_cancel_right
tff(fact_777_add__diff__cancel__left_H,axiom,
! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),A) = B ).
% add_diff_cancel_left'
tff(fact_778_add__diff__cancel__left_H,axiom,
! [A: rat,B: rat] : minus_minus_rat(plus_plus_rat(A,B),A) = B ).
% add_diff_cancel_left'
tff(fact_779_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),A) = B ).
% add_diff_cancel_left'
tff(fact_780_add__diff__cancel__left_H,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),A) = B ).
% add_diff_cancel_left'
tff(fact_781_add__diff__cancel__left,axiom,
! [C: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ).
% add_diff_cancel_left
tff(fact_782_add__diff__cancel__left,axiom,
! [C: rat,A: rat,B: rat] : minus_minus_rat(plus_plus_rat(C,A),plus_plus_rat(C,B)) = minus_minus_rat(A,B) ).
% add_diff_cancel_left
tff(fact_783_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).
% add_diff_cancel_left
tff(fact_784_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).
% add_diff_cancel_left
tff(fact_785_diff__add__cancel,axiom,
! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),B) = A ).
% diff_add_cancel
tff(fact_786_diff__add__cancel,axiom,
! [A: rat,B: rat] : plus_plus_rat(minus_minus_rat(A,B),B) = A ).
% diff_add_cancel
tff(fact_787_diff__add__cancel,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = A ).
% diff_add_cancel
tff(fact_788_add__diff__cancel,axiom,
! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B) = A ).
% add_diff_cancel
tff(fact_789_add__diff__cancel,axiom,
! [A: rat,B: rat] : minus_minus_rat(plus_plus_rat(A,B),B) = A ).
% add_diff_cancel
tff(fact_790_add__diff__cancel,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = A ).
% add_diff_cancel
tff(fact_791_lessI,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,N))) ).
% lessI
tff(fact_792_Suc__mono,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) ) ).
% Suc_mono
tff(fact_793_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% Suc_less_eq
tff(fact_794_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),aa_nat_nat(suc,M)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ).
% Suc_le_mono
tff(fact_795_add__Suc__right,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).
% add_Suc_right
tff(fact_796_mod__add__self2,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).
% mod_add_self2
tff(fact_797_mod__add__self2,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).
% mod_add_self2
tff(fact_798_mod__add__self2,axiom,
! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),B) = modulo364778990260209775nteger(A,B) ).
% mod_add_self2
tff(fact_799_mod__add__self1,axiom,
! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).
% mod_add_self1
tff(fact_800_mod__add__self1,axiom,
! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).
% mod_add_self1
tff(fact_801_mod__add__self1,axiom,
! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(B,A),B) = modulo364778990260209775nteger(A,B) ).
% mod_add_self1
tff(fact_802_minus__mod__self2,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).
% minus_mod_self2
tff(fact_803_minus__mod__self2,axiom,
! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),B) = modulo364778990260209775nteger(A,B) ).
% minus_mod_self2
tff(fact_804_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),K) ).
% Suc_diff_diff
tff(fact_805_diff__Suc__Suc,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ).
% diff_Suc_Suc
tff(fact_806_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),I)) = I ) ) ).
% diff_diff_cancel
tff(fact_807_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ).
% diff_diff_left
tff(fact_808_mod__less,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = M ) ) ).
% mod_less
tff(fact_809_le__add__diff__inverse2,axiom,
! [B: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),B) = A ) ) ).
% le_add_diff_inverse2
tff(fact_810_le__add__diff__inverse2,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
=> ( plus_plus_rat(minus_minus_rat(A,B),B) = A ) ) ).
% le_add_diff_inverse2
tff(fact_811_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),B) = A ) ) ).
% le_add_diff_inverse2
tff(fact_812_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = A ) ) ).
% le_add_diff_inverse2
tff(fact_813_le__add__diff__inverse,axiom,
! [B: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = A ) ) ).
% le_add_diff_inverse
tff(fact_814_le__add__diff__inverse,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
=> ( plus_plus_rat(B,minus_minus_rat(A,B)) = A ) ) ).
% le_add_diff_inverse
tff(fact_815_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ) ) ).
% le_add_diff_inverse
tff(fact_816_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = A ) ) ).
% le_add_diff_inverse
tff(fact_817_right__diff__distrib__numeral,axiom,
! [V: num,B: complex,C: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(minus_minus_complex(B),C)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),B)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),C)) ).
% right_diff_distrib_numeral
tff(fact_818_right__diff__distrib__numeral,axiom,
! [V: num,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),C)) ).
% right_diff_distrib_numeral
tff(fact_819_right__diff__distrib__numeral,axiom,
! [V: num,B: rat,C: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),B),aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),C)) ).
% right_diff_distrib_numeral
tff(fact_820_right__diff__distrib__numeral,axiom,
! [V: num,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),C)) ).
% right_diff_distrib_numeral
tff(fact_821_left__diff__distrib__numeral,axiom,
! [A: complex,B: complex,V: num] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(A),B)),numera6690914467698888265omplex(V)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(B),numera6690914467698888265omplex(V))) ).
% left_diff_distrib_numeral
tff(fact_822_left__diff__distrib__numeral,axiom,
! [A: real,B: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),numeral_numeral_real(V)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),numeral_numeral_real(V))) ).
% left_diff_distrib_numeral
tff(fact_823_left__diff__distrib__numeral,axiom,
! [A: rat,B: rat,V: num] : aa_rat_rat(times_times_rat(minus_minus_rat(A,B)),numeral_numeral_rat(V)) = minus_minus_rat(aa_rat_rat(times_times_rat(A),numeral_numeral_rat(V)),aa_rat_rat(times_times_rat(B),numeral_numeral_rat(V))) ).
% left_diff_distrib_numeral
tff(fact_824_left__diff__distrib__numeral,axiom,
! [A: int,B: int,V: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_num_int(numeral_numeral_int,V))) ).
% left_diff_distrib_numeral
tff(fact_825_mod__mult__self4,axiom,
! [B: nat,C: nat,A: nat] : aa_nat_nat(aa_nat_fun_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,B),C)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).
% mod_mult_self4
tff(fact_826_mod__mult__self4,axiom,
! [B: int,C: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).
% mod_mult_self4
tff(fact_827_mod__mult__self4,axiom,
! [B: code_integer,C: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C),A),B) = modulo364778990260209775nteger(A,B) ).
% mod_mult_self4
tff(fact_828_mod__mult__self3,axiom,
! [C: nat,B: nat,A: nat] : aa_nat_nat(aa_nat_fun_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,C),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).
% mod_mult_self3
tff(fact_829_mod__mult__self3,axiom,
! [C: int,B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).
% mod_mult_self3
tff(fact_830_mod__mult__self3,axiom,
! [C: code_integer,B: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B),A),B) = modulo364778990260209775nteger(A,B) ).
% mod_mult_self3
tff(fact_831_mod__mult__self2,axiom,
! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).
% mod_mult_self2
tff(fact_832_mod__mult__self2,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).
% mod_mult_self2
tff(fact_833_mod__mult__self2,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),B) = modulo364778990260209775nteger(A,B) ).
% mod_mult_self2
tff(fact_834_mod__mult__self1,axiom,
! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).
% mod_mult_self1
tff(fact_835_mod__mult__self1,axiom,
! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).
% mod_mult_self1
tff(fact_836_mod__mult__self1,axiom,
! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)),B) = modulo364778990260209775nteger(A,B) ).
% mod_mult_self1
tff(fact_837_mult__Suc__right,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) ).
% mult_Suc_right
tff(fact_838_diff__Suc__1,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,N)),one_one_nat) = N ).
% diff_Suc_1
tff(fact_839_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_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_840_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I) = aa_nat_nat(aa_nat_fun_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_841_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_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_842_both__member__options__from__chilf__to__complete__tree,axiom,
! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),Deg))
=> ( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),X)) ) ) ) ).
% both_member_options_from_chilf_to_complete_tree
tff(fact_843_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))),I) = aa_nat_nat(aa_nat_fun_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_844_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))) = aa_nat_nat(aa_nat_fun_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_845_Suc__numeral,axiom,
! [N: num] : aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).
% Suc_numeral
tff(fact_846_Suc__mod__mult__self4,axiom,
! [N: nat,K: nat,M: nat] : aa_nat_nat(aa_nat_fun_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,N),K)),M))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).
% Suc_mod_mult_self4
tff(fact_847_Suc__mod__mult__self3,axiom,
! [K: nat,N: nat,M: nat] : aa_nat_nat(aa_nat_fun_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),N)),M))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).
% Suc_mod_mult_self3
tff(fact_848_Suc__mod__mult__self2,axiom,
! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_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,N),K)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).
% Suc_mod_mult_self2
tff(fact_849_Suc__mod__mult__self1,axiom,
! [M: nat,K: nat,N: nat] : aa_nat_nat(aa_nat_fun_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),N)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).
% Suc_mod_mult_self1
tff(fact_850_bits__one__mod__two__eq__one,axiom,
aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).
% bits_one_mod_two_eq_one
tff(fact_851_bits__one__mod__two__eq__one,axiom,
aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).
% bits_one_mod_two_eq_one
tff(fact_852_bits__one__mod__two__eq__one,axiom,
modulo364778990260209775nteger(one_one_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).
% bits_one_mod_two_eq_one
tff(fact_853_one__mod__two__eq__one,axiom,
aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).
% one_mod_two_eq_one
tff(fact_854_one__mod__two__eq__one,axiom,
aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).
% one_mod_two_eq_one
tff(fact_855_one__mod__two__eq__one,axiom,
modulo364778990260209775nteger(one_one_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).
% one_mod_two_eq_one
tff(fact_856_add__2__eq__Suc_H,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,N)) ).
% add_2_eq_Suc'
tff(fact_857_add__2__eq__Suc,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) = aa_nat_nat(suc,aa_nat_nat(suc,N)) ).
% add_2_eq_Suc
tff(fact_858_Suc__1,axiom,
aa_nat_nat(suc,one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).
% Suc_1
tff(fact_859_div2__Suc__Suc,axiom,
! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).
% div2_Suc_Suc
tff(fact_860_mod2__Suc__Suc,axiom,
! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% mod2_Suc_Suc
tff(fact_861_Suc__times__numeral__mod__eq,axiom,
! [K: num,N: nat] :
( ( aa_num_nat(numeral_numeral_nat,K) != one_one_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,K)),N))),aa_num_nat(numeral_numeral_nat,K)) = one_one_nat ) ) ).
% Suc_times_numeral_mod_eq
tff(fact_862_Suc__div__eq__add3__div__numeral,axiom,
! [M: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),aa_num_nat(numeral_numeral_nat,V)) ).
% Suc_div_eq_add3_div_numeral
tff(fact_863_div__Suc__eq__div__add3,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N)))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N)) ).
% div_Suc_eq_div_add3
tff(fact_864_Suc__mod__eq__add3__mod__numeral,axiom,
! [M: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),aa_num_nat(numeral_numeral_nat,V)) ).
% Suc_mod_eq_add3_mod_numeral
tff(fact_865_mod__Suc__eq__mod__add3,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N)))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N)) ).
% mod_Suc_eq_mod_add3
tff(fact_866_diff__right__commute,axiom,
! [A: rat,C: rat,B: rat] : minus_minus_rat(minus_minus_rat(A,C),B) = minus_minus_rat(minus_minus_rat(A,B),C) ).
% diff_right_commute
tff(fact_867_diff__right__commute,axiom,
! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C)),B) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C) ).
% diff_right_commute
tff(fact_868_diff__right__commute,axiom,
! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).
% diff_right_commute
tff(fact_869_diff__eq__diff__eq,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( minus_minus_rat(A,B) = minus_minus_rat(C,D) )
=> ( ( A = B )
<=> ( C = D ) ) ) ).
% diff_eq_diff_eq
tff(fact_870_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
=> ( ( A = B )
<=> ( C = D ) ) ) ).
% diff_eq_diff_eq
tff(fact_871_mod__Suc__Suc__eq,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),N) ).
% mod_Suc_Suc_eq
tff(fact_872_mod__Suc__eq,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).
% mod_Suc_eq
tff(fact_873_mod__diff__right__eq,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).
% mod_diff_right_eq
tff(fact_874_mod__diff__right__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(A,modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) ).
% mod_diff_right_eq
tff(fact_875_mod__diff__left__eq,axiom,
! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).
% mod_diff_left_eq
tff(fact_876_mod__diff__left__eq,axiom,
! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(modulo364778990260209775nteger(A,C),B),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) ).
% mod_diff_left_eq
tff(fact_877_mod__diff__cong,axiom,
! [A: int,C: int,A3: int,B: int,B3: int] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A3),C) )
=> ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B3),C) )
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A3),B3)),C) ) ) ) ).
% mod_diff_cong
tff(fact_878_mod__diff__cong,axiom,
! [A: code_integer,C: code_integer,A3: code_integer,B: code_integer,B3: code_integer] :
( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A3,C) )
=> ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B3,C) )
=> ( modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A3,B3),C) ) ) ) ).
% mod_diff_cong
tff(fact_879_mod__diff__eq,axiom,
! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).
% mod_diff_eq
tff(fact_880_mod__diff__eq,axiom,
! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) ).
% mod_diff_eq
tff(fact_881_zero__induct__lemma,axiom,
! [P: fun_nat_bool,K: nat,I: nat] :
( pp(aa_nat_bool(P,K))
=> ( ! [N2: nat] :
( pp(aa_nat_bool(P,aa_nat_nat(suc,N2)))
=> pp(aa_nat_bool(P,N2)) )
=> pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),I))) ) ) ).
% zero_induct_lemma
tff(fact_882_diff__commute,axiom,
! [I: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),K)),J) ).
% diff_commute
tff(fact_883_n__not__Suc__n,axiom,
! [N: nat] : N != aa_nat_nat(suc,N) ).
% n_not_Suc_n
tff(fact_884_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( aa_nat_nat(suc,X) = aa_nat_nat(suc,Y) )
=> ( X = Y ) ) ).
% Suc_inject
tff(fact_885_mod__geq,axiom,
! [M: nat,N: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ) ).
% mod_geq
tff(fact_886_mod__if,axiom,
! [M: nat,N: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = M ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ) ) ).
% mod_if
tff(fact_887_mod__induct,axiom,
! [P: fun_nat_bool,N: nat,P2: nat,M: nat] :
( pp(aa_nat_bool(P,N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),P2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),P2))
=> ( ! [N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),P2))
=> ( pp(aa_nat_bool(P,N2))
=> pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N2)),P2))) ) )
=> pp(aa_nat_bool(P,M)) ) ) ) ) ).
% mod_induct
tff(fact_888_le__mod__geq,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ) ).
% le_mod_geq
tff(fact_889_mod__Suc__le__divisor,axiom,
! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,N))),N)) ).
% mod_Suc_le_divisor
tff(fact_890_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
=> ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ) ) ).
% Suc_diff_Suc
tff(fact_891_diff__less__Suc,axiom,
! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),aa_nat_nat(suc,M))) ).
% diff_less_Suc
tff(fact_892_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ) ) ).
% Suc_diff_le
tff(fact_893_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N) ).
% diff_Suc_eq_diff_pred
tff(fact_894_minus__div__mult__eq__mod,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).
% minus_div_mult_eq_mod
tff(fact_895_minus__div__mult__eq__mod,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).
% minus_div_mult_eq_mod
tff(fact_896_minus__div__mult__eq__mod,axiom,
! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B)) = modulo364778990260209775nteger(A,B) ).
% minus_div_mult_eq_mod
tff(fact_897_minus__mod__eq__div__mult,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B) ).
% minus_mod_eq_div_mult
tff(fact_898_minus__mod__eq__div__mult,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B) ).
% minus_mod_eq_div_mult
tff(fact_899_minus__mod__eq__div__mult,axiom,
! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B) ).
% minus_mod_eq_div_mult
tff(fact_900_minus__mod__eq__mult__div,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ).
% minus_mod_eq_mult_div
tff(fact_901_minus__mod__eq__mult__div,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ).
% minus_mod_eq_mult_div
tff(fact_902_minus__mod__eq__mult__div,axiom,
! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B)) ).
% minus_mod_eq_mult_div
tff(fact_903_minus__mult__div__eq__mod,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).
% minus_mult_div_eq_mod
tff(fact_904_minus__mult__div__eq__mod,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).
% minus_mult_div_eq_mod
tff(fact_905_minus__mult__div__eq__mod,axiom,
! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B))) = modulo364778990260209775nteger(A,B) ).
% minus_mult_div_eq_mod
tff(fact_906_size__neq__size__imp__neq,axiom,
! [X: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
( ( size_s6755466524823107622T_VEBT(X) != size_s6755466524823107622T_VEBT(Y) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
tff(fact_907_size__neq__size__imp__neq,axiom,
! [X: list_bool,Y: list_bool] :
( ( size_size_list_bool(X) != size_size_list_bool(Y) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
tff(fact_908_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_909_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_910_size__neq__size__imp__neq,axiom,
! [X: num,Y: num] :
( ( aa_num_nat(size_size_num,X) != aa_num_nat(size_size_num,Y) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
tff(fact_911_modulo__nat__def,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),N)) ).
% modulo_nat_def
tff(fact_912_mod__mult__right__eq,axiom,
! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).
% mod_mult_right_eq
tff(fact_913_mod__mult__right__eq,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ).
% mod_mult_right_eq
tff(fact_914_mod__mult__right__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ).
% mod_mult_right_eq
tff(fact_915_mod__mult__left__eq,axiom,
! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).
% mod_mult_left_eq
tff(fact_916_mod__mult__left__eq,axiom,
! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ).
% mod_mult_left_eq
tff(fact_917_mod__mult__left__eq,axiom,
! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,C)),B),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ).
% mod_mult_left_eq
tff(fact_918_mult__mod__right,axiom,
! [C: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ).
% mult_mod_right
tff(fact_919_mult__mod__right,axiom,
! [C: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,C),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ).
% mult_mod_right
tff(fact_920_mult__mod__right,axiom,
! [C: code_integer,A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),modulo364778990260209775nteger(A,B)) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ).
% mult_mod_right
tff(fact_921_mod__mult__mult2,axiom,
! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),C) ).
% mod_mult_mult2
tff(fact_922_mod__mult__mult2,axiom,
! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),C) ).
% mod_mult_mult2
tff(fact_923_mod__mult__mult2,axiom,
! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,B)),C) ).
% mod_mult_mult2
tff(fact_924_mod__mult__cong,axiom,
! [A: nat,C: nat,A3: nat,B: nat,B3: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A3),C) )
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B3),C) )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A3),B3)),C) ) ) ) ).
% mod_mult_cong
tff(fact_925_mod__mult__cong,axiom,
! [A: int,C: int,A3: int,B: int,B3: int] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A3),C) )
=> ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B3),C) )
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A3),B3)),C) ) ) ) ).
% mod_mult_cong
tff(fact_926_mod__mult__cong,axiom,
! [A: code_integer,C: code_integer,A3: code_integer,B: code_integer,B3: code_integer] :
( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A3,C) )
=> ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B3,C) )
=> ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A3),B3),C) ) ) ) ).
% mod_mult_cong
tff(fact_927_mod__mult__eq,axiom,
! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).
% mod_mult_eq
tff(fact_928_mod__mult__eq,axiom,
! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ).
% mod_mult_eq
tff(fact_929_mod__mult__eq,axiom,
! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,C)),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ).
% mod_mult_eq
tff(fact_930_mod__add__right__eq,axiom,
! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).
% mod_add_right_eq
tff(fact_931_mod__add__right__eq,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).
% mod_add_right_eq
tff(fact_932_mod__add__right__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) ).
% mod_add_right_eq
tff(fact_933_mod__add__left__eq,axiom,
! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).
% mod_add_left_eq
tff(fact_934_mod__add__left__eq,axiom,
! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).
% mod_add_left_eq
tff(fact_935_mod__add__left__eq,axiom,
! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),B),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) ).
% mod_add_left_eq
tff(fact_936_mod__add__cong,axiom,
! [A: nat,C: nat,A3: nat,B: nat,B3: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A3),C) )
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B3),C) )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),B3)),C) ) ) ) ).
% mod_add_cong
tff(fact_937_mod__add__cong,axiom,
! [A: int,C: int,A3: int,B: int,B3: int] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A3),C) )
=> ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B3),C) )
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A3),B3)),C) ) ) ) ).
% mod_add_cong
tff(fact_938_mod__add__cong,axiom,
! [A: code_integer,C: code_integer,A3: code_integer,B: code_integer,B3: code_integer] :
( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A3,C) )
=> ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B3,C) )
=> ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A3,B3),C) ) ) ) ).
% mod_add_cong
tff(fact_939_mod__add__eq,axiom,
! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).
% mod_add_eq
tff(fact_940_mod__add__eq,axiom,
! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).
% mod_add_eq
tff(fact_941_mod__add__eq,axiom,
! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) ).
% mod_add_eq
tff(fact_942_power__mod,axiom,
! [A: nat,B: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),N)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),B) ).
% power_mod
tff(fact_943_power__mod,axiom,
! [A: int,B: int,N: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),N)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(power_power_int(A),N)),B) ).
% power_mod
tff(fact_944_power__mod,axiom,
! [A: code_integer,B: code_integer,N: nat] : modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(modulo364778990260209775nteger(A,B)),N),B) = modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),N),B) ).
% power_mod
tff(fact_945_diff__eq__diff__less__eq,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( minus_minus_rat(A,B) = minus_minus_rat(C,D) )
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D)) ) ) ).
% diff_eq_diff_less_eq
tff(fact_946_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D)) ) ) ).
% diff_eq_diff_less_eq
tff(fact_947_diff__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,minus_minus_rat(A,C)),minus_minus_rat(B,C))) ) ).
% diff_right_mono
tff(fact_948_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C))) ) ).
% diff_right_mono
tff(fact_949_diff__left__mono,axiom,
! [B: rat,A: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,minus_minus_rat(C,A)),minus_minus_rat(C,B))) ) ).
% diff_left_mono
tff(fact_950_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))) ) ).
% diff_left_mono
tff(fact_951_diff__mono,axiom,
! [A: rat,B: rat,D: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,D),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,minus_minus_rat(A,C)),minus_minus_rat(B,D))) ) ) ).
% diff_mono
tff(fact_952_diff__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D))) ) ) ).
% diff_mono
tff(fact_953_diff__strict__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D))) ) ) ).
% diff_strict_mono
tff(fact_954_diff__strict__mono,axiom,
! [A: rat,B: rat,D: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,D),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(A,C)),minus_minus_rat(B,D))) ) ) ).
% diff_strict_mono
tff(fact_955_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,D),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D))) ) ) ).
% diff_strict_mono
tff(fact_956_diff__eq__diff__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D)) ) ) ).
% diff_eq_diff_less
tff(fact_957_diff__eq__diff__less,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( minus_minus_rat(A,B) = minus_minus_rat(C,D) )
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D)) ) ) ).
% diff_eq_diff_less
tff(fact_958_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D)) ) ) ).
% diff_eq_diff_less
tff(fact_959_diff__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B))) ) ).
% diff_strict_left_mono
tff(fact_960_diff__strict__left__mono,axiom,
! [B: rat,A: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(C,A)),minus_minus_rat(C,B))) ) ).
% diff_strict_left_mono
tff(fact_961_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))) ) ).
% diff_strict_left_mono
tff(fact_962_diff__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C))) ) ).
% diff_strict_right_mono
tff(fact_963_diff__strict__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(A,C)),minus_minus_rat(B,C))) ) ).
% diff_strict_right_mono
tff(fact_964_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C))) ) ).
% diff_strict_right_mono
tff(fact_965_right__diff__distrib_H,axiom,
! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).
% right_diff_distrib'
tff(fact_966_right__diff__distrib_H,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(times_times_rat(A),B),aa_rat_rat(times_times_rat(A),C)) ).
% right_diff_distrib'
tff(fact_967_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).
% right_diff_distrib'
tff(fact_968_right__diff__distrib_H,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).
% right_diff_distrib'
tff(fact_969_left__diff__distrib_H,axiom,
! [B: real,C: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)),A) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)) ).
% left_diff_distrib'
tff(fact_970_left__diff__distrib_H,axiom,
! [B: rat,C: rat,A: rat] : aa_rat_rat(times_times_rat(minus_minus_rat(B,C)),A) = minus_minus_rat(aa_rat_rat(times_times_rat(B),A),aa_rat_rat(times_times_rat(C),A)) ).
% left_diff_distrib'
tff(fact_971_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)) ).
% left_diff_distrib'
tff(fact_972_left__diff__distrib_H,axiom,
! [B: int,C: int,A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A)) ).
% left_diff_distrib'
tff(fact_973_right__diff__distrib,axiom,
! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).
% right_diff_distrib
tff(fact_974_right__diff__distrib,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),minus_minus_rat(B,C)) = minus_minus_rat(aa_rat_rat(times_times_rat(A),B),aa_rat_rat(times_times_rat(A),C)) ).
% right_diff_distrib
tff(fact_975_right__diff__distrib,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).
% right_diff_distrib
tff(fact_976_left__diff__distrib,axiom,
! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ).
% left_diff_distrib
tff(fact_977_left__diff__distrib,axiom,
! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(minus_minus_rat(A,B)),C) = minus_minus_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(B),C)) ).
% left_diff_distrib
tff(fact_978_left__diff__distrib,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).
% left_diff_distrib
tff(fact_979_add__diff__add,axiom,
! [A: real,C: real,B: real,D: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D)) ).
% add_diff_add
tff(fact_980_add__diff__add,axiom,
! [A: rat,C: rat,B: rat,D: rat] : minus_minus_rat(plus_plus_rat(A,C),plus_plus_rat(B,D)) = plus_plus_rat(minus_minus_rat(A,B),minus_minus_rat(C,D)) ).
% add_diff_add
tff(fact_981_add__diff__add,axiom,
! [A: int,C: int,B: int,D: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D)) ).
% add_diff_add
tff(fact_982_diff__diff__eq,axiom,
! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) ).
% diff_diff_eq
tff(fact_983_diff__diff__eq,axiom,
! [A: rat,B: rat,C: rat] : minus_minus_rat(minus_minus_rat(A,B),C) = minus_minus_rat(A,plus_plus_rat(B,C)) ).
% diff_diff_eq
tff(fact_984_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).
% diff_diff_eq
tff(fact_985_diff__diff__eq,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).
% diff_diff_eq
tff(fact_986_add__implies__diff,axiom,
! [C: real,B: real,A: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B) = A )
=> ( C = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) ) ) ).
% add_implies_diff
tff(fact_987_add__implies__diff,axiom,
! [C: rat,B: rat,A: rat] :
( ( plus_plus_rat(C,B) = A )
=> ( C = minus_minus_rat(A,B) ) ) ).
% add_implies_diff
tff(fact_988_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B) = A )
=> ( C = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ) ).
% add_implies_diff
tff(fact_989_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B) = A )
=> ( C = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ) ) ).
% add_implies_diff
tff(fact_990_diff__add__eq__diff__diff__swap,axiom,
! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),B) ).
% diff_add_eq_diff_diff_swap
tff(fact_991_diff__add__eq__diff__diff__swap,axiom,
! [A: rat,B: rat,C: rat] : minus_minus_rat(A,plus_plus_rat(B,C)) = minus_minus_rat(minus_minus_rat(A,C),B) ).
% diff_add_eq_diff_diff_swap
tff(fact_992_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),B) ).
% diff_add_eq_diff_diff_swap
tff(fact_993_diff__add__eq,axiom,
! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ).
% diff_add_eq
tff(fact_994_diff__add__eq,axiom,
! [A: rat,B: rat,C: rat] : plus_plus_rat(minus_minus_rat(A,B),C) = minus_minus_rat(plus_plus_rat(A,C),B) ).
% diff_add_eq
tff(fact_995_diff__add__eq,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ).
% diff_add_eq
tff(fact_996_diff__diff__eq2,axiom,
! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B) ).
% diff_diff_eq2
tff(fact_997_diff__diff__eq2,axiom,
! [A: rat,B: rat,C: rat] : minus_minus_rat(A,minus_minus_rat(B,C)) = minus_minus_rat(plus_plus_rat(A,C),B) ).
% diff_diff_eq2
tff(fact_998_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ).
% diff_diff_eq2
tff(fact_999_add__diff__eq,axiom,
! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C) ).
% add_diff_eq
tff(fact_1000_add__diff__eq,axiom,
! [A: rat,B: rat,C: rat] : plus_plus_rat(A,minus_minus_rat(B,C)) = minus_minus_rat(plus_plus_rat(A,B),C) ).
% add_diff_eq
tff(fact_1001_add__diff__eq,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).
% add_diff_eq
tff(fact_1002_eq__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( A = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B) )
<=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = C ) ) ).
% eq_diff_eq
tff(fact_1003_eq__diff__eq,axiom,
! [A: rat,C: rat,B: rat] :
( ( A = minus_minus_rat(C,B) )
<=> ( plus_plus_rat(A,B) = C ) ) ).
% eq_diff_eq
tff(fact_1004_eq__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( A = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B) )
<=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = C ) ) ).
% eq_diff_eq
tff(fact_1005_diff__eq__eq,axiom,
! [A: real,B: real,C: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = C )
<=> ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B) ) ) ).
% diff_eq_eq
tff(fact_1006_diff__eq__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( minus_minus_rat(A,B) = C )
<=> ( A = plus_plus_rat(C,B) ) ) ).
% diff_eq_eq
tff(fact_1007_diff__eq__eq,axiom,
! [A: int,B: int,C: int] :
( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = C )
<=> ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B) ) ) ).
% diff_eq_eq
tff(fact_1008_group__cancel_Osub1,axiom,
! [A2: real,K: real,A: real,B: real] :
( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ) ) ).
% group_cancel.sub1
tff(fact_1009_group__cancel_Osub1,axiom,
! [A2: rat,K: rat,A: rat,B: rat] :
( ( A2 = plus_plus_rat(K,A) )
=> ( minus_minus_rat(A2,B) = plus_plus_rat(K,minus_minus_rat(A,B)) ) ) ).
% group_cancel.sub1
tff(fact_1010_group__cancel_Osub1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).
% group_cancel.sub1
tff(fact_1011_mod__less__eq__dividend,axiom,
! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)),M)) ).
% mod_less_eq_dividend
tff(fact_1012_diff__divide__distrib,axiom,
! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(A),B)),C) = aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),C)),aa_complex_complex(divide1717551699836669952omplex(B),C)) ).
% diff_divide_distrib
tff(fact_1013_diff__divide__distrib,axiom,
! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) ).
% diff_divide_distrib
tff(fact_1014_diff__divide__distrib,axiom,
! [A: rat,B: rat,C: rat] : divide_divide_rat(minus_minus_rat(A,B),C) = minus_minus_rat(divide_divide_rat(A,C),divide_divide_rat(B,C)) ).
% diff_divide_distrib
tff(fact_1015_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),L))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M))) ) ) ).
% diff_less_mono2
tff(fact_1016_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),K))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),N)),K)) ) ).
% less_imp_diff_less
tff(fact_1017_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M))) ) ).
% diff_le_mono2
tff(fact_1018_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),C))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),B)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ) ) ).
% le_diff_iff'
tff(fact_1019_diff__le__self,axiom,
! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),M)) ).
% diff_le_self
tff(fact_1020_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),L)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),L))) ) ).
% diff_le_mono
tff(fact_1021_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ) ) ) ).
% Nat.diff_diff_eq
tff(fact_1022_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ) ).
% le_diff_iff
tff(fact_1023_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K) )
<=> ( M = N ) ) ) ) ).
% eq_diff_iff
tff(fact_1024_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K))
=> ( ( K != aa_nat_nat(suc,I) )
=> ~ ! [J2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
=> ( K != aa_nat_nat(suc,J2) ) ) ) ) ).
% Nat.lessE
tff(fact_1025_Suc__lessD,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% Suc_lessD
tff(fact_1026_Suc__lessE,axiom,
! [I: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),K))
=> ~ ! [J2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J2))
=> ( K != aa_nat_nat(suc,J2) ) ) ) ).
% Suc_lessE
tff(fact_1027_Suc__lessI,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> ( ( aa_nat_nat(suc,M) != N )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),N)) ) ) ).
% Suc_lessI
tff(fact_1028_less__SucE,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> ( M = N ) ) ) ).
% less_SucE
tff(fact_1029_less__SucI,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N))) ) ).
% less_SucI
tff(fact_1030_Ex__less__Suc,axiom,
! [N: nat,P: fun_nat_bool] :
( ? [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(suc,N)))
& pp(aa_nat_bool(P,I4)) )
<=> ( pp(aa_nat_bool(P,N))
| ? [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
& pp(aa_nat_bool(P,I4)) ) ) ) ).
% Ex_less_Suc
tff(fact_1031_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
| ( M = N ) ) ) ).
% less_Suc_eq
tff(fact_1032_not__less__eq,axiom,
! [M: nat,N: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M))) ) ).
% not_less_eq
tff(fact_1033_All__less__Suc,axiom,
! [N: nat,P: fun_nat_bool] :
( ! [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(suc,N)))
=> pp(aa_nat_bool(P,I4)) )
<=> ( pp(aa_nat_bool(P,N))
& ! [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
=> pp(aa_nat_bool(P,I4)) ) ) ) ).
% All_less_Suc
tff(fact_1034_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
<=> ? [M4: nat] :
( ( M = aa_nat_nat(suc,M4) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M4)) ) ) ).
% Suc_less_eq2
tff(fact_1035_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
=> ( M = N ) ) ) ).
% less_antisym
tff(fact_1036_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% Suc_less_SucD
tff(fact_1037_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),K))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),K)) ) ) ).
% less_trans_Suc
tff(fact_1038_less__Suc__induct,axiom,
! [I: nat,J: nat,P: fun_nat_fun_nat_bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
=> ( ! [I2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I2),aa_nat_nat(suc,I2)))
=> ( ! [I2: nat,J2: nat,K3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),K3))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I2),J2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,J2),K3))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I2),K3)) ) ) ) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I),J)) ) ) ) ).
% less_Suc_induct
tff(fact_1039_strict__inc__induct,axiom,
! [I: nat,J: nat,P: fun_nat_bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
=> ( ! [I2: nat] :
( ( J = aa_nat_nat(suc,I2) )
=> pp(aa_nat_bool(P,I2)) )
=> ( ! [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),J))
=> ( pp(aa_nat_bool(P,aa_nat_nat(suc,I2)))
=> pp(aa_nat_bool(P,I2)) ) )
=> pp(aa_nat_bool(P,I)) ) ) ) ).
% strict_inc_induct
tff(fact_1040_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
<=> ( N = M ) ) ) ).
% not_less_less_Suc_eq
tff(fact_1041_diff__add__inverse2,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),N) = M ).
% diff_add_inverse2
tff(fact_1042_diff__add__inverse,axiom,
! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),N) = M ).
% diff_add_inverse
tff(fact_1043_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] : aa_nat_nat(aa_nat_fun_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,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ).
% diff_cancel2
tff(fact_1044_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_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),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ).
% Nat.diff_cancel
tff(fact_1045_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: fun_nat_fun_nat_bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( ! [X3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,X3),X3))
=> ( ! [X3: nat,Y2: nat,Z3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,X3),Y2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,Y2),Z3))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,X3),Z3)) ) )
=> ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,N2),aa_nat_nat(suc,N2)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,M),N)) ) ) ) ) ).
% transitive_stepwise_le
tff(fact_1046_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: fun_nat_bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( pp(aa_nat_bool(P,M))
=> ( ! [N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N2))
=> ( pp(aa_nat_bool(P,N2))
=> pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) )
=> pp(aa_nat_bool(P,N)) ) ) ) ).
% nat_induct_at_least
tff(fact_1047_full__nat__induct,axiom,
! [P: fun_nat_bool,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M2)),N2))
=> pp(aa_nat_bool(P,M2)) )
=> pp(aa_nat_bool(P,N2)) )
=> pp(aa_nat_bool(P,N)) ) ).
% full_nat_induct
tff(fact_1048_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M)) ) ).
% not_less_eq_eq
tff(fact_1049_Suc__n__not__le__n,axiom,
! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),N)) ).
% Suc_n_not_le_n
tff(fact_1050_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
| ( M = aa_nat_nat(suc,N) ) ) ) ).
% le_Suc_eq
tff(fact_1051_Suc__le__D,axiom,
! [N: nat,M5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M5))
=> ? [M3: nat] : M5 = aa_nat_nat(suc,M3) ) ).
% Suc_le_D
tff(fact_1052_le__SucI,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N))) ) ).
% le_SucI
tff(fact_1053_le__SucE,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( M = aa_nat_nat(suc,N) ) ) ) ).
% le_SucE
tff(fact_1054_Suc__leD,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% Suc_leD
tff(fact_1055_add__Suc__shift,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,N)) ).
% add_Suc_shift
tff(fact_1056_add__Suc,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).
% add_Suc
tff(fact_1057_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
=> ( aa_nat_nat(suc,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(suc,A)) ) ) ).
% nat_arith.suc1
tff(fact_1058_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_nat_nat(aa_nat_fun_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),N)) ).
% diff_mult_distrib2
tff(fact_1059_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),K) = aa_nat_nat(aa_nat_fun_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,N),K)) ).
% diff_mult_distrib
tff(fact_1060_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: 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)),N) )
<=> ( M = N ) ) ).
% Suc_mult_cancel1
tff(fact_1061_Suc__mod__eq__add3__mod,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),N) ).
% Suc_mod_eq_add3_mod
tff(fact_1062_cong__exp__iff__simps_I9_J,axiom,
! [M: num,Q2: num,N: num] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q2)) ) ) ).
% cong_exp_iff_simps(9)
tff(fact_1063_cong__exp__iff__simps_I9_J,axiom,
! [M: num,Q2: num,N: num] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
<=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q2)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q2)) ) ) ).
% cong_exp_iff_simps(9)
tff(fact_1064_cong__exp__iff__simps_I9_J,axiom,
! [M: num,Q2: num,N: num] :
( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,M)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,N)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) )
<=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Q2)) = modulo364778990260209775nteger(numera6620942414471956472nteger(N),numera6620942414471956472nteger(Q2)) ) ) ).
% cong_exp_iff_simps(9)
tff(fact_1065_cong__exp__iff__simps_I4_J,axiom,
! [M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,one)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,one)) ).
% cong_exp_iff_simps(4)
tff(fact_1066_cong__exp__iff__simps_I4_J,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,one)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,one)) ).
% cong_exp_iff_simps(4)
tff(fact_1067_cong__exp__iff__simps_I4_J,axiom,
! [M: num,N: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(one)) = modulo364778990260209775nteger(numera6620942414471956472nteger(N),numera6620942414471956472nteger(one)) ).
% cong_exp_iff_simps(4)
tff(fact_1068_mod__eqE,axiom,
! [A: int,C: int,B: int] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) )
=> ~ ! [D2: int] : B != aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),D2)) ) ).
% mod_eqE
tff(fact_1069_mod__eqE,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(B,C) )
=> ~ ! [D2: code_integer] : B != plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D2)) ) ).
% mod_eqE
tff(fact_1070_div__add1__eq,axiom,
! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = 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(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C)) ).
% div_add1_eq
tff(fact_1071_div__add1__eq,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C)) ).
% div_add1_eq
tff(fact_1072_div__add1__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)),divide6298287555418463151nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B,C)),C)) ).
% div_add1_eq
tff(fact_1073_add__le__add__imp__diff__le,axiom,
! [I: real,K: real,N: real,J: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,N),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K)))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,N),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K)))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,N),K)),J)) ) ) ) ) ).
% add_le_add_imp_diff_le
tff(fact_1074_add__le__add__imp__diff__le,axiom,
! [I: rat,K: rat,N: rat,J: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(I,K)),N))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,N),plus_plus_rat(J,K)))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(I,K)),N))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,N),plus_plus_rat(J,K)))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,minus_minus_rat(N,K)),J)) ) ) ) ) ).
% add_le_add_imp_diff_le
tff(fact_1075_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),J)) ) ) ) ) ).
% add_le_add_imp_diff_le
tff(fact_1076_add__le__add__imp__diff__le,axiom,
! [I: int,K: int,N: int,J: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),N))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),N))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K)))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,N),K)),J)) ) ) ) ) ).
% add_le_add_imp_diff_le
tff(fact_1077_add__le__imp__le__diff,axiom,
! [I: real,K: real,N: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),N))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,N),K))) ) ).
% add_le_imp_le_diff
tff(fact_1078_add__le__imp__le__diff,axiom,
! [I: rat,K: rat,N: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(I,K)),N))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),minus_minus_rat(N,K))) ) ).
% add_le_imp_le_diff
tff(fact_1079_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K))) ) ).
% add_le_imp_le_diff
tff(fact_1080_add__le__imp__le__diff,axiom,
! [I: int,K: int,N: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),aa_int_int(aa_int_fun_int_int(minus_minus_int,N),K))) ) ).
% add_le_imp_le_diff
tff(fact_1081_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A) = C )
<=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_1082_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_1083_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_1084_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_1085_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_1086_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_1087_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_1088_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B)) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_1089_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A))) ) ).
% le_add_diff
tff(fact_1090_diff__add,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),A) = B ) ) ).
% diff_add
tff(fact_1091_le__diff__eq,axiom,
! [A: real,C: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C)) ) ).
% le_diff_eq
tff(fact_1092_le__diff__eq,axiom,
! [A: rat,C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),minus_minus_rat(C,B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(A,B)),C)) ) ).
% le_diff_eq
tff(fact_1093_le__diff__eq,axiom,
! [A: int,C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C)) ) ).
% le_diff_eq
tff(fact_1094_diff__le__eq,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))) ) ).
% diff_le_eq
tff(fact_1095_diff__le__eq,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,minus_minus_rat(A,B)),C))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),plus_plus_rat(C,B))) ) ).
% diff_le_eq
tff(fact_1096_diff__le__eq,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))) ) ).
% diff_le_eq
tff(fact_1097_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: real,B: real] :
( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = A ) ) ).
% linordered_semidom_class.add_diff_inverse
tff(fact_1098_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: rat,B: rat] :
( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( plus_plus_rat(B,minus_minus_rat(A,B)) = A ) ) ).
% linordered_semidom_class.add_diff_inverse
tff(fact_1099_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ) ) ).
% linordered_semidom_class.add_diff_inverse
tff(fact_1100_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = A ) ) ).
% linordered_semidom_class.add_diff_inverse
tff(fact_1101_less__diff__eq,axiom,
! [A: real,C: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),C)) ) ).
% less_diff_eq
tff(fact_1102_less__diff__eq,axiom,
! [A: rat,C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),minus_minus_rat(C,B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(A,B)),C)) ) ).
% less_diff_eq
tff(fact_1103_less__diff__eq,axiom,
! [A: int,C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C)) ) ).
% less_diff_eq
tff(fact_1104_diff__less__eq,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),C))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B))) ) ).
% diff_less_eq
tff(fact_1105_diff__less__eq,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(A,B)),C))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),plus_plus_rat(C,B))) ) ).
% diff_less_eq
tff(fact_1106_diff__less__eq,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))) ) ).
% diff_less_eq
tff(fact_1107_mult__diff__mult,axiom,
! [X: real,Y: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),B))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),A)),B)) ).
% mult_diff_mult
tff(fact_1108_mult__diff__mult,axiom,
! [X: rat,Y: rat,A: rat,B: rat] : minus_minus_rat(aa_rat_rat(times_times_rat(X),Y),aa_rat_rat(times_times_rat(A),B)) = plus_plus_rat(aa_rat_rat(times_times_rat(X),minus_minus_rat(Y,B)),aa_rat_rat(times_times_rat(minus_minus_rat(X,A)),B)) ).
% mult_diff_mult
tff(fact_1109_mult__diff__mult,axiom,
! [X: int,Y: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),A)),B)) ).
% mult_diff_mult
tff(fact_1110_square__diff__square__factored,axiom,
! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)) ).
% square_diff_square_factored
tff(fact_1111_square__diff__square__factored,axiom,
! [X: rat,Y: rat] : minus_minus_rat(aa_rat_rat(times_times_rat(X),X),aa_rat_rat(times_times_rat(Y),Y)) = aa_rat_rat(times_times_rat(plus_plus_rat(X,Y)),minus_minus_rat(X,Y)) ).
% square_diff_square_factored
tff(fact_1112_square__diff__square__factored,axiom,
! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)) ).
% square_diff_square_factored
tff(fact_1113_eq__add__iff2,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D) )
<=> ( C = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E)),D) ) ) ).
% eq_add_iff2
tff(fact_1114_eq__add__iff2,axiom,
! [A: rat,E: rat,C: rat,B: rat,D: rat] :
( ( plus_plus_rat(aa_rat_rat(times_times_rat(A),E),C) = plus_plus_rat(aa_rat_rat(times_times_rat(B),E),D) )
<=> ( C = plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(B,A)),E),D) ) ) ).
% eq_add_iff2
tff(fact_1115_eq__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D) )
<=> ( C = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D) ) ) ).
% eq_add_iff2
tff(fact_1116_eq__add__iff1,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D) )
<=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E)),C) = D ) ) ).
% eq_add_iff1
tff(fact_1117_eq__add__iff1,axiom,
! [A: rat,E: rat,C: rat,B: rat,D: rat] :
( ( plus_plus_rat(aa_rat_rat(times_times_rat(A),E),C) = plus_plus_rat(aa_rat_rat(times_times_rat(B),E),D) )
<=> ( plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(A,B)),E),C) = D ) ) ).
% eq_add_iff1
tff(fact_1118_eq__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D) )
<=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E)),C) = D ) ) ).
% eq_add_iff1
tff(fact_1119_nat__mod__eq__iff,axiom,
! [X: nat,N: nat,Y: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),N) )
<=> ? [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,N),Q1)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q22)) ) ).
% nat_mod_eq_iff
tff(fact_1120_lift__Suc__mono__less,axiom,
! [F: fun_nat_real,N: nat,N4: nat] :
( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2))))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,N4))) ) ) ).
% lift_Suc_mono_less
tff(fact_1121_lift__Suc__mono__less,axiom,
! [F: fun_nat_rat,N: nat,N4: nat] :
( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2))))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,N4))) ) ) ).
% lift_Suc_mono_less
tff(fact_1122_lift__Suc__mono__less,axiom,
! [F: fun_nat_num,N: nat,N4: nat] :
( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2))))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,N4))) ) ) ).
% lift_Suc_mono_less
tff(fact_1123_lift__Suc__mono__less,axiom,
! [F: fun_nat_nat,N: nat,N4: nat] :
( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2))))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,N4))) ) ) ).
% lift_Suc_mono_less
tff(fact_1124_lift__Suc__mono__less,axiom,
! [F: fun_nat_int,N: nat,N4: nat] :
( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2))))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N4))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,N4))) ) ) ).
% lift_Suc_mono_less
tff(fact_1125_lift__Suc__mono__less__iff,axiom,
! [F: fun_nat_real,N: nat,M: nat] :
( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2))))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,M)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).
% lift_Suc_mono_less_iff
tff(fact_1126_lift__Suc__mono__less__iff,axiom,
! [F: fun_nat_rat,N: nat,M: nat] :
( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2))))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,M)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).
% lift_Suc_mono_less_iff
tff(fact_1127_lift__Suc__mono__less__iff,axiom,
! [F: fun_nat_num,N: nat,M: nat] :
( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2))))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,M)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).
% lift_Suc_mono_less_iff
tff(fact_1128_lift__Suc__mono__less__iff,axiom,
! [F: fun_nat_nat,N: nat,M: nat] :
( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2))))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,M)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).
% lift_Suc_mono_less_iff
tff(fact_1129_lift__Suc__mono__less__iff,axiom,
! [F: fun_nat_int,N: nat,M: nat] :
( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2))))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,M)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ).
% lift_Suc_mono_less_iff
tff(fact_1130_power__Suc,axiom,
! [A: complex,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,N)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(A),N)) ).
% power_Suc
tff(fact_1131_power__Suc,axiom,
! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ).
% power_Suc
tff(fact_1132_power__Suc,axiom,
! [A: rat,N: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),N)) ).
% power_Suc
tff(fact_1133_power__Suc,axiom,
! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ).
% power_Suc
tff(fact_1134_power__Suc,axiom,
! [A: int,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N)) ).
% power_Suc
tff(fact_1135_power__Suc2,axiom,
! [A: complex,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,N)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),N)),A) ).
% power_Suc2
tff(fact_1136_power__Suc2,axiom,
! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),A) ).
% power_Suc2
tff(fact_1137_power__Suc2,axiom,
! [A: rat,N: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),N)),A) ).
% power_Suc2
tff(fact_1138_power__Suc2,axiom,
! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),A) ).
% power_Suc2
tff(fact_1139_power__Suc2,axiom,
! [A: int,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),N)),A) ).
% power_Suc2
tff(fact_1140_lift__Suc__antimono__le,axiom,
! [F: fun_nat_set_nat,N: nat,N4: nat] :
( ! [N2: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(F,aa_nat_nat(suc,N2))),aa_nat_set_nat(F,N2)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(F,N4)),aa_nat_set_nat(F,N))) ) ) ).
% lift_Suc_antimono_le
tff(fact_1141_lift__Suc__antimono__le,axiom,
! [F: fun_nat_rat,N: nat,N4: nat] :
( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,aa_nat_nat(suc,N2))),aa_nat_rat(F,N2)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,N4)),aa_nat_rat(F,N))) ) ) ).
% lift_Suc_antimono_le
tff(fact_1142_lift__Suc__antimono__le,axiom,
! [F: fun_nat_num,N: nat,N4: nat] :
( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,aa_nat_nat(suc,N2))),aa_nat_num(F,N2)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N4)),aa_nat_num(F,N))) ) ) ).
% lift_Suc_antimono_le
tff(fact_1143_lift__Suc__antimono__le,axiom,
! [F: fun_nat_nat,N: nat,N4: nat] :
( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,aa_nat_nat(suc,N2))),aa_nat_nat(F,N2)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N4)),aa_nat_nat(F,N))) ) ) ).
% lift_Suc_antimono_le
tff(fact_1144_lift__Suc__antimono__le,axiom,
! [F: fun_nat_int,N: nat,N4: nat] :
( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,aa_nat_nat(suc,N2))),aa_nat_int(F,N2)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N4)),aa_nat_int(F,N))) ) ) ).
% lift_Suc_antimono_le
tff(fact_1145_lift__Suc__mono__le,axiom,
! [F: fun_nat_set_nat,N: nat,N4: nat] :
( ! [N2: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(F,N2)),aa_nat_set_nat(F,aa_nat_nat(suc,N2))))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(F,N)),aa_nat_set_nat(F,N4))) ) ) ).
% lift_Suc_mono_le
tff(fact_1146_lift__Suc__mono__le,axiom,
! [F: fun_nat_rat,N: nat,N4: nat] :
( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2))))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,N)),aa_nat_rat(F,N4))) ) ) ).
% lift_Suc_mono_le
tff(fact_1147_lift__Suc__mono__le,axiom,
! [F: fun_nat_num,N: nat,N4: nat] :
( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2))))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,N)),aa_nat_num(F,N4))) ) ) ).
% lift_Suc_mono_le
tff(fact_1148_lift__Suc__mono__le,axiom,
! [F: fun_nat_nat,N: nat,N4: nat] :
( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2))))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(F,N4))) ) ) ).
% lift_Suc_mono_le
tff(fact_1149_lift__Suc__mono__le,axiom,
! [F: fun_nat_int,N: nat,N4: nat] :
( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2))))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N4))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(F,N4))) ) ) ).
% lift_Suc_mono_le
tff(fact_1150_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C))) ) ) ).
% diff_less_mono
tff(fact_1151_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ) ).
% less_diff_iff
tff(fact_1152_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = M ) ) ).
% add_diff_inverse_nat
tff(fact_1153_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J)) ) ).
% less_diff_conv
tff(fact_1154_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> ( ( aa_nat_nat(aa_nat_fun_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_1155_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
=> ( aa_nat_nat(aa_nat_fun_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(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I) ) ) ).
% Nat.diff_add_assoc2
tff(fact_1156_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
=> ( aa_nat_nat(aa_nat_fun_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(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) ) ) ).
% Nat.diff_add_assoc
tff(fact_1157_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J)) ) ) ).
% Nat.le_diff_conv2
tff(fact_1158_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K))) ) ).
% le_diff_conv
tff(fact_1159_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N))) ) ).
% le_imp_less_Suc
tff(fact_1160_less__eq__Suc__le,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M)) ) ).
% less_eq_Suc_le
tff(fact_1161_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% less_Suc_eq_le
tff(fact_1162_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
<=> ( N = M ) ) ) ).
% le_less_Suc_eq
tff(fact_1163_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% Suc_le_lessD
tff(fact_1164_inc__induct,axiom,
! [I: nat,J: nat,P: fun_nat_bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> ( pp(aa_nat_bool(P,J))
=> ( ! [N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),J))
=> ( pp(aa_nat_bool(P,aa_nat_nat(suc,N2)))
=> pp(aa_nat_bool(P,N2)) ) ) )
=> pp(aa_nat_bool(P,I)) ) ) ) ).
% inc_induct
tff(fact_1165_dec__induct,axiom,
! [I: nat,J: nat,P: fun_nat_bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> ( pp(aa_nat_bool(P,I))
=> ( ! [N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),J))
=> ( pp(aa_nat_bool(P,N2))
=> pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) ) )
=> pp(aa_nat_bool(P,J)) ) ) ) ).
% dec_induct
tff(fact_1166_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% Suc_le_eq
tff(fact_1167_Suc__leI,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N)) ) ).
% Suc_leI
tff(fact_1168_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> ? [K3: nat] : N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3)) ) ).
% less_imp_Suc_add
tff(fact_1169_less__iff__Suc__add,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
<=> ? [K2: nat] : N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2)) ) ).
% less_iff_Suc_add
tff(fact_1170_less__add__Suc2,axiom,
! [I: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1171_less__add__Suc1,axiom,
! [I: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(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_1172_less__natE,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> ~ ! [Q3: nat] : N != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q3)) ) ).
% less_natE
tff(fact_1173_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% Suc_mult_less_cancel1
tff(fact_1174_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% Suc_mult_le_cancel1
tff(fact_1175_Suc__eq__plus1__left,axiom,
! [N: nat] : aa_nat_nat(suc,N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N) ).
% Suc_eq_plus1_left
tff(fact_1176_plus__1__eq__Suc,axiom,
aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).
% plus_1_eq_Suc
tff(fact_1177_Suc__eq__plus1,axiom,
! [N: nat] : aa_nat_nat(suc,N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat) ).
% Suc_eq_plus1
tff(fact_1178_Suc__div__le__mono,axiom,
! [M: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,M)),N))) ).
% Suc_div_le_mono
tff(fact_1179_mult__Suc,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) ).
% mult_Suc
tff(fact_1180_mult__exp__mod__exp__eq,axiom,
! [M: nat,N: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ).
% mult_exp_mod_exp_eq
tff(fact_1181_mult__exp__mod__exp__eq,axiom,
! [M: nat,N: nat,A: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) ) ) ).
% mult_exp_mod_exp_eq
tff(fact_1182_mult__exp__mod__exp__eq,axiom,
! [M: nat,N: nat,A: code_integer] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)) ) ) ).
% mult_exp_mod_exp_eq
tff(fact_1183_cong__exp__iff__simps_I6_J,axiom,
! [Q2: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) ).
% cong_exp_iff_simps(6)
tff(fact_1184_cong__exp__iff__simps_I6_J,axiom,
! [Q2: num,N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).
% cong_exp_iff_simps(6)
tff(fact_1185_cong__exp__iff__simps_I6_J,axiom,
! [Q2: num,N: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) != modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,N)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) ).
% cong_exp_iff_simps(6)
tff(fact_1186_cong__exp__iff__simps_I8_J,axiom,
! [M: num,Q2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) ).
% cong_exp_iff_simps(8)
tff(fact_1187_cong__exp__iff__simps_I8_J,axiom,
! [M: num,Q2: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).
% cong_exp_iff_simps(8)
tff(fact_1188_cong__exp__iff__simps_I8_J,axiom,
! [M: num,Q2: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,M)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) != modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) ).
% cong_exp_iff_simps(8)
tff(fact_1189_mult__div__mod__eq,axiom,
! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = A ).
% mult_div_mod_eq
tff(fact_1190_mult__div__mod__eq,axiom,
! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = A ).
% mult_div_mod_eq
tff(fact_1191_mult__div__mod__eq,axiom,
! [B: code_integer,A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B)),modulo364778990260209775nteger(A,B)) = A ).
% mult_div_mod_eq
tff(fact_1192_mod__mult__div__eq,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))) = A ).
% mod_mult_div_eq
tff(fact_1193_mod__mult__div__eq,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) = A ).
% mod_mult_div_eq
tff(fact_1194_mod__mult__div__eq,axiom,
! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B))) = A ).
% mod_mult_div_eq
tff(fact_1195_mod__div__mult__eq,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)) = A ).
% mod_div_mult_eq
tff(fact_1196_mod__div__mult__eq,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)) = A ).
% mod_div_mult_eq
tff(fact_1197_mod__div__mult__eq,axiom,
! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B)) = A ).
% mod_div_mult_eq
tff(fact_1198_div__mult__mod__eq,axiom,
! [A: nat,B: 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(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = A ).
% div_mult_mod_eq
tff(fact_1199_div__mult__mod__eq,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = A ).
% div_mult_mod_eq
tff(fact_1200_div__mult__mod__eq,axiom,
! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)) = A ).
% div_mult_mod_eq
tff(fact_1201_mod__div__decomp,axiom,
! [A: nat,B: 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(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ).
% mod_div_decomp
tff(fact_1202_mod__div__decomp,axiom,
! [A: int,B: int] : A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ).
% mod_div_decomp
tff(fact_1203_mod__div__decomp,axiom,
! [A: code_integer,B: code_integer] : A = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)) ).
% mod_div_decomp
tff(fact_1204_cancel__div__mod__rules_I1_J,axiom,
! [A: nat,B: 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_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).
% cancel_div_mod_rules(1)
tff(fact_1205_cancel__div__mod__rules_I1_J,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ).
% cancel_div_mod_rules(1)
tff(fact_1206_cancel__div__mod__rules_I1_J,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)),C) = plus_p5714425477246183910nteger(A,C) ).
% cancel_div_mod_rules(1)
tff(fact_1207_cancel__div__mod__rules_I2_J,axiom,
! [B: nat,A: 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_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).
% cancel_div_mod_rules(2)
tff(fact_1208_cancel__div__mod__rules_I2_J,axiom,
! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ).
% cancel_div_mod_rules(2)
tff(fact_1209_cancel__div__mod__rules_I2_J,axiom,
! [B: code_integer,A: code_integer,C: code_integer] : plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B)),modulo364778990260209775nteger(A,B)),C) = plus_p5714425477246183910nteger(A,C) ).
% cancel_div_mod_rules(2)
tff(fact_1210_div__mult1__eq,axiom,
! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C)) ).
% div_mult1_eq
tff(fact_1211_div__mult1__eq,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C)) ).
% div_mult1_eq
tff(fact_1212_div__mult1__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] : divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,C)),divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),modulo364778990260209775nteger(B,C)),C)) ).
% div_mult1_eq
tff(fact_1213_cong__exp__iff__simps_I10_J,axiom,
! [M: num,Q2: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) ).
% cong_exp_iff_simps(10)
tff(fact_1214_cong__exp__iff__simps_I10_J,axiom,
! [M: num,Q2: num,N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).
% cong_exp_iff_simps(10)
tff(fact_1215_cong__exp__iff__simps_I10_J,axiom,
! [M: num,Q2: num,N: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,M)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) != modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,N)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) ).
% cong_exp_iff_simps(10)
tff(fact_1216_cong__exp__iff__simps_I12_J,axiom,
! [M: num,Q2: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) ).
% cong_exp_iff_simps(12)
tff(fact_1217_cong__exp__iff__simps_I12_J,axiom,
! [M: num,Q2: num,N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).
% cong_exp_iff_simps(12)
tff(fact_1218_cong__exp__iff__simps_I12_J,axiom,
! [M: num,Q2: num,N: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,M)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) != modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,N)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) ).
% cong_exp_iff_simps(12)
tff(fact_1219_cong__exp__iff__simps_I13_J,axiom,
! [M: num,Q2: num,N: num] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q2)) ) ) ).
% cong_exp_iff_simps(13)
tff(fact_1220_cong__exp__iff__simps_I13_J,axiom,
! [M: num,Q2: num,N: num] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
<=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q2)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q2)) ) ) ).
% cong_exp_iff_simps(13)
tff(fact_1221_cong__exp__iff__simps_I13_J,axiom,
! [M: num,Q2: num,N: num] :
( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,M)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,N)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) )
<=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Q2)) = modulo364778990260209775nteger(numera6620942414471956472nteger(N),numera6620942414471956472nteger(Q2)) ) ) ).
% cong_exp_iff_simps(13)
tff(fact_1222_ordered__ring__class_Ole__add__iff2,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E)),D))) ) ).
% ordered_ring_class.le_add_iff2
tff(fact_1223_ordered__ring__class_Ole__add__iff2,axiom,
! [A: rat,E: rat,C: rat,B: rat,D: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(aa_rat_rat(times_times_rat(A),E),C)),plus_plus_rat(aa_rat_rat(times_times_rat(B),E),D)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(B,A)),E),D))) ) ).
% ordered_ring_class.le_add_iff2
tff(fact_1224_ordered__ring__class_Ole__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D))) ) ).
% ordered_ring_class.le_add_iff2
tff(fact_1225_ordered__ring__class_Ole__add__iff1,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E)),C)),D)) ) ).
% ordered_ring_class.le_add_iff1
tff(fact_1226_ordered__ring__class_Ole__add__iff1,axiom,
! [A: rat,E: rat,C: rat,B: rat,D: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(aa_rat_rat(times_times_rat(A),E),C)),plus_plus_rat(aa_rat_rat(times_times_rat(B),E),D)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(A,B)),E),C)),D)) ) ).
% ordered_ring_class.le_add_iff1
tff(fact_1227_ordered__ring__class_Ole__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E)),C)),D)) ) ).
% ordered_ring_class.le_add_iff1
tff(fact_1228_less__add__iff2,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),E)),D))) ) ).
% less_add_iff2
tff(fact_1229_less__add__iff2,axiom,
! [A: rat,E: rat,C: rat,B: rat,D: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(aa_rat_rat(times_times_rat(A),E),C)),plus_plus_rat(aa_rat_rat(times_times_rat(B),E),D)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(B,A)),E),D))) ) ).
% less_add_iff2
tff(fact_1230_less__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D))) ) ).
% less_add_iff2
tff(fact_1231_less__add__iff1,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),E)),D)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),E)),C)),D)) ) ).
% less_add_iff1
tff(fact_1232_less__add__iff1,axiom,
! [A: rat,E: rat,C: rat,B: rat,D: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(aa_rat_rat(times_times_rat(A),E),C)),plus_plus_rat(aa_rat_rat(times_times_rat(B),E),D)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(A,B)),E),C)),D)) ) ).
% less_add_iff1
tff(fact_1233_less__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E)),C)),D)) ) ).
% less_add_iff1
tff(fact_1234_square__diff__one__factored,axiom,
! [X: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),X)),one_one_complex) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(X),one_one_complex)),aa_complex_complex(minus_minus_complex(X),one_one_complex)) ).
% square_diff_one_factored
tff(fact_1235_square__diff__one__factored,axiom,
! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real)) ).
% square_diff_one_factored
tff(fact_1236_square__diff__one__factored,axiom,
! [X: rat] : minus_minus_rat(aa_rat_rat(times_times_rat(X),X),one_one_rat) = aa_rat_rat(times_times_rat(plus_plus_rat(X,one_one_rat)),minus_minus_rat(X,one_one_rat)) ).
% square_diff_one_factored
tff(fact_1237_square__diff__one__factored,axiom,
! [X: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),one_one_int)) ).
% square_diff_one_factored
tff(fact_1238_nat__mod__eq__lemma,axiom,
! [X: nat,N: nat,Y: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),N) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
=> ? [Q3: 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,N),Q3)) ) ) ).
% nat_mod_eq_lemma
tff(fact_1239_mod__eq__nat2E,axiom,
! [M: nat,Q2: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q2) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ~ ! [S3: nat] : N != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S3)) ) ) ).
% mod_eq_nat2E
tff(fact_1240_mod__eq__nat1E,axiom,
! [M: nat,Q2: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q2) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ~ ! [S3: nat] : M != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S3)) ) ) ).
% mod_eq_nat1E
tff(fact_1241_mod__mult2__eq,axiom,
! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),Q2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) ).
% mod_mult2_eq
tff(fact_1242_power__gt1,axiom,
! [A: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,N)))) ) ).
% power_gt1
tff(fact_1243_power__gt1,axiom,
! [A: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N)))) ) ).
% power_gt1
tff(fact_1244_power__gt1,axiom,
! [A: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)))) ) ).
% power_gt1
tff(fact_1245_power__gt1,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)))) ) ).
% power_gt1
tff(fact_1246_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),J))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K))) ) ) ).
% less_diff_conv2
tff(fact_1247_eval__nat__numeral_I3_J,axiom,
! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))) ).
% eval_nat_numeral(3)
tff(fact_1248_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> ( aa_nat_nat(aa_nat_fun_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)),N)) = aa_nat_nat(aa_nat_fun_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(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N)) ) ) ).
% nat_diff_add_eq2
tff(fact_1249_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
=> ( aa_nat_nat(aa_nat_fun_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)),N)) = aa_nat_nat(aa_nat_fun_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(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),N) ) ) ).
% nat_diff_add_eq1
tff(fact_1250_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N))) ) ) ).
% nat_le_add_iff2
tff(fact_1251_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),N)) ) ) ).
% nat_le_add_iff1
tff(fact_1252_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)),N) )
<=> ( 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(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N) ) ) ) ).
% nat_eq_add_iff2
tff(fact_1253_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)),N) )
<=> ( 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(minus_minus_nat,I),J)),U)),M) = N ) ) ) ).
% nat_eq_add_iff1
tff(fact_1254_power2__commute,axiom,
! [X: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% power2_commute
tff(fact_1255_power2__commute,axiom,
! [X: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(Y),X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% power2_commute
tff(fact_1256_power2__commute,axiom,
! [X: rat,Y: rat] : aa_nat_rat(power_power_rat(minus_minus_rat(X,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(minus_minus_rat(Y,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% power2_commute
tff(fact_1257_power2__commute,axiom,
! [X: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% power2_commute
tff(fact_1258_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N))) ) ) ).
% nat_less_add_iff2
tff(fact_1259_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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)),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),N)) ) ) ).
% nat_less_add_iff1
tff(fact_1260_Suc3__eq__add__3,axiom,
! [N: nat] : aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N) ).
% Suc3_eq_add_3
tff(fact_1261_div__nat__eqI,axiom,
! [N: nat,Q2: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,Q2))))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = Q2 ) ) ) ).
% div_nat_eqI
tff(fact_1262_Suc__nat__number__of__add,axiom,
! [V: num,N: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),one))),N) ).
% Suc_nat_number_of_add
tff(fact_1263_diff__le__diff__pow,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),N)))) ) ).
% diff_le_diff_pow
tff(fact_1264_Suc__div__eq__add3__div,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),N) ).
% Suc_div_eq_add3_div
tff(fact_1265_div__exp__mod__exp__eq,axiom,
! [A: nat,N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).
% div_exp_mod_exp_eq
tff(fact_1266_div__exp__mod__exp__eq,axiom,
! [A: int,N: nat,M: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).
% div_exp_mod_exp_eq
tff(fact_1267_div__exp__mod__exp__eq,axiom,
! [A: code_integer,N: nat,M: nat] : modulo364778990260209775nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)) = divide6298287555418463151nteger(modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) ).
% div_exp_mod_exp_eq
tff(fact_1268_insersimp,axiom,
! [T: vEBT_VEBT,N: nat,Y: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X_1))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t(T),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))) ) ) ).
% insersimp
tff(fact_1269_power__odd__eq,axiom,
! [A: complex,N: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).
% power_odd_eq
tff(fact_1270_power__odd__eq,axiom,
! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).
% power_odd_eq
tff(fact_1271_power__odd__eq,axiom,
! [A: rat,N: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).
% power_odd_eq
tff(fact_1272_power__odd__eq,axiom,
! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).
% power_odd_eq
tff(fact_1273_power__odd__eq,axiom,
! [A: int,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).
% power_odd_eq
tff(fact_1274_power2__diff,axiom,
! [X: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)),Y)) ).
% power2_diff
tff(fact_1275_power2__diff,axiom,
! [X: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),X)),Y)) ).
% power2_diff
tff(fact_1276_power2__diff,axiom,
! [X: rat,Y: rat] : aa_nat_rat(power_power_rat(minus_minus_rat(X,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = minus_minus_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),X)),Y)) ).
% power2_diff
tff(fact_1277_power2__diff,axiom,
! [X: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X)),Y)) ).
% power2_diff
tff(fact_1278_in__children__def,axiom,
! [N: nat,TreeList: list_VEBT_VEBT,X: nat] :
( pp(aa_nat_bool(vEBT_V5917875025757280293ildren(N,TreeList),X))
<=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N))) ) ).
% in_children_def
tff(fact_1279_invar__vebt_Ointros_I5_J,axiom,
! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X3: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList)))
=> pp(aa_nat_bool(vEBT_invar_vebt(X3),N)) )
=> ( pp(aa_nat_bool(vEBT_invar_vebt(Summary),M))
=> ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
=> ( ( M = aa_nat_nat(suc,N) )
=> ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
=> ( ! [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
=> ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I2)),X_13))
<=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),I2)) ) )
=> ( ( ( Mi = Ma )
=> ! [X3: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList)))
=> ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_1)) ) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),Ma))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)))
=> ( ( ( Mi != Ma )
=> ! [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
=> ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I2 )
=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N))) )
& ! [X3: nat] :
( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N) = I2 )
& pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N))) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X3))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Ma)) ) ) ) ) )
=> pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),Deg)) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(5)
tff(fact_1280_invar__vebt_Ointros_I4_J,axiom,
! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X3: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList)))
=> pp(aa_nat_bool(vEBT_invar_vebt(X3),N)) )
=> ( pp(aa_nat_bool(vEBT_invar_vebt(Summary),M))
=> ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
=> ( ( M = N )
=> ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
=> ( ! [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
=> ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I2)),X_13))
<=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),I2)) ) )
=> ( ( ( Mi = Ma )
=> ! [X3: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList)))
=> ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_1)) ) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),Ma))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)))
=> ( ( ( Mi != Ma )
=> ! [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
=> ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I2 )
=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N))) )
& ! [X3: nat] :
( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N) = I2 )
& pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N))) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X3))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Ma)) ) ) ) ) )
=> pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),Deg)) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(4)
tff(fact_1281_set__vebt__set__vebt_H__valid,axiom,
! [T: vEBT_VEBT,N: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( vEBT_set_vebt(T) = vEBT_VEBT_set_vebt(T) ) ) ).
% set_vebt_set_vebt'_valid
tff(fact_1282_divmod__step__eq,axiom,
! [L: num,R: nat,Q2: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,L)),R))
=> ( unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q2),R)) = 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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Q2)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,R),aa_num_nat(numeral_numeral_nat,L))) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,L)),R))
=> ( unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q2),R)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Q2)),R) ) ) ) ).
% divmod_step_eq
tff(fact_1283_divmod__step__eq,axiom,
! [L: num,R: int,Q2: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R))
=> ( aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q2)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,R),aa_num_int(numeral_numeral_int,L))) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R))
=> ( aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q2)),R) ) ) ) ).
% divmod_step_eq
tff(fact_1284_divmod__step__eq,axiom,
! [L: num,R: code_integer,Q2: code_integer] :
( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(L)),R))
=> ( unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q2,R)) = produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Q2),one_one_Code_integer),minus_8373710615458151222nteger(R,numera6620942414471956472nteger(L))) ) )
& ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(L)),R))
=> ( unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q2,R)) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Q2),R) ) ) ) ).
% divmod_step_eq
tff(fact_1285_double__not__eq__Suc__double,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).
% double_not_eq_Suc_double
tff(fact_1286_Suc__double__not__eq__double,axiom,
! [M: nat,N: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).
% Suc_double_not_eq_double
tff(fact_1287_div__mod__decomp,axiom,
! [A2: nat,N: nat] : A2 = 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(divide_divide_nat,A2),N)),N)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),N)) ).
% div_mod_decomp
tff(fact_1288_unset__bit__Suc,axiom,
! [N: nat,A: code_integer] : bit_se8260200283734997820nteger(aa_nat_nat(suc,N),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se8260200283734997820nteger(N,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).
% unset_bit_Suc
tff(fact_1289_unset__bit__Suc,axiom,
! [N: nat,A: int] : aa_int_int(bit_se4203085406695923979it_int(aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se4203085406695923979it_int(N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).
% unset_bit_Suc
tff(fact_1290_unset__bit__Suc,axiom,
! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).
% unset_bit_Suc
tff(fact_1291_flip__bit__Suc,axiom,
! [N: nat,A: code_integer] : bit_se1345352211410354436nteger(aa_nat_nat(suc,N),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1345352211410354436nteger(N,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).
% flip_bit_Suc
tff(fact_1292_flip__bit__Suc,axiom,
! [N: nat,A: int] : aa_int_int(bit_se2159334234014336723it_int(aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se2159334234014336723it_int(N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).
% flip_bit_Suc
tff(fact_1293_flip__bit__Suc,axiom,
! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).
% flip_bit_Suc
tff(fact_1294_set__bit__Suc,axiom,
! [N: nat,A: code_integer] : bit_se2793503036327961859nteger(aa_nat_nat(suc,N),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se2793503036327961859nteger(N,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).
% set_bit_Suc
tff(fact_1295_set__bit__Suc,axiom,
! [N: nat,A: int] : aa_int_int(bit_se7879613467334960850it_int(aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se7879613467334960850it_int(N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).
% set_bit_Suc
tff(fact_1296_set__bit__Suc,axiom,
! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).
% set_bit_Suc
tff(fact_1297_verit__eq__simplify_I8_J,axiom,
! [X23: num,Y22: num] :
( ( aa_num_num(bit0,X23) = aa_num_num(bit0,Y22) )
<=> ( X23 = Y22 ) ) ).
% verit_eq_simplify(8)
tff(fact_1298_verit__eq__simplify_I9_J,axiom,
! [X33: num,Y32: num] :
( ( aa_num_num(bit1,X33) = aa_num_num(bit1,Y32) )
<=> ( X33 = Y32 ) ) ).
% verit_eq_simplify(9)
tff(fact_1299_VEBT_Oinject_I1_J,axiom,
! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
( ( vEBT_Node(X11,X12,X13,X14) = vEBT_Node(Y11,Y12,Y13,Y14) )
<=> ( ( X11 = Y11 )
& ( X12 = Y12 )
& ( X13 = Y13 )
& ( X14 = Y14 ) ) ) ).
% VEBT.inject(1)
tff(fact_1300_zmod__numeral__Bit0,axiom,
! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W))) ).
% zmod_numeral_Bit0
tff(fact_1301_zmod__numeral__Bit1,axiom,
! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(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_1302_pred__member,axiom,
! [T: vEBT_VEBT,X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(T)),X),Y))
<=> ( pp(aa_nat_bool(vEBT_vebt_member(T),Y))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
& ! [Z4: nat] :
( ( pp(aa_nat_bool(vEBT_vebt_member(T),Z4))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z4),X)) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z4),Y)) ) ) ) ).
% pred_member
tff(fact_1303_succ__member,axiom,
! [T: vEBT_VEBT,X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(T)),X),Y))
<=> ( pp(aa_nat_bool(vEBT_vebt_member(T),Y))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
& ! [Z4: nat] :
( ( pp(aa_nat_bool(vEBT_vebt_member(T),Z4))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z4)) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z4)) ) ) ) ).
% succ_member
tff(fact_1304_div__mod__decomp__int,axiom,
! [A2: int,N: int] : A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),N)),N)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),N)) ).
% div_mod_decomp_int
tff(fact_1305_add__diff__assoc__enat,axiom,
! [Z: extended_enat,Y: extended_enat,X: extended_enat] :
( pp(aa_Ext2772979874139974381t_bool(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_1306_xor__num_Ocases,axiom,
! [X: product_prod_num_num] :
( ( X != product_Pair_num_num(one,one) )
=> ( ! [N2: num] : X != product_Pair_num_num(one,aa_num_num(bit0,N2))
=> ( ! [N2: num] : X != product_Pair_num_num(one,aa_num_num(bit1,N2))
=> ( ! [M3: num] : X != product_Pair_num_num(aa_num_num(bit0,M3),one)
=> ( ! [M3: num,N2: num] : X != product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit0,N2))
=> ( ! [M3: num,N2: num] : X != product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit1,N2))
=> ( ! [M3: num] : X != product_Pair_num_num(aa_num_num(bit1,M3),one)
=> ( ! [M3: num,N2: num] : X != product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit0,N2))
=> ~ ! [M3: num,N2: num] : X != product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit1,N2)) ) ) ) ) ) ) ) ) ).
% xor_num.cases
tff(fact_1307_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(A),A)) ).
% verit_comp_simplify1(2)
tff(fact_1308_verit__comp__simplify1_I2_J,axiom,
! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),A)) ).
% verit_comp_simplify1(2)
tff(fact_1309_verit__comp__simplify1_I2_J,axiom,
! [A: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),A)) ).
% verit_comp_simplify1(2)
tff(fact_1310_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),A)) ).
% verit_comp_simplify1(2)
tff(fact_1311_verit__comp__simplify1_I2_J,axiom,
! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A)) ).
% verit_comp_simplify1(2)
tff(fact_1312_verit__la__disequality,axiom,
! [A: rat,B: rat] :
( ( A = B )
| ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
| ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).
% verit_la_disequality
tff(fact_1313_verit__la__disequality,axiom,
! [A: num,B: num] :
( ( A = B )
| ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
| ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A)) ) ).
% verit_la_disequality
tff(fact_1314_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ).
% verit_la_disequality
tff(fact_1315_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
| ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ).
% verit_la_disequality
tff(fact_1316_verit__comp__simplify1_I1_J,axiom,
! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),A)) ).
% verit_comp_simplify1(1)
tff(fact_1317_verit__comp__simplify1_I1_J,axiom,
! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),A)) ).
% verit_comp_simplify1(1)
tff(fact_1318_verit__comp__simplify1_I1_J,axiom,
! [A: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),A)) ).
% verit_comp_simplify1(1)
tff(fact_1319_verit__comp__simplify1_I1_J,axiom,
! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),A)) ).
% verit_comp_simplify1(1)
tff(fact_1320_verit__comp__simplify1_I1_J,axiom,
! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),A)) ).
% verit_comp_simplify1(1)
tff(fact_1321_verit__comp__simplify1_I3_J,axiom,
! [B3: real,A3: real] :
( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B3),A3))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A3),B3)) ) ).
% verit_comp_simplify1(3)
tff(fact_1322_verit__comp__simplify1_I3_J,axiom,
! [B3: rat,A3: rat] :
( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B3),A3))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A3),B3)) ) ).
% verit_comp_simplify1(3)
tff(fact_1323_verit__comp__simplify1_I3_J,axiom,
! [B3: num,A3: num] :
( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B3),A3))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A3),B3)) ) ).
% verit_comp_simplify1(3)
tff(fact_1324_verit__comp__simplify1_I3_J,axiom,
! [B3: nat,A3: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B3),A3))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A3),B3)) ) ).
% verit_comp_simplify1(3)
tff(fact_1325_verit__comp__simplify1_I3_J,axiom,
! [B3: int,A3: int] :
( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),A3))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A3),B3)) ) ).
% verit_comp_simplify1(3)
tff(fact_1326_verit__eq__simplify_I10_J,axiom,
! [X23: num] : one != aa_num_num(bit0,X23) ).
% verit_eq_simplify(10)
tff(fact_1327_verit__eq__simplify_I14_J,axiom,
! [X23: num,X33: num] : aa_num_num(bit0,X23) != aa_num_num(bit1,X33) ).
% verit_eq_simplify(14)
tff(fact_1328_verit__eq__simplify_I12_J,axiom,
! [X33: num] : one != aa_num_num(bit1,X33) ).
% verit_eq_simplify(12)
tff(fact_1329_real__average__minus__second,axiom,
! [B: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),numeral_numeral_real(aa_num_num(bit0,one)))),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),numeral_numeral_real(aa_num_num(bit0,one))) ).
% real_average_minus_second
tff(fact_1330_real__average__minus__first,axiom,
! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),numeral_numeral_real(aa_num_num(bit0,one)))),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),numeral_numeral_real(aa_num_num(bit0,one))) ).
% real_average_minus_first
tff(fact_1331_all__set__conv__all__nth,axiom,
! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool] :
( ! [X2: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(Xs)))
=> pp(aa_VEBT_VEBT_bool(P,X2)) )
<=> ! [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s6755466524823107622T_VEBT(Xs)))
=> pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,I4))) ) ) ).
% all_set_conv_all_nth
tff(fact_1332_all__set__conv__all__nth,axiom,
! [Xs: list_bool,P: fun_bool_bool] :
( ! [X2: bool] :
( member_bool(X2,set_bool2(Xs))
=> pp(aa_bool_bool(P,X2)) )
<=> ! [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_bool(Xs)))
=> pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),I4))) ) ) ).
% all_set_conv_all_nth
tff(fact_1333_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P: fun_nat_bool] :
( ! [X2: nat] :
( pp(aa_set_nat_bool(member_nat(X2),set_nat2(Xs)))
=> pp(aa_nat_bool(P,X2)) )
<=> ! [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_nat(Xs)))
=> pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),I4))) ) ) ).
% all_set_conv_all_nth
tff(fact_1334_all__set__conv__all__nth,axiom,
! [Xs: list_int,P: fun_int_bool] :
( ! [X2: int] :
( pp(aa_set_int_bool(member_int(X2),set_int2(Xs)))
=> pp(aa_int_bool(P,X2)) )
<=> ! [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_int(Xs)))
=> pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),I4))) ) ) ).
% all_set_conv_all_nth
tff(fact_1335_all__nth__imp__all__set,axiom,
! [Xs: list_P6011104703257516679at_nat,P: fun_Pr7757163693017731481t_bool,X: product_prod_nat_nat] :
( ! [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s5460976970255530739at_nat(Xs)))
=> pp(aa_Pro2232559814647311250t_bool(P,nth_Pr7617993195940197384at_nat(Xs,I2))) )
=> ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs)))
=> pp(aa_Pro2232559814647311250t_bool(P,X)) ) ) ).
% all_nth_imp_all_set
tff(fact_1336_all__nth__imp__all__set,axiom,
! [Xs: list_complex,P: fun_complex_bool,X: complex] :
( ! [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s3451745648224563538omplex(Xs)))
=> pp(aa_complex_bool(P,aa_nat_complex(nth_complex(Xs),I2))) )
=> ( pp(aa_set_complex_bool(member_complex(X),set_complex2(Xs)))
=> pp(aa_complex_bool(P,X)) ) ) ).
% all_nth_imp_all_set
tff(fact_1337_all__nth__imp__all__set,axiom,
! [Xs: list_real,P: fun_real_bool,X: real] :
( ! [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_real(Xs)))
=> pp(aa_real_bool(P,aa_nat_real(nth_real(Xs),I2))) )
=> ( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
=> pp(aa_real_bool(P,X)) ) ) ).
% all_nth_imp_all_set
tff(fact_1338_all__nth__imp__all__set,axiom,
! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool,X: vEBT_VEBT] :
( ! [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
=> pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,I2))) )
=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
=> pp(aa_VEBT_VEBT_bool(P,X)) ) ) ).
% all_nth_imp_all_set
tff(fact_1339_all__nth__imp__all__set,axiom,
! [Xs: list_bool,P: fun_bool_bool,X: bool] :
( ! [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_bool(Xs)))
=> pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),I2))) )
=> ( member_bool(X,set_bool2(Xs))
=> pp(aa_bool_bool(P,X)) ) ) ).
% all_nth_imp_all_set
tff(fact_1340_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P: fun_nat_bool,X: nat] :
( ! [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_nat(Xs)))
=> pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),I2))) )
=> ( pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
=> pp(aa_nat_bool(P,X)) ) ) ).
% all_nth_imp_all_set
tff(fact_1341_all__nth__imp__all__set,axiom,
! [Xs: list_int,P: fun_int_bool,X: int] :
( ! [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_int(Xs)))
=> pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),I2))) )
=> ( pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
=> pp(aa_int_bool(P,X)) ) ) ).
% all_nth_imp_all_set
tff(fact_1342_in__set__conv__nth,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs)))
<=> ? [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s5460976970255530739at_nat(Xs)))
& ( nth_Pr7617993195940197384at_nat(Xs,I4) = X ) ) ) ).
% in_set_conv_nth
tff(fact_1343_in__set__conv__nth,axiom,
! [X: complex,Xs: list_complex] :
( pp(aa_set_complex_bool(member_complex(X),set_complex2(Xs)))
<=> ? [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s3451745648224563538omplex(Xs)))
& ( aa_nat_complex(nth_complex(Xs),I4) = X ) ) ) ).
% in_set_conv_nth
tff(fact_1344_in__set__conv__nth,axiom,
! [X: real,Xs: list_real] :
( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
<=> ? [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_real(Xs)))
& ( aa_nat_real(nth_real(Xs),I4) = X ) ) ) ).
% in_set_conv_nth
tff(fact_1345_in__set__conv__nth,axiom,
! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
<=> ? [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s6755466524823107622T_VEBT(Xs)))
& ( nth_VEBT_VEBT(Xs,I4) = X ) ) ) ).
% in_set_conv_nth
tff(fact_1346_in__set__conv__nth,axiom,
! [X: bool,Xs: list_bool] :
( member_bool(X,set_bool2(Xs))
<=> ? [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_bool(Xs)))
& ( pp(aa_nat_bool(nth_bool(Xs),I4))
<=> pp(X) ) ) ) ).
% in_set_conv_nth
tff(fact_1347_in__set__conv__nth,axiom,
! [X: nat,Xs: list_nat] :
( pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
<=> ? [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_nat(Xs)))
& ( aa_nat_nat(nth_nat(Xs),I4) = X ) ) ) ).
% in_set_conv_nth
tff(fact_1348_in__set__conv__nth,axiom,
! [X: int,Xs: list_int] :
( pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
<=> ? [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_int(Xs)))
& ( aa_nat_int(nth_int(Xs),I4) = X ) ) ) ).
% in_set_conv_nth
tff(fact_1349_list__ball__nth,axiom,
! [N: nat,Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
=> ( ! [X3: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(Xs)))
=> pp(aa_VEBT_VEBT_bool(P,X3)) )
=> pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(Xs,N))) ) ) ).
% list_ball_nth
tff(fact_1350_list__ball__nth,axiom,
! [N: nat,Xs: list_bool,P: fun_bool_bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
=> ( ! [X3: bool] :
( member_bool(X3,set_bool2(Xs))
=> pp(aa_bool_bool(P,X3)) )
=> pp(aa_bool_bool(P,aa_nat_bool(nth_bool(Xs),N))) ) ) ).
% list_ball_nth
tff(fact_1351_list__ball__nth,axiom,
! [N: nat,Xs: list_nat,P: fun_nat_bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_nat(Xs)))
=> ( ! [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),set_nat2(Xs)))
=> pp(aa_nat_bool(P,X3)) )
=> pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),N))) ) ) ).
% list_ball_nth
tff(fact_1352_list__ball__nth,axiom,
! [N: nat,Xs: list_int,P: fun_int_bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),set_int2(Xs)))
=> pp(aa_int_bool(P,X3)) )
=> pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),N))) ) ) ).
% list_ball_nth
tff(fact_1353_nth__mem,axiom,
! [N: nat,Xs: list_P6011104703257516679at_nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s5460976970255530739at_nat(Xs)))
=> pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(nth_Pr7617993195940197384at_nat(Xs,N)),set_Pr5648618587558075414at_nat(Xs))) ) ).
% nth_mem
tff(fact_1354_nth__mem,axiom,
! [N: nat,Xs: list_complex] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3451745648224563538omplex(Xs)))
=> pp(aa_set_complex_bool(member_complex(aa_nat_complex(nth_complex(Xs),N)),set_complex2(Xs))) ) ).
% nth_mem
tff(fact_1355_nth__mem,axiom,
! [N: nat,Xs: list_real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
=> pp(aa_set_real_bool(member_real(aa_nat_real(nth_real(Xs),N)),set_real2(Xs))) ) ).
% nth_mem
tff(fact_1356_nth__mem,axiom,
! [N: nat,Xs: list_VEBT_VEBT] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
=> pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,nth_VEBT_VEBT(Xs,N)),set_VEBT_VEBT2(Xs))) ) ).
% nth_mem
tff(fact_1357_nth__mem,axiom,
! [N: nat,Xs: list_bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
=> member_bool(aa_nat_bool(nth_bool(Xs),N),set_bool2(Xs)) ) ).
% nth_mem
tff(fact_1358_nth__mem,axiom,
! [N: nat,Xs: list_nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_nat(Xs)))
=> pp(aa_set_nat_bool(member_nat(aa_nat_nat(nth_nat(Xs),N)),set_nat2(Xs))) ) ).
% nth_mem
tff(fact_1359_nth__mem,axiom,
! [N: nat,Xs: list_int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
=> pp(aa_set_int_bool(member_int(aa_nat_int(nth_int(Xs),N)),set_int2(Xs))) ) ).
% nth_mem
tff(fact_1360_nested__mint,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),N))
=> ( ( N = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),Mi))
=> ( ( Ma != Mi )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ).
% nested_mint
tff(fact_1361_invar__vebt_Ointros_I3_J,axiom,
! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
( ! [X3: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList)))
=> pp(aa_nat_bool(vEBT_invar_vebt(X3),N)) )
=> ( pp(aa_nat_bool(vEBT_invar_vebt(Summary),M))
=> ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
=> ( ( M = aa_nat_nat(suc,N) )
=> ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
=> ( ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_1))
=> ( ! [X3: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList)))
=> ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_1)) )
=> pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary)),Deg)) ) ) ) ) ) ) ) ).
% invar_vebt.intros(3)
tff(fact_1362_dbl__inc__simps_I3_J,axiom,
aa_complex_complex(neg_nu8557863876264182079omplex,one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit1,one)) ).
% dbl_inc_simps(3)
tff(fact_1363_dbl__inc__simps_I3_J,axiom,
aa_real_real(neg_nu8295874005876285629c_real,one_one_real) = numeral_numeral_real(aa_num_num(bit1,one)) ).
% dbl_inc_simps(3)
tff(fact_1364_dbl__inc__simps_I3_J,axiom,
neg_nu5219082963157363817nc_rat(one_one_rat) = numeral_numeral_rat(aa_num_num(bit1,one)) ).
% dbl_inc_simps(3)
tff(fact_1365_dbl__inc__simps_I3_J,axiom,
aa_int_int(neg_nu5851722552734809277nc_int,one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ).
% dbl_inc_simps(3)
tff(fact_1366_dbl__inc__simps_I5_J,axiom,
! [K: num] : aa_complex_complex(neg_nu8557863876264182079omplex,numera6690914467698888265omplex(K)) = numera6690914467698888265omplex(aa_num_num(bit1,K)) ).
% dbl_inc_simps(5)
tff(fact_1367_dbl__inc__simps_I5_J,axiom,
! [K: num] : aa_real_real(neg_nu8295874005876285629c_real,numeral_numeral_real(K)) = numeral_numeral_real(aa_num_num(bit1,K)) ).
% dbl_inc_simps(5)
tff(fact_1368_dbl__inc__simps_I5_J,axiom,
! [K: num] : neg_nu5219082963157363817nc_rat(numeral_numeral_rat(K)) = numeral_numeral_rat(aa_num_num(bit1,K)) ).
% dbl_inc_simps(5)
tff(fact_1369_dbl__inc__simps_I5_J,axiom,
! [K: num] : aa_int_int(neg_nu5851722552734809277nc_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)) ).
% dbl_inc_simps(5)
tff(fact_1370_vebt__member_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X: nat] : ~ pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)),X)) ).
% vebt_member.simps(2)
tff(fact_1371_VEBT__internal_OminNull_Osimps_I4_J,axiom,
! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux,Uy))) ).
% VEBT_internal.minNull.simps(4)
tff(fact_1372_subset__code_I1_J,axiom,
! [Xs: list_P6011104703257516679at_nat,B2: set_Pr1261947904930325089at_nat] :
( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(Xs)),B2))
<=> ! [X2: product_prod_nat_nat] :
( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X2),set_Pr5648618587558075414at_nat(Xs)))
=> pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X2),B2)) ) ) ).
% subset_code(1)
tff(fact_1373_subset__code_I1_J,axiom,
! [Xs: list_complex,B2: set_complex] :
( pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Xs)),B2))
<=> ! [X2: complex] :
( pp(aa_set_complex_bool(member_complex(X2),set_complex2(Xs)))
=> pp(aa_set_complex_bool(member_complex(X2),B2)) ) ) ).
% subset_code(1)
tff(fact_1374_subset__code_I1_J,axiom,
! [Xs: list_real,B2: set_real] :
( pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(Xs)),B2))
<=> ! [X2: real] :
( pp(aa_set_real_bool(member_real(X2),set_real2(Xs)))
=> pp(aa_set_real_bool(member_real(X2),B2)) ) ) ).
% subset_code(1)
tff(fact_1375_subset__code_I1_J,axiom,
! [Xs: list_VEBT_VEBT,B2: set_VEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs)),B2))
<=> ! [X2: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(Xs)))
=> pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),B2)) ) ) ).
% subset_code(1)
tff(fact_1376_subset__code_I1_J,axiom,
! [Xs: list_int,B2: set_int] :
( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Xs)),B2))
<=> ! [X2: int] :
( pp(aa_set_int_bool(member_int(X2),set_int2(Xs)))
=> pp(aa_set_int_bool(member_int(X2),B2)) ) ) ).
% subset_code(1)
tff(fact_1377_subset__code_I1_J,axiom,
! [Xs: list_nat,B2: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(Xs)),B2))
<=> ! [X2: nat] :
( pp(aa_set_nat_bool(member_nat(X2),set_nat2(Xs)))
=> pp(aa_set_nat_bool(member_nat(X2),B2)) ) ) ).
% subset_code(1)
tff(fact_1378_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_VEBT_VEBT] : size_s6755466524823107622T_VEBT(Xs2) = N ).
% Ex_list_of_length
tff(fact_1379_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_bool] : size_size_list_bool(Xs2) = N ).
% Ex_list_of_length
tff(fact_1380_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_nat] : size_size_list_nat(Xs2) = N ).
% Ex_list_of_length
tff(fact_1381_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_int] : size_size_list_int(Xs2) = N ).
% Ex_list_of_length
tff(fact_1382_neq__if__length__neq,axiom,
! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
( ( size_s6755466524823107622T_VEBT(Xs) != size_s6755466524823107622T_VEBT(Ys) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
tff(fact_1383_neq__if__length__neq,axiom,
! [Xs: list_bool,Ys: list_bool] :
( ( size_size_list_bool(Xs) != size_size_list_bool(Ys) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
tff(fact_1384_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( size_size_list_nat(Xs) != size_size_list_nat(Ys) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
tff(fact_1385_neq__if__length__neq,axiom,
! [Xs: list_int,Ys: list_int] :
( ( size_size_list_int(Xs) != size_size_list_int(Ys) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
tff(fact_1386_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I4_J,axiom,
! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux,Uy)) = one_one_nat ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(4)
tff(fact_1387_dbl__inc__def,axiom,
! [X: complex] : aa_complex_complex(neg_nu8557863876264182079omplex,X) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(X),X)),one_one_complex) ).
% dbl_inc_def
tff(fact_1388_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_1389_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_1390_dbl__inc__def,axiom,
! [X: int] : aa_int_int(neg_nu5851722552734809277nc_int,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),X)),one_one_int) ).
% dbl_inc_def
tff(fact_1391_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)),X) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
tff(fact_1392_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I4_J,axiom,
! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList,Summary)),X) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(4)
tff(fact_1393_length__induct,axiom,
! [P: fun_li6006399982103069984T_bool,Xs: list_VEBT_VEBT] :
( ! [Xs2: list_VEBT_VEBT] :
( ! [Ys2: list_VEBT_VEBT] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s6755466524823107622T_VEBT(Ys2)),size_s6755466524823107622T_VEBT(Xs2)))
=> pp(aa_lis2822252094023462681T_bool(P,Ys2)) )
=> pp(aa_lis2822252094023462681T_bool(P,Xs2)) )
=> pp(aa_lis2822252094023462681T_bool(P,Xs)) ) ).
% length_induct
tff(fact_1394_length__induct,axiom,
! [P: fun_list_bool_bool,Xs: list_bool] :
( ! [Xs2: list_bool] :
( ! [Ys2: list_bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_bool(Ys2)),size_size_list_bool(Xs2)))
=> pp(aa_list_bool_bool(P,Ys2)) )
=> pp(aa_list_bool_bool(P,Xs2)) )
=> pp(aa_list_bool_bool(P,Xs)) ) ).
% length_induct
tff(fact_1395_length__induct,axiom,
! [P: fun_list_nat_bool,Xs: list_nat] :
( ! [Xs2: list_nat] :
( ! [Ys2: list_nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_nat(Ys2)),size_size_list_nat(Xs2)))
=> pp(aa_list_nat_bool(P,Ys2)) )
=> pp(aa_list_nat_bool(P,Xs2)) )
=> pp(aa_list_nat_bool(P,Xs)) ) ).
% length_induct
tff(fact_1396_length__induct,axiom,
! [P: fun_list_int_bool,Xs: list_int] :
( ! [Xs2: list_int] :
( ! [Ys2: list_int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_int(Ys2)),size_size_list_int(Xs2)))
=> pp(aa_list_int_bool(P,Ys2)) )
=> pp(aa_list_int_bool(P,Xs2)) )
=> pp(aa_list_int_bool(P,Xs)) ) ).
% length_induct
tff(fact_1397_nth__equalityI,axiom,
! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
=> ( ! [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs)))
=> ( nth_VEBT_VEBT(Xs,I2) = nth_VEBT_VEBT(Ys,I2) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
tff(fact_1398_nth__equalityI,axiom,
! [Xs: list_bool,Ys: list_bool] :
( ( size_size_list_bool(Xs) = size_size_list_bool(Ys) )
=> ( ! [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_bool(Xs)))
=> ( pp(aa_nat_bool(nth_bool(Xs),I2))
<=> pp(aa_nat_bool(nth_bool(Ys),I2)) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
tff(fact_1399_nth__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
=> ( ! [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_nat(Xs)))
=> ( aa_nat_nat(nth_nat(Xs),I2) = aa_nat_nat(nth_nat(Ys),I2) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
tff(fact_1400_nth__equalityI,axiom,
! [Xs: list_int,Ys: list_int] :
( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
=> ( ! [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),size_size_list_int(Xs)))
=> ( aa_nat_int(nth_int(Xs),I2) = aa_nat_int(nth_int(Ys),I2) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
tff(fact_1401_Skolem__list__nth,axiom,
! [K: nat,P: fun_na2351039247863542899T_bool] :
( ! [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
=> ? [X_13: vEBT_VEBT] : pp(aa_VEBT_VEBT_bool(aa_nat8029987864780243636T_bool(P,I4),X_13)) )
<=> ? [Xs3: list_VEBT_VEBT] :
( ( size_s6755466524823107622T_VEBT(Xs3) = K )
& ! [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
=> pp(aa_VEBT_VEBT_bool(aa_nat8029987864780243636T_bool(P,I4),nth_VEBT_VEBT(Xs3,I4))) ) ) ) ).
% Skolem_list_nth
tff(fact_1402_Skolem__list__nth,axiom,
! [K: nat,P: fun_na9137082442855928016l_bool] :
( ! [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
=> ? [X_13: bool] : pp(aa_bool_bool(aa_nat_fun_bool_bool(P,I4),X_13)) )
<=> ? [Xs3: list_bool] :
( ( size_size_list_bool(Xs3) = K )
& ! [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
=> pp(aa_bool_bool(aa_nat_fun_bool_bool(P,I4),aa_nat_bool(nth_bool(Xs3),I4))) ) ) ) ).
% Skolem_list_nth
tff(fact_1403_Skolem__list__nth,axiom,
! [K: nat,P: fun_nat_fun_nat_bool] :
( ! [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
=> ? [X_13: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I4),X_13)) )
<=> ? [Xs3: list_nat] :
( ( size_size_list_nat(Xs3) = K )
& ! [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I4),aa_nat_nat(nth_nat(Xs3),I4))) ) ) ) ).
% Skolem_list_nth
tff(fact_1404_Skolem__list__nth,axiom,
! [K: nat,P: fun_nat_fun_int_bool] :
( ! [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
=> ? [X_13: int] : pp(aa_int_bool(aa_nat_fun_int_bool(P,I4),X_13)) )
<=> ? [Xs3: list_int] :
( ( size_size_list_int(Xs3) = K )
& ! [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K))
=> pp(aa_int_bool(aa_nat_fun_int_bool(P,I4),aa_nat_int(nth_int(Xs3),I4))) ) ) ) ).
% Skolem_list_nth
tff(fact_1405_list__eq__iff__nth__eq,axiom,
! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
( ( Xs = Ys )
<=> ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
& ! [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s6755466524823107622T_VEBT(Xs)))
=> ( nth_VEBT_VEBT(Xs,I4) = nth_VEBT_VEBT(Ys,I4) ) ) ) ) ).
% list_eq_iff_nth_eq
tff(fact_1406_list__eq__iff__nth__eq,axiom,
! [Xs: list_bool,Ys: list_bool] :
( ( Xs = Ys )
<=> ( ( size_size_list_bool(Xs) = size_size_list_bool(Ys) )
& ! [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_bool(Xs)))
=> ( pp(aa_nat_bool(nth_bool(Xs),I4))
<=> pp(aa_nat_bool(nth_bool(Ys),I4)) ) ) ) ) ).
% list_eq_iff_nth_eq
tff(fact_1407_list__eq__iff__nth__eq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs = Ys )
<=> ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
& ! [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_nat(Xs)))
=> ( aa_nat_nat(nth_nat(Xs),I4) = aa_nat_nat(nth_nat(Ys),I4) ) ) ) ) ).
% list_eq_iff_nth_eq
tff(fact_1408_list__eq__iff__nth__eq,axiom,
! [Xs: list_int,Ys: list_int] :
( ( Xs = Ys )
<=> ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
& ! [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_int(Xs)))
=> ( aa_nat_int(nth_int(Xs),I4) = aa_nat_int(nth_int(Ys),I4) ) ) ) ) ).
% list_eq_iff_nth_eq
tff(fact_1409_invar__vebt_Ointros_I2_J,axiom,
! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
( ! [X3: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList)))
=> pp(aa_nat_bool(vEBT_invar_vebt(X3),N)) )
=> ( pp(aa_nat_bool(vEBT_invar_vebt(Summary),M))
=> ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
=> ( ( M = N )
=> ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
=> ( ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary),X_1))
=> ( ! [X3: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList)))
=> ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X3),X_1)) )
=> pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary)),Deg)) ) ) ) ) ) ) ) ).
% invar_vebt.intros(2)
tff(fact_1410_summaxma,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),Deg))
=> ( ( Mi != Ma )
=> ( the_nat(vEBT_vebt_maxt(Summary)) = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).
% summaxma
tff(fact_1411_mintlistlength,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),N))
=> ( ( Mi != Ma )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),Ma))
& ? [M3: nat] :
( ( some_nat(M3) = vEBT_vebt_mint(Summary) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ) ) ) ).
% mintlistlength
tff(fact_1412_option_Ocollapse,axiom,
! [Option: option4927543243414619207at_nat] :
( ( Option != none_P5556105721700978146at_nat )
=> ( some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) = Option ) ) ).
% option.collapse
tff(fact_1413_option_Ocollapse,axiom,
! [Option: option_nat] :
( ( Option != none_nat )
=> ( some_nat(the_nat(Option)) = Option ) ) ).
% option.collapse
tff(fact_1414_option_Ocollapse,axiom,
! [Option: option_num] :
( ( Option != none_num )
=> ( some_num(the_num(Option)) = Option ) ) ).
% option.collapse
tff(fact_1415_vebt__insert_Osimps_I4_J,axiom,
! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),X)),aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList,Summary) ).
% vebt_insert.simps(4)
tff(fact_1416_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z)) ) ).
% zle_add1_eq_le
tff(fact_1417_not__Some__eq,axiom,
! [X: option4927543243414619207at_nat] :
( ! [Y4: product_prod_nat_nat] : X != some_P7363390416028606310at_nat(Y4)
<=> ( X = none_P5556105721700978146at_nat ) ) ).
% not_Some_eq
tff(fact_1418_not__Some__eq,axiom,
! [X: option_nat] :
( ! [Y4: nat] : X != some_nat(Y4)
<=> ( X = none_nat ) ) ).
% not_Some_eq
tff(fact_1419_not__Some__eq,axiom,
! [X: option_num] :
( ! [Y4: num] : X != some_num(Y4)
<=> ( X = none_num ) ) ).
% not_Some_eq
tff(fact_1420_not__None__eq,axiom,
! [X: option4927543243414619207at_nat] :
( ( X != none_P5556105721700978146at_nat )
<=> ? [Y4: product_prod_nat_nat] : X = some_P7363390416028606310at_nat(Y4) ) ).
% not_None_eq
tff(fact_1421_not__None__eq,axiom,
! [X: option_nat] :
( ( X != none_nat )
<=> ? [Y4: nat] : X = some_nat(Y4) ) ).
% not_None_eq
tff(fact_1422_not__None__eq,axiom,
! [X: option_num] :
( ( X != none_num )
<=> ? [Y4: num] : X = some_num(Y4) ) ).
% not_None_eq
tff(fact_1423_misiz,axiom,
! [T: vEBT_VEBT,N: nat,M: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( ( some_nat(M) = vEBT_vebt_mint(T) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ) ).
% misiz
tff(fact_1424_mint__corr,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( ( vEBT_vebt_mint(T) = some_nat(X) )
=> pp(aa_nat_bool(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(T)),X)) ) ) ).
% mint_corr
tff(fact_1425_mint__sound,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( pp(aa_nat_bool(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(T)),X))
=> ( vEBT_vebt_mint(T) = some_nat(X) ) ) ) ).
% mint_sound
tff(fact_1426_minminNull,axiom,
! [T: vEBT_VEBT] :
( ( vEBT_vebt_mint(T) = none_nat )
=> pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T)) ) ).
% minminNull
tff(fact_1427_minNullmin,axiom,
! [T: vEBT_VEBT] :
( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
=> ( vEBT_vebt_mint(T) = none_nat ) ) ).
% minNullmin
tff(fact_1428_maxbmo,axiom,
! [T: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_maxt(T) = some_nat(X) )
=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X)) ) ).
% maxbmo
tff(fact_1429_power__shift,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),Y) = Z )
<=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_power,some_nat(X)),some_nat(Y)) = some_nat(Z) ) ) ).
% power_shift
tff(fact_1430_mint__member,axiom,
! [T: vEBT_VEBT,N: nat,Maxi: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( ( vEBT_vebt_mint(T) = some_nat(Maxi) )
=> pp(aa_nat_bool(vEBT_vebt_member(T),Maxi)) ) ) ).
% mint_member
tff(fact_1431_maxt__member,axiom,
! [T: vEBT_VEBT,N: nat,Maxi: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( ( vEBT_vebt_maxt(T) = some_nat(Maxi) )
=> pp(aa_nat_bool(vEBT_vebt_member(T),Maxi)) ) ) ).
% maxt_member
tff(fact_1432_option_Oinject,axiom,
! [X23: product_prod_nat_nat,Y22: product_prod_nat_nat] :
( ( some_P7363390416028606310at_nat(X23) = some_P7363390416028606310at_nat(Y22) )
<=> ( X23 = Y22 ) ) ).
% option.inject
tff(fact_1433_option_Oinject,axiom,
! [X23: nat,Y22: nat] :
( ( some_nat(X23) = some_nat(Y22) )
<=> ( X23 = Y22 ) ) ).
% option.inject
tff(fact_1434_option_Oinject,axiom,
! [X23: num,Y22: num] :
( ( some_num(X23) = some_num(Y22) )
<=> ( X23 = Y22 ) ) ).
% option.inject
tff(fact_1435_mint__corr__help,axiom,
! [T: vEBT_VEBT,N: nat,Mini: nat,X: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( ( vEBT_vebt_mint(T) = some_nat(Mini) )
=> ( pp(aa_nat_bool(vEBT_vebt_member(T),X))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mini),X)) ) ) ) ).
% mint_corr_help
tff(fact_1436_maxt__corr__help,axiom,
! [T: vEBT_VEBT,N: nat,Maxi: nat,X: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( ( vEBT_vebt_maxt(T) = some_nat(Maxi) )
=> ( pp(aa_nat_bool(vEBT_vebt_member(T),X))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Maxi)) ) ) ) ).
% maxt_corr_help
tff(fact_1437_maxt__sound,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( pp(aa_nat_bool(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(T)),X))
=> ( vEBT_vebt_maxt(T) = some_nat(X) ) ) ) ).
% maxt_sound
tff(fact_1438_maxt__corr,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( ( vEBT_vebt_maxt(T) = some_nat(X) )
=> pp(aa_nat_bool(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(T)),X)) ) ) ).
% maxt_corr
tff(fact_1439_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),one_one_int)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ).
% zle_diff1_eq
tff(fact_1440_lesseq__shift,axiom,
! [X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
<=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_lesseq,some_nat(X)),some_nat(Y))) ) ).
% lesseq_shift
tff(fact_1441_set__bit__greater__eq,axiom,
! [K: int,N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(bit_se7879613467334960850it_int(N),K))) ).
% set_bit_greater_eq
tff(fact_1442_unset__bit__less__eq,axiom,
! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se4203085406695923979it_int(N),K)),K)) ).
% unset_bit_less_eq
tff(fact_1443_int__le__induct,axiom,
! [I: int,K: int,P: fun_int_bool] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),K))
=> ( pp(aa_int_bool(P,K))
=> ( ! [I2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),K))
=> ( pp(aa_int_bool(P,I2))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int))) ) )
=> pp(aa_int_bool(P,I)) ) ) ) ).
% int_le_induct
tff(fact_1444_int__less__induct,axiom,
! [I: int,K: int,P: fun_int_bool] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K))
=> ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int)))
=> ( ! [I2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I2),K))
=> ( pp(aa_int_bool(P,I2))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int))) ) )
=> pp(aa_int_bool(P,I)) ) ) ) ).
% int_less_induct
tff(fact_1445_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] : aa_int_int(aa_int_fun_int_int(times_times_int,W),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z1),Z22)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z1)),aa_int_int(aa_int_fun_int_int(times_times_int,W),Z22)) ).
% int_distrib(4)
tff(fact_1446_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z1),Z22)),W) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z1),W)),aa_int_int(aa_int_fun_int_int(times_times_int,Z22),W)) ).
% int_distrib(3)
tff(fact_1447_option_Odistinct_I1_J,axiom,
! [X23: product_prod_nat_nat] : none_P5556105721700978146at_nat != some_P7363390416028606310at_nat(X23) ).
% option.distinct(1)
tff(fact_1448_option_Odistinct_I1_J,axiom,
! [X23: nat] : none_nat != some_nat(X23) ).
% option.distinct(1)
tff(fact_1449_option_Odistinct_I1_J,axiom,
! [X23: num] : none_num != some_num(X23) ).
% option.distinct(1)
tff(fact_1450_option_OdiscI,axiom,
! [Option: option4927543243414619207at_nat,X23: product_prod_nat_nat] :
( ( Option = some_P7363390416028606310at_nat(X23) )
=> ( Option != none_P5556105721700978146at_nat ) ) ).
% option.discI
tff(fact_1451_option_OdiscI,axiom,
! [Option: option_nat,X23: nat] :
( ( Option = some_nat(X23) )
=> ( Option != none_nat ) ) ).
% option.discI
tff(fact_1452_option_OdiscI,axiom,
! [Option: option_num,X23: num] :
( ( Option = some_num(X23) )
=> ( Option != none_num ) ) ).
% option.discI
tff(fact_1453_option_Oexhaust,axiom,
! [Y: option4927543243414619207at_nat] :
( ( Y != none_P5556105721700978146at_nat )
=> ~ ! [X22: product_prod_nat_nat] : Y != some_P7363390416028606310at_nat(X22) ) ).
% option.exhaust
tff(fact_1454_option_Oexhaust,axiom,
! [Y: option_nat] :
( ( Y != none_nat )
=> ~ ! [X22: nat] : Y != some_nat(X22) ) ).
% option.exhaust
tff(fact_1455_option_Oexhaust,axiom,
! [Y: option_num] :
( ( Y != none_num )
=> ~ ! [X22: num] : Y != some_num(X22) ) ).
% option.exhaust
tff(fact_1456_split__option__ex,axiom,
! [P: fun_op2627489687811228755t_bool] :
( ? [X_13: option4927543243414619207at_nat] : pp(aa_opt6800557945374683724t_bool(P,X_13))
<=> ( pp(aa_opt6800557945374683724t_bool(P,none_P5556105721700978146at_nat))
| ? [X2: product_prod_nat_nat] : pp(aa_opt6800557945374683724t_bool(P,some_P7363390416028606310at_nat(X2))) ) ) ).
% split_option_ex
tff(fact_1457_split__option__ex,axiom,
! [P: fun_option_nat_bool] :
( ? [X_13: option_nat] : pp(aa_option_nat_bool(P,X_13))
<=> ( pp(aa_option_nat_bool(P,none_nat))
| ? [X2: nat] : pp(aa_option_nat_bool(P,some_nat(X2))) ) ) ).
% split_option_ex
tff(fact_1458_split__option__ex,axiom,
! [P: fun_option_num_bool] :
( ? [X_13: option_num] : pp(aa_option_num_bool(P,X_13))
<=> ( pp(aa_option_num_bool(P,none_num))
| ? [X2: num] : pp(aa_option_num_bool(P,some_num(X2))) ) ) ).
% split_option_ex
tff(fact_1459_split__option__all,axiom,
! [P: fun_op2627489687811228755t_bool] :
( ! [X_13: option4927543243414619207at_nat] : pp(aa_opt6800557945374683724t_bool(P,X_13))
<=> ( pp(aa_opt6800557945374683724t_bool(P,none_P5556105721700978146at_nat))
& ! [X2: product_prod_nat_nat] : pp(aa_opt6800557945374683724t_bool(P,some_P7363390416028606310at_nat(X2))) ) ) ).
% split_option_all
tff(fact_1460_split__option__all,axiom,
! [P: fun_option_nat_bool] :
( ! [X_13: option_nat] : pp(aa_option_nat_bool(P,X_13))
<=> ( pp(aa_option_nat_bool(P,none_nat))
& ! [X2: nat] : pp(aa_option_nat_bool(P,some_nat(X2))) ) ) ).
% split_option_all
tff(fact_1461_split__option__all,axiom,
! [P: fun_option_num_bool] :
( ! [X_13: option_num] : pp(aa_option_num_bool(P,X_13))
<=> ( pp(aa_option_num_bool(P,none_num))
& ! [X2: num] : pp(aa_option_num_bool(P,some_num(X2))) ) ) ).
% split_option_all
tff(fact_1462_combine__options__cases,axiom,
! [X: option4927543243414619207at_nat,P: fun_op5682630191948829869t_bool,Y: option4927543243414619207at_nat] :
( ( ( X = none_P5556105721700978146at_nat )
=> pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(P,X),Y)) )
=> ( ( ( Y = none_P5556105721700978146at_nat )
=> pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(P,X),Y)) )
=> ( ! [A4: product_prod_nat_nat,B4: product_prod_nat_nat] :
( ( X = some_P7363390416028606310at_nat(A4) )
=> ( ( Y = some_P7363390416028606310at_nat(B4) )
=> pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(P,X),Y)) ) )
=> pp(aa_opt6800557945374683724t_bool(aa_opt7606875699174042974t_bool(P,X),Y)) ) ) ) ).
% combine_options_cases
tff(fact_1463_combine__options__cases,axiom,
! [X: option4927543243414619207at_nat,P: fun_op9100744243108201484t_bool,Y: option_nat] :
( ( ( X = none_P5556105721700978146at_nat )
=> pp(aa_option_nat_bool(aa_opt5915962183577952389t_bool(P,X),Y)) )
=> ( ( ( Y = none_nat )
=> pp(aa_option_nat_bool(aa_opt5915962183577952389t_bool(P,X),Y)) )
=> ( ! [A4: product_prod_nat_nat,B4: nat] :
( ( X = some_P7363390416028606310at_nat(A4) )
=> ( ( Y = some_nat(B4) )
=> pp(aa_option_nat_bool(aa_opt5915962183577952389t_bool(P,X),Y)) ) )
=> pp(aa_option_nat_bool(aa_opt5915962183577952389t_bool(P,X),Y)) ) ) ) ).
% combine_options_cases
tff(fact_1464_combine__options__cases,axiom,
! [X: option4927543243414619207at_nat,P: fun_op775672374849087042m_bool,Y: option_num] :
( ( ( X = none_P5556105721700978146at_nat )
=> pp(aa_option_num_bool(aa_opt9110142220287573435m_bool(P,X),Y)) )
=> ( ( ( Y = none_num )
=> pp(aa_option_num_bool(aa_opt9110142220287573435m_bool(P,X),Y)) )
=> ( ! [A4: product_prod_nat_nat,B4: num] :
( ( X = some_P7363390416028606310at_nat(A4) )
=> ( ( Y = some_num(B4) )
=> pp(aa_option_num_bool(aa_opt9110142220287573435m_bool(P,X),Y)) ) )
=> pp(aa_option_num_bool(aa_opt9110142220287573435m_bool(P,X),Y)) ) ) ) ).
% combine_options_cases
tff(fact_1465_combine__options__cases,axiom,
! [X: option_nat,P: fun_op5698422665285572208t_bool,Y: option4927543243414619207at_nat] :
( ( ( X = none_nat )
=> pp(aa_opt6800557945374683724t_bool(aa_opt9009324110926966577t_bool(P,X),Y)) )
=> ( ( ( Y = none_P5556105721700978146at_nat )
=> pp(aa_opt6800557945374683724t_bool(aa_opt9009324110926966577t_bool(P,X),Y)) )
=> ( ! [A4: nat,B4: product_prod_nat_nat] :
( ( X = some_nat(A4) )
=> ( ( Y = some_P7363390416028606310at_nat(B4) )
=> pp(aa_opt6800557945374683724t_bool(aa_opt9009324110926966577t_bool(P,X),Y)) ) )
=> pp(aa_opt6800557945374683724t_bool(aa_opt9009324110926966577t_bool(P,X),Y)) ) ) ) ).
% combine_options_cases
tff(fact_1466_combine__options__cases,axiom,
! [X: option_nat,P: fun_op7792301606653096457t_bool,Y: option_nat] :
( ( ( X = none_nat )
=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(P,X),Y)) )
=> ( ( ( Y = none_nat )
=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(P,X),Y)) )
=> ( ! [A4: nat,B4: nat] :
( ( X = some_nat(A4) )
=> ( ( Y = some_nat(B4) )
=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(P,X),Y)) ) )
=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(P,X),Y)) ) ) ) ).
% combine_options_cases
tff(fact_1467_combine__options__cases,axiom,
! [X: option_nat,P: fun_op8690601775248757823m_bool,Y: option_num] :
( ( ( X = none_nat )
=> pp(aa_option_num_bool(aa_opt1914333813751119784m_bool(P,X),Y)) )
=> ( ( ( Y = none_num )
=> pp(aa_option_num_bool(aa_opt1914333813751119784m_bool(P,X),Y)) )
=> ( ! [A4: nat,B4: num] :
( ( X = some_nat(A4) )
=> ( ( Y = some_num(B4) )
=> pp(aa_option_num_bool(aa_opt1914333813751119784m_bool(P,X),Y)) ) )
=> pp(aa_option_num_bool(aa_opt1914333813751119784m_bool(P,X),Y)) ) ) ) ).
% combine_options_cases
tff(fact_1468_combine__options__cases,axiom,
! [X: option_num,P: fun_op7732454363416259258t_bool,Y: option4927543243414619207at_nat] :
( ( ( X = none_num )
=> pp(aa_opt6800557945374683724t_bool(aa_opt1819983772202877819t_bool(P,X),Y)) )
=> ( ( ( Y = none_P5556105721700978146at_nat )
=> pp(aa_opt6800557945374683724t_bool(aa_opt1819983772202877819t_bool(P,X),Y)) )
=> ( ! [A4: num,B4: product_prod_nat_nat] :
( ( X = some_num(A4) )
=> ( ( Y = some_P7363390416028606310at_nat(B4) )
=> pp(aa_opt6800557945374683724t_bool(aa_opt1819983772202877819t_bool(P,X),Y)) ) )
=> pp(aa_opt6800557945374683724t_bool(aa_opt1819983772202877819t_bool(P,X),Y)) ) ) ) ).
% combine_options_cases
tff(fact_1469_combine__options__cases,axiom,
! [X: option_num,P: fun_op1064277265382982207t_bool,Y: option_nat] :
( ( ( X = none_num )
=> pp(aa_option_nat_bool(aa_opt1215501472626160296t_bool(P,X),Y)) )
=> ( ( ( Y = none_nat )
=> pp(aa_option_nat_bool(aa_opt1215501472626160296t_bool(P,X),Y)) )
=> ( ! [A4: num,B4: nat] :
( ( X = some_num(A4) )
=> ( ( Y = some_nat(B4) )
=> pp(aa_option_nat_bool(aa_opt1215501472626160296t_bool(P,X),Y)) ) )
=> pp(aa_option_nat_bool(aa_opt1215501472626160296t_bool(P,X),Y)) ) ) ) ).
% combine_options_cases
tff(fact_1470_combine__options__cases,axiom,
! [X: option_num,P: fun_op1962577433978643573m_bool,Y: option_num] :
( ( ( X = none_num )
=> pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(P,X),Y)) )
=> ( ( ( Y = none_num )
=> pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(P,X),Y)) )
=> ( ! [A4: num,B4: num] :
( ( X = some_num(A4) )
=> ( ( Y = some_num(B4) )
=> pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(P,X),Y)) ) )
=> pp(aa_option_num_bool(aa_opt4409681509335781342m_bool(P,X),Y)) ) ) ) ).
% combine_options_cases
tff(fact_1471_add1__zle__eq,axiom,
! [W: int,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),one_one_int)),Z))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ).
% add1_zle_eq
tff(fact_1472_int__gr__induct,axiom,
! [K: int,I: int,P: fun_int_bool] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I))
=> ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)))
=> ( ! [I2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I2))
=> ( pp(aa_int_bool(P,I2))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int))) ) )
=> pp(aa_int_bool(P,I)) ) ) ) ).
% int_gr_induct
tff(fact_1473_zless__add1__eq,axiom,
! [W: int,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int)))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z))
| ( W = Z ) ) ) ).
% zless_add1_eq
tff(fact_1474_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),one_one_int)),Z)) ) ).
% zless_imp_add1_zle
tff(fact_1475_int__induct,axiom,
! [P: fun_int_bool,K: int,I: int] :
( pp(aa_int_bool(P,K))
=> ( ! [I2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I2))
=> ( pp(aa_int_bool(P,I2))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int))) ) )
=> ( ! [I2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I2),K))
=> ( pp(aa_int_bool(P,I2))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int))) ) )
=> pp(aa_int_bool(P,I)) ) ) ) ).
% int_induct
tff(fact_1476_option_Osel,axiom,
! [X23: product_prod_nat_nat] : the_Pr8591224930841456533at_nat(some_P7363390416028606310at_nat(X23)) = X23 ).
% option.sel
tff(fact_1477_option_Osel,axiom,
! [X23: nat] : the_nat(some_nat(X23)) = X23 ).
% option.sel
tff(fact_1478_option_Osel,axiom,
! [X23: num] : the_num(some_num(X23)) = X23 ).
% option.sel
tff(fact_1479_option_Oexhaust__sel,axiom,
! [Option: option4927543243414619207at_nat] :
( ( Option != none_P5556105721700978146at_nat )
=> ( Option = some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) ) ) ).
% option.exhaust_sel
tff(fact_1480_option_Oexhaust__sel,axiom,
! [Option: option_nat] :
( ( Option != none_nat )
=> ( Option = some_nat(the_nat(Option)) ) ) ).
% option.exhaust_sel
tff(fact_1481_option_Oexhaust__sel,axiom,
! [Option: option_num] :
( ( Option != none_num )
=> ( Option = some_num(the_num(Option)) ) ) ).
% option.exhaust_sel
tff(fact_1482_pred__list__to__short,axiom,
! [Deg: nat,X: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Ma))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeList)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
=> ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = none_nat ) ) ) ) ).
% pred_list_to_short
tff(fact_1483_succ__list__to__short,axiom,
! [Deg: nat,Mi: nat,X: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),X))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeList)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
=> ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = none_nat ) ) ) ) ).
% succ_list_to_short
tff(fact_1484_pred__max,axiom,
! [Deg: nat,Ma: nat,X: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X))
=> ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = some_nat(Ma) ) ) ) ).
% pred_max
tff(fact_1485_succ__min,axiom,
! [Deg: nat,X: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
=> ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = some_nat(Mi) ) ) ) ).
% succ_min
tff(fact_1486_less__shift,axiom,
! [X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
<=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(X)),some_nat(Y))) ) ).
% less_shift
tff(fact_1487_greater__shift,axiom,
! [Y: nat,X: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
<=> pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(X)),some_nat(Y))) ) ).
% greater_shift
tff(fact_1488_helpyd,axiom,
! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( ( vEBT_vebt_succ(T,X) = some_nat(Y) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ) ).
% helpyd
tff(fact_1489_helpypredd,axiom,
! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( ( vEBT_vebt_pred(T,X) = some_nat(Y) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ) ).
% helpypredd
tff(fact_1490_succ__correct,axiom,
! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( ( vEBT_vebt_succ(T,X) = some_nat(Sx) )
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(vEBT_set_vebt(T)),X),Sx)) ) ) ).
% succ_correct
tff(fact_1491_pred__correct,axiom,
! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( ( vEBT_vebt_pred(T,X) = some_nat(Sx) )
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(vEBT_set_vebt(T)),X),Sx)) ) ) ).
% pred_correct
tff(fact_1492_succ__corr,axiom,
! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( ( vEBT_vebt_succ(T,X) = some_nat(Sx) )
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(T)),X),Sx)) ) ) ).
% succ_corr
tff(fact_1493_pred__corr,axiom,
! [T: vEBT_VEBT,N: nat,X: nat,Px: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( ( vEBT_vebt_pred(T,X) = some_nat(Px) )
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(T)),X),Px)) ) ) ).
% pred_corr
tff(fact_1494_geqmaxNone,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary)),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Ma),X))
=> ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = none_nat ) ) ) ).
% geqmaxNone
tff(fact_1495_local_Opower__def,axiom,
vEBT_VEBT_power = vEBT_V4262088993061758097ft_nat(power_power_nat) ).
% local.power_def
tff(fact_1496_vebt__maxt_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy,Uz)) = some_nat(Ma) ).
% vebt_maxt.simps(3)
tff(fact_1497_vebt__mint_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy,Uz)) = some_nat(Mi) ).
% vebt_mint.simps(3)
tff(fact_1498_mul__shift,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y) = Z )
<=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(X)),some_nat(Y)) = some_nat(Z) ) ) ).
% mul_shift
tff(fact_1499_add__shift,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = Z )
<=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,some_nat(X)),some_nat(Y)) = some_nat(Z) ) ) ).
% add_shift
tff(fact_1500_vebt__maxt_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)) = none_nat ).
% vebt_maxt.simps(2)
tff(fact_1501_vebt__pred_Osimps_I4_J,axiom,
! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] : vEBT_vebt_pred(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz,Va),Vb) = none_nat ).
% vebt_pred.simps(4)
tff(fact_1502_vebt__succ_Osimps_I3_J,axiom,
! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] : vEBT_vebt_succ(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy,Uz),Va) = none_nat ).
% vebt_succ.simps(3)
tff(fact_1503_vebt__mint_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)) = none_nat ).
% vebt_mint.simps(2)
tff(fact_1504_maxt__corr__help__empty,axiom,
! [T: vEBT_VEBT,N: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( ( vEBT_vebt_maxt(T) = none_nat )
=> ( vEBT_VEBT_set_vebt(T) = bot_bot_set_nat ) ) ) ).
% maxt_corr_help_empty
tff(fact_1505_mint__corr__help__empty,axiom,
! [T: vEBT_VEBT,N: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( ( vEBT_vebt_mint(T) = none_nat )
=> ( vEBT_VEBT_set_vebt(T) = bot_bot_set_nat ) ) ) ).
% mint_corr_help_empty
tff(fact_1506_add__def,axiom,
vEBT_VEBT_add = vEBT_V4262088993061758097ft_nat(plus_plus_nat) ).
% add_def
tff(fact_1507_mul__def,axiom,
vEBT_VEBT_mul = vEBT_V4262088993061758097ft_nat(times_times_nat) ).
% mul_def
tff(fact_1508_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
! [F: fun_Pr2477239776880934807at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] : vEBT_V1502963449132264192at_nat(F,some_P7363390416028606310at_nat(A),some_P7363390416028606310at_nat(B)) = some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(F,A),B)) ).
% VEBT_internal.option_shift.simps(3)
tff(fact_1509_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
! [F: fun_num_fun_num_num,A: num,B: num] : vEBT_V819420779217536731ft_num(F,some_num(A),some_num(B)) = some_num(aa_num_num(aa_num_fun_num_num(F,A),B)) ).
% VEBT_internal.option_shift.simps(3)
tff(fact_1510_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
! [F: fun_nat_fun_nat_nat,A: nat,B: nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(F),some_nat(A)),some_nat(B)) = some_nat(aa_nat_nat(aa_nat_fun_nat_nat(F,A),B)) ).
% VEBT_internal.option_shift.simps(3)
tff(fact_1511_VEBT__internal_Ooption__shift_Oelims,axiom,
! [X: fun_Pr2477239776880934807at_nat,Xa2: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
( ( vEBT_V1502963449132264192at_nat(X,Xa2,Xb) = Y )
=> ( ( ( Xa2 = none_P5556105721700978146at_nat )
=> ( Y != none_P5556105721700978146at_nat ) )
=> ( ( ? [V2: product_prod_nat_nat] : Xa2 = some_P7363390416028606310at_nat(V2)
=> ( ( Xb = none_P5556105721700978146at_nat )
=> ( Y != none_P5556105721700978146at_nat ) ) )
=> ~ ! [A4: product_prod_nat_nat] :
( ( Xa2 = some_P7363390416028606310at_nat(A4) )
=> ! [B4: product_prod_nat_nat] :
( ( Xb = some_P7363390416028606310at_nat(B4) )
=> ( Y != some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(X,A4),B4)) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.elims
tff(fact_1512_VEBT__internal_Ooption__shift_Oelims,axiom,
! [X: fun_num_fun_num_num,Xa2: option_num,Xb: option_num,Y: option_num] :
( ( vEBT_V819420779217536731ft_num(X,Xa2,Xb) = Y )
=> ( ( ( Xa2 = none_num )
=> ( Y != none_num ) )
=> ( ( ? [V2: num] : Xa2 = some_num(V2)
=> ( ( Xb = none_num )
=> ( Y != none_num ) ) )
=> ~ ! [A4: num] :
( ( Xa2 = some_num(A4) )
=> ! [B4: num] :
( ( Xb = some_num(B4) )
=> ( Y != some_num(aa_num_num(aa_num_fun_num_num(X,A4),B4)) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.elims
tff(fact_1513_VEBT__internal_Ooption__shift_Oelims,axiom,
! [X: fun_nat_fun_nat_nat,Xa2: option_nat,Xb: option_nat,Y: option_nat] :
( ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(X),Xa2),Xb) = Y )
=> ( ( ( Xa2 = none_nat )
=> ( Y != none_nat ) )
=> ( ( ? [V2: nat] : Xa2 = some_nat(V2)
=> ( ( Xb = none_nat )
=> ( Y != none_nat ) ) )
=> ~ ! [A4: nat] :
( ( Xa2 = some_nat(A4) )
=> ! [B4: nat] :
( ( Xb = some_nat(B4) )
=> ( Y != some_nat(aa_nat_nat(aa_nat_fun_nat_nat(X,A4),B4)) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.elims
tff(fact_1514_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
! [Uw: fun_Pr2477239776880934807at_nat,V: product_prod_nat_nat] : vEBT_V1502963449132264192at_nat(Uw,some_P7363390416028606310at_nat(V),none_P5556105721700978146at_nat) = none_P5556105721700978146at_nat ).
% VEBT_internal.option_shift.simps(2)
tff(fact_1515_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
! [Uw: fun_num_fun_num_num,V: num] : vEBT_V819420779217536731ft_num(Uw,some_num(V),none_num) = none_num ).
% VEBT_internal.option_shift.simps(2)
tff(fact_1516_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
! [Uw: fun_nat_fun_nat_nat,V: nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Uw),some_nat(V)),none_nat) = none_nat ).
% VEBT_internal.option_shift.simps(2)
tff(fact_1517_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
! [X: produc533602882674032673at_nat] :
( ! [Uu2: fun_Pr248683899324676269t_bool,Uv2: option4927543243414619207at_nat] : X != produc5961576194287291153at_nat(Uu2,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv2))
=> ( ! [Uw2: fun_Pr248683899324676269t_bool,V2: product_prod_nat_nat] : X != produc5961576194287291153at_nat(Uw2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat))
=> ~ ! [F2: fun_Pr248683899324676269t_bool,X3: product_prod_nat_nat,Y2: product_prod_nat_nat] : X != produc5961576194287291153at_nat(F2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(X3),some_P7363390416028606310at_nat(Y2))) ) ) ).
% VEBT_internal.option_comp_shift.cases
tff(fact_1518_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
! [X: produc3700678068921409119on_nat] :
( ! [Uu2: fun_nat_fun_nat_bool,Uv2: option_nat] : X != produc2004789812759622991on_nat(Uu2,produc5098337634421038937on_nat(none_nat,Uv2))
=> ( ! [Uw2: fun_nat_fun_nat_bool,V2: nat] : X != produc2004789812759622991on_nat(Uw2,produc5098337634421038937on_nat(some_nat(V2),none_nat))
=> ~ ! [F2: fun_nat_fun_nat_bool,X3: nat,Y2: nat] : X != produc2004789812759622991on_nat(F2,produc5098337634421038937on_nat(some_nat(X3),some_nat(Y2))) ) ) ).
% VEBT_internal.option_comp_shift.cases
tff(fact_1519_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
! [X: produc6167219322466932359on_num] :
( ! [Uu2: fun_num_fun_num_bool,Uv2: option_num] : X != produc8433281988922931063on_num(Uu2,produc8585076106096196333on_num(none_num,Uv2))
=> ( ! [Uw2: fun_num_fun_num_bool,V2: num] : X != produc8433281988922931063on_num(Uw2,produc8585076106096196333on_num(some_num(V2),none_num))
=> ~ ! [F2: fun_num_fun_num_bool,X3: num,Y2: num] : X != produc8433281988922931063on_num(F2,produc8585076106096196333on_num(some_num(X3),some_num(Y2))) ) ) ).
% VEBT_internal.option_comp_shift.cases
tff(fact_1520_VEBT__internal_Ooption__shift_Ocases,axiom,
! [X: produc1260151622200656695at_nat] :
( ! [Uu2: fun_Pr2477239776880934807at_nat,Uv2: option4927543243414619207at_nat] : X != produc7814092408141220007at_nat(Uu2,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv2))
=> ( ! [Uw2: fun_Pr2477239776880934807at_nat,V2: product_prod_nat_nat] : X != produc7814092408141220007at_nat(Uw2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat))
=> ~ ! [F2: fun_Pr2477239776880934807at_nat,A4: product_prod_nat_nat,B4: product_prod_nat_nat] : X != produc7814092408141220007at_nat(F2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(A4),some_P7363390416028606310at_nat(B4))) ) ) ).
% VEBT_internal.option_shift.cases
tff(fact_1521_VEBT__internal_Ooption__shift_Ocases,axiom,
! [X: produc7004362771917097900on_nat] :
( ! [Uu2: fun_nat_fun_nat_nat,Uv2: option_nat] : X != produc6699827698519632094on_nat(Uu2,produc5098337634421038937on_nat(none_nat,Uv2))
=> ( ! [Uw2: fun_nat_fun_nat_nat,V2: nat] : X != produc6699827698519632094on_nat(Uw2,produc5098337634421038937on_nat(some_nat(V2),none_nat))
=> ~ ! [F2: fun_nat_fun_nat_nat,A4: nat,B4: nat] : X != produc6699827698519632094on_nat(F2,produc5098337634421038937on_nat(some_nat(A4),some_nat(B4))) ) ) ).
% VEBT_internal.option_shift.cases
tff(fact_1522_VEBT__internal_Ooption__shift_Ocases,axiom,
! [X: produc2703417854250991842on_num] :
( ! [Uu2: fun_num_fun_num_num,Uv2: option_num] : X != produc6360833703471310868on_num(Uu2,produc8585076106096196333on_num(none_num,Uv2))
=> ( ! [Uw2: fun_num_fun_num_num,V2: num] : X != produc6360833703471310868on_num(Uw2,produc8585076106096196333on_num(some_num(V2),none_num))
=> ~ ! [F2: fun_num_fun_num_num,A4: num,B4: num] : X != produc6360833703471310868on_num(F2,produc8585076106096196333on_num(some_num(A4),some_num(B4))) ) ) ).
% VEBT_internal.option_shift.cases
tff(fact_1523_is__pred__in__set__def,axiom,
! [Xs: set_nat,X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(Xs),X),Y))
<=> ( pp(aa_set_nat_bool(member_nat(Y),Xs))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
& ! [X2: nat] :
( pp(aa_set_nat_bool(member_nat(X2),Xs))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),X))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y)) ) ) ) ) ).
% is_pred_in_set_def
tff(fact_1524_is__succ__in__set__def,axiom,
! [Xs: set_nat,X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(Xs),X),Y))
<=> ( pp(aa_set_nat_bool(member_nat(Y),Xs))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
& ! [X2: nat] :
( pp(aa_set_nat_bool(member_nat(X2),Xs))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),X2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X2)) ) ) ) ) ).
% is_succ_in_set_def
tff(fact_1525_buildup__gives__empty,axiom,
! [N: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(N)) = bot_bot_set_nat ).
% buildup_gives_empty
tff(fact_1526_Diff__eq__empty__iff,axiom,
! [A2: set_int,B2: set_int] :
( ( minus_minus_set_int(A2,B2) = bot_bot_set_int )
<=> pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2)) ) ).
% Diff_eq_empty_iff
tff(fact_1527_Diff__eq__empty__iff,axiom,
! [A2: set_real,B2: set_real] :
( ( minus_minus_set_real(A2,B2) = bot_bot_set_real )
<=> pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2)) ) ).
% Diff_eq_empty_iff
tff(fact_1528_Diff__eq__empty__iff,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( minus_1356011639430497352at_nat(A2,B2) = bot_bo2099793752762293965at_nat )
<=> pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),B2)) ) ).
% Diff_eq_empty_iff
tff(fact_1529_Diff__eq__empty__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( minus_minus_set_nat(A2,B2) = bot_bot_set_nat )
<=> pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),B2)) ) ).
% Diff_eq_empty_iff
tff(fact_1530_succ__less__length__list,axiom,
! [Deg: nat,Mi: nat,X: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),X))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
=> ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))) ) ) ) ) ).
% succ_less_length_list
tff(fact_1531_succ__greatereq__min,axiom,
! [Deg: nat,Mi: nat,X: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi),X))
=> ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ).
% succ_greatereq_min
tff(fact_1532_pred__lesseq__max,axiom,
! [Deg: nat,X: nat,Ma: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Ma))
=> ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X),some_nat(Mi),none_nat),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ).
% pred_lesseq_max
tff(fact_1533_pred__less__length__list,axiom,
! [Deg: nat,X: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Ma))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
=> ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X),some_nat(Mi),none_nat),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))) ) ) ) ) ).
% pred_less_length_list
tff(fact_1534_product__nth,axiom,
! [N: nat,Xs: list_num,Ys: list_num] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_num(Xs)),size_size_list_num(Ys))))
=> ( nth_Pr6456567536196504476um_num(product_num_num(Xs,Ys),N) = product_Pair_num_num(aa_nat_num(nth_num(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_num(Ys))),aa_nat_num(nth_num(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_num(Ys)))) ) ) ).
% product_nth
tff(fact_1535_product__nth,axiom,
! [N: nat,Xs: list_Code_integer,Ys: list_bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s3445333598471063425nteger(Xs)),size_size_list_bool(Ys))))
=> ( nth_Pr136253548056570280r_bool(produc2469108709554126195r_bool(Xs,Ys),N) = produc5135875973608410785r_bool(aa_nat_Code_integer(nth_Code_integer(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_bool(Ys))),aa_nat_bool(nth_bool(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_bool(Ys)))) ) ) ).
% product_nth
tff(fact_1536_product__nth,axiom,
! [N: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys))))
=> ( nth_Pr4953567300277697838T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys),N) = produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).
% product_nth
tff(fact_1537_product__nth,axiom,
! [N: nat,Xs: list_VEBT_VEBT,Ys: list_bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_bool(Ys))))
=> ( nth_Pr1289831921885633937T_bool(produc5466675335756156456T_bool(Xs,Ys),N) = produc3035007008238225530T_bool(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_bool(Ys))),aa_nat_bool(nth_bool(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_bool(Ys)))) ) ) ).
% product_nth
tff(fact_1538_product__nth,axiom,
! [N: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_nat(Ys))))
=> ( nth_Pr1791586995822124652BT_nat(produc7295137177222721919BT_nat(Xs,Ys),N) = produc738532404422230701BT_nat(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_nat(Ys))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_nat(Ys)))) ) ) ).
% product_nth
tff(fact_1539_product__nth,axiom,
! [N: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_int(Ys))))
=> ( nth_Pr6837108013167703752BT_int(produc7292646706713671643BT_int(Xs,Ys),N) = produc736041933913180425BT_int(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_int(Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_int(Ys)))) ) ) ).
% product_nth
tff(fact_1540_product__nth,axiom,
! [N: nat,Xs: list_bool,Ys: list_VEBT_VEBT] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_s6755466524823107622T_VEBT(Ys))))
=> ( nth_Pr3250499561091541003T_VEBT(produc2282853104913622818T_VEBT(Xs,Ys),N) = produc9074556814250467700T_VEBT(aa_nat_bool(nth_bool(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).
% product_nth
tff(fact_1541_product__nth,axiom,
! [N: nat,Xs: list_bool,Ys: list_bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_size_list_bool(Ys))))
=> ( nth_Pr2856333732135533678l_bool(product_bool_bool(Xs,Ys),N) = produc8297744024737662807l_bool(aa_nat_bool(nth_bool(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_bool(Ys))),aa_nat_bool(nth_bool(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_bool(Ys)))) ) ) ).
% product_nth
tff(fact_1542_product__nth,axiom,
! [N: nat,Xs: list_bool,Ys: list_nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_size_list_nat(Ys))))
=> ( nth_Pr2186422209488468047ol_nat(product_bool_nat(Xs,Ys),N) = produc8874003862241262736ol_nat(aa_nat_bool(nth_bool(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_nat(Ys))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_nat(Ys)))) ) ) ).
% product_nth
tff(fact_1543_product__nth,axiom,
! [N: nat,Xs: list_bool,Ys: list_int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_size_list_int(Ys))))
=> ( nth_Pr7231943226834047147ol_int(product_bool_int(Xs,Ys),N) = produc8871513391732212460ol_int(aa_nat_bool(nth_bool(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_int(Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_int(Ys)))) ) ) ).
% product_nth
tff(fact_1544_obtain__set__succ,axiom,
! [X: nat,Z: nat,A2: set_nat,B2: set_nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z))
=> ( pp(aa_nat_bool(vEBT_VEBT_max_in_set(A2),Z))
=> ( pp(aa_set_nat_bool(finite_finite_nat,B2))
=> ( ( A2 = B2 )
=> ? [X_1: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(A2),X),X_1)) ) ) ) ) ).
% obtain_set_succ
tff(fact_1545_obtain__set__pred,axiom,
! [Z: nat,X: nat,A2: set_nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X))
=> ( pp(aa_nat_bool(vEBT_VEBT_min_in_set(A2),Z))
=> ( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ? [X_1: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(A2),X),X_1)) ) ) ) ).
% obtain_set_pred
tff(fact_1546_vebt__succ_Osimps_I6_J,axiom,
! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
=> ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary),X) = some_nat(Mi) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
=> ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary),X) = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_succ(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ).
% vebt_succ.simps(6)
tff(fact_1547_set__vebt_H__def,axiom,
! [T: vEBT_VEBT] : vEBT_VEBT_set_vebt(T) = collect_nat(vEBT_vebt_member(T)) ).
% set_vebt'_def
tff(fact_1548_set__vebt__finite,axiom,
! [T: vEBT_VEBT,N: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> pp(aa_set_nat_bool(finite_finite_nat,vEBT_VEBT_set_vebt(T))) ) ).
% set_vebt_finite
tff(fact_1549_pred__none__empty,axiom,
! [Xs: set_nat,A: nat] :
( ~ ? [X_1: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_pred_in_set(Xs),A),X_1))
=> ( pp(aa_set_nat_bool(finite_finite_nat,Xs))
=> ~ ? [X4: nat] :
( pp(aa_set_nat_bool(member_nat(X4),Xs))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),A)) ) ) ) ).
% pred_none_empty
tff(fact_1550_succ__none__empty,axiom,
! [Xs: set_nat,A: nat] :
( ~ ? [X_1: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(vEBT_is_succ_in_set(Xs),A),X_1))
=> ( pp(aa_set_nat_bool(finite_finite_nat,Xs))
=> ~ ? [X4: nat] :
( pp(aa_set_nat_bool(member_nat(X4),Xs))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),X4)) ) ) ) ).
% succ_none_empty
tff(fact_1551_psubsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),B2))
=> ( ( A2 != B2 )
=> pp(aa_set_nat_bool(ord_less_set_nat(A2),B2)) ) ) ).
% psubsetI
tff(fact_1552_DiffI,axiom,
! [C: complex,A2: set_complex,B2: set_complex] :
( pp(aa_set_complex_bool(member_complex(C),A2))
=> ( ~ pp(aa_set_complex_bool(member_complex(C),B2))
=> pp(aa_set_complex_bool(member_complex(C),minus_811609699411566653omplex(A2,B2))) ) ) ).
% DiffI
tff(fact_1553_DiffI,axiom,
! [C: real,A2: set_real,B2: set_real] :
( pp(aa_set_real_bool(member_real(C),A2))
=> ( ~ pp(aa_set_real_bool(member_real(C),B2))
=> pp(aa_set_real_bool(member_real(C),minus_minus_set_real(A2,B2))) ) ) ).
% DiffI
tff(fact_1554_DiffI,axiom,
! [C: int,A2: set_int,B2: set_int] :
( pp(aa_set_int_bool(member_int(C),A2))
=> ( ~ pp(aa_set_int_bool(member_int(C),B2))
=> pp(aa_set_int_bool(member_int(C),minus_minus_set_int(A2,B2))) ) ) ).
% DiffI
tff(fact_1555_DiffI,axiom,
! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),A2))
=> ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),B2))
=> pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),minus_1356011639430497352at_nat(A2,B2))) ) ) ).
% DiffI
tff(fact_1556_DiffI,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( pp(aa_set_nat_bool(member_nat(C),A2))
=> ( ~ pp(aa_set_nat_bool(member_nat(C),B2))
=> pp(aa_set_nat_bool(member_nat(C),minus_minus_set_nat(A2,B2))) ) ) ).
% DiffI
tff(fact_1557_Diff__iff,axiom,
! [C: complex,A2: set_complex,B2: set_complex] :
( pp(aa_set_complex_bool(member_complex(C),minus_811609699411566653omplex(A2,B2)))
<=> ( pp(aa_set_complex_bool(member_complex(C),A2))
& ~ pp(aa_set_complex_bool(member_complex(C),B2)) ) ) ).
% Diff_iff
tff(fact_1558_Diff__iff,axiom,
! [C: real,A2: set_real,B2: set_real] :
( pp(aa_set_real_bool(member_real(C),minus_minus_set_real(A2,B2)))
<=> ( pp(aa_set_real_bool(member_real(C),A2))
& ~ pp(aa_set_real_bool(member_real(C),B2)) ) ) ).
% Diff_iff
tff(fact_1559_Diff__iff,axiom,
! [C: int,A2: set_int,B2: set_int] :
( pp(aa_set_int_bool(member_int(C),minus_minus_set_int(A2,B2)))
<=> ( pp(aa_set_int_bool(member_int(C),A2))
& ~ pp(aa_set_int_bool(member_int(C),B2)) ) ) ).
% Diff_iff
tff(fact_1560_Diff__iff,axiom,
! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),minus_1356011639430497352at_nat(A2,B2)))
<=> ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),A2))
& ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),B2)) ) ) ).
% Diff_iff
tff(fact_1561_Diff__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( pp(aa_set_nat_bool(member_nat(C),minus_minus_set_nat(A2,B2)))
<=> ( pp(aa_set_nat_bool(member_nat(C),A2))
& ~ pp(aa_set_nat_bool(member_nat(C),B2)) ) ) ).
% Diff_iff
tff(fact_1562_Diff__idemp,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(minus_1356011639430497352at_nat(A2,B2),B2) = minus_1356011639430497352at_nat(A2,B2) ).
% Diff_idemp
tff(fact_1563_Diff__idemp,axiom,
! [A2: set_nat,B2: set_nat] : minus_minus_set_nat(minus_minus_set_nat(A2,B2),B2) = minus_minus_set_nat(A2,B2) ).
% Diff_idemp
tff(fact_1564_succ__empty,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( ( vEBT_vebt_succ(T,X) = none_nat )
<=> ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_af(T),X)) = bot_bot_set_nat ) ) ) ).
% succ_empty
tff(fact_1565_pred__empty,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( ( vEBT_vebt_pred(T,X) = none_nat )
<=> ( collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ag(T),X)) = bot_bot_set_nat ) ) ) ).
% pred_empty
tff(fact_1566_List_Ofinite__set,axiom,
! [Xs: list_VEBT_VEBT] : pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,set_VEBT_VEBT2(Xs))) ).
% List.finite_set
tff(fact_1567_List_Ofinite__set,axiom,
! [Xs: list_nat] : pp(aa_set_nat_bool(finite_finite_nat,set_nat2(Xs))) ).
% List.finite_set
tff(fact_1568_List_Ofinite__set,axiom,
! [Xs: list_int] : pp(aa_set_int_bool(finite_finite_int,set_int2(Xs))) ).
% List.finite_set
tff(fact_1569_List_Ofinite__set,axiom,
! [Xs: list_complex] : pp(aa_set_complex_bool(finite3207457112153483333omplex,set_complex2(Xs))) ).
% List.finite_set
tff(fact_1570_Diff__empty,axiom,
! [A2: set_int] : minus_minus_set_int(A2,bot_bot_set_int) = A2 ).
% Diff_empty
tff(fact_1571_Diff__empty,axiom,
! [A2: set_real] : minus_minus_set_real(A2,bot_bot_set_real) = A2 ).
% Diff_empty
tff(fact_1572_Diff__empty,axiom,
! [A2: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(A2,bot_bo2099793752762293965at_nat) = A2 ).
% Diff_empty
tff(fact_1573_Diff__empty,axiom,
! [A2: set_nat] : minus_minus_set_nat(A2,bot_bot_set_nat) = A2 ).
% Diff_empty
tff(fact_1574_empty__Diff,axiom,
! [A2: set_int] : minus_minus_set_int(bot_bot_set_int,A2) = bot_bot_set_int ).
% empty_Diff
tff(fact_1575_empty__Diff,axiom,
! [A2: set_real] : minus_minus_set_real(bot_bot_set_real,A2) = bot_bot_set_real ).
% empty_Diff
tff(fact_1576_empty__Diff,axiom,
! [A2: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(bot_bo2099793752762293965at_nat,A2) = bot_bo2099793752762293965at_nat ).
% empty_Diff
tff(fact_1577_empty__Diff,axiom,
! [A2: set_nat] : minus_minus_set_nat(bot_bot_set_nat,A2) = bot_bot_set_nat ).
% empty_Diff
tff(fact_1578_Diff__cancel,axiom,
! [A2: set_int] : minus_minus_set_int(A2,A2) = bot_bot_set_int ).
% Diff_cancel
tff(fact_1579_Diff__cancel,axiom,
! [A2: set_real] : minus_minus_set_real(A2,A2) = bot_bot_set_real ).
% Diff_cancel
tff(fact_1580_Diff__cancel,axiom,
! [A2: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(A2,A2) = bot_bo2099793752762293965at_nat ).
% Diff_cancel
tff(fact_1581_Diff__cancel,axiom,
! [A2: set_nat] : minus_minus_set_nat(A2,A2) = bot_bot_set_nat ).
% Diff_cancel
tff(fact_1582_length__product,axiom,
! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] : size_s7466405169056248089T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).
% length_product
tff(fact_1583_length__product,axiom,
! [Xs: list_VEBT_VEBT,Ys: list_bool] : size_s1343096544515248124T_bool(produc5466675335756156456T_bool(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_bool(Ys)) ).
% length_product
tff(fact_1584_length__product,axiom,
! [Xs: list_VEBT_VEBT,Ys: list_nat] : size_s6152045936467909847BT_nat(produc7295137177222721919BT_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_nat(Ys)) ).
% length_product
tff(fact_1585_length__product,axiom,
! [Xs: list_VEBT_VEBT,Ys: list_int] : size_s3661962791536183091BT_int(produc7292646706713671643BT_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_int(Ys)) ).
% length_product
tff(fact_1586_length__product,axiom,
! [Xs: list_bool,Ys: list_VEBT_VEBT] : size_s7211631430880329078T_VEBT(produc2282853104913622818T_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).
% length_product
tff(fact_1587_length__product,axiom,
! [Xs: list_bool,Ys: list_bool] : size_s6988337448923680857l_bool(product_bool_bool(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_size_list_bool(Ys)) ).
% length_product
tff(fact_1588_length__product,axiom,
! [Xs: list_bool,Ys: list_nat] : size_s3243772863075988794ol_nat(product_bool_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_size_list_nat(Ys)) ).
% length_product
tff(fact_1589_length__product,axiom,
! [Xs: list_bool,Ys: list_int] : size_s753689718144262038ol_int(product_bool_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_bool(Xs)),size_size_list_int(Ys)) ).
% length_product
tff(fact_1590_length__product,axiom,
! [Xs: list_nat,Ys: list_VEBT_VEBT] : size_s4762443039079500285T_VEBT(produc7156399406898700509T_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).
% length_product
tff(fact_1591_length__product,axiom,
! [Xs: list_nat,Ys: list_bool] : size_s783543119881238752t_bool(product_nat_bool(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_bool(Ys)) ).
% length_product
tff(fact_1592_mult__commute__abs,axiom,
! [C: real] : aa_rea7340605660300375871l_real(aTP_Lamm_ah,C) = aa_rea7340605660300375871l_real(times_times_real,C) ).
% mult_commute_abs
tff(fact_1593_mult__commute__abs,axiom,
! [C: rat] : aTP_Lamm_ai(C) = times_times_rat(C) ).
% mult_commute_abs
tff(fact_1594_mult__commute__abs,axiom,
! [C: nat] : aa_nat_fun_nat_nat(aTP_Lamm_aj,C) = aa_nat_fun_nat_nat(times_times_nat,C) ).
% mult_commute_abs
tff(fact_1595_mult__commute__abs,axiom,
! [C: int] : aa_int_fun_int_int(aTP_Lamm_ak,C) = aa_int_fun_int_int(times_times_int,C) ).
% mult_commute_abs
tff(fact_1596_DiffE,axiom,
! [C: complex,A2: set_complex,B2: set_complex] :
( pp(aa_set_complex_bool(member_complex(C),minus_811609699411566653omplex(A2,B2)))
=> ~ ( pp(aa_set_complex_bool(member_complex(C),A2))
=> pp(aa_set_complex_bool(member_complex(C),B2)) ) ) ).
% DiffE
tff(fact_1597_DiffE,axiom,
! [C: real,A2: set_real,B2: set_real] :
( pp(aa_set_real_bool(member_real(C),minus_minus_set_real(A2,B2)))
=> ~ ( pp(aa_set_real_bool(member_real(C),A2))
=> pp(aa_set_real_bool(member_real(C),B2)) ) ) ).
% DiffE
tff(fact_1598_DiffE,axiom,
! [C: int,A2: set_int,B2: set_int] :
( pp(aa_set_int_bool(member_int(C),minus_minus_set_int(A2,B2)))
=> ~ ( pp(aa_set_int_bool(member_int(C),A2))
=> pp(aa_set_int_bool(member_int(C),B2)) ) ) ).
% DiffE
tff(fact_1599_DiffE,axiom,
! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),minus_1356011639430497352at_nat(A2,B2)))
=> ~ ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),A2))
=> pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),B2)) ) ) ).
% DiffE
tff(fact_1600_DiffE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( pp(aa_set_nat_bool(member_nat(C),minus_minus_set_nat(A2,B2)))
=> ~ ( pp(aa_set_nat_bool(member_nat(C),A2))
=> pp(aa_set_nat_bool(member_nat(C),B2)) ) ) ).
% DiffE
tff(fact_1601_DiffD1,axiom,
! [C: complex,A2: set_complex,B2: set_complex] :
( pp(aa_set_complex_bool(member_complex(C),minus_811609699411566653omplex(A2,B2)))
=> pp(aa_set_complex_bool(member_complex(C),A2)) ) ).
% DiffD1
tff(fact_1602_DiffD1,axiom,
! [C: real,A2: set_real,B2: set_real] :
( pp(aa_set_real_bool(member_real(C),minus_minus_set_real(A2,B2)))
=> pp(aa_set_real_bool(member_real(C),A2)) ) ).
% DiffD1
tff(fact_1603_DiffD1,axiom,
! [C: int,A2: set_int,B2: set_int] :
( pp(aa_set_int_bool(member_int(C),minus_minus_set_int(A2,B2)))
=> pp(aa_set_int_bool(member_int(C),A2)) ) ).
% DiffD1
tff(fact_1604_DiffD1,axiom,
! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),minus_1356011639430497352at_nat(A2,B2)))
=> pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),A2)) ) ).
% DiffD1
tff(fact_1605_DiffD1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( pp(aa_set_nat_bool(member_nat(C),minus_minus_set_nat(A2,B2)))
=> pp(aa_set_nat_bool(member_nat(C),A2)) ) ).
% DiffD1
tff(fact_1606_DiffD2,axiom,
! [C: complex,A2: set_complex,B2: set_complex] :
( pp(aa_set_complex_bool(member_complex(C),minus_811609699411566653omplex(A2,B2)))
=> ~ pp(aa_set_complex_bool(member_complex(C),B2)) ) ).
% DiffD2
tff(fact_1607_DiffD2,axiom,
! [C: real,A2: set_real,B2: set_real] :
( pp(aa_set_real_bool(member_real(C),minus_minus_set_real(A2,B2)))
=> ~ pp(aa_set_real_bool(member_real(C),B2)) ) ).
% DiffD2
tff(fact_1608_DiffD2,axiom,
! [C: int,A2: set_int,B2: set_int] :
( pp(aa_set_int_bool(member_int(C),minus_minus_set_int(A2,B2)))
=> ~ pp(aa_set_int_bool(member_int(C),B2)) ) ).
% DiffD2
tff(fact_1609_DiffD2,axiom,
! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),minus_1356011639430497352at_nat(A2,B2)))
=> ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),B2)) ) ).
% DiffD2
tff(fact_1610_DiffD2,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( pp(aa_set_nat_bool(member_nat(C),minus_minus_set_nat(A2,B2)))
=> ~ pp(aa_set_nat_bool(member_nat(C),B2)) ) ).
% DiffD2
tff(fact_1611_set__diff__eq,axiom,
! [A2: set_complex,B2: set_complex] : minus_811609699411566653omplex(A2,B2) = collect_complex(aa_set3741564773079358752x_bool(aTP_Lamm_al(A2),B2)) ).
% set_diff_eq
tff(fact_1612_set__diff__eq,axiom,
! [A2: set_real,B2: set_real] : minus_minus_set_real(A2,B2) = collect_real(aa_set8565328311858951076l_bool(aTP_Lamm_am(A2),B2)) ).
% set_diff_eq
tff(fact_1613_set__diff__eq,axiom,
! [A2: set_list_nat,B2: set_list_nat] : minus_7954133019191499631st_nat(A2,B2) = collect_list_nat(aa_set257547205302447420t_bool(aTP_Lamm_an(A2),B2)) ).
% set_diff_eq
tff(fact_1614_set__diff__eq,axiom,
! [A2: set_int,B2: set_int] : minus_minus_set_int(A2,B2) = collect_int(aa_set7012339369581096484t_bool(aTP_Lamm_ao(A2),B2)) ).
% set_diff_eq
tff(fact_1615_set__diff__eq,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(A2,B2) = collec3392354462482085612at_nat(aa_set5223363028666124222t_bool(aTP_Lamm_ap(A2),B2)) ).
% set_diff_eq
tff(fact_1616_set__diff__eq,axiom,
! [A2: set_nat,B2: set_nat] : minus_minus_set_nat(A2,B2) = collect_nat(aa_set3744802623701456604t_bool(aTP_Lamm_aq(A2),B2)) ).
% set_diff_eq
tff(fact_1617_minus__set__def,axiom,
! [A2: set_complex,B2: set_complex] : minus_811609699411566653omplex(A2,B2) = collect_complex(minus_5423061390740260265x_bool(aTP_Lamm_aa(A2),aTP_Lamm_aa(B2))) ).
% minus_set_def
tff(fact_1618_minus__set__def,axiom,
! [A2: set_real,B2: set_real] : minus_minus_set_real(A2,B2) = collect_real(minus_6582619742941512235l_bool(aTP_Lamm_ab(A2),aTP_Lamm_ab(B2))) ).
% minus_set_def
tff(fact_1619_minus__set__def,axiom,
! [A2: set_list_nat,B2: set_list_nat] : minus_7954133019191499631st_nat(A2,B2) = collect_list_nat(minus_8064998606961450743t_bool(aTP_Lamm_ac(A2),aTP_Lamm_ac(B2))) ).
% minus_set_def
tff(fact_1620_minus__set__def,axiom,
! [A2: set_int,B2: set_int] : minus_minus_set_int(A2,B2) = collect_int(minus_7727386867957376939t_bool(aTP_Lamm_ae(A2),aTP_Lamm_ae(B2))) ).
% minus_set_def
tff(fact_1621_minus__set__def,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(A2,B2) = collec3392354462482085612at_nat(minus_5597117856863188850t_bool(aTP_Lamm_a(A2),aTP_Lamm_a(B2))) ).
% minus_set_def
tff(fact_1622_minus__set__def,axiom,
! [A2: set_nat,B2: set_nat] : minus_minus_set_nat(A2,B2) = collect_nat(minus_1628305797571225351t_bool(aTP_Lamm_ad(A2),aTP_Lamm_ad(B2))) ).
% minus_set_def
tff(fact_1623_psubset__imp__ex__mem,axiom,
! [A2: set_complex,B2: set_complex] :
( pp(aa_set_complex_bool(ord_less_set_complex(A2),B2))
=> ? [B4: complex] : pp(aa_set_complex_bool(member_complex(B4),minus_811609699411566653omplex(B2,A2))) ) ).
% psubset_imp_ex_mem
tff(fact_1624_psubset__imp__ex__mem,axiom,
! [A2: set_real,B2: set_real] :
( pp(aa_set_real_bool(ord_less_set_real(A2),B2))
=> ? [B4: real] : pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(B2,A2))) ) ).
% psubset_imp_ex_mem
tff(fact_1625_psubset__imp__ex__mem,axiom,
! [A2: set_int,B2: set_int] :
( pp(aa_set_int_bool(ord_less_set_int(A2),B2))
=> ? [B4: int] : pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(B2,A2))) ) ).
% psubset_imp_ex_mem
tff(fact_1626_psubset__imp__ex__mem,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(A2),B2))
=> ? [B4: product_prod_nat_nat] : pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(B4),minus_1356011639430497352at_nat(B2,A2))) ) ).
% psubset_imp_ex_mem
tff(fact_1627_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B2: set_nat] :
( pp(aa_set_nat_bool(ord_less_set_nat(A2),B2))
=> ? [B4: nat] : pp(aa_set_nat_bool(member_nat(B4),minus_minus_set_nat(B2,A2))) ) ).
% psubset_imp_ex_mem
tff(fact_1628_finite__lists__length__eq,axiom,
! [A2: set_complex,N: nat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> finite8712137658972009173omplex(collect_list_complex(aa_nat8039161444313734436x_bool(aTP_Lamm_ar(A2),N))) ) ).
% finite_lists_length_eq
tff(fact_1629_finite__lists__length__eq,axiom,
! [A2: set_VEBT_VEBT,N: nat] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
=> finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1297483763028037422T_bool(aTP_Lamm_as(A2),N))) ) ).
% finite_lists_length_eq
tff(fact_1630_finite__lists__length__eq,axiom,
! [A2: set_bool,N: nat] :
( finite_finite_bool(A2)
=> finite7866827932528546246t_bool(collect_list_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_at(A2),N))) ) ).
% finite_lists_length_eq
tff(fact_1631_finite__lists__length__eq,axiom,
! [A2: set_int,N: nat] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> finite3922522038869484883st_int(collect_list_int(aa_nat1619170893822700070t_bool(aTP_Lamm_au(A2),N))) ) ).
% finite_lists_length_eq
tff(fact_1632_finite__lists__length__eq,axiom,
! [A2: set_nat,N: nat] :
( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> finite8100373058378681591st_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_av(A2),N))) ) ).
% finite_lists_length_eq
tff(fact_1633_finite__lists__length__le,axiom,
! [A2: set_complex,N: nat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> finite8712137658972009173omplex(collect_list_complex(aa_nat8039161444313734436x_bool(aTP_Lamm_aw(A2),N))) ) ).
% finite_lists_length_le
tff(fact_1634_finite__lists__length__le,axiom,
! [A2: set_VEBT_VEBT,N: nat] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
=> finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1297483763028037422T_bool(aTP_Lamm_ax(A2),N))) ) ).
% finite_lists_length_le
tff(fact_1635_finite__lists__length__le,axiom,
! [A2: set_bool,N: nat] :
( finite_finite_bool(A2)
=> finite7866827932528546246t_bool(collect_list_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_ay(A2),N))) ) ).
% finite_lists_length_le
tff(fact_1636_finite__lists__length__le,axiom,
! [A2: set_int,N: nat] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> finite3922522038869484883st_int(collect_list_int(aa_nat1619170893822700070t_bool(aTP_Lamm_az(A2),N))) ) ).
% finite_lists_length_le
tff(fact_1637_finite__lists__length__le,axiom,
! [A2: set_nat,N: nat] :
( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> finite8100373058378681591st_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_ba(A2),N))) ) ).
% finite_lists_length_le
tff(fact_1638_lambda__one,axiom,
aTP_Lamm_bb = times_times_complex(one_one_complex) ).
% lambda_one
tff(fact_1639_lambda__one,axiom,
aTP_Lamm_bc = aa_rea7340605660300375871l_real(times_times_real,one_one_real) ).
% lambda_one
tff(fact_1640_lambda__one,axiom,
aTP_Lamm_bd = times_times_rat(one_one_rat) ).
% lambda_one
tff(fact_1641_lambda__one,axiom,
aTP_Lamm_be = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).
% lambda_one
tff(fact_1642_lambda__one,axiom,
aTP_Lamm_bf = aa_int_fun_int_int(times_times_int,one_one_int) ).
% lambda_one
tff(fact_1643_finite__list,axiom,
! [A2: set_VEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
=> ? [Xs2: list_VEBT_VEBT] : set_VEBT_VEBT2(Xs2) = A2 ) ).
% finite_list
tff(fact_1644_finite__list,axiom,
! [A2: set_nat] :
( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ? [Xs2: list_nat] : set_nat2(Xs2) = A2 ) ).
% finite_list
tff(fact_1645_finite__list,axiom,
! [A2: set_int] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ? [Xs2: list_int] : set_int2(Xs2) = A2 ) ).
% finite_list
tff(fact_1646_finite__list,axiom,
! [A2: set_complex] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ? [Xs2: list_complex] : set_complex2(Xs2) = A2 ) ).
% finite_list
tff(fact_1647_set__vebt__def,axiom,
! [T: vEBT_VEBT] : vEBT_set_vebt(T) = collect_nat(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T)) ).
% set_vebt_def
tff(fact_1648_numeral__code_I2_J,axiom,
! [N: num] : numera6690914467698888265omplex(aa_num_num(bit0,N)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),numera6690914467698888265omplex(N)) ).
% numeral_code(2)
tff(fact_1649_numeral__code_I2_J,axiom,
! [N: num] : numeral_numeral_real(aa_num_num(bit0,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(N)),numeral_numeral_real(N)) ).
% numeral_code(2)
tff(fact_1650_numeral__code_I2_J,axiom,
! [N: num] : numeral_numeral_rat(aa_num_num(bit0,N)) = plus_plus_rat(numeral_numeral_rat(N),numeral_numeral_rat(N)) ).
% numeral_code(2)
tff(fact_1651_numeral__code_I2_J,axiom,
! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N)) ).
% numeral_code(2)
tff(fact_1652_numeral__code_I2_J,axiom,
! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N)) ).
% numeral_code(2)
tff(fact_1653_numeral__code_I3_J,axiom,
! [N: num] : numera6690914467698888265omplex(aa_num_num(bit1,N)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(N)),numera6690914467698888265omplex(N))),one_one_complex) ).
% numeral_code(3)
tff(fact_1654_numeral__code_I3_J,axiom,
! [N: num] : numeral_numeral_real(aa_num_num(bit1,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(N)),numeral_numeral_real(N))),one_one_real) ).
% numeral_code(3)
tff(fact_1655_numeral__code_I3_J,axiom,
! [N: num] : numeral_numeral_rat(aa_num_num(bit1,N)) = plus_plus_rat(plus_plus_rat(numeral_numeral_rat(N),numeral_numeral_rat(N)),one_one_rat) ).
% numeral_code(3)
tff(fact_1656_numeral__code_I3_J,axiom,
! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N))),one_one_nat) ).
% numeral_code(3)
tff(fact_1657_numeral__code_I3_J,axiom,
! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N))),one_one_int) ).
% numeral_code(3)
tff(fact_1658_power__numeral__even,axiom,
! [Z: complex,W: num] : aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,W))),aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,W))) ).
% power_numeral_even
tff(fact_1659_power__numeral__even,axiom,
! [Z: real,W: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,W))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,W))) ).
% power_numeral_even
tff(fact_1660_power__numeral__even,axiom,
! [Z: rat,W: num] : aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W))),aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W))) ).
% power_numeral_even
tff(fact_1661_power__numeral__even,axiom,
! [Z: nat,W: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W))) ).
% power_numeral_even
tff(fact_1662_power__numeral__even,axiom,
! [Z: int,W: num] : aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,W))),aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,W))) ).
% power_numeral_even
tff(fact_1663_power__numeral__odd,axiom,
! [Z: complex,W: num] : aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(Z),aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,W)))),aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,W))) ).
% power_numeral_odd
tff(fact_1664_power__numeral__odd,axiom,
! [Z: real,W: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,W)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,W))) ).
% power_numeral_odd
tff(fact_1665_power__numeral__odd,axiom,
! [Z: rat,W: num] : aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(Z),aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W)))),aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W))) ).
% power_numeral_odd
tff(fact_1666_power__numeral__odd,axiom,
! [Z: nat,W: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W))) ).
% power_numeral_odd
tff(fact_1667_power__numeral__odd,axiom,
! [Z: int,W: num] : aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z),aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,W)))),aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,W))) ).
% power_numeral_odd
tff(fact_1668_not__psubset__empty,axiom,
! [A2: set_nat] : ~ pp(aa_set_nat_bool(ord_less_set_nat(A2),bot_bot_set_nat)) ).
% not_psubset_empty
tff(fact_1669_not__psubset__empty,axiom,
! [A2: set_int] : ~ pp(aa_set_int_bool(ord_less_set_int(A2),bot_bot_set_int)) ).
% not_psubset_empty
tff(fact_1670_not__psubset__empty,axiom,
! [A2: set_real] : ~ pp(aa_set_real_bool(ord_less_set_real(A2),bot_bot_set_real)) ).
% not_psubset_empty
tff(fact_1671_subset__iff__psubset__eq,axiom,
! [A2: set_nat,B2: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),B2))
<=> ( pp(aa_set_nat_bool(ord_less_set_nat(A2),B2))
| ( A2 = B2 ) ) ) ).
% subset_iff_psubset_eq
tff(fact_1672_subset__psubset__trans,axiom,
! [A2: set_nat,B2: set_nat,C4: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),B2))
=> ( pp(aa_set_nat_bool(ord_less_set_nat(B2),C4))
=> pp(aa_set_nat_bool(ord_less_set_nat(A2),C4)) ) ) ).
% subset_psubset_trans
tff(fact_1673_subset__not__subset__eq,axiom,
! [A2: set_nat,B2: set_nat] :
( pp(aa_set_nat_bool(ord_less_set_nat(A2),B2))
<=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),B2))
& ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(B2),A2)) ) ) ).
% subset_not_subset_eq
tff(fact_1674_psubset__subset__trans,axiom,
! [A2: set_nat,B2: set_nat,C4: set_nat] :
( pp(aa_set_nat_bool(ord_less_set_nat(A2),B2))
=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B2),C4))
=> pp(aa_set_nat_bool(ord_less_set_nat(A2),C4)) ) ) ).
% psubset_subset_trans
tff(fact_1675_psubset__imp__subset,axiom,
! [A2: set_nat,B2: set_nat] :
( pp(aa_set_nat_bool(ord_less_set_nat(A2),B2))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),B2)) ) ).
% psubset_imp_subset
tff(fact_1676_psubset__eq,axiom,
! [A2: set_nat,B2: set_nat] :
( pp(aa_set_nat_bool(ord_less_set_nat(A2),B2))
<=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),B2))
& ( A2 != B2 ) ) ) ).
% psubset_eq
tff(fact_1677_psubsetE,axiom,
! [A2: set_nat,B2: set_nat] :
( pp(aa_set_nat_bool(ord_less_set_nat(A2),B2))
=> ~ ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),B2))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(B2),A2)) ) ) ).
% psubsetE
tff(fact_1678_Diff__mono,axiom,
! [A2: set_Pr1261947904930325089at_nat,C4: set_Pr1261947904930325089at_nat,D3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),C4))
=> ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(D3),B2))
=> pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(minus_1356011639430497352at_nat(A2,B2)),minus_1356011639430497352at_nat(C4,D3))) ) ) ).
% Diff_mono
tff(fact_1679_Diff__mono,axiom,
! [A2: set_nat,C4: set_nat,D3: set_nat,B2: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),C4))
=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(D3),B2))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(minus_minus_set_nat(A2,B2)),minus_minus_set_nat(C4,D3))) ) ) ).
% Diff_mono
tff(fact_1680_Diff__subset,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(minus_1356011639430497352at_nat(A2,B2)),A2)) ).
% Diff_subset
tff(fact_1681_Diff__subset,axiom,
! [A2: set_nat,B2: set_nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(minus_minus_set_nat(A2,B2)),A2)) ).
% Diff_subset
tff(fact_1682_double__diff,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C4: set_Pr1261947904930325089at_nat] :
( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),B2))
=> ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(B2),C4))
=> ( minus_1356011639430497352at_nat(B2,minus_1356011639430497352at_nat(C4,A2)) = A2 ) ) ) ).
% double_diff
tff(fact_1683_double__diff,axiom,
! [A2: set_nat,B2: set_nat,C4: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),B2))
=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B2),C4))
=> ( minus_minus_set_nat(B2,minus_minus_set_nat(C4,A2)) = A2 ) ) ) ).
% double_diff
tff(fact_1684_vebt__member_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary)),X))
<=> ( ( X != Mi )
=> ( ( X != Ma )
=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X))
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X))
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
=> pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ) ) ) ) ) ).
% vebt_member.simps(5)
tff(fact_1685_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary)),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_m_e_m_b_e_r(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_nat)))))))))) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(5)
tff(fact_1686_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary)),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma))))),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(vEBT_T_i_n_s_e_r_t(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(vEBT_T_i_n_s_e_r_t(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat)) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(5)
tff(fact_1687_vebt__pred_Osimps_I7_J,axiom,
! [Ma: nat,X: nat,Mi: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X))
=> ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary),X) = some_nat(Ma) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X))
=> ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary),X) = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X),some_nat(Mi),none_nat),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_pred(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ).
% vebt_pred.simps(7)
tff(fact_1688_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t2(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary)),X) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t2(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(5)
tff(fact_1689_finite__Collect__le__nat,axiom,
! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bg,K)))) ).
% finite_Collect_le_nat
tff(fact_1690_finite__Collect__less__nat,axiom,
! [K: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bh,K)))) ).
% finite_Collect_less_nat
tff(fact_1691_finite__roots__unity,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
=> pp(aa_set_real_bool(finite_finite_real,collect_real(aTP_Lamm_bi(N)))) ) ).
% finite_roots_unity
tff(fact_1692_finite__roots__unity,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),N))
=> pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aTP_Lamm_bj(N)))) ) ).
% finite_roots_unity
tff(fact_1693_finite__Diff,axiom,
! [A2: set_int,B2: set_int] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(A2,B2))) ) ).
% finite_Diff
tff(fact_1694_finite__Diff,axiom,
! [A2: set_complex,B2: set_complex] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A2,B2))) ) ).
% finite_Diff
tff(fact_1695_finite__Diff,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A2))
=> pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(A2,B2))) ) ).
% finite_Diff
tff(fact_1696_finite__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(A2,B2))) ) ).
% finite_Diff
tff(fact_1697_finite__Diff2,axiom,
! [B2: set_int,A2: set_int] :
( pp(aa_set_int_bool(finite_finite_int,B2))
=> ( pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(A2,B2)))
<=> pp(aa_set_int_bool(finite_finite_int,A2)) ) ) ).
% finite_Diff2
tff(fact_1698_finite__Diff2,axiom,
! [B2: set_complex,A2: set_complex] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
=> ( pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A2,B2)))
<=> pp(aa_set_complex_bool(finite3207457112153483333omplex,A2)) ) ) ).
% finite_Diff2
tff(fact_1699_finite__Diff2,axiom,
! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B2))
=> ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(A2,B2)))
<=> pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A2)) ) ) ).
% finite_Diff2
tff(fact_1700_finite__Diff2,axiom,
! [B2: set_nat,A2: set_nat] :
( pp(aa_set_nat_bool(finite_finite_nat,B2))
=> ( pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(A2,B2)))
<=> pp(aa_set_nat_bool(finite_finite_nat,A2)) ) ) ).
% finite_Diff2
tff(fact_1701_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
( pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V),TreeList,Vc)),X))
<=> ( ( X = Mi )
| ( X = Ma )
| ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
=> pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ) ).
% VEBT_internal.membermima.simps(4)
tff(fact_1702_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
( pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList,S)),X))
<=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
=> pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ).
% VEBT_internal.naive_member.simps(3)
tff(fact_1703_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
! [V: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,X: nat] :
( pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList,Vd)),X))
<=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
=> pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))) ) ) ).
% VEBT_internal.membermima.simps(5)
tff(fact_1704_buildup__nothing__in__leaf,axiom,
! [N: nat,X: nat] : ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_vebt_buildup(N)),X)) ).
% buildup_nothing_in_leaf
tff(fact_1705_buildup__nothing__in__min__max,axiom,
! [N: nat,X: nat] : ~ pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_vebt_buildup(N)),X)) ).
% buildup_nothing_in_min_max
tff(fact_1706_both__member__options__def,axiom,
! [T: vEBT_VEBT,X: nat] :
( pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X))
<=> ( pp(aa_nat_bool(vEBT_V5719532721284313246member(T),X))
| pp(aa_nat_bool(vEBT_VEBT_membermima(T),X)) ) ) ).
% both_member_options_def
tff(fact_1707_member__valid__both__member__options,axiom,
! [Tree: vEBT_VEBT,N: nat,X: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(Tree),N))
=> ( pp(aa_nat_bool(vEBT_vebt_member(Tree),X))
=> ( pp(aa_nat_bool(vEBT_V5719532721284313246member(Tree),X))
| pp(aa_nat_bool(vEBT_VEBT_membermima(Tree),X)) ) ) ) ).
% member_valid_both_member_options
tff(fact_1708_finite__interval__int4,axiom,
! [A: int,B: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_bk(A),B)))) ).
% finite_interval_int4
tff(fact_1709_finite__interval__int2,axiom,
! [A: int,B: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_bl(A),B)))) ).
% finite_interval_int2
tff(fact_1710_finite__interval__int3,axiom,
! [A: int,B: int] : pp(aa_set_int_bool(finite_finite_int,collect_int(aa_int_fun_int_bool(aTP_Lamm_bm(A),B)))) ).
% finite_interval_int3
tff(fact_1711_less__set__def,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(A2),B2))
<=> pp(aa_fun8663595511526100224l_bool(ord_le7158584875777748973t_bool(aTP_Lamm_a(A2)),aTP_Lamm_a(B2))) ) ).
% less_set_def
tff(fact_1712_less__set__def,axiom,
! [A2: set_complex,B2: set_complex] :
( pp(aa_set_complex_bool(ord_less_set_complex(A2),B2))
<=> ord_le3776645642988763374x_bool(aTP_Lamm_aa(A2),aTP_Lamm_aa(B2)) ) ).
% less_set_def
tff(fact_1713_less__set__def,axiom,
! [A2: set_real,B2: set_real] :
( pp(aa_set_real_bool(ord_less_set_real(A2),B2))
<=> ord_le7519368973566061808l_bool(aTP_Lamm_ab(A2),aTP_Lamm_ab(B2)) ) ).
% less_set_def
tff(fact_1714_less__set__def,axiom,
! [A2: set_nat,B2: set_nat] :
( pp(aa_set_nat_bool(ord_less_set_nat(A2),B2))
<=> ord_le6841235830588351564t_bool(aTP_Lamm_ad(A2),aTP_Lamm_ad(B2)) ) ).
% less_set_def
tff(fact_1715_less__set__def,axiom,
! [A2: set_int,B2: set_int] :
( pp(aa_set_int_bool(ord_less_set_int(A2),B2))
<=> ord_le3716944864119727344t_bool(aTP_Lamm_ae(A2),aTP_Lamm_ae(B2)) ) ).
% less_set_def
tff(fact_1716_psubsetD,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
( pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(A2),B2))
=> ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),A2))
=> pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),B2)) ) ) ).
% psubsetD
tff(fact_1717_psubsetD,axiom,
! [A2: set_complex,B2: set_complex,C: complex] :
( pp(aa_set_complex_bool(ord_less_set_complex(A2),B2))
=> ( pp(aa_set_complex_bool(member_complex(C),A2))
=> pp(aa_set_complex_bool(member_complex(C),B2)) ) ) ).
% psubsetD
tff(fact_1718_psubsetD,axiom,
! [A2: set_real,B2: set_real,C: real] :
( pp(aa_set_real_bool(ord_less_set_real(A2),B2))
=> ( pp(aa_set_real_bool(member_real(C),A2))
=> pp(aa_set_real_bool(member_real(C),B2)) ) ) ).
% psubsetD
tff(fact_1719_psubsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( pp(aa_set_nat_bool(ord_less_set_nat(A2),B2))
=> ( pp(aa_set_nat_bool(member_nat(C),A2))
=> pp(aa_set_nat_bool(member_nat(C),B2)) ) ) ).
% psubsetD
tff(fact_1720_psubsetD,axiom,
! [A2: set_int,B2: set_int,C: int] :
( pp(aa_set_int_bool(ord_less_set_int(A2),B2))
=> ( pp(aa_set_int_bool(member_int(C),A2))
=> pp(aa_set_int_bool(member_int(C),B2)) ) ) ).
% psubsetD
tff(fact_1721_finite__maxlen,axiom,
! [M6: set_list_VEBT_VEBT] :
( finite3004134309566078307T_VEBT(M6)
=> ? [N2: nat] :
! [X4: list_VEBT_VEBT] :
( member2936631157270082147T_VEBT(X4,M6)
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s6755466524823107622T_VEBT(X4)),N2)) ) ) ).
% finite_maxlen
tff(fact_1722_finite__maxlen,axiom,
! [M6: set_list_bool] :
( finite7866827932528546246t_bool(M6)
=> ? [N2: nat] :
! [X4: list_bool] :
( member_list_bool(X4,M6)
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_bool(X4)),N2)) ) ) ).
% finite_maxlen
tff(fact_1723_finite__maxlen,axiom,
! [M6: set_list_nat] :
( finite8100373058378681591st_nat(M6)
=> ? [N2: nat] :
! [X4: list_nat] :
( member_list_nat(X4,M6)
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_nat(X4)),N2)) ) ) ).
% finite_maxlen
tff(fact_1724_finite__maxlen,axiom,
! [M6: set_list_int] :
( finite3922522038869484883st_int(M6)
=> ? [N2: nat] :
! [X4: list_int] :
( member_list_int(X4,M6)
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_int(X4)),N2)) ) ) ).
% finite_maxlen
tff(fact_1725_finite__psubset__induct,axiom,
! [A2: set_nat,P: fun_set_nat_bool] :
( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ( ! [A5: set_nat] :
( pp(aa_set_nat_bool(finite_finite_nat,A5))
=> ( ! [B5: set_nat] :
( pp(aa_set_nat_bool(ord_less_set_nat(B5),A5))
=> pp(aa_set_nat_bool(P,B5)) )
=> pp(aa_set_nat_bool(P,A5)) ) )
=> pp(aa_set_nat_bool(P,A2)) ) ) ).
% finite_psubset_induct
tff(fact_1726_finite__psubset__induct,axiom,
! [A2: set_int,P: fun_set_int_bool] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( ! [A5: set_int] :
( pp(aa_set_int_bool(finite_finite_int,A5))
=> ( ! [B5: set_int] :
( pp(aa_set_int_bool(ord_less_set_int(B5),A5))
=> pp(aa_set_int_bool(P,B5)) )
=> pp(aa_set_int_bool(P,A5)) ) )
=> pp(aa_set_int_bool(P,A2)) ) ) ).
% finite_psubset_induct
tff(fact_1727_finite__psubset__induct,axiom,
! [A2: set_complex,P: fun_set_complex_bool] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( ! [A5: set_complex] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A5))
=> ( ! [B5: set_complex] :
( pp(aa_set_complex_bool(ord_less_set_complex(B5),A5))
=> pp(aa_set_complex_bool(P,B5)) )
=> pp(aa_set_complex_bool(P,A5)) ) )
=> pp(aa_set_complex_bool(P,A2)) ) ) ).
% finite_psubset_induct
tff(fact_1728_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I4_J,axiom,
! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t2(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList,Summary)),X) = one_one_nat ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(4)
tff(fact_1729_insersimp_H,axiom,
! [T: vEBT_VEBT,N: nat,Y: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( ~ ? [X_1: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,T),X_1))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t2(T),Y)),one_one_nat)) ) ) ).
% insersimp'
tff(fact_1730_finite__has__maximal2,axiom,
! [A2: set_real,A: real] :
( pp(aa_set_real_bool(finite_finite_real,A2))
=> ( pp(aa_set_real_bool(member_real(A),A2))
=> ? [X3: real] :
( pp(aa_set_real_bool(member_real(X3),A2))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X3))
& ! [Xa: real] :
( pp(aa_set_real_bool(member_real(Xa),A2))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Xa))
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
tff(fact_1731_finite__has__maximal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( finite1152437895449049373et_nat(A2)
=> ( member_set_nat(A,A2)
=> ? [X3: set_nat] :
( member_set_nat(X3,A2)
& pp(aa_set_nat_bool(ord_less_eq_set_nat(A),X3))
& ! [Xa: set_nat] :
( member_set_nat(Xa,A2)
=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X3),Xa))
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
tff(fact_1732_finite__has__maximal2,axiom,
! [A2: set_rat,A: rat] :
( pp(aa_set_rat_bool(finite_finite_rat,A2))
=> ( pp(aa_set_rat_bool(member_rat(A),A2))
=> ? [X3: rat] :
( pp(aa_set_rat_bool(member_rat(X3),A2))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),X3))
& ! [Xa: rat] :
( pp(aa_set_rat_bool(member_rat(Xa),A2))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Xa))
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
tff(fact_1733_finite__has__maximal2,axiom,
! [A2: set_num,A: num] :
( pp(aa_set_num_bool(finite_finite_num,A2))
=> ( pp(aa_set_num_bool(member_num(A),A2))
=> ? [X3: num] :
( pp(aa_set_num_bool(member_num(X3),A2))
& pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),X3))
& ! [Xa: num] :
( pp(aa_set_num_bool(member_num(Xa),A2))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Xa))
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
tff(fact_1734_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ( pp(aa_set_nat_bool(member_nat(A),A2))
=> ? [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),A2))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),X3))
& ! [Xa: nat] :
( pp(aa_set_nat_bool(member_nat(Xa),A2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Xa))
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
tff(fact_1735_finite__has__maximal2,axiom,
! [A2: set_int,A: int] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( pp(aa_set_int_bool(member_int(A),A2))
=> ? [X3: int] :
( pp(aa_set_int_bool(member_int(X3),A2))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),X3))
& ! [Xa: int] :
( pp(aa_set_int_bool(member_int(Xa),A2))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Xa))
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
tff(fact_1736_finite__has__minimal2,axiom,
! [A2: set_real,A: real] :
( pp(aa_set_real_bool(finite_finite_real,A2))
=> ( pp(aa_set_real_bool(member_real(A),A2))
=> ? [X3: real] :
( pp(aa_set_real_bool(member_real(X3),A2))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),A))
& ! [Xa: real] :
( pp(aa_set_real_bool(member_real(Xa),A2))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Xa),X3))
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
tff(fact_1737_finite__has__minimal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( finite1152437895449049373et_nat(A2)
=> ( member_set_nat(A,A2)
=> ? [X3: set_nat] :
( member_set_nat(X3,A2)
& pp(aa_set_nat_bool(ord_less_eq_set_nat(X3),A))
& ! [Xa: set_nat] :
( member_set_nat(Xa,A2)
=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Xa),X3))
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
tff(fact_1738_finite__has__minimal2,axiom,
! [A2: set_rat,A: rat] :
( pp(aa_set_rat_bool(finite_finite_rat,A2))
=> ( pp(aa_set_rat_bool(member_rat(A),A2))
=> ? [X3: rat] :
( pp(aa_set_rat_bool(member_rat(X3),A2))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),A))
& ! [Xa: rat] :
( pp(aa_set_rat_bool(member_rat(Xa),A2))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Xa),X3))
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
tff(fact_1739_finite__has__minimal2,axiom,
! [A2: set_num,A: num] :
( pp(aa_set_num_bool(finite_finite_num,A2))
=> ( pp(aa_set_num_bool(member_num(A),A2))
=> ? [X3: num] :
( pp(aa_set_num_bool(member_num(X3),A2))
& pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),A))
& ! [Xa: num] :
( pp(aa_set_num_bool(member_num(Xa),A2))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Xa),X3))
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
tff(fact_1740_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ( pp(aa_set_nat_bool(member_nat(A),A2))
=> ? [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),A2))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),A))
& ! [Xa: nat] :
( pp(aa_set_nat_bool(member_nat(Xa),A2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa),X3))
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
tff(fact_1741_finite__has__minimal2,axiom,
! [A2: set_int,A: int] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( pp(aa_set_int_bool(member_int(A),A2))
=> ? [X3: int] :
( pp(aa_set_int_bool(member_int(X3),A2))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),A))
& ! [Xa: int] :
( pp(aa_set_int_bool(member_int(Xa),A2))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Xa),X3))
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
tff(fact_1742_insertsimp_H,axiom,
! [T: vEBT_VEBT,N: nat,L: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,T))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t2(T),L)),one_one_nat)) ) ) ).
% insertsimp'
tff(fact_1743_Diff__infinite__finite,axiom,
! [T2: set_int,S2: set_int] :
( pp(aa_set_int_bool(finite_finite_int,T2))
=> ( ~ pp(aa_set_int_bool(finite_finite_int,S2))
=> ~ pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(S2,T2))) ) ) ).
% Diff_infinite_finite
tff(fact_1744_Diff__infinite__finite,axiom,
! [T2: set_complex,S2: set_complex] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
=> ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
=> ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(S2,T2))) ) ) ).
% Diff_infinite_finite
tff(fact_1745_Diff__infinite__finite,axiom,
! [T2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,T2))
=> ( ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,S2))
=> ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(S2,T2))) ) ) ).
% Diff_infinite_finite
tff(fact_1746_Diff__infinite__finite,axiom,
! [T2: set_nat,S2: set_nat] :
( pp(aa_set_nat_bool(finite_finite_nat,T2))
=> ( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
=> ~ pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(S2,T2))) ) ) ).
% Diff_infinite_finite
tff(fact_1747_finite__has__minimal,axiom,
! [A2: set_real] :
( pp(aa_set_real_bool(finite_finite_real,A2))
=> ( ( A2 != bot_bot_set_real )
=> ? [X3: real] :
( pp(aa_set_real_bool(member_real(X3),A2))
& ! [Xa: real] :
( pp(aa_set_real_bool(member_real(Xa),A2))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Xa),X3))
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
tff(fact_1748_finite__has__minimal,axiom,
! [A2: set_set_nat] :
( finite1152437895449049373et_nat(A2)
=> ( ( A2 != bot_bot_set_set_nat )
=> ? [X3: set_nat] :
( member_set_nat(X3,A2)
& ! [Xa: set_nat] :
( member_set_nat(Xa,A2)
=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(Xa),X3))
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
tff(fact_1749_finite__has__minimal,axiom,
! [A2: set_rat] :
( pp(aa_set_rat_bool(finite_finite_rat,A2))
=> ( ( A2 != bot_bot_set_rat )
=> ? [X3: rat] :
( pp(aa_set_rat_bool(member_rat(X3),A2))
& ! [Xa: rat] :
( pp(aa_set_rat_bool(member_rat(Xa),A2))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Xa),X3))
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
tff(fact_1750_finite__has__minimal,axiom,
! [A2: set_num] :
( pp(aa_set_num_bool(finite_finite_num,A2))
=> ( ( A2 != bot_bot_set_num )
=> ? [X3: num] :
( pp(aa_set_num_bool(member_num(X3),A2))
& ! [Xa: num] :
( pp(aa_set_num_bool(member_num(Xa),A2))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Xa),X3))
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
tff(fact_1751_finite__has__minimal,axiom,
! [A2: set_nat] :
( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),A2))
& ! [Xa: nat] :
( pp(aa_set_nat_bool(member_nat(Xa),A2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa),X3))
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
tff(fact_1752_finite__has__minimal,axiom,
! [A2: set_int] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( ( A2 != bot_bot_set_int )
=> ? [X3: int] :
( pp(aa_set_int_bool(member_int(X3),A2))
& ! [Xa: int] :
( pp(aa_set_int_bool(member_int(Xa),A2))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Xa),X3))
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
tff(fact_1753_finite__has__maximal,axiom,
! [A2: set_real] :
( pp(aa_set_real_bool(finite_finite_real,A2))
=> ( ( A2 != bot_bot_set_real )
=> ? [X3: real] :
( pp(aa_set_real_bool(member_real(X3),A2))
& ! [Xa: real] :
( pp(aa_set_real_bool(member_real(Xa),A2))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),Xa))
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
tff(fact_1754_finite__has__maximal,axiom,
! [A2: set_set_nat] :
( finite1152437895449049373et_nat(A2)
=> ( ( A2 != bot_bot_set_set_nat )
=> ? [X3: set_nat] :
( member_set_nat(X3,A2)
& ! [Xa: set_nat] :
( member_set_nat(Xa,A2)
=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X3),Xa))
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
tff(fact_1755_finite__has__maximal,axiom,
! [A2: set_rat] :
( pp(aa_set_rat_bool(finite_finite_rat,A2))
=> ( ( A2 != bot_bot_set_rat )
=> ? [X3: rat] :
( pp(aa_set_rat_bool(member_rat(X3),A2))
& ! [Xa: rat] :
( pp(aa_set_rat_bool(member_rat(Xa),A2))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X3),Xa))
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
tff(fact_1756_finite__has__maximal,axiom,
! [A2: set_num] :
( pp(aa_set_num_bool(finite_finite_num,A2))
=> ( ( A2 != bot_bot_set_num )
=> ? [X3: num] :
( pp(aa_set_num_bool(member_num(X3),A2))
& ! [Xa: num] :
( pp(aa_set_num_bool(member_num(Xa),A2))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X3),Xa))
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
tff(fact_1757_finite__has__maximal,axiom,
! [A2: set_nat] :
( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),A2))
& ! [Xa: nat] :
( pp(aa_set_nat_bool(member_nat(Xa),A2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),Xa))
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
tff(fact_1758_finite__has__maximal,axiom,
! [A2: set_int] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( ( A2 != bot_bot_set_int )
=> ? [X3: int] :
( pp(aa_set_int_bool(member_int(X3),A2))
& ! [Xa: int] :
( pp(aa_set_int_bool(member_int(Xa),A2))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X3),Xa))
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
tff(fact_1759_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2,Vb2)
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2)
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
=> pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) )
=> ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2)
=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
=> pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(2)
tff(fact_1760_vebt__succ_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
( ( vEBT_vebt_succ(X,Xa2) = Y )
=> ( ! [Uu2: bool,B4: bool] :
( ( X = vEBT_Leaf(Uu2,B4) )
=> ( ( Xa2 = zero_zero_nat )
=> ~ ( ( pp(B4)
=> ( Y = some_nat(one_one_nat) ) )
& ( ~ pp(B4)
=> ( Y = none_nat ) ) ) ) )
=> ( ( ? [Uv2: bool,Uw2: bool] : X = vEBT_Leaf(Uv2,Uw2)
=> ( ? [N2: nat] : Xa2 = aa_nat_nat(suc,N2)
=> ( Y != none_nat ) ) )
=> ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2)
=> ( Y != none_nat ) )
=> ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2)
=> ( Y != none_nat ) )
=> ( ( ? [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg,Vh)
=> ( Y != none_nat ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2) )
=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
=> ( Y = some_nat(Mi2) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
=> ( Y = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ) ) ) ) ) ) ) ).
% vebt_succ.elims
tff(fact_1761_vebt__pred_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
( ( vEBT_vebt_pred(X,Xa2) = Y )
=> ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
=> ( ( Xa2 = zero_zero_nat )
=> ( Y != none_nat ) ) )
=> ( ! [A4: bool] :
( ? [Uw2: bool] : X = vEBT_Leaf(A4,Uw2)
=> ( ( Xa2 = aa_nat_nat(suc,zero_zero_nat) )
=> ~ ( ( pp(A4)
=> ( Y = some_nat(zero_zero_nat) ) )
& ( ~ pp(A4)
=> ( Y = none_nat ) ) ) ) )
=> ( ! [A4: bool,B4: bool] :
( ( X = vEBT_Leaf(A4,B4) )
=> ( ? [Va3: nat] : Xa2 = aa_nat_nat(suc,aa_nat_nat(suc,Va3))
=> ~ ( ( pp(B4)
=> ( Y = some_nat(one_one_nat) ) )
& ( ~ pp(B4)
=> ( ( pp(A4)
=> ( Y = some_nat(zero_zero_nat) ) )
& ( ~ pp(A4)
=> ( Y = none_nat ) ) ) ) ) ) )
=> ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va2)
=> ( Y != none_nat ) )
=> ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve)
=> ( Y != none_nat ) )
=> ( ( ? [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh,Vi)
=> ( Y != none_nat ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2) )
=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
=> ( Y = some_nat(Ma2) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
=> ( Y = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),Xa2),some_nat(Mi2),none_nat),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_pred.elims
tff(fact_1762_buildup__gives__valid,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_nat_bool(vEBT_invar_vebt(vEBT_vebt_buildup(N)),N)) ) ).
% buildup_gives_valid
tff(fact_1763_prod_Ofinite__Collect__op,axiom,
! [I5: set_real,X: fun_real_complex,Y: fun_real_complex] :
( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_bn(I5),X))))
=> ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_bn(I5),Y))))
=> pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aa_fun5520078544946397361l_bool(aTP_Lamm_bo(I5),X),Y)))) ) ) ).
% prod.finite_Collect_op
tff(fact_1764_prod_Ofinite__Collect__op,axiom,
! [I5: set_nat,X: fun_nat_complex,Y: fun_nat_complex] :
( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_bp(I5),X))))
=> ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_bp(I5),Y))))
=> pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aa_fun5430245816583273541t_bool(aTP_Lamm_bq(I5),X),Y)))) ) ) ).
% prod.finite_Collect_op
tff(fact_1765_prod_Ofinite__Collect__op,axiom,
! [I5: set_int,X: fun_int_complex,Y: fun_int_complex] :
( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_br(I5),X))))
=> ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_br(I5),Y))))
=> pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aa_fun3850298374753145905t_bool(aTP_Lamm_bs(I5),X),Y)))) ) ) ).
% prod.finite_Collect_op
tff(fact_1766_prod_Ofinite__Collect__op,axiom,
! [I5: set_complex,X: fun_complex_complex,Y: fun_complex_complex] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_bt(I5),X))))
=> ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_bt(I5),Y))))
=> pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aa_fun6641798613628721963x_bool(aTP_Lamm_bu(I5),X),Y)))) ) ) ).
% prod.finite_Collect_op
tff(fact_1767_prod_Ofinite__Collect__op,axiom,
! [I5: set_real,X: fun_real_real,Y: fun_real_real] :
( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_bv(I5),X))))
=> ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_bv(I5),Y))))
=> pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_bw(I5),X),Y)))) ) ) ).
% prod.finite_Collect_op
tff(fact_1768_prod_Ofinite__Collect__op,axiom,
! [I5: set_nat,X: fun_nat_real,Y: fun_nat_real] :
( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_bx(I5),X))))
=> ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_bx(I5),Y))))
=> pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_by(I5),X),Y)))) ) ) ).
% prod.finite_Collect_op
tff(fact_1769_prod_Ofinite__Collect__op,axiom,
! [I5: set_int,X: fun_int_real,Y: fun_int_real] :
( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_bz(I5),X))))
=> ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_bz(I5),Y))))
=> pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_ca(I5),X),Y)))) ) ) ).
% prod.finite_Collect_op
tff(fact_1770_prod_Ofinite__Collect__op,axiom,
! [I5: set_complex,X: fun_complex_real,Y: fun_complex_real] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_cb(I5),X))))
=> ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_cb(I5),Y))))
=> pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aa_fun2094807616088678831x_bool(aTP_Lamm_cc(I5),X),Y)))) ) ) ).
% prod.finite_Collect_op
tff(fact_1771_prod_Ofinite__Collect__op,axiom,
! [I5: set_real,X: fun_real_rat,Y: fun_real_rat] :
( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_cd(I5),X))))
=> ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_cd(I5),Y))))
=> pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aa_fun8530242357055030877l_bool(aTP_Lamm_ce(I5),X),Y)))) ) ) ).
% prod.finite_Collect_op
tff(fact_1772_prod_Ofinite__Collect__op,axiom,
! [I5: set_nat,X: fun_nat_rat,Y: fun_nat_rat] :
( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aTP_Lamm_cf(I5),X))))
=> ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aTP_Lamm_cf(I5),Y))))
=> pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aa_fun2107527232086395633t_bool(aTP_Lamm_cg(I5),X),Y)))) ) ) ).
% prod.finite_Collect_op
tff(fact_1773_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
( ( aa_nat_nat(vEBT_T_i_n_s_e_r_t(X),Xa2) = Y )
=> ( ( ? [A4: bool,B4: bool] : X = vEBT_Leaf(A4,B4)
=> ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),zero_zero_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
=> ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] : X = vEBT_Node(Info2,zero_zero_nat,Ts,S3)
=> ( Y != one_one_nat ) )
=> ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] : X = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S3)
=> ( Y != one_one_nat ) )
=> ( ( ? [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2)
=> ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2) )
=> ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Ma2))))),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(vEBT_T_i_n_s_e_r_t(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(vEBT_T_i_n_s_e_r_t(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat)) ) ) ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.elims
tff(fact_1774_set__bit__0,axiom,
! [A: int] : aa_int_int(bit_se7879613467334960850it_int(zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).
% set_bit_0
tff(fact_1775_set__bit__0,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,zero_zero_nat),A) = 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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).
% set_bit_0
tff(fact_1776_valid__0__not,axiom,
! [T: vEBT_VEBT] : ~ pp(aa_nat_bool(vEBT_invar_vebt(T),zero_zero_nat)) ).
% valid_0_not
tff(fact_1777_valid__tree__deg__neq__0,axiom,
! [T: vEBT_VEBT] : ~ pp(aa_nat_bool(vEBT_invar_vebt(T),zero_zero_nat)) ).
% valid_tree_deg_neq_0
tff(fact_1778_deg__not__0,axiom,
! [T: vEBT_VEBT,N: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).
% deg_not_0
tff(fact_1779_Leaf__0__not,axiom,
! [A: bool,B: bool] : ~ pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Leaf(A,B)),zero_zero_nat)) ).
% Leaf_0_not
tff(fact_1780_deg__1__Leafy,axiom,
! [T: vEBT_VEBT,N: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( ( N = one_one_nat )
=> ? [A4: bool,B4: bool] : T = vEBT_Leaf(A4,B4) ) ) ).
% deg_1_Leafy
tff(fact_1781_deg__1__Leaf,axiom,
! [T: vEBT_VEBT] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),one_one_nat))
=> ? [A4: bool,B4: bool] : T = vEBT_Leaf(A4,B4) ) ).
% deg_1_Leaf
tff(fact_1782_deg1Leaf,axiom,
! [T: vEBT_VEBT] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),one_one_nat))
<=> ? [A6: bool,B6: bool] : T = vEBT_Leaf(A6,B6) ) ).
% deg1Leaf
tff(fact_1783_le__zero__eq,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),zero_zero_nat))
<=> ( N = zero_zero_nat ) ) ).
% le_zero_eq
tff(fact_1784_not__gr__zero,axiom,
! [N: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
<=> ( N = zero_zero_nat ) ) ).
% not_gr_zero
tff(fact_1785_mult__cancel__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(times_times_complex(B),C) )
<=> ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% mult_cancel_right
tff(fact_1786_mult__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
<=> ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
tff(fact_1787_mult__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( aa_rat_rat(times_times_rat(A),C) = aa_rat_rat(times_times_rat(B),C) )
<=> ( ( C = zero_zero_rat )
| ( A = B ) ) ) ).
% mult_cancel_right
tff(fact_1788_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
<=> ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
tff(fact_1789_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
<=> ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
tff(fact_1790_mult__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( aa_complex_complex(times_times_complex(C),A) = aa_complex_complex(times_times_complex(C),B) )
<=> ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% mult_cancel_left
tff(fact_1791_mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
<=> ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
tff(fact_1792_mult__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( aa_rat_rat(times_times_rat(C),A) = aa_rat_rat(times_times_rat(C),B) )
<=> ( ( C = zero_zero_rat )
| ( A = B ) ) ) ).
% mult_cancel_left
tff(fact_1793_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
<=> ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
tff(fact_1794_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
<=> ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
tff(fact_1795_mult__eq__0__iff,axiom,
! [A: complex,B: complex] :
( ( aa_complex_complex(times_times_complex(A),B) = zero_zero_complex )
<=> ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% mult_eq_0_iff
tff(fact_1796_mult__eq__0__iff,axiom,
! [A: real,B: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = zero_zero_real )
<=> ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
tff(fact_1797_mult__eq__0__iff,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(times_times_rat(A),B) = zero_zero_rat )
<=> ( ( A = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% mult_eq_0_iff
tff(fact_1798_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = zero_zero_nat )
<=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
tff(fact_1799_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = zero_zero_int )
<=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
tff(fact_1800_mult__zero__right,axiom,
! [A: complex] : aa_complex_complex(times_times_complex(A),zero_zero_complex) = zero_zero_complex ).
% mult_zero_right
tff(fact_1801_mult__zero__right,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),zero_zero_real) = zero_zero_real ).
% mult_zero_right
tff(fact_1802_mult__zero__right,axiom,
! [A: rat] : aa_rat_rat(times_times_rat(A),zero_zero_rat) = zero_zero_rat ).
% mult_zero_right
tff(fact_1803_mult__zero__right,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),zero_zero_nat) = zero_zero_nat ).
% mult_zero_right
tff(fact_1804_mult__zero__right,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),zero_zero_int) = zero_zero_int ).
% mult_zero_right
tff(fact_1805_mult__zero__left,axiom,
! [A: complex] : aa_complex_complex(times_times_complex(zero_zero_complex),A) = zero_zero_complex ).
% mult_zero_left
tff(fact_1806_mult__zero__left,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,zero_zero_real),A) = zero_zero_real ).
% mult_zero_left
tff(fact_1807_mult__zero__left,axiom,
! [A: rat] : aa_rat_rat(times_times_rat(zero_zero_rat),A) = zero_zero_rat ).
% mult_zero_left
tff(fact_1808_mult__zero__left,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),A) = zero_zero_nat ).
% mult_zero_left
tff(fact_1809_mult__zero__left,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,zero_zero_int),A) = zero_zero_int ).
% mult_zero_left
tff(fact_1810_add__0,axiom,
! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).
% add_0
tff(fact_1811_add__0,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).
% add_0
tff(fact_1812_add__0,axiom,
! [A: rat] : plus_plus_rat(zero_zero_rat,A) = A ).
% add_0
tff(fact_1813_add__0,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A) = A ).
% add_0
tff(fact_1814_add__0,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).
% add_0
tff(fact_1815_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) )
<=> ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
tff(fact_1816_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
<=> ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
tff(fact_1817_add__cancel__right__right,axiom,
! [A: complex,B: complex] :
( ( A = aa_complex_complex(plus_plus_complex(A),B) )
<=> ( B = zero_zero_complex ) ) ).
% add_cancel_right_right
tff(fact_1818_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) )
<=> ( B = zero_zero_real ) ) ).
% add_cancel_right_right
tff(fact_1819_add__cancel__right__right,axiom,
! [A: rat,B: rat] :
( ( A = plus_plus_rat(A,B) )
<=> ( B = zero_zero_rat ) ) ).
% add_cancel_right_right
tff(fact_1820_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) )
<=> ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
tff(fact_1821_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) )
<=> ( B = zero_zero_int ) ) ).
% add_cancel_right_right
tff(fact_1822_add__cancel__right__left,axiom,
! [A: complex,B: complex] :
( ( A = aa_complex_complex(plus_plus_complex(B),A) )
<=> ( B = zero_zero_complex ) ) ).
% add_cancel_right_left
tff(fact_1823_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) )
<=> ( B = zero_zero_real ) ) ).
% add_cancel_right_left
tff(fact_1824_add__cancel__right__left,axiom,
! [A: rat,B: rat] :
( ( A = plus_plus_rat(B,A) )
<=> ( B = zero_zero_rat ) ) ).
% add_cancel_right_left
tff(fact_1825_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) )
<=> ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
tff(fact_1826_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) )
<=> ( B = zero_zero_int ) ) ).
% add_cancel_right_left
tff(fact_1827_add__cancel__left__right,axiom,
! [A: complex,B: complex] :
( ( aa_complex_complex(plus_plus_complex(A),B) = A )
<=> ( B = zero_zero_complex ) ) ).
% add_cancel_left_right
tff(fact_1828_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = A )
<=> ( B = zero_zero_real ) ) ).
% add_cancel_left_right
tff(fact_1829_add__cancel__left__right,axiom,
! [A: rat,B: rat] :
( ( plus_plus_rat(A,B) = A )
<=> ( B = zero_zero_rat ) ) ).
% add_cancel_left_right
tff(fact_1830_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = A )
<=> ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
tff(fact_1831_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = A )
<=> ( B = zero_zero_int ) ) ).
% add_cancel_left_right
tff(fact_1832_add__cancel__left__left,axiom,
! [B: complex,A: complex] :
( ( aa_complex_complex(plus_plus_complex(B),A) = A )
<=> ( B = zero_zero_complex ) ) ).
% add_cancel_left_left
tff(fact_1833_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) = A )
<=> ( B = zero_zero_real ) ) ).
% add_cancel_left_left
tff(fact_1834_add__cancel__left__left,axiom,
! [B: rat,A: rat] :
( ( plus_plus_rat(B,A) = A )
<=> ( B = zero_zero_rat ) ) ).
% add_cancel_left_left
tff(fact_1835_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = A )
<=> ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
tff(fact_1836_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = A )
<=> ( B = zero_zero_int ) ) ).
% add_cancel_left_left
tff(fact_1837_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A) )
<=> ( A = zero_zero_real ) ) ).
% double_zero_sym
tff(fact_1838_double__zero__sym,axiom,
! [A: rat] :
( ( zero_zero_rat = plus_plus_rat(A,A) )
<=> ( A = zero_zero_rat ) ) ).
% double_zero_sym
tff(fact_1839_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) )
<=> ( A = zero_zero_int ) ) ).
% double_zero_sym
tff(fact_1840_add_Oright__neutral,axiom,
! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).
% add.right_neutral
tff(fact_1841_add_Oright__neutral,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ).
% add.right_neutral
tff(fact_1842_add_Oright__neutral,axiom,
! [A: rat] : plus_plus_rat(A,zero_zero_rat) = A ).
% add.right_neutral
tff(fact_1843_add_Oright__neutral,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ).
% add.right_neutral
tff(fact_1844_add_Oright__neutral,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).
% add.right_neutral
tff(fact_1845_double__eq__0__iff,axiom,
! [A: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A) = zero_zero_real )
<=> ( A = zero_zero_real ) ) ).
% double_eq_0_iff
tff(fact_1846_double__eq__0__iff,axiom,
! [A: rat] :
( ( plus_plus_rat(A,A) = zero_zero_rat )
<=> ( A = zero_zero_rat ) ) ).
% double_eq_0_iff
tff(fact_1847_double__eq__0__iff,axiom,
! [A: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) = zero_zero_int )
<=> ( A = zero_zero_int ) ) ).
% double_eq_0_iff
tff(fact_1848_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: complex] : aa_complex_complex(minus_minus_complex(A),A) = zero_zero_complex ).
% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1849_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),A) = zero_zero_real ).
% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1850_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: rat] : minus_minus_rat(A,A) = zero_zero_rat ).
% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1851_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),A) = zero_zero_nat ).
% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1852_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),A) = zero_zero_int ).
% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1853_diff__zero,axiom,
! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).
% diff_zero
tff(fact_1854_diff__zero,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),zero_zero_real) = A ).
% diff_zero
tff(fact_1855_diff__zero,axiom,
! [A: rat] : minus_minus_rat(A,zero_zero_rat) = A ).
% diff_zero
tff(fact_1856_diff__zero,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),zero_zero_nat) = A ).
% diff_zero
tff(fact_1857_diff__zero,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),zero_zero_int) = A ).
% diff_zero
tff(fact_1858_zero__diff,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),A) = zero_zero_nat ).
% zero_diff
tff(fact_1859_diff__0__right,axiom,
! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).
% diff_0_right
tff(fact_1860_diff__0__right,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),zero_zero_real) = A ).
% diff_0_right
tff(fact_1861_diff__0__right,axiom,
! [A: rat] : minus_minus_rat(A,zero_zero_rat) = A ).
% diff_0_right
tff(fact_1862_diff__0__right,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),zero_zero_int) = A ).
% diff_0_right
tff(fact_1863_diff__self,axiom,
! [A: complex] : aa_complex_complex(minus_minus_complex(A),A) = zero_zero_complex ).
% diff_self
tff(fact_1864_diff__self,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),A) = zero_zero_real ).
% diff_self
tff(fact_1865_diff__self,axiom,
! [A: rat] : minus_minus_rat(A,A) = zero_zero_rat ).
% diff_self
tff(fact_1866_diff__self,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),A) = zero_zero_int ).
% diff_self
tff(fact_1867_div__0,axiom,
! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(zero_zero_complex),A) = zero_zero_complex ).
% div_0
tff(fact_1868_div__0,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,zero_zero_real),A) = zero_zero_real ).
% div_0
tff(fact_1869_div__0,axiom,
! [A: rat] : divide_divide_rat(zero_zero_rat,A) = zero_zero_rat ).
% div_0
tff(fact_1870_div__0,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),A) = zero_zero_nat ).
% div_0
tff(fact_1871_div__0,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A) = zero_zero_int ).
% div_0
tff(fact_1872_divide__eq__0__iff,axiom,
! [A: complex,B: complex] :
( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = zero_zero_complex )
<=> ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% divide_eq_0_iff
tff(fact_1873_divide__eq__0__iff,axiom,
! [A: real,B: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = zero_zero_real )
<=> ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_eq_0_iff
tff(fact_1874_divide__eq__0__iff,axiom,
! [A: rat,B: rat] :
( ( divide_divide_rat(A,B) = zero_zero_rat )
<=> ( ( A = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% divide_eq_0_iff
tff(fact_1875_div__by__0,axiom,
! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).
% div_by_0
tff(fact_1876_div__by__0,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),zero_zero_real) = zero_zero_real ).
% div_by_0
tff(fact_1877_div__by__0,axiom,
! [A: rat] : divide_divide_rat(A,zero_zero_rat) = zero_zero_rat ).
% div_by_0
tff(fact_1878_div__by__0,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),zero_zero_nat) = zero_zero_nat ).
% div_by_0
tff(fact_1879_div__by__0,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),zero_zero_int) = zero_zero_int ).
% div_by_0
tff(fact_1880_divide__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( aa_complex_complex(divide1717551699836669952omplex(C),A) = aa_complex_complex(divide1717551699836669952omplex(C),B) )
<=> ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% divide_cancel_left
tff(fact_1881_divide__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B) )
<=> ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_left
tff(fact_1882_divide__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( divide_divide_rat(C,A) = divide_divide_rat(C,B) )
<=> ( ( C = zero_zero_rat )
| ( A = B ) ) ) ).
% divide_cancel_left
tff(fact_1883_divide__cancel__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
<=> ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% divide_cancel_right
tff(fact_1884_divide__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
<=> ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_right
tff(fact_1885_divide__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( divide_divide_rat(A,C) = divide_divide_rat(B,C) )
<=> ( ( C = zero_zero_rat )
| ( A = B ) ) ) ).
% divide_cancel_right
tff(fact_1886_bits__div__0,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),A) = zero_zero_nat ).
% bits_div_0
tff(fact_1887_bits__div__0,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A) = zero_zero_int ).
% bits_div_0
tff(fact_1888_bits__div__by__0,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),zero_zero_nat) = zero_zero_nat ).
% bits_div_by_0
tff(fact_1889_bits__div__by__0,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),zero_zero_int) = zero_zero_int ).
% bits_div_by_0
tff(fact_1890_division__ring__divide__zero,axiom,
! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).
% division_ring_divide_zero
tff(fact_1891_division__ring__divide__zero,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),zero_zero_real) = zero_zero_real ).
% division_ring_divide_zero
tff(fact_1892_division__ring__divide__zero,axiom,
! [A: rat] : divide_divide_rat(A,zero_zero_rat) = zero_zero_rat ).
% division_ring_divide_zero
tff(fact_1893_bits__mod__0,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,zero_zero_nat),A) = zero_zero_nat ).
% bits_mod_0
tff(fact_1894_bits__mod__0,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,zero_zero_int),A) = zero_zero_int ).
% bits_mod_0
tff(fact_1895_bits__mod__0,axiom,
! [A: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).
% bits_mod_0
tff(fact_1896_mod__self,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),A) = zero_zero_nat ).
% mod_self
tff(fact_1897_mod__self,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),A) = zero_zero_int ).
% mod_self
tff(fact_1898_mod__self,axiom,
! [A: code_integer] : modulo364778990260209775nteger(A,A) = zero_z3403309356797280102nteger ).
% mod_self
tff(fact_1899_mod__by__0,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),zero_zero_nat) = A ).
% mod_by_0
tff(fact_1900_mod__by__0,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),zero_zero_int) = A ).
% mod_by_0
tff(fact_1901_mod__by__0,axiom,
! [A: code_integer] : modulo364778990260209775nteger(A,zero_z3403309356797280102nteger) = A ).
% mod_by_0
tff(fact_1902_mod__0,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,zero_zero_nat),A) = zero_zero_nat ).
% mod_0
tff(fact_1903_mod__0,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,zero_zero_int),A) = zero_zero_int ).
% mod_0
tff(fact_1904_mod__0,axiom,
! [A: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).
% mod_0
tff(fact_1905_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A)) ) ).
% bot_nat_0.not_eq_extremum
tff(fact_1906_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).
% neq0_conv
tff(fact_1907_less__nat__zero__code,axiom,
! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).
% less_nat_zero_code
tff(fact_1908_le0,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),N)) ).
% le0
tff(fact_1909_bot__nat__0_Oextremum,axiom,
! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A)) ).
% bot_nat_0.extremum
tff(fact_1910_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_1911_add__is__0,axiom,
! [M: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = zero_zero_nat )
<=> ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
tff(fact_1912_diff__self__eq__0,axiom,
! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),M) = zero_zero_nat ).
% diff_self_eq_0
tff(fact_1913_diff__0__eq__0,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),N) = zero_zero_nat ).
% diff_0_eq_0
tff(fact_1914_mult__cancel2,axiom,
! [M: nat,K: nat,N: 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,N),K) )
<=> ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
tff(fact_1915_mult__cancel1,axiom,
! [K: nat,M: nat,N: 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),N) )
<=> ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
tff(fact_1916_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_1917_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = zero_zero_nat )
<=> ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
tff(fact_1918_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).
% zero_le_double_add_iff_zero_le_single_add
tff(fact_1919_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),plus_plus_rat(A,A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).
% zero_le_double_add_iff_zero_le_single_add
tff(fact_1920_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).
% zero_le_double_add_iff_zero_le_single_add
tff(fact_1921_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).
% double_add_le_zero_iff_single_add_le_zero
tff(fact_1922_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(A,A)),zero_zero_rat))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).
% double_add_le_zero_iff_single_add_le_zero
tff(fact_1923_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).
% double_add_le_zero_iff_single_add_le_zero
tff(fact_1924_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) ).
% le_add_same_cancel2
tff(fact_1925_le__add__same__cancel2,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),plus_plus_rat(B,A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ).
% le_add_same_cancel2
tff(fact_1926_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B)) ) ).
% le_add_same_cancel2
tff(fact_1927_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) ) ).
% le_add_same_cancel2
tff(fact_1928_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) ).
% le_add_same_cancel1
tff(fact_1929_le__add__same__cancel1,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),plus_plus_rat(A,B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ).
% le_add_same_cancel1
tff(fact_1930_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B)) ) ).
% le_add_same_cancel1
tff(fact_1931_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) ) ).
% le_add_same_cancel1
tff(fact_1932_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).
% add_le_same_cancel2
tff(fact_1933_add__le__same__cancel2,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(A,B)),B))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).
% add_le_same_cancel2
tff(fact_1934_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat)) ) ).
% add_le_same_cancel2
tff(fact_1935_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).
% add_le_same_cancel2
tff(fact_1936_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).
% add_le_same_cancel1
tff(fact_1937_add__le__same__cancel1,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(B,A)),B))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).
% add_le_same_cancel1
tff(fact_1938_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat)) ) ).
% add_le_same_cancel1
tff(fact_1939_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).
% add_le_same_cancel1
tff(fact_1940_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).
% zero_less_double_add_iff_zero_less_single_add
tff(fact_1941_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),plus_plus_rat(A,A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).
% zero_less_double_add_iff_zero_less_single_add
tff(fact_1942_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).
% zero_less_double_add_iff_zero_less_single_add
tff(fact_1943_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A)),zero_zero_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).
% double_add_less_zero_iff_single_add_less_zero
tff(fact_1944_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(A,A)),zero_zero_rat))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).
% double_add_less_zero_iff_single_add_less_zero
tff(fact_1945_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).
% double_add_less_zero_iff_single_add_less_zero
tff(fact_1946_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ).
% less_add_same_cancel2
tff(fact_1947_less__add__same__cancel2,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),plus_plus_rat(B,A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ).
% less_add_same_cancel2
tff(fact_1948_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B)) ) ).
% less_add_same_cancel2
tff(fact_1949_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ).
% less_add_same_cancel2
tff(fact_1950_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ).
% less_add_same_cancel1
tff(fact_1951_less__add__same__cancel1,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),plus_plus_rat(A,B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ).
% less_add_same_cancel1
tff(fact_1952_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B)) ) ).
% less_add_same_cancel1
tff(fact_1953_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ).
% less_add_same_cancel1
tff(fact_1954_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),B))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).
% add_less_same_cancel2
tff(fact_1955_add__less__same__cancel2,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(A,B)),B))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).
% add_less_same_cancel2
tff(fact_1956_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat)) ) ).
% add_less_same_cancel2
tff(fact_1957_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).
% add_less_same_cancel2
tff(fact_1958_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)),B))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).
% add_less_same_cancel1
tff(fact_1959_add__less__same__cancel1,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(B,A)),B))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).
% add_less_same_cancel1
tff(fact_1960_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat)) ) ).
% add_less_same_cancel1
tff(fact_1961_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).
% add_less_same_cancel1
tff(fact_1962_diff__ge__0__iff__ge,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ).
% diff_ge_0_iff_ge
tff(fact_1963_diff__ge__0__iff__ge,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),minus_minus_rat(A,B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).
% diff_ge_0_iff_ge
tff(fact_1964_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ).
% diff_ge_0_iff_ge
tff(fact_1965_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ).
% diff_gt_0_iff_gt
tff(fact_1966_diff__gt__0__iff__gt,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),minus_minus_rat(A,B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ).
% diff_gt_0_iff_gt
tff(fact_1967_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ).
% diff_gt_0_iff_gt
tff(fact_1968_mult__cancel__right2,axiom,
! [A: complex,C: complex] :
( ( aa_complex_complex(times_times_complex(A),C) = C )
<=> ( ( C = zero_zero_complex )
| ( A = one_one_complex ) ) ) ).
% mult_cancel_right2
tff(fact_1969_mult__cancel__right2,axiom,
! [A: real,C: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = C )
<=> ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_right2
tff(fact_1970_mult__cancel__right2,axiom,
! [A: rat,C: rat] :
( ( aa_rat_rat(times_times_rat(A),C) = C )
<=> ( ( C = zero_zero_rat )
| ( A = one_one_rat ) ) ) ).
% mult_cancel_right2
tff(fact_1971_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = C )
<=> ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
tff(fact_1972_mult__cancel__right1,axiom,
! [C: complex,B: complex] :
( ( C = aa_complex_complex(times_times_complex(B),C) )
<=> ( ( C = zero_zero_complex )
| ( B = one_one_complex ) ) ) ).
% mult_cancel_right1
tff(fact_1973_mult__cancel__right1,axiom,
! [C: real,B: real] :
( ( C = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
<=> ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
tff(fact_1974_mult__cancel__right1,axiom,
! [C: rat,B: rat] :
( ( C = aa_rat_rat(times_times_rat(B),C) )
<=> ( ( C = zero_zero_rat )
| ( B = one_one_rat ) ) ) ).
% mult_cancel_right1
tff(fact_1975_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
<=> ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
tff(fact_1976_mult__cancel__left2,axiom,
! [C: complex,A: complex] :
( ( aa_complex_complex(times_times_complex(C),A) = C )
<=> ( ( C = zero_zero_complex )
| ( A = one_one_complex ) ) ) ).
% mult_cancel_left2
tff(fact_1977_mult__cancel__left2,axiom,
! [C: real,A: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = C )
<=> ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_left2
tff(fact_1978_mult__cancel__left2,axiom,
! [C: rat,A: rat] :
( ( aa_rat_rat(times_times_rat(C),A) = C )
<=> ( ( C = zero_zero_rat )
| ( A = one_one_rat ) ) ) ).
% mult_cancel_left2
tff(fact_1979_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = C )
<=> ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
tff(fact_1980_mult__cancel__left1,axiom,
! [C: complex,B: complex] :
( ( C = aa_complex_complex(times_times_complex(C),B) )
<=> ( ( C = zero_zero_complex )
| ( B = one_one_complex ) ) ) ).
% mult_cancel_left1
tff(fact_1981_mult__cancel__left1,axiom,
! [C: real,B: real] :
( ( C = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
<=> ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
tff(fact_1982_mult__cancel__left1,axiom,
! [C: rat,B: rat] :
( ( C = aa_rat_rat(times_times_rat(C),B) )
<=> ( ( C = zero_zero_rat )
| ( B = one_one_rat ) ) ) ).
% mult_cancel_left1
tff(fact_1983_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
<=> ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
tff(fact_1984_sum__squares__eq__zero__iff,axiom,
! [X: real,Y: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)) = zero_zero_real )
<=> ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
tff(fact_1985_sum__squares__eq__zero__iff,axiom,
! [X: rat,Y: rat] :
( ( plus_plus_rat(aa_rat_rat(times_times_rat(X),X),aa_rat_rat(times_times_rat(Y),Y)) = zero_zero_rat )
<=> ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_squares_eq_zero_iff
tff(fact_1986_sum__squares__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = zero_zero_int )
<=> ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
tff(fact_1987_diff__numeral__special_I9_J,axiom,
aa_complex_complex(minus_minus_complex(one_one_complex),one_one_complex) = zero_zero_complex ).
% diff_numeral_special(9)
tff(fact_1988_diff__numeral__special_I9_J,axiom,
aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),one_one_real) = zero_zero_real ).
% diff_numeral_special(9)
tff(fact_1989_diff__numeral__special_I9_J,axiom,
minus_minus_rat(one_one_rat,one_one_rat) = zero_zero_rat ).
% diff_numeral_special(9)
tff(fact_1990_diff__numeral__special_I9_J,axiom,
aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),one_one_int) = zero_zero_int ).
% diff_numeral_special(9)
tff(fact_1991_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(C),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
tff(fact_1992_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
tff(fact_1993_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( divide_divide_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(C),B)) = divide_divide_rat(A,B) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
tff(fact_1994_nonzero__mult__div__cancel__right,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B)),B) = A ) ) ).
% nonzero_mult_div_cancel_right
tff(fact_1995_nonzero__mult__div__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),B) = A ) ) ).
% nonzero_mult_div_cancel_right
tff(fact_1996_nonzero__mult__div__cancel__right,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( divide_divide_rat(aa_rat_rat(times_times_rat(A),B),B) = A ) ) ).
% nonzero_mult_div_cancel_right
tff(fact_1997_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),B) = A ) ) ).
% nonzero_mult_div_cancel_right
tff(fact_1998_nonzero__mult__div__cancel__right,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),B) = A ) ) ).
% nonzero_mult_div_cancel_right
tff(fact_1999_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).
% nonzero_mult_divide_mult_cancel_right
tff(fact_2000_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).
% nonzero_mult_divide_mult_cancel_right
tff(fact_2001_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( divide_divide_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(B),C)) = divide_divide_rat(A,B) ) ) ).
% nonzero_mult_divide_mult_cancel_right
tff(fact_2002_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
tff(fact_2003_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
tff(fact_2004_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( divide_divide_rat(aa_rat_rat(times_times_rat(C),A),aa_rat_rat(times_times_rat(B),C)) = divide_divide_rat(A,B) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
tff(fact_2005_nonzero__mult__div__cancel__left,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B)),A) = B ) ) ).
% nonzero_mult_div_cancel_left
tff(fact_2006_nonzero__mult__div__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),A) = B ) ) ).
% nonzero_mult_div_cancel_left
tff(fact_2007_nonzero__mult__div__cancel__left,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( divide_divide_rat(aa_rat_rat(times_times_rat(A),B),A) = B ) ) ).
% nonzero_mult_div_cancel_left
tff(fact_2008_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),A) = B ) ) ).
% nonzero_mult_div_cancel_left
tff(fact_2009_nonzero__mult__div__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),A) = B ) ) ).
% nonzero_mult_div_cancel_left
tff(fact_2010_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).
% nonzero_mult_divide_mult_cancel_left
tff(fact_2011_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).
% nonzero_mult_divide_mult_cancel_left
tff(fact_2012_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( divide_divide_rat(aa_rat_rat(times_times_rat(C),A),aa_rat_rat(times_times_rat(C),B)) = divide_divide_rat(A,B) ) ) ).
% nonzero_mult_divide_mult_cancel_left
tff(fact_2013_mult__divide__mult__cancel__left__if,axiom,
! [C: complex,A: complex,B: complex] :
( ( ( C = zero_zero_complex )
=> ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)) = zero_zero_complex ) )
& ( ( C != zero_zero_complex )
=> ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ) ).
% mult_divide_mult_cancel_left_if
tff(fact_2014_mult__divide__mult__cancel__left__if,axiom,
! [C: real,A: real,B: real] :
( ( ( C = zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = zero_zero_real ) )
& ( ( C != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ) ).
% mult_divide_mult_cancel_left_if
tff(fact_2015_mult__divide__mult__cancel__left__if,axiom,
! [C: rat,A: rat,B: rat] :
( ( ( C = zero_zero_rat )
=> ( divide_divide_rat(aa_rat_rat(times_times_rat(C),A),aa_rat_rat(times_times_rat(C),B)) = zero_zero_rat ) )
& ( ( C != zero_zero_rat )
=> ( divide_divide_rat(aa_rat_rat(times_times_rat(C),A),aa_rat_rat(times_times_rat(C),B)) = divide_divide_rat(A,B) ) ) ) ).
% mult_divide_mult_cancel_left_if
tff(fact_2016_div__mult__mult1__if,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( C = zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).
% div_mult_mult1_if
tff(fact_2017_div__mult__mult1__if,axiom,
! [C: int,A: int,B: int] :
( ( ( C = zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = zero_zero_int ) )
& ( ( C != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ).
% div_mult_mult1_if
tff(fact_2018_div__mult__mult2,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).
% div_mult_mult2
tff(fact_2019_div__mult__mult2,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ).
% div_mult_mult2
tff(fact_2020_div__mult__mult1,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).
% div_mult_mult1
tff(fact_2021_div__mult__mult1,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ).
% div_mult_mult1
tff(fact_2022_diff__add__zero,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = zero_zero_nat ).
% diff_add_zero
tff(fact_2023_zero__eq__1__divide__iff,axiom,
! [A: real] :
( ( zero_zero_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) )
<=> ( A = zero_zero_real ) ) ).
% zero_eq_1_divide_iff
tff(fact_2024_zero__eq__1__divide__iff,axiom,
! [A: rat] :
( ( zero_zero_rat = divide_divide_rat(one_one_rat,A) )
<=> ( A = zero_zero_rat ) ) ).
% zero_eq_1_divide_iff
tff(fact_2025_one__divide__eq__0__iff,axiom,
! [A: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) = zero_zero_real )
<=> ( A = zero_zero_real ) ) ).
% one_divide_eq_0_iff
tff(fact_2026_one__divide__eq__0__iff,axiom,
! [A: rat] :
( ( divide_divide_rat(one_one_rat,A) = zero_zero_rat )
<=> ( A = zero_zero_rat ) ) ).
% one_divide_eq_0_iff
tff(fact_2027_eq__divide__eq__1,axiom,
! [B: real,A: real] :
( ( one_one_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A) )
<=> ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% eq_divide_eq_1
tff(fact_2028_eq__divide__eq__1,axiom,
! [B: rat,A: rat] :
( ( one_one_rat = divide_divide_rat(B,A) )
<=> ( ( A != zero_zero_rat )
& ( A = B ) ) ) ).
% eq_divide_eq_1
tff(fact_2029_divide__eq__eq__1,axiom,
! [B: real,A: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A) = one_one_real )
<=> ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_eq_1
tff(fact_2030_divide__eq__eq__1,axiom,
! [B: rat,A: rat] :
( ( divide_divide_rat(B,A) = one_one_rat )
<=> ( ( A != zero_zero_rat )
& ( A = B ) ) ) ).
% divide_eq_eq_1
tff(fact_2031_divide__self__if,axiom,
! [A: complex] :
( ( ( A = zero_zero_complex )
=> ( aa_complex_complex(divide1717551699836669952omplex(A),A) = zero_zero_complex ) )
& ( ( A != zero_zero_complex )
=> ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ) ).
% divide_self_if
tff(fact_2032_divide__self__if,axiom,
! [A: real] :
( ( ( A = zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = zero_zero_real ) )
& ( ( A != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = one_one_real ) ) ) ).
% divide_self_if
tff(fact_2033_divide__self__if,axiom,
! [A: rat] :
( ( ( A = zero_zero_rat )
=> ( divide_divide_rat(A,A) = zero_zero_rat ) )
& ( ( A != zero_zero_rat )
=> ( divide_divide_rat(A,A) = one_one_rat ) ) ) ).
% divide_self_if
tff(fact_2034_divide__self,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).
% divide_self
tff(fact_2035_divide__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = one_one_real ) ) ).
% divide_self
tff(fact_2036_divide__self,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( divide_divide_rat(A,A) = one_one_rat ) ) ).
% divide_self
tff(fact_2037_one__eq__divide__iff,axiom,
! [A: complex,B: complex] :
( ( one_one_complex = aa_complex_complex(divide1717551699836669952omplex(A),B) )
<=> ( ( B != zero_zero_complex )
& ( A = B ) ) ) ).
% one_eq_divide_iff
tff(fact_2038_one__eq__divide__iff,axiom,
! [A: real,B: real] :
( ( one_one_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) )
<=> ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% one_eq_divide_iff
tff(fact_2039_one__eq__divide__iff,axiom,
! [A: rat,B: rat] :
( ( one_one_rat = divide_divide_rat(A,B) )
<=> ( ( B != zero_zero_rat )
& ( A = B ) ) ) ).
% one_eq_divide_iff
tff(fact_2040_div__self,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).
% div_self
tff(fact_2041_div__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),A) = one_one_real ) ) ).
% div_self
tff(fact_2042_div__self,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( divide_divide_rat(A,A) = one_one_rat ) ) ).
% div_self
tff(fact_2043_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),A) = one_one_nat ) ) ).
% div_self
tff(fact_2044_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),A) = one_one_int ) ) ).
% div_self
tff(fact_2045_divide__eq__1__iff,axiom,
! [A: complex,B: complex] :
( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = one_one_complex )
<=> ( ( B != zero_zero_complex )
& ( A = B ) ) ) ).
% divide_eq_1_iff
tff(fact_2046_divide__eq__1__iff,axiom,
! [A: real,B: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = one_one_real )
<=> ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_1_iff
tff(fact_2047_divide__eq__1__iff,axiom,
! [A: rat,B: rat] :
( ( divide_divide_rat(A,B) = one_one_rat )
<=> ( ( B != zero_zero_rat )
& ( A = B ) ) ) ).
% divide_eq_1_iff
tff(fact_2048_power__0__Suc,axiom,
! [N: nat] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_nat_nat(suc,N)) = zero_zero_rat ).
% power_0_Suc
tff(fact_2049_power__0__Suc,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_nat_nat(suc,N)) = zero_zero_nat ).
% power_0_Suc
tff(fact_2050_power__0__Suc,axiom,
! [N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_nat_nat(suc,N)) = zero_zero_real ).
% power_0_Suc
tff(fact_2051_power__0__Suc,axiom,
! [N: nat] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_nat_nat(suc,N)) = zero_zero_complex ).
% power_0_Suc
tff(fact_2052_power__0__Suc,axiom,
! [N: nat] : aa_nat_int(power_power_int(zero_zero_int),aa_nat_nat(suc,N)) = zero_zero_int ).
% power_0_Suc
tff(fact_2053_power__zero__numeral,axiom,
! [K: num] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_rat ).
% power_zero_numeral
tff(fact_2054_power__zero__numeral,axiom,
! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_nat ).
% power_zero_numeral
tff(fact_2055_power__zero__numeral,axiom,
! [K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_real ).
% power_zero_numeral
tff(fact_2056_power__zero__numeral,axiom,
! [K: num] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_complex ).
% power_zero_numeral
tff(fact_2057_power__zero__numeral,axiom,
! [K: num] : aa_nat_int(power_power_int(zero_zero_int),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_int ).
% power_zero_numeral
tff(fact_2058_mod__mult__self2__is__0,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),B) = zero_zero_nat ).
% mod_mult_self2_is_0
tff(fact_2059_mod__mult__self2__is__0,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),B) = zero_zero_int ).
% mod_mult_self2_is_0
tff(fact_2060_mod__mult__self2__is__0,axiom,
! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),B) = zero_z3403309356797280102nteger ).
% mod_mult_self2_is_0
tff(fact_2061_mod__mult__self1__is__0,axiom,
! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),B) = zero_zero_nat ).
% mod_mult_self1_is_0
tff(fact_2062_mod__mult__self1__is__0,axiom,
! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),B) = zero_zero_int ).
% mod_mult_self1_is_0
tff(fact_2063_mod__mult__self1__is__0,axiom,
! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A),B) = zero_z3403309356797280102nteger ).
% mod_mult_self1_is_0
tff(fact_2064_bits__mod__by__1,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),one_one_nat) = zero_zero_nat ).
% bits_mod_by_1
tff(fact_2065_bits__mod__by__1,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),one_one_int) = zero_zero_int ).
% bits_mod_by_1
tff(fact_2066_bits__mod__by__1,axiom,
! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).
% bits_mod_by_1
tff(fact_2067_mod__by__1,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),one_one_nat) = zero_zero_nat ).
% mod_by_1
tff(fact_2068_mod__by__1,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),one_one_int) = zero_zero_int ).
% mod_by_1
tff(fact_2069_mod__by__1,axiom,
! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).
% mod_by_1
tff(fact_2070_power__Suc0__right,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,zero_zero_nat)) = A ).
% power_Suc0_right
tff(fact_2071_power__Suc0__right,axiom,
! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,zero_zero_nat)) = A ).
% power_Suc0_right
tff(fact_2072_power__Suc0__right,axiom,
! [A: complex] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,zero_zero_nat)) = A ).
% power_Suc0_right
tff(fact_2073_power__Suc0__right,axiom,
! [A: int] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,zero_zero_nat)) = A ).
% power_Suc0_right
tff(fact_2074_bits__mod__div__trivial,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = zero_zero_nat ).
% bits_mod_div_trivial
tff(fact_2075_bits__mod__div__trivial,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) = zero_zero_int ).
% bits_mod_div_trivial
tff(fact_2076_bits__mod__div__trivial,axiom,
! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A,B),B) = zero_z3403309356797280102nteger ).
% bits_mod_div_trivial
tff(fact_2077_mod__div__trivial,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = zero_zero_nat ).
% mod_div_trivial
tff(fact_2078_mod__div__trivial,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) = zero_zero_int ).
% mod_div_trivial
tff(fact_2079_mod__div__trivial,axiom,
! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A,B),B) = zero_z3403309356797280102nteger ).
% mod_div_trivial
tff(fact_2080_less__Suc0,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,zero_zero_nat)))
<=> ( N = zero_zero_nat ) ) ).
% less_Suc0
tff(fact_2081_zero__less__Suc,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,N))) ).
% zero_less_Suc
tff(fact_2082_add__gr__0,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
| pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).
% add_gr_0
tff(fact_2083_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = aa_nat_nat(suc,zero_zero_nat) )
<=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
& ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).
% mult_eq_1_iff
tff(fact_2084_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) )
<=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
& ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).
% one_eq_mult_iff
tff(fact_2085_div__by__Suc__0,axiom,
! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(suc,zero_zero_nat)) = M ).
% div_by_Suc_0
tff(fact_2086_less__one,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),one_one_nat))
<=> ( N = zero_zero_nat ) ) ).
% less_one
tff(fact_2087_zero__less__diff,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% zero_less_diff
tff(fact_2088_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).
% nat_0_less_mult_iff
tff(fact_2089_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,N),K)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).
% mult_less_cancel2
tff(fact_2090_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).
% nat_mult_less_cancel_disj
tff(fact_2091_div__less,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat ) ) ).
% div_less
tff(fact_2092_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat )
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% diff_is_0_eq
tff(fact_2093_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat ) ) ).
% diff_is_0_eq'
tff(fact_2094_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),M) = aa_nat_nat(suc,zero_zero_nat) )
<=> ( ( M = zero_zero_nat )
| ( X = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).
% nat_power_eq_Suc_0_iff
tff(fact_2095_power__Suc__0,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(suc,zero_zero_nat) ).
% power_Suc_0
tff(fact_2096_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
tff(fact_2097_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_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),N)) = zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_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),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) ) ) ) ).
% nat_mult_div_cancel_disj
tff(fact_2098_mod__by__Suc__0,axiom,
! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).
% mod_by_Suc_0
tff(fact_2099_dbl__inc__simps_I2_J,axiom,
aa_complex_complex(neg_nu8557863876264182079omplex,zero_zero_complex) = one_one_complex ).
% dbl_inc_simps(2)
tff(fact_2100_dbl__inc__simps_I2_J,axiom,
aa_real_real(neg_nu8295874005876285629c_real,zero_zero_real) = one_one_real ).
% dbl_inc_simps(2)
tff(fact_2101_dbl__inc__simps_I2_J,axiom,
neg_nu5219082963157363817nc_rat(zero_zero_rat) = one_one_rat ).
% dbl_inc_simps(2)
tff(fact_2102_dbl__inc__simps_I2_J,axiom,
aa_int_int(neg_nu5851722552734809277nc_int,zero_zero_int) = one_one_int ).
% dbl_inc_simps(2)
tff(fact_2103_zero__le__divide__1__iff,axiom,
! [A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).
% zero_le_divide_1_iff
tff(fact_2104_zero__le__divide__1__iff,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(one_one_rat,A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).
% zero_le_divide_1_iff
tff(fact_2105_divide__le__0__1__iff,axiom,
! [A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),zero_zero_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).
% divide_le_0_1_iff
tff(fact_2106_divide__le__0__1__iff,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(one_one_rat,A)),zero_zero_rat))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).
% divide_le_0_1_iff
tff(fact_2107_zero__less__divide__1__iff,axiom,
! [A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).
% zero_less_divide_1_iff
tff(fact_2108_zero__less__divide__1__iff,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),divide_divide_rat(one_one_rat,A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).
% zero_less_divide_1_iff
tff(fact_2109_less__divide__eq__1__pos,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).
% less_divide_eq_1_pos
tff(fact_2110_less__divide__eq__1__pos,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),divide_divide_rat(B,A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).
% less_divide_eq_1_pos
tff(fact_2111_less__divide__eq__1__neg,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ).
% less_divide_eq_1_neg
tff(fact_2112_less__divide__eq__1__neg,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),divide_divide_rat(B,A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ).
% less_divide_eq_1_neg
tff(fact_2113_divide__less__eq__1__pos,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ).
% divide_less_eq_1_pos
tff(fact_2114_divide__less__eq__1__pos,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(B,A)),one_one_rat))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ).
% divide_less_eq_1_pos
tff(fact_2115_divide__less__eq__1__neg,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).
% divide_less_eq_1_neg
tff(fact_2116_divide__less__eq__1__neg,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(B,A)),one_one_rat))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).
% divide_less_eq_1_neg
tff(fact_2117_divide__less__0__1__iff,axiom,
! [A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A)),zero_zero_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).
% divide_less_0_1_iff
tff(fact_2118_divide__less__0__1__iff,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(one_one_rat,A)),zero_zero_rat))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).
% divide_less_0_1_iff
tff(fact_2119_eq__divide__eq__numeral1_I1_J,axiom,
! [A: complex,B: complex,W: num] :
( ( A = aa_complex_complex(divide1717551699836669952omplex(B),numera6690914467698888265omplex(W)) )
<=> ( ( ( numera6690914467698888265omplex(W) != zero_zero_complex )
=> ( aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W)) = B ) )
& ( ( numera6690914467698888265omplex(W) = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% eq_divide_eq_numeral1(1)
tff(fact_2120_eq__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W: num] :
( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W)) )
<=> ( ( ( numeral_numeral_real(W) != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W)) = B ) )
& ( ( numeral_numeral_real(W) = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral1(1)
tff(fact_2121_eq__divide__eq__numeral1_I1_J,axiom,
! [A: rat,B: rat,W: num] :
( ( A = divide_divide_rat(B,numeral_numeral_rat(W)) )
<=> ( ( ( numeral_numeral_rat(W) != zero_zero_rat )
=> ( aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W)) = B ) )
& ( ( numeral_numeral_rat(W) = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral1(1)
tff(fact_2122_divide__eq__eq__numeral1_I1_J,axiom,
! [B: complex,W: num,A: complex] :
( ( aa_complex_complex(divide1717551699836669952omplex(B),numera6690914467698888265omplex(W)) = A )
<=> ( ( ( numera6690914467698888265omplex(W) != zero_zero_complex )
=> ( B = aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W)) ) )
& ( ( numera6690914467698888265omplex(W) = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% divide_eq_eq_numeral1(1)
tff(fact_2123_divide__eq__eq__numeral1_I1_J,axiom,
! [B: real,W: num,A: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),numeral_numeral_real(W)) = A )
<=> ( ( ( numeral_numeral_real(W) != zero_zero_real )
=> ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W)) ) )
& ( ( numeral_numeral_real(W) = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral1(1)
tff(fact_2124_divide__eq__eq__numeral1_I1_J,axiom,
! [B: rat,W: num,A: rat] :
( ( divide_divide_rat(B,numeral_numeral_rat(W)) = A )
<=> ( ( ( numeral_numeral_rat(W) != zero_zero_rat )
=> ( B = aa_rat_rat(times_times_rat(A),numeral_numeral_rat(W)) ) )
& ( ( numeral_numeral_rat(W) = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral1(1)
tff(fact_2125_nonzero__divide__mult__cancel__right,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A) ) ) ).
% nonzero_divide_mult_cancel_right
tff(fact_2126_nonzero__divide__mult__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) ) ) ).
% nonzero_divide_mult_cancel_right
tff(fact_2127_nonzero__divide__mult__cancel__right,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( divide_divide_rat(B,aa_rat_rat(times_times_rat(A),B)) = divide_divide_rat(one_one_rat,A) ) ) ).
% nonzero_divide_mult_cancel_right
tff(fact_2128_nonzero__divide__mult__cancel__left,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),B) ) ) ).
% nonzero_divide_mult_cancel_left
tff(fact_2129_nonzero__divide__mult__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),B) ) ) ).
% nonzero_divide_mult_cancel_left
tff(fact_2130_nonzero__divide__mult__cancel__left,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( divide_divide_rat(A,aa_rat_rat(times_times_rat(A),B)) = divide_divide_rat(one_one_rat,B) ) ) ).
% nonzero_divide_mult_cancel_left
tff(fact_2131_div__mult__self4,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_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,B),C)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).
% div_mult_self4
tff(fact_2132_div__mult__self4,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).
% div_mult_self4
tff(fact_2133_div__mult__self3,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_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,C),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).
% div_mult_self3
tff(fact_2134_div__mult__self3,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).
% div_mult_self3
tff(fact_2135_div__mult__self2,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).
% div_mult_self2
tff(fact_2136_div__mult__self2,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).
% div_mult_self2
tff(fact_2137_div__mult__self1,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).
% div_mult_self1
tff(fact_2138_div__mult__self1,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).
% div_mult_self1
tff(fact_2139_power__eq__0__iff,axiom,
! [A: rat,N: nat] :
( ( aa_nat_rat(power_power_rat(A),N) = zero_zero_rat )
<=> ( ( A = zero_zero_rat )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).
% power_eq_0_iff
tff(fact_2140_power__eq__0__iff,axiom,
! [A: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) = zero_zero_nat )
<=> ( ( A = zero_zero_nat )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).
% power_eq_0_iff
tff(fact_2141_power__eq__0__iff,axiom,
! [A: real,N: nat] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) = zero_zero_real )
<=> ( ( A = zero_zero_real )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).
% power_eq_0_iff
tff(fact_2142_power__eq__0__iff,axiom,
! [A: complex,N: nat] :
( ( aa_nat_complex(power_power_complex(A),N) = zero_zero_complex )
<=> ( ( A = zero_zero_complex )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).
% power_eq_0_iff
tff(fact_2143_power__eq__0__iff,axiom,
! [A: int,N: nat] :
( ( aa_nat_int(power_power_int(A),N) = zero_zero_int )
<=> ( ( A = zero_zero_int )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).
% power_eq_0_iff
tff(fact_2144_Suc__pred,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))) = N ) ) ).
% Suc_pred
tff(fact_2145_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),M))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),N)) ) ) ).
% one_le_mult_iff
tff(fact_2146_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(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,N),K)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).
% mult_le_cancel2
tff(fact_2147_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).
% nat_mult_le_cancel_disj
tff(fact_2148_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M)),N) = M ) ) ).
% div_mult_self1_is_m
tff(fact_2149_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),N) = M ) ) ).
% div_mult_self_is_m
tff(fact_2150_le__divide__eq__1__pos,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).
% le_divide_eq_1_pos
tff(fact_2151_le__divide__eq__1__pos,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),divide_divide_rat(B,A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).
% le_divide_eq_1_pos
tff(fact_2152_le__divide__eq__1__neg,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ).
% le_divide_eq_1_neg
tff(fact_2153_le__divide__eq__1__neg,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),divide_divide_rat(B,A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).
% le_divide_eq_1_neg
tff(fact_2154_divide__le__eq__1__pos,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ).
% divide_le_eq_1_pos
tff(fact_2155_divide__le__eq__1__pos,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(B,A)),one_one_rat))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).
% divide_le_eq_1_pos
tff(fact_2156_divide__le__eq__1__neg,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).
% divide_le_eq_1_neg
tff(fact_2157_divide__le__eq__1__neg,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(B,A)),one_one_rat))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).
% divide_le_eq_1_neg
tff(fact_2158_power__strict__decreasing__iff,axiom,
! [B: real,M: nat,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),one_one_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).
% power_strict_decreasing_iff
tff(fact_2159_power__strict__decreasing__iff,axiom,
! [B: rat,M: nat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),one_one_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(B),M)),aa_nat_rat(power_power_rat(B),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).
% power_strict_decreasing_iff
tff(fact_2160_power__strict__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),one_one_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).
% power_strict_decreasing_iff
tff(fact_2161_power__strict__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),one_one_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),M)),aa_nat_int(power_power_int(B),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M)) ) ) ) ).
% power_strict_decreasing_iff
tff(fact_2162_zero__eq__power2,axiom,
! [A: rat] :
( ( aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_rat )
<=> ( A = zero_zero_rat ) ) ).
% zero_eq_power2
tff(fact_2163_zero__eq__power2,axiom,
! [A: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat )
<=> ( A = zero_zero_nat ) ) ).
% zero_eq_power2
tff(fact_2164_zero__eq__power2,axiom,
! [A: real] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real )
<=> ( A = zero_zero_real ) ) ).
% zero_eq_power2
tff(fact_2165_zero__eq__power2,axiom,
! [A: complex] :
( ( aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_complex )
<=> ( A = zero_zero_complex ) ) ).
% zero_eq_power2
tff(fact_2166_zero__eq__power2,axiom,
! [A: int] :
( ( aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_int )
<=> ( A = zero_zero_int ) ) ).
% zero_eq_power2
tff(fact_2167_power__mono__iff,axiom,
! [A: real,B: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ) ) ).
% power_mono_iff
tff(fact_2168_power__mono__iff,axiom,
! [A: rat,B: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ) ) ).
% power_mono_iff
tff(fact_2169_power__mono__iff,axiom,
! [A: nat,B: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ) ) ).
% power_mono_iff
tff(fact_2170_power__mono__iff,axiom,
! [A: int,B: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ) ) ).
% power_mono_iff
tff(fact_2171_Suc__diff__1,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = N ) ) ).
% Suc_diff_1
tff(fact_2172_bits__1__div__2,axiom,
aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).
% bits_1_div_2
tff(fact_2173_bits__1__div__2,axiom,
aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).
% bits_1_div_2
tff(fact_2174_one__div__two__eq__zero,axiom,
aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).
% one_div_two_eq_zero
tff(fact_2175_one__div__two__eq__zero,axiom,
aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).
% one_div_two_eq_zero
tff(fact_2176_power__decreasing__iff,axiom,
! [B: real,M: nat,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),one_one_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).
% power_decreasing_iff
tff(fact_2177_power__decreasing__iff,axiom,
! [B: rat,M: nat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),one_one_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),M)),aa_nat_rat(power_power_rat(B),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).
% power_decreasing_iff
tff(fact_2178_power__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),one_one_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).
% power_decreasing_iff
tff(fact_2179_power__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),one_one_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),M)),aa_nat_int(power_power_int(B),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)) ) ) ) ).
% power_decreasing_iff
tff(fact_2180_power2__eq__iff__nonneg,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
=> ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
<=> ( X = Y ) ) ) ) ).
% power2_eq_iff_nonneg
tff(fact_2181_power2__eq__iff__nonneg,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
=> ( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
<=> ( X = Y ) ) ) ) ).
% power2_eq_iff_nonneg
tff(fact_2182_power2__eq__iff__nonneg,axiom,
! [X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
<=> ( X = Y ) ) ) ) ).
% power2_eq_iff_nonneg
tff(fact_2183_power2__eq__iff__nonneg,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
=> ( ( aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
<=> ( X = Y ) ) ) ) ).
% power2_eq_iff_nonneg
tff(fact_2184_power2__less__eq__zero__iff,axiom,
! [A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real))
<=> ( A = zero_zero_real ) ) ).
% power2_less_eq_zero_iff
tff(fact_2185_power2__less__eq__zero__iff,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat))
<=> ( A = zero_zero_rat ) ) ).
% power2_less_eq_zero_iff
tff(fact_2186_power2__less__eq__zero__iff,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int))
<=> ( A = zero_zero_int ) ) ).
% power2_less_eq_zero_iff
tff(fact_2187_zero__less__power2,axiom,
! [A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
<=> ( A != zero_zero_real ) ) ).
% zero_less_power2
tff(fact_2188_zero__less__power2,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
<=> ( A != zero_zero_rat ) ) ).
% zero_less_power2
tff(fact_2189_zero__less__power2,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
<=> ( A != zero_zero_int ) ) ).
% zero_less_power2
tff(fact_2190_sum__power2__eq__zero__iff,axiom,
! [X: rat,Y: rat] :
( ( plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_rat )
<=> ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_power2_eq_zero_iff
tff(fact_2191_sum__power2__eq__zero__iff,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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real )
<=> ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_power2_eq_zero_iff
tff(fact_2192_sum__power2__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_int )
<=> ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_power2_eq_zero_iff
tff(fact_2193_not__mod__2__eq__1__eq__0,axiom,
! [A: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != one_one_nat )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).
% not_mod_2_eq_1_eq_0
tff(fact_2194_not__mod__2__eq__1__eq__0,axiom,
! [A: int] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int )
<=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).
% not_mod_2_eq_1_eq_0
tff(fact_2195_not__mod__2__eq__1__eq__0,axiom,
! [A: code_integer] :
( ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != one_one_Code_integer )
<=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).
% not_mod_2_eq_1_eq_0
tff(fact_2196_not__mod__2__eq__0__eq__1,axiom,
! [A: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != zero_zero_nat )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).
% not_mod_2_eq_0_eq_1
tff(fact_2197_not__mod__2__eq__0__eq__1,axiom,
! [A: int] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int )
<=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).
% not_mod_2_eq_0_eq_1
tff(fact_2198_not__mod__2__eq__0__eq__1,axiom,
! [A: code_integer] :
( ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != zero_z3403309356797280102nteger )
<=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).
% not_mod_2_eq_0_eq_1
tff(fact_2199_not__mod2__eq__Suc__0__eq__0,axiom,
! [N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != aa_nat_nat(suc,zero_zero_nat) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).
% not_mod2_eq_Suc_0_eq_0
tff(fact_2200_add__self__mod__2,axiom,
! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),M)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).
% add_self_mod_2
tff(fact_2201_mod2__gr__0,axiom,
! [M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).
% mod2_gr_0
tff(fact_2202_unset__bit__0,axiom,
! [A: int] : aa_int_int(bit_se4203085406695923979it_int(zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).
% unset_bit_0
tff(fact_2203_unset__bit__0,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).
% unset_bit_0
tff(fact_2204_VEBT_Osize_I4_J,axiom,
! [X21: bool,X222: bool] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,vEBT_Leaf(X21,X222)) = zero_zero_nat ).
% VEBT.size(4)
tff(fact_2205_zero__reorient,axiom,
! [X: complex] :
( ( zero_zero_complex = X )
<=> ( X = zero_zero_complex ) ) ).
% zero_reorient
tff(fact_2206_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
<=> ( X = zero_zero_real ) ) ).
% zero_reorient
tff(fact_2207_zero__reorient,axiom,
! [X: rat] :
( ( zero_zero_rat = X )
<=> ( X = zero_zero_rat ) ) ).
% zero_reorient
tff(fact_2208_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
<=> ( X = zero_zero_nat ) ) ).
% zero_reorient
tff(fact_2209_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
<=> ( X = zero_zero_int ) ) ).
% zero_reorient
tff(fact_2210_VEBT__internal_Onaive__member_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [A4: bool,B4: bool,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),X3)
=> ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Ux2)
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S3),X3) ) ) ).
% VEBT_internal.naive_member.cases
tff(fact_2211_invar__vebt_Ointros_I1_J,axiom,
! [A: bool,B: bool] : pp(aa_nat_bool(vEBT_invar_vebt(vEBT_Leaf(A,B)),aa_nat_nat(suc,zero_zero_nat))) ).
% invar_vebt.intros(1)
tff(fact_2212_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_rat(power_power_rat(zero_zero_rat),N) = one_one_rat ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_rat(power_power_rat(zero_zero_rat),N) = zero_zero_rat ) ) ) ).
% power_0_left
tff(fact_2213_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = zero_zero_nat ) ) ) ).
% power_0_left
tff(fact_2214_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),N) = one_one_real ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),N) = zero_zero_real ) ) ) ).
% power_0_left
tff(fact_2215_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_complex(power_power_complex(zero_zero_complex),N) = one_one_complex ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_complex(power_power_complex(zero_zero_complex),N) = zero_zero_complex ) ) ) ).
% power_0_left
tff(fact_2216_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_int(power_power_int(zero_zero_int),N) = one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_int(power_power_int(zero_zero_int),N) = zero_zero_int ) ) ) ).
% power_0_left
tff(fact_2217_zero__power,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_rat(power_power_rat(zero_zero_rat),N) = zero_zero_rat ) ) ).
% zero_power
tff(fact_2218_zero__power,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = zero_zero_nat ) ) ).
% zero_power
tff(fact_2219_zero__power,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),N) = zero_zero_real ) ) ).
% zero_power
tff(fact_2220_zero__power,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_complex(power_power_complex(zero_zero_complex),N) = zero_zero_complex ) ) ).
% zero_power
tff(fact_2221_zero__power,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_int(power_power_int(zero_zero_int),N) = zero_zero_int ) ) ).
% zero_power
tff(fact_2222_vebt__member_Osimps_I1_J,axiom,
! [A: bool,B: bool,X: nat] :
( pp(aa_nat_bool(vEBT_vebt_member(vEBT_Leaf(A,B)),X))
<=> ( ( ( X = zero_zero_nat )
=> pp(A) )
& ( ( X != zero_zero_nat )
=> ( ( ( X = one_one_nat )
=> pp(B) )
& ( X = one_one_nat ) ) ) ) ) ).
% vebt_member.simps(1)
tff(fact_2223_VEBT_Oexhaust,axiom,
! [Y: vEBT_VEBT] :
( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] : Y != vEBT_Node(X112,X122,X132,X142)
=> ~ ! [X212: bool,X223: bool] : Y != vEBT_Leaf(X212,X223) ) ).
% VEBT.exhaust
tff(fact_2224_VEBT_Odistinct_I1_J,axiom,
! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: bool,X222: bool] : vEBT_Node(X11,X12,X13,X14) != vEBT_Leaf(X21,X222) ).
% VEBT.distinct(1)
tff(fact_2225_VEBT__internal_Ovalid_H_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [Uu2: bool,Uv2: bool,D2: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),D2)
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList2,Summary2),Deg3) ) ).
% VEBT_internal.valid'.cases
tff(fact_2226_vebt__buildup_Osimps_I2_J,axiom,
vEBT_vebt_buildup(aa_nat_nat(suc,zero_zero_nat)) = vEBT_Leaf(fFalse,fFalse) ).
% vebt_buildup.simps(2)
tff(fact_2227_vebt__insert_Osimps_I1_J,axiom,
! [X: nat,A: bool,B: bool] :
( ( ( X = zero_zero_nat )
=> ( vEBT_vebt_insert(vEBT_Leaf(A,B),X) = vEBT_Leaf(fTrue,B) ) )
& ( ( X != zero_zero_nat )
=> ( ( ( X = one_one_nat )
=> ( vEBT_vebt_insert(vEBT_Leaf(A,B),X) = vEBT_Leaf(A,fTrue) ) )
& ( ( X != one_one_nat )
=> ( vEBT_vebt_insert(vEBT_Leaf(A,B),X) = vEBT_Leaf(A,B) ) ) ) ) ) ).
% vebt_insert.simps(1)
tff(fact_2228_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
! [A: bool,B: bool,X: nat] :
( pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_Leaf(A,B)),X))
<=> ( ( ( X = zero_zero_nat )
=> pp(A) )
& ( ( X != zero_zero_nat )
=> ( ( ( X = one_one_nat )
=> pp(B) )
& ( X = one_one_nat ) ) ) ) ) ).
% VEBT_internal.naive_member.simps(1)
tff(fact_2229_le__numeral__extra_I3_J,axiom,
pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),zero_zero_real)) ).
% le_numeral_extra(3)
tff(fact_2230_le__numeral__extra_I3_J,axiom,
pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),zero_zero_rat)) ).
% le_numeral_extra(3)
tff(fact_2231_le__numeral__extra_I3_J,axiom,
pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),zero_zero_nat)) ).
% le_numeral_extra(3)
tff(fact_2232_le__numeral__extra_I3_J,axiom,
pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),zero_zero_int)) ).
% le_numeral_extra(3)
tff(fact_2233_zero__le,axiom,
! [X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X)) ).
% zero_le
tff(fact_2234_field__lbound__gt__zero,axiom,
! [D1: real,D22: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D1))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D22))
=> ? [E2: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,E2),D1))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,E2),D22)) ) ) ) ).
% field_lbound_gt_zero
tff(fact_2235_field__lbound__gt__zero,axiom,
! [D1: rat,D22: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),D1))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),D22))
=> ? [E2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E2))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,E2),D1))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,E2),D22)) ) ) ) ).
% field_lbound_gt_zero
tff(fact_2236_less__numeral__extra_I3_J,axiom,
~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),zero_zero_real)) ).
% less_numeral_extra(3)
tff(fact_2237_less__numeral__extra_I3_J,axiom,
~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),zero_zero_rat)) ).
% less_numeral_extra(3)
tff(fact_2238_less__numeral__extra_I3_J,axiom,
~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),zero_zero_nat)) ).
% less_numeral_extra(3)
tff(fact_2239_less__numeral__extra_I3_J,axiom,
~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),zero_zero_int)) ).
% less_numeral_extra(3)
tff(fact_2240_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).
% gr_zeroI
tff(fact_2241_not__less__zero,axiom,
! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).
% not_less_zero
tff(fact_2242_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
tff(fact_2243_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
<=> ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
tff(fact_2244_zero__neq__numeral,axiom,
! [N: num] : zero_zero_complex != numera6690914467698888265omplex(N) ).
% zero_neq_numeral
tff(fact_2245_zero__neq__numeral,axiom,
! [N: num] : zero_zero_real != numeral_numeral_real(N) ).
% zero_neq_numeral
tff(fact_2246_zero__neq__numeral,axiom,
! [N: num] : zero_zero_rat != numeral_numeral_rat(N) ).
% zero_neq_numeral
tff(fact_2247_zero__neq__numeral,axiom,
! [N: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,N) ).
% zero_neq_numeral
tff(fact_2248_zero__neq__numeral,axiom,
! [N: num] : zero_zero_int != aa_num_int(numeral_numeral_int,N) ).
% zero_neq_numeral
tff(fact_2249_mult__right__cancel,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(times_times_complex(B),C) )
<=> ( A = B ) ) ) ).
% mult_right_cancel
tff(fact_2250_mult__right__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C) )
<=> ( A = B ) ) ) ).
% mult_right_cancel
tff(fact_2251_mult__right__cancel,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( aa_rat_rat(times_times_rat(A),C) = aa_rat_rat(times_times_rat(B),C) )
<=> ( A = B ) ) ) ).
% mult_right_cancel
tff(fact_2252_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
<=> ( A = B ) ) ) ).
% mult_right_cancel
tff(fact_2253_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
<=> ( A = B ) ) ) ).
% mult_right_cancel
tff(fact_2254_mult__left__cancel,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( aa_complex_complex(times_times_complex(C),A) = aa_complex_complex(times_times_complex(C),B) )
<=> ( A = B ) ) ) ).
% mult_left_cancel
tff(fact_2255_mult__left__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) )
<=> ( A = B ) ) ) ).
% mult_left_cancel
tff(fact_2256_mult__left__cancel,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( aa_rat_rat(times_times_rat(C),A) = aa_rat_rat(times_times_rat(C),B) )
<=> ( A = B ) ) ) ).
% mult_left_cancel
tff(fact_2257_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
<=> ( A = B ) ) ) ).
% mult_left_cancel
tff(fact_2258_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
<=> ( A = B ) ) ) ).
% mult_left_cancel
tff(fact_2259_no__zero__divisors,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( B != zero_zero_complex )
=> ( aa_complex_complex(times_times_complex(A),B) != zero_zero_complex ) ) ) ).
% no_zero_divisors
tff(fact_2260_no__zero__divisors,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) != zero_zero_real ) ) ) ).
% no_zero_divisors
tff(fact_2261_no__zero__divisors,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( B != zero_zero_rat )
=> ( aa_rat_rat(times_times_rat(A),B) != zero_zero_rat ) ) ) ).
% no_zero_divisors
tff(fact_2262_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) != zero_zero_nat ) ) ) ).
% no_zero_divisors
tff(fact_2263_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) != zero_zero_int ) ) ) ).
% no_zero_divisors
tff(fact_2264_divisors__zero,axiom,
! [A: complex,B: complex] :
( ( aa_complex_complex(times_times_complex(A),B) = zero_zero_complex )
=> ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% divisors_zero
tff(fact_2265_divisors__zero,axiom,
! [A: real,B: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) = zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
tff(fact_2266_divisors__zero,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(times_times_rat(A),B) = zero_zero_rat )
=> ( ( A = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% divisors_zero
tff(fact_2267_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
tff(fact_2268_divisors__zero,axiom,
! [A: int,B: int] :
( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
tff(fact_2269_mult__not__zero,axiom,
! [A: complex,B: complex] :
( ( aa_complex_complex(times_times_complex(A),B) != zero_zero_complex )
=> ( ( A != zero_zero_complex )
& ( B != zero_zero_complex ) ) ) ).
% mult_not_zero
tff(fact_2270_mult__not__zero,axiom,
! [A: real,B: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) != zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
tff(fact_2271_mult__not__zero,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(times_times_rat(A),B) != zero_zero_rat )
=> ( ( A != zero_zero_rat )
& ( B != zero_zero_rat ) ) ) ).
% mult_not_zero
tff(fact_2272_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) != zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
tff(fact_2273_mult__not__zero,axiom,
! [A: int,B: int] :
( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) != zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
tff(fact_2274_zero__neq__one,axiom,
zero_zero_complex != one_one_complex ).
% zero_neq_one
tff(fact_2275_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
tff(fact_2276_zero__neq__one,axiom,
zero_zero_rat != one_one_rat ).
% zero_neq_one
tff(fact_2277_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
tff(fact_2278_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
tff(fact_2279_add_Ogroup__left__neutral,axiom,
! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).
% add.group_left_neutral
tff(fact_2280_add_Ogroup__left__neutral,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).
% add.group_left_neutral
tff(fact_2281_add_Ogroup__left__neutral,axiom,
! [A: rat] : plus_plus_rat(zero_zero_rat,A) = A ).
% add.group_left_neutral
tff(fact_2282_add_Ogroup__left__neutral,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).
% add.group_left_neutral
tff(fact_2283_add_Ocomm__neutral,axiom,
! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).
% add.comm_neutral
tff(fact_2284_add_Ocomm__neutral,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ).
% add.comm_neutral
tff(fact_2285_add_Ocomm__neutral,axiom,
! [A: rat] : plus_plus_rat(A,zero_zero_rat) = A ).
% add.comm_neutral
tff(fact_2286_add_Ocomm__neutral,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ).
% add.comm_neutral
tff(fact_2287_add_Ocomm__neutral,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).
% add.comm_neutral
tff(fact_2288_comm__monoid__add__class_Oadd__0,axiom,
! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).
% comm_monoid_add_class.add_0
tff(fact_2289_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).
% comm_monoid_add_class.add_0
tff(fact_2290_comm__monoid__add__class_Oadd__0,axiom,
! [A: rat] : plus_plus_rat(zero_zero_rat,A) = A ).
% comm_monoid_add_class.add_0
tff(fact_2291_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A) = A ).
% comm_monoid_add_class.add_0
tff(fact_2292_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).
% comm_monoid_add_class.add_0
tff(fact_2293_verit__sum__simplify,axiom,
! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).
% verit_sum_simplify
tff(fact_2294_verit__sum__simplify,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),zero_zero_real) = A ).
% verit_sum_simplify
tff(fact_2295_verit__sum__simplify,axiom,
! [A: rat] : plus_plus_rat(A,zero_zero_rat) = A ).
% verit_sum_simplify
tff(fact_2296_verit__sum__simplify,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ).
% verit_sum_simplify
tff(fact_2297_verit__sum__simplify,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).
% verit_sum_simplify
tff(fact_2298_eq__iff__diff__eq__0,axiom,
! [A: complex,B: complex] :
( ( A = B )
<=> ( aa_complex_complex(minus_minus_complex(A),B) = zero_zero_complex ) ) ).
% eq_iff_diff_eq_0
tff(fact_2299_eq__iff__diff__eq__0,axiom,
! [A: real,B: real] :
( ( A = B )
<=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = zero_zero_real ) ) ).
% eq_iff_diff_eq_0
tff(fact_2300_eq__iff__diff__eq__0,axiom,
! [A: rat,B: rat] :
( ( A = B )
<=> ( minus_minus_rat(A,B) = zero_zero_rat ) ) ).
% eq_iff_diff_eq_0
tff(fact_2301_eq__iff__diff__eq__0,axiom,
! [A: int,B: int] :
( ( A = B )
<=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = zero_zero_int ) ) ).
% eq_iff_diff_eq_0
tff(fact_2302_power__not__zero,axiom,
! [A: rat,N: nat] :
( ( A != zero_zero_rat )
=> ( aa_nat_rat(power_power_rat(A),N) != zero_zero_rat ) ) ).
% power_not_zero
tff(fact_2303_power__not__zero,axiom,
! [A: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) != zero_zero_nat ) ) ).
% power_not_zero
tff(fact_2304_power__not__zero,axiom,
! [A: real,N: nat] :
( ( A != zero_zero_real )
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) != zero_zero_real ) ) ).
% power_not_zero
tff(fact_2305_power__not__zero,axiom,
! [A: complex,N: nat] :
( ( A != zero_zero_complex )
=> ( aa_nat_complex(power_power_complex(A),N) != zero_zero_complex ) ) ).
% power_not_zero
tff(fact_2306_power__not__zero,axiom,
! [A: int,N: nat] :
( ( A != zero_zero_int )
=> ( aa_nat_int(power_power_int(A),N) != zero_zero_int ) ) ).
% power_not_zero
tff(fact_2307_num_Osize_I4_J,axiom,
aa_num_nat(size_size_num,one) = zero_zero_nat ).
% num.size(4)
tff(fact_2308_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] : N = aa_nat_nat(suc,M3) ) ).
% not0_implies_Suc
tff(fact_2309_Zero__not__Suc,axiom,
! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).
% Zero_not_Suc
tff(fact_2310_Zero__neq__Suc,axiom,
! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).
% Zero_neq_Suc
tff(fact_2311_Suc__neq__Zero,axiom,
! [M: nat] : aa_nat_nat(suc,M) != zero_zero_nat ).
% Suc_neq_Zero
tff(fact_2312_zero__induct,axiom,
! [P: fun_nat_bool,K: nat] :
( pp(aa_nat_bool(P,K))
=> ( ! [N2: nat] :
( pp(aa_nat_bool(P,aa_nat_nat(suc,N2)))
=> pp(aa_nat_bool(P,N2)) )
=> pp(aa_nat_bool(P,zero_zero_nat)) ) ) ).
% zero_induct
tff(fact_2313_diff__induct,axiom,
! [P: fun_nat_fun_nat_bool,M: nat,N: nat] :
( ! [X3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,X3),zero_zero_nat))
=> ( ! [Y2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,zero_zero_nat),aa_nat_nat(suc,Y2)))
=> ( ! [X3: nat,Y2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,X3),Y2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,aa_nat_nat(suc,X3)),aa_nat_nat(suc,Y2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M),N)) ) ) ) ).
% diff_induct
tff(fact_2314_nat__induct,axiom,
! [P: fun_nat_bool,N: nat] :
( pp(aa_nat_bool(P,zero_zero_nat))
=> ( ! [N2: nat] :
( pp(aa_nat_bool(P,N2))
=> pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) )
=> pp(aa_nat_bool(P,N)) ) ) ).
% nat_induct
tff(fact_2315_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] : Y != aa_nat_nat(suc,Nat3) ) ).
% old.nat.exhaust
tff(fact_2316_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat = aa_nat_nat(suc,X23) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
tff(fact_2317_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).
% old.nat.distinct(1)
tff(fact_2318_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).
% old.nat.distinct(2)
tff(fact_2319_nat_Odistinct_I1_J,axiom,
! [X23: nat] : zero_zero_nat != aa_nat_nat(suc,X23) ).
% nat.distinct(1)
tff(fact_2320_vebt__buildup_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ( ( X != aa_nat_nat(suc,zero_zero_nat) )
=> ~ ! [Va3: nat] : X != aa_nat_nat(suc,aa_nat_nat(suc,Va3)) ) ) ).
% vebt_buildup.cases
tff(fact_2321_VEBT__internal_OminNull_Osimps_I3_J,axiom,
! [Uu: bool] : ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Leaf(Uu,fTrue))) ).
% VEBT_internal.minNull.simps(3)
tff(fact_2322_VEBT__internal_OminNull_Osimps_I2_J,axiom,
! [Uv: bool] : ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Leaf(fTrue,Uv))) ).
% VEBT_internal.minNull.simps(2)
tff(fact_2323_VEBT__internal_OminNull_Osimps_I1_J,axiom,
pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,vEBT_Leaf(fFalse,fFalse))) ).
% VEBT_internal.minNull.simps(1)
tff(fact_2324_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat)) ).
% bot_nat_0.extremum_strict
tff(fact_2325_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).
% gr0I
tff(fact_2326_not__gr0,axiom,
! [N: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
<=> ( N = zero_zero_nat ) ) ).
% not_gr0
tff(fact_2327_not__less0,axiom,
! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).
% not_less0
tff(fact_2328_less__zeroE,axiom,
! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).
% less_zeroE
tff(fact_2329_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
tff(fact_2330_infinite__descent0,axiom,
! [P: fun_nat_bool,N: nat] :
( pp(aa_nat_bool(P,zero_zero_nat))
=> ( ! [N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
=> ( ~ pp(aa_nat_bool(P,N2))
=> ? [M2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N2))
& ~ pp(aa_nat_bool(P,M2)) ) ) )
=> pp(aa_nat_bool(P,N)) ) ) ).
% infinite_descent0
tff(fact_2331_le__0__eq,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),zero_zero_nat))
<=> ( N = zero_zero_nat ) ) ).
% le_0_eq
tff(fact_2332_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
tff(fact_2333_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
<=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
tff(fact_2334_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),N)) ).
% less_eq_nat.simps(1)
tff(fact_2335_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
tff(fact_2336_plus__nat_Oadd__0,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),N) = N ).
% plus_nat.add_0
tff(fact_2337_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat )
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M) = zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
tff(fact_2338_minus__nat_Odiff__0,axiom,
! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),zero_zero_nat) = M ).
% minus_nat.diff_0
tff(fact_2339_mult__0,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),N) = zero_zero_nat ).
% mult_0
tff(fact_2340_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: 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),N) )
<=> ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
tff(fact_2341_power__eq__iff__eq__base,axiom,
! [N: nat,A: real,B: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
=> ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N) )
<=> ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
tff(fact_2342_power__eq__iff__eq__base,axiom,
! [N: nat,A: rat,B: rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
=> ( ( aa_nat_rat(power_power_rat(A),N) = aa_nat_rat(power_power_rat(B),N) )
<=> ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
tff(fact_2343_power__eq__iff__eq__base,axiom,
! [N: nat,A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N) )
<=> ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
tff(fact_2344_power__eq__iff__eq__base,axiom,
! [N: nat,A: int,B: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
=> ( ( aa_nat_int(power_power_int(A),N) = aa_nat_int(power_power_int(B),N) )
<=> ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
tff(fact_2345_power__eq__imp__eq__base,axiom,
! [A: real,N: nat,B: real] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N) )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
tff(fact_2346_power__eq__imp__eq__base,axiom,
! [A: rat,N: nat,B: rat] :
( ( aa_nat_rat(power_power_rat(A),N) = aa_nat_rat(power_power_rat(B),N) )
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
tff(fact_2347_power__eq__imp__eq__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
tff(fact_2348_power__eq__imp__eq__base,axiom,
! [A: int,N: nat,B: int] :
( ( aa_nat_int(power_power_int(A),N) = aa_nat_int(power_power_int(B),N) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
tff(fact_2349_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I1_J,axiom,
! [A: bool,B: bool,X: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t(vEBT_Leaf(A,B)),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),zero_zero_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(1)
tff(fact_2350_lambda__zero,axiom,
aTP_Lamm_ch = times_times_complex(zero_zero_complex) ).
% lambda_zero
tff(fact_2351_lambda__zero,axiom,
aTP_Lamm_ci = aa_rea7340605660300375871l_real(times_times_real,zero_zero_real) ).
% lambda_zero
tff(fact_2352_lambda__zero,axiom,
aTP_Lamm_cj = times_times_rat(zero_zero_rat) ).
% lambda_zero
tff(fact_2353_lambda__zero,axiom,
aTP_Lamm_ck = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).
% lambda_zero
tff(fact_2354_lambda__zero,axiom,
aTP_Lamm_cl = aa_int_fun_int_int(times_times_int,zero_zero_int) ).
% lambda_zero
tff(fact_2355_sum_Ofinite__Collect__op,axiom,
! [I5: set_real,X: fun_real_complex,Y: fun_real_complex] :
( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_cm(I5),X))))
=> ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aTP_Lamm_cm(I5),Y))))
=> pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun3813300650643916655l_bool(aa_fun5520078544946397361l_bool(aTP_Lamm_cn(I5),X),Y)))) ) ) ).
% sum.finite_Collect_op
tff(fact_2356_sum_Ofinite__Collect__op,axiom,
! [I5: set_nat,X: fun_nat_complex,Y: fun_nat_complex] :
( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_co(I5),X))))
=> ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aTP_Lamm_co(I5),Y))))
=> pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun8004580481347240231t_bool(aa_fun5430245816583273541t_bool(aTP_Lamm_cp(I5),X),Y)))) ) ) ).
% sum.finite_Collect_op
tff(fact_2357_sum_Ofinite__Collect__op,axiom,
! [I5: set_int,X: fun_int_complex,Y: fun_int_complex] :
( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_cq(I5),X))))
=> ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aTP_Lamm_cq(I5),Y))))
=> pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun8146511558608327279t_bool(aa_fun3850298374753145905t_bool(aTP_Lamm_cr(I5),X),Y)))) ) ) ).
% sum.finite_Collect_op
tff(fact_2358_sum_Ofinite__Collect__op,axiom,
! [I5: set_complex,X: fun_complex_complex,Y: fun_complex_complex] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_cs(I5),X))))
=> ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aTP_Lamm_cs(I5),Y))))
=> pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun9073502203726719595x_bool(aa_fun6641798613628721963x_bool(aTP_Lamm_ct(I5),X),Y)))) ) ) ).
% sum.finite_Collect_op
tff(fact_2359_sum_Ofinite__Collect__op,axiom,
! [I5: set_real,X: fun_real_real,Y: fun_real_real] :
( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_cu(I5),X))))
=> ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aTP_Lamm_cu(I5),Y))))
=> pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_cv(I5),X),Y)))) ) ) ).
% sum.finite_Collect_op
tff(fact_2360_sum_Ofinite__Collect__op,axiom,
! [I5: set_nat,X: fun_nat_real,Y: fun_nat_real] :
( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_cw(I5),X))))
=> ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aTP_Lamm_cw(I5),Y))))
=> pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_cx(I5),X),Y)))) ) ) ).
% sum.finite_Collect_op
tff(fact_2361_sum_Ofinite__Collect__op,axiom,
! [I5: set_int,X: fun_int_real,Y: fun_int_real] :
( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_cy(I5),X))))
=> ( pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aTP_Lamm_cy(I5),Y))))
=> pp(aa_set_int_bool(finite_finite_int,collect_int(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_cz(I5),X),Y)))) ) ) ).
% sum.finite_Collect_op
tff(fact_2362_sum_Ofinite__Collect__op,axiom,
! [I5: set_complex,X: fun_complex_real,Y: fun_complex_real] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_da(I5),X))))
=> ( pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aTP_Lamm_da(I5),Y))))
=> pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_fun2898354424061222765x_bool(aa_fun2094807616088678831x_bool(aTP_Lamm_db(I5),X),Y)))) ) ) ).
% sum.finite_Collect_op
tff(fact_2363_sum_Ofinite__Collect__op,axiom,
! [I5: set_real,X: fun_real_rat,Y: fun_real_rat] :
( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_dc(I5),X))))
=> ( pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aTP_Lamm_dc(I5),Y))))
=> pp(aa_set_real_bool(finite_finite_real,collect_real(aa_fun8987621170725249221l_bool(aa_fun8530242357055030877l_bool(aTP_Lamm_dd(I5),X),Y)))) ) ) ).
% sum.finite_Collect_op
tff(fact_2364_sum_Ofinite__Collect__op,axiom,
! [I5: set_nat,X: fun_nat_rat,Y: fun_nat_rat] :
( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aTP_Lamm_de(I5),X))))
=> ( pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aTP_Lamm_de(I5),Y))))
=> pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_fun5637262893798792573t_bool(aa_fun2107527232086395633t_bool(aTP_Lamm_df(I5),X),Y)))) ) ) ).
% sum.finite_Collect_op
tff(fact_2365_power__strict__mono,axiom,
! [A: real,B: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N))) ) ) ) ).
% power_strict_mono
tff(fact_2366_power__strict__mono,axiom,
! [A: rat,B: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N))) ) ) ) ).
% power_strict_mono
tff(fact_2367_power__strict__mono,axiom,
! [A: nat,B: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N))) ) ) ) ).
% power_strict_mono
tff(fact_2368_power__strict__mono,axiom,
! [A: int,B: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N))) ) ) ) ).
% power_strict_mono
tff(fact_2369_vebt__pred_Osimps_I2_J,axiom,
! [A: bool,Uw: bool] :
( ( pp(A)
=> ( vEBT_vebt_pred(vEBT_Leaf(A,Uw),aa_nat_nat(suc,zero_zero_nat)) = some_nat(zero_zero_nat) ) )
& ( ~ pp(A)
=> ( vEBT_vebt_pred(vEBT_Leaf(A,Uw),aa_nat_nat(suc,zero_zero_nat)) = none_nat ) ) ) ).
% vebt_pred.simps(2)
tff(fact_2370_vebt__mint_Osimps_I1_J,axiom,
! [A: bool,B: bool] :
( ( pp(A)
=> ( vEBT_vebt_mint(vEBT_Leaf(A,B)) = some_nat(zero_zero_nat) ) )
& ( ~ pp(A)
=> ( ( pp(B)
=> ( vEBT_vebt_mint(vEBT_Leaf(A,B)) = some_nat(one_one_nat) ) )
& ( ~ pp(B)
=> ( vEBT_vebt_mint(vEBT_Leaf(A,B)) = none_nat ) ) ) ) ) ).
% vebt_mint.simps(1)
tff(fact_2371_vebt__succ_Osimps_I1_J,axiom,
! [B: bool,Uu: bool] :
( ( pp(B)
=> ( vEBT_vebt_succ(vEBT_Leaf(Uu,B),zero_zero_nat) = some_nat(one_one_nat) ) )
& ( ~ pp(B)
=> ( vEBT_vebt_succ(vEBT_Leaf(Uu,B),zero_zero_nat) = none_nat ) ) ) ).
% vebt_succ.simps(1)
tff(fact_2372_vebt__maxt_Osimps_I1_J,axiom,
! [B: bool,A: bool] :
( ( pp(B)
=> ( vEBT_vebt_maxt(vEBT_Leaf(A,B)) = some_nat(one_one_nat) ) )
& ( ~ pp(B)
=> ( ( pp(A)
=> ( vEBT_vebt_maxt(vEBT_Leaf(A,B)) = some_nat(zero_zero_nat) ) )
& ( ~ pp(A)
=> ( vEBT_vebt_maxt(vEBT_Leaf(A,B)) = none_nat ) ) ) ) ) ).
% vebt_maxt.simps(1)
tff(fact_2373_not__numeral__le__zero,axiom,
! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(N)),zero_zero_real)) ).
% not_numeral_le_zero
tff(fact_2374_not__numeral__le__zero,axiom,
! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,numeral_numeral_rat(N)),zero_zero_rat)) ).
% not_numeral_le_zero
tff(fact_2375_not__numeral__le__zero,axiom,
! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N)),zero_zero_nat)) ).
% not_numeral_le_zero
tff(fact_2376_not__numeral__le__zero,axiom,
! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),zero_zero_int)) ).
% not_numeral_le_zero
tff(fact_2377_zero__le__numeral,axiom,
! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),numeral_numeral_real(N))) ).
% zero_le_numeral
tff(fact_2378_zero__le__numeral,axiom,
! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),numeral_numeral_rat(N))) ).
% zero_le_numeral
tff(fact_2379_zero__le__numeral,axiom,
! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,N))) ).
% zero_le_numeral
tff(fact_2380_zero__le__numeral,axiom,
! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_num_int(numeral_numeral_int,N))) ).
% zero_le_numeral
tff(fact_2381_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2382_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2383_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2384_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2385_zero__le__mult__iff,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) )
| ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) ) ) ) ).
% zero_le_mult_iff
tff(fact_2386_zero__le__mult__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) )
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) ) ) ) ).
% zero_le_mult_iff
tff(fact_2387_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) )
| ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) ) ) ) ).
% zero_le_mult_iff
tff(fact_2388_mult__nonneg__nonpos2,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real)) ) ) ).
% mult_nonneg_nonpos2
tff(fact_2389_mult__nonneg__nonpos2,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(B),A)),zero_zero_rat)) ) ) ).
% mult_nonneg_nonpos2
tff(fact_2390_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat)) ) ) ).
% mult_nonneg_nonpos2
tff(fact_2391_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int)) ) ) ).
% mult_nonneg_nonpos2
tff(fact_2392_mult__nonpos__nonneg,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ) ).
% mult_nonpos_nonneg
tff(fact_2393_mult__nonpos__nonneg,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)) ) ) ).
% mult_nonpos_nonneg
tff(fact_2394_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat)) ) ) ).
% mult_nonpos_nonneg
tff(fact_2395_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)) ) ) ).
% mult_nonpos_nonneg
tff(fact_2396_mult__nonneg__nonpos,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ) ).
% mult_nonneg_nonpos
tff(fact_2397_mult__nonneg__nonpos,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)) ) ) ).
% mult_nonneg_nonpos
tff(fact_2398_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat)) ) ) ).
% mult_nonneg_nonpos
tff(fact_2399_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)) ) ) ).
% mult_nonneg_nonpos
tff(fact_2400_mult__nonneg__nonneg,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ) ).
% mult_nonneg_nonneg
tff(fact_2401_mult__nonneg__nonneg,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))) ) ) ).
% mult_nonneg_nonneg
tff(fact_2402_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))) ) ) ).
% mult_nonneg_nonneg
tff(fact_2403_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ) ) ).
% mult_nonneg_nonneg
tff(fact_2404_split__mult__neg__le,axiom,
! [A: real,B: real] :
( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) )
| ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ).
% split_mult_neg_le
tff(fact_2405_split__mult__neg__le,axiom,
! [A: rat,B: rat] :
( ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) )
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)) ) ).
% split_mult_neg_le
tff(fact_2406_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat)) )
| ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B)) ) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat)) ) ).
% split_mult_neg_le
tff(fact_2407_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) )
| ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) ) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)) ) ).
% split_mult_neg_le
tff(fact_2408_mult__le__0__iff,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) )
| ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) ) ) ).
% mult_le_0_iff
tff(fact_2409_mult__le__0__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) )
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ) ) ).
% mult_le_0_iff
tff(fact_2410_mult__le__0__iff,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) )
| ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) ) ) ) ).
% mult_le_0_iff
tff(fact_2411_mult__right__mono,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ) ).
% mult_right_mono
tff(fact_2412_mult__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))) ) ) ).
% mult_right_mono
tff(fact_2413_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))) ) ) ).
% mult_right_mono
tff(fact_2414_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ) ).
% mult_right_mono
tff(fact_2415_mult__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ) ).
% mult_right_mono_neg
tff(fact_2416_mult__right__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))) ) ) ).
% mult_right_mono_neg
tff(fact_2417_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ) ).
% mult_right_mono_neg
tff(fact_2418_mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).
% mult_left_mono
tff(fact_2419_mult__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).
% mult_left_mono
tff(fact_2420_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ).
% mult_left_mono
tff(fact_2421_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).
% mult_left_mono
tff(fact_2422_mult__nonpos__nonpos,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ) ).
% mult_nonpos_nonpos
tff(fact_2423_mult__nonpos__nonpos,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))) ) ) ).
% mult_nonpos_nonpos
tff(fact_2424_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ) ) ).
% mult_nonpos_nonpos
tff(fact_2425_mult__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).
% mult_left_mono_neg
tff(fact_2426_mult__left__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).
% mult_left_mono_neg
tff(fact_2427_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).
% mult_left_mono_neg
tff(fact_2428_split__mult__pos__le,axiom,
! [A: real,B: real] :
( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) )
| ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) ) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ).
% split_mult_pos_le
tff(fact_2429_split__mult__pos__le,axiom,
! [A: rat,B: rat] :
( ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) )
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) ) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))) ) ).
% split_mult_pos_le
tff(fact_2430_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B)) )
| ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) ) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ) ).
% split_mult_pos_le
tff(fact_2431_zero__le__square,axiom,
! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A))) ).
% zero_le_square
tff(fact_2432_zero__le__square,axiom,
! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),A))) ).
% zero_le_square
tff(fact_2433_zero__le__square,axiom,
! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),A))) ).
% zero_le_square
tff(fact_2434_mult__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).
% mult_mono'
tff(fact_2435_mult__mono_H,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).
% mult_mono'
tff(fact_2436_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).
% mult_mono'
tff(fact_2437_mult__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).
% mult_mono'
tff(fact_2438_mult__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).
% mult_mono
tff(fact_2439_mult__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).
% mult_mono
tff(fact_2440_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).
% mult_mono
tff(fact_2441_mult__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).
% mult_mono
tff(fact_2442_not__numeral__less__zero,axiom,
! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(N)),zero_zero_real)) ).
% not_numeral_less_zero
tff(fact_2443_not__numeral__less__zero,axiom,
! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,numeral_numeral_rat(N)),zero_zero_rat)) ).
% not_numeral_less_zero
tff(fact_2444_not__numeral__less__zero,axiom,
! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,N)),zero_zero_nat)) ).
% not_numeral_less_zero
tff(fact_2445_not__numeral__less__zero,axiom,
! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),zero_zero_int)) ).
% not_numeral_less_zero
tff(fact_2446_zero__less__numeral,axiom,
! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),numeral_numeral_real(N))) ).
% zero_less_numeral
tff(fact_2447_zero__less__numeral,axiom,
! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),numeral_numeral_rat(N))) ).
% zero_less_numeral
tff(fact_2448_zero__less__numeral,axiom,
! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,N))) ).
% zero_less_numeral
tff(fact_2449_zero__less__numeral,axiom,
! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_num_int(numeral_numeral_int,N))) ).
% zero_less_numeral
tff(fact_2450_not__one__le__zero,axiom,
~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),zero_zero_real)) ).
% not_one_le_zero
tff(fact_2451_not__one__le__zero,axiom,
~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),zero_zero_rat)) ).
% not_one_le_zero
tff(fact_2452_not__one__le__zero,axiom,
~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),zero_zero_nat)) ).
% not_one_le_zero
tff(fact_2453_not__one__le__zero,axiom,
~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),zero_zero_int)) ).
% not_one_le_zero
tff(fact_2454_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),one_one_real)) ).
% linordered_nonzero_semiring_class.zero_le_one
tff(fact_2455_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),one_one_rat)) ).
% linordered_nonzero_semiring_class.zero_le_one
tff(fact_2456_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),one_one_nat)) ).
% linordered_nonzero_semiring_class.zero_le_one
tff(fact_2457_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).
% linordered_nonzero_semiring_class.zero_le_one
tff(fact_2458_zero__less__one__class_Ozero__le__one,axiom,
pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),one_one_real)) ).
% zero_less_one_class.zero_le_one
tff(fact_2459_zero__less__one__class_Ozero__le__one,axiom,
pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),one_one_rat)) ).
% zero_less_one_class.zero_le_one
tff(fact_2460_zero__less__one__class_Ozero__le__one,axiom,
pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),one_one_nat)) ).
% zero_less_one_class.zero_le_one
tff(fact_2461_zero__less__one__class_Ozero__le__one,axiom,
pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).
% zero_less_one_class.zero_le_one
tff(fact_2462_add__nonpos__eq__0__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
=> ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y) = zero_zero_real )
<=> ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
tff(fact_2463_add__nonpos__eq__0__iff,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),zero_zero_rat))
=> ( ( plus_plus_rat(X,Y) = zero_zero_rat )
<=> ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
tff(fact_2464_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),zero_zero_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),zero_zero_nat))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
<=> ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
tff(fact_2465_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),zero_zero_int))
=> ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y) = zero_zero_int )
<=> ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
tff(fact_2466_add__nonneg__eq__0__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
=> ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y) = zero_zero_real )
<=> ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
tff(fact_2467_add__nonneg__eq__0__iff,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
=> ( ( plus_plus_rat(X,Y) = zero_zero_rat )
<=> ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
tff(fact_2468_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
<=> ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
tff(fact_2469_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
=> ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y) = zero_zero_int )
<=> ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
tff(fact_2470_add__nonpos__nonpos,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real)) ) ) ).
% add_nonpos_nonpos
tff(fact_2471_add__nonpos__nonpos,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(A,B)),zero_zero_rat)) ) ) ).
% add_nonpos_nonpos
tff(fact_2472_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat)) ) ) ).
% add_nonpos_nonpos
tff(fact_2473_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int)) ) ) ).
% add_nonpos_nonpos
tff(fact_2474_add__nonneg__nonneg,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).
% add_nonneg_nonneg
tff(fact_2475_add__nonneg__nonneg,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),plus_plus_rat(A,B))) ) ) ).
% add_nonneg_nonneg
tff(fact_2476_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).
% add_nonneg_nonneg
tff(fact_2477_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).
% add_nonneg_nonneg
tff(fact_2478_add__increasing2,axiom,
! [C: real,B: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).
% add_increasing2
tff(fact_2479_add__increasing2,axiom,
! [C: rat,B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),plus_plus_rat(A,C))) ) ) ).
% add_increasing2
tff(fact_2480_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).
% add_increasing2
tff(fact_2481_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).
% add_increasing2
tff(fact_2482_add__decreasing2,axiom,
! [C: real,A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B)) ) ) ).
% add_decreasing2
tff(fact_2483_add__decreasing2,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(A,C)),B)) ) ) ).
% add_decreasing2
tff(fact_2484_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),zero_zero_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B)) ) ) ).
% add_decreasing2
tff(fact_2485_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B)) ) ) ).
% add_decreasing2
tff(fact_2486_add__increasing,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).
% add_increasing
tff(fact_2487_add__increasing,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),plus_plus_rat(A,C))) ) ) ).
% add_increasing
tff(fact_2488_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).
% add_increasing
tff(fact_2489_add__increasing,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).
% add_increasing
tff(fact_2490_add__decreasing,axiom,
! [A: real,C: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),B))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B)) ) ) ).
% add_decreasing
tff(fact_2491_add__decreasing,axiom,
! [A: rat,C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(A,C)),B)) ) ) ).
% add_decreasing
tff(fact_2492_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B)) ) ) ).
% add_decreasing
tff(fact_2493_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B)) ) ) ).
% add_decreasing
tff(fact_2494_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2495_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2496_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2497_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2498_mult__less__cancel__right__disj,axiom,
! [A: real,C: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
| ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).
% mult_less_cancel_right_disj
tff(fact_2499_mult__less__cancel__right__disj,axiom,
! [A: rat,C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).
% mult_less_cancel_right_disj
tff(fact_2500_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) )
| ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).
% mult_less_cancel_right_disj
tff(fact_2501_mult__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ) ).
% mult_strict_right_mono
tff(fact_2502_mult__strict__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))) ) ) ).
% mult_strict_right_mono
tff(fact_2503_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))) ) ) ).
% mult_strict_right_mono
tff(fact_2504_mult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ) ).
% mult_strict_right_mono
tff(fact_2505_mult__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ) ).
% mult_strict_right_mono_neg
tff(fact_2506_mult__strict__right__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))) ) ) ).
% mult_strict_right_mono_neg
tff(fact_2507_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ) ).
% mult_strict_right_mono_neg
tff(fact_2508_mult__less__cancel__left__disj,axiom,
! [C: real,A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
| ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).
% mult_less_cancel_left_disj
tff(fact_2509_mult__less__cancel__left__disj,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).
% mult_less_cancel_left_disj
tff(fact_2510_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) )
| ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).
% mult_less_cancel_left_disj
tff(fact_2511_mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).
% mult_strict_left_mono
tff(fact_2512_mult__strict__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).
% mult_strict_left_mono
tff(fact_2513_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ).
% mult_strict_left_mono
tff(fact_2514_mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).
% mult_strict_left_mono
tff(fact_2515_mult__strict__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B))) ) ) ).
% mult_strict_left_mono_neg
tff(fact_2516_mult__strict__left__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))) ) ) ).
% mult_strict_left_mono_neg
tff(fact_2517_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ).
% mult_strict_left_mono_neg
tff(fact_2518_mult__less__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).
% mult_less_cancel_left_pos
tff(fact_2519_mult__less__cancel__left__pos,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).
% mult_less_cancel_left_pos
tff(fact_2520_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).
% mult_less_cancel_left_pos
tff(fact_2521_mult__less__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ).
% mult_less_cancel_left_neg
tff(fact_2522_mult__less__cancel__left__neg,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ).
% mult_less_cancel_left_neg
tff(fact_2523_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ).
% mult_less_cancel_left_neg
tff(fact_2524_zero__less__mult__pos2,axiom,
! [B: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ) ).
% zero_less_mult_pos2
tff(fact_2525_zero__less__mult__pos2,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(B),A)))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ) ).
% zero_less_mult_pos2
tff(fact_2526_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B)) ) ) ).
% zero_less_mult_pos2
tff(fact_2527_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,B),A)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ) ).
% zero_less_mult_pos2
tff(fact_2528_zero__less__mult__pos,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ) ).
% zero_less_mult_pos
tff(fact_2529_zero__less__mult__pos,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ) ).
% zero_less_mult_pos
tff(fact_2530_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B)) ) ) ).
% zero_less_mult_pos
tff(fact_2531_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ) ).
% zero_less_mult_pos
tff(fact_2532_zero__less__mult__iff,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) )
| ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real)) ) ) ) ).
% zero_less_mult_iff
tff(fact_2533_zero__less__mult__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) )
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat)) ) ) ) ).
% zero_less_mult_iff
tff(fact_2534_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) )
| ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int)) ) ) ) ).
% zero_less_mult_iff
tff(fact_2535_mult__pos__neg2,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A)),zero_zero_real)) ) ) ).
% mult_pos_neg2
tff(fact_2536_mult__pos__neg2,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(B),A)),zero_zero_rat)) ) ) ).
% mult_pos_neg2
tff(fact_2537_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),zero_zero_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat)) ) ) ).
% mult_pos_neg2
tff(fact_2538_mult__pos__neg2,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int)) ) ) ).
% mult_pos_neg2
tff(fact_2539_mult__pos__pos,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ) ).
% mult_pos_pos
tff(fact_2540_mult__pos__pos,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))) ) ) ).
% mult_pos_pos
tff(fact_2541_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))) ) ) ).
% mult_pos_pos
tff(fact_2542_mult__pos__pos,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ) ) ).
% mult_pos_pos
tff(fact_2543_mult__pos__neg,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ) ).
% mult_pos_neg
tff(fact_2544_mult__pos__neg,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)) ) ) ).
% mult_pos_neg
tff(fact_2545_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),zero_zero_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat)) ) ) ).
% mult_pos_neg
tff(fact_2546_mult__pos__neg,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)) ) ) ).
% mult_pos_neg
tff(fact_2547_mult__neg__pos,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real)) ) ) ).
% mult_neg_pos
tff(fact_2548_mult__neg__pos,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)) ) ) ).
% mult_neg_pos
tff(fact_2549_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat)) ) ) ).
% mult_neg_pos
tff(fact_2550_mult__neg__pos,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)) ) ) ).
% mult_neg_pos
tff(fact_2551_mult__less__0__iff,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),zero_zero_real))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real)) )
| ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ) ) ).
% mult_less_0_iff
tff(fact_2552_mult__less__0__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat)) )
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ) ) ).
% mult_less_0_iff
tff(fact_2553_mult__less__0__iff,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int)) )
| ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ) ) ).
% mult_less_0_iff
tff(fact_2554_not__square__less__zero,axiom,
! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A)),zero_zero_real)) ).
% not_square_less_zero
tff(fact_2555_not__square__less__zero,axiom,
! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),A)),zero_zero_rat)) ).
% not_square_less_zero
tff(fact_2556_not__square__less__zero,axiom,
! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),A)),zero_zero_int)) ).
% not_square_less_zero
tff(fact_2557_mult__neg__neg,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ) ) ).
% mult_neg_neg
tff(fact_2558_mult__neg__neg,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))) ) ) ).
% mult_neg_neg
tff(fact_2559_mult__neg__neg,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ) ) ).
% mult_neg_neg
tff(fact_2560_not__one__less__zero,axiom,
~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),zero_zero_real)) ).
% not_one_less_zero
tff(fact_2561_not__one__less__zero,axiom,
~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),zero_zero_rat)) ).
% not_one_less_zero
tff(fact_2562_not__one__less__zero,axiom,
~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),zero_zero_nat)) ).
% not_one_less_zero
tff(fact_2563_not__one__less__zero,axiom,
~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),zero_zero_int)) ).
% not_one_less_zero
tff(fact_2564_zero__less__one,axiom,
pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),one_one_real)) ).
% zero_less_one
tff(fact_2565_zero__less__one,axiom,
pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),one_one_rat)) ).
% zero_less_one
tff(fact_2566_zero__less__one,axiom,
pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).
% zero_less_one
tff(fact_2567_zero__less__one,axiom,
pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),one_one_int)) ).
% zero_less_one
tff(fact_2568_less__numeral__extra_I1_J,axiom,
pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),one_one_real)) ).
% less_numeral_extra(1)
tff(fact_2569_less__numeral__extra_I1_J,axiom,
pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),one_one_rat)) ).
% less_numeral_extra(1)
tff(fact_2570_less__numeral__extra_I1_J,axiom,
pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).
% less_numeral_extra(1)
tff(fact_2571_less__numeral__extra_I1_J,axiom,
pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),one_one_int)) ).
% less_numeral_extra(1)
tff(fact_2572_add__less__zeroD,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
| pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real)) ) ) ).
% add_less_zeroD
tff(fact_2573_add__less__zeroD,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(X,Y)),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),zero_zero_rat))
| pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat)) ) ) ).
% add_less_zeroD
tff(fact_2574_add__less__zeroD,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),zero_zero_int))
| pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),zero_zero_int)) ) ) ).
% add_less_zeroD
tff(fact_2575_pos__add__strict,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).
% pos_add_strict
tff(fact_2576_pos__add__strict,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),plus_plus_rat(A,C))) ) ) ).
% pos_add_strict
tff(fact_2577_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).
% pos_add_strict
tff(fact_2578_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).
% pos_add_strict
tff(fact_2579_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ~ ! [C2: nat] :
( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
tff(fact_2580_add__pos__pos,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).
% add_pos_pos
tff(fact_2581_add__pos__pos,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),plus_plus_rat(A,B))) ) ) ).
% add_pos_pos
tff(fact_2582_add__pos__pos,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).
% add_pos_pos
tff(fact_2583_add__pos__pos,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).
% add_pos_pos
tff(fact_2584_add__neg__neg,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real)) ) ) ).
% add_neg_neg
tff(fact_2585_add__neg__neg,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(A,B)),zero_zero_rat)) ) ) ).
% add_neg_neg
tff(fact_2586_add__neg__neg,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),zero_zero_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat)) ) ) ).
% add_neg_neg
tff(fact_2587_add__neg__neg,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int)) ) ) ).
% add_neg_neg
tff(fact_2588_le__iff__diff__le__0,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),zero_zero_real)) ) ).
% le_iff_diff_le_0
tff(fact_2589_le__iff__diff__le__0,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,minus_minus_rat(A,B)),zero_zero_rat)) ) ).
% le_iff_diff_le_0
tff(fact_2590_le__iff__diff__le__0,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int)) ) ).
% le_iff_diff_le_0
tff(fact_2591_divide__right__mono__neg,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C))) ) ) ).
% divide_right_mono_neg
tff(fact_2592_divide__right__mono__neg,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(B,C)),divide_divide_rat(A,C))) ) ) ).
% divide_right_mono_neg
tff(fact_2593_divide__nonpos__nonpos,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).
% divide_nonpos_nonpos
tff(fact_2594_divide__nonpos__nonpos,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(X,Y))) ) ) ).
% divide_nonpos_nonpos
tff(fact_2595_divide__nonpos__nonneg,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).
% divide_nonpos_nonneg
tff(fact_2596_divide__nonpos__nonneg,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(X,Y)),zero_zero_rat)) ) ) ).
% divide_nonpos_nonneg
tff(fact_2597_divide__nonneg__nonpos,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).
% divide_nonneg_nonpos
tff(fact_2598_divide__nonneg__nonpos,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(X,Y)),zero_zero_rat)) ) ) ).
% divide_nonneg_nonpos
tff(fact_2599_divide__nonneg__nonneg,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).
% divide_nonneg_nonneg
tff(fact_2600_divide__nonneg__nonneg,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(X,Y))) ) ) ).
% divide_nonneg_nonneg
tff(fact_2601_zero__le__divide__iff,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) )
| ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) ) ) ) ).
% zero_le_divide_iff
tff(fact_2602_zero__le__divide__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(A,B)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) )
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) ) ) ) ).
% zero_le_divide_iff
tff(fact_2603_divide__right__mono,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))) ) ) ).
% divide_right_mono
tff(fact_2604_divide__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C))) ) ) ).
% divide_right_mono
tff(fact_2605_divide__le__0__iff,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),zero_zero_real))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) )
| ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B)) ) ) ) ).
% divide_le_0_iff
tff(fact_2606_divide__le__0__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(A,B)),zero_zero_rat))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) )
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B)) ) ) ) ).
% divide_le_0_iff
tff(fact_2607_less__iff__diff__less__0,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)),zero_zero_real)) ) ).
% less_iff_diff_less_0
tff(fact_2608_less__iff__diff__less__0,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(A,B)),zero_zero_rat)) ) ).
% less_iff_diff_less_0
tff(fact_2609_less__iff__diff__less__0,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int)) ) ).
% less_iff_diff_less_0
tff(fact_2610_divide__neg__neg,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).
% divide_neg_neg
tff(fact_2611_divide__neg__neg,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),divide_divide_rat(X,Y))) ) ) ).
% divide_neg_neg
tff(fact_2612_divide__neg__pos,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).
% divide_neg_pos
tff(fact_2613_divide__neg__pos,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(X,Y)),zero_zero_rat)) ) ) ).
% divide_neg_pos
tff(fact_2614_divide__pos__neg,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).
% divide_pos_neg
tff(fact_2615_divide__pos__neg,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(X,Y)),zero_zero_rat)) ) ) ).
% divide_pos_neg
tff(fact_2616_divide__pos__pos,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).
% divide_pos_pos
tff(fact_2617_divide__pos__pos,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),divide_divide_rat(X,Y))) ) ) ).
% divide_pos_pos
tff(fact_2618_divide__less__0__iff,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)),zero_zero_real))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real)) )
| ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) ) ) ) ).
% divide_less_0_iff
tff(fact_2619_divide__less__0__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(A,B)),zero_zero_rat))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat)) )
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) ) ) ) ).
% divide_less_0_iff
tff(fact_2620_divide__less__cancel,axiom,
! [A: real,C: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
& ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) )
& ( C != zero_zero_real ) ) ) ).
% divide_less_cancel
tff(fact_2621_divide__less__cancel,axiom,
! [A: rat,C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
& ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) )
& ( C != zero_zero_rat ) ) ) ).
% divide_less_cancel
tff(fact_2622_zero__less__divide__iff,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B)) )
| ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real)) ) ) ) ).
% zero_less_divide_iff
tff(fact_2623_zero__less__divide__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),divide_divide_rat(A,B)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B)) )
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat)) ) ) ) ).
% zero_less_divide_iff
tff(fact_2624_divide__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))) ) ) ).
% divide_strict_right_mono
tff(fact_2625_divide__strict__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C))) ) ) ).
% divide_strict_right_mono
tff(fact_2626_divide__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))) ) ) ).
% divide_strict_right_mono_neg
tff(fact_2627_divide__strict__right__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C))) ) ) ).
% divide_strict_right_mono_neg
tff(fact_2628_power__mono,axiom,
! [A: real,B: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N))) ) ) ).
% power_mono
tff(fact_2629_power__mono,axiom,
! [A: rat,B: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N))) ) ) ).
% power_mono
tff(fact_2630_power__mono,axiom,
! [A: nat,B: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N))) ) ) ).
% power_mono
tff(fact_2631_power__mono,axiom,
! [A: int,B: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N))) ) ) ).
% power_mono
tff(fact_2632_zero__le__power,axiom,
! [A: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ).
% zero_le_power
tff(fact_2633_zero__le__power,axiom,
! [A: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N))) ) ).
% zero_le_power
tff(fact_2634_zero__le__power,axiom,
! [A: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).
% zero_le_power
tff(fact_2635_zero__le__power,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),N))) ) ).
% zero_le_power
tff(fact_2636_zero__less__power,axiom,
! [A: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ).
% zero_less_power
tff(fact_2637_zero__less__power,axiom,
! [A: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N))) ) ).
% zero_less_power
tff(fact_2638_zero__less__power,axiom,
! [A: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).
% zero_less_power
tff(fact_2639_zero__less__power,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),N))) ) ).
% zero_less_power
tff(fact_2640_nonzero__eq__divide__eq,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) )
<=> ( aa_complex_complex(times_times_complex(A),C) = B ) ) ) ).
% nonzero_eq_divide_eq
tff(fact_2641_nonzero__eq__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
<=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B ) ) ) ).
% nonzero_eq_divide_eq
tff(fact_2642_nonzero__eq__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( A = divide_divide_rat(B,C) )
<=> ( aa_rat_rat(times_times_rat(A),C) = B ) ) ) ).
% nonzero_eq_divide_eq
tff(fact_2643_nonzero__divide__eq__eq,axiom,
! [C: complex,B: complex,A: complex] :
( ( C != zero_zero_complex )
=> ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A )
<=> ( B = aa_complex_complex(times_times_complex(A),C) ) ) ) ).
% nonzero_divide_eq_eq
tff(fact_2644_nonzero__divide__eq__eq,axiom,
! [C: real,B: real,A: real] :
( ( C != zero_zero_real )
=> ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = A )
<=> ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) ) ) ) ).
% nonzero_divide_eq_eq
tff(fact_2645_nonzero__divide__eq__eq,axiom,
! [C: rat,B: rat,A: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat(B,C) = A )
<=> ( B = aa_rat_rat(times_times_rat(A),C) ) ) ) ).
% nonzero_divide_eq_eq
tff(fact_2646_eq__divide__imp,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( aa_complex_complex(times_times_complex(A),C) = B )
=> ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) ) ) ) ).
% eq_divide_imp
tff(fact_2647_eq__divide__imp,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B )
=> ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) ) ) ) ).
% eq_divide_imp
tff(fact_2648_eq__divide__imp,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( aa_rat_rat(times_times_rat(A),C) = B )
=> ( A = divide_divide_rat(B,C) ) ) ) ).
% eq_divide_imp
tff(fact_2649_divide__eq__imp,axiom,
! [C: complex,B: complex,A: complex] :
( ( C != zero_zero_complex )
=> ( ( B = aa_complex_complex(times_times_complex(A),C) )
=> ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A ) ) ) ).
% divide_eq_imp
tff(fact_2650_divide__eq__imp,axiom,
! [C: real,B: real,A: real] :
( ( C != zero_zero_real )
=> ( ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) )
=> ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = A ) ) ) ).
% divide_eq_imp
tff(fact_2651_divide__eq__imp,axiom,
! [C: rat,B: rat,A: rat] :
( ( C != zero_zero_rat )
=> ( ( B = aa_rat_rat(times_times_rat(A),C) )
=> ( divide_divide_rat(B,C) = A ) ) ) ).
% divide_eq_imp
tff(fact_2652_eq__divide__eq,axiom,
! [A: complex,B: complex,C: complex] :
( ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) )
<=> ( ( ( C != zero_zero_complex )
=> ( aa_complex_complex(times_times_complex(A),C) = B ) )
& ( ( C = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% eq_divide_eq
tff(fact_2653_eq__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
<=> ( ( ( C != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq
tff(fact_2654_eq__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( A = divide_divide_rat(B,C) )
<=> ( ( ( C != zero_zero_rat )
=> ( aa_rat_rat(times_times_rat(A),C) = B ) )
& ( ( C = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_divide_eq
tff(fact_2655_divide__eq__eq,axiom,
! [B: complex,C: complex,A: complex] :
( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A )
<=> ( ( ( C != zero_zero_complex )
=> ( B = aa_complex_complex(times_times_complex(A),C) ) )
& ( ( C = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% divide_eq_eq
tff(fact_2656_divide__eq__eq,axiom,
! [B: real,C: real,A: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = A )
<=> ( ( ( C != zero_zero_real )
=> ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq
tff(fact_2657_divide__eq__eq,axiom,
! [B: rat,C: rat,A: rat] :
( ( divide_divide_rat(B,C) = A )
<=> ( ( ( C != zero_zero_rat )
=> ( B = aa_rat_rat(times_times_rat(A),C) ) )
& ( ( C = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% divide_eq_eq
tff(fact_2658_frac__eq__eq,axiom,
! [Y: complex,Z: complex,X: complex,W: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z != zero_zero_complex )
=> ( ( aa_complex_complex(divide1717551699836669952omplex(X),Y) = aa_complex_complex(divide1717551699836669952omplex(W),Z) )
<=> ( aa_complex_complex(times_times_complex(X),Z) = aa_complex_complex(times_times_complex(W),Y) ) ) ) ) ).
% frac_eq_eq
tff(fact_2659_frac__eq__eq,axiom,
! [Y: real,Z: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z) )
<=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y) ) ) ) ) ).
% frac_eq_eq
tff(fact_2660_frac__eq__eq,axiom,
! [Y: rat,Z: rat,X: rat,W: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( ( divide_divide_rat(X,Y) = divide_divide_rat(W,Z) )
<=> ( aa_rat_rat(times_times_rat(X),Z) = aa_rat_rat(times_times_rat(W),Y) ) ) ) ) ).
% frac_eq_eq
tff(fact_2661_right__inverse__eq,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = one_one_complex )
<=> ( A = B ) ) ) ).
% right_inverse_eq
tff(fact_2662_right__inverse__eq,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = one_one_real )
<=> ( A = B ) ) ) ).
% right_inverse_eq
tff(fact_2663_right__inverse__eq,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( divide_divide_rat(A,B) = one_one_rat )
<=> ( A = B ) ) ) ).
% right_inverse_eq
tff(fact_2664_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,modulo364778990260209775nteger(A,B)),A)) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_2665_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),A)) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_2666_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),A)) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_2667_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B)) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_2668_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B)) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_2669_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,B)),B)) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_2670_power__0,axiom,
! [A: rat] : aa_nat_rat(power_power_rat(A),zero_zero_nat) = one_one_rat ).
% power_0
tff(fact_2671_power__0,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),zero_zero_nat) = one_one_nat ).
% power_0
tff(fact_2672_power__0,axiom,
! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),zero_zero_nat) = one_one_real ).
% power_0
tff(fact_2673_power__0,axiom,
! [A: complex] : aa_nat_complex(power_power_complex(A),zero_zero_nat) = one_one_complex ).
% power_0
tff(fact_2674_power__0,axiom,
! [A: int] : aa_nat_int(power_power_int(A),zero_zero_nat) = one_one_int ).
% power_0
tff(fact_2675_mod__eq__self__iff__div__eq__0,axiom,
! [A: nat,B: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = A )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat ) ) ).
% mod_eq_self_iff_div_eq_0
tff(fact_2676_mod__eq__self__iff__div__eq__0,axiom,
! [A: int,B: int] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = A )
<=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int ) ) ).
% mod_eq_self_iff_div_eq_0
tff(fact_2677_mod__eq__self__iff__div__eq__0,axiom,
! [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger(A,B) = A )
<=> ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger ) ) ).
% mod_eq_self_iff_div_eq_0
tff(fact_2678_Ex__less__Suc2,axiom,
! [N: nat,P: fun_nat_bool] :
( ? [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(suc,N)))
& pp(aa_nat_bool(P,I4)) )
<=> ( pp(aa_nat_bool(P,zero_zero_nat))
| ? [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
& pp(aa_nat_bool(P,aa_nat_nat(suc,I4))) ) ) ) ).
% Ex_less_Suc2
tff(fact_2679_gr0__conv__Suc,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
<=> ? [M7: nat] : N = aa_nat_nat(suc,M7) ) ).
% gr0_conv_Suc
tff(fact_2680_All__less__Suc2,axiom,
! [N: nat,P: fun_nat_bool] :
( ! [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(suc,N)))
=> pp(aa_nat_bool(P,I4)) )
<=> ( pp(aa_nat_bool(P,zero_zero_nat))
& ! [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
=> pp(aa_nat_bool(P,aa_nat_nat(suc,I4))) ) ) ) ).
% All_less_Suc2
tff(fact_2681_gr0__implies__Suc,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ? [M3: nat] : N = aa_nat_nat(suc,M3) ) ).
% gr0_implies_Suc
tff(fact_2682_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
<=> ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M = aa_nat_nat(suc,J3) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) ).
% less_Suc_eq_0_disj
tff(fact_2683_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I1_J,axiom,
! [A: bool,B: bool,X: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t2(vEBT_Leaf(A,B)),X) = one_one_nat ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(1)
tff(fact_2684_add__is__1,axiom,
! [M: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = aa_nat_nat(suc,zero_zero_nat) )
<=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).
% add_is_1
tff(fact_2685_one__is__add,axiom,
! [M: nat,N: nat] :
( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) )
<=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).
% one_is_add
tff(fact_2686_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I1_J,axiom,
aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,vEBT_Leaf(fFalse,fFalse)) = one_one_nat ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(1)
tff(fact_2687_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I2_J,axiom,
! [Uv: bool] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,vEBT_Leaf(fTrue,Uv)) = one_one_nat ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(2)
tff(fact_2688_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I3_J,axiom,
! [Uu: bool] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,vEBT_Leaf(Uu,fTrue)) = one_one_nat ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(3)
tff(fact_2689_option_Osize_I4_J,axiom,
! [X23: product_prod_nat_nat] : size_s170228958280169651at_nat(some_P7363390416028606310at_nat(X23)) = aa_nat_nat(suc,zero_zero_nat) ).
% option.size(4)
tff(fact_2690_option_Osize_I4_J,axiom,
! [X23: nat] : size_size_option_nat(some_nat(X23)) = aa_nat_nat(suc,zero_zero_nat) ).
% option.size(4)
tff(fact_2691_option_Osize_I4_J,axiom,
! [X23: num] : size_size_option_num(some_num(X23)) = aa_nat_nat(suc,zero_zero_nat) ).
% option.size(4)
tff(fact_2692_ex__least__nat__le,axiom,
! [P: fun_nat_bool,N: nat] :
( pp(aa_nat_bool(P,N))
=> ( ~ pp(aa_nat_bool(P,zero_zero_nat))
=> ? [K3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N))
& ! [I3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),K3))
=> ~ pp(aa_nat_bool(P,I3)) )
& pp(aa_nat_bool(P,K3)) ) ) ) ).
% ex_least_nat_le
tff(fact_2693_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
=> ? [K3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K3))
& ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K3) = J ) ) ) ).
% less_imp_add_positive
tff(fact_2694_One__nat__def,axiom,
one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).
% One_nat_def
tff(fact_2695_option_Osize_I3_J,axiom,
size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).
% option.size(3)
tff(fact_2696_option_Osize_I3_J,axiom,
size_size_option_nat(none_nat) = aa_nat_nat(suc,zero_zero_nat) ).
% option.size(3)
tff(fact_2697_option_Osize_I3_J,axiom,
size_size_option_num(none_num) = aa_nat_nat(suc,zero_zero_nat) ).
% option.size(3)
tff(fact_2698_diff__less,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),M)) ) ) ).
% diff_less
tff(fact_2699_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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_2700_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(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_2701_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).
% nat_mult_less_cancel1
tff(fact_2702_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N) )
<=> ( M = N ) ) ) ).
% nat_mult_eq_cancel1
tff(fact_2703_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat )
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
tff(fact_2704_diff__add__0,axiom,
! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = zero_zero_nat ).
% diff_add_0
tff(fact_2705_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),I))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ) ).
% nat_power_less_imp_less
tff(fact_2706_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
tff(fact_2707_mod__Suc,axiom,
! [M: nat,N: nat] :
( ( ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = N )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) = zero_zero_nat ) )
& ( ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) != N )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) ) ) ) ).
% mod_Suc
tff(fact_2708_mod__less__divisor,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)),N)) ) ).
% mod_less_divisor
tff(fact_2709_mod__eq__0D,axiom,
! [M: nat,D: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),D) = zero_zero_nat )
=> ? [Q3: nat] : M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q3) ) ).
% mod_eq_0D
tff(fact_2710_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [A4: bool,B4: bool,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),X3)
=> ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),X3)
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),X3)
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),X3)
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2),X3) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
tff(fact_2711_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [A4: bool,B4: bool,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),X3)
=> ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts,S3),X3)
=> ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S3),X3)
=> ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2),X3)
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2),X3) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
tff(fact_2712_VEBT__internal_Omembermima_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [Uu2: bool,Uv2: bool,Uw2: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Uw2)
=> ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Uz2)
=> ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2,Vb2),X3)
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2),X3)
=> ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2),X3) ) ) ) ) ).
% VEBT_internal.membermima.cases
tff(fact_2713_vebt__pred_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [Uu2: bool,Uv2: bool] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),zero_zero_nat)
=> ( ! [A4: bool,Uw2: bool] : X != produc738532404422230701BT_nat(vEBT_Leaf(A4,Uw2),aa_nat_nat(suc,zero_zero_nat))
=> ( ! [A4: bool,B4: bool,Va3: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),aa_nat_nat(suc,aa_nat_nat(suc,Va3)))
=> ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va2),Vb2)
=> ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve),Vf)
=> ( ! [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh,Vi),Vj)
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2),X3) ) ) ) ) ) ) ).
% vebt_pred.cases
tff(fact_2714_vebt__succ_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [Uu2: bool,B4: bool] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uu2,B4),zero_zero_nat)
=> ( ! [Uv2: bool,Uw2: bool,N2: nat] : X != produc738532404422230701BT_nat(vEBT_Leaf(Uv2,Uw2),aa_nat_nat(suc,N2))
=> ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va2: nat] : X != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2),Va2)
=> ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2),Ve)
=> ( ! [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg,Vh),Vi)
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] : X != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2),X3) ) ) ) ) ) ).
% vebt_succ.cases
tff(fact_2715_vebt__insert_Osimps_I2_J,axiom,
! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(Info,zero_zero_nat,Ts2,S),X) = vEBT_Node(Info,zero_zero_nat,Ts2,S) ).
% vebt_insert.simps(2)
tff(fact_2716_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] : ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(vEBT_Node(Uu,zero_zero_nat,Uv,Uw)),Ux)) ).
% VEBT_internal.naive_member.simps(2)
tff(fact_2717_vebt__pred_Osimps_I3_J,axiom,
! [B: bool,A: bool,Va: nat] :
( ( pp(B)
=> ( vEBT_vebt_pred(vEBT_Leaf(A,B),aa_nat_nat(suc,aa_nat_nat(suc,Va))) = some_nat(one_one_nat) ) )
& ( ~ pp(B)
=> ( ( pp(A)
=> ( vEBT_vebt_pred(vEBT_Leaf(A,B),aa_nat_nat(suc,aa_nat_nat(suc,Va))) = some_nat(zero_zero_nat) ) )
& ( ~ pp(A)
=> ( vEBT_vebt_pred(vEBT_Leaf(A,B),aa_nat_nat(suc,aa_nat_nat(suc,Va))) = none_nat ) ) ) ) ) ).
% vebt_pred.simps(3)
tff(fact_2718_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Ocases,axiom,
! [X: vEBT_VEBT] :
( ( X != vEBT_Leaf(fFalse,fFalse) )
=> ( ! [Uv2: bool] : X != vEBT_Leaf(fTrue,Uv2)
=> ( ! [Uu2: bool] : X != vEBT_Leaf(Uu2,fTrue)
=> ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)
=> ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2,Vc2) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.cases
tff(fact_2719_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I1_J,axiom,
! [A: bool,B: bool,X: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r(vEBT_Leaf(A,B)),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),zero_zero_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(1)
tff(fact_2720_VEBT__internal_OminNull_Oelims_I3_J,axiom,
! [X: vEBT_VEBT] :
( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
=> ( ! [Uv2: bool] : X != vEBT_Leaf(fTrue,Uv2)
=> ( ! [Uu2: bool] : X != vEBT_Leaf(Uu2,fTrue)
=> ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2,Vc2) ) ) ) ).
% VEBT_internal.minNull.elims(3)
tff(fact_2721_vebt__succ_Osimps_I2_J,axiom,
! [Uv: bool,Uw: bool,N: nat] : vEBT_vebt_succ(vEBT_Leaf(Uv,Uw),aa_nat_nat(suc,N)) = none_nat ).
% vebt_succ.simps(2)
tff(fact_2722_VEBT__internal_OminNull_Oelims_I2_J,axiom,
! [X: vEBT_VEBT] :
( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
=> ( ( X != vEBT_Leaf(fFalse,fFalse) )
=> ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) ) ) ).
% VEBT_internal.minNull.elims(2)
tff(fact_2723_mult__less__le__imp__less,axiom,
! [A: real,B: real,C: real,D: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).
% mult_less_le_imp_less
tff(fact_2724_mult__less__le__imp__less,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).
% mult_less_le_imp_less
tff(fact_2725_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).
% mult_less_le_imp_less
tff(fact_2726_mult__less__le__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).
% mult_less_le_imp_less
tff(fact_2727_mult__le__less__imp__less,axiom,
! [A: real,B: real,C: real,D: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).
% mult_le_less_imp_less
tff(fact_2728_mult__le__less__imp__less,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).
% mult_le_less_imp_less
tff(fact_2729_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).
% mult_le_less_imp_less
tff(fact_2730_mult__le__less__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).
% mult_le_less_imp_less
tff(fact_2731_mult__right__le__imp__le,axiom,
! [A: real,C: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).
% mult_right_le_imp_le
tff(fact_2732_mult__right__le__imp__le,axiom,
! [A: rat,C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).
% mult_right_le_imp_le
tff(fact_2733_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).
% mult_right_le_imp_le
tff(fact_2734_mult__right__le__imp__le,axiom,
! [A: int,C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).
% mult_right_le_imp_le
tff(fact_2735_mult__left__le__imp__le,axiom,
! [C: real,A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).
% mult_left_le_imp_le
tff(fact_2736_mult__left__le__imp__le,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).
% mult_left_le_imp_le
tff(fact_2737_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).
% mult_left_le_imp_le
tff(fact_2738_mult__left__le__imp__le,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).
% mult_left_le_imp_le
tff(fact_2739_mult__le__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).
% mult_le_cancel_left_pos
tff(fact_2740_mult__le__cancel__left__pos,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).
% mult_le_cancel_left_pos
tff(fact_2741_mult__le__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).
% mult_le_cancel_left_pos
tff(fact_2742_mult__le__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ).
% mult_le_cancel_left_neg
tff(fact_2743_mult__le__cancel__left__neg,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ).
% mult_le_cancel_left_neg
tff(fact_2744_mult__le__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ).
% mult_le_cancel_left_neg
tff(fact_2745_mult__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
& ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).
% mult_less_cancel_right
tff(fact_2746_mult__less__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
& ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).
% mult_less_cancel_right
tff(fact_2747_mult__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).
% mult_less_cancel_right
tff(fact_2748_mult__strict__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).
% mult_strict_mono'
tff(fact_2749_mult__strict__mono_H,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).
% mult_strict_mono'
tff(fact_2750_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).
% mult_strict_mono'
tff(fact_2751_mult__strict__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).
% mult_strict_mono'
tff(fact_2752_mult__right__less__imp__less,axiom,
! [A: real,C: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).
% mult_right_less_imp_less
tff(fact_2753_mult__right__less__imp__less,axiom,
! [A: rat,C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).
% mult_right_less_imp_less
tff(fact_2754_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).
% mult_right_less_imp_less
tff(fact_2755_mult__right__less__imp__less,axiom,
! [A: int,C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).
% mult_right_less_imp_less
tff(fact_2756_mult__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
& ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).
% mult_less_cancel_left
tff(fact_2757_mult__less__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
& ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).
% mult_less_cancel_left
tff(fact_2758_mult__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ) ).
% mult_less_cancel_left
tff(fact_2759_mult__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),D))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ) ) ).
% mult_strict_mono
tff(fact_2760_mult__strict__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),D))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ) ) ).
% mult_strict_mono
tff(fact_2761_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ) ) ).
% mult_strict_mono
tff(fact_2762_mult__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ) ) ).
% mult_strict_mono
tff(fact_2763_mult__left__less__imp__less,axiom,
! [C: real,A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).
% mult_left_less_imp_less
tff(fact_2764_mult__left__less__imp__less,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).
% mult_left_less_imp_less
tff(fact_2765_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).
% mult_left_less_imp_less
tff(fact_2766_mult__left__less__imp__less,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).
% mult_left_less_imp_less
tff(fact_2767_mult__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
& ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).
% mult_le_cancel_right
tff(fact_2768_mult__le__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
& ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).
% mult_le_cancel_right
tff(fact_2769_mult__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ) ).
% mult_le_cancel_right
tff(fact_2770_mult__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
& ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).
% mult_le_cancel_left
tff(fact_2771_mult__le__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
& ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).
% mult_le_cancel_left
tff(fact_2772_mult__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ) ).
% mult_le_cancel_left
tff(fact_2773_field__le__epsilon,axiom,
! [X: real,Y: real] :
( ! [E2: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),E2))) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).
% field_le_epsilon
tff(fact_2774_field__le__epsilon,axiom,
! [X: rat,Y: rat] :
( ! [E2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),plus_plus_rat(Y,E2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).
% field_le_epsilon
tff(fact_2775_add__strict__increasing2,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).
% add_strict_increasing2
tff(fact_2776_add__strict__increasing2,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),plus_plus_rat(A,C))) ) ) ).
% add_strict_increasing2
tff(fact_2777_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).
% add_strict_increasing2
tff(fact_2778_add__strict__increasing2,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).
% add_strict_increasing2
tff(fact_2779_add__strict__increasing,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C))) ) ) ).
% add_strict_increasing
tff(fact_2780_add__strict__increasing,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),plus_plus_rat(A,C))) ) ) ).
% add_strict_increasing
tff(fact_2781_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C))) ) ) ).
% add_strict_increasing
tff(fact_2782_add__strict__increasing,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C))) ) ) ).
% add_strict_increasing
tff(fact_2783_add__pos__nonneg,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).
% add_pos_nonneg
tff(fact_2784_add__pos__nonneg,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),plus_plus_rat(A,B))) ) ) ).
% add_pos_nonneg
tff(fact_2785_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).
% add_pos_nonneg
tff(fact_2786_add__pos__nonneg,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).
% add_pos_nonneg
tff(fact_2787_add__nonpos__neg,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real)) ) ) ).
% add_nonpos_neg
tff(fact_2788_add__nonpos__neg,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(A,B)),zero_zero_rat)) ) ) ).
% add_nonpos_neg
tff(fact_2789_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),zero_zero_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),zero_zero_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat)) ) ) ).
% add_nonpos_neg
tff(fact_2790_add__nonpos__neg,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int)) ) ) ).
% add_nonpos_neg
tff(fact_2791_add__nonneg__pos,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).
% add_nonneg_pos
tff(fact_2792_add__nonneg__pos,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),plus_plus_rat(A,B))) ) ) ).
% add_nonneg_pos
tff(fact_2793_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).
% add_nonneg_pos
tff(fact_2794_add__nonneg__pos,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).
% add_nonneg_pos
tff(fact_2795_add__neg__nonpos,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),zero_zero_real)) ) ) ).
% add_neg_nonpos
tff(fact_2796_add__neg__nonpos,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(A,B)),zero_zero_rat)) ) ) ).
% add_neg_nonpos
tff(fact_2797_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),zero_zero_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),zero_zero_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat)) ) ) ).
% add_neg_nonpos
tff(fact_2798_add__neg__nonpos,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int)) ) ) ).
% add_neg_nonpos
tff(fact_2799_frac__le,axiom,
! [Y: real,X: real,W: real,Z: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),W))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,W),Z))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W))) ) ) ) ) ).
% frac_le
tff(fact_2800_frac__le,axiom,
! [Y: rat,X: rat,W: rat,Z: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),W))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,W),Z))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(X,Z)),divide_divide_rat(Y,W))) ) ) ) ) ).
% frac_le
tff(fact_2801_frac__less,axiom,
! [X: real,Y: real,W: real,Z: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),W))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,W),Z))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W))) ) ) ) ) ).
% frac_less
tff(fact_2802_frac__less,axiom,
! [X: rat,Y: rat,W: rat,Z: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),W))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,W),Z))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(X,Z)),divide_divide_rat(Y,W))) ) ) ) ) ).
% frac_less
tff(fact_2803_frac__less2,axiom,
! [X: real,Y: real,W: real,Z: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),W))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,W),Z))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W))) ) ) ) ) ).
% frac_less2
tff(fact_2804_frac__less2,axiom,
! [X: rat,Y: rat,W: rat,Z: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),W))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,W),Z))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(X,Z)),divide_divide_rat(Y,W))) ) ) ) ) ).
% frac_less2
tff(fact_2805_divide__le__cancel,axiom,
! [A: real,C: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
& ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).
% divide_le_cancel
tff(fact_2806_divide__le__cancel,axiom,
! [A: rat,C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(A,C)),divide_divide_rat(B,C)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
& ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).
% divide_le_cancel
tff(fact_2807_divide__nonneg__neg,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).
% divide_nonneg_neg
tff(fact_2808_divide__nonneg__neg,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(X,Y)),zero_zero_rat)) ) ) ).
% divide_nonneg_neg
tff(fact_2809_divide__nonneg__pos,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).
% divide_nonneg_pos
tff(fact_2810_divide__nonneg__pos,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(X,Y))) ) ) ).
% divide_nonneg_pos
tff(fact_2811_divide__nonpos__neg,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).
% divide_nonpos_neg
tff(fact_2812_divide__nonpos__neg,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(X,Y))) ) ) ).
% divide_nonpos_neg
tff(fact_2813_divide__nonpos__pos,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),zero_zero_real)) ) ) ).
% divide_nonpos_pos
tff(fact_2814_divide__nonpos__pos,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(X,Y)),zero_zero_rat)) ) ) ).
% divide_nonpos_pos
tff(fact_2815_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_less
tff(fact_2816_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_less
tff(fact_2817_div__positive,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))) ) ) ).
% div_positive
tff(fact_2818_div__positive,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) ) ) ).
% div_positive
tff(fact_2819_mult__left__le,axiom,
! [C: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),one_one_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),A)) ) ) ).
% mult_left_le
tff(fact_2820_mult__left__le,axiom,
! [C: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),one_one_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),A)) ) ) ).
% mult_left_le
tff(fact_2821_mult__left__le,axiom,
! [C: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),one_one_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),A)) ) ) ).
% mult_left_le
tff(fact_2822_mult__left__le,axiom,
! [C: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),one_one_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),A)) ) ) ).
% mult_left_le
tff(fact_2823_mult__le__one,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),one_one_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),one_one_real)) ) ) ) ).
% mult_le_one
tff(fact_2824_mult__le__one,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),one_one_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),one_one_rat)) ) ) ) ).
% mult_le_one
tff(fact_2825_mult__le__one,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),one_one_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat)) ) ) ) ).
% mult_le_one
tff(fact_2826_mult__le__one,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),one_one_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int)) ) ) ) ).
% mult_le_one
tff(fact_2827_mult__right__le__one__le,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)),X)) ) ) ) ).
% mult_right_le_one_le
tff(fact_2828_mult__right__le__one__le,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),one_one_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(X),Y)),X)) ) ) ) ).
% mult_right_le_one_le
tff(fact_2829_mult__right__le__one__le,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),one_one_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)),X)) ) ) ) ).
% mult_right_le_one_le
tff(fact_2830_mult__left__le__one__le,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),X)),X)) ) ) ) ).
% mult_left_le_one_le
tff(fact_2831_mult__left__le__one__le,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),one_one_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(Y),X)),X)) ) ) ) ).
% mult_left_le_one_le
tff(fact_2832_mult__left__le__one__le,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),one_one_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),X)),X)) ) ) ) ).
% mult_left_le_one_le
tff(fact_2833_sum__squares__ge__zero,axiom,
! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)))) ).
% sum_squares_ge_zero
tff(fact_2834_sum__squares__ge__zero,axiom,
! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),plus_plus_rat(aa_rat_rat(times_times_rat(X),X),aa_rat_rat(times_times_rat(Y),Y)))) ).
% sum_squares_ge_zero
tff(fact_2835_sum__squares__ge__zero,axiom,
! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)))) ).
% sum_squares_ge_zero
tff(fact_2836_sum__squares__le__zero__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real))
<=> ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_le_zero_iff
tff(fact_2837_sum__squares__le__zero__iff,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(aa_rat_rat(times_times_rat(X),X),aa_rat_rat(times_times_rat(Y),Y))),zero_zero_rat))
<=> ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_squares_le_zero_iff
tff(fact_2838_sum__squares__le__zero__iff,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int))
<=> ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
tff(fact_2839_not__sum__squares__lt__zero,axiom,
! [X: real,Y: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real)) ).
% not_sum_squares_lt_zero
tff(fact_2840_not__sum__squares__lt__zero,axiom,
! [X: rat,Y: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(aa_rat_rat(times_times_rat(X),X),aa_rat_rat(times_times_rat(Y),Y))),zero_zero_rat)) ).
% not_sum_squares_lt_zero
tff(fact_2841_not__sum__squares__lt__zero,axiom,
! [X: int,Y: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int)) ).
% not_sum_squares_lt_zero
tff(fact_2842_sum__squares__gt__zero__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))))
<=> ( ( X != zero_zero_real )
| ( Y != zero_zero_real ) ) ) ).
% sum_squares_gt_zero_iff
tff(fact_2843_sum__squares__gt__zero__iff,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),plus_plus_rat(aa_rat_rat(times_times_rat(X),X),aa_rat_rat(times_times_rat(Y),Y))))
<=> ( ( X != zero_zero_rat )
| ( Y != zero_zero_rat ) ) ) ).
% sum_squares_gt_zero_iff
tff(fact_2844_sum__squares__gt__zero__iff,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))))
<=> ( ( X != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
tff(fact_2845_power__less__imp__less__base,axiom,
! [A: real,N: nat,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).
% power_less_imp_less_base
tff(fact_2846_power__less__imp__less__base,axiom,
! [A: rat,N: nat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N)))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ) ).
% power_less_imp_less_base
tff(fact_2847_power__less__imp__less__base,axiom,
! [A: nat,N: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B)) ) ) ).
% power_less_imp_less_base
tff(fact_2848_power__less__imp__less__base,axiom,
! [A: int,N: nat,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).
% power_less_imp_less_base
tff(fact_2849_zero__less__two,axiom,
pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))) ).
% zero_less_two
tff(fact_2850_zero__less__two,axiom,
pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),plus_plus_rat(one_one_rat,one_one_rat))) ).
% zero_less_two
tff(fact_2851_zero__less__two,axiom,
pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).
% zero_less_two
tff(fact_2852_zero__less__two,axiom,
pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int))) ).
% zero_less_two
tff(fact_2853_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [C: nat,A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_2854_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_2855_divide__less__eq,axiom,
! [B: real,C: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ) ) ).
% divide_less_eq
tff(fact_2856_divide__less__eq,axiom,
! [B: rat,C: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(B,C)),A))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B)) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ) ) ) ) ).
% divide_less_eq
tff(fact_2857_less__divide__eq,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ) ) ) ).
% less_divide_eq
tff(fact_2858_less__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),divide_divide_rat(B,C)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B)) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ) ) ) ).
% less_divide_eq
tff(fact_2859_neg__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) ) ) ).
% neg_divide_less_eq
tff(fact_2860_neg__divide__less__eq,axiom,
! [C: rat,B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(B,C)),A))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B)) ) ) ).
% neg_divide_less_eq
tff(fact_2861_neg__less__divide__eq,axiom,
! [C: real,A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).
% neg_less_divide_eq
tff(fact_2862_neg__less__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),divide_divide_rat(B,C)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C))) ) ) ).
% neg_less_divide_eq
tff(fact_2863_pos__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).
% pos_divide_less_eq
tff(fact_2864_pos__divide__less__eq,axiom,
! [C: rat,B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(B,C)),A))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C))) ) ) ).
% pos_divide_less_eq
tff(fact_2865_pos__less__divide__eq,axiom,
! [C: real,A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) ) ) ).
% pos_less_divide_eq
tff(fact_2866_pos__less__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),divide_divide_rat(B,C)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B)) ) ) ).
% pos_less_divide_eq
tff(fact_2867_mult__imp__div__pos__less,axiom,
! [Y: real,X: real,Z: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),Z)) ) ) ).
% mult_imp_div_pos_less
tff(fact_2868_mult__imp__div__pos__less,axiom,
! [Y: rat,X: rat,Z: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_rat_rat(times_times_rat(Z),Y)))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(X,Y)),Z)) ) ) ).
% mult_imp_div_pos_less
tff(fact_2869_mult__imp__less__div__pos,axiom,
! [Y: real,Z: real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).
% mult_imp_less_div_pos
tff(fact_2870_mult__imp__less__div__pos,axiom,
! [Y: rat,Z: rat,X: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(Z),Y)),X))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),divide_divide_rat(X,Y))) ) ) ).
% mult_imp_less_div_pos
tff(fact_2871_divide__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B))) ) ) ) ).
% divide_strict_left_mono
tff(fact_2872_divide__strict__left__mono,axiom,
! [B: rat,A: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(C,A)),divide_divide_rat(C,B))) ) ) ) ).
% divide_strict_left_mono
tff(fact_2873_divide__strict__left__mono__neg,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B))) ) ) ) ).
% divide_strict_left_mono_neg
tff(fact_2874_divide__strict__left__mono__neg,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(C,A)),divide_divide_rat(C,B))) ) ) ) ).
% divide_strict_left_mono_neg
tff(fact_2875_divide__less__eq__1,axiom,
! [B: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) )
| ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
| ( A = zero_zero_real ) ) ) ).
% divide_less_eq_1
tff(fact_2876_divide__less__eq__1,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(B,A)),one_one_rat))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) )
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
| ( A = zero_zero_rat ) ) ) ).
% divide_less_eq_1
tff(fact_2877_less__divide__eq__1,axiom,
! [B: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) )
| ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A)) ) ) ) ).
% less_divide_eq_1
tff(fact_2878_less__divide__eq__1,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),divide_divide_rat(B,A)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) )
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A)) ) ) ) ).
% less_divide_eq_1
tff(fact_2879_power__le__one,axiom,
! [A: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),one_one_real)) ) ) ).
% power_le_one
tff(fact_2880_power__le__one,axiom,
! [A: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),one_one_rat)) ) ) ).
% power_le_one
tff(fact_2881_power__le__one,axiom,
! [A: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),one_one_nat)) ) ) ).
% power_le_one
tff(fact_2882_power__le__one,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),one_one_int)) ) ) ).
% power_le_one
tff(fact_2883_eq__divide__eq__numeral_I1_J,axiom,
! [W: num,B: complex,C: complex] :
( ( numera6690914467698888265omplex(W) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
<=> ( ( ( C != zero_zero_complex )
=> ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),C) = B ) )
& ( ( C = zero_zero_complex )
=> ( numera6690914467698888265omplex(W) = zero_zero_complex ) ) ) ) ).
% eq_divide_eq_numeral(1)
tff(fact_2884_eq__divide__eq__numeral_I1_J,axiom,
! [W: num,B: real,C: real] :
( ( numeral_numeral_real(W) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
<=> ( ( ( C != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C) = B ) )
& ( ( C = zero_zero_real )
=> ( numeral_numeral_real(W) = zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral(1)
tff(fact_2885_eq__divide__eq__numeral_I1_J,axiom,
! [W: num,B: rat,C: rat] :
( ( numeral_numeral_rat(W) = divide_divide_rat(B,C) )
<=> ( ( ( C != zero_zero_rat )
=> ( aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C) = B ) )
& ( ( C = zero_zero_rat )
=> ( numeral_numeral_rat(W) = zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral(1)
tff(fact_2886_divide__eq__eq__numeral_I1_J,axiom,
! [B: complex,C: complex,W: num] :
( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = numera6690914467698888265omplex(W) )
<=> ( ( ( C != zero_zero_complex )
=> ( B = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),C) ) )
& ( ( C = zero_zero_complex )
=> ( numera6690914467698888265omplex(W) = zero_zero_complex ) ) ) ) ).
% divide_eq_eq_numeral(1)
tff(fact_2887_divide__eq__eq__numeral_I1_J,axiom,
! [B: real,C: real,W: num] :
( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = numeral_numeral_real(W) )
<=> ( ( ( C != zero_zero_real )
=> ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C) ) )
& ( ( C = zero_zero_real )
=> ( numeral_numeral_real(W) = zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral(1)
tff(fact_2888_divide__eq__eq__numeral_I1_J,axiom,
! [B: rat,C: rat,W: num] :
( ( divide_divide_rat(B,C) = numeral_numeral_rat(W) )
<=> ( ( ( C != zero_zero_rat )
=> ( B = aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C) ) )
& ( ( C = zero_zero_rat )
=> ( numeral_numeral_rat(W) = zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral(1)
tff(fact_2889_divide__add__eq__iff,axiom,
! [Z: complex,X: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Z)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).
% divide_add_eq_iff
tff(fact_2890_divide__add__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z)),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).
% divide_add_eq_iff
tff(fact_2891_divide__add__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( plus_plus_rat(divide_divide_rat(X,Z),Y) = divide_divide_rat(plus_plus_rat(X,aa_rat_rat(times_times_rat(Y),Z)),Z) ) ) ).
% divide_add_eq_iff
tff(fact_2892_add__divide__eq__iff,axiom,
! [Z: complex,X: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( aa_complex_complex(plus_plus_complex(X),aa_complex_complex(divide1717551699836669952omplex(Y),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(X),Z)),Y)),Z) ) ) ).
% add_divide_eq_iff
tff(fact_2893_add__divide__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),Y)),Z) ) ) ).
% add_divide_eq_iff
tff(fact_2894_add__divide__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( plus_plus_rat(X,divide_divide_rat(Y,Z)) = divide_divide_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),Z),Y),Z) ) ) ).
% add_divide_eq_iff
tff(fact_2895_add__num__frac,axiom,
! [Y: complex,Z: complex,X: complex] :
( ( Y != zero_zero_complex )
=> ( aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X),aa_complex_complex(times_times_complex(Z),Y))),Y) ) ) ).
% add_num_frac
tff(fact_2896_add__num__frac,axiom,
! [Y: real,Z: real,X: real] :
( ( Y != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))),Y) ) ) ).
% add_num_frac
tff(fact_2897_add__num__frac,axiom,
! [Y: rat,Z: rat,X: rat] :
( ( Y != zero_zero_rat )
=> ( plus_plus_rat(Z,divide_divide_rat(X,Y)) = divide_divide_rat(plus_plus_rat(X,aa_rat_rat(times_times_rat(Z),Y)),Y) ) ) ).
% add_num_frac
tff(fact_2898_add__frac__num,axiom,
! [Y: complex,X: complex,Z: complex] :
( ( Y != zero_zero_complex )
=> ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),Z) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X),aa_complex_complex(times_times_complex(Z),Y))),Y) ) ) ).
% add_frac_num
tff(fact_2899_add__frac__num,axiom,
! [Y: real,X: real,Z: real] :
( ( Y != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))),Y) ) ) ).
% add_frac_num
tff(fact_2900_add__frac__num,axiom,
! [Y: rat,X: rat,Z: rat] :
( ( Y != zero_zero_rat )
=> ( plus_plus_rat(divide_divide_rat(X,Y),Z) = divide_divide_rat(plus_plus_rat(X,aa_rat_rat(times_times_rat(Z),Y)),Y) ) ) ).
% add_frac_num
tff(fact_2901_add__frac__eq,axiom,
! [Y: complex,Z: complex,X: complex,W: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z != zero_zero_complex )
=> ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(W),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(X),Z)),aa_complex_complex(times_times_complex(W),Y))),aa_complex_complex(times_times_complex(Y),Z)) ) ) ) ).
% add_frac_eq
tff(fact_2902_add__frac__eq,axiom,
! [Y: real,Z: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ) ) ) ).
% add_frac_eq
tff(fact_2903_add__frac__eq,axiom,
! [Y: rat,Z: rat,X: rat,W: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( plus_plus_rat(divide_divide_rat(X,Y),divide_divide_rat(W,Z)) = divide_divide_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),Z),aa_rat_rat(times_times_rat(W),Y)),aa_rat_rat(times_times_rat(Y),Z)) ) ) ) ).
% add_frac_eq
tff(fact_2904_add__divide__eq__if__simps_I1_J,axiom,
! [Z: complex,A: complex,B: complex] :
( ( ( Z = zero_zero_complex )
=> ( aa_complex_complex(plus_plus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = A ) )
& ( ( Z != zero_zero_complex )
=> ( aa_complex_complex(plus_plus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),Z)),B)),Z) ) ) ) ).
% add_divide_eq_if_simps(1)
tff(fact_2905_add__divide__eq__if__simps_I1_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z)) = A ) )
& ( ( Z != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),Z)),B)),Z) ) ) ) ).
% add_divide_eq_if_simps(1)
tff(fact_2906_add__divide__eq__if__simps_I1_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( plus_plus_rat(A,divide_divide_rat(B,Z)) = A ) )
& ( ( Z != zero_zero_rat )
=> ( plus_plus_rat(A,divide_divide_rat(B,Z)) = divide_divide_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),Z),B),Z) ) ) ) ).
% add_divide_eq_if_simps(1)
tff(fact_2907_add__divide__eq__if__simps_I2_J,axiom,
! [Z: complex,A: complex,B: complex] :
( ( ( Z = zero_zero_complex )
=> ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B) = B ) )
& ( ( Z != zero_zero_complex )
=> ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),aa_complex_complex(times_times_complex(B),Z))),Z) ) ) ) ).
% add_divide_eq_if_simps(2)
tff(fact_2908_add__divide__eq__if__simps_I2_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = B ) )
& ( ( Z != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z) ) ) ) ).
% add_divide_eq_if_simps(2)
tff(fact_2909_add__divide__eq__if__simps_I2_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( plus_plus_rat(divide_divide_rat(A,Z),B) = B ) )
& ( ( Z != zero_zero_rat )
=> ( plus_plus_rat(divide_divide_rat(A,Z),B) = divide_divide_rat(plus_plus_rat(A,aa_rat_rat(times_times_rat(B),Z)),Z) ) ) ) ).
% add_divide_eq_if_simps(2)
tff(fact_2910_power__inject__base,axiom,
! [A: real,N: nat,B: real] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,N)) )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
=> ( A = B ) ) ) ) ).
% power_inject_base
tff(fact_2911_power__inject__base,axiom,
! [A: rat,N: nat,B: rat] :
( ( aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N)) = aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,N)) )
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
=> ( A = B ) ) ) ) ).
% power_inject_base
tff(fact_2912_power__inject__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,N)) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
=> ( A = B ) ) ) ) ).
% power_inject_base
tff(fact_2913_power__inject__base,axiom,
! [A: int,N: nat,B: int] :
( ( aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)) = aa_nat_int(power_power_int(B),aa_nat_nat(suc,N)) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
=> ( A = B ) ) ) ) ).
% power_inject_base
tff(fact_2914_power__le__imp__le__base,axiom,
! [A: real,N: nat,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,N))))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).
% power_le_imp_le_base
tff(fact_2915_power__le__imp__le__base,axiom,
! [A: rat,N: nat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N))),aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,N))))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ) ).
% power_le_imp_le_base
tff(fact_2916_power__le__imp__le__base,axiom,
! [A: nat,N: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,N))))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ) ).
% power_le_imp_le_base
tff(fact_2917_power__le__imp__le__base,axiom,
! [A: int,N: nat,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,N))),aa_nat_int(power_power_int(B),aa_nat_nat(suc,N))))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).
% power_le_imp_le_base
tff(fact_2918_div__add__self2,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).
% div_add_self2
tff(fact_2919_div__add__self2,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).
% div_add_self2
tff(fact_2920_div__add__self1,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).
% div_add_self1
tff(fact_2921_div__add__self1,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).
% div_add_self1
tff(fact_2922_add__divide__eq__if__simps_I4_J,axiom,
! [Z: complex,A: complex,B: complex] :
( ( ( Z = zero_zero_complex )
=> ( aa_complex_complex(minus_minus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = A ) )
& ( ( Z != zero_zero_complex )
=> ( aa_complex_complex(minus_minus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),Z)),B)),Z) ) ) ) ).
% add_divide_eq_if_simps(4)
tff(fact_2923_add__divide__eq__if__simps_I4_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z)) = A ) )
& ( ( Z != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),Z)),B)),Z) ) ) ) ).
% add_divide_eq_if_simps(4)
tff(fact_2924_add__divide__eq__if__simps_I4_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( minus_minus_rat(A,divide_divide_rat(B,Z)) = A ) )
& ( ( Z != zero_zero_rat )
=> ( minus_minus_rat(A,divide_divide_rat(B,Z)) = divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),Z),B),Z) ) ) ) ).
% add_divide_eq_if_simps(4)
tff(fact_2925_diff__frac__eq,axiom,
! [Y: complex,Z: complex,X: complex,W: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z != zero_zero_complex )
=> ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y)),aa_complex_complex(divide1717551699836669952omplex(W),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),Z)),aa_complex_complex(times_times_complex(W),Y))),aa_complex_complex(times_times_complex(Y),Z)) ) ) ) ).
% diff_frac_eq
tff(fact_2926_diff__frac__eq,axiom,
! [Y: real,Z: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ) ) ) ).
% diff_frac_eq
tff(fact_2927_diff__frac__eq,axiom,
! [Y: rat,Z: rat,X: rat,W: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( minus_minus_rat(divide_divide_rat(X,Y),divide_divide_rat(W,Z)) = divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z),aa_rat_rat(times_times_rat(W),Y)),aa_rat_rat(times_times_rat(Y),Z)) ) ) ) ).
% diff_frac_eq
tff(fact_2928_diff__divide__eq__iff,axiom,
! [Z: complex,X: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( aa_complex_complex(minus_minus_complex(X),aa_complex_complex(divide1717551699836669952omplex(Y),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),Z)),Y)),Z) ) ) ).
% diff_divide_eq_iff
tff(fact_2929_diff__divide__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),Y)),Z) ) ) ).
% diff_divide_eq_iff
tff(fact_2930_diff__divide__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( minus_minus_rat(X,divide_divide_rat(Y,Z)) = divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z),Y),Z) ) ) ).
% diff_divide_eq_iff
tff(fact_2931_divide__diff__eq__iff,axiom,
! [Z: complex,X: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Z)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(X),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).
% divide_diff_eq_iff
tff(fact_2932_divide__diff__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z)),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).
% divide_diff_eq_iff
tff(fact_2933_divide__diff__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( minus_minus_rat(divide_divide_rat(X,Z),Y) = divide_divide_rat(minus_minus_rat(X,aa_rat_rat(times_times_rat(Y),Z)),Z) ) ) ).
% divide_diff_eq_iff
tff(fact_2934_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> ( modulo364778990260209775nteger(A,B) = A ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_2935_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = A ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_2936_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = A ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_2937_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),modulo364778990260209775nteger(A,B))) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_2938_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_2939_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_2940_cong__exp__iff__simps_I2_J,axiom,
! [N: num,Q2: num] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = zero_zero_nat )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).
% cong_exp_iff_simps(2)
tff(fact_2941_cong__exp__iff__simps_I2_J,axiom,
! [N: num,Q2: num] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = zero_zero_int )
<=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).
% cong_exp_iff_simps(2)
tff(fact_2942_cong__exp__iff__simps_I2_J,axiom,
! [N: num,Q2: num] :
( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,N)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) = zero_z3403309356797280102nteger )
<=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(N),numera6620942414471956472nteger(Q2)) = zero_z3403309356797280102nteger ) ) ).
% cong_exp_iff_simps(2)
tff(fact_2943_cong__exp__iff__simps_I1_J,axiom,
! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,one)) = zero_zero_nat ).
% cong_exp_iff_simps(1)
tff(fact_2944_cong__exp__iff__simps_I1_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,one)) = zero_zero_int ).
% cong_exp_iff_simps(1)
tff(fact_2945_cong__exp__iff__simps_I1_J,axiom,
! [N: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(N),numera6620942414471956472nteger(one)) = zero_z3403309356797280102nteger ).
% cong_exp_iff_simps(1)
tff(fact_2946_numeral__1__eq__Suc__0,axiom,
aa_num_nat(numeral_numeral_nat,one) = aa_nat_nat(suc,zero_zero_nat) ).
% numeral_1_eq_Suc_0
tff(fact_2947_num_Osize_I5_J,axiom,
! [X23: num] : aa_num_nat(size_size_num,aa_num_num(bit0,X23)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X23)),aa_nat_nat(suc,zero_zero_nat)) ).
% num.size(5)
tff(fact_2948_ex__least__nat__less,axiom,
! [P: fun_nat_bool,N: nat] :
( pp(aa_nat_bool(P,N))
=> ( ~ pp(aa_nat_bool(P,zero_zero_nat))
=> ? [K3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),N))
& ! [I3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I3),K3))
=> ~ pp(aa_nat_bool(P,I3)) )
& pp(aa_nat_bool(P,aa_nat_nat(suc,K3))) ) ) ) ).
% ex_least_nat_less
tff(fact_2949_nat__induct__non__zero,axiom,
! [N: nat,P: fun_nat_bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_nat_bool(P,one_one_nat))
=> ( ! [N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
=> ( pp(aa_nat_bool(P,N2))
=> pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) )
=> pp(aa_nat_bool(P,N)) ) ) ) ).
% nat_induct_non_zero
tff(fact_2950_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,I))),N)) ) ).
% diff_Suc_less
tff(fact_2951_num_Osize_I6_J,axiom,
! [X33: num] : aa_num_nat(size_size_num,aa_num_num(bit1,X33)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X33)),aa_nat_nat(suc,zero_zero_nat)) ).
% num.size(6)
tff(fact_2952_one__less__mult,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))) ) ) ).
% one_less_mult
tff(fact_2953_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))) ) ) ).
% n_less_m_mult_n
tff(fact_2954_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M))) ) ) ).
% n_less_n_mult_m
tff(fact_2955_length__pos__if__in__set,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s5460976970255530739at_nat(Xs))) ) ).
% length_pos_if_in_set
tff(fact_2956_length__pos__if__in__set,axiom,
! [X: complex,Xs: list_complex] :
( pp(aa_set_complex_bool(member_complex(X),set_complex2(Xs)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s3451745648224563538omplex(Xs))) ) ).
% length_pos_if_in_set
tff(fact_2957_length__pos__if__in__set,axiom,
! [X: real,Xs: list_real] :
( pp(aa_set_real_bool(member_real(X),set_real2(Xs)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_real(Xs))) ) ).
% length_pos_if_in_set
tff(fact_2958_length__pos__if__in__set,axiom,
! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(Xs)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs))) ) ).
% length_pos_if_in_set
tff(fact_2959_length__pos__if__in__set,axiom,
! [X: bool,Xs: list_bool] :
( member_bool(X,set_bool2(Xs))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_bool(Xs))) ) ).
% length_pos_if_in_set
tff(fact_2960_length__pos__if__in__set,axiom,
! [X: nat,Xs: list_nat] :
( pp(aa_set_nat_bool(member_nat(X),set_nat2(Xs)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_nat(Xs))) ) ).
% length_pos_if_in_set
tff(fact_2961_length__pos__if__in__set,axiom,
! [X: int,Xs: list_int] :
( pp(aa_set_int_bool(member_int(X),set_int2(Xs)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_int(Xs))) ) ).
% length_pos_if_in_set
tff(fact_2962_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).
% nat_mult_le_cancel1
tff(fact_2963_nat__diff__split,axiom,
! [P: fun_nat_bool,A: nat,B: nat] :
( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)))
<=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> pp(aa_nat_bool(P,zero_zero_nat)) )
& ! [D4: nat] :
( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D4) )
=> pp(aa_nat_bool(P,D4)) ) ) ) ).
% nat_diff_split
tff(fact_2964_nat__diff__split__asm,axiom,
! [P: fun_nat_bool,A: nat,B: nat] :
( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)))
<=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
& ~ pp(aa_nat_bool(P,zero_zero_nat)) )
| ? [D4: nat] :
( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D4) )
& ~ pp(aa_nat_bool(P,D4)) ) ) ) ).
% nat_diff_split_asm
tff(fact_2965_power__gt__expt,axiom,
! [N: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),K))) ) ).
% power_gt_expt
tff(fact_2966_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).
% div_greater_zero_iff
tff(fact_2967_div__le__mono2,axiom,
! [M: nat,N: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),M))) ) ) ).
% div_le_mono2
tff(fact_2968_nat__one__le__power,axiom,
! [I: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),I))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N))) ) ).
% nat_one_le_power
tff(fact_2969_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = M )
<=> ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
tff(fact_2970_div__less__dividend,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),M)) ) ) ).
% div_less_dividend
tff(fact_2971_div__less__iff__less__mult,axiom,
! [Q2: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Q2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Q2)),N))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2))) ) ) ).
% div_less_iff_less_mult
tff(fact_2972_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> ( aa_nat_nat(aa_nat_fun_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),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) ) ) ).
% nat_mult_div_cancel1
tff(fact_2973_mod__le__divisor,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)),N)) ) ).
% mod_le_divisor
tff(fact_2974_div__less__mono,axiom,
! [A2: nat,B2: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),N) = zero_zero_nat )
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),N) = zero_zero_nat )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),N))) ) ) ) ) ).
% div_less_mono
tff(fact_2975_vebt__insert_Osimps_I3_J,axiom,
! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S),X) = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S) ).
% vebt_insert.simps(3)
tff(fact_2976_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I2_J,axiom,
! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t(vEBT_Node(Info,zero_zero_nat,Ts2,S)),X) = one_one_nat ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
tff(fact_2977_vebt__member_Osimps_I3_J,axiom,
! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] : ~ pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy,Uz)),X)) ).
% vebt_member.simps(3)
tff(fact_2978_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] : ~ pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy)),Uz)) ).
% VEBT_internal.membermima.simps(2)
tff(fact_2979_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I2_J,axiom,
! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t2(vEBT_Node(Info,zero_zero_nat,Ts2,S)),X) = one_one_nat ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
tff(fact_2980_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
! [X: vEBT_VEBT] :
( ! [A4: bool,B4: bool] : X != vEBT_Leaf(A4,B4)
=> ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.cases
tff(fact_2981_VEBT__internal_OminNull_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Y: bool] :
( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
<=> pp(Y) )
=> ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
=> ~ pp(Y) )
=> ( ( ? [Uv2: bool] : X = vEBT_Leaf(fTrue,Uv2)
=> pp(Y) )
=> ( ( ? [Uu2: bool] : X = vEBT_Leaf(Uu2,fTrue)
=> pp(Y) )
=> ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)
=> ~ pp(Y) )
=> ~ ( ? [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2,Vc2)
=> pp(Y) ) ) ) ) ) ) ).
% VEBT_internal.minNull.elims(1)
tff(fact_2982_field__le__mult__one__interval,axiom,
! [X: real,Y: real] :
( ! [Z3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),one_one_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z3),X)),Y)) ) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).
% field_le_mult_one_interval
tff(fact_2983_field__le__mult__one__interval,axiom,
! [X: rat,Y: rat] :
( ! [Z3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z3))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),one_one_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z3),X)),Y)) ) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).
% field_le_mult_one_interval
tff(fact_2984_mult__less__cancel__right2,axiom,
! [A: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),C))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real)) )
& ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A)) ) ) ) ).
% mult_less_cancel_right2
tff(fact_2985_mult__less__cancel__right2,axiom,
! [A: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),C))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat)) )
& ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A)) ) ) ) ).
% mult_less_cancel_right2
tff(fact_2986_mult__less__cancel__right2,axiom,
! [A: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int)) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A)) ) ) ) ).
% mult_less_cancel_right2
tff(fact_2987_mult__less__cancel__right1,axiom,
! [C: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B)) )
& ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),one_one_real)) ) ) ) ).
% mult_less_cancel_right1
tff(fact_2988_mult__less__cancel__right1,axiom,
! [C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(times_times_rat(B),C)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B)) )
& ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),one_one_rat)) ) ) ) ).
% mult_less_cancel_right1
tff(fact_2989_mult__less__cancel__right1,axiom,
! [C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B)) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),one_one_int)) ) ) ) ).
% mult_less_cancel_right1
tff(fact_2990_mult__less__cancel__left2,axiom,
! [C: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),C))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real)) )
& ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A)) ) ) ) ).
% mult_less_cancel_left2
tff(fact_2991_mult__less__cancel__left2,axiom,
! [C: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),C))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat)) )
& ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A)) ) ) ) ).
% mult_less_cancel_left2
tff(fact_2992_mult__less__cancel__left2,axiom,
! [C: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int)) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A)) ) ) ) ).
% mult_less_cancel_left2
tff(fact_2993_mult__less__cancel__left1,axiom,
! [C: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B)) )
& ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),one_one_real)) ) ) ) ).
% mult_less_cancel_left1
tff(fact_2994_mult__less__cancel__left1,axiom,
! [C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),aa_rat_rat(times_times_rat(C),B)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),B)) )
& ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),one_one_rat)) ) ) ) ).
% mult_less_cancel_left1
tff(fact_2995_mult__less__cancel__left1,axiom,
! [C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B)) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),one_one_int)) ) ) ) ).
% mult_less_cancel_left1
tff(fact_2996_mult__le__cancel__right2,axiom,
! [A: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),C))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real)) )
& ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A)) ) ) ) ).
% mult_le_cancel_right2
tff(fact_2997_mult__le__cancel__right2,axiom,
! [A: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),C))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat)) )
& ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A)) ) ) ) ).
% mult_le_cancel_right2
tff(fact_2998_mult__le__cancel__right2,axiom,
! [A: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int)) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A)) ) ) ) ).
% mult_le_cancel_right2
tff(fact_2999_mult__le__cancel__right1,axiom,
! [C: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),B)) )
& ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),one_one_real)) ) ) ) ).
% mult_le_cancel_right1
tff(fact_3000_mult__le__cancel__right1,axiom,
! [C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(times_times_rat(B),C)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),B)) )
& ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),one_one_rat)) ) ) ) ).
% mult_le_cancel_right1
tff(fact_3001_mult__le__cancel__right1,axiom,
! [C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),B)) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),one_one_int)) ) ) ) ).
% mult_le_cancel_right1
tff(fact_3002_mult__le__cancel__left2,axiom,
! [C: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),C))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real)) )
& ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A)) ) ) ) ).
% mult_le_cancel_left2
tff(fact_3003_mult__le__cancel__left2,axiom,
! [C: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),C))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat)) )
& ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A)) ) ) ) ).
% mult_le_cancel_left2
tff(fact_3004_mult__le__cancel__left2,axiom,
! [C: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int)) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A)) ) ) ) ).
% mult_le_cancel_left2
tff(fact_3005_mult__le__cancel__left1,axiom,
! [C: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),B)) )
& ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),one_one_real)) ) ) ) ).
% mult_le_cancel_left1
tff(fact_3006_mult__le__cancel__left1,axiom,
! [C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),aa_rat_rat(times_times_rat(C),B)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),B)) )
& ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),one_one_rat)) ) ) ) ).
% mult_le_cancel_left1
tff(fact_3007_mult__le__cancel__left1,axiom,
! [C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),B)) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),one_one_int)) ) ) ) ).
% mult_le_cancel_left1
tff(fact_3008_divide__left__mono__neg,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B))) ) ) ) ).
% divide_left_mono_neg
tff(fact_3009_divide__left__mono__neg,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(C,A)),divide_divide_rat(C,B))) ) ) ) ).
% divide_left_mono_neg
tff(fact_3010_mult__imp__le__div__pos,axiom,
! [Y: real,Z: real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y))) ) ) ).
% mult_imp_le_div_pos
tff(fact_3011_mult__imp__le__div__pos,axiom,
! [Y: rat,Z: rat,X: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z),Y)),X))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),divide_divide_rat(X,Y))) ) ) ).
% mult_imp_le_div_pos
tff(fact_3012_mult__imp__div__pos__le,axiom,
! [Y: real,X: real,Z: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),Z)) ) ) ).
% mult_imp_div_pos_le
tff(fact_3013_mult__imp__div__pos__le,axiom,
! [Y: rat,X: rat,Z: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_rat_rat(times_times_rat(Z),Y)))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(X,Y)),Z)) ) ) ).
% mult_imp_div_pos_le
tff(fact_3014_pos__le__divide__eq,axiom,
! [C: real,A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) ) ) ).
% pos_le_divide_eq
tff(fact_3015_pos__le__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),divide_divide_rat(B,C)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B)) ) ) ).
% pos_le_divide_eq
tff(fact_3016_pos__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).
% pos_divide_le_eq
tff(fact_3017_pos__divide__le__eq,axiom,
! [C: rat,B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(B,C)),A))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C))) ) ) ).
% pos_divide_le_eq
tff(fact_3018_neg__le__divide__eq,axiom,
! [C: real,A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).
% neg_le_divide_eq
tff(fact_3019_neg__le__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),divide_divide_rat(B,C)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C))) ) ) ).
% neg_le_divide_eq
tff(fact_3020_neg__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) ) ) ).
% neg_divide_le_eq
tff(fact_3021_neg__divide__le__eq,axiom,
! [C: rat,B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(B,C)),A))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B)) ) ) ).
% neg_divide_le_eq
tff(fact_3022_divide__left__mono,axiom,
! [B: real,A: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B))) ) ) ) ).
% divide_left_mono
tff(fact_3023_divide__left__mono,axiom,
! [B: rat,A: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),C))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(C,A)),divide_divide_rat(C,B))) ) ) ) ).
% divide_left_mono
tff(fact_3024_le__divide__eq,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ) ) ) ) ).
% le_divide_eq
tff(fact_3025_le__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),divide_divide_rat(B,C)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B)) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ) ) ) ) ).
% le_divide_eq
tff(fact_3026_divide__le__eq,axiom,
! [B: real,C: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),A))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B)) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ) ) ).
% divide_le_eq
tff(fact_3027_divide__le__eq,axiom,
! [B: rat,C: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(B,C)),A))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B)) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ) ) ) ).
% divide_le_eq
tff(fact_3028_le__divide__eq__1,axiom,
! [B: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
| ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) ) ) ) ).
% le_divide_eq_1
tff(fact_3029_le__divide__eq__1,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),divide_divide_rat(B,A)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ) ) ).
% le_divide_eq_1
tff(fact_3030_divide__le__eq__1,axiom,
! [B: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),A)),one_one_real))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),A)) )
| ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) )
| ( A = zero_zero_real ) ) ) ).
% divide_le_eq_1
tff(fact_3031_divide__le__eq__1,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(B,A)),one_one_rat))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) )
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) )
| ( A = zero_zero_rat ) ) ) ).
% divide_le_eq_1
tff(fact_3032_convex__bound__le,axiom,
! [X: real,A: real,Y: real,U: real,V: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),U))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),V))
=> ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A)) ) ) ) ) ) ).
% convex_bound_le
tff(fact_3033_convex__bound__le,axiom,
! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),U))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),V))
=> ( ( plus_plus_rat(U,V) = one_one_rat )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(aa_rat_rat(times_times_rat(U),X),aa_rat_rat(times_times_rat(V),Y))),A)) ) ) ) ) ) ).
% convex_bound_le
tff(fact_3034_convex__bound__le,axiom,
! [X: int,A: int,Y: int,U: int,V: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),U))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),V))
=> ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A)) ) ) ) ) ) ).
% convex_bound_le
tff(fact_3035_less__divide__eq__numeral_I1_J,axiom,
! [W: num,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(W)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),B)) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C))) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(W)),zero_zero_real)) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
tff(fact_3036_less__divide__eq__numeral_I1_J,axiom,
! [W: num,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,numeral_numeral_rat(W)),divide_divide_rat(B,C)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),B)) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,numeral_numeral_rat(W)),zero_zero_rat)) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
tff(fact_3037_divide__less__eq__numeral_I1_J,axiom,
! [B: real,C: real,W: num] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),numeral_numeral_real(W)))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C))) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),B)) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),numeral_numeral_real(W))) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
tff(fact_3038_divide__less__eq__numeral_I1_J,axiom,
! [B: rat,C: rat,W: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(B,C)),numeral_numeral_rat(W)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),B)) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),numeral_numeral_rat(W))) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
tff(fact_3039_frac__le__eq,axiom,
! [Y: real,Z: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),zero_zero_real)) ) ) ) ).
% frac_le_eq
tff(fact_3040_frac__le__eq,axiom,
! [Y: rat,Z: rat,X: rat,W: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(X,Y)),divide_divide_rat(W,Z)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z),aa_rat_rat(times_times_rat(W),Y)),aa_rat_rat(times_times_rat(Y),Z))),zero_zero_rat)) ) ) ) ).
% frac_le_eq
tff(fact_3041_power__Suc__less,axiom,
! [A: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ) ).
% power_Suc_less
tff(fact_3042_power__Suc__less,axiom,
! [A: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),N))),aa_nat_rat(power_power_rat(A),N))) ) ) ).
% power_Suc_less
tff(fact_3043_power__Suc__less,axiom,
! [A: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),one_one_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ).
% power_Suc_less
tff(fact_3044_power__Suc__less,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N))),aa_nat_int(power_power_int(A),N))) ) ) ).
% power_Suc_less
tff(fact_3045_frac__less__eq,axiom,
! [Y: real,Z: real,X: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),zero_zero_real)) ) ) ) ).
% frac_less_eq
tff(fact_3046_frac__less__eq,axiom,
! [Y: rat,Z: rat,X: rat,W: rat] :
( ( Y != zero_zero_rat )
=> ( ( Z != zero_zero_rat )
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(X,Y)),divide_divide_rat(W,Z)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z),aa_rat_rat(times_times_rat(W),Y)),aa_rat_rat(times_times_rat(Y),Z))),zero_zero_rat)) ) ) ) ).
% frac_less_eq
tff(fact_3047_vebt__mint_Oelims,axiom,
! [X: vEBT_VEBT,Y: option_nat] :
( ( vEBT_vebt_mint(X) = Y )
=> ( ! [A4: bool,B4: bool] :
( ( X = vEBT_Leaf(A4,B4) )
=> ~ ( ( pp(A4)
=> ( Y = some_nat(zero_zero_nat) ) )
& ( ~ pp(A4)
=> ( ( pp(B4)
=> ( Y = some_nat(one_one_nat) ) )
& ( ~ pp(B4)
=> ( Y = none_nat ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
=> ( Y != none_nat ) )
=> ~ ! [Mi2: nat] :
( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2)
=> ( Y != some_nat(Mi2) ) ) ) ) ) ).
% vebt_mint.elims
tff(fact_3048_power__Suc__le__self,axiom,
! [A: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,N))),A)) ) ) ).
% power_Suc_le_self
tff(fact_3049_power__Suc__le__self,axiom,
! [A: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N))),A)) ) ) ).
% power_Suc_le_self
tff(fact_3050_power__Suc__le__self,axiom,
! [A: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N))),A)) ) ) ).
% power_Suc_le_self
tff(fact_3051_power__Suc__le__self,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,N))),A)) ) ) ).
% power_Suc_le_self
tff(fact_3052_power__Suc__less__one,axiom,
! [A: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,N))),one_one_real)) ) ) ).
% power_Suc_less_one
tff(fact_3053_power__Suc__less__one,axiom,
! [A: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,N))),one_one_rat)) ) ) ).
% power_Suc_less_one
tff(fact_3054_power__Suc__less__one,axiom,
! [A: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),one_one_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N))),one_one_nat)) ) ) ).
% power_Suc_less_one
tff(fact_3055_power__Suc__less__one,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,N))),one_one_int)) ) ) ).
% power_Suc_less_one
tff(fact_3056_vebt__maxt_Oelims,axiom,
! [X: vEBT_VEBT,Y: option_nat] :
( ( vEBT_vebt_maxt(X) = Y )
=> ( ! [A4: bool,B4: bool] :
( ( X = vEBT_Leaf(A4,B4) )
=> ~ ( ( pp(B4)
=> ( Y = some_nat(one_one_nat) ) )
& ( ~ pp(B4)
=> ( ( pp(A4)
=> ( Y = some_nat(zero_zero_nat) ) )
& ( ~ pp(A4)
=> ( Y = none_nat ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
=> ( Y != none_nat ) )
=> ~ ! [Mi2: nat,Ma2: nat] :
( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2)
=> ( Y != some_nat(Ma2) ) ) ) ) ) ).
% vebt_maxt.elims
tff(fact_3057_power__strict__decreasing,axiom,
! [N: nat,N3: nat,A: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),one_one_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N3)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ) ) ).
% power_strict_decreasing
tff(fact_3058_power__strict__decreasing,axiom,
! [N: nat,N3: nat,A: rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),one_one_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N3)),aa_nat_rat(power_power_rat(A),N))) ) ) ) ).
% power_strict_decreasing
tff(fact_3059_power__strict__decreasing,axiom,
! [N: nat,N3: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),one_one_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N3)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ) ).
% power_strict_decreasing
tff(fact_3060_power__strict__decreasing,axiom,
! [N: nat,N3: nat,A: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),one_one_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N3)),aa_nat_int(power_power_int(A),N))) ) ) ) ).
% power_strict_decreasing
tff(fact_3061_power__decreasing,axiom,
! [N: nat,N3: nat,A: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),one_one_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N3)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ) ) ).
% power_decreasing
tff(fact_3062_power__decreasing,axiom,
! [N: nat,N3: nat,A: rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),one_one_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N3)),aa_nat_rat(power_power_rat(A),N))) ) ) ) ).
% power_decreasing
tff(fact_3063_power__decreasing,axiom,
! [N: nat,N3: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),one_one_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N3)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ) ).
% power_decreasing
tff(fact_3064_power__decreasing,axiom,
! [N: nat,N3: nat,A: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),one_one_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N3)),aa_nat_int(power_power_int(A),N))) ) ) ) ).
% power_decreasing
tff(fact_3065_zero__power2,axiom,
aa_nat_rat(power_power_rat(zero_zero_rat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_rat ).
% zero_power2
tff(fact_3066_zero__power2,axiom,
aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).
% zero_power2
tff(fact_3067_zero__power2,axiom,
aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real ).
% zero_power2
tff(fact_3068_zero__power2,axiom,
aa_nat_complex(power_power_complex(zero_zero_complex),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_complex ).
% zero_power2
tff(fact_3069_zero__power2,axiom,
aa_nat_int(power_power_int(zero_zero_int),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_int ).
% zero_power2
tff(fact_3070_self__le__power,axiom,
! [A: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ) ).
% self_le_power
tff(fact_3071_self__le__power,axiom,
! [A: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),aa_nat_rat(power_power_rat(A),N))) ) ) ).
% self_le_power
tff(fact_3072_self__le__power,axiom,
! [A: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ).
% self_le_power
tff(fact_3073_self__le__power,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(A),N))) ) ) ).
% self_le_power
tff(fact_3074_cong__exp__iff__simps_I3_J,axiom,
! [N: num,Q2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != zero_zero_nat ).
% cong_exp_iff_simps(3)
tff(fact_3075_cong__exp__iff__simps_I3_J,axiom,
! [N: num,Q2: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != zero_zero_int ).
% cong_exp_iff_simps(3)
tff(fact_3076_cong__exp__iff__simps_I3_J,axiom,
! [N: num,Q2: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,N)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) != zero_z3403309356797280102nteger ).
% cong_exp_iff_simps(3)
tff(fact_3077_one__less__power,axiom,
! [A: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ) ).
% one_less_power
tff(fact_3078_one__less__power,axiom,
! [A: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),N))) ) ) ).
% one_less_power
tff(fact_3079_one__less__power,axiom,
! [A: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ) ).
% one_less_power
tff(fact_3080_one__less__power,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),N))) ) ) ).
% one_less_power
tff(fact_3081_numeral__2__eq__2,axiom,
aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) = aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat)) ).
% numeral_2_eq_2
tff(fact_3082_pos2,axiom,
pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).
% pos2
tff(fact_3083_power__diff,axiom,
! [A: complex,N: nat,M: nat] :
( ( A != zero_zero_complex )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),M)),aa_nat_complex(power_power_complex(A),N)) ) ) ) ).
% power_diff
tff(fact_3084_power__diff,axiom,
! [A: real,N: nat,M: nat] :
( ( A != zero_zero_real )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ) ) ) ).
% power_diff
tff(fact_3085_power__diff,axiom,
! [A: rat,N: nat,M: nat] :
( ( A != zero_zero_rat )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = divide_divide_rat(aa_nat_rat(power_power_rat(A),M),aa_nat_rat(power_power_rat(A),N)) ) ) ) ).
% power_diff
tff(fact_3086_power__diff,axiom,
! [A: nat,N: nat,M: nat] :
( ( A != zero_zero_nat )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ) ) ).
% power_diff
tff(fact_3087_power__diff,axiom,
! [A: int,N: nat,M: nat] :
( ( A != zero_zero_int )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)) ) ) ) ).
% power_diff
tff(fact_3088_numeral__3__eq__3,axiom,
aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) = aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))) ).
% numeral_3_eq_3
tff(fact_3089_Suc__pred_H,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).
% Suc_pred'
tff(fact_3090_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).
% Suc_diff_eq_diff_pred
tff(fact_3091_div__geq,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ) ) ) ).
% div_geq
tff(fact_3092_div__if,axiom,
! [M: nat,N: nat] :
( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
| ( N = zero_zero_nat ) )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat ) )
& ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
| ( N = zero_zero_nat ) )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ) ) ) ).
% div_if
tff(fact_3093_add__eq__if,axiom,
! [M: nat,N: nat] :
( ( ( M = zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = N ) )
& ( ( M != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N)) ) ) ) ).
% add_eq_if
tff(fact_3094_less__eq__div__iff__mult__less__eq,axiom,
! [Q2: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Q2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),Q2)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)),N)) ) ) ).
% less_eq_div_iff_mult_less_eq
tff(fact_3095_dividend__less__times__div,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N))))) ) ).
% dividend_less_times_div
tff(fact_3096_dividend__less__div__times,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),N)))) ) ).
% dividend_less_div_times
tff(fact_3097_split__div,axiom,
! [P: fun_nat_bool,M: nat,N: nat] :
( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)))
<=> ( ( ( N = zero_zero_nat )
=> pp(aa_nat_bool(P,zero_zero_nat)) )
& ( ( N != zero_zero_nat )
=> ! [I4: nat,J3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N))
=> ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),I4)),J3) )
=> pp(aa_nat_bool(P,I4)) ) ) ) ) ) ).
% split_div
tff(fact_3098_mult__eq__if,axiom,
! [M: nat,N: nat] :
( ( ( M = zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = zero_zero_nat ) )
& ( ( M != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N)) ) ) ) ).
% mult_eq_if
tff(fact_3099_split__mod,axiom,
! [P: fun_nat_bool,M: nat,N: nat] :
( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))
<=> ( ( ( N = zero_zero_nat )
=> pp(aa_nat_bool(P,M)) )
& ( ( N != zero_zero_nat )
=> ! [I4: nat,J3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N))
=> ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),I4)),J3) )
=> pp(aa_nat_bool(P,J3)) ) ) ) ) ) ).
% split_mod
tff(fact_3100_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I3_J,axiom,
! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S)),X) = one_one_nat ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
tff(fact_3101_vebt__member_Osimps_I4_J,axiom,
! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] : ~ pp(aa_nat_bool(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb,Vc)),X)) ).
% vebt_member.simps(4)
tff(fact_3102_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I3_J,axiom,
! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t2(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S)),X) = one_one_nat ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
tff(fact_3103_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
( pp(aa_nat_bool(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va,Vb)),X))
<=> ( ( X = Mi )
| ( X = Ma ) ) ) ).
% VEBT_internal.membermima.simps(3)
tff(fact_3104_vebt__pred_Osimps_I5_J,axiom,
! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] : vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd,Ve2),Vf2) = none_nat ).
% vebt_pred.simps(5)
tff(fact_3105_vebt__succ_Osimps_I4_J,axiom,
! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve2: nat] : vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc,Vd),Ve2) = none_nat ).
% vebt_succ.simps(4)
tff(fact_3106_convex__bound__lt,axiom,
! [X: real,A: real,Y: real,U: real,V: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),U))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),V))
=> ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A)) ) ) ) ) ) ).
% convex_bound_lt
tff(fact_3107_convex__bound__lt,axiom,
! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),U))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),V))
=> ( ( plus_plus_rat(U,V) = one_one_rat )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(aa_rat_rat(times_times_rat(U),X),aa_rat_rat(times_times_rat(V),Y))),A)) ) ) ) ) ) ).
% convex_bound_lt
tff(fact_3108_convex__bound__lt,axiom,
! [X: int,A: int,Y: int,U: int,V: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),U))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),V))
=> ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A)) ) ) ) ) ) ).
% convex_bound_lt
tff(fact_3109_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Oelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,X) = Y )
=> ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
=> ( Y != one_one_nat ) )
=> ( ( ? [Uv2: bool] : X = vEBT_Leaf(fTrue,Uv2)
=> ( Y != one_one_nat ) )
=> ( ( ? [Uu2: bool] : X = vEBT_Leaf(Uu2,fTrue)
=> ( Y != one_one_nat ) )
=> ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)
=> ( Y != one_one_nat ) )
=> ~ ( ? [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2,Vc2)
=> ( Y != one_one_nat ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.elims
tff(fact_3110_le__divide__eq__numeral_I1_J,axiom,
! [W: num,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(W)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),B)) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C))) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(W)),zero_zero_real)) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
tff(fact_3111_le__divide__eq__numeral_I1_J,axiom,
! [W: num,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,numeral_numeral_rat(W)),divide_divide_rat(B,C)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),B)) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,numeral_numeral_rat(W)),zero_zero_rat)) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
tff(fact_3112_divide__le__eq__numeral_I1_J,axiom,
! [B: real,C: real,W: num] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),numeral_numeral_real(W)))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C))) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),C)),B)) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),numeral_numeral_real(W))) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
tff(fact_3113_divide__le__eq__numeral_I1_J,axiom,
! [B: rat,C: rat,W: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(B,C)),numeral_numeral_rat(W)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),B)) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),numeral_numeral_rat(W))) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
tff(fact_3114_half__gt__zero__iff,axiom,
! [A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),numeral_numeral_real(aa_num_num(bit0,one)))))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).
% half_gt_zero_iff
tff(fact_3115_half__gt__zero__iff,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),divide_divide_rat(A,numeral_numeral_rat(aa_num_num(bit0,one)))))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).
% half_gt_zero_iff
tff(fact_3116_half__gt__zero,axiom,
! [A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),numeral_numeral_real(aa_num_num(bit0,one))))) ) ).
% half_gt_zero
tff(fact_3117_half__gt__zero,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),divide_divide_rat(A,numeral_numeral_rat(aa_num_num(bit0,one))))) ) ).
% half_gt_zero
tff(fact_3118_scaling__mono,axiom,
! [U: real,V: real,R: real,S: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,U),V))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),R))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,R),S))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,V),U))),S))),V)) ) ) ) ).
% scaling_mono
tff(fact_3119_scaling__mono,axiom,
! [U: rat,V: rat,R: rat,S: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,U),V))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),R))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,R),S))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(U,divide_divide_rat(aa_rat_rat(times_times_rat(R),minus_minus_rat(V,U)),S))),V)) ) ) ) ).
% scaling_mono
tff(fact_3120_power2__le__imp__le,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).
% power2_le_imp_le
tff(fact_3121_power2__le__imp__le,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ) ).
% power2_le_imp_le
tff(fact_3122_power2__le__imp__le,axiom,
! [X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).
% power2_le_imp_le
tff(fact_3123_power2__le__imp__le,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ).
% power2_le_imp_le
tff(fact_3124_power2__eq__imp__eq,axiom,
! [X: real,Y: real] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
=> ( X = Y ) ) ) ) ).
% power2_eq_imp_eq
tff(fact_3125_power2__eq__imp__eq,axiom,
! [X: rat,Y: rat] :
( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
=> ( X = Y ) ) ) ) ).
% power2_eq_imp_eq
tff(fact_3126_power2__eq__imp__eq,axiom,
! [X: nat,Y: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
=> ( X = Y ) ) ) ) ).
% power2_eq_imp_eq
tff(fact_3127_power2__eq__imp__eq,axiom,
! [X: int,Y: int] :
( ( aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
=> ( X = Y ) ) ) ) ).
% power2_eq_imp_eq
tff(fact_3128_zero__le__power2,axiom,
! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).
% zero_le_power2
tff(fact_3129_zero__le__power2,axiom,
! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).
% zero_le_power2
tff(fact_3130_zero__le__power2,axiom,
! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).
% zero_le_power2
tff(fact_3131_power2__less__0,axiom,
! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real)) ).
% power2_less_0
tff(fact_3132_power2__less__0,axiom,
! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat)) ).
% power2_less_0
tff(fact_3133_power2__less__0,axiom,
! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int)) ).
% power2_less_0
tff(fact_3134_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
=> ( modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),modulo364778990260209775nteger(divide6298287555418463151nteger(A,B),C)),modulo364778990260209775nteger(A,B)) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3135_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [C: nat,A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3136_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3137_exp__add__not__zero__imp__right,axiom,
! [M: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) != zero_zero_nat ) ) ).
% exp_add_not_zero_imp_right
tff(fact_3138_exp__add__not__zero__imp__right,axiom,
! [M: nat,N: nat] :
( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_zero_int )
=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) != zero_zero_int ) ) ).
% exp_add_not_zero_imp_right
tff(fact_3139_exp__add__not__zero__imp__left,axiom,
! [M: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) != zero_zero_nat ) ) ).
% exp_add_not_zero_imp_left
tff(fact_3140_exp__add__not__zero__imp__left,axiom,
! [M: nat,N: nat] :
( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_zero_int )
=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M) != zero_zero_int ) ) ).
% exp_add_not_zero_imp_left
tff(fact_3141_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [N: nat,M: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) != zero_zero_nat ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
tff(fact_3142_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [N: nat,M: nat] :
( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) != zero_zero_int )
=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) != zero_zero_int ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
tff(fact_3143_cong__exp__iff__simps_I7_J,axiom,
! [Q2: num,N: num] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).
% cong_exp_iff_simps(7)
tff(fact_3144_cong__exp__iff__simps_I7_J,axiom,
! [Q2: num,N: num] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
<=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).
% cong_exp_iff_simps(7)
tff(fact_3145_cong__exp__iff__simps_I7_J,axiom,
! [Q2: num,N: num] :
( ( modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,N)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) )
<=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(N),numera6620942414471956472nteger(Q2)) = zero_z3403309356797280102nteger ) ) ).
% cong_exp_iff_simps(7)
tff(fact_3146_cong__exp__iff__simps_I11_J,axiom,
! [M: num,Q2: num] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).
% cong_exp_iff_simps(11)
tff(fact_3147_cong__exp__iff__simps_I11_J,axiom,
! [M: num,Q2: num] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
<=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).
% cong_exp_iff_simps(11)
tff(fact_3148_cong__exp__iff__simps_I11_J,axiom,
! [M: num,Q2: num] :
( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,M)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) = modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) )
<=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Q2)) = zero_z3403309356797280102nteger ) ) ).
% cong_exp_iff_simps(11)
tff(fact_3149_power__diff__power__eq,axiom,
! [A: nat,N: nat,M: nat] :
( ( A != zero_zero_nat )
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ) ) ).
% power_diff_power_eq
tff(fact_3150_power__diff__power__eq,axiom,
! [A: int,N: nat,M: nat] :
( ( A != zero_zero_int )
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)) = aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ) ) ).
% power_diff_power_eq
tff(fact_3151_less__2__cases__iff,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
<=> ( ( N = zero_zero_nat )
| ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).
% less_2_cases_iff
tff(fact_3152_less__2__cases,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
=> ( ( N = zero_zero_nat )
| ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).
% less_2_cases
tff(fact_3153_nat__induct2,axiom,
! [P: fun_nat_bool,N: nat] :
( pp(aa_nat_bool(P,zero_zero_nat))
=> ( pp(aa_nat_bool(P,one_one_nat))
=> ( ! [N2: nat] :
( pp(aa_nat_bool(P,N2))
=> pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
=> pp(aa_nat_bool(P,N)) ) ) ) ).
% nat_induct2
tff(fact_3154_power__eq__if,axiom,
! [M: nat,P2: complex] :
( ( ( M = zero_zero_nat )
=> ( aa_nat_complex(power_power_complex(P2),M) = one_one_complex ) )
& ( ( M != zero_zero_nat )
=> ( aa_nat_complex(power_power_complex(P2),M) = aa_complex_complex(times_times_complex(P2),aa_nat_complex(power_power_complex(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).
% power_eq_if
tff(fact_3155_power__eq__if,axiom,
! [M: nat,P2: real] :
( ( ( M = zero_zero_nat )
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,P2),M) = one_one_real ) )
& ( ( M != zero_zero_nat )
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,P2),M) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,P2),aa_nat_real(aa_real_fun_nat_real(power_power_real,P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).
% power_eq_if
tff(fact_3156_power__eq__if,axiom,
! [M: nat,P2: rat] :
( ( ( M = zero_zero_nat )
=> ( aa_nat_rat(power_power_rat(P2),M) = one_one_rat ) )
& ( ( M != zero_zero_nat )
=> ( aa_nat_rat(power_power_rat(P2),M) = aa_rat_rat(times_times_rat(P2),aa_nat_rat(power_power_rat(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).
% power_eq_if
tff(fact_3157_power__eq__if,axiom,
! [M: nat,P2: nat] :
( ( ( M = zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P2),M) = one_one_nat ) )
& ( ( M != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P2),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).
% power_eq_if
tff(fact_3158_power__eq__if,axiom,
! [M: nat,P2: int] :
( ( ( M = zero_zero_nat )
=> ( aa_nat_int(power_power_int(P2),M) = one_one_int ) )
& ( ( M != zero_zero_nat )
=> ( aa_nat_int(power_power_int(P2),M) = aa_int_int(aa_int_fun_int_int(times_times_int,P2),aa_nat_int(power_power_int(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).
% power_eq_if
tff(fact_3159_power__minus__mult,axiom,
! [N: nat,A: complex] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_complex(power_power_complex(A),N) ) ) ).
% power_minus_mult
tff(fact_3160_power__minus__mult,axiom,
! [N: nat,A: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) ) ) ).
% power_minus_mult
tff(fact_3161_power__minus__mult,axiom,
! [N: nat,A: rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_rat(power_power_rat(A),N) ) ) ).
% power_minus_mult
tff(fact_3162_power__minus__mult,axiom,
! [N: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) ) ) ).
% power_minus_mult
tff(fact_3163_power__minus__mult,axiom,
! [N: nat,A: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_int(power_power_int(A),N) ) ) ).
% power_minus_mult
tff(fact_3164_le__div__geq,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ) ) ) ).
% le_div_geq
tff(fact_3165_split__div_H,axiom,
! [P: fun_nat_bool,M: nat,N: nat] :
( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)))
<=> ( ( ( N = zero_zero_nat )
& pp(aa_nat_bool(P,zero_zero_nat)) )
| ? [Q4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q4)),M))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,Q4))))
& pp(aa_nat_bool(P,Q4)) ) ) ) ).
% split_div'
tff(fact_3166_Suc__times__mod__eq,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))),M) = one_one_nat ) ) ).
% Suc_times_mod_eq
tff(fact_3167_vebt__pred_Osimps_I6_J,axiom,
! [V: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] : vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2),Vj2) = none_nat ).
% vebt_pred.simps(6)
tff(fact_3168_vebt__succ_Osimps_I5_J,axiom,
! [V: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] : vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2),Vi2) = none_nat ).
% vebt_succ.simps(5)
tff(fact_3169_power2__less__imp__less,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ).
% power2_less_imp_less
tff(fact_3170_power2__less__imp__less,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ) ).
% power2_less_imp_less
tff(fact_3171_power2__less__imp__less,axiom,
! [X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).
% power2_less_imp_less
tff(fact_3172_power2__less__imp__less,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ) ).
% power2_less_imp_less
tff(fact_3173_sum__power2__le__zero__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real))
<=> ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_power2_le_zero_iff
tff(fact_3174_sum__power2__le__zero__iff,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat))
<=> ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
% sum_power2_le_zero_iff
tff(fact_3175_sum__power2__le__zero__iff,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int))
<=> ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_power2_le_zero_iff
tff(fact_3176_sum__power2__ge__zero,axiom,
! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).
% sum_power2_ge_zero
tff(fact_3177_sum__power2__ge__zero,axiom,
! [X: rat,Y: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).
% sum_power2_ge_zero
tff(fact_3178_sum__power2__ge__zero,axiom,
! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).
% sum_power2_ge_zero
tff(fact_3179_sum__power2__gt__zero__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
<=> ( ( X != zero_zero_real )
| ( Y != zero_zero_real ) ) ) ).
% sum_power2_gt_zero_iff
tff(fact_3180_sum__power2__gt__zero__iff,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
<=> ( ( X != zero_zero_rat )
| ( Y != zero_zero_rat ) ) ) ).
% sum_power2_gt_zero_iff
tff(fact_3181_sum__power2__gt__zero__iff,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
<=> ( ( X != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_power2_gt_zero_iff
tff(fact_3182_not__sum__power2__lt__zero,axiom,
! [X: real,Y: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real)) ).
% not_sum_power2_lt_zero
tff(fact_3183_not__sum__power2__lt__zero,axiom,
! [X: rat,Y: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat)) ).
% not_sum_power2_lt_zero
tff(fact_3184_not__sum__power2__lt__zero,axiom,
! [X: int,Y: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int)) ).
% not_sum_power2_lt_zero
tff(fact_3185_divmod__digit__0_I2_J,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ).
% divmod_digit_0(2)
tff(fact_3186_divmod__digit__0_I2_J,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ) ) ) ).
% divmod_digit_0(2)
tff(fact_3187_divmod__digit__0_I2_J,axiom,
! [B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))),B))
=> ( modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)) = modulo364778990260209775nteger(A,B) ) ) ) ).
% divmod_digit_0(2)
tff(fact_3188_bits__stable__imp__add__self,axiom,
! [A: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_nat ) ) ).
% bits_stable_imp_add_self
tff(fact_3189_bits__stable__imp__add__self,axiom,
! [A: int] :
( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = zero_zero_int ) ) ).
% bits_stable_imp_add_self
tff(fact_3190_bits__stable__imp__add__self,axiom,
! [A: code_integer] :
( ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A )
=> ( plus_p5714425477246183910nteger(A,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) = zero_z3403309356797280102nteger ) ) ).
% bits_stable_imp_add_self
tff(fact_3191_zero__le__even__power_H,axiom,
! [A: real,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).
% zero_le_even_power'
tff(fact_3192_zero__le__even__power_H,axiom,
! [A: rat,N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).
% zero_le_even_power'
tff(fact_3193_zero__le__even__power_H,axiom,
! [A: int,N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).
% zero_le_even_power'
tff(fact_3194_nat__bit__induct,axiom,
! [P: fun_nat_bool,N: nat] :
( pp(aa_nat_bool(P,zero_zero_nat))
=> ( ! [N2: nat] :
( pp(aa_nat_bool(P,N2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
=> pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))) ) )
=> ( ! [N2: nat] :
( pp(aa_nat_bool(P,N2))
=> pp(aa_nat_bool(P,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)))) )
=> pp(aa_nat_bool(P,N)) ) ) ) ).
% nat_bit_induct
tff(fact_3195_Suc__n__div__2__gt__zero,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).
% Suc_n_div_2_gt_zero
tff(fact_3196_div__2__gt__zero,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).
% div_2_gt_zero
tff(fact_3197_verit__le__mono__div,axiom,
! [A2: nat,B2: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),N)),zero_zero_nat)),one_one_nat),zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),N))) ) ) ).
% verit_le_mono_div
tff(fact_3198_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
=> ( ! [A4: bool,B4: bool] :
( ( X = vEBT_Leaf(A4,B4) )
=> ( ( ( Xa2 = zero_zero_nat )
=> pp(A4) )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> pp(B4) )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X != vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [S3: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S3)
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
=> pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(3)
tff(fact_3199_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
=> ( ! [A4: bool,B4: bool] :
( ( X = vEBT_Leaf(A4,B4) )
=> ~ ( ( ( Xa2 = zero_zero_nat )
=> pp(A4) )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> pp(B4) )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [S3: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S3)
=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
=> pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ).
% VEBT_internal.naive_member.elims(2)
tff(fact_3200_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
( ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
<=> pp(Y) )
=> ( ! [A4: bool,B4: bool] :
( ( X = vEBT_Leaf(A4,B4) )
=> ( pp(Y)
<=> ~ ( ( ( Xa2 = zero_zero_nat )
=> pp(A4) )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> pp(B4) )
& ( Xa2 = one_one_nat ) ) ) ) ) )
=> ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
=> pp(Y) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [S3: vEBT_VEBT] : X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S3)
=> ( pp(Y)
<=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
=> pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(1)
tff(fact_3201_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I3_J,axiom,
! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy,Uz)),X) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(3)
tff(fact_3202_divmod__digit__0_I1_J,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).
% divmod_digit_0(1)
tff(fact_3203_divmod__digit__0_I1_J,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B))
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ).
% divmod_digit_0(1)
tff(fact_3204_divmod__digit__0_I1_J,axiom,
! [B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))),B))
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))) = divide6298287555418463151nteger(A,B) ) ) ) ).
% divmod_digit_0(1)
tff(fact_3205_odd__0__le__power__imp__0__le,axiom,
! [A: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).
% odd_0_le_power_imp_0_le
tff(fact_3206_odd__0__le__power__imp__0__le,axiom,
! [A: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).
% odd_0_le_power_imp_0_le
tff(fact_3207_odd__0__le__power__imp__0__le,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).
% odd_0_le_power_imp_0_le
tff(fact_3208_odd__power__less__zero,axiom,
! [A: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),zero_zero_real)) ) ).
% odd_power_less_zero
tff(fact_3209_odd__power__less__zero,axiom,
! [A: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),zero_zero_rat)) ) ).
% odd_power_less_zero
tff(fact_3210_odd__power__less__zero,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),zero_zero_int)) ) ).
% odd_power_less_zero
tff(fact_3211_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
! [X: nat,N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))) ) ) ) ).
% VEBT_internal.exp_split_high_low(1)
tff(fact_3212_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
! [X: nat,N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ) ) ).
% VEBT_internal.exp_split_high_low(2)
tff(fact_3213_vebt__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
=> ( ! [A4: bool,B4: bool] :
( ( X = vEBT_Leaf(A4,B4) )
=> ~ ( ( ( Xa2 = zero_zero_nat )
=> pp(A4) )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> pp(B4) )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2)
=> ~ ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
=> pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(2)
tff(fact_3214_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
=> ( ! [Uu2: bool,Uv2: bool] : X != vEBT_Leaf(Uu2,Uv2)
=> ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2,Vb2)
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2)
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
=> pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) )
=> ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2)
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
=> pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(3)
tff(fact_3215_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
( ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
<=> pp(Y) )
=> ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
=> pp(Y) )
=> ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
=> pp(Y) )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2,Vb2)
=> ( pp(Y)
<=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2)
=> ( pp(Y)
<=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
=> pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) )
=> ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2)
=> ( pp(Y)
<=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
=> pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(1)
tff(fact_3216_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I4_J,axiom,
! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb,Vc)),X) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
tff(fact_3217_mod__double__modulus,axiom,
! [M: code_integer,X: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),M))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X))
=> ( ( modulo364778990260209775nteger(X,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),M)) = modulo364778990260209775nteger(X,M) )
| ( modulo364778990260209775nteger(X,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),M)) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(X,M),M) ) ) ) ) ).
% mod_double_modulus
tff(fact_3218_mod__double__modulus,axiom,
! [M: nat,X: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),M) )
| ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),M)),M) ) ) ) ) ).
% mod_double_modulus
tff(fact_3219_mod__double__modulus,axiom,
! [M: int,X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),M))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
=> ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),M) )
| ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),M)),M) ) ) ) ) ).
% mod_double_modulus
tff(fact_3220_divmod__digit__1_I2_J,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))))
=> ( minus_8373710615458151222nteger(modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)),B) = modulo364778990260209775nteger(A,B) ) ) ) ) ).
% divmod_digit_1(2)
tff(fact_3221_divmod__digit__1_I2_J,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ) ).
% divmod_digit_1(2)
tff(fact_3222_divmod__digit__1_I2_J,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))))
=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ) ) ) ) ).
% divmod_digit_1(2)
tff(fact_3223_vebt__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
( ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
<=> pp(Y) )
=> ( ! [A4: bool,B4: bool] :
( ( X = vEBT_Leaf(A4,B4) )
=> ( pp(Y)
<=> ~ ( ( ( Xa2 = zero_zero_nat )
=> pp(A4) )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> pp(B4) )
& ( Xa2 = one_one_nat ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
=> pp(Y) )
=> ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
=> pp(Y) )
=> ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
=> pp(Y) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2)
=> ( pp(Y)
<=> ~ ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
=> pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(1)
tff(fact_3224_vebt__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
=> ( ! [A4: bool,B4: bool] :
( ( X = vEBT_Leaf(A4,B4) )
=> ( ( ( Xa2 = zero_zero_nat )
=> pp(A4) )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> pp(B4) )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X != vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X != vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2)
=> ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
=> pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(3)
tff(fact_3225_arith__geo__mean,axiom,
! [U: real,X: real,Y: real] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y) )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,U),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),numeral_numeral_real(aa_num_num(bit0,one))))) ) ) ) ).
% arith_geo_mean
tff(fact_3226_arith__geo__mean,axiom,
! [U: rat,X: rat,Y: rat] :
( ( aa_nat_rat(power_power_rat(U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(times_times_rat(X),Y) )
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,U),divide_divide_rat(plus_plus_rat(X,Y),numeral_numeral_rat(aa_num_num(bit0,one))))) ) ) ) ).
% arith_geo_mean
tff(fact_3227_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
( ( aa_nat_nat(vEBT_T_m_e_m_b_e_r(X),Xa2) = Y )
=> ( ( ? [A4: bool,B4: bool] : X = vEBT_Leaf(A4,B4)
=> ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),zero_zero_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
=> ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
=> ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
=> ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
=> ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
=> ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
=> ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2)
=> ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Ma2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_m_e_m_b_e_r(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_nat)))))))))) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.elims
tff(fact_3228_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
( ( aa_nat_nat(vEBT_T_i_n_s_e_r_t2(X),Xa2) = Y )
=> ( ( ? [A4: bool,B4: bool] : X = vEBT_Leaf(A4,B4)
=> ( Y != one_one_nat ) )
=> ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] : X = vEBT_Node(Info2,zero_zero_nat,Ts,S3)
=> ( Y != one_one_nat ) )
=> ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] : X = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S3)
=> ( Y != one_one_nat ) )
=> ( ( ? [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2)
=> ( Y != one_one_nat ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2) )
=> ( Y != aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Ma2))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t2(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat) ) ) ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.elims
tff(fact_3229_invar__vebt_Osimps,axiom,
! [A1: vEBT_VEBT,A22: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(A1),A22))
<=> ( ( ? [A6: bool,B6: bool] : A1 = vEBT_Leaf(A6,B6)
& ( A22 = aa_nat_nat(suc,zero_zero_nat) ) )
| ? [TreeList3: list_VEBT_VEBT,N5: nat,Summary3: vEBT_VEBT] :
( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3,Summary3) )
& ! [X2: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList3)))
=> pp(aa_nat_bool(vEBT_invar_vebt(X2),N5)) )
& pp(aa_nat_bool(vEBT_invar_vebt(Summary3),N5))
& ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5) )
& ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N5),N5) )
& ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),X_13))
& ! [X2: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList3)))
=> ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_13)) ) )
| ? [TreeList3: list_VEBT_VEBT,N5: nat,Summary3: vEBT_VEBT] :
( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3,Summary3) )
& ! [X2: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList3)))
=> pp(aa_nat_bool(vEBT_invar_vebt(X2),N5)) )
& pp(aa_nat_bool(vEBT_invar_vebt(Summary3),aa_nat_nat(suc,N5)))
& ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N5)) )
& ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N5),aa_nat_nat(suc,N5)) )
& ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),X_13))
& ! [X2: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList3)))
=> ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_13)) ) )
| ? [TreeList3: list_VEBT_VEBT,N5: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),A22,TreeList3,Summary3) )
& ! [X2: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList3)))
=> pp(aa_nat_bool(vEBT_invar_vebt(X2),N5)) )
& pp(aa_nat_bool(vEBT_invar_vebt(Summary3),N5))
& ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5) )
& ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N5),N5) )
& ! [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5)))
=> ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),X_13))
<=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),I4)) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList3)))
=> ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_13)) ) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi3),Ma3))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22)))
& ( ( Mi3 != Ma3 )
=> ! [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5)))
=> ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N5) = I4 )
=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N5))) )
& ! [X2: nat] :
( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),N5) = I4 )
& pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),N5))) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi3),X2))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Ma3)) ) ) ) ) ) )
| ? [TreeList3: list_VEBT_VEBT,N5: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),A22,TreeList3,Summary3) )
& ! [X2: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList3)))
=> pp(aa_nat_bool(vEBT_invar_vebt(X2),N5)) )
& pp(aa_nat_bool(vEBT_invar_vebt(Summary3),aa_nat_nat(suc,N5)))
& ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N5)) )
& ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N5),aa_nat_nat(suc,N5)) )
& ! [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N5))))
=> ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),X_13))
<=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary3),I4)) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList3)))
=> ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_13)) ) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi3),Ma3))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22)))
& ( ( Mi3 != Ma3 )
=> ! [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N5))))
=> ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N5) = I4 )
=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N5))) )
& ! [X2: nat] :
( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),N5) = I4 )
& pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),N5))) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi3),X2))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Ma3)) ) ) ) ) ) ) ) ) ).
% invar_vebt.simps
tff(fact_3230_invar__vebt_Ocases,axiom,
! [A1: vEBT_VEBT,A22: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(A1),A22))
=> ( ( ? [A4: bool,B4: bool] : A1 = vEBT_Leaf(A4,B4)
=> ( A22 != aa_nat_nat(suc,zero_zero_nat) ) )
=> ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary2) )
=> ( ( A22 = Deg2 )
=> ( ! [X4: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
=> pp(aa_nat_bool(vEBT_invar_vebt(X4),N2)) )
=> ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M3))
=> ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M3) )
=> ( ( M3 = N2 )
=> ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M3) )
=> ( ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),X_12))
=> ~ ! [X4: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
=> ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_12)) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary2) )
=> ( ( A22 = Deg2 )
=> ( ! [X4: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
=> pp(aa_nat_bool(vEBT_invar_vebt(X4),N2)) )
=> ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M3))
=> ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M3) )
=> ( ( M3 = aa_nat_nat(suc,N2) )
=> ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M3) )
=> ( ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),X_12))
=> ~ ! [X4: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
=> ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_12)) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg2,TreeList2,Summary2) )
=> ( ( A22 = Deg2 )
=> ( ! [X4: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
=> pp(aa_nat_bool(vEBT_invar_vebt(X4),N2)) )
=> ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M3))
=> ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M3) )
=> ( ( M3 = N2 )
=> ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M3) )
=> ( ! [I3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M3)))
=> ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I3)),X_13))
<=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),I3)) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X4: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
=> ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_12)) ) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi2),Ma2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)))
=> ~ ( ( Mi2 != Ma2 )
=> ! [I3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M3)))
=> ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N2) = I3 )
=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N2))) )
& ! [X4: nat] :
( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N2) = I3 )
& pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N2))) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),X4))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg2,TreeList2,Summary2) )
=> ( ( A22 = Deg2 )
=> ( ! [X4: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
=> pp(aa_nat_bool(vEBT_invar_vebt(X4),N2)) )
=> ( pp(aa_nat_bool(vEBT_invar_vebt(Summary2),M3))
=> ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M3) )
=> ( ( M3 = aa_nat_nat(suc,N2) )
=> ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M3) )
=> ( ! [I3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M3)))
=> ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I3)),X_13))
<=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2),I3)) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X4: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
=> ~ ? [X_12: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X4),X_12)) ) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Mi2),Ma2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)))
=> ~ ( ( Mi2 != Ma2 )
=> ! [I3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M3)))
=> ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N2) = I3 )
=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N2))) )
& ! [X4: nat] :
( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N2) = I3 )
& pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N2))) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),X4))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Ma2)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.cases
tff(fact_3231_divmod__digit__1_I1_J,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))))
=> ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))),one_one_Code_integer) = divide6298287555418463151nteger(A,B) ) ) ) ) ).
% divmod_digit_1(1)
tff(fact_3232_divmod__digit__1_I1_J,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)))),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ) ).
% divmod_digit_1(1)
tff(fact_3233_divmod__digit__1_I1_J,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))))
=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))),one_one_int) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ) ).
% divmod_digit_1(1)
tff(fact_3234_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
( ( aa_nat_nat(vEBT_T_m_e_m_b_e_r2(X),Xa2) = Y )
=> ( ( ? [A4: bool,B4: bool] : X = vEBT_Leaf(A4,B4)
=> ( Y != one_one_nat ) )
=> ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
=> ( Y != one_one_nat ) )
=> ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
=> ( Y != one_one_nat ) )
=> ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
=> ( Y != one_one_nat ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2)
=> ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Ma2)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),Xa2)),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Ma2))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(vEBT_T_m_e_m_b_e_r2(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),zero_zero_nat)),zero_zero_nat)))))) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.elims
tff(fact_3235_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r2(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary)),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma),X)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X)),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Ma))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_nat_nat(vEBT_T_m_e_m_b_e_r2(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),zero_zero_nat)),zero_zero_nat)))))) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(5)
tff(fact_3236_mod__exhaust__less__4,axiom,
! [M: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = zero_zero_nat )
| ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = one_one_nat )
| ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
| ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) ) ).
% mod_exhaust_less_4
tff(fact_3237_arcosh__1,axiom,
aa_real_real(arcosh_real,one_one_real) = zero_zero_real ).
% arcosh_1
tff(fact_3238_inrange,axiom,
! [T: vEBT_VEBT,N: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(vEBT_VEBT_set_vebt(T)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat)))) ) ).
% inrange
tff(fact_3239_finite__nth__roots,axiom,
! [N: nat,C: complex] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_set_complex_bool(finite3207457112153483333omplex,collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_dg(N),C)))) ) ).
% finite_nth_roots
tff(fact_3240_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,X) = Y )
=> ( ! [A4: bool] :
( ? [B4: bool] : X = vEBT_Leaf(A4,B4)
=> ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(A4),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
=> ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
=> ( Y != one_one_nat ) )
=> ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2)
=> ( Y != one_one_nat ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
tff(fact_3241_vebt__succ_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
( ( vEBT_vebt_succ(X,Xa2) = Y )
=> ( accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(X,Xa2))
=> ( ! [Uu2: bool,B4: bool] :
( ( X = vEBT_Leaf(Uu2,B4) )
=> ( ( Xa2 = zero_zero_nat )
=> ( ( ( pp(B4)
=> ( Y = some_nat(one_one_nat) ) )
& ( ~ pp(B4)
=> ( Y = none_nat ) ) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,B4),zero_zero_nat)) ) ) )
=> ( ! [Uv2: bool,Uw2: bool] :
( ( X = vEBT_Leaf(Uv2,Uw2) )
=> ! [N2: nat] :
( ( Xa2 = aa_nat_nat(suc,N2) )
=> ( ( Y = none_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uv2,Uw2),aa_nat_nat(suc,N2))) ) ) )
=> ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X = vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2) )
=> ( ( Y = none_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2,Uz2),Xa2)) ) )
=> ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2) )
=> ( ( Y = none_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2,Vd2),Xa2)) ) )
=> ( ! [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg,Vh) )
=> ( ( Y = none_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg,Vh),Xa2)) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2) )
=> ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
=> ( Y = some_nat(Mi2) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
=> ( Y = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_less,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_succ(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_succ(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2),Xa2)) ) ) ) ) ) ) ) ) ) ).
% vebt_succ.pelims
tff(fact_3242_vebt__pred_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
( ( vEBT_vebt_pred(X,Xa2) = Y )
=> ( accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(X,Xa2))
=> ( ! [Uu2: bool,Uv2: bool] :
( ( X = vEBT_Leaf(Uu2,Uv2) )
=> ( ( Xa2 = zero_zero_nat )
=> ( ( Y = none_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),zero_zero_nat)) ) ) )
=> ( ! [A4: bool,Uw2: bool] :
( ( X = vEBT_Leaf(A4,Uw2) )
=> ( ( Xa2 = aa_nat_nat(suc,zero_zero_nat) )
=> ( ( ( pp(A4)
=> ( Y = some_nat(zero_zero_nat) ) )
& ( ~ pp(A4)
=> ( Y = none_nat ) ) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,Uw2),aa_nat_nat(suc,zero_zero_nat))) ) ) )
=> ( ! [A4: bool,B4: bool] :
( ( X = vEBT_Leaf(A4,B4) )
=> ! [Va3: nat] :
( ( Xa2 = aa_nat_nat(suc,aa_nat_nat(suc,Va3)) )
=> ( ( ( pp(B4)
=> ( Y = some_nat(one_one_nat) ) )
& ( ~ pp(B4)
=> ( ( pp(A4)
=> ( Y = some_nat(zero_zero_nat) ) )
& ( ~ pp(A4)
=> ( Y = none_nat ) ) ) ) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),aa_nat_nat(suc,aa_nat_nat(suc,Va3)))) ) ) )
=> ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] :
( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va2) )
=> ( ( Y = none_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va2),Xa2)) ) )
=> ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve) )
=> ( ( Y = none_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve),Xa2)) ) )
=> ( ! [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh,Vi) )
=> ( ( Y = none_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh,Vi),Xa2)) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2) )
=> ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
=> ( Y = some_nat(Ma2) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
=> ( Y = if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)),if_option_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),none_nat))),aa_option_nat_bool(aa_opt7943525813896274546t_bool(vEBT_VEBT_greater,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_option_nat(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),none_nat),if_option_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),Xa2),some_nat(Mi2),none_nat),aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_pred(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),none_nat) ) ) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2),Xa2)) ) ) ) ) ) ) ) ) ) ) ).
% vebt_pred.pelims
tff(fact_3243_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,X) = Y )
=> ( ! [A4: bool,B4: bool] :
( ( X = vEBT_Leaf(A4,B4) )
=> ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(B4),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
=> ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
=> ( Y != one_one_nat ) )
=> ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2)
=> ( Y != one_one_nat ) ) ) ) ) ).
% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
tff(fact_3244_flip__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se2159334234014336723it_int(N),K)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).
% flip_bit_nonnegative_int_iff
tff(fact_3245_flip__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se2159334234014336723it_int(N),K)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).
% flip_bit_negative_int_iff
tff(fact_3246_i0__less,axiom,
! [N: extended_enat] :
( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),N))
<=> ( N != zero_z5237406670263579293d_enat ) ) ).
% i0_less
tff(fact_3247_idiff__0,axiom,
! [N: extended_enat] : minus_3235023915231533773d_enat(zero_z5237406670263579293d_enat,N) = zero_z5237406670263579293d_enat ).
% idiff_0
tff(fact_3248_idiff__0__right,axiom,
! [N: extended_enat] : minus_3235023915231533773d_enat(N,zero_z5237406670263579293d_enat) = N ).
% idiff_0_right
tff(fact_3249_not__real__square__gt__zero,axiom,
! [X: real] :
( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)))
<=> ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
tff(fact_3250_unset__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se4203085406695923979it_int(N),K)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).
% unset_bit_nonnegative_int_iff
tff(fact_3251_unset__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se4203085406695923979it_int(N),K)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).
% unset_bit_negative_int_iff
tff(fact_3252_set__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se7879613467334960850it_int(N),K)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).
% set_bit_nonnegative_int_iff
tff(fact_3253_set__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se7879613467334960850it_int(N),K)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).
% set_bit_negative_int_iff
tff(fact_3254_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(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_3255_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(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_3256_mod__pos__pos__trivial,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),L))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = K ) ) ) ).
% mod_pos_pos_trivial
tff(fact_3257_mod__neg__neg__trivial,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),K))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = K ) ) ) ).
% mod_neg_neg_trivial
tff(fact_3258_half__nonnegative__int__iff,axiom,
! [K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(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,aa_num_num(bit0,one)))))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).
% half_nonnegative_int_iff
tff(fact_3259_half__negative__int__iff,axiom,
! [K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).
% half_negative_int_iff
tff(fact_3260_less__int__code_I1_J,axiom,
~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),zero_zero_int)) ).
% less_int_code(1)
tff(fact_3261_minus__int__code_I1_J,axiom,
! [K: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,K),zero_zero_int) = K ).
% minus_int_code(1)
tff(fact_3262_enat__0__less__mult__iff,axiom,
! [M: extended_enat,N: extended_enat] :
( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),times_7803423173614009249d_enat(M,N)))
<=> ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),M))
& pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),N)) ) ) ).
% enat_0_less_mult_iff
tff(fact_3263_not__iless0,axiom,
! [N: extended_enat] : ~ pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(N),zero_z5237406670263579293d_enat)) ).
% not_iless0
tff(fact_3264_iadd__is__0,axiom,
! [M: extended_enat,N: extended_enat] :
( ( plus_p3455044024723400733d_enat(M,N) = zero_z5237406670263579293d_enat )
<=> ( ( M = zero_z5237406670263579293d_enat )
& ( N = zero_z5237406670263579293d_enat ) ) ) ).
% iadd_is_0
tff(fact_3265_i0__lb,axiom,
! [N: extended_enat] : pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(zero_z5237406670263579293d_enat),N)) ).
% i0_lb
tff(fact_3266_ile0__eq,axiom,
! [N: extended_enat] :
( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(N),zero_z5237406670263579293d_enat))
<=> ( N = zero_z5237406670263579293d_enat ) ) ).
% ile0_eq
tff(fact_3267_all__nat__less,axiom,
! [N: nat,P: fun_nat_bool] :
( ! [M7: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N))
=> pp(aa_nat_bool(P,M7)) )
<=> ! [X2: nat] :
( pp(aa_set_nat_bool(member_nat(X2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)))
=> pp(aa_nat_bool(P,X2)) ) ) ).
% all_nat_less
tff(fact_3268_ex__nat__less,axiom,
! [N: nat,P: fun_nat_bool] :
( ? [M7: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N))
& pp(aa_nat_bool(P,M7)) )
<=> ? [X2: nat] :
( pp(aa_set_nat_bool(member_nat(X2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)))
& pp(aa_nat_bool(P,X2)) ) ) ).
% ex_nat_less
tff(fact_3269_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I)),aa_int_int(aa_int_fun_int_int(times_times_int,K),J))) ) ) ).
% zmult_zless_mono2
tff(fact_3270_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).
% pos_imp_zdiv_neg_iff
tff(fact_3271_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ) ).
% neg_imp_zdiv_neg_iff
tff(fact_3272_div__neg__pos__less0,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int)) ) ) ).
% div_neg_pos_less0
tff(fact_3273_zmod__le__nonneg__dividend,axiom,
! [M: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),M))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M),K)),M)) ) ).
% zmod_le_nonneg_dividend
tff(fact_3274_neg__mod__bound,axiom,
! [L: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))) ) ).
% neg_mod_bound
tff(fact_3275_Euclidean__Division_Opos__mod__bound,axiom,
! [L: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)),L)) ) ).
% Euclidean_Division.pos_mod_bound
tff(fact_3276_zmod__eq__0__iff,axiom,
! [M: int,D: int] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M),D) = zero_zero_int )
<=> ? [Q4: int] : M = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q4) ) ).
% zmod_eq_0_iff
tff(fact_3277_zmod__eq__0D,axiom,
! [M: int,D: int] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M),D) = zero_zero_int )
=> ? [Q3: int] : M = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q3) ) ).
% zmod_eq_0D
tff(fact_3278_realpow__pos__nth2,axiom,
! [A: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ? [R3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
& ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R3),aa_nat_nat(suc,N)) = A ) ) ) ).
% realpow_pos_nth2
tff(fact_3279_real__arch__pow__inv,axiom,
! [Y: real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
=> ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N2)),Y)) ) ) ).
% real_arch_pow_inv
tff(fact_3280_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).
% int_one_le_iff_zero_less
tff(fact_3281_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),M))
=> ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),N) = one_one_int )
<=> ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
tff(fact_3282_odd__less__0__iff,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)),Z)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),zero_zero_int)) ) ).
% odd_less_0_iff
tff(fact_3283_nonneg1__imp__zdiv__pos__iff,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B)) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
tff(fact_3284_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ).
% pos_imp_zdiv_nonneg_iff
tff(fact_3285_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ) ).
% neg_imp_zdiv_nonneg_iff
tff(fact_3286_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,I),K)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),I)) ) ) ).
% pos_imp_zdiv_pos_iff
tff(fact_3287_div__nonpos__pos__le0,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int)) ) ) ).
% div_nonpos_pos_le0
tff(fact_3288_div__nonneg__neg__le0,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int)) ) ) ).
% div_nonneg_neg_le0
tff(fact_3289_div__positive__int,axiom,
! [L: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),K))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L))) ) ) ).
% div_positive_int
tff(fact_3290_div__int__pos__iff,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(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 )
| ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) )
| ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ) ).
% div_int_pos_iff
tff(fact_3291_zdiv__mono2__neg,axiom,
! [A: int,B3: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B3)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) ) ) ) ).
% zdiv_mono2_neg
tff(fact_3292_zdiv__mono1__neg,axiom,
! [A: int,A3: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A3))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) ) ) ).
% zdiv_mono1_neg
tff(fact_3293_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 )
| ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K)) )
| ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),zero_zero_int))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I)) ) ) ) ).
% zdiv_eq_0_iff
tff(fact_3294_zdiv__mono2,axiom,
! [A: int,B3: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B3))) ) ) ) ).
% zdiv_mono2
tff(fact_3295_zdiv__mono1,axiom,
! [A: int,A3: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),A3))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),B))) ) ) ).
% zdiv_mono1
tff(fact_3296_int__div__less__self,axiom,
! [X: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),K))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),K)),X)) ) ) ).
% int_div_less_self
tff(fact_3297_zdiv__zmult2__eq,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).
% zdiv_zmult2_eq
tff(fact_3298_Euclidean__Division_Opos__mod__sign,axiom,
! [L: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))) ) ).
% Euclidean_Division.pos_mod_sign
tff(fact_3299_neg__mod__sign,axiom,
! [L: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)),zero_zero_int)) ) ).
% neg_mod_sign
tff(fact_3300_zmod__trivial__iff,axiom,
! [I: int,K: int] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,I),K) = I )
<=> ( ( K = zero_zero_int )
| ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K)) )
| ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),zero_zero_int))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),I)) ) ) ) ).
% zmod_trivial_iff
tff(fact_3301_pos__mod__conj,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B)) ) ) ).
% pos_mod_conj
tff(fact_3302_neg__mod__conj,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),zero_zero_int))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))) ) ) ).
% neg_mod_conj
tff(fact_3303_zdiv__mono__strict,axiom,
! [A2: int,B2: int,N: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
=> ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),N) = zero_zero_int )
=> ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),N) = zero_zero_int )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),N)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),N))) ) ) ) ) ).
% zdiv_mono_strict
tff(fact_3304_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I1_J,axiom,
! [A: bool,B: bool,X: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r2(vEBT_Leaf(A,B)),X) = one_one_nat ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(1)
tff(fact_3305_not__exp__less__eq__0__int,axiom,
! [N: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),zero_zero_int)) ).
% not_exp_less_eq_0_int
tff(fact_3306_realpow__pos__nth__unique,axiom,
! [N: nat,A: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ? [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
& ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X3),N) = A )
& ! [Y3: real] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y3))
& ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),N) = A ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
tff(fact_3307_realpow__pos__nth,axiom,
! [N: nat,A: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ? [R3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
& ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R3),N) = A ) ) ) ) ).
% realpow_pos_nth
tff(fact_3308_le__imp__0__less,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z))) ) ).
% le_imp_0_less
tff(fact_3309_q__pos__lemma,axiom,
! [B3: int,Q5: int,R4: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),Q5)),R4)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),B3))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Q5)) ) ) ) ).
% q_pos_lemma
tff(fact_3310_zdiv__mono2__lemma,axiom,
! [B: int,Q2: int,R: int,B3: int,Q5: int,R4: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),Q5)),R4) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),Q5)),R4)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),B3))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q2),Q5)) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
tff(fact_3311_zdiv__mono2__neg__lemma,axiom,
! [B: int,Q2: int,R: int,B3: int,Q5: int,R4: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),Q5)),R4) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),Q5)),R4)),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R4))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B3))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B3),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q5),Q2)) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
tff(fact_3312_unique__quotient__lemma,axiom,
! [B: int,Q5: int,R4: int,Q2: int,R: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q5)),R4)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R4))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R4),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q5),Q2)) ) ) ) ) ).
% unique_quotient_lemma
tff(fact_3313_unique__quotient__lemma__neg,axiom,
! [B: int,Q5: int,R4: int,Q2: int,R: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q5)),R4)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R4))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Q2),Q5)) ) ) ) ) ).
% unique_quotient_lemma_neg
tff(fact_3314_mod__pos__neg__trivial,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L) ) ) ) ).
% mod_pos_neg_trivial
tff(fact_3315_mod__pos__geq,axiom,
! [L: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),K))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)),L) ) ) ) ).
% mod_pos_geq
tff(fact_3316_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r2(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)),X) = one_one_nat ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(2)
tff(fact_3317_int__div__pos__eq,axiom,
! [A: int,B: int,Q2: int,R: int] :
( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R),B))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q2 ) ) ) ) ).
% int_div_pos_eq
tff(fact_3318_int__div__neg__eq,axiom,
! [A: int,B: int,Q2: int,R: int] :
( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q2 ) ) ) ) ).
% int_div_neg_eq
tff(fact_3319_split__zdiv,axiom,
! [P: fun_int_bool,N: int,K: int] :
( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),K)))
<=> ( ( ( K = zero_zero_int )
=> pp(aa_int_bool(P,zero_zero_int)) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
=> ! [I4: int,J3: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),J3))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J3),K))
& ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
=> pp(aa_int_bool(P,I4)) ) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
=> ! [I4: int,J3: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),J3))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J3),zero_zero_int))
& ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
=> pp(aa_int_bool(P,I4)) ) ) ) ) ).
% split_zdiv
tff(fact_3320_int__mod__pos__eq,axiom,
! [A: int,B: int,Q2: int,R: int] :
( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R),B))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = R ) ) ) ) ).
% int_mod_pos_eq
tff(fact_3321_int__mod__neg__eq,axiom,
! [A: int,B: int,Q2: int,R: int] :
( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),R))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = R ) ) ) ) ).
% int_mod_neg_eq
tff(fact_3322_split__zmod,axiom,
! [P: fun_int_bool,N: int,K: int] :
( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),K)))
<=> ( ( ( K = zero_zero_int )
=> pp(aa_int_bool(P,N)) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
=> ! [I4: int,J3: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),J3))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J3),K))
& ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
=> pp(aa_int_bool(P,J3)) ) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
=> ! [I4: int,J3: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),J3))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J3),zero_zero_int))
& ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
=> pp(aa_int_bool(P,J3)) ) ) ) ) ).
% split_zmod
tff(fact_3323_zmod__zmult2__eq,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).
% zmod_zmult2_eq
tff(fact_3324_int__power__div__base,axiom,
! [M: nat,K: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(K),M)),K) = aa_nat_int(power_power_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,zero_zero_nat))) ) ) ) ).
% int_power_div_base
tff(fact_3325_div__pos__geq,axiom,
! [L: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),K))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)),L)),one_one_int) ) ) ) ).
% div_pos_geq
tff(fact_3326_verit__le__mono__div__int,axiom,
! [A2: int,B2: int,N: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),N)),aa_int_int(aa_int_fun_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),N)),zero_zero_int)),one_one_int),zero_zero_int))),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),N))) ) ) ).
% verit_le_mono_div_int
tff(fact_3327_split__pos__lemma,axiom,
! [K: int,P: fun_int_fun_int_bool,N: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),K)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),K)))
<=> ! [I4: int,J3: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),J3))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J3),K))
& ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
=> pp(aa_int_bool(aa_int_fun_int_bool(P,I4),J3)) ) ) ) ).
% split_pos_lemma
tff(fact_3328_split__neg__lemma,axiom,
! [K: int,P: fun_int_fun_int_bool,N: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),K)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),K)))
<=> ! [I4: int,J3: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),J3))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J3),zero_zero_int))
& ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
=> pp(aa_int_bool(aa_int_fun_int_bool(P,I4),J3)) ) ) ) ).
% split_neg_lemma
tff(fact_3329_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I1_J,axiom,
! [A: bool,B: bool] : aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,vEBT_Leaf(A,B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(B),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).
% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(1)
tff(fact_3330_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I3_J,axiom,
! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r2(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy,Uz)),X) = one_one_nat ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
tff(fact_3331_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)) = one_one_nat ).
% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(2)
tff(fact_3332_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)) = one_one_nat ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(2)
tff(fact_3333_eq__diff__eq_H,axiom,
! [X: real,Y: real,Z: real] :
( ( X = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z) )
<=> ( Y = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Z) ) ) ).
% eq_diff_eq'
tff(fact_3334_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I4_J,axiom,
! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r2(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb,Vc)),X) = one_one_nat ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
tff(fact_3335_pos__zdiv__mult__2,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) ) ) ).
% pos_zdiv_mult_2
tff(fact_3336_neg__zdiv__mult__2,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)),A) ) ) ).
% neg_zdiv_mult_2
tff(fact_3337_maxt__bound,axiom,
! [T: vEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,T)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))) ).
% maxt_bound
tff(fact_3338_mint__bound,axiom,
! [T: vEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,T)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))) ).
% mint_bound
tff(fact_3339_pos__zmod__mult__2,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),A))) ) ) ).
% pos_zmod_mult_2
tff(fact_3340_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I1_J,axiom,
! [A: bool,B: bool] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,vEBT_Leaf(A,B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(1)
tff(fact_3341_member__bound__height_H,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_T_m_e_m_b_e_r2(T),X)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)))) ) ).
% member_bound_height'
tff(fact_3342_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy,Uz)) = one_one_nat ).
% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(3)
tff(fact_3343_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy,Uz)) = one_one_nat ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(3)
tff(fact_3344_neg__zmod__mult__2,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)),A))),one_one_int) ) ) ).
% neg_zmod_mult_2
tff(fact_3345_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
( ( aa_nat_nat(vEBT_T_i_n_s_e_r_t(X),Xa2) = Y )
=> ( accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(X,Xa2))
=> ( ! [A4: bool,B4: bool] :
( ( X = vEBT_Leaf(A4,B4) )
=> ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),zero_zero_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),Xa2)) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
( ( X = vEBT_Node(Info2,zero_zero_nat,Ts,S3) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts,S3),Xa2)) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
( ( X = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S3) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S3),Xa2)) ) )
=> ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2) )
=> ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2),Xa2)) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2) )
=> ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Ma2))))),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(vEBT_T_i_n_s_e_r_t(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(vEBT_T_i_n_s_e_r_t(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat)) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2),Xa2)) ) ) ) ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
tff(fact_3346_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
( ( aa_nat_nat(vEBT_T_i_n_s_e_r_t2(X),Xa2) = Y )
=> ( accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(X,Xa2))
=> ( ! [A4: bool,B4: bool] :
( ( X = vEBT_Leaf(A4,B4) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),Xa2)) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
( ( X = vEBT_Node(Info2,zero_zero_nat,Ts,S3) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts,S3),Xa2)) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
( ( X = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S3) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S3),Xa2)) ) )
=> ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2),Xa2)) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2) )
=> ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Ma2))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t2(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat))),one_one_nat) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2),Xa2)) ) ) ) ) ) ) ) ) ).
% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
tff(fact_3347_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
( ( aa_nat_nat(vEBT_T_m_e_m_b_e_r(X),Xa2) = Y )
=> ( accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(X,Xa2))
=> ( ! [A4: bool,B4: bool] :
( ( X = vEBT_Leaf(A4,B4) )
=> ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),zero_zero_nat)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),Xa2)) ) )
=> ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
=> ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xa2)) ) )
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
=> ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xa2)) ) )
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
=> ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xa2)) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2) )
=> ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Ma2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_m_e_m_b_e_r(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_nat)))))))))) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2),Xa2)) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
tff(fact_3348_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: nat] :
( ( aa_nat_nat(vEBT_T_m_e_m_b_e_r2(X),Xa2) = Y )
=> ( accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(X,Xa2))
=> ( ! [A4: bool,B4: bool] :
( ( X = vEBT_Leaf(A4,B4) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),Xa2)) ) )
=> ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xa2)) ) )
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xa2)) ) )
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
=> ( ( Y = one_one_nat )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xa2)) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2) )
=> ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Ma2)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2)),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi2),Xa2)),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Ma2))),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(vEBT_T_m_e_m_b_e_r2(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),zero_zero_nat)),zero_zero_nat)))))) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2),Xa2)) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
tff(fact_3349_vebt__member_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
( ( pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
<=> pp(Y) )
=> ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa2))
=> ( ! [A4: bool,B4: bool] :
( ( X = vEBT_Leaf(A4,B4) )
=> ( ( pp(Y)
<=> ( ( ( Xa2 = zero_zero_nat )
=> pp(A4) )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> pp(B4) )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),Xa2)) ) )
=> ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
=> ( ~ pp(Y)
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xa2)) ) )
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
=> ( ~ pp(Y)
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xa2)) ) )
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
=> ( ~ pp(Y)
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xa2)) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2) )
=> ( ( pp(Y)
<=> ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
=> pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2),Xa2)) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(1)
tff(fact_3350_vebt__member_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
=> ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa2))
=> ( ! [A4: bool,B4: bool] :
( ( X = vEBT_Leaf(A4,B4) )
=> ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),Xa2))
=> ( ( ( Xa2 = zero_zero_nat )
=> pp(A4) )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> pp(B4) )
& ( Xa2 = one_one_nat ) ) ) ) ) )
=> ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xa2)) )
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xa2)) )
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xa2)) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2) )
=> ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2),Xa2))
=> ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
=> pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(3)
tff(fact_3351_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
( ( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
<=> pp(Y) )
=> ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa2))
=> ( ! [A4: bool,B4: bool] :
( ( X = vEBT_Leaf(A4,B4) )
=> ( ( pp(Y)
<=> ( ( ( Xa2 = zero_zero_nat )
=> pp(A4) )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> pp(B4) )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),Xa2)) ) )
=> ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
=> ( ~ pp(Y)
=> ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xa2)) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S3) )
=> ( ( pp(Y)
<=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
=> pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S3),Xa2)) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(1)
tff(fact_3352_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
=> ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa2))
=> ( ! [A4: bool,B4: bool] :
( ( X = vEBT_Leaf(A4,B4) )
=> ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),Xa2))
=> ~ ( ( ( Xa2 = zero_zero_nat )
=> pp(A4) )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> pp(B4) )
& ( Xa2 = one_one_nat ) ) ) ) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S3) )
=> ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S3),Xa2))
=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
=> pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(2)
tff(fact_3353_zero__one__enat__neq_I1_J,axiom,
zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% zero_one_enat_neq(1)
tff(fact_3354_imult__is__0,axiom,
! [M: extended_enat,N: extended_enat] :
( ( times_7803423173614009249d_enat(M,N) = zero_z5237406670263579293d_enat )
<=> ( ( M = zero_z5237406670263579293d_enat )
| ( N = zero_z5237406670263579293d_enat ) ) ) ).
% imult_is_0
tff(fact_3355_vebt__member_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( pp(aa_nat_bool(vEBT_vebt_member(X),Xa2))
=> ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(X,Xa2))
=> ( ! [A4: bool,B4: bool] :
( ( X = vEBT_Leaf(A4,B4) )
=> ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),Xa2))
=> ~ ( ( ( Xa2 = zero_zero_nat )
=> pp(A4) )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> pp(B4) )
& ( Xa2 = one_one_nat ) ) ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2) )
=> ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2),Xa2))
=> ~ ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2))
=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Ma2),Xa2))
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
=> pp(aa_nat_bool(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(2)
tff(fact_3356_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ pp(aa_nat_bool(vEBT_V5719532721284313246member(X),Xa2))
=> ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(X,Xa2))
=> ( ! [A4: bool,B4: bool] :
( ( X = vEBT_Leaf(A4,B4) )
=> ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),Xa2))
=> ( ( ( Xa2 = zero_zero_nat )
=> pp(A4) )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> pp(B4) )
& ( Xa2 = one_one_nat ) ) ) ) ) )
=> ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xa2)) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
( ( X = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S3) )
=> ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S3),Xa2))
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
=> pp(aa_nat_bool(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(3)
tff(fact_3357_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
=> ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa2))
=> ( ! [Uu2: bool,Uv2: bool] :
( ( X = vEBT_Leaf(Uu2,Uv2) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa2)) )
=> ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Xa2)) )
=> ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2,Vb2) )
=> ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2,Vb2),Xa2))
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2) )
=> ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2),Xa2))
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
=> pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) )
=> ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2) )
=> ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2),Xa2))
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
=> pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(3)
tff(fact_3358_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
( ( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
<=> pp(Y) )
=> ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa2))
=> ( ! [Uu2: bool,Uv2: bool] :
( ( X = vEBT_Leaf(Uu2,Uv2) )
=> ( ~ pp(Y)
=> ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa2)) ) )
=> ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
=> ( ~ pp(Y)
=> ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Xa2)) ) )
=> ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2,Vb2) )
=> ( ( pp(Y)
<=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2,Vb2),Xa2)) ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2) )
=> ( ( pp(Y)
<=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
=> pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2),Xa2)) ) )
=> ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2) )
=> ( ( pp(Y)
<=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
=> pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2),Xa2)) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(1)
tff(fact_3359_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( pp(aa_nat_bool(vEBT_VEBT_membermima(X),Xa2))
=> ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(X,Xa2))
=> ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2,Vb2) )
=> ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),zero_zero_nat,Va2,Vb2),Xa2))
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2) )
=> ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2,Vc2),Xa2))
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
=> pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) )
=> ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2) )
=> ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2),Xa2))
=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2)))
=> pp(aa_nat_bool(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(2)
tff(fact_3360_infinite__Icc__iff,axiom,
! [A: rat,B: rat] :
( ~ pp(aa_set_rat_bool(finite_finite_rat,set_or633870826150836451st_rat(A,B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).
% infinite_Icc_iff
tff(fact_3361_infinite__Icc__iff,axiom,
! [A: real,B: real] :
( ~ pp(aa_set_real_bool(finite_finite_real,set_or1222579329274155063t_real(A,B)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).
% infinite_Icc_iff
tff(fact_3362_atLeastatMost__empty,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
=> ( set_or633870826150836451st_rat(A,B) = bot_bot_set_rat ) ) ).
% atLeastatMost_empty
tff(fact_3363_atLeastatMost__empty,axiom,
! [B: num,A: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
=> ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num ) ) ).
% atLeastatMost_empty
tff(fact_3364_atLeastatMost__empty,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
=> ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = bot_bot_set_nat ) ) ).
% atLeastatMost_empty
tff(fact_3365_atLeastatMost__empty,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
=> ( set_or1266510415728281911st_int(A,B) = bot_bot_set_int ) ) ).
% atLeastatMost_empty
tff(fact_3366_atLeastatMost__empty,axiom,
! [B: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
=> ( set_or1222579329274155063t_real(A,B) = bot_bot_set_real ) ) ).
% atLeastatMost_empty
tff(fact_3367_atLeastatMost__subset__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
( ord_le6893508408891458716et_nat(set_or4548717258645045905et_nat(A,B),set_or4548717258645045905et_nat(C,D))
<=> ( ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
| ( pp(aa_set_nat_bool(ord_less_eq_set_nat(C),A))
& pp(aa_set_nat_bool(ord_less_eq_set_nat(B),D)) ) ) ) ).
% atLeastatMost_subset_iff
tff(fact_3368_atLeastatMost__subset__iff,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_or633870826150836451st_rat(A,B)),set_or633870826150836451st_rat(C,D)))
<=> ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),D)) ) ) ) ).
% atLeastatMost_subset_iff
tff(fact_3369_atLeastatMost__subset__iff,axiom,
! [A: num,B: num,C: num,D: num] :
( pp(aa_set_num_bool(ord_less_eq_set_num(set_or7049704709247886629st_num(A,B)),set_or7049704709247886629st_num(C,D)))
<=> ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
| ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A))
& pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),D)) ) ) ) ).
% atLeastatMost_subset_iff
tff(fact_3370_atLeastatMost__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)),aa_nat_set_nat(set_or1269000886237332187st_nat(C),D)))
<=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
| ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),D)) ) ) ) ).
% atLeastatMost_subset_iff
tff(fact_3371_atLeastatMost__subset__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( pp(aa_set_int_bool(ord_less_eq_set_int(set_or1266510415728281911st_int(A,B)),set_or1266510415728281911st_int(C,D)))
<=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
| ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),D)) ) ) ) ).
% atLeastatMost_subset_iff
tff(fact_3372_atLeastatMost__subset__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( pp(aa_set_real_bool(ord_less_eq_set_real(set_or1222579329274155063t_real(A,B)),set_or1222579329274155063t_real(C,D)))
<=> ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
| ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),A))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),D)) ) ) ) ).
% atLeastatMost_subset_iff
tff(fact_3373_atLeastatMost__empty__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( set_or4548717258645045905et_nat(A,B) = bot_bot_set_set_nat )
<=> ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B)) ) ).
% atLeastatMost_empty_iff
tff(fact_3374_atLeastatMost__empty__iff,axiom,
! [A: rat,B: rat] :
( ( set_or633870826150836451st_rat(A,B) = bot_bot_set_rat )
<=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).
% atLeastatMost_empty_iff
tff(fact_3375_atLeastatMost__empty__iff,axiom,
! [A: num,B: num] :
( ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num )
<=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B)) ) ).
% atLeastatMost_empty_iff
tff(fact_3376_atLeastatMost__empty__iff,axiom,
! [A: nat,B: nat] :
( ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = bot_bot_set_nat )
<=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).
% atLeastatMost_empty_iff
tff(fact_3377_atLeastatMost__empty__iff,axiom,
! [A: int,B: int] :
( ( set_or1266510415728281911st_int(A,B) = bot_bot_set_int )
<=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).
% atLeastatMost_empty_iff
tff(fact_3378_atLeastatMost__empty__iff,axiom,
! [A: real,B: real] :
( ( set_or1222579329274155063t_real(A,B) = bot_bot_set_real )
<=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).
% atLeastatMost_empty_iff
tff(fact_3379_atLeastatMost__empty__iff2,axiom,
! [A: set_nat,B: set_nat] :
( ( bot_bot_set_set_nat = set_or4548717258645045905et_nat(A,B) )
<=> ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B)) ) ).
% atLeastatMost_empty_iff2
tff(fact_3380_atLeastatMost__empty__iff2,axiom,
! [A: rat,B: rat] :
( ( bot_bot_set_rat = set_or633870826150836451st_rat(A,B) )
<=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).
% atLeastatMost_empty_iff2
tff(fact_3381_atLeastatMost__empty__iff2,axiom,
! [A: num,B: num] :
( ( bot_bot_set_num = set_or7049704709247886629st_num(A,B) )
<=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B)) ) ).
% atLeastatMost_empty_iff2
tff(fact_3382_atLeastatMost__empty__iff2,axiom,
! [A: nat,B: nat] :
( ( bot_bot_set_nat = aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) )
<=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B)) ) ).
% atLeastatMost_empty_iff2
tff(fact_3383_atLeastatMost__empty__iff2,axiom,
! [A: int,B: int] :
( ( bot_bot_set_int = set_or1266510415728281911st_int(A,B) )
<=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).
% atLeastatMost_empty_iff2
tff(fact_3384_atLeastatMost__empty__iff2,axiom,
! [A: real,B: real] :
( ( bot_bot_set_real = set_or1222579329274155063t_real(A,B) )
<=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).
% atLeastatMost_empty_iff2
tff(fact_3385_Icc__eq__Icc,axiom,
! [L: set_nat,H: set_nat,L2: set_nat,H2: set_nat] :
( ( set_or4548717258645045905et_nat(L,H) = set_or4548717258645045905et_nat(L2,H2) )
<=> ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(L),H))
& ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(L2),H2)) ) ) ) ).
% Icc_eq_Icc
tff(fact_3386_Icc__eq__Icc,axiom,
! [L: rat,H: rat,L2: rat,H2: rat] :
( ( set_or633870826150836451st_rat(L,H) = set_or633870826150836451st_rat(L2,H2) )
<=> ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,L),H))
& ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,L2),H2)) ) ) ) ).
% Icc_eq_Icc
tff(fact_3387_Icc__eq__Icc,axiom,
! [L: num,H: num,L2: num,H2: num] :
( ( set_or7049704709247886629st_num(L,H) = set_or7049704709247886629st_num(L2,H2) )
<=> ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L),H))
& ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L2),H2)) ) ) ) ).
% Icc_eq_Icc
tff(fact_3388_Icc__eq__Icc,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( aa_nat_set_nat(set_or1269000886237332187st_nat(L),H) = aa_nat_set_nat(set_or1269000886237332187st_nat(L2),H2) )
<=> ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),H))
& ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L2),H2)) ) ) ) ).
% Icc_eq_Icc
tff(fact_3389_Icc__eq__Icc,axiom,
! [L: int,H: int,L2: int,H2: int] :
( ( set_or1266510415728281911st_int(L,H) = set_or1266510415728281911st_int(L2,H2) )
<=> ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),H))
& ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L2),H2)) ) ) ) ).
% Icc_eq_Icc
tff(fact_3390_Icc__eq__Icc,axiom,
! [L: real,H: real,L2: real,H2: real] :
( ( set_or1222579329274155063t_real(L,H) = set_or1222579329274155063t_real(L2,H2) )
<=> ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),H))
& ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L2),H2)) ) ) ) ).
% Icc_eq_Icc
tff(fact_3391_atLeastAtMost__iff,axiom,
! [I: set_nat,L: set_nat,U: set_nat] :
( member_set_nat(I,set_or4548717258645045905et_nat(L,U))
<=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(L),I))
& pp(aa_set_nat_bool(ord_less_eq_set_nat(I),U)) ) ) ).
% atLeastAtMost_iff
tff(fact_3392_atLeastAtMost__iff,axiom,
! [I: rat,L: rat,U: rat] :
( pp(aa_set_rat_bool(member_rat(I),set_or633870826150836451st_rat(L,U)))
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,L),I))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),U)) ) ) ).
% atLeastAtMost_iff
tff(fact_3393_atLeastAtMost__iff,axiom,
! [I: num,L: num,U: num] :
( pp(aa_set_num_bool(member_num(I),set_or7049704709247886629st_num(L,U)))
<=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L),I))
& pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,I),U)) ) ) ).
% atLeastAtMost_iff
tff(fact_3394_atLeastAtMost__iff,axiom,
! [I: nat,L: nat,U: nat] :
( pp(aa_set_nat_bool(member_nat(I),aa_nat_set_nat(set_or1269000886237332187st_nat(L),U)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),I))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),U)) ) ) ).
% atLeastAtMost_iff
tff(fact_3395_atLeastAtMost__iff,axiom,
! [I: int,L: int,U: int] :
( pp(aa_set_int_bool(member_int(I),set_or1266510415728281911st_int(L,U)))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),I))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),U)) ) ) ).
% atLeastAtMost_iff
tff(fact_3396_atLeastAtMost__iff,axiom,
! [I: real,L: real,U: real] :
( pp(aa_set_real_bool(member_real(I),set_or1222579329274155063t_real(L,U)))
<=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),I))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),U)) ) ) ).
% atLeastAtMost_iff
tff(fact_3397_bounded__Max__nat,axiom,
! [P: fun_nat_bool,X: nat,M6: nat] :
( pp(aa_nat_bool(P,X))
=> ( ! [X3: nat] :
( pp(aa_nat_bool(P,X3))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X3),M6)) )
=> ~ ! [M3: nat] :
( pp(aa_nat_bool(P,M3))
=> ~ ! [X4: nat] :
( pp(aa_nat_bool(P,X4))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),M3)) ) ) ) ) ).
% bounded_Max_nat
tff(fact_3398_fold__atLeastAtMost__nat_Ocases,axiom,
! [X: produc8114805097830739791at_num] :
~ ! [F2: fun_nat_fun_num_num,A4: nat,B4: nat,Acc: num] : X != produc460779904311030791at_num(F2,produc1195630363706982562at_num(A4,product_Pair_nat_num(B4,Acc))) ).
% fold_atLeastAtMost_nat.cases
tff(fact_3399_fold__atLeastAtMost__nat_Ocases,axiom,
! [X: produc9217583074051984497at_nat] :
~ ! [F2: fun_nat_fun_nat_nat,A4: nat,B4: nat,Acc: nat] : X != produc2818902965508115497at_nat(F2,produc487386426758144856at_nat(A4,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B4),Acc))) ).
% fold_atLeastAtMost_nat.cases
tff(fact_3400_finite__nat__set__iff__bounded,axiom,
! [N3: set_nat] :
( pp(aa_set_nat_bool(finite_finite_nat,N3))
<=> ? [M7: nat] :
! [X2: nat] :
( pp(aa_set_nat_bool(member_nat(X2),N3))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),M7)) ) ) ).
% finite_nat_set_iff_bounded
tff(fact_3401_bounded__nat__set__is__finite,axiom,
! [N3: set_nat,N: nat] :
( ! [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),N3))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),N)) )
=> pp(aa_set_nat_bool(finite_finite_nat,N3)) ) ).
% bounded_nat_set_is_finite
tff(fact_3402_finite__nat__set__iff__bounded__le,axiom,
! [N3: set_nat] :
( pp(aa_set_nat_bool(finite_finite_nat,N3))
<=> ? [M7: nat] :
! [X2: nat] :
( pp(aa_set_nat_bool(member_nat(X2),N3))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),M7)) ) ) ).
% finite_nat_set_iff_bounded_le
tff(fact_3403_finite__M__bounded__by__nat,axiom,
! [P: fun_nat_bool,I: nat] : pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_dh(P),I)))) ).
% finite_M_bounded_by_nat
tff(fact_3404_finite__less__ub,axiom,
! [F: fun_nat_nat,U: nat] :
( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),aa_nat_nat(F,N2)))
=> pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_di(F),U)))) ) ).
% finite_less_ub
tff(fact_3405_infinite__Icc,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ~ pp(aa_set_rat_bool(finite_finite_rat,set_or633870826150836451st_rat(A,B))) ) ).
% infinite_Icc
tff(fact_3406_infinite__Icc,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> ~ pp(aa_set_real_bool(finite_finite_real,set_or1222579329274155063t_real(A,B))) ) ).
% infinite_Icc
tff(fact_3407_atLeastatMost__psubset__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
( ord_less_set_set_nat(set_or4548717258645045905et_nat(A,B),set_or4548717258645045905et_nat(C,D))
<=> ( ( ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(A),B))
| ( pp(aa_set_nat_bool(ord_less_eq_set_nat(C),A))
& pp(aa_set_nat_bool(ord_less_eq_set_nat(B),D))
& ( pp(aa_set_nat_bool(ord_less_set_nat(C),A))
| pp(aa_set_nat_bool(ord_less_set_nat(B),D)) ) ) )
& pp(aa_set_nat_bool(ord_less_eq_set_nat(C),D)) ) ) ).
% atLeastatMost_psubset_iff
tff(fact_3408_atLeastatMost__psubset__iff,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( pp(aa_set_rat_bool(ord_less_set_rat(set_or633870826150836451st_rat(A,B)),set_or633870826150836451st_rat(C,D)))
<=> ( ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
| ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),D))
& ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A))
| pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),D)) ) ) )
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),D)) ) ) ).
% atLeastatMost_psubset_iff
tff(fact_3409_atLeastatMost__psubset__iff,axiom,
! [A: num,B: num,C: num,D: num] :
( pp(aa_set_num_bool(ord_less_set_num(set_or7049704709247886629st_num(A,B)),set_or7049704709247886629st_num(C,D)))
<=> ( ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
| ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A))
& pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),D))
& ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A))
| pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),D)) ) ) )
& pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),D)) ) ) ).
% atLeastatMost_psubset_iff
tff(fact_3410_atLeastatMost__psubset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( pp(aa_set_nat_bool(ord_less_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)),aa_nat_set_nat(set_or1269000886237332187st_nat(C),D)))
<=> ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
| ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),D))
& ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A))
| pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),D)) ) ) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D)) ) ) ).
% atLeastatMost_psubset_iff
tff(fact_3411_atLeastatMost__psubset__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( pp(aa_set_int_bool(ord_less_set_int(set_or1266510415728281911st_int(A,B)),set_or1266510415728281911st_int(C,D)))
<=> ( ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
| ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),D))
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A))
| pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),D)) ) ) )
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D)) ) ) ).
% atLeastatMost_psubset_iff
tff(fact_3412_atLeastatMost__psubset__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( pp(aa_set_real_bool(ord_less_set_real(set_or1222579329274155063t_real(A,B)),set_or1222579329274155063t_real(C,D)))
<=> ( ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
| ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),A))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),D))
& ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A))
| pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),D)) ) ) )
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),D)) ) ) ).
% atLeastatMost_psubset_iff
tff(fact_3413_cppi,axiom,
! [D3: int,P: fun_int_bool,P3: fun_int_bool,A2: set_int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
=> ( ? [Z2: int] :
! [X3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X3))
=> ( pp(aa_int_bool(P,X3))
<=> pp(aa_int_bool(P3,X3)) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D3)))
=> ! [Xb2: int] :
( pp(aa_set_int_bool(member_int(Xb2),A2))
=> ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
=> ( pp(aa_int_bool(P,X3))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D3))) ) )
=> ( ! [X3: int,K3: int] :
( pp(aa_int_bool(P3,X3))
<=> pp(aa_int_bool(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D3)))) )
=> ( ? [X_13: int] : pp(aa_int_bool(P,X_13))
<=> ( ? [X2: int] :
( pp(aa_set_int_bool(member_int(X2),set_or1266510415728281911st_int(one_one_int,D3)))
& pp(aa_int_bool(P3,X2)) )
| ? [X2: int] :
( pp(aa_set_int_bool(member_int(X2),set_or1266510415728281911st_int(one_one_int,D3)))
& ? [Xa3: int] :
( pp(aa_set_int_bool(member_int(Xa3),A2))
& pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa3),X2))) ) ) ) ) ) ) ) ) ).
% cppi
tff(fact_3414_cpmi,axiom,
! [D3: int,P: fun_int_bool,P3: fun_int_bool,B2: set_int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
=> ( ? [Z2: int] :
! [X3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z2))
=> ( pp(aa_int_bool(P,X3))
<=> pp(aa_int_bool(P3,X3)) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D3)))
=> ! [Xb2: int] :
( pp(aa_set_int_bool(member_int(Xb2),B2))
=> ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
=> ( pp(aa_int_bool(P,X3))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D3))) ) )
=> ( ! [X3: int,K3: int] :
( pp(aa_int_bool(P3,X3))
<=> pp(aa_int_bool(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D3)))) )
=> ( ? [X_13: int] : pp(aa_int_bool(P,X_13))
<=> ( ? [X2: int] :
( pp(aa_set_int_bool(member_int(X2),set_or1266510415728281911st_int(one_one_int,D3)))
& pp(aa_int_bool(P3,X2)) )
| ? [X2: int] :
( pp(aa_set_int_bool(member_int(X2),set_or1266510415728281911st_int(one_one_int,D3)))
& ? [Xa3: int] :
( pp(aa_set_int_bool(member_int(Xa3),B2))
& pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa3),X2))) ) ) ) ) ) ) ) ) ).
% cpmi
tff(fact_3415_bset_I6_J,axiom,
! [D3: int,B2: set_int,T: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
=> ! [X4: int] :
( ! [Xa4: int] :
( pp(aa_set_int_bool(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3)))
=> ! [Xb3: int] :
( pp(aa_set_int_bool(member_int(Xb3),B2))
=> ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),T))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D3)),T)) ) ) ) ).
% bset(6)
tff(fact_3416_bset_I8_J,axiom,
! [D3: int,T: int,B2: set_int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
=> ( pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T),one_one_int)),B2))
=> ! [X4: int] :
( ! [Xa4: int] :
( pp(aa_set_int_bool(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3)))
=> ! [Xb3: int] :
( pp(aa_set_int_bool(member_int(Xb3),B2))
=> ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X4))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D3))) ) ) ) ) ).
% bset(8)
tff(fact_3417_aset_I6_J,axiom,
! [D3: int,T: int,A2: set_int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
=> ( pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T),one_one_int)),A2))
=> ! [X4: int] :
( ! [Xa4: int] :
( pp(aa_set_int_bool(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3)))
=> ! [Xb3: int] :
( pp(aa_set_int_bool(member_int(Xb3),A2))
=> ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),T))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D3)),T)) ) ) ) ) ).
% aset(6)
tff(fact_3418_aset_I8_J,axiom,
! [D3: int,A2: set_int,T: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
=> ! [X4: int] :
( ! [Xa4: int] :
( pp(aa_set_int_bool(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3)))
=> ! [Xb3: int] :
( pp(aa_set_int_bool(member_int(Xb3),A2))
=> ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X4))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D3))) ) ) ) ).
% aset(8)
tff(fact_3419_bset_I3_J,axiom,
! [D3: int,T: int,B2: set_int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
=> ( pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T),one_one_int)),B2))
=> ! [X4: int] :
( ! [Xa4: int] :
( pp(aa_set_int_bool(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3)))
=> ! [Xb3: int] :
( pp(aa_set_int_bool(member_int(Xb3),B2))
=> ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
=> ( ( X4 = T )
=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D3) = T ) ) ) ) ) ).
% bset(3)
tff(fact_3420_bset_I4_J,axiom,
! [D3: int,T: int,B2: set_int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
=> ( pp(aa_set_int_bool(member_int(T),B2))
=> ! [X4: int] :
( ! [Xa4: int] :
( pp(aa_set_int_bool(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3)))
=> ! [Xb3: int] :
( pp(aa_set_int_bool(member_int(Xb3),B2))
=> ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
=> ( ( X4 != T )
=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D3) != T ) ) ) ) ) ).
% bset(4)
tff(fact_3421_minf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T),X4)) ) ).
% minf(7)
tff(fact_3422_minf_I7_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X4: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z3))
=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,T),X4)) ) ).
% minf(7)
tff(fact_3423_minf_I7_J,axiom,
! [T: num] :
? [Z3: num] :
! [X4: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z3))
=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,T),X4)) ) ).
% minf(7)
tff(fact_3424_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,T),X4)) ) ).
% minf(7)
tff(fact_3425_minf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X4)) ) ).
% minf(7)
tff(fact_3426_minf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),T)) ) ).
% minf(5)
tff(fact_3427_minf_I5_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X4: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z3))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),T)) ) ).
% minf(5)
tff(fact_3428_minf_I5_J,axiom,
! [T: num] :
? [Z3: num] :
! [X4: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z3))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),T)) ) ).
% minf(5)
tff(fact_3429_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),T)) ) ).
% minf(5)
tff(fact_3430_minf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),T)) ) ).
% minf(5)
tff(fact_3431_minf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
=> ( X4 != T ) ) ).
% minf(4)
tff(fact_3432_minf_I4_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X4: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z3))
=> ( X4 != T ) ) ).
% minf(4)
tff(fact_3433_minf_I4_J,axiom,
! [T: num] :
? [Z3: num] :
! [X4: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z3))
=> ( X4 != T ) ) ).
% minf(4)
tff(fact_3434_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
=> ( X4 != T ) ) ).
% minf(4)
tff(fact_3435_minf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
=> ( X4 != T ) ) ).
% minf(4)
tff(fact_3436_minf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
=> ( X4 != T ) ) ).
% minf(3)
tff(fact_3437_minf_I3_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X4: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z3))
=> ( X4 != T ) ) ).
% minf(3)
tff(fact_3438_minf_I3_J,axiom,
! [T: num] :
? [Z3: num] :
! [X4: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z3))
=> ( X4 != T ) ) ).
% minf(3)
tff(fact_3439_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
=> ( X4 != T ) ) ).
% minf(3)
tff(fact_3440_minf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
=> ( X4 != T ) ) ).
% minf(3)
tff(fact_3441_minf_I2_J,axiom,
! [P: fun_real_bool,P3: fun_real_bool,Q: fun_real_bool,Q6: fun_real_bool] :
( ? [Z2: real] :
! [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z2))
=> ( pp(aa_real_bool(P,X3))
<=> pp(aa_real_bool(P3,X3)) ) )
=> ( ? [Z2: real] :
! [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z2))
=> ( pp(aa_real_bool(Q,X3))
<=> pp(aa_real_bool(Q6,X3)) ) )
=> ? [Z3: real] :
! [X4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
=> ( ( pp(aa_real_bool(P,X4))
| pp(aa_real_bool(Q,X4)) )
<=> ( pp(aa_real_bool(P3,X4))
| pp(aa_real_bool(Q6,X4)) ) ) ) ) ) ).
% minf(2)
tff(fact_3442_minf_I2_J,axiom,
! [P: fun_rat_bool,P3: fun_rat_bool,Q: fun_rat_bool,Q6: fun_rat_bool] :
( ? [Z2: rat] :
! [X3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z2))
=> ( pp(aa_rat_bool(P,X3))
<=> pp(aa_rat_bool(P3,X3)) ) )
=> ( ? [Z2: rat] :
! [X3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z2))
=> ( pp(aa_rat_bool(Q,X3))
<=> pp(aa_rat_bool(Q6,X3)) ) )
=> ? [Z3: rat] :
! [X4: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z3))
=> ( ( pp(aa_rat_bool(P,X4))
| pp(aa_rat_bool(Q,X4)) )
<=> ( pp(aa_rat_bool(P3,X4))
| pp(aa_rat_bool(Q6,X4)) ) ) ) ) ) ).
% minf(2)
tff(fact_3443_minf_I2_J,axiom,
! [P: fun_num_bool,P3: fun_num_bool,Q: fun_num_bool,Q6: fun_num_bool] :
( ? [Z2: num] :
! [X3: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z2))
=> ( pp(aa_num_bool(P,X3))
<=> pp(aa_num_bool(P3,X3)) ) )
=> ( ? [Z2: num] :
! [X3: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z2))
=> ( pp(aa_num_bool(Q,X3))
<=> pp(aa_num_bool(Q6,X3)) ) )
=> ? [Z3: num] :
! [X4: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z3))
=> ( ( pp(aa_num_bool(P,X4))
| pp(aa_num_bool(Q,X4)) )
<=> ( pp(aa_num_bool(P3,X4))
| pp(aa_num_bool(Q6,X4)) ) ) ) ) ) ).
% minf(2)
tff(fact_3444_minf_I2_J,axiom,
! [P: fun_nat_bool,P3: fun_nat_bool,Q: fun_nat_bool,Q6: fun_nat_bool] :
( ? [Z2: nat] :
! [X3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z2))
=> ( pp(aa_nat_bool(P,X3))
<=> pp(aa_nat_bool(P3,X3)) ) )
=> ( ? [Z2: nat] :
! [X3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z2))
=> ( pp(aa_nat_bool(Q,X3))
<=> pp(aa_nat_bool(Q6,X3)) ) )
=> ? [Z3: nat] :
! [X4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
=> ( ( pp(aa_nat_bool(P,X4))
| pp(aa_nat_bool(Q,X4)) )
<=> ( pp(aa_nat_bool(P3,X4))
| pp(aa_nat_bool(Q6,X4)) ) ) ) ) ) ).
% minf(2)
tff(fact_3445_minf_I2_J,axiom,
! [P: fun_int_bool,P3: fun_int_bool,Q: fun_int_bool,Q6: fun_int_bool] :
( ? [Z2: int] :
! [X3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z2))
=> ( pp(aa_int_bool(P,X3))
<=> pp(aa_int_bool(P3,X3)) ) )
=> ( ? [Z2: int] :
! [X3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z2))
=> ( pp(aa_int_bool(Q,X3))
<=> pp(aa_int_bool(Q6,X3)) ) )
=> ? [Z3: int] :
! [X4: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
=> ( ( pp(aa_int_bool(P,X4))
| pp(aa_int_bool(Q,X4)) )
<=> ( pp(aa_int_bool(P3,X4))
| pp(aa_int_bool(Q6,X4)) ) ) ) ) ) ).
% minf(2)
tff(fact_3446_minf_I1_J,axiom,
! [P: fun_real_bool,P3: fun_real_bool,Q: fun_real_bool,Q6: fun_real_bool] :
( ? [Z2: real] :
! [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z2))
=> ( pp(aa_real_bool(P,X3))
<=> pp(aa_real_bool(P3,X3)) ) )
=> ( ? [Z2: real] :
! [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Z2))
=> ( pp(aa_real_bool(Q,X3))
<=> pp(aa_real_bool(Q6,X3)) ) )
=> ? [Z3: real] :
! [X4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
=> ( ( pp(aa_real_bool(P,X4))
& pp(aa_real_bool(Q,X4)) )
<=> ( pp(aa_real_bool(P3,X4))
& pp(aa_real_bool(Q6,X4)) ) ) ) ) ) ).
% minf(1)
tff(fact_3447_minf_I1_J,axiom,
! [P: fun_rat_bool,P3: fun_rat_bool,Q: fun_rat_bool,Q6: fun_rat_bool] :
( ? [Z2: rat] :
! [X3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z2))
=> ( pp(aa_rat_bool(P,X3))
<=> pp(aa_rat_bool(P3,X3)) ) )
=> ( ? [Z2: rat] :
! [X3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Z2))
=> ( pp(aa_rat_bool(Q,X3))
<=> pp(aa_rat_bool(Q6,X3)) ) )
=> ? [Z3: rat] :
! [X4: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z3))
=> ( ( pp(aa_rat_bool(P,X4))
& pp(aa_rat_bool(Q,X4)) )
<=> ( pp(aa_rat_bool(P3,X4))
& pp(aa_rat_bool(Q6,X4)) ) ) ) ) ) ).
% minf(1)
tff(fact_3448_minf_I1_J,axiom,
! [P: fun_num_bool,P3: fun_num_bool,Q: fun_num_bool,Q6: fun_num_bool] :
( ? [Z2: num] :
! [X3: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z2))
=> ( pp(aa_num_bool(P,X3))
<=> pp(aa_num_bool(P3,X3)) ) )
=> ( ? [Z2: num] :
! [X3: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Z2))
=> ( pp(aa_num_bool(Q,X3))
<=> pp(aa_num_bool(Q6,X3)) ) )
=> ? [Z3: num] :
! [X4: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z3))
=> ( ( pp(aa_num_bool(P,X4))
& pp(aa_num_bool(Q,X4)) )
<=> ( pp(aa_num_bool(P3,X4))
& pp(aa_num_bool(Q6,X4)) ) ) ) ) ) ).
% minf(1)
tff(fact_3449_minf_I1_J,axiom,
! [P: fun_nat_bool,P3: fun_nat_bool,Q: fun_nat_bool,Q6: fun_nat_bool] :
( ? [Z2: nat] :
! [X3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z2))
=> ( pp(aa_nat_bool(P,X3))
<=> pp(aa_nat_bool(P3,X3)) ) )
=> ( ? [Z2: nat] :
! [X3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Z2))
=> ( pp(aa_nat_bool(Q,X3))
<=> pp(aa_nat_bool(Q6,X3)) ) )
=> ? [Z3: nat] :
! [X4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
=> ( ( pp(aa_nat_bool(P,X4))
& pp(aa_nat_bool(Q,X4)) )
<=> ( pp(aa_nat_bool(P3,X4))
& pp(aa_nat_bool(Q6,X4)) ) ) ) ) ) ).
% minf(1)
tff(fact_3450_minf_I1_J,axiom,
! [P: fun_int_bool,P3: fun_int_bool,Q: fun_int_bool,Q6: fun_int_bool] :
( ? [Z2: int] :
! [X3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z2))
=> ( pp(aa_int_bool(P,X3))
<=> pp(aa_int_bool(P3,X3)) ) )
=> ( ? [Z2: int] :
! [X3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z2))
=> ( pp(aa_int_bool(Q,X3))
<=> pp(aa_int_bool(Q6,X3)) ) )
=> ? [Z3: int] :
! [X4: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
=> ( ( pp(aa_int_bool(P,X4))
& pp(aa_int_bool(Q,X4)) )
<=> ( pp(aa_int_bool(P3,X4))
& pp(aa_int_bool(Q6,X4)) ) ) ) ) ) ).
% minf(1)
tff(fact_3451_pinf_I7_J,axiom,
! [T: real] :
? [Z3: real] :
! [X4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T),X4)) ) ).
% pinf(7)
tff(fact_3452_pinf_I7_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X4: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X4))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,T),X4)) ) ).
% pinf(7)
tff(fact_3453_pinf_I7_J,axiom,
! [T: num] :
? [Z3: num] :
! [X4: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X4))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,T),X4)) ) ).
% pinf(7)
tff(fact_3454_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,T),X4)) ) ).
% pinf(7)
tff(fact_3455_pinf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X4)) ) ).
% pinf(7)
tff(fact_3456_pinf_I5_J,axiom,
! [T: real] :
? [Z3: real] :
! [X4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),T)) ) ).
% pinf(5)
tff(fact_3457_pinf_I5_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X4: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X4))
=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),T)) ) ).
% pinf(5)
tff(fact_3458_pinf_I5_J,axiom,
! [T: num] :
? [Z3: num] :
! [X4: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X4))
=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),T)) ) ).
% pinf(5)
tff(fact_3459_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),T)) ) ).
% pinf(5)
tff(fact_3460_pinf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),T)) ) ).
% pinf(5)
tff(fact_3461_pinf_I4_J,axiom,
! [T: real] :
? [Z3: real] :
! [X4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
=> ( X4 != T ) ) ).
% pinf(4)
tff(fact_3462_pinf_I4_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X4: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X4))
=> ( X4 != T ) ) ).
% pinf(4)
tff(fact_3463_pinf_I4_J,axiom,
! [T: num] :
? [Z3: num] :
! [X4: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X4))
=> ( X4 != T ) ) ).
% pinf(4)
tff(fact_3464_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
=> ( X4 != T ) ) ).
% pinf(4)
tff(fact_3465_pinf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
=> ( X4 != T ) ) ).
% pinf(4)
tff(fact_3466_pinf_I3_J,axiom,
! [T: real] :
? [Z3: real] :
! [X4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
=> ( X4 != T ) ) ).
% pinf(3)
tff(fact_3467_pinf_I3_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X4: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X4))
=> ( X4 != T ) ) ).
% pinf(3)
tff(fact_3468_pinf_I3_J,axiom,
! [T: num] :
? [Z3: num] :
! [X4: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X4))
=> ( X4 != T ) ) ).
% pinf(3)
tff(fact_3469_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
=> ( X4 != T ) ) ).
% pinf(3)
tff(fact_3470_pinf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
=> ( X4 != T ) ) ).
% pinf(3)
tff(fact_3471_pinf_I2_J,axiom,
! [P: fun_real_bool,P3: fun_real_bool,Q: fun_real_bool,Q6: fun_real_bool] :
( ? [Z2: real] :
! [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X3))
=> ( pp(aa_real_bool(P,X3))
<=> pp(aa_real_bool(P3,X3)) ) )
=> ( ? [Z2: real] :
! [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X3))
=> ( pp(aa_real_bool(Q,X3))
<=> pp(aa_real_bool(Q6,X3)) ) )
=> ? [Z3: real] :
! [X4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
=> ( ( pp(aa_real_bool(P,X4))
| pp(aa_real_bool(Q,X4)) )
<=> ( pp(aa_real_bool(P3,X4))
| pp(aa_real_bool(Q6,X4)) ) ) ) ) ) ).
% pinf(2)
tff(fact_3472_pinf_I2_J,axiom,
! [P: fun_rat_bool,P3: fun_rat_bool,Q: fun_rat_bool,Q6: fun_rat_bool] :
( ? [Z2: rat] :
! [X3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X3))
=> ( pp(aa_rat_bool(P,X3))
<=> pp(aa_rat_bool(P3,X3)) ) )
=> ( ? [Z2: rat] :
! [X3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X3))
=> ( pp(aa_rat_bool(Q,X3))
<=> pp(aa_rat_bool(Q6,X3)) ) )
=> ? [Z3: rat] :
! [X4: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X4))
=> ( ( pp(aa_rat_bool(P,X4))
| pp(aa_rat_bool(Q,X4)) )
<=> ( pp(aa_rat_bool(P3,X4))
| pp(aa_rat_bool(Q6,X4)) ) ) ) ) ) ).
% pinf(2)
tff(fact_3473_pinf_I2_J,axiom,
! [P: fun_num_bool,P3: fun_num_bool,Q: fun_num_bool,Q6: fun_num_bool] :
( ? [Z2: num] :
! [X3: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X3))
=> ( pp(aa_num_bool(P,X3))
<=> pp(aa_num_bool(P3,X3)) ) )
=> ( ? [Z2: num] :
! [X3: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X3))
=> ( pp(aa_num_bool(Q,X3))
<=> pp(aa_num_bool(Q6,X3)) ) )
=> ? [Z3: num] :
! [X4: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X4))
=> ( ( pp(aa_num_bool(P,X4))
| pp(aa_num_bool(Q,X4)) )
<=> ( pp(aa_num_bool(P3,X4))
| pp(aa_num_bool(Q6,X4)) ) ) ) ) ) ).
% pinf(2)
tff(fact_3474_pinf_I2_J,axiom,
! [P: fun_nat_bool,P3: fun_nat_bool,Q: fun_nat_bool,Q6: fun_nat_bool] :
( ? [Z2: nat] :
! [X3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X3))
=> ( pp(aa_nat_bool(P,X3))
<=> pp(aa_nat_bool(P3,X3)) ) )
=> ( ? [Z2: nat] :
! [X3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X3))
=> ( pp(aa_nat_bool(Q,X3))
<=> pp(aa_nat_bool(Q6,X3)) ) )
=> ? [Z3: nat] :
! [X4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
=> ( ( pp(aa_nat_bool(P,X4))
| pp(aa_nat_bool(Q,X4)) )
<=> ( pp(aa_nat_bool(P3,X4))
| pp(aa_nat_bool(Q6,X4)) ) ) ) ) ) ).
% pinf(2)
tff(fact_3475_pinf_I2_J,axiom,
! [P: fun_int_bool,P3: fun_int_bool,Q: fun_int_bool,Q6: fun_int_bool] :
( ? [Z2: int] :
! [X3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X3))
=> ( pp(aa_int_bool(P,X3))
<=> pp(aa_int_bool(P3,X3)) ) )
=> ( ? [Z2: int] :
! [X3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X3))
=> ( pp(aa_int_bool(Q,X3))
<=> pp(aa_int_bool(Q6,X3)) ) )
=> ? [Z3: int] :
! [X4: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
=> ( ( pp(aa_int_bool(P,X4))
| pp(aa_int_bool(Q,X4)) )
<=> ( pp(aa_int_bool(P3,X4))
| pp(aa_int_bool(Q6,X4)) ) ) ) ) ) ).
% pinf(2)
tff(fact_3476_pinf_I1_J,axiom,
! [P: fun_real_bool,P3: fun_real_bool,Q: fun_real_bool,Q6: fun_real_bool] :
( ? [Z2: real] :
! [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X3))
=> ( pp(aa_real_bool(P,X3))
<=> pp(aa_real_bool(P3,X3)) ) )
=> ( ? [Z2: real] :
! [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z2),X3))
=> ( pp(aa_real_bool(Q,X3))
<=> pp(aa_real_bool(Q6,X3)) ) )
=> ? [Z3: real] :
! [X4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
=> ( ( pp(aa_real_bool(P,X4))
& pp(aa_real_bool(Q,X4)) )
<=> ( pp(aa_real_bool(P3,X4))
& pp(aa_real_bool(Q6,X4)) ) ) ) ) ) ).
% pinf(1)
tff(fact_3477_pinf_I1_J,axiom,
! [P: fun_rat_bool,P3: fun_rat_bool,Q: fun_rat_bool,Q6: fun_rat_bool] :
( ? [Z2: rat] :
! [X3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X3))
=> ( pp(aa_rat_bool(P,X3))
<=> pp(aa_rat_bool(P3,X3)) ) )
=> ( ? [Z2: rat] :
! [X3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z2),X3))
=> ( pp(aa_rat_bool(Q,X3))
<=> pp(aa_rat_bool(Q6,X3)) ) )
=> ? [Z3: rat] :
! [X4: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X4))
=> ( ( pp(aa_rat_bool(P,X4))
& pp(aa_rat_bool(Q,X4)) )
<=> ( pp(aa_rat_bool(P3,X4))
& pp(aa_rat_bool(Q6,X4)) ) ) ) ) ) ).
% pinf(1)
tff(fact_3478_pinf_I1_J,axiom,
! [P: fun_num_bool,P3: fun_num_bool,Q: fun_num_bool,Q6: fun_num_bool] :
( ? [Z2: num] :
! [X3: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X3))
=> ( pp(aa_num_bool(P,X3))
<=> pp(aa_num_bool(P3,X3)) ) )
=> ( ? [Z2: num] :
! [X3: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X3))
=> ( pp(aa_num_bool(Q,X3))
<=> pp(aa_num_bool(Q6,X3)) ) )
=> ? [Z3: num] :
! [X4: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X4))
=> ( ( pp(aa_num_bool(P,X4))
& pp(aa_num_bool(Q,X4)) )
<=> ( pp(aa_num_bool(P3,X4))
& pp(aa_num_bool(Q6,X4)) ) ) ) ) ) ).
% pinf(1)
tff(fact_3479_pinf_I1_J,axiom,
! [P: fun_nat_bool,P3: fun_nat_bool,Q: fun_nat_bool,Q6: fun_nat_bool] :
( ? [Z2: nat] :
! [X3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X3))
=> ( pp(aa_nat_bool(P,X3))
<=> pp(aa_nat_bool(P3,X3)) ) )
=> ( ? [Z2: nat] :
! [X3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X3))
=> ( pp(aa_nat_bool(Q,X3))
<=> pp(aa_nat_bool(Q6,X3)) ) )
=> ? [Z3: nat] :
! [X4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
=> ( ( pp(aa_nat_bool(P,X4))
& pp(aa_nat_bool(Q,X4)) )
<=> ( pp(aa_nat_bool(P3,X4))
& pp(aa_nat_bool(Q6,X4)) ) ) ) ) ) ).
% pinf(1)
tff(fact_3480_pinf_I1_J,axiom,
! [P: fun_int_bool,P3: fun_int_bool,Q: fun_int_bool,Q6: fun_int_bool] :
( ? [Z2: int] :
! [X3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X3))
=> ( pp(aa_int_bool(P,X3))
<=> pp(aa_int_bool(P3,X3)) ) )
=> ( ? [Z2: int] :
! [X3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X3))
=> ( pp(aa_int_bool(Q,X3))
<=> pp(aa_int_bool(Q6,X3)) ) )
=> ? [Z3: int] :
! [X4: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
=> ( ( pp(aa_int_bool(P,X4))
& pp(aa_int_bool(Q,X4)) )
<=> ( pp(aa_int_bool(P3,X4))
& pp(aa_int_bool(Q6,X4)) ) ) ) ) ) ).
% pinf(1)
tff(fact_3481_pinf_I6_J,axiom,
! [T: real] :
? [Z3: real] :
! [X4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),T)) ) ).
% pinf(6)
tff(fact_3482_pinf_I6_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X4: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X4))
=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),T)) ) ).
% pinf(6)
tff(fact_3483_pinf_I6_J,axiom,
! [T: num] :
? [Z3: num] :
! [X4: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X4))
=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),T)) ) ).
% pinf(6)
tff(fact_3484_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),T)) ) ).
% pinf(6)
tff(fact_3485_pinf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),T)) ) ).
% pinf(6)
tff(fact_3486_pinf_I8_J,axiom,
! [T: real] :
? [Z3: real] :
! [X4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),X4)) ) ).
% pinf(8)
tff(fact_3487_pinf_I8_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X4: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X4))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),X4)) ) ).
% pinf(8)
tff(fact_3488_pinf_I8_J,axiom,
! [T: num] :
? [Z3: num] :
! [X4: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X4))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,T),X4)) ) ).
% pinf(8)
tff(fact_3489_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),X4)) ) ).
% pinf(8)
tff(fact_3490_pinf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X4)) ) ).
% pinf(8)
tff(fact_3491_minf_I6_J,axiom,
! [T: real] :
? [Z3: real] :
! [X4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),T)) ) ).
% minf(6)
tff(fact_3492_minf_I6_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X4: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z3))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),T)) ) ).
% minf(6)
tff(fact_3493_minf_I6_J,axiom,
! [T: num] :
? [Z3: num] :
! [X4: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z3))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),T)) ) ).
% minf(6)
tff(fact_3494_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),T)) ) ).
% minf(6)
tff(fact_3495_minf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),T)) ) ).
% minf(6)
tff(fact_3496_minf_I8_J,axiom,
! [T: real] :
? [Z3: real] :
! [X4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),X4)) ) ).
% minf(8)
tff(fact_3497_minf_I8_J,axiom,
! [T: rat] :
? [Z3: rat] :
! [X4: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z3))
=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),X4)) ) ).
% minf(8)
tff(fact_3498_minf_I8_J,axiom,
! [T: num] :
? [Z3: num] :
! [X4: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),Z3))
=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,T),X4)) ) ).
% minf(8)
tff(fact_3499_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),X4)) ) ).
% minf(8)
tff(fact_3500_minf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X4: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X4)) ) ).
% minf(8)
tff(fact_3501_inf__period_I1_J,axiom,
! [P: fun_real_bool,D3: real,Q: fun_real_bool] :
( ! [X3: real,K3: real] :
( pp(aa_real_bool(P,X3))
<=> pp(aa_real_bool(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K3),D3)))) )
=> ( ! [X3: real,K3: real] :
( pp(aa_real_bool(Q,X3))
<=> pp(aa_real_bool(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K3),D3)))) )
=> ! [X4: real,K4: real] :
( ( pp(aa_real_bool(P,X4))
& pp(aa_real_bool(Q,X4)) )
<=> ( pp(aa_real_bool(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D3))))
& pp(aa_real_bool(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D3)))) ) ) ) ) ).
% inf_period(1)
tff(fact_3502_inf__period_I1_J,axiom,
! [P: fun_rat_bool,D3: rat,Q: fun_rat_bool] :
( ! [X3: rat,K3: rat] :
( pp(aa_rat_bool(P,X3))
<=> pp(aa_rat_bool(P,minus_minus_rat(X3,aa_rat_rat(times_times_rat(K3),D3)))) )
=> ( ! [X3: rat,K3: rat] :
( pp(aa_rat_bool(Q,X3))
<=> pp(aa_rat_bool(Q,minus_minus_rat(X3,aa_rat_rat(times_times_rat(K3),D3)))) )
=> ! [X4: rat,K4: rat] :
( ( pp(aa_rat_bool(P,X4))
& pp(aa_rat_bool(Q,X4)) )
<=> ( pp(aa_rat_bool(P,minus_minus_rat(X4,aa_rat_rat(times_times_rat(K4),D3))))
& pp(aa_rat_bool(Q,minus_minus_rat(X4,aa_rat_rat(times_times_rat(K4),D3)))) ) ) ) ) ).
% inf_period(1)
tff(fact_3503_inf__period_I1_J,axiom,
! [P: fun_int_bool,D3: int,Q: fun_int_bool] :
( ! [X3: int,K3: int] :
( pp(aa_int_bool(P,X3))
<=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D3)))) )
=> ( ! [X3: int,K3: int] :
( pp(aa_int_bool(Q,X3))
<=> pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D3)))) )
=> ! [X4: int,K4: int] :
( ( pp(aa_int_bool(P,X4))
& pp(aa_int_bool(Q,X4)) )
<=> ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3))))
& pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3)))) ) ) ) ) ).
% inf_period(1)
tff(fact_3504_inf__period_I2_J,axiom,
! [P: fun_real_bool,D3: real,Q: fun_real_bool] :
( ! [X3: real,K3: real] :
( pp(aa_real_bool(P,X3))
<=> pp(aa_real_bool(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K3),D3)))) )
=> ( ! [X3: real,K3: real] :
( pp(aa_real_bool(Q,X3))
<=> pp(aa_real_bool(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K3),D3)))) )
=> ! [X4: real,K4: real] :
( ( pp(aa_real_bool(P,X4))
| pp(aa_real_bool(Q,X4)) )
<=> ( pp(aa_real_bool(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D3))))
| pp(aa_real_bool(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D3)))) ) ) ) ) ).
% inf_period(2)
tff(fact_3505_inf__period_I2_J,axiom,
! [P: fun_rat_bool,D3: rat,Q: fun_rat_bool] :
( ! [X3: rat,K3: rat] :
( pp(aa_rat_bool(P,X3))
<=> pp(aa_rat_bool(P,minus_minus_rat(X3,aa_rat_rat(times_times_rat(K3),D3)))) )
=> ( ! [X3: rat,K3: rat] :
( pp(aa_rat_bool(Q,X3))
<=> pp(aa_rat_bool(Q,minus_minus_rat(X3,aa_rat_rat(times_times_rat(K3),D3)))) )
=> ! [X4: rat,K4: rat] :
( ( pp(aa_rat_bool(P,X4))
| pp(aa_rat_bool(Q,X4)) )
<=> ( pp(aa_rat_bool(P,minus_minus_rat(X4,aa_rat_rat(times_times_rat(K4),D3))))
| pp(aa_rat_bool(Q,minus_minus_rat(X4,aa_rat_rat(times_times_rat(K4),D3)))) ) ) ) ) ).
% inf_period(2)
tff(fact_3506_inf__period_I2_J,axiom,
! [P: fun_int_bool,D3: int,Q: fun_int_bool] :
( ! [X3: int,K3: int] :
( pp(aa_int_bool(P,X3))
<=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D3)))) )
=> ( ! [X3: int,K3: int] :
( pp(aa_int_bool(Q,X3))
<=> pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D3)))) )
=> ! [X4: int,K4: int] :
( ( pp(aa_int_bool(P,X4))
| pp(aa_int_bool(Q,X4)) )
<=> ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3))))
| pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3)))) ) ) ) ) ).
% inf_period(2)
tff(fact_3507_aset_I2_J,axiom,
! [D3: int,A2: set_int,P: fun_int_bool,Q: fun_int_bool] :
( ! [X3: int] :
( ! [Xa: int] :
( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D3)))
=> ! [Xb2: int] :
( pp(aa_set_int_bool(member_int(Xb2),A2))
=> ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
=> ( pp(aa_int_bool(P,X3))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D3))) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D3)))
=> ! [Xb2: int] :
( pp(aa_set_int_bool(member_int(Xb2),A2))
=> ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
=> ( pp(aa_int_bool(Q,X3))
=> pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D3))) ) )
=> ! [X4: int] :
( ! [Xa4: int] :
( pp(aa_set_int_bool(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3)))
=> ! [Xb3: int] :
( pp(aa_set_int_bool(member_int(Xb3),A2))
=> ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
=> ( ( pp(aa_int_bool(P,X4))
| pp(aa_int_bool(Q,X4)) )
=> ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D3)))
| pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D3))) ) ) ) ) ) ).
% aset(2)
tff(fact_3508_aset_I1_J,axiom,
! [D3: int,A2: set_int,P: fun_int_bool,Q: fun_int_bool] :
( ! [X3: int] :
( ! [Xa: int] :
( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D3)))
=> ! [Xb2: int] :
( pp(aa_set_int_bool(member_int(Xb2),A2))
=> ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
=> ( pp(aa_int_bool(P,X3))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D3))) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D3)))
=> ! [Xb2: int] :
( pp(aa_set_int_bool(member_int(Xb2),A2))
=> ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
=> ( pp(aa_int_bool(Q,X3))
=> pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D3))) ) )
=> ! [X4: int] :
( ! [Xa4: int] :
( pp(aa_set_int_bool(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3)))
=> ! [Xb3: int] :
( pp(aa_set_int_bool(member_int(Xb3),A2))
=> ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
=> ( ( pp(aa_int_bool(P,X4))
& pp(aa_int_bool(Q,X4)) )
=> ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D3)))
& pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D3))) ) ) ) ) ) ).
% aset(1)
tff(fact_3509_bset_I2_J,axiom,
! [D3: int,B2: set_int,P: fun_int_bool,Q: fun_int_bool] :
( ! [X3: int] :
( ! [Xa: int] :
( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D3)))
=> ! [Xb2: int] :
( pp(aa_set_int_bool(member_int(Xb2),B2))
=> ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
=> ( pp(aa_int_bool(P,X3))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D3))) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D3)))
=> ! [Xb2: int] :
( pp(aa_set_int_bool(member_int(Xb2),B2))
=> ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
=> ( pp(aa_int_bool(Q,X3))
=> pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D3))) ) )
=> ! [X4: int] :
( ! [Xa4: int] :
( pp(aa_set_int_bool(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3)))
=> ! [Xb3: int] :
( pp(aa_set_int_bool(member_int(Xb3),B2))
=> ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
=> ( ( pp(aa_int_bool(P,X4))
| pp(aa_int_bool(Q,X4)) )
=> ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D3)))
| pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D3))) ) ) ) ) ) ).
% bset(2)
tff(fact_3510_bset_I1_J,axiom,
! [D3: int,B2: set_int,P: fun_int_bool,Q: fun_int_bool] :
( ! [X3: int] :
( ! [Xa: int] :
( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D3)))
=> ! [Xb2: int] :
( pp(aa_set_int_bool(member_int(Xb2),B2))
=> ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
=> ( pp(aa_int_bool(P,X3))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D3))) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( pp(aa_set_int_bool(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D3)))
=> ! [Xb2: int] :
( pp(aa_set_int_bool(member_int(Xb2),B2))
=> ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
=> ( pp(aa_int_bool(Q,X3))
=> pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D3))) ) )
=> ! [X4: int] :
( ! [Xa4: int] :
( pp(aa_set_int_bool(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3)))
=> ! [Xb3: int] :
( pp(aa_set_int_bool(member_int(Xb3),B2))
=> ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
=> ( ( pp(aa_int_bool(P,X4))
& pp(aa_int_bool(Q,X4)) )
=> ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D3)))
& pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D3))) ) ) ) ) ) ).
% bset(1)
tff(fact_3511_plusinfinity,axiom,
! [D: int,P3: fun_int_bool,P: fun_int_bool] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
=> ( ! [X3: int,K3: int] :
( pp(aa_int_bool(P3,X3))
<=> pp(aa_int_bool(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D)))) )
=> ( ? [Z2: int] :
! [X3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X3))
=> ( pp(aa_int_bool(P,X3))
<=> pp(aa_int_bool(P3,X3)) ) )
=> ( ? [X_12: int] : pp(aa_int_bool(P3,X_12))
=> ? [X_1: int] : pp(aa_int_bool(P,X_1)) ) ) ) ) ).
% plusinfinity
tff(fact_3512_minusinfinity,axiom,
! [D: int,P1: fun_int_bool,P: fun_int_bool] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
=> ( ! [X3: int,K3: int] :
( pp(aa_int_bool(P1,X3))
<=> pp(aa_int_bool(P1,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D)))) )
=> ( ? [Z2: int] :
! [X3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Z2))
=> ( pp(aa_int_bool(P,X3))
<=> pp(aa_int_bool(P1,X3)) ) )
=> ( ? [X_12: int] : pp(aa_int_bool(P1,X_12))
=> ? [X_1: int] : pp(aa_int_bool(P,X_1)) ) ) ) ) ).
% minusinfinity
tff(fact_3513_incr__mult__lemma,axiom,
! [D: int,P: fun_int_bool,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
=> ( ! [X3: int] :
( pp(aa_int_bool(P,X3))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D))) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
=> ! [X4: int] :
( pp(aa_int_bool(P,X4))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D)))) ) ) ) ) ).
% incr_mult_lemma
tff(fact_3514_decr__mult__lemma,axiom,
! [D: int,P: fun_int_bool,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
=> ( ! [X3: int] :
( pp(aa_int_bool(P,X3))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D))) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
=> ! [X4: int] :
( pp(aa_int_bool(P,X4))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D)))) ) ) ) ) ).
% decr_mult_lemma
tff(fact_3515_periodic__finite__ex,axiom,
! [D: int,P: fun_int_bool] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
=> ( ! [X3: int,K3: int] :
( pp(aa_int_bool(P,X3))
<=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D)))) )
=> ( ? [X_13: int] : pp(aa_int_bool(P,X_13))
<=> ? [X2: int] :
( pp(aa_set_int_bool(member_int(X2),set_or1266510415728281911st_int(one_one_int,D)))
& pp(aa_int_bool(P,X2)) ) ) ) ) ).
% periodic_finite_ex
tff(fact_3516_aset_I7_J,axiom,
! [D3: int,A2: set_int,T: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
=> ! [X4: int] :
( ! [Xa4: int] :
( pp(aa_set_int_bool(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3)))
=> ! [Xb3: int] :
( pp(aa_set_int_bool(member_int(Xb3),A2))
=> ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X4))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D3))) ) ) ) ).
% aset(7)
tff(fact_3517_aset_I5_J,axiom,
! [D3: int,T: int,A2: set_int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
=> ( pp(aa_set_int_bool(member_int(T),A2))
=> ! [X4: int] :
( ! [Xa4: int] :
( pp(aa_set_int_bool(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3)))
=> ! [Xb3: int] :
( pp(aa_set_int_bool(member_int(Xb3),A2))
=> ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),T))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D3)),T)) ) ) ) ) ).
% aset(5)
tff(fact_3518_aset_I4_J,axiom,
! [D3: int,T: int,A2: set_int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
=> ( pp(aa_set_int_bool(member_int(T),A2))
=> ! [X4: int] :
( ! [Xa4: int] :
( pp(aa_set_int_bool(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3)))
=> ! [Xb3: int] :
( pp(aa_set_int_bool(member_int(Xb3),A2))
=> ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
=> ( ( X4 != T )
=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D3) != T ) ) ) ) ) ).
% aset(4)
tff(fact_3519_aset_I3_J,axiom,
! [D3: int,T: int,A2: set_int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
=> ( pp(aa_set_int_bool(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T),one_one_int)),A2))
=> ! [X4: int] :
( ! [Xa4: int] :
( pp(aa_set_int_bool(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3)))
=> ! [Xb3: int] :
( pp(aa_set_int_bool(member_int(Xb3),A2))
=> ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
=> ( ( X4 = T )
=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D3) = T ) ) ) ) ) ).
% aset(3)
tff(fact_3520_bset_I7_J,axiom,
! [D3: int,T: int,B2: set_int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
=> ( pp(aa_set_int_bool(member_int(T),B2))
=> ! [X4: int] :
( ! [Xa4: int] :
( pp(aa_set_int_bool(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3)))
=> ! [Xb3: int] :
( pp(aa_set_int_bool(member_int(Xb3),B2))
=> ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X4))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D3))) ) ) ) ) ).
% bset(7)
tff(fact_3521_bset_I5_J,axiom,
! [D3: int,B2: set_int,T: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D3))
=> ! [X4: int] :
( ! [Xa4: int] :
( pp(aa_set_int_bool(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3)))
=> ! [Xb3: int] :
( pp(aa_set_int_bool(member_int(Xb3),B2))
=> ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),T))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D3)),T)) ) ) ) ).
% bset(5)
tff(fact_3522_Bolzano,axiom,
! [A: real,B: real,P: fun_re1255401033344146673l_bool] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
=> ( ! [A4: real,B4: real,C2: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A4),B4))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,B4),C2))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A4),B4))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B4),C2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A4),C2)) ) ) ) )
=> ( ! [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),B))
=> ? [D5: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D5))
& ! [A4: real,B4: real] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A4),X3))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),B4))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B4),A4)),D5)) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A4),B4)) ) ) ) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,A),B)) ) ) ) ).
% Bolzano
tff(fact_3523_divmod__algorithm__code_I7_J,axiom,
! [M: num,N: num] :
( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
=> ( unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit1,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))) ) )
& ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
=> ( unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit1,N)) = unique5026877609467782581ep_nat(aa_num_num(bit1,N),unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).
% divmod_algorithm_code(7)
tff(fact_3524_divmod__algorithm__code_I7_J,axiom,
! [M: num,N: num] :
( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
=> ( unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit1,N)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))) ) )
& ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
=> ( unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit1,N)) = aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(aa_num_num(bit1,N)),unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).
% divmod_algorithm_code(7)
tff(fact_3525_divmod__algorithm__code_I7_J,axiom,
! [M: num,N: num] :
( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
=> ( unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit1,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit0,M))) ) )
& ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,M),N))
=> ( unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit1,N)) = unique4921790084139445826nteger(aa_num_num(bit1,N),unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).
% divmod_algorithm_code(7)
tff(fact_3526_divmod__algorithm__code_I8_J,axiom,
! [M: num,N: num] :
( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
=> ( unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit1,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))) ) )
& ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
=> ( unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit1,N)) = unique5026877609467782581ep_nat(aa_num_num(bit1,N),unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).
% divmod_algorithm_code(8)
tff(fact_3527_divmod__algorithm__code_I8_J,axiom,
! [M: num,N: num] :
( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
=> ( unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit1,N)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))) ) )
& ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
=> ( unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit1,N)) = aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(aa_num_num(bit1,N)),unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).
% divmod_algorithm_code(8)
tff(fact_3528_divmod__algorithm__code_I8_J,axiom,
! [M: num,N: num] :
( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
=> ( unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit1,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit1,M))) ) )
& ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
=> ( unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit1,N)) = unique4921790084139445826nteger(aa_num_num(bit1,N),unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N)))) ) ) ) ).
% divmod_algorithm_code(8)
tff(fact_3529_mult__le__cancel__iff2,axiom,
! [Z: real,X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).
% mult_le_cancel_iff2
tff(fact_3530_mult__le__cancel__iff2,axiom,
! [Z: rat,X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z),X)),aa_rat_rat(times_times_rat(Z),Y)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ) ).
% mult_le_cancel_iff2
tff(fact_3531_mult__le__cancel__iff2,axiom,
! [Z: int,X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Z),Y)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ).
% mult_le_cancel_iff2
tff(fact_3532_mult__le__cancel__iff1,axiom,
! [Z: real,X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).
% mult_le_cancel_iff1
tff(fact_3533_mult__le__cancel__iff1,axiom,
! [Z: rat,X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(X),Z)),aa_rat_rat(times_times_rat(Y),Z)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ) ).
% mult_le_cancel_iff1
tff(fact_3534_mult__le__cancel__iff1,axiom,
! [Z: int,X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ).
% mult_le_cancel_iff1
tff(fact_3535_divides__aux__eq,axiom,
! [Q2: nat,R: nat] :
( pp(aa_Pro2232559814647311250t_bool(unique6322359934112328802ux_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q2),R)))
<=> ( R = zero_zero_nat ) ) ).
% divides_aux_eq
tff(fact_3536_divides__aux__eq,axiom,
! [Q2: int,R: int] :
( pp(aa_Pro4501881432599481050t_bool(unique6319869463603278526ux_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R)))
<=> ( R = zero_zero_int ) ) ).
% divides_aux_eq
tff(fact_3537_divmod__algorithm__code_I2_J,axiom,
! [M: num] : unique5052692396658037445od_int(M,one) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,M)),zero_zero_int) ).
% divmod_algorithm_code(2)
tff(fact_3538_divmod__algorithm__code_I2_J,axiom,
! [M: num] : unique5055182867167087721od_nat(M,one) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_num_nat(numeral_numeral_nat,M)),zero_zero_nat) ).
% divmod_algorithm_code(2)
tff(fact_3539_divmod__algorithm__code_I2_J,axiom,
! [M: num] : unique3479559517661332726nteger(M,one) = produc1086072967326762835nteger(numera6620942414471956472nteger(M),zero_z3403309356797280102nteger) ).
% divmod_algorithm_code(2)
tff(fact_3540_divmod__algorithm__code_I3_J,axiom,
! [N: num] : unique5052692396658037445od_int(one,aa_num_num(bit0,N)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,one)) ).
% divmod_algorithm_code(3)
tff(fact_3541_divmod__algorithm__code_I3_J,axiom,
! [N: num] : unique5055182867167087721od_nat(one,aa_num_num(bit0,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).
% divmod_algorithm_code(3)
tff(fact_3542_divmod__algorithm__code_I3_J,axiom,
! [N: num] : unique3479559517661332726nteger(one,aa_num_num(bit0,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(one)) ).
% divmod_algorithm_code(3)
tff(fact_3543_divmod__algorithm__code_I4_J,axiom,
! [N: num] : unique5052692396658037445od_int(one,aa_num_num(bit1,N)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,one)) ).
% divmod_algorithm_code(4)
tff(fact_3544_divmod__algorithm__code_I4_J,axiom,
! [N: num] : unique5055182867167087721od_nat(one,aa_num_num(bit1,N)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).
% divmod_algorithm_code(4)
tff(fact_3545_divmod__algorithm__code_I4_J,axiom,
! [N: num] : unique3479559517661332726nteger(one,aa_num_num(bit1,N)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(one)) ).
% divmod_algorithm_code(4)
tff(fact_3546_divmod__int__def,axiom,
! [M: num,N: num] : unique5052692396658037445od_int(M,N) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_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,N))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))) ).
% divmod_int_def
tff(fact_3547_divmod__def,axiom,
! [M: num,N: num] : unique5052692396658037445od_int(M,N) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_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,N))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))) ).
% divmod_def
tff(fact_3548_divmod__def,axiom,
! [M: num,N: num] : unique5055182867167087721od_nat(M,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))) ).
% divmod_def
tff(fact_3549_divmod__def,axiom,
! [M: num,N: num] : unique3479559517661332726nteger(M,N) = produc1086072967326762835nteger(divide6298287555418463151nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(N)),modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(N))) ).
% divmod_def
tff(fact_3550_divmod_H__nat__def,axiom,
! [M: num,N: num] : unique5055182867167087721od_nat(M,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))) ).
% divmod'_nat_def
tff(fact_3551_divmod__divmod__step,axiom,
! [M: num,N: num] :
( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
=> ( unique5055182867167087721od_nat(M,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,M)) ) )
& ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
=> ( unique5055182867167087721od_nat(M,N) = unique5026877609467782581ep_nat(N,unique5055182867167087721od_nat(M,aa_num_num(bit0,N))) ) ) ) ).
% divmod_divmod_step
tff(fact_3552_divmod__divmod__step,axiom,
! [M: num,N: num] :
( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
=> ( unique5052692396658037445od_int(M,N) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,M)) ) )
& ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
=> ( unique5052692396658037445od_int(M,N) = aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(N),unique5052692396658037445od_int(M,aa_num_num(bit0,N))) ) ) ) ).
% divmod_divmod_step
tff(fact_3553_divmod__divmod__step,axiom,
! [M: num,N: num] :
( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
=> ( unique3479559517661332726nteger(M,N) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(M)) ) )
& ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N))
=> ( unique3479559517661332726nteger(M,N) = unique4921790084139445826nteger(N,unique3479559517661332726nteger(M,aa_num_num(bit0,N))) ) ) ) ).
% divmod_divmod_step
tff(fact_3554_mult__less__iff1,axiom,
! [Z: real,X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ).
% mult_less_iff1
tff(fact_3555_mult__less__iff1,axiom,
! [Z: rat,X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Z))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(X),Z)),aa_rat_rat(times_times_rat(Y),Z)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ) ).
% mult_less_iff1
tff(fact_3556_mult__less__iff1,axiom,
! [Z: int,X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ) ).
% mult_less_iff1
tff(fact_3557_neg__eucl__rel__int__mult__2,axiom,
! [B: int,A: int,Q2: int,R: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int))
=> ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int),B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R)))
=> pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R)),one_one_int)))) ) ) ).
% neg_eucl_rel_int_mult_2
tff(fact_3558_old_Oprod_Oinject,axiom,
! [A: code_integer,B: bool,A3: code_integer,B3: bool] :
( ( produc5135875973608410785r_bool(A,B) = produc5135875973608410785r_bool(A3,B3) )
<=> ( ( A = A3 )
& ( pp(B)
<=> pp(B3) ) ) ) ).
% old.prod.inject
tff(fact_3559_old_Oprod_Oinject,axiom,
! [A: num,B: num,A3: num,B3: num] :
( ( product_Pair_num_num(A,B) = product_Pair_num_num(A3,B3) )
<=> ( ( A = A3 )
& ( B = B3 ) ) ) ).
% old.prod.inject
tff(fact_3560_old_Oprod_Oinject,axiom,
! [A: nat,B: num,A3: nat,B3: num] :
( ( product_Pair_nat_num(A,B) = product_Pair_nat_num(A3,B3) )
<=> ( ( A = A3 )
& ( B = B3 ) ) ) ).
% old.prod.inject
tff(fact_3561_old_Oprod_Oinject,axiom,
! [A: nat,B: nat,A3: nat,B3: nat] :
( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3) )
<=> ( ( A = A3 )
& ( B = B3 ) ) ) ).
% old.prod.inject
tff(fact_3562_old_Oprod_Oinject,axiom,
! [A: int,B: int,A3: int,B3: int] :
( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) )
<=> ( ( A = A3 )
& ( B = B3 ) ) ) ).
% old.prod.inject
tff(fact_3563_prod_Oinject,axiom,
! [X1: code_integer,X23: bool,Y1: code_integer,Y22: bool] :
( ( produc5135875973608410785r_bool(X1,X23) = produc5135875973608410785r_bool(Y1,Y22) )
<=> ( ( X1 = Y1 )
& ( pp(X23)
<=> pp(Y22) ) ) ) ).
% prod.inject
tff(fact_3564_prod_Oinject,axiom,
! [X1: num,X23: num,Y1: num,Y22: num] :
( ( product_Pair_num_num(X1,X23) = product_Pair_num_num(Y1,Y22) )
<=> ( ( X1 = Y1 )
& ( X23 = Y22 ) ) ) ).
% prod.inject
tff(fact_3565_prod_Oinject,axiom,
! [X1: nat,X23: num,Y1: nat,Y22: num] :
( ( product_Pair_nat_num(X1,X23) = product_Pair_nat_num(Y1,Y22) )
<=> ( ( X1 = Y1 )
& ( X23 = Y22 ) ) ) ).
% prod.inject
tff(fact_3566_prod_Oinject,axiom,
! [X1: nat,X23: nat,Y1: nat,Y22: nat] :
( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X23) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y1),Y22) )
<=> ( ( X1 = Y1 )
& ( X23 = Y22 ) ) ) ).
% prod.inject
tff(fact_3567_prod_Oinject,axiom,
! [X1: int,X23: int,Y1: int,Y22: int] :
( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X23) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Y1),Y22) )
<=> ( ( X1 = Y1 )
& ( X23 = Y22 ) ) ) ).
% prod.inject
tff(fact_3568_unique__quotient,axiom,
! [A: int,B: int,Q2: int,R: int,Q5: int,R4: int] :
( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R)))
=> ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q5),R4)))
=> ( Q2 = Q5 ) ) ) ).
% unique_quotient
tff(fact_3569_unique__remainder,axiom,
! [A: int,B: int,Q2: int,R: int,Q5: int,R4: int] :
( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R)))
=> ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q5),R4)))
=> ( R = R4 ) ) ) ).
% unique_remainder
tff(fact_3570_eucl__rel__int__by0,axiom,
! [K: int] : pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K,zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),K))) ).
% eucl_rel_int_by0
tff(fact_3571_div__int__unique,axiom,
! [K: int,L: int,Q2: int,R: int] :
( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R)))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = Q2 ) ) ).
% div_int_unique
tff(fact_3572_mod__int__unique,axiom,
! [K: int,L: int,Q2: int,R: int] :
( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R)))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = R ) ) ).
% mod_int_unique
tff(fact_3573_eucl__rel__int__dividesI,axiom,
! [L: int,K: int,Q2: int] :
( ( L != zero_zero_int )
=> ( ( K = aa_int_int(aa_int_fun_int_int(times_times_int,Q2),L) )
=> pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),zero_zero_int))) ) ) ).
% eucl_rel_int_dividesI
tff(fact_3574_Pair__inject,axiom,
! [A: code_integer,B: bool,A3: code_integer,B3: bool] :
( ( produc5135875973608410785r_bool(A,B) = produc5135875973608410785r_bool(A3,B3) )
=> ~ ( ( A = A3 )
=> ( pp(B)
<=> ~ pp(B3) ) ) ) ).
% Pair_inject
tff(fact_3575_Pair__inject,axiom,
! [A: num,B: num,A3: num,B3: num] :
( ( product_Pair_num_num(A,B) = product_Pair_num_num(A3,B3) )
=> ~ ( ( A = A3 )
=> ( B != B3 ) ) ) ).
% Pair_inject
tff(fact_3576_Pair__inject,axiom,
! [A: nat,B: num,A3: nat,B3: num] :
( ( product_Pair_nat_num(A,B) = product_Pair_nat_num(A3,B3) )
=> ~ ( ( A = A3 )
=> ( B != B3 ) ) ) ).
% Pair_inject
tff(fact_3577_Pair__inject,axiom,
! [A: nat,B: nat,A3: nat,B3: nat] :
( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3) )
=> ~ ( ( A = A3 )
=> ( B != B3 ) ) ) ).
% Pair_inject
tff(fact_3578_Pair__inject,axiom,
! [A: int,B: int,A3: int,B3: int] :
( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) )
=> ~ ( ( A = A3 )
=> ( B != B3 ) ) ) ).
% Pair_inject
tff(fact_3579_prod__cases,axiom,
! [P: fun_Pr8281063099935739201l_bool,P2: produc7417142470700560111r_bool] :
( ! [A4: code_integer,B4: bool] : pp(aa_Pro6456355659118895914l_bool(P,produc5135875973608410785r_bool(A4,B4)))
=> pp(aa_Pro6456355659118895914l_bool(P,P2)) ) ).
% prod_cases
tff(fact_3580_prod__cases,axiom,
! [P: fun_Pr6287530515171585797m_bool,P2: product_prod_num_num] :
( ! [A4: num,B4: num] : pp(aa_Pro762926636801165566m_bool(P,product_Pair_num_num(A4,B4)))
=> pp(aa_Pro762926636801165566m_bool(P,P2)) ) ).
% prod_cases
tff(fact_3581_prod__cases,axiom,
! [P: fun_Pr1727971692872576719m_bool,P2: product_prod_nat_num] :
( ! [A4: nat,B4: num] : pp(aa_Pro5426739851356932296m_bool(P,product_Pair_nat_num(A4,B4)))
=> pp(aa_Pro5426739851356932296m_bool(P,P2)) ) ).
% prod_cases
tff(fact_3582_prod__cases,axiom,
! [P: fun_Pr7757163693017731481t_bool,P2: product_prod_nat_nat] :
( ! [A4: nat,B4: nat] : pp(aa_Pro2232559814647311250t_bool(P,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A4),B4)))
=> pp(aa_Pro2232559814647311250t_bool(P,P2)) ) ).
% prod_cases
tff(fact_3583_prod__cases,axiom,
! [P: fun_Pr803113274115125473t_bool,P2: product_prod_int_int] :
( ! [A4: int,B4: int] : pp(aa_Pro4501881432599481050t_bool(P,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A4),B4)))
=> pp(aa_Pro4501881432599481050t_bool(P,P2)) ) ).
% prod_cases
tff(fact_3584_surj__pair,axiom,
! [P2: produc7417142470700560111r_bool] :
? [X3: code_integer,Y2: bool] : P2 = produc5135875973608410785r_bool(X3,Y2) ).
% surj_pair
tff(fact_3585_surj__pair,axiom,
! [P2: product_prod_num_num] :
? [X3: num,Y2: num] : P2 = product_Pair_num_num(X3,Y2) ).
% surj_pair
tff(fact_3586_surj__pair,axiom,
! [P2: product_prod_nat_num] :
? [X3: nat,Y2: num] : P2 = product_Pair_nat_num(X3,Y2) ).
% surj_pair
tff(fact_3587_surj__pair,axiom,
! [P2: product_prod_nat_nat] :
? [X3: nat,Y2: nat] : P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y2) ).
% surj_pair
tff(fact_3588_surj__pair,axiom,
! [P2: product_prod_int_int] :
? [X3: int,Y2: int] : P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y2) ).
% surj_pair
tff(fact_3589_old_Oprod_Oexhaust,axiom,
! [Y: produc7417142470700560111r_bool] :
~ ! [A4: code_integer,B4: bool] : Y != produc5135875973608410785r_bool(A4,B4) ).
% old.prod.exhaust
tff(fact_3590_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_num_num] :
~ ! [A4: num,B4: num] : Y != product_Pair_num_num(A4,B4) ).
% old.prod.exhaust
tff(fact_3591_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_nat_num] :
~ ! [A4: nat,B4: num] : Y != product_Pair_nat_num(A4,B4) ).
% old.prod.exhaust
tff(fact_3592_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_nat_nat] :
~ ! [A4: nat,B4: nat] : Y != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A4),B4) ).
% old.prod.exhaust
tff(fact_3593_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_int_int] :
~ ! [A4: int,B4: int] : Y != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A4),B4) ).
% old.prod.exhaust
tff(fact_3594_eucl__rel__int,axiom,
! [K: int,L: int] : pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)))) ).
% eucl_rel_int
tff(fact_3595_eucl__rel__int__iff,axiom,
! [K: int,L: int,Q2: int,R: int] :
( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R)))
<=> ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),Q2)),R) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R),L)) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),R))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R),zero_zero_int)) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
=> ( Q2 = zero_zero_int ) ) ) ) ) ) ).
% eucl_rel_int_iff
tff(fact_3596_pos__eucl__rel__int__mult__2,axiom,
! [B: int,A: int,Q2: int,R: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
=> ( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R)))
=> pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R))))) ) ) ).
% pos_eucl_rel_int_mult_2
tff(fact_3597_VEBT__internal_Oheight_Osimps_I1_J,axiom,
! [A: bool,B: bool] : aa_VEBT_VEBT_nat(vEBT_VEBT_height,vEBT_Leaf(A,B)) = zero_zero_nat ).
% VEBT_internal.height.simps(1)
tff(fact_3598_divmod__BitM__2__eq,axiom,
! [M: num] : unique5052692396658037445od_int(bitM(M),aa_num_num(bit0,one)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),one_one_int)),one_one_int) ).
% divmod_BitM_2_eq
tff(fact_3599_insert__simp__norm,axiom,
! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Mi),X))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
=> ( ( X != Ma )
=> ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summary)) ) ) ) ) ) ).
% insert_simp_norm
tff(fact_3600_insert__simp__excp,axiom,
! [Mi: nat,Deg: nat,TreeList: list_VEBT_VEBT,X: nat,Ma: nat,Summary: vEBT_VEBT] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
=> ( ( X != Ma )
=> ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary),X) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mi),Ma))),Deg,list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summary)) ) ) ) ) ) ).
% insert_simp_excp
tff(fact_3601_gcd__nat__induct,axiom,
! [P: fun_nat_fun_nat_bool,M: nat,N: nat] :
( ! [M3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M3),zero_zero_nat))
=> ( ! [M3: nat,N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,N2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M3),N2)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M3),N2)) ) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M),N)) ) ) ).
% gcd_nat_induct
tff(fact_3602_concat__bit__Suc,axiom,
! [N: nat,K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(suc,N)),K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),L))) ).
% concat_bit_Suc
tff(fact_3603_dbl__simps_I3_J,axiom,
aa_complex_complex(neg_nu7009210354673126013omplex,one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).
% dbl_simps(3)
tff(fact_3604_dbl__simps_I3_J,axiom,
aa_real_real(neg_numeral_dbl_real,one_one_real) = numeral_numeral_real(aa_num_num(bit0,one)) ).
% dbl_simps(3)
tff(fact_3605_dbl__simps_I3_J,axiom,
neg_numeral_dbl_rat(one_one_rat) = numeral_numeral_rat(aa_num_num(bit0,one)) ).
% dbl_simps(3)
tff(fact_3606_dbl__simps_I3_J,axiom,
aa_int_int(neg_numeral_dbl_int,one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).
% dbl_simps(3)
tff(fact_3607_max_Oabsorb1,axiom,
! [B: extended_enat,A: extended_enat] :
( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),A))
=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = A ) ) ).
% max.absorb1
tff(fact_3608_max_Oabsorb1,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
=> ( ord_max_rat(A,B) = A ) ) ).
% max.absorb1
tff(fact_3609_max_Oabsorb1,axiom,
! [B: num,A: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ).
% max.absorb1
tff(fact_3610_max_Oabsorb1,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).
% max.absorb1
tff(fact_3611_max_Oabsorb1,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).
% max.absorb1
tff(fact_3612_max_Oabsorb2,axiom,
! [A: extended_enat,B: extended_enat] :
( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(A),B))
=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = B ) ) ).
% max.absorb2
tff(fact_3613_max_Oabsorb2,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( ord_max_rat(A,B) = B ) ) ).
% max.absorb2
tff(fact_3614_max_Oabsorb2,axiom,
! [A: num,B: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) ) ).
% max.absorb2
tff(fact_3615_max_Oabsorb2,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).
% max.absorb2
tff(fact_3616_max_Oabsorb2,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).
% max.absorb2
tff(fact_3617_max_Obounded__iff,axiom,
! [B: extended_enat,C: extended_enat,A: extended_enat] :
( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),C)),A))
<=> ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),A))
& pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C),A)) ) ) ).
% max.bounded_iff
tff(fact_3618_max_Obounded__iff,axiom,
! [B: rat,C: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,ord_max_rat(B,C)),A))
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A)) ) ) ).
% max.bounded_iff
tff(fact_3619_max_Obounded__iff,axiom,
! [B: num,C: num,A: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A))
<=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
& pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A)) ) ) ).
% max.bounded_iff
tff(fact_3620_max_Obounded__iff,axiom,
! [B: nat,C: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A)) ) ) ).
% max.bounded_iff
tff(fact_3621_max_Obounded__iff,axiom,
! [B: int,C: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A)) ) ) ).
% max.bounded_iff
tff(fact_3622_max__less__iff__conj,axiom,
! [X: extended_enat,Y: extended_enat,Z: extended_enat] :
( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,X),Y)),Z))
<=> ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(X),Z))
& pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(Y),Z)) ) ) ).
% max_less_iff_conj
tff(fact_3623_max__less__iff__conj,axiom,
! [X: real,Y: real,Z: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,X),Y)),Z))
<=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Z))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),Z)) ) ) ).
% max_less_iff_conj
tff(fact_3624_max__less__iff__conj,axiom,
! [X: rat,Y: rat,Z: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,ord_max_rat(X,Y)),Z))
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Z))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),Z)) ) ) ).
% max_less_iff_conj
tff(fact_3625_max__less__iff__conj,axiom,
! [X: num,Y: num,Z: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(aa_num_fun_num_num(ord_max_num,X),Y)),Z))
<=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z))
& pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),Z)) ) ) ).
% max_less_iff_conj
tff(fact_3626_max__less__iff__conj,axiom,
! [X: nat,Y: nat,Z: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)),Z))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z)) ) ) ).
% max_less_iff_conj
tff(fact_3627_max__less__iff__conj,axiom,
! [X: int,Y: int,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)),Z))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z)) ) ) ).
% max_less_iff_conj
tff(fact_3628_max_Oabsorb4,axiom,
! [A: extended_enat,B: extended_enat] :
( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(A),B))
=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = B ) ) ).
% max.absorb4
tff(fact_3629_max_Oabsorb4,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) = B ) ) ).
% max.absorb4
tff(fact_3630_max_Oabsorb4,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> ( ord_max_rat(A,B) = B ) ) ).
% max.absorb4
tff(fact_3631_max_Oabsorb4,axiom,
! [A: num,B: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A),B))
=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) ) ).
% max.absorb4
tff(fact_3632_max_Oabsorb4,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).
% max.absorb4
tff(fact_3633_max_Oabsorb4,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).
% max.absorb4
tff(fact_3634_max_Oabsorb3,axiom,
! [B: extended_enat,A: extended_enat] :
( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(B),A))
=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = A ) ) ).
% max.absorb3
tff(fact_3635_max_Oabsorb3,axiom,
! [B: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
=> ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) = A ) ) ).
% max.absorb3
tff(fact_3636_max_Oabsorb3,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
=> ( ord_max_rat(A,B) = A ) ) ).
% max.absorb3
tff(fact_3637_max_Oabsorb3,axiom,
! [B: num,A: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ).
% max.absorb3
tff(fact_3638_max_Oabsorb3,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).
% max.absorb3
tff(fact_3639_max_Oabsorb3,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).
% max.absorb3
tff(fact_3640_length__list__update,axiom,
! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] : size_s6755466524823107622T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X)) = size_s6755466524823107622T_VEBT(Xs) ).
% length_list_update
tff(fact_3641_length__list__update,axiom,
! [Xs: list_bool,I: nat,X: bool] : size_size_list_bool(list_update_bool(Xs,I,X)) = size_size_list_bool(Xs) ).
% length_list_update
tff(fact_3642_length__list__update,axiom,
! [Xs: list_nat,I: nat,X: nat] : size_size_list_nat(list_update_nat(Xs,I,X)) = size_size_list_nat(Xs) ).
% length_list_update
tff(fact_3643_length__list__update,axiom,
! [Xs: list_int,I: nat,X: int] : size_size_list_int(list_update_int(Xs,I,X)) = size_size_list_int(Xs) ).
% length_list_update
tff(fact_3644_max__Suc__Suc,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)) ).
% max_Suc_Suc
tff(fact_3645_max__nat_Oeq__neutr__iff,axiom,
! [A: nat,B: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = zero_zero_nat )
<=> ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.eq_neutr_iff
tff(fact_3646_max__nat_Oleft__neutral,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),A) = A ).
% max_nat.left_neutral
tff(fact_3647_max__nat_Oneutr__eq__iff,axiom,
! [A: nat,B: nat] :
( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
<=> ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.neutr_eq_iff
tff(fact_3648_max__nat_Oright__neutral,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),zero_zero_nat) = A ).
% max_nat.right_neutral
tff(fact_3649_max__0L,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),N) = N ).
% max_0L
tff(fact_3650_max__0R,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),zero_zero_nat) = N ).
% max_0R
tff(fact_3651_nth__list__update__neq,axiom,
! [I: nat,J: nat,Xs: list_nat,X: nat] :
( ( I != J )
=> ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X)),J) = aa_nat_nat(nth_nat(Xs),J) ) ) ).
% nth_list_update_neq
tff(fact_3652_nth__list__update__neq,axiom,
! [I: nat,J: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
( ( I != J )
=> ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),J) = nth_VEBT_VEBT(Xs,J) ) ) ).
% nth_list_update_neq
tff(fact_3653_list__update__id,axiom,
! [Xs: list_nat,I: nat] : list_update_nat(Xs,I,aa_nat_nat(nth_nat(Xs),I)) = Xs ).
% list_update_id
tff(fact_3654_list__update__id,axiom,
! [Xs: list_VEBT_VEBT,I: nat] : list_u1324408373059187874T_VEBT(Xs,I,nth_VEBT_VEBT(Xs,I)) = Xs ).
% list_update_id
tff(fact_3655_concat__bit__0,axiom,
! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(zero_zero_nat),K),L) = L ).
% concat_bit_0
tff(fact_3656_dbl__simps_I2_J,axiom,
aa_complex_complex(neg_nu7009210354673126013omplex,zero_zero_complex) = zero_zero_complex ).
% dbl_simps(2)
tff(fact_3657_dbl__simps_I2_J,axiom,
aa_real_real(neg_numeral_dbl_real,zero_zero_real) = zero_zero_real ).
% dbl_simps(2)
tff(fact_3658_dbl__simps_I2_J,axiom,
neg_numeral_dbl_rat(zero_zero_rat) = zero_zero_rat ).
% dbl_simps(2)
tff(fact_3659_dbl__simps_I2_J,axiom,
aa_int_int(neg_numeral_dbl_int,zero_zero_int) = zero_zero_int ).
% dbl_simps(2)
tff(fact_3660_max__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)))
=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)) = numera1916890842035813515d_enat(V) ) )
& ( ~ pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)))
=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)) = numera1916890842035813515d_enat(U) ) ) ) ).
% max_number_of(1)
tff(fact_3661_max__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(U)),numeral_numeral_real(V)))
=> ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,numeral_numeral_real(U)),numeral_numeral_real(V)) = numeral_numeral_real(V) ) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(U)),numeral_numeral_real(V)))
=> ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,numeral_numeral_real(U)),numeral_numeral_real(V)) = numeral_numeral_real(U) ) ) ) ).
% max_number_of(1)
tff(fact_3662_max__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,numeral_numeral_rat(U)),numeral_numeral_rat(V)))
=> ( ord_max_rat(numeral_numeral_rat(U),numeral_numeral_rat(V)) = numeral_numeral_rat(V) ) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,numeral_numeral_rat(U)),numeral_numeral_rat(V)))
=> ( ord_max_rat(numeral_numeral_rat(U),numeral_numeral_rat(V)) = numeral_numeral_rat(U) ) ) ) ).
% max_number_of(1)
tff(fact_3663_max__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)) = aa_num_nat(numeral_numeral_nat,V) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)) = aa_num_nat(numeral_numeral_nat,U) ) ) ) ).
% max_number_of(1)
tff(fact_3664_max__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)))
=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)) = aa_num_int(numeral_numeral_int,V) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)))
=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)) = aa_num_int(numeral_numeral_int,U) ) ) ) ).
% max_number_of(1)
tff(fact_3665_max__0__1_I4_J,axiom,
! [X: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,numera1916890842035813515d_enat(X)),zero_z5237406670263579293d_enat) = numera1916890842035813515d_enat(X) ).
% max_0_1(4)
tff(fact_3666_max__0__1_I4_J,axiom,
! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,numeral_numeral_real(X)),zero_zero_real) = numeral_numeral_real(X) ).
% max_0_1(4)
tff(fact_3667_max__0__1_I4_J,axiom,
! [X: num] : ord_max_rat(numeral_numeral_rat(X),zero_zero_rat) = numeral_numeral_rat(X) ).
% max_0_1(4)
tff(fact_3668_max__0__1_I4_J,axiom,
! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,X)),zero_zero_nat) = aa_num_nat(numeral_numeral_nat,X) ).
% max_0_1(4)
tff(fact_3669_max__0__1_I4_J,axiom,
! [X: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,X)),zero_zero_int) = aa_num_int(numeral_numeral_int,X) ).
% max_0_1(4)
tff(fact_3670_max__0__1_I3_J,axiom,
! [X: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(X)) = numera1916890842035813515d_enat(X) ).
% max_0_1(3)
tff(fact_3671_max__0__1_I3_J,axiom,
! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,zero_zero_real),numeral_numeral_real(X)) = numeral_numeral_real(X) ).
% max_0_1(3)
tff(fact_3672_max__0__1_I3_J,axiom,
! [X: num] : ord_max_rat(zero_zero_rat,numeral_numeral_rat(X)) = numeral_numeral_rat(X) ).
% max_0_1(3)
tff(fact_3673_max__0__1_I3_J,axiom,
! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,X)) = aa_num_nat(numeral_numeral_nat,X) ).
% max_0_1(3)
tff(fact_3674_max__0__1_I3_J,axiom,
! [X: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,zero_zero_int),aa_num_int(numeral_numeral_int,X)) = aa_num_int(numeral_numeral_int,X) ).
% max_0_1(3)
tff(fact_3675_max__0__1_I2_J,axiom,
aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,one_one_real),zero_zero_real) = one_one_real ).
% max_0_1(2)
tff(fact_3676_max__0__1_I2_J,axiom,
ord_max_rat(one_one_rat,zero_zero_rat) = one_one_rat ).
% max_0_1(2)
tff(fact_3677_max__0__1_I2_J,axiom,
aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),zero_zero_nat) = one_one_nat ).
% max_0_1(2)
tff(fact_3678_max__0__1_I2_J,axiom,
aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,one_on7984719198319812577d_enat),zero_z5237406670263579293d_enat) = one_on7984719198319812577d_enat ).
% max_0_1(2)
tff(fact_3679_max__0__1_I2_J,axiom,
aa_int_int(aa_int_fun_int_int(ord_max_int,one_one_int),zero_zero_int) = one_one_int ).
% max_0_1(2)
tff(fact_3680_max__0__1_I1_J,axiom,
aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,zero_zero_real),one_one_real) = one_one_real ).
% max_0_1(1)
tff(fact_3681_max__0__1_I1_J,axiom,
ord_max_rat(zero_zero_rat,one_one_rat) = one_one_rat ).
% max_0_1(1)
tff(fact_3682_max__0__1_I1_J,axiom,
aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),one_one_nat) = one_one_nat ).
% max_0_1(1)
tff(fact_3683_max__0__1_I1_J,axiom,
aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,zero_z5237406670263579293d_enat),one_on7984719198319812577d_enat) = one_on7984719198319812577d_enat ).
% max_0_1(1)
tff(fact_3684_max__0__1_I1_J,axiom,
aa_int_int(aa_int_fun_int_int(ord_max_int,zero_zero_int),one_one_int) = one_one_int ).
% max_0_1(1)
tff(fact_3685_max__0__1_I6_J,axiom,
! [X: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,numera1916890842035813515d_enat(X)),one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(X) ).
% max_0_1(6)
tff(fact_3686_max__0__1_I6_J,axiom,
! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,numeral_numeral_real(X)),one_one_real) = numeral_numeral_real(X) ).
% max_0_1(6)
tff(fact_3687_max__0__1_I6_J,axiom,
! [X: num] : ord_max_rat(numeral_numeral_rat(X),one_one_rat) = numeral_numeral_rat(X) ).
% max_0_1(6)
tff(fact_3688_max__0__1_I6_J,axiom,
! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,X)),one_one_nat) = aa_num_nat(numeral_numeral_nat,X) ).
% max_0_1(6)
tff(fact_3689_max__0__1_I6_J,axiom,
! [X: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,X)),one_one_int) = aa_num_int(numeral_numeral_int,X) ).
% max_0_1(6)
tff(fact_3690_max__0__1_I5_J,axiom,
! [X: num] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,one_on7984719198319812577d_enat),numera1916890842035813515d_enat(X)) = numera1916890842035813515d_enat(X) ).
% max_0_1(5)
tff(fact_3691_max__0__1_I5_J,axiom,
! [X: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,one_one_real),numeral_numeral_real(X)) = numeral_numeral_real(X) ).
% max_0_1(5)
tff(fact_3692_max__0__1_I5_J,axiom,
! [X: num] : ord_max_rat(one_one_rat,numeral_numeral_rat(X)) = numeral_numeral_rat(X) ).
% max_0_1(5)
tff(fact_3693_max__0__1_I5_J,axiom,
! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,X)) = aa_num_nat(numeral_numeral_nat,X) ).
% max_0_1(5)
tff(fact_3694_max__0__1_I5_J,axiom,
! [X: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,one_one_int),aa_num_int(numeral_numeral_int,X)) = aa_num_int(numeral_numeral_int,X) ).
% max_0_1(5)
tff(fact_3695_list__update__beyond,axiom,
! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),I))
=> ( list_u1324408373059187874T_VEBT(Xs,I,X) = Xs ) ) ).
% list_update_beyond
tff(fact_3696_list__update__beyond,axiom,
! [Xs: list_bool,I: nat,X: bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_bool(Xs)),I))
=> ( list_update_bool(Xs,I,X) = Xs ) ) ).
% list_update_beyond
tff(fact_3697_list__update__beyond,axiom,
! [Xs: list_nat,I: nat,X: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_nat(Xs)),I))
=> ( list_update_nat(Xs,I,X) = Xs ) ) ).
% list_update_beyond
tff(fact_3698_list__update__beyond,axiom,
! [Xs: list_int,I: nat,X: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_int(Xs)),I))
=> ( list_update_int(Xs,I,X) = Xs ) ) ).
% list_update_beyond
tff(fact_3699_concat__bit__nonnegative__iff,axiom,
! [N: nat,K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),L)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ).
% concat_bit_nonnegative_iff
tff(fact_3700_concat__bit__negative__iff,axiom,
! [N: nat,K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),L)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ).
% concat_bit_negative_iff
tff(fact_3701_dbl__simps_I5_J,axiom,
! [K: num] : aa_complex_complex(neg_nu7009210354673126013omplex,numera6690914467698888265omplex(K)) = numera6690914467698888265omplex(aa_num_num(bit0,K)) ).
% dbl_simps(5)
tff(fact_3702_dbl__simps_I5_J,axiom,
! [K: num] : aa_real_real(neg_numeral_dbl_real,numeral_numeral_real(K)) = numeral_numeral_real(aa_num_num(bit0,K)) ).
% dbl_simps(5)
tff(fact_3703_dbl__simps_I5_J,axiom,
! [K: num] : neg_numeral_dbl_rat(numeral_numeral_rat(K)) = numeral_numeral_rat(aa_num_num(bit0,K)) ).
% dbl_simps(5)
tff(fact_3704_dbl__simps_I5_J,axiom,
! [K: num] : aa_int_int(neg_numeral_dbl_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)) ).
% dbl_simps(5)
tff(fact_3705_nth__list__update__eq,axiom,
! [I: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
=> ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),I) = X ) ) ).
% nth_list_update_eq
tff(fact_3706_nth__list__update__eq,axiom,
! [I: nat,Xs: list_bool,X: bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
=> ( pp(aa_nat_bool(nth_bool(list_update_bool(Xs,I,X)),I))
<=> pp(X) ) ) ).
% nth_list_update_eq
tff(fact_3707_nth__list__update__eq,axiom,
! [I: nat,Xs: list_nat,X: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
=> ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X)),I) = X ) ) ).
% nth_list_update_eq
tff(fact_3708_nth__list__update__eq,axiom,
! [I: nat,Xs: list_int,X: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
=> ( aa_nat_int(nth_int(list_update_int(Xs,I,X)),I) = X ) ) ).
% nth_list_update_eq
tff(fact_3709_set__swap,axiom,
! [I: nat,Xs: list_VEBT_VEBT,J: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_s6755466524823107622T_VEBT(Xs)))
=> ( set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,nth_VEBT_VEBT(Xs,J)),J,nth_VEBT_VEBT(Xs,I))) = set_VEBT_VEBT2(Xs) ) ) ) ).
% set_swap
tff(fact_3710_set__swap,axiom,
! [I: nat,Xs: list_bool,J: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_bool(Xs)))
=> ( set_bool2(list_update_bool(list_update_bool(Xs,I,aa_nat_bool(nth_bool(Xs),J)),J,aa_nat_bool(nth_bool(Xs),I))) = set_bool2(Xs) ) ) ) ).
% set_swap
tff(fact_3711_set__swap,axiom,
! [I: nat,Xs: list_nat,J: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_nat(Xs)))
=> ( set_nat2(list_update_nat(list_update_nat(Xs,I,aa_nat_nat(nth_nat(Xs),J)),J,aa_nat_nat(nth_nat(Xs),I))) = set_nat2(Xs) ) ) ) ).
% set_swap
tff(fact_3712_set__swap,axiom,
! [I: nat,Xs: list_int,J: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_int(Xs)))
=> ( set_int2(list_update_int(list_update_int(Xs,I,aa_nat_int(nth_int(Xs),J)),J,aa_nat_int(nth_int(Xs),I))) = set_int2(Xs) ) ) ) ).
% set_swap
tff(fact_3713_max_Omono,axiom,
! [C: extended_enat,A: extended_enat,D: extended_enat,B: extended_enat] :
( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C),A))
=> ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(D),B))
=> pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,C),D)),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B))) ) ) ).
% max.mono
tff(fact_3714_max_Omono,axiom,
! [C: rat,A: rat,D: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,D),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,ord_max_rat(C,D)),ord_max_rat(A,B))) ) ) ).
% max.mono
tff(fact_3715_max_Omono,axiom,
! [C: num,A: num,D: num,B: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,D),B))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,C),D)),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ) ) ).
% max.mono
tff(fact_3716_max_Omono,axiom,
! [C: nat,A: nat,D: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,D),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,C),D)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ) ).
% max.mono
tff(fact_3717_max_Omono,axiom,
! [C: int,A: int,D: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,C),D)),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ) ) ).
% max.mono
tff(fact_3718_max_OorderE,axiom,
! [B: extended_enat,A: extended_enat] :
( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),A))
=> ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) ) ) ).
% max.orderE
tff(fact_3719_max_OorderE,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
=> ( A = ord_max_rat(A,B) ) ) ).
% max.orderE
tff(fact_3720_max_OorderE,axiom,
! [B: num,A: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
=> ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) ) ) ).
% max.orderE
tff(fact_3721_max_OorderE,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).
% max.orderE
tff(fact_3722_max_OorderE,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
=> ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ) ) ).
% max.orderE
tff(fact_3723_max_OorderI,axiom,
! [A: extended_enat,B: extended_enat] :
( ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) )
=> pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),A)) ) ).
% max.orderI
tff(fact_3724_max_OorderI,axiom,
! [A: rat,B: rat] :
( ( A = ord_max_rat(A,B) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A)) ) ).
% max.orderI
tff(fact_3725_max_OorderI,axiom,
! [A: num,B: num] :
( ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) )
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A)) ) ).
% max.orderI
tff(fact_3726_max_OorderI,axiom,
! [A: nat,B: nat] :
( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A)) ) ).
% max.orderI
tff(fact_3727_max_OorderI,axiom,
! [A: int,B: int] :
( ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ).
% max.orderI
tff(fact_3728_max_OboundedE,axiom,
! [B: extended_enat,C: extended_enat,A: extended_enat] :
( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),C)),A))
=> ~ ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),A))
=> ~ pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C),A)) ) ) ).
% max.boundedE
tff(fact_3729_max_OboundedE,axiom,
! [B: rat,C: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,ord_max_rat(B,C)),A))
=> ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A)) ) ) ).
% max.boundedE
tff(fact_3730_max_OboundedE,axiom,
! [B: num,C: num,A: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A))
=> ~ ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A)) ) ) ).
% max.boundedE
tff(fact_3731_max_OboundedE,axiom,
! [B: nat,C: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A))
=> ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A)) ) ) ).
% max.boundedE
tff(fact_3732_max_OboundedE,axiom,
! [B: int,C: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A))
=> ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A)) ) ) ).
% max.boundedE
tff(fact_3733_max_OboundedI,axiom,
! [B: extended_enat,A: extended_enat,C: extended_enat] :
( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),A))
=> ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C),A))
=> pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),C)),A)) ) ) ).
% max.boundedI
tff(fact_3734_max_OboundedI,axiom,
! [B: rat,A: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,ord_max_rat(B,C)),A)) ) ) ).
% max.boundedI
tff(fact_3735_max_OboundedI,axiom,
! [B: num,A: num,C: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A)) ) ) ).
% max.boundedI
tff(fact_3736_max_OboundedI,axiom,
! [B: nat,A: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A)) ) ) ).
% max.boundedI
tff(fact_3737_max_OboundedI,axiom,
! [B: int,A: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A)) ) ) ).
% max.boundedI
tff(fact_3738_max_Oorder__iff,axiom,
! [B: extended_enat,A: extended_enat] :
( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),A))
<=> ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) ) ) ).
% max.order_iff
tff(fact_3739_max_Oorder__iff,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
<=> ( A = ord_max_rat(A,B) ) ) ).
% max.order_iff
tff(fact_3740_max_Oorder__iff,axiom,
! [B: num,A: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
<=> ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) ) ) ).
% max.order_iff
tff(fact_3741_max_Oorder__iff,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
<=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).
% max.order_iff
tff(fact_3742_max_Oorder__iff,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
<=> ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ) ) ).
% max.order_iff
tff(fact_3743_max_Ocobounded1,axiom,
! [A: extended_enat,B: extended_enat] : pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(A),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B))) ).
% max.cobounded1
tff(fact_3744_max_Ocobounded1,axiom,
! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),ord_max_rat(A,B))) ).
% max.cobounded1
tff(fact_3745_max_Ocobounded1,axiom,
! [A: num,B: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ).
% max.cobounded1
tff(fact_3746_max_Ocobounded1,axiom,
! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ).
% max.cobounded1
tff(fact_3747_max_Ocobounded1,axiom,
! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ).
% max.cobounded1
tff(fact_3748_max_Ocobounded2,axiom,
! [B: extended_enat,A: extended_enat] : pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B))) ).
% max.cobounded2
tff(fact_3749_max_Ocobounded2,axiom,
! [B: rat,A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),ord_max_rat(A,B))) ).
% max.cobounded2
tff(fact_3750_max_Ocobounded2,axiom,
! [B: num,A: num] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ).
% max.cobounded2
tff(fact_3751_max_Ocobounded2,axiom,
! [B: nat,A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ).
% max.cobounded2
tff(fact_3752_max_Ocobounded2,axiom,
! [B: int,A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ).
% max.cobounded2
tff(fact_3753_le__max__iff__disj,axiom,
! [Z: extended_enat,X: extended_enat,Y: extended_enat] :
( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(Z),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,X),Y)))
<=> ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(Z),X))
| pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(Z),Y)) ) ) ).
% le_max_iff_disj
tff(fact_3754_le__max__iff__disj,axiom,
! [Z: rat,X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),ord_max_rat(X,Y)))
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),X))
| pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Z),Y)) ) ) ).
% le_max_iff_disj
tff(fact_3755_le__max__iff__disj,axiom,
! [Z: num,X: num,Y: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),aa_num_num(aa_num_fun_num_num(ord_max_num,X),Y)))
<=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),X))
| pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Z),Y)) ) ) ).
% le_max_iff_disj
tff(fact_3756_le__max__iff__disj,axiom,
! [Z: nat,X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),X))
| pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),Y)) ) ) ).
% le_max_iff_disj
tff(fact_3757_le__max__iff__disj,axiom,
! [Z: int,X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),X))
| pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),Y)) ) ) ).
% le_max_iff_disj
tff(fact_3758_max_Oabsorb__iff1,axiom,
! [B: extended_enat,A: extended_enat] :
( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(B),A))
<=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = A ) ) ).
% max.absorb_iff1
tff(fact_3759_max_Oabsorb__iff1,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),A))
<=> ( ord_max_rat(A,B) = A ) ) ).
% max.absorb_iff1
tff(fact_3760_max_Oabsorb__iff1,axiom,
! [B: num,A: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B),A))
<=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = A ) ) ).
% max.absorb_iff1
tff(fact_3761_max_Oabsorb__iff1,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),A))
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).
% max.absorb_iff1
tff(fact_3762_max_Oabsorb__iff1,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A))
<=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).
% max.absorb_iff1
tff(fact_3763_max_Oabsorb__iff2,axiom,
! [A: extended_enat,B: extended_enat] :
( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(A),B))
<=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) = B ) ) ).
% max.absorb_iff2
tff(fact_3764_max_Oabsorb__iff2,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
<=> ( ord_max_rat(A,B) = B ) ) ).
% max.absorb_iff2
tff(fact_3765_max_Oabsorb__iff2,axiom,
! [A: num,B: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A),B))
<=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) = B ) ) ).
% max.absorb_iff2
tff(fact_3766_max_Oabsorb__iff2,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).
% max.absorb_iff2
tff(fact_3767_max_Oabsorb__iff2,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
<=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).
% max.absorb_iff2
tff(fact_3768_max_OcoboundedI1,axiom,
! [C: extended_enat,A: extended_enat,B: extended_enat] :
( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C),A))
=> pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B))) ) ).
% max.coboundedI1
tff(fact_3769_max_OcoboundedI1,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),ord_max_rat(A,B))) ) ).
% max.coboundedI1
tff(fact_3770_max_OcoboundedI1,axiom,
! [C: num,A: num,B: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),A))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ) ).
% max.coboundedI1
tff(fact_3771_max_OcoboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ).
% max.coboundedI1
tff(fact_3772_max_OcoboundedI1,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ) ).
% max.coboundedI1
tff(fact_3773_max_OcoboundedI2,axiom,
! [C: extended_enat,B: extended_enat,A: extended_enat] :
( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C),B))
=> pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(C),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B))) ) ).
% max.coboundedI2
tff(fact_3774_max_OcoboundedI2,axiom,
! [C: rat,B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,C),ord_max_rat(A,B))) ) ).
% max.coboundedI2
tff(fact_3775_max_OcoboundedI2,axiom,
! [C: num,B: num,A: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),B))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ) ).
% max.coboundedI2
tff(fact_3776_max_OcoboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ).
% max.coboundedI2
tff(fact_3777_max_OcoboundedI2,axiom,
! [C: int,B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ) ).
% max.coboundedI2
tff(fact_3778_less__max__iff__disj,axiom,
! [Z: extended_enat,X: extended_enat,Y: extended_enat] :
( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(Z),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,X),Y)))
<=> ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(Z),X))
| pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(Z),Y)) ) ) ).
% less_max_iff_disj
tff(fact_3779_less__max__iff__disj,axiom,
! [Z: real,X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,X),Y)))
<=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),X))
| pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),Y)) ) ) ).
% less_max_iff_disj
tff(fact_3780_less__max__iff__disj,axiom,
! [Z: rat,X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),ord_max_rat(X,Y)))
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),X))
| pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z),Y)) ) ) ).
% less_max_iff_disj
tff(fact_3781_less__max__iff__disj,axiom,
! [Z: num,X: num,Y: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),aa_num_num(aa_num_fun_num_num(ord_max_num,X),Y)))
<=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),X))
| pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),Y)) ) ) ).
% less_max_iff_disj
tff(fact_3782_less__max__iff__disj,axiom,
! [Z: nat,X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X))
| pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),Y)) ) ) ).
% less_max_iff_disj
tff(fact_3783_less__max__iff__disj,axiom,
! [Z: int,X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),X))
| pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),Y)) ) ) ).
% less_max_iff_disj
tff(fact_3784_max_Ostrict__boundedE,axiom,
! [B: extended_enat,C: extended_enat,A: extended_enat] :
( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,B),C)),A))
=> ~ ( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(B),A))
=> ~ pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(C),A)) ) ) ).
% max.strict_boundedE
tff(fact_3785_max_Ostrict__boundedE,axiom,
! [B: real,C: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,B),C)),A))
=> ~ ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A)) ) ) ).
% max.strict_boundedE
tff(fact_3786_max_Ostrict__boundedE,axiom,
! [B: rat,C: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,ord_max_rat(B,C)),A))
=> ~ ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A)) ) ) ).
% max.strict_boundedE
tff(fact_3787_max_Ostrict__boundedE,axiom,
! [B: num,C: num,A: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B),C)),A))
=> ~ ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
=> ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A)) ) ) ).
% max.strict_boundedE
tff(fact_3788_max_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A))
=> ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A)) ) ) ).
% max.strict_boundedE
tff(fact_3789_max_Ostrict__boundedE,axiom,
! [B: int,C: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A))
=> ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A)) ) ) ).
% max.strict_boundedE
tff(fact_3790_max_Ostrict__order__iff,axiom,
! [B: extended_enat,A: extended_enat] :
( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(B),A))
<=> ( ( A = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B) )
& ( A != B ) ) ) ).
% max.strict_order_iff
tff(fact_3791_max_Ostrict__order__iff,axiom,
! [B: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
<=> ( ( A = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B) )
& ( A != B ) ) ) ).
% max.strict_order_iff
tff(fact_3792_max_Ostrict__order__iff,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),A))
<=> ( ( A = ord_max_rat(A,B) )
& ( A != B ) ) ) ).
% max.strict_order_iff
tff(fact_3793_max_Ostrict__order__iff,axiom,
! [B: num,A: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B),A))
<=> ( ( A = aa_num_num(aa_num_fun_num_num(ord_max_num,A),B) )
& ( A != B ) ) ) ).
% max.strict_order_iff
tff(fact_3794_max_Ostrict__order__iff,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B),A))
<=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
& ( A != B ) ) ) ).
% max.strict_order_iff
tff(fact_3795_max_Ostrict__order__iff,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A))
<=> ( ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) )
& ( A != B ) ) ) ).
% max.strict_order_iff
tff(fact_3796_max_Ostrict__coboundedI1,axiom,
! [C: extended_enat,A: extended_enat,B: extended_enat] :
( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(C),A))
=> pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(C),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B))) ) ).
% max.strict_coboundedI1
tff(fact_3797_max_Ostrict__coboundedI1,axiom,
! [C: real,A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),A))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B))) ) ).
% max.strict_coboundedI1
tff(fact_3798_max_Ostrict__coboundedI1,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),ord_max_rat(A,B))) ) ).
% max.strict_coboundedI1
tff(fact_3799_max_Ostrict__coboundedI1,axiom,
! [C: num,A: num,B: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ) ).
% max.strict_coboundedI1
tff(fact_3800_max_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ).
% max.strict_coboundedI1
tff(fact_3801_max_Ostrict__coboundedI1,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ) ).
% max.strict_coboundedI1
tff(fact_3802_max_Ostrict__coboundedI2,axiom,
! [C: extended_enat,B: extended_enat,A: extended_enat] :
( pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(C),B))
=> pp(aa_Ext2772979874139974381t_bool(ord_le72135733267957522d_enat(C),aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,A),B))) ) ).
% max.strict_coboundedI2
tff(fact_3803_max_Ostrict__coboundedI2,axiom,
! [C: real,B: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A),B))) ) ).
% max.strict_coboundedI2
tff(fact_3804_max_Ostrict__coboundedI2,axiom,
! [C: rat,B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),ord_max_rat(A,B))) ) ).
% max.strict_coboundedI2
tff(fact_3805_max_Ostrict__coboundedI2,axiom,
! [C: num,B: num,A: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),B))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A),B))) ) ).
% max.strict_coboundedI2
tff(fact_3806_max_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B))) ) ).
% max.strict_coboundedI2
tff(fact_3807_max_Ostrict__coboundedI2,axiom,
! [C: int,B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B))) ) ).
% max.strict_coboundedI2
tff(fact_3808_max__add__distrib__left,axiom,
! [X: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,X),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Z)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),Z)) ).
% max_add_distrib_left
tff(fact_3809_max__add__distrib__left,axiom,
! [X: rat,Y: rat,Z: rat] : plus_plus_rat(ord_max_rat(X,Y),Z) = ord_max_rat(plus_plus_rat(X,Z),plus_plus_rat(Y,Z)) ).
% max_add_distrib_left
tff(fact_3810_max__add__distrib__left,axiom,
! [X: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Z)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),Z)) ).
% max_add_distrib_left
tff(fact_3811_max__add__distrib__left,axiom,
! [X: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Z)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Y),Z)) ).
% max_add_distrib_left
tff(fact_3812_max__add__distrib__right,axiom,
! [X: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Y),Z)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Z)) ).
% max_add_distrib_right
tff(fact_3813_max__add__distrib__right,axiom,
! [X: rat,Y: rat,Z: rat] : plus_plus_rat(X,ord_max_rat(Y,Z)) = ord_max_rat(plus_plus_rat(X,Y),plus_plus_rat(X,Z)) ).
% max_add_distrib_right
tff(fact_3814_max__add__distrib__right,axiom,
! [X: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Z)) ).
% max_add_distrib_right
tff(fact_3815_max__add__distrib__right,axiom,
! [X: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(ord_max_int,Y),Z)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Z)) ).
% max_add_distrib_right
tff(fact_3816_max__diff__distrib__left,axiom,
! [X: rat,Y: rat,Z: rat] : minus_minus_rat(ord_max_rat(X,Y),Z) = ord_max_rat(minus_minus_rat(X,Z),minus_minus_rat(Y,Z)) ).
% max_diff_distrib_left
tff(fact_3817_max__diff__distrib__left,axiom,
! [X: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Z)) ).
% max_diff_distrib_left
tff(fact_3818_nat__add__max__left,axiom,
! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),Q2)) ).
% nat_add_max_left
tff(fact_3819_nat__add__max__right,axiom,
! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q2)) ).
% nat_add_max_right
tff(fact_3820_nat__mult__max__left,axiom,
! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) ).
% nat_mult_max_left
tff(fact_3821_nat__mult__max__right,axiom,
! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)) ).
% nat_mult_max_right
tff(fact_3822_semiring__norm_I26_J,axiom,
bitM(one) = one ).
% semiring_norm(26)
tff(fact_3823_max__def__raw,axiom,
! [X4: extended_enat,Xa: extended_enat] :
( ( pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(X4),Xa))
=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,X4),Xa) = Xa ) )
& ( ~ pp(aa_Ext2772979874139974381t_bool(ord_le2932123472753598470d_enat(X4),Xa))
=> ( aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,X4),Xa) = X4 ) ) ) ).
% max_def_raw
tff(fact_3824_max__def__raw,axiom,
! [X4: set_nat,Xa: set_nat] :
( ( pp(aa_set_nat_bool(ord_less_eq_set_nat(X4),Xa))
=> ( ord_max_set_nat(X4,Xa) = Xa ) )
& ( ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(X4),Xa))
=> ( ord_max_set_nat(X4,Xa) = X4 ) ) ) ).
% max_def_raw
tff(fact_3825_max__def__raw,axiom,
! [X4: rat,Xa: rat] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Xa))
=> ( ord_max_rat(X4,Xa) = Xa ) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X4),Xa))
=> ( ord_max_rat(X4,Xa) = X4 ) ) ) ).
% max_def_raw
tff(fact_3826_max__def__raw,axiom,
! [X4: num,Xa: num] :
( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Xa))
=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,X4),Xa) = Xa ) )
& ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X4),Xa))
=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,X4),Xa) = X4 ) ) ) ).
% max_def_raw
tff(fact_3827_max__def__raw,axiom,
! [X4: nat,Xa: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Xa))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X4),Xa) = Xa ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Xa))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X4),Xa) = X4 ) ) ) ).
% max_def_raw
tff(fact_3828_max__def__raw,axiom,
! [X4: int,Xa: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Xa))
=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,X4),Xa) = Xa ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X4),Xa))
=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,X4),Xa) = X4 ) ) ) ).
% max_def_raw
tff(fact_3829_concat__bit__assoc,axiom,
! [N: nat,K: int,M: nat,L: int,R: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),aa_int_int(aa_int_fun_int_int(bit_concat_bit(M),L),R)) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),L)),R) ).
% concat_bit_assoc
tff(fact_3830_nat__minus__add__max,axiom,
! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)),M) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),M) ).
% nat_minus_add_max
tff(fact_3831_set__update__subsetI,axiom,
! [Xs: list_P6011104703257516679at_nat,A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,I: nat] :
( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(Xs)),A2))
=> ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A2))
=> pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(list_u6180841689913720943at_nat(Xs,I,X))),A2)) ) ) ).
% set_update_subsetI
tff(fact_3832_set__update__subsetI,axiom,
! [Xs: list_complex,A2: set_complex,X: complex,I: nat] :
( pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Xs)),A2))
=> ( pp(aa_set_complex_bool(member_complex(X),A2))
=> pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(list_update_complex(Xs,I,X))),A2)) ) ) ).
% set_update_subsetI
tff(fact_3833_set__update__subsetI,axiom,
! [Xs: list_real,A2: set_real,X: real,I: nat] :
( pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(Xs)),A2))
=> ( pp(aa_set_real_bool(member_real(X),A2))
=> pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(list_update_real(Xs,I,X))),A2)) ) ) ).
% set_update_subsetI
tff(fact_3834_set__update__subsetI,axiom,
! [Xs: list_int,A2: set_int,X: int,I: nat] :
( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Xs)),A2))
=> ( pp(aa_set_int_bool(member_int(X),A2))
=> pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(list_update_int(Xs,I,X))),A2)) ) ) ).
% set_update_subsetI
tff(fact_3835_set__update__subsetI,axiom,
! [Xs: list_VEBT_VEBT,A2: set_VEBT_VEBT,X: vEBT_VEBT,I: nat] :
( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs)),A2))
=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),A2))
=> pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,X))),A2)) ) ) ).
% set_update_subsetI
tff(fact_3836_set__update__subsetI,axiom,
! [Xs: list_nat,A2: set_nat,X: nat,I: nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(Xs)),A2))
=> ( pp(aa_set_nat_bool(member_nat(X),A2))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(list_update_nat(Xs,I,X))),A2)) ) ) ).
% set_update_subsetI
tff(fact_3837_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_3838_dbl__def,axiom,
! [X: rat] : neg_numeral_dbl_rat(X) = plus_plus_rat(X,X) ).
% dbl_def
tff(fact_3839_dbl__def,axiom,
! [X: int] : aa_int_int(neg_numeral_dbl_int,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,X),X) ).
% dbl_def
tff(fact_3840_semiring__norm_I28_J,axiom,
! [N: num] : bitM(aa_num_num(bit1,N)) = aa_num_num(bit1,aa_num_num(bit0,N)) ).
% semiring_norm(28)
tff(fact_3841_semiring__norm_I27_J,axiom,
! [N: num] : bitM(aa_num_num(bit0,N)) = aa_num_num(bit1,bitM(N)) ).
% semiring_norm(27)
tff(fact_3842_set__update__memI,axiom,
! [N: nat,Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s5460976970255530739at_nat(Xs)))
=> pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(list_u6180841689913720943at_nat(Xs,N,X)))) ) ).
% set_update_memI
tff(fact_3843_set__update__memI,axiom,
! [N: nat,Xs: list_complex,X: complex] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s3451745648224563538omplex(Xs)))
=> pp(aa_set_complex_bool(member_complex(X),set_complex2(list_update_complex(Xs,N,X)))) ) ).
% set_update_memI
tff(fact_3844_set__update__memI,axiom,
! [N: nat,Xs: list_real,X: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_real(Xs)))
=> pp(aa_set_real_bool(member_real(X),set_real2(list_update_real(Xs,N,X)))) ) ).
% set_update_memI
tff(fact_3845_set__update__memI,axiom,
! [N: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs)))
=> pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,N,X)))) ) ).
% set_update_memI
tff(fact_3846_set__update__memI,axiom,
! [N: nat,Xs: list_bool,X: bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
=> member_bool(X,set_bool2(list_update_bool(Xs,N,X))) ) ).
% set_update_memI
tff(fact_3847_set__update__memI,axiom,
! [N: nat,Xs: list_nat,X: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_nat(Xs)))
=> pp(aa_set_nat_bool(member_nat(X),set_nat2(list_update_nat(Xs,N,X)))) ) ).
% set_update_memI
tff(fact_3848_set__update__memI,axiom,
! [N: nat,Xs: list_int,X: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
=> pp(aa_set_int_bool(member_int(X),set_int2(list_update_int(Xs,N,X)))) ) ).
% set_update_memI
tff(fact_3849_list__update__same__conv,axiom,
! [I: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
=> ( ( list_u1324408373059187874T_VEBT(Xs,I,X) = Xs )
<=> ( nth_VEBT_VEBT(Xs,I) = X ) ) ) ).
% list_update_same_conv
tff(fact_3850_list__update__same__conv,axiom,
! [I: nat,Xs: list_bool,X: bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
=> ( ( list_update_bool(Xs,I,X) = Xs )
<=> ( pp(aa_nat_bool(nth_bool(Xs),I))
<=> pp(X) ) ) ) ).
% list_update_same_conv
tff(fact_3851_list__update__same__conv,axiom,
! [I: nat,Xs: list_nat,X: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
=> ( ( list_update_nat(Xs,I,X) = Xs )
<=> ( aa_nat_nat(nth_nat(Xs),I) = X ) ) ) ).
% list_update_same_conv
tff(fact_3852_list__update__same__conv,axiom,
! [I: nat,Xs: list_int,X: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
=> ( ( list_update_int(Xs,I,X) = Xs )
<=> ( aa_nat_int(nth_int(Xs),I) = X ) ) ) ).
% list_update_same_conv
tff(fact_3853_nth__list__update,axiom,
! [I: nat,Xs: list_VEBT_VEBT,J: nat,X: vEBT_VEBT] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs)))
=> ( ( ( I = J )
=> ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),J) = X ) )
& ( ( I != J )
=> ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,X),J) = nth_VEBT_VEBT(Xs,J) ) ) ) ) ).
% nth_list_update
tff(fact_3854_nth__list__update,axiom,
! [I: nat,Xs: list_bool,X: bool,J: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(Xs)))
=> ( pp(aa_nat_bool(nth_bool(list_update_bool(Xs,I,X)),J))
<=> ( ( ( I = J )
=> pp(X) )
& ( ( I != J )
=> pp(aa_nat_bool(nth_bool(Xs),J)) ) ) ) ) ).
% nth_list_update
tff(fact_3855_nth__list__update,axiom,
! [I: nat,Xs: list_nat,J: nat,X: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
=> ( ( ( I = J )
=> ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X)),J) = X ) )
& ( ( I != J )
=> ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X)),J) = aa_nat_nat(nth_nat(Xs),J) ) ) ) ) ).
% nth_list_update
tff(fact_3856_nth__list__update,axiom,
! [I: nat,Xs: list_int,J: nat,X: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
=> ( ( ( I = J )
=> ( aa_nat_int(nth_int(list_update_int(Xs,I,X)),J) = X ) )
& ( ( I != J )
=> ( aa_nat_int(nth_int(list_update_int(Xs,I,X)),J) = aa_nat_int(nth_int(Xs),J) ) ) ) ) ).
% nth_list_update
tff(fact_3857_eval__nat__numeral_I2_J,axiom,
! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,bitM(N))) ).
% eval_nat_numeral(2)
tff(fact_3858_BitM__plus__one,axiom,
! [N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,bitM(N)),one) = aa_num_num(bit0,N) ).
% BitM_plus_one
tff(fact_3859_one__plus__BitM,axiom,
! [N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),bitM(N)) = aa_num_num(bit0,N) ).
% one_plus_BitM
tff(fact_3860_numeral__BitM,axiom,
! [N: num] : numera6690914467698888265omplex(bitM(N)) = aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(aa_num_num(bit0,N))),one_one_complex) ).
% numeral_BitM
tff(fact_3861_numeral__BitM,axiom,
! [N: num] : numeral_numeral_real(bitM(N)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,numeral_numeral_real(aa_num_num(bit0,N))),one_one_real) ).
% numeral_BitM
tff(fact_3862_numeral__BitM,axiom,
! [N: num] : numeral_numeral_rat(bitM(N)) = minus_minus_rat(numeral_numeral_rat(aa_num_num(bit0,N)),one_one_rat) ).
% numeral_BitM
tff(fact_3863_numeral__BitM,axiom,
! [N: num] : aa_num_int(numeral_numeral_int,bitM(N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),one_one_int) ).
% numeral_BitM
tff(fact_3864_Euclid__induct,axiom,
! [P: fun_nat_fun_nat_bool,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A4),B4))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,B4),A4)) )
=> ( ! [A4: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A4),zero_zero_nat))
=> ( ! [A4: nat,B4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A4),B4))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A4),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A4),B4))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A),B)) ) ) ) ).
% Euclid_induct
tff(fact_3865_VEBT__internal_Oheight_Ocases,axiom,
! [X: vEBT_VEBT] :
( ! [A4: bool,B4: bool] : X != vEBT_Leaf(A4,B4)
=> ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] : X != vEBT_Node(Uu2,Deg2,TreeList2,Summary2) ) ).
% VEBT_internal.height.cases
tff(fact_3866_vebt__insert_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] : vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary),X) = if_VEBT_VEBT(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Mi)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Ma)))),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),X),Mi)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Mi)),Mi),X)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summary)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary)) ).
% vebt_insert.simps(5)
tff(fact_3867_vebt__insert_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
( ( vEBT_vebt_insert(X,Xa2) = Y )
=> ( ! [A4: bool,B4: bool] :
( ( X = vEBT_Leaf(A4,B4) )
=> ~ ( ( ( Xa2 = zero_zero_nat )
=> ( Y = vEBT_Leaf(fTrue,B4) ) )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> ( Y = vEBT_Leaf(A4,fTrue) ) )
& ( ( Xa2 != one_one_nat )
=> ( Y = vEBT_Leaf(A4,B4) ) ) ) ) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
( ( X = vEBT_Node(Info2,zero_zero_nat,Ts,S3) )
=> ( Y != vEBT_Node(Info2,zero_zero_nat,Ts,S3) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
( ( X = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S3) )
=> ( Y != vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S3) ) )
=> ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2) )
=> ( Y != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),Xa2)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2) )
=> ( Y != if_VEBT_VEBT(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Ma2)))),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Xa2),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summary2)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2)) ) ) ) ) ) ) ) ).
% vebt_insert.elims
tff(fact_3868_vebt__insert_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
( ( vEBT_vebt_insert(X,Xa2) = Y )
=> ( accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(X,Xa2))
=> ( ! [A4: bool,B4: bool] :
( ( X = vEBT_Leaf(A4,B4) )
=> ( ( ( ( Xa2 = zero_zero_nat )
=> ( Y = vEBT_Leaf(fTrue,B4) ) )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> ( Y = vEBT_Leaf(A4,fTrue) ) )
& ( ( Xa2 != one_one_nat )
=> ( Y = vEBT_Leaf(A4,B4) ) ) ) ) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Leaf(A4,B4),Xa2)) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
( ( X = vEBT_Node(Info2,zero_zero_nat,Ts,S3) )
=> ( ( Y = vEBT_Node(Info2,zero_zero_nat,Ts,S3) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts,S3),Xa2)) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
( ( X = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S3) )
=> ( ( Y = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S3) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S3),Xa2)) ) )
=> ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2) )
=> ( ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),Xa2)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2),Xa2)) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2) )
=> ( ( Y = if_VEBT_VEBT(aa_bool_bool(fconj(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList2))),aa_bool_bool(fNot,aa_bool_bool(fdisj(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Mi2)),aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Xa2),Ma2)))),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Xa2),Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),list_u1324408373059187874T_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),if_VEBT_VEBT(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa2),Mi2)),Mi2),Xa2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summary2)),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2)) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va3)),TreeList2,Summary2),Xa2)) ) ) ) ) ) ) ) ) ).
% vebt_insert.pelims
tff(fact_3869_even__succ__mod__exp,axiom,
! [A: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ) ) ).
% even_succ_mod_exp
tff(fact_3870_even__succ__mod__exp,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ) ).
% even_succ_mod_exp
tff(fact_3871_even__succ__mod__exp,axiom,
! [A: code_integer,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = plus_p5714425477246183910nteger(one_one_Code_integer,modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N))) ) ) ) ).
% even_succ_mod_exp
tff(fact_3872_divmod__algorithm__code_I6_J,axiom,
! [M: num,N: num] : unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit0,N)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_dj),unique5052692396658037445od_int(M,N)) ).
% divmod_algorithm_code(6)
tff(fact_3873_divmod__algorithm__code_I6_J,axiom,
! [M: num,N: num] : unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit0,N)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_dk),unique5055182867167087721od_nat(M,N)) ).
% divmod_algorithm_code(6)
tff(fact_3874_divmod__algorithm__code_I6_J,axiom,
! [M: num,N: num] : unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit0,N)) = produc6916734918728496179nteger(aTP_Lamm_dl,unique3479559517661332726nteger(M,N)) ).
% divmod_algorithm_code(6)
tff(fact_3875_even__succ__div__exp,axiom,
! [A: code_integer,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) ) ) ) ).
% even_succ_div_exp
tff(fact_3876_even__succ__div__exp,axiom,
! [A: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ) ) ).
% even_succ_div_exp
tff(fact_3877_even__succ__div__exp,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ) ) ) ).
% even_succ_div_exp
tff(fact_3878_option_Osize__gen_I2_J,axiom,
! [X: fun_Pr2602117396471556196at_nat,X23: product_prod_nat_nat] : size_o8335143837870341156at_nat(X,some_P7363390416028606310at_nat(X23)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Pro4243173770044397973at_nat(X,X23)),aa_nat_nat(suc,zero_zero_nat)) ).
% option.size_gen(2)
tff(fact_3879_option_Osize__gen_I2_J,axiom,
! [X: fun_nat_nat,X23: nat] : size_option_nat(X,some_nat(X23)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(X,X23)),aa_nat_nat(suc,zero_zero_nat)) ).
% option.size_gen(2)
tff(fact_3880_option_Osize__gen_I2_J,axiom,
! [X: fun_num_nat,X23: num] : size_option_num(X,some_num(X23)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(X,X23)),aa_nat_nat(suc,zero_zero_nat)) ).
% option.size_gen(2)
tff(fact_3881_signed__take__bit__Suc,axiom,
! [N: nat,A: code_integer] : bit_ri6519982836138164636nteger(aa_nat_nat(suc,N),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(N,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).
% signed_take_bit_Suc
tff(fact_3882_signed__take__bit__Suc,axiom,
! [N: nat,A: int] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).
% signed_take_bit_Suc
tff(fact_3883_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),one_one_nat))
<=> ( M = one_one_nat ) ) ).
% nat_dvd_1_iff_1
tff(fact_3884_max__enat__simps_I2_J,axiom,
! [Q2: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Q2),zero_z5237406670263579293d_enat) = Q2 ).
% max_enat_simps(2)
tff(fact_3885_max__enat__simps_I3_J,axiom,
! [Q2: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,zero_z5237406670263579293d_enat),Q2) = Q2 ).
% max_enat_simps(3)
tff(fact_3886_dvd__0__left__iff,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),A))
<=> ( A = zero_z3403309356797280102nteger ) ) ).
% dvd_0_left_iff
tff(fact_3887_dvd__0__left__iff,axiom,
! [A: complex] :
( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,zero_zero_complex),A))
<=> ( A = zero_zero_complex ) ) ).
% dvd_0_left_iff
tff(fact_3888_dvd__0__left__iff,axiom,
! [A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,zero_zero_real),A))
<=> ( A = zero_zero_real ) ) ).
% dvd_0_left_iff
tff(fact_3889_dvd__0__left__iff,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,zero_zero_rat),A))
<=> ( A = zero_zero_rat ) ) ).
% dvd_0_left_iff
tff(fact_3890_dvd__0__left__iff,axiom,
! [A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
<=> ( A = zero_zero_nat ) ) ).
% dvd_0_left_iff
tff(fact_3891_dvd__0__left__iff,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,zero_zero_int),A))
<=> ( A = zero_zero_int ) ) ).
% dvd_0_left_iff
tff(fact_3892_dvd__0__right,axiom,
! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),zero_z3403309356797280102nteger)) ).
% dvd_0_right
tff(fact_3893_dvd__0__right,axiom,
! [A: complex] : pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),zero_zero_complex)) ).
% dvd_0_right
tff(fact_3894_dvd__0__right,axiom,
! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),zero_zero_real)) ).
% dvd_0_right
tff(fact_3895_dvd__0__right,axiom,
! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),zero_zero_rat)) ).
% dvd_0_right
tff(fact_3896_dvd__0__right,axiom,
! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),zero_zero_nat)) ).
% dvd_0_right
tff(fact_3897_dvd__0__right,axiom,
! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),zero_zero_int)) ).
% dvd_0_right
tff(fact_3898_dvd__1__iff__1,axiom,
! [M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_3899_dvd__1__left,axiom,
! [K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(suc,zero_zero_nat)),K)) ).
% dvd_1_left
tff(fact_3900_dvd__add__triv__right__iff,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,A)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B)) ) ).
% dvd_add_triv_right_iff
tff(fact_3901_dvd__add__triv__right__iff,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).
% dvd_add_triv_right_iff
tff(fact_3902_dvd__add__triv__right__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),plus_plus_rat(B,A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ).
% dvd_add_triv_right_iff
tff(fact_3903_dvd__add__triv__right__iff,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).
% dvd_add_triv_right_iff
tff(fact_3904_dvd__add__triv__right__iff,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).
% dvd_add_triv_right_iff
tff(fact_3905_dvd__add__triv__left__iff,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(A,B)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B)) ) ).
% dvd_add_triv_left_iff
tff(fact_3906_dvd__add__triv__left__iff,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).
% dvd_add_triv_left_iff
tff(fact_3907_dvd__add__triv__left__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),plus_plus_rat(A,B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ).
% dvd_add_triv_left_iff
tff(fact_3908_dvd__add__triv__left__iff,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).
% dvd_add_triv_left_iff
tff(fact_3909_dvd__add__triv__left__iff,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).
% dvd_add_triv_left_iff
tff(fact_3910_div__dvd__div,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),C))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,divide6298287555418463151nteger(B,A)),divide6298287555418463151nteger(C,A)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),C)) ) ) ) ).
% div_dvd_div
tff(fact_3911_div__dvd__div,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C)) ) ) ) ).
% div_dvd_div
tff(fact_3912_div__dvd__div,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C)) ) ) ) ).
% div_dvd_div
tff(fact_3913_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N)))
<=> ( ( K = zero_zero_nat )
| pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).
% nat_mult_dvd_cancel_disj
tff(fact_3914_signed__take__bit__of__0,axiom,
! [N: nat] : aa_int_int(bit_ri631733984087533419it_int(N),zero_zero_int) = zero_zero_int ).
% signed_take_bit_of_0
tff(fact_3915_case__prod__conv,axiom,
! [F: fun_na1668100129766719005at_nat,A: nat,B: nat] : aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,A),B) ).
% case_prod_conv
tff(fact_3916_case__prod__conv,axiom,
! [F: fun_na4786964484885125043t_bool,A: nat,B: nat] : aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)) = aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(F,A),B) ).
% case_prod_conv
tff(fact_3917_case__prod__conv,axiom,
! [F: fun_in1696471567231710611nt_int,A: int,B: int] : aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,A),B) ).
% case_prod_conv
tff(fact_3918_case__prod__conv,axiom,
! [F: fun_int_fun_int_bool,A: int,B: int] :
( pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(F,A),B)) ) ).
% case_prod_conv
tff(fact_3919_case__prod__conv,axiom,
! [F: fun_int_fun_int_int,A: int,B: int] : aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) = aa_int_int(aa_int_fun_int_int(F,A),B) ).
% case_prod_conv
tff(fact_3920_dvd__mult__cancel__left,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)))
<=> ( ( C = zero_z3403309356797280102nteger )
| pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B)) ) ) ).
% dvd_mult_cancel_left
tff(fact_3921_dvd__mult__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)))
<=> ( ( C = zero_zero_complex )
| pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B)) ) ) ).
% dvd_mult_cancel_left
tff(fact_3922_dvd__mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B)))
<=> ( ( C = zero_zero_real )
| pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ) ).
% dvd_mult_cancel_left
tff(fact_3923_dvd__mult__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)))
<=> ( ( C = zero_zero_rat )
| pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ) ).
% dvd_mult_cancel_left
tff(fact_3924_dvd__mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
<=> ( ( C = zero_zero_int )
| pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ) ).
% dvd_mult_cancel_left
tff(fact_3925_dvd__mult__cancel__right,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)))
<=> ( ( C = zero_z3403309356797280102nteger )
| pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B)) ) ) ).
% dvd_mult_cancel_right
tff(fact_3926_dvd__mult__cancel__right,axiom,
! [A: complex,C: complex,B: complex] :
( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),C)))
<=> ( ( C = zero_zero_complex )
| pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B)) ) ) ).
% dvd_mult_cancel_right
tff(fact_3927_dvd__mult__cancel__right,axiom,
! [A: real,C: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)))
<=> ( ( C = zero_zero_real )
| pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ) ).
% dvd_mult_cancel_right
tff(fact_3928_dvd__mult__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)))
<=> ( ( C = zero_zero_rat )
| pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ) ).
% dvd_mult_cancel_right
tff(fact_3929_dvd__mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
<=> ( ( C = zero_zero_int )
| pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ) ).
% dvd_mult_cancel_right
tff(fact_3930_dvd__times__left__cancel__iff,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),C)) ) ) ).
% dvd_times_left_cancel_iff
tff(fact_3931_dvd__times__left__cancel__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C)) ) ) ).
% dvd_times_left_cancel_iff
tff(fact_3932_dvd__times__left__cancel__iff,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C)) ) ) ).
% dvd_times_left_cancel_iff
tff(fact_3933_dvd__times__right__cancel__iff,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),C)) ) ) ).
% dvd_times_right_cancel_iff
tff(fact_3934_dvd__times__right__cancel__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C)) ) ) ).
% dvd_times_right_cancel_iff
tff(fact_3935_dvd__times__right__cancel__iff,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C)) ) ) ).
% dvd_times_right_cancel_iff
tff(fact_3936_unit__prod,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer)) ) ) ).
% unit_prod
tff(fact_3937_unit__prod,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat)) ) ) ).
% unit_prod
tff(fact_3938_unit__prod,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int)) ) ) ).
% unit_prod
tff(fact_3939_dvd__add__times__triv__right__iff,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A))))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B)) ) ).
% dvd_add_times_triv_right_iff
tff(fact_3940_dvd__add__times__triv__right__iff,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A))))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).
% dvd_add_times_triv_right_iff
tff(fact_3941_dvd__add__times__triv__right__iff,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),plus_plus_rat(B,aa_rat_rat(times_times_rat(C),A))))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ).
% dvd_add_times_triv_right_iff
tff(fact_3942_dvd__add__times__triv__right__iff,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A))))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).
% dvd_add_times_triv_right_iff
tff(fact_3943_dvd__add__times__triv__right__iff,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,C),A))))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).
% dvd_add_times_triv_right_iff
tff(fact_3944_dvd__add__times__triv__left__iff,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A),B)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B)) ) ).
% dvd_add_times_triv_left_iff
tff(fact_3945_dvd__add__times__triv__left__iff,axiom,
! [A: real,C: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)),B)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).
% dvd_add_times_triv_left_iff
tff(fact_3946_dvd__add__times__triv__left__iff,axiom,
! [A: rat,C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),plus_plus_rat(aa_rat_rat(times_times_rat(C),A),B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ).
% dvd_add_times_triv_left_iff
tff(fact_3947_dvd__add__times__triv__left__iff,axiom,
! [A: nat,C: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),B)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).
% dvd_add_times_triv_left_iff
tff(fact_3948_dvd__add__times__triv__left__iff,axiom,
! [A: int,C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).
% dvd_add_times_triv_left_iff
tff(fact_3949_dvd__mult__div__cancel,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B))
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,A)) = B ) ) ).
% dvd_mult_div_cancel
tff(fact_3950_dvd__mult__div__cancel,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)) = B ) ) ).
% dvd_mult_div_cancel
tff(fact_3951_dvd__mult__div__cancel,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)) = B ) ) ).
% dvd_mult_div_cancel
tff(fact_3952_dvd__div__mult__self,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B))
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B,A)),A) = B ) ) ).
% dvd_div_mult_self
tff(fact_3953_dvd__div__mult__self,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),A) = B ) ) ).
% dvd_div_mult_self
tff(fact_3954_dvd__div__mult__self,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)),A) = B ) ) ).
% dvd_div_mult_self
tff(fact_3955_unit__div__1__div__1,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
=> ( divide6298287555418463151nteger(one_one_Code_integer,divide6298287555418463151nteger(one_one_Code_integer,A)) = A ) ) ).
% unit_div_1_div_1
tff(fact_3956_unit__div__1__div__1,axiom,
! [A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)) = A ) ) ).
% unit_div_1_div_1
tff(fact_3957_unit__div__1__div__1,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)) = A ) ) ).
% unit_div_1_div_1
tff(fact_3958_unit__div__1__unit,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,divide6298287555418463151nteger(one_one_Code_integer,A)),one_one_Code_integer)) ) ).
% unit_div_1_unit
tff(fact_3959_unit__div__1__unit,axiom,
! [A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)),one_one_nat)) ) ).
% unit_div_1_unit
tff(fact_3960_unit__div__1__unit,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)),one_one_int)) ) ).
% unit_div_1_unit
tff(fact_3961_unit__div,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B)),one_one_Code_integer)) ) ) ).
% unit_div
tff(fact_3962_unit__div,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat)) ) ) ).
% unit_div
tff(fact_3963_unit__div,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int)) ) ) ).
% unit_div
tff(fact_3964_div__add,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),B))
=> ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ) ).
% div_add
tff(fact_3965_div__add,axiom,
! [C: nat,A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ) ).
% div_add
tff(fact_3966_div__add,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ) ).
% div_add
tff(fact_3967_div__diff,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),B))
=> ( divide6298287555418463151nteger(minus_8373710615458151222nteger(A,B),C) = minus_8373710615458151222nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ) ).
% div_diff
tff(fact_3968_div__diff,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ) ).
% div_diff
tff(fact_3969_dvd__imp__mod__0,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),A) = zero_zero_nat ) ) ).
% dvd_imp_mod_0
tff(fact_3970_dvd__imp__mod__0,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),A) = zero_zero_int ) ) ).
% dvd_imp_mod_0
tff(fact_3971_dvd__imp__mod__0,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B))
=> ( modulo364778990260209775nteger(B,A) = zero_z3403309356797280102nteger ) ) ).
% dvd_imp_mod_0
tff(fact_3972_signed__take__bit__Suc__1,axiom,
! [N: nat] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),one_one_int) = one_one_int ).
% signed_take_bit_Suc_1
tff(fact_3973_signed__take__bit__numeral__of__1,axiom,
! [K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,K)),one_one_int) = one_one_int ).
% signed_take_bit_numeral_of_1
tff(fact_3974_even__Suc__Suc__iff,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,N))))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ).
% even_Suc_Suc_iff
tff(fact_3975_even__Suc,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N)))
<=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ).
% even_Suc
tff(fact_3976_unit__div__mult__self,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B,A)),A) = B ) ) ).
% unit_div_mult_self
tff(fact_3977_unit__div__mult__self,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),A) = B ) ) ).
% unit_div_mult_self
tff(fact_3978_unit__div__mult__self,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)),A) = B ) ) ).
% unit_div_mult_self
tff(fact_3979_unit__mult__div__div,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(one_one_Code_integer,A)) = divide6298287555418463151nteger(B,A) ) ) ).
% unit_mult_div_div
tff(fact_3980_unit__mult__div__div,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) ) ) ).
% unit_mult_div_div
tff(fact_3981_unit__mult__div__div,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) ) ) ).
% unit_mult_div_div
tff(fact_3982_pow__divides__pow__iff,axiom,
! [N: nat,A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ) ).
% pow_divides_pow_iff
tff(fact_3983_pow__divides__pow__iff,axiom,
! [N: nat,A: int,B: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ) ).
% pow_divides_pow_iff
tff(fact_3984_even__mult__iff,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)))
<=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
| pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)) ) ) ).
% even_mult_iff
tff(fact_3985_even__mult__iff,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
| pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) ) ) ).
% even_mult_iff
tff(fact_3986_even__mult__iff,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
| pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) ) ) ).
% even_mult_iff
tff(fact_3987_even__add,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B)))
<=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)) ) ) ).
% even_add
tff(fact_3988_even__add,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) ) ) ).
% even_add
tff(fact_3989_even__add,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) ) ) ).
% even_add
tff(fact_3990_odd__add,axiom,
! [A: code_integer,B: code_integer] :
( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B)))
<=> ~ ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
<=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)) ) ) ).
% odd_add
tff(fact_3991_odd__add,axiom,
! [A: nat,B: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
<=> ~ ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
<=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) ) ) ).
% odd_add
tff(fact_3992_odd__add,axiom,
! [A: int,B: int] :
( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)))
<=> ~ ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
<=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) ) ) ).
% odd_add
tff(fact_3993_even__mod__2__iff,axiom,
! [A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)) ) ).
% even_mod_2_iff
tff(fact_3994_even__mod__2__iff,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) ) ).
% even_mod_2_iff
tff(fact_3995_even__mod__2__iff,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)) ) ).
% even_mod_2_iff
tff(fact_3996_even__Suc__div__two,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).
% even_Suc_div_two
tff(fact_3997_odd__Suc__div__two,axiom,
! [N: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).
% odd_Suc_div_two
tff(fact_3998_signed__take__bit__Suc__bit0,axiom,
! [N: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(N),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).
% signed_take_bit_Suc_bit0
tff(fact_3999_dvd__numeral__simp,axiom,
! [M: num,N: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)))
<=> pp(aa_Pro4501881432599481050t_bool(unique6319869463603278526ux_int,unique5052692396658037445od_int(N,M))) ) ).
% dvd_numeral_simp
tff(fact_4000_dvd__numeral__simp,axiom,
! [M: num,N: num] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
<=> pp(aa_Pro2232559814647311250t_bool(unique6322359934112328802ux_nat,unique5055182867167087721od_nat(N,M))) ) ).
% dvd_numeral_simp
tff(fact_4001_dvd__numeral__simp,axiom,
! [M: num,N: num] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(M)),numera6620942414471956472nteger(N)))
<=> unique5706413561485394159nteger(unique3479559517661332726nteger(N,M)) ) ).
% dvd_numeral_simp
tff(fact_4002_zero__le__power__eq__numeral,axiom,
! [A: real,W: num] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W))))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
| ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ).
% zero_le_power_eq_numeral
tff(fact_4003_zero__le__power__eq__numeral,axiom,
! [A: rat,W: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W))))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
| ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ) ).
% zero_le_power_eq_numeral
tff(fact_4004_zero__le__power__eq__numeral,axiom,
! [A: int,W: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W))))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
| ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ) ).
% zero_le_power_eq_numeral
tff(fact_4005_power__less__zero__eq__numeral,axiom,
! [A: real,W: num] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_real))
<=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ).
% power_less_zero_eq_numeral
tff(fact_4006_power__less__zero__eq__numeral,axiom,
! [A: rat,W: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_rat))
<=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ).
% power_less_zero_eq_numeral
tff(fact_4007_power__less__zero__eq__numeral,axiom,
! [A: int,W: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_int))
<=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).
% power_less_zero_eq_numeral
tff(fact_4008_power__less__zero__eq,axiom,
! [A: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),zero_zero_real))
<=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ).
% power_less_zero_eq
tff(fact_4009_power__less__zero__eq,axiom,
! [A: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(A),N)),zero_zero_rat))
<=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ).
% power_less_zero_eq
tff(fact_4010_power__less__zero__eq,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(A),N)),zero_zero_int))
<=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).
% power_less_zero_eq
tff(fact_4011_even__plus__one__iff,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,one_one_Code_integer)))
<=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)) ) ).
% even_plus_one_iff
tff(fact_4012_even__plus__one__iff,axiom,
! [A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)))
<=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)) ) ).
% even_plus_one_iff
tff(fact_4013_even__plus__one__iff,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)))
<=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) ) ).
% even_plus_one_iff
tff(fact_4014_even__diff,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),minus_8373710615458151222nteger(A,B)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B))) ) ).
% even_diff
tff(fact_4015_even__diff,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ).
% even_diff
tff(fact_4016_odd__Suc__minus__one,axiom,
! [N: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))) = N ) ) ).
% odd_Suc_minus_one
tff(fact_4017_even__diff__nat,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
| pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ).
% even_diff_nat
tff(fact_4018_zero__less__power__eq__numeral,axiom,
! [A: real,W: num] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W))))
<=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
| ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
& ( A != zero_zero_real ) )
| ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ).
% zero_less_power_eq_numeral
tff(fact_4019_zero__less__power__eq__numeral,axiom,
! [A: rat,W: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W))))
<=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
| ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
& ( A != zero_zero_rat ) )
| ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ) ) ).
% zero_less_power_eq_numeral
tff(fact_4020_zero__less__power__eq__numeral,axiom,
! [A: int,W: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W))))
<=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
| ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
& ( A != zero_zero_int ) )
| ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ) ) ).
% zero_less_power_eq_numeral
tff(fact_4021_even__succ__div__2,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
=> ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),numera6620942414471956472nteger(aa_num_num(bit0,one))) = divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ) ) ).
% even_succ_div_2
tff(fact_4022_even__succ__div__2,axiom,
! [A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).
% even_succ_div_2
tff(fact_4023_even__succ__div__2,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).
% even_succ_div_2
tff(fact_4024_even__succ__div__two,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
=> ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ) ) ).
% even_succ_div_two
tff(fact_4025_even__succ__div__two,axiom,
! [A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).
% even_succ_div_two
tff(fact_4026_even__succ__div__two,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).
% even_succ_div_two
tff(fact_4027_odd__succ__div__two,axiom,
! [A: code_integer] :
( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
=> ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),one_one_Code_integer) ) ) ).
% odd_succ_div_two
tff(fact_4028_odd__succ__div__two,axiom,
! [A: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ) ) ).
% odd_succ_div_two
tff(fact_4029_odd__succ__div__two,axiom,
! [A: int] :
( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ) ) ).
% odd_succ_div_two
tff(fact_4030_even__power,axiom,
! [A: code_integer,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)))
<=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).
% even_power
tff(fact_4031_even__power,axiom,
! [A: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).
% even_power
tff(fact_4032_even__power,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(A),N)))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).
% even_power
tff(fact_4033_odd__two__times__div__two__nat,axiom,
! [N: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat) ) ) ).
% odd_two_times_div_two_nat
tff(fact_4034_divmod__algorithm__code_I5_J,axiom,
! [M: num,N: num] : unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit0,N)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_dm),unique5052692396658037445od_int(M,N)) ).
% divmod_algorithm_code(5)
tff(fact_4035_divmod__algorithm__code_I5_J,axiom,
! [M: num,N: num] : unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit0,N)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_dn),unique5055182867167087721od_nat(M,N)) ).
% divmod_algorithm_code(5)
tff(fact_4036_divmod__algorithm__code_I5_J,axiom,
! [M: num,N: num] : unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit0,N)) = produc6916734918728496179nteger(aTP_Lamm_do,unique3479559517661332726nteger(M,N)) ).
% divmod_algorithm_code(5)
tff(fact_4037_odd__two__times__div__two__succ,axiom,
! [A: code_integer] :
( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
=> ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))),one_one_Code_integer) = A ) ) ).
% odd_two_times_div_two_succ
tff(fact_4038_odd__two__times__div__two__succ,axiom,
! [A: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_nat) = A ) ) ).
% odd_two_times_div_two_succ
tff(fact_4039_odd__two__times__div__two__succ,axiom,
! [A: int] :
( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_int) = A ) ) ).
% odd_two_times_div_two_succ
tff(fact_4040_power__le__zero__eq__numeral,axiom,
! [A: real,W: num] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_real))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W)))
& ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) )
| ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
& ( A = zero_zero_real ) ) ) ) ) ).
% power_le_zero_eq_numeral
tff(fact_4041_power__le__zero__eq__numeral,axiom,
! [A: rat,W: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_rat))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W)))
& ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) )
| ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
& ( A = zero_zero_rat ) ) ) ) ) ).
% power_le_zero_eq_numeral
tff(fact_4042_power__le__zero__eq__numeral,axiom,
! [A: int,W: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_int))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W)))
& ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) )
| ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
& ( A = zero_zero_int ) ) ) ) ) ).
% power_le_zero_eq_numeral
tff(fact_4043_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N),one_one_Code_integer)))
<=> ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
tff(fact_4044_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat)))
<=> ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
tff(fact_4045_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int)))
<=> ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
tff(fact_4046_signed__take__bit__Suc__bit1,axiom,
! [N: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(N),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).
% signed_take_bit_Suc_bit1
tff(fact_4047_dvd__trans,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).
% dvd_trans
tff(fact_4048_dvd__trans,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).
% dvd_trans
tff(fact_4049_dvd__trans,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),C)) ) ) ).
% dvd_trans
tff(fact_4050_dvd__refl,axiom,
! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),A)) ).
% dvd_refl
tff(fact_4051_dvd__refl,axiom,
! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),A)) ).
% dvd_refl
tff(fact_4052_dvd__refl,axiom,
! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),A)) ).
% dvd_refl
tff(fact_4053_dvd__0__left,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),A))
=> ( A = zero_z3403309356797280102nteger ) ) ).
% dvd_0_left
tff(fact_4054_dvd__0__left,axiom,
! [A: complex] :
( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,zero_zero_complex),A))
=> ( A = zero_zero_complex ) ) ).
% dvd_0_left
tff(fact_4055_dvd__0__left,axiom,
! [A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,zero_zero_real),A))
=> ( A = zero_zero_real ) ) ).
% dvd_0_left
tff(fact_4056_dvd__0__left,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,zero_zero_rat),A))
=> ( A = zero_zero_rat ) ) ).
% dvd_0_left
tff(fact_4057_dvd__0__left,axiom,
! [A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),A))
=> ( A = zero_zero_nat ) ) ).
% dvd_0_left
tff(fact_4058_dvd__0__left,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,zero_zero_int),A))
=> ( A = zero_zero_int ) ) ).
% dvd_0_left
tff(fact_4059_dvd__field__iff,axiom,
! [A: complex,B: complex] :
( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B))
<=> ( ( A = zero_zero_complex )
=> ( B = zero_zero_complex ) ) ) ).
% dvd_field_iff
tff(fact_4060_dvd__field__iff,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
<=> ( ( A = zero_zero_real )
=> ( B = zero_zero_real ) ) ) ).
% dvd_field_iff
tff(fact_4061_dvd__field__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B))
<=> ( ( A = zero_zero_rat )
=> ( B = zero_zero_rat ) ) ) ).
% dvd_field_iff
tff(fact_4062_dvd__triv__right,axiom,
! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A))) ).
% dvd_triv_right
tff(fact_4063_dvd__triv__right,axiom,
! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),A))) ).
% dvd_triv_right
tff(fact_4064_dvd__triv__right,axiom,
! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),A))) ).
% dvd_triv_right
tff(fact_4065_dvd__triv__right,axiom,
! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A))) ).
% dvd_triv_right
tff(fact_4066_dvd__triv__right,axiom,
! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),A))) ).
% dvd_triv_right
tff(fact_4067_dvd__mult__right,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),C)) ) ).
% dvd_mult_right
tff(fact_4068_dvd__mult__right,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),C)) ) ).
% dvd_mult_right
tff(fact_4069_dvd__mult__right,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),B)),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),C)) ) ).
% dvd_mult_right
tff(fact_4070_dvd__mult__right,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C)) ) ).
% dvd_mult_right
tff(fact_4071_dvd__mult__right,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C)) ) ).
% dvd_mult_right
tff(fact_4072_mult__dvd__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),D))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D))) ) ) ).
% mult_dvd_mono
tff(fact_4073_mult__dvd__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C),D))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))) ) ) ).
% mult_dvd_mono
tff(fact_4074_mult__dvd__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,C),D))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D))) ) ) ).
% mult_dvd_mono
tff(fact_4075_mult__dvd__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),D))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D))) ) ) ).
% mult_dvd_mono
tff(fact_4076_mult__dvd__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),D))
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))) ) ) ).
% mult_dvd_mono
tff(fact_4077_dvd__triv__left,axiom,
! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B))) ).
% dvd_triv_left
tff(fact_4078_dvd__triv__left,axiom,
! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))) ).
% dvd_triv_left
tff(fact_4079_dvd__triv__left,axiom,
! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(A),B))) ).
% dvd_triv_left
tff(fact_4080_dvd__triv__left,axiom,
! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))) ).
% dvd_triv_left
tff(fact_4081_dvd__triv__left,axiom,
! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ).
% dvd_triv_left
tff(fact_4082_dvd__mult__left,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),C)) ) ).
% dvd_mult_left
tff(fact_4083_dvd__mult__left,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C)) ) ).
% dvd_mult_left
tff(fact_4084_dvd__mult__left,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),B)),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C)) ) ).
% dvd_mult_left
tff(fact_4085_dvd__mult__left,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ).
% dvd_mult_left
tff(fact_4086_dvd__mult__left,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ).
% dvd_mult_left
tff(fact_4087_dvd__mult2,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))) ) ).
% dvd_mult2
tff(fact_4088_dvd__mult2,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ).
% dvd_mult2
tff(fact_4089_dvd__mult2,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),C))) ) ).
% dvd_mult2
tff(fact_4090_dvd__mult2,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))) ) ).
% dvd_mult2
tff(fact_4091_dvd__mult2,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ).
% dvd_mult2
tff(fact_4092_dvd__mult,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))) ) ).
% dvd_mult
tff(fact_4093_dvd__mult,axiom,
! [A: real,C: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ) ).
% dvd_mult
tff(fact_4094_dvd__mult,axiom,
! [A: rat,C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),C))) ) ).
% dvd_mult
tff(fact_4095_dvd__mult,axiom,
! [A: nat,C: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))) ) ).
% dvd_mult
tff(fact_4096_dvd__mult,axiom,
! [A: int,C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))) ) ).
% dvd_mult
tff(fact_4097_dvd__def,axiom,
! [B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A))
<=> ? [K2: code_integer] : A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K2) ) ).
% dvd_def
tff(fact_4098_dvd__def,axiom,
! [B: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A))
<=> ? [K2: real] : A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K2) ) ).
% dvd_def
tff(fact_4099_dvd__def,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
<=> ? [K2: rat] : A = aa_rat_rat(times_times_rat(B),K2) ) ).
% dvd_def
tff(fact_4100_dvd__def,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
<=> ? [K2: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K2) ) ).
% dvd_def
tff(fact_4101_dvd__def,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
<=> ? [K2: int] : A = aa_int_int(aa_int_fun_int_int(times_times_int,B),K2) ) ).
% dvd_def
tff(fact_4102_dvdI,axiom,
! [A: code_integer,B: code_integer,K: code_integer] :
( ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A)) ) ).
% dvdI
tff(fact_4103_dvdI,axiom,
! [A: real,B: real,K: real] :
( ( A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A)) ) ).
% dvdI
tff(fact_4104_dvdI,axiom,
! [A: rat,B: rat,K: rat] :
( ( A = aa_rat_rat(times_times_rat(B),K) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A)) ) ).
% dvdI
tff(fact_4105_dvdI,axiom,
! [A: nat,B: nat,K: nat] :
( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ).
% dvdI
tff(fact_4106_dvdI,axiom,
! [A: int,B: int,K: int] :
( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B),K) )
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ).
% dvdI
tff(fact_4107_dvdE,axiom,
! [B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A))
=> ~ ! [K3: code_integer] : A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K3) ) ).
% dvdE
tff(fact_4108_dvdE,axiom,
! [B: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A))
=> ~ ! [K3: real] : A != aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),K3) ) ).
% dvdE
tff(fact_4109_dvdE,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
=> ~ ! [K3: rat] : A != aa_rat_rat(times_times_rat(B),K3) ) ).
% dvdE
tff(fact_4110_dvdE,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
=> ~ ! [K3: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K3) ) ).
% dvdE
tff(fact_4111_dvdE,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
=> ~ ! [K3: int] : A != aa_int_int(aa_int_fun_int_int(times_times_int,B),K3) ) ).
% dvdE
tff(fact_4112_division__decomp,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)))
=> ? [B7: nat,C5: nat] :
( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B7),C5) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B7),B))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C5),C)) ) ) ).
% division_decomp
tff(fact_4113_division__decomp,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
=> ? [B7: int,C5: int] :
( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B7),C5) )
& pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B7),B))
& pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C5),C)) ) ) ).
% division_decomp
tff(fact_4114_dvd__productE,axiom,
! [P2: nat,A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)))
=> ~ ! [X3: nat,Y2: nat] :
( ( P2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X3),Y2) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X3),A))
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Y2),B)) ) ) ) ).
% dvd_productE
tff(fact_4115_dvd__productE,axiom,
! [P2: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,P2),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)))
=> ~ ! [X3: int,Y2: int] :
( ( P2 = aa_int_int(aa_int_fun_int_int(times_times_int,X3),Y2) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X3),A))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Y2),B)) ) ) ) ).
% dvd_productE
tff(fact_4116_dvd__unit__imp__unit,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer)) ) ) ).
% dvd_unit_imp_unit
tff(fact_4117_dvd__unit__imp__unit,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat)) ) ) ).
% dvd_unit_imp_unit
tff(fact_4118_dvd__unit__imp__unit,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int)) ) ) ).
% dvd_unit_imp_unit
tff(fact_4119_unit__imp__dvd,axiom,
! [B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A)) ) ).
% unit_imp_dvd
tff(fact_4120_unit__imp__dvd,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ).
% unit_imp_dvd
tff(fact_4121_unit__imp__dvd,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ).
% unit_imp_dvd
tff(fact_4122_one__dvd,axiom,
! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,one_one_Code_integer),A)) ).
% one_dvd
tff(fact_4123_one__dvd,axiom,
! [A: complex] : pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,one_one_complex),A)) ).
% one_dvd
tff(fact_4124_one__dvd,axiom,
! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,one_one_real),A)) ).
% one_dvd
tff(fact_4125_one__dvd,axiom,
! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,one_one_rat),A)) ).
% one_dvd
tff(fact_4126_one__dvd,axiom,
! [A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,one_one_nat),A)) ).
% one_dvd
tff(fact_4127_one__dvd,axiom,
! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,one_one_int),A)) ).
% one_dvd
tff(fact_4128_dvd__add__right__iff,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,C)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),C)) ) ) ).
% dvd_add_right_iff
tff(fact_4129_dvd__add__right__iff,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C)) ) ) ).
% dvd_add_right_iff
tff(fact_4130_dvd__add__right__iff,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),plus_plus_rat(B,C)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C)) ) ) ).
% dvd_add_right_iff
tff(fact_4131_dvd__add__right__iff,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).
% dvd_add_right_iff
tff(fact_4132_dvd__add__right__iff,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).
% dvd_add_right_iff
tff(fact_4133_dvd__add__left__iff,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),C))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,C)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B)) ) ) ).
% dvd_add_left_iff
tff(fact_4134_dvd__add__left__iff,axiom,
! [A: real,C: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ) ).
% dvd_add_left_iff
tff(fact_4135_dvd__add__left__iff,axiom,
! [A: rat,C: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),plus_plus_rat(B,C)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B)) ) ) ).
% dvd_add_left_iff
tff(fact_4136_dvd__add__left__iff,axiom,
! [A: nat,C: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ) ).
% dvd_add_left_iff
tff(fact_4137_dvd__add__left__iff,axiom,
! [A: int,C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ) ).
% dvd_add_left_iff
tff(fact_4138_dvd__add,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),C))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,C))) ) ) ).
% dvd_add
tff(fact_4139_dvd__add,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),C))) ) ) ).
% dvd_add
tff(fact_4140_dvd__add,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,A),plus_plus_rat(B,C))) ) ) ).
% dvd_add
tff(fact_4141_dvd__add,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))) ) ) ).
% dvd_add
tff(fact_4142_dvd__add,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C))
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))) ) ) ).
% dvd_add
tff(fact_4143_dvd__diff__commute,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),minus_8373710615458151222nteger(C,B)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),minus_8373710615458151222nteger(B,C))) ) ).
% dvd_diff_commute
tff(fact_4144_dvd__diff__commute,axiom,
! [A: int,C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C))) ) ).
% dvd_diff_commute
tff(fact_4145_dvd__diff,axiom,
! [X: code_integer,Y: code_integer,Z: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),Y))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),Z))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),minus_8373710615458151222nteger(Y,Z))) ) ) ).
% dvd_diff
tff(fact_4146_dvd__diff,axiom,
! [X: rat,Y: rat,Z: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),Y))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),Z))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),minus_minus_rat(Y,Z))) ) ) ).
% dvd_diff
tff(fact_4147_dvd__diff,axiom,
! [X: int,Y: int,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Z))
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Z))) ) ) ).
% dvd_diff
tff(fact_4148_dvd__div__eq__iff,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),B))
=> ( ( divide6298287555418463151nteger(A,C) = divide6298287555418463151nteger(B,C) )
<=> ( A = B ) ) ) ) ).
% dvd_div_eq_iff
tff(fact_4149_dvd__div__eq__iff,axiom,
! [C: complex,A: complex,B: complex] :
( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,C),A))
=> ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,C),B))
=> ( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
<=> ( A = B ) ) ) ) ).
% dvd_div_eq_iff
tff(fact_4150_dvd__div__eq__iff,axiom,
! [C: real,A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C),B))
=> ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
<=> ( A = B ) ) ) ) ).
% dvd_div_eq_iff
tff(fact_4151_dvd__div__eq__iff,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,C),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,C),B))
=> ( ( divide_divide_rat(A,C) = divide_divide_rat(B,C) )
<=> ( A = B ) ) ) ) ).
% dvd_div_eq_iff
tff(fact_4152_dvd__div__eq__iff,axiom,
! [C: nat,A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C) )
<=> ( A = B ) ) ) ) ).
% dvd_div_eq_iff
tff(fact_4153_dvd__div__eq__iff,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
=> ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C) )
<=> ( A = B ) ) ) ) ).
% dvd_div_eq_iff
tff(fact_4154_dvd__div__eq__cancel,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( divide6298287555418463151nteger(A,C) = divide6298287555418463151nteger(B,C) )
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),B))
=> ( A = B ) ) ) ) ).
% dvd_div_eq_cancel
tff(fact_4155_dvd__div__eq__cancel,axiom,
! [A: complex,C: complex,B: complex] :
( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
=> ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,C),A))
=> ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,C),B))
=> ( A = B ) ) ) ) ).
% dvd_div_eq_cancel
tff(fact_4156_dvd__div__eq__cancel,axiom,
! [A: real,C: real,B: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,C),B))
=> ( A = B ) ) ) ) ).
% dvd_div_eq_cancel
tff(fact_4157_dvd__div__eq__cancel,axiom,
! [A: rat,C: rat,B: rat] :
( ( divide_divide_rat(A,C) = divide_divide_rat(B,C) )
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,C),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,C),B))
=> ( A = B ) ) ) ) ).
% dvd_div_eq_cancel
tff(fact_4158_dvd__div__eq__cancel,axiom,
! [A: nat,C: nat,B: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
=> ( A = B ) ) ) ) ).
% dvd_div_eq_cancel
tff(fact_4159_dvd__div__eq__cancel,axiom,
! [A: int,C: int,B: int] :
( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
=> ( A = B ) ) ) ) ).
% dvd_div_eq_cancel
tff(fact_4160_div__div__div__same,axiom,
! [D: code_integer,B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A))
=> ( divide6298287555418463151nteger(divide6298287555418463151nteger(A,D),divide6298287555418463151nteger(B,D)) = divide6298287555418463151nteger(A,B) ) ) ) ).
% div_div_div_same
tff(fact_4161_div__div__div__same,axiom,
! [D: nat,B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),D)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).
% div_div_div_same
tff(fact_4162_div__div__div__same,axiom,
! [D: int,B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),D)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),D)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ).
% div_div_div_same
tff(fact_4163_dvd__power__same,axiom,
! [X: code_integer,Y: code_integer,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),Y))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(X),N)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),N))) ) ).
% dvd_power_same
tff(fact_4164_dvd__power__same,axiom,
! [X: nat,Y: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),Y))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),N))) ) ).
% dvd_power_same
tff(fact_4165_dvd__power__same,axiom,
! [X: real,Y: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),N))) ) ).
% dvd_power_same
tff(fact_4166_dvd__power__same,axiom,
! [X: complex,Y: complex,N: nat] :
( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X),Y))
=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(power_power_complex(X),N)),aa_nat_complex(power_power_complex(Y),N))) ) ).
% dvd_power_same
tff(fact_4167_dvd__power__same,axiom,
! [X: int,Y: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y))
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(X),N)),aa_nat_int(power_power_int(Y),N))) ) ).
% dvd_power_same
tff(fact_4168_old_Oprod_Ocase,axiom,
! [F: fun_na1668100129766719005at_nat,X1: nat,X23: nat] : aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X23)) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X1),X23) ).
% old.prod.case
tff(fact_4169_old_Oprod_Ocase,axiom,
! [F: fun_na4786964484885125043t_bool,X1: nat,X23: nat] : aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X23)) = aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(F,X1),X23) ).
% old.prod.case
tff(fact_4170_old_Oprod_Ocase,axiom,
! [F: fun_in1696471567231710611nt_int,X1: int,X23: int] : aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X23)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X1),X23) ).
% old.prod.case
tff(fact_4171_old_Oprod_Ocase,axiom,
! [F: fun_int_fun_int_bool,X1: int,X23: int] :
( pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X23)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(F,X1),X23)) ) ).
% old.prod.case
tff(fact_4172_old_Oprod_Ocase,axiom,
! [F: fun_int_fun_int_int,X1: int,X23: int] : aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X23)) = aa_int_int(aa_int_fun_int_int(F,X1),X23) ).
% old.prod.case
tff(fact_4173_mod__mod__cancel,axiom,
! [C: nat,B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) ) ) ).
% mod_mod_cancel
tff(fact_4174_mod__mod__cancel,axiom,
! [C: int,B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) ) ) ).
% mod_mod_cancel
tff(fact_4175_mod__mod__cancel,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),B))
=> ( modulo364778990260209775nteger(modulo364778990260209775nteger(A,B),C) = modulo364778990260209775nteger(A,C) ) ) ).
% mod_mod_cancel
tff(fact_4176_dvd__mod,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))) ) ) ).
% dvd_mod
tff(fact_4177_dvd__mod,axiom,
! [K: int,M: int,N: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),M))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M),N))) ) ) ).
% dvd_mod
tff(fact_4178_dvd__mod,axiom,
! [K: code_integer,M: code_integer,N: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,K),M))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,K),N))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,K),modulo364778990260209775nteger(M,N))) ) ) ).
% dvd_mod
tff(fact_4179_dvd__mod__imp__dvd,axiom,
! [C: nat,A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A)) ) ) ).
% dvd_mod_imp_dvd
tff(fact_4180_dvd__mod__imp__dvd,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A)) ) ) ).
% dvd_mod_imp_dvd
tff(fact_4181_dvd__mod__imp__dvd,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),modulo364778990260209775nteger(A,B)))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),A)) ) ) ).
% dvd_mod_imp_dvd
tff(fact_4182_dvd__mod__iff,axiom,
! [C: nat,B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A)) ) ) ).
% dvd_mod_iff
tff(fact_4183_dvd__mod__iff,axiom,
! [C: int,B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A)) ) ) ).
% dvd_mod_iff
tff(fact_4184_dvd__mod__iff,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),modulo364778990260209775nteger(A,B)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),A)) ) ) ).
% dvd_mod_iff
tff(fact_4185_signed__take__bit__mult,axiom,
! [N: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),aa_int_int(bit_ri631733984087533419it_int(N),L))) = aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(aa_int_fun_int_int(times_times_int,K),L)) ).
% signed_take_bit_mult
tff(fact_4186_signed__take__bit__add,axiom,
! [N: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),aa_int_int(bit_ri631733984087533419it_int(N),L))) = aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)) ).
% signed_take_bit_add
tff(fact_4187_dvd__diff__nat,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ) ).
% dvd_diff_nat
tff(fact_4188_signed__take__bit__diff,axiom,
! [N: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),aa_int_int(bit_ri631733984087533419it_int(N),L))) = aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)) ).
% signed_take_bit_diff
tff(fact_4189_zdvd__zdiffD,axiom,
! [K: int,M: int,N: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,M),N)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,K),M)) ) ) ).
% zdvd_zdiffD
tff(fact_4190_subset__divisors__dvd,axiom,
! [A: complex,B: complex] :
( pp(aa_set_complex_bool(ord_le211207098394363844omplex(collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_dp,A))),collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_dp,B))))
<=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B)) ) ).
% subset_divisors_dvd
tff(fact_4191_subset__divisors__dvd,axiom,
! [A: real,B: real] :
( pp(aa_set_real_bool(ord_less_eq_set_real(collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_dq,A))),collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_dq,B))))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B)) ) ).
% subset_divisors_dvd
tff(fact_4192_subset__divisors__dvd,axiom,
! [A: int,B: int] :
( pp(aa_set_int_bool(ord_less_eq_set_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_dr,A))),collect_int(aa_int_fun_int_bool(aTP_Lamm_dr,B))))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B)) ) ).
% subset_divisors_dvd
tff(fact_4193_subset__divisors__dvd,axiom,
! [A: code_integer,B: code_integer] :
( ord_le7084787975880047091nteger(collect_Code_integer(aa_Cod7250484176784686200r_bool(aTP_Lamm_ds,A)),collect_Code_integer(aa_Cod7250484176784686200r_bool(aTP_Lamm_ds,B)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B)) ) ).
% subset_divisors_dvd
tff(fact_4194_subset__divisors__dvd,axiom,
! [A: nat,B: nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_dt,A))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_dt,B))))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B)) ) ).
% subset_divisors_dvd
tff(fact_4195_case__prodE2,axiom,
! [Q: fun_fu7919257609373989805t_bool,P: fun_na1668100129766719005at_nat,Z: product_prod_nat_nat] :
( pp(aa_fun4519828053337421206t_bool(Q,aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(P),Z)))
=> ~ ! [X3: nat,Y2: nat] :
( ( Z = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y2) )
=> ~ pp(aa_fun4519828053337421206t_bool(Q,aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(P,X3),Y2))) ) ) ).
% case_prodE2
tff(fact_4196_case__prodE2,axiom,
! [Q: fun_fu5641360819561228055l_bool,P: fun_na4786964484885125043t_bool,Z: product_prod_nat_nat] :
( pp(aa_fun8663595511526100224l_bool(Q,aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(P),Z)))
=> ~ ! [X3: nat,Y2: nat] :
( ( Z = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y2) )
=> ~ pp(aa_fun8663595511526100224l_bool(Q,aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(P,X3),Y2))) ) ) ).
% case_prodE2
tff(fact_4197_case__prodE2,axiom,
! [Q: fun_Pr803113274115125473t_bool,P: fun_in1696471567231710611nt_int,Z: product_prod_int_int] :
( pp(aa_Pro4501881432599481050t_bool(Q,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(P),Z)))
=> ~ ! [X3: int,Y2: int] :
( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y2) )
=> ~ pp(aa_Pro4501881432599481050t_bool(Q,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(P,X3),Y2))) ) ) ).
% case_prodE2
tff(fact_4198_case__prodE2,axiom,
! [Q: fun_bool_bool,P: fun_int_fun_int_bool,Z: product_prod_int_int] :
( pp(aa_bool_bool(Q,aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(P),Z)))
=> ~ ! [X3: int,Y2: int] :
( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y2) )
=> ~ pp(aa_bool_bool(Q,aa_int_bool(aa_int_fun_int_bool(P,X3),Y2))) ) ) ).
% case_prodE2
tff(fact_4199_case__prodE2,axiom,
! [Q: fun_int_bool,P: fun_int_fun_int_int,Z: product_prod_int_int] :
( pp(aa_int_bool(Q,aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(P),Z)))
=> ~ ! [X3: int,Y2: int] :
( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y2) )
=> ~ pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(P,X3),Y2))) ) ) ).
% case_prodE2
tff(fact_4200_case__prod__eta,axiom,
! [F: fun_Pr2477239776880934807at_nat] : produc4528391901023682385at_nat(aTP_Lamm_du(F)) = F ).
% case_prod_eta
tff(fact_4201_case__prod__eta,axiom,
! [F: fun_Pr248683899324676269t_bool] : produc5023667602222594023t_bool(aTP_Lamm_dv(F)) = F ).
% case_prod_eta
tff(fact_4202_case__prod__eta,axiom,
! [F: fun_Pr2478283820680522243nt_int] : produc4245557441103728435nt_int(aTP_Lamm_dw(F)) = F ).
% case_prod_eta
tff(fact_4203_case__prod__eta,axiom,
! [F: fun_Pr803113274115125473t_bool] : produc8560705615207772305t_bool(aTP_Lamm_dx(F)) = F ).
% case_prod_eta
tff(fact_4204_case__prod__eta,axiom,
! [F: fun_Pr982841511892225912nt_int] : produc8211389475949308722nt_int(aTP_Lamm_dy(F)) = F ).
% case_prod_eta
tff(fact_4205_cond__case__prod__eta,axiom,
! [F: fun_na1668100129766719005at_nat,G: fun_Pr2477239776880934807at_nat] :
( ! [X3: nat,Y2: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X3),Y2) = aa_Pro8023792847356852296at_nat(G,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y2))
=> ( produc4528391901023682385at_nat(F) = G ) ) ).
% cond_case_prod_eta
tff(fact_4206_cond__case__prod__eta,axiom,
! [F: fun_na4786964484885125043t_bool,G: fun_Pr248683899324676269t_bool] :
( ! [X3: nat,Y2: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(F,X3),Y2) = aa_Pro595088089689268830t_bool(G,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y2))
=> ( produc5023667602222594023t_bool(F) = G ) ) ).
% cond_case_prod_eta
tff(fact_4207_cond__case__prod__eta,axiom,
! [F: fun_in1696471567231710611nt_int,G: fun_Pr2478283820680522243nt_int] :
( ! [X3: int,Y2: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X3),Y2) = aa_Pro6970504127699319548nt_int(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y2))
=> ( produc4245557441103728435nt_int(F) = G ) ) ).
% cond_case_prod_eta
tff(fact_4208_cond__case__prod__eta,axiom,
! [F: fun_int_fun_int_bool,G: fun_Pr803113274115125473t_bool] :
( ! [X3: int,Y2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(F,X3),Y2))
<=> pp(aa_Pro4501881432599481050t_bool(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y2))) )
=> ( produc8560705615207772305t_bool(F) = G ) ) ).
% cond_case_prod_eta
tff(fact_4209_cond__case__prod__eta,axiom,
! [F: fun_int_fun_int_int,G: fun_Pr982841511892225912nt_int] :
( ! [X3: int,Y2: int] : aa_int_int(aa_int_fun_int_int(F,X3),Y2) = aa_Pro6799258434465214121nt_int(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y2))
=> ( produc8211389475949308722nt_int(F) = G ) ) ).
% cond_case_prod_eta
tff(fact_4210_strict__subset__divisors__dvd,axiom,
! [A: complex,B: complex] :
( pp(aa_set_complex_bool(ord_less_set_complex(collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_dp,A))),collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_dp,B))))
<=> ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,A),B))
& ~ pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,B),A)) ) ) ).
% strict_subset_divisors_dvd
tff(fact_4211_strict__subset__divisors__dvd,axiom,
! [A: real,B: real] :
( pp(aa_set_real_bool(ord_less_set_real(collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_dq,A))),collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_dq,B))))
<=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,A),B))
& ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A)) ) ) ).
% strict_subset_divisors_dvd
tff(fact_4212_strict__subset__divisors__dvd,axiom,
! [A: nat,B: nat] :
( pp(aa_set_nat_bool(ord_less_set_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_dt,A))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_dt,B))))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
& ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ) ).
% strict_subset_divisors_dvd
tff(fact_4213_strict__subset__divisors__dvd,axiom,
! [A: int,B: int] :
( pp(aa_set_int_bool(ord_less_set_int(collect_int(aa_int_fun_int_bool(aTP_Lamm_dr,A))),collect_int(aa_int_fun_int_bool(aTP_Lamm_dr,B))))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
& ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ) ).
% strict_subset_divisors_dvd
tff(fact_4214_strict__subset__divisors__dvd,axiom,
! [A: code_integer,B: code_integer] :
( ord_le1307284697595431911nteger(collect_Code_integer(aa_Cod7250484176784686200r_bool(aTP_Lamm_ds,A)),collect_Code_integer(aa_Cod7250484176784686200r_bool(aTP_Lamm_ds,B)))
<=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B))
& ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A)) ) ) ).
% strict_subset_divisors_dvd
tff(fact_4215_even__signed__take__bit__iff,axiom,
! [M: nat,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(M,A)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)) ) ).
% even_signed_take_bit_iff
tff(fact_4216_even__signed__take__bit__iff,axiom,
! [M: nat,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(M),A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) ) ).
% even_signed_take_bit_iff
tff(fact_4217_not__is__unit__0,axiom,
~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),one_one_Code_integer)) ).
% not_is_unit_0
tff(fact_4218_not__is__unit__0,axiom,
~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,zero_zero_nat),one_one_nat)) ).
% not_is_unit_0
tff(fact_4219_not__is__unit__0,axiom,
~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,zero_zero_int),one_one_int)) ).
% not_is_unit_0
tff(fact_4220_pinf_I9_J,axiom,
! [D: code_integer,S: code_integer] :
? [Z3: code_integer] :
! [X4: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z3),X4))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X4,S)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X4,S))) ) ) ).
% pinf(9)
tff(fact_4221_pinf_I9_J,axiom,
! [D: real,S: real] :
? [Z3: real] :
! [X4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S))) ) ) ).
% pinf(9)
tff(fact_4222_pinf_I9_J,axiom,
! [D: rat,S: rat] :
? [Z3: rat] :
! [X4: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X4))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(X4,S)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(X4,S))) ) ) ).
% pinf(9)
tff(fact_4223_pinf_I9_J,axiom,
! [D: nat,S: nat] :
? [Z3: nat] :
! [X4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S))) ) ) ).
% pinf(9)
tff(fact_4224_pinf_I9_J,axiom,
! [D: int,S: int] :
? [Z3: int] :
! [X4: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),S)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),S))) ) ) ).
% pinf(9)
tff(fact_4225_pinf_I10_J,axiom,
! [D: code_integer,S: code_integer] :
? [Z3: code_integer] :
! [X4: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z3),X4))
=> ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X4,S)))
<=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X4,S))) ) ) ).
% pinf(10)
tff(fact_4226_pinf_I10_J,axiom,
! [D: real,S: real] :
? [Z3: real] :
! [X4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),X4))
=> ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S)))
<=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S))) ) ) ).
% pinf(10)
tff(fact_4227_pinf_I10_J,axiom,
! [D: rat,S: rat] :
? [Z3: rat] :
! [X4: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Z3),X4))
=> ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(X4,S)))
<=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(X4,S))) ) ) ).
% pinf(10)
tff(fact_4228_pinf_I10_J,axiom,
! [D: nat,S: nat] :
? [Z3: nat] :
! [X4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X4))
=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S)))
<=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S))) ) ) ).
% pinf(10)
tff(fact_4229_pinf_I10_J,axiom,
! [D: int,S: int] :
? [Z3: int] :
! [X4: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X4))
=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),S)))
<=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),S))) ) ) ).
% pinf(10)
tff(fact_4230_minf_I9_J,axiom,
! [D: code_integer,S: code_integer] :
? [Z3: code_integer] :
! [X4: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X4),Z3))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X4,S)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X4,S))) ) ) ).
% minf(9)
tff(fact_4231_minf_I9_J,axiom,
! [D: real,S: real] :
? [Z3: real] :
! [X4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S))) ) ) ).
% minf(9)
tff(fact_4232_minf_I9_J,axiom,
! [D: rat,S: rat] :
? [Z3: rat] :
! [X4: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z3))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(X4,S)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(X4,S))) ) ) ).
% minf(9)
tff(fact_4233_minf_I9_J,axiom,
! [D: nat,S: nat] :
? [Z3: nat] :
! [X4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S))) ) ) ).
% minf(9)
tff(fact_4234_minf_I9_J,axiom,
! [D: int,S: int] :
? [Z3: int] :
! [X4: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),S)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),S))) ) ) ).
% minf(9)
tff(fact_4235_minf_I10_J,axiom,
! [D: code_integer,S: code_integer] :
? [Z3: code_integer] :
! [X4: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X4),Z3))
=> ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X4,S)))
<=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X4,S))) ) ) ).
% minf(10)
tff(fact_4236_minf_I10_J,axiom,
! [D: real,S: real] :
? [Z3: real] :
! [X4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),Z3))
=> ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S)))
<=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S))) ) ) ).
% minf(10)
tff(fact_4237_minf_I10_J,axiom,
! [D: rat,S: rat] :
? [Z3: rat] :
! [X4: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),Z3))
=> ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(X4,S)))
<=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(X4,S))) ) ) ).
% minf(10)
tff(fact_4238_minf_I10_J,axiom,
! [D: nat,S: nat] :
? [Z3: nat] :
! [X4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Z3))
=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S)))
<=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S))) ) ) ).
% minf(10)
tff(fact_4239_minf_I10_J,axiom,
! [D: int,S: int] :
? [Z3: int] :
! [X4: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),Z3))
=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),S)))
<=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),S))) ) ) ).
% minf(10)
tff(fact_4240_dvd__div__eq__0__iff,axiom,
! [B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A))
=> ( ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger )
<=> ( A = zero_z3403309356797280102nteger ) ) ) ).
% dvd_div_eq_0_iff
tff(fact_4241_dvd__div__eq__0__iff,axiom,
! [B: complex,A: complex] :
( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,B),A))
=> ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = zero_zero_complex )
<=> ( A = zero_zero_complex ) ) ) ).
% dvd_div_eq_0_iff
tff(fact_4242_dvd__div__eq__0__iff,axiom,
! [B: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A))
=> ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = zero_zero_real )
<=> ( A = zero_zero_real ) ) ) ).
% dvd_div_eq_0_iff
tff(fact_4243_dvd__div__eq__0__iff,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
=> ( ( divide_divide_rat(A,B) = zero_zero_rat )
<=> ( A = zero_zero_rat ) ) ) ).
% dvd_div_eq_0_iff
tff(fact_4244_dvd__div__eq__0__iff,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat )
<=> ( A = zero_zero_nat ) ) ) ).
% dvd_div_eq_0_iff
tff(fact_4245_dvd__div__eq__0__iff,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
=> ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int )
<=> ( A = zero_zero_int ) ) ) ).
% dvd_div_eq_0_iff
tff(fact_4246_unit__mult__right__cancel,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
=> ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) )
<=> ( B = C ) ) ) ).
% unit_mult_right_cancel
tff(fact_4247_unit__mult__right__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) )
<=> ( B = C ) ) ) ).
% unit_mult_right_cancel
tff(fact_4248_unit__mult__right__cancel,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
=> ( ( aa_int_int(aa_int_fun_int_int(times_times_int,B),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),A) )
<=> ( B = C ) ) ) ).
% unit_mult_right_cancel
tff(fact_4249_unit__mult__left__cancel,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
=> ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C) )
<=> ( B = C ) ) ) ).
% unit_mult_left_cancel
tff(fact_4250_unit__mult__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) )
<=> ( B = C ) ) ) ).
% unit_mult_left_cancel
tff(fact_4251_unit__mult__left__cancel,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
=> ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = aa_int_int(aa_int_fun_int_int(times_times_int,A),C) )
<=> ( B = C ) ) ) ).
% unit_mult_left_cancel
tff(fact_4252_mult__unit__dvd__iff_H,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),C)) ) ) ).
% mult_unit_dvd_iff'
tff(fact_4253_mult__unit__dvd__iff_H,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),C)) ) ) ).
% mult_unit_dvd_iff'
tff(fact_4254_mult__unit__dvd__iff_H,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),C)) ) ) ).
% mult_unit_dvd_iff'
tff(fact_4255_dvd__mult__unit__iff_H,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),C)) ) ) ).
% dvd_mult_unit_iff'
tff(fact_4256_dvd__mult__unit__iff_H,axiom,
! [B: nat,A: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).
% dvd_mult_unit_iff'
tff(fact_4257_dvd__mult__unit__iff_H,axiom,
! [B: int,A: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).
% dvd_mult_unit_iff'
tff(fact_4258_mult__unit__dvd__iff,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),C)) ) ) ).
% mult_unit_dvd_iff
tff(fact_4259_mult__unit__dvd__iff,axiom,
! [B: nat,A: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).
% mult_unit_dvd_iff
tff(fact_4260_mult__unit__dvd__iff,axiom,
! [B: int,A: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).
% mult_unit_dvd_iff
tff(fact_4261_dvd__mult__unit__iff,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),C)) ) ) ).
% dvd_mult_unit_iff
tff(fact_4262_dvd__mult__unit__iff,axiom,
! [B: nat,A: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).
% dvd_mult_unit_iff
tff(fact_4263_dvd__mult__unit__iff,axiom,
! [B: int,A: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).
% dvd_mult_unit_iff
tff(fact_4264_is__unit__mult__iff,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer))
<=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
& pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer)) ) ) ).
% is_unit_mult_iff
tff(fact_4265_is__unit__mult__iff,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat)) ) ) ).
% is_unit_mult_iff
tff(fact_4266_is__unit__mult__iff,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
& pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int)) ) ) ).
% is_unit_mult_iff
tff(fact_4267_div__mult__div__if__dvd,axiom,
! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),C))
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),divide6298287555418463151nteger(C,D)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D)) ) ) ) ).
% div_mult_div_if_dvd
tff(fact_4268_div__mult__div__if__dvd,axiom,
! [B: nat,A: nat,D: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),C))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ).
% div_mult_div_if_dvd
tff(fact_4269_div__mult__div__if__dvd,axiom,
! [B: int,A: int,D: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),C))
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),D)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ).
% div_mult_div_if_dvd
tff(fact_4270_dvd__mult__imp__div,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(B,C))) ) ).
% dvd_mult_imp_div
tff(fact_4271_dvd__mult__imp__div,axiom,
! [A: nat,C: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))) ) ).
% dvd_mult_imp_div
tff(fact_4272_dvd__mult__imp__div,axiom,
! [A: int,C: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))) ) ).
% dvd_mult_imp_div
tff(fact_4273_dvd__div__mult2__eq,axiom,
! [B: code_integer,C: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),A))
=> ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ).
% dvd_div_mult2_eq
tff(fact_4274_dvd__div__mult2__eq,axiom,
! [B: nat,C: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).
% dvd_div_mult2_eq
tff(fact_4275_dvd__div__mult2__eq,axiom,
! [B: int,C: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).
% dvd_div_mult2_eq
tff(fact_4276_div__div__eq__right,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A))
=> ( divide6298287555418463151nteger(A,divide6298287555418463151nteger(B,C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),C) ) ) ) ).
% div_div_eq_right
tff(fact_4277_div__div__eq__right,axiom,
! [C: nat,B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).
% div_div_eq_right
tff(fact_4278_div__div__eq__right,axiom,
! [C: int,B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).
% div_div_eq_right
tff(fact_4279_div__mult__swap,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),B))
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,C)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ) ) ).
% div_mult_swap
tff(fact_4280_div__mult__swap,axiom,
! [C: nat,B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).
% div_mult_swap
tff(fact_4281_div__mult__swap,axiom,
! [C: int,B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ) ) ).
% div_mult_swap
tff(fact_4282_dvd__div__mult,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),B))
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B,C)),A) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A),C) ) ) ).
% dvd_div_mult
tff(fact_4283_dvd__div__mult,axiom,
! [C: nat,B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),C) ) ) ).
% dvd_div_mult
tff(fact_4284_dvd__div__mult,axiom,
! [C: int,B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),C) ) ) ).
% dvd_div_mult
tff(fact_4285_dvd__div__unit__iff,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(C,B)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),C)) ) ) ).
% dvd_div_unit_iff
tff(fact_4286_dvd__div__unit__iff,axiom,
! [B: nat,A: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).
% dvd_div_unit_iff
tff(fact_4287_dvd__div__unit__iff,axiom,
! [B: int,A: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).
% dvd_div_unit_iff
tff(fact_4288_div__unit__dvd__iff,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B)),C))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),C)) ) ) ).
% div_unit_dvd_iff
tff(fact_4289_div__unit__dvd__iff,axiom,
! [B: nat,A: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),C)) ) ) ).
% div_unit_dvd_iff
tff(fact_4290_div__unit__dvd__iff,axiom,
! [B: int,A: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),C)) ) ) ).
% div_unit_dvd_iff
tff(fact_4291_unit__div__cancel,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
=> ( ( divide6298287555418463151nteger(B,A) = divide6298287555418463151nteger(C,A) )
<=> ( B = C ) ) ) ).
% unit_div_cancel
tff(fact_4292_unit__div__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) )
<=> ( B = C ) ) ) ).
% unit_div_cancel
tff(fact_4293_unit__div__cancel,axiom,
! [A: int,B: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
=> ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A) )
<=> ( B = C ) ) ) ).
% unit_div_cancel
tff(fact_4294_div__plus__div__distrib__dvd__right,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),B))
=> ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ).
% div_plus_div_distrib_dvd_right
tff(fact_4295_div__plus__div__distrib__dvd__right,axiom,
! [C: nat,B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ).
% div_plus_div_distrib_dvd_right
tff(fact_4296_div__plus__div__distrib__dvd__right,axiom,
! [C: int,B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ).
% div_plus_div_distrib_dvd_right
tff(fact_4297_div__plus__div__distrib__dvd__left,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),A))
=> ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ).
% div_plus_div_distrib_dvd_left
tff(fact_4298_div__plus__div__distrib__dvd__left,axiom,
! [C: nat,A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),A))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ).
% div_plus_div_distrib_dvd_left
tff(fact_4299_div__plus__div__distrib__dvd__left,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),A))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ).
% div_plus_div_distrib_dvd_left
tff(fact_4300_div__power,axiom,
! [B: code_integer,A: code_integer,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A))
=> ( aa_nat_Code_integer(power_8256067586552552935nteger(divide6298287555418463151nteger(A,B)),N) = divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),N),aa_nat_Code_integer(power_8256067586552552935nteger(B),N)) ) ) ).
% div_power
tff(fact_4301_div__power,axiom,
! [B: nat,A: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)) ) ) ).
% div_power
tff(fact_4302_div__power,axiom,
! [B: int,A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
=> ( aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),N) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N)) ) ) ).
% div_power
tff(fact_4303_mod__0__imp__dvd,axiom,
! [A: nat,B: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ).
% mod_0_imp_dvd
tff(fact_4304_mod__0__imp__dvd,axiom,
! [A: int,B: int] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ).
% mod_0_imp_dvd
tff(fact_4305_mod__0__imp__dvd,axiom,
! [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A)) ) ).
% mod_0_imp_dvd
tff(fact_4306_dvd__eq__mod__eq__0,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),A) = zero_zero_nat ) ) ).
% dvd_eq_mod_eq_0
tff(fact_4307_dvd__eq__mod__eq__0,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
<=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),A) = zero_zero_int ) ) ).
% dvd_eq_mod_eq_0
tff(fact_4308_dvd__eq__mod__eq__0,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B))
<=> ( modulo364778990260209775nteger(B,A) = zero_z3403309356797280102nteger ) ) ).
% dvd_eq_mod_eq_0
tff(fact_4309_mod__eq__0__iff__dvd,axiom,
! [A: nat,B: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat )
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A)) ) ).
% mod_eq_0_iff_dvd
tff(fact_4310_mod__eq__0__iff__dvd,axiom,
! [A: int,B: int] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A)) ) ).
% mod_eq_0_iff_dvd
tff(fact_4311_mod__eq__0__iff__dvd,axiom,
! [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger )
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A)) ) ).
% mod_eq_0_iff_dvd
tff(fact_4312_dvd__power__le,axiom,
! [X: code_integer,Y: code_integer,N: nat,M: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),Y))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(X),N)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),M))) ) ) ).
% dvd_power_le
tff(fact_4313_dvd__power__le,axiom,
! [X: nat,Y: nat,N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),Y))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),M))) ) ) ).
% dvd_power_le
tff(fact_4314_dvd__power__le,axiom,
! [X: real,Y: real,N: nat,M: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Y))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),M))) ) ) ).
% dvd_power_le
tff(fact_4315_dvd__power__le,axiom,
! [X: complex,Y: complex,N: nat,M: nat] :
( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X),Y))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(power_power_complex(X),N)),aa_nat_complex(power_power_complex(Y),M))) ) ) ).
% dvd_power_le
tff(fact_4316_dvd__power__le,axiom,
! [X: int,Y: int,N: nat,M: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(X),N)),aa_nat_int(power_power_int(Y),M))) ) ) ).
% dvd_power_le
tff(fact_4317_power__le__dvd,axiom,
! [A: code_integer,N: nat,B: code_integer,M: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),B)) ) ) ).
% power_le_dvd
tff(fact_4318_power__le__dvd,axiom,
! [A: nat,N: nat,B: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),B)) ) ) ).
% power_le_dvd
tff(fact_4319_power__le__dvd,axiom,
! [A: real,N: nat,B: real,M: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),B)) ) ) ).
% power_le_dvd
tff(fact_4320_power__le__dvd,axiom,
! [A: complex,N: nat,B: complex,M: nat] :
( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),N)),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),M)),B)) ) ) ).
% power_le_dvd
tff(fact_4321_power__le__dvd,axiom,
! [A: int,N: nat,B: int,M: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),N)),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),M)),B)) ) ) ).
% power_le_dvd
tff(fact_4322_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A: code_integer] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N))) ) ).
% le_imp_power_dvd
tff(fact_4323_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).
% le_imp_power_dvd
tff(fact_4324_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ).
% le_imp_power_dvd
tff(fact_4325_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A: complex] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_nat_complex(power_power_complex(A),M)),aa_nat_complex(power_power_complex(A),N))) ) ).
% le_imp_power_dvd
tff(fact_4326_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),M)),aa_nat_int(power_power_int(A),N))) ) ).
% le_imp_power_dvd
tff(fact_4327_mod__eq__dvd__iff,axiom,
! [A: int,C: int,B: int] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) )
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))) ) ).
% mod_eq_dvd_iff
tff(fact_4328_mod__eq__dvd__iff,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(B,C) )
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),minus_8373710615458151222nteger(A,B))) ) ).
% mod_eq_dvd_iff
tff(fact_4329_dvd__minus__mod,axiom,
! [B: nat,A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)))) ).
% dvd_minus_mod
tff(fact_4330_dvd__minus__mod,axiom,
! [B: int,A: int] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)))) ).
% dvd_minus_mod
tff(fact_4331_dvd__minus__mod,axiom,
! [B: code_integer,A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B)))) ).
% dvd_minus_mod
tff(fact_4332_nat__dvd__not__less,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)) ) ) ).
% nat_dvd_not_less
tff(fact_4333_dvd__pos__nat,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M)) ) ) ).
% dvd_pos_nat
tff(fact_4334_dvd__minus__self,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
| pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).
% dvd_minus_self
tff(fact_4335_dvd__diffD,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M)) ) ) ) ).
% dvd_diffD
tff(fact_4336_dvd__diffD1,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N)) ) ) ) ).
% dvd_diffD1
tff(fact_4337_less__eq__dvd__minus,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).
% less_eq_dvd_minus
tff(fact_4338_zdvd__not__zless,axiom,
! [M: int,N: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),M))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M),N))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,N),M)) ) ) ).
% zdvd_not_zless
tff(fact_4339_bezout__lemma__nat,axiom,
! [D: nat,A: nat,B: nat,X: nat,Y: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),B))
=> ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y)),D) )
| ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y)),D) ) )
=> ? [X3: nat,Y2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),A))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)))
& ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3) = 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,A),B)),Y2)),D) )
| ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y2)),D) ) ) ) ) ) ) ).
% bezout_lemma_nat
tff(fact_4340_bezout__add__nat,axiom,
! [A: nat,B: nat] :
? [D2: nat,X3: nat,Y2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),A))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),B))
& ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)),D2) )
| ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y2)),D2) ) ) ) ).
% bezout_add_nat
tff(fact_4341_bezout1__nat,axiom,
! [A: nat,B: nat] :
? [D2: nat,X3: nat,Y2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),A))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),B))
& ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)) = D2 )
| ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y2)) = D2 ) ) ) ).
% bezout1_nat
tff(fact_4342_div2__even__ext__nat,axiom,
! [X: nat,Y: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Y)) )
=> ( X = Y ) ) ) ).
% div2_even_ext_nat
tff(fact_4343_unit__dvdE,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
=> ~ ( ( A != zero_z3403309356797280102nteger )
=> ! [C2: code_integer] : B != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C2) ) ) ).
% unit_dvdE
tff(fact_4344_unit__dvdE,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
=> ~ ( ( A != zero_zero_nat )
=> ! [C2: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C2) ) ) ).
% unit_dvdE
tff(fact_4345_unit__dvdE,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
=> ~ ( ( A != zero_zero_int )
=> ! [C2: int] : B != aa_int_int(aa_int_fun_int_int(times_times_int,A),C2) ) ) ).
% unit_dvdE
tff(fact_4346_unity__coeff__ex,axiom,
! [P: fun_Co8822979462182099027r_bool,L: code_integer] :
( ? [X2: code_integer] : pp(aa_Code_integer_bool(P,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,L),X2)))
<=> ? [X2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,L),plus_p5714425477246183910nteger(X2,zero_z3403309356797280102nteger)))
& pp(aa_Code_integer_bool(P,X2)) ) ) ).
% unity_coeff_ex
tff(fact_4347_unity__coeff__ex,axiom,
! [P: fun_complex_bool,L: complex] :
( ? [X2: complex] : pp(aa_complex_bool(P,aa_complex_complex(times_times_complex(L),X2)))
<=> ? [X2: complex] :
( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,L),aa_complex_complex(plus_plus_complex(X2),zero_zero_complex)))
& pp(aa_complex_bool(P,X2)) ) ) ).
% unity_coeff_ex
tff(fact_4348_unity__coeff__ex,axiom,
! [P: fun_real_bool,L: real] :
( ? [X2: real] : pp(aa_real_bool(P,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,L),X2)))
<=> ? [X2: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),zero_zero_real)))
& pp(aa_real_bool(P,X2)) ) ) ).
% unity_coeff_ex
tff(fact_4349_unity__coeff__ex,axiom,
! [P: fun_rat_bool,L: rat] :
( ? [X2: rat] : pp(aa_rat_bool(P,aa_rat_rat(times_times_rat(L),X2)))
<=> ? [X2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,L),plus_plus_rat(X2,zero_zero_rat)))
& pp(aa_rat_bool(P,X2)) ) ) ).
% unity_coeff_ex
tff(fact_4350_unity__coeff__ex,axiom,
! [P: fun_nat_bool,L: nat] :
( ? [X2: nat] : pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,L),X2)))
<=> ? [X2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),zero_zero_nat)))
& pp(aa_nat_bool(P,X2)) ) ) ).
% unity_coeff_ex
tff(fact_4351_unity__coeff__ex,axiom,
! [P: fun_int_bool,L: int] :
( ? [X2: int] : pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(times_times_int,L),X2)))
<=> ? [X2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),zero_zero_int)))
& pp(aa_int_bool(P,X2)) ) ) ).
% unity_coeff_ex
tff(fact_4352_dvd__div__eq__mult,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B))
=> ( ( divide6298287555418463151nteger(B,A) = C )
<=> ( B = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) ) ) ) ) ).
% dvd_div_eq_mult
tff(fact_4353_dvd__div__eq__mult,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = C )
<=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) ) ) ) ) ).
% dvd_div_eq_mult
tff(fact_4354_dvd__div__eq__mult,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
=> ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = C )
<=> ( B = aa_int_int(aa_int_fun_int_int(times_times_int,C),A) ) ) ) ) ).
% dvd_div_eq_mult
tff(fact_4355_div__dvd__iff__mult,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B)),C))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))) ) ) ) ).
% div_dvd_iff_mult
tff(fact_4356_div__dvd__iff__mult,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))) ) ) ) ).
% div_dvd_iff_mult
tff(fact_4357_div__dvd__iff__mult,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))) ) ) ) ).
% div_dvd_iff_mult
tff(fact_4358_dvd__div__iff__mult,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( ( C != zero_z3403309356797280102nteger )
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(B,C)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B)) ) ) ) ).
% dvd_div_iff_mult
tff(fact_4359_dvd__div__iff__mult,axiom,
! [C: nat,B: nat,A: nat] :
( ( C != zero_zero_nat )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B)) ) ) ) ).
% dvd_div_iff_mult
tff(fact_4360_dvd__div__iff__mult,axiom,
! [C: int,B: int,A: int] :
( ( C != zero_zero_int )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B)) ) ) ) ).
% dvd_div_iff_mult
tff(fact_4361_dvd__div__div__eq__mult,axiom,
! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( C != zero_z3403309356797280102nteger )
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),D))
=> ( ( divide6298287555418463151nteger(B,A) = divide6298287555418463151nteger(D,C) )
<=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),D) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
tff(fact_4362_dvd__div__div__eq__mult,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( A != zero_zero_nat )
=> ( ( C != zero_zero_nat )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),D))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,D),C) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
tff(fact_4363_dvd__div__div__eq__mult,axiom,
! [A: int,C: int,B: int,D: int] :
( ( A != zero_zero_int )
=> ( ( C != zero_zero_int )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),D))
=> ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,D),C) )
<=> ( aa_int_int(aa_int_fun_int_int(times_times_int,B),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),D) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
tff(fact_4364_unit__div__eq__0__iff,axiom,
! [B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
=> ( ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger )
<=> ( A = zero_z3403309356797280102nteger ) ) ) ).
% unit_div_eq_0_iff
tff(fact_4365_unit__div__eq__0__iff,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat )
<=> ( A = zero_zero_nat ) ) ) ).
% unit_div_eq_0_iff
tff(fact_4366_unit__div__eq__0__iff,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
=> ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int )
<=> ( A = zero_zero_int ) ) ) ).
% unit_div_eq_0_iff
tff(fact_4367_even__numeral,axiom,
! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),numera6620942414471956472nteger(aa_num_num(bit0,N)))) ).
% even_numeral
tff(fact_4368_even__numeral,axiom,
! [N: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N)))) ).
% even_numeral
tff(fact_4369_even__numeral,axiom,
! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) ).
% even_numeral
tff(fact_4370_inf__period_I3_J,axiom,
! [D: code_integer,D3: code_integer,T: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),D3))
=> ! [X4: code_integer,K4: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X4,T)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(minus_8373710615458151222nteger(X4,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D3)),T))) ) ) ).
% inf_period(3)
tff(fact_4371_inf__period_I3_J,axiom,
! [D: real,D3: real,T: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),D3))
=> ! [X4: real,K4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),T)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D3))),T))) ) ) ).
% inf_period(3)
tff(fact_4372_inf__period_I3_J,axiom,
! [D: rat,D3: rat,T: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),D3))
=> ! [X4: rat,K4: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(X4,T)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(minus_minus_rat(X4,aa_rat_rat(times_times_rat(K4),D3)),T))) ) ) ).
% inf_period(3)
tff(fact_4373_inf__period_I3_J,axiom,
! [D: int,D3: int,T: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D3))
=> ! [X4: int,K4: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),T)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3))),T))) ) ) ).
% inf_period(3)
tff(fact_4374_inf__period_I4_J,axiom,
! [D: code_integer,D3: code_integer,T: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),D3))
=> ! [X4: code_integer,K4: code_integer] :
( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X4,T)))
<=> ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(minus_8373710615458151222nteger(X4,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D3)),T))) ) ) ).
% inf_period(4)
tff(fact_4375_inf__period_I4_J,axiom,
! [D: real,D3: real,T: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),D3))
=> ! [X4: real,K4: real] :
( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),T)))
<=> ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D3))),T))) ) ) ).
% inf_period(4)
tff(fact_4376_inf__period_I4_J,axiom,
! [D: rat,D3: rat,T: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),D3))
=> ! [X4: rat,K4: rat] :
( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(X4,T)))
<=> ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,D),plus_plus_rat(minus_minus_rat(X4,aa_rat_rat(times_times_rat(K4),D3)),T))) ) ) ).
% inf_period(4)
tff(fact_4377_inf__period_I4_J,axiom,
! [D: int,D3: int,T: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D3))
=> ! [X4: int,K4: int] :
( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),T)))
<=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3))),T))) ) ) ).
% inf_period(4)
tff(fact_4378_is__unit__div__mult2__eq,axiom,
! [B: code_integer,C: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),one_one_Code_integer))
=> ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ) ).
% is_unit_div_mult2_eq
tff(fact_4379_is__unit__div__mult2__eq,axiom,
! [B: nat,C: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),one_one_nat))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).
% is_unit_div_mult2_eq
tff(fact_4380_is__unit__div__mult2__eq,axiom,
! [B: int,C: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),one_one_int))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).
% is_unit_div_mult2_eq
tff(fact_4381_unit__div__mult__swap,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),one_one_Code_integer))
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,C)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ) ) ).
% unit_div_mult_swap
tff(fact_4382_unit__div__mult__swap,axiom,
! [C: nat,A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),one_one_nat))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).
% unit_div_mult_swap
tff(fact_4383_unit__div__mult__swap,axiom,
! [C: int,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),one_one_int))
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ) ) ).
% unit_div_mult_swap
tff(fact_4384_unit__div__commute,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),C) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),B) ) ) ).
% unit_div_commute
tff(fact_4385_unit__div__commute,axiom,
! [B: nat,A: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ) ) ).
% unit_div_commute
tff(fact_4386_unit__div__commute,axiom,
! [B: int,A: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B) ) ) ).
% unit_div_commute
tff(fact_4387_div__mult__unit2,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,C),one_one_Code_integer))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A))
=> ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ) ).
% div_mult_unit2
tff(fact_4388_div__mult__unit2,axiom,
! [C: nat,B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,C),one_one_nat))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),A))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).
% div_mult_unit2
tff(fact_4389_div__mult__unit2,axiom,
! [C: int,B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,C),one_one_int))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).
% div_mult_unit2
tff(fact_4390_unit__eq__div2,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
=> ( ( A = divide6298287555418463151nteger(C,B) )
<=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = C ) ) ) ).
% unit_eq_div2
tff(fact_4391_unit__eq__div2,axiom,
! [B: nat,A: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = C ) ) ) ).
% unit_eq_div2
tff(fact_4392_unit__eq__div2,axiom,
! [B: int,A: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
=> ( ( A = aa_int_int(aa_int_fun_int_int(divide_divide_int,C),B) )
<=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = C ) ) ) ).
% unit_eq_div2
tff(fact_4393_unit__eq__div1,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
=> ( ( divide6298287555418463151nteger(A,B) = C )
<=> ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) ) ) ) ).
% unit_eq_div1
tff(fact_4394_unit__eq__div1,axiom,
! [B: nat,A: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = C )
<=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) ) ) ) ).
% unit_eq_div1
tff(fact_4395_unit__eq__div1,axiom,
! [B: int,A: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
=> ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = C )
<=> ( A = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) ) ) ) ).
% unit_eq_div1
tff(fact_4396_is__unit__power__iff,axiom,
! [A: code_integer,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),one_one_Code_integer))
<=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
tff(fact_4397_is__unit__power__iff,axiom,
! [A: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),one_one_nat))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
tff(fact_4398_is__unit__power__iff,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(A),N)),one_one_int))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
tff(fact_4399_unit__imp__mod__eq__0,axiom,
! [B: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat ) ) ).
% unit_imp_mod_eq_0
tff(fact_4400_unit__imp__mod__eq__0,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int ) ) ).
% unit_imp_mod_eq_0
tff(fact_4401_unit__imp__mod__eq__0,axiom,
! [B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
=> ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger ) ) ).
% unit_imp_mod_eq_0
tff(fact_4402_dvd__imp__le,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ) ).
% dvd_imp_le
tff(fact_4403_dvd__mult__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).
% dvd_mult_cancel
tff(fact_4404_nat__mult__dvd__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(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),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ) ).
% nat_mult_dvd_cancel1
tff(fact_4405_bezout__add__strong__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ? [D2: nat,X3: nat,Y2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),A))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D2),B))
& ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)),D2) ) ) ) ).
% bezout_add_strong_nat
tff(fact_4406_zdvd__imp__le,axiom,
! [Z: int,N: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Z),N))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),N)) ) ) ).
% zdvd_imp_le
tff(fact_4407_mod__greater__zero__iff__not__dvd,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))
<=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)) ) ).
% mod_greater_zero_iff_not_dvd
tff(fact_4408_mod__eq__dvd__iff__nat,axiom,
! [N: nat,M: nat,Q2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q2) )
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ) ).
% mod_eq_dvd_iff_nat
tff(fact_4409_prod__decode__aux_Ocases,axiom,
! [X: product_prod_nat_nat] :
~ ! [K3: nat,M3: nat] : X != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(K3),M3) ).
% prod_decode_aux.cases
tff(fact_4410_finite__divisors__nat,axiom,
! [M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> pp(aa_set_nat_bool(finite_finite_nat,collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_dt,M)))) ) ).
% finite_divisors_nat
tff(fact_4411_even__zero,axiom,
pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_z3403309356797280102nteger)) ).
% even_zero
tff(fact_4412_even__zero,axiom,
pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),zero_zero_nat)) ).
% even_zero
tff(fact_4413_even__zero,axiom,
pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),zero_zero_int)) ).
% even_zero
tff(fact_4414_is__unit__div__mult__cancel__right,axiom,
! [A: code_integer,B: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
=> ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)) = divide6298287555418463151nteger(one_one_Code_integer,B) ) ) ) ).
% is_unit_div_mult_cancel_right
tff(fact_4415_is__unit__div__mult__cancel__right,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) ) ) ) ).
% is_unit_div_mult_cancel_right
tff(fact_4416_is__unit__div__mult__cancel__right,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B) ) ) ) ).
% is_unit_div_mult_cancel_right
tff(fact_4417_is__unit__div__mult__cancel__left,axiom,
! [A: code_integer,B: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),one_one_Code_integer))
=> ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = divide6298287555418463151nteger(one_one_Code_integer,B) ) ) ) ).
% is_unit_div_mult_cancel_left
tff(fact_4418_is__unit__div__mult__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B),one_one_nat))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) ) ) ) ).
% is_unit_div_mult_cancel_left
tff(fact_4419_is__unit__div__mult__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),one_one_int))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B) ) ) ) ).
% is_unit_div_mult_cancel_left
tff(fact_4420_is__unitE,axiom,
! [A: code_integer,C: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,A),one_one_Code_integer))
=> ~ ( ( A != zero_z3403309356797280102nteger )
=> ! [B4: code_integer] :
( ( B4 != zero_z3403309356797280102nteger )
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B4),one_one_Code_integer))
=> ( ( divide6298287555418463151nteger(one_one_Code_integer,A) = B4 )
=> ( ( divide6298287555418463151nteger(one_one_Code_integer,B4) = A )
=> ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B4) = one_one_Code_integer )
=> ( divide6298287555418463151nteger(C,A) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B4) ) ) ) ) ) ) ) ) ).
% is_unitE
tff(fact_4421_is__unitE,axiom,
! [A: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,A),one_one_nat))
=> ~ ( ( A != zero_zero_nat )
=> ! [B4: nat] :
( ( B4 != zero_zero_nat )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,B4),one_one_nat))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A) = B4 )
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B4) = A )
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B4) = one_one_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B4) ) ) ) ) ) ) ) ) ).
% is_unitE
tff(fact_4422_is__unitE,axiom,
! [A: int,C: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,A),one_one_int))
=> ~ ( ( A != zero_zero_int )
=> ! [B4: int] :
( ( B4 != zero_zero_int )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B4),one_one_int))
=> ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A) = B4 )
=> ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B4) = A )
=> ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B4) = one_one_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A) != aa_int_int(aa_int_fun_int_int(times_times_int,C),B4) ) ) ) ) ) ) ) ) ).
% is_unitE
tff(fact_4423_evenE,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
=> ~ ! [B4: code_integer] : A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B4) ) ).
% evenE
tff(fact_4424_evenE,axiom,
! [A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
=> ~ ! [B4: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B4) ) ).
% evenE
tff(fact_4425_evenE,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
=> ~ ! [B4: int] : A != aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B4) ) ).
% evenE
tff(fact_4426_odd__one,axiom,
~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),one_one_Code_integer)) ).
% odd_one
tff(fact_4427_odd__one,axiom,
~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),one_one_nat)) ).
% odd_one
tff(fact_4428_odd__one,axiom,
~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),one_one_int)) ).
% odd_one
tff(fact_4429_odd__even__add,axiom,
! [A: code_integer,B: code_integer] :
( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
=> ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B))) ) ) ).
% odd_even_add
tff(fact_4430_odd__even__add,axiom,
! [A: nat,B: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).
% odd_even_add
tff(fact_4431_odd__even__add,axiom,
! [A: int,B: int] :
( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).
% odd_even_add
tff(fact_4432_bit__eq__rec,axiom,
! [A: code_integer,B: code_integer] :
( ( A = B )
<=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)) )
& ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = divide6298287555418463151nteger(B,numera6620942414471956472nteger(aa_num_num(bit0,one))) ) ) ) ).
% bit_eq_rec
tff(fact_4433_bit__eq__rec,axiom,
! [A: nat,B: nat] :
( ( A = B )
<=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) )
& ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ) ).
% bit_eq_rec
tff(fact_4434_bit__eq__rec,axiom,
! [A: int,B: int] :
( ( A = B )
<=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) )
& ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ) ).
% bit_eq_rec
tff(fact_4435_dvd__power__iff,axiom,
! [X: code_integer,M: nat,N: nat] :
( ( X != zero_z3403309356797280102nteger )
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(X),M)),aa_nat_Code_integer(power_8256067586552552935nteger(X),N)))
<=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),one_one_Code_integer))
| pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ) ).
% dvd_power_iff
tff(fact_4436_dvd__power__iff,axiom,
! [X: nat,M: nat,N: nat] :
( ( X != zero_zero_nat )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),one_one_nat))
| pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ) ).
% dvd_power_iff
tff(fact_4437_dvd__power__iff,axiom,
! [X: int,M: nat,N: nat] :
( ( X != zero_zero_int )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(X),M)),aa_nat_int(power_power_int(X),N)))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),one_one_int))
| pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ) ).
% dvd_power_iff
tff(fact_4438_odd__numeral,axiom,
! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),numera6620942414471956472nteger(aa_num_num(bit1,N)))) ).
% odd_numeral
tff(fact_4439_odd__numeral,axiom,
! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)))) ).
% odd_numeral
tff(fact_4440_odd__numeral,axiom,
! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) ).
% odd_numeral
tff(fact_4441_dvd__power,axiom,
! [N: nat,X: code_integer] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
| ( X = one_one_Code_integer ) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),aa_nat_Code_integer(power_8256067586552552935nteger(X),N))) ) ).
% dvd_power
tff(fact_4442_dvd__power,axiom,
! [N: nat,X: rat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
| ( X = one_one_rat ) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),aa_nat_rat(power_power_rat(X),N))) ) ).
% dvd_power
tff(fact_4443_dvd__power,axiom,
! [N: nat,X: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
| ( X = one_one_nat ) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N))) ) ).
% dvd_power
tff(fact_4444_dvd__power,axiom,
! [N: nat,X: real] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
| ( X = one_one_real ) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ).
% dvd_power
tff(fact_4445_dvd__power,axiom,
! [N: nat,X: complex] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
| ( X = one_one_complex ) )
=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X),aa_nat_complex(power_power_complex(X),N))) ) ).
% dvd_power
tff(fact_4446_dvd__power,axiom,
! [N: nat,X: int] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
| ( X = one_one_int ) )
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),aa_nat_int(power_power_int(X),N))) ) ).
% dvd_power
tff(fact_4447_even__even__mod__4__iff,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))))) ) ).
% even_even_mod_4_iff
tff(fact_4448_dvd__mult__cancel2,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M)),M))
<=> ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel2
tff(fact_4449_dvd__mult__cancel1,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),M))
<=> ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel1
tff(fact_4450_odd__numeral__BitM,axiom,
! [W: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),numera6620942414471956472nteger(bitM(W)))) ).
% odd_numeral_BitM
tff(fact_4451_odd__numeral__BitM,axiom,
! [W: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,bitM(W)))) ).
% odd_numeral_BitM
tff(fact_4452_odd__numeral__BitM,axiom,
! [W: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,bitM(W)))) ).
% odd_numeral_BitM
tff(fact_4453_dvd__minus__add,axiom,
! [Q2: nat,N: nat,R: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Q2),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),M)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),Q2)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),M)),Q2)))) ) ) ) ).
% dvd_minus_add
tff(fact_4454_power__dvd__imp__le,axiom,
! [I: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),I))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).
% power_dvd_imp_le
tff(fact_4455_mod__nat__eqI,axiom,
! [R: nat,N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,R),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),R)))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = R ) ) ) ) ).
% mod_nat_eqI
tff(fact_4456_mod__int__pos__iff,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
| ( ( L = zero_zero_int )
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) )
| pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L)) ) ) ).
% mod_int_pos_iff
tff(fact_4457_aset_I10_J,axiom,
! [D: int,D3: int,A2: set_int,T: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D3))
=> ! [X4: int] :
( ! [Xa4: int] :
( pp(aa_set_int_bool(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3)))
=> ! [Xb3: int] :
( pp(aa_set_int_bool(member_int(Xb3),A2))
=> ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),T)))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D3)),T))) ) ) ) ).
% aset(10)
tff(fact_4458_aset_I9_J,axiom,
! [D: int,D3: int,A2: set_int,T: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D3))
=> ! [X4: int] :
( ! [Xa4: int] :
( pp(aa_set_int_bool(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3)))
=> ! [Xb3: int] :
( pp(aa_set_int_bool(member_int(Xb3),A2))
=> ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),T)))
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D3)),T))) ) ) ) ).
% aset(9)
tff(fact_4459_bset_I10_J,axiom,
! [D: int,D3: int,B2: set_int,T: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D3))
=> ! [X4: int] :
( ! [Xa4: int] :
( pp(aa_set_int_bool(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3)))
=> ! [Xb3: int] :
( pp(aa_set_int_bool(member_int(Xb3),B2))
=> ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),T)))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D3)),T))) ) ) ) ).
% bset(10)
tff(fact_4460_bset_I9_J,axiom,
! [D: int,D3: int,B2: set_int,T: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),D3))
=> ! [X4: int] :
( ! [Xa4: int] :
( pp(aa_set_int_bool(member_int(Xa4),set_or1266510415728281911st_int(one_one_int,D3)))
=> ! [Xb3: int] :
( pp(aa_set_int_bool(member_int(Xb3),B2))
=> ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),T)))
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D3)),T))) ) ) ) ).
% bset(9)
tff(fact_4461_even__two__times__div__two,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) = A ) ) ).
% even_two_times_div_two
tff(fact_4462_even__two__times__div__two,axiom,
! [A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = A ) ) ).
% even_two_times_div_two
tff(fact_4463_even__two__times__div__two,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = A ) ) ).
% even_two_times_div_two
tff(fact_4464_even__iff__mod__2__eq__zero,axiom,
! [A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).
% even_iff_mod_2_eq_zero
tff(fact_4465_even__iff__mod__2__eq__zero,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
<=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).
% even_iff_mod_2_eq_zero
tff(fact_4466_even__iff__mod__2__eq__zero,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
<=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).
% even_iff_mod_2_eq_zero
tff(fact_4467_odd__iff__mod__2__eq__one,axiom,
! [A: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).
% odd_iff_mod_2_eq_one
tff(fact_4468_odd__iff__mod__2__eq__one,axiom,
! [A: int] :
( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
<=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).
% odd_iff_mod_2_eq_one
tff(fact_4469_odd__iff__mod__2__eq__one,axiom,
! [A: code_integer] :
( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
<=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).
% odd_iff_mod_2_eq_one
tff(fact_4470_power__mono__odd,axiom,
! [N: nat,A: real,B: real] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N))) ) ) ).
% power_mono_odd
tff(fact_4471_power__mono__odd,axiom,
! [N: nat,A: rat,B: rat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N))) ) ) ).
% power_mono_odd
tff(fact_4472_power__mono__odd,axiom,
! [N: nat,A: int,B: int] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N))) ) ) ).
% power_mono_odd
tff(fact_4473_odd__pos,axiom,
! [N: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).
% odd_pos
tff(fact_4474_dvd__power__iff__le,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).
% dvd_power_iff_le
tff(fact_4475_signed__take__bit__int__less__exp,axiom,
! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).
% signed_take_bit_int_less_exp
tff(fact_4476_even__unset__bit__iff,axiom,
! [M: nat,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se8260200283734997820nteger(M,A)))
<=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
| ( M = zero_zero_nat ) ) ) ).
% even_unset_bit_iff
tff(fact_4477_even__unset__bit__iff,axiom,
! [M: nat,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se4203085406695923979it_int(M),A)))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
| ( M = zero_zero_nat ) ) ) ).
% even_unset_bit_iff
tff(fact_4478_even__unset__bit__iff,axiom,
! [M: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),A)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
| ( M = zero_zero_nat ) ) ) ).
% even_unset_bit_iff
tff(fact_4479_even__set__bit__iff,axiom,
! [M: nat,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se2793503036327961859nteger(M,A)))
<=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
& ( M != zero_zero_nat ) ) ) ).
% even_set_bit_iff
tff(fact_4480_even__set__bit__iff,axiom,
! [M: nat,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se7879613467334960850it_int(M),A)))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
& ( M != zero_zero_nat ) ) ) ).
% even_set_bit_iff
tff(fact_4481_even__set__bit__iff,axiom,
! [M: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),A)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
& ( M != zero_zero_nat ) ) ) ).
% even_set_bit_iff
tff(fact_4482_even__flip__bit__iff,axiom,
! [M: nat,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1345352211410354436nteger(M,A)))
<=> ~ ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
<=> ( M = zero_zero_nat ) ) ) ).
% even_flip_bit_iff
tff(fact_4483_even__flip__bit__iff,axiom,
! [M: nat,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se2159334234014336723it_int(M),A)))
<=> ~ ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
<=> ( M = zero_zero_nat ) ) ) ).
% even_flip_bit_iff
tff(fact_4484_even__flip__bit__iff,axiom,
! [M: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),A)))
<=> ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
<=> ( M = zero_zero_nat ) ) ) ).
% even_flip_bit_iff
tff(fact_4485_even__diff__iff,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L))) ) ).
% even_diff_iff
tff(fact_4486_oddE,axiom,
! [A: code_integer] :
( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
=> ~ ! [B4: code_integer] : A != plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B4),one_one_Code_integer) ) ).
% oddE
tff(fact_4487_oddE,axiom,
! [A: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
=> ~ ! [B4: 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_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B4)),one_one_nat) ) ).
% oddE
tff(fact_4488_oddE,axiom,
! [A: int] :
( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
=> ~ ! [B4: int] : A != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B4)),one_one_int) ) ).
% oddE
tff(fact_4489_parity__cases,axiom,
! [A: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != zero_zero_nat ) )
=> ~ ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != one_one_nat ) ) ) ).
% parity_cases
tff(fact_4490_parity__cases,axiom,
! [A: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int ) )
=> ~ ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int ) ) ) ).
% parity_cases
tff(fact_4491_parity__cases,axiom,
! [A: code_integer] :
( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != zero_z3403309356797280102nteger ) )
=> ~ ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != one_one_Code_integer ) ) ) ).
% parity_cases
tff(fact_4492_mod2__eq__if,axiom,
! [A: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ) ).
% mod2_eq_if
tff(fact_4493_mod2__eq__if,axiom,
! [A: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ) ).
% mod2_eq_if
tff(fact_4494_mod2__eq__if,axiom,
! [A: code_integer] :
( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) )
& ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ) ) ) ).
% mod2_eq_if
tff(fact_4495_zero__le__even__power,axiom,
! [N: nat,A: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N))) ) ).
% zero_le_even_power
tff(fact_4496_zero__le__even__power,axiom,
! [N: nat,A: rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N))) ) ).
% zero_le_even_power
tff(fact_4497_zero__le__even__power,axiom,
! [N: nat,A: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),N))) ) ).
% zero_le_even_power
tff(fact_4498_zero__le__odd__power,axiom,
! [N: nat,A: real] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ).
% zero_le_odd_power
tff(fact_4499_zero__le__odd__power,axiom,
! [N: nat,A: rat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).
% zero_le_odd_power
tff(fact_4500_zero__le__odd__power,axiom,
! [N: nat,A: int] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ).
% zero_le_odd_power
tff(fact_4501_zero__le__power__eq,axiom,
! [A: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
| ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ).
% zero_le_power_eq
tff(fact_4502_zero__le__power__eq,axiom,
! [A: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
| ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ) ).
% zero_le_power_eq
tff(fact_4503_zero__le__power__eq,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
| ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ) ).
% zero_le_power_eq
tff(fact_4504_signed__take__bit__int__less__self__iff,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),K))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K)) ) ).
% signed_take_bit_int_less_self_iff
tff(fact_4505_signed__take__bit__int__greater__eq__self__iff,axiom,
! [K: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(bit_ri631733984087533419it_int(N),K)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ).
% signed_take_bit_int_greater_eq_self_iff
tff(fact_4506_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N2: nat] : X != aa_nat_nat(suc,N2) ) ).
% list_decode.cases
tff(fact_4507_zero__less__power__eq,axiom,
! [A: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)))
<=> ( ( N = zero_zero_nat )
| ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
& ( A != zero_zero_real ) )
| ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ).
% zero_less_power_eq
tff(fact_4508_zero__less__power__eq,axiom,
! [A: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),N)))
<=> ( ( N = zero_zero_nat )
| ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
& ( A != zero_zero_rat ) )
| ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ) ) ).
% zero_less_power_eq
tff(fact_4509_zero__less__power__eq,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),N)))
<=> ( ( N = zero_zero_nat )
| ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
& ( A != zero_zero_int ) )
| ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ) ) ).
% zero_less_power_eq
tff(fact_4510_signed__take__bit__int__less__eq,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N))))) ) ).
% signed_take_bit_int_less_eq
tff(fact_4511_even__mask__div__iff_H,axiom,
! [M: nat,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N))))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% even_mask_div_iff'
tff(fact_4512_even__mask__div__iff_H,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% even_mask_div_iff'
tff(fact_4513_even__mask__div__iff_H,axiom,
! [M: nat,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% even_mask_div_iff'
tff(fact_4514_power__le__zero__eq,axiom,
! [A: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),zero_zero_real))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
& ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) )
| ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
& ( A = zero_zero_real ) ) ) ) ) ).
% power_le_zero_eq
tff(fact_4515_power__le__zero__eq,axiom,
! [A: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),zero_zero_rat))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
& ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) )
| ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
& ( A = zero_zero_rat ) ) ) ) ) ).
% power_le_zero_eq
tff(fact_4516_power__le__zero__eq,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),zero_zero_int))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
& ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) )
| ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
& ( A = zero_zero_int ) ) ) ) ) ).
% power_le_zero_eq
tff(fact_4517_option_Osize__gen_I1_J,axiom,
! [X: fun_Pr2602117396471556196at_nat] : size_o8335143837870341156at_nat(X,none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).
% option.size_gen(1)
tff(fact_4518_option_Osize__gen_I1_J,axiom,
! [X: fun_nat_nat] : size_option_nat(X,none_nat) = aa_nat_nat(suc,zero_zero_nat) ).
% option.size_gen(1)
tff(fact_4519_option_Osize__gen_I1_J,axiom,
! [X: fun_num_nat] : size_option_num(X,none_num) = aa_nat_nat(suc,zero_zero_nat) ).
% option.size_gen(1)
tff(fact_4520_even__mod__4__div__2,axiom,
! [N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(suc,zero_zero_nat) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).
% even_mod_4_div_2
tff(fact_4521_divmod__step__nat__def,axiom,
! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_dz(L)),Qr) ).
% divmod_step_nat_def
tff(fact_4522_even__mask__div__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N))))
<=> ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N) = zero_z3403309356797280102nteger )
| pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).
% even_mask_div_iff
tff(fact_4523_even__mask__div__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))
<=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) = zero_zero_nat )
| pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).
% even_mask_div_iff
tff(fact_4524_even__mask__div__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))))
<=> ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) = zero_zero_int )
| pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ) ).
% even_mask_div_iff
tff(fact_4525_divmod__step__int__def,axiom,
! [L: num,Qr: product_prod_int_int] : aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),Qr) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_ea(L)),Qr) ).
% divmod_step_int_def
tff(fact_4526_odd__mod__4__div__2,axiom,
! [N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).
% odd_mod_4_div_2
tff(fact_4527_even__mult__exp__div__exp__iff,axiom,
! [A: code_integer,M: nat,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N))))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
| ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N) = zero_z3403309356797280102nteger )
| ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
& pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))))) ) ) ) ).
% even_mult_exp_div_exp_iff
tff(fact_4528_even__mult__exp__div__exp__iff,axiom,
! [A: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
| ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) = zero_zero_nat )
| ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))))) ) ) ) ).
% even_mult_exp_div_exp_iff
tff(fact_4529_even__mult__exp__div__exp__iff,axiom,
! [A: int,M: nat,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
| ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) = zero_zero_int )
| ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
& pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))))) ) ) ) ).
% even_mult_exp_div_exp_iff
tff(fact_4530_divmod__step__def,axiom,
! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_dz(L)),Qr) ).
% divmod_step_def
tff(fact_4531_divmod__step__def,axiom,
! [L: num,Qr: product_prod_int_int] : aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),Qr) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_ea(L)),Qr) ).
% divmod_step_def
tff(fact_4532_divmod__step__def,axiom,
! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_eb(L),Qr) ).
% divmod_step_def
tff(fact_4533_ex__min__if__finite,axiom,
! [S2: set_real] :
( pp(aa_set_real_bool(finite_finite_real,S2))
=> ( ( S2 != bot_bot_set_real )
=> ? [X3: real] :
( pp(aa_set_real_bool(member_real(X3),S2))
& ~ ? [Xa: real] :
( pp(aa_set_real_bool(member_real(Xa),S2))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Xa),X3)) ) ) ) ) ).
% ex_min_if_finite
tff(fact_4534_ex__min__if__finite,axiom,
! [S2: set_rat] :
( pp(aa_set_rat_bool(finite_finite_rat,S2))
=> ( ( S2 != bot_bot_set_rat )
=> ? [X3: rat] :
( pp(aa_set_rat_bool(member_rat(X3),S2))
& ~ ? [Xa: rat] :
( pp(aa_set_rat_bool(member_rat(Xa),S2))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Xa),X3)) ) ) ) ) ).
% ex_min_if_finite
tff(fact_4535_ex__min__if__finite,axiom,
! [S2: set_num] :
( pp(aa_set_num_bool(finite_finite_num,S2))
=> ( ( S2 != bot_bot_set_num )
=> ? [X3: num] :
( pp(aa_set_num_bool(member_num(X3),S2))
& ~ ? [Xa: num] :
( pp(aa_set_num_bool(member_num(Xa),S2))
& pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Xa),X3)) ) ) ) ) ).
% ex_min_if_finite
tff(fact_4536_ex__min__if__finite,axiom,
! [S2: set_nat] :
( pp(aa_set_nat_bool(finite_finite_nat,S2))
=> ( ( S2 != bot_bot_set_nat )
=> ? [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),S2))
& ~ ? [Xa: nat] :
( pp(aa_set_nat_bool(member_nat(Xa),S2))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),X3)) ) ) ) ) ).
% ex_min_if_finite
tff(fact_4537_ex__min__if__finite,axiom,
! [S2: set_int] :
( pp(aa_set_int_bool(finite_finite_int,S2))
=> ( ( S2 != bot_bot_set_int )
=> ? [X3: int] :
( pp(aa_set_int_bool(member_int(X3),S2))
& ~ ? [Xa: int] :
( pp(aa_set_int_bool(member_int(Xa),S2))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Xa),X3)) ) ) ) ) ).
% ex_min_if_finite
tff(fact_4538_infinite__growing,axiom,
! [X5: set_real] :
( ( X5 != bot_bot_set_real )
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),X5))
=> ? [Xa: real] :
( pp(aa_set_real_bool(member_real(Xa),X5))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Xa)) ) )
=> ~ pp(aa_set_real_bool(finite_finite_real,X5)) ) ) ).
% infinite_growing
tff(fact_4539_infinite__growing,axiom,
! [X5: set_rat] :
( ( X5 != bot_bot_set_rat )
=> ( ! [X3: rat] :
( pp(aa_set_rat_bool(member_rat(X3),X5))
=> ? [Xa: rat] :
( pp(aa_set_rat_bool(member_rat(Xa),X5))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X3),Xa)) ) )
=> ~ pp(aa_set_rat_bool(finite_finite_rat,X5)) ) ) ).
% infinite_growing
tff(fact_4540_infinite__growing,axiom,
! [X5: set_num] :
( ( X5 != bot_bot_set_num )
=> ( ! [X3: num] :
( pp(aa_set_num_bool(member_num(X3),X5))
=> ? [Xa: num] :
( pp(aa_set_num_bool(member_num(Xa),X5))
& pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X3),Xa)) ) )
=> ~ pp(aa_set_num_bool(finite_finite_num,X5)) ) ) ).
% infinite_growing
tff(fact_4541_infinite__growing,axiom,
! [X5: set_nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),X5))
=> ? [Xa: nat] :
( pp(aa_set_nat_bool(member_nat(Xa),X5))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X3),Xa)) ) )
=> ~ pp(aa_set_nat_bool(finite_finite_nat,X5)) ) ) ).
% infinite_growing
tff(fact_4542_infinite__growing,axiom,
! [X5: set_int] :
( ( X5 != bot_bot_set_int )
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),X5))
=> ? [Xa: int] :
( pp(aa_set_int_bool(member_int(Xa),X5))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X3),Xa)) ) )
=> ~ pp(aa_set_int_bool(finite_finite_int,X5)) ) ) ).
% infinite_growing
tff(fact_4543_triangle__def,axiom,
! [N: nat] : aa_nat_nat(nat_triangle,N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% triangle_def
tff(fact_4544_vebt__buildup_Oelims,axiom,
! [X: nat,Y: vEBT_VEBT] :
( ( vEBT_vebt_buildup(X) = Y )
=> ( ( ( X = zero_zero_nat )
=> ( Y != vEBT_Leaf(fFalse,fFalse) ) )
=> ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
=> ( Y != vEBT_Leaf(fFalse,fFalse) ) )
=> ~ ! [Va3: nat] :
( ( X = aa_nat_nat(suc,aa_nat_nat(suc,Va3)) )
=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va3))))
=> ( Y = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va3))))
=> ( Y = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ).
% vebt_buildup.elims
tff(fact_4545_divmod__nat__if,axiom,
! [N: nat,M: nat] :
( ( ( ( N = zero_zero_nat )
| pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) )
=> ( divmod_nat(M,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),M) ) )
& ( ~ ( ( N = zero_zero_nat )
| pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) )
=> ( divmod_nat(M,N) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_ec),divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N),N)) ) ) ) ).
% divmod_nat_if
tff(fact_4546_signed__take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(aa_int_fun_int_int(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,aa_num_num(bit0,one)))),one_one_int) ).
% signed_take_bit_Suc_minus_bit1
tff(fact_4547_signed__take__bit__rec,axiom,
! [N: nat,A: code_integer] :
( ( ( N = zero_zero_nat )
=> ( bit_ri6519982836138164636nteger(N,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) )
& ( ( N != zero_zero_nat )
=> ( bit_ri6519982836138164636nteger(N,A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ) ) ) ).
% signed_take_bit_rec
tff(fact_4548_signed__take__bit__rec,axiom,
! [N: nat,A: int] :
( ( ( N = zero_zero_nat )
=> ( aa_int_int(bit_ri631733984087533419it_int(N),A) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ) )
& ( ( N != zero_zero_nat )
=> ( aa_int_int(bit_ri631733984087533419it_int(N),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).
% signed_take_bit_rec
tff(fact_4549_signed__take__bit__numeral__bit1,axiom,
! [L: num,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).
% signed_take_bit_numeral_bit1
tff(fact_4550_flip__bit__0,axiom,
! [A: code_integer] : bit_se1345352211410354436nteger(zero_zero_nat,A) = plus_p5714425477246183910nteger(zero_n356916108424825756nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).
% flip_bit_0
tff(fact_4551_flip__bit__0,axiom,
! [A: int] : aa_int_int(bit_se2159334234014336723it_int(zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).
% flip_bit_0
tff(fact_4552_flip__bit__0,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).
% flip_bit_0
tff(fact_4553_intind,axiom,
! [I: nat,N: nat,P: fun_nat_bool,X: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
=> ( pp(aa_nat_bool(P,X))
=> pp(aa_nat_bool(P,aa_nat_nat(nth_nat(replicate_nat(N,X)),I))) ) ) ).
% intind
tff(fact_4554_intind,axiom,
! [I: nat,N: nat,P: fun_VEBT_VEBT_bool,X: vEBT_VEBT] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
=> ( pp(aa_VEBT_VEBT_bool(P,X))
=> pp(aa_VEBT_VEBT_bool(P,nth_VEBT_VEBT(replicate_VEBT_VEBT(N,X),I))) ) ) ).
% intind
tff(fact_4555_add_Oinverse__inverse,axiom,
! [A: int] : aa_int_int(uminus_uminus_int,aa_int_int(uminus_uminus_int,A)) = A ).
% add.inverse_inverse
tff(fact_4556_add_Oinverse__inverse,axiom,
! [A: real] : aa_real_real(uminus_uminus_real,aa_real_real(uminus_uminus_real,A)) = A ).
% add.inverse_inverse
tff(fact_4557_add_Oinverse__inverse,axiom,
! [A: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = A ).
% add.inverse_inverse
tff(fact_4558_add_Oinverse__inverse,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = A ).
% add.inverse_inverse
tff(fact_4559_add_Oinverse__inverse,axiom,
! [A: rat] : uminus_uminus_rat(uminus_uminus_rat(A)) = A ).
% add.inverse_inverse
tff(fact_4560_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( aa_int_int(uminus_uminus_int,A) = aa_int_int(uminus_uminus_int,B) )
<=> ( A = B ) ) ).
% neg_equal_iff_equal
tff(fact_4561_neg__equal__iff__equal,axiom,
! [A: real,B: real] :
( ( aa_real_real(uminus_uminus_real,A) = aa_real_real(uminus_uminus_real,B) )
<=> ( A = B ) ) ).
% neg_equal_iff_equal
tff(fact_4562_neg__equal__iff__equal,axiom,
! [A: complex,B: complex] :
( ( aa_complex_complex(uminus1482373934393186551omplex,A) = aa_complex_complex(uminus1482373934393186551omplex,B) )
<=> ( A = B ) ) ).
% neg_equal_iff_equal
tff(fact_4563_neg__equal__iff__equal,axiom,
! [A: code_integer,B: code_integer] :
( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
<=> ( A = B ) ) ).
% neg_equal_iff_equal
tff(fact_4564_neg__equal__iff__equal,axiom,
! [A: rat,B: rat] :
( ( uminus_uminus_rat(A) = uminus_uminus_rat(B) )
<=> ( A = B ) ) ).
% neg_equal_iff_equal
tff(fact_4565_case__prodI2,axiom,
! [P2: produc7417142470700560111r_bool,C: fun_Co3748208068438171869l_bool] :
( ! [A4: code_integer,B4: bool] :
( ( P2 = produc5135875973608410785r_bool(A4,B4) )
=> pp(aa_bool_bool(aa_Cod8494480694608518558l_bool(C,A4),B4)) )
=> pp(aa_Pro6456355659118895914l_bool(produc2506095642196730057l_bool(C),P2)) ) ).
% case_prodI2
tff(fact_4566_case__prodI2,axiom,
! [P2: product_prod_num_num,C: fun_num_fun_num_bool] :
( ! [A4: num,B4: num] :
( ( P2 = product_Pair_num_num(A4,B4) )
=> pp(aa_num_bool(aa_num_fun_num_bool(C,A4),B4)) )
=> pp(aa_Pro762926636801165566m_bool(produc8237051459014760373m_bool(C),P2)) ) ).
% case_prodI2
tff(fact_4567_case__prodI2,axiom,
! [P2: product_prod_nat_num,C: fun_nat_fun_num_bool] :
( ! [A4: nat,B4: num] :
( ( P2 = product_Pair_nat_num(A4,B4) )
=> pp(aa_num_bool(aa_nat_fun_num_bool(C,A4),B4)) )
=> pp(aa_Pro5426739851356932296m_bool(produc3677492636715751295m_bool(C),P2)) ) ).
% case_prodI2
tff(fact_4568_case__prodI2,axiom,
! [P2: product_prod_nat_nat,C: fun_nat_fun_nat_bool] :
( ! [A4: nat,B4: nat] :
( ( P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A4),B4) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(C,A4),B4)) )
=> pp(aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(C),P2)) ) ).
% case_prodI2
tff(fact_4569_case__prodI2,axiom,
! [P2: product_prod_int_int,C: fun_int_fun_int_bool] :
( ! [A4: int,B4: int] :
( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A4),B4) )
=> pp(aa_int_bool(aa_int_fun_int_bool(C,A4),B4)) )
=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(C),P2)) ) ).
% case_prodI2
tff(fact_4570_case__prodI,axiom,
! [F: fun_Co3748208068438171869l_bool,A: code_integer,B: bool] :
( pp(aa_bool_bool(aa_Cod8494480694608518558l_bool(F,A),B))
=> pp(aa_Pro6456355659118895914l_bool(produc2506095642196730057l_bool(F),produc5135875973608410785r_bool(A,B))) ) ).
% case_prodI
tff(fact_4571_case__prodI,axiom,
! [F: fun_num_fun_num_bool,A: num,B: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(F,A),B))
=> pp(aa_Pro762926636801165566m_bool(produc8237051459014760373m_bool(F),product_Pair_num_num(A,B))) ) ).
% case_prodI
tff(fact_4572_case__prodI,axiom,
! [F: fun_nat_fun_num_bool,A: nat,B: num] :
( pp(aa_num_bool(aa_nat_fun_num_bool(F,A),B))
=> pp(aa_Pro5426739851356932296m_bool(produc3677492636715751295m_bool(F),product_Pair_nat_num(A,B))) ) ).
% case_prodI
tff(fact_4573_case__prodI,axiom,
! [F: fun_nat_fun_nat_bool,A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(F,A),B))
=> pp(aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B))) ) ).
% case_prodI
tff(fact_4574_case__prodI,axiom,
! [F: fun_int_fun_int_bool,A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(F,A),B))
=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))) ) ).
% case_prodI
tff(fact_4575_mem__case__prodI2,axiom,
! [P2: produc7417142470700560111r_bool,Z: complex,C: fun_Co3393511490457469770omplex] :
( ! [A4: code_integer,B4: bool] :
( ( P2 = produc5135875973608410785r_bool(A4,B4) )
=> pp(aa_set_complex_bool(member_complex(Z),aa_bool_set_complex(aa_Cod7104299617959582899omplex(C,A4),B4))) )
=> pp(aa_set_complex_bool(member_complex(Z),produc4797875024747972146omplex(C,P2))) ) ).
% mem_case_prodI2
tff(fact_4576_mem__case__prodI2,axiom,
! [P2: produc7417142470700560111r_bool,Z: real,C: fun_Co2315910201472382152t_real] :
( ! [A4: code_integer,B4: bool] :
( ( P2 = produc5135875973608410785r_bool(A4,B4) )
=> pp(aa_set_real_bool(member_real(Z),aa_bool_set_real(aa_Cod7081859000793828145t_real(C,A4),B4))) )
=> pp(aa_set_real_bool(member_real(Z),produc4532933866054709936t_real(C,P2))) ) ).
% mem_case_prodI2
tff(fact_4577_mem__case__prodI2,axiom,
! [P2: produc7417142470700560111r_bool,Z: nat,C: fun_Co3833597871255845228et_nat] :
( ! [A4: code_integer,B4: bool] :
( ( P2 = produc5135875973608410785r_bool(A4,B4) )
=> pp(aa_set_nat_bool(member_nat(Z),aa_bool_set_nat(aa_Cod5518294568803216341et_nat(C,A4),B4))) )
=> pp(aa_set_nat_bool(member_nat(Z),produc9066999194162305620et_nat(C,P2))) ) ).
% mem_case_prodI2
tff(fact_4578_mem__case__prodI2,axiom,
! [P2: produc7417142470700560111r_bool,Z: int,C: fun_Co8243421489322702280et_int] :
( ! [A4: code_integer,B4: bool] :
( ( P2 = produc5135875973608410785r_bool(A4,B4) )
=> pp(aa_set_int_bool(member_int(Z),aa_bool_set_int(aa_Cod3028211423871489585et_int(C,A4),B4))) )
=> pp(aa_set_int_bool(member_int(Z),produc4889148174653108912et_int(C,P2))) ) ).
% mem_case_prodI2
tff(fact_4579_mem__case__prodI2,axiom,
! [P2: product_prod_num_num,Z: complex,C: fun_nu9041852840018610706omplex] :
( ! [A4: num,B4: num] :
( ( P2 = product_Pair_num_num(A4,B4) )
=> pp(aa_set_complex_bool(member_complex(Z),aa_num_set_complex(aa_num241437687621652691omplex(C,A4),B4))) )
=> pp(aa_set_complex_bool(member_complex(Z),produc2866383454006189126omplex(C,P2))) ) ).
% mem_case_prodI2
tff(fact_4580_mem__case__prodI2,axiom,
! [P2: product_prod_num_num,Z: real,C: fun_nu4698968662094489488t_real] :
( ! [A4: num,B4: num] :
( ( P2 = product_Pair_num_num(A4,B4) )
=> pp(aa_set_real_bool(member_real(Z),aa_num_set_real(aa_num2469771427700027217t_real(C,A4),B4))) )
=> pp(aa_set_real_bool(member_real(Z),produc8296048397933160132t_real(C,P2))) ) ).
% mem_case_prodI2
tff(fact_4581_mem__case__prodI2,axiom,
! [P2: product_prod_num_num,Z: nat,C: fun_nu530747996811530292et_nat] :
( ! [A4: num,B4: num] :
( ( P2 = product_Pair_num_num(A4,B4) )
=> pp(aa_set_nat_bool(member_nat(Z),aa_num_set_nat(aa_num5617506412915843061et_nat(C,A4),B4))) )
=> pp(aa_set_nat_bool(member_nat(Z),produc1361121860356118632et_nat(C,P2))) ) ).
% mem_case_prodI2
tff(fact_4582_mem__case__prodI2,axiom,
! [P2: product_prod_num_num,Z: int,C: fun_nu4940571614878387344et_int] :
( ! [A4: num,B4: num] :
( ( P2 = product_Pair_num_num(A4,B4) )
=> pp(aa_set_int_bool(member_int(Z),aa_num_set_int(aa_num3127423267984116305et_int(C,A4),B4))) )
=> pp(aa_set_int_bool(member_int(Z),produc6406642877701697732et_int(C,P2))) ) ).
% mem_case_prodI2
tff(fact_4583_mem__case__prodI2,axiom,
! [P2: product_prod_nat_num,Z: complex,C: fun_na3925014923085203400omplex] :
( ! [A4: nat,B4: num] :
( ( P2 = product_Pair_nat_num(A4,B4) )
=> pp(aa_set_complex_bool(member_complex(Z),aa_num_set_complex(aa_nat4347971807543021193omplex(C,A4),B4))) )
=> pp(aa_set_complex_bool(member_complex(Z),produc6231982587499038204omplex(C,P2))) ) ).
% mem_case_prodI2
tff(fact_4584_mem__case__prodI2,axiom,
! [P2: product_prod_nat_num,Z: real,C: fun_na3666575854997362246t_real] :
( ! [A4: nat,B4: num] :
( ( P2 = product_Pair_nat_num(A4,B4) )
=> pp(aa_set_real_bool(member_real(Z),aa_num_set_real(aa_nat1437378620602899975t_real(C,A4),B4))) )
=> pp(aa_set_real_bool(member_real(Z),produc1435849484188172666t_real(C,P2))) ) ).
% mem_case_prodI2
tff(fact_4585_mem__case__prodI,axiom,
! [Z: complex,C: fun_Co3393511490457469770omplex,A: code_integer,B: bool] :
( pp(aa_set_complex_bool(member_complex(Z),aa_bool_set_complex(aa_Cod7104299617959582899omplex(C,A),B)))
=> pp(aa_set_complex_bool(member_complex(Z),produc4797875024747972146omplex(C,produc5135875973608410785r_bool(A,B)))) ) ).
% mem_case_prodI
tff(fact_4586_mem__case__prodI,axiom,
! [Z: real,C: fun_Co2315910201472382152t_real,A: code_integer,B: bool] :
( pp(aa_set_real_bool(member_real(Z),aa_bool_set_real(aa_Cod7081859000793828145t_real(C,A),B)))
=> pp(aa_set_real_bool(member_real(Z),produc4532933866054709936t_real(C,produc5135875973608410785r_bool(A,B)))) ) ).
% mem_case_prodI
tff(fact_4587_mem__case__prodI,axiom,
! [Z: nat,C: fun_Co3833597871255845228et_nat,A: code_integer,B: bool] :
( pp(aa_set_nat_bool(member_nat(Z),aa_bool_set_nat(aa_Cod5518294568803216341et_nat(C,A),B)))
=> pp(aa_set_nat_bool(member_nat(Z),produc9066999194162305620et_nat(C,produc5135875973608410785r_bool(A,B)))) ) ).
% mem_case_prodI
tff(fact_4588_mem__case__prodI,axiom,
! [Z: int,C: fun_Co8243421489322702280et_int,A: code_integer,B: bool] :
( pp(aa_set_int_bool(member_int(Z),aa_bool_set_int(aa_Cod3028211423871489585et_int(C,A),B)))
=> pp(aa_set_int_bool(member_int(Z),produc4889148174653108912et_int(C,produc5135875973608410785r_bool(A,B)))) ) ).
% mem_case_prodI
tff(fact_4589_mem__case__prodI,axiom,
! [Z: complex,C: fun_nu9041852840018610706omplex,A: num,B: num] :
( pp(aa_set_complex_bool(member_complex(Z),aa_num_set_complex(aa_num241437687621652691omplex(C,A),B)))
=> pp(aa_set_complex_bool(member_complex(Z),produc2866383454006189126omplex(C,product_Pair_num_num(A,B)))) ) ).
% mem_case_prodI
tff(fact_4590_mem__case__prodI,axiom,
! [Z: real,C: fun_nu4698968662094489488t_real,A: num,B: num] :
( pp(aa_set_real_bool(member_real(Z),aa_num_set_real(aa_num2469771427700027217t_real(C,A),B)))
=> pp(aa_set_real_bool(member_real(Z),produc8296048397933160132t_real(C,product_Pair_num_num(A,B)))) ) ).
% mem_case_prodI
tff(fact_4591_mem__case__prodI,axiom,
! [Z: nat,C: fun_nu530747996811530292et_nat,A: num,B: num] :
( pp(aa_set_nat_bool(member_nat(Z),aa_num_set_nat(aa_num5617506412915843061et_nat(C,A),B)))
=> pp(aa_set_nat_bool(member_nat(Z),produc1361121860356118632et_nat(C,product_Pair_num_num(A,B)))) ) ).
% mem_case_prodI
tff(fact_4592_mem__case__prodI,axiom,
! [Z: int,C: fun_nu4940571614878387344et_int,A: num,B: num] :
( pp(aa_set_int_bool(member_int(Z),aa_num_set_int(aa_num3127423267984116305et_int(C,A),B)))
=> pp(aa_set_int_bool(member_int(Z),produc6406642877701697732et_int(C,product_Pair_num_num(A,B)))) ) ).
% mem_case_prodI
tff(fact_4593_mem__case__prodI,axiom,
! [Z: complex,C: fun_na3925014923085203400omplex,A: nat,B: num] :
( pp(aa_set_complex_bool(member_complex(Z),aa_num_set_complex(aa_nat4347971807543021193omplex(C,A),B)))
=> pp(aa_set_complex_bool(member_complex(Z),produc6231982587499038204omplex(C,product_Pair_nat_num(A,B)))) ) ).
% mem_case_prodI
tff(fact_4594_mem__case__prodI,axiom,
! [Z: real,C: fun_na3666575854997362246t_real,A: nat,B: num] :
( pp(aa_set_real_bool(member_real(Z),aa_num_set_real(aa_nat1437378620602899975t_real(C,A),B)))
=> pp(aa_set_real_bool(member_real(Z),produc1435849484188172666t_real(C,product_Pair_nat_num(A,B)))) ) ).
% mem_case_prodI
tff(fact_4595_case__prodI2_H,axiom,
! [P2: product_prod_nat_nat,C: fun_na4786964484885125043t_bool,X: product_prod_nat_nat] :
( ! [A4: nat,B4: nat] :
( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A4),B4) = P2 )
=> pp(aa_Pro2232559814647311250t_bool(aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(C,A4),B4),X)) )
=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(C),P2),X)) ) ).
% case_prodI2'
tff(fact_4596_neg__le__iff__le,axiom,
! [B: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).
% neg_le_iff_le
tff(fact_4597_neg__le__iff__le,axiom,
! [B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) ) ).
% neg_le_iff_le
tff(fact_4598_neg__le__iff__le,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(B)),uminus_uminus_rat(A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).
% neg_le_iff_le
tff(fact_4599_neg__le__iff__le,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).
% neg_le_iff_le
tff(fact_4600_add_Oinverse__neutral,axiom,
aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).
% add.inverse_neutral
tff(fact_4601_add_Oinverse__neutral,axiom,
aa_real_real(uminus_uminus_real,zero_zero_real) = zero_zero_real ).
% add.inverse_neutral
tff(fact_4602_add_Oinverse__neutral,axiom,
aa_complex_complex(uminus1482373934393186551omplex,zero_zero_complex) = zero_zero_complex ).
% add.inverse_neutral
tff(fact_4603_add_Oinverse__neutral,axiom,
aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).
% add.inverse_neutral
tff(fact_4604_add_Oinverse__neutral,axiom,
uminus_uminus_rat(zero_zero_rat) = zero_zero_rat ).
% add.inverse_neutral
tff(fact_4605_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int = aa_int_int(uminus_uminus_int,A) )
<=> ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
tff(fact_4606_neg__0__equal__iff__equal,axiom,
! [A: real] :
( ( zero_zero_real = aa_real_real(uminus_uminus_real,A) )
<=> ( zero_zero_real = A ) ) ).
% neg_0_equal_iff_equal
tff(fact_4607_neg__0__equal__iff__equal,axiom,
! [A: complex] :
( ( zero_zero_complex = aa_complex_complex(uminus1482373934393186551omplex,A) )
<=> ( zero_zero_complex = A ) ) ).
% neg_0_equal_iff_equal
tff(fact_4608_neg__0__equal__iff__equal,axiom,
! [A: code_integer] :
( ( zero_z3403309356797280102nteger = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
<=> ( zero_z3403309356797280102nteger = A ) ) ).
% neg_0_equal_iff_equal
tff(fact_4609_neg__0__equal__iff__equal,axiom,
! [A: rat] :
( ( zero_zero_rat = uminus_uminus_rat(A) )
<=> ( zero_zero_rat = A ) ) ).
% neg_0_equal_iff_equal
tff(fact_4610_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( aa_int_int(uminus_uminus_int,A) = zero_zero_int )
<=> ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
tff(fact_4611_neg__equal__0__iff__equal,axiom,
! [A: real] :
( ( aa_real_real(uminus_uminus_real,A) = zero_zero_real )
<=> ( A = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
tff(fact_4612_neg__equal__0__iff__equal,axiom,
! [A: complex] :
( ( aa_complex_complex(uminus1482373934393186551omplex,A) = zero_zero_complex )
<=> ( A = zero_zero_complex ) ) ).
% neg_equal_0_iff_equal
tff(fact_4613_neg__equal__0__iff__equal,axiom,
! [A: code_integer] :
( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = zero_z3403309356797280102nteger )
<=> ( A = zero_z3403309356797280102nteger ) ) ).
% neg_equal_0_iff_equal
tff(fact_4614_neg__equal__0__iff__equal,axiom,
! [A: rat] :
( ( uminus_uminus_rat(A) = zero_zero_rat )
<=> ( A = zero_zero_rat ) ) ).
% neg_equal_0_iff_equal
tff(fact_4615_equal__neg__zero,axiom,
! [A: int] :
( ( A = aa_int_int(uminus_uminus_int,A) )
<=> ( A = zero_zero_int ) ) ).
% equal_neg_zero
tff(fact_4616_equal__neg__zero,axiom,
! [A: real] :
( ( A = aa_real_real(uminus_uminus_real,A) )
<=> ( A = zero_zero_real ) ) ).
% equal_neg_zero
tff(fact_4617_equal__neg__zero,axiom,
! [A: code_integer] :
( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
<=> ( A = zero_z3403309356797280102nteger ) ) ).
% equal_neg_zero
tff(fact_4618_equal__neg__zero,axiom,
! [A: rat] :
( ( A = uminus_uminus_rat(A) )
<=> ( A = zero_zero_rat ) ) ).
% equal_neg_zero
tff(fact_4619_neg__equal__zero,axiom,
! [A: int] :
( ( aa_int_int(uminus_uminus_int,A) = A )
<=> ( A = zero_zero_int ) ) ).
% neg_equal_zero
tff(fact_4620_neg__equal__zero,axiom,
! [A: real] :
( ( aa_real_real(uminus_uminus_real,A) = A )
<=> ( A = zero_zero_real ) ) ).
% neg_equal_zero
tff(fact_4621_neg__equal__zero,axiom,
! [A: code_integer] :
( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = A )
<=> ( A = zero_z3403309356797280102nteger ) ) ).
% neg_equal_zero
tff(fact_4622_neg__equal__zero,axiom,
! [A: rat] :
( ( uminus_uminus_rat(A) = A )
<=> ( A = zero_zero_rat ) ) ).
% neg_equal_zero
tff(fact_4623_neg__less__iff__less,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ).
% neg_less_iff_less
tff(fact_4624_neg__less__iff__less,axiom,
! [B: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ).
% neg_less_iff_less
tff(fact_4625_neg__less__iff__less,axiom,
! [B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B)) ) ).
% neg_less_iff_less
tff(fact_4626_neg__less__iff__less,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(B)),uminus_uminus_rat(A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B)) ) ).
% neg_less_iff_less
tff(fact_4627_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( 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,N)) )
<=> ( M = N ) ) ).
% neg_numeral_eq_iff
tff(fact_4628_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(M)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(N)) )
<=> ( M = N ) ) ).
% neg_numeral_eq_iff
tff(fact_4629_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) )
<=> ( M = N ) ) ).
% neg_numeral_eq_iff
tff(fact_4630_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)) )
<=> ( M = N ) ) ).
% neg_numeral_eq_iff
tff(fact_4631_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( uminus_uminus_rat(numeral_numeral_rat(M)) = uminus_uminus_rat(numeral_numeral_rat(N)) )
<=> ( M = N ) ) ).
% neg_numeral_eq_iff
tff(fact_4632_mult__minus__left,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ).
% mult_minus_left
tff(fact_4633_mult__minus__left,axiom,
! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).
% mult_minus_left
tff(fact_4634_mult__minus__left,axiom,
! [A: complex,B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(A),B)) ).
% mult_minus_left
tff(fact_4635_mult__minus__left,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).
% mult_minus_left
tff(fact_4636_mult__minus__left,axiom,
! [A: rat,B: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(A)),B) = uminus_uminus_rat(aa_rat_rat(times_times_rat(A),B)) ).
% mult_minus_left
tff(fact_4637_minus__mult__minus,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),B) ).
% minus_mult_minus
tff(fact_4638_minus__mult__minus,axiom,
! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B) ).
% minus_mult_minus
tff(fact_4639_minus__mult__minus,axiom,
! [A: complex,B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(times_times_complex(A),B) ).
% minus_mult_minus
tff(fact_4640_minus__mult__minus,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) ).
% minus_mult_minus
tff(fact_4641_minus__mult__minus,axiom,
! [A: rat,B: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(A)),uminus_uminus_rat(B)) = aa_rat_rat(times_times_rat(A),B) ).
% minus_mult_minus
tff(fact_4642_mult__minus__right,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ).
% mult_minus_right
tff(fact_4643_mult__minus__right,axiom,
! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).
% mult_minus_right
tff(fact_4644_mult__minus__right,axiom,
! [A: complex,B: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(A),B)) ).
% mult_minus_right
tff(fact_4645_mult__minus__right,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).
% mult_minus_right
tff(fact_4646_mult__minus__right,axiom,
! [A: rat,B: rat] : aa_rat_rat(times_times_rat(A),uminus_uminus_rat(B)) = uminus_uminus_rat(aa_rat_rat(times_times_rat(A),B)) ).
% mult_minus_right
tff(fact_4647_add__minus__cancel,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),B)) = B ).
% add_minus_cancel
tff(fact_4648_add__minus__cancel,axiom,
! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),B)) = B ).
% add_minus_cancel
tff(fact_4649_add__minus__cancel,axiom,
! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B)) = B ).
% add_minus_cancel
tff(fact_4650_add__minus__cancel,axiom,
! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(A,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B)) = B ).
% add_minus_cancel
tff(fact_4651_add__minus__cancel,axiom,
! [A: rat,B: rat] : plus_plus_rat(A,plus_plus_rat(uminus_uminus_rat(A),B)) = B ).
% add_minus_cancel
tff(fact_4652_minus__add__cancel,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = B ).
% minus_add_cancel
tff(fact_4653_minus__add__cancel,axiom,
! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = B ).
% minus_add_cancel
tff(fact_4654_minus__add__cancel,axiom,
! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(plus_plus_complex(A),B)) = B ).
% minus_add_cancel
tff(fact_4655_minus__add__cancel,axiom,
! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),plus_p5714425477246183910nteger(A,B)) = B ).
% minus_add_cancel
tff(fact_4656_minus__add__cancel,axiom,
! [A: rat,B: rat] : plus_plus_rat(uminus_uminus_rat(A),plus_plus_rat(A,B)) = B ).
% minus_add_cancel
tff(fact_4657_minus__add__distrib,axiom,
! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) ).
% minus_add_distrib
tff(fact_4658_minus__add__distrib,axiom,
! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) ).
% minus_add_distrib
tff(fact_4659_minus__add__distrib,axiom,
! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) ).
% minus_add_distrib
tff(fact_4660_minus__add__distrib,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).
% minus_add_distrib
tff(fact_4661_minus__add__distrib,axiom,
! [A: rat,B: rat] : uminus_uminus_rat(plus_plus_rat(A,B)) = plus_plus_rat(uminus_uminus_rat(A),uminus_uminus_rat(B)) ).
% minus_add_distrib
tff(fact_4662_minus__diff__eq,axiom,
! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A) ).
% minus_diff_eq
tff(fact_4663_minus__diff__eq,axiom,
! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A) ).
% minus_diff_eq
tff(fact_4664_minus__diff__eq,axiom,
! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(minus_minus_complex(A),B)) = aa_complex_complex(minus_minus_complex(B),A) ).
% minus_diff_eq
tff(fact_4665_minus__diff__eq,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,minus_8373710615458151222nteger(A,B)) = minus_8373710615458151222nteger(B,A) ).
% minus_diff_eq
tff(fact_4666_minus__diff__eq,axiom,
! [A: rat,B: rat] : uminus_uminus_rat(minus_minus_rat(A,B)) = minus_minus_rat(B,A) ).
% minus_diff_eq
tff(fact_4667_div__minus__minus,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ).
% div_minus_minus
tff(fact_4668_div__minus__minus,axiom,
! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = divide6298287555418463151nteger(A,B) ).
% div_minus_minus
tff(fact_4669_dvd__minus__iff,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),aa_int_int(uminus_uminus_int,Y)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y)) ) ).
% dvd_minus_iff
tff(fact_4670_dvd__minus__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),aa_real_real(uminus_uminus_real,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Y)) ) ).
% dvd_minus_iff
tff(fact_4671_dvd__minus__iff,axiom,
! [X: complex,Y: complex] :
( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X),aa_complex_complex(uminus1482373934393186551omplex,Y)))
<=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X),Y)) ) ).
% dvd_minus_iff
tff(fact_4672_dvd__minus__iff,axiom,
! [X: code_integer,Y: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),Y)) ) ).
% dvd_minus_iff
tff(fact_4673_dvd__minus__iff,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),uminus_uminus_rat(Y)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),Y)) ) ).
% dvd_minus_iff
tff(fact_4674_minus__dvd__iff,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(uminus_uminus_int,X)),Y))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),Y)) ) ).
% minus_dvd_iff
tff(fact_4675_minus__dvd__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(uminus_uminus_real,X)),Y))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,X),Y)) ) ).
% minus_dvd_iff
tff(fact_4676_minus__dvd__iff,axiom,
! [X: complex,Y: complex] :
( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,aa_complex_complex(uminus1482373934393186551omplex,X)),Y))
<=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,X),Y)) ) ).
% minus_dvd_iff
tff(fact_4677_minus__dvd__iff,axiom,
! [X: code_integer,Y: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)),Y))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,X),Y)) ) ).
% minus_dvd_iff
tff(fact_4678_minus__dvd__iff,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,uminus_uminus_rat(X)),Y))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,X),Y)) ) ).
% minus_dvd_iff
tff(fact_4679_mod__minus__minus,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ).
% mod_minus_minus
tff(fact_4680_mod__minus__minus,axiom,
! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,B)) ).
% mod_minus_minus
tff(fact_4681_of__bool__less__eq__iff,axiom,
! [P: bool,Q: bool] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_n2052037380579107095ol_rat(P)),zero_n2052037380579107095ol_rat(Q)))
<=> ( pp(P)
=> pp(Q) ) ) ).
% of_bool_less_eq_iff
tff(fact_4682_of__bool__less__eq__iff,axiom,
! [P: bool,Q: bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_n2687167440665602831ol_nat(P)),zero_n2687167440665602831ol_nat(Q)))
<=> ( pp(P)
=> pp(Q) ) ) ).
% of_bool_less_eq_iff
tff(fact_4683_of__bool__less__eq__iff,axiom,
! [P: bool,Q: bool] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_bool_int(zero_n2684676970156552555ol_int,P)),aa_bool_int(zero_n2684676970156552555ol_int,Q)))
<=> ( pp(P)
=> pp(Q) ) ) ).
% of_bool_less_eq_iff
tff(fact_4684_of__bool__less__eq__iff,axiom,
! [P: bool,Q: bool] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_n356916108424825756nteger(P)),zero_n356916108424825756nteger(Q)))
<=> ( pp(P)
=> pp(Q) ) ) ).
% of_bool_less_eq_iff
tff(fact_4685_of__bool__eq_I1_J,axiom,
zero_n1201886186963655149omplex(fFalse) = zero_zero_complex ).
% of_bool_eq(1)
tff(fact_4686_of__bool__eq_I1_J,axiom,
zero_n3304061248610475627l_real(fFalse) = zero_zero_real ).
% of_bool_eq(1)
tff(fact_4687_of__bool__eq_I1_J,axiom,
zero_n2052037380579107095ol_rat(fFalse) = zero_zero_rat ).
% of_bool_eq(1)
tff(fact_4688_of__bool__eq_I1_J,axiom,
zero_n2687167440665602831ol_nat(fFalse) = zero_zero_nat ).
% of_bool_eq(1)
tff(fact_4689_of__bool__eq_I1_J,axiom,
aa_bool_int(zero_n2684676970156552555ol_int,fFalse) = zero_zero_int ).
% of_bool_eq(1)
tff(fact_4690_of__bool__eq_I1_J,axiom,
zero_n356916108424825756nteger(fFalse) = zero_z3403309356797280102nteger ).
% of_bool_eq(1)
tff(fact_4691_of__bool__eq__0__iff,axiom,
! [P: bool] :
( ( zero_n1201886186963655149omplex(P) = zero_zero_complex )
<=> ~ pp(P) ) ).
% of_bool_eq_0_iff
tff(fact_4692_of__bool__eq__0__iff,axiom,
! [P: bool] :
( ( zero_n3304061248610475627l_real(P) = zero_zero_real )
<=> ~ pp(P) ) ).
% of_bool_eq_0_iff
tff(fact_4693_of__bool__eq__0__iff,axiom,
! [P: bool] :
( ( zero_n2052037380579107095ol_rat(P) = zero_zero_rat )
<=> ~ pp(P) ) ).
% of_bool_eq_0_iff
tff(fact_4694_of__bool__eq__0__iff,axiom,
! [P: bool] :
( ( zero_n2687167440665602831ol_nat(P) = zero_zero_nat )
<=> ~ pp(P) ) ).
% of_bool_eq_0_iff
tff(fact_4695_of__bool__eq__0__iff,axiom,
! [P: bool] :
( ( aa_bool_int(zero_n2684676970156552555ol_int,P) = zero_zero_int )
<=> ~ pp(P) ) ).
% of_bool_eq_0_iff
tff(fact_4696_of__bool__eq__0__iff,axiom,
! [P: bool] :
( ( zero_n356916108424825756nteger(P) = zero_z3403309356797280102nteger )
<=> ~ pp(P) ) ).
% of_bool_eq_0_iff
tff(fact_4697_real__add__minus__iff,axiom,
! [X: real,A: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(uminus_uminus_real,A)) = zero_zero_real )
<=> ( X = A ) ) ).
% real_add_minus_iff
tff(fact_4698_of__bool__less__iff,axiom,
! [P: bool,Q: bool] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_n3304061248610475627l_real(P)),zero_n3304061248610475627l_real(Q)))
<=> ( ~ pp(P)
& pp(Q) ) ) ).
% of_bool_less_iff
tff(fact_4699_of__bool__less__iff,axiom,
! [P: bool,Q: bool] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_n2052037380579107095ol_rat(P)),zero_n2052037380579107095ol_rat(Q)))
<=> ( ~ pp(P)
& pp(Q) ) ) ).
% of_bool_less_iff
tff(fact_4700_of__bool__less__iff,axiom,
! [P: bool,Q: bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_n2687167440665602831ol_nat(P)),zero_n2687167440665602831ol_nat(Q)))
<=> ( ~ pp(P)
& pp(Q) ) ) ).
% of_bool_less_iff
tff(fact_4701_of__bool__less__iff,axiom,
! [P: bool,Q: bool] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_bool_int(zero_n2684676970156552555ol_int,P)),aa_bool_int(zero_n2684676970156552555ol_int,Q)))
<=> ( ~ pp(P)
& pp(Q) ) ) ).
% of_bool_less_iff
tff(fact_4702_of__bool__less__iff,axiom,
! [P: bool,Q: bool] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_n356916108424825756nteger(P)),zero_n356916108424825756nteger(Q)))
<=> ( ~ pp(P)
& pp(Q) ) ) ).
% of_bool_less_iff
tff(fact_4703_of__bool__eq_I2_J,axiom,
zero_n1201886186963655149omplex(fTrue) = one_one_complex ).
% of_bool_eq(2)
tff(fact_4704_of__bool__eq_I2_J,axiom,
zero_n3304061248610475627l_real(fTrue) = one_one_real ).
% of_bool_eq(2)
tff(fact_4705_of__bool__eq_I2_J,axiom,
zero_n2052037380579107095ol_rat(fTrue) = one_one_rat ).
% of_bool_eq(2)
tff(fact_4706_of__bool__eq_I2_J,axiom,
zero_n2687167440665602831ol_nat(fTrue) = one_one_nat ).
% of_bool_eq(2)
tff(fact_4707_of__bool__eq_I2_J,axiom,
aa_bool_int(zero_n2684676970156552555ol_int,fTrue) = one_one_int ).
% of_bool_eq(2)
tff(fact_4708_of__bool__eq_I2_J,axiom,
zero_n356916108424825756nteger(fTrue) = one_one_Code_integer ).
% of_bool_eq(2)
tff(fact_4709_of__bool__eq__1__iff,axiom,
! [P: bool] :
( ( zero_n1201886186963655149omplex(P) = one_one_complex )
<=> pp(P) ) ).
% of_bool_eq_1_iff
tff(fact_4710_of__bool__eq__1__iff,axiom,
! [P: bool] :
( ( zero_n3304061248610475627l_real(P) = one_one_real )
<=> pp(P) ) ).
% of_bool_eq_1_iff
tff(fact_4711_of__bool__eq__1__iff,axiom,
! [P: bool] :
( ( zero_n2052037380579107095ol_rat(P) = one_one_rat )
<=> pp(P) ) ).
% of_bool_eq_1_iff
tff(fact_4712_of__bool__eq__1__iff,axiom,
! [P: bool] :
( ( zero_n2687167440665602831ol_nat(P) = one_one_nat )
<=> pp(P) ) ).
% of_bool_eq_1_iff
tff(fact_4713_of__bool__eq__1__iff,axiom,
! [P: bool] :
( ( aa_bool_int(zero_n2684676970156552555ol_int,P) = one_one_int )
<=> pp(P) ) ).
% of_bool_eq_1_iff
tff(fact_4714_of__bool__eq__1__iff,axiom,
! [P: bool] :
( ( zero_n356916108424825756nteger(P) = one_one_Code_integer )
<=> pp(P) ) ).
% of_bool_eq_1_iff
tff(fact_4715_length__replicate,axiom,
! [N: nat,X: vEBT_VEBT] : size_s6755466524823107622T_VEBT(replicate_VEBT_VEBT(N,X)) = N ).
% length_replicate
tff(fact_4716_length__replicate,axiom,
! [N: nat,X: bool] : size_size_list_bool(replicate_bool(N,X)) = N ).
% length_replicate
tff(fact_4717_length__replicate,axiom,
! [N: nat,X: nat] : size_size_list_nat(replicate_nat(N,X)) = N ).
% length_replicate
tff(fact_4718_length__replicate,axiom,
! [N: nat,X: int] : size_size_list_int(replicate_int(N,X)) = N ).
% length_replicate
tff(fact_4719_of__bool__or__iff,axiom,
! [P: bool,Q: bool] : zero_n2687167440665602831ol_nat(aa_bool_bool(fdisj(P),Q)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_n2687167440665602831ol_nat(P)),zero_n2687167440665602831ol_nat(Q)) ).
% of_bool_or_iff
tff(fact_4720_of__bool__or__iff,axiom,
! [P: bool,Q: bool] : aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fdisj(P),Q)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_bool_int(zero_n2684676970156552555ol_int,P)),aa_bool_int(zero_n2684676970156552555ol_int,Q)) ).
% of_bool_or_iff
tff(fact_4721_of__bool__or__iff,axiom,
! [P: bool,Q: bool] : zero_n356916108424825756nteger(aa_bool_bool(fdisj(P),Q)) = ord_max_Code_integer(zero_n356916108424825756nteger(P),zero_n356916108424825756nteger(Q)) ).
% of_bool_or_iff
tff(fact_4722_neg__less__eq__nonneg,axiom,
! [A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),A))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).
% neg_less_eq_nonneg
tff(fact_4723_neg__less__eq__nonneg,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)) ) ).
% neg_less_eq_nonneg
tff(fact_4724_neg__less__eq__nonneg,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(A)),A))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).
% neg_less_eq_nonneg
tff(fact_4725_neg__less__eq__nonneg,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),A))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).
% neg_less_eq_nonneg
tff(fact_4726_less__eq__neg__nonpos,axiom,
! [A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,A)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).
% less_eq_neg_nonpos
tff(fact_4727_less__eq__neg__nonpos,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) ) ).
% less_eq_neg_nonpos
tff(fact_4728_less__eq__neg__nonpos,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),uminus_uminus_rat(A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).
% less_eq_neg_nonpos
tff(fact_4729_less__eq__neg__nonpos,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).
% less_eq_neg_nonpos
tff(fact_4730_neg__le__0__iff__le,axiom,
! [A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).
% neg_le_0_iff_le
tff(fact_4731_neg__le__0__iff__le,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)) ) ).
% neg_le_0_iff_le
tff(fact_4732_neg__le__0__iff__le,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(A)),zero_zero_rat))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).
% neg_le_0_iff_le
tff(fact_4733_neg__le__0__iff__le,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).
% neg_le_0_iff_le
tff(fact_4734_neg__0__le__iff__le,axiom,
! [A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,A)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ).
% neg_0_le_iff_le
tff(fact_4735_neg__0__le__iff__le,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) ) ).
% neg_0_le_iff_le
tff(fact_4736_neg__0__le__iff__le,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),uminus_uminus_rat(A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ).
% neg_0_le_iff_le
tff(fact_4737_neg__0__le__iff__le,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ).
% neg_0_le_iff_le
tff(fact_4738_less__neg__neg,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(uminus_uminus_int,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).
% less_neg_neg
tff(fact_4739_less__neg__neg,axiom,
! [A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,A)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).
% less_neg_neg
tff(fact_4740_less__neg__neg,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ).
% less_neg_neg
tff(fact_4741_less__neg__neg,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),uminus_uminus_rat(A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).
% less_neg_neg
tff(fact_4742_neg__less__pos,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),A))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).
% neg_less_pos
tff(fact_4743_neg__less__pos,axiom,
! [A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A)),A))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).
% neg_less_pos
tff(fact_4744_neg__less__pos,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)) ) ).
% neg_less_pos
tff(fact_4745_neg__less__pos,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(A)),A))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).
% neg_less_pos
tff(fact_4746_neg__0__less__iff__less,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ).
% neg_0_less_iff_less
tff(fact_4747_neg__0__less__iff__less,axiom,
! [A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,A)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ).
% neg_0_less_iff_less
tff(fact_4748_neg__0__less__iff__less,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)) ) ).
% neg_0_less_iff_less
tff(fact_4749_neg__0__less__iff__less,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),uminus_uminus_rat(A)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ).
% neg_0_less_iff_less
tff(fact_4750_neg__less__0__iff__less,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ).
% neg_less_0_iff_less
tff(fact_4751_neg__less__0__iff__less,axiom,
! [A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ).
% neg_less_0_iff_less
tff(fact_4752_neg__less__0__iff__less,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)) ) ).
% neg_less_0_iff_less
tff(fact_4753_neg__less__0__iff__less,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(A)),zero_zero_rat))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ).
% neg_less_0_iff_less
tff(fact_4754_add_Oright__inverse,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,A)) = zero_zero_int ).
% add.right_inverse
tff(fact_4755_add_Oright__inverse,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,A)) = zero_zero_real ).
% add.right_inverse
tff(fact_4756_add_Oright__inverse,axiom,
! [A: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,A)) = zero_zero_complex ).
% add.right_inverse
tff(fact_4757_add_Oright__inverse,axiom,
! [A: code_integer] : plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = zero_z3403309356797280102nteger ).
% add.right_inverse
tff(fact_4758_add_Oright__inverse,axiom,
! [A: rat] : plus_plus_rat(A,uminus_uminus_rat(A)) = zero_zero_rat ).
% add.right_inverse
tff(fact_4759_ab__left__minus,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),A) = zero_zero_int ).
% ab_left_minus
tff(fact_4760_ab__left__minus,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),A) = zero_zero_real ).
% ab_left_minus
tff(fact_4761_ab__left__minus,axiom,
! [A: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),A) = zero_zero_complex ).
% ab_left_minus
tff(fact_4762_ab__left__minus,axiom,
! [A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) = zero_z3403309356797280102nteger ).
% ab_left_minus
tff(fact_4763_ab__left__minus,axiom,
! [A: rat] : plus_plus_rat(uminus_uminus_rat(A),A) = zero_zero_rat ).
% ab_left_minus
tff(fact_4764_verit__minus__simplify_I3_J,axiom,
! [B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),B) = aa_int_int(uminus_uminus_int,B) ).
% verit_minus_simplify(3)
tff(fact_4765_verit__minus__simplify_I3_J,axiom,
! [B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,zero_zero_real),B) = aa_real_real(uminus_uminus_real,B) ).
% verit_minus_simplify(3)
tff(fact_4766_verit__minus__simplify_I3_J,axiom,
! [B: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ).
% verit_minus_simplify(3)
tff(fact_4767_verit__minus__simplify_I3_J,axiom,
! [B: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).
% verit_minus_simplify(3)
tff(fact_4768_verit__minus__simplify_I3_J,axiom,
! [B: rat] : minus_minus_rat(zero_zero_rat,B) = uminus_uminus_rat(B) ).
% verit_minus_simplify(3)
tff(fact_4769_diff__0,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),A) = aa_int_int(uminus_uminus_int,A) ).
% diff_0
tff(fact_4770_diff__0,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,zero_zero_real),A) = aa_real_real(uminus_uminus_real,A) ).
% diff_0
tff(fact_4771_diff__0,axiom,
! [A: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),A) = aa_complex_complex(uminus1482373934393186551omplex,A) ).
% diff_0
tff(fact_4772_diff__0,axiom,
! [A: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).
% diff_0
tff(fact_4773_diff__0,axiom,
! [A: rat] : minus_minus_rat(zero_zero_rat,A) = uminus_uminus_rat(A) ).
% diff_0
tff(fact_4774_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_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,N))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))) ).
% add_neg_numeral_simps(3)
tff(fact_4775_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(M)),numeral_numeral_real(N))) ).
% add_neg_numeral_simps(3)
tff(fact_4776_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(M)),numera6690914467698888265omplex(N))) ).
% add_neg_numeral_simps(3)
tff(fact_4777_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(N))) ).
% add_neg_numeral_simps(3)
tff(fact_4778_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] : plus_plus_rat(uminus_uminus_rat(numeral_numeral_rat(M)),uminus_uminus_rat(numeral_numeral_rat(N))) = uminus_uminus_rat(plus_plus_rat(numeral_numeral_rat(M),numeral_numeral_rat(N))) ).
% add_neg_numeral_simps(3)
tff(fact_4779_mult__minus1,axiom,
! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,one_one_int)),Z) = aa_int_int(uminus_uminus_int,Z) ).
% mult_minus1
tff(fact_4780_mult__minus1,axiom,
! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,one_one_real)),Z) = aa_real_real(uminus_uminus_real,Z) ).
% mult_minus1
tff(fact_4781_mult__minus1,axiom,
! [Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Z) = aa_complex_complex(uminus1482373934393186551omplex,Z) ).
% mult_minus1
tff(fact_4782_mult__minus1,axiom,
! [Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Z) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Z) ).
% mult_minus1
tff(fact_4783_mult__minus1,axiom,
! [Z: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(one_one_rat)),Z) = uminus_uminus_rat(Z) ).
% mult_minus1
tff(fact_4784_mult__minus1__right,axiom,
! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Z),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,Z) ).
% mult_minus1_right
tff(fact_4785_mult__minus1__right,axiom,
! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,Z) ).
% mult_minus1_right
tff(fact_4786_mult__minus1__right,axiom,
! [Z: complex] : aa_complex_complex(times_times_complex(Z),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,Z) ).
% mult_minus1_right
tff(fact_4787_mult__minus1__right,axiom,
! [Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Z) ).
% mult_minus1_right
tff(fact_4788_mult__minus1__right,axiom,
! [Z: rat] : aa_rat_rat(times_times_rat(Z),uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(Z) ).
% mult_minus1_right
tff(fact_4789_diff__minus__eq__add,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) ).
% diff_minus_eq_add
tff(fact_4790_diff__minus__eq__add,axiom,
! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) ).
% diff_minus_eq_add
tff(fact_4791_diff__minus__eq__add,axiom,
! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(plus_plus_complex(A),B) ).
% diff_minus_eq_add
tff(fact_4792_diff__minus__eq__add,axiom,
! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = plus_p5714425477246183910nteger(A,B) ).
% diff_minus_eq_add
tff(fact_4793_diff__minus__eq__add,axiom,
! [A: rat,B: rat] : minus_minus_rat(A,uminus_uminus_rat(B)) = plus_plus_rat(A,B) ).
% diff_minus_eq_add
tff(fact_4794_uminus__add__conv__diff,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A) ).
% uminus_add_conv_diff
tff(fact_4795_uminus__add__conv__diff,axiom,
! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A) ).
% uminus_add_conv_diff
tff(fact_4796_uminus__add__conv__diff,axiom,
! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(minus_minus_complex(B),A) ).
% uminus_add_conv_diff
tff(fact_4797_uminus__add__conv__diff,axiom,
! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) = minus_8373710615458151222nteger(B,A) ).
% uminus_add_conv_diff
tff(fact_4798_uminus__add__conv__diff,axiom,
! [A: rat,B: rat] : plus_plus_rat(uminus_uminus_rat(A),B) = minus_minus_rat(B,A) ).
% uminus_add_conv_diff
tff(fact_4799_div__minus1__right,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,A) ).
% div_minus1_right
tff(fact_4800_div__minus1__right,axiom,
! [A: code_integer] : divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).
% div_minus1_right
tff(fact_4801_divide__minus1,axiom,
! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,X) ).
% divide_minus1
tff(fact_4802_divide__minus1,axiom,
! [X: complex] : aa_complex_complex(divide1717551699836669952omplex(X),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,X) ).
% divide_minus1
tff(fact_4803_divide__minus1,axiom,
! [X: rat] : divide_divide_rat(X,uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(X) ).
% divide_minus1
tff(fact_4804_minus__mod__self1,axiom,
! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) ).
% minus_mod_self1
tff(fact_4805_minus__mod__self1,axiom,
! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(B,A),B) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).
% minus_mod_self1
tff(fact_4806_zero__less__of__bool__iff,axiom,
! [P: bool] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),zero_n3304061248610475627l_real(P)))
<=> pp(P) ) ).
% zero_less_of_bool_iff
tff(fact_4807_zero__less__of__bool__iff,axiom,
! [P: bool] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),zero_n2052037380579107095ol_rat(P)))
<=> pp(P) ) ).
% zero_less_of_bool_iff
tff(fact_4808_zero__less__of__bool__iff,axiom,
! [P: bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),zero_n2687167440665602831ol_nat(P)))
<=> pp(P) ) ).
% zero_less_of_bool_iff
tff(fact_4809_zero__less__of__bool__iff,axiom,
! [P: bool] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_bool_int(zero_n2684676970156552555ol_int,P)))
<=> pp(P) ) ).
% zero_less_of_bool_iff
tff(fact_4810_zero__less__of__bool__iff,axiom,
! [P: bool] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_n356916108424825756nteger(P)))
<=> pp(P) ) ).
% zero_less_of_bool_iff
tff(fact_4811_of__bool__less__one__iff,axiom,
! [P: bool] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_n3304061248610475627l_real(P)),one_one_real))
<=> ~ pp(P) ) ).
% of_bool_less_one_iff
tff(fact_4812_of__bool__less__one__iff,axiom,
! [P: bool] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_n2052037380579107095ol_rat(P)),one_one_rat))
<=> ~ pp(P) ) ).
% of_bool_less_one_iff
tff(fact_4813_of__bool__less__one__iff,axiom,
! [P: bool] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_n2687167440665602831ol_nat(P)),one_one_nat))
<=> ~ pp(P) ) ).
% of_bool_less_one_iff
tff(fact_4814_of__bool__less__one__iff,axiom,
! [P: bool] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_bool_int(zero_n2684676970156552555ol_int,P)),one_one_int))
<=> ~ pp(P) ) ).
% of_bool_less_one_iff
tff(fact_4815_of__bool__less__one__iff,axiom,
! [P: bool] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_n356916108424825756nteger(P)),one_one_Code_integer))
<=> ~ pp(P) ) ).
% of_bool_less_one_iff
tff(fact_4816_of__bool__not__iff,axiom,
! [P: bool] : zero_n1201886186963655149omplex(aa_bool_bool(fNot,P)) = aa_complex_complex(minus_minus_complex(one_one_complex),zero_n1201886186963655149omplex(P)) ).
% of_bool_not_iff
tff(fact_4817_of__bool__not__iff,axiom,
! [P: bool] : zero_n3304061248610475627l_real(aa_bool_bool(fNot,P)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),zero_n3304061248610475627l_real(P)) ).
% of_bool_not_iff
tff(fact_4818_of__bool__not__iff,axiom,
! [P: bool] : zero_n2052037380579107095ol_rat(aa_bool_bool(fNot,P)) = minus_minus_rat(one_one_rat,zero_n2052037380579107095ol_rat(P)) ).
% of_bool_not_iff
tff(fact_4819_of__bool__not__iff,axiom,
! [P: bool] : aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,P)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_bool_int(zero_n2684676970156552555ol_int,P)) ).
% of_bool_not_iff
tff(fact_4820_of__bool__not__iff,axiom,
! [P: bool] : zero_n356916108424825756nteger(aa_bool_bool(fNot,P)) = minus_8373710615458151222nteger(one_one_Code_integer,zero_n356916108424825756nteger(P)) ).
% of_bool_not_iff
tff(fact_4821_Suc__0__mod__eq,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),N) = zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),aa_nat_nat(suc,zero_zero_nat)))) ).
% Suc_0_mod_eq
tff(fact_4822_signed__take__bit__of__minus__1,axiom,
! [N: nat] : bit_ri6519982836138164636nteger(N,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% signed_take_bit_of_minus_1
tff(fact_4823_signed__take__bit__of__minus__1,axiom,
! [N: nat] : aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).
% signed_take_bit_of_minus_1
tff(fact_4824_Ball__set__replicate,axiom,
! [N: nat,A: int,P: fun_int_bool] :
( ! [X2: int] :
( pp(aa_set_int_bool(member_int(X2),set_int2(replicate_int(N,A))))
=> pp(aa_int_bool(P,X2)) )
<=> ( pp(aa_int_bool(P,A))
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
tff(fact_4825_Ball__set__replicate,axiom,
! [N: nat,A: nat,P: fun_nat_bool] :
( ! [X2: nat] :
( pp(aa_set_nat_bool(member_nat(X2),set_nat2(replicate_nat(N,A))))
=> pp(aa_nat_bool(P,X2)) )
<=> ( pp(aa_nat_bool(P,A))
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
tff(fact_4826_Ball__set__replicate,axiom,
! [N: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_bool] :
( ! [X2: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(replicate_VEBT_VEBT(N,A))))
=> pp(aa_VEBT_VEBT_bool(P,X2)) )
<=> ( pp(aa_VEBT_VEBT_bool(P,A))
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
tff(fact_4827_Bex__set__replicate,axiom,
! [N: nat,A: int,P: fun_int_bool] :
( ? [X2: int] :
( pp(aa_set_int_bool(member_int(X2),set_int2(replicate_int(N,A))))
& pp(aa_int_bool(P,X2)) )
<=> ( pp(aa_int_bool(P,A))
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
tff(fact_4828_Bex__set__replicate,axiom,
! [N: nat,A: nat,P: fun_nat_bool] :
( ? [X2: nat] :
( pp(aa_set_nat_bool(member_nat(X2),set_nat2(replicate_nat(N,A))))
& pp(aa_nat_bool(P,X2)) )
<=> ( pp(aa_nat_bool(P,A))
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
tff(fact_4829_Bex__set__replicate,axiom,
! [N: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_bool] :
( ? [X2: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(replicate_VEBT_VEBT(N,A))))
& pp(aa_VEBT_VEBT_bool(P,X2)) )
<=> ( pp(aa_VEBT_VEBT_bool(P,A))
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
tff(fact_4830_in__set__replicate,axiom,
! [X: product_prod_nat_nat,N: nat,Y: product_prod_nat_nat] :
( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(replic4235873036481779905at_nat(N,Y))))
<=> ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
tff(fact_4831_in__set__replicate,axiom,
! [X: complex,N: nat,Y: complex] :
( pp(aa_set_complex_bool(member_complex(X),set_complex2(replicate_complex(N,Y))))
<=> ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
tff(fact_4832_in__set__replicate,axiom,
! [X: real,N: nat,Y: real] :
( pp(aa_set_real_bool(member_real(X),set_real2(replicate_real(N,Y))))
<=> ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
tff(fact_4833_in__set__replicate,axiom,
! [X: int,N: nat,Y: int] :
( pp(aa_set_int_bool(member_int(X),set_int2(replicate_int(N,Y))))
<=> ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
tff(fact_4834_in__set__replicate,axiom,
! [X: nat,N: nat,Y: nat] :
( pp(aa_set_nat_bool(member_nat(X),set_nat2(replicate_nat(N,Y))))
<=> ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
tff(fact_4835_in__set__replicate,axiom,
! [X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),set_VEBT_VEBT2(replicate_VEBT_VEBT(N,Y))))
<=> ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
tff(fact_4836_pred__numeral__simps_I1_J,axiom,
aa_num_nat(pred_numeral,one) = zero_zero_nat ).
% pred_numeral_simps(1)
tff(fact_4837_eq__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,N) )
<=> ( aa_num_nat(pred_numeral,K) = N ) ) ).
% eq_numeral_Suc
tff(fact_4838_Suc__eq__numeral,axiom,
! [N: nat,K: num] :
( ( aa_nat_nat(suc,N) = aa_num_nat(numeral_numeral_nat,K) )
<=> ( N = aa_num_nat(pred_numeral,K) ) ) ).
% Suc_eq_numeral
tff(fact_4839_nth__replicate,axiom,
! [I: nat,N: nat,X: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
=> ( aa_nat_nat(nth_nat(replicate_nat(N,X)),I) = X ) ) ).
% nth_replicate
tff(fact_4840_nth__replicate,axiom,
! [I: nat,N: nat,X: vEBT_VEBT] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),N))
=> ( nth_VEBT_VEBT(replicate_VEBT_VEBT(N,X),I) = X ) ) ).
% nth_replicate
tff(fact_4841_dbl__simps_I1_J,axiom,
! [K: num] : aa_int_int(neg_numeral_dbl_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_numeral_dbl_int,aa_num_int(numeral_numeral_int,K))) ).
% dbl_simps(1)
tff(fact_4842_dbl__simps_I1_J,axiom,
! [K: num] : aa_real_real(neg_numeral_dbl_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_numeral_dbl_real,numeral_numeral_real(K))) ).
% dbl_simps(1)
tff(fact_4843_dbl__simps_I1_J,axiom,
! [K: num] : aa_complex_complex(neg_nu7009210354673126013omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu7009210354673126013omplex,numera6690914467698888265omplex(K))) ).
% dbl_simps(1)
tff(fact_4844_dbl__simps_I1_J,axiom,
! [K: num] : neg_nu8804712462038260780nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu8804712462038260780nteger(numera6620942414471956472nteger(K))) ).
% dbl_simps(1)
tff(fact_4845_dbl__simps_I1_J,axiom,
! [K: num] : neg_numeral_dbl_rat(uminus_uminus_rat(numeral_numeral_rat(K))) = uminus_uminus_rat(neg_numeral_dbl_rat(numeral_numeral_rat(K))) ).
% dbl_simps(1)
tff(fact_4846_dbl__inc__simps_I4_J,axiom,
aa_int_int(neg_nu5851722552734809277nc_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).
% dbl_inc_simps(4)
tff(fact_4847_dbl__inc__simps_I4_J,axiom,
aa_real_real(neg_nu8295874005876285629c_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,one_one_real) ).
% dbl_inc_simps(4)
tff(fact_4848_dbl__inc__simps_I4_J,axiom,
aa_complex_complex(neg_nu8557863876264182079omplex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).
% dbl_inc_simps(4)
tff(fact_4849_dbl__inc__simps_I4_J,axiom,
neg_nu5831290666863070958nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% dbl_inc_simps(4)
tff(fact_4850_dbl__inc__simps_I4_J,axiom,
neg_nu5219082963157363817nc_rat(uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(one_one_rat) ).
% dbl_inc_simps(4)
tff(fact_4851_triangle__Suc,axiom,
! [N: nat] : aa_nat_nat(nat_triangle,aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,N)),aa_nat_nat(suc,N)) ).
% triangle_Suc
tff(fact_4852_add__neg__numeral__special_I8_J,axiom,
aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = zero_zero_int ).
% add_neg_numeral_special(8)
tff(fact_4853_add__neg__numeral__special_I8_J,axiom,
aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) = zero_zero_real ).
% add_neg_numeral_special(8)
tff(fact_4854_add__neg__numeral__special_I8_J,axiom,
aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),one_one_complex) = zero_zero_complex ).
% add_neg_numeral_special(8)
tff(fact_4855_add__neg__numeral__special_I8_J,axiom,
plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) = zero_z3403309356797280102nteger ).
% add_neg_numeral_special(8)
tff(fact_4856_add__neg__numeral__special_I8_J,axiom,
plus_plus_rat(uminus_uminus_rat(one_one_rat),one_one_rat) = zero_zero_rat ).
% add_neg_numeral_special(8)
tff(fact_4857_add__neg__numeral__special_I7_J,axiom,
aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).
% add_neg_numeral_special(7)
tff(fact_4858_add__neg__numeral__special_I7_J,axiom,
aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) = zero_zero_real ).
% add_neg_numeral_special(7)
tff(fact_4859_add__neg__numeral__special_I7_J,axiom,
aa_complex_complex(plus_plus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = zero_zero_complex ).
% add_neg_numeral_special(7)
tff(fact_4860_add__neg__numeral__special_I7_J,axiom,
plus_p5714425477246183910nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).
% add_neg_numeral_special(7)
tff(fact_4861_add__neg__numeral__special_I7_J,axiom,
plus_plus_rat(one_one_rat,uminus_uminus_rat(one_one_rat)) = zero_zero_rat ).
% add_neg_numeral_special(7)
tff(fact_4862_diff__numeral__special_I12_J,axiom,
aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).
% diff_numeral_special(12)
tff(fact_4863_diff__numeral__special_I12_J,axiom,
aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = zero_zero_real ).
% diff_numeral_special(12)
tff(fact_4864_diff__numeral__special_I12_J,axiom,
aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = zero_zero_complex ).
% diff_numeral_special(12)
tff(fact_4865_diff__numeral__special_I12_J,axiom,
minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).
% diff_numeral_special(12)
tff(fact_4866_diff__numeral__special_I12_J,axiom,
minus_minus_rat(uminus_uminus_rat(one_one_rat),uminus_uminus_rat(one_one_rat)) = zero_zero_rat ).
% diff_numeral_special(12)
tff(fact_4867_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( aa_int_int(uminus_uminus_int,one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) )
<=> ( N = one ) ) ).
% neg_one_eq_numeral_iff
tff(fact_4868_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( aa_real_real(uminus_uminus_real,one_one_real) = aa_real_real(uminus_uminus_real,numeral_numeral_real(N)) )
<=> ( N = one ) ) ).
% neg_one_eq_numeral_iff
tff(fact_4869_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) )
<=> ( N = one ) ) ).
% neg_one_eq_numeral_iff
tff(fact_4870_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)) )
<=> ( N = one ) ) ).
% neg_one_eq_numeral_iff
tff(fact_4871_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( uminus_uminus_rat(one_one_rat) = uminus_uminus_rat(numeral_numeral_rat(N)) )
<=> ( N = one ) ) ).
% neg_one_eq_numeral_iff
tff(fact_4872_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,one_one_int) )
<=> ( N = one ) ) ).
% numeral_eq_neg_one_iff
tff(fact_4873_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(N)) = aa_real_real(uminus_uminus_real,one_one_real) )
<=> ( N = one ) ) ).
% numeral_eq_neg_one_iff
tff(fact_4874_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
<=> ( N = one ) ) ).
% numeral_eq_neg_one_iff
tff(fact_4875_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
<=> ( N = one ) ) ).
% numeral_eq_neg_one_iff
tff(fact_4876_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( uminus_uminus_rat(numeral_numeral_rat(N)) = uminus_uminus_rat(one_one_rat) )
<=> ( N = one ) ) ).
% numeral_eq_neg_one_iff
tff(fact_4877_minus__one__mult__self,axiom,
! [N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)) = one_one_int ).
% minus_one_mult_self
tff(fact_4878_minus__one__mult__self,axiom,
! [N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N)) = one_one_real ).
% minus_one_mult_self
tff(fact_4879_minus__one__mult__self,axiom,
! [N: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)) = one_one_complex ).
% minus_one_mult_self
tff(fact_4880_minus__one__mult__self,axiom,
! [N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)) = one_one_Code_integer ).
% minus_one_mult_self
tff(fact_4881_minus__one__mult__self,axiom,
! [N: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N)),aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N)) = one_one_rat ).
% minus_one_mult_self
tff(fact_4882_left__minus__one__mult__self,axiom,
! [N: nat,A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),A)) = A ).
% left_minus_one_mult_self
tff(fact_4883_left__minus__one__mult__self,axiom,
! [N: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N)),A)) = A ).
% left_minus_one_mult_self
tff(fact_4884_left__minus__one__mult__self,axiom,
! [N: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),A)) = A ).
% left_minus_one_mult_self
tff(fact_4885_left__minus__one__mult__self,axiom,
! [N: nat,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),A)) = A ).
% left_minus_one_mult_self
tff(fact_4886_left__minus__one__mult__self,axiom,
! [N: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N)),aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N)),A)) = A ).
% left_minus_one_mult_self
tff(fact_4887_mod__minus1__right,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).
% mod_minus1_right
tff(fact_4888_mod__minus1__right,axiom,
! [A: code_integer] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).
% mod_minus1_right
tff(fact_4889_max__number__of_I4_J,axiom,
! [U: num,V: num] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))))
=> ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(V)) ) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))))
=> ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(U)) ) ) ) ).
% max_number_of(4)
tff(fact_4890_max__number__of_I4_J,axiom,
! [U: num,V: num] :
( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))))
=> ( ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)) ) )
& ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))))
=> ( ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)) ) ) ) ).
% max_number_of(4)
tff(fact_4891_max__number__of_I4_J,axiom,
! [U: num,V: num] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(U))),uminus_uminus_rat(numeral_numeral_rat(V))))
=> ( ord_max_rat(uminus_uminus_rat(numeral_numeral_rat(U)),uminus_uminus_rat(numeral_numeral_rat(V))) = uminus_uminus_rat(numeral_numeral_rat(V)) ) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(U))),uminus_uminus_rat(numeral_numeral_rat(V))))
=> ( ord_max_rat(uminus_uminus_rat(numeral_numeral_rat(U)),uminus_uminus_rat(numeral_numeral_rat(V))) = uminus_uminus_rat(numeral_numeral_rat(U)) ) ) ) ).
% max_number_of(4)
tff(fact_4892_max__number__of_I4_J,axiom,
! [U: num,V: num] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U)) ) ) ) ).
% max_number_of(4)
tff(fact_4893_max__number__of_I3_J,axiom,
! [U: num,V: num] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),numeral_numeral_real(V)))
=> ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),numeral_numeral_real(V)) = numeral_numeral_real(V) ) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),numeral_numeral_real(V)))
=> ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),numeral_numeral_real(V)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(U)) ) ) ) ).
% max_number_of(3)
tff(fact_4894_max__number__of_I3_J,axiom,
! [U: num,V: num] :
( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),numera6620942414471956472nteger(V)))
=> ( ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)) = numera6620942414471956472nteger(V) ) )
& ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),numera6620942414471956472nteger(V)))
=> ( ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)) ) ) ) ).
% max_number_of(3)
tff(fact_4895_max__number__of_I3_J,axiom,
! [U: num,V: num] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(U))),numeral_numeral_rat(V)))
=> ( ord_max_rat(uminus_uminus_rat(numeral_numeral_rat(U)),numeral_numeral_rat(V)) = numeral_numeral_rat(V) ) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(U))),numeral_numeral_rat(V)))
=> ( ord_max_rat(uminus_uminus_rat(numeral_numeral_rat(U)),numeral_numeral_rat(V)) = uminus_uminus_rat(numeral_numeral_rat(U)) ) ) ) ).
% max_number_of(3)
tff(fact_4896_max__number__of_I3_J,axiom,
! [U: num,V: num] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)))
=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)) = aa_num_int(numeral_numeral_int,V) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)))
=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U)) ) ) ) ).
% max_number_of(3)
tff(fact_4897_max__number__of_I2_J,axiom,
! [U: num,V: num] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(U)),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))))
=> ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,numeral_numeral_real(U)),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(V)) ) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(U)),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))))
=> ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,numeral_numeral_real(U)),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = numeral_numeral_real(U) ) ) ) ).
% max_number_of(2)
tff(fact_4898_max__number__of_I2_J,axiom,
! [U: num,V: num] :
( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))))
=> ( ord_max_Code_integer(numera6620942414471956472nteger(U),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)) ) )
& ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))))
=> ( ord_max_Code_integer(numera6620942414471956472nteger(U),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = numera6620942414471956472nteger(U) ) ) ) ).
% max_number_of(2)
tff(fact_4899_max__number__of_I2_J,axiom,
! [U: num,V: num] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,numeral_numeral_rat(U)),uminus_uminus_rat(numeral_numeral_rat(V))))
=> ( ord_max_rat(numeral_numeral_rat(U),uminus_uminus_rat(numeral_numeral_rat(V))) = uminus_uminus_rat(numeral_numeral_rat(V)) ) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,numeral_numeral_rat(U)),uminus_uminus_rat(numeral_numeral_rat(V))))
=> ( ord_max_rat(numeral_numeral_rat(U),uminus_uminus_rat(numeral_numeral_rat(V))) = numeral_numeral_rat(U) ) ) ) ).
% max_number_of(2)
tff(fact_4900_max__number__of_I2_J,axiom,
! [U: num,V: num] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = aa_num_int(numeral_numeral_int,U) ) ) ) ).
% max_number_of(2)
tff(fact_4901_semiring__norm_I168_J,axiom,
! [V: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y) ).
% semiring_norm(168)
tff(fact_4902_semiring__norm_I168_J,axiom,
! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y) ).
% semiring_norm(168)
tff(fact_4903_semiring__norm_I168_J,axiom,
! [V: num,W: num,Y: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y) ).
% semiring_norm(168)
tff(fact_4904_semiring__norm_I168_J,axiom,
! [V: num,W: num,Y: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W)),Y)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Y) ).
% semiring_norm(168)
tff(fact_4905_semiring__norm_I168_J,axiom,
! [V: num,W: num,Y: rat] : plus_plus_rat(uminus_uminus_rat(numeral_numeral_rat(V)),plus_plus_rat(uminus_uminus_rat(numeral_numeral_rat(W)),Y)) = plus_plus_rat(uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Y) ).
% semiring_norm(168)
tff(fact_4906_diff__numeral__simps_I2_J,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).
% diff_numeral_simps(2)
tff(fact_4907_diff__numeral__simps_I2_J,axiom,
! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).
% diff_numeral_simps(2)
tff(fact_4908_diff__numeral__simps_I2_J,axiom,
! [M: num,N: num] : aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(M)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).
% diff_numeral_simps(2)
tff(fact_4909_diff__numeral__simps_I2_J,axiom,
! [M: num,N: num] : minus_8373710615458151222nteger(numera6620942414471956472nteger(M),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))) = numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).
% diff_numeral_simps(2)
tff(fact_4910_diff__numeral__simps_I2_J,axiom,
! [M: num,N: num] : minus_minus_rat(numeral_numeral_rat(M),uminus_uminus_rat(numeral_numeral_rat(N))) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).
% diff_numeral_simps(2)
tff(fact_4911_diff__numeral__simps_I3_J,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).
% diff_numeral_simps(3)
tff(fact_4912_diff__numeral__simps_I3_J,axiom,
! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),numeral_numeral_real(N)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).
% diff_numeral_simps(3)
tff(fact_4913_diff__numeral__simps_I3_J,axiom,
! [M: num,N: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),numera6690914467698888265omplex(N)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).
% diff_numeral_simps(3)
tff(fact_4914_diff__numeral__simps_I3_J,axiom,
! [M: num,N: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)),numera6620942414471956472nteger(N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).
% diff_numeral_simps(3)
tff(fact_4915_diff__numeral__simps_I3_J,axiom,
! [M: num,N: num] : minus_minus_rat(uminus_uminus_rat(numeral_numeral_rat(M)),numeral_numeral_rat(N)) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).
% diff_numeral_simps(3)
tff(fact_4916_semiring__norm_I172_J,axiom,
! [V: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).
% semiring_norm(172)
tff(fact_4917_semiring__norm_I172_J,axiom,
! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).
% semiring_norm(172)
tff(fact_4918_semiring__norm_I172_J,axiom,
! [V: num,W: num,Y: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).
% semiring_norm(172)
tff(fact_4919_semiring__norm_I172_J,axiom,
! [V: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W))),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).
% semiring_norm(172)
tff(fact_4920_semiring__norm_I172_J,axiom,
! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(V))),aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),Y)) = aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).
% semiring_norm(172)
tff(fact_4921_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).
% semiring_norm(171)
tff(fact_4922_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).
% semiring_norm(171)
tff(fact_4923_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).
% semiring_norm(171)
tff(fact_4924_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(V)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W))),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).
% semiring_norm(171)
tff(fact_4925_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),Y)) = aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).
% semiring_norm(171)
tff(fact_4926_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W)),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).
% semiring_norm(170)
tff(fact_4927_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).
% semiring_norm(170)
tff(fact_4928_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).
% semiring_norm(170)
tff(fact_4929_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(W)),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).
% semiring_norm(170)
tff(fact_4930_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(V))),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),Y)) = aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).
% semiring_norm(170)
tff(fact_4931_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).
% mult_neg_numeral_simps(3)
tff(fact_4932_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).
% mult_neg_numeral_simps(3)
tff(fact_4933_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(M)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).
% mult_neg_numeral_simps(3)
tff(fact_4934_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).
% mult_neg_numeral_simps(3)
tff(fact_4935_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] : aa_rat_rat(times_times_rat(numeral_numeral_rat(M)),uminus_uminus_rat(numeral_numeral_rat(N))) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).
% mult_neg_numeral_simps(3)
tff(fact_4936_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).
% mult_neg_numeral_simps(2)
tff(fact_4937_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),numeral_numeral_real(N)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).
% mult_neg_numeral_simps(2)
tff(fact_4938_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),numera6690914467698888265omplex(N)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).
% mult_neg_numeral_simps(2)
tff(fact_4939_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),numera6620942414471956472nteger(N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).
% mult_neg_numeral_simps(2)
tff(fact_4940_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] : aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(M))),numeral_numeral_rat(N)) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).
% mult_neg_numeral_simps(2)
tff(fact_4941_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_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,N))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).
% mult_neg_numeral_simps(1)
tff(fact_4942_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).
% mult_neg_numeral_simps(1)
tff(fact_4943_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).
% mult_neg_numeral_simps(1)
tff(fact_4944_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))) = numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).
% mult_neg_numeral_simps(1)
tff(fact_4945_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] : aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(M))),uminus_uminus_rat(numeral_numeral_rat(N))) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).
% mult_neg_numeral_simps(1)
tff(fact_4946_less__numeral__Suc,axiom,
! [K: num,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(pred_numeral,K)),N)) ) ).
% less_numeral_Suc
tff(fact_4947_less__Suc__numeral,axiom,
! [N: nat,K: num] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(pred_numeral,K))) ) ).
% less_Suc_numeral
tff(fact_4948_pred__numeral__simps_I3_J,axiom,
! [K: num] : aa_num_nat(pred_numeral,aa_num_num(bit1,K)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K)) ).
% pred_numeral_simps(3)
tff(fact_4949_le__Suc__numeral,axiom,
! [N: nat,K: num] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_num_nat(pred_numeral,K))) ) ).
% le_Suc_numeral
tff(fact_4950_le__numeral__Suc,axiom,
! [K: num,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(pred_numeral,K)),N)) ) ).
% le_numeral_Suc
tff(fact_4951_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),M)) ) ).
% neg_numeral_le_iff
tff(fact_4952_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),M)) ) ).
% neg_numeral_le_iff
tff(fact_4953_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(M))),uminus_uminus_rat(numeral_numeral_rat(N))))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),M)) ) ).
% neg_numeral_le_iff
tff(fact_4954_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(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,N))))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,N),M)) ) ).
% neg_numeral_le_iff
tff(fact_4955_diff__Suc__numeral,axiom,
! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_num_nat(pred_numeral,K)) ).
% diff_Suc_numeral
tff(fact_4956_diff__numeral__Suc,axiom,
! [K: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(pred_numeral,K)),N) ).
% diff_numeral_Suc
tff(fact_4957_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_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,N))))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N),M)) ) ).
% neg_numeral_less_iff
tff(fact_4958_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N),M)) ) ).
% neg_numeral_less_iff
tff(fact_4959_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N),M)) ) ).
% neg_numeral_less_iff
tff(fact_4960_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(M))),uminus_uminus_rat(numeral_numeral_rat(N))))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,N),M)) ) ).
% neg_numeral_less_iff
tff(fact_4961_max__Suc__numeral,axiom,
! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),aa_num_nat(pred_numeral,K))) ).
% max_Suc_numeral
tff(fact_4962_max__numeral__Suc,axiom,
! [K: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(pred_numeral,K)),N)) ).
% max_numeral_Suc
tff(fact_4963_pred__numeral__simps_I2_J,axiom,
! [K: num] : aa_num_nat(pred_numeral,aa_num_num(bit0,K)) = aa_num_nat(numeral_numeral_nat,bitM(K)) ).
% pred_numeral_simps(2)
tff(fact_4964_not__neg__one__le__neg__numeral__iff,axiom,
! [M: num] :
( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,numeral_numeral_real(M))))
<=> ( M != one ) ) ).
% not_neg_one_le_neg_numeral_iff
tff(fact_4965_not__neg__one__le__neg__numeral__iff,axiom,
! [M: num] :
( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))))
<=> ( M != one ) ) ).
% not_neg_one_le_neg_numeral_iff
tff(fact_4966_not__neg__one__le__neg__numeral__iff,axiom,
! [M: num] :
( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(one_one_rat)),uminus_uminus_rat(numeral_numeral_rat(M))))
<=> ( M != one ) ) ).
% not_neg_one_le_neg_numeral_iff
tff(fact_4967_not__neg__one__le__neg__numeral__iff,axiom,
! [M: num] :
( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))))
<=> ( M != one ) ) ).
% not_neg_one_le_neg_numeral_iff
tff(fact_4968_neg__numeral__less__neg__one__iff,axiom,
! [M: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int)))
<=> ( M != one ) ) ).
% neg_numeral_less_neg_one_iff
tff(fact_4969_neg__numeral__less__neg__one__iff,axiom,
! [M: num] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,one_one_real)))
<=> ( M != one ) ) ).
% neg_numeral_less_neg_one_iff
tff(fact_4970_neg__numeral__less__neg__one__iff,axiom,
! [M: num] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)))
<=> ( M != one ) ) ).
% neg_numeral_less_neg_one_iff
tff(fact_4971_neg__numeral__less__neg__one__iff,axiom,
! [M: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(M))),uminus_uminus_rat(one_one_rat)))
<=> ( M != one ) ) ).
% neg_numeral_less_neg_one_iff
tff(fact_4972_divide__le__eq__numeral1_I2_J,axiom,
! [B: real,W: num,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))),A))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))),B)) ) ).
% divide_le_eq_numeral1(2)
tff(fact_4973_divide__le__eq__numeral1_I2_J,axiom,
! [B: rat,W: num,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W)))),A))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),uminus_uminus_rat(numeral_numeral_rat(W)))),B)) ) ).
% divide_le_eq_numeral1(2)
tff(fact_4974_le__divide__eq__numeral1_I2_J,axiom,
! [A: real,B: real,W: num] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))))) ) ).
% le_divide_eq_numeral1(2)
tff(fact_4975_le__divide__eq__numeral1_I2_J,axiom,
! [A: rat,B: rat,W: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W)))))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),uminus_uminus_rat(numeral_numeral_rat(W))))) ) ).
% le_divide_eq_numeral1(2)
tff(fact_4976_divide__eq__eq__numeral1_I2_J,axiom,
! [B: real,W: num,A: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = A )
<=> ( ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) != zero_zero_real )
=> ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) ) )
& ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral1(2)
tff(fact_4977_divide__eq__eq__numeral1_I2_J,axiom,
! [B: complex,W: num,A: complex] :
( ( aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = A )
<=> ( ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) != zero_zero_complex )
=> ( B = aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) ) )
& ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% divide_eq_eq_numeral1(2)
tff(fact_4978_divide__eq__eq__numeral1_I2_J,axiom,
! [B: rat,W: num,A: rat] :
( ( divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W))) = A )
<=> ( ( ( uminus_uminus_rat(numeral_numeral_rat(W)) != zero_zero_rat )
=> ( B = aa_rat_rat(times_times_rat(A),uminus_uminus_rat(numeral_numeral_rat(W))) ) )
& ( ( uminus_uminus_rat(numeral_numeral_rat(W)) = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral1(2)
tff(fact_4979_eq__divide__eq__numeral1_I2_J,axiom,
! [A: real,B: real,W: num] :
( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) )
<=> ( ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = B ) )
& ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral1(2)
tff(fact_4980_eq__divide__eq__numeral1_I2_J,axiom,
! [A: complex,B: complex,W: num] :
( ( A = aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) )
<=> ( ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) != zero_zero_complex )
=> ( aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = B ) )
& ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% eq_divide_eq_numeral1(2)
tff(fact_4981_eq__divide__eq__numeral1_I2_J,axiom,
! [A: rat,B: rat,W: num] :
( ( A = divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W))) )
<=> ( ( ( uminus_uminus_rat(numeral_numeral_rat(W)) != zero_zero_rat )
=> ( aa_rat_rat(times_times_rat(A),uminus_uminus_rat(numeral_numeral_rat(W))) = B ) )
& ( ( uminus_uminus_rat(numeral_numeral_rat(W)) = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral1(2)
tff(fact_4982_divide__less__eq__numeral1_I2_J,axiom,
! [B: real,W: num,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))),A))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))),B)) ) ).
% divide_less_eq_numeral1(2)
tff(fact_4983_divide__less__eq__numeral1_I2_J,axiom,
! [B: rat,W: num,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W)))),A))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),uminus_uminus_rat(numeral_numeral_rat(W)))),B)) ) ).
% divide_less_eq_numeral1(2)
tff(fact_4984_less__divide__eq__numeral1_I2_J,axiom,
! [A: real,B: real,W: num] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))))) ) ).
% less_divide_eq_numeral1(2)
tff(fact_4985_less__divide__eq__numeral1_I2_J,axiom,
! [A: rat,B: rat,W: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),divide_divide_rat(B,uminus_uminus_rat(numeral_numeral_rat(W)))))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(A),uminus_uminus_rat(numeral_numeral_rat(W))))) ) ).
% less_divide_eq_numeral1(2)
tff(fact_4986_power2__minus,axiom,
! [A: int] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% power2_minus
tff(fact_4987_power2__minus,axiom,
! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% power2_minus
tff(fact_4988_power2__minus,axiom,
! [A: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% power2_minus
tff(fact_4989_power2__minus,axiom,
! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% power2_minus
tff(fact_4990_power2__minus,axiom,
! [A: rat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% power2_minus
tff(fact_4991_odd__of__bool__self,axiom,
! [P2: bool] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),zero_n2687167440665602831ol_nat(P2)))
<=> pp(P2) ) ).
% odd_of_bool_self
tff(fact_4992_odd__of__bool__self,axiom,
! [P2: bool] :
( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_bool_int(zero_n2684676970156552555ol_int,P2)))
<=> pp(P2) ) ).
% odd_of_bool_self
tff(fact_4993_odd__of__bool__self,axiom,
! [P2: bool] :
( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger(P2)))
<=> pp(P2) ) ).
% odd_of_bool_self
tff(fact_4994_add__neg__numeral__special_I9_J,axiom,
aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).
% add_neg_numeral_special(9)
tff(fact_4995_add__neg__numeral__special_I9_J,axiom,
aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit0,one))) ).
% add_neg_numeral_special(9)
tff(fact_4996_add__neg__numeral__special_I9_J,axiom,
aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit0,one))) ).
% add_neg_numeral_special(9)
tff(fact_4997_add__neg__numeral__special_I9_J,axiom,
plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).
% add_neg_numeral_special(9)
tff(fact_4998_add__neg__numeral__special_I9_J,axiom,
plus_plus_rat(uminus_uminus_rat(one_one_rat),uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(bit0,one))) ).
% add_neg_numeral_special(9)
tff(fact_4999_diff__numeral__special_I11_J,axiom,
aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).
% diff_numeral_special(11)
tff(fact_5000_diff__numeral__special_I11_J,axiom,
aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) = numeral_numeral_real(aa_num_num(bit0,one)) ).
% diff_numeral_special(11)
tff(fact_5001_diff__numeral__special_I11_J,axiom,
aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).
% diff_numeral_special(11)
tff(fact_5002_diff__numeral__special_I11_J,axiom,
minus_8373710615458151222nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = numera6620942414471956472nteger(aa_num_num(bit0,one)) ).
% diff_numeral_special(11)
tff(fact_5003_diff__numeral__special_I11_J,axiom,
minus_minus_rat(one_one_rat,uminus_uminus_rat(one_one_rat)) = numeral_numeral_rat(aa_num_num(bit0,one)) ).
% diff_numeral_special(11)
tff(fact_5004_diff__numeral__special_I10_J,axiom,
aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).
% diff_numeral_special(10)
tff(fact_5005_diff__numeral__special_I10_J,axiom,
aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit0,one))) ).
% diff_numeral_special(10)
tff(fact_5006_diff__numeral__special_I10_J,axiom,
aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit0,one))) ).
% diff_numeral_special(10)
tff(fact_5007_diff__numeral__special_I10_J,axiom,
minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).
% diff_numeral_special(10)
tff(fact_5008_diff__numeral__special_I10_J,axiom,
minus_minus_rat(uminus_uminus_rat(one_one_rat),one_one_rat) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(bit0,one))) ).
% diff_numeral_special(10)
tff(fact_5009_minus__1__div__2__eq,axiom,
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,aa_num_num(bit0,one))) = aa_int_int(uminus_uminus_int,one_one_int) ).
% minus_1_div_2_eq
tff(fact_5010_minus__1__div__2__eq,axiom,
divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% minus_1_div_2_eq
tff(fact_5011_minus__1__mod__2__eq,axiom,
aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).
% minus_1_mod_2_eq
tff(fact_5012_minus__1__mod__2__eq,axiom,
modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).
% minus_1_mod_2_eq
tff(fact_5013_bits__minus__1__mod__2__eq,axiom,
aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).
% bits_minus_1_mod_2_eq
tff(fact_5014_bits__minus__1__mod__2__eq,axiom,
modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).
% bits_minus_1_mod_2_eq
tff(fact_5015_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: int,N: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).
% Power.ring_1_class.power_minus_even
tff(fact_5016_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).
% Power.ring_1_class.power_minus_even
tff(fact_5017_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: complex,N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).
% Power.ring_1_class.power_minus_even
tff(fact_5018_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).
% Power.ring_1_class.power_minus_even
tff(fact_5019_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: rat,N: nat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).
% Power.ring_1_class.power_minus_even
tff(fact_5020_of__bool__half__eq__0,axiom,
! [B: bool] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_n2687167440665602831ol_nat(B)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).
% of_bool_half_eq_0
tff(fact_5021_of__bool__half__eq__0,axiom,
! [B: bool] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_bool_int(zero_n2684676970156552555ol_int,B)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).
% of_bool_half_eq_0
tff(fact_5022_of__bool__half__eq__0,axiom,
! [B: bool] : divide6298287555418463151nteger(zero_n356916108424825756nteger(B),numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).
% of_bool_half_eq_0
tff(fact_5023_power__minus__odd,axiom,
! [N: nat,A: int] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(A),N)) ) ) ).
% power_minus_odd
tff(fact_5024_power__minus__odd,axiom,
! [N: nat,A: real] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),N) = aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ) ) ).
% power_minus_odd
tff(fact_5025_power__minus__odd,axiom,
! [N: nat,A: complex] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(A),N)) ) ) ).
% power_minus_odd
tff(fact_5026_power__minus__odd,axiom,
! [N: nat,A: code_integer] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ) ).
% power_minus_odd
tff(fact_5027_power__minus__odd,axiom,
! [N: nat,A: rat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),N) = uminus_uminus_rat(aa_nat_rat(power_power_rat(A),N)) ) ) ).
% power_minus_odd
tff(fact_5028_Parity_Oring__1__class_Opower__minus__even,axiom,
! [N: nat,A: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N) = aa_nat_int(power_power_int(A),N) ) ) ).
% Parity.ring_1_class.power_minus_even
tff(fact_5029_Parity_Oring__1__class_Opower__minus__even,axiom,
! [N: nat,A: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) ) ) ).
% Parity.ring_1_class.power_minus_even
tff(fact_5030_Parity_Oring__1__class_Opower__minus__even,axiom,
! [N: nat,A: complex] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N) = aa_nat_complex(power_power_complex(A),N) ) ) ).
% Parity.ring_1_class.power_minus_even
tff(fact_5031_Parity_Oring__1__class_Opower__minus__even,axiom,
! [N: nat,A: code_integer] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N) = aa_nat_Code_integer(power_8256067586552552935nteger(A),N) ) ) ).
% Parity.ring_1_class.power_minus_even
tff(fact_5032_Parity_Oring__1__class_Opower__minus__even,axiom,
! [N: nat,A: rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),N) = aa_nat_rat(power_power_rat(A),N) ) ) ).
% Parity.ring_1_class.power_minus_even
tff(fact_5033_diff__numeral__special_I4_J,axiom,
! [M: num] : aa_int_int(aa_int_fun_int_int(minus_minus_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,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).
% diff_numeral_special(4)
tff(fact_5034_diff__numeral__special_I4_J,axiom,
! [M: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),one_one_real) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).
% diff_numeral_special(4)
tff(fact_5035_diff__numeral__special_I4_J,axiom,
! [M: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).
% diff_numeral_special(4)
tff(fact_5036_diff__numeral__special_I4_J,axiom,
! [M: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).
% diff_numeral_special(4)
tff(fact_5037_diff__numeral__special_I4_J,axiom,
! [M: num] : minus_minus_rat(uminus_uminus_rat(numeral_numeral_rat(M)),one_one_rat) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).
% diff_numeral_special(4)
tff(fact_5038_diff__numeral__special_I3_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).
% diff_numeral_special(3)
tff(fact_5039_diff__numeral__special_I3_J,axiom,
! [N: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).
% diff_numeral_special(3)
tff(fact_5040_diff__numeral__special_I3_J,axiom,
! [N: num] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).
% diff_numeral_special(3)
tff(fact_5041_diff__numeral__special_I3_J,axiom,
! [N: num] : minus_8373710615458151222nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))) = numera6620942414471956472nteger(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).
% diff_numeral_special(3)
tff(fact_5042_diff__numeral__special_I3_J,axiom,
! [N: num] : minus_minus_rat(one_one_rat,uminus_uminus_rat(numeral_numeral_rat(N))) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).
% diff_numeral_special(3)
tff(fact_5043_dbl__simps_I4_J,axiom,
aa_int_int(neg_numeral_dbl_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).
% dbl_simps(4)
tff(fact_5044_dbl__simps_I4_J,axiom,
aa_real_real(neg_numeral_dbl_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit0,one))) ).
% dbl_simps(4)
tff(fact_5045_dbl__simps_I4_J,axiom,
aa_complex_complex(neg_nu7009210354673126013omplex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit0,one))) ).
% dbl_simps(4)
tff(fact_5046_dbl__simps_I4_J,axiom,
neg_nu8804712462038260780nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).
% dbl_simps(4)
tff(fact_5047_dbl__simps_I4_J,axiom,
neg_numeral_dbl_rat(uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(bit0,one))) ).
% dbl_simps(4)
tff(fact_5048_power__minus1__even,axiom,
! [N: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = one_one_int ).
% power_minus1_even
tff(fact_5049_power__minus1__even,axiom,
! [N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = one_one_real ).
% power_minus1_even
tff(fact_5050_power__minus1__even,axiom,
! [N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = one_one_complex ).
% power_minus1_even
tff(fact_5051_power__minus1__even,axiom,
! [N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = one_one_Code_integer ).
% power_minus1_even
tff(fact_5052_power__minus1__even,axiom,
! [N: nat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = one_one_rat ).
% power_minus1_even
tff(fact_5053_neg__one__odd__power,axiom,
! [N: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).
% neg_one_odd_power
tff(fact_5054_neg__one__odd__power,axiom,
! [N: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).
% neg_one_odd_power
tff(fact_5055_neg__one__odd__power,axiom,
! [N: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ).
% neg_one_odd_power
tff(fact_5056_neg__one__odd__power,axiom,
! [N: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).
% neg_one_odd_power
tff(fact_5057_neg__one__odd__power,axiom,
! [N: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N) = uminus_uminus_rat(one_one_rat) ) ) ).
% neg_one_odd_power
tff(fact_5058_neg__one__even__power,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N) = one_one_int ) ) ).
% neg_one_even_power
tff(fact_5059_neg__one__even__power,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) = one_one_real ) ) ).
% neg_one_even_power
tff(fact_5060_neg__one__even__power,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N) = one_one_complex ) ) ).
% neg_one_even_power
tff(fact_5061_neg__one__even__power,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N) = one_one_Code_integer ) ) ).
% neg_one_even_power
tff(fact_5062_neg__one__even__power,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N) = one_one_rat ) ) ).
% neg_one_even_power
tff(fact_5063_signed__take__bit__0,axiom,
! [A: code_integer] : bit_ri6519982836138164636nteger(zero_zero_nat,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ).
% signed_take_bit_0
tff(fact_5064_signed__take__bit__0,axiom,
! [A: int] : aa_int_int(bit_ri631733984087533419it_int(zero_zero_nat),A) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).
% signed_take_bit_0
tff(fact_5065_signed__take__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).
% signed_take_bit_Suc_minus_bit0
tff(fact_5066_signed__take__bit__numeral__bit0,axiom,
! [L: num,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).
% signed_take_bit_numeral_bit0
tff(fact_5067_one__div__2__pow__eq,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).
% one_div_2_pow_eq
tff(fact_5068_one__div__2__pow__eq,axiom,
! [N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).
% one_div_2_pow_eq
tff(fact_5069_one__div__2__pow__eq,axiom,
! [N: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = zero_n356916108424825756nteger(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).
% one_div_2_pow_eq
tff(fact_5070_bits__1__div__exp,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).
% bits_1_div_exp
tff(fact_5071_bits__1__div__exp,axiom,
! [N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).
% bits_1_div_exp
tff(fact_5072_bits__1__div__exp,axiom,
! [N: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = zero_n356916108424825756nteger(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).
% bits_1_div_exp
tff(fact_5073_signed__take__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).
% signed_take_bit_numeral_minus_bit0
tff(fact_5074_one__mod__2__pow__eq,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).
% one_mod_2_pow_eq
tff(fact_5075_one__mod__2__pow__eq,axiom,
! [N: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).
% one_mod_2_pow_eq
tff(fact_5076_one__mod__2__pow__eq,axiom,
! [N: nat] : modulo364778990260209775nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = zero_n356916108424825756nteger(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).
% one_mod_2_pow_eq
tff(fact_5077_signed__take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_int_int(aa_int_fun_int_int(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,aa_num_num(bit0,one)))),one_one_int) ).
% signed_take_bit_numeral_minus_bit1
tff(fact_5078_dvd__antisym,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M))
=> ( M = N ) ) ) ).
% dvd_antisym
tff(fact_5079_signed__take__bit__minus,axiom,
! [N: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(uminus_uminus_int,aa_int_int(bit_ri631733984087533419it_int(N),K))) = aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(uminus_uminus_int,K)) ).
% signed_take_bit_minus
tff(fact_5080_of__bool__eq__iff,axiom,
! [P2: bool,Q2: bool] :
( ( zero_n2687167440665602831ol_nat(P2) = zero_n2687167440665602831ol_nat(Q2) )
<=> ( pp(P2)
<=> pp(Q2) ) ) ).
% of_bool_eq_iff
tff(fact_5081_of__bool__eq__iff,axiom,
! [P2: bool,Q2: bool] :
( ( aa_bool_int(zero_n2684676970156552555ol_int,P2) = aa_bool_int(zero_n2684676970156552555ol_int,Q2) )
<=> ( pp(P2)
<=> pp(Q2) ) ) ).
% of_bool_eq_iff
tff(fact_5082_of__bool__eq__iff,axiom,
! [P2: bool,Q2: bool] :
( ( zero_n356916108424825756nteger(P2) = zero_n356916108424825756nteger(Q2) )
<=> ( pp(P2)
<=> pp(Q2) ) ) ).
% of_bool_eq_iff
tff(fact_5083_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A = aa_int_int(uminus_uminus_int,B) )
<=> ( B = aa_int_int(uminus_uminus_int,A) ) ) ).
% equation_minus_iff
tff(fact_5084_equation__minus__iff,axiom,
! [A: real,B: real] :
( ( A = aa_real_real(uminus_uminus_real,B) )
<=> ( B = aa_real_real(uminus_uminus_real,A) ) ) ).
% equation_minus_iff
tff(fact_5085_equation__minus__iff,axiom,
! [A: complex,B: complex] :
( ( A = aa_complex_complex(uminus1482373934393186551omplex,B) )
<=> ( B = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ).
% equation_minus_iff
tff(fact_5086_equation__minus__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
<=> ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).
% equation_minus_iff
tff(fact_5087_equation__minus__iff,axiom,
! [A: rat,B: rat] :
( ( A = uminus_uminus_rat(B) )
<=> ( B = uminus_uminus_rat(A) ) ) ).
% equation_minus_iff
tff(fact_5088_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( aa_int_int(uminus_uminus_int,A) = B )
<=> ( aa_int_int(uminus_uminus_int,B) = A ) ) ).
% minus_equation_iff
tff(fact_5089_minus__equation__iff,axiom,
! [A: real,B: real] :
( ( aa_real_real(uminus_uminus_real,A) = B )
<=> ( aa_real_real(uminus_uminus_real,B) = A ) ) ).
% minus_equation_iff
tff(fact_5090_minus__equation__iff,axiom,
! [A: complex,B: complex] :
( ( aa_complex_complex(uminus1482373934393186551omplex,A) = B )
<=> ( aa_complex_complex(uminus1482373934393186551omplex,B) = A ) ) ).
% minus_equation_iff
tff(fact_5091_minus__equation__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B )
<=> ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) = A ) ) ).
% minus_equation_iff
tff(fact_5092_minus__equation__iff,axiom,
! [A: rat,B: rat] :
( ( uminus_uminus_rat(A) = B )
<=> ( uminus_uminus_rat(B) = A ) ) ).
% minus_equation_iff
tff(fact_5093_of__bool__conj,axiom,
! [P: bool,Q: bool] : zero_n3304061248610475627l_real(aa_bool_bool(fconj(P),Q)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,zero_n3304061248610475627l_real(P)),zero_n3304061248610475627l_real(Q)) ).
% of_bool_conj
tff(fact_5094_of__bool__conj,axiom,
! [P: bool,Q: bool] : zero_n2052037380579107095ol_rat(aa_bool_bool(fconj(P),Q)) = aa_rat_rat(times_times_rat(zero_n2052037380579107095ol_rat(P)),zero_n2052037380579107095ol_rat(Q)) ).
% of_bool_conj
tff(fact_5095_of__bool__conj,axiom,
! [P: bool,Q: bool] : zero_n2687167440665602831ol_nat(aa_bool_bool(fconj(P),Q)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat(P)),zero_n2687167440665602831ol_nat(Q)) ).
% of_bool_conj
tff(fact_5096_of__bool__conj,axiom,
! [P: bool,Q: bool] : aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(P),Q)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_bool_int(zero_n2684676970156552555ol_int,P)),aa_bool_int(zero_n2684676970156552555ol_int,Q)) ).
% of_bool_conj
tff(fact_5097_of__bool__conj,axiom,
! [P: bool,Q: bool] : zero_n356916108424825756nteger(aa_bool_bool(fconj(P),Q)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(P)),zero_n356916108424825756nteger(Q)) ).
% of_bool_conj
tff(fact_5098_mem__case__prodE,axiom,
! [Z: complex,C: fun_Co3393511490457469770omplex,P2: produc7417142470700560111r_bool] :
( pp(aa_set_complex_bool(member_complex(Z),produc4797875024747972146omplex(C,P2)))
=> ~ ! [X3: code_integer,Y2: bool] :
( ( P2 = produc5135875973608410785r_bool(X3,Y2) )
=> ~ pp(aa_set_complex_bool(member_complex(Z),aa_bool_set_complex(aa_Cod7104299617959582899omplex(C,X3),Y2))) ) ) ).
% mem_case_prodE
tff(fact_5099_mem__case__prodE,axiom,
! [Z: real,C: fun_Co2315910201472382152t_real,P2: produc7417142470700560111r_bool] :
( pp(aa_set_real_bool(member_real(Z),produc4532933866054709936t_real(C,P2)))
=> ~ ! [X3: code_integer,Y2: bool] :
( ( P2 = produc5135875973608410785r_bool(X3,Y2) )
=> ~ pp(aa_set_real_bool(member_real(Z),aa_bool_set_real(aa_Cod7081859000793828145t_real(C,X3),Y2))) ) ) ).
% mem_case_prodE
tff(fact_5100_mem__case__prodE,axiom,
! [Z: nat,C: fun_Co3833597871255845228et_nat,P2: produc7417142470700560111r_bool] :
( pp(aa_set_nat_bool(member_nat(Z),produc9066999194162305620et_nat(C,P2)))
=> ~ ! [X3: code_integer,Y2: bool] :
( ( P2 = produc5135875973608410785r_bool(X3,Y2) )
=> ~ pp(aa_set_nat_bool(member_nat(Z),aa_bool_set_nat(aa_Cod5518294568803216341et_nat(C,X3),Y2))) ) ) ).
% mem_case_prodE
tff(fact_5101_mem__case__prodE,axiom,
! [Z: int,C: fun_Co8243421489322702280et_int,P2: produc7417142470700560111r_bool] :
( pp(aa_set_int_bool(member_int(Z),produc4889148174653108912et_int(C,P2)))
=> ~ ! [X3: code_integer,Y2: bool] :
( ( P2 = produc5135875973608410785r_bool(X3,Y2) )
=> ~ pp(aa_set_int_bool(member_int(Z),aa_bool_set_int(aa_Cod3028211423871489585et_int(C,X3),Y2))) ) ) ).
% mem_case_prodE
tff(fact_5102_mem__case__prodE,axiom,
! [Z: complex,C: fun_nu9041852840018610706omplex,P2: product_prod_num_num] :
( pp(aa_set_complex_bool(member_complex(Z),produc2866383454006189126omplex(C,P2)))
=> ~ ! [X3: num,Y2: num] :
( ( P2 = product_Pair_num_num(X3,Y2) )
=> ~ pp(aa_set_complex_bool(member_complex(Z),aa_num_set_complex(aa_num241437687621652691omplex(C,X3),Y2))) ) ) ).
% mem_case_prodE
tff(fact_5103_mem__case__prodE,axiom,
! [Z: real,C: fun_nu4698968662094489488t_real,P2: product_prod_num_num] :
( pp(aa_set_real_bool(member_real(Z),produc8296048397933160132t_real(C,P2)))
=> ~ ! [X3: num,Y2: num] :
( ( P2 = product_Pair_num_num(X3,Y2) )
=> ~ pp(aa_set_real_bool(member_real(Z),aa_num_set_real(aa_num2469771427700027217t_real(C,X3),Y2))) ) ) ).
% mem_case_prodE
tff(fact_5104_mem__case__prodE,axiom,
! [Z: nat,C: fun_nu530747996811530292et_nat,P2: product_prod_num_num] :
( pp(aa_set_nat_bool(member_nat(Z),produc1361121860356118632et_nat(C,P2)))
=> ~ ! [X3: num,Y2: num] :
( ( P2 = product_Pair_num_num(X3,Y2) )
=> ~ pp(aa_set_nat_bool(member_nat(Z),aa_num_set_nat(aa_num5617506412915843061et_nat(C,X3),Y2))) ) ) ).
% mem_case_prodE
tff(fact_5105_mem__case__prodE,axiom,
! [Z: int,C: fun_nu4940571614878387344et_int,P2: product_prod_num_num] :
( pp(aa_set_int_bool(member_int(Z),produc6406642877701697732et_int(C,P2)))
=> ~ ! [X3: num,Y2: num] :
( ( P2 = product_Pair_num_num(X3,Y2) )
=> ~ pp(aa_set_int_bool(member_int(Z),aa_num_set_int(aa_num3127423267984116305et_int(C,X3),Y2))) ) ) ).
% mem_case_prodE
tff(fact_5106_mem__case__prodE,axiom,
! [Z: complex,C: fun_na3925014923085203400omplex,P2: product_prod_nat_num] :
( pp(aa_set_complex_bool(member_complex(Z),produc6231982587499038204omplex(C,P2)))
=> ~ ! [X3: nat,Y2: num] :
( ( P2 = product_Pair_nat_num(X3,Y2) )
=> ~ pp(aa_set_complex_bool(member_complex(Z),aa_num_set_complex(aa_nat4347971807543021193omplex(C,X3),Y2))) ) ) ).
% mem_case_prodE
tff(fact_5107_mem__case__prodE,axiom,
! [Z: real,C: fun_na3666575854997362246t_real,P2: product_prod_nat_num] :
( pp(aa_set_real_bool(member_real(Z),produc1435849484188172666t_real(C,P2)))
=> ~ ! [X3: nat,Y2: num] :
( ( P2 = product_Pair_nat_num(X3,Y2) )
=> ~ pp(aa_set_real_bool(member_real(Z),aa_num_set_real(aa_nat1437378620602899975t_real(C,X3),Y2))) ) ) ).
% mem_case_prodE
tff(fact_5108_le__minus__iff,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,B)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(uminus_uminus_real,A))) ) ).
% le_minus_iff
tff(fact_5109_le__minus__iff,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))) ) ).
% le_minus_iff
tff(fact_5110_le__minus__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),uminus_uminus_rat(B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),uminus_uminus_rat(A))) ) ).
% le_minus_iff
tff(fact_5111_le__minus__iff,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),aa_int_int(uminus_uminus_int,A))) ) ).
% le_minus_iff
tff(fact_5112_minus__le__iff,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),A)) ) ).
% minus_le_iff
tff(fact_5113_minus__le__iff,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A)) ) ).
% minus_le_iff
tff(fact_5114_minus__le__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(A)),B))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(B)),A)) ) ).
% minus_le_iff
tff(fact_5115_minus__le__iff,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),A)) ) ).
% minus_le_iff
tff(fact_5116_le__imp__neg__le,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))) ) ).
% le_imp_neg_le
tff(fact_5117_le__imp__neg__le,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))) ) ).
% le_imp_neg_le
tff(fact_5118_le__imp__neg__le,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(B)),uminus_uminus_rat(A))) ) ).
% le_imp_neg_le
tff(fact_5119_le__imp__neg__le,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))) ) ).
% le_imp_neg_le
tff(fact_5120_verit__negate__coefficient_I2_J,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))) ) ).
% verit_negate_coefficient(2)
tff(fact_5121_verit__negate__coefficient_I2_J,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))) ) ).
% verit_negate_coefficient(2)
tff(fact_5122_verit__negate__coefficient_I2_J,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))) ) ).
% verit_negate_coefficient(2)
tff(fact_5123_verit__negate__coefficient_I2_J,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(B)),uminus_uminus_rat(A))) ) ).
% verit_negate_coefficient(2)
tff(fact_5124_minus__less__iff,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),B))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,B)),A)) ) ).
% minus_less_iff
tff(fact_5125_minus__less__iff,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A)),B))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),A)) ) ).
% minus_less_iff
tff(fact_5126_minus__less__iff,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A)) ) ).
% minus_less_iff
tff(fact_5127_minus__less__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(A)),B))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(B)),A)) ) ).
% minus_less_iff
tff(fact_5128_less__minus__iff,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_int_int(uminus_uminus_int,B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),aa_int_int(uminus_uminus_int,A))) ) ).
% less_minus_iff
tff(fact_5129_less__minus__iff,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,B)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(uminus_uminus_real,A))) ) ).
% less_minus_iff
tff(fact_5130_less__minus__iff,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))) ) ).
% less_minus_iff
tff(fact_5131_less__minus__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),uminus_uminus_rat(B)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),uminus_uminus_rat(A))) ) ).
% less_minus_iff
tff(fact_5132_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] : aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)) != aa_num_int(numeral_numeral_int,N) ).
% neg_numeral_neq_numeral
tff(fact_5133_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] : aa_real_real(uminus_uminus_real,numeral_numeral_real(M)) != numeral_numeral_real(N) ).
% neg_numeral_neq_numeral
tff(fact_5134_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] : aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M)) != numera6690914467698888265omplex(N) ).
% neg_numeral_neq_numeral
tff(fact_5135_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)) != numera6620942414471956472nteger(N) ).
% neg_numeral_neq_numeral
tff(fact_5136_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] : uminus_uminus_rat(numeral_numeral_rat(M)) != numeral_numeral_rat(N) ).
% neg_numeral_neq_numeral
tff(fact_5137_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] : aa_num_int(numeral_numeral_int,M) != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).
% numeral_neq_neg_numeral
tff(fact_5138_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] : numeral_numeral_real(M) != aa_real_real(uminus_uminus_real,numeral_numeral_real(N)) ).
% numeral_neq_neg_numeral
tff(fact_5139_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] : numera6690914467698888265omplex(M) != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) ).
% numeral_neq_neg_numeral
tff(fact_5140_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] : numera6620942414471956472nteger(M) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)) ).
% numeral_neq_neg_numeral
tff(fact_5141_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] : numeral_numeral_rat(M) != uminus_uminus_rat(numeral_numeral_rat(N)) ).
% numeral_neq_neg_numeral
tff(fact_5142_square__eq__iff,axiom,
! [A: int,B: int] :
( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),A) = aa_int_int(aa_int_fun_int_int(times_times_int,B),B) )
<=> ( ( A = B )
| ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ).
% square_eq_iff
tff(fact_5143_square__eq__iff,axiom,
! [A: real,B: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),B) )
<=> ( ( A = B )
| ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ).
% square_eq_iff
tff(fact_5144_square__eq__iff,axiom,
! [A: complex,B: complex] :
( ( aa_complex_complex(times_times_complex(A),A) = aa_complex_complex(times_times_complex(B),B) )
<=> ( ( A = B )
| ( A = aa_complex_complex(uminus1482373934393186551omplex,B) ) ) ) ).
% square_eq_iff
tff(fact_5145_square__eq__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),B) )
<=> ( ( A = B )
| ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ).
% square_eq_iff
tff(fact_5146_square__eq__iff,axiom,
! [A: rat,B: rat] :
( ( aa_rat_rat(times_times_rat(A),A) = aa_rat_rat(times_times_rat(B),B) )
<=> ( ( A = B )
| ( A = uminus_uminus_rat(B) ) ) ) ).
% square_eq_iff
tff(fact_5147_minus__mult__commute,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(uminus_uminus_int,B)) ).
% minus_mult_commute
tff(fact_5148_minus__mult__commute,axiom,
! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,B)) ).
% minus_mult_commute
tff(fact_5149_minus__mult__commute,axiom,
! [A: complex,B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).
% minus_mult_commute
tff(fact_5150_minus__mult__commute,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).
% minus_mult_commute
tff(fact_5151_minus__mult__commute,axiom,
! [A: rat,B: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(A)),B) = aa_rat_rat(times_times_rat(A),uminus_uminus_rat(B)) ).
% minus_mult_commute
tff(fact_5152_one__neq__neg__one,axiom,
one_one_int != aa_int_int(uminus_uminus_int,one_one_int) ).
% one_neq_neg_one
tff(fact_5153_one__neq__neg__one,axiom,
one_one_real != aa_real_real(uminus_uminus_real,one_one_real) ).
% one_neq_neg_one
tff(fact_5154_one__neq__neg__one,axiom,
one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).
% one_neq_neg_one
tff(fact_5155_one__neq__neg__one,axiom,
one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% one_neq_neg_one
tff(fact_5156_one__neq__neg__one,axiom,
one_one_rat != uminus_uminus_rat(one_one_rat) ).
% one_neq_neg_one
tff(fact_5157_group__cancel_Oneg1,axiom,
! [A2: int,K: int,A: int] :
( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
=> ( aa_int_int(uminus_uminus_int,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K)),aa_int_int(uminus_uminus_int,A)) ) ) ).
% group_cancel.neg1
tff(fact_5158_group__cancel_Oneg1,axiom,
! [A2: real,K: real,A: real] :
( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
=> ( aa_real_real(uminus_uminus_real,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K)),aa_real_real(uminus_uminus_real,A)) ) ) ).
% group_cancel.neg1
tff(fact_5159_group__cancel_Oneg1,axiom,
! [A2: complex,K: complex,A: complex] :
( ( A2 = aa_complex_complex(plus_plus_complex(K),A) )
=> ( aa_complex_complex(uminus1482373934393186551omplex,A2) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(uminus1482373934393186551omplex,A)) ) ) ).
% group_cancel.neg1
tff(fact_5160_group__cancel_Oneg1,axiom,
! [A2: code_integer,K: code_integer,A: code_integer] :
( ( A2 = plus_p5714425477246183910nteger(K,A) )
=> ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ) ).
% group_cancel.neg1
tff(fact_5161_group__cancel_Oneg1,axiom,
! [A2: rat,K: rat,A: rat] :
( ( A2 = plus_plus_rat(K,A) )
=> ( uminus_uminus_rat(A2) = plus_plus_rat(uminus_uminus_rat(K),uminus_uminus_rat(A)) ) ) ).
% group_cancel.neg1
tff(fact_5162_add_Oinverse__distrib__swap,axiom,
! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ).
% add.inverse_distrib_swap
tff(fact_5163_add_Oinverse__distrib__swap,axiom,
! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ).
% add.inverse_distrib_swap
tff(fact_5164_add_Oinverse__distrib__swap,axiom,
! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,B)),aa_complex_complex(uminus1482373934393186551omplex,A)) ).
% add.inverse_distrib_swap
tff(fact_5165_add_Oinverse__distrib__swap,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ).
% add.inverse_distrib_swap
tff(fact_5166_add_Oinverse__distrib__swap,axiom,
! [A: rat,B: rat] : uminus_uminus_rat(plus_plus_rat(A,B)) = plus_plus_rat(uminus_uminus_rat(B),uminus_uminus_rat(A)) ).
% add.inverse_distrib_swap
tff(fact_5167_is__num__normalize_I8_J,axiom,
! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ).
% is_num_normalize(8)
tff(fact_5168_is__num__normalize_I8_J,axiom,
! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ).
% is_num_normalize(8)
tff(fact_5169_is__num__normalize_I8_J,axiom,
! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,B)),aa_complex_complex(uminus1482373934393186551omplex,A)) ).
% is_num_normalize(8)
tff(fact_5170_is__num__normalize_I8_J,axiom,
! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ).
% is_num_normalize(8)
tff(fact_5171_is__num__normalize_I8_J,axiom,
! [A: rat,B: rat] : uminus_uminus_rat(plus_plus_rat(A,B)) = plus_plus_rat(uminus_uminus_rat(B),uminus_uminus_rat(A)) ).
% is_num_normalize(8)
tff(fact_5172_minus__diff__minus,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ).
% minus_diff_minus
tff(fact_5173_minus__diff__minus,axiom,
! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ).
% minus_diff_minus
tff(fact_5174_minus__diff__minus,axiom,
! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(minus_minus_complex(A),B)) ).
% minus_diff_minus
tff(fact_5175_minus__diff__minus,axiom,
! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,minus_8373710615458151222nteger(A,B)) ).
% minus_diff_minus
tff(fact_5176_minus__diff__minus,axiom,
! [A: rat,B: rat] : minus_minus_rat(uminus_uminus_rat(A),uminus_uminus_rat(B)) = uminus_uminus_rat(minus_minus_rat(A,B)) ).
% minus_diff_minus
tff(fact_5177_minus__diff__commute,axiom,
! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,B)),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,A)),B) ).
% minus_diff_commute
tff(fact_5178_minus__diff__commute,axiom,
! [B: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,B)),A) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A)),B) ).
% minus_diff_commute
tff(fact_5179_minus__diff__commute,axiom,
! [B: complex,A: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,B)),A) = aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) ).
% minus_diff_commute
tff(fact_5180_minus__diff__commute,axiom,
! [B: code_integer,A: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B),A) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).
% minus_diff_commute
tff(fact_5181_minus__diff__commute,axiom,
! [B: rat,A: rat] : minus_minus_rat(uminus_uminus_rat(B),A) = minus_minus_rat(uminus_uminus_rat(A),B) ).
% minus_diff_commute
tff(fact_5182_div__minus__right,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) ).
% div_minus_right
tff(fact_5183_div__minus__right,axiom,
! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).
% div_minus_right
tff(fact_5184_minus__divide__left,axiom,
! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),B) ).
% minus_divide_left
tff(fact_5185_minus__divide__left,axiom,
! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) ).
% minus_divide_left
tff(fact_5186_minus__divide__left,axiom,
! [A: rat,B: rat] : uminus_uminus_rat(divide_divide_rat(A,B)) = divide_divide_rat(uminus_uminus_rat(A),B) ).
% minus_divide_left
tff(fact_5187_minus__divide__divide,axiom,
! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ).
% minus_divide_divide
tff(fact_5188_minus__divide__divide,axiom,
! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ).
% minus_divide_divide
tff(fact_5189_minus__divide__divide,axiom,
! [A: rat,B: rat] : divide_divide_rat(uminus_uminus_rat(A),uminus_uminus_rat(B)) = divide_divide_rat(A,B) ).
% minus_divide_divide
tff(fact_5190_minus__divide__right,axiom,
! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(uminus_uminus_real,B)) ).
% minus_divide_right
tff(fact_5191_minus__divide__right,axiom,
! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).
% minus_divide_right
tff(fact_5192_minus__divide__right,axiom,
! [A: rat,B: rat] : uminus_uminus_rat(divide_divide_rat(A,B)) = divide_divide_rat(A,uminus_uminus_rat(B)) ).
% minus_divide_right
tff(fact_5193_mod__minus__eq,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) ).
% mod_minus_eq
tff(fact_5194_mod__minus__eq,axiom,
! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,B)),B) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).
% mod_minus_eq
tff(fact_5195_mod__minus__cong,axiom,
! [A: int,B: int,A3: int] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A3),B) )
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A3)),B) ) ) ).
% mod_minus_cong
tff(fact_5196_mod__minus__cong,axiom,
! [A: code_integer,B: code_integer,A3: code_integer] :
( ( modulo364778990260209775nteger(A,B) = modulo364778990260209775nteger(A3,B) )
=> ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A3),B) ) ) ).
% mod_minus_cong
tff(fact_5197_mod__minus__right,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B)) ).
% mod_minus_right
tff(fact_5198_mod__minus__right,axiom,
! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B)) ).
% mod_minus_right
tff(fact_5199_case__prodE,axiom,
! [C: fun_Co3748208068438171869l_bool,P2: produc7417142470700560111r_bool] :
( pp(aa_Pro6456355659118895914l_bool(produc2506095642196730057l_bool(C),P2))
=> ~ ! [X3: code_integer,Y2: bool] :
( ( P2 = produc5135875973608410785r_bool(X3,Y2) )
=> ~ pp(aa_bool_bool(aa_Cod8494480694608518558l_bool(C,X3),Y2)) ) ) ).
% case_prodE
tff(fact_5200_case__prodE,axiom,
! [C: fun_num_fun_num_bool,P2: product_prod_num_num] :
( pp(aa_Pro762926636801165566m_bool(produc8237051459014760373m_bool(C),P2))
=> ~ ! [X3: num,Y2: num] :
( ( P2 = product_Pair_num_num(X3,Y2) )
=> ~ pp(aa_num_bool(aa_num_fun_num_bool(C,X3),Y2)) ) ) ).
% case_prodE
tff(fact_5201_case__prodE,axiom,
! [C: fun_nat_fun_num_bool,P2: product_prod_nat_num] :
( pp(aa_Pro5426739851356932296m_bool(produc3677492636715751295m_bool(C),P2))
=> ~ ! [X3: nat,Y2: num] :
( ( P2 = product_Pair_nat_num(X3,Y2) )
=> ~ pp(aa_num_bool(aa_nat_fun_num_bool(C,X3),Y2)) ) ) ).
% case_prodE
tff(fact_5202_case__prodE,axiom,
! [C: fun_nat_fun_nat_bool,P2: product_prod_nat_nat] :
( pp(aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(C),P2))
=> ~ ! [X3: nat,Y2: nat] :
( ( P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y2) )
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(C,X3),Y2)) ) ) ).
% case_prodE
tff(fact_5203_case__prodE,axiom,
! [C: fun_int_fun_int_bool,P2: product_prod_int_int] :
( pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(C),P2))
=> ~ ! [X3: int,Y2: int] :
( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y2) )
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(C,X3),Y2)) ) ) ).
% case_prodE
tff(fact_5204_case__prodD,axiom,
! [F: fun_Co3748208068438171869l_bool,A: code_integer,B: bool] :
( pp(aa_Pro6456355659118895914l_bool(produc2506095642196730057l_bool(F),produc5135875973608410785r_bool(A,B)))
=> pp(aa_bool_bool(aa_Cod8494480694608518558l_bool(F,A),B)) ) ).
% case_prodD
tff(fact_5205_case__prodD,axiom,
! [F: fun_num_fun_num_bool,A: num,B: num] :
( pp(aa_Pro762926636801165566m_bool(produc8237051459014760373m_bool(F),product_Pair_num_num(A,B)))
=> pp(aa_num_bool(aa_num_fun_num_bool(F,A),B)) ) ).
% case_prodD
tff(fact_5206_case__prodD,axiom,
! [F: fun_nat_fun_num_bool,A: nat,B: num] :
( pp(aa_Pro5426739851356932296m_bool(produc3677492636715751295m_bool(F),product_Pair_nat_num(A,B)))
=> pp(aa_num_bool(aa_nat_fun_num_bool(F,A),B)) ) ).
% case_prodD
tff(fact_5207_case__prodD,axiom,
! [F: fun_nat_fun_nat_bool,A: nat,B: nat] :
( pp(aa_Pro2232559814647311250t_bool(produc8955392012077580105t_bool(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(F,A),B)) ) ).
% case_prodD
tff(fact_5208_case__prodD,axiom,
! [F: fun_int_fun_int_bool,A: int,B: int] :
( pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)))
=> pp(aa_int_bool(aa_int_fun_int_bool(F,A),B)) ) ).
% case_prodD
tff(fact_5209_case__prodE_H,axiom,
! [C: fun_na4786964484885125043t_bool,P2: product_prod_nat_nat,Z: product_prod_nat_nat] :
( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(C),P2),Z))
=> ~ ! [X3: nat,Y2: nat] :
( ( P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y2) )
=> ~ pp(aa_Pro2232559814647311250t_bool(aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(C,X3),Y2),Z)) ) ) ).
% case_prodE'
tff(fact_5210_case__prodD_H,axiom,
! [R2: fun_na4786964484885125043t_bool,A: nat,B: nat,C: product_prod_nat_nat] :
( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(R2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),C))
=> pp(aa_Pro2232559814647311250t_bool(aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(R2,A),B),C)) ) ).
% case_prodD'
tff(fact_5211_zero__less__eq__of__bool,axiom,
! [P: bool] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),zero_n3304061248610475627l_real(P))) ).
% zero_less_eq_of_bool
tff(fact_5212_zero__less__eq__of__bool,axiom,
! [P: bool] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),zero_n2052037380579107095ol_rat(P))) ).
% zero_less_eq_of_bool
tff(fact_5213_zero__less__eq__of__bool,axiom,
! [P: bool] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),zero_n2687167440665602831ol_nat(P))) ).
% zero_less_eq_of_bool
tff(fact_5214_zero__less__eq__of__bool,axiom,
! [P: bool] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_bool_int(zero_n2684676970156552555ol_int,P))) ).
% zero_less_eq_of_bool
tff(fact_5215_zero__less__eq__of__bool,axiom,
! [P: bool] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),zero_n356916108424825756nteger(P))) ).
% zero_less_eq_of_bool
tff(fact_5216_of__bool__less__eq__one,axiom,
! [P: bool] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_n3304061248610475627l_real(P)),one_one_real)) ).
% of_bool_less_eq_one
tff(fact_5217_of__bool__less__eq__one,axiom,
! [P: bool] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_n2052037380579107095ol_rat(P)),one_one_rat)) ).
% of_bool_less_eq_one
tff(fact_5218_of__bool__less__eq__one,axiom,
! [P: bool] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_n2687167440665602831ol_nat(P)),one_one_nat)) ).
% of_bool_less_eq_one
tff(fact_5219_of__bool__less__eq__one,axiom,
! [P: bool] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_bool_int(zero_n2684676970156552555ol_int,P)),one_one_int)) ).
% of_bool_less_eq_one
tff(fact_5220_of__bool__less__eq__one,axiom,
! [P: bool] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_n356916108424825756nteger(P)),one_one_Code_integer)) ).
% of_bool_less_eq_one
tff(fact_5221_of__bool__def,axiom,
! [P2: bool] :
( ( pp(P2)
=> ( zero_n1201886186963655149omplex(P2) = one_one_complex ) )
& ( ~ pp(P2)
=> ( zero_n1201886186963655149omplex(P2) = zero_zero_complex ) ) ) ).
% of_bool_def
tff(fact_5222_of__bool__def,axiom,
! [P2: bool] :
( ( pp(P2)
=> ( zero_n3304061248610475627l_real(P2) = one_one_real ) )
& ( ~ pp(P2)
=> ( zero_n3304061248610475627l_real(P2) = zero_zero_real ) ) ) ).
% of_bool_def
tff(fact_5223_of__bool__def,axiom,
! [P2: bool] :
( ( pp(P2)
=> ( zero_n2052037380579107095ol_rat(P2) = one_one_rat ) )
& ( ~ pp(P2)
=> ( zero_n2052037380579107095ol_rat(P2) = zero_zero_rat ) ) ) ).
% of_bool_def
tff(fact_5224_of__bool__def,axiom,
! [P2: bool] :
( ( pp(P2)
=> ( zero_n2687167440665602831ol_nat(P2) = one_one_nat ) )
& ( ~ pp(P2)
=> ( zero_n2687167440665602831ol_nat(P2) = zero_zero_nat ) ) ) ).
% of_bool_def
tff(fact_5225_of__bool__def,axiom,
! [P2: bool] :
( ( pp(P2)
=> ( aa_bool_int(zero_n2684676970156552555ol_int,P2) = one_one_int ) )
& ( ~ pp(P2)
=> ( aa_bool_int(zero_n2684676970156552555ol_int,P2) = zero_zero_int ) ) ) ).
% of_bool_def
tff(fact_5226_of__bool__def,axiom,
! [P2: bool] :
( ( pp(P2)
=> ( zero_n356916108424825756nteger(P2) = one_one_Code_integer ) )
& ( ~ pp(P2)
=> ( zero_n356916108424825756nteger(P2) = zero_z3403309356797280102nteger ) ) ) ).
% of_bool_def
tff(fact_5227_split__of__bool,axiom,
! [P: fun_complex_bool,P2: bool] :
( pp(aa_complex_bool(P,zero_n1201886186963655149omplex(P2)))
<=> ( ( pp(P2)
=> pp(aa_complex_bool(P,one_one_complex)) )
& ( ~ pp(P2)
=> pp(aa_complex_bool(P,zero_zero_complex)) ) ) ) ).
% split_of_bool
tff(fact_5228_split__of__bool,axiom,
! [P: fun_real_bool,P2: bool] :
( pp(aa_real_bool(P,zero_n3304061248610475627l_real(P2)))
<=> ( ( pp(P2)
=> pp(aa_real_bool(P,one_one_real)) )
& ( ~ pp(P2)
=> pp(aa_real_bool(P,zero_zero_real)) ) ) ) ).
% split_of_bool
tff(fact_5229_split__of__bool,axiom,
! [P: fun_rat_bool,P2: bool] :
( pp(aa_rat_bool(P,zero_n2052037380579107095ol_rat(P2)))
<=> ( ( pp(P2)
=> pp(aa_rat_bool(P,one_one_rat)) )
& ( ~ pp(P2)
=> pp(aa_rat_bool(P,zero_zero_rat)) ) ) ) ).
% split_of_bool
tff(fact_5230_split__of__bool,axiom,
! [P: fun_nat_bool,P2: bool] :
( pp(aa_nat_bool(P,zero_n2687167440665602831ol_nat(P2)))
<=> ( ( pp(P2)
=> pp(aa_nat_bool(P,one_one_nat)) )
& ( ~ pp(P2)
=> pp(aa_nat_bool(P,zero_zero_nat)) ) ) ) ).
% split_of_bool
tff(fact_5231_split__of__bool,axiom,
! [P: fun_int_bool,P2: bool] :
( pp(aa_int_bool(P,aa_bool_int(zero_n2684676970156552555ol_int,P2)))
<=> ( ( pp(P2)
=> pp(aa_int_bool(P,one_one_int)) )
& ( ~ pp(P2)
=> pp(aa_int_bool(P,zero_zero_int)) ) ) ) ).
% split_of_bool
tff(fact_5232_split__of__bool,axiom,
! [P: fun_Co8822979462182099027r_bool,P2: bool] :
( pp(aa_Code_integer_bool(P,zero_n356916108424825756nteger(P2)))
<=> ( ( pp(P2)
=> pp(aa_Code_integer_bool(P,one_one_Code_integer)) )
& ( ~ pp(P2)
=> pp(aa_Code_integer_bool(P,zero_z3403309356797280102nteger)) ) ) ) ).
% split_of_bool
tff(fact_5233_split__of__bool__asm,axiom,
! [P: fun_complex_bool,P2: bool] :
( pp(aa_complex_bool(P,zero_n1201886186963655149omplex(P2)))
<=> ~ ( ( pp(P2)
& ~ pp(aa_complex_bool(P,one_one_complex)) )
| ( ~ pp(P2)
& ~ pp(aa_complex_bool(P,zero_zero_complex)) ) ) ) ).
% split_of_bool_asm
tff(fact_5234_split__of__bool__asm,axiom,
! [P: fun_real_bool,P2: bool] :
( pp(aa_real_bool(P,zero_n3304061248610475627l_real(P2)))
<=> ~ ( ( pp(P2)
& ~ pp(aa_real_bool(P,one_one_real)) )
| ( ~ pp(P2)
& ~ pp(aa_real_bool(P,zero_zero_real)) ) ) ) ).
% split_of_bool_asm
tff(fact_5235_split__of__bool__asm,axiom,
! [P: fun_rat_bool,P2: bool] :
( pp(aa_rat_bool(P,zero_n2052037380579107095ol_rat(P2)))
<=> ~ ( ( pp(P2)
& ~ pp(aa_rat_bool(P,one_one_rat)) )
| ( ~ pp(P2)
& ~ pp(aa_rat_bool(P,zero_zero_rat)) ) ) ) ).
% split_of_bool_asm
tff(fact_5236_split__of__bool__asm,axiom,
! [P: fun_nat_bool,P2: bool] :
( pp(aa_nat_bool(P,zero_n2687167440665602831ol_nat(P2)))
<=> ~ ( ( pp(P2)
& ~ pp(aa_nat_bool(P,one_one_nat)) )
| ( ~ pp(P2)
& ~ pp(aa_nat_bool(P,zero_zero_nat)) ) ) ) ).
% split_of_bool_asm
tff(fact_5237_split__of__bool__asm,axiom,
! [P: fun_int_bool,P2: bool] :
( pp(aa_int_bool(P,aa_bool_int(zero_n2684676970156552555ol_int,P2)))
<=> ~ ( ( pp(P2)
& ~ pp(aa_int_bool(P,one_one_int)) )
| ( ~ pp(P2)
& ~ pp(aa_int_bool(P,zero_zero_int)) ) ) ) ).
% split_of_bool_asm
tff(fact_5238_split__of__bool__asm,axiom,
! [P: fun_Co8822979462182099027r_bool,P2: bool] :
( pp(aa_Code_integer_bool(P,zero_n356916108424825756nteger(P2)))
<=> ~ ( ( pp(P2)
& ~ pp(aa_Code_integer_bool(P,one_one_Code_integer)) )
| ( ~ pp(P2)
& ~ pp(aa_Code_integer_bool(P,zero_z3403309356797280102nteger)) ) ) ) ).
% split_of_bool_asm
tff(fact_5239_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),numeral_numeral_real(N))) ).
% neg_numeral_le_numeral
tff(fact_5240_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),numera6620942414471956472nteger(N))) ).
% neg_numeral_le_numeral
tff(fact_5241_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(M))),numeral_numeral_rat(N))) ).
% neg_numeral_le_numeral
tff(fact_5242_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N))) ).
% neg_numeral_le_numeral
tff(fact_5243_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,numeral_numeral_real(N)))) ).
% not_numeral_le_neg_numeral
tff(fact_5244_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)))) ).
% not_numeral_le_neg_numeral
tff(fact_5245_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,numeral_numeral_rat(M)),uminus_uminus_rat(numeral_numeral_rat(N)))) ).
% not_numeral_le_neg_numeral
tff(fact_5246_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ).
% not_numeral_le_neg_numeral
tff(fact_5247_zero__neq__neg__numeral,axiom,
! [N: num] : zero_zero_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).
% zero_neq_neg_numeral
tff(fact_5248_zero__neq__neg__numeral,axiom,
! [N: num] : zero_zero_real != aa_real_real(uminus_uminus_real,numeral_numeral_real(N)) ).
% zero_neq_neg_numeral
tff(fact_5249_zero__neq__neg__numeral,axiom,
! [N: num] : zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) ).
% zero_neq_neg_numeral
tff(fact_5250_zero__neq__neg__numeral,axiom,
! [N: num] : zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)) ).
% zero_neq_neg_numeral
tff(fact_5251_zero__neq__neg__numeral,axiom,
! [N: num] : zero_zero_rat != uminus_uminus_rat(numeral_numeral_rat(N)) ).
% zero_neq_neg_numeral
tff(fact_5252_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N))) ).
% neg_numeral_less_numeral
tff(fact_5253_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),numeral_numeral_real(N))) ).
% neg_numeral_less_numeral
tff(fact_5254_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),numera6620942414471956472nteger(N))) ).
% neg_numeral_less_numeral
tff(fact_5255_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(M))),numeral_numeral_rat(N))) ).
% neg_numeral_less_numeral
tff(fact_5256_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ).
% not_numeral_less_neg_numeral
tff(fact_5257_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,numeral_numeral_real(N)))) ).
% not_numeral_less_neg_numeral
tff(fact_5258_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)))) ).
% not_numeral_less_neg_numeral
tff(fact_5259_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,numeral_numeral_rat(M)),uminus_uminus_rat(numeral_numeral_rat(N)))) ).
% not_numeral_less_neg_numeral
tff(fact_5260_le__minus__one__simps_I4_J,axiom,
~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real))) ).
% le_minus_one_simps(4)
tff(fact_5261_le__minus__one__simps_I4_J,axiom,
~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).
% le_minus_one_simps(4)
tff(fact_5262_le__minus__one__simps_I4_J,axiom,
~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),uminus_uminus_rat(one_one_rat))) ).
% le_minus_one_simps(4)
tff(fact_5263_le__minus__one__simps_I4_J,axiom,
~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int))) ).
% le_minus_one_simps(4)
tff(fact_5264_le__minus__one__simps_I2_J,axiom,
pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real)) ).
% le_minus_one_simps(2)
tff(fact_5265_le__minus__one__simps_I2_J,axiom,
pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer)) ).
% le_minus_one_simps(2)
tff(fact_5266_le__minus__one__simps_I2_J,axiom,
pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(one_one_rat)),one_one_rat)) ).
% le_minus_one_simps(2)
tff(fact_5267_le__minus__one__simps_I2_J,axiom,
pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int)) ).
% le_minus_one_simps(2)
tff(fact_5268_zero__neq__neg__one,axiom,
zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).
% zero_neq_neg_one
tff(fact_5269_zero__neq__neg__one,axiom,
zero_zero_real != aa_real_real(uminus_uminus_real,one_one_real) ).
% zero_neq_neg_one
tff(fact_5270_zero__neq__neg__one,axiom,
zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).
% zero_neq_neg_one
tff(fact_5271_zero__neq__neg__one,axiom,
zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% zero_neq_neg_one
tff(fact_5272_zero__neq__neg__one,axiom,
zero_zero_rat != uminus_uminus_rat(one_one_rat) ).
% zero_neq_neg_one
tff(fact_5273_neg__eq__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( aa_int_int(uminus_uminus_int,A) = B )
<=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
tff(fact_5274_neg__eq__iff__add__eq__0,axiom,
! [A: real,B: real] :
( ( aa_real_real(uminus_uminus_real,A) = B )
<=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real ) ) ).
% neg_eq_iff_add_eq_0
tff(fact_5275_neg__eq__iff__add__eq__0,axiom,
! [A: complex,B: complex] :
( ( aa_complex_complex(uminus1482373934393186551omplex,A) = B )
<=> ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex ) ) ).
% neg_eq_iff_add_eq_0
tff(fact_5276_neg__eq__iff__add__eq__0,axiom,
! [A: code_integer,B: code_integer] :
( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B )
<=> ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger ) ) ).
% neg_eq_iff_add_eq_0
tff(fact_5277_neg__eq__iff__add__eq__0,axiom,
! [A: rat,B: rat] :
( ( uminus_uminus_rat(A) = B )
<=> ( plus_plus_rat(A,B) = zero_zero_rat ) ) ).
% neg_eq_iff_add_eq_0
tff(fact_5278_eq__neg__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( A = aa_int_int(uminus_uminus_int,B) )
<=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
tff(fact_5279_eq__neg__iff__add__eq__0,axiom,
! [A: real,B: real] :
( ( A = aa_real_real(uminus_uminus_real,B) )
<=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real ) ) ).
% eq_neg_iff_add_eq_0
tff(fact_5280_eq__neg__iff__add__eq__0,axiom,
! [A: complex,B: complex] :
( ( A = aa_complex_complex(uminus1482373934393186551omplex,B) )
<=> ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex ) ) ).
% eq_neg_iff_add_eq_0
tff(fact_5281_eq__neg__iff__add__eq__0,axiom,
! [A: code_integer,B: code_integer] :
( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
<=> ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger ) ) ).
% eq_neg_iff_add_eq_0
tff(fact_5282_eq__neg__iff__add__eq__0,axiom,
! [A: rat,B: rat] :
( ( A = uminus_uminus_rat(B) )
<=> ( plus_plus_rat(A,B) = zero_zero_rat ) ) ).
% eq_neg_iff_add_eq_0
tff(fact_5283_add_Oinverse__unique,axiom,
! [A: int,B: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int )
=> ( aa_int_int(uminus_uminus_int,A) = B ) ) ).
% add.inverse_unique
tff(fact_5284_add_Oinverse__unique,axiom,
! [A: real,B: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real )
=> ( aa_real_real(uminus_uminus_real,A) = B ) ) ).
% add.inverse_unique
tff(fact_5285_add_Oinverse__unique,axiom,
! [A: complex,B: complex] :
( ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex )
=> ( aa_complex_complex(uminus1482373934393186551omplex,A) = B ) ) ).
% add.inverse_unique
tff(fact_5286_add_Oinverse__unique,axiom,
! [A: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B ) ) ).
% add.inverse_unique
tff(fact_5287_add_Oinverse__unique,axiom,
! [A: rat,B: rat] :
( ( plus_plus_rat(A,B) = zero_zero_rat )
=> ( uminus_uminus_rat(A) = B ) ) ).
% add.inverse_unique
tff(fact_5288_ab__group__add__class_Oab__left__minus,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),A) = zero_zero_int ).
% ab_group_add_class.ab_left_minus
tff(fact_5289_ab__group__add__class_Oab__left__minus,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),A) = zero_zero_real ).
% ab_group_add_class.ab_left_minus
tff(fact_5290_ab__group__add__class_Oab__left__minus,axiom,
! [A: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),A) = zero_zero_complex ).
% ab_group_add_class.ab_left_minus
tff(fact_5291_ab__group__add__class_Oab__left__minus,axiom,
! [A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) = zero_z3403309356797280102nteger ).
% ab_group_add_class.ab_left_minus
tff(fact_5292_ab__group__add__class_Oab__left__minus,axiom,
! [A: rat] : plus_plus_rat(uminus_uminus_rat(A),A) = zero_zero_rat ).
% ab_group_add_class.ab_left_minus
tff(fact_5293_add__eq__0__iff,axiom,
! [A: int,B: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int )
<=> ( B = aa_int_int(uminus_uminus_int,A) ) ) ).
% add_eq_0_iff
tff(fact_5294_add__eq__0__iff,axiom,
! [A: real,B: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B) = zero_zero_real )
<=> ( B = aa_real_real(uminus_uminus_real,A) ) ) ).
% add_eq_0_iff
tff(fact_5295_add__eq__0__iff,axiom,
! [A: complex,B: complex] :
( ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex )
<=> ( B = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ).
% add_eq_0_iff
tff(fact_5296_add__eq__0__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger )
<=> ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).
% add_eq_0_iff
tff(fact_5297_add__eq__0__iff,axiom,
! [A: rat,B: rat] :
( ( plus_plus_rat(A,B) = zero_zero_rat )
<=> ( B = uminus_uminus_rat(A) ) ) ).
% add_eq_0_iff
tff(fact_5298_less__minus__one__simps_I4_J,axiom,
~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int))) ).
% less_minus_one_simps(4)
tff(fact_5299_less__minus__one__simps_I4_J,axiom,
~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real))) ).
% less_minus_one_simps(4)
tff(fact_5300_less__minus__one__simps_I4_J,axiom,
~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).
% less_minus_one_simps(4)
tff(fact_5301_less__minus__one__simps_I4_J,axiom,
~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),uminus_uminus_rat(one_one_rat))) ).
% less_minus_one_simps(4)
tff(fact_5302_less__minus__one__simps_I2_J,axiom,
pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int)) ).
% less_minus_one_simps(2)
tff(fact_5303_less__minus__one__simps_I2_J,axiom,
pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real)) ).
% less_minus_one_simps(2)
tff(fact_5304_less__minus__one__simps_I2_J,axiom,
pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer)) ).
% less_minus_one_simps(2)
tff(fact_5305_less__minus__one__simps_I2_J,axiom,
pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(one_one_rat)),one_one_rat)) ).
% less_minus_one_simps(2)
tff(fact_5306_numeral__times__minus__swap,axiom,
! [W: num,X: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W)),aa_int_int(uminus_uminus_int,X)) = aa_int_int(aa_int_fun_int_int(times_times_int,X),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))) ).
% numeral_times_minus_swap
tff(fact_5307_numeral__times__minus__swap,axiom,
! [W: num,X: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),aa_real_real(uminus_uminus_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) ).
% numeral_times_minus_swap
tff(fact_5308_numeral__times__minus__swap,axiom,
! [W: num,X: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),aa_complex_complex(uminus1482373934393186551omplex,X)) = aa_complex_complex(times_times_complex(X),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) ).
% numeral_times_minus_swap
tff(fact_5309_numeral__times__minus__swap,axiom,
! [W: num,X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(W)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W))) ).
% numeral_times_minus_swap
tff(fact_5310_numeral__times__minus__swap,axiom,
! [W: num,X: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),uminus_uminus_rat(X)) = aa_rat_rat(times_times_rat(X),uminus_uminus_rat(numeral_numeral_rat(W))) ).
% numeral_times_minus_swap
tff(fact_5311_nonzero__minus__divide__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(uminus_uminus_real,B)) ) ) ).
% nonzero_minus_divide_right
tff(fact_5312_nonzero__minus__divide__right,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ) ) ).
% nonzero_minus_divide_right
tff(fact_5313_nonzero__minus__divide__right,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( uminus_uminus_rat(divide_divide_rat(A,B)) = divide_divide_rat(A,uminus_uminus_rat(B)) ) ) ).
% nonzero_minus_divide_right
tff(fact_5314_nonzero__minus__divide__divide,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) ) ) ).
% nonzero_minus_divide_divide
tff(fact_5315_nonzero__minus__divide__divide,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).
% nonzero_minus_divide_divide
tff(fact_5316_nonzero__minus__divide__divide,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( divide_divide_rat(uminus_uminus_rat(A),uminus_uminus_rat(B)) = divide_divide_rat(A,B) ) ) ).
% nonzero_minus_divide_divide
tff(fact_5317_numeral__neq__neg__one,axiom,
! [N: num] : aa_num_int(numeral_numeral_int,N) != aa_int_int(uminus_uminus_int,one_one_int) ).
% numeral_neq_neg_one
tff(fact_5318_numeral__neq__neg__one,axiom,
! [N: num] : numeral_numeral_real(N) != aa_real_real(uminus_uminus_real,one_one_real) ).
% numeral_neq_neg_one
tff(fact_5319_numeral__neq__neg__one,axiom,
! [N: num] : numera6690914467698888265omplex(N) != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).
% numeral_neq_neg_one
tff(fact_5320_numeral__neq__neg__one,axiom,
! [N: num] : numera6620942414471956472nteger(N) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% numeral_neq_neg_one
tff(fact_5321_numeral__neq__neg__one,axiom,
! [N: num] : numeral_numeral_rat(N) != uminus_uminus_rat(one_one_rat) ).
% numeral_neq_neg_one
tff(fact_5322_one__neq__neg__numeral,axiom,
! [N: num] : one_one_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).
% one_neq_neg_numeral
tff(fact_5323_one__neq__neg__numeral,axiom,
! [N: num] : one_one_real != aa_real_real(uminus_uminus_real,numeral_numeral_real(N)) ).
% one_neq_neg_numeral
tff(fact_5324_one__neq__neg__numeral,axiom,
! [N: num] : one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) ).
% one_neq_neg_numeral
tff(fact_5325_one__neq__neg__numeral,axiom,
! [N: num] : one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)) ).
% one_neq_neg_numeral
tff(fact_5326_one__neq__neg__numeral,axiom,
! [N: num] : one_one_rat != uminus_uminus_rat(numeral_numeral_rat(N)) ).
% one_neq_neg_numeral
tff(fact_5327_square__eq__1__iff,axiom,
! [X: int] :
( ( aa_int_int(aa_int_fun_int_int(times_times_int,X),X) = one_one_int )
<=> ( ( X = one_one_int )
| ( X = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).
% square_eq_1_iff
tff(fact_5328_square__eq__1__iff,axiom,
! [X: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X) = one_one_real )
<=> ( ( X = one_one_real )
| ( X = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).
% square_eq_1_iff
tff(fact_5329_square__eq__1__iff,axiom,
! [X: complex] :
( ( aa_complex_complex(times_times_complex(X),X) = one_one_complex )
<=> ( ( X = one_one_complex )
| ( X = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).
% square_eq_1_iff
tff(fact_5330_square__eq__1__iff,axiom,
! [X: code_integer] :
( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X) = one_one_Code_integer )
<=> ( ( X = one_one_Code_integer )
| ( X = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).
% square_eq_1_iff
tff(fact_5331_square__eq__1__iff,axiom,
! [X: rat] :
( ( aa_rat_rat(times_times_rat(X),X) = one_one_rat )
<=> ( ( X = one_one_rat )
| ( X = uminus_uminus_rat(one_one_rat) ) ) ) ).
% square_eq_1_iff
tff(fact_5332_group__cancel_Osub2,axiom,
! [B2: int,K: int,B: int,A: int] :
( ( B2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B) )
=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).
% group_cancel.sub2
tff(fact_5333_group__cancel_Osub2,axiom,
! [B2: real,K: real,B: real,A: real] :
( ( B2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B) )
=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ) ) ).
% group_cancel.sub2
tff(fact_5334_group__cancel_Osub2,axiom,
! [B2: complex,K: complex,B: complex,A: complex] :
( ( B2 = aa_complex_complex(plus_plus_complex(K),B) )
=> ( aa_complex_complex(minus_minus_complex(A),B2) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(minus_minus_complex(A),B)) ) ) ).
% group_cancel.sub2
tff(fact_5335_group__cancel_Osub2,axiom,
! [B2: code_integer,K: code_integer,B: code_integer,A: code_integer] :
( ( B2 = plus_p5714425477246183910nteger(K,B) )
=> ( minus_8373710615458151222nteger(A,B2) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),minus_8373710615458151222nteger(A,B)) ) ) ).
% group_cancel.sub2
tff(fact_5336_group__cancel_Osub2,axiom,
! [B2: rat,K: rat,B: rat,A: rat] :
( ( B2 = plus_plus_rat(K,B) )
=> ( minus_minus_rat(A,B2) = plus_plus_rat(uminus_uminus_rat(K),minus_minus_rat(A,B)) ) ) ).
% group_cancel.sub2
tff(fact_5337_diff__conv__add__uminus,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,B)) ).
% diff_conv_add_uminus
tff(fact_5338_diff__conv__add__uminus,axiom,
! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,B)) ).
% diff_conv_add_uminus
tff(fact_5339_diff__conv__add__uminus,axiom,
! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(A),B) = aa_complex_complex(plus_plus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).
% diff_conv_add_uminus
tff(fact_5340_diff__conv__add__uminus,axiom,
! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,B) = plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).
% diff_conv_add_uminus
tff(fact_5341_diff__conv__add__uminus,axiom,
! [A: rat,B: rat] : minus_minus_rat(A,B) = plus_plus_rat(A,uminus_uminus_rat(B)) ).
% diff_conv_add_uminus
tff(fact_5342_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,B)) ).
% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5343_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,B)) ).
% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5344_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(A),B) = aa_complex_complex(plus_plus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).
% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5345_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,B) = plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).
% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5346_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: rat,B: rat] : minus_minus_rat(A,B) = plus_plus_rat(A,uminus_uminus_rat(B)) ).
% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5347_replicate__eqI,axiom,
! [Xs: list_P6011104703257516679at_nat,N: nat,X: product_prod_nat_nat] :
( ( size_s5460976970255530739at_nat(Xs) = N )
=> ( ! [Y2: product_prod_nat_nat] :
( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y2),set_Pr5648618587558075414at_nat(Xs)))
=> ( Y2 = X ) )
=> ( Xs = replic4235873036481779905at_nat(N,X) ) ) ) ).
% replicate_eqI
tff(fact_5348_replicate__eqI,axiom,
! [Xs: list_complex,N: nat,X: complex] :
( ( size_s3451745648224563538omplex(Xs) = N )
=> ( ! [Y2: complex] :
( pp(aa_set_complex_bool(member_complex(Y2),set_complex2(Xs)))
=> ( Y2 = X ) )
=> ( Xs = replicate_complex(N,X) ) ) ) ).
% replicate_eqI
tff(fact_5349_replicate__eqI,axiom,
! [Xs: list_real,N: nat,X: real] :
( ( size_size_list_real(Xs) = N )
=> ( ! [Y2: real] :
( pp(aa_set_real_bool(member_real(Y2),set_real2(Xs)))
=> ( Y2 = X ) )
=> ( Xs = replicate_real(N,X) ) ) ) ).
% replicate_eqI
tff(fact_5350_replicate__eqI,axiom,
! [Xs: list_VEBT_VEBT,N: nat,X: vEBT_VEBT] :
( ( size_s6755466524823107622T_VEBT(Xs) = N )
=> ( ! [Y2: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Y2),set_VEBT_VEBT2(Xs)))
=> ( Y2 = X ) )
=> ( Xs = replicate_VEBT_VEBT(N,X) ) ) ) ).
% replicate_eqI
tff(fact_5351_replicate__eqI,axiom,
! [Xs: list_bool,N: nat,X: bool] :
( ( size_size_list_bool(Xs) = N )
=> ( ! [Y2: bool] :
( member_bool(Y2,set_bool2(Xs))
=> ( pp(Y2)
<=> pp(X) ) )
=> ( Xs = replicate_bool(N,X) ) ) ) ).
% replicate_eqI
tff(fact_5352_replicate__eqI,axiom,
! [Xs: list_nat,N: nat,X: nat] :
( ( size_size_list_nat(Xs) = N )
=> ( ! [Y2: nat] :
( pp(aa_set_nat_bool(member_nat(Y2),set_nat2(Xs)))
=> ( Y2 = X ) )
=> ( Xs = replicate_nat(N,X) ) ) ) ).
% replicate_eqI
tff(fact_5353_replicate__eqI,axiom,
! [Xs: list_int,N: nat,X: int] :
( ( size_size_list_int(Xs) = N )
=> ( ! [Y2: int] :
( pp(aa_set_int_bool(member_int(Y2),set_int2(Xs)))
=> ( Y2 = X ) )
=> ( Xs = replicate_int(N,X) ) ) ) ).
% replicate_eqI
tff(fact_5354_replicate__length__same,axiom,
! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
( ! [X3: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(Xs)))
=> ( X3 = X ) )
=> ( replicate_VEBT_VEBT(size_s6755466524823107622T_VEBT(Xs),X) = Xs ) ) ).
% replicate_length_same
tff(fact_5355_replicate__length__same,axiom,
! [Xs: list_bool,X: bool] :
( ! [X3: bool] :
( member_bool(X3,set_bool2(Xs))
=> ( pp(X3)
<=> pp(X) ) )
=> ( replicate_bool(size_size_list_bool(Xs),X) = Xs ) ) ).
% replicate_length_same
tff(fact_5356_replicate__length__same,axiom,
! [Xs: list_nat,X: nat] :
( ! [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),set_nat2(Xs)))
=> ( X3 = X ) )
=> ( replicate_nat(size_size_list_nat(Xs),X) = Xs ) ) ).
% replicate_length_same
tff(fact_5357_replicate__length__same,axiom,
! [Xs: list_int,X: int] :
( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),set_int2(Xs)))
=> ( X3 = X ) )
=> ( replicate_int(size_size_list_int(Xs),X) = Xs ) ) ).
% replicate_length_same
tff(fact_5358_dvd__div__neg,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).
% dvd_div_neg
tff(fact_5359_dvd__div__neg,axiom,
! [B: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A))
=> ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) ) ) ).
% dvd_div_neg
tff(fact_5360_dvd__div__neg,axiom,
! [B: complex,A: complex] :
( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,B),A))
=> ( aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) ) ) ).
% dvd_div_neg
tff(fact_5361_dvd__div__neg,axiom,
! [B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A))
=> ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,divide6298287555418463151nteger(A,B)) ) ) ).
% dvd_div_neg
tff(fact_5362_dvd__div__neg,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
=> ( divide_divide_rat(A,uminus_uminus_rat(B)) = uminus_uminus_rat(divide_divide_rat(A,B)) ) ) ).
% dvd_div_neg
tff(fact_5363_dvd__neg__div,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,B),A))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).
% dvd_neg_div
tff(fact_5364_dvd__neg__div,axiom,
! [B: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,B),A))
=> ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) ) ) ).
% dvd_neg_div
tff(fact_5365_dvd__neg__div,axiom,
! [B: complex,A: complex] :
( pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,B),A))
=> ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) ) ) ).
% dvd_neg_div
tff(fact_5366_dvd__neg__div,axiom,
! [B: code_integer,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,B),A))
=> ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,divide6298287555418463151nteger(A,B)) ) ) ).
% dvd_neg_div
tff(fact_5367_dvd__neg__div,axiom,
! [B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,B),A))
=> ( divide_divide_rat(uminus_uminus_rat(A),B) = uminus_uminus_rat(divide_divide_rat(A,B)) ) ) ).
% dvd_neg_div
tff(fact_5368_real__minus__mult__self__le,axiom,
! [U: real,X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),U))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X))) ).
% real_minus_mult_self_le
tff(fact_5369_minus__int__code_I2_J,axiom,
! [L: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),L) = aa_int_int(uminus_uminus_int,L) ).
% minus_int_code(2)
tff(fact_5370_numeral__eq__Suc,axiom,
! [K: num] : aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,aa_num_nat(pred_numeral,K)) ).
% numeral_eq_Suc
tff(fact_5371_zmod__zminus1__not__zero,axiom,
! [K: int,L: int] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,K)),L) != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) != zero_zero_int ) ) ).
% zmod_zminus1_not_zero
tff(fact_5372_zmod__zminus2__not__zero,axiom,
! [K: int,L: int] :
( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_int_int(uminus_uminus_int,L)) != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) != zero_zero_int ) ) ).
% zmod_zminus2_not_zero
tff(fact_5373_minus__real__def,axiom,
! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_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_5374_neg__numeral__le__zero,axiom,
! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(N))),zero_zero_real)) ).
% neg_numeral_le_zero
tff(fact_5375_neg__numeral__le__zero,axiom,
! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))),zero_z3403309356797280102nteger)) ).
% neg_numeral_le_zero
tff(fact_5376_neg__numeral__le__zero,axiom,
! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(N))),zero_zero_rat)) ).
% neg_numeral_le_zero
tff(fact_5377_neg__numeral__le__zero,axiom,
! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),zero_zero_int)) ).
% neg_numeral_le_zero
tff(fact_5378_not__zero__le__neg__numeral,axiom,
! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(N)))) ).
% not_zero_le_neg_numeral
tff(fact_5379_not__zero__le__neg__numeral,axiom,
! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)))) ).
% not_zero_le_neg_numeral
tff(fact_5380_not__zero__le__neg__numeral,axiom,
! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),uminus_uminus_rat(numeral_numeral_rat(N)))) ).
% not_zero_le_neg_numeral
tff(fact_5381_not__zero__le__neg__numeral,axiom,
! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ).
% not_zero_le_neg_numeral
tff(fact_5382_not__zero__less__neg__numeral,axiom,
! [N: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))) ).
% not_zero_less_neg_numeral
tff(fact_5383_not__zero__less__neg__numeral,axiom,
! [N: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(N)))) ).
% not_zero_less_neg_numeral
tff(fact_5384_not__zero__less__neg__numeral,axiom,
! [N: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)))) ).
% not_zero_less_neg_numeral
tff(fact_5385_not__zero__less__neg__numeral,axiom,
! [N: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),uminus_uminus_rat(numeral_numeral_rat(N)))) ).
% not_zero_less_neg_numeral
tff(fact_5386_neg__numeral__less__zero,axiom,
! [N: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),zero_zero_int)) ).
% neg_numeral_less_zero
tff(fact_5387_neg__numeral__less__zero,axiom,
! [N: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(N))),zero_zero_real)) ).
% neg_numeral_less_zero
tff(fact_5388_neg__numeral__less__zero,axiom,
! [N: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))),zero_z3403309356797280102nteger)) ).
% neg_numeral_less_zero
tff(fact_5389_neg__numeral__less__zero,axiom,
! [N: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(N))),zero_zero_rat)) ).
% neg_numeral_less_zero
tff(fact_5390_le__minus__one__simps_I3_J,axiom,
~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real))) ).
% le_minus_one_simps(3)
tff(fact_5391_le__minus__one__simps_I3_J,axiom,
~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).
% le_minus_one_simps(3)
tff(fact_5392_le__minus__one__simps_I3_J,axiom,
~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),uminus_uminus_rat(one_one_rat))) ).
% le_minus_one_simps(3)
tff(fact_5393_le__minus__one__simps_I3_J,axiom,
~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int))) ).
% le_minus_one_simps(3)
tff(fact_5394_le__minus__one__simps_I1_J,axiom,
pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real)) ).
% le_minus_one_simps(1)
tff(fact_5395_le__minus__one__simps_I1_J,axiom,
pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger)) ).
% le_minus_one_simps(1)
tff(fact_5396_le__minus__one__simps_I1_J,axiom,
pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(one_one_rat)),zero_zero_rat)) ).
% le_minus_one_simps(1)
tff(fact_5397_le__minus__one__simps_I1_J,axiom,
pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int)) ).
% le_minus_one_simps(1)
tff(fact_5398_less__minus__one__simps_I1_J,axiom,
pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int)) ).
% less_minus_one_simps(1)
tff(fact_5399_less__minus__one__simps_I1_J,axiom,
pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real)) ).
% less_minus_one_simps(1)
tff(fact_5400_less__minus__one__simps_I1_J,axiom,
pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger)) ).
% less_minus_one_simps(1)
tff(fact_5401_less__minus__one__simps_I1_J,axiom,
pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(one_one_rat)),zero_zero_rat)) ).
% less_minus_one_simps(1)
tff(fact_5402_less__minus__one__simps_I3_J,axiom,
~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int))) ).
% less_minus_one_simps(3)
tff(fact_5403_less__minus__one__simps_I3_J,axiom,
~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real))) ).
% less_minus_one_simps(3)
tff(fact_5404_less__minus__one__simps_I3_J,axiom,
~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).
% less_minus_one_simps(3)
tff(fact_5405_less__minus__one__simps_I3_J,axiom,
~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),uminus_uminus_rat(one_one_rat))) ).
% less_minus_one_simps(3)
tff(fact_5406_neg__numeral__le__one,axiom,
! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),one_one_real)) ).
% neg_numeral_le_one
tff(fact_5407_neg__numeral__le__one,axiom,
! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),one_one_Code_integer)) ).
% neg_numeral_le_one
tff(fact_5408_neg__numeral__le__one,axiom,
! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(M))),one_one_rat)) ).
% neg_numeral_le_one
tff(fact_5409_neg__numeral__le__one,axiom,
! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int)) ).
% neg_numeral_le_one
tff(fact_5410_neg__one__le__numeral,axiom,
! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),numeral_numeral_real(M))) ).
% neg_one_le_numeral
tff(fact_5411_neg__one__le__numeral,axiom,
! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(M))) ).
% neg_one_le_numeral
tff(fact_5412_neg__one__le__numeral,axiom,
! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(one_one_rat)),numeral_numeral_rat(M))) ).
% neg_one_le_numeral
tff(fact_5413_neg__one__le__numeral,axiom,
! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,M))) ).
% neg_one_le_numeral
tff(fact_5414_neg__numeral__le__neg__one,axiom,
! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,one_one_real))) ).
% neg_numeral_le_neg_one
tff(fact_5415_neg__numeral__le__neg__one,axiom,
! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).
% neg_numeral_le_neg_one
tff(fact_5416_neg__numeral__le__neg__one,axiom,
! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(M))),uminus_uminus_rat(one_one_rat))) ).
% neg_numeral_le_neg_one
tff(fact_5417_neg__numeral__le__neg__one,axiom,
! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int))) ).
% neg_numeral_le_neg_one
tff(fact_5418_not__numeral__le__neg__one,axiom,
! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,one_one_real))) ).
% not_numeral_le_neg_one
tff(fact_5419_not__numeral__le__neg__one,axiom,
! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).
% not_numeral_le_neg_one
tff(fact_5420_not__numeral__le__neg__one,axiom,
! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,numeral_numeral_rat(M)),uminus_uminus_rat(one_one_rat))) ).
% not_numeral_le_neg_one
tff(fact_5421_not__numeral__le__neg__one,axiom,
! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int))) ).
% not_numeral_le_neg_one
tff(fact_5422_not__one__le__neg__numeral,axiom,
! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(M)))) ).
% not_one_le_neg_numeral
tff(fact_5423_not__one__le__neg__numeral,axiom,
! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)))) ).
% not_one_le_neg_numeral
tff(fact_5424_not__one__le__neg__numeral,axiom,
! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),uminus_uminus_rat(numeral_numeral_rat(M)))) ).
% not_one_le_neg_numeral
tff(fact_5425_not__one__le__neg__numeral,axiom,
! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)))) ).
% not_one_le_neg_numeral
tff(fact_5426_neg__numeral__less__one,axiom,
! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int)) ).
% neg_numeral_less_one
tff(fact_5427_neg__numeral__less__one,axiom,
! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),one_one_real)) ).
% neg_numeral_less_one
tff(fact_5428_neg__numeral__less__one,axiom,
! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M))),one_one_Code_integer)) ).
% neg_numeral_less_one
tff(fact_5429_neg__numeral__less__one,axiom,
! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(M))),one_one_rat)) ).
% neg_numeral_less_one
tff(fact_5430_neg__one__less__numeral,axiom,
! [M: num] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,M))) ).
% neg_one_less_numeral
tff(fact_5431_neg__one__less__numeral,axiom,
! [M: num] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),numeral_numeral_real(M))) ).
% neg_one_less_numeral
tff(fact_5432_neg__one__less__numeral,axiom,
! [M: num] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(M))) ).
% neg_one_less_numeral
tff(fact_5433_neg__one__less__numeral,axiom,
! [M: num] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(one_one_rat)),numeral_numeral_rat(M))) ).
% neg_one_less_numeral
tff(fact_5434_not__numeral__less__neg__one,axiom,
! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int))) ).
% not_numeral_less_neg_one
tff(fact_5435_not__numeral__less__neg__one,axiom,
! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,one_one_real))) ).
% not_numeral_less_neg_one
tff(fact_5436_not__numeral__less__neg__one,axiom,
! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))) ).
% not_numeral_less_neg_one
tff(fact_5437_not__numeral__less__neg__one,axiom,
! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,numeral_numeral_rat(M)),uminus_uminus_rat(one_one_rat))) ).
% not_numeral_less_neg_one
tff(fact_5438_not__one__less__neg__numeral,axiom,
! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)))) ).
% not_one_less_neg_numeral
tff(fact_5439_not__one__less__neg__numeral,axiom,
! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(M)))) ).
% not_one_less_neg_numeral
tff(fact_5440_not__one__less__neg__numeral,axiom,
! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)))) ).
% not_one_less_neg_numeral
tff(fact_5441_not__one__less__neg__numeral,axiom,
! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),uminus_uminus_rat(numeral_numeral_rat(M)))) ).
% not_one_less_neg_numeral
tff(fact_5442_not__neg__one__less__neg__numeral,axiom,
! [M: num] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)))) ).
% not_neg_one_less_neg_numeral
tff(fact_5443_not__neg__one__less__neg__numeral,axiom,
! [M: num] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,numeral_numeral_real(M)))) ).
% not_neg_one_less_neg_numeral
tff(fact_5444_not__neg__one__less__neg__numeral,axiom,
! [M: num] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)))) ).
% not_neg_one_less_neg_numeral
tff(fact_5445_not__neg__one__less__neg__numeral,axiom,
! [M: num] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(one_one_rat)),uminus_uminus_rat(numeral_numeral_rat(M)))) ).
% not_neg_one_less_neg_numeral
tff(fact_5446_nonzero__neg__divide__eq__eq2,axiom,
! [B: real,C: real,A: real] :
( ( B != zero_zero_real )
=> ( ( C = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) )
<=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) = aa_real_real(uminus_uminus_real,A) ) ) ) ).
% nonzero_neg_divide_eq_eq2
tff(fact_5447_nonzero__neg__divide__eq__eq2,axiom,
! [B: complex,C: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( C = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) )
<=> ( aa_complex_complex(times_times_complex(C),B) = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ) ).
% nonzero_neg_divide_eq_eq2
tff(fact_5448_nonzero__neg__divide__eq__eq2,axiom,
! [B: rat,C: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( C = uminus_uminus_rat(divide_divide_rat(A,B)) )
<=> ( aa_rat_rat(times_times_rat(C),B) = uminus_uminus_rat(A) ) ) ) ).
% nonzero_neg_divide_eq_eq2
tff(fact_5449_nonzero__neg__divide__eq__eq,axiom,
! [B: real,A: real,C: real] :
( ( B != zero_zero_real )
=> ( ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = C )
<=> ( aa_real_real(uminus_uminus_real,A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B) ) ) ) ).
% nonzero_neg_divide_eq_eq
tff(fact_5450_nonzero__neg__divide__eq__eq,axiom,
! [B: complex,A: complex,C: complex] :
( ( B != zero_zero_complex )
=> ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = C )
<=> ( aa_complex_complex(uminus1482373934393186551omplex,A) = aa_complex_complex(times_times_complex(C),B) ) ) ) ).
% nonzero_neg_divide_eq_eq
tff(fact_5451_nonzero__neg__divide__eq__eq,axiom,
! [B: rat,A: rat,C: rat] :
( ( B != zero_zero_rat )
=> ( ( uminus_uminus_rat(divide_divide_rat(A,B)) = C )
<=> ( uminus_uminus_rat(A) = aa_rat_rat(times_times_rat(C),B) ) ) ) ).
% nonzero_neg_divide_eq_eq
tff(fact_5452_minus__divide__eq__eq,axiom,
! [B: real,C: real,A: real] :
( ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) = A )
<=> ( ( ( C != zero_zero_real )
=> ( aa_real_real(uminus_uminus_real,B) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% minus_divide_eq_eq
tff(fact_5453_minus__divide__eq__eq,axiom,
! [B: complex,C: complex,A: complex] :
( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B),C)) = A )
<=> ( ( ( C != zero_zero_complex )
=> ( aa_complex_complex(uminus1482373934393186551omplex,B) = aa_complex_complex(times_times_complex(A),C) ) )
& ( ( C = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% minus_divide_eq_eq
tff(fact_5454_minus__divide__eq__eq,axiom,
! [B: rat,C: rat,A: rat] :
( ( uminus_uminus_rat(divide_divide_rat(B,C)) = A )
<=> ( ( ( C != zero_zero_rat )
=> ( uminus_uminus_rat(B) = aa_rat_rat(times_times_rat(A),C) ) )
& ( ( C = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% minus_divide_eq_eq
tff(fact_5455_eq__minus__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( A = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)) )
<=> ( ( ( C != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(uminus_uminus_real,B) ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_minus_divide_eq
tff(fact_5456_eq__minus__divide__eq,axiom,
! [A: complex,B: complex,C: complex] :
( ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B),C)) )
<=> ( ( ( C != zero_zero_complex )
=> ( aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(uminus1482373934393186551omplex,B) ) )
& ( ( C = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% eq_minus_divide_eq
tff(fact_5457_eq__minus__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( A = uminus_uminus_rat(divide_divide_rat(B,C)) )
<=> ( ( ( C != zero_zero_rat )
=> ( aa_rat_rat(times_times_rat(A),C) = uminus_uminus_rat(B) ) )
& ( ( C = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_minus_divide_eq
tff(fact_5458_divide__eq__minus__1__iff,axiom,
! [A: real,B: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B) = aa_real_real(uminus_uminus_real,one_one_real) )
<=> ( ( B != zero_zero_real )
& ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ).
% divide_eq_minus_1_iff
tff(fact_5459_divide__eq__minus__1__iff,axiom,
! [A: complex,B: complex] :
( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
<=> ( ( B != zero_zero_complex )
& ( A = aa_complex_complex(uminus1482373934393186551omplex,B) ) ) ) ).
% divide_eq_minus_1_iff
tff(fact_5460_divide__eq__minus__1__iff,axiom,
! [A: rat,B: rat] :
( ( divide_divide_rat(A,B) = uminus_uminus_rat(one_one_rat) )
<=> ( ( B != zero_zero_rat )
& ( A = uminus_uminus_rat(B) ) ) ) ).
% divide_eq_minus_1_iff
tff(fact_5461_mult__1s__ring__1_I2_J,axiom,
! [B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))) = aa_int_int(uminus_uminus_int,B) ).
% mult_1s_ring_1(2)
tff(fact_5462_mult__1s__ring__1_I2_J,axiom,
! [B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_real_real(uminus_uminus_real,numeral_numeral_real(one))) = aa_real_real(uminus_uminus_real,B) ).
% mult_1s_ring_1(2)
tff(fact_5463_mult__1s__ring__1_I2_J,axiom,
! [B: complex] : aa_complex_complex(times_times_complex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one))) = aa_complex_complex(uminus1482373934393186551omplex,B) ).
% mult_1s_ring_1(2)
tff(fact_5464_mult__1s__ring__1_I2_J,axiom,
! [B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).
% mult_1s_ring_1(2)
tff(fact_5465_mult__1s__ring__1_I2_J,axiom,
! [B: rat] : aa_rat_rat(times_times_rat(B),uminus_uminus_rat(numeral_numeral_rat(one))) = uminus_uminus_rat(B) ).
% mult_1s_ring_1(2)
tff(fact_5466_mult__1s__ring__1_I1_J,axiom,
! [B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))),B) = aa_int_int(uminus_uminus_int,B) ).
% mult_1s_ring_1(1)
tff(fact_5467_mult__1s__ring__1_I1_J,axiom,
! [B: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(one))),B) = aa_real_real(uminus_uminus_real,B) ).
% mult_1s_ring_1(1)
tff(fact_5468_mult__1s__ring__1_I1_J,axiom,
! [B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one))),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ).
% mult_1s_ring_1(1)
tff(fact_5469_mult__1s__ring__1_I1_J,axiom,
! [B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one))),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).
% mult_1s_ring_1(1)
tff(fact_5470_mult__1s__ring__1_I1_J,axiom,
! [B: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(one))),B) = uminus_uminus_rat(B) ).
% mult_1s_ring_1(1)
tff(fact_5471_uminus__numeral__One,axiom,
aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one)) = aa_int_int(uminus_uminus_int,one_one_int) ).
% uminus_numeral_One
tff(fact_5472_uminus__numeral__One,axiom,
aa_real_real(uminus_uminus_real,numeral_numeral_real(one)) = aa_real_real(uminus_uminus_real,one_one_real) ).
% uminus_numeral_One
tff(fact_5473_uminus__numeral__One,axiom,
aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).
% uminus_numeral_One
tff(fact_5474_uminus__numeral__One,axiom,
aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% uminus_numeral_One
tff(fact_5475_uminus__numeral__One,axiom,
uminus_uminus_rat(numeral_numeral_rat(one)) = uminus_uminus_rat(one_one_rat) ).
% uminus_numeral_One
tff(fact_5476_power__minus,axiom,
! [A: int,N: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_nat_int(power_power_int(A),N)) ).
% power_minus
tff(fact_5477_power__minus,axiom,
! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),N) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ).
% power_minus
tff(fact_5478_power__minus,axiom,
! [A: complex,N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),aa_nat_complex(power_power_complex(A),N)) ).
% power_minus
tff(fact_5479_power__minus,axiom,
! [A: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ).
% power_minus
tff(fact_5480_power__minus,axiom,
! [A: rat,N: nat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),N) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N)),aa_nat_rat(power_power_rat(A),N)) ).
% power_minus
tff(fact_5481_power__minus__Bit0,axiom,
! [X: int,K: num] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).
% power_minus_Bit0
tff(fact_5482_power__minus__Bit0,axiom,
! [X: real,K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).
% power_minus_Bit0
tff(fact_5483_power__minus__Bit0,axiom,
! [X: complex,K: num] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).
% power_minus_Bit0
tff(fact_5484_power__minus__Bit0,axiom,
! [X: code_integer,K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).
% power_minus_Bit0
tff(fact_5485_power__minus__Bit0,axiom,
! [X: rat,K: num] : aa_nat_rat(power_power_rat(uminus_uminus_rat(X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).
% power_minus_Bit0
tff(fact_5486_power__minus__Bit1,axiom,
! [X: int,K: num] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).
% power_minus_Bit1
tff(fact_5487_power__minus__Bit1,axiom,
! [X: real,K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).
% power_minus_Bit1
tff(fact_5488_power__minus__Bit1,axiom,
! [X: complex,K: num] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).
% power_minus_Bit1
tff(fact_5489_power__minus__Bit1,axiom,
! [X: code_integer,K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).
% power_minus_Bit1
tff(fact_5490_power__minus__Bit1,axiom,
! [X: rat,K: num] : aa_nat_rat(power_power_rat(uminus_uminus_rat(X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = uminus_uminus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).
% power_minus_Bit1
tff(fact_5491_real__0__less__add__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,X)),Y)) ) ).
% real_0_less_add_iff
tff(fact_5492_real__add__less__0__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),zero_zero_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(uminus_uminus_real,X))) ) ).
% real_add_less_0_iff
tff(fact_5493_real__add__le__0__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),zero_zero_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(uminus_uminus_real,X))) ) ).
% real_add_le_0_iff
tff(fact_5494_real__0__le__add__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,X)),Y)) ) ).
% real_0_le_add_iff
tff(fact_5495_zmod__zminus1__eq__if,axiom,
! [A: int,B: int] :
( ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) = zero_zero_int ) )
& ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ) ).
% zmod_zminus1_eq_if
tff(fact_5496_zmod__zminus2__eq__if,axiom,
! [A: int,B: int] :
( ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,B)) = zero_zero_int ) )
& ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) ) ) ) ).
% zmod_zminus2_eq_if
tff(fact_5497_pred__numeral__def,axiom,
! [K: num] : aa_num_nat(pred_numeral,K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),one_one_nat) ).
% pred_numeral_def
tff(fact_5498_pos__minus__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).
% pos_minus_divide_less_eq
tff(fact_5499_pos__minus__divide__less__eq,axiom,
! [C: rat,B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(divide_divide_rat(B,C))),A))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C))) ) ) ).
% pos_minus_divide_less_eq
tff(fact_5500_pos__less__minus__divide__eq,axiom,
! [C: real,A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) ) ) ).
% pos_less_minus_divide_eq
tff(fact_5501_pos__less__minus__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),uminus_uminus_rat(divide_divide_rat(B,C))))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B))) ) ) ).
% pos_less_minus_divide_eq
tff(fact_5502_neg__minus__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) ) ) ).
% neg_minus_divide_less_eq
tff(fact_5503_neg__minus__divide__less__eq,axiom,
! [C: rat,B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(divide_divide_rat(B,C))),A))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B))) ) ) ).
% neg_minus_divide_less_eq
tff(fact_5504_neg__less__minus__divide__eq,axiom,
! [C: real,A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).
% neg_less_minus_divide_eq
tff(fact_5505_neg__less__minus__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),uminus_uminus_rat(divide_divide_rat(B,C))))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C))) ) ) ).
% neg_less_minus_divide_eq
tff(fact_5506_minus__divide__less__eq,axiom,
! [B: real,C: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A)) ) ) ) ) ) ).
% minus_divide_less_eq
tff(fact_5507_minus__divide__less__eq,axiom,
! [B: rat,C: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(divide_divide_rat(B,C))),A))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A)) ) ) ) ) ) ).
% minus_divide_less_eq
tff(fact_5508_less__minus__divide__eq,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real)) ) ) ) ) ) ).
% less_minus_divide_eq
tff(fact_5509_less__minus__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),uminus_uminus_rat(divide_divide_rat(B,C))))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat)) ) ) ) ) ) ).
% less_minus_divide_eq
tff(fact_5510_divide__eq__eq__numeral_I2_J,axiom,
! [B: real,C: real,W: num] :
( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) = aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) )
<=> ( ( ( C != zero_zero_real )
=> ( B = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C) ) )
& ( ( C = zero_zero_real )
=> ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) = zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral(2)
tff(fact_5511_divide__eq__eq__numeral_I2_J,axiom,
! [B: complex,C: complex,W: num] :
( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) )
<=> ( ( ( C != zero_zero_complex )
=> ( B = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),C) ) )
& ( ( C = zero_zero_complex )
=> ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex ) ) ) ) ).
% divide_eq_eq_numeral(2)
tff(fact_5512_divide__eq__eq__numeral_I2_J,axiom,
! [B: rat,C: rat,W: num] :
( ( divide_divide_rat(B,C) = uminus_uminus_rat(numeral_numeral_rat(W)) )
<=> ( ( ( C != zero_zero_rat )
=> ( B = aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C) ) )
& ( ( C = zero_zero_rat )
=> ( uminus_uminus_rat(numeral_numeral_rat(W)) = zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral(2)
tff(fact_5513_eq__divide__eq__numeral_I2_J,axiom,
! [W: num,B: real,C: real] :
( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C) )
<=> ( ( ( C != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C) = B ) )
& ( ( C = zero_zero_real )
=> ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) = zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral(2)
tff(fact_5514_eq__divide__eq__numeral_I2_J,axiom,
! [W: num,B: complex,C: complex] :
( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
<=> ( ( ( C != zero_zero_complex )
=> ( aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),C) = B ) )
& ( ( C = zero_zero_complex )
=> ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex ) ) ) ) ).
% eq_divide_eq_numeral(2)
tff(fact_5515_eq__divide__eq__numeral_I2_J,axiom,
! [W: num,B: rat,C: rat] :
( ( uminus_uminus_rat(numeral_numeral_rat(W)) = divide_divide_rat(B,C) )
<=> ( ( ( C != zero_zero_rat )
=> ( aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C) = B ) )
& ( ( C = zero_zero_rat )
=> ( uminus_uminus_rat(numeral_numeral_rat(W)) = zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral(2)
tff(fact_5516_add__divide__eq__if__simps_I3_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z))),B) = B ) )
& ( ( Z != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z))),B) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z) ) ) ) ).
% add_divide_eq_if_simps(3)
tff(fact_5517_add__divide__eq__if__simps_I3_J,axiom,
! [Z: complex,A: complex,B: complex] :
( ( ( Z = zero_zero_complex )
=> ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B) = B ) )
& ( ( Z != zero_zero_complex )
=> ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(times_times_complex(B),Z))),Z) ) ) ) ).
% add_divide_eq_if_simps(3)
tff(fact_5518_add__divide__eq__if__simps_I3_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( plus_plus_rat(uminus_uminus_rat(divide_divide_rat(A,Z)),B) = B ) )
& ( ( Z != zero_zero_rat )
=> ( plus_plus_rat(uminus_uminus_rat(divide_divide_rat(A,Z)),B) = divide_divide_rat(plus_plus_rat(uminus_uminus_rat(A),aa_rat_rat(times_times_rat(B),Z)),Z) ) ) ) ).
% add_divide_eq_if_simps(3)
tff(fact_5519_minus__divide__add__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z))),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).
% minus_divide_add_eq_iff
tff(fact_5520_minus__divide__add__eq__iff,axiom,
! [Z: complex,X: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(X),Z))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).
% minus_divide_add_eq_iff
tff(fact_5521_minus__divide__add__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( plus_plus_rat(uminus_uminus_rat(divide_divide_rat(X,Z)),Y) = divide_divide_rat(plus_plus_rat(uminus_uminus_rat(X),aa_rat_rat(times_times_rat(Y),Z)),Z) ) ) ).
% minus_divide_add_eq_iff
tff(fact_5522_minus__divide__diff__eq__iff,axiom,
! [Z: real,X: real,Y: real] :
( ( Z != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Z))),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).
% minus_divide_diff_eq_iff
tff(fact_5523_minus__divide__diff__eq__iff,axiom,
! [Z: complex,X: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(X),Z))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),aa_complex_complex(times_times_complex(Y),Z))),Z) ) ) ).
% minus_divide_diff_eq_iff
tff(fact_5524_minus__divide__diff__eq__iff,axiom,
! [Z: rat,X: rat,Y: rat] :
( ( Z != zero_zero_rat )
=> ( minus_minus_rat(uminus_uminus_rat(divide_divide_rat(X,Z)),Y) = divide_divide_rat(minus_minus_rat(uminus_uminus_rat(X),aa_rat_rat(times_times_rat(Y),Z)),Z) ) ) ).
% minus_divide_diff_eq_iff
tff(fact_5525_add__divide__eq__if__simps_I5_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = aa_real_real(uminus_uminus_real,B) ) )
& ( ( Z != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z)),B) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z) ) ) ) ).
% add_divide_eq_if_simps(5)
tff(fact_5526_add__divide__eq__if__simps_I5_J,axiom,
! [Z: complex,A: complex,B: complex] :
( ( ( Z = zero_zero_complex )
=> ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ) )
& ( ( Z != zero_zero_complex )
=> ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z)),B) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(A),aa_complex_complex(times_times_complex(B),Z))),Z) ) ) ) ).
% add_divide_eq_if_simps(5)
tff(fact_5527_add__divide__eq__if__simps_I5_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( minus_minus_rat(divide_divide_rat(A,Z),B) = uminus_uminus_rat(B) ) )
& ( ( Z != zero_zero_rat )
=> ( minus_minus_rat(divide_divide_rat(A,Z),B) = divide_divide_rat(minus_minus_rat(A,aa_rat_rat(times_times_rat(B),Z)),Z) ) ) ) ).
% add_divide_eq_if_simps(5)
tff(fact_5528_add__divide__eq__if__simps_I6_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z))),B) = aa_real_real(uminus_uminus_real,B) ) )
& ( ( Z != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Z))),B) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),Z))),Z) ) ) ) ).
% add_divide_eq_if_simps(6)
tff(fact_5529_add__divide__eq__if__simps_I6_J,axiom,
! [Z: complex,A: complex,B: complex] :
( ( ( Z = zero_zero_complex )
=> ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ) )
& ( ( Z != zero_zero_complex )
=> ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z))),B) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(times_times_complex(B),Z))),Z) ) ) ) ).
% add_divide_eq_if_simps(6)
tff(fact_5530_add__divide__eq__if__simps_I6_J,axiom,
! [Z: rat,A: rat,B: rat] :
( ( ( Z = zero_zero_rat )
=> ( minus_minus_rat(uminus_uminus_rat(divide_divide_rat(A,Z)),B) = uminus_uminus_rat(B) ) )
& ( ( Z != zero_zero_rat )
=> ( minus_minus_rat(uminus_uminus_rat(divide_divide_rat(A,Z)),B) = divide_divide_rat(minus_minus_rat(uminus_uminus_rat(A),aa_rat_rat(times_times_rat(B),Z)),Z) ) ) ) ).
% add_divide_eq_if_simps(6)
tff(fact_5531_even__minus,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(uminus_uminus_int,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) ) ).
% even_minus
tff(fact_5532_even__minus,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)) ) ).
% even_minus
tff(fact_5533_power2__eq__iff,axiom,
! [X: int,Y: int] :
( ( aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
<=> ( ( X = Y )
| ( X = aa_int_int(uminus_uminus_int,Y) ) ) ) ).
% power2_eq_iff
tff(fact_5534_power2__eq__iff,axiom,
! [X: real,Y: real] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
<=> ( ( X = Y )
| ( X = aa_real_real(uminus_uminus_real,Y) ) ) ) ).
% power2_eq_iff
tff(fact_5535_power2__eq__iff,axiom,
! [X: complex,Y: complex] :
( ( aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
<=> ( ( X = Y )
| ( X = aa_complex_complex(uminus1482373934393186551omplex,Y) ) ) ) ).
% power2_eq_iff
tff(fact_5536_power2__eq__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
<=> ( ( X = Y )
| ( X = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y) ) ) ) ).
% power2_eq_iff
tff(fact_5537_power2__eq__iff,axiom,
! [X: rat,Y: rat] :
( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
<=> ( ( X = Y )
| ( X = uminus_uminus_rat(Y) ) ) ) ).
% power2_eq_iff
tff(fact_5538_uminus__power__if,axiom,
! [N: nat,A: int] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N) = aa_nat_int(power_power_int(A),N) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(A),N)) ) ) ) ).
% uminus_power_if
tff(fact_5539_uminus__power__if,axiom,
! [N: nat,A: real] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),N) = aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ) ) ) ).
% uminus_power_if
tff(fact_5540_uminus__power__if,axiom,
! [N: nat,A: complex] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N) = aa_nat_complex(power_power_complex(A),N) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(A),N)) ) ) ) ).
% uminus_power_if
tff(fact_5541_uminus__power__if,axiom,
! [N: nat,A: code_integer] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N) = aa_nat_Code_integer(power_8256067586552552935nteger(A),N) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ) ) ) ).
% uminus_power_if
tff(fact_5542_uminus__power__if,axiom,
! [N: nat,A: rat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),N) = aa_nat_rat(power_power_rat(A),N) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),N) = uminus_uminus_rat(aa_nat_rat(power_power_rat(A),N)) ) ) ) ).
% uminus_power_if
tff(fact_5543_verit__less__mono__div__int2,axiom,
! [A2: int,B2: int,N: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,N)))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),N)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),N))) ) ) ).
% verit_less_mono_div_int2
tff(fact_5544_div__eq__minus1,axiom,
! [B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).
% div_eq_minus1
tff(fact_5545_of__bool__odd__eq__mod__2,axiom,
! [A: nat] : zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% of_bool_odd_eq_mod_2
tff(fact_5546_of__bool__odd__eq__mod__2,axiom,
! [A: int] : aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).
% of_bool_odd_eq_mod_2
tff(fact_5547_of__bool__odd__eq__mod__2,axiom,
! [A: code_integer] : zero_n356916108424825756nteger(aa_bool_bool(fNot,aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))) = modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).
% of_bool_odd_eq_mod_2
tff(fact_5548_le__minus__divide__eq,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) ) ) ) ) ) ).
% le_minus_divide_eq
tff(fact_5549_le__minus__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),uminus_uminus_rat(divide_divide_rat(B,C))))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) ) ) ) ) ) ).
% le_minus_divide_eq
tff(fact_5550_minus__divide__le__eq,axiom,
! [B: real,C: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ) ) ) ) ).
% minus_divide_le_eq
tff(fact_5551_minus__divide__le__eq,axiom,
! [B: rat,C: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(divide_divide_rat(B,C))),A))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ) ) ) ) ).
% minus_divide_le_eq
tff(fact_5552_neg__le__minus__divide__eq,axiom,
! [C: real,A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).
% neg_le_minus_divide_eq
tff(fact_5553_neg__le__minus__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),uminus_uminus_rat(divide_divide_rat(B,C))))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C))) ) ) ).
% neg_le_minus_divide_eq
tff(fact_5554_neg__minus__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) ) ) ).
% neg_minus_divide_le_eq
tff(fact_5555_neg__minus__divide__le__eq,axiom,
! [C: rat,B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(divide_divide_rat(B,C))),A))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B))) ) ) ).
% neg_minus_divide_le_eq
tff(fact_5556_pos__le__minus__divide__eq,axiom,
! [C: real,A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(uminus_uminus_real,B))) ) ) ).
% pos_le_minus_divide_eq
tff(fact_5557_pos__le__minus__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),uminus_uminus_rat(divide_divide_rat(B,C))))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),uminus_uminus_rat(B))) ) ) ).
% pos_le_minus_divide_eq
tff(fact_5558_pos__minus__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C))),A))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ) ).
% pos_minus_divide_le_eq
tff(fact_5559_pos__minus__divide__le__eq,axiom,
! [C: rat,B: rat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(divide_divide_rat(B,C))),A))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(B)),aa_rat_rat(times_times_rat(A),C))) ) ) ).
% pos_minus_divide_le_eq
tff(fact_5560_divide__less__eq__numeral_I2_J,axiom,
! [B: real,C: real,W: num] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C))) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),B)) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
tff(fact_5561_divide__less__eq__numeral_I2_J,axiom,
! [B: rat,C: rat,W: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(B,C)),uminus_uminus_rat(numeral_numeral_rat(W))))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C)),B)) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),uminus_uminus_rat(numeral_numeral_rat(W)))) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
tff(fact_5562_less__divide__eq__numeral_I2_J,axiom,
! [W: num,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),B)) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C))) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),zero_zero_real)) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
tff(fact_5563_less__divide__eq__numeral_I2_J,axiom,
! [W: num,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(W))),divide_divide_rat(B,C)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C)),B)) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B),aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(W))),zero_zero_rat)) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
tff(fact_5564_power2__eq__1__iff,axiom,
! [A: int] :
( ( aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int )
<=> ( ( A = one_one_int )
| ( A = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).
% power2_eq_1_iff
tff(fact_5565_power2__eq__1__iff,axiom,
! [A: real] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
<=> ( ( A = one_one_real )
| ( A = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).
% power2_eq_1_iff
tff(fact_5566_power2__eq__1__iff,axiom,
! [A: complex] :
( ( aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex )
<=> ( ( A = one_one_complex )
| ( A = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).
% power2_eq_1_iff
tff(fact_5567_power2__eq__1__iff,axiom,
! [A: code_integer] :
( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer )
<=> ( ( A = one_one_Code_integer )
| ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).
% power2_eq_1_iff
tff(fact_5568_power2__eq__1__iff,axiom,
! [A: rat] :
( ( aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
<=> ( ( A = one_one_rat )
| ( A = uminus_uminus_rat(one_one_rat) ) ) ) ).
% power2_eq_1_iff
tff(fact_5569_minus__one__power__iff,axiom,
! [N: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N) = one_one_int ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).
% minus_one_power_iff
tff(fact_5570_minus__one__power__iff,axiom,
! [N: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) = one_one_real ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).
% minus_one_power_iff
tff(fact_5571_minus__one__power__iff,axiom,
! [N: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N) = one_one_complex ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).
% minus_one_power_iff
tff(fact_5572_minus__one__power__iff,axiom,
! [N: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N) = one_one_Code_integer ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).
% minus_one_power_iff
tff(fact_5573_minus__one__power__iff,axiom,
! [N: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N) = one_one_rat ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N) = uminus_uminus_rat(one_one_rat) ) ) ) ).
% minus_one_power_iff
tff(fact_5574_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5575_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5576_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5577_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5578_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5579_realpow__square__minus__le,axiom,
! [U: real,X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).
% realpow_square_minus_le
tff(fact_5580_signed__take__bit__int__greater__eq__minus__exp,axiom,
! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),aa_int_int(bit_ri631733984087533419it_int(N),K))) ).
% signed_take_bit_int_greater_eq_minus_exp
tff(fact_5581_signed__take__bit__int__less__eq__self__iff,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),K))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),K)) ) ).
% signed_take_bit_int_less_eq_self_iff
tff(fact_5582_signed__take__bit__int__greater__self__iff,axiom,
! [K: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(bit_ri631733984087533419it_int(N),K)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))) ) ).
% signed_take_bit_int_greater_self_iff
tff(fact_5583_minus__mod__int__eq,axiom,
! [L: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,K)),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,L),one_one_int)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int)),L)) ) ) ).
% minus_mod_int_eq
tff(fact_5584_zmod__minus1,axiom,
! [B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),one_one_int) ) ) ).
% zmod_minus1
tff(fact_5585_zdiv__zminus1__eq__if,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) )
& ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),one_one_int) ) ) ) ) ).
% zdiv_zminus1_eq_if
tff(fact_5586_zdiv__zminus2__eq__if,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) )
& ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),one_one_int) ) ) ) ) ).
% zdiv_zminus2_eq_if
tff(fact_5587_zminus1__lemma,axiom,
! [A: int,B: int,Q2: int,R: int] :
( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R)))
=> ( ( B != zero_zero_int )
=> pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(aa_int_int(uminus_uminus_int,A),B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,R),zero_zero_int)),aa_int_int(uminus_uminus_int,Q2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,Q2)),one_one_int))),aa_int_int(aa_int_fun_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,R),zero_zero_int)),zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),R))))) ) ) ).
% zminus1_lemma
tff(fact_5588_bits__induct,axiom,
! [P: fun_nat_bool,A: nat] :
( ! [A4: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A4 )
=> pp(aa_nat_bool(P,A4)) )
=> ( ! [A4: nat,B4: bool] :
( pp(aa_nat_bool(P,A4))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat(B4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A4))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A4 )
=> pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat(B4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A4)))) ) )
=> pp(aa_nat_bool(P,A)) ) ) ).
% bits_induct
tff(fact_5589_bits__induct,axiom,
! [P: fun_int_bool,A: int] :
( ! [A4: int] :
( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A4),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A4 )
=> pp(aa_int_bool(P,A4)) )
=> ( ! [A4: int,B4: bool] :
( pp(aa_int_bool(P,A4))
=> ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,B4)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A4))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A4 )
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,B4)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A4)))) ) )
=> pp(aa_int_bool(P,A)) ) ) ).
% bits_induct
tff(fact_5590_bits__induct,axiom,
! [P: fun_Co8822979462182099027r_bool,A: code_integer] :
( ! [A4: code_integer] :
( ( divide6298287555418463151nteger(A4,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A4 )
=> pp(aa_Code_integer_bool(P,A4)) )
=> ( ! [A4: code_integer,B4: bool] :
( pp(aa_Code_integer_bool(P,A4))
=> ( ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(zero_n356916108424825756nteger(B4),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),A4)),numera6620942414471956472nteger(aa_num_num(bit0,one))) = A4 )
=> pp(aa_Code_integer_bool(P,plus_p5714425477246183910nteger(zero_n356916108424825756nteger(B4),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),A4)))) ) )
=> pp(aa_Code_integer_bool(P,A)) ) ) ).
% bits_induct
tff(fact_5591_divide__le__eq__numeral_I2_J,axiom,
! [B: real,C: real,W: num] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C))) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),B)) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
tff(fact_5592_divide__le__eq__numeral_I2_J,axiom,
! [B: rat,C: rat,W: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(B,C)),uminus_uminus_rat(numeral_numeral_rat(W))))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C)),B)) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),uminus_uminus_rat(numeral_numeral_rat(W)))) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
tff(fact_5593_le__divide__eq__numeral_I2_J,axiom,
! [W: num,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B),C)))
<=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),B)) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C))) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),zero_zero_real)) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
tff(fact_5594_le__divide__eq__numeral_I2_J,axiom,
! [W: num,B: rat,C: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(W))),divide_divide_rat(B,C)))
<=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C)),B)) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),C))
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C))) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,C),zero_zero_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(W))),zero_zero_rat)) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
tff(fact_5595_square__le__1,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).
% square_le_1
tff(fact_5596_square__le__1,axiom,
! [X: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),X))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),one_one_Code_integer))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer)) ) ) ).
% square_le_1
tff(fact_5597_square__le__1,axiom,
! [X: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(one_one_rat)),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),one_one_rat))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat)) ) ) ).
% square_le_1
tff(fact_5598_square__le__1,axiom,
! [X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),X))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),one_one_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int)) ) ) ).
% square_le_1
tff(fact_5599_minus__power__mult__self,axiom,
! [A: int,N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N)) = aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).
% minus_power_mult_self
tff(fact_5600_minus__power__mult__self,axiom,
! [A: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).
% minus_power_mult_self
tff(fact_5601_minus__power__mult__self,axiom,
! [A: complex,N: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),N)) = aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).
% minus_power_mult_self
tff(fact_5602_minus__power__mult__self,axiom,
! [A: code_integer,N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).
% minus_power_mult_self
tff(fact_5603_minus__power__mult__self,axiom,
! [A: rat,N: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),N)),aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),N)) = aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).
% minus_power_mult_self
tff(fact_5604_signed__take__bit__int__eq__self,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),K))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
=> ( aa_int_int(bit_ri631733984087533419it_int(N),K) = K ) ) ) ).
% signed_take_bit_int_eq_self
tff(fact_5605_signed__take__bit__int__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( aa_int_int(bit_ri631733984087533419it_int(N),K) = K )
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),K))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ).
% signed_take_bit_int_eq_self_iff
tff(fact_5606_minus__1__div__exp__eq__int,axiom,
! [N: 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,aa_num_num(bit0,one))),N)) = aa_int_int(uminus_uminus_int,one_one_int) ).
% minus_1_div_exp_eq_int
tff(fact_5607_div__pos__neg__trivial,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_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_5608_exp__mod__exp,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).
% exp_mod_exp
tff(fact_5609_exp__mod__exp,axiom,
! [M: nat,N: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) ).
% exp_mod_exp
tff(fact_5610_exp__mod__exp,axiom,
! [M: nat,N: nat] : modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)) ).
% exp_mod_exp
tff(fact_5611_divmod__nat__def,axiom,
! [M: nat,N: nat] : divmod_nat(M,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) ).
% divmod_nat_def
tff(fact_5612_power__minus1__odd,axiom,
! [N: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_int_int(uminus_uminus_int,one_one_int) ).
% power_minus1_odd
tff(fact_5613_power__minus1__odd,axiom,
! [N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_real_real(uminus_uminus_real,one_one_real) ).
% power_minus1_odd
tff(fact_5614_power__minus1__odd,axiom,
! [N: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).
% power_minus1_odd
tff(fact_5615_power__minus1__odd,axiom,
! [N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% power_minus1_odd
tff(fact_5616_power__minus1__odd,axiom,
! [N: nat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = uminus_uminus_rat(one_one_rat) ).
% power_minus1_odd
tff(fact_5617_int__bit__induct,axiom,
! [P: fun_int_bool,K: int] :
( pp(aa_int_bool(P,zero_zero_int))
=> ( pp(aa_int_bool(P,aa_int_int(uminus_uminus_int,one_one_int)))
=> ( ! [K3: int] :
( pp(aa_int_bool(P,K3))
=> ( ( K3 != zero_zero_int )
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(times_times_int,K3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ) )
=> ( ! [K3: int] :
( pp(aa_int_bool(P,K3))
=> ( ( K3 != aa_int_int(uminus_uminus_int,one_one_int) )
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,K3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) )
=> pp(aa_int_bool(P,K)) ) ) ) ) ).
% int_bit_induct
tff(fact_5618_signed__take__bit__int__greater__eq,axiom,
! [K: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N)))),aa_int_int(bit_ri631733984087533419it_int(N),K))) ) ).
% signed_take_bit_int_greater_eq
tff(fact_5619_exp__div__exp__eq,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),zero_zero_nat))),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ).
% exp_div_exp_eq
tff(fact_5620_exp__div__exp__eq,axiom,
! [M: nat,N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),zero_zero_int))),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ).
% exp_div_exp_eq
tff(fact_5621_exp__div__exp__eq,axiom,
! [M: nat,N: nat] : divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)),zero_z3403309356797280102nteger))),aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M)))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ).
% exp_div_exp_eq
tff(fact_5622_vebt__buildup_Osimps_I3_J,axiom,
! [Va: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
=> ( vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(suc,Va))) = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))))
=> ( vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(suc,Va))) = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ).
% vebt_buildup.simps(3)
tff(fact_5623_one__div__minus__numeral,axiom,
! [N: 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,N))) = aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(adjust_div,unique5052692396658037445od_int(one,N))) ).
% one_div_minus_numeral
tff(fact_5624_minus__one__div__numeral,axiom,
! [N: 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,N)) = aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(adjust_div,unique5052692396658037445od_int(one,N))) ).
% minus_one_div_numeral
tff(fact_5625_compl__le__compl__iff,axiom,
! [X: set_nat,Y: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(uminus5710092332889474511et_nat(X)),uminus5710092332889474511et_nat(Y)))
<=> pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),X)) ) ).
% compl_le_compl_iff
tff(fact_5626_numeral__div__minus__numeral,axiom,
! [M: num,N: 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,N))) = aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(adjust_div,unique5052692396658037445od_int(M,N))) ).
% numeral_div_minus_numeral
tff(fact_5627_minus__numeral__div__numeral,axiom,
! [M: num,N: 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,N)) = aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(adjust_div,unique5052692396658037445od_int(M,N))) ).
% minus_numeral_div_numeral
tff(fact_5628_dbl__dec__simps_I4_J,axiom,
aa_int_int(neg_nu3811975205180677377ec_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))) ).
% dbl_dec_simps(4)
tff(fact_5629_dbl__dec__simps_I4_J,axiom,
aa_real_real(neg_nu6075765906172075777c_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit1,one))) ).
% dbl_dec_simps(4)
tff(fact_5630_dbl__dec__simps_I4_J,axiom,
aa_complex_complex(neg_nu6511756317524482435omplex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit1,one))) ).
% dbl_dec_simps(4)
tff(fact_5631_dbl__dec__simps_I4_J,axiom,
neg_nu7757733837767384882nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit1,one))) ).
% dbl_dec_simps(4)
tff(fact_5632_dbl__dec__simps_I4_J,axiom,
neg_nu3179335615603231917ec_rat(uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(bit1,one))) ).
% dbl_dec_simps(4)
tff(fact_5633_dbl__dec__simps_I3_J,axiom,
aa_complex_complex(neg_nu6511756317524482435omplex,one_one_complex) = one_one_complex ).
% dbl_dec_simps(3)
tff(fact_5634_dbl__dec__simps_I3_J,axiom,
aa_real_real(neg_nu6075765906172075777c_real,one_one_real) = one_one_real ).
% dbl_dec_simps(3)
tff(fact_5635_dbl__dec__simps_I3_J,axiom,
neg_nu3179335615603231917ec_rat(one_one_rat) = one_one_rat ).
% dbl_dec_simps(3)
tff(fact_5636_dbl__dec__simps_I3_J,axiom,
aa_int_int(neg_nu3811975205180677377ec_int,one_one_int) = one_one_int ).
% dbl_dec_simps(3)
tff(fact_5637_dbl__dec__simps_I5_J,axiom,
! [K: num] : aa_complex_complex(neg_nu6511756317524482435omplex,numera6690914467698888265omplex(K)) = numera6690914467698888265omplex(bitM(K)) ).
% dbl_dec_simps(5)
tff(fact_5638_dbl__dec__simps_I5_J,axiom,
! [K: num] : aa_real_real(neg_nu6075765906172075777c_real,numeral_numeral_real(K)) = numeral_numeral_real(bitM(K)) ).
% dbl_dec_simps(5)
tff(fact_5639_dbl__dec__simps_I5_J,axiom,
! [K: num] : neg_nu3179335615603231917ec_rat(numeral_numeral_rat(K)) = numeral_numeral_rat(bitM(K)) ).
% dbl_dec_simps(5)
tff(fact_5640_dbl__dec__simps_I5_J,axiom,
! [K: num] : aa_int_int(neg_nu3811975205180677377ec_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,bitM(K)) ).
% dbl_dec_simps(5)
tff(fact_5641_dbl__dec__simps_I2_J,axiom,
aa_int_int(neg_nu3811975205180677377ec_int,zero_zero_int) = aa_int_int(uminus_uminus_int,one_one_int) ).
% dbl_dec_simps(2)
tff(fact_5642_dbl__dec__simps_I2_J,axiom,
aa_real_real(neg_nu6075765906172075777c_real,zero_zero_real) = aa_real_real(uminus_uminus_real,one_one_real) ).
% dbl_dec_simps(2)
tff(fact_5643_dbl__dec__simps_I2_J,axiom,
aa_complex_complex(neg_nu6511756317524482435omplex,zero_zero_complex) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).
% dbl_dec_simps(2)
tff(fact_5644_dbl__dec__simps_I2_J,axiom,
neg_nu7757733837767384882nteger(zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).
% dbl_dec_simps(2)
tff(fact_5645_dbl__dec__simps_I2_J,axiom,
neg_nu3179335615603231917ec_rat(zero_zero_rat) = uminus_uminus_rat(one_one_rat) ).
% dbl_dec_simps(2)
tff(fact_5646_Divides_Oadjust__div__eq,axiom,
! [Q2: int,R: int] : aa_Pro6799258434465214121nt_int(adjust_div,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Q2),aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,R),zero_zero_int)))) ).
% Divides.adjust_div_eq
tff(fact_5647_dbl__dec__simps_I1_J,axiom,
! [K: num] : aa_int_int(neg_nu3811975205180677377ec_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_nu5851722552734809277nc_int,aa_num_int(numeral_numeral_int,K))) ).
% dbl_dec_simps(1)
tff(fact_5648_dbl__dec__simps_I1_J,axiom,
! [K: num] : aa_real_real(neg_nu6075765906172075777c_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu8295874005876285629c_real,numeral_numeral_real(K))) ).
% dbl_dec_simps(1)
tff(fact_5649_dbl__dec__simps_I1_J,axiom,
! [K: num] : aa_complex_complex(neg_nu6511756317524482435omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu8557863876264182079omplex,numera6690914467698888265omplex(K))) ).
% dbl_dec_simps(1)
tff(fact_5650_dbl__dec__simps_I1_J,axiom,
! [K: num] : neg_nu7757733837767384882nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu5831290666863070958nteger(numera6620942414471956472nteger(K))) ).
% dbl_dec_simps(1)
tff(fact_5651_dbl__dec__simps_I1_J,axiom,
! [K: num] : neg_nu3179335615603231917ec_rat(uminus_uminus_rat(numeral_numeral_rat(K))) = uminus_uminus_rat(neg_nu5219082963157363817nc_rat(numeral_numeral_rat(K))) ).
% dbl_dec_simps(1)
tff(fact_5652_dbl__inc__simps_I1_J,axiom,
! [K: num] : aa_int_int(neg_nu5851722552734809277nc_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_nu3811975205180677377ec_int,aa_num_int(numeral_numeral_int,K))) ).
% dbl_inc_simps(1)
tff(fact_5653_dbl__inc__simps_I1_J,axiom,
! [K: num] : aa_real_real(neg_nu8295874005876285629c_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu6075765906172075777c_real,numeral_numeral_real(K))) ).
% dbl_inc_simps(1)
tff(fact_5654_dbl__inc__simps_I1_J,axiom,
! [K: num] : aa_complex_complex(neg_nu8557863876264182079omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu6511756317524482435omplex,numera6690914467698888265omplex(K))) ).
% dbl_inc_simps(1)
tff(fact_5655_dbl__inc__simps_I1_J,axiom,
! [K: num] : neg_nu5831290666863070958nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu7757733837767384882nteger(numera6620942414471956472nteger(K))) ).
% dbl_inc_simps(1)
tff(fact_5656_dbl__inc__simps_I1_J,axiom,
! [K: num] : neg_nu5219082963157363817nc_rat(uminus_uminus_rat(numeral_numeral_rat(K))) = uminus_uminus_rat(neg_nu3179335615603231917ec_rat(numeral_numeral_rat(K))) ).
% dbl_inc_simps(1)
tff(fact_5657_Divides_Oadjust__div__def,axiom,
! [Qr: product_prod_int_int] : aa_Pro6799258434465214121nt_int(adjust_div,Qr) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(aTP_Lamm_ed),Qr) ).
% Divides.adjust_div_def
tff(fact_5658_dbl__dec__def,axiom,
! [X: complex] : aa_complex_complex(neg_nu6511756317524482435omplex,X) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(X),X)),one_one_complex) ).
% dbl_dec_def
tff(fact_5659_dbl__dec__def,axiom,
! [X: real] : aa_real_real(neg_nu6075765906172075777c_real,X) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),X)),one_one_real) ).
% dbl_dec_def
tff(fact_5660_dbl__dec__def,axiom,
! [X: rat] : neg_nu3179335615603231917ec_rat(X) = minus_minus_rat(plus_plus_rat(X,X),one_one_rat) ).
% dbl_dec_def
tff(fact_5661_dbl__dec__def,axiom,
! [X: int] : aa_int_int(neg_nu3811975205180677377ec_int,X) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),X)),one_one_int) ).
% dbl_dec_def
tff(fact_5662_compl__le__swap2,axiom,
! [Y: set_nat,X: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Y)),X))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(uminus5710092332889474511et_nat(X)),Y)) ) ).
% compl_le_swap2
tff(fact_5663_compl__le__swap1,axiom,
! [Y: set_nat,X: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(Y),uminus5710092332889474511et_nat(X)))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(X),uminus5710092332889474511et_nat(Y))) ) ).
% compl_le_swap1
tff(fact_5664_compl__mono,axiom,
! [X: set_nat,Y: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Y)),uminus5710092332889474511et_nat(X))) ) ).
% compl_mono
tff(fact_5665_diff__shunt__var,axiom,
! [X: set_int,Y: set_int] :
( ( minus_minus_set_int(X,Y) = bot_bot_set_int )
<=> pp(aa_set_int_bool(ord_less_eq_set_int(X),Y)) ) ).
% diff_shunt_var
tff(fact_5666_diff__shunt__var,axiom,
! [X: set_real,Y: set_real] :
( ( minus_minus_set_real(X,Y) = bot_bot_set_real )
<=> pp(aa_set_real_bool(ord_less_eq_set_real(X),Y)) ) ).
% diff_shunt_var
tff(fact_5667_diff__shunt__var,axiom,
! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
( ( minus_1356011639430497352at_nat(X,Y) = bot_bo2099793752762293965at_nat )
<=> pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(X),Y)) ) ).
% diff_shunt_var
tff(fact_5668_diff__shunt__var,axiom,
! [X: set_nat,Y: set_nat] :
( ( minus_minus_set_nat(X,Y) = bot_bot_set_nat )
<=> pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y)) ) ).
% diff_shunt_var
tff(fact_5669_ln__one__minus__pos__lower__bound,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)))) ) ) ).
% ln_one_minus_pos_lower_bound
tff(fact_5670_of__int__code__if,axiom,
! [K: int] :
( ( ( K = zero_zero_int )
=> ( aa_int_int(ring_1_of_int_int,K) = zero_zero_int ) )
& ( ( K != zero_zero_int )
=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
=> ( aa_int_int(ring_1_of_int_int,K) = aa_int_int(uminus_uminus_int,aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,K))) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
=> ( aa_int_int(ring_1_of_int_int,K) = aa_int_int(aa_int_fun_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_int)) ) ) ) ) ) ).
% of_int_code_if
tff(fact_5671_of__int__code__if,axiom,
! [K: int] :
( ( ( K = zero_zero_int )
=> ( aa_int_real(ring_1_of_int_real,K) = zero_zero_real ) )
& ( ( K != zero_zero_int )
=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
=> ( aa_int_real(ring_1_of_int_real,K) = aa_real_real(uminus_uminus_real,aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,K))) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
=> ( aa_int_real(ring_1_of_int_real,K) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_real)) ) ) ) ) ) ).
% of_int_code_if
tff(fact_5672_of__int__code__if,axiom,
! [K: int] :
( ( ( K = zero_zero_int )
=> ( aa_int_complex(ring_17405671764205052669omplex,K) = zero_zero_complex ) )
& ( ( K != zero_zero_int )
=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
=> ( aa_int_complex(ring_17405671764205052669omplex,K) = aa_complex_complex(uminus1482373934393186551omplex,aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,K))) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
=> ( aa_int_complex(ring_17405671764205052669omplex,K) = aa_complex_complex(if_complex(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_int_complex(ring_17405671764205052669omplex,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_int_complex(ring_17405671764205052669omplex,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),one_one_complex)) ) ) ) ) ) ).
% of_int_code_if
tff(fact_5673_of__int__code__if,axiom,
! [K: int] :
( ( ( K = zero_zero_int )
=> ( aa_int_Code_integer(ring_18347121197199848620nteger,K) = zero_z3403309356797280102nteger ) )
& ( ( K != zero_zero_int )
=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
=> ( aa_int_Code_integer(ring_18347121197199848620nteger,K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,K))) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
=> ( aa_int_Code_integer(ring_18347121197199848620nteger,K) = if_Code_integer(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_Code_integer)) ) ) ) ) ) ).
% of_int_code_if
tff(fact_5674_of__int__code__if,axiom,
! [K: int] :
( ( ( K = zero_zero_int )
=> ( aa_int_rat(ring_1_of_int_rat,K) = zero_zero_rat ) )
& ( ( K != zero_zero_int )
=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
=> ( aa_int_rat(ring_1_of_int_rat,K) = uminus_uminus_rat(aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,K))) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
=> ( aa_int_rat(ring_1_of_int_rat,K) = if_rat(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int),aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),plus_plus_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_rat)) ) ) ) ) ) ).
% of_int_code_if
tff(fact_5675_vebt__buildup_Opelims,axiom,
! [X: nat,Y: vEBT_VEBT] :
( ( vEBT_vebt_buildup(X) = Y )
=> ( pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),X))
=> ( ( ( X = zero_zero_nat )
=> ( ( Y = vEBT_Leaf(fFalse,fFalse) )
=> ~ pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),zero_zero_nat)) ) )
=> ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
=> ( ( Y = vEBT_Leaf(fFalse,fFalse) )
=> ~ pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,zero_zero_nat))) ) )
=> ~ ! [Va3: nat] :
( ( X = aa_nat_nat(suc,aa_nat_nat(suc,Va3)) )
=> ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va3))))
=> ( Y = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va3))))
=> ( Y = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) )
=> ~ pp(aa_nat_bool(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va3)))) ) ) ) ) ) ) ).
% vebt_buildup.pelims
tff(fact_5676_int__ge__less__than__def,axiom,
! [D: int] : int_ge_less_than(D) = collec213857154873943460nt_int(produc8560705615207772305t_bool(aTP_Lamm_ee(D))) ).
% int_ge_less_than_def
tff(fact_5677_int__ge__less__than2__def,axiom,
! [D: int] : int_ge_less_than2(D) = collec213857154873943460nt_int(produc8560705615207772305t_bool(aTP_Lamm_ef(D))) ).
% int_ge_less_than2_def
tff(fact_5678_of__int__numeral,axiom,
! [K: num] : aa_int_complex(ring_17405671764205052669omplex,aa_num_int(numeral_numeral_int,K)) = numera6690914467698888265omplex(K) ).
% of_int_numeral
tff(fact_5679_of__int__numeral,axiom,
! [K: num] : aa_int_real(ring_1_of_int_real,aa_num_int(numeral_numeral_int,K)) = numeral_numeral_real(K) ).
% of_int_numeral
tff(fact_5680_of__int__numeral,axiom,
! [K: num] : aa_int_rat(ring_1_of_int_rat,aa_num_int(numeral_numeral_int,K)) = numeral_numeral_rat(K) ).
% of_int_numeral
tff(fact_5681_of__int__numeral,axiom,
! [K: num] : aa_int_int(ring_1_of_int_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,K) ).
% of_int_numeral
tff(fact_5682_of__int__eq__numeral__iff,axiom,
! [Z: int,N: num] :
( ( aa_int_complex(ring_17405671764205052669omplex,Z) = numera6690914467698888265omplex(N) )
<=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).
% of_int_eq_numeral_iff
tff(fact_5683_of__int__eq__numeral__iff,axiom,
! [Z: int,N: num] :
( ( aa_int_real(ring_1_of_int_real,Z) = numeral_numeral_real(N) )
<=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).
% of_int_eq_numeral_iff
tff(fact_5684_of__int__eq__numeral__iff,axiom,
! [Z: int,N: num] :
( ( aa_int_rat(ring_1_of_int_rat,Z) = numeral_numeral_rat(N) )
<=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).
% of_int_eq_numeral_iff
tff(fact_5685_of__int__eq__numeral__iff,axiom,
! [Z: int,N: num] :
( ( aa_int_int(ring_1_of_int_int,Z) = aa_num_int(numeral_numeral_int,N) )
<=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).
% of_int_eq_numeral_iff
tff(fact_5686_of__int__le__iff,axiom,
! [W: int,Z: int] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z)) ) ).
% of_int_le_iff
tff(fact_5687_of__int__le__iff,axiom,
! [W: int,Z: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z)) ) ).
% of_int_le_iff
tff(fact_5688_of__int__le__iff,axiom,
! [W: int,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z)) ) ).
% of_int_le_iff
tff(fact_5689_of__int__less__iff,axiom,
! [W: int,Z: int] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ).
% of_int_less_iff
tff(fact_5690_of__int__less__iff,axiom,
! [W: int,Z: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ).
% of_int_less_iff
tff(fact_5691_of__int__less__iff,axiom,
! [W: int,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ).
% of_int_less_iff
tff(fact_5692_ln__one,axiom,
aa_real_real(ln_ln_real,one_one_real) = zero_zero_real ).
% ln_one
tff(fact_5693_of__int__1,axiom,
aa_int_complex(ring_17405671764205052669omplex,one_one_int) = one_one_complex ).
% of_int_1
tff(fact_5694_of__int__1,axiom,
aa_int_int(ring_1_of_int_int,one_one_int) = one_one_int ).
% of_int_1
tff(fact_5695_of__int__1,axiom,
aa_int_real(ring_1_of_int_real,one_one_int) = one_one_real ).
% of_int_1
tff(fact_5696_of__int__1,axiom,
aa_int_rat(ring_1_of_int_rat,one_one_int) = one_one_rat ).
% of_int_1
tff(fact_5697_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( aa_int_complex(ring_17405671764205052669omplex,Z) = one_one_complex )
<=> ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
tff(fact_5698_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( aa_int_int(ring_1_of_int_int,Z) = one_one_int )
<=> ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
tff(fact_5699_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( aa_int_real(ring_1_of_int_real,Z) = one_one_real )
<=> ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
tff(fact_5700_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( aa_int_rat(ring_1_of_int_rat,Z) = one_one_rat )
<=> ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
tff(fact_5701_of__int__mult,axiom,
! [W: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)) ).
% of_int_mult
tff(fact_5702_of__int__mult,axiom,
! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)) = aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)) ).
% of_int_mult
tff(fact_5703_of__int__mult,axiom,
! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).
% of_int_mult
tff(fact_5704_of__int__add,axiom,
! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).
% of_int_add
tff(fact_5705_of__int__add,axiom,
! [W: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)) ).
% of_int_add
tff(fact_5706_of__int__add,axiom,
! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),Z)) = plus_plus_rat(aa_int_rat(ring_1_of_int_rat,W),aa_int_rat(ring_1_of_int_rat,Z)) ).
% of_int_add
tff(fact_5707_of__int__diff,axiom,
! [W: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(minus_minus_int,W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)) ).
% of_int_diff
tff(fact_5708_of__int__diff,axiom,
! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(minus_minus_int,W),Z)) = minus_minus_rat(aa_int_rat(ring_1_of_int_rat,W),aa_int_rat(ring_1_of_int_rat,Z)) ).
% of_int_diff
tff(fact_5709_of__int__diff,axiom,
! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,W),Z)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).
% of_int_diff
tff(fact_5710_ln__less__cancel__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ).
% ln_less_cancel_iff
tff(fact_5711_ln__inj__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
=> ( ( aa_real_real(ln_ln_real,X) = aa_real_real(ln_ln_real,Y) )
<=> ( X = Y ) ) ) ) ).
% ln_inj_iff
tff(fact_5712_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( aa_int_rat(ring_1_of_int_rat,X) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W) )
<=> ( X = aa_nat_int(power_power_int(B),W) ) ) ).
% of_int_power_eq_of_int_cancel_iff
tff(fact_5713_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( aa_int_real(ring_1_of_int_real,X) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W) )
<=> ( X = aa_nat_int(power_power_int(B),W) ) ) ).
% of_int_power_eq_of_int_cancel_iff
tff(fact_5714_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( aa_int_complex(ring_17405671764205052669omplex,X) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B)),W) )
<=> ( X = aa_nat_int(power_power_int(B),W) ) ) ).
% of_int_power_eq_of_int_cancel_iff
tff(fact_5715_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( aa_int_int(ring_1_of_int_int,X) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W) )
<=> ( X = aa_nat_int(power_power_int(B),W) ) ) ).
% of_int_power_eq_of_int_cancel_iff
tff(fact_5716_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W) = aa_int_rat(ring_1_of_int_rat,X) )
<=> ( aa_nat_int(power_power_int(B),W) = X ) ) ).
% of_int_eq_of_int_power_cancel_iff
tff(fact_5717_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W) = aa_int_real(ring_1_of_int_real,X) )
<=> ( aa_nat_int(power_power_int(B),W) = X ) ) ).
% of_int_eq_of_int_power_cancel_iff
tff(fact_5718_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B)),W) = aa_int_complex(ring_17405671764205052669omplex,X) )
<=> ( aa_nat_int(power_power_int(B),W) = X ) ) ).
% of_int_eq_of_int_power_cancel_iff
tff(fact_5719_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W) = aa_int_int(ring_1_of_int_int,X) )
<=> ( aa_nat_int(power_power_int(B),W) = X ) ) ).
% of_int_eq_of_int_power_cancel_iff
tff(fact_5720_of__int__power,axiom,
! [Z: int,N: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(power_power_int(Z),N)) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,Z)),N) ).
% of_int_power
tff(fact_5721_of__int__power,axiom,
! [Z: int,N: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(power_power_int(Z),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,Z)),N) ).
% of_int_power
tff(fact_5722_of__int__power,axiom,
! [Z: int,N: nat] : aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(power_power_int(Z),N)) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,Z)),N) ).
% of_int_power
tff(fact_5723_of__int__power,axiom,
! [Z: int,N: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(power_power_int(Z),N)) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,Z)),N) ).
% of_int_power
tff(fact_5724_ln__le__cancel__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ).
% ln_le_cancel_iff
tff(fact_5725_ln__less__zero__iff,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X)),zero_zero_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ) ).
% ln_less_zero_iff
tff(fact_5726_ln__gt__zero__iff,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ) ).
% ln_gt_zero_iff
tff(fact_5727_ln__eq__zero__iff,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( ( aa_real_real(ln_ln_real,X) = zero_zero_real )
<=> ( X = one_one_real ) ) ) ).
% ln_eq_zero_iff
tff(fact_5728_of__int__0__le__iff,axiom,
! [Z: int] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) ) ).
% of_int_0_le_iff
tff(fact_5729_of__int__0__le__iff,axiom,
! [Z: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) ) ).
% of_int_0_le_iff
tff(fact_5730_of__int__0__le__iff,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) ) ).
% of_int_0_le_iff
tff(fact_5731_of__int__le__0__iff,axiom,
! [Z: int] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),zero_zero_real))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int)) ) ).
% of_int_le_0_iff
tff(fact_5732_of__int__le__0__iff,axiom,
! [Z: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int)) ) ).
% of_int_le_0_iff
tff(fact_5733_of__int__le__0__iff,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int)) ) ).
% of_int_le_0_iff
tff(fact_5734_of__int__less__0__iff,axiom,
! [Z: int] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),zero_zero_real))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),zero_zero_int)) ) ).
% of_int_less_0_iff
tff(fact_5735_of__int__less__0__iff,axiom,
! [Z: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),zero_zero_int)) ) ).
% of_int_less_0_iff
tff(fact_5736_of__int__less__0__iff,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),zero_zero_int)) ) ).
% of_int_less_0_iff
tff(fact_5737_of__int__0__less__iff,axiom,
! [Z: int] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).
% of_int_0_less_iff
tff(fact_5738_of__int__0__less__iff,axiom,
! [Z: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).
% of_int_0_less_iff
tff(fact_5739_of__int__0__less__iff,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).
% of_int_0_less_iff
tff(fact_5740_of__int__le__numeral__iff,axiom,
! [Z: int,N: num] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),numeral_numeral_real(N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).
% of_int_le_numeral_iff
tff(fact_5741_of__int__le__numeral__iff,axiom,
! [Z: int,N: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),numeral_numeral_rat(N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).
% of_int_le_numeral_iff
tff(fact_5742_of__int__le__numeral__iff,axiom,
! [Z: int,N: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).
% of_int_le_numeral_iff
tff(fact_5743_of__int__numeral__le__iff,axiom,
! [N: num,Z: int] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(N)),aa_int_real(ring_1_of_int_real,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).
% of_int_numeral_le_iff
tff(fact_5744_of__int__numeral__le__iff,axiom,
! [N: num,Z: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,numeral_numeral_rat(N)),aa_int_rat(ring_1_of_int_rat,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).
% of_int_numeral_le_iff
tff(fact_5745_of__int__numeral__le__iff,axiom,
! [N: num,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),aa_int_int(ring_1_of_int_int,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).
% of_int_numeral_le_iff
tff(fact_5746_of__int__numeral__less__iff,axiom,
! [N: num,Z: int] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(N)),aa_int_real(ring_1_of_int_real,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).
% of_int_numeral_less_iff
tff(fact_5747_of__int__numeral__less__iff,axiom,
! [N: num,Z: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,numeral_numeral_rat(N)),aa_int_rat(ring_1_of_int_rat,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).
% of_int_numeral_less_iff
tff(fact_5748_of__int__numeral__less__iff,axiom,
! [N: num,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),aa_int_int(ring_1_of_int_int,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,N)),Z)) ) ).
% of_int_numeral_less_iff
tff(fact_5749_of__int__less__numeral__iff,axiom,
! [Z: int,N: num] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),numeral_numeral_real(N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).
% of_int_less_numeral_iff
tff(fact_5750_of__int__less__numeral__iff,axiom,
! [Z: int,N: num] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),numeral_numeral_rat(N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).
% of_int_less_numeral_iff
tff(fact_5751_of__int__less__numeral__iff,axiom,
! [Z: int,N: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_num_int(numeral_numeral_int,N))) ) ).
% of_int_less_numeral_iff
tff(fact_5752_of__int__1__le__iff,axiom,
! [Z: int] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_int_real(ring_1_of_int_real,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z)) ) ).
% of_int_1_le_iff
tff(fact_5753_of__int__1__le__iff,axiom,
! [Z: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z)) ) ).
% of_int_1_le_iff
tff(fact_5754_of__int__1__le__iff,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_int_int(ring_1_of_int_int,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),Z)) ) ).
% of_int_1_le_iff
tff(fact_5755_of__int__le__1__iff,axiom,
! [Z: int] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),one_one_int)) ) ).
% of_int_le_1_iff
tff(fact_5756_of__int__le__1__iff,axiom,
! [Z: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),one_one_int)) ) ).
% of_int_le_1_iff
tff(fact_5757_of__int__le__1__iff,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),one_one_int)) ) ).
% of_int_le_1_iff
tff(fact_5758_of__int__less__1__iff,axiom,
! [Z: int] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),one_one_int)) ) ).
% of_int_less_1_iff
tff(fact_5759_of__int__less__1__iff,axiom,
! [Z: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),one_one_int)) ) ).
% of_int_less_1_iff
tff(fact_5760_of__int__less__1__iff,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),one_one_int)) ) ).
% of_int_less_1_iff
tff(fact_5761_of__int__1__less__iff,axiom,
! [Z: int] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_int_real(ring_1_of_int_real,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).
% of_int_1_less_iff
tff(fact_5762_of__int__1__less__iff,axiom,
! [Z: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).
% of_int_1_less_iff
tff(fact_5763_of__int__1__less__iff,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(ring_1_of_int_int,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).
% of_int_1_less_iff
tff(fact_5764_ln__ge__zero__iff,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ) ).
% ln_ge_zero_iff
tff(fact_5765_ln__le__zero__iff,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),zero_zero_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ) ).
% ln_le_zero_iff
tff(fact_5766_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X)),N) )
<=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ) ) ).
% of_int_eq_numeral_power_cancel_iff
tff(fact_5767_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(X)),N) )
<=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ) ) ).
% of_int_eq_numeral_power_cancel_iff
tff(fact_5768_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),N) )
<=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ) ) ).
% of_int_eq_numeral_power_cancel_iff
tff(fact_5769_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) )
<=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ) ) ).
% of_int_eq_numeral_power_cancel_iff
tff(fact_5770_numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X)),N) = aa_int_complex(ring_17405671764205052669omplex,Y) )
<=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = Y ) ) ).
% numeral_power_eq_of_int_cancel_iff
tff(fact_5771_numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(X)),N) = aa_int_real(ring_1_of_int_real,Y) )
<=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = Y ) ) ).
% numeral_power_eq_of_int_cancel_iff
tff(fact_5772_numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),N) = aa_int_rat(ring_1_of_int_rat,Y) )
<=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = Y ) ) ).
% numeral_power_eq_of_int_cancel_iff
tff(fact_5773_numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = aa_int_int(ring_1_of_int_int,Y) )
<=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = Y ) ) ).
% numeral_power_eq_of_int_cancel_iff
tff(fact_5774_of__int__power__le__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(power_power_int(B),W))) ) ).
% of_int_power_le_of_int_cancel_iff
tff(fact_5775_of__int__power__le__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(power_power_int(B),W))) ) ).
% of_int_power_le_of_int_cancel_iff
tff(fact_5776_of__int__power__le__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,X)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(power_power_int(B),W))) ) ).
% of_int_power_le_of_int_cancel_iff
tff(fact_5777_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W)),aa_int_real(ring_1_of_int_real,X)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),X)) ) ).
% of_int_le_of_int_power_cancel_iff
tff(fact_5778_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W)),aa_int_rat(ring_1_of_int_rat,X)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),X)) ) ).
% of_int_le_of_int_power_cancel_iff
tff(fact_5779_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W)),aa_int_int(ring_1_of_int_int,X)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),X)) ) ).
% of_int_le_of_int_power_cancel_iff
tff(fact_5780_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(B),W))) ) ).
% of_int_power_less_of_int_cancel_iff
tff(fact_5781_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,X)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(B),W))) ) ).
% of_int_power_less_of_int_cancel_iff
tff(fact_5782_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,X)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(B),W))) ) ).
% of_int_power_less_of_int_cancel_iff
tff(fact_5783_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W)),aa_int_real(ring_1_of_int_real,X)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),W)),X)) ) ).
% of_int_less_of_int_power_cancel_iff
tff(fact_5784_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W)),aa_int_rat(ring_1_of_int_rat,X)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),W)),X)) ) ).
% of_int_less_of_int_power_cancel_iff
tff(fact_5785_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W)),aa_int_int(ring_1_of_int_int,X)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(B),W)),X)) ) ).
% of_int_less_of_int_power_cancel_iff
tff(fact_5786_numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(X)),N)),aa_int_real(ring_1_of_int_real,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).
% numeral_power_le_of_int_cancel_iff
tff(fact_5787_numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),N)),aa_int_rat(ring_1_of_int_rat,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).
% numeral_power_le_of_int_cancel_iff
tff(fact_5788_numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),aa_int_int(ring_1_of_int_int,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).
% numeral_power_le_of_int_cancel_iff
tff(fact_5789_of__int__le__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(X)),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).
% of_int_le_numeral_power_cancel_iff
tff(fact_5790_of__int__le__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).
% of_int_le_numeral_power_cancel_iff
tff(fact_5791_of__int__le__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).
% of_int_le_numeral_power_cancel_iff
tff(fact_5792_of__int__less__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(X)),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).
% of_int_less_numeral_power_cancel_iff
tff(fact_5793_of__int__less__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).
% of_int_less_numeral_power_cancel_iff
tff(fact_5794_of__int__less__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).
% of_int_less_numeral_power_cancel_iff
tff(fact_5795_numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(X)),N)),aa_int_real(ring_1_of_int_real,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).
% numeral_power_less_of_int_cancel_iff
tff(fact_5796_numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),N)),aa_int_rat(ring_1_of_int_rat,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).
% numeral_power_less_of_int_cancel_iff
tff(fact_5797_numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),aa_int_int(ring_1_of_int_int,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).
% numeral_power_less_of_int_cancel_iff
tff(fact_5798_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) )
<=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5799_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(X))),N) )
<=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5800_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(X))),N) )
<=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5801_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( aa_int_Code_integer(ring_18347121197199848620nteger,Y) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),N) )
<=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5802_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(X))),N) )
<=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5803_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = aa_int_int(ring_1_of_int_int,Y) )
<=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5804_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(X))),N) = aa_int_real(ring_1_of_int_real,Y) )
<=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5805_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(X))),N) = aa_int_complex(ring_17405671764205052669omplex,Y) )
<=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5806_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),N) = aa_int_Code_integer(ring_18347121197199848620nteger,Y) )
<=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5807_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(X))),N) = aa_int_rat(ring_1_of_int_rat,Y) )
<=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N) = Y ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5808_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(X))),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).
% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5809_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).
% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5810_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(X))),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).
% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5811_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).
% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5812_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(X))),N)),aa_int_real(ring_1_of_int_real,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).
% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5813_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),N)),aa_int_Code_integer(ring_18347121197199848620nteger,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).
% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5814_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(X))),N)),aa_int_rat(ring_1_of_int_rat,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).
% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5815_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),aa_int_int(ring_1_of_int_int,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).
% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5816_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).
% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5817_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(X))),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).
% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5818_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).
% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5819_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(X))),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N))) ) ).
% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5820_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),aa_int_int(ring_1_of_int_int,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).
% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5821_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(X))),N)),aa_int_real(ring_1_of_int_real,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).
% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5822_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),N)),aa_int_Code_integer(ring_18347121197199848620nteger,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).
% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5823_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(X))),N)),aa_int_rat(ring_1_of_int_rat,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),N)),A)) ) ).
% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5824_mult__of__int__commute,axiom,
! [X: int,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,X)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_int_real(ring_1_of_int_real,X)) ).
% mult_of_int_commute
tff(fact_5825_mult__of__int__commute,axiom,
! [X: int,Y: rat] : aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,X)),Y) = aa_rat_rat(times_times_rat(Y),aa_int_rat(ring_1_of_int_rat,X)) ).
% mult_of_int_commute
tff(fact_5826_mult__of__int__commute,axiom,
! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,X)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_int_int(ring_1_of_int_int,X)) ).
% mult_of_int_commute
tff(fact_5827_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
tff(fact_5828_bot__enat__def,axiom,
bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).
% bot_enat_def
tff(fact_5829_ln__less__self,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X)),X)) ) ).
% ln_less_self
tff(fact_5830_ln__bound,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),X)) ) ).
% ln_bound
tff(fact_5831_ln__gt__zero__imp__gt__one,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X)))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ) ).
% ln_gt_zero_imp_gt_one
tff(fact_5832_ln__less__zero,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,X)),zero_zero_real)) ) ) ).
% ln_less_zero
tff(fact_5833_ln__gt__zero,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X))) ) ).
% ln_gt_zero
tff(fact_5834_ln__ge__zero,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X))) ) ).
% ln_ge_zero
tff(fact_5835_real__of__int__div4,axiom,
! [N: int,X: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),X))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,N)),aa_int_real(ring_1_of_int_real,X)))) ).
% real_of_int_div4
tff(fact_5836_real__of__int__div,axiom,
! [D: int,N: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D),N))
=> ( aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),D)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,N)),aa_int_real(ring_1_of_int_real,D)) ) ) ).
% real_of_int_div
tff(fact_5837_ln__ge__zero__imp__ge__one,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X)))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ) ).
% ln_ge_zero_imp_ge_one
tff(fact_5838_ln__add__one__self__le__self,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_5839_ln__mult,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
=> ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_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_5840_ln__eq__minus__one,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( ( aa_real_real(ln_ln_real,X) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real) )
=> ( X = one_one_real ) ) ) ).
% ln_eq_minus_one
tff(fact_5841_ln__div,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
=> ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y)) ) ) ) ).
% ln_div
tff(fact_5842_of__int__nonneg,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z))) ) ).
% of_int_nonneg
tff(fact_5843_of__int__nonneg,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))) ) ).
% of_int_nonneg
tff(fact_5844_of__int__nonneg,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))) ) ).
% of_int_nonneg
tff(fact_5845_of__int__pos,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z))) ) ).
% of_int_pos
tff(fact_5846_of__int__pos,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))) ) ).
% of_int_pos
tff(fact_5847_of__int__pos,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))) ) ).
% of_int_pos
tff(fact_5848_of__int__neg__numeral,axiom,
! [K: num] : aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)) ).
% of_int_neg_numeral
tff(fact_5849_of__int__neg__numeral,axiom,
! [K: num] : aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(K)) ).
% of_int_neg_numeral
tff(fact_5850_of__int__neg__numeral,axiom,
! [K: num] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K)) ).
% of_int_neg_numeral
tff(fact_5851_of__int__neg__numeral,axiom,
! [K: num] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K)) ).
% of_int_neg_numeral
tff(fact_5852_of__int__neg__numeral,axiom,
! [K: num] : aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = uminus_uminus_rat(numeral_numeral_rat(K)) ).
% of_int_neg_numeral
tff(fact_5853_int__le__real__less,axiom,
! [N: int,M: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),M))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,M)),one_one_real))) ) ).
% int_le_real_less
tff(fact_5854_int__less__real__le,axiom,
! [N: int,M: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,N),M))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N)),one_one_real)),aa_int_real(ring_1_of_int_real,M))) ) ).
% int_less_real_le
tff(fact_5855_real__of__int__div__aux,axiom,
! [X: int,D: int] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,X)),aa_int_real(ring_1_of_int_real,D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),D))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),D))),aa_int_real(ring_1_of_int_real,D))) ).
% real_of_int_div_aux
tff(fact_5856_ln__2__less__1,axiom,
pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(ln_ln_real,numeral_numeral_real(aa_num_num(bit0,one)))),one_one_real)) ).
% ln_2_less_1
tff(fact_5857_ln__le__minus__one,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real))) ) ).
% ln_le_minus_one
tff(fact_5858_ln__diff__le,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),Y))) ) ) ).
% ln_diff_le
tff(fact_5859_ln__add__one__self__le__self2,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_5860_real__of__int__div2,axiom,
! [N: int,X: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,N)),aa_int_real(ring_1_of_int_real,X))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),X))))) ).
% real_of_int_div2
tff(fact_5861_real__of__int__div3,axiom,
! [N: int,X: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,N)),aa_int_real(ring_1_of_int_real,X))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),X)))),one_one_real)) ).
% real_of_int_div3
tff(fact_5862_ln__one__minus__pos__upper__bound,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X))),aa_real_real(uminus_uminus_real,X))) ) ) ).
% ln_one_minus_pos_upper_bound
tff(fact_5863_even__of__int__iff,axiom,
! [K: int] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(ring_18347121197199848620nteger,K)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)) ) ).
% even_of_int_iff
tff(fact_5864_even__of__int__iff,axiom,
! [K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(ring_1_of_int_int,K)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)) ) ).
% even_of_int_iff
tff(fact_5865_ln__one__plus__pos__lower__bound,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(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_5866_artanh__def,axiom,
! [X: real] : aa_real_real(artanh_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),X)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)))),numeral_numeral_real(aa_num_num(bit0,one))) ).
% artanh_def
tff(fact_5867_floor__exists1,axiom,
! [X: real] :
? [X3: int] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X3)),X))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),one_one_int))))
& ! [Y3: int] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y3)),X))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y3),one_one_int)))) )
=> ( Y3 = X3 ) ) ) ).
% floor_exists1
tff(fact_5868_floor__exists1,axiom,
! [X: rat] :
? [X3: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X3)),X))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),one_one_int))))
& ! [Y3: int] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y3)),X))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y3),one_one_int)))) )
=> ( Y3 = X3 ) ) ) ).
% floor_exists1
tff(fact_5869_floor__exists,axiom,
! [X: real] :
? [Z3: int] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z3)),X))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z3),one_one_int)))) ) ).
% floor_exists
tff(fact_5870_floor__exists,axiom,
! [X: rat] :
? [Z3: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z3)),X))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z3),one_one_int)))) ) ).
% floor_exists
tff(fact_5871_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).
% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_5872_tanh__ln__real,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( aa_real_real(tanh_real,aa_real_real(ln_ln_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).
% tanh_ln_real
tff(fact_5873_abs__ln__one__plus__x__minus__x__bound,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).
% abs_ln_one_plus_x_minus_x_bound
tff(fact_5874_abs__idempotent,axiom,
! [A: real] : aa_real_real(abs_abs_real,aa_real_real(abs_abs_real,A)) = aa_real_real(abs_abs_real,A) ).
% abs_idempotent
tff(fact_5875_abs__idempotent,axiom,
! [A: int] : aa_int_int(abs_abs_int,aa_int_int(abs_abs_int,A)) = aa_int_int(abs_abs_int,A) ).
% abs_idempotent
tff(fact_5876_abs__idempotent,axiom,
! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).
% abs_idempotent
tff(fact_5877_abs__idempotent,axiom,
! [A: rat] : abs_abs_rat(abs_abs_rat(A)) = abs_abs_rat(A) ).
% abs_idempotent
tff(fact_5878_abs__abs,axiom,
! [A: real] : aa_real_real(abs_abs_real,aa_real_real(abs_abs_real,A)) = aa_real_real(abs_abs_real,A) ).
% abs_abs
tff(fact_5879_abs__abs,axiom,
! [A: int] : aa_int_int(abs_abs_int,aa_int_int(abs_abs_int,A)) = aa_int_int(abs_abs_int,A) ).
% abs_abs
tff(fact_5880_abs__abs,axiom,
! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).
% abs_abs
tff(fact_5881_abs__abs,axiom,
! [A: rat] : abs_abs_rat(abs_abs_rat(A)) = abs_abs_rat(A) ).
% abs_abs
tff(fact_5882_abs__zero,axiom,
abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).
% abs_zero
tff(fact_5883_abs__zero,axiom,
aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).
% abs_zero
tff(fact_5884_abs__zero,axiom,
abs_abs_rat(zero_zero_rat) = zero_zero_rat ).
% abs_zero
tff(fact_5885_abs__zero,axiom,
aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).
% abs_zero
tff(fact_5886_abs__eq__0,axiom,
! [A: code_integer] :
( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
<=> ( A = zero_z3403309356797280102nteger ) ) ).
% abs_eq_0
tff(fact_5887_abs__eq__0,axiom,
! [A: real] :
( ( aa_real_real(abs_abs_real,A) = zero_zero_real )
<=> ( A = zero_zero_real ) ) ).
% abs_eq_0
tff(fact_5888_abs__eq__0,axiom,
! [A: rat] :
( ( abs_abs_rat(A) = zero_zero_rat )
<=> ( A = zero_zero_rat ) ) ).
% abs_eq_0
tff(fact_5889_abs__eq__0,axiom,
! [A: int] :
( ( aa_int_int(abs_abs_int,A) = zero_zero_int )
<=> ( A = zero_zero_int ) ) ).
% abs_eq_0
tff(fact_5890_abs__0__eq,axiom,
! [A: code_integer] :
( ( zero_z3403309356797280102nteger = abs_abs_Code_integer(A) )
<=> ( A = zero_z3403309356797280102nteger ) ) ).
% abs_0_eq
tff(fact_5891_abs__0__eq,axiom,
! [A: real] :
( ( zero_zero_real = aa_real_real(abs_abs_real,A) )
<=> ( A = zero_zero_real ) ) ).
% abs_0_eq
tff(fact_5892_abs__0__eq,axiom,
! [A: rat] :
( ( zero_zero_rat = abs_abs_rat(A) )
<=> ( A = zero_zero_rat ) ) ).
% abs_0_eq
tff(fact_5893_abs__0__eq,axiom,
! [A: int] :
( ( zero_zero_int = aa_int_int(abs_abs_int,A) )
<=> ( A = zero_zero_int ) ) ).
% abs_0_eq
tff(fact_5894_abs__0,axiom,
abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).
% abs_0
tff(fact_5895_abs__0,axiom,
aa_complex_complex(abs_abs_complex,zero_zero_complex) = zero_zero_complex ).
% abs_0
tff(fact_5896_abs__0,axiom,
aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).
% abs_0
tff(fact_5897_abs__0,axiom,
abs_abs_rat(zero_zero_rat) = zero_zero_rat ).
% abs_0
tff(fact_5898_abs__0,axiom,
aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).
% abs_0
tff(fact_5899_abs__numeral,axiom,
! [N: num] : abs_abs_Code_integer(numera6620942414471956472nteger(N)) = numera6620942414471956472nteger(N) ).
% abs_numeral
tff(fact_5900_abs__numeral,axiom,
! [N: num] : aa_real_real(abs_abs_real,numeral_numeral_real(N)) = numeral_numeral_real(N) ).
% abs_numeral
tff(fact_5901_abs__numeral,axiom,
! [N: num] : abs_abs_rat(numeral_numeral_rat(N)) = numeral_numeral_rat(N) ).
% abs_numeral
tff(fact_5902_abs__numeral,axiom,
! [N: num] : aa_int_int(abs_abs_int,aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,N) ).
% abs_numeral
tff(fact_5903_abs__mult__self__eq,axiom,
! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(A)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A) ).
% abs_mult_self_eq
tff(fact_5904_abs__mult__self__eq,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A) ).
% abs_mult_self_eq
tff(fact_5905_abs__mult__self__eq,axiom,
! [A: rat] : aa_rat_rat(times_times_rat(abs_abs_rat(A)),abs_abs_rat(A)) = aa_rat_rat(times_times_rat(A),A) ).
% abs_mult_self_eq
tff(fact_5906_abs__mult__self__eq,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,A)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),A) ).
% abs_mult_self_eq
tff(fact_5907_abs__1,axiom,
abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).
% abs_1
tff(fact_5908_abs__1,axiom,
aa_complex_complex(abs_abs_complex,one_one_complex) = one_one_complex ).
% abs_1
tff(fact_5909_abs__1,axiom,
aa_real_real(abs_abs_real,one_one_real) = one_one_real ).
% abs_1
tff(fact_5910_abs__1,axiom,
abs_abs_rat(one_one_rat) = one_one_rat ).
% abs_1
tff(fact_5911_abs__1,axiom,
aa_int_int(abs_abs_int,one_one_int) = one_one_int ).
% abs_1
tff(fact_5912_abs__add__abs,axiom,
! [A: code_integer,B: code_integer] : abs_abs_Code_integer(plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))) = plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)) ).
% abs_add_abs
tff(fact_5913_abs__add__abs,axiom,
! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).
% abs_add_abs
tff(fact_5914_abs__add__abs,axiom,
! [A: rat,B: rat] : abs_abs_rat(plus_plus_rat(abs_abs_rat(A),abs_abs_rat(B))) = plus_plus_rat(abs_abs_rat(A),abs_abs_rat(B)) ).
% abs_add_abs
tff(fact_5915_abs__add__abs,axiom,
! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ).
% abs_add_abs
tff(fact_5916_abs__divide,axiom,
! [A: complex,B: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(abs_abs_complex,A)),aa_complex_complex(abs_abs_complex,B)) ).
% abs_divide
tff(fact_5917_abs__divide,axiom,
! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).
% abs_divide
tff(fact_5918_abs__divide,axiom,
! [A: rat,B: rat] : abs_abs_rat(divide_divide_rat(A,B)) = divide_divide_rat(abs_abs_rat(A),abs_abs_rat(B)) ).
% abs_divide
tff(fact_5919_abs__minus__cancel,axiom,
! [A: int] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,A)) = aa_int_int(abs_abs_int,A) ).
% abs_minus_cancel
tff(fact_5920_abs__minus__cancel,axiom,
! [A: real] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(abs_abs_real,A) ).
% abs_minus_cancel
tff(fact_5921_abs__minus__cancel,axiom,
! [A: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = abs_abs_Code_integer(A) ).
% abs_minus_cancel
tff(fact_5922_abs__minus__cancel,axiom,
! [A: rat] : abs_abs_rat(uminus_uminus_rat(A)) = abs_abs_rat(A) ).
% abs_minus_cancel
tff(fact_5923_abs__minus,axiom,
! [A: int] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,A)) = aa_int_int(abs_abs_int,A) ).
% abs_minus
tff(fact_5924_abs__minus,axiom,
! [A: real] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(abs_abs_real,A) ).
% abs_minus
tff(fact_5925_abs__minus,axiom,
! [A: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(uminus1482373934393186551omplex,A)) = aa_complex_complex(abs_abs_complex,A) ).
% abs_minus
tff(fact_5926_abs__minus,axiom,
! [A: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = abs_abs_Code_integer(A) ).
% abs_minus
tff(fact_5927_abs__minus,axiom,
! [A: rat] : abs_abs_rat(uminus_uminus_rat(A)) = abs_abs_rat(A) ).
% abs_minus
tff(fact_5928_dvd__abs__iff,axiom,
! [M: real,K: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,M),aa_real_real(abs_abs_real,K)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,M),K)) ) ).
% dvd_abs_iff
tff(fact_5929_dvd__abs__iff,axiom,
! [M: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),aa_int_int(abs_abs_int,K)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),K)) ) ).
% dvd_abs_iff
tff(fact_5930_dvd__abs__iff,axiom,
! [M: code_integer,K: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,M),abs_abs_Code_integer(K)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,M),K)) ) ).
% dvd_abs_iff
tff(fact_5931_dvd__abs__iff,axiom,
! [M: rat,K: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,M),abs_abs_rat(K)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,M),K)) ) ).
% dvd_abs_iff
tff(fact_5932_abs__dvd__iff,axiom,
! [M: real,K: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(abs_abs_real,M)),K))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,M),K)) ) ).
% abs_dvd_iff
tff(fact_5933_abs__dvd__iff,axiom,
! [M: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(abs_abs_int,M)),K))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M),K)) ) ).
% abs_dvd_iff
tff(fact_5934_abs__dvd__iff,axiom,
! [M: code_integer,K: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,abs_abs_Code_integer(M)),K))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,M),K)) ) ).
% abs_dvd_iff
tff(fact_5935_abs__dvd__iff,axiom,
! [M: rat,K: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,abs_abs_rat(M)),K))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,M),K)) ) ).
% abs_dvd_iff
tff(fact_5936_abs__bool__eq,axiom,
! [P: bool] : aa_real_real(abs_abs_real,zero_n3304061248610475627l_real(P)) = zero_n3304061248610475627l_real(P) ).
% abs_bool_eq
tff(fact_5937_abs__bool__eq,axiom,
! [P: bool] : abs_abs_rat(zero_n2052037380579107095ol_rat(P)) = zero_n2052037380579107095ol_rat(P) ).
% abs_bool_eq
tff(fact_5938_abs__bool__eq,axiom,
! [P: bool] : aa_int_int(abs_abs_int,aa_bool_int(zero_n2684676970156552555ol_int,P)) = aa_bool_int(zero_n2684676970156552555ol_int,P) ).
% abs_bool_eq
tff(fact_5939_abs__bool__eq,axiom,
! [P: bool] : abs_abs_Code_integer(zero_n356916108424825756nteger(P)) = zero_n356916108424825756nteger(P) ).
% abs_bool_eq
tff(fact_5940_tanh__real__less__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tanh_real,X)),aa_real_real(tanh_real,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).
% tanh_real_less_iff
tff(fact_5941_tanh__real__le__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(tanh_real,X)),aa_real_real(tanh_real,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).
% tanh_real_le_iff
tff(fact_5942_abs__le__zero__iff,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),zero_z3403309356797280102nteger))
<=> ( A = zero_z3403309356797280102nteger ) ) ).
% abs_le_zero_iff
tff(fact_5943_abs__le__zero__iff,axiom,
! [A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),zero_zero_real))
<=> ( A = zero_zero_real ) ) ).
% abs_le_zero_iff
tff(fact_5944_abs__le__zero__iff,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(A)),zero_zero_rat))
<=> ( A = zero_zero_rat ) ) ).
% abs_le_zero_iff
tff(fact_5945_abs__le__zero__iff,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),zero_zero_int))
<=> ( A = zero_zero_int ) ) ).
% abs_le_zero_iff
tff(fact_5946_abs__le__self__iff,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),A))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)) ) ).
% abs_le_self_iff
tff(fact_5947_abs__le__self__iff,axiom,
! [A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),A))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A)) ) ).
% abs_le_self_iff
tff(fact_5948_abs__le__self__iff,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(A)),A))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A)) ) ).
% abs_le_self_iff
tff(fact_5949_abs__le__self__iff,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),A))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ).
% abs_le_self_iff
tff(fact_5950_abs__of__nonneg,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
=> ( abs_abs_Code_integer(A) = A ) ) ).
% abs_of_nonneg
tff(fact_5951_abs__of__nonneg,axiom,
! [A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
=> ( aa_real_real(abs_abs_real,A) = A ) ) ).
% abs_of_nonneg
tff(fact_5952_abs__of__nonneg,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( abs_abs_rat(A) = A ) ) ).
% abs_of_nonneg
tff(fact_5953_abs__of__nonneg,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
=> ( aa_int_int(abs_abs_int,A) = A ) ) ).
% abs_of_nonneg
tff(fact_5954_zero__less__abs__iff,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A)))
<=> ( A != zero_z3403309356797280102nteger ) ) ).
% zero_less_abs_iff
tff(fact_5955_zero__less__abs__iff,axiom,
! [A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,A)))
<=> ( A != zero_zero_real ) ) ).
% zero_less_abs_iff
tff(fact_5956_zero__less__abs__iff,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),abs_abs_rat(A)))
<=> ( A != zero_zero_rat ) ) ).
% zero_less_abs_iff
tff(fact_5957_zero__less__abs__iff,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(abs_abs_int,A)))
<=> ( A != zero_zero_int ) ) ).
% zero_less_abs_iff
tff(fact_5958_abs__neg__numeral,axiom,
! [N: num] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,N) ).
% abs_neg_numeral
tff(fact_5959_abs__neg__numeral,axiom,
! [N: num] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = numeral_numeral_real(N) ).
% abs_neg_numeral
tff(fact_5960_abs__neg__numeral,axiom,
! [N: num] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))) = numera6620942414471956472nteger(N) ).
% abs_neg_numeral
tff(fact_5961_abs__neg__numeral,axiom,
! [N: num] : abs_abs_rat(uminus_uminus_rat(numeral_numeral_rat(N))) = numeral_numeral_rat(N) ).
% abs_neg_numeral
tff(fact_5962_abs__neg__one,axiom,
aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,one_one_int)) = one_one_int ).
% abs_neg_one
tff(fact_5963_abs__neg__one,axiom,
aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,one_one_real)) = one_one_real ).
% abs_neg_one
tff(fact_5964_abs__neg__one,axiom,
abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = one_one_Code_integer ).
% abs_neg_one
tff(fact_5965_abs__neg__one,axiom,
abs_abs_rat(uminus_uminus_rat(one_one_rat)) = one_one_rat ).
% abs_neg_one
tff(fact_5966_abs__power__minus,axiom,
! [A: int,N: nat] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N)) = aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A),N)) ).
% abs_power_minus
tff(fact_5967_abs__power__minus,axiom,
! [A: real,N: nat] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),N)) = aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) ).
% abs_power_minus
tff(fact_5968_abs__power__minus,axiom,
! [A: code_integer,N: nat] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),N)) = abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) ).
% abs_power_minus
tff(fact_5969_abs__power__minus,axiom,
! [A: rat,N: nat] : abs_abs_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(A)),N)) = abs_abs_rat(aa_nat_rat(power_power_rat(A),N)) ).
% abs_power_minus
tff(fact_5970_tanh__real__pos__iff,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(tanh_real,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).
% tanh_real_pos_iff
tff(fact_5971_tanh__real__neg__iff,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tanh_real,X)),zero_zero_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).
% tanh_real_neg_iff
tff(fact_5972_tanh__real__nonneg__iff,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(tanh_real,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).
% tanh_real_nonneg_iff
tff(fact_5973_tanh__real__nonpos__iff,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(tanh_real,X)),zero_zero_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).
% tanh_real_nonpos_iff
tff(fact_5974_divide__le__0__abs__iff,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(abs_abs_real,B))),zero_zero_real))
<=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
| ( B = zero_zero_real ) ) ) ).
% divide_le_0_abs_iff
tff(fact_5975_divide__le__0__abs__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(A,abs_abs_rat(B))),zero_zero_rat))
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
| ( B = zero_zero_rat ) ) ) ).
% divide_le_0_abs_iff
tff(fact_5976_zero__le__divide__abs__iff,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(abs_abs_real,B))))
<=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
| ( B = zero_zero_real ) ) ) ).
% zero_le_divide_abs_iff
tff(fact_5977_zero__le__divide__abs__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(A,abs_abs_rat(B))))
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
| ( B = zero_zero_rat ) ) ) ).
% zero_le_divide_abs_iff
tff(fact_5978_abs__of__nonpos,axiom,
! [A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
=> ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) ) ).
% abs_of_nonpos
tff(fact_5979_abs__of__nonpos,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger))
=> ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).
% abs_of_nonpos
tff(fact_5980_abs__of__nonpos,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat))
=> ( abs_abs_rat(A) = uminus_uminus_rat(A) ) ) ).
% abs_of_nonpos
tff(fact_5981_abs__of__nonpos,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int))
=> ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).
% abs_of_nonpos
tff(fact_5982_artanh__minus__real,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
=> ( aa_real_real(artanh_real,aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(artanh_real,X)) ) ) ).
% artanh_minus_real
tff(fact_5983_zero__less__power__abs__iff,axiom,
! [A: code_integer,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),N)))
<=> ( ( A != zero_z3403309356797280102nteger )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
tff(fact_5984_zero__less__power__abs__iff,axiom,
! [A: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),N)))
<=> ( ( A != zero_zero_real )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
tff(fact_5985_zero__less__power__abs__iff,axiom,
! [A: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(abs_abs_rat(A)),N)))
<=> ( ( A != zero_zero_rat )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
tff(fact_5986_zero__less__power__abs__iff,axiom,
! [A: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),N)))
<=> ( ( A != zero_zero_int )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
tff(fact_5987_power2__abs,axiom,
! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% power2_abs
tff(fact_5988_power2__abs,axiom,
! [A: rat] : aa_nat_rat(power_power_rat(abs_abs_rat(A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% power2_abs
tff(fact_5989_power2__abs,axiom,
! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% power2_abs
tff(fact_5990_power2__abs,axiom,
! [A: int] : aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% power2_abs
tff(fact_5991_abs__power2,axiom,
! [A: code_integer] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% abs_power2
tff(fact_5992_abs__power2,axiom,
! [A: rat] : abs_abs_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% abs_power2
tff(fact_5993_abs__power2,axiom,
! [A: real] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% abs_power2
tff(fact_5994_abs__power2,axiom,
! [A: int] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% abs_power2
tff(fact_5995_power__even__abs__numeral,axiom,
! [W: num,A: code_integer] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
=> ( aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).
% power_even_abs_numeral
tff(fact_5996_power__even__abs__numeral,axiom,
! [W: num,A: rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
=> ( aa_nat_rat(power_power_rat(abs_abs_rat(A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).
% power_even_abs_numeral
tff(fact_5997_power__even__abs__numeral,axiom,
! [W: num,A: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).
% power_even_abs_numeral
tff(fact_5998_power__even__abs__numeral,axiom,
! [W: num,A: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W)))
=> ( aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).
% power_even_abs_numeral
tff(fact_5999_abs__ge__self,axiom,
! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),aa_real_real(abs_abs_real,A))) ).
% abs_ge_self
tff(fact_6000_abs__ge__self,axiom,
! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),abs_abs_Code_integer(A))) ).
% abs_ge_self
tff(fact_6001_abs__ge__self,axiom,
! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),abs_abs_rat(A))) ).
% abs_ge_self
tff(fact_6002_abs__ge__self,axiom,
! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_int_int(abs_abs_int,A))) ).
% abs_ge_self
tff(fact_6003_abs__le__D1,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ).
% abs_le_D1
tff(fact_6004_abs__le__D1,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B)) ) ).
% abs_le_D1
tff(fact_6005_abs__le__D1,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(A)),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B)) ) ).
% abs_le_D1
tff(fact_6006_abs__le__D1,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ).
% abs_le_D1
tff(fact_6007_abs__eq__0__iff,axiom,
! [A: code_integer] :
( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
<=> ( A = zero_z3403309356797280102nteger ) ) ).
% abs_eq_0_iff
tff(fact_6008_abs__eq__0__iff,axiom,
! [A: complex] :
( ( aa_complex_complex(abs_abs_complex,A) = zero_zero_complex )
<=> ( A = zero_zero_complex ) ) ).
% abs_eq_0_iff
tff(fact_6009_abs__eq__0__iff,axiom,
! [A: real] :
( ( aa_real_real(abs_abs_real,A) = zero_zero_real )
<=> ( A = zero_zero_real ) ) ).
% abs_eq_0_iff
tff(fact_6010_abs__eq__0__iff,axiom,
! [A: rat] :
( ( abs_abs_rat(A) = zero_zero_rat )
<=> ( A = zero_zero_rat ) ) ).
% abs_eq_0_iff
tff(fact_6011_abs__eq__0__iff,axiom,
! [A: int] :
( ( aa_int_int(abs_abs_int,A) = zero_zero_int )
<=> ( A = zero_zero_int ) ) ).
% abs_eq_0_iff
tff(fact_6012_abs__mult,axiom,
! [A: code_integer,B: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B)) ).
% abs_mult
tff(fact_6013_abs__mult,axiom,
! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).
% abs_mult
tff(fact_6014_abs__mult,axiom,
! [A: rat,B: rat] : abs_abs_rat(aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(abs_abs_rat(A)),abs_abs_rat(B)) ).
% abs_mult
tff(fact_6015_abs__mult,axiom,
! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ).
% abs_mult
tff(fact_6016_abs__one,axiom,
abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).
% abs_one
tff(fact_6017_abs__one,axiom,
aa_real_real(abs_abs_real,one_one_real) = one_one_real ).
% abs_one
tff(fact_6018_abs__one,axiom,
abs_abs_rat(one_one_rat) = one_one_rat ).
% abs_one
tff(fact_6019_abs__one,axiom,
aa_int_int(abs_abs_int,one_one_int) = one_one_int ).
% abs_one
tff(fact_6020_abs__minus__commute,axiom,
! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)) ).
% abs_minus_commute
tff(fact_6021_abs__minus__commute,axiom,
! [A: code_integer,B: code_integer] : abs_abs_Code_integer(minus_8373710615458151222nteger(A,B)) = abs_abs_Code_integer(minus_8373710615458151222nteger(B,A)) ).
% abs_minus_commute
tff(fact_6022_abs__minus__commute,axiom,
! [A: rat,B: rat] : abs_abs_rat(minus_minus_rat(A,B)) = abs_abs_rat(minus_minus_rat(B,A)) ).
% abs_minus_commute
tff(fact_6023_abs__minus__commute,axiom,
! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)) ).
% abs_minus_commute
tff(fact_6024_abs__eq__iff,axiom,
! [X: int,Y: int] :
( ( aa_int_int(abs_abs_int,X) = aa_int_int(abs_abs_int,Y) )
<=> ( ( X = Y )
| ( X = aa_int_int(uminus_uminus_int,Y) ) ) ) ).
% abs_eq_iff
tff(fact_6025_abs__eq__iff,axiom,
! [X: real,Y: real] :
( ( aa_real_real(abs_abs_real,X) = aa_real_real(abs_abs_real,Y) )
<=> ( ( X = Y )
| ( X = aa_real_real(uminus_uminus_real,Y) ) ) ) ).
% abs_eq_iff
tff(fact_6026_abs__eq__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( abs_abs_Code_integer(X) = abs_abs_Code_integer(Y) )
<=> ( ( X = Y )
| ( X = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y) ) ) ) ).
% abs_eq_iff
tff(fact_6027_abs__eq__iff,axiom,
! [X: rat,Y: rat] :
( ( abs_abs_rat(X) = abs_abs_rat(Y) )
<=> ( ( X = Y )
| ( X = uminus_uminus_rat(Y) ) ) ) ).
% abs_eq_iff
tff(fact_6028_power__abs,axiom,
! [A: code_integer,N: nat] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),N) ).
% power_abs
tff(fact_6029_power__abs,axiom,
! [A: rat,N: nat] : abs_abs_rat(aa_nat_rat(power_power_rat(A),N)) = aa_nat_rat(power_power_rat(abs_abs_rat(A)),N) ).
% power_abs
tff(fact_6030_power__abs,axiom,
! [A: real,N: nat] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),N) ).
% power_abs
tff(fact_6031_power__abs,axiom,
! [A: int,N: nat] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A),N)) = aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),N) ).
% power_abs
tff(fact_6032_dvd__if__abs__eq,axiom,
! [L: real,K: real] :
( ( aa_real_real(abs_abs_real,L) = aa_real_real(abs_abs_real,K) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,L),K)) ) ).
% dvd_if_abs_eq
tff(fact_6033_dvd__if__abs__eq,axiom,
! [L: int,K: int] :
( ( aa_int_int(abs_abs_int,L) = aa_int_int(abs_abs_int,K) )
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K)) ) ).
% dvd_if_abs_eq
tff(fact_6034_dvd__if__abs__eq,axiom,
! [L: code_integer,K: code_integer] :
( ( abs_abs_Code_integer(L) = abs_abs_Code_integer(K) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,L),K)) ) ).
% dvd_if_abs_eq
tff(fact_6035_dvd__if__abs__eq,axiom,
! [L: rat,K: rat] :
( ( abs_abs_rat(L) = abs_abs_rat(K) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,L),K)) ) ).
% dvd_if_abs_eq
tff(fact_6036_abs__ge__zero,axiom,
! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A))) ).
% abs_ge_zero
tff(fact_6037_abs__ge__zero,axiom,
! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(abs_abs_real,A))) ).
% abs_ge_zero
tff(fact_6038_abs__ge__zero,axiom,
! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),abs_abs_rat(A))) ).
% abs_ge_zero
tff(fact_6039_abs__ge__zero,axiom,
! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(abs_abs_int,A))) ).
% abs_ge_zero
tff(fact_6040_abs__of__pos,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A))
=> ( abs_abs_Code_integer(A) = A ) ) ).
% abs_of_pos
tff(fact_6041_abs__of__pos,axiom,
! [A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ( aa_real_real(abs_abs_real,A) = A ) ) ).
% abs_of_pos
tff(fact_6042_abs__of__pos,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( abs_abs_rat(A) = A ) ) ).
% abs_of_pos
tff(fact_6043_abs__of__pos,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
=> ( aa_int_int(abs_abs_int,A) = A ) ) ).
% abs_of_pos
tff(fact_6044_abs__not__less__zero,axiom,
! [A: code_integer] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),zero_z3403309356797280102nteger)) ).
% abs_not_less_zero
tff(fact_6045_abs__not__less__zero,axiom,
! [A: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,A)),zero_zero_real)) ).
% abs_not_less_zero
tff(fact_6046_abs__not__less__zero,axiom,
! [A: rat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,abs_abs_rat(A)),zero_zero_rat)) ).
% abs_not_less_zero
tff(fact_6047_abs__not__less__zero,axiom,
! [A: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,A)),zero_zero_int)) ).
% abs_not_less_zero
tff(fact_6048_abs__triangle__ineq,axiom,
! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(plus_p5714425477246183910nteger(A,B))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)))) ).
% abs_triangle_ineq
tff(fact_6049_abs__triangle__ineq,axiom,
! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))) ).
% abs_triangle_ineq
tff(fact_6050_abs__triangle__ineq,axiom,
! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(plus_plus_rat(A,B))),plus_plus_rat(abs_abs_rat(A),abs_abs_rat(B)))) ).
% abs_triangle_ineq
tff(fact_6051_abs__triangle__ineq,axiom,
! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))) ).
% abs_triangle_ineq
tff(fact_6052_abs__mult__less,axiom,
! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),C))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(B)),D))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D))) ) ) ).
% abs_mult_less
tff(fact_6053_abs__mult__less,axiom,
! [A: real,C: real,B: real,D: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,A)),C))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,B)),D))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),D))) ) ) ).
% abs_mult_less
tff(fact_6054_abs__mult__less,axiom,
! [A: rat,C: rat,B: rat,D: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,abs_abs_rat(A)),C))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,abs_abs_rat(B)),D))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(abs_abs_rat(A)),abs_abs_rat(B))),aa_rat_rat(times_times_rat(C),D))) ) ) ).
% abs_mult_less
tff(fact_6055_abs__mult__less,axiom,
! [A: int,C: int,B: int,D: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,A)),C))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,B)),D))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(aa_int_fun_int_int(times_times_int,C),D))) ) ) ).
% abs_mult_less
tff(fact_6056_abs__triangle__ineq2,axiom,
! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)))) ).
% abs_triangle_ineq2
tff(fact_6057_abs__triangle__ineq2,axiom,
! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))),abs_abs_Code_integer(minus_8373710615458151222nteger(A,B)))) ).
% abs_triangle_ineq2
tff(fact_6058_abs__triangle__ineq2,axiom,
! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,minus_minus_rat(abs_abs_rat(A),abs_abs_rat(B))),abs_abs_rat(minus_minus_rat(A,B)))) ).
% abs_triangle_ineq2
tff(fact_6059_abs__triangle__ineq2,axiom,
! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)))) ).
% abs_triangle_ineq2
tff(fact_6060_abs__triangle__ineq3,axiom,
! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)))) ).
% abs_triangle_ineq3
tff(fact_6061_abs__triangle__ineq3,axiom,
! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)))),abs_abs_Code_integer(minus_8373710615458151222nteger(A,B)))) ).
% abs_triangle_ineq3
tff(fact_6062_abs__triangle__ineq3,axiom,
! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(abs_abs_rat(A),abs_abs_rat(B)))),abs_abs_rat(minus_minus_rat(A,B)))) ).
% abs_triangle_ineq3
tff(fact_6063_abs__triangle__ineq3,axiom,
! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)))) ).
% abs_triangle_ineq3
tff(fact_6064_abs__triangle__ineq2__sym,axiom,
! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)))) ).
% abs_triangle_ineq2_sym
tff(fact_6065_abs__triangle__ineq2__sym,axiom,
! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))),abs_abs_Code_integer(minus_8373710615458151222nteger(B,A)))) ).
% abs_triangle_ineq2_sym
tff(fact_6066_abs__triangle__ineq2__sym,axiom,
! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,minus_minus_rat(abs_abs_rat(A),abs_abs_rat(B))),abs_abs_rat(minus_minus_rat(B,A)))) ).
% abs_triangle_ineq2_sym
tff(fact_6067_abs__triangle__ineq2__sym,axiom,
! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)))) ).
% abs_triangle_ineq2_sym
tff(fact_6068_nonzero__abs__divide,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).
% nonzero_abs_divide
tff(fact_6069_nonzero__abs__divide,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( abs_abs_rat(divide_divide_rat(A,B)) = divide_divide_rat(abs_abs_rat(A),abs_abs_rat(B)) ) ) ).
% nonzero_abs_divide
tff(fact_6070_abs__ge__minus__self,axiom,
! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(abs_abs_real,A))) ).
% abs_ge_minus_self
tff(fact_6071_abs__ge__minus__self,axiom,
! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),abs_abs_Code_integer(A))) ).
% abs_ge_minus_self
tff(fact_6072_abs__ge__minus__self,axiom,
! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(A)),abs_abs_rat(A))) ).
% abs_ge_minus_self
tff(fact_6073_abs__ge__minus__self,axiom,
! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(abs_abs_int,A))) ).
% abs_ge_minus_self
tff(fact_6074_abs__le__iff,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B))
<=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)) ) ) ).
% abs_le_iff
tff(fact_6075_abs__le__iff,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B))
<=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
& pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)) ) ) ).
% abs_le_iff
tff(fact_6076_abs__le__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(A)),B))
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(A)),B)) ) ) ).
% abs_le_iff
tff(fact_6077_abs__le__iff,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)) ) ) ).
% abs_le_iff
tff(fact_6078_abs__le__D2,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)) ) ).
% abs_le_D2
tff(fact_6079_abs__le__D2,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)) ) ).
% abs_le_D2
tff(fact_6080_abs__le__D2,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(A)),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(A)),B)) ) ).
% abs_le_D2
tff(fact_6081_abs__le__D2,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)) ) ).
% abs_le_D2
tff(fact_6082_abs__leI,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)) ) ) ).
% abs_leI
tff(fact_6083_abs__leI,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),B))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)) ) ) ).
% abs_leI
tff(fact_6084_abs__leI,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(A)),B))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(A)),B)) ) ) ).
% abs_leI
tff(fact_6085_abs__leI,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)) ) ) ).
% abs_leI
tff(fact_6086_abs__less__iff,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,A)),B))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,A)),B)) ) ) ).
% abs_less_iff
tff(fact_6087_abs__less__iff,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,A)),B))
<=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,A)),B)) ) ) ).
% abs_less_iff
tff(fact_6088_abs__less__iff,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),B))
<=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),B))
& pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)) ) ) ).
% abs_less_iff
tff(fact_6089_abs__less__iff,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,abs_abs_rat(A)),B))
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),B))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(A)),B)) ) ) ).
% abs_less_iff
tff(fact_6090_tanh__real__lt__1,axiom,
! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tanh_real,X)),one_one_real)) ).
% tanh_real_lt_1
tff(fact_6091_dense__eq0__I,axiom,
! [X: real] :
( ! [E2: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),E2)) )
=> ( X = zero_zero_real ) ) ).
% dense_eq0_I
tff(fact_6092_dense__eq0__I,axiom,
! [X: rat] :
( ! [E2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(X)),E2)) )
=> ( X = zero_zero_rat ) ) ).
% dense_eq0_I
tff(fact_6093_abs__eq__mult,axiom,
! [A: code_integer,B: code_integer] :
( ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
| pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)) )
& ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
| pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger)) ) )
=> ( abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B)) ) ) ).
% abs_eq_mult
tff(fact_6094_abs__eq__mult,axiom,
! [A: real,B: real] :
( ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
| pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real)) )
& ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
| pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),zero_zero_real)) ) )
=> ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).
% abs_eq_mult
tff(fact_6095_abs__eq__mult,axiom,
! [A: rat,B: rat] :
( ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
| pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,A),zero_zero_rat)) )
& ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
| pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,B),zero_zero_rat)) ) )
=> ( abs_abs_rat(aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(abs_abs_rat(A)),abs_abs_rat(B)) ) ) ).
% abs_eq_mult
tff(fact_6096_abs__eq__mult,axiom,
! [A: int,B: int] :
( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
| pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
| pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),zero_zero_int)) ) )
=> ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ) ) ).
% abs_eq_mult
tff(fact_6097_abs__mult__pos,axiom,
! [X: code_integer,Y: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X))
=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(Y)),X) = abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),X)) ) ) ).
% abs_mult_pos
tff(fact_6098_abs__mult__pos,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,Y)),X) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),X)) ) ) ).
% abs_mult_pos
tff(fact_6099_abs__mult__pos,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X))
=> ( aa_rat_rat(times_times_rat(abs_abs_rat(Y)),X) = abs_abs_rat(aa_rat_rat(times_times_rat(Y),X)) ) ) ).
% abs_mult_pos
tff(fact_6100_abs__mult__pos,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Y)),X) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),X)) ) ) ).
% abs_mult_pos
tff(fact_6101_abs__eq__iff_H,axiom,
! [A: real,B: real] :
( ( aa_real_real(abs_abs_real,A) = B )
<=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
& ( ( A = B )
| ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ) ).
% abs_eq_iff'
tff(fact_6102_abs__eq__iff_H,axiom,
! [A: code_integer,B: code_integer] :
( ( abs_abs_Code_integer(A) = B )
<=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B))
& ( ( A = B )
| ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ) ).
% abs_eq_iff'
tff(fact_6103_abs__eq__iff_H,axiom,
! [A: rat,B: rat] :
( ( abs_abs_rat(A) = B )
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
& ( ( A = B )
| ( A = uminus_uminus_rat(B) ) ) ) ) ).
% abs_eq_iff'
tff(fact_6104_abs__eq__iff_H,axiom,
! [A: int,B: int] :
( ( aa_int_int(abs_abs_int,A) = B )
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
& ( ( A = B )
| ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ) ).
% abs_eq_iff'
tff(fact_6105_eq__abs__iff_H,axiom,
! [A: real,B: real] :
( ( A = aa_real_real(abs_abs_real,B) )
<=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
& ( ( B = A )
| ( B = aa_real_real(uminus_uminus_real,A) ) ) ) ) ).
% eq_abs_iff'
tff(fact_6106_eq__abs__iff_H,axiom,
! [A: code_integer,B: code_integer] :
( ( A = abs_abs_Code_integer(B) )
<=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A))
& ( ( B = A )
| ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ) ) ).
% eq_abs_iff'
tff(fact_6107_eq__abs__iff_H,axiom,
! [A: rat,B: rat] :
( ( A = abs_abs_rat(B) )
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
& ( ( B = A )
| ( B = uminus_uminus_rat(A) ) ) ) ) ).
% eq_abs_iff'
tff(fact_6108_eq__abs__iff_H,axiom,
! [A: int,B: int] :
( ( A = aa_int_int(abs_abs_int,B) )
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A))
& ( ( B = A )
| ( B = aa_int_int(uminus_uminus_int,A) ) ) ) ) ).
% eq_abs_iff'
tff(fact_6109_abs__minus__le__zero,axiom,
! [A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(abs_abs_real,A))),zero_zero_real)) ).
% abs_minus_le_zero
tff(fact_6110_abs__minus__le__zero,axiom,
! [A: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,abs_abs_Code_integer(A))),zero_z3403309356797280102nteger)) ).
% abs_minus_le_zero
tff(fact_6111_abs__minus__le__zero,axiom,
! [A: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(abs_abs_rat(A))),zero_zero_rat)) ).
% abs_minus_le_zero
tff(fact_6112_abs__minus__le__zero,axiom,
! [A: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_int_int(abs_abs_int,A))),zero_zero_int)) ).
% abs_minus_le_zero
tff(fact_6113_abs__div__pos,axiom,
! [Y: real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
=> ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,X)),Y) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) ) ) ).
% abs_div_pos
tff(fact_6114_abs__div__pos,axiom,
! [Y: rat,X: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Y))
=> ( divide_divide_rat(abs_abs_rat(X),Y) = abs_abs_rat(divide_divide_rat(X,Y)) ) ) ).
% abs_div_pos
tff(fact_6115_zero__le__power__abs,axiom,
! [A: code_integer,N: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),N))) ).
% zero_le_power_abs
tff(fact_6116_zero__le__power__abs,axiom,
! [A: real,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),N))) ).
% zero_le_power_abs
tff(fact_6117_zero__le__power__abs,axiom,
! [A: rat,N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(abs_abs_rat(A)),N))) ).
% zero_le_power_abs
tff(fact_6118_zero__le__power__abs,axiom,
! [A: int,N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),N))) ).
% zero_le_power_abs
tff(fact_6119_abs__if__raw,axiom,
! [X4: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),zero_zero_int))
=> ( aa_int_int(abs_abs_int,X4) = aa_int_int(uminus_uminus_int,X4) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),zero_zero_int))
=> ( aa_int_int(abs_abs_int,X4) = X4 ) ) ) ).
% abs_if_raw
tff(fact_6120_abs__if__raw,axiom,
! [X4: real] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),zero_zero_real))
=> ( aa_real_real(abs_abs_real,X4) = aa_real_real(uminus_uminus_real,X4) ) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),zero_zero_real))
=> ( aa_real_real(abs_abs_real,X4) = X4 ) ) ) ).
% abs_if_raw
tff(fact_6121_abs__if__raw,axiom,
! [X4: code_integer] :
( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X4),zero_z3403309356797280102nteger))
=> ( abs_abs_Code_integer(X4) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X4) ) )
& ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X4),zero_z3403309356797280102nteger))
=> ( abs_abs_Code_integer(X4) = X4 ) ) ) ).
% abs_if_raw
tff(fact_6122_abs__if__raw,axiom,
! [X4: rat] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),zero_zero_rat))
=> ( abs_abs_rat(X4) = uminus_uminus_rat(X4) ) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),zero_zero_rat))
=> ( abs_abs_rat(X4) = X4 ) ) ) ).
% abs_if_raw
tff(fact_6123_abs__if,axiom,
! [A: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
=> ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
=> ( aa_int_int(abs_abs_int,A) = A ) ) ) ).
% abs_if
tff(fact_6124_abs__if,axiom,
! [A: real] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
=> ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
=> ( aa_real_real(abs_abs_real,A) = A ) ) ) ).
% abs_if
tff(fact_6125_abs__if,axiom,
! [A: code_integer] :
( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
=> ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) )
& ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
=> ( abs_abs_Code_integer(A) = A ) ) ) ).
% abs_if
tff(fact_6126_abs__if,axiom,
! [A: rat] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
=> ( abs_abs_rat(A) = uminus_uminus_rat(A) ) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
=> ( abs_abs_rat(A) = A ) ) ) ).
% abs_if
tff(fact_6127_abs__of__neg,axiom,
! [A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int))
=> ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).
% abs_of_neg
tff(fact_6128_abs__of__neg,axiom,
! [A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
=> ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) ) ).
% abs_of_neg
tff(fact_6129_abs__of__neg,axiom,
! [A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger))
=> ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).
% abs_of_neg
tff(fact_6130_abs__of__neg,axiom,
! [A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
=> ( abs_abs_rat(A) = uminus_uminus_rat(A) ) ) ).
% abs_of_neg
tff(fact_6131_abs__diff__le__iff,axiom,
! [X: code_integer,A: code_integer,R: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(X,A))),R))
<=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(A,R)),X))
& pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),plus_p5714425477246183910nteger(A,R))) ) ) ).
% abs_diff_le_iff
tff(fact_6132_abs__diff__le__iff,axiom,
! [X: real,A: real,R: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),A))),R))
<=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),R)),X))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R))) ) ) ).
% abs_diff_le_iff
tff(fact_6133_abs__diff__le__iff,axiom,
! [X: rat,A: rat,R: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(X,A))),R))
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,minus_minus_rat(A,R)),X))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),plus_plus_rat(A,R))) ) ) ).
% abs_diff_le_iff
tff(fact_6134_abs__diff__le__iff,axiom,
! [X: int,A: int,R: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),A))),R))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R)),X))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R))) ) ) ).
% abs_diff_le_iff
tff(fact_6135_abs__diff__triangle__ineq,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(plus_p5714425477246183910nteger(A,B),plus_p5714425477246183910nteger(C,D)))),plus_p5714425477246183910nteger(abs_abs_Code_integer(minus_8373710615458151222nteger(A,C)),abs_abs_Code_integer(minus_8373710615458151222nteger(B,D))))) ).
% abs_diff_triangle_ineq
tff(fact_6136_abs__diff__triangle__ineq,axiom,
! [A: real,B: real,C: real,D: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D))))) ).
% abs_diff_triangle_ineq
tff(fact_6137_abs__diff__triangle__ineq,axiom,
! [A: rat,B: rat,C: rat,D: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(plus_plus_rat(A,B),plus_plus_rat(C,D)))),plus_plus_rat(abs_abs_rat(minus_minus_rat(A,C)),abs_abs_rat(minus_minus_rat(B,D))))) ).
% abs_diff_triangle_ineq
tff(fact_6138_abs__diff__triangle__ineq,axiom,
! [A: int,B: int,C: int,D: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),D)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D))))) ).
% abs_diff_triangle_ineq
tff(fact_6139_abs__triangle__ineq4,axiom,
! [A: code_integer,B: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(A,B))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)))) ).
% abs_triangle_ineq4
tff(fact_6140_abs__triangle__ineq4,axiom,
! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))) ).
% abs_triangle_ineq4
tff(fact_6141_abs__triangle__ineq4,axiom,
! [A: rat,B: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(A,B))),plus_plus_rat(abs_abs_rat(A),abs_abs_rat(B)))) ).
% abs_triangle_ineq4
tff(fact_6142_abs__triangle__ineq4,axiom,
! [A: int,B: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))) ).
% abs_triangle_ineq4
tff(fact_6143_abs__diff__less__iff,axiom,
! [X: code_integer,A: code_integer,R: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(X,A))),R))
<=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(A,R)),X))
& pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),plus_p5714425477246183910nteger(A,R))) ) ) ).
% abs_diff_less_iff
tff(fact_6144_abs__diff__less__iff,axiom,
! [X: real,A: real,R: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),A))),R))
<=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),R)),X))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R))) ) ) ).
% abs_diff_less_iff
tff(fact_6145_abs__diff__less__iff,axiom,
! [X: rat,A: rat,R: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,abs_abs_rat(minus_minus_rat(X,A))),R))
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(A,R)),X))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),plus_plus_rat(A,R))) ) ) ).
% abs_diff_less_iff
tff(fact_6146_abs__diff__less__iff,axiom,
! [X: int,A: int,R: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),A))),R))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R)),X))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R))) ) ) ).
% abs_diff_less_iff
tff(fact_6147_abs__real__def,axiom,
! [A: real] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
=> ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
=> ( aa_real_real(abs_abs_real,A) = A ) ) ) ).
% abs_real_def
tff(fact_6148_lemma__interval__lt,axiom,
! [A: real,X: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),B))
=> ? [D2: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
& ! [Y3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y3))),D2))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Y3))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),B)) ) ) ) ) ) ).
% lemma_interval_lt
tff(fact_6149_sin__bound__lemma,axiom,
! [X: real,Y: real,U: real,V: real] :
( ( X = Y )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,U)),V))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),U)),Y))),V)) ) ) ).
% sin_bound_lemma
tff(fact_6150_tanh__real__gt__neg1,axiom,
! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(tanh_real,X))) ).
% tanh_real_gt_neg1
tff(fact_6151_abs__add__one__gt__zero,axiom,
! [X: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(one_one_Code_integer,abs_abs_Code_integer(X)))) ).
% abs_add_one_gt_zero
tff(fact_6152_abs__add__one__gt__zero,axiom,
! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(abs_abs_real,X)))) ).
% abs_add_one_gt_zero
tff(fact_6153_abs__add__one__gt__zero,axiom,
! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),plus_plus_rat(one_one_rat,abs_abs_rat(X)))) ).
% abs_add_one_gt_zero
tff(fact_6154_abs__add__one__gt__zero,axiom,
! [X: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(abs_abs_int,X)))) ).
% abs_add_one_gt_zero
tff(fact_6155_of__int__leD,axiom,
! [N: int,X: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,N))),X))
=> ( ( N = zero_zero_int )
| pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),X)) ) ) ).
% of_int_leD
tff(fact_6156_of__int__leD,axiom,
! [N: int,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,N))),X))
=> ( ( N = zero_zero_int )
| pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ) ).
% of_int_leD
tff(fact_6157_of__int__leD,axiom,
! [N: int,X: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(aa_int_rat(ring_1_of_int_rat,N))),X))
=> ( ( N = zero_zero_int )
| pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),X)) ) ) ).
% of_int_leD
tff(fact_6158_of__int__leD,axiom,
! [N: int,X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,N))),X))
=> ( ( N = zero_zero_int )
| pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),X)) ) ) ).
% of_int_leD
tff(fact_6159_of__int__lessD,axiom,
! [N: int,X: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,N))),X))
=> ( ( N = zero_zero_int )
| pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),X)) ) ) ).
% of_int_lessD
tff(fact_6160_of__int__lessD,axiom,
! [N: int,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,N))),X))
=> ( ( N = zero_zero_int )
| pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ) ).
% of_int_lessD
tff(fact_6161_of__int__lessD,axiom,
! [N: int,X: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,abs_abs_rat(aa_int_rat(ring_1_of_int_rat,N))),X))
=> ( ( N = zero_zero_int )
| pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),X)) ) ) ).
% of_int_lessD
tff(fact_6162_of__int__lessD,axiom,
! [N: int,X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,N))),X))
=> ( ( N = zero_zero_int )
| pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),X)) ) ) ).
% of_int_lessD
tff(fact_6163_lemma__interval,axiom,
! [A: real,X: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),B))
=> ? [D2: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
& ! [Y3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y3))),D2))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Y3))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),B)) ) ) ) ) ) ).
% lemma_interval
tff(fact_6164_abs__le__square__iff,axiom,
! [X: code_integer,Y: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(X)),abs_abs_Code_integer(Y)))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).
% abs_le_square_iff
tff(fact_6165_abs__le__square__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(abs_abs_real,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).
% abs_le_square_iff
tff(fact_6166_abs__le__square__iff,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(X)),abs_abs_rat(Y)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).
% abs_le_square_iff
tff(fact_6167_abs__le__square__iff,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,X)),aa_int_int(abs_abs_int,Y)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).
% abs_le_square_iff
tff(fact_6168_abs__square__eq__1,axiom,
! [X: code_integer] :
( ( aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer )
<=> ( abs_abs_Code_integer(X) = one_one_Code_integer ) ) ).
% abs_square_eq_1
tff(fact_6169_abs__square__eq__1,axiom,
! [X: rat] :
( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
<=> ( abs_abs_rat(X) = one_one_rat ) ) ).
% abs_square_eq_1
tff(fact_6170_abs__square__eq__1,axiom,
! [X: real] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
<=> ( aa_real_real(abs_abs_real,X) = one_one_real ) ) ).
% abs_square_eq_1
tff(fact_6171_abs__square__eq__1,axiom,
! [X: int] :
( ( aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int )
<=> ( aa_int_int(abs_abs_int,X) = one_one_int ) ) ).
% abs_square_eq_1
tff(fact_6172_power__even__abs,axiom,
! [N: nat,A: code_integer] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),N) = aa_nat_Code_integer(power_8256067586552552935nteger(A),N) ) ) ).
% power_even_abs
tff(fact_6173_power__even__abs,axiom,
! [N: nat,A: rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_rat(power_power_rat(abs_abs_rat(A)),N) = aa_nat_rat(power_power_rat(A),N) ) ) ).
% power_even_abs
tff(fact_6174_power__even__abs,axiom,
! [N: nat,A: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N) ) ) ).
% power_even_abs
tff(fact_6175_power__even__abs,axiom,
! [N: nat,A: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),N) = aa_nat_int(power_power_int(A),N) ) ) ).
% power_even_abs
tff(fact_6176_power2__le__iff__abs__le,axiom,
! [Y: code_integer,X: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(X)),Y)) ) ) ).
% power2_le_iff_abs_le
tff(fact_6177_power2__le__iff__abs__le,axiom,
! [Y: real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),Y)) ) ) ).
% power2_le_iff_abs_le
tff(fact_6178_power2__le__iff__abs__le,axiom,
! [Y: rat,X: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),Y))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(X)),Y)) ) ) ).
% power2_le_iff_abs_le
tff(fact_6179_power2__le__iff__abs__le,axiom,
! [Y: int,X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,X)),Y)) ) ) ).
% power2_le_iff_abs_le
tff(fact_6180_abs__sqrt__wlog,axiom,
! [P: fun_Co8310303586393106575r_bool,X: code_integer] :
( ! [X3: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X3))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(P,X3),aa_nat_Code_integer(power_8256067586552552935nteger(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(P,abs_abs_Code_integer(X)),aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).
% abs_sqrt_wlog
tff(fact_6181_abs__sqrt__wlog,axiom,
! [P: fun_re1255401033344146673l_bool,X: real] :
( ! [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,X3),aa_nat_real(aa_real_fun_nat_real(power_power_real,X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(P,aa_real_real(abs_abs_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).
% abs_sqrt_wlog
tff(fact_6182_abs__sqrt__wlog,axiom,
! [P: fun_rat_fun_rat_bool,X: rat] :
( ! [X3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X3))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(P,X3),aa_nat_rat(power_power_rat(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(P,abs_abs_rat(X)),aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).
% abs_sqrt_wlog
tff(fact_6183_abs__sqrt__wlog,axiom,
! [P: fun_int_fun_int_bool,X: int] :
( ! [X3: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X3))
=> pp(aa_int_bool(aa_int_fun_int_bool(P,X3),aa_nat_int(power_power_int(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
=> pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(abs_abs_int,X)),aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).
% abs_sqrt_wlog
tff(fact_6184_abs__square__le__1,axiom,
! [X: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(X)),one_one_Code_integer)) ) ).
% abs_square_le_1
tff(fact_6185_abs__square__le__1,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)) ) ).
% abs_square_le_1
tff(fact_6186_abs__square__le__1,axiom,
! [X: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(X)),one_one_rat)) ) ).
% abs_square_le_1
tff(fact_6187_abs__square__le__1,axiom,
! [X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,X)),one_one_int)) ) ).
% abs_square_le_1
tff(fact_6188_abs__square__less__1,axiom,
! [X: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,abs_abs_Code_integer(X)),one_one_Code_integer)) ) ).
% abs_square_less_1
tff(fact_6189_abs__square__less__1,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real)) ) ).
% abs_square_less_1
tff(fact_6190_abs__square__less__1,axiom,
! [X: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,abs_abs_rat(X)),one_one_rat)) ) ).
% abs_square_less_1
tff(fact_6191_abs__square__less__1,axiom,
! [X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,X)),one_one_int)) ) ).
% abs_square_less_1
tff(fact_6192_power__mono__even,axiom,
! [N: nat,A: code_integer,B: code_integer] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B)))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B),N))) ) ) ).
% power_mono_even
tff(fact_6193_power__mono__even,axiom,
! [N: nat,A: real,B: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N))) ) ) ).
% power_mono_even
tff(fact_6194_power__mono__even,axiom,
! [N: nat,A: rat,B: rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(A)),abs_abs_rat(B)))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N)),aa_nat_rat(power_power_rat(B),N))) ) ) ).
% power_mono_even
tff(fact_6195_power__mono__even,axiom,
! [N: nat,A: int,B: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N))) ) ) ).
% power_mono_even
tff(fact_6196_exists__least__lemma,axiom,
! [P: fun_nat_bool] :
( ~ pp(aa_nat_bool(P,zero_zero_nat))
=> ( ? [X_12: nat] : pp(aa_nat_bool(P,X_12))
=> ? [N2: nat] :
( ~ pp(aa_nat_bool(P,N2))
& pp(aa_nat_bool(P,aa_nat_nat(suc,N2))) ) ) ) ).
% exists_least_lemma
tff(fact_6197_ex__le__of__int,axiom,
! [X: real] :
? [Z3: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,Z3))) ).
% ex_le_of_int
tff(fact_6198_ex__le__of__int,axiom,
! [X: rat] :
? [Z3: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z3))) ).
% ex_le_of_int
tff(fact_6199_ex__of__int__less,axiom,
! [X: real] :
? [Z3: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z3)),X)) ).
% ex_of_int_less
tff(fact_6200_ex__of__int__less,axiom,
! [X: rat] :
? [Z3: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z3)),X)) ).
% ex_of_int_less
tff(fact_6201_ex__less__of__int,axiom,
! [X: real] :
? [Z3: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,Z3))) ).
% ex_less_of_int
tff(fact_6202_ex__less__of__int,axiom,
! [X: rat] :
? [Z3: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,Z3))) ).
% ex_less_of_int
tff(fact_6203_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).
% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_6204_round__unique,axiom,
! [X: real,Y: int] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,Y)))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))))
=> ( aa_real_int(archim8280529875227126926d_real,X) = Y ) ) ) ).
% round_unique
tff(fact_6205_round__unique,axiom,
! [X: rat,Y: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(X,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,Y)))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y)),plus_plus_rat(X,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))))
=> ( archim7778729529865785530nd_rat(X) = Y ) ) ) ).
% round_unique
tff(fact_6206_round__unique_H,axiom,
! [X: real,N: int] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_int_real(ring_1_of_int_real,N)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))))
=> ( aa_real_int(archim8280529875227126926d_real,X) = N ) ) ).
% round_unique'
tff(fact_6207_round__unique_H,axiom,
! [X: rat,N: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,abs_abs_rat(minus_minus_rat(X,aa_int_rat(ring_1_of_int_rat,N)))),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one)))))
=> ( archim7778729529865785530nd_rat(X) = N ) ) ).
% round_unique'
tff(fact_6208_of__int__round__abs__le,axiom,
! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X))),X))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))) ).
% of_int_round_abs_le
tff(fact_6209_of__int__round__abs__le,axiom,
! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(X)),X))),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))) ).
% of_int_round_abs_le
tff(fact_6210_arctan__double,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(arctan,X)) = aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),X)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).
% arctan_double
tff(fact_6211_of__int__round__gt,axiom,
! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X)))) ).
% of_int_round_gt
tff(fact_6212_of__int__round__gt,axiom,
! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(X,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(X)))) ).
% of_int_round_gt
tff(fact_6213_of__int__round__ge,axiom,
! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X)))) ).
% of_int_round_ge
tff(fact_6214_of__int__round__ge,axiom,
! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,minus_minus_rat(X,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(X)))) ).
% of_int_round_ge
tff(fact_6215_zabs__less__one__iff,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,Z)),one_one_int))
<=> ( Z = zero_zero_int ) ) ).
% zabs_less_one_iff
tff(fact_6216_arctan__less__zero__iff,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,X)),zero_zero_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).
% arctan_less_zero_iff
tff(fact_6217_zero__less__arctan__iff,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(arctan,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).
% zero_less_arctan_iff
tff(fact_6218_zero__le__arctan__iff,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(arctan,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).
% zero_le_arctan_iff
tff(fact_6219_arctan__le__zero__iff,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arctan,X)),zero_zero_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).
% arctan_le_zero_iff
tff(fact_6220_round__numeral,axiom,
! [N: num] : aa_real_int(archim8280529875227126926d_real,numeral_numeral_real(N)) = aa_num_int(numeral_numeral_int,N) ).
% round_numeral
tff(fact_6221_round__numeral,axiom,
! [N: num] : archim7778729529865785530nd_rat(numeral_numeral_rat(N)) = aa_num_int(numeral_numeral_int,N) ).
% round_numeral
tff(fact_6222_round__1,axiom,
aa_real_int(archim8280529875227126926d_real,one_one_real) = one_one_int ).
% round_1
tff(fact_6223_round__1,axiom,
archim7778729529865785530nd_rat(one_one_rat) = one_one_int ).
% round_1
tff(fact_6224_round__neg__numeral,axiom,
! [N: num] : aa_real_int(archim8280529875227126926d_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).
% round_neg_numeral
tff(fact_6225_round__neg__numeral,axiom,
! [N: num] : archim7778729529865785530nd_rat(uminus_uminus_rat(numeral_numeral_rat(N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).
% round_neg_numeral
tff(fact_6226_arctan__less__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).
% arctan_less_iff
tff(fact_6227_arctan__monotone,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y))) ) ).
% arctan_monotone
tff(fact_6228_arctan__monotone_H,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y))) ) ).
% arctan_monotone'
tff(fact_6229_arctan__le__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).
% arctan_le_iff
tff(fact_6230_abs__div,axiom,
! [Y: int,X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Y),X))
=> ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,X)),aa_int_int(abs_abs_int,Y)) ) ) ).
% abs_div
tff(fact_6231_round__mono,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,archim7778729529865785530nd_rat(X)),archim7778729529865785530nd_rat(Y))) ) ).
% round_mono
tff(fact_6232_zabs__def,axiom,
! [I: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),zero_zero_int))
=> ( aa_int_int(abs_abs_int,I) = aa_int_int(uminus_uminus_int,I) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),zero_zero_int))
=> ( aa_int_int(abs_abs_int,I) = I ) ) ) ).
% zabs_def
tff(fact_6233_abs__mod__less,axiom,
! [L: int,K: int] :
( ( L != zero_zero_int )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))),aa_int_int(abs_abs_int,L))) ) ).
% abs_mod_less
tff(fact_6234_even__add__abs__iff,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(abs_abs_int,L))))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L))) ) ).
% even_add_abs_iff
tff(fact_6235_even__abs__add__iff,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,K)),L)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L))) ) ).
% even_abs_add_iff
tff(fact_6236_round__diff__minimal,axiom,
! [Z: real,M: int] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Z))))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),aa_int_real(ring_1_of_int_real,M))))) ).
% round_diff_minimal
tff(fact_6237_round__diff__minimal,axiom,
! [Z: rat,M: int] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(Z,aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(Z))))),abs_abs_rat(minus_minus_rat(Z,aa_int_rat(ring_1_of_int_rat,M))))) ).
% round_diff_minimal
tff(fact_6238_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F: fun_nat_int,K: int] :
( ! [I2: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),I2))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N)) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int)) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,M)),K))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(F,N)))
=> ? [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),I2))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),N))
& ( aa_nat_int(F,I2) = K ) ) ) ) ) ) ).
% nat_intermed_int_val
tff(fact_6239_incr__lemma,axiom,
! [D: int,Z: int,X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z))),one_one_int)),D)))) ) ).
% incr_lemma
tff(fact_6240_decr__lemma,axiom,
! [D: int,X: int,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z))),one_one_int)),D))),Z)) ) ).
% decr_lemma
tff(fact_6241_nat__ivt__aux,axiom,
! [N: nat,F: fun_nat_int,K: int] :
( ! [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int)) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(F,N)))
=> ? [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),N))
& ( aa_nat_int(F,I2) = K ) ) ) ) ) ).
% nat_ivt_aux
tff(fact_6242_nat0__intermed__int__val,axiom,
! [N: nat,F: fun_nat_int,K: int] :
( ! [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_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)) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(F,N)))
=> ? [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),N))
& ( aa_nat_int(F,I2) = K ) ) ) ) ) ).
% nat0_intermed_int_val
tff(fact_6243_arctan__add,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)))) ) ) ) ).
% arctan_add
tff(fact_6244_of__int__round__le,axiom,
! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))))) ).
% of_int_round_le
tff(fact_6245_of__int__round__le,axiom,
! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(X))),plus_plus_rat(X,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one)))))) ).
% of_int_round_le
tff(fact_6246_set__decode__0,axiom,
! [X: nat] :
( pp(aa_set_nat_bool(member_nat(zero_zero_nat),aa_nat_set_nat(nat_set_decode,X)))
<=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X)) ) ).
% set_decode_0
tff(fact_6247_set__decode__Suc,axiom,
! [N: nat,X: nat] :
( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,N)),aa_nat_set_nat(nat_set_decode,X)))
<=> pp(aa_set_nat_bool(member_nat(N),aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).
% set_decode_Suc
tff(fact_6248_add__scale__eq__noteq,axiom,
! [R: complex,A: complex,B: complex,C: complex,D: complex] :
( ( R != zero_zero_complex )
=> ( ( ( A = B )
& ( C != D ) )
=> ( aa_complex_complex(plus_plus_complex(A),aa_complex_complex(times_times_complex(R),C)) != aa_complex_complex(plus_plus_complex(B),aa_complex_complex(times_times_complex(R),D)) ) ) ) ).
% add_scale_eq_noteq
tff(fact_6249_add__scale__eq__noteq,axiom,
! [R: real,A: real,B: real,C: real,D: real] :
( ( R != zero_zero_real )
=> ( ( ( A = B )
& ( C != D ) )
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),C)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),D)) ) ) ) ).
% add_scale_eq_noteq
tff(fact_6250_add__scale__eq__noteq,axiom,
! [R: rat,A: rat,B: rat,C: rat,D: rat] :
( ( R != zero_zero_rat )
=> ( ( ( A = B )
& ( C != D ) )
=> ( plus_plus_rat(A,aa_rat_rat(times_times_rat(R),C)) != plus_plus_rat(B,aa_rat_rat(times_times_rat(R),D)) ) ) ) ).
% add_scale_eq_noteq
tff(fact_6251_add__scale__eq__noteq,axiom,
! [R: nat,A: nat,B: nat,C: nat,D: nat] :
( ( R != zero_zero_nat )
=> ( ( ( A = B )
& ( C != D ) )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),C)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),D)) ) ) ) ).
% add_scale_eq_noteq
tff(fact_6252_add__scale__eq__noteq,axiom,
! [R: int,A: int,B: int,C: int,D: int] :
( ( R != zero_zero_int )
=> ( ( ( A = B )
& ( C != D ) )
=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,R),C)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,R),D)) ) ) ) ).
% add_scale_eq_noteq
tff(fact_6253_Sum__Icc__int,axiom,
! [M: int,N: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M),N))
=> ( groups4538972089207619220nt_int(aTP_Lamm_bf,set_or1266510415728281911st_int(M,N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,N),one_one_int))),aa_int_int(aa_int_fun_int_int(times_times_int,M),aa_int_int(aa_int_fun_int_int(minus_minus_int,M),one_one_int)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).
% Sum_Icc_int
tff(fact_6254_set__decode__def,axiom,
! [X: nat] : aa_nat_set_nat(nat_set_decode,X) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_eg,X)) ).
% set_decode_def
tff(fact_6255_even__set__encode__iff,axiom,
! [A2: set_nat] :
( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(nat_set_encode,A2)))
<=> ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) ) ) ).
% even_set_encode_iff
tff(fact_6256_sum__abs,axiom,
! [F: fun_int_int,A2: set_int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,groups4538972089207619220nt_int(F,A2))),groups4538972089207619220nt_int(aTP_Lamm_eh(F),A2))) ).
% sum_abs
tff(fact_6257_sum__abs,axiom,
! [F: fun_nat_real,A2: set_nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,groups6591440286371151544t_real(F,A2))),groups6591440286371151544t_real(aTP_Lamm_ei(F),A2))) ).
% sum_abs
tff(fact_6258_sum__abs__ge__zero,axiom,
! [F: fun_int_int,A2: set_int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),groups4538972089207619220nt_int(aTP_Lamm_eh(F),A2))) ).
% sum_abs_ge_zero
tff(fact_6259_sum__abs__ge__zero,axiom,
! [F: fun_nat_real,A2: set_nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups6591440286371151544t_real(aTP_Lamm_ei(F),A2))) ).
% sum_abs_ge_zero
tff(fact_6260_sum__mono,axiom,
! [K5: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
( ! [I2: complex] :
( pp(aa_set_complex_bool(member_complex(I2),K5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I2)),aa_complex_rat(G,I2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5058264527183730370ex_rat(F,K5)),groups5058264527183730370ex_rat(G,K5))) ) ).
% sum_mono
tff(fact_6261_sum__mono,axiom,
! [K5: set_real,F: fun_real_rat,G: fun_real_rat] :
( ! [I2: real] :
( pp(aa_set_real_bool(member_real(I2),K5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I2)),aa_real_rat(G,I2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1300246762558778688al_rat(F,K5)),groups1300246762558778688al_rat(G,K5))) ) ).
% sum_mono
tff(fact_6262_sum__mono,axiom,
! [K5: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
( ! [I2: nat] :
( pp(aa_set_nat_bool(member_nat(I2),K5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I2)),aa_nat_rat(G,I2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,K5)),groups2906978787729119204at_rat(G,K5))) ) ).
% sum_mono
tff(fact_6263_sum__mono,axiom,
! [K5: set_int,F: fun_int_rat,G: fun_int_rat] :
( ! [I2: int] :
( pp(aa_set_int_bool(member_int(I2),K5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I2)),aa_int_rat(G,I2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,K5)),groups3906332499630173760nt_rat(G,K5))) ) ).
% sum_mono
tff(fact_6264_sum__mono,axiom,
! [K5: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
( ! [I2: complex] :
( pp(aa_set_complex_bool(member_complex(I2),K5))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,I2)),aa_complex_nat(G,I2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups5693394587270226106ex_nat(F,K5)),groups5693394587270226106ex_nat(G,K5))) ) ).
% sum_mono
tff(fact_6265_sum__mono,axiom,
! [K5: set_real,F: fun_real_nat,G: fun_real_nat] :
( ! [I2: real] :
( pp(aa_set_real_bool(member_real(I2),K5))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I2)),aa_real_nat(G,I2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1935376822645274424al_nat(F,K5)),groups1935376822645274424al_nat(G,K5))) ) ).
% sum_mono
tff(fact_6266_sum__mono,axiom,
! [K5: set_int,F: fun_int_nat,G: fun_int_nat] :
( ! [I2: int] :
( pp(aa_set_int_bool(member_int(I2),K5))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,I2)),aa_int_nat(G,I2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4541462559716669496nt_nat(F,K5)),groups4541462559716669496nt_nat(G,K5))) ) ).
% sum_mono
tff(fact_6267_sum__mono,axiom,
! [K5: set_complex,F: fun_complex_int,G: fun_complex_int] :
( ! [I2: complex] :
( pp(aa_set_complex_bool(member_complex(I2),K5))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_complex_int(F,I2)),aa_complex_int(G,I2))) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups5690904116761175830ex_int(F,K5)),groups5690904116761175830ex_int(G,K5))) ) ).
% sum_mono
tff(fact_6268_sum__mono,axiom,
! [K5: set_real,F: fun_real_int,G: fun_real_int] :
( ! [I2: real] :
( pp(aa_set_real_bool(member_real(I2),K5))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,I2)),aa_real_int(G,I2))) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups1932886352136224148al_int(F,K5)),groups1932886352136224148al_int(G,K5))) ) ).
% sum_mono
tff(fact_6269_sum__mono,axiom,
! [K5: set_nat,F: fun_nat_int,G: fun_nat_int] :
( ! [I2: nat] :
( pp(aa_set_nat_bool(member_nat(I2),K5))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,I2)),aa_nat_int(G,I2))) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,K5)),groups3539618377306564664at_int(G,K5))) ) ).
% sum_mono
tff(fact_6270_sum__distrib__left,axiom,
! [R: int,F: fun_int_int,A2: set_int] : aa_int_int(aa_int_fun_int_int(times_times_int,R),groups4538972089207619220nt_int(F,A2)) = groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ej(R),F),A2) ).
% sum_distrib_left
tff(fact_6271_sum__distrib__left,axiom,
! [R: complex,F: fun_complex_complex,A2: set_complex] : aa_complex_complex(times_times_complex(R),groups7754918857620584856omplex(F,A2)) = groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_ek(R),F),A2) ).
% sum_distrib_left
tff(fact_6272_sum__distrib__left,axiom,
! [R: nat,F: fun_nat_nat,A2: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_el(R),F)),A2) ).
% sum_distrib_left
tff(fact_6273_sum__distrib__left,axiom,
! [R: real,F: fun_nat_real,A2: set_nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),groups6591440286371151544t_real(F,A2)) = groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_em(R),F),A2) ).
% sum_distrib_left
tff(fact_6274_sum__distrib__right,axiom,
! [F: fun_int_int,A2: set_int,R: int] : aa_int_int(aa_int_fun_int_int(times_times_int,groups4538972089207619220nt_int(F,A2)),R) = groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_en(F),R),A2) ).
% sum_distrib_right
tff(fact_6275_sum__distrib__right,axiom,
! [F: fun_complex_complex,A2: set_complex,R: complex] : aa_complex_complex(times_times_complex(groups7754918857620584856omplex(F,A2)),R) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_eo(F),R),A2) ).
% sum_distrib_right
tff(fact_6276_sum__distrib__right,axiom,
! [F: fun_nat_nat,A2: set_nat,R: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),R) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ep(F),R)),A2) ).
% sum_distrib_right
tff(fact_6277_sum__distrib__right,axiom,
! [F: fun_nat_real,A2: set_nat,R: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups6591440286371151544t_real(F,A2)),R) = groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_eq(F),R),A2) ).
% sum_distrib_right
tff(fact_6278_sum__product,axiom,
! [F: fun_int_int,A2: set_int,G: fun_int_int,B2: set_int] : aa_int_int(aa_int_fun_int_int(times_times_int,groups4538972089207619220nt_int(F,A2)),groups4538972089207619220nt_int(G,B2)) = groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aa_fun8076508010538293575nt_int(aTP_Lamm_es(F),G),B2),A2) ).
% sum_product
tff(fact_6279_sum__product,axiom,
! [F: fun_complex_complex,A2: set_complex,G: fun_complex_complex,B2: set_complex] : aa_complex_complex(times_times_complex(groups7754918857620584856omplex(F,A2)),groups7754918857620584856omplex(G,B2)) = groups7754918857620584856omplex(aa_set7743095458595710203omplex(aa_fun1583801116089702097omplex(aTP_Lamm_eu(F),G),B2),A2) ).
% sum_product
tff(fact_6280_sum__product,axiom,
! [F: fun_nat_nat,A2: set_nat,G: fun_nat_nat,B2: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(G),B2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aa_fun4561837934398617467at_nat(aTP_Lamm_ew(F),G),B2)),A2) ).
% sum_product
tff(fact_6281_sum__product,axiom,
! [F: fun_nat_real,A2: set_nat,G: fun_nat_real,B2: set_nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups6591440286371151544t_real(F,A2)),groups6591440286371151544t_real(G,B2)) = groups6591440286371151544t_real(aa_set1166323925324055613t_real(aa_fun1400677770452585267t_real(aTP_Lamm_ey(F),G),B2),A2) ).
% sum_product
tff(fact_6282_sum_Odistrib,axiom,
! [G: fun_int_int,H: fun_int_int,A2: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ez(G),H),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,A2)),groups4538972089207619220nt_int(H,A2)) ).
% sum.distrib
tff(fact_6283_sum_Odistrib,axiom,
! [G: fun_complex_complex,H: fun_complex_complex,A2: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_fa(G),H),A2) = aa_complex_complex(plus_plus_complex(groups7754918857620584856omplex(G,A2)),groups7754918857620584856omplex(H,A2)) ).
% sum.distrib
tff(fact_6284_sum_Odistrib,axiom,
! [G: fun_nat_nat,H: fun_nat_nat,A2: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_fb(G),H)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(H),A2)) ).
% sum.distrib
tff(fact_6285_sum_Odistrib,axiom,
! [G: fun_nat_real,H: fun_nat_real,A2: set_nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_fc(G),H),A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,A2)),groups6591440286371151544t_real(H,A2)) ).
% sum.distrib
tff(fact_6286_sum__subtractf,axiom,
! [F: fun_int_int,G: fun_int_int,A2: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_fd(F),G),A2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(F,A2)),groups4538972089207619220nt_int(G,A2)) ).
% sum_subtractf
tff(fact_6287_sum__subtractf,axiom,
! [F: fun_complex_complex,G: fun_complex_complex,A2: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_fe(F),G),A2) = aa_complex_complex(minus_minus_complex(groups7754918857620584856omplex(F,A2)),groups7754918857620584856omplex(G,A2)) ).
% sum_subtractf
tff(fact_6288_sum__subtractf,axiom,
! [F: fun_nat_real,G: fun_nat_real,A2: set_nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ff(F),G),A2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups6591440286371151544t_real(F,A2)),groups6591440286371151544t_real(G,A2)) ).
% sum_subtractf
tff(fact_6289_sum__divide__distrib,axiom,
! [F: fun_complex_complex,A2: set_complex,R: complex] : aa_complex_complex(divide1717551699836669952omplex(groups7754918857620584856omplex(F,A2)),R) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_fg(F),R),A2) ).
% sum_divide_distrib
tff(fact_6290_sum__divide__distrib,axiom,
! [F: fun_nat_real,A2: set_nat,R: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,groups6591440286371151544t_real(F,A2)),R) = groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_fh(F),R),A2) ).
% sum_divide_distrib
tff(fact_6291_mod__sum__eq,axiom,
! [F: fun_int_int,A: int,A2: set_int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_fi(F),A),A2)),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups4538972089207619220nt_int(F,A2)),A) ).
% mod_sum_eq
tff(fact_6292_mod__sum__eq,axiom,
! [F: fun_nat_nat,A: nat,A2: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_fj(F),A)),A2)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),A) ).
% mod_sum_eq
tff(fact_6293_sum__nonneg,axiom,
! [A2: set_complex,F: fun_complex_real] :
( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),A2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X3))) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups5808333547571424918x_real(F,A2))) ) ).
% sum_nonneg
tff(fact_6294_sum__nonneg,axiom,
! [A2: set_real,F: fun_real_real] :
( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),A2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X3))) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups8097168146408367636l_real(F,A2))) ) ).
% sum_nonneg
tff(fact_6295_sum__nonneg,axiom,
! [A2: set_int,F: fun_int_real] :
( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),A2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X3))) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),groups8778361861064173332t_real(F,A2))) ) ).
% sum_nonneg
tff(fact_6296_sum__nonneg,axiom,
! [A2: set_complex,F: fun_complex_rat] :
( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),A2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X3))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,A2))) ) ).
% sum_nonneg
tff(fact_6297_sum__nonneg,axiom,
! [A2: set_real,F: fun_real_rat] :
( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),A2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X3))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups1300246762558778688al_rat(F,A2))) ) ).
% sum_nonneg
tff(fact_6298_sum__nonneg,axiom,
! [A2: set_nat,F: fun_nat_rat] :
( ! [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),A2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X3))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups2906978787729119204at_rat(F,A2))) ) ).
% sum_nonneg
tff(fact_6299_sum__nonneg,axiom,
! [A2: set_int,F: fun_int_rat] :
( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),A2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X3))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,A2))) ) ).
% sum_nonneg
tff(fact_6300_sum__nonneg,axiom,
! [A2: set_complex,F: fun_complex_nat] :
( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),A2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X3))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,A2))) ) ).
% sum_nonneg
tff(fact_6301_sum__nonneg,axiom,
! [A2: set_real,F: fun_real_nat] :
( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),A2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X3))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),groups1935376822645274424al_nat(F,A2))) ) ).
% sum_nonneg
tff(fact_6302_sum__nonneg,axiom,
! [A2: set_int,F: fun_int_nat] :
( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),A2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X3))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,A2))) ) ).
% sum_nonneg
tff(fact_6303_sum__nonpos,axiom,
! [A2: set_complex,F: fun_complex_real] :
( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),A2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X3)),zero_zero_real)) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,A2)),zero_zero_real)) ) ).
% sum_nonpos
tff(fact_6304_sum__nonpos,axiom,
! [A2: set_real,F: fun_real_real] :
( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),A2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X3)),zero_zero_real)) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8097168146408367636l_real(F,A2)),zero_zero_real)) ) ).
% sum_nonpos
tff(fact_6305_sum__nonpos,axiom,
! [A2: set_int,F: fun_int_real] :
( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),A2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X3)),zero_zero_real)) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,A2)),zero_zero_real)) ) ).
% sum_nonpos
tff(fact_6306_sum__nonpos,axiom,
! [A2: set_complex,F: fun_complex_rat] :
( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),A2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,X3)),zero_zero_rat)) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A2)),zero_zero_rat)) ) ).
% sum_nonpos
tff(fact_6307_sum__nonpos,axiom,
! [A2: set_real,F: fun_real_rat] :
( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),A2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,X3)),zero_zero_rat)) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1300246762558778688al_rat(F,A2)),zero_zero_rat)) ) ).
% sum_nonpos
tff(fact_6308_sum__nonpos,axiom,
! [A2: set_nat,F: fun_nat_rat] :
( ! [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),A2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X3)),zero_zero_rat)) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,A2)),zero_zero_rat)) ) ).
% sum_nonpos
tff(fact_6309_sum__nonpos,axiom,
! [A2: set_int,F: fun_int_rat] :
( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),A2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X3)),zero_zero_rat)) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A2)),zero_zero_rat)) ) ).
% sum_nonpos
tff(fact_6310_sum__nonpos,axiom,
! [A2: set_complex,F: fun_complex_nat] :
( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),A2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,X3)),zero_zero_nat)) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups5693394587270226106ex_nat(F,A2)),zero_zero_nat)) ) ).
% sum_nonpos
tff(fact_6311_sum__nonpos,axiom,
! [A2: set_real,F: fun_real_nat] :
( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),A2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,X3)),zero_zero_nat)) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1935376822645274424al_nat(F,A2)),zero_zero_nat)) ) ).
% sum_nonpos
tff(fact_6312_sum__nonpos,axiom,
! [A2: set_int,F: fun_int_nat] :
( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),A2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,X3)),zero_zero_nat)) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4541462559716669496nt_nat(F,A2)),zero_zero_nat)) ) ).
% sum_nonpos
tff(fact_6313_sum__mono__inv,axiom,
! [F: fun_real_rat,I5: set_real,G: fun_real_rat,I: real] :
( ( groups1300246762558778688al_rat(F,I5) = groups1300246762558778688al_rat(G,I5) )
=> ( ! [I2: real] :
( pp(aa_set_real_bool(member_real(I2),I5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I2)),aa_real_rat(G,I2))) )
=> ( pp(aa_set_real_bool(member_real(I),I5))
=> ( pp(aa_set_real_bool(finite_finite_real,I5))
=> ( aa_real_rat(F,I) = aa_real_rat(G,I) ) ) ) ) ) ).
% sum_mono_inv
tff(fact_6314_sum__mono__inv,axiom,
! [F: fun_nat_rat,I5: set_nat,G: fun_nat_rat,I: nat] :
( ( groups2906978787729119204at_rat(F,I5) = groups2906978787729119204at_rat(G,I5) )
=> ( ! [I2: nat] :
( pp(aa_set_nat_bool(member_nat(I2),I5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I2)),aa_nat_rat(G,I2))) )
=> ( pp(aa_set_nat_bool(member_nat(I),I5))
=> ( pp(aa_set_nat_bool(finite_finite_nat,I5))
=> ( aa_nat_rat(F,I) = aa_nat_rat(G,I) ) ) ) ) ) ).
% sum_mono_inv
tff(fact_6315_sum__mono__inv,axiom,
! [F: fun_int_rat,I5: set_int,G: fun_int_rat,I: int] :
( ( groups3906332499630173760nt_rat(F,I5) = groups3906332499630173760nt_rat(G,I5) )
=> ( ! [I2: int] :
( pp(aa_set_int_bool(member_int(I2),I5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I2)),aa_int_rat(G,I2))) )
=> ( pp(aa_set_int_bool(member_int(I),I5))
=> ( pp(aa_set_int_bool(finite_finite_int,I5))
=> ( aa_int_rat(F,I) = aa_int_rat(G,I) ) ) ) ) ) ).
% sum_mono_inv
tff(fact_6316_sum__mono__inv,axiom,
! [F: fun_complex_rat,I5: set_complex,G: fun_complex_rat,I: complex] :
( ( groups5058264527183730370ex_rat(F,I5) = groups5058264527183730370ex_rat(G,I5) )
=> ( ! [I2: complex] :
( pp(aa_set_complex_bool(member_complex(I2),I5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I2)),aa_complex_rat(G,I2))) )
=> ( pp(aa_set_complex_bool(member_complex(I),I5))
=> ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
=> ( aa_complex_rat(F,I) = aa_complex_rat(G,I) ) ) ) ) ) ).
% sum_mono_inv
tff(fact_6317_sum__mono__inv,axiom,
! [F: fun_real_nat,I5: set_real,G: fun_real_nat,I: real] :
( ( groups1935376822645274424al_nat(F,I5) = groups1935376822645274424al_nat(G,I5) )
=> ( ! [I2: real] :
( pp(aa_set_real_bool(member_real(I2),I5))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I2)),aa_real_nat(G,I2))) )
=> ( pp(aa_set_real_bool(member_real(I),I5))
=> ( pp(aa_set_real_bool(finite_finite_real,I5))
=> ( aa_real_nat(F,I) = aa_real_nat(G,I) ) ) ) ) ) ).
% sum_mono_inv
tff(fact_6318_sum__mono__inv,axiom,
! [F: fun_int_nat,I5: set_int,G: fun_int_nat,I: int] :
( ( groups4541462559716669496nt_nat(F,I5) = groups4541462559716669496nt_nat(G,I5) )
=> ( ! [I2: int] :
( pp(aa_set_int_bool(member_int(I2),I5))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,I2)),aa_int_nat(G,I2))) )
=> ( pp(aa_set_int_bool(member_int(I),I5))
=> ( pp(aa_set_int_bool(finite_finite_int,I5))
=> ( aa_int_nat(F,I) = aa_int_nat(G,I) ) ) ) ) ) ).
% sum_mono_inv
tff(fact_6319_sum__mono__inv,axiom,
! [F: fun_complex_nat,I5: set_complex,G: fun_complex_nat,I: complex] :
( ( groups5693394587270226106ex_nat(F,I5) = groups5693394587270226106ex_nat(G,I5) )
=> ( ! [I2: complex] :
( pp(aa_set_complex_bool(member_complex(I2),I5))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,I2)),aa_complex_nat(G,I2))) )
=> ( pp(aa_set_complex_bool(member_complex(I),I5))
=> ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
=> ( aa_complex_nat(F,I) = aa_complex_nat(G,I) ) ) ) ) ) ).
% sum_mono_inv
tff(fact_6320_sum__mono__inv,axiom,
! [F: fun_real_int,I5: set_real,G: fun_real_int,I: real] :
( ( groups1932886352136224148al_int(F,I5) = groups1932886352136224148al_int(G,I5) )
=> ( ! [I2: real] :
( pp(aa_set_real_bool(member_real(I2),I5))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,I2)),aa_real_int(G,I2))) )
=> ( pp(aa_set_real_bool(member_real(I),I5))
=> ( pp(aa_set_real_bool(finite_finite_real,I5))
=> ( aa_real_int(F,I) = aa_real_int(G,I) ) ) ) ) ) ).
% sum_mono_inv
tff(fact_6321_sum__mono__inv,axiom,
! [F: fun_nat_int,I5: set_nat,G: fun_nat_int,I: nat] :
( ( groups3539618377306564664at_int(F,I5) = groups3539618377306564664at_int(G,I5) )
=> ( ! [I2: nat] :
( pp(aa_set_nat_bool(member_nat(I2),I5))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,I2)),aa_nat_int(G,I2))) )
=> ( pp(aa_set_nat_bool(member_nat(I),I5))
=> ( pp(aa_set_nat_bool(finite_finite_nat,I5))
=> ( aa_nat_int(F,I) = aa_nat_int(G,I) ) ) ) ) ) ).
% sum_mono_inv
tff(fact_6322_sum__mono__inv,axiom,
! [F: fun_complex_int,I5: set_complex,G: fun_complex_int,I: complex] :
( ( groups5690904116761175830ex_int(F,I5) = groups5690904116761175830ex_int(G,I5) )
=> ( ! [I2: complex] :
( pp(aa_set_complex_bool(member_complex(I2),I5))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_complex_int(F,I2)),aa_complex_int(G,I2))) )
=> ( pp(aa_set_complex_bool(member_complex(I),I5))
=> ( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
=> ( aa_complex_int(F,I) = aa_complex_int(G,I) ) ) ) ) ) ).
% sum_mono_inv
tff(fact_6323_sum__le__included,axiom,
! [S: set_int,T: set_int,G: fun_int_real,I: fun_int_int,F: fun_int_real] :
( pp(aa_set_int_bool(finite_finite_int,S))
=> ( pp(aa_set_int_bool(finite_finite_int,T))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),T))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(G,X3))) )
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),S))
=> ? [Xa: int] :
( pp(aa_set_int_bool(member_int(Xa),T))
& ( aa_int_int(I,Xa) = X3 )
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X3)),aa_int_real(G,Xa))) ) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,S)),groups8778361861064173332t_real(G,T))) ) ) ) ) ).
% sum_le_included
tff(fact_6324_sum__le__included,axiom,
! [S: set_int,T: set_complex,G: fun_complex_real,I: fun_complex_int,F: fun_int_real] :
( pp(aa_set_int_bool(finite_finite_int,S))
=> ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T))
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),T))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X3))) )
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),S))
=> ? [Xa: complex] :
( pp(aa_set_complex_bool(member_complex(Xa),T))
& ( aa_complex_int(I,Xa) = X3 )
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X3)),aa_complex_real(G,Xa))) ) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,S)),groups5808333547571424918x_real(G,T))) ) ) ) ) ).
% sum_le_included
tff(fact_6325_sum__le__included,axiom,
! [S: set_complex,T: set_int,G: fun_int_real,I: fun_int_complex,F: fun_complex_real] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
=> ( pp(aa_set_int_bool(finite_finite_int,T))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),T))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(G,X3))) )
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),S))
=> ? [Xa: int] :
( pp(aa_set_int_bool(member_int(Xa),T))
& ( aa_int_complex(I,Xa) = X3 )
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X3)),aa_int_real(G,Xa))) ) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,S)),groups8778361861064173332t_real(G,T))) ) ) ) ) ).
% sum_le_included
tff(fact_6326_sum__le__included,axiom,
! [S: set_complex,T: set_complex,G: fun_complex_real,I: fun_complex_complex,F: fun_complex_real] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
=> ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T))
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),T))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X3))) )
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),S))
=> ? [Xa: complex] :
( pp(aa_set_complex_bool(member_complex(Xa),T))
& ( aa_complex_complex(I,Xa) = X3 )
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X3)),aa_complex_real(G,Xa))) ) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,S)),groups5808333547571424918x_real(G,T))) ) ) ) ) ).
% sum_le_included
tff(fact_6327_sum__le__included,axiom,
! [S: set_nat,T: set_nat,G: fun_nat_rat,I: fun_nat_nat,F: fun_nat_rat] :
( pp(aa_set_nat_bool(finite_finite_nat,S))
=> ( pp(aa_set_nat_bool(finite_finite_nat,T))
=> ( ! [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),T))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X3))) )
=> ( ! [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),S))
=> ? [Xa: nat] :
( pp(aa_set_nat_bool(member_nat(Xa),T))
& ( aa_nat_nat(I,Xa) = X3 )
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(G,Xa))) ) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,S)),groups2906978787729119204at_rat(G,T))) ) ) ) ) ).
% sum_le_included
tff(fact_6328_sum__le__included,axiom,
! [S: set_nat,T: set_int,G: fun_int_rat,I: fun_int_nat,F: fun_nat_rat] :
( pp(aa_set_nat_bool(finite_finite_nat,S))
=> ( pp(aa_set_int_bool(finite_finite_int,T))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),T))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(G,X3))) )
=> ( ! [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),S))
=> ? [Xa: int] :
( pp(aa_set_int_bool(member_int(Xa),T))
& ( aa_int_nat(I,Xa) = X3 )
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_int_rat(G,Xa))) ) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,S)),groups3906332499630173760nt_rat(G,T))) ) ) ) ) ).
% sum_le_included
tff(fact_6329_sum__le__included,axiom,
! [S: set_nat,T: set_complex,G: fun_complex_rat,I: fun_complex_nat,F: fun_nat_rat] :
( pp(aa_set_nat_bool(finite_finite_nat,S))
=> ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T))
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),T))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(G,X3))) )
=> ( ! [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),S))
=> ? [Xa: complex] :
( pp(aa_set_complex_bool(member_complex(Xa),T))
& ( aa_complex_nat(I,Xa) = X3 )
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_complex_rat(G,Xa))) ) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,S)),groups5058264527183730370ex_rat(G,T))) ) ) ) ) ).
% sum_le_included
tff(fact_6330_sum__le__included,axiom,
! [S: set_int,T: set_nat,G: fun_nat_rat,I: fun_nat_int,F: fun_int_rat] :
( pp(aa_set_int_bool(finite_finite_int,S))
=> ( pp(aa_set_nat_bool(finite_finite_nat,T))
=> ( ! [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),T))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X3))) )
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),S))
=> ? [Xa: nat] :
( pp(aa_set_nat_bool(member_nat(Xa),T))
& ( aa_nat_int(I,Xa) = X3 )
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X3)),aa_nat_rat(G,Xa))) ) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,S)),groups2906978787729119204at_rat(G,T))) ) ) ) ) ).
% sum_le_included
tff(fact_6331_sum__le__included,axiom,
! [S: set_int,T: set_int,G: fun_int_rat,I: fun_int_int,F: fun_int_rat] :
( pp(aa_set_int_bool(finite_finite_int,S))
=> ( pp(aa_set_int_bool(finite_finite_int,T))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),T))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(G,X3))) )
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),S))
=> ? [Xa: int] :
( pp(aa_set_int_bool(member_int(Xa),T))
& ( aa_int_int(I,Xa) = X3 )
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X3)),aa_int_rat(G,Xa))) ) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,S)),groups3906332499630173760nt_rat(G,T))) ) ) ) ) ).
% sum_le_included
tff(fact_6332_sum__le__included,axiom,
! [S: set_int,T: set_complex,G: fun_complex_rat,I: fun_complex_int,F: fun_int_rat] :
( pp(aa_set_int_bool(finite_finite_int,S))
=> ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T))
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),T))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(G,X3))) )
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),S))
=> ? [Xa: complex] :
( pp(aa_set_complex_bool(member_complex(Xa),T))
& ( aa_complex_int(I,Xa) = X3 )
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X3)),aa_complex_rat(G,Xa))) ) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,S)),groups5058264527183730370ex_rat(G,T))) ) ) ) ) ).
% sum_le_included
tff(fact_6333_sum__nonneg__eq__0__iff,axiom,
! [A2: set_real,F: fun_real_real] :
( pp(aa_set_real_bool(finite_finite_real,A2))
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),A2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X3))) )
=> ( ( groups8097168146408367636l_real(F,A2) = zero_zero_real )
<=> ! [X2: real] :
( pp(aa_set_real_bool(member_real(X2),A2))
=> ( aa_real_real(F,X2) = zero_zero_real ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
tff(fact_6334_sum__nonneg__eq__0__iff,axiom,
! [A2: set_int,F: fun_int_real] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),A2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X3))) )
=> ( ( groups8778361861064173332t_real(F,A2) = zero_zero_real )
<=> ! [X2: int] :
( pp(aa_set_int_bool(member_int(X2),A2))
=> ( aa_int_real(F,X2) = zero_zero_real ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
tff(fact_6335_sum__nonneg__eq__0__iff,axiom,
! [A2: set_complex,F: fun_complex_real] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),A2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X3))) )
=> ( ( groups5808333547571424918x_real(F,A2) = zero_zero_real )
<=> ! [X2: complex] :
( pp(aa_set_complex_bool(member_complex(X2),A2))
=> ( aa_complex_real(F,X2) = zero_zero_real ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
tff(fact_6336_sum__nonneg__eq__0__iff,axiom,
! [A2: set_real,F: fun_real_rat] :
( pp(aa_set_real_bool(finite_finite_real,A2))
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),A2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X3))) )
=> ( ( groups1300246762558778688al_rat(F,A2) = zero_zero_rat )
<=> ! [X2: real] :
( pp(aa_set_real_bool(member_real(X2),A2))
=> ( aa_real_rat(F,X2) = zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
tff(fact_6337_sum__nonneg__eq__0__iff,axiom,
! [A2: set_nat,F: fun_nat_rat] :
( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ( ! [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),A2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X3))) )
=> ( ( groups2906978787729119204at_rat(F,A2) = zero_zero_rat )
<=> ! [X2: nat] :
( pp(aa_set_nat_bool(member_nat(X2),A2))
=> ( aa_nat_rat(F,X2) = zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
tff(fact_6338_sum__nonneg__eq__0__iff,axiom,
! [A2: set_int,F: fun_int_rat] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),A2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X3))) )
=> ( ( groups3906332499630173760nt_rat(F,A2) = zero_zero_rat )
<=> ! [X2: int] :
( pp(aa_set_int_bool(member_int(X2),A2))
=> ( aa_int_rat(F,X2) = zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
tff(fact_6339_sum__nonneg__eq__0__iff,axiom,
! [A2: set_complex,F: fun_complex_rat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),A2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X3))) )
=> ( ( groups5058264527183730370ex_rat(F,A2) = zero_zero_rat )
<=> ! [X2: complex] :
( pp(aa_set_complex_bool(member_complex(X2),A2))
=> ( aa_complex_rat(F,X2) = zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
tff(fact_6340_sum__nonneg__eq__0__iff,axiom,
! [A2: set_real,F: fun_real_nat] :
( pp(aa_set_real_bool(finite_finite_real,A2))
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),A2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X3))) )
=> ( ( groups1935376822645274424al_nat(F,A2) = zero_zero_nat )
<=> ! [X2: real] :
( pp(aa_set_real_bool(member_real(X2),A2))
=> ( aa_real_nat(F,X2) = zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
tff(fact_6341_sum__nonneg__eq__0__iff,axiom,
! [A2: set_int,F: fun_int_nat] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),A2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X3))) )
=> ( ( groups4541462559716669496nt_nat(F,A2) = zero_zero_nat )
<=> ! [X2: int] :
( pp(aa_set_int_bool(member_int(X2),A2))
=> ( aa_int_nat(F,X2) = zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
tff(fact_6342_sum__nonneg__eq__0__iff,axiom,
! [A2: set_complex,F: fun_complex_nat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),A2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X3))) )
=> ( ( groups5693394587270226106ex_nat(F,A2) = zero_zero_nat )
<=> ! [X2: complex] :
( pp(aa_set_complex_bool(member_complex(X2),A2))
=> ( aa_complex_nat(F,X2) = zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
tff(fact_6343_sum__strict__mono__ex1,axiom,
! [A2: set_int,F: fun_int_real,G: fun_int_real] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),A2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,X3)),aa_int_real(G,X3))) )
=> ( ? [X4: int] :
( pp(aa_set_int_bool(member_int(X4),A2))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X4)),aa_int_real(G,X4))) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(G,A2))) ) ) ) ).
% sum_strict_mono_ex1
tff(fact_6344_sum__strict__mono__ex1,axiom,
! [A2: set_complex,F: fun_complex_real,G: fun_complex_real] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),A2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,X3)),aa_complex_real(G,X3))) )
=> ( ? [X4: complex] :
( pp(aa_set_complex_bool(member_complex(X4),A2))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F,X4)),aa_complex_real(G,X4))) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(G,A2))) ) ) ) ).
% sum_strict_mono_ex1
tff(fact_6345_sum__strict__mono__ex1,axiom,
! [A2: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ( ! [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),A2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(G,X3))) )
=> ( ? [X4: nat] :
( pp(aa_set_nat_bool(member_nat(X4),A2))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(G,X4))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups2906978787729119204at_rat(F,A2)),groups2906978787729119204at_rat(G,A2))) ) ) ) ).
% sum_strict_mono_ex1
tff(fact_6346_sum__strict__mono__ex1,axiom,
! [A2: set_int,F: fun_int_rat,G: fun_int_rat] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),A2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,X3)),aa_int_rat(G,X3))) )
=> ( ? [X4: int] :
( pp(aa_set_int_bool(member_int(X4),A2))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(G,X4))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(G,A2))) ) ) ) ).
% sum_strict_mono_ex1
tff(fact_6347_sum__strict__mono__ex1,axiom,
! [A2: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),A2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,X3)),aa_complex_rat(G,X3))) )
=> ( ? [X4: complex] :
( pp(aa_set_complex_bool(member_complex(X4),A2))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_complex_rat(F,X4)),aa_complex_rat(G,X4))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(G,A2))) ) ) ) ).
% sum_strict_mono_ex1
tff(fact_6348_sum__strict__mono__ex1,axiom,
! [A2: set_int,F: fun_int_nat,G: fun_int_nat] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),A2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,X3)),aa_int_nat(G,X3))) )
=> ( ? [X4: int] :
( pp(aa_set_int_bool(member_int(X4),A2))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(F,X4)),aa_int_nat(G,X4))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(G,A2))) ) ) ) ).
% sum_strict_mono_ex1
tff(fact_6349_sum__strict__mono__ex1,axiom,
! [A2: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),A2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,X3)),aa_complex_nat(G,X3))) )
=> ( ? [X4: complex] :
( pp(aa_set_complex_bool(member_complex(X4),A2))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_complex_nat(F,X4)),aa_complex_nat(G,X4))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(G,A2))) ) ) ) ).
% sum_strict_mono_ex1
tff(fact_6350_sum__strict__mono__ex1,axiom,
! [A2: set_nat,F: fun_nat_int,G: fun_nat_int] :
( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ( ! [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),A2))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,X3)),aa_nat_int(G,X3))) )
=> ( ? [X4: nat] :
( pp(aa_set_nat_bool(member_nat(X4),A2))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F,X4)),aa_nat_int(G,X4))) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups3539618377306564664at_int(F,A2)),groups3539618377306564664at_int(G,A2))) ) ) ) ).
% sum_strict_mono_ex1
tff(fact_6351_sum__strict__mono__ex1,axiom,
! [A2: set_complex,F: fun_complex_int,G: fun_complex_int] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),A2))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_complex_int(F,X3)),aa_complex_int(G,X3))) )
=> ( ? [X4: complex] :
( pp(aa_set_complex_bool(member_complex(X4),A2))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_complex_int(F,X4)),aa_complex_int(G,X4))) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups5690904116761175830ex_int(F,A2)),groups5690904116761175830ex_int(G,A2))) ) ) ) ).
% sum_strict_mono_ex1
tff(fact_6352_sum__strict__mono__ex1,axiom,
! [A2: set_int,F: fun_int_int,G: fun_int_int] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),A2))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(F,X3)),aa_int_int(G,X3))) )
=> ( ? [X4: int] :
( pp(aa_set_int_bool(member_int(X4),A2))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(F,X4)),aa_int_int(G,X4))) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups4538972089207619220nt_int(F,A2)),groups4538972089207619220nt_int(G,A2))) ) ) ) ).
% sum_strict_mono_ex1
tff(fact_6353_sum_Orelated,axiom,
! [R2: fun_co1378794544173401837x_bool,S2: set_nat,H: fun_nat_complex,G: fun_nat_complex] :
( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,zero_zero_complex),zero_zero_complex))
=> ( ! [X15: complex,Y15: complex,X22: complex,Y23: complex] :
( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,X15),X22))
& pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,Y15),Y23)) )
=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,aa_complex_complex(plus_plus_complex(X15),Y15)),aa_complex_complex(plus_plus_complex(X22),Y23))) )
=> ( pp(aa_set_nat_bool(finite_finite_nat,S2))
=> ( ! [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),S2))
=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,aa_nat_complex(H,X3)),aa_nat_complex(G,X3))) )
=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,groups2073611262835488442omplex(H,S2)),groups2073611262835488442omplex(G,S2))) ) ) ) ) ).
% sum.related
tff(fact_6354_sum_Orelated,axiom,
! [R2: fun_co1378794544173401837x_bool,S2: set_int,H: fun_int_complex,G: fun_int_complex] :
( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,zero_zero_complex),zero_zero_complex))
=> ( ! [X15: complex,Y15: complex,X22: complex,Y23: complex] :
( ( pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,X15),X22))
& pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,Y15),Y23)) )
=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,aa_complex_complex(plus_plus_complex(X15),Y15)),aa_complex_complex(plus_plus_complex(X22),Y23))) )
=> ( pp(aa_set_int_bool(finite_finite_int,S2))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),S2))
=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,aa_int_complex(H,X3)),aa_int_complex(G,X3))) )
=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(R2,groups3049146728041665814omplex(H,S2)),groups3049146728041665814omplex(G,S2))) ) ) ) ) ).
% sum.related
tff(fact_6355_sum_Orelated,axiom,
! [R2: fun_re1255401033344146673l_bool,S2: set_int,H: fun_int_real,G: fun_int_real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,zero_zero_real),zero_zero_real))
=> ( ! [X15: real,Y15: real,X22: real,Y23: real] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,X15),X22))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,Y15),Y23)) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X22),Y23))) )
=> ( pp(aa_set_int_bool(finite_finite_int,S2))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),S2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,aa_int_real(H,X3)),aa_int_real(G,X3))) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,groups8778361861064173332t_real(H,S2)),groups8778361861064173332t_real(G,S2))) ) ) ) ) ).
% sum.related
tff(fact_6356_sum_Orelated,axiom,
! [R2: fun_re1255401033344146673l_bool,S2: set_complex,H: fun_complex_real,G: fun_complex_real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,zero_zero_real),zero_zero_real))
=> ( ! [X15: real,Y15: real,X22: real,Y23: real] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,X15),X22))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,Y15),Y23)) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X22),Y23))) )
=> ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),S2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,aa_complex_real(H,X3)),aa_complex_real(G,X3))) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(R2,groups5808333547571424918x_real(H,S2)),groups5808333547571424918x_real(G,S2))) ) ) ) ) ).
% sum.related
tff(fact_6357_sum_Orelated,axiom,
! [R2: fun_rat_fun_rat_bool,S2: set_nat,H: fun_nat_rat,G: fun_nat_rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,zero_zero_rat),zero_zero_rat))
=> ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,X15),X22))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,Y15),Y23)) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,plus_plus_rat(X15,Y15)),plus_plus_rat(X22,Y23))) )
=> ( pp(aa_set_nat_bool(finite_finite_nat,S2))
=> ( ! [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),S2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,aa_nat_rat(H,X3)),aa_nat_rat(G,X3))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,groups2906978787729119204at_rat(H,S2)),groups2906978787729119204at_rat(G,S2))) ) ) ) ) ).
% sum.related
tff(fact_6358_sum_Orelated,axiom,
! [R2: fun_rat_fun_rat_bool,S2: set_int,H: fun_int_rat,G: fun_int_rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,zero_zero_rat),zero_zero_rat))
=> ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,X15),X22))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,Y15),Y23)) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,plus_plus_rat(X15,Y15)),plus_plus_rat(X22,Y23))) )
=> ( pp(aa_set_int_bool(finite_finite_int,S2))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),S2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,aa_int_rat(H,X3)),aa_int_rat(G,X3))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,groups3906332499630173760nt_rat(H,S2)),groups3906332499630173760nt_rat(G,S2))) ) ) ) ) ).
% sum.related
tff(fact_6359_sum_Orelated,axiom,
! [R2: fun_rat_fun_rat_bool,S2: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,zero_zero_rat),zero_zero_rat))
=> ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,X15),X22))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,Y15),Y23)) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,plus_plus_rat(X15,Y15)),plus_plus_rat(X22,Y23))) )
=> ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),S2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,aa_complex_rat(H,X3)),aa_complex_rat(G,X3))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(R2,groups5058264527183730370ex_rat(H,S2)),groups5058264527183730370ex_rat(G,S2))) ) ) ) ) ).
% sum.related
tff(fact_6360_sum_Orelated,axiom,
! [R2: fun_nat_fun_nat_bool,S2: set_int,H: fun_int_nat,G: fun_int_nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,zero_zero_nat),zero_zero_nat))
=> ( ! [X15: nat,Y15: nat,X22: nat,Y23: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,X15),X22))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,Y15),Y23)) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X15),Y15)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X22),Y23))) )
=> ( pp(aa_set_int_bool(finite_finite_int,S2))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),S2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,aa_int_nat(H,X3)),aa_int_nat(G,X3))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,groups4541462559716669496nt_nat(H,S2)),groups4541462559716669496nt_nat(G,S2))) ) ) ) ) ).
% sum.related
tff(fact_6361_sum_Orelated,axiom,
! [R2: fun_nat_fun_nat_bool,S2: set_complex,H: fun_complex_nat,G: fun_complex_nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,zero_zero_nat),zero_zero_nat))
=> ( ! [X15: nat,Y15: nat,X22: nat,Y23: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,X15),X22))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,Y15),Y23)) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X15),Y15)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X22),Y23))) )
=> ( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),S2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,aa_complex_nat(H,X3)),aa_complex_nat(G,X3))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,groups5693394587270226106ex_nat(H,S2)),groups5693394587270226106ex_nat(G,S2))) ) ) ) ) ).
% sum.related
tff(fact_6362_sum_Orelated,axiom,
! [R2: fun_int_fun_int_bool,S2: set_nat,H: fun_nat_int,G: fun_nat_int] :
( pp(aa_int_bool(aa_int_fun_int_bool(R2,zero_zero_int),zero_zero_int))
=> ( ! [X15: int,Y15: int,X22: int,Y23: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(R2,X15),X22))
& pp(aa_int_bool(aa_int_fun_int_bool(R2,Y15),Y23)) )
=> pp(aa_int_bool(aa_int_fun_int_bool(R2,aa_int_int(aa_int_fun_int_int(plus_plus_int,X15),Y15)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X22),Y23))) )
=> ( pp(aa_set_nat_bool(finite_finite_nat,S2))
=> ( ! [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),S2))
=> pp(aa_int_bool(aa_int_fun_int_bool(R2,aa_nat_int(H,X3)),aa_nat_int(G,X3))) )
=> pp(aa_int_bool(aa_int_fun_int_bool(R2,groups3539618377306564664at_int(H,S2)),groups3539618377306564664at_int(G,S2))) ) ) ) ) ).
% sum.related
tff(fact_6363_sum__strict__mono,axiom,
! [A2: set_complex,F: fun_complex_real,G: fun_complex_real] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( ( A2 != bot_bot_set_complex )
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),A2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(F,X3)),aa_complex_real(G,X3))) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(G,A2))) ) ) ) ).
% sum_strict_mono
tff(fact_6364_sum__strict__mono,axiom,
! [A2: set_int,F: fun_int_real,G: fun_int_real] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( ( A2 != bot_bot_set_int )
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),A2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(F,X3)),aa_int_real(G,X3))) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(G,A2))) ) ) ) ).
% sum_strict_mono
tff(fact_6365_sum__strict__mono,axiom,
! [A2: set_real,F: fun_real_real,G: fun_real_real] :
( pp(aa_set_real_bool(finite_finite_real,A2))
=> ( ( A2 != bot_bot_set_real )
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),A2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X3)),aa_real_real(G,X3))) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8097168146408367636l_real(F,A2)),groups8097168146408367636l_real(G,A2))) ) ) ) ).
% sum_strict_mono
tff(fact_6366_sum__strict__mono,axiom,
! [A2: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( ( A2 != bot_bot_set_complex )
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),A2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_complex_rat(F,X3)),aa_complex_rat(G,X3))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(G,A2))) ) ) ) ).
% sum_strict_mono
tff(fact_6367_sum__strict__mono,axiom,
! [A2: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),A2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(F,X3)),aa_nat_rat(G,X3))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups2906978787729119204at_rat(F,A2)),groups2906978787729119204at_rat(G,A2))) ) ) ) ).
% sum_strict_mono
tff(fact_6368_sum__strict__mono,axiom,
! [A2: set_int,F: fun_int_rat,G: fun_int_rat] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( ( A2 != bot_bot_set_int )
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),A2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(F,X3)),aa_int_rat(G,X3))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(G,A2))) ) ) ) ).
% sum_strict_mono
tff(fact_6369_sum__strict__mono,axiom,
! [A2: set_real,F: fun_real_rat,G: fun_real_rat] :
( pp(aa_set_real_bool(finite_finite_real,A2))
=> ( ( A2 != bot_bot_set_real )
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),A2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_real_rat(F,X3)),aa_real_rat(G,X3))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups1300246762558778688al_rat(F,A2)),groups1300246762558778688al_rat(G,A2))) ) ) ) ).
% sum_strict_mono
tff(fact_6370_sum__strict__mono,axiom,
! [A2: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( ( A2 != bot_bot_set_complex )
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),A2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_complex_nat(F,X3)),aa_complex_nat(G,X3))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(G,A2))) ) ) ) ).
% sum_strict_mono
tff(fact_6371_sum__strict__mono,axiom,
! [A2: set_int,F: fun_int_nat,G: fun_int_nat] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( ( A2 != bot_bot_set_int )
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),A2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(F,X3)),aa_int_nat(G,X3))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(G,A2))) ) ) ) ).
% sum_strict_mono
tff(fact_6372_sum__strict__mono,axiom,
! [A2: set_real,F: fun_real_nat,G: fun_real_nat] :
( pp(aa_set_real_bool(finite_finite_real,A2))
=> ( ( A2 != bot_bot_set_real )
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),A2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_real_nat(F,X3)),aa_real_nat(G,X3))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups1935376822645274424al_nat(F,A2)),groups1935376822645274424al_nat(G,A2))) ) ) ) ).
% sum_strict_mono
tff(fact_6373_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_real,T3: set_real,S2: set_real,I: fun_real_real,J: fun_real_real,T2: set_real,G: fun_real_complex,H: fun_real_complex] :
( pp(aa_set_real_bool(finite_finite_real,S4))
=> ( pp(aa_set_real_bool(finite_finite_real,T3))
=> ( ! [A4: real] :
( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(S2,S4)))
=> ( aa_real_real(I,aa_real_real(J,A4)) = A4 ) )
=> ( ! [A4: real] :
( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(S2,S4)))
=> pp(aa_set_real_bool(member_real(aa_real_real(J,A4)),minus_minus_set_real(T2,T3))) )
=> ( ! [B4: real] :
( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(T2,T3)))
=> ( aa_real_real(J,aa_real_real(I,B4)) = B4 ) )
=> ( ! [B4: real] :
( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(T2,T3)))
=> pp(aa_set_real_bool(member_real(aa_real_real(I,B4)),minus_minus_set_real(S2,S4))) )
=> ( ! [A4: real] :
( pp(aa_set_real_bool(member_real(A4),S4))
=> ( aa_real_complex(G,A4) = zero_zero_complex ) )
=> ( ! [B4: real] :
( pp(aa_set_real_bool(member_real(B4),T3))
=> ( aa_real_complex(H,B4) = zero_zero_complex ) )
=> ( ! [A4: real] :
( pp(aa_set_real_bool(member_real(A4),S2))
=> ( aa_real_complex(H,aa_real_real(J,A4)) = aa_real_complex(G,A4) ) )
=> ( groups5754745047067104278omplex(G,S2) = groups5754745047067104278omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
tff(fact_6374_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_real,T3: set_int,S2: set_real,I: fun_int_real,J: fun_real_int,T2: set_int,G: fun_real_complex,H: fun_int_complex] :
( pp(aa_set_real_bool(finite_finite_real,S4))
=> ( pp(aa_set_int_bool(finite_finite_int,T3))
=> ( ! [A4: real] :
( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(S2,S4)))
=> ( aa_int_real(I,aa_real_int(J,A4)) = A4 ) )
=> ( ! [A4: real] :
( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(S2,S4)))
=> pp(aa_set_int_bool(member_int(aa_real_int(J,A4)),minus_minus_set_int(T2,T3))) )
=> ( ! [B4: int] :
( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(T2,T3)))
=> ( aa_real_int(J,aa_int_real(I,B4)) = B4 ) )
=> ( ! [B4: int] :
( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(T2,T3)))
=> pp(aa_set_real_bool(member_real(aa_int_real(I,B4)),minus_minus_set_real(S2,S4))) )
=> ( ! [A4: real] :
( pp(aa_set_real_bool(member_real(A4),S4))
=> ( aa_real_complex(G,A4) = zero_zero_complex ) )
=> ( ! [B4: int] :
( pp(aa_set_int_bool(member_int(B4),T3))
=> ( aa_int_complex(H,B4) = zero_zero_complex ) )
=> ( ! [A4: real] :
( pp(aa_set_real_bool(member_real(A4),S2))
=> ( aa_int_complex(H,aa_real_int(J,A4)) = aa_real_complex(G,A4) ) )
=> ( groups5754745047067104278omplex(G,S2) = groups3049146728041665814omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
tff(fact_6375_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_int,T3: set_real,S2: set_int,I: fun_real_int,J: fun_int_real,T2: set_real,G: fun_int_complex,H: fun_real_complex] :
( pp(aa_set_int_bool(finite_finite_int,S4))
=> ( pp(aa_set_real_bool(finite_finite_real,T3))
=> ( ! [A4: int] :
( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(S2,S4)))
=> ( aa_real_int(I,aa_int_real(J,A4)) = A4 ) )
=> ( ! [A4: int] :
( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(S2,S4)))
=> pp(aa_set_real_bool(member_real(aa_int_real(J,A4)),minus_minus_set_real(T2,T3))) )
=> ( ! [B4: real] :
( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(T2,T3)))
=> ( aa_int_real(J,aa_real_int(I,B4)) = B4 ) )
=> ( ! [B4: real] :
( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(T2,T3)))
=> pp(aa_set_int_bool(member_int(aa_real_int(I,B4)),minus_minus_set_int(S2,S4))) )
=> ( ! [A4: int] :
( pp(aa_set_int_bool(member_int(A4),S4))
=> ( aa_int_complex(G,A4) = zero_zero_complex ) )
=> ( ! [B4: real] :
( pp(aa_set_real_bool(member_real(B4),T3))
=> ( aa_real_complex(H,B4) = zero_zero_complex ) )
=> ( ! [A4: int] :
( pp(aa_set_int_bool(member_int(A4),S2))
=> ( aa_real_complex(H,aa_int_real(J,A4)) = aa_int_complex(G,A4) ) )
=> ( groups3049146728041665814omplex(G,S2) = groups5754745047067104278omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
tff(fact_6376_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_int,T3: set_int,S2: set_int,I: fun_int_int,J: fun_int_int,T2: set_int,G: fun_int_complex,H: fun_int_complex] :
( pp(aa_set_int_bool(finite_finite_int,S4))
=> ( pp(aa_set_int_bool(finite_finite_int,T3))
=> ( ! [A4: int] :
( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(S2,S4)))
=> ( aa_int_int(I,aa_int_int(J,A4)) = A4 ) )
=> ( ! [A4: int] :
( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(S2,S4)))
=> pp(aa_set_int_bool(member_int(aa_int_int(J,A4)),minus_minus_set_int(T2,T3))) )
=> ( ! [B4: int] :
( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(T2,T3)))
=> ( aa_int_int(J,aa_int_int(I,B4)) = B4 ) )
=> ( ! [B4: int] :
( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(T2,T3)))
=> pp(aa_set_int_bool(member_int(aa_int_int(I,B4)),minus_minus_set_int(S2,S4))) )
=> ( ! [A4: int] :
( pp(aa_set_int_bool(member_int(A4),S4))
=> ( aa_int_complex(G,A4) = zero_zero_complex ) )
=> ( ! [B4: int] :
( pp(aa_set_int_bool(member_int(B4),T3))
=> ( aa_int_complex(H,B4) = zero_zero_complex ) )
=> ( ! [A4: int] :
( pp(aa_set_int_bool(member_int(A4),S2))
=> ( aa_int_complex(H,aa_int_int(J,A4)) = aa_int_complex(G,A4) ) )
=> ( groups3049146728041665814omplex(G,S2) = groups3049146728041665814omplex(H,T2) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
tff(fact_6377_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_real,T3: set_real,S2: set_real,I: fun_real_real,J: fun_real_real,T2: set_real,G: fun_real_real,H: fun_real_real] :
( pp(aa_set_real_bool(finite_finite_real,S4))
=> ( pp(aa_set_real_bool(finite_finite_real,T3))
=> ( ! [A4: real] :
( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(S2,S4)))
=> ( aa_real_real(I,aa_real_real(J,A4)) = A4 ) )
=> ( ! [A4: real] :
( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(S2,S4)))
=> pp(aa_set_real_bool(member_real(aa_real_real(J,A4)),minus_minus_set_real(T2,T3))) )
=> ( ! [B4: real] :
( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(T2,T3)))
=> ( aa_real_real(J,aa_real_real(I,B4)) = B4 ) )
=> ( ! [B4: real] :
( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(T2,T3)))
=> pp(aa_set_real_bool(member_real(aa_real_real(I,B4)),minus_minus_set_real(S2,S4))) )
=> ( ! [A4: real] :
( pp(aa_set_real_bool(member_real(A4),S4))
=> ( aa_real_real(G,A4) = zero_zero_real ) )
=> ( ! [B4: real] :
( pp(aa_set_real_bool(member_real(B4),T3))
=> ( aa_real_real(H,B4) = zero_zero_real ) )
=> ( ! [A4: real] :
( pp(aa_set_real_bool(member_real(A4),S2))
=> ( aa_real_real(H,aa_real_real(J,A4)) = aa_real_real(G,A4) ) )
=> ( groups8097168146408367636l_real(G,S2) = groups8097168146408367636l_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
tff(fact_6378_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_real,T3: set_int,S2: set_real,I: fun_int_real,J: fun_real_int,T2: set_int,G: fun_real_real,H: fun_int_real] :
( pp(aa_set_real_bool(finite_finite_real,S4))
=> ( pp(aa_set_int_bool(finite_finite_int,T3))
=> ( ! [A4: real] :
( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(S2,S4)))
=> ( aa_int_real(I,aa_real_int(J,A4)) = A4 ) )
=> ( ! [A4: real] :
( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(S2,S4)))
=> pp(aa_set_int_bool(member_int(aa_real_int(J,A4)),minus_minus_set_int(T2,T3))) )
=> ( ! [B4: int] :
( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(T2,T3)))
=> ( aa_real_int(J,aa_int_real(I,B4)) = B4 ) )
=> ( ! [B4: int] :
( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(T2,T3)))
=> pp(aa_set_real_bool(member_real(aa_int_real(I,B4)),minus_minus_set_real(S2,S4))) )
=> ( ! [A4: real] :
( pp(aa_set_real_bool(member_real(A4),S4))
=> ( aa_real_real(G,A4) = zero_zero_real ) )
=> ( ! [B4: int] :
( pp(aa_set_int_bool(member_int(B4),T3))
=> ( aa_int_real(H,B4) = zero_zero_real ) )
=> ( ! [A4: real] :
( pp(aa_set_real_bool(member_real(A4),S2))
=> ( aa_int_real(H,aa_real_int(J,A4)) = aa_real_real(G,A4) ) )
=> ( groups8097168146408367636l_real(G,S2) = groups8778361861064173332t_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
tff(fact_6379_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_real,T3: set_complex,S2: set_real,I: fun_complex_real,J: fun_real_complex,T2: set_complex,G: fun_real_real,H: fun_complex_real] :
( pp(aa_set_real_bool(finite_finite_real,S4))
=> ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
=> ( ! [A4: real] :
( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(S2,S4)))
=> ( aa_complex_real(I,aa_real_complex(J,A4)) = A4 ) )
=> ( ! [A4: real] :
( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(S2,S4)))
=> pp(aa_set_complex_bool(member_complex(aa_real_complex(J,A4)),minus_811609699411566653omplex(T2,T3))) )
=> ( ! [B4: complex] :
( pp(aa_set_complex_bool(member_complex(B4),minus_811609699411566653omplex(T2,T3)))
=> ( aa_real_complex(J,aa_complex_real(I,B4)) = B4 ) )
=> ( ! [B4: complex] :
( pp(aa_set_complex_bool(member_complex(B4),minus_811609699411566653omplex(T2,T3)))
=> pp(aa_set_real_bool(member_real(aa_complex_real(I,B4)),minus_minus_set_real(S2,S4))) )
=> ( ! [A4: real] :
( pp(aa_set_real_bool(member_real(A4),S4))
=> ( aa_real_real(G,A4) = zero_zero_real ) )
=> ( ! [B4: complex] :
( pp(aa_set_complex_bool(member_complex(B4),T3))
=> ( aa_complex_real(H,B4) = zero_zero_real ) )
=> ( ! [A4: real] :
( pp(aa_set_real_bool(member_real(A4),S2))
=> ( aa_complex_real(H,aa_real_complex(J,A4)) = aa_real_real(G,A4) ) )
=> ( groups8097168146408367636l_real(G,S2) = groups5808333547571424918x_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
tff(fact_6380_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_int,T3: set_real,S2: set_int,I: fun_real_int,J: fun_int_real,T2: set_real,G: fun_int_real,H: fun_real_real] :
( pp(aa_set_int_bool(finite_finite_int,S4))
=> ( pp(aa_set_real_bool(finite_finite_real,T3))
=> ( ! [A4: int] :
( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(S2,S4)))
=> ( aa_real_int(I,aa_int_real(J,A4)) = A4 ) )
=> ( ! [A4: int] :
( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(S2,S4)))
=> pp(aa_set_real_bool(member_real(aa_int_real(J,A4)),minus_minus_set_real(T2,T3))) )
=> ( ! [B4: real] :
( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(T2,T3)))
=> ( aa_int_real(J,aa_real_int(I,B4)) = B4 ) )
=> ( ! [B4: real] :
( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(T2,T3)))
=> pp(aa_set_int_bool(member_int(aa_real_int(I,B4)),minus_minus_set_int(S2,S4))) )
=> ( ! [A4: int] :
( pp(aa_set_int_bool(member_int(A4),S4))
=> ( aa_int_real(G,A4) = zero_zero_real ) )
=> ( ! [B4: real] :
( pp(aa_set_real_bool(member_real(B4),T3))
=> ( aa_real_real(H,B4) = zero_zero_real ) )
=> ( ! [A4: int] :
( pp(aa_set_int_bool(member_int(A4),S2))
=> ( aa_real_real(H,aa_int_real(J,A4)) = aa_int_real(G,A4) ) )
=> ( groups8778361861064173332t_real(G,S2) = groups8097168146408367636l_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
tff(fact_6381_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_int,T3: set_int,S2: set_int,I: fun_int_int,J: fun_int_int,T2: set_int,G: fun_int_real,H: fun_int_real] :
( pp(aa_set_int_bool(finite_finite_int,S4))
=> ( pp(aa_set_int_bool(finite_finite_int,T3))
=> ( ! [A4: int] :
( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(S2,S4)))
=> ( aa_int_int(I,aa_int_int(J,A4)) = A4 ) )
=> ( ! [A4: int] :
( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(S2,S4)))
=> pp(aa_set_int_bool(member_int(aa_int_int(J,A4)),minus_minus_set_int(T2,T3))) )
=> ( ! [B4: int] :
( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(T2,T3)))
=> ( aa_int_int(J,aa_int_int(I,B4)) = B4 ) )
=> ( ! [B4: int] :
( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(T2,T3)))
=> pp(aa_set_int_bool(member_int(aa_int_int(I,B4)),minus_minus_set_int(S2,S4))) )
=> ( ! [A4: int] :
( pp(aa_set_int_bool(member_int(A4),S4))
=> ( aa_int_real(G,A4) = zero_zero_real ) )
=> ( ! [B4: int] :
( pp(aa_set_int_bool(member_int(B4),T3))
=> ( aa_int_real(H,B4) = zero_zero_real ) )
=> ( ! [A4: int] :
( pp(aa_set_int_bool(member_int(A4),S2))
=> ( aa_int_real(H,aa_int_int(J,A4)) = aa_int_real(G,A4) ) )
=> ( groups8778361861064173332t_real(G,S2) = groups8778361861064173332t_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
tff(fact_6382_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_int,T3: set_complex,S2: set_int,I: fun_complex_int,J: fun_int_complex,T2: set_complex,G: fun_int_real,H: fun_complex_real] :
( pp(aa_set_int_bool(finite_finite_int,S4))
=> ( pp(aa_set_complex_bool(finite3207457112153483333omplex,T3))
=> ( ! [A4: int] :
( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(S2,S4)))
=> ( aa_complex_int(I,aa_int_complex(J,A4)) = A4 ) )
=> ( ! [A4: int] :
( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(S2,S4)))
=> pp(aa_set_complex_bool(member_complex(aa_int_complex(J,A4)),minus_811609699411566653omplex(T2,T3))) )
=> ( ! [B4: complex] :
( pp(aa_set_complex_bool(member_complex(B4),minus_811609699411566653omplex(T2,T3)))
=> ( aa_int_complex(J,aa_complex_int(I,B4)) = B4 ) )
=> ( ! [B4: complex] :
( pp(aa_set_complex_bool(member_complex(B4),minus_811609699411566653omplex(T2,T3)))
=> pp(aa_set_int_bool(member_int(aa_complex_int(I,B4)),minus_minus_set_int(S2,S4))) )
=> ( ! [A4: int] :
( pp(aa_set_int_bool(member_int(A4),S4))
=> ( aa_int_real(G,A4) = zero_zero_real ) )
=> ( ! [B4: complex] :
( pp(aa_set_complex_bool(member_complex(B4),T3))
=> ( aa_complex_real(H,B4) = zero_zero_real ) )
=> ( ! [A4: int] :
( pp(aa_set_int_bool(member_int(A4),S2))
=> ( aa_complex_real(H,aa_int_complex(J,A4)) = aa_int_real(G,A4) ) )
=> ( groups8778361861064173332t_real(G,S2) = groups5808333547571424918x_real(H,T2) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
tff(fact_6383_sum__nonneg__0,axiom,
! [S: set_real,F: fun_real_real,I: real] :
( pp(aa_set_real_bool(finite_finite_real,S))
=> ( ! [I2: real] :
( pp(aa_set_real_bool(member_real(I2),S))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2))) )
=> ( ( groups8097168146408367636l_real(F,S) = zero_zero_real )
=> ( pp(aa_set_real_bool(member_real(I),S))
=> ( aa_real_real(F,I) = zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
tff(fact_6384_sum__nonneg__0,axiom,
! [S: set_int,F: fun_int_real,I: int] :
( pp(aa_set_int_bool(finite_finite_int,S))
=> ( ! [I2: int] :
( pp(aa_set_int_bool(member_int(I2),S))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2))) )
=> ( ( groups8778361861064173332t_real(F,S) = zero_zero_real )
=> ( pp(aa_set_int_bool(member_int(I),S))
=> ( aa_int_real(F,I) = zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
tff(fact_6385_sum__nonneg__0,axiom,
! [S: set_complex,F: fun_complex_real,I: complex] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
=> ( ! [I2: complex] :
( pp(aa_set_complex_bool(member_complex(I2),S))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2))) )
=> ( ( groups5808333547571424918x_real(F,S) = zero_zero_real )
=> ( pp(aa_set_complex_bool(member_complex(I),S))
=> ( aa_complex_real(F,I) = zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
tff(fact_6386_sum__nonneg__0,axiom,
! [S: set_real,F: fun_real_rat,I: real] :
( pp(aa_set_real_bool(finite_finite_real,S))
=> ( ! [I2: real] :
( pp(aa_set_real_bool(member_real(I2),S))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2))) )
=> ( ( groups1300246762558778688al_rat(F,S) = zero_zero_rat )
=> ( pp(aa_set_real_bool(member_real(I),S))
=> ( aa_real_rat(F,I) = zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_0
tff(fact_6387_sum__nonneg__0,axiom,
! [S: set_nat,F: fun_nat_rat,I: nat] :
( pp(aa_set_nat_bool(finite_finite_nat,S))
=> ( ! [I2: nat] :
( pp(aa_set_nat_bool(member_nat(I2),S))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2))) )
=> ( ( groups2906978787729119204at_rat(F,S) = zero_zero_rat )
=> ( pp(aa_set_nat_bool(member_nat(I),S))
=> ( aa_nat_rat(F,I) = zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_0
tff(fact_6388_sum__nonneg__0,axiom,
! [S: set_int,F: fun_int_rat,I: int] :
( pp(aa_set_int_bool(finite_finite_int,S))
=> ( ! [I2: int] :
( pp(aa_set_int_bool(member_int(I2),S))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2))) )
=> ( ( groups3906332499630173760nt_rat(F,S) = zero_zero_rat )
=> ( pp(aa_set_int_bool(member_int(I),S))
=> ( aa_int_rat(F,I) = zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_0
tff(fact_6389_sum__nonneg__0,axiom,
! [S: set_complex,F: fun_complex_rat,I: complex] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
=> ( ! [I2: complex] :
( pp(aa_set_complex_bool(member_complex(I2),S))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2))) )
=> ( ( groups5058264527183730370ex_rat(F,S) = zero_zero_rat )
=> ( pp(aa_set_complex_bool(member_complex(I),S))
=> ( aa_complex_rat(F,I) = zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_0
tff(fact_6390_sum__nonneg__0,axiom,
! [S: set_real,F: fun_real_nat,I: real] :
( pp(aa_set_real_bool(finite_finite_real,S))
=> ( ! [I2: real] :
( pp(aa_set_real_bool(member_real(I2),S))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I2))) )
=> ( ( groups1935376822645274424al_nat(F,S) = zero_zero_nat )
=> ( pp(aa_set_real_bool(member_real(I),S))
=> ( aa_real_nat(F,I) = zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
tff(fact_6391_sum__nonneg__0,axiom,
! [S: set_int,F: fun_int_nat,I: int] :
( pp(aa_set_int_bool(finite_finite_int,S))
=> ( ! [I2: int] :
( pp(aa_set_int_bool(member_int(I2),S))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I2))) )
=> ( ( groups4541462559716669496nt_nat(F,S) = zero_zero_nat )
=> ( pp(aa_set_int_bool(member_int(I),S))
=> ( aa_int_nat(F,I) = zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
tff(fact_6392_sum__nonneg__0,axiom,
! [S: set_complex,F: fun_complex_nat,I: complex] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
=> ( ! [I2: complex] :
( pp(aa_set_complex_bool(member_complex(I2),S))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I2))) )
=> ( ( groups5693394587270226106ex_nat(F,S) = zero_zero_nat )
=> ( pp(aa_set_complex_bool(member_complex(I),S))
=> ( aa_complex_nat(F,I) = zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
tff(fact_6393_sum__nonneg__leq__bound,axiom,
! [S: set_real,F: fun_real_real,B2: real,I: real] :
( pp(aa_set_real_bool(finite_finite_real,S))
=> ( ! [I2: real] :
( pp(aa_set_real_bool(member_real(I2),S))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2))) )
=> ( ( groups8097168146408367636l_real(F,S) = B2 )
=> ( pp(aa_set_real_bool(member_real(I),S))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,I)),B2)) ) ) ) ) ).
% sum_nonneg_leq_bound
tff(fact_6394_sum__nonneg__leq__bound,axiom,
! [S: set_int,F: fun_int_real,B2: real,I: int] :
( pp(aa_set_int_bool(finite_finite_int,S))
=> ( ! [I2: int] :
( pp(aa_set_int_bool(member_int(I2),S))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2))) )
=> ( ( groups8778361861064173332t_real(F,S) = B2 )
=> ( pp(aa_set_int_bool(member_int(I),S))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,I)),B2)) ) ) ) ) ).
% sum_nonneg_leq_bound
tff(fact_6395_sum__nonneg__leq__bound,axiom,
! [S: set_complex,F: fun_complex_real,B2: real,I: complex] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
=> ( ! [I2: complex] :
( pp(aa_set_complex_bool(member_complex(I2),S))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2))) )
=> ( ( groups5808333547571424918x_real(F,S) = B2 )
=> ( pp(aa_set_complex_bool(member_complex(I),S))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,I)),B2)) ) ) ) ) ).
% sum_nonneg_leq_bound
tff(fact_6396_sum__nonneg__leq__bound,axiom,
! [S: set_real,F: fun_real_rat,B2: rat,I: real] :
( pp(aa_set_real_bool(finite_finite_real,S))
=> ( ! [I2: real] :
( pp(aa_set_real_bool(member_real(I2),S))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2))) )
=> ( ( groups1300246762558778688al_rat(F,S) = B2 )
=> ( pp(aa_set_real_bool(member_real(I),S))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I)),B2)) ) ) ) ) ).
% sum_nonneg_leq_bound
tff(fact_6397_sum__nonneg__leq__bound,axiom,
! [S: set_nat,F: fun_nat_rat,B2: rat,I: nat] :
( pp(aa_set_nat_bool(finite_finite_nat,S))
=> ( ! [I2: nat] :
( pp(aa_set_nat_bool(member_nat(I2),S))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2))) )
=> ( ( groups2906978787729119204at_rat(F,S) = B2 )
=> ( pp(aa_set_nat_bool(member_nat(I),S))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I)),B2)) ) ) ) ) ).
% sum_nonneg_leq_bound
tff(fact_6398_sum__nonneg__leq__bound,axiom,
! [S: set_int,F: fun_int_rat,B2: rat,I: int] :
( pp(aa_set_int_bool(finite_finite_int,S))
=> ( ! [I2: int] :
( pp(aa_set_int_bool(member_int(I2),S))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2))) )
=> ( ( groups3906332499630173760nt_rat(F,S) = B2 )
=> ( pp(aa_set_int_bool(member_int(I),S))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I)),B2)) ) ) ) ) ).
% sum_nonneg_leq_bound
tff(fact_6399_sum__nonneg__leq__bound,axiom,
! [S: set_complex,F: fun_complex_rat,B2: rat,I: complex] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
=> ( ! [I2: complex] :
( pp(aa_set_complex_bool(member_complex(I2),S))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2))) )
=> ( ( groups5058264527183730370ex_rat(F,S) = B2 )
=> ( pp(aa_set_complex_bool(member_complex(I),S))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I)),B2)) ) ) ) ) ).
% sum_nonneg_leq_bound
tff(fact_6400_sum__nonneg__leq__bound,axiom,
! [S: set_real,F: fun_real_nat,B2: nat,I: real] :
( pp(aa_set_real_bool(finite_finite_real,S))
=> ( ! [I2: real] :
( pp(aa_set_real_bool(member_real(I2),S))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I2))) )
=> ( ( groups1935376822645274424al_nat(F,S) = B2 )
=> ( pp(aa_set_real_bool(member_real(I),S))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(F,I)),B2)) ) ) ) ) ).
% sum_nonneg_leq_bound
tff(fact_6401_sum__nonneg__leq__bound,axiom,
! [S: set_int,F: fun_int_nat,B2: nat,I: int] :
( pp(aa_set_int_bool(finite_finite_int,S))
=> ( ! [I2: int] :
( pp(aa_set_int_bool(member_int(I2),S))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I2))) )
=> ( ( groups4541462559716669496nt_nat(F,S) = B2 )
=> ( pp(aa_set_int_bool(member_int(I),S))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F,I)),B2)) ) ) ) ) ).
% sum_nonneg_leq_bound
tff(fact_6402_sum__nonneg__leq__bound,axiom,
! [S: set_complex,F: fun_complex_nat,B2: nat,I: complex] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,S))
=> ( ! [I2: complex] :
( pp(aa_set_complex_bool(member_complex(I2),S))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I2))) )
=> ( ( groups5693394587270226106ex_nat(F,S) = B2 )
=> ( pp(aa_set_complex_bool(member_complex(I),S))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(F,I)),B2)) ) ) ) ) ).
% sum_nonneg_leq_bound
tff(fact_6403_sum_Osetdiff__irrelevant,axiom,
! [A2: set_real,G: fun_real_complex] :
( pp(aa_set_real_bool(finite_finite_real,A2))
=> ( groups5754745047067104278omplex(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_fk(G)))) = groups5754745047067104278omplex(G,A2) ) ) ).
% sum.setdiff_irrelevant
tff(fact_6404_sum_Osetdiff__irrelevant,axiom,
! [A2: set_int,G: fun_int_complex] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( groups3049146728041665814omplex(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_fl(G)))) = groups3049146728041665814omplex(G,A2) ) ) ).
% sum.setdiff_irrelevant
tff(fact_6405_sum_Osetdiff__irrelevant,axiom,
! [A2: set_real,G: fun_real_real] :
( pp(aa_set_real_bool(finite_finite_real,A2))
=> ( groups8097168146408367636l_real(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_fm(G)))) = groups8097168146408367636l_real(G,A2) ) ) ).
% sum.setdiff_irrelevant
tff(fact_6406_sum_Osetdiff__irrelevant,axiom,
! [A2: set_int,G: fun_int_real] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( groups8778361861064173332t_real(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_fn(G)))) = groups8778361861064173332t_real(G,A2) ) ) ).
% sum.setdiff_irrelevant
tff(fact_6407_sum_Osetdiff__irrelevant,axiom,
! [A2: set_complex,G: fun_complex_real] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_fo(G)))) = groups5808333547571424918x_real(G,A2) ) ) ).
% sum.setdiff_irrelevant
tff(fact_6408_sum_Osetdiff__irrelevant,axiom,
! [A2: set_real,G: fun_real_rat] :
( pp(aa_set_real_bool(finite_finite_real,A2))
=> ( groups1300246762558778688al_rat(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_fp(G)))) = groups1300246762558778688al_rat(G,A2) ) ) ).
% sum.setdiff_irrelevant
tff(fact_6409_sum_Osetdiff__irrelevant,axiom,
! [A2: set_int,G: fun_int_rat] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_fq(G)))) = groups3906332499630173760nt_rat(G,A2) ) ) ).
% sum.setdiff_irrelevant
tff(fact_6410_sum_Osetdiff__irrelevant,axiom,
! [A2: set_complex,G: fun_complex_rat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,collect_complex(aTP_Lamm_fr(G)))) = groups5058264527183730370ex_rat(G,A2) ) ) ).
% sum.setdiff_irrelevant
tff(fact_6411_sum_Osetdiff__irrelevant,axiom,
! [A2: set_real,G: fun_real_nat] :
( pp(aa_set_real_bool(finite_finite_real,A2))
=> ( groups1935376822645274424al_nat(G,minus_minus_set_real(A2,collect_real(aTP_Lamm_fs(G)))) = groups1935376822645274424al_nat(G,A2) ) ) ).
% sum.setdiff_irrelevant
tff(fact_6412_sum_Osetdiff__irrelevant,axiom,
! [A2: set_int,G: fun_int_nat] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( groups4541462559716669496nt_nat(G,minus_minus_set_int(A2,collect_int(aTP_Lamm_ft(G)))) = groups4541462559716669496nt_nat(G,A2) ) ) ).
% sum.setdiff_irrelevant
tff(fact_6413_sum__pos2,axiom,
! [I5: set_real,I: real,F: fun_real_real] :
( pp(aa_set_real_bool(finite_finite_real,I5))
=> ( pp(aa_set_real_bool(member_real(I),I5))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,I)))
=> ( ! [I2: real] :
( pp(aa_set_real_bool(member_real(I2),I5))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2))) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F,I5))) ) ) ) ) ).
% sum_pos2
tff(fact_6414_sum__pos2,axiom,
! [I5: set_int,I: int,F: fun_int_real] :
( pp(aa_set_int_bool(finite_finite_int,I5))
=> ( pp(aa_set_int_bool(member_int(I),I5))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(F,I)))
=> ( ! [I2: int] :
( pp(aa_set_int_bool(member_int(I2),I5))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2))) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F,I5))) ) ) ) ) ).
% sum_pos2
tff(fact_6415_sum__pos2,axiom,
! [I5: set_complex,I: complex,F: fun_complex_real] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
=> ( pp(aa_set_complex_bool(member_complex(I),I5))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(F,I)))
=> ( ! [I2: complex] :
( pp(aa_set_complex_bool(member_complex(I2),I5))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2))) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F,I5))) ) ) ) ) ).
% sum_pos2
tff(fact_6416_sum__pos2,axiom,
! [I5: set_real,I: real,F: fun_real_rat] :
( pp(aa_set_real_bool(finite_finite_real,I5))
=> ( pp(aa_set_real_bool(member_real(I),I5))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_real_rat(F,I)))
=> ( ! [I2: real] :
( pp(aa_set_real_bool(member_real(I2),I5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(F,I5))) ) ) ) ) ).
% sum_pos2
tff(fact_6417_sum__pos2,axiom,
! [I5: set_nat,I: nat,F: fun_nat_rat] :
( pp(aa_set_nat_bool(finite_finite_nat,I5))
=> ( pp(aa_set_nat_bool(member_nat(I),I5))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I)))
=> ( ! [I2: nat] :
( pp(aa_set_nat_bool(member_nat(I2),I5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(F,I5))) ) ) ) ) ).
% sum_pos2
tff(fact_6418_sum__pos2,axiom,
! [I5: set_int,I: int,F: fun_int_rat] :
( pp(aa_set_int_bool(finite_finite_int,I5))
=> ( pp(aa_set_int_bool(member_int(I),I5))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(F,I)))
=> ( ! [I2: int] :
( pp(aa_set_int_bool(member_int(I2),I5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,I5))) ) ) ) ) ).
% sum_pos2
tff(fact_6419_sum__pos2,axiom,
! [I5: set_complex,I: complex,F: fun_complex_rat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
=> ( pp(aa_set_complex_bool(member_complex(I),I5))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I)))
=> ( ! [I2: complex] :
( pp(aa_set_complex_bool(member_complex(I2),I5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,I5))) ) ) ) ) ).
% sum_pos2
tff(fact_6420_sum__pos2,axiom,
! [I5: set_real,I: real,F: fun_real_nat] :
( pp(aa_set_real_bool(finite_finite_real,I5))
=> ( pp(aa_set_real_bool(member_real(I),I5))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_real_nat(F,I)))
=> ( ! [I2: real] :
( pp(aa_set_real_bool(member_real(I2),I5))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups1935376822645274424al_nat(F,I5))) ) ) ) ) ).
% sum_pos2
tff(fact_6421_sum__pos2,axiom,
! [I5: set_int,I: int,F: fun_int_nat] :
( pp(aa_set_int_bool(finite_finite_int,I5))
=> ( pp(aa_set_int_bool(member_int(I),I5))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(F,I)))
=> ( ! [I2: int] :
( pp(aa_set_int_bool(member_int(I2),I5))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,I5))) ) ) ) ) ).
% sum_pos2
tff(fact_6422_sum__pos2,axiom,
! [I5: set_complex,I: complex,F: fun_complex_nat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
=> ( pp(aa_set_complex_bool(member_complex(I),I5))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_complex_nat(F,I)))
=> ( ! [I2: complex] :
( pp(aa_set_complex_bool(member_complex(I2),I5))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,I5))) ) ) ) ) ).
% sum_pos2
tff(fact_6423_sum__pos,axiom,
! [I5: set_complex,F: fun_complex_real] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
=> ( ( I5 != bot_bot_set_complex )
=> ( ! [I2: complex] :
( pp(aa_set_complex_bool(member_complex(I2),I5))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(F,I2))) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F,I5))) ) ) ) ).
% sum_pos
tff(fact_6424_sum__pos,axiom,
! [I5: set_int,F: fun_int_real] :
( pp(aa_set_int_bool(finite_finite_int,I5))
=> ( ( I5 != bot_bot_set_int )
=> ( ! [I2: int] :
( pp(aa_set_int_bool(member_int(I2),I5))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(F,I2))) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F,I5))) ) ) ) ).
% sum_pos
tff(fact_6425_sum__pos,axiom,
! [I5: set_real,F: fun_real_real] :
( pp(aa_set_real_bool(finite_finite_real,I5))
=> ( ( I5 != bot_bot_set_real )
=> ( ! [I2: real] :
( pp(aa_set_real_bool(member_real(I2),I5))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,I2))) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F,I5))) ) ) ) ).
% sum_pos
tff(fact_6426_sum__pos,axiom,
! [I5: set_complex,F: fun_complex_rat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
=> ( ( I5 != bot_bot_set_complex )
=> ( ! [I2: complex] :
( pp(aa_set_complex_bool(member_complex(I2),I5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,I5))) ) ) ) ).
% sum_pos
tff(fact_6427_sum__pos,axiom,
! [I5: set_nat,F: fun_nat_rat] :
( pp(aa_set_nat_bool(finite_finite_nat,I5))
=> ( ( I5 != bot_bot_set_nat )
=> ( ! [I2: nat] :
( pp(aa_set_nat_bool(member_nat(I2),I5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(F,I5))) ) ) ) ).
% sum_pos
tff(fact_6428_sum__pos,axiom,
! [I5: set_int,F: fun_int_rat] :
( pp(aa_set_int_bool(finite_finite_int,I5))
=> ( ( I5 != bot_bot_set_int )
=> ( ! [I2: int] :
( pp(aa_set_int_bool(member_int(I2),I5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(F,I2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,I5))) ) ) ) ).
% sum_pos
tff(fact_6429_sum__pos,axiom,
! [I5: set_real,F: fun_real_rat] :
( pp(aa_set_real_bool(finite_finite_real,I5))
=> ( ( I5 != bot_bot_set_real )
=> ( ! [I2: real] :
( pp(aa_set_real_bool(member_real(I2),I5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_real_rat(F,I2))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(F,I5))) ) ) ) ).
% sum_pos
tff(fact_6430_sum__pos,axiom,
! [I5: set_complex,F: fun_complex_nat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,I5))
=> ( ( I5 != bot_bot_set_complex )
=> ( ! [I2: complex] :
( pp(aa_set_complex_bool(member_complex(I2),I5))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_complex_nat(F,I2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,I5))) ) ) ) ).
% sum_pos
tff(fact_6431_sum__pos,axiom,
! [I5: set_int,F: fun_int_nat] :
( pp(aa_set_int_bool(finite_finite_int,I5))
=> ( ( I5 != bot_bot_set_int )
=> ( ! [I2: int] :
( pp(aa_set_int_bool(member_int(I2),I5))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(F,I2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,I5))) ) ) ) ).
% sum_pos
tff(fact_6432_sum__pos,axiom,
! [I5: set_real,F: fun_real_nat] :
( pp(aa_set_real_bool(finite_finite_real,I5))
=> ( ( I5 != bot_bot_set_real )
=> ( ! [I2: real] :
( pp(aa_set_real_bool(member_real(I2),I5))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_real_nat(F,I2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),groups1935376822645274424al_nat(F,I5))) ) ) ) ).
% sum_pos
tff(fact_6433_sum_Osame__carrier,axiom,
! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_complex,H: fun_real_complex] :
( pp(aa_set_real_bool(finite_finite_real,C4))
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C4))
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C4))
=> ( ! [A4: real] :
( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(C4,A2)))
=> ( aa_real_complex(G,A4) = zero_zero_complex ) )
=> ( ! [B4: real] :
( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(C4,B2)))
=> ( aa_real_complex(H,B4) = zero_zero_complex ) )
=> ( ( groups5754745047067104278omplex(G,A2) = groups5754745047067104278omplex(H,B2) )
<=> ( groups5754745047067104278omplex(G,C4) = groups5754745047067104278omplex(H,C4) ) ) ) ) ) ) ) ).
% sum.same_carrier
tff(fact_6434_sum_Osame__carrier,axiom,
! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_complex,H: fun_int_complex] :
( pp(aa_set_int_bool(finite_finite_int,C4))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C4))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C4))
=> ( ! [A4: int] :
( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(C4,A2)))
=> ( aa_int_complex(G,A4) = zero_zero_complex ) )
=> ( ! [B4: int] :
( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(C4,B2)))
=> ( aa_int_complex(H,B4) = zero_zero_complex ) )
=> ( ( groups3049146728041665814omplex(G,A2) = groups3049146728041665814omplex(H,B2) )
<=> ( groups3049146728041665814omplex(G,C4) = groups3049146728041665814omplex(H,C4) ) ) ) ) ) ) ) ).
% sum.same_carrier
tff(fact_6435_sum_Osame__carrier,axiom,
! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_real,H: fun_real_real] :
( pp(aa_set_real_bool(finite_finite_real,C4))
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C4))
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C4))
=> ( ! [A4: real] :
( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(C4,A2)))
=> ( aa_real_real(G,A4) = zero_zero_real ) )
=> ( ! [B4: real] :
( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(C4,B2)))
=> ( aa_real_real(H,B4) = zero_zero_real ) )
=> ( ( groups8097168146408367636l_real(G,A2) = groups8097168146408367636l_real(H,B2) )
<=> ( groups8097168146408367636l_real(G,C4) = groups8097168146408367636l_real(H,C4) ) ) ) ) ) ) ) ).
% sum.same_carrier
tff(fact_6436_sum_Osame__carrier,axiom,
! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_real,H: fun_int_real] :
( pp(aa_set_int_bool(finite_finite_int,C4))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C4))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C4))
=> ( ! [A4: int] :
( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(C4,A2)))
=> ( aa_int_real(G,A4) = zero_zero_real ) )
=> ( ! [B4: int] :
( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(C4,B2)))
=> ( aa_int_real(H,B4) = zero_zero_real ) )
=> ( ( groups8778361861064173332t_real(G,A2) = groups8778361861064173332t_real(H,B2) )
<=> ( groups8778361861064173332t_real(G,C4) = groups8778361861064173332t_real(H,C4) ) ) ) ) ) ) ) ).
% sum.same_carrier
tff(fact_6437_sum_Osame__carrier,axiom,
! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_real,H: fun_complex_real] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C4))
=> ( ! [A4: complex] :
( pp(aa_set_complex_bool(member_complex(A4),minus_811609699411566653omplex(C4,A2)))
=> ( aa_complex_real(G,A4) = zero_zero_real ) )
=> ( ! [B4: complex] :
( pp(aa_set_complex_bool(member_complex(B4),minus_811609699411566653omplex(C4,B2)))
=> ( aa_complex_real(H,B4) = zero_zero_real ) )
=> ( ( groups5808333547571424918x_real(G,A2) = groups5808333547571424918x_real(H,B2) )
<=> ( groups5808333547571424918x_real(G,C4) = groups5808333547571424918x_real(H,C4) ) ) ) ) ) ) ) ).
% sum.same_carrier
tff(fact_6438_sum_Osame__carrier,axiom,
! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_rat,H: fun_real_rat] :
( pp(aa_set_real_bool(finite_finite_real,C4))
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C4))
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C4))
=> ( ! [A4: real] :
( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(C4,A2)))
=> ( aa_real_rat(G,A4) = zero_zero_rat ) )
=> ( ! [B4: real] :
( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(C4,B2)))
=> ( aa_real_rat(H,B4) = zero_zero_rat ) )
=> ( ( groups1300246762558778688al_rat(G,A2) = groups1300246762558778688al_rat(H,B2) )
<=> ( groups1300246762558778688al_rat(G,C4) = groups1300246762558778688al_rat(H,C4) ) ) ) ) ) ) ) ).
% sum.same_carrier
tff(fact_6439_sum_Osame__carrier,axiom,
! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_rat,H: fun_int_rat] :
( pp(aa_set_int_bool(finite_finite_int,C4))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C4))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C4))
=> ( ! [A4: int] :
( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(C4,A2)))
=> ( aa_int_rat(G,A4) = zero_zero_rat ) )
=> ( ! [B4: int] :
( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(C4,B2)))
=> ( aa_int_rat(H,B4) = zero_zero_rat ) )
=> ( ( groups3906332499630173760nt_rat(G,A2) = groups3906332499630173760nt_rat(H,B2) )
<=> ( groups3906332499630173760nt_rat(G,C4) = groups3906332499630173760nt_rat(H,C4) ) ) ) ) ) ) ) ).
% sum.same_carrier
tff(fact_6440_sum_Osame__carrier,axiom,
! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C4))
=> ( ! [A4: complex] :
( pp(aa_set_complex_bool(member_complex(A4),minus_811609699411566653omplex(C4,A2)))
=> ( aa_complex_rat(G,A4) = zero_zero_rat ) )
=> ( ! [B4: complex] :
( pp(aa_set_complex_bool(member_complex(B4),minus_811609699411566653omplex(C4,B2)))
=> ( aa_complex_rat(H,B4) = zero_zero_rat ) )
=> ( ( groups5058264527183730370ex_rat(G,A2) = groups5058264527183730370ex_rat(H,B2) )
<=> ( groups5058264527183730370ex_rat(G,C4) = groups5058264527183730370ex_rat(H,C4) ) ) ) ) ) ) ) ).
% sum.same_carrier
tff(fact_6441_sum_Osame__carrier,axiom,
! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_nat,H: fun_real_nat] :
( pp(aa_set_real_bool(finite_finite_real,C4))
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C4))
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C4))
=> ( ! [A4: real] :
( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(C4,A2)))
=> ( aa_real_nat(G,A4) = zero_zero_nat ) )
=> ( ! [B4: real] :
( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(C4,B2)))
=> ( aa_real_nat(H,B4) = zero_zero_nat ) )
=> ( ( groups1935376822645274424al_nat(G,A2) = groups1935376822645274424al_nat(H,B2) )
<=> ( groups1935376822645274424al_nat(G,C4) = groups1935376822645274424al_nat(H,C4) ) ) ) ) ) ) ) ).
% sum.same_carrier
tff(fact_6442_sum_Osame__carrier,axiom,
! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_nat,H: fun_int_nat] :
( pp(aa_set_int_bool(finite_finite_int,C4))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C4))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C4))
=> ( ! [A4: int] :
( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(C4,A2)))
=> ( aa_int_nat(G,A4) = zero_zero_nat ) )
=> ( ! [B4: int] :
( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(C4,B2)))
=> ( aa_int_nat(H,B4) = zero_zero_nat ) )
=> ( ( groups4541462559716669496nt_nat(G,A2) = groups4541462559716669496nt_nat(H,B2) )
<=> ( groups4541462559716669496nt_nat(G,C4) = groups4541462559716669496nt_nat(H,C4) ) ) ) ) ) ) ) ).
% sum.same_carrier
tff(fact_6443_sum_Osame__carrierI,axiom,
! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_complex,H: fun_real_complex] :
( pp(aa_set_real_bool(finite_finite_real,C4))
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C4))
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C4))
=> ( ! [A4: real] :
( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(C4,A2)))
=> ( aa_real_complex(G,A4) = zero_zero_complex ) )
=> ( ! [B4: real] :
( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(C4,B2)))
=> ( aa_real_complex(H,B4) = zero_zero_complex ) )
=> ( ( groups5754745047067104278omplex(G,C4) = groups5754745047067104278omplex(H,C4) )
=> ( groups5754745047067104278omplex(G,A2) = groups5754745047067104278omplex(H,B2) ) ) ) ) ) ) ) ).
% sum.same_carrierI
tff(fact_6444_sum_Osame__carrierI,axiom,
! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_complex,H: fun_int_complex] :
( pp(aa_set_int_bool(finite_finite_int,C4))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C4))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C4))
=> ( ! [A4: int] :
( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(C4,A2)))
=> ( aa_int_complex(G,A4) = zero_zero_complex ) )
=> ( ! [B4: int] :
( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(C4,B2)))
=> ( aa_int_complex(H,B4) = zero_zero_complex ) )
=> ( ( groups3049146728041665814omplex(G,C4) = groups3049146728041665814omplex(H,C4) )
=> ( groups3049146728041665814omplex(G,A2) = groups3049146728041665814omplex(H,B2) ) ) ) ) ) ) ) ).
% sum.same_carrierI
tff(fact_6445_sum_Osame__carrierI,axiom,
! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_real,H: fun_real_real] :
( pp(aa_set_real_bool(finite_finite_real,C4))
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C4))
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C4))
=> ( ! [A4: real] :
( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(C4,A2)))
=> ( aa_real_real(G,A4) = zero_zero_real ) )
=> ( ! [B4: real] :
( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(C4,B2)))
=> ( aa_real_real(H,B4) = zero_zero_real ) )
=> ( ( groups8097168146408367636l_real(G,C4) = groups8097168146408367636l_real(H,C4) )
=> ( groups8097168146408367636l_real(G,A2) = groups8097168146408367636l_real(H,B2) ) ) ) ) ) ) ) ).
% sum.same_carrierI
tff(fact_6446_sum_Osame__carrierI,axiom,
! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_real,H: fun_int_real] :
( pp(aa_set_int_bool(finite_finite_int,C4))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C4))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C4))
=> ( ! [A4: int] :
( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(C4,A2)))
=> ( aa_int_real(G,A4) = zero_zero_real ) )
=> ( ! [B4: int] :
( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(C4,B2)))
=> ( aa_int_real(H,B4) = zero_zero_real ) )
=> ( ( groups8778361861064173332t_real(G,C4) = groups8778361861064173332t_real(H,C4) )
=> ( groups8778361861064173332t_real(G,A2) = groups8778361861064173332t_real(H,B2) ) ) ) ) ) ) ) ).
% sum.same_carrierI
tff(fact_6447_sum_Osame__carrierI,axiom,
! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_real,H: fun_complex_real] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C4))
=> ( ! [A4: complex] :
( pp(aa_set_complex_bool(member_complex(A4),minus_811609699411566653omplex(C4,A2)))
=> ( aa_complex_real(G,A4) = zero_zero_real ) )
=> ( ! [B4: complex] :
( pp(aa_set_complex_bool(member_complex(B4),minus_811609699411566653omplex(C4,B2)))
=> ( aa_complex_real(H,B4) = zero_zero_real ) )
=> ( ( groups5808333547571424918x_real(G,C4) = groups5808333547571424918x_real(H,C4) )
=> ( groups5808333547571424918x_real(G,A2) = groups5808333547571424918x_real(H,B2) ) ) ) ) ) ) ) ).
% sum.same_carrierI
tff(fact_6448_sum_Osame__carrierI,axiom,
! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_rat,H: fun_real_rat] :
( pp(aa_set_real_bool(finite_finite_real,C4))
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C4))
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C4))
=> ( ! [A4: real] :
( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(C4,A2)))
=> ( aa_real_rat(G,A4) = zero_zero_rat ) )
=> ( ! [B4: real] :
( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(C4,B2)))
=> ( aa_real_rat(H,B4) = zero_zero_rat ) )
=> ( ( groups1300246762558778688al_rat(G,C4) = groups1300246762558778688al_rat(H,C4) )
=> ( groups1300246762558778688al_rat(G,A2) = groups1300246762558778688al_rat(H,B2) ) ) ) ) ) ) ) ).
% sum.same_carrierI
tff(fact_6449_sum_Osame__carrierI,axiom,
! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_rat,H: fun_int_rat] :
( pp(aa_set_int_bool(finite_finite_int,C4))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C4))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C4))
=> ( ! [A4: int] :
( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(C4,A2)))
=> ( aa_int_rat(G,A4) = zero_zero_rat ) )
=> ( ! [B4: int] :
( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(C4,B2)))
=> ( aa_int_rat(H,B4) = zero_zero_rat ) )
=> ( ( groups3906332499630173760nt_rat(G,C4) = groups3906332499630173760nt_rat(H,C4) )
=> ( groups3906332499630173760nt_rat(G,A2) = groups3906332499630173760nt_rat(H,B2) ) ) ) ) ) ) ) ).
% sum.same_carrierI
tff(fact_6450_sum_Osame__carrierI,axiom,
! [C4: set_complex,A2: set_complex,B2: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,C4))
=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),C4))
=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),C4))
=> ( ! [A4: complex] :
( pp(aa_set_complex_bool(member_complex(A4),minus_811609699411566653omplex(C4,A2)))
=> ( aa_complex_rat(G,A4) = zero_zero_rat ) )
=> ( ! [B4: complex] :
( pp(aa_set_complex_bool(member_complex(B4),minus_811609699411566653omplex(C4,B2)))
=> ( aa_complex_rat(H,B4) = zero_zero_rat ) )
=> ( ( groups5058264527183730370ex_rat(G,C4) = groups5058264527183730370ex_rat(H,C4) )
=> ( groups5058264527183730370ex_rat(G,A2) = groups5058264527183730370ex_rat(H,B2) ) ) ) ) ) ) ) ).
% sum.same_carrierI
tff(fact_6451_sum_Osame__carrierI,axiom,
! [C4: set_real,A2: set_real,B2: set_real,G: fun_real_nat,H: fun_real_nat] :
( pp(aa_set_real_bool(finite_finite_real,C4))
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),C4))
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(B2),C4))
=> ( ! [A4: real] :
( pp(aa_set_real_bool(member_real(A4),minus_minus_set_real(C4,A2)))
=> ( aa_real_nat(G,A4) = zero_zero_nat ) )
=> ( ! [B4: real] :
( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(C4,B2)))
=> ( aa_real_nat(H,B4) = zero_zero_nat ) )
=> ( ( groups1935376822645274424al_nat(G,C4) = groups1935376822645274424al_nat(H,C4) )
=> ( groups1935376822645274424al_nat(G,A2) = groups1935376822645274424al_nat(H,B2) ) ) ) ) ) ) ) ).
% sum.same_carrierI
tff(fact_6452_sum_Osame__carrierI,axiom,
! [C4: set_int,A2: set_int,B2: set_int,G: fun_int_nat,H: fun_int_nat] :
( pp(aa_set_int_bool(finite_finite_int,C4))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),C4))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),C4))
=> ( ! [A4: int] :
( pp(aa_set_int_bool(member_int(A4),minus_minus_set_int(C4,A2)))
=> ( aa_int_nat(G,A4) = zero_zero_nat ) )
=> ( ! [B4: int] :
( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(C4,B2)))
=> ( aa_int_nat(H,B4) = zero_zero_nat ) )
=> ( ( groups4541462559716669496nt_nat(G,C4) = groups4541462559716669496nt_nat(H,C4) )
=> ( groups4541462559716669496nt_nat(G,A2) = groups4541462559716669496nt_nat(H,B2) ) ) ) ) ) ) ) ).
% sum.same_carrierI
tff(fact_6453_sum_Omono__neutral__left,axiom,
! [T2: set_int,S2: set_int,G: fun_int_complex] :
( pp(aa_set_int_bool(finite_finite_int,T2))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(T2,S2)))
=> ( aa_int_complex(G,X3) = zero_zero_complex ) )
=> ( groups3049146728041665814omplex(G,S2) = groups3049146728041665814omplex(G,T2) ) ) ) ) ).
% sum.mono_neutral_left
tff(fact_6454_sum_Omono__neutral__left,axiom,
! [T2: set_int,S2: set_int,G: fun_int_real] :
( pp(aa_set_int_bool(finite_finite_int,T2))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(T2,S2)))
=> ( aa_int_real(G,X3) = zero_zero_real ) )
=> ( groups8778361861064173332t_real(G,S2) = groups8778361861064173332t_real(G,T2) ) ) ) ) ).
% sum.mono_neutral_left
tff(fact_6455_sum_Omono__neutral__left,axiom,
! [T2: set_complex,S2: set_complex,G: fun_complex_real] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(T2,S2)))
=> ( aa_complex_real(G,X3) = zero_zero_real ) )
=> ( groups5808333547571424918x_real(G,S2) = groups5808333547571424918x_real(G,T2) ) ) ) ) ).
% sum.mono_neutral_left
tff(fact_6456_sum_Omono__neutral__left,axiom,
! [T2: set_int,S2: set_int,G: fun_int_rat] :
( pp(aa_set_int_bool(finite_finite_int,T2))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(T2,S2)))
=> ( aa_int_rat(G,X3) = zero_zero_rat ) )
=> ( groups3906332499630173760nt_rat(G,S2) = groups3906332499630173760nt_rat(G,T2) ) ) ) ) ).
% sum.mono_neutral_left
tff(fact_6457_sum_Omono__neutral__left,axiom,
! [T2: set_complex,S2: set_complex,G: fun_complex_rat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(T2,S2)))
=> ( aa_complex_rat(G,X3) = zero_zero_rat ) )
=> ( groups5058264527183730370ex_rat(G,S2) = groups5058264527183730370ex_rat(G,T2) ) ) ) ) ).
% sum.mono_neutral_left
tff(fact_6458_sum_Omono__neutral__left,axiom,
! [T2: set_int,S2: set_int,G: fun_int_nat] :
( pp(aa_set_int_bool(finite_finite_int,T2))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(T2,S2)))
=> ( aa_int_nat(G,X3) = zero_zero_nat ) )
=> ( groups4541462559716669496nt_nat(G,S2) = groups4541462559716669496nt_nat(G,T2) ) ) ) ) ).
% sum.mono_neutral_left
tff(fact_6459_sum_Omono__neutral__left,axiom,
! [T2: set_complex,S2: set_complex,G: fun_complex_nat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(T2,S2)))
=> ( aa_complex_nat(G,X3) = zero_zero_nat ) )
=> ( groups5693394587270226106ex_nat(G,S2) = groups5693394587270226106ex_nat(G,T2) ) ) ) ) ).
% sum.mono_neutral_left
tff(fact_6460_sum_Omono__neutral__left,axiom,
! [T2: set_complex,S2: set_complex,G: fun_complex_int] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(T2,S2)))
=> ( aa_complex_int(G,X3) = zero_zero_int ) )
=> ( groups5690904116761175830ex_int(G,S2) = groups5690904116761175830ex_int(G,T2) ) ) ) ) ).
% sum.mono_neutral_left
tff(fact_6461_sum_Omono__neutral__left,axiom,
! [T2: set_nat,S2: set_nat,G: fun_nat_complex] :
( pp(aa_set_nat_bool(finite_finite_nat,T2))
=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(S2),T2))
=> ( ! [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),minus_minus_set_nat(T2,S2)))
=> ( aa_nat_complex(G,X3) = zero_zero_complex ) )
=> ( groups2073611262835488442omplex(G,S2) = groups2073611262835488442omplex(G,T2) ) ) ) ) ).
% sum.mono_neutral_left
tff(fact_6462_sum_Omono__neutral__left,axiom,
! [T2: set_nat,S2: set_nat,G: fun_nat_rat] :
( pp(aa_set_nat_bool(finite_finite_nat,T2))
=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(S2),T2))
=> ( ! [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),minus_minus_set_nat(T2,S2)))
=> ( aa_nat_rat(G,X3) = zero_zero_rat ) )
=> ( groups2906978787729119204at_rat(G,S2) = groups2906978787729119204at_rat(G,T2) ) ) ) ) ).
% sum.mono_neutral_left
tff(fact_6463_sum_Omono__neutral__right,axiom,
! [T2: set_int,S2: set_int,G: fun_int_complex] :
( pp(aa_set_int_bool(finite_finite_int,T2))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(T2,S2)))
=> ( aa_int_complex(G,X3) = zero_zero_complex ) )
=> ( groups3049146728041665814omplex(G,T2) = groups3049146728041665814omplex(G,S2) ) ) ) ) ).
% sum.mono_neutral_right
tff(fact_6464_sum_Omono__neutral__right,axiom,
! [T2: set_int,S2: set_int,G: fun_int_real] :
( pp(aa_set_int_bool(finite_finite_int,T2))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(T2,S2)))
=> ( aa_int_real(G,X3) = zero_zero_real ) )
=> ( groups8778361861064173332t_real(G,T2) = groups8778361861064173332t_real(G,S2) ) ) ) ) ).
% sum.mono_neutral_right
tff(fact_6465_sum_Omono__neutral__right,axiom,
! [T2: set_complex,S2: set_complex,G: fun_complex_real] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(T2,S2)))
=> ( aa_complex_real(G,X3) = zero_zero_real ) )
=> ( groups5808333547571424918x_real(G,T2) = groups5808333547571424918x_real(G,S2) ) ) ) ) ).
% sum.mono_neutral_right
tff(fact_6466_sum_Omono__neutral__right,axiom,
! [T2: set_int,S2: set_int,G: fun_int_rat] :
( pp(aa_set_int_bool(finite_finite_int,T2))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(T2,S2)))
=> ( aa_int_rat(G,X3) = zero_zero_rat ) )
=> ( groups3906332499630173760nt_rat(G,T2) = groups3906332499630173760nt_rat(G,S2) ) ) ) ) ).
% sum.mono_neutral_right
tff(fact_6467_sum_Omono__neutral__right,axiom,
! [T2: set_complex,S2: set_complex,G: fun_complex_rat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(T2,S2)))
=> ( aa_complex_rat(G,X3) = zero_zero_rat ) )
=> ( groups5058264527183730370ex_rat(G,T2) = groups5058264527183730370ex_rat(G,S2) ) ) ) ) ).
% sum.mono_neutral_right
tff(fact_6468_sum_Omono__neutral__right,axiom,
! [T2: set_int,S2: set_int,G: fun_int_nat] :
( pp(aa_set_int_bool(finite_finite_int,T2))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(T2,S2)))
=> ( aa_int_nat(G,X3) = zero_zero_nat ) )
=> ( groups4541462559716669496nt_nat(G,T2) = groups4541462559716669496nt_nat(G,S2) ) ) ) ) ).
% sum.mono_neutral_right
tff(fact_6469_sum_Omono__neutral__right,axiom,
! [T2: set_complex,S2: set_complex,G: fun_complex_nat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(T2,S2)))
=> ( aa_complex_nat(G,X3) = zero_zero_nat ) )
=> ( groups5693394587270226106ex_nat(G,T2) = groups5693394587270226106ex_nat(G,S2) ) ) ) ) ).
% sum.mono_neutral_right
tff(fact_6470_sum_Omono__neutral__right,axiom,
! [T2: set_complex,S2: set_complex,G: fun_complex_int] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(T2,S2)))
=> ( aa_complex_int(G,X3) = zero_zero_int ) )
=> ( groups5690904116761175830ex_int(G,T2) = groups5690904116761175830ex_int(G,S2) ) ) ) ) ).
% sum.mono_neutral_right
tff(fact_6471_sum_Omono__neutral__right,axiom,
! [T2: set_nat,S2: set_nat,G: fun_nat_complex] :
( pp(aa_set_nat_bool(finite_finite_nat,T2))
=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(S2),T2))
=> ( ! [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),minus_minus_set_nat(T2,S2)))
=> ( aa_nat_complex(G,X3) = zero_zero_complex ) )
=> ( groups2073611262835488442omplex(G,T2) = groups2073611262835488442omplex(G,S2) ) ) ) ) ).
% sum.mono_neutral_right
tff(fact_6472_sum_Omono__neutral__right,axiom,
! [T2: set_nat,S2: set_nat,G: fun_nat_rat] :
( pp(aa_set_nat_bool(finite_finite_nat,T2))
=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(S2),T2))
=> ( ! [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),minus_minus_set_nat(T2,S2)))
=> ( aa_nat_rat(G,X3) = zero_zero_rat ) )
=> ( groups2906978787729119204at_rat(G,T2) = groups2906978787729119204at_rat(G,S2) ) ) ) ) ).
% sum.mono_neutral_right
tff(fact_6473_sum_Omono__neutral__cong__left,axiom,
! [T2: set_real,S2: set_real,H: fun_real_complex,G: fun_real_complex] :
( pp(aa_set_real_bool(finite_finite_real,T2))
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),minus_minus_set_real(T2,S2)))
=> ( aa_real_complex(H,X3) = zero_zero_complex ) )
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),S2))
=> ( aa_real_complex(G,X3) = aa_real_complex(H,X3) ) )
=> ( groups5754745047067104278omplex(G,S2) = groups5754745047067104278omplex(H,T2) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
tff(fact_6474_sum_Omono__neutral__cong__left,axiom,
! [T2: set_int,S2: set_int,H: fun_int_complex,G: fun_int_complex] :
( pp(aa_set_int_bool(finite_finite_int,T2))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(T2,S2)))
=> ( aa_int_complex(H,X3) = zero_zero_complex ) )
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),S2))
=> ( aa_int_complex(G,X3) = aa_int_complex(H,X3) ) )
=> ( groups3049146728041665814omplex(G,S2) = groups3049146728041665814omplex(H,T2) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
tff(fact_6475_sum_Omono__neutral__cong__left,axiom,
! [T2: set_real,S2: set_real,H: fun_real_real,G: fun_real_real] :
( pp(aa_set_real_bool(finite_finite_real,T2))
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),minus_minus_set_real(T2,S2)))
=> ( aa_real_real(H,X3) = zero_zero_real ) )
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),S2))
=> ( aa_real_real(G,X3) = aa_real_real(H,X3) ) )
=> ( groups8097168146408367636l_real(G,S2) = groups8097168146408367636l_real(H,T2) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
tff(fact_6476_sum_Omono__neutral__cong__left,axiom,
! [T2: set_int,S2: set_int,H: fun_int_real,G: fun_int_real] :
( pp(aa_set_int_bool(finite_finite_int,T2))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(T2,S2)))
=> ( aa_int_real(H,X3) = zero_zero_real ) )
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),S2))
=> ( aa_int_real(G,X3) = aa_int_real(H,X3) ) )
=> ( groups8778361861064173332t_real(G,S2) = groups8778361861064173332t_real(H,T2) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
tff(fact_6477_sum_Omono__neutral__cong__left,axiom,
! [T2: set_complex,S2: set_complex,H: fun_complex_real,G: fun_complex_real] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(T2,S2)))
=> ( aa_complex_real(H,X3) = zero_zero_real ) )
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),S2))
=> ( aa_complex_real(G,X3) = aa_complex_real(H,X3) ) )
=> ( groups5808333547571424918x_real(G,S2) = groups5808333547571424918x_real(H,T2) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
tff(fact_6478_sum_Omono__neutral__cong__left,axiom,
! [T2: set_real,S2: set_real,H: fun_real_rat,G: fun_real_rat] :
( pp(aa_set_real_bool(finite_finite_real,T2))
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),minus_minus_set_real(T2,S2)))
=> ( aa_real_rat(H,X3) = zero_zero_rat ) )
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),S2))
=> ( aa_real_rat(G,X3) = aa_real_rat(H,X3) ) )
=> ( groups1300246762558778688al_rat(G,S2) = groups1300246762558778688al_rat(H,T2) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
tff(fact_6479_sum_Omono__neutral__cong__left,axiom,
! [T2: set_int,S2: set_int,H: fun_int_rat,G: fun_int_rat] :
( pp(aa_set_int_bool(finite_finite_int,T2))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(T2,S2)))
=> ( aa_int_rat(H,X3) = zero_zero_rat ) )
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),S2))
=> ( aa_int_rat(G,X3) = aa_int_rat(H,X3) ) )
=> ( groups3906332499630173760nt_rat(G,S2) = groups3906332499630173760nt_rat(H,T2) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
tff(fact_6480_sum_Omono__neutral__cong__left,axiom,
! [T2: set_complex,S2: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(T2,S2)))
=> ( aa_complex_rat(H,X3) = zero_zero_rat ) )
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),S2))
=> ( aa_complex_rat(G,X3) = aa_complex_rat(H,X3) ) )
=> ( groups5058264527183730370ex_rat(G,S2) = groups5058264527183730370ex_rat(H,T2) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
tff(fact_6481_sum_Omono__neutral__cong__left,axiom,
! [T2: set_real,S2: set_real,H: fun_real_nat,G: fun_real_nat] :
( pp(aa_set_real_bool(finite_finite_real,T2))
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),minus_minus_set_real(T2,S2)))
=> ( aa_real_nat(H,X3) = zero_zero_nat ) )
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),S2))
=> ( aa_real_nat(G,X3) = aa_real_nat(H,X3) ) )
=> ( groups1935376822645274424al_nat(G,S2) = groups1935376822645274424al_nat(H,T2) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
tff(fact_6482_sum_Omono__neutral__cong__left,axiom,
! [T2: set_int,S2: set_int,H: fun_int_nat,G: fun_int_nat] :
( pp(aa_set_int_bool(finite_finite_int,T2))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(T2,S2)))
=> ( aa_int_nat(H,X3) = zero_zero_nat ) )
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),S2))
=> ( aa_int_nat(G,X3) = aa_int_nat(H,X3) ) )
=> ( groups4541462559716669496nt_nat(G,S2) = groups4541462559716669496nt_nat(H,T2) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
tff(fact_6483_sum_Omono__neutral__cong__right,axiom,
! [T2: set_real,S2: set_real,G: fun_real_complex,H: fun_real_complex] :
( pp(aa_set_real_bool(finite_finite_real,T2))
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),minus_minus_set_real(T2,S2)))
=> ( aa_real_complex(G,X3) = zero_zero_complex ) )
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),S2))
=> ( aa_real_complex(G,X3) = aa_real_complex(H,X3) ) )
=> ( groups5754745047067104278omplex(G,T2) = groups5754745047067104278omplex(H,S2) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
tff(fact_6484_sum_Omono__neutral__cong__right,axiom,
! [T2: set_int,S2: set_int,G: fun_int_complex,H: fun_int_complex] :
( pp(aa_set_int_bool(finite_finite_int,T2))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(T2,S2)))
=> ( aa_int_complex(G,X3) = zero_zero_complex ) )
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),S2))
=> ( aa_int_complex(G,X3) = aa_int_complex(H,X3) ) )
=> ( groups3049146728041665814omplex(G,T2) = groups3049146728041665814omplex(H,S2) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
tff(fact_6485_sum_Omono__neutral__cong__right,axiom,
! [T2: set_real,S2: set_real,G: fun_real_real,H: fun_real_real] :
( pp(aa_set_real_bool(finite_finite_real,T2))
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),minus_minus_set_real(T2,S2)))
=> ( aa_real_real(G,X3) = zero_zero_real ) )
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),S2))
=> ( aa_real_real(G,X3) = aa_real_real(H,X3) ) )
=> ( groups8097168146408367636l_real(G,T2) = groups8097168146408367636l_real(H,S2) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
tff(fact_6486_sum_Omono__neutral__cong__right,axiom,
! [T2: set_int,S2: set_int,G: fun_int_real,H: fun_int_real] :
( pp(aa_set_int_bool(finite_finite_int,T2))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(T2,S2)))
=> ( aa_int_real(G,X3) = zero_zero_real ) )
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),S2))
=> ( aa_int_real(G,X3) = aa_int_real(H,X3) ) )
=> ( groups8778361861064173332t_real(G,T2) = groups8778361861064173332t_real(H,S2) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
tff(fact_6487_sum_Omono__neutral__cong__right,axiom,
! [T2: set_complex,S2: set_complex,G: fun_complex_real,H: fun_complex_real] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(T2,S2)))
=> ( aa_complex_real(G,X3) = zero_zero_real ) )
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),S2))
=> ( aa_complex_real(G,X3) = aa_complex_real(H,X3) ) )
=> ( groups5808333547571424918x_real(G,T2) = groups5808333547571424918x_real(H,S2) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
tff(fact_6488_sum_Omono__neutral__cong__right,axiom,
! [T2: set_real,S2: set_real,G: fun_real_rat,H: fun_real_rat] :
( pp(aa_set_real_bool(finite_finite_real,T2))
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),minus_minus_set_real(T2,S2)))
=> ( aa_real_rat(G,X3) = zero_zero_rat ) )
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),S2))
=> ( aa_real_rat(G,X3) = aa_real_rat(H,X3) ) )
=> ( groups1300246762558778688al_rat(G,T2) = groups1300246762558778688al_rat(H,S2) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
tff(fact_6489_sum_Omono__neutral__cong__right,axiom,
! [T2: set_int,S2: set_int,G: fun_int_rat,H: fun_int_rat] :
( pp(aa_set_int_bool(finite_finite_int,T2))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(T2,S2)))
=> ( aa_int_rat(G,X3) = zero_zero_rat ) )
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),S2))
=> ( aa_int_rat(G,X3) = aa_int_rat(H,X3) ) )
=> ( groups3906332499630173760nt_rat(G,T2) = groups3906332499630173760nt_rat(H,S2) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
tff(fact_6490_sum_Omono__neutral__cong__right,axiom,
! [T2: set_complex,S2: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,T2))
=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(S2),T2))
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(T2,S2)))
=> ( aa_complex_rat(G,X3) = zero_zero_rat ) )
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),S2))
=> ( aa_complex_rat(G,X3) = aa_complex_rat(H,X3) ) )
=> ( groups5058264527183730370ex_rat(G,T2) = groups5058264527183730370ex_rat(H,S2) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
tff(fact_6491_sum_Omono__neutral__cong__right,axiom,
! [T2: set_real,S2: set_real,G: fun_real_nat,H: fun_real_nat] :
( pp(aa_set_real_bool(finite_finite_real,T2))
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(S2),T2))
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),minus_minus_set_real(T2,S2)))
=> ( aa_real_nat(G,X3) = zero_zero_nat ) )
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),S2))
=> ( aa_real_nat(G,X3) = aa_real_nat(H,X3) ) )
=> ( groups1935376822645274424al_nat(G,T2) = groups1935376822645274424al_nat(H,S2) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
tff(fact_6492_sum_Omono__neutral__cong__right,axiom,
! [T2: set_int,S2: set_int,G: fun_int_nat,H: fun_int_nat] :
( pp(aa_set_int_bool(finite_finite_int,T2))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(S2),T2))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(T2,S2)))
=> ( aa_int_nat(G,X3) = zero_zero_nat ) )
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),S2))
=> ( aa_int_nat(G,X3) = aa_int_nat(H,X3) ) )
=> ( groups4541462559716669496nt_nat(G,T2) = groups4541462559716669496nt_nat(H,S2) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
tff(fact_6493_sum_Osubset__diff,axiom,
! [B2: set_int,A2: set_int,G: fun_int_real] :
( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
=> ( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( groups8778361861064173332t_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8778361861064173332t_real(G,minus_minus_set_int(A2,B2))),groups8778361861064173332t_real(G,B2)) ) ) ) ).
% sum.subset_diff
tff(fact_6494_sum_Osubset__diff,axiom,
! [B2: set_complex,A2: set_complex,G: fun_complex_real] :
( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
=> ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( groups5808333547571424918x_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,B2))),groups5808333547571424918x_real(G,B2)) ) ) ) ).
% sum.subset_diff
tff(fact_6495_sum_Osubset__diff,axiom,
! [B2: set_int,A2: set_int,G: fun_int_rat] :
( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
=> ( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( groups3906332499630173760nt_rat(G,A2) = plus_plus_rat(groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,B2)),groups3906332499630173760nt_rat(G,B2)) ) ) ) ).
% sum.subset_diff
tff(fact_6496_sum_Osubset__diff,axiom,
! [B2: set_complex,A2: set_complex,G: fun_complex_rat] :
( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
=> ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( groups5058264527183730370ex_rat(G,A2) = plus_plus_rat(groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,B2)),groups5058264527183730370ex_rat(G,B2)) ) ) ) ).
% sum.subset_diff
tff(fact_6497_sum_Osubset__diff,axiom,
! [B2: set_int,A2: set_int,G: fun_int_nat] :
( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
=> ( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( groups4541462559716669496nt_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G,minus_minus_set_int(A2,B2))),groups4541462559716669496nt_nat(G,B2)) ) ) ) ).
% sum.subset_diff
tff(fact_6498_sum_Osubset__diff,axiom,
! [B2: set_complex,A2: set_complex,G: fun_complex_nat] :
( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
=> ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( groups5693394587270226106ex_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,B2))),groups5693394587270226106ex_nat(G,B2)) ) ) ) ).
% sum.subset_diff
tff(fact_6499_sum_Osubset__diff,axiom,
! [B2: set_complex,A2: set_complex,G: fun_complex_int] :
( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
=> ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( groups5690904116761175830ex_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,B2))),groups5690904116761175830ex_int(G,B2)) ) ) ) ).
% sum.subset_diff
tff(fact_6500_sum_Osubset__diff,axiom,
! [B2: set_nat,A2: set_nat,G: fun_nat_rat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(B2),A2))
=> ( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ( groups2906978787729119204at_rat(G,A2) = plus_plus_rat(groups2906978787729119204at_rat(G,minus_minus_set_nat(A2,B2)),groups2906978787729119204at_rat(G,B2)) ) ) ) ).
% sum.subset_diff
tff(fact_6501_sum_Osubset__diff,axiom,
! [B2: set_nat,A2: set_nat,G: fun_nat_int] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(B2),A2))
=> ( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ( groups3539618377306564664at_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,minus_minus_set_nat(A2,B2))),groups3539618377306564664at_int(G,B2)) ) ) ) ).
% sum.subset_diff
tff(fact_6502_sum_Osubset__diff,axiom,
! [B2: set_int,A2: set_int,G: fun_int_int] :
( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
=> ( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( groups4538972089207619220nt_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,minus_minus_set_int(A2,B2))),groups4538972089207619220nt_int(G,B2)) ) ) ) ).
% sum.subset_diff
tff(fact_6503_sum__diff,axiom,
! [A2: set_int,B2: set_int,F: fun_int_rat] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
=> ( groups3906332499630173760nt_rat(F,minus_minus_set_int(A2,B2)) = minus_minus_rat(groups3906332499630173760nt_rat(F,A2),groups3906332499630173760nt_rat(F,B2)) ) ) ) ).
% sum_diff
tff(fact_6504_sum__diff,axiom,
! [A2: set_complex,B2: set_complex,F: fun_complex_rat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
=> ( groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A2,B2)) = minus_minus_rat(groups5058264527183730370ex_rat(F,A2),groups5058264527183730370ex_rat(F,B2)) ) ) ) ).
% sum_diff
tff(fact_6505_sum__diff,axiom,
! [A2: set_complex,B2: set_complex,F: fun_complex_int] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
=> ( groups5690904116761175830ex_int(F,minus_811609699411566653omplex(A2,B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups5690904116761175830ex_int(F,A2)),groups5690904116761175830ex_int(F,B2)) ) ) ) ).
% sum_diff
tff(fact_6506_sum__diff,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,F: fun_Pr8317260874932935276at_rat] :
( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A2))
=> ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(B2),A2))
=> ( groups342789780944988191at_rat(F,minus_1356011639430497352at_nat(A2,B2)) = minus_minus_rat(groups342789780944988191at_rat(F,A2),groups342789780944988191at_rat(F,B2)) ) ) ) ).
% sum_diff
tff(fact_6507_sum__diff,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,F: fun_Pr7647638413817135296at_int] :
( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A2))
=> ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(B2),A2))
=> ( groups975429370522433651at_int(F,minus_1356011639430497352at_nat(A2,B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups975429370522433651at_int(F,A2)),groups975429370522433651at_int(F,B2)) ) ) ) ).
% sum_diff
tff(fact_6508_sum__diff,axiom,
! [A2: set_nat,B2: set_nat,F: fun_nat_rat] :
( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B2),A2))
=> ( groups2906978787729119204at_rat(F,minus_minus_set_nat(A2,B2)) = minus_minus_rat(groups2906978787729119204at_rat(F,A2),groups2906978787729119204at_rat(F,B2)) ) ) ) ).
% sum_diff
tff(fact_6509_sum__diff,axiom,
! [A2: set_nat,B2: set_nat,F: fun_nat_int] :
( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B2),A2))
=> ( groups3539618377306564664at_int(F,minus_minus_set_nat(A2,B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(F,A2)),groups3539618377306564664at_int(F,B2)) ) ) ) ).
% sum_diff
tff(fact_6510_sum__diff,axiom,
! [A2: set_int,B2: set_int,F: fun_int_int] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
=> ( groups4538972089207619220nt_int(F,minus_minus_set_int(A2,B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(F,A2)),groups4538972089207619220nt_int(F,B2)) ) ) ) ).
% sum_diff
tff(fact_6511_sum__diff,axiom,
! [A2: set_complex,B2: set_complex,F: fun_complex_complex] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
=> ( groups7754918857620584856omplex(F,minus_811609699411566653omplex(A2,B2)) = aa_complex_complex(minus_minus_complex(groups7754918857620584856omplex(F,A2)),groups7754918857620584856omplex(F,B2)) ) ) ) ).
% sum_diff
tff(fact_6512_sum__diff,axiom,
! [A2: set_nat,B2: set_nat,F: fun_nat_real] :
( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B2),A2))
=> ( groups6591440286371151544t_real(F,minus_minus_set_nat(A2,B2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups6591440286371151544t_real(F,A2)),groups6591440286371151544t_real(F,B2)) ) ) ) ).
% sum_diff
tff(fact_6513_subset__decode__imp__le,axiom,
! [M: nat,N: nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(nat_set_decode,M)),aa_nat_set_nat(nat_set_decode,N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% subset_decode_imp_le
tff(fact_6514_sum__mono2,axiom,
! [B2: set_real,A2: set_real,F: fun_real_real] :
( pp(aa_set_real_bool(finite_finite_real,B2))
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
=> ( ! [B4: real] :
( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(B2,A2)))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,B4))) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8097168146408367636l_real(F,A2)),groups8097168146408367636l_real(F,B2))) ) ) ) ).
% sum_mono2
tff(fact_6515_sum__mono2,axiom,
! [B2: set_int,A2: set_int,F: fun_int_real] :
( pp(aa_set_int_bool(finite_finite_int,B2))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
=> ( ! [B4: int] :
( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(B2,A2)))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,B4))) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(F,B2))) ) ) ) ).
% sum_mono2
tff(fact_6516_sum__mono2,axiom,
! [B2: set_complex,A2: set_complex,F: fun_complex_real] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
=> ( ! [B4: complex] :
( pp(aa_set_complex_bool(member_complex(B4),minus_811609699411566653omplex(B2,A2)))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,B4))) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(F,B2))) ) ) ) ).
% sum_mono2
tff(fact_6517_sum__mono2,axiom,
! [B2: set_real,A2: set_real,F: fun_real_rat] :
( pp(aa_set_real_bool(finite_finite_real,B2))
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
=> ( ! [B4: real] :
( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(B2,A2)))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,B4))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups1300246762558778688al_rat(F,A2)),groups1300246762558778688al_rat(F,B2))) ) ) ) ).
% sum_mono2
tff(fact_6518_sum__mono2,axiom,
! [B2: set_int,A2: set_int,F: fun_int_rat] :
( pp(aa_set_int_bool(finite_finite_int,B2))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
=> ( ! [B4: int] :
( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(B2,A2)))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,B4))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(F,B2))) ) ) ) ).
% sum_mono2
tff(fact_6519_sum__mono2,axiom,
! [B2: set_complex,A2: set_complex,F: fun_complex_rat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
=> ( ! [B4: complex] :
( pp(aa_set_complex_bool(member_complex(B4),minus_811609699411566653omplex(B2,A2)))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,B4))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(F,B2))) ) ) ) ).
% sum_mono2
tff(fact_6520_sum__mono2,axiom,
! [B2: set_real,A2: set_real,F: fun_real_nat] :
( pp(aa_set_real_bool(finite_finite_real,B2))
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
=> ( ! [B4: real] :
( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(B2,A2)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,B4))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups1935376822645274424al_nat(F,A2)),groups1935376822645274424al_nat(F,B2))) ) ) ) ).
% sum_mono2
tff(fact_6521_sum__mono2,axiom,
! [B2: set_int,A2: set_int,F: fun_int_nat] :
( pp(aa_set_int_bool(finite_finite_int,B2))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
=> ( ! [B4: int] :
( pp(aa_set_int_bool(member_int(B4),minus_minus_set_int(B2,A2)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,B4))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(F,B2))) ) ) ) ).
% sum_mono2
tff(fact_6522_sum__mono2,axiom,
! [B2: set_complex,A2: set_complex,F: fun_complex_nat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
=> ( ! [B4: complex] :
( pp(aa_set_complex_bool(member_complex(B4),minus_811609699411566653omplex(B2,A2)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,B4))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(F,B2))) ) ) ) ).
% sum_mono2
tff(fact_6523_sum__mono2,axiom,
! [B2: set_real,A2: set_real,F: fun_real_int] :
( pp(aa_set_real_bool(finite_finite_real,B2))
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
=> ( ! [B4: real] :
( pp(aa_set_real_bool(member_real(B4),minus_minus_set_real(B2,A2)))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(F,B4))) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups1932886352136224148al_int(F,A2)),groups1932886352136224148al_int(F,B2))) ) ) ) ).
% sum_mono2
tff(fact_6524_sum__strict__mono2,axiom,
! [B2: set_real,A2: set_real,B: real,F: fun_real_real] :
( pp(aa_set_real_bool(finite_finite_real,B2))
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
=> ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(B2,A2)))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,B)))
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),B2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X3))) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8097168146408367636l_real(F,A2)),groups8097168146408367636l_real(F,B2))) ) ) ) ) ) ).
% sum_strict_mono2
tff(fact_6525_sum__strict__mono2,axiom,
! [B2: set_int,A2: set_int,B: int,F: fun_int_real] :
( pp(aa_set_int_bool(finite_finite_int,B2))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
=> ( pp(aa_set_int_bool(member_int(B),minus_minus_set_int(B2,A2)))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_int_real(F,B)))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),B2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X3))) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups8778361861064173332t_real(F,A2)),groups8778361861064173332t_real(F,B2))) ) ) ) ) ) ).
% sum_strict_mono2
tff(fact_6526_sum__strict__mono2,axiom,
! [B2: set_complex,A2: set_complex,B: complex,F: fun_complex_real] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
=> ( pp(aa_set_complex_bool(member_complex(B),minus_811609699411566653omplex(B2,A2)))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(F,B)))
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),B2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X3))) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups5808333547571424918x_real(F,A2)),groups5808333547571424918x_real(F,B2))) ) ) ) ) ) ).
% sum_strict_mono2
tff(fact_6527_sum__strict__mono2,axiom,
! [B2: set_real,A2: set_real,B: real,F: fun_real_rat] :
( pp(aa_set_real_bool(finite_finite_real,B2))
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
=> ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(B2,A2)))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_real_rat(F,B)))
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),B2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X3))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups1300246762558778688al_rat(F,A2)),groups1300246762558778688al_rat(F,B2))) ) ) ) ) ) ).
% sum_strict_mono2
tff(fact_6528_sum__strict__mono2,axiom,
! [B2: set_int,A2: set_int,B: int,F: fun_int_rat] :
( pp(aa_set_int_bool(finite_finite_int,B2))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
=> ( pp(aa_set_int_bool(member_int(B),minus_minus_set_int(B2,A2)))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(F,B)))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),B2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X3))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups3906332499630173760nt_rat(F,A2)),groups3906332499630173760nt_rat(F,B2))) ) ) ) ) ) ).
% sum_strict_mono2
tff(fact_6529_sum__strict__mono2,axiom,
! [B2: set_complex,A2: set_complex,B: complex,F: fun_complex_rat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
=> ( pp(aa_set_complex_bool(member_complex(B),minus_811609699411566653omplex(B2,A2)))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_complex_rat(F,B)))
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),B2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X3))) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,groups5058264527183730370ex_rat(F,A2)),groups5058264527183730370ex_rat(F,B2))) ) ) ) ) ) ).
% sum_strict_mono2
tff(fact_6530_sum__strict__mono2,axiom,
! [B2: set_real,A2: set_real,B: real,F: fun_real_nat] :
( pp(aa_set_real_bool(finite_finite_real,B2))
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
=> ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(B2,A2)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_real_nat(F,B)))
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),B2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X3))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups1935376822645274424al_nat(F,A2)),groups1935376822645274424al_nat(F,B2))) ) ) ) ) ) ).
% sum_strict_mono2
tff(fact_6531_sum__strict__mono2,axiom,
! [B2: set_int,A2: set_int,B: int,F: fun_int_nat] :
( pp(aa_set_int_bool(finite_finite_int,B2))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2))
=> ( pp(aa_set_int_bool(member_int(B),minus_minus_set_int(B2,A2)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(F,B)))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),B2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X3))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(F,B2))) ) ) ) ) ) ).
% sum_strict_mono2
tff(fact_6532_sum__strict__mono2,axiom,
! [B2: set_complex,A2: set_complex,B: complex,F: fun_complex_nat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2))
=> ( pp(aa_set_complex_bool(member_complex(B),minus_811609699411566653omplex(B2,A2)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_complex_nat(F,B)))
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),B2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X3))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(F,B2))) ) ) ) ) ) ).
% sum_strict_mono2
tff(fact_6533_sum__strict__mono2,axiom,
! [B2: set_real,A2: set_real,B: real,F: fun_real_int] :
( pp(aa_set_real_bool(finite_finite_real,B2))
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2))
=> ( pp(aa_set_real_bool(member_real(B),minus_minus_set_real(B2,A2)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_real_int(F,B)))
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),B2))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(F,X3))) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups1932886352136224148al_int(F,A2)),groups1932886352136224148al_int(F,B2))) ) ) ) ) ) ).
% sum_strict_mono2
tff(fact_6534_convex__sum__bound__le,axiom,
! [I5: set_complex,X: fun_co8153928105989009288nteger,A: fun_co8153928105989009288nteger,B: code_integer,Delta: code_integer] :
( ! [I2: complex] :
( pp(aa_set_complex_bool(member_complex(I2),I5))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_com4217951221296830281nteger(X,I2))) )
=> ( ( groups6621422865394947399nteger(X,I5) = one_one_Code_integer )
=> ( ! [I2: complex] :
( pp(aa_set_complex_bool(member_complex(I2),I5))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_com4217951221296830281nteger(A,I2),B))),Delta)) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups6621422865394947399nteger(aa_fun5267417455630836558nteger(aTP_Lamm_fu(X),A),I5),B))),Delta)) ) ) ) ).
% convex_sum_bound_le
tff(fact_6535_convex__sum__bound__le,axiom,
! [I5: set_real,X: fun_re6978948838062957830nteger,A: fun_re6978948838062957830nteger,B: code_integer,Delta: code_integer] :
( ! [I2: real] :
( pp(aa_set_real_bool(member_real(I2),I5))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_real_Code_integer(X,I2))) )
=> ( ( groups7713935264441627589nteger(X,I5) = one_one_Code_integer )
=> ( ! [I2: real] :
( pp(aa_set_real_bool(member_real(I2),I5))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_real_Code_integer(A,I2),B))),Delta)) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7713935264441627589nteger(aa_fun7923231501384339402nteger(aTP_Lamm_fv(X),A),I5),B))),Delta)) ) ) ) ).
% convex_sum_bound_le
tff(fact_6536_convex__sum__bound__le,axiom,
! [I5: set_nat,X: fun_nat_Code_integer,A: fun_nat_Code_integer,B: code_integer,Delta: code_integer] :
( ! [I2: nat] :
( pp(aa_set_nat_bool(member_nat(I2),I5))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(X,I2))) )
=> ( ( groups7501900531339628137nteger(X,I5) = one_one_Code_integer )
=> ( ! [I2: nat] :
( pp(aa_set_nat_bool(member_nat(I2),I5))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_nat_Code_integer(A,I2),B))),Delta)) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7501900531339628137nteger(aa_fun589076786913273746nteger(aTP_Lamm_fw(X),A),I5),B))),Delta)) ) ) ) ).
% convex_sum_bound_le
tff(fact_6537_convex__sum__bound__le,axiom,
! [I5: set_int,X: fun_int_Code_integer,A: fun_int_Code_integer,B: code_integer,Delta: code_integer] :
( ! [I2: int] :
( pp(aa_set_int_bool(member_int(I2),I5))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(X,I2))) )
=> ( ( groups7873554091576472773nteger(X,I5) = one_one_Code_integer )
=> ( ! [I2: int] :
( pp(aa_set_int_bool(member_int(I2),I5))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(aa_int_Code_integer(A,I2),B))),Delta)) )
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(groups7873554091576472773nteger(aa_fun5304498221708291146nteger(aTP_Lamm_fx(X),A),I5),B))),Delta)) ) ) ) ).
% convex_sum_bound_le
tff(fact_6538_convex__sum__bound__le,axiom,
! [I5: set_complex,X: fun_complex_real,A: fun_complex_real,B: real,Delta: real] :
( ! [I2: complex] :
( pp(aa_set_complex_bool(member_complex(I2),I5))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(X,I2))) )
=> ( ( groups5808333547571424918x_real(X,I5) = one_one_real )
=> ( ! [I2: complex] :
( pp(aa_set_complex_bool(member_complex(I2),I5))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(A,I2)),B))),Delta)) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_fy(X),A),I5)),B))),Delta)) ) ) ) ).
% convex_sum_bound_le
tff(fact_6539_convex__sum__bound__le,axiom,
! [I5: set_real,X: fun_real_real,A: fun_real_real,B: real,Delta: real] :
( ! [I2: real] :
( pp(aa_set_real_bool(member_real(I2),I5))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(X,I2))) )
=> ( ( groups8097168146408367636l_real(X,I5) = one_one_real )
=> ( ! [I2: real] :
( pp(aa_set_real_bool(member_real(I2),I5))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(A,I2)),B))),Delta)) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_fz(X),A),I5)),B))),Delta)) ) ) ) ).
% convex_sum_bound_le
tff(fact_6540_convex__sum__bound__le,axiom,
! [I5: set_int,X: fun_int_real,A: fun_int_real,B: real,Delta: real] :
( ! [I2: int] :
( pp(aa_set_int_bool(member_int(I2),I5))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(X,I2))) )
=> ( ( groups8778361861064173332t_real(X,I5) = one_one_real )
=> ( ! [I2: int] :
( pp(aa_set_int_bool(member_int(I2),I5))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(A,I2)),B))),Delta)) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_ga(X),A),I5)),B))),Delta)) ) ) ) ).
% convex_sum_bound_le
tff(fact_6541_convex__sum__bound__le,axiom,
! [I5: set_complex,X: fun_complex_rat,A: fun_complex_rat,B: rat,Delta: rat] :
( ! [I2: complex] :
( pp(aa_set_complex_bool(member_complex(I2),I5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(X,I2))) )
=> ( ( groups5058264527183730370ex_rat(X,I5) = one_one_rat )
=> ( ! [I2: complex] :
( pp(aa_set_complex_bool(member_complex(I2),I5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(aa_complex_rat(A,I2),B))),Delta)) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_gb(X),A),I5),B))),Delta)) ) ) ) ).
% convex_sum_bound_le
tff(fact_6542_convex__sum__bound__le,axiom,
! [I5: set_real,X: fun_real_rat,A: fun_real_rat,B: rat,Delta: rat] :
( ! [I2: real] :
( pp(aa_set_real_bool(member_real(I2),I5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(X,I2))) )
=> ( ( groups1300246762558778688al_rat(X,I5) = one_one_rat )
=> ( ! [I2: real] :
( pp(aa_set_real_bool(member_real(I2),I5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(aa_real_rat(A,I2),B))),Delta)) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_gc(X),A),I5),B))),Delta)) ) ) ) ).
% convex_sum_bound_le
tff(fact_6543_convex__sum__bound__le,axiom,
! [I5: set_nat,X: fun_nat_rat,A: fun_nat_rat,B: rat,Delta: rat] :
( ! [I2: nat] :
( pp(aa_set_nat_bool(member_nat(I2),I5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(X,I2))) )
=> ( ( groups2906978787729119204at_rat(X,I5) = one_one_rat )
=> ( ! [I2: nat] :
( pp(aa_set_nat_bool(member_nat(I2),I5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(aa_nat_rat(A,I2),B))),Delta)) )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_gd(X),A),I5),B))),Delta)) ) ) ) ).
% convex_sum_bound_le
tff(fact_6544_add__0__iff,axiom,
! [B: complex,A: complex] :
( ( B = aa_complex_complex(plus_plus_complex(B),A) )
<=> ( A = zero_zero_complex ) ) ).
% add_0_iff
tff(fact_6545_add__0__iff,axiom,
! [B: real,A: real] :
( ( B = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),A) )
<=> ( A = zero_zero_real ) ) ).
% add_0_iff
tff(fact_6546_add__0__iff,axiom,
! [B: rat,A: rat] :
( ( B = plus_plus_rat(B,A) )
<=> ( A = zero_zero_rat ) ) ).
% add_0_iff
tff(fact_6547_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) )
<=> ( A = zero_zero_nat ) ) ).
% add_0_iff
tff(fact_6548_add__0__iff,axiom,
! [B: int,A: int] :
( ( B = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) )
<=> ( A = zero_zero_int ) ) ).
% add_0_iff
tff(fact_6549_crossproduct__noteq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( A != B )
& ( C != D ) )
<=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),D)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C)) ) ) ).
% crossproduct_noteq
tff(fact_6550_crossproduct__noteq,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ( A != B )
& ( C != D ) )
<=> ( plus_plus_rat(aa_rat_rat(times_times_rat(A),C),aa_rat_rat(times_times_rat(B),D)) != plus_plus_rat(aa_rat_rat(times_times_rat(A),D),aa_rat_rat(times_times_rat(B),C)) ) ) ).
% crossproduct_noteq
tff(fact_6551_crossproduct__noteq,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( A != B )
& ( C != D ) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).
% crossproduct_noteq
tff(fact_6552_crossproduct__noteq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( A != B )
& ( C != D ) )
<=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).
% crossproduct_noteq
tff(fact_6553_crossproduct__eq,axiom,
! [W: real,Y: real,X: real,Z: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) )
<=> ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
tff(fact_6554_crossproduct__eq,axiom,
! [W: rat,Y: rat,X: rat,Z: rat] :
( ( plus_plus_rat(aa_rat_rat(times_times_rat(W),Y),aa_rat_rat(times_times_rat(X),Z)) = plus_plus_rat(aa_rat_rat(times_times_rat(W),Z),aa_rat_rat(times_times_rat(X),Y)) )
<=> ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
tff(fact_6555_crossproduct__eq,axiom,
! [W: nat,Y: nat,X: nat,Z: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Y)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y)) )
<=> ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
tff(fact_6556_crossproduct__eq,axiom,
! [W: int,Y: int,X: int,Z: int] :
( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,X),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)) )
<=> ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
tff(fact_6557_infinite__int__iff__unbounded,axiom,
! [S2: set_int] :
( ~ pp(aa_set_int_bool(finite_finite_int,S2))
<=> ! [M7: int] :
? [N5: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M7),aa_int_int(abs_abs_int,N5)))
& pp(aa_set_int_bool(member_int(N5),S2)) ) ) ).
% infinite_int_iff_unbounded
tff(fact_6558_mask__numeral,axiom,
! [N: num] : aa_nat_nat(bit_se2002935070580805687sk_nat,aa_num_nat(numeral_numeral_nat,N)) = 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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(bit_se2002935070580805687sk_nat,aa_num_nat(pred_numeral,N)))) ).
% mask_numeral
tff(fact_6559_mask__numeral,axiom,
! [N: num] : aa_nat_int(bit_se2000444600071755411sk_int,aa_num_nat(numeral_numeral_nat,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(bit_se2000444600071755411sk_int,aa_num_nat(pred_numeral,N)))) ).
% mask_numeral
tff(fact_6560_num_Osize__gen_I3_J,axiom,
! [X33: num] : aa_num_nat(size_num,aa_num_num(bit1,X33)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X33)),aa_nat_nat(suc,zero_zero_nat)) ).
% num.size_gen(3)
tff(fact_6561_take__bit__rec,axiom,
! [N: nat,A: code_integer] :
( ( ( N = zero_zero_nat )
=> ( bit_se1745604003318907178nteger(N,A) = zero_z3403309356797280102nteger ) )
& ( ( N != zero_zero_nat )
=> ( bit_se1745604003318907178nteger(N,A) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,bit_se1745604003318907178nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))),numera6620942414471956472nteger(aa_num_num(bit0,one))),modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) ) ) ).
% take_bit_rec
tff(fact_6562_take__bit__rec,axiom,
! [N: nat,A: int] :
( ( ( N = zero_zero_nat )
=> ( aa_int_int(bit_se2923211474154528505it_int(N),A) = zero_zero_int ) )
& ( ( N != zero_zero_nat )
=> ( aa_int_int(bit_se2923211474154528505it_int(N),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ) ) ) ).
% take_bit_rec
tff(fact_6563_take__bit__rec,axiom,
! [N: nat,A: nat] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),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(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).
% take_bit_rec
tff(fact_6564_num_Osize__gen_I2_J,axiom,
! [X23: num] : aa_num_nat(size_num,aa_num_num(bit0,X23)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X23)),aa_nat_nat(suc,zero_zero_nat)) ).
% num.size_gen(2)
tff(fact_6565_tanh__real__altdef,axiom,
! [X: real] : aa_real_real(tanh_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit0,one)))),X)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit0,one)))),X)))) ).
% tanh_real_altdef
tff(fact_6566_mask__nat__positive__iff,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).
% mask_nat_positive_iff
tff(fact_6567_take__bit__of__0,axiom,
! [N: nat] : aa_int_int(bit_se2923211474154528505it_int(N),zero_zero_int) = zero_zero_int ).
% take_bit_of_0
tff(fact_6568_take__bit__of__0,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),zero_zero_nat) = zero_zero_nat ).
% take_bit_of_0
tff(fact_6569_exp__less__mono,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y))) ) ).
% exp_less_mono
tff(fact_6570_exp__less__cancel__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).
% exp_less_cancel_iff
tff(fact_6571_exp__le__cancel__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).
% exp_le_cancel_iff
tff(fact_6572_concat__bit__of__zero__2,axiom,
! [N: nat,K: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),zero_zero_int) = aa_int_int(bit_se2923211474154528505it_int(N),K) ).
% concat_bit_of_zero_2
tff(fact_6573_exp__zero,axiom,
aa_complex_complex(exp_complex,zero_zero_complex) = one_one_complex ).
% exp_zero
tff(fact_6574_exp__zero,axiom,
aa_real_real(exp_real,zero_zero_real) = one_one_real ).
% exp_zero
tff(fact_6575_take__bit__0,axiom,
! [A: int] : aa_int_int(bit_se2923211474154528505it_int(zero_zero_nat),A) = zero_zero_int ).
% take_bit_0
tff(fact_6576_take__bit__0,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,zero_zero_nat),A) = zero_zero_nat ).
% take_bit_0
tff(fact_6577_take__bit__Suc__1,axiom,
! [N: nat] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),one_one_int) = one_one_int ).
% take_bit_Suc_1
tff(fact_6578_take__bit__Suc__1,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N)),one_one_nat) = one_one_nat ).
% take_bit_Suc_1
tff(fact_6579_take__bit__numeral__1,axiom,
! [L: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),one_one_int) = one_one_int ).
% take_bit_numeral_1
tff(fact_6580_take__bit__numeral__1,axiom,
! [L: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,L)),one_one_nat) = one_one_nat ).
% take_bit_numeral_1
tff(fact_6581_mask__0,axiom,
aa_nat_nat(bit_se2002935070580805687sk_nat,zero_zero_nat) = zero_zero_nat ).
% mask_0
tff(fact_6582_mask__0,axiom,
aa_nat_int(bit_se2000444600071755411sk_int,zero_zero_nat) = zero_zero_int ).
% mask_0
tff(fact_6583_mask__eq__0__iff,axiom,
! [N: nat] :
( ( aa_nat_nat(bit_se2002935070580805687sk_nat,N) = zero_zero_nat )
<=> ( N = zero_zero_nat ) ) ).
% mask_eq_0_iff
tff(fact_6584_mask__eq__0__iff,axiom,
! [N: nat] :
( ( aa_nat_int(bit_se2000444600071755411sk_int,N) = zero_zero_int )
<=> ( N = zero_zero_nat ) ) ).
% mask_eq_0_iff
tff(fact_6585_exp__eq__one__iff,axiom,
! [X: real] :
( ( aa_real_real(exp_real,X) = one_one_real )
<=> ( X = zero_zero_real ) ) ).
% exp_eq_one_iff
tff(fact_6586_take__bit__of__1__eq__0__iff,axiom,
! [N: nat] :
( ( aa_int_int(bit_se2923211474154528505it_int(N),one_one_int) = zero_zero_int )
<=> ( N = zero_zero_nat ) ) ).
% take_bit_of_1_eq_0_iff
tff(fact_6587_take__bit__of__1__eq__0__iff,axiom,
! [N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),one_one_nat) = zero_zero_nat )
<=> ( N = zero_zero_nat ) ) ).
% take_bit_of_1_eq_0_iff
tff(fact_6588_mask__Suc__0,axiom,
aa_nat_nat(bit_se2002935070580805687sk_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).
% mask_Suc_0
tff(fact_6589_mask__Suc__0,axiom,
aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).
% mask_Suc_0
tff(fact_6590_one__less__exp__iff,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(exp_real,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).
% one_less_exp_iff
tff(fact_6591_exp__less__one__iff,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X)),one_one_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).
% exp_less_one_iff
tff(fact_6592_one__le__exp__iff,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(exp_real,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).
% one_le_exp_iff
tff(fact_6593_exp__le__one__iff,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X)),one_one_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).
% exp_le_one_iff
tff(fact_6594_take__bit__minus__one__eq__mask,axiom,
! [N: nat] : bit_se1745604003318907178nteger(N,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_nat_Code_integer(bit_se2119862282449309892nteger,N) ).
% take_bit_minus_one_eq_mask
tff(fact_6595_take__bit__minus__one__eq__mask,axiom,
! [N: nat] : aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(uminus_uminus_int,one_one_int)) = aa_nat_int(bit_se2000444600071755411sk_int,N) ).
% take_bit_minus_one_eq_mask
tff(fact_6596_exp__ln__iff,axiom,
! [X: real] :
( ( aa_real_real(exp_real,aa_real_real(ln_ln_real,X)) = X )
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).
% exp_ln_iff
tff(fact_6597_exp__ln,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( aa_real_real(exp_real,aa_real_real(ln_ln_real,X)) = X ) ) ).
% exp_ln
tff(fact_6598_take__bit__of__Suc__0,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).
% take_bit_of_Suc_0
tff(fact_6599_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: fun_nat_complex] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
=> ( groups2073611262835488442omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = zero_zero_complex ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
=> ( groups2073611262835488442omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_complex_complex(plus_plus_complex(groups2073611262835488442omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_complex(G,aa_nat_nat(suc,N))) ) ) ) ).
% sum.cl_ivl_Suc
tff(fact_6600_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: fun_nat_rat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
=> ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = zero_zero_rat ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
=> ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)),aa_nat_rat(G,aa_nat_nat(suc,N))) ) ) ) ).
% sum.cl_ivl_Suc
tff(fact_6601_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: fun_nat_int] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
=> ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = zero_zero_int ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
=> ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_int(G,aa_nat_nat(suc,N))) ) ) ) ).
% sum.cl_ivl_Suc
tff(fact_6602_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: fun_nat_nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
=> ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = zero_zero_nat ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
=> ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ) ) ) ).
% sum.cl_ivl_Suc
tff(fact_6603_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: fun_nat_real] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
=> ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = zero_zero_real ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
=> ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_real(G,aa_nat_nat(suc,N))) ) ) ) ).
% sum.cl_ivl_Suc
tff(fact_6604_take__bit__of__1,axiom,
! [N: nat] : bit_se1745604003318907178nteger(N,one_one_Code_integer) = zero_n356916108424825756nteger(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).
% take_bit_of_1
tff(fact_6605_take__bit__of__1,axiom,
! [N: nat] : aa_int_int(bit_se2923211474154528505it_int(N),one_one_int) = aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).
% take_bit_of_1
tff(fact_6606_take__bit__of__1,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),one_one_nat) = zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).
% take_bit_of_1
tff(fact_6607_sum__zero__power,axiom,
! [A2: set_nat,C: fun_nat_complex] :
( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
& pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
=> ( groups2073611262835488442omplex(aTP_Lamm_ge(C),A2) = aa_nat_complex(C,zero_zero_nat) ) )
& ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
& pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
=> ( groups2073611262835488442omplex(aTP_Lamm_ge(C),A2) = zero_zero_complex ) ) ) ).
% sum_zero_power
tff(fact_6608_sum__zero__power,axiom,
! [A2: set_nat,C: fun_nat_rat] :
( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
& pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
=> ( groups2906978787729119204at_rat(aTP_Lamm_gf(C),A2) = aa_nat_rat(C,zero_zero_nat) ) )
& ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
& pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
=> ( groups2906978787729119204at_rat(aTP_Lamm_gf(C),A2) = zero_zero_rat ) ) ) ).
% sum_zero_power
tff(fact_6609_sum__zero__power,axiom,
! [A2: set_nat,C: fun_nat_real] :
( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
& pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
=> ( groups6591440286371151544t_real(aTP_Lamm_gg(C),A2) = aa_nat_real(C,zero_zero_nat) ) )
& ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
& pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
=> ( groups6591440286371151544t_real(aTP_Lamm_gg(C),A2) = zero_zero_real ) ) ) ).
% sum_zero_power
tff(fact_6610_even__take__bit__eq,axiom,
! [N: nat,A: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1745604003318907178nteger(N,A)))
<=> ( ( N = zero_zero_nat )
| pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)) ) ) ).
% even_take_bit_eq
tff(fact_6611_even__take__bit__eq,axiom,
! [N: nat,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se2923211474154528505it_int(N),A)))
<=> ( ( N = zero_zero_nat )
| pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) ) ) ).
% even_take_bit_eq
tff(fact_6612_even__take__bit__eq,axiom,
! [N: nat,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)))
<=> ( ( N = zero_zero_nat )
| pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)) ) ) ).
% even_take_bit_eq
tff(fact_6613_sum__zero__power_H,axiom,
! [A2: set_nat,C: fun_nat_complex,D: fun_nat_complex] :
( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
& pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
=> ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_gh(C),D),A2) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(C,zero_zero_nat)),aa_nat_complex(D,zero_zero_nat)) ) )
& ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
& pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
=> ( groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_gh(C),D),A2) = zero_zero_complex ) ) ) ).
% sum_zero_power'
tff(fact_6614_sum__zero__power_H,axiom,
! [A2: set_nat,C: fun_nat_rat,D: fun_nat_rat] :
( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
& pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
=> ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_gi(C),D),A2) = divide_divide_rat(aa_nat_rat(C,zero_zero_nat),aa_nat_rat(D,zero_zero_nat)) ) )
& ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
& pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
=> ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_gi(C),D),A2) = zero_zero_rat ) ) ) ).
% sum_zero_power'
tff(fact_6615_sum__zero__power_H,axiom,
! [A2: set_nat,C: fun_nat_real,D: fun_nat_real] :
( ( ( pp(aa_set_nat_bool(finite_finite_nat,A2))
& pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
=> ( groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_gj(C),D),A2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(C,zero_zero_nat)),aa_nat_real(D,zero_zero_nat)) ) )
& ( ~ ( pp(aa_set_nat_bool(finite_finite_nat,A2))
& pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2)) )
=> ( groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_gj(C),D),A2) = zero_zero_real ) ) ) ).
% sum_zero_power'
tff(fact_6616_take__bit__Suc__0,axiom,
! [A: code_integer] : bit_se1745604003318907178nteger(aa_nat_nat(suc,zero_zero_nat),A) = modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).
% take_bit_Suc_0
tff(fact_6617_take__bit__Suc__0,axiom,
! [A: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,zero_zero_nat)),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).
% take_bit_Suc_0
tff(fact_6618_take__bit__Suc__0,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,zero_zero_nat)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% take_bit_Suc_0
tff(fact_6619_take__bit__of__exp,axiom,
! [M: nat,N: nat] : bit_se1745604003318907178nteger(M,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) ).
% take_bit_of_exp
tff(fact_6620_take__bit__of__exp,axiom,
! [M: nat,N: nat] : aa_int_int(bit_se2923211474154528505it_int(M),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).
% take_bit_of_exp
tff(fact_6621_take__bit__of__exp,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).
% take_bit_of_exp
tff(fact_6622_take__bit__of__2,axiom,
! [N: nat] : bit_se1745604003318907178nteger(N,numera6620942414471956472nteger(aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).
% take_bit_of_2
tff(fact_6623_take__bit__of__2,axiom,
! [N: nat] : aa_int_int(bit_se2923211474154528505it_int(N),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).
% take_bit_of_2
tff(fact_6624_take__bit__of__2,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% take_bit_of_2
tff(fact_6625_of__int__mask__eq,axiom,
! [N: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(bit_se2000444600071755411sk_int,N)) = aa_nat_int(bit_se2000444600071755411sk_int,N) ).
% of_int_mask_eq
tff(fact_6626_take__bit__of__int,axiom,
! [N: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(ring_1_of_int_int,K)) = aa_int_int(ring_1_of_int_int,aa_int_int(bit_se2923211474154528505it_int(N),K)) ).
% take_bit_of_int
tff(fact_6627_take__bit__add,axiom,
! [N: nat,A: int,B: int] : aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_se2923211474154528505it_int(N),A)),aa_int_int(bit_se2923211474154528505it_int(N),B))) = aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ).
% take_bit_add
tff(fact_6628_take__bit__add,axiom,
! [N: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),B))) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).
% take_bit_add
tff(fact_6629_take__bit__tightened,axiom,
! [N: nat,A: int,B: int,M: nat] :
( ( aa_int_int(bit_se2923211474154528505it_int(N),A) = aa_int_int(bit_se2923211474154528505it_int(N),B) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_int_int(bit_se2923211474154528505it_int(M),A) = aa_int_int(bit_se2923211474154528505it_int(M),B) ) ) ) ).
% take_bit_tightened
tff(fact_6630_take__bit__tightened,axiom,
! [N: nat,A: nat,B: nat,M: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),B) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),B) ) ) ) ).
% take_bit_tightened
tff(fact_6631_take__bit__nat__less__eq__self,axiom,
! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M)),M)) ).
% take_bit_nat_less_eq_self
tff(fact_6632_take__bit__tightened__less__eq__nat,axiom,
! [M: nat,N: nat,Q2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),Q2))) ) ).
% take_bit_tightened_less_eq_nat
tff(fact_6633_exp__less__cancel,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).
% exp_less_cancel
tff(fact_6634_take__bit__minus,axiom,
! [N: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(uminus_uminus_int,aa_int_int(bit_se2923211474154528505it_int(N),K))) = aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(uminus_uminus_int,K)) ).
% take_bit_minus
tff(fact_6635_exp__times__arg__commute,axiom,
! [A2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,A2)),A2) = aa_complex_complex(times_times_complex(A2),aa_complex_complex(exp_complex,A2)) ).
% exp_times_arg_commute
tff(fact_6636_exp__times__arg__commute,axiom,
! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,A2)),A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(exp_real,A2)) ).
% exp_times_arg_commute
tff(fact_6637_take__bit__mult,axiom,
! [N: nat,K: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),aa_int_int(bit_se2923211474154528505it_int(N),L))) = aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(times_times_int,K),L)) ).
% take_bit_mult
tff(fact_6638_take__bit__diff,axiom,
! [N: nat,K: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),aa_int_int(bit_se2923211474154528505it_int(N),L))) = aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)) ).
% take_bit_diff
tff(fact_6639_concat__bit__eq__iff,axiom,
! [N: nat,K: int,L: int,R: int,S: int] :
( ( aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),L) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),R),S) )
<=> ( ( aa_int_int(bit_se2923211474154528505it_int(N),K) = aa_int_int(bit_se2923211474154528505it_int(N),R) )
& ( L = S ) ) ) ).
% concat_bit_eq_iff
tff(fact_6640_concat__bit__take__bit__eq,axiom,
! [N: nat,B: int] : aa_int_fun_int_int(bit_concat_bit(N),aa_int_int(bit_se2923211474154528505it_int(N),B)) = aa_int_fun_int_int(bit_concat_bit(N),B) ).
% concat_bit_take_bit_eq
tff(fact_6641_less__eq__mask,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(bit_se2002935070580805687sk_nat,N))) ).
% less_eq_mask
tff(fact_6642_take__bit__eq__mask__iff,axiom,
! [N: nat,K: int] :
( ( aa_int_int(bit_se2923211474154528505it_int(N),K) = aa_nat_int(bit_se2000444600071755411sk_int,N) )
<=> ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)) = zero_zero_int ) ) ).
% take_bit_eq_mask_iff
tff(fact_6643_exp__total,axiom,
! [Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
=> ? [X3: real] : aa_real_real(exp_real,X3) = Y ) ).
% exp_total
tff(fact_6644_exp__gt__zero,axiom,
! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(exp_real,X))) ).
% exp_gt_zero
tff(fact_6645_not__exp__less__zero,axiom,
! [X: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(exp_real,X)),zero_zero_real)) ).
% not_exp_less_zero
tff(fact_6646_not__exp__le__zero,axiom,
! [X: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,X)),zero_zero_real)) ).
% not_exp_le_zero
tff(fact_6647_exp__ge__zero,axiom,
! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(exp_real,X))) ).
% exp_ge_zero
tff(fact_6648_sum__cong__Suc,axiom,
! [A2: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2))
=> ( ! [X3: nat] :
( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,X3)),A2))
=> ( aa_nat_nat(F,aa_nat_nat(suc,X3)) = aa_nat_nat(G,aa_nat_nat(suc,X3)) ) )
=> ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_set_nat_nat(groups3542108847815614940at_nat(G),A2) ) ) ) ).
% sum_cong_Suc
tff(fact_6649_sum__cong__Suc,axiom,
! [A2: set_nat,F: fun_nat_real,G: fun_nat_real] :
( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),A2))
=> ( ! [X3: nat] :
( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,X3)),A2))
=> ( aa_nat_real(F,aa_nat_nat(suc,X3)) = aa_nat_real(G,aa_nat_nat(suc,X3)) ) )
=> ( groups6591440286371151544t_real(F,A2) = groups6591440286371151544t_real(G,A2) ) ) ) ).
% sum_cong_Suc
tff(fact_6650_take__bit__tightened__less__eq__int,axiom,
! [M: nat,N: nat,K: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(M),K)),aa_int_int(bit_se2923211474154528505it_int(N),K))) ) ).
% take_bit_tightened_less_eq_int
tff(fact_6651_signed__take__bit__eq__iff__take__bit__eq,axiom,
! [N: nat,A: int,B: int] :
( ( aa_int_int(bit_ri631733984087533419it_int(N),A) = aa_int_int(bit_ri631733984087533419it_int(N),B) )
<=> ( aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),A) = aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),B) ) ) ).
% signed_take_bit_eq_iff_take_bit_eq
tff(fact_6652_take__bit__nonnegative,axiom,
! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se2923211474154528505it_int(N),K))) ).
% take_bit_nonnegative
tff(fact_6653_take__bit__int__less__eq__self__iff,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),K))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).
% take_bit_int_less_eq_self_iff
tff(fact_6654_not__take__bit__negative,axiom,
! [N: nat,K: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),zero_zero_int)) ).
% not_take_bit_negative
tff(fact_6655_take__bit__int__greater__self__iff,axiom,
! [K: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_int_int(bit_se2923211474154528505it_int(N),K)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).
% take_bit_int_greater_self_iff
tff(fact_6656_signed__take__bit__take__bit,axiom,
! [M: nat,N: nat,A: int] : aa_int_int(bit_ri631733984087533419it_int(M),aa_int_int(bit_se2923211474154528505it_int(N),A)) = aa_int_int(if_fun_int_int(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M),bit_se2923211474154528505it_int(N),bit_ri631733984087533419it_int(M)),A) ).
% signed_take_bit_take_bit
tff(fact_6657_mult__exp__exp,axiom,
! [X: complex,Y: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,Y)) = aa_complex_complex(exp_complex,aa_complex_complex(plus_plus_complex(X),Y)) ).
% mult_exp_exp
tff(fact_6658_mult__exp__exp,axiom,
! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)) = aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) ).
% mult_exp_exp
tff(fact_6659_exp__add__commuting,axiom,
! [X: complex,Y: complex] :
( ( aa_complex_complex(times_times_complex(X),Y) = aa_complex_complex(times_times_complex(Y),X) )
=> ( aa_complex_complex(exp_complex,aa_complex_complex(plus_plus_complex(X),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,Y)) ) ) ).
% exp_add_commuting
tff(fact_6660_exp__add__commuting,axiom,
! [X: real,Y: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),X) )
=> ( aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)) ) ) ).
% exp_add_commuting
tff(fact_6661_exp__diff,axiom,
! [X: complex,Y: complex] : aa_complex_complex(exp_complex,aa_complex_complex(minus_minus_complex(X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,Y)) ).
% exp_diff
tff(fact_6662_exp__diff,axiom,
! [X: real,Y: real] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y)) ).
% exp_diff
tff(fact_6663_take__bit__unset__bit__eq,axiom,
! [N: nat,M: nat,A: int] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(bit_se4203085406695923979it_int(M),A)) = aa_int_int(bit_se2923211474154528505it_int(N),A) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(bit_se4203085406695923979it_int(M),A)) = aa_int_int(bit_se4203085406695923979it_int(M),aa_int_int(bit_se2923211474154528505it_int(N),A)) ) ) ) ).
% take_bit_unset_bit_eq
tff(fact_6664_take__bit__unset__bit__eq,axiom,
! [N: nat,M: nat,A: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)) ) ) ) ).
% take_bit_unset_bit_eq
tff(fact_6665_take__bit__set__bit__eq,axiom,
! [N: nat,M: nat,A: int] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(bit_se7879613467334960850it_int(M),A)) = aa_int_int(bit_se2923211474154528505it_int(N),A) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(bit_se7879613467334960850it_int(M),A)) = aa_int_int(bit_se7879613467334960850it_int(M),aa_int_int(bit_se2923211474154528505it_int(N),A)) ) ) ) ).
% take_bit_set_bit_eq
tff(fact_6666_take__bit__set__bit__eq,axiom,
! [N: nat,M: nat,A: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)) ) ) ) ).
% take_bit_set_bit_eq
tff(fact_6667_take__bit__flip__bit__eq,axiom,
! [N: nat,M: nat,A: int] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(bit_se2159334234014336723it_int(M),A)) = aa_int_int(bit_se2923211474154528505it_int(N),A) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(bit_se2159334234014336723it_int(M),A)) = aa_int_int(bit_se2159334234014336723it_int(M),aa_int_int(bit_se2923211474154528505it_int(N),A)) ) ) ) ).
% take_bit_flip_bit_eq
tff(fact_6668_take__bit__flip__bit__eq,axiom,
! [N: nat,M: nat,A: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)) ) ) ) ).
% take_bit_flip_bit_eq
tff(fact_6669_sum__subtractf__nat,axiom,
! [A2: set_Pr1261947904930325089at_nat,G: fun_Pr2602117396471556196at_nat,F: fun_Pr2602117396471556196at_nat] :
( ! [X3: product_prod_nat_nat] :
( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),A2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_Pro4243173770044397973at_nat(G,X3)),aa_Pro4243173770044397973at_nat(F,X3))) )
=> ( groups977919841031483927at_nat(aa_fun2351638177499875132at_nat(aTP_Lamm_gk(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups977919841031483927at_nat(F,A2)),groups977919841031483927at_nat(G,A2)) ) ) ).
% sum_subtractf_nat
tff(fact_6670_sum__subtractf__nat,axiom,
! [A2: set_complex,G: fun_complex_nat,F: fun_complex_nat] :
( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),A2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_complex_nat(G,X3)),aa_complex_nat(F,X3))) )
=> ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_gl(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(G,A2)) ) ) ).
% sum_subtractf_nat
tff(fact_6671_sum__subtractf__nat,axiom,
! [A2: set_real,G: fun_real_nat,F: fun_real_nat] :
( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),A2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(G,X3)),aa_real_nat(F,X3))) )
=> ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_gm(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(F,A2)),groups1935376822645274424al_nat(G,A2)) ) ) ).
% sum_subtractf_nat
tff(fact_6672_sum__subtractf__nat,axiom,
! [A2: set_int,G: fun_int_nat,F: fun_int_nat] :
( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),A2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(G,X3)),aa_int_nat(F,X3))) )
=> ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_gn(G),F),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(G,A2)) ) ) ).
% sum_subtractf_nat
tff(fact_6673_sum__subtractf__nat,axiom,
! [A2: set_nat,G: fun_nat_nat,F: fun_nat_nat] :
( ! [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),A2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(G,X3)),aa_nat_nat(F,X3))) )
=> ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_go(G),F)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(G),A2)) ) ) ).
% sum_subtractf_nat
tff(fact_6674_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_gp(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).
% sum.shift_bounds_cl_Suc_ivl
tff(fact_6675_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: fun_nat_real,M: nat,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = groups6591440286371151544t_real(aTP_Lamm_gq(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).
% sum.shift_bounds_cl_Suc_ivl
tff(fact_6676_mask__nonnegative__int,axiom,
! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(bit_se2000444600071755411sk_int,N))) ).
% mask_nonnegative_int
tff(fact_6677_not__mask__negative__int,axiom,
! [N: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(bit_se2000444600071755411sk_int,N)),zero_zero_int)) ).
% not_mask_negative_int
tff(fact_6678_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [G: fun_nat_nat,M: nat,K: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_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,N),K))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_gr(G),K)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).
% sum.shift_bounds_cl_nat_ivl
tff(fact_6679_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [G: fun_nat_real,M: nat,K: nat,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_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,N),K))) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_gs(G),K),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).
% sum.shift_bounds_cl_nat_ivl
tff(fact_6680_sum__eq__Suc0__iff,axiom,
! [A2: set_int,F: fun_int_nat] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( ( groups4541462559716669496nt_nat(F,A2) = aa_nat_nat(suc,zero_zero_nat) )
<=> ? [X2: int] :
( pp(aa_set_int_bool(member_int(X2),A2))
& ( aa_int_nat(F,X2) = aa_nat_nat(suc,zero_zero_nat) )
& ! [Xa3: int] :
( pp(aa_set_int_bool(member_int(Xa3),A2))
=> ( ( X2 != Xa3 )
=> ( aa_int_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
tff(fact_6681_sum__eq__Suc0__iff,axiom,
! [A2: set_complex,F: fun_complex_nat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( ( groups5693394587270226106ex_nat(F,A2) = aa_nat_nat(suc,zero_zero_nat) )
<=> ? [X2: complex] :
( pp(aa_set_complex_bool(member_complex(X2),A2))
& ( aa_complex_nat(F,X2) = aa_nat_nat(suc,zero_zero_nat) )
& ! [Xa3: complex] :
( pp(aa_set_complex_bool(member_complex(Xa3),A2))
=> ( ( X2 != Xa3 )
=> ( aa_complex_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
tff(fact_6682_sum__eq__Suc0__iff,axiom,
! [A2: set_nat,F: fun_nat_nat] :
( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_nat_nat(suc,zero_zero_nat) )
<=> ? [X2: nat] :
( pp(aa_set_nat_bool(member_nat(X2),A2))
& ( aa_nat_nat(F,X2) = aa_nat_nat(suc,zero_zero_nat) )
& ! [Xa3: nat] :
( pp(aa_set_nat_bool(member_nat(Xa3),A2))
=> ( ( X2 != Xa3 )
=> ( aa_nat_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
tff(fact_6683_sum__SucD,axiom,
! [F: fun_nat_nat,A2: set_nat,N: nat] :
( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = aa_nat_nat(suc,N) )
=> ? [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),A2))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X3))) ) ) ).
% sum_SucD
tff(fact_6684_sum__eq__1__iff,axiom,
! [A2: set_int,F: fun_int_nat] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( ( groups4541462559716669496nt_nat(F,A2) = one_one_nat )
<=> ? [X2: int] :
( pp(aa_set_int_bool(member_int(X2),A2))
& ( aa_int_nat(F,X2) = one_one_nat )
& ! [Xa3: int] :
( pp(aa_set_int_bool(member_int(Xa3),A2))
=> ( ( X2 != Xa3 )
=> ( aa_int_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).
% sum_eq_1_iff
tff(fact_6685_sum__eq__1__iff,axiom,
! [A2: set_complex,F: fun_complex_nat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( ( groups5693394587270226106ex_nat(F,A2) = one_one_nat )
<=> ? [X2: complex] :
( pp(aa_set_complex_bool(member_complex(X2),A2))
& ( aa_complex_nat(F,X2) = one_one_nat )
& ! [Xa3: complex] :
( pp(aa_set_complex_bool(member_complex(Xa3),A2))
=> ( ( X2 != Xa3 )
=> ( aa_complex_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).
% sum_eq_1_iff
tff(fact_6686_sum__eq__1__iff,axiom,
! [A2: set_nat,F: fun_nat_nat] :
( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) = one_one_nat )
<=> ? [X2: nat] :
( pp(aa_set_nat_bool(member_nat(X2),A2))
& ( aa_nat_nat(F,X2) = one_one_nat )
& ! [Xa3: nat] :
( pp(aa_set_nat_bool(member_nat(Xa3),A2))
=> ( ( X2 != Xa3 )
=> ( aa_nat_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).
% sum_eq_1_iff
tff(fact_6687_exp__gt__one,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(exp_real,X))) ) ).
% exp_gt_one
tff(fact_6688_take__bit__signed__take__bit,axiom,
! [M: nat,N: nat,A: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
=> ( aa_int_int(bit_se2923211474154528505it_int(M),aa_int_int(bit_ri631733984087533419it_int(N),A)) = aa_int_int(bit_se2923211474154528505it_int(M),A) ) ) ).
% take_bit_signed_take_bit
tff(fact_6689_exp__ge__add__one__self,axiom,
! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_6690_exp__minus__inverse,axiom,
! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X))) = one_one_real ).
% exp_minus_inverse
tff(fact_6691_exp__minus__inverse,axiom,
! [X: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,X))) = one_one_complex ).
% exp_minus_inverse
tff(fact_6692_take__bit__decr__eq,axiom,
! [N: nat,K: int] :
( ( aa_int_int(bit_se2923211474154528505it_int(N),K) != zero_zero_int )
=> ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),one_one_int) ) ) ).
% take_bit_decr_eq
tff(fact_6693_sum__power__add,axiom,
! [X: complex,M: nat,I5: set_nat] : groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_gt(X),M),I5) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),M)),groups2073611262835488442omplex(power_power_complex(X),I5)) ).
% sum_power_add
tff(fact_6694_sum__power__add,axiom,
! [X: rat,M: nat,I5: set_nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_gu(X),M),I5) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),M)),groups2906978787729119204at_rat(power_power_rat(X),I5)) ).
% sum_power_add
tff(fact_6695_sum__power__add,axiom,
! [X: int,M: nat,I5: set_nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_gv(X),M),I5) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X),M)),groups3539618377306564664at_int(power_power_int(X),I5)) ).
% sum_power_add
tff(fact_6696_sum__power__add,axiom,
! [X: real,M: nat,I5: set_nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_gw(X),M),I5) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),M)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),I5)) ).
% sum_power_add
tff(fact_6697_sum_OatLeastAtMost__rev,axiom,
! [G: fun_nat_nat,N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_gx(G),N),M)),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) ).
% sum.atLeastAtMost_rev
tff(fact_6698_sum_OatLeastAtMost__rev,axiom,
! [G: fun_nat_real,N: nat,M: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_gy(G),N),M),aa_nat_set_nat(set_or1269000886237332187st_nat(N),M)) ).
% sum.atLeastAtMost_rev
tff(fact_6699_less__mask,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(bit_se2002935070580805687sk_nat,N))) ) ).
% less_mask
tff(fact_6700_sum__nth__roots,axiom,
! [N: nat,C: complex] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
=> ( groups7754918857620584856omplex(aTP_Lamm_bb,collect_complex(aa_com1553036987509291606x_bool(aTP_Lamm_dg(N),C))) = zero_zero_complex ) ) ).
% sum_nth_roots
tff(fact_6701_sum__roots__unity,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
=> ( groups7754918857620584856omplex(aTP_Lamm_bb,collect_complex(aTP_Lamm_bj(N))) = zero_zero_complex ) ) ).
% sum_roots_unity
tff(fact_6702_take__bit__eq__mask__iff__exp__dvd,axiom,
! [N: nat,K: int] :
( ( aa_int_int(bit_se2923211474154528505it_int(N),K) = aa_nat_int(bit_se2000444600071755411sk_int,N) )
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int))) ) ).
% take_bit_eq_mask_iff_exp_dvd
tff(fact_6703_sum__diff__nat,axiom,
! [B2: set_int,A2: set_int,F: fun_int_nat] :
( pp(aa_set_int_bool(finite_finite_int,B2))
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(B2),A2))
=> ( groups4541462559716669496nt_nat(F,minus_minus_set_int(A2,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A2)),groups4541462559716669496nt_nat(F,B2)) ) ) ) ).
% sum_diff_nat
tff(fact_6704_sum__diff__nat,axiom,
! [B2: set_complex,A2: set_complex,F: fun_complex_nat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(B2),A2))
=> ( groups5693394587270226106ex_nat(F,minus_811609699411566653omplex(A2,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5693394587270226106ex_nat(F,A2)),groups5693394587270226106ex_nat(F,B2)) ) ) ) ).
% sum_diff_nat
tff(fact_6705_sum__diff__nat,axiom,
! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B2))
=> ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(B2),A2))
=> ( groups977919841031483927at_nat(F,minus_1356011639430497352at_nat(A2,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups977919841031483927at_nat(F,A2)),groups977919841031483927at_nat(F,B2)) ) ) ) ).
% sum_diff_nat
tff(fact_6706_sum__diff__nat,axiom,
! [B2: set_nat,A2: set_nat,F: fun_nat_nat] :
( pp(aa_set_nat_bool(finite_finite_nat,B2))
=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(B2),A2))
=> ( aa_set_nat_nat(groups3542108847815614940at_nat(F),minus_minus_set_nat(A2,B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_set_nat_nat(groups3542108847815614940at_nat(F),B2)) ) ) ) ).
% sum_diff_nat
tff(fact_6707_exp__ge__add__one__self__aux,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_6708_sum__shift__lb__Suc0__0,axiom,
! [F: fun_nat_complex,K: nat] :
( ( aa_nat_complex(F,zero_zero_nat) = zero_zero_complex )
=> ( groups2073611262835488442omplex(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = groups2073611262835488442omplex(F,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).
% sum_shift_lb_Suc0_0
tff(fact_6709_sum__shift__lb__Suc0__0,axiom,
! [F: fun_nat_rat,K: nat] :
( ( aa_nat_rat(F,zero_zero_nat) = zero_zero_rat )
=> ( groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).
% sum_shift_lb_Suc0_0
tff(fact_6710_sum__shift__lb__Suc0__0,axiom,
! [F: fun_nat_int,K: nat] :
( ( aa_nat_int(F,zero_zero_nat) = zero_zero_int )
=> ( groups3539618377306564664at_int(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = groups3539618377306564664at_int(F,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).
% sum_shift_lb_Suc0_0
tff(fact_6711_sum__shift__lb__Suc0__0,axiom,
! [F: fun_nat_nat,K: nat] :
( ( aa_nat_nat(F,zero_zero_nat) = zero_zero_nat )
=> ( aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).
% sum_shift_lb_Suc0_0
tff(fact_6712_sum__shift__lb__Suc0__0,axiom,
! [F: fun_nat_real,K: nat] :
( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
=> ( groups6591440286371151544t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),K)) = groups6591440286371151544t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K)) ) ) ).
% sum_shift_lb_Suc0_0
tff(fact_6713_sum_OatLeast0__atMost__Suc,axiom,
! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)),aa_nat_rat(G,aa_nat_nat(suc,N))) ).
% sum.atLeast0_atMost_Suc
tff(fact_6714_sum_OatLeast0__atMost__Suc,axiom,
! [G: fun_nat_int,N: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_int(G,aa_nat_nat(suc,N))) ).
% sum.atLeast0_atMost_Suc
tff(fact_6715_sum_OatLeast0__atMost__Suc,axiom,
! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ).
% sum.atLeast0_atMost_Suc
tff(fact_6716_sum_OatLeast0__atMost__Suc,axiom,
! [G: fun_nat_real,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))),aa_nat_real(G,aa_nat_nat(suc,N))) ).
% sum.atLeast0_atMost_Suc
tff(fact_6717_lemma__exp__total,axiom,
! [Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),Y))
=> ? [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),one_one_real)))
& ( aa_real_real(exp_real,X3) = Y ) ) ) ).
% lemma_exp_total
tff(fact_6718_sum_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G: fun_nat_rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = plus_plus_rat(aa_nat_rat(G,M),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).
% sum.atLeast_Suc_atMost
tff(fact_6719_sum_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G: fun_nat_int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,M)),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).
% sum.atLeast_Suc_atMost
tff(fact_6720_sum_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G: fun_nat_nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).
% sum.atLeast_Suc_atMost
tff(fact_6721_sum_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G: fun_nat_real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,M)),groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N))) ) ) ).
% sum.atLeast_Suc_atMost
tff(fact_6722_sum_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G: fun_nat_rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
=> ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = plus_plus_rat(aa_nat_rat(G,aa_nat_nat(suc,N)),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).
% sum.nat_ivl_Suc'
tff(fact_6723_sum_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G: fun_nat_int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
=> ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,aa_nat_nat(suc,N))),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).
% sum.nat_ivl_Suc'
tff(fact_6724_sum_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G: fun_nat_nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
=> ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,aa_nat_nat(suc,N))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).
% sum.nat_ivl_Suc'
tff(fact_6725_sum_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G: fun_nat_real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
=> ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,aa_nat_nat(suc,N))),groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).
% sum.nat_ivl_Suc'
tff(fact_6726_ln__ge__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(ln_ln_real,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,Y)),X)) ) ) ).
% ln_ge_iff
tff(fact_6727_ln__x__over__x__mono,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Y)),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,X)),X))) ) ) ).
% ln_x_over_x_mono
tff(fact_6728_sum_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G: fun_nat_rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)),aa_nat_rat(G,aa_nat_nat(suc,N))) = plus_plus_rat(aa_nat_rat(G,M),groups2906978787729119204at_rat(aTP_Lamm_gz(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).
% sum.Suc_reindex_ivl
tff(fact_6729_sum_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G: fun_nat_int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_int(G,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,M)),groups3539618377306564664at_int(aTP_Lamm_ha(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).
% sum.Suc_reindex_ivl
tff(fact_6730_sum_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G: fun_nat_nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_gp(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).
% sum.Suc_reindex_ivl
tff(fact_6731_sum_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G: fun_nat_real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_real(G,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,M)),groups6591440286371151544t_real(aTP_Lamm_gq(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) ) ) ).
% sum.Suc_reindex_ivl
tff(fact_6732_sum__Suc__diff,axiom,
! [M: nat,N: nat,F: fun_nat_rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
=> ( groups2906978787729119204at_rat(aTP_Lamm_hb(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = minus_minus_rat(aa_nat_rat(F,aa_nat_nat(suc,N)),aa_nat_rat(F,M)) ) ) ).
% sum_Suc_diff
tff(fact_6733_sum__Suc__diff,axiom,
! [M: nat,N: nat,F: fun_nat_int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
=> ( groups3539618377306564664at_int(aTP_Lamm_hc(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,N))),aa_nat_int(F,M)) ) ) ).
% sum_Suc_diff
tff(fact_6734_sum__Suc__diff,axiom,
! [M: nat,N: nat,F: fun_nat_real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
=> ( groups6591440286371151544t_real(aTP_Lamm_hd(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,aa_nat_nat(suc,N))),aa_nat_real(F,M)) ) ) ).
% sum_Suc_diff
tff(fact_6735_take__bit__Suc__bit0,axiom,
! [N: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(N),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).
% take_bit_Suc_bit0
tff(fact_6736_take__bit__Suc__bit0,axiom,
! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% take_bit_Suc_bit0
tff(fact_6737_take__bit__eq__mod,axiom,
! [N: nat,A: code_integer] : bit_se1745604003318907178nteger(N,A) = modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) ).
% take_bit_eq_mod
tff(fact_6738_take__bit__eq__mod,axiom,
! [N: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(N),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).
% take_bit_eq_mod
tff(fact_6739_take__bit__eq__mod,axiom,
! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).
% take_bit_eq_mod
tff(fact_6740_take__bit__nat__eq__self,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M) = M ) ) ).
% take_bit_nat_eq_self
tff(fact_6741_take__bit__nat__less__exp,axiom,
! [N: nat,M: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).
% take_bit_nat_less_exp
tff(fact_6742_take__bit__nat__eq__self__iff,axiom,
! [N: nat,M: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M) = M )
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ).
% take_bit_nat_eq_self_iff
tff(fact_6743_take__bit__nat__def,axiom,
! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).
% take_bit_nat_def
tff(fact_6744_exp__le,axiom,
pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),numeral_numeral_real(aa_num_num(bit1,one)))) ).
% exp_le
tff(fact_6745_sum_Oub__add__nat,axiom,
! [M: nat,N: nat,G: fun_nat_rat,P2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
=> ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2))) = plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2)))) ) ) ).
% sum.ub_add_nat
tff(fact_6746_sum_Oub__add__nat,axiom,
! [M: nat,N: nat,G: fun_nat_int,P2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
=> ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2)))) ) ) ).
% sum.ub_add_nat
tff(fact_6747_sum_Oub__add__nat,axiom,
! [M: nat,N: nat,G: fun_nat_nat,P2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
=> ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2)))) ) ) ).
% sum.ub_add_nat
tff(fact_6748_sum_Oub__add__nat,axiom,
! [M: nat,N: nat,G: fun_nat_real,P2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
=> ( groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2)))) ) ) ).
% sum.ub_add_nat
tff(fact_6749_take__bit__int__less__exp,axiom,
! [N: nat,K: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).
% take_bit_int_less_exp
tff(fact_6750_take__bit__int__def,axiom,
! [N: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(N),K) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).
% take_bit_int_def
tff(fact_6751_set__encode__def,axiom,
nat_set_encode = groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).
% set_encode_def
tff(fact_6752_tanh__altdef,axiom,
! [X: real] : aa_real_real(tanh_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X)))) ).
% tanh_altdef
tff(fact_6753_tanh__altdef,axiom,
! [X: complex] : aa_complex_complex(tanh_complex,X) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,X)))),aa_complex_complex(plus_plus_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,X)))) ).
% tanh_altdef
tff(fact_6754_num_Osize__gen_I1_J,axiom,
aa_num_nat(size_num,one) = zero_zero_nat ).
% num.size_gen(1)
tff(fact_6755_take__bit__eq__0__iff,axiom,
! [N: nat,A: code_integer] :
( ( bit_se1745604003318907178nteger(N,A) = zero_z3403309356797280102nteger )
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)),A)) ) ).
% take_bit_eq_0_iff
tff(fact_6756_take__bit__eq__0__iff,axiom,
! [N: nat,A: int] :
( ( aa_int_int(bit_se2923211474154528505it_int(N),A) = zero_zero_int )
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),A)) ) ).
% take_bit_eq_0_iff
tff(fact_6757_take__bit__eq__0__iff,axiom,
! [N: nat,A: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = zero_zero_nat )
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),A)) ) ).
% take_bit_eq_0_iff
tff(fact_6758_take__bit__numeral__bit0,axiom,
! [L: num,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).
% take_bit_numeral_bit0
tff(fact_6759_take__bit__numeral__bit0,axiom,
! [L: num,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,L)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(pred_numeral,L)),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% take_bit_numeral_bit0
tff(fact_6760_take__bit__nat__less__self__iff,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M)),M))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),M)) ) ).
% take_bit_nat_less_self_iff
tff(fact_6761_Suc__mask__eq__exp,axiom,
! [N: nat] : aa_nat_nat(suc,aa_nat_nat(bit_se2002935070580805687sk_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).
% Suc_mask_eq_exp
tff(fact_6762_mask__nat__less__exp,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).
% mask_nat_less_exp
tff(fact_6763_exp__half__le2,axiom,
pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))),numeral_numeral_real(aa_num_num(bit0,one)))) ).
% exp_half_le2
tff(fact_6764_take__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).
% take_bit_Suc_minus_bit0
tff(fact_6765_take__bit__int__greater__eq__self__iff,axiom,
! [K: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(bit_se2923211474154528505it_int(N),K)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ).
% take_bit_int_greater_eq_self_iff
tff(fact_6766_take__bit__int__less__self__iff,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),K))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K)) ) ).
% take_bit_int_less_self_iff
tff(fact_6767_exp__double,axiom,
! [Z: complex] : aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Z)) = aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% exp_double
tff(fact_6768_exp__double,axiom,
! [Z: real] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),Z)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% exp_double
tff(fact_6769_sum__natinterval__diff,axiom,
! [M: nat,N: nat,F: fun_nat_complex] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( groups2073611262835488442omplex(aTP_Lamm_he(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_complex_complex(minus_minus_complex(aa_nat_complex(F,M)),aa_nat_complex(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( groups2073611262835488442omplex(aTP_Lamm_he(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_complex ) ) ) ).
% sum_natinterval_diff
tff(fact_6770_sum__natinterval__diff,axiom,
! [M: nat,N: nat,F: fun_nat_rat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( groups2906978787729119204at_rat(aTP_Lamm_hf(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = minus_minus_rat(aa_nat_rat(F,M),aa_nat_rat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( groups2906978787729119204at_rat(aTP_Lamm_hf(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_rat ) ) ) ).
% sum_natinterval_diff
tff(fact_6771_sum__natinterval__diff,axiom,
! [M: nat,N: nat,F: fun_nat_int] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( groups3539618377306564664at_int(aTP_Lamm_hg(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,M)),aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( groups3539618377306564664at_int(aTP_Lamm_hg(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_int ) ) ) ).
% sum_natinterval_diff
tff(fact_6772_sum__natinterval__diff,axiom,
! [M: nat,N: nat,F: fun_nat_real] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( groups6591440286371151544t_real(aTP_Lamm_hh(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,M)),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( groups6591440286371151544t_real(aTP_Lamm_hh(F),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_real ) ) ) ).
% sum_natinterval_diff
tff(fact_6773_sum__telescope_H_H,axiom,
! [M: nat,N: nat,F: fun_nat_rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( groups2906978787729119204at_rat(aTP_Lamm_hi(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N)) = minus_minus_rat(aa_nat_rat(F,N),aa_nat_rat(F,M)) ) ) ).
% sum_telescope''
tff(fact_6774_sum__telescope_H_H,axiom,
! [M: nat,N: nat,F: fun_nat_int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( groups3539618377306564664at_int(aTP_Lamm_hj(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,N)),aa_nat_int(F,M)) ) ) ).
% sum_telescope''
tff(fact_6775_sum__telescope_H_H,axiom,
! [M: nat,N: nat,F: fun_nat_real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( groups6591440286371151544t_real(aTP_Lamm_hk(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,N)),aa_nat_real(F,M)) ) ) ).
% sum_telescope''
tff(fact_6776_semiring__bit__operations__class_Oeven__mask__iff,axiom,
! [N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(bit_se2119862282449309892nteger,N)))
<=> ( N = zero_zero_nat ) ) ).
% semiring_bit_operations_class.even_mask_iff
tff(fact_6777_semiring__bit__operations__class_Oeven__mask__iff,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(bit_se2002935070580805687sk_nat,N)))
<=> ( N = zero_zero_nat ) ) ).
% semiring_bit_operations_class.even_mask_iff
tff(fact_6778_semiring__bit__operations__class_Oeven__mask__iff,axiom,
! [N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(bit_se2000444600071755411sk_int,N)))
<=> ( N = zero_zero_nat ) ) ).
% semiring_bit_operations_class.even_mask_iff
tff(fact_6779_take__bit__int__eq__self,axiom,
! [K: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
=> ( aa_int_int(bit_se2923211474154528505it_int(N),K) = K ) ) ) ).
% take_bit_int_eq_self
tff(fact_6780_take__bit__int__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( aa_int_int(bit_se2923211474154528505it_int(N),K) = K )
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ).
% take_bit_int_eq_self_iff
tff(fact_6781_mask__nat__def,axiom,
! [N: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat) ).
% mask_nat_def
tff(fact_6782_mask__half__int,axiom,
! [N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(bit_se2000444600071755411sk_int,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ).
% mask_half_int
tff(fact_6783_take__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).
% take_bit_numeral_minus_bit0
tff(fact_6784_take__bit__incr__eq,axiom,
! [N: nat,K: int] :
( ( aa_int_int(bit_se2923211474154528505it_int(N),K) != aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int) )
=> ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(bit_se2923211474154528505it_int(N),K)) ) ) ).
% take_bit_incr_eq
tff(fact_6785_mask__eq__sum__exp,axiom,
! [N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int) = groups3539618377306564664at_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bh,N))) ).
% mask_eq_sum_exp
tff(fact_6786_mask__eq__sum__exp,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bh,N))) ).
% mask_eq_sum_exp
tff(fact_6787_mask__int__def,axiom,
! [N: nat] : aa_nat_int(bit_se2000444600071755411sk_int,N) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int) ).
% mask_int_def
tff(fact_6788_sum__gp__multiplied,axiom,
! [M: nat,N: nat,X: complex] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) = aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),M)),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N))) ) ) ).
% sum_gp_multiplied
tff(fact_6789_sum__gp__multiplied,axiom,
! [M: nat,N: nat,X: rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_rat_rat(times_times_rat(minus_minus_rat(one_one_rat,X)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) = minus_minus_rat(aa_nat_rat(power_power_rat(X),M),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N))) ) ) ).
% sum_gp_multiplied
tff(fact_6790_sum__gp__multiplied,axiom,
! [M: nat,N: nat,X: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X)),groups3539618377306564664at_int(power_power_int(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X),M)),aa_nat_int(power_power_int(X),aa_nat_nat(suc,N))) ) ) ).
% sum_gp_multiplied
tff(fact_6791_sum__gp__multiplied,axiom,
! [M: nat,N: nat,X: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,N))) ) ) ).
% sum_gp_multiplied
tff(fact_6792_sum_Oin__pairs,axiom,
! [G: fun_nat_rat,M: nat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups2906978787729119204at_rat(aTP_Lamm_hl(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).
% sum.in_pairs
tff(fact_6793_sum_Oin__pairs,axiom,
! [G: fun_nat_int,M: nat,N: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups3539618377306564664at_int(aTP_Lamm_hm(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).
% sum.in_pairs
tff(fact_6794_sum_Oin__pairs,axiom,
! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_hn(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).
% sum.in_pairs
tff(fact_6795_sum_Oin__pairs,axiom,
! [G: fun_nat_real,M: nat,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) = groups6591440286371151544t_real(aTP_Lamm_ho(G),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ).
% sum.in_pairs
tff(fact_6796_take__bit__Suc__minus__1__eq,axiom,
! [N: nat] : bit_se1745604003318907178nteger(aa_nat_nat(suc,N),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(suc,N)),one_one_Code_integer) ).
% take_bit_Suc_minus_1_eq
tff(fact_6797_take__bit__Suc__minus__1__eq,axiom,
! [N: nat] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N))),one_one_int) ).
% take_bit_Suc_minus_1_eq
tff(fact_6798_take__bit__Suc__bit1,axiom,
! [N: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(N),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).
% take_bit_Suc_bit1
tff(fact_6799_take__bit__Suc__bit1,axiom,
! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,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(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ).
% take_bit_Suc_bit1
tff(fact_6800_take__bit__numeral__minus__1__eq,axiom,
! [K: num] : bit_se1745604003318907178nteger(aa_num_nat(numeral_numeral_nat,K),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,K)),one_one_Code_integer) ).
% take_bit_numeral_minus_1_eq
tff(fact_6801_take__bit__numeral__minus__1__eq,axiom,
! [K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,K)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,K))),one_one_int) ).
% take_bit_numeral_minus_1_eq
tff(fact_6802_take__bit__Suc,axiom,
! [N: nat,A: code_integer] : bit_se1745604003318907178nteger(aa_nat_nat(suc,N),A) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,bit_se1745604003318907178nteger(N,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))),numera6620942414471956472nteger(aa_num_num(bit0,one))),modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ).
% take_bit_Suc
tff(fact_6803_take__bit__Suc,axiom,
! [N: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).
% take_bit_Suc
tff(fact_6804_take__bit__Suc,axiom,
! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N)),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(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).
% take_bit_Suc
tff(fact_6805_mask__eq__exp__minus__1,axiom,
! [N: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat) ).
% mask_eq_exp_minus_1
tff(fact_6806_mask__eq__exp__minus__1,axiom,
! [N: nat] : aa_nat_int(bit_se2000444600071755411sk_int,N) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int) ).
% mask_eq_exp_minus_1
tff(fact_6807_infinite__nat__iff__unbounded,axiom,
! [S2: set_nat] :
( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
<=> ! [M7: nat] :
? [N5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M7),N5))
& pp(aa_set_nat_bool(member_nat(N5),S2)) ) ) ).
% infinite_nat_iff_unbounded
tff(fact_6808_unbounded__k__infinite,axiom,
! [K: nat,S2: set_nat] :
( ! [M3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),M3))
=> ? [N6: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N6))
& pp(aa_set_nat_bool(member_nat(N6),S2)) ) )
=> ~ pp(aa_set_nat_bool(finite_finite_nat,S2)) ) ).
% unbounded_k_infinite
tff(fact_6809_infinite__nat__iff__unbounded__le,axiom,
! [S2: set_nat] :
( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
<=> ! [M7: nat] :
? [N5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N5))
& pp(aa_set_nat_bool(member_nat(N5),S2)) ) ) ).
% infinite_nat_iff_unbounded_le
tff(fact_6810_exp__bound,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).
% exp_bound
tff(fact_6811_take__bit__int__less__eq,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))) ) ) ).
% take_bit_int_less_eq
tff(fact_6812_take__bit__int__greater__eq,axiom,
! [K: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),aa_int_int(bit_se2923211474154528505it_int(N),K))) ) ).
% take_bit_int_greater_eq
tff(fact_6813_signed__take__bit__eq__take__bit__shift,axiom,
! [N: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(N),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).
% signed_take_bit_eq_take_bit_shift
tff(fact_6814_mask__eq__sum__exp__nat,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),aa_nat_nat(suc,zero_zero_nat)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bh,N))) ).
% mask_eq_sum_exp_nat
tff(fact_6815_stable__imp__take__bit__eq,axiom,
! [A: code_integer,N: nat] :
( ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A )
=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
=> ( bit_se1745604003318907178nteger(N,A) = zero_z3403309356797280102nteger ) )
& ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
=> ( bit_se1745604003318907178nteger(N,A) = minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N),one_one_Code_integer) ) ) ) ) ).
% stable_imp_take_bit_eq
tff(fact_6816_stable__imp__take__bit__eq,axiom,
! [A: int,N: nat] :
( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
=> ( aa_int_int(bit_se2923211474154528505it_int(N),A) = zero_zero_int ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
=> ( aa_int_int(bit_se2923211474154528505it_int(N),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int) ) ) ) ) ).
% stable_imp_take_bit_eq
tff(fact_6817_stable__imp__take__bit__eq,axiom,
! [A: nat,N: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A )
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = zero_zero_nat ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat) ) ) ) ) ).
% stable_imp_take_bit_eq
tff(fact_6818_gauss__sum__nat,axiom,
! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_be),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% gauss_sum_nat
tff(fact_6819_take__bit__numeral__bit1,axiom,
! [L: num,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).
% take_bit_numeral_bit1
tff(fact_6820_take__bit__numeral__bit1,axiom,
! [L: num,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,L)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,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(bit_se2925701944663578781it_nat,aa_num_nat(pred_numeral,L)),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ).
% take_bit_numeral_bit1
tff(fact_6821_real__exp__bound__lemma,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(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(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),X)))) ) ) ).
% real_exp_bound_lemma
tff(fact_6822_take__bit__minus__small__eq,axiom,
! [K: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
=> ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(uminus_uminus_int,K)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K) ) ) ) ).
% take_bit_minus_small_eq
tff(fact_6823_arith__series__nat,axiom,
! [A: nat,D: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_hp(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% arith_series_nat
tff(fact_6824_Sum__Icc__nat,axiom,
! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_be),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% Sum_Icc_nat
tff(fact_6825_exp__lower__Taylor__quadratic,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(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(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(exp_real,X))) ) ).
% exp_lower_Taylor_quadratic
tff(fact_6826_take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).
% take_bit_numeral_minus_bit1
tff(fact_6827_take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).
% take_bit_Suc_minus_bit1
tff(fact_6828_sum__gp,axiom,
! [N: nat,M: nat,X: complex] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
=> ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_complex ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
=> ( ( ( X = one_one_complex )
=> ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M)) ) )
& ( ( X != one_one_complex )
=> ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),M)),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N)))),aa_complex_complex(minus_minus_complex(one_one_complex),X)) ) ) ) ) ) ).
% sum_gp
tff(fact_6829_sum__gp,axiom,
! [N: nat,M: nat,X: rat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
=> ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_rat ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
=> ( ( ( X = one_one_rat )
=> ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M)) ) )
& ( ( X != one_one_rat )
=> ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = divide_divide_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X),M),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N))),minus_minus_rat(one_one_rat,X)) ) ) ) ) ) ).
% sum_gp
tff(fact_6830_sum__gp,axiom,
! [N: nat,M: nat,X: real] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
=> ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = zero_zero_real ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
=> ( ( ( X = one_one_real )
=> ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M)) ) )
& ( ( X != one_one_real )
=> ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,N)))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)) ) ) ) ) ) ).
% sum_gp
tff(fact_6831_log__base__10__eq1,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(ln_ln_real,X)) ) ) ).
% log_base_10_eq1
tff(fact_6832_gauss__sum__from__Suc__0,axiom,
! [N: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).
% gauss_sum_from_Suc_0
tff(fact_6833_gauss__sum__from__Suc__0,axiom,
! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% gauss_sum_from_Suc_0
tff(fact_6834_and__int__unfold,axiom,
! [K: int,L: int] :
( ( ( ( K = zero_zero_int )
| ( L = zero_zero_int ) )
=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = zero_zero_int ) )
& ( ~ ( ( K = zero_zero_int )
| ( L = zero_zero_int ) )
=> ( ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = L ) )
& ( ( K != aa_int_int(uminus_uminus_int,one_one_int) )
=> ( ( ( L = aa_int_int(uminus_uminus_int,one_one_int) )
=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = K ) )
& ( ( L != aa_int_int(uminus_uminus_int,one_one_int) )
=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ).
% and_int_unfold
tff(fact_6835_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_nat_int(semiri1314217659103216013at_int,N) )
<=> ( M = N ) ) ).
% of_nat_eq_iff
tff(fact_6836_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(semiri5074537144036343181t_real,N) )
<=> ( M = N ) ) ).
% of_nat_eq_iff
tff(fact_6837_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( aa_nat_nat(semiri1316708129612266289at_nat,M) = aa_nat_nat(semiri1316708129612266289at_nat,N) )
<=> ( M = N ) ) ).
% of_nat_eq_iff
tff(fact_6838_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( aa_nat_complex(semiri8010041392384452111omplex,M) = aa_nat_complex(semiri8010041392384452111omplex,N) )
<=> ( M = N ) ) ).
% of_nat_eq_iff
tff(fact_6839_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( aa_nat_rat(semiri681578069525770553at_rat,M) = aa_nat_rat(semiri681578069525770553at_rat,N) )
<=> ( M = N ) ) ).
% of_nat_eq_iff
tff(fact_6840_and_Oright__idem,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),B) ).
% and.right_idem
tff(fact_6841_and_Oright__idem,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B) ).
% and.right_idem
tff(fact_6842_and_Oleft__idem,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),B)) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),B) ).
% and.left_idem
tff(fact_6843_and_Oleft__idem,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B) ).
% and.left_idem
tff(fact_6844_and_Oidem,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),A) = A ).
% and.idem
tff(fact_6845_and_Oidem,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),A) = A ).
% and.idem
tff(fact_6846_int__eq__iff__numeral,axiom,
! [M: nat,V: num] :
( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_num_int(numeral_numeral_int,V) )
<=> ( M = aa_num_nat(numeral_numeral_nat,V) ) ) ).
% int_eq_iff_numeral
tff(fact_6847_bit_Oconj__zero__right,axiom,
! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),zero_zero_int) = zero_zero_int ).
% bit.conj_zero_right
tff(fact_6848_bit_Oconj__zero__left,axiom,
! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,zero_zero_int),X) = zero_zero_int ).
% bit.conj_zero_left
tff(fact_6849_zero__and__eq,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,zero_zero_int),A) = zero_zero_int ).
% zero_and_eq
tff(fact_6850_zero__and__eq,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,zero_zero_nat),A) = zero_zero_nat ).
% zero_and_eq
tff(fact_6851_and__zero__eq,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),zero_zero_int) = zero_zero_int ).
% and_zero_eq
tff(fact_6852_and__zero__eq,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),zero_zero_nat) = zero_zero_nat ).
% and_zero_eq
tff(fact_6853_abs__of__nat,axiom,
! [N: nat] : abs_abs_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,N)) = aa_nat_Code_integer(semiri4939895301339042750nteger,N) ).
% abs_of_nat
tff(fact_6854_abs__of__nat,axiom,
! [N: nat] : aa_int_int(abs_abs_int,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).
% abs_of_nat
tff(fact_6855_abs__of__nat,axiom,
! [N: nat] : aa_real_real(abs_abs_real,aa_nat_real(semiri5074537144036343181t_real,N)) = aa_nat_real(semiri5074537144036343181t_real,N) ).
% abs_of_nat
tff(fact_6856_abs__of__nat,axiom,
! [N: nat] : abs_abs_rat(aa_nat_rat(semiri681578069525770553at_rat,N)) = aa_nat_rat(semiri681578069525770553at_rat,N) ).
% abs_of_nat
tff(fact_6857_take__bit__and,axiom,
! [N: nat,A: int,B: int] : aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),B)) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(bit_se2923211474154528505it_int(N),A)),aa_int_int(bit_se2923211474154528505it_int(N),B)) ).
% take_bit_and
tff(fact_6858_take__bit__and,axiom,
! [N: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),B)) ).
% take_bit_and
tff(fact_6859_height__double__log__univ__size,axiom,
! [U: real,Deg: nat,T: vEBT_VEBT] :
( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),Deg) )
=> ( pp(aa_nat_bool(vEBT_invar_vebt(T),Deg))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_height,T))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),U))))) ) ) ).
% height_double_log_univ_size
tff(fact_6860_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( aa_nat_int(semiri1314217659103216013at_int,M) = zero_zero_int )
<=> ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
tff(fact_6861_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( aa_nat_real(semiri5074537144036343181t_real,M) = zero_zero_real )
<=> ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
tff(fact_6862_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( aa_nat_nat(semiri1316708129612266289at_nat,M) = zero_zero_nat )
<=> ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
tff(fact_6863_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( aa_nat_complex(semiri8010041392384452111omplex,M) = zero_zero_complex )
<=> ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
tff(fact_6864_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( aa_nat_rat(semiri681578069525770553at_rat,M) = zero_zero_rat )
<=> ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
tff(fact_6865_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int = aa_nat_int(semiri1314217659103216013at_int,N) )
<=> ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
tff(fact_6866_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_real = aa_nat_real(semiri5074537144036343181t_real,N) )
<=> ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
tff(fact_6867_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat = aa_nat_nat(semiri1316708129612266289at_nat,N) )
<=> ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
tff(fact_6868_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_complex = aa_nat_complex(semiri8010041392384452111omplex,N) )
<=> ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
tff(fact_6869_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_rat = aa_nat_rat(semiri681578069525770553at_rat,N) )
<=> ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
tff(fact_6870_of__nat__0,axiom,
aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).
% of_nat_0
tff(fact_6871_of__nat__0,axiom,
aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).
% of_nat_0
tff(fact_6872_of__nat__0,axiom,
aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).
% of_nat_0
tff(fact_6873_of__nat__0,axiom,
aa_nat_complex(semiri8010041392384452111omplex,zero_zero_nat) = zero_zero_complex ).
% of_nat_0
tff(fact_6874_of__nat__0,axiom,
aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).
% of_nat_0
tff(fact_6875_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% of_nat_less_iff
tff(fact_6876_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% of_nat_less_iff
tff(fact_6877_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% of_nat_less_iff
tff(fact_6878_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% of_nat_less_iff
tff(fact_6879_of__nat__numeral,axiom,
! [N: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,N)) = aa_num_int(numeral_numeral_int,N) ).
% of_nat_numeral
tff(fact_6880_of__nat__numeral,axiom,
! [N: num] : aa_nat_real(semiri5074537144036343181t_real,aa_num_nat(numeral_numeral_nat,N)) = numeral_numeral_real(N) ).
% of_nat_numeral
tff(fact_6881_of__nat__numeral,axiom,
! [N: num] : aa_nat_nat(semiri1316708129612266289at_nat,aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,N) ).
% of_nat_numeral
tff(fact_6882_of__nat__numeral,axiom,
! [N: num] : aa_nat_complex(semiri8010041392384452111omplex,aa_num_nat(numeral_numeral_nat,N)) = numera6690914467698888265omplex(N) ).
% of_nat_numeral
tff(fact_6883_of__nat__numeral,axiom,
! [N: num] : aa_nat_rat(semiri681578069525770553at_rat,aa_num_nat(numeral_numeral_nat,N)) = numeral_numeral_rat(N) ).
% of_nat_numeral
tff(fact_6884_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% of_nat_le_iff
tff(fact_6885_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% of_nat_le_iff
tff(fact_6886_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% of_nat_le_iff
tff(fact_6887_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% of_nat_le_iff
tff(fact_6888_of__nat__add,axiom,
! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).
% of_nat_add
tff(fact_6889_of__nat__add,axiom,
! [M: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)) ).
% of_nat_add
tff(fact_6890_of__nat__add,axiom,
! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).
% of_nat_add
tff(fact_6891_of__nat__add,axiom,
! [M: nat,N: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri8010041392384452111omplex,N)) ).
% of_nat_add
tff(fact_6892_of__nat__add,axiom,
! [M: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,M),aa_nat_rat(semiri681578069525770553at_rat,N)) ).
% of_nat_add
tff(fact_6893_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( aa_nat_int(semiri1314217659103216013at_int,N) = one_one_int )
<=> ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
tff(fact_6894_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( aa_nat_real(semiri5074537144036343181t_real,N) = one_one_real )
<=> ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
tff(fact_6895_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( aa_nat_nat(semiri1316708129612266289at_nat,N) = one_one_nat )
<=> ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
tff(fact_6896_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( aa_nat_complex(semiri8010041392384452111omplex,N) = one_one_complex )
<=> ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
tff(fact_6897_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( aa_nat_rat(semiri681578069525770553at_rat,N) = one_one_rat )
<=> ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
tff(fact_6898_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,N) )
<=> ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
tff(fact_6899_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_real = aa_nat_real(semiri5074537144036343181t_real,N) )
<=> ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
tff(fact_6900_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,N) )
<=> ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
tff(fact_6901_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_complex = aa_nat_complex(semiri8010041392384452111omplex,N) )
<=> ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
tff(fact_6902_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_rat = aa_nat_rat(semiri681578069525770553at_rat,N) )
<=> ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
tff(fact_6903_of__nat__1,axiom,
aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).
% of_nat_1
tff(fact_6904_of__nat__1,axiom,
aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).
% of_nat_1
tff(fact_6905_of__nat__1,axiom,
aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).
% of_nat_1
tff(fact_6906_of__nat__1,axiom,
aa_nat_complex(semiri8010041392384452111omplex,one_one_nat) = one_one_complex ).
% of_nat_1
tff(fact_6907_of__nat__1,axiom,
aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).
% of_nat_1
tff(fact_6908_of__nat__mult,axiom,
! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).
% of_nat_mult
tff(fact_6909_of__nat__mult,axiom,
! [M: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)) ).
% of_nat_mult
tff(fact_6910_of__nat__mult,axiom,
! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).
% of_nat_mult
tff(fact_6911_of__nat__mult,axiom,
! [M: nat,N: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri8010041392384452111omplex,N)) ).
% of_nat_mult
tff(fact_6912_of__nat__mult,axiom,
! [M: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)) ).
% of_nat_mult
tff(fact_6913_bit_Oconj__one__right,axiom,
! [X: code_integer] : bit_se3949692690581998587nteger(X,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = X ).
% bit.conj_one_right
tff(fact_6914_bit_Oconj__one__right,axiom,
! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),aa_int_int(uminus_uminus_int,one_one_int)) = X ).
% bit.conj_one_right
tff(fact_6915_and_Oright__neutral,axiom,
! [A: code_integer] : bit_se3949692690581998587nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = A ).
% and.right_neutral
tff(fact_6916_and_Oright__neutral,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),aa_int_int(uminus_uminus_int,one_one_int)) = A ).
% and.right_neutral
tff(fact_6917_and_Oleft__neutral,axiom,
! [A: code_integer] : bit_se3949692690581998587nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),A) = A ).
% and.left_neutral
tff(fact_6918_and_Oleft__neutral,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,one_one_int)),A) = A ).
% and.left_neutral
tff(fact_6919_of__nat__power,axiom,
! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,M)),N) ).
% of_nat_power
tff(fact_6920_of__nat__power,axiom,
! [M: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,M)),N) ).
% of_nat_power
tff(fact_6921_of__nat__power,axiom,
! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),N) ).
% of_nat_power
tff(fact_6922_of__nat__power,axiom,
! [M: nat,N: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),N) ).
% of_nat_power
tff(fact_6923_of__nat__power,axiom,
! [M: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),N) ).
% of_nat_power
tff(fact_6924_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W) = aa_nat_int(semiri1314217659103216013at_int,X) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
tff(fact_6925_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W) = aa_nat_real(semiri5074537144036343181t_real,X) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
tff(fact_6926_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W) = aa_nat_nat(semiri1316708129612266289at_nat,X) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
tff(fact_6927_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,B)),W) = aa_nat_complex(semiri8010041392384452111omplex,X) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
tff(fact_6928_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W) = aa_nat_rat(semiri681578069525770553at_rat,X) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
tff(fact_6929_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( aa_nat_int(semiri1314217659103216013at_int,X) = aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W) )
<=> ( X = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
tff(fact_6930_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( aa_nat_real(semiri5074537144036343181t_real,X) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W) )
<=> ( X = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
tff(fact_6931_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( aa_nat_nat(semiri1316708129612266289at_nat,X) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W) )
<=> ( X = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
tff(fact_6932_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( aa_nat_complex(semiri8010041392384452111omplex,X) = aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,B)),W) )
<=> ( X = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
tff(fact_6933_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( aa_nat_rat(semiri681578069525770553at_rat,X) = aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W) )
<=> ( X = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
tff(fact_6934_negative__zless,axiom,
! [N: nat,M: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)))),aa_nat_int(semiri1314217659103216013at_int,M))) ).
% negative_zless
tff(fact_6935_log__one,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(log,A),one_one_real) = zero_zero_real ).
% log_one
tff(fact_6936_and__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
| pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ) ).
% and_nonnegative_int_iff
tff(fact_6937_and__negative__int__iff,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)),zero_zero_int))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ).
% and_negative_int_iff
tff(fact_6938_pred__numeral__inc,axiom,
! [K: num] : aa_num_nat(pred_numeral,inc(K)) = aa_num_nat(numeral_numeral_nat,K) ).
% pred_numeral_inc
tff(fact_6939_of__nat__of__bool,axiom,
! [P: bool] : aa_nat_real(semiri5074537144036343181t_real,zero_n2687167440665602831ol_nat(P)) = zero_n3304061248610475627l_real(P) ).
% of_nat_of_bool
tff(fact_6940_of__nat__of__bool,axiom,
! [P: bool] : aa_nat_complex(semiri8010041392384452111omplex,zero_n2687167440665602831ol_nat(P)) = zero_n1201886186963655149omplex(P) ).
% of_nat_of_bool
tff(fact_6941_of__nat__of__bool,axiom,
! [P: bool] : aa_nat_rat(semiri681578069525770553at_rat,zero_n2687167440665602831ol_nat(P)) = zero_n2052037380579107095ol_rat(P) ).
% of_nat_of_bool
tff(fact_6942_of__nat__of__bool,axiom,
! [P: bool] : aa_nat_nat(semiri1316708129612266289at_nat,zero_n2687167440665602831ol_nat(P)) = zero_n2687167440665602831ol_nat(P) ).
% of_nat_of_bool
tff(fact_6943_of__nat__of__bool,axiom,
! [P: bool] : aa_nat_int(semiri1314217659103216013at_int,zero_n2687167440665602831ol_nat(P)) = aa_bool_int(zero_n2684676970156552555ol_int,P) ).
% of_nat_of_bool
tff(fact_6944_of__nat__of__bool,axiom,
! [P: bool] : aa_nat_Code_integer(semiri4939895301339042750nteger,zero_n2687167440665602831ol_nat(P)) = zero_n356916108424825756nteger(P) ).
% of_nat_of_bool
tff(fact_6945_of__nat__le__0__iff,axiom,
! [M: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real))
<=> ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
tff(fact_6946_of__nat__le__0__iff,axiom,
! [M: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat))
<=> ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
tff(fact_6947_of__nat__le__0__iff,axiom,
! [M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat))
<=> ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
tff(fact_6948_of__nat__le__0__iff,axiom,
! [M: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int))
<=> ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
tff(fact_6949_of__nat__Suc,axiom,
! [M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_nat_int(semiri1314217659103216013at_int,M)) ).
% of_nat_Suc
tff(fact_6950_of__nat__Suc,axiom,
! [M: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,M)) ).
% of_nat_Suc
tff(fact_6951_of__nat__Suc,axiom,
! [M: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(semiri1316708129612266289at_nat,M)) ).
% of_nat_Suc
tff(fact_6952_of__nat__Suc,axiom,
! [M: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,M)) = aa_complex_complex(plus_plus_complex(one_one_complex),aa_nat_complex(semiri8010041392384452111omplex,M)) ).
% of_nat_Suc
tff(fact_6953_of__nat__Suc,axiom,
! [M: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M)) = plus_plus_rat(one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,M)) ).
% of_nat_Suc
tff(fact_6954_and__numerals_I2_J,axiom,
! [Y: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = one_one_int ).
% and_numerals(2)
tff(fact_6955_and__numerals_I2_J,axiom,
! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = one_one_nat ).
% and_numerals(2)
tff(fact_6956_and__numerals_I8_J,axiom,
! [X: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),one_one_int) = one_one_int ).
% and_numerals(8)
tff(fact_6957_and__numerals_I8_J,axiom,
! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),one_one_nat) = one_one_nat ).
% and_numerals(8)
tff(fact_6958_numeral__less__real__of__nat__iff,axiom,
! [W: num,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(W)),aa_nat_real(semiri5074537144036343181t_real,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,W)),N)) ) ).
% numeral_less_real_of_nat_iff
tff(fact_6959_real__of__nat__less__numeral__iff,axiom,
! [N: nat,W: num] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),numeral_numeral_real(W)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,W))) ) ).
% real_of_nat_less_numeral_iff
tff(fact_6960_numeral__le__real__of__nat__iff,axiom,
! [N: num,M: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(N)),aa_nat_real(semiri5074537144036343181t_real,M)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N)),M)) ) ).
% numeral_le_real_of_nat_iff
tff(fact_6961_log__eq__one,axiom,
! [A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ( ( A != one_one_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(log,A),A) = one_one_real ) ) ) ).
% log_eq_one
tff(fact_6962_log__less__cancel__iff,axiom,
! [A: real,X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ) ).
% log_less_cancel_iff
tff(fact_6963_log__less__one__cancel__iff,axiom,
! [A: real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),one_one_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),A)) ) ) ) ).
% log_less_one_cancel_iff
tff(fact_6964_one__less__log__cancel__iff,axiom,
! [A: real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X)) ) ) ) ).
% one_less_log_cancel_iff
tff(fact_6965_log__less__zero__cancel__iff,axiom,
! [A: real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),zero_zero_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ) ) ).
% log_less_zero_cancel_iff
tff(fact_6966_zero__less__log__cancel__iff,axiom,
! [A: real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ) ) ).
% zero_less_log_cancel_iff
tff(fact_6967_of__nat__0__less__iff,axiom,
! [N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).
% of_nat_0_less_iff
tff(fact_6968_of__nat__0__less__iff,axiom,
! [N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).
% of_nat_0_less_iff
tff(fact_6969_of__nat__0__less__iff,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).
% of_nat_0_less_iff
tff(fact_6970_of__nat__0__less__iff,axiom,
! [N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).
% of_nat_0_less_iff
tff(fact_6971_and__numerals_I1_J,axiom,
! [Y: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = zero_zero_int ).
% and_numerals(1)
tff(fact_6972_and__numerals_I1_J,axiom,
! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = zero_zero_nat ).
% and_numerals(1)
tff(fact_6973_and__numerals_I5_J,axiom,
! [X: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),one_one_int) = zero_zero_int ).
% and_numerals(5)
tff(fact_6974_and__numerals_I5_J,axiom,
! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),one_one_nat) = zero_zero_nat ).
% and_numerals(5)
tff(fact_6975_and__numerals_I3_J,axiom,
! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y))) ).
% and_numerals(3)
tff(fact_6976_and__numerals_I3_J,axiom,
! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y))) ).
% and_numerals(3)
tff(fact_6977_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W)),aa_nat_int(semiri1314217659103216013at_int,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).
% of_nat_less_of_nat_power_cancel_iff
tff(fact_6978_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W)),aa_nat_real(semiri5074537144036343181t_real,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).
% of_nat_less_of_nat_power_cancel_iff
tff(fact_6979_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W)),aa_nat_nat(semiri1316708129612266289at_nat,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).
% of_nat_less_of_nat_power_cancel_iff
tff(fact_6980_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)),aa_nat_rat(semiri681578069525770553at_rat,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).
% of_nat_less_of_nat_power_cancel_iff
tff(fact_6981_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).
% of_nat_power_less_of_nat_cancel_iff
tff(fact_6982_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).
% of_nat_power_less_of_nat_cancel_iff
tff(fact_6983_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).
% of_nat_power_less_of_nat_cancel_iff
tff(fact_6984_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).
% of_nat_power_less_of_nat_cancel_iff
tff(fact_6985_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N: nat] :
( ( aa_nat_int(semiri1314217659103216013at_int,Y) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) )
<=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_6986_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N: nat] :
( ( aa_nat_real(semiri5074537144036343181t_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(X)),N) )
<=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_6987_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N: nat] :
( ( aa_nat_nat(semiri1316708129612266289at_nat,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) )
<=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_6988_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N: nat] :
( ( aa_nat_complex(semiri8010041392384452111omplex,Y) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X)),N) )
<=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_6989_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y: nat,X: num,N: nat] :
( ( aa_nat_rat(semiri681578069525770553at_rat,Y) = aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),N) )
<=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_6990_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y: nat] :
( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = aa_nat_int(semiri1314217659103216013at_int,Y) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
tff(fact_6991_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y: nat] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(X)),N) = aa_nat_real(semiri5074537144036343181t_real,Y) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
tff(fact_6992_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = aa_nat_nat(semiri1316708129612266289at_nat,Y) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
tff(fact_6993_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y: nat] :
( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(X)),N) = aa_nat_complex(semiri8010041392384452111omplex,Y) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
tff(fact_6994_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y: nat] :
( ( aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),N) = aa_nat_rat(semiri681578069525770553at_rat,Y) )
<=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = Y ) ) ).
% numeral_power_eq_of_nat_cancel_iff
tff(fact_6995_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).
% of_nat_power_le_of_nat_cancel_iff
tff(fact_6996_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).
% of_nat_power_le_of_nat_cancel_iff
tff(fact_6997_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).
% of_nat_power_le_of_nat_cancel_iff
tff(fact_6998_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W))) ) ).
% of_nat_power_le_of_nat_cancel_iff
tff(fact_6999_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W)),aa_nat_real(semiri5074537144036343181t_real,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).
% of_nat_le_of_nat_power_cancel_iff
tff(fact_7000_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)),aa_nat_rat(semiri681578069525770553at_rat,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).
% of_nat_le_of_nat_power_cancel_iff
tff(fact_7001_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W)),aa_nat_nat(semiri1316708129612266289at_nat,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).
% of_nat_le_of_nat_power_cancel_iff
tff(fact_7002_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W)),aa_nat_int(semiri1314217659103216013at_int,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X)) ) ).
% of_nat_le_of_nat_power_cancel_iff
tff(fact_7003_zero__le__log__cancel__iff,axiom,
! [A: real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ) ) ).
% zero_le_log_cancel_iff
tff(fact_7004_log__le__zero__cancel__iff,axiom,
! [A: real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),zero_zero_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ) ) ).
% log_le_zero_cancel_iff
tff(fact_7005_one__le__log__cancel__iff,axiom,
! [A: real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X)) ) ) ) ).
% one_le_log_cancel_iff
tff(fact_7006_log__le__one__cancel__iff,axiom,
! [A: real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),one_one_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),A)) ) ) ) ).
% log_le_one_cancel_iff
tff(fact_7007_log__le__cancel__iff,axiom,
! [A: real,X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ) ).
% log_le_cancel_iff
tff(fact_7008_add__neg__numeral__special_I5_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(N))) ).
% add_neg_numeral_special(5)
tff(fact_7009_add__neg__numeral__special_I5_J,axiom,
! [N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(inc(N))) ).
% add_neg_numeral_special(5)
tff(fact_7010_add__neg__numeral__special_I5_J,axiom,
! [N: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(inc(N))) ).
% add_neg_numeral_special(5)
tff(fact_7011_add__neg__numeral__special_I5_J,axiom,
! [N: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(inc(N))) ).
% add_neg_numeral_special(5)
tff(fact_7012_add__neg__numeral__special_I5_J,axiom,
! [N: num] : plus_plus_rat(uminus_uminus_rat(one_one_rat),uminus_uminus_rat(numeral_numeral_rat(N))) = uminus_uminus_rat(numeral_numeral_rat(inc(N))) ).
% add_neg_numeral_special(5)
tff(fact_7013_add__neg__numeral__special_I6_J,axiom,
! [M: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(M))) ).
% add_neg_numeral_special(6)
tff(fact_7014_add__neg__numeral__special_I6_J,axiom,
! [M: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(inc(M))) ).
% add_neg_numeral_special(6)
tff(fact_7015_add__neg__numeral__special_I6_J,axiom,
! [M: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(inc(M))) ).
% add_neg_numeral_special(6)
tff(fact_7016_add__neg__numeral__special_I6_J,axiom,
! [M: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(inc(M))) ).
% add_neg_numeral_special(6)
tff(fact_7017_add__neg__numeral__special_I6_J,axiom,
! [M: num] : plus_plus_rat(uminus_uminus_rat(numeral_numeral_rat(M)),uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(numeral_numeral_rat(inc(M))) ).
% add_neg_numeral_special(6)
tff(fact_7018_diff__numeral__special_I5_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(N))) ).
% diff_numeral_special(5)
tff(fact_7019_diff__numeral__special_I5_J,axiom,
! [N: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,one_one_real)),numeral_numeral_real(N)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(inc(N))) ).
% diff_numeral_special(5)
tff(fact_7020_diff__numeral__special_I5_J,axiom,
! [N: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),numera6690914467698888265omplex(N)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(inc(N))) ).
% diff_numeral_special(5)
tff(fact_7021_diff__numeral__special_I5_J,axiom,
! [N: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),numera6620942414471956472nteger(N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(inc(N))) ).
% diff_numeral_special(5)
tff(fact_7022_diff__numeral__special_I5_J,axiom,
! [N: num] : minus_minus_rat(uminus_uminus_rat(one_one_rat),numeral_numeral_rat(N)) = uminus_uminus_rat(numeral_numeral_rat(inc(N))) ).
% diff_numeral_special(5)
tff(fact_7023_diff__numeral__special_I6_J,axiom,
! [M: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_num_int(numeral_numeral_int,inc(M)) ).
% diff_numeral_special(6)
tff(fact_7024_diff__numeral__special_I6_J,axiom,
! [M: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,one_one_real)) = numeral_numeral_real(inc(M)) ).
% diff_numeral_special(6)
tff(fact_7025_diff__numeral__special_I6_J,axiom,
! [M: num] : aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(M)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = numera6690914467698888265omplex(inc(M)) ).
% diff_numeral_special(6)
tff(fact_7026_diff__numeral__special_I6_J,axiom,
! [M: num] : minus_8373710615458151222nteger(numera6620942414471956472nteger(M),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = numera6620942414471956472nteger(inc(M)) ).
% diff_numeral_special(6)
tff(fact_7027_diff__numeral__special_I6_J,axiom,
! [M: num] : minus_minus_rat(numeral_numeral_rat(M),uminus_uminus_rat(one_one_rat)) = numeral_numeral_rat(inc(M)) ).
% diff_numeral_special(6)
tff(fact_7028_log__pow__cancel,axiom,
! [A: real,B: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ( ( A != one_one_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),B)) = aa_nat_real(semiri5074537144036343181t_real,B) ) ) ) ).
% log_pow_cancel
tff(fact_7029_and__minus__numerals_I2_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = one_one_int ).
% and_minus_numerals(2)
tff(fact_7030_and__minus__numerals_I6_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))),one_one_int) = one_one_int ).
% and_minus_numerals(6)
tff(fact_7031_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,X)),N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
tff(fact_7032_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,X)),N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
tff(fact_7033_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
tff(fact_7034_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,X)),N)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
tff(fact_7035_and__numerals_I4_J,axiom,
! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y))) ).
% and_numerals(4)
tff(fact_7036_and__numerals_I4_J,axiom,
! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y))) ).
% and_numerals(4)
tff(fact_7037_and__numerals_I6_J,axiom,
! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y))) ).
% and_numerals(6)
tff(fact_7038_and__numerals_I6_J,axiom,
! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y))) ).
% and_numerals(6)
tff(fact_7039_and__minus__numerals_I1_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = zero_zero_int ).
% and_minus_numerals(1)
tff(fact_7040_and__minus__numerals_I5_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))),one_one_int) = zero_zero_int ).
% and_minus_numerals(5)
tff(fact_7041_even__of__nat,axiom,
! [N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(semiri4939895301339042750nteger,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ).
% even_of_nat
tff(fact_7042_even__of__nat,axiom,
! [N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(semiri1314217659103216013at_int,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ).
% even_of_nat
tff(fact_7043_even__of__nat,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(semiri1316708129612266289at_nat,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ).
% even_of_nat
tff(fact_7044_numeral__power__less__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),N)),aa_nat_int(semiri1314217659103216013at_int,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).
% numeral_power_less_of_nat_cancel_iff
tff(fact_7045_numeral__power__less__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(I)),N)),aa_nat_real(semiri5074537144036343181t_real,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).
% numeral_power_less_of_nat_cancel_iff
tff(fact_7046_numeral__power__less__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),aa_nat_nat(semiri1316708129612266289at_nat,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).
% numeral_power_less_of_nat_cancel_iff
tff(fact_7047_numeral__power__less__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(I)),N)),aa_nat_rat(semiri681578069525770553at_rat,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).
% numeral_power_less_of_nat_cancel_iff
tff(fact_7048_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).
% of_nat_less_numeral_power_cancel_iff
tff(fact_7049_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(I)),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).
% of_nat_less_numeral_power_cancel_iff
tff(fact_7050_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).
% of_nat_less_numeral_power_cancel_iff
tff(fact_7051_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(numeral_numeral_rat(I)),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).
% of_nat_less_numeral_power_cancel_iff
tff(fact_7052_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(I)),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).
% of_nat_le_numeral_power_cancel_iff
tff(fact_7053_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(numeral_numeral_rat(I)),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).
% of_nat_le_numeral_power_cancel_iff
tff(fact_7054_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).
% of_nat_le_numeral_power_cancel_iff
tff(fact_7055_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I: num,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))) ) ).
% of_nat_le_numeral_power_cancel_iff
tff(fact_7056_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(I)),N)),aa_nat_real(semiri5074537144036343181t_real,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).
% numeral_power_le_of_nat_cancel_iff
tff(fact_7057_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(I)),N)),aa_nat_rat(semiri681578069525770553at_rat,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).
% numeral_power_le_of_nat_cancel_iff
tff(fact_7058_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),aa_nat_nat(semiri1316708129612266289at_nat,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).
% numeral_power_le_of_nat_cancel_iff
tff(fact_7059_numeral__power__le__of__nat__cancel__iff,axiom,
! [I: num,N: nat,X: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),N)),aa_nat_int(semiri1314217659103216013at_int,X)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),X)) ) ).
% numeral_power_le_of_nat_cancel_iff
tff(fact_7060_and__numerals_I7_J,axiom,
! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y)))) ).
% and_numerals(7)
tff(fact_7061_and__numerals_I7_J,axiom,
! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = 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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y)))) ).
% and_numerals(7)
tff(fact_7062_and_Oleft__commute,axiom,
! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,B),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),C)) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,B),C)) ).
% and.left_commute
tff(fact_7063_and_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,B),C)) ).
% and.left_commute
tff(fact_7064_of__nat__and__eq,axiom,
! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N)) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).
% of_nat_and_eq
tff(fact_7065_of__nat__and__eq,axiom,
! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).
% of_nat_and_eq
tff(fact_7066_and_Ocommute,axiom,
! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),B) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,B),A) ).
% and.commute
tff(fact_7067_and_Ocommute,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,B),A) ).
% and.commute
tff(fact_7068_and_Oassoc,axiom,
! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,B),C)) ).
% and.assoc
tff(fact_7069_and_Oassoc,axiom,
! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,B),C)) ).
% and.assoc
tff(fact_7070_of__int__and__eq,axiom,
! [K: int,L: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(ring_1_of_int_int,K)),aa_int_int(ring_1_of_int_int,L)) ).
% of_int_and_eq
tff(fact_7071_real__arch__simple,axiom,
! [X: real] :
? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,N2))) ).
% real_arch_simple
tff(fact_7072_real__arch__simple,axiom,
! [X: rat] :
? [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_nat_rat(semiri681578069525770553at_rat,N2))) ).
% real_arch_simple
tff(fact_7073_reals__Archimedean2,axiom,
! [X: real] :
? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,N2))) ).
% reals_Archimedean2
tff(fact_7074_reals__Archimedean2,axiom,
! [X: rat] :
? [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_nat_rat(semiri681578069525770553at_rat,N2))) ).
% reals_Archimedean2
tff(fact_7075_mult__of__nat__commute,axiom,
! [X: nat,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(semiri1314217659103216013at_int,X)) ).
% mult_of_nat_commute
tff(fact_7076_mult__of__nat__commute,axiom,
! [X: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_nat_real(semiri5074537144036343181t_real,X)) ).
% mult_of_nat_commute
tff(fact_7077_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_7078_mult__of__nat__commute,axiom,
! [X: nat,Y: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,X)),Y) = aa_complex_complex(times_times_complex(Y),aa_nat_complex(semiri8010041392384452111omplex,X)) ).
% mult_of_nat_commute
tff(fact_7079_mult__of__nat__commute,axiom,
! [X: nat,Y: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,X)),Y) = aa_rat_rat(times_times_rat(Y),aa_nat_rat(semiri681578069525770553at_rat,X)) ).
% mult_of_nat_commute
tff(fact_7080_take__bit__of__nat,axiom,
! [N: nat,M: nat] : aa_int_int(bit_se2923211474154528505it_int(N),aa_nat_int(semiri1314217659103216013at_int,M)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M)) ).
% take_bit_of_nat
tff(fact_7081_take__bit__of__nat,axiom,
! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(semiri1316708129612266289at_nat,M)) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M)) ).
% take_bit_of_nat
tff(fact_7082_int__diff__cases,axiom,
! [Z: int] :
~ ! [M3: nat,N2: nat] : Z != aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M3)),aa_nat_int(semiri1314217659103216013at_int,N2)) ).
% int_diff_cases
tff(fact_7083_of__nat__mask__eq,axiom,
! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,N)) = aa_nat_nat(bit_se2002935070580805687sk_nat,N) ).
% of_nat_mask_eq
tff(fact_7084_of__nat__mask__eq,axiom,
! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(bit_se2002935070580805687sk_nat,N)) = aa_nat_int(bit_se2000444600071755411sk_int,N) ).
% of_nat_mask_eq
tff(fact_7085_less__log__of__power,axiom,
! [B: real,N: nat,M: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)),M))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,B),M))) ) ) ).
% less_log_of_power
tff(fact_7086_log__of__power__eq,axiom,
! [M: nat,B: real,N: nat] :
( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N) )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
=> ( aa_nat_real(semiri5074537144036343181t_real,N) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,M)) ) ) ) ).
% log_of_power_eq
tff(fact_7087_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(ring_1_of_int_int,X)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),X)) ) ).
% of_nat_less_of_int_iff
tff(fact_7088_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_int_real(ring_1_of_int_real,X)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),X)) ) ).
% of_nat_less_of_int_iff
tff(fact_7089_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_int_rat(ring_1_of_int_rat,X)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),X)) ) ).
% of_nat_less_of_int_iff
tff(fact_7090_num__induct,axiom,
! [P: fun_num_bool,X: num] :
( pp(aa_num_bool(P,one))
=> ( ! [X3: num] :
( pp(aa_num_bool(P,X3))
=> pp(aa_num_bool(P,inc(X3))) )
=> pp(aa_num_bool(P,X)) ) ) ).
% num_induct
tff(fact_7091_add__inc,axiom,
! [X: num,Y: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,X),inc(Y)) = inc(aa_num_num(aa_num_fun_num_num(plus_plus_num,X),Y)) ).
% add_inc
tff(fact_7092_le__log__of__power,axiom,
! [B: real,N: nat,M: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)),M))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,B),M))) ) ) ).
% le_log_of_power
tff(fact_7093_log__base__pow,axiom,
! [A: real,N: nat,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N)),X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ).
% log_base_pow
tff(fact_7094_log__nat__power,axiom,
! [X: real,B: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) ) ) ).
% log_nat_power
tff(fact_7095_and__eq__minus__1__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( bit_se3949692690581998587nteger(A,B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
<=> ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
& ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).
% and_eq_minus_1_iff
tff(fact_7096_and__eq__minus__1__iff,axiom,
! [A: int,B: int] :
( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),B) = aa_int_int(uminus_uminus_int,one_one_int) )
<=> ( ( A = aa_int_int(uminus_uminus_int,one_one_int) )
& ( B = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).
% and_eq_minus_1_iff
tff(fact_7097_of__nat__0__le__iff,axiom,
! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,N))) ).
% of_nat_0_le_iff
tff(fact_7098_of__nat__0__le__iff,axiom,
! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,N))) ).
% of_nat_0_le_iff
tff(fact_7099_of__nat__0__le__iff,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,N))) ).
% of_nat_0_le_iff
tff(fact_7100_of__nat__0__le__iff,axiom,
! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,N))) ).
% of_nat_0_le_iff
tff(fact_7101_of__nat__less__0__iff,axiom,
! [M: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int)) ).
% of_nat_less_0_iff
tff(fact_7102_of__nat__less__0__iff,axiom,
! [M: nat] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real)) ).
% of_nat_less_0_iff
tff(fact_7103_of__nat__less__0__iff,axiom,
! [M: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat)) ).
% of_nat_less_0_iff
tff(fact_7104_of__nat__less__0__iff,axiom,
! [M: nat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat)) ).
% of_nat_less_0_iff
tff(fact_7105_of__nat__neq__0,axiom,
! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)) != zero_zero_int ).
% of_nat_neq_0
tff(fact_7106_of__nat__neq__0,axiom,
! [N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N)) != zero_zero_real ).
% of_nat_neq_0
tff(fact_7107_of__nat__neq__0,axiom,
! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,N)) != zero_zero_nat ).
% of_nat_neq_0
tff(fact_7108_of__nat__neq__0,axiom,
! [N: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,N)) != zero_zero_complex ).
% of_nat_neq_0
tff(fact_7109_of__nat__neq__0,axiom,
! [N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N)) != zero_zero_rat ).
% of_nat_neq_0
tff(fact_7110_div__mult2__eq_H,axiom,
! [A: int,M: nat,N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_int(semiri1314217659103216013at_int,N)) ).
% div_mult2_eq'
tff(fact_7111_div__mult2__eq_H,axiom,
! [A: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).
% div_mult2_eq'
tff(fact_7112_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).
% less_imp_of_nat_less
tff(fact_7113_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N))) ) ).
% less_imp_of_nat_less
tff(fact_7114_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))) ) ).
% less_imp_of_nat_less
tff(fact_7115_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N))) ) ).
% less_imp_of_nat_less
tff(fact_7116_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% of_nat_less_imp_less
tff(fact_7117_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% of_nat_less_imp_less
tff(fact_7118_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% of_nat_less_imp_less
tff(fact_7119_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% of_nat_less_imp_less
tff(fact_7120_of__nat__mono,axiom,
! [I: nat,J: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,I)),aa_nat_real(semiri5074537144036343181t_real,J))) ) ).
% of_nat_mono
tff(fact_7121_of__nat__mono,axiom,
! [I: nat,J: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,I)),aa_nat_rat(semiri681578069525770553at_rat,J))) ) ).
% of_nat_mono
tff(fact_7122_of__nat__mono,axiom,
! [I: nat,J: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,I)),aa_nat_nat(semiri1316708129612266289at_nat,J))) ) ).
% of_nat_mono
tff(fact_7123_of__nat__mono,axiom,
! [I: nat,J: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,I)),aa_nat_int(semiri1314217659103216013at_int,J))) ) ).
% of_nat_mono
tff(fact_7124_AND__upper2_H,axiom,
! [Y: int,Z: int,X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y)),Z)) ) ) ).
% AND_upper2'
tff(fact_7125_AND__upper1_H,axiom,
! [Y: int,Z: int,Ya: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Y),Ya)),Z)) ) ) ).
% AND_upper1'
tff(fact_7126_AND__upper2,axiom,
! [Y: int,X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y)),Y)) ) ).
% AND_upper2
tff(fact_7127_AND__upper1,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y)),X)) ) ).
% AND_upper1
tff(fact_7128_AND__lower,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y))) ) ).
% AND_lower
tff(fact_7129_log__def,axiom,
! [A: real,X: real] : aa_real_real(aa_rea7340605660300375871l_real(log,A),X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,A)) ).
% log_def
tff(fact_7130_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_7131_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_7132_of__nat__dvd__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ).
% of_nat_dvd_iff
tff(fact_7133_of__nat__dvd__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ).
% of_nat_dvd_iff
tff(fact_7134_of__nat__dvd__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),N)) ) ).
% of_nat_dvd_iff
tff(fact_7135_take__bit__eq__mask,axiom,
! [N: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(N),A) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),aa_nat_int(bit_se2000444600071755411sk_int,N)) ).
% take_bit_eq_mask
tff(fact_7136_take__bit__eq__mask,axiom,
! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(bit_se2002935070580805687sk_nat,N)) ).
% take_bit_eq_mask
tff(fact_7137_int__ops_I3_J,axiom,
! [N: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,N)) = aa_num_int(numeral_numeral_int,N) ).
% int_ops(3)
tff(fact_7138_int__cases,axiom,
! [Z: int] :
( ! [N2: nat] : Z != aa_nat_int(semiri1314217659103216013at_int,N2)
=> ~ ! [N2: nat] : Z != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))) ) ).
% int_cases
tff(fact_7139_int__of__nat__induct,axiom,
! [P: fun_int_bool,Z: int] :
( ! [N2: nat] : pp(aa_int_bool(P,aa_nat_int(semiri1314217659103216013at_int,N2)))
=> ( ! [N2: nat] : pp(aa_int_bool(P,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2)))))
=> pp(aa_int_bool(P,Z)) ) ) ).
% int_of_nat_induct
tff(fact_7140_nat__int__comparison_I2_J,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) ) ).
% nat_int_comparison(2)
tff(fact_7141_nat__int__comparison_I3_J,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) ) ).
% nat_int_comparison(3)
tff(fact_7142_zle__int,axiom,
! [M: nat,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% zle_int
tff(fact_7143_of__nat__mod,axiom,
! [M: nat,N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = modulo364778990260209775nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ).
% of_nat_mod
tff(fact_7144_of__nat__mod,axiom,
! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).
% of_nat_mod
tff(fact_7145_of__nat__mod,axiom,
! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).
% of_nat_mod
tff(fact_7146_int__ops_I2_J,axiom,
aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).
% int_ops(2)
tff(fact_7147_int__ops_I5_J,axiom,
! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).
% int_ops(5)
tff(fact_7148_int__plus,axiom,
! [N: nat,M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(semiri1314217659103216013at_int,M)) ).
% int_plus
tff(fact_7149_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))),Z) ).
% zadd_int_left
tff(fact_7150_int__ops_I7_J,axiom,
! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).
% int_ops(7)
tff(fact_7151_not__int__zless__negative,axiom,
! [N: nat,M: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M)))) ).
% not_int_zless_negative
tff(fact_7152_zdiv__int,axiom,
! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).
% zdiv_int
tff(fact_7153_of__nat__max,axiom,
! [X: nat,Y: nat] : semiri4216267220026989637d_enat(aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,semiri4216267220026989637d_enat(X)),semiri4216267220026989637d_enat(Y)) ).
% of_nat_max
tff(fact_7154_of__nat__max,axiom,
! [X: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Y)) ).
% of_nat_max
tff(fact_7155_of__nat__max,axiom,
! [X: nat,Y: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(semiri5074537144036343181t_real,Y)) ).
% of_nat_max
tff(fact_7156_of__nat__max,axiom,
! [X: nat,Y: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(semiri1316708129612266289at_nat,Y)) ).
% of_nat_max
tff(fact_7157_of__nat__max,axiom,
! [X: nat,Y: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)) = ord_max_rat(aa_nat_rat(semiri681578069525770553at_rat,X),aa_nat_rat(semiri681578069525770553at_rat,Y)) ).
% of_nat_max
tff(fact_7158_log2__of__power__eq,axiom,
! [M: nat,N: nat] :
( ( M = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) )
=> ( aa_nat_real(semiri5074537144036343181t_real,N) = aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M)) ) ) ).
% log2_of_power_eq
tff(fact_7159_log__of__power__less,axiom,
! [M: nat,B: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).
% log_of_power_less
tff(fact_7160_zmod__int,axiom,
! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).
% zmod_int
tff(fact_7161_nat__less__as__int,axiom,
! [X4: nat,Xa: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),Xa))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ) ).
% nat_less_as_int
tff(fact_7162_nat__leq__as__int,axiom,
! [X4: nat,Xa: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X4),Xa))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ) ).
% nat_leq_as_int
tff(fact_7163_inc_Osimps_I1_J,axiom,
inc(one) = aa_num_num(bit0,one) ).
% inc.simps(1)
tff(fact_7164_inc_Osimps_I2_J,axiom,
! [X: num] : inc(aa_num_num(bit0,X)) = aa_num_num(bit1,X) ).
% inc.simps(2)
tff(fact_7165_inc_Osimps_I3_J,axiom,
! [X: num] : inc(aa_num_num(bit1,X)) = aa_num_num(bit0,inc(X)) ).
% inc.simps(3)
tff(fact_7166_log__of__power__le,axiom,
! [M: nat,B: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),N)))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).
% log_of_power_le
tff(fact_7167_ex__less__of__nat__mult,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),X))) ) ).
% ex_less_of_nat_mult
tff(fact_7168_ex__less__of__nat__mult,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
=> ? [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Y),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N2)),X))) ) ).
% ex_less_of_nat_mult
tff(fact_7169_add__One,axiom,
! [X: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,X),one) = inc(X) ).
% add_One
tff(fact_7170_AND__upper2_H_H,axiom,
! [Y: int,Z: int,X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y)),Z)) ) ) ).
% AND_upper2''
tff(fact_7171_AND__upper1_H_H,axiom,
! [Y: int,Z: int,Ya: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Y),Ya)),Z)) ) ) ).
% AND_upper1''
tff(fact_7172_and__less__eq,axiom,
! [L: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)),K)) ) ).
% and_less_eq
tff(fact_7173_inc__BitM__eq,axiom,
! [N: num] : inc(bitM(N)) = aa_num_num(bit0,N) ).
% inc_BitM_eq
tff(fact_7174_of__nat__diff,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ) ) ).
% of_nat_diff
tff(fact_7175_of__nat__diff,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ).
% of_nat_diff
tff(fact_7176_of__nat__diff,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ) ) ).
% of_nat_diff
tff(fact_7177_of__nat__diff,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_complex_complex(minus_minus_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri8010041392384452111omplex,N)) ) ) ).
% of_nat_diff
tff(fact_7178_of__nat__diff,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,M),aa_nat_rat(semiri681578069525770553at_rat,N)) ) ) ).
% of_nat_diff
tff(fact_7179_BitM__inc__eq,axiom,
! [N: num] : bitM(inc(N)) = aa_num_num(bit1,N) ).
% BitM_inc_eq
tff(fact_7180_exp__of__nat2__mult,axiom,
! [X: real,N: nat] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,X)),N) ).
% exp_of_nat2_mult
tff(fact_7181_exp__of__nat2__mult,axiom,
! [X: complex,N: nat] : aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(X),aa_nat_complex(semiri8010041392384452111omplex,N))) = aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,X)),N) ).
% exp_of_nat2_mult
tff(fact_7182_exp__of__nat__mult,axiom,
! [N: nat,X: real] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),X)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,X)),N) ).
% exp_of_nat_mult
tff(fact_7183_exp__of__nat__mult,axiom,
! [N: nat,X: complex] : aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),X)) = aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,X)),N) ).
% exp_of_nat_mult
tff(fact_7184_log__ln,axiom,
! [X: real] : aa_real_real(ln_ln_real,X) = aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(exp_real,one_one_real)),X) ).
% log_ln
tff(fact_7185_reals__Archimedean3,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ! [Y3: real] :
? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),X))) ) ).
% reals_Archimedean3
tff(fact_7186_int__cases4,axiom,
! [M: int] :
( ! [N2: nat] : M != aa_nat_int(semiri1314217659103216013at_int,N2)
=> ~ ! [N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
=> ( M != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) ) ) ) ).
% int_cases4
tff(fact_7187_real__of__nat__div4,axiom,
! [N: nat,X: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),X))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,X)))) ).
% real_of_nat_div4
tff(fact_7188_int__ops_I4_J,axiom,
! [A: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,A)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),one_one_int) ).
% int_ops(4)
tff(fact_7189_int__Suc,axiom,
! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int) ).
% int_Suc
tff(fact_7190_zless__iff__Suc__zadd,axiom,
! [W: int,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z))
<=> ? [N5: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N5))) ) ).
% zless_iff_Suc_zadd
tff(fact_7191_real__of__nat__div,axiom,
! [D: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,D),N))
=> ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),D)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,D)) ) ) ).
% real_of_nat_div
tff(fact_7192_mult__inc,axiom,
! [X: num,Y: num] : aa_num_num(aa_num_fun_num_num(times_times_num,X),inc(Y)) = aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(times_times_num,X),Y)),X) ).
% mult_inc
tff(fact_7193_less__log2__of__power,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),M))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M)))) ) ).
% less_log2_of_power
tff(fact_7194_le__log2__of__power,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),M))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M)))) ) ).
% le_log2_of_power
tff(fact_7195_even__and__iff,axiom,
! [A: code_integer,B: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se3949692690581998587nteger(A,B)))
<=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))
| pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)) ) ) ).
% even_and_iff
tff(fact_7196_even__and__iff,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),B)))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))
| pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) ) ) ).
% even_and_iff
tff(fact_7197_even__and__iff,axiom,
! [A: nat,B: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B)))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))
| pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) ) ) ).
% even_and_iff
tff(fact_7198_log2__of__power__less,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ).
% log2_of_power_less
tff(fact_7199_log__base__change,axiom,
! [A: real,B: real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ( ( A != one_one_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(log,B),X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A),B)) ) ) ) ).
% log_base_change
tff(fact_7200_even__and__iff__int,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))
| pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L)) ) ) ).
% even_and_iff_int
tff(fact_7201_numeral__inc,axiom,
! [X: num] : numera6690914467698888265omplex(inc(X)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(X)),one_one_complex) ).
% numeral_inc
tff(fact_7202_numeral__inc,axiom,
! [X: num] : numeral_numeral_real(inc(X)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(X)),one_one_real) ).
% numeral_inc
tff(fact_7203_numeral__inc,axiom,
! [X: num] : numeral_numeral_rat(inc(X)) = plus_plus_rat(numeral_numeral_rat(X),one_one_rat) ).
% numeral_inc
tff(fact_7204_numeral__inc,axiom,
! [X: num] : aa_num_nat(numeral_numeral_nat,inc(X)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,X)),one_one_nat) ).
% numeral_inc
tff(fact_7205_numeral__inc,axiom,
! [X: num] : aa_num_int(numeral_numeral_int,inc(X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,X)),one_one_int) ).
% numeral_inc
tff(fact_7206_mod__mult2__eq_H,axiom,
! [A: code_integer,M: nat,N: nat] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N))) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),modulo364778990260209775nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N))),modulo364778990260209775nteger(A,aa_nat_Code_integer(semiri4939895301339042750nteger,M))) ).
% mod_mult2_eq'
tff(fact_7207_mod__mult2__eq_H,axiom,
! [A: int,M: nat,N: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_int(semiri1314217659103216013at_int,N)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(semiri1314217659103216013at_int,M))) ).
% mod_mult2_eq'
tff(fact_7208_mod__mult2__eq_H,axiom,
! [A: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))) = 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(semiri1316708129612266289at_nat,M)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(semiri1316708129612266289at_nat,N)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,M))) ).
% mod_mult2_eq'
tff(fact_7209_field__char__0__class_Oof__nat__div,axiom,
! [M: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))),aa_nat_real(semiri5074537144036343181t_real,N)) ).
% field_char_0_class.of_nat_div
tff(fact_7210_field__char__0__class_Oof__nat__div,axiom,
! [M: nat,N: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))),aa_nat_complex(semiri8010041392384452111omplex,N)) ).
% field_char_0_class.of_nat_div
tff(fact_7211_field__char__0__class_Oof__nat__div,axiom,
! [M: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = divide_divide_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,M),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))),aa_nat_rat(semiri681578069525770553at_rat,N)) ).
% field_char_0_class.of_nat_div
tff(fact_7212_pos__int__cases,axiom,
! [K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
=> ~ ! [N2: nat] :
( ( K = aa_nat_int(semiri1314217659103216013at_int,N2) )
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ).
% pos_int_cases
tff(fact_7213_zero__less__imp__eq__int,axiom,
! [K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K))
=> ? [N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
& ( K = aa_nat_int(semiri1314217659103216013at_int,N2) ) ) ) ).
% zero_less_imp_eq_int
tff(fact_7214_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N2: nat] :
( ( K = aa_nat_int(semiri1314217659103216013at_int,N2) )
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) )
=> ~ ! [N2: nat] :
( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) )
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ) ).
% int_cases3
tff(fact_7215_nat__less__real__le,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,M))) ) ).
% nat_less_real_le
tff(fact_7216_nat__le__real__less,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),one_one_real))) ) ).
% nat_le_real_less
tff(fact_7217_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),I)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),J))) ) ) ).
% zmult_zless_mono2_lemma
tff(fact_7218_not__zle__0__negative,axiom,
! [N: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))))) ).
% not_zle_0_negative
tff(fact_7219_negD,axiom,
! [X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),zero_zero_int))
=> ? [N2: nat] : X = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))) ) ).
% negD
tff(fact_7220_negative__zless__0,axiom,
! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)))),zero_zero_int)) ).
% negative_zless_0
tff(fact_7221_log2__of__power__le,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ).
% log2_of_power_le
tff(fact_7222_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)))
=> ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = zero_zero_int ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)))
=> ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ) ) ).
% int_ops(6)
tff(fact_7223_real__of__nat__div__aux,axiom,
! [X: nat,D: nat] : aa_real_real(aa_rea7340605660300375871l_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(aa_nat_fun_nat_nat(divide_divide_nat,X),D))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),D))),aa_nat_real(semiri5074537144036343181t_real,D))) ).
% real_of_nat_div_aux
tff(fact_7224_one__and__eq,axiom,
! [A: code_integer] : bit_se3949692690581998587nteger(one_one_Code_integer,A) = modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).
% one_and_eq
tff(fact_7225_one__and__eq,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).
% one_and_eq
tff(fact_7226_one__and__eq,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% one_and_eq
tff(fact_7227_and__one__eq,axiom,
! [A: code_integer] : bit_se3949692690581998587nteger(A,one_one_Code_integer) = modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).
% and_one_eq
tff(fact_7228_and__one__eq,axiom,
! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),one_one_int) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).
% and_one_eq
tff(fact_7229_and__one__eq,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% and_one_eq
tff(fact_7230_log__mult,axiom,
! [A: real,X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ( ( A != one_one_real )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
=> ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)) ) ) ) ) ) ).
% log_mult
tff(fact_7231_log__divide,axiom,
! [A: real,X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ( ( A != one_one_real )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
=> ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)) ) ) ) ) ) ).
% log_divide
tff(fact_7232_nat__approx__posE,axiom,
! [E: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
=> ~ ! [N2: nat] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N2)))),E)) ) ).
% nat_approx_posE
tff(fact_7233_nat__approx__posE,axiom,
! [E: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),E))
=> ~ ! [N2: nat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,divide_divide_rat(one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N2)))),E)) ) ).
% nat_approx_posE
tff(fact_7234_of__nat__less__two__power,axiom,
! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).
% of_nat_less_two_power
tff(fact_7235_of__nat__less__two__power,axiom,
! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),N))) ).
% of_nat_less_two_power
tff(fact_7236_of__nat__less__two__power,axiom,
! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_nat_rat(power_power_rat(numeral_numeral_rat(aa_num_num(bit0,one))),N))) ).
% of_nat_less_two_power
tff(fact_7237_inverse__of__nat__le,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( ( N != zero_zero_nat )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,M))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,N)))) ) ) ).
% inverse_of_nat_le
tff(fact_7238_inverse__of__nat__le,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( ( N != zero_zero_nat )
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,divide_divide_rat(one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,M))),divide_divide_rat(one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,N)))) ) ) ).
% inverse_of_nat_le
tff(fact_7239_member__bound__size__univ,axiom,
! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),N) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_T_m_e_m_b_e_r(T),X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),U)))))) ) ) ).
% member_bound_size_univ
tff(fact_7240_exp__divide__power__eq,axiom,
! [N: nat,X: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_nat_real(semiri5074537144036343181t_real,N)))),N) = aa_real_real(exp_real,X) ) ) ).
% exp_divide_power_eq
tff(fact_7241_exp__divide__power__eq,axiom,
! [N: nat,X: complex] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,aa_complex_complex(divide1717551699836669952omplex(X),aa_nat_complex(semiri8010041392384452111omplex,N)))),N) = aa_complex_complex(exp_complex,X) ) ) ).
% exp_divide_power_eq
tff(fact_7242_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),C))
=> ( ! [M3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M3))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M3)),X)),C)) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
tff(fact_7243_neg__int__cases,axiom,
! [K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
=> ~ ! [N2: nat] :
( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) )
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2)) ) ) ).
% neg_int_cases
tff(fact_7244_zdiff__int__split,axiom,
! [P: fun_int_bool,X: nat,Y: nat] :
( pp(aa_int_bool(P,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Y))))
<=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Y)))) )
& ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
=> pp(aa_int_bool(P,zero_zero_int)) ) ) ) ).
% zdiff_int_split
tff(fact_7245_real__of__nat__div2,axiom,
! [N: nat,X: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,X))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),X))))) ).
% real_of_nat_div2
tff(fact_7246_real__of__nat__div3,axiom,
! [N: nat,X: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,X))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),X)))),one_one_real)) ).
% real_of_nat_div3
tff(fact_7247_ln__realpow,axiom,
! [X: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(ln_ln_real,X)) ) ) ).
% ln_realpow
tff(fact_7248_log__eq__div__ln__mult__log,axiom,
! [A: real,B: real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ( ( A != one_one_real )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
=> ( ( B != one_one_real )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( aa_real_real(aa_rea7340605660300375871l_real(log,A),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B)),aa_real_real(ln_ln_real,A))),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) ) ) ) ) ) ) ).
% log_eq_div_ln_mult_log
tff(fact_7249_linear__plus__1__le__power,axiom,
! [X: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),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)),N))) ) ).
% linear_plus_1_le_power
tff(fact_7250_Bernoulli__inequality,axiom,
! [X: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),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)),N))) ) ).
% Bernoulli_inequality
tff(fact_7251_and__int__rec,axiom,
! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).
% and_int_rec
tff(fact_7252_double__arith__series,axiom,
! [A: int,D: int,N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_hq(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),D))) ).
% double_arith_series
tff(fact_7253_double__arith__series,axiom,
! [A: complex,D: complex,N: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_hr(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)),aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),A)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),D))) ).
% double_arith_series
tff(fact_7254_double__arith__series,axiom,
! [A: rat,D: rat,N: nat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_hs(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_rat_rat(times_times_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,N),one_one_rat)),plus_plus_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),A),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),D))) ).
% double_arith_series
tff(fact_7255_double__arith__series,axiom,
! [A: nat,D: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ht(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),D))) ).
% double_arith_series
tff(fact_7256_double__arith__series,axiom,
! [A: real,D: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_hu(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),D))) ).
% double_arith_series
tff(fact_7257_double__gauss__sum,axiom,
! [N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)) ).
% double_gauss_sum
tff(fact_7258_double__gauss__sum,axiom,
! [N: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(semiri8010041392384452111omplex,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)) ).
% double_gauss_sum
tff(fact_7259_double__gauss__sum,axiom,
! [N: nat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,N),one_one_rat)) ).
% double_gauss_sum
tff(fact_7260_double__gauss__sum,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat)) ).
% double_gauss_sum
tff(fact_7261_double__gauss__sum,axiom,
! [N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),groups6591440286371151544t_real(semiri5074537144036343181t_real,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)) ).
% double_gauss_sum
tff(fact_7262_arith__series,axiom,
! [A: int,D: int,N: nat] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_hq(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),D)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).
% arith_series
tff(fact_7263_arith__series,axiom,
! [A: nat,D: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ht(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% arith_series
tff(fact_7264_gauss__sum,axiom,
! [N: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).
% gauss_sum
tff(fact_7265_gauss__sum,axiom,
! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% gauss_sum
tff(fact_7266_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int)) ).
% double_gauss_sum_from_Suc_0
tff(fact_7267_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),groups2073611262835488442omplex(semiri8010041392384452111omplex,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)) ).
% double_gauss_sum_from_Suc_0
tff(fact_7268_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,N),one_one_rat)) ).
% double_gauss_sum_from_Suc_0
tff(fact_7269_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),one_one_nat)) ).
% double_gauss_sum_from_Suc_0
tff(fact_7270_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),groups6591440286371151544t_real(semiri5074537144036343181t_real,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)) ).
% double_gauss_sum_from_Suc_0
tff(fact_7271_sum__gp__offset,axiom,
! [X: complex,M: nat,N: nat] :
( ( ( X = one_one_complex )
=> ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex) ) )
& ( ( X != one_one_complex )
=> ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(X),M)),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N))))),aa_complex_complex(minus_minus_complex(one_one_complex),X)) ) ) ) ).
% sum_gp_offset
tff(fact_7272_sum__gp__offset,axiom,
! [X: rat,M: nat,N: nat] :
( ( ( X = one_one_rat )
=> ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,N),one_one_rat) ) )
& ( ( X != one_one_rat )
=> ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),M)),minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N)))),minus_minus_rat(one_one_rat,X)) ) ) ) ).
% sum_gp_offset
tff(fact_7273_sum__gp__offset,axiom,
! [X: real,M: nat,N: nat] :
( ( ( X = one_one_real )
=> ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real) ) )
& ( ( X != one_one_real )
=> ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),M)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,N))))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)) ) ) ) ).
% sum_gp_offset
tff(fact_7274_Bernoulli__inequality__even,axiom,
! [N: nat,X: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),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)),N))) ) ).
% Bernoulli_inequality_even
tff(fact_7275_exp__ge__one__plus__x__over__n__power__n,axiom,
! [N: nat,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),X))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(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(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_nat_real(semiri5074537144036343181t_real,N)))),N)),aa_real_real(exp_real,X))) ) ) ).
% exp_ge_one_plus_x_over_n_power_n
tff(fact_7276_exp__ge__one__minus__x__over__n__power__n,axiom,
! [X: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,N)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_nat_real(semiri5074537144036343181t_real,N)))),N)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X)))) ) ) ).
% exp_ge_one_minus_x_over_n_power_n
tff(fact_7277_log__base__10__eq2,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,X)) ) ) ).
% log_base_10_eq2
tff(fact_7278_of__nat__code__if,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_int(semiri1314217659103216013at_int,N) = zero_zero_int ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_int(semiri1314217659103216013at_int,N) = aa_Pro4240683299535347697at_int(produc6840382203811409530at_int(aTP_Lamm_hv),divmod_nat(N,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).
% of_nat_code_if
tff(fact_7279_of__nat__code__if,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_real(semiri5074537144036343181t_real,N) = zero_zero_real ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_real(semiri5074537144036343181t_real,N) = aa_Pro6168199777849092593t_real(produc1703576794950452218t_real(aTP_Lamm_hw),divmod_nat(N,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).
% of_nat_code_if
tff(fact_7280_of__nat__code__if,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_nat(semiri1316708129612266289at_nat,N) = zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_nat(semiri1316708129612266289at_nat,N) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aTP_Lamm_hx),divmod_nat(N,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).
% of_nat_code_if
tff(fact_7281_of__nat__code__if,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_complex(semiri8010041392384452111omplex,N) = zero_zero_complex ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_complex(semiri8010041392384452111omplex,N) = aa_Pro82899191565270131omplex(produc1917071388513777916omplex(aTP_Lamm_hy),divmod_nat(N,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).
% of_nat_code_if
tff(fact_7282_of__nat__code__if,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_rat(semiri681578069525770553at_rat,N) = zero_zero_rat ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_rat(semiri681578069525770553at_rat,N) = aa_Pro3608043709957902237at_rat(produc6207742614233964070at_rat(aTP_Lamm_hz),divmod_nat(N,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).
% of_nat_code_if
tff(fact_7283_monoseq__arctan__series,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
=> topolo6980174941875973593q_real(aa_real_fun_nat_real(aTP_Lamm_ia,X)) ) ).
% monoseq_arctan_series
tff(fact_7284_lemma__termdiff3,axiom,
! [H: real,Z: real,K5: real,N: nat] :
( ( H != zero_zero_real )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),K5))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H))),K5))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H)),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),N))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(real_V7735802525324610683m_real,H)))) ) ) ) ).
% lemma_termdiff3
tff(fact_7285_lemma__termdiff3,axiom,
! [H: complex,Z: complex,K5: real,N: nat] :
( ( H != zero_zero_complex )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),K5))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Z),H))),K5))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Z),H)),N)),aa_nat_complex(power_power_complex(Z),N))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_nat_complex(power_power_complex(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_complex_real(real_V1022390504157884413omplex,H)))) ) ) ) ).
% lemma_termdiff3
tff(fact_7286_log__ceil__idem,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
=> ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),X)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X)))) ) ) ).
% log_ceil_idem
tff(fact_7287_ln__series,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),numeral_numeral_real(aa_num_num(bit0,one))))
=> ( aa_real_real(ln_ln_real,X) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_ib,X)) ) ) ) ).
% ln_series
tff(fact_7288_lemma__termdiff2,axiom,
! [H: complex,Z: complex,N: nat] :
( ( H != zero_zero_complex )
=> ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Z),H)),N)),aa_nat_complex(power_power_complex(Z),N))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_nat_complex(power_power_complex(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) = aa_complex_complex(times_times_complex(H),groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_id(H),Z),N),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).
% lemma_termdiff2
tff(fact_7289_lemma__termdiff2,axiom,
! [H: rat,Z: rat,N: nat] :
( ( H != zero_zero_rat )
=> ( minus_minus_rat(divide_divide_rat(minus_minus_rat(aa_nat_rat(power_power_rat(plus_plus_rat(Z,H)),N),aa_nat_rat(power_power_rat(Z),N)),H),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),aa_nat_rat(power_power_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) = aa_rat_rat(times_times_rat(H),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_if(H),Z),N),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).
% lemma_termdiff2
tff(fact_7290_lemma__termdiff2,axiom,
! [H: real,Z: real,N: nat] :
( ( H != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H)),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),N))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,H),groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ih(H),Z),N),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).
% lemma_termdiff2
tff(fact_7291_lessThan__iff,axiom,
! [I: rat,K: rat] :
( pp(aa_set_rat_bool(member_rat(I),set_ord_lessThan_rat(K)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,I),K)) ) ).
% lessThan_iff
tff(fact_7292_lessThan__iff,axiom,
! [I: num,K: num] :
( pp(aa_set_num_bool(member_num(I),set_ord_lessThan_num(K)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,I),K)) ) ).
% lessThan_iff
tff(fact_7293_lessThan__iff,axiom,
! [I: int,K: int] :
( pp(aa_set_int_bool(member_int(I),set_ord_lessThan_int(K)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),K)) ) ).
% lessThan_iff
tff(fact_7294_lessThan__iff,axiom,
! [I: nat,K: nat] :
( pp(aa_set_nat_bool(member_nat(I),aa_nat_set_nat(set_ord_lessThan_nat,K)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K)) ) ).
% lessThan_iff
tff(fact_7295_lessThan__iff,axiom,
! [I: real,K: real] :
( pp(aa_set_real_bool(member_real(I),set_or5984915006950818249n_real(K)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,I),K)) ) ).
% lessThan_iff
tff(fact_7296_of__int__ceiling__cancel,axiom,
! [X: rat] :
( ( aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(X)) = X )
<=> ? [N5: int] : X = aa_int_rat(ring_1_of_int_rat,N5) ) ).
% of_int_ceiling_cancel
tff(fact_7297_of__int__ceiling__cancel,axiom,
! [X: real] :
( ( aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X)) = X )
<=> ? [N5: int] : X = aa_int_real(ring_1_of_int_real,N5) ) ).
% of_int_ceiling_cancel
tff(fact_7298_heigt__uplog__rel,axiom,
! [T: vEBT_VEBT,N: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),N))
=> ( aa_nat_int(semiri1314217659103216013at_int,aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ).
% heigt_uplog_rel
tff(fact_7299_lessThan__subset__iff,axiom,
! [X: rat,Y: rat] :
( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_ord_lessThan_rat(X)),set_ord_lessThan_rat(Y)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).
% lessThan_subset_iff
tff(fact_7300_lessThan__subset__iff,axiom,
! [X: num,Y: num] :
( pp(aa_set_num_bool(ord_less_eq_set_num(set_ord_lessThan_num(X)),set_ord_lessThan_num(Y)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) ) ).
% lessThan_subset_iff
tff(fact_7301_lessThan__subset__iff,axiom,
! [X: int,Y: int] :
( pp(aa_set_int_bool(ord_less_eq_set_int(set_ord_lessThan_int(X)),set_ord_lessThan_int(Y)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ).
% lessThan_subset_iff
tff(fact_7302_lessThan__subset__iff,axiom,
! [X: nat,Y: nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,X)),aa_nat_set_nat(set_ord_lessThan_nat,Y)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ).
% lessThan_subset_iff
tff(fact_7303_lessThan__subset__iff,axiom,
! [X: real,Y: real] :
( pp(aa_set_real_bool(ord_less_eq_set_real(set_or5984915006950818249n_real(X)),set_or5984915006950818249n_real(Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).
% lessThan_subset_iff
tff(fact_7304_ceiling__numeral,axiom,
! [V: num] : aa_real_int(archim7802044766580827645g_real,numeral_numeral_real(V)) = aa_num_int(numeral_numeral_int,V) ).
% ceiling_numeral
tff(fact_7305_ceiling__numeral,axiom,
! [V: num] : archim2889992004027027881ng_rat(numeral_numeral_rat(V)) = aa_num_int(numeral_numeral_int,V) ).
% ceiling_numeral
tff(fact_7306_ceiling__one,axiom,
archim2889992004027027881ng_rat(one_one_rat) = one_one_int ).
% ceiling_one
tff(fact_7307_ceiling__one,axiom,
aa_real_int(archim7802044766580827645g_real,one_one_real) = one_one_int ).
% ceiling_one
tff(fact_7308_sum_OlessThan__Suc,axiom,
! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,N)),aa_nat_rat(G,N)) ).
% sum.lessThan_Suc
tff(fact_7309_sum_OlessThan__Suc,axiom,
! [G: fun_nat_int,N: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_int(G,N)) ).
% sum.lessThan_Suc
tff(fact_7310_sum_OlessThan__Suc,axiom,
! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_nat(G,N)) ).
% sum.lessThan_Suc
tff(fact_7311_sum_OlessThan__Suc,axiom,
! [G: fun_nat_real,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_real(G,N)) ).
% sum.lessThan_Suc
tff(fact_7312_ceiling__add__of__int,axiom,
! [X: rat,Z: int] : archim2889992004027027881ng_rat(plus_plus_rat(X,aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,archim2889992004027027881ng_rat(X)),Z) ).
% ceiling_add_of_int
tff(fact_7313_ceiling__add__of__int,axiom,
! [X: real,Z: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_int_real(ring_1_of_int_real,Z))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X)),Z) ).
% ceiling_add_of_int
tff(fact_7314_ceiling__diff__of__int,axiom,
! [X: rat,Z: int] : archim2889992004027027881ng_rat(minus_minus_rat(X,aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,archim2889992004027027881ng_rat(X)),Z) ).
% ceiling_diff_of_int
tff(fact_7315_ceiling__diff__of__int,axiom,
! [X: real,Z: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_int_real(ring_1_of_int_real,Z))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,X)),Z) ).
% ceiling_diff_of_int
tff(fact_7316_ceiling__le__zero,axiom,
! [X: real] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),zero_zero_int))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).
% ceiling_le_zero
tff(fact_7317_ceiling__le__zero,axiom,
! [X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,archim2889992004027027881ng_rat(X)),zero_zero_int))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat)) ) ).
% ceiling_le_zero
tff(fact_7318_zero__less__ceiling,axiom,
! [X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),archim2889992004027027881ng_rat(X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X)) ) ).
% zero_less_ceiling
tff(fact_7319_zero__less__ceiling,axiom,
! [X: real] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).
% zero_less_ceiling
tff(fact_7320_ceiling__le__numeral,axiom,
! [X: real,V: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),aa_num_int(numeral_numeral_int,V)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),numeral_numeral_real(V))) ) ).
% ceiling_le_numeral
tff(fact_7321_ceiling__le__numeral,axiom,
! [X: rat,V: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,archim2889992004027027881ng_rat(X)),aa_num_int(numeral_numeral_int,V)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),numeral_numeral_rat(V))) ) ).
% ceiling_le_numeral
tff(fact_7322_ceiling__less__one,axiom,
! [X: real] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),one_one_int))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).
% ceiling_less_one
tff(fact_7323_ceiling__less__one,axiom,
! [X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,archim2889992004027027881ng_rat(X)),one_one_int))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),zero_zero_rat)) ) ).
% ceiling_less_one
tff(fact_7324_one__le__ceiling,axiom,
! [X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),archim2889992004027027881ng_rat(X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X)) ) ).
% one_le_ceiling
tff(fact_7325_one__le__ceiling,axiom,
! [X: real] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_real_int(archim7802044766580827645g_real,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).
% one_le_ceiling
tff(fact_7326_numeral__less__ceiling,axiom,
! [V: num,X: real] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,numeral_numeral_real(V)),X)) ) ).
% numeral_less_ceiling
tff(fact_7327_numeral__less__ceiling,axiom,
! [V: num,X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,V)),archim2889992004027027881ng_rat(X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,numeral_numeral_rat(V)),X)) ) ).
% numeral_less_ceiling
tff(fact_7328_ceiling__le__one,axiom,
! [X: real] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),one_one_int))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ).
% ceiling_le_one
tff(fact_7329_ceiling__le__one,axiom,
! [X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,archim2889992004027027881ng_rat(X)),one_one_int))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),one_one_rat)) ) ).
% ceiling_le_one
tff(fact_7330_one__less__ceiling,axiom,
! [X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),archim2889992004027027881ng_rat(X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),X)) ) ).
% one_less_ceiling
tff(fact_7331_one__less__ceiling,axiom,
! [X: real] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_real_int(archim7802044766580827645g_real,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X)) ) ).
% one_less_ceiling
tff(fact_7332_ceiling__add__numeral,axiom,
! [X: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),numeral_numeral_real(V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X)),aa_num_int(numeral_numeral_int,V)) ).
% ceiling_add_numeral
tff(fact_7333_ceiling__add__numeral,axiom,
! [X: rat,V: num] : archim2889992004027027881ng_rat(plus_plus_rat(X,numeral_numeral_rat(V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,archim2889992004027027881ng_rat(X)),aa_num_int(numeral_numeral_int,V)) ).
% ceiling_add_numeral
tff(fact_7334_ceiling__neg__numeral,axiom,
! [V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).
% ceiling_neg_numeral
tff(fact_7335_ceiling__neg__numeral,axiom,
! [V: num] : archim2889992004027027881ng_rat(uminus_uminus_rat(numeral_numeral_rat(V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).
% ceiling_neg_numeral
tff(fact_7336_ceiling__add__one,axiom,
! [X: rat] : archim2889992004027027881ng_rat(plus_plus_rat(X,one_one_rat)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,archim2889992004027027881ng_rat(X)),one_one_int) ).
% ceiling_add_one
tff(fact_7337_ceiling__add__one,axiom,
! [X: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X)),one_one_int) ).
% ceiling_add_one
tff(fact_7338_ceiling__diff__numeral,axiom,
! [X: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),numeral_numeral_real(V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,X)),aa_num_int(numeral_numeral_int,V)) ).
% ceiling_diff_numeral
tff(fact_7339_ceiling__diff__numeral,axiom,
! [X: rat,V: num] : archim2889992004027027881ng_rat(minus_minus_rat(X,numeral_numeral_rat(V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,archim2889992004027027881ng_rat(X)),aa_num_int(numeral_numeral_int,V)) ).
% ceiling_diff_numeral
tff(fact_7340_ceiling__diff__one,axiom,
! [X: rat] : archim2889992004027027881ng_rat(minus_minus_rat(X,one_one_rat)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,archim2889992004027027881ng_rat(X)),one_one_int) ).
% ceiling_diff_one
tff(fact_7341_ceiling__diff__one,axiom,
! [X: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,X)),one_one_int) ).
% ceiling_diff_one
tff(fact_7342_ceiling__numeral__power,axiom,
! [X: num,N: nat] : aa_real_int(archim7802044766580827645g_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(X)),N)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ).
% ceiling_numeral_power
tff(fact_7343_ceiling__numeral__power,axiom,
! [X: num,N: nat] : archim2889992004027027881ng_rat(aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),N)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ).
% ceiling_numeral_power
tff(fact_7344_and__nat__numerals_I3_J,axiom,
! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).
% and_nat_numerals(3)
tff(fact_7345_and__nat__numerals_I1_J,axiom,
! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = zero_zero_nat ).
% and_nat_numerals(1)
tff(fact_7346_powser__zero,axiom,
! [F: fun_nat_complex] : suminf_complex(aTP_Lamm_ge(F)) = aa_nat_complex(F,zero_zero_nat) ).
% powser_zero
tff(fact_7347_powser__zero,axiom,
! [F: fun_nat_real] : suminf_real(aTP_Lamm_gg(F)) = aa_nat_real(F,zero_zero_nat) ).
% powser_zero
tff(fact_7348_ceiling__less__zero,axiom,
! [X: real] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),zero_zero_int))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(uminus_uminus_real,one_one_real))) ) ).
% ceiling_less_zero
tff(fact_7349_ceiling__less__zero,axiom,
! [X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,archim2889992004027027881ng_rat(X)),zero_zero_int))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),uminus_uminus_rat(one_one_rat))) ) ).
% ceiling_less_zero
tff(fact_7350_zero__le__ceiling,axiom,
! [X: real] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)) ) ).
% zero_le_ceiling
tff(fact_7351_zero__le__ceiling,axiom,
! [X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),archim2889992004027027881ng_rat(X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(one_one_rat)),X)) ) ).
% zero_le_ceiling
tff(fact_7352_and__nat__numerals_I4_J,axiom,
! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).
% and_nat_numerals(4)
tff(fact_7353_and__nat__numerals_I2_J,axiom,
! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = one_one_nat ).
% and_nat_numerals(2)
tff(fact_7354_ceiling__divide__eq__div__numeral,axiom,
! [A: num,B: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(A)),numeral_numeral_real(B))) = 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,A))),aa_num_int(numeral_numeral_int,B))) ).
% ceiling_divide_eq_div_numeral
tff(fact_7355_ceiling__less__numeral,axiom,
! [X: real,V: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),aa_num_int(numeral_numeral_int,V)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,numeral_numeral_real(V)),one_one_real))) ) ).
% ceiling_less_numeral
tff(fact_7356_ceiling__less__numeral,axiom,
! [X: rat,V: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,archim2889992004027027881ng_rat(X)),aa_num_int(numeral_numeral_int,V)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),minus_minus_rat(numeral_numeral_rat(V),one_one_rat))) ) ).
% ceiling_less_numeral
tff(fact_7357_numeral__le__ceiling,axiom,
! [V: num,X: real] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,numeral_numeral_real(V)),one_one_real)),X)) ) ).
% numeral_le_ceiling
tff(fact_7358_numeral__le__ceiling,axiom,
! [V: num,X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),archim2889992004027027881ng_rat(X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(numeral_numeral_rat(V),one_one_rat)),X)) ) ).
% numeral_le_ceiling
tff(fact_7359_ceiling__le__neg__numeral,axiom,
! [X: real,V: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(uminus_uminus_real,numeral_numeral_real(V)))) ) ).
% ceiling_le_neg_numeral
tff(fact_7360_ceiling__le__neg__numeral,axiom,
! [X: rat,V: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,archim2889992004027027881ng_rat(X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),uminus_uminus_rat(numeral_numeral_rat(V)))) ) ).
% ceiling_le_neg_numeral
tff(fact_7361_neg__numeral__less__ceiling,axiom,
! [V: num,X: real] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim7802044766580827645g_real,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),X)) ) ).
% neg_numeral_less_ceiling
tff(fact_7362_neg__numeral__less__ceiling,axiom,
! [V: num,X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),archim2889992004027027881ng_rat(X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(V))),X)) ) ).
% neg_numeral_less_ceiling
tff(fact_7363_Suc__0__and__eq,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% Suc_0_and_eq
tff(fact_7364_and__Suc__0__eq,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% and_Suc_0_eq
tff(fact_7365_ceiling__minus__divide__eq__div__numeral,axiom,
! [A: num,B: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(A)),numeral_numeral_real(B)))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,A)),aa_num_int(numeral_numeral_int,B))) ).
% ceiling_minus_divide_eq_div_numeral
tff(fact_7366_ceiling__less__neg__numeral,axiom,
! [X: real,V: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),one_one_real))) ) ).
% ceiling_less_neg_numeral
tff(fact_7367_ceiling__less__neg__numeral,axiom,
! [X: rat,V: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,archim2889992004027027881ng_rat(X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),minus_minus_rat(uminus_uminus_rat(numeral_numeral_rat(V)),one_one_rat))) ) ).
% ceiling_less_neg_numeral
tff(fact_7368_neg__numeral__le__ceiling,axiom,
! [V: num,X: real] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim7802044766580827645g_real,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),one_one_real)),X)) ) ).
% neg_numeral_le_ceiling
tff(fact_7369_neg__numeral__le__ceiling,axiom,
! [V: num,X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),archim2889992004027027881ng_rat(X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(uminus_uminus_rat(numeral_numeral_rat(V)),one_one_rat)),X)) ) ).
% neg_numeral_le_ceiling
tff(fact_7370_lessThan__def,axiom,
! [U: rat] : set_ord_lessThan_rat(U) = collect_rat(aa_rat_fun_rat_bool(aTP_Lamm_ii,U)) ).
% lessThan_def
tff(fact_7371_lessThan__def,axiom,
! [U: num] : set_ord_lessThan_num(U) = collect_num(aa_num_fun_num_bool(aTP_Lamm_ij,U)) ).
% lessThan_def
tff(fact_7372_lessThan__def,axiom,
! [U: int] : set_ord_lessThan_int(U) = collect_int(aa_int_fun_int_bool(aTP_Lamm_ik,U)) ).
% lessThan_def
tff(fact_7373_lessThan__def,axiom,
! [U: nat] : aa_nat_set_nat(set_ord_lessThan_nat,U) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bh,U)) ).
% lessThan_def
tff(fact_7374_lessThan__def,axiom,
! [U: real] : set_or5984915006950818249n_real(U) = collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_il,U)) ).
% lessThan_def
tff(fact_7375_ceiling__mono,axiom,
! [Y: real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Y)),aa_real_int(archim7802044766580827645g_real,X))) ) ).
% ceiling_mono
tff(fact_7376_ceiling__mono,axiom,
! [Y: rat,X: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Y),X))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,archim2889992004027027881ng_rat(Y)),archim2889992004027027881ng_rat(X))) ) ).
% ceiling_mono
tff(fact_7377_le__of__int__ceiling,axiom,
! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X)))) ).
% le_of_int_ceiling
tff(fact_7378_le__of__int__ceiling,axiom,
! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(X)))) ).
% le_of_int_ceiling
tff(fact_7379_ceiling__less__cancel,axiom,
! [X: rat,Y: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,archim2889992004027027881ng_rat(X)),archim2889992004027027881ng_rat(Y)))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ).
% ceiling_less_cancel
tff(fact_7380_ceiling__less__cancel,axiom,
! [X: real,Y: real] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),aa_real_int(archim7802044766580827645g_real,Y)))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).
% ceiling_less_cancel
tff(fact_7381_lessThan__strict__subset__iff,axiom,
! [M: rat,N: rat] :
( pp(aa_set_rat_bool(ord_less_set_rat(set_ord_lessThan_rat(M)),set_ord_lessThan_rat(N)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,M),N)) ) ).
% lessThan_strict_subset_iff
tff(fact_7382_lessThan__strict__subset__iff,axiom,
! [M: num,N: num] :
( pp(aa_set_num_bool(ord_less_set_num(set_ord_lessThan_num(M)),set_ord_lessThan_num(N)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M),N)) ) ).
% lessThan_strict_subset_iff
tff(fact_7383_lessThan__strict__subset__iff,axiom,
! [M: int,N: int] :
( pp(aa_set_int_bool(ord_less_set_int(set_ord_lessThan_int(M)),set_ord_lessThan_int(N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,M),N)) ) ).
% lessThan_strict_subset_iff
tff(fact_7384_lessThan__strict__subset__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_set_nat_bool(ord_less_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,M)),aa_nat_set_nat(set_ord_lessThan_nat,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% lessThan_strict_subset_iff
tff(fact_7385_lessThan__strict__subset__iff,axiom,
! [M: real,N: real] :
( pp(aa_set_real_bool(ord_less_set_real(set_or5984915006950818249n_real(M)),set_or5984915006950818249n_real(N)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,M),N)) ) ).
% lessThan_strict_subset_iff
tff(fact_7386_complex__mod__minus__le__complex__mod,axiom,
! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_complex_real(real_V1022390504157884413omplex,X))),aa_complex_real(real_V1022390504157884413omplex,X))) ).
% complex_mod_minus_le_complex_mod
tff(fact_7387_complex__mod__triangle__ineq2,axiom,
! [B: complex,A: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(B),A))),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,A))) ).
% complex_mod_triangle_ineq2
tff(fact_7388_ceiling__le,axiom,
! [X: real,A: int] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,A)))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),A)) ) ).
% ceiling_le
tff(fact_7389_ceiling__le,axiom,
! [X: rat,A: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,A)))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,archim2889992004027027881ng_rat(X)),A)) ) ).
% ceiling_le
tff(fact_7390_ceiling__le__iff,axiom,
! [X: real,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),Z))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,Z))) ) ).
% ceiling_le_iff
tff(fact_7391_ceiling__le__iff,axiom,
! [X: rat,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,archim2889992004027027881ng_rat(X)),Z))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z))) ) ).
% ceiling_le_iff
tff(fact_7392_less__ceiling__iff,axiom,
! [Z: int,X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),archim2889992004027027881ng_rat(X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),X)) ) ).
% less_ceiling_iff
tff(fact_7393_less__ceiling__iff,axiom,
! [Z: int,X: real] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_real_int(archim7802044766580827645g_real,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),X)) ) ).
% less_ceiling_iff
tff(fact_7394_ceiling__add__le,axiom,
! [X: rat,Y: rat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,archim2889992004027027881ng_rat(plus_plus_rat(X,Y))),aa_int_int(aa_int_fun_int_int(plus_plus_int,archim2889992004027027881ng_rat(X)),archim2889992004027027881ng_rat(Y)))) ).
% ceiling_add_le
tff(fact_7395_ceiling__add__le,axiom,
! [X: real,Y: real] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X)),aa_real_int(archim7802044766580827645g_real,Y)))) ).
% ceiling_add_le
tff(fact_7396_norm__exp,axiom,
! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(exp_real,X))),aa_real_real(exp_real,aa_real_real(real_V7735802525324610683m_real,X)))) ).
% norm_exp
tff(fact_7397_norm__exp,axiom,
! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,X))),aa_real_real(exp_real,aa_complex_real(real_V1022390504157884413omplex,X)))) ).
% norm_exp
tff(fact_7398_sum_Onat__diff__reindex,axiom,
! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_im(G),N)),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).
% sum.nat_diff_reindex
tff(fact_7399_sum_Onat__diff__reindex,axiom,
! [G: fun_nat_real,N: nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_in(G),N),aa_nat_set_nat(set_ord_lessThan_nat,N)) = groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,N)) ).
% sum.nat_diff_reindex
tff(fact_7400_sum__diff__distrib,axiom,
! [Q: fun_real_nat,P: fun_real_nat,N: real] :
( ! [X3: real] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_real_nat(Q,X3)),aa_real_nat(P,X3)))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(P,set_or5984915006950818249n_real(N))),groups1935376822645274424al_nat(Q,set_or5984915006950818249n_real(N))) = groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_gm(Q),P),set_or5984915006950818249n_real(N)) ) ) ).
% sum_diff_distrib
tff(fact_7401_sum__diff__distrib,axiom,
! [Q: fun_nat_nat,P: fun_nat_nat,N: nat] :
( ! [X3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(Q,X3)),aa_nat_nat(P,X3)))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(P),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_set_nat_nat(groups3542108847815614940at_nat(Q),aa_nat_set_nat(set_ord_lessThan_nat,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_go(Q),P)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ) ) ).
% sum_diff_distrib
tff(fact_7402_of__int__ceiling__le__add__one,axiom,
! [R: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,R))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R),one_one_real))) ).
% of_int_ceiling_le_add_one
tff(fact_7403_of__int__ceiling__le__add__one,axiom,
! [R: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(R))),plus_plus_rat(R,one_one_rat))) ).
% of_int_ceiling_le_add_one
tff(fact_7404_of__int__ceiling__diff__one__le,axiom,
! [R: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,R))),one_one_real)),R)) ).
% of_int_ceiling_diff_one_le
tff(fact_7405_of__int__ceiling__diff__one__le,axiom,
! [R: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(R)),one_one_rat)),R)) ).
% of_int_ceiling_diff_one_le
tff(fact_7406_ceiling__divide__eq__div,axiom,
! [A: int,B: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,A)),aa_int_real(ring_1_of_int_real,B))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B)) ).
% ceiling_divide_eq_div
tff(fact_7407_ceiling__divide__eq__div,axiom,
! [A: int,B: int] : archim2889992004027027881ng_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,A),aa_int_rat(ring_1_of_int_rat,B))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B)) ).
% ceiling_divide_eq_div
tff(fact_7408_sum_OlessThan__Suc__shift,axiom,
! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = plus_plus_rat(aa_nat_rat(G,zero_zero_nat),groups2906978787729119204at_rat(aTP_Lamm_gz(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).
% sum.lessThan_Suc_shift
tff(fact_7409_sum_OlessThan__Suc__shift,axiom,
! [G: fun_nat_int,N: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_ha(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).
% sum.lessThan_Suc_shift
tff(fact_7410_sum_OlessThan__Suc__shift,axiom,
! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_gp(G)),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).
% sum.lessThan_Suc_shift
tff(fact_7411_sum_OlessThan__Suc__shift,axiom,
! [G: fun_nat_real,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),groups6591440286371151544t_real(aTP_Lamm_gq(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).
% sum.lessThan_Suc_shift
tff(fact_7412_sum__lessThan__telescope_H,axiom,
! [F: fun_nat_rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_io(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = minus_minus_rat(aa_nat_rat(F,zero_zero_nat),aa_nat_rat(F,M)) ).
% sum_lessThan_telescope'
tff(fact_7413_sum__lessThan__telescope_H,axiom,
! [F: fun_nat_int,M: nat] : groups3539618377306564664at_int(aTP_Lamm_ip(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,zero_zero_nat)),aa_nat_int(F,M)) ).
% sum_lessThan_telescope'
tff(fact_7414_sum__lessThan__telescope_H,axiom,
! [F: fun_nat_real,M: nat] : groups6591440286371151544t_real(aTP_Lamm_iq(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,M)) ).
% sum_lessThan_telescope'
tff(fact_7415_sum__lessThan__telescope,axiom,
! [F: fun_nat_rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_hb(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = minus_minus_rat(aa_nat_rat(F,M),aa_nat_rat(F,zero_zero_nat)) ).
% sum_lessThan_telescope
tff(fact_7416_sum__lessThan__telescope,axiom,
! [F: fun_nat_int,M: nat] : groups3539618377306564664at_int(aTP_Lamm_hc(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,M)),aa_nat_int(F,zero_zero_nat)) ).
% sum_lessThan_telescope
tff(fact_7417_sum__lessThan__telescope,axiom,
! [F: fun_nat_real,M: nat] : groups6591440286371151544t_real(aTP_Lamm_hd(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,M)),aa_nat_real(F,zero_zero_nat)) ).
% sum_lessThan_telescope
tff(fact_7418_sumr__diff__mult__const2,axiom,
! [F: fun_nat_int,N: nat,R: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),R)) = groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_ir(F),R),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).
% sumr_diff_mult_const2
tff(fact_7419_sumr__diff__mult__const2,axiom,
! [F: fun_nat_complex,N: nat,R: complex] : aa_complex_complex(minus_minus_complex(groups2073611262835488442omplex(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),R)) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_is(F),R),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).
% sumr_diff_mult_const2
tff(fact_7420_sumr__diff__mult__const2,axiom,
! [F: fun_nat_rat,N: nat,R: rat] : minus_minus_rat(groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_lessThan_nat,N)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),R)) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_it(F),R),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).
% sumr_diff_mult_const2
tff(fact_7421_sumr__diff__mult__const2,axiom,
! [F: fun_nat_real,N: nat,R: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),R)) = groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_iu(F),R),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).
% sumr_diff_mult_const2
tff(fact_7422_sum_OatLeast1__atMost__eq,axiom,
! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_gp(G)),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).
% sum.atLeast1_atMost_eq
tff(fact_7423_sum_OatLeast1__atMost__eq,axiom,
! [G: fun_nat_real,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = groups6591440286371151544t_real(aTP_Lamm_gq(G),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).
% sum.atLeast1_atMost_eq
tff(fact_7424_ceiling__correct,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X))),one_one_real)),X))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X)))) ) ).
% ceiling_correct
tff(fact_7425_ceiling__correct,axiom,
! [X: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(X)),one_one_rat)),X))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(X)))) ) ).
% ceiling_correct
tff(fact_7426_ceiling__unique,axiom,
! [Z: int,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,Z)))
=> ( aa_real_int(archim7802044766580827645g_real,X) = Z ) ) ) ).
% ceiling_unique
tff(fact_7427_ceiling__unique,axiom,
! [Z: int,X: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat)),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z)))
=> ( archim2889992004027027881ng_rat(X) = Z ) ) ) ).
% ceiling_unique
tff(fact_7428_ceiling__eq__iff,axiom,
! [X: real,A: int] :
( ( aa_real_int(archim7802044766580827645g_real,X) = A )
<=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,A)),one_one_real)),X))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,A))) ) ) ).
% ceiling_eq_iff
tff(fact_7429_ceiling__eq__iff,axiom,
! [X: rat,A: int] :
( ( archim2889992004027027881ng_rat(X) = A )
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,A),one_one_rat)),X))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,A))) ) ) ).
% ceiling_eq_iff
tff(fact_7430_ceiling__split,axiom,
! [P: fun_int_bool,T: real] :
( pp(aa_int_bool(P,aa_real_int(archim7802044766580827645g_real,T)))
<=> ! [I4: int] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,I4)),one_one_real)),T))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T),aa_int_real(ring_1_of_int_real,I4))) )
=> pp(aa_int_bool(P,I4)) ) ) ).
% ceiling_split
tff(fact_7431_ceiling__split,axiom,
! [P: fun_int_bool,T: rat] :
( pp(aa_int_bool(P,archim2889992004027027881ng_rat(T)))
<=> ! [I4: int] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,I4),one_one_rat)),T))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,T),aa_int_rat(ring_1_of_int_rat,I4))) )
=> pp(aa_int_bool(P,I4)) ) ) ).
% ceiling_split
tff(fact_7432_mult__ceiling__le,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,A)),aa_real_int(archim7802044766580827645g_real,B)))) ) ) ).
% mult_ceiling_le
tff(fact_7433_mult__ceiling__le,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,archim2889992004027027881ng_rat(aa_rat_rat(times_times_rat(A),B))),aa_int_int(aa_int_fun_int_int(times_times_int,archim2889992004027027881ng_rat(A)),archim2889992004027027881ng_rat(B)))) ) ) ).
% mult_ceiling_le
tff(fact_7434_ceiling__less__iff,axiom,
! [X: real,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),Z))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real))) ) ).
% ceiling_less_iff
tff(fact_7435_ceiling__less__iff,axiom,
! [X: rat,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,archim2889992004027027881ng_rat(X)),Z))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat))) ) ).
% ceiling_less_iff
tff(fact_7436_le__ceiling__iff,axiom,
! [Z: int,X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),archim2889992004027027881ng_rat(X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat)),X)) ) ).
% le_ceiling_iff
tff(fact_7437_le__ceiling__iff,axiom,
! [Z: int,X: real] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_real_int(archim7802044766580827645g_real,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)),X)) ) ).
% le_ceiling_iff
tff(fact_7438_power__diff__1__eq,axiom,
! [X: complex,N: nat] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),N)),one_one_complex) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),one_one_complex)),groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).
% power_diff_1_eq
tff(fact_7439_power__diff__1__eq,axiom,
! [X: rat,N: nat] : minus_minus_rat(aa_nat_rat(power_power_rat(X),N),one_one_rat) = aa_rat_rat(times_times_rat(minus_minus_rat(X,one_one_rat)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).
% power_diff_1_eq
tff(fact_7440_power__diff__1__eq,axiom,
! [X: int,N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X),N)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),one_one_int)),groups3539618377306564664at_int(power_power_int(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).
% power_diff_1_eq
tff(fact_7441_power__diff__1__eq,axiom,
! [X: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).
% power_diff_1_eq
tff(fact_7442_one__diff__power__eq,axiom,
! [X: complex,N: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).
% one_diff_power_eq
tff(fact_7443_one__diff__power__eq,axiom,
! [X: rat,N: nat] : minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(X),N)) = aa_rat_rat(times_times_rat(minus_minus_rat(one_one_rat,X)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).
% one_diff_power_eq
tff(fact_7444_one__diff__power__eq,axiom,
! [X: int,N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(X),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X)),groups3539618377306564664at_int(power_power_int(X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).
% one_diff_power_eq
tff(fact_7445_one__diff__power__eq,axiom,
! [X: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).
% one_diff_power_eq
tff(fact_7446_geometric__sum,axiom,
! [X: complex,N: nat] :
( ( X != one_one_complex )
=> ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),N)),one_one_complex)),aa_complex_complex(minus_minus_complex(X),one_one_complex)) ) ) ).
% geometric_sum
tff(fact_7447_geometric__sum,axiom,
! [X: rat,N: nat] :
( ( X != one_one_rat )
=> ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = divide_divide_rat(minus_minus_rat(aa_nat_rat(power_power_rat(X),N),one_one_rat),minus_minus_rat(X,one_one_rat)) ) ) ).
% geometric_sum
tff(fact_7448_geometric__sum,axiom,
! [X: real,N: nat] :
( ( X != one_one_real )
=> ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real)) ) ) ).
% geometric_sum
tff(fact_7449_ceiling__divide__upper,axiom,
! [Q2: real,P2: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Q2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,P2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P2),Q2)))),Q2))) ) ).
% ceiling_divide_upper
tff(fact_7450_ceiling__divide__upper,axiom,
! [Q2: rat,P2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Q2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,P2),aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(divide_divide_rat(P2,Q2)))),Q2))) ) ).
% ceiling_divide_upper
tff(fact_7451_monoseq__realpow,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
=> topolo6980174941875973593q_real(aa_real_fun_nat_real(power_power_real,X)) ) ) ).
% monoseq_realpow
tff(fact_7452_sum__gp__strict,axiom,
! [X: complex,N: nat] :
( ( ( X = one_one_complex )
=> ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_complex(semiri8010041392384452111omplex,N) ) )
& ( ( X != one_one_complex )
=> ( groups2073611262835488442omplex(power_power_complex(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),N))),aa_complex_complex(minus_minus_complex(one_one_complex),X)) ) ) ) ).
% sum_gp_strict
tff(fact_7453_sum__gp__strict,axiom,
! [X: rat,N: nat] :
( ( ( X = one_one_rat )
=> ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_rat(semiri681578069525770553at_rat,N) ) )
& ( ( X != one_one_rat )
=> ( groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = divide_divide_rat(minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(X),N)),minus_minus_rat(one_one_rat,X)) ) ) ) ).
% sum_gp_strict
tff(fact_7454_sum__gp__strict,axiom,
! [X: real,N: nat] :
( ( ( X = one_one_real )
=> ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_real(semiri5074537144036343181t_real,N) ) )
& ( ( X != one_one_real )
=> ( groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)) ) ) ) ).
% sum_gp_strict
tff(fact_7455_lemma__termdiff1,axiom,
! [Z: complex,H: complex,M: nat] : groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_iv(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_iw(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).
% lemma_termdiff1
tff(fact_7456_lemma__termdiff1,axiom,
! [Z: rat,H: rat,M: nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_ix(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_iy(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).
% lemma_termdiff1
tff(fact_7457_lemma__termdiff1,axiom,
! [Z: int,H: int,M: nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_iz(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_ja(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).
% lemma_termdiff1
tff(fact_7458_lemma__termdiff1,axiom,
! [Z: real,H: real,M: nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_jb(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_jc(Z),H),M),aa_nat_set_nat(set_ord_lessThan_nat,M)) ).
% lemma_termdiff1
tff(fact_7459_power__diff__sumr2,axiom,
! [X: complex,N: nat,Y: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),N)),aa_nat_complex(power_power_complex(Y),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_jd(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).
% power_diff_sumr2
tff(fact_7460_power__diff__sumr2,axiom,
! [X: rat,N: nat,Y: rat] : minus_minus_rat(aa_nat_rat(power_power_rat(X),N),aa_nat_rat(power_power_rat(Y),N)) = aa_rat_rat(times_times_rat(minus_minus_rat(X,Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_je(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).
% power_diff_sumr2
tff(fact_7461_power__diff__sumr2,axiom,
! [X: int,N: nat,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X),N)),aa_nat_int(power_power_int(Y),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_jf(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).
% power_diff_sumr2
tff(fact_7462_power__diff__sumr2,axiom,
! [X: real,N: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_jg(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).
% power_diff_sumr2
tff(fact_7463_diff__power__eq__sum,axiom,
! [X: complex,N: nat,Y: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(X),aa_nat_nat(suc,N))),aa_nat_complex(power_power_complex(Y),aa_nat_nat(suc,N))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),Y)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_jh(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).
% diff_power_eq_sum
tff(fact_7464_diff__power__eq__sum,axiom,
! [X: rat,N: nat,Y: rat] : minus_minus_rat(aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,N)),aa_nat_rat(power_power_rat(Y),aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(minus_minus_rat(X,Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_ji(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).
% diff_power_eq_sum
tff(fact_7465_diff__power__eq__sum,axiom,
! [X: int,N: nat,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(X),aa_nat_nat(suc,N))),aa_nat_int(power_power_int(Y),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_jj(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).
% diff_power_eq_sum
tff(fact_7466_diff__power__eq__sum,axiom,
! [X: real,N: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)),groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_jk(X),N),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)))) ).
% diff_power_eq_sum
tff(fact_7467_ceiling__divide__lower,axiom,
! [Q2: real,P2: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Q2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P2),Q2)))),one_one_real)),Q2)),P2)) ) ).
% ceiling_divide_lower
tff(fact_7468_ceiling__divide__lower,axiom,
! [Q2: rat,P2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Q2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_rat_rat(times_times_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(divide_divide_rat(P2,Q2))),one_one_rat)),Q2)),P2)) ) ).
% ceiling_divide_lower
tff(fact_7469_ceiling__eq,axiom,
! [N: int,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,N)),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N)),one_one_real)))
=> ( aa_real_int(archim7802044766580827645g_real,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,N),one_one_int) ) ) ) ).
% ceiling_eq
tff(fact_7470_ceiling__eq,axiom,
! [N: int,X: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(ring_1_of_int_rat,N)),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),plus_plus_rat(aa_int_rat(ring_1_of_int_rat,N),one_one_rat)))
=> ( archim2889992004027027881ng_rat(X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,N),one_one_int) ) ) ) ).
% ceiling_eq
tff(fact_7471_real__sum__nat__ivl__bounded2,axiom,
! [N: nat,F: fun_nat_rat,K5: rat,K: nat] :
( ! [P4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,P4),N))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,P4)),K5)) )
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),K5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),K5))) ) ) ).
% real_sum_nat_ivl_bounded2
tff(fact_7472_real__sum__nat__ivl__bounded2,axiom,
! [N: nat,F: fun_nat_int,K5: int,K: nat] :
( ! [P4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,P4),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,P4)),K5)) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K5))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),K5))) ) ) ).
% real_sum_nat_ivl_bounded2
tff(fact_7473_real__sum__nat__ivl__bounded2,axiom,
! [N: nat,F: fun_nat_nat,K5: nat,K: nat] :
( ! [P4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,P4),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,P4)),K5)) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),K5))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),K5))) ) ) ).
% real_sum_nat_ivl_bounded2
tff(fact_7474_real__sum__nat__ivl__bounded2,axiom,
! [N: nat,F: fun_nat_real,K5: real,K: nat] :
( ! [P4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,P4),N))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,P4)),K5)) )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),K5))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),K5))) ) ) ).
% real_sum_nat_ivl_bounded2
tff(fact_7475_exp__bound__half,axiom,
! [Z: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(exp_real,Z))),numeral_numeral_real(aa_num_num(bit0,one)))) ) ).
% exp_bound_half
tff(fact_7476_exp__bound__half,axiom,
! [Z: complex] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,Z))),numeral_numeral_real(aa_num_num(bit0,one)))) ) ).
% exp_bound_half
tff(fact_7477_one__diff__power__eq_H,axiom,
! [X: complex,N: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(X),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_jl(X),N),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).
% one_diff_power_eq'
tff(fact_7478_one__diff__power__eq_H,axiom,
! [X: rat,N: nat] : minus_minus_rat(one_one_rat,aa_nat_rat(power_power_rat(X),N)) = aa_rat_rat(times_times_rat(minus_minus_rat(one_one_rat,X)),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_jm(X),N),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).
% one_diff_power_eq'
tff(fact_7479_one__diff__power__eq_H,axiom,
! [X: int,N: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(X),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X)),groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_jn(X),N),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).
% one_diff_power_eq'
tff(fact_7480_one__diff__power__eq_H,axiom,
! [X: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),X)),groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_jo(X),N),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).
% one_diff_power_eq'
tff(fact_7481_and__nat__unfold,axiom,
! [M: nat,N: nat] :
( ( ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = zero_zero_nat ) )
& ( ~ ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = 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(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ).
% and_nat_unfold
tff(fact_7482_and__nat__rec,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).
% and_nat_rec
tff(fact_7483_sum__split__even__odd,axiom,
! [F: fun_nat_real,G: fun_nat_real,N: nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_jp(F),G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aTP_Lamm_jq(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),groups6591440286371151544t_real(aTP_Lamm_jr(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).
% sum_split_even_odd
tff(fact_7484_exp__bound__lemma,axiom,
! [Z: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(exp_real,Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(real_V7735802525324610683m_real,Z))))) ) ).
% exp_bound_lemma
tff(fact_7485_exp__bound__lemma,axiom,
! [Z: complex] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_complex_real(real_V1022390504157884413omplex,Z))))) ) ).
% exp_bound_lemma
tff(fact_7486_ceiling__log__nat__eq__if,axiom,
! [B: nat,N: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
=> ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int) ) ) ) ) ).
% ceiling_log_nat_eq_if
tff(fact_7487_ceiling__log__nat__eq__powr__iff,axiom,
! [B: nat,K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int) )
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))) ) ) ) ) ).
% ceiling_log_nat_eq_powr_iff
tff(fact_7488_ceiling__log2__div2,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))))),one_one_int) ) ) ).
% ceiling_log2_div2
tff(fact_7489_arctan__series,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_js,X)) ) ) ).
% arctan_series
tff(fact_7490_norm__divide__numeral,axiom,
! [A: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),numeral_numeral_real(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A)),numeral_numeral_real(W)) ).
% norm_divide_numeral
tff(fact_7491_norm__divide__numeral,axiom,
! [A: complex,W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A)),numeral_numeral_real(W)) ).
% norm_divide_numeral
tff(fact_7492_norm__mult__numeral1,axiom,
! [W: num,A: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),aa_real_real(real_V7735802525324610683m_real,A)) ).
% norm_mult_numeral1
tff(fact_7493_norm__mult__numeral1,axiom,
! [W: num,A: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(W)),aa_complex_real(real_V1022390504157884413omplex,A)) ).
% norm_mult_numeral1
tff(fact_7494_norm__mult__numeral2,axiom,
! [A: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),numeral_numeral_real(W))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,A)),numeral_numeral_real(W)) ).
% norm_mult_numeral2
tff(fact_7495_norm__mult__numeral2,axiom,
! [A: complex,W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,A)),numeral_numeral_real(W)) ).
% norm_mult_numeral2
tff(fact_7496_norm__neg__numeral,axiom,
! [W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = numeral_numeral_real(W) ).
% norm_neg_numeral
tff(fact_7497_norm__neg__numeral,axiom,
! [W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = numeral_numeral_real(W) ).
% norm_neg_numeral
tff(fact_7498_norm__le__zero__iff,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,X)),zero_zero_real))
<=> ( X = zero_zero_real ) ) ).
% norm_le_zero_iff
tff(fact_7499_norm__le__zero__iff,axiom,
! [X: complex] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,X)),zero_zero_real))
<=> ( X = zero_zero_complex ) ) ).
% norm_le_zero_iff
tff(fact_7500_norm__one,axiom,
aa_real_real(real_V7735802525324610683m_real,one_one_real) = one_one_real ).
% norm_one
tff(fact_7501_norm__one,axiom,
aa_complex_real(real_V1022390504157884413omplex,one_one_complex) = one_one_real ).
% norm_one
tff(fact_7502_norm__numeral,axiom,
! [W: num] : aa_real_real(real_V7735802525324610683m_real,numeral_numeral_real(W)) = numeral_numeral_real(W) ).
% norm_numeral
tff(fact_7503_norm__numeral,axiom,
! [W: num] : aa_complex_real(real_V1022390504157884413omplex,numera6690914467698888265omplex(W)) = numeral_numeral_real(W) ).
% norm_numeral
tff(fact_7504_zero__less__norm__iff,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(real_V7735802525324610683m_real,X)))
<=> ( X != zero_zero_real ) ) ).
% zero_less_norm_iff
tff(fact_7505_zero__less__norm__iff,axiom,
! [X: complex] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,X)))
<=> ( X != zero_zero_complex ) ) ).
% zero_less_norm_iff
tff(fact_7506_norm__minus__commute,axiom,
! [A: complex,B: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),B)) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(B),A)) ).
% norm_minus_commute
tff(fact_7507_norm__not__less__zero,axiom,
! [X: complex] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),zero_zero_real)) ).
% norm_not_less_zero
tff(fact_7508_norm__ge__zero,axiom,
! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,X))) ).
% norm_ge_zero
tff(fact_7509_norm__mult,axiom,
! [X: real,Y: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y)) ).
% norm_mult
tff(fact_7510_norm__mult,axiom,
! [X: complex,Y: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y)) ).
% norm_mult
tff(fact_7511_norm__divide,axiom,
! [A: real,B: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B)) ).
% norm_divide
tff(fact_7512_norm__divide,axiom,
! [A: complex,B: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)) ).
% norm_divide
tff(fact_7513_sum__norm__le,axiom,
! [S2: set_Pr1261947904930325089at_nat,F: fun_Pr4050773816208733250omplex,G: fun_Pr672294278902479040t_real] :
( ! [X3: product_prod_nat_nat] :
( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X3),S2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_Pro82899191565270131omplex(F,X3))),aa_Pro6168199777849092593t_real(G,X3))) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups6381953495645901045omplex(F,S2))),groups4567486121110086003t_real(G,S2))) ) ).
% sum_norm_le
tff(fact_7514_sum__norm__le,axiom,
! [S2: set_real,F: fun_real_complex,G: fun_real_real] :
( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),S2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(F,X3))),aa_real_real(G,X3))) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups5754745047067104278omplex(F,S2))),groups8097168146408367636l_real(G,S2))) ) ).
% sum_norm_le
tff(fact_7515_sum__norm__le,axiom,
! [S2: set_int,F: fun_int_complex,G: fun_int_real] :
( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),S2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_int_complex(F,X3))),aa_int_real(G,X3))) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups3049146728041665814omplex(F,S2))),groups8778361861064173332t_real(G,S2))) ) ).
% sum_norm_le
tff(fact_7516_sum__norm__le,axiom,
! [S2: set_nat,F: fun_nat_complex,G: fun_nat_real] :
( ! [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),S2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,X3))),aa_nat_real(G,X3))) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups2073611262835488442omplex(F,S2))),groups6591440286371151544t_real(G,S2))) ) ).
% sum_norm_le
tff(fact_7517_sum__norm__le,axiom,
! [S2: set_complex,F: fun_complex_complex,G: fun_complex_real] :
( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),S2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(F,X3))),aa_complex_real(G,X3))) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups7754918857620584856omplex(F,S2))),groups5808333547571424918x_real(G,S2))) ) ).
% sum_norm_le
tff(fact_7518_sum__norm__le,axiom,
! [S2: set_nat,F: fun_nat_real,G: fun_nat_real] :
( ! [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),S2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,X3))),aa_nat_real(G,X3))) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,groups6591440286371151544t_real(F,S2))),groups6591440286371151544t_real(G,S2))) ) ).
% sum_norm_le
tff(fact_7519_norm__power,axiom,
! [X: real,N: nat] : aa_real_real(real_V7735802525324610683m_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,X)),N) ).
% norm_power
tff(fact_7520_norm__power,axiom,
! [X: complex,N: nat] : aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(power_power_complex(X),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,X)),N) ).
% norm_power
tff(fact_7521_norm__sum,axiom,
! [F: fun_nat_complex,A2: set_nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups2073611262835488442omplex(F,A2))),groups6591440286371151544t_real(aTP_Lamm_jt(F),A2))) ).
% norm_sum
tff(fact_7522_norm__sum,axiom,
! [F: fun_complex_complex,A2: set_complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups7754918857620584856omplex(F,A2))),groups5808333547571424918x_real(aTP_Lamm_ju(F),A2))) ).
% norm_sum
tff(fact_7523_norm__sum,axiom,
! [F: fun_nat_real,A2: set_nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,groups6591440286371151544t_real(F,A2))),groups6591440286371151544t_real(aTP_Lamm_jv(F),A2))) ).
% norm_sum
tff(fact_7524_norm__uminus__minus,axiom,
! [X: real,Y: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,X)),Y)) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) ).
% norm_uminus_minus
tff(fact_7525_norm__uminus__minus,axiom,
! [X: complex,Y: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),Y)) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y)) ).
% norm_uminus_minus
tff(fact_7526_nonzero__norm__divide,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B)) ) ) ).
% nonzero_norm_divide
tff(fact_7527_nonzero__norm__divide,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)) ) ) ).
% nonzero_norm_divide
tff(fact_7528_power__eq__imp__eq__norm,axiom,
! [W: real,N: nat,Z: real] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),N) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_real_real(real_V7735802525324610683m_real,W) = aa_real_real(real_V7735802525324610683m_real,Z) ) ) ) ).
% power_eq_imp_eq_norm
tff(fact_7529_power__eq__imp__eq__norm,axiom,
! [W: complex,N: nat,Z: complex] :
( ( aa_nat_complex(power_power_complex(W),N) = aa_nat_complex(power_power_complex(Z),N) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_complex_real(real_V1022390504157884413omplex,W) = aa_complex_real(real_V1022390504157884413omplex,Z) ) ) ) ).
% power_eq_imp_eq_norm
tff(fact_7530_norm__mult__less,axiom,
! [X: real,R: real,Y: real,S: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),R))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),S))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),S))) ) ) ).
% norm_mult_less
tff(fact_7531_norm__mult__less,axiom,
! [X: complex,R: real,Y: complex,S: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),R))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y)),S))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),S))) ) ) ).
% norm_mult_less
tff(fact_7532_norm__mult__ineq,axiom,
! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y)))) ).
% norm_mult_ineq
tff(fact_7533_norm__mult__ineq,axiom,
! [X: complex,Y: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y)))) ).
% norm_mult_ineq
tff(fact_7534_norm__triangle__lt,axiom,
! [X: real,Y: real,E: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y))),E))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),E)) ) ).
% norm_triangle_lt
tff(fact_7535_norm__triangle__lt,axiom,
! [X: complex,Y: complex,E: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y))),E))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),E)) ) ).
% norm_triangle_lt
tff(fact_7536_norm__add__less,axiom,
! [X: real,R: real,Y: real,S: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),R))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),S))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R),S))) ) ) ).
% norm_add_less
tff(fact_7537_norm__add__less,axiom,
! [X: complex,R: real,Y: complex,S: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),R))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y)),S))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R),S))) ) ) ).
% norm_add_less
tff(fact_7538_norm__add__leD,axiom,
! [A: real,B: real,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,A)),C))) ) ).
% norm_add_leD
tff(fact_7539_norm__add__leD,axiom,
! [A: complex,B: complex,C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B))),C))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,A)),C))) ) ).
% norm_add_leD
tff(fact_7540_norm__triangle__le,axiom,
! [X: real,Y: real,E: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y))),E))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),E)) ) ).
% norm_triangle_le
tff(fact_7541_norm__triangle__le,axiom,
! [X: complex,Y: complex,E: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y))),E))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),E)) ) ).
% norm_triangle_le
tff(fact_7542_norm__triangle__ineq,axiom,
! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y)))) ).
% norm_triangle_ineq
tff(fact_7543_norm__triangle__ineq,axiom,
! [X: complex,Y: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y)))) ).
% norm_triangle_ineq
tff(fact_7544_norm__triangle__mono,axiom,
! [A: real,R: real,B: real,S: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,A)),R))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B)),S))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R),S))) ) ) ).
% norm_triangle_mono
tff(fact_7545_norm__triangle__mono,axiom,
! [A: complex,R: real,B: complex,S: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,A)),R))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B)),S))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R),S))) ) ) ).
% norm_triangle_mono
tff(fact_7546_norm__power__ineq,axiom,
! [X: real,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,X)),N))) ).
% norm_power_ineq
tff(fact_7547_norm__power__ineq,axiom,
! [X: complex,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(power_power_complex(X),N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,X)),N))) ).
% norm_power_ineq
tff(fact_7548_norm__diff__triangle__less,axiom,
! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Y))),E1))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Y),Z))),E22))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22))) ) ) ).
% norm_diff_triangle_less
tff(fact_7549_norm__triangle__sub,axiom,
! [X: complex,Y: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Y)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Y))))) ).
% norm_triangle_sub
tff(fact_7550_norm__triangle__ineq4,axiom,
! [A: complex,B: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),B))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)))) ).
% norm_triangle_ineq4
tff(fact_7551_norm__diff__triangle__le,axiom,
! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Y))),E1))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Y),Z))),E22))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22))) ) ) ).
% norm_diff_triangle_le
tff(fact_7552_norm__triangle__le__diff,axiom,
! [X: complex,Y: complex,E: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y))),E))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Y))),E)) ) ).
% norm_triangle_le_diff
tff(fact_7553_norm__diff__ineq,axiom,
! [A: real,B: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)))) ).
% norm_diff_ineq
tff(fact_7554_norm__diff__ineq,axiom,
! [A: complex,B: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B)))) ).
% norm_diff_ineq
tff(fact_7555_norm__triangle__ineq2,axiom,
! [A: complex,B: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),B)))) ).
% norm_triangle_ineq2
tff(fact_7556_power__eq__1__iff,axiom,
! [W: real,N: nat] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W),N) = one_one_real )
=> ( ( aa_real_real(real_V7735802525324610683m_real,W) = one_one_real )
| ( N = zero_zero_nat ) ) ) ).
% power_eq_1_iff
tff(fact_7557_power__eq__1__iff,axiom,
! [W: complex,N: nat] :
( ( aa_nat_complex(power_power_complex(W),N) = one_one_complex )
=> ( ( aa_complex_real(real_V1022390504157884413omplex,W) = one_one_real )
| ( N = zero_zero_nat ) ) ) ).
% power_eq_1_iff
tff(fact_7558_norm__diff__triangle__ineq,axiom,
! [A: real,B: real,C: real,D: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D))))) ).
% norm_diff_triangle_ineq
tff(fact_7559_norm__diff__triangle__ineq,axiom,
! [A: complex,B: complex,C: complex,D: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),B)),aa_complex_complex(plus_plus_complex(C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),C))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(B),D))))) ).
% norm_diff_triangle_ineq
tff(fact_7560_norm__triangle__ineq3,axiom,
! [A: complex,B: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),B)))) ).
% norm_triangle_ineq3
tff(fact_7561_square__norm__one,axiom,
! [X: real] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
=> ( aa_real_real(real_V7735802525324610683m_real,X) = one_one_real ) ) ).
% square_norm_one
tff(fact_7562_square__norm__one,axiom,
! [X: complex] :
( ( aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex )
=> ( aa_complex_real(real_V1022390504157884413omplex,X) = one_one_real ) ) ).
% square_norm_one
tff(fact_7563_norm__power__diff,axiom,
! [Z: real,W: real,M: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),one_one_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,W)),one_one_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,W),M)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),W))))) ) ) ).
% norm_power_diff
tff(fact_7564_norm__power__diff,axiom,
! [Z: complex,W: complex,M: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),one_one_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,W)),one_one_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Z),M)),aa_nat_complex(power_power_complex(W),M)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Z),W))))) ) ) ).
% norm_power_diff
tff(fact_7565_suminf__geometric,axiom,
! [C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real))
=> ( suminf_real(aa_real_fun_nat_real(power_power_real,C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),C)) ) ) ).
% suminf_geometric
tff(fact_7566_suminf__geometric,axiom,
! [C: complex] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real))
=> ( suminf_complex(power_power_complex(C)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_complex_complex(minus_minus_complex(one_one_complex),C)) ) ) ).
% suminf_geometric
tff(fact_7567_sum__bounds__lt__plus1,axiom,
! [F: fun_nat_nat,Mm: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_gp(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mm)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Mm)) ).
% sum_bounds_lt_plus1
tff(fact_7568_sum__bounds__lt__plus1,axiom,
! [F: fun_nat_real,Mm: nat] : groups6591440286371151544t_real(aTP_Lamm_gq(F),aa_nat_set_nat(set_ord_lessThan_nat,Mm)) = groups6591440286371151544t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Mm)) ).
% sum_bounds_lt_plus1
tff(fact_7569_sumr__cos__zero__one,axiom,
! [N: nat] : groups6591440286371151544t_real(aTP_Lamm_jw,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = one_one_real ).
% sumr_cos_zero_one
tff(fact_7570_pi__series,axiom,
aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) = suminf_real(aTP_Lamm_jx) ).
% pi_series
tff(fact_7571_summable__arctan__series,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
=> summable_real(aa_real_fun_nat_real(aTP_Lamm_js,X)) ) ).
% summable_arctan_series
tff(fact_7572_summable__iff__shift,axiom,
! [F: fun_nat_real,K: nat] :
( summable_real(aa_nat_fun_nat_real(aTP_Lamm_gs(F),K))
<=> summable_real(F) ) ).
% summable_iff_shift
tff(fact_7573_cos__coeff__0,axiom,
aa_nat_real(cos_coeff,zero_zero_nat) = one_one_real ).
% cos_coeff_0
tff(fact_7574_summable__cmult__iff,axiom,
! [C: complex,F: fun_nat_complex] :
( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_jy(C),F))
<=> ( ( C = zero_zero_complex )
| summable_complex(F) ) ) ).
% summable_cmult_iff
tff(fact_7575_summable__cmult__iff,axiom,
! [C: real,F: fun_nat_real] :
( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_em(C),F))
<=> ( ( C = zero_zero_real )
| summable_real(F) ) ) ).
% summable_cmult_iff
tff(fact_7576_summable__divide__iff,axiom,
! [F: fun_nat_complex,C: complex] :
( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_jz(F),C))
<=> ( ( C = zero_zero_complex )
| summable_complex(F) ) ) ).
% summable_divide_iff
tff(fact_7577_summable__divide__iff,axiom,
! [F: fun_nat_real,C: real] :
( summable_real(aa_real_fun_nat_real(aTP_Lamm_fh(F),C))
<=> ( ( C = zero_zero_real )
| summable_real(F) ) ) ).
% summable_divide_iff
tff(fact_7578_summable__geometric__iff,axiom,
! [C: real] :
( summable_real(aa_real_fun_nat_real(power_power_real,C))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real)) ) ).
% summable_geometric_iff
tff(fact_7579_summable__geometric__iff,axiom,
! [C: complex] :
( summable_complex(power_power_complex(C))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real)) ) ).
% summable_geometric_iff
tff(fact_7580_summable__comparison__test,axiom,
! [F: fun_nat_real,G: fun_nat_real] :
( ? [N7: nat] :
! [N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N2))),aa_nat_real(G,N2))) )
=> ( summable_real(G)
=> summable_real(F) ) ) ).
% summable_comparison_test
tff(fact_7581_summable__comparison__test,axiom,
! [F: fun_nat_complex,G: fun_nat_real] :
( ? [N7: nat] :
! [N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N2))),aa_nat_real(G,N2))) )
=> ( summable_real(G)
=> summable_complex(F) ) ) ).
% summable_comparison_test
tff(fact_7582_summable__comparison__test_H,axiom,
! [G: fun_nat_real,N3: nat,F: fun_nat_real] :
( summable_real(G)
=> ( ! [N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N3),N2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N2))),aa_nat_real(G,N2))) )
=> summable_real(F) ) ) ).
% summable_comparison_test'
tff(fact_7583_summable__comparison__test_H,axiom,
! [G: fun_nat_real,N3: nat,F: fun_nat_complex] :
( summable_real(G)
=> ( ! [N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N3),N2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N2))),aa_nat_real(G,N2))) )
=> summable_complex(F) ) ) ).
% summable_comparison_test'
tff(fact_7584_summable__ignore__initial__segment,axiom,
! [F: fun_nat_real,K: nat] :
( summable_real(F)
=> summable_real(aa_nat_fun_nat_real(aTP_Lamm_gs(F),K)) ) ).
% summable_ignore_initial_segment
tff(fact_7585_summable__add,axiom,
! [F: fun_nat_real,G: fun_nat_real] :
( summable_real(F)
=> ( summable_real(G)
=> summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_fc(F),G)) ) ) ).
% summable_add
tff(fact_7586_summable__add,axiom,
! [F: fun_nat_nat,G: fun_nat_nat] :
( summable_nat(F)
=> ( summable_nat(G)
=> summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_fb(F),G)) ) ) ).
% summable_add
tff(fact_7587_summable__add,axiom,
! [F: fun_nat_int,G: fun_nat_int] :
( summable_int(F)
=> ( summable_int(G)
=> summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_ka(F),G)) ) ) ).
% summable_add
tff(fact_7588_summable__mult,axiom,
! [F: fun_nat_real,C: real] :
( summable_real(F)
=> summable_real(aa_real_fun_nat_real(aTP_Lamm_kb(F),C)) ) ).
% summable_mult
tff(fact_7589_summable__mult2,axiom,
! [F: fun_nat_real,C: real] :
( summable_real(F)
=> summable_real(aa_real_fun_nat_real(aTP_Lamm_eq(F),C)) ) ).
% summable_mult2
tff(fact_7590_summable__diff,axiom,
! [F: fun_nat_real,G: fun_nat_real] :
( summable_real(F)
=> ( summable_real(G)
=> summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_ff(F),G)) ) ) ).
% summable_diff
tff(fact_7591_summable__divide,axiom,
! [F: fun_nat_complex,C: complex] :
( summable_complex(F)
=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_jz(F),C)) ) ).
% summable_divide
tff(fact_7592_summable__divide,axiom,
! [F: fun_nat_real,C: real] :
( summable_real(F)
=> summable_real(aa_real_fun_nat_real(aTP_Lamm_fh(F),C)) ) ).
% summable_divide
tff(fact_7593_summable__Suc__iff,axiom,
! [F: fun_nat_real] :
( summable_real(aTP_Lamm_gq(F))
<=> summable_real(F) ) ).
% summable_Suc_iff
tff(fact_7594_suminf__le,axiom,
! [F: fun_nat_real,G: fun_nat_real] :
( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(G,N2)))
=> ( summable_real(F)
=> ( summable_real(G)
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,suminf_real(F)),suminf_real(G))) ) ) ) ).
% suminf_le
tff(fact_7595_suminf__le,axiom,
! [F: fun_nat_nat,G: fun_nat_nat] :
( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(G,N2)))
=> ( summable_nat(F)
=> ( summable_nat(G)
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,suminf_nat(F)),suminf_nat(G))) ) ) ) ).
% suminf_le
tff(fact_7596_suminf__le,axiom,
! [F: fun_nat_int,G: fun_nat_int] :
( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(G,N2)))
=> ( summable_int(F)
=> ( summable_int(G)
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,suminf_int(F)),suminf_int(G))) ) ) ) ).
% suminf_le
tff(fact_7597_summable__mult__D,axiom,
! [C: complex,F: fun_nat_complex] :
( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_jy(C),F))
=> ( ( C != zero_zero_complex )
=> summable_complex(F) ) ) ).
% summable_mult_D
tff(fact_7598_summable__mult__D,axiom,
! [C: real,F: fun_nat_real] :
( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_em(C),F))
=> ( ( C != zero_zero_real )
=> summable_real(F) ) ) ).
% summable_mult_D
tff(fact_7599_summable__zero__power,axiom,
summable_real(aa_real_fun_nat_real(power_power_real,zero_zero_real)) ).
% summable_zero_power
tff(fact_7600_summable__zero__power,axiom,
summable_complex(power_power_complex(zero_zero_complex)) ).
% summable_zero_power
tff(fact_7601_summable__zero__power,axiom,
summable_int(power_power_int(zero_zero_int)) ).
% summable_zero_power
tff(fact_7602_suminf__mult2,axiom,
! [F: fun_nat_real,C: real] :
( summable_real(F)
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(F)),C) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_eq(F),C)) ) ) ).
% suminf_mult2
tff(fact_7603_suminf__mult,axiom,
! [F: fun_nat_real,C: real] :
( summable_real(F)
=> ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_kb(F),C)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),suminf_real(F)) ) ) ).
% suminf_mult
tff(fact_7604_suminf__add,axiom,
! [F: fun_nat_real,G: fun_nat_real] :
( summable_real(F)
=> ( summable_real(G)
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,suminf_real(F)),suminf_real(G)) = suminf_real(aa_fun1839206346889086768t_real(aTP_Lamm_fc(F),G)) ) ) ) ).
% suminf_add
tff(fact_7605_suminf__add,axiom,
! [F: fun_nat_nat,G: fun_nat_nat] :
( summable_nat(F)
=> ( summable_nat(G)
=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,suminf_nat(F)),suminf_nat(G)) = suminf_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_fb(F),G)) ) ) ) ).
% suminf_add
tff(fact_7606_suminf__add,axiom,
! [F: fun_nat_int,G: fun_nat_int] :
( summable_int(F)
=> ( summable_int(G)
=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,suminf_int(F)),suminf_int(G)) = suminf_int(aa_fun5565043122120104496at_int(aTP_Lamm_ka(F),G)) ) ) ) ).
% suminf_add
tff(fact_7607_suminf__diff,axiom,
! [F: fun_nat_real,G: fun_nat_real] :
( summable_real(F)
=> ( summable_real(G)
=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(F)),suminf_real(G)) = suminf_real(aa_fun1839206346889086768t_real(aTP_Lamm_ff(F),G)) ) ) ) ).
% suminf_diff
tff(fact_7608_suminf__divide,axiom,
! [F: fun_nat_complex,C: complex] :
( summable_complex(F)
=> ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_jz(F),C)) = aa_complex_complex(divide1717551699836669952omplex(suminf_complex(F)),C) ) ) ).
% suminf_divide
tff(fact_7609_suminf__divide,axiom,
! [F: fun_nat_real,C: real] :
( summable_real(F)
=> ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_fh(F),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,suminf_real(F)),C) ) ) ).
% suminf_divide
tff(fact_7610_powser__insidea,axiom,
! [F: fun_nat_real,X: real,Z: real] :
( summable_real(aa_real_fun_nat_real(aTP_Lamm_kc(F),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(real_V7735802525324610683m_real,X)))
=> summable_real(aa_real_fun_nat_real(aTP_Lamm_kd(F),Z)) ) ) ).
% powser_insidea
tff(fact_7611_powser__insidea,axiom,
! [F: fun_nat_complex,X: complex,Z: complex] :
( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ke(F),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(real_V1022390504157884413omplex,X)))
=> summable_real(aa_com20334259897932645t_real(aTP_Lamm_kf(F),Z)) ) ) ).
% powser_insidea
tff(fact_7612_suminf__nonneg,axiom,
! [F: fun_nat_real] :
( summable_real(F)
=> ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),suminf_real(F))) ) ) ).
% suminf_nonneg
tff(fact_7613_suminf__nonneg,axiom,
! [F: fun_nat_nat] :
( summable_nat(F)
=> ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),suminf_nat(F))) ) ) ).
% suminf_nonneg
tff(fact_7614_suminf__nonneg,axiom,
! [F: fun_nat_int] :
( summable_int(F)
=> ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),suminf_int(F))) ) ) ).
% suminf_nonneg
tff(fact_7615_suminf__eq__zero__iff,axiom,
! [F: fun_nat_real] :
( summable_real(F)
=> ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
=> ( ( suminf_real(F) = zero_zero_real )
<=> ! [N5: nat] : aa_nat_real(F,N5) = zero_zero_real ) ) ) ).
% suminf_eq_zero_iff
tff(fact_7616_suminf__eq__zero__iff,axiom,
! [F: fun_nat_nat] :
( summable_nat(F)
=> ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
=> ( ( suminf_nat(F) = zero_zero_nat )
<=> ! [N5: nat] : aa_nat_nat(F,N5) = zero_zero_nat ) ) ) ).
% suminf_eq_zero_iff
tff(fact_7617_suminf__eq__zero__iff,axiom,
! [F: fun_nat_int] :
( summable_int(F)
=> ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
=> ( ( suminf_int(F) = zero_zero_int )
<=> ! [N5: nat] : aa_nat_int(F,N5) = zero_zero_int ) ) ) ).
% suminf_eq_zero_iff
tff(fact_7618_suminf__pos,axiom,
! [F: fun_nat_real] :
( summable_real(F)
=> ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,N2)))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),suminf_real(F))) ) ) ).
% suminf_pos
tff(fact_7619_suminf__pos,axiom,
! [F: fun_nat_nat] :
( summable_nat(F)
=> ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,N2)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),suminf_nat(F))) ) ) ).
% suminf_pos
tff(fact_7620_suminf__pos,axiom,
! [F: fun_nat_int] :
( summable_int(F)
=> ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,N2)))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),suminf_int(F))) ) ) ).
% suminf_pos
tff(fact_7621_summable__zero__power_H,axiom,
! [F: fun_nat_complex] : summable_complex(aTP_Lamm_ge(F)) ).
% summable_zero_power'
tff(fact_7622_summable__zero__power_H,axiom,
! [F: fun_nat_real] : summable_real(aTP_Lamm_gg(F)) ).
% summable_zero_power'
tff(fact_7623_summable__zero__power_H,axiom,
! [F: fun_nat_int] : summable_int(aTP_Lamm_kg(F)) ).
% summable_zero_power'
tff(fact_7624_summable__0__powser,axiom,
! [F: fun_nat_complex] : summable_complex(aTP_Lamm_ge(F)) ).
% summable_0_powser
tff(fact_7625_summable__0__powser,axiom,
! [F: fun_nat_real] : summable_real(aTP_Lamm_gg(F)) ).
% summable_0_powser
tff(fact_7626_powser__split__head_I3_J,axiom,
! [F: fun_nat_complex,Z: complex] :
( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ke(F),Z))
=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_kh(F),Z)) ) ).
% powser_split_head(3)
tff(fact_7627_powser__split__head_I3_J,axiom,
! [F: fun_nat_real,Z: real] :
( summable_real(aa_real_fun_nat_real(aTP_Lamm_kc(F),Z))
=> summable_real(aa_real_fun_nat_real(aTP_Lamm_ki(F),Z)) ) ).
% powser_split_head(3)
tff(fact_7628_summable__powser__split__head,axiom,
! [F: fun_nat_complex,Z: complex] :
( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_kh(F),Z))
<=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ke(F),Z)) ) ).
% summable_powser_split_head
tff(fact_7629_summable__powser__split__head,axiom,
! [F: fun_nat_real,Z: real] :
( summable_real(aa_real_fun_nat_real(aTP_Lamm_ki(F),Z))
<=> summable_real(aa_real_fun_nat_real(aTP_Lamm_kc(F),Z)) ) ).
% summable_powser_split_head
tff(fact_7630_summable__powser__ignore__initial__segment,axiom,
! [F: fun_nat_complex,M: nat,Z: complex] :
( summable_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_kj(F),M),Z))
<=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ke(F),Z)) ) ).
% summable_powser_ignore_initial_segment
tff(fact_7631_summable__powser__ignore__initial__segment,axiom,
! [F: fun_nat_real,M: nat,Z: real] :
( summable_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_kk(F),M),Z))
<=> summable_real(aa_real_fun_nat_real(aTP_Lamm_kc(F),Z)) ) ).
% summable_powser_ignore_initial_segment
tff(fact_7632_pi__gt__zero,axiom,
pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),pi)) ).
% pi_gt_zero
tff(fact_7633_pi__not__less__zero,axiom,
~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,pi),zero_zero_real)) ).
% pi_not_less_zero
tff(fact_7634_pi__ge__zero,axiom,
pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),pi)) ).
% pi_ge_zero
tff(fact_7635_summable__norm__comparison__test,axiom,
! [F: fun_nat_complex,G: fun_nat_real] :
( ? [N7: nat] :
! [N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N2))),aa_nat_real(G,N2))) )
=> ( summable_real(G)
=> summable_real(aTP_Lamm_jt(F)) ) ) ).
% summable_norm_comparison_test
tff(fact_7636_summable__rabs__comparison__test,axiom,
! [F: fun_nat_real,G: fun_nat_real] :
( ? [N7: nat] :
! [N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(F,N2))),aa_nat_real(G,N2))) )
=> ( summable_real(G)
=> summable_real(aTP_Lamm_ei(F)) ) ) ).
% summable_rabs_comparison_test
tff(fact_7637_summable__rabs,axiom,
! [F: fun_nat_real] :
( summable_real(aTP_Lamm_ei(F))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,suminf_real(F))),suminf_real(aTP_Lamm_ei(F)))) ) ).
% summable_rabs
tff(fact_7638_suminf__pos2,axiom,
! [F: fun_nat_real,I: nat] :
( summable_real(F)
=> ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,I)))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),suminf_real(F))) ) ) ) ).
% suminf_pos2
tff(fact_7639_suminf__pos2,axiom,
! [F: fun_nat_nat,I: nat] :
( summable_nat(F)
=> ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),suminf_nat(F))) ) ) ) ).
% suminf_pos2
tff(fact_7640_suminf__pos2,axiom,
! [F: fun_nat_int,I: nat] :
( summable_int(F)
=> ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,I)))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),suminf_int(F))) ) ) ) ).
% suminf_pos2
tff(fact_7641_suminf__pos__iff,axiom,
! [F: fun_nat_real] :
( summable_real(F)
=> ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),suminf_real(F)))
<=> ? [I4: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,I4))) ) ) ) ).
% suminf_pos_iff
tff(fact_7642_suminf__pos__iff,axiom,
! [F: fun_nat_nat] :
( summable_nat(F)
=> ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),suminf_nat(F)))
<=> ? [I4: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I4))) ) ) ) ).
% suminf_pos_iff
tff(fact_7643_suminf__pos__iff,axiom,
! [F: fun_nat_int] :
( summable_int(F)
=> ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),suminf_int(F)))
<=> ? [I4: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,I4))) ) ) ) ).
% suminf_pos_iff
tff(fact_7644_suminf__le__const,axiom,
! [F: fun_nat_int,X: int] :
( summable_int(F)
=> ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N2))),X))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,suminf_int(F)),X)) ) ) ).
% suminf_le_const
tff(fact_7645_suminf__le__const,axiom,
! [F: fun_nat_nat,X: nat] :
( summable_nat(F)
=> ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N2))),X))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,suminf_nat(F)),X)) ) ) ).
% suminf_le_const
tff(fact_7646_suminf__le__const,axiom,
! [F: fun_nat_real,X: real] :
( summable_real(F)
=> ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,N2))),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,suminf_real(F)),X)) ) ) ).
% suminf_le_const
tff(fact_7647_summableI__nonneg__bounded,axiom,
! [F: fun_nat_int,X: int] :
( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2)))
=> ( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N2))),X))
=> summable_int(F) ) ) ).
% summableI_nonneg_bounded
tff(fact_7648_summableI__nonneg__bounded,axiom,
! [F: fun_nat_nat,X: nat] :
( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2)))
=> ( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N2))),X))
=> summable_nat(F) ) ) ).
% summableI_nonneg_bounded
tff(fact_7649_summableI__nonneg__bounded,axiom,
! [F: fun_nat_real,X: real] :
( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2)))
=> ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,N2))),X))
=> summable_real(F) ) ) ).
% summableI_nonneg_bounded
tff(fact_7650_summable__geometric,axiom,
! [C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real))
=> summable_real(aa_real_fun_nat_real(power_power_real,C)) ) ).
% summable_geometric
tff(fact_7651_summable__geometric,axiom,
! [C: complex] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real))
=> summable_complex(power_power_complex(C)) ) ).
% summable_geometric
tff(fact_7652_complete__algebra__summable__geometric,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),one_one_real))
=> summable_real(aa_real_fun_nat_real(power_power_real,X)) ) ).
% complete_algebra_summable_geometric
tff(fact_7653_complete__algebra__summable__geometric,axiom,
! [X: complex] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),one_one_real))
=> summable_complex(power_power_complex(X)) ) ).
% complete_algebra_summable_geometric
tff(fact_7654_suminf__split__head,axiom,
! [F: fun_nat_real] :
( summable_real(F)
=> ( suminf_real(aTP_Lamm_gq(F)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(F)),aa_nat_real(F,zero_zero_nat)) ) ) ).
% suminf_split_head
tff(fact_7655_summable__norm,axiom,
! [F: fun_nat_real] :
( summable_real(aTP_Lamm_jv(F))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,suminf_real(F))),suminf_real(aTP_Lamm_jv(F)))) ) ).
% summable_norm
tff(fact_7656_summable__norm,axiom,
! [F: fun_nat_complex] :
( summable_real(aTP_Lamm_jt(F))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,suminf_complex(F))),suminf_real(aTP_Lamm_jt(F)))) ) ).
% summable_norm
tff(fact_7657_sum__le__suminf,axiom,
! [F: fun_nat_int,I5: set_nat] :
( summable_int(F)
=> ( pp(aa_set_nat_bool(finite_finite_nat,I5))
=> ( ! [N2: nat] :
( pp(aa_set_nat_bool(member_nat(N2),uminus5710092332889474511et_nat(I5)))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N2))) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups3539618377306564664at_int(F,I5)),suminf_int(F))) ) ) ) ).
% sum_le_suminf
tff(fact_7658_sum__le__suminf,axiom,
! [F: fun_nat_nat,I5: set_nat] :
( summable_nat(F)
=> ( pp(aa_set_nat_bool(finite_finite_nat,I5))
=> ( ! [N2: nat] :
( pp(aa_set_nat_bool(member_nat(N2),uminus5710092332889474511et_nat(I5)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N2))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),I5)),suminf_nat(F))) ) ) ) ).
% sum_le_suminf
tff(fact_7659_sum__le__suminf,axiom,
! [F: fun_nat_real,I5: set_nat] :
( summable_real(F)
=> ( pp(aa_set_nat_bool(finite_finite_nat,I5))
=> ( ! [N2: nat] :
( pp(aa_set_nat_bool(member_nat(N2),uminus5710092332889474511et_nat(I5)))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N2))) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups6591440286371151544t_real(F,I5)),suminf_real(F))) ) ) ) ).
% sum_le_suminf
tff(fact_7660_suminf__split__initial__segment,axiom,
! [F: fun_nat_real,K: nat] :
( summable_real(F)
=> ( suminf_real(F) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,suminf_real(aa_nat_fun_nat_real(aTP_Lamm_gs(F),K))),groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,K))) ) ) ).
% suminf_split_initial_segment
tff(fact_7661_suminf__minus__initial__segment,axiom,
! [F: fun_nat_real,K: nat] :
( summable_real(F)
=> ( suminf_real(aa_nat_fun_nat_real(aTP_Lamm_gs(F),K)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(F)),groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,K))) ) ) ).
% suminf_minus_initial_segment
tff(fact_7662_powser__inside,axiom,
! [F: fun_nat_real,X: real,Z: real] :
( summable_real(aa_real_fun_nat_real(aTP_Lamm_kc(F),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(real_V7735802525324610683m_real,X)))
=> summable_real(aa_real_fun_nat_real(aTP_Lamm_kc(F),Z)) ) ) ).
% powser_inside
tff(fact_7663_powser__inside,axiom,
! [F: fun_nat_complex,X: complex,Z: complex] :
( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ke(F),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(real_V1022390504157884413omplex,X)))
=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ke(F),Z)) ) ) ).
% powser_inside
tff(fact_7664_sum__less__suminf,axiom,
! [F: fun_nat_int,N: nat] :
( summable_int(F)
=> ( ! [M3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M3))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,M3))) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_int(F))) ) ) ).
% sum_less_suminf
tff(fact_7665_sum__less__suminf,axiom,
! [F: fun_nat_nat,N: nat] :
( summable_nat(F)
=> ( ! [M3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M3))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,M3))) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_nat(F))) ) ) ).
% sum_less_suminf
tff(fact_7666_sum__less__suminf,axiom,
! [F: fun_nat_real,N: nat] :
( summable_real(F)
=> ( ! [M3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M3))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,M3))) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_real(F))) ) ) ).
% sum_less_suminf
tff(fact_7667_pi__less__4,axiom,
pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one))))) ).
% pi_less_4
tff(fact_7668_pi__ge__two,axiom,
pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)) ).
% pi_ge_two
tff(fact_7669_pi__half__neq__two,axiom,
aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))) != numeral_numeral_real(aa_num_num(bit0,one)) ).
% pi_half_neq_two
tff(fact_7670_powser__split__head_I1_J,axiom,
! [F: fun_nat_complex,Z: complex] :
( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ke(F),Z))
=> ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_ke(F),Z)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(F,zero_zero_nat)),aa_complex_complex(times_times_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_kh(F),Z))),Z)) ) ) ).
% powser_split_head(1)
tff(fact_7671_powser__split__head_I1_J,axiom,
! [F: fun_nat_real,Z: real] :
( summable_real(aa_real_fun_nat_real(aTP_Lamm_kc(F),Z))
=> ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_kc(F),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,zero_zero_nat)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_ki(F),Z))),Z)) ) ) ).
% powser_split_head(1)
tff(fact_7672_powser__split__head_I2_J,axiom,
! [F: fun_nat_complex,Z: complex] :
( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ke(F),Z))
=> ( aa_complex_complex(times_times_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_kh(F),Z))),Z) = aa_complex_complex(minus_minus_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_ke(F),Z))),aa_nat_complex(F,zero_zero_nat)) ) ) ).
% powser_split_head(2)
tff(fact_7673_powser__split__head_I2_J,axiom,
! [F: fun_nat_real,Z: real] :
( summable_real(aa_real_fun_nat_real(aTP_Lamm_kc(F),Z))
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_ki(F),Z))),Z) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_kc(F),Z))),aa_nat_real(F,zero_zero_nat)) ) ) ).
% powser_split_head(2)
tff(fact_7674_summable__partial__sum__bound,axiom,
! [F: fun_nat_complex,E: real] :
( summable_complex(F)
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
=> ~ ! [N8: nat] :
~ ! [M2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),M2))
=> ! [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,groups2073611262835488442omplex(F,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N6)))),E)) ) ) ) ).
% summable_partial_sum_bound
tff(fact_7675_summable__partial__sum__bound,axiom,
! [F: fun_nat_real,E: real] :
( summable_real(F)
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
=> ~ ! [N8: nat] :
~ ! [M2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),M2))
=> ! [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),N6)))),E)) ) ) ) ).
% summable_partial_sum_bound
tff(fact_7676_suminf__exist__split,axiom,
! [R: real,F: fun_nat_real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R))
=> ( summable_real(F)
=> ? [N8: nat] :
! [N6: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),N6))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,suminf_real(aa_nat_fun_nat_real(aTP_Lamm_gs(F),N6)))),R)) ) ) ) ).
% suminf_exist_split
tff(fact_7677_suminf__exist__split,axiom,
! [R: real,F: fun_nat_complex] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R))
=> ( summable_complex(F)
=> ? [N8: nat] :
! [N6: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N8),N6))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,suminf_complex(aa_nat6454746962644518025omplex(aTP_Lamm_kl(F),N6)))),R)) ) ) ) ).
% suminf_exist_split
tff(fact_7678_summable__power__series,axiom,
! [F: fun_nat_real,Z: real] :
( ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,I2)),one_one_real))
=> ( ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I2)))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Z))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z),one_one_real))
=> summable_real(aa_real_fun_nat_real(aTP_Lamm_kc(F),Z)) ) ) ) ) ).
% summable_power_series
tff(fact_7679_Abel__lemma,axiom,
! [R: real,R0: real,A: fun_nat_complex,M6: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),R))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,R),R0))
=> ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(A,N2))),aa_nat_real(aa_real_fun_nat_real(power_power_real,R0),N2))),M6))
=> summable_real(aa_fun7054779078796324018t_real(aTP_Lamm_km(R),A)) ) ) ) ).
% Abel_lemma
tff(fact_7680_summable__ratio__test,axiom,
! [C: real,N3: nat,F: fun_nat_real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),one_one_real))
=> ( ! [N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N3),N2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,aa_nat_nat(suc,N2)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N2))))) )
=> summable_real(F) ) ) ).
% summable_ratio_test
tff(fact_7681_summable__ratio__test,axiom,
! [C: real,N3: nat,F: fun_nat_complex] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),one_one_real))
=> ( ! [N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N3),N2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,aa_nat_nat(suc,N2)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N2))))) )
=> summable_complex(F) ) ) ).
% summable_ratio_test
tff(fact_7682_sum__less__suminf2,axiom,
! [F: fun_nat_int,N: nat,I: nat] :
( summable_int(F)
=> ( ! [M3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M3))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,M3))) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),I))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(F,I)))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_int(F))) ) ) ) ) ).
% sum_less_suminf2
tff(fact_7683_sum__less__suminf2,axiom,
! [F: fun_nat_nat,N: nat,I: nat] :
( summable_nat(F)
=> ( ! [M3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M3))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,M3))) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),I))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_nat(F))) ) ) ) ) ).
% sum_less_suminf2
tff(fact_7684_sum__less__suminf2,axiom,
! [F: fun_nat_real,N: nat,I: nat] :
( summable_real(F)
=> ( ! [M3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M3))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,M3))) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),I))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(F,I)))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,N))),suminf_real(F))) ) ) ) ) ).
% sum_less_suminf2
tff(fact_7685_pi__half__neq__zero,axiom,
aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))) != zero_zero_real ).
% pi_half_neq_zero
tff(fact_7686_pi__half__less__two,axiom,
pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one)))) ).
% pi_half_less_two
tff(fact_7687_pi__half__le__two,axiom,
pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one)))) ).
% pi_half_le_two
tff(fact_7688_pi__half__gt__zero,axiom,
pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))) ).
% pi_half_gt_zero
tff(fact_7689_pi__half__ge__zero,axiom,
pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))) ).
% pi_half_ge_zero
tff(fact_7690_m2pi__less__pi,axiom,
pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))),pi)) ).
% m2pi_less_pi
tff(fact_7691_arctan__ubound,axiom,
! [Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))) ).
% arctan_ubound
tff(fact_7692_arctan__one,axiom,
aa_real_real(arctan,one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) ).
% arctan_one
tff(fact_7693_minus__pi__half__less__zero,axiom,
pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),zero_zero_real)) ).
% minus_pi_half_less_zero
tff(fact_7694_arctan__bounded,axiom,
! [Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arctan,Y)))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))) ) ).
% arctan_bounded
tff(fact_7695_arctan__lbound,axiom,
! [Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arctan,Y))) ).
% arctan_lbound
tff(fact_7696_machin__Euler,axiom,
aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit0,one)))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(aa_num_num(bit1,one))),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) ).
% machin_Euler
tff(fact_7697_machin,axiom,
aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one))))))))))) ).
% machin
tff(fact_7698_sum__pos__lt__pair,axiom,
! [F: fun_nat_real,K: nat] :
( summable_real(F)
=> ( ! [D2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))),D2)))),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))),D2)),one_one_nat))))))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,K))),suminf_real(F))) ) ) ).
% sum_pos_lt_pair
tff(fact_7699_sin__cos__npi,axiom,
! [N: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),pi)),numeral_numeral_real(aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) ).
% sin_cos_npi
tff(fact_7700_and__int_Oelims,axiom,
! [X: int,Xa2: int,Y: int] :
( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Xa2) = Y )
=> ( ( ( pp(aa_set_int_bool(member_int(X),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
& pp(aa_set_int_bool(member_int(Xa2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
=> ( Y = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa2))))) ) )
& ( ~ ( pp(aa_set_int_bool(member_int(X),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
& pp(aa_set_int_bool(member_int(Xa2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
=> ( Y = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa2))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ).
% and_int.elims
tff(fact_7701_and__int_Osimps,axiom,
! [K: int,L: int] :
( ( ( pp(aa_set_int_bool(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
& pp(aa_set_int_bool(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(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))) ) )
& ( ~ ( pp(aa_set_int_bool(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
& pp(aa_set_int_bool(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(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).
% and_int.simps
tff(fact_7702_cos__pi__eq__zero,axiom,
! [M: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))))),numeral_numeral_real(aa_num_num(bit0,one)))) = zero_zero_real ).
% cos_pi_eq_zero
tff(fact_7703_Maclaurin__exp__lt,axiom,
! [X: real,N: nat] :
( ( X != zero_zero_real )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ? [T4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T4)))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,T4)),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_kn,X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T4)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).
% Maclaurin_exp_lt
tff(fact_7704_Diff__insert0,axiom,
! [X: vEBT_VEBT,A2: set_VEBT_VEBT,B2: set_VEBT_VEBT] :
( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),A2))
=> ( minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(X,B2)) = minus_5127226145743854075T_VEBT(A2,B2) ) ) ).
% Diff_insert0
tff(fact_7705_Diff__insert0,axiom,
! [X: complex,A2: set_complex,B2: set_complex] :
( ~ pp(aa_set_complex_bool(member_complex(X),A2))
=> ( minus_811609699411566653omplex(A2,insert_complex(X,B2)) = minus_811609699411566653omplex(A2,B2) ) ) ).
% Diff_insert0
tff(fact_7706_Diff__insert0,axiom,
! [X: real,A2: set_real,B2: set_real] :
( ~ pp(aa_set_real_bool(member_real(X),A2))
=> ( minus_minus_set_real(A2,insert_real(X,B2)) = minus_minus_set_real(A2,B2) ) ) ).
% Diff_insert0
tff(fact_7707_Diff__insert0,axiom,
! [X: int,A2: set_int,B2: set_int] :
( ~ pp(aa_set_int_bool(member_int(X),A2))
=> ( minus_minus_set_int(A2,insert_int(X,B2)) = minus_minus_set_int(A2,B2) ) ) ).
% Diff_insert0
tff(fact_7708_Diff__insert0,axiom,
! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A2))
=> ( minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(X,B2)) = minus_1356011639430497352at_nat(A2,B2) ) ) ).
% Diff_insert0
tff(fact_7709_Diff__insert0,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ~ pp(aa_set_nat_bool(member_nat(X),A2))
=> ( minus_minus_set_nat(A2,insert_nat(X,B2)) = minus_minus_set_nat(A2,B2) ) ) ).
% Diff_insert0
tff(fact_7710_insert__Diff1,axiom,
! [X: vEBT_VEBT,B2: set_VEBT_VEBT,A2: set_VEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),B2))
=> ( minus_5127226145743854075T_VEBT(insert_VEBT_VEBT(X,A2),B2) = minus_5127226145743854075T_VEBT(A2,B2) ) ) ).
% insert_Diff1
tff(fact_7711_insert__Diff1,axiom,
! [X: complex,B2: set_complex,A2: set_complex] :
( pp(aa_set_complex_bool(member_complex(X),B2))
=> ( minus_811609699411566653omplex(insert_complex(X,A2),B2) = minus_811609699411566653omplex(A2,B2) ) ) ).
% insert_Diff1
tff(fact_7712_insert__Diff1,axiom,
! [X: real,B2: set_real,A2: set_real] :
( pp(aa_set_real_bool(member_real(X),B2))
=> ( minus_minus_set_real(insert_real(X,A2),B2) = minus_minus_set_real(A2,B2) ) ) ).
% insert_Diff1
tff(fact_7713_insert__Diff1,axiom,
! [X: int,B2: set_int,A2: set_int] :
( pp(aa_set_int_bool(member_int(X),B2))
=> ( minus_minus_set_int(insert_int(X,A2),B2) = minus_minus_set_int(A2,B2) ) ) ).
% insert_Diff1
tff(fact_7714_insert__Diff1,axiom,
! [X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B2))
=> ( minus_1356011639430497352at_nat(insert8211810215607154385at_nat(X,A2),B2) = minus_1356011639430497352at_nat(A2,B2) ) ) ).
% insert_Diff1
tff(fact_7715_insert__Diff1,axiom,
! [X: nat,B2: set_nat,A2: set_nat] :
( pp(aa_set_nat_bool(member_nat(X),B2))
=> ( minus_minus_set_nat(insert_nat(X,A2),B2) = minus_minus_set_nat(A2,B2) ) ) ).
% insert_Diff1
tff(fact_7716_cos__zero,axiom,
aa_complex_complex(cos_complex,zero_zero_complex) = one_one_complex ).
% cos_zero
tff(fact_7717_cos__zero,axiom,
aa_real_real(cos_real,zero_zero_real) = one_one_real ).
% cos_zero
tff(fact_7718_insert__Diff__single,axiom,
! [A: vEBT_VEBT,A2: set_VEBT_VEBT] : insert_VEBT_VEBT(A,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) = insert_VEBT_VEBT(A,A2) ).
% insert_Diff_single
tff(fact_7719_insert__Diff__single,axiom,
! [A: int,A2: set_int] : insert_int(A,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = insert_int(A,A2) ).
% insert_Diff_single
tff(fact_7720_insert__Diff__single,axiom,
! [A: real,A2: set_real] : insert_real(A,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = insert_real(A,A2) ).
% insert_Diff_single
tff(fact_7721_insert__Diff__single,axiom,
! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(A,minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))) = insert8211810215607154385at_nat(A,A2) ).
% insert_Diff_single
tff(fact_7722_insert__Diff__single,axiom,
! [A: nat,A2: set_nat] : insert_nat(A,minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = insert_nat(A,A2) ).
% insert_Diff_single
tff(fact_7723_finite__Diff__insert,axiom,
! [A2: set_VEBT_VEBT,A: vEBT_VEBT,B2: set_VEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,B2))))
<=> pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,minus_5127226145743854075T_VEBT(A2,B2))) ) ).
% finite_Diff_insert
tff(fact_7724_finite__Diff__insert,axiom,
! [A2: set_real,A: real,B2: set_real] :
( pp(aa_set_real_bool(finite_finite_real,minus_minus_set_real(A2,insert_real(A,B2))))
<=> pp(aa_set_real_bool(finite_finite_real,minus_minus_set_real(A2,B2))) ) ).
% finite_Diff_insert
tff(fact_7725_finite__Diff__insert,axiom,
! [A2: set_int,A: int,B2: set_int] :
( pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(A2,insert_int(A,B2))))
<=> pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(A2,B2))) ) ).
% finite_Diff_insert
tff(fact_7726_finite__Diff__insert,axiom,
! [A2: set_complex,A: complex,B2: set_complex] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A2,insert_complex(A,B2))))
<=> pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A2,B2))) ) ).
% finite_Diff_insert
tff(fact_7727_finite__Diff__insert,axiom,
! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(A,B2))))
<=> pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(A2,B2))) ) ).
% finite_Diff_insert
tff(fact_7728_finite__Diff__insert,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(A2,insert_nat(A,B2))))
<=> pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(A2,B2))) ) ).
% finite_Diff_insert
tff(fact_7729_fact__0,axiom,
aa_nat_complex(semiri5044797733671781792omplex,zero_zero_nat) = one_one_complex ).
% fact_0
tff(fact_7730_fact__0,axiom,
aa_nat_rat(semiri773545260158071498ct_rat,zero_zero_nat) = one_one_rat ).
% fact_0
tff(fact_7731_fact__0,axiom,
aa_nat_int(semiri1406184849735516958ct_int,zero_zero_nat) = one_one_int ).
% fact_0
tff(fact_7732_fact__0,axiom,
aa_nat_real(semiri2265585572941072030t_real,zero_zero_nat) = one_one_real ).
% fact_0
tff(fact_7733_fact__0,axiom,
aa_nat_nat(semiri1408675320244567234ct_nat,zero_zero_nat) = one_one_nat ).
% fact_0
tff(fact_7734_fact__1,axiom,
aa_nat_complex(semiri5044797733671781792omplex,one_one_nat) = one_one_complex ).
% fact_1
tff(fact_7735_fact__1,axiom,
aa_nat_rat(semiri773545260158071498ct_rat,one_one_nat) = one_one_rat ).
% fact_1
tff(fact_7736_fact__1,axiom,
aa_nat_int(semiri1406184849735516958ct_int,one_one_nat) = one_one_int ).
% fact_1
tff(fact_7737_fact__1,axiom,
aa_nat_real(semiri2265585572941072030t_real,one_one_nat) = one_one_real ).
% fact_1
tff(fact_7738_fact__1,axiom,
aa_nat_nat(semiri1408675320244567234ct_nat,one_one_nat) = one_one_nat ).
% fact_1
tff(fact_7739_sin__pi__minus,axiom,
! [X: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),X)) = aa_real_real(sin_real,X) ).
% sin_pi_minus
tff(fact_7740_sum_Oinsert,axiom,
! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: fun_VEBT_VEBT_real] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
=> ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),A2))
=> ( groups2240296850493347238T_real(G,insert_VEBT_VEBT(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(G,X)),groups2240296850493347238T_real(G,A2)) ) ) ) ).
% sum.insert
tff(fact_7741_sum_Oinsert,axiom,
! [A2: set_real,X: real,G: fun_real_real] :
( pp(aa_set_real_bool(finite_finite_real,A2))
=> ( ~ pp(aa_set_real_bool(member_real(X),A2))
=> ( groups8097168146408367636l_real(G,insert_real(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,X)),groups8097168146408367636l_real(G,A2)) ) ) ) ).
% sum.insert
tff(fact_7742_sum_Oinsert,axiom,
! [A2: set_int,X: int,G: fun_int_real] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( ~ pp(aa_set_int_bool(member_int(X),A2))
=> ( groups8778361861064173332t_real(G,insert_int(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,X)),groups8778361861064173332t_real(G,A2)) ) ) ) ).
% sum.insert
tff(fact_7743_sum_Oinsert,axiom,
! [A2: set_complex,X: complex,G: fun_complex_real] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( ~ pp(aa_set_complex_bool(member_complex(X),A2))
=> ( groups5808333547571424918x_real(G,insert_complex(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,X)),groups5808333547571424918x_real(G,A2)) ) ) ) ).
% sum.insert
tff(fact_7744_sum_Oinsert,axiom,
! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: fun_VEBT_VEBT_rat] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
=> ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),A2))
=> ( groups136491112297645522BT_rat(G,insert_VEBT_VEBT(X,A2)) = plus_plus_rat(aa_VEBT_VEBT_rat(G,X),groups136491112297645522BT_rat(G,A2)) ) ) ) ).
% sum.insert
tff(fact_7745_sum_Oinsert,axiom,
! [A2: set_real,X: real,G: fun_real_rat] :
( pp(aa_set_real_bool(finite_finite_real,A2))
=> ( ~ pp(aa_set_real_bool(member_real(X),A2))
=> ( groups1300246762558778688al_rat(G,insert_real(X,A2)) = plus_plus_rat(aa_real_rat(G,X),groups1300246762558778688al_rat(G,A2)) ) ) ) ).
% sum.insert
tff(fact_7746_sum_Oinsert,axiom,
! [A2: set_nat,X: nat,G: fun_nat_rat] :
( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
=> ( groups2906978787729119204at_rat(G,insert_nat(X,A2)) = plus_plus_rat(aa_nat_rat(G,X),groups2906978787729119204at_rat(G,A2)) ) ) ) ).
% sum.insert
tff(fact_7747_sum_Oinsert,axiom,
! [A2: set_int,X: int,G: fun_int_rat] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( ~ pp(aa_set_int_bool(member_int(X),A2))
=> ( groups3906332499630173760nt_rat(G,insert_int(X,A2)) = plus_plus_rat(aa_int_rat(G,X),groups3906332499630173760nt_rat(G,A2)) ) ) ) ).
% sum.insert
tff(fact_7748_sum_Oinsert,axiom,
! [A2: set_complex,X: complex,G: fun_complex_rat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( ~ pp(aa_set_complex_bool(member_complex(X),A2))
=> ( groups5058264527183730370ex_rat(G,insert_complex(X,A2)) = plus_plus_rat(aa_complex_rat(G,X),groups5058264527183730370ex_rat(G,A2)) ) ) ) ).
% sum.insert
tff(fact_7749_sum_Oinsert,axiom,
! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: fun_VEBT_VEBT_nat] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
=> ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),A2))
=> ( groups771621172384141258BT_nat(G,insert_VEBT_VEBT(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(G,X)),groups771621172384141258BT_nat(G,A2)) ) ) ) ).
% sum.insert
tff(fact_7750_fact__Suc__0,axiom,
aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,zero_zero_nat)) = one_one_complex ).
% fact_Suc_0
tff(fact_7751_fact__Suc__0,axiom,
aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,zero_zero_nat)) = one_one_rat ).
% fact_Suc_0
tff(fact_7752_fact__Suc__0,axiom,
aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).
% fact_Suc_0
tff(fact_7753_fact__Suc__0,axiom,
aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,zero_zero_nat)) = one_one_real ).
% fact_Suc_0
tff(fact_7754_fact__Suc__0,axiom,
aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).
% fact_Suc_0
tff(fact_7755_fact__Suc,axiom,
! [N: nat] : aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))),aa_nat_int(semiri1406184849735516958ct_int,N)) ).
% fact_Suc
tff(fact_7756_fact__Suc,axiom,
! [N: nat] : aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,N)) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,N))),aa_nat_complex(semiri5044797733671781792omplex,N)) ).
% fact_Suc
tff(fact_7757_fact__Suc,axiom,
! [N: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N))),aa_nat_rat(semiri773545260158071498ct_rat,N)) ).
% fact_Suc
tff(fact_7758_fact__Suc,axiom,
! [N: nat] : aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))),aa_nat_real(semiri2265585572941072030t_real,N)) ).
% fact_Suc
tff(fact_7759_fact__Suc,axiom,
! [N: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,N))),aa_nat_nat(semiri1408675320244567234ct_nat,N)) ).
% fact_Suc
tff(fact_7760_single__Diff__lessThan,axiom,
! [K: int] : minus_minus_set_int(insert_int(K,bot_bot_set_int),set_ord_lessThan_int(K)) = insert_int(K,bot_bot_set_int) ).
% single_Diff_lessThan
tff(fact_7761_single__Diff__lessThan,axiom,
! [K: nat] : minus_minus_set_nat(insert_nat(K,bot_bot_set_nat),aa_nat_set_nat(set_ord_lessThan_nat,K)) = insert_nat(K,bot_bot_set_nat) ).
% single_Diff_lessThan
tff(fact_7762_single__Diff__lessThan,axiom,
! [K: real] : minus_minus_set_real(insert_real(K,bot_bot_set_real),set_or5984915006950818249n_real(K)) = insert_real(K,bot_bot_set_real) ).
% single_Diff_lessThan
tff(fact_7763_cos__pi,axiom,
aa_real_real(cos_real,pi) = aa_real_real(uminus_uminus_real,one_one_real) ).
% cos_pi
tff(fact_7764_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_7765_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_7766_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_7767_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_7768_cos__pi__minus,axiom,
! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),X)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,X)) ).
% cos_pi_minus
tff(fact_7769_cos__minus__pi,axiom,
! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,X)) ).
% cos_minus_pi
tff(fact_7770_sin__minus__pi,axiom,
! [X: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X)) ).
% sin_minus_pi
tff(fact_7771_fact__2,axiom,
aa_nat_complex(semiri5044797733671781792omplex,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).
% fact_2
tff(fact_7772_fact__2,axiom,
aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = numeral_numeral_rat(aa_num_num(bit0,one)) ).
% fact_2
tff(fact_7773_fact__2,axiom,
aa_nat_int(semiri1406184849735516958ct_int,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).
% fact_2
tff(fact_7774_fact__2,axiom,
aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = numeral_numeral_real(aa_num_num(bit0,one)) ).
% fact_2
tff(fact_7775_fact__2,axiom,
aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).
% fact_2
tff(fact_7776_sin__cos__squared__add3,axiom,
! [X: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,X))),aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,X)),aa_complex_complex(sin_complex,X))) = one_one_complex ).
% sin_cos_squared_add3
tff(fact_7777_sin__cos__squared__add3,axiom,
! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,X))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,X))) = one_one_real ).
% sin_cos_squared_add3
tff(fact_7778_set__replicate,axiom,
! [N: nat,X: vEBT_VEBT] :
( ( N != zero_zero_nat )
=> ( set_VEBT_VEBT2(replicate_VEBT_VEBT(N,X)) = insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT) ) ) ).
% set_replicate
tff(fact_7779_set__replicate,axiom,
! [N: nat,X: nat] :
( ( N != zero_zero_nat )
=> ( set_nat2(replicate_nat(N,X)) = insert_nat(X,bot_bot_set_nat) ) ) ).
% set_replicate
tff(fact_7780_set__replicate,axiom,
! [N: nat,X: int] :
( ( N != zero_zero_nat )
=> ( set_int2(replicate_int(N,X)) = insert_int(X,bot_bot_set_int) ) ) ).
% set_replicate
tff(fact_7781_set__replicate,axiom,
! [N: nat,X: real] :
( ( N != zero_zero_nat )
=> ( set_real2(replicate_real(N,X)) = insert_real(X,bot_bot_set_real) ) ) ).
% set_replicate
tff(fact_7782_sin__npi,axiom,
! [N: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),pi)) = zero_zero_real ).
% sin_npi
tff(fact_7783_sin__npi2,axiom,
! [N: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,N))) = zero_zero_real ).
% sin_npi2
tff(fact_7784_sin__npi__int,axiom,
! [N: int] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,N))) = zero_zero_real ).
% sin_npi_int
tff(fact_7785_cos__pi__half,axiom,
aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) = zero_zero_real ).
% cos_pi_half
tff(fact_7786_sin__two__pi,axiom,
aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)) = zero_zero_real ).
% sin_two_pi
tff(fact_7787_sin__pi__half,axiom,
aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) = one_one_real ).
% sin_pi_half
tff(fact_7788_cos__two__pi,axiom,
aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)) = one_one_real ).
% cos_two_pi
tff(fact_7789_cos__periodic,axiom,
! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))) = aa_real_real(cos_real,X) ).
% cos_periodic
tff(fact_7790_sin__periodic,axiom,
! [X: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))) = aa_real_real(sin_real,X) ).
% sin_periodic
tff(fact_7791_cos__2pi__minus,axiom,
! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),X)) = aa_real_real(cos_real,X) ).
% cos_2pi_minus
tff(fact_7792_cos__npi2,axiom,
! [N: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) ).
% cos_npi2
tff(fact_7793_cos__npi,axiom,
! [N: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),pi)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) ).
% cos_npi
tff(fact_7794_sin__cos__squared__add2,axiom,
! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).
% sin_cos_squared_add2
tff(fact_7795_sin__cos__squared__add2,axiom,
! [X: complex] : aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).
% sin_cos_squared_add2
tff(fact_7796_sin__cos__squared__add,axiom,
! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).
% sin_cos_squared_add
tff(fact_7797_sin__cos__squared__add,axiom,
! [X: complex] : aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).
% sin_cos_squared_add
tff(fact_7798_sin__2npi,axiom,
! [N: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)) = zero_zero_real ).
% sin_2npi
tff(fact_7799_cos__2npi,axiom,
! [N: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)) = one_one_real ).
% cos_2npi
tff(fact_7800_sin__2pi__minus,axiom,
! [X: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),X)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X)) ).
% sin_2pi_minus
tff(fact_7801_sin__int__2pin,axiom,
! [N: int] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,N))) = zero_zero_real ).
% sin_int_2pin
tff(fact_7802_cos__int__2pin,axiom,
! [N: int] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,N))) = one_one_real ).
% cos_int_2pin
tff(fact_7803_cos__3over2__pi,axiom,
aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(aa_num_num(bit1,one))),numeral_numeral_real(aa_num_num(bit0,one)))),pi)) = zero_zero_real ).
% cos_3over2_pi
tff(fact_7804_sin__3over2__pi,axiom,
aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(aa_num_num(bit1,one))),numeral_numeral_real(aa_num_num(bit0,one)))),pi)) = aa_real_real(uminus_uminus_real,one_one_real) ).
% sin_3over2_pi
tff(fact_7805_cos__npi__int,axiom,
! [N: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
=> ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,N))) = one_one_real ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
=> ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,N))) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).
% cos_npi_int
tff(fact_7806_polar__Ex,axiom,
! [X: real,Y: real] :
? [R3: real,A4: real] :
( ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R3),aa_real_real(cos_real,A4)) )
& ( Y = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R3),aa_real_real(sin_real,A4)) ) ) ).
% polar_Ex
tff(fact_7807_sin__diff,axiom,
! [X: real,Y: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(sin_real,Y))) ).
% sin_diff
tff(fact_7808_cos__one__sin__zero,axiom,
! [X: complex] :
( ( aa_complex_complex(cos_complex,X) = one_one_complex )
=> ( aa_complex_complex(sin_complex,X) = zero_zero_complex ) ) ).
% cos_one_sin_zero
tff(fact_7809_cos__one__sin__zero,axiom,
! [X: real] :
( ( aa_real_real(cos_real,X) = one_one_real )
=> ( aa_real_real(sin_real,X) = zero_zero_real ) ) ).
% cos_one_sin_zero
tff(fact_7810_sin__add,axiom,
! [X: real,Y: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(sin_real,Y))) ).
% sin_add
tff(fact_7811_insert__Diff__if,axiom,
! [X: vEBT_VEBT,B2: set_VEBT_VEBT,A2: set_VEBT_VEBT] :
( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),B2))
=> ( minus_5127226145743854075T_VEBT(insert_VEBT_VEBT(X,A2),B2) = minus_5127226145743854075T_VEBT(A2,B2) ) )
& ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),B2))
=> ( minus_5127226145743854075T_VEBT(insert_VEBT_VEBT(X,A2),B2) = insert_VEBT_VEBT(X,minus_5127226145743854075T_VEBT(A2,B2)) ) ) ) ).
% insert_Diff_if
tff(fact_7812_insert__Diff__if,axiom,
! [X: complex,B2: set_complex,A2: set_complex] :
( ( pp(aa_set_complex_bool(member_complex(X),B2))
=> ( minus_811609699411566653omplex(insert_complex(X,A2),B2) = minus_811609699411566653omplex(A2,B2) ) )
& ( ~ pp(aa_set_complex_bool(member_complex(X),B2))
=> ( minus_811609699411566653omplex(insert_complex(X,A2),B2) = insert_complex(X,minus_811609699411566653omplex(A2,B2)) ) ) ) ).
% insert_Diff_if
tff(fact_7813_insert__Diff__if,axiom,
! [X: real,B2: set_real,A2: set_real] :
( ( pp(aa_set_real_bool(member_real(X),B2))
=> ( minus_minus_set_real(insert_real(X,A2),B2) = minus_minus_set_real(A2,B2) ) )
& ( ~ pp(aa_set_real_bool(member_real(X),B2))
=> ( minus_minus_set_real(insert_real(X,A2),B2) = insert_real(X,minus_minus_set_real(A2,B2)) ) ) ) ).
% insert_Diff_if
tff(fact_7814_insert__Diff__if,axiom,
! [X: int,B2: set_int,A2: set_int] :
( ( pp(aa_set_int_bool(member_int(X),B2))
=> ( minus_minus_set_int(insert_int(X,A2),B2) = minus_minus_set_int(A2,B2) ) )
& ( ~ pp(aa_set_int_bool(member_int(X),B2))
=> ( minus_minus_set_int(insert_int(X,A2),B2) = insert_int(X,minus_minus_set_int(A2,B2)) ) ) ) ).
% insert_Diff_if
tff(fact_7815_insert__Diff__if,axiom,
! [X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B2))
=> ( minus_1356011639430497352at_nat(insert8211810215607154385at_nat(X,A2),B2) = minus_1356011639430497352at_nat(A2,B2) ) )
& ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B2))
=> ( minus_1356011639430497352at_nat(insert8211810215607154385at_nat(X,A2),B2) = insert8211810215607154385at_nat(X,minus_1356011639430497352at_nat(A2,B2)) ) ) ) ).
% insert_Diff_if
tff(fact_7816_insert__Diff__if,axiom,
! [X: nat,B2: set_nat,A2: set_nat] :
( ( pp(aa_set_nat_bool(member_nat(X),B2))
=> ( minus_minus_set_nat(insert_nat(X,A2),B2) = minus_minus_set_nat(A2,B2) ) )
& ( ~ pp(aa_set_nat_bool(member_nat(X),B2))
=> ( minus_minus_set_nat(insert_nat(X,A2),B2) = insert_nat(X,minus_minus_set_nat(A2,B2)) ) ) ) ).
% insert_Diff_if
tff(fact_7817_cos__diff,axiom,
! [X: real,Y: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y))) ).
% cos_diff
tff(fact_7818_cos__add,axiom,
! [X: real,Y: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y))) ).
% cos_add
tff(fact_7819_sin__zero__norm__cos__one,axiom,
! [X: real] :
( ( aa_real_real(sin_real,X) = zero_zero_real )
=> ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(cos_real,X)) = one_one_real ) ) ).
% sin_zero_norm_cos_one
tff(fact_7820_sin__zero__norm__cos__one,axiom,
! [X: complex] :
( ( aa_complex_complex(sin_complex,X) = zero_zero_complex )
=> ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(cos_complex,X)) = one_one_real ) ) ).
% sin_zero_norm_cos_one
tff(fact_7821_sin__zero__abs__cos__one,axiom,
! [X: real] :
( ( aa_real_real(sin_real,X) = zero_zero_real )
=> ( aa_real_real(abs_abs_real,aa_real_real(cos_real,X)) = one_one_real ) ) ).
% sin_zero_abs_cos_one
tff(fact_7822_sin__double,axiom,
! [X: complex] : aa_complex_complex(sin_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,X))),aa_complex_complex(cos_complex,X)) ).
% sin_double
tff(fact_7823_sin__double,axiom,
! [X: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),X)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(sin_real,X))),aa_real_real(cos_real,X)) ).
% sin_double
tff(fact_7824_sincos__principal__value,axiom,
! [X: real] :
? [Y2: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Y2))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_7825_sin__x__le__x,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,X)),X)) ) ).
% sin_x_le_x
tff(fact_7826_sin__le__one,axiom,
! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,X)),one_one_real)) ).
% sin_le_one
tff(fact_7827_cos__le__one,axiom,
! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,X)),one_one_real)) ).
% cos_le_one
tff(fact_7828_fact__ge__zero,axiom,
! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,N))) ).
% fact_ge_zero
tff(fact_7829_fact__ge__zero,axiom,
! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,N))) ).
% fact_ge_zero
tff(fact_7830_fact__ge__zero,axiom,
! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,N))) ).
% fact_ge_zero
tff(fact_7831_fact__ge__zero,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).
% fact_ge_zero
tff(fact_7832_fact__gt__zero,axiom,
! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,N))) ).
% fact_gt_zero
tff(fact_7833_fact__gt__zero,axiom,
! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,N))) ).
% fact_gt_zero
tff(fact_7834_fact__gt__zero,axiom,
! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,N))) ).
% fact_gt_zero
tff(fact_7835_fact__gt__zero,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).
% fact_gt_zero
tff(fact_7836_fact__not__neg,axiom,
! [N: nat] : ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,N)),zero_zero_rat)) ).
% fact_not_neg
tff(fact_7837_fact__not__neg,axiom,
! [N: nat] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,N)),zero_zero_int)) ).
% fact_not_neg
tff(fact_7838_fact__not__neg,axiom,
! [N: nat] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,N)),zero_zero_real)) ).
% fact_not_neg
tff(fact_7839_fact__not__neg,axiom,
! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),zero_zero_nat)) ).
% fact_not_neg
tff(fact_7840_fact__ge__1,axiom,
! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_nat_rat(semiri773545260158071498ct_rat,N))) ).
% fact_ge_1
tff(fact_7841_fact__ge__1,axiom,
! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_nat_int(semiri1406184849735516958ct_int,N))) ).
% fact_ge_1
tff(fact_7842_fact__ge__1,axiom,
! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_nat_real(semiri2265585572941072030t_real,N))) ).
% fact_ge_1
tff(fact_7843_fact__ge__1,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).
% fact_ge_1
tff(fact_7844_abs__sin__x__le__abs__x,axiom,
! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_7845_fact__mono,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,M)),aa_nat_rat(semiri773545260158071498ct_rat,N))) ) ).
% fact_mono
tff(fact_7846_fact__mono,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,M)),aa_nat_int(semiri1406184849735516958ct_int,N))) ) ).
% fact_mono
tff(fact_7847_fact__mono,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,M)),aa_nat_real(semiri2265585572941072030t_real,N))) ) ).
% fact_mono
tff(fact_7848_fact__mono,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ).
% fact_mono
tff(fact_7849_fact__dvd,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_nat_int(semiri1406184849735516958ct_int,N)),aa_nat_int(semiri1406184849735516958ct_int,M))) ) ).
% fact_dvd
tff(fact_7850_fact__dvd,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_nat_Code_integer(semiri3624122377584611663nteger,N)),aa_nat_Code_integer(semiri3624122377584611663nteger,M))) ) ).
% fact_dvd
tff(fact_7851_fact__dvd,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_nat_real(semiri2265585572941072030t_real,N)),aa_nat_real(semiri2265585572941072030t_real,M))) ) ).
% fact_dvd
tff(fact_7852_fact__dvd,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(semiri1408675320244567234ct_nat,M))) ) ).
% fact_dvd
tff(fact_7853_Diff__insert,axiom,
! [A2: set_VEBT_VEBT,A: vEBT_VEBT,B2: set_VEBT_VEBT] : minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,B2)) = minus_5127226145743854075T_VEBT(minus_5127226145743854075T_VEBT(A2,B2),insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)) ).
% Diff_insert
tff(fact_7854_Diff__insert,axiom,
! [A2: set_int,A: int,B2: set_int] : minus_minus_set_int(A2,insert_int(A,B2)) = minus_minus_set_int(minus_minus_set_int(A2,B2),insert_int(A,bot_bot_set_int)) ).
% Diff_insert
tff(fact_7855_Diff__insert,axiom,
! [A2: set_real,A: real,B2: set_real] : minus_minus_set_real(A2,insert_real(A,B2)) = minus_minus_set_real(minus_minus_set_real(A2,B2),insert_real(A,bot_bot_set_real)) ).
% Diff_insert
tff(fact_7856_Diff__insert,axiom,
! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(A,B2)) = minus_1356011639430497352at_nat(minus_1356011639430497352at_nat(A2,B2),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat)) ).
% Diff_insert
tff(fact_7857_Diff__insert,axiom,
! [A2: set_nat,A: nat,B2: set_nat] : minus_minus_set_nat(A2,insert_nat(A,B2)) = minus_minus_set_nat(minus_minus_set_nat(A2,B2),insert_nat(A,bot_bot_set_nat)) ).
% Diff_insert
tff(fact_7858_insert__Diff,axiom,
! [A: vEBT_VEBT,A2: set_VEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
=> ( insert_VEBT_VEBT(A,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) = A2 ) ) ).
% insert_Diff
tff(fact_7859_insert__Diff,axiom,
! [A: complex,A2: set_complex] :
( pp(aa_set_complex_bool(member_complex(A),A2))
=> ( insert_complex(A,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = A2 ) ) ).
% insert_Diff
tff(fact_7860_insert__Diff,axiom,
! [A: int,A2: set_int] :
( pp(aa_set_int_bool(member_int(A),A2))
=> ( insert_int(A,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = A2 ) ) ).
% insert_Diff
tff(fact_7861_insert__Diff,axiom,
! [A: real,A2: set_real] :
( pp(aa_set_real_bool(member_real(A),A2))
=> ( insert_real(A,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = A2 ) ) ).
% insert_Diff
tff(fact_7862_insert__Diff,axiom,
! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),A2))
=> ( insert8211810215607154385at_nat(A,minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))) = A2 ) ) ).
% insert_Diff
tff(fact_7863_insert__Diff,axiom,
! [A: nat,A2: set_nat] :
( pp(aa_set_nat_bool(member_nat(A),A2))
=> ( insert_nat(A,minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = A2 ) ) ).
% insert_Diff
tff(fact_7864_Diff__insert2,axiom,
! [A2: set_VEBT_VEBT,A: vEBT_VEBT,B2: set_VEBT_VEBT] : minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,B2)) = minus_5127226145743854075T_VEBT(minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)),B2) ).
% Diff_insert2
tff(fact_7865_Diff__insert2,axiom,
! [A2: set_int,A: int,B2: set_int] : minus_minus_set_int(A2,insert_int(A,B2)) = minus_minus_set_int(minus_minus_set_int(A2,insert_int(A,bot_bot_set_int)),B2) ).
% Diff_insert2
tff(fact_7866_Diff__insert2,axiom,
! [A2: set_real,A: real,B2: set_real] : minus_minus_set_real(A2,insert_real(A,B2)) = minus_minus_set_real(minus_minus_set_real(A2,insert_real(A,bot_bot_set_real)),B2) ).
% Diff_insert2
tff(fact_7867_Diff__insert2,axiom,
! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] : minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(A,B2)) = minus_1356011639430497352at_nat(minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat)),B2) ).
% Diff_insert2
tff(fact_7868_Diff__insert2,axiom,
! [A2: set_nat,A: nat,B2: set_nat] : minus_minus_set_nat(A2,insert_nat(A,B2)) = minus_minus_set_nat(minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat)),B2) ).
% Diff_insert2
tff(fact_7869_Diff__insert__absorb,axiom,
! [X: vEBT_VEBT,A2: set_VEBT_VEBT] :
( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),A2))
=> ( minus_5127226145743854075T_VEBT(insert_VEBT_VEBT(X,A2),insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT)) = A2 ) ) ).
% Diff_insert_absorb
tff(fact_7870_Diff__insert__absorb,axiom,
! [X: complex,A2: set_complex] :
( ~ pp(aa_set_complex_bool(member_complex(X),A2))
=> ( minus_811609699411566653omplex(insert_complex(X,A2),insert_complex(X,bot_bot_set_complex)) = A2 ) ) ).
% Diff_insert_absorb
tff(fact_7871_Diff__insert__absorb,axiom,
! [X: int,A2: set_int] :
( ~ pp(aa_set_int_bool(member_int(X),A2))
=> ( minus_minus_set_int(insert_int(X,A2),insert_int(X,bot_bot_set_int)) = A2 ) ) ).
% Diff_insert_absorb
tff(fact_7872_Diff__insert__absorb,axiom,
! [X: real,A2: set_real] :
( ~ pp(aa_set_real_bool(member_real(X),A2))
=> ( minus_minus_set_real(insert_real(X,A2),insert_real(X,bot_bot_set_real)) = A2 ) ) ).
% Diff_insert_absorb
tff(fact_7873_Diff__insert__absorb,axiom,
! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A2))
=> ( minus_1356011639430497352at_nat(insert8211810215607154385at_nat(X,A2),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)) = A2 ) ) ).
% Diff_insert_absorb
tff(fact_7874_Diff__insert__absorb,axiom,
! [X: nat,A2: set_nat] :
( ~ pp(aa_set_nat_bool(member_nat(X),A2))
=> ( minus_minus_set_nat(insert_nat(X,A2),insert_nat(X,bot_bot_set_nat)) = A2 ) ) ).
% Diff_insert_absorb
tff(fact_7875_Compl__insert,axiom,
! [X: vEBT_VEBT,A2: set_VEBT_VEBT] : uminus8041839845116263051T_VEBT(insert_VEBT_VEBT(X,A2)) = minus_5127226145743854075T_VEBT(uminus8041839845116263051T_VEBT(A2),insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT)) ).
% Compl_insert
tff(fact_7876_Compl__insert,axiom,
! [X: int,A2: set_int] : uminus1532241313380277803et_int(insert_int(X,A2)) = minus_minus_set_int(uminus1532241313380277803et_int(A2),insert_int(X,bot_bot_set_int)) ).
% Compl_insert
tff(fact_7877_Compl__insert,axiom,
! [X: real,A2: set_real] : uminus612125837232591019t_real(insert_real(X,A2)) = minus_minus_set_real(uminus612125837232591019t_real(A2),insert_real(X,bot_bot_set_real)) ).
% Compl_insert
tff(fact_7878_Compl__insert,axiom,
! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] : uminus6524753893492686040at_nat(insert8211810215607154385at_nat(X,A2)) = minus_1356011639430497352at_nat(uminus6524753893492686040at_nat(A2),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)) ).
% Compl_insert
tff(fact_7879_Compl__insert,axiom,
! [X: nat,A2: set_nat] : uminus5710092332889474511et_nat(insert_nat(X,A2)) = minus_minus_set_nat(uminus5710092332889474511et_nat(A2),insert_nat(X,bot_bot_set_nat)) ).
% Compl_insert
tff(fact_7880_subset__Diff__insert,axiom,
! [A2: set_VEBT_VEBT,B2: set_VEBT_VEBT,X: vEBT_VEBT,C4: set_VEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),minus_5127226145743854075T_VEBT(B2,insert_VEBT_VEBT(X,C4))))
<=> ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),minus_5127226145743854075T_VEBT(B2,C4)))
& ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),A2)) ) ) ).
% subset_Diff_insert
tff(fact_7881_subset__Diff__insert,axiom,
! [A2: set_complex,B2: set_complex,X: complex,C4: set_complex] :
( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),minus_811609699411566653omplex(B2,insert_complex(X,C4))))
<=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),minus_811609699411566653omplex(B2,C4)))
& ~ pp(aa_set_complex_bool(member_complex(X),A2)) ) ) ).
% subset_Diff_insert
tff(fact_7882_subset__Diff__insert,axiom,
! [A2: set_real,B2: set_real,X: real,C4: set_real] :
( pp(aa_set_real_bool(ord_less_eq_set_real(A2),minus_minus_set_real(B2,insert_real(X,C4))))
<=> ( pp(aa_set_real_bool(ord_less_eq_set_real(A2),minus_minus_set_real(B2,C4)))
& ~ pp(aa_set_real_bool(member_real(X),A2)) ) ) ).
% subset_Diff_insert
tff(fact_7883_subset__Diff__insert,axiom,
! [A2: set_int,B2: set_int,X: int,C4: set_int] :
( pp(aa_set_int_bool(ord_less_eq_set_int(A2),minus_minus_set_int(B2,insert_int(X,C4))))
<=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A2),minus_minus_set_int(B2,C4)))
& ~ pp(aa_set_int_bool(member_int(X),A2)) ) ) ).
% subset_Diff_insert
tff(fact_7884_subset__Diff__insert,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,C4: set_Pr1261947904930325089at_nat] :
( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),minus_1356011639430497352at_nat(B2,insert8211810215607154385at_nat(X,C4))))
<=> ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),minus_1356011639430497352at_nat(B2,C4)))
& ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A2)) ) ) ).
% subset_Diff_insert
tff(fact_7885_subset__Diff__insert,axiom,
! [A2: set_nat,B2: set_nat,X: nat,C4: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),minus_minus_set_nat(B2,insert_nat(X,C4))))
<=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),minus_minus_set_nat(B2,C4)))
& ~ pp(aa_set_nat_bool(member_nat(X),A2)) ) ) ).
% subset_Diff_insert
tff(fact_7886_sin__cos__le1,axiom,
! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))))),one_one_real)) ).
% sin_cos_le1
tff(fact_7887_sin__squared__eq,axiom,
! [X: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).
% sin_squared_eq
tff(fact_7888_sin__squared__eq,axiom,
! [X: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).
% sin_squared_eq
tff(fact_7889_cos__squared__eq,axiom,
! [X: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).
% cos_squared_eq
tff(fact_7890_cos__squared__eq,axiom,
! [X: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).
% cos_squared_eq
tff(fact_7891_sin__gt__zero,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),pi))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,X))) ) ) ).
% sin_gt_zero
tff(fact_7892_sin__x__ge__neg__x,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,X)),aa_real_real(sin_real,X))) ) ).
% sin_x_ge_neg_x
tff(fact_7893_sin__ge__zero,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,X))) ) ) ).
% sin_ge_zero
tff(fact_7894_sin__ge__minus__one,axiom,
! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_7895_cos__monotone__0__pi__le,axiom,
! [Y: real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))) ) ) ) ).
% cos_monotone_0_pi_le
tff(fact_7896_cos__mono__le__eq,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),pi))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ) ) ) ) ).
% cos_mono_le_eq
tff(fact_7897_cos__inj__pi,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_7898_cos__ge__minus__one,axiom,
! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_7899_abs__sin__le__one,axiom,
! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_7900_abs__cos__le__one,axiom,
! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_7901_finite__ranking__induct,axiom,
! [S2: set_VEBT_VEBT,P: fun_se448528567104302406T_bool,F: fun_VEBT_VEBT_rat] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S2))
=> ( pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT))
=> ( ! [X3: vEBT_VEBT,S5: set_VEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S5))
=> ( ! [Y3: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Y3),S5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,Y3)),aa_VEBT_VEBT_rat(F,X3))) )
=> ( pp(aa_set8260762808118238015T_bool(P,S5))
=> pp(aa_set8260762808118238015T_bool(P,insert_VEBT_VEBT(X3,S5))) ) ) )
=> pp(aa_set8260762808118238015T_bool(P,S2)) ) ) ) ).
% finite_ranking_induct
tff(fact_7902_finite__ranking__induct,axiom,
! [S2: set_complex,P: fun_set_complex_bool,F: fun_complex_rat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
=> ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
=> ( ! [X3: complex,S5: set_complex] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,S5))
=> ( ! [Y3: complex] :
( pp(aa_set_complex_bool(member_complex(Y3),S5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,Y3)),aa_complex_rat(F,X3))) )
=> ( pp(aa_set_complex_bool(P,S5))
=> pp(aa_set_complex_bool(P,insert_complex(X3,S5))) ) ) )
=> pp(aa_set_complex_bool(P,S2)) ) ) ) ).
% finite_ranking_induct
tff(fact_7903_finite__ranking__induct,axiom,
! [S2: set_nat,P: fun_set_nat_bool,F: fun_nat_rat] :
( pp(aa_set_nat_bool(finite_finite_nat,S2))
=> ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
=> ( ! [X3: nat,S5: set_nat] :
( pp(aa_set_nat_bool(finite_finite_nat,S5))
=> ( ! [Y3: nat] :
( pp(aa_set_nat_bool(member_nat(Y3),S5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,Y3)),aa_nat_rat(F,X3))) )
=> ( pp(aa_set_nat_bool(P,S5))
=> pp(aa_set_nat_bool(P,insert_nat(X3,S5))) ) ) )
=> pp(aa_set_nat_bool(P,S2)) ) ) ) ).
% finite_ranking_induct
tff(fact_7904_finite__ranking__induct,axiom,
! [S2: set_int,P: fun_set_int_bool,F: fun_int_rat] :
( pp(aa_set_int_bool(finite_finite_int,S2))
=> ( pp(aa_set_int_bool(P,bot_bot_set_int))
=> ( ! [X3: int,S5: set_int] :
( pp(aa_set_int_bool(finite_finite_int,S5))
=> ( ! [Y3: int] :
( pp(aa_set_int_bool(member_int(Y3),S5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,Y3)),aa_int_rat(F,X3))) )
=> ( pp(aa_set_int_bool(P,S5))
=> pp(aa_set_int_bool(P,insert_int(X3,S5))) ) ) )
=> pp(aa_set_int_bool(P,S2)) ) ) ) ).
% finite_ranking_induct
tff(fact_7905_finite__ranking__induct,axiom,
! [S2: set_real,P: fun_set_real_bool,F: fun_real_rat] :
( pp(aa_set_real_bool(finite_finite_real,S2))
=> ( pp(aa_set_real_bool(P,bot_bot_set_real))
=> ( ! [X3: real,S5: set_real] :
( pp(aa_set_real_bool(finite_finite_real,S5))
=> ( ! [Y3: real] :
( pp(aa_set_real_bool(member_real(Y3),S5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,Y3)),aa_real_rat(F,X3))) )
=> ( pp(aa_set_real_bool(P,S5))
=> pp(aa_set_real_bool(P,insert_real(X3,S5))) ) ) )
=> pp(aa_set_real_bool(P,S2)) ) ) ) ).
% finite_ranking_induct
tff(fact_7906_finite__ranking__induct,axiom,
! [S2: set_VEBT_VEBT,P: fun_se448528567104302406T_bool,F: fun_VEBT_VEBT_num] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S2))
=> ( pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT))
=> ( ! [X3: vEBT_VEBT,S5: set_VEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S5))
=> ( ! [Y3: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,Y3),S5))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_VEBT_VEBT_num(F,Y3)),aa_VEBT_VEBT_num(F,X3))) )
=> ( pp(aa_set8260762808118238015T_bool(P,S5))
=> pp(aa_set8260762808118238015T_bool(P,insert_VEBT_VEBT(X3,S5))) ) ) )
=> pp(aa_set8260762808118238015T_bool(P,S2)) ) ) ) ).
% finite_ranking_induct
tff(fact_7907_finite__ranking__induct,axiom,
! [S2: set_complex,P: fun_set_complex_bool,F: fun_complex_num] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
=> ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
=> ( ! [X3: complex,S5: set_complex] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,S5))
=> ( ! [Y3: complex] :
( pp(aa_set_complex_bool(member_complex(Y3),S5))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_complex_num(F,Y3)),aa_complex_num(F,X3))) )
=> ( pp(aa_set_complex_bool(P,S5))
=> pp(aa_set_complex_bool(P,insert_complex(X3,S5))) ) ) )
=> pp(aa_set_complex_bool(P,S2)) ) ) ) ).
% finite_ranking_induct
tff(fact_7908_finite__ranking__induct,axiom,
! [S2: set_nat,P: fun_set_nat_bool,F: fun_nat_num] :
( pp(aa_set_nat_bool(finite_finite_nat,S2))
=> ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
=> ( ! [X3: nat,S5: set_nat] :
( pp(aa_set_nat_bool(finite_finite_nat,S5))
=> ( ! [Y3: nat] :
( pp(aa_set_nat_bool(member_nat(Y3),S5))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F,Y3)),aa_nat_num(F,X3))) )
=> ( pp(aa_set_nat_bool(P,S5))
=> pp(aa_set_nat_bool(P,insert_nat(X3,S5))) ) ) )
=> pp(aa_set_nat_bool(P,S2)) ) ) ) ).
% finite_ranking_induct
tff(fact_7909_finite__ranking__induct,axiom,
! [S2: set_int,P: fun_set_int_bool,F: fun_int_num] :
( pp(aa_set_int_bool(finite_finite_int,S2))
=> ( pp(aa_set_int_bool(P,bot_bot_set_int))
=> ( ! [X3: int,S5: set_int] :
( pp(aa_set_int_bool(finite_finite_int,S5))
=> ( ! [Y3: int] :
( pp(aa_set_int_bool(member_int(Y3),S5))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_int_num(F,Y3)),aa_int_num(F,X3))) )
=> ( pp(aa_set_int_bool(P,S5))
=> pp(aa_set_int_bool(P,insert_int(X3,S5))) ) ) )
=> pp(aa_set_int_bool(P,S2)) ) ) ) ).
% finite_ranking_induct
tff(fact_7910_finite__ranking__induct,axiom,
! [S2: set_real,P: fun_set_real_bool,F: fun_real_num] :
( pp(aa_set_real_bool(finite_finite_real,S2))
=> ( pp(aa_set_real_bool(P,bot_bot_set_real))
=> ( ! [X3: real,S5: set_real] :
( pp(aa_set_real_bool(finite_finite_real,S5))
=> ( ! [Y3: real] :
( pp(aa_set_real_bool(member_real(Y3),S5))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_real_num(F,Y3)),aa_real_num(F,X3))) )
=> ( pp(aa_set_real_bool(P,S5))
=> pp(aa_set_real_bool(P,insert_real(X3,S5))) ) ) )
=> pp(aa_set_real_bool(P,S2)) ) ) ) ).
% finite_ranking_induct
tff(fact_7911_finite__linorder__min__induct,axiom,
! [A2: set_real,P: fun_set_real_bool] :
( pp(aa_set_real_bool(finite_finite_real,A2))
=> ( pp(aa_set_real_bool(P,bot_bot_set_real))
=> ( ! [B4: real,A5: set_real] :
( pp(aa_set_real_bool(finite_finite_real,A5))
=> ( ! [X4: real] :
( pp(aa_set_real_bool(member_real(X4),A5))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B4),X4)) )
=> ( pp(aa_set_real_bool(P,A5))
=> pp(aa_set_real_bool(P,insert_real(B4,A5))) ) ) )
=> pp(aa_set_real_bool(P,A2)) ) ) ) ).
% finite_linorder_min_induct
tff(fact_7912_finite__linorder__min__induct,axiom,
! [A2: set_rat,P: fun_set_rat_bool] :
( pp(aa_set_rat_bool(finite_finite_rat,A2))
=> ( pp(aa_set_rat_bool(P,bot_bot_set_rat))
=> ( ! [B4: rat,A5: set_rat] :
( pp(aa_set_rat_bool(finite_finite_rat,A5))
=> ( ! [X4: rat] :
( pp(aa_set_rat_bool(member_rat(X4),A5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B4),X4)) )
=> ( pp(aa_set_rat_bool(P,A5))
=> pp(aa_set_rat_bool(P,insert_rat(B4,A5))) ) ) )
=> pp(aa_set_rat_bool(P,A2)) ) ) ) ).
% finite_linorder_min_induct
tff(fact_7913_finite__linorder__min__induct,axiom,
! [A2: set_num,P: fun_set_num_bool] :
( pp(aa_set_num_bool(finite_finite_num,A2))
=> ( pp(aa_set_num_bool(P,bot_bot_set_num))
=> ( ! [B4: num,A5: set_num] :
( pp(aa_set_num_bool(finite_finite_num,A5))
=> ( ! [X4: num] :
( pp(aa_set_num_bool(member_num(X4),A5))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B4),X4)) )
=> ( pp(aa_set_num_bool(P,A5))
=> pp(aa_set_num_bool(P,insert_num(B4,A5))) ) ) )
=> pp(aa_set_num_bool(P,A2)) ) ) ) ).
% finite_linorder_min_induct
tff(fact_7914_finite__linorder__min__induct,axiom,
! [A2: set_nat,P: fun_set_nat_bool] :
( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
=> ( ! [B4: nat,A5: set_nat] :
( pp(aa_set_nat_bool(finite_finite_nat,A5))
=> ( ! [X4: nat] :
( pp(aa_set_nat_bool(member_nat(X4),A5))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B4),X4)) )
=> ( pp(aa_set_nat_bool(P,A5))
=> pp(aa_set_nat_bool(P,insert_nat(B4,A5))) ) ) )
=> pp(aa_set_nat_bool(P,A2)) ) ) ) ).
% finite_linorder_min_induct
tff(fact_7915_finite__linorder__min__induct,axiom,
! [A2: set_int,P: fun_set_int_bool] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( pp(aa_set_int_bool(P,bot_bot_set_int))
=> ( ! [B4: int,A5: set_int] :
( pp(aa_set_int_bool(finite_finite_int,A5))
=> ( ! [X4: int] :
( pp(aa_set_int_bool(member_int(X4),A5))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B4),X4)) )
=> ( pp(aa_set_int_bool(P,A5))
=> pp(aa_set_int_bool(P,insert_int(B4,A5))) ) ) )
=> pp(aa_set_int_bool(P,A2)) ) ) ) ).
% finite_linorder_min_induct
tff(fact_7916_finite__linorder__max__induct,axiom,
! [A2: set_real,P: fun_set_real_bool] :
( pp(aa_set_real_bool(finite_finite_real,A2))
=> ( pp(aa_set_real_bool(P,bot_bot_set_real))
=> ( ! [B4: real,A5: set_real] :
( pp(aa_set_real_bool(finite_finite_real,A5))
=> ( ! [X4: real] :
( pp(aa_set_real_bool(member_real(X4),A5))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X4),B4)) )
=> ( pp(aa_set_real_bool(P,A5))
=> pp(aa_set_real_bool(P,insert_real(B4,A5))) ) ) )
=> pp(aa_set_real_bool(P,A2)) ) ) ) ).
% finite_linorder_max_induct
tff(fact_7917_finite__linorder__max__induct,axiom,
! [A2: set_rat,P: fun_set_rat_bool] :
( pp(aa_set_rat_bool(finite_finite_rat,A2))
=> ( pp(aa_set_rat_bool(P,bot_bot_set_rat))
=> ( ! [B4: rat,A5: set_rat] :
( pp(aa_set_rat_bool(finite_finite_rat,A5))
=> ( ! [X4: rat] :
( pp(aa_set_rat_bool(member_rat(X4),A5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X4),B4)) )
=> ( pp(aa_set_rat_bool(P,A5))
=> pp(aa_set_rat_bool(P,insert_rat(B4,A5))) ) ) )
=> pp(aa_set_rat_bool(P,A2)) ) ) ) ).
% finite_linorder_max_induct
tff(fact_7918_finite__linorder__max__induct,axiom,
! [A2: set_num,P: fun_set_num_bool] :
( pp(aa_set_num_bool(finite_finite_num,A2))
=> ( pp(aa_set_num_bool(P,bot_bot_set_num))
=> ( ! [B4: num,A5: set_num] :
( pp(aa_set_num_bool(finite_finite_num,A5))
=> ( ! [X4: num] :
( pp(aa_set_num_bool(member_num(X4),A5))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X4),B4)) )
=> ( pp(aa_set_num_bool(P,A5))
=> pp(aa_set_num_bool(P,insert_num(B4,A5))) ) ) )
=> pp(aa_set_num_bool(P,A2)) ) ) ) ).
% finite_linorder_max_induct
tff(fact_7919_finite__linorder__max__induct,axiom,
! [A2: set_nat,P: fun_set_nat_bool] :
( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
=> ( ! [B4: nat,A5: set_nat] :
( pp(aa_set_nat_bool(finite_finite_nat,A5))
=> ( ! [X4: nat] :
( pp(aa_set_nat_bool(member_nat(X4),A5))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X4),B4)) )
=> ( pp(aa_set_nat_bool(P,A5))
=> pp(aa_set_nat_bool(P,insert_nat(B4,A5))) ) ) )
=> pp(aa_set_nat_bool(P,A2)) ) ) ) ).
% finite_linorder_max_induct
tff(fact_7920_finite__linorder__max__induct,axiom,
! [A2: set_int,P: fun_set_int_bool] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( pp(aa_set_int_bool(P,bot_bot_set_int))
=> ( ! [B4: int,A5: set_int] :
( pp(aa_set_int_bool(finite_finite_int,A5))
=> ( ! [X4: int] :
( pp(aa_set_int_bool(member_int(X4),A5))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X4),B4)) )
=> ( pp(aa_set_int_bool(P,A5))
=> pp(aa_set_int_bool(P,insert_int(B4,A5))) ) ) )
=> pp(aa_set_int_bool(P,A2)) ) ) ) ).
% finite_linorder_max_induct
tff(fact_7921_fact__less__mono,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,M)),aa_nat_rat(semiri773545260158071498ct_rat,N))) ) ) ).
% fact_less_mono
tff(fact_7922_fact__less__mono,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,M)),aa_nat_int(semiri1406184849735516958ct_int,N))) ) ) ).
% fact_less_mono
tff(fact_7923_fact__less__mono,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,M)),aa_nat_real(semiri2265585572941072030t_real,N))) ) ) ).
% fact_less_mono
tff(fact_7924_fact__less__mono,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ) ).
% fact_less_mono
tff(fact_7925_sum_Oinsert__if,axiom,
! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: fun_VEBT_VEBT_real] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
=> ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),A2))
=> ( groups2240296850493347238T_real(G,insert_VEBT_VEBT(X,A2)) = groups2240296850493347238T_real(G,A2) ) )
& ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),A2))
=> ( groups2240296850493347238T_real(G,insert_VEBT_VEBT(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(G,X)),groups2240296850493347238T_real(G,A2)) ) ) ) ) ).
% sum.insert_if
tff(fact_7926_sum_Oinsert__if,axiom,
! [A2: set_real,X: real,G: fun_real_real] :
( pp(aa_set_real_bool(finite_finite_real,A2))
=> ( ( pp(aa_set_real_bool(member_real(X),A2))
=> ( groups8097168146408367636l_real(G,insert_real(X,A2)) = groups8097168146408367636l_real(G,A2) ) )
& ( ~ pp(aa_set_real_bool(member_real(X),A2))
=> ( groups8097168146408367636l_real(G,insert_real(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(G,X)),groups8097168146408367636l_real(G,A2)) ) ) ) ) ).
% sum.insert_if
tff(fact_7927_sum_Oinsert__if,axiom,
! [A2: set_int,X: int,G: fun_int_real] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( ( pp(aa_set_int_bool(member_int(X),A2))
=> ( groups8778361861064173332t_real(G,insert_int(X,A2)) = groups8778361861064173332t_real(G,A2) ) )
& ( ~ pp(aa_set_int_bool(member_int(X),A2))
=> ( groups8778361861064173332t_real(G,insert_int(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,X)),groups8778361861064173332t_real(G,A2)) ) ) ) ) ).
% sum.insert_if
tff(fact_7928_sum_Oinsert__if,axiom,
! [A2: set_complex,X: complex,G: fun_complex_real] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( ( pp(aa_set_complex_bool(member_complex(X),A2))
=> ( groups5808333547571424918x_real(G,insert_complex(X,A2)) = groups5808333547571424918x_real(G,A2) ) )
& ( ~ pp(aa_set_complex_bool(member_complex(X),A2))
=> ( groups5808333547571424918x_real(G,insert_complex(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,X)),groups5808333547571424918x_real(G,A2)) ) ) ) ) ).
% sum.insert_if
tff(fact_7929_sum_Oinsert__if,axiom,
! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: fun_VEBT_VEBT_rat] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
=> ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),A2))
=> ( groups136491112297645522BT_rat(G,insert_VEBT_VEBT(X,A2)) = groups136491112297645522BT_rat(G,A2) ) )
& ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),A2))
=> ( groups136491112297645522BT_rat(G,insert_VEBT_VEBT(X,A2)) = plus_plus_rat(aa_VEBT_VEBT_rat(G,X),groups136491112297645522BT_rat(G,A2)) ) ) ) ) ).
% sum.insert_if
tff(fact_7930_sum_Oinsert__if,axiom,
! [A2: set_real,X: real,G: fun_real_rat] :
( pp(aa_set_real_bool(finite_finite_real,A2))
=> ( ( pp(aa_set_real_bool(member_real(X),A2))
=> ( groups1300246762558778688al_rat(G,insert_real(X,A2)) = groups1300246762558778688al_rat(G,A2) ) )
& ( ~ pp(aa_set_real_bool(member_real(X),A2))
=> ( groups1300246762558778688al_rat(G,insert_real(X,A2)) = plus_plus_rat(aa_real_rat(G,X),groups1300246762558778688al_rat(G,A2)) ) ) ) ) ).
% sum.insert_if
tff(fact_7931_sum_Oinsert__if,axiom,
! [A2: set_nat,X: nat,G: fun_nat_rat] :
( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ( ( pp(aa_set_nat_bool(member_nat(X),A2))
=> ( groups2906978787729119204at_rat(G,insert_nat(X,A2)) = groups2906978787729119204at_rat(G,A2) ) )
& ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
=> ( groups2906978787729119204at_rat(G,insert_nat(X,A2)) = plus_plus_rat(aa_nat_rat(G,X),groups2906978787729119204at_rat(G,A2)) ) ) ) ) ).
% sum.insert_if
tff(fact_7932_sum_Oinsert__if,axiom,
! [A2: set_int,X: int,G: fun_int_rat] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( ( pp(aa_set_int_bool(member_int(X),A2))
=> ( groups3906332499630173760nt_rat(G,insert_int(X,A2)) = groups3906332499630173760nt_rat(G,A2) ) )
& ( ~ pp(aa_set_int_bool(member_int(X),A2))
=> ( groups3906332499630173760nt_rat(G,insert_int(X,A2)) = plus_plus_rat(aa_int_rat(G,X),groups3906332499630173760nt_rat(G,A2)) ) ) ) ) ).
% sum.insert_if
tff(fact_7933_sum_Oinsert__if,axiom,
! [A2: set_complex,X: complex,G: fun_complex_rat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( ( pp(aa_set_complex_bool(member_complex(X),A2))
=> ( groups5058264527183730370ex_rat(G,insert_complex(X,A2)) = groups5058264527183730370ex_rat(G,A2) ) )
& ( ~ pp(aa_set_complex_bool(member_complex(X),A2))
=> ( groups5058264527183730370ex_rat(G,insert_complex(X,A2)) = plus_plus_rat(aa_complex_rat(G,X),groups5058264527183730370ex_rat(G,A2)) ) ) ) ) ).
% sum.insert_if
tff(fact_7934_sum_Oinsert__if,axiom,
! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: fun_VEBT_VEBT_nat] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
=> ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),A2))
=> ( groups771621172384141258BT_nat(G,insert_VEBT_VEBT(X,A2)) = groups771621172384141258BT_nat(G,A2) ) )
& ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),A2))
=> ( groups771621172384141258BT_nat(G,insert_VEBT_VEBT(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(G,X)),groups771621172384141258BT_nat(G,A2)) ) ) ) ) ).
% sum.insert_if
tff(fact_7935_sin__times__sin,axiom,
! [W: complex,Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,W)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(W),Z))),aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(W),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).
% sin_times_sin
tff(fact_7936_sin__times__sin,axiom,
! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)))),numeral_numeral_real(aa_num_num(bit0,one))) ).
% sin_times_sin
tff(fact_7937_sin__times__cos,axiom,
! [W: complex,Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,W)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(W),Z))),aa_complex_complex(sin_complex,aa_complex_complex(minus_minus_complex(W),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).
% sin_times_cos
tff(fact_7938_sin__times__cos,axiom,
! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z)))),numeral_numeral_real(aa_num_num(bit0,one))) ).
% sin_times_cos
tff(fact_7939_cos__times__sin,axiom,
! [W: complex,Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,W)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(W),Z))),aa_complex_complex(sin_complex,aa_complex_complex(minus_minus_complex(W),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).
% cos_times_sin
tff(fact_7940_cos__times__sin,axiom,
! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z)))),numeral_numeral_real(aa_num_num(bit0,one))) ).
% cos_times_sin
tff(fact_7941_sin__plus__sin,axiom,
! [W: complex,Z: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(sin_complex,W)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).
% sin_plus_sin
tff(fact_7942_sin__plus__sin,axiom,
! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sin_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)),numeral_numeral_real(aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z)),numeral_numeral_real(aa_num_num(bit0,one))))) ).
% sin_plus_sin
tff(fact_7943_sin__diff__sin,axiom,
! [W: complex,Z: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(sin_complex,W)),aa_complex_complex(sin_complex,Z)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).
% sin_diff_sin
tff(fact_7944_sin__diff__sin,axiom,
! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z)),numeral_numeral_real(aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)),numeral_numeral_real(aa_num_num(bit0,one))))) ).
% sin_diff_sin
tff(fact_7945_cos__diff__cos,axiom,
! [W: complex,Z: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(cos_complex,W)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(Z),W)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).
% cos_diff_cos
tff(fact_7946_cos__diff__cos,axiom,
! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(cos_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)),numeral_numeral_real(aa_num_num(bit0,one)))))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),W)),numeral_numeral_real(aa_num_num(bit0,one))))) ).
% cos_diff_cos
tff(fact_7947_fact__fact__dvd__fact,axiom,
! [K: nat,N: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,K)),aa_nat_Code_integer(semiri3624122377584611663nteger,N))),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))) ).
% fact_fact_dvd_fact
tff(fact_7948_fact__fact__dvd__fact,axiom,
! [K: nat,N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,N))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))) ).
% fact_fact_dvd_fact
tff(fact_7949_fact__fact__dvd__fact,axiom,
! [K: nat,N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1406184849735516958ct_int,K)),aa_nat_int(semiri1406184849735516958ct_int,N))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))) ).
% fact_fact_dvd_fact
tff(fact_7950_fact__fact__dvd__fact,axiom,
! [K: nat,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))) ).
% fact_fact_dvd_fact
tff(fact_7951_fact__fact__dvd__fact,axiom,
! [K: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,N))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)))) ).
% fact_fact_dvd_fact
tff(fact_7952_fact__mod,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1406184849735516958ct_int,N)),aa_nat_int(semiri1406184849735516958ct_int,M)) = zero_zero_int ) ) ).
% fact_mod
tff(fact_7953_fact__mod,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( modulo364778990260209775nteger(aa_nat_Code_integer(semiri3624122377584611663nteger,N),aa_nat_Code_integer(semiri3624122377584611663nteger,M)) = zero_z3403309356797280102nteger ) ) ).
% fact_mod
tff(fact_7954_fact__mod,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(semiri1408675320244567234ct_nat,M)) = zero_zero_nat ) ) ).
% fact_mod
tff(fact_7955_cos__double,axiom,
! [X: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)) = aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).
% cos_double
tff(fact_7956_cos__double,axiom,
! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),X)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).
% cos_double
tff(fact_7957_finite__empty__induct,axiom,
! [A2: set_VEBT_VEBT,P: fun_se448528567104302406T_bool] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
=> ( pp(aa_set8260762808118238015T_bool(P,A2))
=> ( ! [A4: vEBT_VEBT,A5: set_VEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A5))
=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A4),A5))
=> ( pp(aa_set8260762808118238015T_bool(P,A5))
=> pp(aa_set8260762808118238015T_bool(P,minus_5127226145743854075T_VEBT(A5,insert_VEBT_VEBT(A4,bot_bo8194388402131092736T_VEBT)))) ) ) )
=> pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT)) ) ) ) ).
% finite_empty_induct
tff(fact_7958_finite__empty__induct,axiom,
! [A2: set_complex,P: fun_set_complex_bool] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( pp(aa_set_complex_bool(P,A2))
=> ( ! [A4: complex,A5: set_complex] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A5))
=> ( pp(aa_set_complex_bool(member_complex(A4),A5))
=> ( pp(aa_set_complex_bool(P,A5))
=> pp(aa_set_complex_bool(P,minus_811609699411566653omplex(A5,insert_complex(A4,bot_bot_set_complex)))) ) ) )
=> pp(aa_set_complex_bool(P,bot_bot_set_complex)) ) ) ) ).
% finite_empty_induct
tff(fact_7959_finite__empty__induct,axiom,
! [A2: set_int,P: fun_set_int_bool] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( pp(aa_set_int_bool(P,A2))
=> ( ! [A4: int,A5: set_int] :
( pp(aa_set_int_bool(finite_finite_int,A5))
=> ( pp(aa_set_int_bool(member_int(A4),A5))
=> ( pp(aa_set_int_bool(P,A5))
=> pp(aa_set_int_bool(P,minus_minus_set_int(A5,insert_int(A4,bot_bot_set_int)))) ) ) )
=> pp(aa_set_int_bool(P,bot_bot_set_int)) ) ) ) ).
% finite_empty_induct
tff(fact_7960_finite__empty__induct,axiom,
! [A2: set_real,P: fun_set_real_bool] :
( pp(aa_set_real_bool(finite_finite_real,A2))
=> ( pp(aa_set_real_bool(P,A2))
=> ( ! [A4: real,A5: set_real] :
( pp(aa_set_real_bool(finite_finite_real,A5))
=> ( pp(aa_set_real_bool(member_real(A4),A5))
=> ( pp(aa_set_real_bool(P,A5))
=> pp(aa_set_real_bool(P,minus_minus_set_real(A5,insert_real(A4,bot_bot_set_real)))) ) ) )
=> pp(aa_set_real_bool(P,bot_bot_set_real)) ) ) ) ).
% finite_empty_induct
tff(fact_7961_finite__empty__induct,axiom,
! [A2: set_Pr1261947904930325089at_nat,P: fun_se8074476271188067385t_bool] :
( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A2))
=> ( pp(aa_set3135850010578026034t_bool(P,A2))
=> ( ! [A4: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat] :
( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A5))
=> ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A4),A5))
=> ( pp(aa_set3135850010578026034t_bool(P,A5))
=> pp(aa_set3135850010578026034t_bool(P,minus_1356011639430497352at_nat(A5,insert8211810215607154385at_nat(A4,bot_bo2099793752762293965at_nat)))) ) ) )
=> pp(aa_set3135850010578026034t_bool(P,bot_bo2099793752762293965at_nat)) ) ) ) ).
% finite_empty_induct
tff(fact_7962_finite__empty__induct,axiom,
! [A2: set_nat,P: fun_set_nat_bool] :
( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ( pp(aa_set_nat_bool(P,A2))
=> ( ! [A4: nat,A5: set_nat] :
( pp(aa_set_nat_bool(finite_finite_nat,A5))
=> ( pp(aa_set_nat_bool(member_nat(A4),A5))
=> ( pp(aa_set_nat_bool(P,A5))
=> pp(aa_set_nat_bool(P,minus_minus_set_nat(A5,insert_nat(A4,bot_bot_set_nat)))) ) ) )
=> pp(aa_set_nat_bool(P,bot_bot_set_nat)) ) ) ) ).
% finite_empty_induct
tff(fact_7963_infinite__coinduct,axiom,
! [X5: fun_se448528567104302406T_bool,A2: set_VEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(X5,A2))
=> ( ! [A5: set_VEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(X5,A5))
=> ? [X4: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),A5))
& ( pp(aa_set8260762808118238015T_bool(X5,minus_5127226145743854075T_VEBT(A5,insert_VEBT_VEBT(X4,bot_bo8194388402131092736T_VEBT))))
| ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,minus_5127226145743854075T_VEBT(A5,insert_VEBT_VEBT(X4,bot_bo8194388402131092736T_VEBT)))) ) ) )
=> ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2)) ) ) ).
% infinite_coinduct
tff(fact_7964_infinite__coinduct,axiom,
! [X5: fun_set_complex_bool,A2: set_complex] :
( pp(aa_set_complex_bool(X5,A2))
=> ( ! [A5: set_complex] :
( pp(aa_set_complex_bool(X5,A5))
=> ? [X4: complex] :
( pp(aa_set_complex_bool(member_complex(X4),A5))
& ( pp(aa_set_complex_bool(X5,minus_811609699411566653omplex(A5,insert_complex(X4,bot_bot_set_complex))))
| ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(A5,insert_complex(X4,bot_bot_set_complex)))) ) ) )
=> ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2)) ) ) ).
% infinite_coinduct
tff(fact_7965_infinite__coinduct,axiom,
! [X5: fun_set_int_bool,A2: set_int] :
( pp(aa_set_int_bool(X5,A2))
=> ( ! [A5: set_int] :
( pp(aa_set_int_bool(X5,A5))
=> ? [X4: int] :
( pp(aa_set_int_bool(member_int(X4),A5))
& ( pp(aa_set_int_bool(X5,minus_minus_set_int(A5,insert_int(X4,bot_bot_set_int))))
| ~ pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(A5,insert_int(X4,bot_bot_set_int)))) ) ) )
=> ~ pp(aa_set_int_bool(finite_finite_int,A2)) ) ) ).
% infinite_coinduct
tff(fact_7966_infinite__coinduct,axiom,
! [X5: fun_set_real_bool,A2: set_real] :
( pp(aa_set_real_bool(X5,A2))
=> ( ! [A5: set_real] :
( pp(aa_set_real_bool(X5,A5))
=> ? [X4: real] :
( pp(aa_set_real_bool(member_real(X4),A5))
& ( pp(aa_set_real_bool(X5,minus_minus_set_real(A5,insert_real(X4,bot_bot_set_real))))
| ~ pp(aa_set_real_bool(finite_finite_real,minus_minus_set_real(A5,insert_real(X4,bot_bot_set_real)))) ) ) )
=> ~ pp(aa_set_real_bool(finite_finite_real,A2)) ) ) ).
% infinite_coinduct
tff(fact_7967_infinite__coinduct,axiom,
! [X5: fun_se8074476271188067385t_bool,A2: set_Pr1261947904930325089at_nat] :
( pp(aa_set3135850010578026034t_bool(X5,A2))
=> ( ! [A5: set_Pr1261947904930325089at_nat] :
( pp(aa_set3135850010578026034t_bool(X5,A5))
=> ? [X4: product_prod_nat_nat] :
( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X4),A5))
& ( pp(aa_set3135850010578026034t_bool(X5,minus_1356011639430497352at_nat(A5,insert8211810215607154385at_nat(X4,bot_bo2099793752762293965at_nat))))
| ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(A5,insert8211810215607154385at_nat(X4,bot_bo2099793752762293965at_nat)))) ) ) )
=> ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A2)) ) ) ).
% infinite_coinduct
tff(fact_7968_infinite__coinduct,axiom,
! [X5: fun_set_nat_bool,A2: set_nat] :
( pp(aa_set_nat_bool(X5,A2))
=> ( ! [A5: set_nat] :
( pp(aa_set_nat_bool(X5,A5))
=> ? [X4: nat] :
( pp(aa_set_nat_bool(member_nat(X4),A5))
& ( pp(aa_set_nat_bool(X5,minus_minus_set_nat(A5,insert_nat(X4,bot_bot_set_nat))))
| ~ pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(A5,insert_nat(X4,bot_bot_set_nat)))) ) ) )
=> ~ pp(aa_set_nat_bool(finite_finite_nat,A2)) ) ) ).
% infinite_coinduct
tff(fact_7969_infinite__remove,axiom,
! [S2: set_VEBT_VEBT,A: vEBT_VEBT] :
( ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S2))
=> ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))) ) ).
% infinite_remove
tff(fact_7970_infinite__remove,axiom,
! [S2: set_complex,A: complex] :
( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
=> ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) ).
% infinite_remove
tff(fact_7971_infinite__remove,axiom,
! [S2: set_int,A: int] :
( ~ pp(aa_set_int_bool(finite_finite_int,S2))
=> ~ pp(aa_set_int_bool(finite_finite_int,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int)))) ) ).
% infinite_remove
tff(fact_7972_infinite__remove,axiom,
! [S2: set_real,A: real] :
( ~ pp(aa_set_real_bool(finite_finite_real,S2))
=> ~ pp(aa_set_real_bool(finite_finite_real,minus_minus_set_real(S2,insert_real(A,bot_bot_set_real)))) ) ).
% infinite_remove
tff(fact_7973_infinite__remove,axiom,
! [S2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
( ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,S2))
=> ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,minus_1356011639430497352at_nat(S2,insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat)))) ) ).
% infinite_remove
tff(fact_7974_infinite__remove,axiom,
! [S2: set_nat,A: nat] :
( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
=> ~ pp(aa_set_nat_bool(finite_finite_nat,minus_minus_set_nat(S2,insert_nat(A,bot_bot_set_nat)))) ) ).
% infinite_remove
tff(fact_7975_subset__insert__iff,axiom,
! [A2: set_VEBT_VEBT,X: vEBT_VEBT,B2: set_VEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),insert_VEBT_VEBT(X,B2)))
<=> ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),A2))
=> pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT))),B2)) )
& ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),A2))
=> pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B2)) ) ) ) ).
% subset_insert_iff
tff(fact_7976_subset__insert__iff,axiom,
! [A2: set_complex,X: complex,B2: set_complex] :
( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),insert_complex(X,B2)))
<=> ( ( pp(aa_set_complex_bool(member_complex(X),A2))
=> pp(aa_set_complex_bool(ord_le211207098394363844omplex(minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex))),B2)) )
& ( ~ pp(aa_set_complex_bool(member_complex(X),A2))
=> pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2)) ) ) ) ).
% subset_insert_iff
tff(fact_7977_subset__insert__iff,axiom,
! [A2: set_int,X: int,B2: set_int] :
( pp(aa_set_int_bool(ord_less_eq_set_int(A2),insert_int(X,B2)))
<=> ( ( pp(aa_set_int_bool(member_int(X),A2))
=> pp(aa_set_int_bool(ord_less_eq_set_int(minus_minus_set_int(A2,insert_int(X,bot_bot_set_int))),B2)) )
& ( ~ pp(aa_set_int_bool(member_int(X),A2))
=> pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2)) ) ) ) ).
% subset_insert_iff
tff(fact_7978_subset__insert__iff,axiom,
! [A2: set_real,X: real,B2: set_real] :
( pp(aa_set_real_bool(ord_less_eq_set_real(A2),insert_real(X,B2)))
<=> ( ( pp(aa_set_real_bool(member_real(X),A2))
=> pp(aa_set_real_bool(ord_less_eq_set_real(minus_minus_set_real(A2,insert_real(X,bot_bot_set_real))),B2)) )
& ( ~ pp(aa_set_real_bool(member_real(X),A2))
=> pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2)) ) ) ) ).
% subset_insert_iff
tff(fact_7979_subset__insert__iff,axiom,
! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),insert8211810215607154385at_nat(X,B2)))
<=> ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A2))
=> pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))),B2)) )
& ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A2))
=> pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),B2)) ) ) ) ).
% subset_insert_iff
tff(fact_7980_subset__insert__iff,axiom,
! [A2: set_nat,X: nat,B2: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),insert_nat(X,B2)))
<=> ( ( pp(aa_set_nat_bool(member_nat(X),A2))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(minus_minus_set_nat(A2,insert_nat(X,bot_bot_set_nat))),B2)) )
& ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),B2)) ) ) ) ).
% subset_insert_iff
tff(fact_7981_Diff__single__insert,axiom,
! [A2: set_VEBT_VEBT,X: vEBT_VEBT,B2: set_VEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT))),B2))
=> pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),insert_VEBT_VEBT(X,B2))) ) ).
% Diff_single_insert
tff(fact_7982_Diff__single__insert,axiom,
! [A2: set_int,X: int,B2: set_int] :
( pp(aa_set_int_bool(ord_less_eq_set_int(minus_minus_set_int(A2,insert_int(X,bot_bot_set_int))),B2))
=> pp(aa_set_int_bool(ord_less_eq_set_int(A2),insert_int(X,B2))) ) ).
% Diff_single_insert
tff(fact_7983_Diff__single__insert,axiom,
! [A2: set_real,X: real,B2: set_real] :
( pp(aa_set_real_bool(ord_less_eq_set_real(minus_minus_set_real(A2,insert_real(X,bot_bot_set_real))),B2))
=> pp(aa_set_real_bool(ord_less_eq_set_real(A2),insert_real(X,B2))) ) ).
% Diff_single_insert
tff(fact_7984_Diff__single__insert,axiom,
! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))),B2))
=> pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),insert8211810215607154385at_nat(X,B2))) ) ).
% Diff_single_insert
tff(fact_7985_Diff__single__insert,axiom,
! [A2: set_nat,X: nat,B2: set_nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(minus_minus_set_nat(A2,insert_nat(X,bot_bot_set_nat))),B2))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),insert_nat(X,B2))) ) ).
% Diff_single_insert
tff(fact_7986_fact__le__power,axiom,
! [N: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,N)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),N)))) ).
% fact_le_power
tff(fact_7987_fact__le__power,axiom,
! [N: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,N)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),N)))) ).
% fact_le_power
tff(fact_7988_fact__le__power,axiom,
! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,N)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),N)))) ).
% fact_le_power
tff(fact_7989_fact__le__power,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),N)))) ).
% fact_le_power
tff(fact_7990_set__update__subset__insert,axiom,
! [Xs: list_real,I: nat,X: real] : pp(aa_set_real_bool(ord_less_eq_set_real(set_real2(list_update_real(Xs,I,X))),insert_real(X,set_real2(Xs)))) ).
% set_update_subset_insert
tff(fact_7991_set__update__subset__insert,axiom,
! [Xs: list_int,I: nat,X: int] : pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(list_update_int(Xs,I,X))),insert_int(X,set_int2(Xs)))) ).
% set_update_subset_insert
tff(fact_7992_set__update__subset__insert,axiom,
! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] : pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,X))),insert_VEBT_VEBT(X,set_VEBT_VEBT2(Xs)))) ).
% set_update_subset_insert
tff(fact_7993_set__update__subset__insert,axiom,
! [Xs: list_nat,I: nat,X: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(list_update_nat(Xs,I,X))),insert_nat(X,set_nat2(Xs)))) ).
% set_update_subset_insert
tff(fact_7994_cos__double__sin,axiom,
! [W: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),W)) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).
% cos_double_sin
tff(fact_7995_cos__double__sin,axiom,
! [W: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),W)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).
% cos_double_sin
tff(fact_7996_cos__two__neq__zero,axiom,
aa_real_real(cos_real,numeral_numeral_real(aa_num_num(bit0,one))) != zero_zero_real ).
% cos_two_neq_zero
tff(fact_7997_cos__mono__less__eq,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),pi))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ) ) ) ).
% cos_mono_less_eq
tff(fact_7998_cos__monotone__0__pi,axiom,
! [Y: real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))) ) ) ) ).
% cos_monotone_0_pi
tff(fact_7999_sin__eq__0__pi,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),pi))
=> ( ( aa_real_real(sin_real,X) = zero_zero_real )
=> ( X = zero_zero_real ) ) ) ) ).
% sin_eq_0_pi
tff(fact_8000_sin__zero__pi__iff,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),pi))
=> ( ( aa_real_real(sin_real,X) = zero_zero_real )
<=> ( X = zero_zero_real ) ) ) ).
% sin_zero_pi_iff
tff(fact_8001_cos__monotone__minus__pi__0_H,axiom,
! [Y: real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,X))) ) ) ) ).
% cos_monotone_minus_pi_0'
tff(fact_8002_sin__zero__iff__int2,axiom,
! [X: real] :
( ( aa_real_real(sin_real,X) = zero_zero_real )
<=> ? [I4: int] : X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I4)),pi) ) ).
% sin_zero_iff_int2
tff(fact_8003_choose__dvd,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,K)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_Code_integer(semiri3624122377584611663nteger,N))) ) ).
% choose_dvd
tff(fact_8004_choose__dvd,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(dvd_dvd_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_rat(semiri773545260158071498ct_rat,N))) ) ).
% choose_dvd
tff(fact_8005_choose__dvd,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1406184849735516958ct_int,K)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_int(semiri1406184849735516958ct_int,N))) ) ).
% choose_dvd
tff(fact_8006_choose__dvd,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_real(semiri2265585572941072030t_real,N))) ) ).
% choose_dvd
tff(fact_8007_choose__dvd,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_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(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ).
% choose_dvd
tff(fact_8008_remove__induct,axiom,
! [P: fun_se448528567104302406T_bool,B2: set_VEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT))
=> ( ( ~ pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,B2))
=> pp(aa_set8260762808118238015T_bool(P,B2)) )
=> ( ! [A5: set_VEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A5))
=> ( ( A5 != bot_bo8194388402131092736T_VEBT )
=> ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A5),B2))
=> ( ! [X4: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),A5))
=> pp(aa_set8260762808118238015T_bool(P,minus_5127226145743854075T_VEBT(A5,insert_VEBT_VEBT(X4,bot_bo8194388402131092736T_VEBT)))) )
=> pp(aa_set8260762808118238015T_bool(P,A5)) ) ) ) )
=> pp(aa_set8260762808118238015T_bool(P,B2)) ) ) ) ).
% remove_induct
tff(fact_8009_remove__induct,axiom,
! [P: fun_set_complex_bool,B2: set_complex] :
( pp(aa_set_complex_bool(P,bot_bot_set_complex))
=> ( ( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
=> pp(aa_set_complex_bool(P,B2)) )
=> ( ! [A5: set_complex] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A5))
=> ( ( A5 != bot_bot_set_complex )
=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A5),B2))
=> ( ! [X4: complex] :
( pp(aa_set_complex_bool(member_complex(X4),A5))
=> pp(aa_set_complex_bool(P,minus_811609699411566653omplex(A5,insert_complex(X4,bot_bot_set_complex)))) )
=> pp(aa_set_complex_bool(P,A5)) ) ) ) )
=> pp(aa_set_complex_bool(P,B2)) ) ) ) ).
% remove_induct
tff(fact_8010_remove__induct,axiom,
! [P: fun_set_int_bool,B2: set_int] :
( pp(aa_set_int_bool(P,bot_bot_set_int))
=> ( ( ~ pp(aa_set_int_bool(finite_finite_int,B2))
=> pp(aa_set_int_bool(P,B2)) )
=> ( ! [A5: set_int] :
( pp(aa_set_int_bool(finite_finite_int,A5))
=> ( ( A5 != bot_bot_set_int )
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A5),B2))
=> ( ! [X4: int] :
( pp(aa_set_int_bool(member_int(X4),A5))
=> pp(aa_set_int_bool(P,minus_minus_set_int(A5,insert_int(X4,bot_bot_set_int)))) )
=> pp(aa_set_int_bool(P,A5)) ) ) ) )
=> pp(aa_set_int_bool(P,B2)) ) ) ) ).
% remove_induct
tff(fact_8011_remove__induct,axiom,
! [P: fun_set_real_bool,B2: set_real] :
( pp(aa_set_real_bool(P,bot_bot_set_real))
=> ( ( ~ pp(aa_set_real_bool(finite_finite_real,B2))
=> pp(aa_set_real_bool(P,B2)) )
=> ( ! [A5: set_real] :
( pp(aa_set_real_bool(finite_finite_real,A5))
=> ( ( A5 != bot_bot_set_real )
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(A5),B2))
=> ( ! [X4: real] :
( pp(aa_set_real_bool(member_real(X4),A5))
=> pp(aa_set_real_bool(P,minus_minus_set_real(A5,insert_real(X4,bot_bot_set_real)))) )
=> pp(aa_set_real_bool(P,A5)) ) ) ) )
=> pp(aa_set_real_bool(P,B2)) ) ) ) ).
% remove_induct
tff(fact_8012_remove__induct,axiom,
! [P: fun_se8074476271188067385t_bool,B2: set_Pr1261947904930325089at_nat] :
( pp(aa_set3135850010578026034t_bool(P,bot_bo2099793752762293965at_nat))
=> ( ( ~ pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B2))
=> pp(aa_set3135850010578026034t_bool(P,B2)) )
=> ( ! [A5: set_Pr1261947904930325089at_nat] :
( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A5))
=> ( ( A5 != bot_bo2099793752762293965at_nat )
=> ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A5),B2))
=> ( ! [X4: product_prod_nat_nat] :
( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X4),A5))
=> pp(aa_set3135850010578026034t_bool(P,minus_1356011639430497352at_nat(A5,insert8211810215607154385at_nat(X4,bot_bo2099793752762293965at_nat)))) )
=> pp(aa_set3135850010578026034t_bool(P,A5)) ) ) ) )
=> pp(aa_set3135850010578026034t_bool(P,B2)) ) ) ) ).
% remove_induct
tff(fact_8013_remove__induct,axiom,
! [P: fun_set_nat_bool,B2: set_nat] :
( pp(aa_set_nat_bool(P,bot_bot_set_nat))
=> ( ( ~ pp(aa_set_nat_bool(finite_finite_nat,B2))
=> pp(aa_set_nat_bool(P,B2)) )
=> ( ! [A5: set_nat] :
( pp(aa_set_nat_bool(finite_finite_nat,A5))
=> ( ( A5 != bot_bot_set_nat )
=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A5),B2))
=> ( ! [X4: nat] :
( pp(aa_set_nat_bool(member_nat(X4),A5))
=> pp(aa_set_nat_bool(P,minus_minus_set_nat(A5,insert_nat(X4,bot_bot_set_nat)))) )
=> pp(aa_set_nat_bool(P,A5)) ) ) ) )
=> pp(aa_set_nat_bool(P,B2)) ) ) ) ).
% remove_induct
tff(fact_8014_finite__remove__induct,axiom,
! [B2: set_VEBT_VEBT,P: fun_se448528567104302406T_bool] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,B2))
=> ( pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT))
=> ( ! [A5: set_VEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A5))
=> ( ( A5 != bot_bo8194388402131092736T_VEBT )
=> ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A5),B2))
=> ( ! [X4: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),A5))
=> pp(aa_set8260762808118238015T_bool(P,minus_5127226145743854075T_VEBT(A5,insert_VEBT_VEBT(X4,bot_bo8194388402131092736T_VEBT)))) )
=> pp(aa_set8260762808118238015T_bool(P,A5)) ) ) ) )
=> pp(aa_set8260762808118238015T_bool(P,B2)) ) ) ) ).
% finite_remove_induct
tff(fact_8015_finite__remove__induct,axiom,
! [B2: set_complex,P: fun_set_complex_bool] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,B2))
=> ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
=> ( ! [A5: set_complex] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A5))
=> ( ( A5 != bot_bot_set_complex )
=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(A5),B2))
=> ( ! [X4: complex] :
( pp(aa_set_complex_bool(member_complex(X4),A5))
=> pp(aa_set_complex_bool(P,minus_811609699411566653omplex(A5,insert_complex(X4,bot_bot_set_complex)))) )
=> pp(aa_set_complex_bool(P,A5)) ) ) ) )
=> pp(aa_set_complex_bool(P,B2)) ) ) ) ).
% finite_remove_induct
tff(fact_8016_finite__remove__induct,axiom,
! [B2: set_int,P: fun_set_int_bool] :
( pp(aa_set_int_bool(finite_finite_int,B2))
=> ( pp(aa_set_int_bool(P,bot_bot_set_int))
=> ( ! [A5: set_int] :
( pp(aa_set_int_bool(finite_finite_int,A5))
=> ( ( A5 != bot_bot_set_int )
=> ( pp(aa_set_int_bool(ord_less_eq_set_int(A5),B2))
=> ( ! [X4: int] :
( pp(aa_set_int_bool(member_int(X4),A5))
=> pp(aa_set_int_bool(P,minus_minus_set_int(A5,insert_int(X4,bot_bot_set_int)))) )
=> pp(aa_set_int_bool(P,A5)) ) ) ) )
=> pp(aa_set_int_bool(P,B2)) ) ) ) ).
% finite_remove_induct
tff(fact_8017_finite__remove__induct,axiom,
! [B2: set_real,P: fun_set_real_bool] :
( pp(aa_set_real_bool(finite_finite_real,B2))
=> ( pp(aa_set_real_bool(P,bot_bot_set_real))
=> ( ! [A5: set_real] :
( pp(aa_set_real_bool(finite_finite_real,A5))
=> ( ( A5 != bot_bot_set_real )
=> ( pp(aa_set_real_bool(ord_less_eq_set_real(A5),B2))
=> ( ! [X4: real] :
( pp(aa_set_real_bool(member_real(X4),A5))
=> pp(aa_set_real_bool(P,minus_minus_set_real(A5,insert_real(X4,bot_bot_set_real)))) )
=> pp(aa_set_real_bool(P,A5)) ) ) ) )
=> pp(aa_set_real_bool(P,B2)) ) ) ) ).
% finite_remove_induct
tff(fact_8018_finite__remove__induct,axiom,
! [B2: set_Pr1261947904930325089at_nat,P: fun_se8074476271188067385t_bool] :
( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,B2))
=> ( pp(aa_set3135850010578026034t_bool(P,bot_bo2099793752762293965at_nat))
=> ( ! [A5: set_Pr1261947904930325089at_nat] :
( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,A5))
=> ( ( A5 != bot_bo2099793752762293965at_nat )
=> ( pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A5),B2))
=> ( ! [X4: product_prod_nat_nat] :
( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X4),A5))
=> pp(aa_set3135850010578026034t_bool(P,minus_1356011639430497352at_nat(A5,insert8211810215607154385at_nat(X4,bot_bo2099793752762293965at_nat)))) )
=> pp(aa_set3135850010578026034t_bool(P,A5)) ) ) ) )
=> pp(aa_set3135850010578026034t_bool(P,B2)) ) ) ) ).
% finite_remove_induct
tff(fact_8019_finite__remove__induct,axiom,
! [B2: set_nat,P: fun_set_nat_bool] :
( pp(aa_set_nat_bool(finite_finite_nat,B2))
=> ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
=> ( ! [A5: set_nat] :
( pp(aa_set_nat_bool(finite_finite_nat,A5))
=> ( ( A5 != bot_bot_set_nat )
=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(A5),B2))
=> ( ! [X4: nat] :
( pp(aa_set_nat_bool(member_nat(X4),A5))
=> pp(aa_set_nat_bool(P,minus_minus_set_nat(A5,insert_nat(X4,bot_bot_set_nat)))) )
=> pp(aa_set_nat_bool(P,A5)) ) ) ) )
=> pp(aa_set_nat_bool(P,B2)) ) ) ) ).
% finite_remove_induct
tff(fact_8020_finite__induct__select,axiom,
! [S2: set_VEBT_VEBT,P: fun_se448528567104302406T_bool] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S2))
=> ( pp(aa_set8260762808118238015T_bool(P,bot_bo8194388402131092736T_VEBT))
=> ( ! [T5: set_VEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(ord_le3480810397992357184T_VEBT(T5),S2))
=> ( pp(aa_set8260762808118238015T_bool(P,T5))
=> ? [X4: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),minus_5127226145743854075T_VEBT(S2,T5)))
& pp(aa_set8260762808118238015T_bool(P,insert_VEBT_VEBT(X4,T5))) ) ) )
=> pp(aa_set8260762808118238015T_bool(P,S2)) ) ) ) ).
% finite_induct_select
tff(fact_8021_finite__induct__select,axiom,
! [S2: set_complex,P: fun_set_complex_bool] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
=> ( pp(aa_set_complex_bool(P,bot_bot_set_complex))
=> ( ! [T5: set_complex] :
( pp(aa_set_complex_bool(ord_less_set_complex(T5),S2))
=> ( pp(aa_set_complex_bool(P,T5))
=> ? [X4: complex] :
( pp(aa_set_complex_bool(member_complex(X4),minus_811609699411566653omplex(S2,T5)))
& pp(aa_set_complex_bool(P,insert_complex(X4,T5))) ) ) )
=> pp(aa_set_complex_bool(P,S2)) ) ) ) ).
% finite_induct_select
tff(fact_8022_finite__induct__select,axiom,
! [S2: set_int,P: fun_set_int_bool] :
( pp(aa_set_int_bool(finite_finite_int,S2))
=> ( pp(aa_set_int_bool(P,bot_bot_set_int))
=> ( ! [T5: set_int] :
( pp(aa_set_int_bool(ord_less_set_int(T5),S2))
=> ( pp(aa_set_int_bool(P,T5))
=> ? [X4: int] :
( pp(aa_set_int_bool(member_int(X4),minus_minus_set_int(S2,T5)))
& pp(aa_set_int_bool(P,insert_int(X4,T5))) ) ) )
=> pp(aa_set_int_bool(P,S2)) ) ) ) ).
% finite_induct_select
tff(fact_8023_finite__induct__select,axiom,
! [S2: set_real,P: fun_set_real_bool] :
( pp(aa_set_real_bool(finite_finite_real,S2))
=> ( pp(aa_set_real_bool(P,bot_bot_set_real))
=> ( ! [T5: set_real] :
( pp(aa_set_real_bool(ord_less_set_real(T5),S2))
=> ( pp(aa_set_real_bool(P,T5))
=> ? [X4: real] :
( pp(aa_set_real_bool(member_real(X4),minus_minus_set_real(S2,T5)))
& pp(aa_set_real_bool(P,insert_real(X4,T5))) ) ) )
=> pp(aa_set_real_bool(P,S2)) ) ) ) ).
% finite_induct_select
tff(fact_8024_finite__induct__select,axiom,
! [S2: set_Pr1261947904930325089at_nat,P: fun_se8074476271188067385t_bool] :
( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,S2))
=> ( pp(aa_set3135850010578026034t_bool(P,bot_bo2099793752762293965at_nat))
=> ( ! [T5: set_Pr1261947904930325089at_nat] :
( pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(T5),S2))
=> ( pp(aa_set3135850010578026034t_bool(P,T5))
=> ? [X4: product_prod_nat_nat] :
( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X4),minus_1356011639430497352at_nat(S2,T5)))
& pp(aa_set3135850010578026034t_bool(P,insert8211810215607154385at_nat(X4,T5))) ) ) )
=> pp(aa_set3135850010578026034t_bool(P,S2)) ) ) ) ).
% finite_induct_select
tff(fact_8025_finite__induct__select,axiom,
! [S2: set_nat,P: fun_set_nat_bool] :
( pp(aa_set_nat_bool(finite_finite_nat,S2))
=> ( pp(aa_set_nat_bool(P,bot_bot_set_nat))
=> ( ! [T5: set_nat] :
( pp(aa_set_nat_bool(ord_less_set_nat(T5),S2))
=> ( pp(aa_set_nat_bool(P,T5))
=> ? [X4: nat] :
( pp(aa_set_nat_bool(member_nat(X4),minus_minus_set_nat(S2,T5)))
& pp(aa_set_nat_bool(P,insert_nat(X4,T5))) ) ) )
=> pp(aa_set_nat_bool(P,S2)) ) ) ) ).
% finite_induct_select
tff(fact_8026_psubset__insert__iff,axiom,
! [A2: set_VEBT_VEBT,X: vEBT_VEBT,B2: set_VEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(ord_le3480810397992357184T_VEBT(A2),insert_VEBT_VEBT(X,B2)))
<=> ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),B2))
=> pp(aa_set8260762808118238015T_bool(ord_le3480810397992357184T_VEBT(A2),B2)) )
& ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),B2))
=> ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),A2))
=> pp(aa_set8260762808118238015T_bool(ord_le3480810397992357184T_VEBT(minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT))),B2)) )
& ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),A2))
=> pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(A2),B2)) ) ) ) ) ) ).
% psubset_insert_iff
tff(fact_8027_psubset__insert__iff,axiom,
! [A2: set_complex,X: complex,B2: set_complex] :
( pp(aa_set_complex_bool(ord_less_set_complex(A2),insert_complex(X,B2)))
<=> ( ( pp(aa_set_complex_bool(member_complex(X),B2))
=> pp(aa_set_complex_bool(ord_less_set_complex(A2),B2)) )
& ( ~ pp(aa_set_complex_bool(member_complex(X),B2))
=> ( ( pp(aa_set_complex_bool(member_complex(X),A2))
=> pp(aa_set_complex_bool(ord_less_set_complex(minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex))),B2)) )
& ( ~ pp(aa_set_complex_bool(member_complex(X),A2))
=> pp(aa_set_complex_bool(ord_le211207098394363844omplex(A2),B2)) ) ) ) ) ) ).
% psubset_insert_iff
tff(fact_8028_psubset__insert__iff,axiom,
! [A2: set_int,X: int,B2: set_int] :
( pp(aa_set_int_bool(ord_less_set_int(A2),insert_int(X,B2)))
<=> ( ( pp(aa_set_int_bool(member_int(X),B2))
=> pp(aa_set_int_bool(ord_less_set_int(A2),B2)) )
& ( ~ pp(aa_set_int_bool(member_int(X),B2))
=> ( ( pp(aa_set_int_bool(member_int(X),A2))
=> pp(aa_set_int_bool(ord_less_set_int(minus_minus_set_int(A2,insert_int(X,bot_bot_set_int))),B2)) )
& ( ~ pp(aa_set_int_bool(member_int(X),A2))
=> pp(aa_set_int_bool(ord_less_eq_set_int(A2),B2)) ) ) ) ) ) ).
% psubset_insert_iff
tff(fact_8029_psubset__insert__iff,axiom,
! [A2: set_real,X: real,B2: set_real] :
( pp(aa_set_real_bool(ord_less_set_real(A2),insert_real(X,B2)))
<=> ( ( pp(aa_set_real_bool(member_real(X),B2))
=> pp(aa_set_real_bool(ord_less_set_real(A2),B2)) )
& ( ~ pp(aa_set_real_bool(member_real(X),B2))
=> ( ( pp(aa_set_real_bool(member_real(X),A2))
=> pp(aa_set_real_bool(ord_less_set_real(minus_minus_set_real(A2,insert_real(X,bot_bot_set_real))),B2)) )
& ( ~ pp(aa_set_real_bool(member_real(X),A2))
=> pp(aa_set_real_bool(ord_less_eq_set_real(A2),B2)) ) ) ) ) ) ).
% psubset_insert_iff
tff(fact_8030_psubset__insert__iff,axiom,
! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
( pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(A2),insert8211810215607154385at_nat(X,B2)))
<=> ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B2))
=> pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(A2),B2)) )
& ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B2))
=> ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A2))
=> pp(aa_set3135850010578026034t_bool(ord_le7866589430770878221at_nat(minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))),B2)) )
& ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A2))
=> pp(aa_set3135850010578026034t_bool(ord_le3146513528884898305at_nat(A2),B2)) ) ) ) ) ) ).
% psubset_insert_iff
tff(fact_8031_psubset__insert__iff,axiom,
! [A2: set_nat,X: nat,B2: set_nat] :
( pp(aa_set_nat_bool(ord_less_set_nat(A2),insert_nat(X,B2)))
<=> ( ( pp(aa_set_nat_bool(member_nat(X),B2))
=> pp(aa_set_nat_bool(ord_less_set_nat(A2),B2)) )
& ( ~ pp(aa_set_nat_bool(member_nat(X),B2))
=> ( ( pp(aa_set_nat_bool(member_nat(X),A2))
=> pp(aa_set_nat_bool(ord_less_set_nat(minus_minus_set_nat(A2,insert_nat(X,bot_bot_set_nat))),B2)) )
& ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),B2)) ) ) ) ) ) ).
% psubset_insert_iff
tff(fact_8032_fact__numeral,axiom,
! [K: num] : aa_nat_complex(semiri5044797733671781792omplex,aa_num_nat(numeral_numeral_nat,K)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(K)),aa_nat_complex(semiri5044797733671781792omplex,aa_num_nat(pred_numeral,K))) ).
% fact_numeral
tff(fact_8033_fact__numeral,axiom,
! [K: num] : aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(numeral_numeral_nat,K)) = aa_rat_rat(times_times_rat(numeral_numeral_rat(K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(pred_numeral,K))) ).
% fact_numeral
tff(fact_8034_fact__numeral,axiom,
! [K: num] : aa_nat_int(semiri1406184849735516958ct_int,aa_num_nat(numeral_numeral_nat,K)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,K)),aa_nat_int(semiri1406184849735516958ct_int,aa_num_nat(pred_numeral,K))) ).
% fact_numeral
tff(fact_8035_fact__numeral,axiom,
! [K: num] : aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(numeral_numeral_nat,K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(K)),aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(pred_numeral,K))) ).
% fact_numeral
tff(fact_8036_fact__numeral,axiom,
! [K: num] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(pred_numeral,K))) ).
% fact_numeral
tff(fact_8037_set__replicate__Suc,axiom,
! [N: nat,X: vEBT_VEBT] : set_VEBT_VEBT2(replicate_VEBT_VEBT(aa_nat_nat(suc,N),X)) = insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT) ).
% set_replicate_Suc
tff(fact_8038_set__replicate__Suc,axiom,
! [N: nat,X: nat] : set_nat2(replicate_nat(aa_nat_nat(suc,N),X)) = insert_nat(X,bot_bot_set_nat) ).
% set_replicate_Suc
tff(fact_8039_set__replicate__Suc,axiom,
! [N: nat,X: int] : set_int2(replicate_int(aa_nat_nat(suc,N),X)) = insert_int(X,bot_bot_set_int) ).
% set_replicate_Suc
tff(fact_8040_set__replicate__Suc,axiom,
! [N: nat,X: real] : set_real2(replicate_real(aa_nat_nat(suc,N),X)) = insert_real(X,bot_bot_set_real) ).
% set_replicate_Suc
tff(fact_8041_sincos__total__pi,axiom,
! [Y: real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
=> ? [T4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T4))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T4),pi))
& ( X = aa_real_real(cos_real,T4) )
& ( Y = aa_real_real(sin_real,T4) ) ) ) ) ).
% sincos_total_pi
tff(fact_8042_set__replicate__conv__if,axiom,
! [N: nat,X: vEBT_VEBT] :
( ( ( N = zero_zero_nat )
=> ( set_VEBT_VEBT2(replicate_VEBT_VEBT(N,X)) = bot_bo8194388402131092736T_VEBT ) )
& ( ( N != zero_zero_nat )
=> ( set_VEBT_VEBT2(replicate_VEBT_VEBT(N,X)) = insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT) ) ) ) ).
% set_replicate_conv_if
tff(fact_8043_set__replicate__conv__if,axiom,
! [N: nat,X: nat] :
( ( ( N = zero_zero_nat )
=> ( set_nat2(replicate_nat(N,X)) = bot_bot_set_nat ) )
& ( ( N != zero_zero_nat )
=> ( set_nat2(replicate_nat(N,X)) = insert_nat(X,bot_bot_set_nat) ) ) ) ).
% set_replicate_conv_if
tff(fact_8044_set__replicate__conv__if,axiom,
! [N: nat,X: int] :
( ( ( N = zero_zero_nat )
=> ( set_int2(replicate_int(N,X)) = bot_bot_set_int ) )
& ( ( N != zero_zero_nat )
=> ( set_int2(replicate_int(N,X)) = insert_int(X,bot_bot_set_int) ) ) ) ).
% set_replicate_conv_if
tff(fact_8045_set__replicate__conv__if,axiom,
! [N: nat,X: real] :
( ( ( N = zero_zero_nat )
=> ( set_real2(replicate_real(N,X)) = bot_bot_set_real ) )
& ( ( N != zero_zero_nat )
=> ( set_real2(replicate_real(N,X)) = insert_real(X,bot_bot_set_real) ) ) ) ).
% set_replicate_conv_if
tff(fact_8046_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(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),pi)),numeral_numeral_real(aa_num_num(bit0,one))))) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),pi)),numeral_numeral_real(aa_num_num(bit0,one))))) ).
% sin_expansion_lemma
tff(fact_8047_sum__diff1__nat,axiom,
! [A: vEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
=> ( groups771621172384141258BT_nat(F,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups771621172384141258BT_nat(F,A2)),aa_VEBT_VEBT_nat(F,A)) ) )
& ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
=> ( groups771621172384141258BT_nat(F,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) = groups771621172384141258BT_nat(F,A2) ) ) ) ).
% sum_diff1_nat
tff(fact_8048_sum__diff1__nat,axiom,
! [A: complex,A2: set_complex,F: fun_complex_nat] :
( ( pp(aa_set_complex_bool(member_complex(A),A2))
=> ( groups5693394587270226106ex_nat(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5693394587270226106ex_nat(F,A2)),aa_complex_nat(F,A)) ) )
& ( ~ pp(aa_set_complex_bool(member_complex(A),A2))
=> ( groups5693394587270226106ex_nat(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = groups5693394587270226106ex_nat(F,A2) ) ) ) ).
% sum_diff1_nat
tff(fact_8049_sum__diff1__nat,axiom,
! [A: int,A2: set_int,F: fun_int_nat] :
( ( pp(aa_set_int_bool(member_int(A),A2))
=> ( groups4541462559716669496nt_nat(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A2)),aa_int_nat(F,A)) ) )
& ( ~ pp(aa_set_int_bool(member_int(A),A2))
=> ( groups4541462559716669496nt_nat(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = groups4541462559716669496nt_nat(F,A2) ) ) ) ).
% sum_diff1_nat
tff(fact_8050_sum__diff1__nat,axiom,
! [A: real,A2: set_real,F: fun_real_nat] :
( ( pp(aa_set_real_bool(member_real(A),A2))
=> ( groups1935376822645274424al_nat(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(F,A2)),aa_real_nat(F,A)) ) )
& ( ~ pp(aa_set_real_bool(member_real(A),A2))
=> ( groups1935376822645274424al_nat(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = groups1935376822645274424al_nat(F,A2) ) ) ) ).
% sum_diff1_nat
tff(fact_8051_sum__diff1__nat,axiom,
! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),A2))
=> ( groups977919841031483927at_nat(F,minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups977919841031483927at_nat(F,A2)),aa_Pro4243173770044397973at_nat(F,A)) ) )
& ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A),A2))
=> ( groups977919841031483927at_nat(F,minus_1356011639430497352at_nat(A2,insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))) = groups977919841031483927at_nat(F,A2) ) ) ) ).
% sum_diff1_nat
tff(fact_8052_sum__diff1__nat,axiom,
! [A: nat,A2: set_nat,F: fun_nat_nat] :
( ( pp(aa_set_nat_bool(member_nat(A),A2))
=> ( aa_set_nat_nat(groups3542108847815614940at_nat(F),minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),aa_nat_nat(F,A)) ) )
& ( ~ pp(aa_set_nat_bool(member_nat(A),A2))
=> ( aa_set_nat_nat(groups3542108847815614940at_nat(F),minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = aa_set_nat_nat(groups3542108847815614940at_nat(F),A2) ) ) ) ).
% sum_diff1_nat
tff(fact_8053_simp__from__to,axiom,
! [J: int,I: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
=> ( set_or1266510415728281911st_int(I,J) = bot_bot_set_int ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
=> ( set_or1266510415728281911st_int(I,J) = insert_int(I,set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) ) ) ).
% simp_from_to
tff(fact_8054_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(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),pi)),numeral_numeral_real(aa_num_num(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(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),pi)),numeral_numeral_real(aa_num_num(bit0,one)))))) ).
% cos_expansion_lemma
tff(fact_8055_sin__gt__zero__02,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),numeral_numeral_real(aa_num_num(bit0,one))))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,X))) ) ) ).
% sin_gt_zero_02
tff(fact_8056_cos__two__less__zero,axiom,
pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,numeral_numeral_real(aa_num_num(bit0,one)))),zero_zero_real)) ).
% cos_two_less_zero
tff(fact_8057_cos__is__zero,axiom,
? [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),numeral_numeral_real(aa_num_num(bit0,one))))
& ( aa_real_real(cos_real,X3) = zero_zero_real )
& ! [Y3: real] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),numeral_numeral_real(aa_num_num(bit0,one))))
& ( aa_real_real(cos_real,Y3) = zero_zero_real ) )
=> ( Y3 = X3 ) ) ) ).
% cos_is_zero
tff(fact_8058_cos__two__le__zero,axiom,
pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cos_real,numeral_numeral_real(aa_num_num(bit0,one)))),zero_zero_real)) ).
% cos_two_le_zero
tff(fact_8059_cos__monotone__minus__pi__0,axiom,
! [Y: real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,X))) ) ) ) ).
% cos_monotone_minus_pi_0
tff(fact_8060_cos__total,axiom,
! [Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
=> ? [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),pi))
& ( aa_real_real(cos_real,X3) = Y )
& ! [Y3: real] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),pi))
& ( aa_real_real(cos_real,Y3) = Y ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% cos_total
tff(fact_8061_sum_Oinsert__remove,axiom,
! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,X: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
=> ( groups2240296850493347238T_real(G,insert_VEBT_VEBT(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(G,X)),groups2240296850493347238T_real(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT)))) ) ) ).
% sum.insert_remove
tff(fact_8062_sum_Oinsert__remove,axiom,
! [A2: set_complex,G: fun_complex_real,X: complex] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( groups5808333547571424918x_real(G,insert_complex(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,X)),groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ).
% sum.insert_remove
tff(fact_8063_sum_Oinsert__remove,axiom,
! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,X: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
=> ( groups136491112297645522BT_rat(G,insert_VEBT_VEBT(X,A2)) = plus_plus_rat(aa_VEBT_VEBT_rat(G,X),groups136491112297645522BT_rat(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT)))) ) ) ).
% sum.insert_remove
tff(fact_8064_sum_Oinsert__remove,axiom,
! [A2: set_complex,G: fun_complex_rat,X: complex] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( groups5058264527183730370ex_rat(G,insert_complex(X,A2)) = plus_plus_rat(aa_complex_rat(G,X),groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ).
% sum.insert_remove
tff(fact_8065_sum_Oinsert__remove,axiom,
! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_nat,X: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
=> ( groups771621172384141258BT_nat(G,insert_VEBT_VEBT(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(G,X)),groups771621172384141258BT_nat(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT)))) ) ) ).
% sum.insert_remove
tff(fact_8066_sum_Oinsert__remove,axiom,
! [A2: set_complex,G: fun_complex_nat,X: complex] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( groups5693394587270226106ex_nat(G,insert_complex(X,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,X)),groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ).
% sum.insert_remove
tff(fact_8067_sum_Oinsert__remove,axiom,
! [A2: set_VEBT_VEBT,G: fun_VEBT_VEBT_int,X: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
=> ( groups769130701875090982BT_int(G,insert_VEBT_VEBT(X,A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_VEBT_VEBT_int(G,X)),groups769130701875090982BT_int(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT)))) ) ) ).
% sum.insert_remove
tff(fact_8068_sum_Oinsert__remove,axiom,
! [A2: set_complex,G: fun_complex_int,X: complex] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( groups5690904116761175830ex_int(G,insert_complex(X,A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(G,X)),groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ).
% sum.insert_remove
tff(fact_8069_sum_Oinsert__remove,axiom,
! [A2: set_int,G: fun_int_real,X: int] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( groups8778361861064173332t_real(G,insert_int(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,X)),groups8778361861064173332t_real(G,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int)))) ) ) ).
% sum.insert_remove
tff(fact_8070_sum_Oinsert__remove,axiom,
! [A2: set_int,G: fun_int_rat,X: int] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( groups3906332499630173760nt_rat(G,insert_int(X,A2)) = plus_plus_rat(aa_int_rat(G,X),groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int)))) ) ) ).
% sum.insert_remove
tff(fact_8071_sum_Oremove,axiom,
! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: fun_VEBT_VEBT_real] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),A2))
=> ( groups2240296850493347238T_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(G,X)),groups2240296850493347238T_real(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).
% sum.remove
tff(fact_8072_sum_Oremove,axiom,
! [A2: set_complex,X: complex,G: fun_complex_real] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( pp(aa_set_complex_bool(member_complex(X),A2))
=> ( groups5808333547571424918x_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(G,X)),groups5808333547571424918x_real(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).
% sum.remove
tff(fact_8073_sum_Oremove,axiom,
! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: fun_VEBT_VEBT_rat] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),A2))
=> ( groups136491112297645522BT_rat(G,A2) = plus_plus_rat(aa_VEBT_VEBT_rat(G,X),groups136491112297645522BT_rat(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).
% sum.remove
tff(fact_8074_sum_Oremove,axiom,
! [A2: set_complex,X: complex,G: fun_complex_rat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( pp(aa_set_complex_bool(member_complex(X),A2))
=> ( groups5058264527183730370ex_rat(G,A2) = plus_plus_rat(aa_complex_rat(G,X),groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).
% sum.remove
tff(fact_8075_sum_Oremove,axiom,
! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: fun_VEBT_VEBT_nat] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),A2))
=> ( groups771621172384141258BT_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(G,X)),groups771621172384141258BT_nat(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).
% sum.remove
tff(fact_8076_sum_Oremove,axiom,
! [A2: set_complex,X: complex,G: fun_complex_nat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( pp(aa_set_complex_bool(member_complex(X),A2))
=> ( groups5693394587270226106ex_nat(G,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(G,X)),groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).
% sum.remove
tff(fact_8077_sum_Oremove,axiom,
! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: fun_VEBT_VEBT_int] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
=> ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),A2))
=> ( groups769130701875090982BT_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_VEBT_VEBT_int(G,X)),groups769130701875090982BT_int(G,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(X,bot_bo8194388402131092736T_VEBT)))) ) ) ) ).
% sum.remove
tff(fact_8078_sum_Oremove,axiom,
! [A2: set_complex,X: complex,G: fun_complex_int] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( pp(aa_set_complex_bool(member_complex(X),A2))
=> ( groups5690904116761175830ex_int(G,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(G,X)),groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A2,insert_complex(X,bot_bot_set_complex)))) ) ) ) ).
% sum.remove
tff(fact_8079_sum_Oremove,axiom,
! [A2: set_int,X: int,G: fun_int_real] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( pp(aa_set_int_bool(member_int(X),A2))
=> ( groups8778361861064173332t_real(G,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(G,X)),groups8778361861064173332t_real(G,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int)))) ) ) ) ).
% sum.remove
tff(fact_8080_sum_Oremove,axiom,
! [A2: set_int,X: int,G: fun_int_rat] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( pp(aa_set_int_bool(member_int(X),A2))
=> ( groups3906332499630173760nt_rat(G,A2) = plus_plus_rat(aa_int_rat(G,X),groups3906332499630173760nt_rat(G,minus_minus_set_int(A2,insert_int(X,bot_bot_set_int)))) ) ) ) ).
% sum.remove
tff(fact_8081_sum__diff1,axiom,
! [A2: set_VEBT_VEBT,A: vEBT_VEBT,F: fun_VEBT_VEBT_rat] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
=> ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
=> ( groups136491112297645522BT_rat(F,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) = minus_minus_rat(groups136491112297645522BT_rat(F,A2),aa_VEBT_VEBT_rat(F,A)) ) )
& ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
=> ( groups136491112297645522BT_rat(F,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) = groups136491112297645522BT_rat(F,A2) ) ) ) ) ).
% sum_diff1
tff(fact_8082_sum__diff1,axiom,
! [A2: set_complex,A: complex,F: fun_complex_rat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( ( pp(aa_set_complex_bool(member_complex(A),A2))
=> ( groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = minus_minus_rat(groups5058264527183730370ex_rat(F,A2),aa_complex_rat(F,A)) ) )
& ( ~ pp(aa_set_complex_bool(member_complex(A),A2))
=> ( groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = groups5058264527183730370ex_rat(F,A2) ) ) ) ) ).
% sum_diff1
tff(fact_8083_sum__diff1,axiom,
! [A2: set_int,A: int,F: fun_int_rat] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( ( pp(aa_set_int_bool(member_int(A),A2))
=> ( groups3906332499630173760nt_rat(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = minus_minus_rat(groups3906332499630173760nt_rat(F,A2),aa_int_rat(F,A)) ) )
& ( ~ pp(aa_set_int_bool(member_int(A),A2))
=> ( groups3906332499630173760nt_rat(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = groups3906332499630173760nt_rat(F,A2) ) ) ) ) ).
% sum_diff1
tff(fact_8084_sum__diff1,axiom,
! [A2: set_real,A: real,F: fun_real_rat] :
( pp(aa_set_real_bool(finite_finite_real,A2))
=> ( ( pp(aa_set_real_bool(member_real(A),A2))
=> ( groups1300246762558778688al_rat(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = minus_minus_rat(groups1300246762558778688al_rat(F,A2),aa_real_rat(F,A)) ) )
& ( ~ pp(aa_set_real_bool(member_real(A),A2))
=> ( groups1300246762558778688al_rat(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = groups1300246762558778688al_rat(F,A2) ) ) ) ) ).
% sum_diff1
tff(fact_8085_sum__diff1,axiom,
! [A2: set_VEBT_VEBT,A: vEBT_VEBT,F: fun_VEBT_VEBT_int] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
=> ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
=> ( groups769130701875090982BT_int(F,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups769130701875090982BT_int(F,A2)),aa_VEBT_VEBT_int(F,A)) ) )
& ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),A2))
=> ( groups769130701875090982BT_int(F,minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) = groups769130701875090982BT_int(F,A2) ) ) ) ) ).
% sum_diff1
tff(fact_8086_sum__diff1,axiom,
! [A2: set_complex,A: complex,F: fun_complex_int] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( ( pp(aa_set_complex_bool(member_complex(A),A2))
=> ( groups5690904116761175830ex_int(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups5690904116761175830ex_int(F,A2)),aa_complex_int(F,A)) ) )
& ( ~ pp(aa_set_complex_bool(member_complex(A),A2))
=> ( groups5690904116761175830ex_int(F,minus_811609699411566653omplex(A2,insert_complex(A,bot_bot_set_complex))) = groups5690904116761175830ex_int(F,A2) ) ) ) ) ).
% sum_diff1
tff(fact_8087_sum__diff1,axiom,
! [A2: set_real,A: real,F: fun_real_int] :
( pp(aa_set_real_bool(finite_finite_real,A2))
=> ( ( pp(aa_set_real_bool(member_real(A),A2))
=> ( groups1932886352136224148al_int(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups1932886352136224148al_int(F,A2)),aa_real_int(F,A)) ) )
& ( ~ pp(aa_set_real_bool(member_real(A),A2))
=> ( groups1932886352136224148al_int(F,minus_minus_set_real(A2,insert_real(A,bot_bot_set_real))) = groups1932886352136224148al_int(F,A2) ) ) ) ) ).
% sum_diff1
tff(fact_8088_sum__diff1,axiom,
! [A2: set_nat,A: nat,F: fun_nat_rat] :
( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ( ( pp(aa_set_nat_bool(member_nat(A),A2))
=> ( groups2906978787729119204at_rat(F,minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = minus_minus_rat(groups2906978787729119204at_rat(F,A2),aa_nat_rat(F,A)) ) )
& ( ~ pp(aa_set_nat_bool(member_nat(A),A2))
=> ( groups2906978787729119204at_rat(F,minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = groups2906978787729119204at_rat(F,A2) ) ) ) ) ).
% sum_diff1
tff(fact_8089_sum__diff1,axiom,
! [A2: set_nat,A: nat,F: fun_nat_int] :
( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ( ( pp(aa_set_nat_bool(member_nat(A),A2))
=> ( groups3539618377306564664at_int(F,minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(F,A2)),aa_nat_int(F,A)) ) )
& ( ~ pp(aa_set_nat_bool(member_nat(A),A2))
=> ( groups3539618377306564664at_int(F,minus_minus_set_nat(A2,insert_nat(A,bot_bot_set_nat))) = groups3539618377306564664at_int(F,A2) ) ) ) ) ).
% sum_diff1
tff(fact_8090_sum__diff1,axiom,
! [A2: set_int,A: int,F: fun_int_int] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( ( pp(aa_set_int_bool(member_int(A),A2))
=> ( groups4538972089207619220nt_int(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(F,A2)),aa_int_int(F,A)) ) )
& ( ~ pp(aa_set_int_bool(member_int(A),A2))
=> ( groups4538972089207619220nt_int(F,minus_minus_set_int(A2,insert_int(A,bot_bot_set_int))) = groups4538972089207619220nt_int(F,A2) ) ) ) ) ).
% sum_diff1
tff(fact_8091_sincos__total__pi__half,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
=> ? [T4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T4))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
& ( X = aa_real_real(cos_real,T4) )
& ( Y = aa_real_real(sin_real,T4) ) ) ) ) ) ).
% sincos_total_pi_half
tff(fact_8092_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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
=> ? [T4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T4))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)))
& ( X = aa_real_real(cos_real,T4) )
& ( Y = aa_real_real(sin_real,T4) ) ) ) ).
% sincos_total_2pi_le
tff(fact_8093_sum_Odelta__remove,axiom,
! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_real,C: fun_VEBT_VEBT_real] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S2))
=> ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
=> ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aa_fun910392650362481850T_real(aTP_Lamm_ko(A),B),C),S2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(B,A)),groups2240296850493347238T_real(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))) ) )
& ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
=> ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aa_fun910392650362481850T_real(aTP_Lamm_ko(A),B),C),S2) = groups2240296850493347238T_real(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) ) ) ) ) ).
% sum.delta_remove
tff(fact_8094_sum_Odelta__remove,axiom,
! [S2: set_complex,A: complex,B: fun_complex_real,C: fun_complex_real] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
=> ( ( pp(aa_set_complex_bool(member_complex(A),S2))
=> ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_kp(A),B),C),S2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(B,A)),groups5808333547571424918x_real(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) )
& ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
=> ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_kp(A),B),C),S2) = groups5808333547571424918x_real(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).
% sum.delta_remove
tff(fact_8095_sum_Odelta__remove,axiom,
! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_rat,C: fun_VEBT_VEBT_rat] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S2))
=> ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
=> ( groups136491112297645522BT_rat(aa_fun8105832842879916156BT_rat(aa_fun2693410532875676774BT_rat(aTP_Lamm_kq(A),B),C),S2) = plus_plus_rat(aa_VEBT_VEBT_rat(B,A),groups136491112297645522BT_rat(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))) ) )
& ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
=> ( groups136491112297645522BT_rat(aa_fun8105832842879916156BT_rat(aa_fun2693410532875676774BT_rat(aTP_Lamm_kq(A),B),C),S2) = groups136491112297645522BT_rat(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) ) ) ) ) ).
% sum.delta_remove
tff(fact_8096_sum_Odelta__remove,axiom,
! [S2: set_complex,A: complex,B: fun_complex_rat,C: fun_complex_rat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
=> ( ( pp(aa_set_complex_bool(member_complex(A),S2))
=> ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_kr(A),B),C),S2) = plus_plus_rat(aa_complex_rat(B,A),groups5058264527183730370ex_rat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) )
& ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
=> ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_kr(A),B),C),S2) = groups5058264527183730370ex_rat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).
% sum.delta_remove
tff(fact_8097_sum_Odelta__remove,axiom,
! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_nat,C: fun_VEBT_VEBT_nat] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S2))
=> ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
=> ( groups771621172384141258BT_nat(aa_fun8284231930576791164BT_nat(aa_fun8778301808485728350BT_nat(aTP_Lamm_ks(A),B),C),S2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(B,A)),groups771621172384141258BT_nat(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))) ) )
& ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
=> ( groups771621172384141258BT_nat(aa_fun8284231930576791164BT_nat(aa_fun8778301808485728350BT_nat(aTP_Lamm_ks(A),B),C),S2) = groups771621172384141258BT_nat(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) ) ) ) ) ).
% sum.delta_remove
tff(fact_8098_sum_Odelta__remove,axiom,
! [S2: set_complex,A: complex,B: fun_complex_nat,C: fun_complex_nat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
=> ( ( pp(aa_set_complex_bool(member_complex(A),S2))
=> ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_kt(A),B),C),S2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(B,A)),groups5693394587270226106ex_nat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) )
& ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
=> ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_kt(A),B),C),S2) = groups5693394587270226106ex_nat(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).
% sum.delta_remove
tff(fact_8099_sum_Odelta__remove,axiom,
! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: fun_VEBT_VEBT_int,C: fun_VEBT_VEBT_int] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,S2))
=> ( ( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
=> ( groups769130701875090982BT_int(aa_fun7144222995531728252BT_int(aa_fun7485955581636121530BT_int(aTP_Lamm_ku(A),B),C),S2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_VEBT_VEBT_int(B,A)),groups769130701875090982BT_int(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT)))) ) )
& ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,A),S2))
=> ( groups769130701875090982BT_int(aa_fun7144222995531728252BT_int(aa_fun7485955581636121530BT_int(aTP_Lamm_ku(A),B),C),S2) = groups769130701875090982BT_int(C,minus_5127226145743854075T_VEBT(S2,insert_VEBT_VEBT(A,bot_bo8194388402131092736T_VEBT))) ) ) ) ) ).
% sum.delta_remove
tff(fact_8100_sum_Odelta__remove,axiom,
! [S2: set_complex,A: complex,B: fun_complex_int,C: fun_complex_int] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
=> ( ( pp(aa_set_complex_bool(member_complex(A),S2))
=> ( groups5690904116761175830ex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_kv(A),B),C),S2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_complex_int(B,A)),groups5690904116761175830ex_int(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex)))) ) )
& ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
=> ( groups5690904116761175830ex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_kv(A),B),C),S2) = groups5690904116761175830ex_int(C,minus_811609699411566653omplex(S2,insert_complex(A,bot_bot_set_complex))) ) ) ) ) ).
% sum.delta_remove
tff(fact_8101_sum_Odelta__remove,axiom,
! [S2: set_int,A: int,B: fun_int_real,C: fun_int_real] :
( pp(aa_set_int_bool(finite_finite_int,S2))
=> ( ( pp(aa_set_int_bool(member_int(A),S2))
=> ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_kw(A),B),C),S2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(B,A)),groups8778361861064173332t_real(C,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int)))) ) )
& ( ~ pp(aa_set_int_bool(member_int(A),S2))
=> ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_kw(A),B),C),S2) = groups8778361861064173332t_real(C,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int))) ) ) ) ) ).
% sum.delta_remove
tff(fact_8102_sum_Odelta__remove,axiom,
! [S2: set_int,A: int,B: fun_int_rat,C: fun_int_rat] :
( pp(aa_set_int_bool(finite_finite_int,S2))
=> ( ( pp(aa_set_int_bool(member_int(A),S2))
=> ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_kx(A),B),C),S2) = plus_plus_rat(aa_int_rat(B,A),groups3906332499630173760nt_rat(C,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int)))) ) )
& ( ~ pp(aa_set_int_bool(member_int(A),S2))
=> ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_kx(A),B),C),S2) = groups3906332499630173760nt_rat(C,minus_minus_set_int(S2,insert_int(A,bot_bot_set_int))) ) ) ) ) ).
% sum.delta_remove
tff(fact_8103_square__fact__le__2__fact,axiom,
! [N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,N)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).
% square_fact_le_2_fact
tff(fact_8104_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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
=> ~ ! [T4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T4))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)))
=> ( ( X = aa_real_real(cos_real,T4) )
=> ( Y != aa_real_real(sin_real,T4) ) ) ) ) ) ).
% sincos_total_2pi
tff(fact_8105_sin__pi__divide__n__ge__0,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_nat_real(semiri5074537144036343181t_real,N))))) ) ).
% sin_pi_divide_n_ge_0
tff(fact_8106_member__le__sum,axiom,
! [I: vEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_real] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I),A2))
=> ( ! [X3: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(I,bot_bo8194388402131092736T_VEBT))))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F,X3))) )
=> ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_VEBT_VEBT_real(F,I)),groups2240296850493347238T_real(F,A2))) ) ) ) ).
% member_le_sum
tff(fact_8107_member__le__sum,axiom,
! [I: complex,A2: set_complex,F: fun_complex_real] :
( pp(aa_set_complex_bool(member_complex(I),A2))
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(A2,insert_complex(I,bot_bot_set_complex))))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X3))) )
=> ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(F,I)),groups5808333547571424918x_real(F,A2))) ) ) ) ).
% member_le_sum
tff(fact_8108_member__le__sum,axiom,
! [I: int,A2: set_int,F: fun_int_real] :
( pp(aa_set_int_bool(member_int(I),A2))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(A2,insert_int(I,bot_bot_set_int))))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_int_real(F,X3))) )
=> ( pp(aa_set_int_bool(finite_finite_int,A2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(F,I)),groups8778361861064173332t_real(F,A2))) ) ) ) ).
% member_le_sum
tff(fact_8109_member__le__sum,axiom,
! [I: real,A2: set_real,F: fun_real_real] :
( pp(aa_set_real_bool(member_real(I),A2))
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),minus_minus_set_real(A2,insert_real(I,bot_bot_set_real))))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X3))) )
=> ( pp(aa_set_real_bool(finite_finite_real,A2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,I)),groups8097168146408367636l_real(F,A2))) ) ) ) ).
% member_le_sum
tff(fact_8110_member__le__sum,axiom,
! [I: vEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_rat] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I),A2))
=> ( ! [X3: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(I,bot_bo8194388402131092736T_VEBT))))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(F,X3))) )
=> ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_VEBT_VEBT_rat(F,I)),groups136491112297645522BT_rat(F,A2))) ) ) ) ).
% member_le_sum
tff(fact_8111_member__le__sum,axiom,
! [I: complex,A2: set_complex,F: fun_complex_rat] :
( pp(aa_set_complex_bool(member_complex(I),A2))
=> ( ! [X3: complex] :
( pp(aa_set_complex_bool(member_complex(X3),minus_811609699411566653omplex(A2,insert_complex(I,bot_bot_set_complex))))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X3))) )
=> ( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_complex_rat(F,I)),groups5058264527183730370ex_rat(F,A2))) ) ) ) ).
% member_le_sum
tff(fact_8112_member__le__sum,axiom,
! [I: int,A2: set_int,F: fun_int_rat] :
( pp(aa_set_int_bool(member_int(I),A2))
=> ( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),minus_minus_set_int(A2,insert_int(I,bot_bot_set_int))))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X3))) )
=> ( pp(aa_set_int_bool(finite_finite_int,A2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(F,I)),groups3906332499630173760nt_rat(F,A2))) ) ) ) ).
% member_le_sum
tff(fact_8113_member__le__sum,axiom,
! [I: real,A2: set_real,F: fun_real_rat] :
( pp(aa_set_real_bool(member_real(I),A2))
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),minus_minus_set_real(A2,insert_real(I,bot_bot_set_real))))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X3))) )
=> ( pp(aa_set_real_bool(finite_finite_real,A2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_real_rat(F,I)),groups1300246762558778688al_rat(F,A2))) ) ) ) ).
% member_le_sum
tff(fact_8114_member__le__sum,axiom,
! [I: nat,A2: set_nat,F: fun_nat_rat] :
( pp(aa_set_nat_bool(member_nat(I),A2))
=> ( ! [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),minus_minus_set_nat(A2,insert_nat(I,bot_bot_set_nat))))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X3))) )
=> ( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(F,I)),groups2906978787729119204at_rat(F,A2))) ) ) ) ).
% member_le_sum
tff(fact_8115_member__le__sum,axiom,
! [I: vEBT_VEBT,A2: set_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,I),A2))
=> ( ! [X3: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),minus_5127226145743854075T_VEBT(A2,insert_VEBT_VEBT(I,bot_bo8194388402131092736T_VEBT))))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_VEBT_VEBT_nat(F,X3))) )
=> ( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(F,I)),groups771621172384141258BT_nat(F,A2))) ) ) ) ).
% member_le_sum
tff(fact_8116_fact__num__eq__if,axiom,
! [M: nat] :
( ( ( M = zero_zero_nat )
=> ( aa_nat_int(semiri1406184849735516958ct_int,M) = one_one_int ) )
& ( ( M != zero_zero_nat )
=> ( aa_nat_int(semiri1406184849735516958ct_int,M) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).
% fact_num_eq_if
tff(fact_8117_fact__num__eq__if,axiom,
! [M: nat] :
( ( ( M = zero_zero_nat )
=> ( aa_nat_complex(semiri5044797733671781792omplex,M) = one_one_complex ) )
& ( ( M != zero_zero_nat )
=> ( aa_nat_complex(semiri5044797733671781792omplex,M) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).
% fact_num_eq_if
tff(fact_8118_fact__num__eq__if,axiom,
! [M: nat] :
( ( ( M = zero_zero_nat )
=> ( aa_nat_rat(semiri773545260158071498ct_rat,M) = one_one_rat ) )
& ( ( M != zero_zero_nat )
=> ( aa_nat_rat(semiri773545260158071498ct_rat,M) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).
% fact_num_eq_if
tff(fact_8119_fact__num__eq__if,axiom,
! [M: nat] :
( ( ( M = zero_zero_nat )
=> ( aa_nat_real(semiri2265585572941072030t_real,M) = one_one_real ) )
& ( ( M != zero_zero_nat )
=> ( aa_nat_real(semiri2265585572941072030t_real,M) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).
% fact_num_eq_if
tff(fact_8120_fact__num__eq__if,axiom,
! [M: nat] :
( ( ( M = zero_zero_nat )
=> ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = one_one_nat ) )
& ( ( M != zero_zero_nat )
=> ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) ) ) ) ).
% fact_num_eq_if
tff(fact_8121_fact__reduce,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_int(semiri1406184849735516958ct_int,N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).
% fact_reduce
tff(fact_8122_fact__reduce,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_complex(semiri5044797733671781792omplex,N) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).
% fact_reduce
tff(fact_8123_fact__reduce,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_rat(semiri773545260158071498ct_rat,N) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).
% fact_reduce
tff(fact_8124_fact__reduce,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_real(semiri2265585572941072030t_real,N) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).
% fact_reduce
tff(fact_8125_fact__reduce,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_nat(semiri1408675320244567234ct_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ) ).
% fact_reduce
tff(fact_8126_cos__times__cos,axiom,
! [W: complex,Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,W)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(W),Z))),aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(W),Z)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).
% cos_times_cos
tff(fact_8127_cos__times__cos,axiom,
! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)))),numeral_numeral_real(aa_num_num(bit0,one))) ).
% cos_times_cos
tff(fact_8128_cos__plus__cos,axiom,
! [W: complex,Z: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,W)),aa_complex_complex(cos_complex,Z)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W),Z)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).
% cos_plus_cos
tff(fact_8129_cos__plus__cos,axiom,
! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)),numeral_numeral_real(aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Z)),numeral_numeral_real(aa_num_num(bit0,one))))) ).
% cos_plus_cos
tff(fact_8130_sin__gt__zero2,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,X))) ) ) ).
% sin_gt_zero2
tff(fact_8131_sin__lt__zero,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,pi),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,X)),zero_zero_real)) ) ) ).
% sin_lt_zero
tff(fact_8132_cos__double__less__one,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),numeral_numeral_real(aa_num_num(bit0,one))))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),X))),one_one_real)) ) ) ).
% cos_double_less_one
tff(fact_8133_sin__30,axiom,
aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))) ).
% sin_30
tff(fact_8134_cos__gt__zero,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cos_real,X))) ) ) ).
% cos_gt_zero
tff(fact_8135_sin__monotone__2pi__le,axiom,
! [Y: real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,X))) ) ) ) ).
% sin_monotone_2pi_le
tff(fact_8136_sin__mono__le__eq,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ) ) ).
% sin_mono_le_eq
tff(fact_8137_sin__inj__pi,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
=> ( ( aa_real_real(sin_real,X) = aa_real_real(sin_real,Y) )
=> ( X = Y ) ) ) ) ) ) ).
% sin_inj_pi
tff(fact_8138_cos__60,axiom,
aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit1,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))) ).
% cos_60
tff(fact_8139_cos__one__2pi__int,axiom,
! [X: real] :
( ( aa_real_real(cos_real,X) = one_one_real )
<=> ? [X2: int] : X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,X2)),numeral_numeral_real(aa_num_num(bit0,one)))),pi) ) ).
% cos_one_2pi_int
tff(fact_8140_cos__double__cos,axiom,
! [W: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),W)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_complex) ).
% cos_double_cos
tff(fact_8141_cos__double__cos,axiom,
! [W: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),W)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_real) ).
% cos_double_cos
tff(fact_8142_cos__treble__cos,axiom,
! [X: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit1,one))),X)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))))),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit1,one))),aa_complex_complex(cos_complex,X))) ).
% cos_treble_cos
tff(fact_8143_cos__treble__cos,axiom,
! [X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit1,one))),X)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit1,one))),aa_real_real(cos_real,X))) ).
% cos_treble_cos
tff(fact_8144_Maclaurin__cos__expansion,axiom,
! [X: real,N: nat] :
? [T4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),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_ky,X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ).
% Maclaurin_cos_expansion
tff(fact_8145_sin__le__zero,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,pi),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,X)),zero_zero_real)) ) ) ).
% sin_le_zero
tff(fact_8146_sin__less__zero,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(aa_num_num(bit0,one)))),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,X)),zero_zero_real)) ) ) ).
% sin_less_zero
tff(fact_8147_sin__monotone__2pi,axiom,
! [Y: real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,X))) ) ) ) ).
% sin_monotone_2pi
tff(fact_8148_sin__mono__less__eq,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ) ) ).
% sin_mono_less_eq
tff(fact_8149_sin__total,axiom,
! [Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
=> ? [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X3))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
& ( aa_real_real(sin_real,X3) = Y )
& ! [Y3: real] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y3))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
& ( aa_real_real(sin_real,Y3) = Y ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% sin_total
tff(fact_8150_cos__gt__zero__pi,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cos_real,X))) ) ) ).
% cos_gt_zero_pi
tff(fact_8151_cos__ge__zero,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(cos_real,X))) ) ) ).
% cos_ge_zero
tff(fact_8152_cos__one__2pi,axiom,
! [X: real] :
( ( aa_real_real(cos_real,X) = one_one_real )
<=> ( ? [X2: nat] : X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X2)),numeral_numeral_real(aa_num_num(bit0,one)))),pi)
| ? [X2: nat] : X = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X2)),numeral_numeral_real(aa_num_num(bit0,one)))),pi)) ) ) ).
% cos_one_2pi
tff(fact_8153_Maclaurin__zero,axiom,
! [X: real,N: nat,Diff: fun_na5723442673449079972x_real] :
( ( X = zero_zero_real )
=> ( ( N != zero_zero_nat )
=> ( groups6591440286371151544t_real(aa_fun5746040925223427547t_real(aTP_Lamm_kz(X),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_complex_real(aa_nat480766339200631653x_real(Diff,zero_zero_nat),zero_zero_complex) ) ) ) ).
% Maclaurin_zero
tff(fact_8154_Maclaurin__zero,axiom,
! [X: real,N: nat,Diff: fun_na6724076748692969250l_real] :
( ( X = zero_zero_real )
=> ( ( N != zero_zero_nat )
=> ( groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_la(X),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_real_real(aa_nat_fun_real_real(Diff,zero_zero_nat),zero_zero_real) ) ) ) ).
% Maclaurin_zero
tff(fact_8155_Maclaurin__zero,axiom,
! [X: real,N: nat,Diff: fun_nat_fun_rat_real] :
( ( X = zero_zero_real )
=> ( ( N != zero_zero_nat )
=> ( groups6591440286371151544t_real(aa_fun507281639146497029t_real(aTP_Lamm_lb(X),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_rat_real(aa_nat_fun_rat_real(Diff,zero_zero_nat),zero_zero_rat) ) ) ) ).
% Maclaurin_zero
tff(fact_8156_Maclaurin__zero,axiom,
! [X: real,N: nat,Diff: fun_nat_fun_nat_real] :
( ( X = zero_zero_real )
=> ( ( N != zero_zero_nat )
=> ( groups6591440286371151544t_real(aa_fun7039663905532832253t_real(aTP_Lamm_lc(X),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_real(aa_nat_fun_nat_real(Diff,zero_zero_nat),zero_zero_nat) ) ) ) ).
% Maclaurin_zero
tff(fact_8157_Maclaurin__zero,axiom,
! [X: real,N: nat,Diff: fun_nat_fun_int_real] :
( ( X = zero_zero_real )
=> ( ( N != zero_zero_nat )
=> ( groups6591440286371151544t_real(aa_fun241848180832411225t_real(aTP_Lamm_ld(X),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_int_real(aa_nat_fun_int_real(Diff,zero_zero_nat),zero_zero_int) ) ) ) ).
% Maclaurin_zero
tff(fact_8158_Maclaurin__lemma,axiom,
! [H: real,F: fun_real_real,J: fun_nat_real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H))
=> ? [B8: real] : aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_le(H),J),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B8),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N)),aa_nat_real(semiri2265585572941072030t_real,N)))) ) ).
% Maclaurin_lemma
tff(fact_8159_Maclaurin__minus__cos__expansion,axiom,
! [N: nat,X: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
=> ? [T4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),T4))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T4),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_ky,X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).
% Maclaurin_minus_cos_expansion
tff(fact_8160_Maclaurin__cos__expansion2,axiom,
! [X: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ? [T4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T4))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T4),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_ky,X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).
% Maclaurin_cos_expansion2
tff(fact_8161_sin__pi__divide__n__gt__0,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_nat_real(semiri5074537144036343181t_real,N))))) ) ).
% sin_pi_divide_n_gt_0
tff(fact_8162_Maclaurin__exp__le,axiom,
! [X: real,N: nat] :
? [T4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),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_kn,X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T4)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ).
% Maclaurin_exp_le
tff(fact_8163_sin__zero__iff__int,axiom,
! [X: real] :
( ( aa_real_real(sin_real,X) = zero_zero_real )
<=> ? [I4: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I4))
& ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I4)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ).
% sin_zero_iff_int
tff(fact_8164_cos__zero__iff__int,axiom,
! [X: real] :
( ( aa_real_real(cos_real,X) = zero_zero_real )
<=> ? [I4: int] :
( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I4))
& ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I4)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ).
% cos_zero_iff_int
tff(fact_8165_sin__zero__lemma,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( ( aa_real_real(sin_real,X) = zero_zero_real )
=> ? [N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
& ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ) ).
% sin_zero_lemma
tff(fact_8166_sin__zero__iff,axiom,
! [X: real] :
( ( aa_real_real(sin_real,X) = zero_zero_real )
<=> ( ? [N5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5))
& ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N5)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) )
| ? [N5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5))
& ( X = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N5)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))) ) ) ) ) ).
% sin_zero_iff
tff(fact_8167_cos__coeff__def,axiom,
! [X4: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X4))
=> ( aa_nat_real(cos_coeff,X4) = aa_real_real(aa_rea7340605660300375871l_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(aa_nat_fun_nat_nat(divide_divide_nat,X4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X4)) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X4))
=> ( aa_nat_real(cos_coeff,X4) = zero_zero_real ) ) ) ).
% cos_coeff_def
tff(fact_8168_cos__zero__lemma,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( ( aa_real_real(cos_real,X) = zero_zero_real )
=> ? [N2: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2))
& ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ) ).
% cos_zero_lemma
tff(fact_8169_cos__zero__iff,axiom,
! [X: real] :
( ( aa_real_real(cos_real,X) = zero_zero_real )
<=> ( ? [N5: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5))
& ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N5)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) )
| ? [N5: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5))
& ( X = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N5)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))) ) ) ) ) ).
% cos_zero_iff
tff(fact_8170_Maclaurin__sin__expansion3,axiom,
! [N: nat,X: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ? [T4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T4))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T4),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_lf,X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).
% Maclaurin_sin_expansion3
tff(fact_8171_Maclaurin__sin__expansion4,axiom,
! [X: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ? [T4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T4))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T4),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_lf,X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ).
% Maclaurin_sin_expansion4
tff(fact_8172_Maclaurin__sin__expansion2,axiom,
! [X: real,N: nat] :
? [T4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),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_lf,X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ).
% Maclaurin_sin_expansion2
tff(fact_8173_Maclaurin__sin__expansion,axiom,
! [X: real,N: nat] :
? [T4: 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_lf,X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ).
% Maclaurin_sin_expansion
tff(fact_8174_sin__coeff__def,axiom,
! [X4: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X4))
=> ( aa_nat_real(sin_coeff,X4) = zero_zero_real ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X4))
=> ( aa_nat_real(sin_coeff,X4) = aa_real_real(aa_rea7340605660300375871l_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(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X4),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X4)) ) ) ) ).
% sin_coeff_def
tff(fact_8175_set__encode__insert,axiom,
! [A2: set_nat,N: nat] :
( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ( ~ pp(aa_set_nat_bool(member_nat(N),A2))
=> ( aa_set_nat_nat(nat_set_encode,insert_nat(N,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),aa_set_nat_nat(nat_set_encode,A2)) ) ) ) ).
% set_encode_insert
tff(fact_8176_fact__ge__self,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).
% fact_ge_self
tff(fact_8177_fact__mono__nat,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ).
% fact_mono_nat
tff(fact_8178_fact__less__mono__nat,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ) ).
% fact_less_mono_nat
tff(fact_8179_lessThan__Suc,axiom,
! [K: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K)) = insert_nat(K,aa_nat_set_nat(set_ord_lessThan_nat,K)) ).
% lessThan_Suc
tff(fact_8180_fact__ge__Suc__0__nat,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ).
% fact_ge_Suc_0_nat
tff(fact_8181_dvd__fact,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ) ).
% dvd_fact
tff(fact_8182_atLeast0__atMost__Suc,axiom,
! [N: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N)) = insert_nat(aa_nat_nat(suc,N),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) ).
% atLeast0_atMost_Suc
tff(fact_8183_Icc__eq__insert__lb__nat,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_nat_set_nat(set_or1269000886237332187st_nat(M),N) = insert_nat(M,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N)) ) ) ).
% Icc_eq_insert_lb_nat
tff(fact_8184_atLeastAtMostSuc__conv,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_nat_nat(suc,N)))
=> ( aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N)) = insert_nat(aa_nat_nat(suc,N),aa_nat_set_nat(set_or1269000886237332187st_nat(M),N)) ) ) ).
% atLeastAtMostSuc_conv
tff(fact_8185_atLeastAtMost__insertL,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( insert_nat(M,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),N)) = aa_nat_set_nat(set_or1269000886237332187st_nat(M),N) ) ) ).
% atLeastAtMost_insertL
tff(fact_8186_lessThan__nat__numeral,axiom,
! [K: num] : aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(pred_numeral,K),aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(pred_numeral,K))) ).
% lessThan_nat_numeral
tff(fact_8187_fact__diff__Suc,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M)))
=> ( aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ) ).
% fact_diff_Suc
tff(fact_8188_fact__div__fact__le__pow,axiom,
! [R: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),R)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),R))) ) ).
% fact_div_fact_le_pow
tff(fact_8189_set__decode__plus__power__2,axiom,
! [N: nat,Z: nat] :
( ~ pp(aa_set_nat_bool(member_nat(N),aa_nat_set_nat(nat_set_decode,Z)))
=> ( aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),Z)) = insert_nat(N,aa_nat_set_nat(nat_set_decode,Z)) ) ) ).
% set_decode_plus_power_2
tff(fact_8190_sin__coeff__Suc,axiom,
! [N: nat] : aa_nat_real(sin_coeff,aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(cos_coeff,N)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))) ).
% sin_coeff_Suc
tff(fact_8191_cos__coeff__Suc,axiom,
! [N: nat] : aa_nat_real(cos_coeff,aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_nat_real(sin_coeff,N))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))) ).
% cos_coeff_Suc
tff(fact_8192_tan__double,axiom,
! [X: complex] :
( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
=> ( ( aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)) != zero_zero_complex )
=> ( aa_complex_complex(tan_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(tan_complex,X))),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(tan_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).
% tan_double
tff(fact_8193_tan__double,axiom,
! [X: real] :
( ( aa_real_real(cos_real,X) != zero_zero_real )
=> ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),X)) != zero_zero_real )
=> ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(tan_real,X))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).
% tan_double
tff(fact_8194_complex__unimodular__polar,axiom,
! [Z: complex] :
( ( aa_complex_real(real_V1022390504157884413omplex,Z) = one_one_real )
=> ~ ! [T4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T4))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)))
=> ( Z != aa_real_complex(complex2(aa_real_real(cos_real,T4)),aa_real_real(sin_real,T4)) ) ) ) ) ).
% complex_unimodular_polar
tff(fact_8195_and__int_Opsimps,axiom,
! [K: int,L: int] :
( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K),L)))
=> ( ( ( pp(aa_set_int_bool(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
& pp(aa_set_int_bool(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(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))) ) )
& ( ~ ( pp(aa_set_int_bool(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
& pp(aa_set_int_bool(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(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ).
% and_int.psimps
tff(fact_8196_and__int_Opelims,axiom,
! [X: int,Xa2: int,Y: int] :
( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Xa2) = Y )
=> ( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa2)))
=> ~ ( ( ( ( pp(aa_set_int_bool(member_int(X),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
& pp(aa_set_int_bool(member_int(Xa2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
=> ( Y = aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa2))))) ) )
& ( ~ ( pp(aa_set_int_bool(member_int(X),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
& pp(aa_set_int_bool(member_int(Xa2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
=> ( Y = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fconj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa2))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) )
=> ~ pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa2))) ) ) ) ).
% and_int.pelims
tff(fact_8197_sin__paired,axiom,
! [X: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_lg,X)),aa_real_real(sin_real,X))) ).
% sin_paired
tff(fact_8198_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_8199_tan__npi,axiom,
! [N: nat] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),pi)) = zero_zero_real ).
% tan_npi
tff(fact_8200_tan__periodic__n,axiom,
! [X: real,N: num] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(N)),pi))) = aa_real_real(tan_real,X) ).
% tan_periodic_n
tff(fact_8201_tan__periodic__nat,axiom,
! [X: real,N: nat] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),pi))) = aa_real_real(tan_real,X) ).
% tan_periodic_nat
tff(fact_8202_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(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I)),pi))) = aa_real_real(tan_real,X) ).
% tan_periodic_int
tff(fact_8203_powser__sums__zero__iff,axiom,
! [A: fun_nat_complex,X: complex] :
( pp(aa_complex_bool(sums_complex(aTP_Lamm_ge(A)),X))
<=> ( aa_nat_complex(A,zero_zero_nat) = X ) ) ).
% powser_sums_zero_iff
tff(fact_8204_powser__sums__zero__iff,axiom,
! [A: fun_nat_real,X: real] :
( pp(aa_real_bool(sums_real(aTP_Lamm_gg(A)),X))
<=> ( aa_nat_real(A,zero_zero_nat) = X ) ) ).
% powser_sums_zero_iff
tff(fact_8205_norm__cos__sin,axiom,
! [T: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(complex2(aa_real_real(cos_real,T)),aa_real_real(sin_real,T))) = one_one_real ).
% norm_cos_sin
tff(fact_8206_tan__periodic,axiom,
! [X: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))) = aa_real_real(tan_real,X) ).
% tan_periodic
tff(fact_8207_complex__diff,axiom,
! [A: real,B: real,C: real,D: real] : aa_complex_complex(minus_minus_complex(aa_real_complex(complex2(A),B)),aa_real_complex(complex2(C),D)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),D)) ).
% complex_diff
tff(fact_8208_sums__le,axiom,
! [F: fun_nat_real,G: fun_nat_real,S: real,T: real] :
( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(G,N2)))
=> ( pp(aa_real_bool(sums_real(F),S))
=> ( pp(aa_real_bool(sums_real(G),T))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,S),T)) ) ) ) ).
% sums_le
tff(fact_8209_sums__le,axiom,
! [F: fun_nat_nat,G: fun_nat_nat,S: nat,T: nat] :
( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(G,N2)))
=> ( pp(aa_nat_bool(sums_nat(F),S))
=> ( pp(aa_nat_bool(sums_nat(G),T))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,S),T)) ) ) ) ).
% sums_le
tff(fact_8210_sums__le,axiom,
! [F: fun_nat_int,G: fun_nat_int,S: int,T: int] :
( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(G,N2)))
=> ( pp(aa_int_bool(sums_int(F),S))
=> ( pp(aa_int_bool(sums_int(G),T))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,S),T)) ) ) ) ).
% sums_le
tff(fact_8211_sums__mult2,axiom,
! [F: fun_nat_real,A: real,C: real] :
( pp(aa_real_bool(sums_real(F),A))
=> pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_eq(F),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C))) ) ).
% sums_mult2
tff(fact_8212_sums__mult,axiom,
! [F: fun_nat_real,A: real,C: real] :
( pp(aa_real_bool(sums_real(F),A))
=> pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_kb(F),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A))) ) ).
% sums_mult
tff(fact_8213_sums__add,axiom,
! [F: fun_nat_real,A: real,G: fun_nat_real,B: real] :
( pp(aa_real_bool(sums_real(F),A))
=> ( pp(aa_real_bool(sums_real(G),B))
=> pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_fc(F),G)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B))) ) ) ).
% sums_add
tff(fact_8214_sums__add,axiom,
! [F: fun_nat_nat,A: nat,G: fun_nat_nat,B: nat] :
( pp(aa_nat_bool(sums_nat(F),A))
=> ( pp(aa_nat_bool(sums_nat(G),B))
=> pp(aa_nat_bool(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_fb(F),G)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))) ) ) ).
% sums_add
tff(fact_8215_sums__add,axiom,
! [F: fun_nat_int,A: int,G: fun_nat_int,B: int] :
( pp(aa_int_bool(sums_int(F),A))
=> ( pp(aa_int_bool(sums_int(G),B))
=> pp(aa_int_bool(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_ka(F),G)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))) ) ) ).
% sums_add
tff(fact_8216_sums__diff,axiom,
! [F: fun_nat_real,A: real,G: fun_nat_real,B: real] :
( pp(aa_real_bool(sums_real(F),A))
=> ( pp(aa_real_bool(sums_real(G),B))
=> pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_ff(F),G)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B))) ) ) ).
% sums_diff
tff(fact_8217_sums__divide,axiom,
! [F: fun_nat_complex,A: complex,C: complex] :
( pp(aa_complex_bool(sums_complex(F),A))
=> pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_jz(F),C)),aa_complex_complex(divide1717551699836669952omplex(A),C))) ) ).
% sums_divide
tff(fact_8218_sums__divide,axiom,
! [F: fun_nat_real,A: real,C: real] :
( pp(aa_real_bool(sums_real(F),A))
=> pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_fh(F),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C))) ) ).
% sums_divide
tff(fact_8219_sums__mult2__iff,axiom,
! [C: complex,F: fun_nat_complex,D: complex] :
( ( C != zero_zero_complex )
=> ( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_lh(C),F)),aa_complex_complex(times_times_complex(D),C)))
<=> pp(aa_complex_bool(sums_complex(F),D)) ) ) ).
% sums_mult2_iff
tff(fact_8220_sums__mult2__iff,axiom,
! [C: real,F: fun_nat_real,D: real] :
( ( C != zero_zero_real )
=> ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_li(C),F)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,D),C)))
<=> pp(aa_real_bool(sums_real(F),D)) ) ) ).
% sums_mult2_iff
tff(fact_8221_sums__mult__iff,axiom,
! [C: complex,F: fun_nat_complex,D: complex] :
( ( C != zero_zero_complex )
=> ( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_jy(C),F)),aa_complex_complex(times_times_complex(C),D)))
<=> pp(aa_complex_bool(sums_complex(F),D)) ) ) ).
% sums_mult_iff
tff(fact_8222_sums__mult__iff,axiom,
! [C: real,F: fun_nat_real,D: real] :
( ( C != zero_zero_real )
=> ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_em(C),F)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),D)))
<=> pp(aa_real_bool(sums_real(F),D)) ) ) ).
% sums_mult_iff
tff(fact_8223_Complex__eq__numeral,axiom,
! [A: real,B: real,W: num] :
( ( aa_real_complex(complex2(A),B) = numera6690914467698888265omplex(W) )
<=> ( ( A = numeral_numeral_real(W) )
& ( B = zero_zero_real ) ) ) ).
% Complex_eq_numeral
tff(fact_8224_complex__add,axiom,
! [A: real,B: real,C: real,D: real] : aa_complex_complex(plus_plus_complex(aa_real_complex(complex2(A),B)),aa_real_complex(complex2(C),D)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)) ).
% complex_add
tff(fact_8225_sums__mult__D,axiom,
! [C: complex,F: fun_nat_complex,A: complex] :
( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_jy(C),F)),A))
=> ( ( C != zero_zero_complex )
=> pp(aa_complex_bool(sums_complex(F),aa_complex_complex(divide1717551699836669952omplex(A),C))) ) ) ).
% sums_mult_D
tff(fact_8226_sums__mult__D,axiom,
! [C: real,F: fun_nat_real,A: real] :
( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_em(C),F)),A))
=> ( ( C != zero_zero_real )
=> pp(aa_real_bool(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C))) ) ) ).
% sums_mult_D
tff(fact_8227_sums__Suc__imp,axiom,
! [F: fun_nat_complex,S: complex] :
( ( aa_nat_complex(F,zero_zero_nat) = zero_zero_complex )
=> ( pp(aa_complex_bool(sums_complex(aTP_Lamm_lj(F)),S))
=> pp(aa_complex_bool(sums_complex(F),S)) ) ) ).
% sums_Suc_imp
tff(fact_8228_sums__Suc__imp,axiom,
! [F: fun_nat_real,S: real] :
( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
=> ( pp(aa_real_bool(sums_real(aTP_Lamm_gq(F)),S))
=> pp(aa_real_bool(sums_real(F),S)) ) ) ).
% sums_Suc_imp
tff(fact_8229_sums__Suc,axiom,
! [F: fun_nat_real,L: real] :
( pp(aa_real_bool(sums_real(aTP_Lamm_gq(F)),L))
=> pp(aa_real_bool(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,L),aa_nat_real(F,zero_zero_nat)))) ) ).
% sums_Suc
tff(fact_8230_sums__Suc,axiom,
! [F: fun_nat_nat,L: nat] :
( pp(aa_nat_bool(sums_nat(aTP_Lamm_gp(F)),L))
=> pp(aa_nat_bool(sums_nat(F),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,L),aa_nat_nat(F,zero_zero_nat)))) ) ).
% sums_Suc
tff(fact_8231_sums__Suc,axiom,
! [F: fun_nat_int,L: int] :
( pp(aa_int_bool(sums_int(aTP_Lamm_ha(F)),L))
=> pp(aa_int_bool(sums_int(F),aa_int_int(aa_int_fun_int_int(plus_plus_int,L),aa_nat_int(F,zero_zero_nat)))) ) ).
% sums_Suc
tff(fact_8232_sums__Suc__iff,axiom,
! [F: fun_nat_real,S: real] :
( pp(aa_real_bool(sums_real(aTP_Lamm_gq(F)),S))
<=> pp(aa_real_bool(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S),aa_nat_real(F,zero_zero_nat)))) ) ).
% sums_Suc_iff
tff(fact_8233_sums__zero__iff__shift,axiom,
! [N: nat,F: fun_nat_complex,S: complex] :
( ! [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
=> ( aa_nat_complex(F,I2) = zero_zero_complex ) )
=> ( pp(aa_complex_bool(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_lk(N),F)),S))
<=> pp(aa_complex_bool(sums_complex(F),S)) ) ) ).
% sums_zero_iff_shift
tff(fact_8234_sums__zero__iff__shift,axiom,
! [N: nat,F: fun_nat_real,S: real] :
( ! [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I2),N))
=> ( aa_nat_real(F,I2) = zero_zero_real ) )
=> ( pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_ll(N),F)),S))
<=> pp(aa_real_bool(sums_real(F),S)) ) ) ).
% sums_zero_iff_shift
tff(fact_8235_Complex__eq__neg__numeral,axiom,
! [A: real,B: real,W: num] :
( ( aa_real_complex(complex2(A),B) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) )
<=> ( ( A = aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) )
& ( B = zero_zero_real ) ) ) ).
% Complex_eq_neg_numeral
tff(fact_8236_complex__mult,axiom,
! [A: real,B: real,C: real,D: real] : aa_complex_complex(times_times_complex(aa_real_complex(complex2(A),B)),aa_real_complex(complex2(C),D)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),D))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),D)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),C))) ).
% complex_mult
tff(fact_8237_one__complex_Ocode,axiom,
one_one_complex = aa_real_complex(complex2(one_one_real),zero_zero_real) ).
% one_complex.code
tff(fact_8238_Complex__eq__1,axiom,
! [A: real,B: real] :
( ( aa_real_complex(complex2(A),B) = one_one_complex )
<=> ( ( A = one_one_real )
& ( B = zero_zero_real ) ) ) ).
% Complex_eq_1
tff(fact_8239_powser__sums__if,axiom,
! [M: nat,Z: complex] : pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_lm(M),Z)),aa_nat_complex(power_power_complex(Z),M))) ).
% powser_sums_if
tff(fact_8240_powser__sums__if,axiom,
! [M: nat,Z: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_ln(M),Z)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),M))) ).
% powser_sums_if
tff(fact_8241_powser__sums__if,axiom,
! [M: nat,Z: int] : pp(aa_int_bool(sums_int(aa_int_fun_nat_int(aTP_Lamm_lo(M),Z)),aa_nat_int(power_power_int(Z),M))) ).
% powser_sums_if
tff(fact_8242_powser__sums__zero,axiom,
! [A: fun_nat_complex] : pp(aa_complex_bool(sums_complex(aTP_Lamm_ge(A)),aa_nat_complex(A,zero_zero_nat))) ).
% powser_sums_zero
tff(fact_8243_powser__sums__zero,axiom,
! [A: fun_nat_real] : pp(aa_real_bool(sums_real(aTP_Lamm_gg(A)),aa_nat_real(A,zero_zero_nat))) ).
% powser_sums_zero
tff(fact_8244_tan__def,axiom,
! [X4: complex] : aa_complex_complex(tan_complex,X4) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,X4)),aa_complex_complex(cos_complex,X4)) ).
% tan_def
tff(fact_8245_tan__def,axiom,
! [X4: real] : aa_real_real(tan_real,X4) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,X4)),aa_real_real(cos_real,X4)) ).
% tan_def
tff(fact_8246_sums__iff__shift,axiom,
! [F: fun_nat_real,N: nat,S: real] :
( pp(aa_real_bool(sums_real(aa_nat_fun_nat_real(aTP_Lamm_gs(F),N)),S))
<=> pp(aa_real_bool(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S),groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,N))))) ) ).
% sums_iff_shift
tff(fact_8247_sums__split__initial__segment,axiom,
! [F: fun_nat_real,S: real,N: nat] :
( pp(aa_real_bool(sums_real(F),S))
=> pp(aa_real_bool(sums_real(aa_nat_fun_nat_real(aTP_Lamm_gs(F),N)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,S),groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,N))))) ) ).
% sums_split_initial_segment
tff(fact_8248_sums__iff__shift_H,axiom,
! [F: fun_nat_real,N: nat,S: real] :
( pp(aa_real_bool(sums_real(aa_nat_fun_nat_real(aTP_Lamm_gs(F),N)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,S),groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,N)))))
<=> pp(aa_real_bool(sums_real(F),S)) ) ).
% sums_iff_shift'
tff(fact_8249_sums__If__finite__set_H,axiom,
! [G: fun_nat_real,S2: real,A2: set_nat,S4: real,F: fun_nat_real] :
( pp(aa_real_bool(sums_real(G),S2))
=> ( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ( ( S4 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S2),groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_lp(G),F),A2)) )
=> pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aa_set1316517696749040435t_real(aTP_Lamm_lq(G),A2),F)),S4)) ) ) ) ).
% sums_If_finite_set'
tff(fact_8250_Complex__eq__neg__1,axiom,
! [A: real,B: real] :
( ( aa_real_complex(complex2(A),B) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
<=> ( ( A = aa_real_real(uminus_uminus_real,one_one_real) )
& ( B = zero_zero_real ) ) ) ).
% Complex_eq_neg_1
tff(fact_8251_tan__45,axiom,
aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one))))) = one_one_real ).
% tan_45
tff(fact_8252_geometric__sums,axiom,
! [C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real))
=> pp(aa_real_bool(sums_real(aa_real_fun_nat_real(power_power_real,C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),C)))) ) ).
% geometric_sums
tff(fact_8253_geometric__sums,axiom,
! [C: complex] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real))
=> pp(aa_complex_bool(sums_complex(power_power_complex(C)),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_complex_complex(minus_minus_complex(one_one_complex),C)))) ) ).
% geometric_sums
tff(fact_8254_power__half__series,axiom,
pp(aa_real_bool(sums_real(aTP_Lamm_lr),one_one_real)) ).
% power_half_series
tff(fact_8255_lemma__tan__total,axiom,
! [Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
=> ? [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X3))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(tan_real,X3))) ) ) ).
% lemma_tan_total
tff(fact_8256_tan__gt__zero,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(tan_real,X))) ) ) ).
% tan_gt_zero
tff(fact_8257_lemma__tan__total1,axiom,
! [Y: real] :
? [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X3))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
& ( aa_real_real(tan_real,X3) = Y ) ) ).
% lemma_tan_total1
tff(fact_8258_tan__mono__lt__eq,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ) ) ).
% tan_mono_lt_eq
tff(fact_8259_tan__monotone_H,axiom,
! [Y: real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,X))) ) ) ) ) ) ).
% tan_monotone'
tff(fact_8260_tan__monotone,axiom,
! [Y: real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,X))) ) ) ) ).
% tan_monotone
tff(fact_8261_tan__total,axiom,
! [Y: real] :
? [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X3))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
& ( aa_real_real(tan_real,X3) = Y )
& ! [Y3: real] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y3))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
& ( aa_real_real(tan_real,Y3) = Y ) )
=> ( Y3 = X3 ) ) ) ).
% tan_total
tff(fact_8262_tan__minus__45,axiom,
aa_real_real(tan_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))))) = aa_real_real(uminus_uminus_real,one_one_real) ).
% tan_minus_45
tff(fact_8263_tan__inverse,axiom,
! [Y: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(tan_real,Y)) = aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))),Y)) ).
% tan_inverse
tff(fact_8264_add__tan__eq,axiom,
! [X: complex,Y: complex] :
( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
=> ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
=> ( aa_complex_complex(plus_plus_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(X),Y))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,Y))) ) ) ) ).
% add_tan_eq
tff(fact_8265_add__tan__eq,axiom,
! [X: real,Y: real] :
( ( aa_real_real(cos_real,X) != zero_zero_real )
=> ( ( aa_real_real(cos_real,Y) != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))) ) ) ) ).
% add_tan_eq
tff(fact_8266_sums__if_H,axiom,
! [G: fun_nat_real,X: real] :
( pp(aa_real_bool(sums_real(G),X))
=> pp(aa_real_bool(sums_real(aTP_Lamm_ls(G)),X)) ) ).
% sums_if'
tff(fact_8267_sums__if,axiom,
! [G: fun_nat_real,X: real,F: fun_nat_real,Y: real] :
( pp(aa_real_bool(sums_real(G),X))
=> ( pp(aa_real_bool(sums_real(F),Y))
=> pp(aa_real_bool(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_lt(G),F)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))) ) ) ).
% sums_if
tff(fact_8268_tan__pos__pi2__le,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(tan_real,X))) ) ) ).
% tan_pos_pi2_le
tff(fact_8269_tan__total__pos,axiom,
! [Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
=> ? [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X3))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
& ( aa_real_real(tan_real,X3) = Y ) ) ) ).
% tan_total_pos
tff(fact_8270_tan__less__zero,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(aa_num_num(bit0,one)))),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(tan_real,X)),zero_zero_real)) ) ) ).
% tan_less_zero
tff(fact_8271_tan__mono__le__eq,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ) ) ).
% tan_mono_le_eq
tff(fact_8272_tan__mono__le,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y))) ) ) ) ).
% tan_mono_le
tff(fact_8273_tan__bound__pi2,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one))))))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(tan_real,X))),one_one_real)) ) ).
% tan_bound_pi2
tff(fact_8274_arctan__unique,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
=> ( ( aa_real_real(tan_real,X) = Y )
=> ( aa_real_real(arctan,Y) = X ) ) ) ) ).
% arctan_unique
tff(fact_8275_arctan__tan,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
=> ( aa_real_real(arctan,aa_real_real(tan_real,X)) = X ) ) ) ).
% arctan_tan
tff(fact_8276_arctan,axiom,
! [Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arctan,Y)))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
& ( aa_real_real(tan_real,aa_real_real(arctan,Y)) = Y ) ) ).
% arctan
tff(fact_8277_and__int_Opinduct,axiom,
! [A0: int,A1: int,P: fun_int_fun_int_bool] :
( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A0),A1)))
=> ( ! [K3: int,L3: int] :
( pp(aa_Pro4501881432599481050t_bool(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K3),L3)))
=> ( ( ~ ( pp(aa_set_int_bool(member_int(K3),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))))
& pp(aa_set_int_bool(member_int(L3),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))) )
=> pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,K3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) )
=> pp(aa_int_bool(aa_int_fun_int_bool(P,K3),L3)) ) )
=> pp(aa_int_bool(aa_int_fun_int_bool(P,A0),A1)) ) ) ).
% and_int.pinduct
tff(fact_8278_lemma__tan__add1,axiom,
! [X: complex,Y: complex] :
( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
=> ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
=> ( aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y))) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(X),Y))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,Y))) ) ) ) ).
% lemma_tan_add1
tff(fact_8279_lemma__tan__add1,axiom,
! [X: real,Y: real] :
( ( aa_real_real(cos_real,X) != zero_zero_real )
=> ( ( aa_real_real(cos_real,Y) != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y))) ) ) ) ).
% lemma_tan_add1
tff(fact_8280_tan__diff,axiom,
! [X: complex,Y: complex] :
( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
=> ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
=> ( ( aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(X),Y)) != zero_zero_complex )
=> ( aa_complex_complex(tan_complex,aa_complex_complex(minus_minus_complex(X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y))),aa_complex_complex(plus_plus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y)))) ) ) ) ) ).
% tan_diff
tff(fact_8281_tan__diff,axiom,
! [X: real,Y: real] :
( ( aa_real_real(cos_real,X) != zero_zero_real )
=> ( ( aa_real_real(cos_real,Y) != zero_zero_real )
=> ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)) != zero_zero_real )
=> ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y)))) ) ) ) ) ).
% tan_diff
tff(fact_8282_tan__add,axiom,
! [X: complex,Y: complex] :
( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
=> ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
=> ( ( aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(X),Y)) != zero_zero_complex )
=> ( aa_complex_complex(tan_complex,aa_complex_complex(plus_plus_complex(X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y))),aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y)))) ) ) ) ) ).
% tan_add
tff(fact_8283_tan__add,axiom,
! [X: real,Y: real] :
( ( aa_real_real(cos_real,X) != zero_zero_real )
=> ( ( aa_real_real(cos_real,Y) != zero_zero_real )
=> ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) != zero_zero_real )
=> ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y)))) ) ) ) ) ).
% tan_add
tff(fact_8284_tan__total__pi4,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
=> ? [Z3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))))),Z3))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one))))))
& ( aa_real_real(tan_real,Z3) = X ) ) ) ).
% tan_total_pi4
tff(fact_8285_cos__paired,axiom,
! [X: real] : pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_lu,X)),aa_real_real(cos_real,X))) ).
% cos_paired
tff(fact_8286_tan__half,axiom,
! [X: complex] : aa_complex_complex(tan_complex,X) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X))),aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X))),one_one_complex)) ).
% tan_half
tff(fact_8287_tan__half,axiom,
! [X: real] : aa_real_real(tan_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),X))),one_one_real)) ).
% tan_half
tff(fact_8288_geometric__deriv__sums,axiom,
! [Z: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),one_one_real))
=> pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_lv,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).
% geometric_deriv_sums
tff(fact_8289_geometric__deriv__sums,axiom,
! [Z: complex] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),one_one_real))
=> pp(aa_complex_bool(sums_complex(aTP_Lamm_lw(Z)),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(one_one_complex),Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).
% geometric_deriv_sums
tff(fact_8290_diffs__equiv,axiom,
! [C: fun_nat_real,X: real] :
( summable_real(aa_real_fun_nat_real(aTP_Lamm_lx(C),X))
=> pp(aa_real_bool(sums_real(aa_real_fun_nat_real(aTP_Lamm_ly(C),X)),suminf_real(aa_real_fun_nat_real(aTP_Lamm_lx(C),X)))) ) ).
% diffs_equiv
tff(fact_8291_diffs__equiv,axiom,
! [C: fun_nat_complex,X: complex] :
( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_lz(C),X))
=> pp(aa_complex_bool(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_ma(C),X)),suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_lz(C),X)))) ) ).
% diffs_equiv
tff(fact_8292_ceiling__log__eq__powr__iff,axiom,
! [X: real,B: real,K: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
=> ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,K)),one_one_int) )
<=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_nat_real(semiri5074537144036343181t_real,K))),X))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),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_8293_sin__tan,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
=> ( aa_real_real(sin_real,X) = aa_real_real(aa_rea7340605660300375871l_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)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).
% sin_tan
tff(fact_8294_cos__tan,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
=> ( aa_real_real(cos_real,X) = aa_real_real(aa_rea7340605660300375871l_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)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).
% cos_tan
tff(fact_8295_monoseq__def,axiom,
! [X5: fun_nat_real] :
( topolo6980174941875973593q_real(X5)
<=> ( ! [M7: nat,N5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N5))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X5,M7)),aa_nat_real(X5,N5))) )
| ! [M7: nat,N5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N5))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X5,N5)),aa_nat_real(X5,M7))) ) ) ) ).
% monoseq_def
tff(fact_8296_monoseq__def,axiom,
! [X5: fun_nat_set_nat] :
( topolo7278393974255667507et_nat(X5)
<=> ( ! [M7: nat,N5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N5))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(X5,M7)),aa_nat_set_nat(X5,N5))) )
| ! [M7: nat,N5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N5))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(X5,N5)),aa_nat_set_nat(X5,M7))) ) ) ) ).
% monoseq_def
tff(fact_8297_monoseq__def,axiom,
! [X5: fun_nat_rat] :
( topolo4267028734544971653eq_rat(X5)
<=> ( ! [M7: nat,N5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X5,M7)),aa_nat_rat(X5,N5))) )
| ! [M7: nat,N5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X5,N5)),aa_nat_rat(X5,M7))) ) ) ) ).
% monoseq_def
tff(fact_8298_monoseq__def,axiom,
! [X5: fun_nat_num] :
( topolo1459490580787246023eq_num(X5)
<=> ( ! [M7: nat,N5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N5))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(X5,M7)),aa_nat_num(X5,N5))) )
| ! [M7: nat,N5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N5))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(X5,N5)),aa_nat_num(X5,M7))) ) ) ) ).
% monoseq_def
tff(fact_8299_monoseq__def,axiom,
! [X5: fun_nat_nat] :
( topolo4902158794631467389eq_nat(X5)
<=> ( ! [M7: nat,N5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N5))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X5,M7)),aa_nat_nat(X5,N5))) )
| ! [M7: nat,N5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N5))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X5,N5)),aa_nat_nat(X5,M7))) ) ) ) ).
% monoseq_def
tff(fact_8300_monoseq__def,axiom,
! [X5: fun_nat_int] :
( topolo4899668324122417113eq_int(X5)
<=> ( ! [M7: nat,N5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N5))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X5,M7)),aa_nat_int(X5,N5))) )
| ! [M7: nat,N5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M7),N5))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X5,N5)),aa_nat_int(X5,M7))) ) ) ) ).
% monoseq_def
tff(fact_8301_real__sqrt__eq__iff,axiom,
! [X: real,Y: real] :
( ( aa_real_real(sqrt,X) = aa_real_real(sqrt,Y) )
<=> ( X = Y ) ) ).
% real_sqrt_eq_iff
tff(fact_8302_powr__one__eq__one,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,one_one_real),A) = one_one_real ).
% powr_one_eq_one
tff(fact_8303_real__sqrt__zero,axiom,
aa_real_real(sqrt,zero_zero_real) = zero_zero_real ).
% real_sqrt_zero
tff(fact_8304_real__sqrt__eq__zero__cancel__iff,axiom,
! [X: real] :
( ( aa_real_real(sqrt,X) = zero_zero_real )
<=> ( X = zero_zero_real ) ) ).
% real_sqrt_eq_zero_cancel_iff
tff(fact_8305_real__sqrt__less__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).
% real_sqrt_less_iff
tff(fact_8306_real__sqrt__le__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).
% real_sqrt_le_iff
tff(fact_8307_real__sqrt__eq__1__iff,axiom,
! [X: real] :
( ( aa_real_real(sqrt,X) = one_one_real )
<=> ( X = one_one_real ) ) ).
% real_sqrt_eq_1_iff
tff(fact_8308_real__sqrt__one,axiom,
aa_real_real(sqrt,one_one_real) = one_one_real ).
% real_sqrt_one
tff(fact_8309_powr__zero__eq__one,axiom,
! [X: real] :
( ( ( X = zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),zero_zero_real) = zero_zero_real ) )
& ( ( X != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),zero_zero_real) = one_one_real ) ) ) ).
% powr_zero_eq_one
tff(fact_8310_real__sqrt__gt__0__iff,axiom,
! [Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sqrt,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y)) ) ).
% real_sqrt_gt_0_iff
tff(fact_8311_real__sqrt__lt__0__iff,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),zero_zero_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).
% real_sqrt_lt_0_iff
tff(fact_8312_real__sqrt__le__0__iff,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X)),zero_zero_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).
% real_sqrt_le_0_iff
tff(fact_8313_real__sqrt__ge__0__iff,axiom,
! [Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y)) ) ).
% real_sqrt_ge_0_iff
tff(fact_8314_powr__gt__zero,axiom,
! [X: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)))
<=> ( X != zero_zero_real ) ) ).
% powr_gt_zero
tff(fact_8315_powr__nonneg__iff,axiom,
! [A: real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),X)),zero_zero_real))
<=> ( A = zero_zero_real ) ) ).
% powr_nonneg_iff
tff(fact_8316_real__sqrt__gt__1__iff,axiom,
! [Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(sqrt,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),Y)) ) ).
% real_sqrt_gt_1_iff
tff(fact_8317_real__sqrt__lt__1__iff,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),one_one_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ).
% real_sqrt_lt_1_iff
tff(fact_8318_real__sqrt__le__1__iff,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X)),one_one_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ).
% real_sqrt_le_1_iff
tff(fact_8319_real__sqrt__ge__1__iff,axiom,
! [Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(sqrt,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),Y)) ) ).
% real_sqrt_ge_1_iff
tff(fact_8320_powr__less__cancel__iff,axiom,
! [X: real,A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).
% powr_less_cancel_iff
tff(fact_8321_real__sqrt__abs2,axiom,
! [X: real] : aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)) = aa_real_real(abs_abs_real,X) ).
% real_sqrt_abs2
tff(fact_8322_real__sqrt__mult__self,axiom,
! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sqrt,A)),aa_real_real(sqrt,A)) = aa_real_real(abs_abs_real,A) ).
% real_sqrt_mult_self
tff(fact_8323_real__sqrt__four,axiom,
aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) = numeral_numeral_real(aa_num_num(bit0,one)) ).
% real_sqrt_four
tff(fact_8324_powr__eq__one__iff,axiom,
! [A: real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),A))
=> ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),X) = one_one_real )
<=> ( X = zero_zero_real ) ) ) ).
% powr_eq_one_iff
tff(fact_8325_powr__one__gt__zero__iff,axiom,
! [X: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),one_one_real) = X )
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).
% powr_one_gt_zero_iff
tff(fact_8326_powr__one,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),one_one_real) = X ) ) ).
% powr_one
tff(fact_8327_powr__le__cancel__iff,axiom,
! [X: real,A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B)) ) ) ).
% powr_le_cancel_iff
tff(fact_8328_numeral__powr__numeral__real,axiom,
! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,numeral_numeral_real(M)),numeral_numeral_real(N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(M)),aa_num_nat(numeral_numeral_nat,N)) ).
% numeral_powr_numeral_real
tff(fact_8329_log__powr__cancel,axiom,
! [A: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ( ( A != one_one_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Y)) = Y ) ) ) ).
% log_powr_cancel
tff(fact_8330_powr__log__cancel,axiom,
! [A: real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ( ( A != one_one_real )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),aa_real_real(aa_rea7340605660300375871l_real(log,A),X)) = X ) ) ) ) ).
% powr_log_cancel
tff(fact_8331_real__sqrt__abs,axiom,
! [X: real] : aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,X) ).
% real_sqrt_abs
tff(fact_8332_powr__numeral,axiom,
! [X: real,N: num] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),numeral_numeral_real(N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,N)) ) ) ).
% powr_numeral
tff(fact_8333_real__sqrt__pow2,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = X ) ) ).
% real_sqrt_pow2
tff(fact_8334_real__sqrt__pow2__iff,axiom,
! [X: real] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = X )
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).
% real_sqrt_pow2_iff
tff(fact_8335_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(aa_rea7340605660300375871l_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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).
% real_sqrt_sum_squares_mult_squared_eq
tff(fact_8336_square__powr__half,axiom,
! [X: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,X) ).
% square_powr_half
tff(fact_8337_real__sqrt__power,axiom,
! [X: real,K: nat] : aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,X)),K) ).
% real_sqrt_power
tff(fact_8338_real__sqrt__mult,axiom,
! [X: real,Y: real] : aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y)) ).
% real_sqrt_mult
tff(fact_8339_real__sqrt__divide,axiom,
! [X: real,Y: real] : aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y)) ).
% real_sqrt_divide
tff(fact_8340_powr__powr,axiom,
! [X: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),B) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)) ).
% powr_powr
tff(fact_8341_real__sqrt__le__mono,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y))) ) ).
% real_sqrt_le_mono
tff(fact_8342_real__sqrt__less__mono,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y))) ) ).
% real_sqrt_less_mono
tff(fact_8343_real__sqrt__minus,axiom,
! [X: real] : aa_real_real(sqrt,aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(sqrt,X)) ).
% real_sqrt_minus
tff(fact_8344_powr__non__neg,axiom,
! [A: real,X: real] : ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),X)),zero_zero_real)) ).
% powr_non_neg
tff(fact_8345_powr__less__mono2__neg,axiom,
! [A: real,X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A))) ) ) ) ).
% powr_less_mono2_neg
tff(fact_8346_powr__ge__pzero,axiom,
! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),Y))) ).
% powr_ge_pzero
tff(fact_8347_powr__mono2,axiom,
! [A: real,X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A))) ) ) ) ).
% powr_mono2
tff(fact_8348_real__sqrt__gt__zero,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sqrt,X))) ) ).
% real_sqrt_gt_zero
tff(fact_8349_real__sqrt__ge__zero,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,X))) ) ).
% real_sqrt_ge_zero
tff(fact_8350_real__sqrt__eq__zero__cancel,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_8351_powr__less__mono,axiom,
! [A: real,B: real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B))) ) ) ).
% powr_less_mono
tff(fact_8352_powr__less__cancel,axiom,
! [X: real,A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B)))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B)) ) ) ).
% powr_less_cancel
tff(fact_8353_powr__mono,axiom,
! [A: real,B: real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B))) ) ) ).
% powr_mono
tff(fact_8354_real__sqrt__ge__one,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(sqrt,X))) ) ).
% real_sqrt_ge_one
tff(fact_8355_powr__less__mono2,axiom,
! [A: real,X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A))) ) ) ) ).
% powr_less_mono2
tff(fact_8356_powr__mono2_H,axiom,
! [A: real,X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A))) ) ) ) ).
% powr_mono2'
tff(fact_8357_gr__one__powr,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),Y))) ) ) ).
% gr_one_powr
tff(fact_8358_powr__inj,axiom,
! [A: real,X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ( ( A != one_one_real )
=> ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),X) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Y) )
<=> ( X = Y ) ) ) ) ).
% powr_inj
tff(fact_8359_ge__one__powr__ge__zero,axiom,
! [X: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A))) ) ) ).
% ge_one_powr_ge_zero
tff(fact_8360_powr__mono__both,axiom,
! [A: real,B: real,X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),B))) ) ) ) ) ).
% powr_mono_both
tff(fact_8361_powr__le1,axiom,
! [A: real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),one_one_real)) ) ) ) ).
% powr_le1
tff(fact_8362_powr__divide,axiom,
! [X: real,Y: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
=> ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ) ).
% powr_divide
tff(fact_8363_powr__mult,axiom,
! [X: real,Y: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
=> ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ) ).
% powr_mult
tff(fact_8364_real__div__sqrt,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_real_real(sqrt,X)) = aa_real_real(sqrt,X) ) ) ).
% real_div_sqrt
tff(fact_8365_sqrt__add__le__add__sqrt,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_8366_le__real__sqrt__sumsq,axiom,
! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))))) ).
% le_real_sqrt_sumsq
tff(fact_8367_divide__powr__uminus,axiom,
! [A: real,B: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),C)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_real_real(uminus_uminus_real,C))) ).
% divide_powr_uminus
tff(fact_8368_log__base__powr,axiom,
! [A: real,B: real,X: real] :
( ( A != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),B)),X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)),B) ) ) ).
% log_base_powr
tff(fact_8369_log__powr,axiom,
! [X: real,B: real,Y: real] :
( ( X != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) ) ) ).
% log_powr
tff(fact_8370_ln__powr,axiom,
! [X: real,Y: real] :
( ( X != zero_zero_real )
=> ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_real_real(ln_ln_real,X)) ) ) ).
% ln_powr
tff(fact_8371_powr__add,axiom,
! [X: real,A: real,B: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),B)) ).
% powr_add
tff(fact_8372_powr__diff,axiom,
! [W: real,Z1: real,Z22: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,W),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z1),Z22)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,W),Z1)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,W),Z22)) ).
% powr_diff
tff(fact_8373_powr__half__sqrt,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))) = aa_real_real(sqrt,X) ) ) ).
% powr_half_sqrt
tff(fact_8374_sqrt2__less__2,axiom,
pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one)))) ).
% sqrt2_less_2
tff(fact_8375_powr__realpow,axiom,
! [X: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_nat_real(semiri5074537144036343181t_real,N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N) ) ) ).
% powr_realpow
tff(fact_8376_less__log__iff,axiom,
! [B: real,X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X)) ) ) ) ).
% less_log_iff
tff(fact_8377_log__less__iff,axiom,
! [B: real,X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))) ) ) ) ).
% log_less_iff
tff(fact_8378_less__powr__iff,axiom,
! [B: real,X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y)) ) ) ) ).
% less_powr_iff
tff(fact_8379_powr__less__iff,axiom,
! [B: real,X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X))) ) ) ) ).
% powr_less_iff
tff(fact_8380_diffs__def,axiom,
! [C: fun_nat_int,X4: nat] : aa_nat_int(diffs_int(C),X4) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,X4))),aa_nat_int(C,aa_nat_nat(suc,X4))) ).
% diffs_def
tff(fact_8381_diffs__def,axiom,
! [C: fun_nat_real,X4: nat] : aa_nat_real(diffs_real(C),X4) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,X4))),aa_nat_real(C,aa_nat_nat(suc,X4))) ).
% diffs_def
tff(fact_8382_diffs__def,axiom,
! [C: fun_nat_complex,X4: nat] : aa_nat_complex(diffs_complex(C),X4) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,X4))),aa_nat_complex(C,aa_nat_nat(suc,X4))) ).
% diffs_def
tff(fact_8383_diffs__def,axiom,
! [C: fun_nat_rat,X4: nat] : aa_nat_rat(diffs_rat(C),X4) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,X4))),aa_nat_rat(C,aa_nat_nat(suc,X4))) ).
% diffs_def
tff(fact_8384_powr__minus__divide,axiom,
! [X: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(uminus_uminus_real,A)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)) ).
% powr_minus_divide
tff(fact_8385_real__less__rsqrt,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(sqrt,Y))) ) ).
% real_less_rsqrt
tff(fact_8386_sqrt__le__D,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X)),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).
% sqrt_le_D
tff(fact_8387_real__le__rsqrt,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(sqrt,Y))) ) ).
% real_le_rsqrt
tff(fact_8388_powr__neg__one,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),X) ) ) ).
% powr_neg_one
tff(fact_8389_termdiff__converges__all,axiom,
! [C: fun_nat_complex,X: complex] :
( ! [X3: complex] : summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ke(C),X3))
=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_lz(C),X)) ) ).
% termdiff_converges_all
tff(fact_8390_termdiff__converges__all,axiom,
! [C: fun_nat_real,X: real] :
( ! [X3: real] : summable_real(aa_real_fun_nat_real(aTP_Lamm_kc(C),X3))
=> summable_real(aa_real_fun_nat_real(aTP_Lamm_lx(C),X)) ) ).
% termdiff_converges_all
tff(fact_8391_powr__mult__base,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Y)) ) ) ).
% powr_mult_base
tff(fact_8392_tan__60,axiom,
aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit1,one)))) = aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one))) ).
% tan_60
tff(fact_8393_powr__le__iff,axiom,
! [B: real,X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X))) ) ) ) ).
% powr_le_iff
tff(fact_8394_le__powr__iff,axiom,
! [B: real,X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y)) ) ) ) ).
% le_powr_iff
tff(fact_8395_log__le__iff,axiom,
! [B: real,X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))) ) ) ) ).
% log_le_iff
tff(fact_8396_le__log__iff,axiom,
! [B: real,X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X)) ) ) ) ).
% le_log_iff
tff(fact_8397_real__le__lsqrt,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,X)),Y)) ) ) ) ).
% real_le_lsqrt
tff(fact_8398_real__sqrt__unique,axiom,
! [Y: real,X: real] :
( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = X )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
=> ( aa_real_real(sqrt,X) = Y ) ) ) ).
% real_sqrt_unique
tff(fact_8399_lemma__real__divide__sqrt__less,axiom,
! [U: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),U))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one))))),U)) ) ).
% lemma_real_divide_sqrt_less
tff(fact_8400_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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = X )
=> ( Y = zero_zero_real ) ) ).
% real_sqrt_sum_squares_eq_cancel
tff(fact_8401_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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = Y )
=> ( X = zero_zero_real ) ) ).
% real_sqrt_sum_squares_eq_cancel2
tff(fact_8402_real__sqrt__sum__squares__ge1,axiom,
! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).
% real_sqrt_sum_squares_ge1
tff(fact_8403_real__sqrt__sum__squares__ge2,axiom,
! [Y: real,X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).
% real_sqrt_sum_squares_ge2
tff(fact_8404_real__sqrt__sum__squares__triangle__ineq,axiom,
! [A: real,C: real,B: real,D: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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,A),C)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B),D)),aa_num_nat(numeral_numeral_nat,aa_num_num(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,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ).
% real_sqrt_sum_squares_triangle_ineq
tff(fact_8405_sqrt__ge__absD,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(sqrt,Y)))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y)) ) ).
% sqrt_ge_absD
tff(fact_8406_ln__powr__bound,axiom,
! [X: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(ln_ln_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A)),A))) ) ) ).
% ln_powr_bound
tff(fact_8407_ln__powr__bound2,axiom,
! [X: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(ln_ln_real,X)),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),A)),X))) ) ) ).
% ln_powr_bound2
tff(fact_8408_cos__45,axiom,
aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).
% cos_45
tff(fact_8409_sin__45,axiom,
aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).
% sin_45
tff(fact_8410_add__log__eq__powr,axiom,
! [B: real,X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
=> ( ( B != one_one_real )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X)) ) ) ) ) ).
% add_log_eq_powr
tff(fact_8411_log__add__eq__powr,axiom,
! [B: real,X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
=> ( ( B != one_one_real )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y))) ) ) ) ) ).
% log_add_eq_powr
tff(fact_8412_minus__log__eq__powr,axiom,
! [B: real,X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
=> ( ( B != one_one_real )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),Y)),X)) ) ) ) ) ).
% minus_log_eq_powr
tff(fact_8413_powr__def,axiom,
! [X: real,A: real] :
( ( ( X = zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A) = zero_zero_real ) )
& ( ( X != zero_zero_real )
=> ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),A) = aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(ln_ln_real,X))) ) ) ) ).
% powr_def
tff(fact_8414_real__less__lsqrt,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sqrt,X)),Y)) ) ) ) ).
% real_less_lsqrt
tff(fact_8415_sqrt__sum__squares__le__sum,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))) ) ) ).
% sqrt_sum_squares_le_sum
tff(fact_8416_tan__30,axiom,
aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one)))) ).
% tan_30
tff(fact_8417_sqrt__even__pow2,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).
% sqrt_even_pow2
tff(fact_8418_sqrt__sum__squares__le__sum__abs,axiom,
! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(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_8419_real__sqrt__ge__abs2,axiom,
! [Y: real,X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).
% real_sqrt_ge_abs2
tff(fact_8420_real__sqrt__ge__abs1,axiom,
! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).
% real_sqrt_ge_abs1
tff(fact_8421_ln__sqrt,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( aa_real_real(ln_ln_real,aa_real_real(sqrt,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,X)),numeral_numeral_real(aa_num_num(bit0,one))) ) ) ).
% ln_sqrt
tff(fact_8422_cos__30,axiom,
aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).
% cos_30
tff(fact_8423_sin__60,axiom,
aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit1,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).
% sin_60
tff(fact_8424_log__minus__eq__powr,axiom,
! [B: real,X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
=> ( ( B != one_one_real )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)),Y) = aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_real_real(uminus_uminus_real,Y)))) ) ) ) ) ).
% log_minus_eq_powr
tff(fact_8425_complex__norm,axiom,
! [X: real,Y: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).
% complex_norm
tff(fact_8426_termdiff__converges,axiom,
! [X: real,K5: real,C: fun_nat_real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),K5))
=> ( ! [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X3)),K5))
=> summable_real(aa_real_fun_nat_real(aTP_Lamm_kc(C),X3)) )
=> summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_mb(X),C)) ) ) ).
% termdiff_converges
tff(fact_8427_termdiff__converges,axiom,
! [X: complex,K5: real,C: fun_nat_complex] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),K5))
=> ( ! [X3: complex] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X3)),K5))
=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ke(C),X3)) )
=> summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_mc(X),C)) ) ) ).
% termdiff_converges
tff(fact_8428_arsinh__real__aux,axiom,
! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))))) ).
% arsinh_real_aux
tff(fact_8429_real__sqrt__power__even,axiom,
! [N: nat,X: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,X)),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).
% real_sqrt_power_even
tff(fact_8430_real__sqrt__sum__squares__mult__ge__zero,axiom,
! [X: real,Y: real,Xa2: real,Ya: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ).
% real_sqrt_sum_squares_mult_ge_zero
tff(fact_8431_arith__geo__mean__sqrt,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)),numeral_numeral_real(aa_num_num(bit0,one))))) ) ) ).
% arith_geo_mean_sqrt
tff(fact_8432_powr__neg__numeral,axiom,
! [X: real,N: num] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,N))) ) ) ).
% powr_neg_numeral
tff(fact_8433_cos__x__y__le__one,axiom,
! [X: real,Y: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),one_one_real)) ).
% cos_x_y_le_one
tff(fact_8434_real__sqrt__sum__squares__less,axiom,
! [X: real,U: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one))))))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one))))))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U)) ) ) ).
% real_sqrt_sum_squares_less
tff(fact_8435_arcosh__real__def,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ) ) ).
% arcosh_real_def
tff(fact_8436_cos__arctan,axiom,
! [X: real] : aa_real_real(cos_real,aa_real_real(arctan,X)) = aa_real_real(aa_rea7340605660300375871l_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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).
% cos_arctan
tff(fact_8437_sin__arctan,axiom,
! [X: real] : aa_real_real(sin_real,aa_real_real(arctan,X)) = aa_real_real(aa_rea7340605660300375871l_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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).
% sin_arctan
tff(fact_8438_sqrt__sum__squares__half__less,axiom,
! [X: real,U: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),numeral_numeral_real(aa_num_num(bit0,one)))))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),numeral_numeral_real(aa_num_num(bit0,one)))))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U)) ) ) ) ) ).
% sqrt_sum_squares_half_less
tff(fact_8439_sin__cos__sqrt,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).
% sin_cos_sqrt
tff(fact_8440_arctan__half,axiom,
! [X: real] : aa_real_real(arctan,X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ).
% arctan_half
tff(fact_8441_mono__SucI1,axiom,
! [X5: fun_nat_real] :
( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X5,N2)),aa_nat_real(X5,aa_nat_nat(suc,N2))))
=> topolo6980174941875973593q_real(X5) ) ).
% mono_SucI1
tff(fact_8442_mono__SucI1,axiom,
! [X5: fun_nat_set_nat] :
( ! [N2: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(X5,N2)),aa_nat_set_nat(X5,aa_nat_nat(suc,N2))))
=> topolo7278393974255667507et_nat(X5) ) ).
% mono_SucI1
tff(fact_8443_mono__SucI1,axiom,
! [X5: fun_nat_rat] :
( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X5,N2)),aa_nat_rat(X5,aa_nat_nat(suc,N2))))
=> topolo4267028734544971653eq_rat(X5) ) ).
% mono_SucI1
tff(fact_8444_mono__SucI1,axiom,
! [X5: fun_nat_num] :
( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(X5,N2)),aa_nat_num(X5,aa_nat_nat(suc,N2))))
=> topolo1459490580787246023eq_num(X5) ) ).
% mono_SucI1
tff(fact_8445_mono__SucI1,axiom,
! [X5: fun_nat_nat] :
( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X5,N2)),aa_nat_nat(X5,aa_nat_nat(suc,N2))))
=> topolo4902158794631467389eq_nat(X5) ) ).
% mono_SucI1
tff(fact_8446_mono__SucI1,axiom,
! [X5: fun_nat_int] :
( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X5,N2)),aa_nat_int(X5,aa_nat_nat(suc,N2))))
=> topolo4899668324122417113eq_int(X5) ) ).
% mono_SucI1
tff(fact_8447_mono__SucI2,axiom,
! [X5: fun_nat_real] :
( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X5,aa_nat_nat(suc,N2))),aa_nat_real(X5,N2)))
=> topolo6980174941875973593q_real(X5) ) ).
% mono_SucI2
tff(fact_8448_mono__SucI2,axiom,
! [X5: fun_nat_set_nat] :
( ! [N2: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(X5,aa_nat_nat(suc,N2))),aa_nat_set_nat(X5,N2)))
=> topolo7278393974255667507et_nat(X5) ) ).
% mono_SucI2
tff(fact_8449_mono__SucI2,axiom,
! [X5: fun_nat_rat] :
( ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X5,aa_nat_nat(suc,N2))),aa_nat_rat(X5,N2)))
=> topolo4267028734544971653eq_rat(X5) ) ).
% mono_SucI2
tff(fact_8450_mono__SucI2,axiom,
! [X5: fun_nat_num] :
( ! [N2: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(X5,aa_nat_nat(suc,N2))),aa_nat_num(X5,N2)))
=> topolo1459490580787246023eq_num(X5) ) ).
% mono_SucI2
tff(fact_8451_mono__SucI2,axiom,
! [X5: fun_nat_nat] :
( ! [N2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X5,aa_nat_nat(suc,N2))),aa_nat_nat(X5,N2)))
=> topolo4902158794631467389eq_nat(X5) ) ).
% mono_SucI2
tff(fact_8452_mono__SucI2,axiom,
! [X5: fun_nat_int] :
( ! [N2: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X5,aa_nat_nat(suc,N2))),aa_nat_int(X5,N2)))
=> topolo4899668324122417113eq_int(X5) ) ).
% mono_SucI2
tff(fact_8453_monoseq__Suc,axiom,
! [X5: fun_nat_real] :
( topolo6980174941875973593q_real(X5)
<=> ( ! [N5: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X5,N5)),aa_nat_real(X5,aa_nat_nat(suc,N5))))
| ! [N5: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X5,aa_nat_nat(suc,N5))),aa_nat_real(X5,N5))) ) ) ).
% monoseq_Suc
tff(fact_8454_monoseq__Suc,axiom,
! [X5: fun_nat_set_nat] :
( topolo7278393974255667507et_nat(X5)
<=> ( ! [N5: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(X5,N5)),aa_nat_set_nat(X5,aa_nat_nat(suc,N5))))
| ! [N5: nat] : pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(X5,aa_nat_nat(suc,N5))),aa_nat_set_nat(X5,N5))) ) ) ).
% monoseq_Suc
tff(fact_8455_monoseq__Suc,axiom,
! [X5: fun_nat_rat] :
( topolo4267028734544971653eq_rat(X5)
<=> ( ! [N5: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X5,N5)),aa_nat_rat(X5,aa_nat_nat(suc,N5))))
| ! [N5: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X5,aa_nat_nat(suc,N5))),aa_nat_rat(X5,N5))) ) ) ).
% monoseq_Suc
tff(fact_8456_monoseq__Suc,axiom,
! [X5: fun_nat_num] :
( topolo1459490580787246023eq_num(X5)
<=> ( ! [N5: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(X5,N5)),aa_nat_num(X5,aa_nat_nat(suc,N5))))
| ! [N5: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(X5,aa_nat_nat(suc,N5))),aa_nat_num(X5,N5))) ) ) ).
% monoseq_Suc
tff(fact_8457_monoseq__Suc,axiom,
! [X5: fun_nat_nat] :
( topolo4902158794631467389eq_nat(X5)
<=> ( ! [N5: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X5,N5)),aa_nat_nat(X5,aa_nat_nat(suc,N5))))
| ! [N5: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X5,aa_nat_nat(suc,N5))),aa_nat_nat(X5,N5))) ) ) ).
% monoseq_Suc
tff(fact_8458_monoseq__Suc,axiom,
! [X5: fun_nat_int] :
( topolo4899668324122417113eq_int(X5)
<=> ( ! [N5: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X5,N5)),aa_nat_int(X5,aa_nat_nat(suc,N5))))
| ! [N5: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X5,aa_nat_nat(suc,N5))),aa_nat_int(X5,N5))) ) ) ).
% monoseq_Suc
tff(fact_8459_monoI1,axiom,
! [X5: fun_nat_real] :
( ! [M3: nat,N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X5,M3)),aa_nat_real(X5,N2))) )
=> topolo6980174941875973593q_real(X5) ) ).
% monoI1
tff(fact_8460_monoI1,axiom,
! [X5: fun_nat_set_nat] :
( ! [M3: nat,N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(X5,M3)),aa_nat_set_nat(X5,N2))) )
=> topolo7278393974255667507et_nat(X5) ) ).
% monoI1
tff(fact_8461_monoI1,axiom,
! [X5: fun_nat_rat] :
( ! [M3: nat,N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X5,M3)),aa_nat_rat(X5,N2))) )
=> topolo4267028734544971653eq_rat(X5) ) ).
% monoI1
tff(fact_8462_monoI1,axiom,
! [X5: fun_nat_num] :
( ! [M3: nat,N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(X5,M3)),aa_nat_num(X5,N2))) )
=> topolo1459490580787246023eq_num(X5) ) ).
% monoI1
tff(fact_8463_monoI1,axiom,
! [X5: fun_nat_nat] :
( ! [M3: nat,N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X5,M3)),aa_nat_nat(X5,N2))) )
=> topolo4902158794631467389eq_nat(X5) ) ).
% monoI1
tff(fact_8464_monoI1,axiom,
! [X5: fun_nat_int] :
( ! [M3: nat,N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X5,M3)),aa_nat_int(X5,N2))) )
=> topolo4899668324122417113eq_int(X5) ) ).
% monoI1
tff(fact_8465_monoI2,axiom,
! [X5: fun_nat_real] :
( ! [M3: nat,N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X5,N2)),aa_nat_real(X5,M3))) )
=> topolo6980174941875973593q_real(X5) ) ).
% monoI2
tff(fact_8466_monoI2,axiom,
! [X5: fun_nat_set_nat] :
( ! [M3: nat,N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
=> pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(X5,N2)),aa_nat_set_nat(X5,M3))) )
=> topolo7278393974255667507et_nat(X5) ) ).
% monoI2
tff(fact_8467_monoI2,axiom,
! [X5: fun_nat_rat] :
( ! [M3: nat,N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X5,N2)),aa_nat_rat(X5,M3))) )
=> topolo4267028734544971653eq_rat(X5) ) ).
% monoI2
tff(fact_8468_monoI2,axiom,
! [X5: fun_nat_num] :
( ! [M3: nat,N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(X5,N2)),aa_nat_num(X5,M3))) )
=> topolo1459490580787246023eq_num(X5) ) ).
% monoI2
tff(fact_8469_monoI2,axiom,
! [X5: fun_nat_nat] :
( ! [M3: nat,N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(X5,N2)),aa_nat_nat(X5,M3))) )
=> topolo4902158794631467389eq_nat(X5) ) ).
% monoI2
tff(fact_8470_monoI2,axiom,
! [X5: fun_nat_int] :
( ! [M3: nat,N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M3),N2))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(X5,N2)),aa_nat_int(X5,M3))) )
=> topolo4899668324122417113eq_int(X5) ) ).
% monoI2
tff(fact_8471_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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).
% arsinh_real_def
tff(fact_8472_arcosh__def,axiom,
! [X: real] : 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(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))))) ).
% arcosh_def
tff(fact_8473_cos__arcsin,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).
% cos_arcsin
tff(fact_8474_sin__arccos__abs,axiom,
! [Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).
% sin_arccos_abs
tff(fact_8475_sin__arccos,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).
% sin_arccos
tff(fact_8476_of__real__eq__1__iff,axiom,
! [X: real] :
( ( aa_real_real(real_V1803761363581548252l_real,X) = one_one_real )
<=> ( X = one_one_real ) ) ).
% of_real_eq_1_iff
tff(fact_8477_of__real__eq__1__iff,axiom,
! [X: real] :
( ( aa_real_complex(real_V4546457046886955230omplex,X) = one_one_complex )
<=> ( X = one_one_real ) ) ).
% of_real_eq_1_iff
tff(fact_8478_of__real__1,axiom,
aa_real_real(real_V1803761363581548252l_real,one_one_real) = one_one_real ).
% of_real_1
tff(fact_8479_of__real__1,axiom,
aa_real_complex(real_V4546457046886955230omplex,one_one_real) = one_one_complex ).
% of_real_1
tff(fact_8480_of__real__numeral,axiom,
! [W: num] : aa_real_real(real_V1803761363581548252l_real,numeral_numeral_real(W)) = numeral_numeral_real(W) ).
% of_real_numeral
tff(fact_8481_of__real__numeral,axiom,
! [W: num] : aa_real_complex(real_V4546457046886955230omplex,numeral_numeral_real(W)) = numera6690914467698888265omplex(W) ).
% of_real_numeral
tff(fact_8482_of__real__mult,axiom,
! [X: real,Y: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V1803761363581548252l_real,X)),aa_real_real(real_V1803761363581548252l_real,Y)) ).
% of_real_mult
tff(fact_8483_of__real__mult,axiom,
! [X: real,Y: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,X)),aa_real_complex(real_V4546457046886955230omplex,Y)) ).
% of_real_mult
tff(fact_8484_of__real__divide,axiom,
! [X: real,Y: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,X)),aa_real_real(real_V1803761363581548252l_real,Y)) ).
% of_real_divide
tff(fact_8485_of__real__divide,axiom,
! [X: real,Y: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,X)),aa_real_complex(real_V4546457046886955230omplex,Y)) ).
% of_real_divide
tff(fact_8486_of__real__add,axiom,
! [X: real,Y: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,X)),aa_real_real(real_V1803761363581548252l_real,Y)) ).
% of_real_add
tff(fact_8487_of__real__add,axiom,
! [X: real,Y: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)) = aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,X)),aa_real_complex(real_V4546457046886955230omplex,Y)) ).
% of_real_add
tff(fact_8488_of__real__power,axiom,
! [X: real,N: nat] : aa_real_real(real_V1803761363581548252l_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V1803761363581548252l_real,X)),N) ).
% of_real_power
tff(fact_8489_of__real__power,axiom,
! [X: real,N: nat] : aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = aa_nat_complex(power_power_complex(aa_real_complex(real_V4546457046886955230omplex,X)),N) ).
% of_real_power
tff(fact_8490_of__real__diff,axiom,
! [X: real,Y: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)) = aa_complex_complex(minus_minus_complex(aa_real_complex(real_V4546457046886955230omplex,X)),aa_real_complex(real_V4546457046886955230omplex,Y)) ).
% of_real_diff
tff(fact_8491_arccos__1,axiom,
aa_real_real(arccos,one_one_real) = zero_zero_real ).
% arccos_1
tff(fact_8492_arccos__minus__1,axiom,
aa_real_real(arccos,aa_real_real(uminus_uminus_real,one_one_real)) = pi ).
% arccos_minus_1
tff(fact_8493_of__real__neg__numeral,axiom,
! [W: num] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) ).
% of_real_neg_numeral
tff(fact_8494_of__real__neg__numeral,axiom,
! [W: num] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) ).
% of_real_neg_numeral
tff(fact_8495_cos__of__real__pi,axiom,
aa_real_real(cos_real,aa_real_real(real_V1803761363581548252l_real,pi)) = aa_real_real(uminus_uminus_real,one_one_real) ).
% cos_of_real_pi
tff(fact_8496_cos__of__real__pi,axiom,
aa_complex_complex(cos_complex,aa_real_complex(real_V4546457046886955230omplex,pi)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).
% cos_of_real_pi
tff(fact_8497_cos__arccos,axiom,
! [Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
=> ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ) ).
% cos_arccos
tff(fact_8498_sin__arcsin,axiom,
! [Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
=> ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ).
% sin_arcsin
tff(fact_8499_norm__of__real__add1,axiom,
! [X: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,X)),one_one_real)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)) ).
% norm_of_real_add1
tff(fact_8500_norm__of__real__add1,axiom,
! [X: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,X)),one_one_complex)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)) ).
% norm_of_real_add1
tff(fact_8501_norm__of__real__addn,axiom,
! [X: real,B: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,X)),numeral_numeral_real(B))) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),numeral_numeral_real(B))) ).
% norm_of_real_addn
tff(fact_8502_norm__of__real__addn,axiom,
! [X: real,B: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,X)),numera6690914467698888265omplex(B))) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),numeral_numeral_real(B))) ).
% norm_of_real_addn
tff(fact_8503_arccos__0,axiom,
aa_real_real(arccos,zero_zero_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))) ).
% arccos_0
tff(fact_8504_arcsin__1,axiom,
aa_real_real(arcsin,one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))) ).
% arcsin_1
tff(fact_8505_cos__of__real__pi__half,axiom,
aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,pi)),numeral_numeral_real(aa_num_num(bit0,one)))) = zero_zero_real ).
% cos_of_real_pi_half
tff(fact_8506_cos__of__real__pi__half,axiom,
aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) = zero_zero_complex ).
% cos_of_real_pi_half
tff(fact_8507_sin__of__real__pi__half,axiom,
aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,pi)),numeral_numeral_real(aa_num_num(bit0,one)))) = one_one_real ).
% sin_of_real_pi_half
tff(fact_8508_sin__of__real__pi__half,axiom,
aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) = one_one_complex ).
% sin_of_real_pi_half
tff(fact_8509_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(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) ).
% arcsin_minus_1
tff(fact_8510_nonzero__of__real__divide,axiom,
! [Y: real,X: real] :
( ( Y != zero_zero_real )
=> ( aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,X)),aa_real_real(real_V1803761363581548252l_real,Y)) ) ) ).
% nonzero_of_real_divide
tff(fact_8511_nonzero__of__real__divide,axiom,
! [Y: real,X: real] :
( ( Y != zero_zero_real )
=> ( aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,X)),aa_real_complex(real_V4546457046886955230omplex,Y)) ) ) ).
% nonzero_of_real_divide
tff(fact_8512_Complex__mult__complex__of__real,axiom,
! [X: real,Y: real,R: real] : aa_complex_complex(times_times_complex(aa_real_complex(complex2(X),Y)),aa_real_complex(real_V4546457046886955230omplex,R)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),R)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),R)) ).
% Complex_mult_complex_of_real
tff(fact_8513_complex__of__real__mult__Complex,axiom,
! [R: real,X: real,Y: real] : aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R)),aa_real_complex(complex2(X),Y)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),X)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),Y)) ).
% complex_of_real_mult_Complex
tff(fact_8514_complex__of__real__add__Complex,axiom,
! [R: real,X: real,Y: real] : aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,R)),aa_real_complex(complex2(X),Y)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R),X)),Y) ).
% complex_of_real_add_Complex
tff(fact_8515_Complex__add__complex__of__real,axiom,
! [X: real,Y: real,R: real] : aa_complex_complex(plus_plus_complex(aa_real_complex(complex2(X),Y)),aa_real_complex(real_V4546457046886955230omplex,R)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),R)),Y) ).
% Complex_add_complex_of_real
tff(fact_8516_arccos__le__arccos,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y)),aa_real_real(arccos,X))) ) ) ) ).
% arccos_le_arccos
tff(fact_8517_arccos__eq__iff,axiom,
! [X: real,Y: real] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_8518_arccos__le__mono,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,X)),aa_real_real(arccos,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ) ) ).
% arccos_le_mono
tff(fact_8519_arcsin__minus,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_8520_arcsin__le__arcsin,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y))) ) ) ) ).
% arcsin_le_arcsin
tff(fact_8521_arcsin__eq__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_8522_arcsin__le__mono,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ).
% arcsin_le_mono
tff(fact_8523_norm__less__p1,axiom,
! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,aa_real_real(real_V7735802525324610683m_real,X))),one_one_real)))) ).
% norm_less_p1
tff(fact_8524_norm__less__p1,axiom,
! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(real_V1022390504157884413omplex,X))),one_one_complex)))) ).
% norm_less_p1
tff(fact_8525_arccos__lbound,axiom,
! [Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y))) ) ) ).
% arccos_lbound
tff(fact_8526_arccos__less__arccos,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arccos,Y)),aa_real_real(arccos,X))) ) ) ) ).
% arccos_less_arccos
tff(fact_8527_arccos__less__mono,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arccos,X)),aa_real_real(arccos,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ) ).
% arccos_less_mono
tff(fact_8528_arccos__ubound,axiom,
! [Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y)),pi)) ) ) ).
% arccos_ubound
tff(fact_8529_arccos__cos,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
=> ( aa_real_real(arccos,aa_real_real(cos_real,X)) = X ) ) ) ).
% arccos_cos
tff(fact_8530_arcsin__less__arcsin,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y))) ) ) ) ).
% arcsin_less_arcsin
tff(fact_8531_arcsin__less__mono,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ).
% arcsin_less_mono
tff(fact_8532_cos__arccos__abs,axiom,
! [Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_8533_norm__of__real__diff,axiom,
! [B: real,A: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_real_complex(real_V4546457046886955230omplex,B)),aa_real_complex(real_V4546457046886955230omplex,A)))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)))) ).
% norm_of_real_diff
tff(fact_8534_arccos__cos__eq__abs,axiom,
! [Theta: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_8535_arccos__lt__bounded,axiom,
! [Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),one_one_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(arccos,Y)))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arccos,Y)),pi)) ) ) ) ).
% arccos_lt_bounded
tff(fact_8536_arccos__bounded,axiom,
! [Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y)))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y)),pi)) ) ) ) ).
% arccos_bounded
tff(fact_8537_sin__arccos__nonzero,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
=> ( aa_real_real(sin_real,aa_real_real(arccos,X)) != zero_zero_real ) ) ) ).
% sin_arccos_nonzero
tff(fact_8538_cos__int__times__real,axiom,
! [M: int,X: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,M)),aa_real_real(real_V1803761363581548252l_real,X))) = aa_real_real(real_V1803761363581548252l_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,M)),X))) ).
% cos_int_times_real
tff(fact_8539_cos__int__times__real,axiom,
! [M: int,X: real] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(aa_int_complex(ring_17405671764205052669omplex,M)),aa_real_complex(real_V4546457046886955230omplex,X))) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,M)),X))) ).
% cos_int_times_real
tff(fact_8540_sin__int__times__real,axiom,
! [M: int,X: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,M)),aa_real_real(real_V1803761363581548252l_real,X))) = aa_real_real(real_V1803761363581548252l_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,M)),X))) ).
% sin_int_times_real
tff(fact_8541_sin__int__times__real,axiom,
! [M: int,X: real] : aa_complex_complex(sin_complex,aa_complex_complex(times_times_complex(aa_int_complex(ring_17405671764205052669omplex,M)),aa_real_complex(real_V4546457046886955230omplex,X))) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,M)),X))) ).
% sin_int_times_real
tff(fact_8542_arccos__cos2,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_8543_arccos__minus,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
=> ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),aa_real_real(arccos,X)) ) ) ) ).
% arccos_minus
tff(fact_8544_cos__arcsin__nonzero,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
=> ( aa_real_real(cos_real,aa_real_real(arcsin,X)) != zero_zero_real ) ) ) ).
% cos_arcsin_nonzero
tff(fact_8545_arccos,axiom,
! [Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y)))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y)),pi))
& ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ) ) ).
% arccos
tff(fact_8546_arccos__minus__abs,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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(aa_rea7340605660300375871l_real(minus_minus_real,pi),aa_real_real(arccos,X)) ) ) ).
% arccos_minus_abs
tff(fact_8547_sin__cos__eq,axiom,
! [X: real] : aa_real_real(sin_real,X) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,pi)),numeral_numeral_real(aa_num_num(bit0,one)))),X)) ).
% sin_cos_eq
tff(fact_8548_sin__cos__eq,axiom,
! [X: complex] : aa_complex_complex(sin_complex,X) = aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one)))),X)) ).
% sin_cos_eq
tff(fact_8549_cos__sin__eq,axiom,
! [X: real] : aa_real_real(cos_real,X) = aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,pi)),numeral_numeral_real(aa_num_num(bit0,one)))),X)) ).
% cos_sin_eq
tff(fact_8550_cos__sin__eq,axiom,
! [X: complex] : aa_complex_complex(cos_complex,X) = aa_complex_complex(sin_complex,aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one)))),X)) ).
% cos_sin_eq
tff(fact_8551_arsinh__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(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))))) ).
% arsinh_def
tff(fact_8552_minus__sin__cos__eq,axiom,
! [X: real] : aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X)) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,pi)),numeral_numeral_real(aa_num_num(bit0,one))))) ).
% minus_sin_cos_eq
tff(fact_8553_minus__sin__cos__eq,axiom,
! [X: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(sin_complex,X)) = aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(X),aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).
% minus_sin_cos_eq
tff(fact_8554_arccos__le__pi2,axiom,
! [Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arccos,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))) ) ) ).
% arccos_le_pi2
tff(fact_8555_arcsin__lt__bounded,axiom,
! [Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),one_one_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y)))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))) ) ) ) ).
% arcsin_lt_bounded
tff(fact_8556_arcsin__bounded,axiom,
! [Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y)))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))) ) ) ) ).
% arcsin_bounded
tff(fact_8557_arcsin__ubound,axiom,
! [Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))) ) ) ).
% arcsin_ubound
tff(fact_8558_arcsin__lbound,axiom,
! [Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))) ) ) ).
% arcsin_lbound
tff(fact_8559_arcsin__sin,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
=> ( aa_real_real(arcsin,aa_real_real(sin_real,X)) = X ) ) ) ).
% arcsin_sin
tff(fact_8560_le__arcsin__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(aa_num_num(bit0,one)))),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(arcsin,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sin_real,Y)),X)) ) ) ) ) ) ).
% le_arcsin_iff
tff(fact_8561_arcsin__le__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(aa_num_num(bit0,one)))),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,X)),Y))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(sin_real,Y))) ) ) ) ) ) ).
% arcsin_le_iff
tff(fact_8562_arcsin__pi,axiom,
! [Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y)))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_8563_arcsin,axiom,
! [Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),one_one_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y)))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
& ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ) ).
% arcsin
tff(fact_8564_arccos__cos__eq__abs__2pi,axiom,
! [Theta: real] :
~ ! [K3: int] : aa_real_real(arccos,aa_real_real(cos_real,Theta)) != aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Theta),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,K3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)))) ).
% arccos_cos_eq_abs_2pi
tff(fact_8565_pochhammer__double,axiom,
! [Z: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),N))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))),N)) ).
% pochhammer_double
tff(fact_8566_pochhammer__double,axiom,
! [Z: complex,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_nat_complex(comm_s2602460028002588243omplex(Z),N))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one))))),N)) ).
% pochhammer_double
tff(fact_8567_pochhammer__double,axiom,
! [Z: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),N))),aa_nat_rat(comm_s4028243227959126397er_rat(plus_plus_rat(Z,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),N)) ).
% pochhammer_double
tff(fact_8568_floor__log__nat__eq__powr__iff,axiom,
! [B: nat,K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,N) )
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))) ) ) ) ) ).
% floor_log_nat_eq_powr_iff
tff(fact_8569_of__nat__code,axiom,
! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,N) = aa_int_int(semiri8420488043553186161ux_int(aTP_Lamm_md,N),zero_zero_int) ).
% of_nat_code
tff(fact_8570_of__nat__code,axiom,
! [N: nat] : aa_nat_real(semiri5074537144036343181t_real,N) = aa_real_real(aa_nat_fun_real_real(semiri7260567687927622513x_real(aTP_Lamm_me),N),zero_zero_real) ).
% of_nat_code
tff(fact_8571_of__nat__code,axiom,
! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(semiri8422978514062236437ux_nat(aTP_Lamm_mf),N),zero_zero_nat) ).
% of_nat_code
tff(fact_8572_of__nat__code,axiom,
! [N: nat] : aa_nat_complex(semiri8010041392384452111omplex,N) = aa_complex_complex(semiri2816024913162550771omplex(aTP_Lamm_mg,N),zero_zero_complex) ).
% of_nat_code
tff(fact_8573_of__nat__code,axiom,
! [N: nat] : aa_nat_rat(semiri681578069525770553at_rat,N) = semiri7787848453975740701ux_rat(aTP_Lamm_mh,N,zero_zero_rat) ).
% of_nat_code
tff(fact_8574_gchoose__row__sum__weighted,axiom,
! [R: complex,M: nat] : groups2073611262835488442omplex(aTP_Lamm_mi(R),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),M)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,M))),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(gbinomial_complex(R),aa_nat_nat(suc,M))) ).
% gchoose_row_sum_weighted
tff(fact_8575_gchoose__row__sum__weighted,axiom,
! [R: rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_mj(R),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),M)) = aa_rat_rat(times_times_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M)),numeral_numeral_rat(aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(R),aa_nat_nat(suc,M))) ).
% gchoose_row_sum_weighted
tff(fact_8576_gchoose__row__sum__weighted,axiom,
! [R: real,M: nat] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_mk,R),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,R),aa_nat_nat(suc,M))) ).
% gchoose_row_sum_weighted
tff(fact_8577_of__int__floor__cancel,axiom,
! [X: real] :
( ( aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X)) = X )
<=> ? [N5: int] : X = aa_int_real(ring_1_of_int_real,N5) ) ).
% of_int_floor_cancel
tff(fact_8578_of__int__floor__cancel,axiom,
! [X: rat] :
( ( aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(X)) = X )
<=> ? [N5: int] : X = aa_int_rat(ring_1_of_int_rat,N5) ) ).
% of_int_floor_cancel
tff(fact_8579_gbinomial__0_I2_J,axiom,
! [K: nat] : aa_nat_complex(gbinomial_complex(zero_zero_complex),aa_nat_nat(suc,K)) = zero_zero_complex ).
% gbinomial_0(2)
tff(fact_8580_gbinomial__0_I2_J,axiom,
! [K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,zero_zero_real),aa_nat_nat(suc,K)) = zero_zero_real ).
% gbinomial_0(2)
tff(fact_8581_gbinomial__0_I2_J,axiom,
! [K: nat] : aa_nat_rat(gbinomial_rat(zero_zero_rat),aa_nat_nat(suc,K)) = zero_zero_rat ).
% gbinomial_0(2)
tff(fact_8582_gbinomial__0_I2_J,axiom,
! [K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,zero_zero_nat),aa_nat_nat(suc,K)) = zero_zero_nat ).
% gbinomial_0(2)
tff(fact_8583_gbinomial__0_I2_J,axiom,
! [K: nat] : aa_nat_int(gbinomial_int(zero_zero_int),aa_nat_nat(suc,K)) = zero_zero_int ).
% gbinomial_0(2)
tff(fact_8584_floor__numeral,axiom,
! [V: num] : aa_real_int(archim6058952711729229775r_real,numeral_numeral_real(V)) = aa_num_int(numeral_numeral_int,V) ).
% floor_numeral
tff(fact_8585_floor__numeral,axiom,
! [V: num] : archim3151403230148437115or_rat(numeral_numeral_rat(V)) = aa_num_int(numeral_numeral_int,V) ).
% floor_numeral
tff(fact_8586_gbinomial__0_I1_J,axiom,
! [A: complex] : aa_nat_complex(gbinomial_complex(A),zero_zero_nat) = one_one_complex ).
% gbinomial_0(1)
tff(fact_8587_gbinomial__0_I1_J,axiom,
! [A: real] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),zero_zero_nat) = one_one_real ).
% gbinomial_0(1)
tff(fact_8588_gbinomial__0_I1_J,axiom,
! [A: rat] : aa_nat_rat(gbinomial_rat(A),zero_zero_nat) = one_one_rat ).
% gbinomial_0(1)
tff(fact_8589_gbinomial__0_I1_J,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,A),zero_zero_nat) = one_one_nat ).
% gbinomial_0(1)
tff(fact_8590_gbinomial__0_I1_J,axiom,
! [A: int] : aa_nat_int(gbinomial_int(A),zero_zero_nat) = one_one_int ).
% gbinomial_0(1)
tff(fact_8591_floor__one,axiom,
aa_real_int(archim6058952711729229775r_real,one_one_real) = one_one_int ).
% floor_one
tff(fact_8592_floor__one,axiom,
archim3151403230148437115or_rat(one_one_rat) = one_one_int ).
% floor_one
tff(fact_8593_pochhammer__0,axiom,
! [A: complex] : aa_nat_complex(comm_s2602460028002588243omplex(A),zero_zero_nat) = one_one_complex ).
% pochhammer_0
tff(fact_8594_pochhammer__0,axiom,
! [A: real] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),zero_zero_nat) = one_one_real ).
% pochhammer_0
tff(fact_8595_pochhammer__0,axiom,
! [A: rat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),zero_zero_nat) = one_one_rat ).
% pochhammer_0
tff(fact_8596_pochhammer__0,axiom,
! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),zero_zero_nat) = one_one_nat ).
% pochhammer_0
tff(fact_8597_pochhammer__0,axiom,
! [A: int] : aa_nat_int(comm_s4660882817536571857er_int(A),zero_zero_nat) = one_one_int ).
% pochhammer_0
tff(fact_8598_floor__diff__of__int,axiom,
! [X: real,Z: int] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_int_real(ring_1_of_int_real,Z))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim6058952711729229775r_real,X)),Z) ).
% floor_diff_of_int
tff(fact_8599_floor__diff__of__int,axiom,
! [X: rat,Z: int] : archim3151403230148437115or_rat(minus_minus_rat(X,aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,archim3151403230148437115or_rat(X)),Z) ).
% floor_diff_of_int
tff(fact_8600_zero__le__floor,axiom,
! [X: real] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).
% zero_le_floor
tff(fact_8601_zero__le__floor,axiom,
! [X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),archim3151403230148437115or_rat(X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),X)) ) ).
% zero_le_floor
tff(fact_8602_floor__less__zero,axiom,
! [X: real] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),zero_zero_int))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).
% floor_less_zero
tff(fact_8603_floor__less__zero,axiom,
! [X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,archim3151403230148437115or_rat(X)),zero_zero_int))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),zero_zero_rat)) ) ).
% floor_less_zero
tff(fact_8604_numeral__le__floor,axiom,
! [V: num,X: real] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim6058952711729229775r_real,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(V)),X)) ) ).
% numeral_le_floor
tff(fact_8605_numeral__le__floor,axiom,
! [V: num,X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),archim3151403230148437115or_rat(X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,numeral_numeral_rat(V)),X)) ) ).
% numeral_le_floor
tff(fact_8606_zero__less__floor,axiom,
! [X: real] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ).
% zero_less_floor
tff(fact_8607_zero__less__floor,axiom,
! [X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),archim3151403230148437115or_rat(X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),X)) ) ).
% zero_less_floor
tff(fact_8608_floor__le__zero,axiom,
! [X: real] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),zero_zero_int))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ).
% floor_le_zero
tff(fact_8609_floor__le__zero,axiom,
! [X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,archim3151403230148437115or_rat(X)),zero_zero_int))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),one_one_rat)) ) ).
% floor_le_zero
tff(fact_8610_floor__less__numeral,axiom,
! [X: real,V: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),aa_num_int(numeral_numeral_int,V)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),numeral_numeral_real(V))) ) ).
% floor_less_numeral
tff(fact_8611_floor__less__numeral,axiom,
! [X: rat,V: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,archim3151403230148437115or_rat(X)),aa_num_int(numeral_numeral_int,V)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),numeral_numeral_rat(V))) ) ).
% floor_less_numeral
tff(fact_8612_one__le__floor,axiom,
! [X: real] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),aa_real_int(archim6058952711729229775r_real,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X)) ) ).
% one_le_floor
tff(fact_8613_one__le__floor,axiom,
! [X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),archim3151403230148437115or_rat(X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),X)) ) ).
% one_le_floor
tff(fact_8614_floor__less__one,axiom,
! [X: real] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ).
% floor_less_one
tff(fact_8615_floor__less__one,axiom,
! [X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,archim3151403230148437115or_rat(X)),one_one_int))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),one_one_rat)) ) ).
% floor_less_one
tff(fact_8616_floor__neg__numeral,axiom,
! [V: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).
% floor_neg_numeral
tff(fact_8617_floor__neg__numeral,axiom,
! [V: num] : archim3151403230148437115or_rat(uminus_uminus_rat(numeral_numeral_rat(V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).
% floor_neg_numeral
tff(fact_8618_floor__diff__numeral,axiom,
! [X: real,V: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),numeral_numeral_real(V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim6058952711729229775r_real,X)),aa_num_int(numeral_numeral_int,V)) ).
% floor_diff_numeral
tff(fact_8619_floor__diff__numeral,axiom,
! [X: rat,V: num] : archim3151403230148437115or_rat(minus_minus_rat(X,numeral_numeral_rat(V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,archim3151403230148437115or_rat(X)),aa_num_int(numeral_numeral_int,V)) ).
% floor_diff_numeral
tff(fact_8620_floor__diff__one,axiom,
! [X: real] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),one_one_real)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int) ).
% floor_diff_one
tff(fact_8621_floor__diff__one,axiom,
! [X: rat] : archim3151403230148437115or_rat(minus_minus_rat(X,one_one_rat)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,archim3151403230148437115or_rat(X)),one_one_int) ).
% floor_diff_one
tff(fact_8622_floor__numeral__power,axiom,
! [X: num,N: nat] : aa_real_int(archim6058952711729229775r_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(X)),N)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ).
% floor_numeral_power
tff(fact_8623_floor__numeral__power,axiom,
! [X: num,N: nat] : archim3151403230148437115or_rat(aa_nat_rat(power_power_rat(numeral_numeral_rat(X)),N)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ).
% floor_numeral_power
tff(fact_8624_floor__divide__eq__div__numeral,axiom,
! [A: num,B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(A)),numeral_numeral_real(B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,A)),aa_num_int(numeral_numeral_int,B)) ).
% floor_divide_eq_div_numeral
tff(fact_8625_numeral__less__floor,axiom,
! [V: num,X: real] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim6058952711729229775r_real,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(V)),one_one_real)),X)) ) ).
% numeral_less_floor
tff(fact_8626_numeral__less__floor,axiom,
! [V: num,X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_num_int(numeral_numeral_int,V)),archim3151403230148437115or_rat(X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(numeral_numeral_rat(V),one_one_rat)),X)) ) ).
% numeral_less_floor
tff(fact_8627_floor__le__numeral,axiom,
! [X: real,V: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_num_int(numeral_numeral_int,V)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(V)),one_one_real))) ) ).
% floor_le_numeral
tff(fact_8628_floor__le__numeral,axiom,
! [X: rat,V: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,archim3151403230148437115or_rat(X)),aa_num_int(numeral_numeral_int,V)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),plus_plus_rat(numeral_numeral_rat(V),one_one_rat))) ) ).
% floor_le_numeral
tff(fact_8629_one__less__floor,axiom,
! [X: real] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_real_int(archim6058952711729229775r_real,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(aa_num_num(bit0,one))),X)) ) ).
% one_less_floor
tff(fact_8630_one__less__floor,axiom,
! [X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),archim3151403230148437115or_rat(X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,numeral_numeral_rat(aa_num_num(bit0,one))),X)) ) ).
% one_less_floor
tff(fact_8631_floor__le__one,axiom,
! [X: real] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),numeral_numeral_real(aa_num_num(bit0,one)))) ) ).
% floor_le_one
tff(fact_8632_floor__le__one,axiom,
! [X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,archim3151403230148437115or_rat(X)),one_one_int))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),numeral_numeral_rat(aa_num_num(bit0,one)))) ) ).
% floor_le_one
tff(fact_8633_neg__numeral__le__floor,axiom,
! [V: num,X: real] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim6058952711729229775r_real,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),X)) ) ).
% neg_numeral_le_floor
tff(fact_8634_neg__numeral__le__floor,axiom,
! [V: num,X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),archim3151403230148437115or_rat(X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(V))),X)) ) ).
% neg_numeral_le_floor
tff(fact_8635_floor__less__neg__numeral,axiom,
! [X: real,V: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(uminus_uminus_real,numeral_numeral_real(V)))) ) ).
% floor_less_neg_numeral
tff(fact_8636_floor__less__neg__numeral,axiom,
! [X: rat,V: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,archim3151403230148437115or_rat(X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),uminus_uminus_rat(numeral_numeral_rat(V)))) ) ).
% floor_less_neg_numeral
tff(fact_8637_floor__one__divide__eq__div__numeral,axiom,
! [B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,B)) ).
% floor_one_divide_eq_div_numeral
tff(fact_8638_floor__minus__divide__eq__div__numeral,axiom,
! [A: num,B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,numeral_numeral_real(A)),numeral_numeral_real(B)))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,A))),aa_num_int(numeral_numeral_int,B)) ).
% floor_minus_divide_eq_div_numeral
tff(fact_8639_neg__numeral__less__floor,axiom,
! [V: num,X: real] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim6058952711729229775r_real,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),one_one_real)),X)) ) ).
% neg_numeral_less_floor
tff(fact_8640_neg__numeral__less__floor,axiom,
! [V: num,X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),archim3151403230148437115or_rat(X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(uminus_uminus_rat(numeral_numeral_rat(V)),one_one_rat)),X)) ) ).
% neg_numeral_less_floor
tff(fact_8641_floor__le__neg__numeral,axiom,
! [X: real,V: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),one_one_real))) ) ).
% floor_le_neg_numeral
tff(fact_8642_floor__le__neg__numeral,axiom,
! [X: rat,V: num] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,archim3151403230148437115or_rat(X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),plus_plus_rat(uminus_uminus_rat(numeral_numeral_rat(V)),one_one_rat))) ) ).
% floor_le_neg_numeral
tff(fact_8643_floor__minus__one__divide__eq__div__numeral,axiom,
! [B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(B)))) = 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,B)) ).
% floor_minus_one_divide_eq_div_numeral
tff(fact_8644_floor__mono,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y))) ) ).
% floor_mono
tff(fact_8645_floor__mono,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,archim3151403230148437115or_rat(X)),archim3151403230148437115or_rat(Y))) ) ).
% floor_mono
tff(fact_8646_of__int__floor__le,axiom,
! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X))),X)) ).
% of_int_floor_le
tff(fact_8647_of__int__floor__le,axiom,
! [X: rat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(X))),X)) ).
% of_int_floor_le
tff(fact_8648_floor__less__cancel,axiom,
! [X: real,Y: real] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y)))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).
% floor_less_cancel
tff(fact_8649_floor__less__cancel,axiom,
! [X: rat,Y: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,archim3151403230148437115or_rat(X)),archim3151403230148437115or_rat(Y)))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y)) ) ).
% floor_less_cancel
tff(fact_8650_pochhammer__pos,axiom,
! [X: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,X),N))) ) ).
% pochhammer_pos
tff(fact_8651_pochhammer__pos,axiom,
! [X: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(X),N))) ) ).
% pochhammer_pos
tff(fact_8652_pochhammer__pos,axiom,
! [X: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),N))) ) ).
% pochhammer_pos
tff(fact_8653_pochhammer__pos,axiom,
! [X: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(X),N))) ) ).
% pochhammer_pos
tff(fact_8654_pochhammer__neq__0__mono,axiom,
! [A: complex,M: nat,N: nat] :
( ( aa_nat_complex(comm_s2602460028002588243omplex(A),M) != zero_zero_complex )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) != zero_zero_complex ) ) ) ).
% pochhammer_neq_0_mono
tff(fact_8655_pochhammer__neq__0__mono,axiom,
! [A: real,M: nat,N: nat] :
( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),M) != zero_zero_real )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),N) != zero_zero_real ) ) ) ).
% pochhammer_neq_0_mono
tff(fact_8656_pochhammer__neq__0__mono,axiom,
! [A: rat,M: nat,N: nat] :
( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),M) != zero_zero_rat )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) != zero_zero_rat ) ) ) ).
% pochhammer_neq_0_mono
tff(fact_8657_pochhammer__eq__0__mono,axiom,
! [A: complex,N: nat,M: nat] :
( ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) = zero_zero_complex )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_complex(comm_s2602460028002588243omplex(A),M) = zero_zero_complex ) ) ) ).
% pochhammer_eq_0_mono
tff(fact_8658_pochhammer__eq__0__mono,axiom,
! [A: real,N: nat,M: nat] :
( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),N) = zero_zero_real )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),M) = zero_zero_real ) ) ) ).
% pochhammer_eq_0_mono
tff(fact_8659_pochhammer__eq__0__mono,axiom,
! [A: rat,N: nat,M: nat] :
( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = zero_zero_rat )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_rat(comm_s4028243227959126397er_rat(A),M) = zero_zero_rat ) ) ) ).
% pochhammer_eq_0_mono
tff(fact_8660_pochhammer__fact,axiom,
! [N: nat] : aa_nat_complex(semiri5044797733671781792omplex,N) = aa_nat_complex(comm_s2602460028002588243omplex(one_one_complex),N) ).
% pochhammer_fact
tff(fact_8661_pochhammer__fact,axiom,
! [N: nat] : aa_nat_rat(semiri773545260158071498ct_rat,N) = aa_nat_rat(comm_s4028243227959126397er_rat(one_one_rat),N) ).
% pochhammer_fact
tff(fact_8662_pochhammer__fact,axiom,
! [N: nat] : aa_nat_int(semiri1406184849735516958ct_int,N) = aa_nat_int(comm_s4660882817536571857er_int(one_one_int),N) ).
% pochhammer_fact
tff(fact_8663_pochhammer__fact,axiom,
! [N: nat] : aa_nat_real(semiri2265585572941072030t_real,N) = aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,one_one_real),N) ).
% pochhammer_fact
tff(fact_8664_pochhammer__fact,axiom,
! [N: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,one_one_nat),N) ).
% pochhammer_fact
tff(fact_8665_le__floor__iff,axiom,
! [Z: int,X: real] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),aa_real_int(archim6058952711729229775r_real,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),X)) ) ).
% le_floor_iff
tff(fact_8666_le__floor__iff,axiom,
! [Z: int,X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),archim3151403230148437115or_rat(X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),X)) ) ).
% le_floor_iff
tff(fact_8667_floor__less__iff,axiom,
! [X: real,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),Z))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,Z))) ) ).
% floor_less_iff
tff(fact_8668_floor__less__iff,axiom,
! [X: rat,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,archim3151403230148437115or_rat(X)),Z))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,Z))) ) ).
% floor_less_iff
tff(fact_8669_le__floor__add,axiom,
! [X: real,Y: real] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y))),aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y)))) ).
% le_floor_add
tff(fact_8670_le__floor__add,axiom,
! [X: rat,Y: rat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,archim3151403230148437115or_rat(X)),archim3151403230148437115or_rat(Y))),archim3151403230148437115or_rat(plus_plus_rat(X,Y)))) ).
% le_floor_add
tff(fact_8671_floor__add__int,axiom,
! [X: real,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),Z) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_int_real(ring_1_of_int_real,Z))) ).
% floor_add_int
tff(fact_8672_floor__add__int,axiom,
! [X: rat,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,archim3151403230148437115or_rat(X)),Z) = archim3151403230148437115or_rat(plus_plus_rat(X,aa_int_rat(ring_1_of_int_rat,Z))) ).
% floor_add_int
tff(fact_8673_int__add__floor,axiom,
! [Z: int,X: real] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_real_int(archim6058952711729229775r_real,X)) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),X)) ).
% int_add_floor
tff(fact_8674_int__add__floor,axiom,
! [Z: int,X: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),archim3151403230148437115or_rat(X)) = archim3151403230148437115or_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z),X)) ).
% int_add_floor
tff(fact_8675_floor__divide__of__int__eq,axiom,
! [K: int,L: int] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,K)),aa_int_real(ring_1_of_int_real,L))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) ).
% floor_divide_of_int_eq
tff(fact_8676_floor__divide__of__int__eq,axiom,
! [K: int,L: int] : archim3151403230148437115or_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,K),aa_int_rat(ring_1_of_int_rat,L))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) ).
% floor_divide_of_int_eq
tff(fact_8677_gbinomial__pochhammer,axiom,
! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),K))),aa_nat_complex(semiri5044797733671781792omplex,K)) ).
% gbinomial_pochhammer
tff(fact_8678_gbinomial__pochhammer,axiom,
! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(A)),K)),aa_nat_rat(semiri773545260158071498ct_rat,K)) ).
% gbinomial_pochhammer
tff(fact_8679_gbinomial__pochhammer,axiom,
! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,A)),K))),aa_nat_real(semiri2265585572941072030t_real,K)) ).
% gbinomial_pochhammer
tff(fact_8680_gbinomial__pochhammer_H,axiom,
! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)),aa_nat_complex(semiri5044797733671781792omplex,K)) ).
% gbinomial_pochhammer'
tff(fact_8681_gbinomial__pochhammer_H,axiom,
! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = divide_divide_rat(aa_nat_rat(comm_s4028243227959126397er_rat(plus_plus_rat(minus_minus_rat(A,aa_nat_rat(semiri681578069525770553at_rat,K)),one_one_rat)),K),aa_nat_rat(semiri773545260158071498ct_rat,K)) ).
% gbinomial_pochhammer'
tff(fact_8682_gbinomial__pochhammer_H,axiom,
! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)),aa_nat_real(semiri2265585572941072030t_real,K)) ).
% gbinomial_pochhammer'
tff(fact_8683_floor__power,axiom,
! [X: real,N: nat] :
( ( X = aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X)) )
=> ( aa_real_int(archim6058952711729229775r_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = aa_nat_int(power_power_int(aa_real_int(archim6058952711729229775r_real,X)),N) ) ) ).
% floor_power
tff(fact_8684_floor__power,axiom,
! [X: rat,N: nat] :
( ( X = aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(X)) )
=> ( archim3151403230148437115or_rat(aa_nat_rat(power_power_rat(X),N)) = aa_nat_int(power_power_int(archim3151403230148437115or_rat(X)),N) ) ) ).
% floor_power
tff(fact_8685_gbinomial__Suc__Suc,axiom,
! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(A),K)),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K))) ).
% gbinomial_Suc_Suc
tff(fact_8686_gbinomial__Suc__Suc,axiom,
! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K))) ).
% gbinomial_Suc_Suc
tff(fact_8687_gbinomial__Suc__Suc,axiom,
! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(plus_plus_rat(A,one_one_rat)),aa_nat_nat(suc,K)) = plus_plus_rat(aa_nat_rat(gbinomial_rat(A),K),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K))) ).
% gbinomial_Suc_Suc
tff(fact_8688_gbinomial__of__nat__symmetric,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,N)),K) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).
% gbinomial_of_nat_symmetric
tff(fact_8689_gbinomial__of__nat__symmetric,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),K) = aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).
% gbinomial_of_nat_symmetric
tff(fact_8690_gbinomial__of__nat__symmetric,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),K) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).
% gbinomial_of_nat_symmetric
tff(fact_8691_pochhammer__nonneg,axiom,
! [X: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,X),N))) ) ).
% pochhammer_nonneg
tff(fact_8692_pochhammer__nonneg,axiom,
! [X: rat,N: nat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),X))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(X),N))) ) ).
% pochhammer_nonneg
tff(fact_8693_pochhammer__nonneg,axiom,
! [X: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),N))) ) ).
% pochhammer_nonneg
tff(fact_8694_pochhammer__nonneg,axiom,
! [X: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),X))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(X),N))) ) ).
% pochhammer_nonneg
tff(fact_8695_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_complex(comm_s2602460028002588243omplex(zero_zero_complex),N) = one_one_complex ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_complex(comm_s2602460028002588243omplex(zero_zero_complex),N) = zero_zero_complex ) ) ) ).
% pochhammer_0_left
tff(fact_8696_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,zero_zero_real),N) = one_one_real ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,zero_zero_real),N) = zero_zero_real ) ) ) ).
% pochhammer_0_left
tff(fact_8697_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_rat(comm_s4028243227959126397er_rat(zero_zero_rat),N) = one_one_rat ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_rat(comm_s4028243227959126397er_rat(zero_zero_rat),N) = zero_zero_rat ) ) ) ).
% pochhammer_0_left
tff(fact_8698_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),N) = one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),N) = zero_zero_nat ) ) ) ).
% pochhammer_0_left
tff(fact_8699_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_int(comm_s4660882817536571857er_int(zero_zero_int),N) = one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_int(comm_s4660882817536571857er_int(zero_zero_int),N) = zero_zero_int ) ) ) ).
% pochhammer_0_left
tff(fact_8700_one__add__floor,axiom,
! [X: real] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),one_one_real)) ).
% one_add_floor
tff(fact_8701_one__add__floor,axiom,
! [X: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,archim3151403230148437115or_rat(X)),one_one_int) = archim3151403230148437115or_rat(plus_plus_rat(X,one_one_rat)) ).
% one_add_floor
tff(fact_8702_floor__divide__of__nat__eq,axiom,
! [M: nat,N: nat] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) ).
% floor_divide_of_nat_eq
tff(fact_8703_floor__divide__of__nat__eq,axiom,
! [M: nat,N: nat] : archim3151403230148437115or_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,M),aa_nat_rat(semiri681578069525770553at_rat,N))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) ).
% floor_divide_of_nat_eq
tff(fact_8704_gbinomial__addition__formula,axiom,
! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ).
% gbinomial_addition_formula
tff(fact_8705_gbinomial__addition__formula,axiom,
! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K)) ).
% gbinomial_addition_formula
tff(fact_8706_gbinomial__addition__formula,axiom,
! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)) = plus_plus_rat(aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),aa_nat_nat(suc,K)),aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),K)) ).
% gbinomial_addition_formula
tff(fact_8707_ceiling__diff__floor__le__1,axiom,
! [X: real] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,X)),aa_real_int(archim6058952711729229775r_real,X))),one_one_int)) ).
% ceiling_diff_floor_le_1
tff(fact_8708_ceiling__diff__floor__le__1,axiom,
! [X: rat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,archim2889992004027027881ng_rat(X)),archim3151403230148437115or_rat(X))),one_one_int)) ).
% ceiling_diff_floor_le_1
tff(fact_8709_gbinomial__absorb__comp,axiom,
! [A: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K)) ).
% gbinomial_absorb_comp
tff(fact_8710_gbinomial__absorb__comp,axiom,
! [A: complex,K: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(gbinomial_complex(A),K)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ).
% gbinomial_absorb_comp
tff(fact_8711_gbinomial__absorb__comp,axiom,
! [A: rat,K: nat] : aa_rat_rat(times_times_rat(minus_minus_rat(A,aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(gbinomial_rat(A),K)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),K)) ).
% gbinomial_absorb_comp
tff(fact_8712_gbinomial__ge__n__over__k__pow__k,axiom,
! [K: nat,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,K)),A))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K))) ) ).
% gbinomial_ge_n_over_k_pow_k
tff(fact_8713_gbinomial__ge__n__over__k__pow__k,axiom,
! [K: nat,A: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,K)),A))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(power_power_rat(divide_divide_rat(A,aa_nat_rat(semiri681578069525770553at_rat,K))),K)),aa_nat_rat(gbinomial_rat(A),K))) ) ).
% gbinomial_ge_n_over_k_pow_k
tff(fact_8714_gbinomial__mult__1,axiom,
! [A: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K)))) ).
% gbinomial_mult_1
tff(fact_8715_gbinomial__mult__1,axiom,
! [A: complex,K: nat] : aa_complex_complex(times_times_complex(A),aa_nat_complex(gbinomial_complex(A),K)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,K)),aa_nat_complex(gbinomial_complex(A),K))),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K)))) ).
% gbinomial_mult_1
tff(fact_8716_gbinomial__mult__1,axiom,
! [A: rat,K: nat] : aa_rat_rat(times_times_rat(A),aa_nat_rat(gbinomial_rat(A),K)) = plus_plus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,K)),aa_nat_rat(gbinomial_rat(A),K)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)))) ).
% gbinomial_mult_1
tff(fact_8717_gbinomial__mult__1_H,axiom,
! [A: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K)))) ).
% gbinomial_mult_1'
tff(fact_8718_gbinomial__mult__1_H,axiom,
! [A: complex,K: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),K)),A) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,K)),aa_nat_complex(gbinomial_complex(A),K))),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K)))) ).
% gbinomial_mult_1'
tff(fact_8719_gbinomial__mult__1_H,axiom,
! [A: rat,K: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),K)),A) = plus_plus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,K)),aa_nat_rat(gbinomial_rat(A),K)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)))) ).
% gbinomial_mult_1'
tff(fact_8720_floor__eq,axiom,
! [N: int,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_int_real(ring_1_of_int_real,N)),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N)),one_one_real)))
=> ( aa_real_int(archim6058952711729229775r_real,X) = N ) ) ) ).
% floor_eq
tff(fact_8721_real__of__int__floor__add__one__gt,axiom,
! [R: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,R),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R))),one_one_real))) ).
% real_of_int_floor_add_one_gt
tff(fact_8722_real__of__int__floor__add__one__ge,axiom,
! [R: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,R),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R))),one_one_real))) ).
% real_of_int_floor_add_one_ge
tff(fact_8723_real__of__int__floor__gt__diff__one,axiom,
! [R: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R)))) ).
% real_of_int_floor_gt_diff_one
tff(fact_8724_real__of__int__floor__ge__diff__one,axiom,
! [R: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R)))) ).
% real_of_int_floor_ge_diff_one
tff(fact_8725_pochhammer__rec,axiom,
! [A: complex,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,N)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),N)) ).
% pochhammer_rec
tff(fact_8726_pochhammer__rec,axiom,
! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),N)) ).
% pochhammer_rec
tff(fact_8727_pochhammer__rec,axiom,
! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(comm_s4028243227959126397er_rat(plus_plus_rat(A,one_one_rat)),N)) ).
% pochhammer_rec
tff(fact_8728_pochhammer__rec,axiom,
! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),N)) ).
% pochhammer_rec
tff(fact_8729_pochhammer__rec,axiom,
! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),N)) ).
% pochhammer_rec
tff(fact_8730_pochhammer__Suc,axiom,
! [A: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(A),N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(semiri1314217659103216013at_int,N))) ).
% pochhammer_Suc
tff(fact_8731_pochhammer__Suc,axiom,
! [A: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_nat_real(semiri5074537144036343181t_real,N))) ).
% pochhammer_Suc
tff(fact_8732_pochhammer__Suc,axiom,
! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,N))) ).
% pochhammer_Suc
tff(fact_8733_pochhammer__Suc,axiom,
! [A: complex,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,N)) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(A),N)),aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,N))) ).
% pochhammer_Suc
tff(fact_8734_pochhammer__Suc,axiom,
! [A: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(A),N)),plus_plus_rat(A,aa_nat_rat(semiri681578069525770553at_rat,N))) ).
% pochhammer_Suc
tff(fact_8735_pochhammer__rec_H,axiom,
! [Z: int,N: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,N))),aa_nat_int(comm_s4660882817536571857er_int(Z),N)) ).
% pochhammer_rec'
tff(fact_8736_pochhammer__rec_H,axiom,
! [Z: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(suc,N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,N))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),N)) ).
% pochhammer_rec'
tff(fact_8737_pochhammer__rec_H,axiom,
! [Z: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,N))),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N)) ).
% pochhammer_rec'
tff(fact_8738_pochhammer__rec_H,axiom,
! [Z: complex,N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(suc,N)) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(Z),aa_nat_complex(semiri8010041392384452111omplex,N))),aa_nat_complex(comm_s2602460028002588243omplex(Z),N)) ).
% pochhammer_rec'
tff(fact_8739_pochhammer__rec_H,axiom,
! [Z: rat,N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(plus_plus_rat(Z,aa_nat_rat(semiri681578069525770553at_rat,N))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),N)) ).
% pochhammer_rec'
tff(fact_8740_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
=> ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N))),K) = zero_z3403309356797280102nteger ) ) ).
% pochhammer_of_nat_eq_0_lemma
tff(fact_8741_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
=> ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K) = zero_zero_int ) ) ).
% pochhammer_of_nat_eq_0_lemma
tff(fact_8742_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
=> ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),K) = zero_zero_real ) ) ).
% pochhammer_of_nat_eq_0_lemma
tff(fact_8743_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
=> ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),K) = zero_zero_complex ) ) ).
% pochhammer_of_nat_eq_0_lemma
tff(fact_8744_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
=> ( aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(aa_nat_rat(semiri681578069525770553at_rat,N))),K) = zero_zero_rat ) ) ).
% pochhammer_of_nat_eq_0_lemma
tff(fact_8745_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N))),K) = zero_z3403309356797280102nteger )
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).
% pochhammer_of_nat_eq_0_iff
tff(fact_8746_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K) = zero_zero_int )
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).
% pochhammer_of_nat_eq_0_iff
tff(fact_8747_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),K) = zero_zero_real )
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).
% pochhammer_of_nat_eq_0_iff
tff(fact_8748_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),K) = zero_zero_complex )
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).
% pochhammer_of_nat_eq_0_iff
tff(fact_8749_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(aa_nat_rat(semiri681578069525770553at_rat,N))),K) = zero_zero_rat )
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).
% pochhammer_of_nat_eq_0_iff
tff(fact_8750_pochhammer__eq__0__iff,axiom,
! [A: real,N: nat] :
( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),N) = zero_zero_real )
<=> ? [K2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),N))
& ( A = aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K2)) ) ) ) ).
% pochhammer_eq_0_iff
tff(fact_8751_pochhammer__eq__0__iff,axiom,
! [A: complex,N: nat] :
( ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) = zero_zero_complex )
<=> ? [K2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),N))
& ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K2)) ) ) ) ).
% pochhammer_eq_0_iff
tff(fact_8752_pochhammer__eq__0__iff,axiom,
! [A: rat,N: nat] :
( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = zero_zero_rat )
<=> ? [K2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K2),N))
& ( A = uminus_uminus_rat(aa_nat_rat(semiri681578069525770553at_rat,K2)) ) ) ) ).
% pochhammer_eq_0_iff
tff(fact_8753_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N))),K) != zero_z3403309356797280102nteger ) ) ).
% pochhammer_of_nat_eq_0_lemma'
tff(fact_8754_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),K) != zero_zero_int ) ) ).
% pochhammer_of_nat_eq_0_lemma'
tff(fact_8755_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),K) != zero_zero_real ) ) ).
% pochhammer_of_nat_eq_0_lemma'
tff(fact_8756_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),K) != zero_zero_complex ) ) ).
% pochhammer_of_nat_eq_0_lemma'
tff(fact_8757_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(aa_nat_rat(semiri681578069525770553at_rat,N))),K) != zero_zero_rat ) ) ).
% pochhammer_of_nat_eq_0_lemma'
tff(fact_8758_pochhammer__product_H,axiom,
! [Z: int,N: nat,M: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z),N)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,N))),M)) ).
% pochhammer_product'
tff(fact_8759_pochhammer__product_H,axiom,
! [Z: real,N: nat,M: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),N)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,N))),M)) ).
% pochhammer_product'
tff(fact_8760_pochhammer__product_H,axiom,
! [Z: nat,N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,N))),M)) ).
% pochhammer_product'
tff(fact_8761_pochhammer__product_H,axiom,
! [Z: complex,N: nat,M: nat] : aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z),N)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_nat_complex(semiri8010041392384452111omplex,N))),M)) ).
% pochhammer_product'
tff(fact_8762_pochhammer__product_H,axiom,
! [Z: rat,N: nat,M: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),N)),aa_nat_rat(comm_s4028243227959126397er_rat(plus_plus_rat(Z,aa_nat_rat(semiri681578069525770553at_rat,N))),M)) ).
% pochhammer_product'
tff(fact_8763_floor__split,axiom,
! [P: fun_int_bool,T: real] :
( pp(aa_int_bool(P,aa_real_int(archim6058952711729229775r_real,T)))
<=> ! [I4: int] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,I4)),T))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,I4)),one_one_real))) )
=> pp(aa_int_bool(P,I4)) ) ) ).
% floor_split
tff(fact_8764_floor__split,axiom,
! [P: fun_int_bool,T: rat] :
( pp(aa_int_bool(P,archim3151403230148437115or_rat(T)))
<=> ! [I4: int] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,I4)),T))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,T),plus_plus_rat(aa_int_rat(ring_1_of_int_rat,I4),one_one_rat))) )
=> pp(aa_int_bool(P,I4)) ) ) ).
% floor_split
tff(fact_8765_floor__eq__iff,axiom,
! [X: real,A: int] :
( ( aa_real_int(archim6058952711729229775r_real,X) = A )
<=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),X))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,A)),one_one_real))) ) ) ).
% floor_eq_iff
tff(fact_8766_floor__eq__iff,axiom,
! [X: rat,A: int] :
( ( archim3151403230148437115or_rat(X) = A )
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),X))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),plus_plus_rat(aa_int_rat(ring_1_of_int_rat,A),one_one_rat))) ) ) ).
% floor_eq_iff
tff(fact_8767_floor__unique,axiom,
! [Z: int,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)))
=> ( aa_real_int(archim6058952711729229775r_real,X) = Z ) ) ) ).
% floor_unique
tff(fact_8768_floor__unique,axiom,
! [Z: int,X: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),X))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat)))
=> ( archim3151403230148437115or_rat(X) = Z ) ) ) ).
% floor_unique
tff(fact_8769_le__mult__floor,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),A))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim6058952711729229775r_real,A)),aa_real_int(archim6058952711729229775r_real,B))),aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B)))) ) ) ).
% le_mult_floor
tff(fact_8770_le__mult__floor,axiom,
! [A: rat,B: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),A))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,archim3151403230148437115or_rat(A)),archim3151403230148437115or_rat(B))),archim3151403230148437115or_rat(aa_rat_rat(times_times_rat(A),B)))) ) ) ).
% le_mult_floor
tff(fact_8771_less__floor__iff,axiom,
! [Z: int,X: real] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_real_int(archim6058952711729229775r_real,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)),X)) ) ).
% less_floor_iff
tff(fact_8772_less__floor__iff,axiom,
! [Z: int,X: rat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),archim3151403230148437115or_rat(X)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat)),X)) ) ).
% less_floor_iff
tff(fact_8773_floor__le__iff,axiom,
! [X: real,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),Z))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real))) ) ).
% floor_le_iff
tff(fact_8774_floor__le__iff,axiom,
! [X: rat,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,archim3151403230148437115or_rat(X)),Z))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat))) ) ).
% floor_le_iff
tff(fact_8775_floor__correct,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X))),X))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int)))) ) ).
% floor_correct
tff(fact_8776_floor__correct,axiom,
! [X: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(X))),X))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,archim3151403230148437115or_rat(X)),one_one_int)))) ) ).
% floor_correct
tff(fact_8777_floor__eq2,axiom,
! [N: int,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,N)),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N)),one_one_real)))
=> ( aa_real_int(archim6058952711729229775r_real,X) = N ) ) ) ).
% floor_eq2
tff(fact_8778_Suc__times__gbinomial,axiom,
! [K: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)) ).
% Suc_times_gbinomial
tff(fact_8779_Suc__times__gbinomial,axiom,
! [K: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(gbinomial_complex(A),K)) ).
% Suc_times_gbinomial
tff(fact_8780_Suc__times__gbinomial,axiom,
! [K: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(plus_plus_rat(A,one_one_rat)),aa_nat_nat(suc,K))) = aa_rat_rat(times_times_rat(plus_plus_rat(A,one_one_rat)),aa_nat_rat(gbinomial_rat(A),K)) ).
% Suc_times_gbinomial
tff(fact_8781_gbinomial__absorption,axiom,
! [K: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K)) ).
% gbinomial_absorption
tff(fact_8782_gbinomial__absorption,axiom,
! [K: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K))) = aa_complex_complex(times_times_complex(A),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ).
% gbinomial_absorption
tff(fact_8783_gbinomial__absorption,axiom,
! [K: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K))) = aa_rat_rat(times_times_rat(A),aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),K)) ).
% gbinomial_absorption
tff(fact_8784_floor__divide__real__eq__div,axiom,
! [B: int,A: real] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B))
=> ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_int_real(ring_1_of_int_real,B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_real_int(archim6058952711729229775r_real,A)),B) ) ) ).
% floor_divide_real_eq_div
tff(fact_8785_gbinomial__trinomial__revision,axiom,
! [K: nat,M: nat,A: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),M)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,M)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ).
% gbinomial_trinomial_revision
tff(fact_8786_gbinomial__trinomial__revision,axiom,
! [K: nat,M: nat,A: complex] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
=> ( aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),M)),aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),K)) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),K)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ).
% gbinomial_trinomial_revision
tff(fact_8787_gbinomial__trinomial__revision,axiom,
! [K: nat,M: nat,A: rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
=> ( aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),M)),aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),K)) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),K)),aa_nat_rat(gbinomial_rat(minus_minus_rat(A,aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ).
% gbinomial_trinomial_revision
tff(fact_8788_pochhammer__product,axiom,
! [M: nat,N: nat,Z: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_nat_int(comm_s4660882817536571857er_int(Z),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z),M)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).
% pochhammer_product
tff(fact_8789_pochhammer__product,axiom,
! [M: nat,N: nat,Z: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),N) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),M)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).
% pochhammer_product
tff(fact_8790_pochhammer__product,axiom,
! [M: nat,N: nat,Z: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),M)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).
% pochhammer_product
tff(fact_8791_pochhammer__product,axiom,
! [M: nat,N: nat,Z: complex] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_nat_complex(comm_s2602460028002588243omplex(Z),N) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z),M)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_nat_complex(semiri8010041392384452111omplex,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).
% pochhammer_product
tff(fact_8792_pochhammer__product,axiom,
! [M: nat,N: nat,Z: rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_nat_rat(comm_s4028243227959126397er_rat(Z),N) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),M)),aa_nat_rat(comm_s4028243227959126397er_rat(plus_plus_rat(Z,aa_nat_rat(semiri681578069525770553at_rat,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).
% pochhammer_product
tff(fact_8793_floor__divide__lower,axiom,
! [Q2: real,P2: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Q2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P2),Q2)))),Q2)),P2)) ) ).
% floor_divide_lower
tff(fact_8794_floor__divide__lower,axiom,
! [Q2: rat,P2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Q2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(divide_divide_rat(P2,Q2)))),Q2)),P2)) ) ).
% floor_divide_lower
tff(fact_8795_gbinomial__rec,axiom,
! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K)))) ).
% gbinomial_rec
tff(fact_8796_gbinomial__rec,axiom,
! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),K)),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K)))) ).
% gbinomial_rec
tff(fact_8797_gbinomial__rec,axiom,
! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(plus_plus_rat(A,one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),K)),divide_divide_rat(plus_plus_rat(A,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K)))) ).
% gbinomial_rec
tff(fact_8798_gbinomial__factors,axiom,
! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)) ).
% gbinomial_factors
tff(fact_8799_gbinomial__factors,axiom,
! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K)))),aa_nat_complex(gbinomial_complex(A),K)) ).
% gbinomial_factors
tff(fact_8800_gbinomial__factors,axiom,
! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(plus_plus_rat(A,one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(times_times_rat(divide_divide_rat(plus_plus_rat(A,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K)))),aa_nat_rat(gbinomial_rat(A),K)) ).
% gbinomial_factors
tff(fact_8801_gbinomial__negated__upper,axiom,
! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,K)),A)),one_one_real)),K)) ).
% gbinomial_negated_upper
tff(fact_8802_gbinomial__negated__upper,axiom,
! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(minus_minus_complex(aa_nat_complex(semiri8010041392384452111omplex,K)),A)),one_one_complex)),K)) ).
% gbinomial_negated_upper
tff(fact_8803_gbinomial__negated__upper,axiom,
! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),K)),aa_nat_rat(gbinomial_rat(minus_minus_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,K),A),one_one_rat)),K)) ).
% gbinomial_negated_upper
tff(fact_8804_gbinomial__index__swap,axiom,
! [K: nat,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),one_one_real)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),N)) ).
% gbinomial_index_swap
tff(fact_8805_gbinomial__index__swap,axiom,
! [K: nat,N: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),one_one_complex)),K)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),N)) ).
% gbinomial_index_swap
tff(fact_8806_gbinomial__index__swap,axiom,
! [K: nat,N: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),K)),aa_nat_rat(gbinomial_rat(minus_minus_rat(uminus_uminus_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),one_one_rat)),K)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N)),aa_nat_rat(gbinomial_rat(minus_minus_rat(uminus_uminus_rat(aa_nat_rat(semiri681578069525770553at_rat,K)),one_one_rat)),N)) ).
% gbinomial_index_swap
tff(fact_8807_pochhammer__absorb__comp,axiom,
! [R: code_integer,K: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(R,aa_nat_Code_integer(semiri4939895301339042750nteger,K))),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R)),K)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R),one_one_Code_integer)),K)) ).
% pochhammer_absorb_comp
tff(fact_8808_pochhammer__absorb__comp,axiom,
! [R: int,K: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,R),aa_nat_int(semiri1314217659103216013at_int,K))),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,R)),K)) = aa_int_int(aa_int_fun_int_int(times_times_int,R),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,R)),one_one_int)),K)) ).
% pochhammer_absorb_comp
tff(fact_8809_pochhammer__absorb__comp,axiom,
! [R: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,R)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,R)),one_one_real)),K)) ).
% pochhammer_absorb_comp
tff(fact_8810_pochhammer__absorb__comp,axiom,
! [R: complex,K: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(R),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,R)),K)) = aa_complex_complex(times_times_complex(R),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,R)),one_one_complex)),K)) ).
% pochhammer_absorb_comp
tff(fact_8811_pochhammer__absorb__comp,axiom,
! [R: rat,K: nat] : aa_rat_rat(times_times_rat(minus_minus_rat(R,aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(R)),K)) = aa_rat_rat(times_times_rat(R),aa_nat_rat(comm_s4028243227959126397er_rat(plus_plus_rat(uminus_uminus_rat(R),one_one_rat)),K)) ).
% pochhammer_absorb_comp
tff(fact_8812_floor__divide__upper,axiom,
! [Q2: real,P2: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Q2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,P2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P2),Q2)))),one_one_real)),Q2))) ) ).
% floor_divide_upper
tff(fact_8813_floor__divide__upper,axiom,
! [Q2: rat,P2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),Q2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,P2),aa_rat_rat(times_times_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,archim3151403230148437115or_rat(divide_divide_rat(P2,Q2))),one_one_rat)),Q2))) ) ).
% floor_divide_upper
tff(fact_8814_pochhammer__same,axiom,
! [N: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N))),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_nat_Code_integer(semiri3624122377584611663nteger,N)) ).
% pochhammer_same
tff(fact_8815_pochhammer__same,axiom,
! [N: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_nat_int(semiri1406184849735516958ct_int,N)) ).
% pochhammer_same
tff(fact_8816_pochhammer__same,axiom,
! [N: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,N))),N) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),aa_nat_complex(semiri5044797733671781792omplex,N)) ).
% pochhammer_same
tff(fact_8817_pochhammer__same,axiom,
! [N: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(aa_nat_rat(semiri681578069525770553at_rat,N))),N) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N)),aa_nat_rat(semiri773545260158071498ct_rat,N)) ).
% pochhammer_same
tff(fact_8818_pochhammer__same,axiom,
! [N: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),N) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N)),aa_nat_real(semiri2265585572941072030t_real,N)) ).
% pochhammer_same
tff(fact_8819_round__def,axiom,
! [X: real] : aa_real_int(archim8280529875227126926d_real,X) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))) ).
% round_def
tff(fact_8820_round__def,axiom,
! [X: rat] : archim7778729529865785530nd_rat(X) = archim3151403230148437115or_rat(plus_plus_rat(X,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))) ).
% round_def
tff(fact_8821_gbinomial__minus,axiom,
! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(uminus_uminus_real,A)),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)) ).
% gbinomial_minus
tff(fact_8822_gbinomial__minus,axiom,
! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)) ).
% gbinomial_minus
tff(fact_8823_gbinomial__minus,axiom,
! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(uminus_uminus_rat(A)),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),K)),aa_nat_rat(gbinomial_rat(minus_minus_rat(plus_plus_rat(A,aa_nat_rat(semiri681578069525770553at_rat,K)),one_one_rat)),K)) ).
% gbinomial_minus
tff(fact_8824_gbinomial__reduce__nat,axiom,
! [K: nat,A: complex] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> ( aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ) ) ).
% gbinomial_reduce_nat
tff(fact_8825_gbinomial__reduce__nat,axiom,
! [K: nat,A: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K)) ) ) ).
% gbinomial_reduce_nat
tff(fact_8826_gbinomial__reduce__nat,axiom,
! [K: nat,A: rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> ( aa_nat_rat(gbinomial_rat(A),K) = plus_plus_rat(aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat)),aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),K)) ) ) ).
% gbinomial_reduce_nat
tff(fact_8827_pochhammer__minus_H,axiom,
! [B: code_integer,K: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(minus_8373710615458151222nteger(B,aa_nat_Code_integer(semiri4939895301339042750nteger,K)),one_one_Code_integer)),K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K)) ).
% pochhammer_minus'
tff(fact_8828_pochhammer__minus_H,axiom,
! [B: int,K: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_nat_int(semiri1314217659103216013at_int,K))),one_one_int)),K) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),K)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B)),K)) ).
% pochhammer_minus'
tff(fact_8829_pochhammer__minus_H,axiom,
! [B: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,B)),K)) ).
% pochhammer_minus'
tff(fact_8830_pochhammer__minus_H,axiom,
! [B: complex,K: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(B),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,B)),K)) ).
% pochhammer_minus'
tff(fact_8831_pochhammer__minus_H,axiom,
! [B: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(plus_plus_rat(minus_minus_rat(B,aa_nat_rat(semiri681578069525770553at_rat,K)),one_one_rat)),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(B)),K)) ).
% pochhammer_minus'
tff(fact_8832_pochhammer__minus,axiom,
! [B: code_integer,K: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K)),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(minus_8373710615458151222nteger(B,aa_nat_Code_integer(semiri4939895301339042750nteger,K)),one_one_Code_integer)),K)) ).
% pochhammer_minus
tff(fact_8833_pochhammer__minus,axiom,
! [B: int,K: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B)),K) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),K)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_nat_int(semiri1314217659103216013at_int,K))),one_one_int)),K)) ).
% pochhammer_minus
tff(fact_8834_pochhammer__minus,axiom,
! [B: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,B)),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)) ).
% pochhammer_minus
tff(fact_8835_pochhammer__minus,axiom,
! [B: complex,K: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,B)),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(B),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)) ).
% pochhammer_minus
tff(fact_8836_pochhammer__minus,axiom,
! [B: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(uminus_uminus_rat(B)),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(plus_plus_rat(minus_minus_rat(B,aa_nat_rat(semiri681578069525770553at_rat,K)),one_one_rat)),K)) ).
% pochhammer_minus
tff(fact_8837_gbinomial__sum__up__index,axiom,
! [K: nat,N: nat] : groups2073611262835488442omplex(aTP_Lamm_ml(K),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,N)),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).
% gbinomial_sum_up_index
tff(fact_8838_gbinomial__sum__up__index,axiom,
! [K: nat,N: nat] : groups2906978787729119204at_rat(aTP_Lamm_mm(K),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_rat(gbinomial_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,N),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).
% gbinomial_sum_up_index
tff(fact_8839_gbinomial__sum__up__index,axiom,
! [K: nat,N: nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_mn,K),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N)) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).
% gbinomial_sum_up_index
tff(fact_8840_floor__log__eq__powr__iff,axiom,
! [X: real,B: real,K: int] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
=> ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) = K )
<=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_int_real(ring_1_of_int_real,K))),X))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int))))) ) ) ) ) ).
% floor_log_eq_powr_iff
tff(fact_8841_gbinomial__absorption_H,axiom,
! [K: nat,A: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).
% gbinomial_absorption'
tff(fact_8842_gbinomial__absorption_H,axiom,
! [K: nat,A: complex] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> ( aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).
% gbinomial_absorption'
tff(fact_8843_gbinomial__absorption_H,axiom,
! [K: nat,A: rat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(times_times_rat(divide_divide_rat(A,aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(gbinomial_rat(minus_minus_rat(A,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).
% gbinomial_absorption'
tff(fact_8844_floor__log2__div2,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_int) ) ) ).
% floor_log2_div2
tff(fact_8845_fact__double,axiom,
! [N: nat] : aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one)))),N))),aa_nat_complex(semiri5044797733671781792omplex,N)) ).
% fact_double
tff(fact_8846_fact__double,axiom,
! [N: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(numeral_numeral_rat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),aa_nat_rat(comm_s4028243227959126397er_rat(divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one)))),N))),aa_nat_rat(semiri773545260158071498ct_rat,N)) ).
% fact_double
tff(fact_8847_fact__double,axiom,
! [N: nat] : aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),N))),aa_nat_real(semiri2265585572941072030t_real,N)) ).
% fact_double
tff(fact_8848_floor__log__nat__eq__if,axiom,
! [B: nat,N: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))
=> ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,N) ) ) ) ) ).
% floor_log_nat_eq_if
tff(fact_8849_gbinomial__code,axiom,
! [K: nat,A: complex] :
( ( ( K = zero_zero_nat )
=> ( aa_nat_complex(gbinomial_complex(A),K) = one_one_complex ) )
& ( ( K != zero_zero_nat )
=> ( aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_mo(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat)),one_one_complex)),aa_nat_complex(semiri5044797733671781792omplex,K)) ) ) ) ).
% gbinomial_code
tff(fact_8850_gbinomial__code,axiom,
! [K: nat,A: rat] :
( ( ( K = zero_zero_nat )
=> ( aa_nat_rat(gbinomial_rat(A),K) = one_one_rat ) )
& ( ( K != zero_zero_nat )
=> ( aa_nat_rat(gbinomial_rat(A),K) = divide_divide_rat(set_fo1949268297981939178at_rat(aTP_Lamm_mp(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat),one_one_rat),aa_nat_rat(semiri773545260158071498ct_rat,K)) ) ) ) ).
% gbinomial_code
tff(fact_8851_gbinomial__code,axiom,
! [K: nat,A: real] :
( ( ( K = zero_zero_nat )
=> ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = one_one_real ) )
& ( ( K != zero_zero_nat )
=> ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_mq(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat)),one_one_real)),aa_nat_real(semiri2265585572941072030t_real,K)) ) ) ) ).
% gbinomial_code
tff(fact_8852_pochhammer__times__pochhammer__half,axiom,
! [Z: real,N: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(suc,N))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))),aa_nat_nat(suc,N))) = groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_mr,Z),aa_nat_set_nat(set_or1269000886237332187st_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_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat))) ).
% pochhammer_times_pochhammer_half
tff(fact_8853_pochhammer__times__pochhammer__half,axiom,
! [Z: complex,N: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(suc,N))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one))))),aa_nat_nat(suc,N))) = groups6464643781859351333omplex(aTP_Lamm_ms(Z),aa_nat_set_nat(set_or1269000886237332187st_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_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat))) ).
% pochhammer_times_pochhammer_half
tff(fact_8854_pochhammer__times__pochhammer__half,axiom,
! [Z: rat,N: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,N))),aa_nat_rat(comm_s4028243227959126397er_rat(plus_plus_rat(Z,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),aa_nat_nat(suc,N))) = groups73079841787564623at_rat(aTP_Lamm_mt(Z),aa_nat_set_nat(set_or1269000886237332187st_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_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat))) ).
% pochhammer_times_pochhammer_half
tff(fact_8855_pochhammer__code,axiom,
! [N: nat,A: int] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_int(comm_s4660882817536571857er_int(A),N) = one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_int(comm_s4660882817536571857er_int(A),N) = aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_mu(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_int) ) ) ) ).
% pochhammer_code
tff(fact_8856_pochhammer__code,axiom,
! [N: nat,A: real] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),N) = one_one_real ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),N) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_mv(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_real) ) ) ) ).
% pochhammer_code
tff(fact_8857_pochhammer__code,axiom,
! [N: nat,A: complex] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) = one_one_complex ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_complex(comm_s2602460028002588243omplex(A),N) = aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_mw(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_complex) ) ) ) ).
% pochhammer_code
tff(fact_8858_pochhammer__code,axiom,
! [N: nat,A: rat] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = one_one_rat ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_rat(comm_s4028243227959126397er_rat(A),N) = set_fo1949268297981939178at_rat(aTP_Lamm_mx(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),one_one_rat) ) ) ) ).
% pochhammer_code
tff(fact_8859_pochhammer__code,axiom,
! [N: nat,A: nat] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N) = one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),N) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_my(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),one_one_nat) ) ) ) ).
% pochhammer_code
tff(fact_8860_gbinomial__partial__row__sum,axiom,
! [A: complex,M: nat] : groups2073611262835488442omplex(aTP_Lamm_mi(A),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),one_one_complex)),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),one_one_nat))) ).
% gbinomial_partial_row_sum
tff(fact_8861_gbinomial__partial__row__sum,axiom,
! [A: rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_mj(A),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_rat_rat(times_times_rat(divide_divide_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,M),one_one_rat),numeral_numeral_rat(aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),one_one_nat))) ).
% gbinomial_partial_row_sum
tff(fact_8862_gbinomial__partial__row__sum,axiom,
! [A: real,M: nat] : groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_mk,A),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),one_one_real)),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),one_one_nat))) ).
% gbinomial_partial_row_sum
tff(fact_8863_atMost__iff,axiom,
! [I: real,K: real] :
( pp(aa_set_real_bool(member_real(I),set_ord_atMost_real(K)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,I),K)) ) ).
% atMost_iff
tff(fact_8864_atMost__iff,axiom,
! [I: set_nat,K: set_nat] :
( member_set_nat(I,set_or4236626031148496127et_nat(K))
<=> pp(aa_set_nat_bool(ord_less_eq_set_nat(I),K)) ) ).
% atMost_iff
tff(fact_8865_atMost__iff,axiom,
! [I: rat,K: rat] :
( pp(aa_set_rat_bool(member_rat(I),set_ord_atMost_rat(K)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,I),K)) ) ).
% atMost_iff
tff(fact_8866_atMost__iff,axiom,
! [I: num,K: num] :
( pp(aa_set_num_bool(member_num(I),set_ord_atMost_num(K)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,I),K)) ) ).
% atMost_iff
tff(fact_8867_atMost__iff,axiom,
! [I: int,K: int] :
( pp(aa_set_int_bool(member_int(I),set_ord_atMost_int(K)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),K)) ) ).
% atMost_iff
tff(fact_8868_atMost__iff,axiom,
! [I: nat,K: nat] :
( pp(aa_set_nat_bool(member_nat(I),aa_nat_set_nat(set_ord_atMost_nat,K)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K)) ) ).
% atMost_iff
tff(fact_8869_prod_Oneutral__const,axiom,
! [A2: set_nat] : groups708209901874060359at_nat(aTP_Lamm_mz,A2) = one_one_nat ).
% prod.neutral_const
tff(fact_8870_prod_Oneutral__const,axiom,
! [A2: set_nat] : groups705719431365010083at_int(aTP_Lamm_na,A2) = one_one_int ).
% prod.neutral_const
tff(fact_8871_prod_Oneutral__const,axiom,
! [A2: set_int] : groups1705073143266064639nt_int(aTP_Lamm_nb,A2) = one_one_int ).
% prod.neutral_const
tff(fact_8872_prod_Oempty,axiom,
! [G: fun_nat_complex] : groups6464643781859351333omplex(G,bot_bot_set_nat) = one_one_complex ).
% prod.empty
tff(fact_8873_prod_Oempty,axiom,
! [G: fun_nat_real] : groups129246275422532515t_real(G,bot_bot_set_nat) = one_one_real ).
% prod.empty
tff(fact_8874_prod_Oempty,axiom,
! [G: fun_nat_rat] : groups73079841787564623at_rat(G,bot_bot_set_nat) = one_one_rat ).
% prod.empty
tff(fact_8875_prod_Oempty,axiom,
! [G: fun_int_complex] : groups7440179247065528705omplex(G,bot_bot_set_int) = one_one_complex ).
% prod.empty
tff(fact_8876_prod_Oempty,axiom,
! [G: fun_int_real] : groups2316167850115554303t_real(G,bot_bot_set_int) = one_one_real ).
% prod.empty
tff(fact_8877_prod_Oempty,axiom,
! [G: fun_int_rat] : groups1072433553688619179nt_rat(G,bot_bot_set_int) = one_one_rat ).
% prod.empty
tff(fact_8878_prod_Oempty,axiom,
! [G: fun_int_nat] : groups1707563613775114915nt_nat(G,bot_bot_set_int) = one_one_nat ).
% prod.empty
tff(fact_8879_prod_Oempty,axiom,
! [G: fun_real_complex] : groups713298508707869441omplex(G,bot_bot_set_real) = one_one_complex ).
% prod.empty
tff(fact_8880_prod_Oempty,axiom,
! [G: fun_real_real] : groups1681761925125756287l_real(G,bot_bot_set_real) = one_one_real ).
% prod.empty
tff(fact_8881_prod_Oempty,axiom,
! [G: fun_real_rat] : groups4061424788464935467al_rat(G,bot_bot_set_real) = one_one_rat ).
% prod.empty
tff(fact_8882_prod_Oinfinite,axiom,
! [A2: set_nat,G: fun_nat_complex] :
( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ( groups6464643781859351333omplex(G,A2) = one_one_complex ) ) ).
% prod.infinite
tff(fact_8883_prod_Oinfinite,axiom,
! [A2: set_int,G: fun_int_complex] :
( ~ pp(aa_set_int_bool(finite_finite_int,A2))
=> ( groups7440179247065528705omplex(G,A2) = one_one_complex ) ) ).
% prod.infinite
tff(fact_8884_prod_Oinfinite,axiom,
! [A2: set_complex,G: fun_complex_complex] :
( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( groups3708469109370488835omplex(G,A2) = one_one_complex ) ) ).
% prod.infinite
tff(fact_8885_prod_Oinfinite,axiom,
! [A2: set_nat,G: fun_nat_real] :
( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ( groups129246275422532515t_real(G,A2) = one_one_real ) ) ).
% prod.infinite
tff(fact_8886_prod_Oinfinite,axiom,
! [A2: set_int,G: fun_int_real] :
( ~ pp(aa_set_int_bool(finite_finite_int,A2))
=> ( groups2316167850115554303t_real(G,A2) = one_one_real ) ) ).
% prod.infinite
tff(fact_8887_prod_Oinfinite,axiom,
! [A2: set_complex,G: fun_complex_real] :
( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( groups766887009212190081x_real(G,A2) = one_one_real ) ) ).
% prod.infinite
tff(fact_8888_prod_Oinfinite,axiom,
! [A2: set_nat,G: fun_nat_rat] :
( ~ pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ( groups73079841787564623at_rat(G,A2) = one_one_rat ) ) ).
% prod.infinite
tff(fact_8889_prod_Oinfinite,axiom,
! [A2: set_int,G: fun_int_rat] :
( ~ pp(aa_set_int_bool(finite_finite_int,A2))
=> ( groups1072433553688619179nt_rat(G,A2) = one_one_rat ) ) ).
% prod.infinite
tff(fact_8890_prod_Oinfinite,axiom,
! [A2: set_complex,G: fun_complex_rat] :
( ~ pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( groups225925009352817453ex_rat(G,A2) = one_one_rat ) ) ).
% prod.infinite
tff(fact_8891_prod_Oinfinite,axiom,
! [A2: set_int,G: fun_int_nat] :
( ~ pp(aa_set_int_bool(finite_finite_int,A2))
=> ( groups1707563613775114915nt_nat(G,A2) = one_one_nat ) ) ).
% prod.infinite
tff(fact_8892_atMost__subset__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ord_le6893508408891458716et_nat(set_or4236626031148496127et_nat(X),set_or4236626031148496127et_nat(Y))
<=> pp(aa_set_nat_bool(ord_less_eq_set_nat(X),Y)) ) ).
% atMost_subset_iff
tff(fact_8893_atMost__subset__iff,axiom,
! [X: rat,Y: rat] :
( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_ord_atMost_rat(X)),set_ord_atMost_rat(Y)))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y)) ) ).
% atMost_subset_iff
tff(fact_8894_atMost__subset__iff,axiom,
! [X: num,Y: num] :
( pp(aa_set_num_bool(ord_less_eq_set_num(set_ord_atMost_num(X)),set_ord_atMost_num(Y)))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) ) ).
% atMost_subset_iff
tff(fact_8895_atMost__subset__iff,axiom,
! [X: int,Y: int] :
( pp(aa_set_int_bool(ord_less_eq_set_int(set_ord_atMost_int(X)),set_ord_atMost_int(Y)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ).
% atMost_subset_iff
tff(fact_8896_atMost__subset__iff,axiom,
! [X: nat,Y: nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(set_ord_atMost_nat,X)),aa_nat_set_nat(set_ord_atMost_nat,Y)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ).
% atMost_subset_iff
tff(fact_8897_prod_Odelta_H,axiom,
! [S2: set_real,A: real,B: fun_real_complex] :
( pp(aa_set_real_bool(finite_finite_real,S2))
=> ( ( pp(aa_set_real_bool(member_real(A),S2))
=> ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_nc(A),B),S2) = aa_real_complex(B,A) ) )
& ( ~ pp(aa_set_real_bool(member_real(A),S2))
=> ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_nc(A),B),S2) = one_one_complex ) ) ) ) ).
% prod.delta'
tff(fact_8898_prod_Odelta_H,axiom,
! [S2: set_nat,A: nat,B: fun_nat_complex] :
( pp(aa_set_nat_bool(finite_finite_nat,S2))
=> ( ( pp(aa_set_nat_bool(member_nat(A),S2))
=> ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_nd(A),B),S2) = aa_nat_complex(B,A) ) )
& ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
=> ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_nd(A),B),S2) = one_one_complex ) ) ) ) ).
% prod.delta'
tff(fact_8899_prod_Odelta_H,axiom,
! [S2: set_int,A: int,B: fun_int_complex] :
( pp(aa_set_int_bool(finite_finite_int,S2))
=> ( ( pp(aa_set_int_bool(member_int(A),S2))
=> ( groups7440179247065528705omplex(aa_fun3088484340408535532omplex(aTP_Lamm_ne(A),B),S2) = aa_int_complex(B,A) ) )
& ( ~ pp(aa_set_int_bool(member_int(A),S2))
=> ( groups7440179247065528705omplex(aa_fun3088484340408535532omplex(aTP_Lamm_ne(A),B),S2) = one_one_complex ) ) ) ) ).
% prod.delta'
tff(fact_8900_prod_Odelta_H,axiom,
! [S2: set_complex,A: complex,B: fun_complex_complex] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
=> ( ( pp(aa_set_complex_bool(member_complex(A),S2))
=> ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_nf(A),B),S2) = aa_complex_complex(B,A) ) )
& ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
=> ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_nf(A),B),S2) = one_one_complex ) ) ) ) ).
% prod.delta'
tff(fact_8901_prod_Odelta_H,axiom,
! [S2: set_real,A: real,B: fun_real_real] :
( pp(aa_set_real_bool(finite_finite_real,S2))
=> ( ( pp(aa_set_real_bool(member_real(A),S2))
=> ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_ng(A),B),S2) = aa_real_real(B,A) ) )
& ( ~ pp(aa_set_real_bool(member_real(A),S2))
=> ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_ng(A),B),S2) = one_one_real ) ) ) ) ).
% prod.delta'
tff(fact_8902_prod_Odelta_H,axiom,
! [S2: set_nat,A: nat,B: fun_nat_real] :
( pp(aa_set_nat_bool(finite_finite_nat,S2))
=> ( ( pp(aa_set_nat_bool(member_nat(A),S2))
=> ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_nh(A),B),S2) = aa_nat_real(B,A) ) )
& ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
=> ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_nh(A),B),S2) = one_one_real ) ) ) ) ).
% prod.delta'
tff(fact_8903_prod_Odelta_H,axiom,
! [S2: set_int,A: int,B: fun_int_real] :
( pp(aa_set_int_bool(finite_finite_int,S2))
=> ( ( pp(aa_set_int_bool(member_int(A),S2))
=> ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_ni(A),B),S2) = aa_int_real(B,A) ) )
& ( ~ pp(aa_set_int_bool(member_int(A),S2))
=> ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_ni(A),B),S2) = one_one_real ) ) ) ) ).
% prod.delta'
tff(fact_8904_prod_Odelta_H,axiom,
! [S2: set_complex,A: complex,B: fun_complex_real] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
=> ( ( pp(aa_set_complex_bool(member_complex(A),S2))
=> ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_nj(A),B),S2) = aa_complex_real(B,A) ) )
& ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
=> ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_nj(A),B),S2) = one_one_real ) ) ) ) ).
% prod.delta'
tff(fact_8905_prod_Odelta_H,axiom,
! [S2: set_real,A: real,B: fun_real_rat] :
( pp(aa_set_real_bool(finite_finite_real,S2))
=> ( ( pp(aa_set_real_bool(member_real(A),S2))
=> ( groups4061424788464935467al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_nk(A),B),S2) = aa_real_rat(B,A) ) )
& ( ~ pp(aa_set_real_bool(member_real(A),S2))
=> ( groups4061424788464935467al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_nk(A),B),S2) = one_one_rat ) ) ) ) ).
% prod.delta'
tff(fact_8906_prod_Odelta_H,axiom,
! [S2: set_nat,A: nat,B: fun_nat_rat] :
( pp(aa_set_nat_bool(finite_finite_nat,S2))
=> ( ( pp(aa_set_nat_bool(member_nat(A),S2))
=> ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_nl(A),B),S2) = aa_nat_rat(B,A) ) )
& ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
=> ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_nl(A),B),S2) = one_one_rat ) ) ) ) ).
% prod.delta'
tff(fact_8907_prod_Odelta,axiom,
! [S2: set_real,A: real,B: fun_real_complex] :
( pp(aa_set_real_bool(finite_finite_real,S2))
=> ( ( pp(aa_set_real_bool(member_real(A),S2))
=> ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_nm(A),B),S2) = aa_real_complex(B,A) ) )
& ( ~ pp(aa_set_real_bool(member_real(A),S2))
=> ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_nm(A),B),S2) = one_one_complex ) ) ) ) ).
% prod.delta
tff(fact_8908_prod_Odelta,axiom,
! [S2: set_nat,A: nat,B: fun_nat_complex] :
( pp(aa_set_nat_bool(finite_finite_nat,S2))
=> ( ( pp(aa_set_nat_bool(member_nat(A),S2))
=> ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_nn(A),B),S2) = aa_nat_complex(B,A) ) )
& ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
=> ( groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_nn(A),B),S2) = one_one_complex ) ) ) ) ).
% prod.delta
tff(fact_8909_prod_Odelta,axiom,
! [S2: set_int,A: int,B: fun_int_complex] :
( pp(aa_set_int_bool(finite_finite_int,S2))
=> ( ( pp(aa_set_int_bool(member_int(A),S2))
=> ( groups7440179247065528705omplex(aa_fun3088484340408535532omplex(aTP_Lamm_no(A),B),S2) = aa_int_complex(B,A) ) )
& ( ~ pp(aa_set_int_bool(member_int(A),S2))
=> ( groups7440179247065528705omplex(aa_fun3088484340408535532omplex(aTP_Lamm_no(A),B),S2) = one_one_complex ) ) ) ) ).
% prod.delta
tff(fact_8910_prod_Odelta,axiom,
! [S2: set_complex,A: complex,B: fun_complex_complex] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
=> ( ( pp(aa_set_complex_bool(member_complex(A),S2))
=> ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_np(A),B),S2) = aa_complex_complex(B,A) ) )
& ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
=> ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_np(A),B),S2) = one_one_complex ) ) ) ) ).
% prod.delta
tff(fact_8911_prod_Odelta,axiom,
! [S2: set_real,A: real,B: fun_real_real] :
( pp(aa_set_real_bool(finite_finite_real,S2))
=> ( ( pp(aa_set_real_bool(member_real(A),S2))
=> ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_nq(A),B),S2) = aa_real_real(B,A) ) )
& ( ~ pp(aa_set_real_bool(member_real(A),S2))
=> ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_nq(A),B),S2) = one_one_real ) ) ) ) ).
% prod.delta
tff(fact_8912_prod_Odelta,axiom,
! [S2: set_nat,A: nat,B: fun_nat_real] :
( pp(aa_set_nat_bool(finite_finite_nat,S2))
=> ( ( pp(aa_set_nat_bool(member_nat(A),S2))
=> ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_nr(A),B),S2) = aa_nat_real(B,A) ) )
& ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
=> ( groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_nr(A),B),S2) = one_one_real ) ) ) ) ).
% prod.delta
tff(fact_8913_prod_Odelta,axiom,
! [S2: set_int,A: int,B: fun_int_real] :
( pp(aa_set_int_bool(finite_finite_int,S2))
=> ( ( pp(aa_set_int_bool(member_int(A),S2))
=> ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_ns(A),B),S2) = aa_int_real(B,A) ) )
& ( ~ pp(aa_set_int_bool(member_int(A),S2))
=> ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_ns(A),B),S2) = one_one_real ) ) ) ) ).
% prod.delta
tff(fact_8914_prod_Odelta,axiom,
! [S2: set_complex,A: complex,B: fun_complex_real] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,S2))
=> ( ( pp(aa_set_complex_bool(member_complex(A),S2))
=> ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_nt(A),B),S2) = aa_complex_real(B,A) ) )
& ( ~ pp(aa_set_complex_bool(member_complex(A),S2))
=> ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_nt(A),B),S2) = one_one_real ) ) ) ) ).
% prod.delta
tff(fact_8915_prod_Odelta,axiom,
! [S2: set_real,A: real,B: fun_real_rat] :
( pp(aa_set_real_bool(finite_finite_real,S2))
=> ( ( pp(aa_set_real_bool(member_real(A),S2))
=> ( groups4061424788464935467al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_nu(A),B),S2) = aa_real_rat(B,A) ) )
& ( ~ pp(aa_set_real_bool(member_real(A),S2))
=> ( groups4061424788464935467al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_nu(A),B),S2) = one_one_rat ) ) ) ) ).
% prod.delta
tff(fact_8916_prod_Odelta,axiom,
! [S2: set_nat,A: nat,B: fun_nat_rat] :
( pp(aa_set_nat_bool(finite_finite_nat,S2))
=> ( ( pp(aa_set_nat_bool(member_nat(A),S2))
=> ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_nv(A),B),S2) = aa_nat_rat(B,A) ) )
& ( ~ pp(aa_set_nat_bool(member_nat(A),S2))
=> ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_nv(A),B),S2) = one_one_rat ) ) ) ) ).
% prod.delta
tff(fact_8917_prod_Oinsert,axiom,
! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: fun_VEBT_VEBT_real] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
=> ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),A2))
=> ( groups2703838992350267259T_real(G,insert_VEBT_VEBT(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(G,X)),groups2703838992350267259T_real(G,A2)) ) ) ) ).
% prod.insert
tff(fact_8918_prod_Oinsert,axiom,
! [A2: set_real,X: real,G: fun_real_real] :
( pp(aa_set_real_bool(finite_finite_real,A2))
=> ( ~ pp(aa_set_real_bool(member_real(X),A2))
=> ( groups1681761925125756287l_real(G,insert_real(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(G,X)),groups1681761925125756287l_real(G,A2)) ) ) ) ).
% prod.insert
tff(fact_8919_prod_Oinsert,axiom,
! [A2: set_nat,X: nat,G: fun_nat_real] :
( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
=> ( groups129246275422532515t_real(G,insert_nat(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,X)),groups129246275422532515t_real(G,A2)) ) ) ) ).
% prod.insert
tff(fact_8920_prod_Oinsert,axiom,
! [A2: set_int,X: int,G: fun_int_real] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( ~ pp(aa_set_int_bool(member_int(X),A2))
=> ( groups2316167850115554303t_real(G,insert_int(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(G,X)),groups2316167850115554303t_real(G,A2)) ) ) ) ).
% prod.insert
tff(fact_8921_prod_Oinsert,axiom,
! [A2: set_complex,X: complex,G: fun_complex_real] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( ~ pp(aa_set_complex_bool(member_complex(X),A2))
=> ( groups766887009212190081x_real(G,insert_complex(X,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(G,X)),groups766887009212190081x_real(G,A2)) ) ) ) ).
% prod.insert
tff(fact_8922_prod_Oinsert,axiom,
! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: fun_VEBT_VEBT_rat] :
( pp(aa_set8260762808118238015T_bool(finite5795047828879050333T_VEBT,A2))
=> ( ~ pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X),A2))
=> ( groups5726676334696518183BT_rat(G,insert_VEBT_VEBT(X,A2)) = aa_rat_rat(times_times_rat(aa_VEBT_VEBT_rat(G,X)),groups5726676334696518183BT_rat(G,A2)) ) ) ) ).
% prod.insert
tff(fact_8923_prod_Oinsert,axiom,
! [A2: set_real,X: real,G: fun_real_rat] :
( pp(aa_set_real_bool(finite_finite_real,A2))
=> ( ~ pp(aa_set_real_bool(member_real(X),A2))
=> ( groups4061424788464935467al_rat(G,insert_real(X,A2)) = aa_rat_rat(times_times_rat(aa_real_rat(G,X)),groups4061424788464935467al_rat(G,A2)) ) ) ) ).
% prod.insert
tff(fact_8924_prod_Oinsert,axiom,
! [A2: set_nat,X: nat,G: fun_nat_rat] :
( pp(aa_set_nat_bool(finite_finite_nat,A2))
=> ( ~ pp(aa_set_nat_bool(member_nat(X),A2))
=> ( groups73079841787564623at_rat(G,insert_nat(X,A2)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,X)),groups73079841787564623at_rat(G,A2)) ) ) ) ).
% prod.insert
tff(fact_8925_prod_Oinsert,axiom,
! [A2: set_int,X: int,G: fun_int_rat] :
( pp(aa_set_int_bool(finite_finite_int,A2))
=> ( ~ pp(aa_set_int_bool(member_int(X),A2))
=> ( groups1072433553688619179nt_rat(G,insert_int(X,A2)) = aa_rat_rat(times_times_rat(aa_int_rat(G,X)),groups1072433553688619179nt_rat(G,A2)) ) ) ) ).
% prod.insert
tff(fact_8926_prod_Oinsert,axiom,
! [A2: set_complex,X: complex,G: fun_complex_rat] :
( pp(aa_set_complex_bool(finite3207457112153483333omplex,A2))
=> ( ~ pp(aa_set_complex_bool(member_complex(X),A2))
=> ( groups225925009352817453ex_rat(G,insert_complex(X,A2)) = aa_rat_rat(times_times_rat(aa_complex_rat(G,X)),groups225925009352817453ex_rat(G,A2)) ) ) ) ).
% prod.insert
tff(fact_8927_Icc__subset__Iic__iff,axiom,
! [L: set_nat,H: set_nat,H2: set_nat] :
( ord_le6893508408891458716et_nat(set_or4548717258645045905et_nat(L,H),set_or4236626031148496127et_nat(H2))
<=> ( ~ pp(aa_set_nat_bool(ord_less_eq_set_nat(L),H))
| pp(aa_set_nat_bool(ord_less_eq_set_nat(H),H2)) ) ) ).
% Icc_subset_Iic_iff
tff(fact_8928_Icc__subset__Iic__iff,axiom,
! [L: rat,H: rat,H2: rat] :
( pp(aa_set_rat_bool(ord_less_eq_set_rat(set_or633870826150836451st_rat(L,H)),set_ord_atMost_rat(H2)))
<=> ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,L),H))
| pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,H),H2)) ) ) ).
% Icc_subset_Iic_iff
tff(fact_8929_Icc__subset__Iic__iff,axiom,
! [L: num,H: num,H2: num] :
( pp(aa_set_num_bool(ord_less_eq_set_num(set_or7049704709247886629st_num(L,H)),set_ord_atMost_num(H2)))
<=> ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,L),H))
| pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,H),H2)) ) ) ).
% Icc_subset_Iic_iff
tff(fact_8930_Icc__subset__Iic__iff,axiom,
! [L: nat,H: nat,H2: nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(L),H)),aa_nat_set_nat(set_ord_atMost_nat,H2)))
<=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),H))
| pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,H),H2)) ) ) ).
% Icc_subset_Iic_iff
tff(fact_8931_Icc__subset__Iic__iff,axiom,
! [L: int,H: int,H2: int] :
( pp(aa_set_int_bool(ord_less_eq_set_int(set_or1266510415728281911st_int(L,H)),set_ord_atMost_int(H2)))
<=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,L),H))
| pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,H),H2)) ) ) ).
% Icc_subset_Iic_iff
tff(fact_8932_Icc__subset__Iic__iff,axiom,
! [L: real,H: real,H2: real] :
( pp(aa_set_real_bool(ord_less_eq_set_real(set_or1222579329274155063t_real(L,H)),set_ord_atMost_real(H2)))
<=> ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),H))
| pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,H),H2)) ) ) ).
% Icc_subset_Iic_iff
tff(fact_8933_sum_OatMost__Suc,axiom,
! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,N)),aa_nat_rat(G,aa_nat_nat(suc,N))) ).
% sum.atMost_Suc
tff(fact_8934_sum_OatMost__Suc,axiom,
! [G: fun_nat_int,N: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_int(G,aa_nat_nat(suc,N))) ).
% sum.atMost_Suc
tff(fact_8935_sum_OatMost__Suc,axiom,
! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ).
% sum.atMost_Suc
tff(fact_8936_sum_OatMost__Suc,axiom,
! [G: fun_nat_real,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_real(G,aa_nat_nat(suc,N))) ).
% sum.atMost_Suc
tff(fact_8937_prod_OlessThan__Suc,axiom,
! [G: fun_nat_real,N: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_real(G,N)) ).
% prod.lessThan_Suc
tff(fact_8938_prod_OlessThan__Suc,axiom,
! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_rat(G,N)) ).
% prod.lessThan_Suc
tff(fact_8939_prod_OlessThan__Suc,axiom,
! [G: fun_nat_nat,N: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_nat(G,N)) ).
% prod.lessThan_Suc
tff(fact_8940_prod_OlessThan__Suc,axiom,
! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_nat_int(G,N)) ).
% prod.lessThan_Suc
tff(fact_8941_prod_OatMost__Suc,axiom,
! [G: fun_nat_real,N: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_real(G,aa_nat_nat(suc,N))) ).
% prod.atMost_Suc
tff(fact_8942_prod_OatMost__Suc,axiom,
! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_rat(G,aa_nat_nat(suc,N))) ).
% prod.atMost_Suc
tff(fact_8943_prod_OatMost__Suc,axiom,
! [G: fun_nat_nat,N: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ).
% prod.atMost_Suc
tff(fact_8944_prod_OatMost__Suc,axiom,
! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,N))),aa_nat_int(G,aa_nat_nat(suc,N))) ).
% prod.atMost_Suc
tff(fact_8945_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: fun_nat_complex] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
=> ( groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = one_one_complex ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
=> ( groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_complex(G,aa_nat_nat(suc,N))) ) ) ) ).
% prod.cl_ivl_Suc
tff(fact_8946_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: fun_nat_real] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
=> ( groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = one_one_real ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
=> ( groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_real(G,aa_nat_nat(suc,N))) ) ) ) ).
% prod.cl_ivl_Suc
tff(fact_8947_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: fun_nat_rat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
=> ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = one_one_rat ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
=> ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_rat(G,aa_nat_nat(suc,N))) ) ) ) ).
% prod.cl_ivl_Suc
tff(fact_8948_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: fun_nat_nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
=> ( groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = one_one_nat ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
=> ( groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ) ) ) ).
% prod.cl_ivl_Suc
tff(fact_8949_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: fun_nat_int] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
=> ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = one_one_int ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M))
=> ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(M),N))),aa_nat_int(G,aa_nat_nat(suc,N))) ) ) ) ).
% prod.cl_ivl_Suc
tff(fact_8950_prod_Odistrib,axiom,
! [G: fun_nat_nat,H: fun_nat_nat,A2: set_nat] : groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_nw(G),H),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,A2)),groups708209901874060359at_nat(H,A2)) ).
% prod.distrib
tff(fact_8951_prod_Odistrib,axiom,
! [G: fun_nat_int,H: fun_nat_int,A2: set_nat] : groups705719431365010083at_int(aa_fun5565043122120104496at_int(aTP_Lamm_nx(G),H),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,A2)),groups705719431365010083at_int(H,A2)) ).
% prod.distrib
tff(fact_8952_prod_Odistrib,axiom,
! [G: fun_int_int,H: fun_int_int,A2: set_int] : groups1705073143266064639nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ny(G),H),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,groups1705073143266064639nt_int(G,A2)),groups1705073143266064639nt_int(H,A2)) ).
% prod.distrib
tff(fact_8953_prod_Oneutral,axiom,
! [A2: set_nat,G: fun_nat_nat] :
( ! [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),A2))
=> ( aa_nat_nat(G,X3) = one_one_nat ) )
=> ( groups708209901874060359at_nat(G,A2) = one_one_nat ) ) ).
% prod.neutral
tff(fact_8954_prod_Oneutral,axiom,
! [A2: set_nat,G: fun_nat_int] :
( ! [X3: nat] :
( pp(aa_set_nat_bool(member_nat(X3),A2))
=> ( aa_nat_int(G,X3) = one_one_int ) )
=> ( groups705719431365010083at_int(G,A2) = one_one_int ) ) ).
% prod.neutral
tff(fact_8955_prod_Oneutral,axiom,
! [A2: set_int,G: fun_int_int] :
( ! [X3: int] :
( pp(aa_set_int_bool(member_int(X3),A2))
=> ( aa_int_int(G,X3) = one_one_int ) )
=> ( groups1705073143266064639nt_int(G,A2) = one_one_int ) ) ).
% prod.neutral
tff(fact_8956_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: fun_complex_complex,A2: set_complex] :
( ( groups3708469109370488835omplex(G,A2) != one_one_complex )
=> ~ ! [A4: complex] :
( pp(aa_set_complex_bool(member_complex(A4),A2))
=> ( aa_complex_complex(G,A4) = one_one_complex ) ) ) ).
% prod.not_neutral_contains_not_neutral
tff(fact_8957_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: fun_real_complex,A2: set_real] :
( ( groups713298508707869441omplex(G,A2) != one_one_complex )
=> ~ ! [A4: real] :
( pp(aa_set_real_bool(member_real(A4),A2))
=> ( aa_real_complex(G,A4) = one_one_complex ) ) ) ).
% prod.not_neutral_contains_not_neutral
tff(fact_8958_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: fun_nat_complex,A2: set_nat] :
( ( groups6464643781859351333omplex(G,A2) != one_one_complex )
=> ~ ! [A4: nat] :
( pp(aa_set_nat_bool(member_nat(A4),A2))
=> ( aa_nat_complex(G,A4) = one_one_complex ) ) ) ).
% prod.not_neutral_contains_not_neutral
tff(fact_8959_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: fun_int_complex,A2: set_int] :
( ( groups7440179247065528705omplex(G,A2) != one_one_complex )
=> ~ ! [A4: int] :
( pp(aa_set_int_bool(member_int(A4),A2))
=> ( aa_int_complex(G,A4) = one_one_complex ) ) ) ).
% prod.not_neutral_contains_not_neutral
tff(fact_8960_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: fun_complex_real,A2: set_complex] :
( ( groups766887009212190081x_real(G,A2) != one_one_real )
=> ~ ! [A4: complex] :
( pp(aa_set_complex_bool(member_complex(A4),A2))
=> ( aa_complex_real(G,A4) = one_one_real ) ) ) ).
% prod.not_neutral_contains_not_neutral
tff(fact_8961_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: fun_real_real,A2: set_real] :
( ( groups1681761925125756287l_real(G,A2) != one_one_real )
=> ~ ! [A4: real] :
( pp(aa_set_real_bool(member_real(A4),A2))
=> ( aa_real_real(G,A4) = one_one_real ) ) ) ).
% prod.not_neutral_contains_not_neutral
tff(fact_8962_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: fun_nat_real,A2: set_nat] :
( ( groups129246275422532515t_real(G,A2) != one_one_real )
=> ~ ! [A4: nat] :
( pp(aa_set_nat_bool(member_nat(A4),A2))
=> ( aa_nat_real(G,A4) = one_one_real ) ) ) ).
% prod.not_neutral_contains_not_neutral
tff(fact_8963_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: fun_int_real,A2: set_int] :
( ( groups2316167850115554303t_real(G,A2) != one_one_real )
=> ~ ! [A4: int] :
( pp(aa_set_int_bool(member_int(A4),A2))
=> ( aa_int_real(G,A4) = one_one_real ) ) ) ).
% prod.not_neutral_contains_not_neutral
tff(fact_8964_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: fun_complex_rat,A2: set_complex] :
( ( groups225925009352817453ex_rat(G,A2) != one_one_rat )
=> ~ ! [A4: complex] :
( pp(aa_set_complex_bool(member_complex(A4),A2))
=> ( aa_complex_rat(G,A4) = one_one_rat ) ) ) ).
% prod.not_neutral_contains_not_neutral
tff(fact_8965_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: fun_real_rat,A2: set_real] :
( ( groups4061424788464935467al_rat(G,A2) != one_one_rat )
=> ~ ! [A4: real] :
( pp(aa_set_real_bool(member_real(A4),A2))
=> ( aa_real_rat(G,A4) = one_one_rat ) ) ) ).
% prod.not_neutral_contains_not_neutral
tff(fact_8966_prod__power__distrib,axiom,
! [F: fun_nat_nat,A2: set_nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,groups708209901874060359at_nat(F,A2)),N) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_nz(F),N),A2) ).
% prod_power_distrib
tff(fact_8967_prod__power__distrib,axiom,
! [F: fun_nat_int,A2: set_nat,N: nat] : aa_nat_int(power_power_int(groups705719431365010083at_int(F,A2)),N) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_oa(F),N),A2) ).
% prod_power_distrib
tff(fact_8968_prod__power__distrib,axiom,
! [F: fun_int_int,A2: set_int,N: nat] : aa_nat_int(power_power_int(groups1705073143266064639nt_int(F,A2)),N) = groups1705073143266064639nt_int(aa_nat_fun_int_int(aTP_Lamm_ob(F),N),A2) ).
% prod_power_distrib
tff(fact_8969_mod__prod__eq,axiom,
! [F: fun_nat_nat,A: nat,A2: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_fj(F),A),A2)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,groups708209901874060359at_nat(F,A2)),A) ).
% mod_prod_eq
tff(fact_8970_mod__prod__eq,axiom,
! [F: fun_nat_int,A: int,A2: set_nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_oc(F),A),A2)),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups705719431365010083at_int(F,A2)),A) ).
% mod_prod_eq
tff(fact_8971_mod__prod__eq,axiom,
! [F: fun_int_int,A: int,A2: set_int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups1705073143266064639nt_int(aa_int_fun_int_int(aTP_Lamm_fi(F),A),A2)),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups1705073143266064639nt_int(F,A2)),A) ).
% mod_prod_eq
tff(fact_8972_prod_OatMost__Suc__shift,axiom,
! [G: fun_nat_real,N: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,zero_zero_nat)),groups129246275422532515t_real(aTP_Lamm_gq(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).
% prod.atMost_Suc_shift
tff(fact_8973_prod_OatMost__Suc__shift,axiom,
! [G: fun_nat_rat,N: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_gz(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).
% prod.atMost_Suc_shift
tff(fact_8974_prod_OatMost__Suc__shift,axiom,
! [G: fun_nat_nat,N: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,zero_zero_nat)),groups708209901874060359at_nat(aTP_Lamm_gp(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).
% prod.atMost_Suc_shift
tff(fact_8975_prod_OatMost__Suc__shift,axiom,
! [G: fun_nat_int,N: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_ha(G),aa_nat_set_nat(set_ord_atMost_nat,N))) ).
% prod.atMost_Suc_shift
tff(fact_8976_prod_Onested__swap_H,axiom,
! [A: fun_nat_fun_nat_nat,N: nat] : groups708209901874060359at_nat(aTP_Lamm_od(A),aa_nat_set_nat(set_ord_atMost_nat,N)) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_of(A),N),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).
% prod.nested_swap'
tff(fact_8977_prod_Onested__swap_H,axiom,
! [A: fun_nat_fun_nat_int,N: nat] : groups705719431365010083at_int(aTP_Lamm_og(A),aa_nat_set_nat(set_ord_atMost_nat,N)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_oi(A),N),aa_nat_set_nat(set_ord_lessThan_nat,N)) ).
% prod.nested_swap'
tff(fact_8978_atMost__def,axiom,
! [U: real] : set_ord_atMost_real(U) = collect_real(aa_rea7809428808904475482l_bool(aTP_Lamm_oj,U)) ).
% atMost_def
tff(fact_8979_atMost__def,axiom,
! [U: set_nat] : set_or4236626031148496127et_nat(U) = collect_set_nat(aTP_Lamm_ok(U)) ).
% atMost_def
tff(fact_8980_atMost__def,axiom,
! [U: rat] : set_ord_atMost_rat(U) = collect_rat(aa_rat_fun_rat_bool(aTP_Lamm_ol,U)) ).
% atMost_def
tff(fact_8981_atMost__def,axiom,
! [U: num] : set_ord_atMost_num(U) = collect_num(aa_num_fun_num_bool(aTP_Lamm_om,U)) ).
% atMost_def
tff(fact_8982_atMost__def,axiom,
! [U: int] : set_ord_atMost_int(U) = collect_int(aa_int_fun_int_bool(aTP_Lamm_on,U)) ).
% atMost_def
tff(fact_8983_atMost__def,axiom,
! [U: nat] : aa_nat_set_nat(set_ord_atMost_nat,U) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bg,U)) ).
% atMost_def
tff(fact_8984_prod__mono,axiom,
! [A2: set_complex,F: fun_complex_int,G: fun_complex_int] :
( ! [I2: complex] :
( pp(aa_set_complex_bool(member_complex(I2),A2))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_complex_int(F,I2)))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_complex_int(F,I2)),aa_complex_int(G,I2))) ) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups858564598930262913ex_int(F,A2)),groups858564598930262913ex_int(G,A2))) ) ).
% prod_mono
tff(fact_8985_prod__mono,axiom,
! [A2: set_real,F: fun_real_int,G: fun_real_int] :
( ! [I2: real] :
( pp(aa_set_real_bool(member_real(I2),A2))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_real_int(F,I2)))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_real_int(F,I2)),aa_real_int(G,I2))) ) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups4694064378042380927al_int(F,A2)),groups4694064378042380927al_int(G,A2))) ) ).
% prod_mono
tff(fact_8986_prod__mono,axiom,
! [A2: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
( ! [I2: nat] :
( pp(aa_set_nat_bool(member_nat(I2),A2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,I2)))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F,I2)),aa_nat_nat(G,I2))) ) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,groups708209901874060359at_nat(F,A2)),groups708209901874060359at_nat(G,A2))) ) ).
% prod_mono
tff(fact_8987_prod__mono,axiom,
! [A2: set_nat,F: fun_nat_int,G: fun_nat_int] :
( ! [I2: nat] :
( pp(aa_set_nat_bool(member_nat(I2),A2))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_nat_int(F,I2)))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F,I2)),aa_nat_int(G,I2))) ) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups705719431365010083at_int(F,A2)),groups705719431365010083at_int(G,A2))) ) ).
% prod_mono
tff(fact_8988_prod__mono,axiom,
! [A2: set_int,F: fun_int_int,G: fun_int_int] :
( ! [I2: int] :
( pp(aa_set_int_bool(member_int(I2),A2))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(F,I2)))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(F,I2)),aa_int_int(G,I2))) ) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,groups1705073143266064639nt_int(F,A2)),groups1705073143266064639nt_int(G,A2))) ) ).
% prod_mono
tff(fact_8989_lessThan__Suc__atMost,axiom,
! [K: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K)) = aa_nat_set_nat(set_ord_atMost_nat,K) ).
% lessThan_Suc_atMost
tff(fact_8990_atMost__Suc,axiom,
! [K: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K)) = insert_nat(aa_nat_nat(suc,K),aa_nat_set_nat(set_ord_atMost_nat,K)) ).
% atMost_Suc
tff(fact_8991_atMost__nat__numeral,axiom,
! [K: num] : aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(numeral_numeral_nat,K),aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(pred_numeral,K))) ).
% atMost_nat_numeral
tff(fact_8992_atLeast1__atMost__eq__remove0,axiom,
! [N: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),N) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atMost_nat,N),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).
% atLeast1_atMost_eq_remove0
tff(fact_8993_fact__eq__fact__times,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),groups708209901874060359at_nat(aTP_Lamm_be,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,N)),M))) ) ) ).
% fact_eq_fact_times
tff(fact_8994_fact__div__fact,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N)) = groups708209901874060359at_nat(aTP_Lamm_be,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M)) ) ) ).
% fact_div_fact
tff(fact_8995_polynomial__product__nat,axiom,
! [M: nat,A: fun_nat_nat,N: nat,B: fun_nat_nat,X: nat] :
( ! [I2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),I2))
=> ( aa_nat_nat(A,I2) = zero_zero_nat ) )
=> ( ! [J2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),J2))
=> ( aa_nat_nat(B,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_oo(A),X)),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_oo(B),X)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_oq(A),B),X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ) ).
% polynomial_product_nat
tff(fact_8996_central__binomial__lower__bound,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))),N)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N)))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),N)))) ) ).
% central_binomial_lower_bound
tff(fact_8997_binomial__Suc__n,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),N) = aa_nat_nat(suc,N) ).
% binomial_Suc_n
tff(fact_8998_binomial__n__n,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),N) = one_one_nat ).
% binomial_n_n
tff(fact_8999_binomial__1,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,zero_zero_nat)) = N ).
% binomial_1
tff(fact_9000_binomial__0__Suc,axiom,
! [K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,zero_zero_nat),aa_nat_nat(suc,K)) = zero_zero_nat ).
% binomial_0_Suc
tff(fact_9001_binomial__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = zero_zero_nat )
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K)) ) ).
% binomial_eq_0_iff
tff(fact_9002_binomial__Suc__Suc,axiom,
! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K))) ).
% binomial_Suc_Suc
tff(fact_9003_binomial__n__0,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),zero_zero_nat) = one_one_nat ).
% binomial_n_0
tff(fact_9004_zero__less__binomial__iff,axiom,
! [N: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N)) ) ).
% zero_less_binomial_iff
tff(fact_9005_choose__one,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),one_one_nat) = N ).
% choose_one
tff(fact_9006_binomial__eq__0,axiom,
! [N: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = zero_zero_nat ) ) ).
% binomial_eq_0
tff(fact_9007_Suc__times__binomial,axiom,
! [K: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) ).
% Suc_times_binomial
tff(fact_9008_Suc__times__binomial__eq,axiom,
! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,K))),aa_nat_nat(suc,K)) ).
% Suc_times_binomial_eq
tff(fact_9009_binomial__symmetric,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).
% binomial_symmetric
tff(fact_9010_choose__mult__lemma,axiom,
! [M: nat,R: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_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(aa_nat_fun_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(aa_nat_fun_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(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R)),M)) ).
% choose_mult_lemma
tff(fact_9011_binomial__le__pow,axiom,
! [R: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),R)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),R))) ) ).
% binomial_le_pow
tff(fact_9012_zero__less__binomial,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) ) ).
% zero_less_binomial
tff(fact_9013_Suc__times__binomial__add,axiom,
! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,A)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),aa_nat_nat(suc,A))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,B)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),A)) ).
% Suc_times_binomial_add
tff(fact_9014_binomial__Suc__Suc__eq__times,axiom,
! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))),aa_nat_nat(suc,K)) ).
% binomial_Suc_Suc_eq_times
tff(fact_9015_choose__mult,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),M)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,M),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ) ).
% choose_mult
tff(fact_9016_binomial__absorb__comp,axiom,
! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ).
% binomial_absorb_comp
tff(fact_9017_sum__choose__upper,axiom,
! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_or,M)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,M)) ).
% sum_choose_upper
tff(fact_9018_binomial__absorption,axiom,
! [K: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ).
% binomial_absorption
tff(fact_9019_binomial__fact__lemma,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( 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(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ) ) ).
% binomial_fact_lemma
tff(fact_9020_prod__int__plus__eq,axiom,
! [I: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J))) = groups1705073143266064639nt_int(aTP_Lamm_bf,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_9021_sum__choose__lower,axiom,
! [R: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_os,R)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,R),N))),N) ).
% sum_choose_lower
tff(fact_9022_choose__rising__sum_I2_J,axiom,
! [N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ot,N)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_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,N),M)),one_one_nat)),M) ).
% choose_rising_sum(2)
tff(fact_9023_choose__rising__sum_I1_J,axiom,
! [N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ot,N)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_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,N),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)) ).
% choose_rising_sum(1)
tff(fact_9024_binomial__mono,axiom,
! [K: nat,K6: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),K6))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K6)),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K6))) ) ) ).
% binomial_mono
tff(fact_9025_binomial__maximum_H,axiom,
! [N: nat,K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),N))) ).
% binomial_maximum'
tff(fact_9026_binomial__maximum,axiom,
! [N: nat,K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).
% binomial_maximum
tff(fact_9027_binomial__antimono,axiom,
! [K: nat,K6: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),K6))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),K))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K6),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) ) ) ) ).
% binomial_antimono
tff(fact_9028_binomial__le__pow2,axiom,
! [N: nat,K: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).
% binomial_le_pow2
tff(fact_9029_choose__reduce__nat,axiom,
! [N: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ) ) ) ).
% choose_reduce_nat
tff(fact_9030_times__binomial__minus1__eq,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).
% times_binomial_minus1_eq
tff(fact_9031_binomial__altdef__nat,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),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(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))) ) ) ).
% binomial_altdef_nat
tff(fact_9032_sum__choose__diagonal,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ou(M),N)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),M) ) ) ).
% sum_choose_diagonal
tff(fact_9033_vandermonde,axiom,
! [M: nat,N: nat,R: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ov(M),N),R)),aa_nat_set_nat(set_ord_atMost_nat,R)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),R) ).
% vandermonde
tff(fact_9034_binomial__less__binomial__Suc,axiom,
! [K: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K)))) ) ).
% binomial_less_binomial_Suc
tff(fact_9035_binomial__strict__mono,axiom,
! [K: nat,K6: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),K6))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K6)),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K6))) ) ) ).
% binomial_strict_mono
tff(fact_9036_binomial__strict__antimono,axiom,
! [K: nat,K6: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),K6))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K6),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))) ) ) ) ).
% binomial_strict_antimono
tff(fact_9037_central__binomial__odd,axiom,
! [N: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).
% central_binomial_odd
tff(fact_9038_binomial__addition__formula,axiom,
! [N: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(suc,K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ) ) ).
% binomial_addition_formula
tff(fact_9039_choose__row__sum,axiom,
! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).
% choose_row_sum
tff(fact_9040_binomial,axiom,
! [A: nat,B: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),N) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ow(A),B),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).
% binomial
tff(fact_9041_choose__two,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% choose_two
tff(fact_9042_choose__square__sum,axiom,
! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ox,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),N) ).
% choose_square_sum
tff(fact_9043_binomial__r__part__sum,axiom,
! [M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),one_one_nat))),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).
% binomial_r_part_sum
tff(fact_9044_choose__linear__sum,axiom,
! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_oy,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ).
% choose_linear_sum
tff(fact_9045_binomial__code,axiom,
! [N: nat,K: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = zero_zero_nat ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),one_one_nat)),N),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K)) ) ) ) ) ) ).
% binomial_code
tff(fact_9046_Maclaurin__sin__bound,axiom,
! [X: real,N: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,X)),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_lf,X),aa_nat_set_nat(set_ord_lessThan_nat,N))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,X)),N)))) ).
% Maclaurin_sin_bound
tff(fact_9047_of__nat__id,axiom,
! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,N) = N ).
% of_nat_id
tff(fact_9048_real__sqrt__inverse,axiom,
! [X: real] : aa_real_real(sqrt,aa_real_real(inverse_inverse_real,X)) = aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X)) ).
% real_sqrt_inverse
tff(fact_9049_real__scaleR__def,axiom,
! [A: real,X: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),X) ).
% real_scaleR_def
tff(fact_9050_divide__real__def,axiom,
! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),aa_real_real(inverse_inverse_real,Y)) ).
% divide_real_def
tff(fact_9051_complex__scaleR,axiom,
! [R: real,A: real,B: real] : aa_complex_complex(real_V2046097035970521341omplex(R),aa_real_complex(complex2(A),B)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),B)) ).
% complex_scaleR
tff(fact_9052_inverse__powr,axiom,
! [Y: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
=> ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(inverse_inverse_real,Y)),A) = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ).
% inverse_powr
tff(fact_9053_forall__pos__mono__1,axiom,
! [P: fun_real_bool,E: real] :
( ! [D2: real,E2: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D2),E2))
=> ( pp(aa_real_bool(P,D2))
=> pp(aa_real_bool(P,E2)) ) )
=> ( ! [N2: nat] : pp(aa_real_bool(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N2)))))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
=> pp(aa_real_bool(P,E)) ) ) ) ).
% forall_pos_mono_1
tff(fact_9054_forall__pos__mono,axiom,
! [P: fun_real_bool,E: real] :
( ! [D2: real,E2: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,D2),E2))
=> ( pp(aa_real_bool(P,D2))
=> pp(aa_real_bool(P,E2)) ) )
=> ( ! [N2: nat] :
( ( N2 != zero_zero_nat )
=> pp(aa_real_bool(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N2)))) )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
=> pp(aa_real_bool(P,E)) ) ) ) ).
% forall_pos_mono
tff(fact_9055_real__arch__inverse,axiom,
! [E: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E))
<=> ? [N5: nat] :
( ( N5 != zero_zero_nat )
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N5))))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N5))),E)) ) ) ).
% real_arch_inverse
tff(fact_9056_sqrt__divide__self__eq,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( aa_real_real(aa_rea7340605660300375871l_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_9057_ln__inverse,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( aa_real_real(ln_ln_real,aa_real_real(inverse_inverse_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(ln_ln_real,X)) ) ) ).
% ln_inverse
tff(fact_9058_log__inverse,axiom,
! [A: real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ( ( A != one_one_real )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(inverse_inverse_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),X)) ) ) ) ) ).
% log_inverse
tff(fact_9059_exp__plus__inverse__exp,axiom,
! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(aa_num_num(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_9060_plus__inverse__ge__2,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,numeral_numeral_real(aa_num_num(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_9061_real__inv__sqrt__pow2,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(inverse_inverse_real,X) ) ) ).
% real_inv_sqrt_pow2
tff(fact_9062_tan__cot,axiom,
! [X: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))),X)) = aa_real_real(inverse_inverse_real,aa_real_real(tan_real,X)) ).
% tan_cot
tff(fact_9063_real__le__x__sinh,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_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(aa_num_num(bit0,one))))) ) ).
% real_le_x_sinh
tff(fact_9064_real__le__abs__sinh,axiom,
! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_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(aa_num_num(bit0,one)))))) ).
% real_le_abs_sinh
tff(fact_9065_exp__two__pi__i,axiom,
aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_real_complex(real_V4546457046886955230omplex,pi))),imaginary_unit)) = one_one_complex ).
% exp_two_pi_i
tff(fact_9066_exp__two__pi__i_H,axiom,
aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) = one_one_complex ).
% exp_two_pi_i'
tff(fact_9067_sinh__real__less__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sinh_real,X)),aa_real_real(sinh_real,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ).
% sinh_real_less_iff
tff(fact_9068_sinh__real__le__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sinh_real,X)),aa_real_real(sinh_real,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ).
% sinh_real_le_iff
tff(fact_9069_sinh__real__neg__iff,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sinh_real,X)),zero_zero_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ).
% sinh_real_neg_iff
tff(fact_9070_sinh__real__pos__iff,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(sinh_real,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X)) ) ).
% sinh_real_pos_iff
tff(fact_9071_sinh__real__nonpos__iff,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sinh_real,X)),zero_zero_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).
% sinh_real_nonpos_iff
tff(fact_9072_sinh__real__nonneg__iff,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sinh_real,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).
% sinh_real_nonneg_iff
tff(fact_9073_norm__ii,axiom,
aa_complex_real(real_V1022390504157884413omplex,imaginary_unit) = one_one_real ).
% norm_ii
tff(fact_9074_divide__i,axiom,
! [X: complex] : aa_complex_complex(divide1717551699836669952omplex(X),imaginary_unit) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit)),X) ).
% divide_i
tff(fact_9075_divide__numeral__i,axiom,
! [Z: complex,N: num] : aa_complex_complex(divide1717551699836669952omplex(Z),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(N)),imaginary_unit)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(imaginary_unit),Z))),numera6690914467698888265omplex(N)) ).
% divide_numeral_i
tff(fact_9076_power2__i,axiom,
aa_nat_complex(power_power_complex(imaginary_unit),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).
% power2_i
tff(fact_9077_i__even__power,axiom,
! [N: nat] : aa_nat_complex(power_power_complex(imaginary_unit),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N) ).
% i_even_power
tff(fact_9078_divide__complex__def,axiom,
! [X: complex,Y: complex] : aa_complex_complex(divide1717551699836669952omplex(X),Y) = aa_complex_complex(times_times_complex(X),aa_complex_complex(invers8013647133539491842omplex,Y)) ).
% divide_complex_def
tff(fact_9079_sinh__less__cosh__real,axiom,
! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(sinh_real,X)),aa_real_real(cosh_real,X))) ).
% sinh_less_cosh_real
tff(fact_9080_sinh__le__cosh__real,axiom,
! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sinh_real,X)),aa_real_real(cosh_real,X))) ).
% sinh_le_cosh_real
tff(fact_9081_cosh__real__pos,axiom,
! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cosh_real,X))) ).
% cosh_real_pos
tff(fact_9082_cosh__real__nonpos__le__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),X)) ) ) ) ).
% cosh_real_nonpos_le_iff
tff(fact_9083_cosh__real__nonneg__le__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ) ).
% cosh_real_nonneg_le_iff
tff(fact_9084_cosh__real__nonneg,axiom,
! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(cosh_real,X))) ).
% cosh_real_nonneg
tff(fact_9085_cosh__real__ge__1,axiom,
! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(cosh_real,X))) ).
% cosh_real_ge_1
tff(fact_9086_cosh__real__nonpos__less__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y),zero_zero_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y),X)) ) ) ) ).
% cosh_real_nonpos_less_iff
tff(fact_9087_cosh__real__nonneg__less__iff,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ) ).
% cosh_real_nonneg_less_iff
tff(fact_9088_cosh__real__strict__mono,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y))) ) ) ).
% cosh_real_strict_mono
tff(fact_9089_arcosh__cosh__real,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_9090_Complex__eq__i,axiom,
! [X: real,Y: real] :
( ( aa_real_complex(complex2(X),Y) = imaginary_unit )
<=> ( ( X = zero_zero_real )
& ( Y = one_one_real ) ) ) ).
% Complex_eq_i
tff(fact_9091_imaginary__unit_Ocode,axiom,
imaginary_unit = aa_real_complex(complex2(zero_zero_real),one_one_real) ).
% imaginary_unit.code
tff(fact_9092_complex__inverse,axiom,
! [A: real,B: real] : aa_complex_complex(invers8013647133539491842omplex,aa_real_complex(complex2(A),B)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).
% complex_inverse
tff(fact_9093_cmod__unit__one,axiom,
! [A: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A))))) = one_one_real ).
% cmod_unit_one
tff(fact_9094_cosh__ln__real,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( aa_real_real(cosh_real,aa_real_real(ln_ln_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(inverse_inverse_real,X))),numeral_numeral_real(aa_num_num(bit0,one))) ) ) ).
% cosh_ln_real
tff(fact_9095_sinh__ln__real,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( aa_real_real(sinh_real,aa_real_real(ln_ln_real,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),aa_real_real(inverse_inverse_real,X))),numeral_numeral_real(aa_num_num(bit0,one))) ) ) ).
% sinh_ln_real
tff(fact_9096_Arg__minus__ii,axiom,
aa_complex_real(arg,aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(aa_num_num(bit0,one))) ).
% Arg_minus_ii
tff(fact_9097_csqrt__ii,axiom,
aa_complex_complex(csqrt,imaginary_unit) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(one_one_complex),imaginary_unit)),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one))))) ).
% csqrt_ii
tff(fact_9098_Arg__ii,axiom,
aa_complex_real(arg,imaginary_unit) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))) ).
% Arg_ii
tff(fact_9099_cis__minus__pi__half,axiom,
aa_real_complex(cis,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))) = aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit) ).
% cis_minus_pi_half
tff(fact_9100_norm__cis,axiom,
! [A: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(cis,A)) = one_one_real ).
% norm_cis
tff(fact_9101_power2__csqrt,axiom,
! [Z: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(csqrt,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Z ).
% power2_csqrt
tff(fact_9102_cis__pi__half,axiom,
aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))) = imaginary_unit ).
% cis_pi_half
tff(fact_9103_cis__2pi,axiom,
aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)) = one_one_complex ).
% cis_2pi
tff(fact_9104_cis__divide,axiom,
! [A: real,B: real] : aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(cis,A)),aa_real_complex(cis,B)) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ).
% cis_divide
tff(fact_9105_cis__mult,axiom,
! [A: real,B: real] : aa_complex_complex(times_times_complex(aa_real_complex(cis,A)),aa_real_complex(cis,B)) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ).
% cis_mult
tff(fact_9106_DeMoivre,axiom,
! [A: real,N: nat] : aa_nat_complex(power_power_complex(aa_real_complex(cis,A)),N) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),A)) ).
% DeMoivre
tff(fact_9107_of__real__sqrt,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,X)) = aa_complex_complex(csqrt,aa_real_complex(real_V4546457046886955230omplex,X)) ) ) ).
% of_real_sqrt
tff(fact_9108_Arg__bounded,axiom,
! [Z: complex] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z)))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(arg,Z)),pi)) ) ).
% Arg_bounded
tff(fact_9109_bij__betw__roots__unity,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_set_complex_bool(bij_betw_nat_complex(aTP_Lamm_oz(N),aa_nat_set_nat(set_ord_lessThan_nat,N)),collect_complex(aTP_Lamm_bj(N)))) ) ).
% bij_betw_roots_unity
tff(fact_9110_cot__less__zero,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(aa_num_num(bit0,one)))),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(cot_real,X)),zero_zero_real)) ) ) ).
% cot_less_zero
tff(fact_9111_cot__periodic,axiom,
! [X: real] : aa_real_real(cot_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))) = aa_real_real(cot_real,X) ).
% cot_periodic
tff(fact_9112_arctan__def,axiom,
! [Y: real] : aa_real_real(arctan,Y) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_pa,Y)) ).
% arctan_def
tff(fact_9113_cot__npi,axiom,
! [N: nat] : aa_real_real(cot_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),pi)) = zero_zero_real ).
% cot_npi
tff(fact_9114_ln__neg__is__const,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
=> ( aa_real_real(ln_ln_real,X) = the_real(aTP_Lamm_pb) ) ) ).
% ln_neg_is_const
tff(fact_9115_arccos__def,axiom,
! [Y: real] : aa_real_real(arccos,Y) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_pc,Y)) ).
% arccos_def
tff(fact_9116_pi__half,axiom,
aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))) = the_real(aTP_Lamm_pd) ).
% pi_half
tff(fact_9117_pi__def,axiom,
pi = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),the_real(aTP_Lamm_pd)) ).
% pi_def
tff(fact_9118_cot__gt__zero,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(cot_real,X))) ) ) ).
% cot_gt_zero
tff(fact_9119_tan__cot_H,axiom,
! [X: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))),X)) = aa_real_real(cot_real,X) ).
% tan_cot'
tff(fact_9120_arcsin__def,axiom,
! [Y: real] : aa_real_real(arcsin,Y) = the_real(aa_rea7809428808904475482l_bool(aTP_Lamm_pe,Y)) ).
% arcsin_def
tff(fact_9121_signed__take__bit__eq__take__bit__minus,axiom,
! [N: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(N),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),K)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N))),aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(bit_se1146084159140164899it_int(K),N)))) ).
% signed_take_bit_eq_take_bit_minus
tff(fact_9122_modulo__int__unfold,axiom,
! [L: int,K: int,N: nat,M: nat] :
( ( ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
| ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
| ( N = zero_zero_nat ) )
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M)) ) )
& ( ~ ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
| ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
| ( N = zero_zero_nat ) )
=> ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))) ) )
& ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))) ) ) ) ) ) ).
% modulo_int_unfold
tff(fact_9123_powr__int,axiom,
! [X: real,I: int] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
=> ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_int_real(ring_1_of_int_real,I)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_int_nat(nat2,I)) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),I))
=> ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_int_real(ring_1_of_int_real,I)) = aa_real_real(aa_rea7340605660300375871l_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_9124_nat__numeral,axiom,
! [K: num] : aa_int_nat(nat2,aa_num_int(numeral_numeral_int,K)) = aa_num_nat(numeral_numeral_nat,K) ).
% nat_numeral
tff(fact_9125_nat__1,axiom,
aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).
% nat_1
tff(fact_9126_zless__nat__conj,axiom,
! [W: int,Z: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z)))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ) ).
% zless_nat_conj
tff(fact_9127_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_9128_signed__take__bit__nonnegative__iff,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri631733984087533419it_int(N),K)))
<=> ~ pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N)) ) ).
% signed_take_bit_nonnegative_iff
tff(fact_9129_signed__take__bit__negative__iff,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),zero_zero_int))
<=> pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N)) ) ).
% signed_take_bit_negative_iff
tff(fact_9130_zero__less__nat__eq,axiom,
! [Z: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(nat2,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).
% zero_less_nat_eq
tff(fact_9131_diff__nat__numeral,axiom,
! [V: num,V3: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_num_nat(numeral_numeral_nat,V3)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V3))) ).
% diff_nat_numeral
tff(fact_9132_bit__minus__numeral__Bit0__Suc__iff,axiom,
! [W: num,N: nat] :
( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))),aa_nat_nat(suc,N)))
<=> pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),N)) ) ).
% bit_minus_numeral_Bit0_Suc_iff
tff(fact_9133_bit__minus__numeral__Bit1__Suc__iff,axiom,
! [W: num,N: nat] :
( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))),aa_nat_nat(suc,N)))
<=> ~ pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),N)) ) ).
% bit_minus_numeral_Bit1_Suc_iff
tff(fact_9134_nat__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( aa_int_nat(nat2,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) )
<=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) ) ) ).
% nat_eq_numeral_power_cancel_iff
tff(fact_9135_numeral__power__eq__nat__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N) = aa_int_nat(nat2,Y) )
<=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N) = Y ) ) ).
% numeral_power_eq_nat_cancel_iff
tff(fact_9136_nat__ceiling__le__eq,axiom,
! [X: real,A: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,X))),A))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,A))) ) ).
% nat_ceiling_le_eq
tff(fact_9137_one__less__nat__eq,axiom,
! [Z: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_int_nat(nat2,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).
% one_less_nat_eq
tff(fact_9138_bit__minus__numeral__int_I1_J,axiom,
! [W: num,N: num] :
( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))),aa_num_nat(numeral_numeral_nat,N)))
<=> pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),aa_num_nat(pred_numeral,N))) ) ).
% bit_minus_numeral_int(1)
tff(fact_9139_bit__minus__numeral__int_I2_J,axiom,
! [W: num,N: num] :
( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))),aa_num_nat(numeral_numeral_nat,N)))
<=> ~ pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),aa_num_nat(pred_numeral,N))) ) ).
% bit_minus_numeral_int(2)
tff(fact_9140_nat__numeral__diff__1,axiom,
! [V: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),one_one_nat) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,V)),one_one_int)) ).
% nat_numeral_diff_1
tff(fact_9141_nat__less__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).
% nat_less_numeral_power_cancel_iff
tff(fact_9142_numeral__power__less__nat__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N)),aa_int_nat(nat2,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).
% numeral_power_less_nat_cancel_iff
tff(fact_9143_nat__le__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N))) ) ).
% nat_le_numeral_power_cancel_iff
tff(fact_9144_numeral__power__le__nat__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),N)),aa_int_nat(nat2,A)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,X)),N)),A)) ) ).
% numeral_power_le_nat_cancel_iff
tff(fact_9145_bit__and__int__iff,axiom,
! [K: int,L: int,N: nat] :
( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)),N))
<=> ( pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N))
& pp(aa_nat_bool(bit_se1146084159140164899it_int(L),N)) ) ) ).
% bit_and_int_iff
tff(fact_9146_nat__numeral__as__int,axiom,
! [X4: num] : aa_num_nat(numeral_numeral_nat,X4) = aa_int_nat(nat2,aa_num_int(numeral_numeral_int,X4)) ).
% nat_numeral_as_int
tff(fact_9147_nat__mono,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y))) ) ).
% nat_mono
tff(fact_9148_nat__one__as__int,axiom,
one_one_nat = aa_int_nat(nat2,one_one_int) ).
% nat_one_as_int
tff(fact_9149_div__eq__sgn__abs,axiom,
! [K: int,L: int] :
( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),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_eq_sgn_abs
tff(fact_9150_unset__bit__nat__def,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),N) = aa_int_nat(nat2,aa_int_int(bit_se4203085406695923979it_int(M),aa_nat_int(semiri1314217659103216013at_int,N))) ).
% unset_bit_nat_def
tff(fact_9151_nat__mask__eq,axiom,
! [N: nat] : aa_int_nat(nat2,aa_nat_int(bit_se2000444600071755411sk_int,N)) = aa_nat_nat(bit_se2002935070580805687sk_nat,N) ).
% nat_mask_eq
tff(fact_9152_nat__mono__iff,axiom,
! [Z: int,W: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ) ).
% nat_mono_iff
tff(fact_9153_zless__nat__eq__int__zless,axiom,
! [M: nat,Z: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_int_nat(nat2,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),Z)) ) ).
% zless_nat_eq_int_zless
tff(fact_9154_nat__le__iff,axiom,
! [X: int,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,X)),N))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).
% nat_le_iff
tff(fact_9155_bit__not__int__iff_H,axiom,
! [K: int,N: nat] :
( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,K)),one_one_int)),N))
<=> ~ pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N)) ) ).
% bit_not_int_iff'
tff(fact_9156_nat__int__add,axiom,
! [A: nat,B: nat] : aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) ).
% nat_int_add
tff(fact_9157_sgn__mod,axiom,
! [L: int,K: int] :
( ( L != zero_zero_int )
=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
=> ( aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)) = aa_int_int(sgn_sgn_int,L) ) ) ) ).
% sgn_mod
tff(fact_9158_int__minus,axiom,
! [N: nat,M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(semiri1314217659103216013at_int,M)))) ).
% int_minus
tff(fact_9159_nat__abs__mult__distrib,axiom,
! [W: int,Z: int] : aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_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_9160_real__nat__ceiling__ge,axiom,
! [X: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,X))))) ).
% real_nat_ceiling_ge
tff(fact_9161_and__nat__def,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) ).
% and_nat_def
tff(fact_9162_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,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).
% nat_plus_as_int
tff(fact_9163_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,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).
% nat_times_as_int
tff(fact_9164_nat__minus__as__int,axiom,
! [X4: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X4),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).
% nat_minus_as_int
tff(fact_9165_nat__div__as__int,axiom,
! [X4: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_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_9166_nat__mod__as__int,axiom,
! [X4: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X4),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).
% nat_mod_as_int
tff(fact_9167_zsgn__def,axiom,
! [I: int] :
( ( ( I = zero_zero_int )
=> ( aa_int_int(sgn_sgn_int,I) = zero_zero_int ) )
& ( ( I != zero_zero_int )
=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),I))
=> ( aa_int_int(sgn_sgn_int,I) = one_one_int ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),I))
=> ( aa_int_int(sgn_sgn_int,I) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ) ).
% zsgn_def
tff(fact_9168_nat__less__eq__zless,axiom,
! [W: int,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ) ).
% nat_less_eq_zless
tff(fact_9169_nat__le__eq__zle,axiom,
! [W: int,Z: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),W))
| pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z)) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,W),Z)) ) ) ).
% nat_le_eq_zle
tff(fact_9170_split__nat,axiom,
! [P: fun_nat_bool,I: int] :
( pp(aa_nat_bool(P,aa_int_nat(nat2,I)))
<=> ( ! [N5: nat] :
( ( I = aa_nat_int(semiri1314217659103216013at_int,N5) )
=> pp(aa_nat_bool(P,N5)) )
& ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),zero_zero_int))
=> pp(aa_nat_bool(P,zero_zero_nat)) ) ) ) ).
% split_nat
tff(fact_9171_le__nat__iff,axiom,
! [K: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_int_nat(nat2,K)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,N)),K)) ) ) ).
% le_nat_iff
tff(fact_9172_nat__add__distrib,axiom,
! [Z: int,Z5: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z5))
=> ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ) ).
% nat_add_distrib
tff(fact_9173_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,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,K))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_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_9174_bit__imp__take__bit__positive,axiom,
! [N: nat,M: nat,K: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
=> ( pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(bit_se2923211474154528505it_int(M),K))) ) ) ).
% bit_imp_take_bit_positive
tff(fact_9175_Suc__as__int,axiom,
! [X4: nat] : aa_nat_nat(suc,X4) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X4)),one_one_int)) ).
% Suc_as_int
tff(fact_9176_nat__mult__distrib,axiom,
! [Z: int,Z5: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
=> ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ).
% nat_mult_distrib
tff(fact_9177_nat__diff__distrib,axiom,
! [Z5: int,Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z5))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z5),Z))
=> ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ) ).
% nat_diff_distrib
tff(fact_9178_nat__diff__distrib_H,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
=> ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ) ).
% nat_diff_distrib'
tff(fact_9179_nat__abs__triangle__ineq,axiom,
! [K: int,L: int] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_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_9180_nat__div__distrib_H,axiom,
! [Y: int,X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(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(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ).
% nat_div_distrib'
tff(fact_9181_nat__div__distrib,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(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(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ).
% nat_div_distrib
tff(fact_9182_div__dvd__sgn__abs,axiom,
! [L: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_int_int(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_9183_bit__concat__bit__iff,axiom,
! [M: nat,K: int,L: int,N: nat] :
( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(bit_concat_bit(M),K),L)),N))
<=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M))
& pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N)) )
| ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
& pp(aa_nat_bool(bit_se1146084159140164899it_int(L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ) ).
% bit_concat_bit_iff
tff(fact_9184_nat__power__eq,axiom,
! [Z: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
=> ( aa_int_nat(nat2,aa_nat_int(power_power_int(Z),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z)),N) ) ) ).
% nat_power_eq
tff(fact_9185_nat__floor__neg,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real))
=> ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = zero_zero_nat ) ) ).
% nat_floor_neg
tff(fact_9186_nat__mod__distrib,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
=> ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y)) ) ) ) ).
% nat_mod_distrib
tff(fact_9187_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(aa_nat_fun_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_9188_floor__eq3,axiom,
! [N: nat,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
=> ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = N ) ) ) ).
% floor_eq3
tff(fact_9189_le__nat__floor,axiom,
! [X: nat,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),A))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A)))) ) ).
% le_nat_floor
tff(fact_9190_mod__abs__eq__div__nat,axiom,
! [K: int,L: int] : aa_int_int(aa_int_fun_int_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(aa_nat_fun_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_9191_take__bit__nat__eq,axiom,
! [K: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(N),K)) ) ) ).
% take_bit_nat_eq
tff(fact_9192_nat__take__bit__eq,axiom,
! [K: int,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
=> ( aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_int_nat(nat2,K)) ) ) ).
% nat_take_bit_eq
tff(fact_9193_divide__int__def,axiom,
! [L: int,K: int] :
( ( ( L = zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) )
& ( ( L != zero_zero_int )
=> ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_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(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),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(aa_nat_fun_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_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K)))))) ) ) ) ) ) ).
% divide_int_def
tff(fact_9194_modulo__int__def,axiom,
! [L: int,K: int] :
( ( ( L = zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = K ) )
& ( ( L != zero_zero_int )
=> ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_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))))) ) )
& ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,L)),aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_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_9195_signed__take__bit__eq__concat__bit,axiom,
! [N: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(N),K) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),aa_int_int(uminus_uminus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(bit_se1146084159140164899it_int(K),N)))) ).
% signed_take_bit_eq_concat_bit
tff(fact_9196_nat__2,axiom,
aa_int_nat(nat2,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat)) ).
% nat_2
tff(fact_9197_int__bit__bound,axiom,
! [K: int] :
~ ! [N2: nat] :
( ! [M2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N2),M2))
=> ( pp(aa_nat_bool(bit_se1146084159140164899it_int(K),M2))
<=> pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N2)) ) )
=> ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N2))
=> ( pp(aa_nat_bool(bit_se1146084159140164899it_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat)))
<=> ~ pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N2)) ) ) ) ).
% int_bit_bound
tff(fact_9198_Suc__nat__eq__nat__zadd1,axiom,
! [Z: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Z))
=> ( aa_nat_nat(suc,aa_int_nat(nat2,Z)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)) ) ) ).
% Suc_nat_eq_nat_zadd1
tff(fact_9199_nat__less__iff,axiom,
! [W: int,M: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W)),M))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),aa_nat_int(semiri1314217659103216013at_int,M))) ) ) ).
% nat_less_iff
tff(fact_9200_nat__mult__distrib__neg,axiom,
! [Z: int,Z5: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),zero_zero_int))
=> ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Z5)) = 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,Z5))) ) ) ).
% nat_mult_distrib_neg
tff(fact_9201_nat__abs__int__diff,axiom,
! [A: nat,B: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),B))
=> ( aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ) ) ).
% nat_abs_int_diff
tff(fact_9202_floor__eq4,axiom,
! [N: nat,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N)),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
=> ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = N ) ) ) ).
% floor_eq4
tff(fact_9203_diff__nat__eq__if,axiom,
! [Z5: int,Z: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z5),zero_zero_int))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) = aa_int_nat(nat2,Z) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z5),zero_zero_int))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z5)),zero_zero_int)),zero_zero_nat),aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z5))) ) ) ) ).
% diff_nat_eq_if
tff(fact_9204_bit__int__def,axiom,
! [K: int,N: nat] :
( pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N))
<=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(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,aa_num_num(bit0,one))),N)))) ) ).
% bit_int_def
tff(fact_9205_eucl__rel__int__remainderI,axiom,
! [R: int,L: int,K: int,Q2: int] :
( ( aa_int_int(sgn_sgn_int,R) = aa_int_int(sgn_sgn_int,L) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,R)),aa_int_int(abs_abs_int,L)))
=> ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q2),L)),R) )
=> pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R))) ) ) ) ).
% eucl_rel_int_remainderI
tff(fact_9206_eucl__rel__int_Osimps,axiom,
! [A1: int,A22: int,A32: product_prod_int_int] :
( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A1,A22),A32))
<=> ( ? [K2: int] :
( ( A1 = K2 )
& ( A22 = zero_zero_int )
& ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),K2) ) )
| ? [L4: int,K2: int,Q4: int] :
( ( A1 = K2 )
& ( A22 = L4 )
& ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),zero_zero_int) )
& ( L4 != zero_zero_int )
& ( K2 = aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L4) ) )
| ? [R5: int,L4: int,K2: int,Q4: int] :
( ( A1 = K2 )
& ( A22 = L4 )
& ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),R5) )
& ( aa_int_int(sgn_sgn_int,R5) = aa_int_int(sgn_sgn_int,L4) )
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,R5)),aa_int_int(abs_abs_int,L4)))
& ( K2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L4)),R5) ) ) ) ) ).
% eucl_rel_int.simps
tff(fact_9207_eucl__rel__int_Ocases,axiom,
! [A1: int,A22: int,A32: product_prod_int_int] :
( pp(aa_Pro4501881432599481050t_bool(eucl_rel_int(A1,A22),A32))
=> ( ( ( A22 = zero_zero_int )
=> ( A32 != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),A1) ) )
=> ( ! [Q3: int] :
( ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),zero_zero_int) )
=> ( ( A22 != zero_zero_int )
=> ( A1 != aa_int_int(aa_int_fun_int_int(times_times_int,Q3),A22) ) ) )
=> ~ ! [R3: int,Q3: int] :
( ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R3) )
=> ( ( aa_int_int(sgn_sgn_int,R3) = aa_int_int(sgn_sgn_int,A22) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,R3)),aa_int_int(abs_abs_int,A22)))
=> ( A1 != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q3),A22)),R3) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
tff(fact_9208_div__noneq__sgn__abs,axiom,
! [L: int,K: int] :
( ( L != zero_zero_int )
=> ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_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_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,L),K)))) ) ) ) ).
% div_noneq_sgn_abs
tff(fact_9209_even__nat__iff,axiom,
! [K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_int_nat(nat2,K)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)) ) ) ).
% even_nat_iff
tff(fact_9210_set__bit__eq,axiom,
! [N: nat,K: int] : aa_int_int(bit_se7879613467334960850it_int(N),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(times_times_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_nat_bool(bit_se1146084159140164899it_int(K),N)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).
% set_bit_eq
tff(fact_9211_unset__bit__eq,axiom,
! [N: nat,K: int] : aa_int_int(bit_se4203085406695923979it_int(N),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_int_int(aa_int_fun_int_int(times_times_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(bit_se1146084159140164899it_int(K),N))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).
% unset_bit_eq
tff(fact_9212_take__bit__Suc__from__most,axiom,
! [N: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),aa_bool_int(zero_n2684676970156552555ol_int,aa_nat_bool(bit_se1146084159140164899it_int(K),N)))),aa_int_int(bit_se2923211474154528505it_int(N),K)) ).
% take_bit_Suc_from_most
tff(fact_9213_powr__real__of__int,axiom,
! [X: real,N: int] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
=> ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_int_real(ring_1_of_int_real,N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_int_nat(nat2,N)) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),N))
=> ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_int_real(ring_1_of_int_real,N)) = 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,N)))) ) ) ) ) ).
% powr_real_of_int
tff(fact_9214_divide__int__unfold,axiom,
! [L: int,K: int,N: nat,M: nat] :
( ( ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
| ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
| ( N = zero_zero_nat ) )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = zero_zero_int ) )
& ( ~ ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
| ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
| ( N = zero_zero_nat ) )
=> ( ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) ) )
& ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = 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(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M)))))) ) ) ) ) ) ).
% divide_int_unfold
tff(fact_9215_bij__betw__nth__root__unity,axiom,
! [C: complex,N: nat] :
( ( C != zero_zero_complex )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_set_complex_bool(bij_be1856998921033663316omplex(times_times_complex(aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_nat_fun_real_real(root,N),aa_complex_real(real_V1022390504157884413omplex,C)))),aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(arg,C)),aa_nat_real(semiri5074537144036343181t_real,N))))),collect_complex(aTP_Lamm_bj(N))),collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_pf(C),N)))) ) ) ).
% bij_betw_nth_root_unity
tff(fact_9216_or__int__unfold,axiom,
! [K: int,L: int] :
( ( ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
| ( L = aa_int_int(uminus_uminus_int,one_one_int) ) )
=> ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L) = aa_int_int(uminus_uminus_int,one_one_int) ) )
& ( ~ ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
| ( L = aa_int_int(uminus_uminus_int,one_one_int) ) )
=> ( ( ( K = zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L) = L ) )
& ( ( K != zero_zero_int )
=> ( ( ( L = zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L) = K ) )
& ( ( L != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ).
% or_int_unfold
tff(fact_9217_zero__le__sgn__iff,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(sgn_sgn_real,X)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X)) ) ).
% zero_le_sgn_iff
tff(fact_9218_sgn__le__0__iff,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(sgn_sgn_real,X)),zero_zero_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ).
% sgn_le_0_iff
tff(fact_9219_real__root__zero,axiom,
! [N: nat] : aa_real_real(aa_nat_fun_real_real(root,N),zero_zero_real) = zero_zero_real ).
% real_root_zero
tff(fact_9220_real__root__Suc__0,axiom,
! [X: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(suc,zero_zero_nat)),X) = X ).
% real_root_Suc_0
tff(fact_9221_real__root__eq__iff,axiom,
! [N: nat,X: real,Y: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( ( aa_real_real(aa_nat_fun_real_real(root,N),X) = aa_real_real(aa_nat_fun_real_real(root,N),Y) )
<=> ( X = Y ) ) ) ).
% real_root_eq_iff
tff(fact_9222_root__0,axiom,
! [X: real] : aa_real_real(aa_nat_fun_real_real(root,zero_zero_nat),X) = zero_zero_real ).
% root_0
tff(fact_9223_or__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L)))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ) ).
% or_nonnegative_int_iff
tff(fact_9224_or__negative__int__iff,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L)),zero_zero_int))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
| pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ).
% or_negative_int_iff
tff(fact_9225_real__root__eq__0__iff,axiom,
! [N: nat,X: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( ( aa_real_real(aa_nat_fun_real_real(root,N),X) = zero_zero_real )
<=> ( X = zero_zero_real ) ) ) ).
% real_root_eq_0_iff
tff(fact_9226_real__root__less__iff,axiom,
! [N: nat,X: real,Y: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y)) ) ) ).
% real_root_less_iff
tff(fact_9227_real__root__le__iff,axiom,
! [N: nat,X: real,Y: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y)) ) ) ).
% real_root_le_iff
tff(fact_9228_real__root__eq__1__iff,axiom,
! [N: nat,X: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( ( aa_real_real(aa_nat_fun_real_real(root,N),X) = one_one_real )
<=> ( X = one_one_real ) ) ) ).
% real_root_eq_1_iff
tff(fact_9229_real__root__one,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_real_real(aa_nat_fun_real_real(root,N),one_one_real) = one_one_real ) ) ).
% real_root_one
tff(fact_9230_real__root__gt__0__iff,axiom,
! [N: nat,Y: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y)) ) ) ).
% real_root_gt_0_iff
tff(fact_9231_real__root__lt__0__iff,axiom,
! [N: nat,X: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),zero_zero_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real)) ) ) ).
% real_root_lt_0_iff
tff(fact_9232_real__root__ge__0__iff,axiom,
! [N: nat,Y: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y)) ) ) ).
% real_root_ge_0_iff
tff(fact_9233_real__root__le__0__iff,axiom,
! [N: nat,X: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),zero_zero_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),zero_zero_real)) ) ) ).
% real_root_le_0_iff
tff(fact_9234_real__root__lt__1__iff,axiom,
! [N: nat,X: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),one_one_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real)) ) ) ).
% real_root_lt_1_iff
tff(fact_9235_real__root__gt__1__iff,axiom,
! [N: nat,Y: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),Y)) ) ) ).
% real_root_gt_1_iff
tff(fact_9236_real__root__le__1__iff,axiom,
! [N: nat,X: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),one_one_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real)) ) ) ).
% real_root_le_1_iff
tff(fact_9237_real__root__ge__1__iff,axiom,
! [N: nat,Y: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,N),Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),Y)) ) ) ).
% real_root_ge_1_iff
tff(fact_9238_or__minus__numerals_I6_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))) ).
% or_minus_numerals(6)
tff(fact_9239_or__minus__numerals_I2_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))) ).
% or_minus_numerals(2)
tff(fact_9240_real__root__pow__pos2,axiom,
! [N: nat,X: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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,N),X)),N) = X ) ) ) ).
% real_root_pow_pos2
tff(fact_9241_bit__or__int__iff,axiom,
! [K: int,L: int,N: nat] :
( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L)),N))
<=> ( pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N))
| pp(aa_nat_bool(bit_se1146084159140164899it_int(L),N)) ) ) ).
% bit_or_int_iff
tff(fact_9242_sgn__root,axiom,
! [N: nat,X: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,N),X)) = aa_real_real(sgn_sgn_real,X) ) ) ).
% sgn_root
tff(fact_9243_real__root__inverse,axiom,
! [N: nat,X: real] : aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(inverse_inverse_real,X)) = aa_real_real(inverse_inverse_real,aa_real_real(aa_nat_fun_real_real(root,N),X)) ).
% real_root_inverse
tff(fact_9244_real__root__divide,axiom,
! [N: nat,X: real,Y: real] : aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y)) ).
% real_root_divide
tff(fact_9245_real__root__commute,axiom,
! [M: nat,N: nat,X: real] : aa_real_real(aa_nat_fun_real_real(root,M),aa_real_real(aa_nat_fun_real_real(root,N),X)) = aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(aa_nat_fun_real_real(root,M),X)) ).
% real_root_commute
tff(fact_9246_real__root__minus,axiom,
! [N: nat,X: real] : aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_nat_fun_real_real(root,N),X)) ).
% real_root_minus
tff(fact_9247_real__root__mult__exp,axiom,
! [M: nat,N: nat,X: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),X) = aa_real_real(aa_nat_fun_real_real(root,M),aa_real_real(aa_nat_fun_real_real(root,N),X)) ).
% real_root_mult_exp
tff(fact_9248_real__root__mult,axiom,
! [N: nat,X: real,Y: real] : aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y)) ).
% real_root_mult
tff(fact_9249_not__bit__Suc__0__Suc,axiom,
! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(suc,N))) ).
% not_bit_Suc_0_Suc
tff(fact_9250_bit__Suc__0__iff,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),N))
<=> ( N = zero_zero_nat ) ) ).
% bit_Suc_0_iff
tff(fact_9251_real__root__pos__pos__le,axiom,
! [X: real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ).
% real_root_pos_pos_le
tff(fact_9252_OR__lower,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,X),Y))) ) ) ).
% OR_lower
tff(fact_9253_or__greater__eq,axiom,
! [L: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L))) ) ).
% or_greater_eq
tff(fact_9254_real__sgn__eq,axiom,
! [X: real] : aa_real_real(sgn_sgn_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,X),aa_real_real(abs_abs_real,X)) ).
% real_sgn_eq
tff(fact_9255_plus__and__or,axiom,
! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y)),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,X),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y) ).
% plus_and_or
tff(fact_9256_root__sgn__power,axiom,
! [N: nat,Y: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y)),N))) = Y ) ) ).
% root_sgn_power
tff(fact_9257_sgn__power__root,axiom,
! [N: nat,X: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,N),X))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N),X))),N)) = X ) ) ).
% sgn_power_root
tff(fact_9258_pow_Osimps_I1_J,axiom,
! [X: num] : aa_num_num(aa_num_fun_num_num(pow,X),one) = X ).
% pow.simps(1)
tff(fact_9259_split__root,axiom,
! [P: fun_real_bool,N: nat,X: real] :
( pp(aa_real_bool(P,aa_real_real(aa_nat_fun_real_real(root,N),X)))
<=> ( ( ( N = zero_zero_nat )
=> pp(aa_real_bool(P,zero_zero_real)) )
& ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ! [Y4: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y4)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y4)),N)) = X )
=> pp(aa_real_bool(P,Y4)) ) ) ) ) ).
% split_root
tff(fact_9260_not__bit__Suc__0__numeral,axiom,
! [N: num] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,N))) ).
% not_bit_Suc_0_numeral
tff(fact_9261_real__root__less__mono,axiom,
! [N: nat,X: real,Y: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y))) ) ) ).
% real_root_less_mono
tff(fact_9262_real__root__le__mono,axiom,
! [N: nat,X: real,Y: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N),Y))) ) ) ).
% real_root_le_mono
tff(fact_9263_real__root__power,axiom,
! [N: nat,X: real,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),K) ) ) ).
% real_root_power
tff(fact_9264_real__root__abs,axiom,
! [N: nat,X: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(abs_abs_real,X)) = aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N),X)) ) ) ).
% real_root_abs
tff(fact_9265_sgn__eq,axiom,
! [Z: complex] : aa_complex_complex(sgn_sgn_complex,Z) = aa_complex_complex(divide1717551699836669952omplex(Z),aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(real_V1022390504157884413omplex,Z))) ).
% sgn_eq
tff(fact_9266_real__root__gt__zero,axiom,
! [N: nat,X: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ) ).
% real_root_gt_zero
tff(fact_9267_real__root__strict__decreasing,axiom,
! [N: nat,N3: nat,X: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N3),X)),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ) ) ).
% real_root_strict_decreasing
tff(fact_9268_sqrt__def,axiom,
sqrt = aa_nat_fun_real_real(root,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% sqrt_def
tff(fact_9269_sgn__real__def,axiom,
! [A: real] :
( ( ( A = zero_zero_real )
=> ( aa_real_real(sgn_sgn_real,A) = zero_zero_real ) )
& ( ( A != zero_zero_real )
=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ( aa_real_real(sgn_sgn_real,A) = one_one_real ) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ) ) ).
% sgn_real_def
tff(fact_9270_root__abs__power,axiom,
! [N: nat,Y: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),N))) = aa_real_real(abs_abs_real,Y) ) ) ).
% root_abs_power
tff(fact_9271_bit__nat__iff,axiom,
! [K: int,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_int_nat(nat2,K)),N))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
& pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N)) ) ) ).
% bit_nat_iff
tff(fact_9272_real__root__pos__pos,axiom,
! [N: nat,X: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ) ).
% real_root_pos_pos
tff(fact_9273_real__root__strict__increasing,axiom,
! [N: nat,N3: nat,X: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N3),X))) ) ) ) ) ).
% real_root_strict_increasing
tff(fact_9274_real__root__decreasing,axiom,
! [N: nat,N3: nat,X: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),X))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N3),X)),aa_real_real(aa_nat_fun_real_real(root,N),X))) ) ) ) ).
% real_root_decreasing
tff(fact_9275_real__root__pow__pos,axiom,
! [N: nat,X: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_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,N),X)),N) = X ) ) ) ).
% real_root_pow_pos
tff(fact_9276_real__root__pos__unique,axiom,
! [N: nat,Y: real,X: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y))
=> ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),N) = X )
=> ( aa_real_real(aa_nat_fun_real_real(root,N),X) = Y ) ) ) ) ).
% real_root_pos_unique
tff(fact_9277_real__root__power__cancel,axiom,
! [N: nat,X: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = X ) ) ) ).
% real_root_power_cancel
tff(fact_9278_odd__real__root__pow,axiom,
! [N: nat,X: real] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),N) = X ) ) ).
% odd_real_root_pow
tff(fact_9279_odd__real__root__unique,axiom,
! [N: nat,Y: real,X: real] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),N) = X )
=> ( aa_real_real(aa_nat_fun_real_real(root,N),X) = Y ) ) ) ).
% odd_real_root_unique
tff(fact_9280_odd__real__root__power__cancel,axiom,
! [N: nat,X: real] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) = X ) ) ).
% odd_real_root_power_cancel
tff(fact_9281_sgn__power__injE,axiom,
! [A: real,N: nat,X: real,B: real] :
( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),N)) = X )
=> ( ( X = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,B)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,B)),N)) )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( A = B ) ) ) ) ).
% sgn_power_injE
tff(fact_9282_real__root__increasing,axiom,
! [N: nat,N3: nat,X: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),N3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),one_one_real))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_real_real(aa_nat_fun_real_real(root,N3),X))) ) ) ) ) ).
% real_root_increasing
tff(fact_9283_bit__nat__def,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,M),N))
<=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ) ).
% bit_nat_def
tff(fact_9284_cis__Arg__unique,axiom,
! [Z: complex,X: real] :
( ( aa_complex_complex(sgn_sgn_complex,Z) = aa_real_complex(cis,X) )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),pi))
=> ( aa_complex_real(arg,Z) = X ) ) ) ) ).
% cis_Arg_unique
tff(fact_9285_log__root,axiom,
! [N: nat,A: real,B: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),A))
=> ( aa_real_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_nat_fun_real_real(root,N),A)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,B),A)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ) ).
% log_root
tff(fact_9286_log__base__root,axiom,
! [N: nat,B: real,X: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
=> ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(aa_nat_fun_real_real(root,N),B)),X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(log,B),X)) ) ) ) ).
% log_base_root
tff(fact_9287_ln__root,axiom,
! [N: nat,B: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),B))
=> ( aa_real_real(ln_ln_real,aa_real_real(aa_nat_fun_real_real(root,N),B)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ) ).
% ln_root
tff(fact_9288_OR__upper,axiom,
! [X: int,N: nat,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,X),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ) ).
% OR_upper
tff(fact_9289_Arg__correct,axiom,
! [Z: complex] :
( ( Z != zero_zero_complex )
=> ( ( aa_complex_complex(sgn_sgn_complex,Z) = aa_real_complex(cis,aa_complex_real(arg,Z)) )
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z)))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(arg,Z)),pi)) ) ) ).
% Arg_correct
tff(fact_9290_root__powr__inverse,axiom,
! [N: nat,X: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( aa_real_real(aa_nat_fun_real_real(root,N),X) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).
% root_powr_inverse
tff(fact_9291_or__int__rec,axiom,
! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fdisj(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).
% or_int_rec
tff(fact_9292_arctan__inverse,axiom,
! [X: real] :
( ( X != zero_zero_real )
=> ( aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),X)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,X)),pi)),numeral_numeral_real(aa_num_num(bit0,one)))),aa_real_real(arctan,X)) ) ) ).
% arctan_inverse
tff(fact_9293_or__minus__numerals_I5_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),bitM(N)))) ).
% or_minus_numerals(5)
tff(fact_9294_or__minus__numerals_I1_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),bitM(N)))) ).
% or_minus_numerals(1)
tff(fact_9295_Arg__def,axiom,
! [Z: complex] :
( ( ( Z = zero_zero_complex )
=> ( aa_complex_real(arg,Z) = zero_zero_real ) )
& ( ( Z != zero_zero_complex )
=> ( aa_complex_real(arg,Z) = fChoice_real(aTP_Lamm_pg(Z)) ) ) ) ).
% Arg_def
tff(fact_9296_cis__multiple__2pi,axiom,
! [N: real] :
( pp(aa_set_real_bool(member_real(N),ring_1_Ints_real))
=> ( aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),N)) = one_one_complex ) ) ).
% cis_multiple_2pi
tff(fact_9297_or__nat__numerals_I2_J,axiom,
! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).
% or_nat_numerals(2)
tff(fact_9298_or__nat__numerals_I4_J,axiom,
! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).
% or_nat_numerals(4)
tff(fact_9299_or__nat__numerals_I1_J,axiom,
! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).
% or_nat_numerals(1)
tff(fact_9300_or__nat__numerals_I3_J,axiom,
! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).
% or_nat_numerals(3)
tff(fact_9301_or__minus__numerals_I8_J,axiom,
! [N: num,M: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))),aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),aa_num_num(bit0,N)))) ).
% or_minus_numerals(8)
tff(fact_9302_or__minus__numerals_I4_J,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),aa_num_num(bit0,N)))) ).
% or_minus_numerals(4)
tff(fact_9303_or__minus__numerals_I7_J,axiom,
! [N: num,M: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))),aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),bitM(N)))) ).
% or_minus_numerals(7)
tff(fact_9304_or__minus__numerals_I3_J,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),bitM(N)))) ).
% or_minus_numerals(3)
tff(fact_9305_or__not__num__neg_Osimps_I1_J,axiom,
aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),one) = one ).
% or_not_num_neg.simps(1)
tff(fact_9306_or__not__num__neg_Osimps_I4_J,axiom,
! [N: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit0,N)),one) = aa_num_num(bit0,one) ).
% or_not_num_neg.simps(4)
tff(fact_9307_or__not__num__neg_Osimps_I6_J,axiom,
! [N: num,M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit0,N)),aa_num_num(bit1,M)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) ).
% or_not_num_neg.simps(6)
tff(fact_9308_or__not__num__neg_Osimps_I3_J,axiom,
! [M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bit1,M)) = aa_num_num(bit1,M) ).
% or_not_num_neg.simps(3)
tff(fact_9309_or__not__num__neg_Osimps_I7_J,axiom,
! [N: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit1,N)),one) = one ).
% or_not_num_neg.simps(7)
tff(fact_9310_or__not__num__neg_Osimps_I5_J,axiom,
! [N: num,M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit0,N)),aa_num_num(bit0,M)) = bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) ).
% or_not_num_neg.simps(5)
tff(fact_9311_or__not__num__neg_Osimps_I9_J,axiom,
! [N: num,M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit1,N)),aa_num_num(bit1,M)) = bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) ).
% or_not_num_neg.simps(9)
tff(fact_9312_or__nat__def,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) ).
% or_nat_def
tff(fact_9313_or__not__num__neg_Osimps_I2_J,axiom,
! [M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bit0,M)) = aa_num_num(bit1,M) ).
% or_not_num_neg.simps(2)
tff(fact_9314_or__not__num__neg_Osimps_I8_J,axiom,
! [N: num,M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit1,N)),aa_num_num(bit0,M)) = bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) ).
% or_not_num_neg.simps(8)
tff(fact_9315_sin__times__pi__eq__0,axiom,
! [X: real] :
( ( aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),pi)) = zero_zero_real )
<=> pp(aa_set_real_bool(member_real(X),ring_1_Ints_real)) ) ).
% sin_times_pi_eq_0
tff(fact_9316_or__not__num__neg_Oelims,axiom,
! [X: num,Xa2: num,Y: num] :
( ( aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,X),Xa2) = Y )
=> ( ( ( X = one )
=> ( ( Xa2 = one )
=> ( Y != one ) ) )
=> ( ( ( X = one )
=> ! [M3: num] :
( ( Xa2 = aa_num_num(bit0,M3) )
=> ( Y != aa_num_num(bit1,M3) ) ) )
=> ( ( ( X = one )
=> ! [M3: num] :
( ( Xa2 = aa_num_num(bit1,M3) )
=> ( Y != aa_num_num(bit1,M3) ) ) )
=> ( ( ? [N2: num] : X = aa_num_num(bit0,N2)
=> ( ( Xa2 = one )
=> ( Y != aa_num_num(bit0,one) ) ) )
=> ( ! [N2: num] :
( ( X = aa_num_num(bit0,N2) )
=> ! [M3: num] :
( ( Xa2 = aa_num_num(bit0,M3) )
=> ( Y != bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N2),M3)) ) ) )
=> ( ! [N2: num] :
( ( X = aa_num_num(bit0,N2) )
=> ! [M3: num] :
( ( Xa2 = aa_num_num(bit1,M3) )
=> ( Y != aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N2),M3)) ) ) )
=> ( ( ? [N2: num] : X = aa_num_num(bit1,N2)
=> ( ( Xa2 = one )
=> ( Y != one ) ) )
=> ( ! [N2: num] :
( ( X = aa_num_num(bit1,N2) )
=> ! [M3: num] :
( ( Xa2 = aa_num_num(bit0,M3) )
=> ( Y != bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N2),M3)) ) ) )
=> ~ ! [N2: num] :
( ( X = aa_num_num(bit1,N2) )
=> ! [M3: num] :
( ( Xa2 = aa_num_num(bit1,M3) )
=> ( Y != bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N2),M3)) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_not_num_neg.elims
tff(fact_9317_sin__integer__2pi,axiom,
! [N: real] :
( pp(aa_set_real_bool(member_real(N),ring_1_Ints_real))
=> ( aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),N)) = zero_zero_real ) ) ).
% sin_integer_2pi
tff(fact_9318_cos__integer__2pi,axiom,
! [N: real] :
( pp(aa_set_real_bool(member_real(N),ring_1_Ints_real))
=> ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),N)) = one_one_real ) ) ).
% cos_integer_2pi
tff(fact_9319_or__Suc__0__eq,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).
% or_Suc_0_eq
tff(fact_9320_Suc__0__or__eq,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).
% Suc_0_or_eq
tff(fact_9321_or__nat__rec,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat(aa_bool_bool(fdisj(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).
% or_nat_rec
tff(fact_9322_or__nat__unfold,axiom,
! [M: nat,N: nat] :
( ( ( M = zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = N ) )
& ( ( M != zero_zero_nat )
=> ( ( ( N = zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = M ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = 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(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ) ).
% or_nat_unfold
tff(fact_9323_setceilmax,axiom,
! [S: vEBT_VEBT,M: nat,Listy: list_VEBT_VEBT,N: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(S),M))
=> ( ! [X3: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(Listy)))
=> pp(aa_nat_bool(vEBT_invar_vebt(X3),N)) )
=> ( ( M = aa_nat_nat(suc,N) )
=> ( ! [X3: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(Listy)))
=> ( aa_nat_int(semiri1314217659103216013at_int,aa_VEBT_VEBT_nat(vEBT_VEBT_height,X3)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))) ) )
=> ( ( aa_nat_int(semiri1314217659103216013at_int,aa_VEBT_VEBT_nat(vEBT_VEBT_height,S)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))) )
=> ( aa_nat_int(semiri1314217659103216013at_int,lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(S,set_VEBT_VEBT2(Listy))))) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))) ) ) ) ) ) ) ).
% setceilmax
tff(fact_9324_xor__Suc__0__eq,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).
% xor_Suc_0_eq
tff(fact_9325_Suc__0__xor__eq,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).
% Suc_0_xor_eq
tff(fact_9326_horner__sum__of__bool__2__less,axiom,
! [Bs: list_bool] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,groups4048646594741130792ol_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),Bs)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),size_size_list_bool(Bs)))) ).
% horner_sum_of_bool_2_less
tff(fact_9327_height__compose__list,axiom,
! [T: vEBT_VEBT,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,T),set_VEBT_VEBT2(TreeList)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_height,T)),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summary,set_VEBT_VEBT2(TreeList)))))) ) ).
% height_compose_list
tff(fact_9328_max__ins__scaled,axiom,
! [N: nat,X14: vEBT_VEBT,M: nat,X13: list_VEBT_VEBT] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_VEBT_VEBT_nat(vEBT_VEBT_height,X14))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),lattic8265883725875713057ax_nat(insert_nat(aa_VEBT_VEBT_nat(vEBT_VEBT_height,X14),image_VEBT_VEBT_nat(vEBT_VEBT_height,set_VEBT_VEBT2(X13)))))))) ).
% max_ins_scaled
tff(fact_9329_height__i__max,axiom,
! [I: nat,X13: list_VEBT_VEBT,Foo: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(X13)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_height,nth_VEBT_VEBT(X13,I))),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Foo),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,set_VEBT_VEBT2(X13)))))) ) ).
% height_i_max
tff(fact_9330_max__idx__list,axiom,
! [I: nat,X13: list_VEBT_VEBT,N: nat,X14: vEBT_VEBT] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6755466524823107622T_VEBT(X13)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_VEBT_VEBT_nat(vEBT_VEBT_height,nth_VEBT_VEBT(X13,I)))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_height,X14)),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,set_VEBT_VEBT2(X13))))))))) ) ).
% max_idx_list
tff(fact_9331_xor__nat__numerals_I4_J,axiom,
! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X)) ).
% xor_nat_numerals(4)
tff(fact_9332_xor__nat__numerals_I3_J,axiom,
! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).
% xor_nat_numerals(3)
tff(fact_9333_xor__nat__numerals_I2_J,axiom,
! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y)) ).
% xor_nat_numerals(2)
tff(fact_9334_xor__nat__numerals_I1_J,axiom,
! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).
% xor_nat_numerals(1)
tff(fact_9335_VEBT__internal_Oheight_Osimps_I2_J,axiom,
! [Uu: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_VEBT_height,vEBT_Node(Uu,Deg,TreeList,Summary)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summary,set_VEBT_VEBT2(TreeList))))) ).
% VEBT_internal.height.simps(2)
tff(fact_9336_VEBT__internal_Oheight_Oelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( aa_VEBT_VEBT_nat(vEBT_VEBT_height,X) = Y )
=> ( ( ? [A4: bool,B4: bool] : X = vEBT_Leaf(A4,B4)
=> ( Y != zero_zero_nat ) )
=> ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X = vEBT_Node(Uu2,Deg2,TreeList2,Summary2) )
=> ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summary2,set_VEBT_VEBT2(TreeList2))))) ) ) ) ) ).
% VEBT_internal.height.elims
tff(fact_9337_divide__nat__def,axiom,
! [N: nat,M: nat] :
( ( ( N = zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ph(N),M))) ) ) ) ).
% divide_nat_def
tff(fact_9338_xor__nat__unfold,axiom,
! [M: nat,N: nat] :
( ( ( M = zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = N ) )
& ( ( M != zero_zero_nat )
=> ( ( ( N = zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = M ) )
& ( ( N != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ) ).
% xor_nat_unfold
tff(fact_9339_xor__nat__rec,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat(aa_bool_bool(fNot,aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_bool_bool(fNot,aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).
% xor_nat_rec
tff(fact_9340_bij__betw__Suc,axiom,
! [M6: set_nat,N3: set_nat] :
( pp(aa_set_nat_bool(bij_betw_nat_nat(suc,M6),N3))
<=> ( image_nat_nat(suc,M6) = N3 ) ) ).
% bij_betw_Suc
tff(fact_9341_image__Suc__atLeastAtMost,axiom,
! [I: nat,J: nat] : image_nat_nat(suc,aa_nat_set_nat(set_or1269000886237332187st_nat(I),J)) = aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,I)),aa_nat_nat(suc,J)) ).
% image_Suc_atLeastAtMost
tff(fact_9342_xor__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L)))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),L)) ) ) ).
% xor_nonnegative_int_iff
tff(fact_9343_xor__negative__int__iff,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L)),zero_zero_int))
<=> ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int)) ) ) ).
% xor_negative_int_iff
tff(fact_9344_bit__xor__int__iff,axiom,
! [K: int,L: int,N: nat] :
( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L)),N))
<=> ~ ( pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N))
<=> pp(aa_nat_bool(bit_se1146084159140164899it_int(L),N)) ) ) ).
% bit_xor_int_iff
tff(fact_9345_zero__notin__Suc__image,axiom,
! [A2: set_nat] : ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),image_nat_nat(suc,A2))) ).
% zero_notin_Suc_image
tff(fact_9346_XOR__lower,axiom,
! [X: int,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,X),Y))) ) ) ).
% XOR_lower
tff(fact_9347_xor__nat__def,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) ).
% xor_nat_def
tff(fact_9348_image__Suc__lessThan,axiom,
! [N: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N) ).
% image_Suc_lessThan
tff(fact_9349_image__Suc__atMost,axiom,
! [N: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),aa_nat_nat(suc,N)) ).
% image_Suc_atMost
tff(fact_9350_atLeast0__atMost__Suc__eq__insert__0,axiom,
! [N: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))) ).
% atLeast0_atMost_Suc_eq_insert_0
tff(fact_9351_lessThan__Suc__eq__insert__0,axiom,
! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,N))) ).
% lessThan_Suc_eq_insert_0
tff(fact_9352_atMost__Suc__eq__insert__0,axiom,
! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,N))) ).
% atMost_Suc_eq_insert_0
tff(fact_9353_XOR__upper,axiom,
! [X: int,N: nat,Y: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,X),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ) ).
% XOR_upper
tff(fact_9354_xor__int__rec,axiom,
! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).
% xor_int_rec
tff(fact_9355_xor__int__unfold,axiom,
! [K: int,L: int] :
( ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
=> ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = aa_int_int(bit_ri7919022796975470100ot_int,L) ) )
& ( ( K != aa_int_int(uminus_uminus_int,one_one_int) )
=> ( ( ( L = aa_int_int(uminus_uminus_int,one_one_int) )
=> ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = aa_int_int(bit_ri7919022796975470100ot_int,K) ) )
& ( ( L != aa_int_int(uminus_uminus_int,one_one_int) )
=> ( ( ( K = zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = L ) )
& ( ( K != zero_zero_int )
=> ( ( ( L = zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = K ) )
& ( ( L != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ) ) ) ) ) ) ).
% xor_int_unfold
tff(fact_9356_push__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se545348938243370406it_int(N),K)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).
% push_bit_nonnegative_int_iff
tff(fact_9357_push__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se545348938243370406it_int(N),K)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).
% push_bit_negative_int_iff
tff(fact_9358_concat__bit__of__zero__1,axiom,
! [N: nat,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),zero_zero_int),L) = aa_int_int(bit_se545348938243370406it_int(N),L) ).
% concat_bit_of_zero_1
tff(fact_9359_not__negative__int__iff,axiom,
! [K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).
% not_negative_int_iff
tff(fact_9360_not__nonnegative__int__iff,axiom,
! [K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri7919022796975470100ot_int,K)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).
% not_nonnegative_int_iff
tff(fact_9361_push__bit__of__Suc__0,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).
% push_bit_of_Suc_0
tff(fact_9362_and__minus__minus__numerals,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(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,N))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,N)),one_one_int))) ).
% and_minus_minus_numerals
tff(fact_9363_or__minus__minus__numerals,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(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,N))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,N)),one_one_int))) ).
% or_minus_minus_numerals
tff(fact_9364_unset__bit__int__def,axiom,
! [N: nat,K: int] : aa_int_int(bit_se4203085406695923979it_int(N),K) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se545348938243370406it_int(N),one_one_int))) ).
% unset_bit_int_def
tff(fact_9365_push__bit__nat__eq,axiom,
! [N: nat,K: int] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(bit_se545348938243370406it_int(N),K)) ).
% push_bit_nat_eq
tff(fact_9366_bit__not__int__iff,axiom,
! [K: int,N: nat] :
( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(bit_ri7919022796975470100ot_int,K)),N))
<=> ~ pp(aa_nat_bool(bit_se1146084159140164899it_int(K),N)) ) ).
% bit_not_int_iff
tff(fact_9367_or__int__def,axiom,
! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),aa_int_int(bit_ri7919022796975470100ot_int,L))) ).
% or_int_def
tff(fact_9368_flip__bit__nat__def,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),one_one_nat)) ).
% flip_bit_nat_def
tff(fact_9369_set__bit__nat__def,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),one_one_nat)) ).
% set_bit_nat_def
tff(fact_9370_not__int__def,axiom,
! [K: int] : aa_int_int(bit_ri7919022796975470100ot_int,K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,K)),one_one_int) ).
% not_int_def
tff(fact_9371_and__not__numerals_I1_J,axiom,
aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = zero_zero_int ).
% and_not_numerals(1)
tff(fact_9372_or__not__numerals_I1_J,axiom,
aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).
% or_not_numerals(1)
tff(fact_9373_bit__push__bit__iff__int,axiom,
! [M: nat,K: int,N: nat] :
( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(bit_se545348938243370406it_int(M),K)),N))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
& pp(aa_nat_bool(bit_se1146084159140164899it_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).
% bit_push_bit_iff_int
tff(fact_9374_bit__push__bit__iff__nat,axiom,
! [M: nat,Q2: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),Q2)),N))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(bit_se1148574629649215175it_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).
% bit_push_bit_iff_nat
tff(fact_9375_concat__bit__eq,axiom,
! [N: nat,K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),aa_int_int(bit_se545348938243370406it_int(N),L)) ).
% concat_bit_eq
tff(fact_9376_xor__int__def,axiom,
! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),aa_int_int(bit_ri7919022796975470100ot_int,L))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),L)) ).
% xor_int_def
tff(fact_9377_concat__bit__def,axiom,
! [N: nat,K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),L) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),aa_int_int(bit_se545348938243370406it_int(N),L)) ).
% concat_bit_def
tff(fact_9378_set__bit__int__def,axiom,
! [N: nat,K: int] : aa_int_int(bit_se7879613467334960850it_int(N),K) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),aa_int_int(bit_se545348938243370406it_int(N),one_one_int)) ).
% set_bit_int_def
tff(fact_9379_flip__bit__int__def,axiom,
! [N: nat,K: int] : aa_int_int(bit_se2159334234014336723it_int(N),K) = aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),aa_int_int(bit_se545348938243370406it_int(N),one_one_int)) ).
% flip_bit_int_def
tff(fact_9380_not__int__div__2,axiom,
! [K: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).
% not_int_div_2
tff(fact_9381_even__not__iff__int,axiom,
! [K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,K)))
<=> ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)) ) ).
% even_not_iff_int
tff(fact_9382_and__not__numerals_I4_J,axiom,
! [M: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,M)) ).
% and_not_numerals(4)
tff(fact_9383_and__not__numerals_I2_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = one_one_int ).
% and_not_numerals(2)
tff(fact_9384_or__not__numerals_I2_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))) ).
% or_not_numerals(2)
tff(fact_9385_or__not__numerals_I4_J,axiom,
! [M: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,one_one_int) ).
% or_not_numerals(4)
tff(fact_9386_bit__minus__int__iff,axiom,
! [K: int,N: nat] :
( pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,K)),N))
<=> pp(aa_nat_bool(bit_se1146084159140164899it_int(aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int))),N)) ) ).
% bit_minus_int_iff
tff(fact_9387_int__numeral__or__not__num__neg,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),N))) ).
% int_numeral_or_not_num_neg
tff(fact_9388_int__numeral__not__or__num__neg,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M))) ).
% int_numeral_not_or_num_neg
tff(fact_9389_numeral__or__not__num__eq,axiom,
! [M: num,N: num] : aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),N)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N)))) ).
% numeral_or_not_num_eq
tff(fact_9390_and__not__numerals_I5_J,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N)))) ).
% and_not_numerals(5)
tff(fact_9391_and__not__numerals_I7_J,axiom,
! [M: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,M)) ).
% and_not_numerals(7)
tff(fact_9392_or__not__numerals_I3_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))) ).
% or_not_numerals(3)
tff(fact_9393_and__not__numerals_I3_J,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = zero_zero_int ).
% and_not_numerals(3)
tff(fact_9394_or__not__numerals_I7_J,axiom,
! [M: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).
% or_not_numerals(7)
tff(fact_9395_push__bit__int__def,axiom,
! [N: nat,K: int] : aa_int_int(bit_se545348938243370406it_int(N),K) = aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).
% push_bit_int_def
tff(fact_9396_push__bit__nat__def,axiom,
! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).
% push_bit_nat_def
tff(fact_9397_and__not__numerals_I9_J,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N)))) ).
% and_not_numerals(9)
tff(fact_9398_and__not__numerals_I6_J,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N)))) ).
% and_not_numerals(6)
tff(fact_9399_or__not__numerals_I6_J,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N)))) ).
% or_not_numerals(6)
tff(fact_9400_push__bit__minus__one,axiom,
! [N: nat] : aa_int_int(bit_se545348938243370406it_int(N),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,aa_num_num(bit0,one))),N)) ).
% push_bit_minus_one
tff(fact_9401_or__not__numerals_I5_J,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ).
% or_not_numerals(5)
tff(fact_9402_and__not__numerals_I8_J,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ).
% and_not_numerals(8)
tff(fact_9403_or__not__numerals_I8_J,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ).
% or_not_numerals(8)
tff(fact_9404_or__not__numerals_I9_J,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ).
% or_not_numerals(9)
tff(fact_9405_not__int__rec,axiom,
! [K: int] : aa_int_int(bit_ri7919022796975470100ot_int,K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_bool_int(zero_n2684676970156552555ol_int,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).
% not_int_rec
tff(fact_9406_VEBT__internal_Oheight_Opelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( aa_VEBT_VEBT_nat(vEBT_VEBT_height,X) = Y )
=> ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_height_rel),X))
=> ( ! [A4: bool,B4: bool] :
( ( X = vEBT_Leaf(A4,B4) )
=> ( ( Y = zero_zero_nat )
=> ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_height_rel),vEBT_Leaf(A4,B4))) ) )
=> ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X = vEBT_Node(Uu2,Deg2,TreeList2,Summary2) )
=> ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summary2,set_VEBT_VEBT2(TreeList2))))) )
=> ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_VEBT_height_rel),vEBT_Node(Uu2,Deg2,TreeList2,Summary2))) ) ) ) ) ) ).
% VEBT_internal.height.pelims
tff(fact_9407_vebt__maxt_Opelims,axiom,
! [X: vEBT_VEBT,Y: option_nat] :
( ( vEBT_vebt_maxt(X) = Y )
=> ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),X))
=> ( ! [A4: bool,B4: bool] :
( ( X = vEBT_Leaf(A4,B4) )
=> ( ( ( pp(B4)
=> ( Y = some_nat(one_one_nat) ) )
& ( ~ pp(B4)
=> ( ( pp(A4)
=> ( Y = some_nat(zero_zero_nat) ) )
& ( ~ pp(A4)
=> ( Y = none_nat ) ) ) ) )
=> ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Leaf(A4,B4))) ) )
=> ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
=> ( ( Y = none_nat )
=> ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2))) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2) )
=> ( ( Y = some_nat(Ma2) )
=> ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2))) ) ) ) ) ) ) ).
% vebt_maxt.pelims
tff(fact_9408_vebt__mint_Opelims,axiom,
! [X: vEBT_VEBT,Y: option_nat] :
( ( vEBT_vebt_mint(X) = Y )
=> ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),X))
=> ( ! [A4: bool,B4: bool] :
( ( X = vEBT_Leaf(A4,B4) )
=> ( ( ( pp(A4)
=> ( Y = some_nat(zero_zero_nat) ) )
& ( ~ pp(A4)
=> ( ( pp(B4)
=> ( Y = some_nat(one_one_nat) ) )
& ( ~ pp(B4)
=> ( Y = none_nat ) ) ) ) )
=> ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Leaf(A4,B4))) ) )
=> ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
=> ( ( Y = none_nat )
=> ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2))) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2) )
=> ( ( Y = some_nat(Mi2) )
=> ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2))) ) ) ) ) ) ) ).
% vebt_mint.pelims
tff(fact_9409_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,X) = Y )
=> ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),X))
=> ( ! [A4: bool,B4: bool] :
( ( X = vEBT_Leaf(A4,B4) )
=> ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(A4),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
=> ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),vEBT_Leaf(A4,B4))) ) )
=> ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
=> ( ( Y = one_one_nat )
=> ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2))) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2) )
=> ( ( Y = one_one_nat )
=> ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2))) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
tff(fact_9410_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,X) = Y )
=> ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),X))
=> ( ! [A4: bool,B4: bool] :
( ( X = vEBT_Leaf(A4,B4) )
=> ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(B4),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
=> ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),vEBT_Leaf(A4,B4))) ) )
=> ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
=> ( ( Y = one_one_nat )
=> ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2))) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2) )
=> ( ( Y = one_one_nat )
=> ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux2,Uy2,Uz2))) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
tff(fact_9411_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Opelims,axiom,
! [X: vEBT_VEBT,Y: nat] :
( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,X) = Y )
=> ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),X))
=> ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
=> ( ( Y = one_one_nat )
=> ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),vEBT_Leaf(fFalse,fFalse))) ) )
=> ( ! [Uv2: bool] :
( ( X = vEBT_Leaf(fTrue,Uv2) )
=> ( ( Y = one_one_nat )
=> ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),vEBT_Leaf(fTrue,Uv2))) ) )
=> ( ! [Uu2: bool] :
( ( X = vEBT_Leaf(Uu2,fTrue) )
=> ( ( Y = one_one_nat )
=> ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),vEBT_Leaf(Uu2,fTrue))) ) )
=> ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) )
=> ( ( Y = one_one_nat )
=> ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2))) ) )
=> ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2,Vc2) )
=> ( ( Y = one_one_nat )
=> ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2,Vc2))) ) ) ) ) ) ) ) ) ).
% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.pelims
tff(fact_9412_VEBT__internal_OminNull_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Y: bool] :
( ( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
<=> pp(Y) )
=> ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),X))
=> ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
=> ( pp(Y)
=> ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(fFalse,fFalse))) ) )
=> ( ! [Uv2: bool] :
( ( X = vEBT_Leaf(fTrue,Uv2) )
=> ( ~ pp(Y)
=> ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(fTrue,Uv2))) ) )
=> ( ! [Uu2: bool] :
( ( X = vEBT_Leaf(Uu2,fTrue) )
=> ( ~ pp(Y)
=> ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(Uu2,fTrue))) ) )
=> ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) )
=> ( pp(Y)
=> ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2))) ) )
=> ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2,Vc2) )
=> ( ~ pp(Y)
=> ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2,Vc2))) ) ) ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(1)
tff(fact_9413_VEBT__internal_OminNull_Opelims_I3_J,axiom,
! [X: vEBT_VEBT] :
( ~ pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
=> ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),X))
=> ( ! [Uv2: bool] :
( ( X = vEBT_Leaf(fTrue,Uv2) )
=> ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(fTrue,Uv2))) )
=> ( ! [Uu2: bool] :
( ( X = vEBT_Leaf(Uu2,fTrue) )
=> ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(Uu2,fTrue))) )
=> ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2,Vc2) )
=> ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va2,Vb2,Vc2))) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(3)
tff(fact_9414_VEBT__internal_OminNull_Opelims_I2_J,axiom,
! [X: vEBT_VEBT] :
( pp(aa_VEBT_VEBT_bool(vEBT_VEBT_minNull,X))
=> ( pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),X))
=> ( ( ( X = vEBT_Leaf(fFalse,fFalse) )
=> ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf(fFalse,fFalse))) )
=> ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) )
=> ~ pp(aa_VEBT_VEBT_bool(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2))) ) ) ) ) ).
% VEBT_internal.minNull.pelims(2)
tff(fact_9415_Cauchy__iff2,axiom,
! [X5: fun_nat_real] :
( topolo4055970368930404560y_real(X5)
<=> ! [J3: nat] :
? [M8: nat] :
! [M7: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M8),M7))
=> ! [N5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M8),N5))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(X5,M7)),aa_nat_real(X5,N5)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J3))))) ) ) ) ).
% Cauchy_iff2
tff(fact_9416_Sum__Ico__nat,axiom,
! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_be),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% Sum_Ico_nat
tff(fact_9417_VEBT_Osize_I3_J,axiom,
! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,vEBT_Node(X11,X12,X13,X14)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_list_VEBT_VEBT(size_size_VEBT_VEBT,X13)),aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,X14))),aa_nat_nat(suc,zero_zero_nat)) ).
% VEBT.size(3)
tff(fact_9418_image__Suc__atLeastLessThan,axiom,
! [I: nat,J: nat] : image_nat_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(I),J)) = aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,I)),aa_nat_nat(suc,J)) ).
% image_Suc_atLeastLessThan
tff(fact_9419_atLeastLessThan__singleton,axiom,
! [M: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,M)) = insert_nat(M,bot_bot_set_nat) ).
% atLeastLessThan_singleton
tff(fact_9420_ex__nat__less__eq,axiom,
! [N: nat,P: fun_nat_bool] :
( ? [M7: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M7),N))
& pp(aa_nat_bool(P,M7)) )
<=> ? [X2: nat] :
( pp(aa_set_nat_bool(member_nat(X2),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))
& pp(aa_nat_bool(P,X2)) ) ) ).
% ex_nat_less_eq
tff(fact_9421_all__nat__less__eq,axiom,
! [N: nat,P: fun_nat_bool] :
( ! [M7: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M7),N))
=> pp(aa_nat_bool(P,M7)) )
<=> ! [X2: nat] :
( pp(aa_set_nat_bool(member_nat(X2),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))
=> pp(aa_nat_bool(P,X2)) ) ) ).
% all_nat_less_eq
tff(fact_9422_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L: nat,U: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(L),aa_nat_nat(suc,U)) = aa_nat_set_nat(set_or1269000886237332187st_nat(L),U) ).
% atLeastLessThanSuc_atLeastAtMost
tff(fact_9423_atLeast0__lessThan__Suc,axiom,
! [N: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N)) = insert_nat(N,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)) ).
% atLeast0_lessThan_Suc
tff(fact_9424_atLeastLessThanSuc,axiom,
! [M: nat,N: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N)) = insert_nat(N,aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,N)) = bot_bot_set_nat ) ) ) ).
% atLeastLessThanSuc
tff(fact_9425_atLeast0__lessThan__Suc__eq__insert__0,axiom,
! [N: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))) ).
% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_9426_prod__Suc__Suc__fact,axiom,
! [N: nat] : groups708209901874060359at_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),N)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ).
% prod_Suc_Suc_fact
tff(fact_9427_prod__Suc__fact,axiom,
! [N: nat] : groups708209901874060359at_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ).
% prod_Suc_fact
tff(fact_9428_atLeastLessThan__nat__numeral,axiom,
! [M: nat,K: num] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_num_nat(pred_numeral,K)))
=> ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(pred_numeral,K),aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_num_nat(pred_numeral,K))) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),aa_num_nat(pred_numeral,K)))
=> ( aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_num_nat(numeral_numeral_nat,K)) = bot_bot_set_nat ) ) ) ).
% atLeastLessThan_nat_numeral
tff(fact_9429_image__minus__const__atLeastLessThan__nat,axiom,
! [C: nat,Y: nat,X: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),Y))
=> ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_pi,C),aa_nat_set_nat(set_or4665077453230672383an_nat(X),Y)) = aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Y),C)) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),Y))
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
=> ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_pi,C),aa_nat_set_nat(set_or4665077453230672383an_nat(X),Y)) = insert_nat(zero_zero_nat,bot_bot_set_nat) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
=> ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_pi,C),aa_nat_set_nat(set_or4665077453230672383an_nat(X),Y)) = bot_bot_set_nat ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
tff(fact_9430_atLeast1__lessThan__eq__remove0,axiom,
! [N: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),N) = minus_minus_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,N),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).
% atLeast1_lessThan_eq_remove0
tff(fact_9431_sum__power2,axiom,
! [K: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),one_one_nat) ).
% sum_power2
tff(fact_9432_Chebyshev__sum__upper__nat,axiom,
! [N: nat,A: fun_nat_nat,B: fun_nat_nat] :
( ! [I2: nat,J2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(A,I2)),aa_nat_nat(A,J2))) ) )
=> ( ! [I2: nat,J2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I2),J2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(B,J2)),aa_nat_nat(B,I2))) ) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_nw(A),B)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))),aa_set_nat_nat(groups3542108847815614940at_nat(B),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))))) ) ) ).
% Chebyshev_sum_upper_nat
tff(fact_9433_image__add__int__atLeastLessThan,axiom,
! [L: int,U: int] : image_int_int(aa_int_fun_int_int(aTP_Lamm_pj,L),set_or4662586982721622107an_int(zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L))) = set_or4662586982721622107an_int(L,U) ).
% image_add_int_atLeastLessThan
tff(fact_9434_VEBT_Osize__gen_I1_J,axiom,
! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,vEBT_Node(X11,X12,X13,X14)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_list_VEBT_VEBT(vEBT_size_VEBT,X13)),aa_VEBT_VEBT_nat(vEBT_size_VEBT,X14))),aa_nat_nat(suc,zero_zero_nat)) ).
% VEBT.size_gen(1)
tff(fact_9435_valid__eq2,axiom,
! [T: vEBT_VEBT,D: nat] :
( pp(aa_nat_bool(vEBT_VEBT_valid(T),D))
=> pp(aa_nat_bool(vEBT_invar_vebt(T),D)) ) ).
% valid_eq2
tff(fact_9436_valid__eq1,axiom,
! [T: vEBT_VEBT,D: nat] :
( pp(aa_nat_bool(vEBT_invar_vebt(T),D))
=> pp(aa_nat_bool(vEBT_VEBT_valid(T),D)) ) ).
% valid_eq1
tff(fact_9437_valid__eq,axiom,
! [T: vEBT_VEBT,D: nat] :
( pp(aa_nat_bool(vEBT_VEBT_valid(T),D))
<=> pp(aa_nat_bool(vEBT_invar_vebt(T),D)) ) ).
% valid_eq
tff(fact_9438_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
! [Uu: bool,Uv: bool,D: nat] :
( pp(aa_nat_bool(vEBT_VEBT_valid(vEBT_Leaf(Uu,Uv)),D))
<=> ( D = one_one_nat ) ) ).
% VEBT_internal.valid'.simps(1)
tff(fact_9439_Code__Target__Int_Opositive__def,axiom,
code_Target_positive = numeral_numeral_int ).
% Code_Target_Int.positive_def
tff(fact_9440_csqrt_Osimps_I1_J,axiom,
! [Z: complex] : aa_complex_real(re,aa_complex_complex(csqrt,Z)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),numeral_numeral_real(aa_num_num(bit0,one)))) ).
% csqrt.simps(1)
tff(fact_9441_complex__Re__numeral,axiom,
! [V: num] : aa_complex_real(re,numera6690914467698888265omplex(V)) = numeral_numeral_real(V) ).
% complex_Re_numeral
tff(fact_9442_Re__divide__of__nat,axiom,
! [Z: complex,N: nat] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Z),aa_nat_complex(semiri8010041392384452111omplex,N))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),aa_nat_real(semiri5074537144036343181t_real,N)) ).
% Re_divide_of_nat
tff(fact_9443_Re__divide__of__real,axiom,
! [Z: complex,R: real] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Z),aa_real_complex(real_V4546457046886955230omplex,R))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),R) ).
% Re_divide_of_real
tff(fact_9444_Re__sgn,axiom,
! [Z: complex] : aa_complex_real(re,aa_complex_complex(sgn_sgn_complex,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),aa_complex_real(real_V1022390504157884413omplex,Z)) ).
% Re_sgn
tff(fact_9445_Re__divide__numeral,axiom,
! [Z: complex,W: num] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Z),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),numeral_numeral_real(W)) ).
% Re_divide_numeral
tff(fact_9446_complex__Re__le__cmod,axiom,
! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(re,X)),aa_complex_real(real_V1022390504157884413omplex,X))) ).
% complex_Re_le_cmod
tff(fact_9447_one__complex_Osimps_I1_J,axiom,
aa_complex_real(re,one_one_complex) = one_one_real ).
% one_complex.simps(1)
tff(fact_9448_plus__complex_Osimps_I1_J,axiom,
! [X: complex,Y: complex] : aa_complex_real(re,aa_complex_complex(plus_plus_complex(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(re,X)),aa_complex_real(re,Y)) ).
% plus_complex.simps(1)
tff(fact_9449_scaleR__complex_Osimps_I1_J,axiom,
! [R: real,X: complex] : aa_complex_real(re,aa_complex_complex(real_V2046097035970521341omplex(R),X)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_complex_real(re,X)) ).
% scaleR_complex.simps(1)
tff(fact_9450_minus__complex_Osimps_I1_J,axiom,
! [X: complex,Y: complex] : aa_complex_real(re,aa_complex_complex(minus_minus_complex(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(re,X)),aa_complex_real(re,Y)) ).
% minus_complex.simps(1)
tff(fact_9451_abs__Re__le__cmod,axiom,
! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(re,X))),aa_complex_real(real_V1022390504157884413omplex,X))) ).
% abs_Re_le_cmod
tff(fact_9452_Re__csqrt,axiom,
! [Z: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(csqrt,Z)))) ).
% Re_csqrt
tff(fact_9453_cmod__plus__Re__le__0__iff,axiom,
! [Z: complex] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),zero_zero_real))
<=> ( aa_complex_real(re,Z) = aa_real_real(uminus_uminus_real,aa_complex_real(real_V1022390504157884413omplex,Z)) ) ) ).
% cmod_plus_Re_le_0_iff
tff(fact_9454_cos__n__Re__cis__pow__n,axiom,
! [N: nat,A: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),A)) = aa_complex_real(re,aa_nat_complex(power_power_complex(aa_real_complex(cis,A)),N)) ).
% cos_n_Re_cis_pow_n
tff(fact_9455_csqrt_Ocode,axiom,
! [Z: complex] : aa_complex_complex(csqrt,Z) = aa_real_complex(complex2(aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,aa_complex_real(im,Z)),zero_zero_real)),one_one_real),aa_real_real(sgn_sgn_real,aa_complex_real(im,Z)))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),numeral_numeral_real(aa_num_num(bit0,one)))))) ).
% csqrt.code
tff(fact_9456_csqrt_Osimps_I2_J,axiom,
! [Z: complex] : aa_complex_real(im,aa_complex_complex(csqrt,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,aa_complex_real(im,Z)),zero_zero_real)),one_one_real),aa_real_real(sgn_sgn_real,aa_complex_real(im,Z)))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),numeral_numeral_real(aa_num_num(bit0,one))))) ).
% csqrt.simps(2)
tff(fact_9457_csqrt__of__real__nonpos,axiom,
! [X: complex] :
( ( aa_complex_real(im,X) = zero_zero_real )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(re,X)),zero_zero_real))
=> ( aa_complex_complex(csqrt,X) = aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,aa_real_real(abs_abs_real,aa_complex_real(re,X))))) ) ) ) ).
% csqrt_of_real_nonpos
tff(fact_9458_Im__divide__of__real,axiom,
! [Z: complex,R: real] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Z),aa_real_complex(real_V4546457046886955230omplex,R))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),R) ).
% Im_divide_of_real
tff(fact_9459_Im__sgn,axiom,
! [Z: complex] : aa_complex_real(im,aa_complex_complex(sgn_sgn_complex,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),aa_complex_real(real_V1022390504157884413omplex,Z)) ).
% Im_sgn
tff(fact_9460_Re__power__real,axiom,
! [X: complex,N: nat] :
( ( aa_complex_real(im,X) = zero_zero_real )
=> ( aa_complex_real(re,aa_nat_complex(power_power_complex(X),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,X)),N) ) ) ).
% Re_power_real
tff(fact_9461_Im__divide__numeral,axiom,
! [Z: complex,W: num] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Z),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),numeral_numeral_real(W)) ).
% Im_divide_numeral
tff(fact_9462_Im__divide__of__nat,axiom,
! [Z: complex,N: nat] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Z),aa_nat_complex(semiri8010041392384452111omplex,N))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),aa_nat_real(semiri5074537144036343181t_real,N)) ).
% Im_divide_of_nat
tff(fact_9463_csqrt__of__real__nonneg,axiom,
! [X: complex] :
( ( aa_complex_real(im,X) = zero_zero_real )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(re,X)))
=> ( aa_complex_complex(csqrt,X) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,aa_complex_real(re,X))) ) ) ) ).
% csqrt_of_real_nonneg
tff(fact_9464_csqrt__minus,axiom,
! [X: complex] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(im,X)),zero_zero_real))
| ( ( aa_complex_real(im,X) = zero_zero_real )
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(re,X))) ) )
=> ( aa_complex_complex(csqrt,aa_complex_complex(uminus1482373934393186551omplex,X)) = aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(csqrt,X)) ) ) ).
% csqrt_minus
tff(fact_9465_imaginary__unit_Osimps_I2_J,axiom,
aa_complex_real(im,imaginary_unit) = one_one_real ).
% imaginary_unit.simps(2)
tff(fact_9466_plus__complex_Osimps_I2_J,axiom,
! [X: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(plus_plus_complex(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(im,X)),aa_complex_real(im,Y)) ).
% plus_complex.simps(2)
tff(fact_9467_scaleR__complex_Osimps_I2_J,axiom,
! [R: real,X: complex] : aa_complex_real(im,aa_complex_complex(real_V2046097035970521341omplex(R),X)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_complex_real(im,X)) ).
% scaleR_complex.simps(2)
tff(fact_9468_minus__complex_Osimps_I2_J,axiom,
! [X: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(minus_minus_complex(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(im,X)),aa_complex_real(im,Y)) ).
% minus_complex.simps(2)
tff(fact_9469_abs__Im__le__cmod,axiom,
! [X: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(im,X))),aa_complex_real(real_V1022390504157884413omplex,X))) ).
% abs_Im_le_cmod
tff(fact_9470_times__complex_Osimps_I2_J,axiom,
! [X: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(im,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(re,Y))) ).
% times_complex.simps(2)
tff(fact_9471_cmod__Re__le__iff,axiom,
! [X: complex,Y: complex] :
( ( aa_complex_real(im,X) = aa_complex_real(im,Y) )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(re,X))),aa_real_real(abs_abs_real,aa_complex_real(re,Y)))) ) ) ).
% cmod_Re_le_iff
tff(fact_9472_cmod__Im__le__iff,axiom,
! [X: complex,Y: complex] :
( ( aa_complex_real(re,X) = aa_complex_real(re,Y) )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y)))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(im,X))),aa_real_real(abs_abs_real,aa_complex_real(im,Y)))) ) ) ).
% cmod_Im_le_iff
tff(fact_9473_times__complex_Osimps_I1_J,axiom,
! [X: complex,Y: complex] : aa_complex_real(re,aa_complex_complex(times_times_complex(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(im,Y))) ).
% times_complex.simps(1)
tff(fact_9474_plus__complex_Ocode,axiom,
! [X: complex,Y: complex] : aa_complex_complex(plus_plus_complex(X),Y) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(re,X)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(im,X)),aa_complex_real(im,Y))) ).
% plus_complex.code
tff(fact_9475_scaleR__complex_Ocode,axiom,
! [R: real,X: complex] : aa_complex_complex(real_V2046097035970521341omplex(R),X) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_complex_real(re,X))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_complex_real(im,X))) ).
% scaleR_complex.code
tff(fact_9476_minus__complex_Ocode,axiom,
! [X: complex,Y: complex] : aa_complex_complex(minus_minus_complex(X),Y) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(re,X)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(im,X)),aa_complex_real(im,Y))) ).
% minus_complex.code
tff(fact_9477_csqrt__principal,axiom,
! [Z: complex] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(csqrt,Z))))
| ( ( aa_complex_real(re,aa_complex_complex(csqrt,Z)) = zero_zero_real )
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(csqrt,Z)))) ) ) ).
% csqrt_principal
tff(fact_9478_cmod__le,axiom,
! [Z: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_complex_real(re,Z))),aa_real_real(abs_abs_real,aa_complex_real(im,Z))))) ).
% cmod_le
tff(fact_9479_sin__n__Im__cis__pow__n,axiom,
! [N: nat,A: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),A)) = aa_complex_real(im,aa_nat_complex(power_power_complex(aa_real_complex(cis,A)),N)) ).
% sin_n_Im_cis_pow_n
tff(fact_9480_Re__exp,axiom,
! [Z: complex] : aa_complex_real(re,aa_complex_complex(exp_complex,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,aa_complex_real(re,Z))),aa_real_real(cos_real,aa_complex_real(im,Z))) ).
% Re_exp
tff(fact_9481_Im__exp,axiom,
! [Z: complex] : aa_complex_real(im,aa_complex_complex(exp_complex,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,aa_complex_real(re,Z))),aa_real_real(sin_real,aa_complex_real(im,Z))) ).
% Im_exp
tff(fact_9482_times__complex_Ocode,axiom,
! [X: complex,Y: complex] : aa_complex_complex(times_times_complex(X),Y) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(im,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(re,Y)))) ).
% times_complex.code
tff(fact_9483_cmod__power2,axiom,
! [Z: complex] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).
% cmod_power2
tff(fact_9484_Im__power2,axiom,
! [X: complex] : aa_complex_real(im,aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_complex_real(re,X))),aa_complex_real(im,X)) ).
% Im_power2
tff(fact_9485_Re__power2,axiom,
! [X: complex] : aa_complex_real(re,aa_nat_complex(power_power_complex(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).
% Re_power2
tff(fact_9486_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,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real ) ) ).
% complex_eq_0
tff(fact_9487_norm__complex__def,axiom,
! [Z: complex] : aa_complex_real(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,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).
% norm_complex_def
tff(fact_9488_inverse__complex_Osimps_I1_J,axiom,
! [X: complex] : aa_complex_real(re,aa_complex_complex(invers8013647133539491842omplex,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,X)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).
% inverse_complex.simps(1)
tff(fact_9489_complex__neq__0,axiom,
! [Z: complex] :
( ( Z != zero_zero_complex )
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(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,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).
% complex_neq_0
tff(fact_9490_Re__divide,axiom,
! [X: complex,Y: complex] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).
% Re_divide
tff(fact_9491_csqrt__unique,axiom,
! [W: complex,Z: complex] :
( ( aa_nat_complex(power_power_complex(W),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Z )
=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,W)))
| ( ( aa_complex_real(re,W) = zero_zero_real )
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(im,W))) ) )
=> ( aa_complex_complex(csqrt,Z) = W ) ) ) ).
% csqrt_unique
tff(fact_9492_csqrt__square,axiom,
! [B: complex] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,B)))
| ( ( aa_complex_real(re,B) = zero_zero_real )
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(im,B))) ) )
=> ( aa_complex_complex(csqrt,aa_nat_complex(power_power_complex(B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = B ) ) ).
% csqrt_square
tff(fact_9493_inverse__complex_Osimps_I2_J,axiom,
! [X: complex] : aa_complex_real(im,aa_complex_complex(invers8013647133539491842omplex,X)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_complex_real(im,X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).
% inverse_complex.simps(2)
tff(fact_9494_Im__divide,axiom,
! [X: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(X),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).
% Im_divide
tff(fact_9495_complex__abs__le__norm,axiom,
! [Z: complex] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_complex_real(re,Z))),aa_real_real(abs_abs_real,aa_complex_real(im,Z)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one)))),aa_complex_real(real_V1022390504157884413omplex,Z)))) ).
% complex_abs_le_norm
tff(fact_9496_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(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),aa_complex_real(real_V1022390504157884413omplex,Z))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),aa_complex_real(real_V1022390504157884413omplex,Z))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ) ) ).
% complex_unit_circle
tff(fact_9497_inverse__complex_Ocode,axiom,
! [X: complex] : aa_complex_complex(invers8013647133539491842omplex,X) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,X)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_complex_real(im,X))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).
% inverse_complex.code
tff(fact_9498_Complex__divide,axiom,
! [X: complex,Y: complex] : aa_complex_complex(divide1717551699836669952omplex(X),Y) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,X)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,X)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).
% Complex_divide
tff(fact_9499_Im__Reals__divide,axiom,
! [R: complex,Z: complex] :
( pp(aa_set_complex_bool(member_complex(R),real_V2521375963428798218omplex))
=> ( aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(R),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_complex_real(re,R))),aa_complex_real(im,Z))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).
% Im_Reals_divide
tff(fact_9500_Re__Reals__divide,axiom,
! [R: complex,Z: complex] :
( pp(aa_set_complex_bool(member_complex(R),real_V2521375963428798218omplex))
=> ( aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(R),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,R)),aa_complex_real(re,Z))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).
% Re_Reals_divide
tff(fact_9501_Re__divide__Reals,axiom,
! [R: complex,Z: complex] :
( pp(aa_set_complex_bool(member_complex(R),real_V2521375963428798218omplex))
=> ( aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Z),R)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),aa_complex_real(re,R)) ) ) ).
% Re_divide_Reals
tff(fact_9502_Im__divide__Reals,axiom,
! [R: complex,Z: complex] :
( pp(aa_set_complex_bool(member_complex(R),real_V2521375963428798218omplex))
=> ( aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Z),R)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),aa_complex_real(re,R)) ) ) ).
% Im_divide_Reals
tff(fact_9503_complex__diff__cnj,axiom,
! [Z: complex] : aa_complex_complex(minus_minus_complex(Z),aa_complex_complex(cnj,Z)) = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_complex_real(im,Z)))),imaginary_unit) ).
% complex_diff_cnj
tff(fact_9504_complex__mult__cnj,axiom,
! [Z: complex] : aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).
% complex_mult_cnj
tff(fact_9505_complex__cnj__divide,axiom,
! [X: complex,Y: complex] : aa_complex_complex(cnj,aa_complex_complex(divide1717551699836669952omplex(X),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(cnj,X)),aa_complex_complex(cnj,Y)) ).
% complex_cnj_divide
tff(fact_9506_complex__cnj__diff,axiom,
! [X: complex,Y: complex] : aa_complex_complex(cnj,aa_complex_complex(minus_minus_complex(X),Y)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(cnj,X)),aa_complex_complex(cnj,Y)) ).
% complex_cnj_diff
tff(fact_9507_Re__complex__div__eq__0,axiom,
! [A: complex,B: complex] :
( ( aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B)) = zero_zero_real )
<=> ( aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))) = zero_zero_real ) ) ).
% Re_complex_div_eq_0
tff(fact_9508_Im__complex__div__eq__0,axiom,
! [A: complex,B: complex] :
( ( aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B)) = zero_zero_real )
<=> ( aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))) = zero_zero_real ) ) ).
% Im_complex_div_eq_0
tff(fact_9509_Re__complex__div__lt__0,axiom,
! [A: complex,B: complex] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real)) ) ).
% Re_complex_div_lt_0
tff(fact_9510_Re__complex__div__gt__0,axiom,
! [A: complex,B: complex] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))))) ) ).
% Re_complex_div_gt_0
tff(fact_9511_Re__complex__div__ge__0,axiom,
! [A: complex,B: complex] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))))) ) ).
% Re_complex_div_ge_0
tff(fact_9512_Re__complex__div__le__0,axiom,
! [A: complex,B: complex] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real)) ) ).
% Re_complex_div_le_0
tff(fact_9513_Im__complex__div__lt__0,axiom,
! [A: complex,B: complex] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real)) ) ).
% Im_complex_div_lt_0
tff(fact_9514_Im__complex__div__gt__0,axiom,
! [A: complex,B: complex] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))))) ) ).
% Im_complex_div_gt_0
tff(fact_9515_Im__complex__div__ge__0,axiom,
! [A: complex,B: complex] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))))) ) ).
% Im_complex_div_ge_0
tff(fact_9516_Im__complex__div__le__0,axiom,
! [A: complex,B: complex] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real)) ) ).
% Im_complex_div_le_0
tff(fact_9517_complex__mod__mult__cnj,axiom,
! [Z: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% complex_mod_mult_cnj
tff(fact_9518_complex__div__gt__0,axiom,
! [A: complex,B: complex] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))))) )
& ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))))) ) ) ).
% complex_div_gt_0
tff(fact_9519_complex__norm__square,axiom,
! [Z: complex] : aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z)) ).
% complex_norm_square
tff(fact_9520_complex__add__cnj,axiom,
! [Z: complex] : aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(cnj,Z)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_complex_real(re,Z))) ).
% complex_add_cnj
tff(fact_9521_complex__div__cnj,axiom,
! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))),aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,B)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).
% complex_div_cnj
tff(fact_9522_cnj__add__mult__eq__Re,axiom,
! [Z: complex,W: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,W))),aa_complex_complex(times_times_complex(aa_complex_complex(cnj,Z)),W)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_complex_real(re,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,W))))) ).
% cnj_add_mult_eq_Re
tff(fact_9523_divmod__step__integer__def,axiom,
! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_eb(L),Qr) ).
% divmod_step_integer_def
tff(fact_9524_card__Collect__less__nat,axiom,
! [N: nat] : finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bh,N))) = N ).
% card_Collect_less_nat
tff(fact_9525_card__atMost,axiom,
! [U: nat] : finite_card_nat(aa_nat_set_nat(set_ord_atMost_nat,U)) = aa_nat_nat(suc,U) ).
% card_atMost
tff(fact_9526_card__atLeastLessThan,axiom,
! [L: nat,U: nat] : finite_card_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),L) ).
% card_atLeastLessThan
tff(fact_9527_card__Collect__le__nat,axiom,
! [N: nat] : finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bg,N))) = aa_nat_nat(suc,N) ).
% card_Collect_le_nat
tff(fact_9528_card__atLeastAtMost,axiom,
! [L: nat,U: nat] : finite_card_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,U)),L) ).
% card_atLeastAtMost
tff(fact_9529_card__atLeastLessThan__int,axiom,
! [L: int,U: int] : finite_card_int(set_or4662586982721622107an_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L)) ).
% card_atLeastLessThan_int
tff(fact_9530_card__atLeastAtMost__int,axiom,
! [L: int,U: int] : finite_card_int(set_or1266510415728281911st_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L)),one_one_int)) ).
% card_atLeastAtMost_int
tff(fact_9531_sgn__integer__code,axiom,
! [K: code_integer] :
( ( ( K = zero_z3403309356797280102nteger )
=> ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = zero_z3403309356797280102nteger ) )
& ( ( K != zero_z3403309356797280102nteger )
=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
=> ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) )
& ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
=> ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = one_one_Code_integer ) ) ) ) ) ).
% sgn_integer_code
tff(fact_9532_divmod__integer_H__def,axiom,
! [M: num,N: num] : unique3479559517661332726nteger(M,N) = produc1086072967326762835nteger(divide6298287555418463151nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(N)),modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(N))) ).
% divmod_integer'_def
tff(fact_9533_minus__integer__code_I1_J,axiom,
! [K: code_integer] : minus_8373710615458151222nteger(K,zero_z3403309356797280102nteger) = K ).
% minus_integer_code(1)
tff(fact_9534_minus__integer__code_I2_J,axiom,
! [L: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,L) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,L) ).
% minus_integer_code(2)
tff(fact_9535_card__less,axiom,
! [M6: set_nat,I: nat] :
( pp(aa_set_nat_bool(member_nat(zero_zero_nat),M6))
=> ( finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_pk(M6),I))) != zero_zero_nat ) ) ).
% card_less
tff(fact_9536_card__less__Suc,axiom,
! [M6: set_nat,I: nat] :
( pp(aa_set_nat_bool(member_nat(zero_zero_nat),M6))
=> ( aa_nat_nat(suc,finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_pl(M6),I)))) = finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_pk(M6),I))) ) ) ).
% card_less_Suc
tff(fact_9537_card__less__Suc2,axiom,
! [M6: set_nat,I: nat] :
( ~ pp(aa_set_nat_bool(member_nat(zero_zero_nat),M6))
=> ( finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_pl(M6),I))) = finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_pk(M6),I))) ) ) ).
% card_less_Suc2
tff(fact_9538_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
tff(fact_9539_subset__card__intvl__is__intvl,axiom,
! [A2: set_nat,K: nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(A2),aa_nat_set_nat(set_or4665077453230672383an_nat(K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),finite_card_nat(A2)))))
=> ( A2 = aa_nat_set_nat(set_or4665077453230672383an_nat(K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),finite_card_nat(A2))) ) ) ).
% subset_card_intvl_is_intvl
tff(fact_9540_card__le__Suc__Max,axiom,
! [S2: set_nat] :
( pp(aa_set_nat_bool(finite_finite_nat,S2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_nat(S2)),aa_nat_nat(suc,lattic8265883725875713057ax_nat(S2)))) ) ).
% card_le_Suc_Max
tff(fact_9541_subset__eq__atLeast0__lessThan__card,axiom,
! [N3: set_nat,N: nat] :
( pp(aa_set_nat_bool(ord_less_eq_set_nat(N3),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,finite_card_nat(N3)),N)) ) ).
% subset_eq_atLeast0_lessThan_card
tff(fact_9542_card__sum__le__nat__sum,axiom,
! [S2: set_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_be),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),finite_card_nat(S2)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_be),S2))) ).
% card_sum_le_nat_sum
tff(fact_9543_card__nth__roots,axiom,
! [C: complex,N: nat] :
( ( C != zero_zero_complex )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( finite_card_complex(collect_complex(aa_nat5545486172511849908x_bool(aTP_Lamm_pf(C),N))) = N ) ) ) ).
% card_nth_roots
tff(fact_9544_card__roots__unity__eq,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( finite_card_complex(collect_complex(aTP_Lamm_bj(N))) = N ) ) ).
% card_roots_unity_eq
tff(fact_9545_integer__of__int__code,axiom,
! [K: int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
=> ( aa_int_Code_integer(code_integer_of_int,K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(code_integer_of_int,aa_int_int(uminus_uminus_int,K))) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int))
=> ( ( ( K = zero_zero_int )
=> ( aa_int_Code_integer(code_integer_of_int,K) = zero_z3403309356797280102nteger ) )
& ( ( K != zero_zero_int )
=> ( aa_int_Code_integer(code_integer_of_int,K) = if_Code_integer(aa_int_bool(aa_int_fun_int_bool(fequal_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(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,aa_num_num(bit0,one))))),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(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,aa_num_num(bit0,one))))),one_one_Code_integer)) ) ) ) ) ) ).
% integer_of_int_code
tff(fact_9546_abs__integer__code,axiom,
! [K: code_integer] :
( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
=> ( abs_abs_Code_integer(K) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K) ) )
& ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
=> ( abs_abs_Code_integer(K) = K ) ) ) ).
% abs_integer_code
tff(fact_9547_less__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(code_integer_of_int,Xa2)),aa_int_Code_integer(code_integer_of_int,X)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Xa2),X)) ) ).
% less_integer.abs_eq
tff(fact_9548_less__integer__code_I1_J,axiom,
~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger)) ).
% less_integer_code(1)
tff(fact_9549_divide__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] : divide6298287555418463151nteger(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,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa2),X)) ).
% divide_integer.abs_eq
tff(fact_9550_modulo__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] : modulo364778990260209775nteger(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,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xa2),X)) ).
% modulo_integer.abs_eq
tff(fact_9551_minus__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] : minus_8373710615458151222nteger(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,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa2),X)) ).
% minus_integer.abs_eq
tff(fact_9552_floor__real__def,axiom,
! [X: real] : aa_real_int(archim6058952711729229775r_real,X) = the_int(aTP_Lamm_pm(X)) ).
% floor_real_def
tff(fact_9553_floor__rat__def,axiom,
! [X: rat] : archim3151403230148437115or_rat(X) = the_int(aTP_Lamm_pn(X)) ).
% floor_rat_def
tff(fact_9554_integer__of__num_I3_J,axiom,
! [N: num] : code_integer_of_num(aa_num_num(bit1,N)) = plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(code_integer_of_num(N),code_integer_of_num(N)),one_one_Code_integer) ).
% integer_of_num(3)
tff(fact_9555_less__eq__rat__def,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,X),Y))
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
| ( X = Y ) ) ) ).
% less_eq_rat_def
tff(fact_9556_obtain__pos__sum,axiom,
! [R: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R))
=> ~ ! [S3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),S3))
=> ! [T4: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),T4))
=> ( R != plus_plus_rat(S3,T4) ) ) ) ) ).
% obtain_pos_sum
tff(fact_9557_sgn__rat__def,axiom,
! [A: rat] :
( ( ( A = zero_zero_rat )
=> ( sgn_sgn_rat(A) = zero_zero_rat ) )
& ( ( A != zero_zero_rat )
=> ( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( sgn_sgn_rat(A) = one_one_rat ) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A))
=> ( sgn_sgn_rat(A) = uminus_uminus_rat(one_one_rat) ) ) ) ) ) ).
% sgn_rat_def
tff(fact_9558_abs__rat__def,axiom,
! [A: rat] :
( ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
=> ( abs_abs_rat(A) = uminus_uminus_rat(A) ) )
& ( ~ pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,A),zero_zero_rat))
=> ( abs_abs_rat(A) = A ) ) ) ).
% abs_rat_def
tff(fact_9559_integer__of__num__triv_I1_J,axiom,
code_integer_of_num(one) = one_one_Code_integer ).
% integer_of_num_triv(1)
tff(fact_9560_integer__of__num_I2_J,axiom,
! [N: num] : code_integer_of_num(aa_num_num(bit0,N)) = plus_p5714425477246183910nteger(code_integer_of_num(N),code_integer_of_num(N)) ).
% integer_of_num(2)
tff(fact_9561_integer__of__num__triv_I2_J,axiom,
code_integer_of_num(aa_num_num(bit0,one)) = numera6620942414471956472nteger(aa_num_num(bit0,one)) ).
% integer_of_num_triv(2)
tff(fact_9562_normalize__negative,axiom,
! [Q2: int,P2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Q2),zero_zero_int))
=> ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q2)) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,P2)),aa_int_int(uminus_uminus_int,Q2))) ) ) ).
% normalize_negative
tff(fact_9563_int__of__integer__code,axiom,
! [K: code_integer] :
( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
=> ( code_int_of_integer(K) = aa_int_int(uminus_uminus_int,code_int_of_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K))) ) )
& ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger))
=> ( ( ( K = zero_z3403309356797280102nteger )
=> ( code_int_of_integer(K) = zero_zero_int ) )
& ( ( K != zero_z3403309356797280102nteger )
=> ( code_int_of_integer(K) = produc1553301316500091796er_int(aTP_Lamm_po,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) ) ) ) ) ).
% int_of_integer_code
tff(fact_9564_bit__cut__integer__def,axiom,
! [K: code_integer] : code_bit_cut_integer(K) = produc5135875973608410785r_bool(divide6298287555418463151nteger(K,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_bool_bool(fNot,aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),K))) ).
% bit_cut_integer_def
tff(fact_9565_int__of__integer__numeral,axiom,
! [K: num] : code_int_of_integer(numera6620942414471956472nteger(K)) = aa_num_int(numeral_numeral_int,K) ).
% int_of_integer_numeral
tff(fact_9566_minus__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] : code_int_of_integer(minus_8373710615458151222nteger(X,Xa2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,code_int_of_integer(X)),code_int_of_integer(Xa2)) ).
% minus_integer.rep_eq
tff(fact_9567_divide__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] : code_int_of_integer(divide6298287555418463151nteger(X,Xa2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,code_int_of_integer(X)),code_int_of_integer(Xa2)) ).
% divide_integer.rep_eq
tff(fact_9568_modulo__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] : code_int_of_integer(modulo364778990260209775nteger(X,Xa2)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,code_int_of_integer(X)),code_int_of_integer(Xa2)) ).
% modulo_integer.rep_eq
tff(fact_9569_diff__rat__def,axiom,
! [Q2: rat,R: rat] : minus_minus_rat(Q2,R) = plus_plus_rat(Q2,uminus_uminus_rat(R)) ).
% diff_rat_def
tff(fact_9570_integer__less__iff,axiom,
! [K: code_integer,L: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),L))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,code_int_of_integer(K)),code_int_of_integer(L))) ) ).
% integer_less_iff
tff(fact_9571_less__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Xa2))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,code_int_of_integer(X)),code_int_of_integer(Xa2))) ) ).
% less_integer.rep_eq
tff(fact_9572_normalize__denom__pos,axiom,
! [R: product_prod_int_int,P2: int,Q2: int] :
( ( aa_Pro6970504127699319548nt_int(normalize,R) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q2) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q2)) ) ).
% normalize_denom_pos
tff(fact_9573_divmod__integer__def,axiom,
! [K: code_integer,L: code_integer] : code_divmod_integer(K,L) = produc1086072967326762835nteger(divide6298287555418463151nteger(K,L),modulo364778990260209775nteger(K,L)) ).
% divmod_integer_def
tff(fact_9574_num__of__integer__code,axiom,
! [K: code_integer] :
( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),one_one_Code_integer))
=> ( code_num_of_integer(K) = one ) )
& ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),one_one_Code_integer))
=> ( code_num_of_integer(K) = produc7336495610019696514er_num(aTP_Lamm_pp,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) ) ) ).
% num_of_integer_code
tff(fact_9575_bit__cut__integer__code,axiom,
! [K: code_integer] :
( ( ( K = zero_z3403309356797280102nteger )
=> ( code_bit_cut_integer(K) = produc5135875973608410785r_bool(zero_z3403309356797280102nteger,fFalse) ) )
& ( ( K != zero_z3403309356797280102nteger )
=> ( code_bit_cut_integer(K) = produc6117575242473779287r_bool(aTP_Lamm_pq(K),code_divmod_abs(K,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) ) ) ).
% bit_cut_integer_code
tff(fact_9576_nat__of__integer__code,axiom,
! [K: code_integer] :
( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger))
=> ( code_nat_of_integer(K) = zero_zero_nat ) )
& ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger))
=> ( code_nat_of_integer(K) = produc1555791787009142072er_nat(aTP_Lamm_pr,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ) ) ) ).
% nat_of_integer_code
tff(fact_9577_nat__of__integer__code__post_I3_J,axiom,
! [K: num] : code_nat_of_integer(numera6620942414471956472nteger(K)) = aa_num_nat(numeral_numeral_nat,K) ).
% nat_of_integer_code_post(3)
tff(fact_9578_nat__of__integer__code__post_I2_J,axiom,
code_nat_of_integer(one_one_Code_integer) = one_one_nat ).
% nat_of_integer_code_post(2)
tff(fact_9579_divmod__abs__def,axiom,
! [K: code_integer,L: code_integer] : code_divmod_abs(K,L) = produc1086072967326762835nteger(divide6298287555418463151nteger(abs_abs_Code_integer(K),abs_abs_Code_integer(L)),modulo364778990260209775nteger(abs_abs_Code_integer(K),abs_abs_Code_integer(L))) ).
% divmod_abs_def
tff(fact_9580_divmod__integer__code,axiom,
! [K: code_integer,L: code_integer] :
( ( ( K = zero_z3403309356797280102nteger )
=> ( code_divmod_integer(K,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger) ) )
& ( ( K != zero_z3403309356797280102nteger )
=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L))
=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K))
=> ( code_divmod_integer(K,L) = code_divmod_abs(K,L) ) )
& ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K))
=> ( code_divmod_integer(K,L) = produc6916734918728496179nteger(aTP_Lamm_ps(L),code_divmod_abs(K,L)) ) ) ) )
& ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L))
=> ( ( ( L = zero_z3403309356797280102nteger )
=> ( code_divmod_integer(K,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,K) ) )
& ( ( L != zero_z3403309356797280102nteger )
=> ( code_divmod_integer(K,L) = aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,uminus1351360451143612070nteger),if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_pt(L),code_divmod_abs(K,L)))) ) ) ) ) ) ) ) ).
% divmod_integer_code
tff(fact_9581_rat__minus__code,axiom,
! [P2: rat,Q2: rat] : quotient_of(minus_minus_rat(P2,Q2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_pv(Q2)),quotient_of(P2)) ).
% rat_minus_code
tff(fact_9582_quotient__of__number_I3_J,axiom,
! [K: num] : quotient_of(numeral_numeral_rat(K)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K)),one_one_int) ).
% quotient_of_number(3)
tff(fact_9583_quotient__of__number_I5_J,axiom,
! [K: num] : quotient_of(uminus_uminus_rat(numeral_numeral_rat(K))) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_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_9584_nat_Odisc__eq__case_I2_J,axiom,
! [Nat: nat] :
( ( Nat != zero_zero_nat )
<=> pp(aa_nat_bool(case_nat_bool(fFalse,aTP_Lamm_pw),Nat)) ) ).
% nat.disc_eq_case(2)
tff(fact_9585_nat_Odisc__eq__case_I1_J,axiom,
! [Nat: nat] :
( ( Nat = zero_zero_nat )
<=> pp(aa_nat_bool(case_nat_bool(fTrue,aTP_Lamm_px),Nat)) ) ).
% nat.disc_eq_case(1)
tff(fact_9586_quotient__of__div,axiom,
! [R: rat,N: int,D: int] :
( ( quotient_of(R) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,N),D) )
=> ( R = divide_divide_rat(aa_int_rat(ring_1_of_int_rat,N),aa_int_rat(ring_1_of_int_rat,D)) ) ) ).
% quotient_of_div
tff(fact_9587_less__eq__nat_Osimps_I2_J,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M)),N))
<=> pp(aa_nat_bool(case_nat_bool(fFalse,aa_nat_fun_nat_bool(ord_less_eq_nat,M)),N)) ) ).
% less_eq_nat.simps(2)
tff(fact_9588_quotient__of__denom__pos,axiom,
! [R: rat,P2: int,Q2: int] :
( ( quotient_of(R) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q2) )
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q2)) ) ).
% quotient_of_denom_pos
tff(fact_9589_max__Suc1,axiom,
! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,N)),M) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,N),aa_nat_fun_nat_nat(aTP_Lamm_py,N)),M) ).
% max_Suc1
tff(fact_9590_max__Suc2,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,N),aa_nat_fun_nat_nat(aTP_Lamm_pz,N)),M) ).
% max_Suc2
tff(fact_9591_diff__Suc,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_be),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ).
% diff_Suc
tff(fact_9592_rat__less__code,axiom,
! [P2: rat,Q2: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,P2),Q2))
<=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aTP_Lamm_qb(Q2)),quotient_of(P2))) ) ).
% rat_less_code
tff(fact_9593_rat__divide__code,axiom,
! [P2: rat,Q2: rat] : quotient_of(divide_divide_rat(P2,Q2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_qd(Q2)),quotient_of(P2)) ).
% rat_divide_code
tff(fact_9594_rat__floor__code,axiom,
! [P2: rat] : archim3151403230148437115or_rat(P2) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(divide_divide_int),quotient_of(P2)) ).
% rat_floor_code
tff(fact_9595_divide__rat__def,axiom,
! [Q2: rat,R: rat] : divide_divide_rat(Q2,R) = aa_rat_rat(times_times_rat(Q2),inverse_inverse_rat(R)) ).
% divide_rat_def
tff(fact_9596_pred__def,axiom,
! [Nat: nat] : aa_nat_nat(pred,Nat) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_be),Nat) ).
% pred_def
tff(fact_9597_Frct__code__post_I5_J,axiom,
! [K: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_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_9598_Frct__code__post_I6_J,axiom,
! [K: num,L: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_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_9599_drop__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K)))) ).
% drop_bit_numeral_minus_bit1
tff(fact_9600_drop__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se8568078237143864401it_int(N),K)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K)) ) ).
% drop_bit_nonnegative_int_iff
tff(fact_9601_drop__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(bit_se8568078237143864401it_int(N),K)),zero_zero_int))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K),zero_zero_int)) ) ).
% drop_bit_negative_int_iff
tff(fact_9602_drop__bit__minus__one,axiom,
! [N: nat] : aa_int_int(bit_se8568078237143864401it_int(N),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).
% drop_bit_minus_one
tff(fact_9603_drop__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(bit_se8568078237143864401it_int(N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ).
% drop_bit_Suc_minus_bit0
tff(fact_9604_drop__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ).
% drop_bit_numeral_minus_bit0
tff(fact_9605_drop__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(bit_se8568078237143864401it_int(N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,inc(K)))) ).
% drop_bit_Suc_minus_bit1
tff(fact_9606_drop__bit__push__bit__int,axiom,
! [M: nat,N: nat,K: int] : aa_int_int(bit_se8568078237143864401it_int(M),aa_int_int(bit_se545348938243370406it_int(N),K)) = aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)),K)) ).
% drop_bit_push_bit_int
tff(fact_9607_drop__bit__int__def,axiom,
! [N: nat,K: int] : aa_int_int(bit_se8568078237143864401it_int(N),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,aa_num_num(bit0,one))),N)) ).
% drop_bit_int_def
tff(fact_9608_Frct__code__post_I4_J,axiom,
! [K: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_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_9609_prod__decode__aux_Osimps,axiom,
! [M: nat,K: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),K))
=> ( nat_prod_decode_aux(K,M) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),M)) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),K))
=> ( nat_prod_decode_aux(K,M) = nat_prod_decode_aux(aa_nat_nat(suc,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,K))) ) ) ) ).
% prod_decode_aux.simps
tff(fact_9610_prod__decode__aux_Oelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
( ( nat_prod_decode_aux(X,Xa2) = Y )
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
=> ( Y = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa2)) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
=> ( Y = nat_prod_decode_aux(aa_nat_nat(suc,X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa2),aa_nat_nat(suc,X))) ) ) ) ) ).
% prod_decode_aux.elims
tff(fact_9611_Suc__0__mod__numeral,axiom,
! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,K)) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,unique5055182867167087721od_nat(one,K)) ).
% Suc_0_mod_numeral
tff(fact_9612_drop__bit__of__Suc__0,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,N),zero_zero_nat)) ).
% drop_bit_of_Suc_0
tff(fact_9613_snd__divmod__nat,axiom,
! [M: nat,N: nat] : aa_Pro4243173770044397973at_nat(product_snd_nat_nat,divmod_nat(M,N)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) ).
% snd_divmod_nat
tff(fact_9614_drop__bit__nat__eq,axiom,
! [N: nat,K: int] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(bit_se8568078237143864401it_int(N),K)) ).
% drop_bit_nat_eq
tff(fact_9615_drop__bit__nat__def,axiom,
! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),M) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).
% drop_bit_nat_def
tff(fact_9616_Suc__0__div__numeral,axiom,
! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,K)) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,unique5055182867167087721od_nat(one,K)) ).
% Suc_0_div_numeral
tff(fact_9617_finite__enumerate,axiom,
! [S2: set_nat] :
( pp(aa_set_nat_bool(finite_finite_nat,S2))
=> ? [R3: fun_nat_nat] :
( pp(aa_set_nat_bool(strict1292158309912662752at_nat(R3),aa_nat_set_nat(set_ord_lessThan_nat,finite_card_nat(S2))))
& ! [N6: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N6),finite_card_nat(S2)))
=> pp(aa_set_nat_bool(member_nat(aa_nat_nat(R3,N6)),S2)) ) ) ) ).
% finite_enumerate
tff(fact_9618_divmod__integer__eq__cases,axiom,
! [K: code_integer,L: code_integer] :
( ( ( K = zero_z3403309356797280102nteger )
=> ( code_divmod_integer(K,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger) ) )
& ( ( K != zero_z3403309356797280102nteger )
=> ( ( ( L = zero_z3403309356797280102nteger )
=> ( code_divmod_integer(K,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,K) ) )
& ( ( L != zero_z3403309356797280102nteger )
=> ( code_divmod_integer(K,L) = aa_Pro4693618923521437820nteger(comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(produc6499014454317279255nteger,times_3573771949741848930nteger),sgn_sgn_Code_integer,L),if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,L)),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_qe(L),code_divmod_abs(K,L)))) ) ) ) ) ) ).
% divmod_integer_eq_cases
tff(fact_9619_snd__divmod__integer,axiom,
! [K: code_integer,L: code_integer] : produc6174133586879617921nteger(code_divmod_integer(K,L)) = modulo364778990260209775nteger(K,L) ).
% snd_divmod_integer
tff(fact_9620_snd__divmod__abs,axiom,
! [K: code_integer,L: code_integer] : produc6174133586879617921nteger(code_divmod_abs(K,L)) = modulo364778990260209775nteger(abs_abs_Code_integer(K),abs_abs_Code_integer(L)) ).
% snd_divmod_abs
tff(fact_9621_fst__divmod__nat,axiom,
! [M: nat,N: nat] : aa_Pro4243173770044397973at_nat(product_fst_nat_nat,divmod_nat(M,N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) ).
% fst_divmod_nat
tff(fact_9622_quotient__of__denom__pos_H,axiom,
! [R: rat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_Pro6799258434465214121nt_int(product_snd_int_int,quotient_of(R)))) ).
% quotient_of_denom_pos'
tff(fact_9623_minus__one__mod__numeral,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,N)) = aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,N)),aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(one,N))) ).
% minus_one_mod_numeral
tff(fact_9624_one__mod__minus__numeral,axiom,
! [N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,N)),aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(one,N)))) ).
% one_mod_minus_numeral
tff(fact_9625_minus__numeral__mod__numeral,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,N)),aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(M,N))) ).
% minus_numeral_mod_numeral
tff(fact_9626_fst__divmod__integer,axiom,
! [K: code_integer,L: code_integer] : produc8508995932063986495nteger(code_divmod_integer(K,L)) = divide6298287555418463151nteger(K,L) ).
% fst_divmod_integer
tff(fact_9627_fst__divmod__abs,axiom,
! [K: code_integer,L: code_integer] : produc8508995932063986495nteger(code_divmod_abs(K,L)) = divide6298287555418463151nteger(abs_abs_Code_integer(K),abs_abs_Code_integer(L)) ).
% fst_divmod_abs
tff(fact_9628_numeral__mod__minus__numeral,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,N)),aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(M,N)))) ).
% numeral_mod_minus_numeral
tff(fact_9629_card_Ocomp__fun__commute__on,axiom,
comp_nat_nat_nat(suc,suc) = comp_nat_nat_nat(suc,suc) ).
% card.comp_fun_commute_on
tff(fact_9630_Divides_Oadjust__mod__def,axiom,
! [R: int,L: int] :
( ( ( R = zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(adjust_mod,L),R) = zero_zero_int ) )
& ( ( R != zero_zero_int )
=> ( aa_int_int(aa_int_fun_int_int(adjust_mod,L),R) = aa_int_int(aa_int_fun_int_int(minus_minus_int,L),R) ) ) ) ).
% Divides.adjust_mod_def
tff(fact_9631_bezw_Oelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod_int_int] :
( ( bezw(X,Xa2) = Y )
=> ( ( ( Xa2 = zero_zero_nat )
=> ( Y = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int) ) )
& ( ( Xa2 != zero_zero_nat )
=> ( Y = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa2))))) ) ) ) ) ).
% bezw.elims
tff(fact_9632_bezw__non__0,axiom,
! [Y: nat,X: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Y))
=> ( bezw(X,Y) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y))))) ) ) ).
% bezw_non_0
tff(fact_9633_bezw_Osimps,axiom,
! [Y: nat,X: nat] :
( ( ( Y = zero_zero_nat )
=> ( bezw(X,Y) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int) ) )
& ( ( Y != zero_zero_nat )
=> ( bezw(X,Y) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y))))) ) ) ) ).
% bezw.simps
tff(fact_9634_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_9635_bezw_Opelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod_int_int] :
( ( bezw(X,Xa2) = Y )
=> ( pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2)))
=> ~ ( ( ( ( Xa2 = zero_zero_nat )
=> ( Y = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int) ) )
& ( ( Xa2 != zero_zero_nat )
=> ( Y = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xa2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa2))))) ) ) )
=> ~ pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2))) ) ) ) ).
% bezw.pelims
tff(fact_9636_prod__decode__aux_Opelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
( ( nat_prod_decode_aux(X,Xa2) = Y )
=> ( pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2)))
=> ~ ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
=> ( Y = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa2)) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa2),X))
=> ( Y = nat_prod_decode_aux(aa_nat_nat(suc,X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa2),aa_nat_nat(suc,X))) ) ) )
=> ~ pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2))) ) ) ) ).
% prod_decode_aux.pelims
tff(fact_9637_normalize__def,axiom,
! [P2: product_prod_int_int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)))
=> ( aa_Pro6970504127699319548nt_int(normalize,P2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)))) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)))
=> ( ( ( aa_Pro6799258434465214121nt_int(product_snd_int_int,P2) = zero_zero_int )
=> ( aa_Pro6970504127699319548nt_int(normalize,P2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ) )
& ( ( aa_Pro6799258434465214121nt_int(product_snd_int_int,P2) != zero_zero_int )
=> ( aa_Pro6970504127699319548nt_int(normalize,P2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2))))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2))))) ) ) ) ) ) ).
% normalize_def
tff(fact_9638_gcd__pos__int,axiom,
! [M: int,N: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),N)))
<=> ( ( M != zero_zero_int )
| ( N != zero_zero_int ) ) ) ).
% gcd_pos_int
tff(fact_9639_gcd__neg__numeral__2__int,axiom,
! [X: int,N: num] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),aa_num_int(numeral_numeral_int,N)) ).
% gcd_neg_numeral_2_int
tff(fact_9640_gcd__neg__numeral__1__int,axiom,
! [N: num,X: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),X) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_num_int(numeral_numeral_int,N)),X) ).
% gcd_neg_numeral_1_int
tff(fact_9641_gcd__red__int,axiom,
! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Y),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),Y)) ).
% gcd_red_int
tff(fact_9642_gcd__le1__int,axiom,
! [A: int,B: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),A)) ) ).
% gcd_le1_int
tff(fact_9643_gcd__le2__int,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),B)) ) ).
% gcd_le2_int
tff(fact_9644_gcd__non__0__int,axiom,
! [Y: int,X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Y))
=> ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Y),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),Y)) ) ) ).
% gcd_non_0_int
tff(fact_9645_gcd__code__int,axiom,
! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,K),L) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,L),zero_zero_int)),K),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,L),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L))))) ).
% gcd_code_int
tff(fact_9646_nat__descend__induct,axiom,
! [N: nat,P: fun_nat_bool,M: nat] :
( ! [K3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K3))
=> pp(aa_nat_bool(P,K3)) )
=> ( ! [K3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N))
=> ( ! [I3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),I3))
=> pp(aa_nat_bool(P,I3)) )
=> pp(aa_nat_bool(P,K3)) ) )
=> pp(aa_nat_bool(P,M)) ) ) ).
% nat_descend_induct
tff(fact_9647_card__greaterThanLessThan__int,axiom,
! [L: int,U: int] : finite_card_int(set_or5832277885323065728an_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int))) ).
% card_greaterThanLessThan_int
tff(fact_9648_xor__minus__numerals_I1_J,axiom,
! [N: num,K: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),K) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,neg_numeral_sub_int(N,one)),K)) ).
% xor_minus_numerals(1)
tff(fact_9649_gcd__1__nat,axiom,
! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),one_one_nat) = one_one_nat ).
% gcd_1_nat
tff(fact_9650_gcd__Suc__0,axiom,
! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(suc,zero_zero_nat) ).
% gcd_Suc_0
tff(fact_9651_gcd__pos__nat,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N)))
<=> ( ( M != zero_zero_nat )
| ( N != zero_zero_nat ) ) ) ).
% gcd_pos_nat
tff(fact_9652_xor__minus__numerals_I2_J,axiom,
! [K: int,N: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),neg_numeral_sub_int(N,one))) ).
% xor_minus_numerals(2)
tff(fact_9653_gcd__le1__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),A)) ) ).
% gcd_le1_nat
tff(fact_9654_gcd__le2__nat,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),B)) ) ).
% gcd_le2_nat
tff(fact_9655_gcd__mult__distrib__nat,axiom,
! [K: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N)) = aa_nat_nat(aa_nat_fun_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),N)) ).
% gcd_mult_distrib_nat
tff(fact_9656_gcd__red__nat,axiom,
! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)) ).
% gcd_red_nat
tff(fact_9657_gcd__diff2__nat,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N) ) ) ).
% gcd_diff2_nat
tff(fact_9658_gcd__diff1__nat,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N) ) ) ).
% gcd_diff1_nat
tff(fact_9659_gcd__non__0__nat,axiom,
! [Y: nat,X: nat] :
( ( Y != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)) ) ) ).
% gcd_non_0_nat
tff(fact_9660_gcd__nat_Osimps,axiom,
! [Y: nat,X: nat] :
( ( ( Y = zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = X ) )
& ( ( Y != zero_zero_nat )
=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)) ) ) ) ).
% gcd_nat.simps
tff(fact_9661_gcd__nat_Oelims,axiom,
! [X: nat,Xa2: nat,Y: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Xa2) = Y )
=> ( ( ( Xa2 = zero_zero_nat )
=> ( Y = X ) )
& ( ( Xa2 != zero_zero_nat )
=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)) ) ) ) ) ).
% gcd_nat.elims
tff(fact_9662_bezout__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ? [X3: nat,Y2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)) ) ).
% bezout_nat
tff(fact_9663_bezout__gcd__nat_H,axiom,
! [B: nat,A: nat] :
? [X3: nat,Y2: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3)))
& ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) )
| ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X3)))
& ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y2)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ) ).
% bezout_gcd_nat'
tff(fact_9664_gcd__code__integer,axiom,
! [K: code_integer,L: code_integer] : gcd_gcd_Code_integer(K,L) = abs_abs_Code_integer(if_Code_integer(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,L),zero_z3403309356797280102nteger),K,gcd_gcd_Code_integer(L,modulo364778990260209775nteger(abs_abs_Code_integer(K),abs_abs_Code_integer(L))))) ).
% gcd_code_integer
tff(fact_9665_gcd__is__Max__divisors__nat,axiom,
! [N: nat,M: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N) = lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_qf(N),M))) ) ) ).
% gcd_is_Max_divisors_nat
tff(fact_9666_sub__BitM__One__eq,axiom,
! [N: num] : neg_numeral_sub_int(bitM(N),one) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),neg_numeral_sub_int(N,one)) ).
% sub_BitM_One_eq
tff(fact_9667_gcd__nat_Opelims,axiom,
! [X: nat,Xa2: nat,Y: nat] :
( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Xa2) = Y )
=> ( pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2)))
=> ~ ( ( ( ( Xa2 = zero_zero_nat )
=> ( Y = X ) )
& ( ( Xa2 != zero_zero_nat )
=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa2)) ) ) )
=> ~ pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa2))) ) ) ) ).
% gcd_nat.pelims
tff(fact_9668_card__greaterThanLessThan,axiom,
! [L: nat,U: nat] : finite_card_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),aa_nat_nat(suc,L)) ).
% card_greaterThanLessThan
tff(fact_9669_atLeastSucLessThan__greaterThanLessThan,axiom,
! [L: nat,U: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,L)),U) = aa_nat_set_nat(set_or5834768355832116004an_nat(L),U) ).
% atLeastSucLessThan_greaterThanLessThan
tff(fact_9670_tanh__real__bounds,axiom,
! [X: real] : pp(aa_set_real_bool(member_real(aa_real_real(tanh_real,X)),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real))) ).
% tanh_real_bounds
tff(fact_9671_Suc__funpow,axiom,
! [N: nat] : compow_fun_nat_nat(N,suc) = aa_nat_fun_nat_nat(plus_plus_nat,N) ).
% Suc_funpow
tff(fact_9672_max__nat_Osemilattice__neutr__order__axioms,axiom,
semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_bg,aTP_Lamm_bh) ).
% max_nat.semilattice_neutr_order_axioms
tff(fact_9673_times__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_qh),Xa2),X)) ).
% times_int.abs_eq
tff(fact_9674_eq__Abs__Integ,axiom,
! [Z: int] :
~ ! [X3: nat,Y2: nat] : Z != aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y2)) ).
% eq_Abs_Integ
tff(fact_9675_nat_Oabs__eq,axiom,
! [X: product_prod_nat_nat] : aa_int_nat(nat2,aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),X) ).
% nat.abs_eq
tff(fact_9676_zero__int__def,axiom,
zero_zero_int = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)) ).
% zero_int_def
tff(fact_9677_int__def,axiom,
! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,N) = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N),zero_zero_nat)) ).
% int_def
tff(fact_9678_uminus__int_Oabs__eq,axiom,
! [X: product_prod_nat_nat] : aa_int_int(uminus_uminus_int,aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_qi),X)) ).
% uminus_int.abs_eq
tff(fact_9679_one__int__def,axiom,
one_one_int = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)) ).
% one_int_def
tff(fact_9680_less__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)))
<=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_qk),Xa2),X)) ) ).
% less_int.abs_eq
tff(fact_9681_less__eq__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)))
<=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_qm),Xa2),X)) ) ).
% less_eq_int.abs_eq
tff(fact_9682_plus__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_qo),Xa2),X)) ).
% plus_int.abs_eq
tff(fact_9683_minus__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa2)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_qq),Xa2),X)) ).
% minus_int.abs_eq
tff(fact_9684_Gcd__remove0__nat,axiom,
! [M6: set_nat] :
( pp(aa_set_nat_bool(finite_finite_nat,M6))
=> ( gcd_Gcd_nat(M6) = gcd_Gcd_nat(minus_minus_set_nat(M6,insert_nat(zero_zero_nat,bot_bot_set_nat))) ) ) ).
% Gcd_remove0_nat
tff(fact_9685_Gcd__nat__eq__one,axiom,
! [N3: set_nat] :
( pp(aa_set_nat_bool(member_nat(one_one_nat),N3))
=> ( gcd_Gcd_nat(N3) = one_one_nat ) ) ).
% Gcd_nat_eq_one
tff(fact_9686_less__eq__int_Orep__eq,axiom,
! [X: int,Xa2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa2))
<=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_qm),aa_int1417691210696471063at_nat(rep_Integ,X)),aa_int1417691210696471063at_nat(rep_Integ,Xa2))) ) ).
% less_eq_int.rep_eq
tff(fact_9687_less__int_Orep__eq,axiom,
! [X: int,Xa2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Xa2))
<=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_qk),aa_int1417691210696471063at_nat(rep_Integ,X)),aa_int1417691210696471063at_nat(rep_Integ,Xa2))) ) ).
% less_int.rep_eq
tff(fact_9688_prod__encode__def,axiom,
nat_prod_encode = produc6842872674320459806at_nat(aTP_Lamm_qr) ).
% prod_encode_def
tff(fact_9689_le__prod__encode__1,axiom,
! [A: nat,B: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A),aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)))) ).
% le_prod_encode_1
tff(fact_9690_le__prod__encode__2,axiom,
! [B: nat,A: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B),aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)))) ).
% le_prod_encode_2
tff(fact_9691_nat_Orep__eq,axiom,
! [X: int] : aa_int_nat(nat2,X) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),aa_int1417691210696471063at_nat(rep_Integ,X)) ).
% nat.rep_eq
tff(fact_9692_prod__encode__prod__decode__aux,axiom,
! [K: nat,M: nat] : aa_Pro4243173770044397973at_nat(nat_prod_encode,nat_prod_decode_aux(K,M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,K)),M) ).
% prod_encode_prod_decode_aux
tff(fact_9693_uminus__int__def,axiom,
uminus_uminus_int = aa_fun8399641135911395862nt_int(map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc2626176000494625587at_nat(aTP_Lamm_qi)) ).
% uminus_int_def
tff(fact_9694_num__of__nat_Osimps_I2_J,axiom,
! [N: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_num(num_of_nat,aa_nat_nat(suc,N)) = inc(aa_nat_num(num_of_nat,N)) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_num(num_of_nat,aa_nat_nat(suc,N)) = one ) ) ) ).
% num_of_nat.simps(2)
tff(fact_9695_num__of__nat__numeral__eq,axiom,
! [Q2: num] : aa_nat_num(num_of_nat,aa_num_nat(numeral_numeral_nat,Q2)) = Q2 ).
% num_of_nat_numeral_eq
tff(fact_9696_num__of__nat_Osimps_I1_J,axiom,
aa_nat_num(num_of_nat,zero_zero_nat) = one ).
% num_of_nat.simps(1)
tff(fact_9697_numeral__num__of__nat,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_num_nat(numeral_numeral_nat,aa_nat_num(num_of_nat,N)) = N ) ) ).
% numeral_num_of_nat
tff(fact_9698_num__of__nat__One,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),one_one_nat))
=> ( aa_nat_num(num_of_nat,N) = one ) ) ).
% num_of_nat_One
tff(fact_9699_num__of__nat__double,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N)) = aa_num_num(bit0,aa_nat_num(num_of_nat,N)) ) ) ).
% num_of_nat_double
tff(fact_9700_num__of__nat__plus__distrib,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_nat_num(num_of_nat,M)),aa_nat_num(num_of_nat,N)) ) ) ) ).
% num_of_nat_plus_distrib
tff(fact_9701_times__int__def,axiom,
times_times_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_qh)) ).
% times_int_def
tff(fact_9702_minus__int__def,axiom,
minus_minus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_qq)) ).
% minus_int_def
tff(fact_9703_plus__int__def,axiom,
plus_plus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_qo)) ).
% plus_int_def
tff(fact_9704_pow_Osimps_I3_J,axiom,
! [X: num,Y: num] : aa_num_num(aa_num_fun_num_num(pow,X),aa_num_num(bit1,Y)) = aa_num_num(aa_num_fun_num_num(times_times_num,sqr(aa_num_num(aa_num_fun_num_num(pow,X),Y))),X) ).
% pow.simps(3)
tff(fact_9705_sqr_Osimps_I2_J,axiom,
! [N: num] : sqr(aa_num_num(bit0,N)) = aa_num_num(bit0,aa_num_num(bit0,sqr(N))) ).
% sqr.simps(2)
tff(fact_9706_sqr_Osimps_I1_J,axiom,
sqr(one) = one ).
% sqr.simps(1)
tff(fact_9707_sqr__conv__mult,axiom,
! [X: num] : sqr(X) = aa_num_num(aa_num_fun_num_num(times_times_num,X),X) ).
% sqr_conv_mult
tff(fact_9708_pow_Osimps_I2_J,axiom,
! [X: num,Y: num] : aa_num_num(aa_num_fun_num_num(pow,X),aa_num_num(bit0,Y)) = sqr(aa_num_num(aa_num_fun_num_num(pow,X),Y)) ).
% pow.simps(2)
tff(fact_9709_sqr_Osimps_I3_J,axiom,
! [N: num] : sqr(aa_num_num(bit1,N)) = aa_num_num(bit1,aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,sqr(N)),N))) ).
% sqr.simps(3)
tff(fact_9710_nth__sorted__list__of__set__greaterThanLessThan,axiom,
! [N: nat,J: nat,I: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),aa_nat_nat(suc,I))))
=> ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(I),J))),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),N)) ) ) ).
% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_9711_rat__floor__lemma,axiom,
! [A: int,B: int] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_rat(fract(A),B)))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(fract(A),B)),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int)))) ) ).
% rat_floor_lemma
tff(fact_9712_divide__rat,axiom,
! [A: int,B: int,C: int,D: int] : divide_divide_rat(aa_int_rat(fract(A),B),aa_int_rat(fract(C),D)) = aa_int_rat(fract(aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).
% divide_rat
tff(fact_9713_floor__Fract,axiom,
! [A: int,B: int] : archim3151403230148437115or_rat(aa_int_rat(fract(A),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ).
% floor_Fract
tff(fact_9714_less__rat,axiom,
! [B: int,D: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(fract(A),B)),aa_int_rat(fract(C),D)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)))) ) ) ) ).
% less_rat
tff(fact_9715_diff__rat,axiom,
! [B: int,D: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( minus_minus_rat(aa_int_rat(fract(A),B),aa_int_rat(fract(C),D)) = aa_int_rat(fract(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ).
% diff_rat
tff(fact_9716_Rat__induct__pos,axiom,
! [P: fun_rat_bool,Q2: rat] :
( ! [A4: int,B4: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B4))
=> pp(aa_rat_bool(P,aa_int_rat(fract(A4),B4))) )
=> pp(aa_rat_bool(P,Q2)) ) ).
% Rat_induct_pos
tff(fact_9717_Fract__coprime,axiom,
! [A: int,B: int] : aa_int_rat(fract(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B))),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B))) = aa_int_rat(fract(A),B) ).
% Fract_coprime
tff(fact_9718_Fract__of__int__quotient,axiom,
! [K: int,L: int] : aa_int_rat(fract(K),L) = divide_divide_rat(aa_int_rat(ring_1_of_int_rat,K),aa_int_rat(ring_1_of_int_rat,L)) ).
% Fract_of_int_quotient
tff(fact_9719_rat__number__expand_I3_J,axiom,
! [K: num] : numeral_numeral_rat(K) = aa_int_rat(fract(aa_num_int(numeral_numeral_int,K)),one_one_int) ).
% rat_number_expand(3)
tff(fact_9720_rat__number__collapse_I3_J,axiom,
! [W: num] : aa_int_rat(fract(aa_num_int(numeral_numeral_int,W)),one_one_int) = numeral_numeral_rat(W) ).
% rat_number_collapse(3)
tff(fact_9721_Fract__less__zero__iff,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(fract(A),B)),zero_zero_rat))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),zero_zero_int)) ) ) ).
% Fract_less_zero_iff
tff(fact_9722_zero__less__Fract__iff,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),aa_int_rat(fract(A),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A)) ) ) ).
% zero_less_Fract_iff
tff(fact_9723_one__less__Fract__iff,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,one_one_rat),aa_int_rat(fract(A),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B),A)) ) ) ).
% one_less_Fract_iff
tff(fact_9724_Fract__less__one__iff,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,aa_int_rat(fract(A),B)),one_one_rat))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A),B)) ) ) ).
% Fract_less_one_iff
tff(fact_9725_zero__le__Fract__iff,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,zero_zero_rat),aa_int_rat(fract(A),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A)) ) ) ).
% zero_le_Fract_iff
tff(fact_9726_Fract__le__zero__iff,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(fract(A),B)),zero_zero_rat))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),zero_zero_int)) ) ) ).
% Fract_le_zero_iff
tff(fact_9727_one__le__Fract__iff,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,one_one_rat),aa_int_rat(fract(A),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B),A)) ) ) ).
% one_le_Fract_iff
tff(fact_9728_Fract__le__one__iff,axiom,
! [B: int,A: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(fract(A),B)),one_one_rat))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A),B)) ) ) ).
% Fract_le_one_iff
tff(fact_9729_rat__number__expand_I5_J,axiom,
! [K: num] : uminus_uminus_rat(numeral_numeral_rat(K)) = aa_int_rat(fract(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int) ).
% rat_number_expand(5)
tff(fact_9730_rat__number__collapse_I4_J,axiom,
! [W: num] : aa_int_rat(fract(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),one_one_int) = uminus_uminus_rat(numeral_numeral_rat(W)) ).
% rat_number_collapse(4)
tff(fact_9731_nth__sorted__list__of__set__greaterThanAtMost,axiom,
! [N: nat,J: nat,I: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)))
=> ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(I),J))),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),N)) ) ) ).
% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_9732_take__bit__numeral__minus__numeral__int,axiom,
! [M: num,N: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = case_option_int_num(zero_zero_int,aTP_Lamm_qs(M),bit_take_bit_num(aa_num_nat(numeral_numeral_nat,M),N)) ).
% take_bit_numeral_minus_numeral_int
tff(fact_9733_of__nat__eq__id,axiom,
semiri1316708129612266289at_nat = id_nat ).
% of_nat_eq_id
tff(fact_9734_take__bit__num__simps_I1_J,axiom,
! [M: num] : bit_take_bit_num(zero_zero_nat,M) = none_num ).
% take_bit_num_simps(1)
tff(fact_9735_take__bit__num__simps_I2_J,axiom,
! [N: nat] : bit_take_bit_num(aa_nat_nat(suc,N),one) = some_num(one) ).
% take_bit_num_simps(2)
tff(fact_9736_take__bit__num__simps_I5_J,axiom,
! [R: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R),one) = some_num(one) ).
% take_bit_num_simps(5)
tff(fact_9737_card__greaterThanAtMost,axiom,
! [L: nat,U: nat] : finite_card_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),L) ).
% card_greaterThanAtMost
tff(fact_9738_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
! [N: nat] : bit_take_bit_num(N,one) = case_nat_option_num(none_num,aTP_Lamm_qt,N) ).
% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_9739_atLeastSucAtMost__greaterThanAtMost,axiom,
! [L: nat,U: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,L)),U) = aa_nat_set_nat(set_or6659071591806873216st_nat(L),U) ).
% atLeastSucAtMost_greaterThanAtMost
tff(fact_9740_less__int__def,axiom,
ord_less_int = map_fu2743658913401090412t_bool(rep_Integ,map_fu6608131713810608381l_bool(rep_Integ,id_bool),produc5023667602222594023t_bool(aTP_Lamm_qk)) ).
% less_int_def
tff(fact_9741_less__eq__int__def,axiom,
ord_less_eq_int = map_fu2743658913401090412t_bool(rep_Integ,map_fu6608131713810608381l_bool(rep_Integ,id_bool),produc5023667602222594023t_bool(aTP_Lamm_qm)) ).
% less_eq_int_def
tff(fact_9742_nat__def,axiom,
nat2 = map_fu2345160673673942751at_nat(rep_Integ,id_nat,produc6842872674320459806at_nat(minus_minus_nat)) ).
% nat_def
tff(fact_9743_take__bit__num__def,axiom,
! [N: nat,M: num] :
( ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M)) = zero_zero_nat )
=> ( bit_take_bit_num(N,M) = none_num ) )
& ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M)) != zero_zero_nat )
=> ( bit_take_bit_num(N,M) = some_num(aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M)))) ) ) ) ).
% take_bit_num_def
tff(fact_9744_and__minus__numerals_I3_J,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,bitM(N))) ).
% and_minus_numerals(3)
tff(fact_9745_and__minus__numerals_I7_J,axiom,
! [N: num,M: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))),aa_num_int(numeral_numeral_int,M)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,bitM(N))) ).
% and_minus_numerals(7)
tff(fact_9746_and__minus__numerals_I4_J,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,aa_num_num(bit0,N))) ).
% and_minus_numerals(4)
tff(fact_9747_card__greaterThanAtMost__int,axiom,
! [L: int,U: int] : finite_card_int(set_or6656581121297822940st_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L)) ).
% card_greaterThanAtMost_int
tff(fact_9748_take__bit__num__simps_I4_J,axiom,
! [N: nat,M: num] : bit_take_bit_num(aa_nat_nat(suc,N),aa_num_num(bit1,M)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(N,M))) ).
% take_bit_num_simps(4)
tff(fact_9749_take__bit__num__simps_I3_J,axiom,
! [N: nat,M: num] : bit_take_bit_num(aa_nat_nat(suc,N),aa_num_num(bit0,M)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_qu,bit_take_bit_num(N,M)) ).
% take_bit_num_simps(3)
tff(fact_9750_take__bit__num__simps_I7_J,axiom,
! [R: num,M: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R),aa_num_num(bit1,M)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(aa_num_nat(pred_numeral,R),M))) ).
% take_bit_num_simps(7)
tff(fact_9751_take__bit__num__simps_I6_J,axiom,
! [R: num,M: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R),aa_num_num(bit0,M)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_qu,bit_take_bit_num(aa_num_nat(pred_numeral,R),M)) ).
% take_bit_num_simps(6)
tff(fact_9752_and__minus__numerals_I8_J,axiom,
! [N: num,M: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))),aa_num_int(numeral_numeral_int,M)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,aa_num_num(bit0,N))) ).
% and_minus_numerals(8)
tff(fact_9753_and__not__num_Osimps_I8_J,axiom,
! [M: num,N: num] : bit_and_not_num(aa_num_num(bit1,M),aa_num_num(bit0,N)) = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_qv,bit_and_not_num(M,N)) ).
% and_not_num.simps(8)
tff(fact_9754_and__not__num_Osimps_I1_J,axiom,
bit_and_not_num(one,one) = none_num ).
% and_not_num.simps(1)
tff(fact_9755_and__not__num_Osimps_I4_J,axiom,
! [M: num] : bit_and_not_num(aa_num_num(bit0,M),one) = some_num(aa_num_num(bit0,M)) ).
% and_not_num.simps(4)
tff(fact_9756_and__not__num_Osimps_I2_J,axiom,
! [N: num] : bit_and_not_num(one,aa_num_num(bit0,N)) = some_num(one) ).
% and_not_num.simps(2)
tff(fact_9757_and__not__num_Osimps_I3_J,axiom,
! [N: num] : bit_and_not_num(one,aa_num_num(bit1,N)) = none_num ).
% and_not_num.simps(3)
tff(fact_9758_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
! [N: nat,M: num] : bit_take_bit_num(N,aa_num_num(bit0,M)) = case_nat_option_num(none_num,aTP_Lamm_qw(M),N) ).
% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_9759_and__not__num_Osimps_I7_J,axiom,
! [M: num] : bit_and_not_num(aa_num_num(bit1,M),one) = some_num(aa_num_num(bit0,M)) ).
% and_not_num.simps(7)
tff(fact_9760_and__not__num__eq__Some__iff,axiom,
! [M: num,N: num,Q2: num] :
( ( bit_and_not_num(M,N) = some_num(Q2) )
<=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,Q2) ) ) ).
% and_not_num_eq_Some_iff
tff(fact_9761_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
! [N: nat,M: num] : bit_take_bit_num(N,aa_num_num(bit1,M)) = case_nat_option_num(none_num,aTP_Lamm_qx(M),N) ).
% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_9762_and__not__num__eq__None__iff,axiom,
! [M: num,N: num] :
( ( bit_and_not_num(M,N) = none_num )
<=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))) = zero_zero_int ) ) ).
% and_not_num_eq_None_iff
tff(fact_9763_int__numeral__and__not__num,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(M,N)) ).
% int_numeral_and_not_num
tff(fact_9764_int__numeral__not__and__num,axiom,
! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(N,M)) ).
% int_numeral_not_and_num
tff(fact_9765_Bit__Operations_Otake__bit__num__code,axiom,
! [N: nat,M: num] : bit_take_bit_num(N,M) = produc478579273971653890on_num(aTP_Lamm_rb,product_Pair_nat_num(N,M)) ).
% Bit_Operations.take_bit_num_code
tff(fact_9766_positive__rat,axiom,
! [A: int,B: int] :
( pp(aa_rat_bool(positive,aa_int_rat(fract(A),B)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))) ) ).
% positive_rat
tff(fact_9767_less__rat__def,axiom,
! [X: rat,Y: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,X),Y))
<=> pp(aa_rat_bool(positive,minus_minus_rat(Y,X))) ) ).
% less_rat_def
tff(fact_9768_Rat_Opositive_Orep__eq,axiom,
! [X: rat] :
( pp(aa_rat_bool(positive,X))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_rat7754553326175492643nt_int(rep_Rat,X))),aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_rat7754553326175492643nt_int(rep_Rat,X))))) ) ).
% Rat.positive.rep_eq
tff(fact_9769_and__not__num_Oelims,axiom,
! [X: num,Xa2: num,Y: option_num] :
( ( bit_and_not_num(X,Xa2) = Y )
=> ( ( ( X = one )
=> ( ( Xa2 = one )
=> ( Y != none_num ) ) )
=> ( ( ( X = one )
=> ( ? [N2: num] : Xa2 = aa_num_num(bit0,N2)
=> ( Y != some_num(one) ) ) )
=> ( ( ( X = one )
=> ( ? [N2: num] : Xa2 = aa_num_num(bit1,N2)
=> ( Y != none_num ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ( ( Xa2 = one )
=> ( Y != some_num(aa_num_num(bit0,M3)) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit0,N2) )
=> ( Y != map_option_num_num(bit0,bit_and_not_num(M3,N2)) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit1,N2) )
=> ( Y != map_option_num_num(bit0,bit_and_not_num(M3,N2)) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ( ( Xa2 = one )
=> ( Y != some_num(aa_num_num(bit0,M3)) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit0,N2) )
=> ( Y != case_o6005452278849405969um_num(some_num(one),aTP_Lamm_qv,bit_and_not_num(M3,N2)) ) ) )
=> ~ ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit1,N2) )
=> ( Y != map_option_num_num(bit0,bit_and_not_num(M3,N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.elims
tff(fact_9770_and__not__num_Osimps_I5_J,axiom,
! [M: num,N: num] : bit_and_not_num(aa_num_num(bit0,M),aa_num_num(bit0,N)) = map_option_num_num(bit0,bit_and_not_num(M,N)) ).
% and_not_num.simps(5)
tff(fact_9771_and__not__num_Osimps_I6_J,axiom,
! [M: num,N: num] : bit_and_not_num(aa_num_num(bit0,M),aa_num_num(bit1,N)) = map_option_num_num(bit0,bit_and_not_num(M,N)) ).
% and_not_num.simps(6)
tff(fact_9772_and__not__num_Osimps_I9_J,axiom,
! [M: num,N: num] : bit_and_not_num(aa_num_num(bit1,M),aa_num_num(bit1,N)) = map_option_num_num(bit0,bit_and_not_num(M,N)) ).
% and_not_num.simps(9)
tff(fact_9773_Rat_Opositive__def,axiom,
positive = map_fu5118516228767806217l_bool(rep_Rat,id_bool,aTP_Lamm_rc) ).
% Rat.positive_def
tff(fact_9774_and__num_Oelims,axiom,
! [X: num,Xa2: num,Y: option_num] :
( ( aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,X),Xa2) = Y )
=> ( ( ( X = one )
=> ( ( Xa2 = one )
=> ( Y != some_num(one) ) ) )
=> ( ( ( X = one )
=> ( ? [N2: num] : Xa2 = aa_num_num(bit0,N2)
=> ( Y != none_num ) ) )
=> ( ( ( X = one )
=> ( ? [N2: num] : Xa2 = aa_num_num(bit1,N2)
=> ( Y != some_num(one) ) ) )
=> ( ( ? [M3: num] : X = aa_num_num(bit0,M3)
=> ( ( Xa2 = one )
=> ( Y != none_num ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit0,N2) )
=> ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M3),N2)) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit1,N2) )
=> ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M3),N2)) ) ) )
=> ( ( ? [M3: num] : X = aa_num_num(bit1,M3)
=> ( ( Xa2 = one )
=> ( Y != some_num(one) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit0,N2) )
=> ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M3),N2)) ) ) )
=> ~ ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit1,N2) )
=> ( Y != case_o6005452278849405969um_num(some_num(one),aTP_Lamm_qv,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M3),N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.elims
tff(fact_9775_xor__num_Oelims,axiom,
! [X: num,Xa2: num,Y: option_num] :
( ( aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,X),Xa2) = Y )
=> ( ( ( X = one )
=> ( ( Xa2 = one )
=> ( Y != none_num ) ) )
=> ( ( ( X = one )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit0,N2) )
=> ( Y != some_num(aa_num_num(bit1,N2)) ) ) )
=> ( ( ( X = one )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit1,N2) )
=> ( Y != some_num(aa_num_num(bit0,N2)) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ( ( Xa2 = one )
=> ( Y != some_num(aa_num_num(bit1,M3)) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit0,N2) )
=> ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M3),N2)) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit1,N2) )
=> ( Y != some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M3),N2))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ( ( Xa2 = one )
=> ( Y != some_num(aa_num_num(bit0,M3)) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit0,N2) )
=> ( Y != some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M3),N2))) ) ) )
=> ~ ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit1,N2) )
=> ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M3),N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.elims
tff(fact_9776_and__num_Osimps_I1_J,axiom,
aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,one),one) = some_num(one) ).
% and_num.simps(1)
tff(fact_9777_xor__num_Osimps_I1_J,axiom,
aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,one),one) = none_num ).
% xor_num.simps(1)
tff(fact_9778_xor__num_Osimps_I5_J,axiom,
! [M: num,N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit0,M)),aa_num_num(bit0,N)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M),N)) ).
% xor_num.simps(5)
tff(fact_9779_and__num_Osimps_I5_J,axiom,
! [M: num,N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit0,M)),aa_num_num(bit0,N)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M),N)) ).
% and_num.simps(5)
tff(fact_9780_and__num_Osimps_I3_J,axiom,
! [N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,one),aa_num_num(bit1,N)) = some_num(one) ).
% and_num.simps(3)
tff(fact_9781_and__num_Osimps_I7_J,axiom,
! [M: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit1,M)),one) = some_num(one) ).
% and_num.simps(7)
tff(fact_9782_and__num_Osimps_I2_J,axiom,
! [N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,one),aa_num_num(bit0,N)) = none_num ).
% and_num.simps(2)
tff(fact_9783_and__num_Osimps_I4_J,axiom,
! [M: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit0,M)),one) = none_num ).
% and_num.simps(4)
tff(fact_9784_and__num_Osimps_I8_J,axiom,
! [M: num,N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit1,M)),aa_num_num(bit0,N)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M),N)) ).
% and_num.simps(8)
tff(fact_9785_and__num_Osimps_I6_J,axiom,
! [M: num,N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit0,M)),aa_num_num(bit1,N)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M),N)) ).
% and_num.simps(6)
tff(fact_9786_xor__num_Osimps_I9_J,axiom,
! [M: num,N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit1,M)),aa_num_num(bit1,N)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M),N)) ).
% xor_num.simps(9)
tff(fact_9787_xor__num_Osimps_I2_J,axiom,
! [N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,one),aa_num_num(bit0,N)) = some_num(aa_num_num(bit1,N)) ).
% xor_num.simps(2)
tff(fact_9788_xor__num_Osimps_I3_J,axiom,
! [N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,one),aa_num_num(bit1,N)) = some_num(aa_num_num(bit0,N)) ).
% xor_num.simps(3)
tff(fact_9789_xor__num_Osimps_I4_J,axiom,
! [M: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit0,M)),one) = some_num(aa_num_num(bit1,M)) ).
% xor_num.simps(4)
tff(fact_9790_xor__num_Osimps_I7_J,axiom,
! [M: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit1,M)),one) = some_num(aa_num_num(bit0,M)) ).
% xor_num.simps(7)
tff(fact_9791_and__num_Osimps_I9_J,axiom,
! [M: num,N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit1,M)),aa_num_num(bit1,N)) = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_qv,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M),N)) ).
% and_num.simps(9)
tff(fact_9792_xor__num_Osimps_I6_J,axiom,
! [M: num,N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit0,M)),aa_num_num(bit1,N)) = some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M),N))) ).
% xor_num.simps(6)
tff(fact_9793_xor__num_Osimps_I8_J,axiom,
! [M: num,N: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit1,M)),aa_num_num(bit0,N)) = some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M),N))) ).
% xor_num.simps(8)
tff(fact_9794_xor__num__dict,axiom,
bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).
% xor_num_dict
tff(fact_9795_and__num__dict,axiom,
bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).
% and_num_dict
tff(fact_9796_of__rat__dense,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
=> ? [Q3: rat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),field_7254667332652039916t_real(Q3)))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,field_7254667332652039916t_real(Q3)),Y)) ) ) ).
% of_rat_dense
tff(fact_9797_min__Suc__Suc,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)) ).
% min_Suc_Suc
tff(fact_9798_min__0R,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),zero_zero_nat) = zero_zero_nat ).
% min_0R
tff(fact_9799_min__0L,axiom,
! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),N) = zero_zero_nat ).
% min_0L
tff(fact_9800_min__Suc__numeral,axiom,
! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),aa_num_nat(pred_numeral,K))) ).
% min_Suc_numeral
tff(fact_9801_min__numeral__Suc,axiom,
! [K: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(pred_numeral,K)),N)) ).
% min_numeral_Suc
tff(fact_9802_nat__mult__min__left,axiom,
! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) ).
% nat_mult_min_left
tff(fact_9803_nat__mult__min__right,axiom,
! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)) ).
% nat_mult_min_right
tff(fact_9804_min__diff,axiom,
! [M: nat,I: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),I)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),I)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)),I) ).
% min_diff
tff(fact_9805_concat__bit__assoc__sym,axiom,
! [M: nat,N: nat,K: int,L: int,R: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(M),aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),L)),R) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)),K),aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),L),R)) ).
% concat_bit_assoc_sym
tff(fact_9806_take__bit__concat__bit__eq,axiom,
! [M: nat,N: nat,K: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(M),aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),L)) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)),K),aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),L)) ).
% take_bit_concat_bit_eq
tff(fact_9807_min__Suc2,axiom,
! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_rd,N)),M) ).
% min_Suc2
tff(fact_9808_min__Suc1,axiom,
! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,N)),M) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_re,N)),M) ).
% min_Suc1
tff(fact_9809_min__enat__simps_I2_J,axiom,
! [Q2: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_mi8085742599997312461d_enat,Q2),zero_z5237406670263579293d_enat) = zero_z5237406670263579293d_enat ).
% min_enat_simps(2)
tff(fact_9810_min__enat__simps_I3_J,axiom,
! [Q2: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_mi8085742599997312461d_enat,zero_z5237406670263579293d_enat),Q2) = zero_z5237406670263579293d_enat ).
% min_enat_simps(3)
tff(fact_9811_Inf__real__def,axiom,
! [X5: set_real] : comple4887499456419720421f_real(X5) = aa_real_real(uminus_uminus_real,comple1385675409528146559p_real(image_real_real(uminus_uminus_real,X5))) ).
% Inf_real_def
tff(fact_9812_range__mod,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rf,N),top_top_set_nat) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N) ) ) ).
% range_mod
tff(fact_9813_suminf__eq__SUP__real,axiom,
! [X5: fun_nat_real] :
( summable_real(X5)
=> ( ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(X5,I2)))
=> ( suminf_real(X5) = comple1385675409528146559p_real(image_nat_real(aTP_Lamm_rg(X5),top_top_set_nat)) ) ) ) ).
% suminf_eq_SUP_real
tff(fact_9814_UNIV__nat__eq,axiom,
top_top_set_nat = insert_nat(zero_zero_nat,image_nat_nat(suc,top_top_set_nat)) ).
% UNIV_nat_eq
tff(fact_9815_card__UNIV__unit,axiom,
finite410649719033368117t_unit(top_to1996260823553986621t_unit) = one_one_nat ).
% card_UNIV_unit
tff(fact_9816_card__UNIV__bool,axiom,
finite_card_bool(top_top_set_bool) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).
% card_UNIV_bool
tff(fact_9817_range__mult,axiom,
! [A: real] :
( ( ( A = zero_zero_real )
=> ( image_real_real(aa_rea7340605660300375871l_real(times_times_real,A),top_top_set_real) = insert_real(zero_zero_real,bot_bot_set_real) ) )
& ( ( A != zero_zero_real )
=> ( image_real_real(aa_rea7340605660300375871l_real(times_times_real,A),top_top_set_real) = top_top_set_real ) ) ) ).
% range_mult
tff(fact_9818_root__def,axiom,
! [N: nat,X: real] :
( ( ( N = zero_zero_nat )
=> ( aa_real_real(aa_nat_fun_real_real(root,N),X) = zero_zero_real ) )
& ( ( N != zero_zero_nat )
=> ( aa_real_real(aa_nat_fun_real_real(root,N),X) = aa_real_real(the_in5290026491893676941l_real(top_top_set_real,aa_nat_fun_real_real(aTP_Lamm_rh,N)),X) ) ) ) ).
% root_def
tff(fact_9819_inf__nat__def,axiom,
inf_inf_nat = ord_min_nat ).
% inf_nat_def
tff(fact_9820_inf__enat__def,axiom,
inf_in1870772243966228564d_enat = ord_mi8085742599997312461d_enat ).
% inf_enat_def
tff(fact_9821_card__UNIV__char,axiom,
finite_card_char(top_top_set_char) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))) ).
% card_UNIV_char
tff(fact_9822_UNIV__char__of__nat,axiom,
top_top_set_char = image_nat_char(unique3096191561947761185of_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) ).
% UNIV_char_of_nat
tff(fact_9823_char_Osize_I2_J,axiom,
! [X1: bool,X23: bool,X33: bool,X42: bool,X52: bool,X6: bool,X7: bool,X8: bool] : size_size_char(char2(X1,X23,X33,X42,X52,X6,X7,X8)) = zero_zero_nat ).
% char.size(2)
tff(fact_9824_nat__of__char__less__256,axiom,
! [C: char] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_char_nat(comm_s629917340098488124ar_nat,C)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) ).
% nat_of_char_less_256
tff(fact_9825_range__nat__of__char,axiom,
image_char_nat(comm_s629917340098488124ar_nat,top_top_set_char) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))))) ).
% range_nat_of_char
tff(fact_9826_integer__of__char__code,axiom,
! [B0: bool,B1: bool,B22: bool,B32: bool,B42: bool,B52: bool,B62: bool,B72: bool] : integer_of_char(char2(B0,B1,B22,B32,B42,B52,B62,B72)) = 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(B72)),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger(B62))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger(B52))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger(B42))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger(B32))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger(B22))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger(B1))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger(B0)) ).
% integer_of_char_code
tff(fact_9827_String_Ochar__of__ascii__of,axiom,
! [C: char] : aa_char_nat(comm_s629917340098488124ar_nat,ascii_of(C)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_char_nat(comm_s629917340098488124ar_nat,C)) ).
% String.char_of_ascii_of
tff(fact_9828_sorted__list__of__set__lessThan__Suc,axiom,
! [K: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,K)),cons_nat(K,nil_nat)) ).
% sorted_list_of_set_lessThan_Suc
tff(fact_9829_sorted__list__of__set__atMost__Suc,axiom,
! [K: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,K)),cons_nat(aa_nat_nat(suc,K),nil_nat)) ).
% sorted_list_of_set_atMost_Suc
tff(fact_9830_sorted__list__of__set__greaterThanAtMost,axiom,
! [I: nat,J: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,I)),J))
=> ( linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(I),J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(aa_nat_nat(suc,I)),J))) ) ) ).
% sorted_list_of_set_greaterThanAtMost
tff(fact_9831_sorted__list__of__set__greaterThanLessThan,axiom,
! [I: nat,J: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),J))
=> ( linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(I),J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(aa_nat_nat(suc,I)),J))) ) ) ).
% sorted_list_of_set_greaterThanLessThan
tff(fact_9832_list__encode_Oelims,axiom,
! [X: list_nat,Y: nat] :
( ( nat_list_encode(X) = Y )
=> ( ( ( X = nil_nat )
=> ( Y != zero_zero_nat ) )
=> ~ ! [X3: nat,Xs2: list_nat] :
( ( X = cons_nat(X3,Xs2) )
=> ( Y != aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),nat_list_encode(Xs2)))) ) ) ) ) ).
% list_encode.elims
tff(fact_9833_upto__aux__rec,axiom,
! [J: int,I: int,Js: list_int] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
=> ( upto_aux(I,J,Js) = Js ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
=> ( upto_aux(I,J,Js) = upto_aux(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int),cons_int(J,Js)) ) ) ) ).
% upto_aux_rec
tff(fact_9834_list__encode_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] : nat_list_encode(cons_nat(X,Xs)) = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),nat_list_encode(Xs)))) ).
% list_encode.simps(2)
tff(fact_9835_upto__rec__numeral_I2_J,axiom,
! [M: num,N: num] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
=> ( upto(aa_num_int(numeral_numeral_int,M),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = cons_int(aa_num_int(numeral_numeral_int,M),upto(aa_int_int(aa_int_fun_int_int(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,N)))) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))))
=> ( upto(aa_num_int(numeral_numeral_int,M),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = nil_int ) ) ) ).
% upto_rec_numeral(2)
tff(fact_9836_upto__rec__numeral_I3_J,axiom,
! [M: num,N: num] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)))
=> ( upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),upto(aa_int_int(aa_int_fun_int_int(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,N))) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)))
=> ( upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = nil_int ) ) ) ).
% upto_rec_numeral(3)
tff(fact_9837_upto__Nil,axiom,
! [I: int,J: int] :
( ( upto(I,J) = nil_int )
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I)) ) ).
% upto_Nil
tff(fact_9838_upto__Nil2,axiom,
! [I: int,J: int] :
( ( nil_int = upto(I,J) )
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I)) ) ).
% upto_Nil2
tff(fact_9839_upto__empty,axiom,
! [J: int,I: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
=> ( upto(I,J) = nil_int ) ) ).
% upto_empty
tff(fact_9840_length__upto,axiom,
! [I: int,J: int] : size_size_list_int(upto(I,J)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),I)),one_one_int)) ).
% length_upto
tff(fact_9841_upto__rec__numeral_I1_J,axiom,
! [M: num,N: num] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)))
=> ( upto(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,N)) = cons_int(aa_num_int(numeral_numeral_int,M),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),one_one_int),aa_num_int(numeral_numeral_int,N))) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)))
=> ( upto(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,N)) = nil_int ) ) ) ).
% upto_rec_numeral(1)
tff(fact_9842_upto__rec__numeral_I4_J,axiom,
! [M: num,N: num] :
( ( pp(aa_int_bool(aa_int_fun_int_bool(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,N))))
=> ( 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,N))) = cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),upto(aa_int_int(aa_int_fun_int_int(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,N)))) ) )
& ( ~ pp(aa_int_bool(aa_int_fun_int_bool(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,N))))
=> ( 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,N))) = nil_int ) ) ) ).
% upto_rec_numeral(4)
tff(fact_9843_upto__split1,axiom,
! [I: int,J: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),K))
=> ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),upto(J,K)) ) ) ) ).
% upto_split1
tff(fact_9844_atLeastLessThan__upto,axiom,
! [I: int,J: int] : set_or4662586982721622107an_int(I,J) = set_int2(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).
% atLeastLessThan_upto
tff(fact_9845_upto__rec2,axiom,
! [I: int,J: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
=> ( upto(I,J) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,nil_int)) ) ) ).
% upto_rec2
tff(fact_9846_greaterThanLessThan__upto,axiom,
! [I: int,J: int] : set_or5832277885323065728an_int(I,J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).
% greaterThanLessThan_upto
tff(fact_9847_upto__split3,axiom,
! [I: int,J: int,K: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),K))
=> ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K))) ) ) ) ).
% upto_split3
tff(fact_9848_DERIV__even__real__root,axiom,
! [N: nat,X: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
=> has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).
% DERIV_even_real_root
tff(fact_9849_DERIV__real__root__generic,axiom,
! [N: nat,X: real,D3: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( ( X != zero_zero_real )
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( D3 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) ) )
=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
=> ( D3 = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat)))))) ) ) )
=> ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( D3 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) )
=> has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),D3,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ).
% DERIV_real_root_generic
tff(fact_9850_DERIV__ln,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(inverse_inverse_real,X),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).
% DERIV_ln
tff(fact_9851_has__real__derivative__neg__dec__right,axiom,
! [F: fun_real_real,L: real,X: real,S2: set_real] :
( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S2))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
=> ? [D2: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
& ! [H3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
=> ( pp(aa_set_real_bool(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3)),S2))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3))),aa_real_real(F,X))) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_right
tff(fact_9852_has__real__derivative__pos__inc__right,axiom,
! [F: fun_real_real,L: real,X: real,S2: set_real] :
( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S2))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
=> ? [D2: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
& ! [H3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
=> ( pp(aa_set_real_bool(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3)),S2))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3)))) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_right
tff(fact_9853_has__real__derivative__pos__inc__left,axiom,
! [F: fun_real_real,L: real,X: real,S2: set_real] :
( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S2))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
=> ? [D2: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
& ! [H3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
=> ( pp(aa_set_real_bool(member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3)),S2))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3))),aa_real_real(F,X))) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_left
tff(fact_9854_has__real__derivative__neg__dec__left,axiom,
! [F: fun_real_real,L: real,X: real,S2: set_real] :
( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,S2))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
=> ? [D2: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
& ! [H3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
=> ( pp(aa_set_real_bool(member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3)),S2))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3)))) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_left
tff(fact_9855_DERIV__isconst3,axiom,
! [A: real,B: real,X: real,Y: real,F: fun_real_real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> ( pp(aa_set_real_bool(member_real(X),set_or1633881224788618240n_real(A,B)))
=> ( pp(aa_set_real_bool(member_real(Y),set_or1633881224788618240n_real(A,B)))
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),set_or1633881224788618240n_real(A,B)))
=> has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X3,top_top_set_real)) )
=> ( aa_real_real(F,X) = aa_real_real(F,Y) ) ) ) ) ) ).
% DERIV_isconst3
tff(fact_9856_DERIV__const__ratio__const,axiom,
! [A: real,B: real,F: fun_real_real,K: real] :
( ( A != B )
=> ( ! [X3: real] : has_fi5821293074295781190e_real(F,K,topolo2177554685111907308n_real(X3,top_top_set_real))
=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),K) ) ) ) ).
% DERIV_const_ratio_const
tff(fact_9857_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))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
=> ? [D2: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
& ! [H3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3)))) ) ) ) ) ) ).
% DERIV_pos_inc_right
tff(fact_9858_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))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
=> ? [D2: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
& ! [H3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),H3))),aa_real_real(F,X))) ) ) ) ) ) ).
% DERIV_neg_dec_right
tff(fact_9859_DERIV__pos__inc__left,axiom,
! [F: fun_real_real,L: real,X: real] :
( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
=> ? [D2: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
& ! [H3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3))),aa_real_real(F,X))) ) ) ) ) ) ).
% DERIV_pos_inc_left
tff(fact_9860_DERIV__neg__dec__left,axiom,
! [F: fun_real_real,L: real,X: real] :
( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
=> ? [D2: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D2))
& ! [H3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H3),D2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),H3)))) ) ) ) ) ) ).
% DERIV_neg_dec_left
tff(fact_9861_DERIV__const__ratio__const2,axiom,
! [A: real,B: real,F: fun_real_real,K: real] :
( ( A != B )
=> ( ! [X3: real] : has_fi5821293074295781190e_real(F,K,topolo2177554685111907308n_real(X3,top_top_set_real))
=> ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)) = K ) ) ) ).
% DERIV_const_ratio_const2
tff(fact_9862_DERIV__neg__imp__decreasing,axiom,
! [A: real,B: real,F: fun_real_real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> ( ! [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),B))
=> ? [Y3: real] :
( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X3,top_top_set_real))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),zero_zero_real)) ) ) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,B)),aa_real_real(F,A))) ) ) ).
% DERIV_neg_imp_decreasing
tff(fact_9863_DERIV__pos__imp__increasing,axiom,
! [A: real,B: real,F: fun_real_real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> ( ! [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),B))
=> ? [Y3: real] :
( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X3,top_top_set_real))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y3)) ) ) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A)),aa_real_real(F,B))) ) ) ).
% DERIV_pos_imp_increasing
tff(fact_9864_deriv__nonneg__imp__mono,axiom,
! [A: real,B: real,G: fun_real_real,G2: fun_real_real] :
( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),set_or1222579329274155063t_real(A,B)))
=> has_fi5821293074295781190e_real(G,aa_real_real(G2,X3),topolo2177554685111907308n_real(X3,top_top_set_real)) )
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),set_or1222579329274155063t_real(A,B)))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(G2,X3))) )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(G,A)),aa_real_real(G,B))) ) ) ) ).
% deriv_nonneg_imp_mono
tff(fact_9865_DERIV__nonpos__imp__nonincreasing,axiom,
! [A: real,B: real,F: fun_real_real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
=> ( ! [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),B))
=> ? [Y3: real] :
( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X3,top_top_set_real))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),zero_zero_real)) ) ) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,B)),aa_real_real(F,A))) ) ) ).
% DERIV_nonpos_imp_nonincreasing
tff(fact_9866_DERIV__nonneg__imp__nondecreasing,axiom,
! [A: real,B: real,F: fun_real_real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
=> ( ! [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),B))
=> ? [Y3: real] :
( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X3,top_top_set_real))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Y3)) ) ) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,A)),aa_real_real(F,B))) ) ) ).
% DERIV_nonneg_imp_nondecreasing
tff(fact_9867_DERIV__local__const,axiom,
! [F: fun_real_real,L: real,X: real,D: real] :
( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D))
=> ( ! [Y2: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y2))),D))
=> ( aa_real_real(F,X) = aa_real_real(F,Y2) ) )
=> ( L = zero_zero_real ) ) ) ) ).
% DERIV_local_const
tff(fact_9868_MVT2,axiom,
! [A: real,B: real,F: fun_real_real,F3: fun_real_real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> ( ! [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),B))
=> has_fi5821293074295781190e_real(F,aa_real_real(F3,X3),topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
=> ? [Z3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z3))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),B))
& ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),aa_real_real(F3,Z3)) ) ) ) ) ).
% MVT2
tff(fact_9869_DERIV__const__average,axiom,
! [A: real,B: real,V: fun_real_real,K: real] :
( ( A != B )
=> ( ! [X3: real] : has_fi5821293074295781190e_real(V,K,topolo2177554685111907308n_real(X3,top_top_set_real))
=> ( aa_real_real(V,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)),numeral_numeral_real(aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(V,A)),aa_real_real(V,B))),numeral_numeral_real(aa_num_num(bit0,one))) ) ) ) ).
% DERIV_const_average
tff(fact_9870_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))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D))
=> ( ! [Y2: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y2))),D))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X)),aa_real_real(F,Y2))) )
=> ( L = zero_zero_real ) ) ) ) ).
% DERIV_local_min
tff(fact_9871_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))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D))
=> ( ! [Y2: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y2))),D))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,Y2)),aa_real_real(F,X))) )
=> ( L = zero_zero_real ) ) ) ) ).
% DERIV_local_max
tff(fact_9872_DERIV__ln__divide,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),X),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).
% DERIV_ln_divide
tff(fact_9873_DERIV__pow,axiom,
! [N: nat,X: real,S: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_ri,N),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(X,S)) ).
% DERIV_pow
tff(fact_9874_DERIV__fun__pow,axiom,
! [G: fun_real_real,M: real,X: real,N: nat] :
( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(X,top_top_set_real))
=> has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_rj(G),N),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(G,X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)))),M),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).
% DERIV_fun_pow
tff(fact_9875_has__real__derivative__powr,axiom,
! [Z: real,R: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Z))
=> has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_rk,R),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Z),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R),one_one_real))),topolo2177554685111907308n_real(Z,top_top_set_real)) ) ).
% has_real_derivative_powr
tff(fact_9876_DERIV__log,axiom,
! [X: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(log,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(ln_ln_real,B)),X)),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).
% DERIV_log
tff(fact_9877_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))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(G,X)))
=> has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_rl(G),R),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(G,X)),aa_real_real(aa_rea7340605660300375871l_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_9878_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))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_rm(G),F),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(times_times_real,R),aa_real_real(ln_ln_real,aa_real_real(G,X)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_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_9879_DERIV__real__sqrt,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> has_fi5821293074295781190e_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),numeral_numeral_real(aa_num_num(bit0,one))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).
% DERIV_real_sqrt
tff(fact_9880_DERIV__series_H,axiom,
! [F: fun_re4238438383274954274t_real,F3: fun_re4238438383274954274t_real,X0: real,A: real,B: real,L5: fun_nat_real] :
( ! [N2: nat] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_rn(F),N2),aa_nat_real(aa_real_fun_nat_real(F3,X0),N2),topolo2177554685111907308n_real(X0,top_top_set_real))
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),set_or1633881224788618240n_real(A,B)))
=> summable_real(aa_real_fun_nat_real(F,X3)) )
=> ( pp(aa_set_real_bool(member_real(X0),set_or1633881224788618240n_real(A,B)))
=> ( summable_real(aa_real_fun_nat_real(F3,X0))
=> ( summable_real(L5)
=> ( ! [N2: nat,X3: real,Y2: real] :
( pp(aa_set_real_bool(member_real(X3),set_or1633881224788618240n_real(A,B)))
=> ( pp(aa_set_real_bool(member_real(Y2),set_or1633881224788618240n_real(A,B)))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(F,X3),N2)),aa_nat_real(aa_real_fun_nat_real(F,Y2),N2)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(L5,N2)),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),Y2))))) ) )
=> has_fi5821293074295781190e_real(aTP_Lamm_ro(F),suminf_real(aa_real_fun_nat_real(F3,X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ) ) ) ).
% DERIV_series'
tff(fact_9881_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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(X,top_top_set_real)) ).
% DERIV_arctan
tff(fact_9882_arsinh__real__has__field__derivative,axiom,
! [X: real,A2: set_real] : has_fi5821293074295781190e_real(arsinh_real,aa_real_real(aa_rea7340605660300375871l_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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(X,A2)) ).
% arsinh_real_has_field_derivative
tff(fact_9883_DERIV__real__sqrt__generic,axiom,
! [X: real,D3: real] :
( ( X != zero_zero_real )
=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> ( D3 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),numeral_numeral_real(aa_num_num(bit0,one))) ) )
=> ( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),zero_zero_real))
=> ( D3 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X)))),numeral_numeral_real(aa_num_num(bit0,one))) ) )
=> has_fi5821293074295781190e_real(sqrt,D3,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).
% DERIV_real_sqrt_generic
tff(fact_9884_arcosh__real__has__field__derivative,axiom,
! [X: real,A2: set_real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
=> has_fi5821293074295781190e_real(arcosh_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(X,A2)) ) ).
% arcosh_real_has_field_derivative
tff(fact_9885_artanh__real__has__field__derivative,axiom,
! [X: real,A2: set_real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
=> has_fi5821293074295781190e_real(artanh_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(X,A2)) ) ).
% artanh_real_has_field_derivative
tff(fact_9886_DERIV__power__series_H,axiom,
! [R2: real,F: fun_nat_real,X0: real] :
( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R2),R2)))
=> summable_real(aa_real_fun_nat_real(aTP_Lamm_rp(F),X3)) )
=> ( pp(aa_set_real_bool(member_real(X0),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R2),R2)))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R2))
=> has_fi5821293074295781190e_real(aTP_Lamm_rr(F),suminf_real(aa_real_fun_nat_real(aTP_Lamm_rp(F),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).
% DERIV_power_series'
tff(fact_9887_DERIV__real__root,axiom,
! [N: nat,X: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),X))
=> has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).
% DERIV_real_root
tff(fact_9888_DERIV__arccos,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).
% DERIV_arccos
tff(fact_9889_DERIV__arcsin,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
=> has_fi5821293074295781190e_real(arcsin,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).
% DERIV_arcsin
tff(fact_9890_Maclaurin__all__le__objl,axiom,
! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,X: real,N: nat] :
( ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
& ! [M3: nat,X3: 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)),X3),topolo2177554685111907308n_real(X3,top_top_set_real)) )
=> ? [T4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),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_rs(Diff),X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T4)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ).
% Maclaurin_all_le_objl
tff(fact_9891_Maclaurin__all__le,axiom,
! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,X: real,N: nat] :
( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
=> ( ! [M3: nat,X3: 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)),X3),topolo2177554685111907308n_real(X3,top_top_set_real))
=> ? [T4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),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_rs(Diff),X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T4)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).
% Maclaurin_all_le
tff(fact_9892_DERIV__odd__real__root,axiom,
! [N: nat,X: real] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> ( ( X != zero_zero_real )
=> has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).
% DERIV_odd_real_root
tff(fact_9893_Maclaurin__minus,axiom,
! [H: real,N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H),zero_zero_real))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
=> ( ! [M3: nat,T4: real] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,H),T4))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T4),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)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
=> ? [T4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,H),T4))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T4),zero_zero_real))
& ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_la(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T4)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ) ).
% Maclaurin_minus
tff(fact_9894_Maclaurin2,axiom,
! [H: real,Diff: fun_na6724076748692969250l_real,F: fun_real_real,N: nat] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H))
=> ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
=> ( ! [M3: nat,T4: real] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T4))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T4),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)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
=> ? [T4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T4))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T4),H))
& ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_la(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T4)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ).
% Maclaurin2
tff(fact_9895_Maclaurin,axiom,
! [H: real,N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),H))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
=> ( ! [M3: nat,T4: real] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T4))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T4),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)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
=> ? [T4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),T4))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T4),H))
& ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_la(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T4)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ) ).
% Maclaurin
tff(fact_9896_Maclaurin__all__lt,axiom,
! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,N: nat,X: real] :
( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( ( X != zero_zero_real )
=> ( ! [M3: nat,X3: 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)),X3),topolo2177554685111907308n_real(X3,top_top_set_real))
=> ? [T4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T4)))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,T4)),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_rs(Diff),X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T4)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ) ) ).
% Maclaurin_all_lt
tff(fact_9897_Maclaurin__bi__le,axiom,
! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,N: nat,X: real] :
( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
=> ( ! [M3: nat,T4: real] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),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)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
=> ? [T4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),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_rs(Diff),X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T4)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N))) ) ) ) ) ).
% Maclaurin_bi_le
tff(fact_9898_Taylor__down,axiom,
! [N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
=> ( ! [M3: nat,T4: real] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),T4))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T4),B)) )
=> has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),B))
=> ? [T4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),T4))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T4),C))
& ( aa_real_real(F,A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_rt(Diff),A),C),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T4)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),N))) ) ) ) ) ) ) ) ).
% Taylor_down
tff(fact_9899_Taylor__up,axiom,
! [N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
=> ( ! [M3: nat,T4: real] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),T4))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T4),B)) )
=> has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),C))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),B))
=> ? [T4: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),T4))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T4),B))
& ( aa_real_real(F,B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_rt(Diff),B),C),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T4)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),C)),N))) ) ) ) ) ) ) ) ).
% Taylor_up
tff(fact_9900_Taylor,axiom,
! [N: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real,X: real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
=> ( ! [M3: nat,T4: real] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),T4))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T4),B)) )
=> has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),C))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C),B))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),B))
=> ( ( X != C )
=> ? [T4: real] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),C))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),T4))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T4),C)) ) )
& ( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),C))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C),T4))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,T4),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_ru(Diff),C),X),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,N),T4)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),C)),N))) ) ) ) ) ) ) ) ) ) ) ).
% Taylor
tff(fact_9901_Maclaurin__lemma2,axiom,
! [N: nat,H: real,Diff: fun_na6724076748692969250l_real,K: nat,B2: real] :
( ! [M3: nat,T4: real] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T4))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T4),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)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
=> ( ( N = aa_nat_nat(suc,K) )
=> ! [M2: nat,T6: real] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),T6))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,T6),H)) )
=> has_fi5821293074295781190e_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_rw(N),Diff),B2),M2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T6)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_rx(Diff),M2),T6),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M2))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,T6),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M2)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M2))))))),topolo2177554685111907308n_real(T6,top_top_set_real)) ) ) ) ).
% Maclaurin_lemma2
tff(fact_9902_DERIV__arctan__series,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real))
=> has_fi5821293074295781190e_real(aTP_Lamm_ry,suminf_real(aa_real_fun_nat_real(aTP_Lamm_rz,X)),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).
% DERIV_arctan_series
tff(fact_9903_isCont__Lb__Ub,axiom,
! [A: real,B: real,F: fun_real_real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
=> ( ! [X3: real] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X3))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),B)) )
=> topolo4422821103128117721l_real(topolo2177554685111907308n_real(X3,top_top_set_real),F) )
=> ? [L6: real,M9: real] :
( ! [X4: real] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X4))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X4),B)) )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L6),aa_real_real(F,X4)))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(F,X4)),M9)) ) )
& ! [Y3: real] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L6),Y3))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Y3),M9)) )
=> ? [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X3))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),B))
& ( aa_real_real(F,X3) = Y3 ) ) ) ) ) ) ).
% isCont_Lb_Ub
tff(fact_9904_LIM__fun__less__zero,axiom,
! [F: fun_real_real,L: real,C: real] :
( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,L),zero_zero_real))
=> ? [R3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
& ! [X4: real] :
( ( ( X4 != C )
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),X4))),R3)) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,X4)),zero_zero_real)) ) ) ) ) ).
% LIM_fun_less_zero
tff(fact_9905_LIM__fun__not__zero,axiom,
! [F: fun_real_real,L: real,C: real] :
( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
=> ( ( L != zero_zero_real )
=> ? [R3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
& ! [X4: real] :
( ( ( X4 != C )
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),X4))),R3)) )
=> ( aa_real_real(F,X4) != zero_zero_real ) ) ) ) ) ).
% LIM_fun_not_zero
tff(fact_9906_LIM__fun__gt__zero,axiom,
! [F: fun_real_real,L: real,C: real] :
( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),L))
=> ? [R3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),R3))
& ! [X4: real] :
( ( ( X4 != C )
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),X4))),R3)) )
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F,X4))) ) ) ) ) ).
% LIM_fun_gt_zero
tff(fact_9907_isCont__real__sqrt,axiom,
! [X: real] : topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),sqrt) ).
% isCont_real_sqrt
tff(fact_9908_isCont__real__root,axiom,
! [X: real,N: nat] : topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),aa_nat_fun_real_real(root,N)) ).
% isCont_real_root
tff(fact_9909_isCont__inverse__function2,axiom,
! [A: real,X: real,B: real,G: fun_real_real,F: fun_real_real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),B))
=> ( ! [Z3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Z3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z3),B))
=> ( aa_real_real(G,aa_real_real(F,Z3)) = Z3 ) ) )
=> ( ! [Z3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Z3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z3),B))
=> 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_9910_isCont__arcosh,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
=> topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),arcosh_real) ) ).
% isCont_arcosh
tff(fact_9911_LIM__cos__div__sin,axiom,
filterlim_real_real(aTP_Lamm_sa,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))),top_top_set_real)) ).
% LIM_cos_div_sin
tff(fact_9912_DERIV__inverse__function,axiom,
! [F: fun_real_real,D3: real,G: fun_real_real,X: real,A: real,B: real] :
( has_fi5821293074295781190e_real(F,D3,topolo2177554685111907308n_real(aa_real_real(G,X),top_top_set_real))
=> ( ( D3 != zero_zero_real )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),B))
=> ( ! [Y2: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Y2))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y2),B))
=> ( aa_real_real(F,aa_real_real(G,Y2)) = Y2 ) ) )
=> ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),G)
=> has_fi5821293074295781190e_real(G,aa_real_real(inverse_inverse_real,D3),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ) ).
% DERIV_inverse_function
tff(fact_9913_isCont__arccos,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
=> topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),arccos) ) ) ).
% isCont_arccos
tff(fact_9914_isCont__arcsin,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
=> topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),arcsin) ) ) ).
% isCont_arcsin
tff(fact_9915_LIM__less__bound,axiom,
! [B: real,X: real,F: fun_real_real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),X))
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),set_or1633881224788618240n_real(B,X)))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X3))) )
=> ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),F)
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_real_real(F,X))) ) ) ) ).
% LIM_less_bound
tff(fact_9916_isCont__artanh,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),one_one_real))
=> topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),artanh_real) ) ) ).
% isCont_artanh
tff(fact_9917_isCont__inverse__function,axiom,
! [D: real,X: real,G: fun_real_real,F: fun_real_real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),D))
=> ( ! [Z3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z3),X))),D))
=> ( aa_real_real(G,aa_real_real(F,Z3)) = Z3 ) )
=> ( ! [Z3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_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_9918_GMVT_H,axiom,
! [A: real,B: real,F: fun_real_real,G: fun_real_real,G2: fun_real_real,F3: fun_real_real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> ( ! [Z3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Z3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z3),B))
=> topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),F) ) )
=> ( ! [Z3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),Z3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Z3),B))
=> topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),G) ) )
=> ( ! [Z3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),B))
=> has_fi5821293074295781190e_real(G,aa_real_real(G2,Z3),topolo2177554685111907308n_real(Z3,top_top_set_real)) ) )
=> ( ! [Z3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),B))
=> has_fi5821293074295781190e_real(F,aa_real_real(F3,Z3),topolo2177554685111907308n_real(Z3,top_top_set_real)) ) )
=> ? [C2: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C2))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),B))
& ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A))),aa_real_real(G2,C2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(G,B)),aa_real_real(G,A))),aa_real_real(F3,C2)) ) ) ) ) ) ) ) ).
% GMVT'
tff(fact_9919_summable__Leibniz_I3_J,axiom,
! [A: fun_nat_real] :
( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
=> ( topolo6980174941875973593q_real(A)
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_nat_real(A,zero_zero_nat)),zero_zero_real))
=> ! [N6: nat] : pp(aa_set_real_bool(member_real(suminf_real(aTP_Lamm_sb(A))),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_sb(A),aa_nat_set_nat(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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6)),one_one_nat))),groups6591440286371151544t_real(aTP_Lamm_sb(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6)))))) ) ) ) ).
% summable_Leibniz(3)
tff(fact_9920_summable__Leibniz_I2_J,axiom,
! [A: fun_nat_real] :
( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
=> ( topolo6980174941875973593q_real(A)
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_nat_real(A,zero_zero_nat)))
=> ! [N6: nat] : pp(aa_set_real_bool(member_real(suminf_real(aTP_Lamm_sb(A))),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_sb(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6))),groups6591440286371151544t_real(aTP_Lamm_sb(A),aa_nat_set_nat(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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6)),one_one_nat)))))) ) ) ) ).
% summable_Leibniz(2)
tff(fact_9921_mult__nat__right__at__top,axiom,
! [C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
=> filterlim_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aj,C),at_top_nat,at_top_nat) ) ).
% mult_nat_right_at_top
tff(fact_9922_mult__nat__left__at__top,axiom,
! [C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_9923_monoseq__convergent,axiom,
! [X5: fun_nat_real,B2: real] :
( topolo6980174941875973593q_real(X5)
=> ( ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(X5,I2))),B2))
=> ~ ! [L6: real] : ~ filterlim_nat_real(X5,topolo2815343760600316023s_real(L6),at_top_nat) ) ) ).
% monoseq_convergent
tff(fact_9924_LIMSEQ__root,axiom,
filterlim_nat_real(aTP_Lamm_sc,topolo2815343760600316023s_real(one_one_real),at_top_nat) ).
% LIMSEQ_root
tff(fact_9925_nested__sequence__unique,axiom,
! [F: fun_nat_real,G: fun_nat_real] :
( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2))))
=> ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(G,aa_nat_nat(suc,N2))),aa_nat_real(G,N2)))
=> ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(G,N2)))
=> ( filterlim_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ff(F),G),topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
=> ? [L3: real] :
( ! [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N6)),L3))
& filterlim_nat_real(F,topolo2815343760600316023s_real(L3),at_top_nat)
& ! [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L3),aa_nat_real(G,N6)))
& filterlim_nat_real(G,topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ) ).
% nested_sequence_unique
tff(fact_9926_LIMSEQ__inverse__zero,axiom,
! [X5: fun_nat_real] :
( ! [R3: real] :
? [N7: nat] :
! [N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N7),N2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,R3),aa_nat_real(X5,N2))) )
=> filterlim_nat_real(aTP_Lamm_sd(X5),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).
% LIMSEQ_inverse_zero
tff(fact_9927_lim__inverse__n_H,axiom,
filterlim_nat_real(aTP_Lamm_se,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).
% lim_inverse_n'
tff(fact_9928_LIMSEQ__root__const,axiom,
! [C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),C))
=> filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_sf,C),topolo2815343760600316023s_real(one_one_real),at_top_nat) ) ).
% LIMSEQ_root_const
tff(fact_9929_LIMSEQ__inverse__real__of__nat,axiom,
filterlim_nat_real(aTP_Lamm_sg,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).
% LIMSEQ_inverse_real_of_nat
tff(fact_9930_LIMSEQ__inverse__real__of__nat__add,axiom,
! [R: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_sh,R),topolo2815343760600316023s_real(R),at_top_nat) ).
% LIMSEQ_inverse_real_of_nat_add
tff(fact_9931_increasing__LIMSEQ,axiom,
! [F: fun_nat_real,L: real] :
( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2))))
=> ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(F,N2)),L))
=> ( ! [E2: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),E2))
=> ? [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,N6)),E2))) )
=> filterlim_nat_real(F,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).
% increasing_LIMSEQ
tff(fact_9932_LIMSEQ__realpow__zero,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_9933_LIMSEQ__divide__realpow__zero,axiom,
! [X: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
=> filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_si(X),A),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).
% LIMSEQ_divide_realpow_zero
tff(fact_9934_LIMSEQ__abs__realpow__zero,axiom,
! [C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_9935_LIMSEQ__abs__realpow__zero2,axiom,
! [C: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_9936_LIMSEQ__inverse__realpow__zero,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),X))
=> filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_sj,X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).
% LIMSEQ_inverse_realpow_zero
tff(fact_9937_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
! [R: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_sk,R),topolo2815343760600316023s_real(R),at_top_nat) ).
% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_9938_tendsto__exp__limit__sequentially,axiom,
! [X: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_sl,X),topolo2815343760600316023s_real(aa_real_real(exp_real,X)),at_top_nat) ).
% tendsto_exp_limit_sequentially
tff(fact_9939_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
! [R: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_sm,R),topolo2815343760600316023s_real(R),at_top_nat) ).
% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_9940_summable__Leibniz_I1_J,axiom,
! [A: fun_nat_real] :
( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
=> ( topolo6980174941875973593q_real(A)
=> summable_real(aTP_Lamm_sb(A)) ) ) ).
% summable_Leibniz(1)
tff(fact_9941_summable,axiom,
! [A: fun_nat_real] :
( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
=> ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2)))
=> ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
=> summable_real(aTP_Lamm_sb(A)) ) ) ) ).
% summable
tff(fact_9942_cos__diff__limit__1,axiom,
! [Theta: fun_nat_real,Theta2: real] :
( filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_sn(Theta),Theta2),topolo2815343760600316023s_real(one_one_real),at_top_nat)
=> ~ ! [K3: fun_nat_int] : ~ filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_so(Theta),K3),topolo2815343760600316023s_real(Theta2),at_top_nat) ) ).
% cos_diff_limit_1
tff(fact_9943_cos__limit__1,axiom,
! [Theta: fun_nat_real] :
( filterlim_nat_real(aTP_Lamm_sp(Theta),topolo2815343760600316023s_real(one_one_real),at_top_nat)
=> ? [K3: fun_nat_int] : filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_so(Theta),K3),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).
% cos_limit_1
tff(fact_9944_summable__Leibniz_I4_J,axiom,
! [A: fun_nat_real] :
( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
=> ( topolo6980174941875973593q_real(A)
=> filterlim_nat_real(aTP_Lamm_sq(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_sb(A))),at_top_nat) ) ) ).
% summable_Leibniz(4)
tff(fact_9945_zeroseq__arctan__series,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real))
=> filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_ia,X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).
% zeroseq_arctan_series
tff(fact_9946_summable__Leibniz_H_I2_J,axiom,
! [A: fun_nat_real,N: nat] :
( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
=> ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2)))
=> ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_sb(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),suminf_real(aTP_Lamm_sb(A)))) ) ) ) ).
% summable_Leibniz'(2)
tff(fact_9947_summable__Leibniz_H_I3_J,axiom,
! [A: fun_nat_real] :
( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
=> ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2)))
=> ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
=> filterlim_nat_real(aTP_Lamm_sq(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_sb(A))),at_top_nat) ) ) ) ).
% summable_Leibniz'(3)
tff(fact_9948_sums__alternating__upper__lower,axiom,
! [A: fun_nat_real] :
( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
=> ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2)))
=> ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
=> ? [L3: real] :
( ! [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_sb(A),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6)))),L3))
& filterlim_nat_real(aTP_Lamm_sq(A),topolo2815343760600316023s_real(L3),at_top_nat)
& ! [N6: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L3),groups6591440286371151544t_real(aTP_Lamm_sb(A),aa_nat_set_nat(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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6)),one_one_nat)))))
& filterlim_nat_real(aTP_Lamm_sr(A),topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ).
% sums_alternating_upper_lower
tff(fact_9949_summable__Leibniz_I5_J,axiom,
! [A: fun_nat_real] :
( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
=> ( topolo6980174941875973593q_real(A)
=> filterlim_nat_real(aTP_Lamm_sr(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_sb(A))),at_top_nat) ) ) ).
% summable_Leibniz(5)
tff(fact_9950_summable__Leibniz_H_I4_J,axiom,
! [A: fun_nat_real,N: nat] :
( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
=> ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2)))
=> ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,suminf_real(aTP_Lamm_sb(A))),groups6591440286371151544t_real(aTP_Lamm_sb(A),aa_nat_set_nat(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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat))))) ) ) ) ).
% summable_Leibniz'(4)
tff(fact_9951_summable__Leibniz_H_I5_J,axiom,
! [A: fun_nat_real] :
( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
=> ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2)))
=> ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2)))
=> filterlim_nat_real(aTP_Lamm_sr(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_sb(A))),at_top_nat) ) ) ) ).
% summable_Leibniz'(5)
tff(fact_9952_real__bounded__linear,axiom,
! [F: fun_real_real] :
( real_V5970128139526366754l_real(F)
<=> ? [C3: real] :
! [X2: real] : aa_real_real(F,X2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),C3) ) ).
% real_bounded_linear
tff(fact_9953_filterlim__Suc,axiom,
filterlim_nat_nat(suc,at_top_nat,at_top_nat) ).
% filterlim_Suc
tff(fact_9954_tendsto__exp__limit__at__right,axiom,
! [X: real] : filterlim_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_ss,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_9955_dist__complex__def,axiom,
! [X: complex,Y: complex] : aa_complex_real(real_V3694042436643373181omplex(X),Y) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Y)) ).
% dist_complex_def
tff(fact_9956_dist__real__def,axiom,
! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V975177566351809787t_real,X),Y) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y)) ).
% dist_real_def
tff(fact_9957_tendsto__arcosh__at__left__1,axiom,
filterlim_real_real(arcosh_real,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(one_one_real,set_or5849166863359141190n_real(one_one_real))) ).
% tendsto_arcosh_at_left_1
tff(fact_9958_filterlim__tan__at__right,axiom,
filterlim_real_real(tan_real,at_bot_real,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))),set_or5849166863359141190n_real(aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))))) ).
% filterlim_tan_at_right
tff(fact_9959_greaterThan__0,axiom,
aa_nat_set_nat(set_or1210151606488870762an_nat,zero_zero_nat) = image_nat_nat(suc,top_top_set_nat) ).
% greaterThan_0
tff(fact_9960_greaterThan__Suc,axiom,
! [K: nat] : aa_nat_set_nat(set_or1210151606488870762an_nat,aa_nat_nat(suc,K)) = minus_minus_set_nat(aa_nat_set_nat(set_or1210151606488870762an_nat,K),insert_nat(aa_nat_nat(suc,K),bot_bot_set_nat)) ).
% greaterThan_Suc
tff(fact_9961_tanh__real__at__bot,axiom,
filterlim_real_real(tanh_real,topolo2815343760600316023s_real(aa_real_real(uminus_uminus_real,one_one_real)),at_bot_real) ).
% tanh_real_at_bot
tff(fact_9962_artanh__real__at__right__1,axiom,
filterlim_real_real(artanh_real,at_bot_real,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,one_one_real),set_or5849166863359141190n_real(aa_real_real(uminus_uminus_real,one_one_real)))) ).
% artanh_real_at_right_1
tff(fact_9963_DERIV__pos__imp__increasing__at__bot,axiom,
! [B: real,F: fun_real_real,Flim: real] :
( ! [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),B))
=> ? [Y3: real] :
( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X3,top_top_set_real))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y3)) ) )
=> ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_bot_real)
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Flim),aa_real_real(F,B))) ) ) ).
% DERIV_pos_imp_increasing_at_bot
tff(fact_9964_filterlim__pow__at__bot__odd,axiom,
! [N: nat,F: fun_real_real,F4: filter_real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( filterlim_real_real(F,at_bot_real,F4)
=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_st(N),F),at_bot_real,F4) ) ) ) ).
% filterlim_pow_at_bot_odd
tff(fact_9965_tendsto__arctan__at__bot,axiom,
filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),at_bot_real) ).
% tendsto_arctan_at_bot
tff(fact_9966_filterlim__pow__at__bot__even,axiom,
! [N: nat,F: fun_real_real,F4: filter_real] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> ( filterlim_real_real(F,at_bot_real,F4)
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
=> filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_st(N),F),at_top_real,F4) ) ) ) ).
% filterlim_pow_at_bot_even
tff(fact_9967_atLeast__Suc__greaterThan,axiom,
! [K: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K)) = aa_nat_set_nat(set_or1210151606488870762an_nat,K) ).
% atLeast_Suc_greaterThan
tff(fact_9968_sqrt__at__top,axiom,
filterlim_real_real(sqrt,at_top_real,at_top_real) ).
% sqrt_at_top
tff(fact_9969_tanh__real__at__top,axiom,
filterlim_real_real(tanh_real,topolo2815343760600316023s_real(one_one_real),at_top_real) ).
% tanh_real_at_top
tff(fact_9970_artanh__real__at__left__1,axiom,
filterlim_real_real(artanh_real,at_top_real,topolo2177554685111907308n_real(one_one_real,set_or5984915006950818249n_real(one_one_real))) ).
% artanh_real_at_left_1
tff(fact_9971_atLeast__Suc,axiom,
! [K: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K)) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atLeast_nat,K),insert_nat(K,bot_bot_set_nat)) ).
% atLeast_Suc
tff(fact_9972_ln__x__over__x__tendsto__0,axiom,
filterlim_real_real(aTP_Lamm_su,topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).
% ln_x_over_x_tendsto_0
tff(fact_9973_tendsto__power__div__exp__0,axiom,
! [K: nat] : filterlim_real_real(aa_nat_fun_real_real(aTP_Lamm_sv,K),topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).
% tendsto_power_div_exp_0
tff(fact_9974_tendsto__exp__limit__at__top,axiom,
! [X: real] : filterlim_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_sw,X),topolo2815343760600316023s_real(aa_real_real(exp_real,X)),at_top_real) ).
% tendsto_exp_limit_at_top
tff(fact_9975_filterlim__tan__at__left,axiom,
filterlim_real_real(tan_real,at_top_real,topolo2177554685111907308n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))),set_or5984915006950818249n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))) ).
% filterlim_tan_at_left
tff(fact_9976_DERIV__neg__imp__decreasing__at__top,axiom,
! [B: real,F: fun_real_real,Flim: real] :
( ! [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B),X3))
=> ? [Y3: real] :
( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X3,top_top_set_real))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),zero_zero_real)) ) )
=> ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_top_real)
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Flim),aa_real_real(F,B))) ) ) ).
% DERIV_neg_imp_decreasing_at_top
tff(fact_9977_tendsto__arctan__at__top,axiom,
filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))),at_top_real) ).
% tendsto_arctan_at_top
tff(fact_9978_lhopital__right__at__top,axiom,
! [G: fun_real_real,X: real,G2: fun_real_real,F: fun_real_real,F3: fun_real_real,Y: real] :
( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
=> ( eventually_real(aTP_Lamm_sx(G2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
=> ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_sy(F),F3),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
=> ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_sy(G),G2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
=> ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_sz(G2),F3),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
=> filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_sz(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X))) ) ) ) ) ) ).
% lhopital_right_at_top
tff(fact_9979_lhopital__right__0__at__top,axiom,
! [G: fun_real_real,G2: fun_real_real,F: fun_real_real,F3: 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_sx(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
=> ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_sy(F),F3),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
=> ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_sy(G),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
=> ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_sz(G2),F3),topolo2815343760600316023s_real(X),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
=> filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_sz(G),F),topolo2815343760600316023s_real(X),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ).
% lhopital_right_0_at_top
tff(fact_9980_eventually__at__left__real,axiom,
! [B: real,A: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,B),A))
=> eventually_real(aa_rea7809428808904475482l_bool(aTP_Lamm_ta(B),A),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A))) ) ).
% eventually_at_left_real
tff(fact_9981_eventually__at__right__to__0,axiom,
! [P: fun_real_bool,A: real] :
( eventually_real(P,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
<=> eventually_real(aa_rea7809428808904475482l_bool(aTP_Lamm_tb(P),A),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ).
% eventually_at_right_to_0
tff(fact_9982_eventually__at__right__real,axiom,
! [A: real,B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> eventually_real(aa_rea7809428808904475482l_bool(aTP_Lamm_ta(A),B),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) ) ).
% eventually_at_right_real
tff(fact_9983_lhopital__at__top__at__top,axiom,
! [F: fun_real_real,A: real,G: fun_real_real,F3: fun_real_real,G2: fun_real_real] :
( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,top_top_set_real))
=> ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(A,top_top_set_real))
=> ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_sy(F),F3),topolo2177554685111907308n_real(A,top_top_set_real))
=> ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_sy(G),G2),topolo2177554685111907308n_real(A,top_top_set_real))
=> ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tc(F3),G2),at_top_real,topolo2177554685111907308n_real(A,top_top_set_real))
=> filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tc(F),G),at_top_real,topolo2177554685111907308n_real(A,top_top_set_real)) ) ) ) ) ) ).
% lhopital_at_top_at_top
tff(fact_9984_lhopital__left__at__top__at__top,axiom,
! [F: fun_real_real,A: real,G: fun_real_real,F3: fun_real_real,G2: fun_real_real] :
( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
=> ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
=> ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_sy(F),F3),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
=> ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_sy(G),G2),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
=> ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tc(F3),G2),at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
=> filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tc(F),G),at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A))) ) ) ) ) ) ).
% lhopital_left_at_top_at_top
tff(fact_9985_lhopital,axiom,
! [F: fun_real_real,X: real,G: fun_real_real,G2: fun_real_real,F3: fun_real_real,F4: 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_sx(G),topolo2177554685111907308n_real(X,top_top_set_real))
=> ( eventually_real(aTP_Lamm_sx(G2),topolo2177554685111907308n_real(X,top_top_set_real))
=> ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_sy(F),F3),topolo2177554685111907308n_real(X,top_top_set_real))
=> ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_sy(G),G2),topolo2177554685111907308n_real(X,top_top_set_real))
=> ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_sz(G2),F3),F4,topolo2177554685111907308n_real(X,top_top_set_real))
=> filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tc(F),G),F4,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ) ) ).
% lhopital
tff(fact_9986_lhopital__left,axiom,
! [F: fun_real_real,X: real,G: fun_real_real,G2: fun_real_real,F3: fun_real_real,F4: 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_sx(G),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
=> ( eventually_real(aTP_Lamm_sx(G2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
=> ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_sy(F),F3),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
=> ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_sy(G),G2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
=> ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_sz(G2),F3),F4,topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
=> filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tc(F),G),F4,topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X))) ) ) ) ) ) ) ) ).
% lhopital_left
tff(fact_9987_lhopital__right__at__top__at__top,axiom,
! [F: fun_real_real,A: real,G: fun_real_real,F3: fun_real_real,G2: fun_real_real] :
( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
=> ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
=> ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_sy(F),F3),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
=> ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_sy(G),G2),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
=> ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tc(F3),G2),at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
=> filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tc(F),G),at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) ) ) ) ) ) ).
% lhopital_right_at_top_at_top
tff(fact_9988_lhopital__at__top__at__bot,axiom,
! [F: fun_real_real,A: real,G: fun_real_real,F3: fun_real_real,G2: fun_real_real] :
( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,top_top_set_real))
=> ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(A,top_top_set_real))
=> ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_sy(F),F3),topolo2177554685111907308n_real(A,top_top_set_real))
=> ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_sy(G),G2),topolo2177554685111907308n_real(A,top_top_set_real))
=> ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tc(F3),G2),at_bot_real,topolo2177554685111907308n_real(A,top_top_set_real))
=> filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tc(F),G),at_bot_real,topolo2177554685111907308n_real(A,top_top_set_real)) ) ) ) ) ) ).
% lhopital_at_top_at_bot
tff(fact_9989_lhopital__left__at__top__at__bot,axiom,
! [F: fun_real_real,A: real,G: fun_real_real,F3: fun_real_real,G2: fun_real_real] :
( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
=> ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
=> ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_sy(F),F3),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
=> ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_sy(G),G2),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
=> ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tc(F3),G2),at_bot_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
=> filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tc(F),G),at_bot_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A))) ) ) ) ) ) ).
% lhopital_left_at_top_at_bot
tff(fact_9990_lhopital__at__top,axiom,
! [G: fun_real_real,X: real,G2: fun_real_real,F: fun_real_real,F3: fun_real_real,Y: real] :
( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(X,top_top_set_real))
=> ( eventually_real(aTP_Lamm_sx(G2),topolo2177554685111907308n_real(X,top_top_set_real))
=> ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_sy(F),F3),topolo2177554685111907308n_real(X,top_top_set_real))
=> ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_sy(G),G2),topolo2177554685111907308n_real(X,top_top_set_real))
=> ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_sz(G2),F3),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,top_top_set_real))
=> filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_sz(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ).
% lhopital_at_top
tff(fact_9991_lhospital__at__top__at__top,axiom,
! [G: fun_real_real,G2: fun_real_real,F: fun_real_real,F3: fun_real_real,X: real] :
( filterlim_real_real(G,at_top_real,at_top_real)
=> ( eventually_real(aTP_Lamm_sx(G2),at_top_real)
=> ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_sy(F),F3),at_top_real)
=> ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_sy(G),G2),at_top_real)
=> ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_sz(G2),F3),topolo2815343760600316023s_real(X),at_top_real)
=> filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_sz(G),F),topolo2815343760600316023s_real(X),at_top_real) ) ) ) ) ) ).
% lhospital_at_top_at_top
tff(fact_9992_lhopital__left__at__top,axiom,
! [G: fun_real_real,X: real,G2: fun_real_real,F: fun_real_real,F3: fun_real_real,Y: real] :
( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
=> ( eventually_real(aTP_Lamm_sx(G2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
=> ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_sy(F),F3),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
=> ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_sy(G),G2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
=> ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_sz(G2),F3),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
=> filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_sz(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X))) ) ) ) ) ) ).
% lhopital_left_at_top
tff(fact_9993_lhopital__right__0,axiom,
! [F0: fun_real_real,G0: fun_real_real,G2: fun_real_real,F3: fun_real_real,F4: 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_sx(G0),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
=> ( eventually_real(aTP_Lamm_sx(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
=> ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_sy(F0),F3),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
=> ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_sy(G0),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
=> ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_sz(G2),F3),F4,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
=> filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tc(F0),G0),F4,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ) ) ).
% lhopital_right_0
tff(fact_9994_lhopital__right,axiom,
! [F: fun_real_real,X: real,G: fun_real_real,G2: fun_real_real,F3: fun_real_real,F4: 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_sx(G),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
=> ( eventually_real(aTP_Lamm_sx(G2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
=> ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_sy(F),F3),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
=> ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_sy(G),G2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
=> ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_sz(G2),F3),F4,topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
=> filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tc(F),G),F4,topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X))) ) ) ) ) ) ) ) ).
% lhopital_right
tff(fact_9995_lhopital__right__at__top__at__bot,axiom,
! [F: fun_real_real,A: real,G: fun_real_real,F3: fun_real_real,G2: fun_real_real] :
( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
=> ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
=> ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_sy(F),F3),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
=> ( eventually_real(aa_fun9137271176455094897l_bool(aTP_Lamm_sy(G),G2),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
=> ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tc(F3),G2),at_bot_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
=> filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tc(F),G),at_bot_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) ) ) ) ) ) ).
% lhopital_right_at_top_at_bot
tff(fact_9996_eventually__sequentially__Suc,axiom,
! [P: fun_nat_bool] :
( eventually_nat(aTP_Lamm_td(P),at_top_nat)
<=> eventually_nat(P,at_top_nat) ) ).
% eventually_sequentially_Suc
tff(fact_9997_eventually__sequentially__seg,axiom,
! [P: fun_nat_bool,K: nat] :
( eventually_nat(aa_nat_fun_nat_bool(aTP_Lamm_te(P),K),at_top_nat)
<=> eventually_nat(P,at_top_nat) ) ).
% eventually_sequentially_seg
tff(fact_9998_le__sequentially,axiom,
! [F4: filter_nat] :
( ord_le2510731241096832064er_nat(F4,at_top_nat)
<=> ! [N9: nat] : eventually_nat(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),F4) ) ).
% le_sequentially
tff(fact_9999_eventually__sequentially,axiom,
! [P: fun_nat_bool] :
( eventually_nat(P,at_top_nat)
<=> ? [N9: nat] :
! [N5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),N5))
=> pp(aa_nat_bool(P,N5)) ) ) ).
% eventually_sequentially
tff(fact_10000_eventually__sequentiallyI,axiom,
! [C: nat,P: fun_nat_bool] :
( ! [X3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),X3))
=> pp(aa_nat_bool(P,X3)) )
=> eventually_nat(P,at_top_nat) ) ).
% eventually_sequentiallyI
tff(fact_10001_sequentially__offset,axiom,
! [P: fun_nat_bool,K: nat] :
( eventually_nat(P,at_top_nat)
=> eventually_nat(aa_nat_fun_nat_bool(aTP_Lamm_te(P),K),at_top_nat) ) ).
% sequentially_offset
tff(fact_10002_Bseq__realpow,axiom,
! [X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(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_10003_GreatestI__ex__nat,axiom,
! [P: fun_nat_bool,B: nat] :
( ? [X_12: nat] : pp(aa_nat_bool(P,X_12))
=> ( ! [Y2: nat] :
( pp(aa_nat_bool(P,Y2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y2),B)) )
=> pp(aa_nat_bool(P,order_Greatest_nat(P))) ) ) ).
% GreatestI_ex_nat
tff(fact_10004_Greatest__le__nat,axiom,
! [P: fun_nat_bool,K: nat,B: nat] :
( pp(aa_nat_bool(P,K))
=> ( ! [Y2: nat] :
( pp(aa_nat_bool(P,Y2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y2),B)) )
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),order_Greatest_nat(P))) ) ) ).
% Greatest_le_nat
tff(fact_10005_GreatestI__nat,axiom,
! [P: fun_nat_bool,K: nat,B: nat] :
( pp(aa_nat_bool(P,K))
=> ( ! [Y2: nat] :
( pp(aa_nat_bool(P,Y2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y2),B)) )
=> pp(aa_nat_bool(P,order_Greatest_nat(P))) ) ) ).
% GreatestI_nat
tff(fact_10006_decseq__bounded,axiom,
! [X5: fun_nat_real,B2: real] :
( order_9091379641038594480t_real(X5)
=> ( ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),aa_nat_real(X5,I2)))
=> bfun_nat_real(X5,at_top_nat) ) ) ).
% decseq_bounded
tff(fact_10007_decseq__convergent,axiom,
! [X5: fun_nat_real,B2: real] :
( order_9091379641038594480t_real(X5)
=> ( ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,B2),aa_nat_real(X5,I2)))
=> ~ ! [L6: real] :
( filterlim_nat_real(X5,topolo2815343760600316023s_real(L6),at_top_nat)
=> ~ ! [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,L6),aa_nat_real(X5,I3))) ) ) ) ).
% decseq_convergent
tff(fact_10008_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
=> ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
=> ( Xa2 = one_one_nat ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
=> ( ( Deg2 = Xa2 )
& ! [X3: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
=> pp(aa_nat_bool(vEBT_VEBT_valid(X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
& pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
& ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
& pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_tf(Deg2),TreeList2),Summary2))),Mima)) ) ) ) ) ).
% VEBT_internal.valid'.elims(3)
tff(fact_10009_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
=> ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
=> ( Xa2 != one_one_nat ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
=> ~ ( ( Deg2 = Xa2 )
& ! [X4: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
=> pp(aa_nat_bool(vEBT_VEBT_valid(X4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
& pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
& ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
& pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_tf(Deg2),TreeList2),Summary2))),Mima)) ) ) ) ) ).
% VEBT_internal.valid'.elims(2)
tff(fact_10010_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
( pp(aa_nat_bool(vEBT_VEBT_valid(vEBT_Node(Mima2,Deg,TreeList,Summary)),Deg4))
<=> ( ( Deg = Deg4 )
& ! [X2: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList)))
=> pp(aa_nat_bool(vEBT_VEBT_valid(X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
& pp(aa_nat_bool(vEBT_VEBT_valid(Summary),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
& ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
& pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_tf(Deg),TreeList),Summary))),Mima2)) ) ) ).
% VEBT_internal.valid'.simps(2)
tff(fact_10011_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
( ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
<=> pp(Y) )
=> ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
=> ( pp(Y)
<=> ( Xa2 != one_one_nat ) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
=> ( pp(Y)
<=> ~ ( ( Deg2 = Xa2 )
& ! [X2: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
=> pp(aa_nat_bool(vEBT_VEBT_valid(X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
& pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
& ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
& pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_tf(Deg2),TreeList2),Summary2))),Mima)) ) ) ) ) ) ).
% VEBT_internal.valid'.elims(1)
tff(fact_10012_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: bool] :
( ( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
<=> pp(Y) )
=> ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa2))
=> ( ! [Uu2: bool,Uv2: bool] :
( ( X = vEBT_Leaf(Uu2,Uv2) )
=> ( ( pp(Y)
<=> ( Xa2 = one_one_nat ) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa2)) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
=> ( ( pp(Y)
<=> ( ( Deg2 = Xa2 )
& ! [X2: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(TreeList2)))
=> pp(aa_nat_bool(vEBT_VEBT_valid(X2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
& pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
& ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
& pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_tf(Deg2),TreeList2),Summary2))),Mima)) ) )
=> ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList2,Summary2),Xa2)) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(1)
tff(fact_10013_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
=> ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa2))
=> ( ! [Uu2: bool,Uv2: bool] :
( ( X = vEBT_Leaf(Uu2,Uv2) )
=> ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa2))
=> ( Xa2 != one_one_nat ) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
=> ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList2,Summary2),Xa2))
=> ~ ( ( Deg2 = Xa2 )
& ! [X4: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X4),set_VEBT_VEBT2(TreeList2)))
=> pp(aa_nat_bool(vEBT_VEBT_valid(X4),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
& pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
& ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
& pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_tf(Deg2),TreeList2),Summary2))),Mima)) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(2)
tff(fact_10014_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ pp(aa_nat_bool(vEBT_VEBT_valid(X),Xa2))
=> ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(X,Xa2))
=> ( ! [Uu2: bool,Uv2: bool] :
( ( X = vEBT_Leaf(Uu2,Uv2) )
=> ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf(Uu2,Uv2),Xa2))
=> ( Xa2 = one_one_nat ) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
=> ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList2,Summary2),Xa2))
=> ( ( Deg2 = Xa2 )
& ! [X3: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X3),set_VEBT_VEBT2(TreeList2)))
=> pp(aa_nat_bool(vEBT_VEBT_valid(X3),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
& pp(aa_nat_bool(vEBT_VEBT_valid(Summary2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))
& ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) )
& pp(aa_opt6800557945374683724t_bool(case_o7912701867834613557at_nat(aa_bool_bool(fconj(aa_bool_bool(fNot,aa_fun_nat_bool_bool(fEx_nat,aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Summary2)))),fAll_VEBT_VEBT(cOMBS_6222329278419335549l_bool(cOMBB_3438363824644082366T_VEBT(fimplies,cOMBC_771164880915498576T_bool(member_VEBT_VEBT,set_VEBT_VEBT2(TreeList2))),cOMBB_3745307990312268622T_VEBT(fNot,cOMBB_2885587777813823575T_VEBT(fEx_nat,vEBT_V8194947554948674370ptions))))),produc8955392012077580105t_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_tf(Deg2),TreeList2),Summary2))),Mima)) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(3)
tff(fact_10015_GMVT,axiom,
! [A: real,B: real,F: fun_real_real,G: fun_real_real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> ( ! [X3: real] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X3))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),B)) )
=> topolo4422821103128117721l_real(topolo2177554685111907308n_real(X3,top_top_set_real),F) )
=> ( ! [X3: real] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X3))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),B)) )
=> differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X3,top_top_set_real)) )
=> ( ! [X3: real] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X3))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X3),B)) )
=> topolo4422821103128117721l_real(topolo2177554685111907308n_real(X3,top_top_set_real),G) )
=> ( ! [X3: real] :
( ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X3))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),B)) )
=> differ6690327859849518006l_real(G,topolo2177554685111907308n_real(X3,top_top_set_real)) )
=> ? [G_c: real,F_c: real,C2: real] :
( has_fi5821293074295781190e_real(G,G_c,topolo2177554685111907308n_real(C2,top_top_set_real))
& has_fi5821293074295781190e_real(F,F_c,topolo2177554685111907308n_real(C2,top_top_set_real))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),C2))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,C2),B))
& ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A))),G_c) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(G,B)),aa_real_real(G,A))),F_c) ) ) ) ) ) ) ) ).
% GMVT
tff(fact_10016_MVT,axiom,
! [A: real,B: real,F: fun_real_real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
=> ( ! [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),B))
=> differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
=> ? [L3: real,Z3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z3))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),B))
& has_fi5821293074295781190e_real(F,L3,topolo2177554685111907308n_real(Z3,top_top_set_real))
& ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)),L3) ) ) ) ) ) ).
% MVT
tff(fact_10017_continuous__on__arcosh,axiom,
! [A2: set_real] :
( pp(aa_set_real_bool(ord_less_eq_set_real(A2),set_ord_atLeast_real(one_one_real)))
=> topolo5044208981011980120l_real(A2,arcosh_real) ) ).
% continuous_on_arcosh
tff(fact_10018_continuous__on__arcosh_H,axiom,
! [A2: set_real,F: fun_real_real] :
( topolo5044208981011980120l_real(A2,F)
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),A2))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,one_one_real),aa_real_real(F,X3))) )
=> topolo5044208981011980120l_real(A2,aTP_Lamm_tg(F)) ) ) ).
% continuous_on_arcosh'
tff(fact_10019_continuous__image__closed__interval,axiom,
! [A: real,B: real,F: fun_real_real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),B))
=> ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
=> ? [C2: real,D2: real] :
( ( image_real_real(F,set_or1222579329274155063t_real(A,B)) = set_or1222579329274155063t_real(C2,D2) )
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,C2),D2)) ) ) ) ).
% continuous_image_closed_interval
tff(fact_10020_continuous__on__arccos_H,axiom,
topolo5044208981011980120l_real(set_or1222579329274155063t_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real),arccos) ).
% continuous_on_arccos'
tff(fact_10021_continuous__on__arcsin_H,axiom,
topolo5044208981011980120l_real(set_or1222579329274155063t_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real),arcsin) ).
% continuous_on_arcsin'
tff(fact_10022_continuous__on__artanh,axiom,
! [A2: set_real] :
( pp(aa_set_real_bool(ord_less_eq_set_real(A2),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real)))
=> topolo5044208981011980120l_real(A2,artanh_real) ) ).
% continuous_on_artanh
tff(fact_10023_continuous__on__artanh_H,axiom,
! [A2: set_real,F: fun_real_real] :
( topolo5044208981011980120l_real(A2,F)
=> ( ! [X3: real] :
( pp(aa_set_real_bool(member_real(X3),A2))
=> pp(aa_set_real_bool(member_real(aa_real_real(F,X3)),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real))) )
=> topolo5044208981011980120l_real(A2,aTP_Lamm_th(F)) ) ) ).
% continuous_on_artanh'
tff(fact_10024_Rolle__deriv,axiom,
! [A: real,B: real,F: fun_real_real,F3: fun_re7681423839302404990l_real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> ( ( aa_real_real(F,A) = aa_real_real(F,B) )
=> ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
=> ( ! [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),B))
=> has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F3,X3),topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
=> ? [Z3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z3))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),B))
& ! [X4: real] : aa_real_real(aa_rea7340605660300375871l_real(F3,Z3),X4) = zero_zero_real ) ) ) ) ) ).
% Rolle_deriv
tff(fact_10025_mvt,axiom,
! [A: real,B: real,F: fun_real_real,F3: fun_re7681423839302404990l_real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
=> ( ! [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),B))
=> has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F3,X3),topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
=> ~ ! [Xi: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Xi))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Xi),B))
=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B)),aa_real_real(F,A)) != aa_real_real(aa_rea7340605660300375871l_real(F3,Xi),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B),A)) ) ) ) ) ) ) ).
% mvt
tff(fact_10026_DERIV__isconst__end,axiom,
! [A: real,B: real,F: fun_real_real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
=> ( ! [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),B))
=> has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
=> ( aa_real_real(F,B) = aa_real_real(F,A) ) ) ) ) ).
% DERIV_isconst_end
tff(fact_10027_DERIV__neg__imp__decreasing__open,axiom,
! [A: real,B: real,F: fun_real_real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> ( ! [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),B))
=> ? [Y3: real] :
( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X3,top_top_set_real))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Y3),zero_zero_real)) ) ) )
=> ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,B)),aa_real_real(F,A))) ) ) ) ).
% DERIV_neg_imp_decreasing_open
tff(fact_10028_DERIV__pos__imp__increasing__open,axiom,
! [A: real,B: real,F: fun_real_real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> ( ! [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),B))
=> ? [Y3: real] :
( has_fi5821293074295781190e_real(F,Y3,topolo2177554685111907308n_real(X3,top_top_set_real))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),Y3)) ) ) )
=> ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(F,A)),aa_real_real(F,B))) ) ) ) ).
% DERIV_pos_imp_increasing_open
tff(fact_10029_DERIV__isconst2,axiom,
! [A: real,B: real,F: fun_real_real,X: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
=> ( ! [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),B))
=> has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,A),X))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),B))
=> ( aa_real_real(F,X) = aa_real_real(F,A) ) ) ) ) ) ) ).
% DERIV_isconst2
tff(fact_10030_Rolle,axiom,
! [A: real,B: real,F: fun_real_real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),B))
=> ( ( aa_real_real(F,A) = aa_real_real(F,B) )
=> ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
=> ( ! [X3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),X3))
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),B))
=> differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
=> ? [Z3: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,A),Z3))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Z3),B))
& has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(Z3,top_top_set_real)) ) ) ) ) ) ).
% Rolle
tff(fact_10031_uniformity__real__def,axiom,
topolo1511823702728130853y_real = comple2936214249959783750l_real(image_2178119161166701260l_real(aTP_Lamm_tj,set_or5849166863359141190n_real(zero_zero_real))) ).
% uniformity_real_def
tff(fact_10032_uniformity__complex__def,axiom,
topolo896644834953643431omplex = comple8358262395181532106omplex(image_5971271580939081552omplex(aTP_Lamm_tl,set_or5849166863359141190n_real(zero_zero_real))) ).
% uniformity_complex_def
tff(fact_10033_mono__times__nat,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> order_mono_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N)) ) ).
% mono_times_nat
tff(fact_10034_mono__Suc,axiom,
order_mono_nat_nat(suc) ).
% mono_Suc
tff(fact_10035_incseq__bounded,axiom,
! [X5: fun_nat_real,B2: real] :
( order_mono_nat_real(X5)
=> ( ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X5,I2)),B2))
=> bfun_nat_real(X5,at_top_nat) ) ) ).
% incseq_bounded
tff(fact_10036_incseq__convergent,axiom,
! [X5: fun_nat_real,B2: real] :
( order_mono_nat_real(X5)
=> ( ! [I2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X5,I2)),B2))
=> ~ ! [L6: real] :
( filterlim_nat_real(X5,topolo2815343760600316023s_real(L6),at_top_nat)
=> ~ ! [I3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_nat_real(X5,I3)),L6)) ) ) ) ).
% incseq_convergent
tff(fact_10037_mono__ge2__power__minus__self,axiom,
! [K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
=> order_mono_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tm,K)) ) ).
% mono_ge2_power_minus_self
tff(fact_10038_eventually__prod__sequentially,axiom,
! [P: fun_Pr7757163693017731481t_bool] :
( eventu1038000079068216329at_nat(P,prod_filter_nat_nat(at_top_nat,at_top_nat))
<=> ? [N9: nat] :
! [M7: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),M7))
=> ! [N5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),N5))
=> pp(aa_Pro2232559814647311250t_bool(P,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N5),M7))) ) ) ) ).
% eventually_prod_sequentially
tff(fact_10039_nonneg__incseq__Bseq__subseq__iff,axiom,
! [F: fun_nat_real,G: fun_nat_nat] :
( ! [X3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X3)))
=> ( order_mono_nat_real(F)
=> ( order_5726023648592871131at_nat(G)
=> ( bfun_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_tn(F),G),at_top_nat)
<=> bfun_nat_real(F,at_top_nat) ) ) ) ) ).
% nonneg_incseq_Bseq_subseq_iff
tff(fact_10040_strict__mono__imp__increasing,axiom,
! [F: fun_nat_nat,N: nat] :
( order_5726023648592871131at_nat(F)
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(F,N))) ) ).
% strict_mono_imp_increasing
tff(fact_10041_filtermap__at__right__shift,axiom,
! [D: real,A: real] : filtermap_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_to,D),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) = topolo2177554685111907308n_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),D),set_or5849166863359141190n_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),D))) ).
% filtermap_at_right_shift
tff(fact_10042_at__right__to__0,axiom,
! [A: real] : topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)) = filtermap_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_tp,A),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).
% at_right_to_0
tff(fact_10043_inj__sgn__power,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_set_real_bool(inj_on_real_real(aa_nat_fun_real_real(aTP_Lamm_rh,N)),top_top_set_real)) ) ).
% inj_sgn_power
tff(fact_10044_log__inj,axiom,
! [B: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,one_one_real),B))
=> pp(aa_set_real_bool(inj_on_real_real(aa_rea7340605660300375871l_real(log,B)),set_or5849166863359141190n_real(zero_zero_real))) ) ).
% log_inj
tff(fact_10045_pos__deriv__imp__strict__mono,axiom,
! [F: fun_real_real,F3: fun_real_real] :
( ! [X3: real] : has_fi5821293074295781190e_real(F,aa_real_real(F3,X3),topolo2177554685111907308n_real(X3,top_top_set_real))
=> ( ! [X3: real] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,zero_zero_real),aa_real_real(F3,X3)))
=> order_7092887310737990675l_real(F) ) ) ).
% pos_deriv_imp_strict_mono
tff(fact_10046_inj__on__diff__nat,axiom,
! [N3: set_nat,K: nat] :
( ! [N2: nat] :
( pp(aa_set_nat_bool(member_nat(N2),N3))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N2)) )
=> pp(aa_set_nat_bool(inj_on_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_pi,K)),N3)) ) ).
% inj_on_diff_nat
tff(fact_10047_inj__Suc,axiom,
! [N3: set_nat] : pp(aa_set_nat_bool(inj_on_nat_nat(suc),N3)) ).
% inj_Suc
tff(fact_10048_summable__reindex,axiom,
! [F: fun_nat_real,G: fun_nat_nat] :
( summable_real(F)
=> ( pp(aa_set_nat_bool(inj_on_nat_nat(G),top_top_set_nat))
=> ( ! [X3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X3)))
=> summable_real(comp_nat_real_nat(F,G)) ) ) ) ).
% summable_reindex
tff(fact_10049_suminf__reindex__mono,axiom,
! [F: fun_nat_real,G: fun_nat_nat] :
( summable_real(F)
=> ( pp(aa_set_nat_bool(inj_on_nat_nat(G),top_top_set_nat))
=> ( ! [X3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X3)))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,suminf_real(comp_nat_real_nat(F,G))),suminf_real(F))) ) ) ) ).
% suminf_reindex_mono
tff(fact_10050_inj__on__char__of__nat,axiom,
pp(aa_set_nat_bool(inj_on_nat_char(unique3096191561947761185of_nat),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))))))) ).
% inj_on_char_of_nat
tff(fact_10051_suminf__reindex,axiom,
! [F: fun_nat_real,G: fun_nat_nat] :
( summable_real(F)
=> ( pp(aa_set_nat_bool(inj_on_nat_nat(G),top_top_set_nat))
=> ( ! [X3: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X3)))
=> ( ! [X3: nat] :
( ~ pp(aa_set_nat_bool(member_nat(X3),image_nat_nat(G,top_top_set_nat)))
=> ( aa_nat_real(F,X3) = zero_zero_real ) )
=> ( suminf_real(comp_nat_real_nat(F,G)) = suminf_real(F) ) ) ) ) ) ).
% suminf_reindex
tff(fact_10052_powr__real__of__int_H,axiom,
! [X: real,N: int] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( ( ( X != zero_zero_real )
| pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),N)) )
=> ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,X),aa_int_real(ring_1_of_int_real,N)) = aa_int_real(power_int_real(X),N) ) ) ) ).
% powr_real_of_int'
tff(fact_10053_pred__nat__def,axiom,
pred_nat = collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_tq)) ).
% pred_nat_def
tff(fact_10054_atLeastLessThan__add__Un,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> ( aa_nat_set_nat(set_or4665077453230672383an_nat(I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = sup_sup_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(I),J),aa_nat_set_nat(set_or4665077453230672383an_nat(J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).
% atLeastLessThan_add_Un
tff(fact_10055_sup__nat__def,axiom,
sup_sup_nat = ord_max_nat ).
% sup_nat_def
tff(fact_10056_sup__enat__def,axiom,
sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).
% sup_enat_def
tff(fact_10057_Rats__eq__int__div__nat,axiom,
field_5140801741446780682s_real = collect_real(aTP_Lamm_tr) ).
% Rats_eq_int_div_nat
tff(fact_10058_Rats__abs__iff,axiom,
! [X: real] :
( pp(aa_set_real_bool(member_real(aa_real_real(abs_abs_real,X)),field_5140801741446780682s_real))
<=> pp(aa_set_real_bool(member_real(X),field_5140801741446780682s_real)) ) ).
% Rats_abs_iff
tff(fact_10059_Rats__no__top__le,axiom,
! [X: real] :
? [X3: real] :
( pp(aa_set_real_bool(member_real(X3),field_5140801741446780682s_real))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),X3)) ) ).
% Rats_no_top_le
tff(fact_10060_Rats__no__bot__less,axiom,
! [X: real] :
? [X3: real] :
( pp(aa_set_real_bool(member_real(X3),field_5140801741446780682s_real))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),X)) ) ).
% Rats_no_bot_less
tff(fact_10061_Rats__dense__in__real,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
=> ? [X3: real] :
( pp(aa_set_real_bool(member_real(X3),field_5140801741446780682s_real))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),X3))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X3),Y)) ) ) ).
% Rats_dense_in_real
tff(fact_10062_Rats__eq__int__div__int,axiom,
field_5140801741446780682s_real = collect_real(aTP_Lamm_ts) ).
% Rats_eq_int_div_int
tff(fact_10063_less__eq,axiom,
! [M: nat,N: nat] :
( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),N)),transi6264000038957366511cl_nat(pred_nat)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N)) ) ).
% less_eq
tff(fact_10064_pred__nat__trancl__eq__le,axiom,
! [M: nat,N: nat] :
( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),N)),transi2905341329935302413cl_nat(pred_nat)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N)) ) ).
% pred_nat_trancl_eq_le
tff(fact_10065_tl__upt,axiom,
! [M: nat,N: nat] : tl_nat(upt(M,N)) = upt(aa_nat_nat(suc,M),N) ).
% tl_upt
tff(fact_10066_hd__upt,axiom,
! [I: nat,J: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
=> ( hd_nat(upt(I,J)) = I ) ) ).
% hd_upt
tff(fact_10067_drop__upt,axiom,
! [M: nat,I: nat,J: nat] : drop_nat(M,upt(I,J)) = upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M),J) ).
% drop_upt
tff(fact_10068_length__upt,axiom,
! [I: nat,J: nat] : size_size_list_nat(upt(I,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I) ).
% length_upt
tff(fact_10069_take__upt,axiom,
! [I: nat,M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M)),N))
=> ( take_nat(M,upt(I,N)) = upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M)) ) ) ).
% take_upt
tff(fact_10070_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
=> ( upt(I,J) = nil_nat ) ) ).
% upt_conv_Nil
tff(fact_10071_upt__eq__Nil__conv,axiom,
! [I: nat,J: nat] :
( ( upt(I,J) = nil_nat )
<=> ( ( J = zero_zero_nat )
| pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I)) ) ) ).
% upt_eq_Nil_conv
tff(fact_10072_nth__upt,axiom,
! [I: nat,K: nat,J: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(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_10073_upt__rec__numeral,axiom,
! [M: num,N: num] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
=> ( upt(aa_num_nat(numeral_numeral_nat,M),aa_num_nat(numeral_numeral_nat,N)) = cons_nat(aa_num_nat(numeral_numeral_nat,M),upt(aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)))
=> ( upt(aa_num_nat(numeral_numeral_nat,M),aa_num_nat(numeral_numeral_nat,N)) = nil_nat ) ) ) ).
% upt_rec_numeral
tff(fact_10074_map__Suc__upt,axiom,
! [M: nat,N: nat] : map_nat_nat(suc,upt(M,N)) = upt(aa_nat_nat(suc,M),aa_nat_nat(suc,N)) ).
% map_Suc_upt
tff(fact_10075_map__add__upt,axiom,
! [N: nat,M: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tt,N),upt(zero_zero_nat,M)) = upt(N,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).
% map_add_upt
tff(fact_10076_atMost__upto,axiom,
! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,N) = set_nat2(upt(zero_zero_nat,aa_nat_nat(suc,N))) ).
% atMost_upto
tff(fact_10077_upt__conv__Cons__Cons,axiom,
! [M: nat,N: nat,Ns: list_nat,Q2: nat] :
( ( cons_nat(M,cons_nat(N,Ns)) = upt(M,Q2) )
<=> ( cons_nat(N,Ns) = upt(aa_nat_nat(suc,M),Q2) ) ) ).
% upt_conv_Cons_Cons
tff(fact_10078_greaterThanAtMost__upt,axiom,
! [N: nat,M: nat] : aa_nat_set_nat(set_or6659071591806873216st_nat(N),M) = set_nat2(upt(aa_nat_nat(suc,N),aa_nat_nat(suc,M))) ).
% greaterThanAtMost_upt
tff(fact_10079_greaterThanLessThan__upt,axiom,
! [N: nat,M: nat] : aa_nat_set_nat(set_or5834768355832116004an_nat(N),M) = set_nat2(upt(aa_nat_nat(suc,N),M)) ).
% greaterThanLessThan_upt
tff(fact_10080_atLeastAtMost__upt,axiom,
! [N: nat,M: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(N),M) = set_nat2(upt(N,aa_nat_nat(suc,M))) ).
% atLeastAtMost_upt
tff(fact_10081_upt__conv__Cons,axiom,
! [I: nat,J: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
=> ( upt(I,J) = cons_nat(I,upt(aa_nat_nat(suc,I),J)) ) ) ).
% upt_conv_Cons
tff(fact_10082_map__decr__upt,axiom,
! [M: nat,N: nat] : map_nat_nat(aTP_Lamm_tu,upt(aa_nat_nat(suc,M),aa_nat_nat(suc,N))) = upt(M,N) ).
% map_decr_upt
tff(fact_10083_upt__add__eq__append,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> ( upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = append_nat(upt(I,J),upt(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).
% upt_add_eq_append
tff(fact_10084_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X: nat,Xs: list_nat] :
( ( upt(I,J) = cons_nat(X,Xs) )
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
& ( I = X )
& ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat),J) = Xs ) ) ) ).
% upt_eq_Cons_conv
tff(fact_10085_upt__rec,axiom,
! [I: nat,J: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
=> ( upt(I,J) = cons_nat(I,upt(aa_nat_nat(suc,I),J)) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
=> ( upt(I,J) = nil_nat ) ) ) ).
% upt_rec
tff(fact_10086_upt__Suc__append,axiom,
! [I: nat,J: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> ( upt(I,aa_nat_nat(suc,J)) = append_nat(upt(I,J),cons_nat(J,nil_nat)) ) ) ).
% upt_Suc_append
tff(fact_10087_upt__Suc,axiom,
! [I: nat,J: nat] :
( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> ( upt(I,aa_nat_nat(suc,J)) = append_nat(upt(I,J),cons_nat(J,nil_nat)) ) )
& ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
=> ( upt(I,aa_nat_nat(suc,J)) = nil_nat ) ) ) ).
% upt_Suc
tff(fact_10088_sum__list__upt,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
=> ( groups4561878855575611511st_nat(upt(M,N)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_be),aa_nat_set_nat(set_or4665077453230672383an_nat(M),N)) ) ) ).
% sum_list_upt
tff(fact_10089_card__length__sum__list__rec,axiom,
! [M: nat,N3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),M))
=> ( finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_tv(M),N3))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_tw(M),N3)))),finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_tx(M),N3)))) ) ) ).
% card_length_sum_list_rec
tff(fact_10090_card__length__sum__list,axiom,
! [M: nat,N3: nat] : finite_card_list_nat(collect_list_nat(aa_nat2079453845469346690t_bool(aTP_Lamm_tv(M),N3))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_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_10091_sorted__upt,axiom,
! [M: nat,N: nat] : pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),upt(M,N))) ).
% sorted_upt
tff(fact_10092_sorted__wrt__upt,axiom,
! [M: nat,N: nat] : pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),upt(M,N))) ).
% sorted_wrt_upt
tff(fact_10093_sorted__wrt__less__idx,axiom,
! [Ns: list_nat,I: nat] :
( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),Ns))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Ns)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(nth_nat(Ns),I))) ) ) ).
% sorted_wrt_less_idx
tff(fact_10094_sorted__wrt__upto,axiom,
! [I: int,J: int] : pp(aa_list_int_bool(sorted_wrt_int(ord_less_int),upto(I,J))) ).
% sorted_wrt_upto
tff(fact_10095_list__encode_Opelims,axiom,
! [X: list_nat,Y: nat] :
( ( nat_list_encode(X) = Y )
=> ( pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),X))
=> ( ( ( X = nil_nat )
=> ( ( Y = zero_zero_nat )
=> ~ pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),nil_nat)) ) )
=> ~ ! [X3: nat,Xs2: list_nat] :
( ( X = cons_nat(X3,Xs2) )
=> ( ( Y = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),nat_list_encode(Xs2)))) )
=> ~ pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),cons_nat(X3,Xs2))) ) ) ) ) ) ).
% list_encode.pelims
tff(fact_10096_quotient__of__def,axiom,
! [X: rat] : quotient_of(X) = the_Pr4378521158711661632nt_int(aTP_Lamm_ty(X)) ).
% quotient_of_def
tff(fact_10097_coprime__abs__right__iff,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,K),aa_int_int(abs_abs_int,L)))
<=> pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,K),L)) ) ).
% coprime_abs_right_iff
tff(fact_10098_coprime__abs__left__iff,axiom,
! [K: int,L: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,aa_int_int(abs_abs_int,K)),L))
<=> pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,K),L)) ) ).
% coprime_abs_left_iff
tff(fact_10099_normalize__stable,axiom,
! [Q2: int,P2: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q2))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,P2),Q2))
=> ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q2)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q2) ) ) ) ).
% normalize_stable
tff(fact_10100_coprime__common__divisor__int,axiom,
! [A: int,B: int,X: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,A),B))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),A))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,X),B))
=> ( aa_int_int(abs_abs_int,X) = one_one_int ) ) ) ) ).
% coprime_common_divisor_int
tff(fact_10101_Rat__cases,axiom,
! [Q2: rat] :
~ ! [A4: int,B4: int] :
( ( Q2 = aa_int_rat(fract(A4),B4) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B4))
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,A4),B4)) ) ) ).
% Rat_cases
tff(fact_10102_Rat__induct,axiom,
! [P: fun_rat_bool,Q2: rat] :
( ! [A4: int,B4: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B4))
=> ( pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,A4),B4))
=> pp(aa_rat_bool(P,aa_int_rat(fract(A4),B4))) ) )
=> pp(aa_rat_bool(P,Q2)) ) ).
% Rat_induct
tff(fact_10103_Rat__cases__nonzero,axiom,
! [Q2: rat] :
( ! [A4: int,B4: int] :
( ( Q2 = aa_int_rat(fract(A4),B4) )
=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B4))
=> ( ( A4 != zero_zero_int )
=> ~ pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,A4),B4)) ) ) )
=> ( Q2 = zero_zero_rat ) ) ).
% Rat_cases_nonzero
tff(fact_10104_quotient__of__unique,axiom,
! [R: rat] :
? [X3: product_prod_int_int] :
( ( R = aa_int_rat(fract(aa_Pro6799258434465214121nt_int(product_fst_int_int,X3)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X3)) )
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_Pro6799258434465214121nt_int(product_snd_int_int,X3)))
& pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,X3)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X3)))
& ! [Y3: product_prod_int_int] :
( ( ( R = aa_int_rat(fract(aa_Pro6799258434465214121nt_int(product_fst_int_int,Y3)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Y3)) )
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_Pro6799258434465214121nt_int(product_snd_int_int,Y3)))
& pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Y3)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Y3))) )
=> ( Y3 = X3 ) ) ) ).
% quotient_of_unique
tff(fact_10105_coprime__int__iff,axiom,
! [M: nat,N: nat] :
( pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,M),N)) ) ).
% coprime_int_iff
tff(fact_10106_coprime__nat__abs__right__iff,axiom,
! [N: nat,K: int] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,N),aa_int_nat(nat2,aa_int_int(abs_abs_int,K))))
<=> pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,aa_nat_int(semiri1314217659103216013at_int,N)),K)) ) ).
% coprime_nat_abs_right_iff
tff(fact_10107_coprime__nat__abs__left__iff,axiom,
! [K: int,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),N))
<=> pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,K),aa_nat_int(semiri1314217659103216013at_int,N))) ) ).
% coprime_nat_abs_left_iff
tff(fact_10108_coprime__crossproduct__nat,axiom,
! [A: nat,D: nat,B: nat,C: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,A),D))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,B),C))
=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D) )
<=> ( ( A = B )
& ( C = D ) ) ) ) ) ).
% coprime_crossproduct_nat
tff(fact_10109_coprime__Suc__0__left,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,aa_nat_nat(suc,zero_zero_nat)),N)) ).
% coprime_Suc_0_left
tff(fact_10110_coprime__Suc__0__right,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,N),aa_nat_nat(suc,zero_zero_nat))) ).
% coprime_Suc_0_right
tff(fact_10111_coprime__Suc__right__nat,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,N),aa_nat_nat(suc,N))) ).
% coprime_Suc_right_nat
tff(fact_10112_coprime__Suc__left__nat,axiom,
! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,aa_nat_nat(suc,N)),N)) ).
% coprime_Suc_left_nat
tff(fact_10113_coprime__common__divisor__nat,axiom,
! [A: nat,B: nat,X: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,A),B))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),A))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,X),B))
=> ( X = one_one_nat ) ) ) ) ).
% coprime_common_divisor_nat
tff(fact_10114_coprime__diff__one__left__nat,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),N)) ) ).
% coprime_diff_one_left_nat
tff(fact_10115_coprime__diff__one__right__nat,axiom,
! [N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ).
% coprime_diff_one_right_nat
tff(fact_10116_Rats__abs__nat__div__natE,axiom,
! [X: real] :
( pp(aa_set_real_bool(member_real(X),field_5140801741446780682s_real))
=> ~ ! [M3: nat,N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ( aa_real_real(abs_abs_real,X) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,M3)),aa_nat_real(semiri5074537144036343181t_real,N2)) )
=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,M3),N2)) ) ) ) ).
% Rats_abs_nat_div_natE
tff(fact_10117_vanishes__mult__bounded,axiom,
! [X5: fun_nat_rat,Y5: fun_nat_rat] :
( ? [A7: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),A7))
& ! [N2: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,abs_abs_rat(aa_nat_rat(X5,N2))),A7)) )
=> ( pp(vanishes(Y5))
=> pp(vanishes(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_tz,X5),Y5))) ) ) ).
% vanishes_mult_bounded
tff(fact_10118_vanishes__const,axiom,
! [C: rat] :
( pp(vanishes(aTP_Lamm_ua(C)))
<=> ( C = zero_zero_rat ) ) ).
% vanishes_const
tff(fact_10119_vanishes__diff,axiom,
! [X5: fun_nat_rat,Y5: fun_nat_rat] :
( pp(vanishes(X5))
=> ( pp(vanishes(Y5))
=> pp(vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_ub(X5),Y5))) ) ) ).
% vanishes_diff
tff(fact_10120_vanishes__minus,axiom,
! [X5: fun_nat_rat] :
( pp(vanishes(X5))
=> pp(vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_uc,X5))) ) ).
% vanishes_minus
tff(fact_10121_vanishes__add,axiom,
! [X5: fun_nat_rat,Y5: fun_nat_rat] :
( pp(vanishes(X5))
=> ( pp(vanishes(Y5))
=> pp(vanishes(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ud,X5),Y5))) ) ) ).
% vanishes_add
tff(fact_10122_vanishesD,axiom,
! [X5: fun_nat_rat,R: rat] :
( pp(vanishes(X5))
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R))
=> ? [K3: nat] :
! [N6: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N6))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,abs_abs_rat(aa_nat_rat(X5,N6))),R)) ) ) ) ).
% vanishesD
tff(fact_10123_vanishesI,axiom,
! [X5: fun_nat_rat] :
( ! [R3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R3))
=> ? [K4: nat] :
! [N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K4),N2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,abs_abs_rat(aa_nat_rat(X5,N2))),R3)) ) )
=> pp(vanishes(X5)) ) ).
% vanishesI
tff(fact_10124_vanishes__def,axiom,
! [X5: fun_nat_rat] :
( pp(vanishes(X5))
<=> ! [R5: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
=> ? [K2: nat] :
! [N5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,abs_abs_rat(aa_nat_rat(X5,N5))),R5)) ) ) ) ).
% vanishes_def
tff(fact_10125_and__not__num_Opelims,axiom,
! [X: num,Xa2: num,Y: option_num] :
( ( bit_and_not_num(X,Xa2) = Y )
=> ( pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(X,Xa2)))
=> ( ( ( X = one )
=> ( ( Xa2 = one )
=> ( ( Y = none_num )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(one,one))) ) ) )
=> ( ( ( X = one )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit0,N2) )
=> ( ( Y = some_num(one) )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(one,aa_num_num(bit0,N2)))) ) ) )
=> ( ( ( X = one )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit1,N2) )
=> ( ( Y = none_num )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(one,aa_num_num(bit1,N2)))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ( ( Xa2 = one )
=> ( ( Y = some_num(aa_num_num(bit0,M3)) )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit0,M3),one))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit0,N2) )
=> ( ( Y = map_option_num_num(bit0,bit_and_not_num(M3,N2)) )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit0,N2)))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit1,N2) )
=> ( ( Y = map_option_num_num(bit0,bit_and_not_num(M3,N2)) )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit1,N2)))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ( ( Xa2 = one )
=> ( ( Y = some_num(aa_num_num(bit0,M3)) )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit1,M3),one))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit0,N2) )
=> ( ( Y = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_qv,bit_and_not_num(M3,N2)) )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit0,N2)))) ) ) )
=> ~ ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit1,N2) )
=> ( ( Y = map_option_num_num(bit0,bit_and_not_num(M3,N2)) )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_and_not_num_rel),product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit1,N2)))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.pelims
tff(fact_10126_rcis__inverse,axiom,
! [R: real,A: real] : aa_complex_complex(invers8013647133539491842omplex,aa_real_complex(rcis(R),A)) = aa_real_complex(rcis(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),R)),aa_real_real(uminus_uminus_real,A)) ).
% rcis_inverse
tff(fact_10127_Re__rcis,axiom,
! [R: real,A: real] : aa_complex_real(re,aa_real_complex(rcis(R),A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_real_real(cos_real,A)) ).
% Re_rcis
tff(fact_10128_Im__rcis,axiom,
! [R: real,A: real] : aa_complex_real(im,aa_real_complex(rcis(R),A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R),aa_real_real(sin_real,A)) ).
% Im_rcis
tff(fact_10129_cis__rcis__eq,axiom,
! [A: real] : aa_real_complex(cis,A) = aa_real_complex(rcis(one_one_real),A) ).
% cis_rcis_eq
tff(fact_10130_rcis__mult,axiom,
! [R1: real,A: real,R22: real,B: real] : aa_complex_complex(times_times_complex(aa_real_complex(rcis(R1),A)),aa_real_complex(rcis(R22),B)) = aa_real_complex(rcis(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R1),R22)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B)) ).
% rcis_mult
tff(fact_10131_rcis__divide,axiom,
! [R1: real,A: real,R22: real,B: real] : aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(rcis(R1),A)),aa_real_complex(rcis(R22),B)) = aa_real_complex(rcis(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,R1),R22)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B)) ).
% rcis_divide
tff(fact_10132_DeMoivre2,axiom,
! [R: real,A: real,N: nat] : aa_nat_complex(power_power_complex(aa_real_complex(rcis(R),A)),N) = aa_real_complex(rcis(aa_nat_real(aa_real_fun_nat_real(power_power_real,R),N)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),A)) ).
% DeMoivre2
tff(fact_10133_and__num_Opelims,axiom,
! [X: num,Xa2: num,Y: option_num] :
( ( aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,X),Xa2) = Y )
=> ( pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(X,Xa2)))
=> ( ( ( X = one )
=> ( ( Xa2 = one )
=> ( ( Y = some_num(one) )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(one,one))) ) ) )
=> ( ( ( X = one )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit0,N2) )
=> ( ( Y = none_num )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(one,aa_num_num(bit0,N2)))) ) ) )
=> ( ( ( X = one )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit1,N2) )
=> ( ( Y = some_num(one) )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(one,aa_num_num(bit1,N2)))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ( ( Xa2 = one )
=> ( ( Y = none_num )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit0,M3),one))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit0,N2) )
=> ( ( Y = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M3),N2)) )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit0,N2)))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit1,N2) )
=> ( ( Y = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M3),N2)) )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit1,N2)))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ( ( Xa2 = one )
=> ( ( Y = some_num(one) )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit1,M3),one))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit0,N2) )
=> ( ( Y = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M3),N2)) )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit0,N2)))) ) ) )
=> ~ ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit1,N2) )
=> ( ( Y = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_qv,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M3),N2)) )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel),product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit1,N2)))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.pelims
tff(fact_10134_xor__num_Opelims,axiom,
! [X: num,Xa2: num,Y: option_num] :
( ( aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,X),Xa2) = Y )
=> ( pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(X,Xa2)))
=> ( ( ( X = one )
=> ( ( Xa2 = one )
=> ( ( Y = none_num )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(one,one))) ) ) )
=> ( ( ( X = one )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit0,N2) )
=> ( ( Y = some_num(aa_num_num(bit1,N2)) )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(one,aa_num_num(bit0,N2)))) ) ) )
=> ( ( ( X = one )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit1,N2) )
=> ( ( Y = some_num(aa_num_num(bit0,N2)) )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(one,aa_num_num(bit1,N2)))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ( ( Xa2 = one )
=> ( ( Y = some_num(aa_num_num(bit1,M3)) )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(aa_num_num(bit0,M3),one))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit0,N2) )
=> ( ( Y = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M3),N2)) )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit0,N2)))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit0,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit1,N2) )
=> ( ( Y = some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M3),N2))) )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit1,N2)))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ( ( Xa2 = one )
=> ( ( Y = some_num(aa_num_num(bit0,M3)) )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(aa_num_num(bit1,M3),one))) ) ) )
=> ( ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit0,N2) )
=> ( ( Y = some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M3),N2))) )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit0,N2)))) ) ) )
=> ~ ! [M3: num] :
( ( X = aa_num_num(bit1,M3) )
=> ! [N2: num] :
( ( Xa2 = aa_num_num(bit1,N2) )
=> ( ( Y = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M3),N2)) )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel),product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit1,N2)))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.pelims
tff(fact_10135_or__not__num__neg_Opelims,axiom,
! [X: num,Xa2: num,Y: num] :
( ( aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,X),Xa2) = Y )
=> ( pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(X,Xa2)))
=> ( ( ( X = one )
=> ( ( Xa2 = one )
=> ( ( Y = one )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(one,one))) ) ) )
=> ( ( ( X = one )
=> ! [M3: num] :
( ( Xa2 = aa_num_num(bit0,M3) )
=> ( ( Y = aa_num_num(bit1,M3) )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(one,aa_num_num(bit0,M3)))) ) ) )
=> ( ( ( X = one )
=> ! [M3: num] :
( ( Xa2 = aa_num_num(bit1,M3) )
=> ( ( Y = aa_num_num(bit1,M3) )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(one,aa_num_num(bit1,M3)))) ) ) )
=> ( ! [N2: num] :
( ( X = aa_num_num(bit0,N2) )
=> ( ( Xa2 = one )
=> ( ( Y = aa_num_num(bit0,one) )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit0,N2),one))) ) ) )
=> ( ! [N2: num] :
( ( X = aa_num_num(bit0,N2) )
=> ! [M3: num] :
( ( Xa2 = aa_num_num(bit0,M3) )
=> ( ( Y = bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N2),M3)) )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit0,N2),aa_num_num(bit0,M3)))) ) ) )
=> ( ! [N2: num] :
( ( X = aa_num_num(bit0,N2) )
=> ! [M3: num] :
( ( Xa2 = aa_num_num(bit1,M3) )
=> ( ( Y = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N2),M3)) )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit0,N2),aa_num_num(bit1,M3)))) ) ) )
=> ( ! [N2: num] :
( ( X = aa_num_num(bit1,N2) )
=> ( ( Xa2 = one )
=> ( ( Y = one )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit1,N2),one))) ) ) )
=> ( ! [N2: num] :
( ( X = aa_num_num(bit1,N2) )
=> ! [M3: num] :
( ( Xa2 = aa_num_num(bit0,M3) )
=> ( ( Y = bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N2),M3)) )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit1,N2),aa_num_num(bit0,M3)))) ) ) )
=> ~ ! [N2: num] :
( ( X = aa_num_num(bit1,N2) )
=> ! [M3: num] :
( ( Xa2 = aa_num_num(bit1,M3) )
=> ( ( Y = bitM(aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N2),M3)) )
=> ~ pp(aa_Pro762926636801165566m_bool(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),product_Pair_num_num(aa_num_num(bit1,N2),aa_num_num(bit1,M3)))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_not_num_neg.pelims
tff(fact_10136_and__num__rel__dict,axiom,
bit_un4731106466462545111um_rel = bit_un5425074673868309765um_rel ).
% and_num_rel_dict
tff(fact_10137_xor__num__rel__dict,axiom,
bit_un2901131394128224187um_rel = bit_un3595099601533988841um_rel ).
% xor_num_rel_dict
tff(fact_10138_Field__natLeq__on,axiom,
! [N: nat] : field_nat(collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ue(N)))) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bh,N)) ).
% Field_natLeq_on
tff(fact_10139_natLess__def,axiom,
bNF_Ca8459412986667044542atLess = collec3392354462482085612at_nat(produc8955392012077580105t_bool(ord_less_nat)) ).
% natLess_def
tff(fact_10140_wf__less,axiom,
pp(aa_set3135850010578026034t_bool(wf_nat,collec3392354462482085612at_nat(produc8955392012077580105t_bool(ord_less_nat)))) ).
% wf_less
tff(fact_10141_product__atMost__eq__Un,axiom,
! [A2: set_nat,M: nat] : produc457027306803732586at_nat(A2,aTP_Lamm_uf(M)) = sup_su6327502436637775413at_nat(produc457027306803732586at_nat(A2,aTP_Lamm_ug(M)),produc457027306803732586at_nat(A2,aTP_Lamm_uh(M))) ).
% product_atMost_eq_Un
tff(fact_10142_pairs__le__eq__Sigma,axiom,
! [M: nat] : collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ui(M))) = produc457027306803732586at_nat(aa_nat_set_nat(set_ord_atMost_nat,M),aTP_Lamm_ug(M)) ).
% pairs_le_eq_Sigma
tff(fact_10143_finite__vimage__Suc__iff,axiom,
! [F4: set_nat] :
( pp(aa_set_nat_bool(finite_finite_nat,vimage_nat_nat(suc,F4)))
<=> pp(aa_set_nat_bool(finite_finite_nat,F4)) ) ).
% finite_vimage_Suc_iff
tff(fact_10144_vimage__Suc__insert__Suc,axiom,
! [N: nat,A2: set_nat] : vimage_nat_nat(suc,insert_nat(aa_nat_nat(suc,N),A2)) = insert_nat(N,vimage_nat_nat(suc,A2)) ).
% vimage_Suc_insert_Suc
tff(fact_10145_vimage__Suc__insert__0,axiom,
! [A2: set_nat] : vimage_nat_nat(suc,insert_nat(zero_zero_nat,A2)) = vimage_nat_nat(suc,A2) ).
% vimage_Suc_insert_0
tff(fact_10146_set__decode__div__2,axiom,
! [X: nat] : aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = vimage_nat_nat(suc,aa_nat_set_nat(nat_set_decode,X)) ).
% set_decode_div_2
tff(fact_10147_set__encode__vimage__Suc,axiom,
! [A2: set_nat] : aa_set_nat_nat(nat_set_encode,vimage_nat_nat(suc,A2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(nat_set_encode,A2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% set_encode_vimage_Suc
tff(fact_10148_Restr__natLeq,axiom,
! [N: nat] : inf_in2572325071724192079at_nat(bNF_Ca8665028551170535155natLeq,produc457027306803732586at_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bh,N)),aTP_Lamm_uj(N))) = collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ue(N))) ).
% Restr_natLeq
tff(fact_10149_cauchyD,axiom,
! [X5: fun_nat_rat,R: rat] :
( cauchy(X5)
=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R))
=> ? [K3: nat] :
! [M2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),M2))
=> ! [N6: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N6))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,abs_abs_rat(minus_minus_rat(aa_nat_rat(X5,M2),aa_nat_rat(X5,N6)))),R)) ) ) ) ) ).
% cauchyD
tff(fact_10150_natLeq__def,axiom,
bNF_Ca8665028551170535155natLeq = collec3392354462482085612at_nat(produc8955392012077580105t_bool(ord_less_eq_nat)) ).
% natLeq_def
tff(fact_10151_natLeq__natLess__Id,axiom,
bNF_Ca8459412986667044542atLess = minus_1356011639430497352at_nat(bNF_Ca8665028551170535155natLeq,id_nat2) ).
% natLeq_natLess_Id
tff(fact_10152_cauchy__inverse,axiom,
! [X5: fun_nat_rat] :
( cauchy(X5)
=> ( ~ pp(vanishes(X5))
=> cauchy(aTP_Lamm_uk(X5)) ) ) ).
% cauchy_inverse
tff(fact_10153_cauchy__const,axiom,
! [X: rat] : cauchy(aTP_Lamm_ua(X)) ).
% cauchy_const
tff(fact_10154_cauchy__add,axiom,
! [X5: fun_nat_rat,Y5: fun_nat_rat] :
( cauchy(X5)
=> ( cauchy(Y5)
=> cauchy(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ud,X5),Y5)) ) ) ).
% cauchy_add
tff(fact_10155_cauchy__minus,axiom,
! [X5: fun_nat_rat] :
( cauchy(X5)
=> cauchy(aa_fun765758790203053448at_rat(aTP_Lamm_uc,X5)) ) ).
% cauchy_minus
tff(fact_10156_cauchy__mult,axiom,
! [X5: fun_nat_rat,Y5: fun_nat_rat] :
( cauchy(X5)
=> ( cauchy(Y5)
=> cauchy(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_tz,X5),Y5)) ) ) ).
% cauchy_mult
tff(fact_10157_cauchy__diff,axiom,
! [X5: fun_nat_rat,Y5: fun_nat_rat] :
( cauchy(X5)
=> ( cauchy(Y5)
=> cauchy(aa_fun765758790203053448at_rat(aTP_Lamm_ub(X5),Y5)) ) ) ).
% cauchy_diff
tff(fact_10158_cauchy__imp__bounded,axiom,
! [X5: fun_nat_rat] :
( cauchy(X5)
=> ? [B4: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B4))
& ! [N6: nat] : pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,abs_abs_rat(aa_nat_rat(X5,N6))),B4)) ) ) ).
% cauchy_imp_bounded
tff(fact_10159_vanishes__diff__inverse,axiom,
! [X5: fun_nat_rat,Y5: fun_nat_rat] :
( cauchy(X5)
=> ( ~ pp(vanishes(X5))
=> ( cauchy(Y5)
=> ( ~ pp(vanishes(Y5))
=> ( pp(vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_ub(X5),Y5)))
=> pp(vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_ul(X5),Y5))) ) ) ) ) ) ).
% vanishes_diff_inverse
tff(fact_10160_cauchy__not__vanishes__cases,axiom,
! [X5: fun_nat_rat] :
( cauchy(X5)
=> ( ~ pp(vanishes(X5))
=> ? [B4: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B4))
& ? [K3: nat] :
( ! [N6: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N6))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B4),uminus_uminus_rat(aa_nat_rat(X5,N6)))) )
| ! [N6: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N6))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B4),aa_nat_rat(X5,N6))) ) ) ) ) ) ).
% cauchy_not_vanishes_cases
tff(fact_10161_cauchy__not__vanishes,axiom,
! [X5: fun_nat_rat] :
( cauchy(X5)
=> ( ~ pp(vanishes(X5))
=> ? [B4: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),B4))
& ? [K3: nat] :
! [N6: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N6))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,B4),abs_abs_rat(aa_nat_rat(X5,N6)))) ) ) ) ) ).
% cauchy_not_vanishes
tff(fact_10162_cauchy__def,axiom,
! [X5: fun_nat_rat] :
( cauchy(X5)
<=> ! [R5: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
=> ? [K2: nat] :
! [M7: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),M7))
=> ! [N5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,abs_abs_rat(minus_minus_rat(aa_nat_rat(X5,M7),aa_nat_rat(X5,N5)))),R5)) ) ) ) ) ).
% cauchy_def
tff(fact_10163_cauchyI,axiom,
! [X5: fun_nat_rat] :
( ! [R3: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R3))
=> ? [K4: nat] :
! [M3: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K4),M3))
=> ! [N2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K4),N2))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,abs_abs_rat(minus_minus_rat(aa_nat_rat(X5,M3),aa_nat_rat(X5,N2)))),R3)) ) ) )
=> cauchy(X5) ) ).
% cauchyI
tff(fact_10164_Restr__natLeq2,axiom,
! [N: nat] : inf_in2572325071724192079at_nat(bNF_Ca8665028551170535155natLeq,produc457027306803732586at_nat(aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),N),aTP_Lamm_um(N))) = collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ue(N))) ).
% Restr_natLeq2
tff(fact_10165_le__Real,axiom,
! [X5: fun_nat_rat,Y5: fun_nat_rat] :
( cauchy(X5)
=> ( cauchy(Y5)
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_fun_nat_rat_real(real2,X5)),aa_fun_nat_rat_real(real2,Y5)))
<=> ! [R5: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
=> ? [K2: nat] :
! [N5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X5,N5)),plus_plus_rat(aa_nat_rat(Y5,N5),R5))) ) ) ) ) ) ).
% le_Real
tff(fact_10166_Real__induct,axiom,
! [P: fun_real_bool,X: real] :
( ! [X9: fun_nat_rat] :
( cauchy(X9)
=> pp(aa_real_bool(P,aa_fun_nat_rat_real(real2,X9))) )
=> pp(aa_real_bool(P,X)) ) ).
% Real_induct
tff(fact_10167_natLeq__underS__less,axiom,
! [N: nat] : aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),N) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bh,N)) ).
% natLeq_underS_less
tff(fact_10168_one__real__def,axiom,
one_one_real = aa_fun_nat_rat_real(real2,aTP_Lamm_un) ).
% one_real_def
tff(fact_10169_zero__real__def,axiom,
zero_zero_real = aa_fun_nat_rat_real(real2,aTP_Lamm_uo) ).
% zero_real_def
tff(fact_10170_of__rat__Real,axiom,
! [X: rat] : field_7254667332652039916t_real(X) = aa_fun_nat_rat_real(real2,aTP_Lamm_ua(X)) ).
% of_rat_Real
tff(fact_10171_of__nat__Real,axiom,
! [X: nat] : aa_nat_real(semiri5074537144036343181t_real,X) = aa_fun_nat_rat_real(real2,aTP_Lamm_up(X)) ).
% of_nat_Real
tff(fact_10172_of__int__Real,axiom,
! [X: int] : aa_int_real(ring_1_of_int_real,X) = aa_fun_nat_rat_real(real2,aTP_Lamm_uq(X)) ).
% of_int_Real
tff(fact_10173_less__RealD,axiom,
! [Y5: fun_nat_rat,X: real] :
( cauchy(Y5)
=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),aa_fun_nat_rat_real(real2,Y5)))
=> ? [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),field_7254667332652039916t_real(aa_nat_rat(Y5,N2)))) ) ) ).
% less_RealD
tff(fact_10174_le__RealI,axiom,
! [Y5: fun_nat_rat,X: real] :
( cauchy(Y5)
=> ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),field_7254667332652039916t_real(aa_nat_rat(Y5,N2))))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,X),aa_fun_nat_rat_real(real2,Y5))) ) ) ).
% le_RealI
tff(fact_10175_Real__leI,axiom,
! [X5: fun_nat_rat,Y: real] :
( cauchy(X5)
=> ( ! [N2: nat] : pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,field_7254667332652039916t_real(aa_nat_rat(X5,N2))),Y))
=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_fun_nat_rat_real(real2,X5)),Y)) ) ) ).
% Real_leI
tff(fact_10176_minus__Real,axiom,
! [X5: fun_nat_rat] :
( cauchy(X5)
=> ( aa_real_real(uminus_uminus_real,aa_fun_nat_rat_real(real2,X5)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aTP_Lamm_uc,X5)) ) ) ).
% minus_Real
tff(fact_10177_add__Real,axiom,
! [X5: fun_nat_rat,Y5: fun_nat_rat] :
( cauchy(X5)
=> ( cauchy(Y5)
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_fun_nat_rat_real(real2,X5)),aa_fun_nat_rat_real(real2,Y5)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ud,X5),Y5)) ) ) ) ).
% add_Real
tff(fact_10178_mult__Real,axiom,
! [X5: fun_nat_rat,Y5: fun_nat_rat] :
( cauchy(X5)
=> ( cauchy(Y5)
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_fun_nat_rat_real(real2,X5)),aa_fun_nat_rat_real(real2,Y5)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_tz,X5),Y5)) ) ) ) ).
% mult_Real
tff(fact_10179_diff__Real,axiom,
! [X5: fun_nat_rat,Y5: fun_nat_rat] :
( cauchy(X5)
=> ( cauchy(Y5)
=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_fun_nat_rat_real(real2,X5)),aa_fun_nat_rat_real(real2,Y5)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aTP_Lamm_ub(X5),Y5)) ) ) ) ).
% diff_Real
tff(fact_10180_eq__Real,axiom,
! [X5: fun_nat_rat,Y5: fun_nat_rat] :
( cauchy(X5)
=> ( cauchy(Y5)
=> ( ( aa_fun_nat_rat_real(real2,X5) = aa_fun_nat_rat_real(real2,Y5) )
<=> pp(vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_ub(X5),Y5))) ) ) ) ).
% eq_Real
tff(fact_10181_inverse__Real,axiom,
! [X5: fun_nat_rat] :
( cauchy(X5)
=> ( ( pp(vanishes(X5))
=> ( aa_real_real(inverse_inverse_real,aa_fun_nat_rat_real(real2,X5)) = zero_zero_real ) )
& ( ~ pp(vanishes(X5))
=> ( aa_real_real(inverse_inverse_real,aa_fun_nat_rat_real(real2,X5)) = aa_fun_nat_rat_real(real2,aTP_Lamm_uk(X5)) ) ) ) ) ).
% inverse_Real
tff(fact_10182_not__positive__Real,axiom,
! [X5: fun_nat_rat] :
( cauchy(X5)
=> ( ~ pp(aa_real_bool(positive2,aa_fun_nat_rat_real(real2,X5)))
<=> ! [R5: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
=> ? [K2: nat] :
! [N5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_nat_rat(X5,N5)),R5)) ) ) ) ) ).
% not_positive_Real
tff(fact_10183_positive__Real,axiom,
! [X5: fun_nat_rat] :
( cauchy(X5)
=> ( pp(aa_real_bool(positive2,aa_fun_nat_rat_real(real2,X5)))
<=> ? [R5: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
& ? [K2: nat] :
! [N5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,R5),aa_nat_rat(X5,N5))) ) ) ) ) ).
% positive_Real
tff(fact_10184_less__real__def,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,X),Y))
<=> pp(aa_real_bool(positive2,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),X))) ) ).
% less_real_def
tff(fact_10185_Real_Opositive__minus,axiom,
! [X: real] :
( ~ pp(aa_real_bool(positive2,X))
=> ( ( X != zero_zero_real )
=> pp(aa_real_bool(positive2,aa_real_real(uminus_uminus_real,X))) ) ) ).
% Real.positive_minus
tff(fact_10186_Real_Opositive__zero,axiom,
~ pp(aa_real_bool(positive2,zero_zero_real)) ).
% Real.positive_zero
tff(fact_10187_Real_Opositive__add,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(positive2,X))
=> ( pp(aa_real_bool(positive2,Y))
=> pp(aa_real_bool(positive2,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),Y))) ) ) ).
% Real.positive_add
tff(fact_10188_Real_Opositive__mult,axiom,
! [X: real,Y: real] :
( pp(aa_real_bool(positive2,X))
=> ( pp(aa_real_bool(positive2,Y))
=> pp(aa_real_bool(positive2,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),Y))) ) ) ).
% Real.positive_mult
tff(fact_10189_Real_Opositive_Orep__eq,axiom,
! [X: real] :
( pp(aa_real_bool(positive2,X))
<=> ? [R5: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
& ? [K2: nat] :
! [N5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,R5),aa_nat_rat(aa_real_fun_nat_rat(rep_real,X),N5))) ) ) ) ).
% Real.positive.rep_eq
tff(fact_10190_Real_Opositive__def,axiom,
positive2 = map_fu1722713901878115497l_bool(rep_real,id_bool,aTP_Lamm_ur) ).
% Real.positive_def
tff(fact_10191_inverse__real__def,axiom,
inverse_inverse_real = aa_fun8686829112900759002l_real(map_fu3926683916856352828t_real(rep_real,real2),aTP_Lamm_us) ).
% inverse_real_def
tff(fact_10192_uminus__real__def,axiom,
uminus_uminus_real = aa_fun8686829112900759002l_real(map_fu3926683916856352828t_real(rep_real,real2),aTP_Lamm_uc) ).
% uminus_real_def
tff(fact_10193_times__real__def,axiom,
times_times_real = map_fu2502140041903272327l_real(rep_real,map_fu3926683916856352828t_real(rep_real,real2),aTP_Lamm_tz) ).
% times_real_def
tff(fact_10194_plus__real__def,axiom,
plus_plus_real = map_fu2502140041903272327l_real(rep_real,map_fu3926683916856352828t_real(rep_real,real2),aTP_Lamm_ud) ).
% plus_real_def
tff(fact_10195_inverse__real_Oabs__eq,axiom,
! [X: fun_nat_rat] :
( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X),X))
=> ( aa_real_real(inverse_inverse_real,aa_fun_nat_rat_real(real2,X)) = aa_fun_nat_rat_real(real2,if_fun_nat_rat(vanishes(X),aTP_Lamm_uo,aTP_Lamm_uk(X))) ) ) ).
% inverse_real.abs_eq
tff(fact_10196_real_Oabs__induct,axiom,
! [P: fun_real_bool,X: real] :
( ! [Y2: fun_nat_rat] :
( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,Y2),Y2))
=> pp(aa_real_bool(P,aa_fun_nat_rat_real(real2,Y2))) )
=> pp(aa_real_bool(P,X)) ) ).
% real.abs_induct
tff(fact_10197_one__real_Orsp,axiom,
pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,aTP_Lamm_un),aTP_Lamm_un)) ).
% one_real.rsp
tff(fact_10198_zero__real_Orsp,axiom,
pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,aTP_Lamm_uo),aTP_Lamm_uo)) ).
% zero_real.rsp
tff(fact_10199_le__enumerate,axiom,
! [S2: set_nat,N: nat] :
( ~ pp(aa_set_nat_bool(finite_finite_nat,S2))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(infini8530281810654367211te_nat(S2),N))) ) ).
% le_enumerate
tff(fact_10200_realrel__refl,axiom,
! [X5: fun_nat_rat] :
( cauchy(X5)
=> pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X5),X5)) ) ).
% realrel_refl
tff(fact_10201_uminus__real_Oabs__eq,axiom,
! [X: fun_nat_rat] :
( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X),X))
=> ( aa_real_real(uminus_uminus_real,aa_fun_nat_rat_real(real2,X)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aTP_Lamm_uc,X)) ) ) ).
% uminus_real.abs_eq
tff(fact_10202_plus__real_Oabs__eq,axiom,
! [Xa2: fun_nat_rat,X: fun_nat_rat] :
( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,Xa2),Xa2))
=> ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X),X))
=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_fun_nat_rat_real(real2,Xa2)),aa_fun_nat_rat_real(real2,X)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ud,Xa2),X)) ) ) ) ).
% plus_real.abs_eq
tff(fact_10203_times__real_Oabs__eq,axiom,
! [Xa2: fun_nat_rat,X: fun_nat_rat] :
( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,Xa2),Xa2))
=> ( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X),X))
=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_fun_nat_rat_real(real2,Xa2)),aa_fun_nat_rat_real(real2,X)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_tz,Xa2),X)) ) ) ) ).
% times_real.abs_eq
tff(fact_10204_realrelI,axiom,
! [X5: fun_nat_rat,Y5: fun_nat_rat] :
( cauchy(X5)
=> ( cauchy(Y5)
=> ( pp(vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_ub(X5),Y5)))
=> pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X5),Y5)) ) ) ) ).
% realrelI
tff(fact_10205_realrel__def,axiom,
! [X4: fun_nat_rat,Xa: fun_nat_rat] :
( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X4),Xa))
<=> ( cauchy(X4)
& cauchy(Xa)
& pp(vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_ub(X4),Xa))) ) ) ).
% realrel_def
tff(fact_10206_finite__le__enumerate,axiom,
! [S2: set_nat,N: nat] :
( pp(aa_set_nat_bool(finite_finite_nat,S2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),finite_card_nat(S2)))
=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(infini8530281810654367211te_nat(S2),N))) ) ) ).
% finite_le_enumerate
tff(fact_10207_Real_Opositive_Oabs__eq,axiom,
! [X: fun_nat_rat] :
( pp(aa_fun_nat_rat_bool(aa_fun6650786570619099624t_bool(realrel,X),X))
=> ( pp(aa_real_bool(positive2,aa_fun_nat_rat_real(real2,X)))
<=> ? [R5: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
& ? [K2: nat] :
! [N5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,R5),aa_nat_rat(X,N5))) ) ) ) ) ).
% Real.positive.abs_eq
tff(fact_10208_Real_Opositive_Orsp,axiom,
bNF_re2717234250851183977l_bool(realrel,fequal_bool,aTP_Lamm_ur,aTP_Lamm_ur) ).
% Real.positive.rsp
tff(fact_10209_Least__eq__0,axiom,
! [P: fun_nat_bool] :
( pp(aa_nat_bool(P,zero_zero_nat))
=> ( ord_Least_nat(P) = zero_zero_nat ) ) ).
% Least_eq_0
tff(fact_10210_Least__Suc2,axiom,
! [P: fun_nat_bool,N: nat,Q: fun_nat_bool,M: nat] :
( pp(aa_nat_bool(P,N))
=> ( pp(aa_nat_bool(Q,M))
=> ( ~ pp(aa_nat_bool(P,zero_zero_nat))
=> ( ! [K3: nat] :
( pp(aa_nat_bool(P,aa_nat_nat(suc,K3)))
<=> pp(aa_nat_bool(Q,K3)) )
=> ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(Q)) ) ) ) ) ) ).
% Least_Suc2
tff(fact_10211_Least__Suc,axiom,
! [P: fun_nat_bool,N: nat] :
( pp(aa_nat_bool(P,N))
=> ( ~ pp(aa_nat_bool(P,zero_zero_nat))
=> ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(aTP_Lamm_td(P))) ) ) ) ).
% Least_Suc
tff(fact_10212_ATP_Olambda__1,axiom,
! [Uu: fun_nat_rat] : aa_fun765758790203053448at_rat(aTP_Lamm_us,Uu) = if_fun_nat_rat(vanishes(Uu),aTP_Lamm_uo,aTP_Lamm_uk(Uu)) ).
% ATP.lambda_1
tff(fact_10213_ATP_Olambda__2,axiom,
! [Uu: nat] : aa_nat_real(aTP_Lamm_jx,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).
% ATP.lambda_2
tff(fact_10214_ATP_Olambda__3,axiom,
! [Uu: nat] : aa_nat_real(aTP_Lamm_lr,Uu) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_nat(suc,Uu)) ).
% ATP.lambda_3
tff(fact_10215_ATP_Olambda__4,axiom,
! [Uu: real] :
( pp(aa_real_bool(aTP_Lamm_pd,Uu))
<=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Uu))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uu),numeral_numeral_real(aa_num_num(bit0,one))))
& ( aa_real_real(cos_real,Uu) = zero_zero_real ) ) ) ).
% ATP.lambda_4
tff(fact_10216_ATP_Olambda__5,axiom,
! [Uu: nat] : aa_nat_real(aTP_Lamm_jw,Uu) = aa_real_real(aa_rea7340605660300375871l_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_10217_ATP_Olambda__6,axiom,
! [Uu: real] : aa_real_real(aTP_Lamm_sa,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,Uu)),aa_real_real(sin_real,Uu)) ).
% ATP.lambda_6
tff(fact_10218_ATP_Olambda__7,axiom,
! [Uu: real] : aa_real_real(aTP_Lamm_su,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Uu)),Uu) ).
% ATP.lambda_7
tff(fact_10219_ATP_Olambda__8,axiom,
! [Uu: nat] : aa_nat_real(aTP_Lamm_sc,Uu) = aa_real_real(aa_nat_fun_real_real(root,Uu),aa_nat_real(semiri5074537144036343181t_real,Uu)) ).
% ATP.lambda_8
tff(fact_10220_ATP_Olambda__9,axiom,
! [Uu: nat] : aa_nat_nat(aTP_Lamm_tu,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(suc,zero_zero_nat)) ).
% ATP.lambda_9
tff(fact_10221_ATP_Olambda__10,axiom,
! [Uu: complex] : aa_complex_complex(aTP_Lamm_mg,Uu) = aa_complex_complex(plus_plus_complex(Uu),one_one_complex) ).
% ATP.lambda_10
tff(fact_10222_ATP_Olambda__11,axiom,
! [Uu: real] : aa_real_real(aTP_Lamm_me,Uu) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),one_one_real) ).
% ATP.lambda_11
tff(fact_10223_ATP_Olambda__12,axiom,
! [Uu: rat] : aa_rat_rat(aTP_Lamm_mh,Uu) = plus_plus_rat(Uu,one_one_rat) ).
% ATP.lambda_12
tff(fact_10224_ATP_Olambda__13,axiom,
! [Uu: nat] : aa_nat_nat(aTP_Lamm_mf,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),one_one_nat) ).
% ATP.lambda_13
tff(fact_10225_ATP_Olambda__14,axiom,
! [Uu: int] : aa_int_int(aTP_Lamm_md,Uu) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),one_one_int) ).
% ATP.lambda_14
tff(fact_10226_ATP_Olambda__15,axiom,
! [Uu: product_prod_int_int] :
( pp(aa_Pro4501881432599481050t_bool(aTP_Lamm_rc,Uu))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Uu)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Uu)))) ) ).
% ATP.lambda_15
tff(fact_10227_ATP_Olambda__16,axiom,
! [Uu: nat] : aa_nat_real(aTP_Lamm_se,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Uu)) ).
% ATP.lambda_16
tff(fact_10228_ATP_Olambda__17,axiom,
! [Uu: real] : aa_real_real(aTP_Lamm_ry,Uu) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_js,Uu)) ).
% ATP.lambda_17
tff(fact_10229_ATP_Olambda__18,axiom,
! [Uu: real] : aa_rea5117059003799598681omplex(aTP_Lamm_tl,Uu) = princi3496590319149328850omplex(collec8663557070575231912omplex(produc9214369023355538061x_bool(aTP_Lamm_tk(Uu)))) ).
% ATP.lambda_18
tff(fact_10230_ATP_Olambda__19,axiom,
! [Uu: real] : aa_rea5428163979925208277l_real(aTP_Lamm_tj,Uu) = princi6114159922880469582l_real(collec3799799289383736868l_real(produc5257810830037637649l_bool(aTP_Lamm_ti(Uu)))) ).
% ATP.lambda_19
tff(fact_10231_ATP_Olambda__20,axiom,
! [Uu: nat] : aa_nat_real(aTP_Lamm_sg,Uu) = aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uu))) ).
% ATP.lambda_20
tff(fact_10232_ATP_Olambda__21,axiom,
! [Uu: num] : aa_num_option_num(aTP_Lamm_qv,Uu) = some_num(aa_num_num(bit1,Uu)) ).
% ATP.lambda_21
tff(fact_10233_ATP_Olambda__22,axiom,
! [Uu: num] : aa_num_option_num(aTP_Lamm_qu,Uu) = some_num(aa_num_num(bit0,Uu)) ).
% ATP.lambda_22
tff(fact_10234_ATP_Olambda__23,axiom,
! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_ec,Uu) = product_Pair_nat_nat(aa_nat_nat(suc,Uu)) ).
% ATP.lambda_23
tff(fact_10235_ATP_Olambda__24,axiom,
! [Uu: fun_nat_rat] :
( pp(aa_fun_nat_rat_bool(aTP_Lamm_ur,Uu))
<=> ? [R5: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,zero_zero_rat),R5))
& ? [K2: nat] :
! [N5: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K2),N5))
=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,R5),aa_nat_rat(Uu,N5))) ) ) ) ).
% ATP.lambda_24
tff(fact_10236_ATP_Olambda__25,axiom,
! [Uu: real] :
( pp(aa_real_bool(aTP_Lamm_tr,Uu))
<=> ? [I4: int,N5: nat] :
( ( Uu = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,I4)),aa_nat_real(semiri5074537144036343181t_real,N5)) )
& ( N5 != zero_zero_nat ) ) ) ).
% ATP.lambda_25
tff(fact_10237_ATP_Olambda__26,axiom,
! [Uu: real] :
( pp(aa_real_bool(aTP_Lamm_ts,Uu))
<=> ? [I4: int,J3: int] :
( ( Uu = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,I4)),aa_int_real(ring_1_of_int_real,J3)) )
& ( J3 != zero_zero_int ) ) ) ).
% ATP.lambda_26
tff(fact_10238_ATP_Olambda__27,axiom,
! [Uu: nat] : aa_nat_option_num(aTP_Lamm_qt,Uu) = some_num(one) ).
% ATP.lambda_27
tff(fact_10239_ATP_Olambda__28,axiom,
! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_ra(Uu),Uua) = case_num_option_num(some_num(one),aTP_Lamm_qy(Uua),aTP_Lamm_qz(Uua),Uu) ).
% ATP.lambda_28
tff(fact_10240_ATP_Olambda__29,axiom,
! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ls(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),zero_zero_real),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).
% ATP.lambda_29
tff(fact_10241_ATP_Olambda__30,axiom,
! [Uu: code_integer,Uua: code_integer] : aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_po,Uu),Uua) = aa_int_int(aa_int_fun_int_int(if_int(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uua),zero_z3403309356797280102nteger)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),code_int_of_integer(Uu))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),code_int_of_integer(Uu))),one_one_int)) ).
% ATP.lambda_30
tff(fact_10242_ATP_Olambda__31,axiom,
! [Uu: nat,Uua: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_hy,Uu),Uua) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uua),zero_zero_nat),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(semiri8010041392384452111omplex,Uu))),aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(semiri8010041392384452111omplex,Uu))),one_one_complex)) ).
% ATP.lambda_31
tff(fact_10243_ATP_Olambda__32,axiom,
! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_hw,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uua),zero_zero_nat)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Uu))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Uu))),one_one_real)) ).
% ATP.lambda_32
tff(fact_10244_ATP_Olambda__33,axiom,
! [Uu: nat,Uua: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_hz,Uu),Uua) = if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uua),zero_zero_nat),aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,Uu)),plus_plus_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,Uu)),one_one_rat)) ).
% ATP.lambda_33
tff(fact_10245_ATP_Olambda__34,axiom,
! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hx,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uua),zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(semiri1316708129612266289at_nat,Uu))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(semiri1316708129612266289at_nat,Uu))),one_one_nat)) ).
% ATP.lambda_34
tff(fact_10246_ATP_Olambda__35,axiom,
! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_hv,Uu),Uua) = aa_int_int(aa_int_fun_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uua),zero_zero_nat)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(semiri1314217659103216013at_int,Uu))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(semiri1314217659103216013at_int,Uu))),one_one_int)) ).
% ATP.lambda_35
tff(fact_10247_ATP_Olambda__36,axiom,
! [Uu: code_integer,Uua: code_integer] : aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_pp,Uu),Uua) = aa_num_num(aa_num_fun_num_num(if_num(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uua),zero_z3403309356797280102nteger)),aa_num_num(aa_num_fun_num_num(plus_plus_num,code_num_of_integer(Uu)),code_num_of_integer(Uu))),aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(plus_plus_num,code_num_of_integer(Uu)),code_num_of_integer(Uu))),one)) ).
% ATP.lambda_36
tff(fact_10248_ATP_Olambda__37,axiom,
! [Uu: code_integer,Uua: code_integer] : aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_pr,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uua),zero_z3403309356797280102nteger)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,code_nat_of_integer(Uu)),code_nat_of_integer(Uu))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,code_nat_of_integer(Uu)),code_nat_of_integer(Uu))),one_one_nat)) ).
% ATP.lambda_37
tff(fact_10249_ATP_Olambda__38,axiom,
! [Uu: nat,Uua: num] : aa_num_option_num(aa_nat3713591252005393295on_num(aTP_Lamm_rb,Uu),Uua) = case_nat_option_num(none_num,aTP_Lamm_ra(Uua),Uu) ).
% ATP.lambda_38
tff(fact_10250_ATP_Olambda__39,axiom,
! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_qy(Uu),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_qu,bit_take_bit_num(Uu,Uua)) ).
% ATP.lambda_39
tff(fact_10251_ATP_Olambda__40,axiom,
! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_qw(Uu),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_qu,bit_take_bit_num(Uua,Uu)) ).
% ATP.lambda_40
tff(fact_10252_ATP_Olambda__41,axiom,
! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ib,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(minus_minus_real,Uu),one_one_real)),aa_nat_nat(suc,Uua))) ).
% ATP.lambda_41
tff(fact_10253_ATP_Olambda__42,axiom,
! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_lg,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat))) ).
% ATP.lambda_42
tff(fact_10254_ATP_Olambda__43,axiom,
! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_lu,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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,aa_num_nat(numeral_numeral_nat,aa_num_num(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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ).
% ATP.lambda_43
tff(fact_10255_ATP_Olambda__44,axiom,
! [Uu: real,Uua: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_pe,Uu),Uua))
<=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Uua))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uua),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
& ( aa_real_real(sin_real,Uua) = Uu ) ) ) ).
% ATP.lambda_44
tff(fact_10256_ATP_Olambda__45,axiom,
! [Uu: real,Uua: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_pa,Uu),Uua))
<=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one))))),Uua))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Uua),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),numeral_numeral_real(aa_num_num(bit0,one)))))
& ( aa_real_real(tan_real,Uua) = Uu ) ) ) ).
% ATP.lambda_45
tff(fact_10257_ATP_Olambda__46,axiom,
! [Uu: complex,Uua: real] :
( pp(aa_real_bool(aTP_Lamm_pg(Uu),Uua))
<=> ( ( aa_complex_complex(sgn_sgn_complex,Uu) = aa_real_complex(cis,Uua) )
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Uua))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uua),pi)) ) ) ).
% ATP.lambda_46
tff(fact_10258_ATP_Olambda__47,axiom,
! [Uu: real,Uua: int] :
( pp(aa_int_bool(aTP_Lamm_pm(Uu),Uua))
<=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Uua)),Uu))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Uu),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int)))) ) ) ).
% ATP.lambda_47
tff(fact_10259_ATP_Olambda__48,axiom,
! [Uu: rat,Uua: int] :
( pp(aa_int_bool(aTP_Lamm_pn(Uu),Uua))
<=> ( pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Uua)),Uu))
& pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Uu),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int)))) ) ) ).
% ATP.lambda_48
tff(fact_10260_ATP_Olambda__49,axiom,
! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_js,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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),aa_num_nat(numeral_numeral_nat,aa_num_num(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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))) ).
% ATP.lambda_49
tff(fact_10261_ATP_Olambda__50,axiom,
! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_rz,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).
% ATP.lambda_50
tff(fact_10262_ATP_Olambda__51,axiom,
! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_sb(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_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_51
tff(fact_10263_ATP_Olambda__52,axiom,
! [Uu: rat,Uua: product_prod_int_int] :
( pp(aa_Pro4501881432599481050t_bool(aTP_Lamm_ty(Uu),Uua))
<=> ( ( Uu = aa_int_rat(fract(aa_Pro6799258434465214121nt_int(product_fst_int_int,Uua)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Uua)) )
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_Pro6799258434465214121nt_int(product_snd_int_int,Uua)))
& pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Uua)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Uua))) ) ) ).
% ATP.lambda_52
tff(fact_10264_ATP_Olambda__53,axiom,
! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_mi(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(Uu),Uua)),aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Uu),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).
% ATP.lambda_53
tff(fact_10265_ATP_Olambda__54,axiom,
! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_mk,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Uu),Uua)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uu),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).
% ATP.lambda_54
tff(fact_10266_ATP_Olambda__55,axiom,
! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_mj(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(Uu),Uua)),minus_minus_rat(divide_divide_rat(Uu,numeral_numeral_rat(aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).
% ATP.lambda_55
tff(fact_10267_ATP_Olambda__56,axiom,
! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ox,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).
% ATP.lambda_56
tff(fact_10268_ATP_Olambda__57,axiom,
! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_kn,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_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_57
tff(fact_10269_ATP_Olambda__58,axiom,
! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_sv,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_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_58
tff(fact_10270_ATP_Olambda__59,axiom,
! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_uh(Uu),Uua) = aa_nat_set_nat(set_or6659071591806873216st_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uua)),Uu) ).
% ATP.lambda_59
tff(fact_10271_ATP_Olambda__60,axiom,
! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tm,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),Uua)),Uua) ).
% ATP.lambda_60
tff(fact_10272_ATP_Olambda__61,axiom,
! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ot,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)),Uu) ).
% ATP.lambda_61
tff(fact_10273_ATP_Olambda__62,axiom,
! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_os,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)),Uua) ).
% ATP.lambda_62
tff(fact_10274_ATP_Olambda__63,axiom,
! [Uu: nat,Uua: complex] :
( pp(aa_complex_bool(aTP_Lamm_bj(Uu),Uua))
<=> ( aa_nat_complex(power_power_complex(Uua),Uu) = one_one_complex ) ) ).
% ATP.lambda_63
tff(fact_10275_ATP_Olambda__64,axiom,
! [Uu: nat,Uua: real] :
( pp(aa_real_bool(aTP_Lamm_bi(Uu),Uua))
<=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) = one_one_real ) ) ).
% ATP.lambda_64
tff(fact_10276_ATP_Olambda__65,axiom,
! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sl,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(aa_rea7340605660300375871l_real(divide_divide_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)))),Uua) ).
% ATP.lambda_65
tff(fact_10277_ATP_Olambda__66,axiom,
! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_ss,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),Uua))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Uua)) ).
% ATP.lambda_66
tff(fact_10278_ATP_Olambda__67,axiom,
! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_sw,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uu),Uua))),Uua) ).
% ATP.lambda_67
tff(fact_10279_ATP_Olambda__68,axiom,
! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ia,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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),aa_num_nat(numeral_numeral_nat,aa_num_num(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),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).
% ATP.lambda_68
tff(fact_10280_ATP_Olambda__69,axiom,
! [Uu: real,Uua: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_pc,Uu),Uua))
<=> ( pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,zero_zero_real),Uua))
& pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uua),pi))
& ( aa_real_real(cos_real,Uua) = Uu ) ) ) ).
% ATP.lambda_69
tff(fact_10281_ATP_Olambda__70,axiom,
! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ho(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_real(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).
% ATP.lambda_70
tff(fact_10282_ATP_Olambda__71,axiom,
! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_hl(Uu),Uua) = plus_plus_rat(aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),aa_nat_rat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).
% ATP.lambda_71
tff(fact_10283_ATP_Olambda__72,axiom,
! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_hn(Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_nat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).
% ATP.lambda_72
tff(fact_10284_ATP_Olambda__73,axiom,
! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_hm(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_int(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).
% ATP.lambda_73
tff(fact_10285_ATP_Olambda__74,axiom,
! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_hd(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,aa_nat_nat(suc,Uua))),aa_nat_real(Uu,Uua)) ).
% ATP.lambda_74
tff(fact_10286_ATP_Olambda__75,axiom,
! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_hb(Uu),Uua) = minus_minus_rat(aa_nat_rat(Uu,aa_nat_nat(suc,Uua)),aa_nat_rat(Uu,Uua)) ).
% ATP.lambda_75
tff(fact_10287_ATP_Olambda__76,axiom,
! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_hc(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,aa_nat_nat(suc,Uua))),aa_nat_int(Uu,Uua)) ).
% ATP.lambda_76
tff(fact_10288_ATP_Olambda__77,axiom,
! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ge(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uua)),aa_nat_complex(power_power_complex(zero_zero_complex),Uua)) ).
% ATP.lambda_77
tff(fact_10289_ATP_Olambda__78,axiom,
! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_gg(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uua)) ).
% ATP.lambda_78
tff(fact_10290_ATP_Olambda__79,axiom,
! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_gf(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(power_power_rat(zero_zero_rat),Uua)) ).
% ATP.lambda_79
tff(fact_10291_ATP_Olambda__80,axiom,
! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_kg(Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uua)),aa_nat_int(power_power_int(zero_zero_int),Uua)) ).
% ATP.lambda_80
tff(fact_10292_ATP_Olambda__81,axiom,
! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_hk(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).
% ATP.lambda_81
tff(fact_10293_ATP_Olambda__82,axiom,
! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_hi(Uu),Uua) = minus_minus_rat(aa_nat_rat(Uu,Uua),aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).
% ATP.lambda_82
tff(fact_10294_ATP_Olambda__83,axiom,
! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_hj(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,Uua)),aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).
% ATP.lambda_83
tff(fact_10295_ATP_Olambda__84,axiom,
! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_he(Uu),Uua) = aa_complex_complex(minus_minus_complex(aa_nat_complex(Uu,Uua)),aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).
% ATP.lambda_84
tff(fact_10296_ATP_Olambda__85,axiom,
! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_hh(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).
% ATP.lambda_85
tff(fact_10297_ATP_Olambda__86,axiom,
! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_hf(Uu),Uua) = minus_minus_rat(aa_nat_rat(Uu,Uua),aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).
% ATP.lambda_86
tff(fact_10298_ATP_Olambda__87,axiom,
! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_hg(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,Uua)),aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).
% ATP.lambda_87
tff(fact_10299_ATP_Olambda__88,axiom,
! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_iq(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(suc,Uua))) ).
% ATP.lambda_88
tff(fact_10300_ATP_Olambda__89,axiom,
! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_io(Uu),Uua) = minus_minus_rat(aa_nat_rat(Uu,Uua),aa_nat_rat(Uu,aa_nat_nat(suc,Uua))) ).
% ATP.lambda_89
tff(fact_10301_ATP_Olambda__90,axiom,
! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ip(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,Uua)),aa_nat_int(Uu,aa_nat_nat(suc,Uua))) ).
% ATP.lambda_90
tff(fact_10302_ATP_Olambda__91,axiom,
! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_od(Uu),Uua) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(Uu,Uua),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).
% ATP.lambda_91
tff(fact_10303_ATP_Olambda__92,axiom,
! [Uu: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_og(Uu),Uua) = groups705719431365010083at_int(aa_nat_fun_nat_int(Uu,Uua),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).
% ATP.lambda_92
tff(fact_10304_ATP_Olambda__93,axiom,
! [Uu: fun_complex_real,Uua: complex] :
( pp(aa_complex_bool(aTP_Lamm_fo(Uu),Uua))
<=> ( aa_complex_real(Uu,Uua) = zero_zero_real ) ) ).
% ATP.lambda_93
tff(fact_10305_ATP_Olambda__94,axiom,
! [Uu: fun_complex_rat,Uua: complex] :
( pp(aa_complex_bool(aTP_Lamm_fr(Uu),Uua))
<=> ( aa_complex_rat(Uu,Uua) = zero_zero_rat ) ) ).
% ATP.lambda_94
tff(fact_10306_ATP_Olambda__95,axiom,
! [Uu: fun_real_complex,Uua: real] :
( pp(aa_real_bool(aTP_Lamm_fk(Uu),Uua))
<=> ( aa_real_complex(Uu,Uua) = zero_zero_complex ) ) ).
% ATP.lambda_95
tff(fact_10307_ATP_Olambda__96,axiom,
! [Uu: fun_real_real,Uua: real] :
( pp(aa_real_bool(aTP_Lamm_fm(Uu),Uua))
<=> ( aa_real_real(Uu,Uua) = zero_zero_real ) ) ).
% ATP.lambda_96
tff(fact_10308_ATP_Olambda__97,axiom,
! [Uu: fun_real_rat,Uua: real] :
( pp(aa_real_bool(aTP_Lamm_fp(Uu),Uua))
<=> ( aa_real_rat(Uu,Uua) = zero_zero_rat ) ) ).
% ATP.lambda_97
tff(fact_10309_ATP_Olambda__98,axiom,
! [Uu: fun_real_nat,Uua: real] :
( pp(aa_real_bool(aTP_Lamm_fs(Uu),Uua))
<=> ( aa_real_nat(Uu,Uua) = zero_zero_nat ) ) ).
% ATP.lambda_98
tff(fact_10310_ATP_Olambda__99,axiom,
! [Uu: fun_int_complex,Uua: int] :
( pp(aa_int_bool(aTP_Lamm_fl(Uu),Uua))
<=> ( aa_int_complex(Uu,Uua) = zero_zero_complex ) ) ).
% ATP.lambda_99
tff(fact_10311_ATP_Olambda__100,axiom,
! [Uu: fun_int_real,Uua: int] :
( pp(aa_int_bool(aTP_Lamm_fn(Uu),Uua))
<=> ( aa_int_real(Uu,Uua) = zero_zero_real ) ) ).
% ATP.lambda_100
tff(fact_10312_ATP_Olambda__101,axiom,
! [Uu: fun_int_rat,Uua: int] :
( pp(aa_int_bool(aTP_Lamm_fq(Uu),Uua))
<=> ( aa_int_rat(Uu,Uua) = zero_zero_rat ) ) ).
% ATP.lambda_101
tff(fact_10313_ATP_Olambda__102,axiom,
! [Uu: fun_int_nat,Uua: int] :
( pp(aa_int_bool(aTP_Lamm_ft(Uu),Uua))
<=> ( aa_int_nat(Uu,Uua) = zero_zero_nat ) ) ).
% ATP.lambda_102
tff(fact_10314_ATP_Olambda__103,axiom,
! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_sr(Uu),Uua) = groups6591440286371151544t_real(aTP_Lamm_sb(Uu),aa_nat_set_nat(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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat))) ).
% ATP.lambda_103
tff(fact_10315_ATP_Olambda__104,axiom,
! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_sq(Uu),Uua) = groups6591440286371151544t_real(aTP_Lamm_sb(Uu),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ).
% ATP.lambda_104
tff(fact_10316_ATP_Olambda__105,axiom,
! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qr,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uu) ).
% ATP.lambda_105
tff(fact_10317_ATP_Olambda__106,axiom,
! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_lw(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Uua))),aa_nat_complex(power_power_complex(Uu),Uua)) ).
% ATP.lambda_106
tff(fact_10318_ATP_Olambda__107,axiom,
! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_lv,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).
% ATP.lambda_107
tff(fact_10319_ATP_Olambda__108,axiom,
! [Uu: num,Uua: num] : aa_num_int(aTP_Lamm_qs(Uu),Uua) = aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,Uu)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,Uu))),aa_num_int(numeral_numeral_int,Uua))) ).
% ATP.lambda_108
tff(fact_10320_ATP_Olambda__109,axiom,
! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_rh,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_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_109
tff(fact_10321_ATP_Olambda__110,axiom,
! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_lf,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_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_110
tff(fact_10322_ATP_Olambda__111,axiom,
! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ky,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_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_111
tff(fact_10323_ATP_Olambda__112,axiom,
! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_ml(Uu),Uua) = aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),Uu) ).
% ATP.lambda_112
tff(fact_10324_ATP_Olambda__113,axiom,
! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_mn,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Uua)),Uu) ).
% ATP.lambda_113
tff(fact_10325_ATP_Olambda__114,axiom,
! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_mm(Uu),Uua) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua)),Uu) ).
% ATP.lambda_114
tff(fact_10326_ATP_Olambda__115,axiom,
! [Uu: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_dl,Uu),Uua) = produc1086072967326762835nteger(Uu,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua),one_one_Code_integer)) ).
% ATP.lambda_115
tff(fact_10327_ATP_Olambda__116,axiom,
! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_dk,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)) ).
% ATP.lambda_116
tff(fact_10328_ATP_Olambda__117,axiom,
! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_dj,Uu),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uu),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int)) ).
% ATP.lambda_117
tff(fact_10329_ATP_Olambda__118,axiom,
! [Uu: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_do,Uu),Uua) = produc1086072967326762835nteger(Uu,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua)) ).
% ATP.lambda_118
tff(fact_10330_ATP_Olambda__119,axiom,
! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_dn,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)) ).
% ATP.lambda_119
tff(fact_10331_ATP_Olambda__120,axiom,
! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_dm,Uu),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uu),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)) ).
% ATP.lambda_120
tff(fact_10332_ATP_Olambda__121,axiom,
! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ms(Uu),Uua) = aa_complex_complex(plus_plus_complex(Uu),aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) ).
% ATP.lambda_121
tff(fact_10333_ATP_Olambda__122,axiom,
! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_mr,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Uua)),numeral_numeral_real(aa_num_num(bit0,one)))) ).
% ATP.lambda_122
tff(fact_10334_ATP_Olambda__123,axiom,
! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_mt(Uu),Uua) = plus_plus_rat(Uu,divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua),numeral_numeral_rat(aa_num_num(bit0,one)))) ).
% ATP.lambda_123
tff(fact_10335_ATP_Olambda__124,axiom,
! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_oy,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)) ).
% ATP.lambda_124
tff(fact_10336_ATP_Olambda__125,axiom,
! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sm,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_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_125
tff(fact_10337_ATP_Olambda__126,axiom,
! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_ed,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_bool_int(zero_n2684676970156552555ol_int,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,Uua),zero_zero_int)))) ).
% ATP.lambda_126
tff(fact_10338_ATP_Olambda__127,axiom,
! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sk,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_127
tff(fact_10339_ATP_Olambda__128,axiom,
! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sh,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_128
tff(fact_10340_ATP_Olambda__129,axiom,
! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_rg(Uu),Uua) = groups6591440286371151544t_real(Uu,aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).
% ATP.lambda_129
tff(fact_10341_ATP_Olambda__130,axiom,
! [Uu: nat,Uua: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_tq,Uu),Uua))
<=> ( Uua = aa_nat_nat(suc,Uu) ) ) ).
% ATP.lambda_130
tff(fact_10342_ATP_Olambda__131,axiom,
! [Uu: real,Uua: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_oj,Uu),Uua))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_eq_real,Uua),Uu)) ) ).
% ATP.lambda_131
tff(fact_10343_ATP_Olambda__132,axiom,
! [Uu: set_nat,Uua: set_nat] :
( pp(aa_set_nat_bool(aTP_Lamm_ok(Uu),Uua))
<=> pp(aa_set_nat_bool(ord_less_eq_set_nat(Uua),Uu)) ) ).
% ATP.lambda_132
tff(fact_10344_ATP_Olambda__133,axiom,
! [Uu: rat,Uua: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(aTP_Lamm_ol,Uu),Uua))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_eq_rat,Uua),Uu)) ) ).
% ATP.lambda_133
tff(fact_10345_ATP_Olambda__134,axiom,
! [Uu: num,Uua: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(aTP_Lamm_om,Uu),Uua))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Uua),Uu)) ) ).
% ATP.lambda_134
tff(fact_10346_ATP_Olambda__135,axiom,
! [Uu: nat,Uua: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_bg,Uu),Uua))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uua),Uu)) ) ).
% ATP.lambda_135
tff(fact_10347_ATP_Olambda__136,axiom,
! [Uu: int,Uua: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_on,Uu),Uua))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uua),Uu)) ) ).
% ATP.lambda_136
tff(fact_10348_ATP_Olambda__137,axiom,
! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rf,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).
% ATP.lambda_137
tff(fact_10349_ATP_Olambda__138,axiom,
! [Uu: real,Uua: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_il,Uu),Uua))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,Uua),Uu)) ) ).
% ATP.lambda_138
tff(fact_10350_ATP_Olambda__139,axiom,
! [Uu: rat,Uua: rat] :
( pp(aa_rat_bool(aa_rat_fun_rat_bool(aTP_Lamm_ii,Uu),Uua))
<=> pp(aa_rat_bool(aa_rat_fun_rat_bool(ord_less_rat,Uua),Uu)) ) ).
% ATP.lambda_139
tff(fact_10351_ATP_Olambda__140,axiom,
! [Uu: num,Uua: num] :
( pp(aa_num_bool(aa_num_fun_num_bool(aTP_Lamm_ij,Uu),Uua))
<=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Uua),Uu)) ) ).
% ATP.lambda_140
tff(fact_10352_ATP_Olambda__141,axiom,
! [Uu: nat,Uua: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_bh,Uu),Uua))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uu)) ) ).
% ATP.lambda_141
tff(fact_10353_ATP_Olambda__142,axiom,
! [Uu: int,Uua: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ik,Uu),Uua))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uu)) ) ).
% ATP.lambda_142
tff(fact_10354_ATP_Olambda__143,axiom,
! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_ah,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uua),Uu) ).
% ATP.lambda_143
tff(fact_10355_ATP_Olambda__144,axiom,
! [Uu: rat,Uua: rat] : aa_rat_rat(aTP_Lamm_ai(Uu),Uua) = aa_rat_rat(times_times_rat(Uua),Uu) ).
% ATP.lambda_144
tff(fact_10356_ATP_Olambda__145,axiom,
! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aj,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu) ).
% ATP.lambda_145
tff(fact_10357_ATP_Olambda__146,axiom,
! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_ak,Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uu) ).
% ATP.lambda_146
tff(fact_10358_ATP_Olambda__147,axiom,
! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_to,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uua),Uu) ).
% ATP.lambda_147
tff(fact_10359_ATP_Olambda__148,axiom,
! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_pi,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).
% ATP.lambda_148
tff(fact_10360_ATP_Olambda__149,axiom,
! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_ri,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) ).
% ATP.lambda_149
tff(fact_10361_ATP_Olambda__150,axiom,
! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_tp,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uua),Uu) ).
% ATP.lambda_150
tff(fact_10362_ATP_Olambda__151,axiom,
! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tt,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu) ).
% ATP.lambda_151
tff(fact_10363_ATP_Olambda__152,axiom,
! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_pj,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),Uu) ).
% ATP.lambda_152
tff(fact_10364_ATP_Olambda__153,axiom,
! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_rk,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Uua),Uu) ).
% ATP.lambda_153
tff(fact_10365_ATP_Olambda__154,axiom,
! [Uu: code_integer,Uua: code_integer] :
( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(aTP_Lamm_ds,Uu),Uua))
<=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,Uua),Uu)) ) ).
% ATP.lambda_154
tff(fact_10366_ATP_Olambda__155,axiom,
! [Uu: complex,Uua: complex] :
( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_dp,Uu),Uua))
<=> pp(aa_complex_bool(aa_com1553036987509291606x_bool(dvd_dvd_complex,Uua),Uu)) ) ).
% ATP.lambda_155
tff(fact_10367_ATP_Olambda__156,axiom,
! [Uu: real,Uua: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_dq,Uu),Uua))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(dvd_dvd_real,Uua),Uu)) ) ).
% ATP.lambda_156
tff(fact_10368_ATP_Olambda__157,axiom,
! [Uu: nat,Uua: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_dt,Uu),Uua))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uua),Uu)) ) ).
% ATP.lambda_157
tff(fact_10369_ATP_Olambda__158,axiom,
! [Uu: int,Uua: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_dr,Uu),Uua))
<=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,Uua),Uu)) ) ).
% ATP.lambda_158
tff(fact_10370_ATP_Olambda__159,axiom,
! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_qi,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uu) ).
% ATP.lambda_159
tff(fact_10371_ATP_Olambda__160,axiom,
! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_or,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),Uu) ).
% ATP.lambda_160
tff(fact_10372_ATP_Olambda__161,axiom,
! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sf,Uu),Uua) = aa_real_real(aa_nat_fun_real_real(root,Uua),Uu) ).
% ATP.lambda_161
tff(fact_10373_ATP_Olambda__162,axiom,
! [Uu: set_Pr1261947904930325089at_nat,Uua: product_prod_nat_nat] :
( pp(aa_Pro2232559814647311250t_bool(aTP_Lamm_a(Uu),Uua))
<=> pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Uua),Uu)) ) ).
% ATP.lambda_162
tff(fact_10374_ATP_Olambda__163,axiom,
! [Uu: set_complex,Uua: complex] :
( pp(aa_complex_bool(aTP_Lamm_aa(Uu),Uua))
<=> pp(aa_set_complex_bool(member_complex(Uua),Uu)) ) ).
% ATP.lambda_163
tff(fact_10375_ATP_Olambda__164,axiom,
! [Uu: set_real,Uua: real] :
( pp(aa_real_bool(aTP_Lamm_ab(Uu),Uua))
<=> pp(aa_set_real_bool(member_real(Uua),Uu)) ) ).
% ATP.lambda_164
tff(fact_10376_ATP_Olambda__165,axiom,
! [Uu: set_list_nat,Uua: list_nat] :
( pp(aa_list_nat_bool(aTP_Lamm_ac(Uu),Uua))
<=> member_list_nat(Uua,Uu) ) ).
% ATP.lambda_165
tff(fact_10377_ATP_Olambda__166,axiom,
! [Uu: set_nat,Uua: nat] :
( pp(aa_nat_bool(aTP_Lamm_ad(Uu),Uua))
<=> pp(aa_set_nat_bool(member_nat(Uua),Uu)) ) ).
% ATP.lambda_166
tff(fact_10378_ATP_Olambda__167,axiom,
! [Uu: set_int,Uua: int] :
( pp(aa_int_bool(aTP_Lamm_ae(Uu),Uua))
<=> pp(aa_set_int_bool(member_int(Uua),Uu)) ) ).
% ATP.lambda_167
tff(fact_10379_ATP_Olambda__168,axiom,
! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_um(Uu),Uua) = aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),Uu) ).
% ATP.lambda_168
tff(fact_10380_ATP_Olambda__169,axiom,
! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_jr(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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)) ).
% ATP.lambda_169
tff(fact_10381_ATP_Olambda__170,axiom,
! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_jq(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)) ).
% ATP.lambda_170
tff(fact_10382_ATP_Olambda__171,axiom,
! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_lj(Uu),Uua) = aa_nat_complex(Uu,aa_nat_nat(suc,Uua)) ).
% ATP.lambda_171
tff(fact_10383_ATP_Olambda__172,axiom,
! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_gq(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(suc,Uua)) ).
% ATP.lambda_172
tff(fact_10384_ATP_Olambda__173,axiom,
! [Uu: fun_nat_bool,Uua: nat] :
( pp(aa_nat_bool(aTP_Lamm_td(Uu),Uua))
<=> pp(aa_nat_bool(Uu,aa_nat_nat(suc,Uua))) ) ).
% ATP.lambda_173
tff(fact_10385_ATP_Olambda__174,axiom,
! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_gz(Uu),Uua) = aa_nat_rat(Uu,aa_nat_nat(suc,Uua)) ).
% ATP.lambda_174
tff(fact_10386_ATP_Olambda__175,axiom,
! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_gp(Uu),Uua) = aa_nat_nat(Uu,aa_nat_nat(suc,Uua)) ).
% ATP.lambda_175
tff(fact_10387_ATP_Olambda__176,axiom,
! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ha(Uu),Uua) = aa_nat_int(Uu,aa_nat_nat(suc,Uua)) ).
% ATP.lambda_176
tff(fact_10388_ATP_Olambda__177,axiom,
! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_qz(Uu),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uu,Uua))) ).
% ATP.lambda_177
tff(fact_10389_ATP_Olambda__178,axiom,
! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_qx(Uu),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uua,Uu))) ).
% ATP.lambda_178
tff(fact_10390_ATP_Olambda__179,axiom,
! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_qq,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_qp(Uu),Uua)) ).
% ATP.lambda_179
tff(fact_10391_ATP_Olambda__180,axiom,
! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_qo,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_qn(Uu),Uua)) ).
% ATP.lambda_180
tff(fact_10392_ATP_Olambda__181,axiom,
! [Uu: nat,Uua: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_qm,Uu),Uua) = produc8955392012077580105t_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_ql(Uu),Uua)) ).
% ATP.lambda_181
tff(fact_10393_ATP_Olambda__182,axiom,
! [Uu: nat,Uua: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_qk,Uu),Uua) = produc8955392012077580105t_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_qj(Uu),Uua)) ).
% ATP.lambda_182
tff(fact_10394_ATP_Olambda__183,axiom,
! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_qh,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_qg(Uu),Uua)) ).
% ATP.lambda_183
tff(fact_10395_ATP_Olambda__184,axiom,
! [Uu: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_rr(Uu),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_rq(Uu),Uua)) ).
% ATP.lambda_184
tff(fact_10396_ATP_Olambda__185,axiom,
! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_oz(Uu),Uua) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,Uu))) ).
% ATP.lambda_185
tff(fact_10397_ATP_Olambda__186,axiom,
! [Uu: fun_real_real,Uua: real] :
( pp(aa_real_bool(aTP_Lamm_sx(Uu),Uua))
<=> ( aa_real_real(Uu,Uua) != zero_zero_real ) ) ).
% ATP.lambda_186
tff(fact_10398_ATP_Olambda__187,axiom,
! [Uu: nat,Uua: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_eg,Uu),Uua))
<=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ) ).
% ATP.lambda_187
tff(fact_10399_ATP_Olambda__188,axiom,
! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_mo(Uu),Uua) = times_times_complex(aa_complex_complex(minus_minus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).
% ATP.lambda_188
tff(fact_10400_ATP_Olambda__189,axiom,
! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_mq(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).
% ATP.lambda_189
tff(fact_10401_ATP_Olambda__190,axiom,
! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_mp(Uu),Uua) = times_times_rat(minus_minus_rat(Uu,aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).
% ATP.lambda_190
tff(fact_10402_ATP_Olambda__191,axiom,
! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_mw(Uu),Uua) = times_times_complex(aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).
% ATP.lambda_191
tff(fact_10403_ATP_Olambda__192,axiom,
! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_mv(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).
% ATP.lambda_192
tff(fact_10404_ATP_Olambda__193,axiom,
! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_mx(Uu),Uua) = times_times_rat(plus_plus_rat(Uu,aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).
% ATP.lambda_193
tff(fact_10405_ATP_Olambda__194,axiom,
! [Uu: nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_my(Uu),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua))) ).
% ATP.lambda_194
tff(fact_10406_ATP_Olambda__195,axiom,
! [Uu: int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_mu(Uu),Uua) = aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua))) ).
% ATP.lambda_195
tff(fact_10407_ATP_Olambda__196,axiom,
! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_ug(Uu),Uua) = aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uua)) ).
% ATP.lambda_196
tff(fact_10408_ATP_Olambda__197,axiom,
! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sj,Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).
% ATP.lambda_197
tff(fact_10409_ATP_Olambda__198,axiom,
! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_re,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu),Uua)) ).
% ATP.lambda_198
tff(fact_10410_ATP_Olambda__199,axiom,
! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rd,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu)) ).
% ATP.lambda_199
tff(fact_10411_ATP_Olambda__200,axiom,
! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_py,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu),Uua)) ).
% ATP.lambda_200
tff(fact_10412_ATP_Olambda__201,axiom,
! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_pz,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu)) ).
% ATP.lambda_201
tff(fact_10413_ATP_Olambda__202,axiom,
! [Uu: fun_complex_complex,Uua: complex] : aa_complex_real(aTP_Lamm_ju(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(Uu,Uua)) ).
% ATP.lambda_202
tff(fact_10414_ATP_Olambda__203,axiom,
! [Uu: fun_nat_complex,Uua: nat] : aa_nat_real(aTP_Lamm_jt(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uu,Uua)) ).
% ATP.lambda_203
tff(fact_10415_ATP_Olambda__204,axiom,
! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_jv(Uu),Uua) = aa_real_real(real_V7735802525324610683m_real,aa_nat_real(Uu,Uua)) ).
% ATP.lambda_204
tff(fact_10416_ATP_Olambda__205,axiom,
! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_sd(Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu,Uua)) ).
% ATP.lambda_205
tff(fact_10417_ATP_Olambda__206,axiom,
! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_uk(Uu),Uua) = inverse_inverse_rat(aa_nat_rat(Uu,Uua)) ).
% ATP.lambda_206
tff(fact_10418_ATP_Olambda__207,axiom,
! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_uc,Uu),Uua) = uminus_uminus_rat(aa_nat_rat(Uu,Uua)) ).
% ATP.lambda_207
tff(fact_10419_ATP_Olambda__208,axiom,
! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_th(Uu),Uua) = aa_real_real(artanh_real,aa_real_real(Uu,Uua)) ).
% ATP.lambda_208
tff(fact_10420_ATP_Olambda__209,axiom,
! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_tg(Uu),Uua) = aa_real_real(arcosh_real,aa_real_real(Uu,Uua)) ).
% ATP.lambda_209
tff(fact_10421_ATP_Olambda__210,axiom,
! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ei(Uu),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu,Uua)) ).
% ATP.lambda_210
tff(fact_10422_ATP_Olambda__211,axiom,
! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_eh(Uu),Uua) = aa_int_int(abs_abs_int,aa_int_int(Uu,Uua)) ).
% ATP.lambda_211
tff(fact_10423_ATP_Olambda__212,axiom,
! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_sp(Uu),Uua) = aa_real_real(cos_real,aa_nat_real(Uu,Uua)) ).
% ATP.lambda_212
tff(fact_10424_ATP_Olambda__213,axiom,
! [Uu: fun_re4238438383274954274t_real,Uua: real] : aa_real_real(aTP_Lamm_ro(Uu),Uua) = suminf_real(aa_real_fun_nat_real(Uu,Uua)) ).
% ATP.lambda_213
tff(fact_10425_ATP_Olambda__214,axiom,
! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_uj(Uu),Uua) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_bh,Uu)) ).
% ATP.lambda_214
tff(fact_10426_ATP_Olambda__215,axiom,
! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_uf(Uu),Uua) = aa_nat_set_nat(set_ord_atMost_nat,Uu) ).
% ATP.lambda_215
tff(fact_10427_ATP_Olambda__216,axiom,
! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_up(Uu),Uua) = aa_nat_rat(semiri681578069525770553at_rat,Uu) ).
% ATP.lambda_216
tff(fact_10428_ATP_Olambda__217,axiom,
! [Uu: int,Uua: nat] : aa_nat_rat(aTP_Lamm_uq(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,Uu) ).
% ATP.lambda_217
tff(fact_10429_ATP_Olambda__218,axiom,
! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_lt(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub)),aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).
% ATP.lambda_218
tff(fact_10430_ATP_Olambda__219,axiom,
! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_jp(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub)),aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).
% ATP.lambda_219
tff(fact_10431_ATP_Olambda__220,axiom,
! [Uu: num,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_eb(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Uu)),Uub),produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uub,numera6620942414471956472nteger(Uu))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua),Uub)) ).
% ATP.lambda_220
tff(fact_10432_ATP_Olambda__221,axiom,
! [Uu: num,Uua: nat,Uub: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_dz(Uu),Uua),Uub) = aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(if_Pro6206227464963214023at_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_num_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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,Uu)))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),Uub)) ).
% ATP.lambda_221
tff(fact_10433_ATP_Olambda__222,axiom,
! [Uu: num,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ea(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_num_int(numeral_numeral_int,Uu)),Uub),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Uub),aa_num_int(numeral_numeral_int,Uu)))),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),Uub)) ).
% ATP.lambda_222
tff(fact_10434_ATP_Olambda__223,axiom,
! [Uu: complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_nf(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uu),Uub),aa_complex_complex(Uua,Uub)),one_one_complex) ).
% ATP.lambda_223
tff(fact_10435_ATP_Olambda__224,axiom,
! [Uu: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_nj(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uu),Uub)),aa_complex_real(Uua,Uub)),one_one_real) ).
% ATP.lambda_224
tff(fact_10436_ATP_Olambda__225,axiom,
! [Uu: real,Uua: fun_real_complex,Uub: real] : aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_nc(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub),aa_real_complex(Uua,Uub)),one_one_complex) ).
% ATP.lambda_225
tff(fact_10437_ATP_Olambda__226,axiom,
! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ng(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub)),aa_real_real(Uua,Uub)),one_one_real) ).
% ATP.lambda_226
tff(fact_10438_ATP_Olambda__227,axiom,
! [Uu: real,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_nk(Uu),Uua),Uub) = if_rat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uu),Uub),aa_real_rat(Uua,Uub),one_one_rat) ).
% ATP.lambda_227
tff(fact_10439_ATP_Olambda__228,axiom,
! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_nd(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uu),Uub),aa_nat_complex(Uua,Uub)),one_one_complex) ).
% ATP.lambda_228
tff(fact_10440_ATP_Olambda__229,axiom,
! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_nh(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uu),Uub)),aa_nat_real(Uua,Uub)),one_one_real) ).
% ATP.lambda_229
tff(fact_10441_ATP_Olambda__230,axiom,
! [Uu: nat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_nl(Uu),Uua),Uub) = if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uu),Uub),aa_nat_rat(Uua,Uub),one_one_rat) ).
% ATP.lambda_230
tff(fact_10442_ATP_Olambda__231,axiom,
! [Uu: int,Uua: fun_int_complex,Uub: int] : aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_ne(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu),Uub),aa_int_complex(Uua,Uub)),one_one_complex) ).
% ATP.lambda_231
tff(fact_10443_ATP_Olambda__232,axiom,
! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_ni(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uu),Uub)),aa_int_real(Uua,Uub)),one_one_real) ).
% ATP.lambda_232
tff(fact_10444_ATP_Olambda__233,axiom,
! [Uu: complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_np(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uub),Uu),aa_complex_complex(Uua,Uub)),one_one_complex) ).
% ATP.lambda_233
tff(fact_10445_ATP_Olambda__234,axiom,
! [Uu: complex,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_nt(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uub),Uu)),aa_complex_real(Uua,Uub)),one_one_real) ).
% ATP.lambda_234
tff(fact_10446_ATP_Olambda__235,axiom,
! [Uu: real,Uua: fun_real_complex,Uub: real] : aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_nm(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu),aa_real_complex(Uua,Uub)),one_one_complex) ).
% ATP.lambda_235
tff(fact_10447_ATP_Olambda__236,axiom,
! [Uu: real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_nq(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu)),aa_real_real(Uua,Uub)),one_one_real) ).
% ATP.lambda_236
tff(fact_10448_ATP_Olambda__237,axiom,
! [Uu: real,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_nu(Uu),Uua),Uub) = if_rat(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,Uub),Uu),aa_real_rat(Uua,Uub),one_one_rat) ).
% ATP.lambda_237
tff(fact_10449_ATP_Olambda__238,axiom,
! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_nn(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),aa_nat_complex(Uua,Uub)),one_one_complex) ).
% ATP.lambda_238
tff(fact_10450_ATP_Olambda__239,axiom,
! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_nr(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu)),aa_nat_real(Uua,Uub)),one_one_real) ).
% ATP.lambda_239
tff(fact_10451_ATP_Olambda__240,axiom,
! [Uu: nat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_nv(Uu),Uua),Uub) = if_rat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),aa_nat_rat(Uua,Uub),one_one_rat) ).
% ATP.lambda_240
tff(fact_10452_ATP_Olambda__241,axiom,
! [Uu: int,Uua: fun_int_complex,Uub: int] : aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_no(Uu),Uua),Uub) = aa_complex_complex(if_complex(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uub),Uu),aa_int_complex(Uua,Uub)),one_one_complex) ).
% ATP.lambda_241
tff(fact_10453_ATP_Olambda__242,axiom,
! [Uu: int,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_ns(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uub),Uu)),aa_int_real(Uua,Uub)),one_one_real) ).
% ATP.lambda_242
tff(fact_10454_ATP_Olambda__243,axiom,
! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_pt(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,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_243
tff(fact_10455_ATP_Olambda__244,axiom,
! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_qe(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,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_244
tff(fact_10456_ATP_Olambda__245,axiom,
! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_ps(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,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_245
tff(fact_10457_ATP_Olambda__246,axiom,
! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] :
( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aTP_Lamm_sy(Uu),Uua),Uub))
<=> has_fi5821293074295781190e_real(Uu,aa_real_real(Uua,Uub),topolo2177554685111907308n_real(Uub,top_top_set_real)) ) ).
% ATP.lambda_246
tff(fact_10458_ATP_Olambda__247,axiom,
! [Uu: fun_re4238438383274954274t_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_rn(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(Uu,Uub),Uua) ).
% ATP.lambda_247
tff(fact_10459_ATP_Olambda__248,axiom,
! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_oe(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu,Uub),Uua) ).
% ATP.lambda_248
tff(fact_10460_ATP_Olambda__249,axiom,
! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_oh(Uu),Uua),Uub) = aa_nat_int(aa_nat_fun_nat_int(Uu,Uub),Uua) ).
% ATP.lambda_249
tff(fact_10461_ATP_Olambda__250,axiom,
! [Uu: nat,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_lm(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_complex_complex(if_complex(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu),one_one_complex),zero_zero_complex)),aa_nat_complex(power_power_complex(Uua),Uub)) ).
% ATP.lambda_250
tff(fact_10462_ATP_Olambda__251,axiom,
! [Uu: nat,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ln(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu)),one_one_real),zero_zero_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).
% ATP.lambda_251
tff(fact_10463_ATP_Olambda__252,axiom,
! [Uu: nat,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_lo(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(if_int(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,Uub),Uu)),one_one_int),zero_zero_int)),aa_nat_int(power_power_int(Uua),Uub)) ).
% ATP.lambda_252
tff(fact_10464_ATP_Olambda__253,axiom,
! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod2825494507867081354r_bool(aa_Cod4981436349852597092r_bool(aTP_Lamm_pq(Uu),Uua),Uub) = produc5135875973608410785r_bool(if_Code_integer(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Uu),Uua,minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),Uub)),aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),one_one_Code_integer)) ).
% ATP.lambda_253
tff(fact_10465_ATP_Olambda__254,axiom,
! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_oi(Uu),Uua),Uub) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_oh(Uu),Uub),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).
% ATP.lambda_254
tff(fact_10466_ATP_Olambda__255,axiom,
! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_of(Uu),Uua),Uub) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_oe(Uu),Uub),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).
% ATP.lambda_255
tff(fact_10467_ATP_Olambda__256,axiom,
! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_qd(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_qc(Uua),Uub)),quotient_of(Uu)) ).
% ATP.lambda_256
tff(fact_10468_ATP_Olambda__257,axiom,
! [Uu: rat,Uua: int,Uub: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_qb(Uu),Uua),Uub))
<=> pp(aa_Pro4501881432599481050t_bool(produc8560705615207772305t_bool(aa_int2327297053056786951t_bool(aTP_Lamm_qa(Uua),Uub)),quotient_of(Uu))) ) ).
% ATP.lambda_257
tff(fact_10469_ATP_Olambda__258,axiom,
! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_pv(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_pu(Uua),Uub)),quotient_of(Uu)) ).
% ATP.lambda_258
tff(fact_10470_ATP_Olambda__259,axiom,
! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_rs(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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_259
tff(fact_10471_ATP_Olambda__260,axiom,
! [Uu: real,Uua: fun_na5723442673449079972x_real,Uub: nat] : aa_nat_real(aa_fun5746040925223427547t_real(aTP_Lamm_kz(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(aa_nat480766339200631653x_real(Uua,Uub),zero_zero_complex)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).
% ATP.lambda_260
tff(fact_10472_ATP_Olambda__261,axiom,
! [Uu: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_la(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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_261
tff(fact_10473_ATP_Olambda__262,axiom,
! [Uu: real,Uua: fun_nat_fun_rat_real,Uub: nat] : aa_nat_real(aa_fun507281639146497029t_real(aTP_Lamm_lb(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_rat_real(aa_nat_fun_rat_real(Uua,Uub),zero_zero_rat)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).
% ATP.lambda_262
tff(fact_10474_ATP_Olambda__263,axiom,
! [Uu: real,Uua: fun_nat_fun_nat_real,Uub: nat] : aa_nat_real(aa_fun7039663905532832253t_real(aTP_Lamm_lc(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_nat_fun_nat_real(Uua,Uub),zero_zero_nat)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).
% ATP.lambda_263
tff(fact_10475_ATP_Olambda__264,axiom,
! [Uu: real,Uua: fun_nat_fun_int_real,Uub: nat] : aa_nat_real(aa_fun241848180832411225t_real(aTP_Lamm_ld(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(aa_nat_fun_int_real(Uua,Uub),zero_zero_int)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).
% ATP.lambda_264
tff(fact_10476_ATP_Olambda__265,axiom,
! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_gh(Uu),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(zero_zero_complex),Uub))),aa_nat_complex(Uua,Uub)) ).
% ATP.lambda_265
tff(fact_10477_ATP_Olambda__266,axiom,
! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_gj(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uub))),aa_nat_real(Uua,Uub)) ).
% ATP.lambda_266
tff(fact_10478_ATP_Olambda__267,axiom,
! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_gi(Uu),Uua),Uub) = divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(power_power_rat(zero_zero_rat),Uub)),aa_nat_rat(Uua,Uub)) ).
% ATP.lambda_267
tff(fact_10479_ATP_Olambda__268,axiom,
! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_rp(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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_268
tff(fact_10480_ATP_Olambda__269,axiom,
! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_le(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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_269
tff(fact_10481_ATP_Olambda__270,axiom,
! [Uu: nat,Uua: nat,Uub: list_nat] :
( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_tw(Uu),Uua),Uub))
<=> ( ( size_size_list_nat(Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),one_one_nat) )
& ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).
% ATP.lambda_270
tff(fact_10482_ATP_Olambda__271,axiom,
! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ma(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),aa_nat_complex(Uu,Uub))),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat)))) ).
% ATP.lambda_271
tff(fact_10483_ATP_Olambda__272,axiom,
! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ly(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Uub)),aa_nat_real(Uu,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat)))) ).
% ATP.lambda_272
tff(fact_10484_ATP_Olambda__273,axiom,
! [Uu: nat,Uua: nat,Uub: list_nat] :
( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_tx(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_273
tff(fact_10485_ATP_Olambda__274,axiom,
! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
( pp(aa_lis2822252094023462681T_bool(aa_nat1297483763028037422T_bool(aTP_Lamm_ax(Uu),Uua),Uub))
<=> ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub)),Uu))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Uub)),Uua)) ) ) ).
% ATP.lambda_274
tff(fact_10486_ATP_Olambda__275,axiom,
! [Uu: set_complex,Uua: nat,Uub: list_complex] :
( pp(aa_list_complex_bool(aa_nat8039161444313734436x_bool(aTP_Lamm_aw(Uu),Uua),Uub))
<=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Uub)),Uu))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s3451745648224563538omplex(Uub)),Uua)) ) ) ).
% ATP.lambda_275
tff(fact_10487_ATP_Olambda__276,axiom,
! [Uu: set_bool,Uua: nat,Uub: list_bool] :
( pp(aa_list_bool_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_ay(Uu),Uua),Uub))
<=> ( ord_less_eq_set_bool(set_bool2(Uub),Uu)
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_bool(Uub)),Uua)) ) ) ).
% ATP.lambda_276
tff(fact_10488_ATP_Olambda__277,axiom,
! [Uu: set_nat,Uua: nat,Uub: list_nat] :
( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_ba(Uu),Uua),Uub))
<=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(Uub)),Uu))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_nat(Uub)),Uua)) ) ) ).
% ATP.lambda_277
tff(fact_10489_ATP_Olambda__278,axiom,
! [Uu: set_int,Uua: nat,Uub: list_int] :
( pp(aa_list_int_bool(aa_nat1619170893822700070t_bool(aTP_Lamm_az(Uu),Uua),Uub))
<=> ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Uub)),Uu))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_int(Uub)),Uua)) ) ) ).
% ATP.lambda_278
tff(fact_10490_ATP_Olambda__279,axiom,
! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
( pp(aa_lis2822252094023462681T_bool(aa_nat1297483763028037422T_bool(aTP_Lamm_as(Uu),Uua),Uub))
<=> ( pp(aa_set8260762808118238015T_bool(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub)),Uu))
& ( size_s6755466524823107622T_VEBT(Uub) = Uua ) ) ) ).
% ATP.lambda_279
tff(fact_10491_ATP_Olambda__280,axiom,
! [Uu: set_complex,Uua: nat,Uub: list_complex] :
( pp(aa_list_complex_bool(aa_nat8039161444313734436x_bool(aTP_Lamm_ar(Uu),Uua),Uub))
<=> ( pp(aa_set_complex_bool(ord_le211207098394363844omplex(set_complex2(Uub)),Uu))
& ( size_s3451745648224563538omplex(Uub) = Uua ) ) ) ).
% ATP.lambda_280
tff(fact_10492_ATP_Olambda__281,axiom,
! [Uu: set_bool,Uua: nat,Uub: list_bool] :
( pp(aa_list_bool_bool(aa_nat4862889769322786699l_bool(aTP_Lamm_at(Uu),Uua),Uub))
<=> ( ord_less_eq_set_bool(set_bool2(Uub),Uu)
& ( size_size_list_bool(Uub) = Uua ) ) ) ).
% ATP.lambda_281
tff(fact_10493_ATP_Olambda__282,axiom,
! [Uu: set_nat,Uua: nat,Uub: list_nat] :
( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_av(Uu),Uua),Uub))
<=> ( pp(aa_set_nat_bool(ord_less_eq_set_nat(set_nat2(Uub)),Uu))
& ( size_size_list_nat(Uub) = Uua ) ) ) ).
% ATP.lambda_282
tff(fact_10494_ATP_Olambda__283,axiom,
! [Uu: set_int,Uua: nat,Uub: list_int] :
( pp(aa_list_int_bool(aa_nat1619170893822700070t_bool(aTP_Lamm_au(Uu),Uua),Uub))
<=> ( pp(aa_set_int_bool(ord_less_eq_set_int(set_int2(Uub)),Uu))
& ( size_size_list_int(Uub) = Uua ) ) ) ).
% ATP.lambda_283
tff(fact_10495_ATP_Olambda__284,axiom,
! [Uu: nat,Uua: nat,Uub: list_nat] :
( pp(aa_list_nat_bool(aa_nat2079453845469346690t_bool(aTP_Lamm_tv(Uu),Uua),Uub))
<=> ( ( size_size_list_nat(Uub) = Uu )
& ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).
% ATP.lambda_284
tff(fact_10496_ATP_Olambda__285,axiom,
! [Uu: set_nat,Uua: nat,Uub: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_pl(Uu),Uua),Uub))
<=> ( pp(aa_set_nat_bool(member_nat(aa_nat_nat(suc,Uub)),Uu))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uua)) ) ) ).
% ATP.lambda_285
tff(fact_10497_ATP_Olambda__286,axiom,
! [Uu: nat,Uua: nat,Uub: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ue(Uu),Uua),Uub))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uu))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uu))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uua),Uub)) ) ) ).
% ATP.lambda_286
tff(fact_10498_ATP_Olambda__287,axiom,
! [Uu: set_nat,Uua: nat,Uub: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_pk(Uu),Uua),Uub))
<=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),aa_nat_nat(suc,Uua))) ) ) ).
% ATP.lambda_287
tff(fact_10499_ATP_Olambda__288,axiom,
! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_lz(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(diffs_complex(Uu),Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).
% ATP.lambda_288
tff(fact_10500_ATP_Olambda__289,axiom,
! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_lx(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(diffs_real(Uu),Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).
% ATP.lambda_289
tff(fact_10501_ATP_Olambda__290,axiom,
! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_mc(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(diffs_complex(Uua),Uub)),aa_nat_complex(power_power_complex(Uu),Uub)) ).
% ATP.lambda_290
tff(fact_10502_ATP_Olambda__291,axiom,
! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_mb(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(diffs_real(Uua),Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).
% ATP.lambda_291
tff(fact_10503_ATP_Olambda__292,axiom,
! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ou(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uub)) ).
% ATP.lambda_292
tff(fact_10504_ATP_Olambda__293,axiom,
! [Uu: int,Uua: int,Uub: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ee(Uu),Uua),Uub))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uua))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uub)) ) ) ).
% ATP.lambda_293
tff(fact_10505_ATP_Olambda__294,axiom,
! [Uu: int,Uua: int,Uub: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_ef(Uu),Uua),Uub))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uub))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uua),Uub)) ) ) ).
% ATP.lambda_294
tff(fact_10506_ATP_Olambda__295,axiom,
! [Uu: int,Uua: int,Uub: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_bl(Uu),Uua),Uub))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uu),Uub))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uub),Uua)) ) ) ).
% ATP.lambda_295
tff(fact_10507_ATP_Olambda__296,axiom,
! [Uu: int,Uua: int,Uub: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_bm(Uu),Uua),Uub))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uu),Uub))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Uub),Uua)) ) ) ).
% ATP.lambda_296
tff(fact_10508_ATP_Olambda__297,axiom,
! [Uu: int,Uua: int,Uub: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_bk(Uu),Uua),Uub))
<=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uu),Uub))
& pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Uub),Uua)) ) ) ).
% ATP.lambda_297
tff(fact_10509_ATP_Olambda__298,axiom,
! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_af(Uu),Uua),Uub))
<=> ( pp(aa_nat_bool(vEBT_vebt_member(Uu),Uub))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uub)) ) ) ).
% ATP.lambda_298
tff(fact_10510_ATP_Olambda__299,axiom,
! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ag(Uu),Uua),Uub))
<=> ( pp(aa_nat_bool(vEBT_vebt_member(Uu),Uub))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uua)) ) ) ).
% ATP.lambda_299
tff(fact_10511_ATP_Olambda__300,axiom,
! [Uu: nat,Uua: nat,Uub: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_qf(Uu),Uua),Uub))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uub),Uua))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uub),Uu)) ) ) ).
% ATP.lambda_300
tff(fact_10512_ATP_Olambda__301,axiom,
! [Uu: set_complex,Uua: fun_complex_complex,Uub: complex] :
( pp(aa_complex_bool(aa_fun9073502203726719595x_bool(aTP_Lamm_cs(Uu),Uua),Uub))
<=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
& ( aa_complex_complex(Uua,Uub) != zero_zero_complex ) ) ) ).
% ATP.lambda_301
tff(fact_10513_ATP_Olambda__302,axiom,
! [Uu: set_complex,Uua: fun_complex_real,Uub: complex] :
( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aTP_Lamm_da(Uu),Uua),Uub))
<=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
& ( aa_complex_real(Uua,Uub) != zero_zero_real ) ) ) ).
% ATP.lambda_302
tff(fact_10514_ATP_Olambda__303,axiom,
! [Uu: set_real,Uua: fun_real_complex,Uub: real] :
( pp(aa_real_bool(aa_fun3813300650643916655l_bool(aTP_Lamm_cm(Uu),Uua),Uub))
<=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
& ( aa_real_complex(Uua,Uub) != zero_zero_complex ) ) ) ).
% ATP.lambda_303
tff(fact_10515_ATP_Olambda__304,axiom,
! [Uu: set_real,Uua: fun_real_real,Uub: real] :
( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aTP_Lamm_cu(Uu),Uua),Uub))
<=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
& ( aa_real_real(Uua,Uub) != zero_zero_real ) ) ) ).
% ATP.lambda_304
tff(fact_10516_ATP_Olambda__305,axiom,
! [Uu: set_real,Uua: fun_real_rat,Uub: real] :
( pp(aa_real_bool(aa_fun8987621170725249221l_bool(aTP_Lamm_dc(Uu),Uua),Uub))
<=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
& ( aa_real_rat(Uua,Uub) != zero_zero_rat ) ) ) ).
% ATP.lambda_305
tff(fact_10517_ATP_Olambda__306,axiom,
! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
( pp(aa_nat_bool(aa_fun8004580481347240231t_bool(aTP_Lamm_co(Uu),Uua),Uub))
<=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
& ( aa_nat_complex(Uua,Uub) != zero_zero_complex ) ) ) ).
% ATP.lambda_306
tff(fact_10518_ATP_Olambda__307,axiom,
! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aTP_Lamm_cw(Uu),Uua),Uub))
<=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
& ( aa_nat_real(Uua,Uub) != zero_zero_real ) ) ) ).
% ATP.lambda_307
tff(fact_10519_ATP_Olambda__308,axiom,
! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
( pp(aa_nat_bool(aa_fun5637262893798792573t_bool(aTP_Lamm_de(Uu),Uua),Uub))
<=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
& ( aa_nat_rat(Uua,Uub) != zero_zero_rat ) ) ) ).
% ATP.lambda_308
tff(fact_10520_ATP_Olambda__309,axiom,
! [Uu: set_int,Uua: fun_int_complex,Uub: int] :
( pp(aa_int_bool(aa_fun8146511558608327279t_bool(aTP_Lamm_cq(Uu),Uua),Uub))
<=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
& ( aa_int_complex(Uua,Uub) != zero_zero_complex ) ) ) ).
% ATP.lambda_309
tff(fact_10521_ATP_Olambda__310,axiom,
! [Uu: set_int,Uua: fun_int_real,Uub: int] :
( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aTP_Lamm_cy(Uu),Uua),Uub))
<=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
& ( aa_int_real(Uua,Uub) != zero_zero_real ) ) ) ).
% ATP.lambda_310
tff(fact_10522_ATP_Olambda__311,axiom,
! [Uu: set_complex,Uua: fun_complex_complex,Uub: complex] :
( pp(aa_complex_bool(aa_fun9073502203726719595x_bool(aTP_Lamm_bt(Uu),Uua),Uub))
<=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
& ( aa_complex_complex(Uua,Uub) != one_one_complex ) ) ) ).
% ATP.lambda_311
tff(fact_10523_ATP_Olambda__312,axiom,
! [Uu: set_complex,Uua: fun_complex_real,Uub: complex] :
( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aTP_Lamm_cb(Uu),Uua),Uub))
<=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
& ( aa_complex_real(Uua,Uub) != one_one_real ) ) ) ).
% ATP.lambda_312
tff(fact_10524_ATP_Olambda__313,axiom,
! [Uu: set_real,Uua: fun_real_complex,Uub: real] :
( pp(aa_real_bool(aa_fun3813300650643916655l_bool(aTP_Lamm_bn(Uu),Uua),Uub))
<=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
& ( aa_real_complex(Uua,Uub) != one_one_complex ) ) ) ).
% ATP.lambda_313
tff(fact_10525_ATP_Olambda__314,axiom,
! [Uu: set_real,Uua: fun_real_real,Uub: real] :
( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aTP_Lamm_bv(Uu),Uua),Uub))
<=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
& ( aa_real_real(Uua,Uub) != one_one_real ) ) ) ).
% ATP.lambda_314
tff(fact_10526_ATP_Olambda__315,axiom,
! [Uu: set_real,Uua: fun_real_rat,Uub: real] :
( pp(aa_real_bool(aa_fun8987621170725249221l_bool(aTP_Lamm_cd(Uu),Uua),Uub))
<=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
& ( aa_real_rat(Uua,Uub) != one_one_rat ) ) ) ).
% ATP.lambda_315
tff(fact_10527_ATP_Olambda__316,axiom,
! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
( pp(aa_nat_bool(aa_fun8004580481347240231t_bool(aTP_Lamm_bp(Uu),Uua),Uub))
<=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
& ( aa_nat_complex(Uua,Uub) != one_one_complex ) ) ) ).
% ATP.lambda_316
tff(fact_10528_ATP_Olambda__317,axiom,
! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aTP_Lamm_bx(Uu),Uua),Uub))
<=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
& ( aa_nat_real(Uua,Uub) != one_one_real ) ) ) ).
% ATP.lambda_317
tff(fact_10529_ATP_Olambda__318,axiom,
! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
( pp(aa_nat_bool(aa_fun5637262893798792573t_bool(aTP_Lamm_cf(Uu),Uua),Uub))
<=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
& ( aa_nat_rat(Uua,Uub) != one_one_rat ) ) ) ).
% ATP.lambda_318
tff(fact_10530_ATP_Olambda__319,axiom,
! [Uu: set_int,Uua: fun_int_complex,Uub: int] :
( pp(aa_int_bool(aa_fun8146511558608327279t_bool(aTP_Lamm_br(Uu),Uua),Uub))
<=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
& ( aa_int_complex(Uua,Uub) != one_one_complex ) ) ) ).
% ATP.lambda_319
tff(fact_10531_ATP_Olambda__320,axiom,
! [Uu: set_int,Uua: fun_int_real,Uub: int] :
( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aTP_Lamm_bz(Uu),Uua),Uub))
<=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
& ( aa_int_real(Uua,Uub) != one_one_real ) ) ) ).
% ATP.lambda_320
tff(fact_10532_ATP_Olambda__321,axiom,
! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: product_prod_nat_nat] :
( pp(aa_Pro2232559814647311250t_bool(aa_set5223363028666124222t_bool(aTP_Lamm_ap(Uu),Uua),Uub))
<=> ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Uub),Uu))
& ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Uub),Uua)) ) ) ).
% ATP.lambda_321
tff(fact_10533_ATP_Olambda__322,axiom,
! [Uu: set_complex,Uua: set_complex,Uub: complex] :
( pp(aa_complex_bool(aa_set3741564773079358752x_bool(aTP_Lamm_al(Uu),Uua),Uub))
<=> ( pp(aa_set_complex_bool(member_complex(Uub),Uu))
& ~ pp(aa_set_complex_bool(member_complex(Uub),Uua)) ) ) ).
% ATP.lambda_322
tff(fact_10534_ATP_Olambda__323,axiom,
! [Uu: set_real,Uua: set_real,Uub: real] :
( pp(aa_real_bool(aa_set8565328311858951076l_bool(aTP_Lamm_am(Uu),Uua),Uub))
<=> ( pp(aa_set_real_bool(member_real(Uub),Uu))
& ~ pp(aa_set_real_bool(member_real(Uub),Uua)) ) ) ).
% ATP.lambda_323
tff(fact_10535_ATP_Olambda__324,axiom,
! [Uu: set_list_nat,Uua: set_list_nat,Uub: list_nat] :
( pp(aa_list_nat_bool(aa_set257547205302447420t_bool(aTP_Lamm_an(Uu),Uua),Uub))
<=> ( member_list_nat(Uub,Uu)
& ~ member_list_nat(Uub,Uua) ) ) ).
% ATP.lambda_324
tff(fact_10536_ATP_Olambda__325,axiom,
! [Uu: set_nat,Uua: set_nat,Uub: nat] :
( pp(aa_nat_bool(aa_set3744802623701456604t_bool(aTP_Lamm_aq(Uu),Uua),Uub))
<=> ( pp(aa_set_nat_bool(member_nat(Uub),Uu))
& ~ pp(aa_set_nat_bool(member_nat(Uub),Uua)) ) ) ).
% ATP.lambda_325
tff(fact_10537_ATP_Olambda__326,axiom,
! [Uu: set_int,Uua: set_int,Uub: int] :
( pp(aa_int_bool(aa_set7012339369581096484t_bool(aTP_Lamm_ao(Uu),Uua),Uub))
<=> ( pp(aa_set_int_bool(member_int(Uub),Uu))
& ~ pp(aa_set_int_bool(member_int(Uub),Uua)) ) ) ).
% ATP.lambda_326
tff(fact_10538_ATP_Olambda__327,axiom,
! [Uu: nat,Uua: nat,Uub: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ph(Uu),Uua),Uub))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uu)),Uua)) ) ).
% ATP.lambda_327
tff(fact_10539_ATP_Olambda__328,axiom,
! [Uu: nat,Uua: nat,Uub: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ui(Uu),Uua),Uub))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)),Uu)) ) ).
% ATP.lambda_328
tff(fact_10540_ATP_Olambda__329,axiom,
! [Uu: real,Uua: complex,Uub: complex] :
( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_tk(Uu),Uua),Uub))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_complex_real(real_V3694042436643373181omplex(Uua),Uub)),Uu)) ) ).
% ATP.lambda_329
tff(fact_10541_ATP_Olambda__330,axiom,
! [Uu: real,Uua: real,Uub: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_ti(Uu),Uua),Uub))
<=> pp(aa_real_bool(aa_rea7809428808904475482l_bool(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(real_V975177566351809787t_real,Uua),Uub)),Uu)) ) ).
% ATP.lambda_330
tff(fact_10542_ATP_Olambda__331,axiom,
! [Uu: nat,Uua: complex,Uub: complex] :
( pp(aa_complex_bool(aa_com1553036987509291606x_bool(aTP_Lamm_dg(Uu),Uua),Uub))
<=> ( aa_nat_complex(power_power_complex(Uub),Uu) = Uua ) ) ).
% ATP.lambda_331
tff(fact_10543_ATP_Olambda__332,axiom,
! [Uu: complex,Uua: nat,Uub: complex] :
( pp(aa_complex_bool(aa_nat5545486172511849908x_bool(aTP_Lamm_pf(Uu),Uua),Uub))
<=> ( aa_nat_complex(power_power_complex(Uub),Uua) = Uu ) ) ).
% ATP.lambda_332
tff(fact_10544_ATP_Olambda__333,axiom,
! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_kh(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,aa_nat_nat(suc,Uub))),aa_nat_complex(power_power_complex(Uua),Uub)) ).
% ATP.lambda_333
tff(fact_10545_ATP_Olambda__334,axiom,
! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ki(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,aa_nat_nat(suc,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).
% ATP.lambda_334
tff(fact_10546_ATP_Olambda__335,axiom,
! [Uu: fun_nat_real,Uua: fun_nat_int,Uub: nat] : aa_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_so(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_nat_int(Uua,Uub))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,numeral_numeral_real(aa_num_num(bit0,one))),pi))) ).
% ATP.lambda_335
tff(fact_10547_ATP_Olambda__336,axiom,
! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_rq(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_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_336
tff(fact_10548_ATP_Olambda__337,axiom,
! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ke(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).
% ATP.lambda_337
tff(fact_10549_ATP_Olambda__338,axiom,
! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_kc(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).
% ATP.lambda_338
tff(fact_10550_ATP_Olambda__339,axiom,
! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_oo(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),Uub)) ).
% ATP.lambda_339
tff(fact_10551_ATP_Olambda__340,axiom,
! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_dh(Uu),Uua),Uub))
<=> ( pp(aa_nat_bool(Uu,Uub))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uua)) ) ) ).
% ATP.lambda_340
tff(fact_10552_ATP_Olambda__341,axiom,
! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tc(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).
% ATP.lambda_341
tff(fact_10553_ATP_Olambda__342,axiom,
! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_sz(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(Uua,Uub)),aa_real_real(Uu,Uub)) ).
% ATP.lambda_342
tff(fact_10554_ATP_Olambda__343,axiom,
! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_tz,Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(Uua,Uub)) ).
% ATP.lambda_343
tff(fact_10555_ATP_Olambda__344,axiom,
! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_nw(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_344
tff(fact_10556_ATP_Olambda__345,axiom,
! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_nx(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uub)),aa_nat_int(Uua,Uub)) ).
% ATP.lambda_345
tff(fact_10557_ATP_Olambda__346,axiom,
! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ny(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uub)) ).
% ATP.lambda_346
tff(fact_10558_ATP_Olambda__347,axiom,
! [Uu: fun_co8153928105989009288nteger,Uua: fun_co8153928105989009288nteger,Uub: complex] : aa_com4217951221296830281nteger(aa_fun5267417455630836558nteger(aTP_Lamm_fu(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_com4217951221296830281nteger(Uua,Uub)),aa_com4217951221296830281nteger(Uu,Uub)) ).
% ATP.lambda_347
tff(fact_10559_ATP_Olambda__348,axiom,
! [Uu: fun_complex_real,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_fy(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(Uua,Uub)),aa_complex_real(Uu,Uub)) ).
% ATP.lambda_348
tff(fact_10560_ATP_Olambda__349,axiom,
! [Uu: fun_complex_rat,Uua: fun_complex_rat,Uub: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_gb(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_complex_rat(Uua,Uub)),aa_complex_rat(Uu,Uub)) ).
% ATP.lambda_349
tff(fact_10561_ATP_Olambda__350,axiom,
! [Uu: fun_re6978948838062957830nteger,Uua: fun_re6978948838062957830nteger,Uub: real] : aa_real_Code_integer(aa_fun7923231501384339402nteger(aTP_Lamm_fv(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_real_Code_integer(Uua,Uub)),aa_real_Code_integer(Uu,Uub)) ).
% ATP.lambda_350
tff(fact_10562_ATP_Olambda__351,axiom,
! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_fz(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(Uua,Uub)),aa_real_real(Uu,Uub)) ).
% ATP.lambda_351
tff(fact_10563_ATP_Olambda__352,axiom,
! [Uu: fun_real_rat,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_gc(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_real_rat(Uua,Uub)),aa_real_rat(Uu,Uub)) ).
% ATP.lambda_352
tff(fact_10564_ATP_Olambda__353,axiom,
! [Uu: fun_nat_Code_integer,Uua: fun_nat_Code_integer,Uub: nat] : aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_fw(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uua,Uub)),aa_nat_Code_integer(Uu,Uub)) ).
% ATP.lambda_353
tff(fact_10565_ATP_Olambda__354,axiom,
! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_gd(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uua,Uub)),aa_nat_rat(Uu,Uub)) ).
% ATP.lambda_354
tff(fact_10566_ATP_Olambda__355,axiom,
! [Uu: fun_int_Code_integer,Uua: fun_int_Code_integer,Uub: int] : aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_fx(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(Uua,Uub)),aa_int_Code_integer(Uu,Uub)) ).
% ATP.lambda_355
tff(fact_10567_ATP_Olambda__356,axiom,
! [Uu: fun_int_real,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_ga(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(Uua,Uub)),aa_int_real(Uu,Uub)) ).
% ATP.lambda_356
tff(fact_10568_ATP_Olambda__357,axiom,
! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_fe(Uu),Uua),Uub) = aa_complex_complex(minus_minus_complex(aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uub)) ).
% ATP.lambda_357
tff(fact_10569_ATP_Olambda__358,axiom,
! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ff(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).
% ATP.lambda_358
tff(fact_10570_ATP_Olambda__359,axiom,
! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ub(Uu),Uua),Uub) = minus_minus_rat(aa_nat_rat(Uu,Uub),aa_nat_rat(Uua,Uub)) ).
% ATP.lambda_359
tff(fact_10571_ATP_Olambda__360,axiom,
! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_fd(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uub)) ).
% ATP.lambda_360
tff(fact_10572_ATP_Olambda__361,axiom,
! [Uu: fun_Pr2602117396471556196at_nat,Uua: fun_Pr2602117396471556196at_nat,Uub: product_prod_nat_nat] : aa_Pro4243173770044397973at_nat(aa_fun2351638177499875132at_nat(aTP_Lamm_gk(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_Pro4243173770044397973at_nat(Uua,Uub)),aa_Pro4243173770044397973at_nat(Uu,Uub)) ).
% ATP.lambda_361
tff(fact_10573_ATP_Olambda__362,axiom,
! [Uu: fun_complex_nat,Uua: fun_complex_nat,Uub: complex] : aa_complex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_gl(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_complex_nat(Uua,Uub)),aa_complex_nat(Uu,Uub)) ).
% ATP.lambda_362
tff(fact_10574_ATP_Olambda__363,axiom,
! [Uu: fun_real_nat,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_gm(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_real_nat(Uua,Uub)),aa_real_nat(Uu,Uub)) ).
% ATP.lambda_363
tff(fact_10575_ATP_Olambda__364,axiom,
! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_lp(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uua,Uub)),aa_nat_real(Uu,Uub)) ).
% ATP.lambda_364
tff(fact_10576_ATP_Olambda__365,axiom,
! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_go(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(Uua,Uub)),aa_nat_nat(Uu,Uub)) ).
% ATP.lambda_365
tff(fact_10577_ATP_Olambda__366,axiom,
! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_gn(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(Uua,Uub)),aa_int_nat(Uu,Uub)) ).
% ATP.lambda_366
tff(fact_10578_ATP_Olambda__367,axiom,
! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_fa(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uub)) ).
% ATP.lambda_367
tff(fact_10579_ATP_Olambda__368,axiom,
! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_fc(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).
% ATP.lambda_368
tff(fact_10580_ATP_Olambda__369,axiom,
! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ud,Uu),Uua),Uub) = plus_plus_rat(aa_nat_rat(Uu,Uub),aa_nat_rat(Uua,Uub)) ).
% ATP.lambda_369
tff(fact_10581_ATP_Olambda__370,axiom,
! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_fb(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uub)) ).
% ATP.lambda_370
tff(fact_10582_ATP_Olambda__371,axiom,
! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ka(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uub)),aa_nat_int(Uua,Uub)) ).
% ATP.lambda_371
tff(fact_10583_ATP_Olambda__372,axiom,
! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ez(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uub)) ).
% ATP.lambda_372
tff(fact_10584_ATP_Olambda__373,axiom,
! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_rm(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).
% ATP.lambda_373
tff(fact_10585_ATP_Olambda__374,axiom,
! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_di(Uu),Uua),Uub))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(Uu,Uub)),Uua)) ) ).
% ATP.lambda_374
tff(fact_10586_ATP_Olambda__375,axiom,
! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fj(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(Uu,Uub)),Uua) ).
% ATP.lambda_375
tff(fact_10587_ATP_Olambda__376,axiom,
! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_oc(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(Uu,Uub)),Uua) ).
% ATP.lambda_376
tff(fact_10588_ATP_Olambda__377,axiom,
! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_fi(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(Uu,Uub)),Uua) ).
% ATP.lambda_377
tff(fact_10589_ATP_Olambda__378,axiom,
! [Uu: fun_complex_complex,Uua: complex,Uub: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_fg(Uu),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(Uu,Uub)),Uua) ).
% ATP.lambda_378
tff(fact_10590_ATP_Olambda__379,axiom,
! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_jz(Uu),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(Uu,Uub)),Uua) ).
% ATP.lambda_379
tff(fact_10591_ATP_Olambda__380,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(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(Uu,Uub)),Uua) ).
% ATP.lambda_380
tff(fact_10592_ATP_Olambda__381,axiom,
! [Uu: fun_complex_complex,Uua: complex,Uub: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_eo(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_complex_complex(Uu,Uub)),Uua) ).
% ATP.lambda_381
tff(fact_10593_ATP_Olambda__382,axiom,
! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_eq(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),Uua) ).
% ATP.lambda_382
tff(fact_10594_ATP_Olambda__383,axiom,
! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ep(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),Uua) ).
% ATP.lambda_383
tff(fact_10595_ATP_Olambda__384,axiom,
! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_en(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu,Uub)),Uua) ).
% ATP.lambda_384
tff(fact_10596_ATP_Olambda__385,axiom,
! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_lh(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uua,Uub)),Uu) ).
% ATP.lambda_385
tff(fact_10597_ATP_Olambda__386,axiom,
! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_li(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uua,Uub)),Uu) ).
% ATP.lambda_386
tff(fact_10598_ATP_Olambda__387,axiom,
! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_is(Uu),Uua),Uub) = aa_complex_complex(minus_minus_complex(aa_nat_complex(Uu,Uub)),Uua) ).
% ATP.lambda_387
tff(fact_10599_ATP_Olambda__388,axiom,
! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_iu(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),Uua) ).
% ATP.lambda_388
tff(fact_10600_ATP_Olambda__389,axiom,
! [Uu: fun_nat_rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_it(Uu),Uua),Uub) = minus_minus_rat(aa_nat_rat(Uu,Uub),Uua) ).
% ATP.lambda_389
tff(fact_10601_ATP_Olambda__390,axiom,
! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_ir(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,Uub)),Uua) ).
% ATP.lambda_390
tff(fact_10602_ATP_Olambda__391,axiom,
! [Uu: fun_real_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_rj(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uu,Uub)),Uua) ).
% ATP.lambda_391
tff(fact_10603_ATP_Olambda__392,axiom,
! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_nz(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(Uu,Uub)),Uua) ).
% ATP.lambda_392
tff(fact_10604_ATP_Olambda__393,axiom,
! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_oa(Uu),Uua),Uub) = aa_nat_int(power_power_int(aa_nat_int(Uu,Uub)),Uua) ).
% ATP.lambda_393
tff(fact_10605_ATP_Olambda__394,axiom,
! [Uu: fun_int_int,Uua: nat,Uub: int] : aa_int_int(aa_nat_fun_int_int(aTP_Lamm_ob(Uu),Uua),Uub) = aa_nat_int(power_power_int(aa_int_int(Uu,Uub)),Uua) ).
% ATP.lambda_394
tff(fact_10606_ATP_Olambda__395,axiom,
! [Uu: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_st(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uua,Uub)),Uu) ).
% ATP.lambda_395
tff(fact_10607_ATP_Olambda__396,axiom,
! [Uu: fun_real_real,Uua: real,Uub: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_rl(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),Uua) ).
% ATP.lambda_396
tff(fact_10608_ATP_Olambda__397,axiom,
! [Uu: real,Uua: fun_nat_complex,Uub: nat] : aa_nat_real(aa_fun7054779078796324018t_real(aTP_Lamm_km(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uua,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).
% ATP.lambda_397
tff(fact_10609_ATP_Olambda__398,axiom,
! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ul(Uu),Uua),Uub) = minus_minus_rat(inverse_inverse_rat(aa_nat_rat(Uu,Uub)),inverse_inverse_rat(aa_nat_rat(Uua,Uub))) ).
% ATP.lambda_398
tff(fact_10610_ATP_Olambda__399,axiom,
! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_hr(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(Uu),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),Uua)) ).
% ATP.lambda_399
tff(fact_10611_ATP_Olambda__400,axiom,
! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_hu(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Uub)),Uua)) ).
% ATP.lambda_400
tff(fact_10612_ATP_Olambda__401,axiom,
! [Uu: rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_hs(Uu),Uua),Uub) = plus_plus_rat(Uu,aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Uub)),Uua)) ).
% ATP.lambda_401
tff(fact_10613_ATP_Olambda__402,axiom,
! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ht(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,aa_nat_nat(semiri1316708129612266289at_nat,Uub)),Uua)) ).
% ATP.lambda_402
tff(fact_10614_ATP_Olambda__403,axiom,
! [Uu: int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_hq(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Uub)),Uua)) ).
% ATP.lambda_403
tff(fact_10615_ATP_Olambda__404,axiom,
! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_jl(Uu),Uua),Uub) = aa_nat_complex(power_power_complex(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).
% ATP.lambda_404
tff(fact_10616_ATP_Olambda__405,axiom,
! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_jo(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).
% ATP.lambda_405
tff(fact_10617_ATP_Olambda__406,axiom,
! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_jm(Uu),Uua),Uub) = aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).
% ATP.lambda_406
tff(fact_10618_ATP_Olambda__407,axiom,
! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_jn(Uu),Uua),Uub) = aa_nat_int(power_power_int(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).
% ATP.lambda_407
tff(fact_10619_ATP_Olambda__408,axiom,
! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_si(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uua),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).
% ATP.lambda_408
tff(fact_10620_ATP_Olambda__409,axiom,
! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_gt(Uu),Uua),Uub) = aa_nat_complex(power_power_complex(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).
% ATP.lambda_409
tff(fact_10621_ATP_Olambda__410,axiom,
! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_gw(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).
% ATP.lambda_410
tff(fact_10622_ATP_Olambda__411,axiom,
! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_gu(Uu),Uua),Uub) = aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).
% ATP.lambda_411
tff(fact_10623_ATP_Olambda__412,axiom,
! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_gv(Uu),Uua),Uub) = aa_nat_int(power_power_int(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).
% ATP.lambda_412
tff(fact_10624_ATP_Olambda__413,axiom,
! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hp(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_413
tff(fact_10625_ATP_Olambda__414,axiom,
! [Uu: real,Uua: real,Uub: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_ta(Uu),Uua),Uub))
<=> pp(aa_set_real_bool(member_real(Uub),set_or1633881224788618240n_real(Uu,Uua))) ) ).
% ATP.lambda_414
tff(fact_10626_ATP_Olambda__415,axiom,
! [Uu: complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_ek(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(Uu),aa_complex_complex(Uua,Uub)) ).
% ATP.lambda_415
tff(fact_10627_ATP_Olambda__416,axiom,
! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_jy(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(Uu),aa_nat_complex(Uua,Uub)) ).
% ATP.lambda_416
tff(fact_10628_ATP_Olambda__417,axiom,
! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_em(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),aa_nat_real(Uua,Uub)) ).
% ATP.lambda_417
tff(fact_10629_ATP_Olambda__418,axiom,
! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_el(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),aa_nat_nat(Uua,Uub)) ).
% ATP.lambda_418
tff(fact_10630_ATP_Olambda__419,axiom,
! [Uu: int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_ej(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,Uu),aa_int_int(Uua,Uub)) ).
% ATP.lambda_419
tff(fact_10631_ATP_Olambda__420,axiom,
! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_kb(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uua),aa_nat_real(Uu,Uub)) ).
% ATP.lambda_420
tff(fact_10632_ATP_Olambda__421,axiom,
! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_in(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).
% ATP.lambda_421
tff(fact_10633_ATP_Olambda__422,axiom,
! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_im(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uub))) ).
% ATP.lambda_422
tff(fact_10634_ATP_Olambda__423,axiom,
! [Uu: fun_real_bool,Uua: real,Uub: real] :
( pp(aa_real_bool(aa_rea7809428808904475482l_bool(aTP_Lamm_tb(Uu),Uua),Uub))
<=> pp(aa_real_bool(Uu,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uub),Uua))) ) ).
% ATP.lambda_423
tff(fact_10635_ATP_Olambda__424,axiom,
! [Uu: fun_nat_complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_kl(Uu),Uua),Uub) = aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).
% ATP.lambda_424
tff(fact_10636_ATP_Olambda__425,axiom,
! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_gs(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).
% ATP.lambda_425
tff(fact_10637_ATP_Olambda__426,axiom,
! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_te(Uu),Uua),Uub))
<=> pp(aa_nat_bool(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ) ).
% ATP.lambda_426
tff(fact_10638_ATP_Olambda__427,axiom,
! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_gr(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).
% ATP.lambda_427
tff(fact_10639_ATP_Olambda__428,axiom,
! [Uu: fun_Pr2477239776880934807at_nat,Uua: nat,Uub: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_du(Uu),Uua),Uub) = aa_Pro8023792847356852296at_nat(Uu,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)) ).
% ATP.lambda_428
tff(fact_10640_ATP_Olambda__429,axiom,
! [Uu: fun_Pr248683899324676269t_bool,Uua: nat,Uub: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_dv(Uu),Uua),Uub) = aa_Pro595088089689268830t_bool(Uu,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)) ).
% ATP.lambda_429
tff(fact_10641_ATP_Olambda__430,axiom,
! [Uu: fun_Pr2478283820680522243nt_int,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_dw(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ).
% ATP.lambda_430
tff(fact_10642_ATP_Olambda__431,axiom,
! [Uu: fun_Pr803113274115125473t_bool,Uua: int,Uub: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(aTP_Lamm_dx(Uu),Uua),Uub))
<=> pp(aa_Pro4501881432599481050t_bool(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub))) ) ).
% ATP.lambda_431
tff(fact_10643_ATP_Olambda__432,axiom,
! [Uu: fun_Pr982841511892225912nt_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_dy(Uu),Uua),Uub) = aa_Pro6799258434465214121nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ).
% ATP.lambda_432
tff(fact_10644_ATP_Olambda__433,axiom,
! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_lk(Uu),Uua),Uub) = aa_nat_complex(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).
% ATP.lambda_433
tff(fact_10645_ATP_Olambda__434,axiom,
! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ll(Uu),Uua),Uub) = aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).
% ATP.lambda_434
tff(fact_10646_ATP_Olambda__435,axiom,
! [Uu: fun_nat_real,Uua: fun_nat_nat,Uub: nat] : aa_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_tn(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(Uua,Uub)) ).
% ATP.lambda_435
tff(fact_10647_ATP_Olambda__436,axiom,
! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_real(aa_com20334259897932645t_real(aTP_Lamm_kf(Uu),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(Uua),Uub))) ).
% ATP.lambda_436
tff(fact_10648_ATP_Olambda__437,axiom,
! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_kd(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub))) ).
% ATP.lambda_437
tff(fact_10649_ATP_Olambda__438,axiom,
! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sn(Uu),Uua),Uub) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),Uua)) ).
% ATP.lambda_438
tff(fact_10650_ATP_Olambda__439,axiom,
! [Uu: fun_nat_real,Uua: set_nat,Uub: fun_nat_real,Uuc: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aa_set1316517696749040435t_real(aTP_Lamm_lq(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_set_nat_bool(member_nat(Uuc),Uua)),aa_nat_real(Uub,Uuc)),aa_nat_real(Uu,Uuc)) ).
% ATP.lambda_439
tff(fact_10651_ATP_Olambda__440,axiom,
! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: fun_VEBT_VEBT_real,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aa_fun910392650362481850T_real(aTP_Lamm_ko(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_VEBT_VEBT_bool(fequal_VEBT_VEBT(Uuc),Uu)),aa_VEBT_VEBT_real(Uua,Uuc)),aa_VEBT_VEBT_real(Uub,Uuc)) ).
% ATP.lambda_440
tff(fact_10652_ATP_Olambda__441,axiom,
! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_rat,Uub: fun_VEBT_VEBT_rat,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_rat(aa_fun8105832842879916156BT_rat(aa_fun2693410532875676774BT_rat(aTP_Lamm_kq(Uu),Uua),Uub),Uuc) = if_rat(aa_VEBT_VEBT_bool(fequal_VEBT_VEBT(Uuc),Uu),aa_VEBT_VEBT_rat(Uua,Uuc),aa_VEBT_VEBT_rat(Uub,Uuc)) ).
% ATP.lambda_441
tff(fact_10653_ATP_Olambda__442,axiom,
! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_nat,Uub: fun_VEBT_VEBT_nat,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_nat(aa_fun8284231930576791164BT_nat(aa_fun8778301808485728350BT_nat(aTP_Lamm_ks(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_VEBT_VEBT_bool(fequal_VEBT_VEBT(Uuc),Uu)),aa_VEBT_VEBT_nat(Uua,Uuc)),aa_VEBT_VEBT_nat(Uub,Uuc)) ).
% ATP.lambda_442
tff(fact_10654_ATP_Olambda__443,axiom,
! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_int,Uub: fun_VEBT_VEBT_int,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_int(aa_fun7144222995531728252BT_int(aa_fun7485955581636121530BT_int(aTP_Lamm_ku(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(if_int(aa_VEBT_VEBT_bool(fequal_VEBT_VEBT(Uuc),Uu)),aa_VEBT_VEBT_int(Uua,Uuc)),aa_VEBT_VEBT_int(Uub,Uuc)) ).
% ATP.lambda_443
tff(fact_10655_ATP_Olambda__444,axiom,
! [Uu: complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_kp(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uuc),Uu)),aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) ).
% ATP.lambda_444
tff(fact_10656_ATP_Olambda__445,axiom,
! [Uu: complex,Uua: fun_complex_rat,Uub: fun_complex_rat,Uuc: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_kr(Uu),Uua),Uub),Uuc) = if_rat(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uuc),Uu),aa_complex_rat(Uua,Uuc),aa_complex_rat(Uub,Uuc)) ).
% ATP.lambda_445
tff(fact_10657_ATP_Olambda__446,axiom,
! [Uu: complex,Uua: fun_complex_nat,Uub: fun_complex_nat,Uuc: complex] : aa_complex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_kt(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uuc),Uu)),aa_complex_nat(Uua,Uuc)),aa_complex_nat(Uub,Uuc)) ).
% ATP.lambda_446
tff(fact_10658_ATP_Olambda__447,axiom,
! [Uu: complex,Uua: fun_complex_int,Uub: fun_complex_int,Uuc: complex] : aa_complex_int(aa_fun4286351638413746156ex_int(aa_fun6437630453459604288ex_int(aTP_Lamm_kv(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(if_int(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,Uuc),Uu)),aa_complex_int(Uua,Uuc)),aa_complex_int(Uub,Uuc)) ).
% ATP.lambda_447
tff(fact_10659_ATP_Olambda__448,axiom,
! [Uu: int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] : aa_int_real(aa_fun3252798400008884712t_real(aa_fun6524208927192123834t_real(aTP_Lamm_kw(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(if_real(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uuc),Uu)),aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) ).
% ATP.lambda_448
tff(fact_10660_ATP_Olambda__449,axiom,
! [Uu: int,Uua: fun_int_rat,Uub: fun_int_rat,Uuc: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aa_fun2191249852558612414nt_rat(aTP_Lamm_kx(Uu),Uua),Uub),Uuc) = if_rat(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uuc),Uu),aa_int_rat(Uua,Uuc),aa_int_rat(Uub,Uuc)) ).
% ATP.lambda_449
tff(fact_10661_ATP_Olambda__450,axiom,
! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ih(Uu),Uua),Uub),Uuc) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ig(Uu),Uua),Uub),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).
% ATP.lambda_450
tff(fact_10662_ATP_Olambda__451,axiom,
! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_if(Uu),Uua),Uub),Uuc) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_ie(Uu),Uua),Uub),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).
% ATP.lambda_451
tff(fact_10663_ATP_Olambda__452,axiom,
! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_id(Uu),Uua),Uub),Uuc) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ic(Uu),Uua),Uub),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat))),Uuc))) ).
% ATP.lambda_452
tff(fact_10664_ATP_Olambda__453,axiom,
! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: set_nat,Uuc: nat] : aa_nat_real(aa_set1166323925324055613t_real(aa_fun1400677770452585267t_real(aTP_Lamm_ey(Uu),Uua),Uub),Uuc) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_ex(Uu),Uua),Uuc),Uub) ).
% ATP.lambda_453
tff(fact_10665_ATP_Olambda__454,axiom,
! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: set_nat,Uuc: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aa_fun4561837934398617467at_nat(aTP_Lamm_ew(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_ev(Uu),Uua),Uuc)),Uub) ).
% ATP.lambda_454
tff(fact_10666_ATP_Olambda__455,axiom,
! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: set_complex,Uuc: complex] : aa_complex_complex(aa_set7743095458595710203omplex(aa_fun1583801116089702097omplex(aTP_Lamm_eu(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aa_fun6995112137416684571omplex(aTP_Lamm_et(Uu),Uua),Uuc),Uub) ).
% ATP.lambda_455
tff(fact_10667_ATP_Olambda__456,axiom,
! [Uu: fun_int_int,Uua: fun_int_int,Uub: set_int,Uuc: int] : aa_int_int(aa_set7310040872927613301nt_int(aa_fun8076508010538293575nt_int(aTP_Lamm_es(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_int_fun_int_int(aa_fun5911324870756489105nt_int(aTP_Lamm_er(Uu),Uua),Uuc),Uub) ).
% ATP.lambda_456
tff(fact_10668_ATP_Olambda__457,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_oq(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_op(Uu),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uub),Uuc)) ).
% ATP.lambda_457
tff(fact_10669_ATP_Olambda__458,axiom,
! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_rx(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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_458
tff(fact_10670_ATP_Olambda__459,axiom,
! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_rv(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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_459
tff(fact_10671_ATP_Olambda__460,axiom,
! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_ru(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(minus_minus_real,Uub),Uua)),Uuc)) ).
% ATP.lambda_460
tff(fact_10672_ATP_Olambda__461,axiom,
! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_rt(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(minus_minus_real,Uua),Uub)),Uuc)) ).
% ATP.lambda_461
tff(fact_10673_ATP_Olambda__462,axiom,
! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_iv(Uu),Uua),Uub),Uuc) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_complex(power_power_complex(Uu),Uuc))),aa_nat_complex(power_power_complex(Uu),Uub)) ).
% ATP.lambda_462
tff(fact_10674_ATP_Olambda__463,axiom,
! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_jb(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).
% ATP.lambda_463
tff(fact_10675_ATP_Olambda__464,axiom,
! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_ix(Uu),Uua),Uub),Uuc) = minus_minus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(plus_plus_rat(Uu,Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu),Uuc)),aa_nat_rat(power_power_rat(Uu),Uub)) ).
% ATP.lambda_464
tff(fact_10676_ATP_Olambda__465,axiom,
! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_iz(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_int(power_power_int(Uu),Uuc))),aa_nat_int(power_power_int(Uu),Uub)) ).
% ATP.lambda_465
tff(fact_10677_ATP_Olambda__466,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_qg(Uu),Uua),Uub),Uuc) = 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,Uu),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uub))) ).
% ATP.lambda_466
tff(fact_10678_ATP_Olambda__467,axiom,
! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ic(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uua),Uu)),Uuc)),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Uuc))) ).
% ATP.lambda_467
tff(fact_10679_ATP_Olambda__468,axiom,
! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ig(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uua),Uu)),Uuc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Uuc))) ).
% ATP.lambda_468
tff(fact_10680_ATP_Olambda__469,axiom,
! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_ie(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(plus_plus_rat(Uua,Uu)),Uuc)),aa_nat_rat(power_power_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Uuc))) ).
% ATP.lambda_469
tff(fact_10681_ATP_Olambda__470,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ow(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(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).
% ATP.lambda_470
tff(fact_10682_ATP_Olambda__471,axiom,
! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_jd(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_complex(power_power_complex(Uu),Uuc)) ).
% ATP.lambda_471
tff(fact_10683_ATP_Olambda__472,axiom,
! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_jg(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)) ).
% ATP.lambda_472
tff(fact_10684_ATP_Olambda__473,axiom,
! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_je(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_rat(power_power_rat(Uu),Uuc)) ).
% ATP.lambda_473
tff(fact_10685_ATP_Olambda__474,axiom,
! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_jf(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_int(power_power_int(Uu),Uuc)) ).
% ATP.lambda_474
tff(fact_10686_ATP_Olambda__475,axiom,
! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_iw(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uu),Uuc)),aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_complex(power_power_complex(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).
% ATP.lambda_475
tff(fact_10687_ATP_Olambda__476,axiom,
! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_jc(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).
% ATP.lambda_476
tff(fact_10688_ATP_Olambda__477,axiom,
! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_iy(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uu),Uuc)),minus_minus_rat(aa_nat_rat(power_power_rat(plus_plus_rat(Uu,Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)),aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).
% ATP.lambda_477
tff(fact_10689_ATP_Olambda__478,axiom,
! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_ja(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Uu),Uuc)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_int(power_power_int(Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).
% ATP.lambda_478
tff(fact_10690_ATP_Olambda__479,axiom,
! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_jh(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Uu),Uuc)),aa_nat_complex(power_power_complex(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).
% ATP.lambda_479
tff(fact_10691_ATP_Olambda__480,axiom,
! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_jk(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).
% ATP.lambda_480
tff(fact_10692_ATP_Olambda__481,axiom,
! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_ji(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uu),Uuc)),aa_nat_rat(power_power_rat(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).
% ATP.lambda_481
tff(fact_10693_ATP_Olambda__482,axiom,
! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_jj(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Uu),Uuc)),aa_nat_int(power_power_int(Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).
% ATP.lambda_482
tff(fact_10694_ATP_Olambda__483,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ov(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).
% ATP.lambda_483
tff(fact_10695_ATP_Olambda__484,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_ql(Uu),Uua),Uub),Uuc))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ) ).
% ATP.lambda_484
tff(fact_10696_ATP_Olambda__485,axiom,
! [Uu: int,Uua: int,Uub: int,Uuc: int] :
( pp(aa_int_bool(aa_int_fun_int_bool(aa_int2327297053056786951t_bool(aTP_Lamm_qa(Uu),Uua),Uub),Uuc))
<=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub))) ) ).
% ATP.lambda_485
tff(fact_10697_ATP_Olambda__486,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_qj(Uu),Uua),Uub),Uuc))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ) ).
% ATP.lambda_486
tff(fact_10698_ATP_Olambda__487,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_qn(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)) ).
% ATP.lambda_487
tff(fact_10699_ATP_Olambda__488,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_qp(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).
% ATP.lambda_488
tff(fact_10700_ATP_Olambda__489,axiom,
! [Uu: set_complex,Uua: fun_complex_complex,Uub: fun_complex_complex,Uuc: complex] :
( pp(aa_complex_bool(aa_fun9073502203726719595x_bool(aa_fun6641798613628721963x_bool(aTP_Lamm_bu(Uu),Uua),Uub),Uuc))
<=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
& ( aa_complex_complex(times_times_complex(aa_complex_complex(Uua,Uuc)),aa_complex_complex(Uub,Uuc)) != one_one_complex ) ) ) ).
% ATP.lambda_489
tff(fact_10701_ATP_Olambda__490,axiom,
! [Uu: set_complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aa_fun2094807616088678831x_bool(aTP_Lamm_cc(Uu),Uua),Uub),Uuc))
<=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
& ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) != one_one_real ) ) ) ).
% ATP.lambda_490
tff(fact_10702_ATP_Olambda__491,axiom,
! [Uu: set_real,Uua: fun_real_complex,Uub: fun_real_complex,Uuc: real] :
( pp(aa_real_bool(aa_fun3813300650643916655l_bool(aa_fun5520078544946397361l_bool(aTP_Lamm_bo(Uu),Uua),Uub),Uuc))
<=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
& ( aa_complex_complex(times_times_complex(aa_real_complex(Uua,Uuc)),aa_real_complex(Uub,Uuc)) != one_one_complex ) ) ) ).
% ATP.lambda_491
tff(fact_10703_ATP_Olambda__492,axiom,
! [Uu: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_bw(Uu),Uua),Uub),Uuc))
<=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
& ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != one_one_real ) ) ) ).
% ATP.lambda_492
tff(fact_10704_ATP_Olambda__493,axiom,
! [Uu: set_real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
( pp(aa_real_bool(aa_fun8987621170725249221l_bool(aa_fun8530242357055030877l_bool(aTP_Lamm_ce(Uu),Uua),Uub),Uuc))
<=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
& ( aa_rat_rat(times_times_rat(aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) != one_one_rat ) ) ) ).
% ATP.lambda_493
tff(fact_10705_ATP_Olambda__494,axiom,
! [Uu: set_nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
( pp(aa_nat_bool(aa_fun8004580481347240231t_bool(aa_fun5430245816583273541t_bool(aTP_Lamm_bq(Uu),Uua),Uub),Uuc))
<=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
& ( aa_complex_complex(times_times_complex(aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) != one_one_complex ) ) ) ).
% ATP.lambda_494
tff(fact_10706_ATP_Olambda__495,axiom,
! [Uu: set_nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_by(Uu),Uua),Uub),Uuc))
<=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
& ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != one_one_real ) ) ) ).
% ATP.lambda_495
tff(fact_10707_ATP_Olambda__496,axiom,
! [Uu: set_nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
( pp(aa_nat_bool(aa_fun5637262893798792573t_bool(aa_fun2107527232086395633t_bool(aTP_Lamm_cg(Uu),Uua),Uub),Uuc))
<=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
& ( aa_rat_rat(times_times_rat(aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) != one_one_rat ) ) ) ).
% ATP.lambda_496
tff(fact_10708_ATP_Olambda__497,axiom,
! [Uu: set_int,Uua: fun_int_complex,Uub: fun_int_complex,Uuc: int] :
( pp(aa_int_bool(aa_fun8146511558608327279t_bool(aa_fun3850298374753145905t_bool(aTP_Lamm_bs(Uu),Uua),Uub),Uuc))
<=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
& ( aa_complex_complex(times_times_complex(aa_int_complex(Uua,Uuc)),aa_int_complex(Uub,Uuc)) != one_one_complex ) ) ) ).
% ATP.lambda_497
tff(fact_10709_ATP_Olambda__498,axiom,
! [Uu: set_int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_ca(Uu),Uua),Uub),Uuc))
<=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
& ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != one_one_real ) ) ) ).
% ATP.lambda_498
tff(fact_10710_ATP_Olambda__499,axiom,
! [Uu: set_complex,Uua: fun_complex_complex,Uub: fun_complex_complex,Uuc: complex] :
( pp(aa_complex_bool(aa_fun9073502203726719595x_bool(aa_fun6641798613628721963x_bool(aTP_Lamm_ct(Uu),Uua),Uub),Uuc))
<=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
& ( aa_complex_complex(plus_plus_complex(aa_complex_complex(Uua,Uuc)),aa_complex_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).
% ATP.lambda_499
tff(fact_10711_ATP_Olambda__500,axiom,
! [Uu: set_complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
( pp(aa_complex_bool(aa_fun2898354424061222765x_bool(aa_fun2094807616088678831x_bool(aTP_Lamm_db(Uu),Uua),Uub),Uuc))
<=> ( pp(aa_set_complex_bool(member_complex(Uuc),Uu))
& ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) != zero_zero_real ) ) ) ).
% ATP.lambda_500
tff(fact_10712_ATP_Olambda__501,axiom,
! [Uu: set_real,Uua: fun_real_complex,Uub: fun_real_complex,Uuc: real] :
( pp(aa_real_bool(aa_fun3813300650643916655l_bool(aa_fun5520078544946397361l_bool(aTP_Lamm_cn(Uu),Uua),Uub),Uuc))
<=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
& ( aa_complex_complex(plus_plus_complex(aa_real_complex(Uua,Uuc)),aa_real_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).
% ATP.lambda_501
tff(fact_10713_ATP_Olambda__502,axiom,
! [Uu: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
( pp(aa_real_bool(aa_fun9137271176455094897l_bool(aa_fun334843040499603765l_bool(aTP_Lamm_cv(Uu),Uua),Uub),Uuc))
<=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
& ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != zero_zero_real ) ) ) ).
% ATP.lambda_502
tff(fact_10714_ATP_Olambda__503,axiom,
! [Uu: set_real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
( pp(aa_real_bool(aa_fun8987621170725249221l_bool(aa_fun8530242357055030877l_bool(aTP_Lamm_dd(Uu),Uua),Uub),Uuc))
<=> ( pp(aa_set_real_bool(member_real(Uuc),Uu))
& ( plus_plus_rat(aa_real_rat(Uua,Uuc),aa_real_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).
% ATP.lambda_503
tff(fact_10715_ATP_Olambda__504,axiom,
! [Uu: set_nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
( pp(aa_nat_bool(aa_fun8004580481347240231t_bool(aa_fun5430245816583273541t_bool(aTP_Lamm_cp(Uu),Uua),Uub),Uuc))
<=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
& ( aa_complex_complex(plus_plus_complex(aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).
% ATP.lambda_504
tff(fact_10716_ATP_Olambda__505,axiom,
! [Uu: set_nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
( pp(aa_nat_bool(aa_fun1349230436907296553t_bool(aa_fun849216055071118793t_bool(aTP_Lamm_cx(Uu),Uua),Uub),Uuc))
<=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
& ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != zero_zero_real ) ) ) ).
% ATP.lambda_505
tff(fact_10717_ATP_Olambda__506,axiom,
! [Uu: set_nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
( pp(aa_nat_bool(aa_fun5637262893798792573t_bool(aa_fun2107527232086395633t_bool(aTP_Lamm_df(Uu),Uua),Uub),Uuc))
<=> ( pp(aa_set_nat_bool(member_nat(Uuc),Uu))
& ( plus_plus_rat(aa_nat_rat(Uua,Uuc),aa_nat_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).
% ATP.lambda_506
tff(fact_10718_ATP_Olambda__507,axiom,
! [Uu: set_int,Uua: fun_int_complex,Uub: fun_int_complex,Uuc: int] :
( pp(aa_int_bool(aa_fun8146511558608327279t_bool(aa_fun3850298374753145905t_bool(aTP_Lamm_cr(Uu),Uua),Uub),Uuc))
<=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
& ( aa_complex_complex(plus_plus_complex(aa_int_complex(Uua,Uuc)),aa_int_complex(Uub,Uuc)) != zero_zero_complex ) ) ) ).
% ATP.lambda_507
tff(fact_10719_ATP_Olambda__508,axiom,
! [Uu: set_int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
( pp(aa_int_bool(aa_fun1262781554902806129t_bool(aa_fun6205147705862377397t_bool(aTP_Lamm_cz(Uu),Uua),Uub),Uuc))
<=> ( pp(aa_set_int_bool(member_int(Uuc),Uu))
& ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != zero_zero_real ) ) ) ).
% ATP.lambda_508
tff(fact_10720_ATP_Olambda__509,axiom,
! [Uu: fun_nat_complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_kj(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_complex(power_power_complex(Uub),Uuc)) ).
% ATP.lambda_509
tff(fact_10721_ATP_Olambda__510,axiom,
! [Uu: fun_nat_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_kk(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).
% ATP.lambda_510
tff(fact_10722_ATP_Olambda__511,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_op(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(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).
% ATP.lambda_511
tff(fact_10723_ATP_Olambda__512,axiom,
! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex,Uuc: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aa_fun6995112137416684571omplex(aTP_Lamm_et(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uuc)) ).
% ATP.lambda_512
tff(fact_10724_ATP_Olambda__513,axiom,
! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_ex(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uuc)) ).
% ATP.lambda_513
tff(fact_10725_ATP_Olambda__514,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_ev(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uuc)) ).
% ATP.lambda_514
tff(fact_10726_ATP_Olambda__515,axiom,
! [Uu: fun_int_int,Uua: fun_int_int,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun5911324870756489105nt_int(aTP_Lamm_er(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uuc)) ).
% ATP.lambda_515
tff(fact_10727_ATP_Olambda__516,axiom,
! [Uu: fun_nat_real,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_gy(Uu),Uua),Uub),Uuc) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).
% ATP.lambda_516
tff(fact_10728_ATP_Olambda__517,axiom,
! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_gx(Uu),Uua),Uub),Uuc) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).
% ATP.lambda_517
tff(fact_10729_ATP_Olambda__518,axiom,
! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_pu(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uub),Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).
% ATP.lambda_518
tff(fact_10730_ATP_Olambda__519,axiom,
! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_qc(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub))) ).
% ATP.lambda_519
tff(fact_10731_ATP_Olambda__520,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_rw(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_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_rv(Uua),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uub),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uud),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))))) ).
% ATP.lambda_520
tff(fact_10732_ATP_Olambda__521,axiom,
! [Uu: nat,Uua: list_VEBT_VEBT,Uub: vEBT_VEBT,Uuc: nat,Uud: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_VEB1917483692448707121t_bool(aa_lis1179741779018426550t_bool(aTP_Lamm_tf(Uu),Uua),Uub),Uuc),Uud))
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uuc),Uud))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uud),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu)))
& ! [I4: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))
=> ( ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(Uua,I4)),X_13))
<=> pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,Uub),I4)) ) )
& ( ( Uuc = Uud )
=> ! [X2: vEBT_VEBT] :
( pp(aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(member_VEBT_VEBT,X2),set_VEBT_VEBT2(Uua)))
=> ~ ? [X_13: nat] : pp(aa_nat_bool(aa_VEB4832260192069861672t_bool(vEBT_V8194947554948674370ptions,X2),X_13)) ) )
& ( ( Uuc != Uud )
=> ( pp(aa_nat_bool(vEBT_V5917875025757280293ildren(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),Uud))
& ! [X2: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu)))
=> ( pp(aa_nat_bool(vEBT_V5917875025757280293ildren(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),X2))
=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uuc),X2))
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Uud)) ) ) ) ) ) ) ) ).
% ATP.lambda_521
tff(fact_10733_ATP_Olambda__522,axiom,
! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ua(Uu),Uua) = Uu ).
% ATP.lambda_522
tff(fact_10734_ATP_Olambda__523,axiom,
! [Uu: complex] : aa_complex_complex(aTP_Lamm_bb,Uu) = Uu ).
% ATP.lambda_523
tff(fact_10735_ATP_Olambda__524,axiom,
! [Uu: real] : aa_real_real(aTP_Lamm_bc,Uu) = Uu ).
% ATP.lambda_524
tff(fact_10736_ATP_Olambda__525,axiom,
! [Uu: rat] : aa_rat_rat(aTP_Lamm_bd,Uu) = Uu ).
% ATP.lambda_525
tff(fact_10737_ATP_Olambda__526,axiom,
! [Uu: nat] : aa_nat_nat(aTP_Lamm_be,Uu) = Uu ).
% ATP.lambda_526
tff(fact_10738_ATP_Olambda__527,axiom,
! [Uu: int] : aa_int_int(aTP_Lamm_bf,Uu) = Uu ).
% ATP.lambda_527
tff(fact_10739_ATP_Olambda__528,axiom,
! [Uu: complex] : aa_complex_complex(aTP_Lamm_ch,Uu) = zero_zero_complex ).
% ATP.lambda_528
tff(fact_10740_ATP_Olambda__529,axiom,
! [Uu: real] : aa_real_real(aTP_Lamm_ci,Uu) = zero_zero_real ).
% ATP.lambda_529
tff(fact_10741_ATP_Olambda__530,axiom,
! [Uu: rat] : aa_rat_rat(aTP_Lamm_cj,Uu) = zero_zero_rat ).
% ATP.lambda_530
tff(fact_10742_ATP_Olambda__531,axiom,
! [Uu: nat] : aa_nat_rat(aTP_Lamm_uo,Uu) = zero_zero_rat ).
% ATP.lambda_531
tff(fact_10743_ATP_Olambda__532,axiom,
! [Uu: nat] : aa_nat_nat(aTP_Lamm_ck,Uu) = zero_zero_nat ).
% ATP.lambda_532
tff(fact_10744_ATP_Olambda__533,axiom,
! [Uu: int] : aa_int_int(aTP_Lamm_cl,Uu) = zero_zero_int ).
% ATP.lambda_533
tff(fact_10745_ATP_Olambda__534,axiom,
! [Uu: nat] : aa_nat_rat(aTP_Lamm_un,Uu) = one_one_rat ).
% ATP.lambda_534
tff(fact_10746_ATP_Olambda__535,axiom,
! [Uu: nat] : aa_nat_nat(aTP_Lamm_mz,Uu) = one_one_nat ).
% ATP.lambda_535
tff(fact_10747_ATP_Olambda__536,axiom,
! [Uu: nat] : aa_nat_int(aTP_Lamm_na,Uu) = one_one_int ).
% ATP.lambda_536
tff(fact_10748_ATP_Olambda__537,axiom,
! [Uu: int] : aa_int_int(aTP_Lamm_nb,Uu) = one_one_int ).
% ATP.lambda_537
tff(fact_10749_ATP_Olambda__538,axiom,
! [Uu: real] :
( pp(aa_real_bool(aTP_Lamm_pb,Uu))
<=> $false ) ).
% ATP.lambda_538
tff(fact_10750_ATP_Olambda__539,axiom,
! [Uu: nat] :
( pp(aa_nat_bool(aTP_Lamm_px,Uu))
<=> $false ) ).
% ATP.lambda_539
tff(fact_10751_ATP_Olambda__540,axiom,
! [Uu: nat] :
( pp(aa_nat_bool(aTP_Lamm_pw,Uu))
<=> $true ) ).
% ATP.lambda_540
% Helper facts (66)
tff(help_fNot_2_1_U,axiom,
! [P: bool] :
( pp(P)
| pp(aa_bool_bool(fNot,P)) ) ).
tff(help_fNot_1_1_U,axiom,
! [P: bool] :
( ~ pp(aa_bool_bool(fNot,P))
| ~ pp(P) ) ).
tff(help_fTrue_1_1_U,axiom,
pp(fTrue) ).
tff(help_fconj_3_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(aa_bool_bool(fconj(P),Q))
| pp(Q) ) ).
tff(help_fconj_2_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(aa_bool_bool(fconj(P),Q))
| pp(P) ) ).
tff(help_fconj_1_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(P)
| ~ pp(Q)
| pp(aa_bool_bool(fconj(P),Q)) ) ).
tff(help_fdisj_3_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(aa_bool_bool(fdisj(P),Q))
| pp(P)
| pp(Q) ) ).
tff(help_fdisj_2_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(Q)
| pp(aa_bool_bool(fdisj(P),Q)) ) ).
tff(help_fdisj_1_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(P)
| pp(aa_bool_bool(fdisj(P),Q)) ) ).
tff(help_fFalse_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_fimplies_3_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,P),Q))
| ~ pp(P)
| pp(Q) ) ).
tff(help_fimplies_2_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(Q)
| pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,P),Q)) ) ).
tff(help_fimplies_1_1_U,axiom,
! [P: bool,Q: bool] :
( pp(P)
| pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fimplies,P),Q)) ) ).
tff(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(if_int(fFalse),X),Y) = Y ).
tff(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(if_int(fTrue),X),Y) = X ).
tff(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fFalse),X),Y) = Y ).
tff(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fTrue),X),Y) = X ).
tff(help_If_2_1_If_001t__Num__Onum_T,axiom,
! [X: num,Y: num] : aa_num_num(aa_num_fun_num_num(if_num(fFalse),X),Y) = Y ).
tff(help_If_1_1_If_001t__Num__Onum_T,axiom,
! [X: num,Y: num] : aa_num_num(aa_num_fun_num_num(if_num(fTrue),X),Y) = X ).
tff(help_If_2_1_If_001t__Rat__Orat_T,axiom,
! [X: rat,Y: rat] : if_rat(fFalse,X,Y) = Y ).
tff(help_If_1_1_If_001t__Rat__Orat_T,axiom,
! [X: rat,Y: rat] : if_rat(fTrue,X,Y) = X ).
tff(help_fEx_1_1_fEx_001t__Nat__Onat_U,axiom,
! [P: fun_nat_bool,X: nat] :
( ~ pp(aa_nat_bool(P,X))
| pp(aa_fun_nat_bool_bool(fEx_nat,P)) ) ).
tff(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(if_real(fFalse),X),Y) = Y ).
tff(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(if_real(fTrue),X),Y) = X ).
tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( X != Y )
| pp(aa_int_bool(aa_int_fun_int_bool(fequal_int,X),Y)) ) ).
tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ~ pp(aa_int_bool(aa_int_fun_int_bool(fequal_int,X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
| pp(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Y)) ) ).
tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_fequal_001t__HOL__Obool_T,axiom,
! [X: bool,Y: bool] :
( ( X != Y )
| pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,X),Y)) ) ).
tff(help_fequal_1_1_fequal_001t__HOL__Obool_T,axiom,
! [X: bool,Y: bool] :
( ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_fequal_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( X != Y )
| pp(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,X),Y)) ) ).
tff(help_fequal_1_1_fequal_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ~ pp(aa_real_bool(aa_rea7809428808904475482l_bool(fequal_real,X),Y))
| ( X = Y ) ) ).
tff(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
! [P: fun_real_bool] : aa_real_bool(P,fChoice_real(P)) = aa_fun3352386287069783311l_bool(fEx_real,P) ).
tff(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y: complex] : aa_complex_complex(if_complex(fFalse,X),Y) = Y ).
tff(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y: complex] : aa_complex_complex(if_complex(fTrue,X),Y) = X ).
tff(help_fequal_2_1_fequal_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y: complex] :
( ( X != Y )
| pp(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,X),Y)) ) ).
tff(help_fequal_1_1_fequal_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y: complex] :
( ~ pp(aa_complex_bool(aa_com1553036987509291606x_bool(fequal_complex,X),Y))
| ( X = Y ) ) ).
tff(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
! [X: code_integer,Y: code_integer] : if_Code_integer(fFalse,X,Y) = Y ).
tff(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
! [X: code_integer,Y: code_integer] : if_Code_integer(fTrue,X,Y) = X ).
tff(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
! [X: vEBT_VEBT,Y: vEBT_VEBT] : if_VEBT_VEBT(fFalse,X,Y) = Y ).
tff(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
! [X: vEBT_VEBT,Y: vEBT_VEBT] : if_VEBT_VEBT(fTrue,X,Y) = X ).
tff(help_fAll_1_1_fAll_001t__VEBT____Definitions__OVEBT_U,axiom,
! [P: fun_VEBT_VEBT_bool,X: vEBT_VEBT] :
( ~ pp(fAll_VEBT_VEBT(P))
| pp(aa_VEBT_VEBT_bool(P,X)) ) ).
tff(help_fequal_2_1_fequal_001t__Code____Numeral__Ointeger_T,axiom,
! [X: code_integer,Y: code_integer] :
( ( X != Y )
| pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,X),Y)) ) ).
tff(help_fequal_1_1_fequal_001t__Code____Numeral__Ointeger_T,axiom,
! [X: code_integer,Y: code_integer] :
( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_fequal_001t__VEBT____Definitions__OVEBT_T,axiom,
! [X: vEBT_VEBT,Y: vEBT_VEBT] :
( ( X != Y )
| pp(aa_VEBT_VEBT_bool(fequal_VEBT_VEBT(X),Y)) ) ).
tff(help_fequal_1_1_fequal_001t__VEBT____Definitions__OVEBT_T,axiom,
! [X: vEBT_VEBT,Y: vEBT_VEBT] :
( ~ pp(aa_VEBT_VEBT_bool(fequal_VEBT_VEBT(X),Y))
| ( X = Y ) ) ).
tff(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
! [X: option_nat,Y: option_nat] : if_option_nat(fFalse,X,Y) = Y ).
tff(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
! [X: option_nat,Y: option_nat] : if_option_nat(fTrue,X,Y) = X ).
tff(help_If_2_1_If_001t__fun_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
! [X: fun_int_int,Y: fun_int_int] : if_fun_int_int(fFalse,X,Y) = Y ).
tff(help_If_1_1_If_001t__fun_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
! [X: fun_int_int,Y: fun_int_int] : if_fun_int_int(fTrue,X,Y) = X ).
tff(help_If_2_1_If_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
! [X: fun_nat_rat,Y: fun_nat_rat] : if_fun_nat_rat(fFalse,X,Y) = Y ).
tff(help_If_1_1_If_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
! [X: fun_nat_rat,Y: fun_nat_rat] : if_fun_nat_rat(fTrue,X,Y) = X ).
tff(help_fequal_2_1_fequal_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
! [X: option_nat,Y: option_nat] :
( ( X != Y )
| pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,X),Y)) ) ).
tff(help_fequal_1_1_fequal_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
! [X: option_nat,Y: option_nat] :
( ~ pp(aa_option_nat_bool(aa_opt7943525813896274546t_bool(fequal_option_nat,X),Y))
| ( X = Y ) ) ).
tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
! [X: product_prod_int_int,Y: product_prod_int_int] : aa_Pro6970504127699319548nt_int(if_Pro3027730157355071871nt_int(fFalse,X),Y) = Y ).
tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
! [X: product_prod_int_int,Y: product_prod_int_int] : aa_Pro6970504127699319548nt_int(if_Pro3027730157355071871nt_int(fTrue,X),Y) = X ).
tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(if_Pro6206227464963214023at_nat(fFalse),X),Y) = Y ).
tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(if_Pro6206227464963214023at_nat(fTrue),X),Y) = X ).
tff(help_COMBB_1_1_COMBB_001t__HOL__Obool_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT_U,axiom,
! [P: fun_bool_bool,Q: fun_VEBT_VEBT_bool,R2: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBB_3745307990312268622T_VEBT(P,Q),R2) = aa_bool_bool(P,aa_VEBT_VEBT_bool(Q,R2)) ).
tff(help_COMBS_1_1_COMBS_001t__VEBT____Definitions__OVEBT_001t__HOL__Obool_001t__HOL__Obool_U,axiom,
! [P: fun_VE2066785035971159626l_bool,Q: fun_VEBT_VEBT_bool,R2: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBS_6222329278419335549l_bool(P,Q),R2) = aa_bool_bool(aa_VEB8047906998784800507l_bool(P,R2),aa_VEBT_VEBT_bool(Q,R2)) ).
tff(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] : if_Pro6119634080678213985nteger(fFalse,X,Y) = Y ).
tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] : if_Pro6119634080678213985nteger(fTrue,X,Y) = X ).
tff(help_COMBB_1_1_COMBB_001t__fun_It__Nat__Onat_Mt__HOL__Obool_J_001t__HOL__Obool_001t__VEBT____Definitions__OVEBT_U,axiom,
! [P: fun_fu7168253649841883962l_bool,Q: fun_VE3763956218935105199t_bool,R2: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBB_2885587777813823575T_VEBT(P,Q),R2) = aa_fun_nat_bool_bool(P,aa_VEB4832260192069861672t_bool(Q,R2)) ).
tff(help_COMBB_1_1_COMBB_001t__HOL__Obool_001t__fun_It__HOL__Obool_Mt__HOL__Obool_J_001t__VEBT____Definitions__OVEBT_U,axiom,
! [P: fun_bo1841576699660343469l_bool,Q: fun_VEBT_VEBT_bool,R2: vEBT_VEBT] : aa_VEB8047906998784800507l_bool(cOMBB_3438363824644082366T_VEBT(P,Q),R2) = aa_boo8970175072276269406l_bool(P,aa_VEBT_VEBT_bool(Q,R2)) ).
tff(help_COMBC_1_1_COMBC_001t__VEBT____Definitions__OVEBT_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J_001t__HOL__Obool_U,axiom,
! [P: fun_VE7346385682144900493T_bool,Q: set_VEBT_VEBT,R2: vEBT_VEBT] : aa_VEBT_VEBT_bool(cOMBC_771164880915498576T_bool(P,Q),R2) = aa_set8260762808118238015T_bool(aa_VEB289770264548726718T_bool(P,R2),Q) ).
% Conjectures (1)
tff(conj_0,conjecture,
pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary)),xa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_height,summary)),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))))))) ).
%------------------------------------------------------------------------------